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1301.2698
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11institutetext: University of Chinese Academy of Sciences, Beijing,
P.R.China,
11email: [email protected]
# Evaluating community structure in large network with random walks
Jiankou Li
###### Abstract
Community structure is one of the most important properties of networks. Most
community algorithms are not suitable for large networks because of their time
consuming. In fact there are lots of networks with millons even billons of
nodes. In such case, most algorithms running in time $O(n^{2}logn)$ or even
larger are not practical. What we need are linear or approximately linear time
algorithm. Rising in response to such needs, we propose a quick methods to
evaluate community structure in networks and then put forward a local
community algorithm with nearly linear time based on random walks. Using our
community evaluating measure, we could find some difference results from
measures used before, i.e., the Newman Modularity. Our algorithm are effective
in small benchmark networks with small less accuracy than more complex
algorithms but a great of advantage in time consuming for large networks,
especially super large networks.
###### Keywords:
community structure, random walk, modularity
## 1 Introduction
Networks are important tools to study real systems. Nodes in networks usually
organized into relative densely groups called communities or clusters.
Community structure have become one of the important directions. With the
computer and internet techniques developing, networks we could get become
larger and larger. Take the liveJournal online social network and U.S. patent
dataset for example. LiveJournal is a free on-line community with almost 10
million members [2]. The U.S. patent dataset is maintained by the National
Bureau of Economic Research and includes about 3,923,922 patents [14]. It is
reasonable to believe lots of other real networks are larger and will increase
quickly in future. There is a great need for developing quick community
detection algorithms.
To study community structure in large networks, we should evaluate whether a
given network have community structure and how to find them if there are. So
far, the most accepted measure to evaluate the community structure is
modularity [5] [18]. However, modularity has an intrinsic scale indicting that
modules smaller than that scale may not be resolved [10] and finding the
partition with maximal modularity is not a trivial thing. Local methods which
works independent on the global structure seems more practical in such large
datas [1] [3] [4]. One kind of local algorithms divide the whole network into
two parts [3] [4] [17], a community $C$ and the set of nodes with links to
$C$, say $B$. They usually start from a given node $s$ and then explore $B$
and select one or more nodes to merge into $C$. Such operation is repeated
until some terminal condition is satisfied. Another kind of local algorithms
work in some different ways. Firstly, they calculate a vector around a given
node $s$. This vector includes information that indicates the tendencies to
the $C$ we are to find. Then sort the vector according the score and a new
vector called a support vector is got. Finally, we could take a sweep over
this support vector based on some quality and find the community. One of such
algorithms is [1] which has been used by Leskovec to find lots of interesting
phenomenon in real networks [15] [16].
As we know, random walks have close relationship with community. A random
walker from a given position will be ’trapped’ in a community with high
probability. There are lots of community algorithms proposed inspired by this
idea. A vertices similarity measure and a community similarity measure are
proposed by Latapy and Pons and then communities are get by a agglomerative
procedure [19]. Also based on random walks, Zhou define a distance between
pairs of nodes and use divisive procedure to detect communities with running
time $O(n^{3})$ [26] [27] [28]. Community structure could be related to random
walk through the information theoretic approach where the community detecting
procedure becomes compressing a description of the probability flow of random
walk [20]. Delvenne introduce a quality function indicating the persistence of
clustering over time and unifies the modularity measures [5] [18] as well as
several definition related with random walk [7]. Other community algorithm
based on random walk includes $MarkovClusterAlgorithm(MCL)$ with running time
$O(nk^{2})$[23], methods using signaling process [12] and methods of
minimizing the matrix distance[24]. More detail about these methods could be
found in [9]. All of these algorithm scale $O(n^{2}logn)$ or higher, and are
not practical in super large networks with millons and billons of nodes. In
this paper, we give a quick measure to evaluate the community structure and a
local methods to find communities based on random walks in nearly linear time,
which could be used in very large networks.
We arrange the rest of the paper as follows. In section 2 we propose a
modularity measure based on random walks. Then we give a algorithm and test it
on benchmarks in section 3. The result of some experiments in very large
networks are given in section 4. Finally, we give the conclusion in section 5.
## 2 Random Walk Modularity
As pointed above, the random walk has significant indication of network
structure. We consider the following situation, a random walk is terminated
when it forms a ring and the corresponding number of steps is called random
walk length (RWL for short). A question of interesting is how long the
expectation of RWL is for a given network? We test the relationship between
the RWL in ER random networks [8] where every pair of nodes are linked with
probability $p$ and the planted $l$-partition model [11] which has been used
largely to test a community algorithm’s performance.
Figure 1: The relationship between random walk length and $p$ in ER random
network
In the ER random networks where every pair of nodes are linked with random,
the ring are usually formed by two nodes (the backward walk) as the
probability more than two nodes form a ring is small when the network size
tend to infinite. In every step, the probability a ring is formed by a
backward walk is $\frac{1}{d_{v}}$. We assume every nodes have the same degree
$d$ for the sake of discussion. Let $L_{r}$ be the expectation of the random
walk length, $q_{l}$ is the probability that a random walker ends with l step,
we have the following:
$\begin{split}L_{r}=\sum_{l\in
n}{q_{l}}l=\frac{1}{d}+2*(1-\frac{1}{d})*\frac{1}{d}+...+n*(1-\frac{1}{d})^{n-1}*\frac{1}{d}\\\
=d-(d+n)(1-\frac{1}{d})^{n}\end{split}$ (1)
So when $n\to\infty$ the expectation of RWL is mainly affected by $d$. When
the degree is not constant, the analysis become complex, but there seems a
linear relationship between the average random walk length and $p$ see figure
1.
Figure 2: Average step length a walker have to pass before he encounters a
ring in $l$-partition model [11]. In this model, every node has a fixed
average degree 16, $p$ is the ratio between the its out degree and its total
degree
On the other hand, the ARL has inverse correlation with the community
structure. The planted $l$-partition benchmark is used to illustrate the
relationship between random walk length and community structure. This
benchmark has been very popular to test the performance of community algorithm
since proposed by Condon and Karp [6] and a special case of planted
$l$-partition model is given by Newman[11]. In the Newman model, 128 vertices
are partitioned into 4 groups with each group 32 vertices. Every vertex has
$z_{in}$ links in the same group, and $z_{out}$ links outside of the group.
The average total degree of vertices are fixed to 16. $p$ is the ratio between
$z_{out}$ and average degree of each vertex. So the community structure could
be controlled by $p$. The average RWL has a close relationship with $p$ see
figure 2. This is according with previous idea, a random walker could be
easily trapped in a community. So the chance that a random walk forms a ring
increases in networks with clear community structure. As $p$ gets larger, the
community structure becomes more and more fuzzy. As a result, the random
walker has more probability to escape from the ’trap’ and RWL increases.
From above analysis, we know RWL is mainly affected by the degree sequence and
the community structure. Inspired by this, we propose a simple community
evaluating measure called Random Walk Modularity which could be calculated in
approximate linear time. The definition is as follows:
$Q(G)=1-\frac{L(G)}{L(G_{r})}$ (2)
$L(G)$ means the average random length in graph $G$, and $L(G_{r})$ means the
average random length in graph $G_{r}$ which is the configure model of $G$
with the same degree sequence of $G$. This measure removes the influence of
degree and reflects community structure of different networks.
There are some difference performance in real networks between the Random Walk
Modularity and the modularity [18] given by Newman which we called Newman
Modularity in this paper. Firstly, random network are usually thought to have
no community structure, but when $p$ is small, the Newman modularity and
Conductance Modularity (see section3) can be very large. On the contrary
Random Walk Modularity is not affect by $p$ see figure 3, which is more accord
with our intuition. Secondly, some deterministic networks, eg. the ring and
the lattice, have a high Newman Modularity value, but whether they have
community structure is disputable. In the lattice, every vertex has the same
position, so their community structure even if they have does not interest us.
The Random Walk Modularity all remove such networks by a low value, see table
1. Further more, for networks with clear community structure like the planted
$l$-partition networks, Random Walk Modularity always give high value compared
with other networks with fuzzy community structure, see figure 4. Finally, the
time to calculate Random Walk Modularity is mainly determined by the average
random walk length. Most real networks have average random length less than
10, and all of them less than 20 in our experiment, see section 4. So Random
Walk Modularity can be calculated in a nearly linear time which indicate that
it has a advantage for large networks.
Figure 3: Different modularity measures in ER random network, every pair of vertices are linked with probability p, c_m means conductance modularity, n_m means Newman modularity, r_m means random walk modularity Figure 4: Different modularity measures in planted $l$-partition model, p is the ratio between $z_{out}$ and its average degree, c_m means conductance modularity, n_m means Newman modularity, r_m means random walk modularity Table 1: Different modularity measures in some deterministic networks. The ring is a one dimensional lattice, includes 1000 nodes. The tree includes 1000 nodes with all vertices having the same number of children, 2 in our experiments. The lattice have two dimensions, each dimension has 100 nodes modularity$\backslash$network | ring | tree | lattice
---|---|---|---
RandomWalk | 0.003 | 0.0004 | 0.08
Newman | 0.94 | 0.93 | 0.89
Conductance | 0.97 | 0.93 | 0.9
## 3 Algorithm
In the last section, we propose a quick measure to evaluate the community
structure in large networks. Starting a random walk from a given node, one
could easily be trapped in a community. The average random walk length has a
inverse correlation with the community structure. Owing to the community
structure, the average random walk length becomes shorter. Our community
algorithm are based on this idea. We perform a series of random walk from a
given node $s_{0}$, and ends a walk when this walk forms a ring. At the same
time, we assume the tendencies of nodes to the community of $s_{0}$ has a
positive correlation with the ring position. Our algorithm is given in the
following.
* •
RandomWalkRing($G$, $s_{0}$, n)
1. 1.
Set vector $P$ = $\phi$
2. 2.
Perform a random walk from $s_{0}$, record each node passed, when encounter a
ring ends this walk and record its position $l$. $\forall v$ in the random
walk trail, let $P(v)=P(v)+\frac{1}{l*d(v)}$
3. 3.
repeat step2 $n$ times
4. 4.
Order nodes in $P$ by decreasing value $P(v)$, get a support vector S
5. 5.
Compute the conductance $\phi(S_{i})$ of the first $i$ nodes, for i $\leq|S|$
6. 6.
find the index $k^{*}$ at every local optimal of $\phi(S)$, return
$S_{0}=\\{v_{i}|i\leq k^{*}\\}$
Figure 5: The performance of Random Walk Ring and Local Graph Partitioning
algorithm of Andersen
In the above algorithm, we need a quality function to extract the community
from the support vector. We use conductance as the quality function which has
been proposed in [1] and has been used by Leskovec [25]. Conductance has been
popular to measure community structure in recent years [13][15] [16]. Let the
volume $vol(S)$ of a set $S$ be the total degree of vertices in it, i.e.,
$vol(S)=\sum_{v\in S}\deg(v)$. The conductance $\phi(S)$ of a set $S$ is
defined to be the ratio of the number of edges $e(S,\bar{S})$ coming out of
$S$ with the minimum of the volume of itself and the volume of its complement
$\bar{S}$, i.e., $\phi(S)=e(S,\bar{S})/\min\\{vol(S),vol(\bar{S})\\}$. The
conductance of the graph is the minimum conductance over all sets and it is
extensively studied in computer science, with applications to random walks,
spectral or flow based graph partitioning, and combinatorial object
constructions. Intuitively, a set of low conductance (smaller than some
constant $\phi_{0}$) can be thought of a nice community. Using this
definition, lots of interesting phenomenon have been found. Leskovec finds
most networks seems have a ’core’ contains a constant faction of the nodes
with a periphery consisting of a large number of relatively small ’whiskers’
[15].
Let $C_{1}$ and $C_{2}$ are two communities, we use the following community
similarity measures to evaluate our algorithm, where the planted $l$-partition
model is also used.
$S(C_{1},C_{2})=\frac{|C_{1}\cap C_{2}|}{\sqrt{|C_{1}|*|C_{2}|}}.$ (3)
For each real communities, we find the most similar communities return by our
algorithm, see figure 5 for the performance of our algorithm and the algorithm
of [1]. Our algorithm has similar performance when p is small, and a little
less accuracy when p is larger than 0.25. What we emphasize is that our
algorithms is very quick in large networks as pointed before. So such
sacrificing of accuracy is inevitable. Something should be noticed that the
accuracy of our algorithm is affected by n a lot. Generally, the larger n is,
the more the accuracy of our algorithm is. So there is a compromise between
the performance and speed. In our experiment we set n to 1000.
Conductance is a local definition and could not give a global knowledge to
judge whether a network has good community structure or not. We give another
modularity measure which we called Conductance Modularity to differ from
previous ones. Let $c$ be a real number between $0$ and $1$, $f$ is the
corresponding fraction of nodes in community with smaller conductance than
$c$. then Conductance Modularity is as follows.
$C(G)=max_{c\in[0,1]}\sqrt{(1-c)*f}$ (4)
As we know, usually a smaller conductance indicates the corresponding
community is better. A network with good community structure should have as
many as possible nodes in good communities. The Conductance Modularity
considers both community’s quality and nodes number, which could give us a
intuition whether a network has community structure or not from the point of
conductance.
Figure 4 and figure 3 are an comparison among three modularity measures. All
of them are sensitive with community structure in planted $l$-partition model.
While Conductance Modularity and Newman Modularity are affected by $p$ in ER
random networks a lot, Random Walk Modularity is independent on $p$ and seems
more better.
## 4 Application
We perform our algorithm on 34 networks 111We only consider the corresponding
undirected graphs for all the networks. Except the citation_arnerminer network
[22] [21] is from http://arnetminer.org/citation and the football network is
from http://www-personal.umich.edu/~mejn/netdata, all other networks can be
found from http://snap.standford.edu in a acceptable time including some very
large networks. Using our algorithms, we could find more than one communities
from each node indicating different level of communities just as Leskovec do
[25]. In table 2, all the results are calculated from the first local optimal
community for the sake of discussion. As in most case we are more care about
the smallest group includes us, which is always more compact and has more
influence for us although the conductance is not the optimal in global.
Table 2: The statistics of network. RM means Random Walk Modularity, NM means Newman Modularity found by fast greedy algorithm [5], CM means Conductance Modularity, AvgC means the average conductance of all communities, ARL means average random walk length, AvgS means the average size of all communities network | RM | NM | CM | AvgC | ARL | AvgS
---|---|---|---|---|---|---
amazon0302 | 0.25 | 0.82 | 0.79 | 0.18 | 5.86 | 19.95
amazon0312 | 0.31 | 0.8 | 0.73 | 0.24 | 8.77 | 26.29
amazon0505 | 0.31 | 0.76 | 0.73 | 0.24 | 8.86 | 26.42
amazon0601 | 0.32 | 0.74 | 0.73 | 0.24 | 8.95 | 26.96
cit_arnetminer | 0.06 | 0.65 | 0.63 | 0.42 | 6.63 | 16.8
cit_hepph | 0.25 | 0.56 | 0.55 | 0.48 | 18.5 | 28.7
cit_hepth | 0.27 | 0.53 | 0.59 | 0.42 | 18.2 | 31.6
cit_patents | 0.08 | 0.76 | 0.59 | 0.45 | 8.95 | 18.6
col_astroph | 0.34 | 0.51 | 0.63 | 0.39 | 14 | 22.7
col_condmat | 0.29 | 0.64 | 0.74 | 0.32 | 6.75 | 18.3
col_grqc | 0.31 | 0.79 | 0.83 | 0.33 | 5 | 17.7
col_hepph | 0.42 | 0.58 | 0.71 | 0.33 | 11 | 19.9
col_hepth | 0.19 | 0.69 | 0.75 | 0.33 | 5.4 | 16.7
email_enron | 0.18 | 0.5 | 0.57 | 0.47 | 9.34 | 49.1
email_euall | 0.01 | 0.73 | 0.66 | 0.46 | 3.98 | 409
football | 0.17 | 0.57 | 0.88 | 0.14 | 7.2 | 20.9
livejournal | 0.19 | -1 | 0.53 | 0.48 | 15.2 | 19.3
p2p4 | 0.007 | 0.38 | 0.53 | 0.6 | 8.2 | 11.9
p2p5 | 0.006 | 0.4 | 0.54 | 0.59 | 8.1 | 12.6
p2p6 | 0.006 | 0.39 | 0.54 | 0.59 | 8.1 | 12.5
p2p8 | 0.015 | 0.46 | 0.58 | 0.54 | 7.4 | 12.7
p2p9 | 0.014 | 0.46 | 0.58 | 0.54 | 7.3 | 12.5
p2p24 | 0.002 | 0.47 | 0.62 | 0.48 | 5.9 | 11.1
p2p25 | 0.005 | 0.49 | 0.63 | 0.47 | 5.8 | 11.5
p2p30 | 0.005 | 0.5 | 0.62 | 0.46 | 5.8 | 11.3
p2p31 | 0.003 | 0.5 | 0.63 | 0.46 | 5.7 | 11
roadnet_ca | 0.04 | 0.99 | 0.93 | 0.087 | 3.67 | 26.7
roadnet_pa | 0.04 | 0.99 | 0.93 | 0.087 | 3.68 | 26.9
roadnet_tx | 0.04 | 0.99 | 0.93 | 0.1 | 3.63 | 26.2
vikivote | 0.002 | -1 | 0.58 | 0.5 | 4.89 | 496
web_berkstan | 0.54 | 0.91 | 0.65 | 0.32 | 9 | 44.4
web_google | 0.39 | 0.92 | 0.79 | 0.17 | 6.68 | 30.3
web_notredame | 0.35 | 0.93 | 0.76 | 0.16 | 5 | 88.2
web_stanford | 0.47 | 0.88 | 0.65 | 0.35 | 7.9 | 40.8
The Random Walk Modularity has a different interpretation about the community
structure compared other measures. Before discussion, we give the following
classification of network by their corresponding modularity. Random Walk
Modularity in ER random networks is or very near to 0. So networks with Random
Walk Modularity below 0.05 are thought to have no clear community structure,
between 0.05 and 0.1 are thought to have weak community structure, and above
0.1 are thought to have clear community structure. For Newman Modularity and
Conductance Modularity, the boundary of clear community structure are set to
be 0.3 and 0.5 respectively.
From the point of Newman Modularity and Conductance Modularity, all networks
have clear community structure, except livejournal and vikivote networks whose
Newman Modularity could not be calculated in acceptable time by the fast
greedy algorithm[5]. The road, web, amazon and some collaboration, citation
and email networks have high value, while others networks have relative small
value. The Newman Modularity and Conductance Modularity are usually consistent
with each other, which means when one measure give a high score, the other is
always give a high score.
When consider the Random Walk Modularity, the situation is different. Networks
are divided into three classes as discussed before. The road, p2p, vikivote
and email_euall networks have no clear community structure, even the road
networks have the highest Newman Modularity and Conductance Modularity.
Citation_arnetminer and Citation_ patents networks has weak community
structure. Other networks are thought to have clear community structure. Such
difference could be explained by figure 3 and table 1. Those networks with
high Newman Modularity and Conductance Modularity but small Random Walk
Modularity maybe networks like lattice or with very small average degree whose
community structure are debatable. In all, Random Walk Modularity are more
strict to evaluate the community structure.
In table 2, we also give some other properties. As a whole, most networks
tends to have small RWL, small conductance and small communities size. The
small communities size maybe influenced by our selection of the first optimal
conductance. The short AWL is clear, even the most largest networks the
liveJournal online social network and U.S. patent dataset only need about 15
and 9 steps to form a ring. The RWL seems have a upper bound by the average
degree as analysis before. Owing the influence of community structure, real
RWL is always smaller than that value. The results show Random Walk Modularity
are independent on conductance, RWL and community size. If a network has high
Random Walk Modularity value, we are more believe it has community structure.
## 5 Conclusion
In this paper, we propose a method to evaluate community structure and a local
community algorithm based on random walks with approximately linear running
time. Our experiments show the average random walk length are affected by two
factors, the average degree of the graph and community structure. Average
random walk length are very short in real networks, which is either caused by
networks’ sparseness or community structure or both. Such short average random
walk guarantees Random Walk Modularity could be calculated in near linear
time. We also give a modularity measure from the conductance view, which gives
us a profile about a large networks. Usually the Conductance Modularity and
Newman Modularity are consistent in our experiment, while Random Walk
Modularity could give a different judge. Random Walk Modularity has
advantageous both in evaluating the community structure and speed. Networks
with high Random Walk Modularity are more believable to have good community
structure, while Newman Modularity and Conductance Modularity could also give
some ER random network high value. So Random Walk Modularity should be used
when we cared about the network community structure without the debatable
community.
The running time of random ring algorithm is mainly influenced random walk
length and the random walk number. The former are influence by average degree
and community structure and is usually small, less than 20 in all network in
our experiments. N could be set by user where both accuracy and speed should
be considered. Our results show, with some little accuracy sacrifice we could
improve the algorithm’s speed a lot. The random ring algorithm could be used
on very large networks with millons or billons of nodes.
In the future, we will study the evolution of community structure and explain
why networks form different structures. Methods proposed in this paper could
help disclosed the large network structure a lot.
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|
arxiv-papers
| 2013-01-12T16:49:08 |
2024-09-04T02:49:40.207738
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jiankou Li",
"submitter": "Jiankou Li",
"url": "https://arxiv.org/abs/1301.2698"
}
|
1301.2717
|
# The Spin-down of PSR J0821–4300 and PSR J1210–5226:
Confirmation of Central Compact Objects as Anti-Magnetars
E. V. Gotthelf, J. P. Halpern, and J. Alford Columbia Astrophysics
Laboratory, Columbia University, 550 West 120th Street, New York, NY 10027
###### Abstract
Using XMM–Newton and Chandra, we measure period derivatives for the second and
third known pulsars in the class of Central Compact Objects (CCOs) in
supernova remnants, proving that these young neutron stars have exceptionally
weak dipole magnetic field components. For the 112 ms PSR J0821$-$4300 in
Puppis A, $\dot{P}=(9.28\pm 0.36)\times 10^{-18}$. Its proper motion,
$\mu=61\pm 9$ mas yr-1, was also measured using Chandra. This contributes a
kinematic term to the period derivative via the Shklovskii effect, which is
subtracted from $\dot{P}$ to derive dipole $B_{s}=2.9\times 10^{10}$ G, a
value similar to that of first measured CCO PSR J1852$+$0040 in Kes 79, which
has $B_{s}=3.1\times 10^{10}$ G. Antipodal surface hot spots with different
temperatures and areas are deduced from the X-ray spectrum and pulse profiles.
Paradoxically, such nonuniform surface temperature appears to require strong
crustal magnetic fields, probably toroidal or quadrupolar components much
stronger than the external dipole. A spectral feature, consisting of either an
emission line at $\approx 0.75$ keV or absorption at $\approx 0.46$ keV, is
modulated in strength with the rotation. It may be due to a cyclotron process
in a magnetic field on the surface that is slightly stronger than the dipole
deduced from the spin-down. We also timed anew the 424 ms PSR J1210$-$5226,
resolving previous ambiguities about its spin-down rate. Its $\dot{P}=(2.22\pm
0.02)\times 10^{-17}$, corresponding to $B_{s}=9.8\times 10^{10}$ G. This is
compatible with a cyclotron resonance interpretation of its prominent
absorption line at 0.7 keV and harmonics. These results deepen the mystery of
the origin and evolution of CCOs: why are their numerous descendants not
evident?
###### Subject headings:
ISM: individual (Puppis A) — pulsars: individual (PSR J0821$-$4300, PSR
J1210$-$5226, PSR J1852$+$0040) — stars: neutron
## 1\. Introduction
The class of faint X-ray sources in supernova remnants (SNRs) known as central
compact objects (CCOs) are characterized by steady flux, predominantly surface
thermal X-ray emission, lack of a surrounding pulsar wind nebula, and absence
of detection at any other wavelength. Table 1 lists basic data on the well-
studied CCOs, as well as proposed candidates whose qualifications are not yet
well established. Of the eight most secure CCOs, three are known to be neutron
stars (NSs) with spin periods of 0.105, 0.424, and 0.112 s. Spin-down has been
detected for two of these, the 0.105 s pulsar PSR J1852$+$0040 in Kes 79 and
the 0.424 s pulsar PSR J1210$-$5226 in the SNR PKS 1209$-$51/52. For PSR
J1852$+$0040, the implied surface dipole field is only $B_{s}=3.1\times
10^{10}$ G (Halpern & Gotthelf, 2010a), smaller than that of any other young
known NS. In the case of PSR J1210$-$5226, archival data allow two alternative
timing solutions, with $B_{s}=9.9\times 10^{10}$ or $2.4\times 10^{11}$ G
(Halpern & Gotthelf, 2011).
It is natural to assume that CCOs that have not yet been seen to pulse are
isolated, weakly magnetized NSs of the same class as the CCO pulsars. Where
pulsar searches have been unsuccessful, it is possible that an even weaker
magnetic field, a more uniform surface temperature, or an unfavorable viewing
geometry, prevents detection of rotational modulation. The absence of
pulsations from the youngest known NS, the $\approx 330$ year old CCO in
Cassiopeia A, has been used, in combination with fitting of its X-ray
spectrum, to argue that it is covered with a uniform temperature, non-
magnetized atmosphere of carbon, the product of nuclear burning of H and He
(Ho & Heinke, 2009). Rapid cooling of the NS in Cas A, directly detected by
Chandra (Heinke & Ho, 2010), has been interpreted as evidence for neutron
superfluidity in the core (Page et al., 2011; Shternin et al., 2011).
Table 1Central Compact Objects in Supernova Remnants
CCO | SNR | Age | $d$ | $P$ | $f_{p}$aaUpper limits on pulsed fraction are for a search down to $P=12$ ms or smaller. | $B_{s}$ | $L_{x,\rm bol}$ | References
---|---|---|---|---|---|---|---|---
| | (kyr) | (kpc) | (s) | (%) | ($10^{10}$ G) | (erg s-1) |
RX J0822.0$-$4300 | Puppis A | 4.5 | 2.2 | 0.112 | 11 | 2.9 | $5.6\times 10^{33}$ | 1,2,3,4,5,6
CXOU J085201.4$-$461753 | G266.1$-$1.2 | 1 | 1 | … | $<7$ | … | $2.5\times 10^{32}$ | 7,8,9,10,11
1E 1207.4$-$5209 | PKS 1209$-$51/52 | 7 | 2.2 | 0.424 | 9 | 9.8 | $2.5\times 10^{33}$ | 6,12,13,14,15,16,17
CXOU J160103.1$-$513353 | G330.2$+$1.0 | $\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower 4.0pt\hbox{$\sim$}}\hss}\hbox{$>$}}}3$ | 5 | … | $<40$ | … | $1.5\times 10^{33}$ | 18,19
1WGA J1713.4$-$3949 | G347.3$-$0.5 | 1.6 | 1.3 | … | $<7$ | … | $\sim 1\times 10^{33}$ | 11,20,21
XMMU J172054.5$-$372652 | G350.1$-$0.3 | 0.9 | 4.5 | … | … | … | $3.9\times 10^{33}$ | 22,23
CXOU J185238.6$+$004020 | Kes 79 | 7 | 7 | 0.105 | 64 | 3.1 | $5.3\times 10^{33}$ | 24,25,26,27
CXOU J232327.9$+$584842 | Cas A | 0.33 | 3.4 | … | $<12$ | … | $4.7\times 10^{33}$ | 27,28,29,30,31,32,33
2XMMi J115836.1$-$623516 | G296.8$-$0.3 | 10 | 9.6 | … | … | … | $1.1\times 10^{33}$ | 34
XMMU J173203.3$-$344518 | G353.6$-$0.7 | $\sim 27$ | 3.2 | … | $<9$ | … | $1.3\times 10^{34}$ | 35,36,37,38
CXOU J181852.0$-$150213 | G15.9$+$0.2 | $1-3$ | (8.5) | … | … | … | $\sim 1\times 10^{33}$ | 39
Note. — Above the line are eight well-established CCOs. Below the line are
three candidates.
References. — (1) Hui & Becker 2006a; (2) Gotthelf & Halpern 2009; (3)
Gotthelf et al. 2010; (4) De Luca et al. 2012; (5) Becker et al. 2012; (6)
this paper; (7) Slane et al. 2001; (8) Kargaltsev et al. 2002; (9) Bamba et
al. 2005; (10) Iyudin et al. 2005; (11) De Luca 2008; (12) Zavlin et al. 2000;
(13) Mereghetti et al. 2002; (14) Bignami et al. 2003; (15) De Luca et al.
2004; (16) Gotthelf & Halpern 2007; (17) Halpern & Gotthelf 2011; (18) Park et
al. 2006; (19) Park et al. 2009; (20) Lazendic et al. 2003; (21) Cassam-Chenaï
et al. 2004; (22) Gaensler et al. 2008; (23) Lovchinsky et al. 2011; (24)
Seward et al. 2003; (25) Gotthelf et al. 2005; (26) Halpern et al. 2007; (27)
Halpern & Gotthelf 2010a; (28) Pavlov et al. 2000; (29) Chakrabarty et al.
2001; (30) Mereghetti et al. 2002; (31) Pavlov & Luna 2009; (32) Ho & Heinke
2009; (33) Heinke & Ho 2010; (34) Sánchez-Ayaso et al. 2012; (35) Tian et al.
2008; (36) Abramowski et al. 2011; (37) Halpern & Gotthelf 2010b; (38) Halpern
& Gotthelf 2010c; (39) Reynolds et al. 2006.
The “anti-magnetar” explanation of CCOs, which is motivated by their weak
magnetic fields, absence of variability, and location on the $P-\dot{P}$
diagram, remains incomplete in detail. Specifically, it does not yet account
for the hot spots that are seen on the surfaces of CCO pulsars. Since the
spin-down power of a CCO pulsar is less than its X-ray luminosity, the latter
must be thermal emission from residual cooling, which can only be nonuniform
if there is anisotropic heat conduction. In the absence of strong magnetic
fields or magnetospheric activity, it is difficult to reproduce the light
curve and pulsed fraction of 64% from PSR J1852$+$0040 in Kes 79 (Halpern &
Gotthelf, 2010a; Shabaltas & Lai, 2012), or the two antipodal hot spots of
different temperatures and areas on RX J0822$-$4300 in Puppis A (Gotthelf &
Halpern, 2009; Gotthelf et al., 2010). The latter 0.112 s pulsar, hereafter
PSR J0821$-$4300, is a subject of this paper. Its spectrum is especially
puzzling in also displaying a phase-dependent emission feature at 0.7–0.8 keV
(Gotthelf & Halpern, 2009), which is reported to be variable in the long term
(De Luca et al., 2012).
Here we report the first spin-down measurement for PSR J0821$-$4300, based on
a dedicated program of phase-coherent timing jointly scheduled between
XMM–Newton and Chandra. It was also necessary to incorporate Chandra HRC
observations of the position and proper motion of PSR J0821$-$4300 in order to
determine its small period derivative accurately. The astrometric analysis is
described in Section 2. (The latter work was also carried out, with consistent
results, by Becker et al. 2012.) The results of the timing are presented in
Section 3. The X-ray flux and spectra are discussed in Section 4, with
particular attention paid to the spectral line and the question of its
possible variability. We also obtained new timing observations of PSR
J1210$-$5226 that resolve the prior ambiguity about its spin-down rate in
favor of the smaller value; this definitive result is presented in Section 5.
The nature of CCOs as anti-magnetars, and their possible evolutionary status,
are discussed in Section 6. Conclusions and proposals for future work follow
in Section 7.
## 2\. X-ray Position and Proper Motion
Evidence that PSR J0821$-$4300 has high proper motion from Chandra HRC images
over a 5 year baseline was reported by Hui & Becker (2006b) and Winkler &
Petre (2007), but with slightly disparate measurements of $\mu=107\pm 34$ mas
yr-1 and $\mu=165\pm 25$ mas yr-1, respectively, from the same data. Here, we
are concerned with timing this high-velocity pulsar over an extended period of
time with millisecond accuracy. When trying to measure a small $\dot{P}$ using
X-ray data, position and proper motion can contribute significant errors via
three effects. The first is an instrumental property of the Chandra CCDs when
used in continuous-clocking (CC) mode (see Section 3); the position of the
pulsar must be known a priori to $<0.\\!^{\prime\prime}5$ in order to
determine the time of arrival of each source photon. The second consideration
is the accuracy of the barycentric correction. The third effect is the
magnitude of the proper motion, which contributes a purely kinematic period
derivative via the “train whistle” effect (Shklovskii, 1970). The original
measurements of proper motion are not accurate enough to measure this effect,
which is crucial in the case of PSR J0821$-$4300.
Accordingly, we have reanalyzed the position and proper motion of PSR
J0821$-$4300 using the Chandra HRC-I data listed in Table 2, which now
includes a more recent pair of observations in 2010 August that extends the
baseline to 10.6 yr, enabling higher precision on both the contemporary
position for timing, and the proper motion. We will describe here any
differences between our method and previous work. For example, we did not use
an HRC-S observation (ObsID 1851) because of known systematic differences
between HRC-S and HRC-I. Ultimately, however, our results are consistent with
the recent analysis of the same data (including ObsID 1851) by Becker et al.
(2012).
Table 2Log of Chandra HRC-I Observations of PSR J0821$-$4300 ObsID | Date | Start Epoch | Exposure | Roll angle | Star A | PSR J0821$-$4300
---|---|---|---|---|---|---
| (UT) | (MJD) | (ks) | (∘) | (Counts)aaTotal counts collected in a $1.\\!^{\prime\prime}5$ radius aperture centered on the source. | (Counts)aaTotal counts collected in a $1.\\!^{\prime\prime}5$ radius aperture centered on the source.
749 | 1999 Dec 21 | 51533.95 | 18.0 | 338.7 | 47 | 3257
4612 | 2005 Apr 25 | 53485.31 | 40.2 | 261.9 | 123 | 7260
11819 | 2010 Aug 10 | 55418.72 | 33.7 | 163.4 | 101 | 5455
12201 | 2010 Aug 11 | 55419.13 | 38.7 | 162.9 | 117 | 6296
The data from all epochs were reprocessed and analyzed using the latest
calibration files and software (CIAO 4.4/CALDB 4.4.8). This processing
accounts for the HRC AMP_SF electronic ringing distortions discussed by Hui &
Becker (2006b). The HRC detector is well suited to astrometry, with its
processed pixel size of $0.\\!^{\prime\prime}1318$ that oversamples the on-
axis point spread function (PSF) by a factor of 5. For all observations, the
pulsar was placed close to the optical axis where the PSF is essentially
symmetric. In the following analysis we assume, as there is no evidence to the
contrary, that the HRC focal plane is linear and the aspect reconstruction
introduces no errors in roll angle. The two pointings on consecutive days in
2010 August are sufficiently different in their aspect reconstruction that we
analyze them individually.
The nominal uncertainty in aspect reconstruction for a typical Chandra
observation is $0.\\!^{\prime\prime}6$. It is often possible to remove most of
this systematic error by using nearby X-ray point sources with precisely
measured optical coordinates to correct the absolute astrometry. Hui & Becker
(2006b) used their X-ray detected “star A” as their sole fiducial point, and
fitted a model PSF interpolated from the CIAO library appropriate for its
position and estimated photon energy to determine its position. Winkler &
Petre (2007) used this star and two additional stars, and followed the updated
method outlined in the CIAO thread for generating a Monte Carlo PSF using the
CHaRT/MARX software for input into the CIAO/Sherpa spectral/image fitting
software package.
Figure 1.— Chandra HRC-I images around reference star A (3UC094-058669) used
to define the coordinate system for position and proper motion of PSR
J0821$-$4300\. Each panel shows the observed counts from Star A (left) and the
simulated PSF (right) for its location on the focal plane in native HRC-I
pixels. The plots cover $12^{\prime\prime}\times 12^{\prime\prime}$ in
celestial coordinates. The total counts in a $1.\\!^{\prime\prime}5$ radius
aperture are given in Table 3.
In our analysis, we also follow the CIAO thread to characterize the HRC-I PSF,
but we adopt a simpler approach to measuring source locations, one that is not
dependent on model fitting in the image domain. Our method is guided by the
following observations. First, the on-axis, symmetric image of the pulsar
contains enough counts that a simple centroid calculation is a sufficiently
accurate measurement of its position on the detector. Second, star A of Hui &
Becker (2006b), which lies $2\farcm 7$ from PSR J0821$-$4300, is the only
useful fiducial source for registering the X-ray image. The position and
proper motion of star A are taken from the UCAC3 (Zacharias et al., 2010),
where it is listed as 3UC094-058669 with coordinates (J2000.0) R.A.=$08^{\rm
h}21^{\rm m}46.\\!^{\rm s}2924(16)$,
decl.=$-43^{\circ}02^{\prime}03.\\!^{\prime\prime}640(49)$, and proper motion
$\mu_{\alpha}\,{\rm cos}\,\delta=-14.3(2.0)$, $\mu_{\delta}=-3.6(5.5)$ mas
yr-1. X-ray position measurements of the two weaker, off-axis stars used by
Winkler & Petre (2007) only add to the uncertainty (as quantified below) in
the absolute astrometry. Third, the few X-ray photons from star A, and its
broad off-axis PSF, do not warrant a sophisticated image fitting technique.
Instead, we use a “corrected centroid” method, as described below.
To determine the source location of star A in the X-ray images we start with
the CHaRT/MARX simulation of the PSF, as described in the CIAO user
webpages111http://cxc.harvard.edu/chart, for its respective locations on the
focal plane. Figure 1 shows the distribution of the counts from each HRC image
and corresponding Monte Carlo PSF. It is apparent that star A is poorly
sampled in the data, with total source counts in the range $47\leq N\leq 123$
(Table 2). The maximum number of counts per pixel is typically only $2-4$,
making forward fitting poorly constrained statistically, while sharp features
in the model can cause systematic offsets. Furthermore, the source is immersed
in a substantial diffuse background from the Puppis A SNR which, although it
only contributes a few photons over the source region, adds uncertainty to the
position measurement because of the small count statistics, especially when
fitting over a larger area.
Figure 2.— Four position measurements of PSR J0821$-$4300 spanning 10.6 yr,
after correction using the optical/X-ray star A (3UC094-058669) as a
reference. The two observations in 2010 nearly coincide in time. The positions
(diamonds) and their errors are fitted with a linear model (dashed lines). The
fitted parameters are listed in Table 4.
To sidestep these effects, the coordinates of star A are first found from its
centroid. Photons were extracted from circular aperture of radius
$1.\\!^{\prime\prime}5$, chosen to minimize the background counts. This
aperture encloses essentially all of the signal in that fraction of the PSF
with a finite probability of producing a single count during the observation.
This measurement was made using the CIAO tool dmstat and is iterated to
produce the final coordinates. However, while this results in a well-defined
and statistically meaningful measurement, it does not account for the shape of
the complex off-axis PSF, whose orientation depends on the spacecraft roll
angle (which differs for each observation; see Figure 1), or for the monotonic
off-axis deviations introduced by the flat focal plane. To quantify these
systematic effects we also measure the centroid of the simulated PSF of star A
in sky coordinates in each image and compare it to the coordinates that were
input to CHaRT/MARX for that PSF. The difference constitutes the small, but
critical correction to the centroid of star A.
We estimated the uncertainty in the derived coordinates of star A using a
Monte Carlo method. We generated 500 realizations of star A by sampling the
PSF using a random number generator to match the observed counts, and
accumulated the centroid measurements to build up a distribution in right
ascension and declination. To account for the observed background, we included
a random distribution of photons within the source aperture. The resulting
(Gaussian) width of the distribution of centroids is typically $\sigma\approx
0.\\!^{\prime\prime}06$ in each of the two coordinates (see Table 3). These
reproduce the expected “standard error” for a centroid,
$\approx{\sigma}/\sqrt{N}$. We also simulated the uncertainties for the two
other fiducial sources used in Winkler & Petre (2007) and find that their
inclusion in the position determination would only increase the uncertainty on
the final coordinates.
The pulsar itself is a strong source (see Table 2) whose coordinates are
precisely measured using a centroid calculation with uncertainty an order of
magnitude smaller than those measured for star A. Because of the symmetry of
its on-axis PSF, no systematic correction is required for the pulsar. Its
coordinates are then adjusted by the difference between the optical and X-ray
coordinates of star A at each epoch to produce its final astrometric position
in Table 3. Fitting the position of the pulsar as a function of time then
yields the proper motion. Figure 2 shows the $\chi^{2}$ fit with constant
velocity in each coordinate, and Table 4 lists the formal solution and
quantities derived from it. The derived total proper motion of $61\pm 9$ mas
yr-1 is in agreement with the value determined by Becker et al. (2012), $71\pm
12$ mas yr-1, and is smaller than previously published numbers for reasons
discussed in that paper.
Table 3Position Measurements for PSR J0821$-$4300
Epoch | Star A (Optical) | Star A (X-ray) | PSR J0821$-$4300 (corrected)
---|---|---|---
(year) | R.A. (h m s) | Decl. (${}^{\circ}\ {}^{\prime}\ {}^{\prime\prime}$) | R.A. (h m s) | Decl. (${}^{\circ}\ {}^{\prime}\ {}^{\prime\prime}$) | R.A. (h m s) | Decl. (${}^{\circ}\ {}^{\prime}\ {}^{\prime\prime}$)
1999.975 | 08 21 46.2928(16) | –43 02 03.637(49) | 08 21 46.2882(65) | -43 02 03.397(83) | 08 21 57.4024(67) | -43 00 16.894(96)
2005.316 | 08 21 46.2858(21) | –43 02 03.657(57) | 08 21 46.3011(47) | -43 02 03.756(41) | 08 21 57.3685(52) | -43 00 17.023(71)
2010.610 | 08 21 46.2789(31) | –43 02 03.676(76) | 08 21 46.2621(62) | -43 02 04.113(47) | 08 21 57.3488(69) | -43 00 17.185(89)
2010.611 | 08 21 46.2789(31) | –43 02 03.676(76) | 08 21 46.2575(57) | -43 02 04.068(51) | 08 21 57.3476(65) | -43 00 17.190(92)
Note. — All coordinates are equinox J2000. Optical coordinates for star A
(3UC094-058669) are corrected for the epoch of proper motion. The X-ray
position of star A is determined using the method described in the text. The
pulsar coordinates are corrected by the difference between the optical and
X-ray coordinates of star A. Uncertainties on the last digits are in
parentheses.
The tangential velocity of PSR J0821$-$4300 then depends on a distance
determination for Puppis A, which has ranged from 1 to 2.5 kpc according to
various methods, and is a matter of unresolved debate in the most recent
studies quoted here. One (Reynoso et al., 1995, 2003) is based on 21 cm H I
velocities, and a perceived morphological association of features in H I
surrounding the pulsar and the SNR at $v_{\rm lsr}=+16$ km s-1, the latter
corresponding to $d=2.2\pm 0.3$ kpc. The second method uses spectra of ground-
state hydroxyl lines (Woermann et al., 2000), which show absorption at $v_{\rm
lsr}<+7.6$ km s-1, and emission above this velocity, from which
$d=1.3^{+0.6}_{-0.8}$ kpc is derived. Both sets of authors employ assumptions
that are not mutually accepted, and are beyond the scope of this paper to
evaluate. We will adopt a fiducial distance of $2.2\pm 0.3$ kpc for purposes
of further calculations, while noting the implications of a possible smaller
distance where relevant.
Table 4Ephemeris of PSR J0821$-$4300 Parameter | Value
---|---
Position and Proper Motion
Epoch of position and $\mu$ (MJD) | 53964.0
R.A. (J2000) | $08^{\rm h}21^{\rm m}57.\\!^{\rm s}3653(31)$
Decl. (J2000) | $-43^{\circ}00^{\prime}17.\\!^{\prime\prime}074(43)$
R.A. proper motion, $\mu_{\alpha}\,{\rm cos}\,\delta$ | $-54.1\pm 8.3$ mas yr-1
Decl. proper motion, $\mu_{\delta}$ | $-28.1\pm 10.5$ mas yr-1
Total proper motion, $\mu$ | $61.0\pm 8.8$ mas yr-1
Position angle of proper motion | $242.\\!^{\circ}5\pm 9.\\!^{\circ}5$
Tangential velocityaaAssuming $d=2.2\pm 0.3$ kpc (Reynoso et al., 1995)., $v_{\perp,c}$ | $629\pm 126$ km s-1
Timing Solution
Epoch of ephemeris (MJD TDB)bbEpoch of fitted minima of the $1.5-4.5$ keV pulse profile; phase zero in Figure 5. | 55580.0000006
Span of ephemeris (MJD) | 55,182–56,027
Frequency, $f$ | 8.86529105448(32) Hz
Frequency derivative, $\dot{f}$ | $(-7.29\pm 0.28)\times 10^{-16}$ Hz s-1
Period, $P$ | 0.1127994550720(41) s
Period derivative, $\dot{P}$ | $(9.28\pm 0.36)\times 10^{-18}$
Kinematic period derivativeaaAssuming $d=2.2\pm 0.3$ kpc (Reynoso et al., 1995)., $\dot{P}_{k}$ | $(2.24\pm 0.72)\times 10^{-18}$
Intrinsic period derivativeaaAssuming $d=2.2\pm 0.3$ kpc (Reynoso et al., 1995)., $\dot{P}_{\rm int}$ | $(7.04\pm 0.80)\times 10^{-18}$
Surface dipole magnetic field, $B_{s}$ | $2.9\times 10^{10}$ G
Spin-down luminosity, $\dot{E}$ | $1.9\times 10^{32}$ erg s-1
Characteristic age, $\tau_{c}$ | 254 Myr
The tangential velocity of PSR J0821$-$4300 at $d=2.2\pm 0.3$ kpc is
$v_{\perp}=636\pm 126$ km s-1. This velocity is at the high end of the
distribution of two-dimensional velocities of pulsars measured by Hobbs et al.
(2005), who find mean values of $\bar{v}_{\perp}=246\pm 22$ km s-1 for 121
ordinary (non-recycled) pulsars, and $\bar{v}_{\perp}=307\pm 47$ km s-1 for 46
pulsars whose characteristic ages are $<3$ Myr. The individual pulsar
velocities are corrected for the solar motion with respect to the local
standard of rest and a flat Galactic rotation curve in order to express them
in the frame of the rotating Galactic disk. In the case of PSR J0821$-$4300
this correction is dominated by the solar motion for the range of plausible
distances, and is only $-7$ km s-1, resulting in a corrected tangential
velocity of $v_{\perp,c}=629$ km s-1. If the distance is as small as 1 kpc,
this reduces to an unexceptional 290 km s-1.
The Shklovskii (1970) effect, a purely kinematic contribution to the observed
period derivative, will be significant. For a source moving at constant
velocity the kinematic contribution is
$\dot{P}_{k}\,={\mu^{2}\,P\,d\over c}\ =\ {v^{2}_{\perp}\,P\over d\,c}\ .$
$None$
From the proper motion measurement of PSR J0821$-$4300, $\dot{P}_{k}=(2.24\pm
0.72)\times 10^{-18}$ is calculated, where we have propagated the
uncertainties on both $\mu$ and $d$.
## 3\. X-ray Timing
Previous observations of PSR J0821$-$4300 were only able to set upper limits
on its period derivative (Gotthelf & Halpern, 2009; Gotthelf et al., 2010; De
Luca et al., 2012). Evidently $\dot{P}$ is so small that it can only be
measured by phase-coherent timing. Accordingly, we designed a sequence of
observations coordinated between XMM–Newton and Chandra that would start and
maintain phase connection over a 2 year span, 2010 May – 2012 April. The
scheduling strategy is the same as was used for PSR J1852$+$0040 in Kes 79
(Halpern & Gotthelf, 2010a). By design, the resulting ephemeris was also
connected backward to archival observations that were obtained in 2009
December and 2010 April (De Luca et al., 2012), which extended the time span
to 2.3 years. All of the timing observations used in this analysis are listed
in Table 5. The discovery observations from 2001 (Gotthelf & Halpern, 2009)
are not included, as they are too far removed in time to be reliably
connected.
Table 5Log of X-ray Timing Observations of PSR J0821$-$4300 Mission | Instr/Mode | ObsID | Date | Elapsed time/ | Start Epoch | PeriodaaBarycentric period derived from a $Z^{2}_{1}$ test. The Leahy et al. (1983) uncertainty on the last digits is in parentheses. | $Z^{2}_{1}$
---|---|---|---|---|---|---|---
| | | (UT) | Livetime (ks) | (MJD) | (s) |
XMM | EPIC-pn/SW | 0606280101 | 2009 Dec 17,18 | 85.1/54.9 | 55182.820 | 0.112799488(12) | 173.0
XMM | EPIC-pn/SW | 0606280201 | 2010 Apr 05 | 42.2/29.4 | 55291.377 | 0.112799451(20) | 199.1
XMM | EPIC-pn/SW | 0650220201 | 2010 May 02 | 28.0/19.6 | 55318.782 | 0.112799390(41) | 135.5
Chandra | ACIS-S3/CC | 12108 | 2010 Aug 16 | 34.0/34.0 | 55424.625 | 0.112799470(21) | 192.3
XMM | EPIC-pn/SW | 0650220901 | 2010 Oct 15 | 23.5/16.4 | 55484.109 | 0.112799519(44) | 147.0
XMM | EPIC-pn/SW | 0650221001 | 2010 Oct 15 | 23.5/16.4 | 55484.987 | 0.112799462(39) | 156.7
XMM | EPIC-pn/SW | 0650221101 | 2010 Oct 19 | 26.5/18.6 | 55488.332 | 0.112799518(40) | 150.5
XMM | EPIC-pn/SW | 0650221201 | 2010 Oct 25 | 24.5/17.2 | 55494.228 | 0.112799486(35) | 162.1
XMM | EPIC-pn/SW | 0650221301 | 2010 Nov 12 | 23.5/16.5 | 55512.524 | 0.112799391(52) | 144.7
XMM | EPIC-pn/SW | 0650221401 | 2010 Dec 20 | 27.2/19.0 | 55550.159 | 0.112799450(35) | 163.3
Chandra | ACIS-S3/CC | 12109 | 2011 Feb 04 | 33.0/33.0 | 55596.837 | 0.112799445(27) | 173.4
XMM | EPIC-pn/SW | 0650221501 | 2011 Apr 12 | 30.0/21.0 | 55663.857 | 0.112799449(30) | 156.6
XMM | EPIC-pn/SW | 0657600101 | 2011 May 18 | 36.5/25.6 | 55699.925 | 0.112799480(17) | 195.6
Chandra | ACIS-S3/CC | 12541 | 2011 Aug 11 | 33.0/33.0 | 55784.655 | 0.112799412(22) | 180.4
XMM | EPIC-pn/SW | 0657600201 | 2011 Nov 08 | 37.2/26.1 | 55873.289 | 0.112799459(28) | 142.2
Chandra | ACIS-S3/CC | 12542,14395 | 2012 Feb 18,19 | 33.1/33.1 | 55975.446 | 0.112799481(35) | 165.8
XMM | EPIC-pn/SW | 0657600301 | 2012 Apr 10 | 35.3/24.7 | 56027.022 | 0.112799445(17) | 196.5
The Chandra observations used the Advanced Camera for Imaging and Spectroscopy
(ACIS-S3) in continuous-clocking (CC) mode to provide time resolution of 2.85
ms. All data were reprocessed from the level 1 event files with the
coordinates corrected for the proper motion of PSR J0821$-$4300 given in Table
4, and analyzed using the latest calibration files and software (CIAO
4.4/CALDB 4.4.8). Reprocessing with a source position that is accurate to $<1$
pixel ($<0.\\!^{\prime\prime}5$) ensures that the time assignment is precise
to $\lesssim 3$ ms. All of the XMM–Newton observations used the pn detector of
the European Photon Imaging Camera (EPIC-pn) in “small window” (SW) mode to
achieve 5.7 ms time resolution, and an absolute uncertainty of $\approx 3$ ms
on the arrival time of any photon. Each data set was examined and cleaned of
intervals of high particle background due to solar activity, as necessary. Two
pairs of data sets that were acquired on consecutive days were merged to
improve their statistics. The photon arrival times from all data were
transformed to Barycentric Dynamical Time (TDB) using the coordinates of the
pulsar corrected for proper motion.
For PSR J0821$-$4300, the pulsed signal strength is a strong function of
energy, not only because of its spectrum relative to the background but
because of the cancellation by emission from the opposite pole, as described
in Gotthelf et al. (2010). For each observation listed in Table 5 we extracted
source photons using an aperture centered on the source and optimized for the
signal strength in the hard $1.5-4.5$ keV energy band. For the XMM–Newton
observations we used an aperture of radius of $30^{\prime\prime}$. For the
Chandra CC-mode observations we selected five columns
($2.\\!^{\prime\prime}4$). We also examined the soft, phase shifted $0.5-1.0$
keV band. However, these data are noisier and their use did not in the end
improve the timing results significantly. The phase cancellation effect also
prevents pulsations from being detected by the Chandra HRC, which has
insufficient energy resolution.
As in our previous timing studies of CCOs (Halpern & Gotthelf, 2010a, 2011),
we employed two complementary approaches to fitting an ephemeris. First, we
used the $Z^{2}_{1}$ (Rayleigh) test (Strutt, 1880; Buccheri et al., 1983) in
a coherent analysis of the entire set of 17 observations. Beginning with the
closely spaced set spanning 2010 October 15–19, the $Z^{2}_{1}$ test
determined the pulse frequency with sufficient accuracy to connect in phase
uniquely to the next observation. This procedure was iterated by adding each
subsequent observation, and including a frequency derivative when it became
evident. We also worked backward in time, incorporating all 17 observations in
the resulting unique ephemeris. The fitted frequency derivative is
$\dot{f}=(-6.94\pm 0.28)\times 10^{-16}$ Hz s-1, where the $1\sigma$
uncertainty comes from the $\Delta Z^{2}_{1}=-2.3$ contour around the peak
power in ($f,\dot{f}$) space.
The second method also started with the $Z^{2}_{1}$ test statistic, this time
to find the period and pulse profile separately at each epoch. The 17 profiles
were cross-correlated, shifted, and summed to create a master pulse profile
template. The process was iterated to generate a more accurate template and a
set of time-of-arrival (TOA) measurements and their uncertainties for each
epoch. These TOAs were fitted with a quadratic model in frequency and
frequency derivative using a $\chi^{2}$ fitting routine to minimize their
phase residuals. We searched for a coherent phase-connected solution over a
grid of $f$ and $\dot{f}$ covering the range $f=8.8652906\pm 0.0000016$ Hz (at
epoch MJD 55,580) and $-3.1\times 10^{-14}<\dot{f}<1.9\times 10^{-14}$ Hz s-1,
with an oversampling factor of 10 for accuracy. This range corresponds to the
$3\sigma$ limits of an incoherent fit to all of the measured frequencies,
including the 2001 discovery observations. The resulting frequency derivative
from TOA fitting, $\dot{f}=(-7.29\pm 0.28)\times 10^{-16}$ Hz s-1, is
consistent with the value found above from the coherent $Z^{2}_{1}$ search. We
adopt the TOA result for the final timing solution listed in Table 4.
Figure 3.— $P-\dot{P}$ diagram of isolated pulsars (dots), binary radio
pulsars (circled dots), and other types of isolated X-ray pulsars (colored
symbols). The CCO pulsars (red stars) in Kes 79 and Puppis A have virtually
the same spin parameters. The upper limit on Calvera’s $\dot{P}$ is from
Halpern (2011). The radio pulsar death line $B/P^{2}=1.7\times 10^{11}$ G s-2
of Bhattacharya et al. (1992) is indicated. The van den Heuvel (1987) spin-up
limit for recycled pulsars corresponds to $P({\rm ms})=1.9\,(B/10^{9}\,{\rm
G})^{6/7}$. The exponent in this equation corrects a typographical error in
the caption to Figure 7 of Halpern & Gotthelf (2010a), although the
corresponding line in the Figure was correct.
The observed $\dot{P}=(9.28\pm 0.36)\times 10^{-18}$ can now be split into the
sum of its intrinsic and kinematic contributions, $\dot{P}=\dot{P}_{\rm
int}+\dot{P}_{k}$. Since we determined in Section 2 that $\dot{P}_{k}=(2.24\pm
0.72)\times 10^{-18}$, the intrinsic period derivative is $\dot{P}_{\rm
int}=(7.04\pm 0.80)\times 10^{-18}$. Parenthetically, we note that the small
observed period derivative is independent evidence that the proper motion of
the pulsar is not as large as the value originally quoted by Winkler & Petre
(2007), $\mu=165\pm 25$ mas yr-1. If so, and if $d=2.2$ kpc, $\dot{P}_{k}$
would be $(1.64\pm 0.55)\times 10^{-17}$, requiring $\dot{P}_{\rm int}$ to be
negative, i.e., the pulsar would be spinning up.
In the vacuum dipole spin-down formalism, the values of $P$ and $\dot{P}_{\rm
int}$ imply a surface magnetic field strength $B_{s}=3.2\times
10^{19}(P\dot{P})^{1/2}~{}G=2.9\times 10^{10}$ G, a spin-down luminosity
$\dot{E}=-I\Omega\dot{\Omega}=4\pi^{2}I\dot{P}/P^{3}=1.9\times 10^{32}$ erg
s-1, and characteristic age $\tau_{c}\equiv P/2\dot{P}=254$ Myr. PSR
J0821$-$4300 is nearly identical in its spin properties to PSR J1852$+$0040 in
Kes 79, as shown in Figure 3. The uncertainties in distance and proper motion
have only a small effect on the derived magnetic field. For a smaller distance
of 1 kpc, $\dot{P}_{k}$ is reduced to $1.02\times 10^{-17}$, and
$B_{s}=3.1\times 10^{10}$ G. An absolute upper limit regardless of distance
and proper motion is $B_{s}<3.3\times 10^{10}$ G.
The phase residuals from the best-fit solution are shown in Figure 4. The
weighted rms of the phase residuals is 5.1 ms, or 0.045 pulse cycles, which is
comparable to the individual measurement errors (average $\sigma=3.6$ ms). It
is not clear if there is any real timing noise and/or systematic errors in the
TOAs.
The light curves of PSR J0821$-$4300 in the soft and hard bands are shown in
Figure 5. These were derived by folding all the timing data on the best
fitting ephemeris given in Table 4. As revealed by a cross-correlation, the
soft and hard pulses are out of phase by $0.45\pm 0.02$ cycles, consistent
with that found by De Luca et al. (2012) using the 2009 December and 2010
April XMM–Newton data.
The energy dependence of the pulsar modulation provided an important
diagnostic for modeling the viewing geometry and surface emission of PSR
J0821$-$4300 (Gotthelf et al., 2010). By analyzing the lightcurve in narrower
energy bands than in Figure 5, we can resolve the signal modulation and phase
for PSR J0821$-$4300 over the $0.3-5$ keV range. The data were grouped into 23
energy bands that are at least $100$ eV in width and have a signal-to-noise
$N_{s}/\sqrt{N_{s}+N_{b}}>100$, where $N_{s},N_{b}$ are the source and
background counts, respectively. We used $Z_{1}^{2}$ to provide a model of the
unbinned lightcurve. The first Fourier component is a reasonable estimate as
the lightcurve is sinusoidal in each energy band to within counting
statistics. The error bar for the phase is calculated by cross-correlation,
with the profile of Figure 5 serving as a template.
The result is presented in Figure 6. As the energy dependent modulation
decreases, the phase becomes undefined in two energy bands; these two phase
points are not plotted. The modulation is qualitatively similar to that
predicted by the antipodal model of Gotthelf et al. (2010) (cf. their Figure
6), providing confirmation of the basic model. The prediction was based on
fitting the modulation in only three energy bands, using much less data, and
differs somewhat from the new, resolved data having an order-of-magnitude more
counts. In particular, the energy of the minimum modulation is lower (1.12 vs.
1.28 keV), and the form of the modulation is more complex than predicted.
Furthermore, the phase is seen to drift at lower energies and the transition
is not as sharp compared to the antipodal case, in which the phase was
statistically either 0.0 or 0.5. The observed characteristics likely imply
that the hotspots are offset from a strictly antipodal geometry. The high
quality data presented herein should allow a far more detailed modeling of the
surface emission of PSR J0821$-$4300.
Figure 4.— For the timing observations of PSR J0821$-$4300 listed in Table 5,
pulse-phase residuals from the linear term (dash-dot line) of the phase
ephemeris presented in Table 4. The quadratic term (solid line) contributes
$\approx\pm 0.5$ cycles to the ephemeris over the 2.3 years of timing.
Figure 5.— Summed pulse profiles of PSR J0821$-$4300 in the $1.5-4.5$ keV band
(top) and the $0.5-1.0$ keV band (bottom) using all of the observations listed
in Table 5, folded according to the ephemeris of Table 4. These hard and soft
pulse profiles are out of phase by $\phi=0.45\pm 0.02$ cycles. The intervals
between the vertical lines ($\Delta\phi=0.3$ cycles) correspond to the two
phase regions used in the phase resolved analysis of Section 4.3 and Table 8.
Figure 6.— Modulation and pulse phase as a function of energy for PSR
J0821$-$4300 using all of the observations listed in Table 5, folded according
to the ephemeris of Table 4. The modulation (top) and pulse phase (bottom)
reproduce the form predicted for the antipodal model of Gotthelf et al.
(2010). Two undefined phase points are omitted.
## 4\. Spectral Analysis
Previously, we modelled the original (year 2001) XMM–Newton spectra of PSR
J0821$-$4300 as surface blackbody emission from two antipodal spots of
different temperatures and areas. Crucially, this model is also able to
account for the observed energy-dependent pulse modulation and phase shift
(Gotthelf & Halpern, 2009; Gotthelf et al., 2010). Additionally, we found
evidence of a spectral line feature around 0.8 keV, which more recent data
obtained by De Luca et al. (2012) suggests has a time-variable centroid
energy. With the increased quantity of data now in hand on PSR J0821$-$4300,
we can re-examine this spectral feature, first by combining all 13 XMM–Newton
observations presented in Table 5 plus the two observations obtained in 2001,
and then by testing for any variability among the observations.
### 4.1. Summed XMM–Newton Spectrum
The spectral analysis presented here uses exclusively the data collected with
the EPIC pn. Although background is much reduced in the EPIC MOS, this
instrument is less sensitive at soft energies, collecting 4.7 times fewer
photons than the EPIC pn in the $0.3-1$ keV band. Furthermore, much of the
EPIC MOS data were lost as the high surface brightness of the Puppis A SNR
frequently triggered an automatic shutoff of that detector. We also neglect
the Chandra ACIS spectra here because of their even poorer low-energy
sensitivity and the increased background and other uncertainties involved in
the analyzing the Chandra CC-mode data. Lastly, one ACIS image taken in timed
exposure mode (ObsID 750) suffers from pileup, and was not used.
For the EPIC pn data, the main technical issue is that the SNR background
exceeds the point source counts below 1 keV. Surprisingly, the background
intensity and spectral shape are both strong functions of spacecraft roll
angle. This effect is made evident because the XMM–Newton data sets were
acquired in two narrow ranges of roll angle roughly 180∘ apart associated with
their respective visibility windows, with the time divided nearly equally
between the two. We checked all of our spectral results carefully for features
that might be dependent on roll angle due to systematic errors in background
subtraction. No such systematic effect were found, which indicates that the
background subtraction is reliable in general.
We extracted spectra for each observation from the EPIC pn detector using an
aperture of radius $0\farcm 3$ and a concentric background annulus of $0\farcm
5<r<0\farcm 6$, selecting only events with ${\tt PATTERN}\leq 4$ and FLAG$=0$.
The data were filtered to exclude time intervals of high background identified
by count rate $>0.1$ s-1 in the $10-12$ keV energy band. An inspection of the
pattern distribution of single and double events shows no evidence of pile-up
and suggests that a lower energy bound in the range $0.3-0.4$ keV is
acceptable. Response matrices and effective area files were generated for each
observation using the SAS software suite. We combined data from all
observations using the FTOOL addascaspec to produce a single source spectrum
and associated files. The combined spectrum was grouped to include at least
1000 counts per channel and was fitted using XSPEC v12.21 to a two blackbody
model with interstellar absorption over the energy range $0.3-5$ keV (see
Figure 7 and Table 6).
Table 6Models for the Summed XMM–Newton Spectrum of PSR J0821$-$4300
Model | Two Blackbody | $+$ Emis. line | $+$ Abs. line | $+$ Two Abs. lines | $+$ Cyclabs
---|---|---|---|---|---
$N{\rm{}_{H}}$ ($10^{21}$ cm-2) | $3.8\pm 0.1$ | $4.3\pm 0.3$ | $3.2\pm 0.2$ | $2.9\pm 0.4$ | … | $2.8\pm 1.01$ | …
$k{\rm T}_{w}$ (keV) | $0.26\pm 0.01$ | $0.25\pm 0.01$ | $0.29\pm 0.01$ | $0.29\pm 0.01$ | … | $0.28\pm 0.03$ | …
$k{\rm T}_{h}$ (keV) | $0.46\pm 0.01$ | $0.45\pm 0.01$ | $0.49\pm 0.02$ | $0.49\pm 0.03$ | … | $0.47\pm 0.02$ | …
$L_{w}({\rm bol})$ ($10^{33}$ erg s-1)aaBlackbody bolometric luminosity for a distance of 2.2 kpc. | $3.3\pm 0.1$ | $3.6\pm 0.2$ | $3.1\pm 0.2$ | $3.0\pm 0.2$ | … | $3.2\pm 0.8$ | …
$L_{h}({\rm bol})$ ($10^{33}$ erg s-1)aaBlackbody bolometric luminosity for a distance of 2.2 kpc. | $2.0\pm 0.2$ | $2.0\pm 0.2$ | $1.4\pm 0.3$ | $1.4\pm 0.3$ | … | $1.7\pm 0.4$ | …
$A_{w}$ (km2) | $72\pm 11$ | $89\pm 18$ | $44\pm 6$ | $40\pm 8$ | … | $53\pm 14$ | …
$A_{h}$ (km2) | $4.4\pm 0.9$ | $4.5\pm 0.9$ | $2.5\pm 0.8$ | $2.4\pm 1.0$ | … | $3.4\pm 0.8$ | …
$E_{0}$ (keV) | … | $0.75\pm 0.01$ | $0.46\pm 0.05$ | $0.46\pm 0.01$ | $2E_{o}$ | $0.46\pm 0.01$ | $2E_{o}$
WidthbbGaussian $\sigma$ for the emission or absorption lines, natural width $W$ for the cyclotron absorption model (Makishima et al., 1990; Mihara et al., 1990). (eV) | … | $75\pm 20$ | $85\pm 50$ | $106\pm 20$ | $34-62$ | $53-97$ | $35-290$
EW (eV) | … | $53\pm 10$ | … | … | … | … | …
$\tau_{o}$ccOptical depth at line center. | … | … | $0.1-0.9$ | $0.6-1.3$ | $<0.035$ | $0.9-1.7$ | $<0.14$
$\chi^{2}({\rm DoF})$ | $1.50(359)$ | 1.08(356) | $1.16(356)$ | $1.08(354)$ | $1.12(354)$
Note. — The $1\sigma$ uncertainties for three interesting parameters
($\Delta\chi^{2}=3.53$) are given.
The resulting high signal-to-noise ratio of the fitted spectrum reveals
significant features that are unaccounted for by the two blackbody model. The
best fit model, with $N_{\rm H}=(3.8\pm 0.01)\times 10^{21}$ cm-2,
$kT_{w}=0.26\pm 0.01$ keV, and $kT_{h}=0.46\pm 0.01$ keV, has reduced
$\chi^{2}_{\nu}=1.50$ for 359 degrees of freedom (DoF), which is formally
unacceptable. The deviations are evident in structure in the residuals in
Figure 7a. Adding a Gaussian emission line to the model as suggested by our
previous work improves the fit to $\chi^{2}_{\nu}=1.08$ for 358 DoF (Figure
7b). The centroid energy of the line is $0.75\pm 0.01$ keV, and its equivalent
width is $53\pm 10$ eV.
Figure 7.— (a) EPIC pn spectrum of the 16 summed XMM–Newton observations of
PSR J0821$-$4300 fitted to a double blackbody model. The residuals from the
fit are shown in the bottom panel. (b) The same spectrum fitted with a double
blackbody model plus Gaussian emission line. The parameters of these fits are
given in Table 6.
Considering the shape of the residuals from the two blackbody fit in Figure
7a, an alternative hypothesis is that an absorption feature, or features, is
responsible. Accordingly we applied a Gaussian absorption line, two Gaussian
absorption lines, and finally, the cyclotron absorption model of Makishima et
al. (1990) and Mihara et al. (1990), which is available in XSPEC as cyclabs.
As shown in Table 6 and Figure 8, all of these models gave acceptable fits.
Two absorption lines, when fitted independently, are separated by nearly a
factor of 2 in energy, which is suggestive of the fundamental and first
harmonic in a cyclotron model. Therefore we fixed the ratio of their centroid
energies at 2, with the result that their centroids are at 0.46 keV and 0.92
keV, bracketing the energy previously ascribed to an emission line. The
cyclabs model, which also fixes this ratio, yields the same centroid energies.
We conclude that, from the phase-averaged spectra alone, it is not possible to
distinguish an emission feature from one or two absorption lines, which leaves
the physical interpretation uncertain.
The fundamental energy of the electron cyclotron resonance falls at
$E_{0}=1.16\,(B/10^{11}\,{\rm G})/(1+z)\ {\rm keV},$ $None$
where $z$ is the gravitational redshift. Assuming a typical value of $z=0.3$,
and if $B\approx B_{s}=2.9\times 10^{10}$ G, the equatorial field from the
vacuum dipole spin-down result, then $E_{0}\approx 0.26$ keV is expected. When
compared with $E_{0}=0.46$ keV from the absorption-line fit, or $0.75$ keV
from the emission-line fit, this hints that the local surface field where the
line is formed is larger than the equatorial dipole field, but is perhaps more
compatible with the field at the pole, $B_{p}=2\,B_{s}$. Other factors
possibly affecting a comparison between the dipole spin-down $B$-field and the
surface $B$-field from the cyclotron energy will be explored in Section 6.2.
Figure 8.— Same as Figure 7 but showing the residuals from fits to a double
blackbody model plus, from top to bottom: no line, a Gaussian emission line, a
Gaussian absorption line, two Gaussian absorption lines, and a cyclotron
absorption line model. The parameters of the fits are given in Table 6.
The addition of emission or absorption lines to the model near the low-energy
end of the XMM–Newton spectrum affects the fitted value of the column density
$N_{\rm H}$, with values ranging from $(2.8-4.3)\times 10^{21}$ cm-2 for the
different models in Table 6. Therefore, it is possible that comparison with
independent measurements of $N_{\rm H}$ may suggest a preference for one or
another of these models. For example, the 21 cm H I emission in the foreground
of Puppis A amounts to $N_{\rm HI}=2.5\times 10^{21}$ cm-2 according to
Reynoso et al. (2003). To obtain this value, they integrated the H I line
emission in the radial velocity range $-10$ to $+16$ km s-1, the latter
velocity corresponding to their assumed 2.2 kpc distance of Puppis A.
Comparison with the values of $N_{\rm H}$ in Table 6 tends to favor the X-ray
spectral models that include absorption lines. While this agreement is
encouraging, it is subject to the caveat that the 21 cm column density is
formally a lower limit, assuming as it does that the line is optically thin.
Better support for a low column density comes from the XMM–Newton RGS spectra
of several regions of the Puppis A SNR fitted by Katsuda et al. (2012). These
require X-ray $N_{\rm H}$ in the range $(2.58-2.85)\times 10^{21}$ cm-2, which
also agrees closely with the $N_{\rm H}$ from our absorption-line models for
the pulsar spectrum.
Figure 9.— Blackbody temperatures and flux in the $0.5-5$ keV band for the 16
individual XMM–Newton observations of PSR J0821$-$4300, fitted to the two
blackbody model. Data are taken from Table 7. Errors bars are 1$\sigma$. The
weighted mean values are indicated by the dashed lines.
Figure 10.— Combined XMM–Newton EPIC pn spectrum of PSR J0821$-$4300 in two
pulse-phase intervals defined in Figure 5 ($\Delta\phi_{1}$, blue,
$\Delta\phi_{2}$, red), fitted to a double blackbody model plus Gaussian
emission line. The blackbody temperatures and line centroid energy are linked
between the two spectra. The fitted parameters are given in the last column of
Table 8. The residuals to the fit are shown in the bottom panel.
Finally, we note that the total luminosity and blackbody areas reported in
Table 6 differ from those presented in Gotthelf & Halpern (2009), the areas by
a factor of two or more. This is a consequence of the best fit values for each
model fit. The derived blackbody areas depend on blackbody temperature as
$T^{-4}$, which itself is strongly correlated with the fitted column density.
Future work, simultaneously modeling of the phase dependent spectra, should
better constrain the column density and temperatures, and consequently provide
a more accurate measurement of the blackbody areas.
### 4.2. Search for Variability
To test for long-term variability of PSR J0821$-$4300, we also fitted the
individual XMM–Newton spectra, searching for temperature variations on the
surface, for example. The spectra for each observation are well-fitted by the
two blackbody model, with parameters listed in Table 7; their statistics do
not warrant an additional line component. For this comparison, we held the
absorbing column fixed at the value in Table 6 derived from the fit of the
composite spectrum to the two blackbody model. Figure 9 displays the results.
No significant variation is evident in the flux or in either of the two
blackbody temperatures.
Table 7Individual XMM–Newton Spectra of PSR J0821$-$4300
Group | ObsID | Livetime | $kT_{w}$ | $kT_{h}$ | Flux ($10^{-12}$ | $\chi^{2}_{\nu}({\rm DoF})$
---|---|---|---|---|---|---
| | (ks) | (keV) | (keV) | erg s-1 cm-2) |
1 | 0113020101S | 15.2 | $0.27(0.25,0.29)$ | $0.47(0.44,0.52)$ | $4.23(4.14,4.30)$ | $1.06(154)$
| 0113020301S | 16.1 | $0.27(0.25,0.28)$ | $0.46(0.43,0.50)$ | $4.21(4.12,4.27)$ | $1.05(184)$
2 | 0606280101S | 25.7 | $0.27(0.26,0.28)$ | $0.47(0.45,0.51)$ | $4.15(4.09,4.20)$ | $1.19(282)$
| 0606280101U | 24.5 | $0.25(0.24,0.26)$ | $0.45(0.43,0.47)$ | $4.11(4.05,4.16)$ | $0.91(268)$
| 0606280201S | 24.6 | $0.27(0.26,0.28)$ | $0.47(0.45,0.50)$ | $4.14(4.09,4.20)$ | $1.02(261)$
| 0650220201S | 8.1 | $0.22(0.19,0.26)$ | $0.40(0.38,0.45)$ | $4.11(4.01,4.20)$ | $0.88(94)$
3 | 0650220901S | 16.4 | $0.25(0.24,0.27)$ | $0.45(0.42,0.48)$ | $4.13(4.06,4.19)$ | $1.01(191)$
| 0650221001S | 15.5 | $0.24(0.22,0.25)$ | $0.42(0.40,0.44)$ | $4.13(4.06,4.20)$ | $0.99(183)$
| 0650221101S | 18.6 | $0.26(0.25,0.28)$ | $0.48(0.45,0.51)$ | $4.19(4.12,4.25)$ | $0.97(222)$
| 0650221201S | 17.1 | $0.27(0.26,0.28)$ | $0.49(0.46,0.54)$ | $4.17(4.08,4.23)$ | $1.08(201)$
| 0650221301S | 16.4 | $0.28(0.27,0.29)$ | $0.52(0.48,0.57)$ | $4.06(3.98,4.12)$ | $1.21(182)$
| 0650221401S | 15.5 | $0.24(0.23,0.26)$ | $0.43(0.41,0.46)$ | $4.13(4.05,4.20)$ | $1.00(174)$
4 | 0650221501S | 20.3 | $0.27(0.25,0.28)$ | $0.45(0.43,0.48)$ | $4.26(4.20,4.32)$ | $0.96(225)$
| 0657600101S | 22.4 | $0.26(0.25,0.27)$ | $0.47(0.44,0.50)$ | $4.13(4.07,4.19)$ | $0.99(242)$
| 0657600201S | 26.1 | $0.26(0.25,0.27)$ | $0.45(0.43,0.47)$ | $4.24(4.19,4.30)$ | $1.07(285)$
| 0657600301S | 24.7 | $0.26(0.25,0.28)$ | $0.46(0.44,0.49)$ | $4.22(4.16,4.27)$ | $0.94(262)$
Note. — Results for a fit to an absorbed, two-blackbody model with the column
density held fixed at $N_{\rm H}=3.75\times 10^{21}$ cm-2, the value obtained
for the summed spectrum in Table 6. The absorbed flux in the $0.5-5$ keV range
is tabulated. The range of uncertainty of each value is the $1\sigma$
confidence interval for two interesting parameters.
We next test for pulse-phase dependence of the spectral line feature(s) by
generating spectra in two phase intervals, each of $\Delta\phi=0.3$ cycles in
width centered on the peaks of the soft and hard light curves, respectively
(see Figure 5.) This is motivated by the original finding in Gotthelf &
Halpern (2009) that an emission line is more strongly associated with the warm
region, i.e., the soft phase of the pulse. For simplicity, given the reduced
counts in the phased spectra, we used only the Gaussian emission-line model as
a representative for all of the possible line models. The two phase-resolved
spectra combined from all epochs are shown in Figure 10, where they are fitted
to the two blackbody plus Gaussian emission-line model. In the simultaneous
fit, the two temperatures and the line centroid energy are linked. As shown in
Table 8, the equivalent width of the Gaussian line is indeed about factor of 2
larger in the soft-phase spectrum than in the hard phase.
Figure 11.— XMM–Newton EPIC pn spectra of PSR J0821$-$4300, grouped into the
four sets as listed in Table 7, and fitted to a double blackbody model with
Gaussian emission line. As in Figure 10, red and blue represent the two pulse-
phase intervals defined in Figure 5. The residuals from the fits are shown in
the bottom panels. The fitted parameters are given in Table 8.
De Luca et al. (2012) presented evidence that the emission-line centroid had
decreased from 0.80 keV to 0.73 keV between 2001 and 2009. Here we extend the
examination for variability of the spectral feature, including its phase
dependence, by combining the XMM–Newton spectra into four groups of adjacent
observations as indicated Table 7. We fitted these four sets of spectra to the
two blackbody plus Gaussian emission-line model, with their temperatures and
line width (sigma) linked between the two phase intervals. The four fits are
displayed in Figure 11, and the fitted parameters for each set and their sum
are presented in Table 8. It is clear that the temperatures and their
contributed fluxes are steady in time, consistent with their phase-averaged
behavior and with no time dependence in their phase ratios. On the other hand,
the equivalent width of the fitted Gaussian emission line shows evidence for
having increased between 2001 and 2009, while the line centroid energy
decreased from $0.79^{+0.02}_{-0.03}$ keV to $0.71^{+0.02}_{-0.03}$ keV. While
these results are consistent with the De Luca et al. (2012) analysis of the
same data, there is little if any additional variability between 2009 and
2012.
Table 8XMM–Newton Phase-Resolved Spectra of PSR J0821$-$4300
Model | Group 1 | Group 2 | Group 3 | Group 4 | Sum
---|---|---|---|---|---
Parameter | 2001 Apr–Nov | 2009 Dec – 2010 May | 2010 Oct–Dec | 2011 Apr – 2012 Apr | 2001 Apr – 2012 Apr
$kT_{w}$ (keV) | $0.23(0.21,0.25)$ | $0.25(0.24,0.26)$ | $0.24(0.23,0.25)$ | $0.27(0.25,0.28)$ | $0.25(0.24,0.25)$
$kT_{h}$ (keV) | $0.41(0.39,0.44)$ | $0.45(0.43,0.47)$ | $0.44(0.42,0.46)$ | $0.47(0.44,0.52)$ | $0.44(0.43,0.45)$
$F[{\Delta\phi_{1}]}$aaAbsorbed flux quoted for the $0.5-5.0$ keV band in units of $10^{-12}$ erg s-1 cm-2. | $4.00(3.97,4.08)$ | $3.91(3.88,3.93)$ | $3.94(3.91,3.96)$ | $3.99(3.96,4.01)$ | $3.96(3.94,3.97)$
$F[{\Delta\phi_{2}]}$aaAbsorbed flux quoted for the $0.5-5.0$ keV band in units of $10^{-12}$ erg s-1 cm-2. | $4.33(4.28,4.38)$ | $4.37(4.35,4.40)$ | $4.37(4.34,4.39)$ | $4.45(4.42,4.47)$ | $4.40(4.38,4.41)$
$E_{0}$ (keV) | $0.79(0.76,0.81)$ | $0.71(0.68,0.73)$ | $0.72(0.69,0.74)$ | $0.69(0.63,0.73)$ | $0.72(0.70,0.73)$
$\sigma$ (eV) | $\leq 62$ | $44(21,70)$ | $68(45,95)$ | $133(78,195)$ | $69(52,89)$
$EW[{\Delta\phi_{1}]}$ (eV) | $40(24,51)$ | $77(59,89)$ | $58(46,76)$ | $86(67,107)$ | $61(53,74)$
$EW[{\Delta\phi_{2}]}$ (eV) | $14(3,22)$ | $34(23,42)$ | $42(32,57)$ | $39(26,52)$ | $31(26,39)$
$\chi^{2}_{\nu}{\rm(DoF)}$ | $1.01(87)$ | $1.19(229)$ | $1.05(269)$ | $0.99(250)$ | $1.20(212)$
Note. — Results from simultaneous fits to XMM–Newton spectra of PSR
J0821$-$4300 extracted from two pulse-phase intervals, $\Delta\phi_{1}$,
$\Delta\phi_{2}$, as defined in the Figure 5. Groups numbers are defined in
Table 7. The blackbody temperatures and Gaussian line energy are linked
between the two phases. The column density is fixed at $N_{\rm H}=4.28\times
10^{21}$ cm-2, the phase-averaged value for the summed spectrum in Table 6.
Quoted uncertainties are $1\sigma$ for three interesting parameters.
We also repeated this test for variability with the Gaussian absorption-line
model for the spectral feature (results not tabulated here). For either the
emission-line or the absorption-line model, the measured line centroids for
all epochs but 2001 cluster well within their 1$\sigma$ uncertainty (for two
interesting parameters, $\Delta\chi^{2}=2.3$). The 2001 set deviates from the
mean defined by the other three sets by $\approx 2\sigma$. In terms of
percentage deviation from the mean, the line centroid measured in 2001 was 14%
and 8% higher in the emission line and the absorption line model,
respectively. We conclude that, although the deviation seems large for the
emission line model, it is not inconsistent with the expected variance.
Further observations would be necessary to establish more definite evidence of
long-term variability.
## 5\. A Definitive Spin-Down Measurement for PSR J1210$-$5226
Archival timing observations of PSR J1210$-$5226 spanning the years 2000–2008
were too sparse to securely determine its spin-down rate, as described in
Halpern & Gotthelf (2011). Searching all possible parameter space for a phase-
coherent, quadratic ephemeris, we found two equally acceptable solutions, with
$\dot{f}=-1.243(22)\times 10^{-16}$ Hz s-1 and $\dot{f}=-7.084(22)\times
10^{-16}$ Hz s-1, corresponding to $B_{s}=9.9\times 10^{10}$ G (solution 1)
and $B_{s}=2.4\times 10^{11}$ G (solution 2), respectively. Since such low
$\dot{E}$ pulsars are generally very stable rotators with little timing noise
or glitch activity, it was deemed likely that one of these is the true
solution, and the other one is an alias with an incorrect cycle count. It is
also important that no solutions with smaller dipole $B_{s}$, and no spinning-
up solutions, were found.
Table 9Log of New X-ray Timing Observations of PSR J1210$-$5226 Mission | Instr/Mode | ObsID | Date | Exposure | Start Epoch | FrequencyaaBarycentric frequency derived from a $Z^{2}_{1}$ test. The given uncertainty is for the $1\sigma$ confidence interval. | $Z^{2}_{1}$
---|---|---|---|---|---|---|---
| | | (UT) | (ks) | (MJD) | (s) |
Chandra | ACIS-S3/CC | 14199 | 2011 Nov 25 | 31.0 | 55890.233 | 2.3577625(28) | 48.2
Chandra | ACIS-S3/CC | 14202 | 2012 Apr 10 | 33.0 | 56027.637 | 2.3577709(43) | 22.2
XMM | EPIC-pn/SW | 0679590101 | 2012 Jun 22 | 26.5 | 56100.537 | 2.3577687(25) | 68.3
XMM | EPIC-pn/SW | 0679590201 | 2012 Jun 24 | 22.3 | 56102.752 | 2.3577621(34) | 69.0
XMM | EPIC-pn/SW | 0679590301 | 2012 Jun 28 | 24.9 | 56106.490 | 2.3577636(23) | 109.6
XMM | EPIC-pn/SW | 0679590401 | 2012 Jul 02 | 24.5 | 56110.918 | 2.3577626(28) | 61.4
XMM | EPIC-pn/SW | 0679590501 | 2012 Jul 18 | 27.3 | 56126.553 | 2.3577640(27) | 51.9
XMM | EPIC-pn/SW | 0679590601 | 2012 Aug 11 | 27.3 | 56150.408 | 2.3577637(23) | 81.8
Chandra | ACIS-S3/CC | 14200 | 2012 Dec 01 | 31.1 | 56262.095 | 2.3577634(28) | 39.8
In 2011–2012 we started a new series of observations of PSR J1210$-$5226 with
Chandra and XMM–Newton that was designed to initiate and maintain a unique,
phase-connect timing solution of this pulsar for the first time and eliminate
the prior timing ambiguity. Table 9 is a log of the new observations. The
instrumental setups and analysis methods are the same as those described in
Section 3 for PSR J0821$-$4300, except that the XMM–Newton source photons were
extracted from a $20^{\prime\prime}$ radius aperture instead of
$30^{\prime\prime}$, and the $0.5-2.5$ keV band was selected to optimize the
pulsed signal.
Table 10Ephemeris of PSR J1210$-$5226 Parameter | Value
---|---
R.A. (J2000)aaMeasured from Chandra ACIS-S3 ObsID 3913. Typical uncertainty is $0.\\!^{\prime\prime}6$. | $12^{\rm h}10^{\rm m}00^{\rm s}\\!.91$
Decl. (J2000)aaMeasured from Chandra ACIS-S3 ObsID 3913. Typical uncertainty is $0.\\!^{\prime\prime}6$. | $-52^{\circ}26^{\prime}28^{\prime\prime}\\!.4$
Ephemeris Epoch (MJD TDB)bbEpoch of fitted minima of summed pulse profile; phase zero in Figure 13 | 53562.0000006
Ephemeris Span (MJD) | 51,549–56,262
Frequency, $f$ | 2.357763502865(65) Hz
Frequency derivative, $\dot{f}$ | $(-1.2363\pm 0.0091)\times 10^{-16}$ Hz s-1
Period, $P$ | 0.424130748816(12) s
Period derivative, $\dot{P}$ | $(2.224\pm 0.016)\times 10^{-17}$
Surface dipole dipole field, $B_{s}$ | $9.8\times 10^{10}$ G
Spin-down luminosity, $\dot{E}$ | $1.2\times 10^{31}$ erg s-1
Characteristic age, $\tau_{c}$ | 302 Myr
The coherently measured period from the new observations is precise enough to
reject previous timing solution 2, while solution 1 with the smaller $B_{s}$
extrapolates precisely to the new period. Furthermore, the new pulse phases
are aligned accurately with solution 1, while solution 2 gives random phases.
Thus a unique quadratic ephemeris fits all 12.9 years of Chandra and
XMM–Newton timing. Table 10 gives this global ephemeris, derived from a least-
square fit to the TOAs as described in Section 3, with the phase residuals
shown in Figure 12. The previous uncertainty on the $\dot{f}$ of solution 1 is
reduced by a factor of 2, with the result $\dot{f}=-1.2363(91)\times 10^{-16}$
Hz s-1; the corresponding dipole magnetic field is $B_{s}=9.8\times 10^{10}$
G. Figure 13 shows the pulse profile using data from all the observations
folded on the presented ephemeris.
Figure 12.— Pulse-phase residuals from the linear term (dash-dot line) of the
phase ephemeris of PSR J1210$-$5226 presented in Table 10. Included are the
new observations listed in Table 9 and the archival timing observations from
Table 1 of Halpern & Gotthelf (2011). The quadratic term (solid line)
corresponds to the uniquely determined period derivative spanning the years
2000–2012. The error bars are generally smaller than the symbol size. Figure
13.— Pulse profiles of PSR J1210$-$5226 in the $0.5-2.5$ keV band using data
from all timing observations, folded according to the ephemeris in Table 10.
Phase zero corresponds to the listed TDB epoch of the ephemeris. Included are
the new observations in Table 9 and the archival timing observations from
Table 1 of Halpern & Gotthelf (2011).
Strictly speaking, the derived period derivative is an upper limit to the
intrinsic one, as any proper motion has not been measured and taken into
account. PSR J1210$-$5226 and PSR J0821$-$4300 are at similar distances, and
the location of PSR J1210$-$5226 with respect to the geometric center of SNR
PKS 1209$-$51/52 allows (but does not require, since the kinematics of the SNR
are unknown) a proper motion of $\sim 70$ mas yr-1, or $v_{\perp}\sim 730$ km
s-1 (De Luca et al., 2011), similar to that of PSR J0821$-$4300\. If so, the
Shklovskii effect given by equation (1) would contribute $\dot{P}_{k}\sim
1.1\times 10^{-17}$, or half of the total $\dot{P}$, and the spin-down dipole
field would be reduced somewhat, to $B_{s}\approx 7\times 10^{10}$ G. Of
course, if these two CCOs both have velocities toward the high end of the
distribution of pulsars, it would be a tantalizing physical result in itself.
However, examination of another CCO, the NS in Cas A, doesn’t necessarily
support such high velocities for CCOs, in general; Thorstensen et al. (2001)
and Fesen et al. (2006) find $v_{\perp}\approx 350$ km s-1 with respect to the
explosion center of Cas A.
## 6\. Discussion
### 6.1. On the Age of Puppis A
Becker et al. (2012) discussed the impact of the revised proper motion of PSR
J0821$-$4300 on the inferred age of Puppis A. The age of the SNR had been
derived previously as $3700\pm 300$ yr from the proper motions of optical
filaments that point back to a common center, presumed to be the site of the
explosion, and assuming no deceleration. The motion of the NS also
extrapolates to the same center, but the distance traveled,
$371^{\prime\prime}\pm 31^{\prime\prime}$, corresponds to an age of $5200\pm
1000$ yr (or $6100\pm 1000$ yr for our proper motion of $61\pm 9$ mas yr-1).
Becker et al. (2012) refer to these marginally contradictory measurements as
independent, and choose to average them, giving the value 4.5 kyr listed in
Table 1. However, they are not truly independent, because they assume the same
starting location. Furthermore, the discrepancy is worse if the filaments have
decelerated. Just as the new Chandra observations of the NS have improved the
accuracy of its proper motion, we suggest that a contemporary observation of
the optical filaments of Puppis A may improve the precision of the original
Winkler et al. (1988) study on which the optical proper-motion age is based.
Such an investigation could lead to a more detailed understanding of dynamics
of the filaments, and a more accurate age for Puppis A.
### 6.2. PSR J0821–4300 and PSR J1210$-$5226 as Anti-Magnetars
The spin-down of power PSR J0821$-$4300, $\dot{E}=1.9\times 10^{32}$ erg s-1,
is consistent with being caused by magnetic braking of an isolated neutron
star with a weak dipole field of $B_{s}=2.9\times 10^{10}$ G. Its spin-down
power is much smaller than its observed thermal X-ray luminosity,
$L_{x}\approx 5.6\times 10^{33}\,d_{2.2}^{2}$ erg s-1, which rules out
rotation as a significant power source. As discussed in Halpern & Gotthelf
(2010a) for PSR J1852$+$0040, the very small $\dot{P}$ disfavors propeller
spin-down and accretion as a source of the X-rays. Thus, residual cooling
remains the most plausible source of the X-ray luminosity of PSR J0821$-$4300
and, by extension, of all CCOs. Given the meager spin-down power of PSR
J0821$-$4300, we can also discount the suggestion of Reynoso et al. (2003) and
Castelletti et al. (2006) that structure in the Puppis A SNR and associated H
I is caused by jets emitted by the pulsar, similar to the SS433/W50 system.
Parenthetically, if the H I morphology is not caused by the pulsar, then it
does not provide supporting evidence of its distance.
An unresolved question about PSR J0821$-$4300 is the origin of its phase-
dependent, possibly variable emission or absorption feature. In the absence of
accretion, it would be difficult to understand how an emission line is
generated, or why it would vary in a few years. The indication of variability
is not strong, and there is no other evidence of accretion. Therefore, we will
consider here an absorption-line interpretation, for which there is a good
precedent in PSR J1210$-$5226, the only isolated pulsar to show a series of
strong absorption lines in its spectrum (Sanwal et al., 2002; Mereghetti et
al., 2002; Bignami et al., 2003).
The spectral features in PSR J1210$-$5226 are now widely considered to
comprise the electron cyclotron fundamental at $E_{0}=0.7$ keV and its
harmonics. The surface magnetic field strength where the lines are formed is
inferred to be $B\approx 8\times 10^{10}$ G according to equation (2) and
assuming $z\approx 0.3$. The relative strength of the harmonics is explained
by treating them as resonances in the photospheric free-free opacity in the
presence of the magnetic field (Suleimanov et al., 2010, 2012). PSR
J1210$-$5226 is exceptional in having the largest known spin-down dipole
magnetic field among CCOs, now confirmed as $B_{s}\leq 9.8\times 10^{10}$ G.
This is only slightly larger than $B\approx 8\times 10^{10}$ G inferred from
the spectral features, and is the first case in which such a comparison has
been made between independent methods of measuring surface $B$-field on an
isolated pulsar. Effects that could eliminate the already minor discrepancy
include the unmeasured proper motion, which would decrease the inferred spin-
down dipole field, and the results of numerical models of a force-free
magnetosphere, which imply a different spin-down law from the standard vacuum
dipole expression
$B_{s}=\left(3c^{3}IP\dot{P}\over 8\pi^{2}R^{6}\,{\rm
sin}^{2}\alpha\right)^{1/2}=\ 3.2\times 10^{19}\,\left(P\dot{P}\over{\rm
sin}^{2}\alpha\right)^{1/2}\,{\rm G}$ $None$
such that, more accurately,
$B_{s}\approx\left(c^{3}IP\dot{P}\over 4\pi^{2}R^{6}\,[1+{\rm
sin}^{2}\alpha]\right)^{1/2}=\ 2.6\times 10^{19}\,\left(P\dot{P}\over 1+{\rm
sin}^{2}\alpha\right)^{1/2}\,{\rm G}$ $None$
(Spitkovsky, 2006). This means that a measured spin-down rate is obtained with
$B_{s}$ smaller by a factor $0.58-0.82$ than the conventional approximation
$B_{s}=3.2\times 10^{19}(P\dot{P})^{1/2}$ G.
In the case of PSR J0821$-$4300, by the same logic, its weaker inferred dipole
magnetic field of $B_{s}\leq 2.9\times 10^{10}$ G wouldn’t naturally account
for a spectral feature at $0.7-0.8$ keV, instead predicting $E_{0}\leq 0.26$
keV. The absorption model with the cyclotron fundamental at $E_{0}=0.46$ keV
comes closer to the prediction. However, additional variables can affect the
comparison of a dipole field inferred from spin-down with that from a
cyclotron line. First is the factor of 2 variation of dipole field strength
over the surface, with $B_{p}=2\,B_{s}$. Second, the actual field is not
likely to be a centered dipole, and may be larger than $B_{s}$ or $B_{p}$ in
places where the line is formed. The asymmetric distribution of surface
temperature on PSR J0821$-$4300 already appears to require a complex magnetic
field geometry, such as an off-center dipole or higher multipoles. Third, the
inferred spin-down field depends on the uncertain NS mass and radius as in
equations (3) and (4). $B_{s}$ scales as ${I^{1/2}\,R^{-3}}\propto
M^{1/2}R^{-2}(1+z)$ (Ravenhall & Pethick, 1994), where
$1+z=(1-2GM/Rc^{2})^{-1/2}\propto B/E_{0}$ from equation (2). For an
astrophysically likely $M=1.4\,M_{\odot}$, theoretical NS equations of state
allow $8<R<15$ km, therefore $1.18<1+z<1.44$. The spectroscopic $B$ is
therefore uncertain by $\pm 10\%$ when we adopt $z=0.3$, while the dipole
$B_{s}$ is uncertain by the much larger factor of $\sim 2$. (The gravitational
redshift cancels out in their ratio.) Allowing for these uncertainties, it is
reasonable to adopt the hypothesis that feature(s) in the spectrum of PSR
J0821$-$4300 are due to the cyclotron process.
The continuum X-ray spectrum and pulse profiles of PSR J0821$-$4300 are
indicative of antipodal hot spots of different areas and temperatures
(Gotthelf & Halpern, 2009; Gotthelf et al., 2010), which are difficult to
account for if the magnetic field is weak. A related problem is the high
pulsed fraction of 64% from PSR J1852$+$0040 (Halpern & Gotthelf, 2010a), a
timing twin of PSR J0821$-$4300\. Polar cap heating by any magnetospheric
accelerator must be negligible as a source of surface heating in CCOs, being
only a small fraction of the already insignificant spin-down power. Thermal
X-rays from residual cooling can be nonuniform if there is anisotropic heat
conduction in the star. The effect of different magnetic field configurations
on heat transport in the crust and envelope of NSs has been modelled, most
recently by Geppert et al. (2004, 2006), Pérez-Azorín et al. (2006a), and Pons
et al. (2009). A toroidal field is expected to be the initial configuration
generated by differential rotation in the proto-neutron star dynamo (Thompson
& Duncan, 1993). One of the effects of crustal toroidal field is to insulate
the magnetic equator from heat conduction, resulting in warm spots at the
poles. The warm regions can even be of different sizes due to the antisymmetry
of the poloidal component of the field (Geppert et al., 2006), which is
evocative of the antipodal thermal structure of PSR J0821$-$4300\. To have a
significant effect on the heat transport, the crustal toroidal field strength
required in these models is $\sim 10^{15}$ G, many orders of magnitude greater
than the poloidal field if the latter is measured by the spin-down. Purely
toroidal or poloidal fields are thought to be unstable in an initially fluid
NS (Tayler, 1973; Flowers & Ruderman, 1977), although the toroidal field may
be stabilized by a poloidal field that is several orders of magnitude weaker
(Braithwaite, 2009). We suggest the latter as a viable configuration for a
CCO.
Shabaltas & Lai (2012) tried to model the pulse profile of PSR J1852$+$0040
with anisotropic conduction, and concluded that they needed a toroidal crustal
field of $B_{\phi}>2\times 10^{14}$ G to produce its high pulsed fraction.
Even then, they observe, the shape of the modelled light curve doesn’t match
the observed one. Since the shape of the light curve is not reproduced, the
surface temperature distribution of PSR J1852$+$0040 and its physical origin
are still not known.
Page & Sarmiento (1996), Pérez-Azorín et al. (2006b), Zane & Turolla (2006),
and Zane (2007) investigated NS surface emission patterns using a combination
of star-centered dipole and quadrupole magnetic field components to model
asymmetric pulse profiles. The behavior of PSR J0821$-$4300 may ultimately be
explained by similar models. It remains to be shown that if CCOs can have
field configurations that are strong enough to affect heat transport to the
extent required, while not exceeding the spin-down limits on the external
dipole field.
### 6.3. Origin and Evolution of Anti-magnetars
PSR J0821$-$4300 is nearly a twin of PSR J1852$+$0040 in its spin properties,
and there are no other young neutron stars with measured magnetic fields this
weak. They fall in a region of the $P-\dot{P}$ diagram (Figure 3) that is
devoid of ordinary (non-recycled) radio pulsars (Manchester et al., 2005).
They overlap with the supposed mildly recycled pulsars in this area
(Belczynski et al., 2010). The characteristic age of 254 Myr for PSR
J0821$-$4300 is not meaningful because the pulsar was born spinning at its
current period, as is the case for the other CCO pulsars. If their magnetic
fields remain constant, they will not move in $P-\dot{P}$ space for $>10^{8}$
years (but see below). That CCOs are found in SNRs in comparable numbers to
other classes of NSs implies that they must represent a significant fraction
of NS births. If PSR J0821$-$4300 and PSR J1852$+$0040 are typical CCOs, the
area around them in the $P-\dot{P}$ diagram should be densely populated with
“orphaned CCOs” that remain after their SNRs dissipate in $\sim 10^{5}$ years.
Why there are few ordinary radio pulsars and no older X-ray pulsars near their
location is then a mystery, as emphasized by Kaspi (2010), which may indicate
that radio luminosity is a function of spin-down power.
There are not yet enough CCOs to know whether they are intrinsically radio-
quiet relative to ordinary and recycled pulsars, rather than unfavorably
beamed. It is also possible that some ordinary radio pulsars with
$B_{s}<10^{11}$ G are actually much younger than their characteristic ages,
and may be orphaned CCOs that could be recognized in X-rays. Whether or not
they are radio pulsars, nearby orphaned CCOs should be detectable as thermal
X-ray sources for $10^{5}-10^{6}$ years, similar to the seven ROSAT discovered
isolated NSs (INSs: Haberl, 2007) which, however, have strong magnetic fields
(Kaplan et al., 2009). It is likely that the INSs are kept hot for longer than
CCOs by continuing magnetic field decay (Pons et al., 2007), which would
explain their observed abundance relative to the elusive orphaned CCOs. It may
be difficult to detect and/or recognize orphaned CCOs if they cool faster than
ordinary neutron stars. One effect that can accelerate cooling is an accreted
light-element envelope, which has higher heat conduction than an iron surface
(Kaminker et al., 2006). The newly discovered 59 ms pulsar 1RXS
J141256.0$+$792204 (“Calvera”), originally detected in the ROSAT All-Sky
Survey, may be the first recognized example of an orphaned CCO (Zane et al.,
2011; Rutledge et al., 2008; Halpern, 2011), pending a measurement of its
period derivative. The isolated NS 2XMM J104608.7$-$594306 in the Carina
Nebula (Pires et al., 2012) has been suggested as another orphaned CCO.
It is possible that the weak magnetic field of CCOs is causally related to
slow rotation at birth through the turbulent dynamo (Thompson & Duncan, 1993)
that generates the magnetic field. (See Spruit 2008 for a review of possible
mechanisms for the origin of magnetic fields in neutron stars.) If the dipole
magnetic field $B_{s}$ is simply related to the initial spin period, and
assuming that the present period $P$ is in fact the birth period because the
spin-down time is so much longer than the true age, we may expect an anti-
correlation between $B_{s}$ and $P$. With only three data points to compare,
and with two of them having nearly identical values, there is not much
evidence to examine for a trend. In fact, the CCO with the longest period, PSR
J1210$-$5226, also has the strongest dipole field of the three, which would
not by itself support such a simple correlation. A population analysis of
radio pulsars by Faucher-Giguère & Kaspi (2006) concludes that there is a wide
distribution of birth periods, with a mean of $\sim 300$ ms and a dispersion
of $\sigma\sim 150$ ms. If this is true, the birth periods of CCOs are not in
fact long, and their weak dipole fields may be the effect of some as-yet
unknown parameter.
In an alternative theory for CCOs, a normal ($\sim 10^{12}$ G) magnetic field
is buried in the core or crust of a NS by prompt fall-back of supernova
debris, and takes thousands of years to diffuse back to the surface, during
which time the NS appears as an anti-magnetar. This is assuming that the
accreted matter is itself not magnetized. The timescale for diffusion is
highly dependent on the amount of matter accreted. According to Muslimov &
Page (1995), for accretion of $\sim 10^{-5}\,M_{\odot}$, the regrowth of the
surface field is largely complete after $\sim 10^{3}$ yr, but if
$>0.01\,M_{\odot}$ is accreted, then the diffusion time could be millions of
years. Chevalier (1989) calculated that the neutron star in SN 1987A could
have accreted $\sim 0.1\,M_{\odot}$ of fallback material in the hours after
the SN explosion, aided by a reverse shock from the helium layer of the
progenitor. If so, it may never emerge as a radio pulsar.
Interesting support for the theory of field burial and regrowth is the absence
of evidence for magnetic field strengths $<10^{11}$ G in accreting high-mass
X-ray binary pulsars, as inferred from their pulse periods and period
derivatives (Popov & Turolla, 2013). If intrinsic fields in the range
$10^{10}-10^{11}$ G were common, equilibrium spin periods of $0.1-1$ s should
be frequent in HMXBs, but they are not. If the NSs in these systems are born
in the same way as isolated pulsars, this could imply that birth fields are
never so small; instead, field regrowth has occurred in all cases after it was
initially buried. One caveat here is that $\sim 40\%$ of HMXBs do not have
measured spin periods, although there is no strong selection effect against
detecting $P<1$ s.
Bernal et al. (2010) has revisited the process of hypercritical accretion onto
a magnetized neutron star, while Ho (2011) made new calculations of the
subsequent diffusion to constrain the magnetic fields of CCOs at birth and the
accreted mass, finding $10^{-4}-10^{-5}\,M_{\odot}$ for the latter. These
models are difficult to test using CCOs, because it would involve measuring
the braking index or the change of the dipole magnetic field directly. Ho
(2011) implied that the isolation of CCOs in the $P-\dot{P}$ diagram may be
less problematic in this model because, as their dipole magnetic fields
increase, they will evolve to join the bulk of the pulsar population. However,
for the first $\sim 10^{5}$ yr, field growth can only move a CCO vertically
upward in the $P-\dot{P}$ diagram, as the braking index has a large, negative
value. Orphaned CCOs should still have periods of $\sim 0.1$ s and lie in a
sparse region. For this reason, an important test for an example of field
growth is to determine the $\dot{P}$ of Calvera, which will reveal if its
dipole field is greater than that of PSR J0821$-$4300 and PSR J1852$+$0040.
Previous calculations of field burial and re-emergence for CCOs were one
dimensional. The first two-dimensional calculations for this purpose were
recently reported by Viganò & Pons (2012). They find that the accretion must
be essentially isotropic to bury the field sufficiently to cause the required
orders-of-magnitude reduction of the external dipole. If the accretion were
instead confined to the equator or the magnetic poles, the reduction of the
dipole component would not be significant enough to produce a CCO. Finally, it
remains to be seen if the resulting surface thermal distribution during the
CCO phase can be made compatible with observed spectra and pulse profiles.
Viganò & Pons (2012) briefly presented the temperature distribution from one
of their models that produces hot polar caps at a time when the external
dipole field is $10^{10}$ G. But even in this case, the temperature does not
vary by as much as a factor of 2 over the surface, a range that would be
required to match the properties of PSR J0821$-$4300 and other CCOs.
## 7\. Conclusions and Future Work
Measurement of the spin-down rate of the 112 ms PSR J0821$-$4300 in Puppis A
was achieved using X-ray observations coordinated between XMM–Newton and
Chandra, the resulting phase-connected ephemeris spanning 2.3 yr. We also
measured the proper motion of the pulsar in Chandra HRC images over 10.6
years. The proper motion makes a non-negligible contribution to the period
derivative via the Shklovskii effect, and the uncertainty in proper motion and
distance limits the accuracy of the intrinsic period derivative to $\approx
16\%$. The straightforward interpretation of these results is dipole spin-down
due to a weak surface magnetic field of $B_{s}=2.9\times 10^{10}$ G, the
smallest measured for any young neutron star, and nearly identical to that of
the CCO pulsar PSR J1852$+$0040 in Kes 79.
A phase-dependent spectral feature in PSR J0821$-$4300 can be modelled either
as an emission line of energy $\approx 0.75$ keV, or as a cyclotron absorption
line and its harmonic, with $E_{0}\approx 0.46$ keV. For reasons that are not
clear, it is stronger during the pulse-phase interval in which the continuum
spectrum is softer. There is only marginal evidence for long-term variability
of this feature. The local magnetic field strength in the area where the
spectral feature is produced may have to be larger than the dipole spin-down
value.
The existing spin-down measurements for three CCOs, including a definitive new
result for PSR J1210$-$5226, are compelling evidence that a weak dipole field
component is the physical origin of the CCO class in general. It is reasonable
to assume that the CCOs which have not yet been seen to pulse have magnetic
fields that are similar to or weaker than those of PSR J0821$-$4300 and PSR
J1852$+$0040\. Otherwise, if their fields were stronger, their spectra should
show cyclotron lines similar to PSR J1210$-$5226 and (possibly) PSR
J0821$-$4300\. Deep X-ray timing searches of the remaining members of this
class, to smaller limits on pulsed fraction, could still discover new pulsars
and supply valuable data on their birth properties and evolution.
A remaining theoretical puzzle about CCOs is the origin of their surface
temperature anisotropies, in particular, the one or two warm/hot regions that
are smaller than the full neutron star surface. The measured spin-down power
is too small to contribute to this emission. Continuing accretion is unlikely
because of the small spin-down rates. It appears that any explanation will
have to involve stronger magnetic field components in the crust, with toroidal
or quadrupolar geometries, that do not contribute to the dipole spin-down
torque. A physical model of thermal and magnetic field structure that self-
consistently reproduces the spin-down rates, X-ray spectra, and pulse
profiles, is still needed. If it can be studied with better data, the details
of the phase-dependent and possibly variable spectral feature from PSR
J0821$-$4300 may contribute to a more definite model of its surface magnetic
field geometry.
The CCO pulsars PSR J1852$+$0040 and PSR J0821$-$4300 fall in a region of
$B-P$ space that overlaps with what are assumed to be moderately recycled
pulsars, but is otherwise empty. An understanding of the evolutionary status
of CCOs is critically dependent upon a search for their descendants, the
orphaned CCOs without SNRs. Some single radio pulsars with spin parameters
similar to CCOs may be orphaned CCOs rather than recycled pulsars, and they
may be much younger than their characteristic ages. X-ray observations of
pulsars in this region may find evidence of their relative youth via surface
thermal emission. The spin-down rates of orphaned CCOs would reveal whether
CCOs have intrinsically weak magnetic fields, or if field re-emerges on a
timescale of $\sim 10^{4}$ yr from a normal magnetic field that was buried by
prompt fallback of supernova debris, as is invoked in some theoretical
studies. Such rapid evolution would lead to orphaned CCOs that lie directly
above CCOs on the $P-\dot{P}$ diagram but are still detectable as thermal
X-ray sources. The radio-quiet pulsar Calvera is a possible such candidate.
Whether or not CCOs have buried fields, the paucity of radio pulsars with
similar spin parameters is real and requires an explanation.
We thank Dany Page, and an anonymous referee, for helpful comments. This
investigation is based on observations obtained with XMM–Newton, an ESA
science mission with instruments and contributions directly funded by ESA
Member States and NASA, and Chandra. The opportunity to propose joint,
coordinated observations between XMM–Newton and Chandra was crucial for the
success of this effort. Financial support was provided by NASA grants
NNX11AD19G and NNX12AD41G for the XMM–Newton observations, and by Chandra
awards GO1-12001X and SAO GO1-12071X, issued by the Chandra X-ray Observatory
Center, which is operated by the Smithsonian Astrophysical Observatory for and
on behalf of NASA under contract NAS8-03060.
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|
arxiv-papers
| 2013-01-12T20:25:34 |
2024-09-04T02:49:40.216112
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "E. V. Gotthelf, J. P. Halpern, and J. Alford",
"submitter": "E. V. Gotthelf",
"url": "https://arxiv.org/abs/1301.2717"
}
|
1301.2739
|
# The thermodynamic quantities of a black hole with an $f(R)$ global monopole
Jingyun Man Hongbo Cheng111E-mail address: [email protected]
Department of Physics, East China University of Science and Technology,
Shanghai 200237, China
The Shanghai Key Laboratory of Astrophysics, Shanghai 200234, China
###### Abstract
The thermodynamic quantities such as the local temperature, heat capacity,
off-shell free energy and the stability of a black hole involving a global
monopole within or outside the $f(R)$ gravity are examined. We compare the two
classes of results to show the influence from the generalization of the
general relativity. It is found that the $f(R)$ theory will modify the
thermodynamic properties of black holes, but the shapes of curves for
thermodynamic quantities with respect to the horizon are similar to the
results within the frame of general relativity. In both cases there will exist
a small black hole which will decay and a large stable black hole in the case
that the temperatures are higher than their own critical temperature.
PACS number(s): 04.70Bw, 14.80.Hv
Keywords: Black hole, Global monopole, $f(R)$ gravity
I. Introduction
Recently more contributions were paid to the thermodynamics of various kinds
of black holes. More than thirty years ago, Bekenstein pointed out that the
entropy of a black hole is proportional to its surface area [1-3]. Hawking
also discussed the particle creation around a black holes to show that the
black hole has a thermal radiation with the temperature subject to its surface
gravity [4]. The issues about phase transitions of black holes in the frame of
semiclassical gravity were listed in Ref. [5]. Further the thermodynamic
properties of modified Schwarszchild black holes have been investigated [6,
7]. The thermodynamic behaviours including phase transition in Born-Infeld-
anti-de Sitter black holes were explored by means of various ways [8, 9]. The
phase transition of the quantum-corrected Schwarzschild black hole was
discussed, which fosters the research on the quantum-mechanical aspects of
thermodynamic behaviours [10].
In the process of the vacuum phase transition in the early Universe the
topological defects such as domain walls, cosmic strings and monopoles were
generated from the breakdown of local or global gauge symmetries [11, 12].
Among these topological defects, a global monopole as a spherical symmetric
topological defect occurred in the phase transition of a system composed by a
self-coupling triplet of scalar field whose original global O(3) symmetry is
spontaneously broken to U(1). It has been shown that the metric outside a
monopole has a deficit solid angle [13]. We researched on the strong
gravitational lensing for a massive source with a global monopole and find
that the deficit angle associated with the monopole affect the lensing
properties [14]. H. A. Buchdahl proposed a modified gravity theory named as
$f(R)$ gravity to explain the accelerated expansion of the Universe instead of
adding unknown forms of dark energy or dark matter [15-18]. The metric outside
a gravitational object involving a global monopole in the context of $f(R)$
gravity theory has been studied [19]. Further the classical motion of a
massive test particle around the gravitational source with an $f(R)$ global
monopole is probed [20]. We also examine the gravitational lensing for the
same object in the strong field limit [21].
Here we plan to investigate the thermodynamics of a static and spherically
symmetric black hole swallowing a global monopole or an $f(R)$ global
monopole. It is significant to understand the $f(R)$ theory in a new
direction. We wish to find the influences from the modified gravity on the
thermal properties of black holes. First of all we introduce a black hole
containing a global monopole in the context of $f(R)$ gravity theory and
certainly the black hole metric will recover to be one of the metrics of black
hole with a deficit solid angle. We show the dependence of the horizon on the
mass and parameters describing the monopole and the modification from $f(R)$
theory. We exhibit the thermodynamic characteristics due to the parameters of
the black hole. The thermodynamic stability will also be checked. We are going
to discuss the results in the end.
II. The thermodynamics of black holes involving an $f(R)$ global monopole
We adopt the line element,
$ds^{2}=A(r)dt^{2}-B(r)dr^{2}-r^{2}(d\theta^{2}+\sin^{2}\theta d\varphi^{2})$
(1)
which is static and spherical. In the $f(R)$ gravity theory, the action
introduced is,
$I=\frac{1}{2\kappa}\int d^{4}x\sqrt{-g}f(R)+I_{m}$ (2)
where $f(R)$ is an analytical function of Ricci scalar $R$ and $\kappa=8\pi
G$, $G$ is the Newton constant, $g$ is the determinant of metric and $I_{m}$
is the action of matter fields which can be denoted as,
$I_{m}=\int d^{4}x\sqrt{-g}{\cal L}.$ (3)
Here the Lagrangian for global monopole is [13],
${\cal
L}=\frac{1}{2}(\partial_{\mu}\phi^{a})(\partial^{\mu}\phi^{a})-\frac{1}{4}\lambda(\phi^{a}\phi^{a}-\eta^{2})^{2}$
(4)
where $\lambda$ and $\eta$ are parameters. The ansatz for the triplet of field
configuration showing a monopole is $\phi^{a}=\eta h(r)\frac{x^{a}}{r}$ with
$x^{a}x^{a}=r^{2}$, where $a=1,2,3$. $h(r)$ is a dimensionless function to be
determined by the equation of motion. This model has a global O(3) symmetry,
which is spontaneously broken to U(1). The field equation reads,
$F(R)R_{\mu\nu}-\frac{1}{2}f(R)g_{\mu\nu}-\nabla_{\mu}\nabla_{\nu}F(R)+g_{\mu\nu}\Box
F(R)=\kappa T_{\mu\nu}$ (5)
with $F(R)=\frac{df(R)}{dR}$ and $T_{\mu\nu}$ is the minimally coupled energy-
momentum tensor. The field equation (5) was solved under the weak field
approximation that assumes the components of metric tensor like $A(r)=1+a(r)$
and $B(r)=1+b(r)$ with $|a(r)|$ and $|b(r)|$ being smaller than unity [19].
Here the modified theory of gravity corresponds to a small correction on the
general relativity like $F(R(r))=1+\psi(r)$ with $\psi(r)\ll 1$. It is clear
that $F(R)=1$ is equivalent to the conventional general relativity. Further
the modification can be taken as the simplest analytical function of the
radial coordinate $\psi(r)=\psi_{0}r$. In this case the factor $\psi_{0}$
reflects the deviation of standard general relativity. The external metric of
the black hole with a global monopole is found finally [19, 20],
$A=B^{-1}=1-8\pi G\eta^{2}-\frac{2GM}{r}-\psi_{0}r$ (6)
where $M$ is the mass parameter. It should be pointed out that the parameter
$\eta$ is of the order $10^{16}GeV$ for a typical grand unified theory, which
means $8\pi G\eta^{2}\approx 10^{-5}$. If we choose $\psi_{0}=0$ excluding the
modification from $f(R)$ theory, the metric (6) will recover to be the result
by M. Barriola et.al. [13] like,
$ds^{2}=(1-8\pi G\eta^{2}-\frac{2GM}{r})dt^{2}-\frac{dr^{2}}{1-8\pi
G\eta^{2}-\frac{2GM}{r}}-r^{2}(d\theta^{2}+\sin^{2}\theta d\varphi^{2}).$ (7)
The function $A(r)$ is plotted in Figure 1 for comparison of two metrics for a
global monopole and $f(R)$ monopole black hole. According to the behavior of
$f(R)$ curve, it is clear that the black hole has two horizons, one is the
inner horizon which is supposed to be the event horizon $r_{H}$, the other one
is the outer horizon. The event horizons of two metrics seem to be the same
from Figure 1, because the given parameters $8\pi G\eta^{2}$ is very small,
but the analytic expression of $r_{H}$ are quite distinct. Solving the
equation $A(r_{H})=0$, the event horizon of metric (6) is located at,
$r_{H}=\frac{(1-8\pi G\eta^{2})-\sqrt{(1-8\pi
G\eta^{2})^{2}-8\psi_{0}GM}}{2\psi_{0}}.$ (8)
It also gives the relation between the mass parameter $GM$ and the event
horizon $r_{H}$,
$GM=\frac{1}{2}r_{H}(1-8\pi G\eta^{2}-\psi_{0}r_{H})$ (9)
for $f(R)$ monopole metric. It is seen from Figure 2 that there is a maximun
$GM_{0}=\frac{(1-8\pi G\eta^{2})^{2}}{8\psi_{0}}$ at $r_{H0}=\frac{1-8\pi
G\eta^{2}}{2\psi_{0}}$ where the inner and outer horizon meets. The Hawking
temperature can be obtained,
$\displaystyle T_{H}^{M}=T_{H}^{M}(\eta,\psi_{0})\hskip 28.45274pt$
$\displaystyle=\frac{1}{4\pi}[-g^{tt}g^{rr}g^{\prime}_{tt}]|_{r=r_{H}}\hskip
2.84526pt$ $\displaystyle=\frac{1}{4\pi}(\frac{1-8\pi
G\eta^{2}}{r_{H}}-2\psi_{0})$ (10)
where the prime stands for the derivative with respect to the radial
coordinate $r$. The local temperature is given by [22],
$\displaystyle T_{loc}^{M}=\frac{T_{H}^{M}}{\sqrt{A(r)}}\hskip 284.52756pt$
$\displaystyle=\frac{1}{4\pi}(\frac{1-8\pi
G\eta^{2}}{r_{H}}-2\psi_{0})\sqrt{\frac{r}{\psi_{0}r_{H}^{2}-(1-8\pi
G\eta^{2})r_{H}+(1-8\pi G\eta^{2})r-\psi_{0}r^{2}}}.$ (11)
Having omitted the influence from the modified gravity, we obtain the local
temperature as,
$T_{loc}=\frac{1}{4\pi r_{H}}\sqrt{\frac{(1-8\pi G\eta^{2})r}{r-r_{H}}}.$ (12)
The local temperatures for a global monopole and an $f(R)$ global monopole are
shown respectively in the Figure 3. There is a minimal local temperature
$T_{c}$, which means if the local temperature is below $T_{c}$, no black hole
exists. When the temperatures for the two kinds of black holes are high
enough, there will both exist two black holes, one is small and the other is
large. From the calculation of the extrema of local temperature,
$(\frac{\partial T_{loc}}{\partial r_{H}})_{r}=0$, the minimal local
temperature can be obtained as,
$T_{c}^{M}=\frac{\sqrt{r}}{\pi}(\frac{\psi_{0}}{(1-8\pi
G\eta^{2})^{\frac{2}{3}}-(1-8\pi
G\eta^{2}-2\psi_{0}r)^{\frac{2}{3}}})^{\frac{3}{2}}.$ (13)
The modifying factor $\psi_{0}$ from $f(R)$ leads the minimum of local
temperature lower than the one for global monopole black hole,
$T_{c}=\frac{3\sqrt{3}}{8\pi r}\sqrt{1-8\pi G\eta^{2}}$ (14)
for $8\pi G\eta^{2}=10^{-5}$, $r=10$ and $\psi_{0}=0.02$, then
$T_{c}^{M}=0.01836$, $T_{c}=0.02067$. The Figure 4 shows that the critical
temperature is a decreasing function of the modifying factor $\psi_{0}$
denoting the influence from $f(R)$ gravity here.
According to Bekenstein’s opinion, the entropy is proportional to the area of
event horizon and denoted as [1-3],
$\displaystyle S=\frac{A_{0}}{4}$ $\displaystyle=\pi r_{H}^{2}$ (15)
leading $dS=2\pi r_{H}dr_{H}$.
From the first law of thermodynamics $dE_{loc}=T_{loc}dS$, the thermodynamical
local energy can be derived as,
$\displaystyle E_{loc}^{M}=E_{0}+\int_{S_{0}}^{S}T_{loc}^{M}dS\hskip
199.16928pt$ $\displaystyle=r\sqrt{(1-8\pi
G\eta^{2})-\psi_{0}r}-\sqrt{r(r-r_{H})}\sqrt{(1-8\pi
G\eta^{2})-\psi_{0}(r+r_{H})}$ (16)
Here $S_{0}$ represents that $M=0$. For simplicity $E_{0}=0$. The
thermodynamic local energy in the case not belonging to $f(R)$ theory becomes,
$E_{loc}=\sqrt{(1-8\pi G\eta^{2})r}(\sqrt{r}-\sqrt{r-r_{H}}).$ (17)
For the purpose of checking the stability of the black holes, we should
discuss their heat capacity,
$\displaystyle C^{M}=(\frac{\partial E_{loc}^{M}}{\partial
T_{loc}^{M}})_{r}\hskip 256.0748pt$ $\displaystyle=2\pi(r-r_{H})[(1-8\pi
G\eta^{2})-2\psi_{0}r_{H}][(1-8\pi G\eta^{2})-\psi_{0}(r+r_{H})]\hskip
28.45274pt$ $\displaystyle\times[(1-8\pi
G\eta^{2})(\frac{r^{2}}{r_{H}^{2}}-3)\psi_{0}+(1-8\pi
G\eta^{2})^{2}\frac{3r_{H}-2r}{2r_{H}^{2}}+2\psi_{0}^{2}r_{H}]^{-1}.$ (18)
Similarly we choose the modifying factor $\psi_{0}=0$, the heat capacity of
black holes leads,
$\displaystyle C=(\frac{\partial E_{loc}}{\partial T_{loc}})_{r}\hskip
14.22636pt$ $\displaystyle=4\pi\frac{r_{H}^{2}(r-r_{H})}{3r_{H}-2r}.$ (19)
According to Eq. (18) and Eq. (19), we compare the heat capacities of these
two kinds of black holes in the Figure 5. The shapes of the curves of heat
capacities are similar, but can be recognized explicitly. The expression of
heat capacity for a Schwarzschild black hole with a global monopole is exactly
the same as the one for original Schwarzschild black hole. In Figure 5, the
$f(R)$ curve shifts from the conventional one. In every case the relatively
smaller horizon leads the heat capacity to be negative and the positive
capacity is due to the larger horizon, which meaning that the larger black
holes are stable. It should be emphasized that only huge black holes can
survive for long time within the frame of $f(R)$ theory. When
$r_{H}>\frac{1-8\pi G\eta^{2}-[(1-8\pi G\eta^{2})(2\psi_{0}r-(1-8\pi
G\eta^{2}))^{2}]^{3/2}}{2\psi_{0}}$, the $f(R)$ heat capacity is positive and
a large black hole is stable. A small black hole appears unstable for
$0<r_{H}<\frac{1-8\pi G\eta^{2}-[(1-8\pi G\eta^{2})(2\psi_{0}r-(1-8\pi
G\eta^{2}))^{2}]^{3/2}}{2\psi_{0}}$ since the heat capacity is negative. A
large Schwarzchild black hole with a monopole is stable when
$r_{H}>\frac{2r}{3}$, while it is unstable on $0<r_{H}<\frac{2r}{3}$.
In order to explore the phase transition among the black holes, we should
derive their off-shell free energy. The off-shell free energy can be defined
as,
$F_{off}^{M}=E_{loc}^{M}-TS$ (20)
where $E_{loc}^{M}$ is thermodynamic local energy from Eq. (16), $S$ is the
entropy of the black hole from Eq. (15) and $T$ is an arbitrary temperature.
The off-shell free energy of black holes containing a global monopole governed
by $f(R)$ gravity theory can be calculated as,
$F_{off}^{M}=r\sqrt{(1-8\pi
G\eta^{2})-\psi_{0}r}-\sqrt{r(r-r_{H})}\sqrt{(1-8\pi
G\eta^{2})-\psi_{0}(r+r_{H})}-\pi r_{H}^{2}T.$ (21)
If we are not going to consider the deviation from $f(R)$ theory, the Eq. (21)
will become the expression of the off-shell free energy for the black holes
with only a deficit solid angle like,
$F_{off}=\sqrt{(1-8\pi G\eta^{2})r}(\sqrt{r}-\sqrt{r-r_{H}})-\pi r_{H}^{2}T.$
(22)
In Figure 6 the behaviour of the off-shell free energy of black hole involving
a global monopole proposed by Barriola and Vilenkin [13] is shown as a
function of the horizon under several temperatures. Having considered extrema
of off-shell free energy, $(\frac{\partial F_{off}}{\partial r_{H}})_{r}=0$ or
$(\frac{\partial F^{M}_{off}}{\partial r_{H}})_{r}=0$, we give out the
critical temperatures which has already been mentioned above. When the
temperature is lower than the critical value, no black hole will appear. For
the temperature above the critical one the large black holes are stable but
the smaller ones are unstable. The dependence of the free energy of black
holes with an $f(R)$ global monopole on the horizon is plotted in the Figure 7
when the temperature is chosen to be several values around their own critical
temperature. The shapes of the off-shell free energy are similar to those in
the Figure 6. When the temperature is lower than critical value, there is no
real root from the equation $(\frac{\partial F_{off}}{\partial r_{H}})_{r}=0$
or $(\frac{\partial F^{M}_{off}}{\partial r_{H}})_{r}=0$, therefore no black
hole will appear. There are two equal roots $r_{H1}=r_{H2}=\frac{1-8\pi
G\eta^{2}-[(1-8\pi G\eta^{2})(2\psi_{0}r-(1-8\pi
G\eta^{2}))^{2}]^{3/2}}{2\psi_{0}}$ at critical temperature. When $T>T_{c}$,
two physically meaningful roots exist. An unstable small black hole appears at
event horizon $r_{H}=r_{H1}$, and a stable large black hole appears at
$r_{H}=r_{H2}$.
III. Discussion
We search for the thermodynamic quantities of the black holes with a global
monopole in the context of $f(R)$ gravity theory. We also obtain the
thermodynamic quantities excluding the modifications from $f(R)$ theory. We
compare the two classes of results to show how the generalized gravity
corrects the original thermodynamic quantities such as local temperature, heat
capacity, off-shell free energy. It should be pointed out that the
$f(R)$-modifications on these quantities are manifest. It is interesting that
the shapes of these quantities of black holes controlled by two different
gravity theories are similar although the quantities’ expressions are more
different from each other. In both cases the small black holes are unstable
and the large ones are stable when the temperature is higher than the critical
temperature and no black hole can exist as the temperature is sufficiently
low. The critical temperature has something to do with the modified factor
from $f(R)$ theory. Here we open a new window to explore the $f(R)$ theory
revising the Einstein Gravity.
In this paper, in what we pay more attention to is the thermodynamic
behaviours of a black hole with an $f(R)$ global monopole, as the stablity of
the black hole is dependent on them. There are some methods proposed for
probing a gravitational source which has a solid deflict angle subject to
$f(R)$ global monopole. But one necessary condition is that the black hole has
to be stable enough to be observed. Through calculation of the heat capacity,
only when the event horizon is larger than $\frac{1-8\pi G\eta^{2}-[(1-8\pi
G\eta^{2})(2\psi_{0}r-(1-8\pi G\eta^{2}))^{2}]^{3/2}}{2\psi_{0}}$, the black
hole of an $f(R)$ global monopole can be stable. It is the condition for a
large and more stable black hole given by extrema of off-shell free energy as
well. Moreover, a stable black hole also requires the local temperature higher
than the critical temperature $T_{c}$.
Acknowledge
This work is supported by NSFC No. 10875043 and is partly supported by the
Shanghai Research Foundation No. 07dz22020.
## References
* [1] J. D. Bekenstein, Lett. Nuovo Cim. 4(1972)737
* [2] J. D. Bekenstein, Phys. Rev. D7(1973)2333
* [3] J. D. Bekenstein, Phys. Rev. D9(1974)3292
* [4] S. W. Hawking, Commun. Math. Phys. 43(1975)199
* [5] G. J. Stephens, B. L. Hu, Int. J. Theor. Phys. 40(2001)2183
* [6] W. Kim, E. J. Son, M. Yoon, JHEP 0804(2008)042
* [7] R. G. Cai, L. M. Cao, N. Ohta, JHEP 1004(2010)082
* [8] Y. S. Myung, Y. Kim, Y. Park, Phys. Rev. D78(2008)084002
* [9] A. Lala, D. Roychowdhury, Phys. Rev. D86(2012)084027
* [10] W. Kim, Y. Kim, arXiv: 1207.5318
* [11] T. W. B. Kibble, J. Phys. A9(1976)1387
* [12] A. Vilenkin, Phys. Rep. 121(1985)263
* [13] M. Barriola, A. Vilenkin, Phys. Rev. Lett. 63(1989)341
* [14] H. Cheng, J. Man, Class. Quantum Grav. 28(2011)015001
* [15] H. A. Buchdahl, Non-linear Lagrangians and cosmological theory, MNRAS 150(1970)1
* [16] S. Nojiri, S. D. Odintsov, Phys. Rev. D68(2003)125312
* [17] S. M. Carroll, V. Duvvuri, M. Trodden, M. S. Turner, Phys. Rev. D70(2004)043528
* [18] S. Fay, R. Tavakol, S. Tsujikawa, Phys. Rev. D74(2007)063509
* [19] T. R. P. Carames, E. R. B. de Mello, M. E. X. Guimaraes, Int. J. Mod. Phys. A (2011)
* [20] T. R. P. Carames, E. R. B. de Mello, M. E. X. Guimaraes, arXiv: 1111.1856
* [21] J. Man, H. Cheng, arXiv: 1205.4857
* [22] R. C. Tolman, Phys. Rev. 35(1930)904
Figure 1: The figure shows the function $A(r)$ of $f(R)$ monopole black hole
(solid line) and Schwarzchild black hole with a global monopole (dot line).
Here $8\pi G\eta^{2}\approx 10^{-5}$, $\psi_{0}=0.02$ and $GM=1$.
Figure 2: The figure show the mass parameter $GM$ for $f(R)$ monopole black
hole (solid line) and global monopole black hole (dash line). Here $8\pi
G\eta^{2}\approx 10^{-5}$, $\psi_{0}=0.02$.
Figure 3: The solid and dotted curves of the dependence of the local
temperatures on the horizon with $8\pi G\eta^{2}\approx 10^{-5}$, $r=10$ and
$\psi_{0}=0.02$ for the Schwarzschild black hole with an $f(R)$ global
monopole or a global monopole respectively.
Figure 4: The dependence of the critical temperature for $8\pi
G\eta^{2}\approx 10^{-5}$ and $r=10$ on the modifying factor $\psi_{0}$ from
$f(R)$ gravity.
Figure 5: The solid and dotted curves correspond to the dependence of the heat
capacities on the horizon with $8\pi G\eta^{2}\approx 10^{-5}$, $r=10$ and
$\psi_{0}=0.02$ for the Schwarzschild black hole with an $f(R)$ global
monopole or a global monopole respectively.
Figure 6: The dotted, solid and dashed curves correspond to the dependence of
the off-shell free energy of the Schwarzschild black hole with a global
monopole on the horizon with $8\pi G\eta^{2}\approx 10^{-5}$ and $r=10$ for
$T=0.018,0.021,0.03$ respectively.
Figure 7: The solid, dot and dashed curves correspond to the dependence of the
off-shell free energy of the Schwarzschild black hole with an $f(R)$ global
monopole on the horizon with $8\pi G\eta^{2}\approx 10^{-5}$, $r=10$ and
$\psi_{0}=0.02$ for $T=0.018,0.021,0.03$ respectively.
|
arxiv-papers
| 2013-01-13T02:42:14 |
2024-09-04T02:49:40.230128
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jingyun Man, Hongbo Cheng",
"submitter": "Hongbo Cheng",
"url": "https://arxiv.org/abs/1301.2739"
}
|
1301.2769
|
# The holographic bound in the scalar-tensor and $f(R)$ gravities
J. T. Firouzjaee School of Physics and School of Astronomy, Institute for
Research in Fundamental Sciences (IPM), Tehran, Iran [email protected]
###### Abstract
The holographic bound has been extended to the different theory of gravities
such as scalar-tensor gravity and $f(R)$ gravity according to the Noether
charge definition of the entropy for a black hole surface. We have introduced
some popular examples of the flat FRW cosmology in order to investigate
holographic bound in scalar-tensor and $f(R)$ gravity. Using the holographic
bound, we put an additional constraint on the scalar-tensor gravity and $f(R)$
gravity parameters. We also discuss about the transformation from Jordan frame
to Einstein frame.
###### pacs:
04.50.Kd, 98.80.-k
## I Introduction
Forty years ago Bekenstein made the primary suggestion that black holes have
an entropy equal to a quarter of their surface area in Planck units. He also
conjectured a holographic Bound on the number of localized degrees of freedom
in a gravitational system bekenstein . The extension of this statement to more
general situations leads to the holographic principle thooft ; susskind95 .
Along this way, Fischler and Susskind fischler generalizes and refines a
cosmological bound. Subsequently, the holographic bound has been extended with
a more covariant formulation bousso99 . There were many attempts for extending
the holography bound in the different theory of gravities such as Gong gong
who has extended the holographic bound to the Brans-Dicke cosmology. There was
also another attempt verlinde showing that the closed FRW equation may be
rewritten so as to describe the universe entropy in terms of total energy and
Casimir energy. Surprisingly, it turns out that the corresponding formula has
correspondence with the Cardy formula (Cardy-Verlinde formula) for the entropy
of a two-dimensional conformal field theory. This formula may be written as a
dynamical entropy bound. By the way, the modified $f(R)$ gravity modeled as an
effective fluid and construct the corresponding Cardy-Verlinde formula for it
odinsov .
In the other side, the dark energy problem has led many attentions to modified
theory of gravity. Cosmic acceleration can either be explained by introducing
large amounts of dark energy or considering modifications to gravity such as
the addition of a suitable function $f(R)$ of the Ricci scalar to the
Einstein-Hilbert action reviewf(r) . Much attentions have been drawn by scalar
fields in studies of the early time universe. A variety of scalar potentials
has been investigated and a number of accelerating inflationary cosmology have
been advocated. On the other hand, alternative approaches for the dark energy
problem can be pursued in the scalar-tensor gravity elizalde04 . A good review
for the quintessence model is scalartensor . The prototypical scalar-tensor
alternative to general relativity is Brans-Dicke theory.
To construct the holographic bound in the modified theory of gravities, we
should have more profound look on the entropy of the black holes in these
theories. To this aim, Wald and Visser waldvisser extend the black hole
entropy to the more general theory of gravities which plays a basic role in
the holographic bound. Subsequently, the entropy of black holes in generalized
dilaton theories and in theories with Lagrangians that depend on an arbitrary
function of the Ricci tensor were also examined alex . The difference between
the form of the entropy in the Einstein gravity and the modified gravity leads
us to revisit the black hole area law and the horizon definitions in the
modified gravity. By the way, I examined the holographic bound in generalized
theories of gravity using the Noether charge definition for the black hole
entropy.
The paper is organized as follows. In section II, I extend the holographic
bound for general theory of gravity. Section III is devoted to the Holographic
Bound in Brans-Dicke Cosmology and scalar-tensor gravity. We investigate the
holographic bound in $f(R)$ cosmology in section IV. Section V is devoted to
the discussion about the transformation from the Jordan frame to Einstein
frame. Section VI contains a discussion and the conclusions.
## II Holographic bound for general gravity theories
In this section, the holographic entropy bound in the Susskind process will be
investigated. Let us look at a scalar-tensor gravity with the Lagrangian
density,
${\mathscr{L}}=F(\phi)\frac{R}{16\pi}+\mathrm{other\hskip 2.84544ptterms\hskip
2.84544ptindependent\hskip 2.84544ptof\hskip 2.84544ptRiemann}$ (1)
It has been discussed alex that the non-negative entropy in the closed
surface, $A$, in scalar-tensor theory for spherically symmetric case on the
black horizon is
$S=\frac{F(\phi)A}{4}$ (2)
If we consider a well-separated, non-interacting components which made of some
matter components $C_{i}$ and some black holes with horizon area $A_{i}$, the
total entropy of this system is,
$S_{total}^{initial}=S_{matter}+S_{BH}$ (3)
which $S_{matter}=\sum S(C_{i})$ and $S_{BH}=\sum F(\phi)\frac{A_{i}}{4}$.
Consider the case that this system interact until a new equilibrium is
established. The generalized second law (GSL) states, $S_{total}^{final}\geq
S_{total}^{initial}$ , if $S_{total}^{final}$ be the final entropy of the
system. Now assume that the final state of the system be a black hole with the
entropy $F(\phi)\frac{A}{4}$ (Having this form of the entropy leads that the
entropy bound formalism be invariant by the conformal transformation from the
Jordan frame to Einstein framealex ) this of The GSL state,
$S_{matter}+\sum F(\phi)\frac{A_{i}}{4}\leq F(\phi)\frac{A}{4}$ (4)
Since $\sum F(\phi)\frac{A_{i}}{4}$ is obviously non-negative, therefore
$S_{matter}\leq F(\phi)\frac{A}{4}$. This is the new form of entropy bound for
scalar-tensor theory. Similarly for $f(R)$ gravity with the Lagrangian density
${\mathscr{L}}=\frac{f(R)}{16\pi}$ (5)
the entropy bound will be $S_{matter}\leq f(R)_{,R}\frac{A}{4}$. Note that the
nature of the non-negative entropy leads $f(R)_{,R}\geq 0$. We should notice
that in scalar-tensor and $f(R)$ gravity, we have the non-constant
gravitational coupling. Therefore, we cannot write the entropy of different
parts separately in general. Generally the the area $A$ can be defined for any
two surface but scalar-tensor gravity scalar is a quasi-local function.
Therefore, in this formalism of the entropy bound for the scalar-tensor
gravity, the scalar field must be calculated at the same place in the boundary
with area $A$ .
In the case that there isn’t any symmetry in the surface, $A$, the holographic
bound can be expressed it this way
$S_{matter}\leq\int s_{ab}$ (6)
where the $s_{ab}$ is the entropy two form alex . In the case of scalar-tensor
gravity $s_{ab}=\frac{F(\phi)}{4}\,\varepsilon_{ab}\,$, and in the case $f(R)$
gravity the entropy two form is
$s_{ab}=\frac{f^{\prime}(R)}{4}\varepsilon_{ab}$ (the $\varepsilon_{ab}$ is
the area two form).
It should be stressed that the fact that the requirement of positive black
hole entropy simply avoids the appearance of ghost or tachyon fields in the
corresponding scalar field theory. Then a negative entropy is simply a
footprint of some instabilities in the Einstein frame negativeentropy . As was
shown in elizalde07 , the condition of positive entropy can be used in order
to constrain the viability of modified gravity theories.
The covariant holographic bound for scalar-tensor gravity can be stated as
follows.
Let $A(B)$ be the area of an arbitrary $D-2$ dimensional spatial surface $B$
(which need not be closed) and $F(\phi)|_{B}$ be the scalar-tensor gravity
scalar value at boundary $B$. $A$ $D-1$ dimensional hypersurface $L$ is called
a light-sheet of $B$ if L is generated by light rays which begins at $B$,
extend orthogonally away from B, and have non-positive expansion,
$\theta\leq 0$ (7)
everywhere on $L$. Let S be the entropy on any light-sheet of $B$. Then
$S\leq F(\phi)|_{B}\frac{A(B)}{4}$ (8)
The holographic bound for the $f(R)$ gravity can be stated in the same way,
but the $f(R)_{,R}\frac{A}{4}\geq 0$ is an extra assumption which we assume in
the classical gravities. On the other side the quantity $G_{eff}=G/f(R)_{,R}$
can be regarded as the effective gravitational coupling strength in analogy to
what is done in scalar-tensor gravity reviewf(r) . The positivity of $G_{eff}$
or $f(R)_{,R}>0$ is equivalent to the requirement that the graviton is not a
ghost. The positivity assumption of the entropy is closely related to positive
effective gravitational coupling assumption.
## III Holographic Bound in Brans-Dicke Cosmology and scalar-tensor gravity
The simplest way to incorporate the scalar field as gravitational field is
Brans-Dicke theory in which the gravitational coupling constant is replaced by
a scalar field. The Brans-Dicke Lagrangian in Jordan frame is given by
${\cal
L}_{BD}=-\sqrt{-\gamma}\left[\phi{\tilde{R}}+\omega\,\gamma^{\mu\nu}{\partial_{\mu}\phi\partial_{\nu}\phi\over\phi}\right]-{\cal
L}_{m}(\psi,\,\gamma_{\mu\nu}).$ (9)
The homogeneous and isotropic Friedman-Robertson-Walker (FRW) space-time
metric is
$ds^{2}=-dt^{2}+a^{2}(t)\left[{dr^{2}\over 1-k\,r^{2}}+r^{2}\,d\Omega\right],$
(10)
the above metric can be written as
$ds^{2}=-dt^{2}+a^{2}(t)\left[d\chi^{2}+f^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\right],$ (11)
where
$f=\left\\{\begin{array}[]{lc}\chi&k=0,\\\ \sinh\chi&k=-1,\\\
\sin\chi&k=1.\end{array}\right.$ (12)
Based on the FRW metric and the perfect fluid
$T_{m}^{\mu\nu}=(\rho+p)\,U^{\mu}\,U^{\nu}+p\,g^{\mu\nu}$ as the matter
source, we can get the evolution equations of the universe from the action (9)
$H^{2}+{k\over a^{2}}+H{\dot{\phi}\over\phi}-{\omega\over
6}\left({\dot{\phi}\over\phi}\right)^{2}={8\pi\over 3\phi}\rho,$ (13)
$\ddot{\phi}+3H\dot{\phi}=4\pi\beta^{2}(\rho-3p),$ (14)
$\dot{\rho}+3H(\rho+p)=0.$ (15)
Where $\beta^{2}=2/(2\omega+3)$. If we are given a state equation for the
matter $p=\gamma\rho$, then the solution to Eq. (15) is
$\rho\,a^{3(\gamma+1)}=C_{1}.$ (16)
For the case $k=0$, we can get the power-law solutions to the Eqs. (13) and
(14) with the help of Eq. (16) gong ,
$a(t)=a_{0}\,t^{p},\quad\phi(t)=\phi_{0}\,t^{q},$ (17)
where
$\displaystyle p={2+2\omega(1-\gamma)\over 4+3\omega(1-\gamma^{2})},\quad
q={2(1-3\gamma)\over 4+3\omega(1-\gamma^{2})},\quad$
$\displaystyle-1\leq\gamma<1-{2\over 3+\sqrt{6}/\beta},\hskip 31.2982pt$ (18)
$a_{0}$ and $\phi_{0}$ are integration constants, and
$[q(q-1)+3pq]\phi_{0}=4\pi\beta^{2}(1-3\gamma)C_{1}a_{0}^{-3(\gamma+1)}$.
To calculate the maximum entropy bound we need to define the apparent horizon
surface. The apparent horizon is defined geometrically as a sphere at which at
least one pair of orthogonal null congruences have zero expansion. It
satisfies the condition
${\dot{a}\over a}=\pm{f^{\prime}\over af},$ (19)
In the flat case the apparent horizon radius is $r_{AH}=\frac{1}{\dot{a}}$.
Let us consider the light-sheets which their null generators have negative
expansion (for example we consider past ingoing null generator). At any time,
the spherical area for $r<r_{H}$ is smaller than the apparent horizon surface
area, $A<A_{AH}$.
The holographic bound can be written,
$\frac{4S}{\phi A}\leq\frac{sr_{AH}}{3\phi
a(t)^{2}}=\frac{st^{1-3p-q}}{3p\phi_{0}},$ (20)
where $s$ is the constant comoving entropy density. This term is maximized by
the outermost normal surface at any given time $t$, on the apparent horizon
sphere. The holographic bound in satisfied of $1-3p-q<0$. Therefore, we should
have
$\frac{-3\omega\gamma^{2}+6(w+1)\gamma-3\omega-4}{4+3\omega(1-\gamma^{2})}<0.$
(21)
This is the corrected version of the holographic bound which was expressed in
gong . In the case that we have general relativity limit, $\omega>>1$, we get
the standard causal energy condition (Note that, the current solar system
observation of Cassini spacecraft requires that $\omega$ must exceed 40000
omega .).
This bound will be held for $\gamma>1+\frac{\sqrt{9+6\omega}+3}{3\omega}$ and
$\gamma<1+\frac{\sqrt{9+6\omega}-3}{3\omega}=1-{2\over 3+\sqrt{6}/\beta}$.
Therefore, this holographic bound condition are compatible wtih the condition
(III).
Let us consider an example of scalar-sensor gravity elizalde04 . The scalar-
tensor action can be written as
$\displaystyle S$ $\displaystyle=$ $\displaystyle{1\over\kappa^{2}}\int
d^{d}x\sqrt{-g}\left(F(\Phi)R\mp{1\over
2}\partial_{\mu}\Phi\partial^{\mu}\Phi-U(\Phi)\right)$ $\displaystyle+\int
d^{d}x\sqrt{-g}\left(-{1\over
2}\partial_{\mu}\chi\partial^{\mu}\chi-U(\chi)\right)\ .$
Here
$F(\Phi)={\alpha^{2}\over 4|\gamma|}\Phi^{2}\ ,\quad
U(\Phi)=V_{0}\left({\alpha\Phi\over
2\sqrt{|\gamma|}}\right)^{4\left(1-\sqrt{1+{\gamma\over 3\alpha^{2}}}\right)}\
.$
and the field $\phi$ is given by
$\Phi\equiv{2\sqrt{|\gamma|}\over\alpha}e^{{\alpha\over 2}\phi}$. Here
$\alpha$ and $\gamma$ are constant parameters. We define the new variable
$\varphi=\phi\sqrt{\alpha^{2}+{\gamma\over 3}}.$ (22) $\displaystyle
a=a_{E0}\left({t\over t_{0}}\right)^{{3\over
4}\varphi_{0}^{2}-\beta\varphi_{0}/[1-{\beta\varphi_{0}\over 2}]}$
$\displaystyle\phi={\beta\varphi_{0}\over\alpha\left(1-{\beta\varphi_{0}\over
2}\right)}\ln{t\over t_{0}}\ .~{}~{}~{}~{}$ (23)
Here $\beta\equiv{\alpha\over\sqrt{\alpha^{2}+{\gamma\over 3}}}\ $ and $t_{0}$
is constant. The holographic bound is
$\displaystyle\frac{4S}{F(\Phi)A}\leq\frac{4sr_{AH}}{3F(\Phi)a(t)^{2}}\propto
t^{1-{9\over
4}\varphi_{0}^{2}+\frac{2\beta\varphi_{0}}{[1-{\beta\varphi_{0}\over 2}]}}.$
(24)
To satisfying the holographic bound in any time, we get the following
condition
$1-{9\over
4}\varphi_{0}^{2}+\frac{2\beta\varphi_{0}.}{[1-{\beta\varphi_{0}\over 2}]}<0$
(25)
This gives an extra constraint which enable us to limit the on the $\beta$ and
$\varphi_{0}$ parameters.
## IV Holographic Bound $f(R)$ Cosmology
Recently, $f(R)$ theories have been extensively studied in cosmology and
gravity such as inflation, dark energy, local gravity constraints,
cosmological perturbations, and spherically symmetric solutions in weak and
strong gravitational backgrounds. As discussed in section II, It is reasonable
to assume that the entropy is non-negative quantity in the classical
gravities.
To investigate the holographic bound, we present two famous model in these
gravity. The first example which we will refer to is a model of the
form$f(R)=R-\mu^{2(n+1)}/R^{n}$, where $\mu$ is a suitable parameter carroll .
The Ricci curvature for the flat FRW solution is
$R=6\left[\frac{\ddot{a}}{a}+(\frac{\dot{a}}{a})^{2}\right]$ (26)
In this case, the scale factor is assumed to be a generic power law,
$a(t)\propto t^{p}$ for the flat case, where
$p=\frac{(2n+1)(n+1)}{n+2},$ (27)
The $R$ solution is, $R=-6p/t^{2}$. Therefore,
$f^{\prime}(R)=1+n\mu^{2(n+1)}(-6p)^{-n-1}t^{2n+2}$. The holographic bound
gives
$\displaystyle\frac{4S}{f^{\prime}(R)A}\leq\frac{4sr_{AH}}{3f^{\prime}(R)a(t)^{2}}=\frac{4st^{1-3p}}{3pf^{\prime}}\propto
t^{-1-2n-3p},$ (28)
for $n>-1$. The holographic bound is satisfied if $-1-2n-3p<0$. This condition
will be satisfied for all rang $n>-2$. The best fitting for the astronomical
data bounds $n$ in this range carroll . Note that there is also another
constraint that comes from the positivity of the entropy
$f^{\prime}(R)=1+n\mu^{2(n+1)}(-6p)^{-n-1}t^{2n+2}>0$ elizalde07 .
Another example for the function $f$ is $f(R)\propto R^{n}$ Capozziello2003 .
The solution for the flat FRW universe gives the scale factor as $a(t)\propto
t^{p}$ where
$p=\frac{-2n^{2}+3n-1}{n-2}$ (29)
From the Ricci scalar solution we get $f^{\prime}(R)=(-6p)^{n-1}t^{2-2n}$.
The holographic bound gives
$\frac{4S}{f^{\prime}(R)A}\leq\frac{4sr_{AH}}{3f^{\prime}(R)a(t)^{2}}\propto
t^{-1+2n-3p},$ (30)
The holographic bound leads us to this condition
$-1+2n-3p=\frac{8n^{2}-14n+5}{n-2}<0$. Therefore, the holographic bound
constraint on $n$ becomes
$n<\frac{1}{2}~{},~{}~{}~{}~{}n>\frac{5}{4}$ (31)
The observational data limits the $n$ for $-0.450<n<-0.370$ and
$1.366<n<1.376$. Therefore, the holographic entropy bound is satisfied for
these intervals of $n$ Capozziello2003 .
The holographic bound condition for the FRW model in Einstein gravity leads us
to the causal energy condition bousso02 . Imposing the energy conditions for
the effective stress-energy tensor of f(R) gravity is not very meaningful
because the effective stress-energy terms coming from the geometry (when we
write the field equations of alternative gravities as effective Einstein
equations ) violate all the energy conditions generally faraoni04 .
## V the holographic bound in Einstein frame
Here we recall the conformal transformation which transform the quantities
from the Jordan frame to the Einstein frame which is like the Einstein general
relativity. It has been argued in the literature that, classically, the two
frames are physically equivalent einstein . Since the black hole entropy in
the Einstein frame have the famus form $S=\frac{A}{4}$, the holographic bound
will be the same standard holographic bound which was defined by Bousso
bousso02 . Note that the $\phi$ and $f(R)$ act as a field in the Einstein
equation in Einstein frame. Now look at the conformal transformation which
transfer from Jordan frame to Einstein frame;
$g_{ab}=\Omega^{2}\gamma_{ab}.$ (32)
In this equation $\Omega=\sqrt{F(\Phi)}$ and $\Omega=\sqrt{f^{\prime}(R)}$ in
the case of the scalar-tensor gravity and f(R) theory respectively. Now
consider the FRW cosmology in the Einstein frame;
$ds^{2}=-d\tau^{2}+R^{2}(t)\left[d\chi^{2}+f^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\right],$ (33)
The holographic bound must be written,
$\frac{4S}{A}\leq\frac{sr_{AH}}{3R(t)^{2}}$ (34)
where the apparent horizon radius is $r_{AH}=1/\frac{dR(t)}{d\tau}$. Putting
the $d\tau=\sqrt{F(\Phi)}dt$ and $R(t)=\sqrt{F(\Phi)}a(t)$ in the above
equation, we exactly get the holographic bound for scalar-tensor gravity (24).
Similarly if we insert the $d\tau=\sqrt{f^{\prime}(R))}dt$ and
$R(t)=\sqrt{f^{\prime}(R)}a(t)$ in the above equation, we get the holographic
bound for scalar-tensor gravity (28). Therefore the holographic bounds in
these two frame are equivalent and express the same equation.
## VI discussion
In this paper we have proposed a new version of the holographic bound for
scalar-tensor gravity and $f(R)$ theory of gravity. The key point for this
derivation is the definition of the entropy for a black hole surface in these
theory of gravities waldvisser . I have discussed about the holographic
aspects of flat FRW cosmology for these gravities. As consequence of
holographic bound, some free parameters in these are constrained even in some
case we can rule out some models in scalar-tensor and $f(R)$ gravity. As we
saw, the transformation from the Jordan frame to Einstein frame gives the
standard holographic bound and the holographic bounds in these two frame are
equivalent and express the same equation.
## VII acknowledgment
I would like to thank Alex B. Nielsen for helpful discussion on drafts of this
text.
## References
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|
arxiv-papers
| 2013-01-13T12:32:06 |
2024-09-04T02:49:40.237086
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. T. Firouzjaee",
"submitter": "Javad Taghizadeh firouzjaee",
"url": "https://arxiv.org/abs/1301.2769"
}
|
1301.2957
|
2012 12
# Characters and patterns of communities in networks
ANGSHENG LI and JIANKOU LI Chinese Academy of Sciences
###### Abstract
A community can be seen as a group of vertices with strong cohesion among
themselves and weak cohesion between each other. Community structure is one of
the most remarkable features of many complex networks. There are various kinds
of algorithms for detecting communities. However it is widely open for the
question: what can we do with the communities? In this paper, we propose some
new notions to characterize and analyze the communities. The new notions are
general characters of the communities or local structures of networks. At
first, we introduce the notions of internal dominating set and external
dominating set of a community. We show that most communities in real networks
have a small internal dominating set and a small external dominating set, and
that the internal dominating set of a community keeps much of the information
of the community. Secondly, based on the notions of the internal dominating
set and the external dominating set, we define an internal slope (ISlope, for
short) and an external slope (ESlope, for short) to measure the internal
heterogeneity and external heterogeneity of a community respectively. We show
that the internal slope (ISlope) of a community largely determines the
structure of the community, that most communities in real networks are
heterogeneous, meaning that most of the communities have a core/periphery
structure, and that both ISlopes and ESlopes (reflecting the structure of
communities) of all the communities of a network approximately follow a normal
distribution. Therefore typical values of both ISolpes and ESoples of all the
communities of a given network are in a narrow interval, and there is only a
small number of communities having ISlopes or ESlopes out of the range of
typical values of the ISlopes and ESlopes of the network. Finally, we show
that all the communities of the real networks we studied, have a three degree
separation phenomenon, that is, the average distance of communities is
approximately $3$, implying a general property of true communities for many
real networks, and that good community finding algorithms find communities
that amplify clustering coefficients of the networks, for many real networks.
###### category:
H.2.8 Database Management Database applications
###### keywords:
Data mining
###### keywords:
community, internal dominating set, external dominating set, internal slope,
external slope
††terms: Measurement; Experimentation
The first author is partially supported by the Hundred-Talent Program of the
Chinese Academy of Sciences. All authors are partially supported by the Grand
Project “Network Algorithms and Digital Information” of the Institute of
Software, Chinese Academy of Sciences.
Author’s addresses: Angsheng Li and Jiankou Li, State Key Laboratory of
Computer Science, Institute of Software, Chinese Academy of Sciences, P.O. Box
8718, Beijing, 100190, P.R.China
## 1 Introduction
Real networks differ from random graphs in the way that they are organized
with a high level of order. Such an organization results to remarkable common
phenomena of real networks, for instance: the heavy tail degree distributions,
the high clustering coefficients and the small average distances etc [Barabási
and Albert (1999), Watts and Strogatz (1998)]. In addition, another remarkable
common feature in various networks is the community structure. Community is an
important notion to disclose the structure of networks, playing the role in
bridging the local vertices and the global network. On one hand, we could
extract communities from a network to study its internal structure and its
relationship with the rest of the network from the local point of view. On the
one hand, we could take each community as a unit of the network, to illustrate
the connecting patterns of different communities of real networks through the
distributions of different properties of communities from the global point of
view [De Nooy et al. (2011)].
Massive work has been devoted to the study of communities, including the main
definitions of the community problem, algorithms developing for finding
communities, comparison and tests of different algorithms etc [Fortunato
(2010)]. Leskovec et al. [Leskovec et al. (2009)] analyzed community
structures in large real networks and tried to find the “best” communities at
various sizes. They showed that the “best” communities seem to be
characterized by size of $100$. The distribution of sizes of communities has
also been studied, showing that in some cases, they have the skewed
distribution [Clauset et al. (2004), Newman (2004b)]. The small community
phenomenon was introduced recently, that is, there are models, classical or
new, such that networks from the models are rich in small communities, that
is, quality communities of small sizes [Li and Peng (2011), Li and Peng
(2012)], for which the mechanism is homophyly.
Intuitively speaking, a community of a network can be interpreted as a
relatively independent and stable unit of the network, and the rich
communities of a network are taken as the local structures of the network.
This suggests fundamental questions such as: What can we do with the
communities? Are there some characters of all the communities of a network?
What information of the network can we extract from the communities? What
characters of communities (largely) determine the local patterns of the
network? What are the relationship between the found communities and the true
communities? These questions are widely open in the current state of the art.
This motivates the research in the present paper. For this, we investigate the
following: (1) How to extract central nodes from a community? (2) How to
extract useful information from the communities? (3) How do communities
interact with each other? (4) How to measure the heterogeneity of a community?
(5) What general properties do the communities (found by a reasonably good
algorithm) have?
By using a variant of the local spectral partitioning algorithm [Andersen et
al. (2006)], we find rich communities in real networks. These networks include
collaboration networks, citation networks, email networks and one benchmark
network 111All the data in this paper can be found from the websites:
http://snap.standford.edu, or http://www-personal.umich.edu/~mejn/netdata and
we only consider the corresponding undirected graphs. [Girvan and Newman
(2002), Leskovec et al. (2007)]. In collaboration network a node denotes a
scientist and an edge indicates that the two scientists have coauthored a
paper. In the citation networks a node denotes a paper in some fields and an
edge between two papers indicates that at least one paper has cited the other.
Communities in this networks may correspond to different research groups or
research themes. Two email networks are also used in our study, in which each
node corresponds to an email address and an edge between nodes $i$ and $j$
represents $i$ sending at least one message to $j$ or $j$ sending at least one
message to $i$. A well known benchmark network of American college football
teams complied by Grivan and Newman [Girvan and Newman (2002)] is also used.
Nodes of the network represent teams and an edge between two nodes represents
that the corresponding two teams play against each other. The network contains
$12$ true communities, which correspond to $12$ different conferences that the
teams belong to. All networks above have good community structures so that
they are good candidates for investigating the characters and connecting
patterns of local structures of networks.
We organize the paper as follows. In section 2, we propose the notions of
internal dominating ratio and external dominating ratio to measure the
importance of a subset of a community. Then we give the definition of internal
dominating set (IDS) and external dominating set (EDS). In section 3, we
verify that, the internal dominating set of a community is much more smaller
than the community and keeps largely the information of the community. In
section 4, we define internal slope (ISlope) and external slope (ESlope) of a
community to measure the internal heterogeneity and the external heterogeneity
of the community, respectively. We analyze the relationship between the
structure and the ISlopes and give the distributions of the ISlopes and the
ESlopes of all the communities of the real networks. In Section 5, we analyze
more general properties like average distances, diameters and clustering
coefficients of all the communities for each of the networks. Finally, in
section 6, we summarize the conclusions of the paper.
## 2 Internal and external dominating sets
Statistics of real networks. All the results are calculated by averaging the
corresponding properties of all the communities. The IDR and EDR are the
ratios of centrality of $5$-IDS and $5$-EDS; the IDN and EDN are the sizes of
$0.8$-IDS and $0.8$-EDS Network IDR EDR IDN EDN ISlope ESlope football 0.99
0.61 2.6 9.3 0.19 0.37 cit_hepth 0.75 0.49 10 32 0.41 0.54 cit_hepph 0.73 0.39
12 56 0.5 0.54 col_astroph 0.93 0.79 3.7 8.1 0.36 0.65 col_condmat 0.85 0.79
9.6 16 0.42 0.66 col_grqc 0.94 0.91 3.1 3.9 0.37 0.67 col_hepth 0.69 0.64 23
27 0.38 0.64 col_hepph 0.8 0.7 11 16 0.38 0.64 email_enron 0.93 0.86 3 7.8
0.55 0.68 email_euall 0.98 0.95 1.7 2.4 0.92 0.89
Given a community of a network, we may want to extract a small set of nodes
that are more central to the community than the rest of nodes in the
community. Taking the citation network for an example, we are interested in a
small number, $10$ say, of important papers that are central to the whole
community which usually includes hundreds of papers. In this case, we would
hope that with the short list of key papers, we will not lose any essential
information of the whole community. This analysis of centrality has been
studied for the whole networks, for example, it was shown that a small
fraction of nodes accumulates a large proportion of links in the networks
[Newman (2004a)], and that only $20\%$ of most-linked authors in Economics
account for about $60\%$ of all the links [Goyal et al. (2006)]. So there are
indeed some nodes taking the central position in networks. We believe that
similar centrality phenomena occurs in true communities of many real networks,
and that the main goal of community finding algorithms is to find the true
communities of the networks. The question is: what can we say about the
centrality of the communities found by our algorithms? This would be the first
step to understand the relationship between the true communities and the
communities found by algorithms.
Some centrality measures, initially introduced in social studies, could be
used, for instance, the degree centrality, the closeness centrality, and the
betweenness centrality etc [Freeman (1979)]. These measures assume a
relationship between the structural position and influential power in group
processes [Bavelas (1948)], and are developed and widely used in the
literature [Nicosia et al. (2012)]. The mechanism behind this idea is that the
centrality of a vertex could be predicted from its position and the network
structure in which it was embedded as well as from its own characteristics
[Rogers (1974)]. Except for these centrality measures, vertices could also be
classified according to their roles within their communities.
Guimer$\grave{a}$ and Amaral decide the role of a vertex by a within-module
degree $z_{i}$ and a participation ratio $P_{i}$ and distinguish seven roles
that vertices can play, based on the values of the pair $(z,P)$ [Guimera and
Amaral (2005)].
In this section, we propose the notion of internal and external dominating
sets of a community by modifying the notion of the dominating set. The
dominating set problem is classical in graph algorithms: Given a graph
$G=(V,E)$, we say that a set $S\subseteq V$ is a dominating set if every node
$v\in V$ is either an element of $S$ or adjacent to an element of $S$. The
dominating number is the number of vertices in a smallest dominating set for G
[Allan and Laskar (1978), Haynes et al. (1998)].
For a community, we distinguish two roles that nodes can play in a community,
as an internal role and an external role, measured by links within and outside
of the community respectively. For a subset of a given community, its internal
dominating ratio (IDR, for short) is defined as follows.
Let $C$ be a community, $S$ be a subset of $C$, $N(S,C)$ be the neighbors of
$S$ within community $C$. Then we define the internal dominating ratio of $S$
in $C$, written by IDR, as follows:
${\rm IDR}(S)=\frac{|S\cup N(S,C)|}{|C\cup N(C,C)|}=\frac{|S\cup
N(S,C)|}{|C|}$ (1)
The dominating ratio has been used previously to measure the social centrality
in social networks [Freeman (1979)]. Our internal dominating ratio (IDR)
measures the importance of a group of nodes in a community, and thus it can be
seen as a general format of degree centrality of communities.
Following the definition above, we consider two problems: 1) when given a
number $k$ (usually small), we want to find a subset $S$ of size $k$ with
$\max\\{{\rm IDR}\\}$, in which case, we call this subset a $k$-IDS; 2) when
given a real number $p$ in $[0,1]$, we want to find a subset $S$ whose IDR is
bigger than $p$ with the minimum number of nodes, in which case, we call this
subset a $p$-IDS.
Similarly to IDR, we give the definition of external dominating ratio (EDR).
Let $C$ be a community, $S$ be a subset of $C$, $N(S,\bar{C})$ be the
neighbors of node $v$ that are outside of $C$. Then the external dominating
ratio ( EDR) of $S$ in $C$ is defined as follows:
${\rm EDR}(S)=\frac{|N(S,\bar{C})|}{|N(C,\bar{C})|}$ (2)
Figure 1: A community to illustrate the IDR and EDR. The red nodes are from
one community. Let $S=\\{v_{3},v_{4}\\}$ be a subset of this community,
$N(S,C)=\\{v_{0},v_{2},v_{5}\\}$, $N(S,\bar{C})=\\{v_{6},v_{8}\\}$,
$N(C,\bar{C})=\\{v_{6},v_{7},v_{8}\\}$, so IDR(S) = (2 + 3) / 6 = 5/6, EDR(S)
= 2/3
We also give the notations $k$-EDS and $p$-EDS similarly. Figure 1 is an
example of the IDRs and EDRs. From the definitions, we notice that we are not
using the notion of classic dominating set [Allan and Laskar (1978), Haynes et
al. (1998)], instead, we introduce two parameters $k$ and $p$ to define the
general format of dominating sets. We emphasize that the classification are
based on nodes positions in a community. By definition, it is conceivable that
nodes in the IDS are more important for the function and stability of the
community, and that nodes in the EDS mainly take charge of the communication
between the community and the nodes outside of the community.
The dominating problem is an NP-complete decision problem [Haynes et al.
(1998)]. Here we introduce a simple greedy algorithm to find the $p$-IDS and
$p$-EDS, where $G$ is a graph, $C$ is a community and $p$ is a real number in
$[0,1]$.
Input: Graph $G$, community $C$, and a real number $p\in[0,1]$.
Output: The $p$-internal dominating set $S$.
Let $G_{C}$ be the induced subgraph of vertices $C$ from $G$. Set
$S=\emptyset$;
repeat
Let $v$ be a node in $C\setminus S$ such that $v$ has the maximal number of
neighbors in $C\setminus(S\cup N(S))$, where $N(S)$ is the neighbors of $S$ in
$G_{C}$;
$S\leftarrow S\cup\\{v\\}$ ;
until _${\rm IDR}(S)\leq p$_ ;
ALGORITHM 1 Finding $p$-{IDS}
Input: Graph $G$, community $C$, and a real number $p\in[0,1]$.
Output: The $p$-external dominating set $S$.
Set $S=\emptyset,N(S,\bar{C})=\emptyset$, where $N(S,\bar{C})$ is the number
of neighbors of $S$ outside of $C$;
repeat
Let $v$ be a node in $C\setminus S$ such that $v$ has the maximal number of
neighbors in $N(C,\bar{C})\setminus N(S,\bar{C})$;
$S\leftarrow S\cup\\{v\\}$;
$N(S,\bar{C})\leftarrow N(S,\bar{C})\cup N(v,\bar{C})$;
until _${\rm EDR}(S)\leq p$_ ;
ALGORITHM 2 Finding $p$-{EDS}
Given a number $p$ between $0$ and $1$, we could find the $p$-{IDS} and
$p$-{EDS} by using the above algorithms. Similarly when given a small number
$k$, we could calculate the $k$-{IDS} and $k$-{EDS} by using the same
algorithm with slight modification of the terminating condition. In our
experiment, we set $k=5$ when calculating the $k$-{IDS} and the $k$-{EDS}, and
set $p=0.8$ when calculating the $p$-{IDS} and $p$-{EDS}, see Table 2 for
details.
From Table 2, we observe that only five nodes could dominate most of the
members of the communities from both internal and external sides, that the
internal dominating ratios of $5$ internally central nodes are larger than the
external dominating ratio of $5$ externally central nodes, for each of the
networks, that external connecting patterns of the communities are more
decentralizing than that of the internal connecting patterns, for each of the
networks, that it only needs at most $10$ nodes to internally dominate at
least $80\%$ of the whole community, that it needs at most $32$ nodes to
externally dominate $80\%$ of the outgoing links of the communities, and that
external dominating numbers are larger than the internal dominating numbers
for all communities and for all the networks.
In summary, we have that most communities have a small internal dominating
set, and a small external dominating set, which is slightly larger than the
internal dominating set of the corresponding communities, on the average, for
all the networks.
## 3 Extracting Local Information
In the last section, we verify that most communities have a small internal
dominating set, and a small external dominating set. The questions are: How
much information of a community is preserved in the dominating set of the
community? How to extract essential information of a community from the small
dominating sets?
In this section, we verify that the internal dominating sets (IDSs) indeed
preserve essential information of the communities. We verify this result by
predicting and confirming keywords of papers in a citation network.
We say that a paper has keywords, if its authors have explicitly list its
keywords, and does not have keywords, otherwise.
Keywords of papers play an important role in information retrieval. In many
citation networks, there is a huge number of papers whose keywords are not
listed by their authors, which is an obstacle for people to sufficiently use
the networks222We implement the verification for just one citation network,
because this is the only available network in which titles, abstract of
papers, and keywords of a small number of papers are included. Most networks
have a topological structure with nodes and edges only..
In the citation-hepth networks, there are about $27,770$ papers, in which only
$10\%$ or so have keywords. Predicting and confirming the missing keywords for
the other papers are obviously significant for information retrieval.
Given a community $C$ in a citation network, we predict and confirm keywords
for papers in $C$ by the following procedure.
Input: Graph $G$, community $C$, and keyword dictionary $Dic$
Output: Papers with predicting keywords
Calculate $p$-IDS or $k$-IDS of $C$;
Suppose that $L=\\{k_{1},k_{2},\cdots,k_{i}\\}$ are listed keywords from the
IDS with descending order according to their popularity in $C$. For a given
paper $P$ in $C$ whose keywords are not listed in the network, for each $j\leq
i$, if $k_{j}$ appears in either the title or the abstract of paper $P$, we
say that $k_{j}$ is a predicted and confirmed keyword of $P$;
ALGORITHM 3 Predicting keywords using internal dominating set
We choose parameter $p=0.8$, run the algorithm on the citation network, and
report the results in Table 3. The first column of Table 3 presents the number
of keywords we used for the prediction and confirmation for each communities,
that is, the length $i$ of $L$ in the algorithm, the second column of the
table are numbers of papers whose keywords have been predicted and confirmed
corresponding to different lengths of $L$ in the first column.
From Table 3, taking the first row of the table for example, we know that if
we use the most popular $5$ keywords appearing in the IDS of each of the
communities, then there are $13,283$ papers in the network whose keywords are
predicted and confirmed. As the number of keywords used in the algorithm,
i.e., the lengths of $L$ in the algorithm, becomes larger, we can predict and
confirm keywords for more papers, that is up to $14,691$ papers. The results
show that the IDS is much smaller than the corresponding community and that
the IDS preserves much information of the corresponding community. From the
experiment, it is conceivable that in practical applications, it is sound to
recommend the IDS of a community instead of the whole community which is
usually much larger. The result above is unexpectedly good. We believe that
this property may hold for many other networks other than citation networks,
that is, the internal dominating set of a community keep essential information
of the community. More importantly, the essential information of the internal
dominating set of a community can be easily extracted.
Using 0.8-IDS to predict keywords in citation network hepth Keyword Number
Predicted Paper Number 5 13283 10 13906 15 14375 20 14592 25 14641 30 14647 35
14654 40 14691 45 14691 50 14691
## 4 Internal and External Slopes
In the last section, we show that most communities have a small IDS and a
small EDS, and that the small IDS of a community preserves much information of
the community.
In this section, we will show that the IDS and EDS of a community usually take
the central positions in the community with low degree nodes around them, so
that the community forms a core/periphery structure.
Intuitively speaking, if all nodes in a community have equal position, i.e.,
the regular graph or a random graph, then they are homogeneous; if nodes in a
community form a core/periphery structure, i.e., the star-like graphs, then
they are heterogeneous. Our main question is: How do the IDS and EDS of a
community reflect the homogeneity or the heterogeneity of the community?
Before answering this question, we look at the power law distribution. It was
shown that most networks follow a power law distribution [Barabási and Albert
(1999)], meaning that the number of nodes of degree $k$ is proportional to
$k^{-\beta}$. A power law distribution of power exponent $\beta$, which is
typically lying in the range $2<\beta<3$, measures the heterogeneity of a
network. However it is nontrivial to estimate the exponent $\beta$, especially
for small networks, and not all networks follow the power law distribution
[Clauset et al. (2009)]. Most communities are small, although they may have
heavy tail degree distributions, it is not clear whether they have power law
distributions. More seriously, even if the communities have power law
distributions, fluctuations caused by the small sizes of communities may make
the result inaccurate, and the number of communities is large, it is hard to
characterize the power law distributions of all the communities. Therefore the
power exponent $\beta$ is not suitable to measuring the heterogeneity of all
the communities of a network. Another measure is to notice the relationship
between the number of dominating set and the degree distribution. In fact, it
was shown that the more heterogeneous the degree distribution of a network is,
the smaller the number of dominating set is [Nacher and Akutsu (2012)]. This
suggests that the internal and external dominating sets are closely related to
the heterogeneity of the communities.
We now measure the heterogeneity of communities by the internal and external
dominating sets of communities. See figure 2(a) in which case the community is
homogeneous. All members of the community have equal position, and any single
node could dominate the whole community. From the dominating number, we could
not know the heterogeneity of the community. So the dominating set itself is
insufficient to measure the homogeneity and heterogeneity of a community. To
solve this problem, we use the internal dominating ratio (IDR) of the internal
dominating set (IDS), together with the expectation internal dominating ratio
(IDR) of random selection of nodes of the same size as that of the IDS.
We define the internal slope (ISlope, for short) and external slope (ESlope,
for short) of a community to measure the internal and external heterogeneity
(or the core/periphery structure) of the community. Intuitively, the ISlope of
a community is to measure the distance between the community and regular
graphs or star-like graphs, and the ESlope of a community is to illustrate
whether the community is connected with the rest of the community evenly or
through a small number of nodes like a funnel.
Let $C$ be a community, $p\in[0,1]$ be a real number. Suppose that $K$ is the
size of the $p$-IDS of $C$, that $S$ be the $p$-IDS of $C$, and that
$\mathcal{V}=\\{V_{1},V_{2},\cdots,V_{M}\\}$ is the set of all subsets of $C$
of size $K$. Then define the internal slope of $C$, written by ${\rm
ISlope}(C)$ as follows:
${\rm ISlope}(C)={\rm IDR}(S)-\frac{\sum_{X\in\mathcal{V}}{\rm
IDR}(X)}{|\mathcal{V}|}$ (3)
The ISlope of a community represents the difference between the internal
dominating ratio of the most central nodes and the expectation internal
dominating ratio of random choices of nodes of the same size. It measures the
homogeneity and heterogeneity (core/periphery structure) of the community from
the internal point of view. We extract some communities of real networks found
by our algorithm in Figure 2. From these figures we can observe that the
ISlopes and ESlopes of the communities largely reflect the homogeneity and the
heterogeneity of the corresponding communities.
(a) ISlope = 0
(b) ISlope = 0.41
(c) ISlope = 0.72
(d) ISlope very near 1
Figure 2: Real communities to illustrate ISlope. All of them except (d) are
from collaboration grqc network and (d) is from email_enron network. In each
figure, red nodes come from the same community.
(a) ESlope = 0
(b) ESlope = 0.31
(c) ESlope = 0.52
(d) ESlope = 0.97
Figure 3: Real communities to illustrate ESlope. All of them are from
collaboration grqc network. In each figure, red nodes come from the same
community.
By observing Figure 2, we know that the structures of communities are closely
related to the corresponding ISlopes of the communities. In particular, in
Figure 2(a), all nodes have equal position and a single node could dominate
the whole community; in Figure 2(b), there are some central nodes with
periphery nodes around; in Figure 2(c), the central position of one node is
more obvious, and the structure is a star-like graph; in Figure 2(d), the
community is a star graph with a hub in the center, and the ISlope of the
community is very near $1$. Notice that a star graph is the most heterogeneous
community, in which the hub in its center is the most important node. In
summary, we observe that the smaller the ISlope of a community is, the more
homogeneous a community is, and that on the contrary, the larger the ISlope of
a community is, the more heterogeneous a community is, and that the ISlope of
a community roughly reflects the pattern or structure of the community.
Similarly to ISlope, we define the external slope of a community (ESlope) to
measure the external heterogeneity of the community. By using the ESlope of a
community, we are able to examine the pattern that nodes in a community
connect nodes outside of the community. Whether or not nodes in a community
connect the rest of the community through a small number of representatives or
evenly through most members.
It has been shown that in a collaboration network, most people in the network
(theme, or topic) contact people in the network through just one or two of
their best-connected collaborators [Newman (2004a), Newman et al. (2001)].
Our results show that such a funneling pattern of connections from a community
to outside of the community is very popular in all the communities of a
network, for a wide range of real networks.
Let $C$ be a community, $p\in[0,1]$ be a real number. Suppose that $K$ is the
size of a $p$-EDS of $C$, that $\mathcal{V}=\\{V_{1},V_{2},\cdots,V_{m}\\}$ is
the set of all subsets of $C$ of size $K$. Then we define the external slope
of C (${\rm ESlope}(C)$) as follows:
${\rm ESlope}(C)={\rm EDR}({\rm EDS})-\frac{\sum_{Y\in\mathcal{V}}{\rm
EDR}(Y)}{|\mathcal{V}|}$ (4)
The ESlope of a community represents the difference between the external
dominating ratio of the most central nodes and the expectation external
dominating ratio of random selection of nodes of the same size.
Figure 3 illustrates different connecting patterns of communities with
different ESlopes. In these figures, we also keep the neighbors and the
neighbors of neighbors of the community to highlight their connecting
patterns. In figure 3(a), all members have equal position to connect with
nodes outside of the community. Some nodes only have internal links, while
others have both external and internal links in figure 3(b). Also, some nodes
play the role of bridge in linking nodes in and outside of its community in
figure 3(c). At last, figure 3(d) shows a community in which only one node is
the bridge. All other members communicate with the outside world through this
node. The ESlope indeed identifies different connecting patterns of how
communities connect with each other.
Table 2 gives the average ISlopes and ESlopes of all the communities of
various networks. Except for the football and the email_euall, all other
networks have similar ISlopes and ESlopes with ESlopes larger than ISlopes, on
the average. ISlope and ESlope of a community quantify the core/periphery
structure of the community. Our results indicate that such structures are
universal in real networks and that real networks tend to avoid communities of
either regular or star-like graphs and have structures with ISlopes and
ESlopes in some fixed interval, that is, the ISlopes are roughly in
$[0.35,0.55]$ and the ESlopes in $[0.5,0.7]$.
These results pose a question that why networks tend to have such structures.
We try to explain these as follows: For a community, it is possible that some
key nodes are essential to its formation and evolution. On one hand, it is
unusual to have a community with all members having equal position for a long
period of time. On the other hand, the key nodes of a community should not be
too strong or too weak since otherwise, the community structure may be
fragile. It is intuitive that if the central nodes of a community breakdown,
then the community structure would not exist any more. Therefore too big
ISlopes or ESlopes and too small ISlopes or ESlopes will both go ill with the
evolution of communities. The structures of typical communities of a real
network may be a compromise between the effectiveness and robustness of the
communities. We conjecture that the ESlopes may largely determine the
evolution of communities, which needs to be further investigated ( in our on
going project).
(a) ISlope_football
(b) ISlope_citation
(c) ISlope_collaboration
(d) ISlope_email
Figure 4: Distribution of communities’ ISlope
(a) ESlope_football
(b) ESlope_citation
(c) ESlope_collaboration
(d) ESlope_email
Figure 5: Distribution of communities’ ESlope
(a) ISlope_football
(b) ISlope_citation
(c) ISlope_collaboration
(d) ISlope_email
Figure 6: Cumulative distribution of communities’ ISlope
(a) ESlope_football
(b) ESlope_citation
(c) ESlope_collaboration
(d) ESlope_email
Figure 7: Cumulative distribution of communities’ ESlope
Besides the average values, we also report the distributions of the ISlopes
and ESlopes in figure 4 and figure 5 of all the communities of the real
networks. Figure 6 and figure 7 are the corresponding cumulative distribution.
By observing these figures, we know that:
* •
Most communities have a core/periphery structure, with a small core in central
positions and some low degree nodes in the periphery.
* •
The ISlopes largely determine the structure of the communities.
* •
There are indeed some typical thresholds at which the distribution curve
decreases sharply in most networks.
* •
The typical values of ESlopes are more obvious than that of the ISlopes in the
citation and collaboration networks, in which, the ESlopes of most communities
lie in a very narrow interval.
* •
Communities of the email-euall network have much larger ISlopes and ESlopes in
general.
* •
The ISlopes and ESlopes of all the communities of the citation and
collaboration networks approximately follow a normal distribution.
## 5 More General Properties
Statistics of communities. APL represents average path length, D represents
diameter, CCC represents community clustering coefficient and NCC represents
the network clustering coefficient. All the results except NCC are calculated
by averaging the corresponding property of all communities Network APL D CCC
NCC football 1.8 3.2 0.6 0.41 cit_hepth 2.9 7 0.36 0.12 cit_hepph 2.7 6.7 0.29
0.15 col_astroph 2.2 4.5 0.71 0.32 col_condmat 2.7 5.4 0.53 0.26 col_grqc 2.4
4.7 0.51 0.63 col_hepth 3.3 7.2 0.39 0.28 col_hepph 2.8 5.9 0.65 0.66
email_enron 2.2 4.1 0.39 0.085 email_euall 2.3 3.5 0.0019 0.0042
In the last section, we show that the internal slope (ISlope) of a community
basically determines the structure of the community. In this section, we study
more general properties of the communities. In particular, we consider the
average distances, average diameters and average clustering coefficients of
all the communities in each of the real networks, for which the results are
given in Table 5.
The distance between two nodes is defined as the number of “hops” in the
network one needs to move from one given node to another [Newman (2004a)].
Usually people are interested in the average distances of the whole network
[Milgram (1967), Newman (2001b), Newman et al. (2001), Travers and Milgram
(1969)], showing that most real networks have very short average distances. In
this section, we consider the average distance between two nodes within a
community, which represents the number of “hops” one needs to move from one
node to another only through members of the same community.
From Table 5, we have that, the communities of each network have a small
average distance. In particular, the average distance of all the communities
of the collaboration network hepth reaches $3.3$, which is the largest value
of the average distances of all the communities for all the networks studied
in this paper. Besides, we also give the average diameter of communities. The
average diameter of all the communities for each of the networks is between
$3.2$ and $7.2$. This experiment suggests a conjecture that: there is a three
degree separation property of (true) communities for many real networks. The
conjecture calls for further investigation, which may provide useful
information for understanding both true communities and communities found by
various algorithms.
Clustering coefficient (or transitivity) has been a well studied property for
networks [Newman (2001a), Newman (2001b), Watts and Strogatz (1998)]. It
refers to the phenomenon that the existence of ties between nodes $A$ and $B$
and between nodes $B$ and $C$ implies a tie between $A$ and $C$. Given a graph
$G$, the clustering coefficient of $G$ is defined by:
$C=\frac{3\times\textrm{number of triangles on the graph}}{\textrm{number of
connected triples of vertices}}$ (5)
From table 5, we observe that most communities of the networks have very large
clustering coefficients except for that of the email_euall network, and that
most small communities found by our algorithm have larger clustering
coefficients than that of the corresponding original graphs.
However, in the collaboration network grqc, the clustering coefficient of the
original graph is $0.63$, but many small communities we found have smaller
clustering coefficients. In fact, communities with clustering coefficients
less than $0.6$ take up more than $74\%$ of the communities in this network.
To explain this phenomenon, we count the triangles in the original graph and
its communities respectively. In the original graph, there are $1,350,014$
triangles in all. If we divide the communities into two groups, so that the
first group consists of the ones having clustering coefficients larger than
$0.6$, and the second group consists of the rest of the communities, then we
discover that communities in the first group have $3,306$ triangles on the
average, while communities in the second group contain $60$ triangles on the
average. If we divide communities by clustering coefficient $0.8$ as above,
then the average numbers of triangles appear in the communities in the first
and the second classes are $5,027$ and $147$ respectively. Therefore the
triangles are unevenly distributed in communities with a small number of
communities containing most of triangles of the network. The high clustering
coefficients are mainly caused by the small group of communities which contain
much larger number of triangles.
From table 5, we observe that clustering coefficients of communities vary
among different types of networks. Communities in collaboration networks have
higher clustering coefficients than that of citation and email networks. In
the collaboration networks, two authors having common collaborators are more
likely to collaborate with each other in the future. In the citation networks,
an author citing a paper, tends to cite the references of the paper,
especially when the references are from the same topic. This explains the
reason why collaboration networks and citation networks have higher clustering
coefficients.
Email networks have different patterns. Communities in email_enron network
have average clustering coefficient $0.39$, at the same time, the origin graph
has clustering coefficient only $0.085$. In this case, the communities found
by our algorithm largely amplify the clustering coefficients of the network.
This means that although the network has a small clustering coefficient, there
are also significantly many local structures of the network showing strong
cohesion among themselves. However communities in email_euall network has the
lowest clustering coefficient (only $0.0019$). Both its origin and communities
have very small clustering coefficients. In this case, most communities in
this network are very similar to star-like graphs which have clustering
coefficients near $0$. This local structure of the network is very much
different from other networks.
## 6 Conclusions
In this paper, we propose a methodology to characterize and analyze the local
structures and information of real networks, which includes new notions of
internal dominating set, external dominating set, internal slope and external
slope of a community, and analysis of the distributions of internal and
external slopes, average distances, diameters, and clustering coefficients of
all communities for each of the real networks.
We implement experiments of our method on five collaboration networks, two
citation networks, two email networks and one benchmark network.
The experiments show that: 1) The notions of internal dominating ratio,
external dominating ratio, internal slope and external slope and clustering
coefficients are essential characteristics to understand the patterns and
information of the communities of a real network. 2) Different networks have
different local structures (or patterns). 3) Most communities of a real
network have a small internal dominating set and a small external dominating
set, although the communities may still very large. 4) The small dominating
set of a community keeps much of the information of the community and more
importantly the information of a community can be extracted from the internal
dominating set of the community. 5) Both internal and external slopes of all
the communities of a network approximately follow a normal distribution for
most real networks. This means that typical communities of the networks have
both ISlopes and ESlopes in some small intervals, so that the communities have
similar patterns. 6) The internal slope (ISlope) of a community basically
determines the structure of the community. 7) The result that communities have
average distances less than or equal to $3.3$, implies a general conjecture
that there is a $3$ degree separation phenomenon of true communities of most
real networks. 8) Normally, communities amplify the clustering coefficients of
the corresponding network. 9) If a reasonably good algorithm fails to find
communities that amplify clustering coefficients of the network, then the
communities explore special structures of the network.
The discoveries above are significant in both understanding the structures of
networks, and in practical applications. Most communities in real networks are
not regular or star-like graphs, but they usually appear with some central
nodes with periphery around forming a core/periphery structure. Such structure
favors the evolution of communities. A small set of nodes lead to the
formation and evolution of the communities. Our results also indicate that in
real communities, a single node could rarely take absolute central position as
in star-like graphs, due to the reason that such structures are highly
unstable. Our analysis provides some intuitive pictures of the rich
communities of a network.
In best of our knowledge, this is the first time we can rigorously analyze the
characteristics and patterns, and extract information of the communities of a
real network, although there are already a huge number of community detection
algorithms in the literature. The significance of the research are three
folds: 1) To understand the local structures and connecting patterns of a
network. 2) To extract useful information from the communities of a network.
3) To help to judge the community finding algorithms.
Our future project (in progress) is to understand the roles of the small
internal and external dominating sets in the formation and evolution of
communities, and to understand the mechanisms of the patterns of the
communities.
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|
arxiv-papers
| 2013-01-14T12:50:22 |
2024-09-04T02:49:40.254902
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jiankou Li, Angsheng Li",
"submitter": "Jiankou Li",
"url": "https://arxiv.org/abs/1301.2957"
}
|
1301.3082
|
# $CP$ violation in charm and beauty decays at LHCb
M. Pepe Altarelli111On behalf of the LHCb collaboration
[email protected] CERN, Geneva, Switzerland
###### Abstract
LHCb is a dedicated heavy flavour physics precision experiment at the LHC
searching for New Physics (NP) beyond the Standard Model (SM) through the
study of very rare decays of beauty and charm-flavoured hadrons and precision
measurements of $CP$-violating observables. In this review I will present a
selection of recent precision measurements of $CP$-violating observables in
the decays of beauty and charm-flavoured hadrons. These measurements are based
on an integrated luminosity of up to $1.0~{}\rm{fb}^{-1}$ collected by LHCb in
2011.
###### keywords:
LHCb , Flavour physics, CP violation
LHCb-PROC-2013-002 ; CERN-LHCb-PROC-2013-002xxxxxxxxx
Fourth Workshop on Theory, Phenomenology and Experiments in Flavour
Physicsxxxxxxxxx
11 - 13 June 2012, Anacapri, Italyxxxxxxxxxxxxxxxxxx
††footnotetext: © CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
## 1 Introduction
The LHCb detector has been taking data with high efficiency during the last
three years of operation at the LHC, producing a wealth of exciting physics
results, which have made an impact on the flavour physics landscape and proved
the concept of a dedicated heavy flavour spectrometer in the forward region at
a hadron collider.
The LHC is the world’s most intense source of $b$ hadrons. The $b\overline{b}$
cross-section in proton-proton collisions at $\sqrt{s}=7$ TeV is measured to
be $\sim$$300~{}\mu$b, implying that more than $10^{11}$ $b\overline{b}$ pairs
were produced in LHCb in 2011, when an integrated luminosity of
$1.0\,\rm{fb}^{-1}$ was collected. The $c\overline{c}$ cross-section is about
20 times larger than the $b\overline{b}$ cross-section, giving LHCb great
potential in charm physics studies. As in the case of the Tevatron, a complete
spectrum of $b$ hadrons is available, including $B_{s}$, $B_{c}$ mesons and
$b$ baryons, such as $\Lambda_{\rm b}$.
At the nominal LHC design luminosity of $10^{34}$ cm-2 s-1, multiple $\it pp$
collisions within the same bunch crossing (so-called pile-up) would
significantly complicate the $b$ decay-vertex reconstruction and flavour
tagging, and increase the combinatorial background. For this reason the
detector was designed to operate at a reduced luminosity. The luminosity at
LHCb is locally controlled by transverse displacement of the beams (so-called
“levelling”) to yield $\mathcal{L}=4\times 10^{32}\mbox{cm${}^{-2}$
s${}^{-1}$}$ (approximately a factor of two above the original LHCb design
value) at which the average event pile-up per visible crossing is $\sim$2\.
Running at relatively low luminosity has the additional advantage of reducing
detector occupancy in the tracking systems and limiting radiation damage
effects.
The dominant $b\overline{b}$-production mechanism at the LHC is through gluon-
gluon fusion in which the momenta of the incoming partons are strongly
asymmetric in the $\it pp$ centre-of-mass frame. As a consequence, the
$b\overline{b}$ pair is boosted along the direction of the higher momentum
gluon, and both $b$ hadrons are produced in the same forward (or backward)
direction in the $\it pp$ centre-of-mass frame. The detector is therefore
designed as a single-arm forward spectrometer covering the pseudorapidity
range $2<\eta<5$, which ensures a high geometric efficiency for detecting all
the decay particles from one $b$ hadron together with decay products from the
accompanying $\overline{b}$ hadron to be used as a flavour tag. The key
detector features are a versatile trigger scheme efficient for both leptonic
and hadronic final states, which is able to cope with a variety of modes with
small branching fractions; excellent vertex and proper time resolution;
precise particle identification, specifically for hadron ($\pi/K$) separation;
precise invariant mass reconstruction to reject background efficiently. A full
description of the detector characteristics can be found in Ref.[1].
In this review, I report on a few recent precision measurements of
$CP$-violating observables in decays of beauty and charm-flavoured hadrons,
while the study of rare decays at LHCb is covered by a separate contribution
[2].
## 2 Measurement of the $B_{s}$ mixing phase
The $CP$-violating phase $\phi_{s}$ between $B^{0}_{s}$-${\kern
1.79993pt\overline{\kern-1.79993ptB^{0}_{s}}}$ mixing and the $b\rightarrow
c\overline{c}s$ decay amplitude of the $B_{s}$ meson is determined with a
flavour tagged, angular analysis of the decay $B^{0}_{s}\rightarrow
J\mskip-2.5mu/\mskip-1.0mu\psi\mskip 1.0mu\phi$, with
$J\mskip-2.5mu/\mskip-1.0mu\psi\mskip 1.0mu\to\mu^{+}\mu^{-}$ and $\phi\to
K^{+}K^{-}$. This phase originates from the interference between the
amplitudes for a $B_{s}$ ($\overline{B}_{s}$) to decay directly into the final
state $J\mskip-2.5mu/\mskip-1.0mu\psi\mskip 1.0mu\phi$ or to first mix into
$\overline{B}_{s}$ ($B_{s}$) and then decay. In the SM, $\phi_{s}$ is
predicted to be $\simeq-2\beta_{s}$, where $2\beta_{\rm
s}=2\rm{arg}(-V_{ts}{V^{*}}_{tb}/V_{cs}{V^{*}}_{cb})=(3.6\pm 0.2)\,10^{-2}$
rad [3]. However, NP could significantly modify this prediction if new
particles contribute to the $B^{0}_{s}$-${\kern
1.79993pt\overline{\kern-1.79993ptB^{0}_{s}}}$ box diagram.
The CDF and D0 collaborations [4, 5] have reported measurements of the $B_{s}$
mixing phase based on approximately 11,000 $B^{0}_{s}\rightarrow
J\mskip-2.5mu/\mskip-1.0mu\psi\mskip 1.0mu\phi$ candidates from an integrated
luminosity of $9.6~{}\rm{fb}^{-1}$ (i.e. the full CDF Run II dataset) and
6,500 $B^{0}_{s}\rightarrow J\mskip-2.5mu/\mskip-1.0mu\psi\mskip 1.0mu\phi$
candidates from $8~{}\rm{fb}^{-1}$, respectively. Both results are compatible
with the SM expectation at approximately one standard deviation in the
($\phi_{s},\Delta\Gamma_{s}$) plane.
LHCb has presented results based on a sample that contains approximately
21,200 $B^{0}_{s}\rightarrow J\mskip-2.5mu/\mskip-1.0mu\psi\mskip 1.0mu\phi$
candidates from an integrated luminosity of $1.0\rm\,{fb}^{-1}$ [6]. Besides
having a very large and clean signal yield, LHCb also benefits from an
excellent proper time resolution to resolve fast $B_{s}$ oscillations, which
is measured to be $\sim$45 fs, compared to a $B_{s}$ oscillation period of
$\sim$350 fs. The other key experimental ingredient is flavour tagging, which
is performed by reconstructing the charge of the $b$-hadron accompanying the
$B$ meson under study. The analysis uses an opposite-side flavour tagger based
on four different signatures, namely high $p_{\mathrm{T}}$ muons, electrons
and kaons, and the net charge of an inclusively reconstructed secondary
vertex, with a combined effective tagging power of $\sim$2.4%. The decay
$B^{0}_{s}\rightarrow J\mskip-2.5mu/\mskip-1.0mu\psi\mskip 1.0mu\phi$ is a
pseudoscalar to vector-vector transition. Total angular momentum conservation
implies $l=0,1,2$ and therefore the $J\mskip-2.5mu/\mskip-1.0mu\psi\mskip
1.0mu\phi$ final state is a mixture of $CP$-even ($l=0,2$) and $CP$-odd
($l=1$) eigenstates, which can be disentangled on a statistical basis. This is
accomplished by performing an unbinned maximum likelihood fit to the candidate
invariant mass, decay time, initial $B_{s}$ flavour and the decay angles in
the so-called transversity frame [7]. The fit yields the following result for
the three main observables, namely $\phi_{s}$, the decay width, $\Gamma_{s}$,
and the decay width difference between the light and heavy $B_{s}$ mass
eigenstates, $\Delta\Gamma_{s}$
$\displaystyle\phi_{s}$ $\displaystyle=-0.001$ $\displaystyle\pm\,$
$\displaystyle 0.101(\text{stat})$ $\displaystyle\pm\,$ $\displaystyle
0.027(\text{syst})$ $\displaystyle\text{rad},$ $\displaystyle\Gamma_{s}$
$\displaystyle=0.6580$ $\displaystyle\pm\,$ $\displaystyle
0.0054(\text{stat})$ $\displaystyle\pm\,$ $\displaystyle 0.0066(\text{syst})$
$\displaystyle\text{ps}^{-1},$ $\displaystyle\Delta\Gamma_{s}$
$\displaystyle=0.116$ $\displaystyle\pm\,$ $\displaystyle 0.018(\text{stat})$
$\displaystyle\pm\,$ $\displaystyle 0.006(\text{syst})$
$\displaystyle\text{ps}^{-1}.$
This is the world’s most precise measurement of $\phi_{s}$ and the first
direct observation for a non-zero value for $\Delta\Gamma_{s}$. These results
are fully consistent with the SM, indicating that $CP$ violation in the
$B_{s}$ system is small. Actually, there exists a second mirror solution in
the plane $\Delta\Gamma_{s}$ vs $\phi_{s}$, which arises from the fact that
the time-dependent differential decay rates are invariant under the
transformation
$(\phi_{s},\Delta\Gamma_{s})\rightarrow(\pi-\phi_{s},-\Delta\Gamma_{s})$ (plus
an appropriate transformation of the strong phases). LHCb has recently
resolved this ambiguity [8] by studying the dependence of the strong phase
difference between the $S$-wave and $P$-wave amplitudes on the $K^{+}K^{-}$
mass from $B^{0}_{s}\rightarrow J\mskip-2.5mu/\mskip-1.0mu\psi\mskip
1.0muK^{+}K^{-}$ decays in the region around the $\phi(1020)$ resonance. The
solution with positive $\Delta\Gamma_{s}$ is favoured with a significance of
4.7 standard deviations, indicating that in the $B_{s}$ system the lighter
$CP$ mass eigenstate that is almost $CP$ even decays faster than the state
that is almost $CP$ odd.
The mixing-induced phase $\phi_{s}$ is also measured in the decay $B_{s}\to
J\mskip-2.5mu/\mskip-1.0mu\psi\mskip 1.0mu\pi^{+}\pi^{-}$ [9]. The branching
fraction for this decay is $\sim 25\%$ of $B_{s}\to
J\mskip-2.5mu/\mskip-1.0mu\psi\mskip 1.0mu\phi$, with $\phi\to K^{+}K^{-}$.
However, this final state has been shown to be almost $CP$ pure with a
$CP$-odd fraction larger than 0.977 at 95% CL, and there is no need for an
angular analysis. About 7,400 signal events are selected in
$1.0\,\rm{fb}^{-1}$ of data, yielding the result
$\phi_{s}=-0.019^{+0.173+0.004}_{-0.174-0.003}$ rad. The two datasets are
combined in a simultaneous fit, leading to the preliminary result
$\phi_{s}=-0.002\pm 0.083\pm 0.027$ rad [6], in excellent agreement with the
SM. The precision for this result is completely dominated by the statistical
uncertainty and therefore significant improvements are expected with more
data.
Figure 1 shows 68% confidence-level contours in the
$(\phi_{s},\Delta\Gamma_{s})$ plane for the individual experimental
measurements, their combined contour, as well as the SM predictions [10]. The
combined result is consistent with these predictions at the 0.14 $\sigma$
level.
Figure 1: 68% confidence-level contours in the $(\phi_{s},\Delta\Gamma_{s})$
plane for the individual experimental measurements, their combined contour
(solid line and shaded area), as well as the SM predictions (black rectangle).
## 3 Measurement of the weak phase $\gamma$ from tree-level decays
The angle $\gamma$ is currently the least precisely known parameter of the CKM
unitarity triangle. The direct determination of $\gamma$ via fits to the
experimental data gives $(66\pm 12)^{\circ}$ [11] or $(76\pm 10)^{\circ}$
[12], depending on whether a frequentist or Bayesian treatment is used. In
terms of the elements of the CKM matrix, this weak phase is defined as
$\gamma={\rm arg}(-V_{\rm ud}{V^{*}}_{\rm ub}/V_{\rm cd}{V^{*}}_{\rm cb})$. It
is of particular interest as it can produce direct $CP$ violation in tree-
level decays involving the interference between $b\rightarrow c\overline{u}s$
and $b\rightarrow u\overline{c}s$ transitions that are expected to be
insensitive to NP contributions, thus providing a benchmark against which
measurements sensitive to NP through loop processes can be compared.
One of the most powerful ways to measure the angle $\gamma$ is through charged
$B$ decays to open charm, $B^{\pm}\rightarrow Dh^{\pm}$, where $D$ stands for
a $D^{0}$ or a $\overline{D^{0}}$ and $h$ indicates either a pion or kaon.
Decays in which the hadron $h$ is a kaon carry greater sensitivity to
$\gamma$. The method is based on two key observations: 1. These decays can
produce neutral $D$ mesons of both flavours via colour-favoured or colour-
suppressed decays 2. Neutral $D$ and $\overline{D}$ mesons can decay to a
common final state, for example through Cabibbo-favoured or doubly Cabibbo-
suppressed Feynman diagrams (ADS method [13]) or through decays to $CP$
eigenstates such as $K^{+}K^{-}$ or $\pi^{+}\pi^{-}$ (GLW method [14]). In the
ADS case, the reversed suppression between $B$ and $D$ decays results in very
similar amplitudes leading to a high sensitivity to $\gamma$. The relative
phase between the two interfering amplitudes for $B^{+}\rightarrow DK^{+}$ and
$B^{+}\rightarrow\overline{D}K^{+}$ is the sum of the strong and weak
interaction phases, while in the case of $B^{-}\rightarrow DK^{-}$ and
$B^{-}\rightarrow\overline{D}K^{-}$ it is the difference between the strong
phase and $\gamma$. Therefore both phases can be extracted by measuring the
two charge conjugate modes. In addition, there is a dependence on the ratio
between the magnitude of the suppressed amplitude and the favoured amplitude
$r_{B}$.
LHCb has performed an analysis of the two-body $B^{\pm}\rightarrow Dh^{\pm}$
modes in which the considered $D$ meson final states are $\pi^{+}\pi^{-}$,
$K^{+}K^{-}$, the Cabibbo-favoured $K^{\pm}\pi^{\mp}$ and the Cabibbo-
suppressed $\pi^{\pm}K^{\mp}$ [15]. For both GLW and ADS methods the
observables of interest are $CP$ asymmetries and partial widths. Exploiting
$1.0\,\rm{fb}^{-1}$ of data, the first observation of the suppressed ADS mode
$B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}]_{D}K^{\pm}$ was performed. This is
illustrated in the top plots of Fig 2. A large asymmetry is visible in the
$B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}]_{D}K^{\pm}$ mode (by comparing the
signal yield for $B^{+}$ and $B^{-}$) while for the
$B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}]_{D}\pi^{\pm}$ ADS mode there is the hint
of an (opposite) asymmetry. By combining all the various modes $CP$ violation
in $B^{\pm}\rightarrow DK^{\pm}$ was observed with a $5.8\,\sigma$
significance.
Figure 2: Invariant mass spectra for
$B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}]_{D}h^{\pm}$ events; the left plots are
$B^{-}$ candidates, $B^{+}$ are on the right. The dark (red) curve in the top
plots represents the $B\rightarrow DK^{\pm}$ events, the light (green) curve
in the bottom plots is $B\rightarrow D\pi^{\pm}$.
The three-body final state $D\rightarrow K^{0}_{\rm S}h^{+}h^{-}$, with
$h=(\pi,K$), was also studied [16] through a Dalitz plot analysis. The
strategy relies on comparing the distributions of events in the $D\rightarrow
K^{0}_{s}h^{+}h^{-}$ Dalitz plot for $B^{+}\rightarrow DK^{+}$ and
$B^{-}\rightarrow DK^{-}$ decays. Existing measurements of the CLEO-c
experiment [17] were used to provide input on the $D$ decay strong-phase
parameters. Based on approximately 800 $B^{\pm}\rightarrow DK^{\pm}$ decays
with $D\rightarrow K^{0}_{\rm S}h^{+}h^{-}$ selected from $1.0\,\rm{fb}^{-1}$
of data, the following results were obtained for the weak phase $\gamma$ and
the ratio $r_{B}$ between the suppressed and favoured $B$ decay amplitudes,
$\gamma=(44^{+43}_{-38})^{\circ}$, with a second solution at
$\gamma\rightarrow\gamma+180^{\circ}$, and $r_{B}=0.07\pm 0.04$.
Other flavour specific final states, such as $D\rightarrow K\pi\pi\pi$ are
exploited in a similar manner to the two-body case [18]. However, for multi-
body final states different intermediate resonances can contribute, diluting
$CP$ violation effects. Based on $1.0\,\rm{fb}^{-1}$ of data, the suppressed
ADS modes $B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}\pi^{+}\pi^{-}]_{D}K^{\pm}$ and
$B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}\pi^{+}\pi^{-}]_{D}\pi^{\pm}$ were
observed for the first time with significances of $5.1\,\sigma$ and
$>10\,\sigma$, respectively.
A combination of the $B^{\pm}\rightarrow DK^{\pm}$ results from Refs. [15, 16,
18] was performed to derive an unambiguous best-fit value in $[0,180]^{\circ}$
of $\gamma=(71.1^{+16.6}_{-15.7})^{\circ}$ [19], suggesting very good
prospects for the result based on the full data-set. Additional
$\gamma$-sensitive measurements will also be included in the future. With the
data currently available on tape, LHCb should be able to reduce the error
quoted above by at least a factor of two.
## 4 $CP$ violation in charmless $B$ decays
Charmless $B$ decays represent an interesting family of channels for which a
precise measurement of the charge or time-dependent $CP$ asymmetries can play
an important role in the search for NP. In particular, NP may show up as
virtual contributions of new particles in loop diagrams. A comparison of
results from decays dominated by tree-level diagrams with those that start at
loop level can probe the validity of the SM. In particular, it was pointed out
by Fleischer [20] that $U$-spin related decays (obtained by interchanging $d$
and $s$ quarks) of the type $B_{s,d}\rightarrow h^{+}h^{\prime-}$, with
$h,h^{\prime}=\pi,K$, offer interesting strategies for the measurement of the
angle $\gamma$: in presence of NP in the penguin loops, such a determination
could differ appreciably from that derived from $B$ decays governed by pure
tree amplitudes discussed in Sect. 3.
LHCb has analysed very large samples of $B_{s,d}\rightarrow h^{+}h^{\prime-}$
decays, separating the final state pions and kaons by using the particle
identification provided by the RICH detectors. Figure 3 shows the invariant
$K\pi$ mass spectra for $B_{s,d}\rightarrow K\pi$ events based on an
integrated luminosity of $0.35\,\rm{fb}^{-1}$ [21]. The selection cuts are
optimised for the best sensitivity to $A_{CP}(B_{d}\rightarrow K\pi$) (plots a
and b) and $A_{CP}(B_{s}\rightarrow K\pi$) (plots c and d), where the $CP$
asymmetry in the $B$ decay rate to the final state $f=K\pi$ is defined as
$A_{CP}=\frac{(\Gamma(B\rightarrow
f)-\Gamma(\overline{B}\rightarrow\overline{f})}{(\Gamma(B\rightarrow
f)+\Gamma(\overline{B}\rightarrow\overline{f})}$. LHCb reported [21] the most
precise measurement for $A_{CP}(B_{d}\rightarrow K\pi)$ available to date,
with a significance exceeding 6 $\sigma$, as well as the first evidence, at
the 3.3 $\sigma$ level, for $CP$ violation in the decay of the $B_{s}$ mesons.
The effect of the $CP$ asymmetry is visible in Fig. 3 by comparing the yields
for the $B_{d}\rightarrow K\pi$ in (a) and (b) and those for the
$B_{s}\rightarrow K\pi$ in (c) and (d).
Figure 3: Invariant $K\pi$ mass spectra for $B_{s,d}\rightarrow K\pi$ events;
the selection cuts are optimised for the best sensitivity to
$A_{CP}(B_{d}\rightarrow K\pi$) (plots a and b) and $A_{CP}(B_{s}\rightarrow
K\pi$) (c and d). Plots a and c (b and d) show the $K^{+}\pi^{-}$
($K^{-}\pi^{+}$) invariant mass distribution. The main components of the fit
model are also shown.
The $CP$ asymmetry results need to be corrected for the effect of a possible
$B$ production asymmetry, which is studied by reconstructing a sample of
$B_{d}\rightarrow J/\psi K^{*}$ decays, given that $CP$ violation in
$b\rightarrow c\overline{c}s$ transitions is expected to be small. Effects
related to the instrumental detection efficiencies are evaluated by using data
sets with opposite magnet polarities and reconstructing large samples of
tagged $D^{*\pm}\rightarrow D^{0}(K^{-}\pi^{+})\pi^{\pm}$ and
$D^{*\pm}\rightarrow D^{0}(K^{-}K^{+})\pi^{\pm}$ decays, as well as untagged
$D^{0}\rightarrow K^{-}\pi^{+}$ decays. These corrections are found to be
small.
LHCb has also performed measurements of time-dependent $CP$ violation in
charmless two-body $B$ decays by studying the processes
$B_{d}\rightarrow\pi^{+}\pi^{-}$ and $B_{s}\rightarrow K^{+}K^{-}$ [22]. The
analysis is based on a luminosity of $0.69\,\rm{fb}^{-1}$. Direct and mixing-
induced $CP$ asymmetries are measured in each channel using a tagged, time-
dependent analysis. This analysis is performed for the first time at a hadron
collider, and the $B_{s}\rightarrow K^{+}K^{-}$ decay is studied for the first
time ever. The preliminary results for the $B_{d}\rightarrow\pi^{+}\pi^{-}$
channel are in agreement with the world average from the $B$ factories.
However, more data need to be analysed to be able to extract a measurement of
$\gamma$ from these decays.
## 5 $CP$ violation in charm
The charm sector is an interesting place to probe for the presence of NP
because $CP$ violation is expected to be small in the SM. In particular, in
singly Cabibbo suppressed decays, such as $D\rightarrow\pi^{+}\pi^{-}$ or
$D\rightarrow K^{+}K^{-}$, NP could manifest itself through the interference
between tree-level and penguin diagrams. LHCb has collected very large samples
of charm: one in every ten LHC interaction results in the production of a
charm hadron, of which 1-2 kHz are written to storage and are available for
offline analysis. In 2012, LHCb has collected approximately $5\times 10^{3}$
tagged $D^{*\pm}\rightarrow(D^{0}\rightarrow K^{+}K^{-})\pi^{\pm}$ and
$3\times 10^{5}$ untagged $D\rightarrow K^{-}\pi^{+}$ decays per pb-1 of
integrated luminosity and it has the world’s largest sample of two and three-
body $D_{(s)}$ decays on tape.
The difference in time-integrated asymmetries between
$D\rightarrow\pi^{+}\pi^{-}$ and $D\rightarrow K^{+}K^{-}$ was measured using
$0.6\,\rm{fb}^{-1}$ of data collected in 2011 [23]. The flavour of the charm
meson is determined by requiring a $D^{*\pm}\rightarrow(D^{0}\rightarrow
h^{+}h^{-})\,\pi_{s}^{\pm}$ decay, with $h=\pi$ or $K$, in which the sign of
the slow pion $\pi_{s}$ tags the initial $D^{0}$ or $\overline{D^{0}}$. By
taking the difference of the measured time-integrated asymmetries for
$\pi^{+}\pi^{-}$ and $K^{+}K^{-}$, effects related to the $D^{*}$ production
asymmetry and to the detection asymmetry of the slow pion and $D$ meson in the
final state cancel to first order, so that one can derive the difference of
the $CP$ asymmetries, $\Delta A_{CP}$. Second-order effects are minimised by
performing the analysis in bins of the relevant kinematic variables. A nice
additional advantage of taking this difference is that in the U-spin limit
$A_{CP}(KK)=-A_{CP}(\pi\pi)$ for any direct $CP$ violation [25], so that the
effect is amplified. The final result is
$\Delta A_{CP}=[-0.82\pm 0.21\,(\rm stat)\pm 0.11\,(\rm syst)]\%.$ (1)
This result (subsequently confirmed by CDF [24]) has generated a great deal of
theoretical interest, as it is the first evidence for $CP$ violation in the
charm sector, with a significance of $3.5~{}\sigma$.
The difference in time-integrated asymmetries can be written to first order as
$\Delta
A_{CP}=[a^{dir}_{CP}(K^{+}K^{-})-a^{dir}_{CP}(\pi^{+}\pi^{-})]+\frac{\Delta\langle
t\rangle}{\tau}a^{ind}_{CP}\,,$ (2)
where $a^{dir}_{CP}$ is the asymmetry arising from direct $CP$ violation in
the decay, $\langle t\rangle$ the average decay time of the $D^{0}$ in the
reconstructed sample, and $a^{ind}_{CP}$ the asymmetry from $CP$ violation in
the mixing. In presence of a different time acceptance for the
$\pi^{+}\pi^{-}$ and $K^{+}K^{-}$ final states, a contribution from indirect
$CP$ violation remains.
Figure 4 shows the HFAG world-average combination [26] in the plane
($a^{ind}_{CP},\,a^{dir}_{CP}$). The combined data is consistent with no $CP$
violation at 0.002% CL.
Figure 4: HFAG combination of $\Delta A_{CP}$ and $A_{\Gamma}$ measurements
[26], where the bands represent $\pm 1\sigma$ intervals. The point of no $CP$
violation (0,0) is displayed as a black dot; the ellipses show the two-
dimensional 68% CL, 95% CL and 99.7% CL with the best fit value as a cross.
LHCb is currently completing the analysis of the full 2011 data sample and
pursuing alternative strategies to verify the effect. In particular, an
analysis is being finalised in which the flavour of the $D^{0}$ meson is
tagged using the charge of the muon in semileptonic $B$ decays; furthermore
$CP$ violation is searched for in charged $D$ decays where there is no
possibility of indirect $CP$ violation and a positive signal would indicate
unambiguously the presence of direct $CP$ violation.
## 6 Conclusion
This contribution has reviewed measurements of $CP$ violation in charm and
beauty decays performed by LHCb using up to $1.0\,\rm{fb}^{-1}$ of data
collected in 2011. In the $B^{0}_{s}$ system LHCb has obtained the world’s
most precise measurement of the mixing phase $\phi_{s}$ and the first direct
observation for a non-zero value for $\Delta\Gamma_{s}$. Important milestones
have been reached in the measurement of the weak phase $\gamma$ both from
decays at tree level and from those where new physics could contribute through
loops. The large charm production cross-section at the LHC has allowed for a
dramatic improvement in sensitivity to $CP$ violating effects. During the 2012
run at $\sqrt{s}=8$ TeV, another $2.1\rm{fb}^{-1}$ of data were collected,
which will allow LHCb to increase the precision on the results presented here
and pursue new analyses to understand precisely the nature of $CP$ violation
in charm and beauty and hopefully find signs of new physics.
## Acknowledgement
I would like to thank the organisers of Capri 2012 for their kind invitation
to such a nice and fruitful meeting on a truly splendid island, my LHCb
colleagues for providing the material discussed here and, in particular, Tim
Gershon and Vava Gligorov for their careful reading of this article.
## References
* [1] LHCb collaboration, “The LHCb detector at the LHC”, JINST 3 S08005 (2008).
* [2] J.Albrecht, “Rare decays at LHCb”, these proceedings, arxiv:hep-ex/1209.1208v2.
* [3] A.Lenz, U.Nierste “Numerical updates of lifetimes and mixing parameters of B mesons”, arxiv:hep-ex/1102.4274.
* [4] CDF collaboration, “Measurement of the bottom-strange meson mixing phase with the full CDF dataset”, Phys. Rev. Lett 109, (2012) 171802 ; arxiv:hep-ex/1208.2967.
* [5] D0 collaboration, “Measurement of the $CP$-violating phase $\phi_{s}^{J/\psi\phi}$ using the flavor-tagged decay $B^{0}_{s}\rightarrow J/\psi\phi$ in $8\rm{fb}^{-1}$ of $p\overline{p}$ collisions”, Phys. Rev. D 85, (2012) 032006 ; arxiv:hep-ex/1109.3166v2.
* [6] LHCb collaboration, “Tagged time-dependent angular analysis of $B^{0}_{s}\rightarrow J/\psi\phi$ decays at LHCb”, LHCb-CONF-2012-002.
* [7] A.S.Dighe, I.Dunietz, H.J.Lipkin, and J.L.Rosner, “Angular distributions and lifetime differences in $B^{0}_{s}\rightarrow J/\psi\phi$ decays”, Phys. Lett. B369 (1996) 144, arXiv:hep-ph/9511363.
* [8] LHCb collaboration, “Determination of the sign of the decay width difference in the $B^{0}_{s}$ system”, Phys. Rev. Lett. 108, 241801 (2012).
* [9] LHCb collaboration, “Measurement of the $CP$-violating phase $\phi_{s}$ in $B_{s}$$\to J\mskip-2.5mu/\mskip-1.0mu\psi\mskip 1.0mu\pi^{+}\pi^{-}$ decays”,Phys. Lett. B 713 (2012) 378, arxiv:hep-ph/1204.5675.
* [10] Heavy Flavour Averaging Group, arxiv:hep-ex/1207.1158v1 and www.slac.stanford.edu/xorg/hfag/osc/fall2012.
* [11] CKMfitter group, http://ckmfitter.in2p3.fr/www/results/plots_moriond12.
* [12] UTfit collaboration, http://utfit.org/UTfit/ResultsSummer2012PreICHEP.
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* [14] M. Gronau and D. London, “How to determine all the angles of the unitarity triangle from $B^{0}_{d}\rightarrow DK^{0}_{s}$ and $B^{0}_{s}\rightarrow D\phi$”, Phys. Lett. B253 (1991) 483; M. Gronau and D. Wyler, “On determining a weak phase from $CP$ asymmetries in charged B decay asymmetries”, Phys. Lett. B265 (1991) 172.
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* [17] CLEO collaboration, “Model-independent determination of the strong-phase difference between $D^{0}$ and $\overline{D^{0}}\rightarrow K^{0}_{\rm S,L}h^{+}h^{-}$ ($h=\pi,K$) and its impact on the measurement of the CKM angle $\gamma/\Phi_{3}$”, http://prd.aps.org/abstract/PRD/v82/i11/e112006; arXiv:hep-ex/1010.2817.
* [18] LHCb collaboration, “First observation of the suppressed ADS modes $B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}\pi^{+}\pi^{-}]_{D}K^{\pm}$ and $B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}\pi^{+}\pi^{-}]_{D}\pi^{\pm}$” LHCb-CONF-2012-030.
* [19] LHCb collaboration, “A measurement of $\gamma$ from a combination of $B^{\pm}\rightarrow Dh^{\pm}$ analyses”, LHCb-CONF-2012-032.
* [20] R. Fleischer, “$B_{s,d}\to\pi\pi,\pi K,KK$: Status and Prospects” Eur. Phys. J. C 52 (2007) 267; arXiv:0705.1121 [hep-ph].
* [21] LHCb collaboration, “First evidence of direct $CP$ violation in charmless two-body decays of $B^{0}_{s}$ mesons” Phys. Rev. Lett. 108 (2012) 201601; arXiv:hep-ex/1202.6251.
* [22] LHCb collaboration, “Measurement of time-dependent $CP$ violation in charmless two-body $B$ decays” LHCb-CONF-2012-007.
* [23] LHCb collaboration, “Evidence for $CP$ violation in time-integrated $D^{0}\rightarrow h^{+}h^{-}$ decay rates” Phys. Rev. Lett. 108, (2012) 111602; arXiv:hep-ex/1112.0938.
* [24] CDF collaboration, “Measurement of the difference of $CP$-violating asymmetries in $D^{0}\rightarrow K^{+}K^{-}$ and $D^{0}\rightarrow\pi^{+}\pi^{-}$ decays at CDF” Phys. Rev. Lett. 109, (2012) 111801; arXiv:hep-ex/1207.2158.
* [25] LHCb collaboration and A. Bharuca et al., “Implications of LHCb measurements and future prospects”; arXiv:hep-ex/1208.3355.
* [26] Heavy Flavour Averaging Group, arxiv:hep-ex/1207.1158v1 and www.slac.stanford.edu/xorg/hfag/charm/ICHEP12/.
|
arxiv-papers
| 2013-01-14T18:12:10 |
2024-09-04T02:49:40.266771
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Monica Pepe Altarelli",
"submitter": "Monica Pepe Altarelli",
"url": "https://arxiv.org/abs/1301.3082"
}
|
1301.3102
|
# Estimation de résolvante et construction de quasimode près du bord du
pseudospectre
William Bordeaux Montrieux
Department of Mathematics
University of Toronto
Toronto, Ontario, Canada, M5S 2E4
[email protected]
###### Abstract
Nous considérons un opérateur $h$-pseudodifférentiel non-autoadjoint dans la
limite semiclassique. $p$ désigne le symbole principal. Nous savons que la
résolvante existe à l’intérieur de l’image de $p$ jusqu’à une distance
$\mathcal{O}((h\ln\frac{1}{h})^{\frac{k}{k+1}}),$ de certains points du bord,
où $k\in\\{2,4,\ldots\\}.$ Dans ce travail, nous précisons les estimations de
résolvantes qu’ont obtenues différents auteurs dans le cas $k=2,$ et en
dimension $1.$ Pour la preuve, il s’agit de construire, via un scaling, des
quasimodes pour des valeurs du paramètre spectral très proches du bord de
l’image de $p.$
###### Abstract
We consider a non-self-adjoint pseudodifferential operator in the semi-
classical limit $(h\to 0)$. The principal symbol is given by $p.$ We know that
the resolvent $(z-P)^{-1}$ exists inside the range up to a distance
$\mathcal{O}((h\ln\frac{1}{h})^{\frac{k}{k+1}})$ from certain boundary points,
where $k\in\\{2,4,\ldots\\}.$ In this work, we improve the resolvent estimates
given by different authors in the case $k=2,$ and in dimension one. For the
proof, we will construct quasimodes by a scaling for $z$ very close to the
boundary.
###### ?contentsname?
1. 1 Introduction
2. 2 Modèle sur $\mathbb{R}$ à paramètre
1. 2.1 Ensemble d’énergie pour de petites valeurs du paramètre
2. 2.2 Problème de Grushin.
3. 2.3 Estimation de résolvante près du bord de l’image du symbole.
3. 3 Cas général en dimension 1
1. 3.1 Estimation de résolvante: cas 1
2. 3.2 Estimation de résolvante: cas 2
4. 4 Exemples
1. 4.1 Cas 1: L’opérateur cubique non-autoadjoint
2. 4.2 Cas 2: Oscillateur harmonique non-autoadjoint
3. 4.3 Opérateur d’advection-diffusion
## 1 Introduction
Dans [5], N. Dencker, J. Sjöstrand et M. Zworski obtiennent une estimation de
la résolvante pour certains points du bord de l’image du symbole principal.
Dans le cas d’un point $z_{0}$ de type fini d’ordre $k,$ ils obtiennent que la
résolvante a une croissance en $h^{-\frac{k}{k+1}},$ dans des disques de rayon
$\mathcal{O}(h^{\frac{k}{k+1}})$ centrés en $z_{0}.$ Dans [16], J. Sjöstrand
montre que la résolvante peut s’étendre à des disques de rayon
$\mathcal{O}((h\ln\frac{1}{h})^{\frac{k}{k+1}}),$ et donne une majoration pour
la résolvante. Une majoration similaire avait été obtenue auparavant par
l’auteur pour un opérateur modèle $hD_{x}+g(x),$ $x\in S^{1},$ avec $k=2,$
voir [1] Chapitre 4. On citera aussi J. Martinet [12] qui obtient un
encadrement de la norme de la résolvante pour l’opérateur d’Airy complexe. La
preuve est basée sur une analyse directe du semigroupe après conjugaison par
la transformée de Fourier. Dans ce travail, il s’agit d’améliorer et
généraliser les estimations de la résolvante obtenues par ces différents
auteurs. On se limite au cas d’un point d’ordre $2,$ et à la dimension 1. Les
estimations obtenues nous permettent de préciser l’estimation de résolvante
obtenue par Martinet [12] pour l’opérateur d’Airy complexe sur la droite
réelle $(D_{x})^{2}+ix,$ $x\in\mathbb{R},$ et d’obtenir le comportement à
l’infini des lignes de niveaux de la résolvante de l’oscillateur harmonique
non-autoadjoint. Ce dernier résultat vient compléter ceux publiés par L.
Boulton [2], K. Pravda-Starov [15]. Mentionnons aussi le travail [3] de Davies
et Kuijlaars qui obtiennent le premier terme dans l’asymptotique du logarithme
de la norme des projecteurs spectraux pour les grandes valeurs propres pour
l’oscillateur harmonique non-autoadjoint.
#### Cas modèle sur $\mathbb{R}$ à paramètre.
Nous considérons l’opérateur modèle non-autoadjoint dans $L^{2}(\mathbb{R})$
$P_{\alpha}=hD_{x}+g(x,\alpha)+h^{2}g_{\ast}(x,\alpha;h),\quad h\in(0,1],\quad
D_{x}=\frac{1}{i}\frac{\partial}{\partial x},$
$g$ et $g_{\ast}$ dépendent de manière $C^{\infty}$ de $x$ et d’un paramètre
$\alpha$ strictement positif et borné. On pose
$\widetilde{g}(x,\alpha;h):=g(x,\alpha)+h^{2}g_{\ast}(x,\alpha;h)$ et on
demande qu’il existe $C_{0}>0$ tel que
$\mathrm{Re}\,g=\mathcal{O}(x^{2}+1),$
(cette hypothèse peut facilement être affaiblie). Pour $g_{\ast}$ nous avons
le développement asymptotique en puissances de $h$ suivant
$\displaystyle g_{\ast}(x,\alpha;h)\sim
g^{0}_{\ast}(x,\alpha)+hg^{1}_{\ast}(x,\alpha)+\ldots\mbox{ dans
}C^{\infty}_{0}(\mathbb{R}\times[0,1]).$ (1.1)
L’opérateur $P_{\alpha}$ est muni du domaine $H_{sc}(\mathbb{R}):=\\{u\in
L^{2}(\mathbb{R})|\;\|x^{2}u\|+\|hD_{x}u\|<+\infty\\}.$ On note $p_{\alpha}$
le symbole principale semiclassique de $P_{\alpha},$ soit
$p_{\alpha}=\xi+g(x,\alpha).$ Rappelons que la définition du crochet de
Poisson est
$\\{a,b\\}(x,\xi):=(a^{\prime}_{\xi}b^{\prime}_{x}-a^{\prime}_{x}b^{\prime}_{\xi})(x,\xi)=H_{a}b,$
(1.2)
pour deux fonctions $a(x,\xi),$ $b(x,\xi)$ de classe $C^{1}(\mathbb{R}^{2}).$
Ici $H_{a}=(a_{\xi}\partial_{x}-a_{x}\partial_{\xi})$ désigne le champ
Hamiltonien.
###### Proposition 1.1
Soient $\widetilde{g}$ comme ci-dessus avec $g(x,\alpha)\in
C^{\infty}(\mathbb{R}\times]0,C[),$ et pour tout $h$ fixé
$g_{\ast}(x,\alpha;h)\in C^{\infty}(\mathbb{R}\times[0,1/C])$ pour un $C>0$
donné. Nous rajoutons les hypothèses suivantes:
pour $x\neq 0,$ $g$ satisfait $\mathrm{Im}\,g(x,0)>0,$ et
$\displaystyle g(0,0)$ $\displaystyle=0,\;g^{\prime}_{\alpha}(0,0)=-i,$ (1.3)
$\displaystyle g^{\prime}_{x}(0,0)$
$\displaystyle=0,\;\mathrm{Im}\,g^{\prime\prime}_{xx}(0,0)>0.$ (1.4)
L’opérateur $P_{\alpha}$ est muni du domaine $H_{sc}(\mathbb{R}).$ Dans ces
conditions, la résolvante est définie pour tout $\alpha$
$\displaystyle(hD_{x}+\widetilde{g}(x,\alpha;h))^{-1}:L^{2}(\mathbb{R})\to
L^{2}(\mathbb{R}),$ (1.5)
et vérifie pour $h\ll\alpha^{3/2}$ et $\alpha$ assez petit,
$\|P_{\alpha}^{-1}\|\sim\frac{\sqrt{\pi}\exp(\frac{1}{h}\mathrm{Im}\,\ell_{0}(\alpha))}{h^{1/2}(\frac{1}{2i}\\{p_{\alpha},\overline{p_{\alpha}}\\}(\rho_{+}))^{\frac{1}{4}}(\frac{1}{2i}\\{\overline{p_{\alpha}},p_{\alpha}\\}(\rho_{-}))^{\frac{1}{4}}}(1+\mathcal{O}(\tilde{h}))+\mathcal{O}(\frac{1}{\sqrt{h}\,\alpha^{1/4}}),$
(1.6)
où $\tilde{h}=h/\alpha^{3/2},$ et
$\rho_{\pm}(\alpha):=(x_{\pm}(\alpha),\xi_{\pm}(\alpha))$ sont les solutions
de $p_{\alpha}(\rho_{\pm})=\xi_{\pm}+g(x_{\pm},\alpha)=0$ avec
$\mp\mathrm{Im}\,^{\prime}(x_{\pm},\alpha)>0$ (impliquant
$\mathrm{Im}\,g(x_{\pm},\alpha)=0).$ $\ell_{0}$ est donné par
$\ell_{0}:=-\mathrm{Im}\,\int_{x_{+}}^{x_{-}}g(y,\alpha)dy\asymp\alpha^{3/2}.$
(1.7)
Nous avons pour $\alpha$ assez petit
$x_{\pm}(\alpha)=\mp\alpha^{\frac{1}{2}}\left(\frac{-2}{\\{p_{0},\frac{1}{2i}\\{p_{0},\overline{p_{0}}\\}\\}(0,0)}\right)^{1/2}+\mathcal{O}(\alpha),$
(1.8)
impliquant
$|\frac{1}{2}\\{p_{\alpha},\overline{p_{\alpha}}\\}(\rho_{\pm})|\asymp\alpha^{\frac{1}{2}}.$
Voir la section 2 pour la preuve.
###### Remarque 1.2
Le dénominateur du premier de 1.6 est de l’ordre de $\sqrt{h}\alpha^{1/4}$ et
ce qui le rend dominant dès que $\alpha\gg h^{2/3}.$
###### Remarque 1.3
La proposition reste valable pour $\alpha\in[0,1]$ si on rajoute l’hypothèse
(automatiquement vérifiée pour $\alpha$ petit) que l’équation
$\mathrm{Im}\,g(x,\alpha)=0$ admet exactement deux solution $x=x_{+}(\alpha)$,
$x_{-}(\alpha)$ et de plus
$\mp\mathrm{Im}\,g^{\prime}_{x}(x_{\pm}(\alpha),\alpha)$ $>0.$
Nous considérons l’opérateur d’Airy complexe sur la droite réelle
$\mathcal{A}=D_{x}^{2}+ix,$ étudié par J. Martinet [12], et muni du domaine
$D(\mathcal{A})=\\{u\in H^{2}(\mathbb{R})|\;xu\in L^{2}(\mathbb{R})\\}.$ Nous
savons que le spectre est vide, que la norme de la résolvante
$(\mathcal{A}-z)^{-1}$ ne dépend que de $\mathrm{Re}\,z$ (voir Helffer [11]),
et qu’en conjuguant par une transformée de Fourier, on peut se ramener à
l’étude de l’opérateur
$D_{x}-ix^{2}+iz.$
Pour $z$ dans la zone d’intérêt $\mathrm{Re}\,z>0,$ le changement de variable
$x=(\mathrm{Re}\,z)^{\frac{1}{2}}y,$ donne l’opérateur
$(\mathrm{Re}\,z)\mathcal{Q},$ où $\mathcal{Q}$ est donné par
$\frac{1}{(\mathrm{Re}\,z)^{\frac{3}{2}}}D_{x}-ix^{2}+i-\frac{\mathrm{Im}\,z}{\mathrm{Re}\,z}.$
De la proposition au-dessus et la remarque 1.3, avec
$h=\frac{1}{(\mathrm{Re}\,z)^{\frac{3}{2}}}$ et $\alpha=1,$ on déduit le
corollaire suivant:
###### Corollaire 1.4
Pour $\mathrm{Re}\,z>0$ assez grand, la résolvante de $\mathcal{A}$ satisfait
$\displaystyle\|((D_{x})^{2}+ix-z)^{-1}\|\sim$
$\displaystyle\sqrt{\frac{\pi}{2}}(\mathrm{Re}\,z)^{-\frac{1}{4}}(1+\mathcal{O}(\frac{1}{(\mathrm{Re}\,z)^{3/2}}))\exp(\frac{4}{3}(\mathrm{Re}\,z)^{\frac{3}{2}})$
$\displaystyle+\mathcal{O}(\frac{1}{(\mathrm{Re}\,z)^{1/4}}).$
Ceci précise l’estimation qu’a obtenue J. Martinet dans [12]. La preuve sera
donnée à la fin de sa sous-section 2.2.
#### Cas général en dimension 1.
Soit $p\in S(\mathbb{R}^{2},m)$ indépendant de $h,$ où $m$ désigne une
fonction d’ordre au sens que
$\displaystyle\exists C_{0}\geq 1,\,N_{0}>0\mbox{ tels que }m(\rho)\leq
C_{0}\langle\rho-\mu\rangle^{N_{0}}m(\mu),\forall\rho,\mu\in\mathbb{R}^{2},$
$\displaystyle\mbox{ avec }\langle\rho-\mu\rangle=\sqrt{1+|\rho-\mu|^{2}}.$
L’espace de symboles correspondant est
$S(\mathbb{R}^{2},m)=\\{a\in
C^{\infty}(\mathbb{R}^{2}),\,|\partial^{\alpha}_{\rho}a(\rho)|\leq
C_{\alpha}m(\rho),\,\rho\in\mathbb{R}^{2},\alpha\in\mathbb{N}^{2}\\}.$
Nous désignons par $P=p^{w}$ son $h$-quantifié de Weyl (voir par exemple [4]),
que nous considérons dans $L^{2}(\mathbb{R}).$ Nous faisons une hypothèse
d’ellipticité à l’infini sur $P:$
$\displaystyle|p(x,\xi)|\geq m(x,\xi)/C,\quad|(x,\xi)|\geq C,$ $\displaystyle
m(x,\xi)\to\infty,\,(x,\xi)\to\infty.$ (1.9)
Pour garantir que le spectre de $P$ n’est pas le plan complexe, nous avons
besoin de supposer que
$p(T^{\ast}\mathbb{R}^{2})\neq\mathbb{C}.$ (1.10)
Dans ces conditions, le spectre de $P$ est discret pour $h$ assez petit, voir
par exemple [8] ou [10].
Introduisons l’ensemble
$\Sigma(p)=p(T^{\ast}\mathbb{R}),$ (1.11)
ce dernier est automatiquement fermé à cause de (1). Nous allons faire une
hypothèse sur un point $z_{0}\in\partial\Sigma(p)$:
$p^{-1}(z_{0})=\\{\rho_{0}\\},\quad\\{p,\\{p,\bar{p}\\}\\}(\rho_{0})\neq 0.$
(1.12)
Ceci entraine que $dp(\rho)\neq 0,$ Nous disons aussi $p$ est de type fini
d’ordre $2$ au point $z_{0}.$
Le champ Hamiltonien $H_{\frac{1}{2i}\\{p,\bar{p}\\}}$ est un champ de
vecteurs réel et tangent à l’ensemble
$\\{\rho|\,\frac{1}{2i}\\{p,\bar{p}\\}(\rho)=0\\}.$ Nous avons
$]-T_{1},T_{0}[\ni
s\mapsto\rho(s):=\exp(sH_{\frac{1}{2i}\\{p,\bar{p}\\}})(\rho_{0})\subset\\{\frac{1}{2i}\\{p,\bar{p}\\}=0\\}$
est une courbe orientée. Sa courbe image par $p$; $\gamma=p\circ\rho$ est une
courbe $C^{\infty}$ avec $\dot{\gamma}\neq 0$ et dont l’image coïncide avec
$\partial\Sigma$ près de $z_{0}$. De plus $\Sigma$ se situe à gauche quand on
regarde dans la direction de $\dot{\gamma},$ voir [9]. Près de $z_{0},$ pour
tout complexe $z_{\ast}\in\partial\Sigma(p),$ nous avons
$p^{-1}(z_{\ast})=\\{\rho(s_{\ast})\\}.$ $\gamma=p\circ\rho$ est aussi une
courbe orientée. Le vecteur unitaire tangent à $\partial\Sigma$ dans un
voisinage de $z_{0}$ est alors donné par
$u(s):=\dot{\gamma}(s)/|\dot{\gamma}(s)|,$ lequel vérifie
$u(s)=-\frac{\\{p,\frac{1}{2i}\\{p,\bar{p}\\}\\}(\rho(s))}{|\\{p,\frac{1}{2i}\\{p,\bar{p}\\}\\}(\rho(s))|}.$
Près de $z_{0}$, $\Sigma(p)$ est un ensemble à bord $C^{\infty}$. Il existe un
voisinage $W$ de $z_{0}$ tel que tout $z\in\Sigma(p)\cap W$ peut s’écrire sous
la forme
$z=\gamma(s)+i\alpha u(s),\quad\alpha\geq 0,\quad z_{0}=\gamma(0).$ (1.13)
###### Proposition 1.5
Soit $P$ un opérateur $h$-pseudodifférentiel de symbole indépendant de $h.$
Nous supposons que $(p,z_{0})$ vérifie les conditions (1), (1.10), (1.12). Il
existe un voisinage $W$ de $z_{0}$ tel que pour tout point $z$ de
$(\Sigma(p)\cap W)\setminus\partial\Sigma(p)$ nous avons
$p^{-1}(z)=\\{\rho_{+}(z),\rho_{-}(z)\\},\quad\rho_{\pm}=(x_{\pm},\xi_{\pm}),$
(1.14)
avec
$0<\pm\frac{1}{2i}\\{p,\bar{p}\\}(\rho_{\pm})\asymp\alpha^{1/2}\quad|\rho_{+}-\rho_{-}|\asymp\alpha^{1/2}.$
(le point $\rho_{0}$ se scinde en deux points $\rho_{+}$ et $\rho_{-}$ lorsque
l’on s’éloigne du bord de $\Sigma(p))$.)
Ici on introduit $\ell_{0}(z).$ Dans le cas où $p^{\prime}_{\xi}(\rho_{0})\neq
0$ nous avons la factorisation
$p(\rho)-z=q(x,\xi,\alpha,s)(\xi+g(x,\alpha,s))$ (1.15)
où si $\alpha<0$ alors $\mathrm{Im}\,g(x)>0$, $x>x_{-}$ ou $x<x_{+}$ et
$\mathrm{Im}\,g(x)<0,$ $x_{-}<x<x_{+}.$ Ici $\rho_{\pm}(x_{\pm},\xi_{\pm}),$
$\xi_{\pm}=-g(x_{\pm},\alpha,s).$ Sous ces conditions, on définit
$\ell_{0}(z)$ comme
$\ell_{0}(z):=-\int_{x_{+}(z)}^{x_{-}(z)}g(x)dx.$ (1.16)
Si $p^{\prime}_{\xi}(\rho_{0})=0$ alors $p^{\prime}_{x}(\rho)\neq 0.$ On
applique la discussion précédente à $\widehat{p}(x,\xi)=p(-\xi,x),$ fonction
obtenue de $p$ par composition avec une transformation canonique. On pose
alors
$\widehat{\ell}_{0}(z)=-\int_{x_{+}}^{x_{-}}\widehat{g}(x,\xi)dx$ (1.17)
où
$\widehat{p}(\rho)-z=\widehat{q}(x,\xi,\alpha,s)(\xi+\widehat{g}(x,\alpha,s))$.
On peut prouver que
###### Proposition 1.6
Si $p^{\prime}_{\xi}(\rho_{0})\neq 0$ et $p^{\prime}_{x}(\rho_{0})\neq 0,$
alors nous avons
$\mathrm{Im}\,\ell_{0}(z)=\mathrm{Im}\,\widehat{\ell}_{0}(z)+\mathcal{O}(\alpha^{\infty}).$
###### Théorème 1.7
Il existe une constante $T_{\ast}$ ($<T_{0},T_{1}$) telle que pour toutes
constantes $C_{0},C_{1}>0$ il existe une constante $C_{2}>0$ telle que la
résolvante $(P-z)^{-1}$ est bien définie pour
$|s|<T_{\ast},\;\frac{h^{2/3}}{C_{0}}\leq\alpha\leq
C_{1}(h\ln\frac{1}{h})^{2/3},\;h<\frac{1}{C_{2}},$ (1.18)
et satisfait l’estimation
$\displaystyle\|(P-z)^{-1}\|\sim$
$\displaystyle\frac{\sqrt{\pi}\exp(\frac{1}{h}\mathrm{Im}\,\ell_{0}(z))}{h^{1/2}(\frac{1}{2i}\\{p,\bar{p}\\}(\rho_{+}))^{\frac{1}{4}}(\frac{1}{2i}\\{\bar{p},p\\}(\rho_{-}))^{\frac{1}{4}}}\times(1+\mathcal{O}(\tilde{h}))+\mathcal{O}(\frac{1}{\sqrt{h}\,\alpha^{1/4}}),$
où $\tilde{h}=h/\alpha^{3/2},$ et $\ell_{0}$ (qui est une intégrale d’action)
vérifie
$\ell_{0}(z):=\int_{\gamma\subset p^{-1}(z)}\xi dx,(\gamma\mbox{ relie
}\rho_{-}\mbox{ \\`{a} }\rho_{+}).$ (1.19)
Remerciements: Mes remerciements vont d’abord à Johannes Sjöstrand pour
m’avoir proposé ce sujet, et les aides et les discussions qu’il m’a apportées
ainsi que pour son soutien. Je tiens aussi à remercier Bernard Helffer pour
les échanges sur l’opérateur d’Airy. Ce travail a été soutenu par le Grant à
Vienne sponsorisé par K. Groechenig et H.G. Feichtinger et par les grants
sponsorisés l’un par V. Ivrii et l’autre par M. Sigal. Merci enfin à K.
Groechenig et H.G. Feichtinger à Vienne, et V. Ivrii et M. Sigal. à Toronto
pour leur accueil.
## 2 Modèle sur $\mathbb{R}$ à paramètre
Nous rappelons $p_{\alpha}(x,\xi):=\xi+g(x,\alpha)$ désigne le symbole
principal semiclassique de $P_{\alpha}=hD_{x}+\widetilde{g}(x,\alpha;h).$
Notre opérateur $hD_{x}+\widetilde{g}(x,\alpha;h)$ admet comme domaine naturel
l’espace semiclassique suivant
$H_{sc}(\mathbb{R}):=\\{u\in
L^{2}|\;\|u\|_{H_{sc}}:=\|u\|+\|x^{2}u\|+\|hD_{x}u\|<\infty\\}.$
L’inverse, définie pour tout $\alpha$
$\displaystyle(hD_{x}+\widetilde{g}(x,\alpha;h))^{-1}:L^{2}(\mathbb{R})\to
L^{2}(\mathbb{R}),$ (2.1)
est donnée par la formule suivante
$u(x)=\int_{+\infty}^{x}\frac{i}{h}e^{-\frac{i}{h}\int_{y}^{x}\widetilde{g}(t,\alpha;h)dt}v(y)dy.$
En particulier, quand $\alpha$ est nul, nous avons (voir [5])
$\|(hD_{x}+\widetilde{g}(x,0;h))^{-1}\|\leq Ch^{-2/3},$ (2.2)
### 2.1 Ensemble d’énergie pour de petites valeurs du paramètre
Nous cherchons à décrire l’ensemble $p_{\alpha}^{-1}(0)$ pour $\alpha\ll 1$
assez petit. D’abord remarquons que
$p^{-1}_{0}(0)=\\{(0,0)\\}.$
Résoudre $\xi+g(x,\alpha)=0$ revient à trouver les solutions de l’équation
$\mathrm{Im}\,g(x,\alpha)=0.$
Puisque $\mathrm{Im}\,g^{\prime}_{\alpha}(0)=-1,$ nous avons par la
factorisation $\mathrm{Im}\,g(x,\alpha)=q(x,\alpha)(\alpha-f(x)),$ où $q\neq
0$ dans un voisinage de zéro, avec $f(0)=f^{\prime}(0)=0,$ et
$f^{\prime\prime}(0)=-\mathrm{Im}\,g^{\prime\prime}_{xx}(0)\neq 0.$ Par le
théorème de l’inversion locale, il existe des voisinages $U_{+}:=]0,b_{+}[$ et
$U_{-}:=]b_{-},0[$ tels que $f$ soit un $C^{1}$ diffémorphisme de $U_{+}$ sur
$]0,f(b_{+})[$ et de $U_{-}$ sur $]0,f(b_{-})[.$ Pour avoir les solutions de
l’équation $\alpha-f(x)=0,$ nous inversons simplement une fonction croissante
en décroissante. Nous trouvons alors
$\displaystyle x\in
U_{\pm},\quad\frac{2}{\mathrm{Im}\,g^{\prime\prime}_{xx}(0,0)}\alpha$
$\displaystyle=x^{2}+\mathcal{O}(|x|^{3}),$
$\displaystyle=(x+\mathcal{O}(x^{2}))^{2}.$
Soit
$\displaystyle x\in U_{\pm},\quad x+\mathcal{O}(x^{2})$
$\displaystyle=\pm\alpha^{\frac{1}{2}}\left(\frac{2}{\mathrm{Im}\,g^{\prime\prime}_{xx}(0,0)}\right)^{\frac{1}{2}},$
$\displaystyle x(\alpha)$
$\displaystyle=\pm\alpha^{\frac{1}{2}}\left(\frac{2}{\mathrm{Im}\,g^{\prime\prime}_{xx}(0,0)}\right)^{\frac{1}{2}}+\mathcal{O}(\alpha).$
(2.3)
Nous déduisons alors le résultat suivant:
###### Lemme 2.1
Soit $p_{\alpha}=\xi+g(x,\alpha)$ le symbole principal de $P_{\alpha}.$ (1.3)
et (1.4) sont vérifiés. Nous avons, pour $\alpha>0,$ assez petit,
$p^{-1}_{\alpha}(0)=\\{\rho_{+}(\alpha)\\}\cup\\{\rho_{-}(\alpha)\\},\quad\rho_{\pm}(\alpha)=(x_{\pm},\xi_{\pm}),$
avec
$x_{\pm}(\alpha)=\mp\alpha^{\frac{1}{2}}\left(\frac{-2}{\\{p_{0},\frac{1}{2i}\\{p_{0},\overline{p_{0}}\\}\\}(0,0)}\right)^{1/2}+\mathcal{O}(\alpha),$
(2.4)
tel que
$\pm\frac{1}{2i}\\{p_{\alpha},\overline{p_{\alpha}}\\}(\rho_{\pm}(\alpha))>0.$
(2.5)
L’ensemble $p_{0}^{-1}(0)$ est constitué d’un seul point $(0,0)$ qui va se
“scinder” en deux $\rho_{+}(\alpha)$ et $\rho_{-}(\alpha),$ lorsque $\alpha$
ne s’annule plus.
Preuve. Il faut remarquer que pour le premier crochet et le second crochet de
Poisson, nous avons
$\displaystyle\frac{1}{2i}\\{p_{\alpha},\overline{p_{\alpha}}\\}(\rho_{\pm})$
$\displaystyle=-\mathrm{Im}\,g^{\prime}_{x}(x_{\pm}(\alpha),\alpha),$
$\displaystyle\\{p_{0},\frac{1}{2i}\\{p_{0},\overline{p_{0}}\\}\\}(0,0)$
$\displaystyle=-\mathrm{Im}\,g_{xx}^{\prime\prime}(0,0).$
Dès lors,
$\displaystyle\pm\frac{1}{2i}\\{p_{\alpha},\overline{p_{\alpha}}\\}(\rho_{\pm})$
$\displaystyle=\mp\mathrm{Im}\,g^{\prime}_{x}(x_{\pm}(\alpha),\alpha)$
$\displaystyle=\mp\left(x_{\pm}(\alpha)\mathrm{Im}\,g^{\prime\prime}_{xx}(0,0)+\alpha\mathrm{Im}\,g^{\prime\prime}_{x,\alpha}(0,0)+\mathcal{O}(x_{\pm}^{2}+\alpha^{2})\right)$
(2.6) $\displaystyle>0.$
Le premier terme de (2.6) domine car il se comporte comme $\sqrt{\alpha}$.
$\square$
### 2.2 Problème de Grushin.
Nous nous appuyons sur les sections “solutions locales” et “problème de
Grushin” de Hager [7] et “Enoncé et résolution asymptotique du problème de
Grushin” de Hager [8]. Pour une lecture “plus aisée”, nous reprenons en détail
les grandes lignes, et les calculs dont nous aurons besoin.
On suppose que $\alpha\geq\frac{1}{\mathcal{O}(1)}$ appartient à un intervalle
ouvert $\Omega,$ relativement compact, séparé de l’origine par une constante
indépendante de $h.$ Nous allons d’abord considérer cette situation, puis nous
laisserons $\alpha$ devenir petit et nous ferons des dilations pour se ramener
à ce premier cas.
Avec les hypothèses choisies pour $g,$ nous avons
$\forall\alpha\in\Omega,\;p^{-1}_{\alpha}(0)=\\{\rho_{+}(\alpha),\rho_{-}(\alpha)\\}\;\mbox{
avec }\pm\frac{1}{2i}\\{p_{\alpha},\overline{p_{\alpha}}\\}(\rho_{\pm})>0.$
(2.7)
On peut trouver des intervalles $J_{+}$ et $J_{-}$ disjoints tels que
$\overline{x_{\pm}(\Omega)}\Subset J_{\pm}.$
Considérons $\chi_{\pm}\in C^{\infty}(I_{\pm})$ avec $I_{+}:=]-\infty,\inf
J_{-}[$ et $I_{-}:=]\sup J_{+},+\infty[.$ Il est clair que $I_{+}\cap
I_{-}=]\sup J_{+},\inf J_{-}[.$ On demande que $\chi_{\pm}=1$ sur
$\overline{J_{\pm}}$ tels que
$\mbox{supp}(\chi_{+})\cap\mbox{supp}(\chi_{-})=\emptyset.$ Pour le
comportement de $\chi_{\pm}$ à l’infini, nous imposons que $\chi_{+}=1$ sur
$]-\infty,\inf J_{+}[$ et, $\chi_{-}=1$ sur $]\sup J_{-},+\infty[.$ Les
$\chi_{\pm}$ s’annulent près de $\partial I_{\pm}.$
Considérons sur $I_{+},$ une solution de l’équation $P_{\alpha}e_{+}=0,$
$e_{+}:=c_{+}(\alpha;h)\exp{(-\frac{i}{h}\int_{x_{+}}^{x}\widetilde{g}(y,\alpha;h)dy)}.$
(2.8)
Si nous choisissons le formalisme “BKW”, $e_{+}$ s’écrit
$\displaystyle e_{+}$
$\displaystyle=\left(c_{+}(\alpha;h)e^{-ih\int_{x_{+}}^{x}g_{\ast}(y,\alpha;h)dy}\right)e^{-\frac{i}{h}\int_{x_{+}}^{x}g(y,\alpha)dy},$
$\displaystyle=:a_{+}(x,\alpha;h)e^{\frac{i}{h}\varphi_{+}(x,\alpha)},\quad\varphi_{+}(x,\alpha):=-\int_{x_{+}}^{x}g(y,\alpha)dy,$
avec une phase $\varphi_{+}\in C^{\infty}$, indépendante de $h$, qui vérifie
l’équation eikonale $\partial_{x}\varphi_{+}(x,\alpha)+g(x,\alpha)=0,$ et une
amplitude $a_{+}$ admettant un développement asymptotique en puissances de
$h$,
$a_{+}(x,\alpha;h)\sim\sum_{k\geq 0}a_{+,k}(x,\alpha)h^{k}\mbox{ dans
}C^{\infty}_{b}(\mathbb{R}),\;\forall\alpha\in\Omega.$
Nous avons $\varphi_{+}(x_{+},\alpha)=0,$
$\varphi^{\prime}_{+}(x_{+},\alpha)=\xi_{+}\in\mathbb{R},$ et
$\mathrm{Im}\,\varphi^{\prime\prime}_{+}(x_{+}(\alpha),\alpha)=-\mathrm{Im}\,g^{\prime}_{x}(x_{+},\alpha)=\frac{1}{2i}\\{p_{\alpha},\overline{p_{\alpha}}\\}>0.$
La partie imaginaire de la phase est positive, cela restera vrai globalement
sur $I_{+}:$ $\mathrm{Im}\,\varphi_{+}\geq 0.$
Grâce au choix de $g$ vers $-\infty,$ pour $C>0$ assez grand nous avons
$\|e_{+}\|_{L^{2}(]-\infty,-C])}=\mathcal{O}(e^{-\frac{1}{\widetilde{C}h}}).$
Il est alors possible, grâce au lemme de la phase stationnaire (que nous
rappellerons après), de choisir $c_{+}(\alpha;h)$ de la forme
$c_{+}(\alpha;h)\sim h^{-1/4}(c_{+}^{0}(\alpha)+hc^{1}_{+}(\alpha)+\ldots)>0,$
avec
$\displaystyle c_{+}^{0}(\alpha)$
$\displaystyle=\left(\frac{-2\mathrm{Im}\,g^{\prime}_{x}(x_{+}(\alpha),\alpha)^{1/4}}{2\pi}\right)^{\frac{1}{4}}$
$\displaystyle=\frac{(-\mathrm{Im}\,g^{\prime}_{x}(x_{+},\alpha))^{1/4}}{\pi^{1/4}},$
tel que $e_{+}$ soit normalisé dans $L^{2}$ sur $I_{+}.$ Ce qui implique, dans
le formalisme “BKW”, que
$\displaystyle a_{+}(x,\alpha;h)$
$\displaystyle=c_{+}(\alpha;h)e^{-ih\int_{x_{+}}^{x}g_{\ast}(y,\alpha;h)dy}$
$\displaystyle=c_{+}(\alpha;h)(1-ih\int_{x_{+}}^{x}g^{0}_{\ast}(y,\alpha)dy+\mathcal{O}(h^{2})),$
$\displaystyle=\frac{|\mathrm{Im}\,g^{\prime}_{x}(x_{+},\alpha)|^{1/4}}{(\pi
h)^{1/4}}(1+\mathcal{O}(h))\mbox{ dans }C^{\infty}_{b}(\mathbb{R}).$
###### Lemme 2.2
On désigne par $e_{+}\in H_{sc}(I_{+})$ la solution normalisée dans
$L^{2}(I_{+})$ de $(hD_{x}+\widetilde{g}(x,\alpha;h))e_{+}=0$ sur $I_{+}.$
Nous avons la représentation asymptotique suivante pour $h\ll 1,$ assez petit
$e_{+}\sim\frac{|\mathrm{Im}\,g^{\prime}_{x}(x_{+},\alpha)|^{1/4}}{(\pi
h)^{1/4}}(1+\mathcal{O}(h))e^{-\frac{i}{h}\int_{x_{+}}^{x}\widetilde{g}(y,\alpha;h)dy}.$
De manière analogue nous avons :
###### Lemme 2.3
On désigne par $e_{-}\in H_{sc}(I_{-})$ la solution normalisée dans
$L^{2}(I_{-})$ de $(hD_{x}+\widetilde{g}(x,\alpha;h))^{\ast}e_{-}=0$ sur
$I_{-}.$ $e_{-}$ admet la représentation asymptotique suivante pour $h\ll 1,$
assez petit
$e_{-}\sim\frac{(\mathrm{Im}\,g^{\prime}_{x}(x_{-},\alpha))^{1/4}}{(\pi
h)^{1/4}}(1+\mathcal{O}(h))e^{-\frac{i}{h}\int_{x_{-}}^{x}\overline{\widetilde{g}(y,\alpha;h)}dy}.$
Nous rappelons ici le lemme de la phase stationnaire:
###### Proposition 2.4 (Phase stationnaire)
Soit $a\in C^{\infty}_{0}(\mathbb{R}).$ Supposons que $0\in
K=\mathrm{supp}(a)$ et
$\varphi(0)=\varphi^{\prime}(0)=0,\quad\varphi^{\prime\prime}(0)\neq
0,\quad\mathrm{Im}\,\varphi\geq 0.$
Supposons en plus que $\varphi^{\prime}(x)\neq 0$ sur $K-\\{0\\}.$ Posons
$g(x)=\varphi(x)-\varphi^{\prime\prime}(0)x^{2}/2.$ Alors, nous avons le
développement asymptotique explicite, lorsque $h\to 0,$
$\displaystyle\int_{\mathbb{R}}e^{i\varphi(x)/h}a(x)dx\sim$
$\displaystyle\left(\frac{2\pi
ih}{\varphi^{\prime\prime}(0)}\right)^{\frac{1}{2}}\sum_{k\geq
0}\sum_{\ell\geq 0}$
$\displaystyle\left(\frac{h}{2i\varphi^{\prime\prime}(0)}\right)^{k}\frac{1}{\ell!}\frac{1}{k!}\frac{d^{2k}}{dx^{2k}}((i/h)^{\ell}g^{\ell}a)(0).$
(2.9)
En particulier, nous voyons que le premier terme est $(2\pi
ih)^{1/2}\varphi^{\prime\prime}(0)^{-1/2}a(0).$
Pour une preuve on pourra consulter [19] p.36.
###### Proposition 2.5
Pour $v\in L^{2}(I_{+}),$ $v_{+}\in\mathbb{C},$ le problème de Cauchy
$(hD_{x}+\widetilde{g}(x,\alpha;h))u=v,u(x_{+})=0,$ admet la solution unique
$u=\widetilde{F}v$ avec
$\|\widetilde{F}\|_{L^{2}(I_{+})\to H_{sc}(I_{+})}\leq\frac{C}{\sqrt{h}}.$
Par ailleurs $\widetilde{F}$ admet un noyau intégral $k$ vérifiant l’égalité
$|k(x,y)|\leq\mathcal{O}(1/h)\exp(-|x-y|/(C\sqrt{h})).$
Preuve. $\widetilde{F}$ admet le noyau intégral
$k(x,y)=\frac{i}{h}e^{-\frac{i}{h}\int^{x}_{y}\widetilde{g}(\tilde{x},\alpha;h)\,d\tilde{x}}1_{\\{x_{+}\leq
y\leq
x\\}}-\frac{i}{h}e^{-\frac{i}{h}\int^{x}_{y}\widetilde{g}(\tilde{x},\alpha;h)\,d\tilde{x}}1_{\\{x\leq
y\leq x_{+}\\}}.$ (2.10)
Nous montrerons après qu’un tel noyau vérifie l’inégalité suivante
$|k(x,y)|\leq\mathcal{O}(1/h)\exp(-|x-y|/(C\sqrt{h}))$ (2.11)
pour une constante $C$ donnée.
La norme $L^{2}$ de $\widetilde{F}$ est majorée (lemme de Schur) par
$\left(\sup_{x}\int|k(x,y)|dy\right)^{1/2}\left(\sup_{y}\int|k(x,y)|dx\right)^{1/2}.$
(2.12)
Ce qui nous donne
$\displaystyle\sup_{x\in I_{+}}\int|k(x,y)|dy\leq\frac{C}{\sqrt{h}}\mbox{ et,
}\sup_{y\in I_{+}}\int|k(x,y)|dx\leq\frac{C}{\sqrt{y}}.$ (2.13)
Donc
$\|\widetilde{F}\|_{L^{2}(I_{+})\to L^{2}(I_{+})}\leq\frac{C}{\sqrt{h}}.$
De la même façon on peut montrer que
$\|x^{2}\widetilde{F}\|_{L^{2}(I_{+})\to L^{2}(I_{+})}\leq\frac{C}{\sqrt{h}}.$
Ce qui implique, en utilisant
$(hD_{x}+\widetilde{g}(x,\alpha;h))\widetilde{F}w=w,$ que nous avons
$\|\widetilde{F}\|_{L^{2}(I_{+})\to H_{sc}(I_{+})}\leq\frac{C}{\sqrt{h}}.$
Il nous reste à prouver l’estimation (2.11). Parce que $x_{+}$ est un zéro
non-dégénéré de $\mathrm{Im}\,g,$ et au comportement de $\mathrm{Im}\,g$ vers
$-\infty,$ nous avons pour $h$ assez petit, et $x\in I_{+},$ (rappelons que
$g_{\ast}(x,\alpha;h)$ est uniformément borné)
$\displaystyle\mathrm{Im}\,\widetilde{g}(x,\alpha;h)$
$\displaystyle\leq-\frac{1}{C_{0}}(x-x_{+})+C_{0}h^{2}\mbox{ si }x\geq x_{+},$
(2.14) $\displaystyle\mathrm{Im}\,\widetilde{g}(x,\alpha;h)$
$\displaystyle\geq-\frac{1}{C_{1}}(x-x_{+})-C_{1}h^{2}\mbox{ si }x\leq x_{+}.$
(2.15)
Ce qui implique, si nous considérons le cas $x\geq x_{+},$
$\displaystyle|k(x,y)|$
$\displaystyle\leq\frac{1}{h}e^{\frac{1}{h}\int^{x}_{y}\mathrm{Im}\,\widetilde{g}(\tilde{x},\alpha;h)\,d\tilde{x}}1_{\\{x_{+}\leq
y\leq x\\}}$
$\displaystyle\leq\frac{1}{h}e^{-\frac{1}{C_{0}h}\int^{x}_{y}(\widetilde{x}-x_{+})\,d\tilde{x}}e^{C_{0}h(x-y)}1_{\\{x_{+}\leq
y\leq x\\}}$
$\displaystyle\leq\frac{1}{h}e^{-\frac{1}{2C_{0}h}((x-x_{+})^{2}-(y-x_{+})^{2})}e^{C_{0}h(x-y)}1_{\\{x_{+}\leq
y\leq x\\}},$ (2.16)
Pour $(x-x_{+})\geq\sqrt{h},$ 2.16 s’écrit
$\frac{1}{h}e^{-\frac{1}{2C_{0}h}((x-x_{+})-2(C_{0}h)^{2})(x-y)}1_{\\{x_{+}\leq
y\leq x\\}}=\frac{1}{h}e^{-|x-y|/(C\sqrt{h})},$
et pour $(x-x_{+})\leq\sqrt{h},$
$\frac{1}{h}e^{-\frac{1}{2C_{0}h}((x-x_{+})-2(C_{0}h)^{2})(x-y)}1_{\\{x_{+}\leq
y\leq x\\}}=\mathcal{O}(1/h)e^{-|x-y|/(C\sqrt{h})}.$
Les autres inégalités nécessaires pour déduire (2.11) se calculent de la même
manière. $\square$
###### Proposition 2.6
Pour $P_{\alpha}=hD_{x}+\widetilde{g}(x,\alpha;h),$ $v\in L^{2}(I_{+}),$
$v\in\mathbb{C}$ le problème
$\left\\{\begin{array}[]{cc}P_{\alpha}u=v\\\ R_{+}u=v_{+}\end{array}\right.$
(2.17)
avec
$R_{+}u:=\langle
u,\chi_{+}e_{+}\rangle=\int_{I_{+}}u(x)\chi_{+}(x)\overline{e_{+}(x)}dx,$
admet une solution unique
$u=Fv+F_{+}v_{+}\in H_{sc},$
où
$F_{+}v_{+}:=\frac{1}{\langle
e_{+},\chi_{+}e_{+}\rangle}v_{+}e_{+}=:\frac{1}{D_{+}}v_{+}e_{+},$
avec
$D_{+}=1+\mathcal{O}(e^{-\frac{1}{Ch}}),$
et nous avons
$\displaystyle\|F\|_{L^{2}\to H_{sc}}$ $\displaystyle=\mathcal{O}(h^{-1/2}),$
(2.18) $\displaystyle\|F\|_{\mathbb{C}\to H_{sc}}$
$\displaystyle=\mathcal{O}(1).$ (2.19)
Preuve. Le problème
$\left\\{\begin{array}[]{cc}P_{\alpha}u=0\\\ R_{+}u=v_{+}\end{array}\right.$
admet la solution unique $u=F_{+}v_{+},$ tandis que le problème
$\left\\{\begin{array}[]{cc}P_{\alpha}u=v\\\ R_{+}u=0\end{array}\right.$
admet la solution unique $u=Fv:=(1-F_{+}R_{+})\widetilde{F}v$. $\square$
Nous désignons par $L^{2}_{comp}(I_{-})$ et $H_{sc,comp}(I_{-})$
respectivement l’ensemble des fonctions $f$ appartenant à $L^{2}$ et à
$H_{sc}(I_{-})$ respectivement telles que $f$ est nulle dans un voisinage de
$\inf I_{-}.$
###### Proposition 2.7
Pour $P_{\alpha}=hD_{x}+\widetilde{g}(x,\alpha;h),$ $v\in L^{2}_{comp}(I_{-})$
le problème
$P_{\alpha}u+R_{-}u_{-}=v,$
avec
$R_{-}u_{-}:=u_{-}\chi_{-}e_{-},\;u_{-}\in\mathbb{C},$
admet une solution unique dans $H_{sc,comp}\times\mathbb{C},$ donnée par
$\displaystyle u$ $\displaystyle=Gv,$ (2.20) $\displaystyle u_{-}$
$\displaystyle=G_{-}v:=\frac{1}{\langle\chi_{-}e_{-},e_{-}\rangle}\langle
v,e_{-}\rangle=:\frac{1}{D_{-}}\langle v,e_{-}\rangle,$ (2.21)
où
$D_{-}=1+\mathcal{O}(e^{-\frac{1}{Ch}}),$
avec, en tant qu’opérateur $L^{2}_{comp}(I_{-})\to H_{sc,comp}(I_{-}):$
$\|G\|_{L^{2}\to H_{sc}}\leq\frac{C}{\sqrt{h}}.$
Preuve. Pour $x\leq x_{-},$ nous avons l’unique solution de
$P_{\alpha}u=\tilde{v},$ $\tilde{v}\in L^{2}_{comp}(I_{-}),$ qui est nulle
près de $\inf I_{-}$
$u_{1}(x)=\frac{i}{h}\int_{-\infty}^{x}e^{-\frac{i}{h}\int^{x}_{y}\widetilde{g}(\tilde{x},\alpha;h)d\tilde{x}}\tilde{v}(y)dy:=\widetilde{G}_{1}\tilde{v}(x),$
alors que pour $x\geq x_{-},$ nous avons
$u_{2}(x)=\frac{i}{h}\int_{+\infty}^{x}e^{-\frac{i}{h}\int^{x}_{y}\widetilde{g}(\tilde{x},\alpha;h)d\tilde{x}}\tilde{v}(y)dy:=\widetilde{G}_{2}\tilde{v}(x),$
les deux exposants étant décroissants (à partie réelle strictement négative
loin de $x_{-}$) dans le domaine d’intégration. Afin d’obtenir une solution
continue, il faut imposer
$\displaystyle 0=u_{1}(x_{-})-u_{2}(x_{-})$
$\displaystyle=\frac{i}{h}\int^{+\infty}_{-\infty}e^{-\frac{i}{h}\int^{x_{-}}_{y}\widetilde{g}(\tilde{x},\alpha;h)d\tilde{x}}\tilde{v}(y)dy,$
$\displaystyle=\frac{i}{h\overline{c_{-}}}\langle\tilde{v},e_{-}\rangle.$
Donc, avec
$u_{-}=G_{-}v,$
nous pouvons prendre $\tilde{v}=v-R_{-}u_{-},$ car par construction
$\langle v-R_{-}u_{-},e_{-}\rangle=0.$
Nous avons alors la solution
$\displaystyle u$ $\displaystyle=Gv=\widetilde{G}(I-R_{-}G_{-})v,$
$\displaystyle u_{-}$ $\displaystyle=G_{-}v,$
où $\widetilde{G}$ est donné par $\widetilde{G}_{1,2}$ dans les zones
correspondantes. En observant que $\widetilde{G}$ a le noyau intégral
$k(x,y)=\frac{i}{h}e^{-\frac{i}{h}\int^{x}_{y}\widetilde{g}(\tilde{x},\alpha;h)\,d\tilde{x}}1_{\\{y\leq
x\leq
x_{-}\\}}-\frac{i}{h}e^{-\frac{i}{h}\int^{x}_{y}\widetilde{g}(\tilde{x},\alpha;h)\,d\tilde{y}}1_{\\{x_{-}\leq
x\leq y\\}},$
\- qui est semblable au noyau intégral de $\widetilde{F}$\- et que
$\mathrm{Im}\,g^{\prime}_{x}(x_{+})\sim-\mathrm{Im}\,g^{\prime}_{x}(x_{-}),$
nous pouvons utiliser les estimations du paragraphe précédent pour trouver que
$\|\widetilde{G}\|_{L^{2}(I_{-})\to L^{2}(I_{-})}\leq\frac{C}{\sqrt{h}},$
ainsi que
$\|G\|_{L^{2}(I_{-})\to H_{sc}(I_{-})}\leq\frac{C}{\sqrt{h}}.$
$\square$
#### Problème de Grushin global.
Nous choisissons une partition de l’unité de $\mathbb{R},$ $\psi_{\pm}\in
C^{\infty}(I_{\pm}),$ $\psi_{+}+\psi_{-}=1,$ telle que
$\chi_{\pm}\prec\psi_{\pm},$ où $\psi\prec\phi$ signifie
$\displaystyle\mbox{supp}\,(\psi)\cap\mbox{supp}(1-\phi)=\emptyset.$ (2.22)
Comme les $\chi_{\pm},$ les $\psi_{\pm}$ s’annulent près du bord de
l’intervalle $I_{\pm}$. On suit Hager [7] “Inverse global” : on commence par
résoudre le problème sur $I_{+}$
$\left\\{\begin{array}[]{cc}(hD_{x}+\widetilde{g}(x,\alpha;h))u=\psi_{+}v\\\
R_{+}u=v_{+}.\end{array}\right.$
Sur $I_{+},$ on introduit
$u_{1}:=(1-\chi_{-})F\psi_{+}v+(1-\chi_{-})F_{+}v_{+}.$
Donc, en utilisant $\chi_{+}\prec(1-\chi_{-}),$ et $\psi_{+}\prec(1-\chi_{-})$
(car $\chi_{-}\prec\psi_{-})$
$\displaystyle R_{+}u_{1}=\langle u_{1},\chi_{+}e_{+}\rangle=v_{+},$
$\displaystyle
P_{\alpha}u_{1}=\psi_{+}v-[P_{\alpha},\chi_{-}]F\psi_{+}v-[P_{\alpha},\chi_{-}]F_{+}v_{+},$
en rappelant que $P_{\alpha}:=hD_{x}+\widetilde{g}(x,\alpha;h).$
Comme chez Hager [7], on peut “corriger l’erreur” sur $I_{-}$ en y résolvant
le problème
$(hD_{x}+\widetilde{g}(x,\alpha;h))u_{2}+R_{-}u_{-}=\psi_{-}v+[P_{\alpha},\chi_{-}]F\psi_{+}v+[P_{\alpha},\chi_{-}]F_{+}v_{+}.$
Nous avons la solution
$\displaystyle
u_{2}=G(\psi_{-}v+[P_{\alpha},\chi_{-}]F\psi_{+}v-[P_{\alpha},\chi_{-}]F_{+}v_{+})$
$\displaystyle
u_{-}=G_{-}(\psi_{-}v+[P_{\alpha},\chi_{-}]F\psi_{+}v-[P_{\alpha},\chi_{-}]F_{+}v_{+}).$
Et $R_{+}u_{2}=0$ car $\mbox{supp}(u_{2})\cap\mbox{supp}(\chi_{+})=\emptyset.$
Finalement nous obtenons :
###### Théorème 2.8
Pour tout $\alpha\in\Omega,$
$\mathcal{P}=\left(\begin{array}[]{cc}hD_{x}+\widetilde{g}(x,\alpha;h)&R_{-}\\\
R_{+}&0\end{array}\right):H_{sc}\times\mathbb{C}\to L^{2}\times\mathbb{C},$
est inversible d’inverse
$\mathcal{E}=\left(\begin{array}[]{cc}E&E_{+}\\\
E_{-}&E_{-+}\end{array}\right)$
où
$\displaystyle
E=G\left(\psi_{-}+\frac{h}{i}\chi^{\prime}_{-}F\psi_{+}\right)+(1-\chi_{-})F\psi_{+}$
(2.23) $\displaystyle
E_{+}=(1-\chi_{-})F_{+}+G\frac{h}{i}\chi^{\prime}_{-}F_{+}$ $\displaystyle
E_{-}=G_{-}(\psi_{-}+\frac{h}{i}\chi^{\prime}_{-}F\psi_{+})$ $\displaystyle
E_{-+}=G_{-}\frac{h}{i}\chi^{\prime}_{-}F_{+}=-\frac{h}{iD_{-}D_{+}}\langle\chi^{\prime}_{-}e_{+},e_{-}\rangle.$
Les normes vérifient
$\displaystyle\|E\|_{L^{2}\to
H^{1}_{sc}}=\mathcal{O}(\frac{1}{\sqrt{h}}),\,\|E_{+}\|=\mathcal{O}(1),\,\|E_{-}\|=\mathcal{O}(1),$
(2.24)
$\displaystyle\|E_{-+}\|=\mathcal{O}\left(\sqrt{h}\,e^{-\frac{1}{Ch}}\right).$
(2.25)
Les opérateurs $F,F_{+},G,G_{-}$ ont été définis au-dessus. De plus $E_{+}$ et
$E_{-}$ sont des opérateurs de rang 1, satisfaisant
$\displaystyle E_{+}$
$\displaystyle=\frac{1-\chi_{-}}{D_{+}}e_{+}+\mathcal{O}(h^{\infty}),$ (2.26)
$\displaystyle E_{-}$
$\displaystyle=\langle\bullet,\frac{\psi_{-}}{D_{-}}e_{-}\rangle+\mathcal{O}(h^{\infty}).$
(2.27)
On peut déduire une formule exacte pour $E_{-+},$
$E_{-+}=\frac{hc_{+}\overline{c_{-}}}{iD_{+}D_{-}}\exp{(\frac{i}{h}\int_{x_{-}}^{x_{+}}\widetilde{g}(\tilde{x},\alpha;h)d\tilde{x}}).$
(2.28)
Finalement, en utilisant la relation
$(hD_{x}+\widetilde{g})^{-1}=E-E_{+}(E_{-+})^{-1}E_{-}$, on trouve
l’estimation suivante pour la résolvante:
###### Proposition 2.9
On rappelle que $1/\mathcal{O}(1)\geq\alpha\leq\mathcal{O}(1)$ appartient à un
ouvert $\Omega$, relativement compact et séparé de l’origine par une constante
indépendante de h. Pour tout $\alpha$ dans $\Omega,$ la résolvante vérifie
pour $h$ assez petit
$\displaystyle\|(hD_{x}+\widetilde{g}(x,\alpha;h))^{-1}\|\sim$
$\displaystyle\frac{\sqrt{\pi}e^{\frac{1}{h}\mathrm{Im}\,\ell_{0}(\alpha)}}{h^{1/2}(\mathrm{Im}\,g^{\prime}_{x}(x_{+},\alpha))^{\frac{1}{4}}(\mathrm{Im}\,\bar{g}^{\prime}_{x}(x_{-},\alpha))^{\frac{1}{4}}}(1+\mathcal{O}(h))$
$\displaystyle+\mathcal{O}(\frac{1}{\sqrt{h}}),$ (2.29)
où $\ell_{0}$ vérifie
$\ell_{0}:=-\mathrm{Im}\,\int_{x_{+}}^{x_{-}}g(y,\alpha)dy.$ (2.30)
Preuve. Il suffit de remarquer que $E_{-+}$ s’écrit
$\displaystyle|E_{-+}(\alpha)|\sim\frac{h^{\frac{1}{2}}}{\sqrt{\pi}}\big{(}\mathrm{Im}\,$
$\displaystyle
g^{\prime}_{x}(x_{+},\alpha)\,\mathrm{Im}\,\bar{g}^{\prime}_{x}(x_{-},\alpha)\big{)}^{\frac{1}{4}}$
$\displaystyle\exp{(-\frac{1}{h}\mathrm{Im}\,\int^{x_{+}}_{x_{-}}g(y,\alpha)dy)}\;(1+\mathcal{O}(h)),$
et que
$\|E_{-}E_{+}\|=\|\frac{1-\chi_{-}}{D_{+}}e_{+}\|\,\|\frac{\psi_{-}}{D_{-}}e_{-}\|\sim
1+\mathcal{O}(h^{\infty}).$
$\square$
#### L’opérateur d’Airy complexe sur $\mathbb{R}.$
Considérons l’opérateur $\mathcal{A}=D^{2}_{x}+ix$ sur la droite réelle, muni
du domaine $D(\mathcal{A})=\\{u\in H^{2}(\mathbb{R})|\;xu\in
L^{2}(\mathbb{R})\\}.$ Nous pouvons montrer que pour chaque $z\in\mathbb{C},$
la résolvante satisfait (voir Helffer [11])
$\|(\mathcal{A}-z)^{-1}\|=\|(\mathcal{A}-\mathrm{Re}\,z)^{-1}\|.$
###### Proposition 2.10 (J. Martinet)
Pour $\mathrm{Re}\,z>0$ assez grand, il existe deux constantes positives
$C_{0}$ et $C_{1}$ telles que
$C_{0}|\mathrm{Re}\,z|^{-\frac{1}{4}}\exp\frac{4}{3}(\mathrm{Re}\,z)^{\frac{3}{2}}\leq\|(\mathcal{A}-z)^{-1}\|\leq
C_{1}|\mathrm{Re}\,z|^{-\frac{1}{4}}\exp\frac{4}{3}(\mathrm{Re}\,z)^{\frac{3}{2}}.$
nous voyons que
$\mathcal{F}(D_{x}^{2}+ix-z)\mathcal{F}^{-1}=i(-D_{x}-ix^{2}+iz).$
Après conjugaison par l’opérateur unitaire $u(x)\mapsto u(-x),$ nous notons
$(\mathcal{A}_{0}+iz):=D_{x}-ix^{2}+iz$ ce nouvel opérateur. Nous avons
$\|(\mathcal{A}_{0}-iz)^{-1}\|=\|(\mathcal{A}_{0}-i\mathrm{Re}\,z)^{-1}\|.$ Le
changement de variable $x=(\mathrm{Re}\,z)^{\frac{1}{2}}y,$ nous permet
d’identifier $\mathcal{A}_{0}+iz$ avec $|\mathrm{Re}\,z|\mathcal{Q},$ où
$\mathcal{Q}$ est donné par ($y$ a été remplacé par $x$)
$\frac{1}{(\mathrm{Re}\,z)^{\frac{3}{2}}}D_{x}-ix^{2}+i.$
Nous nous retrouvons alors dans la situation précédente avec
$\mathrm{Im}\,g(x,\alpha)\leq 0,$ plus explicitement $\mathcal{Q}$ est de la
forme $hD_{x}+\widetilde{g}$ avec $\widetilde{g}=g.$ En reprenant les
notations précédentes, nous avons
$\displaystyle h=\frac{1}{(\mathrm{Re}\,z)^{\frac{3}{2}}},$ (2.31)
$\displaystyle g(x,z):=-ix^{2}+i,$ (2.32) $\displaystyle x_{\pm}(z)=\pm
1\mbox{ v\'{e}rifiant }\mathrm{Im}\,g(x_{\pm},z)=0,$ (2.33)
$\displaystyle-\mathrm{Im}\,g^{\prime}_{x}(x_{+}(z),z)=2x_{+}(z)=2,$ (2.34)
$\displaystyle+\mathrm{Im}\,g^{\prime}_{x}(x_{-}(z),z)=-2x_{-}(z)=2.$ (2.35)
De plus
$\displaystyle-\frac{1}{h}\mathrm{Im}\,\int_{x_{+}}^{x_{-}}g(x,z)dx=-\frac{1}{h}\mathrm{Im}\,\int_{1}^{-1}g(x,z)dx$
$\displaystyle=-(-2+\frac{2}{3})(\mathrm{Re}\,z)^{3/2}$
$\displaystyle=\frac{4}{3}(\mathrm{Re}\,z)^{3/2},$ (2.36)
et
$h^{1/2}\big{(}\mathrm{Im}\,g^{\prime}_{x}(x_{+},z)\,\mathrm{Im}\,\bar{g}^{\prime}_{x}(x_{-},z)\big{)}^{\frac{1}{4}}=h^{1/2}(2x_{+}(z))^{1/2}=2^{1/2}(\mathrm{Re}\,z)^{-\frac{3}{4}}.$
On déduit alors le corollaire 1.4.
### 2.3 Estimation de résolvante près du bord de l’image du symbole.
On suppose cette fois ci que $\alpha>0$ tend vers zéro pour le cas modèle à
paramètre $hD_{x}+\widetilde{g}(x,\alpha;h)$ sur la ligne droite. On va se
ramener au cas précédent par “scaling”. Rappelons que (lemme 2.1)
$x_{\pm}(\alpha)=\mp\alpha^{1/2}\left(\frac{-2}{\\{p_{0},\frac{1}{2i}\\{p_{0},\overline{p_{0}}\\}\\}(0,0)}\right)^{1/2}+\mathcal{O}(\alpha).$
(2.37)
Le changement de variable $x=(\alpha)^{1/2}y,$ nous permet d’identifier notre
opérateur avec $\alpha Q$ où $Q$ est donné par
$\displaystyle\frac{h}{\alpha^{1/2}\alpha}D_{y}+\frac{g(\alpha^{1/2}y,\alpha)}{\alpha}+\frac{h^{2}}{\alpha^{3}}(\alpha^{3}g_{\ast}(\alpha^{1/2}y,\alpha;h)=$
$\displaystyle\tilde{h}D_{y}+\frac{g(\alpha^{1/2}y,\alpha)}{\alpha}+\tilde{h}^{2}(\alpha^{3}g_{\ast}(\alpha^{1/2}y,\alpha;h))\mbox{
o\\`{u} }\tilde{h}=h/\alpha^{3/2}.$ (2.38)
On a ainsi remplacé les points $x_{+}(\alpha)$ et $x_{-}(\alpha),$ par des
points $y_{+}(\alpha)=\alpha^{-\frac{1}{2}}x_{+}$ et
$y_{-}(\alpha)=\alpha^{-\frac{1}{2}}x_{-},$ où $y_{\pm}\asymp\mp 1.$ Les
points $y_{\pm}$ vérifient $\mathrm{Im}\,f(y_{\pm},\alpha)=0,$ où $f$ est
défini par $f(x,\alpha):=\frac{g(\alpha^{1/2}x,\alpha)}{\alpha}.$ Nous avons
$\mathrm{Im}\,f^{\prime\prime}_{xx}(0,0)>0,$ et
$\mathrm{Im}\,f(x,\alpha)\asymp(x^{2}-1),$ Nous pouvons alors appliquer la
proposition 2.9 avec $\tilde{h}:=h/\alpha^{3/2}$ comme notre nouveau petit
paramètre semiclassique.
###### Proposition 2.11
Pour tout $\alpha,$ la résolvante vérifie pour $h\ll\alpha^{3/2}$ et $\alpha$
assez petit
$\displaystyle\|(hD_{y}+\widetilde{g}(y,\alpha;h)^{-1}\|\sim$
$\displaystyle\frac{\sqrt{\pi}\exp(\frac{1}{\tilde{h}}\mathrm{Im}\,\widetilde{\ell}(\alpha))}{\alpha\tilde{h}^{1/2}(\mathrm{Im}\,f^{\prime}_{y}(y_{+},\alpha))^{1/4}(\mathrm{Im}\,\overline{f^{\prime}_{y}(y_{-},\alpha)})^{1/4}}(1+\mathcal{O}(\tilde{h}))$
$\displaystyle+\mathcal{O}(\frac{1}{\alpha\sqrt{\tilde{h}}}),$ (2.39)
où $\tilde{h}=h/\alpha^{3/2},$ et $\widetilde{\ell}$ vérifie
$\widetilde{\ell}:=\int_{y_{-}}^{y_{+}}\frac{g(\alpha^{1/2}\tilde{y},\alpha)}{\alpha}d\tilde{y},\quad\mathrm{Im}\,f(y_{\pm},\alpha)=0.$
(2.40)
Rappelons que $f=\frac{g(\alpha^{1/2}y,\alpha)}{\alpha}.$
Ensuite, nous avons les égalités suivantes,
$\displaystyle\frac{1}{\tilde{h}}\widetilde{\ell}$
$\displaystyle=\frac{\alpha^{3/2}}{h}\int_{y_{-}}^{y_{+}}\frac{g(\alpha^{1/2}\tilde{y},\alpha)}{\alpha}d\tilde{y}=\frac{\alpha^{3/2}}{h}\int_{\alpha^{1/2}y_{-}}^{\alpha^{1/2}y_{+}}\frac{g(\tilde{y},\alpha)}{\alpha\alpha^{1/2}}d\tilde{y}$
$\displaystyle=\frac{1}{h}\int_{x_{-}}^{x_{+}}g(\tilde{y},\alpha)d\tilde{y}=:\frac{1}{h}\ell_{0}(\alpha),$
et,
$\displaystyle\alpha\tilde{h}^{1/2}$
$\displaystyle(-\mathrm{Im}\,f^{\prime}_{y}(y_{+},\alpha))^{1/4}\mathrm{Im}\,(f^{\prime}_{y}(y_{-},\alpha))^{1/4}$
$\displaystyle=\alpha^{1/4}h^{1/2}\frac{(\mathrm{Im}\,g^{\prime}_{x}(\alpha^{1/2}y_{+},\alpha))^{\frac{1}{4}}}{\alpha^{1/8}}\frac{\mathrm{Im}\,\overline{({g}^{\prime}_{x}(\alpha^{1/2}y_{-},\alpha)})^{\frac{1}{4}}}{\alpha^{1/8}}$
$\displaystyle=h^{1/2}(\mathrm{Im}\,g^{\prime}_{x}(x_{+},\alpha))^{\frac{1}{4}}(\mathrm{Im}\,\overline{{g}^{\prime}_{x}(x_{-},\alpha)})^{\frac{1}{4}}$
$\displaystyle=h^{1/2}(\frac{1}{2i}\\{p_{\alpha},\overline{p_{\alpha}}\\}(\rho_{+}))^{\frac{1}{4}}(\frac{1}{2i}\\{\overline{p_{\alpha}},p_{\alpha}\\}(\rho_{-}))^{\frac{1}{4}}.$
Nous pouvons ainsi donner une formule invariante pour la résolvante.
###### Corollaire 2.12
Soit $g$ satisfaisant (1.3) et (1.4). Pour tout $\alpha,$ la résolvante
vérifie pour $h\ll\alpha^{3/2}$ avec $\alpha$ assez petit
$\displaystyle\|(hD_{x}+\widetilde{g}(x,\alpha;h))^{-1}\|\sim\frac{\sqrt{\pi}\exp(\frac{1}{h}\mathrm{Im}\,\ell_{0}(\alpha))}{h^{1/2}(\frac{1}{2i}\\{p_{\alpha},\overline{p_{\alpha}}\\}(\rho_{+}))^{\frac{1}{4}}(\frac{1}{2i}\\{\overline{p_{\alpha}},p_{\alpha}\\}(\rho_{-}))^{\frac{1}{4}}}(1+\mathcal{O}(\tilde{h}))$
$\displaystyle+\mathcal{O}(\frac{1}{\sqrt{h}\,\alpha^{1/4}}),$ (2.41)
où $\tilde{h}=h/\alpha^{3/2},$ et $\ell_{0}$ vérifie
$\ell_{0}:=\mathrm{Im}\,\int_{x_{-}}^{x_{+}}g(y,\alpha)dy,\quad\mathrm{Im}\,g(x_{\pm},\alpha)=0.$
(2.42)
###### Remarque 2.13
Puisque $|\alpha_{\pm}(\alpha)|\asymp\alpha^{\frac{1}{2}},$ nous avons les
estimations suivantes :
$\displaystyle\ell_{0}(\alpha)\asymp\alpha\alpha^{\frac{1}{2}}=\alpha^{3/2},$
$\displaystyle\frac{1}{2i}\\{p_{\alpha},\overline{p_{\alpha}}\\}(\rho_{+})=-\mathrm{Im}\,g^{\prime}_{x}(x_{+},\alpha)\asymp\alpha^{1/2},$
$\displaystyle\frac{1}{2i}\\{\overline{p_{\alpha}},p_{\alpha}\\}(\rho_{-})=-\mathrm{Im}\,\bar{g}^{\prime}_{x}(x_{-},\alpha)\asymp\alpha^{1/2}.$
Donc, sous la condition $h\ll\alpha^{3/2}\ll 1$ assez petit, nous avons
$\|(hD_{x}+\widetilde{g}(x,\alpha;h))^{-1}\|\leq\frac{C}{h^{1/2}\alpha^{1/4}}e^{\frac{C\alpha^{3/2}}{h}}.$
Nous retrouvons alors la formule usuelle donnée dans [1] et [16].
Résumons le cas où $\alpha$ tend vers zéro, le théorème 2.8 traite en effet du
point $\alpha\geq\frac{1}{\mathcal{O}(1)}$ :
###### Théorème 2.14
Rappelons que
$(hD_{x}-\widetilde{g}(x,\alpha;h))^{-1}=E-E_{+}E_{-+}^{-1}E_{-},$ avec
$\displaystyle|E_{-+}^{-1}|\sim\frac{\sqrt{\pi}\exp(\frac{1}{h}\mathrm{Im}\,\ell_{0}(\alpha))}{h^{1/2}(\frac{1}{2i}\\{\xi+g,\xi+\bar{g}\\}(\rho_{+}))^{\frac{1}{4}}(\frac{1}{2i}\\{\xi+\bar{g},\xi+g\\}(\rho_{-}))^{\frac{1}{4}}}\times(1+\mathcal{O}(\tilde{h}))$
(2.43)
$\tilde{h},$ $\ell_{0}$ seront rappelés par la suite, et
$E=\mathcal{O}(\frac{1}{\sqrt{h}\,\alpha^{1/4}}).$ $E_{+}$ et $E_{+}$ sont des
opérateurs de rang 1, satisfaisant
$\displaystyle E_{+}$
$\displaystyle=\frac{1-\chi_{-}}{D_{+}}e_{+}+\mathcal{O}(h^{\infty}),$
$\displaystyle E_{-}$
$\displaystyle=\frac{1}{D_{-}}\langle\bullet,\psi_{-}e_{-}\rangle+\mathcal{O}(h^{\infty}),$
où $D_{\pm}=1+\mathcal{O}(h^{\infty}),$ et $e_{+},$ $e_{-}$ sont normalisés,
et vérifient
$e_{+}(x,\alpha;h)=\frac{(\frac{1}{2i}\\{\xi+g,\xi+\bar{g}\\}(\rho_{+}))^{\frac{1}{4}}}{(\pi
h)^{1/4}}(1+\mathcal{O}(\tilde{h}))e^{-\frac{i}{h}\int_{x_{+}}^{x}g(y,\alpha)dy},$
où $(\xi+g)(\rho_{+})=0$ avec
$(\frac{1}{2i}\\{\xi+g,\xi+\bar{g}\\}(\rho_{+}))>0.$ Nous avons une estimation
similaire pour $e_{-}.$ Donc
$\displaystyle\|(hD_{x}-\widetilde{g}(x,\alpha;h))^{-1}\|\sim$
$\displaystyle\frac{\sqrt{\pi}\exp(\frac{1}{h}\mathrm{Im}\,\ell_{0}(\alpha))}{h^{1/2}(\frac{1}{2i}\\{\xi+g,\xi+\bar{g}\\}(\rho_{+}))^{\frac{1}{4}}(\frac{1}{2i}\\{\xi+\bar{g},\xi+g\\}(\rho_{-}))^{\frac{1}{4}}}$
$\displaystyle\times(1+\mathcal{O}(\tilde{h}))+\mathcal{O}(\frac{1}{\sqrt{h}\,\alpha^{1/4}}),$
où $\tilde{h}:=h/\alpha^{3/2},$ et $\ell_{0}$ (est une intégrale d’action)
vérifie
$\ell_{0}:=\mathrm{Im}\,\int_{x_{-}}^{x_{+}}g(x,\alpha)dx.$ (2.44)
## 3 Cas général en dimension 1
Soit $p\in S(\mathbb{R}^{2},m),$ indépendant de $h.$ On choisit un point
$z_{0}$ appartenant au bord de l’image du symbole
$\Sigma(p)=p(T^{\ast}\mathbb{R}),$ qui est fermé grâce à la condition
d’ellipticité sur $p$ et la croissance à l’infini de la fonction d’ordre $m$.
Nous supposons que $p$ est de type principal en $z_{0},$ c’est-à-dire
$p(\rho)-z_{0}=0\Rightarrow dp(\rho)\neq 0,$
et que $p$ est d’ordre 2 en $z_{0}.$
Pour simplifier, nous supposons que $p^{-1}(z_{0})$ ne contient qu’un point
$\rho_{0}$.
En nous appuyant sur Hager ([9]): pour $z_{0}\in\partial\Sigma$ nous avons
$p^{-1}(z_{0})=\\{\rho_{0}\\},$ avec $\\{p,\bar{p}\\}(\rho_{0})=0.$ Soit
$H_{\frac{1}{2i}\\{p,\bar{p}\\}}$ le vecteur réel tangent à l’ensemble
$\\{\rho|\,\frac{1}{2i}\\{p,\bar{p}\\}(\rho)=0\\}.$ Donc
$(-T_{1},T_{0})\ni
s\mapsto\rho(s):=\exp(sH_{\frac{1}{2i}\\{p,\bar{p}\\}})(\rho_{0})\subset\\{\frac{1}{2i}\\{p,\bar{p}\\}=0\\}$
(3.1)
est une courbe orientée, et $\frac{1}{2i}\\{p,\bar{p}\\}$ est positif à la
gauche de cette courbe, et négatif à sa droite. Concernant la ligne (3.1),
nous avons évidemment $\rho(0)=\rho_{0}$, et le sous-ensemble de $\mathbb{C},$
$p\left(\\{\frac{1}{2i}\\{p,\bar{p}\\}=0\\}\right),$ coïncide avec
$\partial\Sigma$ près de $z_{0}$.
$p(\rho(s))$ est aussi une courbe orientée, puisque
$\frac{\partial}{\partial
s}(p(\rho(s)))=(H_{\frac{1}{2i}\\{p,\bar{p}\\}}p)(\rho(s))\\\
=-\\{p,\frac{1}{2i}\\{p,\bar{p}\\}\\}(\rho(s))\neq 0.$
De plus $d\mathrm{Im}\,p\wedge
d\mathrm{Re}\,p=\frac{1}{2i}\\{p,\bar{p}\\}d\xi\wedge dx,$ donc $\rho\to
p(\rho)$ préserve l’orientation à gauche de $\rho(s),$ et la renverse à
droite. Ce qui veut dire, qu’au voisinage de $z_{0},$ si $\rho(s)$ est
parcouru dans le sens positif, on voit que $\Sigma$ se trouve à gauche de la
courbe $p(\rho(s))\in\partial\Sigma.$
On note le chemin $p(\rho(s))$ par $\gamma:]-T_{1},T_{0}[\to\mathbb{C}.$ Un
vecteur tangent à $\partial\Sigma$ est donné par $\dot{\gamma}(s),$ et nous
savons que
$\dot{\gamma}(s)=-\\{p,\frac{1}{2i}\\{p,\bar{p}\\}\\}(\rho(s)).$ (3.2)
Il existe un voisinage de $z_{0}$ où tout point $z$ appartenant à $\Sigma(p)$
s’écrit sous la forme
$z=\gamma(s)+\alpha n(s),\quad 0\leq\alpha<\frac{1}{C_{0}},\quad-
T_{1}<s<T_{0},\quad z_{0}=\gamma(0),\\\ $ (3.3)
pour une constante $C_{0}>0$ donnée, où $n(s)$ est le vecteur unitaire normal
à la courbe $\gamma(s),$ orienté vers l’intérieur de $\Sigma(p).$ $n(s)$
vérifie les relations $n(s)=i\frac{\dot{\gamma}(s)}{|\dot{\gamma}(s)|},$ et
$|n(s)|=1.$ Par la suite le vecteur unitaire tangent à $\gamma(s)$ sera noté
$u(s),$ $u(s)=\frac{\dot{\gamma}(s)}{|\dot{\gamma}(s)|}$ et $|u(s)|=1.$
Puisque $p$ est de type fini d’ordre 2 au point $z_{0}$, nous pouvons alors
supposer sans perte de généralité que $p^{\prime}_{\xi}(\rho_{0})\neq 0$ et
que $p^{\prime}_{x}(\rho_{0})=0.$ Nous avons dans de nouvelles coordonnées
$(x,\xi)$ centrées en $\rho(s)$ \- changement de coordonnées symplectiques
dépendant explicitement de $s$, conjugaison par un opérateur intégral de
Fourier de plus le gradient de $p$ en $(0,0)$ est un multiple d’un vecteur
réel -
$\displaystyle p(\rho,s)-\gamma(s)$ $\displaystyle=p(\rho,s)-p(\rho(s))$
$\displaystyle=u(s)(\frac{p^{\prime}_{\xi}(0,0,s)}{u(s)}\xi+\omega(x,\xi,s)),\mbox{
avec }\omega=\mathcal{O}(x^{2}+\xi^{2}),$
où $\omega$ est $C^{\infty}$ en $s.$ N’oublions pas que $p$ va dépendre de $s$
car les nouvelles coordonnées dépendent de $s$. Aussi
$p^{\prime}_{\xi}(0,0,s)/u(s)$ est réel. Puis, grâce au théorème de
factorisation de Malgrange pour des fonctions $C^{\infty}$, il existe un
ouvert $V$ de $\mathbb{R}^{2}\times\mathbb{R}_{+}\times]-T_{1},T_{0}[$
contenant le point $(0,0,0,0)$ tel que dans $V,$ nous avons la factorisation
$\displaystyle p(\rho,s)-z$ $\displaystyle=p(\rho,s)-\gamma(s)-i\alpha u(s)$
$\displaystyle=u(s)(\frac{p^{\prime}_{\xi}(0,0,s)}{u(s)}\xi+\omega(x,\xi,s)-i\alpha)$
$\displaystyle=u(s)q(x,\xi,\alpha,s)\,(\xi+g(x,\alpha,s)),\quad
q(x,\xi,\alpha,s)\neq 0,$ (3.4)
où $g$ et $q$ sont des fonctions $C^{\infty},$ $g$ s’annulant au point
$(0,0,0).$ De plus pour $x=\xi=0,$ $\alpha=0$ et $s$ non nul, nous avons
$p(0,s)-z=p(0)-\gamma(s)=0,$ impliquant $g(0,0,s)=0.$ Par commodité, nous
supprimons la variable $s$. Pour $x=\xi=0,$ $\alpha=0,$ nous avons alors
$\displaystyle\quad g(0,0)=0,$ (3.5) $\displaystyle\quad
q(0,0,0)=\frac{p^{\prime}_{\xi}(0,0)}{u(s)},$ (3.6) $\displaystyle\quad
g^{\prime}_{\alpha}(0,0)=-i\frac{u(s)}{p^{\prime}_{\xi}(0,0)},$ (3.7)
$\displaystyle\quad g^{\prime}_{x}(0,0)=0.$ (3.8)
Comme $p^{\prime}_{x}(\rho_{0})$ est nul, le rapport
$\frac{p^{\prime}_{\xi}(0,0)}{u(s)}$ est réel et non nul, et sera noté $a$;
$a:=\frac{p^{\prime}_{\xi}(0,0)}{u(s)}=q(0,0,0).$ Puisque
$\omega^{\prime\prime}_{xx}(0,0)=q^{\prime\prime}_{xx}(0,0,0)\,.\,0+2\,q^{\prime}_{x}(0,0,0)\,.\,0+q(0,0,0)g^{\prime\prime}_{xx}(0,0),$
nous avons aussi,
$ag^{\prime\prime}_{xx}(0,0)=\omega^{\prime\prime}_{xx}(0,0).$ (3.9)
Nous avons alors - rappelons que c’est pour $\alpha=0$ -
$\displaystyle\frac{1}{2i}\\{p,\bar{p}\\}(x,\xi)$
$\displaystyle=\frac{|u(s)|^{2}}{2i}\\{a\xi+\omega,a\xi+\bar{\omega}\\}(x,\xi)$
$\displaystyle=\frac{1}{2i}(a\overline{\omega^{\prime}_{x}}-a\omega^{\prime}_{x})$
$\displaystyle=-a\mathrm{Im}\,\omega^{\prime}_{x}(x,\xi),$
ce qui implique
$\displaystyle\\{p,\frac{1}{2i}\\{p,\bar{p}\\}\\}(0,0)$
$\displaystyle=u(s)\\{a\xi+\omega,-\mathrm{Im}\,a\omega^{\prime}_{x}\\}(0,0)$
$\displaystyle=-u(s)a^{2}\mathrm{Im}\,\omega^{\prime\prime}_{xx}(0,0)$
$\displaystyle=-u(s)a^{2}a\mathrm{Im}\,g^{\prime\prime}_{xx}(0,0).$ (3.10)
On déduit alors que
$\mathrm{Im}\,g^{\prime\prime}_{xx}(0,0)\neq 0,$ (3.11)
et que
$\displaystyle\mathrm{Im}\,g^{\prime\prime}_{xx}(0,0)=\frac{\dot{\gamma}(s)}{u(s)a^{2}a}$
$\displaystyle=\frac{\dot{\gamma}(s)^{3}}{|\dot{\gamma}(s)|^{3}}\frac{|\dot{\gamma}(s)|}{p^{\prime}_{\xi}(0,0)^{3}}$
$\displaystyle=\frac{\\{p,\frac{1}{2i}\\{p,\bar{p}\\}\\}(0,0)^{2}}{|\\{p,\frac{1}{2i}\\{p,\bar{p}\\}\\}(0,0)|^{2}}\frac{\\{p,\frac{1}{2i}\\{p,\bar{p}\\}\\}(0,0)}{p^{\prime}_{\xi}(0,0)^{3}}.$
(3.12)
De l’équation (2.3) qui donne le comportement de $x_{\pm}(\alpha)$ dans le cas
modèle $hD_{x}+g(x,\alpha)$ avec l’hypothèse que
$g^{\prime}_{\alpha}(0,0)=-i$, nous obtenons qu’il existe un voisinage de
$z_{0}$ pour lequel nous avons pour tout complexe $z$ appartenant à
$\Sigma(p)$
$z\notin\partial\Sigma(p),\;p^{-1}(z)=\\{\rho_{+}(z),\rho_{-}(z)\\},\quad\rho:=(x,\xi),$
(3.13)
où
$x_{\pm}(z)=x_{0}\mp\epsilon\\-(\alpha|g^{\prime}_{\alpha}(0,0,s)|)^{1/2}\left(\frac{2}{|\mathrm{Im}\,g^{\prime\prime}_{xx}(0,0,s)|}\right)^{1/2}+\mathcal{O}(\alpha),$
$\epsilon$ est le signe de
$g^{\prime}_{\alpha}(0,0,s)\mathrm{Im}\,g^{\prime\prime}_{xx}(0,0,s)$. En
remplaçant $g^{\prime}_{\alpha}(0,0,s)$ et
$\mathrm{Im}\,g^{\prime\prime}_{xx}(0,0,s)$ par leur expression respective
(3.7) et (3.12), nous obtenons
$\displaystyle x_{\pm}(z)$
$\displaystyle=x_{0}\mp\epsilon\,\alpha^{1/2}|p^{\prime}_{\xi}(\rho(s))|\left(\frac{2}{|\\{p,\frac{1}{2i}\\{p,\bar{p}\\}\\}(\rho(s))|}\right)^{1/2}+\mathcal{O}(\alpha),$
(3.14) $\displaystyle z$ $\displaystyle=\gamma(s)+\alpha n(s),\alpha>0,$
où $\epsilon$ est le signe de
$\\{p,\frac{1}{2i}\\{p,\bar{p}\\}\\}(\rho(s))/p^{\prime}_{\xi}(\rho(s))$. Nous
avons que
$\pm\frac{1}{2i}\\{p,\bar{p}\\}(\rho_{\pm})>0.$ (3.15)
On prolonge $g$ sur $\mathbb{R}$ de la même manière que dans le cas modèle à
paramètre.
On distingue, maintenant, deux cas :
* $\bullet$
1er cas, $p^{-1}(z_{0})$ ne contient qu’un point $\\{\rho_{0}\\}$,
* $\bullet$
2ème cas, $p^{-1}(z_{0})$ ne contient que deux points
$\\{\rho_{1},\rho_{2}\\}$.
### 3.1 Estimation de résolvante: cas 1
###### Proposition 3.1
Il existe deux fonctions
$\displaystyle\widetilde{q}$ $\displaystyle\sim\sum_{k\geq
0}\widetilde{q}_{k}h^{k}\in S_{cl}(\mathrm{Vois}(0,0),1),$ (3.16)
$\displaystyle\widetilde{g}$ $\displaystyle\sim\sum_{k\geq
0}\widetilde{g}_{k}h^{k},\quad g_{k}\in
C^{\infty}(\pi_{x}(\mathrm{Vois}(0,0))),$ (3.17)
telles que - nos coordonnées $(x,\xi)$ sont centrées en $\rho(s)$ -
$p(x,\xi)-z\sim\widetilde{q}(x,\xi,\alpha,s;h)\\#(\xi+\widetilde{g}(x,\alpha,s;h))\\#\widetilde{q}(x,\xi,\alpha,s;h),$
(3.18)
dans $S(\mathrm{Vois}(0,0),m)$ uniformément en $z$ ($z$ dépend de $s$ et
$\alpha$), où $z$ appartient à un voisinage de $z_{0}.$ Si l’on tient compte
de la factorisation $(p-z)=n(s)q(x,\xi)(\xi+g(x)),$ nous pouvons choisir
$\widetilde{q}_{0}$ égal à $(n(s)q(x,\xi))^{1/2}$ impliquant que
$\widetilde{g}_{1}=0.$
Pour la suite, par souci de clarté, nous supprimons la dépendance en $\alpha$
et $s$. Preuve. Avec $p(\rho(z))-z=0,$ $p^{\prime}_{\xi}(\rho(z))\neq 0$ nous
obtenons un voisinage $V$ de $\rho(z_{0})$ des fonctions lisses
$\widetilde{q}_{0}$ et $\widetilde{g}_{0}$ définies respectivement sur $V$ et
$\pi_{x}(V)$ telles que
$(p-z)=(n(s)q(x,\xi))^{1/2}(\xi+\widetilde{g}_{0}(x))(n(s)q(x,\xi))^{1/2}$
avec $\widetilde{q}_{0}(x(z_{0}),\xi(z_{0}))\neq 0,$
$\widetilde{g}_{0}(x(z_{0})=-\xi(z_{0})$ On peut rajouter $z$ aux variables et
nous avons toujours une dépendance $C^{\infty}$ de $z$, et les équations ci-
dessus restent valables dans un voisinage de $z_{0}.$ Nous regroupons par
ordre de $h$ les termes de la forme de composition asymptotique:
$\displaystyle 0=$
$\displaystyle\widetilde{q}_{0}^{2}(x,\xi)\widetilde{g}_{N}(x)+2\widetilde{q}_{0}(x,\xi)\widetilde{q}_{N}(x,\xi)(\xi+\widetilde{g}_{0}(x))$
$\displaystyle+\widetilde{G}_{N}(\widetilde{q}_{0},\ldots,\widetilde{q}_{N-1},\widetilde{g}_{0},\dots\widetilde{g}_{N-1},x,\xi,z).$
Etant donné que $\widetilde{q}_{0}$ est non nul, nous avons une équation de la
forme
$G_{N}=2\frac{\widetilde{q}_{N}}{\widetilde{q}_{0}}(\xi+\widetilde{q}_{0}(x))+\widetilde{g}_{N}(x)$
où $G_{N}$ ne dépend que de
$\widetilde{q}_{0},\ldots,\widetilde{q}_{N-1},\widetilde{g}_{0},\dots\widetilde{g}_{N-1}$.
Il est possible de déterminer $\widetilde{g}_{N}$ et $\widetilde{q}_{N}$
inductivement (voir [8]), puisque le théorème de Malgrange nous prouve
l’existence d’un voisinage de $\rho(z_{0})$ et les fonctions
$\widetilde{g}_{N}$ et $\widetilde{q}_{N}$ avec les propriétés demandées. En
itérant, nous obtenons les séries formelles $\widetilde{g}$ et
$\widetilde{q}$.
Pour le dernier point de la proposition, il suffit de remarquer que le terme
en $h$ d’une composition de Weyl symétrique s’annule. $\square$
Nos variables $\rho=(x,\xi)$ sont centrées en $\rho(s):=(x(s),\xi(s))$; si
nous introduisons l’opérateur unitaire $T_{s}$ par
$T_{s}u(y):=e^{iy\xi(s)/h}u(y-x(s)),$
nous trouvons que l’opérateur $\widetilde{P}:=T_{s}PT^{-1}_{s}$ satisfait
$\widetilde{P}=\mathrm{Op}^{w}_{h}(\widetilde{p})\mbox{ o\\`{u}
}\widetilde{p}(\rho)=p(\rho-\rho(s)).$
Il existe alors un symbole $\widetilde{q}$ pour les variables $(x,\xi)$,
$\alpha$ et $s,$ vérifiant
$\widetilde{q}=\chi(x,\xi)q(x,\xi,\alpha,s)^{-\frac{1}{2}}+\mathcal{O}(h),$ où
$\chi=C^{\infty}_{0}(\pi_{x,\xi}(V))$ est une troncature à support compact
indépendant de $z$ (donc de $\alpha$ et $s$) valant $1$ dans un petit
voisinage $V$ de $(0,0),$ tel que
$\displaystyle(\widetilde{q})^{w}T_{s}(P-z)T_{s}^{-1}(\widetilde{q})^{w}=n(s)$
$\displaystyle(hD_{x}+g(x,\alpha,s)+h^{2}g_{\ast}(x,\alpha,s;h))$
$\displaystyle\mbox{ microlocalement pr\\`{e}s de }(0,0),$ (3.19)
(remarque de J. Sjöstrand, pour y arriver, il semblerait qu’il faut aussi une
rotation symplectique) uniformément pour $z$ (qui est une fonction de $\alpha$
et $s,$ voir (3.3)) dans un voisinage donné de $z_{0},$
$\displaystyle g_{\ast}(x,\alpha,s;h)\sim
g^{0}_{\ast}(x,\alpha,s)+hg^{1}_{\ast}(x,\alpha,s)+\ldots\mbox{ dans
}C^{\infty}_{0},$ $\displaystyle\mbox{ uniform\'{e}ment en }\alpha\mbox{ et
}s.$
Par souci de clarté, la variable $s$ est supprimée. On notera
$\widetilde{g}(x,\alpha;h):=g(x,\alpha)+h^{2}g_{\ast}(x,\alpha;h).$
Nous pouvons trouver une partition de l’unité de $\mathbb{R}^{2},$
$\chi_{j}\in S(\mathbb{R}^{2},1),$ $j=0,1$ telle que (voir Faure-Sjöstrand par
exemple)
$\chi_{j}\geq 0,\;\chi_{0}^{2}+\chi_{1}^{2}=1,\mbox{ avec }\chi_{0}\in
C^{\infty}_{0}(\mbox{Vois}(\rho_{0})),\,\chi_{1}\in C^{\infty}_{b}.$
Nous avons
$\|u\|^{2}=\|\chi_{1}u\|^{2}+\|\chi_{0}u\|^{2}.$
En utilisant l’ellipticité de $(P-z)^{-1}$ on peut montrer que pour
$|\alpha|=\mathcal{O}(h\ln(\frac{1}{h})^{2/3})$
$\|(P-z)^{-1}v\|^{2}\leq
M_{0}^{2}\|\chi_{0}v\|^{2}+\mathcal{O}(1)\|\chi_{1}v\|^{2}+\mathcal{O}(h)\|v\|^{2},$
(3.20)
où $M_{0}=\|(P-z)^{-1}\chi_{0}\|.$ Ce qui veut dire que si
$M:=\max(M_{0},\mathcal{O}(1))$ alors
$\|(P-z)^{-1}\|\leq(M^{2}+\mathcal{O}(h))^{\frac{1}{2}}.$
Nous souhaitons maintenant connaître le comportement de $M_{0}$,
$M_{0}=\|(n(s))^{-1}T_{s}^{-1}\widetilde{q}^{w}(hD_{x}+\widetilde{g}(x,\alpha;h))^{-1}\widetilde{q}^{w}T_{s}\chi_{0}\|.$
Du théorème 2.14 et du fait que $E_{-+}$ est scalaire et $T_{s}$ est unitaire,
nous avons que
$\displaystyle
M_{0}=|E_{-+}^{-1}|\times\|\widetilde{q}^{w}E_{+}E_{-}\widetilde{q}^{w}\|+\mathcal{O}(\frac{1}{\sqrt{h}\,\alpha^{1/4}}),$
et nous obtenons une expression pour
$\widetilde{q}^{w}E_{+}E_{-}\widetilde{q}^{w},$ en fonction des quasimodes:
$\widetilde{q}^{w}E_{+}E_{-}\widetilde{q}^{w}=\frac{1-\chi_{-}}{D_{-}D_{+}}(\widetilde{q}^{w}e_{+})\langle\bullet,((\widetilde{q})^{\ast})^{w}\psi_{-}e_{-}\rangle+\mathcal{O}(h^{\infty}).$
(3.21)
Nous avons besoin d’un théorème de Melin, Sjöstrand sur l’action d’un
opérateur pseudodifférentiel sur une fonction BKW avec phase complexe. Dès
lors, puisque $e_{+}$ et $e_{-}$ sont des fonctions BKW normalisées, et
microlocalisées près de $\rho_{\pm},$ nous avons le résultat suivant:
###### Lemme 3.2
Nous avons la formule
$\|\widetilde{q}^{w}E_{+}E_{-}\widetilde{q}^{w}\|\sim\frac{1}{|q(\rho_{+})|^{\frac{1}{2}}|q(\rho_{-})|^{\frac{1}{2}}}(1+\mathcal{O}(\tilde{h})).$
(3.22)
Nous utilisons maintenant l’expression (2.43) de $E_{-+}.$ Puisque
$(p-z)=n(s)q(\xi+g)$, avec $|n(s)|=1$ nous obtenons que
$\frac{1}{|q(\rho_{+})|^{\frac{1}{2}}}\times(\frac{1}{2i}\\{\xi+g,\xi+\bar{g}\\}(\rho_{+}))^{\frac{1}{4}}=(\frac{1}{2i}\\{p,\bar{p}\\}(\rho_{+}))^{\frac{1}{4}},$
et similairement pour $\rho_{-}$. Nous avons donc, pour tout
$h\ll\alpha^{3/2}\ll 1,$ assez petit
$M_{0}=\frac{\sqrt{\pi}\exp(\frac{1}{h}\mathrm{Im}\,\ell_{0}(\alpha))}{h^{1/2}(\frac{1}{2i}\\{p,\bar{p}\\}(\rho_{+}))^{\frac{1}{4}}(\frac{1}{2i}\\{\bar{p},p\\}(\rho_{-}))^{\frac{1}{4}}}(1+\mathcal{O}(\tilde{h}))+\mathcal{O}(\frac{1}{\sqrt{h}\,\alpha^{1/4}}),$
(3.23)
où $\tilde{h}:=h/\alpha^{3/2},$ et $\ell_{0}$ (qui est une intégrale d’action)
vérifie
$\ell_{0}(\alpha):=\mathrm{Im}\,\int_{x_{-}}^{x_{+}}\mathrm{Im}\,g(x,\alpha,s)dx$
(3.24)
(remarquons l’ajout de $s$ qui est important pour la suite).
$\ell_{0}(\alpha)=\ell_{0}(z)$ s’écrit aussi sous la forme
$\displaystyle\ell_{0}(z)$
$\displaystyle=\mathrm{Im}\,\int_{x_{-}(z)}^{x_{+}(z)}\varphi(x,z)dx,\quad
p(x,\varphi(x,z))-z=0,$ $\displaystyle=\mathrm{Im}\,\int_{\gamma\subset
p^{-1}(z)}\xi dx,(\gamma\mbox{ relie }\rho_{-}\mbox{ \\`{a} }\rho_{+}),$
(3.25) $\displaystyle=\mathrm{Im}\,\int_{\gamma\subset p^{-1}(z)}xd\xi.$
Pour que $M_{0}$ ait que croissance tempérée en $h^{-1}$, il faut que
$(\alpha^{3/2}\asymp)\quad\mathrm{Im}\,\int^{x_{+}}_{x_{-}}g(y,z)dy\leq
Ch\ln\frac{1}{h},$
pour un $C>0$ quelconque. Soit si
$\alpha\leq\mathcal{O}(1)\left(h\ln\frac{1}{h}\right)^{2/3}.$ (3.26)
On déduit alors le résultat suivant :
###### Théorème 3.3
Soit $P$ un opérateur $h$-pseudodifférentiel de symbole indépendant de $h.$ On
suppose que $(p,z_{0})$ vérifie les conditions (1), (1.10), (1.12). Un vecteur
tangent à $\partial\Sigma(p)$ dans un voisinage de $z_{0}$ est donné par
$\dot{\gamma}(s).$ Donc tout point $z$ de $\Sigma(p)$ appartenant à un
voisinage de $z_{0}$ peut s’écrire sous la forme
$z=\gamma(s)+\alpha n(s),\quad-T_{1}<s<T_{0},\quad\alpha\geq 0,\quad
z_{0}=\gamma(0),$ (3.27)
où $n(s)=i\dot{\gamma}(s)/|\dot{\gamma}(s)|.$ Il existe un voisinage de
$z_{0},$ pour lequel tout point de $z$ à l’intérieur de $\Sigma(p)$ vérifie
$z\notin\partial\Sigma,\;p^{-1}(z)=\\{\rho_{+}(z),\rho_{-}(z)\\},\quad\rho_{\pm}=(x_{\pm},\xi_{\pm}),$
(3.28)
avec
$\pm\frac{1}{2i}\\{p,\bar{p}\\}(\rho_{\pm})>0.$
Il existe une constante $T_{\ast}$ ($<T_{0},T_{1}$) telle que pour toutes
constantes $C_{0},C_{1}>0,$ il existe une constante $C_{2}>0$ telle que la
résolvante $(P-z)^{-1}$ est bien définie pour
$|s|<T_{\ast},\;\frac{h^{2/3}}{C_{0}}\leq\alpha\leq
C_{1}(h\ln\frac{1}{h})^{2/3},\;h<\frac{1}{C_{2}},$ (3.29)
et satisfait l’estimation
$\displaystyle\|(P-z)^{-1}\|\sim$
$\displaystyle\frac{\sqrt{\pi}\exp(\frac{1}{h}\mathrm{Im}\,\ell_{0}(z))}{h^{1/2}(\frac{1}{2i}\\{p,\bar{p}\\}(\rho_{+}))^{\frac{1}{4}}(\frac{1}{2i}\\{\bar{p},p\\}(\rho_{-}))^{\frac{1}{4}}}\times(1+\mathcal{O}(\tilde{h}))+\mathcal{O}(\frac{1}{\sqrt{h}\,\alpha^{1/4}}),$
où $\tilde{h}=h/\alpha^{3/2},$ et $\ell_{0}$ (qui est une intégrale d’action)
vérifie
$\ell_{0}(z):=\mathrm{Im}\,\int_{\gamma\subset p^{-1}(z)}\xi dx,(\gamma\mbox{
relie }\rho_{-}\mbox{ \\`{a} }\rho_{+}).$ (3.30)
Dans le cas des opérateurs à coefficients analytiques, on peut pousser
$\alpha$ jusqu’à une petite constante.
### 3.2 Estimation de résolvante: cas 2
###### Théorème 3.4
Soit $P$ un opérateur $h$-pseudodifférentiel de symbole indépendant de $h.$ On
demande que pour tout $z$ dans l’intérieur de $\Sigma,$ $p^{-1}(z)$ soit fini
et que $\\{p,\bar{p}\\}(\rho)\neq 0$ pour $\rho\in p^{-1}(z).$
On suppose que p vérifie les conditions (1), (1.10), et que l’hypothèse (1.12)
soit remplacée par
$\displaystyle p^{-1}(z_{0})=$
$\displaystyle\\{\rho_{1},\rho_{2}\\},\,\rho_{j}=(x_{j},\xi_{j}),$
$\displaystyle\\{p,\\{p,\bar{p}\\}\\}(\rho_{1})\neq 0,$
$\displaystyle\\{p,\\{p,\bar{p}\\}\\}(\rho_{2})\neq 0.$ (3.31)
$\Sigma(p)$ sera l’union de deux domaines $\Sigma_{1},$ $\Sigma_{2}$ à bord
$C^{\infty}$ près de $z_{0}$ dont les bords peuvent ne pas coïncider,
$\Sigma_{j}=p(\mbox{vois}(\rho_{j})).$
Soit $H_{\frac{1}{2i}\\{p,\bar{p}\\}}$ le vecteur réel tangent à l’ensemble
$\\{\rho|\,\frac{1}{2i}\\{p,\bar{p}\\}(\rho)=0\\}.$ Donc
$(-T_{1},T_{0})\ni
s\mapsto\rho_{j}(s):=\exp(sH_{\frac{1}{2i}\\{p,\bar{p}\\}})(\rho_{j})\subset\\{\frac{1}{2i}\\{p,\bar{p}\\}=0\\}$
est une courbe orientée. Près de $z_{0},$ pour tout
$z_{\ast}\in\partial\Sigma(p),$ nous avons
$p^{-1}(z)=\\{\rho_{1}(s_{\ast}),\rho_{2}(t_{\ast})\\}.$ $p(\rho_{j}(s))$ est
aussi une courbe orientée notée $\gamma_{j}(s):=p(\rho_{j}(s)).$ Le vecteur
tangent à $\partial\Sigma$ dans un voisinage de $z_{0}$ est donné par
$\dot{\gamma}_{1}(s)$ ou par $\dot{\gamma}_{2}(t).$ Il existe donc un
voisinage de $z_{0}$ tel que tout point $z$ de $\Sigma(p)$ se met sous la
forme :
$\displaystyle z$ $\displaystyle=\gamma_{1}(s)+\alpha_{1}n_{1}(s),\quad-
T_{1}<s<T_{0},\quad\alpha_{1}\geq 0,\quad z_{0}=\gamma_{1}(0),$ (3.32)
$\displaystyle z$
$\displaystyle=\gamma_{2}(s)+\alpha_{2}n_{2}(s)\quad,-T_{1}<s<T_{0},\quad\alpha_{2}\geq
0,\quad z_{0}=\gamma_{2}(0),$ (3.33)
$n_{1}=i\dot{\gamma}_{2}(s)/|\dot{\gamma}_{1}(s)|$ et
$n_{2}=i\dot{\gamma}_{2}(s)/|\dot{\gamma}_{1}(s)|$. Alors pour tout point $z$
de $\Sigma$ dans un voisinage de $z_{0},$ nous avons
$z\notin\partial\Sigma,\;p^{-1}(z)=\\{\rho_{+}^{1}(z),\rho_{-}^{1}(z),\rho_{+}^{2}(z),\rho_{-}^{2}(z)\\},$
où
$\pm\frac{1}{2i}\\{p,\bar{p}\\}(\rho_{\pm}^{j})>0.$
Il existe une constante $T_{\ast}$ ($<T_{0},T_{1}$) telle que pour toutes
constantes $C_{0},C_{1}>0$ il existe une constante $C_{2}>0$ telle que la
résolvante $(P-z)^{-1}$ est bien définie pour
$|s|<T_{\ast},\;\frac{h^{2/3}}{C_{0}}\leq\alpha\leq
C_{1}(h\ln\frac{1}{h})^{2/3},\;h<\frac{1}{C_{2}},\;\alpha=\max(\alpha_{1},\alpha_{2}),$
(3.34)
et satisfait l’estimation pour $z\in\Sigma_{1}\cap\Sigma_{2}:$
$\displaystyle\|(P-z)^{-1}\|\sim\sup_{i=1,2}\bigg{(}$
$\displaystyle\frac{\sqrt{\pi}\exp(\frac{1}{h}\mathrm{Im}\,\ell_{0}^{i}(z))}{h^{1/2}(\frac{1}{2i}\\{p,\bar{p}\\}(\rho_{+}^{i}))^{\frac{1}{4}}(\frac{1}{2i}\\{\bar{p},p\\}(\rho_{-}^{i}))^{\frac{1}{4}}}\bigg{)}\times(1+\mathcal{O}(\tilde{h}))$
$\displaystyle+\mathcal{O}(\frac{1}{\sqrt{h}\,\alpha^{1/4}}),$
pour $z\in\Sigma_{1}\cap\Sigma^{c}_{2}:$
$\displaystyle\|(P-z)^{-1}\|\sim$
$\displaystyle\frac{\sqrt{\pi}\exp(\frac{1}{h}\mathrm{Im}\,\ell_{0}^{1}(z))}{h^{1/2}(\frac{1}{2i}\\{p,\bar{p}\\}(\rho_{+}^{1}))^{\frac{1}{4}}(\frac{1}{2i}\\{\bar{p},p\\}(\rho_{-}^{1}))^{\frac{1}{4}}}\times(1+\mathcal{O}(\tilde{h}))$
$\displaystyle+\mathcal{O}(\frac{1}{\sqrt{h}\,\alpha^{1/4}}),$
pour $z\in\Sigma_{1}^{c}\cap\Sigma_{2}:$
$\displaystyle\|(P-z)^{-1}\|\sim$
$\displaystyle\frac{\sqrt{\pi}\exp(\frac{1}{h}\mathrm{Im}\,\ell_{0}^{2}(z))}{h^{1/2}(\frac{1}{2i}\\{p,\bar{p}\\}(\rho_{+}^{2}))^{\frac{1}{4}}(\frac{1}{2i}\\{\bar{p},p\\}(\rho_{-}^{2}))^{\frac{1}{4}}}\times(1+\mathcal{O}(\tilde{h}))$
$\displaystyle+\mathcal{O}(\frac{1}{\sqrt{h}\,\alpha^{1/4}}),$
où $\tilde{h}=h/\alpha^{3/2},$ et $\ell_{0}^{i}$ (qui est une intégrale
d’action) vérifie
$\ell_{0}^{i}(z):=\mathrm{Im}\,\int_{\gamma\subset p^{-1}(z)}\xi
dx,(\gamma\mbox{ relie }\rho_{-}^{i}\mbox{ \\`{a} }\rho_{+}^{i}).$ (3.35)
Preuve. Fort du cas 1, nous pouvons donner les grandes lignes de la preuve.
Comme précédemment notre raisonnement donne donc deux opérateurs $T_{s}$ et
$T_{s}^{-1}$ tels que
$\displaystyle\widetilde{q}^{w}T_{s}(P-z)T_{s}^{-1}\widetilde{q}^{w}=$
$\displaystyle n_{1}(s)(hD_{x}+\widetilde{g}_{1}(x,\alpha;h))$
$\displaystyle\mbox{ microlocalement pr\\`{e}s de }(0,0),$ (3.36)
$\widetilde{g}_{1}=g_{1}(x,\alpha)+h^{2}g_{\ast,1}(x,\alpha;h),$
De même pour $\rho_{2}.$ Soient $V_{1},V_{2}$ des voisinages arbitrairement
petits près de $\rho_{1},\rho_{2}$. Il existe $\theta_{j}\in
C^{\infty}_{0}(V_{j})$ tels que si
$p_{0}=p+\sum_{j}\theta_{j}$
alors $p_{0}^{-1}(z_{0})$ est vide. Soit $p_{j}=p+\sum_{k,k\neq j}\theta_{k}.$
Donc $p^{-1}_{j}(z_{0})=\\{\rho_{j}\\}.$ Soit $P_{0}=p_{0}^{w}(x,hD_{x})$ et
$P_{j}=p_{j}^{w}(x,hD_{x})$. $P_{0}$ est uniformément elliptique. On suppose
que $\theta_{0}=1$ près de $\rho_{0}.$ Nous écrivons
$R_{0}(z):=\sum_{j}(P_{j}-z)^{-1}\theta_{j}+(P_{0}-z)^{-1}(1-\sum_{j}\theta_{j})$
Nous avons
$\displaystyle(P-z)R_{0}(z)=$ $\displaystyle
1+\sum_{j}(P-P_{j})(P_{j}-z)^{-1}\theta_{j}$
$\displaystyle+(P+P_{0})(P_{0}-z)^{-1}(1-\sum_{j}\theta_{j}),$
où $(P-P_{j})(P_{j}-z)^{-1}\theta_{j}=\mathcal{O}(h^{\infty}),$ (voir [8]) et
aussi pour le dernier terme. Donc $(P-z)=R_{0}(z)(1+K)^{-1}$ où
$K=\mathcal{O}(h^{\infty}).$
Nous pouvons trouver une partition de l’unité de $\mathbb{R}^{2},$
$\chi_{j}\in S(\mathbb{R}^{2},1),$ $j=0,1,2$ telle que
$\chi_{j}\geq 0,\;\chi_{0}^{2}+\ldots+\chi_{2}^{2}=1,\mbox{ avec
}\chi_{j}\prec\theta_{j},\,\chi_{0}\in C^{\infty}_{b}.$
Nous avons
$\|u\|^{2}=\sum_{j=0}^{2}\|\chi_{j}u\|^{2}.$
On peut alors montrer que
$\|(P-z)^{-1}v\|^{2}\leq\sum_{j=1}^{2}M_{j}^{2}\|\chi_{j}v\|^{2}+\mathcal{O}(1)\|\chi_{0}v\|^{2}+\mathcal{O}(h)\|v\|^{2},$
(3.37)
où $M_{j}=\|(P_{j}-z)^{-1}\chi_{j}\|.$ Ce qui veut dire que si
$M:=\max(M_{j},\mathcal{O}(1))$ alors
$\|(p-z)^{-1}\|\leq(M^{2}+\mathcal{O}(h))^{\frac{1}{2}}.$
Nous souhaitons maintenant connaître le comportement de $M_{j}$ qui est aussi
donné par l’expression,
$\displaystyle M_{1}$
$\displaystyle=\|(n_{1}(s))^{-1}T_{s}^{-1}\widetilde{q}^{w}_{1}(hD_{x}+\widetilde{g}_{1}(x,\alpha_{1};h))^{-1}\widetilde{q}^{w}_{1}T_{s}\chi_{1}\|,$
$\displaystyle M_{2}$
$\displaystyle=\|(n_{2}(s))^{-1}S_{s}^{-1}\widetilde{q}^{w}_{2}(hD_{x}+\widetilde{g}_{2}(x,\alpha_{2};h))^{-1}\widetilde{q}^{w}_{2}S_{s}\chi_{2}\|,$
(3.38)
$S_{t}$ a un comportement similaire à $T_{s}$.
De la section précédente, $M_{j},$ $j=1,2$ satisfait
$M_{j}\sim\frac{\sqrt{\pi}\exp(\frac{1}{h}\mathrm{Im}\,\ell_{0}^{j}(z))}{h^{1/2}(\frac{1}{2i}\\{p,\bar{p}\\}(\rho_{+}^{j}))^{\frac{1}{4}}(\frac{1}{2i}\\{\bar{p},p\\}(\rho_{-}^{j}))^{\frac{1}{4}}}(1+\mathcal{O}(\tilde{h}_{j}))+\mathcal{O}(\frac{1}{\sqrt{h}\,\alpha^{1/4}}),$
(3.39)
où $\tilde{h}_{j}=h/\alpha_{j}^{3/2},$ et $\ell_{0}^{j}$ (qui est une
intégrale d’action) vérifie
$\ell_{0}^{j}(z):=\mathrm{Im}\,\int_{\gamma\subset p^{-1}(z)}\xi
dx,(\gamma\mbox{ relie }\rho_{-}^{j}\mbox{ \\`{a} }\rho_{+}^{j}).$ (3.40)
$\square$
## 4 Exemples
Dans ce qui précède, le paramètre $\alpha$ représentait la distance du point
$z$ à la frontière du pseudospectre. Dans ce qui suit, pour les exemples, la
frontière ne sera représentée que par des morceaux de l’axe réel ou
imaginaire. Donc la partie réelle ou imaginaire de $z$ sera notre distance à
la frontière, soit notre paramètre $\alpha.$ On préfère ici changer et
utiliser la lettre $y$ à la place de $\alpha.$
### 4.1 Cas 1: L’opérateur cubique non-autoadjoint
On considère l’opérateur non-autoadjoint sur la droite réelle
$\mathcal{P}=(D_{x})^{2}+ix^{3}.$
L’image du symbole principal est le demi-plan $\mathrm{Re}\,z\geq 0.$ Le
spectre est composé de valeurs propres discrètes situées sur l’axe réel
positif (voir Trepheten, Embree [18]). Le pseudospectre est symétrique par
rapport à l’axe réel. Pour étudier les courbes de niveaux de la résolvante, il
suffit donc de considérer celles situées au-dessus de l’axe réel. $p$ désigne
le symbole de $\mathcal{P}$.
Pour $z$ au dessus de l’axe réel et appartenant à
$\partial\Sigma(p)=i\mathbb{R},$ nous avons
$p^{-1}(z)=\\{\rho\\}=\\{(\mathrm{Im}\,z)^{\frac{1}{3}},0\\}$ et
$\\{p,\frac{1}{2i}\\{p,\bar{p}\\}\\}(\rho)=-2\times
3^{2}(|\mathrm{Im}\,z|)^{4/3}\neq 0.$
Nous sommes donc dans le cas 1.
Pour tout $z$ appartenant à l’intérieur de $\Sigma(p),$ nous avons
$p^{-1}(z)=\\{\rho_{+},\rho_{-}\\},$
où
$\rho_{\pm}(z)=((\mathrm{Im}\,z)^{\frac{1}{3}},\mp(\mathrm{Re}\,z)^{\frac{1}{2}}),$
et
$\frac{1}{2i}\\{p,\bar{p}\\}(\rho_{\pm})=(\mathrm{Im}\,p_{\xi}\bar{p}_{x})(\rho_{\pm})=-2\times
3\xi x^{2}.$
Le changement de variable $x=(\mathrm{Im}\,z)^{\frac{1}{3}}\tilde{x},$ nous
permet d’identifier $(\mathcal{P}-z)$ à
$(\mathrm{Im}\,z)\widetilde{\mathcal{P}}$ où $\widetilde{\mathcal{P}}$ s’écrit
$\big{(}\frac{1}{(\mathrm{Im}\,z)^{\frac{5}{6}}}D_{x}\big{)}^{2}+ix^{3}-\frac{\mathrm{Re}\,z}{\mathrm{Im}\,z}-i,\mbox{
avec
}h:=\frac{1}{(\mathrm{Im}\,z)^{\frac{5}{6}}},\;y:=\frac{\mathrm{Re}\,z}{\mathrm{Im}\,z}.$
(4.1)
$h$ est notre paramètre semiclassique et $y$ est choisi petit. On choisit
$y>0,$ puisque l’on se restreint aux courbes situées au dessus de l’axe réel.
Soit $\varphi(\xi)$ la fonction qui vérifie
$\xi^{2}+i\varphi(\xi)^{3}-y-i=0.$
Nous avons
$\int_{\sqrt{y}}^{-\sqrt{y}}\varphi
d\xi=\int_{\sqrt{y}}^{-\sqrt{y}}(1-iy+i\xi^{2})^{\frac{1}{3}}d\xi$
Puisque $y$ est petit, un développement limité donne
$\displaystyle\mathrm{Im}\,\int_{\sqrt{y}}^{-\sqrt{y}}(1-iy+i\xi^{2})^{\frac{1}{3}}d\xi=\frac{4}{9}y^{\frac{3}{2}}+\mathcal{O}(y^{5/2}).$
Ensuite, remarquons que
$\displaystyle\frac{1}{2i}\\{p,\bar{p}\\}(\rho_{+}(y+i))$
$\displaystyle=\frac{1}{2i}\\{p,\bar{p}\\}(1,-y^{1/2})$
$\displaystyle=(-3x^{2}\xi-3x^{2}\xi)(1,-y^{1/2})=6\sqrt{y},$
il en découle que
$\frac{1}{2i}\\{\bar{p},p\\}(\rho_{-}(y+i))=6\sqrt{y}.$
###### Proposition 4.1
Pour $\mathrm{Im}\,z\gg 1$ et $y:=\frac{\mathrm{Re}\,z}{\mathrm{Im}\,z},$
$h:=\frac{1}{(\mathrm{Im}\,z)^{\frac{5}{6}}},$ tels que
$h\ll y^{3/2}\ll\mathcal{O}(h\ln\frac{1}{h})\mbox{ soit,
}(\mathrm{Im}\,z)^{4/9}\ll\mathrm{Re}\,z\ll\mathrm{Im}\,z,$
nous avons de manière précise,
$\|(\mathcal{P}-z)^{-1}\|\sim\frac{\sqrt{\pi}\exp\left(\frac{1}{h}\mathrm{Im}\,\int_{\sqrt{y}}^{-\sqrt{y}}(1-iy+ix^{2})^{\frac{1}{3}}dx\right)}{6^{1/2}(\mathrm{Im}\,z)^{1/3}(\mathrm{Re}\,z)^{1/4}}+\mathcal{O}(\frac{1}{(\mathrm{Re}\,z)^{\frac{1}{4}}(\mathrm{Im}\,z)^{\frac{1}{3}}}),$
ou bien, sous une forme simplifiée
$\|(\mathcal{P}-z)^{-1}\|\sim\frac{\sqrt{\pi}\exp(\frac{4}{9}\frac{(\mathrm{Re}\,z)^{3/2}}{(\mathrm{Im}\,z)^{2/3}}+\mathcal{O}(\frac{(\mathrm{Re}\,z)^{5/2}}{(\mathrm{Im}\,z)^{5/3}}))}{6^{1/2}(\mathrm{Im}\,z)^{1/3}(\mathrm{Re}\,z)^{1/4}}+\mathcal{O}(\frac{1}{(\mathrm{Re}\,z)^{\frac{1}{4}}(\mathrm{Im}\,z)^{\frac{1}{3}}}).$
(4.2)
#### Lignes de niveaux de la résolvante.
On se sert de l’expression (4.2),
$~\frac{1}{\epsilon}=\|(\mathcal{P}-z)^{-1}\|=(\ref{cub.2}).$
On obtient alors
$-\ln\epsilon=\frac{4}{9}\frac{(\mathrm{Re}\,z)^{3/2}}{(\mathrm{Im}\,z)^{2/3}}+\mathcal{O}(\frac{(\mathrm{Re}\,z)^{5/2}}{(\mathrm{Im}\,z)^{5/3}})+\ln\left(\frac{\sqrt{\pi}}{6^{1/2}}\frac{1}{(\mathrm{Re}\,z)^{1/4}(\mathrm{Im}\,z)^{1/3}}\right).$
(4.3)
(4.3) se met sous la forme
$\displaystyle\mathrm{Re}\,z=$
$\displaystyle(\frac{9}{4})^{2/3}(\mathrm{Im}\,z)^{4/9}\left(\ln(\frac{6^{1/2}(\mathrm{Re}\,z)^{1/4}(\mathrm{Im}\,z)^{1/3}}{\sqrt{\pi}\epsilon})\right)^{2/3}+\mathcal{O}(\frac{(\mathrm{Re}\,z)^{5/3}}{(\mathrm{Im}\,z)^{2/3}})$
$\displaystyle=$
$\displaystyle(\frac{9}{16})^{2/3}(\mathrm{Im}\,z)^{4/9}\left(\ln(\mathrm{Re}\,z(\mathrm{Im}\,z)^{4/3})\right)^{2/3}+\mathcal{O}((\mathrm{Im}\,z)^{4/9})$
$\displaystyle+\mathcal{O}(\frac{(\mathrm{Re}\,z)^{5/3}}{(\mathrm{Im}\,z)^{2/3}}).$
(4.4)
Donc
$\displaystyle\mathrm{Re}\,z=$
$\displaystyle(\frac{9}{16})^{2/3}(\mathrm{Im}\,z)^{4/9}\left(\ln(\mathrm{Im}\,z)^{16/9}(\ln((\mathrm{Re}\,z)(\mathrm{Im}\,z)^{4/3})^{2/3}\right)^{2/3}$
$\displaystyle+\mathcal{O}((\mathrm{Im}\,z)^{4/9}).$ (4.5)
Puisque $\mathrm{Re}\,z<\mathrm{Im}\,z$ et que $\ln(t\ln t)=(1+o(1))\ln t,$ on
obtient
$\displaystyle\mathrm{Re}\,z=(1+o(1))(\frac{9\cdot 16}{16\cdot
9})^{2/3}(\mathrm{Im}\,z)^{4/9}(\ln\mathrm{Im}\,z)^{2/3}.$ (4.6)
Résumons :
###### Proposition 4.2
La partie des lignes de niveaux de l’opérateur non-autoadjoint $\mathcal{P}$
se situant en dessous de l’axe réel, admet la représentation asymptotique
suivante lorsque $\mathrm{Im}\,z\to\infty$
$\displaystyle\mathrm{Re}\,z=(1+o(1))(\mathrm{Im}\,z)^{4/9}(\ln\mathrm{Im}\,z)^{2/3}.$
De manière plus précise, si l’on considère la ligne de niveaux
$\frac{1}{\epsilon}=\|(\mathcal{P}-z)^{-1}\|$ alors nous avons
$\mathrm{Re}\,z=(1+o(1))(\frac{9}{4})^{2/3}(\mathrm{Im}\,z)^{4/9}\left(\ln(\frac{(\mathrm{Im}\,z)^{4/9}}{\epsilon})\right)^{2/3}.$
### 4.2 Cas 2: Oscillateur harmonique non-autoadjoint
Maintenant, considérons l’opérateur harmonique non-autoadjoint,
$Q=(D_{x})^{2}+ix^{2}$ (4.7)
sur la droite réelle.
$\Sigma(p)$ est le premier quadrant supérieur, et
$\partial\Sigma(p)=\mathbb{R}_{+}\cup i\mathbb{R}_{+}.$
Les lignes de niveaux de la résolvante sont symétriques par rapport à l’axe
$e^{i\pi/4}\mathbb{R}.$ Intéressons-nous ici au comportement asymptotique des
lignes de niveaux. Au regard de la symétrie axiale, il suffit de considérer la
partie des lignes qui se situe en dessous de l’axe de symétrie. Nous avons le
résultat suivant de K. Pravda Starov:
###### Proposition 4.3
$S_{i}$ est la symétrie axiale par rapport à l’axe $e^{i\pi/4}\mathbb{R}.$ Il
existe $C>0$ tel que nous avons
$\\{z\in\mathbb{C};\;\mathrm{Re}\,z\geq\nu_{0}\mbox{ et
}0\leq\mathrm{Im}\,z<C(\mathrm{Re}\,z)^{1/3}-\epsilon\\}\cup\\\
S_{i}\left(\\{z\in\mathbb{C};\;\mathrm{Re}\,z\geq\nu_{0}\mbox{ et
}0\leq\mathrm{Im}\,z<C(\mathrm{Re}\,z)^{1/3}-\epsilon\\}\right)\subset\sigma_{\epsilon}(Q)^{c},$
$\sigma_{\epsilon}$ désigne le $\epsilon$-pseudospectre de $Q.$
Pour tout $z$ dans l’image du symbole $q$, nous avons
$\displaystyle q^{-1}$
$\displaystyle(z)=\\{\rho_{-}^{j}(z),\rho_{+}^{j}(z)|\;j=1,2\\},$ (4.8)
$\displaystyle\mbox{o\\`{u} }\frac{1}{2i}\\{q,\bar{q}\\}(\rho_{\pm})>0.$ (4.9)
et si $z_{0}$ un point du bord, appartenant à $\mathbb{R}_{+}$ alors
$q^{-1}(z_{0})=\\{(\xi_{0},0)\\}\cap\\{(-\xi_{0},0)\\},\quad\xi_{0}>0.$
Nous nous trouvons donc dans le cas 2. Dans un voisinage de $z_{0}$ nous avons
$\displaystyle\rho_{+}^{1}(z)=((\mathrm{Im}\,z)^{\frac{1}{2}},-(\mathrm{Re}\,z)^{\frac{1}{2}}),\quad$
$\displaystyle\rho_{-}^{1}(z)=(-(\mathrm{Im}\,z)^{\frac{1}{2}},-(\mathrm{Re}\,z)^{\frac{1}{2}}),$
$\displaystyle\rho_{+}^{2}(z)=(-(\mathrm{Im}\,z)^{\frac{1}{2}},(\mathrm{Re}\,z)^{\frac{1}{2}}),\quad$
$\displaystyle\rho_{-}^{2}(z)=((\mathrm{Im}\,z)^{\frac{1}{2}},(\mathrm{Re}\,z)^{\frac{1}{2}}).$
Le changement de variable $x=(\mathrm{Re}\,z)^{\frac{1}{2}}\tilde{x},$ nous
permet d’identifier $Q-z$ à $(\mathrm{Re}\,z)\widetilde{Q}$ où $\widetilde{Q}$
s’écrit
$\left(\frac{1}{\mathrm{Re}\,z}D_{x}\right)^{2}+ix^{2}-1-i\frac{\mathrm{Im}\,z}{\mathrm{Re}\,z},\mbox{
avec }h:=\frac{1}{\mathrm{Re}\,z},\;y:=\frac{\mathrm{Im}\,z}{\mathrm{Re}\,z}.$
(4.10)
$h$ est notre paramètre semiclassique et $y$ est choisi petit.
Nous avons
$\displaystyle\xi^{2}+ix^{2}-1-iy=$ $\displaystyle(\xi-\sqrt{1+iy-
ix^{2}})(\xi+\sqrt{1+iy-ix^{2}}),$ $\displaystyle=:$
$\displaystyle(\xi-\varphi_{1}(x,y))(\xi-\varphi_{2}(x,y)).$ (4.11)
Nous obtenons
$\displaystyle\int^{\sqrt{y}}_{-\sqrt{y}}\varphi_{1}\,dx=\int^{-\sqrt{y}}_{\sqrt{y}}\varphi_{2}\,dx=\int^{\sqrt{y}}_{-\sqrt{y}}\sqrt{1+iy-
ix^{2}}dx.$ (4.12)
Nous souhaitons calculer cette intégrale.
Soit $F(x)=\frac{1}{2}\arcsin x+\frac{x\sqrt{1-x^{2}}}{2},$
$F^{\prime}(x)=\sqrt{1-x^{2}},$ nous avons
$\displaystyle\frac{d}{dx}F\left(\frac{e^{i\frac{\pi}{4}}x}{\sqrt{1+iy}}\right)=\sqrt{1-\frac{ix^{2}}{1+iy}}\,\frac{e^{i\frac{\pi}{4}}}{\sqrt{1+iy}}.$
(4.13)
Donc
$e^{-\frac{i\pi}{4}}(1+iy)\frac{d}{dx}F\left(\frac{e^{i\frac{\pi}{4}}x}{\sqrt{1+iy}}\right)=\sqrt{1+iy-
ix^{2}}.$
Si bien que
$\displaystyle\int^{-\sqrt{y}}_{\sqrt{y}}\varphi_{2}\,dx$
$\displaystyle=e^{-\frac{i\pi}{4}}(1+iy)\left(F\left(\frac{e^{i\frac{\pi}{4}}\sqrt{y}}{\sqrt{1+iy}}\right)-F\left(\frac{-e^{i\frac{\pi}{4}}\sqrt{y}}{\sqrt{1+iy}}\right)\right)$
$\displaystyle=e^{-\frac{i\pi}{4}}(1+iy)\arcsin(\frac{e^{i\frac{\pi}{4}}\sqrt{y}}{\sqrt{1+iy}}),$
(4.14)
un développement limité donne
$\displaystyle=e^{-\frac{i\pi}{4}}(1+iy)(\frac{e^{i\frac{\pi}{4}}\sqrt{y}}{\sqrt{1+iy}}+\frac{1}{6}\frac{e^{i\frac{3\pi}{4}}y^{3/2}}{(1+iy)^{3/2}}+\ldots),$
(4.15)
$\displaystyle=\sqrt{1+iy}\sqrt{y}+\frac{1}{6}e^{i\frac{\pi}{2}}\frac{1}{(1+iy)^{\frac{1}{2}}}y^{3/2}+\mathcal{O}(y^{5/2}).$
(4.16)
###### Remarque 4.4
On aurait pu aussi traiter l’intégrale (4.12) en remarquant que
$\int\sqrt{x^{2}+a}=\frac{1}{2}(x\sqrt{x^{2}+a}+a\ln(x+\sqrt{x^{2}+a})),$
ce qui donne
$\int^{-\sqrt{y}}_{\sqrt{y}}\varphi_{2}\,dx=\frac{1}{2}e^{-\frac{\pi
i}{4}}(i-y)\ln\frac{\sqrt{y}+\sqrt{i}}{-\sqrt{y}+\sqrt{i}}.$
On retrouve (4.14) par la formule :
$\arcsin(z)=-i\ln\left(iz+(1-z^{2})^{\frac{1}{2}}\right).$
Finalement, en prenant la partie imaginaire dans (4.16), on obtient
$\mathrm{Im}\,\int^{\sqrt{y}}_{-\sqrt{y}}\varphi_{1}\,dx=\mathrm{Im}\,\int^{-\sqrt{y}}_{\sqrt{y}}\varphi_{2}\,dx=\frac{1}{2}y^{3/2}+\frac{1}{6}y^{3/2}=\frac{2}{3}y^{3/2}+\mathcal{O}(y^{5/2}).$
Ensuite, remarquons
$\displaystyle\frac{1}{2i}\\{q,\bar{q}\\}(\rho_{+}^{1}(1+iy))$
$\displaystyle=\frac{1}{2i}\\{q,\bar{q}\\}(\sqrt{y},-1)$ $\displaystyle=(-2\xi
x-2\xi x)(-1,\sqrt{y})=4\sqrt{y},$
$\displaystyle\frac{1}{2i}\\{\bar{q},q\\}(\rho_{-}^{1}(1+iy))$
$\displaystyle=4\sqrt{y}.$
On déduit alors le résultat suivant:
###### Proposition 4.5
Pour $\mathrm{Re}\,z\gg 1$ et $y:=\frac{\mathrm{Im}\,z}{\mathrm{Re}\,z},$
$h:=\frac{1}{\mathrm{Re}\,z},$
$h\ll y^{3/2}\ll 1\mbox{ soit
}(\mathrm{Re}\,z)^{1/3}\ll\mathrm{Im}\,z\ll\mathrm{Re}\,z,$
nous avons de manière précise,
$\displaystyle\|(Q-z)^{-1}\|\sim\frac{\sqrt{\pi}\exp\left(\frac{1}{h}\mathrm{Im}\,(e^{-\frac{i\pi}{4}}(1+iy)\arcsin(\frac{e^{i\frac{\pi}{4}}\sqrt{y}}{\sqrt{1+iy}}))\right)}{2(\mathrm{Re}\,z)^{1/4}(\mathrm{Im}\,z)^{1/4}}$
$\displaystyle+\mathcal{O}(\frac{1}{(\mathrm{Im}\,z)^{\frac{1}{4}}(\mathrm{Re}\,z)^{\frac{1}{4}}}),$
ou bien dans une forme symplifiée,
$\|(Q-z)^{-1}\|\sim\frac{\sqrt{\pi}\exp(\frac{2}{3}\frac{(\mathrm{Im}\,z)^{3/2}}{(\mathrm{Re}\,z)^{1/2}}+\mathcal{O}(\frac{(\mathrm{Im}\,z)^{5/2}}{(\mathrm{Re}\,z)^{3/2}}))}{2(\mathrm{Re}\,z)^{1/4}(\mathrm{Im}\,z)^{1/4}}+\mathcal{O}(\frac{1}{(\mathrm{Im}\,z)^{1/4}(\mathrm{Re}\,z)^{1/4}}).$
(4.17)
#### Lignes de niveaux de la résolvante.
On se sert de l’expression (4.17),
$~\frac{1}{\epsilon}=\|(Q-z)^{-1}\|=(\ref{ex.12}).$
On obtient alors
$-\ln\epsilon=\frac{2}{3}\frac{(\mathrm{Im}\,z)^{3/2}}{(\mathrm{Re}\,z)^{1/2}}+\mathcal{O}(\frac{(\mathrm{Im}\,z)^{5/2}}{(\mathrm{Re}\,z)^{3/2}})+\ln\left(\frac{\sqrt{\pi}}{2}\frac{1}{(\mathrm{Re}\,z)^{1/4}(\mathrm{Im}\,z)^{1/4}}\right).$
(4.18)
En oubliant le facteur logarithmique dans (4.18), on aurait
$\displaystyle-\ln\epsilon$
$\displaystyle=\frac{2}{3}\frac{(\mathrm{Im}\,z)^{3/2}}{(\mathrm{Re}\,z)^{1/2}}+\mathcal{O}(\frac{(\mathrm{Im}\,z)^{5/2}}{(\mathrm{Re}\,z)^{3/2}})=$
$\displaystyle\frac{2}{3}\frac{(\mathrm{Im}\,z)^{3/2}}{(\mathrm{Re}\,z)^{1/2}}(1+\mathcal{O}(\frac{\mathrm{Im}\,z}{\mathrm{Re}\,z}))$
(4.19)
Soit
$(-\ln\epsilon)^{2}\frac{9}{4}=\frac{(\mathrm{Im}\,z)^{3}}{\mathrm{Re}\,z}(1+\mathcal{O}(\frac{\mathrm{Im}\,z}{\mathrm{Re}\,z})),$
ainsi
$\mathrm{Im}\,z=(\mathrm{Re}\,z\,(-\ln\epsilon)^{2}\,\frac{9}{4})^{1/3}(1+\mathcal{O}(\frac{\mathrm{Im}\,z}{\mathrm{Re}\,z})).$
(4.20)
(4.18) se met sous la forme
$\displaystyle\mathrm{Im}\,z$
$\displaystyle=(\frac{3}{2})^{2/3}(\mathrm{Re}\,z)^{1/3}\left(\ln(\frac{2(\mathrm{Re}\,z)^{1/4}(\mathrm{Im}\,z)^{1/4}}{\sqrt{\pi}\epsilon})\right)^{2/3}+\mathcal{O}(\frac{(\mathrm{Im}\,z)^{5/3}}{(\mathrm{Re}\,z)^{2/3}})$
$\displaystyle=(\frac{1}{6})^{2/3}(\mathrm{Re}\,z)^{1/3}\left(\ln(\mathrm{Re}\,z\mathrm{Im}\,z)\right)^{2/3}+\mathcal{O}((\mathrm{Re}\,z)^{1/3})+\mathcal{O}(\frac{(\mathrm{Im}\,z)^{5/3}}{(\mathrm{Re}\,z)^{2/3}}).$
(4.21)
Donc
$\displaystyle\mathrm{Im}\,z=$
$\displaystyle(\frac{1}{6})^{2/3}(\mathrm{Re}\,z)^{1/3}\left(\ln(\mathrm{Re}\,z)^{4/3}(\ln(\mathrm{Re}\,z\mathrm{Im}\,z)^{2/3}\right)^{2/3}$
$\displaystyle+\mathcal{O}((\mathrm{Re}\,z)^{1/3}).$ (4.22)
Puisque $\mathrm{Im}\,z<\mathrm{Re}\,z$ et que $\ln(t\ln t)=(1+o(1))\ln t$ on
obtient
$\displaystyle\mathrm{Im}\,z=(1+o(1))(\frac{1\cdot 4}{6\cdot
3})^{2/3}(\mathrm{Re}\,z)^{1/3}(\ln\mathrm{Re}\,z)^{2/3}.$ (4.23)
Résumons :
###### Proposition 4.6
La partie des lignes de niveaux de l’oscillateur harmonique non-autoadjoint se
situant en dessous de la droite $e^{i\pi/2}\mathbb{R}$ admet la représentation
asymptotique suivante lorsque $\mathrm{Re}\,z\to\infty$
$\displaystyle\mathrm{Im}\,z=(1+o(1))(\frac{1\cdot 4}{6\cdot
3})^{2/3}(\mathrm{Re}\,z)^{1/3}(\ln\mathrm{Re}\,z)^{2/3}.$
De manière plus précise si l’on considère la ligne de niveau
$\frac{1}{\epsilon}=\|(Q-z)^{-1}\|$ alors nous avons
$\mathrm{Im}\,z=(1+o(1))(\frac{3}{2})^{2/3}(\mathrm{Re}\,z)^{1/3}\left(\ln(\frac{(\mathrm{Re}\,z)^{1/3}}{\epsilon})\right)^{2/3}.$
On remarquera que les lignes de niveaux de la résolvante ne se rapprochent pas
à l’infini puisque partant de (4.21), nous avons
$\displaystyle\mathrm{Im}\,z=$
$\displaystyle(\frac{3}{2})^{2/3}(\mathrm{Re}\,z)^{1/3}\left(\ln(\mathrm{Re}\,z)^{4/3}(\ln(\mathrm{Re}\,z\mathrm{Im}\,z)^{2/3}-\ln\epsilon\right)^{2/3}$
$\displaystyle+\mathcal{O}(\frac{(\mathrm{Im}\,z)^{5/3}}{(\mathrm{Re}\,z)^{2/3}}),$
$\displaystyle=$ $\displaystyle
C_{0}(\mathrm{Re}\,z)^{1/3}(\ln\mathrm{Re}\,z)^{2/3}\left(1+C_{1}\frac{-\ln\epsilon}{\ln\mathrm{Re}\,z}+\mathcal{O}(\frac{\ln\epsilon}{\ln\mathrm{Re}\,z})^{2}\right)$
$\displaystyle+\mathcal{O}(\frac{(\mathrm{Im}\,z)^{5/3}}{(\mathrm{Re}\,z)^{2/3}}),$
$\displaystyle=$ $\displaystyle
C_{0}(\mathrm{Re}\,z)^{1/3}(\ln\mathrm{Re}\,z)^{2/3}+C_{0}^{\prime}(\mathrm{Re}\,z)^{1/3}\frac{-\ln\epsilon}{(\ln\mathrm{Re}\,z)^{1/3}}+\ldots$
Le second membre tend vers l’infini pour un $\epsilon$ donné.
Projection spectrale. Ici nous nous intéressons au projection spectrale de
l’oscillateur harmonique non autoadjoint.
###### Théorème 4.7 (Davies, Kuijlaars)
Soit l’opérateur $H=D_{x}^{2}+z^{4}x^{2}$ pour un complexe $z=e^{i\theta}$.
Les valeurs propres sont notées $\lambda_{n}=z^{2}(2n+1).$ On note $N_{n,z}$
la norme du projecteur spectral $\frac{1}{2\pi
i}\int_{D(\lambda_{n},\epsilon)}(z-H)^{-1}dz.$ Si $0<\theta<\pi/4$ alors
$\lim_{n\to\infty}n^{-1}\ln N_{n,z}=2\mathrm{Re}\,(f(r(\theta)e^{i\theta}))$
où
$f(z)=z\sqrt{z^{2}-1}+\ln(z+\sqrt{z^{2}-1}),$
et
$r(\theta)=(2\cos(2\theta))^{-1/2}.$
Nous allons redémontrer ce théorème avec nos estimations.
Preuve. Le changement de variable $x=(2n+1)^{1/2}\tilde{x}$ permet
d’identifier $H-\lambda_{n}$ à $(2n+1)((hD_{x})^{2}+z^{4}x^{2}-z^{2}),$ avec
$h=(2n+1)^{-1}.$ Compte tenu de qui précède, nous avons
$\displaystyle\lim_{n\to\infty}\ln N_{n,z}$
$\displaystyle=-\frac{1}{h}\mathrm{Im}\,\int^{x_{+}}_{x_{-}}\sqrt{(z^{2}-z^{4}x^{2})}\;dx=-\frac{1}{h}\mathrm{Im}\,\int^{x_{+}}_{x_{-}}\sqrt{(1-z^{2}x^{2})}\;dx$
$\displaystyle=\frac{1}{h}\mathrm{Re}\,\int^{x_{+}}_{x_{-}}\sqrt{(z^{2}x^{2}-1)}\;dx,$
ou
$\displaystyle x_{+}(z)$
$\displaystyle=\frac{\mathrm{Im}\,z^{2}}{\mathrm{Im}\,z^{4}}=\sqrt{\frac{\sin(2\theta)}{\sin(4\theta)}}=$
$\displaystyle=\sqrt{\frac{\sin(2\theta)}{2\sin(2\theta)\cos(2\theta)})}=(2\cos(2\theta))^{-1/2},$
$\displaystyle x_{-}(z)$ $\displaystyle=-(2\cos(2\theta))^{-1/2}.$
Etant donné la remarque 4.4, il est facile de conclure. $\square$
### 4.3 Opérateur d’advection-diffusion
Ici nous calculons la résolvante d’un opérateur d’advection-diffusion au point
$1+i.$
On considère l’opérateur non-autoadjoint sur le cercle
$L=-\sin(x)(hD_{x})^{2}-ihD_{x}.$
Fort des informations de [18], nous savons que l’image du symbole à l’infini
est comprise entre les deux paraboles
$\mathrm{Re}\,z=\pm(\mathrm{Im}\,z)^{2}.$ Nous allons montrer que c’est
exactement la frontière de son image. Pour cela, remarquons que
$\displaystyle\ell^{-1}(z)$
$\displaystyle=\\{(\arcsin(-\frac{\mathrm{Re}\,z}{(\mathrm{Im}\,z)^{2}}),-\mathrm{Im}\,z)\\}\cup\\{(\pi-\arcsin(-\frac{\mathrm{Re}\,z}{(\mathrm{Im}\,z)^{2}}),-\mathrm{Im}\,z)\\}$
$\displaystyle=\\{\rho_{+}\\}\cup\\{\rho_{-}\\}$ (4.24)
Nous avons
$\frac{1}{2i}\\{\ell,\bar{\ell}\\}(\rho)=\xi^{2}\cos(x).$ (4.25)
Par souci de simplification, nous notons
$\gamma:=\frac{\mathrm{Re}\,z}{(\mathrm{Im}\,z)^{2}}.$ Puisque $\cos(\arcsin
x)=\sqrt{1-x^{2}}.$ nous voyons que
$\displaystyle\frac{1}{2i}\\{\ell,\bar{\ell}\\}(\rho_{+})$
$\displaystyle=(\mathrm{Im}\,z)^{2}\sqrt{1-\gamma^{2}}>0,$
$\displaystyle\frac{1}{2i}\\{\ell,\bar{\ell}\\}(\rho_{-})$
$\displaystyle=-(\mathrm{Im}\,z)^{2}\sqrt{1-\gamma^{2}}<0.$
En utilisant l’annulation du crochet de Poisson sur le bord de l’image de $l$
nous obtenons:
###### Lemme 4.8
Soit $L=-\sin(x)(hD_{x})^{2}-ihD_{x}$ de symbole $\ell$ alors
$\ell(T^{\ast}{S^{1}})$ est bornée par les paraboles
$\mathrm{Re}\,z=-(\mathrm{Im}\,z)^{2}\mbox{ et
}\mathrm{Re}\,z=+(\mathrm{Im}\,z)^{2}.$
De plus, nous affirmons que les points du bord de $\Sigma(\ell)$ sont tous
d’ordre 2, à l’exception de l’origine.
Preuve. Nous allons calculer le deuxième crochet pour vérifier que les point
du bord sont d’ordre 2
$\displaystyle\\{\ell,\frac{1}{2i}\\{\ell,\bar{\ell}\\}\\}(\rho)=$
$\displaystyle\\{-\sin(x)\xi^{2}-i\xi,\xi^{2}\cos(x))\\}$ $\displaystyle=$
$\displaystyle 2\xi^{3}(\sin x)^{2}+2\xi^{3}(\cos x)^{2}$
$\displaystyle+i\xi^{2}\sin x$ $\displaystyle=$ $\displaystyle
2\xi^{3}+i\xi^{2}(\sin x).$
Dans le cas où $\mathrm{Im}\,z$ ne s’annule pas, il est clair que le bord est
d’ordre 2. $\square$
Soit le point $(1+i)-\alpha e^{i\theta}$ $\theta\in]-\pi/2,\pi/2[$ où
$\cos\theta=-1/\sqrt{5},$ $\sin\theta=2/\sqrt{5}.$ Nous partons pour $\alpha$
petit dans la direction donnée par la normale à la courbe
$\partial\Sigma(\ell)$ au point $1+i.$
Nous avons les formules exactes plus les développements limités suivants (en
utilisant
$\arcsin(1-x^{2})\sim\frac{\pi}{2}+\sqrt{2}|x|+\mathcal{O}(|x|^{3}))$:
$\displaystyle x_{+}(\alpha)$
$\displaystyle=\arcsin(-\frac{1-\alpha\cos\theta}{(1-\alpha\sin\theta)^{2}}),$
$\displaystyle=-\frac{\pi}{2}+\alpha^{1/2}(\sqrt{2}\times(-\cos\theta+2\sin\theta))^{1/2}+\mathcal{O}(\alpha)$
$\displaystyle=-\frac{\pi}{2}+\alpha^{1/2}(\sqrt{2\times
5})^{1/2}+\mathcal{O}(\alpha),$ $\displaystyle x_{-}(\alpha)$
$\displaystyle=\pi-\arcsin(-\frac{1-\alpha\cos\theta}{(1-\alpha\sin\theta)^{2}})$
$\displaystyle=-\frac{\pi}{2}-\alpha^{1/2}(\sqrt{2\times
5})^{1/2}+\mathcal{O}(\alpha).$
En utilisant la formule 3.14 on aurait retrouvé
$x_{\pm}=-\frac{\pi}{2}\pm\sqrt{5}\alpha^{1/2}(\frac{\sqrt{2}}{\sqrt{5}})^{1/2}+\mathcal{O}(\alpha).$
Pour $\xi_{\pm}$ nous avons
$\xi_{\pm}(\alpha)=-1+\alpha\sin\theta.$
Avec les données exactes de $x_{\pm}$ et $\xi_{\pm}$ nous retrouvons
$\frac{1}{2i}\\{\ell,\bar{\ell}\\}(\rho_{\pm})\asymp\alpha^{1/2}.$
Pour l’intégrale d’action, la fonction implicite $\varphi(x,z),$ vérifiant
$\varphi(-\pi/2,1+i)=-1,$ de l’équation $-(\sin x)\xi^{2}-i\xi-z=0$ est
$\frac{1}{2}\times\frac{-i+(-1-4z\sin(x))^{1/2}}{\sin x},$ (4.26)
et
$\displaystyle\varphi^{\prime}_{x}(x,z)=$ $\displaystyle\frac{-i\cos x}{2(\sin
x)^{2}}$ $\displaystyle+\frac{2z(\cos x)(-1-4z\sin x)^{-1/2}-(\cos
x)(-1-4z\sin x)^{1/2}}{(\sin x)^{2}}.$
Comme $\varphi^{\prime}_{x}(-\pi/2,z)=0$, et
$\varphi^{\prime\prime}_{x}(-\pi/2,z)\neq 0$ on peut retrouver
$\ell_{0}(z)=\mathrm{Im}\,\int_{x_{-}}^{x_{+}}\varphi(x,z)dx\asymp\alpha^{3/2}.\\\
$ (4.27)
Nous avons alors redémontré dans le cas de l’opérateur d’advection-diffusion,
la formule que nous connaissions [16]:
###### Proposition 4.9
Soit $\theta$ tel que $\cos\theta=-1/\sqrt{5},$ $\sin\theta=2/\sqrt{5}.$ Pour
$h\ll\alpha^{3/2}<1,$ nous avons
$\displaystyle\|L-((1+i)-\alpha e^{i\theta})\|^{-1}\sim\frac{1}{\sqrt{h}}$
$\displaystyle\frac{\exp(\frac{1}{h}[\mathrm{Im}\,\int\varphi(x,(1+i)-\alpha
e^{i\theta}))]^{x_{+}}_{x_{-}})}{(1-\alpha\sin\theta)\sqrt{1-\frac{1-\alpha\cos\theta}{(1-\alpha\sin\theta)^{2}}}}(1+\mathcal{O}(\frac{h}{\alpha^{3/2}}))$
$\displaystyle+\mathcal{O}(\frac{1}{\sqrt{h}\alpha^{1/4}}),$
où
$\varphi(x,z)=\frac{1}{2}\times\frac{-i+(-1-4z\sin(x))^{1/2}}{\sin x},$
avec
$\displaystyle x_{+}(\alpha)$
$\displaystyle=\arcsin(-\frac{1-\alpha\cos\theta}{(1-\alpha\sin\theta)^{2}}),$
$\displaystyle x_{-}(\alpha)$
$\displaystyle=\pi-\arcsin(-\frac{1-\alpha\cos\theta}{(1-\alpha\sin\theta)^{2}}).$
Ce qui donne, après simplification
$\|L-((1+i)-\alpha
e^{i\theta})\|^{-1}\asymp\frac{C}{h^{1/2}\alpha^{1/4}}e^{\frac{\alpha^{3/2}}{Ch}}.$
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http://xxx.lanl.gov/abs/math.SP/0111293
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|
arxiv-papers
| 2013-01-14T19:47:33 |
2024-09-04T02:49:40.275756
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "William Bordeaux Montrieux",
"submitter": "William Bordeaux Montrieux",
"url": "https://arxiv.org/abs/1301.3102"
}
|
1301.3204
|
††institutetext: Institute of Theoretical Physics, Lanzhou University, Lanzhou
730000, People’s Republic of China
# Resonances of Kalb-Ramond field on symmetric and asymmetric thick branes
Yun-Zhi Du, Li Zhao111Corresponding author, Yi Zhong, Chun-E Fu, Heng Guo
[email protected] [email protected] [email protected] [email protected]
[email protected]
###### Abstract
In this paper, we investigate the localization of the Kalb-Ramond field on
symmetric and asymmetric thick branes, which are generated by a background
scalar field. In order to localize the Kalb-Ramond field, we introduce a
coupling with the background scalar field, and find that there exist some
Kaluza-Klein resonant modes. For the case of symmetric brane, we seek the
resonances by using the relative probability method and transfer matrix
method, and obtain the same result for the two methods. For the asymmetric
case, we use the transfer matrix method. We find that the number of resonances
will decrease with the increase of the asymmetry.
## 1 Introduction
In 1920s, in order to unify electromagnetism with Einstein Gravity, Kaluza and
Klein (KK) considered a five-dimensional space-time with one extra dimension
1921 . And in the string/M-theory, it is required that there are six/seven
extra dimensions. But in these theories, the size of extra dimensions is
restricted to be at the Planck length, which is too small to be detected by
the present experiments. Subsequently, the braneworld scenario was introduced
in a higher-dimensional space-time. The representative brane models are the
Arkani-Hamed-Dimopoulos-Dvali (ADD) model Arkani-Hamed1998 ; Arkani-Hamed1998a
; Antoniadis1998 and the Randall-Sundrum (RS) one Randall1999 ; Randall1999a
. In the brane models, the particles in Standard Model are confined on the
brane, while gravity can propagate in the whole space-time.
In the framework of braneworld scenario, extra dimensions do not have to be so
small as the Planck length; they even can be infinite. So it is possible to
detect the extra dimensions. And the localization of matters on a brane is a
way to realize it. Because the matters from the higher dimensional space-time
carry the information of extra dimensions, so if they can be localized on the
brane, there is a hope for us to find them, and so prove the existence of
extra dimensions. The localization of various matters has been investigated in
the thin brane models George2007 ; Zhang2008 ; Guerrero2010 ; Liu2012a ;
Yang2012 and thick brane models Gremm2000 ; Csaki2000 ; Liu2008 ; Bazeia2009
; Almeida2009 ; Liu2009 ; Liu2010 ; Zhao2010 ; Zhao2011 ; Fu2011 ; Li2011 ;
Guo2011 ; Guo2012 ; Guo2012a ; Fu2012 ; Liu2012b ; Gogberashvili2012 ;
HoffdaSilva2012 . It was found that scalar and gravity fields can be localized
on the RS-like branes in the five-dimensional space-time with one extra
dimension, but it is not true for vector and Kalb-Ramond (KR) tensor fields
Neronov2001 ; Cruz2009 ; Tahim2009 ; Christiansen2012 . However, vector can be
localized on some de sitter thick branes Liu2009 and Weyl thick branes
Liu2008a .
The KR field is an anti-symmetric tensor field, which was first introduced in
the string theory. It is known that in the four-dimensional space-time, the
antisymmetric tensor fields are equivalent to scalar or vector fields by the
symmetry known as duality Quevedo2010 . But they will indicate new types of
particles in extra dimensions. So we are interested in this field. However,
the free KR field cannot be localized on the brane in five-dimensional space-
time Cruz2009 . In Refs. Tahim2009 ; Fu2011 ; Christiansen2012 , the authors
showed that, if the KR field couples to a dilaton field, then the localization
can be realized. While in this paper, we find if we introduce the coupling
with the background kink scalar, the KR field can also be localized on the
brane under some conditions.
Here we consider an interesting brane model, i.e., the symmetric and
asymmetric branes, which was investigated in Refs. Melfo2003 ; Guerrero2005 ;
Liu2009 ; Liu2009a . In this model, there is an asymmetric factor $a$
describing the asymmetry of the brane, and the brane can be split. The
localization of fermion fields in this model has been investigated in Liu2009
; Liu2009a , where the authors mainly focused on the symmetric case. With the
relative probability method applied in Liu2009 ; Liu2009a ; Liu2009c , the
fermion resonances were found for the symmetric case. However, it is seem that
this method is not easy to be extended to the asymmetric case.
In this work, we will use both the relative probability method and the
transfer matrix method to investigate the KR massive KK modes. The transfer
matrix method was used in Refs. Landim2011 ; Landim2012 ; Alencar2012 in the
framework of braneworld, where the authors defined a transmission coefficient
$T$ to find the resonances. And we will use the method to seek the resonances
of the KR field on both symmetric and asymmetric branes, and study the effect
of the asymmetry of the brane on the resonances. For the symmetric case, we
compare this method with the relative probability method, and obtain the same
result.
This paper is organized as follows. Firstly, we simply give in the next
section a review of the construction of symmetric and asymmetric thick branes
in five-dimensional space-time. Then we investigate the resonances of the KR
field on the symmetric case in Sec. 2.1, and on the asymmetric case in Sec.
2.2. Finally, the discussion and conclusion are given in Sec. 3.
## 2 Kalb-Ramond field on symmetric and asymmetric thick branes
We consider the thick brane generated by a single real scalar field $\phi$ in
a five-dimensional space-time. The action of this system is
$S=\int
d^{5}x\sqrt{-g}~{}\Big{[}\frac{1}{2\kappa^{2}_{5}}R-\frac{1}{2}g^{MN}\partial_{M}\phi\partial_{N}\phi-U(\phi)\Big{]},$
(1)
where $\kappa^{2}_{5}=8\pi G_{5}$ with $G_{5}$ the Newton Gravitational
constant, and $U(\phi)$ is the scalar potential. In this paper we set
$\kappa_{5}=1$ for simplicity.
The line-element of the space-time is assumed as
$ds^{2}=e^{2A(z)}(\eta_{\mu\nu}dx^{\mu}dx^{\nu}+dz^{2}).$ (2)
Here $z$ stands for the extra coordinate, and $e^{2A(z)}$ is the warp factor.
The warp factor and the scalar field are both supposed to be functions of $z$
only, i.e., $\phi=\phi(z)$, $A=A(z)$.
A solution for this system was found in Ref. Guerrero2002 :
$\displaystyle\phi$ $\displaystyle=$ $\displaystyle\phi_{0}\arctan(\lambda
z)^{s},$ (3) $\displaystyle e^{2A}$ $\displaystyle=$
$\displaystyle\frac{1}{[1+(\lambda z)^{2s}]^{1/s}G(z)^{2}},$ (4)
$\displaystyle U(\phi)$ $\displaystyle=$
$\displaystyle-\frac{3}{4}\sin^{2}(\phi/\phi_{0})\tan^{-2/s}(\phi/\phi_{0}){\cal
K}(\phi)\times\Big{\\{}16a\tan^{1/s}(\phi/\phi_{0})+\cos^{-2/s}(\phi/\phi_{0})$
(5) $\displaystyle\big{[}5-2s-(3+2s)\cos(2\phi/\phi_{0})\big{]}{\cal
K}(\phi)\Big{\\}}-6a^{2}\cos^{2/s}(2\phi/\phi_{0}),$
where $\phi_{0}=\frac{\sqrt{3(2s-1)}}{s}$ with $s=1,3,5...$, and $G(z)$,
${\cal K}(\phi)$ are defined respectively as
$G(z)\equiv
1+a~{}z~{}{}_{2}F_{1}\left(\frac{1}{2s},\frac{1}{s},1+\frac{1}{2s},-(\lambda
z)^{2s}\right),$ (6) ${\cal
K}(\phi)\equiv\lambda+a\tan^{1/s}\big{(}{\phi}/{\phi_{0}}\big{)}~{}{}_{2}F_{1}\left(\frac{1}{2s},\frac{1}{s},1+\frac{1}{2s},-\tan^{2}\big{(}\frac{\phi}{\phi_{0}}\big{)}\right)$
(7)
with ${}_{2}F_{1}$ the hypergeometric function. This is a three-parameter
($\lambda,s,a$) family of asymmetric thick brane. And $a$ is called asymmetric
factor satisfying
$0\leq a<\frac{2~{}s~{}\lambda~{}\Gamma(1/s)}{\Gamma(1/2s)^{2}}.$ (8)
As $a=0$, we will get a symmetric solution. And for $s=1$ and $s>1$, the
domain wall is a single brane and double one, respectively. Thus, these branes
have very rich inner structure. We will mainly investigate the effect of the
asymmetry on resonance.
Therefore, we consider KR field $B_{\mu\nu}$ in this space-time, and regard
the field as a small perturbation around the background. The action of the
field is given by
$S_{KR}=\int d^{5}x\sqrt{-g}~{}\big{[}f(\phi)H_{MNL}H^{MNL}\big{]},$ (9)
where $H_{MNL}=\partial_{[M}B_{{NL}]}$ is the field strength, and $f(\phi)$ is
the coupling to the background scalar field. The reason we introduce the
coupling is that the free KR field cannot be localized on the brane Cruz2009 ;
Tahim2009 .
From the action (9) and the metric (2), the equations of motion for the KR
field read
$\displaystyle
f(\phi)\partial_{\mu}[\sqrt{-g}H^{\mu\nu\lambda}]+\partial_{z}[\sqrt{-g}f(\phi)H^{z\nu\lambda}]=0,$
(10) $\displaystyle f(\phi)\partial_{\mu}[\sqrt{-g}H^{\mu\nu z}]=0.$ (11)
In the following, we investigate the KK modes of the KR field, which not only
can produce the four-dimensional KR fields, but also carry the information of
the extra dimension. First, we choose a gauge $B_{\mu z}=0$, and decompose the
field as
$B^{\nu\lambda}(x,z)=\sum_{n}b^{\nu\lambda}_{(n)}(x)~{}\chi_{n}(z)~{}e^{pA(z)}$
(12)
with $p$ a coupling constant. Then the field strength becomes
$\displaystyle H_{\mu\nu\lambda}$ $\displaystyle=$
$\displaystyle\sum_{n}e^{4A(z)}~{}h_{\mu\nu\lambda(n)}(x)\chi_{n}(z)~{}e^{pA(z)},$
(13)
where $h_{\mu\nu\lambda(n)}(x)=\partial_{[\mu}b_{\nu\lambda](n)}(x)$ is the
field strength on the brane. Here, we consider the coupling
$f(\phi(z))=e^{(-7-2p)A(z)}$. Then, by substituting (13) into Eq. (10), we
obtain the Schrödinger-like equation for the KK modes $\chi_{n}(z)$:
$\bigg{[}-\frac{d^{2}}{dz^{2}}+V(z)\bigg{]}\chi_{n}(z)=m_{n}^{2}\chi_{n}(z),$
(14)
where $m_{n}$ are the masses of the four-dimensional KR fields, and the
effective potential $V(z)$ takes the form
$V(z)=(4+p)^{2}A^{\prime 2}(z)-(4+p)A^{\prime\prime}(z).$ (15)
Because the Schrödinger-like equation (14) can be written in supersymmetric
quantum mechanics scenario as follows:
$\displaystyle
Q^{\dagger}Q~{}\chi_{n}(z)=\bigg{[}-\frac{d}{dz}+(4+p)A^{\prime}(z)\bigg{]}\bigg{[}\frac{d}{dz}+(4+p)A^{\prime}(z)\bigg{]}\chi_{n}(z)=m_{n}^{2}\chi_{n}(z),$
(16)
we can exclude the existence of tachyonic modes if the KK modes are regular at
the boundaries, and this is an essential condition to keep the stability of
the model.
On the other hand, we need the orthonormality conditions
$\displaystyle\int^{\infty}_{-\infty}dz~{}\chi_{n}(z)\chi_{l}(z)=\delta_{nl},$
(17)
in order to obtain the effective action for the four-dimensional KR fields:
$\displaystyle S_{KR}=\sum_{n}\int
d^{4}x\big{[}h_{\mu\nu\lambda(n)}(x)~{}h^{\mu\nu\lambda}_{(n)}(x)+3~{}m^{2}_{n}~{}b_{\nu\lambda(n)}(x)~{}b^{\nu\lambda}_{(n)}(x)\big{]}.$
(18)
Thus, we can use the orthonormality conditions (17) to judge whether the KK
modes of the KR field can be localized on the symmetric and asymmetric branes.
### 2.1 The symmetric brane case
Firstly, we investigate the case of the symmetric thick brane ($a=0$). The
effective potential (15) is symmetric, and takes the following form
$\displaystyle V^{s}(z)$ $\displaystyle=$ $\displaystyle(4+p)(\lambda
z)^{2s}\frac{(3+p)(\lambda z)^{2s}+(2s-1)}{z^{2}\big{[}1+(\lambda
z)^{2s}\big{]}^{2}}.$ (19)
When $z\rightarrow\infty$, $1+(\lambda z)^{2s}\rightarrow(\lambda z)^{2s}$, so
we have
$\displaystyle
V^{s}(z\rightarrow\infty)\rightarrow\frac{(4+p)(3+p)}{z^{2}}+\frac{(4+p)(2s-1)}{z^{2}(\lambda
z)^{2s}}\rightarrow 0.$ (20)
And when $z\rightarrow 0$, $1+(\lambda z)^{2s}\rightarrow 1$, we find
$\displaystyle V^{s}(z\rightarrow 0)$ $\displaystyle\rightarrow$
$\displaystyle\frac{(4+p)(3+p)(\lambda
z)^{4s}}{z^{2}}+\frac{(4+p)(2s-1)(\lambda z)^{2s}}{z^{2}}$ (23)
$\displaystyle\rightarrow$
$\displaystyle\left\\{\begin{array}[]{ll}\lambda^{2}(4+p)&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{for}~{}~{}~{}~{}~{}s=1\\\
0&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{for}~{}~{}~{}~{}~{}s>1\end{array}\right..~{}~{}~{}~{}~{}$
It can be seen that $V^{s}(z)$ is a volcano potential with a single well for
$s=1$ and two ones for $s\geq 3$. And the distance between the two minima of
$V^{s}(z)$ for $s\geq 3$ increases with $s$. The parameter $p$ mainly effects
the depth of the potential, and with the increase of $p$ the potential well is
deeper. The width of the potential is controlled by the parameter $\lambda$;
when $\lambda$ increases, the potential well becomes narrower. We only plot
the shapes of $V^{s}(z)$ in Fig. 1 for different values of $s$.
Figure 1: The shape of the symmetric potential $V^{s}(z)$ with $\lambda=1$ and
$p=-5$.
For this type of potential, there are a zero mode and continuous massive
modes. Some resonant states may appear. For the zero mode ($m_{0}^{2}=0$), the
wave function is
$\chi_{0}^{s}(z)\propto\bigg{(}1+(\lambda z)^{2s}\bigg{)}^{\frac{4+p}{2s}}.$
(24)
To see whether the zero mode can be localized on the brane, we need to check
the normalization condition (17) for $\chi_{0}^{s}(z)$. Because
$\displaystyle
I^{s}=\int^{\infty}_{-\infty}(\chi_{0}^{s})^{2}dz\propto\int^{\infty}_{-\infty}\bigg{(}1+(\lambda
z)^{2s}\bigg{)}^{\frac{4+p}{s}}dz\rightarrow\int^{\infty}_{-\infty}(\lambda
z)^{2(4+p)}~{}dz~{}~{}~{}~{}\text{for}~{}~{}~{}z\rightarrow\infty,$ (25)
which is finite for $p<-9/2$. So, the zero mode can be localized on the
symmetric thick brane for $p<-9/2$.
In order to find the resonances for this symmetric potential, we can use the
method given in Refs. Liu2009a ; Liu2009c , where the authors defined a
variable, called the relative probability in a box with borders at $\pm
z_{max}$:
$P=\frac{\int_{-z_{b}}^{z_{b}}|\chi_{n}(z)|^{2}dz}{\int_{z_{max}}^{-z_{max}}|\chi_{n}(z)|^{2}dz},$
(26)
and the large relative probability for massive KK modes within a range
$-z_{b}<z<z_{b}$ indicates the existence of resonances. We note here that this
method is the extended version of the one used in Ref. Almeida2009 , which is
only effective for the even KK modes. On the other hand, because the potential
is symmetric, the wave functions are either even-parity or odd-parity. So one
can give two additional boundary conditions to solve the differential equation
(14) numerically:
$\displaystyle\chi_{n}(0)$ $\displaystyle=$ $\displaystyle
0,~{}~{}\chi_{n}^{\prime}(0)=c_{0},~{}~{}~{}~{}~{}~{}\text{for}~{}~{}~{}~{}\text{odd-
parity},$ (27) $\displaystyle\chi_{n}(0)$ $\displaystyle=$ $\displaystyle
c_{1},~{}~{}\chi_{n}^{\prime}(0)=0,~{}~{}~{}~{}~{}~{}\text{for}~{}~{}~{}~{}\text{even-
parity}.$ (28)
Figure 2: The shape of the relative probability $P$ (which is a function of
$m^{2}$) of the massive KK modes for the symmetric case with
$\lambda=1,~{}s=3,~{}p=-15$.
Here, we give an example. We set the parameters as $a=0$, $\lambda=3$, $s=1$,
$p=-15$ and $z_{max}=20z_{0}$, and find some resonances. The shape of the
relative probability is plotted in Fig. 2, where the masses of resonances are
marked, i.e., $m_{n}^{2}=3.997,~{}14.223,~{}27.353,~{}40.84$.
### 2.2 The asymmetric brane case
In this subsection, we consider the case of asymmetric thick brane $(a\neq
0)$. The asymmetric effective potential $V^{a}(z)$ reads
$\displaystyle V^{a}(z)$ $\displaystyle=$ $\displaystyle(4+p)(\lambda
z)^{2s}\frac{(3+p)(\lambda z)^{2s}+(2s-1)}{z^{2}\big{[}1+(\lambda
z)^{2s}\big{]}^{2}}$ (29)
$\displaystyle+(4+p)(3+p)\frac{G^{{}^{\prime}2}}{G^{2}}+(4+p)\frac{G^{{}^{\prime\prime}}}{G}+\frac{2(4+p)^{2}(\lambda
z)^{2}G^{\prime}}{z[1+(\lambda z)^{2s}]G}.$
It is also a volcano-like one. So there are also a zero mode and continuous
massive modes, and some resonances may exist.
With the warp factor (4), the zero mode wave function is given by
$\chi_{0}^{a}(z)\propto[1+(\lambda z)^{2s}]^{\frac{4+p}{2s}}G^{4+p}(z).$ (30)
Then we consider the normalization condition for $\chi_{0}^{a}(z)$:
$\displaystyle
I^{a}=\int_{-\infty}^{\infty}[~{}\chi^{a}_{0}(z)~{}]^{2}~{}dz\propto\int_{-\infty}^{\infty}[~{}\chi^{s}_{0}(z)~{}]^{2}~{}G^{2(4+p)}(z)~{}dz.$
(31)
Noting that, when $z\rightarrow\pm\infty$, $G(z)\rightarrow
1\pm\frac{a~{}\Gamma(\frac{1}{2s})~{}\Gamma(\frac{1}{2s}+1)}{\lambda~{}\Gamma(\frac{1}{s})}$,
which is an finite constant, we get the conclusion that the condition for
localizing the zero mode in this asymmetric brane is the same with that on the
symmetric case, i.e., $p<-9/2$.
Next, we turn to analyze resonances of the KR field on the asymmetric brane.
Because the effective potential is asymmetric, the corresponding wave
functions have no definite parity. When solving the second-order differential
equation (14), we cannot use the initial conditions (27). So the method for
finding the resonances in Sec. 2.1 is not suitable for this asymmetric case.
Here, we can use the transfer matrix method. This method has been used to
investigate the resonances of matter fields in the RS model in Refs.
Landim2011 ; Landim2012 ; Alencar2012 . Here we use this method to solve the
resonances of KR fields on this asymmetric brane.
In this method, a transmission coefficient $T$ is introduced:
$T=\frac{k_{1}}{k_{N}}\frac{1}{\mid M_{22}\mid^{2}},$ (32)
where $N$ is the number of the transfer coefficient matrixes, and $M$ is the
product of the $N$ transfer coefficient matrixes with dimension $2\times 2$.
The relation between $k_{1}$ and $k_{N}$ is
$\displaystyle\frac{k_{1}}{k_{N}}=\sqrt{\frac{\mid V_{1}-E_{n}\mid}{\mid
V_{N}-E_{n}\mid}}$ (33)
with $E_{n}=m_{n}^{2}$.
The transmission coefficient $T$ gives a clear interpretation of what happens
to a plane wave interacting with the branes, no matter whether the brane is
symmetric. For a given mass, we will get a $T(m^{2})$. If the transmission
coefficient has a peak, then it means that the amplitude of the wave function
corresponding to the peak has relative larger value on the brane, namely, we
will have a higher probability to find this KK mode on the brane; hence we
will call this KK mode as a resonant state.
Figure 3: The transmission coefficient $T$ for the KK modes of the KR field
with $p=-15,~{}\lambda=1,~{}s=1,~{}a=0$
Thus, we try to use this transfer matrix method to find the resonances in both
the symmetric and asymmetric branes, and compare the results obtained by the
transfer matrix method with that by the method in Sec. 2.1 for the symmetric
case. Here, we set $N=10^{4}+1$, and $V_{1}=V(-z_{max})$, $V_{N}=V(z_{max})$
with $z_{max}=20$.
We first plot the shape of the transmission coefficient as a function of
$m^{2}$ for the symmetric case in Fig. 3 with
$a=0,~{}\lambda=1,~{}s=3,~{}p=-15$. Comparing Fig. 3 with Fig. 2, we find
that, for the symmetric case, the resonances obtained with the transfer matrix
method is almost the same with the result obtained with the relative
probability method used in Sec. 2.1. In fact, the relative probability $P$ and
the transmission coefficient $T$ have the same physical meaning, i.e., they
indicate the interaction between a plane wave function and the brane. Under
the same condition, the resonances obtained by the two methods should be the
same.
In order to find the effect of the asymmetry factor $a$ on the resonances, we
plot the shapes of the effective potential $V^{a}(z)$ with different values of
$a$ (including $a=0$) in Figs. 4(a), 4(b) and 4(c), for $s=1$, $s=3$ and
$s=5$, respectively. The corresponding logarithm of the transmission
coefficient as a function of $m^{2}$ is also given in these figures. The
masses of the resonances for these cases are listed in Tab. 1.
(a) $p=-30,\lambda=1,s=1$
(b) $p=-30,\lambda=1,s=3$
(c) $p=-30,\lambda=1,s=5$
Figure 4: The logarithm for the KK modes of the KR field on the symmetric and asymmetric thick branes with different parameters. $a=0~{}(m_{n}^{s~{}2})$ | $a=0.3~{}(m_{n}^{a~{}2})$ | $a=0.5~{}(m_{n}^{a~{}2})$
---|---|---
$s=1$ | $s=\\!3$ | $s=\\!5$ | $s=\\!1$ | $s=\\!3$ | $s=\\!5$ | $s=\\!1$ | $s=\\!3$ | $s=\\!5$
$49.08$ | $5.64$ | $14.27$ | $38.24$ | $32.16$ | $14.99$ | $17.28$ | $70.15$ | $50.40$
$91.74$ | $20.07$ | $31.03$ | $66.8$ | $85.17$ | $74.13$ | - | $139.3$ | $128.6$
$127.4$ | $40.27$ | $52.32$ | - | $132.3$ | $126.9$ | - | $200.6$ | $198.1$
- | $64.14$ | $77.68$ | - | $174$ | $174.7$ | - | $254.9$ | $259.8$
- | $91.16$ | $107.1$ | - | $210.3$ | $217.3$ | - | $303.4$ | $316.2$
- | $121.1$ | $138.8$ | - | $239.1$ | $254.1$ | - | $346.2$ | $366.8$
- | $152.3$ | $174.2$ | - | - | $277.3$ | - | $379.5$ | $405$
- | $184.8$ | $211.3$ | - | - | $305.2$ | - | - | $416.1$
- | $217.1$ | $250$ | - | - | $339$ | - | - | -
- | $247.5$ | $289.2$ | - | - | - | - | - | -
- | - | $326.4$ | - | - | - | - | - | -
Table 1: The masses of resonances for the KK modes on symmetric and asymmetric
thick branes. The parameters are set to $p=-30$, $\lambda=1$.
From Fig. 4 and Tab. 1, we find that:
* •
For $s=1$ and $s=5$, comparing with the symmetric case $a=0$, there are less
number of the resonances for the asymmetric case $a\neq 0$. And with the
increase of $a$, the number will decrease.
* •
For $s=3$, the number of the resonances is the most for the symmetric case.
While it does not monotonically decrease with $a$ for the asymmetric case.
And it seems that the $a$ does not decide the number of resonances. While the
degree of the asymmetry plays the role. When the degree of the asymmetry
increases, the number of resonances is less. For $s=3$, the degree of the
asymmetry for the potential seems deeper for $a=0.3$ comparing with $a=0.5$,
so the number of the resonances is less for $a=0.3$.
## 3 Conclusion
In this paper, we investigate the localization of KR field on symmetric and
asymmetric thick branes. There are two important parameters in the brane
model, i.e., $s$ and $a$. The former decides the splitting of the brane, and
the latter leads to the asymmetry of the brane, so it is called as asymmetric
factor. For $a=0$, the brane is symmetric, and for $a\neq 0$ the brane is
asymmetric. We mainly investigate the effect of the asymmetry on the
localization.
We consider the coupling between the KR field and the background scalar field
($f(\phi(z))=e^{(-7-2p)A(z)}$), and find that the KK modes of the KR field
satisfy a Schrödinger-like equation. Because the effective potential is a
volcano-like one for both symmetric and asymmetric cases, there are always a
zero mode and continuous massive modes, and some resonances may exist. The
condition for localizing the zero mode on both the symmetric and asymmetric
branes is the same, i.e., $p<-9/2$.
In order to find the resonances, we first use the relative probability method
for the symmetric case, but it is not suitable for the asymmetric case. Then
we use the transfer matrix method. This method can be used in both cases. And
for the symmetric case, we obtain the same result for the two methods. This
may be because both the relative probability $P$ (in the relative probability
method) and the transmission coefficient $T$ (in the transfer matrix method)
indicate the interaction of a wave function with the brane.
The number of the resonances for the asymmetric case is decided by the degree
of the asymmetry, but not completely the asymmetric factor $a$. The number of
the resonances decreases with the degree of the asymmetry.
## Acknowledgement
We especially thank Prof. Y. X. Liu and Prof. R. R. Landim for sharing some
ideas about resonance state and the codes. We also thank Xiao-Long Du, Xiang-
Nan Zhou and Ke Yang for helpful discussions. This work was supported in part
by the National Natural Science Foundation of China (No. 11075065 and No.
10905027), the Huo Ying-Dong Education Foundation of Chinese Ministry of
Education (No. 121106) and the Fundamental Research Funds for the Central
Universities.
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|
arxiv-papers
| 2013-01-15T02:24:08 |
2024-09-04T02:49:40.294378
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yun-Zhi Du, Li Zhao, Yi Zhong, Chun-E Fu, Heng Guo",
"submitter": "Li Zhao",
"url": "https://arxiv.org/abs/1301.3204"
}
|
1301.3285
|
# CO oxidation on Pd(100) vs. PdO($\sqrt{5}\times\sqrt{5})R27^{\circ}$:
First-Principles Kinetic Phase Diagrams and Bistability Conditions
Max J. Hoffmann Department Chemie, Technische Universität München,
Lichtenbergstr. 4, D-85747 Garching, Germany Karsten Reuter Department
Chemie, Technische Universität München, Lichtenbergstr. 4, D-85747 Garching,
Germany
###### Abstract
We present first-principles kinetic Monte Carlo (1p-kMC) simulations
addressing the CO oxidation reaction at Pd(100) for gas-phase conditions
ranging from ultra-high vacuum (UHV) to ambient pressures and elevated
temperatures. For the latter technologically relevant regime there is a long-
standing debate regarding the nature of the active surface. The pristine
metallic surface, an ultra-thin $(\sqrt{5}\times\sqrt{5})R27^{\circ}$PdO(101)
surface oxide, and thicker oxide layers have each been suggested as _the_
active state. We investigate these hypotheses with 1p-kMC simulations focusing
on either the Pd(100) surface or the PdO(101) surface oxide and intriguingly
obtain a range of $(T,p)$-conditions where both terminations appear
metastable. The predicted bistability regime nicely ties in with oscillatory
behavior reported experimentally by Hendriksen and coworkers [Catal. Today
105, 234 (2005)]. Within this regime we find that both surface terminations
exhibit very similar intrinsic reactivity, which puts doubts on attempts to
assign the catalytic function to just one _active_ state.
## I Introduction
Common use of Pd for catalytic exhaust gas purification gandhi_automotive_2003
has motivated frequent studies on Pd single-crystal model catalysts to
elucidate the underlying molecular-level mechanisms. Notwithstanding, despite
significant efforts the surface structure and composition at _near-ambient_
conditions remains unclear. This is largely due to inherent difficulties in
achieving atomic-scale information/resolution in this technologically relevant
regime of near-ambient pressures and at temperatures of 300-800 K. Generally,
the outcome of such experiments seems strongly dependent on the preparation
conditions and experimental setup. Specifically, there exists a longstanding
and controversial debate on whether reactivity is due to the pristine metal
surfacegao_reply_2010 , a surface oxide filmvan_rijn_comment_2010 , or even a
thicker bulk-like oxide overlayer.hirvi_co_2010
Under ultra-high vacuum (UHV) conditions, different oxidation stages of
Pd(100) zheng_reactivity_2002 have been thoroughly characterized both
experimentally and theoretically. By low-energy electron diffraction (LEED)
Chang and Thiel first identified five distinct ordered structures before the
onset of bulk oxide formation: a $p(2\times 2)$ oxygen adlayer, a $c(2\times
2)$ adlayer, a $(5\times 5)$, and a $(\sqrt{5}\times\sqrt{5})R27^{\circ}$ (for
brevity henceforth denoted as $\sqrt{5}$) reconstruction.chang_oxygen_1988
Todorova _et al._todorova_pd1_2003 established that the latter $\sqrt{5}$
structure corresponds to a single-layer of PdO(101) on top of Pd(100).
At low pressures several experimental studies concluded that formation of this
surface oxide is accompanied with a low catalytic
activity.kiss_deactivation_1985 ; gabasch_comparison_2007 This was primarily
attributed to the low CO binding energy at the surface oxide
zheng_reactivity_2002 , which Gao _et al._ estimated to be around 0.5-0.6
eV.gao_infrared_2008 At elevated pressures the situation is less clear.
Lundgren _et al._lundgren_kinetic_2004 compared structural information from
in situ surface x-ray diffraction measurements in a pure oxygen environment to
a systematic ab initio surface phase diagram rogal_first-principles_2007 ;
rogal_co_2008 , and concluded that the formation of bulk oxide on Pd(100) is
kinetically severely hindered even at temperatures around 675 K. This suggests
that at the $(T,p)$-conditions relevant for near-ambient CO oxidation the most
likely surface terminations are either pristine Pd(100) with some coverage of
O adsorbates, or a $\sqrt{5}$ monolayer. On top of this, reactor scanning
tunneling microscopy (STM) measurements by Hendriksen _et al._ in fact
suggested bistability in this near-ambient regime. hendriksen_oscillatory_2004
; hendriksen_bistability_2005 ; hendriksen_role_2010 In the understanding of
the work by Lundgren et al. this result refers to $(T,p)$ conditions where
both O@Pd(100) and the $\sqrt{5}$ surface oxide are metastable.
Here, we investigate this hypothesis with first-principles kinetic Monte Carlo
(1p-kMC) simulations that focus on either of the two surface states, i.e.
either O@Pd(100) or the $\sqrt{5}$ surface oxide, thereby extending previous
1p-kMC work that had focused exclusively on the $\sqrt{5}$ phase rogal_co_2007
; rogal_first-principles_2007 . We indeed find a range of $(T,p)$-conditions
where both surface models appear metastable. Moreover, in this bistability
regime, both surface terminations yield roughly similar turnover frequencies
(TOFs) for near-ambient pressures. In contrast and consistent with experiment
we find the Pd(100) model to be the much more reactive one in the UHV regime.
This puts experiments gao_reply_2010 ; van_rijn_comment_2010 into perspective
that claim one surface state to be _the_ active one, irrespective of the
specific gas-phase conditions.
## II Methods
### II.1 1p-kMC simulations
Our central goal is to describe the range of gas-phase conditions in which the
surface termination of the Pd(100) model catalyst is characterized by pristine
Pd(100) or by the $\sqrt{5}$ surface oxide. In order to also account for
stabilization effects through the ongoing reaction kinetics this requires
analyzing the steady-state coverages of CO and O in microkinetic models of
these two surface terminations as a function of external feed conditions
$(T,p_{\mathrm{CO}},p_{\mathrm{O}_{2}})$. For this, two factors indicate that
standard microkinetic modeling in terms of mean-field rate equations will not
be sufficient: On pristine Pd(100) lateral interactions between adsorbed O and
CO are known to be rather strongzhang_accuracy_2007 ; behm_adsorption_1980 ;
liu_chemical_2006 ; liu_atomistic_2009 , while on the $\sqrt{5}$ surface oxide
it is its essentially one-dimensional geometric trench structure that will
lead to adlayer inhomogeneities. We thus opt for 1p-kMC simulations as
presently only technique that provides the desired microkinetic information
while fully accounting for the correlations, fluctuations, and explicit
spatial distributions of the chemicals at the catalyst surface.temel_2007 ;
reuter_2011_book ; sabbe_2012
In 1p-kMC the time evolution of the system is coarse-grained to the discrete
rare-event dynamics. Relying on a Markov approximation rejection-free 1p-kMC
algorithms thus generate state-to-state trajectories that in their average
yield the probability-density function $P_{i}(t)$ to find the system at time
$t$ in state $i$ representing the corresponding potential energy surface (PES)
basin $i$. The propagation of this probability density is governed by the
Markovian master equation,
$\frac{dP_{i}(t)}{dt}\;=\;-\sum_{j\neq i}k_{ij}P_{i}(t)\;+\;\sum_{j\neq
i}k_{ji}P_{j}(t)\quad,$ (1)
where the sums run over all system states $j$, and $k_{ij}$ is the rate
constant for going from state $i$ to $j$. The central first-principles
ingredients required for the 1p-kMC simulation are thus the individual rate
constants of the considered elementary processes (adsorption, desorption,
diffusion, reaction). To keep this input tractable, i.e. to arrive at a finite
number of inequivalent processes and corresponding rate constants, 1p-kMC
simulations are commonly performed on lattice models. In the following
sections we will first provide the working equations to determine the first-
principles rate constants $k_{ij}$ and then detail the specific lattice models
employed in the present work for CO oxidation at pristine Pd(100) and at the
$\sqrt{5}$ surface oxide.
For given environmental (gas-phase) conditions (which specify the adsorption
rate constants), the output of 1p-kMC simulations are then the detailed
surface composition and occurrence of each individual elementary process at
any time. Since the latter comprises the surface reaction events, this also
gives the catalytic activity in form of products per surface area and time
(i.e. TOFs), either time-resolved, e.g. during induction, or time-averaged
during steady-state operation.
### II.2 First-principles rate constants
The first-principles rate constants in this work are evaluated following the
approach put forward by Reuter and Scheffler reuter_first-principles_2006 . In
brief, this approach relies on (harmonic) transition state theory (TST) for
bound to bound processes like diffusion, and kinetic gas theory together with
detailed balance to calculate adsorption and desorption rate constants. As the
approach and its derivation have been detailed before, we here restrict
ourselves to the presentation of the working equations for self-containment.
The adsorption rate constant for species $i$ is given by the rate with which
these particles impinge on the unit-cell surface area $A_{uc}$ and the local
sticking coefficient $\widetilde{S}_{st,i}(T)$, which gives the fraction of
the impinging particles that actually stick to a given free site $st$ at
temperature $T$
$\displaystyle k^{\mathrm{ad}}_{st,i}(T,p_{i})$ $\displaystyle=$
$\displaystyle\widetilde{S}_{st,i}(T)\frac{p_{i}A_{uc}}{\sqrt{2\pi
m_{i}k_{B}T}}\quad.$ (2)
Here, $k_{B}$ is the Boltzmann constant, $p_{i}$ the partial pressure of
species $i$, and $m_{i}$ the particle mass. In unactivated adsorption events
the local sticking coefficient merely accounts for the number of inequivalent
sites in the surface unit-cell (vide infra). For activated adsorption events
or Eley-Rideal (ER) type CO oxidation events, it is additionally governed by
the adsorption or reaction barrier $\Delta E_{i,st,j}$, respectively,
$\widetilde{S}_{st,i}(T)\;=\;\left(\frac{A_{st,i}}{A_{uc}}\right)\exp\left(-\frac{\Delta
E_{i,st,j}}{k_{B}T}\right)\quad.$ (3)
where $A_{st,i}$ is a geometrical factor reflecting the relative share with
which molecules impinge on the different inequivalent surface sites $st$.
Desorption of a particle adsorbed on a surface site $st$ is modeled as the
time reversed process of adsorption, and its rate constant thus has to fulfill
detailed balance or microscopic reversibility.
$\displaystyle\frac{k^{\mathrm{ad}}_{st,i}(T,p_{i})}{k^{\mathrm{des}}_{st,i}(T)}$
$\displaystyle=$ $\displaystyle\exp\left(\frac{\Delta
G_{st,i}(T,p_{i})}{k_{B}T}\right)$ (4) $\displaystyle\approx$
$\displaystyle\exp\left(\frac{\mu_{\mathrm{gas},i}(T,p_{i})-E^{\mathrm{bind}}_{st,i}}{k_{B}T}\right)$
where $\Delta G_{st,i}(T,p_{i})$ is the difference in Gibbs free energy
between the particle adsorbed at the surface state and in the gas phase. This
is approximated by the difference between the gas-phase chemical potential
$\mu_{\mathrm{gas},i}(T,p_{i})$ reuter_composition_2001 and the binding
energy of the particle in the adsorbed state $E^{\mathrm{bind}}_{st,i}$.
Following the procedure detailed in Ref. reuter_composition_2001, we
interpolate tabulated values chase_jr_nist-janaf_1998 to determine the gas-
phase chemical potentials at any gas-phase condition.
The diffusion rate constant of an adsorbate from one surface site $st$ to
another site $st^{\prime}$ is approximated as
$k^{\mathrm{diff}}_{st,st^{\prime}\\!\\!,i}(T)\approx\left(\frac{k_{B}T}{h}\right)\exp\left(-\frac{\Delta
E^{\mathrm{diff}}_{st,st^{\prime}\\!\\!,i}}{k_{B}T}\right)\quad,$ (5)
where $\Delta E^{\mathrm{diff}}_{st,st^{\prime}\\!\\!,i}$ is the diffusion
energy barrier. Langmuir-Hinshelwood (LH) type CO oxidation reactions are
described with an equivalent expression containing the reaction barrier. They,
as well as the ER reactions, are generally treated as associative desorption
events though, i.e. the formed CO2 immediately desorbs from the surface.
Within this approach the required first-principles input to determine the rate
constants reduces to the binding energies of the species at the surface sites
(for the desorption rate constant), as well as to their diffusion and reaction
barriers. The latter require the determination of the corresponding transition
states, which is at present commonly achieved with standard transition state
search algorithms like nudged-elastic band henkelman ; henkelman2 .
Unfortunately, we found the low PES corrugation for lateral shifts of the
$\sqrt{5}$ overlayer todorova_pd1_2003 ; kostelnik to severely affect the
performance and reliability of such methods. We therefore instead identified
the transition states through scans along suitable reaction coordinates. For
diffusion barriers the specific reaction coordinate employed was the lateral
coordinate along a straight line connecting the known initial and final state.
For CO oxidation reactions the distance between the C atom of the CO molecule
and the O adsorbate was employed. We carefully checked for hysteresis effects
and estimate the uncertainty in the determined barriers to be of the order of
$\pm 0.1$ eV. The effect of this energetic uncertainty on the presented 1p-kMC
results will be critically discussed below and is not found to affect any of
our conclusions.
All required total energies were obtained from density-functional theory (DFT)
with the generalized gradient approximation functional by Perdew, Becke, and
Ernzerhof (PBE) perdew_generalized_1996 to treat electronic exchange and
correlation. Using the plane-wave code CASTEP clark_first_2005 with standard
library ultrasoft pseudopotentials systematic convergence tests showed that
the quantities of interest (binding energies, diffusion and reaction barriers)
are converged to within 30 meV at the employed energy cut-off of 400 eV and
$k$-point density of 0.4 Å-1. The surfaces were modeled in periodic supercell
geometries, containing vacuum separations of more than 10 Å. For the
$\sqrt{5}$ surface oxide the calculations were done within a $(1\times 1)$
surface unit-cell of surface oxide on top of four layers of Pd(100). For
pristine Pd(100) we employed a $(2\times 2)$ surface unit-cell and also four
metal layer slabs. In both cases, all geometries were fully relaxed to
residual forces below 50 meV/Å, while keeping the bottom two slab layers fixed
at their bulk positions.
### II.3 1p-KMC lattice models
Figure 1: (Color online) Top view of Pd(100) and the $\sqrt{5}$ surface
oxide, illustrating the employed lattice models. Hollow sites are depicted by
yellow circles and bridge sites by green rectangles (see text). Pd(100) top-
layer atoms are represented by large dark green spheres, Pd atoms in the
$\sqrt{5}$ oxide by large light green spheres, and O atoms by small red
spheres.
#### II.3.1 Pd(100)
In setting up the lattice model for Pd(100) we exploit the limited coverage
range, for which the 1p-kMC simulations need to provide a faithful
representation. Detailed experimental work chang_oxygen_1988 indicates the
formation of surface oxides at around a critical O coverage of 0.5 monolayer
(ML, defined as the ratio of O to top layer Pd atoms). As the purpose of this
work is to derive the gas-phase conditions under which no surface oxide is
formed, the developed model only targets the coverage range $\theta<0.5$ ML
and the elementary processes occurring in it. Similarly, CO coverages of more
than 0.5 ML have only been characterized for strongly overstoichiometric CO
pressures and low temperaturesbehm_adsorption_1980 ; stuve_co_1984 ;
uvdal_structure_1988 ; tushaus_understanding_1990 . As this regime is outside
the catalytic context, only CO coverages below 0.5 ML are of interest here,
too.
In this coverage range O atoms bind preferably at the fourfold hollow sites at
Pd(100) rieder_helium_1985 ; zhang_accuracy_2007 , while CO binds at twofold
bridge sites rogal_co_2008 ; bradshaw_chemisorption_1978 . In consequence we
set up the lattice model shown in Fig. 1, in which O can exclusively occupy
hollow sites and CO bridge sites. Lateral interactions between oxygen adatoms
have been systematically calculated by Zhang, Blum, and Reuter and were found
to be strongly repulsive at nearest and moderately repulsive at next-nearest
hollow-hollow distance zhang_accuracy_2007 . For the targeted coverage regime
and intermediate temperatures up to 600 K, this is taken into account in the
1p-kMC model by blocking adsorption or diffusion events into configurations
that would result in O atoms in nearest-neighbor positions. For dissociative
adsorption this implies a required motif of eight empty sites (two next-
nearest neighbor sites for the actual adsorption and their immediately
adjacent six nearest-neigbor sites) known as the 8-site
rulebrundle_summary_1984 ; chang_formation_1987 ; liu_atomistic_2006 ;
liu_atomistic_2009 that is required for an adsorption event to take place.
Similarly repulsive interactions between adsorbed CO molecules are indicated
by the experimentally characterized superstructures behm_adsorption_1980 ;
stuve_co_1984 ; uvdal_structure_1988 ; tushaus_understanding_1990 and DFT
calculations rogal_first-principles_2007 . Accordingly and accounting for the
shorter bridge-bridge distances, we also block CO adsorption and diffusion
events into configurations resulting in CO molecules up to next nearest-
neighbor bridge-bridge distance. We note, however, that the next nearest-
neighbor lateral repulsion is not too strong liu_atomistic_2006 , and might
hence be overestimated by the simple site-blocking in our default model,
henceforth denoted as Pd(100)-2NN. We assess this below by additionally
considering a model, in which CO-CO coadsorption is blocked only at nearest
bridge-bridge distance (Pd(100)-1NN) and find this not to affect our
conclusions towards bistability.
Finally, the absence of mixed O-CO adsorbate structures at low temperatures
stuve_co_1984 suggests O-CO lateral interactions to also be repulsive.
Unfortunately, no systematic first-principles studies characterizing these
interactions have been performed to date. In this situation we note that the
nearest-neighbor hollow-bridge distance at Pd(100) is smaller than the O-CO
distance in the CO oxidation transition states described below. O-CO lateral
interactions are thus modeled by blocking adsorption and diffusion events into
configurations involving O and CO at such close distance.
Within this lattice model and site-blocking rules we consider all non-
concerted O and CO adsorption, desorption, as well as nearest-neighbor site
diffusion and reaction events. Oxygen adsorption is modeled as dissociative
process into two vacant next-nearest hollow sites. CO adsorption occurs
unimolecularly into one free bridge site. Both adsorption processes are
treated as non-activated, i.e. using a local sticking coefficient of 1 for the
dissociative adsorption of O2 and 0.5 for CO, the latter accounting for the
two bridge sites per surface unit-cell. For oxygen adsorption in the low-
coverage range this is supported by explicit DFT sticking coefficient
calculations meyer_first-principles_2011 , while for CO we verified that
lifting the CO molecule vertically up from the bridge adsorption site yields a
path without adsorption barrier. This together with the observation that a CO
lowered above an adsorbed oxygen atom is efficiently steered into a
neighboring vacant bridge site, supports the assumption that all CO molecules
impinging in the vicinity of a vacant bridge site will stick.
Table 1: Summary of DFT binding energies, diffusion and reaction barriers (LH
and ER) used in the Pd(100) and $\sqrt{5}$ 1p-kMC models. All values are in
eV. PdO$\sqrt{5}$ desorption barriers:
---
$E^{0}$ on-site energy, $V$ nearest-neighbor lateral interaction111Rogal _et
al._ , PRB 77, 155410-12, (2008)
$\stackrel{{\scriptstyle E^{0}_{\tiny{\mathrm{O,bridge}}}}}{{-0.51}}$111Rogal _et al._ , PRB 77, 155410-12, (2008) | $\stackrel{{\scriptstyle E^{0}_{\tiny{\mathrm{O,hollow}}}}}{{-1.95}}$111Rogal _et al._ , PRB 77, 155410-12, (2008) | $\stackrel{{\scriptstyle E^{0}_{\tiny{\mathrm{CO,bridge}}}}}{{-1.40}}$111Rogal _et al._ , PRB 77, 155410-12, (2008) | $\stackrel{{\scriptstyle E^{0}_{\tiny{\mathrm{CO,hollow}}}}}{{-1.92}}$111Rogal _et al._ , PRB 77, 155410-12, (2008)
$\stackrel{{\scriptstyle V_{\tiny{\mathrm{O-O,br-br}}}}}{{0.08}}$111Rogal _et al._ , PRB 77, 155410-12, (2008) | $\stackrel{{\scriptstyle V_{\tiny{\mathrm{O-O,hol-hol}}}}}{{0.07}}$111Rogal _et al._ , PRB 77, 155410-12, (2008) | $\stackrel{{\scriptstyle V_{\tiny{\mathrm{O-O,br-hol}}}}}{{0.08}}$111Rogal _et al._ , PRB 77, 155410-12, (2008) |
$\stackrel{{\scriptstyle V_{\tiny{\mathrm{CO-CO,br-br}}}}}{{0.08}}$111Rogal _et al._ , PRB 77, 155410-12, (2008) | $\stackrel{{\scriptstyle V_{\tiny{\mathrm{CO-CO,hol-hol}}}}}{{0.13}}$111Rogal _et al._ , PRB 77, 155410-12, (2008) | $\stackrel{{\scriptstyle V_{\tiny{\mathrm{CO-CO,br-hol}}}}}{{0.14}}$111Rogal _et al._ , PRB 77, 155410-12, (2008) |
$\stackrel{{\scriptstyle V_{\tiny{\mathrm{O-CO,br-br}}}}}{{0.06}}$111Rogal _et al._ , PRB 77, 155410-12, (2008) | $\stackrel{{\scriptstyle V_{\tiny{\mathrm{O-CO,hol-hol}}}}}{{0.11}}$111Rogal _et al._ , PRB 77, 155410-12, (2008) | $\stackrel{{\scriptstyle V_{\tiny{\mathrm{O-CO,br-hol}}}}}{{0.13}}$111Rogal _et al._ , PRB 77, 155410-12, (2008) | $\stackrel{{\scriptstyle V_{\tiny{\mathrm{O-CO,hol-br}}}}}{{0.12}}$111Rogal _et al._ , PRB 77, 155410-12, (2008)
PdO $\sqrt{5}$ diffusion barriers
$\stackrel{{\scriptstyle\rm{CO,br}\rightarrow\rm{br}}}{{0.4}}$111Rogal _et al._ , PRB 77, 155410-12, (2008) | $\stackrel{{\scriptstyle\rm{CO,hol}\rightarrow\rm{hol}}}{{0.6}}$111Rogal _et al._ , PRB 77, 155410-12, (2008) | $\stackrel{{\scriptstyle\rm{CO,br}\rightarrow\rm{hol}}}{{0.3}}$111Rogal _et al._ , PRB 77, 155410-12, (2008) |
$\stackrel{{\scriptstyle\rm{O,br}\rightarrow\rm{br}}}{{1.2}}$111Rogal _et al._ , PRB 77, 155410-12, (2008) | $\stackrel{{\scriptstyle\rm{O,hol}\rightarrow\rm{hol}}}{{1.4}}$111Rogal _et al._ , PRB 77, 155410-12, (2008) | $\stackrel{{\scriptstyle\rm{O,br}\rightarrow\rm{hol}}}{{0.1}}$111Rogal _et al._ , PRB 77, 155410-12, (2008) |
PdO$\sqrt{5}$ reaction barriers (LH)
$\stackrel{{\scriptstyle E_{\rm{Obr,CObr}}}}{{1.0}}$111Rogal _et al._ , PRB 77, 155410-12, (2008) | $\stackrel{{\scriptstyle E_{\rm{Ohol,COhol}}}}{{1.6}}$111Rogal _et al._ , PRB 77, 155410-12, (2008) | $\stackrel{{\scriptstyle E_{\rm{Obr,COhol}}}}{{0.5}}$111Rogal _et al._ , PRB 77, 155410-12, (2008) | $\stackrel{{\scriptstyle E_{\rm{Ohol,CObr}}}}{{0.9}}$111Rogal _et al._ , PRB 77, 155410-12, (2008)
PdO$\sqrt{5}$ reaction barriers (ER)
$\stackrel{{\scriptstyle E_{\rm{Ohol1}}}}{{0.8}}$ | $\stackrel{{\scriptstyle E_{\rm{Ohol2}}}}{{0.5}}$ | $\stackrel{{\scriptstyle E_{\rm{Obr1}}}}{{0.0}}$ | $\stackrel{{\scriptstyle E_{\rm{Obr2}}}}{{0.0}}$
Pd(100) desorption barriers
| $\stackrel{{\scriptstyle E^{0}_{\tiny{\mathrm{O,hollow}}}}}{{-1.25}}$ | $\stackrel{{\scriptstyle E^{0}_{\tiny{\mathrm{CO,bridge}}}}}{{-1.93}}$ |
Pd(100) diffusion barriers
$\stackrel{{\scriptstyle\rm{CO,br}\rightarrow\rm{br}}}{{0.14}}$ | $\stackrel{{\scriptstyle\rm{O,hol}\rightarrow\rm{hol}}}{{0.28}}$ | |
Pd(100) reaction barriers
| | $\stackrel{{\scriptstyle E_{\rm{CObr,Ohol}}}}{{0.9}}$ |
Neglecting any lateral interactions beyond the short-range blocking rules, the
required O and CO binding energies, diffusion and reaction barriers are
independent of the local adsorbate environment. The corresponding values
computed within our DFT setup are summarized in Table I and are generally in
good agreement with previous computations zhang_accuracy_2007 ; rogal_first-
principles_2007 ; todorova_pd1_2003 . The obtained LH reaction barrier of 0.9
eV in particular is in good agreement with previous DFT calculations at
varying coverages (0.76-1.05 eV) hammer_no+co_2001 ; eichler_co_2002 ;
zhang_co_2001-1 . Attempts to calculate an ER type reaction path over a
$(2\times 2)$ oxygen adlayer at Pd(100) showed that CO and O coadsorption is
energetically more favorable and thus an ER reaction is excluded from the
Pd(100) model. As apparent from Table I the O and CO diffusion barriers are
very low. This high mobility severely limits the numerical efficiency of the
1p-kMC simulations, which are completely dominated by frequent executions of
diffusion events at minute time increments. In order to speed up the 1p-kMC
simulations we thus artifically raised the diffusion barriers by as much as
0.5 eV. As diffusion is then still fast enough to achieve full equilibration
of the adlayer in the temperature range of interest to this study, we found
this not to affect the computed coverages or TOFs at all.
#### II.3.2 $\sqrt{5}$ surface oxide
For the $\sqrt{5}$ surface oxide we employ the 1p-kMC model established and
detailed by Rogal, Reuter, and Scheffler rogal_first-principles_2007 ;
rogal_co_2008 . In brief, this model considers two non-equivalent sites named
bridge and hollow as depicted on the right of Fig. 1. Since there is no
process involving sites from adjacent bridge-hollow trenches, the lattice may
be viewed as quasi one-dimensional. In analogy to the just described Pd(100)
model, the elementary process list consists of all non-concerted adsorption,
desorption, diffusion and LH reaction processes involving these sites.
Dissociative O2 adsorption requires two neighboring sites, and is only
hindered by a sizable adsorption barrier of 1.9 eV in the case of adsorption
into two bridge sites. CO adsorption is unimolecular and not hindered by
adsorption barriers.
As only modification of the Rogal model we additionally consider an ER
reaction mechanism, as recently suggested by Hirvi _et al._ for bulk PdO(101)
hirvi_co_2010 . Reaction path calculations vertically impinging a CO molecule
over O atoms adsorbed in the different surface sites indeed also yield rather
low reaction barriers over the $\sqrt{5}$ surface oxide, namely about 0.7 eV
over hollow and essentially zero over bridge. The latter rather astonishing
result has very little consequences for the surface oxide stability and the
catalytic activity in near stoichiometric feeds though. Under corresponding
gas-phase conditions the O coverage of bridge sites is negligible and ER
reaction events involving O in bridge sites correspondinlgy made no
significant contribution to the overall TOF in the simulations.
### II.4 kMC simulation setup
Both 1p-kMC models were implemented using the kmos frameworkhoffmann_kmos_2012
. All simulations were performed in simulation cells containing $(20\times
20)$ unit cells and using periodic boundary conditions. Systematic checks
showed that the quantities of interest here, i.e. the average steady-state
coverages and TOFs, are perfectly converged at these cell sizes. For defined
gas-phase feed conditions $(T,p_{\mathrm{CO}},p_{\mathrm{O}_{2}})$ the 1p-kMC
simulations eventually reach a steady state, with constant TOF and average
surface coverages $\bar{\Theta}_{i,st}$ of species $i$ on site $st$,
$\bar{\Theta}_{i,st}=\frac{\sum_{n}\Theta_{i,st,n}\Delta t_{n}}{\sum_{n}\Delta
t_{n}}\quad,$ (6)
where $n$ denotes the value at the $n$th kMC step after steady state has been
reached. Starting from different initial adsorbate distributions (clean
surface, O-poisoned surface and CO-poisoned surface) and using different
random number seeds we validated that this steady state is well defined, i.e.
we never observed multiple steady-states in this system. In the kinetic phase
diagrams shown in Fig. 2 below the obtained results are summarized in form of
contour plots. All contour plots were interpolated using radial basis
functions in order to reduce numerical noise.
## III Results
### III.1 Kinetic phase diagrams at 600 K
Figure 2: (Color online) 1p-kMC computed O and CO coverage as a function of
partial pressures for $T=600$ K at Pd(100) (upper panels) and at the
$\sqrt{5}$ surface oxide (lower panels). At Pd(100) the coverage refers to O
in hollow sites and CO in bridge sites, at the $\sqrt{5}$ the coverage refers
to O and CO in upper hollow sites (see text). The thick black line in the left
panels indicates the boundary where each phase is expected to be stable: For
Pd(100) this corresponds to the $p\left(2\times 2\right)$ oxygen adlayer, for
PdO$\sqrt{5}$ this is a coverage $>0.9$ ML.
Predicted oxygen and CO coverages on the $\sqrt{5}$, as well as on the Pd(100)
surface are shown in Fig. 2 for a temperature of 600 K. For the Pd(100)
surface the coverages refer to CO in bridge sites and O in hollow sites, as
these are the only available sites for these species in the employed model,
respectively. For the $\sqrt{5}$ surface only the O and CO coverage in the
upper hollow sites is shown, as this is the expected critical descriptor for
the stability of the surface oxide layer rogal_first-principles_2007 . For
Pd(100) a coverage of 1 monolayer (ML) corresponds to one adsorbate (O or CO)
per surface Pd atom. For the $\sqrt{5}$ surface a coverage of 1 ML corresponds
to a complete occupation of the upper hollow rows by the respective species.
Both models yield the intuitively expected coverage variations with partial
pressure. Starting with pure O2 gas conditions (going horizontally along the
bottom of the panels in Fig. 2), both models yield the correct surface
coverages in the respective stable regimes by construction: For Pd(100)
increasing O2 content in the gas phase leads to a gradual coverage increase
starting from the clean surface. For the $\sqrt{5}$ the full ML O coverage in
the bottom right part of the panel in turn reflects a fully intact layer of
the surface oxide in which all upper hollow sites are fully covered by oxygen.
Increasing CO pressure also influences the surface coverages as intuitively
expected: An increasing CO population at the surface defers the stabilization
of surface O at Pd(100) to higher O2 pressures, while it leads to a quicker
depletion of upper hollow O atoms at the $\sqrt{5}$ at decreasing O2
pressures. For highly overstoichiometric CO pressures (upper horizontal line
in the panels) the Pd(100) model displays CO coverages of 0.5 ML. This is less
than the highest CO concentration characterized experimentally, namely 0.75 ML
tushaus_understanding_1990 . This deviation results from the employed
simplified lateral interaction model, which likely features too repulsive CO-
CO interactions. We will scrutinize this with the less repulsive Pd-1NN model
below, but also note that these CO-rich gas-phase conditions (and dense CO
adlayers) are not the focus of our present interest. The same holds for the
incorrect limit of a purely CO-covered $\sqrt{5}$ surface in the upper left
part of the respective panel (where the surface oxide would in reality be
reduced away), as well as for the $c(2\times 2)$ 0.5 ML O coverage in the
lower right part of the Pd(100) panel (where instead a surface or bulk oxide
would be formed).
Figure 3: (Color online) Deduced region of bistability, i.e. gas-phase
conditions where the 1p-kMC models would predict the simultaneous stability of
pristine Pd(100) and the $\sqrt{5}$ surface oxide (see text). All diagram show
the same range of gas phase chemical potentials.
The very transition between CO- and O-covered regimes requires closer
inspection. In the case of the Pd(100) surface a CO-poisoned surface would be
catalytically inactive, but certainly stable. This surface is expected to be
stable at least until the oxygen concentration does not rise above $\sim 0.25$
ML as this corresponds to a coverage regime representative for the
experimentally characterized $p\left(2\times 2\right)$ overlayer
lundgren_kinetic_2004 ; zheng_oxidation_2002 . In Fig. 2 we therefore denote
the stability boundary of the Pd(100) surface at 0.25 ML coverage, i.e. we
would expect a stable Pd(100) surface for any gas-phase conditions to the
upper left of this line. Note that due to the steep coverage rise in the
transition region, this stability boundary would be barely affected on the
scale of Fig. 2 if we had e.g. chosen 0.4 ML coverage as the stability
criterion. In the same spirit Rogal _et al._ have used a coverage exceeding
0.9 ML as stability criterion for the $\sqrt{5}$ surface oxide before
rogal_co_2008 . The corresponding stability boundary for the surface oxide is
also drawn in Fig. 2, and we would expect the surface oxide to be stable
anywhere to the bottom right of this line. Again, on the scale of Fig. 2 it
would make little difference, if a stability criterion of e.g. 0.95 ML or 0.99
ML coverage had been used. With the two stability regions thus quite narrowly
defined, the central and intriguing feature of Fig. 2 is that there is a
finite range of partial pressures where both models are predicted to be
stable. This is highlighted again in Fig. 3, where the resulting bistability
region is marked in green.
### III.2 Variation with temperature
The results presented in the preceding section were obtained for $T=600$ K, a
temperature that falls in the middle of the temperature range from $\sim
300\rm{-}800$ K that is generally most relevant for CO oxidation catalysis.
Not least to make contact with the dedicated reactor STM experiments performed
by Hendriksen et al. at 408-443 K hendriksen_bistability_2005 , an important
next step is to assess the variation of our findings, in particular the
existence of a bistability region, with temperature. Correspondingly and
following the same protocol as before, the two lower panels of Fig. 3
summarize our findings for $T=500$ K and $T=400$ K. Both panels show the same
range of gas-phase chemical potentials as the upper panel for 600 K, allowing
to ascribe all differences between the three panels directly to kinetic
effects, that is deviations from thermodynamic scaling. Intriguingly, these
deviations are rather small, i.e. the differences for the three temperatures
amount primarily to the expected scaling of the pressure axes reuter_first-
principles_2006 . This leaves the central result in form of a bistability over
a finite range of gas-phase pressures essentially untouched within the
temperature range 400-600 K. Particularly intriguing is that certainly for
$T=600$ K the catalytically most relevant near-ambient conditions at
stoichiometric feed fall right within the region of bistability. For the lower
temperatures the bistability region shifts increasingly towards O-rich
conditions, such that e.g. the conditions of the Hendriksen experiments fall
just at the border of this region.
## IV Discussion
### IV.1 Origin and robustness of bistability region
Figure 4: (Color online) ”Constrained” ab initio thermodynamic phase diagram
of the Pd(100)/$\sqrt{5}$ system, considering only ordered structures
consistent with the 1p-kMC lateral interaction models (see text). The thick
dashed line denotes the stability boundary between adsorption phases on
Pd(100) (upper left part) and adsorption phases on the $\sqrt{5}$ surface
oxide (bottom right part). Additionally shown as hatched area is the stability
region of Pd(100) phases with O coverage below 0.5 ML. This is the
thermodynamic equivalent to the bistability region identified in the 1p-kMC
simulations, if the low O coverage defers formation of the (thermodynamically
preferred) surface oxide.
An immediate concern with the observed bistability region is that it is rather
narrow in $(p_{\rm O_{2}},p_{\rm CO})$-space. Considering the range of
uncertainties underlying the 1p-kMC models, prominently the semi-local DFT
energetics and lateral interaction model, this raises doubts as to the
robustness of this finding. Fortunately, in the present case an analysis of
the atomic-scale reason behind the bistability shows that its actual existence
emerges rather independently from these uncertainties, which is why we lead
these two discussions jointly in this section.
A first important step towards an understanding of the atomic-scale origin of
the bistability comes from the observed almost perfect thermodynamic scaling
of the location and extent of the bistability region in $(p_{\rm O_{2}},p_{\rm
CO}$)-space in the temperature range 400-600 K, cf. Fig. 3. This suggests that
the actual reaction kinetics, and the uncertainties in the concomitant
reaction barriers, is not central to its existence. This view is confirmed by
the fact that an equivalent bistability region is already obtained within a
”constrained” ab initio thermodynamics approach reuter_first-principles_2003 ;
reuter_2003 , i.e. an approach that neglects the reaction kinetics completely.
Figure 4 shows the corresponding phase diagram for the same range of chemical
potentials also underlying the panels in Fig. 3. For a comparison to the
1p-kMC results, exactly the same DFT energetics and only ordered structures
consistent with the 1p-kMC lateral interaction models are used. This explains
small differences with respect to the corresponding phase diagram published
before by Rogal, Reuter, and Scheffler rogal_first-principles_2007 , which
e.g. considered additional experimentally characterized CO ordered phases (not
of relevance for the present discussion). Denoted by the thick dashed line is
the stability boundary between adsorption phases on the pristine Pd(100)
surface (upper left part) and adsorption phases on the $\sqrt{5}$ surface
oxide (lower right part). For all catalytically relevant gas-phase conditions,
this boundary runs in fact between CO-covered Pd(100) and different adsorption
phases on the $\sqrt{5}$, i.e. with increasing O-content in the gas phase
constrained thermodynamics predicts an abrupt phase transition from a CO-
poisoned surface directly to the surface oxide. Knowing that a critical local
coverage of about $\sim 0.5$ ML O is necessary to induce the formation of the
surface oxide, it is therefore interesting to also include in the phase
diagram the stability region of O-containing Pd(100) phases with less than 0.5
ML O coverage, i.e. phases where (coming from the pristine metal side) oxide
formation would not yet start. Intriguingly, the resulting region shown in
Fig. 4 extends over similar gas-phase conditions as the bistability region
deduced from the 1p-kMC simulations.
As such we ascribe the bistability region to gas-phase conditions, where the
$\sqrt{5}$ surface oxide is thermodynamically more stable, but where a too low
O coverage at Pd(100) would not readily induce the formation of the oxide.
When crossing the bistability region from O-rich to CO-rich gas-phase
conditions the system will thus prevail in the (thermodynamically preferred)
oxidized state, while in the opposite direction it will prevail in the metal
state as kinetic limitations to stabilize enough oxygen at the surface prevent
the formation of the surface oxide. With this understanding of the atomic-
scale origin, it is primarily the adsorbate binding energies and the lateral
interaction model that are crucial to the robustness of the bistability region
– as they govern the stabilization of oxygen at the surface. More
specifically, it is the relative binding energy differences at the metal and
the $\sqrt{5}$ that will primarily affect the extension of the bistability
region, while the absolute binding energetics will rather shift its location
in $(p_{\rm O_{2}},p_{\rm CO})$-space. We would expect uncertainties in the
employed DFT exchange-correlation functional to rather affect the absolute
binding energetics (i.e. systematic over- or underbinding) and only to a
lesser extent the binding energy differences of O and CO at the two surfaces.
This view is confirmed by systematic tests, in which we increased or decreased
all binding energies by 0.2 eV and in both cases found only small differences
in the extension of the bistability region at more pronounced changes in its
position in $(p_{\rm O_{2}},p_{\rm CO})$-space. Correspondingly, we do not
expect that the predicted gas-phase conditions for the bistability are
accurate to better than some orders of magnitude in pressure. Its actual
existence, however, should be very robust with respect to the uncertainties of
present-day DFT functionals.
Figure 5: (Color online) Comparison of the bistability regions arising from
1p-kMC simulations with different lateral interaction models. Pd(100)-2NN is
the default model employed sofar, where configurations are excluded that would
result in CO adsorbed in nearest-neighbor and next-nearest neighbor positions
(green solid area). Pd(100)-1NN mimics less repulsive interactions and only
excludes configurations resulting in CO adsorbed in nearest-neighbor positions
(hatched area).
This leaves as final important aspect the employed lateral interaction model
on the Pd(100) surface. Already the comparison to the experimentally
determined maximum CO coverage (0.75 ML tushaus_understanding_1990 versus 0.5
ML in Fig. 2) indicated the site-blocking rules in the default Pd(100)-2NN
model to be too repulsive. We assess the consequences for the bistability
region by determining this region with the less repulsive Pd(100)-1NN model,
where now only configurations that would result in CO adsorbed in nearest-
neighbor positions are excluded. The resulting 1p-kMC bistability region at
600 K is shown in Fig. 5 and compared to the corresponding bistability region
deduced before within the default Pd(100)-2NN model. Obviously the eased
stabilization of CO at the Pd(100) surface (maximum coverage now 1 ML) blocks
the O adsorption even more effectively and consequently leads to a much
increased range of gas-phase conditions where both surface states are
predicted to be stable. If we take the comparison to the maximum CO coverage
from experiment as measure, we would expect the true CO-CO repulsion to be
somewhere between the one represented within the Pd(100)-1NN and Pd(100)-2NN
site blocking models. Correspondingly, we would estimate the true extension of
the bistability region to be somewhere between those predicted within the two
models. While thus not fully quantified in its extension, the actual existence
of a bistability region per se is robust against the uncertainty in the
employed lateral interaction model.
### IV.2 Comparison to experiment
The finding of a finite bistability region in our calculations agrees nicely
with the bistability and oscillations reported experimentally by Hendriksen
and coworkers hendriksen_oscillatory_2004 ; hendriksen_bistability_2005 .
Unfortunately, we can not directly compare the temperatures and pressures
where this bistability is found. On the theoretical side this is due to the
aforediscussed uncertainty with respect to the location of this bistability
region in $(p_{\rm O_{2}},p_{\rm CO})$-space caused by the approximate first-
principles parameters and lateral interactions entering the kinetic models. On
the experimental side this is due to the suspected non-negligible mass-
transfer effects in the employed reactor STM setup van_rijn_surface_2011 ;
matera_2009 ; matera_2010 . Nevertheless, the deduced pressure ranges are
intriguingly close, as is the extension of the bistability region. At the
experimental $T\sim 400$ K and $p_{\rm O_{2}}=1$ atm, the measured width of
the bistability region in CO pressure is 0.02 atm hendriksen_bistability_2005
, while it is $\approx 0.5$ atm in the simulations. Also, the finding that at
constant $p_{\rm O_{2}}$, the bistability region shifts with increasing
temperature to increasing $p_{\rm CO}$ agrees with the experimental
observations hendriksen_oscillatory_2004 . As such our interpretation is that
the here obtained bistability between metallic Pd(100) and the $\sqrt{5}$
surface oxide provides an atomic-scale model for the oscillations of the
Hendriksen experiments. This is close to the interpretation arrived at in the
experimental study, with the slight modification that they suspected the
metallic phase to correspond to predominantly O-covered Pd(100), whereas our
simulations show that this is predominantly CO-covered Pd(100).
Figure 6: (Color online) 1p-kMC surface coverages and turn-over-frequencies
(TOFs) at gas-phase conditions comparable to the reactor STM experiments by
Hendriksen et al. hendriksen_bistability_2005 . Shown is the bistability
region in $p_{\rm CO}$ at fixed $T=400$ K and $p_{\rm O_{2}}=1$ atm. Upper
panel: O and CO coverage on the Pd(100) surface; middle panel: O and CO
coverage on the $\sqrt{5}$ surface oxide; lower panel: comparison of the
intrinsic TOFs of the two system states.
The atomic-scale insight into the surface composition provided by our 1p-kMC
simulations also allows to further qualify the reaction mechanism. An
intriguing observation made by Hendriksen and coworkers was that in contrast
to the metal surface, the reaction rate on the oxide did not follow
traditional LH kinetics. They proposed this as a signature of low oxide
stability and a concomitant Mars-van-Krevelen type mechanism, in which the
oxide is continually consumed and reformed. Figure 6 shows that this can
instead be rationalized within a much simpler picture that does not involve
the oxide stability. Shown are the surface coverages on the Pd(100) metal and
$\sqrt{5}$ surface oxide over the $p_{\rm CO}$ pressure range, where we obtain
bistability at fixed $T=400$ K and $p_{\rm O_{2}}=1$ atm, i.e. for gas-phase
conditions comparable to the Hendriksen experiments. At the Pd(100) surface, O
and CO adsorb in inequivalent sites, hollow and bridge, respectively. Due to
the strongly repulsive interactions, mimicked in our simulations by mutual
site blocking, this nevertheless leads effectively to a competition for
adsorption sites between the two adsorbates and in consequence to a LH-type
kinetics. In contrast, on the $\sqrt{5}$ surface oxide, O and CO also adsorb
in inequivalent sites at these gas-phase conditions, namely O almost
exclusively on the hollow sites and CO almost exclusively on the bridge sites,
cf. Fig. 6. At the surface oxide these sites are located further away from
each other and adsorbates even in nearest-neighbor sites experience only very
small lateral interactions, cf. Table I. Consequently, the occupation of these
two site types occurs almost independently of each other at varying gas-phase
conditions and gives rise to a TOF proportional in CO pressure, cf. Fig. 6,
exactly as observed in the experiment. Similarly, we find the TOF to vary only
little when changing the oxygen partial pressure away from the conditions
shown in Fig. 6 – again fully consistent with the experimental findings.
The understanding that the actual reaction mechanism over the $\sqrt{5}$
proceeds by adsorption on the otherwise intact surface oxide suggests that the
continued roughening of the surface observed in the experiments is a by-
product and not essential to the catalytic activity. With respect to the total
activity Fig. 6 shows that over the bistability region both surface states,
Pd(100) and $\sqrt{5}$, exhibit rather similar intrinsic TOFs to within a
factor of three. While the mass-transfer limitations present in the reactor
STM measurements prevent a direct comparison, this finding alone sets the
controversial discussion concerning the active state of the surface into
perspective. At low temperatures and UHV conditions, our 1p-kMC models indeed
yield a significantly higher activity of the Pd(100) surface. This arises
predominantly from the comparatively weak CO binding at the $\sqrt{5}$ surface
oxide and the concomitant limitations in stabilizing it at the surface.
However, at technological gas-phase conditions, which in our simulations do
fall within the bistability region, the TOF differences between the two models
are not large enough and furthermore change sensitively with the detailed gas-
phase conditions, cf. Fig. 6, to support a detailed discussion as to which
state is the more active one. In this respect, the Pd(100) surface differs
qualitatively from the equally prominently discussed Ru/RuO2 system
reuter_review ; over_2003 . On ruthenium the catalytic activity can be clearly
attributed to the oxidized surface, while in the present system two competing
surface states can equally contribute to the catalytic activity, which thus
also explains why a clear signature of palladium (surface) oxide formation may
in some circumstances be more difficult to find.
## V Conclusions
Detailed 1p-kMC simulations on either the Pd(100) surface or the $\sqrt{5}$
surface oxide yield a finite range of CO oxidation gas-phase conditions, where
both surface states appear stable. This finding of a bistability region is
robust against variations in the detailed criteria used to assess the
stability of either state, as well as against the uncertainties arising from
the approximate first-principles energetics and lateral interaction models.
Notwithstanding the latter uncertainties do affect the position where in
$(T,p)$-space the bistability region is found and we expect this to translate
in uncertainties up to the order of 100 K and some orders of magnitude in
pressure.
The bistability region arises from limitations in stabilizing oxygen at the
Pd(100) surface, which extends the stability region of the metallic surface
beyond that predicted by thermodynamics. In some respect this is therefore
reminiscent of the generic Langmuir-Hinshelwood kinetics that lead to
bistability between a CO-poisoned state that effectively blocks O2 adsorption
and an O-rich state, i.e. the bistability of the reaction rate originates
essentially from the inequivalence of the adsorption of carbon monoxide and
oxygen.sales_oscillatory_1982 ; bykov_steady_1981 ; bar_theoretical_1992 ;
zhdanov_kinetic_1994 ; eiswirth_oscillating_1996 . The difference is that in
these classical models the two stable solutions were either supposed to be
different adsorbate phases on the same substrate or did correspond to an
active metal and an inactive oxide state. Instead, in the present system we
have a CO-rich Pd(100) state that blocks oxygen adsorption and the surface
oxide as O-rich state. Furthermore, the bistability does comprise
technological conditions and conditions comparable to the dedicated reactor
STM experiments performed by Hendriksen et al.hendriksen_bistability_2005 ;
hendriksen_role_2010 . Under these conditions, both surface terminations do
show similar intrinsic activity, which sets preceding discussions with respect
to the active state into perspective.
Within the uncertainties of the exact location of the bistability region in
$(T,p)$-space we suggest the here obtained bistability between metallic
Pd(100) and the $\sqrt{5}$ surface oxide as an atomic-scale model for the
oscillations of the Hendriksen experiments. Likewise, our data lends further
support to the view that oxide formation plays an important role in
understanding the catalytic activity of Pd catalysts. If it is even the
oscillations themselves, catalytic activity would again be rationalized as a
kinetic phase transition phenomenon, i.e. the catalyst surface being close to
an instabilityziff_kinetic_1986 ; evans_kinetic_1991 ; reuter_kMC . At present
our simulations performed separately on the two intact surface states,
corresponding to ideal terraces of Pd(100) and the $\sqrt{5}$ surface oxide,
can not address this notion directly. This holds equally for the experimental
interpretation that the continued roughening of the oxidic surface during
reaction, as well as the formation of steps are crucial ingredients to the
oscillatory behavior. Our ongoing work is directed to extend the 1p-kMC
capabilities towards the actual oxide formation and decomposition, which will
then allow us to start scrutinizing these points with predictive-quality
theory.
## VI Acknowledgements
We gratefully acknowledge support from the German Research Council (DFG) and
the TUM Faculty Graduate Center Chemistry, as well as generous computing time
at the Supercomputing Center of the Max-Planck-Society, Garching. We thank
Matthias Scheffler for insightful discussions and his continued support for
this project.
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|
arxiv-papers
| 2013-01-15T10:08:09 |
2024-09-04T02:49:40.305638
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Max J. Hoffmann and Karsten Reuter",
"submitter": "Tongyu Wang",
"url": "https://arxiv.org/abs/1301.3285"
}
|
1301.3395
|
# Predictions for $p+$Pb Collisions at $\sqrt{s_{{}_{NN}}}=5$ TeV
JAVIER L. ALBACETE IPNO, Université Paris-Sud 11, CNRS/IN2P3, 91406 Orsay,
France NESTOR ARMESTO Departamento de Física de Partículas and IGFAE,
Universidade de Santiago de Compostela, 15706 Santiago de Compostela, Galicia,
Spain RUDOLF BAIER Fakultät für Physik, Universität Bielefeld, D-33501,
Bielefeld, Germany GERGELY G. BARNAFÖLDI Institute for Particle and Nuclear
Physics, Wigner Research Centre for Physics, Hungarian Academy of Sciences,
P.O.Box 49, Budapest, 1525, Hungary JEAN BARRETTE McGill University,
Montreal, H3A 2T8, Canada SOMNATH DE Variable Energy Cyclotron Centre, 1/AF,
Bidhan Nagar, Kolkata, 700064, India WEI-TIAN DENG Theory Center, IPNS, KEK,
1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan ADRIAN DUMITRU Department of
Natural Sciences, Baruch College, CUNY, 17 Lexington Avenue, New York, NY
10010, USA RIKEN BNL Research Center, Brookhaven National Laboratory, Upton,
NY 11973, USA KEVIN DUSLING Physics Department, North Carolina State
University, Raleigh, NC 2 7695, USA KARI J. ESKOLA Department of Physics,
P.O. Box 35, FI-40014 University of Jyväskylä, Finland Helsinki Institute of
Physics, P.O. Box 64, FIN-00014 University of Helsinki, Finland RAINER FRIES
Cyclotron Institute and Department of Physics and Astronomy, Texas A&M
University, College Station, TX 77843, USA HIROTSUGU FUJII Institute of
Physics, University of Tokyo, Komaba, Tokyo 153-8902, Japan FRANCOIS GELIS
Institut de Physique Théorique, CEA, 91191 Gif-sur-Yvette Cedex, France
MIKLOS GYULASSY Department of Physics, Columbia University, New York, NY
10027, USA WIGNER RCP, Institute for Particle and Nuclear Physics P.O.Box 49,
Budapest, 1525, Hungary YUNCUN HE Key Laboratory of Quark $\&$ Lepton
Physics (MOE) and Institute of Particle Physics, Central China Normal
University, Wuhan 430079, China ILKKA HELENIUS Department of Physics, P.O.
Box 35, FI-40014 University of Jyväskylä, Finland Helsinki Institute of
Physics, P.O. Box 64, FIN-00014 University of Helsinki, Finland ZHONG-BO KANG
Theoretical Division, MS B283, Los Alamos National Laboratory, Los Alamos, NM
87545, USA BORIS Z. KOPELIOVICH Departamento de Física, Universidad Técnica
Federico Santa María, Avda. España 1680, Casilla 110-V, Valparaíso, Chile
KRZYSZTOF KUTAK Instytut Fizyki Jadrowej im. Henryka Niewodniczańskiego,
Radzikowskiego 152, 31-342 Kraków, Poland PETER LEVAI Institute for Particle
and Nuclear Physics, Wigner Research Centre for Physics, Hungarian Academy of
Sciences, P.O.Box 49, Budapest, 1525, Hungary ZI-WEI LIN C-209 Howell
Science Complex, Department of Physics, East Carolina University, Greenville,
NC 27858, USA ALFRED H. MUELLER Department of Physics, Columbia University,
New York, NY 10027, USA YASUSHI NARA Akita International University, Yuwa,
Akita-city 010-1292, Japan JAN NEMCHIK Czech Technical University in Prague,
FNSPE, Břehová 7, 11519 Prague, Czech Republic GÁBOR PAPP Eötvös Loránd
University, Pázmány Péter sétány 1/A, H-1117, Budapest, Hungary MIHAI
PETROVICI National Institute for Physics and Nuclear Engineering, Horia
Hulubei, R-077125, Bucharest, Romania JIAN-WEI QIU Physics Department,
Brookhaven National Laboratory, Upton, NY 11973, USA C.N. Yang Institute for
Theoretical Physics, Stony Brook University, Stony Brook, NY 11794, USA AMIR
H. REZAEIAN Departamento de Física, Universidad Técnica Federico Santa María,
Avda. España 1680, Casilla 110-V, Valparaíso, Chile PENG RU School of
Physics $\&$ Optoelectronic Technology, Dalian University of Technology,
Dalian, 116024 China Institute of Particle Physics, Central China Normal
University, Wuhan, 430079 China DOMINIQUE SCHIFF LPT, Université Paris-Sud,
Bâtiment 210, F-91405 Orsay, France SEBASTIAN SAPETA Institute for Particle
Physics Phenomenology, Durham University, South Rd, Durham DH1 3LE, UK VASILE
TOPOR POP McGill University, Montreal, H3A 2T8, Canada PRITHWISH TRIBEDY
Variable Energy Cyclotron Centre, 1/AF Bidhan Nagar, Kolkata-70006 4, India
RAJU VENUGOPALAN Physics Department, Brookhaven National Laboratory, Upton,
NY 11973, USA IVAN VITEV Theoretical Division, MS B283, Los Alamos National
Laboratory, Los Alamos, NM 87545, USA RAMONA VOGT Physics Division, Lawrence
Livermore National Laboratory, Livermore, CA 94551, USA Physics Department,
University of California at Davis, Davis, CA 95616, USA
[email protected] ENKE WANG Key Laboratory of Quark $\&$ Lepton
Physics (MOE) and Institute of Particle Physics, Central China Normal
University, Wuhan 430079, China XIN-NIAN WANG Key Laboratory of Quark and
Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal
University, Wuhan 430079, China Nuclear Science Division, MS 70R0319,
Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA HONGXI XING
Institute of Particle Physics, Central China Normal University, Wuhan 430079,
China RONG XU Key Laboratory of Quark and Lepton Physics (MOE) and Institute
of Particle Physics, Central China Normal University, Wuhan 430079, China
BEN-WEI ZHANG Key Laboratory of Quark $\&$ Lepton Physics (MOE) and Institute
of Particle Physics, Central China Normal University, Wuhan 430079, China
Nuclear Science Division, MS 70R0319, Lawrence Berkeley National Laboratory,
Berkeley, CA 94720, USA WEI-NING ZHANG School of Physics $\&$ Optoelectronic
Technology, Dalian University of Technology, Dalian, 116024 China Physics
Department, Harbin Institute of Technology, Harbin 150006, China
###### Abstract
Predictions for charged hadron, identified light hadron, quarkonium, photon,
jet and gauge bosons in $p+$Pb collisions at $\sqrt{s_{{}_{NN}}}=5$ TeV are
compiled and compared. When test run data are available, they are compared to
the model predictions.
###### keywords:
perturbative QCD, hard probes of heavy-ion collisions
12.38.Bx, 25.75.Bh, 25.75.Cj, 13.87.-a
## 1 Introduction
Here predictions for the upcoming $p+$Pb LHC run at $\sqrt{s_{{}_{NN}}}=5$ TeV
compiled by members and friends of the JET Collaboration [1] are presented.
The test run data published by the ALICE Collaboration [2, 3] are compared to
model calculations available before the test run. Most calculations are for
midrapidity and minimum bias collisions. Other results at different rapidities
and centralities are presented when available.
The predictions presented here, as well as the corresponding discussion, were
made assuming that the proton circulated toward positive rapidity and the
nucleus toward negative rapidity, $p+$Pb collisions, similar to fixed-target
configurations. In the fixed-target configuration, the low $x$ nuclear parton
distributions are probed at positive rapidity. Throughout this paper, many of
the results shown have been adjusted to the Pb$+p$ convention of the ALICE
data, as described in Ref. [2], with the nuclear parton density probed at high
$x$ at forward (positive) rapidity and low $x$ at backward (negative)
rapidity. The cases where the results still appear with the assumption that
the proton circulates in the direction of positive rapidity are explicitly
noted.
This paper is organized in the following fashion. Section 2 describes the
models that specifically address charged particle production. These include
saturation approaches, event generators, and perturbative QCD-based
calculations. Section 3 compares results obtained from models described in
Sec. 2 with each other and with the available data. The next several sections
present predictions for specific observables including identified light
hadrons (Sec. 4), quarkonium (Sec. 5), direct photons (Sec. 6), jets (Sec. 7),
and gauge bosons (Sec. 8).
## 2 Model descriptions
In this section, the models used to obtain the results for charged particle
distributions, $dN_{\rm ch}/d\eta$, $dN_{\rm ch}/dp_{T}$ and the nuclear
suppression factor $R_{p{\rm Pb}}$ as a function of $p_{T}$ and $\eta$. The
first calculations described Sec. 2.1 are saturation or Color Glass Condensate
(CGC) based. The next set of calculations are event-generator based with
results from $\mathtt{HIJING}$ in Sec. 2.3, $\mathtt{HIJINGB\overline{B}2.0}$
in Sec. 2.4, and $\mathtt{AMPT}$ in Sec. 2.5. Finally, calculations based on
collinear factorization in perturbative QCD are described in Secs. 2.6 and
2.7.
In the following Section, Sec. 3, the predictions will be compared to the
ALICE Pb$+p$ test run data [2, 3] in September 2012.
### 2.1 Inclusive hadron production in the rcBK-CGC approach (A. Rezaeian)
In the Color Glass Condensate (CGC) approach, gluon jet production in $p+A$
collisions can be described by $k_{T}$-factorization [4],
$\frac{d\sigma}{dy\,d^{2}p_{T}}=\frac{2\alpha_{s}}{C_{F}}\frac{1}{p^{2}_{T}}\int
d^{2}\vec{k}_{T}\phi^{G}_{p}\left(x_{1};\vec{k}_{T}\right)\phi^{G}_{A}\left(x_{2};\vec{p}_{T}-\vec{k}_{T}\right),$
(1)
where $C_{F}=(N_{c}^{2}-1)/2N_{c}$, $N_{c}$ is the number of colors,
$x_{1,2}=(p_{T}/\sqrt{s})e^{\pm y}$, $p_{T}$ and $y$ are the transverse
momentum and rapidity of the produced gluon jet, and $\sqrt{s}$ is the
nucleon-nucleon center-of-mass energy. The unintegrated gluon density,
$\phi^{G}_{A}(x_{i};\vec{k}_{T})$, denotes the probability to find a gluon
that carries fractional energy $x_{i}$ and transverse momentum $k_{T}$ in the
projectile (or target) $A$. The unintegrated gluon density is related to the
color dipole forward scattering amplitude,
$\phi^{G}_{A}\left(x_{i};\vec{k}_{T}\right)=\frac{1}{\alpha_{s}}\frac{C_{F}}{(2\pi)^{3}}\int
d^{2}\vec{b}_{T}\,d^{2}\vec{r}_{T}\,e^{i\vec{k}_{T}\cdot\vec{r}_{T}}\,\nabla^{2}_{T}\mathcal{N}_{A}\left(x_{i};r_{T};b_{T}\right),$
(2)
with
$\mathcal{N}_{A}\left(x_{i};r_{T};b_{T}\right)=2\mathcal{N}_{F}\left(x_{i};r_{T};b_{T}\right)-\mathcal{N}^{2}_{F}\left(x_{i};r_{T};b_{T}\right),$
(3)
where $r_{T}$ is the transverse size of the dipole and $b_{T}$ is the impact
parameter of the collision. The subscript $T$ stands for the transverse
component. The dipole scattering amplitude $\mathcal{N}_{F}$ satisfies the
nonlinear small-$x$ JIMWLK evolution equations [5, 6, 7, 8], see below.
In the $k_{T}$-factorized approach, partons in both the projectile and target
are assumed to be at very small $x$ so that the CGC formalism and small-$x$
resummation is applicable to both the projectile and the target, assuming the
projectile proton moves in the direction of forward rapidity. This approach is
valid away from the projectile fragmentation region. However, to treat the
projectile fragmentation region in the forward region, an alternative approach
developed in Refs. [9, 10], the so-called hybrid approach, is better suited.
In this approach, the projectile is treated perturbatively within the standard
collinear factorization scheme using the standard DGLAP picture while the
target is treated employing CGC methods. The cross section for single
inclusive hadron production at leading twist in asymmetric collisions such as
$p+A$ in the CGC approach is given by [9, 10],
$\displaystyle\frac{dN^{pA\rightarrow hX}}{d\eta d^{2}p_{T}}$ $\displaystyle=$
$\displaystyle\frac{K}{(2\pi)^{2}}\Bigg{[}\int_{x_{F}}^{1}\frac{dz}{z^{2}}\Big{[}x_{1}f_{g}(x_{1},\mu_{F}^{2})N_{A}(x_{2},\frac{p_{T}}{z})D_{h/g}(z,\mu_{\rm
Fr})$ (4) $\displaystyle+$
$\displaystyle\Sigma_{q}x_{1}f_{q}(x_{1},\mu_{F}^{2})N_{F}(x_{2},\frac{p_{T}}{z})D_{h/q}(z,\mu_{\rm
Fr})\Big{]}$ $\displaystyle+$ $\displaystyle\frac{\alpha_{s}^{\rm
in}}{2\pi^{2}}\int_{x_{F}}^{1}\frac{dz}{z^{2}}\frac{z^{4}}{p_{T}^{4}}\int_{k_{T}^{2}<\mu_{F}^{2}}d^{2}k_{T}k_{T}^{2}N_{F}(k_{T},x_{2})\int_{x_{1}}^{1}\frac{d\xi}{\xi}$
$\displaystyle\times$
$\displaystyle\Sigma_{i,j=q,\bar{q},g}w_{i/j}(\xi)P_{i/j}(\xi)x_{1}f_{j}(\frac{x_{1}}{\xi},\mu_{F})D_{h/i}(z,\mu_{\rm
Fr})\Bigg{]}\,\,.$
A $K$-factor has been introduced to effectively incorporate higher-order
corrections. The parton distribution function of the proton,
$f_{j}(x,\mu_{F}^{2})$, depends on the light-cone momentum fractions $x$ and
the hard factorization scale $\mu_{F}$. The function $D_{h/i}(z,\mu_{\rm Fr})$
is the fragmentation function (FF) of parton $i$ to become final-state hadron
$h$ carrying a fraction $z$ of the parent parton momentum at fragmentation
scale $\mu_{\rm Fr}$. The inelastic weight functions, $w_{i/j}$, and the DGLAP
splitting functions, $P_{i/j}$, are given in Ref. [10]. The longitudinal
momentum fractions $x_{1}$ and $x_{2}$ are
$x_{F}\approx\frac{p_{T}}{\sqrt{s}}e^{\eta};\ \ \ \ x_{1}=\frac{x_{F}}{z};\ \
\ \ \ x_{2}=x_{1}e^{-2\eta}.$ (5)
The strong coupling multiplying the inelastic term in Eq. (4), is denoted
$\alpha_{s}^{\rm in}$. The superscript “in” is employed to differentiate it
from the running value of $\alpha_{s}$ in the rcBK equation. In the hybrid
formulation, the strong coupling in the dilute regime of the projectile can
differ from that in the rcBK description of the dense target (or dipole
scattering amplitude). The scale at which $\alpha_{s}^{\rm in}$ should be
evaluated cannot be determined in the current approximation, a full NNLO
calculation is required. The effects of different choices of $\alpha_{s}^{\rm
in}$ will be considered later.
In Eq. (4), the factorization scale $\mu_{F}$ is assumed to be the same in the
fragmentation functions ($\mu_{\rm Fr}=\mu_{F}$) and the parton densities. In
order to investigate the uncertainties associated with choice of $\mu_{F}$,
several values of $\mu_{F}$ are considered: $\mu_{F}=2p_{T}$; $p_{T}$; and
$p_{T}/2$.
In Eq. (4), the amplitude $N_{F}$ ($N_{A}$) is the two-dimensional Fourier
transform of the imaginary part of the forward (proton direction) dipole-
target scattering amplitude, $\mathcal{N}_{A(F)}$, in the fundamental ($F$) or
adjoint ($A$) representations,
$N_{A(F)}(x,k_{T})=\int
d^{2}\vec{r}e^{-i\vec{k}_{T}.\vec{r}}\left[1-\mathcal{N}_{A(F)}(r,Y=\ln\Big{(}\frac{x_{0}}{x}\Big{)})\right],$
(6)
where $r=|\vec{r}|$ is the dipole transverse size. The dipole scattering
amplitude $\mathcal{N}_{A(F)}$ incorporates small-$x$ dynamics and can be
calculated using the JIMWLK evolution equation [5, 6, 7, 8]. In the large
$N_{c}$ limit, the coupled JIMWLK equations are simplified to the Balitsky-
Kovchegov (BK) equation [11, 12, 13, 14], a closed-form equation for the
rapidity evolution of the dipole amplitude. While a numerical solution of the
full next-to-leading logarithmic expressions is not yet available, the running
coupling corrections to the leading log kernel, the so-called running-coupling
BK (rcBK) equation has been very successful in phenomenological applications
[15]. The rcBK equation has the following simple form [11, 12, 13, 14, 16]:
$\displaystyle\frac{\partial\mathcal{N}_{A(F)}(r,x)}{\partial\ln(x_{0}/x)}$
$\displaystyle=$ $\displaystyle\int d^{2}{\vec{r}_{1}}\ K^{{\rm
run}}({\vec{r}},{\vec{r}_{1}},{\vec{r}_{2}})\left[\mathcal{N}_{A(F)}(r_{1},x)+\mathcal{N}_{A(F)}(r_{2},x)\right.$
(7) $\displaystyle-$
$\displaystyle\left.\mathcal{N}_{A(F)}(r,x)-\mathcal{N}_{A(F)}(r_{1},x)\,\mathcal{N}_{A(F)}(r_{2},x)\right],$
where $\vec{r}_{2}\equiv\vec{r}-\vec{r}_{1}$. The rcBK equation only describes
the rapidity/energy evolution of the dipole, the initial profile and
parameters of the dipole still need to be modeled and constrained by
experimental data. The initial condition for the evolution generally takes a
form motivated by the McLerran-Venugopalan model [17, 18, 19],
$\mathcal{N}(r,Y\\!=\\!0)=1-\exp\left[-\frac{\left(r^{2}\,Q_{0s}^{2}\right)^{\gamma}}{4}\,\ln\left(\frac{1}{\Lambda\,r}+e\right)\right],$
(8)
where the onset of small-$x$ evolution is assumed to be at $x_{0}=0.01$, and
the infrared scale is $\Lambda=0.241$ GeV [15]. The only free parameters are
$\gamma$ and the initial saturation scale $Q_{0s}$, with $s=p$ and $A$ for
proton and nuclear targets, respectively. Unfortunately, the current global
set of small-$x$ data are very limited and thus cannot uniquely fix the
initial dipole parameters [15]. This problem is more severe for determining
the dipole scattering amplitude on nuclear targets, leading to rather large
unavoidable theoretical uncertainties on CGC predictions for $p+A$ collisions
at the LHC. In Ref. [20], a simple scheme to test the CGC dynamics at the LHC
was proposed. This scheme will be used to calculate the results shown later
on.
### 2.2 IP-Sat (P. Tribedy and R. Venugopalan)
The impact parameter dependent dipole saturation model (IP-Sat) [21] is a
refinement of the Golec-Biernat–Wusthoff dipole model [22, 23] to give the
right perturbative limit when the dipole radius $r_{T}\rightarrow 0$ [24]. It
is equivalent to the expression derived in the classical effective theory of
the CGC, to leading logarithmic accuracy [25, 26]. The proton dipole cross
section in this model is expressed as
$\displaystyle\frac{d\sigma^{p}_{\rm
dip}}{d^{2}b_{T}}(r_{T},x,b_{T})=2\left[1-\exp\left(-\frac{\pi^{2}}{2N_{c}}r_{T}^{2}\alpha_{s}(\mu^{2})xg(x,\mu^{2})T_{p}(b_{T})\right)\right]\,.$
(9)
Here the scale $\mu^{2}$ is related to dipole radius $r_{T}$ as
$\mu^{2}=\frac{4}{r_{T}^{2}}+\mu_{0}^{2}\,\,,$ (10)
where the leading order expression for the running coupling is
$\alpha_{s}(\mu^{2})=\frac{12\pi}{(33-2n_{f})\log(\mu^{2}/\Lambda_{\rm
QCD}^{2})}$ (11)
with $n_{f}=3$ and $\Lambda_{\rm QCD}=0.2$ GeV. The model includes saturation
as eikonalized power corrections to the DGLAP leading-twist expression and may
be valid in the regime where logs in $Q^{2}$ dominate logs in $x$. For each
value of the dipole radius, the gluon density $xg(x,\mu^{2})$ is evolved from
$\mu_{0}^{2}$ to $\mu^{2}$ using the LO DGLAP evolution equation without
quarks,
$\frac{\partial
xg(x,\mu^{2})}{\partial\log\mu^{2}}=\frac{\alpha_{s}(\mu^{2})}{2\pi}\int\limits_{x}^{1}dzP_{gg}(z)\frac{x}{z}g\left(\frac{x}{z},\mu^{2}\right)\,\,.$
(12)
Here the gluon splitting function is
$P_{gg}(z)=6\left[\frac{z}{(1-z)+}+\frac{1-z}{z}+z(1-z)\right]+\left(\frac{11}{2}-\frac{n_{f}}{3}\right)\delta(1-z)\,\,.$
(13)
The initial gluon density at the scale $\mu^{2}_{0}$ is taken to be of the
form
$xg(x,\mu^{2}_{0})=A_{g}x^{-\lambda_{g}}(1-x)^{5.6}\,\,.$ (14)
An important feature of the IP-Sat model is the $b$-dependence of the dipole
cross section, introduced through the gluon density profile function
$T_{p}(b_{T})$. This profile function, normalized to unity, is chosen to have
the Gaussian form
$T_{p}(b_{T})=\frac{1}{2\pi B_{G}}\exp\left({-b_{T}^{2}\over
2B_{G}}\right)\,\,,$ (15)
where $B_{G}$ is a parameter fit to the HERA diffractive data. It corresponds
to $\langle b^{2}\rangle=2B_{G}$, the average squared gluonic radius of the
proton.
The IP-Sat model parameters are obtained from optimal fits to HERA data [32].
The parameters used in this work are listed in Table 2.2.
Parameters of the IP-Sat model obtained from fits to HERA data [32]. $B_{G}$
(GeV-2) $\mu_{0}$ (GeV2) $A_{g}$ $\lambda_{g}$ 4.0 1.17 2.55 0.020
The parameters of the initial gluon distribution are determined from fits to
the HERA $F_{2}$ data [28, 29] with $\chi^{2}\sim 1$. The value of $B_{G}$ is
determined primarily from the $J/\psi$ $t$-distributions measured by ZEUS [30]
and H1 [31]. With these parameters, excellent agreement with the HERA
exclusive vector meson and DVCS data is obtained. For a detailed comparison of
this model to the HERA data, see Ref. [32]. A more recent fit to the combined
ZEUS and H1 data has been performed in Ref. [33].
The IP-Sat model successfully describes the bulk features of the $p+p$ and
$A+A$ data over a wide range of center-of-mass energies from RHIC to LHC as
well as the features of the RHIC d+Au data [34, 35]. It also provides the
basis for the IP-Glasma model [36, 37] of initial conditions in heavy-ion
collisions.
In the IP-Sat model, the dipole-nucleus cross section in a large nucleus can
be approximated as
$\frac{d\sigma^{A}_{\rm dip}}{d^{2}s_{T}}\approx
2\left[1-\exp\left\\{-\frac{AT_{A}(s_{T})}{2}\sigma_{\rm
dip}^{p}(r_{T},x)\right\\}\right]$ (16)
where $AT_{A}(s_{T})$ is the transverse density of nucleons inside a nucleus
and $\sigma_{\rm dip}^{p}(r_{T},x)$ is obtained by integrating the dipole-
proton cross section in Eq. (9) over the impact parameter distribution in the
proton. This form of the dipole-nucleus cross section was previously shown to
give reasonable fits to the limited available inclusive fixed-target $e+A$
data [38].
In $p+A$ collisions, the LO inclusive gluon distribution can be expressed as
[39]
$\frac{{d}N_{g}^{pA}({b}_{T})}{{d}y~{}{d}^{2}{p}_{T}}=\frac{4\alpha_{s}}{\pi
C_{F}}\frac{1}{p_{T}^{2}}\int\frac{{d}^{2}{k}_{T}}{(2\pi)^{5}}\int{d}^{2}{s}_{T}\frac{{d}\phi_{p}(x_{1},{k}_{T}|{s}_{T})}{{d}^{2}{s}_{T}}\frac{{d}\phi_{A}(x_{2},{p}_{T}-{k}_{T}|{s}_{T}-{b}_{T})}{{d}^{2}{s}_{T}}\,\,.$
(17)
This equation is a generalization of the well known $k_{T}$-factorized
expression for inclusive gluon production [40] to include the impact parameter
dependence of the unintegrated gluon distributions. Here
$C_{F}=(N_{c}^{2}-1)/2N_{c}$ is the Casimir for the fundamental
representation. Using a relation between quark and gluon dipole amplitudes
strictly valid in the large $N_{c}$ limit, the unintegrated gluon distribution
in protons and nuclei can be expressed in terms of the corresponding dipole
cross section measured in DIS as [41]
$\frac{{d}\phi^{p,A}(x,{k}_{T}|{s}_{T})}{{d}^{2}{s}_{T}}=\frac{{k}_{T}^{2}N_{c}}{4\alpha_{s}}\int\limits_{0}^{\infty}{d}^{2}{r}_{T}e^{i\vec{k_{T}}.\vec{r_{T}}}\left[1-\frac{1}{2}\,\frac{d\sigma^{p,A}_{\rm
dip}}{d^{2}{s}_{T}}(r_{T},x,{s}_{T})\right]^{2}\,\,.$ (18)
### 2.3 $\mathtt{HIJING2.1}$ (R. Xu, W.-T. Deng and X.-N. Wang)
The $\mathtt{HIJING}$ [43, 44] Monte Carlo is based on a two-component model
of hadron production in high-energy $p+p$, $p+A$, and $A+A$ collisions. The
soft and hard components are separated by a cutoff momentum $p_{0}$ in the
transverse momentum exchange. Hard parton scatterings with $p_{T}>p_{0}$ are
assumed to be described by perturbative QCD (pQCD), while soft interactions
are approximated by string excitations with an effective cross section
$\sigma_{\mathrm{soft}}$. In $p+A$ collisions, the single jet inclusive cross
section is proportional to the nuclear parton densities
$f_{a/A}(x_{2},p_{T}^{2},b)$,
$\displaystyle\frac{{d}\sigma^{\rm
jet}_{pA}}{{d}y_{1}{d}^{2}p_{T}}=K\int{d}y_{2}\,{d}^{2}b\,T_{A}(b)\sum_{a,b,c}x_{1}f_{a/p}(x_{1},p^{2}_{T})x_{2}f_{a/A}(x_{2},p^{2}_{T},b)\frac{{d}\sigma_{ab\rightarrow
cd}}{{d}t}\,\,.$
Here, $x_{1,2}=p_{T}(e^{\pm y_{1}}+e^{\pm y_{2}})/\sqrt{s}$ are the fractional
momenta of the initial partons while $y_{1,2}$ are the rapidities of the final
parton jets. Higher order corrections are absorbed into the $K$ factor. The
nuclear thickness function is normalized to $A$, $\int{d}^{2}b\,T_{A}(b)=A$.
There are several cold nuclear matter effects that are considered in
$\mathtt{HIJING}$. The first is the shadowing effect. $\mathtt{HIJING2.0}$
[45, 46] employs a factorized form of the parton densities in nuclei[47],
$f_{a/A}(x,\mu_{F}^{2},b)=S_{a/A}(x,\mu_{F}^{2},b)f_{a/A}(x,\mu_{F}^{2})$ (19)
where $S_{a/A}(x,\mu_{F}^{2},b)$ is the impact-parameter dependent nuclear
modification factor. However, the shadowing employed in $\mathtt{HIJING2.0}$
does not include any $\mu_{F}^{2}$ dependence as in e.g. Ref. [48]. Therefore,
shadowing effects in $\mathtt{HIJING}$ should only be valid at low $p_{T}$ and
disappear at larger $p_{T}$.
The second cold matter effect included is the Cronin effect [49], the
enhancement of intermediate $p_{T}$ hadron spectra in $p+A$ collisions.
Multiple scattering inside a nucleus can lead to the transverse momentum
($k_{T}$) broadening of both the initial- and final-state partons. A
$k_{T}$-kick is imparted to both the initial and final-state hard scattered
partons in each binary nucleon-nucleon scattering. The $k_{T}$-kick of each
scattering follows a Gaussian distribution. Fits to the fixed-target $p+A$
data lead to an energy dependence of the Gaussian width,
$\langle k^{2}_{T}\rangle=[0.14{\log}(\sqrt{s}/{\rm GeV})-0.43]\,{\rm
GeV}^{2}/c^{2}\,\,.$ (20)
This $k_{T}$-kick influences the final-state hadron rapidity distribution.
After tuning the gluon shadowing parameter $s_{g}$ [47] in
$\mathtt{HIJING2.1}$, the charged particle rapidity distribution, $dN_{\rm
ch}/d\eta$, in d+Au collisions at $\sqrt{s_{{}_{NN}}}=200$ GeV can be
described. The prediction for LHC energies can be obtained by extrapolation
[50].
In the default $\mathtt{HIJING}$ setting, $p+A$ and $A+A$ collision are
decomposed into independent and sequential nucleon-nucleon collisions. Within
each nucleon-nucleon collision, hard collisions are simulated first, followed
by soft collisions. However, since the time scale for hard scattering is much
shorter than soft interactions, such a sequence of hard and soft interactions
within each binary collision might not be physical. In a revised scheme
denoted by DHC (decoherent hard scattering), all the hard interactions in a
$p+A$ event are simulated first. They are subsequently followed by the soft
interactions. As a consequence, the energy available in each hard scattering
is no longer restricted by soft interactions.
An additional cold matter effect arises from valence quark number conservation
in the proton. In $p+A$ collisions, the projectile proton will suffer multiple
scatterings within the target nucleus. For each binary nucleon-nucleon
collision, there is a finite probability for independent hard parton
scattering involving initial partons from the projectile and target nucleons.
Flavor conservation limits the availability of valence quarks from the
projectile for each of these hard interactions. This effect can change the
relative flavor composition of produced partons per average binary nucleon-
nucleon collision. Since the gluon fragmentation functions are softer than
those of the quarks, the increased fraction of produced gluon jets in $p+A$
collisions can lead to suppression of the final-state high-$p_{T}$ hadron
spectra.
Finally, jet fragmentation can also modify the final hadron spectra in $p+A$
collisions. In the default $\mathtt{HIJING}$ setup, jet shower partons from
initial- and final-state radiation are ordered in rapidity. Gluons are
connected to the valence quark and diquark of the projectile or target
nucleons as kinks to form systems of strings. These strings fragment into
final-state hadrons using the Lund string fragmentation model [51]. In $p+A$
collisions, the projectile can undergo multiple scatterings since its string
systems have many more gluons attached to them than in $p+p$ collisions.
Hadrons produced by the fragmentation of such string systems are softer than
those resulting from independent fragmentation of individual gluons.
### 2.4 $\mathtt{HIJINGB\overline{B}2.0}$ (G. G. Barnaföldi, J. Barette, M.
Gyulassy, P. Levai, M. Petrovici, and V. Topor Pop)
Monte Carlo models such as $\mathtt{HIJING1.0}$ [52, 43], $\mathtt{HIJING2.0}$
[45, 46] and $\mathtt{HIJINGB\overline{B}2.0}$ [53, 54, 55, 56] have been
developed to study hadron production in $p+p$, $p+A$ and $A+A$ collisions.
They are essentially two-component models which describe the production of
hard parton jets and the soft interaction between nucleon remnants. Hard jet
production is calculated employing collinearly-factorized multiple minijet
production within pQCD. A transverse momentum cut-off, $p_{0}$, on the final-
state jet production is introduced so that for $p_{T}<p_{0}$ the interaction
is nonperturbative and is characterized by a finite soft parton cross section
$\sigma_{\rm soft}$. The jet cross sections depend on the parton distribution
functions parameterized from global fits to data [45, 46].
Nucleon remnants interact via soft gluon exchanges described by the string
models [57, 58, 59] and constrained from lower energy $e^{+}+e^{-}$,
$e^{\pm}+p$, and $p+p$ data. The hard jet pairs and the two excited nucleon
remnants are connected by independent strings which fragment to resonances
that decay to the final-state hadrons. Longitudinal beam-jet string
fragmentation depends strongly on the values of the string tensions that
control the quark-antiquark ($q\bar{q}$) and diquark-antidiquark
($qq\overline{qq}$) creation rates and strangeness suppression factors
($\gamma_{s}$).
In $\mathtt{HIJING1.0}$ and $\mathtt{HIJING2.0}$, a constant (vacuum value)
for the effective value of string tension, $\kappa_{0}=1.0$ GeV/fm, is used.
At high initial energy density, the novel nuclear physics is due to the
possibility of overlapping multiple longitudinal flux tubes leading to strong
longitudinal color field (SCF) effects. These effects are modeled in
$\mathtt{HIJINGB\overline{B}2.0}$ by varying the effective string tension.
SCFs also modify the fragmentation processes, resulting in an increase of
(strange)baryons which play an important role in the description of the baryon
to meson anomaly. In order to describe the $p+p$ and central Pb+Pb data at the
LHC, we have shown that the energy and mass dependence of the mean value of
the string tension should be taken into account [53, 54]. Moreover, to better
describe the baryon to meson anomaly seen in the data, a specific
implementation of JJ̄ loops, has to be introduced. For a detailed discussion,
see Refs. [53, 54, 56]. Similar results can be obtained by including extra
diquark-antidiquark production channels from the strong coherent fields formed
in heavy-ion collisions [60].
All $\mathtt{HIJING}$-type models implement nuclear effects such as
modification of the parton distribution functions, shadowing, and jet
quenching via medium-induced parton splitting. (Collisional energy loss is
neglected [52, 43].) In $\mathtt{HIJING1.0}$ and
$\mathtt{HIJINGB\overline{B}2.0}$, the Duke-Owen (DO) parameterizations of the
proton parton densities [61] is used to calculate the jet production cross
section with $p_{T}>p_{0}$. In both codes, a constant cutoff, $p_{0}=2$
GeV/$c$, and a soft parton cross section, $\sigma_{\rm soft}=54$ mb, fit the
experimental $p+p$ data. However, for $A+A$ collisions in
$\mathtt{HIJINGB\overline{B}2.0}$, an energy and mass dependence of the cut-
off parameter, $p_{0}(s,A)=0.416\sqrt{(}s)^{0.191}A^{0.128}$ GeV/$c$, was
introduced [53, 54, 56] at RHIC and LHC energies in order not to violate the
geometrical limit for minijets production per unit transverse area. The $p+p$
cutoff was kept constant at $p_{0}=2$ GeV/$c$. In $\mathtt{HIJING2.0}$ [45,
46], a subsequent version of $\mathtt{HIJING1.0}$ [52, 43] the GRV
parameterization of the proton parton densities [62] is implemented. The GRV
small $x$ gluon density is much higher than that of the DO parameterization.
Here also an energy-dependent cutoff $p_{0}(s)$ and soft cross section
$\sigma_{\rm soft}(s)$ are also assumed in order to better describe the Pb+Pb
data at the LHC. The cutoff used in $\mathtt{HIJINGB\overline{B}2.0}$ varies
from $p_{0}=1.5$ GeV/$c$ at the CERN SPS, $\sqrt{s}=20$ GeV, to 4.2 GeV/$c$ at
$\sqrt{s}=5.5$ TeV while that in $\mathtt{HIJING2.0}$, with a more complex
energy dependence, varies from 1.7 to 3.5 GeV/$c$ in the same energy range.
One of the main uncertainties in the calculation of the charged particle
multiplicity density in Pb+Pb collisions is the nuclear modification of parton
distribution functions, especially gluon distributions at small $x$. In
$\mathtt{HIJING}$-type models, the parton distributions per nucleon in a
nucleus, $f_{a/A}(x,\mu_{F}^{2})$, are factorizable into parton distributions
in a nucleon, $f_{a/N}$, and the shadowing function for parton $a$, $S_{a/A}$,
as in Eq. (19). The shadowing parameterization in $\mathtt{HIJING1.0}$ [52,
43] is employed,
$\displaystyle S_{a/A}(x)$ $\displaystyle\equiv$
$\displaystyle\frac{f_{a/A}(x)}{Af_{a/N}(x)}$ (21) $\displaystyle=$
$\displaystyle 1+1.19\log^{1/6}\\!A\,[x^{3}-1.2x^{2}+0.21x]$ $\displaystyle-
s_{a}(A^{1/3}-1)\left[1-\frac{10.8}{\log(A+1)}\sqrt{x}\right]e^{-x^{2}/0.01}\,\,,$
assuming the same dependence for quarks and gluons. The $\mu_{F}^{2}$
evolution of $S_{a/A}(x,\mu_{F})$ is neglected. The parameter, $s_{a}$, which
determines the shadowing for $x<0.1$, the region with the strongest nuclear
dependence, is $s_{a}=0.1$. For $x>0.1$, the $A$ dependence is rather weak.
The parameterization in Eq. (21) agrees with the $x$ dependence of the quark
structure function at small and medium $x$ [52, 43]. Because the first part of
Eq. (21) has a weak $A$ dependence, impact parameter dependence is only
included on part proportional to $s_{a}$. The impact parameter dependence is
given as
$\displaystyle
s_{a}(b)=s_{a}\frac{5}{3}\bigg{(}1-\frac{b^{2}}{R_{A}^{2}}\bigg{)}\,\,,$ (22)
where $R_{A}$ is the radius of the nucleus and $s_{a}=s_{q}=s_{g}=0.1$.
The LHC Pb+Pb data at $\sqrt{s_{{}_{NN}}}=2.76$ TeV [53, 54] indicate that
impact-parameter dependent shadowing is required to understand the centrality
dependence of the charged particle multiplicity density at midrapidity. These
data place an indirect and model-dependent constraint on quark and gluon
shadowing. Therefore, it is important to directly study quark and gluon
shadowing in $p+A$ collisions at the LHC.
In contrast, in $\mathtt{HIJING2.0}$ [45, 46], the factor ($A^{1/3}-1$) is
raised to the power 0.6 and a stronger impact-parameter dependence, different
for quarks and gluons, $s_{q}=0.1$ and $s_{g}=0.22-0.23$ respectively, is used
to fit the LHC data. This stronger gluon shadowing requires jet quenching to
be neglected [45, 46].
All $\mathtt{HIJING}$-type models assume scale-independent shadowing
(independent of $Q^{2}$). This approximation could break down at sufficiently
large scales due to the dominance of gluon emission in the DGLAP [63]
evolution equation. At $Q=2.0$ and 4.3 GeV/c, typical scales for minijet
production at RHIC and LHC respectively, low $x$ gluon shadowing varies by
$\approx 13\%$ in the EPS09 LO parameterization [48].
### 2.5 $\mathtt{AMPT}$ (Z. Lin)
The multiphase transport model $\mathtt{AMPT}$ [64] was also used to calculate
the yields and $p_{T}$ spectra of particles produced in $p+p$ and $p+$Pb
collisions, as well as the nuclear modification factors $R_{\rm pPb}$ in
$p+$Pb collisions. The flow coefficients have also been calculated. Both the
default, $\mathtt{AMPT-def}$ and the string melting, $\mathtt{AMPT-SM}$,
versions of $\mathtt{AMPT}$ [65] have been employed.
In the default version of $\mathtt{AMPT}$, $\mathtt{AMPT-def}$, only minijet
partons rescatter in the parton stage. After that, Lund string fragmentation
is used for hadronization with the hadron cascade setting in at relatively
high energy density. The cutoff time for the hadron cascade in these
simulations is 30 fm/$c$ ($\mathtt{NTMAX=150}$[65]). On the other hand, the
string-melting version of $\mathtt{AMPT}$, $\mathtt{AMPT-SM}$, converts the
usual initial-state hadronic strings to partonic matter when the energy
density in the collision overlap volume is expected to be higher than that of
the QCD phase transition. $\mathtt{AMPT-SM}$ also uses a simple quark
coalescence model to describe bulk hadronization of the resultant partonic
matter. Thus, secondary interactions are typically dominated by hadron
interactions in $\mathtt{AMPT-def}$ while dominated by parton interactions in
$\mathtt{AMPT-SM}$.
Using the default $\mathtt{HIJING}$ parameters for the Lund symmetric
splitting function gives reasonable charged particle pseudorapidity
distributions, $dN_{\rm ch}/d\eta$ at central values of pseudorapidity for
Pb+Pb collisions at LHC energies [66]. Therefore the same values ($a=0.5$ and
$b=0.9$ GeV-2) are used for both $p+p$ and $p+$Pb collisions, along with the
same values of the strong coupling constant and parton cross section as in
Ref. [66].
In these calculations, $p+p$ events are minimum-bias, including diffractive
events. The MB $p+$Pb events include no restrictions on impact parameter. The
nuclear modification factor, $R_{p\rm Pb}$, as a function of $p_{T}$ is
obtained by dividing the $p+$Pb distribution by the $p+p$ distribution, both
calculated with the same version of $\mathtt{AMPT}$, normalized by the number
of binary $N+N$ collisions, $N_{\rm coll}$. The number of collisions is
assumed to be equal to the number of participant nucleons in the Pb nucleus
($N_{\rm coll}=N_{\rm part}^{\rm Pb}$). The collision centrality is defined
according to the number of charged hadrons within $|\eta|<1$.
### 2.6 Leading-order pQCD calculations (Z.-B. Kang, I. Vitev, H. Xing)
The details of the calculations described here can be found in Ref. [67]. A
summary is given here. To leading order in the framework of factorized
perturbative QCD, single inclusive hadron production in $p+p$ collisions,
$p(p_{1})+p(p_{2})\to h(p_{h})+X$, can be written as [68]
$\displaystyle\frac{d\sigma}{dyd^{2}p_{T}}$ $\displaystyle=$ $\displaystyle
K\frac{\alpha_{s}^{2}}{s}\sum_{a,b,c}\int\frac{dx_{1}}{x_{1}}d^{2}k_{T_{1}}\,f_{a/N}(x_{1},k_{T_{1}}^{2})\int\frac{dx_{2}}{x_{2}}d^{2}k_{T_{2}}\,f_{b/N}(x_{2},k_{T_{2}}^{2})$
(23) $\displaystyle\times\int\frac{dz_{c}}{z_{c}^{2}}\,D_{h/c}(z_{c})H_{ab\to
c}(\hat{s},\hat{t},\hat{u})\delta(\hat{s}+\hat{t}+\hat{u}),$
where $y$ and $p_{T}$ are the rapidity and transverse momentum of the produced
hadron and $\sum_{a,b,c}$ runs over all parton flavors. In Eq. (23),
$s=(p_{1}+p_{2})^{2}$; $D_{h/c}(z_{c})$ is the fragmentation function (FF) of
parton $c$ into hadron $h$; $H_{ab\to c}(\hat{s},\hat{t},\hat{u})$ are hard-
scattering coefficient functions dependent on the partonic Mandelstam
invariants $\hat{s},\hat{t},\hat{u}$ [68]. A phenomenological $K$ factor is
included to account for higher-order QCD contributions. The parton
distribution functions, $f_{a,b/N}(x,k_{T}^{2})$, are dependent on the
longitudinal momentum fraction $x$ and the partonic transverse momentum
$k_{T}$. The $k_{T}$-dependence is included in order to incorporate the Cronin
effect in $p+A$ collisions. A Gaussian form is assumed [68],
$\displaystyle f_{a/N}(x_{1},k_{T_{1}}^{2})=f_{a/N}(x_{1})\frac{1}{\pi\langle
k_{T}^{2}\rangle}e^{-k_{T_{1}}^{2}/\langle k_{T}^{2}\rangle},$ (24)
where $f_{a/N}(x_{1})$ are the usual collinear PDFs in a nucleon. The
factorization scale dependence has been suppressed in the arguments of
$f_{a/N}$.
In $p+p$ collisions, $\langle k_{T}^{2}\rangle_{pp}=1.8$ GeV${}^{2}/c^{2}$.
The CTEQ6L1 PDFs [69] are used with the fDSS parameterization of the parton-
to-hadron fragmentation functions [70]. The factorization and renormalization
scales are fixed to the transverse momentum of the produced particle,
$\mu_{F}=\mu_{R}=p_{T}$, and are suppressed in Eqs. (23) and (24). An
$\mathcal{O}(1)$ $K$-factor is found to give a good description of hadron
production at both RHIC and LHC energies.
#### 2.6.1 Cold nuclear matter effects
The $p+A$ (e.g. d+Au or $p+$Pb) nuclear modification factor, $R_{pA}$, is
typically defined as:
$\displaystyle
R_{pA}=\left[\frac{d\sigma_{pA}}{dyd^{2}p_{T}}\right]\left[\frac{d\sigma_{pp}}{\langle
N_{\rm coll}\rangle dyd^{2}p_{T}}\right]^{-1}\,\,,$ (25)
where $\langle N_{\rm coll}\rangle$ is the average number of binary nucleon-
nucleon collisions. The deviation of $R_{pA}$ from unity reveals the presence
of cold nuclear matter (CNM) effects in $p+A$ collisions.
A variety of CNM effects can affect particle production. This section
describes those that arise from the elastic, inelastic and coherent scattering
of partons in large nuclei [71]. The proton and neutron composition of the
interacting nuclei are also accounted for. In particular, these effects
include isospin, the Cronin effect, cold nuclear matter energy loss and
dynamical shadowing. These effects have been well documented in the
literature. Their implementation is briefly described here.
##### Isospin
The isospin effect can be easily accounted for on average in the nuclear PDFs
for a nucleus with mass number $A$ and charge $Z$ by [72]:
$\displaystyle
f_{a/A}(x)=\frac{Z}{A}f_{a/p}(x)+\left(1-\frac{Z}{A}\right)f_{a/n}(x)\,\,,$
(26)
assuming no modifications of the parton densities. In Eq. (26), $f_{a/p}(x)$
and $f_{a/n}(x)$ are the PDFs in a proton and a neutron, respectively. The
neutron PDFs are related to those in the proton by isospin symmetry.
##### Cronin effect
The Cronin effect has been well documented [73]. It can be modeled by initial-
state multiple parton scatterings in cold nuclei and the corresponding induced
parton transverse momentum broadening [74, 75]. In particular, if the PDFs,
$f_{b/A}(x_{2},k_{T_{2}}^{2})$, have a normalized Gaussian form, the random
elastic scattering induces further $k_{T}$-broadening in the nucleus:
$\displaystyle\langle k_{T_{2}}^{2}\rangle_{pA}=\langle
k_{T_{2}}^{2}\rangle_{pp}+\left\langle\frac{2\mu^{2}L}{\lambda_{q,g}}\right\rangle\zeta\,\,.$
(27)
Here $k_{T_{2}}$ is the transverse momentum component of the parton prior to
the hard scattering, $\zeta=\ln(1+\delta p_{T}^{2})$, $\delta=0.14$
(GeV$/c)^{-2}$, $\mu^{2}=0.12$ (GeV$/c)^{2}$, and
$\lambda_{g}=(C_{F}/C_{A})\lambda_{q}=1$ fm. These parameters describe the
RHIC data reasonably well.
##### Cold nuclear matter energy loss
As the parton from the proton undergoes multiple scattering in the nucleus
before the hard collision, it can lose energy due to medium-induced gluon
bremsstrahlung. This effect can be easily implemented as a shift in the
momentum fraction in the PDFs
$\displaystyle f_{q/p}(x_{1})\to f_{q/p}\left(\frac{x_{1}}{1-\epsilon_{q,\,\rm
eff}}\right)\,\,,\quad f_{g/p}(x_{1})\to
f_{g/p}\left(\frac{x_{1}}{1-\epsilon_{g,\,\rm eff}}\right)\,\,.$ (28)
Ideally, Eq. (28) should include a convolution over the probability of cold
nuclear matter energy loss [76]. However, concurrent implementation of this
distribution together with the Cronin effect and coherent power corrections is
computationally very demanding. The main effect of fluctuations due to
multiple gluon emission is an effective reduction in the fractional energy
loss $\epsilon_{q,g\,\rm eff}$ relative to the mean value
$\langle\epsilon_{q,g}\rangle=\langle\sum_{i}(\Delta E_{i}/E)\rangle$ where
the sum runs over all medium-induced gluons. Here $\epsilon_{q,g\rm
eff}=0.7\langle\epsilon_{q,g}\rangle$. The average cold nuclear matter energy
loss is obtained by integrating the initial-state medium-induced
bremsstrahlung spectrum first derived in Ref. [77]. It also depends on the
typical transverse momentum transfer squared per interaction between the
parton and the medium and the gluon mean-free path $\lambda_{g}$. Therefore,
the parameters are constrained to be the same as in the implementation of the
Cronin effect, $\mu^{2}=0.12$ (GeV$/c)^{2}$ and $\lambda_{g}=1$ fm. This
calculation of initial-state cold nuclear matter energy loss has been shown to
give a good description of the nuclear modification of Drell-Yan production in
fixed-target experiments [76].
##### Dynamical shadowing
Power-suppressed resummed coherent final-state scattering of the struck
partons leads to shadowing effects (suppression of the cross section in the
small-$x$ region) [78]. The effect can be interpreted as the dynamical
generation of parton mass in the background gluon field of the nucleus [79].
Thus
$\displaystyle x\to
x\left(1+C_{d}\frac{\xi^{2}(A^{1/3}-1)}{-\hat{t}}\right)\,\,,$ (29)
where $x$ is the parton momentum fraction in the lead ion,
$C_{d}=C_{F}(C_{A})$ if the parton $d=q(g)$ in the $2\rightarrow 2$ parton
scattering $ab\to cd$, and $\xi^{2}$ represents the characteristic scale of
the multiple scattering per nucleon. At RHIC energies,
$\sqrt{s_{{}_{NN}}}=200$ GeV, $\xi^{2}_{q}=C_{F}/C_{A}\xi^{2}_{g}=0.12$ GeV2
[78] gives a good description of the nuclear modification in d+Au collisions
for both single hadron and dihadron production [80].
### 2.7 Initial-state Shadowing (G. G. Barnaföldi, J. Barette, M. Gyulassy,
P. Levai, G. Papp and V. Topor Pop)
The calculations in this section use the $\mathtt{kTpQCD\\_v2.0}$ code, based
on a phenomenologically-enhanced, perturbative QCD improved parton model
described in detail in Refs. [81, 82]. The main feature of this model is the
phenomenologically-generalized parton distribution function employed to handle
nonperturbative effects at relatively low-$x$ and small $p_{T}$. The model
includes intrinsic $k_{T}$ broadening with the average $k_{T}$ left as a free
parameter to correct for nonperturbative effects. The $k_{T}$ value is
determined from $p+p$ data over a wide range of energies. Within the framework
of this model, the $k_{T}$-broadening in $p+A$ and $A+A$ collisions is related
to nuclear multiple scattering and can generate the Cronin enhancement [49,
83] that appears within $3\leq p_{T}\leq 9$ GeV/$c$ from SPS to RHIC energies.
#### 2.7.1 Theoretical Background
The $\mathtt{kTpQCD\\_v2.0}$ code calculates the invariant cross section for
hadron production in $p+p$, $p+A$ and $A+A$ collisions at LO or NLO in the
$k_{T}$-enhanced pQCD-improved parton model assuming collinear factorization.
The code provides a Monte Carlo-based integration of the convolution [82],
written here for $p+p$ collisions,
$\displaystyle E_{h}\frac{d\sigma_{h}^{pp}}{d^{3}p_{T}}$ $\displaystyle=$
$\displaystyle\frac{1}{s}\sum_{abc}\int^{1-(1-V)/z_{c}}_{VW/z_{c}}\frac{dv}{v(1-v)}\
\int^{1}_{VW/vz_{c}}\frac{dw}{w}\int^{1}{dz_{c}}$
$\displaystyle\times\int{d^{2}{\vec{k}}_{T_{1}}}\
\int{d^{2}{\vec{k}}_{T_{2}}}\,\,f_{a/p}(x_{1},{\vec{k}}_{T_{1}},\mu_{F}^{2})\,f_{b/p}(x_{2},{\vec{k}}_{T_{2}},\mu_{F}^{2})$
$\displaystyle\times\left[\frac{d{\widetilde{\sigma}}}{dv}\delta(1-w)\,+\,\frac{\alpha_{s}(\mu_{R})}{\pi}K_{ab,c}(\hat{s},v,w,\mu_{F},\mu_{R},\mu_{\rm
Fr})\right]\frac{D_{c}^{h}(z_{c},\mu_{\rm Fr}^{2})}{\pi z_{c}^{2}}\,\,.$
Here $d{\widetilde{\sigma}}/dv$ represents the Born cross section of the
partonic subprocess $ab\to cd$ while
$K_{ab,c}(\hat{s},v,w,\mu_{F},\mu_{R},\mu_{\rm Fr})$ is the next order
correction term. The proton and parton level NLO kinematic variables are ($s$,
$V$, and $W$) and ($\hat{s}$, $v$, and $w$) respectively [82, 84, 85, 86]. The
various scales are $\mu_{F}$, the factorization scale; $\mu_{R}$, the
renormalization scale; and $\mu_{\rm Fr}$ the fragmentation scale. The
factorization and the renormalization scales are related to the momentum of
the intermediate jet, $\mu_{F}=\mu_{R}=\kappa p_{q}$ where $\kappa=2/3$,
$p_{q}=p_{T}/z_{c}$ and $z_{c}$ is the fraction of parton $c$ momenta
transferred to the final hadron $h$. The fragmentation scale is related to the
final hadron momentum by $\mu_{\rm Fr}=\kappa p_{T}$.
The $x$-dependent proton parton distribution functions,
$f_{a/p}(x,\mu_{F}^{2})$, defined in the infinite momentum frame, are
generalized to three dimensions by incorporating an initial $k_{T}$
dependence,
$f_{a/p}(x,\vec{k}_{T},\mu_{F}^{2})=g(\vec{k}_{T})f_{a/A}(x,\mu_{F}^{2})\,\,.$
(31)
The two-dimensional initial transverse momentum distribution,
$g({\vec{k}}_{T})$, with intrinsic parton $k_{T}$ employed in these
calculations is described in Refs. [81, 82, 87, 88]. The $k_{T}$ distribution
is described by a Gaussian,
$g({\vec{k}}_{T})\ =\frac{1}{\pi\langle
k^{2}_{T}\rangle}e^{-{k^{2}_{T}}/{\langle k^{2}_{T}\rangle}}\,\,\,.$ (32)
Here $\langle k_{T}^{2}\rangle$ is the width of the $k_{T}$ distribution,
related to the magnitude of the average parton transverse momentum by $\langle
k_{T}^{2}\rangle=4\langle k_{T}\rangle^{2}/\pi$. This treatment was
successfully applied at LO in Ref. [81], along with a $K_{\rm jet}$-based NLO
calculation [89, 90]. In order to reproduce results for $NN$ collisions at
relatively low $x$, $\langle k_{T}^{2}\rangle=2.5$ GeV${}^{2}/c^{2}$ was
required.
The LO or NLO fragmentation functions, $D_{c}^{h}(z_{c},\mu_{\rm Fr}^{2})$,
are the probability for parton $c$ to fragment into hadron $h$ with momentum
fraction $z_{c}$ at fragmentation scale $\mu_{\rm Fr}$. The MRST(cg) [91]
parton densities are used in Eq. (31), along with the KKP parameterization
[42] of the fragmentation functions. Both these sets can be applied at
relatively small scales, $\mu_{F}^{2}=\mu_{\rm Fr}^{2}\approx 1.25$ GeV2. Thus
the results obtained in these calculations are applicable for $p_{T}\geq 2$
GeV/$c$.
#### 2.7.2 Incorporating Initial-State Nuclear Effects in $p+A$ and $A+A$
Collisions
Proton-nucleus and nucleus-nucleus collisions can be described by
incorporating the appropriate collision geometry and and nuclear shadowing. In
the Glauber framework, the cross section for hadron production in an
$A+A^{\prime}$ collision can be written as an integral over impact parameter
$b$:
$E_{h}\frac{d\sigma_{h}^{AA^{\prime}}}{d^{3}p_{T}}=\int
d^{2}b\,d^{2}r\,\,T_{A}(r)\,\,T_{A^{\prime}}(|{\vec{b}}-{\vec{r}}|)\,E_{\pi}\,\frac{d\sigma_{\pi}^{pp}(\langle
k_{T}^{2}\rangle_{pA},\langle k_{T}^{2}\rangle_{pA^{\prime}})}{d^{3}p}\,\,\,.$
(33)
Here the nuclear thickness function, $T_{A}(b)=\int dz\,\rho_{A}(b,z)$,
employing the Woods-Saxon density distribution, is normalized so that $\int
d^{2}b\,T_{A}(b)=A$.
The $p+p$ cross section from Eq. (2.7.1) includes increased $k_{T}$ widths
relative to $p+p$ collisions, Eq. (32), as a consequence of multiple
scattering in nuclei, see Eq. (34). The increased width of the $k_{T}$
distribution is taken into account by adding a function, $h_{pA}(b)$, that
describes the number of effective $N+N$ collisions at impact parameter $b$,
weighted by the average transverse momentum squared imparted by each
collision, to the width in $p+p$ collisions, $\langle k_{T}^{2}\rangle_{pp}$,
$\langle k_{T}^{2}\rangle_{pA}=\langle k_{T}^{2}\rangle_{pp}+Ch_{pA}(b)\,\,.$
(34)
The function $h_{pA}(b)$ is expressed in terms of the number of collisions
suffered by the incoming proton in the target nucleus,
$\nu_{A}(b)=\sigma_{NN}T_{A}(b)$, where $\sigma_{NN}$ is the inelastic $N+N$
cross section:
$h_{pA}(b)=\left\\{\begin{array}[]{cc}\nu_{A}(b)-1&\nu_{A}(b)<\nu_{m}\\\
\nu_{m}-1&\mbox{otherwise}\\\ \end{array}\right.\ .$ (35)
For heavy nuclei, the maximum number of collisions is $3\leq\nu_{m}\leq 4$
with $C=0.4$ GeV${}^{2}/c^{2}$.
Finally, the nuclear PDFs are modified by shadowing [47, 92, 93, 94]. This
effect, as well as the isospin asymmetry, are taken into account on average
using the scale independent parameterization of $S_{a/A}(x)$ adopted from Ref.
[87],
$f_{a/A}(x,\mu_{F}^{2})=S_{a/A}(x)\left[\frac{Z}{A}f_{a/p}(x,\mu_{F}^{2})+\left(1-\frac{Z}{A}\right)f_{a/n}(x,\mu_{F}^{2})\right]\,\,\,\,,$
(36)
where the neutron parton density, $f_{a/n}(x,\mu_{F}^{2})$, is related to that
of the proton. Results are shown with the EKS98 [92], EPS08 [93] and HKN [94]
parameterizations, as well as with the updated $\mathtt{HIJING}$
parameterization [47]. The EKS98, EPS08 and HKN parameterizations differ for
quarks, antiquarks and gluons but are independent of impact parameter. The new
$\mathtt{HIJING}$ parameterization differentiates between quarks and gluons
but can include impact parameter dependence, as in Eq. (22).
## 3 Charged particles
In this section, results on the charged particle multiplicity and $p_{T}$
distributions and the suppression factor $R_{p{\rm Pb}}$ as a function of
$p_{T}$ are compiled. These results are compared with the ALICE test beam data
where available. Other, related, predictions for charged particle observables
are also shown. The upcoming $p+$Pb run at the LHC can place important
constraints on models of the initial state.
Note that the LHC magnet design requires the magnetic rigidity of the beams in
the two rings to be the same. The proton beam, at 4 TeV, circulated in
negative $z$-direction (toward negative rapidity) in the ALICE laboratory
frame while a beam of fully-stripped Pb ions of $(82/208)\times 4$ TeV/nucleon
circulated in the positive $z$-direction (toward positive rapidity), implying
Pb$+p$ collisions if the first-named collision partner travels in the
direction of positive rapidity, rather than $p+$Pb collisions, as the
collisions are referred to throughout this text. This configuration resulted
in a center of mass energy of $\sqrt{s_{{}_{NN}}}=5.02$ TeV, moving with a
rapidity difference, $\Delta y_{{}_{NN}}=0.465$, in the direction of the
proton beam.
It is important to note that the most of the predictions shown here were
orginally made assuming that the proton circulated toward positive rapidity
and the nucleus toward negative rapidity, similar to the fixed-target
configuration and also the convention for d+Au collisions at RHIC. In such
cases, the low $x$ nuclear parton distributions are probed at positive
rapidity. Here and throughout the remainder of this paper, the results shown
have been adjusted to the Pb$+p$ convention of the ALICE data [2] unless
otherwise explicitly noted.
ALICE reported the primary charged particle pseudorapidity density in the
laboratory frame, $dN_{\rm ch}/d\eta_{\rm lab}$ in non single-diffractive
(NSD) $p+$Pb collisions. The lab frame pseudorapidity is defined as $\eta_{\rm
lab}=-\ln\tan(\theta/2)$ where $\theta$ is the polar angle between the
direction of the produced charged particle and the beam axis. The primary
particles are due to both prompt production in the collision and strong
decays.
Calculations are typically performed in the center of mass frame but the
pseudorapidity densities shown here have been also calculated in the lab
frame. In the ALICE paper [2], calculations in the center of mass frame were
shifted by $\Delta y_{{}_{NN}}$ in the lab frame. This is only approximately
correct since, at low $p_{T}$, the rapidity and pseudorapidity are not
identical. The uncertainty on $dN_{\rm ch}/d\eta_{\rm lab}$ due to the choice
of frame was estimated to be less than 6% [2]. It is worth noting that there
is no ambiguity due to the calculational frame for identified particles thus
the frame dependence is not discussed in later sections.
### 3.1 Multiplicity distribution
The calculations of the charged particle multiplicity distributions are
described here. Saturation model predictions are discussed first, followed by
event generator predictions and pQCD calculations with cold matter effects.
#### 3.1.1 Saturation Approaches (J. Albacete, A. Dumitru, H. Fujii, Y. Nara,
A. Rezaeian, and R. Vogt)
In the rcBK approach, used by Albacete and collaborators [95] and Rezaeian for
$R_{p{\rm Pb}}$ [20, 96], the initial condition for the evolution of the
dipole scattering amplitude can be written as (see also Eq. (8)),
$\mathcal{N}(r,x_{0})=1-\exp\left[-\frac{1}{4}\left(r^{2}Q_{s}^{2}(x_{0})\right)^{\gamma}\log\left(e+\frac{1}{r\Lambda}\right)\right]\,\,.$
(37)
There are three sets of unintegrated gluon distributions that are solutions of
the rcBK small-$x$ evolution equations with the AAMQS initial condition in Eq.
(37). The values of $\gamma$ and $Q_{s0}^{2}(x_{0}=0.01)$ for protons that
provide good fits to the $e^{-}+p$ data are given in Table 3.1.1. Albacete et
al. use all three initial conditions [95]. Rezaeian uses g1.119 [96, 97] in
the calculations of $R_{p{\rm Pb}}$ in Sec. 3.3 but employs the b-CGC approach
[20, 27] to calculate $dN_{\rm ch}/d\eta$ here.
The AAMQS initial conditions used in the dipole evolution of the rcBK approach
for protons [95, 96]. Set $Q_{s0}^{2}(x_{0}=0.01)$ (GeV${}^{2}/c^{2}$)
$\gamma$ MV 0.200 1 g1.101 0.157 1.101 g1.119 0.168 1.119
##### rcBK (J. Albacete, A. Dumitru, H. Fujii and Y. Nara)
The calculation of $dN_{\rm ch}/d\eta$ by Albacete et al. [95], employing
$\gamma=1.119$, in both the center of mass and lab frames, is shown in the
dashed cyan curves in Fig. 1.
Figure 1: Charged particle pseudorapidity distributions at
$\sqrt{s_{{}_{NN}}}=5.02$ TeV in the CM (left) and lab (right) frames. The
rcBK (dashed cyan) result is from Ref. [95]. The band labeled Rezaeian (dot-
dot-dash-dashed green with vertical bars outlining the uncertainty), described
in Sec. 2.1, is only shown in the center-of-mass frame. The band showing the
IP-Sat result described in Sec. 2.2 is outlined by the dot-dot-dash-dashed
black curves. The $\mathtt{HIJING2.1}$ result without (NS, dot-dash-dash-
dashed red) and with shadowing ($s_{g}=0.28$, solid red) and the
$\mathtt{HIJINGB\overline{B}2.0}$ result without (dot dashed magenta) and with
shadowing (dotted magenta) are also shown. Finally, the $\mathtt{AMPT-def}$
(dot-dash-dash-dashed blue) and $\mathtt{AMPT-SM}$ (dot-dot-dot-dash-dash-
dashed blue) are given. The ALICE results from Ref. [2] are given on the
right-hand side. The systematic uncertainties are shown in blue, the
statistical uncertainties are too small to be visible on the scale of the
plot.
##### b-CGC (A. Rezaeian)
The results of calculations by Rezaeian [20, 96] in the center of mass frame
are shown by the green curves in Fig. 1. The vertical lines indicate the
uncertainty in the predictions. The dipole forward (proton direction)
amplitude is calculated in the b-CGC saturation model [27]. which has an
explicit impact parameter dependence as well as approximately incorporates all
known features of small-$x$ physics. It describes the small-$x$ HERA data,
including diffractive data [27], and also the RHIC and LHC data at small-$x$
[98, 99, 96, 100, 101, 102]. This framework also provides an excellent
description of the charged hadron multiplicity in d+Au collisions at RHIC in
addition to $p+p$ and $A+A$ collisions over a range of energies [96]. For
other saturation model predictions in minimum bias $p+A$ collisions at
$\sqrt{s_{{}_{NN}}}=4.4$ TeV, see Ref. [103].
##### IP-Sat (P. Tribedy and R. Venugopalan)
The calculation of the minimum-bias charged particle pseudorapidity
distribution was performed in the $k_{T}$-factorization approach using the IP-
Sat model [21] in Eq. (17). For the calculation of $dN_{\rm ch}/d\eta$ in the
lab frame, a constant rapidity shift of 0.46 has been applied. The bands shown
in Fig. 1 arises due to uncertainties in the parameters.
The charged particle multiplicity distributions obtained by the event
generators $\mathtt{HIJING}$, $\mathtt{HIJINGB\overline{B}2.0}$ and
$\mathtt{AMPT}$ are now discussed.
#### 3.1.2 $\mathtt{HIJING2.1}$ (R. Xu, W.-T. Deng and X.-N. Wang)
The $\mathtt{HIJING2.1}$ predictions in Fig. 1 are indicated by the dot-dash-
dash-dashed curves without shadowing and the solid curves with $s_{g}=0.28$.
The difference between results with and without shadowing are largest for this
calculation.
#### 3.1.3 $\mathtt{HIJINGB\overline{B}}2.0$ (G. G. Barnaföldi, J. Barette,
M. Gyulassy, P. Levai, M.Petrovici and V. Topor Pop)
The $\mathtt{HIJINGB\overline{B}2.0}$ predictions of $dN_{\rm ch}/d\eta$ in
minimum bias collisions employ the values for the minijet cutoff and string-
tension parameters of $p_{0}=3.1$ GeV/$c$ and $\kappa=2.0$ GeV/fm. These
values are determined from fits to $p+p$ and $A+A$ systematics from RHIC to
the LHC, see Ref. [53, 54, 56] for details. Note that these calculations
assume no jet quenching.
The absolute normalization of $dN_{\rm ch}/d\eta$ is sensitive to the low
$p_{T}$, $p_{T}<2$ GeV/$c$, nonperturbative hadronization dynamics
encapsulated in the Lund [57, 58] $\mathtt{JETSET}$ [59] string fragmentation
constrained by lower energy $e^{+}+e^{-}$, $e^{\pm}+p$, and $p+p$ data. The
default $\mathtt{HIJING1.0}$ shadowing parametrization leads to substantial
reduction of the global multiplicity at the LHC. The
$\mathtt{HIJINGB\overline{B}2.0}$ results without shadowing are substantially
reduced relative to the same predictions with $\mathtt{HIJING1.0}$ [52, 43]
because both the default minijet cutoff $p_{0}=2$ GeV/$c$ and vacuum string
tension $\kappa_{0}=1$ GeV/fm used in $\mathtt{HIJING1.0}$ are generalized to
vary monotonically with $\sqrt{s_{{}_{NN}}}$ and $A$. As discussed in Ref.
[53, 54, 56], systematics of multi-particle production in $A+A$ collisions
from RHIC to the LHC are used to fix the $\sqrt{s_{{}_{NN}}}$ and $A$
dependence of $p_{0}$ and $\kappa$. The resulting dependencies,
$p_{0}(s,A)=0.416\sqrt{s}^{0.191}A^{0.128}$ GeV/$c$ and
$\kappa(s,A)=\kappa_{0}(s/s0)^{0.04}A^{0.167}$ GeV/fm [56] lead to $p_{0}=3.1$
GeV/$c$ and $\kappa=2.1$ GeV/fm in $p+$Pb collisions at 5.02 TeV.
Constant values of the cutoff are employed in $p+p$ collisions, independent of
the incident energy, $p_{0}^{pp}=2$ GeV/$c$, and string tension,
$\kappa_{pp}=1.9$ GeV/fm. Note that, even without shadowing, the increase of
$p_{0}$ from 2 GeV/$c$ in $p+p$ collisions to 3.1 GeV/$c$ in $p+$Pb collisions
causes a reduction in the minijet cross section and hence the final pion
(charged particle) multiplicity. Such a reduction is also required to fit the
slow growth (by a factor of 2.2) in the $A+A$ charged particle multiplicity
from RHIC to LHC [104]. This reduction could be interpreted as
phenomenological evidence for gluon saturation beyond leading twist shadowing.
#### 3.1.4 $\mathtt{AMPT}$ (Z.-W. Lin)
The $\mathtt{AMPT}$ default and string melting calculations are shown in the
blue curves in Fig. 1. The differences in the two scenarios is not large.
Indeed it is much smaller than models with and without shadowing. These
differences can arise from several sources: the relative rescattering
strengths in the parton and hadron stages; the hadronization models; or a
combination of rescattering and hadronization.
#### 3.1.5 Forward/Backward difference (R. Vogt)
The event generator calculations produce distributions that do not show a
strong forward/backward difference between the lead and proton peaks. The CGC-
based calculations, however, show a much stronger dependence of the results on
pseudorapidity, in both frames.
The charged particle multiplicity results can be further quantified by
comparing the measured to predicted particle density at midrapidity, near the
proton peak, $\eta_{\rm lab}=-2$, and the lead peak, $\eta_{\rm lab}=2$. The
absolute values of $dN_{\rm ch}/d\eta_{\rm lab}$ at $\eta_{\rm lab}=-2$, 0 and
2, along with the ratio $R$ of the multiplicities at $\eta_{\rm lab}=2$ to
$\eta_{\rm lab}=-2)$ are given in Table 3.1.5. The ALICE results as well as
the model results included both here and in the ALICE paper [2] are given.
There are two tabulated values for both $\mathtt{HIJINGB\overline{B}2.0}$ NS
(no shadowing) and WS (with shadowing). The first were shifted from the
center-of-mass to the lab frame by the ALICE collaboration and the second were
direct lab frame calculations.
Comparison of values of $dN_{\rm ch}/d\eta_{\rm lab}$ at $\eta_{\rm lab}=-2$,
0, 2 and the ratio $dN_{\rm ch}/d\eta_{\rm lab}|_{\eta_{{\rm lab}=2}}/dN_{\rm
ch}/d\eta_{\rm lab}|_{\eta_{{\rm lab}=-2}}$, denoted by $R$ below. The
tabulated IP-Sat result is the average of the upper and lower limits depicted
on the right-hand side of Fig. 1. The $*$ on $\mathtt{HIJINGB\overline{B}2.0}$
indicates that the calculations have been shifted to the lab frame by the
ALICE Collaboration while the $\dagger$ are results provided by V. Topor Pop
et al.. Adapted from Ref. [2]. $dN_{\rm ch}/d\eta_{\rm lab}$ $R$ $-2$ 0 2
ALICE $16.65\pm 0.65$ $17.24\pm 0.66$ $19.81\pm 0.78$ $1.19\pm 0.05$
Saturation Models IP-Sat 17.55 20.55 23.11 1.32 KLN 15.96 17.51 22.02 1.38
rcBK 14.27 16.94 22.51 1.58 $\mathtt{HIJING}$-based $\mathtt{HIJING2.1}$ NS
(no shad) 23.58 22.67 24.96 1.06 $\mathtt{HIJING2.1}$ WS ($s_{g}=0.28$) 18.30
17.49 20.21 1.10 $\mathtt{HIJINGB\overline{B}2.0}$ NS∗ 20.03 19.68 23.24 1.16
$\mathtt{HIJINGB\overline{B}2.0}$ NS† 16.84 16.39 19.68 1.16
$\mathtt{HIJINGB\overline{B}2.0}$ WS∗ 12.97 12.09 15.16 1.17
$\mathtt{HIJINGB\overline{B}2.0}$ WS† 13.98 13.71 16.73 1.20 $\mathtt{AMPT}$
$\mathtt{AMPT-Def}$ 19.07 18.56 21.65 1.14 $\mathtt{AMPT-SM}$ 18.14 18.10
20.84 1.15 $\mathtt{DPMJET}$ 17.50 17.61 20.67 1.18
#### 3.1.6 Centrality Dependence of $dN_{\rm ch}/d\eta$
Here we present two calculations of the centrality dependence of the charged
particle multiplicity distributions.
##### $\mathtt{AMPT}$ (Z. Lin)
Figure 2 shows the pseudorapidity rapidity distributions calculated in the
center-of-mass frame by $\mathtt{AMPT-def}$. Results are shown for $p+p$ and
minimum bias $p+$Pb collisions as well as for six different centrality
classes. In $\mathtt{AMPT}$, the $p+p$ events are minimum bias events,
including diffractive events.
Centrality classes of $p+$Pb events from $\mathtt{AMPT-SM}$ The centrality is
determined from the number of charged hadrons within $|\eta|<1$ in the center-
of-mass frame. Bin $\langle b\rangle$ (fm) $b_{\rm min}$ (fm) $b_{\rm max}$
(fm) $N^{\rm Pb}_{\rm part}$ $N^{\rm Pb}_{\rm part,\,in}$ $\langle N_{\rm
ch}(|\eta|<1)\rangle$ MB 5.84 0.0 13.2 7.51 5.37 36.1 0-5% 3.51 0.0 8.8 15.70
12.12 99.9 5-10% 3.76 0.0 9.4 14.19 10.71 79.6 10-20% 4.00 0.0 9.9 12.93 9.59
66.5 20-40% 4.56 0.0 12.0 10.70 7.69 49.6 40-60% 5.65 0.1 13.2 7.30 5.03 31.4
60-80% 7.08 0.1 13.2 3.77 2.50 15.9 80-100% 8.08 0.2 13.2 1.85 1.11 5.5
Since the ratio $R_{p{\rm Pb}}$ is normalized by a factor proportional to the
number of binary nucleon-nucleon collisions, $N_{\rm coll}$, the number of
collisions in the same centrality classes as included in Fig. 2 is shown in
Table 3.1.6. In $\mathtt{AMPT}$, $N_{\rm coll}$ is assumed to be equivalent to
the total number of participant nucleons in the Pb nucleus, $N^{\rm Pb}_{\rm
part}$. Because there is some model dependence in the definition of centrality
bins, it is worthwhile noting that other calculations of this same quantity
may give somewhat different results. Here, the centrality of $p+$Pb collisions
is defined according to the number of charged hadrons within $|\eta|<1$. Table
3.1.6 shows various information for each centrality class in the center-of-
mass frame, including the average, minimum and maximum impact parameters,
$N^{\rm Pb}_{\rm part}$, the number of participant Pb nucleons involved in
inelastic collisions, $N^{\rm Pb}_{\rm part,\,in}$, and the average number of
charged particles within $|\eta|<1$ from $\mathtt{AMPT-SM}$. Note that the
$\mathtt{AMPT-SM}$ values in Table 3.1.6 are essentially the same as those for
$\mathtt{AMPT-def}$ except for the average number of charged particles (last
column).
Figure 2: Charged particle pseudorapidity distributions in the center of mass
frame for $p+p$, $p+$Pb minimum bias and six of the centrality bins defined in
Table 3.1.6 as calculated in $\mathtt{AMPT}$.
In Fig. 2, the $40-60$% centrality bin gives a distribution that is very close
to the min-bias result, in particular on the proton side. The most central bin
is the most asymmetric distribution, as well as the largest in magnitude. On
the other hand, the distribution for the $80-100$% centrality bin is symmetric
around $\eta_{\rm cm}=0$ and lower in magnitude than the $p+p$ distribution.
##### b-CGC (A. Rezaeian)
In Fig. 3, the charged hadron multiplicity obtained from $k_{T}$ factorization
in the b-CGC approach is shown the $0-20\%$, $20-40\%$, $40-60\%$, $60-80\%$
centrality bins as well as minimum-bias collisions [20]. The impact parameter
dependence of the saturation model is crucial for defining the collision
centrality. The largest asymmetry in the multiplicity distribution is observed
in more central collisions while, for peripheral collisions such as the
$60-80\%$ most central, the system becomes more similar to that produced in
$p+p$ collisions. This is reflected in the total charged hadron multiplicity
distribution.
Figure 3: Charged particle pseudorapidity distributions in the center of mass
frame of $p+$Pb collisions at various centralities within the b-CGC saturation
model. The $\sim 5$% theoretical uncertainties arising from fixing the overall
normalization at RHIC are also shown. From Ref. [20].
Note that for all results shown in Fig. 3, a fixed minijet mass equal to
current-quark mass is assumed for all energies/rapidities and centralities.
Since the minijet mass is related to pre-hadronization/hadronization stage and
cannot be obtained from saturation physics, it was fixed by fitting lower
energy minimum-bias data. In very peripheral collisions where the system
becomes more similar to symmetric $p+p$ collisions, this assumption is less
reliable. More importantly, one should also note that the $k_{T}$
factorization employed here is only proven in asymmetric $p+A$ collisions at
small $x$. Therefore, for more peripheral collisions, the current CGC
prescription may be less reliable.
### 3.2 Transverse Momentum distribution
#### 3.2.1 Compilation of midrapidity results (J. Albacete, A. Dumitru, H.
Fujii, Y. Nara, R. Xu, W.-T. Deng, X.-N. Wang, G. G. Barnaföldi, J. Barette,
M. Gyulassy, P. Levai, M. Petrovici, V. Topor Pop, Z. Lin and R. Vogt)
The midrapidity, $|\eta|<0.8$, charged hadron $p_{T}$ distributions are shown
on the left-hand side of Fig. 4 for rcBK [95],
$\mathtt{HIJINGB\overline{B}2.0}$, and $\mathtt{AMPT}$. The
$\mathtt{HIJINGB\overline{B}2.0}$ distributions are similar to the rcBK
results, albeit somewhat higher for $p_{T}>10$ GeV/$c$. The $\mathtt{AMPT}$
distributions, on the other hand, drop faster at low $p_{T}$ than the other
results but then become harder at high $p_{T}$. The $\mathtt{AMPT}$ results
are essentially independent of whether string melting is included or not while
the $\mathtt{HIJINGB\overline{B}2.0}$ results without shadowing lie above
those with shadowing.
Figure 4: (Left) Charged particle $p_{T}$ distributions at
$\sqrt{s_{{}_{NN}}}=5.02$ TeV. The solid and dashed cyan curves outline the
rcBK band calculated by Albacete et al. [95]. The magenta curves, calculated
with $\mathtt{HIJINGB\overline{B}2.0}$ as described in Sec. 2.4 are presented
without (dot-dashed) and with (dotted) shadowing. The $\mathtt{AMPT}$ results,
given by the dot-dash-dash-dashed (default) and dot-dot-dot-dashed (SM) blue
curves, are described in Sec. 2.5. (Right) The charged hadron $p_{T}$
distribution in $p+$Pb collisions with different $\mathtt{HIJING2.1}$ options,
scaled by the indicated factors to separate the curves. The $p+p$ distribution
is shown for comparison.
The right-hand side of Fig. 4 shows several options for cold matter effects in
$\mathtt{HIJING2.1}$ relative to $p+p$ collisions. The $p+p$ result is
unscaled while the $p+$Pb curves with decoherent hard scatterings (DHC)
without shadowing, DHC with shadowing, and shadowing only are separated from
each other, starting from the $p+p$ result, by a factor of 100.
#### 3.2.2 reBK at $y=0$, 2 (J. Albacete, A. Dumitru, H. Fujii and Y. Nara)
Fig. 5 shows the $p_{T}$ spectrum in $p+p$ (left) and $p+$Pb (right)
collisions at different rapidities for the AAMQS initial conditions, see Table
3.1.1. Near central rapidity, $k_{T}$-factorization is employed while at
forward rapidities (in the proton direction) the hybrid formalism is applied.
The bands correspond to uncertainty estimates due to small variations in the
scale entering the coupling and fragmentation functions.
Figure 5: The predicted transverse momentum spectrum in $p+p$ (left) and
minimum-bias $p+$Pb (right) collisions at $\surd s=5$ TeV at different
rapidities (with the convention that the proton beam moves toward forward
rapidity). From Ref. [95].
#### 3.2.3 Flow Coefficients obtained with $\mathtt{AMPT}$ (Z. Lin)
The flow coefficients $v_{n}\\{2\\}$ $(n=2$, $3$, $4)$ in this study are
calculated employing the two-particle cumulant method with
$v_{n}=\sqrt{\left<\cos\left[n(\phi_{i}-\phi_{j})\right]\right>}$, where
$\cos\left[n(\phi_{i}-\phi_{j})\right]$ is averaged over all particle pairs in
the specified phase space with both particles of each pair coming from the
same event.
Figure 6: The flow coefficients $v_{2}\\{2\\}$ (top left), $v_{3}\\{2\\}$ (top
right), and $v_{4}\\{2\\}$ (bottom) of charged particles as a function of
$\eta$ in $p+p$ and minimum-bias $p+$Pb collisions. The results are shown for
both $\mathtt{AMPT-def}$ and $\mathtt{AMPT-SM}$.
Figure 6 shows $v_{2}\\{2\\}$, $v_{3}\\{2\\}$ and $v_{4}\\{2\\}$ for
unidentified charged particles as a function of $\eta$ in $p+p$ and minimum-
bias $p+$Pb collisions from $\mathtt{AMPT-def}$ and $\mathtt{AMPT-SM}$. The
magnitude of the flow coefficients generally decreases with increasing $n$.
There is a local maximum at $\eta\sim 0$. While the $p+p$ results are
approximately symmetric, as expected, the $p+$Pb results exhibit a strong
asymmetry. In the direction of the proton beam, the $p+$Pb coefficients
increase to approach the magnitudes of the $p+p$ coefficients.
Figure 7: The flow coefficients $v_{2}\\{2\\}$ (left) and $v_{3}\\{2\\}$
(right) as a function of centrality in $p+$Pb collisions at
$\langle\eta\rangle=4$, $0$, and $-4$. The results are shown for both
$\mathtt{AMPT-def}$ and $\mathtt{AMPT-SM}$.
Figure 7 shows the centrality dependence of $v_{2}\\{2\\}$ and $v_{3}\\{2\\}$
in $p+$Pb collisions. Three different $\eta$ ranges are shown:
$-5\leq\eta\leq-3$, $|\eta|\leq 1$, and $3\leq\eta\leq 5$
($\langle\eta\rangle=-4$, 0 and 4 respectively. The coefficients increase from
central to peripheral events for both $\mathtt{AMPT-SM}$ and $\mathtt{AMPT-
def}$. The coefficients are largest at $\langle\eta\rangle=0$ while those at
$\langle\eta\rangle=4$, in the direction of the Pb beam, are smallest,
consistent with the results in Fig. 6.
### 3.3 Medium Modification Factor $R_{p{\rm Pb}}$
#### 3.3.1 $R_{p{\rm Pb}}(p_{T})$ at $\eta\sim 0$
We now turn to the medium modification or suppression factors, defined as the
ratio of the $p+A$ to $p+p$ cross section,
$\displaystyle R_{p{\rm Pb}}(p_{T},\eta;b)=\frac{1}{N_{\rm
coll}(b)}\frac{d\sigma_{p{\rm Pb}}/d\eta dp_{T}}{d\sigma_{pp}/d\eta
dp_{T}}\,\,.$ (38)
Results are shown first for $R_{p{\rm Pb}}(p_{T})$ for charged hadrons (pions)
in Fig. 8. Standard shadowing calculation are shown on the left-hand side,
results with event generators in the center, while CGC calculations and
results with energy loss in cold matter are shown on the right-hand side.
Figure 8: Charged particle $R_{p{\rm Pb}}p_{T}$ at $\sqrt{s_{{}_{NN}}}=5.02$
TeV at $\eta\sim 0$. (Top left) Results with more ‘standard’ shadowing
(labeled EKS98, EPS08 and HKN) described in Sec. 2.7, Refs. [105, 106]
(labeled Kopeliovich) and the $\mathtt{HIJING2.1}$ shadowing parameterization
in Eq. (21) are compared. The difference in the $\mathtt{HIJING2.1}$ curves
depends on whether the hard scatterings are coherent or not, see Sec. 2.3.
(Top Right) $\mathtt{HIJINGB\overline{B}2.0}$ with and without shadowing (Sec.
2.4) compared to $\mathtt{AMPT}$ default and with string melting (Sec. 2.5).
(Bottom) The band from rcBK saturation model calculations by Albacete et al.
and Rezaeian with $N=5$ and varying $\alpha_{s}^{\rm in}$ described in Sec.
2.1 are compared to IP-Sat calculations by Tribedy and Venugopalan in Sec. 2.2
and calculations by Vitev et al. discussed in Sec. 2.6. (More detailed results
for the uncertainties in Rezaeian’s calculation at other rapidities can be
found in Fig. 10.) The ALICE results from Ref. [3] are also shown. The
systematic uncertainties are shown in blue, the statistical uncertainties are
in black.
##### Cronin effect expected at the LHC: dipole formulation (B. Z. Kopeliovich
and J. Nemchik)
At LHC energies, hadron production is dominated by gluon fragmentation. The
dipole formalism, on the light cone, is employed in the target rest frame,
leading to the factorized expression for hadron production: a convolution of
the projectile gluon PDF, the gluon fragmentation function, and the gluon
splitting, $g\to gg$, cross section [105]. The gluon splitting cross section
can be written in the dipole representation as
$\displaystyle\frac{d\sigma(gA\to g_{1}gX)}{d^{2}p_{T}\,dy_{1}}=\int
d^{2}b\int
d^{2}r_{1}d^{2}r_{2}\,e^{i\vec{p}_{T}\cdot(\vec{r}_{1}-\vec{r}_{2})}\,\overline{\Psi_{gg}^{*}(\vec{r}_{1},\alpha)\Psi_{gg}(\vec{r}_{2},\alpha)}$
$\displaystyle\times\left[1-e^{-{1\over
2}\sigma^{N}_{3g}(r_{1},x)T_{A}(b)}-e^{-{1\over
2}\sigma^{N}_{3g}(r_{2},x)T_{A}(b)}+e^{-{1\over
2}\sigma^{N}_{3g}(\vec{r}_{1}-\vec{r}_{2},x)T_{A}(b)}\right]\ \,\,,$
where the subscript 1 indicates the hadronizing gluon involved in inclusive
high $p_{T}$ hadron production by gluon fragmentation. The momentum fraction
of radiation gluon $g_{1}$ is defined as $\alpha=p^{+}_{g_{1}}/p^{+}_{g}$, the
transverse coordinates of the emitted gluons are $r_{1}$ and $r_{2}$, $x$ is
the momentum fraction of the gluon in the target nucleus and $T_{A}(b)$ is the
nuclear thickness function. The distribution function,
$\Psi_{gg}(\vec{r},\alpha)$, derived in [106], describes the $|gg\rangle$ Fock
component of the projectile gluon light-cone wavefunction, including the
nonperturbative gluon interaction. It is characterized by the localization of
gluons at a short relative transverse separation, $r_{0}\approx 0.3$ fm. The
$3g$ dipole cross section, $\sigma^{N}_{3g}$, is expressed as a combination of
the $\bar{q}q$ dipole cross sections, extracted from phenomenology and DIS
data, $\sigma^{N}_{3g}(r)={2\over
9}\Bigl{\\{}\sigma_{\bar{q}q}(r)+\sigma_{\bar{q}q}(\alpha
r)+\sigma_{\bar{q}q}[(1-\alpha)r]\Bigr{\\}}$iiiThe $x$ dependence of
$\sigma^{N}_{3g}$ has been suppressed..
Multiple interactions of higher Fock components (containing more than one
gluon) of the incident proton in the nucleus leads to a suppression of the
dipole cross section, known as gluon shadowing. The magnitude of the
suppression factor $R_{g}$ was evaluated in Ref. [106] and, as a consequence
of the small gluon separation, $r_{0}$, the modification was found to be
rather small. Thus, at first order, gluon shadowing can be implemented by the
simple replacement $\sigma^{N}_{3g}\rightarrow R_{g}(x,Q^{2},b)\sigma_{3g}$ in
Eq. (3.3.1). The results Ref. [105] at $\sqrt{s}=5$ TeV are given by the solid
cyan curve on the left-hand side of Fig. 8.
The Cronin enhancement is rather weak with a maximum of $R_{pPb}\sim 1.05$ at
$p_{T}\sim 3$ GeV$/c$. The height of the maximum is extremely sensitive to the
strength of gluon shadowing. The enhancement could completely disappear, or
even change sign, if the strength of gluon shadowing was underestimated in
Ref. [106].
##### Results with standard shadowing parameterizations (G. G. Barnaföldi, J.
Barette, M. Gyulassy, P. Levai, G. Papp and V. Topor Pop)
The left-hand side of Fig. 8 also shows $R_{p{\rm Pb}}(p_{T})$ for several
standard shadowing functions: EKS98 [92] (blue), EPS08 [93] (magenta), and HKN
[94] (green). All three were calculated in the rapidity interval $|\eta|<0.3$,
in the center of mass frame. Since none of these parameterizations include
impact parameter dependence, the results are independent of collision
centrality. For details, see Ref. [55]. For other results employing
$\mathtt{kTpQCD\\_v2.0}$, see Refs. [81, 107, 90, 111, 108].
##### $\mathtt{HIJING2.1}$ (R. Xu, W.-T. Deng and X.-N. Wang)
Two $\mathtt{HIJING2.1}$ calculations including shadowing with parameter
$s_{g}=0.28$ are shown with the standard shadowing parameterizations on the
left-hand side of Fig. 8. They are lower than the other calculations at high
$p_{T}$. Including the decoherent hard scatterings moves $R_{p{\rm Pb}}$
closer to unity over all $p_{T}$.
##### $\mathtt{HIJINGB\overline{B}2.0}$ (G. G. Barnaföldi, J. Barette, M.
Gyulassy, P. Levai, M. Petrovici and V. Topor Pop)
The $\mathtt{HIJINGB\overline{B}2.0}$ results without (NS) and with (WS)
shadowing in the central rapidity region, $|\eta|<0.8$, are shown in the
central panel of Fig. 8. Including shadowing and strong color field effects
reduces $R_{p{\rm Pb}}$ from unity to $\sim 0.7$ for $5<p_{T}<10$ GeV/$c$.
This is similar to CGC predictions in the KKT04 model [109]. Further, similar,
results for $R_{p{\rm Pb}}(p_{T})$ are found in Ref. [110, 111] using LO pQCD
collinear factorization with the $\mathtt{HIJING2.0}$ shadowing
parameterization [47], the GRV proton PDFs [62], and hadron fragmentation
functions from Ref. [42].
##### $\mathtt{AMPT}$ (Z. Lin)
The $\mathtt{AMPT}$ results, evaluated in the center of mass frame
pseudorapidity interval $|\eta|<1$ are also shown in the central panel of Fig.
8. In minimum bias collisions, $N_{\rm coll}=7.51$ in $\mathtt{AMPT}$. There
is an $\approx 50$% suppression of charged hadron production for the entire
$p_{T}$ range shown, considerably lower than that obtained in the other
calculations shown. The statistical uncertainty, not shown in Fig. 8, becomes
large for $p_{T}>10$ GeV/$c$. There is little difference in $R_{p{\rm Pb}}$
between the default and string melting options of $\mathtt{AMPT}$.
The right-hand side of Fig. 8 compares rcBK results from Albacete et al. and
Rezaeian to IP-Sat calculations by Tribedy and Venugopalan and pQCD
calculations by Vitev et al. including energy loss in cold matter.
##### rcBK predictions (J. Albacete, A. Dumitru, H. Fujii, Y. Nara, A.
Rezaeian and R. Vogt)
Rezaeian’s calculation assumes $N_{\rm coll}=6.9$ in minimum bias $p+$Pb
collisions [112]. To compare these calculations to data, it is necessary to
rescale the results to match the normalization $N_{\rm coll}$ to the
experimental value. The NLO MSTW proton PDFs [113, 114] and the NLO KKP
fragmentation functions [42] are employed. The rcBK equation, Eq. (7),
accounts for the rapidity/energy evolution of the dipole but does not include
impact parameter dependence explicitly. In the minimum-bias analysis here the
initial saturation scale, $Q_{0s}(x_{0}=0.01)$, is considered as an impact-
parameter averaged value and extracted from minimum-bias data. Thus,
fluctuations in particle production and any other possible nonperturbative
effects are incorporated into this average value of $Q_{0s}$. It was found
that $Q_{0p}^{2}(x_{0}=0.01)\approx 0.168$ GeV${}^{2}/c^{2}$ with
$\gamma\approx 1.119$ provides a good description of small-$x$ proton data
from the LHC, HERA and RHIC [15, 115], see also Ref. [97]. These values are
employed in the rcBK description of the projectile proton.
Here the difference between protons and nuclei originates from the different
initial saturation scales, $Q_{0s}$, in the rcBK equation, see Eq. (8). The
initial nuclear saturation scale $Q_{0A}$ at a given centrality is generally
less constrained than that in the proton because the small-$x$ data from d+Au
collisions at RHIC and DIS data on heavy target are limited with rather large
uncertainties, leading to correspondingly larger uncertainties on $Q_{0A}$
[20],
$(3-4)~{}Q_{0p}^{2}\leq Q_{0A}^{2}(x_{0}=0.01)\leq(6-7)\,Q_{0p}^{2}\,\,.$ (39)
The role of fluctuations on particle production and other nonperturbative
effect, including the impact-parameter dependece of the initial nuclear
saturation scale are effectively incorporated into the average value of
$Q_{0A}^{2}$ assuming $Q_{0A}^{2}=NQ_{0p}^{2}$ is constrained by experimental
data. Figure 8 shows results at $\eta=0$ with $N=5$ and varying
$\alpha_{s}^{\rm in}$. For results over the full range of $N$ in Eq. (39), see
Fig. 10.
On the other hand, Albacete et al let the initial nuclear saturation scale be
proportional to the nuclear density at each point in the transverse plane.
They compare two different methods [95],
$\displaystyle Q_{0A}^{2}(b,x_{0})$ $\displaystyle=$ $\displaystyle N_{\rm
part}(b)Q_{0p}^{2}(x_{0})\,\,\,{\rm and}$ (40) $\displaystyle=$ $\displaystyle
N_{\rm part}^{1/\gamma}(b)Q_{0p}^{2}(x_{0})\,\,\,.$ (41)
The second option is an ad hoc way of correcting for the violation in the
additivity of the nucleon number in the dipole scattering amplitude at small
dipole sizes resulting from the fact that $\gamma\neq 1$. The position of the
nucleons in the transverse plane is simulated by Monte Carlo methods makes it
possible to account for geometry fluctuations which can have large numerical
impact [95].
##### IP-Sat (P. Tribedy and R. Venugopalan)
The nuclear modification factors for inclusive charged particles in minimum-
bias collisions employing the IP-Sat approach is done using
$k_{T}$-factorization approach at $y=0$ in the center-of-mass frame. The LO
KKP fragmentation functions [42] are used to compute the inclusive charged
hadron spectrum from the gluon distribution. The details of the parameters
used can be found in Refs. [34, 35]. The band shows the uncertainty in the
calculation. Note that $R_{p{\rm Pb}}$ approachs unity with increasing
$p_{T}$.
##### Cold matter energy loss (Z.-B. Kang, I. Vitev and H. Xing)
The red curves on the right-hand side of Fig. 8 are results including cold
matter energy loss at $y=0$, see Sec. 2.6. The upper curve corresponds to
parameters determined for RHIC while the lower curve includes a potential
enhancement in these parameters for LHC energies. There is a very small
“Cronin peak” in the low $p_{T}$ region. The peak is very close to unity and
not as pronounced as that seen in lower energy fixed-target experiments
because dynamical shadowing strongly suppresses particle production in this
region. Initial-state energy loss is also larger because diagrams with
incident gluons make a bigger contribution to the cross section. At high
$p_{T}$, a small, $\sim 15$ %, suppression remains, also due to cold nuclear
matter energy loss.
The preliminary data are most compatible with the saturation model
calculations, albeit the uncertainties of these calculations are still rather
large. The calculation by Kopeliovich is also in relative agreement with the
data. However, we note that none of the calculations available for both
$dN_{\rm ch}/d\eta$ and $R_{p{\rm Pb}}$ at $\eta\sim 0$ agree with both
observables simultaneously.
#### 3.3.2 $\mathtt{HIJING2.1}$ parton vs. hadron $R_{pA}$ (R. Xu, W.-T. Deng
and X.-N. Wang)
On the left-hand side of Fig. 9, the nuclear modification factor of the final-
state parton $p_{T}$ spectra, $R_{pA}(p_{T})$, is shown for $p+$Pb collisions
at the LHC. Here $\langle N_{\rm coll}\rangle$ is the average number of binary
nucleon-nucleon interactions in $p+A$ collisions. It is clear that both
nuclear shadowing and the hard-soft coupling can suppress the $p_{T}$ spectra
of the produced partons. These features will transferred to the final hadron
spectra after hadronization, as shown on the right-hand side of the Figure.
Figure 9: (Left) The nuclear modification factor of the parton $p_{T}$ spectra
in $p+$Pb collisions. (Right) The charged hadron nuclear modification factor
with different $\mathtt{HIJING2.1}$ options. The arrow indicates the most
probable trend of the nuclear modification factor to transition from the low
to the high $p_{T}$ regions.
The predicted charged hadron nuclear modification factors are shown on the
right-hand side of Fig. 9. The enhancement at intermediate $p_{T}$ is due to
the $k_{T}$ broadening arising from multiple scattering. Parton shadowing, the
modified soft-hard coupling and enhanced density of gluon jets due to valence
quark conservation all contribution to suppression of the charged hadron
spectra. In addition, string fragmentation can further suppress the high
$p_{T}$ hadron spectra compared to independent fragmentation.
Since the effects of parton shadowing will disappear at large $p_{T}$ [48] due
to the QCD evolution not yet included here while hard and soft scatterings
will decohere, the nuclear modification factor will likely follow the default
result, including DHC, at low $p_{T}$. At large $p_{T}$, the result should
approach that with DHC effects but no shadowing. There may be further possible
modifications due to the hadronization of multiple jets. In Fig. 9, this
probable trend is indicated by the arrow joining these two results at
intermediate $p_{T}$.
#### 3.3.3 $R_{p{\rm Pb}}(p_{T})$ at $|\eta|\neq 0$
Here we show several results for the charged particle $R_{p{\rm Pb}}$ away
from central rapidity. Two employ the rcBK approach but with different initial
conditions. The last is a pQCD calculation including cold matter energy loss.
##### Rezaeian rcBK (A. Rezaeian)
In Fig. 10, $R_{pA}^{\rm ch}(p_{T})$ is shown for minimum bias collisions at
$\eta=0$, 2, 4, and 6 (with the convention that the proton beam moves toward
forward rapidity) obtained from the hybrid factorization Eq. (4) supplemented
with rcBK evolution. The band labeled CGC-rcBK includes uncertainties
associated with the variation of initial nuclear saturation scale,
$Q_{0A}^{2}$, constrained in Eq. (39), as well as uncertainties due to the
choice of factorization scale, $\mu_{F}=2p_{T}$, $p_{T}$, and $p_{T}/2$, in
Eq. (4). The values of $N$ scale the nuclear saturation scale relative to that
in the proton, $Q_{0A}^{2}=NQ_{0p}^{2}$, with $3<N<7$. The variation of the
results with $N$ represents an effective variation with centrality and
incorporating possible effects of fluctuations. Going forward in proton
rapidity reduces the range of the results and lessens the dependence on $N$.
|
---|---
|
Figure 10: The nuclear modification factor $R_{pA}^{\rm ch}$ for charged
hadron production in minimum bias $p+$Pb collisions at $\eta=0$, $2$, $4$, and
$6$ (with the convention that the proton beam moves toward forward rapidity)
obtained from the hybrid factorization in Eq. (39) assuming different values
of the saturation scale in the nucleus, $Q_{0A}^{2}$. The lines labeled by a
given value of $N$, for $3<N<7$, are results with fixed factorization scale
$\mu_{F}=p_{T}$ and fixed saturation scale $Q_{0A}^{2}=NQ_{0p}^{2}$ and
$Q_{0p}^{2}=0.168$ GeV${}^{2}/c^{2}$. The bands shown the variation in the
results with the choice of factorization scale. Two panels are shown for each
rapidity. The upper panel shows results obtained by taking $\alpha_{s}^{in}=0$
(assuming only elastic contribution) while the bottom panel shows the
variation of $\alpha_{s}^{\rm in}$ in the range $0.09\geq\alpha_{s}^{in}\geq
0.3$. In the bottom panels for $\eta=0$ and 2, results with both
$\alpha_{s}^{\rm in}=0.1$ and 0.2 are shown, while for $\eta=4$ and 6, only
$\alpha_{s}^{\rm in}=0.1$ is shown. The plots are taken from Ref. [20].
As shown in Ref. [97], the RHIC data unfortunately cannot fix the value of
$\alpha_{s}^{\rm in}$ in Eq. (4): with a reasonable $K$-factor,
$0\leq\alpha_{s}^{\rm in}\leq 0.3$ is consistent with hadron production at
RHIC both in $p+p$ and d$+A$ collisions. Figure 10 shows the effect of varying
$\alpha_{s}^{\rm in}$ in Eq. (4). For each value of $\eta$, results are shown
both for $\alpha_{s}^{\rm in}=0$ and variation within the range
$0.09\geq\alpha_{s}^{\rm in}\geq 0.3$. For $\eta=0$ and 2, both
$\alpha_{s}^{\rm in}=0.1$ and 0.2 are shown. There is only a small difference
at $\eta=0$ while a larger difference can be seen at forward proton rapidity,
especially for $N>5$. With $\eta=4$ and 6, only $\alpha_{s}^{\rm in}=0.1$ is
shown. Note that taking $\alpha_{s}^{\rm in}>0$ changes the $p_{T}$ slope
considerably, as may be expected.
In order to quantify the uncertainties due to different initial nuclear
saturation scales, in Fig. 10 $R_{pA}^{\rm ch}$ is shown for different initial
saturation scales $Q_{0A}^{2}=NQ_{0p}^{2}$ for $3<N<7$ with factorization
scale $Q=p_{T}$. Unfortunately, $R^{\rm ch}_{pA}$ is very sensitive to
$Q_{0A}^{2}$. However, by comparing to measurements of $R^{\rm ch}_{pA}$ at
one rapidity, the predictions in Fig. 10 can be used to determine $Q_{0A}^{2}$
at that rapidity. Then, if the measured $R_{pA}^{\rm ch}$ at one rapidity lies
between two lines labeled $N_{1}$ and $N_{2}$, the results at other rapidities
should correspond to the same values of $N$ and thus the same $Q_{0A}^{2}$.
Only these results should be considered to be true CGC predictions. In this
way, knowing $R_{pA}^{\rm ch}$ at one rapidity significantly reduces
theoretical uncertainties associated with $Q_{0A}^{2}$ at other rapidities.
Therefore, despite rather large theoretical uncertainties, it is still
possible to test the main CGC/saturation dynamics at the LHC.
It is clear that CGC approaches predict more suppression at forward proton
rapidities compared to collinear factorization [116, 117]. Moreover, the
small-$x$ evolution washes away the Cronin-type peak at low $p_{T}$ at all
rapidities. Therefore, observation of a Cronin peak in $p+$Pb collisions at
the LHC, regardless of whether it is accompanied by overall enhancement or
suppression, can potentially be a signal of non-CGC physics since it is
difficult to accommodate this feature in the current accuracy of the CGC
approach [20].
##### Albacete et al rcBK (J. Albacete, A. Dumitru, H. Fujii and Y. Nara)
Figure 11 shows bands for $R_{pA}(p_{T})$ at $\eta=0$ and 2 with rcBK-MC. The
upper edges of the bands are calculated with $\gamma=1.119$ and KKP-LO
fragmentation functions. the lower limit of the bands corresponds to the
McLerran-Venugopalan initial condition ($\gamma=1$) with DSS-NLO fragmentation
functions, taken from Ref. [95].
Figure 11: The nuclear modification factor for three different centrality
classes assuming $k_{T}$-factorization. The $\eta=2$ result is obtained with
the convention that the proton beam moves toward forward rapidity.
There is suppression of $R_{p{\rm Pb}}(p_{T}\sim 1\,{\rm GeV}/c)=0.6\pm 0.1$
at midrapidity. The ratio increases monotonically to $p_{T}\sim 3$ GeV/$c$
where it flattens. There is no clear Cronin peak. For all unintegrated gluon
densities, $R_{p{\rm Pb}}$ decreases with increasing rapidity. In addition to
the minimum bias results, two separate centrality classes are also shown, a
central bin, $N_{\rm part}>10$, and the most peripheral bin, $N_{\rm part}<5$.
There is a stronger effect for the most central collisions, as expected, for
low $p_{T}$ at both $\eta=0$ and 2. (Note that there is a smaller difference
between minimum bias and the most central collisions at $\eta=2$. The overall
effect is very weak for the peripheral bin with suppression persisting only to
$p_{T}\sim 2-3$ GeV/$c$. For more details, see Ref. [95].
##### Cold matter energy loss (Z.-B. Kang, I. Vitev and H. Xing)
Figure 12 presents model predictions for $R_{p{\rm Pb}}(p_{T})$ in minimum
bias $p+$Pb collisions at $y=0$ (top), $y=2$ (middle), and $y=4$ (bottom). The
upper edge of the bands corresponds to the RHIC scattering parameters while
The lower edge allows for a potential high-energy enhancement of the
parameters.
Figure 12: Predictions for the nuclear modification factor $R_{pPb}$ as a
function of $p_{T}$ for charged hadron production in minimum bias $p+$Pb
collisions. Results are shown for three rapidities: $y=0$ (top), $y=2$
(center), and $y=4$ (bottom) with the convention that the proton beam moves
toward forward rapidity, see Ref. [67].
The $y=0$ results were already presented in Fig. 8. At large proton rapidity
the CNM effects are all amplified due to the larger values of the proton
momentum fraction $x_{2}$ (relevent for cold nuclear matter energy loss with
the proton moving in the direction of forward rapidity), the smaller values of
the nuclear momentum fraction $x_{1}$ (relevent for dynamical shadowing with
the nucleus moving in the direction of backward rapidity) and more steeply-
falling $p_{T}$ spectra (relevent for the Cronin effect). As a result, at low
$p_{T}$ and large ion rapidity, dynamical shadowing can dominate and lead to
stronger suppression of inclusive particle production (note the disappearance
of the small Cronin enhancement at $y>0$). At high $p_{T}$ the suppression is
a combined effect of cold nuclear matter energy loss and the Cronin effect. As
will be seen later, the behavior of $R_{p{\rm Pb}}$ for $\pi^{0}$ and direct
photon production is qualitatively similar to the $R_{p{\rm Pb}}$ dependence
for charged hadrons shown here.
Away from midrapidity, these predictions suggest a stronger effect than that
found with EPS09 parton shadowing alone. On the other hand, the effect shown
here is weaker than the results using the rcBK-MC approach also shown in this
section.
##### $\mathtt{AMPT}$ at $\eta\neq 0$ (Z. Lin)
Figure 13 shows results with the $\mathtt{AMPT-def}$ calculation in minimum
bias $p+$Pb collisions at five different $\eta$ values. At $\eta=4$, the ratio
is larger than at lower rapidities. The ratios are very similar for $\eta=0$
and 2. The backward rapidity results have lower values of $R_{p{\rm Pb}}$.
While the ratios are indepedent of $p_{T}$ within the uncertainties for
$\eta=-2$, 0 and 2, the ratios increase with $p_{T}$ for $\eta=4$ and $-4$
with the largest increase for $\eta=-4$.
Figure 13: Predictions for the nuclear modification factor $R_{pPb}$ as a
function of $p_{T}$ for charged hadron production in minimum bias $p+$Pb
collisions for $\eta=-4$, $-2$, 0, 2 and 4 calculated using $\mathtt{AMPT-
def}$.
#### 3.3.4 Forward-backward asymmetry(G. G. Barnaföldi, J.Barette, M.
Gyulassy, P. Levai, G. Papp, M. Petrovici, V. Topor Pop, Z. Lin and R. Vogt)
One method of determining the difference between the proton and lead rapidity
regions is studying the forward-backward asymmetry in charged hadron
production, given by
$\displaystyle Y_{\rm asym}^{h}(p_{T})$ $\displaystyle=$
$\displaystyle\frac{E_{h}d^{3}\sigma^{h}_{p\rm
Pb}/d^{2}p_{T}d\eta|_{\eta>0}}{E_{h}d^{3}\sigma^{h}_{p{\rm
Pb}}/d^{2}p_{T}d\eta|_{\eta<0}}\,\,,$ (42) $\displaystyle=$
$\displaystyle\frac{R^{h}_{p{\rm Pb}}(p_{T},\eta>0)}{R^{h}_{p{\rm
Pb}}(p_{T},\eta<0)}\ .$
where the ‘forward’ direction, $\eta>0$, is that of the lead beam, toward
positive rapidity while the ‘backward’ direction, $\eta<0$ is that of the
proton beam, toward negative rapidity.
The asymmetries, calculated in the center of mass frame in the range
$0.3<|\eta|<0.8$ [108], both for standard shadowing and event generators, are
shown in Fig. 14. Two calculations of the forward-backward asymmetry in $p+$Pb
collisions are shown. The first result is in the $\mathtt{kTpQCD\\_v2.0}$
approach with several standard shadowing parameterizations. The second is
obtained with $\mathtt{AMPT}$.
Results with $\mathtt{kTpQCD\\_v2.0}$ including the $\mathtt{HIJING2.0}$ [47],
EKS98 [92], EPS08 [93], and HKN [94] shadowing parameterizations are
presented. Since the EKS98, EPS08 and HKN parameterizations are independent of
impact parameter, there is no difference between minimum bias results and
results with a centrality cut. Thus, these are all labeled MB for minimum
bias. With the HKN parameterization, $Y_{\rm asym}\approx 1$ independent of
$p_{T}$. The EKS98 and EPS08 parameterizations, producing identical ratios,
predict a small enhancement, $Y_{\rm asym}>1$.
Figure 14: Predictions for the forward-backward asymmetry, $Y_{\rm
asym}^{h}(p_{T})$, from Refs. [110, 111]. Centrality independent results are
shown for the HKN, EKS98 and EPS08 parameterizations (labeled MB). Minimum
bias results are also shown for $\mathtt{HIJINGB\overline{B}2.0}$ and
$\mathtt{HIJING2.0}$ with multiple scattering. In addition,
$\mathtt{HIJING2.0}$ results in MB collisions and for the 20% most central
collisions (using the parameter $s_{a}$ in Eq. (22)) are also shown. The blue
points are the $\mathtt{AMPT-def}$ results.
Results with the $\mathtt{HIJING2.0}$ shadowing parameterization with and
without multiple scattering differ for $p_{T}<5$ GeV/$c$ but converge at
higher $p_{T}$. When the collision centrality is taken into account, the
forward-backward asymmetry is enhanced by the centrality cut alone. Chaning
the strength of low $x$ shadowing, as in Eq. (22), does not change the shape
of $Y_{\rm asym}(p_{T})$ for $p_{T}>5$ GeV/$c$.
Finally, results for the minimum bias asymmetry is also shown for
$\mathtt{HIJINGB\overline{B}2.0}$ [53]. These calculations include both
shadowing and strong color field effects (indicated “+ SCF”). Here the ratio
rises from near unity at $p_{T}\sim 0$ to a peak at $p_{T}\sim 5.5$ GeV/$c$.
It then decreases to $Y_{\rm asym}\sim 0.75$ for $p_{T}\geq 10$ GeV/$c$.
The forward-backward asymmetry of $R_{p{\rm Pb}}$, calculated in the center of
mass frame with $\mathtt{AMPT-def}$ in the range $0.3<|\eta|<0.8$, is shown by
the blue points in Fig. 14 for charged particles in minimum-bias events with
simulations employing $\mathtt{AMPT-def}$. The initial enhancement first
increases with $p_{T}$ and then decreases. Although the statistical errors
become very large above $p_{T}\sim 10$ GeV/$c$, it is clear that the result is
similar to the general behavior of the $\mathtt{HIJINGB\overline{B}2.0}$
result.
The low $p_{T}$ enhancement may arise because the asymmetry in $dN/d\eta$ in
$p+$Pb relative to $p+p$ in the lead direction (positive rapidity) introduces
a natural enhancement in $R_{p{\rm Pb}}$ in the forward direction. The
enhanced particle production in $p+$Pb is most likely at low $p_{T}$, see e.g.
Fig. 13.
The $\mathtt{AMPT-def}$ results include the impact-parameter dependent nuclear
shadowing parameterization implemented in $\mathtt{HIJING1.0}$. Therefore the
asymmetry at other centralities would differ from that in minimum-bias events.
#### 3.3.5 $R_{p{\rm Pb}}(\eta)$ (G. G. Barnaföldi, J. Barette, M. Gyulassy,
P. Levai, M. Petrovici and V. Topor Pop)
Finally, we end this section with the predicted ratio $R_{p{\rm Pb}}$ as a
function of pseudorapidity in $\mathtt{HIJINGB\overline{B}2.0}$. As discussed
previously, increasing the minijet cutoff parameter $p_{0}$ from 2 GeV/$c$ in
$p+p$ collisions to 3.1 GeV$/c$ in $p+$Pb causes a slow growth in the per
nucleon multiplicity which could be interpreted as evidence for gluon
saturation. It is difficult to directly relate $p_{0}$ to the saturation scale
$Q_{s}$ because in $\mathtt{HIJING}$ hadronization proceeds through
longitudinal string fragmentation. The low $p_{T}$ part of the minijet
spectrum is particularly sensitive to the $\sqrt{s_{{}_{NN}}}$ and $A$
dependence of minijet suppression while the $p_{T}>5$ GeV/$c$ minijet tails
are unaffected. The energy and $A$ dependence of the string tension arises
from strong color field (color rope) effects not included in CGC phenomenology
that assumes hadronization by $k_{T}$-factorized gluon fusion.
Figure 15: The suppression factor $R_{p{\rm Pb}}$ as a function of
pseudorapidity, $\eta$, for $\mathtt{HIJINGB\overline{B}2.0}$ with strong
color field effects, without (solid) and with (dashed) shadowing.
$\mathtt{HIJING}$ minijet hadronization does not proceed by independent
fragmentation, as in $\mathtt{PYTHIA}$ [59], but through string fragmentation
with gluon minijets represented as kinks in the strings. The interplay between
longitudinal string fragmentation dynamics and minijets is a nonperturbative
feature of $\mathtt{HIJING}$-type models. The effect of string fragmentation
on the multiplicity is manifested in the behavior of $R_{p{\rm Pb}}(\eta)$ in
the fragmentation regions $|\eta|>5$. In the nuclear beam fragmentation
region, the ratio is approximately linearly increasing from $\eta=5$ to the
kinematic limit. There is an approximate plateau in the ratio over $0<\eta<5$,
followed by a decrease toward the value $1/N_{\rm coll}$ in the proton
fragmentation region, a Glauber geometric effect first explained in Refs.
[118, 119] and a feature of string fragmentation in $\mathtt{HIJING}$.
## 4 Identified Particles
### 4.1 $\mathtt{AMPT}$ (Z. Lin)
Some representative $\mathtt{AMPT}$ results for identified particles are shown
here. The rapidity distributions for $K^{+}$ and $K^{-}$ production in the
default and string melting versions are shown in Fig. 16. The default and
string melting versions of the rapidity distributions of protons and
antiprotons are shown in Fig. 17.
There are differences between the default and string melting versions, even in
$p+p$ collisions. However, these differences are typically small. Note that
the bump at midrapidity in the proton and antiproton rapidity distributions in
the string-melting version is likely a result of the simple quark coalescence
model implemented in $\mathtt{AMPT}$ (see e.g. Fig. 38 of Ref. [65]).
Figure 16: Rapidity distribution, $dN/dy$, of $K^{+}$ (left) and $K^{-}$
(right) mesons.
Figure 17: Rapidity distribution, $dN/dy$, of and $p$ and $\overline{p}$ in
the default (left) and SM (right) scenarios.
The left-hand side of Fig. 18 shows $R_{p{\rm Pb}}$ for protons, antiprotons,
neutral pions and charged particles. The identified particles are shown for
$y=0$ while the charged particle ratio is for $\eta=0$. The ratios are all
very similar except for protons which grows almost linearly with $p_{T}$.
The right-hand side of Fig. 18 presents results for the suppression factor for
charged particles within $|\eta|<1$ in three centrality bins: minimum bias;
the 10% most central collisions; and a peripheral bin, the $60-80$% most
central collisions. Interestingly, the ratio is smallest and decreasing with
$p_{T}$ for the more peripheral bin while the largest value of $R_{p{\rm Pb}}$
is the most central bin.
Figure 18: (Left) The minimum bias results at $y=0$ for $\pi^{0}$, $p$ and
$\overline{p}$ are shown with the charged hadron ratio for $\eta=0$. All
results are calculated with $\mathtt{AMPT-def}$. (Right) The charged hadron
results at midrapidity, $|\eta|<1$, are shown for minimum bias collisions as
well as in two centrality bins: the 10% most central and the 60-80% most
central. All results are calculated employing $\mathtt{AMPT-def}$.
### 4.2 Centrality dependent $R_{p{\rm Pb}}^{\pi^{0}}(p_{T},y=0)$ with EPS09s
nPDFs (I. Helenius and K. J. Eskola)
Predictions of the centrality dependence of $\pi^{0}$ production at midrapidty
are presented here. Two spatially-dependent nuclear PDF (nPDF) sets, EPS09s
and EKS98s were determined in Ref. [120]. The key component is a power series
ansatz of the nuclear thickness function ($T_{A}(\vec{s})$) for the spatial
dependence of the nPDF modifications,
$r_{i}^{A}(x,Q^{2},\vec{s})=1+\sum\limits_{j=1}^{n}c^{i}_{j}(x,Q^{2})\left[T_{A}(\vec{s})\right]^{j}\,\,.$
(43)
The $A$ dependence of the earlier global nPDF fits EPS09 [48] and EKS98 [92]
was exploited to obtain the values of the $A$-independent coefficients
$c^{i}_{j}(x,Q^{2})$. It was found that $n=4$ is sufficient for reproducing
the $A$ systematics of the globally-fitted nPDFs.
Predictions are presented for the nuclear modification factor $R_{p{\rm
Pb}}^{\pi^{0}}$ of inclusive neutral pion production at the LHC using the new
EPS09s NLO nPDF set. As in Ref. [120], the centrality classes are defined in
terms of impact parameter intervals. For a given centrality class, $R_{p{\rm
Pb}}^{\pi^{0}}$ is defined as
$R_{p{\rm
Pb}}^{\pi^{0}}(p_{T},y;b_{1},b_{2})\equiv\dfrac{\left\langle\dfrac{d^{2}N_{p{\rm
Pb}}^{\pi^{0}}}{dp_{T}dy}\right\rangle_{b_{1},b_{2}}}{\dfrac{\langle N_{\rm
coll}^{p{\rm Pb}}\rangle_{b_{1},b_{2}}}{\sigma^{NN}_{\rm
in}}\dfrac{d^{2}\sigma_{pp}^{\pi^{0}}}{dp_{T}dy}}=\dfrac{\int_{b_{1}}^{b_{2}}d^{2}\vec{b}\dfrac{d^{2}N_{p{\rm
Pb}}^{\pi^{0}}(\vec{b})}{dp_{T}dy}}{\int_{b_{1}}^{b_{2}}d^{2}\vec{b}\,T_{p{\rm
Pb}}(\vec{b})\dfrac{d^{2}\sigma_{pp}^{\pi^{0}}}{dp_{T}dy}},$ (44)
where the impact parameter limits $b_{1}$ and $b_{2}$ are calculated from the
optical Glauber model. The proton is assumed to be point-like, i.e. $T_{p{\rm
Pb}}(\vec{b})=T_{\rm Pb}(\vec{b})$. The minimum bias ratio $R_{p{\rm
Pb}}^{\pi^{0}}$ is obtained with $b_{1}=0$ and $b_{2}\rightarrow\infty$. The
cross sections are calculated in NLO using the $\mathtt{INCNLO}$
packageiiiiiihttp://lapth.in2p3.fr/PHOX_FAMILY/readme_inc.html [84] with the
CTEQ6M proton PDFs [69] along with three different fragmentation functions:
KKP [42]; AKK [121]; and fDSS [70]. The uncertainty band reflecting the nPDF
uncertainties is calculated using the error sets of EPS09s with the fDSS
fragmentation functions according to the prescription described in the
original EPS09 paper [48].
In Fig. 19 the minimum bias $R_{p{\rm Pb}}^{\pi^{0}}$ is shown as a function
of $p_{T}$ at midrapidity ($y=0$) in the nucleon-nucleon center-of-mass frame.
Both logarithmic and linear $p_{T}$ scales are shown. In the region
$p_{T}<10\text{ GeV}/c$ a suppression due to small-$x$ shadowing in the nPDFs
is observed while at $10<p_{T}<200\text{ GeV}/c$ there is a small enchancement
due to the antishadowing. Even though the different fragmentation functions
may yield different absolute cross sections, these differences cancel in the
ratio $R_{p{\rm Pb}}^{\pi^{0}}$.
Figure 19: The nuclear modification factor for inclusive $\pi^{0}$ production
in $p+$Pb collisions at $y=0$ for minimum bias collisions calculated with the
EPS09s NLO nPDFs, plotted on logarithmic (left) and linear (right) $p_{T}$
scales. The blue error band is calculated employing the EPS09s error sets. All
scales are fixed to the pion $p_{T}$. Based on Ref. [120].
In Fig. 20 $R_{p{\rm Pb}}^{\pi^{0}}$ is presented in four different centrality
classes, $(0-20)~{}\%$, $(20-40)~{}\%$, $(40-60)~{}\%$ and $(60-80)~{}\%$. The
impact parameter limits for these centrality classes can be found in Ref.
[120]. For comparison, the minimum bias results are also shown.
Figure 20: The nuclear modification factor for inclusive $\pi^{0}$ production
in $p+$Pb collisions at $y=0$ in four centrality classes calculated with the
EPS09s NLO nPDFs. The blue error band is calculated employing the EPS09s error
sets. All scales are fixed to the pion $p_{T}$. For comparison, the minimum
bias result is also shown. Based on Ref. [120].
Slightly larger nuclear effects are observed in the most central collisions
than in the minimum bias collisions. Nuclear effects in the most peripheral
collisions are about a factor of two smaller than those in the most central
collisions.
### 4.3 Cold matter effects with energy loss (Z.-B.Kang, I. Vitev and H.
Xing)
The predictions for $R_{p{\rm Pb}}^{\pi^{0}}$, like those for charged hadrons
shown in Fig. 12 are based on perturbative QCD factorization. Cold nuclear
matter effects are implemented separately. The advantage of this approach is
that all CNM effects have a clear physical origin, generally centered around
multiple parton scattering. The implementation of these calculable CNM effects
is well documented, see Sec. 2.6. Isospin effects, the Cronin effect, cold
nuclear matter energy loss, and dynamical shadowing are all included.
The neutral pion results in Fig. 21 are rather similar to those shown in Fig.
12. The upper edges of the bands at $y=0$ (top), $y=2$ (middle), and $y=4$
(bottom) correspond to the RHIC scattering parameters while the lower edges
correspond to potential enhancement of these parameters.
Figure 21: Predictions for the nuclear modification factor $R_{p{\rm
Pb}}^{\pi^{0}}(p_{T})$ as a function in minimum bias collisions at
$\sqrt{s_{{}_{NN}}}=5$ TeV. Results for three rapidities: $y=0$ (top), $y=2$
(middle), and $y=4$ (bottom) are shown. The calculations have been made with
the convention that the proton beam moves toward forward rapidity, see Ref.
[67].
As seen by comparison with the EPS09 minimum bias results for $y=0$ in Fig.
19, the combined effects included here are stronger than with shadowing alone.
Antishadowing in EPS09 produces a ratio larger than unity for $10<p_{T}<200$
GeV/$c$ while the ratio is smaller than unity over the entire range calculated
here ($p_{T}<50$ GeV/$c$).
## 5 Quarkonium (R. Vogt)
The predictions for the $J/\psi$ suppression factor, considering only
shadowing effects on the parton densities are described in this section. There
are a number of possible cold matter effects on $J/\psi$ production, including
modifications of the parton densities in nuclei (shadowing); breakup of the
quarkonium state due to inelastic interactions with nucleons (absorption); and
energy loss in cold matter. Since the quarkonium absorption cross section
decreases with center-of-mass energy, we can expect that shadowing is the most
important cold matter effect at midrapidity, see Refs. [122, 123]. Here we
show results for the rapidity and $p_{T}$ dependence of shadowing at
$\sqrt{s_{{}_{NN}}}=200$ GeV for d+Au collisions at RHIC and the rapidity
dependence at $\sqrt{s_{{}_{NN}}}=5$ TeV $p+$Pb collisions, neglecting
absorption.
The results are obtained in the color evaporation model (CEM) at next-to-
leading order in the total cross section. In the CEM, the quarkonium
production cross section is some fraction, $F_{C}$, of all $Q\overline{Q}$
pairs below the $H\overline{H}$ threshold where $H$ is the lowest mass heavy-
flavor hadron,
$\displaystyle\sigma_{C}^{\rm
CEM}(s)=F_{C}\sum_{i,j}\int_{4m^{2}}^{4m_{H}^{2}}ds\int
dx_{1}\,dx_{2}~{}f_{i}^{p}(x_{1},\mu_{F}^{2})~{}f_{j}^{p}(x_{2},\mu_{F}^{2})~{}\hat{\sigma}_{ij}(\hat{s},\mu_{F}^{2},\mu_{R}^{2})\,\,,$
(45)
where $ij=q\overline{q}$ or $gg$ and $\hat{\sigma}_{ij}(\hat{s})$ is the
$ij\rightarrow Q\overline{Q}$ subprocess cross section. The normalization
factor $F_{C}$ is fit to the forward (integrated over $x_{F}>0$) $J/\psi$
cross section data on only $p$, Be, Li, C, and Si targets. In this way,
uncertainties due to ignoring any cold nuclear matter effects which are on the
order of a few percent in light targets are avoided. The fits are restricted
to the forward cross sections only.
The same values of the central charm quark mass and scale parameters are
employed as those found for open charm, $m=1.27\pm 0.09$ GeV/$c^{2}$,
$\mu_{F}/m=2.10^{+2.55}_{-0.85}$, and $\mu_{R}/m=1.60^{+0.11}_{-0.12}$ [124].
The normalization $F_{C}$ is obtained for the central set,
$(m,\mu_{F}/m,\mu_{R}/m)=(1.27\,{\rm GeV},2.1,1.6)$. The calculations for the
extent of the mass and scale uncertainties are multiplied by the same value of
$F_{C}$ to obtain the extent of the $J/\psi$ uncertainty band [124]. The
results shown here are not the same as those calculated at leading order [125]
because the LO and NLO gluon shadowing parameterizations differ significantly
at low $x$ [48].
|
---|---
|
Figure 22: The ratio $R_{\rm dAu}$ (left) and $R_{\rm AuAu}$ (right) at
$\sqrt{s}=5$ TeV. The dashed red histogram shows the EPS09 uncertainties while
the dot-dashed blue histogram shows the dependence on mass and scale. The
$p+p$ denominator is also calculated at 5 TeV and does not include any
rapidity shift in $p+$Pb collisions. For a discussion about normalizing the
results to $p+p$ collisions at different energies, see Ref. [125].
Figure 22 shows the uncertainty in the shadowing effect due to uncertainties
in the EPS09 shadowing parameterization [48] (red) as well as those due to the
mass and scale uncertainties obtained in the fit to the total charm cross
section (blue) calculated with the EPS09 central set. All the calculations are
NLO in the total cross section and assume that the intrinsic $k_{T}$
broadening is the same in $p+p$ as in $p+$Pb. Note that the rapidity-dependent
ratios are adjusted so that the lead nucleus moves toward negative rapidity
and the small $x$ region of the nucleus is at large negative rapidity, as is
the case for pseudorapidity distributions and the rapidity-dependent ratios
shown previously.
The mass and scale uncertainties are calculated based on results using the one
standard deviation uncertainties on the quark mass and scale parameters. If
the central, upper and lower limits of $\mu_{R,F}/m$ are denoted as $C$, $H$,
and $L$ respectively, then the seven sets corresponding to the scale
uncertainty are $\\{(\mu_{F}/m,\mu_{F}/m)\\}$ = {$(C,C)$, $(H,H)$, $(L,L)$,
$(C,L)$, $(L,C)$, $(C,H)$, $(H,C)$}. The uncertainty band can be obtained for
the best fit sets by adding the uncertainties from the mass and scale
variations in quadrature. The envelope containing the resulting curves,
$\displaystyle\sigma_{\rm max}$ $\displaystyle=$ $\displaystyle\sigma_{\rm
cent}+\sqrt{(\sigma_{\mu,{\rm max}}-\sigma_{\rm cent})^{2}+(\sigma_{m,{\rm
max}}-\sigma_{\rm cent})^{2}}\,\,,$ (46) $\displaystyle\sigma_{\rm min}$
$\displaystyle=$ $\displaystyle\sigma_{\rm cent}-\sqrt{(\sigma_{\mu,{\rm
min}}-\sigma_{\rm cent})^{2}+(\sigma_{m,{\rm min}}-\sigma_{\rm
cent})^{2}}\,\,,$ (47)
defines the uncertainty. The EPS09 band is obtained by calculating the
deviations from the central value for the 15 parameter variations on either
side of the central set and adding them in quadrature. With the new
uncertainties on the charm cross section, the band obtained with the mass and
scale variation is narrower than that with the EPS09 variations.
## 6 Photons
Photons are ideal probes in heavy ion collisions due to their lack of final-
state interactions with either a quark-gluon plasma or a hot hadron gas.
Baseline measurements in $p+p$ and $p+$Pb collisions are very important to
subtract photons from hard initial parton scatterings which are unrelated to
QGP and, in particular, to determine cold nuclear matter effects on photon
production. Measured nuclear modification factors for photons at high
transverse momenta ($p_{T}>5$ GeV/$c$) measured in Pb+Pb collisions at the LHC
and Au+Au collisions at RHIC, $R_{AA}\approx 1$, have demonstrated that hard
processes scale with the number of binary nucleon-nucleon collisions in
nucleus-nucleus collisions. An especially important question for heavy-ion
physics at the LHC is the magnitude of low-$x$ effects on the parton
densities, particularly for gluons. The presence of shadowing, i.e.
suppression of the number of low-$x$ partons, has a significant effect on
benchmarking hard processes at intermediate $p_{T}$. Direct photons and other
electromagnetic probes are in particular needed for a quantitatively precise
determination of the nuclear parton distributions.
This section includes an NLO calculation of direct photon production in $p+p$
and $p+$Pb collisions; LO calculations of cold matter effects on photon
production at several rapidities; saturation effects on direct and inclusive
prompt photon production and photon-hadron correlations; and a discussion on
gluon saturation and shadowing in dilepton and photon production.
### 6.1 Direct photon cross sections (R. J. Fries and S. De)
Predictions of the $p_{T}$ and $y$ dependence of direct photon production are
given here. The impact-parameter averaged EPS09 shadowing parameterizations
[48] are used with the CTEQ6.6 parton densities [126] to calculate results in
minimum bias $p+$Pb collisions. These calculations can thus be employed to
check the validity of the EPS09 modifications. The per nucleon cross sections
given here can be directly compared to the minimum bias $p+$Pb data by
converting the cross sections to differential yields scaling the results by
$\sigma_{\rm in}=67$ mb and the average number of $NN$ collisions in a given
centrality bin.
These NLO in $\alpha_{s}$ calculations are performed with
$\mathtt{JETPHOX1.3.1}$ [127, 128]. Hard prompt photons and fragmentation
photons (obtained with the BFG-II fragmentation functions [129]) are both
included. An isolation cut of $E_{T,\mathrm{hadron}}<5$ GeV on hadronic energy
is imposed within a $R=0.4$ isolation cone around the photon, similar to the
CMS analysis method for $p+p$ collisions at 7 TeV.
Figure 23 shows the rapidity dependence, $d\sigma/dy$, of direct photon
production in $p+p$ and $p+$Pb collisions, normalized per nucleon. A $p_{T}$
cut, $p_{T}>4$ GeV/$c$ is imposed. The results are presented in the lab frame.
Recall that there is a difference, $\Delta y=0.465$, between the center-of-
mass and lab frames at the current energy.
Figure 23: The direct photon cross section, $d\sigma/dy$, as a function of
rapidity in the laboratory frame for photons with $p_{T}>4$ GeV/$c$ and an
isolation cut $E_{T,\mathrm{hadron}}<5$ GeV. Both $p+p$ and $p+$Pb results are
shown. The $p+$Pb results are normalized to the per nucleon cross section.
On the left-hand side of Fig. 24, the $p_{T}$ spectrum at $y=0$,
$d\sigma/d^{2}p_{T}dy$, is shown for $p+p$ and $p+$Pb collisions. The $p+p$
result is scaled down by a factor of 100 to separate the two curves. The
corresponding nuclear modification factor is shown on the right-hand side.
Note that a logarithmic scale is used on the $x$-axis here to highlight the
modification of the low $p_{T}$ part of the photon spectrum.
Figure 24: (Left) The direct photon $p_{T}$ distribution at $y=0$ in the lab
frame. The $p+p$ distribution is scaled down by two orders of magnitude.
(Right) The corresponding modification factor $R_{p{\rm Pb}}(p_{T})$. Note the
logarithmic $p_{T}$ scale.
### 6.2 Cold matter effects on photon production (Z.-B. Kang, I. Vitev and H.
Xing)
Prompt photon production in $p+p$ collisions has two components, the direct
and fragmentation contributions [68]:
$\displaystyle\frac{d\sigma}{dyd^{2}p_{T}}=\frac{d\sigma_{\rm
dir}}{dyd^{2}p_{T}}+\frac{d\sigma_{\rm frag}}{dyd^{2}p_{T}}.$ (48)
The fragmentation contribution is given by:
$\displaystyle\frac{d\sigma_{\rm frag}}{dyd^{2}p_{T}}$ $\displaystyle=$
$\displaystyle
K\frac{\alpha_{s}^{2}}{s}\sum_{a,b,c}\int\frac{dx_{1}}{x_{1}}d^{2}k_{T_{1}}\,f_{a/N}(x_{1},k_{T_{1}}^{2})\int\frac{dx_{2}}{x_{2}}d^{2}k_{T_{2}}\,f_{b/N}(x_{2},k_{T_{2}}^{2})$
(49)
$\displaystyle\times\int\frac{dz_{c}}{z_{c}^{2}}\,D_{\gamma/c}(z_{c})H_{ab\to
c}(\hat{s},\hat{t},\hat{u})\delta(\hat{s}+\hat{t}+\hat{u}).$
The expression is exactly the same as Eq. (23) if the parton-to-hadron
fragmentation function, $D_{h/c}(z_{c})$, is replaced by the parton-to-photon
fragmentation function, $D_{\gamma/c}(z_{c})$. The direct contribution can be
written as:
$\displaystyle\frac{d\sigma_{\rm dir}}{dyd^{2}p_{T}}$ $\displaystyle=$
$\displaystyle K\frac{\alpha_{\rm
em}\alpha_{s}}{s}\sum_{a,b}\int\frac{dx_{1}}{x_{1}}d^{2}k_{T_{1}}f_{a/N}(x_{1},k_{T_{1}}^{2})\,\int\frac{dx_{2}}{x_{2}}d^{2}k_{T_{2}}f_{b/N}(x_{2},k_{T_{2}}^{2})$
(50) $\displaystyle\times
H_{ab\to\gamma}(\hat{s},\hat{t},\hat{u})\delta\left(\hat{s}+\hat{t}+\hat{u}\right),$
where $H_{ab\to\gamma}$ are the partonic hard-scattering functions for direct
photon production [68, 130].
In $p+p$ collisions, $\langle k_{T}^{2}\rangle_{pp}=1.8$ GeV${}^{2}/c^{2}$,
along with a $K$ factor of $\mathcal{O}(1)$, which gives a good description of
production at RHIC and LHC energies. The CTEQ6L1 PDFs [69] are employed with
the GRV parametrization for parton-to-photon fragmentation functions [131].
The factorization and renormalization scales are fixed to the transverse
momentum of the produced photon, $\mu_{F}=\mu_{R}=p_{T}$.
The results shown here are calculated employing the same cold matter effects
described in Sec. 2.6. Figure 25 presents predictions for the nuclear
modification factor in prompt photon production as a function of $p_{T}$ for
$y=0$ (top), $y=2$ (middle), and $y=4$ (bottom). The upper edge of the bands
corresponds to the RHIC scattering parameters The lower edge represents a
high-energy enhancement of the parameters. The behavior of $R_{pPb}$ for
direct photons is qualitatively the same as the $\pi^{0}$s and charged hadron
results shown earlier.
Figure 25: Predictions of the nuclear modification factor $R_{p{\rm Pb}}$ as a
function of $p_{T}$ for prompt photon production in minimum bias $p+$Pb
collisions for $y=0$ (top), $y=2$ (middle), and $y=4$ (bottom), see Ref. [67].
### 6.3 rcBK calculation of photon production (A. Rezaeian)
The cross section for semi-inclusive prompt photon-quark production in $p+A$
collisions at leading twist in the CGC formalism is [132, 133],
$\displaystyle{d\sigma^{q\,A\rightarrow q(l)\,\gamma(p^{\gamma})\,X}\over
d^{2}\vec{b_{T}}\,d^{2}\vec{p_{T}}^{\gamma}\,d^{2}\vec{l_{T}}\,d\eta_{\gamma}\,d\eta_{h}}={Ke_{q}^{2}\,\alpha_{\rm
em}\over\sqrt{2}(4\pi^{4})}\,{p^{-}\over(p_{T}^{\gamma})^{2}\sqrt{s}}\,{1+(l^{-}/k^{-})^{2}\over[p^{-}\,\vec{l_{T}}-l^{-}\vec{p_{T}}^{\gamma}]^{2}}$
$\displaystyle\times\delta\bigg{[}x_{q}-{l_{T}\over\sqrt{s}}e^{\eta_{h}}-{p_{T}^{\gamma}\over\sqrt{s}}e^{\eta_{\gamma}}\bigg{]}\,[2l^{-}p^{-}\,\vec{l_{T}}\cdot\vec{p_{T}}^{\gamma}+p^{-}(k^{-}-p^{-})\,l_{T}^{2}+l^{-}(k^{-}-l^{-})\,(p_{T}^{\gamma})^{2}]$
$\displaystyle\times N_{F}(|\vec{l_{T}}+\vec{p_{T}}^{\gamma}|,x_{g})\,,$ (51)
where $p^{\gamma}$, $l$, and $k$ are the $4$-momenta of the produced prompt
photon, the outgoing $q$ ($\overline{q}$) and the incident $q$
($\overline{q}$), respectively. A $K$ factor was introduced to absorb higher-
order corrections. The light-cone fraction $x_{q}$ is the ratio of the
incoming quark and proton energies, $x_{q}=k^{-}/\sqrt{s/2}$. The
pseudorapidities of the outgoing prompt photon, $\eta_{\gamma}$, and quark,
$\eta_{h}$, are defined as $p^{-}=(p_{T}^{\gamma}/\sqrt{2})e^{\eta_{\gamma}}$
and $l^{-}=(l_{T}/\sqrt{2})e^{\eta_{h}}$. The angle between the final-state
quark and the prompt photon, $\Delta\phi$, is defined as
$\cos(\Delta\phi)\equiv(\vec{l}_{T}\cdot\vec{p}_{T}^{\gamma})/(l_{T}p_{T}^{\gamma})$.
Only high $p_{T}$ light hadron production is considered here. Therefore
rapidity and pseudorapidity are equivalent. The semi-inclusive photon-hadron
production cross section in proton-nucleus collisions can be obtained by
convoluting the partonic cross section, Eq. (51), with the quark distribution
functions of the proton and the quark-hadron fragmentation function,
$\displaystyle\frac{d\sigma^{p\,A\rightarrow
h(p^{h})\,\gamma(p^{\gamma})\,X}}{d^{2}\vec{b_{T}}\,d^{2}\vec{p_{T}}^{\gamma}\,d^{2}\vec{p_{T}}^{h}\,d\eta_{\gamma}\,d\eta_{h}}=$
$\displaystyle\int^{1}_{z_{f}^{\rm
min}}\frac{dz_{f}}{z_{f}^{2}}\,\int\,dx_{q}\,f_{q}(x_{q},Q^{2})\frac{d\sigma^{q\,A\rightarrow
q(l)\,\gamma(p^{\gamma})\,X}}{d^{2}\vec{b_{T}}\,d^{2}\vec{p_{T}}^{\gamma}\,d^{2}\vec{l_{T}}\,d\eta_{\gamma}\,d\eta_{h}}D_{h/q}(z_{f},Q^{2})\,,$
where $p^{h}_{T}$ is the transverse momentum of the produced hadron. The sum
over quark and antiquark flavors in Eq. (6.3) is implied. The light-cone
momentum fractions $x_{q}$, $x_{\bar{q}}$, and $x_{g}$ in Eqs. (51) and (6.3)
are related to the transverse momenta and rapidities of the produced hadron
and prompt photon [134],
$\begin{array}[]{ll}\normalsize
x_{q}=x_{\bar{q}}=\frac{1}{\sqrt{s}}\left(p_{T}^{\gamma}\,e^{\eta_{\gamma}}+\frac{p_{T}^{h}}{z_{f}}\,e^{\eta_{h}}\right)\,\,,&x_{g}=\frac{1}{\sqrt{s}}\left(p_{T}^{\gamma}\,e^{-\eta_{\gamma}}+\frac{p_{T}^{h}}{z_{f}}\,e^{-\eta_{h}}\right)\,\,,\\\
z_{f}=\frac{p_{T}^{h}}{l_{T}}\,\,,&z_{f}^{\rm
min}=\frac{p_{T}^{h}}{\sqrt{s}}\left(\frac{e^{\eta_{h}}}{1-(p_{T}^{\gamma}/\sqrt{s})\,e^{\eta_{\gamma}}}\,\right)\,\,.\end{array}$
The single inclusive prompt photon cross section in the CGC framework can be
obtained from Eq. (51) by integrating over the momenta of the final state
quark or antiquark. The corresponding cross section can be then divided into
two contributions: fragmentation (first term) and direct (second term) photons
[134],
$\displaystyle\frac{d\sigma^{p\,A\rightarrow\gamma(p^{\gamma})\,X}}{d^{2}\vec{b_{T}}d^{2}\vec{p_{T}}^{\gamma}d\eta_{\gamma}}=\frac{K}{(2\pi)^{2}}\Big{[}\int_{x_{q}^{\rm
min}}^{1}dx_{q}f_{q}(x_{q},Q^{2})\frac{1}{z}\,N_{F}(x_{g},p_{T}^{\gamma}/z)D_{\gamma/q}(z,Q^{2})$
$\displaystyle+\frac{e_{q}^{2}\alpha_{\rm
em}}{2\pi^{2}(p_{T}^{\gamma})^{4}}\int_{x_{q}^{\rm
min}}^{1}dx_{q}f_{q}(x_{q},Q^{2})z^{2}[1+(1-z)^{2}]\int_{l_{T}^{2}<Q^{2}}d^{2}\vec{l_{T}}\,l_{T}^{2}\,N_{F}(\bar{x}_{g},l_{T})\Big{]},\
$
where $D_{\gamma/q}(z,Q^{2})$ is the leading order quark-photon fragmentation
function [68, 129, 131]. Similar to the hybrid formalism for inclusive hadron
production, Eq. (4), $Q$ is a hard scale. Although the cross sections given in
Eq. (4) and Eq. (6.3) describe different final-state particle production, they
are strikingly similar. The light-cone momentum fractions $x_{g}$,
$\bar{x}_{g}$, and $z$ above are related to the transverse momentum and
rapidity of the prompt photon [134],
$\begin{array}[]{ll}\normalsize
x_{g}=x_{q}\,e^{-2\,\eta_{\gamma}}\,\,,&\bar{x}_{g}=\frac{1}{x_{q}\,s}\left[{(p_{T}^{\gamma})^{2}\over
z}+\frac{(l_{T}-p_{T}^{\gamma})^{2}}{1-z}\right]\,\,,\\\ x_{q}^{\rm
min}=\frac{p_{T}^{\gamma}}{\sqrt{s}}e^{\eta_{\gamma}}\,\,,&z=\frac{p_{T}^{\gamma}}{x_{q}\,\sqrt{s}}e^{\eta_{\gamma}}\,\,.\end{array}$
The expression in Eq. (6.3) was obtained using a hard cutoff to subtract the
collinear singularity [134]. The use of a cutoff may result in a mismatch
between the finite corrections to Eq. (6.3) and those included in
parameterizations of the photon fragmentation function. However, this mismatch
is higher-order in the coupling constant and its proper treatment requires a
full NLO calculation.
In Eqs. (6.3) and (6.3), the factorization scale $\mu_{F}$ is assumed to be
the same in the fragmentation functions and the parton densities. In order to
investigate the uncertainties associated with choice of $\mu_{F}$, several
values of $\mu_{F}$ are considered: $\mu_{F}=2p^{\gamma}_{T}$;
$p^{\gamma}_{T}$; and $p^{\gamma}_{T}/2$. The amplitude $N_{F}$ in Eqs. (51)
and (6.3) is defined in Eq. (6).
Figure 26: The nuclear modification factor $R_{pA}^{\gamma}$ for direct photon
production in minimum bias $p+$Pb collisions at rapidities $\eta_{\gamma}=0$,
$2$, $4$, and $6$ (with the convention that the proton beam moves toward
forward rapidity) obtained from Eq. (6.3) with solutions of the rcBK equation
with different initial nuclear saturation scales. The band labeled CGC-rcBK
includes uncertainties due to the variation of the nuclear saturation scale
and the factorization scale $\mu_{F}$. Similar to Fig. 10, the lines labeled
$N$ are results with a fixed factorization scale, $\mu_{F}=p^{\gamma}_{T}$,
and a fixed saturation scale $Q_{0A}^{2}=NQ_{0p}^{2}$ with
$Q_{0p}^{2}=0.168\,\text{GeV}^{2}/c^{2}$. The range $3<N<7$ is constrained in
Eq. (39). See Ref. [20].
In Figs. 26 and 27, predictions of the direct photon and single inclusive
prompt photon production nuclear modification factors $R_{pA}^{\gamma}(p_{T})$
in minimum bias $p+$Pb collisions at several rapidities. The solutions of the
rcBK equation were obtained using Eq. (6.3) with different initial nuclear
saturation scales. The band labeled CGC-rcBK includes uncertainties due to the
variation of $Q_{0A}^{2}$ within the range given in Eq. (39) with
factorization scales $\mu_{F}=2p_{T}^{\gamma}$, $p_{T}^{\gamma}$, and
$p_{T}^{\gamma}/2$. In Fig. 27, the $\eta_{\gamma}=0$ results include
inclusive prompt photon production calculated in Ref. [135] employing the
Iancu-Itakura-Munier (IIM) saturation model which also provides a good
description of the HERA data [136]. In the IIM method, saturation is
approached from the BFKL region. Therefore the IIM small-$x$ evolution is
different from that obtained with the rcBK equation.
Figure 27: The nuclear modification factor $R_{pA}^{\gamma}$ for single
inclusive prompt photon production in minimum bias $p+$Pb collisions at
$\eta_{\gamma}=0$ and 2 (with the convention that the proton beam moves toward
forward rapidity). The curves are described in Fig. 26. The dashed red line
labeled CGC-IIM was calculated according to the IIM dipole saturation model
[135]. See Ref. [20].
As discussed in Sec. 3.3.3, the results in Figs. 26 and 27 on
$R_{pA}^{\gamma}$, together with those on $R_{pA}^{\rm ch}$ in Fig. 10, can be
used to fix the nuclear saturation scale. Once $Q_{0A}^{2}$ has been
established at one rapidity, the predictions at other rapidities are fixed and
the CGC/saturation dynamics can be tested.
#### 6.3.1 Photon-hadron correlations
Photon-hadron azimuthal correlations in $p+A$ and $p+p$ collisions could be an
excellent probe of small-$x$ dynamics [134, 137]. This correlation can be
defined as [134, 137],
$P(\Delta\phi)=\bigg{[}{d\sigma^{p\,A\rightarrow
h(p_{T}^{h})\,\gamma(p_{T}^{\gamma})\,X}[\Delta\phi]\over
d^{2}\vec{b_{T}}\,p_{T}^{h}dp_{T}^{h}\,p_{T}^{\gamma}dp_{T}^{\gamma}\,d\eta_{\gamma}\,d\eta_{h}\,d\phi}\bigg{]}\bigg{[}{d\sigma^{p\,A\rightarrow
h(p_{T}^{h})\,\gamma(p_{T}^{\gamma})\,X}[\Delta\phi=\Delta\phi_{c}]\over
d^{2}\vec{b_{T}}\,p_{T}^{h}dp_{T}^{h}\,p_{T}^{\gamma}dp_{T}^{\gamma}\,d\eta_{\gamma}\,d\eta_{h}\,d\phi}\bigg{]}^{-1}\,\,.$
(52)
The correlation function $P(\Delta\phi)$ is the probability of semi-inclusive
photon-hadron pair production in a certain kinematic region at angle
$\Delta\phi$ relative to production in the same kinematics at fixed reference
angle, $\Delta\phi_{c}=\pi/2$ [134, 137].
Fig. 28 shows the predicted $P(\Delta\phi)$ in minimum bias $p+p$ and $p+$Pb
collisions at 5 TeV for $p^{h}_{T}<p^{\gamma}_{T}$ (left) and
$p^{h}_{T}>p^{\gamma}_{T}$ (right). Given the $p_{T}$ of the produced photon
and hadron, the corresponding correlation can have either a double or single
peak structure. In Ref. [137] it was shown that this feature is related to
saturation physics and is governed by the ratio $p_{T}^{h}/p_{T}^{\gamma}$.
The change from a double to a single peak correlation, depending on the
relative $p_{T}$, is unique to semi-inclusive photon-hadron production: since
the trigger particle in dihadron correlations is always a hadron, it
consistently exhibits a single-peak structure.
Photon-hadron correlations can also be quantified by the coincidence
probability. In contrast to production of a more symmetric final state such as
dihadron production, in photon-hadron production the trigger particle can be
either the prompt photon or the hadron [137]. When the photon is used as the
trigger, the coincidence probability is defined as
$CP_{h}(\Delta\phi)=N^{\text{pair}}_{h}(\Delta\phi)/N_{\gamma}$ where
$N^{\text{pair}}_{h}(\Delta\phi)$ is the photon-hadron yield. The momentum of
the photon trigger, the leading ($L$) particle, is denoted $p^{\gamma}_{T,L}$
while the momentum of the associated ($S$) hadron (typically a $\pi^{0}$) is
denoted $p^{h}_{T,S}$. The azimuthal angle between the photon and the
$\pi^{0}$ is $\Delta\phi$ [137],
$\displaystyle
CP_{h}(\Delta\phi;p^{h}_{T,S},p^{\gamma}_{T,L};\eta_{\gamma},\eta_{h})$
$\displaystyle=$
$\displaystyle\frac{2\pi\int_{p^{\gamma}_{T,L}}dp_{T}^{\gamma}p_{T}^{\gamma}\int_{p^{h}_{T,S}}dp_{T}^{h}p_{T}^{h}\frac{dN^{p\,A\rightarrow
h(p_{T}^{h})\,\gamma(p_{T}^{\gamma})\,X}}{d^{2}\vec{p_{T}}^{\gamma}\,d^{2}\vec{p_{T}}^{h}\,d\eta^{\gamma}\,d\eta^{h}}}{\int_{p^{\gamma}_{T,L}}d^{2}\vec{p_{T}}^{\gamma}\,\frac{dN^{p\,A\rightarrow\gamma(p_{T}^{\gamma})\,X}}{d^{2}\vec{p_{T}}^{\gamma}\,d\eta_{\gamma}}}\,\,.$
(53)
The integrals are carried out within momentum intervals defined by
$p^{\gamma}_{T,L}$ and $p^{h}_{T,S}$. The yields in the above expression are
defined in Eqs. (51) and (6.3). The correlation defined in Eq. (52) can be
considered as a snapshot of the integrand in the coincidence probability
defined in Eq. (53). In the same fashion, one can define a hadron-triggered
coincidence probability [137]. If the $\pi^{0}$ is the trigger particle, the
momenta are instead denoted by $p^{\gamma}_{T,S}$ and $p^{h}_{T,L}$ which then
become the lower limits on the integrals in Eq. (53). The away-side
coincidence probability for the azimuthal correlation of photon-hadron pairs
can also have a double-peak or single-peak structure, depending on the trigger
particle selection and kinematics [137].
Figure 28: The $\gamma-\pi^{0}$ correlation $P(\Delta\phi)$ defined in Eq.
(52) in minimum bias $p+p$ and $p+$Pb collisions at $\sqrt{S}=5$ TeV and
$\eta_{h}=\eta_{\gamma}=3$ (with the convention that the proton beam moves
toward forward rapidity) obtained via the rcBK evolution equation with several
initial saturation scales, $Q^{2}_{0A}=NQ^{2}_{0p}$ with $N=3$, 5, and 7, in
two different transverse momentum regions: $p^{h}_{T}<p^{\gamma}_{T}$ (left)
and $p^{h}_{T}>p^{\gamma}_{T}$ (right). Taken from Ref. [20, 137].
In Fig. 29, predictions of the azimuthal correlations between the produced
prompt photon and hadron, calculated employing the coincidence probability
$CP_{h}$ in $p+$Pb collisions are shown for $\sqrt{s}=0.2$, $5$, and $8.8$
TeV. Equation (6.3) is used in the denominator of Eq. (53). The collinear
divergence was removed from Eq. (53) by introducing the photon fragmentation
function. The numerator is calculated using Eq. (51). An overall normalization
problem may result given that the away-side correlation at $\Delta\phi=\pi/2$
is not sensitive to the collinear singularity. Proper treatment of this
problem requires a full NLO calculation which is currently unavailable.
However, choosing a different photon fragmentation function [68, 129, 131]
will only slightly change the results given the freedom to choose the
fragmentation scale and the rather large uncertainties due to $Q_{0A}^{2}$.
This possible normalization problem is not present in the correlation defined
via Eq. (52).
Figure 29: The $\gamma-\pi^{0}$ coincident probability, $CP_{h}(\Delta\phi)$,
defined in Eq. (53) in minimum-bias $p+A$ collisions at
$\eta_{h}=\eta_{\gamma}=3$ (with the convention that the proton beam moves
toward forward rapidity) for $\sqrt{s}=0.2$, 5 and 8.8 TeV. Taken from Ref.
[20, 137].
Although there are theoretical uncertainties associated with the strength of
the photon-hadron correlations, due to both higher-order corrections and the
less constrained saturation scale $Q_{0A}^{2}$. Nevertheless, the
decorrelation of away-side photon-hadron production increases with energy,
rapidity and density. This decorrelation, together with the appearance of a
double or single-peak structure, are robust predictions of CGC/saturation
effects in the leading-log approximation.
### 6.4 Proton-nucleus dilepton and photon production at the LHC: gluon
saturation and shadowing (R. Baier, F. Gelis, A. H. Mueller and D. Schiff)
Here, a qualitative discussion on saturation and shadowing effects on photon
and dilepton production at very large rapidity in high-energy proton-nucleus
collisions is given. The aim is to describe the various steps of assumptions
leading to the present understanding.
In $\sqrt{s_{{}_{NN}}}=200$ GeV d+Au data on high-$p_{T}$ hadron production at
large rapidity (on the deuteron side) there is a significant suppression of
hadron production compared to the expectation from $p+p$ collisions. This
result suggests that there may be a significant amount of leading-twist gluon
shadowing in the nuclear wavefunction in the region probed by forward hard
scattering at RHIC, with even more expected at the LHC.
In many ways hard photons (or dileptons arising from virtual photons) are a
better probe than high-$p_{T}$ hadrons. Photons are less sensitive to
fragmentation effects while final-state effects are almost absent. Thus, at
$p_{T}\sim 2-3$ GeV/$c$, leading-twist factorization is expected to be
accurate. Hence the nuclear gluon distribution probes $x$ values somewhat
smaller than $10^{-4}$ in $p+$Pb collisions at the LHC.
Increasing the photon rapidity into the forward (proton rapidity) region,
$y>0$, the gluon $x$ probed decreases rapidly, e.g. for virtual photons of
mass $M$ and rapidity $y$ produced in the process $q+g\rightarrow\gamma^{*}$,
$x\simeq(M/\sqrt{s})\exp{(-y)}$. With $\sqrt{s}=5500$ GeV, $M=5.0$
GeV/$c^{2}$, and $y=3.5$, $x=2.7\times 10^{-5}$, so that indeed a fast quark
in the direction of proton rapidity produces virtual photons that probe the
small $x$ gluon distribution in the nucleus.
It is extremely interesting to explore, at least qualitatively, the size of
the suppression one might expect in such reactions. The treatment here follows
Ref. [133] which can be examined for further details. This discussion is based
on a picture derived from the Color Glass Condensate (CGC) effective theory
for the gluon distributioniiiiiiiiiFor recent reviews, see Refs. [138, 139]
and references therein. together with BFKL evolution (see e.g. Ref. [140]) to
reach higher values of $y$.
#### 6.4.1 Factorized formula for the inclusive $\gamma^{*}$ cross section
A dimensionless observable, $\sigma(\vec{Q},Y)$, can be written in the
$k_{\bot}$-factorized form
$\sigma(\vec{\mu},Y)=\int\frac{d^{2}q_{T}}{\pi
q^{2}_{T}}\,H(\vec{q}_{T},\vec{\mu})\,\phi_{G}(\vec{q}_{T},Y)\,,$ (54)
where $\mu$ is the hard scale of the reaction, equal to the transverse
momentum, $k_{T}$, of the (massive) photon, i.e.
$\sigma(\vec{k}_{T},Y)\equiv\sigma(\vec{b}_{T},\vec{k}_{T},Y)=\frac{k^{~{}2}_{T}d\sigma}{d^{2}b_{T}d^{2}k_{T}d\ln
z}\,.$ (55)
For simplicity incident quarks are considered instead of protons,
$q+A\rightarrow\gamma^{*}X$, see Fig. 30. Here $k_{T}$ is the transverse
momentum of the $\gamma^{*}$, and $z$ is the longitudinal momentum fraction of
the $\gamma^{*}$ with respect to the incident quark momentum, $z=k_{+}/p_{+}$,
where $p_{+}\rightarrow\infty$. The impact parameter of the $q+A$ collisions
is denoted $\vec{b}$ and $Y=\ln{1/x}$.
bbllx=0,bblly=0,bburx=340,bbury=100, file=Graph.eps,width=11cm
Figure 30: Two gluon exchange graph for quark ($p$) + nucleus ($P$) production
of a real or virtual photon, $\rightarrow~{}\gamma^{(*)}(k)~{}X$.
The leading-twist contribution involves only two exchanged gluons, as shown in
Fig. 30. $H$ is the hard part of the interaction in the $k_{T}$-factorized
form while $\phi_{G}(\vec{q}_{T},Y)/q^{2}_{T}$ is considered to be
proportional to the differential high energy $q+A\rightarrow q+A$ cross
section.
#### 6.4.2 Anomalous scaling and shadowing - specific predictions
While there are theoretical uncertainties, there are also robust
characteristic qualitative results. The function $\phi_{G}(k_{T},Y)$ can be
approximated by the scaling function,
$\phi_{G}(k_{T},Y)=\phi_{G}(k_{T}/Q_{s}(\vec{b},Y))\approx\left(k^{2}_{T}/Q^{2}_{s}(\vec{b},Y)\right)^{\lambda_{0}-1},$
(56)
with anomalous dimension $\lambda_{0}\simeq 0.37$ and saturation scale
$Q_{s}$. In $\vec{b}=0$ collisions at fixed $k_{T}$ and $Y$, an anomalous $A$
dependence is predicted, together with shadowing/suppression of the ratio
$R_{pA}$,
$R_{pA}\approx A^{-\lambda_{0}/3}~{}.$ (57)
In order to obtain results at large photon rapidities based on the BFKL
evolution in the presence of saturation, the rapidity dependence of the
saturation scale is chosen to be compatible with phenomenology,
$Q^{2}_{s}(\vec{b}=0,Y=y)=Q^{2}_{s}(Y=0)\exp(\lambda\,y),\,\,~{}~{}\lambda\simeq
0.3\,\,.$ (58)
There is significant suppression expected, $R_{pA}\simeq 0.5$, especially when
the $\gamma^{(*)}$ is produced at forward (in the proton direction)
rapidities, e.g. $y>3$ and $k_{T}>2$ GeV/$c$. More detailed predictions of
forward inclusive prompt photon production and semi-inclusive photon-hadron
correlations in high energy proton-nucleus collisions at the LHC using the CGC
formalism are presented in Refs. [134, 137].
Measurements of photons and dileptons, in addition to hadrons with moderate
transverse momenta but at large rapidities in $p+$Pb collisions at the LHC
will provide important information supporting the saturation picture of high
density gluon dynamics at high energies and small values of $x$.
## 7 Jets
In this section, several results for jet production and modification in media
are presented. Multi-jet production is discussed and predictions for the jet
yields are shown in Sec. 7.1. Cold nuclear matter effects on jet and dijet
production are presented in Sec. 7.2. The angular decorrelation of dijets due
to saturation effects is discussed in Sec. 7.3. Predictions of long-range
near-side azimuthal collimation event with high multiplicity are given in Sec.
7.4. Finally, enhanced transverse momentum broadening in medium is described
in Sec. 7.5.
### 7.1 Multi-jet Production (N. Armesto)
The study of jet production in Pb+Pb collisions at the LHC [141, 142, 143,
144, 145, 146] and their medium modifications, are important for probing the
properties of the hot and dense matter formed in heavy-ion collisions and is
thus a very hot topic in heavy-ion physics. Therefore, studies of (multi-)jet
production in $p+A$ collisions are of great importance as a cold QCD matter
benchmark. Here the jet rates in minimum bias $p+$Pb collisions at the LHC
(4+1.58 TeV per nucleon) are computed using the Monte Carlo code in Refs.
[147, 148, 149] that implements fixed-order perturbative QCD up to next-lo-
leading order. This code produces at most three jets and contains neither
parton cascades nor hadronization corrections.
In the computation, the renormalization and factorization scales have been set
equal and fixed to $\mu=\mu_{F}=\mu_{R}=E_{T}$, where $E_{T}$ is the total
transverse energy in the event. The central set of the NLO MSTW2008 parton
densities (MSTW2008nlo68cl) [114] have been used for the unmodified nucleons.
The nuclear modification of parton densities was examined using the EKS98
[150, 92] and EPS09NLO [48] sets. The standard Hessian method was employed to
estimate the uncertainties coming from nuclear parton densities. The precision
of the computation, limited by CPU time, gives statistical uncertainties
smaller than 10% for the bin with the highest $E_{T}$ in the results shown
here. The anti-$k_{T}$ jet finding algorithm with $R=0.5$ [151] was used. Only
jets with $E_{T}>20$ GeV are considered. The uncertainties due to the choice
of nucleon parton densities, isospin corrections, and scale fixing, together
with the influence of the jet-finding algorithm and the choice of nuclear
targets and collision energies, were discussed elsewhere [152, 153] and are
not considered here.
Figure 31 shows the sum of the 1-, 2- and 3-jet yields, as well as the
individual 2- and 3-jet yields within four pseudorapidity windows (two
central, one backward and one forward) in the lab frame as a function of the
$E_{T}$ of the hardest jet within the acceptance, $E_{T\,_{\rm hardest}}$. The
yields, computed for luminosity ${\cal L}=0.25\times 10^{29}$ cm-2s-1
integrated over a one month ($10^{6}$ s) run (corresponding to an integrated
luminosity of 25 nb-1) [154], are quite large. The yields are given on the
right-hand vertical axes in Fig. 31. High yields are required to study cold
matter effects in high $p_{T}$ multi-jet production. Figure 31 demonstrates
that such studies are feasible. For example, in the backward region,
$-4.75<\eta<-3$, sufficiently high rates for the sum per GeV of 1+2+3-jet
events can be achieved for $E_{T_{\rm hardest}}<50$ GeV. The only regions
where the yields from certain channels are too low for statistically
significant results are those for 3-jet events in the forward region and 2-
and 3-jets in the backward region (only the yields for 2-jet events are
shown). Note that the turnover in the 2-jet yields when $E_{T_{\rm
hardest}}\rightarrow 20$ GeV comes from singularities that appear in the NLO
calculation due to large logarithms because $E_{T_{1}}\sim E_{T_{2}}(\sim
E_{T_{3}})$. Resummation techniques are required to obtain reliable results in
this kinematic region.
Nuclear modifications of the PDFs are very small, maximum ${\cal O}$(20%),
hardly visible in the yields in Fig. 31. Note that the wide uncertainty bands
in the largest $E_{T_{\rm hardest}}$ bins in the forward and backward regions
are due to statistical fluctuations in the Monte Carlo and do not have a
physical origin. The corresponding hot nuclear matter effects in Pb+Pb
collisions are expected to be much larger.
Further work to compare these yields with experimental data would require
consideration of hadronization corrections and the effects of background
subtraction such as in the well-established jet area method [155]. These
should be the subject of future studies.
Figure 31: The 1+2+3- (black), 2- (red) and 3-jet (green) cross sections as a
function of the $E_{T}$ of the hardest jet within the acceptance. Results in
different pseudorapidity windows in the laboratory frame computed for minimum
bias $p+$Pb collisions are shown. The dashed lines are the results without
nuclear modifications of the PDFs. Results with EPS09NLO [48] (solid) and
EKS98 (dotted) [150, 92] are also shown. The bands correspond to uncertainties
computed using the Hessian method for EPS09 [48]. The right vertical axes
gives the scale for the corresponding yields with an integrated luminosity of
25 nb-1. See the text for further details.
### 7.2 Cold matter effects on jet and dijet production (Y. He, B.-W. Zhang
and E. Wang)
Inclusive NLO [156, 157, 158, 159, 160, 161] jet and dijet production are
studied in $p+$Pb collisions. Cold nuclear matter effects are included by
employing the EPS09 [48], DS11 [162, 163] and HKN07 [164] parametrizations of
nuclear parton densities. The numerical results for the inclusive jet
spectrum, the dijet transverse energy and mass spectra, and the dijet triply-
differential cross sections, all using a jet cone size $R=0.4$ in minimum bias
$p+$Pb collisions are shown.
The left-hand side of Fig. 32 illustrates the inclusive jet spectra scaled by
$N_{\rm coll}$ including the nuclear modifications in the central rapidity
region, $|y|<1$. Results for $R_{p{\rm Pb}}$ are also shown. The spectra with
EPS09 and HKN07 show an enhancement in the transverse energy range
$30<E_{T}<200$ GeV. However, the CNM effects are negligible over the entire
region of jet $E_{T}$ with DS11.
Figure 32: The inclusive jet (left) and dijet (right) results in $p$+Pb
collisions at $\sqrt{s}=5$ TeV. The dijet results, presented as a function of
$E_{T2}$ are given for fixed energy $E_{T1}=100$ GeV for jet 1. The rapidity
acceptance for all jets is $|y|<1$. The $E_{T}$ spectra are shown in the upper
part of the plots while the nuclear modification factors are shown in the
lower panels of the figures.
The right-hand side of Fig. 32 displays the rescaled dijet $E_{T}$ spectra as
a function of $E_{T2}$ with jet 1 at a fixed transverse energy of $E_{T1}=100$
GeV employing the same modifications of the parton densities as before. Both
jets are within the central rapidity region, $|y_{1}|<1$ and $|y_{2}|<1$. The
cross sections in the range $10<E_{T}<200$ GeV exhibit a peak near $E_{T}=100$
GeV. Again the spectra with EPS09 and HKN07 are enhanced. On the other hand,
the DS11 set shows very small suppression and enhancement as shown in Fig 32.
Figure 33 gives the rescaled dijet invariant mass $M_{JJ}$ spectra and the
corresponding nuclear modification factors. Here $M_{JJ}$ is defined as
$[(\sum p^{\mu}_{n})^{2}]^{1/2}$ where the sum is over all particles in the
two jets [165]. The maximum rapidity of the two leading jets,
$|y|_{\max}=\max(|y_{1}|,|y_{2}|)$ is defined so that $|y|_{\max}<1$. Jets
with energies greater than 40 GeV in the rapidity range $|y|<2.8$ are
selected. The CNM effects with EPS09 and HKN07 also enhance the $M_{JJ}$
spectra over a wide $M_{JJ}$ region while the effect with DS11 is modest.
Figure 33 also shows the rescaled dijet triply differential cross sections
[166] and their nuclear modifications. The momentum fractions $x_{1}$ and
$x_{2}$ are defined as
$x_{1}=\sum\limits_{i\in{\rm
jet}}\frac{E_{Ti}}{\sqrt{s}}e^{y_{i}}\,\,,~{}~{}x_{2}=\sum\limits_{i\in{\rm
jet}}\frac{E_{Ti}}{\sqrt{s}}e^{-y_{i}}\,\,.$ (59)
where the sums run over all particles in the jets. The results in Fig. 33 are
shown for the rapidity difference between the jets, $y_{*}=0.5(|y_{1}-y_{2}|)$
where $y_{1}$ and $y_{2}$ are the rapidities of the two leading jets. The
results as a function of $x$ show obvious deviations between nuclear
modification factors which implies that the dijet triply-differential cross
sections could be a good observable to distinguish between different shadowing
parameterizations [167].
Figure 33: (Left) The dijet invariant mass spectra in $p+$Pb collisions.
(Right) The dijet triply differential cross sections in $p+$Pb collisions. The
nuclear modification factors, $R_{p{\rm Pb}}$, are also shown.
### 7.3 Angular decorrelation of dijets as a signature of gluon saturation
(K. Kutak and S. Sapeta)
Here predictions for the emergence of saturation [168] effects on dijet
production in $p+$Pb scattering at the LHC. The results are based on a study
performed in Refs. [169, 170]. The saturation scale characterizes formation of
a dense system of partons. There is growing evidence that the phenomenon of
gluon saturation indeed occurs [171, 172]. This calculation employs a high-
energy factorization formalism which accounts for both the high energy scale
of the scattering and the hard momentum scale $p_{T}$ provided by produced
hard system [173]. Dijets separated by large rapidity gaps [174] are
considered here. More specifically, the focus is on a case where one jet is
measured in the central rapidity region of the detectors while the other is at
large rapidity, see Fig. 34. Such final states probe the parton density in one
of the hadrons at low momentum fraction $x$ while the other is at relatively
large $x$. Predictions of azimuthal decorrelations of dijets production in
$p+$Pb relative to $p+p$ collisions are presented.
Figure 34: Jet production in the forward (assuming the proton moves toward
positive rapidity) region in hadron-hadron collisions.
Assuming, without loss of generality, that $x_{1}\simeq 1$ and $x_{2}\ll 1$
(if the proton moves in the direction of forward rapidity), the cross section
takes the form
$\displaystyle\frac{d\sigma}{dy_{1}dy_{2}dp_{T1}dp_{T2}d\Delta\phi}$
$\displaystyle=$
$\displaystyle\sum_{a,c,d}\frac{p_{T1}p_{T2}}{8\pi^{2}(x_{1}x_{2}s)^{2}}$ (60)
$\displaystyle\mbox{}\times{\cal M}_{ag\to
cd}x_{1}f_{a/A}(x_{1},\mu^{2})\,\phi_{g/B}(x_{2},k^{2}_{T})\frac{1}{1+\delta_{cd}}\,\,,$
where
$k_{T}^{2}=p_{T1}^{2}+p_{T2}^{2}+2p_{T1}p_{T2}\cos\Delta\phi\,\,,$ (61)
and $\Delta\phi$ is the azimuthal distance between the outgoing partons. The
squared matrix element ${\cal M}_{ag\to cd}$ includes $2\to 2$ process with
one off-shell initial-state gluon and three on-shell partons $a$, $c$, and
$d$. The following partonic subprocesses contribute to the production of the
dijet system: $qg\to qg$; $gg\to q\bar{q}$; and $gg\to gg$ [175]. The off-
shell gluon in Eq. (60) is described by the unintegrated gluon density
$\phi_{g/B}(x_{2},k^{2}_{T})$, a solution of a nonlinear evolution equation
[176, 177] that depends on the longitudinal momentum fraction $x_{2}$, and the
transverse momentum of the off-shell gluon, $k_{T}$. On the side of the on-
shell parton, probed at high proton $x_{1}$, the collinear parton density
$f_{a/A}(x_{1},\mu^{2})$ is appropriate.
Figure 35: Ratio of differential cross sections for central-forward dijet
production at $\sqrt{s}=5\text{ TeV}$ as functions of azimuthal distance
between the jets $\Delta\phi$ for three different jet $p_{T}$ cuts. On the
left, the selected rapidities correspond to the CMS detector while, on the
right, the rapidity range appropriate for ALICE is shown. The calculations
have been made with the convention that the proton beam moves toward forward
rapidity.
Figure 35 shows the ratios of the differential cross sections for central-
forward dijet production in $p+$Pb relative to $p+p$ as a function of the
azimuthal distance between the jets, $\Delta\phi$. In the region
$\Delta\phi\sim\pi$, the gluon density is probed at small $k_{T}$, where it is
strongly suppressed due to nonlinear effects. The ratio is a signal of
saturation that is sensitive to the enhancement of saturation effects afforded
by the larger $A$ of the Pb nucleus.
The left-hand side of Fig. 35 shows the results for various jet $p_{T}$ cuts:
$p_{T\,{\rm jet}}>15$, 25, and 35 GeV/$c$ while the jet rapidities are
restricted to positive values within the coverage of the central and forward
CMS detectors. On the right-hand side of Fig. 35, lower $p_{T}$ cuts,
$p_{T}>10$ and 20 GeV/$c$, are used and the rapidity is restricted to positive
values within the central and larger rapidity range of the ALICE detector. The
results show that, when the two jets are back-to-back, the cross section ratio
is significantly smaller than one. This is a consequence of stronger gluon
saturation in the Pb nucleus than in a proton. Thus the unintegrated gluon
distribution in the region of small and medium $k_{T}$ is more suppressed in
Pb than in protons, as shown in Fig. 35. In addition to the effects shown
here, the region near $\Delta\phi\simeq\pi$ is sensitive to Sudakov effects
which further suppress the cross section. Therefore, refinement along those
lines could be envisaged in the future. However, the ratio shown here is less
sensitive to these corrections. It is worth emphasizing that the suppression
due to saturation predicted in Fig. 35 is strong and extends over a large
enough $\Delta\phi$ range for experimental observation, even if the region
near $\Delta\phi=\pi$ could be further refined.
### 7.4 Predictions for long-range near-side azimuthal collimation in high
multiplicity p+Pb collisions (K. Dusling and R. Venugopalan)
Rapidity-separated dihadron correlation measurements can provide valuable
insight into the gluon dynamics of the nuclear wavefunction. For example, it
was recently realized [172] that a novel “near-side” azimuthal collimation in
high multiplicity $p+p$ events [178] is a consequence of nonlinear gluon
interactions and was found to be in excellent agreement with computations in
the Color Glass Condensate (CGC) effective field theory [179, 180].
The situation in $p+$Pb collisions is more attractive. The degree of near-side
collimation is highly sensitive to the fine structure of the unintegrated
gluon distribution. It can be shown that the collimated yield is enhanced by
$\left(Q_{s}^{\rm Pb}/Q_{s}^{p}\right)^{2}$ in asymmetric collisions. Here
$Q_{s}^{\rm Pb}$ is the saturation scale probed in the lead nucleus and can be
estimated to scale with the number of participants, $Q_{s}^{2,\rm Pb}\approx
N_{\rm part}Q_{s}^{2,p}$, where $Q_{s}^{p}$ is the saturation scale for the
proton. Therefore, for high multiplicity events where many nucleons in the
lead nucleus participate, a significant near-side collimation is observed
[181]. Similar to the $p+p$ data, calculations within the CGC effective theory
are able to explain the data [182].
This last study combined the physics of saturation and BFKL dynamics in order
to understand the near-side collimation and the recoiling away-side jet. Both
are required for a full understanding of the azimuthal structure of dihadron
correlations. The per-trigger yields shown in Fig. 36 are well described by
this framework for the symmetric $p_{T}^{\rm trig}\sim p_{T}^{\rm assoc}$
windows where data are currently available. Predictions for the asymmetric
$p_{T}^{\rm trig}\neq p_{T}^{\rm assoc}$ windows are also shown.
Figure 36: Correlated yield $1/N_{\rm trig}d^{2}N/d\Delta\phi$ after ZYAM
(zero yield at minimum, used to remove the $\Delta\Phi$-independent pedestal)
as a function of $|\Delta\phi|$ integrated over $2\leq|\Delta\eta|\leq 4$ for
the most central multiplicity bin $N_{\rm trk}^{\rm offline}\geq 110$. The CMS
data [181] have currently only been provided for the diagonal components,
$p_{T}^{\rm trig}\sim p_{T}^{\rm assoc}$, of the correlation matrix. The
theory curves are the result of adding the BFKL contribution responsible for
the away-side jet and the “dipole”-like Glasma contribution. The solid black
curve is the result for $Q_{0p}^{2}=0.504$ GeV2 on $N_{\rm part}^{\rm Pb}=14$
and the dashed green is for $Q_{0p}^{2}=0.336$ GeV2 on $N_{\rm part}^{\rm
Pb}=16$.
### 7.5 Enhancement of transverse momentum broadening (H. Xing, Z.-B. Kang,
I. Vitev and E. Wang)
Both initial-state and final-state multiple scattering lead to acoplanarity,
or momentum imbalance of the two leading final-state particles. To quantify
this effect, the transverse momentum imbalance $\vec{q}_{T}$ is defined as:
$\displaystyle\vec{q}_{T}=\vec{p}_{T1}+\vec{p}_{T2},$ (62)
with the average squared transverse momentum imbalance
$\displaystyle\langle q_{T}^{2}\rangle=\left(\int
d^{2}\vec{q}_{T}q_{T}^{2}\frac{d\sigma}{d\mathcal{PS}\,d^{2}\vec{q}_{T}}\right)\left(\frac{d\sigma}{d\mathcal{PS}}\right)^{-1}\,\,.$
(63)
Here, $d\sigma/d\mathcal{PS}$ is the differential cross section with the
appropriate phase space factor, determined separately for each process. For
example, in $p+A$ collisions, $d\mathcal{PS}=dy_{1}dy_{2}dp_{T}^{2}$ for
photon+jet production and $d\mathcal{PS}=dy_{1}dy_{2}dp_{T1}dp_{T2}$ for
photon+hadron production.
The enhancement of the transverse momentum imbalance (or nuclear broadening)
in $h+A$ ($h=p$, $\gamma^{*}$) collisions relative to $h+p$ collisions can be
quantified by the difference:
$\displaystyle\Delta\langle q_{T}^{2}\rangle=\langle
q_{T}^{2}\rangle_{hA}-\langle q_{T}^{2}\rangle_{hp}\,\,.$ (64)
The broadening $\Delta\langle q_{T}^{2}\rangle$ is a result of multiple quark
and gluon scattering and is a direct probe of the properties of the nuclear
medium. Both initial-state and final-state multiple parton interactions are
taken into account to calculate the nuclear broadening $\Delta\langle
q_{T}^{2}\rangle$ for photon+jet (photon+hadron) and heavy quark (heavy meson)
pair production in $p+A$ collisions.
The nuclear broadening $\Delta\langle q_{T}^{2}\rangle$ can be calculated in
perturbative QCD. A specific method based on double parton scattering has been
discussed in detail in Refs. [183, 80]. The derivation discussed here closely
follows Ref. [80]. The leading contribution to the nuclear broadening comes
from double scattering: either in the initial-state or the final-state. The
contributions from these diagrams in the covariant gauge can be calculated to
obtain the following expression for the nuclear broadening of photon+jet
production in $p+A$ collisions:
$\displaystyle\Delta\langle q_{T}^{2}\rangle$ $\displaystyle=$
$\displaystyle\left(\frac{8\pi^{2}\alpha_{s}}{N_{c}^{2}-1}\right)$ (65)
$\displaystyle\mbox{}\times\frac{\sum_{a,b}f_{a/p}(x^{\prime})\left[T_{b/A}^{(I)}(x)H^{I}_{ab\to\gamma
d}(\hat{s},\hat{t},\hat{u})+T_{b/A}^{(F)}(x)H^{F}_{ab\to\gamma
d}(\hat{s},\hat{t},\hat{u})\right]}{\sum_{a,b}f_{a/p}(x^{\prime})f_{b/A}(x)H^{U}_{ab\to\gamma
d}(\hat{s},\hat{t},\hat{u})}\,\,.$
Here $T_{b/A}^{(I)}(x)=T_{q/A}^{(I)}(x)$ (or $T_{g/A}^{(I)}(x)$) are twist-4
quark-gluon (or gluon-gluon) correlation functions associated with initial-
state multiple scattering, defined as [183, 80]
$\displaystyle T_{q/A}^{(I)}(x)$ $\displaystyle=$
$\displaystyle\int\frac{dy^{-}}{2\pi}\,e^{ixp^{+}y^{-}}\int\frac{dy_{1}^{-}dy_{2}^{-}}{2\pi}\,\theta(y^{-}-y_{1}^{-})\,\theta(-y_{2}^{-})$
(66) $\displaystyle\mbox{}\times\frac{1}{2}\,\langle p_{A}|F_{\alpha}^{\
+}(y_{2}^{-})\bar{\psi}_{q}(0)\gamma^{+}\psi_{q}(y^{-})F^{+\alpha}(y_{1}^{-})|p_{A}\rangle\,\,,$
$\displaystyle T_{g/A}^{(I)}(x)$ $\displaystyle=$
$\displaystyle\int\frac{dy^{-}}{2\pi}\,e^{ixp^{+}y^{-}}\int\frac{dy_{1}^{-}dy_{2}^{-}}{2\pi}\,\theta(y^{-}-y_{1}^{-})\,\theta(-y_{2}^{-})$
(67) $\displaystyle\mbox{}\times\frac{1}{xp^{+}}\,\langle
p_{A}|F_{\alpha}^{~{}+}(y_{2}^{-})F^{\sigma+}(0)F^{+}_{~{}\sigma}(y^{-})F^{+\alpha}(y_{1}^{-})|p_{A}\rangle\,\,.$
The corresponding twist-4 correlation functions associated with final-state
multiple scatter are $T_{q/A}^{(F)}(x)$ and $T_{g/A}^{(F)}(x)$. They are
defined as in Eqs. (66) and (67), except the $\theta$-functions are replaced
such that [183, 80],
$\displaystyle\theta(y^{-}-y_{1}^{-})\,\theta(-y_{2}^{-})\to\theta(y_{1}^{-}-y^{-})\,\theta(y_{2}^{-})\,\,.$
(68)
The broadening in photon + hadron production is calculated similarly.
Nuclear broadening is evaluated in a formalism where multiple scattering
contributes to the cross section via higher-twist matrix elements in the
nuclear state. This framework follows a well-established QCD factorization
formalism for particle production in $p+A$ collisions and has previously been
used to describe cold nuclear matter effects such as energy loss, dynamical
shadowing and broadening. This work differs from more generic parton
broadening phenomenology in because the color and kinematic structures of the
hard part are evaluated exactly. In particular, the nuclear enhancement of the
transverse momentum imbalance is studied in dijet and photon+jet production.
Figure 37: Nuclear broadening, $\Delta\langle q_{T}^{2}\rangle$, is shown for
dijet (left) and photon+jet (right) production at specified rapidities in
$p+A$ collisions as a function of $N_{\rm coll}$. Results are shown for
$y_{1}=y_{2}=2$ in $\sqrt{s}=5$ TeV $p+$Pb collisions and $y_{1}=y_{2}=1$ at
$\sqrt{s}=200$ GeV d+Au collisions. (The calculations have been made with the
convention that the proton beam moves toward forward rapidity.) The jet
transverse momentum is integrated over $30<p_{T}<40$ GeV/$c$ at $\sqrt{s}=5$
TeV and $15<p_{T}<25$ GeV/$c$ at $\sqrt{s}=200$ GeV. The red line shows the
result for RHIC kinematics with scattering parameter $\xi^{2}=0.12$ GeV2 while
the yellow band represents the variation of $\xi^{2}$ in the LHC kinematics.
The results are shown in Fig. 37. The line gives the baseline $\Delta\langle
q_{T}^{2}\rangle$ determined from RHIC as a function of $N_{\rm coll}$. The
band is the result in the LHC kinematics for a plausible range of the
scattering parameter $\xi^{2}$, defined in Eq. (29). The band is broader for
dijet production and $\Delta\langle q_{T}^{2}\rangle$ is also larger. There is
also a somewhat larger deviation of the bottom edge of the band from the RHIC
result for dijet production.
## 8 Gauge Bosons
Gauge boson production is discussed in this section. Section 8.1 makes
predictions for $W$ and $Z^{0}$ $p_{T}$ and rapidity distributions in $p+p$
and $p+$Pb collisions. The $W^{\pm}$ charge asymmetry as a function of the
decay lepton rapidity is also shown. Section 8.2 describes a calculation of
the resummed $Z^{0}$ $p_{T}$ distribution while transverse broadening of
vector boson production is discussed in Sec. 8.3.
### 8.1 $W$ and $Z$ production and $W^{\pm}$ charge asymmetry (P. Ru, E.
Wang, B.-W. Zhang and W.-N. Zhang)
Production of the gauge bosons, $W^{+},W^{-}$ and $Z^{0}$, is discussed. The
transverse momentum and the rapidity distributions in min-bias $p+$Pb
collisions at $\sqrt{s}=5$ TeV are calculated at next-to-leading order and
next-to-next-to-leading order. The fiducial cross sections $\sigma_{\rm fid}$
are calculated in the fiducial phase space for vector boson production used by
ATLAS Collaboration [184, 185] in both $p+$Pb and $p+p$ collisions.
The fiducial $Z^{0}$ cross section is the inclusive cross section $p+\,{\rm
Pb}\to Z^{0}/\gamma^{*}+X$ multiplied by the branching ratio for
$Z^{0}/\gamma^{*}\to l^{+}l^{-}$ within the fiducial acceptance. Here $X$
denotes the underlying event and the recoil system. The fiducial acceptance is
assumed to be the same as that defined by ATLAS in $p+p$ collisions [184] with
the following cuts on the lepton transverse momentum and pseudorapidity, and
the dilepton invariant mass: $p_{T}^{l}>20$ GeV/$c$; $|\eta^{l}|<2.4$; and
$66<m_{ll}<116$ GeV/$c^{2}$.
The fiducial $W^{\pm}$ cross sections are the inclusive cross sections
$p+\,{\rm Pb}\to W^{\pm}+X$ multiplied by the branching ratios for $W^{\pm}\to
l\nu$ within the fiducial acceptance. Following the ATLAS definition [185],
the acceptance cuts on lepton and neutrino transverse momentum and
pseudorapidity as well as the $W$ transverse mass are: $p_{T}^{l}>20$ GeV/$c$;
$|\eta_{l,\nu}|<2.4$; $p_{T}^{\nu}>25$ GeV/$c$; and
$m_{T}=\sqrt{2p_{T}^{l}p_{T}^{\nu}(1-\cos(\phi^{l}-\phi^{\nu}))}>40$
GeV/$c^{2}$.
Results are simulated to NLO ($\mathcal{O}(\alpha_{s})$) and NNLO
($\mathcal{O}(\alpha_{s}^{2})$) in the total cross section employing
$\mathtt{DYNNLO}$ for Drell-Yan-like production in hadron-hadron collisions
[186]. The MSTWNLO and MSTWNNLO proton parton densities are used, along with
the EPS09 [48] and DSSZ [163] shadowing parameterizations.
Tables 8.1 and 8.1 shows the fiducial cross sections, normalized to their per
nucleon values for better comparison. The NLO and NNLO values are shown for
comparison, both for $p+$Pb and $p+p$ collisions at the same energy. The ratio
between the NNLO and NLO cross section, giving some indication of the
theoretical uncertainty and the convergence of the perturbative expansion for
gauge boson production, is $\sim 1.02$ showing that the higher order
corrections are small. When the NLO and NNLO results are compared in $p+$Pb
collisions with the EPS09 shadowing parameterization, a similar correction is
found. The difference between the EPS09 and DSSZ parameterizations, both
calculated at NLO, is also quite small, on the order of 1%. There is a slight
decrease in the total fiducial $Z^{0}$ cross section. There is a larger
decrease for $W^{+}$ and an enhancement in the fiducial cross section in
$p+$Pb collisions. This is less an effect of shadowing than it is of isospin
since $u\overline{d}\to W^{+}$ and $\overline{u}d\to W^{-}$ and there are more
$d$ quarks in the lead nucleus, causing the enhanced cross section.
The total vector boson production cross sections in the fiducial phase space,
$\sigma_{\rm fid}$, in units of nb in $p+p$ collisions. The results at NLO and
NNLO are compared. $p+p$ $\sigma_{\rm fid}$ (nb) Decay channel MSTWNNLO
MSTWNLO $Z\to e^{+}e^{-}$ 0.339 0.332 $W^{+}\to e^{+}\nu$ 2.35 2.30 $W^{-}\to
e^{-}\nu$ 1.47 1.44
The total vector boson production cross sections per nucleon in the fiducial
phase space, $\sigma_{\rm fid}/\langle N_{\rm bin}\rangle$ in $p+$Pb
collisions. Columns 2 and 3 compare the results at NLO and NNLO calculated
with EPS09 while columns 3 and 4 compare the EPS09 and DSSZ shadowing
parameterizations at NLO. $p+$Pb $\sigma_{\rm fid}/\langle N_{\rm
coll}\rangle$ (nb) MSTWNNLO MSTWNLO MSTWNLO Decay channel EPS09 EPS09 DSSZ
$Z\to e^{+}e^{-}$ 0.338 0.328 0.329 $W^{+}\to e^{+}\nu$ 2.06 2.02 2.05
$W^{-}\to e^{-}\nu$ 1.54 1.52 1.53
Figure 38 shows the $Z^{0}$ $p_{T}$ distributions in $p+$Pb collisions at both
NLO and NNLO. The NNLO result is somewhat higher and not as smooth as the NLO
calculations which appear independent of the choice of shadowing
parameterization. Differences between the results with EPS09 and DSSZ, which
can be attributed to shadowing effects rather than isospin, are only apparent
when the ratio $R_{p\,{\rm Pb}}(p_{T})$ is formed. The slight decrease in per
nucleon yield in $p+$Pb relative to $p+p$ seen in the total cross sections in
Tables 8.1 and 8.1 are due to the lowest $p_{T}$ bin, $p_{T}<20$ GeV/$c$. At
higher $p_{T}$, the ratio increases above unity. However, the effect is not
significantly larger than 5% over the entire $p_{T}$ range.
Figure 38: (Left) Normalized $Z^{0}$ differential cross section
$(1/\sigma_{\rm fid})(d\sigma_{\rm fid}/dp_{T})$. (Right) The suppression
factor $R_{pA}(p_{T})$ in 20 GeV/$c$ $p_{T}$ bins.
The individual $W^{+}$ and $W^{-}$ $p_{T}$ distributions are shown at NLO and
NNLO on the left-hand side of Fig. 39. Again, the NLO results are smoother
than the NNLO calculations. The difference in the overall cross sections are
clearly observable: in the fiducial range of the calculations, the $W^{+}$
cross section is $\approx 35$% greater than the $W^{-}$ cross section. The
ratio of $p+$Pb to $p+p$ is shown on the right-hand side of Fig. 39. The
$W^{-}$ ratio is larger than unity and increasing strongly with $p_{T}$. The
effect is due to the greater abundance of $d$ quarks in the Pb nucleus (126
neutrons vs. 82 protons). The valence $d$ quark distributions in the neutrons,
equivalent to the valuence $u$ quark distributions in the protons, have a
larger density at relatively high $x$, causing the observed increase.
Conversely, the lower density of valence $u$ quarks in the Pb nucleus causes
$R_{p{\rm Pb}}<1$ over the entire $p_{T}$ range. The sum of the two charged
gauge bosons shows a trend very similar to that of the $Z^{0}$ in Fig. 38,
revealing a result closer to the true shadowing effect. Even though the
$W^{+}$ cross section is greater than that of the $W^{-}$, the isospin effect
on the $W^{-}$ is large enough to make the ratio larger than unity at high
$p_{T}$.
Figure 39: (Left) The normalized $W=W^{+}+W^{-}$ differential cross section
$(1/\sigma_{\rm fid})(d\sigma_{\rm fid}/dp_{T})$. (Right) The suppression
factor $R_{pA}(p_{T})$ for $W$ production given in 20 GeV/$c$ $p_{T}$ bins.
Figures 40 and 41 show the $Z^{0}$, $W^{+}$ and $W^{-}$ rapidity
distributions, normalized per nucleon. For comparison the $p+p$ and $p+$Pb
distributions are shown both at NLO and NNLO. Aside from numerical
fluctuations at NNLO, the order of the calculation makes little difference in
either the shape or the magnitude of the rapidity distributions. The $p+p$
distributions are all symmetric around $y=0$ while the $p+$Pb distributions
are peaked in the direction of the Pb nucleus.
Figure 40: The $Z^{0}$ rapidity distribution $(1/\langle N_{\rm
coll}\rangle)(d\sigma_{\rm fid}/dy)$. Results are shown for both $p+$Pb and
$p+p$ collisions in the center-of-mass frame for both systems.
There are significant differences between the $W^{+}$ and $W^{-}$
distributions even for $p+p$ collisions. The $W^{+}$ distribution is
considerably broader with peaks away from $y=0$, at $|y|\sim 1.7$ due to the
larger average momentum fraction $x$ of the valence $u$ quarks in the proton.
The greater density of valence $u$ quarks in the proton leads to the larger
overall cross section. The isospin effect tends to make the $W^{+}$ rapidity
distribution more symmetric since the valence $u$ quark distribution in the
neutron, equivalent to the valence $d$ distribution in the proton, has a
smaller average $x$ and lower density which reduces the cross section while
removing the peaks away from midrapidity. The $W^{-}$ distribution in $p+p$
collisions is both smaller and narrower than the $W^{+}$. This distribution
shows the strongest isospin effect with rapidity.
Figure 41: The $W^{+}$ (left) and $W^{-}$ (right) rapidity distributions
$(1/\langle N_{\rm coll}\rangle)(d\sigma_{\rm fid}/dy)$. Results are shown for
both $p+$Pb and $p+p$ collisions in the center-of-mass frame for both systems.
These differences are reflected in the $W^{\pm}$ charge asymmetry, defined as
$(N_{W^{+}}-N_{W^{-}})/(N_{W^{+}}+N_{W^{-}})$ and shown as a function of the
decay lepton pseudorapidity. The results are given in Fig. 42. The $p+p$
asymmetry is symmetric around $y=0$ with a strong dip at midrapidity. The
origin is clear from the individual $W^{+}$ and $W^{-}$ rapidity
distributions. There is, however, a strong forward/backward asymmetry in
$p+$Pb collisions. In the direction of the proton beam, the asymmetry follows
that of the $p+p$ result. It falls off and becomes negative in the direction
of the lead beam. This trend is independent of the order of the calculation
and the shadowing parameterization used.
Figure 42: The $W^{\pm}$ charge asymmetry,
$(N_{W^{+}}-N_{W^{-}})/(N_{W^{+}}+N_{W^{-}})$, as a function of the charged
lepton pseudorapidity in both $p+p$ and $p+$Pb collisions in the center-of-
mass frame for both systems.
### 8.2 Nuclear modification of the transverse momentum spectrum of $Z^{0}$
production (Z.-B. Kang and J.-W. Qiu)
The Collins-Soper-Sterman formalism [187] is used to calculate $Z^{0}$
production over the full $p_{T}$ range at the LHC,
$\displaystyle\frac{d\sigma_{A+B\rightarrow Z^{0}+X}}{dy\,dp_{T}^{2}}$
$\displaystyle=$ $\displaystyle\frac{1}{(2\pi)^{2}}\int
d^{2}b\,e^{i\vec{p}_{T}\cdot\vec{b}}\,\tilde{W}(b,M_{Z},x_{1},x_{2})+Y(p_{T},M_{Z},x_{1},x_{2})\,.$
(69)
The $\tilde{W}$ term gives the dominant contribution when $p_{T}\ll M_{Z}$
while the $Y$ term is perturbatively calculable, see Ref. [188], allowing a
smooth transition from the resummed low $p_{T}$ region to $p_{T}\sim M_{Z}$
where the fixed-order perturbative QCD calculations work well. In Eq. (69),
$x_{1}=e^{y}\,M_{Z}/\sqrt{s}$ and $x_{2}=e^{-y}\,M_{Z}/\sqrt{s}$ while
$\tilde{W}$ is given by [188]
$\tilde{W}(b,M_{Z},x_{1},x_{2})=\left\\{\begin{array}[]{ll}\tilde{W}^{\rm
P}(b,M_{Z},x_{1},x_{2})&\quad\mbox{$b\leq b_{\rm max}$}\\\ \tilde{W}^{\rm
P}(b_{\rm max},M_{Z},x_{1},x_{2})\,\tilde{F}^{\rm
NP}(b,M_{Z},x_{1},x_{2};b_{\rm max})&\quad\mbox{$b>b_{\rm
max}$}\end{array}\right.$ (70)
where $b_{\rm max}\sim 1/$(few GeV) is a parameter that specifies the region
in which $\tilde{W}^{\rm P}$ is perturbatively valid, and $\tilde{F}^{\rm NP}$
is a nonperturbative function determining the large $b$ behavior of
$\tilde{W}$ and is defined below. In Eq. (70), $\tilde{W}^{\rm
P}(b,M_{Z},x_{1},x_{2})$ includes all powers of large perturbative logarithms
resummed from $\ln(1/b^{2})$ to $\ln(M_{Z}^{2})$ [187]
$\tilde{W}^{\rm P}(b,M_{Z},x_{1},x_{2})={\rm e}^{-S(b,M_{Z})}\,\tilde{W}^{\rm
P}(b,c/b,x_{1},x_{2})\,,$ (71)
where $c$ is a constant of order one [187, 188], and
$S(b,M_{Z})=\int_{c^{2}/b^{2}}^{M_{Z}^{2}}\,\frac{d{\mu}^{2}}{{\mu}^{2}}\left[\ln\left(\frac{M_{Z}^{2}}{{\mu}^{2}}\right)A(\alpha_{s}({\mu}))+B(\alpha_{s}({\mu}))\right],$
(72)
with perturbatively-calculated coefficients $A(\alpha_{s})$ and
$B(\alpha_{s})$ given in Ref. [188] and references therein. The perturbative
factor in Eq. (71), $\tilde{W}^{\rm P}(b,c/b,x_{1},x_{2})$, has no large
logarithms. It is expressed as
$\tilde{W}^{\rm
P}(b,c/b,x_{1},x_{2})=\sigma_{0}\sum_{i=q,\bar{q}}f_{i/A}(x_{1},\mu=c/b)\,f_{\bar{i}/B}(x_{2},\mu=c/b)\,$
(73)
where $\sigma_{0}$ is the leading order $q\overline{q}\rightarrow Z^{0}$
partonic cross section [188]. The functions $f_{i/A}$ and $f_{\bar{i}/B}$ are
the modified parton distributions given by [187]
$f_{i/A}(x_{1},\mu)=\sum_{a}\int_{x_{1}}^{1}\frac{d\xi}{\xi}\,C_{i/a}(x_{1}/\xi,\mu)\,\phi_{a/A}(\xi,\mu)$
(74)
where $\sum_{a}$ is over $a=q,\bar{q},g$, $\phi_{a/A}(\xi,\mu)$ are the normal
proton or effective nuclear parton distribution functions (PDFs) and
$C_{i/a}=\sum_{n=0}C_{i/a}^{(n)}(\alpha_{s}/\pi)^{n}$ are perturbatively
calculable coefficient functions for finding a parton $i$ from a parton $a$,
given in Ref. [188].
The non-perturbative function $\tilde{F}^{\rm NP}$ in Eq. (70) has the form,
$\displaystyle F^{\rm NP}(b,M_{Z},x_{1},x_{2};b_{\rm max})$ $\displaystyle=$
$\displaystyle\exp\Bigg{\\{}-\ln\left(\frac{M_{Z}^{2}b_{\rm
max}^{2}}{c^{2}}\right)\left[g_{1}\left((b^{2})^{\alpha}-(b_{\rm
max}^{2})^{\alpha}\right)\right.$ (75) $\displaystyle+$
$\displaystyle\left.g_{2}\left(b^{2}-b_{\rm
max}^{2}\right)\right]-\bar{g}_{2}\left(b^{2}-b_{\rm
max}^{2}\right)\Bigg{\\}}\,.$
where the explicit logarithmic dependence, $\ln(M_{Z}^{2}\,b_{\rm
max}^{2}/c^{2})$, was derived by solving the Collins-Soper equation [187]. The
${g}_{2}$ term is a result of adding a general power correction to the
renormalization group equation while the $\bar{g}_{2}$ term represents the
size of the intrinsic transverse momentum of active partons [188].
The coefficients of the two terms proportional to $b^{2}$ in Eq. (75) can be
combined [189],
$\displaystyle G_{2}=\ln\left(\frac{M^{2}b_{\rm
max}^{2}}{c^{2}}\right)\,g_{2}+\bar{g}_{2}\,,$ (76)
to sum the dynamical and intrinsic power corrections. By requiring the first
and second derivatives of $\tilde{W}$ to be continuous at $b=b_{\rm max}$, the
parameters $\alpha$ and $g_{1}$ in Eq. (75) can be uniquely fixed, leaving
only one parameter, $G_{2}$, sensitive to the power corrections and other
nonperturbative effects. Taking $\bar{g}_{2}=0.25\pm 0.05$ GeV2 and
$g_{2}=0.01\pm 0.005$ GeV2, $G_{2}^{pp}=0.324$ GeV2. Predictions employing Eq.
(69) are consistent with all $p+\overline{p}$ and $p+p$ data from the Tevatron
and the LHC [188].
The EPS09 NLO parameterization [48] is used to account for the leading-twist
nuclear effects on the parton densities in $p+A$ collisions. Following the
method proposed in Ref. [189], the nuclear-size-enhanced multiple scattering
effects are accounted for by choosing $g_{2}\to g_{2}A^{1/3}$. Then for
$Z^{0}$ production, $G_{2}^{p{\rm Pb}}=0.689{\rm~{}GeV}^{2}$.
In Fig. 43, the predictions for $Z^{0}$ production are shown. The cross
section including resummation in Eq. (69) is evaluated employing the CTEQ6M
parton densities at factorization scale
$\mu=M_{T}/2=0.5\sqrt{M_{Z}^{2}+p_{T}^{2}}$. The $Y$ term is calculated at NLO
in $\alpha_{s}$ [188].
Figure 43: $Z^{0}$ boson production in $p+p$ and $p+$Pb collisions at
$\sqrt{s}=5$ TeV and $y=0$ [190].
The upper panel shows the $Z^{0}$ production cross section as a function of
$p_{T}$. The black dashed curve is the $p+p$ baseline while the red solid
curve shows the minimum bias $p+$Pb result. The blue dotted curve is the
minimum bias $p+$Pb result without the $A^{1/3}$ enhancement of $g_{2}$ so
that the nuclear-size-enhanced dynamical power corrections from multiple
scattering are absent. The lower panel presents the nuclear modification
factor $R_{pA}$.
In Fig. 43, the red solid curves are almost indistinguishable from the dotted
curves. Thus, power corrections are not important for $Z^{0}$ production at
LHC energies. Therefore, $Z^{0}$ production in $p+A$ collisions is an ideal
probe of the modification of the parton densities in nuclei as well as of the
high energy “isospin” effect.
The $Z^{0}$ cross section at $y=0$ is dominated by gluon-initiated
subprocesses for $p_{T}>20$ GeV/$c$. That is, $R_{pA}$ is an excellent
observable to study nuclear modifications of the gluon distribution,
heretofore effectively unknown, especially at the values of $x$ and $\mu$
probed by $Z^{0}$ production. At factorization scale $\mu=M_{Z}$, the EPS09
gluon shadowing factor [48] is less than unity (shadowing) for $x<0.005$ and
greater than unity (antishadowing) over a sufficiently large range,
$0.005<x<0.2$, as shown in Fig. 44.
Figure 44: The EPS09 shadowing ratio at scale $Q=M_{Z}$:
$R_{i}^{A}=f_{i/A}(x,Q^{2})/f_{i/p}(x,Q^{2})$. The red solid curve shows the
gluon ratio while the blue dashed curve shows the valence $u$-quark ratio.
As indicated in the lower half of Fig. 43, the nuclear modification factor
$R_{pA}$ for $Z^{0}$ production is suppressed at $p_{T}<10$ GeV/$c$ and
enhanced at high $p_{T}$, up to $p_{T}\sim 70$ GeV/$c$. The low $p_{T}$
suppression is a result of low $x$ shadowing in EPS09. However, the strong
enhancement over such a large $p_{T}$ range was unexpected. After a careful
examination of the kinematics, it was found that, at $y=0$, $x\sim
M_{T}/\sqrt{s}\sim 0.013$, already in the EPS09 antishadowing region. Thus,
the clear enhancement of $R_{pA}$ in the large $p_{T}$ region in Fig. 43 can
be explained by the broad EPS09 antishadowing region at the $Z^{0}$ scale. The
large $p_{T}$ enhancement of $R_{pA}$ vanishes for other shadowing
parameterizations with smaller gluon antishadowing. Therefore, the measurement
of $R_{pA}$ for $Z^{0}$ production in $p+$Pb collisions at the LHC provides a
clean and unique test of gluon antishadowing as proposed in the EPS09
parameterization. Furthermore, $R_{pA}$ is a direct measurement of nuclear
gluon distribution since the cross section is dominated by gluon-initiated
subprocesses for $p_{T}>20$ GeV/$c$.
### 8.3 Transverse momentum broadening of vector boson production (Z.-B. Kang
and J.-W. Qiu)
Finally, transverse momentum broadening of inclusive vector boson production,
$A(p_{1})+B(p_{2})\to V[{\rm J/}\psi,\,\Upsilon,\,Z^{0}](q)+X$ at the LHC is
discussed. The average squared transverse momentum of vector boson production
is
$\displaystyle\langle q_{T}^{2}\rangle(y)_{AB}\equiv\int
dq_{T}^{2}\,q_{T}^{2}\,\frac{d\sigma_{AB\to V}}{dy\,dq_{T}^{2}}\left[\int
dq_{T}^{2}\,\frac{d\sigma_{AB\to V}}{dy\,dq_{T}^{2}}\right]^{-1}\,.$ (77)
The transverse momentum broadening in $p+$Pb collisions is defined as
$\displaystyle\Delta\langle q_{T}^{2}\rangle_{p\,{\rm Pb}}(y)\equiv\langle
q_{T}^{2}\rangle(y)_{p\,{\rm Pb}}-\langle q_{T}^{2}\rangle(y)_{pp}\,.$ (78)
Following the derivation in Refs. [183, 130], the first nonvanishing
contribution to the transverse momentum broadening of heavy quarkonium
production is
$\displaystyle\Delta\langle q_{T}^{2}\rangle_{\rm HQ}^{\rm
CEM}=\left(\frac{8\pi^{2}\alpha_{s}}{N_{c}^{2}-1}\,\lambda^{2}\,A^{1/3}\right)\frac{(C_{F}+C_{A})\,\sigma_{q\bar{q}}+2\,C_{A}\,\sigma_{gg}+\Delta\sigma_{gg}}{\sigma_{q\bar{q}}+\sigma_{gg}}$
(79)
where the superscript “CEM” indicates that heavy quarkonium production is
evaluated in the Color Evaporation Model (CEM). A similar result was derived
in the NRQCD approach [183]. The $\sigma_{q\bar{q}}$ and $\sigma_{gg}$
partonic cross sections are contributions from quark-antiquark and gluon-gluon
subprocesses, respectively [183]. The $\Delta\sigma_{gg}$ term is a small,
color-suppressed correction to the $gg$ subprocess derived in Ref. [130]. In
the region where the $gg$ subprocess dominates heavy quarkonium production,
$\sigma_{gg}\gg\sigma_{q\bar{q}},\,\Delta\sigma_{gg}$, heavy quarkonium
broadening is further simplified as [183]
$\displaystyle\Delta\langle q_{T}^{2}\rangle_{\rm HQ}^{\rm CEM}\approx
2\,C_{A}\left(\frac{8\pi^{2}\alpha_{s}}{N_{c}^{2}-1}\,\lambda^{2}\,A^{1/3}\right)\,.$
(80)
In Fig. 45, the predictions of transverse momentum broadening of Drell-Yan
type vector boson production in $p+$Pb collisions at the LHC at $y=0$ are
shown as a function of $N_{\rm coll}$. To determine the effective dependence
on $N_{\rm coll}$, the $A^{1/3}$ in Eq. (79) is replaced by $A^{1/3}N_{\rm
coll}(b)/N_{\rm coll}(b_{\rm min\,bias})$. In $p+$Pb collisions at the LHC, a
Glauber-model calculation with $\sigma_{\rm NN}^{\rm in}=70$ mb at
$\sqrt{s}=5$ TeV gives $N_{\rm coll}(b_{\rm min\,bias})\sim 7$. In addition to
heavy quarkonium production, the broadening of $W/Z^{0}$ production,
calculated using the formalism derived in Ref. [183], is also shown in Fig.
45. The dramatic difference in the magnitude of the broadening between heavy
quarkonium and $W/Z^{0}$ production in Fig. 45 should be a signature QCD
prediction.
Figure 45: The transverse momentum broadening of vector boson production in
$p+$Pb collisions at $y=0$, shown as a function of $N_{\rm coll}$. The
$\Upsilon$ (red solid), $J/\psi$ (red dashed), $W^{\pm}$ (black solid), and
$Z^{0}$ (black dashed) results are given [190].
## 9 Acknowledgments
The research of J. L. Albacete is supported by a fellowship from the Théorie
LHC France initiative funded by the IN2P3. The work of N. Armesto was
supported by the European Research Council grant HotLHC ERC-2001-StG-279579;
by Ministerio de Ciencia e Innovación of Spain grants FPA2008-01177,
FPA2009-06867-E and Consolider-Ingenio 2010 CPAN CSD2007-00042; by Xunta de
Galicia grant PGIDIT10PXIB 206017PR; and by FEDER. G. G. Barnafóldi was
partially supported by the János Bolyai Research Scholarship of the Hungarian
Academy of Sciences. G. G. Barnafóldi, M. Gyulassy, and P. Levai also
acknowledge Hungarian grants OTKA PD73596, NK77816, NK106119, NIH
TET_10-1_2011-0061 and ZA-15/2009. J. Barrette and V. Topor Pop are supported
by the Natural Sciences and Engineering Research Council of Canada. S. De is
grateful to the Department of Atomic Energy of India for financial support. A.
Dumitru is supported by the DOE Office of Nuclear Physics through Grant No.
DE-FG02-09ER41620 and by The City University of New York through the PSC-CUNY
Research Award Program, grant 65041-0043. K. Dusling is supported by the US
Department of Energy under DOE Contract No. DE-FG02-03ER41260. K. J. Eskola is
supported by the Academy of Finland, Project 133005. R. J. Fries would like to
acknowledge support by NSF CAREER Award PHY-0847538 and by the JET
Collaboration and DOE grant DE-FG02-10ER41682. H. Fujii and Y. Nara are
supported in part by Grant-in-Aid for Scientific Research (B) 22340064. F.
Gelis is supported by the Agence Nationale de la Recherche project
11-BS04-015-01. M. Gyulassy is supported by the Division of Nuclear Science,
U.S. Department of Energy, under Contract No. DE-AC03-76SF00098 and DE-
FG02-93ER-40764 (associated with the JET Topical Collaboration Project). I.
Helenius is supported by the Magnus Ehrnrooth Foundation. The work of B. Z.
Kopeliovich was partially supported by Fondecyt (Chile) grant No. 1090291. The
work of K. Kutak and S. Sapeta was partially supported by the Foundation for
Polish Science with the grant Homing Plus/2010-2/6. The work of A. H. Mueller
is supported in part by the US Department of Energy under contract No. DE-
FG02-92ER-40699. The work of J. Nemchik was supported by grants VZ MŠMT
6840770039 and LA 08015 (Ministry of Education of the Czech Republic). M.
Petrovici is supported by the Romanian Authority for Scientific Research,
CNCS-UEFIS-CDI project number PN-II-ID-2011-3-0368. The work of A. H. Rezaeian
was partially supported by Fondecyt (Chile) grant No. 1110781. R. Venugopalan
was supported by US Department of Energy under DOE Contract No. DE-
AC02-98CH10886. The work of R. Vogt was performed under the auspices of the
U.S. Department of Energy by Lawrence Livermore National Laboratory under
Contract DE-AC52-07NA27344 and within the framework of the JET Collaboration.
The work of X.-N. Wang was performed under the auspices of the U.S. Department
of Energy under Contract No. DE-AC02-05CH11231, by the National Natural
Science Foundation of China under grant No. 11221504, and within the framework
of the JET Collaboration. I. Vitev is supported by the US Department of
Energy, Office of Science, under Contract No. DE-AC52-06NA25396 and by the
LDRD program at LANL.
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|
arxiv-papers
| 2013-01-15T16:13:20 |
2024-09-04T02:49:40.323242
|
{
"license": "Public Domain",
"authors": "J. L. Albacete, N. Armesto, R. Baier, G. G. Barnafoldi, J. Barrette,\n S. De, W.-T. Deng, A. Dumitru, K. Dusling, K. J. Eskola, R. Fries, H. Fujii,\n F. Gelis, M. Gyulassy, I. Helenius, Z.-B. Kang, B. Z. Kopeliovich, K. Kutak,\n P. Levai, Z. Lin, A. H. Mueller, Y. Nara, J. Nemchik, G. Papp, M. Petrovici,\n J.-W. Qiu, A. H. Rezaeian, P. Ru, D. Schiff, S. Sapeta, V. Topor Pop, P.\n Tribedy, R. Venugopalan, I. Vitev, R. Vogt, E. Wang, X.-N. Wang, H. Xing, R.\n Xu, B.-W. Zhang and W.-N. Zhang",
"submitter": "Ramona Vogt",
"url": "https://arxiv.org/abs/1301.3395"
}
|
1301.3511
|
Comment on “Unified Formalism of Andreev Reflection at a
Ferromagnet/Superconductor Interface” by T. Y. Chen, Z. Tesanovic, and C. L.
Chien
A recent paper of Chen et al. Chen , claims to have derived an allegedly,
previously unavailable “unified” Andreev Reflection (AR) formalism for an
arbitrary spin polarization $P$ that recovers earlier results as its special
limits. In this Comment we show that, contrary to this claim, there are
numerous works correctly solving the problem formulated in Ref. Chen for an
arbitrary $P$ deJong ; Mazin01 ; Kashiwaya ; Zutic99 ; Chalsani ; Eschrig ,
while Ref. Chen fails to correctly incorporate $P\neq 0$ effects and violate
basic physical principles.
In the $P=0$ limit the approach of Ref. Chen is identical to the one
dimensional (1D) BTK model Blonder for nonmagnetic metals, while for $P\neq
0$ the only difference is postulating an AR wavefunction that has an
additional evanescent wave contribution parameterized by a decay constant
$\alpha$, $\psi_{AR}=a\begin{Bmatrix}0\\\
1\end{Bmatrix}\exp[(\alpha+i)q^{+}x]$, with $\alpha=2\sqrt{P/(1-P)}$. However,
this wavefunction violates the charge conservation: the corresponding charge
current density is
$j_{AR}(x)\propto\mathrm{Im}[\psi_{AR}^{\ast}\nabla\psi_{AR}]\propto
q^{+}|a|^{2}\exp[2\alpha q^{+}x]$. In the F-region, for any $0<\alpha<\infty$,
i.e., $0<P<1$, the divergence of the _total_ current is finite and thus the
total charge is not conserved (in particular, for $0<x\lesssim{1}/{2\alpha
q^{+}}$), signaling that $\psi_{AR}$ is unphysical. Even for the $P=1$ half-
metal (HM) state, where the above expression correctly gives $j_{AR}=0,$ it is
still incorrect. For a HM the AR electron decays with a finite (not infinitely
small, as postulated in Ref.Chen ) penetration depth, which depends on the
electronic structure details, primarily the size of the gap in the nonmetallic
spin channel note1 .
The rationale for inventing a new $\psi_{AR}$ Chen was based on the premise
that: (a) the result in Ref. Mazin01 was derived only in the extreme case of
a fully-polarized HM with $\tilde{\psi}_{AR}=a\begin{Bmatrix}0\\\
1\end{Bmatrix}\exp(\kappa x$) and a nonmagnetic metal with
$\tilde{\psi}_{AR}=a\begin{Bmatrix}0\\\ 1\end{Bmatrix}\exp(ikx$), while
postulating that for an intermediate $P$ the current will be a linear
combination of these cases, (b) no prior work had treated a $0<P<1$ case, and
(c) one can meaningfully define $P$ of an individual electron. Regarding
(a-b), the derivation in Ref. Mazin01 , as well as in other works Kashiwaya ;
Zutic99 ; Chalsani ; Eschrig , is rigorous for an arbitrary $P$. The last
point, (c) is particularly misleading. In fact, one cannot define a BTK model
in 1D with an arbitrary $P$ because any given electron in a metal Andreev-
reflects either into a propagating, or into an evanescent wave. Finite $P$
simply means that some current-carrying electrons belong to the former group
and the others to the latter; one can only define transport spin polarization
in a multielectron system, where the numbers of electron states at the Fermi
surface (conductivity channels) for the two spin directions differ. If the
electron wavevector $\mathbf{k}$ is decomposed into a non-conserved $k_{x}$
(normal to the F/S interface) and conserved note2 2D $\mathbf{{k}_{\|}}$,
then, after the usual quantization of $\mathbf{{k}_{\parallel},}$ one finds
that the total number of states available for transport in the $x$ direction
(i.e., the number of the conductivity channels) for a given spin direction is
proportional to the total area of the Fermi surface projection on the
interface plane, given by the Fermi surface average of the Fermi velocity,
$n=\left\langle N(E_{F})v_{Fx}\right\rangle$ Mazin01 . After summation over
$all$ electronic states the total current $turns$ $out$ to be a linear
combination (with the weights defined by $P),$ of the solutions of the 1D BTK
model with $P=0$ and $P=1.$ This was not postulated, but derived, in numerous
papers (see the discussion in Ref. Eschrig, ).
In contrast to the “universal” $P$ in the Ref. Chen , independent of
electronic mass, Fermi wavevectors, or any band structure at all, the real
transport spin polarization for AR spectroscopy depends on the _overall_ Fermi
surface properties. Moreover, there is no unique spin polarization (even for a
uniform bulk metal), as it depends on an experimental probe. In fact, the
definition used in Ref. Chen (neglecting the Fermi velocities) does not
correspond to the AR experiments, but rather to spin-polarized photoemission.
To summarize, Ref. Chen has misinterpreted or ignored previous works where
the posed problem has been correctly solved for an arbitrary $P,$ and
attempted to supplant the existing solution with an incorrect formula,
postulating an unphysical wavefunction for the reflected electron, which does
not conserve charge and has an incorrect HM limit. They proceeded by
calculating the current due to $\psi_{AR}$ at $x=0,$ even though the actual
current is measured far away from the interface (where they would have gotten
zero Andreev contribution for any $P$ and grave disagreement with the
experiment). Furthermore, they have arbitrarily decided that the penetration
depth for an electron inside the band gap in the transport-spin-minority
channel is uniquely defined by the spin polarization (these two quantities are
in fact unrelated). As a result, they arrived at a formula that for their own
experimental data provides a fit that is essentially identical to that
obtained by using Ref. Mazin01 (the difference is below potential errors
introduced by the BTK approximations of a step-shaped pairing potential and
spin-independent $\delta$-shaped barrier note3 ).
Finally, we note that the inclusion of inelastic scattering using a
phenomenological finite $\Gamma\neq 0$, another claimed novelty of Ref. Chen
has already been previously employed for AR with arbitrary $P$ Chalsani ;
Cohen .
It may be that the formulas contrived in Ref. Chen fit a particular
experiment. However, there is a maxim attributed to Niels Bohr, that there
exists an infinite number of incorrect theories that correctly describe the
finite number of experiments.
M. Eschrig1, A. A. Golubov2, I. I. Mazin3, B. Nadgorny4, Y. Tanaka5, O. T.
Valls6, Igor Žutić7
1 Department of Physics, Royal Holloway, University of London, Egham, Surrey
TW20 0EX, UK, 2Faculty of Science and Technology and MESA+ Institute of
Nanotechnology, University of Twente, The Netherlands, 3Naval Research
Laboratory, Washington, DC 20375, USA, 4Department of Physics and Astronomy,
Wayne State University, Detroit, MI 48201, USA, 5Department of Applied
Physics, Nagoya University, Aichi 464-8603, Japan, 6School of Physics and
Astronomy, University of Minnesota, Minneapolis, MN 55455, USA, 7Department of
Physics, University at Buffalo, SUNY, Buffalo, NY 14260, USA
##
* (1) T. Y. Chen, Z. Tesanovic, and C. L. Chien, Phys. Rev. Lett. 109, 146602 (2012).
* (2) M. J. M. de Jong and C. W. J. Beenakker, Phys. Rev. Lett. 74, 1657 (1995).
* (3) I. I. Mazin, A. A. Golubov, and B. Nadgorny, J. Appl. Phys. 89, 7576 (2001); G. T. Woods, R. J. Soulen, Jr., I. I. Mazin, B. Nadgorny, M. S. Osofsky, J. Sanders, H. Srikanth, W. F. Egelhoff, and R. Datla, Phys. Rev. B 70, 054416 (2004).
* (4) S. Kashiwaya, Y. Tanaka, N. Yoshida, and M. R. Beasley, Phys. Rev. B 60, 3572 (1999).
* (5) I. Žutić and O. T. Valls, Phys. Rev. B 61, 1555 (2000), see Eq. (2.8); I. Žutić and S. Das Sarma, Phys. Rev. B 60, R16322 (1999).
* (6) P. Chalsani, S. K. Upadhyay, O. Ozatay, and R. A. Buhrman, Phys. Rev. B 75, 094417 (2007).
* (7) R. Grein, T. Lofwander, G. Metalidis, and M. Eschrig, Phys. Rev. B 81, 094508 (2010).
* (8) G. E. Blonder and M. Tinkham and T. M. Klapwijk, Phys. Rev. B 25, 4515 (1982).
* (9) Ref. Mazin01 , as well as some others, eventually uses the limit of the infinitely small penetration depth, because the final result depends on this parameter very little, but it is important to use in the derivation a physically meaningful wavefunction that allows for a finite penetration depth. This inconsistency in Ref. Chen is a part of a bigger problem of using the same $k_{F}$ for both spin channels, despite the fact that as $P\rightarrow 1$ one of the two $k_{F}^{\pm}$ gradually vanishes deJong ; Kashiwaya ; Zutic99 ; Eschrig .
* (10) For a ballistic flat interface. The generalization onto a diffusive case, where only the total number of the conductivity channels for each spin direction, but not the individual $\mathbf{k}_{\parallel}$ are conserved, is straightforward Mazin01 .
* (11) For other parameters [see their Fig. 2(c)] their approach leads to unphysical kinks at zero bias and unphysical notches near the gap voltage.
* (12) Other authors observed that instead of adding a $\Gamma$, at an extra computational cost, one can simply artificially increase the temperature in the original expressions, since the effect of the two types of broadening on the conductance curves is essentially the same, see, $e.g.$,Y. Bugoslavsky, Y. Miyoshi, S. K. Clowes, W. R. Branford, M. Lake, I. Brown, A. D. Caplin, and L. F. Cohen, Phys. Rev. B 71, 104523 (2005)
|
arxiv-papers
| 2013-01-15T21:53:52 |
2024-09-04T02:49:40.350160
|
{
"license": "Public Domain",
"authors": "M. Eschrig, A. A. Golubov, I. I. Mazin, B. Nadgorny, Y. Tanaka, O. T.\n Valls, Igor Zutic",
"submitter": "Igor Mazin",
"url": "https://arxiv.org/abs/1301.3511"
}
|
1301.3517
|
∎
e3e-mail: [email protected]
11institutetext: Mathematical Sciences Institute, The Australian National
University, ACT 0200, Australia
# Galactic Escape Speeds in Mirror and Cold Dark Matter Models
Alvin J. K. Chuaaddr1 D. T. Wickramasingheaddr1 Lilia Ferrarioe3,addr1
(Received: date / Accepted: date)
###### Abstract
The mirror dark matter (MDM) model of Berezhiani et al. has been shown to
reproduce observed galactic rotational curves for a variety of spiral
galaxies, and has been presented as an alternative to cold dark matter (CDM)
models. We investigate possible additional tests involving the properties of
stellar orbits, which may be used to discriminate between the two models. We
demonstrate that in MDM and CDM models fitted equally well to a galactic
rotational curve, one generally expects predictable differences in escape
speeds from the disc.
The recent radial velocity (RAVE) survey of the Milky Way has pinned down the
escape speed from the solar neighbourhood to $v_{\rm esc}=544^{+64}_{-46}$ km
s-1, placing an additional constraint on dark matter models. We have
constructed an MDM model for the Milky Way based on its rotational curve, and
find an escape speed that is just consistent with the observed value given the
current errors, which lends credence to the viability of the MDM model. The
Gaia-ESO spectroscopic survey is expected to lead to an even more precise
estimate of the escape speed that will further constrain dark matter models.
However, the largest differences in stellar escape speeds between both models
are predicted for dark matter dominated dwarf galaxies such as DDO 154, and
kinematical studies of such galaxies could prove key in establishing, or
abolishing, the validity of the MDM model.
††journal: Eur. Phys. J. C
## 1 Introduction
Over the last decade, mirror matter has emerged as a promising candidate for
dark matter. The idea that there might be a hidden “mirror sector” of the
Universe dates back to Lee1956 , was subsequently expanded upon by
Kobzarev1966 , and owes its modern form (in the context of gauge theories) to
Foot1991 . It is based on the observation that parity symmetry is violated by
the weak interaction in the “ordinary” sector we observe, but may be restored
by the existence of a duplicate sector with the same fundamental particles and
microphysics (apart from the opposite handedness of the weak interaction).
Support for the existence of a mirror world may also come from the recent
experiments of Berezhiani2012 on neutron oscillations.
Mirror dark matter (MDM) models linked to superstring inspired unifications of
the four forces in the early universe have been discussed by Kolb1985 . The
ordinary and mirror sectors are coupled through gravity but not the other
fundamental interactions, which explains the intrinsic suitability of mirror
matter as a dark matter candidate. Much research has been done on cosmological
signatures of mirror matter; these turn up in the context of early Universe
thermodynamics, Big Bang nucleosynthesis, primordial structure
formation/evolution, cosmic microwave background (CMB) radiation and large-
scale structure power spectra. Detailed reviews of this work have been
provided by Berezhiani2004 and Ciarcelluti2010 .
Astrophysical signatures of mirror matter have also been looked at in the
context of galactic rotational curves, using a specific model for mirror
gravity developed by Rossi2008 , Berezhiani2009 and Berezhiani2010
(henceforth BPR). In this model, the ordinary and mirror sectors are assumed
to have separate metric tensors (which are coupled to a third metric tensor),
leading to a Yukawa-like modification of the gravitational interaction between
both sectors. An ordinary matter test particle located at a radial distance
$r$ from a point-like source of ordinary mass $M_{1}$ and mirror mass $M_{2}$
will feel a Yukawa-like potential
$\phi\left(r\right)=-\frac{G}{2r}\left(M_{1}+M_{2}+\left(M_{1}-M_{2}\right)e^{-\frac{r}{r_{m}}}\right),$
(1)
where _G_ is the Newtonian gravitational constant and $r_{m}$ is the Yukawa
radius. The free-fall acceleration of the particle follows as
$g\left(r\right)=\frac{G}{2r^{2}}\left[M_{1}+M_{2}+\left(M_{1}-M_{2}\right)\left(1+\frac{r}{r_{m}}\right)e^{-\frac{r}{r_{m}}}\right].$
(2)
Although mirror matter is unable to form extended halos around galaxies due to
its collisional and dissipative nature, the combined model of mirror matter
and mirror gravity has been successfully fitted to the observed rotational
curves of several disc galaxies by BPR, on the assumption that ordinary matter
has a similar density profile to its mirror counterpart. This makes it the
most viable theory of MDM that is presently available.
In this paper, we investigate some implications of the BPR model over and
above its demonstrated success in modelling rotation curves, and suggest
astrophysical signatures that may be observable and potentially used to
discriminate between the MDM and CDM models. In Section 2, we use point-like
source models to investigate the orbits of ordinary matter particles in the
presence of mirror matter, and show that significant differences are expected
in the nature of orbits near the Yukawa radius. We then construct in Section 3
CDM models for the galaxies NGC 2403, DDO 170 and DDO 154 (whose rotational
curves have previously been modelled using MDM), and calculate and compare
escape speeds from the stellar disc as a function of galactic radius for both
types of model. We show that the MDM models predict specific values for the
escape speeds that are generally different from those in the CDM models. In
Section 4, we construct MDM and CDM models for the Milky Way, and find that
the escape speed from the solar neighbourhood in the MDM model is in good
agreement with the current observed value. We discuss and present our main
conclusions in Section 5.
## 2 Point-like source orbits
Figure 1: Mirror gravity orbits (solid) for ordinary test body about point-
like source with $M_{2}/M_{1}=10$, initial radius $r_{0}=\alpha r_{m}$ and
initial velocity
$\mathbf{v}_{0}=\sqrt{GM_{1}/r_{0}}{\boldsymbol{\hat{\theta}}}$. The
corresponding Newtonian orbits (dashed) are circular. All orbits are shown up
to five revolutions.
The deviation of mirror gravity from an inverse-square law leads to
distinctive orbital dynamics in the gravitational field of a point-like
source, especially at distances $r\sim r_{m}$. Two prominent features are
orbital precession and a varying escape speed to circular speed ratio, neither
of which are present for point-like sources in classical gravity.
Orbits for an ordinary test body in the gravitational field given by Equations
(1) and (2) are described by its Binet equation
$\begin{split}&\frac{d^{2}u}{d\theta^{2}}+u=\\\
&\frac{G}{2h^{2}}\left[M_{1}+M_{2}+\left(M_{1}-M_{2}\right)\left(1+\frac{1}{r_{m}u}\right)e^{-\frac{1}{r_{m}u}}\right],\end{split}$
(3)
where the reciprocal radial distance $u=1/r$ is a function of the plane polar
angle $\theta$, and the specific angular momentum $h=L/m=r^{2}\dot{\theta}$ is
a conserved quantity. Taking the Yukawa radius $r_{m}=10\;\mathrm{kpc}$ (as
assumed by BPR) here and henceforth, we solve Equation (3) numerically for
orbits about a point-like source with $M_{2}/M_{1}=10$ (see Figure 1).
The test body is given an initial radial distance of $r_{0}=\alpha r_{m}$ and
an initial transverse velocity of
$\mathbf{v}_{0}=\sqrt{GM_{1}/r_{0}}{\boldsymbol{\hat{\theta}}}$, which is the
circular velocity in the absence of mirror matter and mirror gravity. Its
orbit is essentially Newtonian for $\alpha\ll 1$, as gravitational interaction
between ordinary and mirror matter is negligible at small distances, and
Equation (2) reduces to
$g\left(r\right)=\frac{GM_{1}}{r^{2}}$
in the limit $r\rightarrow 0$.
For $\alpha\gg 1$, gravitational interaction between ordinary and mirror
matter becomes universal, and Equation (2) reduces to
$g\left(r\right)=\frac{G\left(M_{1}+M_{2}\right)}{2r^{2}}$
in the limit $r\rightarrow\infty$. The test body feels the combined inverse-
square field of both matter types (but with an effective gravitational
constant of $G/2$), resulting in an elliptical orbit.
Precessing orbits are manifest for $\alpha\sim 0.1$ to $\alpha\sim 10$, and
are explicitly non-classical. They may be expected to play a role in galactic
stellar dynamics at distance scales on the order of the Yukawa radius from the
centre of a galaxy.
A more useful signature of mirror gravity, which may allow it to be
observationally corroborated or falsified, lies in the escape speeds that it
predicts. For a point-like source with gravitational field given by Equations
(1) and (2), the escape and circular speeds are given respectively by
$\begin{split}v_{\mathrm{esc}}\left(r\right)&=\sqrt{-2\phi\left(r\right)}\\\
&=\sqrt{\frac{G}{r}\left(M_{1}+M_{2}+\left(M_{1}-M_{2}\right)e^{-\frac{r}{r_{m}}}\right)},\end{split}$
(4)
and
$\begin{split}&v_{\mathrm{cir}}\left(r\right)=\sqrt{rg\left(r\right)}\\\
&~{}~{}=\sqrt{\frac{G}{2r}\left[M_{1}+M_{2}+\left(M_{1}-M_{2}\right)\left(1+\frac{r}{r_{m}}\right)e^{-\frac{r}{r_{m}}}\right]}.\end{split}$
(5)
In general, the ratio $v_{\mathrm{esc}}/v_{\mathrm{cir}}\neq\sqrt{2}$ and is a
function of _r_ , in contrast to the Newtonian case. A plot of
$v_{\mathrm{esc}}/v_{\mathrm{cir}}$ for different values of $M_{2}/M_{1}\geq
1$ shows that $v_{\mathrm{esc}}/v_{\mathrm{cir}}\geq\sqrt{2}$, with the
discrepancy being more pronounced for larger mass ratios (see Figure 2).
111The cosmological ratio between the dark and visible energy densities is
$\rho_{2}/\rho_{1}\approx 10$ in the BPR model. $M_{2}/M_{1}$ should approach
the cosmological ratio for larger galaxies, while smaller galaxies are
expected to be more dark matter-dominated. In any case, only values of
$M_{2}/M_{1}>1$ will be considered in this paper. This suggests that mirror
gravity may predict significantly larger escape speeds compared to classical
gravity when both models are fitted to give similar circular speeds, as is
done in the modelling of galactic rotational curves.
Figure 2: Plot of $v_{\mathrm{esc}}/v_{\mathrm{cir}}$ against $r/r_{m}$ for
$M_{2}/M_{1}=1000$ (solid), $M_{2}/M_{1}=100$ (dashed) and $M_{2}/M_{1}=10$
(dot-dashed). The Newtonian ratio $v_{\mathrm{esc}}/v_{\mathrm{cir}}=\sqrt{2}$
is recovered for $M_{2}/M_{1}=1$ (dotted).
## 3 Escape speeds in disc galaxies
Mirror matter and mirror gravity have been put forth by BPR as a collisional
and dissipative dark matter candidate capable of explaining flat galactic
rotational curves. They have successfully fitted this MDM model to the
observed rotational curves of several disc galaxies, using simplified and
similar mass density profiles for the visible and dark matter distributions.
In this section, we obtain comparable fits using a CDM model of the same
complexity, and generate the corresponding galactic escape speed curves for
both models.
The distribution of visible matter in a typical disc galaxy is approximated
most simply as a two-dimensional disc with the exponential surface mass
density
$\Sigma_{1,d}\left(r\right)=\frac{M_{1,d}}{2\pi
r_{1,d}^{2}}e^{-\frac{r}{r_{1,d}}},$ (6)
where $M_{1,d}$ and $r_{1,d}$ are the total mass and characteristic length
scale of the disc respectively. In the BPR model, the distribution of mirror
dark matter is assumed to take on a similar profile (i.e. another exponential
disc concentric to the visible disc); this is justifiable since mirror matter
has the same microphysics as ordinary matter and can be expected to undergo
the same process of galaxy formation, up to a rescaling of parameters. Hence
its surface mass density is given by
$\Sigma_{2,d}\left(r\right)=\frac{M_{2,d}}{2\pi
r_{2,d}^{2}}e^{-\frac{r}{r_{2,d}}},$ (7)
where $M_{2,d}$ and $r_{2,d}$ are the corresponding parameters for the mirror
dark disc.
For an ordinary test body with position vector r in the plane of the two
discs, the acceleration vector follows as
$\begin{split}&\mathbf{g}\left(\mathbf{r}\right)=-\frac{G}{2}\int_{\mathrm{disc}}d^{2}\mathbf{r}^{\prime}\frac{\mathbf{r}-\mathbf{r}^{\prime}}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|^{3}}\left[\vphantom{\frac{1}{2}}\Sigma_{1,d}\left(r^{\prime}\right)+\Sigma_{2,d}\left(r^{\prime}\right)\right.\\\
&+\left.\left(\Sigma_{1,d}\left(r^{\prime}\right)-\Sigma_{2,d}\left(r^{\prime}\right)\right)\left(1+\frac{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}{r_{m}}\right)e^{-\frac{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}{r_{m}}}\right],\end{split}$
(8)
where $\mathbf{g}\left(\mathbf{r}\right)=-g\left(r\right)\hat{\mathbf{r}}$.
The integration can be performed semi-analytically in this case, and yields
the rotational curve $v\left(r\right)=\sqrt{rg\left(r\right)}$ of the model.
Assuming that the mass-to-light ratio $\gamma=M_{1,d}/M_{L}$ of the visible
matter (where the luminous mass $M_{L}$ is an observed fixed parameter) is
constant for each galaxy, the dimensionless quantities $\gamma$,
$\beta=M_{2,d}/M_{1,d}$ and $\alpha=r_{2,d}/r_{1,d}$ have been taken by BPR as
fit parameters for the rotational curves of four disc galaxies. 222Rotational
curve fits for the large spiral galaxy NGC 7331 require the contribution of an
additional bulge component, and will be omitted from the MDM–CDM(IT)
comparisons in this section. We address the issue of a galactic bulge when
modelling the Milky Way in Section 4. The visible length scale $r_{1,d}$ for
each galaxy is deduced from and fixed by observations. The resulting three-
parameter fits are demonstrably good, with $\chi_{\mathrm{red}}^{2}\approx 1$
for each galaxy (see Figure 3 and Table 1).
Numerous galactic rotational curve fits for CDM models exist and are well
documented in the literature. These are of varying complexity, but nearly all
of them incorporate components such as a galactic bulge or a gaseous
contribution, in addition to the standard galactic disc and dark halo
components. Indeed, the rotational curves of the galaxies considered by BPR
have been fitted for a multi-component CDM model by Begeman1991 (henceforth
BBS).
We adopt a simplified two-component version of the BBS model in this paper,
allowing direct comparison between the MDM model and a CDM model of the same
complexity. The distribution of visible matter is again an exponential disc
with surface mass density given by Equation (6), while a pseudo-isothermal
truncated sphere (which we shall refer to from now on as CDM(IT)) is assumed
for the distribution of dark matter.
The volume mass density of the dark halo is given by
$\rho_{h}\left(r\right)=\frac{\rho_{s}}{1+\frac{r^{2}}{r_{s}^{2}}},$ (9)
where $\rho_{s}$ and $r_{s}$ are the central density and characteristic length
scale of the halo respectively. This profile avoids the central cusps inherent
in those produced by numerical _n_ -body simulations (e.g. the model by
Navarro1995 ), but has a divergent total mass and must be truncated at a
suitable distance from the galactic centre.
An analytical expression is available for the rotational curve of this
simplified BBS model. We can write
$v\left(r\right)=\sqrt{v_{\mathrm{disc}}^{2}\left(r\right)+v_{\mathrm{halo}}^{2}\left(r\right)},$
(10)
where $v_{\mathrm{disc}}\left(r\right)$ and $v_{\mathrm{halo}}\left(r\right)$
are rotational curves for the visible disc and dark halo respectively. The
contribution from the disc has been obtained by Freeman1970 , using a method
by Toomre1963 , as
$\begin{split}v_{\mathrm{disc}}^{2}\left(r\right)&=\frac{GM_{1,d}r^{2}}{2r_{1,d}^{3}}\left[\vphantom{\frac{1}{2}}I_{0}\left(\frac{r}{2r_{1,d}}\right)K_{0}\left(\frac{r}{2r_{1,d}}\right)\right.\\\
&-\left.I_{1}\left(\frac{r}{2r_{1,d}}\right)K_{1}\left(\frac{r}{2r_{1,d}}\right)\right],\end{split}$
(11)
where the $I_{n}$ and $K_{n}$ are modified Bessel functions evaluated at
$r/2r_{1,d}$. Begeman1987 gives the contribution from the halo as
$v_{\mathrm{halo}}^{2}\left(r\right)=v_{\mathrm{max}}^{2}\left(1-\frac{r_{s}}{r}\tan^{-1}\frac{r}{r_{s}}\right),$
(12)
where $v_{\mathrm{max}}=\sqrt{4\pi G\rho_{s}r_{s}^{2}}$ is the asymptotic
rotational speed for the halo at $r\gg r_{s}$.
As in the BPR model, we use data from Begeman1987 , Carignan1989 , Lake1990
and Begeman1991 for the intermediate spiral galaxy NGC 2403 and the dwarf
galaxies DDO 170 and DDO 154. Again assuming a constant mass-to-light ratio
for each galaxy and taking $M_{L}$ and $r_{1,d}$ as fixed parameters, we fit
the rotational curves of these three disc galaxies by using $\gamma$,
$v_{\mathrm{max}}$ and $r_{s}$ as fit parameters.
Furthermore, we have generated synthetic rotational curves for two disc
galaxies, one whose physics is dictated by MDM and the other by CDM(IT).
Typical observational errors (Gaussian noise) have been assigned to these
synthetic galaxies, which are then treated as actual observed galaxies and
fitted using the procedures outlined in this section. We find that the MDM fit
recovers best-fit values in good agreement with the actual values for the
synthetic MDM galaxy, while the CDM(IT) fit matches the parameters of the
synthetic CDM(IT) galaxy as well; this verifies the accuracy of our fitting
algorithms. Finally, a side-by-side comparison of all the MDM and CDM(IT) fits
shows that both models are able to reproduce both the observed and synthetic
galactic rotational curves, and equally well (see Figure 3 and Table 1).
Figure 3: Observed and synthetic rotational curves for different disc galaxies, with both MDM (solid) and CDM(IT) (dashed) three-parameter fits. Parameters and values of $\chi_{\mathrm{red}}^{2}$ are given in Table 1. Table 1: MDM and CDM(IT) best-fit parameters for the rotational curves of different disc galaxies. $M_{L}$ and $r_{1,d}$ are taken as fixed parameters, while $\gamma$, $\beta$, $\alpha$, $v_{\mathrm{max}}$ and $r_{s}$ are taken as fit parameters. Both models provide realistic (all values of $\gamma\sim 1$) and good (all values of $\chi_{\mathrm{red}}^{2}\approx 1$) fits. The fitting algorithms recover best-fit values in good agreement with the actual values for the synthetic disc galaxies (given in parentheses). | | NGC 2403 | DDO 170 | DDO 154 | Synthetic (MDM) | Synthetic (CDM(IT))
---|---|---|---|---|---|---
Fixed parameters | $M_{L}\left(10^{9}M_{\odot}\right)$ | 7.9 | 0.18 | 0.05 | 8.0 | 0.10
| $r_{1,d}\left(\mathrm{kpc}\right)$ | 2.0 | 1.1 | 0.5 | 1.6 | 0.8
MDM fit | $\gamma=M_{1,d}/M_{L}$ | 1.6 | 2.2 | 0.6 | 1.0 (1.0) | 1.2
| $M_{1,d}\left(10^{9}M_{\odot}\right)$ | 12.6 | 0.4 | 0.03 | 8.2 (8.0) | 0.12
| $\beta=M_{2,d}/M_{1,d}$ | 20.6 | 190 | 2000 | 22.3 (23.0) | 567
| $M_{2,d}\left(10^{9}M_{\odot}\right)$ | 260 | 76 | 60 | 183 (184) | 68
| $\alpha=r_{2,d}/r_{1,d}$ | 1.6 | 0.8 | 1.3 | 1.6 (1.6) | 1.0
| $r_{2,d}\left(\mathrm{kpc}\right)$ | 3.2 | 0.87 | 0.66 | 2.5 (2.6) | 0.8
| $\chi_{\mathrm{red}}^{2}$ | 1.3 | 1.5 | 0.6 | 1.3 | 1.3
CDM(IT) fit | $\gamma=M_{1,d}/M_{L}$ | 1.5 | 1.1 | 0.2 | 1.0 | 0.9 (1.0)
| $M_{1,d}\left(10^{9}M_{\odot}\right)$ | 12.1 | 0.2 | 0.01 | 7.7 | 0.09 (0.10)
| $v_{\mathrm{max}}\left(\mathrm{kms^{-1}}\right)$ | 155 | 77 | 61 | 129 | 70 (70)
| $r_{s}\left(\mathrm{kpc}\right)$ | 5.0 | 2.1 | 1.7 | 4.1 | 2.0 (2.0)
| $\chi_{\mathrm{red}}^{2}$ | 1.6 | 1.2 | 0.8 | 1.3 | 1.0
Now, for the MDM model, the gravitational potential felt by an ordinary test
body with position vector r in the plane of the two discs is given by
$\begin{split}\phi_{\mathrm{MDM}}\left(r\right)&=-\frac{G}{2}\int_{\mathrm{disc}}d^{2}\mathbf{r}^{\prime}\frac{1}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}\big{[}\Sigma_{1,d}\left(r^{\prime}\right)+\Sigma_{2,d}\left(r^{\prime}\right)\\\
&+\left.\left(\Sigma_{1,d}\left(r^{\prime}\right)-\Sigma_{2,d}\left(r^{\prime}\right)\right)e^{-\frac{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}{r_{m}}}\right],\end{split}$
(13)
where $\phi_{\mathrm{MDM}}\left(r\right)$ depends only on _r_ due to
cylindrical symmetry. For the CDM(IT) model, the gravitational potential felt
by a test body with position vector r in the plane of the disc is given by
$\phi_{\mathrm{CDM(IT)}}\left(r\right)=-G\int_{\mathrm{disc}}\\!\\!\\!d^{2}\mathbf{r}^{\prime}\frac{\Sigma_{1,d}\left(r^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}-G\int_{\mathrm{halo}}\\!\\!\\!d^{3}\mathbf{r}^{\prime}\frac{\rho_{h}\left(r^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|},$
(14)
where $\phi_{\mathrm{CDM(IT)}}\left(r\right)$ depends only on _r_ as well.
Using the parameters given in Table 1, we obtain MDM and CDM(IT) escape speed
curves $v\left(r\right)=\sqrt{-2\phi\left(r\right)}$ for the five disc
galaxies by numerical integration, truncating the discs and halos at a value
of _r_ just beyond the farthest data point for each galaxy (see Figure 4).
Figure 4: Predicted MDM (solid) and CDM(IT) (dashed) escape speed curves for
stars in the galactic plane of observed and synthetic disc galaxies, using
parameters given in 1. The actual (dotted) escape speed curves for the
synthetic galaxies have been included as well.
As anticipated from our discussion of point-like source models in Section 2,
escape speeds for stars in the galactic plane are mostly higher in the MDM
model. The offset between the two models appears to be nearly constant out to
the truncation distance. It is both absolutely and relatively larger for the
dwarf galaxies ($\Delta v\geq 30\;\mathrm{kms^{-1}}$ and $\Delta
v/v_{\mathrm{CDM(IT)}}\geq 0.2$) than for NGC 2403 ($\Delta v\approx
10\;\mathrm{kms^{-1}}$ and $\Delta v/v_{\mathrm{CDM(IT)}}\approx 0.03$).
This systematic escape speed discrepancy may be attributed to the increasing
importance of mirror dark matter over ordinary matter in the smaller galaxies
(as indicated by the higher values of $\beta$), which in turn is due to
typical distance scales in these galaxies being smaller than the Yukawa scale
$r_{m}$ ($\sim\rm{10kpc}$) such that dark matter is less effective in
gravitationally binding to ordinary matter. In the largest galaxies such as
the Milky Way (see Section 4), the ratio of the energy density of mirror dark
matter to visible matter $\rho_{2}/\rho_{1}$ approaches the cosmological value
$\sim 10$ as required by the BPR model.
Crucially, we have also observed in our analysis that escape speeds for the
MDM model do not depend significantly on the choice of truncation radius, and
are constrained by the rotational curve fits. Thus, observations of the
stellar escape speeds in disc galaxies of different sizes could prove key in
establishing—or abolishing—the validity of the MDM model.
## 4 Escape speeds in the Milky Way
Although the rotational curves of numerous external disc galaxies have been
obtained to a good degree of precision, there are no direct estimates of their
escape speeds at the present time. Conversely, while the rotational curve of
our own galaxy is less established, efforts have been made by Smith2007 to
constrain the escape speed at the solar neighbourhood through the Radial
Velocity Experiment (RAVE) survey. The current estimate is significantly in
excess of $\sqrt{2}v_{\mathrm{cir}}$ (where $v_{\mathrm{cir}}\approx
220\;\mathrm{kms^{-1}}$ is the local circular speed), and has been taken as
independent evidence in favour of CDM models. In this section, we repeat the
MDM–CDM(IT) comparison for the Milky Way, and show that the MDM model is also
able to predict a local escape speed comparable to the currently observed
value.
As in Section 3, the visible disc of the Galaxy in both models is taken as an
exponential disc and described by Equation (6). Likewise, the density profiles
of the mirror dark disc in the MDM model and the dark halo in the CDM(IT)
model are given respectively by Equations (7) and (9). In both models,
however, an additional bulge component is required to reproduce the observed
Galactic rotational curve.
We employ the tractable Plummer sphere for this purpose, in lieu of the
truncated power-law bulges used in more detailed mass models. Hence the
gravitational potential of the visible bulge is given by
$\phi_{\mathrm{bulge}}\left(r\right)=-\frac{GM_{1,b}}{\sqrt{r^{2}+r_{1,b}^{2}}},$
(15)
where $M_{1,b}$ and $r_{1,b}$ are the total mass and characteristic length
scale of the bulge respectively. Its volume mass density follows as
$\rho_{1,b}\left(r\right)=\frac{3M_{1,b}}{4\pi
r_{1,b}^{3}}\left(1+\frac{r^{2}}{r_{1,b}^{2}}\right)^{-\frac{5}{2}}$ (16)
from Poisson’s equation.
By assumption, the presence of a visible bulge necessitates the addition of a
mirror dark bulge component to the MDM model. This is similarly described by
$\rho_{2,b}\left(r\right)=\frac{3M_{2,b}}{4\pi
r_{2,b}^{3}}\left(1+\frac{r^{2}}{r_{2,b}^{2}}\right)^{-\frac{5}{2}},$ (17)
where $M_{2,b}$ and $r_{2,b}$ are the corresponding parameters for the dark
bulge. The acceleration vector of an ordinary test body with position vector r
in the Galactic plane follows as
$\mathbf{g}\left(\mathbf{r}\right)=\mathbf{g}_{\mathrm{disc}}\left(\mathbf{r}\right)+\mathbf{g}_{\mathrm{bulge}}\left(\mathbf{r}\right),$
(18)
where $\mathbf{g}_{\mathrm{disc}}\left(\mathbf{r}\right)$ is given by Equation
(8) and $\mathbf{g}_{\mathrm{bulge}}\left(\mathbf{r}\right)$ is obtained in
identical fashion by numerically integrating Equations (16) and (17) over the
bulge.
With the addition of the visible bulge, the rotational curve of the CDM(IT)
model becomes
$v\left(r\right)=\sqrt{v_{\mathrm{disc}}^{2}\left(r\right)+v_{\mathrm{halo}}^{2}\left(r\right)+v_{\mathrm{bulge}}^{2}\left(r\right)},$
(19)
where $v_{\mathrm{disc}}^{2}\left(r\right)$ and
$v_{\mathrm{halo}}^{2}\left(r\right)$ are given by Equations (11) and (12) as
before, and $v_{\mathrm{bulge}}^{2}\left(r\right)$ follows from Equation (15)
as
$v_{\mathrm{bulge}}^{2}\left(r\right)=r\frac{d}{dr}\phi_{\mathrm{bulge}}\left(r\right)=\frac{GM_{1,b}r^{2}}{r_{1,b}^{3}}\left(1+\frac{r^{2}}{r_{1,b}^{2}}\right)^{-\frac{3}{2}}.$
(20)
For the MDM model, we assume a constant dark-to-visible mass ratio (such that
$\beta=M_{2,i}/M_{1,i}$) and a constant dark-to-visible length scale ratio
(such that $\alpha=r_{2,i}/r_{1,i}$) in the Galaxy. We use $M_{1,d}$,
$M_{1,b}$, $\beta$ and $\alpha$ as fit parameters, and take
$r_{1,d}=3\;\mathrm{kpc}$ and $r_{1,b}=0.3\;\mathrm{kpc}$ in rough accordance
with observation and other mass models (e.g. McMillan2011 ).
For the CDM(IT) model, we use $M_{1,d}$, $M_{1,b}$, $v_{\mathrm{max}}$ and
$r_{s}$ as fit parameters, and take $r_{1,d}$ and $r_{1,b}$ as fixed
parameters with the MDM values. With data out to $2R_{\odot}$ (where
$R_{\odot}=8.5\;\mathrm{kpc}$) from Burton1978 , Blitz1982 and Clemens1985 ,
we are able to generate good rotational curve fits for both models (see Figure
5 and Table 2).
Figure 5: Observed Galactic rotational curve out to $2R_{\odot}$, with both MDM (solid) and CDM(IT) (dashed) four-parameter fits. Parameters and values of $\chi_{\mathrm{red}}^{2}$ are given in Table 2. Table 2: MDM and CDM(IT) best-fit parameters for the Galactic rotational curve out to $2R_{\odot}$. $r_{1,d}$ and $r_{1,b}$ are taken as fixed parameters, while $M_{1,d}$, $M_{1,b}$, $\beta$, $\alpha$, $v_{\mathrm{max}}$ and $r_{s}$ are taken as fit parameters. Both models provide good ($\chi_{\mathrm{red}}^{2}\approx 1$) fits that agree with observation and other mass models. | Parameters | Values
---|---|---
Fixed parameters | $r_{1,d}\left(\mathrm{kpc}\right)$ | 3
| $r_{1,b}\left(\mathrm{kpc}\right)$ | 0.3
MDM fit | $M_{1,d}\left(10^{9}M_{\odot}\right)$ | 48
| $M_{1,b}\left(10^{9}M_{\odot}\right)$ | 13
| $\beta=M_{2,i}/M_{1,i}$ | 12
| $M_{2,d}\left(10^{9}M_{\odot}\right)$ | 576
| $M_{2,b}\left(10^{9}M_{\odot}\right)$ | 156
| $\alpha=r_{2,i}/r_{1,i}$ | 1.7
| $r_{2,d}\left(\mathrm{kpc}\right)$ | 5.1
| $r_{2,b}\left(\mathrm{kpc}\right)$ | 0.51
| $\chi_{\mathrm{red}}^{2}$ | 1.3
CDM(IT) fit | $M_{1,d}\left(10^{9}M_{\odot}\right)$ | 54
| $M_{1,b}\left(10^{9}M_{\odot}\right)$ | 11.8
| $v_{\mathrm{max}}\left(\mathrm{kms^{-1}}\right)$ | 241
| $r_{s}\left(\mathrm{kpc}\right)$ | 6.6
| $\chi_{\mathrm{red}}^{2}$ | 1.1
Although the fits are simplistic, they are realistic enough to serve as
adequate models of the Galaxy. The total visible (stellar) mass for the MDM
fit is $6.1\times 10^{10}\;M_{\odot}$ with a bulge contribution of $1.3\times
10^{10}\;M_{\odot}$; these values are comparable to estimates by Gerhard2002 ,
Flynn2006 , Widrow2008 and McMillan2011 . The dark-to-visible mass ratio is
$\beta=12$, which approaches the cosmological ratio $\rho_{2}/\rho_{1}\approx
10$ in support of the hypothesis by BPR. The local surface mass density (at
$R_{\odot}=8.5\;\mathrm{kpc}$) of the disc is $50\;M_{\odot}\mathrm{pc}^{-2}$,
in good agreement with estimates by Kuijken1989 , Flynn1994 , Bienayme2006
and Flynn2006 .
The total visible mass for the CDM(IT) fit is $6.6\times 10^{10}\;M_{\odot}$
with a bulge contribution of $1.2\times 10^{10}\;M_{\odot}$, while the local
surface mass density of the disc is $56\;M_{\odot}$ $\mathrm{pc}^{-2}$. Again,
these values match up reasonably well with other estimates. The local dark
matter density is $0.35\;\mathrm{GeVcm}^{-3}$, which is towards the low end of
bounds given by Catena2010 and McMillan2011 .
Using the parameters given in Table 2, we obtain MDM and CDM(IT) escape speed
curves for the Galaxy as in Section 3, but with a bulge term added to the
gravitational potential (see Figure 6). For the MDM model,
$\phi_{\mathrm{bulge}}\left(r\right)$ is obtained in identical fashion to
Equation (13) by numerically integrating Equations (16) and (17) over the
bulge; for the CDM(IT) model, it is given by Equation (15).
Figure 6: Predicted MDM (solid) and CDM(IT) (dashed) escape speed curves out
to $2R_{\odot}$ for stars in the Galactic plane, using the parameters given in
Tab. 2. The RAVE value $v_{\mathrm{esc}}=544_{-46}^{+64}\;\mathrm{kms^{-1}}$
for the local escape speed (at $R_{\odot}=8.5\;\mathrm{kpc}$) is included as a
reference point.
Again, escape speeds for stars in the Galactic plane are higher in the MDM
model, with an offset that grows from $\approx 20\;\mathrm{kms^{-1}}$ to
$\approx 50\;\mathrm{kms^{-1}}$ but remains nearly constant beyond
$r=R_{\odot}$. The relative offset at $r=R_{\odot}$ is $\Delta
v/v_{\mathrm{CDM(IT)}}\approx 0.09$. Both models predict values for the local
escape speed that fall just short of the lower bound on the RAVE value
$v_{\mathrm{esc}}=544_{-46}^{+64}\;\mathrm{kms^{-1}}$; in the CDM(IT) case,
the fairly large discrepancy is due to the truncation of the dark halo at
$r=2R_{\odot}$. Indeed, by extending the truncation radius of the halo to 45
kpc, we find an escape speed for the CDM(IT) model that matches the observed
RAVE value. Regardless, the fact that the MDM model is able to predict a local
value of $v_{\mathrm{esc}}$ significantly greater than
$\sqrt{2}v_{\mathrm{cir}}$ corroborates the RAVE measurement, and lends
credence to its own viability as a dark matter model.
## 5 Discussion and conclusions
The addition of a non-luminous component of matter distributed spherically
about the centre of a galaxy and extending well beyond the luminous disc
component (i.e. a dark matter halo) explained the flat rotational curves of
disc galaxies in the framework of Newtonian gravity, and led to the CDM
paradigm in cosmology. In these models, a significant fraction of the
gravitating matter in the Universe is in a dark collisionless form, with a
density parameter ratio between dark and baryonic matter of
$\Omega_{d}/\Omega_{b}\sim 5$. The CDM model has been extremely successful in
explaining early Universe thermodynamics, through to fluctuations in the CMB
and the formation of structures on cosmological scales. Any viable alternative
model must not only explain galactic rotational curves, but also match the
successes of CDM in large-scale cosmology.
A much-discussed alternative for explaining galaxy rotational curves is the
modified Newtonian dynamics (MOND) model of Milgrom1983 , which entertains the
possibility of an acceleration due to gravity that departs from the Newtonian
form in regimes where the dynamical accelerations are small. An in-depth
review of the theory is provided by Sanders2002 . First proposed as a
phenomenological model, the essence of MOND has subsequently been incorporated
in a theory of modified gravity (MOG) by Moffat2005 , which is based on a
covariant generalisation of Einstein’s theory with the addition of auxiliary
gravitational fields to the metric. MOG has been successfully used to model a
large sample of galactic rotational curves Brownstein2006a and galaxy
clusters Brownstein2006b . However, these models that focus on the
gravitational force appear not to be as well-placed to address questions on
early Universe thermodynamics and large-scale cosmology.
On the other hand, the MDM model of BPR is based on a fundamental theory of
all particles and forces; as with the CDM model, it appears to have the
potential to not only explain galaxy rotational curves, but also to address
problems relating to the early history of the Universe. In proposing their
model, BPR point out that to date there has been no observational evidence for
the dark matter postulated in CDM models. They note that one of the major
unanswered questions in CDM cosmology, namely the reason for the closeness of
the observed ratio $\Omega_{d}/\Omega_{b}$ to unity, may find an easier
solution in MDM models—where the dark (mirror) and baryonic (ordinary) sectors
are expected to be closely linked. They also note that the density profiles
predicted by _n_ -body simulations for the CDM halos in galaxies show a
central cusp, for which there is no observational evidence from rotational
curves of low surface brightness galaxies. While acknowledging that this may
be a problem with the numerical simulations of galaxy formation in the CDM
scenario, they point out that dark matter is collisional in the MDM model, and
such cusps are not expected.
Rotational curves, being based on the circular speed (usually of gas and
sometimes augmented by that of stars) in galaxies, cannot by themselves
provide strong constraints on dark matter models or gravity models. Indeed,
CDM, MDM and MOG models have all been successfully used to model rotational
curves of both low and high surface brightness galaxies. In this paper, we
have sought to highlight some of the differences that may be expected in the
dynamics of stars in galactic discs, which may potentially be used to
discriminate between the CDM and MDM models. In particular, we have focused on
stellar escape speeds from the galactic disc. Following BPR, our basic premise
is that due to the collisional and dissipative nature of MDM, matter in the
two sectors will have similar density distributions up to a scaling factor. A
consequence is that the MDM model has a very definite prediction on escape
speeds that is readily falsifiable. Notwithstanding the uncertainties inherent
in modelling the structure and extent of the CDM halos, we predict systematic
and measurable differences in escape speed between the two models.
We have taken the galaxies NGC 2403, DDO 170 and DDO 154 (whose rotational
curves have been modelled by BPR using MDM) along with a couple of synthetic
galaxies as representative of spiral galaxies with a range of sizes, and used
them to calculate and compare stellar escape speeds between the MDM and
CDM(IT) models. We find that the escape speeds from the galactic disc are
typically higher in the MDM model, with the percentage increase in escape
speed relative to what is expected from an equivalent CDM(IT) model
approaching $\Delta v/v_{\mathrm{CDM}}\geq 20$ percent in the smaller low
surface brightness galaxies. A similar procedure has been adopted for the
Milky Way, where in addition to the MDM stellar disc, we have also included an
MDM bulge component.
In the case of the Milky Way, recent estimates from the RAVE survey yield an
escape speed from the solar neighbourhood of
$v_{\mathrm{esc}}=544_{-46}^{+64}\;\mathrm{kms^{-1}}$, and an escape to
circular speed ratio $v_{\mathrm{esc}}/v_{\mathrm{cir}}\sim 2.5$. If one
assumes normal gravity and axially symmetric density structures, the circular
speed gives a measure of the amount of matter within the solar circle, while
the escape speed gives a measure of the amount of matter outside the solar
circle. A measurement of $v_{\mathrm{esc}}/v_{\mathrm{cir}}$ that is
significantly larger than $\sqrt{2}$ could then be taken as independent
evidence that there is a large proportion of mass outside the solar circle.
However, we have shown that our MDM model of the Milky Way yields a similar
value $v_{\mathrm{esc}}/v_{\mathrm{cir}}\sim 2.2$, and that given the errors,
this particular test cannot be used to establish the presence of a dark halo
or to distinguish between the two models at the present time.
This situation is expected to change when the Gaia–ESO public spectroscopic
survey using FLAMES becomes available in 2016 Gilmore2012 ; the survey will
yield accurate position and velocity information of a large number of
individual stars in the thin and thick discs, the bulge and the halo. Detailed
studies of the motions of stars in the solar neighbourhood (as has been done
with RAVE) will allow a more precise measurement of the local escape speed.
With the new set of data, the study could be extended to establish the radial
dependence of the escape velocity, which will provide even stronger
constraints on the different dark matter models. We may expect substantial
differences under the MDM model in the kinematics of stars in the galactic
halo, which may also be measurable in future surveys.
The measurement of escape speeds from discs in external galaxies may also soon
become possible. This will require the accumulation of position and velocity
data for a large number of individual stars using multi-object optical and
infrared spectroscopy with large optical telescopes, along the lines of the
recent spectroscopic study of the nearby dwarf irregular galaxy WLM using the
VLT and Keck II Leaman2012 . Statistical studies of the resulting stellar
velocity distributions of a number of such galaxies viewed at different
inclinations could then be used to estimate escape speeds and to test the MDM
paradigm.
###### Acknowledgements.
We wish to thank Ken Freeman for helpful comments and discussion. We also wish
to thank the anonymous Referee for his/her constructive suggestions which have
greatly improved the manuscript.
## References
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|
arxiv-papers
| 2013-01-15T22:23:05 |
2024-09-04T02:49:40.355852
|
{
"license": "Public Domain",
"authors": "Alvin J. K. Chua, D. T. Wickramasinghe and Lilia Ferrario",
"submitter": "Lilia Ferrario",
"url": "https://arxiv.org/abs/1301.3517"
}
|
1301.3532
|
# ANL-HEP-CP-13-3
Order-$\bm{v^{4}}$ Relativistic Corrections to
Gluon Fragmentation into $\bm{{}^{3}S_{1}}$ Quarkonium
Geoffrey T. Bodwin
HEP Division, Argonne National Laboratory
E-mail [email protected] U-Rae Kim
Department of Physics, Korea University
E-mail [email protected]
Department of Physics, Korea University
E-mail
###### Abstract:
We compute the relativistic corrections to the color-singlet contribution to
gluon fragmentation into a $J/\psi$ at relative order $v^{4}$, making use of
the nonrelativistic QCD (NRQCD) factorization approach. The corresponding
full-QCD process exhibits infrared divergences that manifest themselves as
single and double poles in $\epsilon$ in $4-2\epsilon$ dimensions. We isolate
the infrared-divergent contributions and treat them analytically. In the
matching of full QCD to NRQCD, the pole contributions are absorbed into long-
distance NRQCD matrix elements. The renormalizations of the ultraviolet
divergences of the long-distance NRQCD matrix elements involve Born and one-
loop single-pole counterterm contributions and Born double-pole counterterm
contributions. While the order-$v^{4}$ contribution enhances the $J/\psi$
hadroproduction rate for the color-singlet channel substantially, this
contribution is not important numerically in comparison with the color-octet
contributions. We also find that the ${}^{3}P_{J}$ color-octet channel in the
gluon fragmentation function contributes to $J/\psi$ hadroproduction
significantly in comparison with the complete contribution of next-to-leading
order in $\alpha_{s}$ in that channel.
## 1 Introduction
In this proceedings contribution, we summarize the computation of the
fragmentation of a gluon into a spin-triplet $S$-wave quarkonium via the
${}^{3}S_{1}^{[1]}$ channel in relative order $v^{4}$, where $v$ is the
velocity of the heavy quark $Q$ or heavy antiquark $\bar{Q}$ in the quarkonium
rest frame. We refer the reader to Ref. [1] for details of this calculation.
The expression ${}^{2s+1}l_{j}^{[c]}$ is the standard spectroscopic notation
for the spin $s$, orbital angular momentum $l$, total angular momentum $j$,
and color $c=1$ (singlet) or 8 (octet) of the $Q\bar{Q}$ pair that is created
at short distances and that evolves into the heavy quarkonium.
Our computation make use of the nonrelativistic QCD (NRQCD) factorization
approach [2]. That is, we express the fragmentation functions as sums of
products of NRQCD long-distance matrix elements (LDMEs) and short-distance
coefficients. The focus of our calculation is the short-distance coefficient
that appears in the order $v^{4}$ contribution to the fragmentation function
for the ${}^{3}S_{1}^{[1]}$ channel. The short-distance coefficients for gluon
fragmentation in this channel in relative orders $v^{0}$ and $v^{2}$ have
already been computed in Refs. [3, 4] and [5], respectively. The contribution
of relative order $v^{4}$ is interesting theoretically because it is at this
order that the ${}^{3}S_{1}^{[1]}$ fragmentation channel first develops soft
divergences. These soft divergences are ultimately absorbed into the NRQCD
LDME for the evolution of a ${}^{3}S_{1}^{[8]}$ or ${}^{3}P_{J}^{[8]}$
$Q\bar{Q}$ pair into a spin-triplet $S$-wave quarkonium state. Consequently,
the ${}^{3}S_{1}^{[1]}$ contribution in order $v^{4}$ and the
${}^{3}S_{1}^{[8]}$ and ${}^{3}P_{J}^{[8]}$ contributions at the leading
nontrivial order in $v$ are related by logarithms of the factorization scale.
Since the ${}^{3}S_{1}^{[8]}$ contribution to the fragmentation of a gluon
into a $J/\psi$ is known to be significant phenomenologically, it is important
to compute the ${}^{3}S_{1}^{[1]}$ contribution in order $v^{4}$, as well.
An important technical issue in this calculation is the appearance of both
single and double soft poles in the dimensional-regularization parameter
$\epsilon$. This is the first NRQCD factorization calculation in which double
soft poles have appeared. In order to effect the matching between NRQCD and
full QCD that removes these poles, it is necessary to work out, for the first
time, two-loop corrections to NRQCD operators and the corresponding
ultraviolet renormalizations.
We employ the Collins-Soper definition [6] of the fragmentation function for a
gluon fragmenting into a quarkonium. We compute the full-QCD fragmentation
functions for a gluon fragmenting into free $Q\bar{Q}$ states with various
quantum numbers. We then determine the NRQCD short-distance coefficients by
comparing the full-QCD fragmentation functions with the corresponding NRQCD
expressions, making use of the NRQCD factorization formulas.
The remainder of this paper is organized as follows. In Sec. 2, we present the
NRQCD factorization formulas through relative order $v^{4}$ for the
fragmentation functions for a gluon fragmenting into a $J/\psi$. The results
of the full-QCD calculation for the corresponding fragmentation functions for
free $Q\bar{Q}$ states are given in Sec. 3. In Sec. 4, we compute the relevant
NRQCD LDMEs for free $Q\bar{Q}$ states analytically in dimensional
regularization, and we determine the evolution equations for the LDMEs. In
Sec. 5, we compute the short-distance coefficients by matching the NRQCD and
full-QCD results. Sec. 6 contains estimates of the relative sizes of the
various fragmentation contributions to the $J/\psi$ hadroproduction cross
section. We give a summary of our results in Sec. 7.
## 2 Factorization formulas
We denote by $D[g\to{J/\psi}](z,\mu_{\Lambda})$ the fragmentation function for
a gluon fragmenting into a $J/\psi$. Here, $\mu_{\Lambda}$ is the
factorization scale, $z=P^{+}/k^{+}$, and $P$ and $k$ are the momenta of the
${J/\psi}$ and the fragmenting gluon, respectively. We define light-cone
coordinates for a four-vector $V=(V^{+},V^{-},\bm{V}_{\bot})$ by
$V^{\pm}\equiv(V^{0}\pm V^{3})/\sqrt{2}$. We make use of the Collins-Soper
definition of $D[g\to{J/\psi}](z,\mu_{\Lambda})$ [6], which is gauge
invariant, and we regularize soft and ultraviolet divergences dimensionally,
taking $d=4-2\epsilon$ space-time dimensions.
The NRQCD factorization formula for $D[g\to{J/\psi}](z,\mu_{\Lambda})$ is
given by
$D[g\to J/\psi](z)=\sum_{n}d_{n}(z)\langle
0|\mathcal{O}_{n}^{J/\psi}|0\rangle,$ (1)
where the $\langle 0|\mathcal{O}_{n}^{J/\psi}|0\rangle$ are NRQCD LDMEs, the
$d_{n}(z)$ are short-distance coefficients, and we have suppressed the
$\mu_{\Lambda}$ dependences in $d_{n}(z)$ and $\langle
0|\mathcal{O}_{n}^{J/\psi}|0\rangle$. Since the $d_{n}(z)$ are independent of
the hadronic final state, we can write the fragmentation function for a gluon
fragmenting into a free $Q\bar{Q}$ state as
$D[g\to Q\bar{Q}](z)=\sum_{n}d_{n}(z)\langle
0|\mathcal{O}_{n}^{Q\bar{Q}}|0\rangle.$ (2)
By computing the left side of Eq. (2) in full QCD and comparing it with the
right side, we can determine the $d_{n}(z)$. In the remainder of this paper,
we denote the order-$v^{k}$ contribution to $D[g\to H]$ by $D_{k}[g\to H]$,
where $H$ can be either a quarkonium or a free-$Q\bar{Q}$ final state.
The operator LDMEs that we use in this paper are
$\displaystyle\langle 0|\mathcal{O}_{0}^{H}({}^{3}S_{1}^{[1]})|0\rangle$
$\displaystyle=$ $\displaystyle\langle 0|\chi^{\dagger}\sigma^{i}\psi\;{\cal
P}_{H}\;\psi^{\dagger}\sigma^{i}\chi|0\rangle,$ (3) $\displaystyle\langle
0|\mathcal{O}_{2}^{H}({}^{3}S_{1}^{[1]})|0\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}\langle
0|\chi^{\dagger}\sigma^{i}(-\frac{i}{2}\\!\\!\stackrel{{\scriptstyle\leftrightarrow}}{{\bm{D}}})^{2}\psi\;{\cal
P}_{H}\;\psi^{\dagger}\sigma^{i}\chi+\textrm{H.~{}c.}|0\rangle,$ (4)
$\displaystyle\langle 0|{\cal O}_{0}^{H}({}^{1}S_{0}^{[8]})|0\rangle$
$\displaystyle=$ $\displaystyle\langle 0|\chi^{\dagger}T^{a}\psi\;{\cal
P}_{H}\;\psi^{\dagger}T^{a}\chi|0\rangle,$ (5) $\displaystyle\langle 0|{\cal
O}_{4,1}^{H}({}^{3}S_{1}^{[1]})|0\rangle$ $\displaystyle=$
$\displaystyle\langle
0|\chi^{\dagger}\sigma^{i}(-\frac{i}{2}\\!\\!\stackrel{{\scriptstyle\leftrightarrow}}{{\bm{D}}})^{2}\psi\;{\cal
P}_{H}\;\psi^{\dagger}\sigma^{i}(-\frac{i}{2}\\!\\!\stackrel{{\scriptstyle\leftrightarrow}}{{\bm{D}}})^{2}\chi|0\rangle,$
(6) $\displaystyle\langle 0|\mathcal{O}_{4,2}^{H}({}^{3}S_{1}^{[1]})|0\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{2}\langle
0|\chi^{\dagger}\sigma^{i}(-\frac{i}{2}\\!\\!\stackrel{{\scriptstyle\leftrightarrow}}{{\bm{D}}})^{4}\psi\;{\cal
P}_{H}\;\psi^{\dagger}\sigma^{i}\chi+\textrm{H.~{}c.}|0\rangle,$ (7)
$\displaystyle\langle 0|\mathcal{O}_{4,3}^{H}({}^{3}S_{1}^{[1]})|0\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{2}\langle
0|\chi^{\dagger}\sigma^{i}\psi\;{\cal
P}_{H}\;\psi^{\dagger}\sigma^{i}(\stackrel{{\scriptstyle\leftrightarrow}}{{\bm{D}}}\\!\cdot
g_{s}\bm{E}+g_{s}\bm{E}\cdot\\!\stackrel{{\scriptstyle\leftrightarrow}}{{\bm{D}}})\chi$
(8)
$\displaystyle\qquad-\chi^{\dagger}\sigma^{i}(\stackrel{{\scriptstyle\leftrightarrow}}{{\bm{D}}}\\!\cdot
g_{s}\bm{E}+g_{s}\bm{E}\cdot\\!\stackrel{{\scriptstyle\leftrightarrow}}{{\bm{D}}})\psi\;{\cal
P}_{H}\;\psi^{\dagger}\sigma^{i}\chi|0\rangle,$ $\displaystyle\langle 0|{\cal
O}_{0}^{H}({}^{3}S_{1}^{[8]})|0\rangle$ $\displaystyle=$ $\displaystyle\langle
0|\chi^{\dagger}\sigma^{i}T^{a}\psi\;{\cal
P}_{H}\;\psi^{\dagger}\sigma^{i}T^{a}\chi|0\rangle,$ (9) $\displaystyle\langle
0|{\cal O}_{0}^{H}({}^{3}P^{[8]})|0\rangle$ $\displaystyle=$
$\displaystyle\langle
0|\chi^{\dagger}(-\frac{i}{2}\\!\\!\stackrel{{\scriptstyle\leftrightarrow}}{{\bm{D}}})^{r}\sigma^{n}T^{a}\psi\;{\cal
P}_{H}\;\psi^{\dagger}(-\frac{i}{2}\\!\\!\stackrel{{\scriptstyle\leftrightarrow}}{{\bm{D}}})^{r}\sigma^{n}T^{a}\chi|0\rangle,$
(10)
where $g_{s}=\sqrt{4\pi\alpha_{s}}$, $\psi^{\dagger}$ and $\chi$ are two-
component (Pauli) fields that create a heavy quark and a heavy antiquark,
$\bm{D}$ is the gauge-covariant derivative, and $\bm{E}$ is the chromoelectric
field operator. ${\cal P}_{H(P)}=\sum_{X}|H(P)+X\rangle\langle H(P)+X|$ is a
projection onto a state consisting of a quarkonium $H$, with four-momentum
$P$, plus anything. ${\cal P}_{H(P)}$ contains a sum over any quarkonium
polarization quantum numbers that are not specified explicitly. The symmetric
traceless product is defined by
$A^{(i}B^{j)}=\frac{1}{2}(A^{i}B^{j}+A^{j}B^{i})-\frac{1}{d-1}\,\delta^{ij}A^{k}B^{k}$,
and the antisymmetric product is defined by
$A^{[i}B^{j]}=\frac{1}{2}(A^{i}B^{j}-A^{j}B^{i})$.
We now give the NRQCD factorization formulas for gluon fragmentation into a
$J/\psi$ through relative order $v^{4}$. The contributions to $D[g\to J/\psi]$
in relative orders $v^{0}$, $v^{2}$, and $v^{3}$ are
$\displaystyle D_{0}[g\to J/\psi]$ $\displaystyle=$ $\displaystyle d_{0}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]\langle
0|\mathcal{O}_{0}^{J/\psi}({}^{3}S_{1}^{[1]})|0\rangle,$ (11) $\displaystyle
D_{2}[g\to J/\psi]$ $\displaystyle=$ $\displaystyle d_{2}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]\langle
0|\mathcal{O}_{2}^{J/\psi}({}^{3}S^{[1]})|0\rangle,$ (12) $\displaystyle
D_{3}[g\to J/\psi]$ $\displaystyle=$ $\displaystyle d_{0}[g\to
Q\bar{Q}({}^{1}S_{0}^{[8]})]\langle
0|\mathcal{O}_{0}^{J/\psi}({}^{1}S_{0}^{[8]})|0\rangle.$ (13)
The short-distance coefficients $d_{0}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]$ and
$d_{2}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]$ were calculated previously in Refs.
[3, 4] and Ref. [5], respectively. The short-distance coefficient $d_{0}[g\to
Q\bar{Q}({}^{1}S_{0}^{[8]})]$ is related by an overall color factor to
$d_{0}[g\to Q\bar{Q}({}^{1}S_{0}^{[1]})]$, which was calculated in Ref. [7].
The contribution to $D[g\to J/\psi]$ in relative order $v^{4}$ is
$\displaystyle D_{4}[g\to J/\psi]$ $\displaystyle=$
$\displaystyle\big{\\{}\,d_{4,1}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]+d_{4,2}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]\,\big{\\}}\,\langle
0|\mathcal{O}_{4}^{J/\psi}({}^{3}S_{1}^{[1]})|0\rangle$ (14) $\displaystyle+$
$\displaystyle d_{0}[g\to Q\bar{Q}({}^{3}P^{[8]})]\langle
0|\mathcal{O}_{0}^{J/\psi}({}^{3}P^{[8]})|0\rangle$ $\displaystyle+$
$\displaystyle d_{0}[g\to Q\bar{Q}({}^{3}S_{1}^{[8]})]\langle
0|\mathcal{O}_{0}^{J/\psi}({}^{3}S_{1}^{[8]})|0\rangle.$
The short-distance coefficient $d_{0}[g\to Q\bar{Q}({}^{3}S_{1}^{[8]})]$ was
calculated previously in Refs. [5, 7, 8, 9]. The short-distance coefficient
$d_{0}[g\to Q\bar{Q}({}^{3}P^{[8]})]$ is related by an overall color factor to
the sum over $J$ of the short-distance coefficients $d_{0}[g\to
Q\bar{Q}({}^{3}P_{J}^{[1]})]$ that were calculated in Ref. [7]. The
computation of the combination of short-distance coefficients $d_{4,1}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]+d_{4,2}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]$ is the
main goal of this work. 111In fact, the color-singlet contribution to
$D_{4}[g\to J/\psi]$ is a linear combination of the LDMEs $\langle
0|\mathcal{O}_{4,i}^{J/\psi}({}^{3}S_{1}^{[1]})|0\rangle$, for $i=1$–$3$. We
express the contribution of $\langle
0|\mathcal{O}_{4,3}^{J/\psi}({}^{3}S_{1}^{[1]})|0\rangle$ in terms of the
other two LDMEs by making use of the NRQCD equations of motion, and we use the
vacuum-saturation approximation to replace the $\langle
0|\mathcal{O}_{4,i}^{J/\psi}({}^{3}S_{1}^{[1]})|0\rangle$ with a single LDME,
$\langle 0|\mathcal{O}_{4}^{J/\psi}({}^{3}S_{1}^{[1]})|0\rangle$, making an
error of relative order $v^{2}$ [10].
## 3 Full-QCD calculations
We compute the fragmentation functions for free $Q\bar{Q}$ states in full QCD
in $d=4-2\epsilon$ dimensions, multiplying the square of the dimensional-
regularization scale $\mu$ by the factor $e^{\gamma_{\rm\,E}}/(4\pi)$ that is
appropriate to the modified-minimal-subtraction ($\overline{\rm MS}$) scheme.
Here, ${{\gamma}}_{\rm E}$ is the Euler-Mascheroni constant.
The results for $D_{0}[g\to Q\bar{Q}({}^{3}S_{1}^{[8]})]$ and $D_{2}[g\to
Q\bar{Q}({}^{3}P^{[8]})]$ at leading order (LO) in $\alpha_{s}$ and $v$ are
$\displaystyle D_{0}[g\to Q\bar{Q}({}^{3}S_{1}^{[8]})]$ $\displaystyle=$
$\displaystyle\frac{\pi\alpha_{s}}{m^{3}}\left(\frac{\mu^{2}}{4\pi}e^{{{\gamma}}_{\rm
E}}\right)^{\epsilon}\delta(1-z),$ (15) $\displaystyle D_{2}[g\to
Q\bar{Q}({}^{3}P^{[8]})]$ $\displaystyle=$
$\displaystyle\frac{8\alpha_{s}^{2}\,\bm{q}^{2}}{(d-1)m^{5}}\,\frac{N_{c}^{2}-4}{4N_{c}}(1-\epsilon)\Gamma(1+\epsilon)\left(\frac{\mu^{2}}{4\pi}e^{{{\gamma}}_{\rm
E}}\right)^{\epsilon}\left(\frac{\mu^{2}}{4m^{2}}e^{{{\gamma}}_{\rm
E}}\right)^{\epsilon}$ (16) $\displaystyle\times\left[-\frac{1}{2\epsilon_{\rm
IR}}\,\delta(1-z)+f(z)\right],$
where $N_{c}$ is the number of colors, $\bm{q}$ is half the relative momentum
of the $Q$ and $\bar{Q}$ in the $Q\bar{Q}$ rest frame, and the finite function
$f(z)$ is defined in Eq. (5.20) of Ref.[1]. The subscript “IR” in
$\epsilon_{\rm IR}^{-1}$ indicates that the pole is infrared in origin.
The result for $D_{4}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]$ at LO in $\alpha_{s}$
is
$D_{4}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]=D_{4}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{\rm finite}+I[S_{12}]+I[S_{1}]+I[S_{2}],$ (17)
where $I[S_{12}]$, $I[S_{1}]$, and $I[S_{2}]$ contain the double- and single-
pole contributions:
$\displaystyle I[S_{12}]$ $\displaystyle=$
$\displaystyle\left\\{\frac{1}{8\epsilon_{\rm
IR}^{2}}\,\delta(1-z)-\frac{1}{2\epsilon_{\rm
IR}}\left[\frac{1}{(1-z)^{1+4\epsilon}}\right]_{+}+\frac{1-z^{1+2\epsilon}}{2\epsilon_{\rm
IR}(1-z)^{1+4\epsilon}}\right\\}$ (18)
$\displaystyle\times\left(\frac{8\alpha_{s}}{3\pi
m^{2}}\right)^{2}\frac{N_{c}^{2}-4}{16N_{c}^{2}}\,\frac{\pi\alpha_{s}}{(d-1)m^{3}}\left(\frac{\mu^{2}}{4\pi}e^{{{\gamma}}_{\rm
E}}\right)^{\epsilon}\frac{\bm{q}^{4}}{d-1}$
$\displaystyle\times\left(\frac{\mu^{2}}{4m^{2}}e^{{{\gamma}}_{\rm
E}}\right)^{2\epsilon}\frac{\Gamma^{2}(1+\epsilon)\Gamma^{2}(1-2\epsilon)}{\Gamma(1-4\epsilon)}(1-\epsilon)(6-2\epsilon-\epsilon^{2}-2\epsilon^{3}),$
$\displaystyle I[S_{1}]$ $\displaystyle=$ $\displaystyle
I[S_{2}]=\left(-\frac{\tau_{1}}{2\epsilon_{\rm
IR}}+\tau_{0}\right)\left(\frac{\mu^{2}}{4m^{2}}e^{{{\gamma}}_{\rm
E}}\right)^{2\epsilon}\frac{z^{-2+2\epsilon}(1-z)^{-4\epsilon}\Gamma^{2}(1+\epsilon)}{48(1-\epsilon)}$
(19) $\displaystyle\times$ $\displaystyle\left(\frac{8\alpha_{s}}{3\pi
m^{2}}\right)^{2}\frac{N_{c}^{2}-4}{16N_{c}^{2}}\left(\frac{\mu^{2}}{4\pi}e^{{{\gamma}}_{\rm
E}}\right)^{\epsilon}\frac{\pi\alpha_{s}}{(d-1)m^{3}}\frac{\bm{q}^{4}}{d-1}+O(\epsilon).$
Here, $\tau_{1}$ and $\tau_{0}$ are functions of $z$ that are defined in Eq.
(5.29) of Ref.[1]. $D_{4}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]^{\rm finite}$ in
Eq. (17) is a finite contribution, which we evaluate numerically in $d=4$
dimensions.
## 4 NRQCD LDMEs
In this section we tabulate our results for the NRQCD LDMEs for free
$Q\bar{Q}({}^{3}S_{1}^{[1]})$ states that are relevant through relative order
$v^{4}$.
The free $Q\bar{Q}$ matrix elements at order $\alpha_{s}^{0}$ are normalized
as
$\displaystyle\langle
0|\mathcal{O}_{0}^{Q\bar{Q}({}^{1}S_{0}^{[8]})}({}^{1}S_{0}^{[8]})|0\rangle^{(0)}$
$\displaystyle=$ $\displaystyle(N_{c}^{2}-1),$ (20) $\displaystyle\langle
0|\mathcal{O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[1]})|0\rangle^{(0)}$
$\displaystyle=$ $\displaystyle 2(d-1)N_{c},$ (21) $\displaystyle\langle
0|\mathcal{O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[8]})}({}^{3}S_{1}^{[8]})|0\rangle^{(0)}$
$\displaystyle=$ $\displaystyle(d-1)(N_{c}^{2}-1),$ (22) $\displaystyle\langle
0|\mathcal{O}_{2}^{Q\bar{Q}({}^{3}S_{1}^{[n]})}({}^{3}S_{1}^{[n]})|0\rangle^{(0)}$
$\displaystyle=$ $\displaystyle\bm{q}^{2}\langle
0|\mathcal{O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[n]})}({}^{3}S_{1}^{[n]})|0\rangle^{(0)},$
(23) $\displaystyle\langle
0|\mathcal{O}_{4}^{Q\bar{Q}({}^{3}S_{1}^{[n]})}({}^{3}S_{1}^{[n]})|0\rangle^{(0)}$
$\displaystyle=$ $\displaystyle\bm{q}^{4}\langle
0|\mathcal{O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[n]})}({}^{3}S_{1}^{[n]})|0\rangle^{(0)},$
(24) $\displaystyle\langle
0|\mathcal{O}_{0}^{Q\bar{Q}({}^{3}P^{[8]})}({}^{3}P^{[8]})|0\rangle^{(0)}$
$\displaystyle=$ $\displaystyle\bm{q}^{2}(d-1)(N_{c}^{2}-1),$ (25)
where there is an implied sum over final-state polarizations, and the
superscript $(k)$ indicates the order in $\alpha_{s}$.
In order $\alpha_{s}$ and order $\alpha_{s}^{2}$, the relevant LDMEs mix:
$\displaystyle\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}P^{[8]})|0\rangle_{\overline{\rm
MS}}^{(1)}$ $\displaystyle=$ $\displaystyle\frac{8\alpha_{s}}{3\pi
m^{2}}\left(\frac{-1}{2\epsilon_{\rm
IR}}\right)\frac{N_{c}^{2}-1}{4N_{c}^{2}}\,\frac{1}{d-1}\langle 0|{\cal
O}_{4,1}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[1]})|0\rangle^{(0)},$
(26) $\displaystyle\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}P^{[8]})}({}^{3}S_{1}^{[8]})|0\rangle_{\overline{\rm
MS}}^{(1)}$ $\displaystyle=$ $\displaystyle\frac{8\alpha_{s}}{3\pi
m^{2}}\left(\frac{-1}{2\epsilon_{\rm
IR}}\right)\frac{N_{c}^{2}-4}{4N_{c}}\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}P^{[8]})}({}^{3}P^{[8]})|0\rangle^{(0)},\phantom{xxx}$
(27) $\displaystyle\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[8]})|0\rangle_{\overline{\rm
MS}}^{(2)}$ $\displaystyle=$ $\displaystyle\frac{1}{8\epsilon_{\rm
IR}^{2}}\left(\frac{8\alpha_{s}}{3\pi
m^{2}}\right)^{2}\frac{(N_{c}^{2}-1)(N_{c}^{2}-4)}{16N_{c}^{3}}\frac{1}{d-1}\langle
0|{\cal
O}_{4,1}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[1]})|0\rangle^{(0)}.$
Here, the subscript $\overline{\rm MS}$ indicates that we have removed the
ultraviolet poles in $\varepsilon$ by using the minimal-subtraction procedure,
with the choice of scale that is appropriate to $\overline{\rm MS}$
subtraction. The renormalization subtractions for $\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[8]})|0\rangle^{(2)}$
involve both Born diagrams for the double-pole counterterm and one-loop
diagrams for the single-pole counterterm. In the minimal subtraction
procedure, one subtracts only pure pole contributions in the ultraviolet-
divergent subdiagrams. It is essential for the consistency of the minimal-
subtraction program to treat factors that are external to those divergent
subdiagrams, such as angular-momentum projections involving external momenta,
exactly in $d=4-2\varepsilon$ dimensions. A failure to follow this procedure
in the presence of poles in $d=4-2\varepsilon$ of order two or higher can
result in the appearance of uncanceled poles in $\varepsilon$ in the NRQCD
short-distance coefficients. We refer the reader to Ref. [1] for details.
The dimensional-regularization scale $\mu$ can be identified with the NRQCD
factorization scale $\mu_{\Lambda}$. It then follows from
$d\alpha_{s}/d\log(\mu_{\Lambda})=-2\varepsilon\alpha_{s}+O(\alpha_{s}^{2})$
that the renormalization-group evolution equations for the LDMEs are
$\displaystyle\frac{d}{d\log\mu_{\Lambda}}\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}P^{[8]})|0\rangle_{\overline{\rm
MS}}^{(1)}$ $\displaystyle=$ $\displaystyle\frac{8\alpha_{s}}{3\pi
m^{2}}\,\frac{N_{c}^{2}-1}{4N_{c}^{2}}\,\frac{1}{d-1}\langle 0|{\cal
O}_{4,1}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[1]})|0\rangle^{(0)},$
(29) $\displaystyle\frac{d}{d\log\mu_{\Lambda}}\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}P^{[8]})}({}^{3}S_{1}^{[8]})|0\rangle_{\overline{\rm
MS}}^{(1)}$ $\displaystyle=$ $\displaystyle\frac{8\alpha_{s}}{3\pi
m^{2}}\frac{N_{c}^{2}-4}{4N_{c}}\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}P^{[8]})}({}^{3}P^{[8]})|0\rangle^{(0)},\phantom{xxx}$
(30) $\displaystyle\frac{d}{d\log\mu_{\Lambda}}\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[8]})|0\rangle_{\overline{\rm
MS}}^{(2)}$ $\displaystyle=$ $\displaystyle\frac{8\alpha_{s}}{3\pi
m^{2}}\,\frac{N_{c}^{2}-4}{4N_{c}}\langle 0|{\cal
O}_{0}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}P^{[8]})|0\rangle_{\overline{\rm
MS}}^{(1)}.$ (31)
The result in Eq. (29) agrees with the corresponding results of Refs. [11,
12]. Equations (30) and (31) agree with the result in Eq. (B19b) of Ref. [2]
at the leading nontrivial order in $v$ and with the corresponding result in
Ref [12], but disagree with the corresponding result in Ref. [11].
## 5 Short-distance coefficients
By making use of the results of Secs. 3 and 4 and the free-$Q\bar{Q}$ versions
of the NRQCD factorization (matching) equations (11), (12), and (14) in Sec.
2, we obtain
$\displaystyle d_{4,1}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}+d_{4,2}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}$ $\displaystyle\qquad=\,\,d_{4}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{\rm
finite}\,\,+\,\,\frac{2\alpha_{s}^{3}(N_{c}^{2}-4)}{3\pi(d-1)^{3}N_{c}^{3}m^{7}}\,\,\bigg{\\{}\,\,\delta(1-z)\,\bigg{(}\frac{1}{24}-\frac{\pi^{2}}{6}$
$\displaystyle\phantom{xxxxx}-\frac{1}{3}\log\frac{\mu_{\Lambda}}{2m}+\log^{2}\frac{\mu_{\Lambda}}{2m}\bigg{)}+\left(\frac{1}{1-z}\right)_{\\!\\!+}\\!\\!\bigg{(}\frac{1}{3}-2\log\frac{\mu_{\Lambda}}{2m}\bigg{)}+2\left[\frac{\log(1-z)}{1-z}\right]_{\\!+}\\!$
$\displaystyle\phantom{xxxxx}-\frac{104-29z-10z^{2}}{24}+\frac{7[z+(1+z)\log(1-z)]}{2z^{2}}+\frac{(1-2z)(8-5z)}{4}\log\frac{\mu_{\Lambda}}{2m}$
$\displaystyle\phantom{xxxxx}+\frac{1+z}{4}\left(31-6z-\frac{36}{z}\right)\log(1-z)-\frac{z}{4}\left(\\!39-6z+\frac{8}{1-z}\\!\right)\log
z$
$\displaystyle\phantom{xxxxx}+\frac{13-7z}{2}\bigg{[}\,\bigg{(}\\!\log\frac{1-z}{z^{2}}-\log\frac{\mu_{\Lambda}}{2m}\bigg{)}\log(1-z)-\textrm{Li}_{2}(z)\,\bigg{]}\,\,\bigg{\\}}+O(\epsilon).$
(32)
Here, $d_{4}[g\to Q\bar{Q}({}^{3}S_{1}^{[1]})]^{\rm finite}={D_{4}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{\rm finite}}/{\langle
0|\mathcal{O}_{4}^{Q\bar{Q}({}^{3}S_{1}^{[1]})}({}^{3}S_{1}^{[1]})|0\rangle^{(0)}}$
is evaluated by carrying out the integrations over the phase space
numerically. The results of the numerical integration are shown in Fig. 1. One
can find details of the matching procedure in Sec. 7 of Ref. [1].
Figure 1: The color-singlet short-distance coefficients $d_{0}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}(z)$, $d_{2}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}(z)$ and $d_{4}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{\rm finite}(z)$, which are defined in Eqs. (11),
(12) and (5), respectively, as functions of $z$. The scaling factors are
$(\mathcal{N}_{0},\mathcal{N}_{2},\mathcal{N}_{4})=(10^{-3}\times\alpha_{s}^{3}/m^{3},10^{-2}\times\alpha_{s}^{3}/m^{5},10^{-2}\times\alpha_{s}^{3}/m^{7})$.
## 6 Relative sizes of the fragmentation contributions
We now make estimates of the relative sizes of the contributions of the
various fragmentation functions to the cross section for $J/\psi$ production
in hadron-hadron collisions. At large $p_{T}$, the differential cross section
for a gluon can be approximated as ${d\sigma_{g}}/{d{{p}}_{T}}\propto
1/{{p}}_{T}^{\kappa}$, where $\kappa$ is a fixed power. Then, the
fragmentation contribution to the $J/\psi$ production rate becomes
$d\sigma_{J/\psi}^{\rm frag}/d{{p}}_{T}\propto I_{\kappa}(D)$, where
$I_{\kappa}(D)=\int_{0}^{1}\\!dz\,z^{\kappa}\,D(z)$ [1]. Hence, we can obtain
a rough estimate of the relative contribution of a fragmentation process to
the cross section by computing $I_{\kappa}(D)$. At large $p_{{}_{T}}$, the
cross section at LO in $\alpha_{s}$ is dominated by fragmentation of a gluon
into a $J/\psi$ with $z=1$ [13]. The next-to-leading order (NLO) $k$ factor
for this channel is essentially independent of $p_{{}_{T}}$ for the cross
sections that were measured at the Tevatron [14]. Making use of these facts
and the result for the ${}^{3}S_{1}^{[8]}$ contribution to
$d\sigma/dp_{{}_{T}}$ at NLO in $\alpha_{s}$ that appears in Fig. 1(c) of Ref.
[15], we have determined that $\kappa\approx 5.2$.
We can compare the relative contributions to $d\sigma_{J/\psi}^{\rm frag}$ of
each of the three $Q\bar{Q}$ channels that contribute to $D[g\to J/\psi]$ in
order $v^{4}$ [Eq. (14)]. For each channel, we compute $I_{5.2}(D)$. Then, we
multiply by the following LDMEs: $\langle
0|\mathcal{O}_{4}^{J/\psi}({}^{3}S_{1}^{[1]})|0\rangle=m^{4}\langle
v^{2}\rangle^{2}\times 1.32~{}{\rm GeV}^{3}$, $\langle
0|\mathcal{O}_{0}^{J/\psi}({}^{3}P^{[8]})|0\rangle=-0.109~{}{\rm GeV}^{5}$,
$\langle 0|\mathcal{O}_{0}^{J/\psi}({}^{3}S_{1}^{[8]})|0\rangle=3.12\times
10^{-3}~{}{\rm GeV}^{3}$, $\langle
0|\mathcal{O}_{0}^{J/\psi}({}^{1}S_{0}^{[8]})|0\rangle=4.50\times
10^{-2}~{}{\rm GeV}^{3}$. We have obtained these LDMEs from Ref. [15]. In the
case of the first LDME, we have used the generalized Gremm-Kapustin relation
[16] $\langle
0|\mathcal{O}_{4}^{J/\psi}({}^{3}S_{1}^{[1]})|0\rangle=m^{4}\langle
v^{2}\rangle^{2}\langle
0|\mathcal{O}_{0}^{J/\psi}({}^{3}S_{1}^{[1]})|0\rangle$ to obtain the matrix
element of order $v^{4}$ from the matrix element of order $v^{0}$ that appears
in Ref. [15]. We use the value of $m^{2}\langle v^{2}\rangle$ from Table I of
Ref. [16]: $m^{2}\langle v^{2}\rangle=0.437\,\textrm{GeV}^{2}$ at
$m=1.5\,\textrm{GeV}$. The results of this computation are shown in Table 1.
We see from the second row of Table 1 that the $Q\bar{Q}({}^{3}S_{1}^{[1]})$
channel makes a small contribution to $D_{4}[g\to J/\psi]$ at ${p}_{T}=20$
GeV, confirming that this channel is not important phenomenologically at the
current level of precision. We also see that the $Q\bar{Q}({}^{3}S_{1}^{[8]})$
and $Q\bar{Q}({}^{3}P^{[8]})$ channels give comparable contributions to
$D_{4}[g\to J/\psi]$ at ${p}_{T}=20$ GeV. We estimate that the fragmentation
contribution to $d\sigma/d{p}_{T}\times B(J/\psi\to\mu\mu)$ at ${p}_{T}=20$
GeV from the $Q\bar{Q}({}^{3}P^{[8]})$ channel is about $6\times 10^{-3}$
nb/GeV [1], which is comparable to (about a factor of $2$ larger than) the
total NLO contribution in the $Q\bar{Q}({}^{3}P^{[8]})$ channel in Fig. 1(c)
of Ref. [15]. A more precise calculation of the fragmentation contribution in
the $Q\bar{Q}({}^{3}P^{[8]})$ channel will be necessary in order to determine
whether it is the dominant contribution in that channel at NLO in $\alpha_{s}$
at high $p_{{}_{T}}$.
Table 1: Relative contributions to $d\sigma_{J/\psi}^{\rm frag}$ in order $v^{4}$. The first and second rows give $I_{5.2}(d)$ times the LDMEs that are given in the text. We take $m=m_{c}=1.5\,\textrm{GeV}$. For compatibility with Ref. [15], we take $\alpha_{s}=\alpha_{s}({{m}}_{T})$, where ${{m}}_{T}=\sqrt{{{p}}_{T}^{2}+4m_{c}^{2}}$. We choose the point ${{p}}_{T}=20$ GeV, which implies that $\alpha_{s}({{m}}_{T})=0.154$. $I_{\kappa}(d)\,\backslash\,$channel | $\\!\\!\\!\\!\mathcal{O}_{0}^{J/\psi}({}^{1}S_{0}^{[8]})$ | | $\\!\\!\\!\\!\mathcal{O}_{0}^{J/\psi}({}^{3}S_{1}^{[8]})$ | $\mathcal{O}_{0}^{J/\psi}({}^{3}P^{[8]})$ | $\mathcal{O}_{4}^{J/\psi}({}^{3}S_{1}^{[1]})$
---|---|---|---|---|---
$I_{5.2}(d)|_{\mu_{\Lambda}=m\phantom{2}}\times\,10^{6}\,\textrm{LDME}$ | $\phantom{12}1.65$ | | $\phantom{59}18.6$ | $\phantom{-1}32.7$ | $\phantom{-}0.192$
$I_{5.2}(d)|_{\mu_{\Lambda}=2m}\times\,10^{6}\,\textrm{LDME}$ | $\phantom{12}1.65$ | | $\phantom{59}18.6$ | $\phantom{-1}43.6$ | $\phantom{-}0.455$
## 7 Summary
We have computed NRQCD short-distance coefficients for gluon fragmentation
into a $J/\psi$ through relative order $v^{4}$. Our main result is the
expression in Eq. (5) for $d_{4,1}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}+d_{4,2}[g\to
Q\bar{Q}({}^{3}S_{1}^{[1]})]^{(3)}$, which is the sum of short-distance
coefficients for the ${}^{3}S_{1}^{[1]}$ NRQCD LDMEs of relative order
$v^{4}$. This is the first time that double soft divergences, as well as
single soft divergences, have appeared in intermediate steps in the
calculation of NRQCD short-distance coefficients. We have also computed the
free $Q\bar{Q}$ NRQCD LDMEs that appear when one uses the matching equations
between full QCD and NRQCD to compute the NRQCD short-distance coefficients.
Our perturbative calculations of these LDMEs involve both one-loop and two-
loop corrections. The ultraviolet renormalization-subtractions for these LDMEs
include Born and one-loop single-pole counterterm contributions and Born
double-pole counterterm contributions. We have worked out the renormalization-
group evolution of the renormalized LDMEs and confirm some previous results in
Refs. [2, 11, 12], but disagree with a result in Ref. [11].
We have estimated the relative sizes of the contributions of the various
channels to gluon fragmentation into a $J/\psi$ through relative order
$v^{4}$. The contribution in order $v^{4}$ of the ${}^{3}S_{1}^{[1]}$ channel
to the cross section at $p_{{}_{T}}=20$ GeV is about a factor of $2$ $(4)$
larger than the contribution in order $v^{0}$ when $\mu_{\Lambda}=m_{c}$
$(2m_{c})$. In spite of this large enhancement of the fragmentation
contribution in order $v^{4}$, the corresponding contribution to the $J/\psi$
production cross section at the Tevatron or the LHC is not important at the
current level of precision.
In the ${}^{3}S_{1}^{[8]}$ channel, most of the contribution to the $J/\psi$
production cross section at hadron-hadron colliders at large $p_{{}_{T}}$
arises from the fragmentation contribution. This is true at LO in $\alpha_{s}$
[13] and at NLO in $\alpha_{s}$ [14]. Our estimate of the fragmentation
contribution to the ${}^{3}P_{J}^{[8]}$ channel indicates that it is an
important part of the contribution in that channel to the high-$p_{{}_{T}}$
$J/\psi$ production cross section at hadron-hadron colliders at NLO in
$\alpha_{s}$. However, it will be necessary to carry out a more precise
calculation of the fragmentation contribution in the ${}^{3}P_{J}^{[8]}$
channel in order to see whether it is actually dominant at high $p_{{}_{T}}$.
###### Acknowledgments.
The work of G.T.B. in the High Energy Physics Division at Argonne National
Laboratory was supported by the U. S. Department of Energy, Division of High
Energy Physics, under Contract No. DE-AC02-06CH11357.
## References
* [1] G. T. Bodwin, U-R. Kim and J. Lee, JHEP 1211 (2012) 020 [arXiv:1208.5301 [hep-ph]].
* [2] G.T. Bodwin, E. Braaten and G.P. Lepage, Phys. Rev. D 51(1995) 1125; 55 (1997) 5853 (E) [hep-ph/9407339].
* [3] E. Braaten and T.C. Yuan, Phys. Rev. Lett. 71 (1993) 1673 [hep-ph/9303205].
* [4] E. Braaten and T.C. Yuan, Phys. Rev. D 52 (1995) 6627 [hep-ph/9507398].
* [5] G.T. Bodwin and J. Lee, Phys. Rev. D 69 (2004) 054003 [hep-ph/0308016].
* [6] J.C. Collins and D.E. Soper, Nucl. Phys. B 194 (1982) 445.
* [7] E. Braaten and Y.-Q. Chen, Phys. Rev. D 55 (1997) 2693 [hep-ph/9610401].
* [8] E. Braaten and J. Lee, Nucl. Phys. B 586 (2000) 427 [hep-ph/0004228].
* [9] J. Lee, Phys. Rev. D 71 (2005) 094007 [hep-ph/0504285]
* [10] G.T. Bodwin and A. Petrelli, Phys. Rev. D 66 (2002) 094011. [hep-ph/0205210].
* [11] M. Gremm and A. Kapustin, Phys. Lett. B 407 (1997) 323 [hep-ph/9701353].
* [12] Zhi-Guo He, private communication.
* [13] P. Cho and A.K. Leibovich, Phys. Rev. D 53 (1996) 150 [hep-ph/9505329].
* [14] B. Gong, X.Q. Li and J.-X. Wang, Phys. Lett. B 673 (2009) 197; 693 (2010) 612 (E) [arXiv:0805.4751 [hep-ph]].
* [15] M. Butenschoen and B.A. Kniehl, Phys. Rev. Lett. 106 (2011) 022003 [arXiv:1009.5662 [hep-ph]].
* [16] G. T. Bodwin, H. S. Chung, D. Kang, J. Lee and C. Yu, Phys. Rev. D 77 (2008) 094017 [arXiv:0710.0994 [hep-ph]].
|
arxiv-papers
| 2013-01-16T00:01:56 |
2024-09-04T02:49:40.364085
|
{
"license": "Public Domain",
"authors": "Geoffrey T. Bodwin (Argonne), U-Rae Kim, Jungil Lee (Korea U.)",
"submitter": "Jungil Lee",
"url": "https://arxiv.org/abs/1301.3532"
}
|
1301.3559
|
# Separation of variables in an asymmetric cyclidic coordinate system
H S Cohl1 and H Volkmer2 1Applied and Computational Mathematics Division,
National Institute of Standards and Technology, Gaithersburg, MD 20899-8910,
USA 2Department of Mathematical Sciences, University of Wisconsin–Milwaukee,
P. O. Box 413, Milwaukee, WI 53201, USA
###### Abstract
A global analysis is presented of solutions for Laplace’s equation on three-
dimensional Euclidean space in one of the most general orthogonal asymmetric
confocal cyclidic coordinate systems which admit solutions through separation
of variables. We refer to this coordinate system as five-cyclide coordinates
since the coordinate surfaces are given by two cyclides of genus zero which
represent the inversion at the unit sphere of each other, a cyclide of genus
one, and two disconnected cyclides of genus zero. This coordinate system is
obtained by stereographic projection of sphero-conal coordinates on four-
dimensional Euclidean space. The harmonics in this coordinate system are given
by products of solutions of second-order Fuchsian ordinary differential
equations with five elementary singularities. The Dirichlet problem for the
global harmonics in this coordinate system is solved using multiparameter
spectral theory in the regions bounded by the asymmetric confocal cyclidic
coordinate surfaces.
###### pacs:
41.20.Cv, 02.30.Em, 02.30.Hq, 02.30.Jr
###### ams:
35J05, 42C15, 34L05, 35J15, 34B30
## 1 Introduction
In 1894 Maxime Bôcher’s book “Ueber die Reihenentwicklungen der
Potentialtheorie” was published [2]. It took its origin from lectures given by
Felix Klein in Göttingen (see for instance [7, 8]). In Bôcher’s book, the
author gives a list of 17 inequivalent coordinate systems in three dimensions
in which the Laplace equation admits separated solutions of the form
$U(x,y,z)=R(x,y,z)w_{1}(s_{1})w_{2}(s_{2})w_{3}(s_{3}),$ (1)
where the modulation factor $R(x,y,z)$ [11, p. 519] is a known and fixed
function, and $s_{1},s_{2},s_{3}$ are curvilinear coordinates of $x,y,z$. The
functions $w_{1},w_{2},w_{3}$ are solutions of second order ordinary
differential equations. The symmetry group of Laplace’s equation is the
conformal group and equivalence between various separable coordinate systems
is established by the existence of a conformal transformation which maps one
separable coordinate system to another.
In general, the coordinate surfaces (called confocal cyclides) are given by
the zero sets of polynomials in $x,y,z$ of degree at most $4$ which can be
broken up into several different subclasses. For instance, eleven of these
coordinate systems have coordinate surfaces which are given by confocal
quadrics [9, Systems 1–11 on p. 164], nine are rotationally-invariant [9,
Systems 2,5–8 on p. 164 and 14–17 on p. 210], four are cylindrical [9, Systems
1–4], and five which are the most general, are of the asymmetric type namely,
confocal ellipsoidal, paraboloidal, sphero-conal, and two cyclidic coordinate
systems [9, Systems 9,10,11 on p. 164 and Systems 12,13 on p. 210]). Bôcher
[2] showed how to solve the Dirichlet problem for harmonic functions on
regions bounded by such confocal cyclides. However, it is stated repeatedly in
Bôcher’s book that the presentation lacked convergence proofs, for instance,
this is mentioned in the preface written by Felix Klein.
It is the purpose of this paper to supply the missing proofs for one of the
asymmetric cyclidic coordinate systems which is listed as number 12 in
Miller’s list [9, page 210] (see also Boyer, Kalnins & Miller (1976) [3, Table
2] and Boyer, Kalnins & Miller (1978) [4] for a more general setting). For
lack of a better name we call it 5-cyclide coordinates. This asymmetric
orthogonal curvilinear coordinate system has coordinates
$s_{i}\in{\mathbf{R}}$ $(i=1,2,3)$ with $s_{i}$ in $(a_{0},a_{1})$,
$(a_{1},a_{2})$ or $(a_{2},a_{3})$, respectively where
$a_{0}<a_{1}<a_{2}<a_{3}$ are given numbers. This coordinate system is
described by coordinate surfaces $s_{i}=const$ which are five compact
cyclides. The surfaces $s_{1}=const$ for $s_{1}\in(a_{0},a_{1})$ are two
cyclides of genus zero representing inversions at the unit sphere of each
other. The surface $s_{2}=const$ for $s_{2}\in(a_{1},a_{2})$ represents a ring
cyclide of genus one and the surfaces $s_{3}=const$ for
$s_{3}\in(a_{2},a_{3})$ represent two disconnected cyclides of genus zero with
reflection symmetry about the $x,y$-plane. The asymptotic behavior of this
coordinate system as the size of these compact cyclides increases without
limit is 6-sphere coordinates (see Moon & Spencer (1961) [10, p. 122]), the
inversion of Cartesian coordinates.
In our notation the coordinate surfaces of this system are given by the
variety
$\frac{(x^{2}+y^{2}+z^{2}-1)^{2}}{s-a_{0}}+\frac{4x^{2}}{s-a_{1}}+\frac{4y^{2}}{s-a_{2}}+\frac{4z^{2}}{s-a_{3}}=0,$
(2)
where $s=s_{i}$ is either in $(a_{0},a_{1})$, $(a_{1},a_{2})$ or
$(a_{2},a_{3})$ respectively.
Figure 1: Surfaces $s_{1,2,3}=const$ for $a_{i}=i,$ where only the component
of the cyclide $s_{1}=const$ inside the ball $x^{2}+y^{2}+z^{2}<1$ is shown.
See Figure 1(a,b) for a graphical illustration of these triply-orthogonal
coordinate surfaces, where we have selected one of the confocal cyclides for
$s_{1}=const$. This is a very general coordinate system containing the
parameters $a_{0},a_{1},a_{2},a_{3}$ which generates many other coordinate
systems by limiting processes. Since the book by Bôcher is quite old and uses
very geometrical methods, we will present our results independently of
Bôcher’s book. We supply convergence proofs based on general multiparameter
spectral theory [1, 13] which was created with such applications in mind. As
far as we know this general theory has never before been applied to the
Dirichlet problems considered by Bôcher.
We start with the observation that 5-cyclide coordinates are the stereographic
image of sphero-conal coordinates in four dimensions (or, expressed in another
way, of ellipsoidal coordinates on the hypersphere $\mathbb{S}^{3}$). We take
the sphero-conal coordinate system as known but we present the needed facts in
Section 2. The well-known stereographic projection is dealt with in Section 3
which also explains the appearance of the factor $R$ in (1). The 5-cyclide
coordinate system is introduced in Section 4. The solution of the Dirichlet
problem on regions bounded by surfaces (2) with $s\in(a_{1},a_{2})$ is
presented in Section 6. The preceding Section 5 provides the needed
convergence proofs based on multiparameter spectral theory. The remaining
sections treat the Dirichlet problem on regions bounded by the surfaces (2)
when $s\in(a_{1},a_{2})$ (ring cyclides) and $s\in(a_{2},a_{3})$.
## 2 Sphero-conal coordinates on ${\mathbf{R}}^{k+1}$
Let $k\in{\mathbf{N}}$. In order to introduce sphero-conal coordinates on
${\mathbf{R}}^{k+1}$, fix real numbers
$a_{0}<a_{1}<a_{2}<\dots<a_{k}.$ (3)
Let $(x_{0},x_{1},\dots,x_{k})$ be in the positive cone of
${\mathbf{R}}^{k+1}$
$x_{0}>0,\ldots,x_{k}>0.$ (4)
Its sphero-conal coordinates $r,s_{1},\dots,s_{k}$ are determined in the
intervals
$r>0,\;a_{i-1}<s_{i}<a_{i},\quad i=1,\dots,k$ (5)
by the equations
$r^{2}=\sum_{j=0}^{k}x_{j}^{2}$ (6)
and
$\sum_{j=0}^{k}\frac{x_{j}^{2}}{s_{i}-a_{j}}=0\quad\mbox{for\ }i=1,\dots,k.$
(7)
The latter equation determines $s_{1},s_{2},\dots,s_{k}$ as the zeros of a
polynomial of degree $k$ with coefficients which are polynomials in
$x_{0}^{2},\dots,x_{k}^{2}$.
In this way we obtain a bijective (real-)analytic map from the positive cone
in ${\mathbf{R}}^{k+1}$ to the set of points $(r,s_{1},\dots,s_{k})$
satisfying (5). The inverse map is found by solving a linear system. It is
also analytic, and it is given by
$x_{j}^{2}=r^{2}\frac{\prod_{i=1}^{k}(s_{i}-a_{j})}{\prod_{j\neq
i=0}^{k}(a_{i}-a_{j})}.$ (8)
Sphero-conal coordinates are orthogonal, and its scale factors (metric
coefficients) are given by $H_{r}=1$, and
$H_{s_{i}}^{2}=\frac{1}{4}\sum_{j=0}^{k}\frac{x_{j}^{2}}{(s_{i}-a_{j})^{2}}=-\frac{1}{4}r^{2}\frac{\prod_{i\neq
j=1}^{k}(s_{i}-s_{j})}{\prod_{j=0}^{k}(s_{i}-a_{j})},\quad i=1,2,\dots,k.$ (9)
Consider the Laplace equation
$\Delta U=\sum_{i=0}^{k}\frac{\partial^{2}U}{\partial x_{i}^{2}}=0$ (10)
for a function $U(x_{0},x_{1},\dots,x_{k})$. Using (9) we transform this
equation to sphero-conal coordinates, and then we apply the method of
separation of variables [12]
$U(x_{0},x_{1},\dots,x_{k})=w_{0}(r)w_{1}(s_{1})w_{2}(s_{2})\dots
w_{k}(s_{k}).$ (11)
For the variable $r$ we obtain the Euler equation
$w_{0}^{\prime\prime}+\frac{k}{r}w_{0}^{\prime}+\frac{4\lambda_{0}}{r^{2}}w_{0}=0$
(12)
while for each of the variables $s_{1},s_{2},\dots,s_{k}$ we obtain the
Fuchsian equation
$\prod_{j=0}^{k}(s-a_{j})\left[w^{\prime\prime}+\frac{1}{2}\sum_{j=0}^{k}\frac{1}{s-a_{j}}w^{\prime}\right]+\left[\sum_{i=0}^{k-1}\lambda_{i}s^{k-1-i}\right]w=0.$
(13)
More precisely, if $\lambda_{0},\ldots,\lambda_{k-1}$ are any given numbers
(separation constants), and if $w_{0}(r)$, $r>0$, solves (12) and
$w_{i}(s_{i})$, $a_{i-1}<s_{i}<a_{i}$, solve (13) for each $i=1,\dots,k$, then
$U$ defined by (11) solves (10) in the positive cone of ${\mathbf{R}}^{k+1}$
(4).
Equation (13) has only regular points except for $k+2$ regular singular points
at $s=a_{0},a_{1},\dots,a_{k}$ and $s=\infty$. The exponents at each finite
singularity $s=a_{j}$ are $0$ and $\frac{1}{2}$. Therefore, for each choice of
parameters $\lambda_{0},\dots,\lambda_{k-1}$, there is a nontrivial analytic
solution at $s=a_{j}$ and another one of the form $w(s)=(s-a_{j})^{1/2}v(s)$,
where $v$ is analytic at $a_{j}$. If $\nu,\mu$ denote the exponents at
$s=\infty$ then
$\mu\nu=\lambda_{0},\quad\mu+\nu=\frac{k-1}{2}.$ (14)
The polynomial $\sum_{i=0}^{k-1}\lambda_{i}s^{k-1-i}$ appearing in (13) is
known as van Vleck polynomial. If $k=1$ then (13) is the hypergeometric
differential equation (up to a linear substitution). If $k=2$ then (13) is the
Heun equation. We will use this equation for $k=3$. According to Miller (1977)
[9, p. 209] (see also [3, p. 71]) in reference to the $k=3$ case, “Very little
is known about the solutions.”
## 3 Stereographic projection
We consider the stereographic projection
$P:\mathbb{S}^{3}\setminus\\{(1,0,0,0)\\}\to{\mathbf{R}}^{3}$ given by
$P(x_{0},x_{1},x_{2},x_{3})=\frac{1}{1-x_{0}}(x_{1},x_{2},x_{3}).$
The inverse map is
$P^{-1}(x,y,z)=\frac{1}{x^{2}+y^{2}+z^{2}+1}(x^{2}+y^{2}+z^{2}-1,2x,2y,2z).$
We extend $P^{-1}$ to a bijective map
$Q:(0,\infty)\times{\mathbf{R}}^{3}\to{\mathbf{R}}^{4}\setminus\\{(x_{0},0,0,0):x_{0}\geq
0\\}$
by defining
$Q(r,x,y,z):=rP^{-1}(x,y,z).$
If we set $(x_{0},x_{1},x_{2},x_{3})=Q(r,x,y,z)$ we may consider $r,x,y,z$ as
curvilinear coordinates on ${\mathbf{R}}^{4}$ with Cartesian coordinates
$x_{0},x_{1},x_{2},x_{3}$. We note that
$x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=r^{2}$ so $r$ is just the distance
between $(x_{0},x_{1},x_{2},x_{3})$ and the origin. Moreover, $(x,y,z)$ is the
stereographic projection of the point
$(x_{0}/r,x_{1}/r,x_{2}/r,x_{3}/r)\in\mathbb{S}^{3}$. It is easy to check that
the coordinate system is orthogonal and scale factors are
$h_{r}=1,\quad h_{x}=h_{y}=h_{z}=2rh,\quad\mbox{where\
}h:=\frac{1}{x^{2}+y^{2}+z^{2}+1}.$
Let $U(x_{0},x_{1},x_{2},x_{3})=V(r,x,y,z)$. Then
$\Delta
U=\frac{1}{8r^{3}h^{3}}\left((2rhV_{x})_{x}+(2rhV_{y})_{y}+(2rhV_{z})_{z}+(8r^{3}h^{3}V_{r})_{r}\right).$
(15)
Suppose that $U$ is homogeneous of degree $\alpha$:
$U(tx_{0},tx_{1},tx_{2},tx_{3})=t^{\alpha}U(x_{0},x_{1},x_{2},x_{3}),\quad
t>0.$
Then $V$ can be written in the form
$V(r,x,y,z)=r^{\alpha}w(x,y,z),$
and (15) implies
$\Delta
U=\frac{r^{\alpha-2}}{4h^{3}}\left((hw_{x})_{x}+(hw_{y})_{y}+(hw_{z})_{z}+4\alpha(\alpha+2)h^{3}w\right).$
(16)
We now introduce the function
$u(x,y,z)=w(x,y,z)(x^{2}+y^{2}+z^{2}+1)^{-1/2}.$
Then a direct calculation changes (16) to
$\Delta
U=\frac{r^{\alpha-2}}{4h^{5/2}}\left(u_{xx}+u_{yy}+u_{zz}+(3+4\alpha(\alpha+2))h^{2}u\right).$
(17)
If $3+4\alpha(\alpha+2)=0$ then $U$ is harmonic if and only if $u$ is
harmonic. Noting that $3+4\alpha(\alpha+2)=(2\alpha+1)(2\alpha+3)$, we obtain
the following theorem.
###### Theorem 3.1.
Let $D$ be an open subset of $\mathbb{S}^{3}$ not containing $(1,0,0,0)$, let
$E=\\{(rx_{0},rx_{1},rx_{2},rx_{3}):r>0,(x_{0},x_{1},x_{2},x_{3})\in D\\}$,
and let $F=P(D)$ be the stereographic image of $D$. Let the function
$U:E\to{\mathbf{R}}$ be homogeneous of degree $-\frac{1}{2}$ or
$-\frac{3}{2}$, and let $w:F\to{\mathbf{R}}$ satisfy $U=w\circ P$ on $D$. Then
$U$ is harmonic on $E$ if and only if $w(x,y,z)(x^{2}+y^{2}+z^{2}+1)^{-1/2}$
is harmonic on $F$.
## 4 Five-cyclide coordinate system on ${\mathbf{R}}^{3}$
We introduce sphero-conal coordinates
$r>0,\quad a_{0}<s_{1}<a_{1}<s_{2}<a_{2}<s_{3}<a_{3},$
on ${\mathbf{R}}^{4}$ as explained in Section 2 with $k=3$. Then
$s_{1},s_{2},s_{3}$ form a coordinate system for the intersection of the
hypersphere $\mathbb{S}^{3}$ with the positive cone in ${\mathbf{R}}^{4}$.
Using the stereographic projection $P$ from Section 3 we project these
coordinates to ${\mathbf{R}}^{3}$. We obtain a coordinate system for the set
$T=\\{(x,y,z):x,y,z>0,\,x^{2}+y^{2}+z^{2}>1\\}.$ (18)
Explicitly,
$x=\frac{x_{1}}{1-x_{0}},\quad y=\frac{x_{2}}{1-x_{0}},\quad
z=\frac{x_{3}}{1-x_{0}},$ (19)
where
$x_{j}^{2}=\frac{\prod_{i=1}^{3}(s_{i}-a_{j})}{\prod_{j\neq
i=0}^{3}(a_{i}-a_{j})},\quad j=0,1,2,3.$ (20)
Conversely, the coordinates $s_{1},s_{2},s_{3}$ of a point $(x,y,z)\in T$ are
the solutions of
$\frac{(x^{2}+y^{2}+z^{2}-1)^{2}}{s-a_{0}}+\frac{4x^{2}}{s-a_{1}}+\frac{4y^{2}}{s-a_{2}}+\frac{4z^{2}}{s-a_{3}}=0.$
(21)
Since sphero-conal coordinates are orthogonal and the stereographic projection
preserves angles, 5-cyclide coordinates are orthogonal, too. This is the
twelfth coordinate system in Miller (1977) [9, page 210]. Miller uses a
slightly different notation: $a_{0}=0$, $a_{1}=1$, $a_{2}=b$, $a_{3}=a$, and
$s_{1}=\rho$, $s_{2}=\nu$, $s_{3}=\mu$. Also, $x,z$ are interchanged.
In order to calculate the scale factors for the 5-cyclide coordinate system we
proceed as follows. We start with
$\frac{\partial x}{\partial s_{i}}=\frac{1}{1-x_{0}}\frac{\partial
x_{1}}{\partial s_{i}}+\frac{x_{1}}{(1-x_{0})^{2}}\frac{\partial
x_{0}}{\partial s_{i}},$
and similar formulas for the derivatives of $y$ and $z$. Then using
$x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1,\quad x_{0}\frac{\partial
x_{0}}{\partial s_{i}}+x_{1}\frac{\partial x_{1}}{\partial
s_{i}}+x_{2}\frac{\partial x_{2}}{\partial s_{i}}+x_{3}\frac{\partial
x_{3}}{\partial s_{i}}=0$
a short calculation gives
$\frac{\partial x}{\partial s_{i}}\frac{\partial x}{\partial
s_{j}}+\frac{\partial y}{\partial s_{i}}\frac{\partial y}{\partial
s_{j}}+\frac{\partial z}{\partial s_{i}}\frac{\partial z}{\partial
s_{j}}=\frac{1}{(1-x_{0})^{2}}\sum_{\ell=0}^{3}\frac{\partial
x_{\ell}}{\partial s_{i}}\frac{\partial x_{\ell}}{\partial s_{j}}.$
This confirms that 5-cyclide coordinates are orthogonal and from (9) we obtain
the squares of their scale factors
$h_{i}^{2}=\frac{1}{16}\left(\frac{(\rho^{2}-1)^{2}}{(s_{i}-a_{0})^{2}}+\frac{4x^{2}}{(s_{i}-a_{1})^{2}}+\frac{4y^{2}}{(s_{i}-a_{2})^{2}}+\frac{4z^{2}}{(s_{i}-a_{3})^{2}}\right),$
(22)
where $\rho^{2}=x^{2}+y^{2}+z^{2}$, or, equivalently,
$\displaystyle h_{1}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{16}(\rho^{2}+1)^{2}\frac{(s_{3}-s_{1})(s_{2}-s_{1})}{(s_{1}-a_{0})(a_{1}-s_{1})(a_{2}-s_{1})(a_{3}-s_{1})},$
(23) $\displaystyle h_{2}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{16}(\rho^{2}+1)^{2}\frac{(s_{2}-s_{1})(s_{3}-s_{2})}{(s_{2}-a_{0})(s_{2}-a_{1})(a_{2}-s_{2})(a_{3}-s_{2})},$
(24) $\displaystyle h_{3}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{16}(\rho^{2}+1)^{2}\frac{(s_{3}-s_{1})(s_{3}-s_{2})}{(s_{3}-a_{0})(s_{3}-a_{1})(s_{3}-a_{2})(a_{3}-s_{3})}.$
(25)
We find harmonic functions by separation of variables in 5-cyclide coordinates
as follows.
###### Theorem 4.1.
Let $w_{1}:(a_{0},a_{1})\to{\mathbf{C}}$,
$w_{2}:(a_{1},a_{2})\to{\mathbf{C}}$, $w_{3}:(a_{2},a_{3})\to{\mathbf{C}}$ be
solutions of the Fuchsian equation
$\prod_{j=0}^{3}(s-a_{j})\left[w^{\prime\prime}+\frac{1}{2}\sum_{j=0}^{3}\frac{1}{s-a_{j}}w^{\prime}\right]+\left(\frac{3}{16}s^{2}+\lambda_{1}s+\lambda_{2}\right)w=0,$
(26)
where $\lambda_{1},\lambda_{2}$ are given (separation) constants. Then the
function
$u(x,y,z)=(x^{2}+y^{2}+z^{2}+1)^{-1/2}w_{1}(s_{1})w_{2}(s_{2})w_{3}(s_{3})$
(27)
is a harmonic function on the set (18).
###### Proof.
Using sphero-conal coordinates $r,s_{1},s_{2},s_{3}$ on ${\mathbf{R}}^{4}$, we
define a function $U$ in the positive cone of ${\mathbf{R}}^{4}$ by
$U(x_{0},x_{1},x_{2},x_{3})=r^{-1/2}w_{1}(s_{1})w_{2}(s_{2})w_{3}(s_{3}).$
The function $r^{-1/2}$ is a solution of (12) when $k=3$,
$\lambda_{0}=\frac{3}{16}$. The results from Section 2 imply that $U$ is
harmonic, and, of course, $U$ is homogeneous of degree $-\frac{1}{2}$. The
function $w$ defined on the set (18) by $U=w\circ P$ is given in 5-cyclide
coordinates by
$w(x,y,z)=w_{1}(s_{1})w_{2}(s_{2})w_{3}(s_{3}).$
Therefore, Theorem 3.1 gives the statement of the theorem. ∎
Equation (26) has five regular singularities at
$s=a_{0},a_{1},a_{2},a_{3},\infty$. The exponents at the finite singularities
are $0$ and $\frac{1}{2}$. Using (14), we find that the exponents at infinity
are $\frac{1}{4}$ and $\frac{3}{4}$. So all five singularities are elementary
in the sense of Ince [5]. Equation (26) is one of the standard equations in
the classification of Ince [5, page 500].
We define the 5-cyclide coordinates $s_{1},s_{2},s_{3}$ for an arbitrary point
$(x,y,z)\in{\mathbf{R}}^{3}$ as the zeros $s_{1}\leq s_{2}\leq s_{3}$ of the
cubic equation
$\prod_{j=0}^{3}(s-a_{j})\left[\frac{(x^{2}+y^{2}+z^{2}-1)^{2}}{s-a_{0}}+\frac{4x^{2}}{s-a_{1}}+\frac{4y^{2}}{s-a_{2}}+\frac{4z^{2}}{s-a_{3}}\right]=0.$
(28)
For example, $s_{j}(0,0,0)=a_{j}$ for $j=1,2,3$. Each function
$s_{j}:{\mathbf{R}}^{3}\to[a_{j-1},a_{j}]$ is continuous. We observe that, in
general, there are 16 different points in ${\mathbf{R}}^{3}$ which have the
same coordinates $s_{1},s_{2},s_{3}$. If $(x,y,z)$ is one of these points the
other ones are obtained by applying the group generated by inversion at
$\mathbb{S}^{2}$
$\sigma_{0}(x,y,z)=\rho^{-2}(x,y,z)$ (29)
and reflections at the coordinate planes
$\sigma_{1}(x,y,z)=(-x,y,z),\,\sigma_{2}(x,y,z)=(x,-y,z),\,\sigma_{3}(x,y,z)=(x,y,-z).$
(30)
It is of interest to determine the sets where $s_{j}=a_{j-1}$ or
$s_{j}=a_{j}$. We obtain
$\displaystyle s_{1}$ $\displaystyle=$ $\displaystyle a_{0}\mbox{\ iff\
}x^{2}+y^{2}+z^{2}=1,$ (31) $\displaystyle s_{1}$ $\displaystyle=$
$\displaystyle a_{1}\mbox{\ iff\ }x=0\mbox{\ and\
}\frac{(\rho^{2}-1)^{2}}{a_{1}-a_{0}}+\frac{4y^{2}}{a_{1}-a_{2}}+\frac{4z^{2}}{a_{1}-a_{3}}\geq
0,$ (32) $\displaystyle s_{2}$ $\displaystyle=$ $\displaystyle a_{1}\mbox{\
iff\ }x=0\mbox{\ and\
}\frac{(\rho^{2}-1)^{2}}{a_{1}-a_{0}}+\frac{4y^{2}}{a_{1}-a_{2}}+\frac{4z^{2}}{a_{1}-a_{3}}\leq
0,$ (33) $\displaystyle s_{2}$ $\displaystyle=$ $\displaystyle a_{2}\mbox{\
iff\ }y=0\mbox{\ and\
}\frac{(\rho^{2}-1)^{2}}{a_{2}-a_{0}}+\frac{4x^{2}}{a_{2}-a_{1}}+\frac{4z^{2}}{a_{2}-a_{3}}\geq
0,$ (34) $\displaystyle s_{3}$ $\displaystyle=$ $\displaystyle a_{2}\mbox{\
iff\ }y=0\mbox{\ and\
}\frac{(\rho^{2}-1)^{2}}{a_{2}-a_{0}}+\frac{4x^{2}}{a_{2}-a_{1}}+\frac{4z^{2}}{a_{2}-a_{3}}\leq
0,$ (35) $\displaystyle s_{3}$ $\displaystyle=$ $\displaystyle a_{3}\mbox{\
iff\ }z=0.$ (36)
We define the sets (consisting each of two closed curves)
$\displaystyle A_{1}$ $\displaystyle:=$
$\displaystyle\\{(x,y,z)\in{\mathbf{R}}^{3}:s_{1}=s_{2}=a_{1}\\}$
$\displaystyle=$
$\displaystyle\\{(x,y,z):x=0,\frac{(\rho^{2}-1)^{2}}{a_{1}-a_{0}}+\frac{4y^{2}}{a_{1}-a_{2}}+\frac{4z^{2}}{a_{1}-a_{3}}=0\\},$
see Figure 2, and
$\displaystyle A_{2}$ $\displaystyle:=$
$\displaystyle\\{(x,y,z)\in{\mathbf{R}}^{3}:s_{2}=s_{3}=a_{2}\\}$
$\displaystyle=$
$\displaystyle\\{(x,y,z):y=0,\frac{(\rho^{2}-1)^{2}}{a_{2}-a_{0}}+\frac{4x^{2}}{a_{2}-a_{1}}+\frac{4z^{2}}{a_{2}-a_{3}}=0\\},$
see Figure 3.
Figure 2: The set $A_{1}$ in the $y,z$-plane for $a_{i}=i$. Figure 3: The set
$A_{2}$ in the $x,z$-plane for $a_{i}=i$.
Clearly, $s_{j}$ is analytic at all points $(x,y,z)$ at which $s_{j}$ is a
simple zero of the cubic equation (28). Therefore, $s_{1}$ is analytic on
${\mathbf{R}}^{3}\setminus A_{1}$, $s_{2}$ is analytic on
${\mathbf{R}}^{3}\setminus(A_{1}\cup A_{2})$, and $s_{3}$ is analytic on
${\mathbf{R}}^{3}\setminus A_{2}$.
We may use (27) to define $u(x,y,z)$ for all $(x,y,z)\in{\mathbf{R}}^{3}$.
Since the solutions $w_{1},w_{2},w_{3}$ of (26) have limits at the end points
of their intervals of definition (because the exponents are $0$ and
$\frac{1}{2}$ there), we see that $u$ is a continuous functions on
${\mathbf{R}}^{3}$. The function $(x^{2}+y^{2}+z^{2}+1)^{1/2}u(x,y,z)$ is
invariant under $\sigma_{i}$, $i=0,1,2,3$. In general, $u$ is harmonic only
away from the coordinate planes and the unit sphere. In fact, we observe that
$u$ is a bounded function which converges to $0$ at infinity, so, by
Liouville’s theorem, $u$ cannot be harmonic on all of ${\mathbf{R}}^{3}$
unless it is identically zero.
## 5 First two-parameter Sturm-Liouville problem
We consider equation (26) on the intervals $(a_{1},a_{2})$ and $(a_{2},a_{3})$
and write it in formally self-adjoint form. Setting
$\omega(s):=\left|(s-a_{0})(s-a_{1})(s-a_{2})(s-a_{3})\right|^{1/2},$ (39)
we obtain two Sturm-Liouville equations involving two parameters
$\displaystyle(\omega(s_{2})w_{2}^{\prime})^{\prime}+\frac{1}{\omega(s_{2})}\left(\frac{3}{16}s_{2}^{2}+\lambda_{1}s_{2}+\lambda_{2}\right)w_{2}=0,\quad
a_{1}<s_{2}<a_{2},$ (40)
$\displaystyle(\omega(s_{3})w_{3}^{\prime})^{\prime}-\frac{1}{\omega(s_{3})}\left(\frac{3}{16}s_{3}^{2}+\lambda_{1}s_{3}+\lambda_{2}\right)w_{3}=0,\quad
a_{2}<s_{3}<a_{3}.$ (41)
In (40) $w_{2}$ is a function of $s_{2}$ and the derivatives are taken with
respect to $s_{2}$. In (41) $w_{3}$ is a function of $s_{3}$ and the
derivatives are taken with respect to $s_{3}$. We simplify the equations by
substituting $t_{j}=\Omega(s_{j})$, $u_{j}(t_{j})=w_{j}(s_{j})$, where
$\Omega(s)$ is the elliptic integral (see for instance [8])
$\Omega(s):=\int_{a_{0}}^{s}\frac{d\sigma}{\omega(\sigma)}.$ (42)
This is an increasing absolutely continuous function
$\Omega:[a_{0},a_{3}]\to[0,b_{3}]$, where $b_{j}:=\Omega(a_{j})$. Let
$\phi:[0,b_{3}]\to[a_{0},a_{3}]$ be the inverse function of $\Omega$. Then
(40), (41) become
$\displaystyle
u_{2}^{\prime\prime}+\left(\frac{3}{16}\\{\phi(t_{2})\\}^{2}+\lambda_{1}\phi(t_{2})+\lambda_{2}\right)u_{2}$
$\displaystyle=$ $\displaystyle 0,\quad b_{1}\leq t_{2}\leq b_{2},$ (43)
$\displaystyle
u_{3}^{\prime\prime}-\left(\frac{3}{16}\\{\phi(t_{3})\\}^{2}+\lambda_{1}\phi(t_{3})+\lambda_{2}\right)u_{3}$
$\displaystyle=$ $\displaystyle 0,\quad b_{2}\leq t_{3}\leq b_{3}.$ (44)
We add the boundary conditions
$u_{2}^{\prime}(b_{1})=u_{2}^{\prime}(b_{2})=u_{3}^{\prime}(b_{2})=u_{3}^{\prime}(b_{3})=0.$
(45)
Differential equations (43), (44) together with boundary conditions (45) pose
a two-parameter Sturm-Liouville eigenvalue problem. For the theory of such
multiparameter problems we refer to the books [1, 13] and the references
therein. A pair $(\lambda_{1},\lambda_{2})$ is called an eigenvalue if there
exist (nontrivial) eigenfunctions $u_{2}(t_{2})$ and $u_{3}(t_{3})$ which
satisfy (43), (44), (45). The two-parameter problem is right-definite in the
sense that
$\left|\begin{array}[]{cc}\phi(t_{2})&1\\\
-\phi(t_{3})&-1\end{array}\right|=\phi(t_{3})-\phi(t_{2})>0\quad\mbox{\ for\
}b_{1}<t_{2}<b_{2}<t_{3}<b_{3}.$
However, this determinant is not positive on the closed rectangle
$[b_{1},b_{2}]\times[b_{2},b_{3}]$. This lack of uniform right-definiteness
make some proofs in this section a little longer than they would be otherwise.
We have the following Klein oscillation theorem; see [1, Theorem 5.5.1].
###### Theorem 5.1.
For every $\mathbf{n}=(n_{2},n_{3})\in{\mathbf{N}}_{0}^{2}$, there exists a
uniquely determined eigenvalue
$(\lambda_{1,\mathbf{n}},\lambda_{2,\mathbf{n}})\in{\mathbf{R}}^{2}$ admitting
an eigenfunction $u_{2}$ with exactly $n_{2}$ zeros in $(b_{1},b_{2})$ and an
eigenfunction $u_{3}$ with exactly $n_{3}$ zeros in $(b_{2},b_{3})$.
We state a result on the distribution of eigenvalues; compare with [1, Chapter
8].
###### Theorem 5.2.
There are positive constants $A_{1},A_{2},A_{3},A_{4}$ such that, for all
$\mathbf{n}\in{\mathbf{N}}_{0}^{2}$,
$\displaystyle-A_{1}(n_{2}^{2}+n_{3}^{2}+1)\leq\lambda_{1,\mathbf{n}}\leq-
A_{2}(n_{2}^{2}+n_{3}^{2})+A_{3},$ (46)
$\displaystyle|\lambda_{2,\mathbf{n}}|\leq A_{4}(n_{2}^{2}+n_{3}^{2}+1).$ (47)
###### Proof.
If a differential equation $u^{\prime\prime}+q(t)u=0$ with continuous
$q:[a,b]\to{\mathbf{R}}$ admits a solution $u$ satisfying
$u^{\prime}(a)=u^{\prime}(b)=0$ and having exactly $m$ zeros in $(a,b)$, then
there is $t\in(a,b)$ such that $q(t)=\frac{\pi^{2}m^{2}}{(b-a)^{2}}$. This is
shown by comparing with the eigenvalue problem $u^{\prime\prime}+\lambda u=0$,
$u^{\prime}(a)=u^{\prime}(b)=0$. Applying this fact, we find
$t_{2}\in(b_{1},b_{2})$ and $t_{3}\in(b_{2},b_{3})$ such that
$\displaystyle\frac{3}{16}\\{\phi(t_{2})\\}^{2}+\lambda_{1}\phi(t_{2})+\lambda_{2}$
$\displaystyle=$ $\displaystyle\frac{\pi^{2}n_{2}^{2}}{(b_{2}-b_{1})^{2}},$
(48)
$\displaystyle\frac{3}{16}\\{\phi(t_{3})\\}^{2}+\lambda_{1}\phi(t_{3})+\lambda_{2}$
$\displaystyle=$ $\displaystyle-\frac{\pi^{2}n_{3}^{2}}{(b_{3}-b_{2})^{2}},$
(49)
where we abbreviated $\lambda_{j}=\lambda_{j,\mathbf{n}}$. By subtracting (48)
from (49), we obtain
$\frac{3}{16}\left(\\{\phi(t_{3})\\}^{2}-\\{\phi(t_{2})\\}^{2}\right)+\lambda_{1}(\phi(t_{3})-\phi(t_{2}))=-\frac{\pi^{2}n_{2}^{2}}{(b_{2}-b_{1})^{2}}-\frac{\pi^{2}n_{3}^{2}}{(b_{3}-b_{2})^{2}}\leq
0.$
Dividing by $\phi(t_{3})-\phi(t_{2})$ and using $0<\phi(t_{3})-\phi(t_{2})\leq
a_{3}-a_{1}$, we obtain the second inequality in (46).
To prove the first inequality in (46), suppose that
$\lambda_{1}<-\frac{3}{8}a_{3}$. Then the van Vleck polynomial
$Q(s):=\frac{3}{16}s^{2}+\lambda_{1}s+\lambda_{2}$ (50)
satisfies $Q^{\prime}(s)=\frac{3}{8}s+\lambda_{1}<0$ for $s\leq a_{3}$. Let
$c\in(b_{1},b_{2})$ be determined by $\phi(c)=\frac{1}{2}(a_{1}+a_{2})$. If
$Q(a_{2})\geq 0$ then, for $t\in[b_{1},c]$,
$Q(\phi(t))\geq
Q(\frac{1}{2}(a_{1}+a_{2}))\geq\frac{1}{2}(a_{2}-a_{1})\left(-\lambda_{1}-\frac{3}{8}a_{3}\right).$
By Sturm’s comparison theorem applied to equation (43), we get
$(c-b_{1})^{2}(a_{2}-a_{1})\left(-\lambda_{1}-\frac{3}{8}a_{3}\right)\leq
4\pi^{2}(n_{2}+1)^{2},$
which gives the desired inequality. If $Q(a_{2})<0$ we argue similarly working
with (44) instead.
Finally, (47) follows from (46) and (48). ∎
Let $u_{2,\mathbf{n}}$ and $u_{3,\mathbf{n}}$ denote real-valued
eigenfunctions corresponding to the eigenvalue
$(\lambda_{1,\mathbf{n}},\lambda_{2,\mathbf{n}})$. It is known [1, section
3.5] (and easy to prove) that the system of products
$u_{2,\mathbf{n}}(t_{2})u_{3,\mathbf{n}}(t_{3})$,
$\mathbf{n}\in{\mathbf{N}}_{0}^{2}$, is orthogonal in the Hilbert space
$H_{1}$ consisting of measurable functions
$f:(b_{1},b_{2})\times(b_{2},b_{3})\to{\mathbf{C}}$ satisfying
$\int_{b_{2}}^{b_{3}}\int_{b_{1}}^{b_{2}}(\phi(t_{3})-\phi(t_{2}))\left|f(t_{2},t_{3})\right|^{2}\,dt_{2}\,dt_{3}<\infty$
with inner product
$\int_{b_{2}}^{b_{3}}\int_{b_{1}}^{b_{2}}(\phi(t_{3})-\phi(t_{2}))f(t_{2},t_{3})\overline{g(t_{2},t_{3})}\,dt_{2}\,dt_{3}.$
We normalize the eigenfunctions so that
$\int_{b_{2}}^{b_{3}}\int_{b_{1}}^{b_{2}}(\phi(t_{3})-\phi(t_{2}))\left\\{u_{2,\mathbf{n}}(t_{2})\right\\}^{2}\left\\{u_{3,\mathbf{n}}(t_{3})\right\\}^{2}\,dt_{2}\,dt_{3}=1.$
(51)
We have the following completeness theorem; see [13, Theorem 6.8.3].
###### Theorem 5.3.
The double sequence of functions
$u_{2,\mathbf{n}}(t_{2})u_{3,\mathbf{n}}(t_{3}),\quad\mathbf{n}\in{\mathbf{N}}_{0}^{2},$
forms an orthonormal basis in the Hilbert space $H_{1}$.
The normalization (51) leads to a bound on the values of eigenfunctions.
###### Theorem 5.4.
There is a constant $B>0$ such that, for all
$\mathbf{n}\in{\mathbf{N}}_{0}^{2}$ and all $t_{2}\in[b_{1},b_{2}]$,
$t_{3}\in[b_{2},b_{3}]$,
$|u_{2,\mathbf{n}}(t_{2})u_{3,\mathbf{n}}(t_{3})|\leq
B(n_{2}^{2}+n_{3}^{2}+1).$
###### Proof.
We abbreviate $u_{j}=u_{j,\mathbf{n}}$, $\lambda_{j}=\lambda_{j,\mathbf{n}}$.
Condition (51) is a normalization for the product $u_{2}(t_{2})u_{3}(t_{3})$
but not for each factor separately, so we may assume that, additionally,
$\int_{b_{1}}^{b_{2}}\left\\{u_{2}(t_{2})\right\\}^{2}\,dt=1.$ (52)
Now (51), (52) imply that
$\int_{b_{2}}^{b_{3}}(\phi(t_{3})-\phi(b_{2}))\left\\{u_{3}(t_{3})\right\\}^{2}\,dt_{3}\leq
1.$ (53)
We multiply equations (43), (44) by $u_{2}$ and $u_{3}$, respectively, and
integrate by parts to obtain
$\displaystyle\int_{b_{1}}^{b_{2}}u_{2}^{\prime 2}$ $\displaystyle=$
$\displaystyle\frac{3}{16}\int_{b_{1}}^{b_{2}}\phi^{2}u_{2}^{2}+\lambda_{1}\int_{b_{1}}^{b_{2}}\phi
u_{2}^{2}+\lambda_{2}\int_{b_{1}}^{b_{2}}u_{2}^{2},$ (54)
$\displaystyle\int_{b_{2}}^{b_{3}}u_{3}^{\prime 2}$ $\displaystyle=$
$\displaystyle-\frac{3}{16}\int_{b_{2}}^{b_{3}}\phi^{2}u_{3}^{2}-\lambda_{1}\int_{b_{2}}^{b_{3}}\phi
u_{3}^{2}-\lambda_{2}\int_{b_{2}}^{b_{3}}u_{3}^{2}.$ (55)
It follows from (52), (54) and Theorem 5.2 that there is a constant $B_{1}>0$
such that, for all $\mathbf{n}\in{\mathbf{N}}_{0}^{2}$,
$\int_{b_{1}}^{b_{2}}u_{2}^{\prime 2}\leq B_{1}(n_{2}^{2}+n_{3}^{2}+1).$ (56)
Unfortunately, we cannot argue the same way for $u_{3}$ because we do not have
an upper bound for $\int_{b_{2}}^{b_{3}}u_{3}^{2}$. Instead, we multiply (54)
by $\int u_{3}^{2}$ and (55) by $\int u_{2}^{2}$ and add the equations. Then,
noting (51), we find
$\int_{b_{1}}^{b_{2}}u_{2}^{\prime
2}\int_{b_{2}}^{b_{3}}u_{3}^{2}+\int_{b_{1}}^{b_{2}}u_{2}^{2}\int_{b_{2}}^{b_{3}}u_{3}^{\prime
2}\\\ \leq-\lambda_{1}+\frac{3}{8}\max_{t\in[b_{1},b_{3}]}|\phi(t)|.$
Using Theorem 5.2 and (52), we find a constant $B_{2}>0$ such that, for all
$\mathbf{n}\in{\mathbf{N}}_{0}^{2}$,
$\int_{b_{2}}^{b_{3}}u_{3}^{\prime 2}\leq B_{2}(n_{2}^{2}+n_{3}^{2}+1).$ (57)
We apply the following Lemma 5.5 (noting (52), (53), (56), (57)) and obtain
the desired result. ∎
###### Lemma 5.5.
Let $u:[a,b]\to{\mathbf{R}}$ be a continuously differentiable function, and
let $a\leq c<d\leq b$. Then, for all $t\in[a,b]$,
$(d-c)\left|u(t)\right|^{2}\leq
2\int_{c}^{d}\left|u(r)\right|^{2}\,dr+2(b-a)(d-c)\int_{a}^{b}\left|u^{\prime}(r)\right|^{2}\,dr.$
###### Proof.
For $s,t\in[a,b]$ we have
$|u(t)-u(s)|=\left|\int_{s}^{t}u^{\prime}(r)\,dr\right|\leq|t-s|^{1/2}\left(\int_{a}^{b}\left|u^{\prime}(r)\right|^{2}\,dr\right)^{1/2}.$
This implies
$|u(t)|^{2}\leq
2|u(s)|^{2}+2|t-s|\int_{a}^{b}\left|u^{\prime}(r)\right|^{2}\,dr.$
We integrate from $s=c$ to $s=d$ and obtain the desired inequality. ∎
Let $u_{1,\mathbf{n}}$ be the solution of
$u_{1}^{\prime\prime}-\left(\frac{3}{16}\\{\phi(t_{1})\\}^{2}+\lambda_{1,\mathbf{n}}\phi(t_{1})+\lambda_{2,\mathbf{n}}\right)u_{1}=0,\quad
b_{0}\leq t_{1}\leq b_{1},$ (58)
determined by the initial conditions
$u_{1}(b_{1})=1,\quad u_{1}^{\prime}(b_{1})=0.$
The following estimate on $u_{1,\mathbf{n}}$ will be useful in Section 6.
###### Theorem 5.6.
We have $u_{1,\mathbf{n}}(t_{1})>0$ for all $t_{1}\in[b_{0},b_{1}]$. If
$0=b_{0}\leq c_{1}<c_{2}<b_{1}$, then there are constants $C>0$ and $0<r<1$
such that, for all $\mathbf{n}\in{\mathbf{N}}_{0}^{2}$ and
$t_{1}\in[c_{2},b_{1}]$,
$\frac{u_{1,\mathbf{n}}(t_{1})}{u_{1,\mathbf{n}}(c_{1})}\leq
Cr^{n_{2}+n_{3}}.$
###### Proof.
We abbreviate $u_{1}=u_{1,\mathbf{n}}$ and
$\lambda_{j}=\lambda_{j,\mathbf{n}}$. By definition, $u_{1}$ satisfies the
differential equation
$u_{1}^{\prime\prime}=Q(\phi(t_{1}))u_{1},\quad t_{1}\in[b_{0},b_{1}],$
where $Q$ is given by (50). According to (48), (49), there are
$s_{2}\in(a_{1},a_{2})$ and $s_{3}\in(a_{2},a_{3})$ such that
$Q(s_{2})=\frac{\pi^{2}n_{2}^{2}}{(b_{2}-b_{1})^{2}},\quad
Q(s_{3})=-\frac{\pi^{2}n_{3}^{2}}{(b_{3}-b_{2})^{2}}.$
If $s\leq s_{2}$ then $Q(s)\geq L(s)$, where $L(s)$ is the linear function
with $L(s_{j})=Q(s_{j})$, $j=2,3$. It follows that $Q(s)\geq 0$ for
$s\in[a_{0},a_{1}]$ and
$Q(\phi(t_{1}))\geq C_{1}(n_{2}+n_{3})^{2}\quad\mbox{\ for\
}t_{1}\in[b_{0},c_{2}],$ (59)
where $C_{1}$ is a positive constant independent of $\mathbf{n}$. We now apply
the following Lemma 5.7 to complete the proof. Note that the interval $[a,b]$
in the lemma is $[c_{1},b_{1}]$ but with the end points interchanged. ∎
###### Lemma 5.7.
Let $u:[a,b]\to{\mathbf{R}}$ be a solution of the differential equation
$u^{\prime\prime}(t)=q(t)u(t),\quad t\in[a,b],$
determined by the initial conditions $u(a)=1$, $u^{\prime}(a)=0$, where
$q:[a,b]\to{\mathbf{R}}$ is a continuous function. Suppose that $q(t)\geq 0$
on $[a,b]$ and $q(t)\geq\lambda^{2}$ on $[c,b]$ for some $\lambda>0$ and
$c\in[a,b)$. Then $u(t)>0$ for all $t\in[a,b]$, and
$\frac{u(b)}{u(t)}\geq\frac{1}{2}e^{\lambda(b-c)}\ \mbox{for\ all}\
t\in[0,c].$
###### Proof.
Since $q(t)\geq 0$, $u(t)>0$ and $u^{\prime}(t)\geq 0$ for $t\in[a,b]$. The
function $z=u^{\prime}/u$ satisfies the Riccati equation
$z^{\prime}+z^{2}=q(t),$
and the initial condition $z(a)=0$. It follows that
$z(t)\geq\lambda\tanh(\lambda(t-c))\quad\mbox{\ for\ }t\in[c,b].$
Integrating from $t=c$ to $t=b$ gives
$\ln\frac{u(b)}{u(c)}\geq\ln\cosh\lambda(b-c)\geq\ln\frac{1}{2}e^{\lambda(b-c)}$
which yields the claim since $u$ is nondecreasing. ∎
We now introduce a systematic notation for our eigenvalues and eigenfunctions.
First of all, we note that the results of this section remain valid for other
sets of boundary conditions. We will need eight sets of boundary conditions
labeled by $\mathbf{p}=(p_{1},p_{2},p_{3})\in\\{0,1\\}^{3}$. These boundary
conditions are
$\begin{array}[]{llll}u_{2}^{\prime}(b_{1})=0&\ \mbox{\ if\ }p_{1}=0,&\quad
u_{2}(b_{1})=0&\ \mbox{\ if\ }p_{1}=1,\\\
u_{2}^{\prime}(b_{2})=u_{3}^{\prime}(b_{2})=0&\ \mbox{\ if\ }p_{2}=0,&\quad
u_{2}(b_{2})=u_{3}(b_{2})=0&\ \mbox{\ if\ }p_{2}=1,\\\
u_{3}^{\prime}(b_{3})=0&\ \mbox{\ if\ }p_{3}=0,&\quad u_{3}(b_{3})=0&\ \mbox{\
if\ }p_{3}=1.\end{array}$ (60)
The initial conditions for $u_{1}$ are
$u_{1}(b_{1})=1,u_{1}^{\prime}(b_{1})=0\,\ \mbox{\ if\ }p_{1}=0,\quad
u_{1}(b_{1})=0,u_{1}^{\prime}(b_{1})=1\,\ \mbox{\ if\ }p_{1}=1.$ (61)
We denote the corresponding eigenvalues by
$(\lambda^{(1)}_{1,{\mathbf{n},\mathbf{p}}},\lambda^{(1)}_{2,{\mathbf{n},\mathbf{p}}})$.
For the notation of eigenfunctions we return to the $s_{i}$-variable connected
to $t_{i}$ by $t_{i}=\Omega(s_{i})$. The eigenfunctions will be denoted by
$E^{(1)}_{i,{\mathbf{n},\mathbf{p}}}(s_{i})=u_{i,\mathbf{n}}(t_{i})$,
$i=1,2,3$. The superscript $(1)$ is used to distinguish from eigenvalues and
eigenfunctions introduced in Sections 7 and 9. The subscript
$\mathbf{n}=(n_{2},n_{3})$ indicates the number of zeros of
$E^{(1)}_{2,{\mathbf{n},\mathbf{p}}}(s_{2})$,
$E^{(1)}_{3,{\mathbf{n},\mathbf{p}}}(s_{3})$ in $(a_{1},a_{2})$,
$(a_{2},a_{3})$, respectively. The subscript $\mathbf{p}$ indicates the
boundary conditions used to determine eigenvalues and eigenfunctions. By using
the letter $E$ for eigenfunctions we follow Bôcher [2]. In our notation we
suppressed the dependence of eigenvalues and eigenfunctions on
$a_{0},a_{1},a_{2},a_{3}$.
Summarizing, for $i=1,2,3$, $E^{(1)}_{i,{\mathbf{n},\mathbf{p}}}$ is a
solution of (26) on $(a_{i-1},a_{i})$ with
$(\lambda_{1},\lambda_{2})=(\lambda^{(1)}_{1,{\mathbf{n},\mathbf{p}}},\lambda^{(2)}_{2,{\mathbf{n},\mathbf{p}}})$.
The solution $E^{(1)}_{1,{\mathbf{n},\mathbf{p}}}(s_{1})$ has exponent
$\frac{1}{2}p_{1}$ at $a_{1}$ and it has no zeros in $(a_{0},a_{1})$. The
solution $E^{(1)}_{2,{\mathbf{n},\mathbf{p}}}(s_{2})$ has exponent
$\frac{1}{2}p_{1}$ at $a_{1}$, exponent $\frac{1}{2}p_{2}$ at $a_{2}$, and its
has $n_{2}$ zeros in $(a_{1},a_{2})$. The solution
$E^{(1)}_{3,{\mathbf{n},\mathbf{p}}}(s_{3})$ has exponent $\frac{1}{2}p_{2}$
at $a_{2}$, exponent $\frac{1}{2}p_{3}$ at $a_{3}$, and it has $n_{3}$ zeros
in $(a_{2},a_{3})$.
## 6 First Dirichlet problem
Consider the coordinate surface (21) for fixed $s=d_{1}\in(a_{0},a_{1})$. See
Figure 4 for a graphical depiction of the shape of this surface.
Figure 4: Coordinate surfaces $s_{1}=const$ for $a_{i}=i$ with (a), (b) inside
$B_{1}(\mathbf{0})$, (d), (e), (f) outside $B_{1}(\mathbf{0})$, and (c) the
unit sphere.
Let $(x^{\prime},y^{\prime},z^{\prime})\in\mathbb{S}^{2}$. The ray
$(x,y,z)=t(x^{\prime},y^{\prime},z^{\prime})$, $t>0$, intersects the surface
if
$\frac{(t^{2}-1)^{2}}{d_{1}-a_{0}}=ct^{2},$ (62)
where
$c=\frac{4x^{\prime 2}}{a_{1}-d_{1}}+\frac{4y^{\prime
2}}{a_{2}-d_{1}}+\frac{4z^{\prime 2}}{a_{3}-d_{1}}>0.$
Equation (62) has two positive solutions $t=t_{1},t_{2}$ such that
$0<t_{1}<1<t_{2}$ and $t_{1}t_{2}=1$. Therefore, the coordinate surface
$s_{1}=d_{1}$ consists of two disjoint closed surfaces of genus $0$. One lies
inside the unit ball $B_{1}(\mathbf{0})$ centered at the origin and the other
one is the image of it under the inversion (29). Let $D_{1}$ be the region
interior to the first surface, that is,
$D_{1}=\\{(x,y,z)\in B_{1}(\mathbf{0}):s_{1}>d_{1}\\},$ (63)
or, equivalently,
$D_{1}=\\{(x,y,z)\in
B_{1}(\mathbf{0}):\frac{(\rho^{2}-1)^{2}}{d_{1}-a_{0}}+\frac{4x^{2}}{d_{1}-a_{1}}+\frac{4y^{2}}{d_{1}-a_{2}}+\frac{4z^{2}}{d_{1}-a_{3}}>0\\}.$
We showed that $D_{1}$ is star-shaped with respect to the origin. We now solve
the Dirichlet problem for harmonic functions in $D_{1}$ by the method of
separation of variables.
Let $\mathbf{p}=(p_{1},p_{2},p_{3})\in\\{0,1\\}^{3}$ and
$\mathbf{n}=(n_{2},n_{3})\in{\mathbf{N}}_{0}^{2}$. Using the functions
$E^{(1)}_{i,{\mathbf{n},\mathbf{p}}}$ introduced in Section 5 we define the
internal 5-cyclidic harmonic of the first kind
$G^{(1)}_{\mathbf{n},\mathbf{p}}(x,y,z)=(x^{2}+y^{2}+z^{2}+1)^{-1/2}E^{(1)}_{1,{\mathbf{n},\mathbf{p}}}(s_{1})E^{(1)}_{2,{\mathbf{n},\mathbf{p}}}(s_{2})E^{(1)}_{3,{\mathbf{n},\mathbf{p}}}(s_{3})$
(64)
for $x,y,z\in B_{1}(\mathbf{0})$ with $x,y,z\geq 0$. We extend this function
to $B_{1}(\mathbf{0})$ as a function of parity $\mathbf{p}$. We call a
function $f$ of parity $\mathbf{p}$ if
$f(\sigma_{i}(x,y,z))=(-1)^{p_{i}}f(x,y,z),\quad\mbox{\ for\ }i=1,2,3$ (65)
using the reflections (30).
###### Lemma 6.1.
The function $G^{(1)}_{\mathbf{n},\mathbf{p}}$ is harmonic on
$B_{1}(\mathbf{0})$.
###### Proof.
By Theorem 4.1, $G^{(1)}_{\mathbf{n},\mathbf{p}}$ is harmonic on
$B_{1}(\mathbf{0})$ away from the coordinate planes. Therefore, it is enough
to show that $G^{(1)}_{\mathbf{n},\mathbf{p}}$ is analytic on
$B_{1}(\mathbf{0})$.
Consider first $\mathbf{p}=(0,0,0)$. Then (64) holds on $B_{1}(\mathbf{0})$.
Since $s_{1}\neq a_{0}$ on $B_{1}(\mathbf{0})$,
$G^{(1)}_{\mathbf{n},\mathbf{p}}$ is analytic on
$B_{1}(\mathbf{0})\setminus(A_{1}\cup A_{2})$ as a composition of analytic
functions. In order to show that $G^{(1)}_{\mathbf{n},\mathbf{p}}$ is also
analytic at the points of $A_{1}\cup A_{2}$, one may refer to a classical
result on “singular curves” of harmonic functions; see Kellogg (1967) [6,
Theorem XIII, page 271] but we will argue more directly. Since $A_{1}$ and
$A_{2}$ are disjoint sets, it is clear that
$E^{(1)}_{3,{\mathbf{n},\mathbf{p}}}(s_{3})$ is analytic at every point in
$B_{1}(\mathbf{0})\cap A_{1}$. In order to show that
$E^{(1)}_{1,{\mathbf{n},\mathbf{p}}}(s_{1})E^{(1)}_{2,{\mathbf{n},\mathbf{p}}}(s_{2})$
is analytic at $(x^{\prime},y^{\prime},z^{\prime})\in B_{1}(\mathbf{0})\cap
A_{1}$, we argue as follows. We may assume that there is an analytic function
$w:(a_{1},a_{3})\to{\mathbf{R}}$ such that
$E^{(1)}_{1,{\mathbf{n},\mathbf{p}}}$ and
$E^{(1)}_{2,{\mathbf{n},\mathbf{p}}}$ are restrictions of this function to
$(a_{1},a_{2})$ and $(a_{2},a_{3})$, respectively. Now
$(s_{1}-a_{1})+(s_{2}-a_{1})$ and $(s_{1}-a_{1})(s_{2}-a_{1})$ are analytic
functions of $(x,y,z)$ in a neighborhood of
$(x^{\prime},y^{\prime},z^{\prime})$. Lemma 6.2 implies that
$E^{(1)}_{1,{\mathbf{n},\mathbf{p}}}(s_{1})E^{(1)}_{2,{\mathbf{n},\mathbf{p}}}(s_{2})$
as a function of $(x,y,z)$ is analytic at
$(x^{\prime},y^{\prime},z^{\prime})$. It follows that
$G^{(1)}_{\mathbf{n},\mathbf{p}}$ is analytic at every point in
$B_{1}(\mathbf{0})\cap A_{1}$. In the same way, we show that
$G^{(1)}_{\mathbf{n},\mathbf{p}}$ is analytic at every point in
$B_{1}(\mathbf{0})\cap A_{2}$.
If $\mathbf{p}=(0,0,1)$ then we introduce the function
$\chi:=\begin{array}[]{ll}\sqrt{a_{3}-s_{3}}&\mbox{\ if\ }z\geq
0\\\\[2.84544pt] -\sqrt{a_{3}-s_{3}}&\mbox{otherwise}.\end{array}$
It follows from (19), (20) that $\chi$ is analytic on
${\mathbf{R}}^{3}\setminus A_{2}$. Then
$G^{(1)}_{\mathbf{n},\mathbf{p}}(x,y,z)=(x^{2}+y^{2}+z^{2}+1)^{-1/2}E^{(1)}_{1,{\mathbf{n},\mathbf{p}}}(s_{1})E^{(1)}_{2,{\mathbf{n},\mathbf{p}}}(s_{2})\chi(x,y,z)w_{3}(s_{3})$
on $B_{1}(\mathbf{0})$, where $w_{3}$ is analytic at $s_{3}=a_{3}$. We then
argue as above.
The other parity vectors $\mathbf{p}$ are treated similarly. ∎
###### Lemma 6.2.
Let $f:(B_{\epsilon})^{2}\to{\mathbf{C}}$,
$B_{\epsilon}=\\{s\in{\mathbf{C}}:|s|<\epsilon\\}$, be an analytic function
which is symmetric: $f(s,t)=f(t,s)$. Let $g,h:(B_{\delta})^{3}\to
B_{\epsilon}$ be functions such that $g+h$ and $gh$ are analytic. Then the
function $f(g(x,y,z),h(x,y,z))$ is analytic on $(B_{\delta})^{3}$.
Substituting $t_{j}=\Omega(s_{j})$, $j=2,3$, the Hilbert space $H_{1}$ from
Section 5 transforms to the Hilbert space $\tilde{H}_{1}$ consisting of
measurable functions $g:(a_{1},a_{2})\times(a_{2},a_{3})\to{\mathbf{C}}$ for
which
$\|g\|^{2}:=\int_{a_{2}}^{a_{3}}\int_{a_{1}}^{a_{2}}\frac{s_{3}-s_{2}}{\omega(s_{2})\omega(s_{3})}|g(s_{2},s_{3})|^{2}\,ds_{2}\,ds_{3}<\infty.$
(66)
By Theorem 5.3, for $g\in\tilde{H}_{1}$ and fixed $\mathbf{p}$, we have the
Fourier expansion
$g(s_{2},s_{3})\sim\sum_{\mathbf{n}}c_{{\mathbf{n},\mathbf{p}}}E^{(1)}_{2,{\mathbf{n},\mathbf{p}}}(s_{2})E^{(1)}_{3,{\mathbf{n},\mathbf{p}}}(s_{3}),$
(67)
where the Fourier coefficients are given by
$c_{{\mathbf{n},\mathbf{p}}}=\int_{a_{2}}^{a_{3}}\int_{a_{1}}^{a_{2}}\frac{s_{3}-s_{2}}{\omega(s_{2})\omega(s_{3})}g(s_{2},s_{3})E^{(1)}_{2,{\mathbf{n},\mathbf{p}}}(s_{2})E^{(1)}_{3,{\mathbf{n},\mathbf{p}}}(s_{3})\,ds_{2}\,ds_{3}.$
(68)
###### Theorem 6.3.
Consider the region $D_{1}$ defined by (63) for some fixed
$d_{1}\in(a_{0},a_{1})$. Let $e$ be a function defined on its boundary
$\partial D_{1}$ of parity $\mathbf{p}\in\\{0,1\\}^{3}$, and let
$g(s_{2},s_{3})$ be the representation of
$f(x,y,z):=(x^{2}+y^{2}+z^{2}+1)^{1/2}e(x,y,z)$ (69)
in 5-cyclide coordinates for $(x,y,z)\in\partial D_{1}$ with $x,y,z>0$.
Suppose $g\in\tilde{H}_{1}$ and expand $g$ in the series (67). Then the
function $u(x,y,z)$ given by
$u(x,y,z)=\sum_{\mathbf{n}}\frac{c_{{\mathbf{n},\mathbf{p}}}}{E^{(1)}_{1,{\mathbf{n},\mathbf{p}}}(d_{1})}G^{(1)}_{\mathbf{n},\mathbf{p}}(x,y,z)$
(70)
is harmonic in $D_{1}$ and assumes the values $e$ on the boundary of $D_{1}$
in the $L^{2}$-sense explained below.
###### Proof.
Let $d_{1}<d<a_{1}$ and $s_{1}\in[d,a_{1}]$. Using Theorems 5.4, 5.6 we
estimate
$\left|c_{{\mathbf{n},\mathbf{p}}}\frac{E^{(1)}_{1,{\mathbf{n},\mathbf{p}}}(s_{1})}{E^{(1)}_{1,{\mathbf{n},\mathbf{p}}}(d_{1})}E^{(1)}_{2,{\mathbf{n},\mathbf{p}}}(s_{2})E^{(1)}_{3,{\mathbf{n},\mathbf{p}}}(s_{3})\right|\leq|c_{{\mathbf{n},\mathbf{p}}}|Cr^{n_{2}+n_{3}}B(n_{2}^{2}+n_{3}^{2}+1),$
where the constants $B,C>0$ and $r\in(0,1)$ are independent of $\mathbf{n}$
and $s_{1}\in[d,a_{1}]$, $s_{2}\in[a_{1},a_{2}]$, $s_{3}\in[a_{2},a_{3}]$.
Since $c_{{\mathbf{n},\mathbf{p}}}$ is a bounded double sequence, this proves
that the series in (70) is absolutely and uniformly convergent on compact
subsets of $D_{1}$. Consequently, by Lemma 6.1, $u(x,y,z)$ is harmonic in
$D_{1}$. If we consider $u$ for fixed $s_{1}\in(d_{1},a_{1})$ and compute the
norm $\|u-e\|$ in the Hilbert space $\tilde{H}_{1}$ by the Parseval equality,
we obtain
$\|u-e\|^{2}\leq\sum_{\mathbf{n}}|c_{{\mathbf{n},\mathbf{p}}}|^{2}\left(1-\frac{E^{(1)}_{1,{\mathbf{n},\mathbf{p}}}(s_{1})}{E^{(1)}_{1,{\mathbf{n},\mathbf{p}}}(d_{1})}\right)^{2}.$
It is easy to see that the right-hand side converges to $0$ as $s_{1}\to
d_{1}$. Taking into account that $e$ and $u$ have the same parity, it follows
that $u$ assumes the boundary values $e$ in this $L^{2}$-sense. ∎
If $e$ is a function on $\partial D_{1}$ without parity, we write the function
$f$ from (69) as a sum of eight functions
$f=\sum_{\mathbf{p}}f_{\mathbf{p}},$
where $f_{\mathbf{p}}$ is of parity $\mathbf{p}$. Then the solution of the
corresponding Dirichlet problem is given by
$u(x,y,z)=\sum_{\mathbf{n},\mathbf{p}}\frac{c_{{\mathbf{n},\mathbf{p}}}}{E^{(1)}_{1,{\mathbf{n},\mathbf{p}}}(d_{1})}G^{(1)}_{\mathbf{n},\mathbf{p}}(x,y,z),$
(71)
where
$c_{\mathbf{n},\mathbf{p}}=\int_{a_{2}}^{a_{3}}\int_{a_{1}}^{a_{2}}\frac{s_{3}-s_{2}}{\omega(s_{2})\omega(s_{3})}g_{\mathbf{p}}(s_{2},s_{3})E^{(1)}_{2,{\mathbf{n},\mathbf{p}}}(s_{2})E^{(1)}_{3,{\mathbf{n},\mathbf{p}}}(s_{3})\,ds_{2}\,ds_{3}$
(72)
and $g_{\mathbf{p}}(s_{2},s_{3})$ is the representation of $f_{\mathbf{p}}$ in
5-cyclide coordinates.
We may write the coefficient $c_{\mathbf{n},\mathbf{p}}$ as an integral over
the surface $\partial D_{1}$ itself. The surface element is
$dS=h_{2}h_{3}\,ds_{2}\,ds_{3}$ with the scale factors $h_{2},h_{3}$ given in
(24), (25). Using
$\frac{h_{2}h_{3}}{h_{1}}=\frac{1}{4}(x^{2}+y^{2}+z^{2}+1)\frac{\omega(s_{1})}{\omega(s_{2})\omega(s_{3})}(s_{3}-s_{2}),$
we obtain from (72)
$c_{\mathbf{n},\mathbf{p}}=\frac{1}{2\omega(d_{1})E^{(1)}_{1,{\mathbf{n},\mathbf{p}}}(d_{1})}\int_{\partial
D_{1}}\frac{e}{h_{1}}G^{(1)}_{\mathbf{n},\mathbf{p}}\,dS,$ (73)
where
$h_{1}^{2}=\frac{1}{16}\left(\frac{(x^{2}+y^{2}+z^{2}-1)^{2}}{(d_{1}-a_{0})^{2}}+\frac{4x^{2}}{(d_{1}-a_{1})^{2}}+\frac{4y^{2}}{(d_{1}-a_{2})^{2}}+\frac{4z^{2}}{(d_{1}-a_{3})^{2}}\right).$
## 7 Second two-parameter Sturm-Liouville problem
We treat the two-parameter eigenvalue problem that appears when we wish to
solve the Dirichlet problem in ring cyclides. It is quite similar to the one
considered in Section 5, however, there are also some interesting differences.
Consider equation (26) on the intervals $(a_{0},a_{1})$ and $(a_{2},a_{3})$.
We obtain two Sturm-Liouville equations involving two parameters
$\displaystyle(\omega(s_{1})w_{1}^{\prime})^{\prime}-\frac{1}{\omega(s_{1})}\left(\frac{3}{16}s_{1}^{2}+\lambda_{1}s_{1}+\lambda_{2}\right)w_{1}=0,\quad
a_{0}<s_{1}<a_{1},$ (74)
$\displaystyle(\omega(s_{3})w_{3}^{\prime})^{\prime}-\frac{1}{\omega(s_{3})}\left(\frac{3}{16}s_{3}^{2}+\lambda_{1}s_{3}+\lambda_{2}\right)w_{3}=0,\quad
a_{2}<s_{3}<a_{3}.$ (75)
We again simplify by substituting $t_{j}=\Omega(s_{j})$,
$u_{j}(t_{j})=w_{j}(s_{j})$. Then (74), (75) become
$\displaystyle
u_{1}^{\prime\prime}-\left(\frac{3}{16}\\{\phi(t_{1})\\}^{2}+\lambda_{1}\phi(t_{1})+\lambda_{2}\right)u_{1}$
$\displaystyle=$ $\displaystyle 0,\quad b_{0}\leq t_{1}\leq b_{1},$ (76)
$\displaystyle
u_{3}^{\prime\prime}-\left(\frac{3}{16}\\{\phi(t_{3})\\}^{2}+\lambda_{1}\phi(t_{3})+\lambda_{2}\right)u_{3}$
$\displaystyle=$ $\displaystyle 0,\quad b_{2}\leq t_{3}\leq b_{3}.$ (77)
We add boundary conditions
$u_{1}^{\prime}(b_{0})=u_{1}^{\prime}(b_{1})=u_{3}^{\prime}(b_{2})=u_{3}^{\prime}(b_{3})=0.$
(78)
Differential equations (76), (77) together with boundary conditions (78) pose
a two-parameter Sturm-Liouville eigenvalue problem. In contrast to Section 5,
we now have a uniformly right-definite problem:
$-\left|\begin{array}[]{cc}\phi(t_{1})&1\\\
\phi(t_{3})&1\end{array}\right|=\phi(t_{3})-\phi(t_{1})\geq
a_{2}-a_{1}>0\quad\mbox{\ for\ }b_{0}\leq t_{1}\leq b_{1}\leq t_{3}\leq
b_{3}.$
We again have Klein’s oscillation theorem.
###### Theorem 7.1.
For every $\mathbf{n}=(n_{1},n_{3})\in{\mathbf{N}}_{0}^{2}$, there exists a
uniquely determined eigenvalue
$(\lambda_{1,\mathbf{n}},\lambda_{2,\mathbf{n}})\in{\mathbf{R}}^{2}$ admitting
an eigenfunction $u_{1}$ with exactly $n_{1}$ zeros in $(b_{0},b_{1})$ and an
eigenfunction $u_{3}$ with exactly $n_{3}$ zeros in $(b_{2},b_{3})$.
We state a result on the distribution of eigenvalues.
###### Theorem 7.2.
There are constants $A_{1},A_{2},A_{3}>0$ such that, for all
$\mathbf{n}\in{\mathbf{N}}_{0}^{2}$,
$\displaystyle-A_{1}(n_{3}^{2}+1)\leq\lambda_{1,\mathbf{n}}\leq
A_{2}(n_{1}^{2}+1),$ (79) $\displaystyle|\lambda_{2,\mathbf{n}}|\leq
A_{3}(n_{1}^{2}+n_{3}^{2}+1).$ (80)
###### Proof.
We abbreviate $\lambda_{j}=\lambda_{j,\mathbf{n}}$. Arguing as in the proof of
Theorem 5.2, there are $t_{1}\in[b_{0},b_{1}]$ and $t_{3}\in[b_{2},b_{3}]$
such that
$\displaystyle\frac{3}{16}\\{\phi(t_{1})\\}^{2}+\lambda_{1}\phi(t_{1})+\lambda_{2}$
$\displaystyle=$ $\displaystyle-\frac{\pi^{2}n_{1}^{2}}{(b_{1}-b_{0})^{2}},$
(81)
$\displaystyle\frac{3}{16}\\{\phi(t_{3})\\}^{2}+\lambda_{1}\phi(t_{3})+\lambda_{2}$
$\displaystyle=$ $\displaystyle-\frac{\pi^{2}n_{3}^{2}}{(b_{3}-b_{2})^{2}}.$
(82)
By subtracting (81) from (82), we obtain
$\frac{3}{16}\left(\\{\phi(t_{3})\\}^{2}-\\{\phi(t_{1})\\}^{2}\right)+\lambda_{1}(\phi(t_{3})-\phi(t_{1}))=\frac{\pi^{2}n_{1}^{2}}{(b_{1}-b_{0})^{2}}-\frac{\pi^{2}n_{3}^{2}}{(b_{3}-b_{2})^{2}}$
which implies (79). Now (80) follows from (79) and (81). ∎
Let $u_{1,\mathbf{n}}$ and $u_{3,\mathbf{n}}$ denote eigenfunctions
corresponding to the eigenvalue
$(\lambda_{1,\mathbf{n}},\lambda_{2,\mathbf{n}})$. The system of products
$u_{1,\mathbf{n}}(t_{1})u_{3,\mathbf{n}}(t_{3})$,
$\mathbf{n}\in{\mathbf{N}}_{0}^{2}$, is orthogonal in the Hilbert space
$H_{2}$ consisting of measurable functions
$f:(b_{0},b_{1})\times(b_{2},b_{3})\to{\mathbf{C}}$ satisfying
$\int_{b_{2}}^{b_{3}}\int_{b_{0}}^{b_{1}}(\phi(t_{3})-\phi(t_{1}))\left|f(t_{1},t_{3})\right|^{2}\,dt_{1}\,dt_{3}<\infty$
with inner product
$\int_{b_{2}}^{b_{3}}\int_{b_{0}}^{b_{1}}(\phi(t_{3})-\phi(t_{1}))f(t_{1},t_{3})\overline{g(t_{1},t_{3})}\,dt_{1}\,dt_{3}.$
We normalize the eigenfunctions so that
$\int_{b_{2}}^{b_{3}}\int_{b_{0}}^{b_{1}}(\phi(t_{3})-\phi(t_{1}))\left\\{u_{1,\mathbf{n}}(t_{1})\right\\}^{2}\left\\{u_{3,\mathbf{n}}(t_{3})\right\\}^{2}\,dt_{1}\,dt_{3}=1.$
(83)
We have the following completeness theorem.
###### Theorem 7.3.
The double sequence of functions
$u_{1,\mathbf{n}}(t_{1})u_{3,\mathbf{n}}(t_{3}),\quad\mathbf{n}\in{\mathbf{N}}_{0}^{2},$
forms an orthonormal basis in the Hilbert space $H_{2}$.
The normalization (83) leads to a bound on the values of eigenfunctions. Since
we have uniform right-definiteness, the proof is simpler than the proof of
Theorem 5.4.
###### Theorem 7.4.
There is a constant $B>0$ such that, for all
$\mathbf{n}\in{\mathbf{N}}_{0}^{2}$ and all $t_{1}\in[b_{0},b_{1}]$,
$t_{3}\in[b_{2},b_{3}]$,
$|u_{1,\mathbf{n}}(t_{1})u_{3,\mathbf{n}}(t_{3})|\leq
B(n_{1}^{2}+n_{3}^{2}+1).$
Let $u_{2,\mathbf{n}}$ be the solution of
$u_{2}^{\prime\prime}+\left(\frac{3}{16}\\{\phi(t_{2})\\}^{2}+\lambda_{1,\mathbf{n}}\phi(t_{2})+\lambda_{2,\mathbf{n}}\right)u_{2}=0,\quad
b_{1}\leq t_{2}\leq b_{2}$ (84)
determined by initial conditions
$u_{2}(b_{1})=1,\quad u_{2}^{\prime}(b_{1})=0.$
###### Theorem 7.5.
We have $u_{2,\mathbf{n}}(t_{2})>0$ for all $t_{2}\in[b_{1},b_{2}]$. If
$b_{1}<c_{1}<c_{2}<b_{2}$, then there are constants $C>0$ and $0<r<1$ such
that, for all $\mathbf{n}\in{\mathbf{N}}_{0}^{2}$ and $t_{2}\in[b_{1},c_{1}]$,
$\left|\frac{u_{2,\mathbf{n}}(t_{2})}{u_{2,\mathbf{n}}(c_{2})}\right|\leq
Cr^{n_{1}+n_{3}}.$
###### Proof.
We abbreviate $u_{2}=u_{2,\mathbf{n}}$ and
$\lambda_{j}=\lambda_{j,\mathbf{n}}$. We write (84) in the form
$u_{2}^{\prime\prime}+Q(\phi(t_{2}))u_{2}=0,\quad t_{2}\in[b_{1},b_{2}],$
where $Q$ is given by (50). According to (81), (82), there are
$s_{1}\in(a_{0},a_{1})$ and $s_{3}\in(a_{2},a_{3})$ such that
$Q(s_{1})=-\frac{\pi^{2}n_{1}^{2}}{(b_{1}-b_{0})^{2}},\quad
Q(s_{3})=-\frac{\pi^{2}n_{3}^{2}}{(b_{3}-b_{2})^{2}}.$
If $s\in[s_{1},s_{3}]$, then $Q(s)\leq L(s)$, where $L(s)$ is the linear
function with $L(s_{j})=Q(s_{j})$, $j=1,3$. It follows that
$Q(\phi(t_{2}))\leq-C(n_{1}+n_{3})^{2}\quad\mbox{\ for\
}t_{2}\in[c_{1},c_{2}].$ (85)
Again, we apply Lemma 5.7 to complete the proof. ∎
The results of this section remain valid for other boundary conditions. This
time we will need sixteen sets of boundary conditions labeled by
$\mathbf{p}=(p_{0},p_{1},p_{2},p_{3})\in\\{0,1\\}^{4}$. These boundary
conditions are
$\begin{array}[]{llll}u_{1}^{\prime}(b_{0})=0&\mbox{\ if\ }p_{0}=0,&\quad
u_{1}(b_{0})=0&\mbox{\ if\ }p_{0}=1,\\\ u_{1}^{\prime}(b_{1})=0&\mbox{\ if\
}p_{1}=0,&\quad u_{1}(b_{1})=0&\mbox{\ if\ }p_{1}=1,\\\
u_{3}^{\prime}(b_{2})=0&\mbox{\ if\ }p_{2}=0,&\quad u_{3}(b_{2})=0&\mbox{\ if\
}p_{2}=1,\\\ u_{3}^{\prime}(b_{3})=0&\mbox{\ if\ }p_{3}=0,&\quad
u_{3}(b_{3})=0&\mbox{\ if\ }p_{3}=1.\end{array}$ (86)
The initial conditions for $u_{2}$ are
$u_{2}(b_{1})=1,u_{2}^{\prime}(b_{1})=0\,\mbox{\ if\ }p_{1}=0,\quad
u_{2}(b_{1})=0,u_{2}^{\prime}(b_{1})=1\,\mbox{\ if\ }p_{1}=1.$ (87)
We denote the corresponding eigenvalues by
$(\lambda^{(2)}_{1,{\mathbf{n},\mathbf{p}}},\lambda^{(2)}_{2,{\mathbf{n},\mathbf{p}}})$.
The eigenfunctions will be denoted by
$E^{(2)}_{i,{\mathbf{n},\mathbf{p}}}(s_{i})=u_{i,\mathbf{n}}(t_{i})$,
$i=1,2,3$.
Summarizing, for $i=1,2,3$, $E^{(2)}_{i,{\mathbf{n},\mathbf{p}}}$ is a
solution of (26) on $(a_{i-1},a_{i})$ with
$(\lambda_{1},\lambda_{2})=(\lambda^{(2)}_{1,{\mathbf{n},\mathbf{p}}},\lambda^{(2)}_{2,{\mathbf{n},\mathbf{p}}})$.
The solution $E^{(2)}_{1,{\mathbf{n},\mathbf{p}}}(s_{1})$ has exponent
$\frac{1}{2}p_{0}$ at $a_{0}$, exponent $\frac{1}{2}p_{1}$ at $a_{1}$, and it
has $n_{1}$ zeros in $(a_{0},a_{1})$. The solution
$E^{(2)}_{2,{\mathbf{n},\mathbf{p}}}(s_{2})$ has exponent $\frac{1}{2}p_{1}$
at $a_{1}$, and its has no zeros in $(a_{1},a_{2})$. The solution
$E^{(2)}_{3,{\mathbf{n},\mathbf{p}}}(s_{3})$ has exponent $\frac{1}{2}p_{2}$
at $a_{2}$, exponent $\frac{1}{2}p_{3}$ at $a_{3}$, and it has $n_{3}$ zeros
in $(a_{2},a_{3})$.
## 8 Second Dirichlet problem
Consider the coordinate surface (21) for fixed $s=d_{2}\in(a_{1},a_{2})$. See
Figure 5(a,b,c) for a graphical depiction of the shape of this surface.
Figure 5: Coordinate surfaces $s_{2,3}=const$ for $a_{i}=i$.
If $(x^{\prime},y^{\prime},z^{\prime})\in\mathbb{S}_{2}$ then the ray
$(x,y,z)=t(x^{\prime},y^{\prime},z^{\prime})$ , $t>0$, is tangent to the
surface if and only if $(x^{\prime},y^{\prime},z^{\prime})$ is on the surface.
If $(x^{\prime},y^{\prime},z^{\prime})$ is in the elliptical cone
$\frac{4x^{\prime 2}}{d_{2}-a_{1}}+\frac{4y^{\prime
2}}{d_{2}-a_{2}}+\frac{4z^{\prime 2}}{d_{2}-a_{3}}>0,$
then the ray does not intersect the surface. Otherwise we have two
intersections $t=t_{1},t_{2}$ and $t_{1}t_{2}=1$. It follows from these
considerations that $s_{2}=d_{2}$ describes a connected surface of genus $1$.
The region interior to this surface is
$D_{2}=\\{(x,y,z)\in{\mathbf{R}}^{3}:s_{2}<d_{2}\\},$ (88)
or, equivalently,
$D_{2}=\\{(x,y,z):\frac{(x^{2}+y^{2}+z^{2}-1)^{2}}{d_{2}-a_{0}}+\frac{4x^{2}}{d_{2}-a_{1}}+\frac{4y^{2}}{d_{2}-a_{2}}+\frac{4z^{2}}{d_{2}-a_{3}}<0\\}.$
In this section we solve the Dirichlet problem for harmonic functions in
$D_{2}$ by the method of separation of variables.
Let $\mathbf{p}=(p_{0},p_{1},p_{2},p_{3})\in\\{0,1\\}^{4}$ and
$\mathbf{n}=(n_{1},n_{3})\in{\mathbf{N}}_{0}^{2}$. Using the functions
$E^{(2)}_{i,{\mathbf{n},\mathbf{p}}}$ introduced in Section 7 we define the
internal 5-cyclidic harmonic of the second kind
$G^{(2)}_{\mathbf{n},\mathbf{p}}(x,y,z)=(x^{2}+y^{2}+z^{2}+1)^{-1/2}E^{(2)}_{1,{\mathbf{n},\mathbf{p}}}(s_{1})E^{(2)}_{2,{\mathbf{n},\mathbf{p}}}(s_{2})E^{(2)}_{3,{\mathbf{n},\mathbf{p}}}(s_{3})$
(89)
for $x,y,z\in B_{1}(\mathbf{0})$ with $x,y,z\geq 0$. We extend the function
$(x^{2}+y^{2}+z^{2}+1)^{1/2}G^{(2)}_{\mathbf{n},\mathbf{p}}(x,y,z)$
to ${\mathbf{R}}^{3}$ as a function of parity $\mathbf{p}$. We call a function
$f$ of parity $\mathbf{p}=(p_{0},p_{1},p_{2},p_{3})$ if
$f(\sigma_{i}(x,y,z))=(-1)^{p_{i}}f(x,y,z),\quad\mbox{\ for\ }i=0,1,2,3,$ (90)
using inversion (29) and reflections (30).
We omit the proof of the following lemma which is similar to the proof of
Lemma 6.1.
###### Lemma 8.1.
The function $G^{(2)}_{\mathbf{n},\mathbf{p}}$ is harmonic at all points
$(x,y,z)\in{\mathbf{R}}^{3}$ at which $s_{2}\neq a_{2}$; see (34).
Note that $s_{2}<d_{2}<a_{2}$ in $D_{2}$. Therefore,
$G^{(2)}_{\mathbf{n},\mathbf{p}}$ is harmonic in an open set containing the
closure of $D_{2}$. Geometrically speaking, the set $s_{2}=a_{2}$ consists of
the part of the plane $y=0$ “outside” the two closed curves in Figure 3. The
ring cyclide $D_{2}$ passes through the $y=0$ plane inside those two closed
curves.
Substituting $t_{j}=\Omega(s_{j})$, $j=1,3$, the Hilbert space $H_{2}$ from
Section 7 transforms to the Hilbert space $\tilde{H}_{2}$ consisting of
measurable functions $g:(a_{0},a_{1})\times(a_{2},a_{3})\to{\mathbf{C}}$ for
which
$\|g\|^{2}:=\int_{a_{2}}^{a_{3}}\int_{a_{0}}^{a_{1}}\frac{s_{3}-s_{1}}{\omega(s_{1})\omega(s_{3})}|g(s_{1},s_{3})|^{2}\,ds_{1}\,ds_{3}<\infty.$
(91)
By Theorem 7.3, for $g\in\tilde{H}_{2}$ and fixed $\mathbf{p}$, we have the
Fourier expansion
$g(s_{1},s_{3})\sim\sum_{\mathbf{n}}c_{{\mathbf{n},\mathbf{p}}}E^{(2)}_{1,{\mathbf{n},\mathbf{p}}}(s_{1})E^{(2)}_{3,{\mathbf{n},\mathbf{p}}}(s_{3}),$
(92)
where the Fourier coefficients are given by
$c_{{\mathbf{n},\mathbf{p}}}=\int_{a_{2}}^{a_{3}}\int_{a_{0}}^{a_{1}}\frac{s_{3}-s_{1}}{\omega(s_{1})\omega(s_{3})}g(s_{1},s_{3})E^{(2)}_{1,{\mathbf{n},\mathbf{p}}}(s_{1})E^{(2)}_{3,{\mathbf{n},\mathbf{p}}}(s_{3})\,ds_{1}\,ds_{3}.$
###### Theorem 8.2.
Consider the region $D_{2}$ defined by (88) for some fixed
$d_{2}\in(a_{1},a_{2})$. Let $e$ be a function defined on its boundary
$\partial D_{2}$, and set
$f(x,y,z):=(x^{2}+y^{2}+z^{2}+1)^{1/2}e(x,y,z).$ (93)
Suppose that $f$ has parity $\mathbf{p}\in\\{0,1\\}^{4}$, and its
representation $g(s_{1},s_{3})$ in 5-cyclide coordinates is in
$\tilde{H}_{2}$. Expand $g$ in the series (92). Then the function $u(x,y,z)$
given by
$u(x,y,z)=\sum_{\mathbf{n}}\frac{c_{{\mathbf{n},\mathbf{p}}}}{E^{(2)}_{2,{\mathbf{n},\mathbf{p}}}(d_{2})}G^{(2)}_{\mathbf{n},\mathbf{p}}(x,y,z)$
(94)
is harmonic in $D_{2}$ and assumes the values $e$ on the boundary of $D_{2}$
in the $L^{2}$-sense explained below.
###### Proof.
The proof is similar to the proof of Theorem 6.3. It uses Theorems 7.4 and 7.5
to show that the series in (94) is absolutely and uniformly convergent on
compact subsets of $D_{2}$. Consequently, by Lemma 8.1, $u(x,y,z)$ is harmonic
in $D_{2}$. If we consider $u$ for fixed $s_{2}\in(a_{1},d_{2})$ and compute
the norm $\|u-e\|$ in the Hilbert space $\tilde{H}_{2}$, we obtain
$\|u-e\|^{2}\leq\sum_{\mathbf{n}}|c_{{\mathbf{n},\mathbf{p}}}|^{2}\left(1-\frac{E^{(2)}_{2,{\mathbf{n},\mathbf{p}}}(s_{2})}{E^{(2)}_{2,{\mathbf{n},\mathbf{p}}}(d_{2})}\right)^{2}.$
The right-hand side converges to $0$ as $s_{2}\to d_{2}$. Hence $u$ assumes
the boundary values $e$ in this $L^{2}$-sense. ∎
If $f$ is a function without parity, we write $f$ as a sum of sixteen
functions
$f=\sum_{\mathbf{p}\in\\{0,1\\}^{4}}f_{\mathbf{p}},$
where $f_{\mathbf{p}}$ is of parity $\mathbf{p}$. Then the solution of the
corresponding Dirichlet problem is given by
$u(x,y,z)=\sum_{\mathbf{n},\mathbf{p}}\frac{c_{{\mathbf{n},\mathbf{p}}}}{E^{(2)}_{2,{\mathbf{n},\mathbf{p}}}(d_{2})}G^{(2)}_{\mathbf{n},\mathbf{p}}(x,y,z),$
(95)
where
$c_{\mathbf{n},\mathbf{p}}=\int_{a_{2}}^{a_{3}}\int_{a_{0}}^{a_{1}}\frac{s_{3}-s_{1}}{\omega(s_{1})\omega(s_{3})}g_{\mathbf{p}}(s_{1},s_{3})E^{(2)}_{1,{\mathbf{n},\mathbf{p}}}(s_{1})E^{(2)}_{3,{\mathbf{n},\mathbf{p}}}(s_{3})\,ds_{1}\,ds_{3}$
(96)
and $g_{\mathbf{p}}(s_{1},s_{3})$ is the representation of $f_{\mathbf{p}}$ in
5-cyclide coordinates. We may also write $c_{\mathbf{n},\mathbf{p}}$ as a
surface integral
$c_{\mathbf{n},\mathbf{p}}=\frac{1}{4\omega(d_{2})E^{(2)}_{2,{\mathbf{n},\mathbf{p}}}(d_{2})}\int_{\partial
D_{2}}\frac{e}{h_{2}}G^{(2)}_{\mathbf{n},\mathbf{p}}\,dS,$ (97)
where
$h_{2}^{2}=\frac{1}{16}\left(\frac{(x^{2}+y^{2}+z^{2}-1)^{2}}{(d_{2}-a_{0})^{2}}+\frac{4x^{2}}{(d_{2}-a_{1})^{2}}+\frac{4y^{2}}{(d_{2}-a_{2})^{2}}+\frac{4z^{2}}{(d_{2}-a_{3})^{2}}\right).$
## 9 Third two-parameter Sturm-Liouville problem
If we write (26) on the intervals $(a_{0},a_{1})$ and $(a_{1},a_{2})$ in
formally self-adjoint form, we obtain
$\displaystyle(\omega(s_{1})w_{1}^{\prime})^{\prime}-\frac{1}{\omega(s_{1})}\left(\frac{3}{16}s_{1}^{2}+\lambda_{1}s_{1}+\lambda_{2}\right)w_{1}=0,\quad
a_{0}<s_{1}<a_{1},$ (98)
$\displaystyle(\omega(s_{2})w_{2}^{\prime})^{\prime}+\frac{1}{\omega(s_{2})}\left(\frac{3}{16}s_{2}^{2}+\lambda_{1}s_{2}+\lambda_{2}\right)w_{2}=0,\quad
a_{1}<s_{2}<a_{2}.$ (99)
We simplify the equations by substituting $t_{j}=\Omega(s_{j})$,
$u_{j}(t_{j})=w_{j}(s_{j})$, where $\Omega(s)$ is the elliptic integral (42).
Then (98), (99) become
$\displaystyle
u_{1}^{\prime\prime}-\left(\frac{3}{16}\\{\phi(t_{1})\\}^{2}+\lambda_{1}\phi(t_{1})+\lambda_{2}\right)u_{1}$
$\displaystyle=$ $\displaystyle 0,\quad b_{0}\leq t_{1}\leq b_{1},$ (100)
$\displaystyle
u_{2}^{\prime\prime}+\left(\frac{3}{16}\\{\phi(t_{2})\\}^{2}+\lambda_{1}\phi(t_{2})+\lambda_{2}\right)u_{2}$
$\displaystyle=$ $\displaystyle 0,\quad b_{1}\leq t_{2}\leq b_{2}.$ (101)
Of course, this system is very similar to the one considered in Section 5.
Therefore, we will be brief in this section. For a given
$\mathbf{p}=(p_{0},p_{1},p_{2})\in\\{0,1\\}^{3}$ we consider the boundary
conditions
$\begin{array}[]{llll}u_{1}^{\prime}(b_{0})=0&\mbox{\ if\ }p_{0}=0,&\quad
u_{1}(b_{0})=0&\mbox{\ if\ }p_{0}=1,\\\
u_{1}^{\prime}(b_{1})=u_{2}^{\prime}(b_{1})=0&\mbox{\ if\ }p_{1}=0,&\quad
u_{1}(b_{1})=u_{2}(b_{1})=0&\mbox{\ if\ }p_{1}=1,\\\
u_{2}^{\prime}(b_{2})=0&\mbox{\ if\ }p_{2}=0,&\quad u_{2}(b_{2})=0&\mbox{\ if\
}p_{2}=1.\end{array}$ (102)
The initial conditions for $u_{3}$ are
$u_{3}(b_{2})=1,u_{3}^{\prime}(b_{2})=0\,\mbox{\ if\ }p_{2}=0,\quad
u_{3}(b_{2})=0,u_{3}^{\prime}(b_{2})=1\,\mbox{\ if\ }p_{2}=1.$ (103)
We denote the corresponding eigenvalues by
$(\lambda^{(3)}_{1,{\mathbf{n},\mathbf{p}}},\lambda^{(3)}_{2,{\mathbf{n},\mathbf{p}}})$,
where $\mathbf{n}=(n_{1},n_{2})\in{\mathbf{N}}_{0}^{2}$. The eigenfunctions
will be denoted by
$E^{(3)}_{i,{\mathbf{n},\mathbf{p}}}(s_{i})=u_{i,\mathbf{n}}(t_{i})$,
$i=1,2,3$.
Summarizing, for $i=1,2,3$, $E^{(3)}_{i,{\mathbf{n},\mathbf{p}}}$ is a
solution of (26) on $(a_{i-1},a_{i})$ with
$(\lambda_{1},\lambda_{2})=(\lambda^{(3)}_{1,{\mathbf{n},\mathbf{p}}},\lambda^{(3)}_{2,{\mathbf{n},\mathbf{p}}})$.
The solution $E^{(3)}_{1,{\mathbf{n},\mathbf{p}}}(s_{1})$ has exponent
$\frac{1}{2}p_{0}$ at $a_{0}$, exponent $\frac{1}{2}p_{1}$ at $a_{1}$, and it
has $n_{1}$ zeros in $(a_{0},a_{1})$. The solution
$E^{(3)}_{2,{\mathbf{n},\mathbf{p}}}(s_{2})$ has exponent $\frac{1}{2}p_{1}$
at $a_{1}$, exponent $\frac{1}{2}p_{2}$ at $a_{2}$, and it has $n_{2}$ zeros
in $(a_{1},a_{2})$. The solution $E^{(3)}_{3,{\mathbf{n},\mathbf{p}}}(s_{3})$
has exponent $\frac{1}{2}p_{2}$ at $a_{2}$, and it has no zeros in
$(a_{2},a_{3})$.
## 10 Third Dirichlet problem
Consider the coordinate surface (21) for fixed $s=d_{3}\in(a_{2},a_{3})$. See
Figure 5(d,e,f) for a graphical depiction of the shape of this surface. If
$(x^{\prime},y^{\prime},z^{\prime})\in\mathbb{S}_{2}$ then the ray
$(x,y,z)=t(x^{\prime},y^{\prime},z^{\prime})$ , $t>0$, is tangent to the
surface if and only if $(x^{\prime},y^{\prime},z^{\prime})$ is on the surface.
If $(x^{\prime},y^{\prime},z^{\prime})$ is in the elliptical cone
$\frac{4x^{\prime 2}}{d_{2}-a_{1}}+\frac{4y^{\prime
2}}{d_{2}-a_{2}}+\frac{4z^{\prime 2}}{d_{2}-a_{3}}<0,$
then the ray intersects the surface twice at $t=t_{1},t_{2}$ with
$t_{1}t_{2}=1$. Otherwise, there is no intersection. Therefore, the coordinate
surface $s_{3}=d_{3}$ consists of two disjoint closed surfaces of genus $0$
separated by the plane $z=0$, and they are mirror images of each other under
the reflection $\sigma_{3}$.
We consider the region inside the surface $s_{3}=d_{3}$ with $z>0$
$D_{3}=\\{(x,y,z):z>0,s_{3}<d_{3}\\},$ (104)
or, equivalently,
$D_{3}=\\{(x,y,z):z>0,\frac{(x^{2}+y^{2}+z^{2}-1)^{2}}{d_{3}-a_{0}}+\frac{4x^{2}}{d_{3}-a_{1}}+\frac{4y^{2}}{d_{3}-a_{2}}+\frac{4z^{2}}{d_{3}-a_{3}}<0\\}.$
Next, we solve the Dirichlet problem for harmonic functions in $D_{3}$ by the
method of separation of variables.
Let $\mathbf{p}=(p_{0},p_{1},p_{2})\in\\{0,1\\}^{3}$ and
$\mathbf{n}=(n_{1},n_{2})\in{\mathbf{N}}_{0}^{2}$. Using the functions
$E^{(3)}_{i,{\mathbf{n},\mathbf{p}}}$ introduced in Section 9 we define the
internal 5-cyclidic harmonic of the third kind
$G^{(3)}_{\mathbf{n},\mathbf{p}}(x,y,z)=(x^{2}+y^{2}+z^{2}+1)^{-1/2}E^{(3)}_{1,{\mathbf{n},\mathbf{p}}}(s_{1})E^{(3)}_{2,{\mathbf{n},\mathbf{p}}}(s_{2})E^{(3)}_{3,{\mathbf{n},\mathbf{p}}}(s_{3})$
(105)
for $(x,y,z)\in B_{1}(\mathbf{0})$ with $x,y,z\geq 0$. We extend the function
$(x^{2}+y^{2}+z^{2}+1)^{1/2}G^{(3)}_{\mathbf{n},\mathbf{p}}(x,y,z)$
to the half-space $\\{(x,y,z):z>0\\}$ as a function of parity $\mathbf{p}$. We
call a function $f$ of parity $\mathbf{p}=(p_{0},p_{1},p_{2})$ if
$f(\sigma_{i}(x,y,z))=(-1)^{p_{i}}f(x,y,z),\quad\mbox{\ for\ }i=0,1,2$ (106)
using the inversion $\sigma_{0}$ and the reflections $\sigma_{1},\sigma_{2}$.
As before we have the following lemma.
###### Lemma 10.1.
The function $G^{(3)}_{\mathbf{n},\mathbf{p}}$ is harmonic on
$\\{(x,y,z):z>0\\}$.
We have the Hilbert space $\tilde{H}_{3}$ consisting of measurable functions
$g:(a_{0},a_{1})\times(a_{1},a_{2})\to{\mathbf{C}}$ for which
$\|g\|^{2}:=\int_{a_{1}}^{a_{2}}\int_{a_{0}}^{a_{1}}\frac{s_{2}-s_{1}}{\omega(s_{1})\omega(s_{1})}|g(s_{1},s_{2})|^{2}\,ds_{1}\,ds_{2}<\infty.$
(107)
For $g\in\tilde{H}_{3}$ and fixed $\mathbf{p}$, we have the Fourier expansion
$g(s_{1},s_{2})\sim\sum_{\mathbf{n}}c_{{\mathbf{n},\mathbf{p}}}E^{(3)}_{1,{\mathbf{n},\mathbf{p}}}(s_{1})E^{(3)}_{2,{\mathbf{n},\mathbf{p}}}(s_{2}),$
(108)
where the Fourier coefficients are given by
$c_{{\mathbf{n},\mathbf{p}}}=\int_{a_{1}}^{a_{2}}\int_{a_{0}}^{a_{1}}\frac{s_{2}-s_{1}}{\omega(s_{1})\omega(s_{2})}g(s_{1},s_{2})E^{(3)}_{1,{\mathbf{n},\mathbf{p}}}(s_{1})E^{(3)}_{2,{\mathbf{n},\mathbf{p}}}(s_{2})\,ds_{1}\,ds_{2}.$
(109)
###### Theorem 10.2.
Consider the region $D_{3}$ defined by (104) for some fixed
$d_{3}\in(a_{2},a_{3})$. Let $e$ be a function defined on its boundary
$\partial D_{3}$, and set
$f(x,y,z):=(x^{2}+y^{2}+z^{2}+1)^{1/2}e(x,y,z).$ (110)
Suppose that $f$ has parity $\mathbf{p}\in\\{0,1\\}^{3}$, and its
representation $g(s_{1},s_{2})$ in 5-cyclide coordinates is in
$\tilde{H}_{3}$. Expand $g$ in the series (108). Then the function $u(x,y,z)$
given by
$u(x,y,z)=\sum_{\mathbf{n}}\frac{c_{{\mathbf{n},\mathbf{p}}}}{E^{(3)}_{3,{\mathbf{n},\mathbf{p}}}(d_{3})}G^{(3)}_{\mathbf{n},\mathbf{p}}(x,y,z)$
(111)
is harmonic in $D_{3}$ and assumes the values $e$ on the boundary of $D_{3}$
in an $L^{2}$-sense.
If $f$ is a function without parity, we write $f$ as a sum of eight functions
$f=\sum_{\mathbf{p}\in\\{0,1\\}^{3}}f_{\mathbf{p}},$
where $f_{\mathbf{p}}$ is of parity $\mathbf{p}$. Then the solution of the
corresponding Dirichlet problem is given by
$u(x,y,z)=\sum_{\mathbf{n},\mathbf{p}}\frac{c_{{\mathbf{n},\mathbf{p}}}}{E^{(3)}_{3,{\mathbf{n},\mathbf{p}}}(d_{3})}G^{(3)}_{\mathbf{n},\mathbf{p}}(x,y,z),$
(112)
where
$c_{\mathbf{n},\mathbf{p}}=\int_{a_{1}}^{a_{2}}\int_{a_{0}}^{a_{1}}\frac{s_{2}-s_{1}}{\omega(s_{1})\omega(s_{2})}g_{\mathbf{p}}(s_{1},s_{2})E^{(3)}_{1,{\mathbf{n},\mathbf{p}}}(s_{1})E^{(3)}_{2,{\mathbf{n},\mathbf{p}}}(s_{2})\,ds_{1}\,ds_{2}$
(113)
and $g_{\mathbf{p}}(s_{1},s_{2})$ is the representation of $f_{\mathbf{p}}$ in
5-cyclide coordinates. Alternatively, we have
$c_{\mathbf{n},\mathbf{p}}=\frac{1}{2\omega(d_{3})E^{(3)}_{3,{\mathbf{n},\mathbf{p}}}(d_{3})}\int_{\partial
D_{3}}\frac{e}{h_{3}}G^{(3)}_{\mathbf{n},\mathbf{p}}\,dS,$ (114)
where
$h_{3}^{2}=\frac{1}{16}\left(\frac{(x^{2}+y^{2}+z^{2}-1)^{2}}{(d_{3}-a_{0})^{2}}+\frac{4x^{2}}{(d_{3}-a_{1})^{2}}+\frac{4y^{2}}{(d_{3}-a_{2})^{2}}+\frac{4z^{2}}{(d_{3}-a_{3})^{2}}\right).$
## Acknowledgements
This work was conducted while H. S. Cohl was a National Research Council
Research Postdoctoral Associate in the Applied and Computational Mathematics
Division at the National Institute of Standards and Technology, Gaithersburg,
Maryland, U.S.A.
## References
## References
* [1] F. V. Atkinson and A. B. Mingarelli. Multiparameter eigenvalue problems. Sturm-Liouville theory. CRC Press, Boca Raton, FL, 2011.
* [2] M. Bôcher. Ueber die Reihenentwickelungen der Potentialtheorie. B. G. Teubner, Leipzig, 1894.
* [3] C. P. Boyer, E. G. Kalnins, and W. Miller, Jr. Symmetry and separation of variables for the Helmholtz and Laplace equations. Nagoya Mathematical Journal, 60:35–80, 1976.
* [4] C. P. Boyer, E. G. Kalnins, and W. Miller, Jr. $R$-separable coordinates for three-dimensional complex Riemannian spaces. Transactions of the American Mathematical Society, 242:355–376, 1978.
* [5] E. L. Ince. Ordinary differential equations. Dover Publications, New York, 1956.
* [6] O. D. Kellogg. Foundations of potential theory. Reprint from the first edition of 1929. Die Grundlehren der Mathematischen Wissenschaften, Band 31. Springer-Verlag, Berlin, 1967.
* [7] F. Klein. Ueber Lamé’sche Functionen. Mathematische Annalen, 18(2):237–246, 1881.
* [8] F. Klein. Ueber Normirung der linearen Differentialgleichungen zweiter Ordnung. Mathematische Annalen, 38(1):144–152, 1891.
* [9] W. Miller, Jr. Symmetry and separation of variables. Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1977. With a foreword by Richard Askey, Encyclopedia of Mathematics and its Applications, Vol. 4.
* [10] P. Moon and D. E. Spencer. Field theory handbook, including coordinate systems, differential equations and their solutions. Springer-Verlag, Berlin, 1961.
* [11] P. M. Morse and H. Feshbach. Methods of theoretical physics. 2 volumes. McGraw-Hill Book Co., Inc., New York, 1953.
* [12] D. Schmidt and G. Wolf. A method of generating integral relations by the simultaneous separability of generalized Schrödinger equations. SIAM Journal on Mathematical Analysis, 10(4):823–838, 1979.
* [13] H. Volkmer. Multiparameter eigenvalue problems and expansion theorems, volume 1356 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1988.
|
arxiv-papers
| 2013-01-16T02:25:45 |
2024-09-04T02:49:40.372838
|
{
"license": "Public Domain",
"authors": "Howard S. Cohl and Hans Volkmer",
"submitter": "Howard Cohl",
"url": "https://arxiv.org/abs/1301.3559"
}
|
1301.3635
|
Deconvoluting GPD from an explicit DVCS amplitude
K. Bondarenko a⋄ and L. Jenkovszky b⋆
a Taras Shevchenko National University of Kyiv
b Bogolubov Institute for Theoretical Physics, National Academy of Sciences of
Ukraine
Kiev, 03680 Ukraine
###### Abstract
By assuming factorization of the GPD under the deconvolution integral for the
hand-bag diagram, we develop a method of solving this integral beyond the
cross-over line. As examples we use explicit models of deeply virtual Compton
scattering (DVCS) amplitudes to get solution for relevant GPDs.
$\begin{array}[]{ll}{}^{\diamond}\mbox{{\it e-mail
address:}}&\mbox{[email protected]}\\\ {}^{\star}\mbox{{\it e-mail
address:}}&\mbox{[email protected]}\\\ \end{array}$
## 1 Introduction
Deeply virtual Compton scattering (DVCS) combines the features of inelastic
processes with those of an elastic one. It has been realized [1, 2, 3] that a
straightforward generalization of the ordinary parton densities arises in
exclusive two-photon processes in the Bjorken region, e.g. in Compton
scattering with a highly virtual incoming photon, and in hard photoproduction
of mesons.
Generalized parton distributions (GPD) combine our knowledge about the one-
dimensional parton distribution in the longitudinal momentum with the impact-
parameter, or transverse distribution of matter in a hadron or nucleus[1, 2,
3, 4]. GPD cannot be measured directly, instead they appear as convolution
integrals of the form
$A(\xi,t,Q^{2})=\int\limits_{-1}^{1}\frac{\text{GPD}(x,\xi,t,Q^{2})}{x-\xi+i\varepsilon}dx,$
(1)
where $\xi\approx\dfrac{x_{Bj}}{2-x_{Bj}}$ is called skewness and $x$ is the
average longitudinal momentum fraction of the struck parton in the initial and
final states. This integral corresponds to deeply virtual Compton scattering
(DVCS) in the ”hand-bag” approximation, see Fig. 1 a), and $x$ is the
integration variable, not to be confused with the Bjorken variable $x_{Bj}$.
Eq. (1) is an integral equation for the unknown function GPD. The solution is
well known along the cross-over line $x=\xi$, where (see e.g. [4])
$\text{GPD}(x=\xi,\xi,t,Q^{2})=-\frac{1}{\pi}\text{Im}A(\xi,t,Q^{2}).$ (2)
Figure 1: a) Hand-bag diagram for DVCS; b) Hand-bag diagram for vector meson
production
GPD (the nominator in Eq. (1)) is assumed to be universal for all exclusive
processes of the type $\gamma^{*}p\rightarrow Vp,$ where $V$ stands for a real
photon or vector meson, the dependence on the processes coming from the
”handle”, calculable perturbatively. If so, many different processes of
exclusive vector particle production can be described, according to Eq. (1),
by a universal GPD with different integration kernels, see Fig. 1 b). Since
the GPD is not known apriori, one seeks for models of GPD (see e.g. [5]) based
on general constraints on its analytic and asymptotic behavior. The calculated
scattering amplitudes (cross sections) are than compared with the data to
confirm, modify or reject the chosen form of the GPD.
In this paper we use two explicit models of DVCS and vector meson production
(VMP) to obtain exact analytic expressions for GPD. To this end we propose a
method of deconvolution of integral equations with a singular kernel of the
Cauchy type to enable to go beyond the crossover trajectory and explore the
hitherto unknown properties of the GPD.
Regge pole models provide an adequate framework to describe high-energy, low
$t$ scattering phenomena. Being part of the $S$ matrix theory, however,
strictly speaking, they are valid only for the scattering of on-mass-shall
particles. Still, the successful application of the Regge pole models in
describing the off mass-shall HERA data opened the way to their use deep-
inelastic scattering, DVCS and vector meson production at HERA. The appearance
of the ”forbidden” $Q^{2}$ dependence was circumvent by calling the
$Q^{2}$-dependent Regge trajectories ”effective” ones. A particularly simple
and efficient Regge pole model [7] with $Q^{2}$-dependent residues (vertices)
will be used below to guide our deconvolution procedure.
In a more advanced Regge-pole model [8], to be used below, the Regge
trajectories and the residues do not depend on virtuality. Instead, the
amplitude contains two (or more) Regge-pole terms, whose relative weight
depends on $Q^{2}$, mimicking the multi-pole nature of the so-called QCD
Pomeron. Of great interest is the use of alternative models [6]. It is
important to note that the Regge pole amplitudes are complex functions, their
phase being fixed by the residue function. Since these model satisfy the basic
properties of the theory, yet they fit the data, we shall use them to
demonstrate the merits of our deconvolution procedure.
The paper is organized as follows: in Section 2 (Simple DVCS amplitude) we
introduce an explicit DVCS amplitude whose merit it simplicity, necessary in
testing the capacity of our deconvolution procedure; in Section 3
(Deconvolution) we describe the mathematical basis for our method; in Section
4 (Factorization) we introduce the factorization hypothesis, essential in
deriving GPD. In Sections 5 (Reggeometric model of DVCS) we apply our method
to a more complicated model for the amplitude.
## 2 A simple DVCS amplitude
For the sake of clarity we start with a very simply but efficient model of the
DVCS scattering amplitude [7]. This model accumulate in a compact way the main
properties of the expected DVCS amplitude. Namely, it is Regge-behaved, has
the required behavior in s, t and $Q^{2}$, yet it fits the DVCS data measured
at HERA by the Hi and ZEUS Collaborations. The kinematics of those data are
such that they correspond to diffractive scattering, and consequently the $t$
channel exchange is dominated by a single Pomeron trajectory.
The explicit forms of the relevant scattering amplitude is [7]:
$A(s,t,Q^{2})_{\gamma^{*}p\rightarrow\gamma
p}=-A_{0}e^{b_{1}\alpha(t)}e^{b_{2}\beta(z)}(-is/s_{0})^{\alpha(t)}=-A_{0}e^{(b_{1}+L)\alpha(t)+b_{2}\beta(z)},$
(3)
where $L\equiv\ln(-is/s_{0})$,
$\alpha(t)=\alpha(0)-\alpha_{1}\ln(1-\alpha_{2}t),$ (4)
and
$\beta(z)=\alpha(0)-\alpha_{1}\ln(1-\alpha_{2}z),$ (5)
where $z=t-Q^{2}$ is a new variable introduced in Ref. [7].
From Eq. (3) we get the real and imaginary parts of the DVCS amplitude:
$\text{Im}A_{DVCS}(x_{Bj},t,Q_{0}^{2})=\sin\Bigl{(}\frac{\pi\alpha(t)}{2}\Bigr{)}G(t,Q_{0}^{2})\Bigl{(}\frac{Q_{0}^{2}}{s_{0}x_{Bj}}\Bigr{)}^{\alpha(t)},$
(6)
$\text{Re}A_{DVCS}(x_{Bj},t,Q_{0}^{2})=-\cos\Bigl{(}\frac{\pi\alpha(t)}{2}\Bigr{)}G(t,Q_{0}^{2})\Bigl{(}\frac{Q_{0}^{2}}{s_{0}x_{Bj}}\Bigr{)}^{\alpha(t)},$
(7)
where
$G(t,Q^{2}_{0})=e^{b(\alpha(t)+\beta(t,Q^{2}_{0}))}.$ (8)
Skewness is defined in terms of the Bjorken variable $x_{Bj}$ as
$\xi\simeq\frac{x_{Bj}}{2-x_{Bj}}$ or v.v., $x_{Bj}\simeq\frac{2\xi}{1+\xi}$.
Since the $Q^{2}$ dependence in the model of Ref. [7] may not follow QCD
evolution (ambiguous in DVCS), we keep it fixed at some value $Q^{2}$ that may
be associated with the ”frozen” QCD coupling constant. It has no effect on the
deconvolution procedure in question.
In the deconvolution procedure, Eq. (1), the variables $t$ and $Q^{2}$ appear
merely as ”parameters”, therefore, for simplicity, we rewrite $A_{DVCS}$ as:
$A_{DVCS}(\xi,t,Q^{2})=-e^{-\frac{i\pi\alpha(t)}{2}}B(t,Q^{2})\left(\frac{1+\xi}{\xi}\right)^{\alpha(t)},$
(9)
where
$B(t,Q^{2})=G(t,Q_{0}^{2})\Bigl{(}\frac{Q_{0}^{2}}{2s_{0}}\Bigr{)}^{\alpha(t)}$
(10)
is a real-valued function.
With these ingredients we can now proceed to the deconvolution procedure.
## 3 Deconvolution
By substituting Eq. (9) in Eq. (1) we get (the dependence on the ”parameters”
is suppressed):
$-e^{-\frac{i\pi\alpha}{2}}B\left(\frac{1+\xi}{\xi}\right)^{\alpha}=\int\limits_{-1}^{1}\frac{\text{GPD}(x,\xi)}{x-\xi+i\varepsilon}dx,$
(11)
where $\alpha<1$ and $B$ is defined by Eq. (10). This is a singular integral
equation with a Cauchy-type unknown kernel. By using the Sokhotski rule
($\frac{1}{x+i\varepsilon}=\text{p.v.}\left(\frac{1}{x}\right)-i\pi\delta(x)$),
we obtain:
$-e^{-\frac{i\pi\alpha}{2}}B\left(\frac{1+\xi}{\xi}\right)^{\alpha}=\text{p.v.}\int\limits^{1}_{-1}\frac{\text{GPD}(x,\xi)}{x-\xi}dx-i\pi\text{GPD}(\xi,\xi).$
(12)
Since GPD is a real-valued function, we obtain its important property, called
the cross-over trajectory condition:
$\text{GPD}(x=\xi,\xi,t,Q^{2})=-\frac{1}{\pi}\text{Im}A_{DVCS}(\xi,t,Q^{2}).$
(13)
At this moment we need to make approximation to proceed. Assume that GPD
function could be factorized
$\text{GPD}(x,\xi)=f(\xi)\varphi(x),$ (14)
we get
$-\frac{e^{-\frac{i\pi\alpha}{2}}B}{f(\xi)}\left(\frac{1+\xi}{\xi}\right)^{\alpha}=\text{p.v.}\int\limits_{-1}^{1}\frac{\varphi(x)}{x-\xi}dx-i\pi\varphi(\xi).$
(15)
This is an integral equation for $\phi(x)$ to be resolved. As we know from the
Sokhotski-Plemelj theorem [9], there exist functions $\Phi^{+}(\xi)$ and
$\Phi^{-}(\xi)$ in the complex plain of the $\xi$ variable such that:
$\Phi^{+}(\xi)$ is defined for $\text{Im}(\xi)\geq 0$ and is analytic for
$\text{Im}(\xi)>0$; $\Phi^{-}(\xi)$ is defined for $\text{Im}(\xi)\leq 0$ and
is analytic for $\text{Im}(\xi)<0$, $\Phi^{+}(\xi)$, while $\Phi^{-}(\xi)$ are
continuous on $\mathbb{R}$ (except fir a few points of the integrable
discontinuity), and
$\Phi^{+}(\xi)+\Phi^{-}(\xi)=\frac{1}{\pi
i}\text{p.v.}\int\limits_{-1}^{1}\frac{\varphi(x)}{x-\xi}dx,\hphantom{\varphi(x)}\xi\in(-1,1),\hphantom{\mathbb{R}\setminus[-1,1]}$
(16) $\Phi^{+}(x)-\Phi^{-}(x)=\varphi(x),\hphantom{\frac{1}{\pi
i}\text{p.v.}\int\limits_{-1}^{1}\frac{\varphi(x)}{x-\xi}dx}x\in(-1,1),\hphantom{\mathbb{R}\setminus[-1,1]}$
(17) $\Phi^{+}(\xi)=\Phi^{-}(\xi),\hphantom{-\varphi(x),\frac{1}{\pi
i}\text{p.v.}\int\limits_{-1}^{1}\frac{\varphi(x)}{x-\xi}dx}\xi\in\mathbb{R}\setminus[-1,1].\hphantom{(-1,1)}$
(18)
Here Eq. (18) is a self-consistency condition for open-loop contours, see [9],
Sec. 14.3-11 (for a more profound treatment of the problem see [10]). This
condition means (Eq. (17)) that $\varphi(x)=0$ for
$x\in\mathbb{R}\setminus[-1,1]$. An alternative interpretation of this
condition we will discussed below.
By substituting Eqs. (16) and (17) into Eq. (15,) we arrive at the Riemann
problem:
$\Phi^{-}(\xi)=-\frac{e^{-\frac{i\pi\alpha}{2}}B}{2\pi
if(\xi)}\left(\frac{1+\xi}{\xi}\right)^{\alpha},\text{\quad}\xi\in(-1,1).$
(19)
The analytic continuation of $\Phi^{-}(\xi)$ for $\text{Im}(\xi)<0$ if very
simple: $\Phi^{-}(\xi)=\frac{e^{-\frac{i\pi\alpha}{2}}B}{2\pi
if(\xi)}\left(\frac{1+\xi}{\xi}\right)^{\alpha}$. From Eq. (18) we can also
find $\Phi^{+}(\xi)$ as an analytic continuation of $\Phi^{-}(\xi)$:
$\Phi^{+}(\xi)=-\frac{e^{-\frac{i\pi\alpha}{2}}B}{2\pi
if(\xi)}\left(\frac{1+\xi}{\xi}\right)^{\alpha},\text{\quad}\text{Im}(\xi)>0.$
(20)
Suppose $f(\xi)$ has no zeros or poles within the interval $(-1,1)$. In this
case we can write the solution even without any knowledge of the exact form of
$f(\xi)$. The function $\Phi^{+}(\xi)$ has an order $\alpha$ zero at the point
$-1$ and an order $\alpha$ pole at the point $0$. The cut from $-1$ to $0$
along $\mathbb{R}$ and Eq. (18) almost define $\Phi^{+}(\xi)$ on
$(-1,1)\setminus\\{0\\}$:
$\Phi^{+}(\xi)=-e^{-2\pi i\alpha}\frac{e^{-\frac{i\pi\alpha}{2}}B}{2\pi
if(\xi)}\left(\frac{1+\xi}{\xi}\right)^{\alpha},\text{\quad}\xi\in(-1,0),$
(21) $\Phi^{+}(\xi)=-\frac{e^{-\frac{i\pi\alpha}{2}}B}{2\pi
if(\xi)}\left(\frac{1+\xi}{\xi}\right)^{\alpha},\hphantom{e^{-2\pi
i\alpha}}\text{\quad}\xi\in(0,1).\hphantom{-}$ (22)
$\text{Im}\xi$$\text{Re}\xi$$0$$-\alpha$$\alpha$$-1$$1$
$\mathcal{A}$$\mathcal{B}$ Figure 2: The function $\Phi$ on the complex $\xi$
plane
For a better understanding of this result, look at this problem from a
different perspective: Eq. (18) may be interpret as if there existed only one
complex function on the $\xi$ plane — $\Phi$, while $\Phi^{+}$ and $\Phi^{-}$
are values of this function for $\text{Im}\xi>0$ and $\text{Im}\xi<0$,
respectively. $\Phi$ is a holomorphic function with branching points at $-1$
and $0$, where it has an order $\alpha$ zero and an order $\alpha$ pole. To
determine its values unambiguously on all complex plane, we cut the plane from
$-1$ to $0$ (see Fig. (2)). By encircling the point $-1$, initiating in the
segment $(-1;0)$ (contour $\mathcal{A}$ in the figure), the phase of function
$\Phi$ will change by a factor $e^{-2\pi i\alpha}$. However, if starting point
will be in the segment $(0;1)$ (contour $\mathcal{B}$ on the figure), the
phase of $\Phi$ will not change since the factors coming from the zero and the
pole will cancel; each other.
From Eq. (17) we have:
$\varphi(x)=\left(1-e^{-2\pi
i\alpha}\right)\frac{e^{-\frac{i\pi\alpha}{2}}B}{2\pi
if(x)}\left(\frac{1+x}{x}\right)^{\alpha},\hphantom{0}x\in(-1,0),$ (23)
$\varphi(x)=0,\hphantom{1-\left(e^{-2\pi
i\alpha}\right)\frac{e^{-\frac{i\pi\alpha}{2}}B}{2\pi
if(x)}\left(\frac{1+x}{x}\right)^{\alpha}}x\in(0,1).\hphantom{-}$ (24)
Finally, for the GPD we obtain the solution:
$\text{GPD}(x,\xi)=\left(1-e^{-2\pi
i\alpha}\right)\frac{e^{-\frac{i\pi\alpha}{2}}Bf(\xi)}{2\pi
if(x)}\left(\frac{1+x}{x}\right)^{\alpha},\hphantom{0}x\in(-1,0),$ (25)
$\text{GPD}(x,\xi)=0,\hphantom{1-\left(e^{-2\pi
i\alpha}\right)\frac{e^{-\frac{i\pi\alpha}{2}}Bf(\xi)}{2\pi
if(x)}\left(\frac{1+x}{x}\right)^{\alpha}}x\in(0,1).\hphantom{-}$ (26)
We see that this solution is unphysical, because GPD must be a real-valued
function for physical values of $\xi$ ($0<\xi<1$, see [3]). In the next
section we discuss how to satisfy this condition.
## 4 Factorization
In the previous section we have used the factorization assumption for the
deconvolution of Eq. (1). At this point it is hard to say how good is this
assumption. We shall estimate it by its consequences.
As we have seen in the previous section, the solution for GPD critically
depends on the analytic properties of the left-hand side of the Eq. (15) for
$\xi\in(-1;1)$. Our degree of freedom is the function $f(\xi)$ in Eq. (14).
Its zeros and poles may help us in getting physical values for GPD.
Let us construct a solution that will be a real-valued function for
$x\in(a,b)$, where $(a,b)$ is an arbitrary segment along the real axis. It is
easy to check, that the function
$f(\xi)=\left(\frac{1+\xi}{\xi}\right)^{\alpha}(\xi-a)^{\frac{\alpha}{2}}(b-\xi)^{-\frac{\alpha}{2}}g(\xi),$
(27)
solves this problem. Here $g(\xi)$ is an arbitrary real-valued function
without zeros or discontinuities of fractional order along $[-1,1]$. Let us
check this.
By substituting (27) into (19), we get:
$\Phi^{-}(\xi)=-\frac{e^{-\frac{i\pi\alpha}{2}}B}{2\pi
ig(\xi)}(\xi-a)^{-\frac{\alpha}{2}}(b-\xi)^{\frac{\alpha}{2}},\text{\quad}\xi\in(-1,1).$
(28)
By repeating the procedure from the previous section, we see that the
expression for $\Phi^{+}$ ($g(\xi)$ has only zeros or discontinuities of
integer order, so it changes phase only by integer number of $2\pi i$):
$\Phi^{+}(\xi)=-\frac{e^{\frac{i\pi\alpha}{2}}B}{2\pi
ig(\xi)}(\xi-a)^{-\frac{\alpha}{2}}(b-\xi)^{\frac{\alpha}{2}},\text{\quad}\xi\in(a,b),\hphantom{\mathbb{R}\setminus[a,b]}$
(29) $\Phi^{+}(\xi)=-\frac{e^{-\frac{i\pi\alpha}{2}}B}{2\pi
ig(\xi)}(\xi-a)^{-\frac{\alpha}{2}}(b-\xi)^{\frac{\alpha}{2}},\text{\quad}\xi\in\mathbb{R}\setminus[a,b].\hphantom{(a,b)}$
(30)
From Eq. (17) we have:
$\varphi(x)=-\sin\left(\frac{\pi\alpha}{2}\right)\frac{B}{\pi
g(x)}(x-a)^{-\frac{\alpha}{2}}(b-x)^{\frac{\alpha}{2}},\hphantom{0}x\in(a,b),\hphantom{\mathbb{R}\setminus[a,b]}$
(31) $\varphi(x)=0,\hphantom{-\sin\left(\frac{\pi\alpha}{2}\right)\frac{B}{\pi
g(x)}(x-a)^{-\frac{\alpha}{2}}(b-x)^{\frac{\alpha}{2}}}x\in\mathbb{R}\setminus[a,b].\hphantom{(a,b)}$
(32)
Accordingly, for GPD we have:
$\text{GPD}(x,\xi)=-\sin\left(\frac{\pi\alpha}{2}\right)\frac{B}{\pi}\frac{g(\xi)}{g(x)}\left(\frac{(\xi-a)(b-x)}{(x-a)(b-\xi)}\right)^{\frac{\alpha}{2}}\left(\frac{1+\xi}{\xi}\right)^{\alpha},\hphantom{0}x\in(a,b),\hphantom{\mathbb{R}\setminus[a,b]}$
(33)
$\text{GPD}(x,\xi)=0,\hphantom{-\sin\left(\frac{\pi\alpha}{2}\right)\frac{B}{\pi}\frac{g(\xi)}{g(x)}\left(\frac{(\xi-a)(b-x)}{(x-a)(b-\xi)}\right)^{\frac{\alpha}{2}}\left(\frac{1+\xi}{\xi}\right)^{\alpha}}x\in\mathbb{R}\setminus[a,b].\hphantom{(a,b)}$
(34)
The most interesting case for us is that of $a=-1$, $b=1$, then GPD is given
by the expression:
$\text{GPD}(x,\xi,t,Q^{2})=-\sin\left(\frac{\pi\alpha(t)}{2}\right)\frac{B(t,Q^{2})}{\pi}\frac{g(\xi)}{g(x)}\left(\frac{(1+\xi)(1-x)}{(1+x)(1-\xi)}\right)^{\frac{\alpha(t)}{2}}\left(\frac{1+\xi}{\xi}\right)^{\alpha(t)}=\\\
=-\frac{1}{\pi}\frac{g(\xi)}{g(x)}\left(\frac{(1+\xi)(1-x)}{(1+x)(1-\xi)}\right)^{\frac{\alpha(t)}{2}}\text{Im}A_{DVCS}(\xi,t,Q^{2}).$
(35)
This GPD satisfies the cross-over trajectory condition (13). It is easy to see
that any other function $f$ except (27) will violate the reality of GPD. So,
this is a general solution.
## 5 Reggeometric model of DVCS
Another simple and explicit albeit more advanced model for DVCS, as well as
for vector meson production (VMP) was proposed recently [8].
Here the Pomeron is also unique for all reactions, but the scattering
amplitude contains two terms, a ”soft” ($s$) and ”hard” ($h$) one, ”weighted”
by $\widetilde{Q^{2}}$-dependent pre-factors:
$A(s,t,Q^{2},M_{v}^{2})=\frac{\tilde{A_{s}}}{\Bigl{(}1+\frac{\widetilde{Q^{2}}}{\widetilde{Q_{s}^{2}}}\Bigr{)}^{n_{s}}}e^{-i\frac{\pi}{2}\alpha_{s}(t)}\Bigl{(}\frac{s}{s_{0s}}\Bigr{)}^{\alpha_{s}(t)}e^{2\Bigl{(}\frac{a_{s}}{\widetilde{Q^{2}}}+\frac{b_{s}}{2m_{p}^{2}}\Bigr{)}t}+\\\
+\frac{\tilde{A_{h}}\Bigl{(}\frac{\widetilde{Q^{2}}}{\widetilde{Q_{h}^{2}}}\Bigr{)}}{{\Bigl{(}1+\frac{\widetilde{Q^{2}}}{\widetilde{Q_{h}^{2}}}\Bigr{)}}^{n_{h}+1}}e^{-i\frac{\pi}{2}\alpha_{h}(t)}\Bigl{(}\frac{s}{s_{0h}}\Bigr{)}^{\alpha_{h}(t)}e^{2\Bigl{(}\frac{a_{h}}{\widetilde{Q^{2}}}+\frac{b_{h}}{2m_{p}^{2}}\Bigr{)}t}.$
(36)
In a sense, it mimics the multi(infinite)-component QCD Pomeron and, in
principle, is applicable to any exclusive reactions, be it ”soft” or ”hard”.
The parameters of the model are fixed from ”first principles” and/or from the
fits to the data, see [8].
Similarly to Sec. 3, and by using the relation $s=\frac{Q^{2}(1+\xi)}{2\xi}$,
we rewrite this amplitude in terms of the variable characteristic of GPD:
$A(\xi,t,Q^{2},M_{v}^{2})=e^{-\frac{i\pi\alpha_{s}(t)}{2}}B_{s}(t,Q^{2},M_{v}^{2})\left(\frac{1+\xi}{\xi}\right)^{\alpha_{s}(t)}+e^{-\frac{i\pi\alpha_{h}(t)}{2}}B_{h}(t,Q^{2},M_{v}^{2})\left(\frac{1+\xi}{\xi}\right)^{\alpha_{h}(t)},$
(37)
where
$B_{i}(t,Q^{2},M_{v}^{2})=\frac{\tilde{A_{i}}}{\Bigl{(}1+\frac{\widetilde{Q^{2}}}{\widetilde{Q_{i}^{2}}}\Bigr{)}^{n_{i}}}\Bigl{(}\frac{Q^{2}}{2s_{0i}}\Bigr{)}^{\alpha_{i}(t)}e^{2\Bigl{(}\frac{a_{i}}{\widetilde{Q^{2}}}+\frac{b_{i}}{2m_{p}^{2}}\Bigr{)}t},\text{\quad}i=s,h.$
(38)
It is easy to see that our simple factorization approach is not working here
any more. Really, let assume factorization to be of the form
$\text{GPD}(x,\xi,t,Q^{2},M_{v}^{2})=f(\xi,t,Q^{2},M_{v}^{2})\varphi(x,t,Q^{2},M_{v}^{2}),$
with an arbitrary function $f$. The relevant solution will appear in the form
(the dependence on the ”parameters” $t,Q^{2},M_{v}^{2}$ is again hidden):
$\text{GPD}(x,\xi)=\frac{f(\xi)}{2\pi if(x)}\left(\left(1-e^{-2\pi
i(\alpha_{s}+\alpha_{f}(\xi))}\right)e^{-\frac{i\pi\alpha_{s}}{2}}B_{s}\left(\frac{1+x}{x}\right)^{\alpha_{s}}+\right.\\\
+\left.\left(1-e^{-2\pi
i(\alpha_{h}+\alpha_{f}(\xi))}\right)e^{-\frac{i\pi\alpha_{h}}{2}}B_{h}\left(\frac{1+x}{x}\right)^{\alpha_{h}}\right),$
(39)
where $\alpha_{f}(\xi)$ is an additive phase shift depending on the analytic
properties of $f$. Let $\alpha_{s}\neq\alpha_{h}$ and their values be
independent. Then we can arrange the GPD to be a real-valued function for a
certain point $x=x_{0},\xi=x_{0}$, both terms in (39) complex, their being
real. However, by changing the value of $x$, we break this condition, since
$\alpha_{f}$ cannot depend on $x$.
We may proceed in the following way: the equation for GPD (1) is linear, so we
can split our amplitude (37) in two parts and search for the solution as the
sum of solutions for both parts.
Let
$A(\xi,t,Q^{2},M_{v}^{2})=A_{s}(\xi,t,Q^{2},M_{v}^{2})+A_{h}(\xi,t,Q^{2},M_{v}^{2})$,
where:
$A_{s}(\xi,t,Q^{2},M_{v}^{2})=e^{-\frac{i\pi\alpha_{s}(t)}{2}}B_{s}(t,Q^{2},M_{v}^{2})\left(\frac{1+\xi}{\xi}\right)^{\alpha_{s}(t)},$
(40)
$A_{h}(\xi,t,Q^{2},M_{v}^{2})=e^{-\frac{i\pi\alpha_{h}(t)}{2}}B_{h}(t,Q^{2},M_{v}^{2})\left(\frac{1+\xi}{\xi}\right)^{\alpha_{h}(t)}.$
(41)
By using the known solution (35) for (9) we write:
$\text{GPD}_{s}(x,\xi,\dots)=\sin\left(\frac{\pi\alpha_{s}}{2}\right)\frac{B_{s}}{\pi}\frac{g_{s}(\xi)}{g_{s}(x)}\left(\frac{(1+\xi)(1-x)}{(1+x)(1-\xi)}\right)^{\frac{\alpha_{s}}{2}}\left(\frac{1+\xi}{\xi}\right)^{\alpha_{s}},$
(42)
$\text{GPD}_{h}(x,\xi,\dots)=\sin\left(\frac{\pi\alpha_{h}}{2}\right)\frac{B_{h}}{\pi}\frac{g_{h}(\xi)}{g_{h}(x)}\left(\frac{(1+\xi)(1-x)}{(1+x)(1-\xi)}\right)^{\frac{\alpha_{h}}{2}}\left(\frac{1+\xi}{\xi}\right)^{\alpha_{h}},$
(43)
where $g_{s}(\xi,t,Q^{2},M_{v}^{2})$ and $g_{h}(\xi,t,Q^{2},M_{v}^{2})$ are
two independent arbitrary real-valued functions that have only integer order
zeros and/or singularities for $\xi\in[-1,1]$.
The resulting GPD will be the sum of (42) and (43).
## 6 Conclusions
We have suggested a method for deconvoluting general parton distributions
beyond the cross-over line based on the assumed factorization properties of
the GPD under the convolution integral. We test the method by using two
simple, explicit Regge-pole based models of the DVCS amplitude. It is
important that our scattering amplitudes, due to the presence of the Regge
phases, are complex-valued functions. Otherwise, the phase can be recovered
either by dispersion relation methods or from the interference between the
DVCS amplitude with Bethe-Heitler processes, see [11]. We hope that this
method can be applied to other models of DVCS (or vector meson production
(VMP)). Comparison with alternative models of DVCS (VMP) and the resulting
GPDs is of great interest.
Although the models [7] and [8], used in the present paper, contain some
$Q^{2}$ variation (introdeced phenomenologically), its value was fixed for
simplicity at some $Q_{0}^{2}$ that can be interpreted as a fixed QCD running
constant.
## Acknowledgements
We thank V. Magas and F. Paccanoni for discussions. The work of L.J. was
supported by the Program ”Fundamental properties of matter under extreme
conditions” of the National Academy of Sciences of Ukraine.
## References
* [1] F. M. Dittes, D. Mueller, D. Robaschik, B. Geyer and J. Horejsi, Phys. Lett. B 209 (1988) 325;
D. Mueller, D. Robaschik, B. Geyer, F. M. Dittes and J. Horejsi, Fortsch.
Phys. 42, (1994) 101.
* [2] A.V. Radyushkin, Phys. Rev. D 56 (1997) 5524.
* [3] X. Ji, J. Phys. G24, 1181 (1998), hep-ph/9807358.
* [4] K. Kumericki, D. Müller and K. Passek-Kumericki, Nucl. Phys. B 794 (2008) 244, hep-ph/0703179.
* [5] M. Guidal, M.V. Polyakov, A.V. Radyushkin and M. Vanderhaeghen, Nucleon From Factors from Generalized Parton Distributions, hep-ph/0410251; Matthias Burkardt, Generalized Parton Distributions for Large $x$, hep-ph/0401159; P. Kroll, Generalized Parton Distributions from Nucleon Form Factors, hep-ph/0512229; Charles E. Hyde, Michel Guidal, and Anatoly Radyushkin, Deeply Virtual Exclusive Processes and Generalized Parton Distributions, hep-ph/1101.2482v1; L. Jenkovszky, Phys. Rev. D 74 (2006) 114026; hep-ph/0607340.
* [6] D. Muller, Pomeron dominance in deeply virtual Compton scattering and the femtoholographic image of the proton, hep-ph/0605013.
* [7] M. Capua, R. Fiore, S. Fazio, L. Jenkovszky, and F. Paccanoni, Phys. Lett. B 645 (2007) 161;
S. Fazio, R. Fiore, L Jenkovszky, A. Lavorini, Exclusive diffractive
production of real photons and vector mesons in a factorized Regge-pole model
with non-linear Pomeron trajectory, Phys. Rev. D 85 (2011) 054009; hep-
ph/1109.6374.
* [8] S. Fazio, R. Fiore, L. Jenkovszky, and A. Saliy, Reggeometry of Deeply Virtual Compton Scattering and Exclusive Diffractive Vector Meson Production at HERA, In: Proceedings of the 14th Workshop on Elastic and Diffractive Scattering (EDS Blois Workshop), Frontiers of QCD: From Puzzles to Discoveries, Quy Nhon, Vietnam, December 15-21, 2011, www.slac.stanford.edu/C111215/.
S. Fazio et al., In the Procedings of ”Diffraction-2012”, held in Lanzarote,
Canarias Islands, September 2012;
* [9] A. D. Polyanin, A. V. Manzhirov, Handbook of Integral Equations, 2nd ed., Chapman&Hall/CRC Press, 2008.
* [10] F.D. Gakhov, Kraevye zadachi (Boundary problems). Gosudarstvennoe Izdatelstvo Fiz.-Mat. Literatury, Moscow, 1958 (in Russian).
* [11] K. Kumericki, D. Müller and M. Murray, HERMES impact for the access of Compton form factors, hep-ph/1301.1230.
|
arxiv-papers
| 2013-01-16T09:39:44 |
2024-09-04T02:49:40.387182
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "K. Bondarenko, L. Jenkovszky",
"submitter": "Kyrylo Bondarenko",
"url": "https://arxiv.org/abs/1301.3635"
}
|
1301.3776
|
# Thermodynamics of the apparent horizon in infrared modified Horava-Lifshitz
gravity
Ahmad [email protected] Physics Department and Biruni
Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran
Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O. Box
55134-441, Maragha, Iran
###### Abstract
It is well known that by applying the first law of thermodynamics to the
apparent horizon of a Friedmann-Robertson-Walker universe, one can derive the
corresponding Friedmann equations in Einstein, Gauss-Bonnet, and more general
Lovelock gravity. Is this a generic feature of any gravitational theory? Is
the prescription applicable to other gravities? In this paper we would like to
address the above questions by examining the same procedure for Horava-
Lifshitz gravity. We find that in Horava-Lifshitz gravity, this approach does
not work and we fail to reproduce a corresponding Friedmann equation in this
theory by applying the first law of thermodynamics on the apparent horizon,
together with the appropriate expression for the entropy in Horava-Lifshitz
gravity. The reason for this failure seems to be due to the fact that Horava-
Lifshitz gravity is not diffeomorphism invariant, and thus, the corresponding
field equation cannot be derived from the first law around horizon in the
spacetime. Without this, it implies that the specific gravitational theory is
not consistent, which shows an additional problematic feature of Horrava-
Lifshitz gravity. Nevertheless, if we still take the area formula of geometric
entropy and regard Horava-Lifshitz sector in the Friedmann equation as an
effective dark radiation, we are able to extract the corresponding Friedmann
equation from the first law of thermodynamics.
PACS numbers:04.50.Kd, 04.50.-h.
## I Introduction
Inspired by Lifshitz theory in solid state physics, recently, Horava proposed
a field theory model for a UV complete theory of gravity Horava . This theory
is a nonrelativistic renormalizable theory of gravity and reduces to
Einstein’s general relativity at large scales. The theory is usually referred
to as the Horava-Lifshitz (HL) theory. It also has manifest three- dimensional
spatial general covariance and time reparametrization invariance. Various
aspects of HL gravity have been investigated in the literature. A specific
direction of the research is the investigation of the thermodynamic properties
of HL gravity. In particular, black hole solutions in this gravity theory have
been attracted much attention. Here we review some thermodynamical properties
of HL gravity investigated previously, though our list is not complete.
Thermodynamics and stability of black holes in HL gravity have been studied in
Alex ; My ; cast ; Wang . It was shown that HL black holes are
thermodynamically stable in some parameter space and an unstable phase also
exists in other parameter spaces CaiHL0 . In Jamil ; Sama , the validity of
the generalized second law of thermodynamics has been explored in a universe
governed by HL gravity. In CaiHL the relationship between the first law of
thermodynamics and the gravitational field equation of a static, spherically
symmetric black hole in HL gravity has been explored. It was shown that,
gravitational field equations of static, spherically symmetric black holes in
HL theory can be written as the first law of thermodynamics on the black hole
horizon CaiHL . It was argued that this approach can lead to extracting an
expressions for the entropy and mass of HL black holes which are consistent
with those obtained from other approaches CaiHL . In the cosmological setups,
some attempts have been done to disclose the connection between thermodynamics
and gravitational field equations of the Friedmann-Robertson-Walker (FRW)
universe in HL theory. For example, following the entropic interpretation of
gravity proposed by Verlinde Ver , a modified Friedmann equation in HL gravity
was obtained in Shao . Following Shao , the connection between the Debye model
for the entropic force scenario and the modified Friedmann equations in HL
gravity was also studied Bin , although the results obtained in Shao ; Bin
seem to be incorrect. The reason is that the Friedman-like equation obtained
in these papers from the entropic force scenario is not the same as the one
directly derived from the field equations of HL gravity Alex ; Jamil . This
indicates that one cannot derive the Friedmann equation in HL cosmology from
the entropic gravity perspective. Other studies on HL gravity have been
carried out in otherHL .
On the other side, it was shown that the gravitational field equation of a
static spherically symmetric spacetime in Einstein, Gauss-Bonnet, and more
general Lovelock gravity can be transformed as the first law of thermodynamics
Par . The studies were also extended to other gravity theories such as $f(R)$
gravity Cai0 and scalar-tensor gravity Cai1 . In the cosmological setup, it
was shown that the differential form of the Friedmann equation of FRW universe
can be transformed to the first law of thermodynamics on the apparent horizon
Cai2 ; Cai3 . The extension of this connection has also been carried out in
the braneworld cosmology Cai4 ; Shey1 ; Shey2 . The deep connection between
the gravitational equation describing the gravity in the bulk and the first
law of thermodynamics on the apparent horizon reflects some deep ideas of
holography.
Is the inverse procedure also always possible? (that is extracting the general
field equations from the first law of thermodynamics?) Jacobson Jac was the
first who disclosed that the Einstein field equation of general relativity can
be derived from the relation between the horizon area and entropy, together
with the Clausius relation $\delta Q=T\delta S$. For so called $f(R)$ gravity,
Eling et al. Elin argued that the corresponding field equation describing
gravity can be derived from thermodynamics by using the procedure in Jac , but
a treatment with nonequilibrium thermodynamics of spacetime is needed. By
using the entropy expression associated with the horizon of the static
spherically symmetric black hole solutions in Einstein gravity, and replacing
the horizon radius $r_{+}$ with the apparent horizon radius, $\tilde{r}_{A}$,
and taking the ansatz for the temperature of the apparent horizon, it was
shown that the Friedmann equations of the FRW universe can be derived by
applying the first law of thermodynamics to the apparent horizon for a FRW
universe with any spatial curvature CaiKim . Employing the entropy relation of
black holes to apparent horizon in Gauss-Bonnet gravity and in the more
general Lovelock gravity, one also can get the corresponding Friedmann
equations in these theories CaiKim . Also, starting from the first law of
thermodynamics, $dE=T_{h}dS_{h}+WdV$, at the apparent horizon of a FRW
universe, and assuming that the associated entropy with the apparent horizon
has a quantum corrected relation,
$S=\frac{A}{4G}-\alpha\ln\frac{A}{4G}+\beta\frac{4G}{A}$, one is able to
derive the modified Friedmann equations describing the dynamics of the
universe with any spatial curvature Shey3 . These results indicate the
holographic properties of the gravitational field equations in a wide range of
gravity theories. For a recent review on the thermodynamical aspects of
gravity see Padrev .
In the present work, we would like to address the question on the connection
between thermodynamics and HL gravity, by applying the first law of
thermodynamics on the apparent horizon of a FRW universe and examine whether
we can extract the corresponding Friedmann equation in this gravity theory or
not. Our strategy here is to pick up the entropy expression associated with
the black hole horizon in HL gravity, assuming that the entropy formula also
holds for the apparent horizon of a FRW universe in HL gravity. In other
words, the apparent horizon has the same expression for entropy but we replace
the black hole horizon radius $r_{+}$ by the apparent horizon radius
$\tilde{r}_{A}$. We find out that the resulting Friedmann equation from the
first law of thermodynamics differs from one obtained directly by varying the
action of HL gravity with respect to the FRW metric. This shows an
inconsistency in HL gravity that originates from the fact that this theory is
not diffeomorphism invariant, and thus, the corresponding field equation
cannot derive from the first law around the horizon Pad2 .
This paper is structured as follows. In the next section we review the IR
modified HL theory and derive directly the corresponding Friedman equation by
varying the action. In Sec. III, we apply the the first law of thermodynamics
on the apparent horizon, together with the appropriate expressions for the
entropy and temperature, and extract the Friedmann-like equation of the
modified HL cosmology. Only in the IR limit does the result of this section
coincide with the Friedmann equation obtained in Sec. II. In Sec. IV, we
assume the area law for the apparent horizon and derive an effective Friedmann
equation in modified HL gravity by applying the first law of thermodynamics on
the apparent horizon. We finish our paper with a summary and discussion in
Sec. V.
## II Friedman Equation in IR modified HL cosmology
In this section we first review the cosmological model which is governed by HL
gravity. The dynamical variables are the lapse and shift functions, $N$ and
$N_{i}$, respectively, and the spatial metric $g_{ij}$. Using the Arnowitt-
Deser-Misner (ADM) formalism, the metric is parametrized as
$ds^{2}=-N^{2}dt^{2}+g_{ij}(dx^{i}-N^{i}dt)(dx^{j}-N^{j}dt).$ (1)
The Einstein-Hilbert action can be expressed as
$I_{EH}=\frac{1}{16\pi
G}\int{d^{4}x\sqrt{g}N\left[(K_{ij}K^{ij}-K^{2})+R-2\Lambda\right]},$ (2)
where $K_{ij}$ is the extrinsic curvature which takes the form
$K_{ij}=\frac{1}{2N}(\dot{g}_{ij}-\nabla_{i}N_{j}-\nabla_{j}N_{i}),$ (3)
and a dot denotes a derivative with respect to $t$ and covariant derivatives
defined with respect to the spatial metric $g_{ij}$. The action of HL gravity
is given by Horava
$\displaystyle
I_{SH}=\int{dtd^{3}x(\mathcal{L}_{0}+\tilde{\mathcal{L}}_{1}+\mathcal{L}_{m})},$
(4)
$\displaystyle\mathcal{L}_{0}=\sqrt{g}N\Big{\\{}\frac{2}{\kappa^{2}}(K_{ij}K^{ij}-\lambda
K^{2})+\frac{\kappa^{2}\mu^{2}(\Lambda_{W}R-3\Lambda_{W}^{2})}{8(1-3\lambda)}\Big{\\}},$
(5)
$\displaystyle\tilde{\mathcal{L}}_{1}=\sqrt{g}N\Big{\\{}\frac{\kappa^{2}\mu^{2}(1-4\lambda)}{32(1-3\lambda)}R^{2}-\frac{\kappa^{2}}{2\omega^{4}}\left(C_{ij}-\frac{\mu\omega^{2}}{2}R_{ij}\right)\left(C^{ij}-\frac{\mu\omega^{2}}{2}R^{ij}\right)\Big{\\}},$
(6)
where $\kappa^{2}$, $\lambda$, and $\omega$ are dimensionless constant
parameters, while $\mu$ and $\Lambda_{W}$ are constant parameters with mass
dimensions. Here $\mathcal{L}_{m}$ stands for the Lagrangian of the matter
field, $R$ and $R_{ij}$ are a three-dimensional spatial Ricci scalar and Ricci
tensor, and $C_{ij}$ is the Cotton tensor defined as
$C^{ij}=\epsilon^{ikl}\nabla_{k}\left(R^{j}_{\
l}-\frac{1}{4}R\delta^{j}_{l}\right)=\epsilon^{ikl}\nabla_{k}R^{j}_{\
l}-\frac{1}{4}\epsilon^{ikj}\partial_{k}R,$ (7)
where $\epsilon^{ikl}$ is the totally antisymmetric unit tensor. It is worth
mentioning that the IR vacuum of this theory is anti de-Sitter (AdS)
spacetimes. Hence, it is interesting to take a limit of the theory, which may
lead to a Minkowskian spacetime in the IR sector. For this purpose, one may
modify the theory by introducing $\mu^{4}R$ and then, taking the
$\Lambda_{W}\rightarrow 0$ limit Alex . This does not alter the UV properties
of the theory, but it changes the IR properties. That is, there exists a
Minkowski vacuum, instead of an AdS vacuum. We will now consider the limit of
this theory such that $\Lambda_{W}\rightarrow 0$. The deformed action of the
nonrelativistic renormalizable gravitational theory is hence given by Alex
$\displaystyle
I_{SH}=\int{dtd^{3}x\sqrt{g}N\Big{\\{}\frac{2}{\kappa^{2}}(K_{ij}K^{ij}-\lambda
K^{2})-\frac{\kappa^{2}}{2\omega^{4}}C_{ij}C^{ij}-\frac{\mu^{2}\kappa^{2}}{8}R_{ij}R^{ij}+\frac{\kappa^{2}\mu}{2\omega^{2}}\epsilon^{ijk}R_{il}\nabla_{j}R^{l}_{\
k}+\frac{\kappa^{2}\mu^{2}(1-4\lambda)}{32(1-3\lambda)}R^{2}+\mu^{4}R\Big{\\}}}.$
(8)
In the IR limit, action (8) can be written as the standard Einstein-Hilbert
action in the ADM formalism given in Eq. (2), provided Alex
$\lambda=1,\ \ c^{2}=\frac{\kappa^{2}\mu^{4}}{2},\ \ \
G=\frac{\kappa^{2}}{32\pi c}.$ (9)
The constant $\omega$ is given by cast
$\omega=\frac{8\mu^{2}(3\lambda-1)}{\kappa^{2}}.$ (10)
Besides, for $\omega\rightarrow\infty$ (equivalently, $\kappa^{2}\rightarrow
0$), action (8) reduces to the action of Einstein gravity cast ; Wang and
hence we expect all its solutions also recover to their respective ones in
general relativity.
We now consider a homogeneous and isotropic cosmological solution to the
theory (8) with the standard FRW geometry
$ds^{2}=-c^{2}dt^{2}+a^{2}(t)\left[\frac{dr^{2}}{1-kr^{2}}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\Omega^{2})\right].$ (11)
As usual, $k=-1,0,+1$ corresponds to an open, flat, or closed universe,
respectively. Suppose that the energy-momentum tensor of the total matter and
energy in the universe has the form of a perfect fluid
$T_{\mu\nu}=pg_{\mu\nu}+(\rho+p)U_{\mu}U_{\nu}$ where $U^{\nu}$ denotes the
four-velocity of the fluid and $\rho$ and $p$ are the total energy density and
pressure, respectively. The Friedmann equation, resulting from a variation of
action (8) with respect to the FRW metric, turns out to be Alex
$\displaystyle
H^{2}=\frac{\kappa^{2}}{6(3\lambda-1)}\left(\rho-\frac{6k\mu^{4}}{a^{2}}-\frac{3k^{2}\kappa^{2}\mu^{2}}{8(3\lambda-1)a^{4}}\right),$
(12)
where $H=\dot{a}/a$ is the Hubble parameter and we have imposed so called
projectability condition Char and set $N=1$ and $N^{i}=0$. The term
proportional to $a^{-4}$ in (12) is the usual “dark radiation” . The energy
conservation law $\nabla_{\mu}T^{\mu\nu}=0$ leads to the continuity equation
in the form
$\dot{\rho}+3H(\rho+p)=0.$ (13)
For $k=0$, there is no contribution from the higher order derivative terms in
the action. However, for $k\neq 0$, the higher derivative terms are
significant for small volumes i.e., for small $a$, and become insignificant
for large $a$, where they agree with general relativity. The standard
Friedmann equation is recovered, in units where $c=1$, provided we define
$\displaystyle G_{\rm cosm}=\frac{\kappa^{2}}{16\pi(3\lambda-1)},$ (14)
and
$\displaystyle\frac{\kappa^{2}\mu^{4}}{3\lambda-1}=1,$ (15)
where condition (15) also agrees for $\lambda=1=c$ with the second relation in
(9). Here $G_{\rm cosm}$ is the “cosmological” Newton’s constant. It is worth
mentioning that in theories such as HL, where the Lorentz invariance is
broken, the “gravitational” Newton’s constant $G$ (that is, the one that is
present in the gravitational action) does not coincide with the cosmological
Newton’s constant $G_{\rm cosm}$ (that is, the one that is present in
Friedmann equations), unless Lorentz invariance is restored Jamil . In the IR
limit where $\lambda=1$, the Lorentz invariance is restored, and hence $G_{\rm
cosm}=G$. Using the above identifications, as well as definition (10), the
Friedmann equation (12) can be rewritten as
$\displaystyle H^{2}+\frac{k}{a^{2}}=\frac{8\pi G_{\rm
cosm}}{3}\rho+\frac{k^{2}}{2\omega a^{4}}.$ (16)
One can easily see that in the limit $\omega\rightarrow\infty$, the dark
radiation term vanishes and the standard Friedmann equation is restored for
$\lambda=1$ ($G_{\rm cosm}=G$), as expected. On the other hand, one may also
further rewrite the Friedmann equation (16) in the form
$\displaystyle H^{2}+\frac{k}{a^{2}}=\frac{8\pi G_{\rm
cosm}}{3}\left(\rho+\rho_{\rm rad}\right),$ (17)
where in general $\rho=\rho_{m}+\rho_{D}$ is the total energy density of
matter and dark energy and we have defined
$\rho_{\rm rad}=\frac{3k^{2}}{16\pi\omega a^{4}G_{\rm cosm}}.$ (18)
Therefore, by regarding the dark radiation term which incorporates the effect
of HL gravity in the cosmological equation, we see that the Friedmann equation
of the modified HL cosmology can be written completely in the form of standard
cosmology. In such a case, one does not take into account the richness of the
HL gravity. Note that in the modified HL cosmology, $\rho_{\rm rad}$ cannot be
interpreted as the effective dark energy fluid, since at the late time
($a\rightarrow\infty$) where the dark energy should be dominated, it goes to
zero. Besides, for a universe with spatial curvature, it also has no
contribution, which is not reasonable. This is in contrast to HL cosmology in
the presence of a cosmological constant, where one can interpret the
contribution from $\Lambda$ and $\rho_{\rm rad}$ as an effective dark energy
Jamil ; Sama .
## III Friedman Equation in HL cosmology from the First law
In order to extract the cosmological equation governing the evolution of the
FRW universe in HL gravity, we need to have the entropy expression of static
spherically symmetric black holes in this theory. It was argued that there is
a deep connection between the entropy expression and gravitational field
equations CaiKim ; Shey3 ; CL . Having the entropy expression at hand, one can
derive the corresponding cosmological equations in a wide range of gravity
theories including Einstein, Gauss Bonnet and Lovelock gravity CaiKim . The
entropy expression depends on the gravity theory and takes different forms for
different gravity theories. In the deformed HL gravity, the entropy associated
with the event horizon of the static spherically symmetric black holes has the
form Wang
$S_{h}=\frac{A}{4G}+\frac{\pi}{\omega}\ln\frac{A}{G},$ (19)
where $A=4\pi r_{+}^{2}$ is the area of the black hole horizon. Throughout
this paper, we set $k_{B}=c=\hbar=1$ for simplicity. As one can see the
entropy formula has a logarithmic term, which is a characteristic of HL
gravity theory Wang . The parameter $\omega$ can be regarded as a
characteristic parameter in the deformed HL gravity and the entropy relation
will recover to the well-known area law as $\omega\rightarrow\infty$. In
general, the entropy expression for HL black holes may have an additional
constant term $S_{0}$. However, as the HL parameter $\omega\rightarrow\infty$,
the HL gravity should be reduced to the Einstein gravity; thus, we fix the
constant term in the entropy expression equal to zero.
We further assume the entropy expression (19) is also valid for the apparent
horizon of the FRW universe in HL gravity. Replacing the horizon radius
$r_{+}$ with the apparent horizon radius $\tilde{r}_{A}$, we have
$A=4\pi\tilde{r}_{A}^{2}$ for the apparent horizon area in Eq. (19). Now we
take the differential of the entropy (19),
$dS=\frac{\partial S}{\partial
A}dA=\frac{2\pi\tilde{r}_{A}}{G}\left(1+\frac{G}{\omega\tilde{r}_{A}^{2}}\right)d\tilde{r}_{A}.$
(20)
The line element of the FRW universe can be written as
$ds^{2}={h}_{ab}dx^{a}dx^{b}+\tilde{r}^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (21)
where $x^{0}=t,x^{1}=r$, $\tilde{r}=a(t)r$, and $h_{ab}$=diag
$(-1,a^{2}/(1-kr^{2}))$ represents the two-dimensional metric. Here $k$
specify the curvature of the spatial part of the metric. Solving equation
$h^{ab}\partial_{a}\tilde{r}\partial_{b}\tilde{r}=0$, we obtain the dynamical
apparent horizon radius of the FRW universe as
$\tilde{r}_{A}=\frac{1}{\sqrt{H^{2}+k/a^{2}}}.$ (22)
The temperature associated with the apparent horizon is defined as
$T_{h}=\kappa/2\pi$, where
$\kappa=\frac{1}{2\sqrt{-h}}\partial_{a}\left(\sqrt{-h}h^{ab}\partial_{b}\tilde{r}\right)$
is the surface gravity. It is easy to show that the surface gravity at the
apparent horizon of FRW universe can be written as Shey1
$\kappa=-\frac{1}{\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right).$
(23)
Since for $\dot{\tilde{r}}_{A}<2H\tilde{r}_{A}$, we have $\kappa<0$, which
leads to the negative temperature, one may, in general, define the temperature
on the apparent horizon as $T_{h}=|\kappa|/2\pi$. In addition, since we
associate a temperature with the apparent horizon, one may expect that the
apparent horizon has a kind of Hawking radiation just like a black hole event
horizon. This issue was previously addressed cao by showing the connection
between temperature on the apparent horizon and the Hawking radiation. This
study gives more solid physical implication of the temperature associated with
the apparent horizon.
The next quantity we need to have is the work density. In our case it can be
calculated as Hay2
$W=-\frac{1}{2}T^{\mu\nu}h_{\mu\nu}=\frac{1}{2}(\rho-p).$ (24)
Then, we suppose the first law of thermodynamics on the apparent horizon of
the universe in HL gravity holds and has the form
$dE=T_{h}dS_{h}+WdV,$ (25)
where $S_{h}$ is the entropy associated with the apparent horizon in HL
cosmology which has the form Eq. (19). The term $WdV$ in the first law comes
from the fact that we have a volume change for the total system enveloped by
the apparent horizon. As a result, the work term should be considered in the
first law. For a pure de Sitter space, $\rho=-p$, and the work term reduces to
the standard $-pdV$; thus, we obtain exactly the standard first law of
thermodynamics, $dE=TdS-pdV$.
Assume the total energy content of the universe inside a three-sphere of
radius $\tilde{r}_{A}$ is $E=\rho V$, where
$V=\frac{4\pi}{3}\tilde{r}_{A}^{3}$ is the volume enveloped by three-
dimensional sphere with the area of apparent horizon
$A=4\pi\tilde{r}_{A}^{2}$. Then we have
$\displaystyle dE$ $\displaystyle=$ $\displaystyle 4\pi\tilde{r}_{A}^{2}\rho
d\tilde{r}_{A}+\frac{4\pi}{3}\tilde{r}_{A}^{3}\dot{\rho}dt$ (26)
$\displaystyle=$ $\displaystyle 4\pi\tilde{r}_{A}^{2}\rho d\tilde{r}_{A}-4\pi
H\tilde{r}_{A}^{3}(\rho+p)dt.$
where we have used the continuity equation (13) in the last step. Substituting
Eqs. (20), (24) and (26) in the first law (25) and using the definition of the
temperature associated with the apparent horizon, we can get the differential
form of the Friedmann-like equation
$\frac{1}{4\pi
G}\frac{d\tilde{r}_{A}}{\tilde{r}_{A}^{3}}\left(1+\frac{G}{\omega{\tilde{r}_{A}}^{2}}\right)=H(\rho+p)dt.$
(27)
After using the continuity equation (13), we reach
$\frac{-2d\tilde{r}_{A}}{\tilde{r}_{A}^{3}}\left(1+\frac{G}{\omega{\tilde{r}_{A}}^{2}}\right)=\frac{8\pi
G}{3}d\rho.$ (28)
Integrating (28) yields
$\frac{1}{{\tilde{r}_{A}}^{2}}+\frac{G}{2\omega{\tilde{r}_{A}}^{4}}=\frac{8\pi
G}{3}\rho,$ (29)
where an integration constant has been absorbed into the energy density
$\rho$. Substituting $\tilde{r}_{A}$ from Eq.(22) we obtain a Friedmann-like
equation of a FRW universe in deformed HL gravity,
$H^{2}+\frac{k}{a^{2}}+\frac{G}{2\omega}\left(H^{2}+\frac{k}{a^{2}}\right)^{2}=\frac{8\pi
G}{3}\rho.$ (30)
As one can see the obtained Friedmann equation from the first law of
thermodynamics differs from the one obtained in (16) from varying the action
of the modified HL theory. This shows that at least for the HL cosmology, the
general prescription for deriving the field equation from the first law of
thermodynamics does not work. This indicates a problematic feature of HL
gravity. In the IR limit of the theory where $\omega\rightarrow\infty$, one
recovers the standard Friedmann equation
$H^{2}+\frac{k}{a^{2}}=\frac{8\pi G}{3}\rho.$ (31)
Also for the late time cosmology where the scale factor $a$ becomes large,
Eqs. (16) and (30) coincide. This is also consistent with the fact that the HL
theory of gravity reduces to Einstein’s general relativity at large scales.
## IV An effective approach to HL thermodynamics
In this section, we are going to use the effective approach and try to
reproduce the effective Friedmann equation of the deformed HL cosmology given
in (16), by applying the first law of thermodynamics on the apparent horizon.
In the effective approach all extra information of HL gravity can be absorbed
in an effective energy density and so, we consider that the universe contains
matter (and possible dark energy), plus this dark radiation. Thus, the total
energy density in the effective approach can be written as $\rho_{\rm
tot}=\rho+\rho_{\rm rad}$, where $\rho=\rho_{m}+\rho_{D}$. The total energy
density as well as the dark radiation sector also satisfy the continuity
equations
$\displaystyle\dot{\rho}_{\rm tot}+3H(\rho_{\rm tot}+p_{\rm tot})=0,$ (32)
$\displaystyle\dot{\rho}_{\rm rad}+3H(\rho_{\rm rad}+p_{\rm rad})=0.$ (33)
Since the effective gravitational sector is now just the standard general
relativity, the entropy associated with the apparent horizon obeys the well-
known area formula,
$S_{h}=\frac{A}{4G_{\rm cosm}},$ (34)
where $A$ is the horizon area and $G_{\rm cosm}$ is the cosmological Newton’s
constant given in (14). Taking the differential form of the above entropy, one
finds
$dS_{h}=\frac{2\pi\tilde{r}_{A}}{G_{\rm cosm}}d\tilde{r}_{A}.$ (35)
Now we assume that the total effective energy content of the universe is
$E_{\rm eff}=\rho_{\rm tot}V$. Differentiating we get
$\displaystyle dE_{\rm eff}$ $\displaystyle=$ $\displaystyle
4\pi\tilde{r}_{A}^{2}\rho_{\rm
tot}d\tilde{r}_{A}+\frac{4\pi}{3}\tilde{r}_{A}^{3}\dot{\rho}_{\rm tot}dt$ (36)
$\displaystyle=$ $\displaystyle 4\pi\tilde{r}_{A}^{2}\rho_{\rm
tot}d\tilde{r}_{A}-4\pi H\tilde{r}_{A}^{3}(\rho_{\rm tot}+p_{\rm tot})dt.$
where we have used the continuity equation (32) in the last step. Inserting
Eqs. (24), (35) and (36) in the first law of the form $dE_{\rm
eff}=T_{h}dS_{h}+WdV,$ and using the definition of the temperature associated
with the apparent horizon, we find the following equation:
$\frac{1}{4\pi G_{\rm
cosm}}\frac{d\tilde{r}_{A}}{\tilde{r}_{A}^{3}}=H(\rho_{\rm tot}+p_{\rm
tot})dt.$ (37)
After using the continuity equation (32), we reach
$\frac{-2d\tilde{r}_{A}}{\tilde{r}_{A}^{3}}=\frac{8\pi G_{\rm
cosm}}{3}d\rho_{\rm tot}.$ (38)
Integrating and then substituting $\tilde{r}_{A}$ from Eq.(22), we obtain the
effective Friedmann equation of the FRW universe in deformed HL gravity,
$H^{2}+\frac{k}{a^{2}}=\frac{8\pi G_{\rm cosm}}{3}(\rho+\rho_{\rm rad}).$ (39)
If we use the definition for $\rho_{\rm rad}$ given in Eq. (18), we can
further rewrite the above equation as
$\displaystyle H^{2}+\frac{k}{a^{2}}=\frac{8\pi G_{\rm
cosm}}{3}\rho+\frac{k^{2}}{2\omega a^{4}},$ (40)
which is exactly the result obtained in Eq. (16). This is an expected result,
since in the effective approach, all extra information of HL gravity has been
absorbed in the energy sector and the gravity sector is just standard
Einstein’s gravity. However, in this case one does not take into account the
richness of the effects of gravity. The fact that the results of two
approaches coincide in the IR limit ($\omega\rightarrow\infty$) can be easily
understood, since in this case the correction term in the entropy expression
(19) vanishes.
## V Summary and discussion
In summary, by applying the first law of thermodynamics to the apparent
horizon of a FRW universe one can derive the gravitational equations governing
the dynamics of the universe in a wide range of gravitational theories
including Einstein Jac , Gauss-Bonnet, Lovelock CaiKim , and $f(R)$ gravity
Elin . Can this prescription be applied to other gravitational theories? In
this paper we have shown that the answer is not always positive. Having the
entropy expression associated with the horizon of spherically symmetric black
holes in infrared modified HL gravity, and applying it to the apparent
horizon, we failed to extract the corresponding Friedmann equations in the
modified HL gravity. This is the main result we found in this paper, which
originates from the fact that HL gravity is not diffeomorphism invariant, and
thus the corresponding field equation cannot be derived from the first law
around horizon in the spacetime Pad2 . In order to justify this result, here
are some comments as follows.
(i) It was proved Pad2 that the field equations of any theory of gravity
which is diffeomorphism invariant must be expressible as a thermodynamic
identity, $TdS=dE$, around any event in the spacetime. Also, in Pad2 it was
shown that if the theory is not diffeomorphism invariant, then the field
equation cannot be reexpressed as the first law.
(ii) The action of HL gravity is invariant under a restricted class of
diffeomorphism. The fundamental symmetry of HL theory is the invariance under
space-independent time reparametrization and time-dependent spatial
diffeomorphism $t\rightarrow t^{\prime}(t),\ \
\vec{x}\rightarrow\vec{x}^{\prime}(t,x)$ Kiri . The time-dependent spatial
diffeomorphism allows an arbitrary change of spatial coordinates on each
constant time surface. However, the time reparametrization here is not allowed
to depend on spatial coordinates. As a result, unlike general relativity, in
HL gravity the foliation of spacetime by constant time hypersurfaces is not
just a choice of coordinates but is a physical entity.
(iii) Also HL gravity treats space and time in a different way. For the case
of isolated black holes, the metric is time independent and hence the
spacetime is invariant under infinitesimal diffeomorphism transformations. As
a result, according to the argument of Pad2 , the field equations of the
theory can be expressible as a thermodynamic identity, $TdS=dE$ around the
event horizon as it was shown in CaiHL .
(iv) In the background of a FRW universe, the metric is time dependent and
hence the metric is not invariant under the diffeomorphism transformation
$t\rightarrow t^{\prime}(t),\ \ \vec{x}\rightarrow\vec{x}^{\prime}(t,x)$. As a
result, the field equation cannot be expressed as the thermodynamics identity
$TdS=dE$. Without this, it implies that the specific gravitational theory
seems to be inconsistent and shows an additional problematic feature of HL
gravity. Our result in this paper confirms the general discussion given in
Pad2 .
It is important to note that if we still take the area formula of geometric
entropy and regard the HL sector in the Friedmann equation as an effective
dark radiation, we are able to derive the corresponding Friedmann equation
from the first law of thermodynamics. Indeed, we have the freedom to choose
two approaches: the first is known as the robust approach and the second is
called the effective approach. In the robust approach, we consider that the
universe contains only matter (and possible dark energy) and the effect of the
gravitational sector of HL gravity is incorporated through the modified
entropy expression on the horizon. In the effective approach, all extra
information of HL gravity can be absorbed in an effective energy density and
so we consider that the universe contains matter (and possible dark energy),
plus dark radiation. This approach is essentially same as Einstein’s gravity
theory. We have shown that in the robust approach, we failed to reproduce the
corresponding Friedmann equations in modified HL gravity. Nevertheless, in the
effective approach, we successfully derived the effective Friedmann equation
of modified HL gravity. Note that the results of the two approaches coincide
in the IR limit where $\omega\rightarrow\infty$. This can be easily
understood, since in this limit HL gravity reduces to Einstein’s gravity.
The results obtained in this paper, combined with those in Jamil ; Shao ; Bin
indicate that HL gravity is not a consistent gravitational theory from a
thermodynamic perspective. Is it possible to modify the HL theory in such a
way that it can be consistent with thermodynamics? This is quite an
interesting question, which deserves further investigation.
###### Acknowledgements.
I thank the referee for very constructive comments which helped me to improve
the paper significantly. I am also grateful to R. G. Cai, M. H. Dehghani and
M. H. Vahidinia for useful comments and helpful discussions. This work has
been supported financially by Research Institute for Astronomy and
Astrophysics of Maragha (RIAAM), Iran.
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|
arxiv-papers
| 2013-01-16T18:08:01 |
2024-09-04T02:49:40.399532
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ahmad Sheykhi",
"submitter": "Ahmad Sheykhi",
"url": "https://arxiv.org/abs/1301.3776"
}
|
1301.3811
|
$Id:espcrc1.tex,v1.22004/02/2411:22:11speppingExp$ Interval non-edge-colorable
bipartite graphs and multigraphsPetros A. Petrosyan, Hrant H. Khachatrian
# Interval non-edge-colorable bipartite graphs and multigraphs
Petros A. Petrosyan, Hrant H. Khachatrian[MCSD] email:
pet_petros@{ipia.sci.am, ysu.am, yahoo.com}email: [email protected] Department
of Informatics and Applied Mathematics,
Yerevan State University, 0025, Armenia Institute for Informatics and
Automation Problems,
National Academy of Sciences, 0014, Armenia
###### Abstract
An edge-coloring of a graph $G$ with colors $1,\ldots,t$ is called an interval
$t$-coloring if all colors are used, and the colors of edges incident to any
vertex of $G$ are distinct and form an interval of integers. In 1991 Erdős
constructed a bipartite graph with $27$ vertices and maximum degree $13$ which
has no interval coloring. Erdős’s counterexample is the smallest (in a sense
of maximum degree) known bipartite graph which is not interval colorable. On
the other hand, in 1992 Hansen showed that all bipartite graphs with maximum
degree at most $3$ have an interval coloring. In this paper we give some
methods for constructing of interval non-edge-colorable bipartite graphs. In
particular, by these methods, we construct three bipartite graphs which have
no interval coloring, contain $20,19,21$ vertices and have maximum degree
$11,12,13$, respectively. This partially answers a question that arose in
[T.R. Jensen, B. Toft, Graph coloring problems, Wiley Interscience Series in
Discrete Mathematics and Optimization, 1995, p. 204]. We also consider similar
problems for bipartite multigraphs.
Keywords: edge-coloring, interval coloring, bipartite graph, bipartite
multigraph
## 1 Introduction
In this paper we consider graphs which are finite, undirected, and have no
loops or multiple edges and multigraphs which may contain multiple edges but
no loops. Let $V(G)$ and $E(G)$ denote the sets of vertices and edges of a
multigraph $G$, respectively. For two distinct vertices $u$ and $v$ of a
multigraph $G$, let $E(uv)$ denote the set of all edges of $G$ joining $u$
with $v$, and let $\mu(uv)$ denote the number of edges joining $u$ with $v$
(i.e. $\mu(uv)=|E(uv)|$). The degree of a vertex $v\in V(G)$ is denoted by
$d_{G}(v)$ (or $d(v)$), the maximum degree of $G$ by $\Delta(G)$, and the
edge-chromatic number of $G$ by $\chi^{\prime}\left(G\right)$. The terms and
concepts that we do not define can be found in [23].
Let $G$ be a connected graph and $V(G)=\\{v_{1},\ldots,v_{n}\\}$, $n\geq 2$.
Let $P(v_{i},v_{j})$ be a simple path joining $v_{i}$ and $v_{j}$,
$VP(v_{i},v_{j})$ and $EP(v_{i},v_{j})$ denote the sets of vertices and edges
of this path, respectively.
A proper edge-coloring of a multigraph $G$ is a coloring of the edges of $G$
such that no two adjacent edges receive the same color. If $\alpha$ is a
proper edge-coloring of $G$ and $v\in V(G)$, then $S\left(v,\alpha\right)$
denotes the set of colors of edges incident to $v$. A proper edge-coloring of
a multigraph $G$ with colors $1,\ldots,t$ is called an interval $t$-coloring
if all colors are used, and for any vertex $v$ of $G$, the set
$S\left(v,\alpha\right)$ is an interval of integers. A multigraph $G$ is
interval colorable if it has an interval $t$-coloring for some positive
integer $t$. The set of all interval colorable multigraphs is denoted by
$\mathfrak{N}$. For a multigraph $G\in\mathfrak{N}$, the least value of $t$
for which $G$ has an interval $t$-coloring is denoted by $w\left(G\right)$.
The concept of interval edge-coloring of multigraphs was introduced by
Asratian and Kamalian [1, 2]. In [1, 2] they proved that if $G$ is interval
colorable, then $\chi^{\prime}\left(G\right)=\Delta(G)$. Moreover, if $G$ is
$r$-regular, then $G$ has an interval coloring if and only if $G$ has a proper
$r$-edge-coloring. This implies that the problem “Is a given $r$-regular
($r\geq 3$) graph interval colorable or not?”is $NP$-complete. Asratian and
Kamalian also proved [1, 2] that if a triangle-free graph $G$ has an interval
$t$-coloring, then $t\leq\left|V(G)\right|-1$. In [13] Kamalian investigated
interval colorings of complete bipartite graphs and trees. In particular, he
proved that the complete bipartite graph $K_{m,n}$ has an interval
$t$-coloring if and only if $m+n-\gcd(m,n)\leq t\leq m+n-1$, where $\gcd(m,n)$
is the greatest common divisor of $m$ and $n$. In [18] Petrosyan investigated
interval colorings of complete graphs and $n$-dimensional cubes. In
particular, he proved that if $n\leq t\leq\frac{n\left(n+1\right)}{2}$, then
the $n$-dimensional cube $Q_{n}$ has an interval $t$-coloring. In [20]
Sevast’janov proved that it is an $NP$-complete problem to decide whether a
bipartite graph has an interval coloring or not. On the other hand, computer
search in [5] showed that the following result holds.
###### Theorem 1
All bipartite graphs of order at most $14$ are interval colorable.
For subcubic bipartite graphs, Hansen proved the following
###### Theorem 2
[10]. If $G$ is a bipartite graph with $\Delta(G)\leq 3$, then
$G\in\mathfrak{N}$ and $w(G)\leq 4$.
For bipartite graphs with maximum degree $4$, Giaro proved the following two
results:
###### Theorem 3
[6]. If $G$ is a bipartite graph with $\Delta(G)=4$ and without a vertex of
degree $3$, then $G\in\mathfrak{N}$ and $w(G)=4$.
###### Theorem 4
[6]. The problem of deciding the existence of interval $\Delta(G)$-coloring of
a bipartite graph $G$ can be solved in polynomial time if $\Delta(G)\leq 4$
and is $NP$-complete if $\Delta(G)\geq 5$.
For bipartite graphs where one of the parts is small, the following theorem
was proved in [8].
###### Theorem 5
If $G$ is a bipartite graph with a bipartition $(U,V)$ and
$\min\\{|U|,|V|\\}\leq 3$, then $G\in\mathfrak{N}$.
Also, it is known that all regular bipartite graphs [1, 2], doubly convex
bipartite graphs [3, 14], grids [7], outerplanar bipartite graphs [9],
$\left(2,b\right)$-biregular bipartite graphs [11, 15, 16] and some classes of
$\left(3,4\right)$-biregular bipartite graphs [4, 19, 24] have interval
colorings. However, there are bipartite graphs which have no interval
colorings. First example of a bipartite graph that is not interval colorable
was obtained by Mirumyan [17] in 1989, but it was not published. The graph
which was found by Mirumyan has $19$ vertices and maximum degree $15$. First
published example was given by Sevast’janov [20] and it has $28$ vertices and
maximum degree $21$ (see Fig. 1). Other examples were obtained by Erdős ($27$
vertices and maximum degree $13$), by Hertz and de Werra ($21$ vertices and
maximum degree $14$), and by Malafiejski ($19$ vertices and maximum degree
$15$). In [12], Jensen and Toft posed the following question:
Figure 1: The Sevast’janov graph.
###### Problem 1
Is there a bipartite graph G with $4\leq\Delta(G)\leq 12$ and
$G\notin\mathfrak{N}$?
In the present paper we describe some methods for constructing of interval
non-edge-colorable bipartite graphs. In particular, by these methods, we
construct two bipartite graphs $G$ and $H$ with $\Delta(G)=11$, $\Delta(H)=12$
which have no interval coloring. This partially answers a question of Jensen
and Toft. In this paper we also consider similar problems for bipartite
multigraphs.
## 2 Interval non-edge-colorable bipartite graphs
### 2.1 Counterexamples by fat triangles
In 1949 Shannon [21] proved that
$\chi^{\prime}(G)\leq\left\lfloor\frac{3}{2}\Delta(G)\right\rfloor$ for any
multigraph $G$. Also, he showed that this upper bound is sharp for special
multigraphs which are called fat triangles. The fat triangle is a multigraph
with three vertices $x,y,z$ and $r$ edges between each pair of vertices, that
is, $\mu(xy)=\mu(yz)=\mu(xz)=r$. Later, Vizing [22] proved that if a
multigraph $G$ has
$\chi^{\prime}(G)=\left\lfloor\frac{3}{2}\Delta(G)\right\rfloor$ and
$\Delta(G)\geq 4$, then $G$ has a fat triangle as a subgraph. In this
paragraph we use fat triangles for constructing of interval non-edge-colorable
bipartite graphs. First note that the graph obtained by subdividing every edge
of a fat triangle is bipartite. Moreover, a new graph obtained from the
subdivided graph by connecting every inserted vertex to a new vertex is also
bipartite. Now let us define the graph $\Delta_{r,s,t}$ ($1\leq r\leq s\leq
t$) as follows:
$V(\Delta_{r,s,t})=\\{v,x,y,z\\}\cup\\{a_{1},\ldots,a_{r},b_{1},\ldots,b_{s},c_{1},\ldots,c_{t}\\}$,
$E(\Delta_{r,s,t})=\\{va_{i},xa_{i},ya_{i}:1\leq i\leq
r\\}\cup\\{vb_{j},xb_{j},zb_{j}:1\leq j\leq
s\\}\cup\\{vc_{k},yc_{k},zc_{k}:1\leq k\leq t\\}$.
Clearly, $\Delta_{r,s,t}$ is a connected bipartite graph with
$|V(\Delta_{r,s,t})|=r+s+t+4$, $d(x)=r+s$, $d(y)=r+t$, $d(z)=s+t$, and
$\Delta(\Delta_{r,s,t})=r+s+t$. Note that our $\Delta_{r,s,t}$ graphs
generalize Malafiejski’s rosettes $M_{k}$ given in [8], since
$M_{k}=\Delta_{k,k,k}$ for any $k\in\mathbf{N}$.
###### Theorem 6
If $r\geq 5$, then $\Delta_{r,s,t}\notin\mathfrak{N}$.
* Proof.
Suppose, to the contrary, that the graph $\Delta_{r,s,t}$ has an interval
$q$-coloring $\alpha$ for some $q\geq r+s+t$.
Consider the vertex $v$. Let $u$ and $w$ be two vertices adjacent to $v$ such
that $\alpha(vu)={\min}~{}S(v,\alpha)=p$ and
$\alpha(vw)={\max}~{}S(v,\alpha)=p+r+s+t-1$. By the construction of
$\Delta_{r,s,t}$, there is a path $P(u,w)$ in $\Delta_{r,s,t}-v$ of length two
joining $u$ with $w$, where
$P(u,w)=(u,uv^{\prime},v^{\prime},v^{\prime}w,w)$.
Since $d(u)=3$ and $d(v^{\prime})\leq s+t$, we have
$\alpha(uv^{\prime})\leq p+d(u)-1=p+2$ and thus
$\alpha(v^{\prime}w)\leq p+2+d(v^{\prime})-1=p+1+s+t$.
On the other hand, since $d(w)=3$, we have
$p+r+s+t-1=\alpha(vw)=\max S(v,\alpha)\leq p+1+s+t+d(w)-1=p+s+t+3$.
Hence, $r\leq 4$, which is a contradiction. $\square$
Figure 2: The graph $\Delta_{5,5,5}$.
Theorem 6 implies that the graph $\Delta_{5,5,5}$ with
$|V(\Delta_{5,5,5})|=19$ and $\Delta(\Delta_{5,5,5})=15$ shown in Fig. 2 has
no interval coloring. In fact, this is the example of an interval non-edge-
colorable bipartite graph that first was constructed by Mirumyan. This example
first appeared in [3] and [8], and currently is known as Malafiejski’s rosette
$M_{5}$.
###### Corollary 7
For any positive integer $\Delta\geq 15$, there is a connected bipartite graph
$G$ with $G\notin\mathfrak{N}$ and $\Delta(G)=\Delta$.
### 2.2 Counterexamples by finite projective planes
In this paragraph we use finite projective planes for constructing of interval
non-edge-colorable bipartite graphs. A finite projective plane $\pi(n)$ of
order $n$ ($n\geq 2$) has $n^{2}+n+1$ points and $n^{2}+n+1$ lines, and
satisfies the following properties:
P1
any two points determine a line;
P2
any two lines determine a point;
P3
every point is incident to $n+1$ lines;
P4
every line is incident to $n+1$ points.
Let $\\{1,\ldots,n^{2}+n+1\\}$ be the set of points and $L$ be the set of
lines of $\pi(n)$. Define the graph $Erd(r_{1},\ldots,r_{n^{2}+n+1})$
($r_{1}\geq\ldots\geq r_{n^{2}+n+1}\geq 1$) as follows:
$V(Erd(r_{1},\ldots,r_{n^{2}+n+1}))=\\{u\\}\cup\\{1,\ldots,n^{2}+n+1\\}$
$\cup\left\\{v^{(l_{i})}_{1},\ldots,v^{(l_{i})}_{r_{i}}:l_{i}\in L,1\leq i\leq
n^{2}+n+1\right\\}$,
$E(Erd(r_{1},\ldots,r_{n^{2}+n+1}))=\left\\{uv^{(l_{i})}_{1},\ldots,uv^{(l_{i})}_{r_{i}}:l_{i}\in
L,1\leq i\leq n^{2}+n+1\right\\}\cup$
$\bigcup_{i=1}^{n^{2}+n+1}\left\\{v^{(l_{i})}_{1}k,\ldots,v^{(l_{i})}_{r_{i}}k:l_{i}\in
L,k\in l_{i},1\leq k\leq n^{2}+n+1\right\\}$.
Clearly, $Erd(r_{1},\ldots,r_{n^{2}+n+1})$ is a connected bipartite graph with
$\Delta(Erd(r_{1},\ldots,r_{n^{2}+n+1}))=\underset{i=1}{\overset{n^{2}+n+1}{\sum}}r_{i}$
and
$|V(Erd(r_{1},\ldots,r_{n^{2}+n+1}))|=\underset{i=1}{\overset{n^{2}+n+1}{\sum}}r_{i}+n^{2}+n+2$.
Note that the graph $Erd(1,1,1,1,1,1,1,1,1,1,1,1,1)$ was described by Erdős in
1991 [12]. This graph has $27$ vertices and maximum degree $13$.
###### Theorem 8
If $\underset{i=n+2}{\overset{n^{2}+n+1}{\sum}}r_{i}>2(n+1)$, then
$Erd(r_{1},\ldots,r_{n^{2}+n+1})\notin\mathfrak{N}$.
* Proof.
Suppose, to the contrary, that the graph $Erd(r_{1},\ldots,r_{n^{2}+n+1})$ has
an interval $t$-coloring $\alpha$ for some
$t\geq\underset{i=1}{\overset{n^{2}+n+1}{\sum}}r_{i}$.
Consider the vertex $u$. Let $v_{p}^{(l_{i_{0}})}$ and $v_{q}^{(l_{j_{0}})}$
be two vertices adjacent to $u$ such that
$\alpha\left(uv_{p}^{(l_{i_{0}})}\right)={\min}~{}S(u,\alpha)=s$ and
$\alpha\left(uv_{q}^{(l_{j_{0}})}\right)={\max}~{}S(u,\alpha)=s+\underset{i=1}{\overset{n^{2}+n+1}{\sum}}r_{i}-1$.
If $l_{i_{0}}=l_{j_{0}}$, then, by the construction of
$Erd(r_{1},\ldots,r_{n^{2}+n+1})$, there exists $k_{0}$ such that
$k_{0}v_{p}^{(l_{i_{0}})}$, $k_{0}v_{q}^{(l_{j_{0}})}\in
E(Erd(r_{1},\ldots,r_{n^{2}+n+1}))$. If $l_{i_{0}}\neq l_{j_{0}}$, then, by
the construction of $Erd(r_{1},\ldots,r_{n^{2}+n+1})$ and the property P2,
there exists $k_{0}$ such that $k_{0}v_{p}^{(l_{i_{0}})}$,
$k_{0}v_{q}^{(l_{j_{0}})}\in E(Erd(r_{1},\ldots,r_{n^{2}+n+1}))$.
By the construction of $Erd(r_{1},\ldots,r_{n^{2}+n+1})$ and properties P3 and
P4, we have
$d\left(v_{p}^{(l_{i_{0}})}\right)=d\left(v_{q}^{(l_{j_{0}})}\right)=n+2$ and
$\alpha\left(k_{0}v_{p}^{(l_{i_{0}})}\right)\leq
s+d\left(v_{p}^{(l_{i_{0}})}\right)-1=s+n+1$ and thus
$\alpha\left(k_{0}v_{q}^{(l_{j_{0}})}\right)\leq s+n+1+d(k_{0})-1\leq
s+n+\underset{i=1}{\overset{n+1}{\sum}}r_{i}$.
This implies that
$s+\underset{i=1}{\overset{n^{2}+n+1}{\sum}}r_{i}-1=\alpha\left(uv_{q}^{(l_{j_{0}})}\right)={\max}~{}S(u,\alpha)\leq
s+n+\underset{i=1}{\overset{n+1}{\sum}}r_{i}+d\left(v_{q}^{(l_{j_{0}})}\right)-1=s+2n+1+\underset{i=1}{\overset{n+1}{\sum}}r_{i}$.
Hence, $\underset{i=n+2}{\overset{n^{2}+n+1}{\sum}}r_{i}\leq 2(n+1)$, which is
a contradiction. $\square$
###### Corollary 9
For any positive integer $\Delta\geq 13$, there is a connected bipartite graph
$G$ with $G\notin\mathfrak{N}$ and $\Delta(G)=\Delta$.
Figure 3: The graph $Erd(2,2,2,2,2,2,1)$.
Theorem 8 implies that the graph $Erd(2,2,2,2,2,2,1)$ with
$|V(Erd(2,2,2,2,2,2,1))|=21$ and $\Delta(Erd(2,2,2,2,2,2,1))=13$ shown in Fig.
3 has no interval coloring. Also, Theorem 8 implies that the graph
$Erd(2,2,2,2,2,2,2)$ with $|V(Erd(2,2,2,2,2,2,2))|=22$ and
$\Delta(Erd(2,2,2,2,2,2,1))=14$ has no interval coloring. In the next section
we show that there is a connected bipartite graph $G$ with $|V(G)|=21$ and
$\Delta(G)=14$ which is not interval colorable.
### 2.3 Counterexamples by trees
Let $T$ be a tree and $V(T)=\\{v_{1},\ldots,v_{n}\\}$, $n\geq 2$. For a simple
path $P(v_{i},v_{j})$, define $L(v_{i},v_{j})$ as follows:
$L(v_{i},v_{j})=|EP(v_{i},v_{j})|+|\left\\{uw:uw\in E(T),u\in
VP(v_{i},v_{j}),w\notin VP(v_{i},v_{j})\right\\}|$.
Define:
$M(T)={\max}_{1\leq i\leq n,1\leq j\leq n}L(v_{i},v_{j})$.
In [14], Kamalian proved the following result.
###### Theorem 10
If $T$ is a tree, then $T$ has an interval $t$-coloring if and only if
$\Delta(T)\leq t\leq M(T)$.
Now let $T$ be a tree in which the distance between any two pendant vertices
is even and $F(T)=\\{v:v\in V(T)\wedge d_{T}(v)=1\\}$.
Let us define the graph $\widetilde{T}$ as follows:
$V(\widetilde{T})=V(T)\cup\\{u\\}$, $u\notin V(T)$,
$E(\widetilde{T})=E(T)\cup\\{uv:v\in F(T)\\}$.
Clearly, $\widetilde{T}$ is a connected bipartite graph with
$\Delta(\widetilde{T})=|F(T)|$.
###### Theorem 11
If $T$ is a tree in which the distance between any two pendant vertices is
even and $|F(T)|>M(T)+2$, then $\widetilde{T}\notin\mathfrak{N}$.
* Proof.
Suppose, to the contrary, that $\widetilde{T}$ has an interval $t$-coloring
$\alpha$ for some $t\geq|F(T)|$.
Consider the vertex $u$. Let $v$ and $v^{\prime}$ be two vertices adjacent to
$u$ such that $\alpha(uv)={\min}~{}S(u,\alpha)=s$ and
$\alpha(uv^{\prime})={\max}~{}S(u,\alpha)=s+|F(T)|-1$. Since $\widetilde{T}-u$
is a tree, there is a unique path $P(v,v^{\prime})$ in $\widetilde{T}-u$
joining $v$ with $v^{\prime}$, where
$P(v,v^{\prime})=(v_{1},e_{1},v_{2},\ldots,v_{i},e_{i},v_{i+1},\ldots,v_{k},e_{k},v_{k+1})$,
$v_{1}=v$, $v_{k+1}=v^{\prime}$.
Note that
$\alpha(v_{i}v_{i+1})\leq
s+1+\underset{j=1}{\overset{i}{\sum}}(d_{T}(v_{j})-1)$ for $1\leq i\leq k$.
From this, we have
$\alpha(v_{k}v_{k+1})=\alpha(v_{k}v^{\prime})\leq
s+1+\underset{j=1}{\overset{k}{\sum}}(d_{T}(v_{j})-1)=s+L(v,v^{\prime})\leq
s+M(T)$.
Hence
$s+|F(T)|-1={\max}~{}S(u,\alpha)=\alpha(uv^{\prime})\leq s+1+M(T)$ and thus
$|F(T)|\leq M(T)+2$,
which is a contradiction. $\square$
Now let us consider the tree $T$ shown in Fig. 4.
Figure 4: The tree $T$.
Since $M(T)=11$ and $|F(T)|=14$, the graph $\widetilde{T}$ with
$|V(\widetilde{T})|=21$ and $\Delta(\widetilde{T})=14$ has no interval
coloring. Our constructions by trees generalize Hertz’s graphs $H_{p,q}$ given
in [8]. Moreover, the aforementioned example obtained by the method described
above is smaller than the smallest Hertz’s graph $H_{7,2}$.
### 2.4 Counterexamples by subdivisions
In this section we also need a definition of the interval of positive
integers. For positive integers $a$ and $b$, we denote by $\left[a,b\right]$,
the set of all positive integers $c$ with $a\leq c\leq b$.
Let $G$ be a graph and $V(G)=\\{v_{1},\ldots,v_{n}\\}$. Define graphs $S(G)$
and $\widehat{G}$ as follows:
$V(S(G))=\\{v_{1},\ldots,v_{n}\\}\cup\\{w_{ij}:v_{i}v_{j}\in E(G)\\}$,
$E(S(G))=\\{v_{i}w_{ij},v_{j}w_{ij}:v_{i}v_{j}\in E(G)\\}$,
$V(\widehat{G})=V(S(G))\cup\\{u\\}$, $u\notin V(S(G))$,
$E(\widehat{G})=E(S(G))\cup\\{uw_{ij}:v_{i}v_{j}\in E(G)\\}$.
In other words, $S(G)$ is the graph obtained by subdividing every edge of $G$,
and $\widehat{G}$ is the graph obtained from $S(G)$ by connecting every
inserted vertex to a new vertex $u$. Clearly, $S(G)$ and $\widehat{G}$ are
bipartite graphs.
###### Proposition 12
If $G$ is a bipartite graph and $G\in\mathfrak{N}$, then
$S(G)\in\mathfrak{N}$.
* Proof.
Let $G$ be a bipartite graph with a bipartition $(U,V)$, where
$U=\\{u_{1},\ldots,u_{r}\\}$, $V=\\{v_{1},\ldots,v_{s}\\}$. Also, let $\alpha$
be an interval $t$-coloring of the graph $G$.
Define an edge-coloring $\beta$ of the graph $S(G)$ as follows:
$\beta(u_{i}w_{ij})=\alpha(u_{i}v_{j})$ and
$\beta(v_{j}w_{ij})=\alpha(u_{i}v_{j})+1$ for every $u_{i}v_{j}\in E(G)$.
It is easy to see that $\beta$ is an interval $(t+1)$-coloring of the graph
$S(G)$. $\square$
In [11, 15, 16], it was proved that if $G$ is a regular graph, then
$S(G)\in\mathfrak{N}$. It would be interesting to generalize the last two
statements to general graphs. In other words, we would like to suggest the
following
###### Conjecture 13
If $G$ is a simple graph and $G\in\mathfrak{N}$, then $S(G)\in\mathfrak{N}$.
###### Theorem 14
If $G$ is a connected graph and
$|E(G)|>1+{\max\limits_{P\in\mathbf{P}}}{\sum\limits_{v\in V(P)}}\
\left(d_{\widehat{G}}(v)-1\right)$,
where $\mathbf{P}$ is a set of all shortest paths in $S(G)$ connecting
vertices $w_{ij}$, then $\widehat{G}\notin\mathfrak{N}$.
* Proof.
Suppose, to the contrary, that $\widehat{G}$ has an interval $t$-coloring
$\alpha$ for some $t\geq|E(G)|$.
Consider the vertex $u$. Let $w$ and $w^{\prime}$ be two vertices adjacent to
$u$ such that $\alpha(uw)={\min}~{}S(u,\alpha)=s$ and
$\alpha(uw^{\prime})={\max}~{}S(u,\alpha)=s+|E(G)|-1$. Since $\widehat{G}-u$
is isomorphic to $S(G)$ and connected, there is a shortest path
$P(w,w^{\prime})$ in $\widehat{G}-u$ joining $w$ with $w^{\prime}$, where
$P(w,w^{\prime})=(v_{1},e_{1},v_{2},\ldots,v_{i},e_{i},v_{i+1},\ldots,v_{k},e_{k},v_{k+1})$,
$v_{1}=w$, $v_{k+1}=w^{\prime}$.
Note that
$\alpha(v_{i}v_{i+1})\leq
s+\underset{j=1}{\overset{i}{\sum}}(d_{\widehat{G}}(v_{j})-1)$ for $1\leq
i\leq k$
and
$\alpha(v_{k+1}u)=\alpha(w^{\prime}u)\leq
s+\underset{j=1}{\overset{k+1}{\sum}}(d_{\widehat{G}}(v_{j})-1)$.
Hence
$s+|E(G)|-1={\max}~{}S(u,\alpha)=\alpha(uw^{\prime})\leq
s+\underset{j=1}{\overset{k+1}{\sum}}(d_{\widehat{G}}(v_{j})-1)\leq
s+{\max\limits_{P\in\mathbf{P}}}{\sum\limits_{v\in V(P)}}\
\left(d_{\widehat{G}}(v)-1\right)$
and thus
$|E(G)|\leq 1+{\max\limits_{P\in\mathbf{P}}}{\sum\limits_{v\in V(P)}}\
\left(d_{\widehat{G}}(v)-1\right)$,
which is a contradiction. $\square$
###### Corollary 15
If $n\geq 7$, then $\widehat{K}_{n}\notin\mathfrak{N}$.
###### Corollary 16
If $mn-m-n>5$, then $\widehat{K}_{m,n}\notin\mathfrak{N}$.
Figure 5: The graph $\widehat{K}_{3,4}$.
Now we show that the graph $\widehat{K}_{3,4}$ shown in Fig. 5 has no interval
coloring.
###### Theorem 17
$\widehat{K}_{3,4}\notin\mathfrak{N}$.
* Proof.
Let
$V(\widehat{K}_{3,4})=\\{u,v_{1},v_{2},v_{3},v_{4},v_{5},v_{6},v_{7}\\}\cup\\{w_{ij}:1\leq
i\leq 3,4\leq j\leq 7\\}$ and
$E(\widehat{K}_{3,4})=\\{v_{i}w_{ij},v_{j}w_{ij},uw_{ij}:1\leq i\leq 3,4\leq
j\leq 7\\}$.
Suppose that $\widehat{K}_{3,4}$ has an interval $t$-coloring $\alpha$ for
some $t\geq 12$.
Consider the vertex $u$. Let $w_{i_{0}j_{0}}$ and $w_{i_{1}j_{1}}$ be two
vertices adjacent to $u$ such that
$\alpha(uw_{i_{0}j_{0}})={\min}~{}S(u,\alpha)=s$ and
$\alpha(uw_{i_{1}j_{1}})={\max}~{}S(u,\alpha)=s+11$. We consider two cases.
Case 1: $i_{0}=i_{1}$ or $j_{0}=j_{1}$.
If $i_{0}=i_{1}$, then $v_{i_{0}}w_{i_{0}j_{0}},v_{i_{0}}w_{i_{0}j_{1}}\in
E(\widehat{K}_{3,4})$. This implies that
$\alpha(v_{i_{0}}w_{i_{0}j_{0}})\leq s+2$ and
$\alpha(v_{i_{0}}w_{i_{0}j_{1}})\leq s+5$.
Hence,
$s+11={\max}~{}S(u,\alpha)=\alpha(uw_{i_{0}j_{1}})\leq s+7$,
which is impossible.
If $j_{0}=j_{1}$, then $v_{j_{0}}w_{i_{0}j_{0}},v_{j_{0}}w_{i_{1}j_{0}}\in
E(\widehat{K}_{3,4})$. This implies that
$\alpha(v_{j_{0}}w_{i_{0}j_{0}})\leq s+2$ and
$\alpha(v_{j_{0}}w_{i_{1}j_{0}})\leq s+4$.
Hence,
$s+11={\max}~{}S(u,\alpha)=\alpha(uw_{i_{1}j_{0}})\leq s+6$,
which is impossible.
Case 2: $i_{0}\neq i_{1}$ and $j_{0}\neq j_{1}$.
In this case the edges $v_{i_{0}}v_{j_{0}}$ and $v_{i_{1}}v_{j_{1}}$ are
independent in $K_{3,4}$. Clearly, any two independent edges in $K_{3,4}$ lie
on the cycle of a length four. Hence, there is a cycle
$C=w_{i_{0}j_{0}}v_{j_{0}}w_{i_{1}j_{0}}v_{i_{1}}w_{i_{1}j_{1}}v_{j_{1}}w_{i_{0}j_{1}}v_{i_{0}}w_{i_{0}j_{0}}$
in $\widehat{K}_{3,4}$, which is consists of paths $P$ and $Q$, where
$P=\left(w_{i_{0}j_{0}},v_{j_{0}}w_{i_{0}j_{0}},v_{j_{0}},v_{j_{0}}w_{i_{1}j_{0}},w_{i_{1}j_{0}},v_{i_{1}}w_{i_{1}j_{0}},v_{i_{1}},v_{i_{1}}w_{i_{1}j_{1}},w_{i_{1}j_{1}}\right)$
and
$Q=\left(w_{i_{0}j_{0}},v_{i_{0}}w_{i_{0}j_{0}},v_{i_{0}},v_{i_{0}}w_{i_{0}j_{1}},w_{i_{0}j_{1}},v_{j_{1}}w_{i_{0}j_{1}},v_{j_{1}},v_{j_{1}}w_{i_{1}j_{1}},w_{i_{1}j_{1}}\right)$.
If $\alpha(v_{j_{0}}w_{i_{0}j_{0}})=s+1$, then, by considering the path $P$,
we have $\alpha(v_{i_{1}}w_{i_{1}j_{1}})\leq s+8$ and
${\max}~{}S(w_{i_{1}j_{1}},\alpha)\leq s+10$, a contradiction.
If $\alpha(v_{i_{0}}w_{i_{0}j_{0}})=s+1$, then, by considering the path $Q$,
we have $\alpha(v_{j_{1}}w_{i_{1}j_{1}})\leq s+8$ and
${\max}~{}S(w_{i_{1}j_{1}},\alpha)\leq s+10$, a contradiction.
Hence, $\alpha(v_{j_{0}}w_{i_{0}j_{0}})=\alpha(v_{i_{0}}w_{i_{0}j_{0}})=s+2$,
which is a contradiction. $\square$
Figure 6: The graph $\widehat{K}_{2,2,2}$.
Note that the graph $\widehat{K}_{3,4}$ has $20$ vertices and maximum degree
$12$. Now we show that there is a connected bipartite graph $G$ with
$|V(G)|=19$ and $\Delta(G)=12$ which is not interval colorable. Let
$K_{2,2,2}$ be a complete $3$-partite graph with two vertices in each part.
Then the graph $\widehat{K}_{2,2,2}$ shown in Fig. 6 is not interval
colorable.
###### Theorem 18
$\widehat{K}_{2,2,2}\notin\mathfrak{N}$.
* Proof.
Let
$V(\widehat{K}_{2,2,2})=\\{u,v_{1},v_{2},v_{3},v_{4},v_{5},v_{6}\\}\cup\\{w_{ij}:1\leq
i<j\leq 6,(i,j)\notin\\{(1,2),(3,4),\\\ (5,6)\\}\\}$ and
$E(\widehat{K}_{2,2,2})=\\{v_{i}w_{ij},v_{j}w_{ij},uw_{ij}:1\leq i<j\leq
6,(i,j)\notin\\{(1,2),(3,4),(5,6)\\}\\}$.
Suppose that $\widehat{K}_{2,2,2}$ has an interval $t$-coloring $\alpha$ for
some $t\geq 12$.
Consider the vertex $u$. Let $w_{i_{0}j_{0}}$ and $w_{i_{1}j_{1}}$ be two
vertices adjacent to $u$ such that
$\alpha(uw_{i_{0}j_{0}})={\min}~{}S(u,\alpha)=s$ and
$\alpha(uw_{i_{1}j_{1}})={\max}~{}S(u,\alpha)=s+11$. By the symmetry of the
graph $\widehat{K}_{2,2,2}$, we may assume that $(i_{0},j_{0})=(1,6)$. We
consider two cases.
Case 1: $v_{1}v_{6}$ and $v_{i_{1}}v_{j_{1}}$ are adjacent in $K_{2,2,2}$.
By the symmetry of the graph $\widehat{K}_{2,2,2}$, it suffices to consider
$(i_{1},j_{1})=(1,4)$ and $(i_{1},j_{1})=(1,5)$.
If $\alpha(uw_{14})=s+11$ or $\alpha(uw_{15})=s+11$, then
$\alpha(v_{1}w_{16})\leq s+2$ and $\alpha(v_{1}w_{1j_{1}})\leq s+5$. Hence,
$\alpha(uw_{1j_{1}})\leq s+7$, which is impossible.
Case 2: $v_{1}v_{6}$ and $v_{i_{1}}v_{j_{1}}$ are independent in $K_{2,2,2}$.
By the symmetry of the graph $\widehat{K}_{2,2,2}$, it suffices to consider
$(i_{1},j_{1})=(4,5)$ and $(i_{1},j_{1})=(2,5)$.
If $\alpha(uw_{45})=s+11$, then either $\alpha(v_{1}w_{16})=s+1$ or
$\alpha(v_{1}w_{16})=s+2$, and in both cases the colors of all edges along the
cycle $C=w_{16}v_{1}w_{15}v_{5}w_{45}v_{4}w_{46}v_{6}w_{16}$ are known. This
implies that $\alpha(v_{1}w_{14})\leq s+4$, but $\alpha(v_{4}w_{14})\geq s+7$,
which is a contradiction.
If $\alpha(uw_{25})=s+11$, then, by the symmetry of the graph
$\widehat{K}_{2,2,2}$, we may assume that $\alpha(v_{1}w_{16})=s+1$. Clearly,
in this case the colors of all edges along the cycle
$C=w_{16}v_{1}w_{15}v_{5}w_{25}v_{2}w_{26}v_{6}w_{16}$ are known. This implies
that $\alpha(uw_{26})=s+6$. By the symmetry of the graph
$\widehat{K}_{2,2,2}$, we may assume that $\alpha(v_{1}w_{13})=s+2$ and
$\alpha(v_{1}w_{14})=s+3$. It is easy to see that $\alpha(uw_{13})=s+1$.
Hence, $\alpha(v_{3}w_{13})=s+3$ and taking into account that
$\alpha(v_{2}w_{25})=s+10$, we have $\alpha(v_{3}w_{23})=s+6$. This implies
that $\alpha(v_{3}w_{35})\leq s+5$. On the other hand, since
$\alpha(v_{5}w_{25})=s+9$, we have $\alpha(v_{5}w_{35})\geq s+7$ and thus
$\alpha(uw_{35})=s+6=\alpha(uw_{26})$, which is a contradiction. $\square$
Also, we investigate bipartite graphs which are close to the graph
$\widehat{K}_{3,4}$. In particular, we observe that the graph obtained from
$\widehat{K}_{3,4}$ by deleting any edge incident to the vertex of maximum
degree is not interval colorable, too. Let $\widehat{K}_{3,4}^{\prime}$ be a
graph obtained from $\widehat{K}_{3,4}$ by deleting any edge incident to the
vertex $u$. Clearly, $\widehat{K}_{3,4}^{\prime}$ has $20$ vertices and
maximum degree $11$.
###### Theorem 19
$\widehat{K}_{3,4}^{\prime}\notin\mathfrak{N}$.
* Proof.
Let $V(\widehat{K}_{3,4}^{\prime})=V(\widehat{K}_{3,4})$ and
$E(\widehat{K}_{3,4}^{\prime})=E(\widehat{K}_{3,4})\setminus{uw_{l_{0}m_{0}}}$.
Suppose that $\widehat{K}_{3,4}^{\prime}$ has an interval $t$-coloring
$\alpha$ for some $t\geq 11$.
Consider the vertex $u$. Let $w_{i_{0}j_{0}}$ and $w_{i_{1}j_{1}}$ be two
vertices adjacent to $u$ such that
$\alpha(uw_{i_{0}j_{0}})={\min}~{}S(u,\alpha)=s$ and
$\alpha(uw_{i_{1}j_{1}})={\max}~{}S(u,\alpha)=s+10$.
Let $S(w_{l_{0}m_{0}},\alpha)=\\{c,c+1\\}$. Now we add the edge
$uw_{l_{0}m_{0}}$ to the graph $\widehat{K}_{3,4}^{\prime}$ and we color it
with color $c+2$. Clearly, we obtained an edge-coloring of the graph
$\widehat{K}_{3,4}$ with colors $1,\ldots,t^{\prime}(t^{\prime}\geq t)$. Let
$\beta$ be this edge-coloring. Note that for each vertex $v\in
V(\widehat{K}_{3,4})\setminus\\{u\\}$, $S(v,\beta)$ is an interval of
integers, and $S(u,\beta)=[s,s+10]\cup\\{c+2\\}$ is a multiset in general.
Similarly as in the proof of the case 1 of Theorem 17 it can be shown that
$i_{0}\neq i_{1}$ and $j_{0}\neq j_{1}$. Clearly, the edges
$v_{i_{0}}v_{j_{0}}$ and $v_{i_{1}}v_{j_{1}}$ are independent in $K_{3,4}$. By
the symmetry of the graph $\widehat{K}_{3,4}$, we may assume that
$(i_{0},j_{0})=(1,4)$ and $(i_{1},j_{1})=(3,7)$.
Consider the edge $v_{1}w_{14}$. Clearly, either $\beta(v_{1}w_{14})=s+1$ or
$\beta(v_{1}w_{14})=s+2$. If $\beta(v_{1}w_{14})=s+1$, then the colors of all
edges along the cycle $C=w_{14}v_{1}w_{17}v_{7}w_{37}v_{3}w_{34}v_{4}w_{14}$
are known. This implies that $\beta(uw_{17})=\beta(uw_{34})=s+5$. Hence, the
added color $c+2$ is $s+5$, but this is a contradiction, since
$S(w_{17},\beta)=S(w_{34},\beta)=[s+4,s+6]$ and in both cases $s+5$ is a
middle color of the sets $S(w_{17},\beta)$ and $S(w_{34},\beta)$.
Now assume that $\beta(v_{1}w_{14})=s+2$. In this case the colors of all edges
along the cycle $C=w_{14}v_{1}w_{17}v_{7}w_{37}v_{3}w_{34}v_{4}w_{14}$ are
also known. By the symmetry of the graph $\widehat{K}_{3,4}$, we may assume
that $\beta(v_{1}w_{15})=s+3$ and $\beta(v_{1}w_{16})=s+4$. Since
$\beta(v_{7}w_{27})=s+8$, we have $\min S(v_{2},\beta)\geq s+3$. On the other
hand, since $\beta(v_{4}w_{24})=s+2$, we have $\min S(v_{2},\beta)\leq s+4$.
We consider two cases.
Case 1: $\min S(v_{2},\beta)=s+3$.
In this case $\beta(v_{2}w_{26})\leq s+5$ and thus $\beta(uw_{36})=s+9$. This
implies that $\beta(v_{3}w_{36})=s+7$ and $\beta(v_{6}w_{36})=s+8$. Since
$\beta(v_{1}w_{16})=s+4$, we have $\beta(v_{6}w_{16})=s+6$ and
$\beta(v_{6}w_{26})=s+7$. Also, since $\beta(v_{2}w_{26})\leq s+5$, we have
$\beta(uw_{26})=s+6$. On the other hand, $\beta(uw_{17})=s+6$, but this is a
contradiction, since $S(w_{17},\beta)=S(w_{26},\beta)=[s+5,s+7]$ and in both
cases $s+6$ is a middle color of the sets $S(w_{17},\beta)$ and
$S(w_{26},\beta)$.
Case 2: $\min S(v_{2},\beta)=s+4$.
In this case $\beta(uw_{24})=s+3$ and $\beta(v_{2}w_{24})=s+4$. This implies
that $\beta(v_{2}w_{25})\geq s+5$ and thus $\beta(uw_{15})=s+1$. Since
$\beta(v_{1}w_{15})=s+3$, we have $\beta(v_{5}w_{15})=s+2$. Also, since
$\beta(v_{3}w_{35})\geq s+6$, we have $\beta(v_{5}w_{35})=s+4$ and
$\beta(v_{5}w_{25})=s+3$. This implies that $\beta(uw_{25})=s+4$. On the other
hand, $\beta(uw_{34})=s+4$, but this is a contradiction, since
$S(w_{34},\beta)=S(w_{25},\beta)=[s+3,s+5]$ and in both cases $s+4$ is a
middle color of the sets $S(w_{34},\beta)$ and $S(w_{25},\beta)$. $\square$
## 3 Interval non-edge-colorable bipartite multigraphs
In this section we consider bipartite multigraphs, and first we show that any
bipartite multigraph $G$ with at most four vertices is interval colorable.
###### Theorem 20
If $G$ is a connected bipartite multigraph with $|V(G)|\leq 4$, then
$G\in\mathfrak{N}$.
* Proof.
The cases $|V(G)|\leq 3$ are trivial. Assume that $|V(G)|=4$. If the
underlying graph of $G$ is a tree, then the proof is trivial, too. Now let
$V(G)=\\{u,v,w,z\\}$ and $E(G)=E(uv)\cup E(vw)\cup E(wz)\cup E(uz)$ with
$\mu(uv)=a$, $\mu(vw)=b$, $\mu(wz)=c$, $\mu(uz)=d$. Without loss of generality
we may assume that ${\max}\\{a,b,c,d\\}=d$. We color edges from $E(uv)$ with
colors $d+1,\ldots,d+a$, edges from $E(vw)$ with colors $d-b+1,\ldots,d$,
edges from $E(wz)$ with colors $d+1,\ldots,d+c$, and edges from $E(uz)$ with
colors $1,\ldots,d$. If $a<c$, then the obtained coloring is an interval
$(d+c)$-coloring of the multigraph $G$; otherwise the obtained coloring is an
interval $(d+a)$-coloring of the multigraph $G$. $\square$
Figure 7: The bipartite multigraph $G$.
Note that the bipartite multigraph $G$ with $|V(G)|=5$ and $\Delta(G)=9$ shown
in Fig. 7 has no interval coloring. Now we show a more general result.
Let us define parachute multigraphs $Par(r_{1},\ldots,r_{n})$
($r_{1}\geq\cdots\geq r_{n}\geq 1$) as follows:
$V(Par(r_{1},\ldots,r_{n}))=\\{u,w,v_{1},\ldots,v_{n}\\}$,
$E(Par(r_{1},\ldots,r_{n}))=\\{uv_{i}:\mu(uv_{i})=r_{i},1\leq i\leq
n\\}\cup\\{v_{j}w:1\leq j\leq n\\}$.
Clearly, $Par(r_{1},\ldots,r_{n})$ is a connected bipartite multigraph with
$|V(Par(r_{1},\ldots,r_{n}))|=n+2$,
$\Delta(Par(r_{1},\ldots,r_{n}))=d(u)=\underset{i=1}{\overset{n}{\sum}}r_{i}$,
and $d(w)=n$, $d(v_{i})=r_{i}+1$, $i=1,2,\ldots,n$.
###### Theorem 21
If $\underset{i=3}{\overset{n}{\sum}}r_{i}\geq n+1$ ($n\geq 3$), then
$Par(r_{1},\ldots,r_{n})\notin\mathfrak{N}$.
* Proof.
Suppose, to the contrary, that the multigraph $Par(r_{1},\ldots,r_{n})$ has an
interval $t$-coloring $\alpha$ for some
$t\geq\underset{i=1}{\overset{n}{\sum}}r_{i}$.
Consider the vertex $u$. Let $v_{i_{0}}$ and $v_{i_{1}}$ be two vertices
adjacent to $u$ such that $\alpha(uv_{i_{0}})={\min}~{}S(u,\alpha)=s$ and
$\alpha(uv_{i_{1}})={\max}~{}S(u,\alpha)=s+\underset{i=1}{\overset{n}{\sum}}r_{i}-1$.
If $i_{0}=i_{1}$, then
$\alpha(uv_{i_{0}})={\max}~{}S(u,\alpha)=s+\underset{i=1}{\overset{n}{\sum}}r_{i}-1\leq
s+d(v_{i_{0}})-1=s+r_{i_{0}}$ and thus
$\underset{i=1}{\overset{n}{\sum}}r_{i}-1\leq r_{i_{0}}$, which is impossible.
If $i_{0}\neq i_{1}$, then, by the construction of the multigraph
$Par(r_{1},\ldots,r_{n})$, we have
$\alpha(v_{i_{0}}w)\leq s+d(v_{i_{0}})-1=s+r_{i_{0}}$ and thus
$\alpha(v_{i_{1}}w)\leq s+r_{i_{0}}+d(w)-1=s+r_{i_{0}}+n-1$.
This implies that
$s+\underset{i=1}{\overset{n}{\sum}}r_{i}-1=\alpha(uv_{i_{1}})={\max}~{}S(u,\alpha)\leq
s+r_{i_{0}}+n-1+d(v_{i_{1}})-1=s+r_{i_{0}}+r_{i_{1}}+n-1$.
Hence
$\underset{i=3}{\overset{n}{\sum}}r_{i}\leq\underset{i=1}{\overset{n}{\sum}}r_{i}-(r_{i_{0}}+r_{i_{1}})\leq
n$,
which is a contradiction. $\square$
By Fig. 7 and Theorem 21, we have
###### Corollary 22
For any positive integer $\Delta\geq 9$, there is a connected bipartite
multigraph $G$ with $G\notin\mathfrak{N}$ and $\Delta(G)=\Delta$.
On the other hand, now we prove that all subcubic bipartite multigraphs have
an interval coloring.
###### Theorem 23
If $G$ is a bipartite multigraph with $\Delta(G)\leq 3$, then
$G\in\mathfrak{N}$ and $w(G)\leq 4$.
* Proof.
First note that if $\Delta(G)\leq 2$, then $G\in\mathfrak{N}$ and $w(G)\leq
2$.
Now suppose that $\Delta(G)=3$. For the proof, it suffices to show that $G$
has either an interval $3$-coloring or an interval $4$-coloring.
We show it by induction on $|E(G)|$. The statement is trivial for the case
$|E(G)|\leq 4$. Assume that $|E(G)|\geq 5$, and the statement is true for all
multigraphs $G^{\prime}$ with $\Delta(G^{\prime})=3$ and
$|E(G^{\prime})|<|E(G)|$.
Let us consider a multigraph $G$. Clearly, $G$ is connected. If $G$ has no
multiple edges, then the statement follows from Theorem 2. Now suppose that
$G$ has multiple edges.
Let $uv\in E(G)$ and $\mu(uv)\geq 2$. We consider two cases.
Case 1: $\mu(uv)=d_{G}(v)=2$ and $d_{G}(u)=\Delta(G)=3$.
Clearly, in this case there is an edge $uw$, which is a bridge in $G$. Let us
consider a multigraph $G^{\prime}=G-E(uv)$, where $E(uv)=\\{e_{1},e_{2}\\}$.
By induction hypothesis, $G^{\prime}$ has either an interval $3$-coloring
$\alpha$ or an interval $4$-coloring $\alpha$.
Subcase 1.1: $\alpha(uw)\leq 2$.
We color the edge $e_{i}$ with color $\alpha(uw)+i$, $i=1,2$. It is not
difficult to see that the obtained coloring is an interval $3$-coloring or an
interval $4$-coloring of the multigraph $G$.
Subcase 1.2: $\alpha(uw)\geq 3$.
We color the edge $e_{i}$ with color $\alpha(uw)-i$, $i=1,2$. It is not
difficult to see that the obtained coloring is an interval $3$-coloring or an
interval $4$-coloring of the multigraph $G$.
Case 2: $\mu(uv)=2$ and $d_{G}(u)=d_{G}(v)=\Delta(G)=3$.
Clearly, in this case there are vertices $x,y$ ($x\neq y$) in $G$ such that
$ux\in E(G)$ and $vy\in E(G)$. Let us consider a multigraph
$G^{\prime}=(G-E(uv)-ux-vy)+xy$, where $E(uv)=\\{e_{1},e_{2}\\}$. By induction
hypothesis, $G^{\prime}$ has either an interval $3$-coloring $\alpha$ or an
interval $4$-coloring $\alpha$.
Subcase 2.1: $\alpha(xy)\leq 2$.
We delete the edge $xy$ and color the edges $ux$ and $vy$ with color
$\alpha(xy)$ and the edge $e_{i}$ with color $\alpha(xy)+i$, $i=1,2$. It is
not difficult to see that the obtained coloring is an interval $3$-coloring or
an interval $4$-coloring of the multigraph $G$.
Subcase 2.2: $\alpha(xy)\geq 3$.
We delete the edge $xy$ and color the edges $ux$ and $vy$ with color
$\alpha(xy)$, and the edge $e_{i}$ with color $\alpha(xy)-i$, $i=1,2$. It is
not difficult to see that the obtained coloring is an interval $3$-coloring or
an interval $4$-coloring of the multigraph $G$. $\square$
## 4 Problems
Finally, we restate the problem posed by Jensen and Toft and formulate a
similar problem for multigraphs. The problems are following:
###### Problem 2
Is there a bipartite graph G with $4\leq\Delta(G)\leq 10$ and
$G\notin\mathfrak{N}$?
###### Problem 3
Is there a bipartite multigraph G with $4\leq\Delta(G)\leq 8$ and
$G\notin\mathfrak{N}$?
Since all bipartite graphs of order at most $14$ are interval colorable [5]
and the bipartite graph $\widehat{K}_{2,2,2}$ with
$|V(\widehat{K}_{2,2,2})|=19$ is not interval colorable, we would like to
suggest the following
###### Problem 4
Is there a bipartite graph G with $15\leq|V(G)|\leq 18$ and
$G\notin\mathfrak{N}$?
* Acknowledgement
We would like to thank Rafayel R. Kamalian for his attention to this work. We
also would like to thank our referees for many useful comments and suggestions
which helped us to improve the presentation of the article.
## References
* [1] A.S. Asratian and R.R. Kamalian, Interval colorings of edges of a multigraph, Appl. Math. 5 (1987), 25-34 (in Russian).
* [2] A.S. Asratian and R.R. Kamalian, Investigation on interval edge-colorings of graphs, J. Combin. Theory Ser. B 62 (1994), 34-43.
* [3] A.S. Asratian, T.M.J. Denley and R. Haggkvist, Bipartite Graphs and their Applications, Cambridge University Press, Cambridge, 1998.
* [4] A.S. Asratian, C.J. Casselgren, J. Vandenbussche and D.B. West, Proper path-factors and interval edge-coloring of $\left(3,4\right)$-biregular bigraphs, J. Graph Theory 61 (2009), 88-97.
* [5] K. Giaro, Compact task scheduling on dedicated processors with no waiting periods, PhD thesis, Technical University of Gdansk, EIT faculty, Gdansk, 1999 (in Polish).
* [6] K. Giaro, The complexity of consecutive $\Delta$-coloring of bipartite graphs: $4$ is easy, $5$ is hard, Ars Combin. 47 (1997), 287-298.
* [7] K. Giaro and M. Kubale, Consecutive edge-colorings of complete and incomplete Cartesian products of graphs, Congr, Numer. 128 (1997), 143-149.
* [8] K. Giaro, M. Kubale and M. Malafiejski, On the deficiency of bipartite graphs, Discrete Appl. Math. 94 (1999), 193-203.
* [9] K. Giaro and M. Kubale, Compact scheduling of zero-one time operations in multi-stage systems, Discrete Appl. Math. 145 (2004), 95-103.
* [10] H.M. Hansen, Scheduling with minimum waiting periods, Master’s Thesis, Odense University, Odense, Denmark, 1992 (in Danish).
* [11] D. Hanson, C.O.M. Loten and B. Toft, On interval colorings of bi-regular bipartite graphs, Ars Combin. 50 (1998), 23-32.
* [12] T.R. Jensen and B. Toft, Graph Coloring Problems, Wiley Interscience Series in Discrete Mathematics and Optimization, 1995.
* [13] R.R. Kamalian, Interval colorings of complete bipartite graphs and trees, preprint, Comp. Cen. of Acad. Sci. of Armenian SSR, Erevan, 1989 (in Russian).
* [14] R.R. Kamalian, Interval edge-colorings of graphs, Doctoral Thesis, Novosibirsk, 1990.
* [15] R.R. Kamalian and A.N. Mirumian, Interval edge-colorings of bipartite graphs of some class, Dokl. NAN RA, 97 (1997), 3-5 (in Russian).
* [16] A.V. Kostochka, unpublished manuscript, 1995.
* [17] A.N. Mirumyan, personal communication, 2005.
* [18] P.A. Petrosyan, Interval edge-colorings of complete graphs and $n$-dimensional cubes, Discrete Math. 310 (2010), 1580-1587.
* [19] A.V. Pyatkin, Interval coloring of $\left(3,4\right)$-biregular bipartite graphs having large cubic subgraphs, J. Graph Theory 47 (2004), 122-128.
* [20] S.V. Sevast’janov, Interval colorability of the edges of a bipartite graph, Metody Diskret. Analiza 50 (1990), 61-72 (in Russian).
* [21] C.E. Shannon, A theorem on colouring the lines of a network, J. Math. Phys. 28 (1949), 148-151.
* [22] V.G. Vizing, The chromatic class of a multigraph, Kibernetika 3 (1965), 29-39 (in Russian).
* [23] D.B. West, Introduction to Graph Theory, Prentice-Hall, New Jersey, 1996.
* [24] F. Yang and X. Li, Interval coloring of $\left(3,4\right)$-biregular bigraphs having two $\left(2,3\right)$-biregular bipartite subgraphs, Appl. Math. Let. 24 (2011), 1574-1577.
|
arxiv-papers
| 2013-01-16T20:11:07 |
2024-09-04T02:49:40.407888
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Petros A. Petrosyan, Hrant H. Khachatrian",
"submitter": "Petros Petrosyan",
"url": "https://arxiv.org/abs/1301.3811"
}
|
1301.3929
|
# NMR Investigation of the Low Temperature Dynamics of solid 4He doped with
3He impurities
S. S. Kim1111Current address: École Polytechnique Fédérale de Lausanne,
Lausanne, Switzerland C. Huan1222Current Address: Georgia Institute of
Technology, Atlanta, Ga, USA L. Yin1 J. S. Xia1 D. Candela2 N. S. Sullivan1
1Department of Physics & National High Magnetic Field Laboratory, University
of Florida, Florida 32611, USA.
2Department of Physics, University of Massachusetts, Amherst, Massachusetts
01003, USA [email protected]
###### Abstract
The lattice dynamics of solid 4He has been explored using pulsed NMR methods
to study the motion of 3He impurities in the temperature range where
experiments have revealed anomalies attributed to superflow or unexpected
viscoelastic properties of the solid 4He lattice. We report the results of
measurements of the nuclear spin-lattice and spin-spin relaxation times that
measure the fluctuation spectrum at high and low frequencies, respectively, of
the 3He motion that results from quantum tunneling in the 4He matrix. The
measurements were made for 3He concentrations $16<x_{3}<2000$ ppm. For 3He
concentrations $x_{3}$ = 16 ppm and 24 ppm, large changes are observed for
both the spin-lattice relaxation time $T_{1}$ and the spin-spin relaxation
time $T_{2}$ at temperatures close to those for which the anomalies are
observed in measurements of torsional oscillator responses and the shear
modulus. These changes in the NMR relaxation rates were not observed for
higher 3He concentrations.
Suggested keywords
###### pacs:
67.80.bd, 67.30.hm, 76.60.-k
333†THE
## I Introduction
The observation of distinctive anomalies in the frequency and dissipation of
torsional oscillators containing solid 4He, often referred to as non-classical
rotational inertia fractions (NCRIFs), by Kim and Chan ISI:000188068100037 ;
ISI:000224136000041 has stimulated enormous activity. This is because the
NCRIFs could be the signature of a supersolid state as outlined by
LeggettPhysRevLett.25.1543 and predicted by Andreev and Lifshitz several
years ago.AndreevandLifshitz A large number of independent experiments
PhysRevLett.100.065301 ; PhysRevLett.97.165301 ; Reppy2010 ; Davis2011 ;
Kojima08 have shown that the NCRIF magnitude and temperature dependence are
strongly dependent on defects such as 3He impuritiesGumann2011 ; Kojima12 and
the quality of the crystals, and can be made very small by very careful
annealing. ISI:000224136000041 The observation of NCRIFs was preceded by
reports of a striking anomaly in the measurement of sound attenuation in solid
4He by Goodkind and colleagues who had compared their observations to that
predicted for a true phase transition of a fraction of the solid to a
supersolid state.Goodkind1 ; Goodkind2 Additional support for the
interpretation of the NCRIF anomalies in terms of a true thermodynamic phase
change to a new state of matter was provided by the observation of a small but
distinct contribution to the heat capacity of the solid at the same
temperatures as those reported for the onset of the NCRIFs.Linetal2007
A straightforward interpretation in terms of a transition of part of the solid
to a coherent superfluid component has, however, been hampered by a number of
puzzling observations. These include the lack of evidence for a critical
exponent, an apparent very low critical velocityPhysRevLett.97.165301 , the
absence of experimental evidence for fourth sound modes Aokietal ; Kwonetal
and the null results of attempts to observe pressure- induced superflow though
small restrictions,DayBeamishPRL06 although mass transport has been observed
with the use of porous vycor glass conduits used to allow mass flow to the
solid via superfluid in the pores. HallockPRL08 ; HallockPRB09 In addition,
measurements of the shear modulus of solid 4He by Beamish and colleagues
PhysRevLett.104.195301 ; PhysRevB.79.214524 have revealed a prominent
frequency dependent change in the elastic shear modulus with an enhanced
dissipation peak having a temperature dependence comparable to that observed
for the NCRIF. These results suggest that the torsional oscillator anomalies
may result from unusual elasticElasticPropsBeamish09 or viscoelasticDorsey09
properties of the solid 4He rather than superfluidity. Other interpretations
attribute the anomalies to macroscopic superflow mediated by defects or
dislocation networksBoninsegni06 or in terms of a vortex modelP.W.Anderson
for which the high temperature tail of the NCRIF is associated with the finite
response time of vortices to the oscillating flow fields in the TOs. Another
set of researchers suggest that the superfluidity arises from non-equilibrium
behavior leading to superflow along defects or the formation of a quantum
“superglass” around extended defectsBalatsky09 with ultra-slow relaxation
dynamics reminiscent of glass dynamics.Graf2011
The need for any interpretation to account for both sample dependent coherent
non-inertial flow and changes in the lattice elastic properties was confirmed
strikingly by Kim et al. DKimetal2011 who observed that the TO response could
be changed significantly by rotating the cryostat, with the resonant frequency
changing with speed, but that at the same time, the TO mode that showed
significant drive dependence was not susceptible to changes in the elastic
modulus of the lattice. DKimetal2011
It is clear that the dynamics of the 4He lattice plays an important role in
the low temperature bulk properties of solid 4He and rather than observing a
phase transition to a supersolid state one may be observing a complex
thermally excited dynamical response, or both, in overlapping temperature
ranges. It is therefore important to study the microscopic dynamics of solid
4He using different techniques and especially non-invasive techniques since
all previous studies have involved the application of macroscopic external
mechanical stimuli. Measurements of the quantum tunneling of the 3He atoms in
the solid provide a unique means of probing microscopic lattice properties
because the 3He -4He exchange rates depend exponentially on the lattice
separation and on the magnitude and dynamics of the crystal field deformation
that surround impurities.Eshelby1956 ; Landesman1975137 The characteristic
NMR relaxation rates are determined by the modulation of the nuclear dipole-
dipole interactions by the tunneling motion and the scattering of the
diffusing atoms by the crystal deformation field around the 3He impurities and
other lattice defects.Landesman1975137 ; PhysRevB.11.3374 The NMR relaxation
rates are therefore very sensitive to the local elastic properties of solid
4He and to any changes in the crystal ground state that would modify the
tunneling rate. We have therefore carried out systematic measurements of the
nuclear spin relaxation rates of 3He impurities in solid 4He for a wide range
of concentrations with an emphasis on low concentrations $x_{3}<30$ ppm for
which one expects well characterized NCRIFs from previous studies and yet
sufficient to obtain good signal to noise ratios for NMR measurements. Also
for these concentrations, the amount of the 3He localized on dislocations
estimated from the expected density of dislocations for well-annealed
samplesIwasa2010 is expected to be a small fraction of the total, less than 1
ppm, so that one can be sure that the experiments probe the motion of the 3He
atoms through bulk solid 4He. Preliminary results of these NMR studies have
been reported elsewhere.SSKimJLTP2010 ; SSKimPRL2011
## II Experimental Arrangement
The NMR cell was designed as a nested cross-coil arrangement with receiving
and transmitting coils orthogonal to each other and the applied magnetic field
(see Fig. 1 ). The inner coil is a cylindrical receiving coil wound around a
polycarbonate sample cell which contained an in situ Straty-Adams pressure
gauge. The outer coil is a thermally isolated orthogonal transmitting coil for
the RF pulses that slides onto the receiving coil. This nested arrangementHuan
provides (i) minimization of the unwanted pick-up of the transmitter pulse by
the receiving coil, and (ii) adequate thermal isolation from the heat
generated in the transmitting coil that is heat sunk to the still of the
dilution refrigerator. Thermalization of the sample was assured by thermal
contact with a silver post extending from the dilution refrigerator and the
temperature was measured using a Cernox resistance thermometer calibrated
against a 3He melting curve thermometer. The sample gas was admitted via a
capillary sealed with epoxy (stycast 2585 GT) at the opposite end of the cell.
The sample cell could withstand pressures up to 100 bar and remain superfluid
leak-tight.
Figure 1: (Color online) Schematic representation of the low temperature NMR
cell. The sample in this NMR cell is cylindrical, 5.0 mm diameter and 7.5 mm
long. The preamplifier and tuning capacitor are located on a 1 K cold plate
located a distance of 1.2 m from the sample cell. The RF transmitting and
receiving coils are simplified in this figure.
Samples were grown by the blocked capillary method. Gas samples of pre-
determined 3He fraction were prepared at room temperature by mixing pure
(99.99%) 3He and 4He, compressing the gas mixture to about 50 bar and filling
the sample cell with the gas mixture to a pressure of 46 bar, after which the
cell was cooled to 1.2 K. When the temperature is cooled to the melting curve,
the helium pressure follows the melting curve while the mixture solidifies,
after which the pressure measured by the gauge in the cell remains constant
indicating that the sample is solid. The samples were prepared to all have the
same final pressure of $27.75\pm 0.05$ bar at 1.2 K. The molar volume of the
samples were determined to be $V_{m}=20.85\pm 0.05$ cm3 from the PVT data of
Grilly and MillsGrillyMills and the formula derived by Mullin.Mullin Finally
the samples were annealed for 24 hours just below the melting point except for
one case for which a minimum annealing of 0.5 hours was carried out to
determine the effect of the crystal quality on the nuclear spin relaxation
times.
The measurements of the nuclear spin-spin and spin-lattice relaxation times
were carried out using a superheterodyne pulsed NMR spectrometer operating at
a Larmor frequency of 2.05MHz. At this frequency the calculated relaxation
times due to exchange motion were of the order of $10^{4}$ s. at 3He
concentrations $x_{3}=20$ ppm, extrapolating from previous measurements.GRZ ;
Grigoriev1973 ; Schratter1984 ; Greenberg1972 The longitudinal relaxation
time is expected to increase rapidly with frequencyGRZ , making studies at
higher Larmor frequencies extremely difficult.
The signal/noise ratio for samples with $x_{3}\approx 20$ ppm is very weak
$(<10^{-2})$ for a 10 kHz bandwidth employing a standard geometry with
resonant NMR circuit connected directly to a room temperature amplifier. In
order to enhance the signal/noise we developed a preamplifier that could be
operated at low temperatures in the applied magnetic field.Huan The device
used a pseudomorphic high electron mobility field effect transistor with
adjustable bias so that the output could be matched to a 50 ohm cable. The
total power dissipation was 0.5 mW and to attain the lowest sample
temperatures the amplifier was separated from the sample cell by a 75 cm
length of cable and anchored to a 1K thermal plate. A typical NMR echo
recorded using a $90_{x}-180_{x}$ RF pulse sequence for a 500 ppm 3He sample
is shown in Fig. 2 after signal averaging for 10 pulse sequences using a 17
kHz bandwidth. As it is necessary to wait several times $T_{1}$ between pulse
sequences, up to 15 hours were required to obtain each data point at the
lowest temperatures. Therefore the total number of data points is modest, but
sufficient we believe to reveal and confirm relevant features in the
temperature dependence of the relaxation times.
Figure 2: (Color online) Example of NMR Hahn echo for a sample of 500 ppm 3He
in solid 4He at T=0.4K, showing the in-phase and out-of-phase components.
One of the most important properties of the sample that must be determined is
the exact 3He concentration, because for the long capillary line used (1.2 m
below the 1K plate) an appreciable fraction of the 3He atoms can plate out on
the capillary walls and lead to changes in the 3He concentration of the solid
sample compared to that of the original gas concentration. Fortunately, the
NMR signal itself allows one to obtain an accurate calibration by measuring
the NMR echo amplitude for fixed spectrometer gain at intermediate
temperatures and comparing to a relatively high concentration sample (2000
ppm) for which any fractional change would be small. In Fig. 3 we show the
temperature dependence of NMR echo amplitudes for several different samples.
All the samples show the characteristic Curie law behavior until reaching
temperatures of the order of 0.1K where phase separation occurs. At these low
temperatures the 3He forms Fermi liquid droplets with temperature independent
NMR amplitudes and with different relaxation times. Using this procedure one
can measure the 3He concentration to approximately 5% accuracy. For example,
for a sample prepared from a nominally 30 ppm gas mixture, the final solid 3He
fraction was measured to be 24 ppm. Note that a small deviation was observed
in the amplitude of the echo signal for the 16 ppm sample at $T=175$ mK and
this is attributed to the sudden and unexpected sharp increase in $T_{1}$ at
that temperature. Waiting times between measurements were set at 5 times
$T_{1}$ to provide 5% accuracy in amplitude measurements but at 175mk this
drops to 2.5 times $T_{1}$ and a small decrease in observed amplitude results.
Figure 3: (Color online) Observed temperature dependence of NMR echo
amplitudes for several different samples measured for the same spectrometer
pulse and gain settings. The broken lines represents the Curie-law behavior.
## III Experimental Results
Figure 4: (Color online) Observed temperature dependence of the spin-lattice
and spin-spin relaxation time for two samples (i) $x_{3}$=16 ppm, and (ii)
$x_{3}$=500 ppm.
Fig. 4 shows the observed temperature dependence of the nuclear spin-lattice
relaxation time ($T_{1}$) and the spin-spin relaxation time ($T_{2}$) for a
sample with $x_{3}$= 16 ppm and the comparison with the results for a sample
with $x_{3}=$ 500 ppm. The temperature independent tunneling induced
relaxation occurs for $0.2<T<0.55$ K and a small upturn in $T_{1}$ is observed
for $T>0.55$ K with a corresponding downturn in $T_{2}$. This high temperature
behavior is attributed to the onset of the effect of thermally activated
vacancies. The most interesting features are the sharp peak in $T_{1}$ at
$T=175$ mK and a less well-defined minimum in $T_{2}$ at the same temperature
for the sample with $x_{3}$=16 ppm. The same behavior as a function of
temperature is observed for a sample with $x_{3}$ =24 ppm prepared with a
considerably reduced annealing time. SSKimPRL2011 . The sharp changes in
$T_{1}$ and $T_{2}$ are not observed for samples with concentrations $x_{3}>$
200 ppm. These features near 175 mK are unexpected in the traditional
interpretations of the relaxation of dilute 3He in solid 4He that attribute
the relaxation to a temperature independent quantum tunneling $J_{34}$ of the
3He atoms through the 4He lattice. The temperatures at which these sharp
features for $T_{1}$ and $T_{2}$ occur coincide with the temperatures for
which anomalies are observed for measurements of the rotational
inertiaISI:000224136000041 ; ISI:000188068100037 and shear modulus
experimentsPhysRevB.79.214524 ; Day_Beamish . On cooling below 175 mK, a
precipitous drop in the value of $T_{1}$ is observed at $T=95$ mK which is the
temperature associated with the phase separation of the solid mixture into 3He
rich droplets in otherwise pure solid 4He. This interpretation is confirmed by
the observed change in the NMR amplitude below the phase separation
temperature as shown in Fig. 4. and by the significant hysteresis on cycling
through the phase separation temperature. SSKimPRL2011
## IV Discussion
The theoretical treatments of the relaxation of dilute 3He atoms in solid
4HeLandesman1975137 ; ZhEhsp.T ; LandesmanWinter ; PhysRevB.11.3374 ;
LandesmanVacancy all take into consideration the lattice deformation
surrounding a 3He impurity due to the increased zero-point motion of the 3He
atoms with respect to the lattice compared to the 4He atomsNSandAL The
resulting deformation has been described in terms of a long range anisotropic
interaction between 3He atoms given by
$K(r_{ij})=K_{0}(3cos^{2}\theta_{ij}-1)(\frac{a_{0}}{r_{ij}})^{3}$ (1)
for the deformation energy at site j due to an impurity at site i, where
$a_{0}$ is the lattice constant.Eshelby1956 ; Slyusarev1977 . The 3He atoms
travel through the lattice and scatter from one another with a closest
approach given by the distance $b_{c}$ for which the kinetic energy of the
tunneling particle ($\sim J_{34}$) becomes comparable to the elastic
deformation energy
$b_{c}=(\frac{J_{34}}{K_{0}})^{1/3}a_{0}.$ (2)
The mean separation of the 3He atoms $r_{m}=a_{0}x_{3}^{-1/3}$ and for
$x_{3}\geq x_{30}=\frac{J_{34}}{K_{0}}\sim 10^{-3}$ the 3He atoms are in
continuous interaction with elastic fields of one another. For $x_{3}<x_{30}$,
the atoms move coherently by tunneling until scattered by other 3He atoms or
other lattice defects. This simple estimation of the concentration $x_{30}$
for which there is a cross-over from simple coherent diffusion to the
continuous interaction regime was analyzed in detail by Landesman and
WinterLandesmanWinter using a moment expansion for the calculation of the
physical diffusion and they found a very different value with
$x_{30}\sim(\frac{J_{34}}{K_{0}})^{2}\sim 10^{-6}$. Huang et
al.PhysRevB.11.3374 re-examined this estimate using a more precise accounting
of of the energetics of the scattering process and estimated $x_{30}\sim
10^{-4}$. As we will show below, the Landesman model gives a good description
of the values of the spin lattice relaxation time measured by different
research groups with a characteristic concentration dependence of
$T_{1}\propto x_{3}^{-2/3}$ for $x_{3}>10^{-4}$ and a very different
concentration dependence below 10 ppm.
In order to understand the possible origins of the experimentally observed
peak in T1 at low temperatures we need to examine how the tunneling motion of
the 3He atoms and their accompanying lattice deformations determine the
nuclear spin-lattice relaxation. LandesmanLandesman1975137 treated the motion
of the 3He atoms in the presence of the deformations using a fictious spin
model. In this model a fictious spin $S_{i}$ takes the values $0$ or $\pm 1$
according to whether a site i is occupied by a 4He atom or a 3He atom with
real nuclear spin $I_{j}^{z}=\pm 1$, respectively. The probability that a site
j is occupied by a 3He atom is represented by $\tau_{j}=(S_{Z}^{j})^{2}$ and
the elastic deformation energy by the Hamiltonian
$H_{K}=-2\hbar\sum_{jk}K_{jk}\tau_{j}\tau_{k}.$ (3)
The tunneling Hamiltonian is given by
$H_{T}=-2\hbar J_{34}\sum_{j,k}P_{jk}$ (4)
where
$P_{jk}=\frac{1}{2}(S_{j}^{z}S_{j}^{+}S_{k}^{-}S_{k}^{z}+S_{j}^{z}S_{j}^{-}S_{k}^{+}S_{k}^{z})+h.c$
is the permutation operator for atoms at sites i and j.
The time dependence of the fictious spin operators $S_{j}$ and $\tau_{j}$ are
given by
$S_{j}^{+}(t)=e^{iH_{K}t}S_{j}^{+}e^{-iH_{K}t}=S_{j}^{+}e^{i\omega_{j}t}$ (5)
where $\omega_{j}=2\sum_{j<k}K_{jk}\tau_{k}$. In this time dependence
Landesman did not include a lattice relaxation term to account for relaxation
of the lattice as the deformation surrounding the impurity atom moves through
the lattice. Given the recent results from elastic studies of solid helium at
low temperatures this could be an important effect. The relaxation times can
now be estimated by calculating the correlation functions of the nuclear
dipole-dipole interactions. We consider the reduced correlation function
$\Gamma_{ijkl}^{m}=\frac{4}{x_{3}}\langle
T_{ij}^{m}(t)T_{kl}^{m}(0)^{\dagger}\rangle$ (6)
where the $T_{ij}$ are the irreducible nuclear spin operators (that transform
analogously to the spherical harmonics $Y_{2}^{m}(\Omega_{ij}$)).
$T_{ij}^{m}(t)=e^{iH_{M}t}T_{ij}^{m}e^{-iH_{M}t}$ with $H_{m}=H_{K}+H_{T}$ the
total Hamiltonian responsible for the motions in the lattice. For the dilute
lattice the tunneling Hamiltonian does not commute with all $T_{ij}$ and one
finds that the derivative
$\ddot{\Gamma}_{ij,ij}(t)/\Gamma(0)=-12x_{3}J_{34}^{2}\sum_{p}F_{ip}(t)cos[(K_{ij}-K_{jp})t](\frac{a_{0}}{r_{ij}})^{6}$
(7)
with $F_{ip}(t)=\prod_{s}cos(\omega_{s}(t)$ where
$\omega_{s}=x_{3}(K_{is}-K_{ps})t$. Evaluating $F(t)$ using the statistical
method of AndersonAnderson1951 yields
$\ddot{\Gamma}(t)/\Gamma(0)=-48\Lambda^{{}^{\prime}}x_{3}J_{34}^{2}exp[-x_{3}\Lambda(K_{0}t)^{3/4}]$
(8)
where $\Lambda{{}^{\prime}}=0.28$ and $\Lambda=8.77$.LandesmanWinter ;
Landesman1975137 Integrating Eq’n (8) LandesmanLandesman1975137 finds a
correlation time $\tau_{c}$ given by
$\tau_{c}^{-1}=B\frac{J_{34}^{2}}{K_{0}}x_{3}^{-1/3}$ (9)
with $B=23$.
Figure 5: (Color online) Observed 3He concentration dependence of the nuclear
spin-lattice relaxation time for dilute 3He in solid 4He. Orange circles,
Schratter et al. Schratter1984 , up triangles, Allen et al. 1982JLTP…47..289A
, diamonds Schuster et al. Schuster1996 , down triangles Hirayoshi , open
squares Greenberg , solid squares SSKimPRL2011 .
The correlation time $\tau_{c}<\omega_{L}^{-1}$ for all the experiments
reported and we therefore find for the calculated nuclear spin-lattice
relaxation time in Landesman’s model
$T_{1}^{L}=\frac{\omega_{L}^{2}}{46M_{2}}\frac{1}{J_{eff}}x_{3}^{-2/3}$ (10)
where $M_{2}$ is the second moment for pure 3He and
$J_{eff}=J_{34}^{2}/K_{0}$. As shown in Fig. 5 this calculated value provides
a good description of the observed values of $T_{1}$ in the temperature-
independent ”plateau” region for 3He concentrations $x_{3}>100$ ppm for
$\frac{J_{eff}}{2\pi}=1.2$ kHz and no other adjustable parameter. At lower
concentrations a much stronger concentration dependence is observed with
$T_{1}\propto x_{3}^{-4/3}$. This behavior at low concentrations is attributed
to the crossover from the continuous interaction regime to a region of
coherent diffusion for which the characteristic time is determined by the time
for a 3He atom to travel the mean distance $r_{m}=x_{3}^{-{1/3}}a_{0}$ between
3He atoms. This time is given by $\tau_{coh}=r_{m}/v_{g}$ where the group
velocity $v_{g}=a_{0}zJ_{34}$. We find $\tau_{coh}=(zJ_{34})^{-1}x_{3}^{1/3}$
leading to a calculated longitudinal relaxation time $T_{1}^{coh}=2.8\times
10^{-3}x_{3}^{4/3}$ shown by the solid red line of Fig. 5 for $J_{34}/(2\pi)$
= 1.2 MHz.
The predicted temperature independent relaxation for $x_{3}$ = 16 ppm is shown
by the broken lines in Fig. 6. While the fit to experimental data is good at
high temperatures the anomalous relaxation observed at 175 mK is more than a
factor of two larger than the Landesman prediction for a simple tunneling
motion of the 3He atoms through the lattice. The anomaly occurs well above the
phase separation temperature (90 mK) and is not attributable to the phase
separation which would lead to a sharp decrease in the relaxation time and not
the sharp peak that was observed. This conclusion was confirmed by studying a
sample with 24 ppm that was never never cooled to the phase separation
temperature (see Fig. 6).
It is important to note that a simple change in the elastic field interaction
$K$ associated with thermal population of excited states with different values
for $K_{0}$ would not explain the observed results and would simply lead to a
broad step in the value of $T_{1}$. The results therefore imply that the
anomalous peak observed for $T_{1}$ results either from a change in the
dynamical properties of the lattice that occur at 175 mK or that there is a
phase transition that occurs at that temperature and the peak observed in
$T_{1}$ results from the fluctuations at the phase transition.
Figure 6: (Color online) Comparison of the values of the observed nuclear
spin-lattice relaxation times for $x_{3}$=16 ppm (red circles) and $x_{3}$=24
ppm (blue triangles) with the theoretical model of Landesman (broken lines a
and b) for tunneling 3He impurities in the 4He lattice with a fixed lattice
distortion. The sharp drop at 90 mK marks the well known phase separation with
the formation of pure 3He nanodroplets. The 24 ppm sample was never cooled
below 120 mK to avoid hysteresis and memory effects associated with the phase
separation. Figure 7: (Color online) Comparison of the observed temperature
dependence of the nuclear spin-lattice relaxation time $T_{1}$ with the
dependence expected for a true phase transition at T=175 mK for a sample with
$x_{3}$ = 16 ppm as described in the text.
### IV.1 Effect of a transition to supersolid state on the NMR relaxation
times
We now consider what would be expected if there was a simple phase transition
to a state in which there was a small fraction of superfluid condensate, and
then we will discuss how the effect of a dynamical relaxation of the lattice
as described by Beamish and colleagues would influence the nuclear spin
lattice relaxation time $T_{1}$.
If there was a sharp phase transition at a fixed temperature $T_{C}$ below
which a superfluid state appears one would expect a sudden change in the
spectral density at $T_{C}$ that determines $T_{1}$ and $T_{2}$ because of the
sharp increase in fluctuations just below $T_{C}$ that would modulate the
spin-spin interactions (assuming that the 3He atoms are carried by the
superfluid component). One of the most significant results of Chan et
al.ISI:000224136000041 is that the critical velocity attributed to the
superfluid interpretation of the NCRIFs is very low, typically $10^{-4}$ m/s.
The fluctuations would therefore add weight to the spectral density at low
frequencies and not at high frequencies ($>$MHz). Because the spectral
densities are determined by the fluctuations of the 3He dipole-dipole
interactions (and no other magnetic interactions are present) they are
normalized and thus the increase in spectral weight at low frequencies would
result in a decrease in the spectral weight at high frequencies with an
increase in $T_{1}$ and a concommittant decrease in $T_{2}$.
If we designate the component of the spectral density due to 3He particle
exchange as $J_{E}(\omega)$ and the component due to a possible superfluid
component as $J_{S}(\omega)$, we have
$\int\frac{1}{T_{1}}d\omega=\int[J_{E}(\omega)+J_{S}(\omega)]d\omega=\pi
M_{2}$ (11)
where $M_{2}$ is the NMR second moment. For liquid 4He the coherence length
scales as $\xi=a_{0}(T_{C}-T)^{-2/3}$ where $T_{C}$ is the critical
temperature, and because of the very low critical velocities (inferred from
torsional oscillator measurements), we anticipate a characteristic relaxation
time that behaves as $\tau_{S}=\tau_{so}(T_{C}-T)^{-2/3}$ for $T<T_{C}$, using
$\tau_{SO}\cong r_{m}/v_{c}$, where $r_{m}=x^{-1/3}a_{0}$ is the mean
separation between atoms, $a_{0}$ is the lattice spacing and $v_{c}$ is the
critical velocity. If the supersolid component component has weight
$\alpha(T)$ then the exchange component must have weight $1-\alpha(T)$ so that
the sum rule $\int\frac{1}{T_{1}}=\pi M_{2}$ is obeyed. Thus the observed
relaxation rate will be given by
$\frac{1}{T_{1Obs}}=\alpha(T)J_{S}(\omega_{L})+(1-\alpha(T))J_{E}(\omega_{L})$
(12)
where $J_{S}(\omega_{L})$ is negligible. If $T_{1E}$ is the calculated value
of $T_{1}$ for exchange motion only, we find
$\frac{T_{1Obs}}{T_{1E}}=\frac{1}{[1-\alpha(T)]}$ (13)
where for a supersolid fraction we take
$\alpha(T)=a_{1}\frac{T_{0}}{|T-T_{C})|^{2/3}}$ for $T<T_{C}$ and
$\alpha(T)=0$ for $T>T_{C}$. $a_{1}$ is an adjustable parameter related to the
”‘supersolid”’ density which is of the order of 0.01 but sample dependent.
An example of a fit using the above expression for $T_{1Obs}$ is shown by the
broken line in Fig. 7 for $T_{C}$= 175 mK and $a_{1}=$ 0.01 with no other
adjustable parameter. In reality we would expect to have a small Gaussian
spread (of about 7 mK) in the critical temperature because of the
inhomogeneities and it would not be difficult to obtain a good fit to $T_{1}$
for reasonable numbers but no other experiment points to a sharp transition
(except for the onset of NCRIFs in very pure samples), e.g. no sharp peak in
the heat capacity is reported. A more stringent test of this interpretation is
provided by examining the results for the $T_{2}$ measurements for the same
sample. The same argument as used for evaluating $T_{1}$ needs to be followed
for $T_{2}$ but this time we need to know the values for $\tau_{s0}$ and the
exchange time $\tau_{E}$.
Figure 8: (Color online) Observed 3He concentration dependence of the nuclear
spin-spin relaxation time for dilute 3He in solid 4He for several
concentrations reported by different authors. The solid lines $T_{2}^{L}$ and
$T_{2}^{H}$ are the predictions of the theories of LandesmanLandesman1975137
and Huang et al.PhysRevB.11.3374 , respectively. $T_{2}^{Coh}$ is the value of
$T_{2}$ for coherent diffusion where the mean separation is greater than the
scattering length. See legend of Fig. 6 for definiton of symbols.
First we consider the theoretical predictions for $T_{2}$ for small $x_{3}$.
The values of the correlation time calculated by Landesman (Eqn. 10) that give
a good fit to the $T_{1}$ data do not lead to values of $T_{2}$ that are in
agreement with the experimental data. This discrepancy has been reviewed by
Kim et al.KimLT26 who showed that the detailed spectral density at low
frequencies cannot be described by a single Lorentzian spectral density. A
better approach to evaluating $T_{2}$ was given by Huang et
al.PhysRevB.11.3374 who calculated the scattering of tunneling 3He impurities
in the elastic crystal fields using detailed energy conservation and found
$T_{2}^{H}=1.69\times 10^{-4}x_{3}$ s. for $x_{3}>$ 100 ppm. This result is
compared with the experimental data and Landesman’s theory in Fig.8̃. For very
dilute concentrations, $x_{3}<$ 100 ppm, the model of coherent diffusion as
discussed above gives $T_{2}^{coh}=1.68$ s.
For these dilute samples the tunneling motion of the 3He impurities would lead
to a temperature independent relaxation, but once again, we observed strong
deviations from temperature independence below 175 mK. However, unlike the
$T_{1}$ results which show a clean peak, there is considerable scatter in the
$T_{2}$ data and the dependence observed is that of a small peak followed by a
strong dip, and then at even lower temperature there is an order of magnitude
drop at the phase separation temperature where Fermi liquid droplets form. We
now calculate what would be expected for a simple superfluid phase transition.
In terms of the spectral densities (at zero frequency) for the superfluid and
normal components we have
$\frac{1}{T_{2Obs}}=\alpha(T)J_{S}(0)+[1-\alpha(T)]J_{E}(0)$ (14)
while the fixed temperature independent tunneling term is
$\frac{1}{T_{2E}}=J_{E}(0)$ (15)
We therefore find
$\frac{T_{2Obs}}{T_{2E}}=\frac{1}{b\alpha(T)+[1-\alpha(T)]}$ (16)
where $b=J_{S}(0)/J_{E}(0)=\tau_{S0}/{\tau_{E}}>1$ and $\tau_{E}$ is the
characteristic tunneling time in the normal solid. Because of the large value
of $b$, the result for $T_{2Obs}$ does not look like the inverted image of
$T_{1}$ (as one would naively expect) but is much wider. An example of an
attempted fit for the adjustable parameters $b=4$ and $a_{1}=0.01$ is shown in
comparison with experimental values of $T_{2}$ in Fig.9̃. Even allowing for
the scatter of the data, this approach does not provide a good description of
the temperature dependence of $T_{2}$. We should note that if the system is
very inhomogeneous with different parts of the sample having different values
of $b$ and $a_{1}$ one would expect a lumpy spectral density at low
frequencies and a corresponding scatter in the results for $T_{2}$, but since
$T_{1}$ measures the spectral weight at MHz frequencies, one could still
observe a sharp peak for $T_{1}$ because it is the integrated weights of the
low frequency components of the spectral densities that must be subtracted
from the high frequency spectral density. In this sense the $T_{1}$
measurements are a much cleaner test for the existence of a phase transition
that the $T_{2}$ measurements.
Figure 9: (Color online) Comparison of the observed temperature dependence of
the nuclear spin-spin relaxation time $T_{2}$ for a sample with $x_{3}$ = 16
ppm with the dependence expected for a true phase transition at T=175 mK
(broken purple line) as described in the text with the high temperature lmit
given by Landesman’s model.
### IV.2 Effect of lattice dynamics on the NMR relaxation rates.
An alternative approach to interpreting the NMR relaxation rates is to assume
that as a result of the tunneling of the 3He atoms there is an internal stress
on the 4He lattice as the distortion around the 3He impurity moves with or
attempts to move with the impurity as it tunnels from site to site. The mean
frequency of this motion is $\omega_{x}=x^{4/3}K_{0}\approx 3.5\times 10^{3}$
Hz. We can surmise that the relaxation of the lattice due to this time
dependent disturbance is long and adds a bottleneck for energy exchange
between the tunneling excitations and the thermal bath of phonons. The
correlation function for the lattice operators $\tau(T)$ of Eq’n (3) can be
written as
$\langle\tau(t)\tau(0)\rangle=\langle\tau(0)^{2}\rangle_{R}exp(-i\omega_{x}+\tau_{S}^{-1})t$
(17)
where $\langle......\rangle_{R}$ represents an average over lattice
coordinates, $\tau_{S}$ is the lattice averaged relaxation time which we
assume corresponds to the Debye relaxation $\tau_{S}=\tau_{0}exp(E_{0}/T)$
used to interpret the results of the shear modulus
measurements.PhysRevLett.104.195301 ; PhysRevB.79.214524 ;
ElasticPropsBeamish09 Integrating Eq’n (17, we find for the additional
lattice relaxation time
$\tau_{x}(T)=r_{1}\frac{\tau_{0}e^{E_{0}/T}}{1+(\omega_{x}\tau_{0}e^{E_{0}/T})^{2}}=\frac{r_{1}}{\omega_{x}}\frac{u}{1+u^{2}}$
(18)
where $u=\omega_{x}\tau_{0}e^{(E_{0}/T)}$. $\tau_{0}$ and $E_{0}$ are
adjustable parameters that we expect to be comparable to the values determined
from the shear modulus experiments .ElasticPropsBeamish09 ; Day_Beamish The
magnitude of the relaxation time $r_{1}$ can be estimated crudely using the
golden rule with $r_{1}=\frac{4\pi}{\hbar}\sum_{E}|\langle
E_{1}|K|E_{2}\rangle|^{2}\rho(E)\delta(E_{1}-E_{2}-E)$ where $K$ is the
lattice distortion and the density of states $\rho(E)\sim E^{2}/E_{D}^{2}$.
For $E_{1}-E_{2}=J$, the exchange energy, and $E_{D}$ the Debye energy we find
$r_{1}/\omega_{x}\sim 4.10^{-4}$. In Fig.1̃0 we show the fit to the relaxation
times $T_{1}$, using $T_{1Obs}=T_{1E}+\tau_{x}(T)$ for $\tau_{0}=8.3\times
10^{-9}$ s. and $E_{0}=1.8$ K. These values are to be compared with those
inferred from the shear modulus experiments,PhysRevLett.104.195301 ;
PhysRevB.79.214524 ; ElasticPropsBeamish09 $\tau_{0}=2.3\times 10^{-9}$ s.
and $E_{0}=1.7$ K.PhysRevLett.104.195301 Although the values of $\tau_{0}$
and $E_{0}$ are a little different from those deduced by Beamish et
al.,PhysRevB.79.214524 ; ElasticPropsBeamish09 the fit is remarkably good
given the approximations that have been made. This interpretation of the
results is consistent with the observations of Sasaki et al.PhysRevB.81.214515
; Sasaki2011 who followed the temperature dependence of the NMR signals of 10
ppm 3He in solid 4He for densities such that the phase separation forms solid
clusters of 3He. While they were able to observe the solid clusters they were
not able to detect the signal from isolated 3He atoms above the phase
transition at low temperatures, and they attributed this to a long spin-
lattice relaxation time that lead to saturation and signal loss at the
temperatures of interest for exploring the anomalies of the 4He lattice.
Figure 10: (Color online) Comparison of the observed temperature dependence
of the nuclear spin-lattice relaxation time $T_{1}$ (for a sample with $x_{3}$
= 16 ppm) with the dependence calculated for lattice relaxation due to the
motion of the distortion of the lattice around impurities, using parameters
inferred from the measurements of the shear modulus.
### IV.3 Effect of dislocations on relaxation times
The most significant effect of the presence of dislocations on the NMR studies
is that the 3He atoms will pin the dislocation lines leaving some of the 3He
atoms located on the dislocation lines and the lines themselves essentially
immobile at low temperatures. The number of 3He atoms that are expected to be
affected by this is very small. The localization of the 3He atoms would lead
to three distinct components of the NMR signals and their relaxation: (i) the
contribution from atoms fixed on the dislocation lines, (ii) the interactions
between fixed localized 3He atoms with those tunneling freely in the solid,
and (iii) the interactions between pairs where both are free to move in the
lattice. The latter has been the focus of the discussions above. For those
atoms that are essentially fixed the line shapes would be very broad leading
to short values of $T_{2}$ and very long values for $T_{1}$. The echoes would
have more than one contribution and if the numbers of fixed atoms were
significant the echo would have a central peak, corresponding to the spin-spin
interactions between immobile atoms. We have not observed such a peak or a
deviation from a single exponential decay for the dilute samples down to the
lowest temperatures studied.
This result is not surprising when one considers the number of dislocations
present even in crystals of average quality. Paalanen et al.Paalanen1981 and
Iwasa et al.Iwasaetal1978 ; Iwasa2010 show that the dislocation density
$\Lambda$ and the dislocation network length $L_{N}$ are related by $\Lambda
L_{N}^{2}=0.2$ and $\Lambda\cong 10^{6}$ cm-2 with $L_{N}\sim(3-6)\times
10^{-4}$ cm. $\sim 10^{4}a_{0}$. The concentration of 3 He atoms that would
saturate the dislocation network is therefore estimated as
$x_{3s}\sim(\frac{a_{0}}{L_{N}})^{2}\sim 10^{-8}$. The dislocation density
would need to be much larger than that expected for even average quality
crystalsIwasaetal1978 ; Iwasa2010 if the pinned 3He atoms were to be
observable.
In addition to dislocation lines, Balibar has pointed out BalibarPrivate that
grain boundaries are also likely to bind 3He impurities - possibly affecting
NMR relaxation times in a temperature-dependent manner. A key parameter
determining the importance of grain-boundary binding is the typical grain size
$L_{G}$. We have no direct information on $L_{G}$ for our blocked-capillary
samples, although, for example, $L_{G}\approx 0.5$ mm was observed by Franck
et al. Franck in annealed helium films.
The grain size $L_{G}$ controls any possible effect of grain-boundary binding
on the NMR response in two distinct ways: (a) the maximum concentration of 3He
that can be bound at grain boundaries is of order $a_{0}/L_{G}$, assuming full
monolayer coverage with bound 3He, and (b) the time scale for 3He atoms to
freely diffuse on and off grain boundaries is of order $L_{G}^{2}/D$, where
$D$ is the diffusion coefficient. For $L_{G}=0.5$ mm we calculate
$a_{0}/L_{G}=0.7$ ppm, much less than the 3He concentrations used in the
present experiments. For a diffusion constant $D=3.10^{-6}$
cm2/s.1982JLTP…47..289A we find $L_{G}^{2}/D\approx$103 s., much shorter than
the relaxation times $T_{1}$ that we observe. The latter implies that the
collisions between the 3He impurities in the bulk sample dominates the
relaxation process. Therefore, it appears that the grain size in our samples
would need to be much smaller than was observed in films in Ref.[59] for the
binding of 3He atoms at grain boundaries to be the cause of the $T_{1}$
anomalies we observe.
### IV.4 NMR Relaxation for Droplets
For all samples studied (except the 24 ppm sample) a sharp phase transition is
observed at low temperatures below which the NMR amplitudes are independent of
temperature (see Fig. 4) as expected for the well-known phase separation into
Fermi liquid droplets.PhysRevB.39.4083 The observed phase separation
temperatures are in good agreement with the values calculated by Edwards and
Balibar.PhysRevB.39.4083 The rate of formation of the droplets is very slow
(typically 2-10 hours)HuanDroplets and great care must be taken to ensure
that equilibrium is reached before measuring amplitudes and relaxation times
near the phase separation. After phase separation, the relaxation times are
observed to become temperature independent.HuanDroplets ; Kingsley1998 The
observed relaxation times are consistent with a relaxation that occurs at the
wall of degenerate Fermi liquid droplets. Huan et al.HuanDroplets showed that
the relaxation time was given by $T_{1}=(N_{core}/N_{wall})\tau_{X}$ where
$N_{core}$ and $N_{wall}$ are the number of atoms in the core of the droplet
and in the wall respectively. $\tau_{X}$ is the intrinsic relaxation time at
the wall due to the 3He tunneling is given by $\tau_{X}=J_{3}/M_{2}$ where
$J_{3}$ is the tunneling rate at the wall and $M_{2}$ is the NMR second
moment. This estimate leads to values of $T_{1}\sim$ 170 s. in good
qualitative agreement with the observations (see Fig. 5).
## V Conclusion
Measurements of the nuclear spin-lattice relaxation times for very dilute 3He
concentrations have shown the existence of pronounced peaks in the relaxation
times at the same temperatures as those for which anomalies are observed in
torsional oscillator and shear modulus measurements of solid 4He. Less well-
defined variations are observed for the nuclear spin-spin relaxation times.
The detailed temperature dependences do not fit a model in which there is a
well-defined phase transition to a supersolid or superfluid state where the
critical fluctuations can induce dramatic changes in the relaxations times
with critical exponents for the temperature dependence near the transition
temperature. The observations are best described by the introduction of an
additional relaxation process in series with the usual tunneling-relaxation
process and caused by the response of the lattice to the motion of the lattice
distortions around the tunneling impurity atoms. The characteristic parameters
for this model, the tunneling rate and the lattice excitation energy are
remarkably close to those values deduced from measurements of the shear
modulus. Further studies are needed at lower 3He concentrations and lower
magnetic fields to to create a better separation between the phase separation
temperature and the temperatures for which the NMR and other anomalies are
observed.
###### Acknowledgements.
This research was carried out at the NHMFL High B/T Facility which is
supported by NSF Grant DMR 0654118 and by the State of Florida. This project
was supported in part by an award from the Collaborative Users Grant Program
of the NHMFL. We gratefully acknowledge useful discussions with Sebastian
Balibar, Moses Chan, Brian Cowan, Alan Dorsey, Izumi Iwasa and Pradeep Kumar.
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|
arxiv-papers
| 2013-01-16T21:35:12 |
2024-09-04T02:49:40.418173
|
{
"license": "Public Domain",
"authors": "S. S. Kim, C. Huan, L. Yin, J. S. Xia, D. Candela and N. S. Sullivan",
"submitter": "Neil Sullivan",
"url": "https://arxiv.org/abs/1301.3929"
}
|
1301.4032
|
Philip Bett
([email protected])
# European wind variability over 140 yr
P. E. Bett Met Office Hadley Centre, Exeter, UK H. E. Thornton Met Office
Hadley Centre, Exeter, UK R. T. Clark Met Office Hadley Centre, Exeter, UK
###### Abstract
We present initial results of a study on the variability of wind speeds across
Europe over the past 140 yr, making use of the recent _Twentieth Century
Reanalysis_ data set, which includes uncertainty estimates from an ensemble
method of reanalysis. Maps of the means and standard deviations of daily wind
speeds, and the Weibull-distribution parameters, show the expected features,
such as the strong, highly-variable wind in the north-east Atlantic. We do not
find any clear, strong long-term trends in wind speeds across Europe, and the
variability between decades is large. We examine how different years and
decades are related in the long-term context, by looking at the ranking of
annual mean wind speeds. Picking a region covering eastern England as an
example, our analyses show that the wind speeds there over the past $\sim 20$
yr are within the range expected from natural variability, but do not span the
full range of variability of the 140-yr data set. The calendar-year 2010 is
however found to have the lowest mean wind speed on record for this region.
Knowing the form of the wind speed distribution is of critical importance when
assessing the wind energy potential at a site. Typically, when wind farm
developers or investors consider a site, they assess it using (at best) the
past 20–30 yr, with data from direct observations, NWP models, and reanalyses.
These recent decades reflect our personal experience of wind speeds, but they
do not show the longer-term historical context. Understanding whether the most
recent decades were more or less windy than normal, or if there are any
significant long-term trends, is key to understanding the range of possible
future windspeeds we might experience over the coming $\sim 5$ yr, or over the
lifetime of a wind farm ($\sim 25$ yr). This information is important not just
for managing wind farms, but also for planning investment in future wind
energy projects.
In this study, we show wind speed distributions for Europe over 140 yr
(1871–2010), utilising the _Twentieth Century Reanalysis_ data set (20CR,
Compo et al., 2011). This reanalysis incorporates observations of sea-level
pressure and surface pressure alone, with sea-surface temperature and sea-ice
concentration data used as boundary conditions. The reanalysis used an
experimental coupled atmosphere–land version of the NOAA NCEP Global Forecast
Model, ran with 56 ensemble members. This allows an estimate of the
uncertainties present due to episodes with less data (for example due to the
reduction in Atlantic shipping during the World Wars, or the overall reduction
in observations as one looks further back in time). We use the daily-mean wind
speeds at the near-surface pressure level where
$P/P_{\mathrm{surface}}=0.995$, on a regular lat–lon grid of resolution
$2\degree\times 2\degree$. Other studies of European wind speed climatologies
have used data sets that assimilate more observations or are at higher
resolution. For example, the studies of Kiss and Jánosi (2008), Pryor et al.
(2006), and Siegismund and Schrum (2001) use the ERA-40 (Uppala et al., 2005)
and NCEP/NCAR reanalyses (Kalnay et al., 1996), which both span $\sim 50$ yr.
The key feature of the use of the 20CR in our study is the unprecedented
length of the time series, which allows us to compare decadal-scale
fluctuations in the speed and variability of the wind.
In this short article, we present an initial sample of our results. We reserve
a more full analysis for a later publication.
## 1 Results
### 1.1 Wind speed and variability
The simplest way of describing the long-term wind speed distribution over
Europe is to map the long-term mean and standard deviation of the daily wind
speeds. The long-term average is simply the mean of the daily wind speeds from
all days from all ensemble members. For the standard deviation, we need to
take account of the structure of the 20CR ensemble: the 56 members of the
reanalysis were run as a series of independent “streams”, each covering
consecutive 5-yr periods111Streams 16 and 17 actually have lengths of 6 and 4
yr respectively, and stream 27 consists of 10 yr (1999–2010). We treat the
whole series as 28 equal periods of 5-yr, for simplicity. (see Table III in
Compo et al., 2011). The result of this is that a time series from an ensemble
member is only temporally continuous for 5 yr, and aggregating over any period
longer than this risks mixing interannual variability with ensemble member-to-
member variability. Therefore, for the long-term standard deviation, we first
calculate standard deviations in 5-yr periods for each ensemble member. We
then take ensemble means, giving us a single standard-deviation time series in
5-yr steps. These are then aggregated into a single long-term ensemble-mean
value, using222Note that equation (1) is for the population standard
deviation, which we use as we are aggregating data covering the complete time
period, rather than estimating $\sigma$ from a sample of years.
$\sigma^{2}=\frac{\sum_{i}N_{i}\left(\sigma^{2}_{i}+\bar{U}_{i}^{2}\right)}{\sum_{i}N_{i}}-\bar{U}^{2},$
(1)
where $N_{i}$ is the number of days in the 5-yr period $i$, the ensemble-means
of the 5-yr mean and standard deviations of the daily wind speeds are
$\bar{U}_{i}$ and $\sigma_{i}$ respectively, and the long-term mean wind speed
is $\bar{U}\equiv\left(\sum_{i}\bar{U}_{i}\right)/28$.
We show the resulting long-term maps in Fig. 1. The areas with the highest
wind speeds, and the greatest day-to-day variability, are over the sea rather
than the land. In particular, the highest-wind regions are in the north
Atlantic, west of Ireland and north of Scotland. Relatively high winds also
cover the rest of the British Isles and the coastal areas of north-western
Europe, as well as the central Mediterranean.
Figure 1: Maps of the 140-yr mean and standard deviation of the daily wind
speeds from 20CR. The white box in the top plot marks the region we refer to
as ‘eastern England’.
The extremely low winds over the areas of more complex orography (in an arc
from the Atlas Mountains, over Spain, the Alps, the Carpathians, and through
to Turkey) are likely to be due to the drag scheme used to model the impact of
unresolved orography. (Note that the horizontal scale of the mountains is much
smaller than the grid size used here.) While such features are often seen in
maps of model wind speeds (e.g. Howard and Clark, 2007), it is important to be
particularly wary of the results in these areas, as it is not clear how to
relate them to the real physical situation ‘on the ground’.
We have tested for the presence of any long-term trends, by performing a
simple linear regression on the ensemble-mean time series of 5-yr means and
standard deviations of the wind speeds in each grid cell. The significance of
any trend (i.e. whether the gradient of the linear fit was significantly
different from zero) was assessed using a $t$ test at the $0.1\%$ level.
Figure 2 shows the long-term trends, shaded out when they are not
statistically significant.
Figure 2: Gradients of the linear regressions to the 140-yr time series of
5-yearly ensemble-mean average wind speeds (top) and standard deviations of
daily wind speeds (bottom). Shaded areas show where the trend is not
statistically significant at the $0.1\%$ level.
While most areas over Europe show no significant trend, there are some areas
of interest: the Atlantic off the north and west coasts of Ireland and
Scotland shows a significant positive trend (with the mean wind speed
increasing at a rate of $\sim 0.03\,\unit{m\,s^{-1}\,decade^{-1}}$), as does
the eastern Mediterranean around Crete and Turkey ($\sim
0.07\,\unit{m\,s^{-1}\,decade^{-1}}$). There is a significant negative trend
in the central Mediterranean around Sicily and Malta
($\sim-0.05\,\unit{m\,s^{-1}\,decade^{-1}}$). Note, however, that even where
the trends are significant, they are still extremely small: a change of
$\sim\pm 1\,\unit{m\,s^{-1}}$ per two centuries. The standard deviation trend
map shows similar spatial patterns, but even fewer and smaller areas where
those trends are significant.
We show an example time series in detail in Fig. 3, to illustrate some general
features of the wind time series over Europe, and to show the benefits of
being able to use such a long data set. It uses daily wind speeds averaged
over a region covering eastern England (see box in Fig. 1). This region has
been selected partly because it is an area of interest for future wind farm
development333See e.g. maps on the UK Government’s RESTATS web site
http://restats.decc.gov.uk/app/pub/map/map/ and the UK Crown Estate’s web site
http://www.thecrownestate.co.uk/energy-infrastructure/downloads/maps-and-gis-
data/, but also because it avoids some of the more exceptional features
already discussed, such as the artificially low winds seen over complex
terrain and the weak trends seen further to the north-west. We reserve a more
detailed discussion of such features for subsequent papers.
Figure 3: Time series of the wind speed distribution averaged over the eastern
England region. Top: Ensemble mean of the 5-yr mean wind speed, with lighter
shading indicating the 10th and 90th percentiles of the ensemble member
distribution (most visible at early times). Second from top: the ensemble mean
of the 5-yr median wind speed (blue line), as well as the other deciles
(coloured regions, from 10th percentile at the bottom to the 90th percentile
at the top). The long-term means of the deciles are shown by dotted lines.
Widths of the distribution are shown in the panel second from bottom: Half the
difference between ensemble-mean deciles (as labelled), and the ensemble mean
of the standard deviation of ensemble members’ wind speeds in 5-yr bins
(labelled $\sigma$). Bottom: 5-yr means of the day-to-day ensemble spread
(i.e. standard deviation of the ensemble members each day), as a fraction of
the ensemble-mean 5-yr mean wind speed.
The 5-yearly mean wind speeds (top panel of Fig. 3; note the magnified scale)
clearly show the peak in the late-1980s up to around 2000, and the subsequent
return to the long-term mean. We can also see that this excursion from the
mean, while strong, is not exceptional in the long-term history of this region
(see e.g. the peak in the 1915–1920 period, and the prolonged reduction over
1885–1910). There is also no clear trend.
By looking at the deciles of the distribution (second panel in Fig. 3), we can
see how different features appear at different levels. For example, there is a
clear trough in the early 1970s in the central region of the distribution; it
doesn’t appear in the 10th and 90th percentiles, and only forms part of a much
broader reduction in the 80th percentile. The length and form of the reduction
around the 1890–1910 period is also different in different deciles, with the
upper deciles showing a longer reduction in wind speed. These features can be
tracked through into the distribution widths (third panel of Fig. 3): for
example, the dip in the widths over 1905–1910 is due to the lower deciles
rising while the upper deciles are still low. The actual meteorological
situations in these periods would need to be investigated in more detail to
understand the underlying physical causes, but these features show the
importance of analysing the whole distribution beyond just the mean.
The day-to-day ensemble spread as a fraction of the mean wind speed is shown
in the bottom panel of Fig. 3. This will always be much greater than the
uncertainty in the 5-yr mean shown in the top panel; nonetheless, it
illustrates a general decrease in the uncertainty in the data, with an average
daily spread between the ensemble members of about 2.5% of the mean wind speed
by the 2000s. This reduction in uncertainty is directly related to the
dramatic increase in the amount of data assimilated over the course of the
20CR period, which constrains the ensemble members reducing their spread (see
discussions in e.g. Compo et al. 2011; Ferguson and Villarini 2012; Krueger et
al. 2012; Wang et al. 2012). It is important to note that this panel shows the
_uncertainty_ in the reanalysis (through the daily ensemble spread), whereas
the $\sigma$ in the panel above refers to the _variability_ of wind speeds
over time in all ensemble members.
### 1.2 The Weibull distribution
Distributions of wind speeds at given locations are commonly modelled as being
drawn from a Weibull distribution (e.g. Conradsen et al., 1984; Carta et al.,
2009, and references therein), which has the probability density function
(PDF)
$P(U)=\frac{k}{\lambda}\left(\frac{U}{\lambda}\right)^{k-1}\exp\left[-\left(\frac{U}{\lambda}\right)^{k}\right],$
(2)
where the random variable $U$ is the wind speed, $k$ is the (dimensionless)
shape parameter, and $\lambda$ is the scale parameter (in the same units as
$U$). The form of the distribution for different values of $k$ and $\lambda$
is shown in Fig. 4. The shape parameter $k$ determines the overall form of the
distribution, including the skewness, and the width for a given $\lambda$; for
wind speeds, $k$ tends to lie between 2 and 3. For a given shape, the scale
parameter $\lambda$ determines the distribution’s peak-location and width;
higher values of $\lambda$ result in a broader distribution with a peak at a
higher wind speed. We can use the Weibull parameters as an alternative way of
characterising the wind speed distribution. Examining changes in the Weibull
parameters allows us to track the behaviour of the distribution as a whole,
rather than e.g. just the mean value.
Figure 4: Demonstrations of Weibull distribution PDFs when varying the scale
parameter ($\lambda$, left) and the shape parameter ($k$, right). Values
chosen for display reflect those seen in our results.
In a given grid cell or region, we find maximum-likelihood estimates of the
Weibull scale and shape for the distribution of daily wind speeds of each
ensemble member in each 5-yr period. This yields ensemble estimates of the
time series of Weibull parameters; an example from the same eastern England
region as before is shown in Fig. 5. As expected, the behaviour of the scale
parameter over time is similar to the mean and $\sigma$ time series (Fig. 3),
whereas this is not seen in the shape parameter. As the shape parameter
governs the skewness of the distribution, the variation in $k$ could be
related to the presence of extreme high-wind events, as these could cause the
form of the distribution to be fit better with a longer tail.
Figure 5: Time series (in 5-yr bins) of the Weibull scale and shape parameters
for our sample region covering eastern England. The individual ensemble
members are shown as grey lines (note that since ensemble members are
discontinuous in these 5-yr steps, the lines do not reflect actual time
series, but are plotted merely to guide the eye). The blue line shows the
ensemble-means of the Weibull parameters in each step, with the error bars
given by the ensemble-means of the estimated standard errors from the fits.
The long-term means of the ensemble-mean time series are plotted as dotted
lines.
Once again, we can extract linear trends from these time series, test their
significance, and map the results along with their long-term values. The long-
term mean and trend of the shape parameter are shown in Fig. 6. The map for
the average $k$ shows very little spatial variability, staying in most areas
between 2 and 3. Very few areas show any significant trend, although there is
again an area of reduction over the central Mediterranean (i.e. tending to a
more skewed distribution) and growth around Crete (towards a more symmetric
distribution). The long-term mean and trend of the scale parameter is shown in
Fig. 7. The mean-$\lambda$ map is very similar to both the mean and standard
deviation of the wind speeds (cf. Fig. 1), and the trend map is again
familiar; the area of significant growth in the north-east Atlantic is in this
case much larger.
The distribution width is related to _both_ $\lambda$ and $k$, but in opposite
senses (e.g. narrower distributions can be obtained by reducing $\lambda$ or
increasing $k$, with corresponding impacts on the peak locations and
skewness). Thus, it is not surprising that (for example) the negative trends
in both $k$ and $\lambda$ over Italy and the Tyrrhenian Sea do not correspond
with a significant trend in the wind speed standard deviation (cf. Fig. 2).
It is important to note again that, regardless of statistical significance,
the trends present in both $k$ and $\lambda$ are still exceedingly small.
Figure 6: Maps of the long-term mean of the ensemble-mean time series of
Weibull shape parameters (top), and the trend from linear regresson on those
time series (bottom). As before, areas where the trend is not significantly
different from zero are shaded.
Figure 7: As Fig. 6, but for the Weibull scale parameters.
### 1.3 Year-ranking by wind speed
An additional way of relating the wind speeds in different years to the long-
term context is to plot the annual mean wind speeds in rank order. This is
shown qualitatively in Fig. 8, for the eastern England region. We can
immediately see that the day-to-day variability in wind speed is much larger
than the year-to-year variability (in annual-mean wind speed), and the mixture
of colours shows the lack of a clear long-term trend.
Figure 8: Annual mean wind speeds (black line) in the eastern England region,
divided by the 140-yr mean, and plotted in rank order. The vertical bars (one
for each year, coloured by decade) show the 10th and 90th percentile of the
daily winds each year.
We can get a more quantitative picture by highlighting individual decades.
Fig. 9 shows how the most recent three decades are positioned with respect to
the 140-yr time series. Here we can see the differences in the spreads of the
different decades: nine of the ten years of the 1990s (i.e. of the years
1990–1999 inclusive) are in the top half of the distribution, but only a
couple are in the top 10. The 1980s includes some very high and very low wind
years. The 2000s are also well-spread (6 yr in the top half, 4 yr in the
bottom half; note 2008 is in the top 10), but it is 2010 that is the extreme
low-wind year. This is because, since we are using calendar years, 2010 is hit
by the low winds (or lack of high winds) associated with the extreme cold
temperatures in January–February and November–December that year (e.g. Prior
and Kendon, 2011b, a).
Figure 9: As Fig. 8, but highlighting the past three decades separately. We
have included 2010 as an additional year in the 2000s’ plot.
[Summary and conclusions]
Using the Twentieth Century Reanalysis, we have analysed the distribution of
wind speeds over Europe, and how it has varied over 140 yr of daily data. The
regions with the highest long-term average wind speeds, and highest day-to-day
variabilities, are in the north-east Atlantic, and the data exhibits a strong
land–sea contrast. Most areas show no clear trends over time in the mean wind
speed or its variability; even in those areas where a trend is statistically
significant, the magnitude is so small that it is still questionable whether
it is physically meaningful or in any way relevant for applications such as
wind energy. Long-term trends are difficult to distinguish from natural
variability, and the variability is much greater than the uncertainties in the
data.
However, the decadal-scale variations in the long time series help us
understand the behaviour of recent decades in a long-term context. For the
region that we have illustrated as an example (covering eastern England), we
can see that the variations seen in wind speeds in the past 20–30 yr are
typical of the natural variability of this region. Indeed, the past thirty
years do not cover the full range of variability seen in the 140-yr time
series, e.g. not capturing the the lower-wind episodes of the 1970s and 1900s
in England. Note that, while the results do not depend qualitatively on the
precise area in question, there is significant country-scale variability
across Europe, both quantitatively and qualitatively.
Overall, we have shown that long-term data sets are essential in understanding
wind speed distributions across Europe. In subsequent papers, we will describe
our analysis and results in more detail, including a comparison with other
data sets.
###### Acknowledgements.
PB acknowledges the support of an EMS Young Scientist Travel Award. Support
for the Twentieth Century Reanalysis Project dataset is provided by the U.S.
Department of Energy, Office of Science Innovative and Novel Computational
Impact on Theory and Experiment (DOE INCITE) program, and Office of Biological
and Environmental Research (BER), and by the National Oceanic and Atmospheric
Administration Climate Program Office. The works published in this journal are
distributed under the Creative Commons Attribution 3.0 License. This license
does not affect the Crown copyright work, which is re-usable under the Open
Government Licence (OGL). The Creative Commons Attribution 3.0 License and the
OGL are interoperable and do not conflict with, reduce or limit each other.
©Crown copyright 2013 Edited by: S.-E. Gryning
Reviewed by: O. S. Rathmann and one anonymous referee
## References
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* Kiss and Jánosi (2008) Kiss, P. and Jánosi, I. M.: Comprehensive empirical analysis of ERA-40 surface wind speed distribution over Europe, Energ. Convers. Manage., 49, 2142–2151, 10.1016/j.enconman.2008.02.003, 2008.
* Krueger et al. (2012) Krueger, O., Schenk, F., Feser, F., and Weisse, R.: Inconsistencies between Long-Term Trends in Storminess Derived from the 20CR Reanalysis and Observations, J. Climate, 26, 868–874, 10.1175/jcli-d-12-00309.1, 2012.
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* Uppala et al. (2005) Uppala, S. M., Kållberg, P. W., Simmons, A. J., Andrae, U., Bechtold, Fiorino, M., Gibson, J. K., Haseler, J., Hernandez, A., Kelly, G. A., Li, X., Onogi, K., Saarinen, S., Sokka, N., Allan, R. P., Andersson, E., Arpe, K., Balmaseda, M. A., Beljaars, A. C. M., Berg, Bidlot, J., Bormann, N., Caires, S., Chevallier, F., Dethof, A., Dragosavac, M., Fisher, M., Fuentes, M., Hagemann, S., Hólm, E., Hoskins, B. J., Isaksen, L., Janssen, P. A. E. M., Jenne, R., Mcnally, A. P., Mahfouf, J. F., Morcrette, J. J., Rayner, N. A., Saunders, R. W., Simon, P., Sterl, A., Trenberth, K. E., Untch, A., Vasiljevic, D., Viterbo, P., and Woollen, J.: The ERA-40 re-analysis, Q. J. Roy. Meteor. Soc., 131, 2961–3012, 10.1256/qj.04.176, 2005.
* Wang et al. (2012) Wang, X., Feng, Y., Compo, G. P., Swail, V. R., Zwiers, F. W., Allan, R. J., and Sardeshmukh, P. D.: Trends and low frequency variability of extra-tropical cyclone activity in the ensemble of twentieth century reanalysis, Clim. Dynam., pp. 1–26, 10.1007/s00382-012-1450-9, 2012.
|
arxiv-papers
| 2013-01-17T10:13:44 |
2024-09-04T02:49:40.429809
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Philip E. Bett, Hazel E. Thornton, Robin T. Clark",
"submitter": "Philip Bett",
"url": "https://arxiv.org/abs/1301.4032"
}
|
1301.4314
|
# Perturbation analysis for the generalized inverses with prescribed
idempotents in Banach algebras
Jianbing Cao
Department of Mathematics, East China Normal University,
Shanghai 200241, P.R. China
Department of mathematics, Henan Institute of Science and Technology
Xinxiang, Henan, 453003, P.R. China
Email: [email protected] Yifeng Xue
Department of Mathematics, East China Normal University,
Shanghai 200241, P.R. China Email: [email protected]; Corresponding
author
###### Abstract
In this paper, we first study the perturbations and expressions for the
generalized inverses $a^{(2)}_{p,q}$, $a^{(1,2)}_{p,q}$, $a^{(2,l)}_{p,q}$ and
$a^{(l)}_{p,q}$ with prescribed idempotents $p$ and $q$. Then, we investigate
the general perturbation analysis and error estimate for some of these
generalized inverses when $p,\,q$ and $a$ also have some small perturbations.
2010 Mathematics Subject Classification: 15A09; 46L05
Key words: gap function, idempotent element, generalized inverse, perturbation
## 1 Introduction
Let $\mathcal{R}$ be a unital ring and let $\mathcal{R}^{\bullet}$ denote the
set of all idempotent elements in $\mathcal{R}$. Given
$p,\,q\in\mathcal{R}^{\bullet}$. Recall that an element $a\in\mathcal{R}$ has
the $(p,q)$–outer generalized inverse $b=a^{(2)}_{p,q}\in\mathcal{R}$ if
$bab=b$, $ba=p$ and $1-ab=q$. If $b=a^{(2)}_{p,q}$ also satisfies the equation
$aba=a$, then we say $a$ has the $(p,q)$–generalized inverse $b$, in this
case, written $b=a^{(1,2)}_{p,q}$. If an outer generalized inverse with
prescribed idempotents exists, it is necessarily unique (cf. [6]). According
to this definition, obviously, we see that the Moore–Penrose inverses in a
$C^{*}$-algebra and (generalized) Drazin inverses in a Banach algebra can be
expressed by some $(p,q)$–outer generalized inverses (cf. [6, 5, 1]).
Based on some results of Djordjević and Wei in [6], Ilic, Liu and Zhong gave
some equivalent conditions for the existence of the $(p,q)$–outer generalized
inverse in a Banach algebra in [5]. But in our recent paper [1], we find that
Theorem 1.4 of [5] is wrong. In [1], we first present a counter–example to [5,
Theorem 1.4], then based on our counter–example, we define a new type of
generalized inverse with prescribed idempotents in a Banach algebra as
follows:
###### Definition 1.1 (see [1]).
Let $a\in\mathscr{A}$ and $p,\;q\in\mathscr{A}^{\bullet}$. An element
$b\in\mathscr{A}$ satisfying
$bab=b,\quad R_{r}(b)=R_{r}(p),\quad K_{r}(b)=R_{r}(q),$
will be called the $(p,q,l)$–outer generalized inverse of $a$, written as
$a^{(2,l)}_{p,q}=b.$
In addition, if $a^{(2,l)}_{p,q}$ satisfies $a=aa^{(2,l)}_{p,q}a$, we call
$a^{(2,l)}_{p,q}$ is the $(p,q,l)$–generalized inverse of $a$, denoted by
$a^{(l)}_{p,q}$.
Perturbation analysis of the generalized inverses is very important in both
theory and applications. In recent years, there are many fruitful results
concerning the perturbation analysis for various types generalized inverses of
operators on Hilbert spaces or Banach spaces. The concept of stable
perturbation of an operator on Hilbert spaces and Banach spaces is introduced
by Chen and Xue in [1]. Later the notation is generalized to the set of Banach
algebras by the second author in [13] and to the set of Hilbert
$C^{*}$–modules by Xu, Wei and Gu in [16]. Using the notation “stable
perturbation”, many important results in perturbation analyses for
Moore–Penrose inverses on Hilbert spaces and Drazin inverses on Banach spaces
or in Banach algebras have been obtained. Please see [2, 3, 4, 14, 13, 15] for
detail.
Let $X,Y$ be Banach spaces over complex field $\mathbb{C}$. Let $T$ (resp.
$S)$ be a given closed space in $X$ (resp, $Y$). Let $A$ be a bounded linear
operator from $X$ to $Y$ such that $A_{T,S}^{(2)}$ exists. The perturbation
analysis of $A_{T,S}^{(2)}$ for small perturbation of $T$, $S$ and $A$ has
been done in [7, 8]. Motivated by some recent results concerning the
perturbation analysis for the generalized inverses of operators, in this
paper, we mainly study the perturbations and expressions for various types of
generalized inverses with prescribed idempotents in Banach algebras. We first
consider the stable perturbation characterizations for $a^{(2)}_{p,q}$,
$a^{(1,2)}_{p,q}$, $a^{(2,l)}_{p,q}$ and $a^{(l)}_{p,q}$ with prescribed
idempotents $p$ and $q$. Then, by using stable perturbation characterizations,
we can investigate the general perturbation analysis and error estimate for
some of these generalized inverses when $p,\,q$ and $a$ also have some small
perturbations. The results obtained in this paper extend and improve many
recent results in this area.
## 2 Preliminaries
In this section, we give some notations in this paper, we also list some
preliminary results which will be frequently used in our main sections.
Throughout the paper, $\mathscr{A}$ is always a complex Banach algebra with
the unit 1.
Let $a\in\mathscr{A}$. If there is $b\in\mathscr{A}$ such that $aba=a$ and
$bab=b$, then $a$ is called to be generalized invertible and $b$ is called the
generalized inverse of $a$, denoted by $b=a^{+}$. Let $Gi(\mathscr{A})$ denote
the set of all generalized invertible elements in
$\mathscr{A}\backslash\\{0\\}$. Let $\mathscr{A}^{\bullet}$ denote the set of
all idempotent elements in $\mathscr{A}$. If $a\in Gi(\mathscr{A})$, then
$a^{+}a$ and $1-aa^{+}$ are all idempotent elements. For $a\in\mathscr{A}$,
set
$\displaystyle K_{r}(a)$ $\displaystyle=\\{x\in\mathscr{A}\;|\;ax=0\\},$
$\displaystyle\quad R_{r}(a)$ $\displaystyle=\\{ax\;|\;x\in\mathscr{A}\\};$
$\displaystyle K_{l}(a)$ $\displaystyle=\\{x\in\mathscr{A}\;|\;xa=0\\},$
$\displaystyle\quad R_{l}(a)$ $\displaystyle=\\{xa\;|\;x\in\mathscr{A}\\}.$
Clearly, if $p\in\mathscr{A}^{\bullet}$, then $\mathscr{A}$ has the direct sum
decompositions:
$\mathscr{A}=K_{r}(p)\dotplus R_{r}(p)\quad
or\quad\mathscr{A}=K_{l}(p)\dotplus R_{l}(p).$
The following useful and well–known lemma can be easily proved.
###### Lemma 2.1.
Let $x\in\mathscr{A}$ and $p\in\mathscr{A}^{\bullet}$. Then
$K_{r}(p)$ and $R_{r}(p)$ are all closed and $K_{r}(p)=R_{r}(1-p),\
R_{r}(p)\mathscr{A}\subset R_{r}(p);$
$px=x$ if and only if $R_{r}(x)\subset R_{r}(p)$ or $K_{l}(p)\subset
K_{l}(x);$
$xp=x$ if and only if $K_{r}(p)\subset K_{r}(x)$ or $R_{l}(x)\subset
R_{l}(p).$
We list some of the necessary and sufficient conditions for the existence of
$a^{(2,l)}_{p,q}$ in the following lemma, which will be frequently used in the
paper. Here we should indicate that $a^{(2,l)}_{p,q}$ is unique if it exists.
Please see [1] for the proofs and more information.
###### Lemma 2.2.
Let $a\in\mathscr{A}$ and $p,q\in\mathscr{A}^{\bullet}$. Then the following
statements are equivalent:
$a^{(2,l)}_{p,\,q}$ exists;
There exists $b\in\mathscr{A}$ such that $bab=b$, $R_{r}(b)=R_{r}(p)$ and
$K_{r}(b)=R_{r}(q)$;
$K_{r}(a)\cap R_{r}(p)=\\{0\\}$ and $\mathscr{A}=aR_{r}(p)\dotplus R_{r}(q)$;
There exists $b\in\mathscr{A}$ satisfying $b=pb$, $p=bap$, $b(1-q)=b$,
$1-q=(1-q)ab$;
$p\in R_{l}((1-q)ap)=\\{x(1-q)ap\;|\;x\in\mathscr{A}\\}$ and $1-q\in
R_{r}((1-q)ap)$;
There exist some $s,t\in\mathscr{A}$ such that $p=t(1-q)ap$, $1-q=(1-q)aps$.
The following lemma gives some equivalent conditions about the existence of
$a^{(l)}_{p,\,q}$. See [1] for more information.
###### Lemma 2.3.
Let $a\in\mathscr{A}$ and $p,q\in\mathscr{A}^{\bullet}$. Then the following
conditions are equivalent:
$a^{(l)}_{p,\,q}$ exists, i.e.,there exists some $b\in\mathscr{A}$ such that
$aba=a,\quad bab=b,\quad R_{r}(b)=R_{r}(p),\quad K_{r}(b)=R_{r}(q),$
$\mathscr{A}=R_{r}(a)\dotplus R_{r}(q)=K_{r}(a)\dotplus R_{r}(p),$
$\mathscr{A}=aR_{r}(p)\dotplus R_{r}(q),\;R_{r}(a)\cap
R_{r}(q)=\\{0\\},\;K_{r}(a)\cap R_{r}(p)=\\{0\\}.$
Let $X$ be a complex Banach space. Let $M,\,N$ be two closed subspaces in $X$.
Set
$\delta(M,N)=\begin{cases}\sup\\{{\rm dist}(x,N)\,|\,x\in
M,\,\|x\|=1\\},\quad&M\not=\\{0\\}\\\ 0\quad&M=\\{0\\}\end{cases},$
where ${\rm dist}(x,N)=\inf\\{\|x-y\|\,|\,y\in N\\}$. The gap
$\hat{\delta}(M,N)$ of $M,\,N$ is given by
$\hat{\delta}(M,N)=\max\\{\delta(M,N),\delta(N,M)\\}$. For convenience, we
list some properties about $\delta(M,N)$ and $\hat{\delta}(M,N)$ which come
from [10] as follows.
###### Proposition 2.4 ([10]).
Let $M,\,N$ be closed subspaces in a Banach space $X$. Then
$\delta(M,N)=0$ if and only if $M\subset N$;
$\hat{\delta}(M,N)=0$ if and only if $M=N$;
$\hat{\delta}(M,N)=\hat{\delta}(N,M)$;
$0\leq\delta(M,N)\leq 1$, $0\leq\hat{\delta}(M,N)\leq 1$.
## 3 Stable perturbations for the (p, q)–generalized inverses
Let $a\in Gi(\mathscr{A})$ and let $\bar{a}=a+\delta a\in\mathscr{A}$. Recall
from [14] that $\bar{a}$ is a stable perturbation of $a$ if
$R_{r}(\bar{a})\cap K_{r}(a^{+})=\\{0\\}$. Obviously, we can define the stable
perturbation for various kind of generalized inverses. In this section, we
concern the stable perturbation problem for various types of
$(p,q)$–generalized inverses in a Banach algebra.
###### Lemma 3.1 ([9, Lemma 2.2]).
Let $a,\,b\in\mathscr{A}$. If $1+ab$ is left invertible, then so is $1+ba$.
###### Lemma 3.2.
Let $a,\delta a\in\mathscr{A}$ and $p,\,q\in\mathscr{A}^{\bullet}$ such that
$a^{(2,l)}_{p,\,q}$ exists. Put $\bar{a}=a+\delta a$. If $1+\delta
aa^{(2,l)}_{p,\,q}$ is invertible, $w=a^{(2,l)}_{p,\,q}(1+\delta
aa^{(2,l)}_{p,\,q})^{-1}$. Then $\bar{a}^{(2,l)}_{p,\,q}$ exists and
$w=\bar{a}^{(2,l)}_{p,\,q}$ .
###### Proof.
We prove our result by showing that
$waw=w,R_{r}(w)=R_{r}(p),K_{r}(w)=R_{r}(q)$. It is easy to check that
$w=a^{(2,l)}_{p,\,q}(1+\delta
aa^{(2,l)}_{p,\,q})^{-1}=(1+a^{(2,l)}_{p,\,q}\delta a)^{-1}a^{(2,l)}_{p,\,q}.$
Then, by using these two equalities, we can show
$R_{r}(w)=R_{r}(a^{(2,l)}_{p,\,q})=R_{r}(p)$ and
$K_{r}(w)=K_{r}(a^{(2,l)}_{p,\,q})=R_{r}(q)$. We can also compute
$\displaystyle w\bar{a}w$ $\displaystyle=a^{(2,l)}_{p,\,q}(1+\delta
aa^{(2,l)}_{p,\,q})^{-1}\bar{a}a^{(2,l)}_{p,\,q}(1+\delta
aa^{(2,l)}_{p,\,q})^{-1}$ $\displaystyle=a^{(2,l)}_{p,\,q}(1+\delta
aa^{(2,l)}_{p,\,q})^{-1}[(aa^{(2,l)}_{p,\,q}-1)+(1+\delta
aa^{(2,l)}_{p,\,q})](1+\delta aa^{(2,l)}_{p,\,q})^{-1}$
$\displaystyle=a^{(2,l)}_{p,\,q}(1+\delta
aa^{(2,l)}_{p,\,q})^{-1}(aa^{(2,l)}_{p,\,q}-1)(1+\delta
aa^{(2,l)}_{p,\,q})^{-1}+w$ $\displaystyle=w.$
By Definition 1.1 and the uniqueness of $a^{(2,l)}_{p,\,q}$, we see
$\bar{a}^{(2,l)}_{p,\,q}$ exists and $w=\bar{a}^{(2,l)}_{p,\,q}$ . ∎
Obviously, from the proof of Lemma 3.2, we see that if $a^{(2,l)}_{p,\,q}$
exists and $1+a^{(2,l)}_{p,\,q}\delta a$ is invertible, set
$v=(1+a^{(2,l)}_{p,\,q}\delta a)^{-1}a^{(2,l)}_{p,\,q}$, then we also have
$v=\bar{a}^{(2,l)}_{p,\,q}$. In order to prove the main results about the
stable perturbation, we need one more characterizations of the existence of
$a^{(2,l)}_{p,\,q}$.
For an element $a\in\mathscr{A}$ and $p,\;q\in\mathscr{A}^{\bullet}$. Let
$R_{a}\colon\mathscr{A}\to\mathscr{A}$ be the right multiplier on
$\mathscr{A}$(i.e., $R_{a}(x)=xa$ for any $x\in\mathscr{A}$). Then it easy to
see that $a^{(2,l)}_{p,q}$ exists in $\mathscr{A}$ if and only if
$(R_{a})^{(2)}_{\mathscr{A}(1-q),\mathscr{A}(1-p)}$ exists in the Banach
algebra $B(\mathscr{A})$. So from the equivalences of (1), (2) and (3) in
Lemma 2.2, dually, we can get the following equivalent conditions for the
existence of $a^{(2,l)}_{p,q}$.
###### Proposition 3.3.
Let $a\in\mathscr{A}$ and $p,\;q\in\mathscr{A}^{\bullet}$. Then the following
statements are equivalent:
$a^{(2,l)}_{p,\,q}$ exists;
There exists $c\in\mathscr{A}$ such that $cac=c$, $R_{l}(c)=R_{l}(1-q)$ and
$K_{l}(c)=R_{l}(1-p)$;
$K_{l}(a)\cap R_{l}(1-q)=\\{0\\}$ and $\mathscr{A}=R_{l}(1-q)a\dotplus
R_{l}(1-p)$.
###### Proof.
$(1)\Leftrightarrow(2)$ Suppose that $a^{(2,l)}_{p,\,q}$ exists. Let
$c=a^{(2,l)}_{p,\,q}$. Then from Definition 1.1, we know that $cac=c$, and
then $ca,\,ac\in\mathscr{A}^{\bullet}$, $R_{r}(ca)=R_{r}(c)=R_{r}(p)$,
$K_{r}(ac)=K_{r}(c)=R_{r}(q)$. Thus, it follows from Lemma 2.1 that
$\displaystyle cap=p,\quad pca=ca,\quad ac(1-q)=ac,\quad(1-q)ac=1-q.$
Then, by using Lemma 2.1 again, we have
$\displaystyle K_{l}(ca)\subset K_{l}(p)\subset K_{l}(ca),\quad
R_{l}(ac)\subset R_{l}(1-q)\subset R_{l}(ac).$ (3.1)
By using $cac=c$, we have $K_{l}(ca)=K_{l}(c)$ and $R_{r}(ac)=R_{r}(c)$. Thus
from Eq. (3.1) we see that (2) holds. If (2) holds, similarly, by using
Definition 1.1 and Lemma 2.1, we can obtain $a^{(2,l)}_{p,\,q}$ exists.
$(2)\Leftrightarrow(3)$ By our remark above this lemma, we see these hold
simply from the equivalences of (2) and (3) in Lemma 2.2. Note that we can
also prove these equivalences directly by using the right multiplier $R_{a}$
on $\mathscr{A}$. Here we omit the detail. ∎
Now we can present one of our main results about the stable perturbation of
the generalized inverse $a^{(2,l)}_{p,\,q}$.
###### Theorem 3.4.
Let $a,\,\delta a\in\mathscr{A}$ and $p,\,q\in\mathscr{A}^{\bullet}$ such that
$a^{(2,l)}_{p,\,q}$ exists. Put $\bar{a}=a+\delta a$. Then the following
statements are equivalent:
$1+\delta aa^{(2,l)}_{p,\,q}$ is invertible;
$1+a^{(2,l)}_{p,\,q}\delta a$ is invertible;
$\bar{a}^{(2,l)}_{p,\,q}$ exists. In this case, we have
$\bar{a}^{(2,l)}_{p,\,q}=a^{(2,l)}_{p,\,q}(1+\delta
aa^{(2,l)}_{p,\,q})^{-1}=(1+a^{(2,l)}_{p,\,q}\delta a)^{-1}a^{(2,l)}_{p,\,q}.$
###### Proof.
$(1)\Leftrightarrow(2)$ follows from the well-known spectral theory in Banach
algebras.
$(2)\Rightarrow(3)$ We prove our result by using Lemma 2.2. Let $x\in
K_{r}(\bar{a})\cap R_{r}(p)=\\{0\\}$. Since
$R_{r}(p)=R_{r}(a^{(2,l)}_{p,\,q})$, then there exists some $t\in\mathscr{A}$
satisfying $x=a^{(2)}_{p,\,q}t$ and $\bar{a}t=0$. Thus we have
$\displaystyle(1+a^{(2,l)}_{p,\,q}\delta a)a^{(2,l)}_{p,\,q}t$
$\displaystyle=a^{(2,l)}_{p,\,q}t+a^{(2,l)}_{p,\,q}\delta aa^{(2,l)}_{p,\,q}t$
$\displaystyle=a^{(2,l)}_{p,\,q}(a+\delta a)a^{(2,l)}_{p,\,q}t$
$\displaystyle=a\bar{a}t=0.$
Since $1+a^{(2,l)}_{p,\,q}\delta a$ is invertible, it follows that
$x=a^{(2,l)}_{p,\,q}t=0$. Therefore,
$\displaystyle K_{r}(\bar{a})\cap R_{r}(p)=\\{0\\}$ (3.2)
Let $s\in\bar{a}R_{r}(p)\cap R_{r}(q)$. Since
$R_{r}(p)=R_{r}(a^{(2,l)}_{p,\,q})$ and $R_{r}(q)=K_{r}(a^{(2,l)}_{p,\,q})$,
then there exists some $z\in\mathscr{A}$ such that
$s=\bar{a}a^{(2,l)}_{p,\,q}z$ and $a^{(2,l)}_{p,\,q}s=0$. Similar to the proof
of Eq. (3.2), we can get $s=a^{(2,l)}_{p,\,q}t=0$, i.e, $\bar{a}R_{r}(p)\cap
R_{r}(q)=\\{0\\}$. Since $1+a^{(2,l)}_{p,\,q}\delta a$ is invertible, then for
any $w\in\mathscr{A}$ there is some $v\in\mathscr{A}$ such that
$a^{(2,l)}_{p,\,q}w=(1+a^{(2,l)}_{p,\,q}\delta a)v$. From $\bar{a}=a+\delta
a$, we have
$(1-a^{(2,l)}_{p,\,q}a)v=a^{(2,l)}_{p,\,q}(w-\bar{a}v)\in
R_{r}(a^{(2,l)}_{p,\,q}a)\cap K_{r}(a^{(2,l)}_{p,\,q}a)=\\{0\\}.$
Thus, $w-\bar{a}v\in K_{r}(a^{(2,l)}_{p,\,q})$ and $v=a^{(2,l)}_{p,\,q}av\in
R_{r}(a^{(2,l)}_{p,\,q})$. Since for $w\in\mathscr{A}$, we also have
$w=\bar{a}v+(w-\bar{a}v)\in\bar{a}R_{r}(p)\dotplus R_{r}(q).$ Thus, we have
$\displaystyle\mathscr{A}=\bar{a}R_{r}(p)\dotplus R_{r}(q).$ (3.3)
Now, from Eqs. (3.2) and (3.3), by using Lemma 2.2, we see that
$\bar{a}^{(2,l)}_{p,\,q}$ exists.
$(3)\Rightarrow(1)$ Suppose that $\bar{a}^{(2,l)}_{p,\,q}$ exists, we want to
prove $1+\delta aa^{(2,l)}_{p,\,q}$ is both left and right invertible. Since
$\bar{a}^{(2,l)}_{p,\,q}$ exists, then from Lemma 2.2,
$\mathscr{A}=\bar{a}R_{r}(p)\dotplus
R_{r}(q)=\bar{a}R_{r}(a^{(2,l)}_{p,\,q})\dotplus K_{r}(a^{(2,l)}_{p,\,q})$.
Thus, for any $x\in\mathscr{A}$, we can write
$x=\bar{a}a^{(2,l)}_{p,\,q}t_{1}+t_{2}$, where $t_{1}\in\mathscr{A}$ and
$t_{2}\in K_{r}(a^{(2,l)}_{p,\,q})$. Set $s=aa^{(2,l)}_{p,\,q}t_{1}+t_{2}$,
then
$\displaystyle(1+\delta aa^{(2,l)}_{p,\,q})s$ $\displaystyle=(1+\delta
aa^{(2,l)}_{p,\,q})(aa^{(2,l)}_{p,\,q}t_{1}+t_{2})$
$\displaystyle=\bar{a}a^{(2,l)}_{p,\,q}t_{1}+t_{2}=x.$
Since $x\in\mathscr{A}$ is arbitrary, let $x=1$, then we see that $1+\delta
aa^{(2,l)}_{p,\,q}$ is right invertible. Now we prove that $1+\delta
aa^{(2,l)}_{p,\,q}$ is also left invertible. In fact, from Proposition 3.3, we
also have $\mathscr{A}=R_{l}(1-q)\bar{a}\dotplus
R_{l}(1-p)=R_{l}(a^{(2,l)}_{p,\,q})\bar{a}\dotplus K_{l}(a^{(2,l)}_{p,\,q})$
for $\bar{a}^{(2,l)}_{p,\,q}$ exists. Then for any $z\in\mathscr{A}$, we can
write $z=s_{1}a^{(2,l)}_{p,\,q}\bar{a}+s_{2}$, where $s_{1}\in
R_{l}(a^{(2,l)}_{p,\,q}a)$ and $s_{2}\in K_{l}(a^{(2,l)}_{p,\,q})$. Let
$t=s_{1}+s_{2}$, then we have
$\displaystyle t(1+a^{(2,l)}_{p,\,q}\delta a)$
$\displaystyle=(s_{1}+s_{2})(1+a^{(2,l)}_{p,\,q}\delta a)$
$\displaystyle=s_{1}+s_{2}+s_{1}a^{(2,l)}_{p,\,q}(\bar{a}-a)$
$\displaystyle=s_{1}a^{(2,l)}_{p,\,q}\bar{a}+s_{2}+s_{1}(1-a^{(2,l)}_{p,\,q}a)$
$\displaystyle=z.$
Since $z\in\mathscr{A}$ is arbitrary, let $z=1$, then we get that
$1+a^{(2,l)}_{p,\,q}\delta a$ is left invertible. But from Lemma 3.1 we see
$1+\delta aa^{(2,l)}_{p,\,q}$ is also left invertible. Thus, $1+\delta
aa^{(2,l)}_{p,\,q}$ is invertible.
Now, from Lemma 3.2, $\bar{a}^{(2,l)}_{p,\,q}=a^{(2,l)}_{p,\,q}(1+\delta
aa^{(2,l)}_{p,\,q})^{-1}=(1+a^{(2,l)}_{p,\,q}\delta a)^{-1}a^{(2,l)}_{p,\,q}.$
This completes the proof. ∎
###### Lemma 3.5.
Let $a,\,\delta a\in\mathscr{A}$ and $p,\,q\in\mathscr{A}^{\bullet}$ such that
$a^{(2,l)}_{p,\,q}$ exists. If $1+a^{(2,l)}_{p,\,q}\delta a$ is invertible.
Put $\bar{a}=a+\delta a$ and $f=(1+a^{(2,l)}_{p,\,q}\delta
a)^{-1}(1-a^{(2,l)}_{p,\,q}a)$. Then
$f\in\mathscr{A}^{\bullet}$ with $K_{r}(\bar{a})\subset R_{r}(f)$;
$K_{r}(\bar{a})=R_{r}(f)$ if and only if $R_{r}(\bar{a})\cap
R_{r}(q)=\\{0\\}$.
###### Proof.
(1) Since $(1-a^{(2,l)}_{p,\,q}a)(1+a^{(2,l)}_{p,\,q}\delta
a)=1-a^{(2,l)}_{p,\,q}a$ and $1+a^{(2,l)}_{p,\,q}\delta a$ is invertible, we
have $1-a^{(2,l)}_{p,\,q}a=(1-a^{(2,l)}_{p,\,q}a)(1+a^{(2,l)}_{p,\,q}\delta
a)^{-1}.$ Thus,
$f^{2}=(1+a^{(2,l)}_{p,\,q}\delta
a)^{-1}(1-a^{(2,l)}_{p,\,q}a)(1+a^{(2,l)}_{p,\,q}\delta
a)^{-1}(1-a^{(2,l)}_{p,\,q}a)=f.$
Now for any $x\in\mathscr{A}$, from
$(1-aa^{(2,l)}_{p,\,q})x=(1+a^{(2,l)}_{p,\,q}\delta
a)x-a^{(2,l)}_{p,\,q}\bar{a}x$, we have
$fx=(1+a^{(2,l)}_{p,\,q}\delta
a)^{-1}(1-a^{(2,l)}_{p,\,q}a)x=x-(1+a^{(2,l)}_{p,\,q}\delta
a)^{-1}a^{(2,l)}_{p,\,q}\bar{a}x.$ (3.4)
Eq. (3.4) implies that $K_{r}(\bar{a})\subset R_{r}(f)$.
(2) $(\Rightarrow)$ Let $t\in R_{r}(\bar{a})\cap R_{r}(q)=\\{0\\}$. Since
$R_{r}(q)=K_{r}(a^{(2,l)}_{p,\,q})$, then there is some $x\in\mathscr{A}$ such
that $t=\bar{a}x$ and $a^{(2,l)}_{p,\,q}\bar{a}x=a^{(2,l)}_{p,\,q}t=0$. Thus,
$x=fx$ by Eq. (3.4). So, $x\in R_{r}(f)=K_{r}(\bar{a})$ and $t=\bar{a}x=0$,
i.e., $R_{r}(\bar{a})\cap R_{r}(q)=\\{0\\}$.
$(\Leftarrow)$ Thanks to (1), we need only to prove $R_{r}(f)\subset
K_{r}(\bar{a})$. Let $t\in R_{r}(f)$. Since $f\in\mathscr{A}^{\bullet}$, we
have $t=ft$. So by Eq. (3.4), we get $(1+a^{(2,l)}_{p,\,q}\delta
a)^{-1}a^{(2,l)}_{p,\,q}\bar{a}t=0$ and then $a^{(2,l)}_{p,\,q}\bar{a}t=0$.
Hence $\bar{a}t\in R_{r}(\bar{a})\cap R_{r}(q)=\\{0\\}$ and
$K_{r}(\bar{a})=R_{r}(f)$. ∎
Similar to [13, Proposition 2.2] or [14, Theorem 2.4.7], but by using some of
our characterizations for $a^{(2,l)}_{p,\,q}$ and $a^{(l)}_{p,\,q}$, we can
obtain the following results about the stable perturbations for these two
kinds of generalized inverses.
###### Theorem 3.6.
Let $a,\,\delta a\in\mathscr{A}$ and $p,\,q\in\mathscr{A}^{\bullet}$ such that
$a^{(2,l)}_{p,\,q}$ exists. Suppose that $1+a^{(2,l)}_{p,\,q}\delta a$ is
invertible. Put $\bar{a}=a+\delta a$ and $w=(1+a^{(2,l)}_{p,\,q}\delta
a)^{-1}a^{(2,l)}_{p,\,q}$. Then the following statements are equivalent:
$w=\bar{a}^{(l)}_{p,\,q}$, i.e.,
$\bar{a}^{(2,l)}_{p,\,q}=\bar{a}^{(l)}_{p,\,q}$;
$R_{r}(\bar{a})\cap R_{r}(q)=\\{0\\}$, i.e., $\bar{a}$ is a stable
perturbation of $a$;
$\bar{a}(1+a^{(2,l)}_{p,\,q}\delta a)^{-1}(1-a^{(2,l)}_{p,\,q}a)=0$;
$(1-aa^{(2,l)}_{p,\,q})(1+\delta aa^{(2,l)}_{p,\,q})^{-1}\bar{a}=0$.
###### Proof.
The implication $(1)\Leftrightarrow(2)$ comes from Lemma 2.2, Lemma 2.3 and
Theorem 3.4. The implication $(2)\Leftrightarrow(3)$ comes from Lemma 3.5.
$(3)\Leftrightarrow(4)$ we can compute in the following way,
$\displaystyle\bar{a}(1+a^{(2,l)}_{p,\,q}\delta a)^{-1}(1-a^{(2,l)}_{p,\,q}a)$
$\displaystyle=\bar{a}(1+a^{(2,l)}_{p,\,q}\delta
a)^{-1}[(1+a^{(2,l)}_{p,\,q}\delta a)-a^{(2,l)}_{p,\,q}\bar{a}]$
$\displaystyle=\bar{a}-\bar{a}a^{(2,l)}_{p,\,q}(1+\delta
aa^{(2,l)}_{p,\,q})^{-1}\bar{a}$ $\displaystyle=\bar{a}-[(1+\delta
aa^{(2,l)}_{p,\,q})-(1-aa^{(2,l)}_{p,\,q})](1+\delta
aa^{(2,l)}_{p,\,q})^{-1}\bar{a}$
$\displaystyle=(1-aa^{(2,l)}_{p,\,q})(1+\delta
aa^{(2,l)}_{p,\,q})^{-1}\bar{a}.$
This completes the proof. ∎
Furthermore, by using the above theorem, we have the following results.
###### Corollary 3.7.
Let $a,\,\delta a\in\mathscr{A}$ and $p,\,q\in\mathscr{A}^{\bullet}$ such that
$a^{(2,l)}_{p,\,q}$ exists. Put $\bar{a}=a+\delta a$. If
$1+a^{(2,l)}_{p,\,q}\delta a$ is invertible. Then the following statements are
equivalent:
$R_{r}(\bar{a})\cap R_{r}(q)=\\{0\\}$, i.e., $\bar{a}$ is stable perturbation
of $a$;
$(1+a^{(2,l)}_{p,\,q}\delta a)^{-1}K_{r}(a^{(2,l)}_{p,\,q}a)=K_{r}(\bar{a})$;
$(1+\delta aa^{(2,l)}_{p,\,q})^{-1}R_{r}(\bar{a})=R_{r}(aa^{(2,l)}_{p,\,q})$.
###### Proof.
Note that we have $K_{r}(a^{(2,l)}_{p,\,q}a)=R_{r}(1-a^{(2,l)}_{p,\,q}a)$ and
$R_{r}(aa^{(2,l)}_{p,\,q})=K_{r}(1-aa^{(2,l)}_{p,\,q})$. So we can get the
assertions by using Theorem 3.6. ∎
###### Theorem 3.8.
Let $a,\,\delta a\in\mathscr{A}$ and $p,\,q\in\mathscr{A}^{\bullet}$ such that
$a^{(l)}_{p,\,q}$ exists. Put $\bar{a}=a+\delta a$. Then the following
statements are equivalent:
$1+a^{(l)}_{p,\,q}\delta a$ is invertible, $R_{r}(\bar{a})=K_{r}(q)$ and
$\bar{a}^{(l)}_{p,\,q}=a^{(l)}_{p,\,q}(1+\delta aa^{(l)}_{p,\,q})^{-1}$.
$R_{r}(\bar{a})\cap R_{r}(q)=\\{0\\}$, $K_{r}(\bar{a})\cap R_{r}(p)=\\{0\\}$
and $\bar{a}R_{r}(p)=K_{r}(q)$.
###### Proof.
$(1)\Rightarrow(2)$ Suppose that (1) holds. Since $a^{(l)}_{p,\,q}$ exists, we
obtain that $a^{(2,l)}_{p,\,q}$ exists and
$a^{(2,l)}_{p,\,q}=a^{(l)}_{p,\,q}$. Thus, from our assumption, by using
Theorem 3.6 and Lemma 2.3, we have
$R_{r}(\bar{a})\cap R_{r}(q)=\\{0\\},\quad K_{r}(\bar{a})\cap
R_{r}(p)=\\{0\\}.$
Now we need to show $\bar{a}R_{r}(p)=K_{r}(q)$. But since
$R_{r}(\bar{a})=K_{r}(q)$, so we can prove our result by showing that
$\bar{a}R_{r}(p)=R_{r}(\bar{a})$. Obviously, $\bar{a}R_{r}(p)\subset
R_{r}(\bar{a})$. On the other hand, since $\bar{a}^{(l)}_{p,\,q}$ exists, then
by Lemma 2.3 again, we have $\mathscr{A}=\bar{a}R_{r}(p)\dotplus R_{r}(q)$.
Now for any $x\in R_{r}(\bar{a})$, we can write $x=x_{1}+x_{2}$ with
$x_{1}\in\bar{a}R_{r}(p)$ and $x_{2}\in R_{r}(q)$. From
$\bar{a}R_{r}(p)\subset R_{r}(\bar{a})$, we get $x_{1}\in R_{r}(\bar{a})$.
Thus,
$x_{2}=x-x_{1}\in R_{r}(\bar{a})\cap R_{r}(q)=\\{0\\}.$
Therefore, $x_{2}=0$ and then $x=x_{1}\in\bar{a}R_{r}(p)$. Hence,
$\bar{a}R_{r}(p)=R_{r}(\bar{a})=K_{r}(q)$.
$(2)\Rightarrow(1)$ Since $q\in\mathscr{A}^{\bullet}$ and
$\bar{a}R_{r}(p)=K_{r}(q)$, we can write $\mathscr{A}=K_{r}(q)\dotplus
R_{r}(q)=\bar{a}R_{r}(p)\dotplus R_{r}(q)$. Note that $R_{r}(\bar{a})\cap
R_{r}(q)=\\{0\\}$, $K_{r}(\bar{a})\cap R_{r}(p)=\\{0\\}$, then by using Lemma
2.3, we get $a^{(l)}_{p,\,q}$ exists, then $a^{(2,l)}_{p,\,q}$ also exists and
$a^{(2,l)}_{p,\,q}=a^{(l)}_{p,\,q}$. Thus, from Theorem 3.4, we see
$1+a^{(l)}_{p,\,q}\delta a$ is invertible and
$\bar{a}^{(l)}_{p,\,q}=a^{(l)}_{p,\,q}(1+\delta aa^{(l)}_{p,\,q})^{-1}$. Now,
by using Lemma 2.3, we can get $\mathscr{A}=\bar{a}R_{r}(p)\dotplus R_{r}(q)$.
Similarly, as in $(1)\Rightarrow(2)$, by using $R_{r}(\bar{a})\cap
R_{r}(q)=\\{0\\}$, we can show that $\bar{a}R_{r}(p)=R_{r}(\bar{a})$ and then
$R_{r}(\bar{a})=K_{r}(q)$. This completes the proof. ∎
The first result in the following lemma has been proved for generalized
inverse $a^{+}$ by the second author (see [13, Proposition 2.5]). By using the
same method, we can prove the following results for the general inverse
$a^{(l)}_{p,\,q}$.
###### Lemma 3.9 ([13, Proposition 2.5]).
Let $a,\delta a\in\mathscr{A}$ and $p,\,q\in\mathscr{A}^{\bullet}$ such that
$a^{(l)}_{p,\,q}$ exists. Put $\bar{a}=a+\delta a$.
If $\delta(R_{r}(\bar{a}),R_{r}(a))<\|1-aa^{(l)}_{p,\,q}\|^{-1}$, then
$R_{r}(\bar{a})\cap R_{r}(q)=\\{0\\}$;
If $\delta(K_{r}(\bar{a}),K_{r}(a))<\|a^{(l)}_{p,\,q}a\|^{-1}$, then
$K_{r}(\bar{a})\cap R_{r}(p)=\\{0\\}$.
By using Theorem 3.6, Theorem 3.8 and Lemma 3.9, we have the following:
###### Corollary 3.10.
Let $a,\delta a\in\mathscr{A}$ and $p,\,q\in\mathscr{A}^{\bullet}$ such that
$a^{(l)}_{p,\,q}$ exists. Put $\bar{a}=a+\delta a$. If one of the following
condition holds,
$\delta(K_{r}(\bar{a}),K_{r}(a))<\|a^{(l)}_{p,\,q}a\|^{-1}$,
$\delta(R_{r}(\bar{a}),R_{r}(a))<\|1-aa^{(l)}_{p,\,q}\|^{-1}$ and
$\bar{a}R_{r}(p)=K_{r}(q)$.
$1+a^{(l)}_{p,\,q}\delta a$ is invertible and
$\delta(R_{r}(\bar{a}),R_{r}(a))<\|1-aa^{(l)}_{p,\,q}\|^{-1}$. Then
$\bar{a}^{(l)}_{p,\,q}$ exists and
$\bar{a}^{(l)}_{p,\,q}=a^{(l)}_{p,\,q}(1+\delta aa^{(l)}_{p,\,q})^{-1}$.
Finally, we present some perturbation results for $a^{(2)}_{p,\,q}$.
###### Theorem 3.11.
Let $a,\,\delta a\in\mathscr{A}$ and $p,\,q\in\mathscr{A}^{\bullet}$ such that
$a^{(2)}_{p,\,q}$ exists. Put $\bar{a}=a+\delta a$. If
$1+a^{(2)}_{p,\,q}\delta a$ is invertible. Then the following statements are
equivalent:
$\bar{a}^{(2)}_{p,\,q}$ exists and
$\bar{a}^{(2)}_{p,\,q}=a^{(2)}_{p,\,q}(1+\delta aa^{(2)}_{p,\,q})^{-1}$;
$\bar{a}p=(1-q)\bar{a}$;
$\bar{a}a^{(2)}_{p,\,q}=(1-q)\bar{a}a^{(2)}_{p,\,q}$ and
$a^{(2)}_{p,\,q}\bar{a}=a^{(2)}_{p,\,q}\bar{a}p$.
###### Proof.
$(1)\Leftrightarrow(2)$ comes from [6, Theorem 4.1]. We show that (2) and (3)
are equivalent. If $\bar{a}p=(1-q)\bar{a}$, then
$\bar{a}a^{(2)}_{p,\,q}=\bar{a}pa^{(2)}_{p,\,q}=(1-q)\bar{a}a^{(2)}_{p,\,q}\quad
and\quad
a^{(2)}_{p,\,q}\bar{a}=a^{(2)}_{p,\,q}(1-q)\bar{a}=a^{(2)}_{p,\,q}\bar{a}p.$
Conversely, if (3) holds, then
$\bar{a}p=\bar{a}a^{(2)}_{p,\,q}a=(1-q)\bar{a}a^{(2)}_{p,\,q}a=aa^{(2)}_{p,\,q}\bar{a}p=(1-q)\bar{a}$.
∎
###### Corollary 3.12.
Let $a,\,\delta a\in\mathscr{A}$ and $p,\,q\in\mathscr{A}^{\bullet}$ such that
$a^{(2)}_{p,\,q}$ exists. Put $\bar{a}=a+\delta a$. If
$1+a^{(2)}_{p,\,q}\delta a$ is invertible and $\delta a=(1-q)\delta a=\delta
ap$. Then $\bar{a}^{(2)}_{p,\,q}$ exists and
$\bar{a}^{(2)}_{p,\,q}=a^{(2)}_{p,\,q}(1+\delta
aa^{(2)}_{p,\,q})^{-1}=(1+a^{(2)}_{p,\,q}\delta a)^{-1}a^{(2)}_{p,\,q}.$
###### Proof.
If $\delta a=(1-q)\delta a=\delta ap$, then it is easy to check that
$\bar{a}p=(1-q)\bar{a}$. Thus, Theorem 3.11 shows that our results hold. ∎
## 4 Perturbation analysis for the $(p,q)$–generalized inverses
In this section, we mainly investigate the general perturbations problem for
the $(p,q)$–generalized inverses $a^{(2,l)}_{p,\,q}$ and $a^{(1,2)}_{p,\,q}$.
Let $\kappa=\|a\|\|a^{(2,l)}_{p,\,q}\|$, which is the generalized condition
number of the generalized inverse $a^{(2,l)}_{p,\,q}$.
###### Lemma 4.1.
Let $a\in\mathscr{A}$ and $p\in\mathscr{A}^{\bullet}$ with
$R_{r}(p)=R_{r}(a)$. Let $c\in\mathscr{A}$ with $R_{r}(c)$ closed and
$\hat{\delta}(R_{r}(c),\,R_{r}(a))<\dfrac{1}{1+\|p\|}$. Then
$\mathscr{A}=R_{r}(c)\dotplus K_{r}(p)$.
###### Proof.
Let $L_{p}\,x=p\,x$, $\forall\,x\in\mathscr{A}$. Then $L_{p}$ is an idempotent
operator on $\mathscr{A}$ with $\|L_{p}\|=\|p\|$ and
$\\{L_{p}\,x|\,x\in\mathscr{A}\\}=R_{r}(p)$. By [11, Theorem 11] or [14, Lemma
4.4.4], $\hat{\delta}(R_{r}(c),\,R_{r}(a))<\dfrac{1}{1+\|L_{p}\|}$ implies
that $\mathscr{A}=R_{r}(c)\dotplus K_{r}(p)$. ∎
###### Lemma 4.2 ([13, Lemma 2.4]).
For any $p,\,q\in\mathscr{A}^{\bullet}$ we have
$\hat{\delta}(R_{r}(p),R_{r}(q))\leq\|p-q\|$.
###### Lemma 4.3.
Let $a\in\mathscr{A}$ and $p,\,q\in\mathscr{A}^{\bullet}$ such that
$a^{(2,l)}_{p,\,q}$ exists. Suppose that $p^{\prime}\in\mathscr{A}^{\bullet}$
satisfying $\|p-p^{\prime}\|<\dfrac{1}{1+\kappa}$. Then
$\hat{\delta}(aR_{r}(p),aR_{r}(p^{\prime}))\leq\dfrac{\kappa\|p-p^{\prime}\|}{1-(1+\kappa)\|p-p^{\prime}\|}$;
$aR_{r}(p^{\prime})\subset\mathscr{A}$ is closed and $K_{r}(a)\cap
R_{r}(p^{\prime})=\\{0\\}$.
###### Proof.
(1) Set $b=a^{(2,l)}_{p,\,q}$. For any $t^{\prime}\in R_{r}(p^{\prime})$, we
have
$\displaystyle{\text{dist}}(at^{\prime},aR_{r}(p))$ $\displaystyle=\inf_{t\in
R_{r}(p)}\|at^{\prime}-at\|\leq\|a\|\inf_{t\in R_{r}(p)}\|t-t^{\prime}\|$
$\displaystyle\leq\|a\|{\text{dist}}(t^{\prime},R_{r}(p))$
$\displaystyle\leq\|a\|\|t^{\prime}\|\hat{\delta}(R_{r}(p^{\prime}),R_{r}(p)).$
Thus, we get
$\displaystyle\delta(at^{\prime},aR_{r}(p))\leq\|a\|\|t^{\prime}\|\hat{\delta}(R_{r}(p^{\prime}),R_{r}(p)).$
(4.1)
But for any $t^{\prime}\in R_{r}(p^{\prime})$ and $t\in R_{r}(p)$, we have
$\displaystyle\|b\|\|at^{\prime}\|$
$\displaystyle=\|b\|\|a(t^{\prime}-t+t)\|\geq\|b\|\|at\|-\|b\|\|a\|\|t^{\prime}-t\|$
$\displaystyle\geq\|t\|-\|b\|\|a\|\|t^{\prime}-t\|$
$\displaystyle\geq\|t^{\prime}\|-(1+\|b\|\|a\|)\|t^{\prime}-t\|.$
Thus, $\|t^{\prime}\|-\|b\|\|at^{\prime}\|\leq(1+\|b\|\|a\|)\|t^{\prime}-t\|,$
and then
$\displaystyle\|t^{\prime}\|-\|b\|\|at^{\prime}\|$
$\displaystyle\leq(1+\|b\|\|a\|){\text{dist}}(t^{\prime},R_{r}(p))$
$\displaystyle\leq(1+\|b\|\|a\|)\|t^{\prime}\|\hat{\delta}(R_{r}(p^{\prime}),R_{r}(p)).$
Therefore, we have
$\displaystyle\|t^{\prime}\|\leq\frac{\|b\|\|at^{\prime}\|}{1-(1+\|b\|\|a\|)\hat{\delta}(R_{r}(p^{\prime}),R_{r}(p))}.$
(4.2)
Then by Eq. (4.1) and Eq. (4.2), we get
$\displaystyle\delta(at^{\prime},aR_{r}(p))$
$\displaystyle\leq\frac{\|b\|\|a\|\|at^{\prime}\|\hat{\delta}(R_{r}(p^{\prime}),R_{r}(p))}{1-(1+\|b\|\|a\|)\hat{\delta}(R_{r}(p^{\prime}),R_{r}(p))}.$
Now, by Lemma 4.2 and the Definition of gap–function, we have
$\displaystyle\delta(aR_{r}(p^{\prime}),aR_{r}(p))$
$\displaystyle\leq\frac{\kappa\|p-p^{\prime}\|}{1-(1+\kappa)\|p-p^{\prime}\|}.$
(4.3)
On the other hand, for any $t\in R_{r}(p)$, by Lemma 2.2, we have
$t=pt=bapt=bat,$ then
$\displaystyle{\rm dist}(at,aR_{r}(p^{\prime}))$ $\displaystyle=\inf_{s\in
R_{r}(p^{\prime})}\|at-as\|\leq\|a\|\inf_{s\in R_{r}(p^{\prime})}\|t-s\|$
$\displaystyle=\|a\|{\text{dist}}(t,R_{r}(p^{\prime}))\leq\|a\|\|t\|\delta(R_{r}(p),R_{r}(p^{\prime}))$
$\displaystyle=\|a\|\|bat\|\delta(R_{r}(p),R_{r}(p^{\prime}))$
$\displaystyle\leq\|a\|\|b\|\|at\|\delta(R_{r}(p),R_{r}(p^{\prime})).$
Thus we have
$\delta(aR_{r}(p),aR_{r}(p^{\prime}))\leq\kappa\hat{\delta}(R_{r}(p),R_{r}(p^{\prime}))$.
So by Lemma 4.2,
$\displaystyle\delta(aR_{r}(p),aR_{r}(p^{\prime}))\leq\kappa\|p-p^{\prime}\|.$
(4.4)
Consequently, from Eq. (4.3) and Eq. (4.4), we have
$\displaystyle\hat{\delta}(aR_{r}(p),aR_{r}(p^{\prime}))$
$\displaystyle=\max\\{\delta(aR_{r}(p^{\prime}),aR_{r}(p)),\delta(aR_{r}(p),aR_{r}(p^{\prime}))\\}$
$\displaystyle\leq\dfrac{\kappa\|p-p^{\prime}\|}{1-(1+\kappa)\|p-p^{\prime}\|}.$
(2) Obviously, by Eq. (4.2), we get $aR_{r}(p^{\prime})\subset\mathscr{A}$ is
closed and $K_{r}(a)\cap R_{r}(p^{\prime})=\\{0\\}.$ This completes the proof.
∎
Now we can give the following perturbation result for $a^{(2,l)}_{p,\,q}$ when
$p$ has a small perturbation.
###### Theorem 4.4.
Let $a\in\mathscr{A}$ and $p,\,q\in\mathscr{A}^{\bullet}$ such that
$a^{(2,l)}_{p,\,q}$ exists. Suppose that $p^{\prime}\in\mathscr{A}^{\bullet}$
with $\|p-p^{\prime}\|<\dfrac{1}{(1+\kappa)^{2}}$. Then
$a_{p^{\prime},q}^{(2,l)}$ exists and
$\frac{\|a^{(2,l)}_{p^{\prime},q}-a^{(2,l)}_{p,\,q}\|}{\|a^{(2,l)}_{p,\,q}\|}\leq\dfrac{(1+\kappa)\|p^{\prime}-p\|}{1-(1+\kappa)\|p^{\prime}-p\|}\
\text{and}\
\|a^{(2,l)}_{p^{\prime},q}\|\leq\frac{\|a^{(2,l)}_{p,\,q}\|}{1-(1+\kappa)\|p^{\prime}-p\|}.$
###### Proof.
Let $b=a^{(2,l)}_{p,\,q}$, then by Definition 1.1, we know that
$ab\in\mathscr{A}^{\bullet}$ and $\kappa=\|b\|\|a\|\geq\|ab\|$. By using Lemma
4.3 and note that $\|p-p^{\prime}\|<\dfrac{1}{(1+\kappa)^{2}}$, we have
$\hat{\delta}(aR_{r}(p),aR_{r}(p^{\prime}))\leq\dfrac{\kappa\|p-p^{\prime}\|}{1-(1+\kappa)\|p-p^{\prime}\|}<\frac{1}{1+\|ab\|}.$
From Lemma 4.3 (2), we know that $K_{r}(a)\cap R_{r}(p^{\prime})=\\{0\\}$ and
$aR_{r}(p^{\prime})\subset\mathscr{A}$ is closed. Thus, by Lemma 4.1,
$aR_{r}(p^{\prime})$ is complemented and
$\mathscr{A}=aR_{r}(p^{\prime})\dotplus R_{r}(q)$. Therefore, by Lemma 2.2, we
know $a^{(2)}_{p^{\prime},\,q}$ exists.
For convenience we write $a^{(2,l)}_{p^{\prime},q}=b^{\prime}$. Since we have
proved that $a^{(2,l)}_{p^{\prime},q}$ exists, then by Theorem 2.2,
$\mathscr{A}=aR_{r}(p^{\prime})\dotplus R_{r}(q)=aR_{r}(p^{\prime})\dotplus
K_{r}(b^{\prime})$ and $K_{r}(b^{\prime})=R_{r}(q)=K_{r}(b)$. Thus for any
$x\in\mathscr{A}$, we can write $x=t+t^{\prime}$ with $t=ab^{\prime}z$ for
some $z\in\mathscr{A}$ and $t^{\prime}\in R_{r}(q)$.
Since
dist$(b^{\prime}z,R_{r}(p))\leq\|b^{\prime}z\|\delta(R_{r}(p^{\prime}),R_{r}(p))\leq|b^{\prime}z\|\|p^{\prime}-p\|$,
then for every $\epsilon>0$, we can choose $y\in\mathscr{A}$ such that
$\|b^{\prime}z-by\|<\|b^{\prime}z\|\|p^{\prime}-p\|+\epsilon.$ Put $s=aby$.
Then we have
$\displaystyle\|t-s\|$
$\displaystyle=\|ab^{\prime}z-aby\|<\|a\|\|b^{\prime}z\||p^{\prime}-p\|+\|a\|\epsilon.$
$\displaystyle\|(b^{\prime}-b)x\|$
$\displaystyle=\|(b^{\prime}-b)(t+t^{\prime})\|=\|(b^{\prime}-b)t\|$
$\displaystyle\leq\|b^{\prime}ab^{\prime}t-babs\|+\|bs-bt\|$
$\displaystyle\leq\|b^{\prime}z-by\|+\|b\|\|s-t\|$
$\displaystyle\leq(1+\kappa)\|b^{\prime}z\|\|p^{\prime}-p\|+(1+\kappa)\epsilon.$
(4.5)
From $t=ab^{\prime}z$, we get $b^{\prime}t=b^{\prime}z$ and therefore
$\|b^{\prime}z\|=\|b^{\prime}t\|=\|(b^{\prime}-b)x+bx\|\leq\|(b^{\prime}-b)x\|+\|b\|\|x\|.$
(4.6)
Thus by using Eq. (4) and Eq. (4.6), we get
$\|(b^{\prime}-b)x\|\leq(1+\kappa)(\|b\|\|x\|+\|(b^{\prime}-b)x\|)\|p^{\prime}-p\|+(1+\kappa)\epsilon.$
(4.7)
Letting $\epsilon\to 0^{+}$ in Eq. (4.7), we can get
$\frac{\|a^{(2,l)}_{p^{\prime},q}-a^{(2,l)}_{p,\,q}\|}{\|a^{(2,l)}_{p,\,q}\|}\leq\dfrac{(1+\kappa)\|p^{\prime}-p\|}{1-(1+\kappa)\|p^{\prime}-p\|},\quad\|a^{(2,l)}_{p^{\prime},q}\|\leq\frac{\|a^{(2,l)}_{p,\,q}\|}{1-(1+\kappa)\|p^{\prime}-p\|}.$
This completes the proof. ∎
Some representations for the generalized inverse $a^{(2,l)}_{p,q}$ have been
presented in [1]. The following result gives a representation of
$a^{(2,l)}_{p,q}$ based on (1,5) inverse. Note that this result is also an
improvement of the group inverses representation of $a^{(2,l)}_{p,q}$(see
[1]), which removes the existence of the group inverses of $wa$ or $aw$.
###### Lemma 4.5 ([1, Theorem 5.6]).
Let $a,w\in\mathscr{A}$ and $p,\;q\in\mathscr{A}^{\bullet}$ such that
$R_{r}(w)=R_{r}(p)$ and $K_{r}(w)=R_{r}(q)$. Then the following statements are
equivalent:
$a^{(2,l)}_{p,q}$ exists;
$(aw)^{(1,5)}$ exists and $K_{r}(a)\cap R_{r}(w)=\\{0\\};$
$(wa)^{(1,5)}$ exists and $R_{r}(w)=R_{r}(wa).$ In this case, $waw$ is inner
regular and
$a^{(2,l)}_{p,\,q}=(wa)^{(1,5)}w=w(aw)^{(1,5)}=w(waw)^{-}w.$
Now we give the result when $q$ has a small perturbation. By using our above
lemma 4.5, we can also give a new representation for the generalized inverse
of the perturbed operator.
###### Theorem 4.6.
Let $a\in\mathscr{A}$ and $p,\,q\in\mathscr{A}^{\bullet}$ such that
$a^{(2,l)}_{p,\,q}$ exists. Suppose that $q^{\prime}\in\mathscr{A}^{\bullet}$
with $\|q-q^{\prime}\|<\dfrac{1}{2+\kappa}$. Then $a_{p,\,q\prime}^{(2,l)}$
exists and
$\dfrac{\|(a^{(2,l)}_{p,\,q^{\prime}}-a^{(2,l)}_{p,\,q})\|}{\|a^{(2,l)}_{p,\,q}\|}\leq\dfrac{(1+\kappa)\|q-q^{\prime}\|}{1-\kappa\|q-q^{\prime}\|}$
and
$\|a^{(2,l)}_{p,\,q^{\prime}}\|\leq\dfrac{1+\|q^{\prime}-q\|}{1-\kappa\|q^{\prime}-q\|}\,\|a^{(2,l)}_{p,\,q}\|$.
If there are some $w,\,v\in\mathscr{A}$ with $R_{r}(w)=R_{r}(p)$,
$K_{r}(w)=K_{r}(v)=R_{r}(q)$ and $R_{r}(v)=R_{r}(q^{\prime})$. Then
$a_{p,\,q^{\prime}}^{(2,l)}=a^{(2,l)}_{p,\,q}+a^{(2,l)}_{p,\,q}(av)^{(1,5)}a(v-w)(1-aa^{(2,l)}_{p,\,q}).$
###### Proof.
Since $1+\|1-ab\|\leq 2+\kappa$, then by Lemma 4.2, we have
$\hat{\delta}(R_{r}(q),R_{r}(q^{\prime}))\leq\|q-q^{\prime}\|<\frac{1}{2+\kappa}\leq\frac{1}{1+\|1-ab\|}.$
So $\mathscr{A}=aR_{r}(p)\dotplus R_{r}(q^{\prime})$ by Lemma 4.1. Note that
$K_{r}(a)\cap R_{r}(p)=\\{0\\}$. So $a^{(1,2)}_{p,\,q^{\prime}}$ exists.
(1) From
$a^{(2,l)}_{p,\,q^{\prime}}=pa^{(2,l)}_{p,\,q^{\prime}}=a^{(2,l)}_{p,\,q}aa^{(2,l)}_{p,\,q^{\prime}}$,
we get
$a^{(2,l)}_{p,\,q^{\prime}}-a^{(2,l)}_{p,\,q}=a^{(2,l)}_{p,\,q}(aa^{(2,l)}_{p,\,q^{\prime}}-aa^{(2,l)}_{p,\,q}).$
Since $(1-aa^{(2,l)}_{p,\,q^{\prime}})x\in R_{r}(q^{\prime})$ for any
$x\in\mathscr{A}$, so we have
$\displaystyle{\rm dist}((1-aa^{(2,l)}_{p,\,q^{\prime}})x,R_{r}(q))$
$\displaystyle\leq\hat{\delta}(R_{r}(q^{\prime}),R_{r}(q))\|(1-aa^{(2,l)}_{p,\,q^{\prime}})x\|$
$\displaystyle\leq\|q-q^{\prime}\|\|(1-aa^{(2,l)}_{p,\,q^{\prime}})x\|.$
Since $K_{r}(a^{(2,l)}_{p,\,q})=R_{r}(q)$, then, for any $\epsilon>0$, there
is $z\in\mathscr{A}$ such that
$\|(1-aa^{(2,l)}_{p,\,q^{\prime}})x-(1-aa^{(2,l)}_{p,\,q})z\|\leq\|q-q^{\prime}\|\|(1-aa^{(2,l)}_{p,\,q^{\prime}})x\|+\epsilon.$
and then we have
$\displaystyle\|(a^{(2,l)}_{p,\,q^{\prime}}-a^{(2,l)}_{p,\,q})x\|$
$\displaystyle=\|a^{(2,l)}_{p,\,q}(aa^{(2,l)}_{p,\,q^{\prime}}-aa^{(2,l)}_{p,\,q})x\|$
$\displaystyle\leq\|a^{(2,l)}_{p,\,q}\\{(1-aa^{(2,l)}_{p,\,q^{\prime}})x-(1-aa^{(2,l)}_{p,\,q})z\\}\|$
$\displaystyle\leq\left(\|q-q^{\prime}\|\|(1-aa^{(2,l)}_{p,\,q^{\prime}})x\|+\epsilon\right)\|a^{(2,l)}_{p,\,q}\|.$
(4.8)
Since we also have
$\displaystyle\|(1-aa^{(2,l)}_{p,\,q^{\prime}})x\|$
$\displaystyle\leq\|x\|+\|a\|\|a^{(2,l)}_{p,\,q}x-(a^{(2,l)}_{p,\,q^{\prime}}-a^{(2,l)}_{p,\,q})x\|$
$\displaystyle\leq(1+\kappa)\|x\|+\|a\|\|(a^{(2,l)}_{p,\,q^{\prime}}-a^{(2,l)}_{p,\,q})x\|.$
(4.9)
Now from Eqs. (4) and (4), we can compute
$\|(a^{(2,l)}_{p,\,q^{\prime}}-a^{(2,l)}_{p,\,q})x\|\leq\dfrac{(\|q-q^{\prime}\|(1+\kappa)\|x\|+\epsilon)\|a^{(2,l)}_{p,\,q}\|}{1-\kappa\|q-q^{\prime}\|}.$
Let $\epsilon\to 0^{+}$ in the above inequality, we obtain that
$\frac{\|a^{(2,l)}_{p,\,q^{\prime}}-a^{(2,l)}_{p,\,q}\|}{\|a^{(2,l)}_{p,\,q}\|}\leq\dfrac{(1+\kappa)\|q^{\prime}-q\|}{1-\kappa\|q^{\prime}-q\|},\quad\|a^{(2,l)}_{p,\,q^{\prime}}\|\leq\frac{1+\|q^{\prime}-q\|}{1-\kappa\|q^{\prime}-q\|}\,\|a^{(2,l)}_{p,\,q}\|.$
(2) By Lemma 4.5, $a^{(2,l)}_{p,q^{\prime}}=(va)^{(1,5)}v=v(av)^{(1,5)}$. For
convenience, we write $b=a^{(2,l)}_{p,\,q}$,
$b^{\prime}=a_{p,\,q^{\prime}}^{(2,l)}$ and
$x=a^{(2,l)}_{p,\,q}+a^{(2,l)}_{p,\,q}(av)^{(1,5)}a(v-w)(1-aa^{(2,l)}_{p,\,q})$.
Now we prove that $x=b^{\prime}$ by using Lemma 2.2 (4). Obviously, we have
$px=x$. Note that $p=a^{(2,l)}_{p,\,q}ap$, so
$\displaystyle xap$
$\displaystyle=\\{a^{(2,l)}_{p,\,q}+a^{(2,l)}_{p,\,q}(av)^{(1,5)}a(v-w)(1-aa^{(2,l)}_{p,\,q})\\}ap$
$\displaystyle=a^{(2,l)}_{p,\,q}ap+a^{(2,l)}_{p,\,q}(av)^{(1,5)}a(v-w)(ap-ap)$
$\displaystyle=p.$
Since $b=w(aw)^{(1,5)}$. we have
$\displaystyle xq^{\prime}$
$\displaystyle=\\{a^{(2,l)}_{p,\,q}+a^{(2,l)}_{p,\,q}(av)^{(1,5)}a(v-w)(1-aa^{(2,l)}_{p,\,q})\\}q^{\prime}$
$\displaystyle=bq^{\prime}+\\{b(av)^{(1,5)}av-b(av)^{(1,5)}avab-b(av)^{(1,5)}aw+b(av)^{(1,5)}awab\\}q^{\prime}.$
$\displaystyle=bq^{\prime}+bab^{\prime}q^{\prime}-bab^{\prime}abq^{\prime}-b(av)^{(1,5)}awq^{\prime}+b(av)^{(1,5)}awabq^{\prime}.$
$\displaystyle=bq^{\prime}+0-bq^{\prime}-b(av)^{(1,5)}awq^{\prime}+b(av)^{(1,5)}awaw(aw)^{(1,5)}q^{\prime}.$
$\displaystyle=0.$
Thus, we have $x(1-q^{\prime})=x$. Finally, since $aw=awaw(aw)^{(1,5)}=awab$,
we have
$\displaystyle(1-q^{\prime})ax$
$\displaystyle=(1-q^{\prime})a\\{a^{(2,l)}_{p,\,q}+a^{(2,l)}_{p,\,q}(av)^{(1,5)}a(v-w)(1-aa^{(2,l)}_{p,\,q})\\}$
$\displaystyle=(1-q^{\prime})ab^{\prime}a\\{a^{(2,l)}_{p,\,q}+a^{(2,l)}_{p,\,q}(av)^{(1,5)}a(v-w)(1-aa^{(2,l)}_{p,\,q})\\}$
$\displaystyle=(1-q^{\prime})\\{ab^{\prime}ab+ab^{\prime}ab(av)^{(1,5)}(av-w)(1-ab)\\}$
$\displaystyle=(1-q^{\prime})(ab+ab^{\prime}-ab-
ab^{\prime}ab(av)^{(1,5)}aw+ab^{\prime}ab(av)^{(1,5)}awab)$
$\displaystyle=(1-q^{\prime})ab^{\prime}$ $\displaystyle=1-q^{\prime}.$
Therefore by Lemma 2.2 and the uniqueness of $a^{(2,l)}_{p,\,q^{\prime}}$, we
get that $x=a^{(2,l)}_{p,\,q^{\prime}}$. ∎
When the idempotents $p$ and $q$ both have some small perturbations, we have
the following result.
###### Theorem 4.7.
Let $a\in\mathscr{A}$ and
$p,\,q,\,p^{\prime},\,q^{\prime}\in\mathscr{A}^{\bullet}$ such that
$a^{(2,l)}_{p,\,q}$ exists. If $\|p-p^{\prime}\|<\dfrac{1}{(1+\kappa)^{2}}$
and $\|q-q^{\prime}\|<\dfrac{1}{3+\kappa}$. Then
$a_{p^{\prime},\,q^{\prime}}^{(2,l)}$ exists and
$\displaystyle\frac{\|a^{(2,l)}_{p^{\prime},\,q^{\prime}}-a^{(2,l)}_{p,\,q}\|}{\|a_{p,\,q}^{(2,l)}\|}$
$\displaystyle\leq\frac{(1+\kappa)(\|p-p^{\prime}\|+\|q-q^{\prime}\|)}{1-(1+\kappa)\|p-p^{\prime}\|-\kappa\|q-q^{\prime}\|}$
$\displaystyle\|a_{p^{\prime},\,q^{\prime}}^{(2,l)}\|$
$\displaystyle\leq\dfrac{(1+\|q-q^{\prime}\|)\|a_{p,\,q}^{(2,l)}\|}{1-(1+\kappa)\|p-p^{\prime}\|-\kappa\|q-q^{\prime}\|}.$
###### Proof.
By Theorem 4.4, $a_{p^{\prime},\,q}^{(2,l)}$ exists when
$\|p-p^{\prime}\|<\dfrac{1}{(1+\kappa)^{2}}$ and in this case,
$\|a^{(2,l)}_{p^{\prime},q}\|\leq\frac{\|a^{(2,l)}_{p,\,q}\|}{1-(1+\kappa)\|p^{\prime}-p\|}\leq\frac{1+\kappa}{\kappa}\|a^{(2,l)}_{p,\,q}\|.$
So
$\|q-q^{\prime}\|<\dfrac{1}{3+\kappa}<\dfrac{1}{2+\|a\|\|a^{(2,l)}_{p^{\prime},\,q}\|}$
and consequently, $a^{(2,l)}_{p^{\prime},\,q^{\prime}}$ exists by Theorem 4.6.
Finally, by Theorem 4.4 and Theorem 4.6, we have
$\displaystyle\|a_{p^{\prime},\,q^{\prime}}^{(2,l)}-a_{p,\,q}^{(2,l)}\|$
$\displaystyle\leq\|a_{p^{\prime},\,q^{\prime}}^{(2,l)}-a_{p^{\prime},\,q}^{(2,l)}\|+\|a_{p^{\prime},\,q}^{(2,l)}-a^{(2,l)}_{p,\,q}\|$
$\displaystyle\leq\frac{(1+\|a\|\|a_{p^{\prime},\,q}^{(2,l)})\|q-q^{\prime}\|}{1-\|a\|\|a_{p^{\prime},\,q}^{(2,l)}\|\|q-q^{\prime}\|}\|a_{p^{\prime},\,q}^{(2,l)}\|+\frac{(1+\kappa)\|p-p^{\prime}\|}{1-(1+\kappa)\|p-p^{\prime}\|}\|a_{p^{\prime},\,q}^{(2,l)}\|$
$\displaystyle\leq\frac{(1+\kappa)(1-\|p-p^{\prime}\|)\|q-q^{\prime}\|}{1-(1+\kappa)\|p-p^{\prime}\|-\kappa\|q-q^{\prime}\|}\frac{\|a_{p^{\prime},\,q}^{(2,l)}\|}{1-(1+\kappa)\|p-p^{\prime}\|}$
$\displaystyle\
+\frac{(1+\kappa)\|p-p^{\prime}\|}{1-(1+\kappa)\|p-p^{\prime}\|}\|a_{p^{\prime},\,q}^{(2,l)}\|$
$\displaystyle=\frac{(1+\kappa)(\|p-p^{\prime}\|+\|q-q^{\prime}\|)}{1-(1+\kappa)\|p-p^{\prime}\|-\kappa\|q-q^{\prime}\|}\|a_{p,\,q}^{(2,l)}\|$
and
$\|a_{p^{\prime},\,q^{\prime}}^{(2,l)}\|\leq\|a_{p^{\prime},\,q^{\prime}}^{(2,l)}-a_{p,\,q}^{(2,l)}\|+\|a_{p,\,q}^{(2,l)}\|\leq\dfrac{(1+\|q-q^{\prime}\|)\|a_{p,\,q}^{(2,l)}\|}{1-(1+\kappa)\|p-p^{\prime}\|-\kappa\|q-q^{\prime}\|}.$
∎
Now, we consider the case when the elements $a,p,q\in\mathscr{A}^{\bullet}$
all have some small perturbations.
###### Theorem 4.8.
Let $a,\delta a\in\mathscr{A}$ and
$p,\,q,\,p^{\prime},\,q^{\prime}\in\mathscr{A}^{\bullet}$ such that
$a^{(2,l)}_{p,\,q}$ exists. If $\|p-p^{\prime}\|<\dfrac{1}{(\kappa+1)^{2}}$,
$\|q-q^{\prime}\|<\dfrac{1}{\kappa+3}$ and $\|a^{(2,l)}_{p,\,q}\|\|\delta
a\|<\dfrac{2\kappa}{(\kappa+1)(\kappa+4)}$. Then
$\bar{a}_{p^{\prime},\,q^{\prime}}^{(2,l)}$ exists and
$\displaystyle\|\bar{a}_{p^{\prime},\,q^{\prime}}^{(2,l)}\|$
$\displaystyle\leq\frac{(1+\|q-q^{\prime}\|)\|a_{p,\,q}^{(2,l)}\|}{1-(1+\kappa)\|p-p^{\prime}\|-\kappa\|q-q^{\prime}\|-(1+\|q-q^{\prime}\|)\|a_{p,\,q}^{(2,l)}\|\|\delta
a\|}$
$\displaystyle\|\bar{a}_{p^{\prime},\,q^{\prime}}^{(2,l)}-a_{p,q}^{(2,l)}\|$
$\displaystyle\leq\frac{\|a_{p,\,q}^{(2,l)}\|}{1-(1+\kappa)\|p-p^{\prime}\|-\kappa\|q-q^{\prime}\|}\Big{[}(1+\kappa)(\|p-p^{\prime}\|+\|q-q^{\prime}\|)$
$\displaystyle\ +\frac{(1+\|q-q^{\prime}\|)^{2}\|\delta
a\|\|a_{p,\,q}^{(2,l)}\|}{1-(1+\kappa)\|p-p^{\prime}\|-\kappa\|q-q^{\prime}\|-(1+\|q-q^{\prime}\|)\|a_{p,\,q}^{(2,l)}\|\|\delta
a\|}\Big{]}.$
###### Proof.
Theorem 4.7 indicates that $a_{p^{\prime},\,q^{\prime}}^{(2,l)}$ exists and
$\|a^{(2,l)}_{p^{\prime}\,,q^{\prime}}\|\leq\dfrac{(1+\|q-q^{\prime}\|)\|a_{p,\,q}^{(2,l)}\|}{1-(1+\kappa)\|p-p^{\prime}\|-\kappa\|q-q^{\prime}\|}<\dfrac{(1+\kappa)(4+\kappa)}{2\kappa}\|a^{(2,l)}_{p,\,q}\|.$
Thus, $\|a^{(2,l)}_{p^{\prime},q^{\prime}}\|\|\delta a\|<1$ and hence
$1+a^{(2,l)}_{p^{\prime},q^{\prime}}\delta a$ is invertible. Therefore,
$a^{(2,l)}_{p^{\prime},q^{\prime}}$ exists and
$\bar{a}^{(2,l)}_{p^{\prime},q^{\prime}}=a^{(2,l)}_{p^{\prime},q^{\prime}}(1+\delta
aa^{(2,l)}_{p^{\prime},q^{\prime}})^{-1}$ by Theorem 3.4. Now by Theorem 4.7,
we have
$\|\bar{a}_{p^{\prime},q^{\prime}}^{(2,l)}\|\\!\leq\\!\dfrac{\|a_{p^{\prime},q^{\prime}}^{(2,l)}\|}{1-\|a_{p^{\prime},q^{\prime}}^{(2,l)}\|\|\delta
a\|}\\!\leq\\!\frac{[1+\|q-q^{\prime}\|]\|a_{p,\,q}^{(2,l)}\|}{1-[1+\kappa]\|p-p^{\prime}\|-\kappa\|q-q^{\prime}\|-[1+\|q-q^{\prime}\|]\|a_{p,\,q}^{(2,l)}\|\|\delta
a\|}$
and
$\displaystyle\|\bar{a}_{p^{\prime},\,q^{\prime}}^{(2,l)}-a_{p,q}^{(2,l)}\|$
$\displaystyle\leq\|(1+{a}_{p^{\prime},\,q^{\prime}}^{(2,l)}\delta
a)^{-1}{a}_{p^{\prime},\,q^{\prime}}^{(2,l)}-a_{p^{\prime},\,q^{\prime}}^{(2,l)}\|+\|a_{p^{\prime},\,q^{\prime}}^{(2,l)}-a_{p,\,q}^{(2,l)}\|$
$\displaystyle\leq\frac{\|\delta
a\|\|a_{p^{\prime},\,q^{\prime}}^{(2,l)}\|^{2}}{1-\|\delta
a\|\|a_{p^{\prime},\,q^{\prime}}^{(2,l)}\|}+\|a_{p^{\prime},\,q^{\prime}}^{(2,l)}-a_{p,\,q}^{(2,l)}\|$
$\displaystyle\leq\frac{\|a_{p,\,q}^{(2,l)}\|}{1-(1+\kappa)\|p-p^{\prime}\|-\kappa\|q-q^{\prime}\|}\Big{[}(1+\kappa)(\|p-p^{\prime}\|+\|q-q^{\prime}\|)$
$\displaystyle\ +\frac{(1+\|q-q^{\prime}\|)^{2}\|\delta
a\|\|a_{p,\,q}^{(2,l)}\|}{1-(1+\kappa)\|p-p^{\prime}\|-\kappa\|q-q^{\prime}\|-(1+\|q-q^{\prime}\|)\|a_{p,\,q}^{(2,l)}\|\|\delta
a\|}\Big{]}.$
This completes the proof. ∎
By using perturbation theorems for the generalized inverse
$a^{(2,l)}_{p,\,q}$, we can also investigate the perturbation analysis for the
generalized inverse $a^{(1,2)}_{p,\,q}$ under some conditions.
###### Corollary 4.9.
Let $a\in\mathscr{A}$ and $p,\,q\in\mathscr{A}^{\bullet}$ such that
$a^{(1,2)}_{p,\,q}$ exists. Suppose that $p^{\prime}\in\mathscr{A}^{\bullet}$
with $\|p-p^{\prime}\|<\dfrac{1}{(1+\kappa)^{2}}$ and $ap^{\prime}=a$. Then
$a_{p^{\prime},\,q}^{(1,2)}$ exists and
$\frac{\|a^{(1,2)}_{p^{\prime},q}-a^{(1,2)}_{p,\,q}\|}{\|a^{(2,l)}_{p,\,q}\|}\leq\dfrac{(1+\kappa)\|p^{\prime}-p\|}{1-(1+\kappa)\|p^{\prime}-p\|}\
\text{and}\
\|a^{(1,2)}_{p^{\prime},q}\|\leq\frac{\|a^{(1,2)}_{p,\,q}\|}{1-(1+\kappa)\|p^{\prime}-p\|}.$
###### Proof.
Set $b=a^{(1,2)}_{p,\,q}$. Then $bab=b,\ aba=a,\ ba=p,\ 1-ab=q$ and
$a^{(2,l)}_{p,\,q}=a^{(1,2)}_{p,\,q}=b$. By Theorem 4.4,
$a^{(2,l)}_{p^{\prime},\,q}$ is exists and
$\frac{\|a^{(2,l)}_{p^{\prime},q}-a^{(1,2)}_{p,\,q}\|}{\|a^{(2,l)}_{p,\,q}\|}\leq\dfrac{(1+\kappa)\|p^{\prime}-p\|}{1-(1+\kappa)\|p^{\prime}-p\|}\
\text{and}\
\|a^{(2,l)}_{p^{\prime},q}\|\leq\frac{\|a^{(1,2)}_{p,\,q}\|}{1-(1+\kappa)\|p^{\prime}-p\|}.$
We need only to show that
$a^{(1,2)}_{p^{\prime},\,q}=a^{(2,l)}_{p^{\prime},\,q}$ in this case. Put
$b^{\prime}=a^{(2,l)}_{p^{\prime},\,q}$. Then
$b^{\prime}ab^{\prime}=b^{\prime}$,
$(b^{\prime}a)\mathscr{A}=p^{\prime}\mathscr{A}$,
$(1-ab^{\prime})\mathscr{A}=q\mathscr{A}$. Thus, $(1-q)(1-ab^{\prime})=0$ and
hence $1-q=(1-q)ab^{\prime}=abab^{\prime}=ab^{\prime}$. Furthermore,
$ab^{\prime}a=(1-q)a=aba=a$. From
$(b^{\prime}a)\mathscr{A}=p^{\prime}\mathscr{A}$, we get that
$(1-b^{\prime}a)p^{\prime}=0$ and
$p^{\prime}=b^{\prime}ap^{\prime}=b^{\prime}a$. Therefore,
$b^{\prime}=a^{(1,2)}_{p^{\prime},\,q}$. ∎
We need the following easy representation lemma for $a^{(1,2)}_{p,\,q}$.
###### Lemma 4.10.
Let $a\in\mathscr{A}$ and $p,\,q\in\mathscr{A}^{\bullet}$ such that
$a^{(1,2)}_{p,\,q}$ exists. Let $w\in\mathscr{A}$ such that $wa=p$ and
$aw=1-q$. Then $a^{(1,2)}_{p,\,q}=(wa)^{\\#}w=w(aw)^{\\#}$.
###### Proof.
Obviously, $wa,\ aw\in\mathscr{A}^{g}$ for $wa=p$ and $aw=1-q$. We also have
$(wa)^{\\#}=p$ and $(aw)^{\\#}=1-q$. Then by using the uniqueness of
$a^{(1,2)}_{p,\,q}$, we can prove our lemma by simple computation. ∎
###### Corollary 4.11.
Let $a\in\mathscr{A}$ and $p,\,q\in\mathscr{A}^{\bullet}$ such that
$a^{(1,2)}_{p,\,q}$ exists. Suppose that $q^{\prime}\in\mathscr{A}^{\bullet}$
with $\|q-q^{\prime}\|<\dfrac{1}{2+\kappa}$ and $a=(1-q^{\prime})a$. Then
$a^{(1,2)}_{p,q^{\prime}}$ exists and
$\dfrac{\|(a^{(1,2)}_{p,\,q^{\prime}}-a^{(1,2)}_{p,\,q})\|}{\|a^{(1,2)}_{p,\,q}\|}\leq\dfrac{(1+\kappa)\|q-q^{\prime}\|}{1-\kappa\|q-q^{\prime}\|}$
and
$\|a^{(1,2)}_{p,\,q^{\prime}}\|\leq\dfrac{1+\|q^{\prime}-q\|}{1-\kappa\|q^{\prime}-q\|}\,\|a^{(1,2)}_{p,\,q}\|$.
If there are some $w,\,v\in\mathscr{A}$ with $wa=p=va$, $aw=1-q$ and
$av=1-q^{\prime}$. Then
$a_{p,\,q^{\prime}}^{(1,2)}=a^{(1,2)}_{p,\,q}+a^{(1,2)}_{p,\,q}(av)^{\\#}a(v-w)q.$
###### Proof.
$a^{(2,l)}_{p,\,q^{\prime}}$ exists by Theorem 4.6. From $a=(1-q^{\prime})a$
and Lemma 2.2, we can obtain that $a^{(1.2)}_{p,\,q^{\prime}}$ exists and
$a^{(2,l)}_{p,\,q^{\prime}}=a^{(1.2)}_{p,\,q^{\prime}}$.
Now the estimates in (1) and the representation for
$a^{(1,2)}_{p,\,q^{\prime}}$ in (2) follow from Theorem 4.6 and Lemma 4.10. ∎
Finally, by Corollary 4.9, Corollary 4.11 and Theorem 4.7, we have
###### Corollary 4.12.
Let $a\in\mathscr{A}$ and
$p,\,q,\,p^{\prime},\,q^{\prime}\in\mathscr{A}^{\bullet}$ with
$a^{(1,2)}_{p,\,q}$ exists. If $\|p-p^{\prime}\|<\dfrac{1}{(1+\kappa)^{2}}$,
$\|q-q^{\prime}\|<\dfrac{1}{3+\kappa}$ and $ap^{\prime}=a=(1-q^{\prime})a$.
Then $a_{p^{\prime},\,q^{\prime}}^{(1,2)}$ exists and
$\displaystyle\frac{\|a^{(1,2)}_{p^{\prime},\,q^{\prime}}-a^{(1,2)}_{p,\,q}\|}{\|a_{p,\,q}^{(1,2)}\|}$
$\displaystyle\leq\frac{(1+\kappa)(\|p-p^{\prime}\|+\|q-q^{\prime}\|)}{1-(1+\kappa)\|p-p^{\prime}\|-\kappa\|q-q^{\prime}\|}$
$\displaystyle\|a_{p^{\prime},\,q^{\prime}}^{(1,2)}\|$
$\displaystyle\leq\dfrac{(1+\|q-q^{\prime}\|)\|a_{p,\,q}^{(1,2)}\|}{1-(1+\kappa)\|p-p^{\prime}\|-\kappa\|q-q^{\prime}\|}.$
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|
arxiv-papers
| 2013-01-18T07:41:33 |
2024-09-04T02:49:40.442887
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jianbing Cao and Yifeng Xue",
"submitter": "Yifeng Xue",
"url": "https://arxiv.org/abs/1301.4314"
}
|
1301.4335
|
Optimal bilinear control of nonlinear Schrödinger equations with singular
potentials
Binhua Feng∗, Dun Zhao, Pengyu Chen
School of Mathematics and Statistics, Lanzhou University
Lanzhou, 730000, P.R. China
000 E-mail: [email protected](Binhua Feng)
∗Corresponding author
This work is supported by the Program for the Fundamental Research Funds for
the Central Universities, NSFC Grants 11031003 and 11171028, and the Program
for NCET.
###### Abstract
In this paper, we consider an optimal bilinear control problem for the
nonlinear Schrödinger equations with singular potentials. We show well-
posedness of the problem and existence of an optimal control. In addition, the
first order optimality system is rigorously derived. Our results generalize
the ones in [12] in several aspects.
Keywords: Optimal bilinear control; Nonlinear Schrödinger equation; Optimal
condition
## 1 Introduction
This paper is devoted to study an optimal bilinear control problem for the
following nonlinear Schrödinger equation(NLS):
$\left\\{\begin{array}[]{l}iu_{t}+\Delta
u+\lambda|u|^{2\sigma}u+\phi(t)V(x)u=0,\text{
}(t,x)\in[0,\infty)\times\mathbb{R}^{N},\\\
u(0,x)=u_{0}(x),\end{array}\right.$ (1.1)
where $u_{0}\in H^{1}(\mathbb{R}^{N})$, $\phi(t)$ denotes the control
parameter and $V(x)$ is a given potential. The problem of quantum control via
external potentials $\phi(t)V(x)$, has attracted a great deal of attention
from physicians, see [2, 3, 4]. From the mathematical point of view, quantum
control problems are a specific example of optimal control problems, see [5],
which consist in minimizing a cost functional depending on the solution of a
state equation (here, equation (1.1)) and to characterize the minimum of the
functional by an optimality condition.
The mathematical research for optimal bilinear control of systems governed by
partial differential equations has a long history, see [6, 7] for a general
overview. However, there are only a few rigorous mathematical results about
optimal bilinear control of Schrödinger equations. Recently, optimal control
problems for linear Schrödinger equations have been investigated in [8, 9,
10]. Moreover, those results have been tested numerically in [10, 11]. In
particular, a mathematical framework for optimal bilinear control of abstract
linear Schrödinger equations was presented in [8]. In [9], the authors
considered the optimal bilinear control for the linear Schrödinger equations
including coulombian and electric potentials. For the following NLS of Gross-
Pitaevskii type:
$\left\\{\begin{array}[]{l}iu_{t}+\Delta
u-U(x)u-\lambda|u|^{2\sigma}u-\phi(t)V(x)u=0,\text{
}(t,x)\in[0,\infty)\times\mathbb{R}^{N},\\\
u(0,x)=u_{0}(x),\end{array}\right.$ (1.2)
where $\lambda\geq 0$, $U(x)$ is a subquadratic potential, consequently
restricting initial data $u_{0}\in\Sigma:=\\{u\in
H^{1}(\mathbb{R}^{N}),~{}and~{}~{}xu\in L^{2}(\mathbb{R}^{N})\\}$. The authors
in [12] have presented a novel choice for the cost term, which is based on the
corresponding physical work performed throughout the control process. The
proof of the existence of an optimal control relies heavily on the compact
embedding $\Sigma\hookrightarrow L^{2}(\mathbb{R}^{N})$. In contrast with
(1.2), due to absence of $U(x)u$ in (1.1), we consider (1.1) in
$H^{1}(\mathbb{R}^{N})$. Therefore, how to overcome the difficulty that
embedding $H^{1}(\mathbb{R}^{N})\hookrightarrow L^{2}(\mathbb{R}^{N})$ is not
compact, which is of particular interest, is one of main technique challenges
in this paper.
Borrowing the idea of [12], we now define our optimal control problem. The
natural candidate for an energy corresponding to (1.1) is
$E(t)=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u(t,x)|^{2}dx-\frac{\lambda}{2\sigma+2}\int_{\mathbb{R}^{N}}|u(t,x)|^{2\sigma+2}dx-\frac{\phi(t)}{2}\int_{\mathbb{R}^{N}}V(x)|u(t,x)|^{2}dx.$
(1.3)
Although equation (1.1) enjoys mass conservation, i.e.,
$\|u(t,\cdot)\|_{L^{2}}=\|u_{0}\|_{L^{2}}$ for all $t\in\mathbb{R}$, the
energy $E(t)$ is not conserved. Indeed, its evolution is given by
$\frac{dE(t)}{dt}=-\frac{1}{2}\phi^{\prime}(t)\int_{\mathbb{R}^{N}}V(x)|u(t,x)|^{2}dx.$
(1.4)
Integrating this equality over the compact interval $[0,T]$, we obtain
$E(T)-E(0)=\frac{1}{2}\int_{0}^{T}\phi^{\prime}(t)\int_{\mathbb{R}^{N}}V(x)|u(t,x)|^{2}dxdt.$
(1.5)
For any given $T>0$, we consider $H^{1}(0,T)$ as the real vector space of
control parameters $\phi$. Set
$X(0,T):=L^{2}((0,T),H_{0}^{1})\cap W^{1,2}((0,T),H^{-1}),$ (1.6)
. For some $M_{1}>0$ and $M_{2}>0$, set $B_{1}:=\\{u_{0}\in
H^{1}~{}and~{}\|u_{0}\|_{H^{1}}\leq M_{1}\\}$ and
$B_{2}:=\\{\phi_{0}\in\mathbb{R}~{}and~{}|\phi_{0}|\leq M_{2}\\}$
$\Lambda(0,T):=\\{(u,\phi)\in X(0,T)\times
H^{1}(0,T):~{}u~{}is~{}the~{}solution~{}of~{}\eqref{1.1}~{}with~{}u(0)\in
B_{1}~{}and~{}\phi(0)\in B_{2}\\}.$
Thanks to Lemma 2.3, the set $\Lambda(0,T)$ is not empty. We consequently
define the objective functional $F=F(u,\phi)$ on $\Lambda(0,T)$ by
$F(u,\phi):=\langle
u(T,\cdot),Au(T,\cdot)\rangle_{L^{2}}^{2}+\gamma_{1}\int_{0}^{T}(E^{\prime}(t))^{2}dt+\gamma_{2}\int_{0}^{T}(\phi^{\prime}(t))^{2}dt,$
(1.7)
where parameters $\gamma_{1}\geq 0$ and $\gamma_{2}>0$, $A:H^{1}\rightarrow
L^{2}$ is a bounded linear operator, essentially self-adjoint on $L^{2}$ and
localizing, i.e., there exists $R>0$, such that for all $\psi\in H^{1}$:
$supp_{x\in\mathbb{R}^{N}}(A\psi(x))\subseteq B(R)$.
Now, we can define the following minimizing problem:
$F_{*}=\inf_{(u,\phi)\in\Lambda(0,T)}F(u,\phi).$ (1.8)
Firstly, we consider the existence of a minimizer for the above minimizing
problem. This is what the following theorem shows:
###### Theorem 1.1.
Assume $0<\sigma<\frac{2}{N-2}$ if $\lambda<0$, or $0<\sigma<\frac{2}{N}$ if
$\lambda>0$. Let $V\in L^{p}+L^{\infty}$ for some $p\geq 1$, $p>N/2$. Then,
for any $T>0$, $M_{1}>0$, $M_{2}>0$, $\gamma_{1}\geq 0$ and $\gamma_{2}>0$,
the optimal control problem (1.8) has a minimizer
$(u_{*},\phi_{*})\in\Lambda(0,T)$.
Remarks. (1) In contrast with the results in [12], our results hold for
unbounded potential $V$, both focusing and defocusing nonlinearities. A
typical example satisfying our assumption on $V$ is $\frac{1}{|x|^{\alpha}}$
for some $0<\alpha<1$.
(2) Since the embedding $H^{1}(\mathbb{R}^{N})\hookrightarrow
L^{2}(\mathbb{R}^{N})$ is not compact, the method in [12] fails to work in our
situation. We can derive the compactness of a minimizing sequence by
Propositions 1.1.2 and 1.3.14 in [13].
Thanks to global well-posedness of equation (1.1), for any given initial data
$u_{0}\in H^{1}$, we can define a mapping by
$u:~{}H^{1}(0,T)\rightarrow X(0,T):~{}~{}~{}\phi\mapsto u(\phi).$
Using this mapping we introduce the unconstrained functional
$\mathcal{F}:H^{1}(0,T)\rightarrow\mathbb{R},~{}~{}~{}\phi\mapsto\mathcal{F}(\phi):=F(u(\phi),\phi).$
In the following theorem, we investigate the differentiability of
unconstrained functional $\mathcal{F}$, and consequently obtain the first
order optimality system.
###### Theorem 1.2.
Let $N\leq 3$, $u_{0}\in H^{2}$, $V,\nabla V\in L^{p}+L^{\infty}$ and $V\in
L^{2p}$ for some $p\geq 2$. Assume $\frac{1}{2}\leq\sigma<\frac{2}{N-2}$ if
$\lambda<0$, or $\frac{1}{2}\leq\sigma<\frac{2}{N}$ if $\lambda>0$. Then the
functional $\mathcal{F}(\phi)$ is Gâteaux differentiable and
$\mathcal{F}^{\prime}(\phi)=Re\int_{\mathbb{R}^{N}}\bar{\varphi}(t,x)V(x)u(t,x)dx-2\frac{d}{dt}(\phi^{\prime}(t)(\gamma_{2}+\gamma_{1}\omega^{2}(t))),$
(1.9)
in the sense of distributions, where
$\omega(t)=\int_{\mathbb{R}^{N}}V(x)|u(t,x)|^{2}dx$ and $\varphi\in
C([0,T],L^{2})$ is the solution of the adjoint equation (4.2).
Remarks. (1) Under the assumptions on $u_{0}$ and $V$, it follows from Lemma
2.5 that the solution $u\in L^{\infty}((0,T),H^{2})$ of (1.1). Hence, we
deduce from the inequality
$\|Vu\|_{L^{2}}\leq\|V\|_{L^{p}}\|u\|_{L^{\frac{2p}{p-2}}}$ and $\varphi\in
C([0,T],L^{2})$ that the right hand side of (1.9) is well-defined.
(2) Because control potential $V$ is unbounded, we cannot follow the method in
[12] to obtain sufficiently high regularity of $u$, the solution of the NLS
equation (1.1). We resume the idea due to T.Kato, (see, [13]), based on the
general idea for Schrödinger equations, that two space derivative cost the
same as one time derivative.
(3) In contrast with the assumption $\sigma\in\mathbb{N}$ in [12], our results
follow for $\frac{1}{2}\leq\sigma<\frac{2}{N-2}$ if $\lambda<0$, or
$\frac{1}{2}\leq\sigma<\frac{2}{N}$ if $\lambda>0$.
As an immediate corollary of Theorem 1.2, we derive the precise
characterization for the critical points $\phi_{*}$ of functional
$\mathcal{F}$. The proof is the same as that of Corollary 4.8 in [12], so we
omit it.
###### Corollary 1.3.
Let $u_{*}$ be the solution of (1.1) with control $\phi_{*}$, and
$\varphi_{*}$ be the solution of corresponding adjoint equation (4.2). Then
$\phi_{*}\in C^{2}(0,T)$ is a classical solution of the following ordinary
differential equation
$\frac{d}{dt}(\phi^{\prime}_{*}(t)(\gamma_{2}+\gamma_{1}\omega_{*}^{2}(t)))=\frac{1}{2}Re\int_{\mathbb{R}^{N}}\bar{\varphi_{*}}(t,x)V(x)u_{*}(t,x)dx.$
(1.10)
subject to the initial data $\phi_{*}(0)=\phi_{0}$ and
$\phi^{\prime}_{*}(T)=0$.
This paper is organized as follows: in Section 2, we will collect some
preliminaries such as compactness results, global existence and regularity of
(1.1). In section 3, we will show Theorem 1.1. In section 4, we firstly
analyze well-posedness of the adjoint equation. Next, the Lipschitz continuity
of solution $u=u(\phi)$ with respect to control parameter $\phi$ is obtained.
Finally, we give the proof of Theorem 1.2. Some of the steps of the proof
follow [12], to avoid repetitions we will mainly focus on the differences with
respect to [12].
Notation. Throughout this paper, we use the following notation. $C>0$ will
stand for a constant that may different from line to line when it does not
cause any confusion. Since we exclusively deal with $\mathbb{R}^{N}$, we often
use the abbreviations $L^{r}=L^{r}(\mathbb{R}^{N})$,
$H^{s}=H^{s}(\mathbb{R}^{N})$. Given any interval $I\subset\mathbb{R}$, the
norms of mixed spaces $L^{q}(I,L^{r}(\mathbb{R}^{N}))$ and
$L^{q}(I,H^{s}(\mathbb{R}^{N}))$ are denoted by $\|\cdot\|_{L^{q}(I,L^{r})}$
and $\|\cdot\|_{L^{q}(I,H^{s})}$ respectively. We denote by
$U(t):=e^{it\triangle}$ the free Schrödinger propagator, which is isometric on
$H^{s}$ for every $s\geq 0$, see [13]. We recall that a pair of exponents
$(q,r)$ is Schrödinger-admissible if $\frac{2}{q}=N(\frac{1}{2}-\frac{1}{r})$
and $2\leq r\leq\frac{2N}{N-2}$, ($2\leq r\leq\infty\,\text{ if }\,N=1$;
$2\leq r<\infty\,\text{ if }\,N=2$).
## 2 Preliminaries
In this section, we recall some useful results. First, we recall the following
two compactness lemmas which is vital in our paper, see [13] for detailed
presentation.
###### Lemma 2.1.
[13] Let $X\hookrightarrow Y$ be two Banach spaces, $I$ be a bounded, open
interval of $\mathbb{R}$, and $(u_{n})_{n\in\mathbb{N}}$ be a bounded sequence
in $C(\bar{I},Y)$. Assume that $u_{n}(t)\in X$ for all
$(n,t)\in\mathbb{N}\times I$ and that
$sup\\{\|u_{n}(t)\|_{X},(n,t)\in\mathbb{N}\times I\\}=K<\infty$. Assume
further that $u_{n}$ is uniformly equicontinuous in $Y$. If $X$ is reflexive,
then there exist a function $u\in C(\bar{I},Y)$ which is weakly continuous
$\bar{I}\rightarrow X$ and some subsequence $(u_{n_{k}})_{k\in\mathbb{N}}$
such that for every $t\in\bar{I}$, $u_{n_{k}}(t)\rightharpoonup u(t)$ in $X$
as $k\rightarrow\infty$.
###### Lemma 2.2.
[13] Let $I$ be a bounded interval of $\mathbb{R}$, and
$(u_{n})_{n\in\mathbb{N}}$ be a bounded sequence of
$L^{\infty}(I,H^{1}_{0})\cap W^{1,\infty}(I,H^{-1})$. Then, there exist $u\in
L^{\infty}(I,H^{1}_{0})\cap W^{1,\infty}(I,H^{-1})$ and some subsequence
$(u_{n_{k}})_{k\in\mathbb{N}}$ such that for every $t\in\bar{I}$,
$u_{n_{k}}(t)\rightharpoonup u(t)$ in $H^{1}_{0}$ as $k\rightarrow\infty$.
In the following lemma, we establish some existence results of equation (1.1).
###### Lemma 2.3.
Let $u_{0}\in H^{1}$ and $V\in L^{p}+L^{\infty}$ for some $p\geq 1$, $p>N/2$.
Assume $0<\sigma<\frac{2}{N-2}$ if $\lambda<0$, or $0<\sigma<\frac{2}{N}$ if
$\lambda>0$. For any given $T>0$, $\phi\in H^{1}(0,T)$, there exists a unique
mild solution $u\in C([0,T],H^{1})$ of problem (1.1).
###### Proof.
When $\phi$ is a constant, the author in [13] showed that the solution of
(1.1) is local well-posedness. For our case, since $\phi\in
H^{1}(0,T)\hookrightarrow L^{\infty}(0,T)$, we only need to take the
$L^{\infty}$ norm of $\phi$ when the term $\phi Vu$ has to be estimated in
some norms. Keeping this in mind and applying the method in [13], one can show
the local well-posedness of (1.1). Hence, in order to prove this lemma, it
suffices to show
$\|u(t)\|_{H^{1}}\leq C(T,\|u_{0}\|_{H^{1}},\phi).$ (2.1)
Indeed, we deduce from (1.4) and mass conservation that
$\|E^{\prime}\|_{L^{2}(0,T)}\leq
C\|\phi^{\prime}\|_{L^{2}(0,T)}(\|V_{1}\|_{L^{p}}\|u\|_{L^{\frac{2p}{p-1}}}^{2}+\|V_{2}\|_{L^{\infty}}\|u_{0}\|_{L^{2}}^{2}).$
This implies
$\displaystyle E(t)=$ $\displaystyle E(0)+\int_{0}^{t}E^{\prime}(s)ds\leq
E(0)+\left(T\int_{0}^{T}(E^{\prime}(s))^{2}ds\right)^{1/2}$
$\displaystyle\leq$ $\displaystyle
E(0)+CT^{1/2}\|\phi^{\prime}\|_{L^{2}(0,T)}(\|V_{1}\|_{L^{p}}\|u\|_{L^{\frac{2p}{p-1}}}^{2}+\|V_{2}\|_{L^{\infty}}\|u_{0}\|_{L^{2}}^{2}).$
When $\lambda\leq 0$, it follows from (1.3) that
$\displaystyle\|\nabla u(t)\|_{L^{2}}^{2}\leq$ $\displaystyle
CE(t)+C\|\phi\|_{L^{\infty}(0,T)}\int V|u|^{2}dx$ $\displaystyle\leq$
$\displaystyle
CE(0)+CT^{1/2}\|\phi^{\prime}\|_{L^{2}(0,T)}(\|V_{1}\|_{L^{p}}\|u\|_{L^{\frac{2p}{p-1}}}^{2}+\|V_{2}\|_{L^{\infty}}\|u_{0}\|_{L^{2}}^{2})$
$\displaystyle+C\|\phi\|_{L^{\infty}(0,T)}(\|V_{1}\|_{L^{p}}\|u\|_{L^{\frac{2p}{p-1}}}^{2}+\|V_{2}\|_{L^{\infty}}\|u_{0}\|_{L^{2}}^{2}),$
(2.2)
which, together with the embedding $H^{1}\hookrightarrow L^{\frac{2p}{p-1}}$
and Young’s inequality with $\varepsilon$, implies (2.1).
When $\lambda>0$, by the same argument as above, we have
$\displaystyle\|\nabla u(t)\|_{L^{2}}^{2}\leq$ $\displaystyle
CE(0)+CT^{1/2}\|\phi^{\prime}\|_{L^{2}(0,T)}(\|V_{1}\|_{L^{p}}\|u\|_{L^{\frac{2p}{p-1}}}^{2}+\|V_{2}\|_{L^{\infty}}\|u_{0}\|_{L^{2}}^{2})$
$\displaystyle+C\|\phi\|_{L^{\infty}(0,T)}(\|V_{1}\|_{L^{p}}\|u\|_{L^{\frac{2p}{p-1}}}^{2}+\|V_{2}\|_{L^{\infty}}\|u_{0}\|_{L^{2}}^{2})+C\|u\|_{L^{2\sigma+2}}^{2\sigma+2},$
(2.3)
It follows from Gagliardo-Nirenberg’s inequality that
$\|u\|_{L^{2\sigma+2}}^{2\sigma+2}\leq
C\|u\|_{H^{1}}^{N\sigma}\|u\|_{L^{2}}^{2\sigma+2-N\sigma}.$ (2.4)
Since $N\sigma<2$, (2.1) follows from Young’s inequality with $\varepsilon$. ∎
###### Lemma 2.4.
[13] Let $J\ni 0$ be a bounded interval, $(\gamma,\rho)$ be an admissible pair
and consider $f\in L^{\infty}(J,L^{2})$ such that $f_{t}\in
L^{\gamma^{\prime}}(J,L^{\rho^{\prime}})$. If
$v(t)=i\int_{0}^{t}U(t-s)f(s)ds~{}~{}~{}for~{}all~{}t\in J,$
then $v\in L^{\infty}(J,H^{2})\cap C^{1}(J,L^{2})\cap W^{1,a}(J,L^{b})$ for
every admissible pair $(a,b)$ and
$\|\Delta
v\|_{L^{\infty}(J,L^{2})}\leq\|f\|_{L^{\infty}(J,L^{2})}+\|f(0)\|_{L^{2}}+C\|f\|_{L^{\gamma^{\prime}}(J,L^{\rho^{\prime}})},$
where $C$ is independent of $J$ and $f$.
###### Lemma 2.5.
Let $u_{0}\in H^{2}$, $\phi\in H^{1}(0,T)$ and $V,\nabla V\in
L^{p}+L^{\infty}$ for some $p\geq 2$, $p>N/2$. Assume $0<\sigma<\frac{2}{N-2}$
if $\lambda<0$ or $0<\sigma<\frac{2}{N}$ if $\lambda>0$. Then the mild
solution of (1.1) satisfies $u\in L^{\infty}((0,T),H^{2})$.
This lemma can be proved by applying Remarks 5.3.3 and 5.3.5 in [13]. When
$0<\sigma<\frac{2}{N}$, for this lemma, it suffices to require $V\in
L^{p}+L^{\infty}$ for some $p\geq 1$, $p>N/2$, see Remark 5.3.5 in [13].
## 3 Existence of Minimizers
Our goal in this section is to prove Theorem 1.1.
Proof of Theorem 1.1. The proof proceeds in three steps.
Step 1. Estimates of $(u_{n},\phi_{n})_{n\in\mathbb{N}}$. Let $\phi\in
H^{1}(0,T)$, there exists a unique mild solution $u\in C([0,T],H^{1})$ of
(1.1) by Lemma 2.3. Hence, the set $\Lambda(0,T)$ is nonempty, and there
exists a minimizing sequence $(u_{n},\phi_{n})_{n\in\mathbb{N}}$ such that
$\lim_{n\rightarrow\infty}F(u_{n},\phi_{n})=F_{*}.$
We deduce from $\gamma_{2}>0$ that there exists a constant $C$ such that for
every $n\in\mathbb{N}$
$\int_{0}^{T}(\phi^{\prime}_{n}(t))^{2}dt\leq C<+\infty.$
By using the embedding $H^{1}(0,T)\hookrightarrow C[0,T]$ and $\phi_{n}(0)\in
B_{2}$, we have
$\phi_{n}(t)=\phi_{n}(0)+\int_{0}^{t}\phi^{\prime}_{n}(s)ds\leq
M_{2}+\left(T\int_{0}^{T}(\phi^{\prime}_{n}(s))^{2}ds\right)^{1/2}<+\infty.$
This implies the sequence $(\phi_{n})_{n\in\mathbb{N}}$ is bounded in
$L^{\infty}(0,T)$, so is in $H^{1}(0,T)$. Thus, there exist a subsequence,
which we still denote by $(\phi_{n})_{n\in\mathbb{N}}$, and $\phi_{*}\in
H^{1}(0,T)$ such that
$\phi_{n}\rightharpoonup\phi_{*}~{}~{}in~{}~{}H^{1}(0,T)~{}~{}and~{}~{}\phi_{n}\rightarrow\phi_{*}~{}~{}in~{}~{}L^{2}(0,T)~{}~{}as~{}~{}n\rightarrow\infty.$
(3.1)
On the other hand, we deduce from (1.4) and mass conservation that
$\|E^{\prime}_{n}\|_{L^{2}(0,T)}\leq
C\|\phi^{\prime}_{n}\|_{L^{2}(0,T)}\|V\|_{L^{\infty}}\|u_{0}\|_{L^{2}}^{2}.$
Using the same argument as Lemma 2.3 and $u_{n}(0)\in B_{2}$, we derive
$\|u_{n}\|_{L^{\infty}((0,T),H^{1})}\leq C.$ (3.2)
Combining this estimate and the fact that $u_{n}$ is the solution of (1.1), we
have
$\|(u_{n})_{t}\|_{L^{\infty}((0,T),H^{-1})}\leq C.$ (3.3)
Step 2. Passage to the limit. By applying (3.2), (3.3), and Lemma 2.2, we
deduce that there exist $u_{*}\in L^{\infty}((0,T),H^{1})\cap
W^{1,\infty}((0,T),H^{-1})$ and a subsequence, still denoted by
$(u_{n})_{n\in\mathbb{N}}$, such that, for all $t\in[0,T]$,
$u_{n}(t)\rightharpoonup u_{*}(t)~{}~{}in~{}H^{1}~{}as~{}n\rightarrow\infty.$
(3.4)
From the embedding $W^{1,\infty}((0,T),H^{-1})\hookrightarrow
C^{0,1}([0,T],H^{-1})$ (see [13], Remark 1.3.11) and the inequality
$\|u\|^{2}_{L^{2}}\leq\|u\|_{H^{1}}\|u\|_{H^{-1}}$, we derive for every $u\in
L^{\infty}((0,T),H^{1})\cap W^{1,\infty}((0,T),H^{-1})$
$\|u(t)-u(s)\|_{L^{2}}\leq
C|t-s|^{\frac{1}{2}},~{}~{}for~{}all~{}t,s\in(0,T).$ (3.5)
Next, we note that for all $z_{1},z_{2}\in\mathbb{C}$, it holds
$||z_{1}|^{2\sigma}z_{1}-|z_{2}|^{2\sigma}z_{2}|\leq
C(|z_{1}|^{2\sigma}+|z_{2}|^{2\sigma})|z_{1}-z_{2}|.$ (3.6)
It follows from (2.4), (3.2), (3.5), (3.6), Hölder’s inequality that
$\displaystyle\||u_{n}(t)|^{2\sigma}u_{n}(t)-|u_{n}(s)|^{2\sigma}u_{n}(s)\|_{L^{r^{\prime}}}\leq$
$\displaystyle
C(\|u_{n}(t)\|_{L^{r}}^{2\sigma}+\|u_{n}(s)\|_{L^{r}}^{2\sigma})\|u_{n}(t)-u_{n}(s)\|_{L^{r}}$
$\displaystyle\leq$ $\displaystyle C\|u_{n}(t)-u_{n}(s)\|_{L^{2}}^{a}\leq
C|t-s|^{\frac{a}{2}},$ (3.7)
where $r=2\sigma+2$ and $a=1-N(\frac{1}{2}-\frac{1}{2\sigma+2})$. This implies
$(|u_{n}|^{2\sigma}u_{n})_{n\in\mathbb{N}}$ is a bounded sequence in
$C^{0,\frac{a}{2}}([0,T],L^{r^{\prime}})$. Therefore, we deduce from Lemma 2.1
that there exist a subsequence, still denoted by
$(|u_{n}|^{2\sigma}u_{n})_{n\in\mathbb{N}}$, and $f\in
C^{0,\frac{a}{2}}([0,T],L^{r^{\prime}})$ such that, for all $t\in[0,T]$,
$|u_{n}(t)|^{2\sigma}u_{n}(t)\rightharpoonup
f(t)~{}~{}in~{}L^{r^{\prime}}~{}as~{}n\rightarrow\infty.$ (3.8)
On the other hand, it follows from $(u_{n},\phi_{n})\in\Lambda(0,T)$ that for
every $\omega\in C_{c}^{\infty}(\mathbb{R}^{N})$ and for every
$\eta\in\mathcal{D}(0,T)$,
$\int_{0}^{T}[-\langle
iu_{n},\omega\rangle_{H^{-1},H^{1}_{0}}\eta^{\prime}(t)+\langle\Delta
u_{n}+|u_{n}|^{2\sigma}u_{n}+\phi_{n}(t)Vu_{n},\omega\rangle_{H^{-1},H^{1}_{0}}\eta(t)]dt=0.$
Applying (3.1), (3.4), (3.8), and the dominated convergence theorem, we deduce
easily that
$\int_{0}^{T}[-\langle
iu_{*},\omega\rangle_{H^{-1},H^{1}_{0}}\eta^{\prime}(t)+\langle\Delta
u_{*}+f+\phi_{*}(t)Vu_{*},\omega\rangle_{H^{-1},H^{1}_{0}}\eta(t)]dt=0.$
This implies that $u_{*}$ satisfies
$i\frac{d}{dt}u_{*}+\Delta
u_{*}+f+\phi_{*}(t)Vu_{*}=0~{}~{}for~{}a.e.~{}t\in[0,T].$ (3.9)
We next show $|u_{*}(t,x)|^{2\sigma}u_{*}(t,x)=f(t,x)$ for a.e.
$(t,x)\in[0,T]\times\mathbb{R}^{N}$. It suffices to show that for any given
$t\in[0,T]$
$\int_{\mathbb{R}^{N}}|u_{*}(t,x)|^{2\sigma}(x)u_{*}(t,x)\varphi(x)dx=\int_{\mathbb{R}^{N}}f(t,x)\varphi(x)dx~{}~{}for~{}any~{}\varphi\in
C_{c}^{\infty}(\mathbb{R}^{N}).$ (3.10)
Let us prove (3.10) by contradiction. If not, there exists $\varphi_{0}\in
C_{c}^{\infty}(\mathbb{R}^{N})$ such that
$\int_{\mathbb{R}^{N}}|u_{*}(t,x)|^{2\sigma}(x)u_{*}(t,x)\varphi_{0}(x)dx\neq\int_{\mathbb{R}^{N}}f(t,x)\varphi_{0}(x)dx.$
(3.11)
It follows from (3.8) that
$\int_{\mathbb{R}^{N}}|u_{n}(t,x)|^{2\sigma}(x)u_{n}(t,x)\varphi_{0}(x)dx\rightarrow\int_{\mathbb{R}^{N}}f(t,x)\varphi_{0}(x)dx~{}~{}as~{}n\rightarrow\infty.$
(3.12)
On the other hand, we deduce from (3.4) that there exists a subsequence, still
denoted by $(u_{n}(t))_{n\in\mathbb{N}}$ such that $u_{n}(t)\rightarrow
u_{*}(t)$ in $L^{2\sigma+2}_{loc}(\mathbb{R}^{N})$ and
$|u_{n}(t)|^{2\sigma}\rightarrow|u_{*}(t)|^{2\sigma}$ in
$L^{\frac{2\sigma+2}{2\sigma}}_{loc}(\mathbb{R}^{N})$. Combining this, (3.2)
and (3.4), we derive
$\displaystyle\left|\int_{\mathbb{R}^{N}}|u_{n}(t,x)|^{2\sigma}u_{n}(t,x)\varphi_{0}(x)dx-\int_{\mathbb{R}^{N}}|u_{*}(t,x)|^{2\sigma}u_{*}(t,x)\varphi_{0}(x)dx\right|$
$\displaystyle\leq$
$\displaystyle\|u_{n}(t)\|_{L^{2\sigma+2}}^{2\sigma}\|u_{n}(t)-u_{*}(t)\|_{L^{2\sigma+2}(\Omega)}\|\varphi_{0}\|_{L^{2\sigma+2}}+\|u_{*}(t)\|_{L^{2\sigma+2}}\|u_{n}(t)-u_{*}(t)\|_{L^{\frac{2\sigma+2}{2\sigma}}(\Omega)}\|\varphi_{0}\|_{L^{2\sigma+2}}$
$\displaystyle\xrightarrow{n\rightarrow\infty}0,$ (3.13)
where $\Omega$ is the compact support of $\varphi_{0}$. This is a
contradiction with (3.11) and (3.12).
In summary, $u_{*}\in L^{\infty}((0,T),H^{1})\cap W^{1,\infty}((0,T),H^{-1})$
and satisfies
$i\frac{d}{dt}u_{*}+\Delta
u_{*}+\lambda|u_{*}|^{2\sigma}u_{*}+\phi_{*}(t)Vu_{*}=0,~{}~{}for~{}a.e.~{}t\in[0,T].$
By using the classical argument based on Strichartz’s estimate, we can obtain
the uniqueness of the weak solution $u_{*}$ of (1.1). Arguing as the proof of
Theorem 3.3.9 in [13], it follows that $u_{*}$ is indeed a mild solution of
(1.1) and $u_{*}\in C((0,T),H^{1})\cap C^{1}((0,T),H^{-1})$.
Step 3. Conclusion. In order to conclude that the pair
$(u_{*},\phi_{*})\in\Lambda(0,T)$ is indeed a minimizer of optimal control
problem (1.8), we need only show
$F_{*}=\lim_{n\rightarrow\infty}F(u_{n},\phi_{n})\geq F(u_{*},\phi_{*}).$
(3.14)
Indeed, in view of the assumption on operator $A$, there exists $R>0$, such
that for every $n\in\mathbb{N}$, $supp_{x\in\mathbb{R}^{3}}(Au(T,x))\subseteq
B(R)$. Therefore, we deduce from $u_{n}(T)\rightarrow u_{*}(T)$ in
$L^{2}_{loc}$ and $Au_{n}(T)\rightharpoonup Au_{*}(T)$ in $L^{2}$ that
$\displaystyle|\langle u_{n}(T),Au_{n}(T)\rangle-\langle
u_{*}(T),Au_{*}(T)\rangle|$ $\displaystyle\leq$ $\displaystyle|\langle
u_{n}(T)-u_{*}(T),Au_{n}(T)\rangle|+|\langle
u_{*}(T),A(u_{n}(T)-u_{*}(T))\rangle|\rightarrow
0~{}~{}as~{}n\rightarrow\infty.$ (3.15)
The same argument as Lemma 2.5 in [12], we have
$\liminf_{n\rightarrow\infty}\int_{0}^{T}(\phi_{n}^{\prime}(t))^{2}\omega_{n}^{2}(t)dt\geq\int_{0}^{T}(\phi^{\prime}_{*}(t))^{2}\omega_{*}^{2}(t)dt,$
(3.16)
where
$\omega_{n}(t)=\int_{\mathbb{R}^{3}}V(x)|u_{n}(t,x)|^{2}dx,~{}~{}\omega_{*}(t)=\int_{\mathbb{R}^{3}}V(x)|u_{*}(t,x)|^{2}dx.$
It follows from the weak lower semicontinuity of the norm that
$\liminf_{n\rightarrow\infty}\int_{0}^{T}(\phi_{n}^{\prime}(t))^{2}dt\geq\int_{0}^{T}(\phi^{\prime}_{*}(t))^{2}dt.$
(3.17)
Collecting (3)-(3.17), we derive (3.14). This completes the proof.
## 4 Rigorous characterization of a minimizer
In order to obtain a rigorous characterization of a minimizer
$(u_{*},\phi_{*})\in\Lambda(0,T)$, we need to derive the first order
optimality conditions for our optimal control problem (1.8). For this aim, we
firstly formally calculate the derivative of the objective functional
$F(u,\phi)$ and analyze the resulting adjoint problem in the next subsection.
### 4.1 Derivation and analysis of the adjoint equation.
To begin with, we rewrite equation (1.1) in a more abstract form, i.e.,
$P(u,\phi)=iu_{t}+\Delta u+\lambda|u|^{2\sigma}u+\phi(t)V(x)u=0.$ (4.1)
Thus, formally computation yields
$\partial_{u}P(u,\phi)\varphi=i\varphi_{t}+\Delta\varphi+\phi(t)V(x)\varphi+\lambda(\sigma+1)|u|^{2\sigma}\varphi+\lambda|u|^{2\sigma-2}u^{2}\bar{\varphi},$
where $\varphi\in L^{2}$. Similarly, we have
$\partial_{\phi}P(u,\phi)=V(x)u.$
The analogue argument as Section 3.1 in [12], we can derive the following
adjoint equation:
$\left\\{\begin{array}[]{l}i\varphi_{t}+\Delta\varphi+\phi(t)V(x)\varphi+\lambda(\sigma+1)|u|^{2\sigma}\varphi+\lambda|u|^{2\sigma-2}u^{2}\bar{\varphi}=\frac{\delta
F(u,\phi)}{\delta u(t)},\\\ \varphi(T)=i\frac{\delta F(u,\phi)}{\delta
u(T)},\end{array}\right.$ (4.2)
where $\frac{\delta F(u,\phi)}{\delta u(t)}$ and $\frac{\delta
F(u,\phi)}{\delta u(T)}$ denote the first variation of $F(u,\phi)$ with
respect to $u(t)$ and $u(T)$ respectively. By straightforward computations, we
have
$\displaystyle\frac{\delta F(u,\phi)}{\delta u(t)}=$ $\displaystyle
4\gamma_{1}(\phi^{\prime}(t))^{2}(\int_{\mathbb{R}^{N}}V(x)|u(t,x)|^{2}dx)V(x)u(t,x)$
$\displaystyle=$ $\displaystyle
4\gamma_{1}(\phi^{\prime}(t))^{2}\omega(t)V(x)u(t,x),$ (4.3)
and
$\frac{\delta F(u,\phi)}{\delta u(T)}=4\langle
u(T),Au(T)\rangle_{L^{2}}Au(T).$ (4.4)
Thus, equation (4.2) defines a Cauchy problem for $\varphi$ with data
$\varphi(T)\in L^{2}$, one can solve (4.2) backwards in time.
In the following proposition, we will analyze the existence of solutions to
(4.2).
###### Proposition 4.1.
Let $N\leq 3$, $u_{0}\in H^{2}$ and $V,\nabla V\in L^{p}+L^{\infty}$ for some
$p\geq 2$. Assume $0<\sigma<\frac{2}{N-2}$ if $\lambda<0$ or
$0<\sigma<\frac{2}{N}$ if $\lambda>0$. Then, for every $T>0$, equation (4.2)
admits a unique mild solution $\varphi\in C([0,T],L^{2})$.
###### Proof.
Under our assumptions on $V$, $u_{0}$, and $A$, we deduce from
$H^{2}\hookrightarrow L^{\infty}$ and Lemma 2.5 that $|u|^{2\sigma}$,
$|u|^{2\sigma-2}u^{2}\in L^{\infty}$, $\frac{\delta F(u,\phi)}{\delta u(t)}\in
L^{1}((0,T),L^{\frac{2p}{p+1}})$, $\frac{\delta F(u,\phi)}{\delta u(T)}\in
L^{2}$. Since $V$ is an unbounded potential, it cannot be treated as a
perturbation. Applying consequently Theorem 4.6.4 and Corollary 4.6.5 in [13],
we can obtain the local well-posedness. The global existence can be derived by
the classical argument for Schrödinger equations and Gronwall’s inequality. ∎
### 4.2 Lipschitz continuity with respect to the control.
This subsection is devoted to derive the solution of (1.1) depends Lipschitz
continuously on the control parameter $\phi$, which is vital for investigating
the differentiability of unconstrained functional $\mathcal{F}$. To begin
with, we study the continuous dependence of the solutions $u=u(\phi)$ with
respect to the control parameter $\phi$. Our result is as follows.
###### Proposition 4.2.
Let $N\leq 3$, $V,\nabla V\in L^{p}+L^{\infty}$ and $V\in L^{2p}$ for some
$p\geq 2$. Assume $0<\sigma<\frac{2}{N-2}$ if $\lambda<0$ or
$0<\sigma<\frac{2}{N}$ if $\lambda>0$. Let $u,\tilde{u}\in
L^{\infty}((0,T),H^{2})$ be two mild solutions of (1.1) with the same initial
data $u_{0}\in H^{2}$, corresponding to control parameters
$\phi,\tilde{\phi}\in H^{1}(0,T)$ respectively. Given a constant $M>0$, if
$\|\phi\|_{H^{1}(0,T)},~{}\|\tilde{\phi}\|_{H^{1}(0,T)},~{}\|u(t)\|_{H^{2}},~{}\|\tilde{u}(t)\|_{H^{2}}\leq
M,$
then, there exist $\tau=\tau(M)>0$ and a constant $C=C(M)$ such that
$\|u-\tilde{u}\|_{L^{\infty}(I_{t},H^{2})}\leq
C(\|u(t)-\tilde{u}(t)\|_{H^{2}}+\|\phi-\tilde{\phi}\|_{L^{2}(I_{t})}),$ (4.5)
where $I_{t}:=[t,t+\tau]\cap[0,T]$. In particular, the solution $u(\phi)$
depends continuously on control parameter $\phi\in H^{1}(0,T)$.
###### Proof.
To simplify notation, let us assume $t+\tau\leq T$. Applying Lemmas 2.3 and
2.4, there is a $\tau>0$ depending only on $M$, such that $u|_{I_{t}}$ is a
fixed point of the operator
$\Phi(u):=U(\cdot-t)u(t)+i\int_{t}^{\cdot}U(\cdot-s)(\lambda|u(s)|^{2\sigma}u(s)+\phi(s)Vu(s))ds,$
which maps the Banach space
$Y=\\{u\in L^{\infty}(I_{t},H^{2}),~{}~{}\|u\|_{L^{\infty}(I_{t},H^{2})}\leq
2M\\}$
into itself. The same holds for $\tilde{u}$, we consequently derive
$\displaystyle\tilde{u}(s)-$ $\displaystyle u(s)=U(s-t)(\tilde{u}(t)-u(t))$
$\displaystyle+$ $\displaystyle
i\int_{t}^{s}U(s-r)(\lambda(|\tilde{u}|^{2\sigma}\tilde{u}-|u|^{2\sigma}u)+V(\tilde{u}\tilde{\phi}-u\phi))(r)dr$
(4.6)
where $s\in[t,t+\tau]$. In the following, we set $r=2\sigma+2$ and
$\rho=\frac{2p}{p-1}$, taking $q$ and $\gamma$ such that $(q,r)$ and
$(\gamma,\rho)$ are two admissible pairs. Applying Strichartz’s estimate to
(4.2), the embedding theorems $\tilde{\phi},\phi\in H^{1}(0,T)\hookrightarrow
L^{\infty}(0,T)$ and $\tilde{u}(t,\cdot),u(t,\cdot)\in
H^{2}(\mathbb{R}^{N})\hookrightarrow L^{\infty}(\mathbb{R}^{N})$ when $N\leq
3$, Hölder’s inequality, (3.6), we derive
$\displaystyle\|\tilde{u}-u\|_{L^{\infty}(I_{t},L^{2})}$ $\displaystyle\leq$
$\displaystyle
C\|\tilde{u}(t)-u(t)\|_{L^{2}}+C\||\tilde{u}|^{2\sigma}\tilde{u}-|u|^{2\sigma}u\|_{L^{q^{\prime}}(I_{t},L^{r^{\prime}})}+C\|V(\tilde{u}\tilde{\phi}-u\phi))\|_{L^{\gamma^{\prime}}(I_{t},L^{\rho^{\prime}})}$
$\displaystyle\leq$ $\displaystyle
C\|\tilde{u}(t)-u(t)\|_{L^{2}}+C\tau^{\frac{q-q^{\prime}}{qq^{\prime}}}\left(\|\tilde{u}\|^{2\sigma}_{L^{\infty}(I_{t},L^{r})}+C\|u\|^{2\sigma}_{L^{\infty}(I_{t},L^{r})}\right)\|\tilde{u}-u\|_{L^{q}(I_{t},L^{r})}$
$\displaystyle+C\|V\|_{L^{p}}\|\tilde{\phi}\|_{L^{\infty}}\|\tilde{u}-u\|_{L^{\gamma^{\prime}}(I_{t},L^{\rho})}+\|V\|_{L^{p}}\|\tilde{\phi}-\phi\|_{L^{\gamma^{\prime}}(I_{t})}\|u\|_{L^{\infty}(I_{t},L^{\rho})}$
$\displaystyle\leq$ $\displaystyle
C\|\tilde{u}(t)-u(t)\|_{L^{2}}+C\tau^{\frac{1}{q^{\prime}}}\|\tilde{u}-u\|_{L^{\infty}(I_{t},H^{2})}+C\tau^{\frac{1}{\gamma^{\prime}}}\|\tilde{u}-u\|_{L^{\infty}(I_{t},H^{2})}+C\|\tilde{\phi}-\phi\|_{H^{1}(I_{t})}.$
(4.7)
Set $f_{1}(t)=\lambda(|\tilde{u}|^{2\sigma}\tilde{u}-|u|^{2\sigma}u)(t)$ and
$f_{2}(t)=V(\tilde{u}\tilde{\phi}-u\phi)(t)$, we deduce from Lemma 2.4 that
$\displaystyle\|\Delta(\tilde{u}-u)\|_{L^{\infty}(I_{t},L^{2})}\leq$
$\displaystyle\|\Delta(\tilde{u}(t)-u(t))\|_{L^{2}}+C\|f_{1}(t)+f_{2}(t)\|_{L^{2}}+\|f_{1}+f_{2}\|_{L^{\infty}(I_{t},L^{2})}$
$\displaystyle+C\|(f_{1})_{t}\|_{L^{q^{\prime}}(I_{t},L^{r^{\prime}})}+C\|(f_{2})_{t}\|_{L^{\gamma^{\prime}}(I_{t},L^{\rho^{\prime}})}.$
(4.8)
Let us estimate these terms. By the similar argument as (4.2), we obtain
$\displaystyle\|f_{1}(t)+f_{2}(t)\|_{L^{2}}\leq$ $\displaystyle
C\|(|\tilde{u}|^{2\sigma}+|u|^{2\sigma})(t)(\tilde{u}-u)(t)\|_{L^{2}}+\|V\tilde{\phi}(t)(\tilde{u}-u)(t)\|_{L^{2}}+\|Vu(t)(\tilde{\phi}-\phi)(t)\|_{L^{2}}$
$\displaystyle\leq$ $\displaystyle
C\|\tilde{u}(t)-u(t)\|_{L^{2}}+\tilde{\phi}(t)\|V\|_{L^{p}}\|(\tilde{u}-u)(t)\|_{L^{\frac{2p}{p-2}}}+|(\tilde{\phi}-\phi)(t)|\|Vu(t)\|_{L^{2}}$
$\displaystyle\leq$ $\displaystyle
C\|\tilde{u}(t)-u(t)\|_{L^{2}}+C\|\tilde{u}(t)-u(t)\|_{H^{2}}+C\|\tilde{\phi}-\phi\|_{H^{1}(I_{t})};$
(4.9)
When $2<p<\infty$, $2<\frac{2p}{p-2}<\infty$, we deduce from interpolation
inequality and Young’s inequality with $\varepsilon$ that
$\displaystyle\|f_{1}+f_{2}\|_{L^{\infty}(I_{t},L^{2})}$ $\displaystyle\leq$
$\displaystyle
C\|(|\tilde{u}|^{2\sigma}+|u|^{2\sigma})(\tilde{u}-u)\|_{L^{\infty}(I_{t},L^{2})}+\|V\tilde{\phi}(\tilde{u}-u)\|_{L^{\infty}(I_{t},L^{2})}+\|Vu(\tilde{\phi}-\phi)\|_{L^{\infty}(I_{t},L^{2})}$
$\displaystyle\leq$ $\displaystyle
C\|\tilde{u}-u\|_{L^{\infty}(I_{t},L^{2})}+C\|V\|_{L^{p}}\|\tilde{u}-u\|_{L^{\infty}(I_{t},L^{\frac{2p}{p-2}})}+\|\tilde{\phi}-\phi\|_{H^{1}(I_{t})}\|V\|_{L^{p}}\|u\|_{L^{\infty}(I_{t},H^{2})}$
$\displaystyle\leq$ $\displaystyle
C\|\tilde{u}-u\|_{L^{\infty}(I_{t},L^{2})}+\varepsilon\|\tilde{u}-u\|_{L^{\infty}(I_{t},H^{2})}+C\|\tilde{\phi}-\phi\|_{H^{1}(I_{t})}.$
(4.10)
When $p=2$, by the similar argument as above, we have
$\displaystyle\|V\tilde{\phi}(\tilde{u}-u)\|_{L^{\infty}(I_{t},L^{2})}\leq$
$\displaystyle\|V\|_{L^{4}}\|\tilde{\phi}(\tilde{u}-u)\|_{L^{\infty}(I_{t},L^{4})}$
$\displaystyle\leq$ $\displaystyle
C\|\tilde{u}-u\|_{L^{\infty}(I_{t},L^{2})}+\varepsilon\|\tilde{u}-u\|_{L^{\infty}(I_{t},H^{2})};$
(4.11)
When $p=\infty$,
$\|V\tilde{\phi}(\tilde{u}-u)\|_{L^{\infty}(I_{t},L^{2})}\leq\|V\|_{L^{\infty}}\|\tilde{\phi}(\tilde{u}-u)\|_{L^{\infty}(I_{t},L^{2})}.$
(4.12)
After some fundamental computations, by the similar argument as (4.2), we
obtain
$\displaystyle\|(f_{1})_{t}\|_{L^{q^{\prime}}(I_{t},L^{r^{\prime}})}\leq$
$\displaystyle
C\||\tilde{u}|^{2\sigma}\tilde{u}_{t}-|u|^{2\sigma}u_{t}\|_{L^{q^{\prime}}(I_{t},L^{r^{\prime}})}+\||\tilde{u}|^{2\sigma-2}\bar{\tilde{u}}_{t}\tilde{u}^{2}-|u|^{2\sigma-2}\bar{u}_{t}u^{2}\|_{L^{q^{\prime}}(I_{t},L^{r^{\prime}})}$
$\displaystyle\leq$ $\displaystyle
C\|\tilde{u}_{t}-u_{t}\|_{L^{q^{\prime}}(I_{t},L^{2})}+\||\tilde{u}|^{2\sigma-2}\tilde{u}^{2}(\bar{\tilde{u}}_{t}-\bar{u}_{t})\|_{L^{q^{\prime}}(I_{t},L^{r^{\prime}})}$
$\displaystyle+\|(|\tilde{u}|^{2\sigma-2}\tilde{u}^{2}-|u|^{2\sigma-2}u^{2})\bar{u}_{t}\|_{L^{q^{\prime}}(I_{t},L^{r^{\prime}})}$
$\displaystyle\leq$ $\displaystyle
C\|\tilde{u}_{t}-u_{t}\|_{L^{q^{\prime}}(I_{t},L^{2})}+C\|(\tilde{u}-u)(|\tilde{u}|^{2\sigma-1}+|u|^{2\sigma-1})\bar{u}_{t}\|_{L^{q^{\prime}}(I_{t},L^{r^{\prime}})}$
$\displaystyle\leq$ $\displaystyle
C\|\tilde{u}_{t}-u_{t}\|_{L^{q^{\prime}}(I_{t},L^{2})}+\tau^{\frac{1}{q^{\prime}}}\|\tilde{u}-u\|_{L^{\infty}(I_{t},H^{2})},$
(4.13)
where
$\displaystyle\|\tilde{u}_{t}-u_{t}\|_{L^{q^{\prime}}(I_{t},L^{2})}$
$\displaystyle\leq$ $\displaystyle
C\|\Delta(\tilde{u}-u)\|_{L^{q^{\prime}}(I_{t},L^{2})}+C\|V(\tilde{u}\tilde{\phi}-u\phi)\|_{L^{q^{\prime}}(I_{t},L^{2})}+C\||\tilde{u}|^{2\sigma}\tilde{u}-|u|^{2\sigma}u\|_{L^{q^{\prime}}(I_{t},L^{2})}$
$\displaystyle\leq$ $\displaystyle
C\tau^{\frac{1}{q^{\prime}}}\|\tilde{u}-u\|_{L^{\infty}(I_{t},H^{2})}+C\|\tilde{\phi}-\phi\|_{H^{1}(I_{t})}+\tau^{\frac{1}{q^{\prime}}}\|\tilde{u}-u\|_{L^{\infty}(I_{t},L^{2})}.$
(4.14)
Similarly,
$\displaystyle\|(V(\tilde{u}\tilde{\phi}-u\phi)))_{t}\|_{L^{\gamma^{\prime}}(I_{t},L^{\rho^{\prime}})}\leq$
$\displaystyle\|V(\tilde{\phi}^{\prime}-\phi^{\prime})\tilde{u})\|_{L^{\gamma^{\prime}}(I_{t},L^{\rho^{\prime}})}+\|V\phi^{\prime}(\tilde{u}-u)\|_{L^{\gamma^{\prime}}(I_{t},L^{\rho^{\prime}})}$
$\displaystyle+\|V(\tilde{\phi}-\phi)\tilde{u}_{t})\|_{L^{\gamma^{\prime}}(I_{t},L^{\rho^{\prime}})}+\|V\phi(\tilde{u}_{t}-u_{t})\|_{L^{\gamma^{\prime}}(I_{t},L^{\rho^{\prime}})}$
$\displaystyle\leq$ $\displaystyle
C\|\tilde{\phi}-\phi\|_{H^{1}(I_{t})}+\tau^{\frac{2-\gamma^{\prime}}{2\gamma^{\prime}}}\|\phi^{\prime}\|_{L^{2}}\|\tilde{u}-u\|_{L^{\infty}(I_{t},H^{2})}$
$\displaystyle+\|V\|_{L^{2p}}\|\tilde{\phi}-\phi\|_{H^{1}(I_{t})}+\|V\|_{L^{2p}}\|\tilde{u}_{t}-u_{t}\|_{L^{\gamma^{\prime}}(I_{t},L^{2})}.$
(4.15)
Notice that the estimates (4.2)-(4.2) hold for $V\in L^{\infty}$. Combining
(4.2)-(4.2), using the equivalent norm of $H^{2}$, i.e.,
$\|\cdot\|_{H^{2}}=\|\cdot\|_{L^{2}}+\|\Delta\cdot\|_{L^{2}}$, we obtain
$\displaystyle\|\tilde{u}-u\|_{L^{\infty}(I_{t},H^{2})}\leq$ $\displaystyle
C\|\tilde{u}(t)-u(t)\|_{H^{2}}+C\tau^{\frac{1}{q^{\prime}}}\|\tilde{u}-u\|_{L^{\infty}(I_{t},H^{2})}+C\tau^{\frac{1}{\gamma^{\prime}}}\|\tilde{u}-u\|_{L^{\infty}(I_{t},H^{2})}$
$\displaystyle+C\|\tilde{\phi}-\phi\|_{H^{1}(I_{t})}+\tau^{\frac{2-\gamma^{\prime}}{2\gamma^{\prime}}}\|\tilde{u}-u\|_{L^{\infty}(I_{t},H^{2})}+\varepsilon\|\tilde{u}-u\|_{L^{\infty}(I_{t},H^{2})}.$
(4.16)
Therefore, the estimate (4.5) holds by choosing $\tau$ and $\varepsilon$
sufficiently small. Due to $\tilde{u}(0)=u(0)$, we deduce from continuity
argument and (4.5) that the mapping $\phi\rightarrow u(\phi)$ is continuous
with respect to $\phi\in H^{1}(0,T)$. ∎
We are now in the position to show Lipschitz continuity of solution $u(\phi)$
with respect to $\phi\in H^{1}(0,T)$. With the estimate (4.5) at hand, the
proof is analogue to that of Proposition 4.5 in [12], so we omit it.
###### Proposition 4.3.
Let $N\leq 3$, $V,\nabla V\in L^{p}+L^{\infty}$ and $V\in L^{2p}$ for some
$p\geq 2$. Assume $0<\sigma<\frac{2}{N-2}$ if $\lambda<0$ or
$0<\sigma<\frac{2}{N}$ if $\lambda>0$. Let $\phi\in H^{1}(0,T)$, and
$u=u(\phi)\in L^{\infty}((0,T),H^{2})$ be the solution of (1.1). Given
$\delta_{\phi}\in H^{1}(0,T)$ with $\delta_{\phi}(0)=0$, for every
$\varepsilon\in[-1,1]$, let $\tilde{u}=u(\phi+\epsilon\delta_{\phi})$ be the
solution of (1.1) with control $\phi+\epsilon\delta_{\phi}$ and the same
initial data as $u(\phi)$. Then, there exists a constant $C>0$ such that
$\|\tilde{u}-u\|_{L^{\infty}((0,T),H^{2})}\leq
C\|\tilde{\phi}-\phi\|_{H^{1}(0,T)}=C|\varepsilon|\|\delta_{\phi}\|_{H^{1}(0,T)}.$
In other words, the mapping $\phi\mapsto u(\phi)$ is Lipschitz continuous with
respect to $\phi$ for each fixed direction $\delta_{\phi}$.
Proof of Theorem 1.2. In view of definition of Gâteaux derivative, let
$u=u(\phi)$, $\tilde{u}=u(\tilde{\phi})$ with
$\tilde{\phi}=\phi+\varepsilon\delta_{\phi}$, we compute
$\mathcal{F}(\tilde{\phi})-\mathcal{F}(\phi)=\mathcal{J}_{1}+\mathcal{J}_{2}+\mathcal{J}_{3},$
where
$\mathcal{J}_{1}:=\langle\tilde{u}(T),A\tilde{u}(T)\rangle^{2}-\langle
u(T),Au(T)\rangle^{2},$
$\mathcal{J}_{2}:=\gamma_{2}\int_{0}^{T}\big{[}(\tilde{\phi}^{\prime}(t))^{2}-(\phi^{\prime}(t))^{2}\big{]}dt,$
and
$\mathcal{J}_{3}:=\gamma_{1}\int_{0}^{T}(\tilde{\phi}^{\prime}(t))^{2}\left(\int_{\mathbb{R}^{N}}V(x)|\tilde{\phi}(t,x)|^{2}\right)^{2}dt-\gamma_{1}\int_{0}^{T}(\phi^{\prime}(t))^{2}\left(\int_{\mathbb{R}^{N}}V(x)|\phi(t,x)|^{2}\right)^{2}dt.$
Because we have obtained Proposition 4.3, $\tilde{u},u\in
L^{\infty}((0,T),H^{2})\hookrightarrow L^{\infty}((0,T)\times\mathbb{R}^{N})$,
ones can prove along the lines of Theorem 4.6 in [12], so we omit it.
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|
arxiv-papers
| 2013-01-18T10:05:32 |
2024-09-04T02:49:40.452500
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Binhua Feng, Dun Zhao, Pengyu Chen",
"submitter": "Binhua Feng",
"url": "https://arxiv.org/abs/1301.4335"
}
|
1301.4647
|
# Violation of Geometrical Scaling in pp Collisions at NA61/SHINE
Michal Praszalowicz M. Smoluchowski Institute of Physics, Jagellonian
University, Reymonta 4, 30-059 Krakow, Poland
###### Abstract
We analyze geometrical scaling (GS) of negative pion multiplicity $p_{\rm T}$
distributions at NA61/SHINE energies. We show that even though NA61/SHINE
energies are low, one may expect to find GS in the particle spectra. We argue
that qualitative behavior of ratios of multiplicities at different energies is
in agreement with a simple picture of GS which is violated for $p_{\rm T}$
smaller than some nonperturbative scale $\Lambda$ and when larger Bjorken $x$
of one of the scattering patrons crosses $x_{\rm max}$ above which gluonic
cloud becomes dilute and quark degrees of freedom become important.
###### pacs:
13.85.Ni,12.38.Lg
Scaling properties of physical observables have been for a long time one of
the most instructive ways to get information about the underlying physics. In
some cases this information could be fully quantified; here the most prominent
example is Bjorken scaling of Deep Inelastic Scattering (DIS) structure
functions Bjorken:1968dy . In some other cases, like the Koba -Nielsen-Olesen
scaling Koba:1972ng for example, the underlying physics is still poorly
understood. Some theoretically conjectured scaling laws, like Feynman scaling
in high energy hadronic collisions Feynman:1969ej , have not been
experimentally confirmed. In any case both the emergence of a scaling law, and
– sometimes more importantly – its violation, test our understanding of
microscopic phenomena that are of importance in given experimental conditions.
Again Bjorken scaling is here a benchmark, since both its emergence and
violation are fully understood in terms of perturbative quantum chromodynamics
(QCD) DGLAP .
In this paper we will focus on another scaling law, called geometrical scaling
(GS), which has been introduced in the context of DIS Stasto:2000er . It has
been shown that GS scaling is present also in high energy pp collisions
McLerran:2010ex ; Praszalowicz:2011tc . An onset of GS in heavy ion collisions
at RHIC energies has been reported in Refs. Praszalowicz:2011rm .
The proton seen at low Bjorken $x<x_{\mathrm{max}}$ is dominated by a gluon
cloud characterized by a typical scale $Q_{\text{s}}(x)$ which is called
saturation scale sat1 ; sat2 ; GolecBiernat:1998js . The phenomenon of
saturation (for a review see Mueller:2001fv ; McLerran:2010ub ) appears due to
the nonlinearities of parton evolution at small $x$ given by so-called JIMWLK
hierarchy equations jimwlk which in the large $N_{\rm c}$ limit reduce to the
Balitsky-Kovchegov equation BK . These equations admit traveling wave
solutions which explicitly exhibit GS Munier:2003vc . The appearance of GS in
the prototype of the BK equation obtained in double logarithmic approximation
by Gribov, Levin and Ryskin sat1 has been studied in Ref. Bartels:1992ix . An
effective theory describing the small $x$ regime is Color Glass Condensate
sat1 ; sat2 ; MLV . For the present study the details of saturation are not of
primary interest, it is the very existence of the saturation scale which plays
the crucial role.
Recently it has been shown Praszalowicz:2012zh that GS in DIS works very well
up to relatively large $x_{\text{max}}\sim 0.1$ (see also Caola:2010cy ). This
should not come as a total surprise since it is known that GS scaling extends
well above the saturation scale both in the DGLAP Kwiecinski:2002ep and BFKL
Iancu:2002tr evolution schemes not only for the boundary conditions that
exhibit GS Caola:2008xr to start with. For $x<x_{\mathrm{max}}$ in the first
approximation the virtual photon cross-section $\sigma_{\gamma^{\ast}p}$ in
DIS is a function of a scaling variable $\tau=-q^{2}/Q_{\text{s}}^{2}(x)$
only, where $q^{2}$ is photon virtuality Stasto:2000er ; GolecBiernat:1998js .
Similarly in hadronic collisions charged particle multiplicity in central
rapidity exhibits GS McLerran:2010ex
$\left.\frac{dN}{dyd^{2}p_{\text{T}}}\right|_{y\simeq
0}=\frac{1}{Q_{0}^{2}}F(\tau)$ (1)
where $F$ is a universal dimensionless function of the scaling variable
$\tau=p_{\text{T}}^{2}/Q_{\text{s}}^{2}(x).$ (2)
In this paper we shall address a question of applicability of Eq. (1) beyond
the central rapidity region. This is an important and interesting question for
the following reasons. First, Bjorken $x$’s of scattering gluons that produce
a particle of a given transverse momentum $p_{\mathrm{T}}$ and rapidity $y$
read
$x_{1,2}=e^{\pm y}\,p_{\text{T}}/W$ (3)
and can be quite different for large rapidities (we shall assume positive $y$,
hence $x_{2}<x_{1}$). Here $W=\sqrt{s}$ denotes pp scattering energy. If
$x_{2}$ and $x_{1}$ are different then two different saturation scales emerge:
$Q_{\text{s}1}$ and $Q_{\text{s}2}$ Kharzeev:2001gp . Whether this leads to
violation of GS depends on the form of the saturation scale (for discussion of
different forms of scaling variable see e.g. Beuf:2008mf and references
therein). Second, and – as we shall argue below – more importantly, when the
larger Bjorken $x=x_{1}$ for some $y$ leaves the domain of GS, _i.e._
$x_{1}>x_{\text{max}}$, violation of GS should appear due to the fact that one
of the scattering gluonic clouds becomes dilute and quark degrees of freedom
become important. We are going to show in a model-independent way that this is
what indeed does happen.
The theoretical formula for the multiplicity density in pp collisions is known
since 1981 Gribov:1981kg :
$\frac{dN}{dyd^{2}p_{\text{T}}}=\frac{C}{p_{\text{T}}^{2}}{\displaystyle\int}d^{2}\vec{k}_{\text{T}}\,\alpha_{\text{s}}\,\varphi_{1}(x_{1},\vec{k}_{\text{T}}^{2})\varphi_{2}(x_{2},(\vec{k}-\vec{p}\,)_{\text{T}}^{2}).$
(4)
Here $1/p_{\text{T}}^{2}$ corresponds to the $gg\rightarrow g$ cross section,
$\varphi_{1,2}$ are unintegrated gluon densities and $\alpha_{\text{s}}$
stands for strong coupling constant. Although in principle (4) describes gluon
production, one uses parton-hadron duality Dokshitzer:1991eq to argue that it
can be also applied to particle multiplicities where constant $C$ takes care
of the hardonization effects.
At high energies and small $x$’s unintegrated gluon densities depend on
$k_{\text{T}}$ through the ratio $k_{\text{T}}^{2}/Q_{\text{s}}^{2}(x)$
GolecBiernat:1998js – which is an underlying mechanism behind geometrical
scaling. Throughout this paper we shall use the following form of the
saturation scale GolecBiernat:1998js :
$Q_{\text{s}}^{2}(x_{1,2})=Q_{0}^{2}(e^{\pm
y}p_{\text{T}}/(x_{0}\sqrt{s}))^{-\lambda}$ (5)
although other possibilities have been also discussed in the literature
Beuf:2008mf . Parameters $Q_{0}\sim 1$ GeV$/c$ and $x_{0}\sim 10^{-3}$ depend
on a particular model of gluon densities, whereas exponent $\lambda$ can be
extracted from the data in a model-independent way. Indeed, by analyzing
multiplicity spectra at CMS we have found that for pp collisions
$\lambda=0.27$ McLerran:2010ex . Note that for $\lambda=0$ saturation scale is
equal to $Q_{0}$ and scaling variable $\tau=p_{\text{T}}^{2}/Q_{0}^{2}$ is
essentially transverse momentum squared in units of $Q_{0}^{2}$.
There are many sources of possible violation of GS in multiplicity spectra
given by Eq.(1). First – as already discussed – one of the Bjorken $x$’s (3)
can be outside the GS domain. Second, gluons of very low $k_{\text{T}}$ will
feel nonperturbative effects of the order of $\Lambda_{\text{QCD}}$. Next,
running of the strong coupling $\alpha_{\text{s}}$ will evidently produce
logarithmic corrections to GS. Finally, gluon fragmentation into physical
particles will introduce dependence on their masses, which should induce GS
violation in spectra of identified particles.
In this paper we shall concentrate on the first two effects which can be
studied by looking at rapidity dependence of the particle multiplicities in pp
collisions. For particles produced with $y\neq 0$ Bjorken $x$’s (3) are
different and we have in fact two different saturation scales Kharzeev:2001gp
. Hence instead of $F(\tau)$ in Eq.(1) we will have some other function
$\tilde{G}(\tau_{1},\tau_{2})$. This by itself does not mean that GS is
necessarily violated since we can always rewrite
$\tilde{G}(\tau_{1},\tau_{2})=G(\tau_{1},\rho)$ with
$\rho=Q_{\text{s}1}/Q_{\text{s}2}$. For fixed rapidity $y$ ratio $\rho$ is
constant for $Q_{\text{s}}$ of the form given by Eq.(5) and universal
dependence of multiplicity distribution upon scaling variable $\tau_{1}$ is
retained. Note that $F(\tau)=G(\tau,1)$.
Figure 1: Kinematic range of NA61/SHINE experiment in the $(W,p_{\mathrm{T}})$
plane. Data points are shown as stars, squares, circles and triangles. Shaded
areas denote regions where GS is expected to be violated for $y=0.1$. For
other rapidities (marked at the r.h.s. of the plot) GS violation takes place
above the respective dashed line.
On the other hand by looking at the spectra with increasing $y$ one can
eventually reach $x_{1}>x_{\mathrm{max}}$ and GS violation should be seen.
Luckily we have at our disposal very recent pp data from the NA61/SHINE
experiment at CERN NA61 where particle spectra at different rapidities
$y=0.1\div 3.5$ and five different energies have been measured for
$p_{\text{T}}=0.025\div 1.375$ GeV$/c$. For the present analysis we shall use
negative pion spectra, which has an advantage that one avoids possible GS
violation due to particle masses. The only problem with this piece of data is
that the scattering energies are rather low:
$W_{1_{,}\ldots_{,}5}=17.28,\;12.36,\;8.77,\;7.75$, and $6.28$ GeV.
Nevertheless, as we will show below, Bjorken $x$’s involved are small enough
for GS to be present. This is mainly due to the before-mentioned fact that in
DIS $x_{\text{max}}\sim 0.1$ Praszalowicz:2012zh ; Caola:2010cy ; for the
purpose of the present analysis we shall take $x_{\text{max}}=0.09$. Then GS
should be seen if $x_{1}<x_{\text{max}}$, _i.e._ for
$p_{\text{T}}<p_{\text{Tmax}}(W,y)=x_{\text{max}}W\,e^{-y}.$ (6)
On the other hand transverse momenta of produced gluons should be larger than
some nonperturbative scale $\Lambda$:
$p_{\text{T}}>p_{\text{Tmin}}=\Lambda$ (7)
for which we take $\Lambda=250$ MeV footnote . Having fixed the kinematical
range where GS should be present, we can have a look at kinematical range of
NA61/SHINE spectra shown in Fig. 1. The shaded area below
$p_{\text{Tmin}}=\Lambda$ is excluded for all rapidities. The shaded region in
an upper part of the plot is excluded for $y=0.1$, and dashed lines represent
$p_{\text{Tmax}}(W,y)$ as functions of $W$ for different rapidities $y$.
Therefore for given rapidity only points below these lines should exhibit GS.
Experimental points for different energies are represented by stars (17.28
GeV), squares (12.36 GeV), circles (8.77 GeV), and triangles (7.75 and 6.28
GeV). One can see from Fig. 1 that for $y=0.1$ GS region extends towards the
smallest NA61/SHINE energy. This is due to the fact that $x_{\rm max}$ is as
large as 0.09. By increasing $y$ we see that some points fall outside the GS
window and finally for $y\gtrsim 1.7$ no GS should be present in NA61/SHINE
data. One may note that both $x_{\text{max}}$ and $\Lambda$ together with
exponent $\lambda$ could be determined in a self-consistent way from the
NA61/SHINE data. This is beyond the scope of the present note and will be
presented elsewhere. Therefore our analysis has to be considered mostly as a
qualitative one.
Figure 2: Ratios $R_{1k}$ as functions of $\sqrt{\tau}$ for the lowest
rapidity $y=0.1$: a) for $\lambda=0$ when $\sqrt{\tau}=p_{\mathrm{T}}$ and b)
for $\lambda=0.27$ which corresponds to GS.
Limits (6) and (7) define GS window $\tau_{\text{min}}<\tau<\tau_{\text{max}}$
in scaling variable $\tau$
$\displaystyle\tau_{\text{max}}$
$\displaystyle=e^{-2y}x_{\text{max}}^{2}W^{2}/Q_{0}^{2}\,(x_{\text{max}}/x_{0})^{\lambda},$
$\displaystyle\tau_{\text{min}}$ $\displaystyle=e^{\lambda
y}\Lambda^{2}/Q_{0}^{2}\,(\Lambda/(x_{0}W))^{\lambda}.$ (8)
We see from (8) that for fixed $W$, $\tau_{\text{max}}$ falls exponentially
with rapidity $y$ much faster than $\tau_{\text{min}}$ is rising with $y$.
When $\tau_{\text{min}}\simeq\tau_{\text{max}}$ window for GS closes, and we
get from (8) that the larger $W$ the larger $y$ when this happens. For fixed
$y$ the GS window is growing with $W$.
Now we come to the formulation of a model-independent criterion that allows
one to asses whether GS is present or not. To this end we follow Refs.
McLerran:2010ex ; Praszalowicz:2012zh and define ratios of multiplicities
$R_{ik}(\tau;y)=\frac{dN}{dyd^{2}p_{\text{T}}}(W_{i},\tau;y)/\frac{dN}{dyd^{2}p_{\text{T}}}(W_{k},\tau;y)$
(9)
which, according to (1), should be equal to unity if GS is present. Since GS
window is the largest for $W_{1}=17.28$ GeV, we form four ratios $R_{1k}$ with
$k=2,\ldots,5$ where index $k$ refers to NA61/SHINE energies mentioned above.
In Fig. 2 we plot ratios $R_{1k}$ as functions of $\sqrt{\tau}$ for the lowest
rapidity $y=0.1$ first for $\lambda=0$, _i.e._ essentially as functions of
$p_{\text{T}}$, and second for $\lambda=0.27$. We see that for $\lambda=0$ all
ratios $R_{1k}$ are larger than $1$ and rise with $\tau$ up to $R_{1k}\sim
1.5\div 6$. In contrast, for $\lambda=0.27$ for all ratios $R_{1k}$ there
exists a domain in $\sqrt{\tau}$ where, within the errors, they are equal to
1. We see that $\tau_{\text{min}}$ for $W_{2}=12.36$ GeV is approximately 0.6
while for $W_{5}=6.28$ GeV it is 0.8 or so. On the other hand for $W_{5}$ the
GS windows closes at $\sqrt{\tau}\simeq 1.5,$ whereas for $W_{2}$ it extends
further than 2.1. This is in qualitative agreement with limits (8), and as we
will see in the following, this trend continues once we increase $y$.
Looking at Fig. 1 we see that there is no qualitative difference between
rapidities $0.1$ and $0.3$, and this is what indeed can be seen in Fig. 3.a.
However, already for $y=0.7$ the lowest energy data are no longer in the
scaling region, and we see this clearly in Fig. 3.a. Further on, for rapidity
$y=0.9$ the second lowest energy data leave the GS window, and this is again
nicely substantiated by the behavior of $R_{14}$ shown in Fig. 3.c. Finally
for $y=1.3$ none of the tested energies $W_{2}\div W_{5}$ should exhibit GS,
and this is clearly confirmed by Fig. 3.d. Note that in all these cases the
reference points at scattering energy $W_{1}$ remain in the GS window,
therefore scaling violations of ratios $R_{1k}$ should be attributed to the
points at lower energies $W_{k}<W_{1}$.
It is somewhat surprising that relatively low energy NA61/SHINE data do show
GS at all. This is due to the fact that GS scaling works up to much higher
$x$’s than originally anticipated: $x_{\text{max}}\sim 0.1$. This has been
confirmed by direct analysis of DIS in Ref. Praszalowicz:2012zh and also
indirectly by the present study.
In this paper we have explored NA61/SHINE NA61 negative pion spectra in pp
collisions taken at different rapidities $y$. In order to study GS in
multiplicity distributions we have formed ratios (9) that should be equal to 1
for scaling variable $\tau$ within the GS window (8). We have argued that
behavior of ratios $R_{1k}$ with increasing $y$ is in qualitative agreement
with the following picture of particle production in hadronic collisions. For
low $p_{\text{T}}$’s GS is violated due to the existence of the
nonperturbative energy scale $\Lambda$. For high transverse momenta, the
larger of two Bjorken $x$’s, namely $x_{1}$, crosses the maximal $x$ for which
GS is present, $x_{1}>x_{\text{max}}$, and GS is again violated. For
rapidities larger than 1.3, the window of GS closes for all energies $W_{k}$
but $W_{1}$, and no GS is seen in ratios $R_{1k}$. This behavior of GS for
different energies with varying $y$ is strikingly in qualitative agreement
with the kinematical constraints shown in Fig.1 and more quantitatively with
the range of the GS window given by Eq.(8).
Figure 3: Ratios $R_{1k}$ as functions of $\sqrt{\tau}$ for $\lambda=0.27$ and
for different rapidities a) $y=0.3$, b) $y=0.7$, c) $y=0.9$ and d) $y=1.3$.
With increase of rapidity, gradual closure of the GS window can be seen.
Our aim here was to point out that model-independent analysis of the
multiplicity distributions at different rapidities can provide an interesting
insight into the production mechanism of low and medium $p_{\text{T}}$ pions
from the point of view of gluon saturation and geometrical scaling. One
obvious reservation is that for higher rapidities one can force ratios
$R_{1k}$ to approach unity at the expense of adjusting $\lambda$. However,
very soon $\lambda$ gets bigger than $1$ and for rapidities of the order of 2
it has to be bigger than $3.$ The quality of such ”geometrical scaling” is
very poor, and – more importantly – there is no physical picture supporting
such big values of $\lambda$. This is one of the reasons for which one should
perform a more quantitative analysis of GS in pp collisions, which will be
presented elsewhere.
Let us finish by a remark that experimental data at high LHC energies would be
best suited for an analysis outlined in this paper. For the LHC energies GS
closes for very high rapidities up to $y\sim 8$. Since one needs data at
different rapidities, perhaps the best detector for this purpose would be
LHC-b.
The author wants to thank Marek Gazdzicki and Szymon Pulawski for the access
to the NA61/SHINE data, and Andrzej Bialas and Marek Gazdzicki for careful
reading of the manuscript and remarks. This work was supported by the Polish
NCN Grant 2011/01/B/ST2/00492.
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|
arxiv-papers
| 2013-01-20T10:25:30 |
2024-09-04T02:49:40.471595
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Michal Praszalowicz",
"submitter": "Michal Praszalowicz",
"url": "https://arxiv.org/abs/1301.4647"
}
|
1301.4652
|
# Lepton flavor violating processes $\tau\rightarrow\mu\gamma$,
$\tau\rightarrow 3\mu$ and $Z\rightarrow\mu\tau$ in the supersymmetric
economical 3-3-1 model
L.T. Hue D.T. Huong H.N. Long Institute of Physics, Vietnam Academy of Science
and Technology, 10 Dao Tan, Ba Dinh, Hanoi, Vietnam
###### Abstract
In this work, we study the charged lepton flavor violating (cLFV) decays
$\tau\rightarrow\mu\gamma$, $\tau\rightarrow 3\mu$ and $Z\rightarrow\mu\tau$
in the framework of the supersymmetric economical 3-3-1 model. Analytic
formulas for branching ratios (BR) of these decays are presented. We assume
that there exist lepton flavor violation (LFV) sources in both right- and
left-handed slepton sectors. This leads to the strong enhancement of cLFV
decay rates. We also show that the effects of the LFV source to the cLFV decay
rates in the left-handed slepton sector are greater than those in the right-
handed slepton sector. By numerical investigation, we show that the model
under consideration contains the relative light mass spectrum of sleptons
which satisfies the current experimental bounds on LFV processes in the limit
of small $\tan\gamma$. The interplay between monopole and dipole operators
also was studied.
###### keywords:
Decays of taus, Decays of $Z$ bosons, Supersymmetric models
###### PACS:
13.35.Dx , 13.38.Dg , 12.60.Jv
††thanks: Email: [email protected]††thanks: Email:
[email protected]††thanks: Email: [email protected]
## 1 Introduction.
It is known that lepton flavor (LF) numbers are strictly conversed in Standard
Model (SM). However, the observation from of neutrino experiments [5] strongly
suggests that there is lepton flavor mixing in lepton sector. It means that
(cLFV) processes may also occur at some level. Many current experiments
especially pay attention to search for LFV processes in the charged lepton
sector such as tau decays [1, 2], $Z\rightarrow\mu\tau$ decays [3, 4], etc.
because LFV observed in experiment is evidence of new physics beyond the SM.
Along with experiments, many models beyond SM that contain LFV processes have
been constructed. One class of the simple extended SM models with LFV is the
class of models with non-zero neutrino masses. This kind of models contains a
new type of Yukawa couplings of right-handed neutrinos which are sources of
cLFV. But cLFV processes in these models have been proved to be very
suppressed [22, 23]. However, in supersymmetric models (SUSY), the situation
can be changed. Besides LFV origin affected by the new Yukawa couplings
involving right-handed neutrinos, SUSY models also contain other sources of
cLFV. Particularly, the large mixings of slepton mass parameters in the soft
term greatly enhance the rates for cLFV processes. Because of the appearance
of new cLFV sources in SUSY models, the cLFV decay rates can reach to
experimental bounds even if the mixing angles in the slepton sector are small.
The cLFV processes caused by this kinds of LFV were studied very early [7].
Other interesting properties supporting the study cLFV phenomenology in SUSY
models were discussed in many recent works, for example [28]. Many
investigations about the cLFV in SUSY models such as [10, 24, 9] indicated
that the values of BR$(\tau\rightarrow 3\mu)$ and
BR$(\tau\rightarrow\mu\gamma)$ may exceed the bounds of present experimental
results. So it is necessary to find regions of parameter space satisfying the
experimental results [2, 3, 4]:
$\mathrm{BR}(\tau^{-}\rightarrow\mu^{-}\gamma)<4.4\times 10^{-8},$ (1)
$\mathrm{BR}(\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-})<2.1\times 10^{-8},$ (2)
$\mathrm{BR}(Z\rightarrow\mu^{+}\tau^{-})<1.2\times 10^{-5}.$ (3)
Supersymmetric unified theories predict large rates for cLFV processes except
the decay rate of $Z\rightarrow\mu\tau$. The branching ratio of this decay in
models beyond SM is predicted to $10^{-9}-10^{-8}$ level [8]. It is very
suppressed compared to the experimental bound (3). In fact, the GigaZ option
will be approaching to the sensitivity of $10^{-8}$ level in the
$Z\rightarrow\mu\tau$ decay mode [6]. If observed in a future experiment, it
will be evidence of physics in SUSY models. One of the SUSY models in which
cLFV processes are thoroughly investigated is the Minimal Supersymmetric
Standard Model (MSSM) [10, 11]. In the mentioned work, the authors have shown
that if there exists LFV in the left-handed sector, in order to get the
experimental bound on the LFV decay rates of muon and tauon decays, the mass
parameters of sleptons should be shifted toward TeV scale, especially in the
case of large $\tan\beta$. Because the model contains many LFV effective
operators, the interplay of different effective operators (dipole, monopole
coming from neutral boson exchanges and Higgs exchanges) creates many
interesting consequences. The detail is discussed in works [10, 11].
The current experimental results [5] show that neutrinos are massive, which
contradict what assumed in the SM. Other words speaking, the SM must be
extended. Among extensions, the models based on the
$\mathrm{SU}(3)_{\mathrm{C}}\otimes\mathrm{SU}(3)_{\mathrm{L}}\otimes\mathrm{U}(1)_{\mathrm{X}}$
(3-3-1) gauge group [13, 14] have the following interesting features:
* •
To be anomaly free, the number of triplets should be equal to number of anti-
triplets. This leads to that number of generations is multiple of the color
number which is three. Combining with condition of the QCD asymptotic freedom
the requiring number of quark generations should be less than five. Thus, in
the 3-3-1 models, number of generations is three.
* •
The models give an explanation of electric charge quantization [15], dark
matter and CP violation.
* •
One of the three quark families has to transform under $\mathrm{SU}(3)_{L}$
differently from the other two. This leads to an explanation why the top quark
is uncharacteristically heavy.
The weakness of the 3-3-1 models is the complication in the Higgs sector: in
the minimal 3-3-1 model [13], content of Higgs sector includes three triplets
and one sextet, while in the 3-3-1 model with right-handed neutrino [14],
there are three triplets. To solve the mentioned weakness, the attempt is
reduction in the scalar sector. The first result is the economical 3-3-1 model
[16, 17] \- the version with right-handed neutrino and only two Higgs
triplets. Scalar sector in the minimal 3-3-1 model has been reduced to minimum
with two Higgs triplets, and such version is called the reduced minimal 3-3-1
model [18]. It is emphasized that the problem on neutrino masses is not
totally solved in the last version. The situation seems better in a
supersymmetric version of the reduced minimal 3-3-1 model [19]. An
supersymmetric version of the economical 3-3-1 model has been established in
[20] and called the Supersymmetric Economical 3-3-1 model (SUSYE331). The
SUSYE331 has the same advantages as its non-supersymmetric version. However,
to generate masses for fermions, the non-supersymmetric models with minimal
Higgs content need effective nonrenormalizable interactions, while the SUSY
versions do not (the interested readers can find in Refs. [20, 26]).
In [21], the LFV decays of neutral Higgs bosons in the SUSYE331 were
considered. In this work, we are interested in the cLFV processes in the
SUSYE331, namely: cLFV decays of the tauon and $Z$ bosons.
We remind that in the SUSYE331, there are more leptons, Higgses and gauge
bosons as well as their supersymmetric partners than those of the MSSM. This
implies that the model contains many sources of LFV in the slepton sector.
This suggests that the branching ratios of LFV Higgs and Z decays are greatly
enhanced and may be detectable in near future. In contrast, bounds from
experiments (such as shown in (1) and (2)) will strictly constrain the values
of parameters causing the cLFV. So it is really important to investigate the
parameter space where LVF decay rates can be satisfied the present bound of
experiments. In many previous works [10, 11], the authors showed that in the
MSSM the interested region of parameter space is set at TeV scale with the
large value of $\tan\beta$. Because of appearance of new LFV sources in the
SUSYE331, we can find the interested region of parameter space even in the
limit of small slepton mass parameters. In this work, we are interested in the
processes: $\tau^{-}\rightarrow\mu^{-}\gamma$,
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ and $Z\rightarrow\mu\tau$. In
particular, we incorporate the mixing of slepton mass, of charginos, Higgsino,
gauginos as well as the interactions between gauge boson, slepton, lepton and
the Yukawa interactions. This leads to many types of enhanced diagrams. We
will consider how large contribution from each type of diagrams can be. We
also will extend previous work in [10] to our considered model such as:
constructing the analytical formulas of effective operators from the diagrams,
from which we obtain the formulas of the branching ratios of decays
BR($\tau\rightarrow\mu\gamma$), BR($\tau\rightarrow\mu\mu\mu$) and
BR($Z\rightarrow\mu\tau$). After that, we investigate some numerical results
of these branching ratios in the limit of small $\tan\gamma$ and the slepton
parameters are set below $1$ TeV which may be detected by current colliders.
As in the previous works on SUSYE331, $\tan\gamma$ is defined as the ratio of
two vacuum expectations of two neutral Higgs components $\rho$ and
$\rho^{\prime}$, namely $\tan\gamma=\langle\rho^{0}\rangle/\langle\rho^{\prime
0}\rangle$. Here, the quantity $\tan\gamma$ in the SUSYE331 plays a similar
role as the $\tan\beta$ in the MSSM [21]. In the MSSM model, the mass of the
lightest Higgs depends on $\tan\beta$ and the mass of the standard gauge
bosons. If we combine the theoretical result for the upper bound on the
lightest Higgs-boson mass with the direct experimental search, it leads to
exclude the limit of small $\tan\beta$ for the case where soft parameters are
set to TeV scale. However, this kind of constraints on the $\tan\gamma$ does
not happen in the SUSYE331 model. On the other hand, the large values of
$\tan\gamma$ do not support the region of the soft parameter space below 1
TeV. Hence, in this work, we concentrate on numerical studying in the limit of
small $\tan\gamma$. In this paper we concentrate on only two aims. First, we
establish analytic formulas to calculating some cLFV processes in the
framework of SUSYE331. Second, we prove that there exist regions satisfying
the current bounds of cLFV experiments. Especially, on the basis of numerical
studying, we find some interested regions of parameter space that satisfy the
experimental bound on the cLFV decay rates. We also discuss on the interplay
between mono and dipole operators which may lead to some interesting
consequences.
This paper is arranged as follows: In section 2 we review the particles
content. The effective operators as well as branching ratios of the mentioned
cLFV decay processes are presented in section 3. Section 4 is devoted for
discussion on results of numerical investigation on parameter space and the
last section is the conclusion. All the analytic formulas of effective
couplings appearing in effective operators and interacting terms needed for
our calculation are shown in Appendices A-E.
## 2 Particles content
In this part, let us quickly review the particle content in the SUSYE331
model, which were given in the previous papers [20, 26, 21]. The fermion
superfields are given by
$\mathaccent 866{L}_{aL}=\left(\mathaccent 866{\nu}_{a},\mathaccent
866{l}_{a},\mathaccent 866{\nu}^{c}_{a}\right)^{T}_{L}\sim(1,3,-1/3),\hskip
14.22636pt\mathaccent 866{l}^{c}_{aL}\sim(1,1,1),\hskip 14.22636pta=1,2,3,$
(4) $\mathaccent 866{Q}_{1L}=\left(\mathaccent 866{u}_{1},\ \mathaccent
866{d}_{1},\ \mathaccent 866{u}^{\prime}\right)^{T}_{L}\sim(3,3,1/3),$
$\mathaccent 866{u}^{c}_{1L},\ \mathaccent 866{u}^{\prime
c}_{L}\sim(3^{*},1,-2/3),\mathaccent 866{d}^{c}_{1L}\sim(3^{*},1,1/3),$ (5)
$\begin{array}[]{ccc}\mathaccent 866{Q}_{\alpha L}=\left(\mathaccent
866{d}_{\alpha},-\mathaccent 866{u}_{\alpha},\mathaccent
866{d^{\prime}}_{\alpha}\right)^{T}_{L}\sim(3,3^{*},0),\hskip
14.22636pt\alpha=2,3,\end{array}$ (6) $\mathaccent 866{u}^{c}_{\alpha
L}\sim\left(3^{*},1,-2/3\right),\hskip 14.22636pt\mathaccent
866{d}^{c}_{\alpha L},\ \mathaccent 866{d}^{\prime c}_{\alpha
L}\sim\left(3^{*},1,1/3\right),$ (7)
and the Higgs superfields are written as
$\mathaccent 866{\chi}=\left(\mathaccent 866{\chi}^{0}_{1},\mathaccent
866{\chi}^{-},\mathaccent 866{\chi}^{0}_{2}\right)^{T}\sim(1,3,-1/3),\hskip
14.22636pt\mathaccent 866{\rho}=\left(\mathaccent
866{\rho}^{+}_{1},\mathaccent 866{\rho}^{0},\mathaccent
866{\rho}^{+}_{2}\right)^{T}\sim(1,3,2/3),$ (8) $\mathaccent
866{\chi}^{\prime}=\left(\mathaccent 866{\chi}^{\prime o}_{1},\mathaccent
866{\chi}^{\prime+},\mathaccent 866{\chi}^{\prime
o}_{2}\right)^{T}\sim(1,3^{*},1/3),\,\mathaccent
866{\rho}^{\prime}=\left(\mathaccent 866{\rho}^{\prime-}_{1},\mathaccent
866{\rho}^{\prime o},\mathaccent
866{\rho}^{\prime-}_{2}\right)^{T}\sim(1,3^{*},-2/3).$ (9)
It is noted that $\mathaccent 866{\psi}^{c}_{L}=(\mathaccent
866{\psi}_{R})^{c}\equiv\mathaccent 866{\psi}_{R}^{\dagger}$ and $u^{\prime}$,
$d^{\prime}$ are exotic quarks. The values in each parenthesis show the
quantum numbers of the $(SU(3)_{c},SU(3)_{L},U(1)_{X})$ group, respectively.
The $\mathrm{SU}(3)_{L}\otimes\mathrm{U}(1)_{X}$ gauge group is broken as
follows:
$\mathrm{SU}(3)_{L}\otimes\mathrm{U}(1)_{X}\stackrel{{\scriptstyle
w,w^{\prime}}}{{\longrightarrow}}\
\mathrm{SU}(2)_{L}\otimes\mathrm{U}(1)_{Y}\stackrel{{\scriptstyle
v,v^{\prime},u,u^{\prime}}}{{\longrightarrow}}\mathrm{U}(1)_{Q},$ (10)
with VEVs given by
$\sqrt{2}\langle\chi\rangle^{T}=\left(u,0,w\right),\hskip 14.22636pt\hskip
14.22636pt\sqrt{2}\langle\chi^{\prime}\rangle^{T}=\left(u^{\prime},0,w^{\prime}\right),$
(11) $\sqrt{2}\langle\rho\rangle^{T}=\left(0,v,0\right),\hskip
14.22636pt\hskip
14.22636pt\sqrt{2}\langle\rho^{\prime}\rangle^{T}=\left(0,v^{\prime},0\right).$
(12)
The SUSYE331 model contains seventeen vector superfields such as $\mathaccent
866{V}_{c}^{a}$, $\mathaccent 866{V^{a}}$ and $\mathaccent 866{V}^{\prime}$.
The vector superfields contain the usual gauge bosons given in [20, 26]. The
total supersymmetric Lagrangian was given in [26]. In this work only terms
relevant to our work are collected in appendix A.
## 3 Effective operators and branching ratios
In this section, we extend the previous work [10] in the model under
consideration. In the SUSYE331, there are six physical vector bosons: the
photon, two charged ($W^{\pm},~{}Y^{\pm}$), two Hermitian neutral
($Z,Z^{\prime}$) and a Non-hermitian neutral $X$. The effective couplings of
$X\mu\tau$ interaction is very small so we ignore them in the calculation. Let
us first write down the cLFV effective operators for muon-tau and photon or
$Z$ bosons, $Z^{\prime}$ bosons or other leptons.
### 3.1 $\tau\mu\gamma$ effective operators
First we write down the LFV operators for $\tau,\mu,\gamma$. These operators
are divided into two terms 111The operator containing $D^{\gamma}$ is
different from [10] a factor $i$. This is because of definition of
$\sigma^{\mu\nu}$ and $\bar{\sigma}^{\mu\nu}$. The definitions in [12] are
used.:
$e\left[C^{\gamma}_{L}\bar{\mu}\bar{\sigma}^{\mu}\tau+C^{\gamma}_{R}\mu^{c}\sigma^{\mu}\bar{\tau}^{c}+\mathrm{h.c.}\right]\mathchar
1027\relax A_{\mu},$ (13)
$e~{}m_{\tau}\left[D^{\gamma}_{L}\bar{\mu}\bar{\sigma}^{\mu\nu}\bar{\tau}^{c}+D^{\gamma}_{R}\mu^{c}\sigma^{\mu\nu}\tau+\mathrm{h.c.}\right]F_{\mu\nu}.$
(14)
The processes associated with external photon line depend only on the
$D^{\gamma}_{L,R}$ while other processes with virtual photon depend on both
$C^{\gamma}$ and $D^{\gamma}$. The Feynman diagrams contributing to
$C^{\gamma}$ and $D^{\gamma}$ are given in appendix B. We would like to
emphasize that the number of diagrams in the considered model is more than
that of the MSSM because the SUSYE331 model contains new Higgs and gauge
bosons. In order to obtain analytical formulas for $C^{\gamma}$ and
$D^{\gamma}$, we have used Feynman rules and some approximate expansion. For
more details, the interested reader can see in appendix B. As in the MSSM, the
$C^{\gamma}_{L(R)}$ does not depend on $\tan\gamma$ and the factor
$D^{\gamma}_{L(R)}$ can be divided into three sub-terms including sub-term
$D^{\gamma(a)}_{L(R)}$ which is independent on $\tan\gamma$ and sub-terms
$D^{\gamma(c,b)}_{L(R)}$ which are not, more specifically
$D^{\gamma}_{L(R)}=D^{\gamma(a)}_{L(R)}+D^{\gamma(b)}_{L(R)}+D^{\gamma(c)}_{L(R)}.$
(15)
We would like to mention that only $D^{\gamma(c)}_{L(R)}$ contains left-right
slepton mixing parameters which, in this paper, are denoted by
$(A_{\tau},A^{L}_{\mu\tau},A^{R}_{\mu\tau})$.
### 3.2 $Z\tau\mu$ and $Z^{\prime}\tau\mu$ effective operator
First, let us consider the effective operator of the $Z\tau\mu$. This kind of
effective operators can be written in the standard form given in [10] as
follows:
$g_{Z}~{}m^{2}_{Z}\left[A^{Z}_{L}\bar{\mu}\bar{\sigma}^{\mu}\tau+A^{Z}_{R}\bar{\mu}^{c}\sigma^{\mu}\bar{\tau}^{c}+\mathrm{h.c.}\right]Z_{\mu},$
(16)
$g_{Z}\left[C^{Z}_{L}\bar{\mu}\bar{\sigma}^{\mu}\tau+C^{Z}_{R}\mu^{c}\sigma^{\mu}\bar{\tau}^{c}+\mathrm{h.c.}\right]\mathchar
1027\relax Z_{\mu},$ (17)
$g_{Z}~{}m_{\tau}\left[D^{Z}_{L}\bar{\mu}\bar{\sigma}^{\mu\nu}\tau^{c}+D^{Z}_{R}\mu^{c}\sigma^{\mu\nu}\tau+\mathrm{h.c.}\right]Z_{\mu\nu}.$
(18)
where the mass of the $Z$ boson in the SUSYE331 is determined as [20]:
$\displaystyle m^{2}_{Z}$ $\displaystyle\simeq$
$\displaystyle\frac{g^{2}\left(v^{2}+v^{\prime 2}\right)}{4c^{2}_{W}},$
$\displaystyle\hskip 14.22636ptg_{Z}$ $\displaystyle\simeq$
$\displaystyle\frac{g}{c_{W}}~{}~{}~{}~{}\mathrm{and}~{}~{}~{}~{}g_{Z^{\prime}}\simeq\frac{g}{\kappa_{1}c_{W}}.$
(19)
($g_{Z}$ and $g_{Z^{\prime}}$ are defined from covariant derivatives, as
explained in Appendices A-E.) The operators related to the factors
$A^{Z}_{L,R}$ are chirality conserving (monopole). The leading contribution to
the monopole operators comes from effective couplings of muon and tauon with
two neutral Higgs bosons. In the model under consideration, we have four
neutral Higgs bosons $\rho^{0},~{}\rho^{\prime 0},~{}\chi^{0}_{1}$ and
$\chi^{\prime 0}_{1}$ which can couple to the $Z$ boson. However,
investigation in [17] has noted the relation $u,u^{\prime}\ll v,v^{\prime}\ll
w,w^{\prime}$. In this limit, we can neglect the coupling of the $Z$ with
$\chi$ and $\chi^{\prime}$. It means that we obtain only the leading
interactions of $\mu,\tau$ with $\rho,\rho^{\prime}$. These leading terms lead
to a consequence that the monopole operators in (16) can be extracted out a
factor $m^{2}_{Z}$. We would like to remind that the class of diagrams
containing $\mu,\tau$, $\rho^{0}$ and $\rho^{\prime 0}$ presented in Appendix
14 give contribution to $A^{Z}_{L,R}$ and this factor can be written as sum of
three parts:
$A^{Z}_{L(R)}=A^{Z(a)}_{L(R)}+A^{Z(b)}_{L(R)}+A^{Z(c)}_{L(R)},$ (20)
where analytical formulas of each term in (20) as well as $C^{Z}_{L(R)}$ and
$D^{Z}_{L(R)}$ are given in Appendix C. The operator related to $C^{Z}_{L(R)}$
also are chirality conserving (monopole) while the operators related to
$D^{Z}_{L(R)}$ are chirality flipping (dipole). More details about origin of
these operators, the interested reader can see in [10].
A big difference compared to the MSSM is that the model under consideration
contains more $\mu\tau$ effective operators such as $\mu\tau Z^{\prime}$.
Because of the couplings of $Z^{\prime}$ with all of neutral
Higgs–$\chi^{0}_{2},~{}\chi^{\prime 0}_{2},~{}\rho^{0}$ and $\rho^{\prime
0}$–the monopole operators of $Z^{\prime}\mu\tau$ relate not only to the
factor $m^{2}_{Z}$ in but also the factor $m^{2}_{Z^{\prime}}$, as indicated
in two formulas (21) and (22). It is noted that $A^{1Z^{\prime}(a)}_{L(R)}$
comes from the leading interactions of $\mu\tau$ with two neutral Higgs $\rho$
and $\rho^{\prime}$ while the $A^{1Z^{\prime}(a)}_{L(R)}$ comes from the
leading interactions of $\mu\tau$ with two neutral Higgs $\chi_{2}^{0}$ and
$\chi^{\prime 0}_{2}$. For all of others effective operators mentioned in this
work-$C^{Z,(Z^{\prime})}_{L,(R)}$ and $D^{Z,(Z^{\prime})}_{L,(R)}$-relate with
only two neutral Higgs $\rho$ and $\rho^{\prime}$. Above comments are enough
for us to write the standard form of $\mu\tau Z^{\prime}$ operators as
follows:
$g_{Z^{\prime}}~{}m^{2}_{Z}\left[A^{(1Z^{\prime})}_{L}\bar{\mu}\bar{\sigma}^{\mu}\tau+A^{(1Z^{\prime})}_{R}\bar{\mu}^{c}\sigma^{\mu}\bar{\tau}^{c}+\mathrm{h.c.}\right]Z^{\prime}_{\mu},$
(21)
$g_{Z^{\prime}}~{}m^{2}_{Z^{\prime}}\left[A^{(2Z^{\prime})}_{L}\bar{\mu}\bar{\sigma}^{\mu}\tau+A^{(2Z^{\prime})}_{R}\bar{\mu}^{c}\sigma^{\mu}\bar{\tau}^{c}+\mathrm{h.c.}\right]Z^{\prime}_{\mu},$
(22)
$g_{Z^{\prime}}\left[C^{Z^{\prime}}_{L}\bar{\mu}\bar{\sigma}^{\mu}\tau+C^{Z^{\prime}}_{R}\mu^{c}\sigma^{\mu}\bar{\tau}^{c}+\mathrm{h.c.}\right]\mathchar
1027\relax Z^{\prime}_{\mu},$ (23)
$g_{Z^{\prime}}~{}m_{\tau}\left[D^{Z^{\prime}}_{L}\bar{\mu}\bar{\sigma}^{\mu\nu}\tau^{c}+D^{Z^{\prime}}_{R}\mu^{c}\sigma^{\mu\nu}\tau+\mathrm{h.c.}\right]Z^{\prime}_{\mu\nu}.$
(24)
Note that in the on-shell condition we have $\mathchar
1027\relax(Z)\rightarrow-m^{2}_{Z}$ and $\mathchar
1027\relax(Z^{\prime})\rightarrow-m^{2}_{Z^{\prime}}$, where:
$m^{2}_{Z^{\prime}}=\frac{g^{2}c^{2}_{W}W^{2}}{4c^{2}_{W}-1},\hskip
14.22636ptW^{2}=w^{2}+w^{\prime 2}.$
The $A^{1Z^{\prime}}_{L(R)}$ and $A^{2Z^{\prime}}_{L(R)}$ are also written in
the form as those for $A^{Z}_{L(R)}$. Forms of the $D^{Z}_{L(R)}$ and
$D^{Z^{\prime}}_{L(R)}$ are the same of $D^{\gamma}_{L(R)}$ in (15). All
analytical formulas of effective operators in this section are given in
Appendix D.
In addition, the SUSYE331 model has two others neutral gauge bosons,
$Z^{\prime}$ and $X$ compared to the MSSM. This appearance may give
significant contribution to the $\tau\rightarrow 3\mu$ decay. The mentioned
contribution may be similar to that of the $Z$ boson in some region of
parameter space. This issue will be discussed in more details in the next
section.
### 3.3 $\tau\mu\mu\mu$ effective operator
Let us write down four-fermions $\tau\mu\mu\mu$ effective operators. The
$\tau\mu\mu\mu$ effective operators can be constructed from the effective
operators of $\mu,\tau$ with photon or $Z,Z^{\prime}$ bosons as well as Higgs
bosons. Besides the contributions coming from photon, $Z$ boson and Higgs
exchanges, there are other contributions to the $\tau\mu\mu\mu$ effective
operators which come from box-diagrams shown in Fig. (20). In this part, we
write down only the standard form of $\tau\mu\mu\mu$ effective operator coming
from box-diagrams as
$\displaystyle\left[(\bar{\mu}\bar{\sigma}^{\mu}\tau)\left(B_{L}^{\mu_{L}}\,\bar{\mu}\bar{\sigma_{\mu}}\mu+B_{L}^{\mu_{R}}\,\mu^{c}\sigma_{\mu}\bar{\mu}^{c}\right)\right.$
(25) $\displaystyle+$
$\displaystyle\left.(\mu^{c}\sigma^{\mu}\bar{\tau}^{c})\left(B_{R}^{\mu_{L}}\,\bar{\mu}\bar{\sigma_{\mu}}\mu+B_{R}^{\mu_{R}}\,\mu^{c}\sigma_{\mu}\bar{\mu}^{c}\right)+{\rm
h.c.}\right].$
Analytical forms of the coefficients $B_{L(R)}^{\mu_{L(R)}}$ are presented in
Appendix E.
### 3.4 Branching ratios
General method to construct the branching ratios for the cLFV decay from
effective operator was written in [10]. Especially, based on the basis of the
effective operators, we write effective Lagrangian of the muon and tauon with
photon, gauge bosons $Z,Z^{\prime}$ as well as lepton. From this effective
Lagrangian, we obtain the branching ratio for each of processes by using the
Feynman rules. In this section, we study the branching ratios for the
considered cLFV decays of the tau and the $Z$ bosons.
#### 3.4.1 $\tau\rightarrow\mu\gamma$
Let us consider the cLFV decay mode $\tau\rightarrow\mu\gamma$. It is not hard
to obtain the decay rate of $\tau\rightarrow\mu\gamma$ from effective
operators given in Eqs. (13,14). The detailed calculation can be seen in [24].
Comparing the branching of LFV decay $\tau\rightarrow\mu\gamma$ with that of
the $\tau\rightarrow\mu\bar{\nu}_{\mu}\nu_{\tau}$ decay given in [4, 12], we
can obtain the relation between the two branching ratios of the two above
processes. The result is entirely consistent with the results given in [10],
namely
$\displaystyle\mathrm{BR}(\tau\rightarrow\mu^{-}\gamma)=\frac{48\pi^{2}\alpha}{G^{2}_{F}}\left[|D^{\gamma}_{L}|^{2}+|D^{\gamma}_{R}|^{2}\right]\mathrm{BR}(\tau\rightarrow\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}),$
(26)
where $\alpha=\frac{e^{2}}{4\pi}$,
$\frac{G_{F}}{\sqrt{2}}=\frac{g^{2}}{8m^{2}_{W}}$ and
$\mathrm{Br}(\tau\rightarrow\mu^{-}\bar{\nu}_{\mu}\nu_{\tau})\simeq 17.41\%$.
#### 3.4.2 $Z\rightarrow\mu\tau$
In this subsection, we consider the decay mode of $Z\rightarrow\mu\tau$. The
decay rate of $Z\rightarrow\ell^{+}\ell^{-}$ in case of the SUSY can also be
determined from the general formula established in [12]. In particular, the
decay rate of $Z\rightarrow\mu\tau$ in the SUSYE331 model can be written as
follows:
$\displaystyle\mathrm{\Gamma}(Z\rightarrow\ell^{+}\ell^{-})=\frac{g^{2}_{Z}m_{Z}}{24\pi}\left(1-\frac{4m_{\ell}^{2}}{m_{Z}^{2}}\right)^{1/2}\left[(a_{\ell}^{2}+b_{\ell}^{2})\left(1-\frac{4m_{\ell}^{2}}{m_{Z}^{2}}\right)+6a_{\ell}b_{\ell}\frac{m_{\ell}^{2}}{m_{Z}^{2}}\right],$
(27)
where
$a_{\ell}\equiv T^{3}_{\ell}-Q_{\ell}s^{2}_{W}=\frac{-1+2s^{2}_{W}}{2}\hskip
14.22636pt\mathrm{and}\hskip 14.22636ptb_{\ell}=-Q_{\ell}s^{2}_{W}=s^{2}_{W}.$
For the decay rate of $Z\rightarrow\mu\tau$, because the form of effective
operators of $Z\mu\tau$ in the considered model is the same as that of the
MSSM, the form of decay rate of $Z\rightarrow\mu\tau$ is the same as that
established in [10]. Especially the result is given as follows:
$\displaystyle\mathrm{\Gamma}(Z\rightarrow\mu^{+}\tau^{-})=\frac{g^{2}_{Z}m^{5}_{Z}}{24\pi}\left[\left|F^{Z}_{L}\right|^{2}+\left|F^{Z}_{R}\right|^{2}+\frac{1}{2}\left|\frac{m_{\tau}}{m_{Z}}D^{Z}_{L}\right|^{2}+\frac{1}{2}\left|\frac{m_{\tau}}{m_{Z}}D^{Z}_{R}\right|^{2}\right],$
(28)
where
$F^{Z}_{L,R}=A^{Z}_{L,R}-C^{Z}_{L,R}.$
Comparing the two results taken from Eqs.(27)and (28), we obtain the
relationship between the two branching ratios corresponding to two processes
$Z\rightarrow ll$ and $Z\rightarrow\mu\tau$. Our calculation is consistent
with results given in [10] such as:
$\displaystyle\mathrm{BR}(Z\rightarrow\mu\tau)$
$\displaystyle=cm^{4}_{Z}\left[\left|F^{Z}_{L}\right|^{2}+\left|F^{Z}_{R}\right|^{2}+\frac{1}{2}\left|\frac{m_{\tau}}{m_{Z}}D^{Z}_{L}\right|^{2}+\frac{1}{2}\left|\frac{m_{\tau}}{m_{Z}}D^{Z}_{R}\right|^{2}\right]\mathrm{BR}(Z\rightarrow\ell^{+}\ell^{-}),$
(29)
where
$c\equiv\left(a^{2}_{\ell}+b^{2}_{\ell}\right)^{-1}=(1/4-s^{2}_{W}+2s^{4}_{W})^{-1}\simeq
7.9$ and $\mathrm{BR}(Z\rightarrow\ell^{+}\ell^{-})\simeq 3.4\%$.
#### 3.4.3 $Z^{\prime}\rightarrow\mu\tau$
Similar to the case of the $Z$ boson, the decay rate of
$Z^{\prime}\rightarrow\ell^{+}\ell^{-}$ can be determined in the formula
below:
$\displaystyle\mathrm{\Gamma}(Z^{\prime}\rightarrow\ell^{+}\ell^{-})$
$\displaystyle=$
$\displaystyle\frac{g^{2}_{Z^{\prime}}m_{Z^{\prime}}}{24\pi}\left(1-\frac{4m_{\ell}^{2}}{m_{Z^{\prime}}^{2}}\right)^{1/2}\left[(a_{\ell}^{\prime
2}+b_{\ell}^{\prime
2})\left(1-\frac{4m_{\ell}^{2}}{m_{Z^{\prime}}^{2}}\right)+6a^{\prime}_{\ell}b^{\prime}_{\ell}\frac{m_{\ell}^{2}}{m_{Z^{\prime}}^{2}}\right]$
(30) $\displaystyle\simeq$
$\displaystyle\frac{g^{2}_{Z^{\prime}}m_{Z^{\prime}}}{24\pi}(a^{2}_{\ell}+b^{2}_{\ell}),$
where $a_{\ell}^{\prime 2}+b_{\ell}^{\prime 2}=a_{\ell}^{2}+b_{\ell}^{2}=1/c$
and
$a^{\prime}_{\ell}=\left(4c^{2}_{W}-1\right)\left(T^{3}_{\ell}-Q_{\ell}\right)+3c^{2}_{W}X_{\ell}=-a_{\ell},$
$b^{\prime}_{\ell}=-\left(4c^{2}_{W}-1\right)Q_{\ell_{R}}+3c^{2}_{W}X_{\ell_{R}}=-b_{\ell}.$
The decay rate $Z^{\prime}\rightarrow\mu\tau$ is similar to that of
$Z\rightarrow\mu\tau$. It leads to the relation between two branchings as
follows:
$\displaystyle\mathrm{BR}(Z^{\prime}\rightarrow\mu\tau)$ $\displaystyle=$
$\displaystyle
c~{}m^{4}_{Z^{\prime}}\left[\left|F^{Z^{\prime}}_{L}\right|^{2}+\left|F^{Z^{\prime}}_{R}\right|^{2}+\frac{1}{2}\left|\frac{m_{\tau}}{m_{Z^{\prime}}}D^{Z^{\prime}}_{L}\right|^{2}+\frac{1}{2}\left|\frac{m_{\tau}}{m_{Z^{\prime}}}D^{Z^{\prime}}_{R}\right|^{2}\right]$
(31) $\displaystyle\times$
$\displaystyle\mathrm{BR}(Z^{\prime}\rightarrow\ell^{+}\ell^{-}),$
where
$F^{Z^{\prime}}_{\mathrm{L,R}}=\frac{m_{Z}^{2}}{m^{2}_{Z^{\prime}}}A^{Z^{\prime}(1)}_{\mathrm{L,R}}+A^{Z^{\prime}(2)}_{\mathrm{L,R}}-C^{Z^{\prime}}_{\mathrm{L,R}}.$
(32)
#### 3.4.4 $\tau\rightarrow\mu\mu\mu$
In the SUSYE331, the effective Lagrangian described the
$\tau\rightarrow\mu\mu\mu$ decay can be deduced from the effective operators
given in Eq.(25) combining with those induced by the effective operators of
$\mu\tau$ with $Z$ or $Z^{\prime}$ or photon as well as Higgs boson. The
general study was presented in [10]. The contributions from the box-diagrams
and vector boson exchanges to the branching of the considered decay rate are
sub-leading ones even when $\tan\gamma$ is large or small. However the
contribution from the Higgs-bosons exchange to decay rate is large for large
$\tan\gamma$ (the interested reader can see in [21]). Hence, in this work we
will split each type of contributions to the considered branching ratio.
First, let us consider the case of absence of Higgs exchange: the effective
Lagrangian can be deduced from the effective operators given in Eqs.(14),
(17), (18), (24) and (25). The explicit formula of effective Lagrangian is
$\displaystyle\mathcal{L}^{\mathrm{eff}}_{\tau\mu\mu\mu}$ $\displaystyle=$
$\displaystyle\left[\left(\bar{\mu}\bar{\sigma}^{\mu}\tau\right)\left(F^{\mu_{L}}_{L}\bar{\mu}\bar{\sigma}_{\mu}\mu+F^{\mu_{R}}_{L}\mu^{c}\sigma_{\mu}\bar{\mu^{c}}\right)\right.$
(33) $\displaystyle+$
$\displaystyle\left.\left(\mu^{c}\sigma^{\mu}\bar{\tau^{c}}\right)\left(F^{\mu_{L}}_{R}\bar{\mu}\bar{\sigma}_{\mu}\mu+F^{\mu_{R}}_{R}\mu^{c}\sigma_{\mu}\bar{\mu^{c}}\right)\right]$
$\displaystyle+$ $\displaystyle
2e^{2}\left(D^{\gamma}_{L}\bar{\mu}\bar{\sigma}^{\mu\nu}\bar{\tau^{c}}+D^{\gamma}_{R}\mu^{c}\sigma^{\mu\nu}\tau\right)\frac{m_{\tau}\partial_{\nu}}{\mathchar
1027\relax}$ $\displaystyle\times$
$\displaystyle\left(\bar{\mu}\bar{\sigma}_{\mu}\mu+\mu^{c}\sigma_{\mu}\bar{\mu^{c}}\right)+\mathrm{h.c.},$
where
$A^{Z^{\prime}}_{L(R)}=\frac{m_{Z}^{2}}{m^{2}_{Z^{\prime}}}A^{Z^{\prime}(1)}_{\mathrm{L,R}}+A^{Z^{\prime}(2)}_{\mathrm{L,R}},$
(34)
$F^{\mu_{L}}_{L(R)}=B^{\mu_{L}}_{L(R)}+\frac{1}{2}g_{Z}^{2}c_{2W}A^{Z}_{L(R)}-\frac{1}{2}g_{Z^{\prime}}^{2}c_{2W}A^{Z^{\prime}}_{L(R)}-e^{2}C^{\gamma}_{L(R)},$
(35)
$F^{\mu_{R}}_{L(R)}=B^{\mu_{R}}_{L(R)}+g_{Z}^{2}s^{2}_{W}A^{Z}_{L(R)}-g_{Z^{\prime}}^{2}s^{2}_{W}A^{Z^{\prime}}_{L(R)}-e^{2}C^{\gamma}_{L(R)}.$
(36)
Here we also assume that, as in the case of MSSM, we ignore the contributions
of $C^{Z}_{L,R},~{}C^{Z^{\prime}}_{L,R},~{}D^{Z}_{L,R}$ and
$D^{Z^{\prime}}_{L,R}$ to the above effective Lagrangian. This leads to the
branching ratios of decay $\tau\rightarrow\mu\mu\mu$ is
$\displaystyle\mathrm{BR}(\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-})$
$\displaystyle=$
$\displaystyle\frac{1}{8G_{F}^{2}}\left[2\left|F^{\mu_{L}}_{L}\right|^{2}+\left|F^{\mu_{R}}_{L}\right|^{2}+\left|F^{\mu_{L}}_{R}\right|^{2}+2\left|F^{\mu_{R}}_{R}\right|^{2}\right.$
(37) $\displaystyle+$
$\displaystyle\left.4e^{2}\mathrm{Re}\left(D^{\gamma}_{L}\left(2\bar{F}^{\mu_{L}}_{L}+\bar{F}^{\mu_{R}}_{L}\right)+D^{\gamma}_{R}\left(\bar{F}^{\mu_{L}}_{R}+2\bar{F}^{\mu_{R}}_{R}\right)\right)\right.$
$\displaystyle+$
$\displaystyle\left.8e^{2}\left(\left|D^{\gamma}_{L}\right|^{2}+\left|D^{\gamma}_{R}\right|^{2}\right)\left(\log\frac{m_{\tau}^{2}}{m_{\mu}^{2}}-\frac{11}{4}\right)\right]$
$\displaystyle\times$
$\displaystyle\mathrm{BR}(\tau^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}).$
### 3.5 $H\mu\tau$ contribution to $\tau\rightarrow\mu\mu\mu$
Contribution of Higgs exchange in the SUSYE331 model was investigated in [21],
where the corresponding effective Lagrangian for this, is given by:
$\displaystyle\mathcal{L}^{eff}_{\tau\mu\mu\mu}$ $\displaystyle=$
$\displaystyle-2\sqrt{2}G_{F}m_{\mu}m_{\tau}\tan\gamma\mathcal{C}(\mu^{c}\mu+\bar{\mu}\bar{\mu}^{c})$
(38) $\displaystyle\times$
$\displaystyle(\Delta^{\rho}_{L}\mu\tau^{c}+\Delta^{\rho}_{R}\mu^{c}\tau)+\mathrm{h.c.},$
where
$\displaystyle\mathcal{C}$ $\displaystyle\equiv$ $\displaystyle
t_{\gamma}\left(\frac{s^{2}_{\alpha}}{m^{2}_{\phi_{Sa36}}}+\frac{c^{2}_{\alpha}}{m^{2}_{\varphi_{Sa36}}}\right).$
(39)
Noting that factor $(m_{\mu}\tan\gamma)$ cannot be ignored if value of
$\tan\gamma$ is large enough. This factor causes a shift to $F^{\mu_{R}}_{L}$
and $F^{\mu_{L}}_{R}$, explicitly
$\delta
F^{\mu_{R}}_{L}=\sqrt{2}G_{F}m_{\mu}m_{\tau}\mathcal{C}\Delta_{R},\hskip
14.22636pt\delta
F^{\mu_{L}}_{R}=\sqrt{2}G_{F}m_{\mu}m_{\tau}\mathcal{C}\Delta_{L}.$ (40)
The individual contribution from Higgs exchange to
$\mathrm{BR}(\tau\rightarrow\mu^{-}\mu^{+}\mu^{-})$ now is:
$\mathrm{BR}(\tau\rightarrow\mu^{-}\mu^{+}\mu^{-})_{\Phi^{*}}=\frac{\left(m_{\mu}m_{\tau}\mathcal{C}\right)^{2}\left(\left|\Delta^{\rho}_{L}\right|^{2}\right)+\left|\Delta^{\rho}_{R}\right|^{2}}{32}\mathrm{BR}(\tau\rightarrow\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}).$
(41)
This contribution will be suppressed in the case of small $\tan\gamma$. We
will concentrate on this case in the numerical calculation in the following
section.
## 4 Numerical results
In this section, let us numerically study the cLFV decays of the tau lepton
$\tau\rightarrow\mu\gamma,~{}\tau\rightarrow\mu\mu\mu$ and
$Z\rightarrow\mu\tau$. For this purpose, we first study some constraints on
values of parameter space in the SUSYE331 model.
### 4.1 Implication on the parameter space in SUSYE331 model
In this section, we pay attention to discussing on constraint of parameter
space caused by experimental cLFV bounds (1)-(3). As mentioned, this topic has
been carefully studied in many models beyond SM. Especially ref.[10] not only
indicated regions of parameter space satisfying experimental bounds but also
discussed in details the correlation between dipole and non-dipole kinds of
contributions to cLFV decays in each region. The most important assumption
here is that sources of cLFV come from only the mass terms of sleptons in the
soft term, namely only left- and right-handed slepton mass matrices have large
$\mu-\tau$ entries. Let us compare the sources of cLFV appearing in the MSSM
with those of the SUSYE331. Because of the absence of the right-handed
neutrinos, the MSSM contains only three sources of cLFV, particularly LFV in
left-handed charged sleptons, left-handed sneutrinos and right-handed charged
leptons. But the left-handed sleptons and their sneutrinos live in the same
doublet of $\mathrm{SU(2)_{L}}$ in the MSSM, the origin of cLFV in the left-
handed charged sleptons and left-handed sneutrinos sectors are the same. As a
consequence, in the MSSM there are only two independent sources of cLFV. Due
to the appearance of the right-handed neutrinos in the SUSYE331, there also
appears one more source of LFV. Furthermore, there exist two Higgs multiplets
in the model, i.e., triplet $\rho$ and antitriplet $\rho^{\prime}$ which
independently generate masses of charged and neutral sleptons at tree level.
These two Higgs multiplets also create at least two new corresponding mass
terms of sleptons in the soft term, as shown in details in [20, 26, 21]. In
general, there are at least four independent sources causing the cLFV in the
SUSYE331 where each source is parameterized by a mixing angle defined in the
last subsection of Appendix A. This parameterization creates the similarity
between the SUSYE331 and the MSSM. We exploit this advantage to make some
prediction for the SUSYE331 basing previous investigation of the MSSM. For
example, if all of these sources are appeared, the branching ratio of decay
$\tau\rightarrow\mu\gamma$ will be much enhanced than that in the MSSM, even
it could greatly exceed the bound of experiment given in (1). As considered in
the MSSM, the existence of maximal LFV mixing in the left-handed slepton
sector means the LFV mixing in both charged and neutral sleptons, and their
contributions to cLFV decays are much larger than contributions caused by
right-handed LFV mixing. In addition, if the superparticle spectrum is
relative light, namely the parameter space are set below 1 TeV, there will
exist very small regions of parameter space satisfying experimental results.
The similar situation also happens for the SUSYE331. However, in the SUSYE331
four cLFV sources are independent then there exists a situation that the model
contains only one left-handed maximal LFV source while others vanish. It is
easy to realize that in this case, the values of branching ratios of cLFV
processes in the SUSYE331 are smaller than those of the MSSM and the satisfied
regions of parameter space will be wider in the scale of $\mathcal{O}(100)$
GeV. On the other hand, if four LFV sources are presented, the predicted
results of the considered model consistent with the experimental results at
the TeV scale. In the next subsections, we will examine the influence of LFV
sources appearing in the soft term of the SUSYE331 on the cLFV decays of the
tau and the $Z$ boson. In the numerical investigation, we just pay attention
to the case of soft parameters at $\mathcal{O}(100)$ GeV scale because this
scale allows the existence of small values of slepton masses which can be
detected by present colliders. In fact, the detailed investigation to
determine different properties of cLFV branching ratios among different
regions of parameter space is really needed, as done in many known SUSY
models. In the SUSYE331, this work is more complicated because of the addition
of many new particles so we will come back this interesting topic in another
work.
We would like to emphasize that the sleptons gain mass through main sources
which come from soft terms and interacting terms of sleptons with Higgs
through $F$-and $D$-terms. As mentioned in [27], the $D$-term gives
contribution to mass of slepton that contribution is proportional to the
quantity $(w^{2}-w^{\prime 2})$ while $F$-term, gives contribution to mass of
slepton that is proportional to $(\lambda_{a}w)$ or $(\lambda_{a}w^{\prime})$.
Because the VEVs $\omega$ and $\omega^{\prime}$ break down
$\mathrm{SU(3)}_{\mathrm{L}}$ to $\mathrm{SU(2)}_{\mathrm{L}}$ so these values
can be set to the TeV scale. We would like to note that the SUSYE331 contains
the Tachyon fields, the removal Tachyon fields leads to a condition [26]:
$|w^{2}-w^{\prime 2}|\simeq|v^{2}-v^{\prime 2}|\leq
246^{2}~{}(\mathrm{GeV}^{2}).$ (42)
However, in the last work [26], we have ignored the $B$-type terms, namely
$B_{\rho}\rho\rho^{\prime},B_{\chi}\chi\chi^{\prime}$. If the $B$-type terms
are included into the Higgs potential, not only the tachyon fields are removed
without any conditions but also the stable vacuums are guaranteed. From now,
we will consider the SUSYE331 which include the above B-terms. It is noted
that the Higgs mass spectrum is different from that presented in [20, 26]. For
example, there are three neutral massive pseudo-scalar, five neutral massive
and four charged massive Higgs. It is interesting that masses of light Higgs,
in general, increase by the presence of B-type terms. Especially, the charged
Higgs mass of $m^{2}_{W}$ as shown in [26] now changes into the new value of
$(m^{2}_{W}+2B_{\rho}/\sin 2\gamma)$. So the $B$-terms will make masses of
Higgs satisfied with current limits of electroweak precision tests such as LEP
limit of charged Higgs boson [30], even without loop corrections. For more
detail, the interested reader can see in Refs. [29, 31]. In the SUSYE331,
where the $B$-type terms are included, the $D$-term generates mass for slepton
at the $\mathrm{SU(3)_{L}}$ scale by sub-terms which are always proportional
to $\left(w^{2}(t_{\beta}^{2}-1)/t^{2}_{\beta}\right)$ [27]. So this
contribution should be at the $\mathcal{O}$(100) GeV if slepton masses are in
range of $\mathcal{O}$(100) GeV. This is similar to the condition (42).
Furthermore, if the R-parity is imposed, the coupling $\lambda_{a}$ must
vanish. It means that the contributions from both $F$\- and $D$-terms to
masses of sleptons depend on the $\mathrm{SU(3)_{L}}$ broken scale and the
value of $t_{\beta}$. So if both slepton masses are in the range of
$\mathcal{O}(1)$ TeV scale and $t_{\beta}$ is close to unity, the dominant
contribution to the slepton masses comes from the soft term.
Next, let us discuss on $\mu_{\rho}$ and $\mu_{\chi}$ which play the same role
as $\mu$ parameter in the MSSM. According to [20, 26], including $B$-type
terms, the requirement of canceling all linear terms at tree level in the
Higgs potential leads to conditions:
$\displaystyle\mu_{\chi}^{2}+4m^{2}_{\chi}-4\frac{B_{\chi}}{t_{\beta}}$
$\displaystyle=$
$\displaystyle\frac{g^{2}}{1-4c_{2W}}\left[2c_{2W}\left(\frac{t^{2}_{\beta}-1}{t_{\beta}^{2}}\right)\left(w^{2}+u^{2}\right)+\left(\frac{t_{\gamma}^{2}-1}{t_{\gamma}^{2}}\right)v^{2}\right],$
$\displaystyle\mu_{\rho}^{2}+4m^{2}_{\rho}-4\frac{B\rho}{t_{\beta}}$
$\displaystyle=$
$\displaystyle\frac{g^{2}}{1-4c_{2W}}\left[\left(\frac{t^{2}_{\beta}-1}{t_{\beta}^{2}}\right)\left(w^{2}+u^{2}\right)-2\left(\frac{t_{\gamma}^{2}-1}{t_{\gamma}^{2}}\right)v^{2}\right],$
$\displaystyle m_{\chi}^{2}+m_{\chi}^{\prime 2}+\frac{\mu^{2}_{\chi}}{2}$
$\displaystyle=$ $\displaystyle B_{\chi}\frac{t_{\beta}^{2}+1}{t_{\beta}},$
$\displaystyle m_{\rho}^{2}+m_{\rho}^{\prime 2}+\frac{\mu^{2}_{\rho}}{2}$
$\displaystyle=$ $\displaystyle B_{\rho}\frac{t_{\gamma}^{2}+1}{t_{\gamma}}.$
(43)
The parameters $\mu_{\chi}^{2},~{}\mu_{\rho}^{2},~{}m^{2}_{\chi}$ and
$m^{2}_{\rho}$ as well as $m_{\rho^{\prime}}^{2},m_{\chi^{\prime}}^{2}$ are
positive. The additional $B$-terms depend on the phases of fields. We can
redefine the phases of $\chi,\chi^{\prime},\rho$ and $\rho^{\prime}$ by such
way that can absorb any phase in the $B$-terms, so we can take
$B_{\chi},B_{\rho}$ to be real and positive. The conditions given in Eqs. (43)
lead to the consequences as follows: The scale of all parameters given in the
left-handed side of Eqs. (43) are the same order. If the value of $\tan\beta$
is closed to the unit, the parameters
$\mu_{\chi}^{2},~{}\mu_{\rho}^{2},~{}m^{2}_{\chi},~{}m^{2}_{\rho},$
$~{}m_{\rho^{\prime}}^{2},~{}m_{\chi^{\prime}}^{2}$ and $B_{\chi},B_{\rho}$
are set to the scale of electroweak symmetry breaking else they are set to the
scale of $\mathrm{SU(3)_{L}}$ symmetry breaking. Furthermore the Higgs mass
spectrum studied in [31] also leads to the conclusions on the limit of
$\tan\gamma$. Specially if the soft given in Higgs potential are considered at
the scale of $\mathrm{SU(3)_{L}}$ symmetry breaking the tree level mass of the
SM Higgs boson is $m_{Z}|\cos 2\gamma|<92.0$ GeV. This result is similar to
that given in the MSSM. In this case, the boundary of the SM mass can be
pushed up to 130 GeV by one-loop correction with large $\tan\gamma$. In
another case if the soft $\mu/B$-terms are considered at the scale of
electroweak symmetry breaking, there is no constraint on the value of
$\tan\gamma$.
It is known that three branching ratios $\tau\rightarrow\mu\gamma$,
$\tau\rightarrow\mu\mu\mu$ and $Z\rightarrow\mu\tau$ depend on some of
following quantities: $A^{Z(Z^{\prime})}_{L(R)}$, $B^{\mu_{L(R)}}_{L(R)}$,
$C^{\gamma(Z,Z^{\prime})}_{L(R)}$ and $D^{\gamma(Z,Z^{\prime})}_{L,R}$. In
different regions of parameter space, where some of these quantities give
dominant contribution, there are precise correlations among branching ratios.
In particular, with three considering branching ratios, we have two cases as
listed below:
$D^{\gamma}$-dominance: It is easy to get the relation concerned in [10]
$\frac{\mathrm{BR}(\tau\rightarrow\mu\mu\mu)}{\mathrm{BR}(\tau\rightarrow\mu\gamma)}\simeq
2.2\times 10^{-3},$ (44)
and a consequence is
$\mathrm{BR}(\tau\rightarrow\mu\mu\mu)<10^{-10}.$ (45)
$A^{Z}$-dominance: In this case, we get
$\mathrm{BR}(\tau\rightarrow\mu\mu\mu)=\left(\begin{array}[]{c}0.53\\\ 0.67\\\
\end{array}\right)\mathrm{BR}(Z\rightarrow\mu\tau).$ (46)
Note that in the SUSYE331 model, although $A^{Z^{\prime}}_{L(R)}$ receives
contribution from diagrams which are similar to those of $A^{Z}_{L(R)}$,
$A^{Z^{\prime}}_{R}\simeq 0$. In the limit
$t_{\beta}=\omega/\omega^{\prime}\simeq 1~{}(c_{2\beta}=0)$ and
$A^{Z^{\prime}}_{L}$ depends on only diagrams containing right-handed
sneutrinos.
In next subsection, we consider numerical computation for cLFV decays of the
tauon and the $Z$ boson. Remember that the effective couplings used for
numerical studying are established in appendices B, C, D and E. The standard
loop integrals are given in [10, 25]. In the following investigation, we use
most of the notations defined in [21]. For example, mass parameters of
gauginos and Higgsinos are listed in the formula (B.2): $m_{\lambda}$ denotes
mass of $\mathrm{SU(3)}_{L}$ gaugino, $\mu_{\rho}$ and $\mu_{\chi}$ are
$\mu$-terms of Higgsinos. Only notation for mass of U(1) gaugino used in this
work is $m_{B}$ instead of $m^{\prime}$.
### 4.2 In the case of small $\tan\gamma$ and light mass spectrum
#### 4.2.1 $\tau\rightarrow\mu\gamma$
Let us consider the numerical studying of branching of
$\tau\rightarrow\mu\gamma$ decay. The analytical result given in Eq.(26)
depends on the effective coupling $D_{L(R)}^{\gamma}$ which can be divided
into three parts,
$D^{\gamma}_{L(R)}=D^{\gamma(a)}_{L(R)}+D^{\gamma(b)}_{L(R)}+D^{\gamma(c)}_{L(R)}$.
The analytical formulas of $D^{\gamma(a)}_{L(R)}$, $D^{\gamma(b)}_{L(R)}$ and
$D^{\gamma(c)}_{L(R)}$ are given in Appendix B in which only
$D^{\gamma(a)}_{L(R)}$ does not depend on $\tan\gamma$. In addition, from the
experimental bound (1), we can obtain the constraint on the effective
couplings, namely $|D^{\gamma}_{L,R}|\leq 2.5\times
10^{-9}~{}[\mathrm{GeV}^{-2}]$.
We would like to remind that the diagrams which contribute to the
$D_{L}^{\gamma}$ are collected from three LFV sources: left-handed charged
slepton, left-handed sneutrinos and right-handed sneutrinos sectors, while the
diagrams contributing to $D_{R}^{\gamma}$ are only collected from the charged
right-handed slepton sector. So the values of $D_{L}^{\gamma}$ are predicted
much larger than those of $D_{R}^{\gamma}.$ Another thing we want to emphasize
that since the SUSYE331 has many additional particles, we expected that the
additional particles can modify the predicted results of cLFV decay in the
model under consideration. For more details, let us consider numerical
studying in this decay mode.
First we study the effects of LFV sources on $D_{L}^{\gamma}$ in the case of
small $\tan\gamma$ as well as the presence of all of the three left-handed LFV
sources, especially we fix $\tan\gamma=3$ and
$\theta_{L}=\theta_{\tilde{\nu}_{L}}=\theta_{\tilde{\nu}_{R}}=\pi/4$. As in
the MSSM, $D^{\gamma(b)}_{L}$ is dominant contribution to $D^{\gamma}_{L}$.
Fig.1 shows the dependence of $D^{\gamma(b)}_{L}$ on soft parameters
$m_{\tilde{L}_{3}}=m_{\tilde{\nu}_{L_{3}}}=m_{\tilde{\nu}_{R_{3}}}$ and
$m_{\lambda}$, while other parameters are fixed. The predicted results given
in left panel of Fig.1 are fully consistent with the experimental results if
the domain of parameter $m_{L3}$ is close to the value of $m_{\tilde{L}2}$ and
all soft parameters are set at $\mathcal{O}(100)$ GeV. We also remind that
mixing mass term between left and right slepton, $m_{\tilde{\psi}_{\mu\tau}}$,
is small if the model under consideration has maximal mixing sources. However
if $\mu_{\rho}$ is set at TeV, the domain of parameter $m_{\tilde{L}2}$ ,
where the values of $D_{L}^{\gamma(b)}$ match experimental bound, is more
extensive. For more details, the reader can see in the right panel of Fig.1.
---
Figure 1: $D^{\gamma(b)}_{L}$ isocontours with $\tan\gamma=3$,
$m_{\tilde{L}_{3}}=m_{\tilde{\nu}_{L_{3}}}=m_{\tilde{\nu}_{R_{3}}}$ and
$m_{\tilde{L}_{2}}=m_{\tilde{\nu}_{L_{2}}}=m_{\tilde{\nu}_{R_{2}}}=300$ GeV,
$\theta_{L}=\theta_{\tilde{\nu}_{L}}=\theta_{\tilde{\nu}_{R}}=\pi/4$ and
$\mu_{\rho}=140$ GeV (1TeV) in the left (right) panel. The solid and dashed
lines correspond to $m_{B}=300$ GeV and $m_{B}=-300$ GeV.
Let us consider the case $m_{\tilde{L}3}=1$ TeV. The results given in Fig. 2
predict that the values of $D^{\gamma(b)}_{L}$ exceed the experimental bound
if the remaining parameters are assumed in region of $\mathcal{O}(100)$ GeV.
Even if the values of $\mu_{\rho}$ are assumed as large as 1 TeV, the
predicted value of $D^{\gamma(b)}_{L}$ is larger than the experimental bound.
In this region of parameter space, the predicted results in the SUSYE331 are
much similar to those in the MSSM [10]. As mentioned in [10], the large values
of $A_{\tau}$ as well as small values of $\mu_{\rho}$ and masses of sleptons
must be required to keep value of the total $|D^{\gamma}|$ below the
experimental bound. However, the Fig.2 displays that the predicted values of
$D_{L}^{\gamma b}$ are much greater than those in the MSSM. It means that in
order to obtain the values of $D^{\gamma(b)}_{L}$ being consistent with
experimental bound, the values of $A_{\tau}$ predicted in the SUSYE331 should
be much larger than those in the MSSM. In the limit of large values of
$A_{\tau}$, the model leads to the appearance of Tachyon sleptons. It means
that the model under consideration does not contain the region of parameter
space such that there exists a large difference between the values of slepton
masses. The Fig.2 also shows that the values of $D_{L}^{\gamma(b)}$ exceed to
the experimental results when parameter $m_{\tilde{L}}$ expands into range of
TeV.
|
---|---
Figure 2: $D^{\gamma(b)}_{L}$ isocontours with $\tan\gamma=3$,
$m_{\tilde{L}_{2}}=m_{\tilde{\nu}_{L_{2}}}=m_{\tilde{\nu}_{R_{2}}}$ and
$m_{\tilde{L}_{2}}=m_{\tilde{\nu}_{L_{2}}}=m_{\tilde{\nu}_{R_{2}}}=1$ TeV,
$\theta_{L}=\theta_{\tilde{\nu}_{L}}=\theta_{\tilde{\nu}_{R}}=\pi/4$ and
$\mu_{\rho}=140$ GeV (1TeV) in the left (right) panel. The solid and dashed
lines correspond to $m_{B}=300$ GeV and $m_{B}=-300$ GeV.
So in the SUSYE331 model with the case of the maximal mixing happening in all
three sources (left-handed slepton, left-handed sneutrino and right-handed
sneutrino sectors), the scale of all slepton masses should be the same order,
in range of TeV or in range of (100) GeV.
Now we consider another situation that happens only in the SUSYE331, not in
the MSSM. It is the case of only one left-handed LFV source appearing in the
model. Looking at analytical formulas of effective couplings, three sources
contributing to left-handed effective coupling are parameterized by three
mixing angels $\theta_{L}$, $\theta_{\tilde{\nu}_{L}}$ and
$\theta_{\tilde{\nu}_{R}}$. Two of them, $\theta_{\tilde{\nu}_{L}}$ and
$\theta_{\tilde{\nu}_{R}}$, relate with diagrams containing sneutrino
propagators while the remain relates to diagrams with charged slepton
propagators. Numerical computation indicates that if the mass spectrum of
superpartner particles is in the range of $\mathcal{O}(100)$ GeV, contribution
from sneutrino exchanges are larger than those of charged slepton ones. So if
two sneutrino mixing angles vanish, there is only one source of left-handed
mixing $\theta_{L}$ which generates relatively small effective couplings. The
experimental bounds then are easily satisfied even in regions of light mass
spectrum. Our numerical investigation will focus on this case. In particular,
mixing angle parameters are fixed as
$\theta_{R}=\theta_{\tilde{\nu}_{L}}=\theta_{\tilde{\nu}_{R}}=0$ and
$\theta_{L}=\pi/4$. The Fig.3 displays $D_{L}^{\gamma b}$ as function of the
$A_{\tau}$ and $\mu_{\rho}$ while others are fixed: $m_{\tilde{L}2}=1$ TeV,
$m_{\tilde{R}_{2}}=m_{\tilde{R}_{3}}=m_{\tilde{\nu}_{L_{2}}}=m_{\tilde{\nu}_{L_{3}}}=m_{\tilde{\nu}_{R_{2}}}=m_{\tilde{\nu}_{R_{3}}}\equiv
m_{\tilde{R}}$. The results given in the Fig.3 illustrate that in the
considered limit, we can find the region of the small absolute values of
$A_{\tau}$ in which we can obtain the values of $|D^{\gamma}_{L}|$ satisfying
the experimental bound, particularly:
\- The value of $m_{B}$ should be smaller than that of $m_{\lambda}$.
\- If the value of $m_{B}$ is closer to that of $m_{\lambda}$, the parameter
space of $A_{\tau}$ and, $\mu_{\rho}$ satisfying the experimental bound of
$|D^{\gamma}_{L}|$ has been expanded.
The predicted results given in Fig.3 show that if in the SUSYE331, only one
source of lepton number violation in the charged slepton sector, the model can
contain the region of parameter space in which the all soft parameters are set
to $\mathcal{O}(100)$ GeV except $m_{\tilde{L}_{2}}$ is set to TeV. In this
region of parameter space, the predicted results on the
$\tau\rightarrow\mu\gamma$ are matched the experimental bound on the decay.
The existence of the soft parameter space below TeV scale leads to hope that
the sleptons can be detected by LHC.
Figure 3: $D^{\gamma}_{L}$ isocontours with $\tan\gamma=3$,
$m_{\tilde{L}_{2}}=1$ TeV,
$\theta_{L}=\pi/4,~{}\theta_{R}=\theta_{\tilde{\nu}_{L}}=\theta_{\tilde{\nu}_{R}}=0$,
$A^{L}_{\tau\mu}=0$ (only LFV in $\\{\tilde{m}_{L},\tilde{\tau}_{L}\\}$
sector). For illustrations, we choose three choices of parameter space
$(m_{B},~{}m_{\lambda},~{}m_{\tilde{L}_{3}},~{}m_{\tilde{R}})$ in GeV: (200,
300, 300, 200) (solid), ( 100, 400, 100, 200) (dashed), ( 100, 500, 300, 100)
(dotted). For each example, the center line corresponds to the value of
$D^{\gamma}_{L}=0$, two other ones limit the region where
$|D^{\gamma}_{L}|\leq 2.5\times 10^{-9}~{}[\mathrm{GeV}^{-2}]$.
Finally, we consider the LFV effect in the right-handed charged slepton sector
to the LFV in the tauon decay. We assume that there is only the maximal mixing
of right-handed slepton, i.e., $s_{R}=c_{R}=1/\sqrt{2}$ and all other mixing
sources are set to be equal to zero. Under this assumption the Feynman
diagrams in the model under consideration contributing to $D^{\gamma}$ are
exactly the same as those in the MSSM [10]. In the Fig.4, absolute values of
$D^{\gamma(a)}_{R}$ and $D^{\gamma(b)}_{R}$ are rather small, even a bit
smaller than those in the MSSM.
|
---|---
Figure 4: Isocontours of $D^{\gamma(a)}_{R}$ (left panel) and isocontours of
$D^{\gamma(b)}_{R}$ (right panel). The parameters are $\tan\gamma=3$
, $m_{\tilde{R}_{2}}=1$ TeV,
$\theta_{L}=\theta_{\tilde{\nu}_{L}}=\theta_{\tilde{\nu}_{R}}=0$,
$\theta_{R}=\pi/4$ and $\mu_{\rho}=150$ GeV .
From this investigation, we see that $A_{\tau}$ should have the same order
with the mass parameters of sleptons.
#### 4.2.2 Correlations
Next, we consider the process $\tau\rightarrow\mu\mu\mu$. The amplitude
consists of the contributions from the effective couplings
$A_{L(R)},B_{L(R)},C_{L(R)}$ and $D_{L(R)}$ which are denoted the coupling of
$Z,~{}Z^{\prime}$ gauge bosons, photon and Higgs to charged leptons
$\mu_{L(R)},\tau_{L(R)}$. Each type of diagrams picking up the each particle
exchange contributing to the LFV in the considered process depends on the
region of parameter space. In next work, we will study the contribution of
each effective coupling to the LFV of the $\tau\rightarrow\mu\mu\mu$ process
in two typical cases.
Only maximal mixing in the charged slepton $(\tilde{\mu},\tilde{\tau})$
Let us consider the process $\tau\rightarrow\mu\mu\mu$ in the limit of small
$\tan\gamma$ and in the case of existing only maximal mixing in the charged
slepton, $s_{L}=c_{L}=\frac{1}{\sqrt{2}}$ and
$s_{R}=s_{\tilde{\nu}_{L}}=s_{\tilde{\nu}_{R}}=0$. This constraint leads to
$A^{2Z^{\prime}}_{L}=A^{2Z^{\prime}}_{R}=0$. As mentioned in the last part, it
is interesting to consider the region of parameter at $\mathcal{O}(100)$[GeV].
One can check that $A^{Z}_{L(R)}$ is dominated by $A^{Z(a)}_{L(R)}$ and total
effective couplings contributing to the branching ratio of the considered
process are $D^{\gamma}_{L(R)}$, $C^{\gamma}_{L(R)}$, $A^{Z(a)}_{L}$ and
$B^{\mu_{L(R)}}_{L}$. Indeed, looking at the analytical expression of these
couplings we found that only the analytical expression of $D^{\gamma}_{L}$ is
affected by left-handed slepton parameters such as:
$A_{\tau},~{}A^{L}_{\mu\tau}$. Hence, in order to look for the effective
coupling giving dominant contribution to the considered branching ratio, we
will study numerically two cases:
\- The $A_{\tau}$, $A_{\mu\tau}$ parameters are fixed and the other soft
parameters are changed.
\- The soft parameters are fixed and the $A_{\tau}$, $A_{\mu\tau}$ parameters
are changed.
First, we assume that $A_{\tau}=A_{\mu\tau}=0$. To assess the contribution of
the each effective coupling $A^{Z}_{L}$ and $D^{\gamma}_{L}$ into the BR
$(\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-})$, we define two factors
$f_{A^{Z}}$ and $f_{D^{\gamma}}$ such as:
$f_{A^{Z}}\equiv\frac{g_{Z}^{4}\left[\left|A^{Z}_{L}\right|^{2}\left(\frac{1}{2}c^{2}_{2W}+s^{4}_{W}\right)+\left|A^{Z}_{R}\right|^{2}\left(\frac{1}{4}c^{2}_{2W}+2s^{4}_{W}\right)\right]}{2\left|F^{\mu_{L}}_{L}\right|^{2}+\left|F^{\mu_{R}}_{L}\right|^{2}+\left|F^{\mu_{L}}_{R}\right|^{2}+2\left|F^{\mu_{R}}_{R}\right|^{2}}$
(47)
and
$f_{D^{\gamma}}\equiv\frac{8e^{4}\left(\left|D^{\gamma}_{L}\right|^{2}+\left|D^{\gamma}_{R}\right|^{2}\right)\left[\ln\left(\frac{m^{2}_{\tau}}{m^{2}_{\mu}}\right)-\frac{11}{4}\right]}{MS}$
(48)
where
$\displaystyle MS$ $\displaystyle=$ $\displaystyle
8e^{4}\left(\left|D^{\gamma}_{L}\right|^{2}+\left|D^{\gamma}_{R}\right|^{2}\right)\left[\ln\left(\frac{m^{2}_{\tau}}{m^{2}_{\mu}}\right)-\frac{11}{4}\right]$
(49) $\displaystyle\hskip
14.22636pt+2\left|F^{\mu_{L}}_{L}\right|^{2}+\left|F^{\mu_{R}}_{L}\right|^{2}+\left|F^{\mu_{L}}_{R}\right|^{2}+2\left|F^{\mu_{R}}_{R}\right|^{2}$
The factors $f_{A^{Z}}$ and $f_{D^{\gamma}}$ given in Eqs. (47), (48)
quantitatively measure contributions of $A^{Z}_{L(R)}$ and $D^{\gamma}_{L(R)}$
to the factor $\left|F^{\mu_{L(R)}}\right|$ and the total branching of
$(\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-})$ decay, respectively.
|
---|---
Figure 5: Correlations among $A^{Z}_{L}$, $F^{\mu_{L(R)}}_{L}$ and
$D^{\gamma}_{L}$ with $A_{\tau}=0$. The contours
$f_{A^{Z}_{L}},~{}f_{D^{\gamma}_{L}}$ and $BR(\tau\rightarrow\mu\gamma)$ are
denoted by dashed, dotted and solid black lines. For illustrations, numerical
values for parameter space
$(m_{B},~{}m_{\lambda},m_{\tilde{L}_{2}},~{}m_{\tilde{R}})$ are chosen as
$(100,300,1000,100)$[GeV] (left panel) and $(100,500,1000,100)$ [GeV] (right
panel).
Looking at the Eqs.(35) and (36), the coefficient $F^{\mu_{L(R)}}_{L(R)}$
depends on the other factors. If $A^{Z}$ gives dominant contribution to
$F^{\mu_{L(R)}}_{L(R)}$, it is convenient to define regions where the factor
$f_{A^{Z}}$ satisfies $1.05\geq f_{A^{Z}}\geq 0.95$ as the $A^{Z}$-domination
regions (over $F^{\mu_{L(R)}}_{L(R)}$). On the other hand, we also make
convention that if $f_{D^{\gamma}}\leq 0.05$ then the dominant contribution to
the branching ratio of $\tau\rightarrow 3\mu$ is given by $F$-type of
couplings and if $1.05\geq f_{D^{\gamma}}\geq 0.95$ then the dominant
contribution to the branching ratio of $\tau\rightarrow 3\mu$ is given by
$D$-type of couplings.
The current experimental upper bound on the branching ratio of the process
$BR(\tau\rightarrow\mu\gamma)$ is smaller than $4.4\times 10^{-8}$. The
results of our calculation to this process is given in Fig. 5. The
experimental bound of the branching is denoted by the solid black lines. One
can see that this process satisfies the experimental bound in regions of large
parameter space of $\mu$ and $m_{\tilde{L_{3}}}$.
The regions of parameter space where $A^{Z}_{L}$ gives dominant contribution
to branching ratio $\tau\rightarrow 3\mu$ must satisfy both conditions:
$1.05\geq|f_{A^{Z}}|\geq 0.95$ and $|f_{D^{\gamma}}|\leq 0.05$. The results
given in Fig.5 show that there is no region of $\mu_{\rho}$ and
$m_{\tilde{L}_{2}}$ parameter space that make $A^{Z}_{L}$ given dominant
contribution to branching ratio $\tau\rightarrow 3\mu$. If $D^{\gamma}$ gives
dominant contribution to the considered decay mode, the region of the
$\mu_{\rho}$ parameter satisfied the condition $1.05\geq|f_{D^{\gamma}}|\geq
0.95$, strongly depends on the value of charged gaugino mass, i.e., if the
charged gaugino mass is larger, then the value of $\mu_{\rho}$ parameter is
larger too.
Let us consider the effects of $A_{\tau}$ on the branching of the considered
decay mode. We would like to emphasize that $A^{Z}$ does not depend on the
$A_{\tau}$ while the value of $|D^{\gamma}_{L}|$ depends on the sign as well
as amplitude of $A_{\tau}$. In particularly $D^{\gamma(c)}_{L}$ is
proportional to $(A_{\tau}+\frac{1}{2}\mu_{\rho}\tan\gamma)$, the values of
$A_{\tau}$ will affect on soft parameter space region where $D^{\gamma}_{L}$
gives dominant contribution to the BR$(\tau\rightarrow 3\mu)$. These regions
of soft parameter can be larger than that in the case of $A_{\tau}=0.$
|
---|---
Figure 6: Branching ratios $Z\rightarrow\mu\tau$ (left panel) and
$\tau\rightarrow 3\mu$ (right panel) as functions of $m_{B}$. Three numerical
values for parameter space
$(m_{\lambda},\mu_{\rho},m_{\tilde{L}_{2}},m_{\tilde{L}_{3}},m_{\tilde{R}})$
[GeV] are chosen: (300, 150, 1000, 100, 100)-black line, (400, 200, 1000, 100,
100)-green line, (500, 150, 1000, 100, 100)-blue
line.
Now let us consider $Z\rightarrow\mu\tau$ and $\tau\rightarrow\mu\mu\mu$
decays. The branching ratios of decays $Z\rightarrow\mu\tau$ and
$\tau^{-}\rightarrow\mu\mu\mu$ are presented in (28) and (46). The predicted
branching is shown in Fig. 6. The numerical branching ratio for
$Z\rightarrow\mu\tau$ has the maximum of $5\times 10^{-10}$ if the soft
parameters of sleptons are set to $\mathcal{O}(100)$ GeV and the charged
gaugino mass is set to few hundreds GeV. This predicted result is very
suppressed with the present experimental bound. However, in the same region of
soft parameter space, the predicted result for $\tau\rightarrow\mu\mu\mu$ can
reach to the experimental result and even one can find some regions of
parameter space that predicted our result exceed the experimental bound. One
can check in the left panel of the Fig. 6, the branching values for
$Z\rightarrow\mu\tau$ hardly change when we change the value of the parameter
$m_{B}$. However, the situation is quite different if the charged gaugino mass
is varied, namely the smaller value of charged gaugino mass is, the larger
value of that branching is predicted. Moreover, as we increase the values of
the soft slepton mass parameters these branching values decreased. Therefore,
to increase the value of the branching of $Z\rightarrow\mu\tau$, we have to
change the parameter $\mu_{\rho}$.
Figure 7: Contour plots for branching ratios
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ (dotted), $Z\rightarrow\mu\tau$
(dashed) and $\tau\rightarrow\mu\gamma$ (solid) with $A_{\tau}=0$ and
$(m_{\lambda},~{}m_{\tilde{L}_{2}},~{}m_{\tilde{L}_{3}},~{}m_{\tilde{R}})=(400,150,1000,100,200)$.
The Fig. 7 shows the values of branching ratios of the decays in the plane
$m_{B}-\mu_{\rho}$. In this case, we choose $A_{\tau}=0$ and other parameters
are chosen so that the experimental branching decay of
BR$(\tau\rightarrow\mu\mu\mu)$ is satisfied. We can see that the bounded
regions of BR$(\tau\rightarrow\mu\gamma)$ supports the small values of both
remain decays. In the case $A_{\tau}\neq 0$, because only
BR$(\tau\rightarrow\mu\gamma)$ depends on the values of $A_{\tau}$ and as we
have shown in the above section, there will exist a possibility where
$D^{\gamma}_{L}$ vanishes. This case allows
BR$(\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-})$ can reach the limits of
experiment of order $\mathcal{O}(10^{-8})$ whilst BR$(Z\rightarrow\mu\tau)$ is
still in maximal order of $\mathcal{O}(10^{-9})$.
Only maximal mixing in the right-handed charged slepton sector
$\tilde{\mu},~{}\tilde{\tau}$.
Now we come to consider another case, only maximal LFV in right-handed sector
of charged sleptons
($s_{R}=1/\sqrt{2},~{}s_{L}=s_{\tilde{\nu}_{L}}=s_{\tilde{\nu}_{R}}=0$) where
regions of parameter space can be available in range of
$\mathcal{O}(100)$[GeV]. Both the SUSYE331 and the MSSM models are similar to
each other in this case. So we just discuss more on correlation among
effective couplings. As shown in the left panel of Fig. 8, the bound of
experiment of BR$(\tau\rightarrow\mu\gamma)$ rules out the large values of
BR$(\tau\rightarrow 3\mu)$. This leads to the result BR$(\tau\rightarrow
3\mu)\leq 10^{-9}$ if $A_{\tau}=0$. The right panel shows that in the case of
only large LFV in right-handed charged slepton sector, the $BR(\tau\rightarrow
3\mu)$ is in maximal order of $10^{-9}$, even in the case of non-vanishing
$A_{\tau}$ and $A_{\tau}$ makes $D^{\gamma}_{R}$ suppressed. For the
BR$(Z\rightarrow\mu\tau)$, this case is much smaller than the previous case.
Also we can see a difference from the case of pure large LFV in left-handed
charged slepton sector: the $D^{\gamma}$-domination regions now lie on the
small values of $\mu_{\rho}$ while the large values of $\mu_{\rho}$ are ruled
out by the condition BR$(\tau\rightarrow\mu\gamma)\leq 4.4\times 10^{-8}$.
|
---|---
Figure 8: Contours in $\mu_{\rho}-m_{\tilde{R_{3}}}$ plane (left panel) and
plots of branching ratio of $\tau\rightarrow 3\mu$ (right panel) in the case
of $\tan\gamma=3$ and $A_{\tau}=0$. The respective contours are
BR$(\tau\rightarrow\mu\gamma)$ (solid lines), $f_{D^{\gamma}}$ (dotted lines)
and BR$(\tau\rightarrow 3\mu)$ (dashed lines) with numerical values of
parameters $(m_{B},m_{\tilde{L}},m_{\tilde{R}_{2}})=(100,100,1000)$
($m_{\tilde{L}_{2}}=m_{\tilde{L}_{3}}\equiv m_{\tilde{L}}$). For the plot of
BR$(\tau\rightarrow 3\mu)$ four choices of parameter space
$(m_{B},~{}\mu_{\rho},m_{\tilde{R}_{2}},m_{\tilde{R}_{3}})$ are:
$(100,100,1000,100)$ (black), $(200,100,1000,100)$ (green),
$(100,200,1000,100)$ (blue) and $(100,300,1000,100)$ (red).
## 5 Conclusions
In present paper, we have studied the LFV decays of the tauon and the $Z$
boson in framework of the SUSYE331 model, and have mainly focused on two-
generations slepton mixing, namely both left- and right-handed
$\tilde{\mu}-\tilde{\tau}$ slepton mixings. In order to obtain the relevant
diagrams, we have combined the mixing of sleptons, that of charginos,
Higgsinos, gauginos as well as the interactions of gauge bosons with leptons
and the Yukawa interactions. From these diagrams, we obtained the effective
operators relating to the considered cLFV decays. This leads to the analytical
expression of the branching ratio of the considered decay processes and the
contacted relation between decay rate of non-LFV decay mode with that of cLFV
decay mode. Our analysis is carried out in the limit of small $\tan\gamma$.
The detailed predictions in our model depend strongly on the SUSY parameters
and left- and right-handed slepton mixing. Consequently, we have firstly
considered the effects of SUSY parameters and the mixing of left- and right-
handed sleptons on the $\tau\rightarrow\mu\gamma$ decay such as
* •
In the case of the maximal LFV mixing, the mixing mass terms between left- and
right-handed sleptons $(m_{\tilde{L}_{\mu\tau}},~{}m_{\tilde{R}_{\mu\tau}})$
are small, our results are only consistent with the experimental bounds if the
domains of parameter $m_{\tilde{L}_{3}}$ are close to those of
$m_{\tilde{L}_{2}}$ whenever they are set to the TeV or $\mathcal{O}(100)$ GeV
scales. It means that in the case of maximal LFV mixing, the slepton mass
parameters are in the same order.
* •
If there is only the LFV in the charged left-handed slepton sector, we can
find some regions of parameter space that allow above cLFV branching ratios
matching with the experimental bounds. Especially the slepton mass
$m_{\tilde{L}_{3}}$ is set at 1 TeV while the other, $m_{\tilde{L}_{2}}$, is
set at $\mathcal{O}(100)$ GeV. Noting that the value of parameter $m_{B}$
should be close to that of $m_{\lambda}$ and if the value of $m_{B}$ is closer
to that of $m_{\lambda}$, the parameter space of $A_{\tau}$ has been more
expanded. In the case of left-handed LFV sector, one important result deduced
from our numerical investigation is that although the SUSYE331 model contains
much more supersymmetric particles as well as LFV sources than the MSSM, there
still exist of some wide regions of parameter space which allow not only
masses of sleptons but also $\mu_{\rho}$ keep the small values enough to be
detected by colliders.
* •
If there is only the LFV in the charged right-handed slepton sector, in order
to match the experimental bound, the value of $A_{\tau}$ should be the same
order as that of other soft parameters. Our result is similar to the predicted
result in the MSSM.
Based on the parameter space satisfying the experimental bound on
$\tau\rightarrow\mu\gamma$ decay rate, we consider the branching ratios of
$\tau\rightarrow\mu\mu\mu$ and $Z\rightarrow\mu\tau$. We have concentrated on
the LFV in the charged left-handed slepton sector with only
$\theta_{L}=\pi/4$. In this case, there is no region of parameter space that
$A^{Z}$ gives dominant contribution to the considered decay mode, while there
is region of parameter space that $D^{\gamma}$ gives dominant contribution to
the considered decay mode. The constraint on the $\mu$ parameter is expanded
toward large values if the large value of the charged gaugino mass is chosen.
If we set the value of $A_{\tau}=0$, the constraint on the $\mu$ parameter can
be expanded. Similarly, by numerical study on the branching ratio of
$\tau\rightarrow\mu\mu\mu$ decay mode in the case there exists LFV only in the
charged right-handed slepton sector. The small value of $\mu_{\rho}$ giving
the dominant contribution to considered decay mode coming from $D^{\gamma}$
was obtained. In both cases, our predicted results of
BR$(Z\rightarrow\mu\gamma)$ are very suppressed.
## Acknowledgments
L.T. Hue would like to thank Dr. A. Brignole for the explanation of the
establishing analytical formulas in [10]. This research is funded by Vietnam
National Foundation for Science and Technology Development (NAFOSTED) under
grant number 103.01-2011.63.
## Appendix A Interacting Lagrangian and notations
Some of the interacting vertices used in this work were given in [21]. In this
appendix we list the rest part of interacting Lagrangian which is necessary
for completion our calculation. Here we use the relation
$\tilde{\ell}_{R}\equiv(\tilde{\ell}^{c}_{L})^{*}$ where
$\tilde{\ell}^{c}_{L}$ is the superpartner of a lepton $\ell^{c}_{L}$ in the
model.
### A.1 Interaction between left-right slepton sector with neutral Higgs
bosson
These terms come from two sources:
1. 1.
from $F$-terms:
$-\frac{1}{2}\left[\mu_{\rho}\rho^{0}\left(Y_{\mu}\tilde{\mu}^{*}_{L}\tilde{\mu}^{c*}_{L}+Y_{\tau}\tilde{\tau}^{*}_{L}\tilde{\tau}^{c*}_{L}\right)+\mathrm{h.c.}\right],$
2. 2.
from soft-breaking term:
$\displaystyle\mathcal{L}^{\mathrm{soft}}_{\tilde{l}\tilde{l}H^{0}}$
$\displaystyle=$
$\displaystyle-h^{\prime}_{ab}\tilde{L}_{aL}\rho^{\prime}\tilde{l}^{c}_{bL}+\mathrm{h.c.}$
(50) $\displaystyle=$
$\displaystyle-h^{\prime}_{ab}\left(\tilde{\nu}_{aL}\rho^{\prime-}_{1}+\tilde{l}_{aL}\rho^{\prime
0}+\tilde{\nu}^{c}_{aL}\rho^{\prime-}_{2}\right)\tilde{l}^{c}_{bL}+\mathrm{h.c.}$
$\displaystyle\rightarrow$ $\displaystyle-Y_{\mu}A_{\mu}\rho^{\prime
0}\tilde{\mu}_{L}\tilde{\mu}^{c}_{L}-Y_{\tau}A_{\tau}\rho^{\prime
0}\tilde{\tau}_{L}\tilde{\tau}^{c}_{L}$ $\displaystyle-
Y_{\tau}A^{L}_{\mu\tau}\rho^{\prime
0}\tilde{\mu}_{L}\tilde{\tau}^{c}_{L}-Y_{\tau}A^{R}_{\mu\tau}\rho^{\prime
0}\tilde{\tau}_{L}\tilde{\mu}^{c}_{L}+\mathrm{h.c.},$
where we use new notations that are identified with those in [10]:
$h^{\prime}_{22}=h^{\prime}_{\mu\mu}\equiv Y_{\mu}A_{\mu},\hskip
14.22636pth^{\prime}_{33}=h^{\prime}_{\tau\tau}\equiv Y_{\tau}A_{\tau},$
$h^{\prime}_{23}=h^{\prime}_{\mu\tau}\equiv Y_{\tau}A^{L}_{\mu\tau},\hskip
14.22636pth^{\prime}_{32}=h^{\prime}_{\tau\mu}\equiv Y_{\tau}A^{R}_{\mu\tau}.$
The total interaction part of ($H^{0}\tilde{l}^{c}_{L}\tilde{l}_{L}$)
interactions is:
$\displaystyle\mathcal{L}^{LR}_{\tilde{l}\tilde{l}H^{0}}$ $\displaystyle=$
$\displaystyle-Y_{\mu}\left(\frac{1}{2}\mu_{\rho}\rho^{0*}+A_{\mu}\rho^{\prime
0}\right)\tilde{\mu}_{L}\tilde{\mu}^{c}_{L}-Y_{\tau}\left(\frac{1}{2}\mu_{\rho}\rho^{0*}+A_{\tau}\rho^{\prime
0}\right)\tilde{\tau}_{L}\tilde{\tau}^{c}_{L}$ (51) $\displaystyle-
Y_{\tau}A^{L}_{\mu\tau}\rho^{\prime
0}\tilde{\mu}_{L}\tilde{\tau}^{c}_{L}-Y_{\tau}A^{R}_{\mu\tau}\rho^{\prime
0}\tilde{\tau}_{L}\tilde{\mu}^{c}_{L}+\mathrm{h.c}.$
### A.2 Gauge boson interactions
This kind of vertex is only contained in gauge invariant kinetics of all
fields in the theory. In this work, we just study on cLFV in lepton sector so
the related part of the Lagrangian is [20]:
$\displaystyle\mathcal{L}_{\mathrm{kinetic}}$ $\displaystyle=$
$\displaystyle(D^{\mu}\rho)^{\dagger}D_{\mu}\rho+(\bar{D}^{\mu}\rho^{\prime})^{\dagger}\bar{D}_{\mu}\rho^{\prime}+i\bar{\tilde{\rho}}\bar{\sigma}^{\mu}D_{\mu}\tilde{\rho}+i\bar{\tilde{\rho}}^{\prime}\bar{\sigma}^{\mu}\bar{D}_{\mu}\tilde{\rho}^{\prime}$
(52) $\displaystyle+$
$\displaystyle(D^{\mu}\tilde{L}_{iL})^{\dagger}D_{\mu}\tilde{L}_{iL}+i\bar{L}_{iL}\bar{\sigma}^{\mu}D_{\mu}L_{iL}$
$\displaystyle+$
$\displaystyle(D_{1\mu}l^{c}_{iL})^{\dagger}D_{1\mu}l^{c}_{iL}+i\bar{l}^{c}_{iL}\bar{\sigma}^{\mu}D_{1\mu}l^{c}_{iL}$
$\displaystyle-$
$\displaystyle\frac{1}{4}F_{a}^{\mu\nu}F_{a,\mu\nu}-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+i\bar{\lambda}^{a}_{V}\bar{\sigma}^{\mu}D^{L}_{\mu}\lambda^{a}_{V}+i\bar{\lambda}_{B}\bar{\sigma}^{\mu}\partial_{\mu}\lambda_{B}$
where $i=1,2,3$ is family index, $a=1,2,...,8$ corresponds to eight gauge
bosons of SU$(3)_{L}$ group. Covariant derivatives
$D_{\mu},~{}\bar{D}_{\mu},D_{1\mu}$ and $D^{L}_{\mu}$ correspond to triplets,
anti-triplets $\mathrm{SU(3)}_{L}$, singlet $\mathrm{SU(3)}_{L}$ and adjoint
presentation of $\mathrm{SU(3)}_{L}$. They are defined as follows:
$\displaystyle F_{a\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}V_{a\nu}-\partial_{\nu}V_{a\mu}-gf^{abc}V_{b\mu}V_{c\nu},\hskip
14.22636ptF_{\mu\nu}=\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu},$
$\displaystyle D_{\mu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}+igT^{a}V_{a\mu}+ig^{\prime}XT^{9}B_{\mu},$
$\displaystyle\bar{D}_{\mu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}-igT^{a*}V_{a\mu}+ig^{\prime}XT^{9}B_{\mu},$
$\displaystyle D_{1\mu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}+ig^{\prime}XT^{9}B_{\mu},$ $\displaystyle
D^{L}_{\mu}\lambda^{a}_{V}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\lambda^{a}_{V}-gf^{abc}V^{b}\lambda^{c}_{V}.$
(53)
Here $X$ denotes $U(1)$ hypercharge, $f^{abc}$ is structure constant of
$\mathrm{SU(3)}$, $T^{9}$ is the generator of $U(1)_{X}$ which is defined by
$T^{9}=1/\sqrt{6}~{}\mathrm{diagonal}(1,1,1)$. We just pay attention to
neutral bosons in covariant derivatives so they can be written as [20, 17].
$\displaystyle D^{N}_{\mu}$ $\displaystyle\equiv$
$\displaystyle\partial_{\mu}+i\mathcal{P}^{\mathrm{NC}}_{\mu}$
$\displaystyle=$
$\displaystyle\partial_{\mu}+ig\left(T^{3}V_{3\mu}+T^{8}V_{8\mu}+tT^{9}XB_{\mu}+T^{4}V_{4\mu}+T^{5}V_{5\mu}\right),$
$\displaystyle\bar{D}^{N}_{\mu}$ $\displaystyle\equiv$
$\displaystyle\partial_{\mu}-i\mathcal{P}^{\mathrm{NC}}_{\mu}$
$\displaystyle=$
$\displaystyle\partial_{\mu}-ig\left(T^{3}V_{3\mu}+T^{8}V_{8\mu}-tT^{9}XB_{\mu}\right)-ig\left(T^{4}V_{4\mu}-T^{5}V_{5\mu}\right),$
$\displaystyle D_{1\mu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}+ig^{\prime}XT^{9}B_{\mu},$ $\displaystyle
D^{L}_{\mu}\lambda^{a}_{V}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\lambda^{a}_{V}-g\left(f^{a3c}V^{3}_{\mu}+f^{a8c}V^{8}_{\mu}+f^{a4c}V^{4}_{\mu}+f^{a5c}V^{5}_{\mu}\right)\lambda^{c}_{V},$
(54)
where gauge bosons $W_{3},W_{8},W_{4}$, and $B$ relate with physical states
according to the transformation:
$\displaystyle\left(\begin{array}[]{c}W_{3}\\\ W_{8}\\\ B\\\ W_{4}\\\
\end{array}\right)=\left(\begin{array}[]{cccc}s_{W}&c_{\varphi}c_{\theta^{\prime}}c_{W}&s_{\varphi}c_{\theta^{\prime}}c_{W}&s_{\theta^{\prime}}c_{W}\\\
-\frac{s_{W}}{\sqrt{3}}&\frac{c_{\varphi}\kappa_{3}-s_{\varphi}\kappa_{1}\kappa_{2}}{\sqrt{3}c_{W}c_{\theta^{\prime}}}&\frac{s_{\varphi}\kappa_{3}+c_{\varphi}\kappa_{1}\kappa_{2}}{\sqrt{3}c_{W}c_{\theta^{\prime}}}&\sqrt{3}s_{\theta^{\prime}}c_{W}\\\
\frac{\kappa_{1}}{\sqrt{3}}&-\frac{t_{W}(c_{\varphi}\kappa_{1}+s_{\varphi}\kappa_{2})}{\sqrt{3}c_{\theta^{\prime}}}&-\frac{t_{W}(s_{\varphi}\kappa_{1}-c_{\varphi}\kappa_{2})}{\sqrt{3}c_{\theta^{\prime}}}&0\\\
0&-t_{\theta^{\prime}}(c_{\varphi}\kappa_{2}-s_{\varphi}\kappa_{1})&-t_{\theta^{\prime}}(s_{\varphi}\kappa_{2}+c_{\varphi}\kappa_{1})&\kappa_{2}\\\
\end{array}\right)\left(\begin{array}[]{c}A\\\ Z\\\ Z^{\prime}\\\
W^{\prime}_{4}\\\ \end{array}\right),$ (67) (68)
where some new notations are used:
$\displaystyle t_{\theta}\equiv\tan\theta$ $\displaystyle=$
$\displaystyle\frac{u}{w},\;t_{2\theta}\equiv\tan(2\theta),\;s_{\theta^{\prime}}=\frac{t_{2\theta}}{c_{W}\sqrt{1+4t^{2}_{2\theta}}}$
$\displaystyle t$ $\displaystyle\equiv$
$\displaystyle\frac{g^{\prime}}{g}=\frac{3\sqrt{2}s_{W}}{\sqrt{3-4s^{2}_{W}}},\;\kappa_{1}\equiv\sqrt{4c^{2}_{W}-1}=\frac{3\sqrt{2}s_{W}}{t}$
$\displaystyle\kappa_{2}$ $\displaystyle\equiv$
$\displaystyle\sqrt{1-4s^{2}_{\theta^{\prime}}c^{2}_{W}},\kappa_{3}=s^{2}_{W}-3c^{2}_{W}s^{2}_{\theta^{\prime}}.$
(69)
In the SUSYE331 model, we have all $\theta,~{}\varphi$ and $\theta^{\prime}\ll
1$. Thus, we can use the approximation
$\sin\theta=\sin\varphi=sin\theta^{\prime}=\tan\theta=\tan\varphi=\tan\theta^{\prime}=0$
to simplify the calculation. We also take the approximation:
$\kappa_{2}\simeq 1,\;\kappa_{3}\simeq s^{2}_{W}.$
In addition, $W_{5}$ and $W^{\prime}_{4}$ make of a physical neutral non-
Hermitian gauge boson $X^{0}$ which is defined by the combination:
$X^{0}_{\mu}\equiv\frac{W^{\prime}_{4\mu}-iW_{5\mu}}{\sqrt{2}}$ (70)
So we can rewrite the above covariance derivatives in the form below:
$\displaystyle D^{N}_{\mu}$ $\displaystyle\simeq$
$\displaystyle\partial_{\mu}+ieQA_{\mu}+ig_{Z}\left(T^{3}-s^{2}_{W}Q\right)Z_{\mu}$
$\displaystyle+$ $\displaystyle
ig_{Z^{\prime}}\left[(4c^{2}_{W}-1)(T^{3}-Q)+3c^{2}_{W}X\right]Z^{\prime}_{\mu},$
$\displaystyle\bar{D}^{N}_{\mu}$ $\displaystyle\simeq$
$\displaystyle\partial_{\mu}+ieQA_{\mu}+ig_{Z}\left(-T^{3}-s^{2}_{W}Q\right)Z_{\mu}$
$\displaystyle+$ $\displaystyle
ig_{Z^{\prime}}\left[(4c^{2}_{W}-1)(-T^{3}-Q)+3c^{2}_{W}X\right]Z^{\prime}_{\mu},$
$\displaystyle D^{N}_{1\mu}$ $\displaystyle\simeq$
$\displaystyle\partial_{\mu}+ieQA_{\mu}-ig_{Z}s^{2}_{W}QZ_{\mu}$ (71)
$\displaystyle+$ $\displaystyle
ig_{Z^{\prime}}\left[-(4c^{2}_{W}-1)Q+3c^{2}_{W}X\right]Z^{\prime}_{\mu}$
where we have defined
$g_{Z}\equiv\frac{gc_{\varphi}}{c_{W}c_{\theta^{\prime}}}\simeq\frac{g}{c_{W}}~{}~{}~{}~{}~{}~{}~{}\mathrm{and}~{}~{}~{}~{}~{}~{}g_{Z^{\prime}}\equiv\frac{gc_{\varphi}\kappa_{2}}{c_{W}c_{\theta^{\prime}}\kappa_{1}}\simeq\frac{g}{c_{W}\kappa_{1}}.$
For the charged gauginos, we have:
$\displaystyle\tilde{W}^{\pm}$ $\displaystyle\equiv$
$\displaystyle\frac{\lambda^{1}_{V}\mp i\lambda_{V}^{2}}{\sqrt{2}},\hskip
14.22636pt\tilde{Y}^{\pm}\equiv\frac{\lambda^{6}_{V}\pm
i\lambda_{V}^{7}}{\sqrt{2}}.$ (72)
This leads to the covariant derivative of charged gauginos:
$\displaystyle D^{L}_{\mu}\tilde{W}^{\pm}$ $\displaystyle\simeq$
$\displaystyle\partial_{\mu}\tilde{W}^{\pm}\pm
i\left(eA_{\mu}+gc_{W}Z_{\mu}\right)\tilde{W}^{\pm}=\partial_{\mu}\tilde{W}^{\pm}\pm
i\left(eA_{\mu}+g_{Z}c^{2}_{W}Z_{\mu}\right)\tilde{W}^{\pm}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\tilde{W}^{\pm}+iQ_{\tilde{W}}\left(eA_{\mu}+g_{Z}c^{2}_{W}Z_{\mu}\right)\tilde{W}^{\pm},$
$\displaystyle D^{L}_{\mu}\tilde{Y}^{\pm}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\tilde{Y}^{\pm}\pm
i\left(eA_{\mu}+g\frac{c_{\varphi}\left(c_{2W}+2c^{2}_{W}s^{2}_{\theta^{\prime}}\right)+s_{\varphi}\kappa_{1}\kappa_{2}}{2c_{W}c_{\theta^{\prime}}}Z_{\mu}\right.$
(73) $\displaystyle+$
$\displaystyle\left.g\frac{s_{\varphi}\left(c_{2W}+2c^{2}_{W}s^{2}_{\theta^{\prime}}\right)-c_{\varphi}\kappa_{1}\kappa_{2}}{2c_{W}c_{\theta^{\prime}}}Z^{\prime}_{\mu}\right)\tilde{Y}^{\pm}$
$\displaystyle\simeq$
$\displaystyle\partial_{\mu}\tilde{Y}^{\pm}+iQ_{\tilde{Y}^{\pm}}\left(eA_{\mu}+\frac{1}{2}g_{Z}c_{2W}Z_{\mu}-\frac{1}{2}g_{Z^{\prime}}\kappa_{1}^{2}Z^{\prime}_{\mu}\right)\tilde{Y}^{\pm}.$
From these two formulas, we can deduce the vertices of neutral gauge boson-
charged gaugino-charged gaugino.
### A.3 Gauge boson-slepton-slepton interactions
This kind of vertex comes from the part [20]:
$\displaystyle\mathcal{L}_{\tilde{l}\tilde{l}V}$ $\displaystyle=$
$\displaystyle\frac{ig}{2}\left[\partial^{\mu}\bar{\tilde{L}}_{i}\lambda^{a}\tilde{L}_{i}-\bar{\tilde{L}}_{i}\lambda^{a}\partial^{\mu}\tilde{L}_{i}\right]V^{a}_{\mu}$
$\displaystyle+$
$\displaystyle\frac{ig^{\prime}}{\sqrt{6}}\left[-\frac{1}{3}\left(\partial^{\mu}\bar{\tilde{L}}_{i}\tilde{L}_{i}-\bar{\tilde{L}}_{i}\partial^{\mu}\tilde{L}_{i}\right)+\left(\partial^{\mu}\bar{\tilde{l^{c}}}\tilde{l}^{c}-\bar{\tilde{l^{c}}}\partial^{\mu}\tilde{l}^{c}\right)\right]B_{\mu}$
where $i=1,2,3$ is the flavor index and $a=1,2,...,8$ is generator index of
$SU(3)$. For the $\\{\tilde{\mu},\tilde{\tau}\\}$ sector with neutral boson we
have:
$\displaystyle\mathcal{L}_{\tilde{l}\tilde{l}V}$ $\displaystyle\simeq$
$\displaystyle\frac{ig}{2}\left[\frac{1}{c_{W}}Z_{\mu}+\frac{c_{2W}}{\kappa_{1}c_{W}}Z^{\prime}_{\mu}\right]\times\left(\partial^{\mu}\bar{\tilde{\nu}}_{\tau}\tilde{\nu}_{\tau}-\bar{\tilde{\nu}}_{\tau}\partial^{\mu}\tilde{\nu}_{\tau}\right)$
(74) $\displaystyle+$
$\displaystyle\frac{ig}{2}\left[-2s_{W}A_{\mu}-\frac{c_{2W}}{c_{W}}Z_{\mu}+\frac{c_{2W}}{\kappa_{1}c_{W}}Z^{\prime}_{\mu}\right]\left(\partial^{\mu}\bar{\tilde{\tau}}\tilde{\tau}-\bar{\tilde{\tau}}\partial^{\mu}\tilde{\tau}\right)$
$\displaystyle+$
$\displaystyle\frac{ig}{2}\left[-\frac{2c_{W}}{\kappa_{1}}Z^{\prime}_{\mu}\right]\left(\partial^{\mu}\bar{\tilde{\nu^{c}}}_{\tau}\tilde{\nu^{c}}_{\tau}-\bar{\tilde{\nu^{c}}}_{\tau}\partial^{\mu}\tilde{\nu^{c}}_{\tau}\right)+(\tau\rightarrow\mu)$
$\displaystyle+$
$\displaystyle\left[i\left(eA_{\mu}-e~{}t_{W}Z_{\mu}+\frac{e~{}t_{W}}{\kappa_{1}}Z^{\prime}_{\mu}\right)\left(\partial^{\mu}\bar{\tilde{\tau^{c}}}\tilde{\tau}^{c}-\bar{\tilde{\tau^{c}}}\partial^{\mu}\tilde{\tau}^{c}\right)+\left(\tau^{c}\rightarrow\mu^{c}\right)\right]$
$\displaystyle=$ $\displaystyle
i\left[\frac{1}{2}g_{Z}Z_{\mu}+\frac{1}{2}g_{Z^{\prime}}c_{2W}Z^{\prime}_{\mu}\right]\times\left(\partial^{\mu}\bar{\tilde{\nu}}_{\tau}\tilde{\nu}_{\tau}-\bar{\tilde{\nu}}_{\tau}\partial^{\mu}\tilde{\nu}_{\tau}\right)$
$\displaystyle-$ $\displaystyle
i\left[eA_{\mu}+\frac{1}{2}g_{Z}c_{2W}Z_{\mu}-\frac{1}{2}g_{Z^{\prime}}c_{2W}Z^{\prime}_{\mu}\right]\left(\partial^{\mu}\bar{\tilde{\tau}}\tilde{\tau}-\bar{\tilde{\tau}}\partial^{\mu}\tilde{\tau}\right)$
$\displaystyle-$ $\displaystyle
i\left[g_{Z^{\prime}}c^{2}_{W}Z^{\prime}_{\mu}\right]\left(\partial^{\mu}\bar{\tilde{\nu^{c}}}_{\tau}\tilde{\nu^{c}}_{\tau}-\bar{\tilde{\nu^{c}}}_{\tau}\partial^{\mu}\tilde{\nu^{c}}_{\tau}\right)+(\tau\rightarrow\mu)$
$\displaystyle+$
$\displaystyle\left[i\left(eA_{\mu}-g_{Z}s^{2}_{W}Z_{\mu}+g_{Z^{\prime}}s^{2}_{W}Z^{\prime}_{\mu}\right)\left(\partial^{\mu}\bar{\tilde{\tau^{c}}}\tilde{\tau}^{c}-\bar{\tilde{\tau^{c}}}\partial^{\mu}\tilde{\tau}^{c}\right)\right.$
$\displaystyle+$
$\displaystyle\left.\left(\tau^{c}\rightarrow\mu^{c}\right)\right]$
$V^{\mu}$$p$$p^{\prime}$
$V^{\mu}$$p$$p^{\prime}$
Figure 9: Notations of directions of scalars and fermions. Here $V^{\mu}$
denotes a photon $A$, $Z$ or $Z^{\prime}$ boson.
Interaction vertices of photon, $Z$ and $Z^{\prime}$ bosons relating with our
calculation are summarized in tables 1, 2 and 3, respectively. We denote
directions of momentums in Fig.9. For simplicity, we omit spinor index in the
formulas of boson-fermion-fermion vertices. The precise formulas of this kind
of vertices is easily deduced using rules concerned in [12].
Table 1: Photon vertices. vertex | factor | vertex | factor
---|---|---|---
Photon-scalar-scalar | | |
(scalar $\varphi:H,\tilde{f}$) | $-ieQ_{\varphi}(p+p^{\prime})^{\mu}$ | |
Photon-spinor-spinor | | |
(spinor $\psi$: fermion, Higgsino) | $-ieQ_{\psi}\bar{\sigma}^{\mu}$ ($ieQ_{\psi}\sigma^{\mu}$) | $\gamma\psi^{c}\psi^{c}$ | $ieQ_{\psi}\bar{\sigma}^{\mu}$ ($-ieQ_{\psi}\sigma^{\mu}$)
Photon-boson-boson | | |
$W^{+\rho}W^{-\mu}A^{\nu}$ | $ie[p_{+\rho},~{}p_{-\mu},~{}p_{A\nu}]$ | $Y^{+\rho}Y^{-\mu}A^{\nu}$ | $-ie[p_{+\rho},~{}p_{-\mu},~{}p_{A\nu}]$
Photon-Higgs-gauge boson | | |
$A^{\mu}W^{\nu}\rho_{1}$ | $\frac{1}{2}(ieg)g_{\mu\nu}$ | $A^{\mu}Y^{\nu}\rho_{2}$ | $\frac{1}{2}(ieg)g_{\mu\nu}$
Photon-gaugino-gaugino | | |
$\overline{\tilde{W}^{+}}A_{\mu}\tilde{W}^{+}$ | $-ie\bar{\sigma}^{\mu}$ (or $ie\sigma^{\mu}$) | $\overline{\tilde{W}^{-}}A_{\mu}\tilde{W}^{-}$ | $ie\bar{\sigma}^{\mu}$ (or $-ie\sigma^{\mu}$)
$\overline{\tilde{Y}^{+}}A_{\mu}\tilde{Y}^{+}$ | $-ie\bar{\sigma}^{\mu}$ (or $ie\sigma^{\mu}$) | $\overline{\tilde{Y}^{-}}A_{\mu}\tilde{Y}^{-}$ | $ie\bar{\sigma}^{\mu}$ (or $-ie\sigma^{\mu}$)
Table 2: $Z$ boson vertices Vertex | Factor | Vertex | Factor
---|---|---|---
$Z_{\mu}\tilde{\nu}^{*}_{L}\tilde{\nu}_{L}$ | $-\frac{i}{2}~{}g_{Z}(p+p^{\prime})^{\mu}$ | |
$Z_{\mu}\tilde{\ell}^{*}_{L}\tilde{\ell}_{L}$ | $\frac{i}{2}c_{2W}g_{Z}(p+p^{\prime})^{\mu}$ | $Z_{\mu}\tilde{\ell}_{R}^{*}\tilde{\ell}_{R}$ | $ig_{Z}s^{2}_{W}(p+p^{\prime})^{\mu}$
$\rho^{0*}\rho^{0}Z_{\mu}$ | $\frac{i}{2}g_{Z}(p+p^{\prime})^{\mu}$ | $\rho^{\prime 0*}\rho^{\prime 0}Z_{\mu}$ | $-\frac{i}{2}g_{Z}(p+p^{\prime})^{\mu}$
$\chi^{0*}_{1}\chi^{0}_{1}Z_{\mu}$ | $-\frac{i}{2}g_{Z}(p+p^{\prime})^{\mu}$ | $\chi^{\prime 0*}_{1}\chi^{\prime 0}_{1}Z_{\mu}$ | $\frac{i}{2}g_{Z}(p+p^{\prime})^{\mu}$
$\overline{\tilde{\rho}^{0}}\tilde{\rho}^{0}Z_{\mu}$ | $\frac{i}{2}g_{Z}\bar{\sigma}^{\mu}$ (or $-\frac{i}{2}g_{Z}\sigma^{\mu}$) | $\overline{\tilde{\rho}^{\prime 0}}\tilde{\rho}^{\prime 0}Z_{\mu}$ | $-\frac{i}{2}g_{Z}\sigma^{\mu}$ (or $\frac{i}{2}g_{Z}\sigma^{\mu}$)
$\overline{\tilde{\rho}^{+}_{1}}\tilde{\rho}^{+}_{1}Z_{\mu}$ | $-\frac{i}{2}g_{Z}c_{2W}\bar{\sigma}^{\mu}$ (or $\frac{i}{2}g_{Z}c_{2W}\sigma^{\mu}$) | $\overline{\tilde{\rho}^{\prime-}_{1}}\tilde{\rho}^{\prime-}_{1}Z_{\mu}$ | $\frac{i}{2}g_{Z}c_{2W}\sigma^{\mu}$ (or $-\frac{i}{2}g_{Z}c_{2W}\sigma^{\mu}$)
$\overline{\tilde{\rho}^{+}_{2}}\tilde{\rho}^{+}_{2}Z_{\mu}$ | $\frac{i}{2}g_{Z}s^{2}_{W}\bar{\sigma}^{\mu}$ (or $-\frac{i}{2}g_{Z}s^{2}_{W}\sigma^{\mu}$) | $\overline{\tilde{\rho}^{\prime-}_{2}}\tilde{\rho}^{\prime-}_{2}Z_{\mu}$ | $-\frac{i}{2}g_{Z}s^{2}_{W}\sigma^{\mu}$ (or $\frac{i}{2}g_{Z}s^{2}_{W}\sigma^{\mu}$)
$\overline{\tilde{W}^{+}}Z_{\mu}\tilde{W}^{+}$ | $-ig_{Z}c^{2}_{W}\bar{\sigma}^{\mu}$ (or $ig_{Z}c^{2}_{W}\sigma^{\mu}$) | $\overline{\tilde{W}^{-}}Z_{\mu}\tilde{W}^{-}$ | $ig_{Z}c^{2}_{W}\bar{\sigma}^{\mu}$ (or $-ig_{Z}c^{2}_{W}\sigma^{\mu}$)
$\overline{\tilde{Y}^{+}}Z_{\mu}\tilde{Y}^{+}$ | $-\frac{i}{2}g_{Z}c_{2W}\bar{\sigma}^{\mu}$ (or $\frac{i}{2}g_{Z}c_{2W}\sigma^{\mu}$) | $\overline{\tilde{Y}^{-}}Z_{\mu}\tilde{Y}^{-}$ | $\frac{i}{2}g_{Z}c_{2W}\bar{\sigma}^{\mu}$ (or $-\frac{i}{2}g_{Z}c_{2W}\sigma^{\mu}$)
Table 3: $Z^{\prime}$ boson vertices Vertex | Factor | Vertex | Factor
---|---|---|---
$Z^{\prime}_{\mu}\tilde{\nu}^{*}_{L}\tilde{\nu}_{L}$ | $-\frac{i}{2}~{}g_{Z^{\prime}}c_{2W}(p+p^{\prime})^{\mu}$ | $Z^{\prime}_{\mu}\tilde{\nu}_{R}^{*}\tilde{\nu}_{R}$ | $-g_{Z^{\prime}}c^{2}_{W}(p+p^{\prime})^{\mu}$
$Z^{\prime}_{\mu}\tilde{\ell}^{*}_{L}\tilde{\ell}_{L}$ | $-\frac{i}{2}c_{2W}g_{Z^{\prime}}(p+p^{\prime})^{\mu}$ | $Z^{\prime}_{\mu}\tilde{\ell}_{R}^{*}\tilde{\ell}_{R}$ | $-ig_{Z^{\prime}}s^{2}_{W}(p+p^{\prime})^{\mu}$
$\rho^{0*}\rho^{0}Z^{\prime}_{\mu}$ | $-\frac{i}{2}g_{Z}(p+p^{\prime})^{\mu}$ | $\rho^{\prime 0*}\rho^{\prime 0}Z^{\prime}_{\mu}$ | $\frac{i}{2}g_{Z}(p+p^{\prime})^{\mu}$
$\chi^{0*}_{1}\chi^{0}_{1}Z^{\prime}_{\mu}$ | $-\frac{i}{2}g_{Z^{\prime}}c_{2W}(p+p^{\prime})^{\mu}$ | $\chi^{\prime 0*}_{1}\chi^{\prime 0}_{1}Z^{\prime}_{\mu}$ | $\frac{i}{2}g_{Z^{\prime}}c_{2W}(p+p^{\prime})^{\mu}$
$\chi^{0*}_{2}\chi^{0}_{2}Z^{\prime}_{\mu}$ | $-\frac{i}{2}g_{Z^{\prime}}c^{2}_{W}(p+p^{\prime})^{\mu}$ | $\chi^{\prime 0*}_{2}\chi^{\prime 0}_{2}Z^{\prime}_{\mu}$ | $\frac{i}{2}g_{Z^{\prime}}c^{2}_{W}(p+p^{\prime})^{\mu}$
$\overline{\tilde{\rho}^{0}}\tilde{\rho}^{0}Z^{\prime}_{\mu}$ | $-\frac{i}{2}g_{Z^{\prime}}\bar{\sigma}^{\mu}$ (or $\frac{i}{2}g_{Z^{\prime}}\sigma^{\mu}$) | $\overline{\tilde{\rho}^{\prime 0}}\tilde{\rho}^{\prime 0}Z_{\mu}$ | $\frac{i}{2}g_{Z^{\prime}}\sigma^{\mu}$ (or $-\frac{i}{2}g_{Z^{\prime}}\sigma^{\mu}$)
$\overline{\tilde{\rho}^{+}_{1}}\tilde{\rho}^{+}_{1}Z^{\prime}_{\mu}$ | $-\frac{i}{2}g_{Z^{\prime}}\bar{\sigma}^{\mu}$ (or $\frac{i}{2}g_{Z^{\prime}}\sigma^{\mu}$) | $\overline{\tilde{\rho}^{\prime-}_{1}}\tilde{\rho}^{\prime-}_{1}Z^{\prime}_{\mu}$ | $\frac{i}{2}g_{Z^{\prime}}\sigma^{\mu}$ (or $-\frac{i}{2}g_{Z^{\prime}}\sigma^{\mu}$)
$\overline{\tilde{\rho}^{+}_{2}}\tilde{\rho}^{+}_{2}Z^{\prime}_{\mu}$ | $\frac{i}{2}g_{Z^{\prime}}\bar{\sigma}^{\mu}$ (or $-\frac{i}{2}g_{Z}\sigma^{\mu}$) | $\overline{\tilde{\rho}^{\prime-}_{2}}\tilde{\rho}^{\prime-}_{2}Z^{\prime}_{\mu}$ | $-\frac{i}{2}g_{Z^{\prime}}c_{2W}\sigma^{\mu}$ (or $\frac{i}{2}g_{Z}c_{2W}\sigma^{\mu}$)
$\overline{\tilde{Y}^{+}}Z^{\prime}_{\mu}\tilde{Y}^{+}$ | $\frac{i}{2}g_{Z^{\prime}}\kappa_{1}^{2}\bar{\sigma}^{\mu}$ (or $-\frac{i}{2}g_{Z^{\prime}}\kappa_{1}^{2}\sigma^{\mu}$) | $\overline{\tilde{Y}^{-}}Z^{\prime}_{\mu}\tilde{Y}^{-}$ | $-\frac{i}{2}g_{Z^{\prime}}\kappa_{1}^{2}\bar{\sigma}^{\mu}$ (or $\frac{i}{2}g_{Z^{\prime}}\kappa_{1}^{2}\sigma^{\mu}$)
### A.4 Mixing in the slepton sector
As we know, in supersymmetric models, in order to keep the conversation of LFV
in the lepton sector at tree level, the sources of LFV are assumed to be from
the slepton mass terms in the soft-breaking part of the Lagrangian [10, 24].
For the SUSYE331, there are three mass terms of left-handed slepton, right-
handed slepton and sneutrinos which may independently be sources of LFV. In
addition, there exists another LFV source original from the Yukawa couplings
between Higgs and neutrinos. Thus in the SUSYE331 model, there are at least
four independent sources of LFV and we will parameterize them as follows. In
each case of supersymmetric particle (sleptons) $\tilde{\psi}$
($\psi=l_{L},~{}l_{R},~{}\nu_{L},\nu_{R}$), we define a corresponding mixing
angle $\theta_{\tilde{\varphi}}$ which was defined in [21]. In what follows we
just remind some general formulas for the review. The mass mixing matrices of
smuon and stau as well as their sneutrinos can be written in the general form
of:
$\mathcal{M}^{2}_{\tilde{\psi}}=\left(\begin{array}[]{cc}m^{2}_{\tilde{\psi}_{\mu\mu}}&m^{2}_{\tilde{\psi}_{\mu\tau}}\\\
m^{2}_{\tilde{\psi}_{\mu\tau}}&m^{2}_{\tilde{\psi}_{\tau\tau}}\\\
\end{array}\right).$ (75)
Mixing angles then can be determined as
$s_{\tilde{\psi}}\equiv\sin\theta_{\tilde{\psi}},\hskip
14.22636ptc_{\tilde{\psi}}\equiv\cos\theta_{\tilde{\psi}}\hskip
14.22636pt\mathrm{where}\hskip
14.22636pts_{\tilde{\psi}}c_{\tilde{\psi}}=\frac{m^{2}_{\tilde{\psi}_{\mu\tau}}}{m^{2}_{\psi_{2}}-m^{2}_{\psi_{3}}},$
(76)
where
$s_{\tilde{\psi}}=\\{s_{L},s_{R},s_{\tilde{\nu}_{L}},s_{\tilde{\nu}_{R}}\\}$
and $\\{m^{2}_{\psi_{2}},~{}m^{2}_{\psi_{3}}\\}$ are eigenvalues of
$\mathcal{M}^{2}_{\tilde{\psi}}$, according to notations in [21]. In addition,
for convenience we denote $m^{2}_{\tilde{\psi}}$ instead of
$\tilde{m}^{2}_{\psi}$. We always choose $m^{2}_{\psi_{3}}<m^{2}_{\psi_{2}}$
to take the positive values of $s_{\tilde{\psi}}$ and $c_{\tilde{\psi}}$. The
mass-eigenstates of sleptons are denoted as
$\\{\tilde{\psi}_{2},~{}\tilde{\psi}_{3}\\}$ while the flavor-eigenstates are
$\\{\tilde{\psi}_{\mu},~{}\tilde{\psi}_{\tau}\\}$. The relation between two
bases are:
$\tilde{\psi}_{\mu}=c_{\tilde{\psi}}\tilde{\psi}_{2}-s_{\tilde{\psi}}\tilde{\psi}_{3}\hskip
14.22636pt\mathrm{and}\hskip
14.22636pt\tilde{\psi}_{\tau}=s_{\tilde{\psi}}\tilde{\psi}_{2}+c_{\tilde{\psi}}\tilde{\psi}_{3}.$
(77)
## Appendix B Contribution to $\tau\rightarrow\mu\gamma$
Diagrams relating to $C^{\gamma}_{L,R}$ are drawn in Fig. 10 with no line of
Higgs insertion.
$\tau$$\mu$$\tilde{W}^{+}$$\tilde{W}^{+}$$\gamma$$\tilde{\nu}_{L_{\alpha}}$(1)
$\tau$$\mu$$\tilde{Y}^{+}$$\tilde{Y}^{+}$$\gamma$$\tilde{\nu}_{R_{\alpha}}$$(2)$
$\tau$$\mu$$\lambda_{B}$$\lambda^{3}_{A},\lambda^{8}_{A}$$\gamma$$\tilde{\ell}_{L_{\alpha}}$$(3)$
$\tau^{c}$$\mu^{c}$$\lambda_{B}$$\gamma$$\tilde{\ell}_{R_{\alpha}}$$(4)$
Figure 10: Diagrams contributing to $C^{\gamma}_{L,R}$.
Formulas of $C^{\gamma}_{L,R}$ are:
$\displaystyle C^{\gamma}_{L}$ $\displaystyle=$
$\displaystyle\frac{(g^{2}c_{L}s_{L})}{16\pi^{2}}\times\frac{1}{9}\left[-K_{5}(m^{2}_{\lambda},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}})\right]$
$\displaystyle+$
$\displaystyle\frac{(g^{2}c_{\nu_{L}}s_{\nu_{L}})}{16\pi^{2}}\times\frac{1}{6}\left[-2K_{5}(m^{2}_{\lambda},m^{2}_{\lambda},m^{2}_{\lambda},m^{2}_{\lambda},\tilde{m}^{2}_{\nu_{L2}})\right.$
$\displaystyle+$
$\displaystyle\left.3m^{2}_{\lambda}J_{5}(m^{2}_{\lambda},m^{2}_{\lambda},m^{2}_{\lambda},m^{2}_{\lambda},\tilde{m}^{2}_{\nu_{L2}})\right]$
$\displaystyle+$
$\displaystyle\frac{(g^{2}c_{\nu_{R}}s_{\nu_{R}})}{16\pi^{2}}\times\frac{1}{6}\left[-2K_{5}(m^{2}_{\lambda},m^{2}_{\lambda},m^{2}_{\lambda},m^{2}_{\lambda},\tilde{m}^{2}_{\nu_{R2}})\right.$
$\displaystyle+$
$\displaystyle\left.3m^{2}_{\lambda}J_{5}(m^{2}_{\lambda},m^{2}_{\lambda},m^{2}_{\lambda},m^{2}_{\lambda},\tilde{m}^{2}_{\nu_{R2}})\right]$
$\displaystyle+$ $\displaystyle\frac{g^{\prime
2}c_{L}s_{L}}{16\pi^{2}}\times\frac{1}{162}\left[-K_{5}(m^{2}_{B},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}})\right]$
$\displaystyle-$ $\displaystyle(L_{2}\rightarrow L_{3},R_{2}\rightarrow
R_{3}),$ $\displaystyle C^{\gamma}_{R}$ $\displaystyle=$
$\displaystyle\frac{g^{\prime
2}c_{R}s_{R}}{16\pi^{2}}\times\frac{1}{18}\left[-K_{5}(m^{2}_{B},m^{2}_{\tilde{l}_{R2}},m^{2}_{\tilde{l}_{R2}},m^{2}_{\tilde{l}_{R2}},m^{2}_{\tilde{l}_{R2}})\right]-[R_{2}\rightarrow
R_{3}].$ (78)
On the other hand, $D^{\gamma}$ gets contributions from diagrams with one line
Higgs insertion, figs. 11,80 and 13. There is another class of LFV sources
relating with neutrino-mediation in which their contributions are very small
[22] so we will ignore them in our investigation.
$\tau^{c}$$\tau$$\mu$$\rho^{\prime
0}$$\tilde{W}^{+}$$\tilde{W}^{+}$$\gamma$$\tilde{\nu_{L}}_{\alpha}$$(1)$
$\tau^{c}$$\tau$$\mu$$\rho^{\prime
0}$$\tilde{Y}^{+}$$\tilde{Y}^{+}$$\gamma$$\tilde{\nu}_{R\alpha}$$(2)$
$\tau^{c}$$\tau$$\mu$$\rho^{\prime
0}$$\lambda_{B}$$\lambda^{3}_{A},\lambda^{8}_{A}$$\gamma$$\tilde{\ell}_{L_{\alpha}}$$(3)$
$\tau$$\tau^{c}$$\mu^{c}$$\rho^{\prime
0}$$\lambda_{B}$$\gamma$$\tilde{\ell}_{R_{\alpha}}$$(4)$
Figure 11: Contribution to $D^{\gamma(a)}_{L}$ [1-3] and $D^{\gamma(a)}_{R}$
[4].
The $D^{\gamma}_{L,R}$ can be separated into three parts:
$D^{\gamma}_{L,R}=D^{\gamma(a)}_{L,R}+D^{\gamma(b)}_{L,R}+D^{\gamma(c)}_{L,R},$
where diagrams involving each part are expressed in three figs.11, 80 and 13.
For $D^{\gamma(a)}$:
$\displaystyle D^{\gamma(a)}_{L}$ $\displaystyle=$
$\displaystyle\frac{g^{2}c_{L}s_{L}}{16\pi^{2}}\times\frac{1}{3}\left[m^{2}_{\tilde{l}_{L2}}J_{5}(m^{2}_{\lambda},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}})\right]$
$\displaystyle-$
$\displaystyle\frac{g^{2}c_{\nu_{L}}s_{\nu_{L}}}{16\pi^{2}}\times\frac{1}{2}\left[m^{2}_{\lambda}~{}J_{5}(m^{2}_{\lambda},m^{2}_{\lambda},m^{2}_{\lambda},m^{2}_{\lambda},m^{2}_{\tilde{\nu}_{L2}})\right]$
$\displaystyle-$
$\displaystyle\frac{g^{2}c_{\nu_{R}}s_{\nu_{R}}}{16\pi^{2}}\times\frac{1}{2}\left[m^{2}_{\lambda}~{}J_{5}(m^{2}_{\lambda},m^{2}_{\lambda},m^{2}_{\lambda},m^{2}_{\lambda},m^{2}_{\tilde{\nu}_{R2}})\right]$
$\displaystyle+$ $\displaystyle\frac{g^{\prime
2}c_{L}s_{L}}{16\pi^{2}}m^{2}_{\tilde{l}_{L2}}\left[\frac{1}{54}J_{5}(m^{2}_{B},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}})\right]$
$\displaystyle-$ $\displaystyle[L_{2}\rightarrow L_{3},R_{2}\rightarrow
R_{3}],$ $\displaystyle D^{\gamma(a)}_{R}$ $\displaystyle=$
$\displaystyle\frac{g^{\prime
2}c_{R}s_{R}}{16\pi^{2}}m^{2}_{\tilde{l}_{R2}}\left[\frac{1}{6}J_{5}(m^{2}_{B},m^{2}_{\tilde{l}_{R2}},m^{2}_{\tilde{l}_{R2}},m^{2}_{\tilde{l}_{R2}},m^{2}_{\tilde{l}_{R2}})\right]-[R_{2}\rightarrow
R_{3}].$ (79)
$\tau^{c}$$\mu$$\rho^{\prime
0}$$\tilde{\rho}^{\prime-}_{1}$$\tilde{W}^{+}$$\tilde{W}^{+}$$\gamma$$\tilde{\nu}_{L\alpha}$$(1)$
$\tau^{c}$$\mu$$\rho^{\prime
0}$$\tilde{\rho}^{\prime-}_{1}$$\tilde{\rho}^{\prime-}_{1}$$\tilde{W}^{+}$$\gamma$$\tilde{\nu}_{L\alpha}$$(2)$
$\tau^{c}$$\mu$$\rho^{\prime
0}$$\tilde{\rho}^{\prime-}_{2}$$\tilde{Y}^{+}$$\tilde{Y}^{+}$$\gamma$$\tilde{\nu}_{R\alpha}$$(3)$
$\tau^{c}$$\mu$$\rho^{\prime
0}$$\tilde{\rho}^{\prime-}_{2}$$\tilde{\rho}^{\prime-}_{2}$$\tilde{Y}^{+}$$\gamma$$\tilde{\nu}_{R\alpha}$$(4)$
$\tau^{c}$$\mu$$\rho^{0}$$\tilde{\rho}^{\prime-}_{1}$$\tilde{\rho}^{+}_{1}$$\tilde{W}^{-}$$\tilde{W}^{+}$$\gamma$$\tilde{\nu}_{L\alpha}$$(5)$
$\tau^{c}$$\mu$$\rho^{0}$$\tilde{\rho}^{\prime-}_{1}$$\tilde{\rho}^{+}_{1}$$\tilde{W}^{-}$$\tilde{W}^{+}$$\gamma$$\tilde{\nu}_{L\alpha}$$(6)$
$\tau^{c}$$\mu$$\rho^{0}$$\tilde{\rho}^{\prime-}_{2}$$\tilde{\rho}^{+}_{2}$$\tilde{Y}^{-}$$\tilde{Y}^{+}$$\gamma$$\tilde{\nu}_{R\alpha}$$(7)$
$\tau^{c}$$\mu$$\rho^{0}$$\tilde{\rho}^{\prime-}_{2}$$\tilde{\rho}^{+}_{2}$$\tilde{Y}^{-}$$\tilde{Y}^{+}$$\gamma$$\tilde{\nu}_{R\alpha}$$(8)$
$\tau^{c}$$\mu$$\rho^{\prime 0}$$\tilde{\rho}^{\prime
0}$$\lambda^{3}_{A},\lambda^{8}_{A}$$\lambda_{B}$$\gamma$$\tilde{\ell}_{L_{\alpha}}$$(9)$
$\tau^{c}$$\mu$$\rho^{0}$$\tilde{\rho}^{\prime
0}$$\tilde{\rho}^{0}$$\lambda^{3}_{A},\lambda^{8}_{A}$$\lambda_{B}$$\gamma$$\tilde{\ell}_{L_{\alpha}}$$(10)$
$\tau$$\mu^{c}$$\rho^{\prime 0}$$\tilde{\rho}^{\prime
0}$$\lambda_{B}$$\gamma$$\tilde{\ell}_{R_{\alpha}}$$(11)$
$\tau$$\mu^{c}$$\rho^{0}$$\tilde{\rho}^{\prime
0}$$\tilde{\rho}^{0}$$\lambda_{B}$$\gamma$$\tilde{\ell}_{R_{\alpha}}$$(12)$
Figure 12: Contribution to $D^{\gamma(b)}_{L}$ [1-10] and $D^{\gamma(b)}_{R}$
[11,12].
For $D^{\gamma(b)}$:
$\displaystyle D^{\gamma(b)}_{L}$ $\displaystyle=$
$\displaystyle-\frac{g^{2}s_{\nu_{L}}c_{\nu_{L}}}{16\pi^{2}}m^{4}_{\tilde{\nu}_{L2}}I_{5}(m_{\lambda}^{2},\mu^{2}_{\rho},m^{2}_{\tilde{\nu}_{L2}},m^{2}_{\tilde{\nu}_{L2}},m^{2}_{\tilde{\nu}_{L2}})$
$\displaystyle-$
$\displaystyle\frac{g^{2}s_{\nu_{R}}c_{\nu_{R}}}{16\pi^{2}}m^{4}_{\tilde{\nu}_{R2}}I_{5}(m_{\lambda}^{2},\mu^{2}_{\rho},m^{2}_{\tilde{\nu}_{R2}},m^{2}_{\tilde{\nu}_{R2}},m^{2}_{\tilde{\nu}_{R2}})$
$\displaystyle+$
$\displaystyle\frac{g^{2}s_{\nu_{L2}}c_{\nu_{L2}}}{16\pi^{2}}\times
m_{\lambda}~{}\mu~{}\tan\gamma\left[J_{5}(m_{\lambda}^{2},m_{\lambda}^{2},\mu^{2}_{\rho},\mu^{2}_{\rho},m^{2}_{\tilde{\nu}_{L2}})\right.$
$\displaystyle+$
$\displaystyle\left.J_{5}(m_{\lambda}^{2},m_{\lambda}^{2},m_{\lambda}^{2},\mu^{2}_{\rho},m^{2}_{\tilde{\nu}_{L2}})+J_{5}(m_{\lambda}^{2},\mu^{2}_{\rho},\mu^{2}_{\rho},\mu^{2}_{\rho},m^{2}_{\tilde{\nu}_{L2}})\right]$
$\displaystyle+$
$\displaystyle\frac{g^{2}s_{\nu_{R2}}c_{\nu_{R2}}}{16\pi^{2}}\times
m_{\lambda}~{}\mu~{}\tan\gamma\left[J_{5}(m_{\lambda}^{2},m_{\lambda}^{2},\mu^{2}_{\rho},\mu^{2}_{\rho},m^{2}_{\tilde{\nu}_{R2}})\right.$
$\displaystyle+$
$\displaystyle\left.J_{5}(m_{\lambda}^{2},m_{\lambda}^{2},m_{\lambda}^{2},\mu^{2}_{\rho},m^{2}_{\tilde{\nu}_{L2}})+J_{5}(m_{\lambda}^{2},\mu^{2}_{\rho},\mu^{2}_{\rho},\mu^{2}_{\rho},m^{2}_{\tilde{\nu}_{R2}})\right]$
$\displaystyle-$
$\displaystyle\frac{g^{2}s_{L}c_{L}}{16\pi^{2}}~{}m^{2}_{\tilde{l}_{L2}}\times\frac{2}{3}\left[J_{5}(m_{\lambda}^{2},\mu^{2}_{\rho},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}})\right.$
$\displaystyle-$
$\displaystyle\left.m_{\lambda}~{}\mu_{\rho}\tan\gamma~{}I_{5}(m_{\lambda}^{2},\mu^{2}_{\rho},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}})\right]$
$\displaystyle+$ $\displaystyle\frac{g^{\prime
2}s_{L}c_{L}}{16\pi^{2}}~{}m^{2}_{\tilde{l}_{L2}}\times\frac{2}{27}\left[J_{5}(m_{B}^{2},\mu^{2}_{\rho},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}})\right.$
$\displaystyle-$
$\displaystyle\left.m_{B}~{}\mu~{}\tan\gamma~{}I_{5}(m_{B}^{2},\mu^{2}_{\rho},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}})\right]$
$\displaystyle-$ $\displaystyle[L_{2}\rightarrow L_{3}],$ $\displaystyle
D^{\gamma(b)}_{R}$ $\displaystyle=$ $\displaystyle\frac{g^{\prime
2}s_{R}c_{R}}{16\pi^{2}}m^{2}_{\tilde{l}_{R2}}\times\frac{2}{9}\left[-J_{5}(m_{B}^{2},\mu^{2}_{\rho},m^{2}_{\tilde{l}_{R2}},m^{2}_{\tilde{l}_{R2}},m^{2}_{\tilde{l}_{R2}})\right.$
(80) $\displaystyle+$ $\displaystyle
m_{B}~{}\mu_{\rho}\tan\gamma~{}I_{5}(m_{B}^{2},\mu^{2}_{\rho},m^{2}_{\tilde{l}_{R2}},m^{2}_{\tilde{l}_{R2}},m^{2}_{\tilde{l}_{R2}})\left.\right]$
$\displaystyle-$ $\displaystyle[R_{2}\rightarrow R_{3}].$
$\tau^{c}$$\mu$$\lambda_{B}$$\gamma$$\rho^{\prime
0}$$\tilde{\ell}_{R_{\alpha}}$$\tilde{\ell}_{L_{\beta}}$$(1)$
$\tau^{c}$$\mu$$\lambda_{B}$$\gamma$$\rho^{0}$$\tilde{\ell}_{R_{\alpha}}$$\tilde{\ell}_{L_{\beta}}$$(2)$
$\tau^{c}$$\mu$$\lambda_{B}$$\gamma$$\rho^{\prime
0}$$\tilde{\ell}_{R_{\alpha}}$$\tilde{\ell}_{L_{\beta}}$$(3)$
$\tau^{c}$$\mu$$\lambda_{B}$$\gamma$$\rho^{0}$$\tilde{\ell}_{R_{\alpha}}$$\tilde{\ell}_{L_{\beta}}$$(4)$
$\tau$$\mu^{c}$$\lambda_{B}$$\gamma$$\rho^{\prime
0}$$\tilde{\ell}_{L_{\alpha}}$$\tilde{\ell}_{R_{\beta}}$$(5)$
$\tau$$\mu^{c}$$\lambda_{B}$$\gamma$$\rho^{0}$$\tilde{\ell}_{L_{\alpha}}$$\tilde{\ell}_{R_{\beta}}$$(6)$
$\tau$$\mu^{c}$$\lambda_{B}$$\gamma$$\rho^{\prime
0}$$\tilde{\ell}_{L_{\alpha}}$$\tilde{\ell}_{R_{\beta}}$$(7)$
$\tau$$\mu^{c}$$\lambda_{B}$$\gamma$$\rho^{0}$$\tilde{\ell}_{L_{\alpha}}$$\tilde{\ell}_{R_{\beta}}$$(8)$
Figure 13: Contribution to $D^{\gamma(c)}_{L}$ [1-6] and $D^{\gamma}_{R}$
[7,8].
For $D^{\gamma(c)}$:
$\displaystyle
D^{\gamma(c)}_{L}=-\frac{g^{\prime}}{16\pi^{2}}~{}\frac{m^{3}_{B}}{9}$
$\displaystyle\times$
$\displaystyle\left\\{\left[s_{L}c_{L}\left(s^{2}_{R}[A_{\tau}+\frac{1}{2}\mu_{\rho}\tan\gamma]+s_{R}c_{R}A^{R}_{\mu\tau}\right)+c^{2}_{L}s^{2}_{R}A^{L}_{\mu\tau}\right]\right.$
$\displaystyle\times$
$\displaystyle\left.I_{5}(m_{B}^{2},m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{R2}})\right.$
$\displaystyle-$
$\displaystyle\left.\left[s_{L}c_{L}\left(s^{2}_{R}[A_{\tau}+\frac{1}{2}\mu_{\rho}\tan\gamma]+s_{R}c_{R}A^{R}_{\mu\tau}\right)-s^{2}_{L}s^{2}_{R}A^{L}_{\mu\tau}\right]\right.$
$\displaystyle\times$
$\displaystyle\left.I_{5}(m_{B}^{2},m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{L3}},m^{2}_{\tilde{l}_{R2}})\right.$
$\displaystyle+$
$\displaystyle\left.\left[s_{L}c_{L}\left(c^{2}_{R}[A_{\tau}+\frac{1}{2}\mu_{\rho}\tan\gamma]-s_{R}c_{R}A^{R}_{\mu\tau}\right)+c^{2}_{L}c^{2}_{R}A^{L}_{\mu\tau}\right]\right.$
$\displaystyle\times$
$\displaystyle\left.I_{5}(m_{B}^{2},m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{R3}})\right.$
$\displaystyle-$
$\displaystyle\left.\left[s_{L}c_{L}\left(c^{2}_{R}[A_{\tau}+\frac{1}{2}\mu_{\rho}\tan\gamma]-s_{R}c_{R}A^{R}_{\mu\tau}\right)-s^{2}_{L}c^{2}_{R}A^{L}_{\mu\tau}\right]\right.$
$\displaystyle\times$
$\displaystyle\left.I_{5}(m_{B}^{2},m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{L3}},m^{2}_{\tilde{l}_{R3}})\right\\},$
$\displaystyle
D^{\gamma(c)}_{R}=-\frac{g^{\prime}}{16\pi^{2}}~{}\frac{m^{3}_{B}}{9}$
$\displaystyle\times$
$\displaystyle\left\\{\left[s_{R}c_{R}\left(s^{2}_{L}[A_{\tau}+\frac{1}{2}\mu_{\rho}\tan\gamma]+s_{L}c_{L}A^{L}_{\mu\tau}\right)+c^{2}_{R}s^{2}_{L}A^{R}_{\mu\tau}\right]\right.$
(81) $\displaystyle\times$
$\displaystyle\left.I_{5}(m_{B}^{2},m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{R2}})\right.$
$\displaystyle-$
$\displaystyle\left.\left[s_{R}c_{R}\left(s^{2}_{L}[A_{\tau}+\frac{1}{2}\mu_{\rho}\tan\gamma]+s_{L}c_{L}A^{L}_{\mu\tau}\right)-s^{2}_{R}s^{2}_{L}A^{R}_{\mu\tau}\right]\right.$
$\displaystyle\times$
$\displaystyle\left.I_{5}(m_{B}^{2},m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{R3}})\right.$
$\displaystyle+$
$\displaystyle\left.\left[s_{R}c_{R}\left(c^{2}_{L}[A_{\tau}+\frac{1}{2}\mu_{\rho}\tan\gamma]-s_{L}c_{L}A^{L}_{\mu\tau}\right)+c^{2}_{R}c^{2}_{L}A^{R}_{\mu\tau}\right]\right.$
$\displaystyle\times$
$\displaystyle\left.I_{5}(m_{B}^{2},m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{L3}},m^{2}_{\tilde{l}_{R2}})\right.$
$\displaystyle-$
$\displaystyle\left.\left[s_{R}c_{R}\left(c^{2}_{L}[A_{\tau}+\frac{1}{2}\mu_{\rho}\tan\gamma]-s_{L}c_{L}A^{L}_{\mu\tau}\right)-s^{2}_{R}c^{2}_{L}A^{R}_{\mu\tau}\right]\right.$
$\displaystyle\times$
$\displaystyle\left.I_{5}(m_{B}^{2},m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{L3}},m^{2}_{\tilde{l}_{R3}})\right\\}.$
## Appendix C Contributions to $Z\rightarrow\mu\tau$
In this appendix, we draw all the possible diagrams which contribute to the
effective operator $Z\rightarrow\mu\tau$ in the limit of assumption given out
in [10]. All of these diagrams can be applied to the case of $Z^{\prime}$
boson.
### C.1 Contributions to $A^{Z}_{L,R}$
Diagrams contributing to $A^{Z(a)}_{L,R}$ are shown in Fig.14.
$\tau$$\mu$$\tilde{W}^{+}$$\tilde{\rho}_{1}^{\prime-}$$\tilde{\rho}_{1}^{+}$$\tilde{W}^{-}$$\tilde{W}^{+}$$\rho{\prime^{0}}$$\rho^{0}$$\tilde{\nu}_{L\alpha}$(1)
$\tau$$\mu$$\tilde{W}^{+}$$\tilde{W}^{-}$$\tilde{\rho}_{1}^{+}$$\tilde{\rho}_{1}^{\prime-}$$\tilde{W}^{+}$$\rho^{0}$$\rho^{\prime
0}$$\tilde{\nu}_{L_{\alpha}}$(2)
$\tau$$\mu$$\tilde{W}^{+}$$\tilde{\rho}_{1}^{\prime-}$$\tilde{W}^{+}$$\rho^{\prime
0}$$\rho^{\prime
0}$$\tilde{\nu}_{L_{\alpha}}$(3)$\tau$$\mu$$\tilde{W}^{+}$$\tilde{W}^{-}$$\tilde{\rho}_{1}^{+}$$\tilde{W}^{-}$$\tilde{W}^{+}$$\rho^{0}$$\rho^{0}$$\tilde{\nu}_{L_{\alpha}}$(4)
$\tau$$\mu$$\tilde{Y}^{+}$$\tilde{\rho}_{2}^{\prime-}$$\tilde{\rho}_{2}^{+}$$\tilde{Y}^{-}$$\tilde{Y}^{+}$$\rho{\prime^{0}}$$\rho^{0}$$\tilde{\nu}_{R_{\alpha}}$(5)
$\tau$$\mu$$\tilde{Y}^{+}$$\tilde{Y}^{-}$$\tilde{\rho}_{2}^{+}$$\tilde{\rho}_{2}^{\prime-}$$\tilde{Y}^{+}$$\rho^{0}$$\rho^{\prime
0}$$\tilde{\nu}_{R_{\alpha}}$(6)
$\tau$$\mu$$\tilde{Y}^{+}$$\tilde{\rho}_{2}^{\prime-}$$\tilde{Y}^{+}$$\rho^{\prime
0}$$\rho^{\prime
0}$$\tilde{\nu}_{R_{\alpha}}$(7)$\tau$$\mu$$\tilde{Y}^{+}$$\tilde{Y}^{-}$$\tilde{\rho}_{2}^{+}$$\tilde{Y}^{-}$$\tilde{Y}^{+}$$\rho^{0}$$\rho^{0}$$\tilde{\nu}_{R_{\alpha}}$(8)
$\tau$$\mu$$\lambda_{i}$$\tilde{H}^{0}_{k}$$\lambda_{i}$$H^{0}_{k}$$H^{0}_{k}$$\tilde{\ell}_{L_{\alpha}}$(9)
$\tau$$\mu$$\lambda_{i}$$\tilde{H}^{0}_{k}$$\lambda_{i}$$H^{0}_{k}$$H^{0}_{k}$$\tilde{\ell}_{L_{\alpha}}$(10)
$\tau$$\mu$$\lambda_{i}$$\tilde{H}^{0}_{k}$$\lambda_{j}$$H^{0}_{k}$$H^{0}_{k}$$\tilde{\ell}_{L_{\alpha}}$(11)$\tau$$\mu$$\lambda_{i}$$\tilde{H}^{0}_{k}$$\lambda_{j}$$H^{0}_{k}$$H^{0}_{k}$$\tilde{\ell}_{L_{\alpha}}$(12)
$\tau^{c}$$\mu^{c}$$\lambda_{B}$$\tilde{H}^{0}_{k}$$\lambda_{B}$$H^{0}_{k}$$H^{0}_{k}$$\tilde{\ell}_{R_{\alpha}}$(13)
$\tau^{c}$$\mu^{c}$$\lambda_{B}$$\tilde{H}^{0}_{k}$$\lambda_{B}$$H^{0}_{k}$$H^{0}_{k}$$\tilde{\ell}_{R_{\alpha}}$(14)
Figure 14: Diagrams contributing to $A^{Z(a)}_{L}$ (or
$A^{1Z^{\prime}(a)}_{L}$) (first, second and third rows) and $A^{Z(a)}_{R}$
(or $A^{1Z^{\prime}(a)}_{R}$) (fourth row). Here we denote
$H^{0}_{k}\in\\{\rho^{0},~{}\rho^{\prime 0}\\}$ while $\lambda_{i,j}$ implies
$i,j=\\{B,3,8\\}$ and $i\neq j$.
The formulas are:
$\displaystyle A^{Z(a)}_{L}$ $\displaystyle=$
$\displaystyle(s_{\nu_{L}}c_{\nu_{L}})\times\frac{g^{2}c^{2}_{W}}{16\pi^{2}}\times\frac{1}{4}(1+c_{2\gamma})$
(82) $\displaystyle\times$
$\displaystyle\left[-\mu^{2}_{\rho}J_{5}(m^{2}_{\lambda},m^{2}_{\lambda},\mu^{2}_{\rho},\mu^{2}_{\rho},m^{2}_{\tilde{\nu}_{L2}})-2J_{4}(m^{2}_{\lambda},m^{2}_{\lambda},\mu^{2}_{\rho},m^{2}_{\tilde{\nu}_{L2}})\right]$
$\displaystyle+$
$\displaystyle(s_{\nu_{R}}c_{\nu_{R}})\times\frac{g^{2}c^{2}_{W}}{16\pi^{2}}\times\frac{1}{4}(1+c_{2\gamma})$
$\displaystyle\times$
$\displaystyle\left[-\mu^{2}_{\rho}J_{5}(m^{2}_{\lambda},m^{2}_{\lambda},\mu^{2}_{\rho},\mu^{2}_{\rho},m^{2}_{\tilde{\nu}_{R2}})-2J_{4}(m^{2}_{\lambda},m^{2}_{\lambda},\mu^{2}_{\rho},m^{2}_{\tilde{\nu}_{R2}})\right]$
$\displaystyle+$
$\displaystyle(s_{L}c_{L})\times\frac{g^{2}c^{2}_{W}}{16\pi^{2}}\times\frac{11}{36}c_{2\gamma}$
$\displaystyle\times$
$\displaystyle\left[-\mu^{2}_{\rho}J_{5}(m^{2}_{\lambda},m^{2}_{\lambda},\mu_{\rho}^{2},\mu_{\rho}^{2},m^{2}_{\tilde{l}_{L2}})-2J_{4}(m^{2}_{\lambda},m^{2}_{\lambda},\mu_{\rho}^{2},m^{2}_{\tilde{l}_{L2}})\right.$
$\displaystyle+$
$\displaystyle\left.m_{\lambda}^{2}\left(I_{4}(m_{\lambda}^{2},m_{\lambda}^{2},\mu_{\rho}^{2},m^{2}_{\tilde{l}_{L2}})-\mu_{\rho}^{2}I_{5}(m_{\lambda}^{2},m_{\lambda}^{2},\mu_{\rho}^{2},\mu_{\rho}^{2},m^{2}_{\tilde{l}_{L2}})\right)\right]$
$\displaystyle-$
$\displaystyle(s_{\nu_{L}}c_{\nu_{L}})\times\frac{g^{2}c^{2}_{W}m^{2}_{\lambda}}{16\pi^{2}}\times\frac{1}{4}(1-c_{2\gamma})$
$\displaystyle\times$
$\displaystyle\left[\mu_{\rho}^{2}I_{5}(m_{\lambda}^{2},m_{\lambda}^{2},\mu_{\rho}^{2},\mu_{\rho}^{2},m^{2}_{\tilde{\nu}_{L2}})-I_{4}(m_{\lambda}^{2},m_{\lambda}^{2},\mu_{\rho}^{2},m^{2}_{\tilde{\nu}_{L2}})\right]$
$\displaystyle-$
$\displaystyle(s_{\nu_{R}}c_{\nu_{R}})\times\frac{g^{2}c^{2}_{W}m^{2}_{\lambda}}{16\pi^{2}}\times\frac{1}{4}(1-c_{2\gamma})$
$\displaystyle\times$
$\displaystyle\left[\mu_{\rho}^{2}I_{5}(m_{\lambda}^{2},m_{\lambda}^{2},\mu_{\rho}^{2},\mu_{\rho}^{2},m^{2}_{\tilde{\nu}_{R2}})-I_{4}(m_{\lambda}^{2},m_{\lambda}^{2},\mu_{\rho}^{2},m^{2}_{\tilde{\nu}_{R2}})\right]$
$\displaystyle+$ $\displaystyle(s_{L}c_{L})\times\frac{g^{\prime
2}c^{2}_{W}}{16\pi^{2}}\times\frac{8}{81}c_{2\gamma}$ $\displaystyle\times$
$\displaystyle\left\\{\mu^{2}_{\rho}J_{5}(m^{2}_{B},m^{2}_{\lambda},\mu_{\rho}^{2},\mu_{\rho}^{2},m^{2}_{\tilde{l}_{L2}})+2J_{4}(m^{2}_{B},m^{2}_{\lambda},\mu_{\rho}^{2},m^{2}_{\tilde{l}_{L2}})\right.$
$\displaystyle-$
$\displaystyle\left.m_{B}m_{\lambda}\left[\mu_{\rho}^{2}I_{5}(m_{B}^{2},m_{\lambda}^{2},\mu_{\rho}^{2},\mu_{\rho}^{2},m^{2}_{\tilde{l}_{L2}})-I_{4}(m_{B}^{2},m_{\lambda}^{2},\mu_{\rho}^{2},m_{\tilde{l}_{L2}})\right]\right\\}$
$\displaystyle+$
$\displaystyle(s_{\nu_{L}}c_{\nu_{L}})\times\frac{g^{2}c^{2}_{W}}{16\pi^{2}}\times\frac{1}{2}s_{2\gamma}\left[\mu_{\rho}~{}m_{\lambda}~{}J_{5}(m_{\lambda}^{2},m_{\lambda}^{2},\mu^{2}_{\rho},\mu^{2}_{\rho},m^{2}_{\tilde{\nu}_{L2}})\right]$
$\displaystyle+$
$\displaystyle(s_{\nu_{R}}c_{\nu_{R}})\times\frac{g^{2}c^{2}_{W}}{16\pi^{2}}\times\frac{1}{2}s_{2\gamma}\left[\mu_{\rho}~{}m_{\lambda}~{}J_{5}(m_{\lambda}^{2},m_{\lambda}^{2},\mu^{2}_{\rho},\mu^{2}_{\rho},m^{2}_{\tilde{\nu}_{R2}})\right]$
$\displaystyle+$ $\displaystyle(s_{L}c_{L})\times\frac{g^{\prime
2}t^{2}c^{2}_{W}}{16\pi^{2}}\times\frac{2}{729}c_{2\gamma}$
$\displaystyle\times$
$\displaystyle\left[-\mu^{2}_{\rho}J_{5}(m_{B}^{2},m_{B}^{2},\mu_{\rho}^{2},\mu_{\rho}^{2},m^{2}_{\tilde{l}_{L2}})-2J_{4}(m_{B}^{2},m_{B}^{2},\mu_{\rho}^{2},m^{2}_{\tilde{l}_{L2}})\right.$
$\displaystyle+$
$\displaystyle\left.m^{2}_{B}\left(\mu_{\rho}^{2}I_{5}(m_{B}^{2},m_{B}^{2},\mu_{\rho}^{2},\mu_{\rho}^{2},m^{2}_{\tilde{l}_{L2}})-I_{4}(m_{B}^{2},m_{B}^{2},\mu_{\rho}^{2},m^{2}_{\tilde{l}_{L2}})\right)\right]$
$\displaystyle-$ $\displaystyle(L_{2}\rightarrow L_{3},R_{2}\rightarrow
R_{3}),$ $\displaystyle A^{Z(a)}_{R}$ $\displaystyle=$
$\displaystyle(s_{R}c_{R})\frac{g^{\prime
2}t^{2}c^{2}_{W}}{16\pi^{2}}\times\frac{2}{81}c_{2\gamma}$ (83)
$\displaystyle\times$
$\displaystyle\left\\{\mu^{2}_{\rho}J_{5}(m_{B}^{2},m_{B}^{2},\mu_{\rho}^{2},\mu_{\rho}^{2},m^{2}_{\tilde{l}_{R2}})+2J_{4}(m_{B}^{2},m_{B}^{2},\mu_{\rho}^{2},m^{2}_{\tilde{l}_{R2}})\right.$
$\displaystyle-$
$\displaystyle\left.m_{B}^{2}\left[\mu_{\rho}^{2}I_{5}(m_{B}^{2},m_{B}^{2},\mu_{\rho}^{2},\mu_{\rho}^{2},m^{2}_{\tilde{l}_{R2}})\right.\right.$
$\displaystyle-$
$\displaystyle\left.\left.I_{4}(m_{B}^{2},m_{B}^{2},\mu_{\rho}^{2},m^{2}_{\tilde{l}_{R2}})\right]\right\\}-\left[R_{2}\rightarrow
R_{3}\right].$
For $A^{Z(b,c)}_{L,R}$, see Fig. 15:
$\tau$$\mu$$\tilde{\rho}^{\prime
0}$$\tilde{\rho}^{0}$$(\lambda_{3},\lambda_{8})$$\lambda_{B}$$\rho^{0}$$\rho^{0}$$\tilde{\ell}_{R_{\alpha}}$$\tilde{\ell}_{L_{\beta}}$(1)
$\tau^{c}$$\mu^{c}$$\tilde{\rho}^{\prime
0}$$\tilde{\rho}^{0}$$\lambda_{B}$$\rho^{0}$$\rho^{0}$$\tilde{\ell}_{L_{\alpha}}$$\tilde{\ell}_{R_{\beta}}$(2)
$\tau$$\mu$$\lambda_{3}$$(\lambda_{B},\lambda_{8})$$\rho^{0}$$\rho^{0}$$\tilde{\ell}_{L_{\alpha}}$$\tilde{\ell}_{L_{\gamma}}$$\tilde{\ell}_{R_{\beta}}$(3)$\tau^{c}$$\mu^{c}$$\lambda_{B}$$\rho^{0}$$\rho^{0}$$\tilde{\ell}_{R_{\alpha}}$$\tilde{\ell}_{R_{\gamma}}$$\tilde{\ell}_{L_{\beta}}$(4)
Figure 15: Diagrams contributing to $A^{Z(b)}_{L,R}$ (left side) and
$A^{Z(c)}_{L,R}$ (right side) in SUSYE331.
Formula for $A^{Z(b)}_{L}$:
$\displaystyle A^{Z(b)}_{L}$ $\displaystyle=$
$\displaystyle(s_{L}c_{L})\times\frac{m^{2}_{\tau}c^{2}_{W}}{16\pi^{2}V^{2}}\times\frac{1}{3}(t^{2}_{\gamma}~{}\mu^{2}_{\rho})\left[s^{2}_{R}\left(J_{5}(m^{2}_{\lambda},\mu^{2}_{\rho},\mu^{2}_{\rho},m^{2}_{\tilde{l}_{L_{2}}},m^{2}_{\tilde{l}_{R_{2}}})\right.\right.$
(84) $\displaystyle+$
$\displaystyle\left.\left.J_{5}(m^{2}_{\lambda},\mu_{\rho}^{2},m^{2}_{\tilde{l}_{L_{2}}},m^{2}_{\tilde{l}_{R_{2}}},m^{2}_{\tilde{l}_{R_{2}}})\right)+c^{2}_{R}\left(J_{5}(m^{2}_{\lambda},\mu^{2}_{\rho},\mu^{2}_{\rho},m^{2}_{\tilde{l}_{L_{2}}},m^{2}_{\tilde{l}_{R_{3}}})\right.\right.$
$\displaystyle+$
$\displaystyle\left.\left.J_{5}(m^{2}_{\lambda},\mu_{\rho}^{2},m^{2}_{\tilde{l}_{L_{2}}},m^{2}_{\tilde{l}_{R_{3}}},m^{2}_{\tilde{l}_{R_{3}}})\right)\right]$
$\displaystyle+$
$\displaystyle(s_{L}c_{L})\times\frac{m^{2}_{\tau}t^{2}c^{2}_{W}}{16\pi^{2}~{}V^{2}}\times\frac{1}{27}(t^{2}_{\gamma}~{}\mu^{2}_{\rho})\left[-s^{2}_{R}\left(J_{5}(m^{2}_{B},\mu^{2}_{\rho},\mu^{2}_{\rho},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{R2}})\right.\right.$
$\displaystyle+$
$\displaystyle\left.\left.J_{5}(m^{2}_{B},\mu_{\rho}^{2},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{R2}},m^{2}_{\tilde{l}_{R2}})\right)-c^{2}_{R}\left(J_{5}(m^{2}_{B},\mu^{2}_{\rho},\mu^{2}_{\rho},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{R3}})\right.\right.$
$\displaystyle+$
$\displaystyle\left.\left.J_{5}(m^{2}_{B},\mu_{\rho}^{2},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{R3}},m^{2}_{\tilde{l}_{R3}})\right)\right]-(L_{2}\rightarrow
L_{3}),$
where $m_{\tau}=Y_{\tau}\times v^{\prime}/\sqrt{2}$ is mass of the tau and
$V\equiv v_{\mathrm{weak}}=\sqrt{v^{2}+v^{{}^{\prime}2}}$ in the SUSYE331
model. We also have formula of $A^{Z(b)}_{R}$:
$\displaystyle A^{Z(b)}_{R}$ $\displaystyle=$
$\displaystyle(s_{R}c_{R})\times\frac{m_{\tau}^{2}t^{2}c^{2}_{W}}{16\pi^{2}~{}V^{2}}\times\frac{1}{9}t^{2}_{\gamma}\mu^{2}_{\rho}$
(85) $\displaystyle\times$
$\displaystyle\left[-s^{2}_{L}\left(J_{5}(m^{2}_{B},\mu^{2}_{\rho},\mu^{2}_{\rho},m^{2}_{\tilde{l}_{L_{2}}},m^{2}_{\tilde{l}_{R_{2}}})\right.\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}+\left.\left.J_{5}(m^{2}_{B},\mu_{\rho}^{2},m^{2}_{\tilde{l}_{L_{2}}},m^{2}_{\tilde{l}_{L_{2}}},m^{2}_{\tilde{l}_{R_{2}}})\right)\right.$
$\displaystyle~{}-\left.c^{2}_{L}\left(J_{5}(m^{2}_{B},\mu^{2}_{\rho},\mu^{2}_{\rho},m^{2}_{\tilde{l}_{L_{3}}},m^{2}_{\tilde{l}_{R_{2}}})\right.\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}+\left.\left.J_{5}(m^{2}_{B},\mu_{\rho}^{2},m^{2}_{\tilde{l}_{L_{3}}},m^{2}_{\tilde{l}_{L_{3}}},m^{2}_{\tilde{l}_{R_{2}}})\right)\right]-\left(R_{2}\rightarrow
R_{3}\right).$
Formulas for $A^{Z(c)}$:
$\displaystyle A^{Z(c)}_{L}$ $\displaystyle=$
$\displaystyle(s_{L}c_{L})\times\frac{m^{2}_{\tau}t^{2}c^{2}_{W}}{16\pi^{2}~{}V^{2}}\times\frac{1}{6}(t^{2}_{\gamma}\mu^{2}_{\rho})$
(86) $\displaystyle\times$
$\displaystyle\left\\{s^{2}_{L}\left[s^{2}_{R}J_{5}(m^{2}_{\lambda},m^{2}_{\tilde{l}_{R_{2}}},m^{2}_{\tilde{l}_{R_{2}}},m^{2}_{\tilde{l}_{L_{2}}},m^{2}_{\tilde{l}_{L_{2}}})\right.\right.$
$\displaystyle+$
$\displaystyle\left.\left.c^{2}_{R}J_{5}(m^{2}_{\lambda},m^{2}_{\tilde{l}_{R_{3}}},m^{2}_{\tilde{l}_{R_{3}}},m^{2}_{\tilde{l}_{L_{2}}},m^{2}_{\tilde{l}_{L_{2}}})\right]\right.$
$\displaystyle-$
$\displaystyle\left.c^{2}_{L}\left[s^{2}_{R}J_{5}(m^{2}_{\lambda},m^{2}_{\tilde{l}_{R_{2}}},m^{2}_{\tilde{l}_{R_{2}}},m^{2}_{\tilde{l}_{L_{3}}},m^{2}_{\tilde{l}_{L_{3}}})\right.\right.$
$\displaystyle+$
$\displaystyle\left.\left.c^{2}_{R}J_{5}(m^{2}_{\lambda},m^{2}_{\tilde{l}_{R_{3}}},m^{2}_{\tilde{l}_{R_{3}}},m^{2}_{\tilde{l}_{L_{3}}},m^{2}_{\tilde{l}_{L_{3}}})\right]\right.$
$\displaystyle-$
$\displaystyle\left.\left(s^{2}_{L}-c^{2}_{L}\right)\left[s^{2}_{R}J_{5}(m^{2}_{\lambda},m^{2}_{\tilde{l}_{R_{2}}},m^{2}_{\tilde{l}_{R_{2}}},m^{2}_{\tilde{l}_{L_{2}}},m^{2}_{\tilde{l}_{L_{3}}})\right.\right.$
$\displaystyle+$
$\displaystyle\left.\left.c^{2}_{R}J_{5}(m^{2}_{\lambda},m^{2}_{\tilde{l}_{R_{3}}},m^{2}_{\tilde{l}_{R_{3}}},m^{2}_{\tilde{l}_{L_{2}}},m^{2}_{\tilde{l}_{L_{3}}})\right]\right\\}$
$\displaystyle+$
$\displaystyle(s_{L}c_{L})\times\frac{m^{2}_{\tau}t^{2}c^{2}_{W}}{16\pi^{2}~{}V^{2}}\times\frac{1}{108}(t^{2}_{\gamma}\mu^{2}_{\rho})$
$\displaystyle\times$
$\displaystyle\left\\{s^{2}_{L}\left[s^{2}_{R}J_{5}(m^{2}_{B},m^{2}_{\tilde{l}_{R_{2}}},m^{2}_{\tilde{l}_{R_{2}}},m^{2}_{\tilde{l}_{L_{2}}},m^{2}_{\tilde{l}_{L_{2}}})\right.\right.$
$\displaystyle+$
$\displaystyle\left.\left.c^{2}_{R}J_{5}(m^{2}_{B},m^{2}_{\tilde{l}_{R_{3}}},m^{2}_{\tilde{l}_{R_{3}}},m^{2}_{\tilde{l}_{L_{2}}},m^{2}_{\tilde{l}_{L_{2}}})\right]\right.$
$\displaystyle-$
$\displaystyle\left.c^{2}_{L}\left[s^{2}_{R}J_{5}(m^{2}_{B},m^{2}_{\tilde{l}_{R_{2}}},m^{2}_{\tilde{l}_{R_{2}}},m^{2}_{\tilde{l}_{L_{3}}},m^{2}_{\tilde{l}_{L_{3}}})\right.\right.$
$\displaystyle+$
$\displaystyle\left.\left.c^{2}_{R}J_{5}(m^{2}_{B},m^{2}_{\tilde{l}_{R_{3}}},m^{2}_{\tilde{l}_{R_{3}}},m^{2}_{\tilde{l}_{L_{3}}},m^{2}_{\tilde{l}_{L_{3}}})\right]\right.$
$\displaystyle-$
$\displaystyle\left.\left(s^{2}_{L}-c^{2}_{L}\right)\left[s^{2}_{R}J_{5}(m^{2}_{B},m^{2}_{\tilde{l}_{R_{2}}},m^{2}_{\tilde{l}_{R_{2}}},m^{2}_{\tilde{l}_{L_{2}}},m^{2}_{\tilde{l}_{L_{3}}})\right.\right.$
$\displaystyle+$
$\displaystyle\left.\left.c^{2}_{R}J_{5}(m^{2}_{B},m^{2}_{\tilde{l}_{R_{3}}},m^{2}_{\tilde{l}_{R_{3}}},m^{2}_{\tilde{l}_{L_{2}}},m^{2}_{\tilde{l}_{L_{3}}})\right]\right\\}$
$\displaystyle A^{Z(c)}_{R}$ $\displaystyle=$
$\displaystyle(s_{R}c_{R})\times\frac{m^{2}_{\tau}t^{2}c^{2}_{W}}{16\pi^{2}~{}V^{2}}\times\frac{1}{12}(t^{2}_{\gamma}\mu^{2}_{\rho})$
(87) $\displaystyle\times$
$\displaystyle\left\\{-s^{2}_{R}\left[s^{2}_{L}J_{5}(m^{2}_{B},m^{2}_{\tilde{l}_{R_{2}}},m^{2}_{\tilde{l}_{R_{2}}},m^{2}_{\tilde{l}_{L_{2}}},m^{2}_{\tilde{l}_{L_{2}}})\right.\right.$
$\displaystyle+$
$\displaystyle\left.\left.c^{2}_{L}J_{5}(m^{2}_{B},m^{2}_{\tilde{l}_{R_{2}}},m^{2}_{\tilde{l}_{R_{2}}},m^{2}_{\tilde{l}_{L_{3}}},m^{2}_{\tilde{l}_{L_{3}}})\right]\right.$
$\displaystyle+$
$\displaystyle\left.c^{2}_{R}\left[s^{2}_{L}J_{5}(m^{2}_{B},m^{2}_{\tilde{l}_{R_{3}}},m^{2}_{\tilde{l}_{R_{3}}},m^{2}_{\tilde{l}_{L_{2}}},m^{2}_{\tilde{l}_{L_{2}}})\right.\right.$
$\displaystyle+$
$\displaystyle\left.\left.c^{2}_{L}J_{5}(m^{2}_{B},m^{2}_{\tilde{l}_{R_{3}}},m^{2}_{\tilde{l}_{R_{3}}},m^{2}_{\tilde{l}_{L_{3}}},m^{2}_{\tilde{l}_{L_{3}}})\right]\right.$
$\displaystyle+$
$\displaystyle\left.\left(s^{2}_{R}-c^{2}_{R}\right)\left[s^{2}_{L}J_{5}(m^{2}_{B},m^{2}_{\tilde{l}_{R_{2}}},m^{2}_{\tilde{l}_{R_{3}}},m^{2}_{\tilde{l}_{L_{2}}},m^{2}_{\tilde{l}_{L_{2}}})\right.\right.$
$\displaystyle+$
$\displaystyle\left.\left.c^{2}_{L}J_{5}(m^{2}_{B},m^{2}_{\tilde{l}_{R_{2}}},m^{2}_{\tilde{l}_{R_{3}}},m^{2}_{\tilde{l}_{L_{3}}},m^{2}_{\tilde{l}_{L_{3}}})\right]\right\\}.$
### C.2 Contributions to $C^{Z}_{L,R}$
For $C^{Z}_{L,R}$, in Fig. 16.
$\tau$$\mu$$\tilde{W}^{+}$$\tilde{W}^{+}$$Z$$\tilde{\nu}_{\alpha}$(1)
$\tau$$\mu$$\tilde{W}^{+}$$Z$($Z^{\prime}$)$\tilde{\nu}_{\alpha}$(2)
$\tau$$\mu$$(\lambda_{3},\lambda_{8})$$\lambda_{B}$$Z$($Z^{\prime}$)$\tilde{\ell}_{L_{\alpha}}$(3)
$\tau$$\mu$$\tilde{Y}^{+}$$\tilde{Y}^{+}$$Z$($Z^{\prime}$)$\tilde{\nu}_{R\alpha}$(4)
$\tau$$\mu$$\tilde{Y}^{+}$$Z^{\prime}$$\tilde{\nu}_{R\alpha}$(5)
$\tau^{c}$$\mu^{c}$$\lambda_{B}$$Z$$(Z^{\prime})$$\tilde{\ell}_{R_{\alpha}}$(6)
Figure 16: Diagrams contributing to $C^{Z}_{L,R}$ ($C^{Z^{\prime}}_{L,R}$ ).
Only the last gives contribution to $C^{Z}_{R}$ ($C^{Z^{\prime}}_{R}$). The
first diagram only contributes to $C^{Z}_{L}$ while the fifth only contributes
to $C^{Z^{\prime}}_{L}$.
The formulas for these two quantities are written as below:
$\displaystyle C^{Z}_{L}$ $\displaystyle=$
$\displaystyle(c_{\nu_{L}}s_{\nu_{L}})\times\frac{g^{2}}{16\pi^{2}}\times\frac{1}{12}\left[-K_{5}(m^{2}_{\lambda},m^{2}_{\tilde{\nu}_{L2}},m^{2}_{\tilde{\nu}_{L2}},m^{2}_{\tilde{\nu}_{L2}},m^{2}_{\tilde{\nu}_{L2}})\right]$
(88) $\displaystyle+$
$\displaystyle(c_{L}s_{L})\times\frac{g^{2}}{16\pi^{2}}\times\frac{1}{18}c_{2W}\left[-K_{5}(m^{2}_{\lambda},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}})\right]$
$\displaystyle+$
$\displaystyle(c_{\nu_{L}}s_{\nu_{L}})\times\frac{g^{2}}{16\pi^{2}}\times\frac{1}{12}c^{2}_{W}\left[-2K_{5}(m^{2}_{\lambda},m^{2}_{\lambda},m^{2}_{\lambda},m^{2}_{\lambda},\tilde{m}^{2}_{\nu_{L2}})\right.$
$\displaystyle+$
$\displaystyle\left.3m^{2}_{\lambda}J_{5}(m^{2}_{\lambda},m^{2}_{\lambda},m^{2}_{\lambda},m^{2}_{\lambda},\tilde{m}^{2}_{\nu_{L2}})\right]$
$\displaystyle+$
$\displaystyle(c_{\nu_{R}}s_{\nu_{R}})\times\frac{g^{2}}{16\pi^{2}}\times\frac{c_{2W}}{12}\left[-2K_{5}(m^{2}_{\lambda},m^{2}_{\lambda},m^{2}_{\lambda},m^{2}_{\lambda},\tilde{m}^{2}_{\nu_{R2}})\right.$
$\displaystyle+$
$\displaystyle\left.3m^{2}_{\lambda}J_{5}(m^{2}_{\lambda},m^{2}_{\lambda},m^{2}_{\lambda},m^{2}_{\lambda},\tilde{m}^{2}_{\nu_{R2}})\right]$
$\displaystyle+$ $\displaystyle(c_{L}s_{L})\times\frac{g^{\prime
2}}{16\pi^{2}}\times\frac{1}{324}(1-2s^{2}_{W})\left[K_{5}(m^{2}_{B},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}})\right]$
$\displaystyle-$ $\displaystyle\left[L_{2}\rightarrow
L_{3},~{}R_{2}\rightarrow R_{3}\right],$ $\displaystyle C^{Z}_{R}$
$\displaystyle=$ $\displaystyle(c_{R}s_{R})\times\frac{g^{\prime
2}}{16\pi^{2}}\times\frac{1}{36}s^{2}_{W}\left[K_{5}(m^{2}_{B},m^{2}_{\tilde{l}_{R2}},m^{2}_{\tilde{l}_{R2}},m^{2}_{\tilde{l}_{R2}},m^{2}_{\tilde{l}_{R2}})\right]$
(89) $\displaystyle-$ $\displaystyle[R_{2}\rightarrow R_{3}].$
We note that because $Z$ boson couples much weakly to right-handed neutrinos
so the diagram 5 in the Fig.16 give suppressed contribution to $C^{Z}_{L}$. In
contrast, the case of $Z^{\prime}$ boson is different, it weakly couples with
$\tilde{W}^{\pm}$ but non-negligible to right-handed neutrinos. So for the
$C^{Z^{\prime}}_{L}$, we neglect the first diagram and keep the fifth. This
conclusion is held in the case of $D^{Z}$ and $D^{Z^{\prime}}$.
### C.3 Contributions to $D^{Z}_{L,R}$
For $D^{Z}_{L,R}$, we have $D^{Z}_{L,R}=D^{Z(b)}_{L,R}+D^{Z(c)}_{L,R}$. They
are presented by diagrams in figs. 17 and 18.
$\tau^{c}$$\mu$$\rho^{0}$$\tilde{\rho}^{\prime-}_{1}$$\tilde{\rho}^{+}_{1}$$\tilde{W}^{-}$$\tilde{W}^{+}$$Z$$\tilde{\nu}_{\alpha}$(1)
$\tau^{c}$$\mu$$\rho^{0}$$\tilde{\rho}^{\prime-}_{1}$$\tilde{\rho}^{+}_{1}$$\tilde{W}^{-}$$\tilde{W}^{+}$$Z$$(Z^{\prime})$$\tilde{\nu}_{\alpha}$(2)
$\tau^{c}$$\mu$$\rho^{0}$$\tilde{\rho}^{\prime-}_{1}$$\tilde{\rho}^{+}_{1}$$\tilde{W}^{-}$$\tilde{W}^{+}$$Z$$(Z^{\prime})$$\tilde{\nu}_{\alpha}$(3)
$\tau^{c}$$\mu$$\rho^{0}$$\tilde{\rho}^{\prime-}_{2}$$\tilde{\rho}^{+}_{2}$$\tilde{Y}^{-}$$\tilde{Y}^{+}$$Z$$(Z^{\prime})$$\tilde{\nu}_{R\alpha}$(4)
$\tau^{c}$$\mu$$\rho^{0}$$\tilde{\rho}^{\prime-}_{2}$$\tilde{\rho}^{+}_{2}$$\tilde{Y}^{-}$$\tilde{Y}^{+}$$Z$$(Z)$$\tilde{\nu}_{R\alpha}$(5)
$\tau^{c}$$\mu$$\rho^{0}$$\tilde{\rho}^{\prime-}_{2}$$\tilde{\rho}^{+}_{2}$$\tilde{Y}^{-}$$\tilde{Y}^{+}$$Z^{\prime}$$\tilde{\nu}_{R\alpha}$(6)
$\tau^{c}$$\mu$$\rho^{0}$$\tilde{\rho}^{\prime
0}$$\tilde{\rho}^{0}$$\lambda_{3}$$(\lambda_{B},\lambda_{8})$$Z$$(Z^{\prime})$$\tilde{\ell}_{L_{\alpha}}$(7)
$\tau^{c}$$\mu$$\rho^{0}$$\tilde{\rho}^{\prime
0}$$\tilde{\rho}^{0}$$\lambda_{3}$$(\lambda_{B},\lambda_{8})$$Z$$(Z^{\prime})$$\tilde{\ell}_{L_{\alpha}}$(8)
$\tau$$\mu^{c}$$\rho^{0}$$\tilde{\rho}^{\prime
0}$$\tilde{\rho}^{0}$$\lambda_{B}$$Z$$(Z^{\prime})$$\tilde{\ell}_{R_{\alpha}}$(9)$\tau$$\mu^{c}$$\rho^{0}$$\tilde{\rho}^{\prime
0}$$\tilde{\rho}^{0}$$\lambda_{B}$$Z$$(Z^{\prime})$$\tilde{\ell}_{R_{\alpha}}$(10)
Figure 17: Diagrams contributing to $D^{Z(b)}_{L}$ ($D^{Z^{\prime}(b)}_{L}$)
(two first lines) and $D^{Z(b)}_{R}$ ($D^{Z^{\prime}(b)}_{R}$) (the last
line). Noting that the first diagram only contributes to $D^{Z(b)}_{L}$ while
the sixth only contributes to $D^{Z^{\prime}(b)}_{L}$.
Formulas for $D^{Z(b)}$:
$\displaystyle D^{Z(b)}_{L}$ $\displaystyle=$
$\displaystyle(s_{L}c_{L})\frac{g^{2}}{16\pi^{2}}\times\frac{1}{6}\times\mu_{\rho}m_{\lambda}\tan\gamma$
(90) $\displaystyle\times$
$\displaystyle\left[2J_{5}(m_{\lambda}^{2},\mu^{2}_{\rho},\mu^{2}_{\rho},\mu^{2}_{\rho},m^{2}_{\tilde{l}_{L2}})+J_{5}(m_{\lambda}^{2},m_{\lambda}^{2},\mu^{2}_{\rho},\mu^{2}_{\rho},m^{2}_{\tilde{l}_{L2}})\right]$
$\displaystyle+$
$\displaystyle(s_{L}c_{L})\frac{g^{2}}{16\pi^{2}}\times\frac{1}{3}\mu_{\rho}m_{\lambda}\tan\gamma$
$\displaystyle\times$ $\displaystyle
c_{2W}\left[m^{2}_{\tilde{l}_{L2}}I_{5}(m_{\lambda}^{2},\mu_{\rho}^{2},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}})\right]$
$\displaystyle+$
$\displaystyle(s_{\nu_{L}}c_{\nu_{L}})\frac{g^{2}}{16\pi^{2}}\times\frac{1}{2}\mu_{\rho}m_{\lambda}\tan\gamma$
$\displaystyle\times$ $\displaystyle
m^{2}_{\tilde{\nu}_{L2}}I_{5}(m_{\lambda}^{2},\mu_{\rho}^{2},m^{2}_{\tilde{\nu}_{L2}},m^{2}_{\tilde{\nu}_{L2}},m^{2}_{\tilde{\nu}_{L2}})$
$\displaystyle-$
$\displaystyle(s_{\nu_{L}}c_{\nu_{L}})\frac{g^{2}}{16\pi^{2}}\times\frac{1}{2}\left(\mu_{\rho}m_{\lambda}\tan\gamma\right)$
$\displaystyle\times$ $\displaystyle
c^{2}_{W}\left[2J_{5}(m_{\lambda}^{2},m_{\lambda}^{2},m_{\lambda}^{2},\mu^{2}_{\rho},m^{2}_{\tilde{\nu}_{L2}})+J_{5}(m_{\lambda}^{2},m_{\lambda}^{2},\mu^{2}_{\rho},\mu^{2}_{\rho},m^{2}_{\tilde{\nu}_{L2}})\right]$
$\displaystyle-$
$\displaystyle(s_{\nu_{R}}c_{\nu_{R}})\frac{g^{2}}{16\pi^{2}}\times\frac{1}{4}\left(\mu_{\rho}m_{\lambda}\tan\gamma\right)\times$
$\displaystyle\times$ $\displaystyle
c_{2W}\left[2J_{5}(m_{\lambda}^{2},m_{\lambda}^{2},m_{\lambda}^{2},\mu^{2}_{\rho},m^{2}_{\tilde{\nu}_{R2}})+J_{5}(m_{\lambda}^{2},m_{\lambda}^{2},\mu^{2}_{\rho},\mu^{2}_{\rho},m^{2}_{\tilde{\nu}_{R2}})\right]$
$\displaystyle+$
$\displaystyle(s_{\nu_{L}}c_{\nu_{L}})\frac{g^{2}}{16\pi^{2}}\times\left(\mu_{\rho}m_{\lambda}\tan\gamma\right)\times\frac{1}{4}$
$\displaystyle\times$
$\displaystyle\left(-1+2s^{2}_{W}\right)\left[2J_{5}(m_{\lambda}^{2},\mu^{2}_{\rho},\mu^{2}_{\rho},\mu^{2}_{\rho},m^{2}_{\tilde{\nu}_{L2}})+J_{5}(m_{\lambda}^{2},m_{\lambda}^{2},\mu^{2}_{\rho},\mu^{2}_{\rho},m^{2}_{\tilde{\nu}_{L2}})\right]$
$\displaystyle+$
$\displaystyle(s_{\nu_{R}}c_{\nu_{R}})\frac{g^{2}}{16\pi^{2}}\times\left(\mu_{\rho}m_{\lambda}\tan\gamma\right)\times\frac{1}{4}$
$\displaystyle\times$ $\displaystyle
s^{2}_{W}\left[2J_{5}(m_{\lambda}^{2},\mu^{2}_{\rho},\mu^{2}_{\rho},\mu^{2}_{\rho},m^{2}_{\tilde{\nu}_{R2}})+J_{5}(m_{\lambda}^{2},m_{\lambda}^{2},\mu^{2}_{\rho},\mu^{2}_{\rho},m^{2}_{\tilde{\nu}_{R2}})\right]$
$\displaystyle+$ $\displaystyle(s_{L}c_{L})\frac{g^{\prime
2}}{16\pi^{2}}\times\frac{1}{27}\times\mu_{\rho}m_{B}\tan\gamma$
$\displaystyle\times$ $\displaystyle
c_{2W}m^{2}_{\tilde{l}_{L2}}I_{5}(m_{B}^{2},\mu_{\rho}^{2},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}})$
$\displaystyle+$ $\displaystyle(s_{L}c_{L})\frac{g^{\prime
2}}{16\pi^{2}}\times\frac{1}{54}\times\mu_{\rho}m_{B}\tan\gamma$
$\displaystyle\times$
$\displaystyle\left[2J_{5}(m_{B}^{2},\mu^{2}_{\rho},\mu^{2}_{\rho},\mu^{2}_{\rho},m^{2}_{\tilde{l}_{L2}})+J_{5}(m_{B}^{2},m_{B}^{2},\mu^{2}_{\rho},\mu^{2}_{\rho},m^{2}_{\tilde{l}_{L2}})\right]$
$\displaystyle-$ $\displaystyle(L_{2}\rightarrow L_{3},~{}R_{2}\rightarrow
R_{3}),$ $\displaystyle D^{Z(b)}_{R}$ $\displaystyle=$
$\displaystyle-(s_{R}c_{R})\frac{g^{\prime
2}}{16\pi^{2}}\times\frac{1}{18}m_{B}\mu_{\rho}\tan\gamma$ (91)
$\displaystyle\times$
$\displaystyle\left[(-4s^{2}_{W})m^{2}_{\tilde{l}_{R2}}I_{5}(m_{B}^{2},\mu_{\rho}^{2},m^{2}_{\tilde{l}_{R2}},m^{2}_{\tilde{l}_{R2}},m^{2}_{\tilde{l}_{R2}})\right.$
$\displaystyle+$
$\displaystyle\left.2J_{5}(m_{B}^{2},\mu_{\rho}^{2},\mu_{\rho}^{2},\mu_{\rho}^{2},m^{2}_{\tilde{l}_{R2}})+J_{5}(m_{B}^{2},m_{B}^{2},\mu_{\rho}^{2},\mu_{\rho}^{2},m^{2}_{\tilde{l}_{R2}})\right]$
$\displaystyle-$ $\displaystyle(R_{2}\rightarrow R_{3}).$
$\tau^{c}$$\mu$$\lambda_{B}$$Z$$(Z^{\prime})$$\rho^{0}$$\tilde{\ell}_{R_{\alpha}}$$\tilde{\ell}_{L_{\beta}}$
$\tau^{c}$$\mu$$\lambda_{B}$$Z$$(Z^{\prime})$$\rho^{0}$$\tilde{\ell}_{R_{\alpha}}$$\tilde{\ell}_{L_{\beta}}$
$\tau$$\mu^{c}$$\lambda_{B}$$Z$$(Z^{\prime})$$\rho^{0}$$\tilde{\ell}_{L_{\alpha}}$$\tilde{\ell}_{R_{\beta}}$$\tau$$\mu^{c}$$\lambda_{B}$$Z$$(Z^{\prime})$$\rho^{0}$$\tilde{\ell}_{L_{\alpha}}$$\tilde{\ell}_{R_{\beta}}$
Figure 18: Diagrams that contribute to $D^{Z(c)}_{L,R}$
($D^{Z^{\prime}(c)}_{L,R}$).
Formulas for $D^{Z(c)}_{L,R}$:
$\displaystyle D^{Z(c)}_{L}$ $\displaystyle=$
$\displaystyle-(s_{L}c_{L})\frac{g^{\prime
2}}{16\pi^{2}}m_{B}\mu_{\rho}\tan\gamma\times\frac{1}{72}$ (92)
$\displaystyle\times$
$\displaystyle\left[(1-2s^{2}_{W})\left(s^{2}_{R}J_{5}(m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{R2}})\right.\right.$
$\displaystyle+$
$\displaystyle\left.c^{2}_{R}J_{5}(m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{R3}})\left.\right)\right.$
$\displaystyle+$
$\displaystyle\left.2s^{2}_{W}\left(s^{2}_{R}J_{5}(m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{R2}},m^{2}_{\tilde{l}_{R2}})\right.\right.$
$\displaystyle+$
$\displaystyle\left.c^{2}_{R}J_{5}(m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{R3}},m^{2}_{\tilde{l}_{R3}})\left.\right)\right]-(L_{2}\rightarrow
L_{3}),$ $\displaystyle D^{Z(c)}_{R}$ $\displaystyle=$
$\displaystyle-(s_{R}c_{R})\times\frac{g^{\prime
2}}{16\pi^{2}}m_{B}\mu_{\rho}\tan\gamma\times\frac{1}{72}$ (93)
$\displaystyle\times$
$\displaystyle\left[(1-2s^{2}_{W})\left(s^{2}_{L}J_{5}(m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{R2}})\right.\right.$
$\displaystyle+$
$\displaystyle\left.c^{2}_{L}J_{5}(m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{L3}},m^{2}_{\tilde{l}_{L3}},m^{2}_{\tilde{l}_{R2}})\left.\right)\right.$
$\displaystyle+$
$\displaystyle\left.2s^{2}_{W}\left(s^{2}_{L}J_{5}(m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{R2}},m^{2}_{\tilde{l}_{R2}})\right.\right.$
$\displaystyle+$
$\displaystyle\left.c^{2}_{L}J_{5}(m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{L3}},m^{2}_{\tilde{l}_{R2}},m^{2}_{\tilde{l}_{R2}})\left.\right)\right]-(R_{2}\rightarrow
R_{3}).$
## Appendix D Contributions to $Z^{\prime}\rightarrow\mu\tau$
### D.1 Contributions to $A^{1Z^{\prime}}_{L,R}$
To determine the values of $A^{Z}_{L,R}$, $A^{1Z^{\prime}}_{L,R}$ and
$A^{2Z^{\prime}}_{L,R}$, we use techniques mentioned in [10]. From formulas of
covariant derivatives of neutral Higgs in Appendix A.2, it is easy to see that
two terms relating with $Z$ and $Z^{\prime}$ bosons appearing in these
covariant derivatives are different from each others one factor $(-1)$. For
$A^{1Z^{\prime}}_{L,R}$ which relates with $\rho^{0}$ and $\rho^{\prime 0}$ we
have
$A^{1Z^{\prime}}_{L,R}=A^{1Z^{\prime}(a)}_{L,R}+A^{Z^{\prime}(b)}_{L,R}+A^{Z^{\prime}(c)}_{L,R}$
and
$A^{Z^{\prime}}_{L(R)}=(m^{2}_{Z}/m^{2}_{Z^{\prime}})A^{1Z^{\prime}}_{L(R)}+A^{2Z^{\prime}}_{L(R)}$.
This leads to the results:
$\displaystyle A^{1Z^{\prime}(a)}_{L,R}=-A^{Z(a)}_{L,R},$ $\displaystyle
A^{Z^{\prime}(b)}_{L,R}=-A^{Z(b)}_{L,R},$ $\displaystyle
A^{Z^{\prime}(c)}_{L,R}=-A^{Z(c)}_{L,R}.$ (94)
### D.2 Contributions to $A^{2Z^{\prime}}_{L,R}$
Diagrams contributing to $A^{2Z^{\prime}}_{L,R}$ are quite similar to those
shown in Fig.14.
$\tau$$\mu$$\tilde{Y}^{+}$$\tilde{\chi}^{-}$$\tilde{\chi}^{\prime+}$$\tilde{Y}^{-}$$\tilde{Y}^{+}$$\chi^{0}_{2}$$\chi^{\prime
0}_{2}$$\tilde{\nu}_{R_{\alpha}}$(1)
$\tau$$\mu$$\tilde{Y}^{+}$$\tilde{Y}^{-}$$\tilde{\chi}^{\prime+}$$\tilde{\chi}^{-}$$\tilde{Y}^{+}$$\chi^{\prime
0}_{2}$$\chi^{0}_{2}$$\tilde{\nu}_{R_{\alpha}}$(2)
$\tau$$\mu$$\tilde{Y}^{+}$$\tilde{\chi}^{-}$$\tilde{Y}^{+}$$\chi^{0}_{2}$$\chi^{0}_{2}$$\tilde{\nu}_{R_{\alpha}}$(3)$\tau$$\mu$$\tilde{Y}^{+}$$\tilde{Y}^{-}$$\tilde{\chi}^{\prime+}$$\tilde{Y}^{-}$$\tilde{Y}^{+}$$\chi^{\prime
0}_{2}$$\chi^{\prime 0}_{2}$$\tilde{\nu}_{R_{\alpha}}$(4)
$\tau$$\mu$$\lambda_{i}$$\tilde{H}^{0}_{k}$$\lambda_{i}$$H^{0}_{k}$$H^{0}_{k}$$\tilde{\ell}_{L_{\alpha}}$(5)
$\tau$$\mu$$\lambda_{i}$$\tilde{H}^{0}_{k}$$\lambda_{i}$$H^{0}_{k}$$H^{0}_{k}$$\tilde{\ell}_{L_{\alpha}}$(6)
$\tau$$\mu$$\lambda_{i}$$\tilde{H}^{0}_{k}$$\lambda_{j}$$H^{0}_{k}$$H^{0}_{k}$$\tilde{\ell}_{L_{\alpha}}$(7)$\tau$$\mu$$\lambda_{i}$$\tilde{H}^{0}_{k}$$\lambda_{j}$$H_{k}^{0}$$H_{k}^{0}$$\tilde{\ell}_{L_{\alpha}}$(8)
$\tau^{c}$$\mu^{c}$$\lambda_{B}$$\tilde{H}^{0}_{k}$$\lambda_{B}$$H^{0}_{k}$$H^{0}_{k}$$\tilde{\ell}_{R_{\alpha}}$(9)
$\tau^{c}$$\mu^{c}$$\lambda_{B}$$\tilde{H}^{0}_{k}$$\lambda_{B}$$H^{0}_{k}$$H^{0}_{k}$$\tilde{\ell}_{R_{\alpha}}$(10)
Figure 19: Diagrams contributing to $A^{(2Z^{\prime})}_{L}$ (first and second
rows) and $A^{(2Z^{\prime})}_{R}$ (third row). Here we denote
$H^{0}_{k}\in\\{\chi^{0}_{2},~{}\chi^{\prime 0}_{2}\\}$ while $\lambda_{i,j}$
implies $i,j=\\{B,8\\}$ and $i\neq j$.
There is a interesting point in the SUSYE331 model that both Higgs $\chi$ and
$\chi^{\prime}$ do not couple to leptons and sleptons. As a consequence,
$A^{(2Z^{\prime})}_{L,R}$ give contributions from only the class of diagrams
depicted in Fig.19 where $\rho^{0}$, $\rho^{\prime 0}$ and $Z$ boson in Fig.14
are correspondingly replaced with $\chi^{0}_{2}$, $\chi^{0}_{2}$ and
$Z^{\prime}$ boson. We use the equivalence role between
$\rho^{0}\leftrightarrow\chi^{0}_{2}$ and $\rho^{\prime
0}\leftrightarrow\chi^{\prime 0}_{2}$ for the calculation ( see App.A.2). The
results are:
$\displaystyle A^{(2Z^{\prime})}_{L}$ $\displaystyle=$
$\displaystyle(s_{\nu_{R}}c_{\nu_{R}})\frac{g^{2}\kappa_{1}^{2}}{16\pi^{2}}\times\frac{1}{4}m_{\lambda}\mu_{\chi}s_{2\beta}~{}J_{5}(m^{2}_{\lambda},m^{2}_{\lambda},\mu^{2}_{\chi},\mu^{2}_{\chi},\tilde{m}^{2}_{\tilde{\nu}_{R2}})$
(95) $\displaystyle-$
$\displaystyle(s_{\nu_{R}}c_{\nu_{R}})\frac{g^{2}\kappa_{1}^{2}}{16\pi^{2}}\times\frac{1}{4}s^{2}_{\beta}\left[~{}~{}2J_{4}(m^{2}_{\lambda},m^{2}_{\lambda},\mu^{2}_{\chi},\tilde{m}^{2}_{\tilde{\nu}_{R2}})\right.$
$\displaystyle\left.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\mu^{2}_{\chi}J_{5}(m^{2}_{\lambda},m^{2}_{\lambda},\mu^{2}_{\chi},\mu^{2}_{\chi},\tilde{m}^{2}_{\tilde{\nu}_{R2}})\right]$
$\displaystyle-$
$\displaystyle(s_{\nu_{R}}c_{\nu_{R}})\frac{g^{2}\kappa_{1}^{2}}{16\pi^{2}}\times\frac{1}{4}m^{2}_{\lambda}c^{2}_{\beta}\left[~{}~{}\mu^{2}_{\chi}I_{5}(m^{2}_{\lambda},m^{2}_{\lambda},\mu^{2}_{\chi},\mu^{2}_{\chi},\tilde{m}^{2}_{\tilde{\nu}_{R2}})\right.$
$\displaystyle\left.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-I_{4}(m^{2}_{\lambda},m^{2}_{\lambda},\mu^{2}_{\chi},\tilde{m}^{2}_{\tilde{\nu}_{R2}})\right]$
$\displaystyle+$ $\displaystyle(s_{L}c_{L})\frac{g^{\prime
2}t^{2}\kappa_{1}^{2}}{16\pi^{2}}\times\frac{1}{2916}c_{2\beta}\left[~{}2J_{4}(m^{2}_{B},m^{2}_{B},\mu^{2}_{\chi},\tilde{m}^{2}_{\tilde{\ell}_{L2}})\right.$
$\displaystyle\left.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\mu^{2}_{\chi}J_{5}(m^{2}_{B},m^{2}_{B},\mu^{2}_{\chi},\mu^{2}_{\chi},\tilde{m}^{2}_{\tilde{\ell}_{L2}})\right]$
$\displaystyle+$
$\displaystyle(s_{L}c_{L})\frac{g^{2}\kappa_{1}^{2}}{16\pi^{2}}\times\frac{1}{9}c_{2\beta}\left[~{}2J_{4}(m^{2}_{\lambda},m^{2}_{\lambda},\mu^{2}_{\chi},\tilde{m}^{2}_{\tilde{\ell}_{L2}})\right.$
$\displaystyle\left.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\mu^{2}_{\chi}J_{5}(m^{2}_{\lambda},m^{2}_{\lambda},\mu^{2}_{\chi},\mu^{2}_{\chi},\tilde{m}^{2}_{\tilde{\ell}_{L2}})\right]$
$\displaystyle-$ $\displaystyle(s_{L}c_{L})\frac{g^{\prime
2}t^{2}\kappa_{1}^{2}}{16\pi^{2}}\times\frac{1}{2916}m^{2}_{B}c_{2\beta}\left[~{}~{}\mu^{2}_{\chi}I_{5}(m^{2}_{B},m^{2}_{B},\mu^{2}_{\chi},\mu^{2}_{\chi},\tilde{m}^{2}_{\tilde{\ell}_{L2}})\right.$
$\displaystyle\left.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-I_{4}(m^{2}_{B},m^{2}_{B},\mu^{2}_{\chi},\tilde{m}^{2}_{\tilde{\ell}_{L2}})\right]$
$\displaystyle-$
$\displaystyle(s_{L}c_{L})\frac{g^{2}\kappa_{1}^{2}}{16\pi^{2}}\times\frac{1}{9}m^{2}_{\lambda}c_{2\beta}\left[~{}~{}\mu^{2}_{\chi}I_{5}(m^{2}_{\lambda},m^{2}_{\lambda},\mu^{2}_{\chi},\mu^{2}_{\chi},\tilde{m}^{2}_{\tilde{\ell}_{L2}})\right.$
$\displaystyle\left.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-I_{4}(m^{2}_{\lambda},m^{2}_{\lambda},\mu^{2}_{\chi},\tilde{m}^{2}_{\tilde{\ell}_{L2}})\right]$
$\displaystyle-$ $\displaystyle(s_{L}c_{L})\frac{g^{\prime
2}\kappa_{1}^{2}}{16\pi^{2}}\times\frac{1}{162}c_{2\beta}\left[~{}2J_{4}(m^{2}_{B},m^{2}_{\lambda},\mu^{2}_{\chi},\tilde{m}^{2}_{\tilde{\ell}_{L2}})\right.$
$\displaystyle\left.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\mu^{2}_{\chi}J_{5}(m^{2}_{B},m^{2}_{\lambda},\mu^{2}_{\chi},\mu^{2}_{\chi},\tilde{m}^{2}_{\tilde{\ell}_{L2}})\right]$
$\displaystyle+$ $\displaystyle(s_{L}c_{L})\frac{g^{\prime
2}\kappa_{1}^{2}}{16\pi^{2}}\times\frac{1}{162}m_{B}m_{\lambda}c_{2\beta}\left[~{}~{}\mu^{2}_{\chi}I_{5}(m^{2}_{B},m^{2}_{\lambda},\mu^{2}_{\chi},\mu^{2}_{\chi},\tilde{m}^{2}_{\tilde{\ell}_{L2}})\right.$
$\displaystyle\left.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-I_{4}(m^{2}_{B},m^{2}_{\lambda},\mu^{2}_{\chi},\tilde{m}^{2}_{\tilde{\ell}_{L2}})\right]$
$\displaystyle-$ $\displaystyle\left(L_{2}\rightarrow L_{3},R_{2}\rightarrow
R_{3}\right),$ $\displaystyle A^{(2Z^{\prime})}_{R}$ $\displaystyle=$
$\displaystyle-(s_{R}c_{R})\frac{g^{\prime
2}t^{2}\kappa_{1}^{2}}{16\pi^{2}}\times\frac{1}{324}c_{2\beta}\left[~{}2J_{4}(m^{2}_{B},m^{2}_{B},\mu^{2}_{\chi},\tilde{m}^{2}_{\tilde{\ell}_{R2}})\right.$
(96)
$\displaystyle\left.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\mu^{2}_{\chi}J_{5}(m^{2}_{B},m^{2}_{B},\mu^{2}_{\chi},\mu^{2}_{\chi},\tilde{m}^{2}_{\tilde{\ell}_{R2}})\right]$
$\displaystyle+$ $\displaystyle(s_{R}c_{R})\frac{g^{\prime
2}\kappa_{1}^{2}}{16\pi^{2}}\times\frac{1}{324}m^{2}_{B}c_{2\beta}\left[~{}~{}\mu^{2}_{\chi}I_{5}(m^{2}_{B},m^{2}_{B},\mu^{2}_{\chi},\mu^{2}_{\chi},\tilde{m}^{2}_{\tilde{\ell}_{R2}})\right.$
$\displaystyle\left.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-I_{4}(m^{2}_{B},m^{2}_{B},\mu^{2}_{\chi},\tilde{m}^{2}_{\tilde{\ell}_{R2}})\right]$
$\displaystyle-$ $\displaystyle\left(R_{2}\rightarrow R_{3}\right).$
### D.3 Contributions to $C^{Z^{\prime}}_{L,R}$
Diagrams contributing to $C^{Z^{\prime}}_{L,R}$ are those from 2-6 in the Fig.
16. Comparing with the case of the $Z$ boson we easily deduce the formulas as:
$\displaystyle C^{Z^{\prime}}_{L}$ $\displaystyle=$
$\displaystyle-(c_{\nu_{L}}s_{\nu_{L}})\times\frac{g^{2}}{16\pi^{2}}\times\frac{1}{12}c_{2W}\left[K_{5}(m^{2}_{\lambda},m^{2}_{\tilde{\nu}_{L2}},m^{2}_{\tilde{\nu}_{L2}},m^{2}_{\tilde{\nu}_{L2}},m^{2}_{\tilde{\nu}_{L2}})\right]$
(97) $\displaystyle-$
$\displaystyle(c_{L}s_{L})\times\frac{g^{2}}{16\pi^{2}}\times\frac{1}{18}c_{2W}\left[-K_{5}(m^{2}_{\lambda},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}})\right]$
$\displaystyle-$
$\displaystyle(c_{\nu_{R}}s_{\nu_{R}})\times\frac{g^{2}}{16\pi^{2}}\times\frac{4c^{2}_{W}-1}{12}\left[-2K_{5}(m^{2}_{\lambda},m^{2}_{\lambda},m^{2}_{\lambda},m^{2}_{\lambda},\tilde{m}^{2}_{\nu_{R2}})\right.$
$\displaystyle+$
$\displaystyle\left.3m^{2}_{\lambda}J_{5}(m^{2}_{\lambda},m^{2}_{\lambda},m^{2}_{\lambda},m^{2}_{\lambda},\tilde{m}^{2}_{\nu_{R2}})\right]$
$\displaystyle+$ $\displaystyle(c_{L}s_{L})\times\frac{g^{\prime
2}}{16\pi^{2}}\times\frac{1}{324}(1-2s^{2}_{W})\left[K_{5}(m^{2}_{B},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}})\right]$
$\displaystyle+$
$\displaystyle(c_{\nu_{R}}s_{\nu_{R}})\times\frac{g^{2}}{16\pi^{2}}\times\frac{1}{12}c^{2}_{W}\left[K_{5}(m^{2}_{\lambda},m^{2}_{\tilde{\nu}_{R2}},m^{2}_{\tilde{\nu}_{R2}},m^{2}_{\tilde{\nu}_{R2}},m^{2}_{\tilde{\nu}_{R2}})\right]$
$\displaystyle-$ $\displaystyle[L_{2}\rightarrow L_{3}],$
$C^{Z^{\prime}}_{R}=-C^{Z}_{R}.$ (98)
### D.4 Contributions to $D^{Z^{\prime}}_{L,R}$
Contribution to $D^{Z^{\prime}}_{L,R}$ can be deduced from diagrams shown for
$D^{Z}_{L,R}$ in figs. 17 and 18. We also write
$D^{Z^{\prime}}_{L,R}=D^{Z^{\prime}(b)}_{L,R}+D^{Z^{\prime}(c)}_{L,R}$. From
Fig. 17 we can deduce formulas to determine $D^{Z^{\prime}(b)}$:
$\displaystyle D^{Z^{\prime}(b)}_{L}$ $\displaystyle=$
$\displaystyle-(s_{\nu_{L}}c_{\nu_{L}})\frac{g^{2}}{16\pi^{2}}\times\left(\mu_{\rho}m_{\lambda}\tan\gamma\right)\times\frac{1}{4}$
(99) $\displaystyle\times$
$\displaystyle\left[2J_{5}(m_{\lambda}^{2},\mu^{2}_{\rho},\mu^{2}_{\rho},\mu^{2}_{\rho},m^{2}_{\tilde{\nu}_{L2}})+J_{5}(m_{\lambda}^{2},m_{\lambda}^{2},\mu^{2}_{\rho},\mu^{2}_{\rho},m^{2}_{\tilde{\nu}_{L2}})\right]$
$\displaystyle+$
$\displaystyle(s_{\nu_{L}}c_{\nu_{L}})\frac{g^{2}}{16\pi^{2}}\times\frac{1}{2}\mu_{\rho}m_{\lambda}\tan\gamma$
$\displaystyle\times$ $\displaystyle
c_{2W}\left[m^{2}_{\tilde{\nu}_{L2}}I_{5}(m_{\lambda}^{2},\mu_{\rho}^{2},m^{2}_{\tilde{\nu}_{L2}},m^{2}_{\tilde{\nu}_{L2}},m^{2}_{\tilde{\nu}_{L2}})\right]$
$\displaystyle+$
$\displaystyle(s_{\nu_{R}}c_{\nu_{R}})\frac{g^{2}}{16\pi^{2}}\times\frac{1}{4}\left(\mu_{\rho}m_{\lambda}\tan\gamma\right)$
$\displaystyle\times$
$\displaystyle\kappa_{1}^{2}\left[2J_{5}(m_{\lambda}^{2},m_{\lambda}^{2},m_{\lambda}^{2},\mu^{2}_{\rho},m^{2}_{\tilde{\nu}_{R2}})+J_{5}(m_{\lambda}^{2},m_{\lambda}^{2},\mu^{2}_{\rho},\mu^{2}_{\rho},m^{2}_{\tilde{\nu}_{R2}})\right]$
$\displaystyle+$
$\displaystyle(s_{\nu_{R}}c_{\nu_{R}})\frac{g^{2}}{16\pi^{2}}\times\left(\mu_{\rho}m_{\lambda}\tan\gamma\right)\times\frac{1}{4}$
$\displaystyle\times$ $\displaystyle
c_{2W}\left[2J_{5}(m_{\lambda}^{2},\mu^{2}_{\rho},\mu^{2}_{\rho},\mu^{2}_{\rho},m^{2}_{\tilde{\nu}_{R2}})+J_{5}(m_{\lambda}^{2},m_{\lambda}^{2},\mu^{2}_{\rho},\mu^{2}_{\rho},m^{2}_{\tilde{\nu}_{R2}})\right]$
$\displaystyle+$
$\displaystyle(s_{\nu_{R}}c_{\nu_{R}})\frac{g^{2}}{16\pi^{2}}\times\mu_{\rho}m_{\lambda}\tan\gamma$
$\displaystyle\times$ $\displaystyle
c^{2}_{W}\left[m^{2}_{\tilde{\nu}_{R2}}I_{5}(m_{\lambda}^{2},\mu_{\rho}^{2},m^{2}_{\tilde{\nu}_{R2}},m^{2}_{\tilde{\nu}_{R2}},m^{2}_{\tilde{\nu}_{R2}})\right]$
$\displaystyle-$ $\displaystyle(s_{L}c_{L})\frac{g^{\prime
2}}{16\pi^{2}}\times\frac{1}{27}\times\mu_{\rho}m_{B}\tan\gamma$
$\displaystyle\times$ $\displaystyle
c_{2W}\left[m^{2}_{\tilde{l}_{L2}}I_{5}(m_{B}^{2},\mu_{\rho}^{2},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}})\right]$
$\displaystyle-$
$\displaystyle(s_{L}c_{L})\frac{g^{2}}{16\pi^{2}}\times\frac{1}{3}\mu_{\rho}m_{\lambda}\tan\gamma$
$\displaystyle\times$ $\displaystyle
c_{2W}\left[m^{2}_{\tilde{l}_{L2}}I_{5}(m_{\lambda}^{2},\mu_{\rho}^{2},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}})\right]$
$\displaystyle-$ $\displaystyle(s_{L}c_{L})\frac{g^{\prime
2}}{16\pi^{2}}\times\frac{1}{54}\times\mu_{\rho}m_{B}\tan\gamma$
$\displaystyle\times$
$\displaystyle\left[2J_{5}(m_{B}^{2},\mu^{2}_{\rho},\mu^{2}_{\rho},\mu^{2}_{\rho},m^{2}_{\tilde{l}_{L2}})+J_{5}(m_{B}^{2},m_{B}^{2},\mu^{2}_{\rho},\mu^{2}_{\rho},m^{2}_{\tilde{l}_{L2}})\right]$
$\displaystyle-$
$\displaystyle(s_{L}c_{L})\frac{g^{2}}{16\pi^{2}}\times\frac{1}{6}\times\mu_{\rho}m_{\lambda}\tan\gamma$
$\displaystyle\times$
$\displaystyle\left[2J_{5}(m_{\lambda}^{2},\mu^{2}_{\rho},\mu^{2}_{\rho},\mu^{2}_{\rho},m^{2}_{\tilde{l}_{L2}})+J_{5}(m_{\lambda}^{2},m_{\lambda}^{2},\mu^{2}_{\rho},\mu^{2}_{\rho},m^{2}_{\tilde{l}_{L2}})\right]$
$\displaystyle-$ $\displaystyle\left(L_{2}\rightarrow
L_{3},~{}R_{2}\rightarrow R_{3}\right),$
$D^{Z^{\prime}(b)}_{R}=-D^{Z(b)}_{R}$ (100)
$D^{Z^{\prime}(c)}_{L}=-D^{Z(c)}_{L}$ (101)
$D^{Z^{\prime}(c)}_{R}=-D^{Z(c)}_{R}$ (102)
## Appendix E Contribution from $B^{\mu_{L,R}}_{L,R}$ to $\tau\rightarrow
3\mu$
Contributions to $B^{\mu_{L,R}}_{L,R}$ arise from the diagrams in Fig. 20.
$\tau$$\mu$$\mu$$\mu$$\tilde{W}^{+}$$\tilde{W}^{+}$$\tilde{\nu}_{L_{\alpha}}$$\tilde{\nu}_{L_{\alpha}}$
$\tau$$\mu$$\mu$$\mu$$\tilde{Y}^{+}$$\tilde{Y}^{+}$$\tilde{\nu}_{R_{\alpha}}$$\tilde{\nu}_{R_{\alpha}}$
$\tau$$\mu$$\mu$$\mu$$\lambda_{i}$$\lambda_{i}$$\tilde{\ell}_{L_{\alpha}}$$\tilde{\ell}_{L_{\beta}}$$\tau$$\mu$$\mu$$\mu$$\lambda_{i}$$\lambda_{i}$$\tilde{\ell}_{L_{\alpha}}$$\tilde{\ell}_{L_{\beta}}$
$\tau$$\mu$$\mu$$\mu$$\lambda_{i}$$\lambda_{j}$$\tilde{\ell}_{L_{\alpha}}$$\tilde{\ell}_{L_{\beta}}$
$\tau$$\mu$$\mu$$\mu$$\lambda_{i}$$\lambda_{j}$$\tilde{\ell}_{L_{\alpha}}$$\tilde{\ell}_{L_{\beta}}$
$\tau$$\mu$$\mu^{c}$$\mu^{c}$$\lambda_{B}$$\lambda_{B}$$\tilde{\ell}_{L_{\alpha}}$$\tilde{\ell}_{R_{\beta}}$$\tau$$\mu$$\mu^{c}$$\mu^{c}$$\lambda_{B}$$\lambda_{B}$$\tilde{\ell}_{L_{\alpha}}$$\tilde{\ell}_{R_{\beta}}$
$\tau^{c}$$\mu^{c}$$\mu$$\mu$$\lambda_{B}$$\lambda_{B}$$\tilde{\ell}_{R_{\alpha}}$$\tilde{\ell}_{L_{\beta}}$
$\tau^{c}$$\mu^{c}$$\mu$$\mu$$\lambda_{B}$$\lambda_{B}$$\tilde{\ell}_{R_{\alpha}}$$\tilde{\ell}_{L_{\beta}}$
$\tau^{c}$$\mu^{c}$$\mu^{c}$$\mu^{c}$$\lambda_{B}$$\lambda_{B}$$\tilde{\ell}_{R_{\alpha}}$$\tilde{\ell}_{R_{\beta}}$$\tau^{c}$$\mu^{c}$$\mu^{c}$$\mu^{c}$$\lambda_{B}$$\lambda_{B}$$\tilde{\ell}_{R_{\alpha}}$$\tilde{\ell}_{R_{\beta}}$
Figure 20: Diagrams contributing to $B^{\mu_{L,R}}_{L}$ (first and second
rows) and $B^{\mu_{L,R}}_{R}$ (third row). $\lambda_{i}$ and $\lambda_{j}$
($i\neq j$ in each above diagram) are gauginos in which $\lambda_{i}$ and
$\lambda_{j}\in\\{\lambda_{B},\lambda_{3},\lambda_{8}\\}$.
The formulas are:
$\displaystyle B^{\mu_{L}}_{L}$ $\displaystyle=$
$\displaystyle(s_{\nu_{L}}c_{\nu_{L}})\times\frac{g^{4}}{16\pi^{2}}\times\frac{1}{8}\left[-c^{2}_{\nu_{L}}J_{4}(m^{2}_{\lambda},m^{2}_{\lambda},m^{2}_{\tilde{\nu}_{L2}},m^{2}_{\tilde{\nu}_{L2}})\right.$
$\displaystyle+$
$\displaystyle\left.s^{2}_{\nu_{L}}J_{4}(m^{2}_{\lambda},m^{2}_{\lambda},m^{2}_{\tilde{\nu}_{L3}},m^{2}_{\tilde{\nu}_{L3}})\right.$
$\displaystyle+$
$\displaystyle\left.(c^{2}_{\nu_{L}}-s^{2}_{\nu_{L}})J_{4}(m^{2}_{\lambda},m^{2}_{\lambda},m^{2}_{\tilde{\nu}_{L2}},m^{2}_{\tilde{\nu}_{L3}})\right]$
$\displaystyle+$
$\displaystyle(s_{\nu_{R}}c_{\nu_{R}})\frac{g^{4}}{16\pi^{2}}\times\frac{1}{8}\left[-c^{2}_{\nu_{R}}J_{4}(m^{2}_{\lambda},m^{2}_{\lambda},m^{2}_{\tilde{\nu}_{R2}},m^{2}_{\tilde{\nu}_{R2}})\right.$
$\displaystyle+$
$\displaystyle\left.s^{2}_{\nu_{R}}J_{4}(m^{2}_{\lambda},m^{2}_{\lambda},m^{2}_{\tilde{\nu}_{R3}},m^{2}_{\tilde{\nu}_{R3}})\right.$
$\displaystyle+$
$\displaystyle\left.(c^{2}_{\nu_{R}}-s^{2}_{\nu_{R}})J_{4}(m^{2}_{\lambda},m^{2}_{\lambda},m^{2}_{\tilde{\nu}_{R2}},m^{2}_{\tilde{\nu}_{R3}})\right]$
$\displaystyle+$
$\displaystyle(s_{L}c_{L})\times\frac{g^{4}}{16\pi^{2}}\times\frac{1}{18}$
$\displaystyle\times$
$\displaystyle\left[-c^{2}_{L}\left(J_{4}(m_{\lambda}^{2},m_{\lambda}^{2},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}})+2m_{\lambda}^{2}I_{4}(m_{\lambda}^{2},m_{\lambda}^{2},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}})\right)\right.$
$\displaystyle+$
$\displaystyle\left.s^{2}_{L}\left(J_{4}(m_{\lambda}^{2},m_{\lambda}^{2},m^{2}_{\tilde{l}_{L3}},m^{2}_{\tilde{l}_{L3}})+2m_{\lambda}^{2}I_{4}(m_{\lambda}^{2},m_{\lambda}^{2},m^{2}_{\tilde{l}_{L3}},m^{2}_{\tilde{l}_{L3}})\right)\right.$
$\displaystyle+$
$\displaystyle\left.(c^{2}_{L}-s^{2}_{L})\left(J_{4}(m_{\lambda}^{2},m_{\lambda}^{2},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L3}})+2m_{\lambda}^{2}I_{4}(m_{\lambda}^{2},m_{\lambda}^{2},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L3}})\right)\right]$
$\displaystyle+$ $\displaystyle(s_{L}c_{L})\times\frac{g^{2}g^{\prime
2}}{16\pi^{2}}\times\frac{1}{162}$ $\displaystyle\times$
$\displaystyle\left[-c^{2}_{L}\left(J_{4}(m_{B}^{2},m_{\lambda}^{2},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}})+2m_{B}~{}m_{\lambda}I_{4}(m_{B}^{2},m_{\lambda}^{2},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}})\right)\right.$
$\displaystyle+$
$\displaystyle\left.s^{2}_{L}\left(J_{4}(m_{B}^{2},m_{\lambda}^{2},m^{2}_{\tilde{l}_{L3}},m^{2}_{\tilde{l}_{L3}})+2m_{B}~{}m_{\lambda}I_{4}(m_{B}^{2},m_{\lambda}^{2},m^{2}_{\tilde{l}_{L3}},m^{2}_{\tilde{l}_{L3}})\right)\right.$
$\displaystyle+$
$\displaystyle\left.(c^{2}_{L}-s^{2}_{L})\left(J_{4}(m_{B}^{2},m_{\lambda}^{2},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L3}})+2m_{B}~{}m_{\lambda}I_{4}(m_{B}^{2},m_{\lambda}^{2},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L3}})\right)\right]$
$\displaystyle+$ $\displaystyle(s_{L}c_{L})\times\frac{g^{\prime
4}}{16\pi^{2}}\times\frac{1}{216}$ $\displaystyle\times$
$\displaystyle\left[-c^{2}_{L}\left(J_{4}(m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}})+2m_{B}^{2}I_{4}(m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L2}})\right)\right.$
$\displaystyle+$
$\displaystyle\left.s^{2}_{L}\left(J_{4}(m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{L3}},m^{2}_{\tilde{l}_{L3}})+2m_{B}^{2}I_{4}(m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{L3}},m^{2}_{\tilde{l}_{L3}})\right)\right.$
$\displaystyle+$
$\displaystyle\left.(c^{2}_{L}-s^{2}_{L})\left(J_{4}(m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L3}})+2m_{B}^{2}I_{4}(m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{L3}})\right)\right],$
$\displaystyle\frac{B^{\mu_{R}}_{L}}{s_{L}c_{L}}$ $\displaystyle=$
$\displaystyle\frac{g^{\prime
4}}{16\pi^{2}}\times\frac{1}{648}\left[c^{2}_{R}\left(J_{4}(m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{R2}},m^{2}_{\tilde{l}_{L2}})+2m_{B}^{2}I_{4}(m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{R2}},m^{2}_{\tilde{l}_{L2}})\right)\right.$
(104) $\displaystyle+$
$\displaystyle\left.s^{2}_{R}\left(J_{4}(m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{R3}},m^{2}_{\tilde{l}_{L2}})+2m_{B}^{2}I_{4}(m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{R3}},m^{2}_{\tilde{l}_{L2}})\right)\right]$
$\displaystyle-$ $\displaystyle(L_{2}\rightarrow L_{3}),$
$\displaystyle\frac{B^{\mu_{L}}_{R}}{s_{R}c_{R}}$ $\displaystyle=$
$\displaystyle\frac{g^{\prime
4}}{16\pi^{2}}\times\frac{1}{648}\left[c^{2}_{L}\left(J_{4}(m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{R2}})+2m_{B}^{2}I_{4}(m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{L2}},m^{2}_{\tilde{l}_{R2}})\right)\right.$
(105) $\displaystyle+$
$\displaystyle\left.s^{2}_{L}\left(J_{4}(m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{L3}},m^{2}_{\tilde{l}_{R2}})+2m_{B}^{2}I_{4}(m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{L3}},m^{2}_{\tilde{l}_{R2}})\right)\right]$
$\displaystyle-$ $\displaystyle(R_{2}\rightarrow R_{3}),$
$\displaystyle\frac{B^{\mu_{R}}_{R}}{s_{R}c_{R}}$ $\displaystyle=$
$\displaystyle\frac{g^{\prime
4}}{16\pi^{2}}\times\frac{1}{18}\left[-c^{2}_{R}\left(J_{4}(m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{R2}},m^{2}_{\tilde{l}_{R2}})+2m_{B}^{2}I_{4}(m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{R2}},m^{2}_{\tilde{l}_{R2}})\right)\right.$
$\displaystyle+$
$\displaystyle\left.s^{2}_{R}\left(J_{4}(m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{R3}},m^{2}_{\tilde{l}_{R3}})+2m_{B}^{2}I_{4}(m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{R3}},m^{2}_{\tilde{l}_{R3}})\right)\right.$
$\displaystyle+$
$\displaystyle\left.\left(c^{2}_{R}-s^{2}_{R}\right)\left(J_{4}(m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{R2}},m^{2}_{\tilde{l}_{R3}})+2m_{B}^{2}I_{4}(m_{B}^{2},m_{B}^{2},m^{2}_{\tilde{l}_{R2}},m^{2}_{\tilde{l}_{R3}})\right)\right].$
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|
arxiv-papers
| 2013-01-20T11:24:23 |
2024-09-04T02:49:40.480593
|
{
"license": "Public Domain",
"authors": "L. T. Hue, D. T. Huong and H. N. Long",
"submitter": "Tho Hue Le",
"url": "https://arxiv.org/abs/1301.4652"
}
|
1301.4733
|
# Exact Results for Random Two-way Selections
Bin Zhoua,b [email protected] Shu-Jia Qinc Xiao-Pu Hand Zhe Hea Jia-
Rong Xiea Bing-Hong Wanga,e a Department of Modern Physics, University of
Science and Technology of China, Hefei, 230026, China
b Jiangsu University of Science and Technology, Zhenjiang, Jiangsu 212003,
China
c State Key Lab of Robotics, Shenyang Institute of Automation, Chinese Academy
of Sciences, Shenyang, 110016, China
d Institute of Information Economy and Alibaba Business College, Hangzhou
Normal University, Hangzhou, 310036, China
e The Research Center for Complex System Science, University of Shanghai for
Science and Technology, Shanghai, 200093 China
###### Abstract
Two-way selections play an important role in human social life. In this paper,
we analyze the minimum model of random two-way selections system and provide
its exact results on the matching ratio. Defining two basic relative ratios:
$\eta$ for the ratio of selection quantities from both sides, and $\xi$ for
the ratio of the total number of states to the smaller number of two sides, we
find the matching rate $P$ of two-way selections trends toward an inverse
proportion to both $\eta$ and $\xi$, in agreement with the numerical
simulations. We also surprisingly notice that, most of the empirical data of
the matchmaking fairs, the typical real-world two-way selections, are
completely included in the range predicted by our model. More interestingly,
by the fitting of empirical data, we find that the average number of
characters in real-world human sexual chosen is about 15, which is generally
in agreement with daily experiences, implying our model would be useful for
the estimation on the main factors of affecting the results of two-way
selections.
###### pacs:
89.75.-k, 89. 65.Ef, 05.45.-a
## I Introduction
Two-way selections between people are very common in daily life, such as the
marriage selections between men and women, the selections between seekers and
recruiters, and trades between buyers and sellers. In a sense, two-way
selections can be regarded as the base in the build of many social
relationships. Generally, the participants in a two-way selection process
usually consider many characters of the people in the other side. For
instance, in the human marriage selections, her/his looks, personality,
wealth, sense of humor, and so on, are usually in the considerations. These
characters, which generally can not clearly be distinguished and are difficult
to be estimated through traditional methods such as psychological test and
social survey, deeply affect the result of selection. Furthermore, two-way
selections are usually affected by many factors, such as, the size of members
on the two sides and the corresponding ratio. For example, the supply-demand
ratio on market would deeply influence on the prices and business practices.
Therefore, it would be significant to study the basic laws dominating the two-
way selections, including the factors affecting the matching rate and the
method for estimating the total number of characters affecting the selection.
The basic laws may be useful for several social services such as the
prediction of friendships and social relations1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ;
9 , seeking marital partners, and so on.
In this paper, we propose a minimum model for two-way selections to
investigate the factors influencing on the matching rate. Based on our model,
we also find an efficient way to estimate the total number of characters
affecting the two-way selections from empirical data.
## II The Model and Results
Generally speaking, two-way selections have at least two parts of members, and
the condition of successful matching is usually that the two sides of pairing
satisfy the characters for each other. The minimum model of random two-way
selections can be described as follows: i). The systems has two types of
agents, $A$ and $B$, respectively with the total number $k_{1}$ and $k_{2}$.
ii). Each agent (the $i$-th agent, say), either in type A or type B, has a own
character signed by $C_{Ai}$ ($i\in A$) or $C_{Bi}$ ($i\in B$). At the same
time, the character which each agent want to select is signed by $S_{Ai}$ or
$S_{Bi}$. Therefore, describing the state of each agent $Ai$ or $Bi$ requires
two characters which are independently chosen from $n$ kinds of characters
following the uniform distribution. iii). The condition of successful match of
two agents ($i\in A$ and $j\in B$) is $C_{Ai}=S_{Bj}$ and $S_{Ai}=C_{Bj}$.
For given $k_{1}$, $k_{2}$, and $n$, the total number of matching pairs $E$
can be written out according to the rules above:
$\displaystyle
E\left(k_{1},k_{2},n\right)=k_{1}\sum\limits_{i=0}^{k_{1}}\left[\sum\limits_{j=i+1}^{k_{2}}C_{k_{1}-1}^{i}C_{k_{2}}^{j}\left(\frac{1}{n^{2}}\right)^{i+j}\left(\frac{n^{2}-1}{n^{2}}\right)^{k_{1}+k_{2}-i-j-1}\right]+k_{2}\sum\limits_{j=0}^{k_{2}}\left[\sum\limits_{i=j}^{k_{1}}C_{k_{1}}^{i+1}C_{k_{2}-1}^{j-1}\left(\frac{1}{n^{2}}\right)^{i+j}\left(\frac{n^{2}-1}{n^{2}}\right)^{k_{1}+k_{2}-i-j-1}\right].$
(1)
According to :
$\lim_{k\rightarrow\infty}C_{k}^{i}x^{i}(1-x)^{k-i}=\frac{e^{-\lambda}\lambda^{i}}{i!},\quad\sum\limits_{i=0}^{\infty}\frac{x^{i}}{(i!)^{2}}=\text{BesselI}(0,2\sqrt{x}),\quad\sum\limits_{i=0}^{\infty}\frac{x^{i}}{(i!)(i-1)!}=\sqrt{x}\text{BesselI}(1,2\sqrt{x}),$
(2)
where $\lambda=xk$, $\text{BesselI}(0,2\sqrt{x})$ and
$\text{BesselI}(1,2\sqrt{x})$ are Bessel functions of the amount of virtual
cases. From Eq.(1) we obtain
$\displaystyle E\approx
k_{1}\left[1-e^{-\lambda_{1}-\lambda_{2}}\left(\text{BesselI}\left(0,2\sqrt{\lambda_{1}\lambda_{2}}\right)+\sqrt{\frac{\lambda_{1}}{\lambda_{2}}}\text{BesselI}\left(1,2\sqrt{\lambda_{1}\lambda_{2}}\right)\right)\right],$
(3)
where $\lambda_{1}=k_{1}/n^{2}$, $\lambda_{2}=k_{2}/n^{2}$.
Due to the symmetry of k1 and k2 in Eq.(1) , it is enough to study the range
$k_{1}\leqslant k_{2}$. We discuss Eq.(1) under three conditions of different
relative magnitude of $k_{1}$,$k_{2}$ and $n^{2}$. When $k_{1}\leqslant
k_{2}\ll n^{2}$, resulting in $\lambda_{1}\rightarrow 0,\lambda_{2}\rightarrow
0$, we only need to calculate the zeroth power term and the first power term
of Eq.(3). We obtain
$E\approx\frac{1}{n^{2}}k_{1}k_{2}.$ (4)
When $k_{1}\leqslant n^{2}\ll k_{2}$ or $n^{2}\ll k_{1}\leqslant k_{2}$,
according to
$\lim_{x\rightarrow\infty}\text{BesselI}(0,2x)=\lim_{x\rightarrow\infty}\text{BesselI}(1,2x)=\frac{e^{2x}}{2\sqrt{\pi
x}},$ (5)
from Eq.(3) we obtain
$\displaystyle E\approx
k_{1}-k_{1}\left[\frac{e^{-(\sqrt{\lambda_{1}}-\sqrt{\lambda_{2}})^{2}}}{\sqrt{\pi\sqrt{\lambda_{1}\lambda_{2}}}}\right].$
(6)
Because Eq.(6) is obtained under the condition of $k_{1}\leqslant n^{2}\ll
k_{2}$ or $n^{2}\ll k_{1}\leqslant k_{2}$, $k_{2}$ is very large, from Eq.(6)
we can obtain
$E\approx k_{1}.$ (7)
We defined that
$\displaystyle\eta=\frac{k_{2}}{k_{1}},\quad\xi=\frac{n^{2}}{k_{1}},\quad
P=\frac{2E}{k_{1}+k_{2}}.$ (8)
Therefore, $\eta$ represents the ratio between $k_{2}$ and $k_{1}$; $\xi$
represents the ratio between $n^{2}$ and $k_{1}$; $P$ represents the
probability of the agents which successfully match. The Eq.(3) is transformed
into
$\displaystyle
P=\frac{2}{1+\eta}\left[1-e^{-\frac{1}{\xi}-\frac{\eta}{\xi}}\left(\text{BesselI}\left(0,\frac{2\sqrt{\eta}}{\xi}\right)+\frac{1}{\sqrt{\eta}}\text{BesselI}\left(1,\frac{2\sqrt{\eta}}{\xi}\right)\right)\right].$
(9)
The Eq.(4) can written as:
$\displaystyle
P\approx\frac{2\eta}{\xi(\eta+1)}\quad\quad\quad(1\leqslant\eta\ll\xi).$ (10)
The Eq.(7) can written as:
$\displaystyle
P\approx\frac{2}{\eta+1}\quad\quad\quad\quad\,(1\leqslant\xi\ll\eta\quad
or\quad\xi\ll 1\leqslant\eta).$ (11)
Figure 1: (Color online) The comparison between analytical predictions and the
simulation results. In the two sub-figures, the parameter $k_{1}=100$; $\eta$
is assigned values from $1$ to $1000$; $\xi$ is assigned values from $0.1$ to
$1000$. (a) shows analytical predictions of Eq.(1); (b) shows the simulation
results.
Figure 2: (Color online) The comparison between analytical predictions and the
simulation results in the log-log plots. In the four sub-figures, the
parameter $k_{1}=100$; the squares indicate the simulation data; the solid
lines indicate analytical predictions. (a), $\eta=2$; $\xi$ is assigned values
from $100$ to $1000$; the solid line is obtained from Eq.(10). (b), $\xi=2$;
$\eta$ is assigned values from $10$ to $1000$; the solid line is obtained from
Eq.(11). (c), $\eta=1$; $\xi$ is assigned values from $10$ to $100$; the solid
line is obtained from Eq.(12). (d), $\xi=0.1$; $\eta$ is assigned values from
$10$ to $1000$; the solid line is obtained from Eq.(11).
Fig 1.(a) and Fig 1.(b) show the comparison between analytical predictions of
Eq.(1) and the simulation results. Fig 2.(a) shows the comparison between
analytical predictions of Eq.(10) and the simulation results under condition
$1\leqslant\eta\ll\xi$ and displays a power-law distribution with the exponent
$-1$ between $P$ and $\xi$. Fig 2.(b) shows the comparison between analytical
predictions of Eq.(11) and the simulation results under condition
$1\leqslant\xi\ll\eta$ and displays a power-law distribution with the exponent
$-1$ between $P$ and $\eta$; Fig 2.(d) shows the comparison between analytical
predictions of Eq.(11) and the simulation results under condition $\xi\ll
1\leqslant\eta$ and displays the same power-law distribution to Fig 2.b. The
above analytical predictions and simulation results are consistent with each
other. It is that to say all analytical results are reliable.
Consider a special case $k_{1}=k_{2}=k$, resulting in $\eta=1$. On the one
hand, from Eq.(10) we obtain
$P\approx\frac{1}{\xi}.$ (12)
The relation between $P$ and $\xi$ approximate to power law with the exponent
$-1$,and this case is well shown in Fig 2.(c); from Eq.(11) we obtain
$P\approx 1$, it means that almost all of agents can match successfully under
condition $\xi\ll 1\leqslant\eta$. On the other hand, according to
$k_{1}=k_{2}=k$, resulting in $\lambda_{1}=\lambda_{2}$, from Eq.(6) we obtain
$E\approx k-\sqrt{\frac{n^{2}}{\pi}}\sqrt{k}.$ (13)
The second term of Eq.(13) is the number of the agents which can not
successfully match in type A or type B. The larger $k$ is, The smaller
$\sqrt{n^{2}/\pi}\sqrt{k}/k$ is. In reality, It’s a result of fluctuation.
There are $n^{2}$ kinds of the states that a agent maybe have in the model. In
theory, the expected times of each state appearing is $k/n^{2}$. However, due
to the fluctuations, almost all frequencies of every states appearing deviate
around $k/n^{2}$. As a result, some agents can not successfully match. The
number of times that each state may appear obeys the binomial distribution.
The fluctuation is close related to the standard deviation, according to the
binomial theorem and standard deviation formula, we can obtain that the
standard deviation equals to $\sqrt{k(1-1/n^{2})/n^{2}}$, which is directly
proportional to $\sqrt{k}$. Thus, the number of the agents which can not
successfully match is also proportional to $\sqrt{k}$. It explains the
relationship between the second item of Eq.(13) and $\sqrt{k}$. From Eq.(13),
we know that the proportionality coefficient is $\sqrt{n^{2}/\pi}$.
Figure 3: (Color online) The relationship between the experimental evidence
and analytical predictions in the log-log plots. The red curve and the olive
curve are obtained from Eq.(9), and The parameters of red curve are $\eta=1$,
$n=15$; The parameters of olive curve are $\eta=356.67$, $n=15$. The round
dots represent the empirical data in APPENDIX A. The $\xi_{max}$ represent the
maximum value $225$ of $\xi$.
Finally, we validate our model in the experimental evidence way. The marriage
selections between men and women is a typical real-world two-way selections
system. Here we collect eighty-two news of real-world matchmaking fairs from
the Internet (see APPENDIX A). But the information of our data are incomplete.
Only the numbers of participants and matching pairs are recorded. The numbers
of men and women joined are omitted. So parameter $\eta$ can not be determined
exactly, but can only be restricted at a range. And $n$ is a internal
parameter need to be measured. We deal with the data and measure $n$ below.
We first consider the parameter $\eta$. Because $k_{1}\leqslant k_{2}$, the
minimum value of $\eta$ is $\eta_{min}=1$. For each set of data, there is a
maximal value. Take the first set of data in TABLE I for example, there are
$13$ participants and $3$ couples matched, thus $k_{1,1}\geqslant 3$,
according to Eq.(8), $\eta\leqslant\eta_{max,1}=(13-3)/3$. And we define two
signs: $\eta_{max,min}=\min_{i=1}^{82}\eta_{max,i}=2$,
$\eta_{max,max}=\max_{i=1}^{82}\eta_{max,i}=365.67$ by calculating eighty-two
sets of data in TABLE I. We estimate $n$ by minimizing
$\sum_{i=1}^{82}(P_{data,i}-P_{analysis,i})^{2}$ (14)
in which $P_{data}$ comes from the experimental data and $P_{analysis}$
calculated from Eq.9. Firstly, we take $\eta=\eta_{min}=1$, and n ranges from
$1$ to $10^{4}$ with a step size of 1, by calculating Eq.(14), we find that
when $n=17$, Eq.(14) is minimal. Then we take $\eta=\eta_{max,min}=2$, the
same to way, we minimize Eq.(14) and find $n=13$. In the Eq.(8), $n^{2}$ and
$\eta$ are inversely proportional relationship ,therefore, we define
$n_{max}=17$ and $n_{min}=13$, and $n$ belongs to $[13,17]$. Finally we
estimate $n$ by $n=(n_{max}+n_{min})/2=15$ and fix it.
Fig 3 shows the relationship between the experimental evidence and analytical
predictions of our model. The red curve and olive curve are obtained from
Eq.(9). The parameters of red curve are $\eta=1$, $n=15$; the parameters of
olive curve are $\eta=356.667$, $n=15$. From the Eq.(8), when $k_{1}$ is equal
to the minimum $1$, the $\xi$ take the maximum value $225$. The error bar of
ordinate $P$ of round dots represents the range of empirical data $P$ in TABLE
I. Because $k_{1}$ is unknown and $\xi$ is undetermined, we can only obtain
$k_{1,min,i}=E_{i}\leqslant k_{1,i}\leqslant k_{1,max,i}=(k_{1}+k_{2})/2$ and
$15^{2}/k_{1,max,i}\leqslant\xi_{i}\leqslant 15^{2}/k_{1,min,i}$. Therefore,
the rang of abscissa $\xi$ of round dots is relatively wide and the middle
points take $\xi_{i}=(15^{2}/k_{1,max,i}+15^{2}/k_{1,min,i})/2$.
We can see from Fig.3 that when $\xi$ is relatively small, corresponding
$k_{1}$ relatively big, all empirical data are enclosed in between the the two
curves; when $\xi$ is relatively big, corresponding $k_{1}$ relatively small,
some empirical data are enclosed in between the two curves, however others
empirical data lie the red curve above and the trend of the empirical data is
opposite to the analytical predictions. We have two reasons to explain them:
one the one hand, organizers of some matchmaking fairs select less guests who
meet their requirements to participate the matchmaking fairs from a lot of
people who enter name and want to participate the matchmaking fairs. In this
case, each guest is easier to find the right opposite sex who can match for
each other; one the other hand, when the number of guests is small in a
matchmaking fair, they know that it is very difficult to find the right
opposite sex who can match for each other, so they are not to find the right
opposite sex but tend to choose a basically mutually acceptable opposite sex.
In this psychological effect, each guest is also easier to find the opposite
sex to pair. The two reasons above cause that the fewer the guests are, the
higher the probability $p$ is. From the above analysis, the experimental data
can be well explained by the model.
## III Conclusions
In summary, we provide the exact solution for random two-way selections
system. The dynamics mechanism of a model is established and the analytical
result is obtained. In several different conditions, we obtain the
approximative and compact analytical results which agree with the simulation
results. In the model, the parameter $n$ is the total number of the characters
from which each agent can chose and directly determine the probability of
successful match. Therefore, $n$ is a very important parameter for random two-
way selections system. We propose a method to estimate the value of $n$ by
fitting empirical data. In this paper, the typical experimental data is
collected by the Internet and the range of parameter $n$ is estimated by the
method we proposed. We find that the average of $n$ is $15$, which is in
agreement with daily experiences, and that most of the experimental evidence
data fall into the range predicted by our model. So the method would be useful
to estimate the parameter $n$, and such simple model indeed catch the core of
random two-way selections system in reality and would be valuable to the real
life and several online social services.
###### Acknowledgements.
This work was funded by the National Important Research Project (Grant No.
91024026), the National Natural Science Foundation of China (No. 11205040,
11105024, 11275186), the Major Important Project Fund for Anhui University
Nature Science Research (Grant No. KJ2011ZD07) and the Specialized Research
Fund for the Doctoral Program of Higher Education of China (Grant No.
20093402110032).
## Appendix A Empirical Data of Matchmaking Fairs
We collect 82 online news of real-world matchmaking fairs and record the
number of guests and matching pairs. Due to some of news did not report exact
numbers, we suppose each of these uncertain numbers has 10% possible ranges.
The reports of these uncertain numbers generally have three types: nearly $x$,
about $x$, and over $x$ (here $x$ is the number). For “’nearly $x$”, we set
the possible range is $0.95x\pm 0.05x$, and $1.00x\pm 0.05x$ for “about $x$”,
and $1.05x\pm 0.05x$ for “over $x$”. All of these information is shown in
Table 1.
## References
* (1) L. Lü, T. Zhou, Link prediction in complex networks: A survey. Physica A 390, (2011) 1150.
* (2) R. Guimer, M. Sales-Pardo, Missing and spurious interactions and the reconstruction of complex networks, Proc. Nat. Acad. Sci. USA,106 (2009) 22073.
* (3) L. Lü, M. Medo, C.-H. Yeung, Y.-C. Zhang, Z.-K. Zhang, T. Zhou, Recommender systems, Phys. Rep., 519 (2012) 1.
* (4) C.-J. Zhang, A. Zeng, Behavior patterns of online users and the effect on information filtering, Physica A, 391 (2012) 1822.
* (5) J.-L. Guillaume, M. Latapy, Bipartite structure of all complex networks, Inform. Proc. Lett., 90 (2004) 215.
* (6) M.-S. Shang, L. Lü, Y.-C. Zhang, T. Zhou, Empirical analysis of web-based user-object bipartite networks, EuoPhys. Lett., 90 (2010) 48006.
* (7) R. Guimer, M. Sales-Pardo, L. Amaral, A Nunes, Module identification in bipartite and directed networks, Phys. Rev. E., 76 (2007) 036102.
* (8) Y.-B. Zhou, L. Lü, M. Li, Quantifying the influence of scientists and their publications: distinguishing between prestige and popularity, New J. Phys., 14 (2012) 033033.
* (9) L. Li, Z. Yang, L. Liu, M. Kitsuregawa, Query-url bipartite based approach to personalized query recommendation, Proceedings of the 23rd national conference on Artificial intelligence. (2008) 1189.
Table 1: The data of matchmaking fairs
Website of the reports of matchmaking fairs Original descriptions The total of
joined people: $k_{1}+k_{2}$ The total number of matched pairs $E$ Matching
ratio $P$ http://www.cdb.org.cn/newsview.php?id=6359 13 joined; 3 matched $13$
$3$ $0.4615$ http://sd.people.com.cn/n/2012/0827/c183718-17407663.html 18
joined; 6 matched $18$ $6$ $0.6667$
http://news.carnoc.com/list/183/183765.html 20 joined; 1 matched $20$ $1$
$0.1000$ http://bbs.tiexue.net/post2_5756757_1.html 21 joined; 6 matched $21$
$6$ $0.5714$ http://www.shxb.net/html/20110516/20110516_278862.shtml 25
joined; 5 matched $25$ $5$ $0.4000$ http://news.qq.com/a/20100511/000472.htm
26 joined; 3 matched $26$ $3$ $0.2308$
http://bbs.ganxianw.com/thread-46316-1-1.html 26 joined; 4 matched $26$ $4$
$0.3077$ http://www.wzrb.com.cn/article321273show.html 30 joined; 2 matched
$30$ $2$ $0.1333$
http://www.wccdaily.com.cn/epaper/hxdsb/html/2012-05/14/content_448239.htm 32
joined; 4 matched $32$ $4$ $0.2500$
http://wbnews.sxrb.com/news/ty/1372966.html 36 joined; 6 matched $36$ $6$
$0.3333$
http://www.nbmz.gov.cn/view.aspx?id=16595&AspxAutoDetectCookieSupport=1 36
joined; 6 matched $36$ $6$ $0.3333$
http://nb.people.com.cn/GB/200892/16491824.html 38 joined; 6 matched $38$ $6$
$0.3158$ http://cq.cqwb.com.cn/NewsFiles/201203/25/921397.shtml 40 joined; 8
matched $40$ $8$ $0.4000$
http://www.sc.chinanews.com.cn/my/data/html/201212/32619.html over 40 joined;
3 matched $42\pm 2$ $3$ $0.1432\pm 0.0068$
http://www.16466.com/info_detail.htm?id=36526 over 50 joined; 3 matched $53\pm
3$ $3$ $0.1136\pm 0.0064$
http://www.ncnews.com.cn/ncxw/shxw/t20121112_943114.htm about 60 joined; 5
matched $60\pm 3$ $5$ $0.1671\pm 0.0084$
http://news.wzsee.com/2012/0502/130061.html over 60 joined; 4 matched $63\pm
3$ $4$ $0.1273\pm 0.0061$ http://news.hexun.com/2012-08-27/145162214.html over
60 joined; 5 matched $63\pm 3$ $5$ $0.1591\pm 0.0076$
http://www.douban.com/group/topic/28145139/ over 60 joined; about 10 matched
$63\pm 3$ $10\pm 1$ $0.3197\pm 0.0470$
http://zhuanti.10yan.com/zt/other/sdcms/html/xqj2012/xiangqindongtai/1377.html
72 joined; 11 matched $72$ $11$ $0.3056$
http://www.wlmqwb.com/3229/syzt/hdzt/seven/201007/t20100719_1287834.shtml 80
joined; 5 matched $80$ $5$ $0.1250$
http://www.dpcm.cn/html/news/shehui/20121211/8a485b96f289e038.htm 80 joined;
10 matched $80$ $10$ $0.2500$ http://www.zhaogejia.com/News/Show/166 80
joined; 13 matched $80$ $13$ $0.3250$
http://cq.cqnews.net/shxw/shwx/200909/t20090928_3635782.htm 80 joined; 18
matched $80$ $18$ $0.4500$
http://a.jiaodong.net/jiaoyou/detail/?/20120717134715.htm nearly 100 joined; 5
matched $95\pm 5$ $5$ $0.1056\pm 0.0056$
http://www.dllake.com/testurl/news/news.asp?id=1874 99 joined; 8 matched $99$
$8$ $0.1616$
http://fj.qq.com/a/20120413/000073.htm?pgv_ref=aio2012&ptlang=2052 100 joined;
5 matched $100$ $5$ $0.1000$
http://epaper.lnd.com.cn/html/bdcb/20110118/bdcb635730.html about 100 joined;
7 matched $100\pm 5$ $7$ $0.1404\pm 0.0070$
http://news.zh853.com/NewsShow-22166.html over 100 joined; 16 matched $110\pm
10$ $16$ $0.2933\pm 0.0267$
http://news.163.com/11/1123/08/7JHJ4PP100014AED.html 150 joined; 7 matched
$150$ $7$ $0.0933$ http://news.xinmin.cn/rollnews/2011/05/03/10539635.html
nearly 200 joined; 8 matched $190\pm 10$ $8$ $0.0844\pm 0.0044$
http://www.0523qq.com/forum.php?mod=viewthread&tid=2779 nearly 200 joined; 22
matched $190\pm 10$ $22$ $0.2322\pm 0.0122$
http://news.ycw.gov.cn/html/2012-04/28/content_15150376.htm about 200 joined;
4 matched $200\pm 10$ $4$ $0.0401\pm 0.0020$
http://epaper.lnd.com.cn/html/bdcb/20110118/bdcb635730.html 206 joined; 10
matched $206$ $10$ $0.0971$
http://www.cqwb.com.cn/NewsFiles/201005/30/20102930062910354716.shtml over 200
joined; 7 matched $210\pm 10$ $7$ $0.0668\pm 0.0032$
http://bddsb.bandao.cn/data/20120827/html/53/content_2.html over 200 joined; 8
matched $210\pm 10$ $8$ $0.0764\pm 0.0036$
http://www.zhaogejia.com/News/Show/150 over 200 joined; 38 matched $210\pm 10$
$38$ $0.3627\pm 0.0173$ http://3g.3xgd.com/news/play.asp?NewsID=80975 216
joined; 19 matched $216$ $19$ $0.1759$
http://wed.cnhan.com/hjb/2012-12-03/3900.html over 240 joined; 22 matched
$252\pm 12$ $22$ $0.1750\pm 0.0083$
http://xt.fangyuan365.com/article/List.asp?ID=8708 about 258 joined; over 10
matched $258\pm 13$ $11\pm 1$ $0.0859\pm 0.0121$
http://cq.cqnews.net/shxw/2012-11/12/content_21432170.htm nearly 300 joined; 4
matched $285\pm 15$ $4$ $0.0282\pm 0.0014$
http://xt.fangyuan365.com/article/List.asp?ID=11694 about 300 joined; 8
matched $300\pm 15$ $8$ $0.0535\pm 0.0027$
http://roll.sohu.com/20120625/n346389451.shtml over 300 joined; over 10
matched $315\pm 15$ $11\pm 1$ $0.0703\pm 0.0097$
http://www.ijxjj.com/article/article_12773.html over 300 joined; 32 matched
$315\pm 15$ $32$ $0.2036\pm 0.0097$
http://news.xinmin.cn/shehui/2013/02/16/18638248_2.html 400 joined ; nearly 20
matched $400$ $19\pm 1$ $0.0950\pm 0.0050$
http://www.sz120.com/xwdt/ynxw/22205/ over 500 joined; 3 matched $525\pm 25$
$3$ $0.0115\pm 0.0005$
http://www.sc.xinhuanet.com/content/2012-02/06/content_24649397.htm over 500
joined; 8 matched $525\pm 25$ $8$ $0.0306\pm 0.0015$
http://dqnews.zjol.com.cn/dqnews/system/2010/08/17/012525402.shtml over 500
joined; over 10 matched $525\pm 25$ $11\pm 1$ $0.0422\pm 0.0058$
http://cheshang.16888.com/newsinfo/2011/1115/141264.html nearly 600 joined;
nearly 40 matched $570\pm 30$ $38\pm 2$ $0.1341\pm 0.0141$
http://bbs.heze.cc/thread-842865-1-1.html over 600 joined; over 78 matched
$630\pm 30$ $78$ $0.2482\pm 0.0118$ http://www.8hy.org/hyjy/hy6240/1 nearly
800 joined; 58 matched $760\pm 40$ $58$ $0.1531\pm 0.0081$
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arxiv-papers
| 2013-01-21T02:44:31 |
2024-09-04T02:49:40.499236
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bin Zhou, Shujia Qin, Xiao-Pu Han, Zhe He, Jia-Rong Xie, Bing-Hong\n Wang",
"submitter": "Bin Zhou",
"url": "https://arxiv.org/abs/1301.4733"
}
|
1301.4829
|
# Extremely long hard bursts observed by Konus-Wind
V. Pal’shin R. Aptekar D. Frederiks S. Golenetskii V. Il’inskii E. Mazets
K. Yamaoka M. Ohno K. Hurley T. Sakamoto P. Oleynik M. Ulanov I. G.
Mitrofanov D. Golovin M. L. Litvak A. B. Sanin W. Boynton C. Fellows K.
Harshman C. Shinohara R. Starr
###### Abstract
We report the observations of the prompt emission of the extremely long hard
burst, GRB 060814B, discovered by Konus-Wind and localized by the IPN. The
observations reveal a smooth, hard, $\sim$40-min long pulse followed by weaker
emission seen several hours after the burst onset. We also present the Konus-
Wind data on similar burst, GRB 971208, localized by BATSE/IPN. And finally we
discuss the different possible origins of these unusual events.
###### Keywords:
gamma-ray bursts
###### :
95.85.Pw
## 1 Introduction
Konus-Wind is a gamma-ray all-sky spectrometer (Aptekar et al., 1995) which
has been successfully operating since November 1994. Wind orbit is far from
the Earth magnetosphere (at distance of $\sim$1–7 light seconds) that enables
nearly uninterrupted observations of all sky under very stable background. In
the waiting mode Konus-Wind measures the count rates in three energy bands
which covers the $\sim$15–1000 keV range with accumulation time of 2.944 s.
This mode enables observations of ultra long gamma-ray bursts with duration
$\geq$500 s. During 13 years of observations Konus-Wind has detected two
extremely long, single pulsed, hard GRBs.
## 2 GRB 060814B
A very long burst was detected by Konus-Wind in the waiting mode at 2006-08-14
T0=37070 s UT (10:17:50). The burst light curve shows a single, smooth, FRED-
like pulse with a duration of $\simeq$2700 s. The most intense part of the
burst (initial several hundreds seconds) was also detected by Ulysses, Mars
Odyssey (HEND), Suzaku-WAM, and INTEGRAL-SPI-ACS. That enabled to localize the
burst to the $\simeq$7.1 deg2 3$\sigma$ IPN error box, whose coordinates are
given in Table 1. The Suzaku-WAM light curve shows a sharp decline at
T-T${}_{0}\simeq$200 s due to source set below the horizon (see Fig. 1). This
occultation step let us to derive much smaller combined IPN/WAM box with the
area of $\simeq$0.6 deg2 (given in Table 1). The galactic coordinates of the
box center are $l$, $b$ = 162.88, +6.06 deg.
Figure 1: Konus-Wind (Left) and Suzaku-WAM (Right) background subtracted light
curves of GRB 060814B in three energy bands and the hardness ratios, T0=37070
s UT (10:17:50).
The Konus-Wind light curve of GRB 060814B in three energy bands is shown in
Fig. 1. The burst demonstrates strong hard-to-soft spectral evolution with the
hardest spectrum at the burst onset. After the end of the main pulse there is
an extended emission seen in the G1 and G2 bands during several hours (4 hours
in the G2 at $>4\sigma$-level and even longer in the G1). The emission was
detected by the same detector which observed the main pulse, so it might be a
burst tail. Unfortunately the localization area occurred in the BAT FoV only
in $\simeq$11 hours after the burst onset. Nothing evident was found in the
BAT data of the 500-s long observation (obsID 00035631001).
The Suzaku-WAM multichannel spectrum accumulated from T0 to T0+200 s is well
fitted (in the 80–2000 keV range) by CPL model: $dN/dE\propto
E^{-\alpha}\exp[-(2-\alpha)E/E_{p}]$ with $\alpha=0.19\pm 0.14$,
$E_{p}=477(-27,+31)$ keV ($\chi^{2}$=21/23 dof). The errors are given at 90%
confidence level. The Konus-Wind 3-channel spectrum for the same interval
yields $\alpha=0.89\pm 0.07$, $E_{p}=544\pm 52$ keV. The errors are estimated
at 1$\sigma$ level by propagation of errors in the observed counts (here and
below). Using the KW 3-channel data we also estimated the spectral parameters
for whole burst, for the hard pulse, and for the possible tail. The results
are given in Table 2. The estimated burst fluence is $(2.35\pm 0.22)\times
10^{-4}$ erg cm-2 (18–1170 keV). The fluence of the tail is $(2.1\pm
0.2)\times 10^{-4}$ erg cm-2 in the same range.
## 3 GRB 971208
An unusually long, smooth, single pulsed gamma-ray burst was detected by BATSE
at 1997-12-08 28092.1295 s UT (trigger 6526: (Connaughton et al., 1997)). This
burst was observed by Konus-Wind in full entirety (see Fig. 2). The total
burst duration is $\simeq$2500 s. One can see in Fig. 2 the BATSE light curve
which shows a similar duration. There is no any sign of extended emission
after T-T${}_{0}\simeq$2500 s.
The burst demonstrates a similar to GRB 060814B hard-to-soft spectral
evolution. Using the KW 3-channel data we estimated the spectral parameters
for the hard pulse, and for the whole burst (see Table 2). The estimated burst
fluence is $(2.55\pm 0.11)\times 10^{-4}$ erg cm-2 (15–1000 keV). The fluence
reported by BATSE for the first 800 s is $(1.86\pm 0.03)\times 10^{-4}$ erg
cm-2 in the 25-1800 keV range (Connaughton et al., 1997).
Figure 2: Left: Konus-Wind (Bottom) and BATSE light curves (Top: background
subtracted, 20–1000 keV range) of GRB 971208. Right: Konus-Wind background
subtracted light curve in three energy bands and the hardness ratios. T0=28092
s UT (07:48:12).
## 4 Discussion
Two obvious factors can make longer the burst: cosmological time dilation and
relativistic curvature effect. Assuming, for example, $z=10$ we would have
$E_{iso}\simeq 2\times 10^{55}$ erg, $\Delta$T${}_{rest}\simeq 230$ s, and
$E_{p,rest}\simeq$3700, and 1600 keV correspondingly for GRB 060814B and
971207. Such $E_{iso}$ is about a factor of 10 greater than the largest known
value. But we should take in account a high probability of lensing for such
redshift. $E_{p,rest}\simeq$3700 keV seems to be a bit large, but it is not
exceptional: Swift GRB 050717 had $E_{p}\simeq 1900$ keV (Krimm et al., 2006),
so even assuming $z\simeq 1$, it would correspond to $E_{p,rest}\simeq$3800
keV. Hence, these bursts might be high redshift GRBs magnified by lensing. The
pulse shapes and the character of spectral evolution indicate that the
curvature effect (e.g. (Kocevski et al., 2003),(Qin et al., 2006)) may play a
main role in forming the pulses. For pulses created purely by the curvature
effect $\Delta$T$\propto(1+z)R_{s}/\Gamma^{2}$. Hence, very long duration may
be due to unusually large shell radius $R_{s}$. That requires some specific
conditions in the circumburst medium and/or in the shock.
| IPN box (J2000) | | Combined IPN/WAM box (J2000)
---|---|---|---
| RA, deg | Dec, deg | | RA, deg | Dec, deg
Center | 81.98 | +47.85 | | 81.19 | +46.60
Corner1 | 83.20 | +46.76 | | 80.73 | +48.66
Corner2 | 80.86 | +44.66 | | 80.76 | +47.55
Corner3 | 80.72 | +48.94 | | 81.42 | +45.19
Corner4 | 83.30 | +51.11 | | 81.66 | +45.40
Table 1: Localization of GRB 060814B GRB 060814B | | GRB 971208
---|---|---
Time interval s | $\alpha$ | $E_{p}$ keV | | Time interval s | $\alpha$ | $E_{p}$ keV
0–700 | 0.91$\pm$0.07 | 374$\pm$30 | | 0–485 | 1.246$\pm$0.033 | 165$\pm$7
0–2700 | 1.46$\pm$0.07 | 341$\pm$61 | | 0–2500 | 1.185$\pm$0.075 | 144$\pm$12
2700–12300111 | 2.7$\pm$0.2 | – | | | |
Table 2: Spectral parameters of GRB 060814B and GRB 971208
We are grateful to Valeri Connaughton for kindly providing us with the figure
of the BATSE light curve of GRB 971208. The Konus-Wind experiment is supported
by a Russian Space Agency contract and RFBR grant 06-02-16070. V.P. and D.F.
thank Neil Gehrels and USRA for the support of their participation in the
conference. KH is grateful for support under the Mars Odyssey Participating
Scientist program (JPL Contract 1282043), the INTEGRAL U.S. Guest Investigator
program (NASA Grant NNX07AJ65G), and the Suzaku U.S. Guest Investigator
program (NASA Grant NNX06AI36G).
## References
* Aptekar et al. (1995) R. L. Aptekar, et al., _Space Science Reviews_ 71, 265–272 (1995).
* Connaughton et al. (1997) V. Connaughton, et al., _IAUC_ 6785, 1 (1997).
* Krimm et al. (2006) H. A. Krimm, et al., _ApJ_ 648, 1117–1124 (2006).
* Kocevski et al. (2003) D. Kocevski, F. Ryde, and E. Liang, _ApJ_ 596, 389–400 (2003).
* Qin et al. (2006) Y.-P. Qin, C.-Y. Su, J. H. Fan, and A. C. Gupta, _Phys. Rev. D_ 74, 063005 (2006).
|
arxiv-papers
| 2013-01-21T11:51:32 |
2024-09-04T02:49:40.509545
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V. Pal'shin, R. Aptekar, D. Frederiks, S. Golenetskii, V. Il'Inskii,\n E. Mazets, K. Yamaoka, M. Ohno, K. Hurley, T. Sakamoto, P. Oleynik, M.\n Ulanov, I. G. Mitrofanov, D. Golovin, M. L. Litvak, A. B. Sanin, W. Boynton,\n C. Fellows, K. Harshman, C. Shinohara, R. Starr",
"submitter": "Valentin Pal'shin",
"url": "https://arxiv.org/abs/1301.4829"
}
|
1301.5018
|
# Gröbner-Shirshov Bases for Commutative Algebras with Multiple Operators and
Free Commutative Rota-Baxter Algebras111Supported by the Young Project on
Natural Science Fund of Zhanjiang Normal University (No.QL0902).
Jianjun Qiu
Mathematics and Computational Science School
Zhanjiang Normal University
Zhanjiang 524048, China
[email protected]
Abstract: In this paper, the Composition-Diamond lemma for commutative
algebras with multiple operators is established. As applications, the Gröbner-
Shirshov bases and linear bases of free commutative Rota-Baxter algebra, free
commutative $\lambda$-differential algebra and free commutative
$\lambda$-differential Rota-Baxter algebra are given, respectively.
Consequently, these three free algebras are constructed directly by
commutative $\Omega$-words.
Key words: commutative, Rota-Baxter algebras, $\lambda$-differential algebras,
$\lambda$-differential Rota-Baxter algebras, commutative algebras with
multiple operators, Gröbner-Shirshov bases.
AMS 2000 Subject Classification: 16S15, 13P10, 16W99, 17A50, 13XX
## 1 Introduction
Let $K$ be a unitary commutative ring and $\lambda\in K$. A Rota-Baxter
algebra of weight $\lambda$ is a $K$-algebra $R$ with a linear operator
$P:R\rightarrow R$ satisfying the Rota-Baxter relation:
$P(u)P(v)=P(uP(v))+P(P(u)v)+\lambda P(uv),\forall u,v\in R.$
The Rota-Baxter operator first occurred in the paper of G. Baxter [1] to solve
an analytic problem, and the algebraic study of this operator was started by
G.-C. Rota [19].
There have been some constructions of free (commutative) Rota-Baxter algebras.
In this aspect, G.-C. Rota [19] and P. Cartier [9] gave the explicit
constructions of free commutative Rota-Baxter algebras of weight $\lambda=1$,
which they called shuffle Baxter and standard Baxter algebras, respectively.
Recently L. Guo and W. Keigher [13, 14] constructed the free commutative Rota-
Baxter algebras (with unit or without unit) for any $\lambda\in K$ using the
mixable shuffle product. These are called the mixable shuffle product
algebras, which generalize the classical construction of shuffle product
algebras. K. Ebrahimi-Fard and L. Guo [11] further constructed the free
associative Rota-Baxter algebras by using the Rota-Baxter words. In [12], K.
Ebrahimi-Fard and L. Guo use rooted trees and forests to give explicit
construction of free noncommutative Rota-Baxter algebras on modules and sets.
A differential algebra of weigh $\lambda$, also called $\lambda$-differential
algebra is a $K$-algebra $R$ with a linear operator $D:R\rightarrow R$ such
that
$D(uv)=D(u)v+uD(v)+\lambda D(u)D(v),\forall u,v\in R.$
Such an operator $D$ is called a $\lambda$-differential operator. E. Kolchin
[16] considered the differential algebra and constructed free differential
algebra of weight $\lambda=0$. L. Guo and W. Keigher [15] dealt with a
generalization of this algebra and using the same way to construct the free
differential algebra of weight $\lambda$ in both commutative and associative
case.
Similar to the relation of integral and differential operators, L. Guo and W.
Keigher [15] introduced the notion of $\lambda$-differential Rota-Baxter
algebra which is a $K$-algebra $R$ with a $\lambda$-differential operator $D$
and a Rota-Baxter operator $P$ such that $DP=Id_{R}$. In the same paper [15],
they construct the free $\lambda$-differential Rota-Baxter (commutative and
associative).
The Gröbner-Shirshov bases theory for Lie algebras was introduced by A. I.
Shirshov [4, 20]. Shirshov [20] defined the composition of two Lie
polynomials, and proved the Composition lemma for the Lie algebras. L. A.
Bokut [3] specialized the approach of Shirshov to associative algebras, see
also Bergman [2]. For commutative polynomials, this lemma is known as the
Buchberger’s Theorem in [7, 8].
The multi-operators algebras ($\Omega$-algebras) were introduced by A. G.
Kurosh [17] and the Gröbner-Shirshov bases for $\Omega$-algebras were given in
the paper by V. Drensky and R. Holtkamp [10]. Composition-Diamond lemma for
associative algebras with multiple linear operators (associative
$\Omega$-algebras) is established in a recent paper by L. A. Bokut, Y. Chen
and J. Qiu [5]. Also, the Gröbner-Shirshov bases for Rota-Baxter algebras is
established by L. A. Bokut, Y. Chen and X. Deng [6] and the Composition-
Diamond lemma for $\lambda$-differential associative algebras with multiple
operators is constructed by J. Qiu and Y. Chen [18].
In this paper, we deal with commutative algebras with multiple linear
operators. We construct free commutative algebras with multiple linear
operators and establish the Composition-Diamond lemma for such algebras. As
applications, we obtain Gröbner-Shirshov bases of free commutative Rota-Baxter
algebra, commutative $\lambda$-differential algebra and commutative
$\lambda$-differential Rota-Baxter algebra, respectively. Then, by using the
Composition-Diamond lemma, linear bases of these three free algebras are
obtained respectively.
The author would like to express his deepest gratitude to Professors L. A.
Bokut and Yuqun Chen for their kind guidance, useful discussions and
enthusiastic encouragements.
## 2 Free commutative algebras with multiple operators
In this section, we construct free commutative algebras with multiple linear
operators.
Let $K$ be a unitary commutative ring. A commutative algebra with multiple
operators is a commutative $K$-algebra $R$ with a set $\Omega$ of multi-linear
operators.
Let $X$ be a set, $CS(X)$ the free commutative semigroup on $X$ and
$\Omega=\bigcup_{t=1}^{\infty}\Omega_{t}$
where $\Omega_{t}$ is the set of $t$-ary operators.
Define
$\Gamma_{0}=X,\ \mathcal{Q}_{0}=CS(\Gamma_{0})$
and
$\Gamma_{1}=X\cup\Omega(\mathcal{Q}_{0}),\ \mathcal{Q}_{1}=CS(\Gamma_{1})$
where
$\Omega(\mathcal{Q}_{0})=\bigcup\limits_{t=1}^{\infty}\\{\theta(u_{1},u_{2},\dots,u_{t})|\theta\in\Omega_{t},u_{i}\in\mathcal{Q}_{0},\
i=1,2,\dots,t\\}.$
For $n>1$, define
$\Gamma_{n}=X\cup\Omega(\mathcal{Q}_{n-1}),\ \mathcal{Q}_{n}=CS(\Gamma_{n})$
where
$\Omega(\mathcal{Q}_{n-1})=\bigcup\limits_{t=1}^{\infty}\\{\theta(u_{1},u_{2},\dots,u_{t})|\theta\in\Omega_{t},u_{i}\in\mathcal{Q}_{n-1},\
i=1,2,\dots,t\\}.$
Let
$\mathcal{Q}(X)=\bigcup_{n\geq 0}\mathcal{Q}_{n}.$
Then it is easy to see that $\mathcal{Q}(X)$ is a commutative semigroup such
that $\Omega(\mathcal{Q}(X))\subseteq\mathcal{Q}(X).$
Let $K[X;\Omega]$ be the commutative $K$-algebra spanned by $\mathcal{Q}(X)$.
Extend linearly each $\sigma\in\Omega_{t}$, $t\geq 1$,
$\sigma:\mathcal{Q}(X)^{t}\rightarrow\mathcal{Q}(X),\
(x_{1},x_{2},\dots,x_{n})\mapsto\sigma(x_{1},x_{2},\dots,x_{n})$
to
$K[X;\Omega]^{t}\rightarrow K[X;\Omega].$
Then, it is easy to see that $K[X;\Omega]$ is a free commutative algebra with
multiple operators $\Omega$ on set $X$.
## 3 Composition-Diamond lemma for commutative algebras with multiple
operators
In this section, we introduce the notions of Gröbner-Shirshov bases for the
commutative algebras with multiple operators and establish the Composition-
Diamond lemma for such algebras.
The element in $\mathcal{Q}(X)$ and $K[X;\Omega]$ are called commutative
$\Omega$-word and commutative $\Omega$-polynomial, respectively. For any
$u\in\mathcal{Q}(X)$, $u$ has a unique expression
$u=u_{1}u_{2}\cdots u_{n}$
where each $u_{i}\in X\cup\Omega(\mathcal{Q}(X))$. If this is the case, we
define $bre(u)=n$. Let $u\in\mathcal{Q}(X)$. Then
$dep(u)=\mbox{min}\\{n|u\in\mathcal{Q}_{n}\\}$
is called the depth of $u$.
Let $\star\notin X$. By a commutative $\star$-$\Omega$-word we mean any
expression in $\mathcal{Q}(X\cup\\{\star\\})$ with only one occurrence of
$\star$. We define the set of all commutative $\star$-$\Omega$-words on $X$ by
$\mathcal{Q}^{\star}(X)$. Let $u$ be a commutative $\star$-$\Omega$-word and
$s\in K[X;\Omega]$. Then we call
$u|_{s}=u|_{\star\ \mapsto s}\ \mbox{(replace $\star$ by $s$ in $u$)}$
is a commutative $s$-$\Omega$-word. For example, if
$u=x_{1}\theta_{3}(x_{2},\theta_{2}(\star
x_{4},x_{5}),x_{6})\in\mathcal{Q}^{\star}(X),$
then
$u|_{s}=u|_{\star\ \mapsto
s}=x_{1}\theta_{3}(x_{2},\theta_{2}(sx_{4},x_{5}),x_{6})$
is a commutative $s$-$\Omega$-word.
Similarly, we can define commutative $(\star_{1},\star_{2})$-$\Omega$-words as
expressions in $\mathcal{Q}(X\cup\\{\star_{1},\star_{2}\\})$ with only one
occurrence of $\star_{1}$ and only one occurrence of $\star_{2}$. Let us
denote by $\mathcal{Q}^{\star_{1},\star_{2}}(X)$ the set of all commutative
$(\star_{1},\star_{2})$-$\Omega$-words. Let
$u\in\mathcal{Q}^{\star_{1},\star_{2}}(X)$, $s_{1},s_{2}\in K[X;\Omega]$. Then
we call
$u|_{s_{1},\ s_{2}}=u|_{\star_{1}\mapsto s_{1},\star_{2}\mapsto s_{2}},$
a commutative $s_{1}$-$s_{2}$-$\Omega$-word.
Now, we assume that $\mathcal{Q}(X)$ is equipped with a monomial order $>$.
This means that $>$ is a well order on $\mathcal{Q}(X)$ such that for any
$w,v\in\mathcal{Q}(X),u\in\mathcal{Q}^{\star}(X)$,
$w>v\Rightarrow u|_{w}>u|_{v}.$
Note that such an order on $\mathcal{Q}(X)$ exists, for example, the order (1)
in the next section.
For every commutative $\Omega$-polynomial $f\in K[X;\Omega]$, let $\bar{f}$ be
the leading term of $f$. If the coefficient of $\bar{f}$ is $1$, then we call
$f$ monic.
Let $f,g$ be two monic $\Omega$-polynomials. Then there are two kinds of
compositions.
1. (I)
If there exists a commutative $\Omega$-word $w=a\bar{f}=b\bar{g}$ for some
$a,b\in\mathcal{Q}(X)$ such that $bre(w)<bre(\bar{f})+bre(\bar{g})$, then we
call $(f,g)_{w}=af-bg$ the intersection composition of $f$ and $g$ with
respect to $w$.
2. (II)
If there exists a commutative $\Omega$-word $w=\bar{f}=u|_{\bar{g}}$ for some
$u\in\mathcal{Q}^{\star}(X)$, then we call $(f,g)_{w}=f-u|_{g}$ the including
composition of $f$ and $g$ with respect to $w$.
In the above definition, $w$ is called the ambiguity of the composition.
Clearly,
$(f,g)_{w}\in Id(f,g)\ \ \ \mbox{ and }\ \ \ \overline{(f,g)_{w}}<w$
where $Id(f,g)$ is the ideal of $K[X;\Omega]$ generated by $f$ and $g$.
Let $f,g$ be commutative $\Omega$-polynomials and $g$ monic with
$\bar{f}=u|_{\bar{g}}$ for some $u\in\mathcal{Q}^{\star}(X)$. Then the
transformation
$f\rightarrow f-\alpha u|_{g}$
is called elimination of the leading commutative $\Omega$-word (ELW) of $f$ by
$g$, where $\alpha$ is the coefficient of the leading commutative
$\Omega$-word of $f$.
Let $S$ be a set of monic commutative $\Omega$-polynomials. Then the
composition $(f,g)_{w}$ is called trivial modulo $(S,w)$, if
$(f,g)_{w}=\sum\alpha_{i}u_{i}|_{s_{i}}$
where each $\alpha_{i}\in K$, $u_{i}\in\mathcal{Q}^{\star}(X)$, $s_{i}\in S$
and $u_{i}|_{\overline{s_{i}}}<w$. If this is the case, we write
$(f,g)_{w}\equiv 0\ \ mod(S,w).$
In general, for any two commutative $\Omega$-polynomials $p$ and $q$, $p\equiv
q\ \ mod(S,w)$ means $p-q=\sum\alpha_{i}u_{i}|_{s_{i}}$ where each
$\alpha_{i}\in K$, $u_{i}\in\mathcal{Q}^{\star}(X)$, $s_{i}\in S$ and
$u_{i}|_{\overline{s_{i}}}<w$.
Then $S$ is called a Gröbner-Shirshov basis in $K[X;\Omega]$ if any
composition $(f,g)_{w}$ of $f,g\in S$ is trivial modulo $(S,w)$.
###### Lemma 3.1
Let $S$ be a Gröbner-Shirshov basis in $K[X;\Omega]$ and $u_{1},\
u_{2}\in\mathcal{Q}^{\star}(X),\ s_{1},s_{2}\in S$. If
$w=u_{1}|_{\overline{s_{1}}}=u_{2}|_{\overline{s_{2}}}$, then
$u_{1}|_{s_{1}}\equiv u_{2}|_{s_{2}}\ mod(S,w).$
Proof: There are three cases to consider.
(I) The commutative $\Omega$-words $\overline{s_{1}}$ and $\overline{s_{2}}$
are disjoint. Then there exits a commutative
$(\star_{1},\star_{2})$-$\Omega$-words $\Pi$ such that
$\Pi|_{\overline{s_{1}},\
\overline{s_{2}}}=u_{1}|_{\overline{s_{1}}}=u_{2}|_{\overline{s_{2}}}.$
Then
$\displaystyle u_{2}|_{s_{2}}-u_{1}|_{s_{1}}$ $\displaystyle=$
$\displaystyle\Pi|_{\overline{s_{1}},\ s_{2}}-\Pi|_{s_{1},\ \overline{s_{2}}}$
$\displaystyle=$ $\displaystyle(-\Pi|_{s_{1}-\overline{s_{1}},\
s_{2}}+\Pi|_{s_{1},\ s_{2}})+(\Pi|_{s_{1},\
s_{2}-\overline{s_{2}}}-\Pi|_{s_{1},\ s_{2}})$ $\displaystyle=$
$\displaystyle-\Pi|_{s_{1}-\overline{s_{1}},\ s_{2}}+\Pi|_{s_{1},\
s_{2}-\overline{s_{2}}}$
Let
$-\Pi|_{s_{1}-\overline{s_{1}},\ s_{2}}=\sum\alpha_{2_{t}}u_{2_{t}}|_{s_{2}}\
\ \mbox{and}\ \ \Pi|_{s_{1},\
s_{2}-\overline{s_{2}}}=\sum\alpha_{1_{l}}u_{1_{l}}|_{s_{1}}.$
Since $\overline{s_{1}-\overline{s_{1}}}<\overline{s_{1}}$ and
$\overline{s_{2}-\overline{s_{2}}}<\overline{s_{2}}$, we have
$u_{2_{t}}|_{\overline{s_{2}}},\ u_{1_{l}}|_{\overline{s_{1}}}<w.$
Therefore
$u_{{}_{2}}|_{s_{2}}-u_{1}|_{s_{1}}=\sum\alpha_{2_{t}}u_{2_{t}}|_{s_{2}}+\sum\alpha_{1_{l}}u_{1_{l}}|_{s_{1}}$
with each $u_{2_{t}}|_{\overline{s_{2}}},\ u_{1_{l}}|_{\overline{s_{1}}}<w$.
It follows that
$u_{1}|_{s_{1}}\equiv u_{2}|_{s_{2}}\ mod(S,w).$
(II) The commutative $\Omega$-words $\overline{s_{1}}$ and $\overline{s_{2}}$
have nonempty intersection but do not include each other. For example,
$a\overline{s_{1}}=b\overline{s_{2}}$
for some commutative $\Omega$-words $a,\ b$. Then there exists a commutative
$\star$-$\Omega$-word $\Pi$ such that
$\Pi|_{a\overline{s_{1}}}=u_{1}|_{\overline{s_{{}_{1}}}}=u_{2}|_{\overline{s_{2}}}=\Pi|_{b\overline{s_{2}}}.$
Then we have
$u_{{}_{2}}|_{s_{{}_{2}}}-u_{1}|_{s_{{}_{1}}}=\Pi|_{bs_{{}_{2}}}-\Pi|_{as_{{}_{1}}}=-\Pi|_{as_{{}_{1}}-bs_{{}_{2}}}.$
Since $S$ is a Gröbner-Shirshov basis in $K[X;\Omega]$, we have
$as_{1}-bs_{2}=\sum\alpha_{j}v_{j}|_{s_{j}}$
where each $\alpha_{j}\in K,\ v_{j}\in\mathcal{Q}^{\star}(X),\ s_{j}\in S$ and
$v_{j}|_{\overline{s_{j}}}<a\overline{s_{1}}$. Let
$\Pi|_{v_{j}|_{s_{j}}}=\Pi_{j}|_{s_{j}}.$
Then
$u_{2}|_{s_{2}}-u_{1}|_{s_{1}}=\sum\alpha_{j}\Pi_{j}|_{s_{j}}$
with
$\Pi_{j}|_{\overline{s_{j}}}<w.$
It follows that
$u_{1}|_{s_{1}}\equiv u_{2}|_{s_{2}}\ mod(S,w).$
(III) One of commutative $\Omega$-words $\overline{s_{1}}$, $\overline{s_{2}}$
is contained in the other. For example, let
$\overline{s_{1}}=u|_{\overline{s_{2}}}$
for some commutative $\star$-$\Omega$-word $u$. Then
$w=u_{2}|_{\overline{s_{2}}}=u_{1}|_{u|_{\overline{s_{2}}}}$
and
$u_{2}|_{s_{2}}-u_{1}|_{s_{1}}=u_{1}|_{u|_{s_{2}}}-u_{1}|_{s_{1}}=-u_{1}|_{s_{1}-u|_{s_{2}}}.$
Similarly to (II), we can obtain the result. $\blacksquare$
The following theorem is an analogy of Shirshov’s composition lemma for Lie
algebras [20], which was specialized to associative algebras by Bokut [3]. For
commutative algebras, this lemma is known as the Buchberger’s Theorem in [7,
8].
###### Theorem 3.2
(Composition-Diamond lemma) Let $S$ be a set of monic commutative
$\Omega$-polynomials in $K[X;\Omega]$ and $>$ a monomial order on
$\mathcal{Q}(X)$. Then the following statement are equivalent:
1. (I)
$S$ is a Gröbner-Shirshov basis in $K[X;\Omega]$.
2. (II)
$f\in Id(S)\Rightarrow\bar{f}=u|_{\overline{s}}$ for some
$u\in\mathcal{Q}^{\star}(X)$ and $s\in S$.
3. (III)
$Irr(S)=\\{w\in\mathcal{Q}(X)|w\neq u|_{\overline{s}}\mbox{ for any}\
u\in\mathcal{Q}^{\star}(X)\ \mbox{and }s\in S\\}$ is a $K$-basis of
$K[X;\Omega]/Id(S)=K[X;\Omega|S]$.
Proof: (I) $\Longrightarrow$ (II) Let $0\neq f\in Id(S)$. Then
$f=\sum\limits_{i=1}^{n}\alpha_{i}u_{i}|_{s_{i}}$
where each $\alpha_{i}\in K$, $u_{i}\in\mathcal{Q}^{\star}(X)$ and $s_{i}\in
S$. Let $w_{i}=u_{i}|_{\overline{s_{i}}}$ and we arrange this leading
commutative $\Omega$-words in non-increasing order by
$w_{1}=w_{2}=\cdots=w_{m}>w_{m+1}\geq\cdots\geq w_{n}.$
We prove the result by induction on $m$.
If $m=1$, then $\bar{f}=u_{1}|_{\overline{s_{1}}}$.
Now we assume that $m\geq 2$. Then
$u_{1}|_{\overline{s_{1}}}=w_{1}=w_{2}=u_{2}|_{\overline{s_{2}}}.$
We prove the result by induction on $w_{1}$. If $\bar{f}=w_{1}$, there is
nothing to prove. Clearly, $w_{1}>\bar{f}$. Since $S$ is a Gröbner-Shirshov
basis in $K[X;\Omega]$, by Lemma 3.1, we have
$u_{2}|_{s_{2}}-u_{1}|_{s_{1}}=\sum\beta_{j}v_{j}|_{s_{j}}$
where $\beta_{j}\in K,\ s_{j}\in S,v_{j}\in\mathcal{Q}^{\star}(X)$ and
$v_{j}|_{\overline{s_{j}}}<w_{1}$. Therefore, since
$\alpha_{1}u_{1}|_{s_{1}}+\alpha_{2}u_{2}|_{s_{2}}=(\alpha_{1}+\alpha_{2})u_{1}|_{s_{1}}+\alpha_{2}(u_{2}|_{s_{2}}-u_{1}|_{s_{1}}),$
we have
$f=(\alpha_{1}+\alpha_{2})u_{1}|_{s_{1}}+\sum\alpha_{2}\beta_{j}v_{j}|_{s_{j}}+\sum\limits_{i=3}^{n}\alpha_{i}u_{i}|_{s_{i}}.$
If either $m>2$ or $\alpha_{1}+\alpha_{2}\neq 0$, then the result follows from
induction on $m$. If $m=2$ and $\alpha_{1}+\alpha_{2}=0$, then the result
follows from induction on $w_{1}$.
(II)$\Longrightarrow$ (III) For any $f\in K[X;\Omega]$, by ELWs, we can obtain
that $f+Id(S)$ can be expressed by the elements of $Irr(S)$. Now suppose
$\alpha_{1}u_{1}+\alpha_{2}u_{2}+\cdots\alpha_{n}u_{n}=0$ in $K[X;\Omega|S]$
with $u_{i}\in Irr(S)$, $u_{1}>u_{2}>\cdots>u_{n}$ and $\alpha_{i}\neq 0$.
Then, in $K[X;\Omega]$,
$g=\alpha_{1}u_{1}+\alpha_{2}u_{2}+\cdots+\alpha_{n}u_{n}\in Id(S).$
By (II), we have $u_{1}=\bar{g}\notin Irr(S)$, a contradiction. So $Irr(S)$ is
$K$-linearly independent. This shows that $Irr(S)$ is a $K$-basis of
$K[X;\Omega|S]$.
(III)$\Longrightarrow$(II) Let $0\neq f\in Id(S)$. Suppose that $\bar{f}\in
Irr(S)$. Then
$f+Id(S)=\alpha(\bar{f}+Id(S))+\sum\alpha_{i}(u_{i}+Id(S))$
where $u_{i}\in Irr(S)$ and $\bar{f}>u_{i}$. Therefore, $f+Id(S)\neq 0$, a
contradiction. So $\bar{f}=u|_{\bar{s}}$ for some $s\in S$ and
$u\in\mathcal{Q}^{\star}(X)$.
(II)$\Longrightarrow$(I) By the definition of the composition, we have
$(f,g)_{w}\in Id(S)$. If $(f,g)_{w}\neq 0$, then by (II),
$\overline{(f,g)_{w}}=u_{1}|_{\overline{s_{1}}}<w$ for some $s_{1}\in S$ and
$u_{1}\in\mathcal{Q}^{\star}(X)$. Let
$h=(f,g)_{w}-\alpha_{1}u_{1}|_{s_{1}}$
where $\alpha_{1}$ is the coefficient of $\overline{(f,g)_{w}}$. Then
$\bar{h}<w$ and $h\in Id(S)$. By induction on $w$, we can get the result.
$\blacksquare$
## 4 Gröbner-Shirshov bases for free commutative Rota-Baxter algebras
In this section, a Gröbner-Shirshov basis for free commutative Rota-Baxter
algebra was obtained. By using the Composition-Diamond lemma (Theorem 3.2), a
linear basis of such algebra was given and the free commutative Rota-Baxter
algebra was directly constructed by commutative $\Omega$-words.
First of all, we define an order on $\mathcal{Q}(X)$, which will be used in
this section. Let $X$ and $\Omega$ be well ordered. We define an order on
$\mathcal{Q}(X)=\bigcup_{n\geq 0}\mathcal{Q}_{n}$ by induction on $n$. For any
$u,v\in\mathcal{Q}_{0}=CS(X)$, we have
$u=x_{1}^{i_{1}}x_{2}^{i_{2}}\cdots x_{t}^{i_{t}}\ \mbox{and}\
v=x_{1}^{j_{1}}x_{2}^{j_{2}}\cdots x_{t}^{j_{t}}$
where each $x_{i}\in X,\ i_{k},j_{l}\geq 0$ and $x_{i}>x_{i+1}$. Then we
define
$u>v\Longleftrightarrow(bre(u),i_{1},i_{2},\dots,i_{t})>(bre(v),j_{1},j_{2},\dots,j_{t})\
\ \mbox{lexicographically}.$
Assume the order on $\mathcal{Q}_{n-1}$ has been defined. Now, we define an
order on $\Gamma_{n}=X\cup\Omega(\mathcal{Q}_{n-1})$. Let
$v_{1},v_{2}\in\Gamma_{n}$. Then $v_{1}>v_{2}$ means one of the following
holds:
(a) $v_{1},v_{2}\in X$ and $v_{1}>v_{2}$;
(b) $v_{1}\in\Omega(\mathcal{Q}_{n-1})$ and $v_{2}\in X$;
(c) $v_{1}=\theta_{k}(v_{1}^{\prime})$,
$v_{2}=\theta_{l}(v_{2}^{\prime}),v_{1}^{\prime}\in\mathcal{Q}_{n-1}^{k},v_{2}^{\prime}\in\mathcal{Q}_{n-1}^{l}$
with $\theta_{k}>\theta_{l}$ or $\theta_{k}=\theta_{l}$,
$v_{1}^{\prime}>v_{2}^{\prime}$ lexicographically.
For any $u,v\in\mathcal{Q}_{n}$, we have
$u=w_{1}^{i_{1}}w_{2}^{i_{2}}\cdots w_{t}^{i_{t}}\ \mbox{and}\
v=w_{1}^{j_{1}}w_{2}^{j_{2}}\cdots w_{t}^{j_{t}}$
where each $w_{i}\in\Gamma_{n}$, $i_{k},j_{l}\geq 0$ and $w_{i}>w_{i+1}$.
Here, $bre(u)=i_{1}+i_{2}+\cdots+i_{t}$. Define
$u>v\Longleftrightarrow(bre(u),i_{1},i_{2},\dots,i_{t})>(bre(v),j_{1},j_{2},\dots,j_{t})\
\ \mbox{lexicographically}$ (1)
Then the Order (1) is a monomial order on $\mathcal{Q}(X)$.
Let $K$ be a commutative ring with unit and $\lambda\in K$. A commutative
Rota-Baxter algebra of weight $\lambda$ ( see [1, 13, 19]) is a commutative
$K$-algebra $R$ with a linear operator $P:R\rightarrow R$ satisfying the Rota-
Baxter relation:
$P(u)P(v)=P(P(u)v)+P(uP(v))+\lambda P(uv),\ \forall u,v\in R.$
It is obvious that any commutative Rota-Baxter algebra is a commutative
algebra with multiple operators $\Omega$, where $\Omega=\\{P\\}$.
In this section, we assume that $\Omega=\\{P\\}$. Let $\mathcal{Q}(X)$ be
defined as before with $\Omega=\\{P\\}$ and $K[X;P]$ be the free commutative
algebra with operator $\Omega=\\{P\\}$ on set $X$.
###### Theorem 4.1
With the order (1) on $\mathcal{Q}(X)$,
$S=\\{P(u)P(v)-P(P(u)v)-P(uP(v))-\lambda P(uv)|\ u,v\in\mathcal{Q}(X)\\}$
is a Gröbner-Shirshov basis in $K[X;P]$.
Proof: The ambiguities of all possible compositions of the commutative
$\Omega$-polynomials in $S$ are:
$(i)\ \ \ \ P(u)P(v)P(w)\ \ \ \ \ \ \ \ (ii)\ \ \ \ P(z|_{P(v)P(w)})P(u)\ \ \
\ $
where $u,v,w\in\mathcal{Q}(X),z\in\mathcal{Q}^{\star}(X)$. It is easy to check
that all these compositions are trivial. Here, for example, we just check
$(i)$. For any $u,v\in\mathcal{Q}(X)$, let
$f(u,v)=P(u)P(v)-P(P(u)v)-P(uP(v))-\lambda P(uv).$
Then
$\displaystyle(f(u,v),f(v,w))_{P(u)P(v)P(w)}$
$\displaystyle=-P(P(u)v)P(w)-P(uP(v))P(w)-\lambda P(uv)P(w)$ $\displaystyle\ \
\ +P(u)P(P(v)w)+P(u)P(vP(w))+\lambda P(u)P(vw)$
$\displaystyle\equiv-P(P(P(u)v)w)-P(P(u)vP(w))-\lambda P(P(u)vw)$
$\displaystyle\ \ \ -P(P(uP(v))w)-P(uP(v)P(w))-\lambda P(uP(v)w)$
$\displaystyle\ \ \ -\lambda P(P(uv)w)-\lambda P(uvP(w))-\lambda^{2}P(uvw)$
$\displaystyle\ \ \ +P(P(u)P(v)w)+P(uP(P(v)w))+\lambda P(uP(v)w)$
$\displaystyle\ \ \ +P(P(u)vP(w))+P(uP(vP(w)))+\lambda P(uvP(w))$
$\displaystyle\ \ \ +\lambda P(P(u)vw)+\lambda P(uP(vw))+\lambda^{2}P(uvw)$
$\displaystyle\equiv-P(P(P(u)v)w)-P(P(uP(v))w)-\lambda P(P(uv)w)$
$\displaystyle\ \ \ -P(uP(P(v)w))-P(uP(vP(w)))-\lambda P(uP(vw))$
$\displaystyle\ \ \ +P(P(P(u)v)w)+P(P(uP(v))w)+\lambda P(P(uv)w)$
$\displaystyle\ \ \ +P(uP(P(v)w))+P(uP(vP(w)))+\lambda P(uP(vw))$
$\displaystyle\equiv 0\ \ mod(S,P(u)P(v)P(w)).\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \blacksquare$
Define
$\displaystyle\Phi_{0}$ $\displaystyle=$ $\displaystyle CS(X),$
$\displaystyle\Phi_{1}$ $\displaystyle=$ $\displaystyle\Phi_{0}\cup
P(\Phi_{0})\cup\Phi_{0}P(\Phi_{0}),$ $\displaystyle\vdots\ \ $ $\displaystyle\
\ \ \ \vdots$ $\displaystyle\Phi_{n}$ $\displaystyle=$
$\displaystyle\Phi_{0}\cup P(\Phi_{n-1})\cup\Phi_{0}P(\Phi_{n-1}),$
$\displaystyle\vdots\ \ $ $\displaystyle\ \ \ \ \vdots$
where
$\Phi_{0}P(\Phi_{n-1})=\\{uP(v)|u\in\Phi_{0},v\in\Phi_{n-1}\\}.$
Let
$\Phi(X)=\bigcup_{n\geq 0}\Phi_{n}.$
The elements in $\Phi(X)$ are called commutative Rota-Baxter words.
By Theorem 4.1 and Theorem 3.2, we have the following theorems.
###### Theorem 4.2
([6]) $Irr(S)=\Phi(X)$ is a basis of $K[X;\Omega|S]$.
###### Theorem 4.3
$K[X;\Omega|S]$ is a free commutative Rota-Baxter algebra of weight $\lambda$
on set $X$ with a basis $\Phi(X)$.
By using ELWs, we have the following algorithm. In fact, it is an algorithm to
compute the product of two commutative Rota-Baxter words in the free
commutative Rota-Baxter algebra $K[X;\Omega|S]$.
Algorithm Let $u,v\in\Phi(X)$. We define $u\diamond v$ by induction on
$n=dep(u)+dep(v)$.
(1) If $n=0$, then $u,v\in CS(X)$ and $u\diamond v=uv$.
(2) If $n\geq 1$, then $u\diamond v=$
$\left\\{\begin{array}[]{l@{\quad\quad}l}uv&\mbox{if}\ u\in CS(X)\ \mbox{or}\
v\in CS(X)\\\ u_{1}v_{1}(P(P(u^{{}^{\prime}})\diamond
v^{{}^{\prime}})+P(u^{{}^{\prime}}\diamond P(v^{{}^{\prime}}))+\lambda
P(u^{{}^{\prime}}\diamond v^{{}^{\prime}}))&\mbox{if}\
u=u_{1}P(u^{{}^{\prime}}),v=v_{1}P(v^{{}^{\prime}})\end{array}\right.$
## 5 Gröbner-Shirshov bases for free commutative $\lambda$-differential
algebras
In this section, we give a Gröbner-Shirshov basis for a free commutative
$\lambda$-differential algebra and then by using the Composition-Diamond lemma
(Theorem 3.2), we obtain a linear basis of such an algebra, which is the same
as the one in [15]. Consequently, we construct the free $\lambda$-differential
algebra on set $X$ directly by commutative $\Omega$-words.
Let $K$ be a commutative unitary ring and $\lambda\in K$. A commutative
$\lambda$-differential algebra ([15, 16]) over $K$ is a commutative
$K$-algebra $R$ together with a linear operator $D:R\rightarrow R$ such that
$D(uv)=D(u)v+uD(v)+\lambda D(u)D(v),\ \forall u,v\in R.$
It is obvious that any commutative $\lambda$-differential algebra is a
commutative algebra with multiple operators $\Omega$ on $X$, where
$\Omega=\\{D\\}$.
In this section, we assume that $\Omega=\\{D\\}$. Let $K[X;D]$ be the free
commutative algebra with operator $\Omega=\\{D\\}$. Let $X$ be well ordered.
For any $u\in\mathcal{Q}(X)$, $u$ has a unique expression
$u=u_{1}u_{2}\cdots u_{n}$
where $n\geq 1$ and each $u_{i}\in X\cup D(\mathcal{Q}(X))$. Define $deg(u)$
the number of $x\in X$ in $u$. For example, if $u=x_{1}D(D(x_{2}))D(x_{3})$,
then $deg(u)=3$.
We define an order on $\mathcal{Q}(X)=\bigcup_{n\geq 0}\mathcal{Q}_{n}$ by
induction on $n$. For any $u,v\in\mathcal{Q}_{0}=CS(X)$, we have
$u=x_{1}^{i_{1}}x_{2}^{i_{2}}\cdots x_{t}^{i_{t}}\ \mbox{and}\
v=x_{1}^{j_{1}}x_{2}^{j_{2}}\cdots x_{t}^{j_{t}}$
where each $x_{i}\in X$, $i_{k},j_{l}\geq 0$ and $x_{i}>x_{i+1}$. Then we
define
$u>v\Longleftrightarrow(deg(u),i_{1},i_{2},\cdots,i_{t})>(deg(v),j_{1},j_{2},\cdots,j_{t})\
\ \mbox{lexicographically}.$
Assume the order on $\mathcal{Q}_{n-1}$ has been defined. Now, we define an
order on $\Gamma_{n}=X\cup\Omega(\mathcal{Q}_{n-1})$ firstly. Let
$v_{1},v_{2}\in\Gamma_{n}$. Then $v_{1}>v_{2}$ means one of the following
holds:
(a) $v_{1},v_{2}\in X$ and $v_{1}>v_{2}$;
(b) $v_{1}\in D(\mathcal{Q}_{n-1})$ and $v_{2}\in X$;
(c) $v_{1}=D(v_{1}^{\prime})$, $v_{2}=D(v_{2}^{\prime})$ and
$v_{1}^{\prime}>v_{2}^{\prime}$.
For any $u,v\in\mathcal{Q}_{n}$, we have
$u=w_{1}^{i_{1}}w_{2}^{i_{2}}\cdots w_{t}^{i_{t}}\ \mbox{and}\
v=w_{1}^{j_{1}}w_{2}^{j_{2}}\cdots w_{t}^{j_{t}}$
where each $w_{i}\in\Gamma_{n}$, $i_{k},j_{l}\geq 0$ and $w_{i}>w_{i+1}$.
Define
$u>v\Longleftrightarrow(deg(u),i_{1},i_{2},\cdots,i_{t})>(deg(v),j_{1},j_{2},\cdots,j_{t})\
\ \ \mbox{lexicographically}$ (2)
Then the Order (2) is a monomial order on $\mathcal{Q}(X)$.
###### Theorem 5.1
With the Order (2) on $\mathcal{Q}(X)$,
$S=\\{D(uv)-D(u)v-uD(v)-\lambda D(u)D(v)|\ u,v\in\mathcal{Q}(X)\\}$
is a Gröbner-Shirshov basis in $K[X;D]$.
Proof: The ambiguities of all possible compositions of the commutative
$\Omega$-polynomials in $S$ are
$D(u|_{{}_{D(xy)}}v)$
where $x,y,v\in\mathcal{Q}(X),u\in\mathcal{Q}^{\star}(X)$. Let
$g(u,v)=D(uv)-D(u)v-uD(v)-\lambda D(u)D(v).$
Then
$\displaystyle(g(u|_{{}_{D(xy)}},v),g(x,y))_{D(u|_{{}_{D(xy)}}v)}$
$\displaystyle=-D(u|_{{}_{D(xy)}})v-u|_{{}_{D(xy)}}D(v)+\lambda
D(u|_{{}_{D(xy)}})D(v)$ $\displaystyle\ \ \
+D(u|_{{}_{D(x)y}}v)+D(u|_{{}_{xD(y)}}v)+\lambda D(u|_{{}_{D(x)D(y)}}v)$
$\displaystyle\equiv-D(u|_{{}_{D(x)y}})v-D(u|_{{}_{xD(y)}})v-\lambda
D(u|_{{}_{D(x)D(y)}})v$ $\displaystyle\ \ \
-u|_{{}_{D(x)y}}D(v)-u|_{{}_{xD(y)}}D(v)-\lambda u|_{{}_{D(x)D(y)}}D(v)$
$\displaystyle\ \ \ -\lambda D(u|_{{}_{D(x)y}})D(v)-\lambda
D(u|_{{}_{xD(y)}})D(v)-\lambda^{2}D(u|_{{}_{D(x)D(y)}})D(v)$ $\displaystyle\ \
\ +D(u|_{{}_{D(x)y}})v+u|_{{}_{D(x)y}}D(v)+\lambda D(u|_{{}_{D(x)y}})D(v)$
$\displaystyle\ \ \ +D(u|_{{}_{xD(y)}})v+u|_{{}_{xD(y)}}D(v)+\lambda
D(u|_{{}_{xD(y)}})D(v)$ $\displaystyle\ \ \ +\lambda
D(u|_{{}_{D(x)D(y)}})v+\lambda
u|_{{}_{D(x)D(y)}}D(v)+\lambda^{2}D(u|_{{}_{D(x)D(y)}})D(v)$
$\displaystyle\equiv 0\ \ mod(S,D(u|_{{}_{D(xy)}}v).\ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \blacksquare$
Let $D^{\omega}(X)=\\{D^{i}(x)|i\geq 0,x\in X\\}$, where $D^{0}(x)=x$ and
$CS(D^{\omega}(X))$ the free commutative semigroup generated by
$D^{\omega}(X)$.
###### Theorem 5.2
([15]) $Irr(S)=CS(D^{\omega}(X))$ is a $K$-basis of $K[X;D|S]$.
Proof: By Theorem 3.2 and Theorem 5.1.
###### Theorem 5.3
([15]) $K[X;D|S]$ is a free commutative $\lambda$-differential algebra on set
$X$ with a basis $CS(D^{\omega}(X))$.
Proof: By Theorem 5.2.
By using ELWs, we have the following algorithm.
Algorithm ([15]) Let $u=u_{1}u_{2}\cdots u_{n}$, where each $u_{k}\in
D^{\omega}(X),n>0$. Define $D(u)$ by induction on $n$.
(1) If $n=1$, i.e., $u=D^{i}(x)$ for some $i\geq 0,x\in X$, then
$D(u)=D^{(i+1)}(x)$.
(2) If $n\geq 1$, then
$D(u)=D(u_{1}u_{2}\cdots u_{n})=D(u_{1})(u_{2}\cdots u_{n})+u_{1}D(u_{2}\cdots
u_{n})+\lambda D(u_{1})D(u_{2}\cdots u_{n}).$
## 6 Gröbner-Shirshov bases for free commutative $\lambda$-differential
Rota-Baxter algebras
In this section, we give a Gröbner-Shirshov basis for a free commutative
$\lambda$-differential Rota-Baxter algebra on a set. By using the Composition-
Diamond lemma for commutative algebras with multiple operators (Theorem 3.2),
we obtain a linear basis of a free commutative $\lambda$-differential Rota-
Baxter algebra on a set. Also, we construct the free commutative
$\lambda$-differential Rota-Baxter algebra on a set directly by commutative
$\Omega$-words.
Let $K$ be a unitary commutative ring and $\lambda\in K$. A commutative
$\lambda$-differential Rota-Baxter algebra ([15]) is a commutative $K$-algebra
$R$ with two linear operators $P,D:R\rightarrow R$ such that, for any $u,v\in
R$,
(I) (Rota-Baxter relation) $P(u)P(v)=P(uP(v))+P(P(u)v)+\lambda P(uv),$
(II) ($\lambda$-differential relation) $D(uv)=D(u)v+uD(v)+\lambda D(u)D(v),$
(III) $D(P(u))=u$.
It is obvious that any commutative $\lambda$-differential Rota-Baxter algebra
is a commutative algebra with multiple operators $\Omega$ where
$\Omega=\\{P,D\\}$.
In this section, we assume that $\Omega=\\{P,D\\}$. Let $K[X;\Omega]$ be the
free commutative algebra with multiple operators $\Omega$ on $X$, where
$\Omega=\\{P,D\\}$.
Let $X$ be well ordered and $D>P$. For any $u\in\mathcal{Q}(X)$, define
$deg_{{}_{P}}(u)$ the number of $P$ in $u$.
Define an order on $\mathcal{Q}(X)=\bigcup_{n\geq 0}\mathcal{Q}_{n}$ by
induction on $n$. For any $u,v\in\mathcal{Q}_{0}=CS(X)$, we have
$u=x_{1}^{i_{1}}x_{2}^{i_{2}}\cdots x_{t}^{i_{t}}\ \mbox{and}\
v=x_{1}^{j_{1}}x_{2}^{j_{2}}\cdots x_{t}^{j_{t}}$
where each $x_{i}\in X$, $i_{k},j_{l}\geq 0$ and $x_{i}>x_{i+1}$. Then we
define
$u>v\Longleftrightarrow(deg(u),i_{1},i_{2},\cdots,i_{t})>(deg(v),j_{1},j_{2},\cdots,j_{t})\
\ \ \ \mbox{lexicographically}.$
Assume the order on $\mathcal{Q}_{n-1}$ has been defined. For any
$u,v\in\mathcal{Q}_{n}$, we have
$u=u_{1}^{k_{1}}\cdots u_{s}^{k_{s}}\ \mbox{and}\ v=u_{1}^{l_{1}}\cdots
u_{s}^{l_{s}}$
where each $u_{i}\in X\cup\Omega(\mathcal{Q}_{n-1})$, $k_{i},l_{i}\geq 0$ and
$u_{i}>u_{i+1}$. Here, $u_{i}>u_{i+1}$ means one of the following holds:
(a) $u_{i},u_{i+1}\in X$ and $u_{i}>u_{i+1}$,
(b) $u_{i}\in D(\mathcal{Q}_{n-1})$ or $u_{i}\in P(\mathcal{Q}_{n-1})$ and
$u_{i+1}\in X$,
(c) $u_{i}=\theta_{1}(u_{i}^{\prime})$, $u_{i+1}=\theta_{2}(u_{i+1}^{\prime})$
where $\theta_{1},\theta_{2}\in\\{D,P\\}$ and
$(deg(u_{i}),deg_{{}_{P}}(u_{i}),\theta_{1},u_{i}^{\prime})>(deg(u_{i}),deg_{{}_{P}}(u_{i}),\theta_{2},u_{i+1}^{\prime})\
\ \ \mbox{lexicographically}$
Then we define $u>v$ if and only if
$(deg(u),deg_{{}_{P}}(u),bre(u),k_{1},\dots,k_{s})>(deg(v),deg_{{}_{P}}(v),bre(v),l_{1},\dots,l_{s})\
\ \mbox{lexicographically}$ (3)
Then the Order (3) is a monomial order on $\mathcal{Q}(X)$.
Let $S$ be a set consisting of the following commutative $\Omega$-polynomials:
1. 1
$P(u)P(v)-P(uP(v))-P(P(u)v)-\lambda P(uv),\ \ u,v\in\mathcal{Q}(X)$;
2. 2
$D(uv)-D(u)v-uD(v)-\lambda D(u)D(v),\ \ \ u,v\in\mathcal{Q}(X)$;
3. 3
$D(P(u))-u,\ \ \ u\in\mathcal{Q}(X).$
###### Theorem 6.1
With the Order (3) on $\mathcal{Q}(X)$, $S$ is a Gröbner-Shirshov basis in
$K[X;\Omega]$.
Proof. Denote by $i\wedge j$ the composition of $\Omega$-polynomials of type
$i$ and type $j$. The ambiguities of all possible compositions of commutative
$\Omega$-polynomials in $S$ are only as below. In the following list, $i\wedge
j\ \ \ w$ means $w$ is the ambiguity of the composition $i\wedge j$.
$3\wedge 3\ \ \ D(P(u|_{{}_{D(P(v))}}))$, | $3\wedge 2\ \ \ D(P(u|_{{}_{D(vw)}}))$
---|---
$3\wedge 1\ \ \ D(P(u|_{{}_{P(v)P(w)}}))$ | $2\wedge 3\ \ \ D(u|_{{}_{D(P(v))}}w)$
$2\wedge 2\ \ \ D(u|_{{}_{D(vw)}}z)$, | $2\wedge 1\ \ \ D(u|_{{}_{P(v)P(w)}}z)$
$1\wedge 3\ \ \ P(u|_{{}_{D(P(v))}})P(w)$ | $1\wedge 2\ \ \ P(u|_{{}_{D(vw)}})P(z)$
$1\wedge 1\ \ \ P(z)P(v)P(w)$, | $1\wedge 1\ \ \ P(v)P(u|_{{}_{P(w)P(z)}})$
where $u\in\mathcal{Q}^{\star}(X),v,w,z\in\mathcal{Q}(X)$. It is easy to check
that all these compositions are trivial. Similar to the proofs in Theorem 4.1
and Theorem 5.1, $1\wedge 1$ and $2\wedge 2$ are trivial. Others are also
easily checked. Here we just check one for examples.
$\displaystyle 2\wedge 3$ $\displaystyle=$
$\displaystyle-D(u|_{{}_{D(P(v))}})w-u|_{{}_{D(P(v))}}D(w)-\lambda
D(u|_{{}_{D(P(v))}})D(w)+D(u|_{{}_{v}}w)$ $\displaystyle\equiv$
$\displaystyle-D(u|_{{}_{v}})w-u|_{{}_{v}}D(w)-\lambda D(u|_{{}_{v}})D(w)$
$\displaystyle+D(u|_{{}_{v}})w+u|_{{}_{v}}D(w)+\lambda D(u|_{{}_{v}})D(w)$
$\displaystyle\equiv$ $\displaystyle 0\ mod(S,D(u|_{{}_{D(P(v))}}w)).\ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \blacksquare$
Let $D^{\omega}(X)=\\{D^{i}(x)|i\geq 0,x\in X\\}$, where $D^{0}(x)=x$. Define
$\displaystyle\Upsilon_{0}$ $\displaystyle=$ $\displaystyle
CS(D^{\omega}(X)),$ $\displaystyle\Upsilon_{1}$ $\displaystyle=$
$\displaystyle\Upsilon_{0}\cup
P(\Upsilon_{0})\cup\Upsilon_{0}P(\Upsilon_{0}),$ $\displaystyle\vdots\ \ $
$\displaystyle\ \ \ \ \vdots$ $\displaystyle\Upsilon_{n}$ $\displaystyle=$
$\displaystyle\Upsilon_{0}\cup
P(\Upsilon_{n-1})\cup\Upsilon_{0}P(\Upsilon_{n-1}),$ $\displaystyle\vdots\ \ $
$\displaystyle\ \ \ \ \vdots$
and
$\Upsilon(D^{\omega}(X))=\bigcup_{n\geq 0}\Upsilon_{n}.$
###### Theorem 6.2
$Irr(S)=\Upsilon(D^{\omega}(X))$ is a basis of $K[X;\Omega|S]$.
Proof: By Theorem 3.2 and Theorem 6.1, we can obtain the result easily.
$\blacksquare$
###### Theorem 6.3
$K[X;\Omega|S]$ is a free commutative $\lambda$-differential Rota-Baxter
algebra on set $X$ with a basis $\Upsilon(D^{\omega}(X))$.
Proof: By Theorem 6.2.
## References
* [1] G. Baxter, An analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math., 10(1960), 731-742.
* [2] G. M. Bergman, The diamond lemma for ring theory, Adv. Math., 29(1978), 178-218.
* [3] L. A. Bokut, Imbeddings into simple associative algebras, Algebra i Logika, 15(1976), 117-142.
* [4] L. A. Bokut, V. Latyshev, I. Shestakov, E. Zelmanov (Eds), Trs M. Bremner, M. Kochetov, Selected Works of A. I. Shirshov, Birkhäuser Verlag, Basel-Boston-Berlin, 2009.
* [5] L. A. Bokut, Y. Chen and J. Qiu, Gröbner-Shirshov bases for associative algebras with multiple operators and free Rota-Baxter algebras, J. Pure Appl. Algebra, 214(2010), 89-100.
* [6] L. A. Bokut, Y. Chen, X. Deng, Gröbner-Shirshov bases for Rota-Baxter algebras, Siberian. Math. J., 51(6)(2010), 978-988.
* [7] B. Buchberger, An algorithm for finding a basis for the residue class ring of a zero-dimensional polynomial ideal [in German], Ph.D. thesis, University of Innsbruck, Austria, (1965).
* [8] B. Buchberger, An algorithmical criteria for the solvability of algebraic systems of equations [in German], Aequationes Math., 4(1970), 374-383.
* [9] P. Cartier, On the structure of free Baxter algebras, Adv. in Math., 9(1972), 253-265.
* [10] V. Drensky, R. Holtkamp, Planar trees, free nonassociative algebras, invariants, and elliptic integrals, Algebra and Discrete Mathmatics, 2(2008), 1-41.
* [11] K. Ebrahimi-Fard, L. Guo, Rota-Baxter algebras and dendriform dialgebras, J. Pure and Applied Algebra, 212(2)(2008), 320-339.
* [12] K. Ebrahimi-Fard, L. Guo, Free Rota-Baxter algebras and rooted trees, J. Algebra and Applications, 7(2008), 167-194.
* [13] L. Guo, W. Keigher, Baxter algebras and Shuffle products, Adv. Math., 150(2000), 117-149.
* [14] L. Guo, W. Keigher, On free Baxter algebras: completions and the internal construction, Adv. Math., 151(2000), 101-127.
* [15] L. Guo, W. Keigher, On differential Rota-Baxter algebras, J. Pure and Applied Algebra, 212(2008), 522-540.
* [16] E. Kolchin, Differential Algebras and Algebraic Groups, Academic Press, NewYork, 1973.
* [17] A. G. Kurosh, Free sums of multiple operator algebras, Siberian. Math. J., 1(1960), 62-70.
* [18] J. Qiu, Y. Chen, Composition-Diamond lemma for $\lambda$-differential associative algebras with multiple operators, J. Algebra and Applications, 9(2)(2010), 223-239.
* [19] G.-C. Rota, Baxter algebras and combinatorial identities I, Bull. Amer. Math. Soc., 5(1969), 325-329.
* [20] A. I. Shirshov, Some algorithmic problem for Lie algebras, Sibirsk. Mat. Z., 3(1962), 292-296 (in Russian). English translation: SIGSAM Bull., 33(2)(1999), 3-6.
|
arxiv-papers
| 2013-01-18T09:51:51 |
2024-09-04T02:49:40.518499
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jianjun Qiu",
"submitter": "Jianjun Qiu",
"url": "https://arxiv.org/abs/1301.5018"
}
|
1301.5103
|
# High-temperature expansion of the one-loop free energy of a scalar field on
a curved background
I.S. Kalinichenko [email protected] Physics Faculty, Tomsk State University,
Tomsk, 634050 Russia P.O. Kazinski [email protected] Physics Faculty, Tomsk
State University, Tomsk, 634050 Russia Department of Higher Mathematics and
Mathematical Physics, Tomsk Polytechnic University, Tomsk, 634050 Russia
###### Abstract
The complete form of the high-temperature expansion of the one-loop
contribution to the free energy of a scalar field on a stationary
gravitational background is derived. The explicit expressions for the
divergent and finite parts of the high-temperature expansion in a three-
dimensional space without boundaries are obtained. These formulas generalize
the known one for the stationary spacetime. In particular, we confirm that for
a massless conformal scalar field the leading correction to the Planck law
proportional to the temperature squared turns out to be nonzero due to non-
static nature of the metric. The explicit expression for the so-called energy-
time anomaly is found. The interrelation between this anomaly and the
conformal (trace) anomaly is established. The natural simplest Lagrangian for
the “Killing vector field” is given.
###### pacs:
04.62.+v
## I Introduction
The high-temperature expansions of the partition functions are the classical
subject of quantum field theory on a curved background. As for the one-loop
contributions to the free energy of quantum fields, we just mention the works
DowKen ; NakFuk ; DowSch ; DowSch1 ; HuCrSt ; Kirsten ; Campor ; GusZeln ;
Fursaev1 ; Fursaev2 where the different approaches to this problem were
implemented. Surprisingly, in spite of the fact that the first attempts to
find the high-temperature expansion were undertaken over about thirty years
ago DowKen , the problem was not completely solved, to our knowledge, in its
general statement for the arbitrary stationary (non-static) gravitational
background. The present paper is aimed to fill this gap. Such a high-
temperature expansion for the free energy will be derived here. In particular,
for a massless conformal scalar field we shall obtain the leading correction
to the Planck law proportional to the temperature squared and confirm the
result of Fursaev1 ; Fursaev2 . This correction is absent for a static
gravitational background DowKen ; NakFuk ; DowSch . Also we shall derive the
explicit expression for the finite part of the high-temperature expansion,
which seems to be a new result.
Apart from the immediate implications for the Casimir effect, astrophysics,
and cosmology, the high-temperature expansion of the free energy for fermions
can be used to analyze the derivative expansion of the one-loop contribution
to the effective action at zero temperature (see, e.g., olopq ; gmse )
regularized by the energy cutoff. According to the general prescriptions of
the renormalization theory Collins , the structures appearing as the
divergencies in the effective action must be included to the initial action to
cancel these divergencies (not necessarily to zero). As a rule, these
divergencies and the finite part depend nontrivially on the Killing vector
$\xi^{\mu}$ defining the stationarity of the background and the vacuum state
of the quantum fields (psfss , see also DeWQFTcspt ; BrOtPa ; FrZel ). We
shall see in the present paper that this is indeed the case. Hence, the
analysis of the high-temperature expansion sheds a light on the low energy
quantum dynamics of the vector field $\xi^{\mu}$ as it was discussed in gmse ;
psfss . To see how this works, one may bear in mind the expansion of the
Heisenberg-Euler effective action HeisEul ; Schwing , where the first non-
trivial term describes the light by light scattering.
Another interesting point following from the results of the present paper is
the interrelation between the conformal (trace) anomaly and the energy-time
anomaly (the notion of the latter was introduced in psfss , see also below).
As we shall see, we cannot renormalize the quantum theory of a massless
conformal scalar field in such a way that both the conformal and energy-time
anomalies vanish. The elimination of the one anomaly results in the appearance
of the other and vice versa. It is noteworthy that the factor at the logarithm
of the temperature entering the high-temperature expansion and determining the
anomaly turns out to be independent of the Killing vector and coincides with
the standard expression for the conformal anomaly DowSch ; Fursaev2 . However,
the finite part of the high-temperature expansion does involve certain
contractions of the Killing vector field. Loosely speaking, the problem of
dependence of the effective action on the Killing vector comes from the
infrared modes of the gravitational field and so it is reasonable that the
renormalization group beta function defining the ultraviolet behaviour of the
theory is independent of the Killing vector. Though, of course, this fact is
not a priori obvious since both in the ultraviolet and the infrared regions we
have the mode functions defined with respect to the same Killing vector (see,
however, Fursaev2 for the possible proof).
The paper is organized as follows. Since the subject of the article is rather
technical, we partition it into the small sections distinguishing the major
successive steps of the derivation, which culminates in Sec. VII where the
results and its implications are discussed. In Sec. II, we provide general
formulas for the one-loop correction to the free energy. The main technical
tool, which we shall employ to derive the high-temperature expansion, is the
heat kernel of a Laplacian operator (see VasilHeatKer for review). Therefore,
in Secs. III, IV, and V we reduce our problem to the evaluation of the heat
kernel expansion coefficients. At first, Sec. III, we reduce the three-
dimensional problem to the four-dimensional one to provide the explicit
general covariance to the expansion. The method, which is used here, is a
finite dimensional analog of the gauge fixing procedure in the functional
integral. Then, in Sec. IV, we resum the heat kernel expansion applying the
theorem proved in Parker . Notice that we do not use the conformal
transformation to derive the high-temperature expansion DowKen ; DowSch ;
DowSch1 ; Kirsten ; Campor ; GusZeln ; Fursaev1 ; Fursaev2 , but apply the
method directly to the Fourier transformed Klein-Gordon operator. So, our
approach is rather close to the one used in NakFuk . After that, the problem
becomes in essence the same as in a flat spacetime. In Sec. V, we evaluate the
expansions of necessary integrals employing the procedure used in olopq .
Then, in Sec. VI, a general formula for the high-temperature expansion of the
free energy is obtained. In Sec. VII, we particularize the general formulas to
the three-dimensional space and single out the divergent and finite parts of
the high-temperature expansion. At this point we essentially employ the
results of Ven , where the heat kernel coefficients were derived up to
$a_{6}$. In Appendix A, we give some useful relations for the metric
possessing the Killing vector. In Appendix B, the relevant parts of the heat
kernel coefficients borrowed from VasilHeatKer ; Ven ; Avramid ; Gilkey are
calculated.
Despite the fact that we study the high-temperature expansion of the free
energy of quantum scalar field on a gravitational background, the results are
easily generalized to higher spins and other stationary backgrounds. The
problem lies only in the amount of calculation, which increases due to the
additional terms in the heat kernel expansion. A non-zero chemical potential
can be also included. We postpone the investigation of these problems for the
future research.
We shall use the following conventions for the curvatures and other structures
appearing in the heat kernel expansion:
$R^{\alpha}_{\
\beta\mu\nu}=\partial_{[\mu}\Gamma^{\alpha}_{\nu]\beta}+\Gamma^{\alpha}_{[\mu\gamma}\Gamma^{\gamma}_{\nu]\beta},\qquad
R_{\mu\nu}=R^{\alpha}_{\ \mu\alpha\nu},\qquad R=R^{\mu}_{\mu}.$ (1)
The square and round brackets at a pair of indices denote antisymmetrization
and symmetrization without $1/2$, respectively. The Greek indices are raised
and lowered by the metric $g_{\mu\nu}$ which has the signature $-2$. Also we
assume that the metric possesses the timelike Killing vector $\xi^{\mu}$:
$\mathcal{L}_{\xi}g_{\mu\nu}=0,\qquad\xi^{2}=g_{\mu\nu}\xi^{\mu}\xi^{\nu}>0.$
(2)
The space dimension will be denoted by $d$ and $d=3$. Nevertheless, we shall
put $d=3$ only in the final result. The system of units is chosen such that
$c=\hbar=1$.
## II General formulas
Consider a scalar quantum field on a stationary gravitational background at a
finite reciprocal temperature $\beta$. The free energy for this system is
defined in the standard way
$e^{-\beta F}:=\operatorname{Tr}e^{-\beta\mathcal{H}},$ (3)
where $\mathcal{H}$ is the Hamiltonian of the scalar field expressed in terms
of the creation-annihilation operators. The mode functions of this field
corresponding to the energy $\omega$ span the kernel of the Klein-Gordon
operator,
$H(x,y)=(-\nabla^{2}_{x}-m^{2})\frac{\delta(x-y)}{|g|^{1/4}(x)|g|^{1/4}(y)}=\\\
=|g|^{-1/4}(x)\biggl{[}-|g|^{-1/4}(x)\partial_{\mu}\sqrt{|g|}g^{\mu\nu}\partial_{\nu}|g|^{-1/4}(x)-m^{2}\biggr{]}\frac{\delta(x-y)}{|g|^{1/4}(y)},$
(4)
where all the time derivatives should be replaced by $-i\omega$. This
operator, which we denote as $H(\omega)$, must be supplemented by the
appropriate boundary conditions. To simplify further calculations we assume
that the system considered is large enough to neglect the boundary effects or
the space represents a compact manifold without boundary. The operator
$H(\omega)$ is Hermitian with respect to the measure $\sqrt{|g|}$ on the
square-integrable functions depending on $\mathbf{x}$.
Our aim is to calculate the one-loop contribution to the free energy (3). To
this end, we use the fact that
$\partial_{\omega}\operatorname{Tr}_{d}\theta(H(\omega))=\sum_{k}\operatorname{sgn}(\varepsilon^{\prime}_{k}(\omega))\delta(\omega-\omega_{k})$
(5)
defines the signed spectral density in the $\omega$-space for the self-adjoint
operator $H(\omega)$ possessing the eigenvalues $\varepsilon_{k}(\omega)$.
Here the Heaviside step function $\theta(H(\omega)-\varepsilon)$ is the
spectral decomposition of unity associated with $H(\omega)$ and $\omega_{k}$
is the solution to the equation
$\varepsilon_{k}(\omega)=0.$ (6)
If this equation has several solutions then the sum over all such solutions
should be taken in (5). Note that $\omega$ may be any parameter of the self-
adjoint operator and not only the energy. In the case when $\omega$ is the
energy as defined above, $\varepsilon^{\prime}_{k}(\omega)$ is usually
positive for $\omega>0$ (the particle branch) and negative for $\omega<0$ (the
antiparticle branch). Differentiating the equations
$H(\omega_{k},m^{2})\psi_{k}(m^{2})=0,\qquad\langle\psi_{k}|\psi_{k}\rangle=1,$
(7)
with respect to $m^{2}$, it is easy to see that the last statement is valid
for the same cases, when the standard prescription $m^{2}\rightarrow m^{2}-i0$
defines the Feynman propagator. It follows from the quasiclassical asymptotic
of the operator $H(\omega)$, where all the derivatives entering it are
replaced by the momenta $-ip_{\mu}$, that $\varepsilon^{\prime}_{k}(\omega)$
can be negative for $\omega>0$ under the ergosphere, i.e., the antiparticle
branch appears in the positive-frequency region. This results in the particle
creation Starobin and the reconstruction of the vacuum state. Therefore we
restrict our consideration by the region out of the ergosphere.
Assuming that the lowest particle energy is strictly positive (this is the
case when the system is placed in a finite “box”), we can write the one-loop
contribution to the free energy from the particles as
$\mp\beta
F=\int_{0}^{\infty}d\omega\partial_{\omega}\operatorname{Tr}_{d}\theta(H(\omega))\ln(1\pm
e^{-\beta\omega})=\pm\beta\int_{0}^{\infty}d\omega\frac{\operatorname{Tr}_{d}\theta(H(\omega))}{e^{\beta\omega}\pm
1},$ (8)
where plus corresponds to fermions and minus is for bosons. The contribution
from the antiparticles is given by the same integral, but with $H(-\omega)$
instead of $H(\omega)$. Hence, the one-loop contribution to the effective
action from one bosonic mode at zero temperature reads (see, e.g., olopq ;
gmse )
$\Gamma^{(1)}_{1b}/T=-\lim_{\beta\rightarrow 0}\partial_{\beta}(\beta
F)=\partial_{\beta}\Big{[}\beta\int_{0}^{\infty}d\omega\frac{\operatorname{Tr}_{d}\theta(H(-\omega))}{e^{\beta\omega}+1}\Big{]}_{\beta\rightarrow
0},$ (9)
where $\beta^{-1}$ plays the role of the energy cutoff and $T$ is the time
interval tending to infinity. The fermionic free energy for a scalar field
(i.e. (8) with the plus sign) can be also used to estimate the partition
function for fermions when the spin-gravity interaction is negligible. For
further purposes it is useful to represent the step function in the integral
form:
$\theta(H(\omega))=\int_{C}\frac{d\tau}{2\pi i}\frac{e^{-\tau
H(\omega)}}{\tau},$ (10)
where the contour $C$ runs along the imaginary axis from top to bottom and
encircles the origin from the left. The exponent on the right-hand side is the
so-called heat kernel. In order to obtain the high-temperature expansion
($\beta\rightarrow 0$), it will be sufficient to know the expansion of the
trace of the heat kernel in a series in $\tau$.
## III Reduction to $d+1$
Of course, we can apply the formulas from VasilHeatKer ; Avramid ; Ven ;
Gilkey directly to (10), but this leads to rather cumbersome expressions that
are not explicitly covariant because of the $(3+1)$ decomposition. Therefore,
we reduce our $3$-dimensional problem to the $4$-dimensional one preserving
explicitly the general covariance. To this end, we introduce the auxiliary
integration variable,
$e^{-\tau H(\omega)}=\int_{-\infty}^{\infty}dp_{0}e^{-\tau
H(p_{0})}\delta(p_{0}-\omega)=\int_{-\infty}^{\infty}dp_{0}e^{-\tau[H(p_{0})+\frac{1}{\xi^{2}}(\omega^{2}-p^{2}_{0})]}\delta(p_{0}-\omega),$
(11)
and employ the standard Gaussian representation of the delta function
$\delta(p_{0}-\omega)=\lim_{\lambda\rightarrow-\infty}\sqrt{\frac{\lambda}{\pi}}e^{-\tau\lambda(p_{0}-\omega)^{2}}\tau^{1/2}.$
(12)
The second term in the exponent in (11) tends to zero when $\lambda$ goes to
minus infinity. The usefulness of such an addition will become clear soon. The
expressions with a finite $\lambda$ correspond to the same system but with a
Gaussian broadening of the spectrum. Denoting
$H_{\lambda}(p_{0},\omega):=H(p_{0})+\lambda(p_{0}-\omega)^{2}+\frac{1}{\xi^{2}}(\omega^{2}-p^{2}_{0}),$
(13)
we have
$\operatorname{Tr}_{d}e^{-\tau
H(\omega)}=\lim_{\lambda\rightarrow-\infty}\sqrt{\frac{\lambda}{\pi}}\int_{-\infty}^{\infty}dp_{0}\operatorname{Tr}_{d}e^{-\tau
H_{\lambda}(p_{0},\omega)}\tau^{1/2}=\frac{1}{T}\lim_{\lambda\rightarrow-\infty}2\sqrt{\pi\lambda}\operatorname{Tr}_{D}e^{-\tau
H_{\lambda}}\tau^{1/2},$ (14)
where $D:=d+1$ is the spacetime dimension. Then we make a similarity transform
such that the diagonal of the heat operator in the $x$-representation and,
consequently, its the trace are left intact, while the Hamiltonian becomes
(see also Appendix A for the notation)
$H_{G}:=(\xi^{2})^{1/4}H_{\lambda}(\xi^{2})^{-1/4}=-G^{\mu\nu}(\tilde{\nabla}_{\mu}-i\omega
g_{\mu})(\tilde{\nabla}_{\nu}-i\omega g_{\nu})-X,$ (15)
where $g_{\mu}:=\xi_{\mu}/\xi^{2}$ and
$\begin{gathered}G_{\mu\nu}=g_{\mu\nu}-\xi^{2}g_{\mu}g_{\nu}+\lambda^{-1}g_{\mu}g_{\nu},\qquad
G^{\mu\nu}=g^{\mu\nu}-\xi^{2}(1-\lambda\xi^{2})g^{\mu}g^{\nu},\qquad\det
G_{\mu\nu}=\lambda^{-1}g^{2}\det g_{\mu\nu},\\\ X=V-E,\qquad
V=\frac{1}{4}h_{\mu}h^{\mu}-\frac{1}{2}\nabla^{\mu}h_{\mu},\qquad
E=\frac{\omega^{2}}{\xi^{2}}-m^{2},\end{gathered}$ (16)
and $h_{\mu}:=\partial_{\mu}\ln\sqrt{\xi^{2}}$. The connection
$\tilde{\nabla}_{\mu}$ is the Levi-Civita connection compatible with the
metric $G_{\mu\nu}$. The latter is negative definite so that the operator
$H_{G}$ is of the Laplacian type.
## IV Resummation of the heat kernel expansion
Now we can use the heat kernel expansion
$\langle x|e^{-\tau
H_{G}}|y\rangle=\sum_{k=0}^{\infty}a_{k}(x,y)\frac{\tau^{k-D/2}}{(4\pi)^{D/2}}.$
(17)
It is also relevant here that the space has no boundary. Otherwise the terms
with the half-integer $k$ appear. To exploit the heat kernel expansion, we
resum it using the theorem proved in Parker
$\langle x|e^{-\tau
H_{G}}|y\rangle=e^{\tau[X(x)+\frac{1}{6}\tilde{R}(x)]}\langle
x|e^{-\tau[X(x)+\frac{1}{6}\tilde{R}(x)]}e^{-\tau
H_{G}}|y\rangle=e^{\tau[X(x)+\frac{1}{6}\tilde{R}(x)]}\sum_{k=0}^{\infty}\tilde{a}_{k}(x,y)\frac{\tau^{k-D/2}}{(4\pi)^{D/2}},$
(18)
where $\tilde{a}_{k}$ are obtained from $a_{k}$ by removing the terms
containing $X$ and $\tilde{R}$ without derivatives. As for the $X$ terms, this
theorem simply follows from the observation that
$\langle x|e^{-\tau[X(x)+\frac{1}{6}\tilde{R}(x)]}e^{-\tau H_{G}}|y\rangle$
(19)
is invariant under the transform $X\rightarrow X+const$ and so its expansion
does not contain $X$ without derivatives. On the other hand, expanding
$\langle x|e^{-\tau[X(x)+\frac{1}{6}\tilde{R}(x)]}e^{-\tau
H_{G}}|y\rangle=e^{-\tau[X(x)+\frac{1}{6}\tilde{R}(x)]}\langle x|e^{-\tau
H_{G}}|y\rangle$ (20)
in $\tau$, we see that at the same power of $\tau$:
$\tilde{a}_{k}=a_{k}+O(X,\tilde{R}),$ (21)
where $O(X,R)$ are the terms containing $X$ and $\tilde{R}$ without
derivatives and $a_{s}$ with $s<k$. If $\tilde{a}_{k}$ does not contain $X$
and $\tilde{R}$ without derivatives in virtue of summation of such terms to
the exponent then $\tilde{a}_{k}$ is obtained by a mere obliteration of the
terms in $a_{k}$ proportional to $X$ or $\tilde{R}$ without derivatives. The
fact that $\tilde{a}_{k}$ do not depend on the scalar curvature $\tilde{R}$
can be proven by a direct inspection of the heat kernel expansion coefficients
Parker or by solving the defining equations for the heat kernel in the
Gaussian approximation (see, e.g., psfss ).
## V Evaluation of the integrals over $\tau$ and $\omega$
For brevity, we introduce the notation
$\begin{gathered}\tilde{m}^{2}:=m^{2}+V+\frac{1}{6}\tilde{R}=m^{2}+\frac{1}{4}h^{2}+\frac{1}{6}(R-\nabla^{\mu}h_{\mu})-\frac{1}{24}(\lambda^{-1}-\xi^{2})f^{2}=\\\
=m^{2}+\frac{1}{4}h^{2}+\frac{1}{6}[R-(\lambda^{-1}-\xi^{2})R_{\mu\nu}g^{\mu}g^{\nu}-\lambda^{-1}g^{2}\nabla^{\mu}h_{\mu}],\end{gathered}$
(22)
where we have used the relations from Appendix A. The quasiclassical expansion
for the free energy (8) after the above transformations becomes
$-F=\frac{-i}{T}\lim_{\lambda\rightarrow-\infty}\sqrt{\frac{\lambda}{\pi}}\sum_{k=0}^{\infty}\int
dx\sqrt{G}\int_{0}^{\infty}\frac{d\omega\tilde{a}_{k}(\omega,x)}{e^{\beta\omega}\pm
1}\int_{C}\frac{d\tau}{(4\pi)^{D/2}}e^{-\tau(\frac{\omega^{2}}{\xi^{2}}-\tilde{m}^{2})}\tau^{k-d/2-1},$
(23)
where $\tilde{a}_{k}(x):=\tilde{a}_{k}(x,x)$. Of course, on using the heat
kernel in the form (18), we neglect all the exponentially suppressed at
$\beta\rightarrow 0$ terms in the high-temperature expansion. For the complete
analytical structure of the heat kernel in the $\tau$-plane see, e.g., psfss .
The integral over $\tau$ can be easily taken
$\int_{C}d\tau
e^{-\tau(\frac{\omega^{2}}{\xi^{2}}-\tilde{m}^{2})}\tau^{k-d/2-1}=(-1)^{k}\frac{2\pi
ie^{-i\pi
d/2}}{\Gamma(d/2-k+1)}\theta\Big{(}\frac{\omega^{2}}{\xi^{2}}-\tilde{m}^{2}\Big{)}\Big{(}\frac{\omega^{2}}{\xi^{2}}-\tilde{m}^{2}\Big{)}^{d/2-k}.$
(24)
Comparing (24), (23), and (16), we see that the imaginary units are all
canceled out and the expression (23) is real as it should be. The formula (24)
ought to be understood in a distributional sense for $k\geq d/2$. The simplest
way to take this into account is to assume that the variable $d$ is complex
and tends to its physical value $3$. This rule can be check by convolving (24)
with a test function single-valued near the real positive semiaxis $\omega\geq
0$. Only for those test functions does the prescription of the analytical
continuation work. In our case this requirement is evidently fulfilled.
Now the integral over $\omega$ has to be evaluated. The heat kernel expansion
coefficients $\tilde{a}_{k}(\omega)$ are polynomial in $\omega$:
$\tilde{a}_{k}(\omega,x)=:\sum_{j=0}^{2k}\tilde{a}_{k}^{(j)}(x)(g^{2}\omega^{2})^{j/2},$
(25)
where the upper summation limit is dictated by dimensional reasons. Observe
that the heat kernel
$\langle x|e^{-\tau H_{G}(\omega)}|y\rangle$ (26)
is Hermitian at the real $\tau$ and therefore its diagonal is real in this
case. On the other hand, from Eq. (15) we see that the complex conjugation
effectively results in a change of the sign of $\omega$. Hence, by the
uniqueness of the analytic continuation in $\tau$, the diagonal of (26) is an
even function of $\omega$ for any $\tau$. Then the polynomial (25) contains
solely the even powers of the energy $\omega$ as long as $E$ is an even
function of $\omega$.
So, we need to take the integral
$I=\int_{0}^{\infty}\frac{\beta d\omega}{e^{\beta\omega}\pm
1}\frac{\theta(\omega^{2}g^{2}-\tilde{m}^{2})}{\Gamma(d/2-k+1)}(g^{2}\omega^{2})^{d/2-k+j/2}\Big{(}1-\frac{\tilde{m}^{2}}{\omega^{2}g^{2}}\Big{)}^{d/2-k}.$
(27)
The further procedure is quite analogous to the case of a flat spacetime (see,
e.g., olopq ; DolJack ; HabWeld ; Kapusta ). We shall follow olopq .
Substituting the expansion
$\Big{(}1-\frac{m^{2}}{\omega^{2}g^{2}}\Big{)}^{d/2-k}=\sum_{n=0}^{\infty}\frac{(-1)^{n}\Gamma(d/2-k+1)}{n!\Gamma(d/2-k-n+1)}\Big{(}\frac{m^{2}}{\omega^{2}g^{2}}\Big{)}^{n}$
(28)
to the integral $I$, we reduce it to a sum of the incomplete zeta functions.
Furthermore, we expand these incomplete zeta functions making use of the
formulas GrRy :
$\begin{split}\int_{a}^{\infty}d\omega\frac{\omega^{\nu-1}}{e^{\omega}-1}&=\Gamma(\nu)\zeta(\nu)-\sum_{n=-1}^{\infty}\frac{(-1)^{n}\zeta(-n)a^{\nu+n}}{\Gamma(n+1)(\nu+n)},\\\
\int_{a}^{\infty}d\omega\frac{\omega^{\nu-1}}{e^{\omega}+1}&=(1-2^{1-\nu})\Gamma(\nu)\zeta(\nu)-\sum_{n=0}^{\infty}(1-2^{1+n})\frac{(-1)^{n}\zeta(-n)a^{\nu+n}}{\Gamma(n+1)(\nu+n)}.\end{split}$
(29)
The term with $n=-1$ is understood as the limit $n\rightarrow-1$. For the
bosonic case we obtain
$I_{b}=\sum_{n=0}^{\infty}\frac{(-1)^{n}\tilde{m}^{d-2k}\beta_{T}^{-j}}{n!\Gamma(d/2-k-n+1)}\Big{[}\Gamma(d+j-2k-2n+1)\zeta(d+j-2k-2n+1)(\beta_{T}\tilde{m})^{2k+2n-d}-\\\
-\sum_{l=-1}^{\infty}\frac{(-1)^{l}\zeta(-l)(\beta_{T}\tilde{m})^{j+l+1}}{\Gamma(l+1)(d+j-2k-2n+l+1)}\Big{]},$
(30)
where $\beta_{T}:=\sqrt{\xi^{2}}\beta$ is the Tolman reciprocal temperature
and we have assumed that $\tilde{m}^{2}\geq 0$. The sum over $n$ of the second
term in the square brackets can be taken and is expressed through the beta
function:
$\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!\Gamma(d/2-k-n+1)(d+j-2k-2n+l+1)}=-\frac{\Gamma(k-(d+j+l+1)/2)}{2\Gamma\big{(}(1-j-l)/2\big{)}}.$
(31)
Whence we get
$I_{b}=\sum_{n=0}^{\infty}\frac{(-1)^{n}\Gamma(d+j-2k-2n+1)\zeta(d+j-2k-2n+1)}{n!\Gamma(d/2-k-n+1)}\tilde{m}^{2n}\beta_{T}^{2k+2n-j-d}+\\\
+\sum_{l=-1}^{\infty}\frac{(-1)^{l}\zeta(-l)\Gamma(k-(d+j+l+1)/2)}{2\Gamma(l+1)\Gamma\big{(}(1-j-l)/2\big{)}}\tilde{m}^{d-2k+j+l+1}\beta_{T}^{l+1}.$
(32)
So, we have all what we need to obtain the high-temperature expansion.
## VI High-temperature expansion
Substituting the expression (32) to (23), we eventually arrive at
$-F_{b}=\sum_{k,j=0}^{\infty}\int
d\mathbf{x}\frac{\sqrt{|g|}\tilde{a}_{k}^{(j)}}{(4\pi)^{d/2}}\Big{[}\sum_{n=0}^{\infty}\frac{(-1)^{n+k}\Gamma(d+j-2k-2n+1)\zeta(d+j-2k-2n+1)}{n!\Gamma(d/2-k-n+1)\beta_{T}^{d+j-2k-2n+1}\tilde{m}^{-2n}}+\\\
+\sum_{l=-1}^{\infty}\frac{(-1)^{l+k}\zeta(-l)\Gamma(k-(d+j+l+1)/2)}{2\Gamma(l+1)\Gamma\big{(}(1-j-l)/2\big{)}}\tilde{m}^{d+j-2k+l+1}\beta_{T}^{l}\Big{]},$
(33)
where the limit of the infinite $\lambda$ is implied. The expression standing
in the square brackets is regular, when $d$ tends to a positive integer, since
(30) is the entire function of $d$. Therefore, we can set $d$ to its physical
value in the factor at this square brackets. Besides, due to the property of
the zeta function a “half” of the terms in the sum over $l$ vanish. These are
the terms with $l=2,4,6,\ldots$ As far as the fermions are concerned, similar
calculations lead to ($\tilde{m}^{2}\geq 0$)
$-F_{f}=\sum_{k,j=0}^{\infty}\int
d\mathbf{x}\frac{\sqrt{|g|}\tilde{a}_{k}^{(j)}}{(4\pi)^{d/2}}\times\\\
\times\Big{[}\sum_{n=0}^{\infty}(1-2^{2n+2k-j-d})\frac{(-1)^{n+k}\Gamma(d+j-2k-2n+1)\zeta(d+j-2k-2n+1)}{n!\Gamma(d/2-k-n+1)\beta_{T}^{d+j-2k-2n+1}\tilde{m}^{-2n}}+\\\
+\sum_{l=0}^{\infty}(1-2^{1+l})\frac{(-1)^{l+k}\zeta(-l)\Gamma(k-(d+j+l+1)/2)}{2\Gamma(l+1)\Gamma\big{(}(1-j-l)/2\big{)}}\tilde{m}^{d+j-2k+l+1}\beta_{T}^{l}\Big{]}.$
(34)
The terms at the fixed power of $\beta$ both for bosons and fermions are the
same up to an overall numeric factor and in that sense are universal. However,
the expansion of the free energy for bosons involves one additional
contribution with $l=-1$.
Let us analyze the first and second terms in the square brackets of (33) and
(34) separately. We see that both for bosons and fermions the second terms in
the square brackets are not expanded in the increasing powers of $\beta$.
Rather, we have a derivative (or large mass) expansion of the terms at the
every fixed power of the reciprocal temperature. It is not difficult to write
out a closed form expression for a whole series at the fixed power of $\beta$.
With this end in view, we introduce the function (cf. olopq ; DowKen )
$\sigma^{l}_{\nu}(m^{2}):=\int_{C}\frac{dss^{\nu-1}}{(e^{2\pi
i\nu}-1)\Gamma(\nu)}\int_{0}^{\infty}d\omega\omega^{l}\operatorname{Tr}_{d}e^{-sH(\omega)}.$
(35)
The function under the integral over $s$ given by the integral over $\omega$
is analytic for $\operatorname{Re}s>0$. It should be analytically continued to
the imaginary axis and to the vicinity of the origin where the contour $C$
lies. Usually, this can be achieved by rotating the integration contour in the
$\omega$-plane (see the high frequency asymptotics in Eq. (23)). Notice that
the function (35) is not the generalized zeta function as long as $H(\omega)$
possesses the negative eigenvalues. Although these functions are closely
related. Repeating all the above calculations, one can convince oneself that
the expansion (33) can be rewritten as
$-F_{b}=\sum_{k,n,j=0}^{\infty}\int
d\mathbf{x}\frac{\sqrt{|g|}\tilde{a}_{k}^{(j)}}{(4\pi)^{d/2}}\frac{(-1)^{k}\Gamma(d+j-2k-2n+1)\zeta(d+j-2k-2n+1)}{n!\Gamma(d/2-k-n+1)\beta_{T}^{d+j-2k-2n+1}(-\tilde{m}^{2})^{-n}}+\\\
+\sum_{l=-1}^{\infty}\frac{(-1)^{l}\zeta(-l)}{\Gamma(l+1)}\sigma_{\epsilon}^{l}\beta^{l},$
(36)
where $\epsilon=(\bar{d}-d)/2$ is the complex number tending to zero and
$\bar{d}$ is the physical dimension ($\bar{d}=3$). The analogous formula holds
for fermions as well. The function $\sigma_{\nu}^{l}(m^{2})$ depends only on
$m^{2}$ and the background fields. Consequently, we need certain additional
assumptions apart from $\beta\rightarrow 0$ in order to obtain an explicit
expression for it. Henceforth we assume that the gravitational field varies
slowly such that the derivative expansions presented in (33) and (34) make
sense and provide reliable approximations for the functions (35). This holds
when the characteristic length of the variation of the gravitational field is
much larger than the Compton wavelength associated with the effective mass
$\tilde{m}$. In any case, these second terms give subleading contributions to
the high-temperature expansion. For fermions, they stand at the nonegative
integer powers of $\beta$, while for bosons the leading contribution from
these terms is proportional to $\beta^{-1}$. The first terms in the square
brackets in the expansions (33) and (34) are much more singular at
$\beta\rightarrow 0$.
Consider the first terms in the square brackets of Eqs. (33) and (34). There
is a finite number of such terms at any fixed power of $\beta$. Indeed, at any
fixed number $k$, the number $j$ must be less or equal to $2k$ by dimensional
reasons. However, inasmuch as we resummed the expansion and collected all the
terms without derivatives of $E$ to the exponent, the latter terms are absent
in $\tilde{a}_{k}$. Consequently, the worst terms, which contain the maximal
power of $\omega$ at the fixed dimension $2k$, are of the form
$\nabla E\nabla E\cdots\nabla E.$ (37)
If the dimension $2k$ is not a multiple of $3$ then the worst terms look like
(37), but with the additional covariant derivative, or with $\Omega^{2}$ (see,
for example, (84)). Thus, we conclude that (see also NakFuk ; BarvMukh )
$j\leq[4k/3],$ (38)
where the square brackets denote the integer part of the number. These worst
terms stand at
$\beta^{2n+2k-[4k/3]-d-1},$ (39)
and so the power of $\beta$ increases. Further, we restrict ourself by the
singular and finite parts of the high-temperature expansion as they give the
leading contribution to the free energy. From (39) we conclude that the
maximal number $k$ that we need is determined by the equality
$2k-[4k/3]=D,$ (40)
and for a three dimensional space $k\leq 6$. Fortunately, the heat kernel
coefficients $a_{k}$ with $k\leq 6$ were found in Ven and we just can borrow
these results. The relevant parts of these coefficients are presented in
Appendix B.
Now we turn again to the second terms in the square brackets of the expansions
(33) and (34). The contributions with the positive even $l$ vanish. Moreover,
the contribution from these terms is zero when $l$ is an odd positive number
and $j$ is an even nonegative number. As we discussed above, the number $j$
must be even and nonegative. The explicit expressions for $\tilde{a}_{k}$
given in Appendix B are found to be even functions of $\omega$ for those
$\tilde{a}_{k}$ which we are interested in and confirm thereby the general
considerations.
Note that these second terms were disregarded in Fursaev1 ; Fursaev2 ; NakFuk
. It is relevant for a correct evaluation of these terms that we did not make
a conformal transformation to the optic metric and resummed the heat kernel
expansion. The resummation gives a non-perturbative expression for the heat
kernel which is more reliable in terms of the original metric rather than the
optic one (see for details psfss ). In principle, the first contributions in
the square brackets of Eqs. (33) and (34) may be evaluated in terms of the
conformally transformed metric. However, in order to provide a transparent way
to the poles cancelation in the $d$-plane they have to be expressed through
the same metric as the second terms.
## VII Results and implications
Bearing in mind these observations, we can write for $d=3$ in the bosonic case
$-F_{b}=\int\frac{d\mathbf{x}\sqrt{|g|}}{(4\pi)^{d/2}}\sum_{k,j=0}^{\infty}(-1)^{k}\Big{\\{}\sideset{}{{}^{\prime}}{\sum}_{n=0}^{\infty}\frac{\Gamma(D+j-2k-2n)\zeta(D+j-2k-2n)\tilde{a}_{k}^{(j)}}{n!\Gamma(d/2-k-n+1)\beta_{T}^{D+j-2k-2n}(-\tilde{m}^{2})^{-n}}\\\
+\frac{(-\tilde{m}^{2})^{(D+j)/2-k}\tilde{a}_{k}^{(j)}}{4((D+j)/2-k)!\Gamma((1-j)/2)}\big{[}\ln\frac{\tilde{m}^{2}\beta_{T}^{2}e^{2\gamma}}{4\pi^{2}}-\psi((D+j)/2-k+1)+\psi((1-j)/2)\big{]}\\\
+\delta_{j,0}\Gamma(k-d/2)\frac{\tilde{a}_{k}^{(0)}}{2\beta_{T}}\tilde{m}^{d-2k}-\tilde{a}_{k}^{(j)}\tilde{m}^{D+j-2k}\frac{\Gamma(k-(D+j)/2)}{4\Gamma((1-j)/2)}\Big{\\}},$
(41)
where $\gamma$ is the Euler constant. The prime at the sum over $n$ says that
the singular terms are discarded, the second term is zero by definition when
the argument of the factorial is negative, and the last term vanishes by
definition when the gamma function entering the numerator tends to infinity.
The last term stands at the negative power of $\tilde{m}$. Later on we shall
cast out such contributions although it is these contributions which are
“protected” from the high energy physics, i.e., the particles with a large
mass (but with $m\ll\beta^{-1}$) give a negligible contribution to these
terms. The high-temperature expansion in the fermionic case looks like
$-F_{f}=\int\frac{d\mathbf{x}\sqrt{|g|}}{(4\pi)^{d/2}}\sum_{k,j=0}^{\infty}(-1)^{k}\Big{\\{}\sideset{}{{}^{\prime}}{\sum}_{n=0}^{\infty}(1-2^{2k+2n-j-d})\frac{\Gamma(D+j-2k-2n)\zeta(D+j-2k-2n)\tilde{a}_{k}^{(j)}}{n!\Gamma(d/2-k-n+1)\beta_{T}^{D+j-2k-2n}(-\tilde{m}^{2})^{-n}}\\\
-\frac{(-\tilde{m}^{2})^{(D+j)/2-k}\tilde{a}_{k}^{(j)}}{4((D+j)/2-k)!\Gamma((1-j)/2)}\big{[}\ln\frac{4\tilde{m}^{2}\beta_{T}^{2}e^{2\gamma}}{\pi^{2}}-\psi((D+j)/2-k+1)+\psi((1-j)/2)\big{]}\\\
+\tilde{a}_{k}^{(j)}\tilde{m}^{D+j-2k}\frac{\Gamma(k-(D+j)/2)}{4\Gamma((1-j)/2)}\Big{\\}}.$
(42)
The digamma functions appearing in the expressions are easily calculated using
the formulas GrRy :
$\psi\Big{(}\frac{1}{2}-n\Big{)}=\sum_{k=1}^{n}\frac{2}{2k-1}-\ln
4-\gamma,\qquad\psi(1+n)=\sum_{k=1}^{n}\frac{1}{k}-\gamma.$ (43)
More specifically, taking into account that $\tilde{a}_{0}=1$ and
$\tilde{a}_{1}=0$, we derive for bosons
$\begin{array}[]{l}-F_{b}=\int
d\mathbf{x}\sqrt{|g|}\Big{\\{}\dfrac{\pi^{2}\beta^{-4}_{T}}{90}-\dfrac{\beta^{-2}_{T}}{24}(\tilde{m}^{2}-\frac{1}{2}\tilde{a}_{2}^{(2)}-\frac{1}{4}\tilde{a}_{3}^{(4)})+\dfrac{\beta_{T}^{-1}}{12\pi}(\tilde{m}^{3}+\frac{3}{4\tilde{m}}\tilde{a}_{2}^{(0)}+\ldots)\\\
+\dfrac{1}{64\pi^{2}}\ln\dfrac{\tilde{m}^{2}\beta_{T}^{2}e^{2\gamma}}{16\pi^{2}}\big{[}\tilde{m}^{4}+\tilde{m}^{2}(\tilde{a}_{2}^{(2)}+\frac{3}{2}\tilde{a}_{3}^{(4)})+2\tilde{a}_{2}^{(0)}+\tilde{a}_{3}^{(2)}+\frac{3}{2}\tilde{a}_{4}^{(4)}+\frac{15}{4}\tilde{a}_{5}^{(6)}+\frac{105}{8}\tilde{a}_{6}^{(8)}\big{]}\\\\[10.00002pt]
-\dfrac{3}{128\pi^{2}}\big{[}\tilde{m}^{4}-\tilde{m}^{2}(\frac{2}{3}\tilde{a}_{2}^{(2)}+\frac{5}{3}\tilde{a}_{3}^{(4)})-\frac{4}{3}(\tilde{a}_{3}^{(2)}+2\tilde{a}_{4}^{(4)}+\frac{23}{4}\tilde{a}_{5}^{(6)}+22\tilde{a}_{6}^{(8)})\big{]}+\ldots\Big{\\}},\end{array}$
(44)
and for fermions
$\begin{array}[]{l}-F_{f}=\int
d\mathbf{x}\sqrt{|g|}\Big{\\{}\dfrac{7\pi^{2}\beta^{-4}_{T}}{720}-\dfrac{\beta^{-2}_{T}}{48}(\tilde{m}^{2}-\frac{1}{2}\tilde{a}_{2}^{(2)}-\frac{1}{4}\tilde{a}_{3}^{(4)})\\\
-\dfrac{1}{64\pi^{2}}\ln\dfrac{\tilde{m}^{2}\beta_{T}^{2}e^{2\gamma}}{\pi^{2}}\big{[}\tilde{m}^{4}+\tilde{m}^{2}(\tilde{a}_{2}^{(2)}+\frac{3}{2}\tilde{a}_{3}^{(4)})+2\tilde{a}_{2}^{(0)}+\tilde{a}_{3}^{(2)}+\frac{3}{2}\tilde{a}_{4}^{(4)}+\frac{15}{4}\tilde{a}_{5}^{(6)}+\frac{105}{8}\tilde{a}_{6}^{(8)}\big{]}\\\\[10.00002pt]
+\dfrac{3}{128\pi^{2}}\big{[}\tilde{m}^{4}-\tilde{m}^{2}(\frac{2}{3}\tilde{a}_{2}^{(2)}+\frac{5}{3}\tilde{a}_{3}^{(4)})-\frac{4}{3}(\tilde{a}_{3}^{(2)}+2\tilde{a}_{4}^{(4)}+\frac{23}{4}\tilde{a}_{5}^{(6)}+22\tilde{a}_{6}^{(8)})\big{]}+\ldots\Big{\\}}.\end{array}$
(45)
The dots in the expressions above denote the terms at the negative powers of
$\tilde{m}$ or the positive powers of $\beta$. The coefficients diverging at
$\beta\rightarrow 0$ provide the energy-time anomaly (psfss and also BrOtPa ;
FrZel ) when the energy cutoff is used for the regularization of the one-loop
contribution to the effective action. Recall that the energy-time anomaly
represents the variance of the renormalized effective action under the uniform
dilatations of the Killing vector (see below). The flat spacetime limit of the
expansions (44) and (45) coincides with the known one DolJack ; Kapusta ;
HabWeld .
Now we consider the coefficients at the different powers of $\beta$ in detail.
First, we observe that
$\int
d\mathbf{x}\sqrt{|g|}\beta^{-2}_{T}(\tilde{m}^{2}-\frac{1}{2}\tilde{a}_{2}^{(2)}-\frac{1}{4}\tilde{a}_{3}^{(4)})=\beta^{-2}\int
d\mathbf{x}\sqrt{|g|}\big{[}(m^{2}+\frac{1}{6}R)\xi^{-2}+\frac{1}{12}f^{2}-\frac{1}{3}\nabla^{\mu}(g^{2}h_{\mu})\big{]}.$
(46)
If we omit the last term representing a total divergence then we confirm the
results of DowKen ; NakFuk ; DowSch that the term at $\beta^{-2}$ is absent
for a massless conformal scalar field on a static gravitational background.
Meanwhile, we see that this is not the case for a stationary spacetime. For
such spacetimes, the leading correction to the Planck law is of the order of
$\beta^{-2}$ and proportional to the Maxwell-like term $f^{2}$. Formula (46)
coincides with Eq. (4.18) of Fursaev1 up to the integral of a total
derivative. Under certain circumstances the last term in (46) reducing to a
boundary term also considerably contributes. However, in this case other
boundary contributions that we discarded earlier must be included into the
free energy too.
In the bosonic case, the next correction is proportional to $\beta^{-1}$. For
a massless scalar field on a curved background satisfying the vacuum Einstein
equations $R_{\mu\nu}=0$, this contribution becomes purely imaginary as long
as the correction to $m^{2}$ is negative in this case psfss . There is also an
imaginary contribution to the free energy due to the logarithmic term. The
imaginary contributions to the effective action signalize about a certain
instability of the vacuum of quantum fields which was initially chosen. As a
result, we may anticipate that a non-zero vacuum expectation value of the
scalar field develops (the Bose-Einstein condensation).
Of course, if we accurately evaluate the high-temperature expansion of (8)
then there are not any imaginary contributions to the free energy as the
expression (8) is real-valued. For $\tilde{m}^{2}<0$, the additional terms
appear in (44) and (45) such that the imaginary part of the free energy
vanishes. In the fermionic case, the contributions to the real part of the
free energy standing at $\beta^{-4}$, $\beta^{-2}$, $\beta^{0}$, and
$\ln\beta$ do not change, i.e., the additional terms cancel exactly the
imaginary part coming from the logarithm and add certain contributions at the
higher powers of $\beta$. At the same time, in the bosonic case, the imaginary
part is also canceled out, but the infrared divergence arises in the term with
$\beta^{-1}$. Other contributions to the singular and finite parts of the
high-temperature expansion remain unchanged. The term at $\beta^{-1}$ entering
the free energy $F_{b}$ becomes
$\int
d\mathbf{x}\frac{\sqrt{|g|}}{(4\pi)^{d/2}}\sum_{k=0}^{\infty}\frac{(-1)^{k}}{2\beta_{T}}\bigg{\\{}\frac{|\tilde{m}|^{d-2k}\tilde{a}_{k}^{(0)}}{\Gamma(d/2-k+1)}\big{[}\ln\frac{\bar{\lambda}^{2}}{\xi^{2}|\tilde{m}|^{2}}+\gamma+\psi(d/2-k+1)\big{]}-\\\
-\sum_{j=1}^{[4k/3]}\tilde{a}_{k}^{(j)}(-1)^{j/2}\frac{|\tilde{m}|^{d+j-2k}\Gamma(j/2)}{\Gamma((d+j)/2-k+1)}\bigg{\\}}=\int\frac{d\mathbf{x}\sqrt{|g|}}{12\pi^{2}\beta_{T}}\Big{[}|\tilde{m}|^{3}\ln\frac{g^{2}\bar{\lambda}^{2}e^{8/3}}{4|\tilde{m}|^{2}}+\frac{3|\tilde{m}|}{2}(\tilde{a}^{(2)}_{2}+\tilde{a}^{(4)}_{3})+\ldots\Big{]},$
(47)
where $\bar{\lambda}$ is the infrared energy cutoff. The contributions
vanishing at $\bar{\lambda}\rightarrow 0$ were neglected and dots denote the
negative powers of $\tilde{m}$.
These corrections to (44) and (45) have their origin in the fact that at
$\tilde{m}^{2}<0$ the integral (27) can be represented as the sum of two
integrals along the contours $[\bar{\lambda},i\sqrt{\xi^{2}}|\tilde{m}|]$ and
$[i\sqrt{\xi^{2}}|\tilde{m}|,+\infty)$. The latter integral was already
evaluated and its contribution to the high-temperature expansion is given by
(41) or (42). The former integral,
$\int_{\bar{\lambda}}^{i\sqrt{\xi^{2}}|\tilde{m}|}\frac{\beta
d\omega}{e^{\beta\omega}\pm
1}\frac{(g^{2}\omega^{2})^{j/2}(g^{2}\omega^{2}-\tilde{m}^{2})^{d/2-k}}{\Gamma(d/2-k+1)},$
(48)
is readily taken if one expands $(e^{\beta\omega}\pm 1)^{-1}$ in a Laurent
series in $\beta\omega$. The infrared cutoff is only needed in the bosonic
case for the contribution at $(\beta\omega)^{-1}$ and $j=0$:
$\int_{\bar{\lambda}}^{i\sqrt{\xi^{2}}|\tilde{m}|}\frac{\beta
d\omega}{e^{\beta\omega}-1}\frac{(g^{2}\omega^{2}-\tilde{m}^{2})^{d/2-k}}{\Gamma(d/2-k+1)}\approx\int_{\bar{\lambda}}^{i\sqrt{\xi^{2}}|\tilde{m}|}\frac{d\omega}{\omega}\frac{(g^{2}\omega^{2}-\tilde{m}^{2})^{d/2-k}}{\Gamma(d/2-k+1)}.$
(49)
This integral, in turn, can be estimated at $\bar{\lambda}\rightarrow 0$
making use of the expansion of the incomplete beta function GrRy :
$\begin{split}\int_{a}^{1}dxx^{\alpha-1}(1-x)^{\beta-1}&=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}-\sum_{n=0}^{\infty}\frac{(-1)^{n}\Gamma(\beta)a^{n+\alpha}}{n!\Gamma(\beta-n)(n+\alpha)},\\\
\int_{a}^{1}\frac{dx}{x}(1-x)^{\beta-1}&=-\gamma-\psi(\beta)-\ln
a-\sum_{n=1}^{\infty}\frac{(-1)^{n}\Gamma(\beta)a^{n}}{n!\Gamma(\beta-n)n}.\end{split}$
(50)
To sum up, we see that at $\tilde{m}^{2}\leq 0$ the infrared divergencies
arise in the free energy. Therefore, a more accurate analysis of the low
energy modes is required. These divergencies may be cured by introducing the
Bose-Einstein condensate and summing the ring diagrams (see, e.g., Kapusta ;
DolJack ) or, as in the flat spacetime, by introducing the boundaries and
accounting for the finite size effects. In the case, when
$l\lesssim m^{-1}\ll L,$ (51)
where $l$ is the characteristic scale of the variations of the gravitational
field and $L$ is the size of the “box”, the boundary effects are negligible
and so the condensation is the only possible mean to remove the infrared
divergencies. Keeping in mind that the free energy obtained above is the
finite-temperature part of the effective action, the appearance of the
infrared divergencies at $m^{2}\rightarrow 0$ for a slowly varying
gravitational field is quite expectable (see, e.g., Kapusta ; DolJack ;
Weinberg ; GrPeYa ; Akhmed ; Polyak ). For the rapidly changing metric field
such divergencies may not appear GusZeln , but this fact has nothing in common
with the infrared divergencies arising in the infrared limit.
The coefficient of the logarithmic term is closely related with the conformal
(trace) anomaly DowKen ; DowSch . Substituting the expressions for
$\tilde{a}_{k}^{(j)}$ from Appendix B, we come to DowKen ; DowSch ; NakFuk ;
GusZeln ; Fursaev2
$\begin{array}[]{l}\tilde{m}^{4}+\tilde{m}^{2}(\tilde{a}_{2}^{(2)}+\frac{3}{2}\tilde{a}_{3}^{(4)})+2\tilde{a}_{2}^{(0)}+\tilde{a}_{3}^{(2)}+\frac{3}{2}\tilde{a}_{4}^{(4)}+\frac{15}{4}\tilde{a}_{5}^{(6)}+\frac{105}{8}\tilde{a}_{6}^{(8)}=\\\
=m^{4}+\frac{1}{3}m^{2}R+\frac{1}{90}(R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}-R^{\mu\nu}R_{\mu\nu}+6\nabla^{2}R)+\frac{1}{36}R^{2}.\end{array}$
(52)
A great number of cancelations happens in calculating this expression. This
fact can be considered as the indirect verification of the correctness of the
coefficients entering the expansion above. Remarkably, this expression does
not depend on the Killing vector field. The nonminimal coupling adds to the
anomaly (52) the term,
$-2m^{2}\kappa
R+(\kappa^{2}-\frac{\kappa}{3})R^{2}-\frac{\kappa}{6}\nabla^{2}R,$ (53)
as one can easily deduce from the substitution rule (76). In the conformal
case $m^{2}=0$, $\kappa=1/6$, the coefficient of the logarithm coincides with
the standard expression for the conformal anomaly.
Some comments on the conformal transformations are in order to caution the
interested reader. In the paper DowSch , the high-temperature expansion of the
free energy of a scalar field on a static gravitational background was
obtained by the use of the conformal transform from the ultrastatic spacetime.
Despite the fact that the final result (Eq. (18), DowSch ) for the terms at
$\beta^{-4}$, $\beta^{-2}$, $\beta^{-1}$, and $\ln\beta$ is correct for the
static gravitational field, the method for its derivation seems to be invalid
at the point where the scaling property of the generalized zeta function (the
equation before Eq. (10) of DowSch ) was employed. This scaling property holds
only for the conformally invariant operator ($m^{2}=0$ and
$\kappa_{D}=(D-2)/4(D-1)$) and does not take place in a general case (see,
e.g., DowHKEscal ). Of course, one can always make a transformation
$\begin{array}[]{l}g_{\mu\nu}\rightarrow\bar{g}_{\mu\nu}=e^{2\rho}g_{\mu\nu},\\\
|\bar{g}|^{1/4}\bar{\nabla}^{2}|\bar{g}|^{-1/4}-\kappa\bar{R}+m^{2}=\\\
=e^{-\rho}\Big{[}|g|^{1/4}\nabla^{2}|g|^{-1/4}-\kappa
R+(D-1)(\kappa-\kappa_{D})\big{(}2\nabla^{2}\rho+(D-2)\nabla_{\mu}\rho\nabla^{\mu}\rho\big{)}+m^{2}e^{2\rho}\Big{]}e^{-\rho},\end{array}$
(54)
with $\rho$ independent of time. Nevertheless, the variation of this
expression with respect to $\rho$ gives not only the term arising in the
scaling property of the zeta function DowSch , but also the contribution from
the variation of the expression standing in the square brackets (54). This
contribution is not trivial in the case $\kappa\neq\kappa_{D}$ and $m^{2}\neq
0$.
On the other hand DowSch1 , one can indeed make the substitution (54) from the
beginning (4) and apply the whole machinery of the heat kernel expansion (17)
to the operator standing in the square brackets. The spectrum associated with
this operator is obviously the same as the spectrum associated with the
initial operator, and so the high-temperature expansions must coincide. If the
function $\rho$ is of a general type, such that, in particular, the
coefficients at $\omega^{2}$ and $m^{2}$ in $H(\omega)$ are not constants,
then this function disappears from the high-temperature expansion upon
restoration of the dependence $g_{\mu\nu}$ on $\rho$. However, for a certain
particular choice of $\rho$ yielding a vanishing of some terms in the
coefficients $a_{k}(\omega,x)$ of the heat kernel expansion, for example, for
$\rho=\ln\sqrt{\xi^{2}}$ leading to the constant coefficient at $\omega^{2}$
in $H(\omega)$, to the optical metric, and to the needlessness of the
resummation of the expansion for a static metric DowSch ; DowSch1 , the
dependence of the free energy on this $\rho$ does not disappear, in general.
In making the conformal transformations, one should take into account that the
coefficients $a^{(j)}_{k}$, $j\geq 2$, become nonvanishing and they also
contribute to the free energy or, in other words, the additional terms arise
in the scaling relation for the generalized zeta function DowHKEscal . This,
of course, is just a manifestation of the fact that the Klein-Gordon equation
is not conformally invariant for the arbitrary $\kappa$ and $m^{2}$.
Now we return to the discussion of the properties of the expansions (44) and
(45). The contribution to the finite part of the free energy at
$\beta\rightarrow 0$, which determines, in particular, the Casimir force at
zero temperature, does depend on the Killing vector. The whole expression is
rather huge, and we do not write it here (see Eq. (88)), but the terms at
$m^{2}$ take a compact form:
$\tilde{m}^{4}-\tilde{m}^{2}(\frac{2}{3}\tilde{a}_{2}^{(2)}+\frac{5}{3}\tilde{a}_{3}^{(4)})+\ldots=m^{4}+\frac{m^{2}}{3}(R+\frac{5}{3}\xi^{2}R_{\mu\nu}g^{\mu}g^{\nu}+\frac{11}{6}h^{2})+\ldots$
(55)
These terms give the leading contribution to the finite part when $m^{-1}$ is
much smaller than the characteristic scale of variations of the gravitational
field. The finite part does not coincide with the answers presented in DowSch
and DowSch1 (compare the coefficient at $h^{2}$). Recalling that the masses
of all the massive particles are generated by the Higgs mechanism, we see that
the term at $m^{2}$ gives a contribution to the effective potential of the
Higgs field. This results in the additional correction to the gravitational
mass-shift effect discussed in gmse .
If one regards the expression (45) as the regularized version of the one-loop
contribution to the effective action at zero temperature with $\beta$ playing
the role of the regularization parameter (see Eq. (9)), then an interesting
relation between the conformal and energy-time anomalies can be observed.
Recall that the corresponding global symmetries lead to the (formal) Noether
theorem
$g_{\mu\nu}(x)\frac{\delta\Gamma}{\delta
g_{\mu\nu}(x)}=2\sqrt{|g|}\nabla_{\mu}D^{\mu},\qquad\xi^{\mu}(x)\frac{\delta\Gamma}{\delta\xi^{\mu}(x)}=\sqrt{|g|}\nabla_{\mu}T^{\mu},$
(56)
where $\Gamma[g_{\mu\nu},\xi^{\mu},\Phi]$ is the renormalized effective
action, and $D^{\mu}$ and $T^{\mu}$ are the corresponding currents. Usually,
these relations are broken by the anomaly terms resulting from the quantum
corrections. In our particular case (the massless conformal scalar field), we
can renormalize the regularized contribution (45) to the effective action in
several ways.
The first one: we add the following counterterms to the initial Lagrangian
$\beta^{-4}(g_{\mu}g^{\mu})^{2},\qquad\beta^{-2}f^{2},\qquad[\tilde{m}^{4}+\tilde{m}^{2}(\tilde{a}_{2}^{(2)}+\frac{3}{2}\tilde{a}_{3}^{(4)})+\cdots]\ln(\beta^{-2}g^{2}).$
(57)
As for the last counterterm, such a choice is unnatural and would be
prohibited if the quantum theory of the vector field $g_{\mu}$ will be
renormalizable (cf. the Coleman-Weinberg potential ColWein containing the
term $\phi^{4}\ln\phi^{2}$). Nevertheless, if we add such counterterms, the
global conformal invariance of the renormalized effective action is broken as
long as this logarithmic counterterm is not conformally invariant. The
conformal anomaly is given by the factor at the logarithm. On the other hand,
the second relation in (56) holds on a quantum level provided the power-like
divergencies are completely canceled out. The finite part and the coefficient
of the logarithm are invariant under the stretching of the Killing vector.
The second way: we add to the initial Lagrangian the counterterms
$\beta^{-4}(g_{\mu}g^{\mu})^{2},\qquad\beta^{-2}f^{2},\qquad[\tilde{m}^{4}+\tilde{m}^{2}(\tilde{a}_{2}^{(2)}+\frac{3}{2}\tilde{a}_{3}^{(4)})+\cdots]\ln(\beta^{-2}),$
(58)
the coefficients at the power-like divergencies being renormalized to finite
constants (not necessarily to zero). Then the second relation in (56) is
broken, while the first one is left untouched DeWQFTcspt . All the
counterterms are invariant even under the local conformal transformations up
to total derivatives.
So, we face with a choice what the symmetry we want to preserve on a quantum
level. Whatever the case, the dependence of the effective action on the
Killing vector cannot be completely removed. The imaginary part of the
effective action, which is responsible for the Hawking particle production
Hawk , depends on the Killing vector.
The field $g_{\mu}$ plays a distinguished role in defining the state of
quantum fields. The structure of the counterterms suggests the simplest form
of the classical action for this field. Its Lagrangian density reads gmse ;
KostelBB ; Will ; DeWQFTcspt
$\mathcal{L}=-\frac{1}{4}f^{2}-\frac{1}{2}(\bar{\alpha}|\phi|^{2}+\bar{\beta}+\bar{\kappa}R)g^{2}-\frac{\bar{\gamma}}{4}g^{4},\qquad\bar{\gamma}>0,$
(59)
where $\phi$ is the Higgs field ($SU(2)$ doublet), $\bar{\alpha}$,
$\bar{\gamma}$, and $\bar{\kappa}$ are the dimensionless constants, and
$\bar{\beta}$ has the mass dimension $2$. This action resembles the simplest
model of a ferromagnetic LandLifshECM near the Curie point in the Landau
theory of phase transitions. The constants $\bar{\alpha}$ and $\bar{\beta}$
are such that the potential possesses the minimum at $g^{2}>0$. It is natural
to suppose that the characteristic magnitude of the vacuum expectation value
of the field $g_{\mu}$ is of the order of the Planck scale. From the
renormalization theory viewpoint, it is also appealing to suppose that
$\bar{\alpha}g^{2}=\mu^{2}$, where $\mu^{2}$ is the parameter of the Higgs
potential (see gmse for the notation). In this case, the constant
$\bar{\beta}$ can be put to zero and all the coupling constants of the model
become dimensionless, apart from the Newton and the cosmological constants. In
particular, we have the estimations for the constants determining the
potential of the field $g_{\mu}$: $\bar{\alpha}\approx\eta^{2}/m_{Pl}^{2}$ and
$\bar{\gamma}\approx\bar{\alpha}^{2}$, where $\eta$ is the vacuum expectation
value of the Higgs field. It should be noted that the dynamics of the field
$g_{\mu}$ are determined not from the least action principle, but from the
self-consistency requirement gmse imposed on the effective action. The field
$g_{\mu}$ is at the minimum of the effective potential only for the Minkowski
spacetime (see gmse ). The exploration of the viability of the above model and
its phenomenological consequences will be given elsewhere. Also note that the
fields with higher spins can induce additional terms in (59). The spinor
fields, for example, add the contribution proportional to $h^{2}$ to the terms
standing in round brackets in (59).
## VIII Conclusion
In this paper, we derived the high-temperature expansion of the free energy in
a complete form (36). The main results of the paper are collected and
discussed in Sec. VII. In particular, the explicit expressions for the
divergent and finite parts of the high-temperature expansions are given in
Eqs. (44), (45) in the case of the four-dimensional spacetime. We generalized
the results of the papers Fursaev1 ; Fursaev2 concerning the high-temperature
expansion of the free energy on stationary spacetimes and found the additional
contributions to the high-temperature expansion that were disregarded in those
papers. In that part of the expansion which was presented in Fursaev1 ;
Fursaev2 our expression for the free energy coincides with given in Fursaev1
; Fursaev2 up to integrals of the total derivatives. The method used in the
present paper is straightforwardly generalized to the case of the higher spin
fields, to the fields with the dispersion law differing from the relativistic
one, and to the grand canonical ensembles.
## Appendix A Useful relations with the Killing vector
In this appendix, we collect some formulas regarding the differential calculus
on the Riemannian manifold with the Killing vector. Let us introduce the
notation:
$g_{\mu}=\frac{\xi_{\mu}}{\xi^{2}},\qquad
h_{\mu}=\partial_{\mu}\ln\sqrt{\xi^{2}},\qquad
f_{\mu\nu}=\partial_{[\mu}g_{\nu]},$ (60)
where $\xi^{\mu}$ is the time-like Killing vector of the metric $g_{\mu\nu}$.
Then the following useful relations hold
$\begin{gathered}f_{\mu\lambda}g^{\lambda}=0,\qquad
g_{\lambda}h^{\lambda}=0,\qquad\nabla_{\mu}g_{\nu}=\frac{1}{2}f_{\mu\nu}-h_{(\mu}g_{\nu)},\\\
\nabla_{\mu}h_{\nu}g^{\nu}=-\frac{1}{2}f_{\mu\nu}h^{\nu}+g_{\mu}h^{2},\qquad
g^{\mu}g^{\nu}\nabla_{\mu}h_{\nu}=g^{2}h^{2},\\\
g^{\lambda}\nabla_{\lambda}f_{\mu\nu}=-g_{[\mu}f_{\nu]\lambda}h^{\lambda},\qquad\nabla_{\mu}f_{\nu\lambda}g^{\lambda}=\frac{1}{2}f^{2}_{\mu\nu}+g_{\mu}f_{\nu\lambda}h^{\lambda},\qquad\nabla^{\lambda}f_{\lambda\mu}g^{\mu}=-\frac{1}{2}f^{2},\end{gathered}$
(61)
and for the curvature
$\begin{gathered}g^{\lambda}R_{\lambda\nu\sigma\mu}=\frac{1}{2}\nabla_{\nu}f_{\sigma\mu}-\frac{1}{2}h_{[\sigma}f_{\mu]\nu}+h_{[\sigma}g_{\mu]}h_{\nu}+f_{\sigma\mu}h_{\nu}-g_{[\sigma}\nabla_{\mu]}h_{\nu},\\\
g^{\lambda}R_{\lambda\mu}=f_{\mu\lambda}h^{\lambda}-\frac{1}{2}\nabla^{\lambda}f_{\lambda\mu}-g_{\mu}\nabla_{\lambda}h^{\lambda},\qquad
R_{\mu\nu}g^{\mu}g^{\nu}=\frac{1}{4}f^{2}-g^{2}\nabla_{\lambda}h^{\lambda},\end{gathered}$
(62)
where $f^{2}:=f_{\mu\nu}f^{\mu\nu}=-\operatorname{Tr}f^{2}$.
In the course of the derivation of the high-temperature expansion, it was
useful to introduce the negative definite metric $G_{\mu\nu}$ (see Eq. (16)).
The corresponding Christoffel symbols are
$\tilde{\Gamma}^{\rho}_{\mu\nu}=\Gamma^{\rho}_{\mu\nu}+\frac{1}{2}(\lambda^{-1}-\xi^{2})g_{(\mu}f_{\nu)}^{\
\rho}-\xi^{2}(h_{(\mu}g_{\nu)}g^{\rho}-g_{\mu}g_{\nu}h^{\rho})=:\Gamma^{\rho}_{\mu\nu}+\gamma^{\rho}_{\mu\nu},$
(63)
where $\Gamma^{\rho}_{\mu\nu}$ are the Christoffel symbols for the metric
$g_{\mu\nu}$. Hereinafter all the quantities associated to the metric
$G_{\mu\nu}$ are designated by tildas. From (63) we see that the connection
$\tilde{\Gamma}^{\rho}_{\mu\nu}$ tends to the projected connection (see, e.g.,
psfss ; LandLifshCTF ; Fursaev1 ; Fursaev2 ) when $\lambda$ goes to infinity.
The curvature tensor for the connection $\tilde{\Gamma}^{\rho}_{\mu\nu}$ is
defined in the standard way
$\tilde{R}^{\rho}_{\ \nu\sigma\mu}=R^{\rho}_{\
\nu\sigma\mu}+\nabla_{[\sigma}\gamma_{\mu]\nu}^{\rho}+\gamma_{[\sigma\lambda}^{\rho}\gamma_{\mu]\nu}^{\lambda}.$
(64)
After tedious calculations, we arrive at
$\tilde{R}^{\rho}_{\ \nu\sigma\mu}=R^{\rho}_{\
\nu\sigma\mu}+\frac{1}{2}(\lambda^{-1}-\xi^{2})\Bigl{(}f_{\sigma\mu}f_{\nu}^{\
\rho}+g_{[\mu}\nabla_{\sigma]}f_{\nu}^{\
\rho}-g_{\nu}\nabla^{\rho}f_{\sigma\mu}-\frac{1}{2}f_{\nu[\sigma}f_{\mu]}^{\
\rho}\Bigr{)}\\\
-\xi^{2}\Bigl{(}g_{[\mu}h_{\sigma]}h_{\nu}g^{\rho}-g_{[\mu}h_{\sigma]}h^{\rho}g_{\nu}+\frac{1}{2}h_{[\mu}f_{\sigma]\nu}g^{\rho}-\frac{1}{2}g_{[\mu}f_{\sigma]\nu}h^{\rho}+g_{[\mu}\nabla_{\sigma]}h_{\nu}g^{\rho}-g_{[\mu}\nabla_{\sigma]}h^{\rho}g_{\nu}+f_{\sigma\mu}(h_{\nu}g^{\rho}-h^{\rho}g_{\nu})+\\\
+f_{\nu}^{\ \rho}h_{[\sigma}g_{\mu]}-\frac{1}{2}h_{[\mu}f_{\sigma]}^{\
\rho}g_{\nu}+\frac{1}{2}g_{[\mu}f_{\sigma]}^{\
\rho}h_{\nu}\Bigr{)}+\frac{1}{4}(\lambda^{-1}-\xi^{2})^{2}g_{\nu}g_{[\sigma}f_{\mu]}^{\
\lambda}f_{\lambda}^{\
\rho}-\frac{1}{2}(\lambda^{-1}-\xi^{2})\xi^{2}g_{[\sigma}f_{\mu]}^{\
\lambda}h_{\lambda}g^{\rho}g_{\nu}.$ (65)
The Ricci curvature becomes
$\tilde{R}_{\mu\nu}=R_{\mu\nu}+\frac{1}{2}(\lambda^{-1}-\xi^{2})[f^{2}_{\mu\nu}-g_{(\mu}\nabla^{\rho}f_{\rho\nu)}-g_{(\mu}f_{\nu)\rho}h^{\rho}]-\xi^{2}[g_{\mu}g_{\nu}(h^{2}-\nabla_{\rho}h^{\rho})+g_{(\mu}f_{\nu)\rho}h^{\rho}]+\\\
+\frac{1}{4}(\lambda^{-1}-\xi^{2})^{2}g_{\mu}g_{\nu}f^{2}+h_{\mu}h_{\nu}+\nabla_{\mu}h_{\nu}.$
(66)
Consequently, the scalar curvature of the metric $G_{\mu\nu}$ is written as
$\tilde{R}=G^{\mu\nu}\tilde{R}_{\mu\nu}=R+2\nabla_{\rho}h^{\rho}-\frac{1}{4}(\lambda^{-1}-\xi^{2})f^{2}.$
(67)
## Appendix B Relevant parts of the heat kernel expansion coefficients
Here we provide the parts of the heat kernel expansion coefficients relevant
to the high-temperature expansion of the one-loop free energy in a three
dimensional space. It is these coefficients that determine the divergent and
finite parts of the high-temperature expansion when $\beta\rightarrow 0$. We
just employ the known expressions VasilHeatKer ; Avramid ; Ven ; Gilkey for
the heat kernel coefficients adapted to the operator $H_{G}$ (15). Notice that
we did not check the general formulas for the coefficients $a_{k}$ presented
in VasilHeatKer ; Avramid ; Ven ; Gilkey save the terms at small $k$.
As follows from Eqs. (33) and (34), the relevant parts of the heat kernel
coefficients $a_{k}$ satisfy the inequality $j\geq k-4$ (for $d=3$), where $j$
is the power of $\omega$ entering the expression. In our case, using the
notation of VasilHeatKer , we should put
$\begin{gathered}E\rightarrow V-E,\qquad\Omega_{\mu\nu}\rightarrow-i\omega
f_{\mu\nu},\\\ V=\frac{1}{4}h^{2}-\frac{1}{2}\nabla^{\mu}h_{\mu},\qquad
E=\frac{\omega^{2}}{\xi^{2}}-m^{2}.\end{gathered}$ (68)
So, $E$ has the second power in $\omega$, the curvature of the gauge fields
$\Omega$ is of the first power, and the other structures are of the zeroth
order in $\omega$. Besides, as we resummed the heat kernel expansion (see Eq.
(18)) and collected the terms $(V-E)$ and $\tilde{R}/6$ without derivatives to
the exponent, such terms in $a_{k}$ must be obliterated. We should to warn the
reader that the Riemann tensor used in VasilHeatKer differs from our one by
the sign, while the Ricci tensors coincide.
Denoting the relevant part of the coefficient by tilde, we have
$\tilde{a}_{0}=1,\qquad\tilde{a}_{1}=0,$ (69)
$\begin{array}[]{l}\tilde{a}_{2}=\frac{1}{2!}\Big{[}\frac{1}{15}\tilde{\nabla}^{2}\tilde{R}-\frac{1}{90}\tilde{R}_{\mu\nu}\tilde{R}^{\mu\nu}+\frac{1}{90}\tilde{R}_{\alpha\beta\mu\nu}\tilde{R}^{\alpha\beta\mu\nu}+\frac{1}{3}\tilde{\nabla}^{2}V-\frac{1}{3}\tilde{\nabla}^{2}E+\frac{1}{6}\Omega^{2}\Big{]},\end{array}$
(70)
where
$\begin{array}[]{l}\tilde{\nabla}^{2}\tilde{R}=(\nabla^{2}-h^{\rho}\nabla_{\rho})[R+2\nabla^{\mu}h_{\mu}-\frac{1}{4}(\lambda^{-1}-\xi^{2})f^{2}],\\\
\tilde{R}^{\mu\nu}\tilde{R}_{\mu\nu}=R^{\mu\nu}R_{\mu\nu}+2R^{\mu\nu}[h_{\mu}h_{\nu}+\nabla_{\mu}h_{\nu}+\frac{1}{2}(\lambda^{-1}-\xi^{2})f^{2}_{\mu\nu}]+\frac{1}{4}(\lambda^{-1}-\xi^{2})^{2}[\operatorname{Tr}f^{4}+\frac{1}{4}(f^{2})^{2}]+\\\
+\frac{1}{2}(\lambda^{-1}-\xi^{2})[(\nabla^{\mu}f_{\mu\nu})^{2}+hf^{2}h+2\nabla^{\mu}h^{\nu}f^{2}_{\mu\nu}]+(\lambda^{-1}+\xi^{2})\nabla^{\rho}f_{\rho\mu}f^{\mu\nu}h_{\nu}+\\\
+\frac{\xi^{2}}{2}f^{2}(\nabla^{\mu}h_{\mu}-h^{2})+h^{\mu}\nabla_{\mu}h^{2}+2h^{2}\nabla^{\mu}h_{\mu}+\nabla_{\mu}h_{\nu}\nabla^{\mu}h^{\nu}-(\nabla^{\mu}h_{\mu})^{2},\\\
\tilde{R}^{\mu\nu\rho\sigma}\tilde{R}_{\mu\nu\rho\sigma}=R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}-\frac{3}{2}(\lambda^{-1}-\xi^{2})R^{\rho\nu\sigma\mu}f_{\rho\nu}f_{\sigma\mu}+(\lambda^{-1}-\xi^{2})(\nabla_{\rho}f_{\mu\sigma}\nabla^{\rho}f^{\mu\sigma}+2hf^{2}h)-\\\
-2\xi^{2}(\nabla^{\mu}h^{\nu}f^{2}_{\mu\nu}-hf^{2}h+\frac{3}{2}h^{\mu}\nabla_{\mu}f^{2}+3h^{2}f^{2}+2g^{2}\nabla_{\mu}h_{\nu}\nabla^{\mu}h^{\nu}+2g^{2}h^{\mu}\nabla_{\mu}h^{2})+\\\
+\frac{1}{8}(\lambda^{-1}-\xi^{2})^{2}[3(f^{2})^{2}+5\operatorname{Tr}f^{4}],\\\
\tilde{\nabla}^{2}V=(\nabla^{2}-h^{\rho}\nabla_{\rho})(\frac{1}{4}h^{2}-\frac{1}{2}\nabla^{\mu}h_{\mu}),\\\
\tilde{\nabla}^{2}E=\omega^{2}g^{2}(6h^{2}-2\nabla^{\mu}h_{\mu}),\\\
\Omega^{2}\equiv\Omega^{\mu\nu}\Omega_{\mu\nu}=-\omega^{2}f^{2}.\end{array}$
(71)
In the expression (70) for $\tilde{a}_{2}$ and further for $\tilde{a}_{k}$
(Eqs. (73), (79), (84), and (87)), the indices are raised and lowered by the
use of the metric $G_{\mu\nu}$. Collecting all the above terms, we obtain
$\begin{array}[]{l}\tilde{a}_{2}=\frac{1}{180}\Big{[}R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}+\frac{3\xi^{2}}{2}R^{\rho\nu\sigma\mu}f_{\rho\nu}f_{\sigma\mu}-R^{\mu\nu}(R_{\mu\nu}+2h_{\mu}h_{\nu}+2\nabla_{\mu}h_{\nu}-\xi^{2}f^{2}_{\mu\nu})+6(\nabla^{2}-h^{\mu}\nabla_{\mu})R\\\
+\frac{\xi^{4}}{16}(5(f^{2})^{2}+6\operatorname{Tr}f^{4})+\frac{\xi^{2}}{2}(\nabla^{\mu}f_{\mu\nu})^{2}-\xi^{2}\nabla_{\mu}f_{\nu\rho}\nabla^{\mu}f^{\nu\rho}-\xi^{2}\nabla^{\rho}f_{\rho\mu}f^{\mu\nu}h_{\nu}+\frac{\xi^{2}}{2}hf^{2}h-\xi^{2}\nabla^{\mu}h^{\nu}f^{2}_{\mu\nu}\\\
+\frac{5\xi^{2}}{2}f^{2}(\nabla^{\mu}h_{\mu}-h^{2})-2h^{2}\nabla^{\mu}h_{\mu}+(\nabla^{\mu}h_{\mu})^{2}+10\nabla_{\mu}h_{\nu}\nabla^{\mu}h^{\nu}-\frac{25}{2}h^{\mu}\nabla_{\mu}h^{2}+15h^{\mu}\nabla^{2}h_{\mu}-3\nabla^{2}\nabla^{\mu}h_{\mu}\\\
+3h^{\mu}\nabla_{\mu}\nabla_{\nu}h^{\nu}+\frac{3\xi^{2}}{2}(h^{\mu}\nabla_{\mu}f^{2}+\nabla^{2}f^{2})\Big{]}+\omega^{2}g^{2}\Big{[}\frac{1}{3}\nabla^{\mu}h_{\mu}-h^{2}-\frac{\xi^{2}}{12}f^{2}\Big{]}.\end{array}$
(72)
The relevant part of the next heat kernel expansion coefficient is (see also
Avramid ; Gilkey )
$\begin{array}[]{l}\tilde{a}_{3}=\frac{1}{3!}\Big{[}\frac{2}{15}\tilde{\nabla}_{\mu}\Omega_{\nu\rho}\tilde{\nabla}^{\mu}\Omega^{\nu\rho}+\frac{1}{30}(\tilde{\nabla}^{\nu}\Omega_{\nu\mu})^{2}+\frac{1}{5}\Omega^{\mu\nu}\tilde{\nabla}^{2}\Omega_{\mu\nu}+\frac{1}{10}\tilde{R}_{\alpha\beta\mu\nu}\Omega^{\alpha\beta}\Omega^{\mu\nu}+\frac{1}{15}\tilde{R}^{\mu\nu}\Omega^{2}_{\mu\nu}-\\\
-\frac{1}{10}(\tilde{\nabla}^{2})^{2}E-\tilde{\nabla}_{\mu}V\tilde{\nabla}^{\mu}E-\frac{1}{15}\tilde{R}_{\mu\nu}\tilde{\nabla}^{\mu}\tilde{\nabla}^{\nu}E-\frac{1}{5}\tilde{\nabla}^{\mu}\tilde{R}\tilde{\nabla}_{\mu}E+\frac{1}{2}\tilde{\nabla}_{\mu}E\tilde{\nabla}^{\mu}E\Big{]},\end{array}$
(73)
where
$\begin{array}[]{l}\tilde{\nabla}_{\mu}\Omega_{\nu\rho}\tilde{\nabla}^{\mu}\Omega^{\nu\rho}=-\omega^{2}[\nabla_{\mu}f_{\nu\rho}\nabla^{\mu}f^{\nu\rho}+2hf^{2}h+\frac{1}{2}(\lambda^{-1}-\xi^{2})\operatorname{Tr}f^{4}],\\\
(\tilde{\nabla}^{\nu}\Omega_{\nu\mu})^{2}=-\omega^{2}[(\nabla^{\nu}f_{\nu\mu})^{2}+\frac{1}{4}(\lambda^{-1}-\xi^{2})(f^{2})^{2}+2\nabla^{\nu}f_{\nu\mu}f^{\mu\rho}h_{\rho}-hf^{2}h],\\\
\Omega^{\mu\nu}\tilde{\nabla}^{2}\Omega_{\mu\nu}=-\omega^{2}[f^{\mu\nu}\nabla^{2}f_{\mu\nu}-\frac{1}{2}(\lambda^{-1}-\xi^{2})\operatorname{Tr}f^{4}-2hf^{2}h-\frac{1}{2}h^{\rho}\nabla_{\rho}f^{2}],\\\
\tilde{R}_{\alpha\beta\mu\nu}\Omega^{\alpha\beta}\Omega^{\mu\nu}=-\omega^{2}[R_{\alpha\beta\mu\nu}f^{\alpha\beta}f^{\mu\nu}-\frac{1}{2}(\lambda^{-1}-\xi^{2})((f^{2})^{2}+\operatorname{Tr}f^{4})],\\\
(\tilde{\nabla}^{2})^{2}E=-2\omega^{2}g^{2}[\nabla^{2}\nabla^{\mu}h_{\mu}-6\nabla_{\mu}h_{\nu}\nabla^{\mu}h^{\nu}-6h^{\mu}\nabla^{2}h_{\mu}+15h^{\mu}\nabla_{\mu}h^{2}-5h^{\mu}\nabla_{\mu}\nabla_{\nu}h^{\nu}-2(\nabla^{\mu}h_{\mu}-3h^{2})^{2}],\\\
\tilde{R}^{\mu\nu}\Omega^{2}_{\mu\nu}=-\omega^{2}[R^{\mu\nu}f^{2}_{\mu\nu}+\frac{1}{2}(\lambda^{-1}-\xi^{2})\operatorname{Tr}f^{4}+hf^{2}h+f^{2}_{\mu\nu}\nabla^{\mu}h^{\nu}],\\\
\tilde{\nabla}_{\mu}V\tilde{\nabla}^{\mu}E=\omega^{2}g^{2}[h^{\mu}\nabla_{\mu}\nabla_{\nu}h^{\nu}-\frac{1}{2}h^{\mu}\nabla_{\mu}h^{2}],\\\
\tilde{R}_{\mu\nu}\tilde{\nabla}^{\mu}\tilde{\nabla}^{\nu}E=\omega^{2}g^{2}[R_{\mu\nu}(4h^{\mu}h^{\nu}-2\nabla^{\mu}h^{\nu})+(3\lambda^{-1}-\xi^{2})hf^{2}h+6h^{4}+h^{\mu}\nabla_{\mu}h^{2}-2\nabla_{\mu}h_{\nu}\nabla^{\mu}h^{\nu}-\\\
-2h^{2}\nabla^{\mu}h_{\mu}-(\lambda^{-1}-\xi^{2})(f^{2}_{\mu\nu}\nabla^{\mu}h^{\nu}+\nabla^{\mu}f_{\mu\nu}f^{\nu\rho}h_{\rho})+\frac{1}{2}h^{2}\xi^{2}f^{2}],\\\
\tilde{\nabla}^{\mu}\tilde{R}\tilde{\nabla}_{\mu}E=-2\omega^{2}g^{2}h^{\nu}\nabla_{\nu}[R+2\nabla^{\mu}h_{\mu}-\frac{1}{4}(\lambda^{-1}-\xi^{2})f^{2}],\\\
\tilde{\nabla}_{\mu}E\tilde{\nabla}^{\mu}E=4\omega^{4}g^{4}h^{2}.\end{array}$
(74)
And so
$\begin{array}[]{l}\tilde{a}_{3}=-\frac{\omega^{2}g^{2}}{90}\Big{[}\frac{3\xi^{2}}{2}R^{\alpha\beta\mu\nu}f_{\alpha\beta}f_{\mu\nu}+R^{\mu\nu}(\xi^{2}f^{2}_{\mu\nu}+4h_{\mu}h_{\nu}-2\nabla_{\mu}h_{\nu})+\frac{3\xi^{4}}{4}(\operatorname{Tr}f^{4}+\frac{5}{6}(f^{2})^{2})\\\
+2\xi^{2}\nabla_{\mu}f_{\nu\rho}\nabla^{\mu}f^{\nu\rho}+\frac{\xi^{2}}{2}(\nabla^{\nu}f_{\nu\mu})^{2}+3\xi^{2}f^{\mu\nu}\nabla^{2}f_{\mu\nu}-\frac{5\xi^{2}}{2}hf^{2}h+2\xi^{2}\nabla^{\nu}f_{\nu\mu}f^{\mu\rho}h_{\rho}+3\xi^{2}h^{\mu}\nabla_{\mu}f^{2}\\\
+2\xi^{2}\nabla^{\mu}h^{\nu}f^{2}_{\mu\nu}-3\nabla^{2}\nabla^{\mu}h_{\mu}+16\nabla^{\mu}h^{\nu}\nabla_{\mu}h_{\nu}+18h^{\mu}\nabla^{2}h_{\mu}-\frac{103}{2}h^{\mu}\nabla_{\mu}h^{2}+18h^{\mu}\nabla_{\mu}\nabla_{\nu}h^{\nu}+6(\nabla^{\mu}h_{\mu})^{2}\\\
-38h^{2}\nabla^{\mu}h_{\mu}+60h^{4}-\frac{5\xi^{2}}{2}h^{2}f^{2}-6h^{\mu}\nabla_{\mu}R\Big{]}+\frac{1}{3}\omega^{4}g^{4}h^{2}.\end{array}$
(75)
If we had included the nonminimal coupling $\kappa R$ of the scalar field with
gravity to the Klein-Gordon equation (4), this just would have resulted in a
change of the coefficients of the scalar curvature entering the expressions
for $\tilde{a}_{2}$ and $\tilde{a}_{3}$ and a redefinition of the effective
mass $\tilde{m}$:
$\tilde{m}^{2}\rightarrow\tilde{m}^{2}-\kappa
R,\qquad\tilde{a}_{2}\rightarrow\tilde{a}_{2}-\frac{\kappa}{6}(\nabla^{2}-h^{\mu}\nabla_{\mu})R,\qquad\tilde{a}_{3}\rightarrow\tilde{a}_{3}-\frac{\omega^{2}g^{2}\kappa}{3}h^{\mu}\nabla_{\mu}R.$
(76)
The higher coefficients are encrypted in Ven . Notice that formula (6.6) for
$\textsf{Z}_{(2)}$ in Ven contains a spurious term $R^{2}/12$. Making use of
the notation from that paper, we write out the relevant parts of the
coefficients:
$\begin{array}[]{l}\tilde{a}_{4}=\frac{1}{4!}\Big{[}\frac{2}{5}\textsf{Z}_{(1}^{\dagger}\hat{\textsf{Z}}\textsf{Z}_{1)}+\frac{2}{5}\\{\textsf{Z}^{\dagger}_{(1}\textsf{Z}_{3)}\\}+\frac{2}{5}\textsf{Z}_{(2}^{\dagger}\textsf{Z}_{2)}+\frac{1}{5}\textsf{Z}^{2}_{(2)}\Big{]},\end{array}$
(77)
where
$\begin{array}[]{l}\textsf{Z}_{(1}^{\dagger}\hat{\textsf{Z}}\textsf{Z}_{1)}\approx-\frac{1}{3}\tilde{R}^{\mu\nu}X_{\mu}X_{\nu}+\frac{2}{3}X^{\mu}\Omega_{\mu\nu}\Omega^{\nu\lambda}_{\
\ ;\lambda},\\\
\\{\textsf{Z}^{\dagger}_{(1}\textsf{Z}_{3)}\\}\approx\frac{4}{3}X^{\mu}X^{\
\nu}_{\mu\ \nu}+\frac{2}{3}X^{\mu}X^{\nu}_{\
\nu\mu}+X^{\mu}(\Omega^{2})_{;\mu}-\frac{4}{3}X^{\mu}\Omega_{\mu\nu}\Omega^{\nu\lambda}_{\
\ ;\lambda},\\\
\textsf{Z}_{(2}^{\dagger}\textsf{Z}_{2)}\approx\frac{1}{3}[(X^{\mu}_{\
\mu})^{2}+2X^{\mu\nu}X_{\mu\nu}-2X^{\mu\nu}\Omega^{2}_{\mu\nu}+X^{\mu}_{\
\mu}\Omega^{2}+\frac{1}{4}(\Omega^{2})^{2}+\frac{1}{2}\operatorname{Tr}\Omega^{4}],\\\
\textsf{Z}^{2}_{(2)}\approx(X^{\mu}_{\ \mu})^{2}+X^{\mu}_{\
\mu}\Omega^{2}+\frac{1}{4}(\Omega^{2})^{2}.\end{array}$ (78)
Here $X=V-E$ as above, the indices of $X$ and semicolon denote the covariant
differentiation with respect to the Levi-Civita connection
$\tilde{\Gamma}^{\rho}_{\mu\nu}$, and the approximate equality means that the
relevant part of the expression is retained. Therefore we have (see also
Avramid )
$\begin{array}[]{l}\tilde{a}_{4}=\frac{1}{4!}\Big{[}\frac{2}{5}\tilde{R}_{\mu\nu}\tilde{\nabla}^{\mu}E\tilde{\nabla}^{\nu}E+\frac{4}{15}\tilde{\nabla}^{\mu}\Omega_{\mu\nu}\Omega^{\nu\rho}\tilde{\nabla}_{\rho}E+\frac{4}{5}\tilde{\nabla}^{\mu}E\tilde{\nabla}_{\mu}\tilde{\nabla}^{2}E-\frac{2}{5}\tilde{\nabla}^{\mu}E\tilde{\nabla}_{\mu}\Omega^{2}+\frac{1}{3}(\tilde{\nabla}^{2}E)^{2}-\\\
-\frac{1}{3}\tilde{\nabla}^{2}E\Omega^{2}+\frac{1}{12}(\Omega^{2})^{2}+\frac{4}{15}\tilde{\nabla}^{\mu}\tilde{\nabla}^{\nu}E\tilde{\nabla}_{\mu}\tilde{\nabla}_{\nu}E+\frac{4}{15}\tilde{\nabla}^{\mu}\tilde{\nabla}^{\nu}E\Omega^{2}_{\mu\nu}+\frac{1}{15}\operatorname{Tr}\Omega^{4}\Big{]},\end{array}$
(79)
where
$\begin{array}[]{l}\tilde{R}_{\mu\nu}\tilde{\nabla}^{\mu}E\tilde{\nabla}^{\nu}E=2\omega^{4}g^{4}[2R_{\mu\nu}h^{\mu}h^{\nu}+(\lambda^{-1}-\xi^{2})hf^{2}h+2h^{4}+h^{\mu}\nabla_{\mu}h^{2}],\\\
\tilde{\nabla}^{\mu}\Omega_{\mu\nu}\Omega^{\nu\rho}\tilde{\nabla}_{\rho}E=2\omega^{4}g^{2}(\nabla^{\mu}f_{\mu\nu}f^{\nu\rho}h_{\rho}-hf^{2}h),\\\
\tilde{\nabla}^{\mu}E\tilde{\nabla}_{\mu}\tilde{\nabla}^{2}E=4\omega^{4}g^{4}(h^{\mu}\nabla_{\mu}\nabla_{\nu}h^{\nu}-2h^{2}\nabla^{\mu}h_{\mu}+6h^{4}-3h^{\mu}\nabla_{\mu}h^{2}),\\\
\tilde{\nabla}^{\mu}E\tilde{\nabla}_{\mu}\Omega^{2}=2\omega^{4}g^{2}h^{\mu}\nabla_{\mu}f^{2},\\\
(\tilde{\nabla}^{2}E)^{2}=4\omega^{4}g^{4}(3h^{2}-\nabla_{\mu}h^{\mu})^{2},\\\
\tilde{\nabla}^{2}E\Omega^{2}=2\omega^{4}g^{2}f^{2}(\nabla_{\mu}h^{\mu}-3h^{2}),\\\
(\Omega^{2})^{2}=\omega^{4}(f^{2})^{2},\\\
\tilde{\nabla}^{\mu}\tilde{\nabla}^{\nu}E\tilde{\nabla}_{\mu}\tilde{\nabla}_{\nu}E=4\omega^{4}g^{4}[\nabla_{\mu}h_{\nu}\nabla^{\mu}h^{\nu}-2h^{\mu}\nabla_{\mu}h^{2}+3h^{4}-\frac{1}{2}(\lambda^{-1}-\xi^{2})hf^{2}h],\\\
\tilde{\nabla}^{\mu}\tilde{\nabla}^{\nu}E\Omega^{2}_{\mu\nu}=2\omega^{4}g^{2}(\nabla^{\mu}h^{\nu}f^{2}_{\mu\nu}-2hf^{2}h),\\\
\operatorname{Tr}\Omega^{4}=\omega^{4}\operatorname{Tr}f^{4}.\end{array}$ (80)
Summing up, we obtain
$\begin{array}[]{l}\tilde{a}_{4}=\frac{\omega^{4}g^{4}}{15}\Big{[}R_{\mu\nu}h^{\mu}h^{\nu}+\frac{\xi^{4}}{96}(5(f^{2})^{2}+4\operatorname{Tr}f^{4})-\frac{7\xi^{2}}{6}hf^{2}h+\frac{\xi^{2}}{3}\nabla^{\mu}h^{\nu}f^{2}_{\mu\nu}+\frac{\xi^{2}}{3}\nabla^{\mu}f_{\mu\nu}f^{\nu\rho}h_{\rho}-\frac{\xi^{2}}{2}h^{\mu}\nabla_{\mu}f^{2}\\\
-\frac{5\xi^{2}}{12}f^{2}(\nabla^{\mu}h_{\mu}-3h^{2})+\frac{45}{2}h^{4}-\frac{41}{6}h^{\mu}\nabla_{\mu}h^{2}+2h^{\mu}\nabla_{\mu}\nabla_{\nu}h^{\nu}-9h^{2}\nabla^{\mu}h_{\mu}+\frac{5}{6}(\nabla^{\mu}h_{\mu})^{2}+\frac{2}{3}\nabla_{\mu}h_{\nu}\nabla^{\mu}h^{\nu}\Big{]}.\end{array}$
(81)
Further,
$\begin{array}[]{l}\tilde{a}_{5}=\frac{1}{5!}\Big{[}\frac{1}{3}\textsf{Z}^{\dagger}_{(1}\hat{\textsf{Z}}{}^{2}\textsf{Z}_{1)}+\frac{9}{7}\textsf{Z}^{\dagger}_{(1}\hat{\textsf{Z}}_{2}\textsf{Z}_{1)}+\frac{6}{7}\\{\textsf{Z}^{\dagger}_{(1}\hat{\textsf{Z}}_{1}\textsf{Z}_{2)}\\}+\frac{1}{3}\\{\textsf{Z}^{\dagger}_{(1}\textsf{Z}_{1)}\textsf{Z}_{(2)}\\}\Big{]},\end{array}$
(82)
where
$\begin{array}[]{l}\textsf{Z}^{\dagger}_{(1}\hat{\textsf{Z}}{}^{2}\textsf{Z}_{1)}\approx
X^{\mu}\Omega^{2}_{\mu\nu}X^{\nu},\\\
\textsf{Z}^{\dagger}_{(1}\hat{\textsf{Z}}_{2}\textsf{Z}_{1)}\approx\frac{2}{3}X^{\mu}X_{\mu\nu}X^{\nu}+\frac{1}{3}X^{\mu}X_{\mu}X^{\nu}_{\
\nu}+\frac{1}{6}X^{\mu}X_{\mu}\Omega^{2}-\frac{1}{3}X^{\mu}\Omega^{2}_{\mu\nu}X^{\nu},\\\
\\{\textsf{Z}^{\dagger}_{(1}\hat{\textsf{Z}}_{1}\textsf{Z}_{2)}\\}\approx
2\textsf{Z}^{\dagger}_{(1}\hat{\textsf{Z}}_{2}\textsf{Z}_{1)},\\\
\\{\textsf{Z}^{\dagger}_{(1}\textsf{Z}_{1)}\textsf{Z}_{(2)}\\}\approx
2[X^{\mu}X_{\mu}X^{\nu}_{\
\nu}+\frac{1}{2}X^{\mu}X_{\mu}\Omega^{2}].\end{array}$ (83)
Hence, substituting (83) to (82), we come to
$\begin{array}[]{l}\tilde{a}_{5}=\frac{1}{5!}\Big{[}-2\tilde{\nabla}^{\mu}E\tilde{\nabla}^{\nu}E\tilde{\nabla}_{\mu}\tilde{\nabla}_{\nu}E-\frac{5}{3}\tilde{\nabla}^{\mu}E\tilde{\nabla}_{\mu}E\tilde{\nabla}^{2}E+\frac{5}{6}\tilde{\nabla}^{\mu}E\tilde{\nabla}_{\mu}E\Omega^{2}-\frac{2}{3}\tilde{\nabla}^{\mu}E\tilde{\nabla}^{\nu}E\Omega^{2}_{\mu\nu}\Big{]},\end{array}$
(84)
where
$\begin{array}[]{l}\tilde{\nabla}^{\mu}E\tilde{\nabla}^{\nu}E\tilde{\nabla}_{\mu}\tilde{\nabla}_{\nu}E=4\omega^{6}g^{6}(4h^{4}-h^{\mu}\nabla_{\mu}h^{2}),\\\
\tilde{\nabla}^{\mu}E\tilde{\nabla}_{\mu}E\tilde{\nabla}^{2}E=8\omega^{6}g^{6}h^{2}(3h^{2}-\nabla^{\mu}h_{\mu}),\\\
\tilde{\nabla}^{\mu}E\tilde{\nabla}_{\mu}E\Omega^{2}=-4\omega^{6}g^{4}h^{2}f^{2},\\\
\tilde{\nabla}^{\mu}E\tilde{\nabla}^{\nu}E\Omega^{2}_{\mu\nu}=-4\omega^{6}g^{4}hf^{2}h.\end{array}$
(85)
Collecting all the terms together, we get
$\begin{array}[]{l}\tilde{a}_{5}=-\frac{4\omega^{6}g^{6}}{5!}\Big{[}18h^{4}-2h^{\mu}\nabla_{\mu}h^{2}-\frac{10}{3}h^{2}\nabla^{\mu}h_{\mu}+\frac{5\xi^{2}}{6}h^{2}f^{2}-\frac{2\xi^{2}}{3}hf^{2}h\Big{]}.\end{array}$
(86)
The last coefficient that we need is
$\begin{array}[]{l}\tilde{a}_{6}=\frac{1}{6!}\Big{[}\frac{4}{7}X_{(1}X_{1)}X_{(1}X_{1)}+\frac{27}{14}X_{(1}X_{1}X_{1}X_{1)}\Big{]}\approx\frac{1}{6!}\frac{5}{2}(\tilde{\nabla}^{\mu}E\tilde{\nabla}_{\mu}E)^{2}=\frac{40}{6!}\omega^{8}g^{8}h^{4}.\end{array}$
(87)
In conclusion we give the explicit expression for the finite part of the high-
temperature expansion of the one-loop contribution to the free energy (see
Eqs. (44), (45)) retaining only the terms at the nonegative powers of $m^{2}$:
$\begin{array}[]{l}\tilde{m}^{4}-\tilde{m}^{2}(\frac{2}{3}\tilde{a}_{2}^{(2)}+\frac{5}{3}\tilde{a}_{3}^{(4)})-\frac{4}{3}(\tilde{a}_{3}^{(2)}+2\tilde{a}_{4}^{(4)}+\frac{23}{4}\tilde{a}_{5}^{(6)}+22\tilde{a}_{6}^{(8)})=m^{4}+m^{2}(\frac{1}{3}R+\frac{5\xi^{2}}{36}f^{2}-\frac{5}{9}\nabla^{\mu}h_{\mu}+\frac{11}{18}h^{2})\\\
+\frac{1}{45}\Big{[}\xi^{2}R^{\alpha\beta\mu\nu}f_{\alpha\beta}f_{\mu\nu}+\frac{2}{3}R^{\mu\nu}(\xi^{2}f^{2}_{\mu\nu}-2\nabla_{\mu}h_{\nu}-2h_{\mu}h_{\nu})+\frac{\xi^{4}}{6}(\operatorname{Tr}f^{4}+\frac{35}{32}(f^{2})^{2})+\frac{\xi^{2}}{3}(\nabla^{\mu}f_{\mu\nu})^{2}\\\
+\frac{4\xi^{2}}{3}(\nabla_{\mu}f_{\nu\rho}\nabla^{\mu}f^{\nu\rho}-\nabla^{\nu}f_{\nu\mu}f^{\mu\rho}h_{\rho}-f^{2}_{\mu\nu}\nabla^{\mu}h^{\nu})+2\xi^{2}(f^{\mu\nu}\nabla^{2}f_{\mu\nu}+h^{\mu}\nabla_{\mu}f^{2})-\frac{5\xi^{2}}{16}h^{2}f^{2}+\frac{15\xi^{2}}{8}f^{2}\nabla^{\mu}h_{\mu}\\\
-2\nabla^{2}\nabla^{\mu}h_{\mu}+\frac{16}{3}\nabla^{\mu}h^{\nu}\nabla_{\mu}h_{\nu}+8h^{\mu}\nabla^{2}h_{\mu}+\frac{1}{4}(\nabla^{\mu}h_{\mu})^{2}+\frac{5}{4}h^{2}\nabla^{\mu}h_{\mu}-\frac{8}{3}h^{\mu}\nabla_{\mu}h^{2}-\frac{109}{48}h^{4}\Big{]}\\\
-\frac{4}{45}h^{\mu}\nabla_{\mu}R+\frac{1}{6}R\Big{(}\frac{11}{18}h^{2}-\frac{5}{9}\nabla^{\mu}h_{\mu}+\frac{5\xi^{2}}{36}f^{2}\Big{)}+\frac{1}{36}R^{2}.\end{array}$
(88)
The nonminimal coupling adds to this part the following terms:
$-\kappa
R(\frac{11}{18}h^{2}-\frac{5}{9}\nabla^{\mu}h_{\mu}+\frac{5\xi^{2}}{36}f^{2}+2m^{2})+(\kappa^{2}-\frac{\kappa}{3})R^{2}+\frac{4\kappa}{9}h^{\mu}\nabla_{\mu}R.$
(89)
In the conformal case, $m^{2}=0$, $\kappa=1/6$, this contribution cancels
completely the terms proportional to the scalar curvature $R$ (without
derivatives) entering the finite part.
###### Acknowledgements.
We are grateful to D. V. Vassilevich for the fruitful correspondence
elucidating certain aspects of the heat kernel expansion technique. ISK
appreciates the Dynasty Foundation for financial support. The work is done
partially under the project 2.3684.2011 of Tomsk State University. It is also
supported by the RFBR grants No. 12-02-31071-mol-a and No. 13-02-00551, and by
the Russian Ministry of Education and Science, contracts No. 14.B37.21.0911
and No. 14.B37.21.1298.
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|
arxiv-papers
| 2013-01-22T08:27:50 |
2024-09-04T02:49:40.530254
|
{
"license": "Public Domain",
"authors": "I. S. Kalinichenko, P. O. Kazinski",
"submitter": "Igor Kalinichenko",
"url": "https://arxiv.org/abs/1301.5103"
}
|
1301.5111
|
# The exponential parameterization of the quark mixing matrix
G. Dattoli [email protected] E. Di Palma [email protected]
ENEA - Centro Richerche Frascati, Via Enrico Fermi 45, 00044, Frascati, Rome,
Italy
###### Abstract
We comment on the exponential parameterization of the quark mixing matrix, by
stressing that it naturally incorporates the Cabibbo structure and the
hierarchical features of the Wolfenstein form. We extend our results to the
neutrino mixing and introduce an exponential generator of the tribimaximal
matrix.
Quark mixing matrix, Kobayashi and Wolfenstein matrix , Cabibbo structure
## I Introduction
The quark mixing matrix can be written in different ways, any of the proposed
forms displays nice features and disadvantages. Whatever form one uses, four
arbitrary parameters and the assumption of its unitarity are necessary to get
physically meaningful results. The models can be roughly grouped in two
categories, the first inspired to Euler like rotation matrices, the second,
containing explicit hierarchical features, employs an expansion, around the
unit matrix, in term of some key parameters. The original Kobayashi and
Maskawa matrixKobayashi had been written in terms of three mixing angles
$\theta_{1,2,3}$ and one CP violating phase $\delta$. In this parameterization
the first family decouples from the others in the limit $\theta_{1}\rightarrow
0$. The particle data groupChau chooses a form in which the CP violating term
is appended to the matrix entries responsible for the coupling of the first
and third generations of quark mass eigenstates. Finally
WolfensteinWolfenstein has proposed a matrix emerging from a kind of
perturbative expansion in terms of the Cabibbo coupling parameter
$\lambda\cong 0.22$Cabibbo . A third model, bridging between the (Kobayashi
,Chau ) and Wolfenstein , is based on the so called exponential
parameterization, which emerges from the request of unitarity, automatically
satisfied by setting Dattoli
$\left.\begin{array}[]{ll}\widehat{V}&=e^{\widehat{A}}\\\
\widehat{A}^{{\dagger}}&=-\widehat{A}\end{array}\right.$ (1)
The second condition in Eq. (1), expressing the anti-hermiticity of the
matrix, is ensured by the following specific choice
$\widehat{A}=\left(\begin{array}[]{ccc}0&\Lambda_{1}&\Lambda_{3}\\\
-\Lambda_{1}&0&\Lambda_{2}\\\ -\Lambda_{3}^{*}&-\Lambda_{2}&0\\\
\end{array}\right)$ (2)
The vanishing of the diagonal entries secures that the matrix $\widehat{V}$ be
unimodular111It would be sufficient to have a matrix with null trace, but for
practical reasons we use the form(2).. The sub-labels 1, 2, 3 determine the
mixing d-s, s-b, d-b respectively, all the entries, except $\Lambda_{3}$, are
real. In the spirit of Wolfeinstein criteria, we use the Cabibbo strength
$\lambda$ as key parameter and make the following identificationsDattoli2
$\left.\begin{array}[]{ll}\Lambda_{1}&=\lambda\\\ \Lambda_{2}&=y\lambda^{2}\\\
\Lambda_{3}&=x\lambda^{3}e^{\imath\delta}\end{array}\right.$ (3)
containing an implicit hierarchical assumption on the coupling between the
different quark families. The vanishing of the $x$, $y$ coefficients allows
the decoupling from the b sector reducing the matrix to the s-d Cabibbo
mixing, namely
$\widehat{V}_{(x,y)\rightarrow 0}=e^{\left(\begin{array}[]{ccc}0&\lambda&0\\\
-\lambda&0&0\\\ 0&0&0\\\
\end{array}\right)}=\left(\begin{array}[]{ccc}\cos(\lambda)&\sin(\lambda)&0\\\
-\sin(\lambda)&\cos(\lambda)&0\\\ 0&0&1\\\ \end{array}\right)$ (4)
It is also to be stressed that $\widehat{V}_{\lambda\rightarrow
0}=\widehat{\mathbb{1}}$, which means that the parameterization in (3)
contains the assumption that the vanishing of the Cabibbo parameter determines
the decoupling of the entire quark matrix. The phase $\delta$ is associated,
as in the particle data group choice, with the smallest coupling term. In this
paper we will see how the exponential parameterization yields a flexible tool
to analyse the quark mixing phenomenology and the relevant consequences.
## II The matrix A and the Wolfenstein parameterization
We will prove that the quark mixing matrix written as in Eq. (1) naturally
contains the Wolfenstein parameterization and the Euler like forms as well. By
keeping the expansion of the exponential in Eq. (1) up to third order in
$\lambda$, namely
$\widehat{V}=\widehat{\mathbb{1}}+\widehat{A}+\frac{\widehat{A}^{2}}{2}+\frac{\widehat{A}^{3}}{3!}+\frac{\widehat{A}^{4}}{4!}+o(\lambda^{5})$
(5)
we obtain the mixing matrix in the form
$\begin{array}[]{l}\widehat{V}\cong\left(\begin{array}[]{ccc}1-\frac{\lambda^{2}}{2}+\frac{\lambda^{4}}{4!}&\lambda-\frac{\lambda^{3}}{3!}&AF\lambda^{3}\\\
-\lambda+\frac{\lambda^{3}}{3!}&1-\frac{\lambda^{2}}{2}+\frac{\lambda^{4}}{4!}-\frac{(A\lambda^{2})^{2}}{2}&\frac{AB\lambda^{2}}{2}\\\
AG\lambda^{3}&\frac{AC\lambda^{2}}{2}&1-\frac{A^{2}\lambda^{4}}{2}\\\
\end{array}\right)\\\ \\\ A=y,\;F=\rho-\imath\eta,\;G=1-\rho-\imath\eta\\\
B=2-\lambda^{2}(\rho-\frac{1}{6}-\imath\eta),\;B+C=-2\lambda^{2}(\rho-\frac{1}{2})\\\
\rho=\frac{x}{y}\cos(\delta)+\frac{1}{2},\;\eta=-\frac{x}{y}\sin(\delta)\end{array}$
(6)
Eq. (6) is recognized as a Wolfenstein-type parameterization, the Taylor
expansion at higher order can provide more accurate expansion in the Cabibbo
coupling parameter, as we will see in the following. The expansion at the
third order allows a one to one correspondence between the Wolfenstein
parameters and those of the matrix $\widehat{A}$, which can be written in the
form
$\widehat{A}=\left(\begin{array}[]{ccc}0&\lambda&A\lambda^{3}(\rho-\imath\eta-\frac{1}{2})\\\
-\lambda&0&A\lambda^{2}\\\
-A\lambda^{3}(\rho+\imath\eta-\frac{1}{2})&-A\lambda^{2}&0\end{array}\right)$
(7)
Using for $A,\;\rho,\;\eta$ the following values, close to those given in the
literature DataGroup :
$\begin{array}[]{ll}\lambda=0.2272\pm
0.0010,&\;\;\;\;\;A=0.818^{+0.007}_{-0.017}\\\ \\\
\rho=0.221^{+0.064}_{-0.028},&\;\;\;\;\;\eta=0.340^{+0.017}_{-0.045}\end{array}$
we find for $x$ and $\delta$ the following values
$x=-0.359^{+0.049}_{-0.052},\;\;\;\delta=0.883^{+0.145}_{-0.118}$
and we get for the mixing matrix222 This result has been obtained by expanding
the matrix at any arbitrary order, namely
$\widehat{V}=\displaystyle{\sum_{0}^{N}}\frac{A^{n}}{n!}$ and by keeping N=50.
We have not included the errors deriving from the experimental and systematic
uncertainties, the relevant analysis would require extreme care for fitting
the data and such an effort is out of the purposes of the present note.
$|A|=\left(\begin{array}[]{ccc}0.97429&0.22523&3.86\cdot 10^{-3}\\\
0.22512&0.97341&0.04215\\\ 8.10\cdot 10^{-3}&0.04154&0.99910\\\
\end{array}\right)$ (8)
in good agreement with the values reported inDataGroup . Higher orders
expansions will be considered in the forthcoming sections.
## III The geometrical meaning of the exponential parameterization and the
Euler like forms
We have so far proved that the exponential parameterization of the mixing
matrix has some nice features which makes its use quite interesting. Before
going further let us speculate on the geometrical (physical) meaning of the
matrix $\widehat{A}$, which can be understood as a kind of Hamiltonian ruling
the process of quark mixing. We introduce, therefore, the Schroedinger
equation
$\imath\partial_{\tau}\underline{\psi}=\widehat{H}\underline{\psi}$ (9)
where $\underline{\psi}|_{\tau=0}$ are the quark mass eigenstates, and
$\widehat{H}\propto\imath\widehat{A}$ (10)
Within such a picture the matrix $\widehat{V}$ is the evolution operator
associated with Eq. (9). In the case of vanishing CP phase $\delta\rightarrow
0$, the Hamiltonian in (10) can be written in terms of SO(3) generators,
namely
$\widehat{H}=\lambda R_{1}+y\lambda^{2}R_{2}+x\lambda^{3}R_{3}$ (11)
with
$\begin{array}[]{l}R_{1}=\imath\left(\begin{array}[]{ccc}0&1&0\\\ -1&0&0\\\
0&0&0\\\ \end{array}\right),\;R_{2}=\imath\left(\begin{array}[]{ccc}0&0&0\\\
0&0&1\\\ 0&-1&0\\\ \end{array}\right)\\\ \\\
R_{3}=\imath\left(\begin{array}[]{ccc}0&0&1\\\ 0&0&0\\\ -1&0&0\\\
\end{array}\right)\end{array}$
The Schroedinger equation (9) can, accordingly, be viewed as a vector equation
of the type
$\left.\begin{array}[]{ll}\partial_{\tau}\overrightarrow{Q}=&\overrightarrow{\Omega}\times\overrightarrow{Q}\\\
\\\
\overrightarrow{\Omega}\equiv&\lambda(-y\lambda,x\lambda^{2},-1)\end{array}\right.$
(12)
where $\overrightarrow{Q}\equiv(\psi_{1},\psi_{2},\psi_{3})$ is the vector
associated with the quark field. The problem of the quark mixing is therefore
understood as a rotation, induced by an Euler-like torque equation.
The torque vector $\overrightarrow{\Omega}$ is reported in Fig. (1) along with
the role played by each vector component.
Figure 1: The Quark mixing torque vector
The quark mixing matrix can be written using the Cayley Hamilton theorem
Babusci (see Sect. IV) as
$\left.\begin{array}[]{l}\widehat{V}=e^{\tau\widehat{A}}|_{\tau=1}=\widehat{\mathbb{1}}+\mbox{sinc}(|\overrightarrow{\Omega}|)\widehat{A}+\displaystyle{\frac{1}{2}(\mbox{sinc}(\frac{|\overrightarrow{\Omega}|}{2}))^{2}\widehat{A}^{2}}\\\
\\\
|\overrightarrow{\Omega}|=\lambda\sqrt{1+y^{2}\lambda^{2}+x^{2}\lambda^{4}}\\\
\\\ \mbox{sinc}(\alpha)=\displaystyle\frac{\sin(\alpha)}{\alpha}\\\
\end{array}\right.$ (13)
Moreover from Eq. (12) the action of the mixing matrix on the initial vector
$\overrightarrow{Q}$ can be specified through the following Rodriguez
rotationBabusci1
$\left.\begin{array}[]{rl}\overrightarrow{Q}=&\cos(|\overrightarrow{\Omega}|)\overrightarrow{Q}_{0}+\sin(|\overrightarrow{\Omega}|)\overrightarrow{n}\times\overrightarrow{Q}_{0}\\\
\\\
&+(1-\cos(|\overrightarrow{\Omega}|))(\overrightarrow{n}\cdot\overrightarrow{Q}_{0})\cdot\overrightarrow{n}\\\
\\\
\overrightarrow{n}=&\frac{\overrightarrow{\Omega}}{|\overrightarrow{\Omega}|}\end{array}\right.$
(14)
The geometrical interpretation is less obvious if we include the CP violating
term. We assume Eq. (11) to be still valid and with a slight abuse of the
notation write
$\left.\begin{array}[]{rll}\overrightarrow{\Omega}&\equiv&\overrightarrow{\Omega}_{1}+\imath\overrightarrow{\Omega}_{2}\\\
\\\
\overrightarrow{\Omega}_{1}&\equiv&(-y\lambda^{2},x\lambda^{3}\cos(\delta),-\lambda)\\\
\\\
\overrightarrow{\Omega}_{2}&\equiv&(0,x\lambda^{3}\sin(\delta),0)\end{array}\right.$
(15)
This assumption contains the bare essence of CP violation from a geometrical
point of view. The vector $\overrightarrow{\Omega}$ splits into a real and
imaginary part, as shown in Fig. (2) where the second component of the torque
vector is composed by two subcomponents:
a)
the coupling vector
$\overrightarrow{\Omega}_{1,3}\equiv(y\lambda^{2},0,-\lambda)$ is the
component of the vector in the 1-3 plane;
b)
the CP violating sector is viewed as the pseudo vector
$(\Omega_{1,3},\mbox{Im}(\Omega_{2}),\mbox{Re}(\Omega_{2}))$.
Figure 2: The real(a) and the imaginary part (b) of the torque vector
$\Omega$.
In terms of the Wolfenstein parameters the modulus of the Torque vector can be
written as
$|\overrightarrow{\Omega}|=\sqrt{\lambda^{2}+(A\lambda^{2})^{2}+\left[\left(\rho-\frac{1}{2}\right)^{2}+\eta^{2}\right](A\lambda^{3})^{2}}$
(16)
or as
$\left.\begin{array}[]{l}|\overrightarrow{\Omega}|=\sqrt{\lambda^{2}(1+\tan^{2}(\phi_{A}))+\lambda^{3}\left(\rho-\frac{1}{2}\right)^{2}}\\\
\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\
\overline{\left[1+\tan^{2}(\delta)\right]\tan^{2}(\phi_{A})}\\\ \\\
\tan(\phi_{A})=A\lambda,\;\;\;\tan(\delta)=\displaystyle{\frac{\eta}{\sqrt{\left(\rho-\frac{1}{2}\right)^{2}}}}\end{array}\right.$
(17)
where $\phi_{A}$ and $\delta$ are indicated in Figs. (2).
The angle $\phi_{A}$ lies in the (1,3) sector and specifies the direction of
the $\overrightarrow{\Omega}$ vector components in this plane. We visualize
the geometric content of our problem as indicated in the second of Figs. (2),
in which the complex vector component lying along the direction of the axis 2
is split into an imaginary and a real part.
In more rigorous mathematical terms we can illustrate the above procedure as
it follows. We first note that
$\widehat{A}=\widehat{A}_{1}+\widehat{A}_{2}$ (18)
with
$\widehat{A}_{2}=\left(\begin{array}[]{ccc}0&0&\Lambda_{3}\\\ 0&0&0\\\
-\Lambda_{3}^{*}&0&0\end{array}\right),\;\;\widehat{A}_{1}=\left(\begin{array}[]{ccc}0&\Lambda_{1}&0\\\
-\Lambda_{1}&0&\Lambda_{2}\\\ 0&-\Lambda_{2}&0\end{array}\right)$
The matrices labelled with 2, 1 are not commuting each other, therefore we
have at the first order in the Zassenhaus disentanglement formula333 The
Zassenhaus formula writes
$e^{\widehat{A}+\widehat{B}}=e^{\widehat{A}}e^{\widehat{B}}\displaystyle{\prod_{n=1}^{\infty}}e^{\widehat{C}_{m}}$
where the operators $\widehat{C}_{m}$ are given in terms of successive
commutators, the first two being
$\widehat{C}_{1}=-\frac{1}{2}[\widehat{A},\widehat{B}]$,
$\widehat{C}_{2}=\frac{1}{3}[\widehat{A},\widehat{B}]+\frac{1}{6}[\widehat{A},[\widehat{A},\widehat{B}]]$
Magnus
$\left.\begin{array}[]{l}\widehat{V}=e^{\widehat{A}_{2}+\widehat{A}_{1}}\cong
e^{\widehat{A}_{2}}e^{\widehat{A}_{1}}e^{\widehat{C}}\\\ \\\
\widehat{C}=-\frac{1}{2}[\widehat{A}_{2},\widehat{A}_{1}]\end{array}\right.$
(19)
Where the (anti-hermitian) matrix $\widehat{C}$ is given by
$\begin{array}[]{rcl}\widehat{C}&=&-\frac{1}{2}\left(\begin{array}[]{ccc}0&-\Lambda_{3}\Lambda_{2}&0\\\
\Lambda_{3}^{*}\Lambda_{2}&0&\Lambda_{1}\Lambda_{3}\\\
0&-\Lambda_{1}\Lambda_{3}^{*}&0\\\ \end{array}\right)=\\\ \\\
&=&-\displaystyle{\frac{\lambda^{4}x}{2}}\left(\begin{array}[]{ccc}0&-\lambda
ye^{\imath\delta}&0\\\ \lambda ye^{-\imath\delta}&0&e^{\imath\delta}\\\
0&e^{-\imath\delta}&0\\\ \end{array}\right)\end{array}$ (20)
Neglecting the matrix $\widehat{C}$, which is of the order $o(\lambda^{4})$,
we find that the CKM matrix can be expressed as
$\widehat{V}\cong e^{\widehat{A}_{2}}e^{\widehat{A}_{1}}$ (21)
With
$e^{\widehat{A}_{2}}=\widehat{V}_{2}=\left(\begin{array}[]{ccc}\cos(|\Lambda_{3}|)&0&\frac{\Lambda_{3}}{|\Lambda_{3}|}\sin(|\Lambda_{3}|)\\\
0&1&0\\\
-\frac{\Lambda_{3}^{*}}{|\Lambda_{3}|}\sin(|\Lambda_{3}|)&0&\cos(|\Lambda_{3}|)\\\
\end{array}\right)$ (22)
and the use of the Cayley Hamilton theorem allows the following (exact) form
of the second exponential
$e^{\widehat{A}_{1}}=\widehat{V}_{1}=C_{0}\widehat{\mathbb{1}}+C_{1}\widehat{A}_{1}+C_{2}\widehat{A}_{1}^{2}$
$\left(\begin{array}[]{c}C_{0}\\\ \\\ C_{1}\\\ \\\ C_{2}\\\
\end{array}\right)=\left(\begin{array}[]{ccc}|\Lambda_{1,2}|^{2}&0&0\\\ \\\
-\Upsilon&\Upsilon&\Upsilon\\\ \\\ \imath&1-\imath&-(1+\imath)\\\
\end{array}\right)\left(\begin{array}[]{c}\displaystyle{\frac{1}{|\Lambda_{1,2}|^{2}}}\\\
\\\ \displaystyle{\frac{e^{\imath|\Lambda_{1,2}|}}{2|\Lambda_{1,2}|^{2}}}\\\
\\\ \displaystyle{\frac{e^{-\imath|\Lambda_{1,2}}|}{2|\Lambda_{1,2}|^{2}}}\\\
\end{array}\right)$ (23)
$\Upsilon=|\Lambda_{1,2}|(1+\imath),\;\;|\Lambda_{1,2}|=\sqrt{\Lambda_{1}^{2}+\Lambda_{2}^{2}}$
The above formulae are a restatement of the tentative geometrical picture of
Fig. (2). The na ve disentanglement has reduced the CKM generation to the
product of two matrices, $\widehat{V}_{1}$ accounting for the mixing, induced
by the vector $\overrightarrow{\Omega}_{1,3}$ , and $\widehat{V}_{2}$
specifying a complex rotation, responsible for the CP violating contributions.
The matrix (21) is an approximation of the exponential form at the order
$o(\lambda^{4})$, but it is not equivalent to Wolfenstein matrix. The matrix
(21), albeit an approximation, since we have neglected higher order
commutators, is unitary at any order in the coupling parameter, while
$\widehat{V}_{W}\widehat{V}_{W}^{{\dagger}}=\widehat{\mathbb{1}}+o(\lambda^{4})$
(where $\widehat{V}_{W}$ is the matrix (6)).
We have stressed that the simple picture in terms of Euler rotation is
hampered by the presence of a complex term, the $\widehat{V}$ matrix cannot be
written in terms of the generators of rotations and indeed we find
$\begin{array}[]{rl}\widehat{V}=&e^{-\imath(\lambda\widehat{R}_{1}+y\lambda^{2}\widehat{R}_{2}+x\lambda^{3}\widehat{T})}\\\
\\\ \widehat{T}=&-\imath\left(\begin{array}[]{ccc}0&0&e^{\imath\delta}\\\
&0&0\\\ -e^{-\imath\delta}&0&0\\\ \end{array}\right)\end{array}$ (24)
The $\widehat{T}$ matrix does not belong to SO(3) and the quark mixing matrix,
written as the product of the exponential matrix correct up to the order
$o(\lambda^{4})$ is
$\widehat{V}=e^{-\imath x\lambda^{3}\widehat{T}}\;e^{-\imath
y\lambda^{2}\widehat{R}_{2}}\;e^{-\imath\lambda^{3}\widehat{R}_{1}}+o(\lambda^{4})$
$\begin{array}[]{rl}\widehat{V}\cong&e^{\left(\begin{array}[]{ccc}0&0&x\lambda^{3}e^{\imath\delta}\\\
0&0&0\\\ -x\lambda^{3}e^{-\imath\delta}&0&0\\\ \end{array}\right)}\\\ \\\
&\;\;\;\;\;e^{\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&y\lambda^{2}\\\
0&-y\lambda^{2}&0\\\
\end{array}\right)}e^{\left(\begin{array}[]{ccc}0&\lambda&0\\\ -\lambda&0&0\\\
0&0&0\\\ \end{array}\right)}=\\\ \\\
=&{\left(\begin{array}[]{ccc}C(x\lambda^{3})&0&e^{\imath\delta}S(x\lambda^{3})\\\
0&1&0\\\ -e^{-\imath\delta}S(x\lambda^{3})&0&C(x\lambda^{3})\\\
\end{array}\right)}\\\ \\\ &\;\;\;{\left(\begin{array}[]{ccc}1&0&0\\\
0&C(y\lambda^{2})&S(y\lambda^{2})\\\ 0&-S(y\lambda^{2})&C(y\lambda^{2})\\\
\end{array}\right)}{\left(\begin{array}[]{ccc}C(\lambda)&S(\lambda)&0\\\
-S(\lambda)&C(\lambda)&0\\\ 0&0&1\\\ \end{array}\right)}\\\ \\\
&C(\phi)=\cos(\phi),\;S(\phi)=\sin(\phi)\end{array}$ (25)
and displays the largely well-known feature that the mixing angles are
proportional to the Cabibbo coupling parameter according to
$\left.\begin{array}[]{ccc}\vartheta_{1,3}&\propto&\lambda^{3}\cong\displaystyle{\sqrt{\frac{m_{d}}{m_{b}}}}\\\
\\\
\vartheta_{2,3}&\propto&\lambda^{2}\cong\displaystyle{\sqrt{\frac{m_{s}}{m_{b}}}}\\\
\\\
\vartheta_{1,2}&\propto&\lambda\cong\displaystyle{\sqrt{\frac{m_{d}}{m_{s}}}}\\\
\\\ \end{array}\right.$ (26)
Furthermore, in full agreement with the particle data group paradigm, we get
$\begin{array}[]{l}e^{\left(\begin{array}[]{ccc}0&0&x\lambda^{3}e^{\imath\delta}\\\
0&0&0\\\ -x\lambda^{3}e^{-\imath\delta}&0&0\\\
\end{array}\right)}=\widehat{U}_{\delta}^{{\dagger}}e^{-\imath
x\lambda^{3}\widehat{R}_{3}}\widehat{U}_{\delta}\\\ \\\
\widehat{U}_{\delta}=e^{\imath\widehat{\delta}},\;\;\mbox{with}\;\;\widehat{\delta}=\left(\begin{array}[]{ccc}0&0&0\\\
0&0&0\\\ 0&0&\delta\\\
\end{array}\right),\;\widehat{R}_{3}=\imath\left(\begin{array}[]{ccc}0&0&1\\\
0&0&0\\\ -1&0&0\\\ \end{array}\right)\end{array}$ (27)
It is also interesting to note that the $\widehat{T}$ matrix can be written as
$\begin{array}[]{l}\widehat{T}=-\imath\cos(\delta)\widehat{R}_{3}+\imath\sin(\delta)\widehat{S}_{3}\\\
\\\ \widehat{S}_{3}=\left(\begin{array}[]{ccc}0&0&1\\\ 0&0&0\\\ 1&0&0\\\
\end{array}\right)\end{array}$ (28)
The na ve disentanglement (order $o(\lambda^{4})$)
$\begin{array}[]{l}\widehat{V}\cong e^{-\imath\widehat{R}}e^{\imath
x\lambda^{3}\sin(\delta)\widehat{S}_{3}}\\\ \\\
\widehat{R}=\lambda\widehat{R}_{1}+y\lambda^{2}\widehat{R}_{2}+x\lambda^{3}\cos(\delta)\widehat{R}_{3}\end{array}$
(29)
Corresponds to the product of two matrices, namely
$\begin{array}[]{l}\widehat{V}\cong\widehat{V}_{R}\widehat{V}_{I}\\\ \\\
e^{\widehat{R}}=\widehat{V}_{R}\\\ \\\
\widehat{V}_{I}=\left(\begin{array}[]{ccc}C(x\lambda^{3}\sin(\delta))&0&\imath
S(x\lambda^{3}\sin(\delta))\\\ 0&1&0\\\ \imath
S(x\lambda^{3}\sin(\delta))&0&C(x\lambda^{3}\sin(\delta))\\\
\end{array}\right)=\\\ \\\
\;\;\;\;=\left(\begin{array}[]{ccc}C(A\lambda^{3}\eta)&0&-\imath
S(A\lambda^{3}\eta)\\\ 0&1&0\\\ -\imath
S(A\lambda^{3}\eta)&0&C(A\lambda^{3}\eta)\\\ \end{array}\right)\end{array}$
(30)
and $\widehat{V}_{R}$ can be written as Eq. 13 with
$\overrightarrow{\Omega}\equiv\lambda(-y\lambda,x\lambda^{2}\cos(\delta),-1)$
(31)
The matrix $\widehat{V}_{I}$ mixes the first and third quark generation mass
eigenstates and is responsible for the CP violation. It is a pseudo rotation
matrix and is generated by a matrix whose determinant is the Jarlskog
invariant Jarlskog , discussed in the forthcoming section.
We have so far shown that the exponential parameterization implicitly contains
Wolfenstein and Euler type forms, in the following sections we will dwell on
its further advantages.
## IV The Cayley Hamilton Theorem and The Quark mixing matrix
The exponential matrix (1) can be treated in different ways.
We have already shown that the use of a Taylor expansion leads to a
Wolfenstein form, which preserves the unitarity of $\widehat{V}$ at the
expansion order (the mixing matrix in Eq. (6) is unitary at the order
$o(\lambda^{4})$).
The method of the exponential disentanglement can be used too and such a
procedure allows an interesting geometrical picture of the mixing dynamics and
albeit an approximation in the Cabibbo coupling parameter, the mixing matrix
written as in Eq. (17) preserves the unitarity at any order in $\lambda$, as
discussed more accurately in the concluding remarks.
The matrix $\widehat{V}$ can, however, be written in an exact form using the
Cayley Hamilton theorem, by setting
$\widehat{V}=C_{0}\widehat{\mathbb{1}}+C_{1}\widehat{A}+C_{2}\widehat{A}^{2}$
(32)
where
$e^{\varepsilon_{j}}=C_{0}+\varepsilon_{j}C_{1}+\varepsilon_{j}^{2}C_{2},\;\;\mbox{with}\;\;j=1,2,3$
(33)
With $\varepsilon_{j}$ being the roots associated with the characteristic
equation of the matrix $\widehat{A}$, namely
$\varepsilon_{j}^{3}+|\overrightarrow{\Omega}|^{2}\varepsilon_{j}+\imath\triangle=0$
(34)
where
$\left.\begin{array}[]{l}\triangle=2xy\lambda^{6}\sin(\delta)=-2A\eta\lambda^{6}\\\
\\\
|\overrightarrow{\Omega}|=\lambda\sqrt{1+y^{2}\lambda^{2}+x^{2}\lambda^{4}}=\\\
\\\
=\lambda\sqrt{1+(A\lambda)^{2}+(A^{2}\lambda^{4})\left[\left(\rho-\frac{1}{2}\right)^{2}+\eta^{2}\right]}\end{array}\right.$
(35)
$\imath\triangle$ is the determinant of the matrix $\widehat{A}$.
A little bit of algebra yields to define the $C_{i}\;\;(i=0,1,2)$ coefficients
as the product of two matrix
$\left(\begin{array}[]{c}C_{0}\\\ \\\ C_{1}\\\ \\\
C_{2}\end{array}\right)=\left(\begin{array}[]{ccc}\varepsilon_{2}\varepsilon_{3}&\varepsilon_{1}\varepsilon_{3}&\varepsilon_{1}\varepsilon_{2}\\\
\\\
-(\varepsilon_{2}+\varepsilon_{3})&-(\varepsilon_{1}+\varepsilon_{3})&-(\varepsilon_{1}+\varepsilon_{2})\\\
\\\ 1&1&1\end{array}\right)\\\
\left(\begin{array}[]{c}\displaystyle{\frac{e^{\varepsilon_{1}}}{(\varepsilon_{2}-\varepsilon_{1})(\varepsilon_{3}-\varepsilon_{1})}}\\\
\\\
\displaystyle{\frac{e^{\varepsilon_{2}}}{(\varepsilon_{1}-\varepsilon_{2})(\varepsilon_{3}-\varepsilon_{2})}}\\\
\\\
\displaystyle{\frac{e^{\varepsilon_{3}}}{(\varepsilon_{1}-\varepsilon_{3})(\varepsilon_{2}-\varepsilon_{3})}}\end{array}\right)$
(36)
Eq. (23) (along with Eqs. (27)) is the most general form of the quark mixing
matrix which can be derived from an exponential parameterization, it is exact
but not easy to remember.
Let us now give an idea of the orders of the numerical values characterizing
the various quantities entering the above equations. The use of the previously
quoted values for the Wolfenstein parameters lead to the following evaluations
for the solution of Eq. (34)
$\begin{array}[]{l}\varepsilon_{1}\cong-0.23171\imath\\\ \\\
\varepsilon_{2}\cong 0.00117\imath\cdot\\\ \\\ \varepsilon_{2}\cong
0.23054\imath\cdot\end{array}$ (37)
It is worth stressing that the matrix $\widehat{D}$ provides the diagonal
forms of either $\widehat{V}$ and $\widehat{A}$. It follows therefore that the
two matrices have the same eigenvectors. They can be determined using
$\widehat{A}$ instead of $\widehat{V}$, because the procedure is significantly
simpler. We find that the eigenvalues are in the form
$|j>=\left(\begin{array}[]{c}1\\\ \\\
-\varepsilon_{j}-xy\lambda^{5}e^{-\imath\delta}\\\ \\\
-y\lambda^{3}-\varepsilon_{j}x\lambda^{3}e^{-\imath\delta}\\\ \\\
\end{array}\right)$ (38)
$\widehat{V}=\left(\begin{array}[]{cc}C(\lambda)+\frac{A^{2}\lambda^{6}}{4!}\Phi&S(\lambda)-\frac{A^{2}\lambda^{5}}{2}(\Pi^{*}-\frac{1}{6})\\\
\\\
-S(\lambda)-\frac{A^{2}\lambda^{5}}{2}(\Pi-\frac{5}{6})&C(\lambda)-(A\lambda^{2})^{2}\left[\frac{1}{2}-\frac{\lambda^{2}}{3}\left(\frac{1}{4}-\imath\eta\right)\right]\\\
\\\
-A\lambda^{3}\left[\left(\Pi-1\right)-\frac{\lambda^{2}}{6}\left(\Pi-\frac{3}{4}\right)\right]&-S(A\lambda^{2})+\frac{A\lambda^{4}}{4!}\left(\lambda^{2}(\Pi-\frac{7}{10})-12(\Pi-\frac{2}{3})\right)\end{array}\right.\\\
\left.\begin{array}[]{c}A\lambda^{3}\left[\Pi^{*}-\frac{\lambda^{2}}{6}\left(\Pi^{*}-\frac{1}{4}\right)\right]\\\
\\\
S(A\lambda^{2})+\frac{A\lambda^{4}}{4!}\left(\lambda^{2}(\Pi^{*}-\frac{3}{10})-12(\Pi^{*}-\frac{1}{3})\right)\\\
\\\ 1-\frac{(A\lambda^{2})^{2}}{2}+\frac{A^{2}\lambda^{6}}{4!}\Phi\\\ \\\
\end{array}\right)$ (39)
$\begin{array}[]{l}\Phi=-12(\rho^{2}+\eta^{2})-8\imath\eta+12\rho-2;\;\;\Pi=\rho+\imath\eta;\;\;\Pi^{*}=\rho-\imath\eta\end{array}$
It is worth mentioning the companion matrix associated with the characteristic
equation (30) Companionmatrix , which writes
$C_{A}=\left(\begin{array}[]{ccc}0&\;\;0&-\varepsilon_{1}\varepsilon_{2}\varepsilon_{3}\\\
\\\
1&\;\;0&-(\varepsilon_{1}\varepsilon_{2}+\varepsilon_{2}\varepsilon_{3}+\varepsilon_{1}\varepsilon_{3})\\\
\\\ 0&\;\;1&-(\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3})\\\ \\\
\end{array}\right)$ (40)
It is accordingly expressed in terms of three invariants444A $3\times 3$
matrix has three invariants given by its determinant, its trace and by the sum
of the determinants of its minors, namely
$\left.\begin{array}[]{l}\displaystyle\varepsilon_{1}\varepsilon_{2}\varepsilon_{3}=-2\imath
x\lambda^{6}\sin(\delta)\\\ \\\
\varepsilon_{1}^{2}+\varepsilon_{2}^{2}+\varepsilon_{3}^{2}=|\overrightarrow{\Omega}|^{2}\\\
\\\ \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}=0\end{array}\right.$ (41)
the first of which is the Jarlskog invariant, a measure of the amount of CP
violations, emerging in quite a natural way in the present analysis.
## V Concluding remarks
We have shown that the exponential parameterization interpolates between
Wolfestein and Euler like forms and could provide a useful and flexible tool
of analysis. Its approximations in terms of the Cabibbo coupling can be either
expressed as Taylor expansions or as unitarity preserving forms bassed on the
Zassenhaus formula.
The Taylor expansion does not meet too much aesthetical criteria, but it can
usefully be exploited to get higher order approximations of Wolfenstein type
parameterizations an example is shown below, where we report the na ve
expansion of the exponential matrix up to the order $o(\lambda^{7})$.
We have reported the matrix (39) (where $C(\lambda),\;S(\lambda)$ denote the
expansion of cosine and sine up to the order $o(\lambda^{7})$) for comparison
purposes with other forms available in literature. The accuracy of this last
matrix is one part over $10^{9}$ and can therefore considered exact for any
expansion purposes.
The extension of the CKM matrix to higher dimensions by the use of the
exponential matrix method is not complicated. In the case of four quark
generations, we define the matrix containing 2 CP violating phases, appended
to the smallest coupling terms. We have furthermore assumed that the coupling
strengths to the fourth family be of the order $\lambda^{3+n},\;\;n=1,2,3$.
$A=\left(\begin{array}[]{cccc}0&\lambda&e^{\imath\delta_{1}}x\lambda^{3}&e^{\imath\delta_{2}}z\lambda^{6}\\\
-\lambda&0&y\lambda^{2}&p\lambda^{5}\\\
-e^{-\imath\delta_{1}}x\lambda^{3}&-y\lambda^{2}&0&u\lambda^{4}\\\
-e^{-\imath\delta_{2}}z\lambda^{6}&-p\lambda^{5}&-u\lambda^{4}&0\end{array}\right)$
(42)
The relevant Wolfenstein like approximation of the mixing matrix is reported
in (43).
$\widehat{V}=\left(\begin{array}[]{cccc}C(\lambda)+\frac{A^{2}\lambda^{6}}{4!}\Phi&S(\lambda)-\frac{A^{2}\lambda^{5}}{2}(\Pi_{1}^{*}-\frac{1}{6})\\\
\\\
-S(\lambda)-\frac{A^{2}\lambda^{5}}{2}(\Pi_{1}-\frac{5}{6})&C(\lambda)-(A\lambda^{2})^{2}\left[\frac{1}{2}-\frac{\lambda^{2}}{3}\left(\frac{1}{4}-\imath\eta\right)\right]\\\
\\\
-A\lambda^{3}\left[\left(\Pi_{1}-1\right)-\frac{\lambda^{2}}{6}\left(\Pi_{1}-\frac{3}{4}\right)\right]&-S(A\lambda^{2})+\frac{A\lambda^{4}}{4!}\left(\lambda^{2}(\Pi_{1}-\frac{7}{10})-12(\Pi_{1}-\frac{2}{3})\right)\\\
\\\
-\lambda^{6}p(\Pi_{2}-1)&-\lambda^{5}\left(-p+Au\frac{\lambda}{2}\right)\end{array}\right.\\\
\\\
\left.\begin{array}[]{cc}A\lambda^{3}\left[\Pi^{*}_{1}-\frac{\lambda^{2}}{6}\left(\Pi^{*}_{1}-\frac{1}{4}\right)\right]&-\Pi^{*}_{2}p\lambda^{6}\\\
\\\
S(A\lambda^{2})+\frac{A\lambda^{4}}{4!}\left(\lambda^{2}(\Pi^{*}_{1}-\frac{3}{10})-12(\Pi^{*}_{1}-\frac{1}{3})\right)&\lambda^{5}\left(p+Au\frac{\lambda}{2}\right)\\\
\\\
1-\frac{(A\lambda^{2})^{2}}{2}+\frac{A^{2}\lambda^{6}}{4!}\Phi&u\lambda^{4}\\\
\\\ -u\lambda^{4}&1\end{array}\right)$ (43)
$\begin{array}[]{l}\Phi_{1}=\rho_{1}+\imath\eta_{1}\;\;\mbox{with}\;\;\displaystyle{\rho_{1}=\frac{x}{y}\cos(\delta_{1})+\frac{1}{2},\;\;\eta_{1}=-\frac{x}{y}\sin(\delta_{1})};\;\;\Phi_{2}=\rho_{2}+\imath\eta_{2}\;\;\mbox{with}\;\;\displaystyle{\rho_{2}=\frac{z}{p}\cos(\delta_{2})+\frac{1}{2},\;\;\eta_{2}=-\frac{z}{p}\sin(\delta_{2})}\\\
\Phi_{j}^{*}=\rho_{j}-\imath\eta_{j}\;\;j=1,2\end{array}$
Furtheremore the invariants (4 for a $4\times 4$ matrix), obtained directly
from (42) read
$\begin{array}[]{rl}J_{2}=&\lambda^{2}f(1,y\lambda,x\lambda^{2})+\lambda^{8}f(u,p\lambda,z\lambda^{2})\\\
&\mbox{where}\;\;f(a,b,c)=a^{2}+b^{2}+c^{2}\\\ \\\
J_{3}=&2\imath\lambda^{6}\left[xy\sin(\delta_{1})+zp\lambda^{6}\sin(\delta_{2})\right]=\\\
=&xy\lambda^{6}(e^{\imath\delta_{1}}-e^{-\imath\delta_{1}})+zp\lambda^{12}(e^{\imath\delta_{2}}-e^{-\imath\delta_{2}})\\\
\\\
J_{4}=&u^{2}\lambda^{10}+[2uyz\cos(\delta_{2})-2pux\cos(\delta_{1})]\lambda^{13}+\\\
&[p^{2}x^{2}-px^{2}ye^{\imath(\delta_{2}-\delta_{1})}-pxyze^{-\imath(\delta_{2}-\delta_{1})}+xy^{2}z]\lambda^{16}\end{array}$
the first invariant, associated with the trace of A, is zero.
It is evident that the $J_{2}$ and $J_{3}$ invariants are just a
generalization of those reported in Eq. (32) while the fourth is completely
new being associated to the full determinant of the matrix. We have reported
this example to show the flexibility of the method it is however evident that
the detection of CP violating effects due to the new phase require an accuracy
at least of the order $\lambda^{6}$.
Before concluding the paper we will address the problems associated with the
exponential forms of the neutrino mixing matrix, which have also been
discussed in Dattoli3 , where the lepton-quark complementarity Minakata has
been reformulated by noting that the relevant rotation occur around axes
forming an angle of $45^{0}$. The present experimental data seem to favor the
tribimaximal (TBM) form Harrison therefore the neutrino mixing matrix reads
$\widehat{U}=\left(\begin{array}[]{ccc}\displaystyle\sqrt{\frac{2}{3}}&\displaystyle\frac{1}{\sqrt{3}}&0\\\
\\\
\displaystyle-\frac{1}{\sqrt{6}}&\displaystyle\frac{1}{\sqrt{3}}&\displaystyle\frac{1}{\sqrt{2}}\\\
\\\
\displaystyle\frac{1}{\sqrt{6}}&\displaystyle-\frac{1}{\sqrt{3}}&\displaystyle\frac{1}{\sqrt{2}}\\\
\\\ \end{array}\right)$ (44)
If we assume that also this form is generated by an exponential matrix (with
all real entries) according to
$\widehat{U}=e^{\widehat{B}};\;\;\;\widehat{B}=\left(\begin{array}[]{ccc}0&\;\;\alpha&\;\;\beta\\\
\\\ -\alpha&\;\;0&\;\;\gamma\\\ \\\ -\beta&\;\;-\gamma&\;\;0\\\ \\\
\end{array}\right)$ (45)
We obtain the following correspondence between the entries of the
$\widehat{B}$ matrix and those of the TBM form
$\begin{array}[]{l}\widehat{B}=\alpha\left(\begin{array}[]{ccc}0&\;\;1&\;\;\displaystyle-\frac{1}{\sqrt{2}+1}\\\
\\\ -1&\;\;0&\;\;\frac{\sqrt{3}+\sqrt{2}}{\sqrt{2}+1}\\\ \\\
\frac{1}{\sqrt{2}+1}&\;\;-\frac{\sqrt{3}+\sqrt{2}}{\sqrt{2}+1}&\;\;0\\\ \\\
\end{array}\right)\\\ \\\
\alpha=\displaystyle{2\frac{\sqrt{2\sqrt{2}+3}}{\sqrt{2\sqrt{2}+2\sqrt{6}+9}}\frac{1}{\sin\left(\frac{\sqrt{3-\left(\sqrt{\frac{2}{3}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{2}}\right)}}{2}\right)}}\end{array}$
(46)
The values of the entries of the TBM matrix do not allow the interpretation of
the neutrino mixing matrix as an expansion around the unit, notwithstanding it
is possible to get a better agreement with experimental by making an
appropriate expansions around the matrix $\widehat{B}$ and then around the
TBM, as it will be shown in a dedicated paper.
In this paper we have provided an extensive account of the possibilities
offered by the exponential form of the CKM matrix, which looks like a
prototype from which all the other forms can be derived, we hope that our
suggestions provide a useful tool in the relevant applications.
###### Acknowledgements.
The authors are deeply indebted to Dr. D. Babusci for stimulating discussions
and comments during any stage of the paper.
## References
* (1) M. Kobayashi, T. Maskawa, Prog. Theor. Phys. 49, 652 (1973)
* (2) L.L. Chau, W. Y. Keung, Phys. Rev. Lett. 53, 1802 (1984)
* (3) L. L. Wolfenstein, Phys. Rev. Lett. 51, 1945 (1983)
* (4) N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963)
* (5) G. Dattoli, K. Zhukowky, Eur. Phys. J. C. 50, 817 (2007) and references therein for earlier works on this subject
* (6) G. Dattoli, K. Zhukowsky, Eur. Phys. C 52, 591 (2007)
* (7) Particle Data Group, W. M. et al. J. Phys. G, Nucl. Part. Phys. 331, 1, (2005)
* (8) D. Babusci, G. Dattoli and M. Del Franco, Lectures on Mathematical Methods For Physics. Thecnical Report 58 ENEA (2010)
* (9) D. Babusci, G. Dattoli and E. Sabia, J. Math. Phys. 3 , P110601(2011)
* (10) W. Magnus, Commun. Pure Appl. Math. 7, 649 (1954)
F. Cassas, A. Murua and Mladen Nadinic, Efficient Computation of the
Zassenhaus formula, [arXiv:math-ph/1204.0389v2], 15 June 2012
* (11) C. Jarlskog, Phys. Rev. Lett. 55, 1039 (1985)
* (12) K. Fujii, H. Oike,[arXiv:quant-ph/0604115vi]
* (13) G. Dattoli, K. Zhukowsky, Eur. Phys. C 55, 547 (2008)
* (14) H. Minakata, A. Y. Smirnov, Phys. Rev. D 70, 073009 (2004)
M. Raidal, Phys. Rev. Lett. 93, 161801 (2004)
* (15) P. F. Harrison, D. H. Perkins and W. G. Scott, Physics Letters B 530: 167 (2002) [arXiv:hep-ph/0202074]
|
arxiv-papers
| 2013-01-22T08:56:28 |
2024-09-04T02:49:40.542941
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "G. Dattoli and E. Di Palma",
"submitter": "Emanuele Di Palma",
"url": "https://arxiv.org/abs/1301.5111"
}
|
1301.5147
|
# A real open book not fillable by a real Lefschetz fibration
Feri̇t Öztürk Boğazi̇çi̇ Üni̇versi̇tesi̇, Department of Mathematics, TR-34342
Bebek, İstanbul, Turkey [email protected] and Nermi̇n Salepci̇
Institut Camille Jordan, Université Lyon I, 43, Boulevard du 11 Novembre 1918
69622 Villeurbanne Cedex, France [email protected]
###### Abstract.
A real 3- or 4-manifold has by definition an orientation preserving smooth
involution acting on it. We consider Lefschetz fibrations of 4-dimensional
manifolds-with-boundary and open book decompositions on their boundary in the
existence of a real structure. We prove that there is a real open book which
cannot be filled by a real Lefschetz fibration, although it is filled by non-
real Lefschetz fibrations.
## 1\. Introduction and basic definitions
There is a very close relation between open book decompositions and Lefschetz
fibrations over $D^{2}$, with fibers surfaces with boundary. It is known that
any open book decomposition is filled by a Lefschetz fibration if its
monodromy can be factorized as a product of positive Dehn twists (work of
Eliashberg, Giroux, Gompf, Loi-Piergallini, Akbulut-Ozbagci etc. See e.g.
[4]). Moreover, two fibrations filling the same open book are isomorphic if
and only if the two such factorizations are Hurwitz equivalent [3].
In the present work we consider Lefschetz fibrations and open book
decompositions on manifolds with a real structure. A real structure on an
oriented $4$-manifold (respectively $3$-manifold) with boundary (possibly
empty) is defined to be an involution which is orientation preserving and the
fixed point set of which is of dimension $2$ (respectively $1$), if it is not
empty. Hence, if $M$ is the oriented boundary of an oriented $4$-manifold $X$,
a real structure on $X$ restricts to a real structure on $M$. We call a
manifold $X$ together with a real structure a real manifold and denote by
$c_{X}$ the real structure on it.
Let $(L,\pi)$ be an open book decomposition of a real 3-manifold $(M,c_{M})$.
Here $L\subset M$ is the binding and $\pi:M-L\rightarrow S^{1}$ is a fibration
with fibers the page surfaces. We say that $(L,\pi)$ is a real open book
decomposition, if $\rho\circ\pi=\pi\circ c_{M}|_{M-L}$ where $\rho:S^{1}\to
S^{1}$ is a reflection [6].
Let $p:E\to B$ be a genus-$g$ Lefschetz fibration of a 4-manifold $E$. A _real
Lefschetz fibration_ is a Lefschetz fibration together with a pair of real
structures $c_{E}:E\to E$ and $c_{B}:B\to B$ commuting with the fiber
structure [8].
In this article we answer a natural question that relates real open books and
real Lefschetz fibrations. We show that there is a Lefschetz fibration that
does not admit a real structure, while the canonical open book on its boundary
is real. We also show that this real open book cannot be filled by any real
Lefschetz fibration with the same fiber topology and with arbitrary number of
singular fibers (Theorem 6).
Acknowledgements. The second author would like to thank A. Degtyarev for
helpful discussions. The work in this article is supported by the Scientific
and Technological Research Council of Turkey [TUBITAK-ARDEB-109T671]. The
second author is supported by the European Community’s Seventh Framework
Programme ([FP7/2007-2013] [FP7/2007-2011]) under grant agreement no [258204].
## 2\. Construction
Let $E$ be an oriented smooth 4-manifold and $B$ an oriented smooth surface. A
_genus- $g$ Lefschetz fibration_ of $E$ is a proper smooth projection $p:E\to
B$ such that $p$ has only finitely many critical points in int($E$) with
pairwise distinct images around which one can choose complex charts such that
the projection takes the form $(z_{1},z_{2})\to z_{1}^{2}+z_{2}^{2}$.
Moreover, the inverse image of a regular value is a closed oriented smooth
surface of genus $g$. It follows from the definition that if $\partial
B\neq\varnothing$, then $\partial E=p^{-1}(\partial B)$ is a fiber bundle over
$\partial B$. The notion of Lefschetz fibration can be slightly generalized to
cover the case of fibers with boundary. Then $E$ turns into a manifold with
corners and its boundary, $\partial E$, becomes naturally divided into two
parts: $p^{-1}(\partial B)$ and $\partial F\times B$. In this case, if, in
particular, $B=D^{2}$, then $\partial E$ admits a canonical open book
decomposition with binding $p^{-1}(0)$ and projection $p|_{\partial E}$. A
_real Lefschetz fibration_ is a fibration together with a pair of real
structures $c_{E}:E\to E$ and $c_{B}:B\to B$ commuting with the fiber
structure.
It is known that around a singular fiber, Lefschetz fibrations are determined
by the monodromy that is a single positive Dehn twist around a simple closed
curve, the _vanishing cycle_. Algebraically, Lefschetz fibrations over
$B=D^{2}$ can be encoded by the factorization $(t_{a_{1}},\dots,t_{a_{n}})$ of
the monodromy $f=t_{a_{n}}\circ\ldots\circ t_{a_{1}}$ (because of the
composition notation the order is reversed) along $\partial D^{2}$ into a
product of positive Dehn twists considered up to Hurwitz equivalence. Namely,
two such factorizations are called _Hurwitz equivalent_ if one can get from
one factorization to the other by a finite sequence of Hurwitz moves:
$\begin{array}[]{lcr}(\ldots,t_{a_{i}},t_{a_{i+1}},\ldots)\to(\ldots,t_{a_{i}}^{-1}\circ
t_{a_{i+1}}\circ t_{a_{i}},t_{a_{i}},\ldots),\\\
(\ldots,t_{a_{i}},t_{a_{i+1}},\ldots)\to(\ldots,t_{a_{i+1}},t_{a_{i+1}}\circ
t_{a_{i}}\circ t_{a_{i+1}}^{-1},\ldots);\end{array}$
and possibly a global conjugation.
In what follows, we consider a genus-$1$ Lefschetz fibration $p:X\to D^{2}$
with exactly two singular fibers. Choose a base point $d\in\partial D^{2}$ and
consider a basis $(\gamma_{1},\gamma_{2})$ of
$\pi_{1}(D^{2}\setminus\\{\mbox{critical values}\\},d)$ obtained by connecting
the base point $d$ to the positively oriented simple loops, each surrounding
the corresponding critical value once. Fix an identification of
$F_{d}=p^{-1}(d)$ with $T^{2}=\mathbb{R}^{2}/\mathbb{Z}^{2}$. Denote by $a$,
the class on $T^{2}$ of $(1,0)\in\mathbb{R}^{2}$ and by $b$, the class of
$(0,1)$ and consider the curves $u,v$ represented respectively by $3a+5b$ and
$a$ on $T^{2}$. We assume that the monodromy along $\gamma_{1}$ and
$\gamma_{2}$ are given respectively by the positive Dehn twists $t_{u}$ and
$t_{v}$. In other words, the corresponding singular fibers are obtained from
$T^{2}$ by pinching the curves $u$ and $v$. The total monodromy of the
fibration, thus, becomes the composition $f=t_{v}\circ t_{u}$.
Recall that $f\mapsto f_{*}$ defines an isomorphism from the mapping class
group Map($T^{2}$) (the identity component of the space of diffeomorphisms) of
the torus to the group of automorphisms of
$H_{1}(T^{2},\mathbb{Z})\cong\mathbb{Z}a\oplus\mathbb{Z}b$. The latter is
isomorphic to
$\textup{SL(2,\,$\mathbb{Z}$)}=\left\\{[t_{a}]=\small{\left[\begin{array}[]{rr}1&1\\\
0&1\end{array}\right]},[t_{b}]=\small{\left[\begin{array}[]{rr}1&0\\\
-1&1\end{array}\right]}:\begin{array}[]{ll}[t_{a}][t_{b}][t_{a}]=[t_{b}][t_{a}][t_{b}]\\\
([t_{a}][t_{b}])^{6}=\textrm{id}\end{array}\right\\}$
where $[t_{\circ}]$ refers to the matrix representation of the automorphism
$t_{\circ*}$. With respect to the above presentation, $[f]=[t_{v}][t_{u}]$ is
given by the matrix $\small{\left[\begin{array}[]{rr}-39&25\\\
-25&16\end{array}\right]}$.
###### Proposition 1.
$p:X\to D^{2}$ does not admit a real structure.
Proof: Suppose that $p:X\to D^{2}$ admits a real structure. That is to say,
there exist real structures $c_{X}:X\to X$ and $c_{D^{2}}:D^{2}\to D^{2}$ such
that $p\circ c_{X}=c_{D^{2}}\circ p$. Therefore, the critical points as well
as their images are invariant under the action of the real structures $c_{X}$
and $c_{D^{2}}$, respectively, so either they are both real or they are
interchanged by the real structures. By definition of real Lefschetz
fibrations, the decomposition $(t_{u},t_{v})$ associated to
$(\gamma_{1},\gamma_{2})$ is Hurwitz equivalent to a decomposition
$(t_{c_{X}(v)},t_{c_{X}(u)})$ associated to
$(c_{D^{2}}(\gamma_{2}),c_{D^{2}}(\gamma_{1}))$. Although, the positions of
$(\gamma_{1},\gamma_{2})$ can be arbitrary with respect to the real structure
$c_{D^{2}}$, by conjugation by a power of $f$ (such a conjugation preserves
Hurwitz classes) and by moving, if necessary, the base point on the boundary,
we can assume that we have only the two positions shown in Figure 1. In the
former case (the case shown on the left in Figure 1), the fiber $F_{d}$ can be
endowed with a real structure $c$ by pulling a real structure on a real fiber
between the two real singular fibers. Up to isotopy, we can assume that
vanishing cycles $u$, $v$ are invariant under the action of the real structure
$c$. Lemma 2 concerns the intersection number of invariant curves and
prohibits this case, (since the intersection number of $u$ and $v$ is equal to
5).
Figure 1. Two possible positions of $(\gamma_{1},\gamma_{2})$ on base $D^{2}$
regarding the real structure reflection with respect to horizontal.
Now assume that critical points are exchanged by the real structure (this is
the case depicted on the right in Figure 1). Let $c:T^{2}\to T^{2}$ be the
real structure conjugate to the inherited real structure on the base fiber
$F_{d}$. Then, by the assumption we have $c(u)=v$; as a consequence, $u+v$ and
$u-v$ are elements of $\pm$ eigenspaces $H^{c}_{+}$ and, respectively,
$H^{c}_{-}$ of $c_{*}:H_{1}(T^{2},\mathbb{Z})\to H_{1}(T^{2},\mathbb{Z})$.
Thus, the primitive forms of the classes of $u+v$ and $u-v$ form a basis,
respectively of $H^{c}_{+}$ and $H^{c}_{-}$. Moreover, for any real structure
$c$, the intersection number of the bases of $H_{+}^{c}$ and $H_{-}^{c}$ is at
most 2 which is violated by our choice of $u$ and $v$. $\Box$
###### Lemma 2.
The number of intersection points of two invariant curves on a real torus can
only be 0, 1 or 2.
Proof: The real structure restricted to an invariant curve defines an action
on the curve. This action can be the identity, a reflection or an antipodal
involution; let us call those curves a real curve, a reflection curve and an
antipodal curve respectively. Recall that up to equivariant diffeomorphisms,
the real structure on a torus are distinguished by the number of real
components that can be 0,1 or 2. for each real structure we have
* •
if $c$ has no real components, then there exist two $c$-equivariant isotopy
classes of antipodal curves and no classes of other types;
* •
if $c$ has one real component, then $T^{2}$ contains a unique $c$-equivariant
isotopy class of non-contractible reflection curves, a unique class of
antipodal curves, and a unique real curve;
* •
if $c$ has two real components, then $T^{2}$ contains no $c$-equivariant
isotopy class of antipodal curves, a unique class of reflection curves, and
two classes of real curves.
The result follows from the following observations: a reflection curve has two
real points, so the intersection with a real curve must be at 0, 1 or 2
points. Moreover, a reflection curve must intersect an antipodal curve at 0 or
2 points, and real curves and antipodal curves are disjoint. $\Box$
###### Proposition 3.
Let $M=\partial X$; then $M$ as a surface bundle is real.
Proof: By [7, Proposition 4], it is enough to check whether its monodromy is
real, i.e. it admits a decomposition into a product of two real structures.
Real structures on $T^{2}$ correspond to involutive elements of determinant
$-1$ in GL(2, $\mathbb{Z}$). Here we have
$[f]=\small{\left[\begin{array}[]{rr}-39&25\\\
-25&16\end{array}\right]}=\small{\left[\begin{array}[]{rr}5&-3\\\
8&-5\end{array}\right]\left[\begin{array}[]{rr}-120&77\\\
-187&120\end{array}\right]}$, a product of two involutive elements of
determinant $-1$. $\Box$
###### Remark 1.
In [7], an explicit classification of real elements of SL(2, $\mathbb{Z}$) as
well as a particular decomposition of a product of two real structures for
each conjugacy class is given. According to [7, Proposition 4], a hyperbolic
element, i.e. an element with absolute value of trace greater than $2$, of
SL(2, $\mathbb{Z}$) is real if and only if its _cutting period cycle_
$[a_{1},\ldots,a_{2k}]$ is odd-bipalindromic. For hyperbolic matrices, the
cutting period cycle together with the sign of the trace is a complete
invariant of conjugacy classes and it can be recovered from the trace and the
periodic tail of the continued fraction expansion of the slope of the
eigenvectors. In our specific example the cutting period cycle of $f$ is found
to be $[1313]$ which is odd-bipalindromic, i.e. up to cyclic ordering, it has
two palindromic pieces $1$, $313$ of odd length.
So far we have constructed a $T^{2}$-fibered Lefschetz fibration cannot be
real while the $T^{2}$ bundle at its boundary is real. Now we show that a
similar construction can be cooked up to obtain a real open book at the
boundary.
Denote by $\breve{p}:\breve{X}\to D^{2}$ the Lefschetz fibration with boundary
obtained from $p:X\to D^{2}$ above by taking out a neighborhood of a section.
Note that such a section always exists for genus 1 Lefschetz fibrations [5].
The regular fibers of $\breve{p}$ are tori with one boundary component. The
fibration $\breve{p}$ has no real structure, since otherwise $p$ would have
one. However, we have:
###### Proposition 4.
The canonical open book of $\breve{M}=\partial\breve{X}$ admits a real
structure.
Proof: As a consequence of Lemma 1 in [6] it is enough to show that the
monodromy of $\breve{p}|_{\breve{M}}$ is real. Note that the monodromy of
$\breve{p}|_{\breve{M}}$ is an element of
$\textup{Map}(T^{2}\smallsetminus\textup{int}{D^{2}},\partial)$, the group of
relative isotopy classes of orientation preserving diffeomorphisms which are
identity on the boundary.
The crucial observation is that being real in Map($T^{2}$) is equivalent to
being real in $\textup{Map}(T^{2}\smallsetminus\textup{int}{D^{2}},\partial)$.
Namely, there is a short exact sequence
$0\to\langle
t_{\partial}\rangle\to\textup{Map}(T^{2}\smallsetminus\textup{int}{D^{2}},\partial)\to\textup{Map($T^{2}$)}\to
0$
given by the central extension of Map($T^{2}$), where $t_{\partial}$ is the
Dehn twist along the boundary component. Obviously, images of real elements of
$\textup{Map}(T^{2}\smallsetminus\textup{int}{D^{2}},\partial)$ are real. For
the converse, suppose im$[f]=[c][c^{\prime}]$ for some
$[f]\in\textup{Map}(T^{2}\smallsetminus\textup{int}{D^{2}},\partial)$ and real
structures $[c],[c^{\prime}]\in\textup{GL(2,\,$\mathbb{Z}$)}$. If necessary,
by replacing $c$ by $c\circ g$ and $c^{\prime}$ by $g^{-1}\circ c^{\prime}$
for some diffeomorphism $g$, we can assume that $c$ and hence $c^{\prime}$
leaves $\partial T^{2}$ invariant. We have
$t_{\partial}=c^{\prime}t_{\partial}^{-1}c^{\prime}$. Thus
$t_{\partial}^{-1}c$ is a real structure which we denote by
$c^{\prime\prime}$. As $[f]=[c][c^{\prime}][t_{\partial}]^{k}$, then for some
integer $k$, $[f]=[c][c^{\prime}][c^{\prime}\circ
c^{\prime\prime}]^{k}=[c][c^{\prime\prime}\circ(c^{\prime}\circ
c^{\prime\prime})^{k-1}]$. Being conjugate to a real structure,
$c^{\prime\prime}\circ(c^{\prime}\circ c^{\prime\prime})^{k-1}$ is a real
structure. Thus, $[f]$ is real. $\Box$
With similar hands-on approach as the one above, we prove the following
further result.
###### Proposition 5.
Up to isomorphism preserving the identification $T^{2}\to F_{d}$, there are
exactly two Lefschetz fibrations with exactly two singular fibers filling the
canonical open book on $\breve{M}$; moreover, neither of the fibrations admit
a real structure.
Proof: Let $u^{\prime},v^{\prime}$ be two simple curves on $T^{2}$ such that
$t_{v^{\prime}}\circ t_{u^{\prime}}=f$. We have
$[f]=[t_{v}][t_{u}]=\small{\left[\begin{array}[]{rr}-39&25\\\
-25&16\end{array}\right]}$. Suppose $v^{\prime}$ is the curve defined by
$\alpha a+\beta b$, then a simple calculation gives
$[t_{v^{\prime}}]=\small{\left[\begin{array}[]{rr}1-\alpha\beta&\alpha^{2}\\\
-\beta^{2}&1+\alpha\beta\end{array}\right]}$. All Dehn twists are represented
by matrices whose traces have absolute value equal to $2$. Therefore, from
$[t_{u^{\prime}}]=[t_{v^{\prime}}^{-1}f]$, we get the identity
$25\alpha^{2}+25\beta^{2}-55\alpha\beta=25$. This quadratic Diophantine
equation has solutions $(\pm 1,0)$, $(0,\pm 1)$ and (the transpose of) these
vectors multiplied from left by the powers of the matrix
$S=\small{\left[\begin{array}[]{rr}-3&5\\\ -5&8\end{array}\right]}$ (see e.g
the step-by-step computation of the online application [2]). Note that $u$ and
$v$ are in the solution set. Note also that $u^{\prime},v^{\prime}$, the
curves with class $b$ and $5a+8b$ respectively, are in the solution set too.
The pairs $(t_{u},t_{v})$ and $(t_{u^{\prime}},t_{v^{\prime}})$ are not
Hurwitz equivalent. Indeed in that case there would be an invertible matrix
$K$ with determinant 1, such that either
$K^{-1}[t^{-1}_{u}][t_{v}][t_{u}]K=[t_{u^{\prime}}]\mbox{ and
}K^{-1}[t_{u}]K=[t_{v^{\prime}}]$
or
$K^{-1}[t^{-1}_{v}][t_{u}][t_{v}]K=[t_{u^{\prime}}]\mbox{ and
}K^{-1}[t_{v}]K=[t_{v^{\prime}}].$
However, by straightforward calculation one can show that there exists no such
$K$.
Note also that the matrices $S$ and $[f]$ commute; hence for any pair $(x,y)$
such that $f=t_{y}\circ t_{x}$, the factorization $(S^{k}x,S^{k}y)$ is Hurwitz
equivalent (with a fixed identification) to the factorization $(x,y)$ for
every $k\in\mathbb{Z}$.
As a consequence, we have two Hurwitz equivalence classes given by the pairs
$(t_{u},t_{v})$ and $(t_{u^{\prime}},t_{v^{\prime}})$. To finish, note that
the number of intersection points of $u^{\prime}$ and $v^{\prime}$ is 5, so
the proof of Proposition 1 applies to $(u^{\prime},v^{\prime})$ to show that
the fibrations defined by the decomposition $(t_{u^{\prime}},t_{v^{\prime}})$
is not real. $\Box$
###### Theorem 6.
The canonical open book on $\breve{M}$ cannot be filled by any real Lefschetz
fibration with the same fiber topology and with arbitrary number of singular
fibers.
Proof: We will show that $\breve{M}$ cannot be filled by a (real or non-real)
Lefschetz fibration with the number of singular fibers different from 2.
Recall that
$\textup{Map}(T^{2}\smallsetminus\textup{int}{D^{2}},\partial)=\left\\{[t_{a}],[t_{b}]:[t_{a}][t_{b}][t_{a}]=[t_{b}][t_{a}][t_{b}]\right\\}.$
Therefore, the homomorphism
$\begin{array}[]{lrcl}deg:&\textup{Map}(T^{2}\smallsetminus\textup{int}{D^{2}},\partial)&\to&\mathbb{Z}\\\
&t_{a},t_{b}&\mapsto&1\end{array}$
is well-defined; hence the number of Dehn twists that may constitute a given
element $h$ equals $deg(h)$. In our case, $deg(f)=2$, so it cannot admit a
factorization as a product of an arbitrary number of positive Dehn twists
other than 2. Moreover, all possible factorizations into a product of two Dehn
twists are shown above to be non-real. $\Box$
###### Remark 2.
Elements admitting a factorization into a product of two Dehn twists will be
studied in [1]. There the classification of such factorizations up to Hurwitz
equivalence are presented and their relation to real structures are also
elaborated.
## References
* [1] Degtyarev A., Salepci N. Products of pairs of Dehn twists and maximal real Lefschetz fibrations. Nagoya Math. J. (to appear)
* [2] Dario Alpern’s two-integer-variable equation solver, http://www.alpertron.com/QUAD:HTM
* [3] Kas, A. On the handlebody decomposition associated to a Lefschetz pencil. Pacific J. Math. 89 (1980), 89-104.
* [4] Loi, A.; Piergallini, R. Compact Stein surfaces with boundary as branched covers of $B^{4}$. Invent. Math. 143 (2001), no. 2, 325–348.
* [5] Moishezon, B. Complex surfaces and connected sums of complex projective planes. Lecture notes in Math. 603 Springer Verlag(1977).
* [6] Öztürk, F., Salepci, N. Real open books and real contact structures, preprint.
* [7] Salepci, N. Real elements in the mapping class group of $T^{2}$, Topology and its Applications 157 (2010) 2580–2590.
* [8] Salepci, N. Real Lefschetz fibrations. Thèse, Université Louis Pasteur, Strasbourg, 2007. Université Louis Pasteur. Institut de Recherche Mathématique Avancée (IRMA), Strasbourg, 2007
|
arxiv-papers
| 2013-01-22T11:15:01 |
2024-09-04T02:49:40.550920
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ferit Ozturk and Nermin Salepci",
"submitter": "Ferit \\\"Ozt\\\"urk",
"url": "https://arxiv.org/abs/1301.5147"
}
|
1301.5171
|
# Depth of some special monomial ideals
Dorin Popescu and Andrei Zarojanu Dorin Popescu, ”Simion Stoilow” Institute
of Mathematics of Romanian Academy, Research unit 5, P.O.Box 1-764, Bucharest
014700, Romania [email protected] Andrei Zarojanu, Faculty of Mathematics
and Computer Sciences, University of Bucharest, Str. Academiei 14, Bucharest,
Romania, and ”Simion Stoilow” Institute of Mathematics of Romanian Academy,
Research group of the project ID-PCE-2011-1023, P.O.Box 1-764, Bucharest
014700, Romania [email protected]
###### Abstract.
Let $I\supsetneq J$ be two squarefree monomial ideals of a polynomial algebra
over a field. Suppose that $I$ is generated by one squarefree monomial of
degree $d>0$, and other squarefree monomials of degrees $\geq d+1$. If the
Stanley depth of $I/J$ is $\leq d+1$ then almost always the usual depth of
$I/J$ is $\leq d+1$ too.
Key words : Monomial Ideals, Depth, Stanley depth.
2000 Mathematics Subject Classification: Primary 13C15, Secondary 13F20,
13F55, 13P10.
The support from grant ID-PCE-2011-1023 of Romanian Ministry of Education,
Research and Innovation is gratefully acknowledged.
## Introduction
Let $S=K[x_{1},\ldots,x_{n}]$ be the polynomial algebra in $n$ variables over
a field $K$ and $I\supsetneq J$ two squarefree monomial ideals of $S$. Suppose
that $I$ is generated by squarefree monomials of degrees $\geq d$ for some
positive integer $d$. Modulo a multigraded isomorphism we may assume either
that $J=0$, or $J$ is generated in degrees $\geq d+1$. Then
$\operatorname{depth}_{S}I/J\geq d$ (see [1, Proposition 3.1], [3, Lemma 1.1])
and upper bounds are given by Stanley’s Conjecture if it holds. Finding such
upper bounds is the subject of several papers [2], [3], [4], [7], [5]. We
remind below the notion of Stanley depth.
Let $P_{I\setminus J}$ be the poset of all squarefree monomials of $I\setminus
J$ with the order given by the divisibility. Let ${\mathcal{P}}$ be a
partition of $P_{I\setminus J}$ in intervals $[u,v]=\\{w\in P_{I\setminus
J}:u|w,w|v\\}$, let us say $P_{I\setminus J}=\cup_{i}[u_{i},v_{i}]$, the union
being disjoint. Define
$\operatorname{sdepth}{\mathcal{P}}=\operatorname{min}_{i}\operatorname{deg}v_{i}$
and the Stanley depth of $I/J$ given by
$\operatorname{sdepth}_{S}I/J=\operatorname{max}_{\mathcal{P}}\operatorname{sdepth}{\mathcal{P}}$,
where ${\mathcal{P}}$ runs over the set of all partitions of $P_{I\setminus
J}$ (see [1], [8]). Stanley’s Conjecture says that
$\operatorname{sdepth}_{S}I/J\geq\operatorname{depth}_{S}I/J$.
Let $r$ be the number of squarefree monomials of degree $d$ of $I$ and $B$
(resp. $C$) be the set of squarefree monomials of degrees $d+1$ (resp. $d+2$)
of $I\setminus J$. Set $s=|B|$, $q=|C|$. If either $s>r+q$, or $r>q$, or
$s<2r$ then $\operatorname{sdepth}_{S}I/J\leq d+1$ and if Stanley’s Conjecture
holds then any of these numerical conditions would imply
$\operatorname{depth}_{S}I/J\leq d+1$, independently of the characteristic of
$K$. In particular this was proved directly in [4] and [7].
Suppose that $r=1$. If $d=1$ we showed in [5, Theorem 1.10] that almost always
Stanley’s Conjecture holds. It is the purpose of this note to complete the
result for $d\geq 1$ in the next form.
Theorem Suppose that $I\subset S$ is minimally generated by a squarefree
monomial $f$ of degree $d$, and a set $E$ of squarefree monomials of degrees
$\geq d+1$. Assume that $s\not=q+1$ and $\operatorname{sdepth}_{S}I/J=d+1$.
Then $\operatorname{depth}_{S}I/J\leq d+1$.
## 1\. Proof of the Theorem
We may assume that $s<q+1$ because if $s>q+1$ then by [4] we get
$\operatorname{depth}_{S}I/J\leq d+1$. Also we may suppose that
$C\subset(f,B)$ by [5, Lemma 1.6]. Induct on $|E|$. Suppose that
$E=\emptyset$. If $C=\emptyset$, then $\operatorname{depth}_{S}I/J\leq d+1$ by
[5, Lemma 1.5]. Otherwise, let $c=fx_{n-1}x_{n}\in C$ and
$I^{\prime}=(B\setminus\\{fx_{n-1},fx_{n}\\})$. In the exact sequence
$0\rightarrow I^{\prime}/J\cap I^{\prime}\rightarrow I/J\rightarrow
I/J+I^{\prime}\rightarrow 0$
the last term has sdepth $d+2$ since $c\not\in I^{\prime}+J$ and so the first
one has sdepth $\leq d+1$ by [6, Lemma 2.2] and even depth $\leq d+1$ by [3,
Theorem 4.3]. Then the Depth Lemma gives $\operatorname{depth}_{S}I/J\leq
d+1$.
Set $I_{n}=(B\setminus\\{fx_{n}\\}),J_{n}=I_{n}\cap J$. In the following exact
sequence
$0\rightarrow I_{n}/J_{n}\rightarrow I/J\rightarrow I/(I_{n}+J)\rightarrow 0$
the last term has sdepth depth $d+1$ since $[f,fx_{n}]$ is the whole poset of
$(f)/(f)\cap(I_{n}+J)$ and $x_{n}\not\in((J+I_{n}):f)$. If the first term has
sdepth $=d+1$ then by [[3], Theorem 4.3.] we get depth $=d+1$ and applying
Depth Lemma the conclusion follows. So we can assume that there exists a
partition ${\mathcal{P}}_{n}$ of $I_{n}/J_{n}$ with sdepth $d+2$. We may
suppose that all intervals of ${\mathcal{P}}_{b_{i}}$ (as well as of other
partitions which we will use) starting with a monomial $v$ of degree $\geq
d+2$ have the form $[v,v]$. In ${\mathcal{P}}_{n}$ we can’t have the interval
$[c,c]$, $c=fx_{n-1}x_{n}$, or the interval $[fx_{n-1},c]$ because otherwise
we can switch it with $[f,c]$ and get a partition of $I/J$ with sdepth $d+2$.
Thus we have in ${\mathcal{P}}_{n}$ the interval $[b_{1},c]$, $b_{1}\in E$.
Switching the interval $[b_{1},c]$ with the interval $[fx_{n},c]$ we get a
partition $P_{B_{b_{1}}}$ for $(B_{b_{1}})/J_{b_{1}}$ where
$B_{b_{1}}=B\setminus\\{b_{1}\\}$ and $J_{b_{1}}=(B_{b_{1}})\cap J$.
In ${\mathcal{P}}_{B_{b_{1}}}$ we have an interval $[\bar{c},\bar{c}]$ because
$s<q+1$. Thus there exists $b_{2}\in E$ such that $\bar{c}\in(b_{2})$. Note
that $\bar{c}\notin(b_{1})$ because otherwise we may replace in
${\mathcal{P}}_{B_{b_{1}}}$ the interval $[\bar{c},\bar{c}]$ with the interval
$[b_{1},\bar{c}]$ and get a partition ${\mathcal{P}}_{B}$ for $(B)/(B)\cap J$
with sdepth $=d+2$. This leads to a contradiction because we may change in
$P_{B}$ two intervals like $[fx_{i},fx_{i}x_{j}]$, $[fx_{j},c^{\prime}]$ for
some $c^{\prime}$ with $[f,fx_{i}x_{j}]$, $[c^{\prime},c^{\prime}]$ and get a
partition ${\mathcal{P}}$ for $I/J$ with sdepth $d+2$.
Let $I_{b_{2}}=(f,E\setminus\\{b_{2}\\})$, $J_{b_{2}}=I_{b_{2}}\cap J$. In the
following exact sequence
$0\rightarrow I_{b_{2}}/J_{b_{2}}\rightarrow I/J\rightarrow
I/(I_{b_{2}}+J)\rightarrow 0$
the last term has depth $\geq d+1$ because it is isomorphic with
$(b_{2})/(b_{2})\cap(I_{b_{2}}+J)$. If
$\operatorname{sdepth}I_{b_{2}}/J_{b_{2}}\leq d+1$ then the first term has
depth $\leq d+1$ by the induction hypothesis, so by the Depth Lemma we get
$\operatorname{depth}I/J\leq d+1$.
Now assume that $\operatorname{sdepth}I_{b_{2}}/J_{b_{2}}\geq d+2$ and let
$P_{b_{2}}$ be a partition on $I_{b_{2}}/J_{b_{2}}$ with sdepth $d+2$. In
${\mathcal{P}}_{b_{2}}$ we have the interval
$[f,fx_{i}x_{j}],i,j\in[n]\setminus\mbox{supp}f$. We have in
${\mathcal{P}}_{b_{2}}$ for all $b\in B\setminus\\{b_{2},fx_{i},fx_{j}\\}$ an
interval $[b,c_{b}]$. We define $h_{2}:(B\setminus\\{b_{2}\\})\rightarrow C$
by $b\rightarrow c_{b}$ and $h_{2}(fx_{i})=h_{2}(fx_{j})=fx_{i}x_{j}$ and let
$g_{2}:\operatorname{Im}h_{2}\rightarrow(B\setminus\\{b_{2}\\})$ defined by
$c_{b}\rightarrow b$, $g_{2}(fx_{i}x_{j})=fx_{i}$. Similarly we define
$h_{1},g_{1}$ for ${\mathcal{P}}_{B_{b_{1}}}$, that is $h_{1}$ is given by
$b^{\prime}\rightarrow c^{\prime}$ if ${\mathcal{P}}_{B_{b_{1}}}$ has the
interval $[b^{\prime},c^{\prime}]$.
We want to show that we can build a partition ${\mathcal{P}}$ with sdepth
$=d+2$ for $I/J$. Consider $a_{0}=b_{1}$ and
$c_{i-1}=h_{2}(a_{i-1}),a_{i}=g_{1}(c_{i-1}),i>0$. The construction stops at
step $e$ if
1) $a_{e}=b_{2}$,
2) $c_{e}\notin\operatorname{Im}h_{1}$,
3)$a_{e}=fx_{j}$ after $a_{u}=fx_{i},u<e$ already appeared. Note that here we
have fixed $a_{u}=fx_{i}$.
In the first case we set $c_{e}=\bar{c}$ and we see that $h_{1}$ gives a
bijection between $\\{a_{1},\ldots,a_{e}\\}$ and $\\{c_{0},\ldots,c_{e-1}\\}$.
But $h_{1}$ also gives a bijection between
$B\setminus\\{b_{1},a_{1},\ldots,a_{e}\\}$ and
$C\setminus\\{\bar{c},c_{0},\ldots,c_{e-1}\\}$. Then the intervals
$[a_{p},c_{p}]$, $0\leq p\leq e$ and the intervals
$[g_{1}({\tilde{c}}),{\tilde{c}}]$, ${\tilde{c}}\in
C\setminus\\{\bar{c},c_{0},\ldots,c_{e-1}\\}$ and some other intervals
starting with monomials of degree $\geq d+2$ give a partition
${\mathcal{P}}_{B}$ of $P_{B/B\cap J}$ with sdepth $\geq d+2$. As before this
is a contradiction with sdepth $I/J=d+1$.
In the second case, as above we see that the intervals $[a_{p},c_{p}]$, $0\leq
p\leq e$ and the intervals $[g_{1}({\tilde{c}}),{\tilde{c}}]$, ${\tilde{c}}\in
C\setminus\\{\bar{c},c_{0},\ldots,c_{e-1}\\}$ and some other intervals
starting with monomials of degree $\geq d+2$ give a partition
${\mathcal{P}}_{B}$ of $P_{B/B\cap J}$ with sdepth $\geq d+2$. Contradiction.
In the last case we see as usual, that $h_{1}$ gives a bijection between
$\\{a_{1},\ldots,a_{e}\\}$ and $\\{c_{0},\ldots,c_{e-1}\\}$. But $h_{1}$ also
gives a bijection between $B\setminus\\{b_{1},a_{1},\ldots,a_{e}\\}$ and
$C\setminus\\{c_{0},\ldots,c_{e-1}\\}$. Then the intervals $[a_{p},c_{p}]$,
$0\leq p\leq e-1,p\neq u$ and the intervals
$[f,c_{u}],[g_{1}({\tilde{c}}),{\tilde{c}}]$, ${\tilde{c}}\in
C\setminus\\{c_{0},\ldots,c_{e-1}\\}$ and some other intervals starting with
monomials of degree $\geq d+2$ give the partition ${\mathcal{P}}$ of $P_{I/J}$
with sdepth $\geq d+2$. Contradiction.
###### Example 1.1.
Let $n=5$, $I=(x_{1}x_{2},x_{3}x_{4}x_{5})$ and
$J=(x_{1}x_{2}x_{3}x_{5},x_{1}x_{2}x_{4}x_{5})$. We see that we have
$\operatorname{sdepth}I/J=d+1=3$ and
$B=\\{x_{1}x_{2}x_{3},x_{1}x_{2}x_{4},x_{1}x_{2}x_{5},x_{3}x_{4}x_{5}\\}$ and
$C=\\{x_{1}x_{2}x_{3}x_{4},x_{1}x_{3}x_{4}x_{5},x_{2}x_{3}x_{4}x_{5}\\}$ so we
are in the case $s=q+1$. We can get $\operatorname{depth}I/J\leq 3$ by using
[5, Lemma 1.5] for $u=x_{1}x_{2}x_{5}$.
###### Remark 1.2.
If in the above example change just one monomial from the generators of $J$,
namely take $J=(x_{1}x_{2}x_{4}x_{5},x_{2}x_{3}x_{4}x_{5})$ then we have
$\operatorname{sdepth}_{S}I/J=4$ because the partition induced by the
intervals $[x_{1}x_{2},x_{1}x_{2}x_{3}x_{4}]$,
$[x_{3}x_{4}x_{5},x_{1}x_{3}x_{4}x_{5}]$,
$[x_{1}x_{2}x_{5},x_{1}x_{2}x_{3}x_{5}]$ has sdepth $d+2=4$. Also we have
$\operatorname{depth}_{S}I/J=4$.
A question is hinted by the following example.
###### Example 1.3.
Let $n=5$, $I=(x_{1},x_{2}x_{3},x_{2}x_{4},x_{2}x_{5},x_{3}x_{4})$ and $J$ the
ideal generated by all squarefree monomials of $I$ of degrees $4$. Then
$E=\\{x_{2}x_{3},x_{2}x_{4},x_{2}x_{5},x_{3}x_{4}\\}$, $f=x_{1}$,
$B=\\{x_{1}x_{2},x_{1}x_{3},x_{1}x_{4},x_{1}x_{5},E\\}$,
$C=\\{x_{1}x_{2}x_{3},x_{1}x_{2}x_{4},x_{1}x_{2}x_{5},x_{1}x_{3}x_{4},$
$x_{2}x_{3}x_{4},x_{2}x_{3}x_{5},x_{2}x_{4}x_{5}\\}$. Thus $s=8=q+1$. We see
that $\operatorname{sdepth}_{S}I/J=d+2=3$ but
$\operatorname{depth}_{S}I/J=d+1=2$. Note that here
$C\subset(\cup_{a\in E}C\cap(f)\cap(a))\cup(\cup_{a,a^{\prime}\in
E,a\not=a^{\prime}}C\cap(a)\cap(a^{\prime}),$
a condition which might imply always $\operatorname{depth}_{S}I/J\leq d+1$,
the inequality being not true for sdepth.
## References
* [1] J. Herzog, M. Vladoiu, X. Zheng, How to compute the Stanley depth of a monomial ideal, J. Algebra, 322 (2009), 3151-3169.
* [2] D. Popescu, Depth and minimal number of generators of square free monomial ideals, An. St. Univ. Ovidius, Constanta, 19 (2), (2011), 187-194.
* [3] D. Popescu, Depth of factors of square free monomial ideals, to appear in Proceedings AMS, arXiv:AC/1110.1963.
* [4] D. Popescu, Upper bounds of depth of monomial ideals, to appear in J. Commutative Alg., (2012),arXiv:AC/1206.3977.
* [5] D. Popescu, A. Zarojanu, Depth of some square free monomial ideals, Bull. Math. Soc. Sci. Math. Roumanie, 56(104), 2013,117-124.
* [6] A. Rauf, Depth and Stanley depth of multigraded modules, Comm. Algebra, 38 (2010),773-784.
* [7] Y.H. Shen, Lexsegment ideals of Hilbert depth 1, (2012), arxiv:AC/1208.1822v1.
* [8] R. P. Stanley, Linear Diophantine equations and local cohomology, Invent. Math. 68 (1982) 175-193.
|
arxiv-papers
| 2013-01-22T12:37:48 |
2024-09-04T02:49:40.557422
|
{
"license": "Public Domain",
"authors": "Dorin Popescu, Andrei Zarojanu",
"submitter": "Dorin Popescu",
"url": "https://arxiv.org/abs/1301.5171"
}
|
1301.5226
|
# Feedback-induced stationary localized patterns in networks of diffusively
coupled bistable elements
Nikos E. Kouvaris and Alexander S. Mikhailov
Department of Physical Chemistry,
Fritz Haber Institute of the Max Planck Society
Faradayweg 4-6, D-14195 Berlin, Germany
([email protected])
###### Abstract
Effects of feedbacks on self-organization phenomena in networks of diffusively
coupled bistable elements are investigated. For regular trees, an approximate
analytical theory for localized stationary patterns under application of
global feedbacks is constructed. Using it, properties of such patterns in
different parts of the parameter space are discussed. Numerical investigations
are performed for large random Erdös-Rényi and scale-free networks. In both
kinds of systems, localized stationary activation patterns have been observed.
The active nodes in such a pattern form a subnetwork, whose size decreases as
the feedback intensity is increased. For strong feedbacks, active subnetworks
are organized as trees. Additionally, local feedbacks affecting only the nodes
with high degrees (i.e. hubs) or the periphery nodes are considered.
## 1 Introduction
Nonequilibrium pattern formation has been extensively studied in distributed
active media [1, 2, 3]. In excitable or bistable media, stationary localized
patterns can develop. They represent domains with high activator density
surrounded by large regions where the density of the activator is much lower.
Such structures have been theoretically considered [4, 5] and experimentally
observed in chemical reactions [6, 7] and in semiconductors [8, 9]. They have
also been studied under the global feedback control [10, 11].
Control of nonequilibrium patterns, as well as their purposeful design, is an
essential issue that has attracted much attention [12, 13, 14]. Global or
local feedback control schemes serve as a standard method used for this
purpose. Typically, global feedback requires a common control signal,
generated by the entire system and applied back to all its elements. In
contrast, local feedback control does not directly affect the entire system,
but is applied only to some of its elements. Various feedback schemes have
been used in theoretical studies [10, 15, 16] and in the experiments [17, 18,
19, 20], either for stabilizing existing unstable patterns or for inducing new
kinds of patterns that do not existing in the absence of feedback.
Self-organization phenomena, such as epidemic spreading [21, 22, 23, 24, 25],
clustering [26] and synchronization [27] of oscillators, Turing patterns [28,
29] or traveling and pinned fronts [30], have been studied in reaction-
diffusion systems organized in complex networks. Moreover, some effects of
control by global feedbacks have been previously investigated for networks
systems. It has been demonstrated, for example, that turbulence in oscillator
networks can be suppressed [31] and hysteresis of Turing network patterns can
be prevented [32] when such feedbacks are applied.
As we show in this Letter, feedback control may also suppress spreading of
activation in networks of bistable elements, and leads to the emergence of
localized stationary patterns which resemble stationary spots in continuous
media. The approximate analytical theory of such phenomena could be
constructed for regular trees and systematic numerical simulations for random
Erdös-Rényi (ER) and scale-free (SF) networks were undertaken. Both global and
local feedbacks, applied to a subset of network nodes, were considered.
Properties of developing stationary patterns, depending on control conditions,
have been explored.
## 2 Bistable systems on networks
Classical continuous one-component bistable media are described by
$\dot{u}(\mathbf{x},t)=f(u,h)+D\nabla^{2}u(\mathbf{x},t)\,,$ where
$u(\mathbf{x},t)$ is the activator density, $D$ is the activation diffusion
constant and function $f(u,h)$ specifies the local bistable dynamics. For
example, this function can be chosen as $f(u,h)=u(h-u)(u-1)\,$, where the
parameter $h$ determines the activation threshold and plays an important role
in front propagation.
In network-organized systems, the activator species occupies the nodes of a
network and can be diffusively transported over network links to other nodes.
The architecture of a network is described in terms of its adjacency matrix
$\mathbf{T}$, whose elements are $T_{ij}=1$, if there is a link connecting the
nodes $i$ and $j$ ($i,j=1,...,N$), and $T_{ij}=0$ otherwise. We will consider
processes in non-directed networks, where the adjacency matrix is symmetric
($T_{ij}=T_{ji}$). Generally, the network analog of one-component bistable
system is given by
$\dot{u}_{i}=f(u_{i},h_{i})+D\sum_{j=1}^{N}\\!\left(T_{ij}u_{j}-T_{ji}u_{i}\right)\,,$
(1)
where $u_{i}$ is the amount of activator in network node $i$ and
$f(u_{i},h_{i})$ describes the local bistable dynamics of the activator. The
last term in eq. (1) takes into account diffusive coupling between the nodes
and the coefficient $D$ characterizes the rate of diffusive transport of the
activator over the network links. An essential property of a node $i$ is its
degree $k_{i}$ (the number of connections) given by $k_{i}=\sum_{j}T_{ji}$.
We introduce the global feedback through the parameter $h$, i.e. as
$h=h_{0}+\mu(S-S_{0})\,,$ (2)
where $\mu>0$ is the intensity of the feedback, $S=\sum_{j=1}^{N}\\!u_{j}\,$
is the total activation of the network and $S_{0}$ is a parameter defining the
size of localized patterns. In our simulations, it was taken equal to the
number of the nodes which were initially activated. Hence, the threshold $h$
depends now on the total activation. It increases when more nodes are
activated, so that a negative feedback is realized.
## 3 Regular tree networks
We first investigate the system (1) without feedback control, where spreading
or retreating activation fronts can be found. Fronts can also get pinned, thus
forming stationary patterns. Previously, we have derived the pinning condition
for regular trees depending on the parameters $k$ and $D$ [30], but now we
would also need to know how the pinning behavior depends on the threshold $h$.
We analyze this behavior and determine how it is affected by the feedback.
Let us consider the system (1) on a regular tree with the branching factor
$k-1$. In such a tree, all nodes, lying at the same distance $l$ from the
origin, can be grouped into a single shell. Suppose that we have taken a node
which belongs to the shell $l$. This node should be diffusively coupled to
$k-1$ nodes in the next shell $l+1$ and to just one node in the previous shell
$l-1$, as described by the equation
$\dot{u}_{l}=f(u_{l},h_{l})+D(u_{l-1}-u_{l})+D(k-1)(u_{l+1}-u_{l})\,,$ (3)
where $u_{l}$ is the density of the activator in the shell $l$. Note that for
$k=2$, eq. (3) corresponds to a one-dimensional chain of coupled bistable
elements, where pinned fronts have been previously investigated [33, 34].
However, the respective approximate analytical theory for the trees ($k>2$)
has been only recently developed [30].
Figure 1: (Color online) Stationary patterns corresponding to pinned fronts of
type I (a) and type II (b) in a regular tree with the branching factor
$k-1=2$. By grouping the nodes with the same distance from the root into a
single shell, the two types of fronts can be displayed as shown on the right
side. The parameters are (a) $h=0.35$ and (b) $h=0.65$; $D=0.02$ Figure 2:
(Color online) Bifurcation diagram in the plane ($h,k$) for $D=0.02$. Red
curves correspond to eqs. (3), while blue curves correspond to eqs. (3). The
black dashed line determines the boundary on which the velocities of both
types of fronts moving towards the tree root become equal. Figure 3: (Color
online) Evolution plots in absence (a–g) and in presence (a′–g′) of global
feedback are shown for different regions of the bifurcation diagram (fig. 2).
Global negative feedback had the intensity $\mu=0.001$ and was characterized
by $S_{0}=100$. The diffusion constant was $D=0.02$. The degrees were (a–e)
$k=4$ and (f,g) $k=13$. The thresholds were chosen as (a) $h=0.1$, (b)
$h=0.25$, (c) $h=0.47$, (d) $h=0.7$, (e) $h=0.9$, (f) $h=0.4$ and (g) $h=0.7$.
The same color coding as in fig. 1 is used.
Generally, both the activation front propagating from the root to the
periphery of a tree (type I) and in the opposite direction, i.e. towards the
tree root (type II), are possible. First, we derive approximate pinning
conditions for these two types of fronts, depending on the branching factor
$k-1$ and the parameter $h$. If diffusion is weak enough (cf. [34, 30]), a
pinned front can be found by setting $\dot{u}_{l}=0$ in eq. (3), so that we
get $f(u_{l},h_{l})+D(u_{l-1}-u_{l})+D(k-1)(u_{l+1}-u_{l})=0\,$. Suppose that
a front of type I is pinned and located at the shell $l=m$. The front is so
sharp that the nodes in the lower $l<m$ shells are all approximately in the
active state. Then, the activation level $u$ at the shell $m$ should
approximately satisfy the condition $g(u)=f(u,h)+D(1-ku)=0\,$. Stationary
localized patterns are found within the parameter region where function $g(u)$
has three real roots. The boundaries of the pinning region in the plane
$(k,h)$ are determined by the equations (see [30])
$\displaystyle k$ $\displaystyle=$
$\displaystyle\frac{u^{4}+2u\left(D-u^{2}\right)+\left(u^{2}-D\right)}{Du^{2}}\,,$
$\displaystyle h$ $\displaystyle=$
$\displaystyle\left(\frac{D}{u^{2}}-1\right)+2u\,.$ (4)
Thus, the fronts of type I are pinned within the region which lies between two
red curves in the bifurcation diagram in fig. 2 and comprises parts b and c.
Stationary patterns around the tree root, corresponding to the pinned front of
type I are shown in fig. 1(a).
Repeating the analysis for a front of type II, one can derive the pinning
condition given in the parametric form by
$\displaystyle k$ $\displaystyle=$
$\displaystyle\frac{u^{4}+\left(D+u^{2}\right)-2u\left(D+u^{2}\right)}{D\left(1-u\right)^{2}}\,,$
$\displaystyle h$ $\displaystyle=$
$\displaystyle-\frac{D}{\left(1-u\right)^{2}}+2u\,.$ (5)
The fronts of type II are pinned between two blue curves in fig. 2, i.e. in
the areas c and d; an example of a stationary pattern in this case is shown in
fig. 1(b).
Thus, our approximate theory has allowed us to identify regions in the
parameter plane ($h,k$) where stationary patterns are localized around the
root or at the periphery of the trees, without the feedback control.
Furthermore, using fig. 2 we can consider evolution of various perturbations
in different regions of the bifurcation diagram.
The evolution plots in figs. 3(a)–(g) show the development of the initial
perturbation, localized in the shells between $l_{1}=150$ and $l_{2}=250$, in
different regions of the bifurcation diagram in fig. 2. In absence of feedback
control, for parameters within the region a, the activation spreads in both
directions, towards the root and the periphery, eventually transferring the
entire tree into the active state (fig. 3(a)). For the parameters within the
regions b and c of the bifurcation diagram stationary patterns are formed. In
region b, the activation spreads only towards the root; once the root gets
activated the pattern becomes stationary and includes all nodes in the shells
$l_{2}\geq l\geq 0$ (fig. 3(b)). In region c, the stationary pattern remains
localized within the initial perturbation interval $l_{2}\geq l\geq l_{1}$
(fig. 3(c)). In region d, the activation gets annihilated, because it is
pinned on the root’s side while retreating from the periphery (fig. 3(d)). In
region e, the perturbation retreats on both sides and disappears (fig. 3(e)).
In region f, the active domain is gradually broadened while traveling in the
root direction, but it is also retreated from the periphery and finally
vanishes. In region g, the local activation is shrinking while traveling in
the same direction.
Thus, in absence of feedback control, local perturbation gives rise to
stationary patterns only for the parameters within the regions b and c of the
bifurcation diagram in fig. 2. As we show below, in presence of global
feedback, the threshold $h$ is not constant but varies with the total
activation, according to eq. 2. Thus, different regions of the bifurcation
diagram can be transversed, until a localized stationary pattern is
established.
Figure 4: (Color online) Localized stationary patterns in an ER network (a–d)
and in a SF network (e–h) with $\langle k\rangle=6$ and $N=300$. The black
square indicates the node at which the activation was initially applied.
Activation patterns of entire networks are shown for $\mu=0.004$ in the upper
row (a,e). The bottom row shows activated subnetworks at the increasing
feedback intensities $\mu=0.004$ (b,f), $\mu=0.008$ (c,g) and $\mu=0.012$
(d,h). Other parameters are $h_{0}=0.1$ and $D=0.02$; the same color coding as
in fig. 1 is used.
When global feedback (eq. (2)) is applied, stationary patterns develop in all
regions of the bifurcation diagram. For the degrees $k$, below the
intersection of curves (3) and (3) at $h=0.5$ (gray shaded region in fig. 2),
such patterns stay localized around the initial perturbation. When the
parameter $h_{0}$ is chosen within region a, the activation starts to spread
in both directions. As a result, the threshold $h$ increases, thus
transferring the dynamics into the region b, where the outer front gets pinned
and spreading continues only inwards to the root. As $h$ increases further,
the region c is entered, where the pattern becomes stationary (fig. 3(a′)). In
fig. 3(b′), a similar scenario takes place, with the feedback gradually
shifting the dynamics from region b to c. In region c, the initial activation
remains frozen (figs. 3(c′)). In region it d, the activation does not spread
in the direction of the root, but it is retreated from the periphery.
Therefore, $h$ decreases and a stationary pattern is formed, once $h$ enters
region c (fig. 3(d′)). Choosing $h_{0}$ within region e, we find that (fig.
3(e′)) the activation first shrinks on both sides. Then, it gets pinned on the
root’s side and after that the same evolution as in fig. 3(d′) takes place.
In the trees with the larger branching factor, i.e. the larger degrees $k$,
feedback induces localized patterns which travel towards the root. Once the
root becomes activated, the size of the patterns changes in such a way that
the threshold $h$ enters region b of the bifurcation diagram and the patterns
become stationary, as shown in figs. 3(f′) and (g′). This behavior, leading to
pattern formation around the tree root, is also observed for the larger
degrees $k$, where pinning of fronts cannot occur.
Summarizing, we have shown that global negative feedback can induce stationary
localized patterns in the trees, even when they do not exist in its absence.
The pattern formation mechanism could be understood by using the bifurcation
diagram in fig. 2.
## 4 Random networks
Effects of global negative feedback have been numerically studied for random
ER and SF networks (fig. 4). As we have seen, the activation that was applied
to one node (marked by a square in fig. 4) started to spread out. This
produced a growing cluster of activated nodes. Its growth was however
accompanied by an increase of the negative feedback. Consequently, the
threshold $h$ increased with the size of the cluster, making the activation
more difficult. As a result, the growth of the activated cluster was slowed
down and finally stopped. Thus, a stationary activation pattern, representing
a small subnetwork embedded in the entire network (see fig. 4) became formed.
By retaining only the nodes with sufficiently high activation level $u>0.7$,
active subnetworks can be identified. A sequence of such subnetworks, obtained
under an increase of the global feedback intensity, is shown in figs. 4(b–d)
for an ER network and in figs. 4(f–h) for a SF network. As the feedback gets
stronger, active subnetworks decrease in size and approach a tree structure.
Our statistical analysis has revealed that, in both ER and SF networks, the
average size of active subnetworks approximately followed the power law
$S\propto\mu^{-1}$.
### 4.1 Local feedback schemes
Figure 5: (Color online) Stationary patterns localized on the hubs (a,c) and
on the periphery nodes (b,d) of the ER (a,b) and the SF (c,d) networks with
$\langle k\rangle=6$ and $N=300$ for $\mu=0.01$. Dependences of the mean
activation $<u>$ on the degree $k$ of the nodes in the localized patterns for
the feedback applied only to the periphery nodes with $k<k_{0}$ where (a)
$k_{0}=9$ and (c) $k_{0}=6$, or for the feedback applied to the hubs with the
degrees exceeding (b) $k_{0}=6$ and (d) $k_{0}=7$. Other parameters are
$h_{0}=0.1$ and $D=0.02$; the same color coding as in fig. 1 is used.
So far, we have assumed that the negative feedback was global. However, it is
also possible that the feedback acts only on a subset of nodes. Suppose for
example that the parameters $h_{i}$, which control the dynamics of individual
nodes $i$, have different sensitivity to the global control signal depending
on the degrees of such nodes, i.e. that
$h_{i}=h_{0}+\mu B(k_{i})(S-S_{0})\,,$ (6)
where $B(k_{i})$ is the step function, $B(k)=1$ if $k<k_{0}$ and $B(k)=0$
otherwise. In this case, the negative feedback is applied only to the
periphery nodes with small degrees $k$, where $k<k_{0}$. By introducing such
local feedback, we can control directly the dynamics of periphery nodes,
whereas hubs remain non-affected and have a constant threshold $h=h_{0}$
sufficient for activation spreading. The simulations show that, in this
situation, the activation applied to a hub node starts to spread forming a
cluster of activated nodes with high degrees $k\geq k_{0}$, whose further
growth to the periphery nodes is suppressed by the feedback. Thus, an active
stationary cluster, localized on the hubs and shown for an ER network in fig.
5(a), becomes formed.
Alternatively, we can choose function $B(k_{i})$ as $B(k)=1$ for $k>k_{0}$ and
$B(k)=0$ otherwise, so that the feedback is applied only to the hubs. Then,
the evolution leads to the formation of a stationary pattern, localized on the
periphery nodes, as shown for the ER network in fig. 5(b).
Similar behavior was observed when local feedbacks were applied to SF
networks. Figure 5(c) shows a stationary pattern, localized at the hubs of a
SF network, with the negative feedback directly affecting the nodes with
degrees smaller than $k_{0}=6$. However, if the feeback is applied to the hubs
with the degrees higher then $k_{0}=7$, the active pattern is localized at the
periphery nodes, as shown in fig. 5(d).
## 5 Discussion
Our study has shown that control and purposeful design of nonequilibrium
patterns is possible on the networks of bistable elements. Stationary
localized patterns were found in the trees as well as in the random ER and SF
networks, by applying different schemes of negative feedback. They consisted
of nodes with high activator density, surrounded by nodes where the density of
activator was much lower, and were representing the network analogs of
stationary spots in classical reaction-diffusion media. Their size and
structure could be controlled by varying the feedback intensity.
In the special case of regular trees, an analytical theory could be
constructed, providing the pinning conditions as a function of control
parameter. As revealed through numerical simulations, the global negative
feedback can drive the system dynamics into the pinning regions, thus giving
rise to the formation of stationary patterns.
For random ER and SF networks, feedback-induced stationary patterns are
localized on subnetworks of the entire system. The structure and the size of
such subnetworks can be controlled by the feedback intensity. Their size
decreases as the feedback becomes stronger and follows a power law. For
sufficiently high feedback intensities $\mu$, the activated subnetwork
approaches a tree.
For large random networks, we have also studied the effects of negative
feedbacks which were local, i.e. acting on the subsets of nodes with small or
large degrees. These feedback schemes resulted in stationary patterns which
were not small subnetworks and consisted of large groups of hub or periphery
nodes.
Thus, the effects of different feedback schemes on dynamics of networks formed
by diffusively coupled bistable elements were investigated. In the future, it
would be interesting to extend this analysis and consider time-delayed
feedback schemes, which may lead to oscillating patterns or other complex
dynamical structures on the networks.
## Acknowledgments
The authors would like to thank Prof. Hiroshi Kori for stimulating
discussions. Financial support from the DFG Collaborative Research Center
SFB910 “Control of Self-Organizing Nonlinear Systems” and from the Volkswagen
Foundation in Germany is gratefully acknowledged.
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|
arxiv-papers
| 2013-01-22T16:27:13 |
2024-09-04T02:49:40.568924
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Nikos E. Kouvaris and Alexander S. Mikhailov",
"submitter": "Nikos Kouvaris",
"url": "https://arxiv.org/abs/1301.5226"
}
|
1301.5228
|
# Analytic classification of families of linear differential systems unfolding
a resonant irregular singularity
Martin Klimeš
###### Abstract
We give a complete classification of analytic equivalence of germs of
parametric families of systems of linear differential equations unfolding a
generic resonant singularity of Poincaré rank 1 in dimension $n\\!=\\!2$. The
moduli space of analytic equivalence classes is described in terms of a tuple
of formal invariants and a single analytic invariant obtained from the trace
of monodromy. Moreover, an universal unfolding is given for each singularity.
To achieve this, we construct a set of two natural domains in a ramified
covering of the complex time–parameter space on which bounded linear
transformations exist between parametric systems with the same formal
invariants.
††footnotetext: _Key words_ : Linear differential equations, irregular
singularity, unfolding, confluence, Stokes matrices, monodromy, analytic
classification, moduli space.
## 1 Introduction
A parametric family of linear differential systems is written as $\,\Delta
y=0$ with
$\Delta(z,m)=h(z,m)\frac{d}{dz}-A(z,m),$ (1)
and $\,y(z,m)\in\mathbb{C}^{n},$ where both the scalar function $h$ and the
$n\\!\times\\!n$-matrix function $A$ depend analytically on a complex time
variable $z$ from an open $Z\subseteq\mathbb{C}$ and a parameter $m$ from an
open $M\subseteq\mathbb{C}^{l}$ for some $l$.
Let $\Delta y=0$ be a family of systems (1) and $y(z,m)=T(z,m)\,u(z,m)$ be a
linear transformation of the dependent variable. We define a transformed
system
$T^{*}\\!\Delta=h\frac{d}{dz}-\left[T^{-1}AT-T^{-1}\frac{dT}{dz}\right],$ (2)
satisfying $\,(T^{*}\\!\Delta)u=0$ if and only if $\Delta y=0$.
###### Definition 1.
Two families of linear systems $\Delta(z,m)$, $\Delta^{\prime}(z,m)$ (1)
depending on the same parameter $m\in M$ are _analytically equivalent_ if
there is an invertible linear transformation
$T(z,m)\in\mathrm{GL}_{n}(\mathbb{C})$ depending analytically on $(z,m)\in
Z\\!\times\\!M$ such that
$h^{\prime-1}\Delta^{\prime}=h^{-1}\,T^{*}\\!\Delta$.
In this article we consider germs of parametric families of systems
$\Delta(z,m)$ (1) in dimension $n=2$ on a neighborhood of $(z,m)=0$, that
unfold a system $\Delta_{0}=\Delta(\cdot,0)$
$\Delta_{0}(z)=z^{2}\frac{d}{dz}-A_{0}(z),\quad\text{with}\quad
A_{0}(0)=\begin{pmatrix}\lambda_{0}^{(0)}&1\\\
0&\lambda_{0}^{(0)}\end{pmatrix},$ (3)
having a resonant singularity of Poincaré rank 1 at the origin, under the
generic condition that the element $a_{21}^{(1)}$ on the position $2,1$ of the
matrix $\tfrac{d}{dz}A_{0}(0)$ is non-null:
$a_{21}^{(1)}=-\tfrac{d}{dz}\det(A_{0}(z)-\lambda_{0}^{(0)}I)\big{|}_{z=0}\neq
0.$ (4)
We shall refer to such a germ $\Delta$ simply as a “parametric system”.
In Theorem I a complete classification of analytical equivalence of these
parametric systems is given together with a description of the moduli space of
the equivalence classes. Theorem II provides universal unfolding for any
system $\Delta_{0}$ (3) satisfying (4).
###### Definition 2 (The invariants).
After a multiplication by a non-vanishing germ of scalar function, any
parametric system $\Delta$ (1) unfolding (3) can be written in a unique way
with
$h(z,m)=z^{2}+h^{(1)}(m)z+h^{(0)}(m).$ (5)
We shall suppose that $h$ is in this form from now on.
We define invariant polynomials $\lambda(z,m),\,\alpha(z,m)$ uniquely by
$\hskip
1.0pt\begin{array}[]{l}\lambda(z,m)=\lambda^{(1)}(m)\,z+\lambda^{(0)}(m)\,\in\,\tfrac{1}{2}\,\mathrm{tr}\,A(z,m)+h(z,m)\mathcal{O}_{0,0},\\\\[6.0pt]
\alpha(z,m)=\alpha^{(1)}(m)\,z+\alpha^{(0)}(m)\in\\!-\det\left(A(z,m)\\!-\\!\lambda(z,m)I\right)+h(z,m)\mathcal{O}_{0,0}\,,\end{array}\hskip-25.0pt$
(6)
where $\mathcal{O}_{0,0}$ designs the ring of germs of analytic functions of
$(z,m)$ at 0.
We define an invariant $\gamma(m)$ by
$\gamma(m)=e^{-2\pi\lambda^{(1)}(m)}\mathrm{tr}\,M(m),$ (7)
where for each fixed value of the parameter $m$, $M(m)$ is a monodromy matrix
of some fundamental solution $\Phi(z,m)$ of the system $\Delta(\cdot,m)$
around the two zeros of $h(z,m)$ in positive direction:
$\Phi(e^{2\pi i}z_{0},m)=\Phi(z_{0},m)M(m).$
The value of $\gamma(m)$ is independent of the choice of the fundamental
solution $\Phi(z,m)$ or the point $z_{0}$, and can be calculated for each
value of $m$ independently.
###### Proposition 3.
$h(z,m)$, $\lambda(z,m)$, $\alpha(z,m)$ and $\gamma(m)$ are all analytic in
$m$ and invariant under analytical equivalence of systems. The genericity
condition (4) means that $\alpha^{(1)}(0)\neq 0$.
###### Proof.
Easy. ∎
###### Remark 4.
Let
$\Delta_{m}:=\Delta(\cdot,m)$
denote the restriction of $\Delta$ to a fixed parameter $m$. The restricted
polynomials $h(\cdot,m)$, $\lambda(\cdot,m)$, $\alpha(\cdot,m)$ are formal
invariants for $\Delta_{m}$ (by which we mean invariants with respect to
formal power series transformations
$\hat{T}(z)=\sum_{l=0}^{\infty}T^{(l)}(z-z_{i})^{l}$ at singularities $z_{i}$
of $\Delta_{m}$). More precisely:
* •
If $h(z,m)$ has a double zero at $z_{1}$, then the values $\lambda(z_{1},m)$,
$\lambda^{(1)}(m)$, $\alpha(z_{1},m)$, $\alpha^{(1)}(m)$ constitute a complete
set of formal invariants of $\Delta_{m}$ at the irregular singularity $z_{1}$.
* •
If $h(z,m)$ has two different zeros $z_{1}\neq z_{2}$, then the system
$\Delta_{m}$ has a regular singularity at each of them. Supposing that
$\Delta_{m}$ is non-resonant at $z_{i}$, i.e. that
$\,2\frac{\sqrt{\alpha(z_{i},m)}}{z_{i}\,-\,z_{j}}\notin\mathbb{Z}$ ($j=3-i$),
then the values of $\,\frac{\lambda(z_{i},m)}{z_{i}\,-\,z_{j}}$ and
$\frac{{\alpha(z_{i},m)}}{(z_{i}\,-\,z_{j})^{2}}\,$ constitute a complete set
of formal invariants for the germ of $\Delta_{m}$ at $z_{i}$.
Cf. [IY], [JLP1],[JLP2].
###### Definition 5.
We call $h(z,m)$, $\lambda(z,m)$ and $\alpha(z,m)$ _the formal invariants_ of
$\Delta$.
The parametric system
$\widehat{\Delta}(z,m)=h(z,m)\frac{d}{dz}-\begin{pmatrix}\lambda(z,m)&1\\\
\alpha(z,m)&\lambda(z,m)\end{pmatrix},$ (8)
is a _“formal model”_ for $\Delta(z,m)$. The restricted system $\Delta_{m}$ to
almost each value of $m$ is formally equivalent to the corresponding
restriction $\widehat{\Delta}_{m}$ of (8) at each zero point of $h$. However,
there is no formal series transformation between the whole parametric families
of systems. Instead, as we will show in Theorem III, there exist a finite
number of special domains covering a full neighborhood of the origin in the
$(z,m)$-space on which bounded normalizing transformations exist that brings
$\Delta$ to $\widehat{\Delta}$. This provides a generalization of the classic
notion of formal equivalence.
###### Theorem I.
* (a)
Two germs of parametric systems $\Delta(z,m)$, $\Delta^{\prime}(z,m)$ are
analytically equivalent if and only if their invariants $h$,
$\lambda,\alpha,\gamma$ are the same:
$\begin{array}[]{rclrcl}h(z,m)&=&h^{\prime}(z,m),&\lambda(z,m)&=&\lambda^{\prime}(z,m),\\\
\alpha(z,m)&=&\alpha^{\prime}(z,m),&\gamma(m)&=&\gamma^{\prime}(m).\end{array}$
* (b)
Any four germs of analytic functions $\,h(z,m)$, $\lambda(z,m)$,
$\alpha(z,m)$, $\gamma(m)$ with $h(z,0)=z^{2}$, $\alpha^{(0)}(0)=0$ and
$\alpha^{(1)}(0)\neq 0\,$ are realizable as invariants of some parametric
system $\Delta(z,m)$.
###### Corollary 6.
Two germs of parametric systems $\Delta(z,m),\Delta^{\prime}(z,m)$ are
analytically equivalent if and only if there exist a small neighborhood
$Z\\!\times\\!M$ of 0 such that for each $m\in M$ the restricted systems
$\Delta_{m}$, $\Delta^{\prime}_{m}$ are analytically equivalent on $Z$.
The Theorem II provides a normal form for any germ of parametric system
unfolding $\Delta_{0}$.
###### Theorem II.
Let $\Delta_{0}$ be a system (3) satisfying the genericity condition (4), and
let $\lambda_{0}(z),\,\alpha_{0}(z),\,\gamma_{0}$ be its invariants. Let
$(h^{(0)},h^{(1)},\lambda^{(0)},\lambda^{(1)},\alpha^{(0)},\alpha^{(1)})$ be a
tuple of parameters taken from a small neighborhood of the point
$(0,0,\lambda_{0}^{(0)},\lambda_{0}^{(1)},0,\alpha_{0}^{(1)})\in\mathbb{C}^{6}$
and denote
$h(z)=z^{2}+h^{(1)}z+h^{(0)},\quad\lambda(z)=\lambda^{(1)}z+\lambda^{(0)},\quad\alpha(z)=\alpha^{(1)}z+\alpha^{(0)}.$
* (i)
If $\gamma_{0}\neq 2$, then the parametric system
$\widetilde{\Delta}(h,\lambda,\alpha,q)=h(z)\frac{d}{dz}-\begin{pmatrix}\lambda(z)&1\\\
\alpha(z)+qh(z)&\lambda(z)\end{pmatrix},$ (9)
with the parameter $q$ from a small neighborhood of some $q_{0}\in\mathbb{C}$
such that $\gamma_{0}=-2\cos\,\pi\sqrt{1+4q_{0}}$, is an universal unfolding
for $\Delta_{0}$. More precisely, any germ of parametric system $\Delta(z,m)$
unfolding $\Delta_{0}$ is analytically equivalent to a unique
$\widetilde{\Delta}(h(z,m),\lambda(z,m),\alpha(z,m),q(m))$
with the germs of functions $h(z,m),\lambda(z,m),\alpha(z,m),\gamma(m)$ given
by (5),(6),(7) and with the unique analytic germ $q(m)$ such that $q(0)=q_{0}$
and
$\gamma(m)=-2\cos\,\pi\sqrt{1+4q(m)}.$ (10)
* (ii)
If $\gamma_{0}\neq-2$, then the parametric system
$\widetilde{\Delta}^{\prime}(h,\lambda,\alpha,b)=h(z)\frac{d}{dz}-\begin{pmatrix}\lambda(z)&1+bz\\\
\beta(z)&\lambda(z)\end{pmatrix},$ (11)
with
$\beta(z)=\alpha^{(0)}-bh^{(0)}\beta^{(1)}+\beta^{(1)}z,\quad\beta^{(1)}=\frac{\alpha^{(1)}+b\alpha^{(0)}}{1+bh^{(1)}+b^{2}h^{(0)}},$
and the parameter $b$ from a small neighborhood of some $b_{0}\in\mathbb{C}$
such that $\gamma_{0}=2\cos\,2\pi\sqrt{\alpha^{(1)}_{0}b_{0}}$, is an
universal unfolding for $\Delta_{0}$. More precisely, any germ of parametric
system $\Delta(z,m)$ unfolding $\Delta_{0}$ is analytically equivalent to a
unique
$\widetilde{\Delta}^{\prime}(h(z,m),\lambda(z,m),\alpha(z,m),b(m)),$
with the germs of functions $h(z,m),\lambda(z,m),\alpha(z,m),\gamma(m)$ given
by (5),(6),(7) and with the unique analytic germ $b(m)$ such that $b(0)=b_{0}$
and
$\gamma(m)=2\cos\,2\pi\sqrt{b\beta^{(1)}}.$ (12)
## 2 Theorem III and proofs
The part (b) of Theorem I follows from Theorem II. To prove Theorem II we will
use the following Proposition 7 and part (a) of Theorem I. The main work is to
prove part (a) of Theorem I. For that, we will construct ramified domains
$\Omega$ in space $(z,m)$ on which an essentially unique bounded normalizing
transformation $T_{\Omega}(z,m)$ exist that brings $\Delta(z,m)$ to the formal
model $\widehat{\Delta}(h(z,m),\lambda(z,m),\alpha(z,m))$ (8) — this is first
done for reduced systems of certain form in a new coordinate $s$ and later
formulated in general in the Theorem III at the end of this section. For the
rest one proceeds in a standard way, by first choosing a canonical system of
fundamental solutions $\Phi_{\Omega}$ on these domains and determining all the
relevant Stokes connection matrices between these solutions, and then showing
that for two germs of parametric systems $\Delta,\Delta^{\prime}$ having the
same sets of invariants, one may chose the fundamental solutions
$\Phi_{\Omega},\Phi^{\prime}_{\Omega}$ in such a way that all the connection
matrices would be identical for the two systems. This will then imply that
$\Phi^{\prime}_{\Omega}\Phi_{\Omega}^{-1}$ agrees on different domains
$\Omega$ and therefore provides an analytic equivalence between the two germs
of parametric systems defined on a full neighborhood of 0 in the
$(z,m)$-space.
###### Proposition 7.
The invariant $\gamma$ defined by (7) of a system
$h(z)\frac{d}{dz}-[A^{(0)}+A^{(1)}z]=0,\quad\text{with}\quad
A^{(k)}=\left(\begin{smallmatrix}a_{11}^{(k)}&a_{12}^{(k)}\\\\[3.0pt]
a_{21}^{(k)}&a_{22}^{(k)}\end{smallmatrix}\right)$ (13)
and $\,h(z)=z^{2}+h^{(1)}z+h^{(0)},\,$ is equal to
$\gamma=2\cos\,2\pi\sqrt{\Big{(}\frac{a_{11}^{(1)}-a_{22}^{(1)}}{2}\Big{)}^{2}\\!+a_{12}^{(1)}a_{21}^{(1)}}.$
(14)
###### Proof.
This system considered on the Riemann sphere $\mathbb{CP}^{1}$ has
singularities only at the zero points of $h(z)$ and the point $z=\infty$.
Therefore in the formula (7)
$\gamma=e^{-2\pi
i\lambda^{(1)}}\mathrm{tr}M,\quad\text{with}\quad\lambda^{(1)}=\frac{a_{11}^{(1)}+a_{22}^{(1)}}{2},$
$M$ is a matrix of monodromy around $z=\infty$ in the negative direction. In
the coordinate $t=z^{-1}$ the system (13) is equivalent to
$t\,(1+h^{(1)}t+h^{(0)}t^{2})\frac{d}{dt}+[A^{(1)}+A^{(0)}t]=0,$ (15)
which has only a regular singularity at $t=0$. The eigenvalues of its
principal matrix $-A^{(1)}$ are $-\lambda^{(1)}\pm\sqrt{D}$ where
$\,D:=(\frac{a_{11}^{(1)}-a_{22}^{(1)}}{2})^{2}+a_{12}^{(1)}a_{21}^{(1)}$.
Suppose first that the singularity is non-resonant, i.e. that
$2\sqrt{D}\notin\mathbb{Z}$, in which case there exists a local analytic
transformation $T(t)$ that brings(15) to the diagonal system
$t\,(1+h^{(1)}t+h^{(0)}t^{2})\frac{d}{dt}+\left(\begin{smallmatrix}\lambda^{(1)}+\sqrt{D}&0\\\\[3.0pt]
0&\lambda^{(1)}-\sqrt{D}\end{smallmatrix}\right)=0,$
for which an associated diagonal fundamental solution has its monodromy matrix
around $t=0$ in the negative direction equal to
$M=e^{2\pi i\lambda^{(1)}}\left(\begin{smallmatrix}e^{2\pi
i\sqrt{D}}&0\\\\[3.0pt] 0&e^{-2\pi i\sqrt{D}}\end{smallmatrix}\right).$
Therefore $\gamma=e^{-2\pi i\lambda^{(1)}}\mathrm{tr}(M)=2\cos\,2\pi\sqrt{D}.$
The resonant case is a limit of non-resonant cases and the formula (14) for
$\gamma$ stays valid, as it is easy to see that $\gamma$ depends analytically
on the coefficients of $A$. ∎
###### Proof of Theorem II.
Use equation (6) to verify that $h(z),\,\lambda(z)$ and $\alpha(z)$ are indeed
the formal invariants of the system $\widetilde{\Delta}(h,\lambda,\alpha,q)$
(resp. $\widetilde{\Delta}^{\prime}(h,\lambda,\alpha,b)$).
To verify (10), set $\ Q:=\frac{1}{2}(-1\pm\sqrt{1+4q}),\,$ so that $\
q=Q^{2}+Q,\,$ and $T(z):=\left(\begin{smallmatrix}1&0\\\\[3.0pt]
Qz&1\end{smallmatrix}\right)$, then
$T^{*}\widetilde{\Delta}(h,\lambda,\alpha,q)=h(z)\frac{d}{dz}-\begin{pmatrix}\lambda(z)+Qz&1\\\
\alpha(z)+(h^{(0)}+h^{(1)}z)Q^{2}&\lambda(z)-Qz\end{pmatrix}.$
Now $\,\gamma=2\cos\,2\pi Q=-2\cos\,\pi\sqrt{1+4q}\ $ (10) (resp.
$\gamma=2\cos\,2\pi\sqrt{b\beta^{(1)}}\,$ (12)) is obtained using Proposition
7. ∎
All we need is to prove part (a) of Theorem I.
###### Lemma 8 (Reduction).
Let $\Delta$, $\Delta^{\prime}$ be two parametric systems with the same formal
invariants $h(z,m),\,\lambda(z,m),\,\alpha(z,m)$. Put
$x(z,m)=\tfrac{1}{\alpha^{(1)}}\left(z+\tfrac{h^{(1)}}{2}\right)\quad\text{and}\quad\epsilon(m)=(\tfrac{1}{\alpha^{(1)}})^{2}\left((\tfrac{h^{(1)}}{2})^{2}-h^{(0)}\right),$
(16)
so that
$h(z,m)=(\alpha^{(1)})^{2}(x^{2}-\epsilon),$
and define reduced systems
$\underline{\Delta\mkern 0.0mu}\mkern
0.0mu(x,m)=(x^{2}-\epsilon)\tfrac{d}{dx}-\tfrac{A(z,m)-\lambda(z,m)I}{\alpha^{(1)}},\quad\underline{\Delta\mkern
0.0mu}\mkern
0.0mu^{\prime}(x,m)=(x^{2}-\epsilon)\tfrac{d}{dx}-\tfrac{A^{\prime}(z,m)-\lambda(z,m)I}{\alpha^{(1)}}$
with formal invariants
$\underline{\lambda\mkern 0.0mu}\mkern
0.0mu(x,m)=0,\quad\underline{\alpha\mkern 0.0mu}\mkern
0.0mu(x,m)=\mu(m)+x,\quad\text{where}\quad\mu=\tfrac{\alpha^{(0)}}{(\alpha^{(1)})^{2}}-\tfrac{h^{(1)}}{2\alpha^{(1)}}.$
(17)
Then for a linear transformation $T(z,m)=\underline{T\mkern 0.0mu}\mkern
0.0mu(x,m)$
$T^{*}\\!\Delta=\Delta^{\prime}\quad\text{if and only
if}\quad\underline{T\mkern 0.0mu}\mkern 0.0mu^{*}\\!\underline{\\!\mkern
0.0mu}\mkern 0.0mu\underline{\Delta\mkern 0.0mu}\mkern
0.0mu=\underline{\Delta\mkern 0.0mu}\mkern 0.0mu^{\prime}.$
In particular, the two parametric systems $\Delta$, $\Delta^{\prime}$ are
analytically equivalent if and only if the reduced parametric systems
$\underline{\Delta\mkern 0.0mu}\mkern 0.0mu$, $\underline{\Delta\mkern
0.0mu}\mkern 0.0mu^{\prime}$ are analytically equivalent.
###### Proof.
The transformation $T(z,m)$ commutes with the scalar matrix function
$\lambda(z,m)I$. ∎
From now on we shall always suppose that a parametric system $\Delta(m)$ has
its formal invariants equal to
$h(x,m)=x^{2}-\epsilon(m),\quad\lambda(x,m)=0,\quad\alpha(x,m)=\mu(m)+x.$ (18)
###### Proposition 9 (Prenormal form).
A germ of a reduced parametric system $\Delta(x,m)$ (with formal invariants
(18)) is analytically equivalent to
$\Delta^{\prime}(x,m)=(x^{2}-\epsilon(m))\frac{d}{dx}-\begin{pmatrix}0&1\\\
\mu(m)+x+(x^{2}\\!-\epsilon(m))r(x,m)&0\end{pmatrix}.$ (19)
###### Proof.
We will bring the system $\Delta$ into the demanded form in four steps.
1) There is a analytic germ of a constant invertible matrices $C(m)$ such that
$\Delta_{1}=:C^{*}\\!\Delta=(x^{2}-\epsilon)\frac{d}{dx}-A(x,m),\quad\text{with}\quad
A(x,m)=\sum_{l=0}^{\infty}A^{(l)}x^{l}$
and $A^{(0)}=\left(\begin{smallmatrix}0&1\\\\[3.0pt]
a_{21}^{(0)}&a_{22}^{(0)}\end{smallmatrix}\right),\,$ see [Arn].
2) Find a transformation in the form of a convergent series
$\,T(x,m)=I+\sum_{l=1}^{\infty}T^{(l)}(m)x^{l},\quad\text{with}\quad
T^{(l)}=\left(\begin{smallmatrix}0&0\\\\[3.0pt]
t_{21}^{(l)}&t_{22}^{(l)}\end{smallmatrix}\right),$
such that
$\,\Delta_{2}=:T^{*}\\!\Delta_{1}=(x^{2}-\epsilon)\frac{d}{dx}-B(x,m)$,
$B(x,m)=\sum_{l=0}^{\infty}B^{(l)}x^{l},\quad\text{with}\quad
B^{(0)}=A^{(0)}\quad\text{and}\quad
B^{(l)}=\left(\begin{smallmatrix}0&0\\\\[3.0pt]
b_{21}^{(l)}&b_{22}^{(l)}\end{smallmatrix}\right).$
This means that
$B^{(l)}=[A^{(0)},T^{(l)}]+A^{(l)}+\sum_{j=1}^{l-1}A^{(j)}T^{(l-j)}-\sum_{j=1}^{l-1}T^{(l-j)}B^{(j)}-(l-1)T^{(l-1)}+\epsilon(l+1)T^{(l+1)},$
with elements in the first line equal to
$0=t^{(l)}_{2,i}+a^{(l)}_{1i}+\sum_{j=1}^{l-1}a^{(j)}_{12}t^{(l-j)}_{2i},\quad
i=1,2,$
giving a recursive formula for the coefficients of $T$. Knowing that $A(x)$ is
convergent, i.e. $|a^{(l)}_{ki}|\leq K^{l}$ for some $K\\!>\\!0$, we shall
find inductively that $|t^{(l)}_{2,i}|\leq(2K)^{l}$: indeed,
$\,|t^{(l)}_{2,i}|\leq\sum_{j=1}^{l}2^{l-j}K^{l}\leq 2^{l}K^{l}.$
3) Put $\,S(x,m)=\left(\begin{smallmatrix}1&0\\\\[3.0pt]
\frac{1}{2}b_{22}&1\end{smallmatrix}\right)$,
$\,b_{22}(x,m)=\sum_{l=0}^{\infty}b_{22}^{(l)}x^{l}$, then
$\Delta_{3}=:S^{*}\\!\Delta_{2}=(x^{2}-\epsilon)\frac{d}{dx}-\left(\begin{smallmatrix}\frac{1}{2}b_{22}(x,m)&1\\\\[3.0pt]
c(x,m)&\frac{1}{2}b_{22}(x,m)\end{smallmatrix}\right).$
By the assumption (18) we know that we can write
$\,b_{22}(x,m)=(x^{2}-\epsilon)g(x,m)\,$ for some analytic germ $g$, and that
$\,c(x,m)=\mu+x+(x^{2}-\epsilon)r(x,m)\,$ for some germ $r$.
4) Finally for $\,R(x,m)=e^{-\int_{0}^{x}\tfrac{1}{2}g(t,m)dt}I$, $\
R^{*}\\!\Delta_{3}=\Delta^{\prime}\,$ is in the demanded form. ∎
###### Remark 10.
One can interpret the two parameter functions $\epsilon(m),\mu(m)$ as being
responsible for the two following phenomena of a qualitative change in the
restricted systems $\Delta_{m}(x):=\Delta(x,m)$:
* •
$\epsilon(m)\neq 0$ separates the irregular singularity of $\Delta_{0}$ at the
origin in two Fuchsian singularities of $\Delta_{m}$ at
$x=\pm\sqrt{\epsilon(m)}$,
* •
for $\epsilon(m)=0$, the quantity $\mu$ is responsible for a change in the
asymptoticity and summability of the formal normalizing transformations for
$\Delta_{m}$ from $\frac{1}{2}$-summable when $\mu(m)=0$ to $1$-summable when
$\mu(m)\neq 0$.
The systems. Let $\Delta(x,\mu,\epsilon)$ be a germ of parametric system in
the prenormal form of Proposition 9 parametrized by $(\mu,\epsilon)$ from a
small neighborhood of the origin in $\mathbb{C}^{2}$:
$\Delta(x,\mu,\epsilon)=(x^{2}\\!-\\!\epsilon)\frac{d}{dx}-\begin{pmatrix}0&1\\\
\mu+x+(x^{2}\\!-\\!\epsilon)r(x,\mu,\epsilon)&0\end{pmatrix}$ (20)
Essentially, the problem of Theorem I (a) is that of proving that two systems
(20) are analytically equivalent if and only if they have the same trace of
monodromy.
Let $s$ be ramified coordinate defined by
$x=s^{2}-\mu,$ (21)
and let
$S(s)=\left(\begin{smallmatrix}1&0\\\\[3.0pt]
0&s\end{smallmatrix}\right),\quad V=\left(\begin{smallmatrix}1&1\\\\[3.0pt]
1&-1\end{smallmatrix}\right).$ (22)
Then the transformed parametric system
$\,\Delta^{\\!s}:=s^{-1}(SV)^{*}\Delta\,$ in the $s$ coordinate is equal
$\Delta^{\\!s}(s,\mu,\epsilon)=\frac{x^{2}\\!-\\!\epsilon}{2s^{2}}\frac{d}{ds}\,-\,\left[\left(\begin{smallmatrix}1&0\\\\[3.0pt]
0&-1\end{smallmatrix}\right)-\frac{x^{2}\\!-\\!\epsilon}{4s^{3}}\left(\begin{smallmatrix}1&-1\\\\[3.0pt]
-1&1\end{smallmatrix}\right)+\frac{x^{2}\\!-\\!\epsilon}{2s^{2}}r\left(\begin{smallmatrix}1&1\\\\[3.0pt]
-1&-1\end{smallmatrix}\right)\right].$ (23)
We will be looking for _diagonalizing transformations_
$F_{\Omega}(s,\mu,\epsilon)$, defined on some domains $\Omega$ in the
$(s,\mu,\epsilon)$-space, such that
$\mkern
2.0mu\overline{\Delta}^{s}:=F_{\Omega}{}^{*}\Delta^{\\!s}=\frac{x^{2}\\!-\\!\epsilon}{2s^{2}}\frac{d}{ds}-\left[\left(\begin{smallmatrix}1&0\\\\[3.0pt]
0&-1\end{smallmatrix}\right)-\frac{x^{2}\\!-\\!\epsilon}{4s^{3}}I\right].$
(24)
This diagonal system $\mkern 2.0mu\overline{\Delta}^{s}$ corresponds to the
system $\mkern 2.0mu\overline{\Delta}=s\,(V^{-1}S^{-1})^{*}\mkern
2.0mu\overline{\Delta}^{s}$ in the $x$ coordinate
$\mkern
2.0mu\overline{\Delta}(x,\mu,\epsilon)=(x^{2}\\!-\\!\epsilon)\frac{d}{dx}-\left[\left(\begin{smallmatrix}0&1\\\\[3.0pt]
\mu+x&0\end{smallmatrix}\right)+\frac{x^{2}\\!-\\!\epsilon}{4(\mu\\!+\\!x)}\left(\begin{smallmatrix}-1&0\\\\[3.0pt]
0&1\end{smallmatrix}\right)\right],$
with an additional singularity at the point $x=\\!-\mu$. (This is not a good
model system.) Let’s take the following sytem
$\widehat{\Delta}(x,\mu,\epsilon)=(x^{2}\\!-\\!\epsilon)\frac{d}{dx}-\begin{pmatrix}0&1\\\
\mu+x&0\end{pmatrix}$ (25)
as the _model system_. If $E_{\Omega}(s,\mu,\epsilon)$ is the normalizing
transformation “$F_{\Omega}(s,\mu,\epsilon)$” for the transform
$\widehat{\Delta}^{s}=s^{-1}(SV)^{*}\widehat{\Delta}$ of (25), on the domain
$\Omega$, then the transformation
$T_{\Omega}(x,\mu,\epsilon)=S(s)VF_{\Omega}(s,\mu,\epsilon)E_{\Omega}^{-1}(s,\mu,\epsilon)V^{-1}S^{-1}(s),$
(26)
defined on the ramified projection of the domain $\Omega$ into the
$x$-coordinate will be non-singular at the point $x=\\!-\mu$ and will bring
$\Delta$ to ${T_{\Omega}\\!}^{*}\Delta=\widehat{\Delta}$.
Fundamental solutions. On a neighborhood of $\infty$ on the Riemann sphere
$\mathbb{CP}^{1}=\mathbb{C}\cup\\{\infty\\}$, define the function
$\theta(s,\mu,\epsilon)$ by
$\frac{d}{ds}\theta(s,\mu,\epsilon)=\frac{2s^{2}}{x^{2}-\epsilon}=\frac{2s^{2}}{(s^{2}\\!-\mu)^{2}\\!-\epsilon},\quad\theta(\infty,\mu,\epsilon)=0.$
(27)
We have
$\hskip 6.0pt\theta(s,\mu,\epsilon)=\left\\{\hskip-6.0pt\begin{array}[]{l
l}\frac{\sqrt{\mu+\sqrt{\epsilon}}}{2\sqrt{\epsilon}}\log\frac{s-\sqrt{\mu+\sqrt{\epsilon}}}{s+\sqrt{\mu+\sqrt{\epsilon}}}-\frac{\sqrt{\mu-\sqrt{\epsilon}}}{2\sqrt{\epsilon}}\log\frac{s-\sqrt{\mu-\sqrt{\epsilon}}}{s+\sqrt{\mu-\sqrt{\epsilon}}},\hskip-6.0pt&\text{if
$\ \epsilon(\mu^{2}\\!-\\!\epsilon)\neq 0$,}\hskip-24.0pt\\\\[6.0pt]
-\frac{s}{s^{2}-\mu}-\frac{1}{2\\!\sqrt{\mu}}\log\frac{s+\\!\sqrt{\mu}}{s-\\!\sqrt{\mu}},&\text{if
$\ \epsilon=0$,}\\\\[6.0pt]
-\frac{1}{\sqrt{2\mu}}\log\frac{s-\sqrt{2\mu}}{s+\sqrt{2\mu}},&\text{if $\
\mu^{2}\\!=\epsilon$,}\\\\[6.0pt] -\frac{2}{s},&\text{if $\
\mu,\epsilon=0$.}\\\ \end{array}\right.$ (28)
which is analytic in
$\,s\in\mathbb{CP}^{1}\,\\!\smallsetminus\\!\,\bigcup_{i=1}^{4}[0,s_{i}]$, if
each $[0,s_{i}]$ denotes the closed segment between the origin and a zero
point $s_{i}(\mu,\epsilon)$ of
$\,x^{2}(s)-\epsilon=(s^{2}\\!-\mu)^{2}\\!-\epsilon$. The function
$\theta(s,\mu,\epsilon)$ is continuous in $(\mu,\epsilon)\in\mathbb{C}^{2}$
and analytic for
$(\mu,\epsilon)\in\mathbb{C}^{2}\\!\smallsetminus\\!\\{\epsilon\,(\mu^{2}\\!-\\!\epsilon)\neq
0\\}$. It is odd in $s$
$\theta(-s,\mu,\epsilon)=-\theta(s,\mu,\epsilon),$
and it satisfies
$\,\theta(s,\mu,\epsilon)=\theta(s,e^{2\pi
i}\mu,\epsilon)=\theta(s,\mu,e^{2\pi i}\epsilon)\,$
for each $s$ in its domain. The function $\theta(s,\mu,\epsilon)$ has a
ramified analytic extension $\theta(\check{s},\check{\mu},\check{\epsilon})$
defined on a covering of the $(s,\mu,\epsilon)$-space ramified at the zero
points $s_{i}(\check{\mu},\check{\epsilon})$.
A simple calculation shows that the matrix function
$\Psi(s,\mu,\epsilon)=\tfrac{\sqrt{2}}{2}i\,s^{-\frac{1}{2}}\left(\begin{matrix}e^{\,\theta(s,\mu,\epsilon)}&0\\\
0&e^{-\theta(s,\mu,\epsilon)}\end{matrix}\right)$ (29)
is a _fundamental solution_ for the diagonal system $\mkern
2.0mu\overline{\Delta}^{s}$ (24).
Subsequently $F_{\Omega}$ (resp. $E_{\Omega}$) are normalizing transformations
for $\Delta^{\\!s}$ (resp. $\widehat{\Delta}^{s}$) as above on some domain
$\Omega$, and $\Psi_{\Omega}$ is a restriction of $\Psi$ on $\Omega$, then the
matrix functions
$SVF_{\Omega}\Psi_{\Omega},\quad\text{(resp.
}SVE_{\Omega}\Psi_{\Omega}\text{)}$ (30)
are _fundamental solutions_ for the parametric systems $\Delta$ (resp.
$\widehat{\Delta}$).
###### Definition 11.
Let $\Omega$ be a connected (ramified) set in the space $(s,m)$ with $m$ being
a parameter. For each value $m$ denote
$\Omega(m)=\\{s\mid(s,m)\in\Omega\\},$
and write
$\begin{array}[]{ll}f\in\mathcal{B}(\Omega)\quad\text{if}&(i)\ \
f\in\mathcal{C}(\Omega)\cap\mathcal{O}(\,\mathrm{int}\,\Omega)\\\ &(ii)\ \
f(\cdot,m)\in\mathcal{O}(\,\mathrm{int}\,\Omega(m))\ \text{ for each
}m.\end{array}$
Our task is to find the right domains $\Omega$ covering a polydisc
neighborhood of 0 in the $(s,\mu,\epsilon)$-space, together with the
normalizing transformations
$F_{\Omega}\in\mathrm{GL}_{2}(\mathcal{B}(\Omega))$. The following remark
summarizes some classical results on local existence of such transformations
near singular points for fixed values of the parameters. These are not the
transformations we are looking for because they do not depend well on
parameters, knowing them shall nevertheless be useful.
###### Remark 12 (Local normalization).
For a fixed parameter $(\mu,\epsilon)$ let $s_{i}(\mu,\epsilon)$ be a zero
point of the function $(s^{2}\\!-\mu)^{2}\\!-\epsilon$ and let
$\Omega_{i}(\mu,\epsilon):=\\{s\in\mathbb{C}\mid|s-s_{i}(\mu,\epsilon)|<\frac{1}{2}\min_{s_{j}\neq
s_{i}}|s_{i}(\mu,\epsilon)-s_{j}(\mu,\epsilon)|\\}$
be a small disc around $s_{i}(\mu,\epsilon)$ not containing any other zero nor
the origin. Clearly, $s_{i}(\mu,\epsilon)$ is a simple zero if and only if the
restriction $\Delta^{\\!s}_{(\mu,\epsilon)}$ of the system in (23) has a
regular singularity at that point.
* (a)
If a regular singularity at $s_{i}(\mu,\epsilon)$ is _non-resonant_ , meaning
that
$\frac{s_{i}}{\sqrt{\epsilon}}\notin\mathbb{Z},$
then there is a unique analytic transformation
$F_{i}(\cdot,\mu,\epsilon)\in\mathrm{GL}_{2}(\mathcal{O}(\Omega_{i}(\mu,\epsilon))$,
with $F_{i}(s_{i})=I$, such that
$F_{i}{}^{*}\Delta^{\\!s}_{(\mu,\epsilon)}=\mkern
2.0mu\overline{\Delta}^{s}_{(\mu,\epsilon)}$ (24). Moreover this
transformation depends analytically on $(\mu,\epsilon)$ as long as the point
$s_{i}(\mu,\epsilon)$ is simple and the singularity stays non-resonant.
* (b)
If $s_{i}(\mu,\epsilon)$ is a resonant regular singularity of (23), i.e. if
$\pm\frac{s_{i}}{\sqrt{\epsilon}}\\!=k$ for some
$k\in\mathbb{N}\\!\smallsetminus\\!\\{0\\}$, then there exists a
transformation
$F_{i}^{\prime}(s)=F_{i0}(s)+(s-s_{i})^{k}\log(s-s_{i})\,F_{i2}(s)$
with $F_{i0}\in\mathrm{GL}_{2}(\mathcal{O}(\Omega_{i}))$ uniquely determined
by $F_{i0}(s_{i})\\!=\\!I$, and $F_{i2}$ an analytic nilpotent matrix function
on $\Omega_{i}$, such that
$(F_{i}^{\prime})^{*}\Delta^{\\!s}_{(\mu,\epsilon)}=\mkern
2.0mu\overline{\Delta}^{s}_{(\mu,\epsilon)}$.
* (c)
If $s_{i}(\mu,\epsilon)=0$ and $(\mu,\epsilon)\neq 0$, then it is a non-
singular point for $\Delta^{\\!s}_{(\mu,\epsilon)}$.
* (d)
If $\epsilon=0$ and $\mu\neq 0$, then $s_{i}(\mu,\epsilon)$ is a double zero.
In that case, there are two sectors $\Omega_{i+}$ and $\Omega_{i-}$ at $s_{i}$
(containing $s_{i}$),
$\Omega_{i\pm}=\\{s\in\Omega_{i}(\mu,\epsilon)\mid|\arg(\tfrac{s-s_{i}}{\sqrt{\mu}})\mp\tfrac{\pi}{2}|<\tfrac{\pi}{2}+\eta\\},\quad\text{for
any }0<\eta<\tfrac{\pi}{2},$
on which unique transformations
$\,F_{i+}\\!\in\mathrm{GL}_{2}(\mathcal{B}(\Omega_{i+}))\,$ and
$\,F_{i-}\\!\in\mathrm{GL}_{2}(\mathcal{B}(\Omega_{i-}))\,$ exist such that
$F_{i\pm}(s_{i})=I$ and $(F_{i\pm})^{*}\Delta^{\\!s}_{(\mu,0)}=\mkern
2.0mu\overline{\Delta}^{s}_{(\mu,0)}$ as in (24).
* (e)
If $(\mu,\epsilon)=0$, then the origin is an irregular singularity of
$\Delta^{\\!s}_{(0,0)}$, and there exists a symmetric pair of sectors
$\Omega_{0}$ and $\Omega_{0}^{\mathsf{P}}\\!:=-\Omega_{0}$ at $0$ (containing
$0$) of radius $\delta(\eta)>0$,
$\Omega_{0}=\\{|s|<\delta(\eta),\
|\arg(s)\mp\tfrac{\pi}{2}|<\tfrac{\pi}{2}+\eta\\},\quad\text{for any
}0<\eta<\tfrac{\pi}{2},$
on which exist unique transformations $F_{0}(s)$ and
$F_{0}^{\mathsf{P}}\\!(s)=\left(\begin{smallmatrix}0&1\\\
1&0\end{smallmatrix}\right)F_{0}(-s)\left(\begin{smallmatrix}0&1\\\
1&0\end{smallmatrix}\right)$ such that
$F_{0}(0)\\!=\\!F_{0}^{\mathsf{P}}\\!(0)\\!=\\!I$ and
${F_{0}}^{*}\Delta^{\\!s}_{(0,0)}=(F_{0}^{\mathsf{P}}\\!\,)^{*}\Delta^{\\!s}_{(0,0)}=\mkern
2.0mu\overline{\Delta}^{s}_{(0,0)}$.
See e.g. [Ba], [IY], [Si] for more details on local sectorial normalization.
The above given local transformations do not behave well on parameters at the
limit when two or more singular points merge together. We need to construct
better behaving normalizing transformations $F_{\Omega}$. The domains $\Omega$
on which such transformations exist are closely related to the function
$\theta$ (28) and to the vector field
$\chi(s,\mu,\epsilon):=\frac{(s^{2}\\!-\mu)^{2}\\!-\epsilon}{4s^{2}}\,\partial_{s},$
(31)
on the Riemann sphere $\mathbb{CP}^{1}=\mathbb{C}\cup\\{\infty\\}$. (It is a
polynomial vector field in $s^{-1}$).
Domains $D_{i}$. Let be given an angular constant $0<\eta<\tfrac{\pi}{2}$,
constants $\delta_{s},\delta_{\mu},\delta_{\epsilon}>0$ and $L\geq 2$,
determining an annular region in the plane $s$
$|s|<\delta_{s},\quad|\frac{(s^{2}-\mu)^{2}-\epsilon}{s^{4}}|<L,$ (32)
and small open discs in the planes of the parameters $\mu,\epsilon$
$\mathsf{M}=\\{|\mu|<\delta_{\mu}\\},\quad\mathsf{E}=\\{|\epsilon|<\delta_{\epsilon}\\}.$
(33)
Let $(\check{\mu},\check{\epsilon})$ be a point of a ramified covering space
of $\,\mathsf{M}\\!\times\\!\mathsf{E}$, ramified over the set
$\\{\epsilon\,(\mu^{2}\\!-\epsilon)=0\\}$, (including the ramification
points), that projects to $(\mu,\epsilon)$. Let
$s_{i}(\check{\mu},\check{\epsilon})$ be a zero of
$\,(s^{2}\\!-\mu)^{2}\\!-\epsilon\,$ depending continuously on
$(\check{\mu},\check{\epsilon})$ and we shall suppose that
$\delta_{\mu},\delta_{\epsilon}$ are small enough so that all the zero points
$s_{i}(\check{\mu},\check{\epsilon})$ fall inside the disc of radius
$\delta_{s}$. So, for each value of $(\check{\mu},\check{\epsilon})$, we
define a ramified simply connected set $D_{i}(\check{\mu},\check{\epsilon})$,
as the union of real positive trajectories of the vector fields
$\,\omega\,\chi(s,\mu,\epsilon),$ while varying $\omega$:
$-\eta<\arg\omega<\eta,\,$
that end in the point $s_{i}(\check{\mu},\check{\epsilon})$ and never leave
the annular region (32) (see Figure 1). Hence $s\in
D_{i}(\check{\mu},\check{\epsilon})$ if there exists a trajectory
$\sigma(\xi),\ \xi\in\mathbb{R}^{+},$ of the field $\omega\chi$ depending
continuously on $s$, such that
* •
$s=\sigma(0),\quad s_{i}=\\!\lim_{\xi\to+\infty}\\!\sigma(\xi),$
* •
$|\sigma(\xi)|<\delta_{s},\quad|\frac{(\sigma(\xi)^{2}-\mu)^{2}-\epsilon}{\sigma(\xi)^{4}}|<L,\text{
for all }\,\xi\in\mathbb{R}^{+},$
where
$\omega\,\xi=2\theta(\sigma(\xi),\mu,\epsilon)-2\theta(s,\mu,\epsilon),$ (34)
for the function $\theta$ (28). For some values of
$(\check{\mu},\check{\epsilon})$ the set $D_{i}(\check{\mu},\check{\epsilon})$
can be empty: when $s_{i}(\check{\mu},\check{\epsilon})$ is repulsive for all
the fields $\omega\chi$, or when $s_{i}(\check{\mu},\check{\epsilon})=0$ and
$\mu\neq 0$, in which case $s_{i}$ is not a singular point of $\chi$.
(a) $0<\epsilon<\mu^{2},\ \mu>0$
(b) $\epsilon=0<\mu$
(c) $\epsilon=\mu=0$
Figure 1: The domains $D_{i}(\mu,\epsilon)$ for selected values of
$\mu,\epsilon$
###### Notation 13.
If $g(s)$ is a function on some domain $Y$ in the $s$-space, denote
$g^{\mathsf{P}}\\!(s):=g(e^{-\pi i}s),\quad s\in Y^{\mathsf{P}}\\!:=e^{\pi
i}Y$
the rotated function on the rotated domain. For a $2\\!\times\\!2$-matrix
function $G(s)$, denote
$G^{\mathsf{P}}\\!(s):=\left(\begin{smallmatrix}0&1\\\\[3.0pt]
1&0\end{smallmatrix}\right)G(e^{-\pi
i}s)\left(\begin{smallmatrix}0&1\\\\[3.0pt] 1&0\end{smallmatrix}\right),$
and for a constant matrix $C$,
$C^{\mathsf{P}}\\!:=\left(\begin{smallmatrix}0&1\\\\[3.0pt]
1&0\end{smallmatrix}\right)C\left(\begin{smallmatrix}0&1\\\\[3.0pt]
1&0\end{smallmatrix}\right).$
###### Proposition 14.
Let a system $\Delta(x,\mu,\epsilon)$ with a function $r(x,\mu,\epsilon)$ be
as in (20), and let the ramified domains $D_{i}(\check{\mu},\check{\epsilon})$
be defined as above with the constants $0<\eta<\tfrac{\pi}{2}$,
$\delta_{s}>0$, $L\geq 2$ independent of $(\check{\mu},\check{\epsilon})$ and
satisfying:
$L\delta_{s}(1+4\sup_{|s|<\delta_{s}}|s\,r|)\leq 2\cos\eta.$ (35)
If $\mathring{\Omega}(\check{\mu},\check{\epsilon})$ is a connected component
of $D_{i}(\check{\mu},\check{\epsilon})\cap
D_{j}^{\mathsf{P}}\\!(\check{\mu},\check{\epsilon})$ for some $i,j$, such that
the two points
$s_{i}(\check{\mu},\check{\epsilon}),-s_{j}(\check{\mu},\check{\epsilon})$ are
in the closure of $\mathring{\Omega}(\check{\mu},\check{\epsilon})$,
$\,\Omega(\check{\mu},\check{\epsilon}):=\mathring{\Omega}(\check{\mu},\check{\epsilon})\cup\\{s_{i}(\check{\mu},\check{\epsilon}),-s_{j}(\check{\mu},\check{\epsilon})\\},$
then there exists a normalizing transformation
$F_{\Omega}\in\mathrm{GL}_{2}(\mathcal{B}(\Omega))$ defined on the domain in
$(s,\check{\mu},\check{\epsilon})$-space
$\,\Omega:=\bigcup_{(\check{\mu},\check{\epsilon})}\Omega(\check{\mu},\check{\epsilon})\times\\{(\check{\mu},\check{\epsilon})\\},$
satisfying ${F_{\Omega}}^{*}\Delta^{\\!s}=\mkern 2.0mu\overline{\Delta}^{s}$
as in (24). This transformation can be uniquely chosen such that
$F_{\Omega}(s_{i},\check{\mu},\check{\epsilon})=\left(\begin{smallmatrix}\kappa_{\Omega}(\check{\mu},\check{\epsilon})&0\\\\[3.0pt]
0&1\end{smallmatrix}\right),\quad
F_{\Omega}(-s_{j},\check{\mu},\check{\epsilon})=\left(\begin{smallmatrix}1&0\\\\[3.0pt]
0&\kappa_{\Omega}(\check{\mu},\check{\epsilon})\end{smallmatrix}\right),$ (36)
with the function $\kappa_{\Omega}$ uniquely determined by the domain $\Omega$
and the system $\Delta$.
###### Proof.
We will be looking for $F_{\Omega}$ written as
$F_{\Omega}(s,\check{\mu},\check{\epsilon})=P_{\Omega}(s,\check{\mu},\check{\epsilon})\begin{pmatrix}e^{\,\int_{-s_{j}}^{s}\\!2\beta_{j}^{\mathsf{P}}\\!}\\!\\!\\!\\!\\!&\
0\\!\\!\\\ \\!\\!0\
&\\!\\!\\!\\!\\!e^{-\\!\\!\int_{s_{i}}^{s}\\!2\beta_{i}}\end{pmatrix},\ \text{
with }\ P_{\Omega}=\left(\begin{smallmatrix}1&p_{i}\\\\[3.0pt]
p_{j}^{\mathsf{P}}\\!&1\end{smallmatrix}\right)$ (37)
that diagonalizes $\Delta^{\\!s}:$
$(P_{\Omega})^{*}\Delta^{\\!s}=\frac{x^{2}\\!-\\!\epsilon}{2s^{2}}\frac{d}{ds}-\left[\left(\begin{smallmatrix}1&0\\\\[3.0pt]
0&-1\end{smallmatrix}\right)-\frac{x^{2}\\!-\\!\epsilon}{4s^{3}}I+\frac{x^{2}\\!-\\!\epsilon}{s^{2}}\left(\begin{smallmatrix}\beta_{j}^{\mathsf{P}}\\!(s)\\!\\!&\
0\\!\\\\[3.0pt] \\!0\ &\\!\\!-\beta_{i}(s)\end{smallmatrix}\right)\right].$
This means that the transformation $P_{\Omega}$ needs to satisfy
$\displaystyle\tfrac{x^{2}\\!-\epsilon}{2s^{2}}\tfrac{d}{ds}P_{\Omega}=\left[\left(\begin{smallmatrix}1&0\\\\[3.0pt]
0&-1\end{smallmatrix}\right),P_{\Omega}\right]+\tfrac{x^{2}\\!-\epsilon}{4s^{3}}\\!\left(\begin{smallmatrix}0&1\\\\[3.0pt]
1&0\end{smallmatrix}\right)\\!P_{\Omega}+r\tfrac{x^{2}\\!-\epsilon}{2s^{2}}\\!\left(\begin{smallmatrix}1&1\\\\[3.0pt]
\\!\\!-1&-1\\!\end{smallmatrix}\right)\\!P_{\Omega}-\tfrac{x^{2}\\!-\epsilon}{s^{2}}P_{\Omega}\\!\left(\begin{smallmatrix}\beta_{j}^{\mathsf{P}}\\!\\!&\
0\\!\\\\[3.0pt] 0\ &\\!-\beta_{i}\\!\end{smallmatrix}\right),$
$\displaystyle\tfrac{x^{2}\\!-\epsilon}{2s^{2}}\tfrac{d}{ds}\left(\begin{smallmatrix}0&p_{i}\\\\[3.0pt]
p_{j}^{\mathsf{P}}\\!&0\end{smallmatrix}\right)=2\\!\left(\begin{smallmatrix}0\,&p_{i}\\\\[3.0pt]
\\!\\!-p_{j}^{\mathsf{P}}\\!&\,0\end{smallmatrix}\right)+\tfrac{x^{2}\\!-\epsilon}{4s^{3}}\\!\left(\begin{smallmatrix}p_{j}^{\mathsf{P}}\\!&\,1\\\\[3.0pt]
1\,&\\!p_{i}\\!\end{smallmatrix}\right)+r\tfrac{x^{2}\\!-\epsilon}{2s^{2}}\\!\left(\begin{smallmatrix}1+p_{j}^{\mathsf{P}}\\!&\
1+p_{i}\\!\\\\[3.0pt]
\\!\\!-1-p_{j}^{\mathsf{P}}\\!&-1-p_{i}\\!\end{smallmatrix}\right)+\tfrac{x^{2}\\!-\epsilon}{s^{2}}\\!\left(\begin{smallmatrix}\\!\\!-\beta_{j}^{\mathsf{P}}\\!\
&\\!p_{i}\beta_{i}\\!\\\\[3.0pt]
\\!\\!-p_{j}^{\mathsf{P}}\\!\beta_{j}^{\mathsf{P}}\\!\\!&\
\beta_{i}\\!\end{smallmatrix}\right).$
The diagonal terms give:
$\begin{split}-\beta_{i}&=\tfrac{1}{4s}p_{i}-\tfrac{r}{2}(1+p_{i})\\\
\beta_{j}^{\mathsf{P}}\\!&=\tfrac{1}{4s}p_{j}^{\mathsf{P}}\\!+\tfrac{r}{2}(1+p_{j}^{\mathsf{P}}\\!\,).\end{split}$
The anti-diagonal terms after substitution of the
$\beta_{i},\,\beta_{j}^{\mathsf{P}}\\!$ and a division by
$\tfrac{x^{2}\\!-\epsilon}{2s^{2}}$ are:
$\displaystyle\tfrac{d}{ds}\,p_{i}$
$\displaystyle=\tfrac{4s^{2}}{x^{2}\\!-\epsilon}\,p_{i}+\tfrac{1}{2s}(1-p_{i}^{2})+(1+p_{i})^{2}r,$
(38) $\displaystyle-\tfrac{d}{ds}\,p_{j}^{\mathsf{P}}\\!$
$\displaystyle=\tfrac{4s^{2}}{x^{2}\\!-\epsilon}\,p_{j}^{\mathsf{P}}\\!-\tfrac{1}{2s}(1-\\!(p_{j}^{\mathsf{P}}\\!\,)^{2})+(1+p_{j}^{\mathsf{P}}\\!\,)^{2}r.$
Therefore both $p_{i}$, $p_{j}$ are solutions of the same ODE (38).
###### Lemma 15.
The equation (38) possesses a unique solution $p_{i}\in\mathcal{B}(D_{i})$
that is bounded in the domain $D_{i}$ and satisfies $p_{i}(s_{i})=0$.
Hence $F_{\Omega}$ is well defined and bounded on the component
$\mathring{\Omega}$ of the intersection $D_{i}\cap D_{j}^{\mathsf{P}}\\!$. It
remains to show that the limits
$F_{\Omega}(s_{k},\check{\mu},\check{\epsilon})=\lim_{s\to
s_{k}}F_{\Omega}(s,\check{\mu},\check{\epsilon})=,\quad s_{k}=s_{i},-s_{j}$
exist and are diagonal. Using the Liouville formula we know that the
determinant of the fundamental solution
$\,\Phi_{\Omega}=SVF_{\Omega}\Psi_{\Omega}$ (30) of $\Delta(x,\mu,\epsilon)$
(20) is constant for each $(\check{\mu},\check{\epsilon})$ fixed since the
trace of the matrix of this system is null; therefore
$\det
F_{\Omega}(s,\check{\mu},\check{\epsilon})=\det\Phi_{\Omega}(x,\check{\mu},\check{\epsilon})=\kappa_{\Omega}(\check{\mu},\check{\epsilon})\in\mathbb{C}^{*}.$
If $F_{\Omega}(s_{i})$ is diagonal, then we also know from the construction
that $F_{\Omega}(s_{i})$ has 1 at second diagonal position, and similarly that
$F_{\Omega}(-s_{j})$ has 1 at the first diagonal position, so we obtain (36).
To see that the limit of $F_{\Omega}$ exists at
$s=s_{k}(\check{\mu},\check{\epsilon})$, we can use the results of Remark 12:
for each fixed value of $(\check{\mu},\check{\epsilon})$ we have one or
several domains on which a normalizing transformation exists that extend
continuously to the point $s_{k}$. A better inspection of our
$\Omega(\check{\mu},\check{\epsilon})$ will show that its restriction to a
small neighborhood of $s_{k}$ is contained in one of the domains from Remark
12, and therefore $F_{\Omega}$ can be written as a corresponding
transformation from Remark 12 multiplied on the right side by a bounded
automorphism of $\mkern 2.0mu\overline{\Delta}^{s}_{(\mu,\epsilon)}$. (An
invertible matrix function $A(s)$ is an _automorphism_ of $\mkern
2.0mu\overline{\Delta}^{s}_{(\mu,\epsilon)}$ if $A^{*}\mkern
2.0mu\overline{\Delta}^{s}_{(\mu,\epsilon)}=\mkern
2.0mu\overline{\Delta}^{s}_{(\mu,\epsilon)}$.) Lemma 16 below shows that this
automorphism extends continuously to the point $s_{k}$ as a diagonal matrix,
therefore
$F_{\Omega}(s_{k}(\check{\mu},\check{\epsilon}),\check{\mu},\check{\epsilon})$
exist and is diagonal. The unicity of this $F_{\Omega}$ follows from Corollary
17 below. ∎
###### Proof of Lemma 15.
We are looking for a solution $p_{i}\in\mathcal{B}(D_{i})$ of (38) satisfying
$\,p_{i}(s_{i})\\!=\\!0$. From the definition of the function $\theta$ (27) we
have
$e^{2\theta}\tfrac{d}{ds}(e^{-2\theta}p_{i})=\tfrac{dp_{i}}{ds}-\tfrac{4s^{2}}{x^{2}-\epsilon}p_{i},$
which we use to rewrite (38) as an integral equation
$p_{i}(s)=\int_{s_{i}}^{\,s}\\!\\!e^{2\theta(s)-2\theta(\sigma)}\\!\left(\tfrac{1-p_{i}^{2}(\sigma)}{2\sigma}+(1+p_{i}(\sigma))^{2}\cdot
r(\sigma^{2}\\!\\!-\\!\mu)\right)d\sigma\,:=\mathcal{K}_{i}p_{i}(s).$
Taking the real trajectories $\sigma(\xi),\ \xi\in\mathbb{R}^{+},$ of the
field $\omega\chi$ (31), $|\omega|=1,\ |\arg\omega|<\eta$, as the integration
trajectories and substituting $\xi$ as in (34), we obtain
$\mathcal{K}_{i}p_{i}(s)=-\omega\int_{0}^{+\infty}\\!\\!\\!e^{-\omega\xi}\,\frac{(\sigma^{2}\\!-\mu)^{2}\\!-\epsilon}{4\sigma^{2}}\left(\frac{1-p_{i}^{2}}{2\sigma}+(1+p_{i})^{2}\\!\cdot
r\right)d\xi$
We are looking for a fixed point $p_{i}$ of $\mathcal{K}_{i}$ in the space of
the functions $f\in\mathcal{B}(D_{i})$ bounded by 1
$||f||:=\sup_{s\in D_{i}}|f(s)|\leq 1.$
Using (32) and (35) we have
$\left|\tfrac{(s^{2}\\!-\mu)^{2}\\!-\epsilon}{4s^{2}}\left(\tfrac{(1-f(s)^{2})}{2s}+(1+f(s))^{2}\\!\cdot
r\right)\right|\leq\tfrac{1}{4}L\delta_{s}\left(1+4||sr||\right)\leq\frac{1}{2}\cos\eta.$
Therefore
$||\mathcal{K}_{i}f||\leq\tfrac{1}{2}\cos\eta\cdot\\!\int_{0}^{+\infty}\\!\\!\\!e^{-\mathrm{Re}\,\omega\,\xi}d\xi\,\leq\,\tfrac{1}{2}\tfrac{\cos\eta}{\mathrm{Re}\,\omega}\,\leq\,\tfrac{1}{2}.$
Similarly
$\begin{split}\left|\tfrac{(s^{2}\\!-\\!\mu)^{2}\\!-\epsilon}{4s^{2}}\\!\left(-\tfrac{{f_{1}}+{f_{2}}}{2s}+(2+\\!{f_{1}}\\!+\\!{f_{2}})\cdot
r\right)\left({f_{1}}-{f_{2}}\right)\right|\,&\leq\,\tfrac{1}{4}L\delta_{s}\left(1+4||sr||\right)||f_{1}-f_{2}||\\\
\,&\leq\,\tfrac{1}{2}\cos\eta\cdot||f_{1}-f_{2}||,\end{split}$
and hence
$||\mathcal{K}_{i}f_{1}-\mathcal{K}_{i}f_{2}||<\tfrac{1}{2}||f_{1}-f_{2}||.$
So $\mathcal{K}_{i}$ is a contraction and the solution $p_{i}$ of (38) on
$D_{i}$ exists, and is unique and bounded by $1$. ∎
###### Lemma 16.
Let $U$ be a simply connected open subset of
$\mathbb{C}\\!\smallsetminus\\!\\{s\mid(s^{2}\\!-\mu)\\!-\epsilon=0\\}\,$ and
let $A_{U}(s)$ be an _automorphism_ of the diagonal system $\mkern
2.0mu\overline{\Delta}^{s}_{(\mu,\epsilon)}$ (24), i.e. ${A_{U}}^{*}\mkern
2.0mu\overline{\Delta}^{s}_{(\mu,\epsilon)}=\mkern
2.0mu\overline{\Delta}^{s}_{(\mu,\epsilon)}$. For a fundamental solution
$\Psi(s)$ (29) of $\mkern 2.0mu\overline{\Delta}^{s}_{(\mu,\epsilon)}$, we can
write
$A_{U}(s)=\Psi(s)C\Psi^{-1}(s)$
for a constant invertible matrix $C=(c_{kl})$. If $A_{U}$ is bounded on $U$,
and if $U$ contains a real-positive trajectory $\sigma_{+}\xi)$ of
$\omega_{+}\chi$ (resp. a real-negative trajectory $\sigma_{-}(\xi)$ of
$\omega_{-}\chi$) for some $\omega_{\pm}$,
$|\arg\omega_{\pm}|<\tfrac{\pi}{2}$, then necessarily $c_{12}=0$ (resp.
$c_{21}=0$) and
$\lim_{\xi\to+\infty}A(\sigma_{+}(\xi))=\left(\begin{smallmatrix}c_{11}&0\\\\[3.0pt]
0&c_{22}\end{smallmatrix}\right)\quad\left(\text{resp. }\
=\lim_{\xi\to-\infty}A(\sigma_{-}(\xi))\ \right).$
###### Proof.
Let $\theta(s)$ be a branch of the function $\theta(s,\mu,\epsilon)$ in (28)
on $U$. We have
$A_{U}(s)=\begin{pmatrix}c_{11}&\\!\\!e^{2\theta(s)}c_{12}\\\
e^{-2\theta(s)}c_{21}\\!\\!&c_{22}\end{pmatrix}.$
If $|\arg\omega_{\pm}|<\frac{\pi}{2}$, then it follows from (34) that
$\mathrm{Re}\,\theta(\sigma_{+}(\xi))\to+\infty$ as $\xi\to+\infty$ and
$\mathrm{Re}\,\theta(\sigma_{-}(\xi))\to-\infty$ as $\xi\to-\infty$. ∎
###### Corollary 17.
An automorphism $A_{\Omega}$ of $\mkern 2.0mu\overline{\Delta}^{s}$ (24) on a
domain $\Omega$ of Proposition 14 is just a constant diagonal matrix
(depending on $(\check{\mu},\check{\epsilon})$).
In order to know the domains $\Omega$ of Proposition 14 better, we need first
understand the vector field $\chi$.
Bifurcations of the vector field $\chi$. If the condition (32) in the
definition of the domains $D_{i}$ was skipped, that is if
$\delta_{s}\\!=\\!L\\!=\\!+\infty$, and if $\omega\\!=\\!1$ was fixed, then
the domains $\mathring{\Omega}(\check{\mu},\check{\epsilon})$ of Proposition
14 would be exactly the regions in $\mathbb{CP}^{1}$ bounded by the real
separatrices of the singularity at the origin of the vector field
$\chi(s,\mu,\epsilon)$ and by its unique real trajectory passing through the
point at infinity. For a generic value of the parameters
$(\check{\mu},\check{\epsilon})$, this gives 4 different regions: a symmetric
pair of inner regions that are bounded solely by the separatrices of the
origin, and a symmetric pair of outer regions bounded by the trajectory
through $\infty$ and the separatrices of the origin, see Figure 3. When
$\epsilon=0$, each of the inner regions splits in two parts; when
$\mu^{2}=\epsilon$, the inner regions are empty.
Let’s take a look how do these regions evolve depending on the parameters
$(\check{\mu},\check{\epsilon})$. There are two possibilities for a
bifurcation:
* $\Sigma_{I}$:
when the stability of a zero point $s_{i}$ of $\chi$ changes between
attractive and repulsive through a center: the dashed lines in Figure 2,
* $\Sigma_{O}$:
when the trajectory passing by infinity changes its end points: the solid
curves (branch of a hyperbola) in Figure 2.
The bifurcation $\Sigma_{I}$ occurs when the multiplier
$\frac{\sqrt{\epsilon}}{s_{i}}$ of the linearization
$\frac{\sqrt{\epsilon}}{s_{i}}(s-s_{i})\partial_{s}$ of vector field $\chi$ at
the point $s_{i}=\pm\sqrt{\mu\pm\sqrt{\epsilon}}$ becomes purely imaginary:
$\frac{\sqrt{\epsilon}}{\sqrt{\mu\pm\sqrt{\epsilon}}}\in i\mathbb{R},$ which
is equivalent to
$\mu\in\mp\sqrt{\epsilon}-\epsilon\,\mathbb{R}^{+}=:\Sigma_{I}(\epsilon).$
(39)
It is well known that a holomorphic vector field in $\mathbb{C}$ is
analytically equivalent to its linearization near each simple zero (see e.q.
[IY]). As a consequence, if $\mu\in\Sigma_{I}(\epsilon)$ then the real phase
portrait of $\chi$ near the point $s_{i}$ with purely imaginary multiplier is
that of a center.
The bifurcation $\Sigma_{O}$ occurs when the trajectory through infinity
passes by the origin, or equivalently if a separatrix of the origin passes
through infinity. This means that $\ \theta(0)-\theta(\infty)\in\mathbb{R}$,
where $\theta$ is as in (28), i.e.
$\pm\tfrac{\sqrt{\mu+\sqrt{\epsilon}}\pm\sqrt{\mu-\sqrt{\epsilon}}}{2\sqrt{\epsilon}}\pi
i\in\mathbb{R},$
which is equivalent to
$\,-\frac{\mu\pm\sqrt{\mu^{2}-\epsilon}}{\epsilon}=a\in\mathbb{R}^{+}\,$ or
$\mu\in\\{-\tfrac{1}{2}(a^{-1}\\!+\epsilon a)\mid
a>0\\}=:\Sigma_{O}(\epsilon).$ (40)
The set $\Sigma_{O}(\epsilon)$ is a branch of a hyperbola.
Note that both of the bifurcations $\Sigma_{I}$, $\Sigma_{O}$ are instances of
the same phenomenon: the appearance of a homoclinic orbit through the origin
in $\mathbb{CP}^{1}$.
###### Remark 18.
Under the transformation: $\ s^{\prime}=\omega
s,\quad\mu^{\prime}=\omega^{2}\mu,\quad\epsilon^{\prime}=\omega^{2}\epsilon,\,$
the vector field $\omega\chi$ becomes the same as $\chi$ of the new
coordinates
$\omega\chi(s,\mu,\epsilon)=\frac{(s^{\prime
2}\\!-\mu^{\prime})^{2}\\!-\epsilon^{\prime}}{s^{\prime
2}}\partial_{s^{\prime}}.$
Figure 2: Bifurcation curves in the $\mu$-plane for the vector field
$\chi(s,\mu,\epsilon)$ according to values of $\epsilon$: dashed lines
$\Sigma_{I}(\epsilon)$ mark change of stability of a singular point, solid
line curve $\Sigma_{O}(\epsilon)$ marks bifurcation of the trajectory passing
through $\infty$.
(0) $\epsilon=0$:
(i) $\epsilon\in\mathbb{R}^{+}$:
(ii) $\epsilon\in i\mathbb{R}^{+}$:
(iii) $\epsilon\in-\mathbb{R}^{+}$:
(iv) $\epsilon\in-i\mathbb{R}^{+}$:
Figure 3: The real phase portrait of the vector field $\chi$ according to
$\mu$ for selected values of $\epsilon$.
The domains $\Omega$. If $\Omega(\check{\mu},\check{\epsilon})$ is a domain
from Proposition 14, then
$s\in\mathring{\Omega}(\check{\mu},\check{\epsilon})$ means that there exist
$\omega_{+}$ (resp. $\omega_{-}$) for which the positive trajectory of
$\omega_{+}\chi$ (resp. negative trajectory of $\omega_{-}\chi$) starting at
$s$ stays within the annular region (32) and connects to the point
$s_{i}(\check{\mu},\check{\epsilon})$ (resp.
$-s_{j}(\check{\mu},\check{\epsilon})$). We will not lose much by restricting
$\mathring{\Omega}(\check{\mu},\check{\epsilon})$ only to the points for which
the same $\,\omega_{+}\\!=\\!\omega_{-}\\!=:\\!\omega\,$ is admissible, in
another words, the whole real trajectory of $\omega\chi$ through $s$ stays
inside the annular region. This shall significantly simplify our further
considerations.
For each $\omega$ take the two symmetric pairs of regions in the $s$-plane
bounded by the separatrices (and the trajectory through infinity) of
$\omega\chi(s,\mu,\epsilon)$, and restrict them to the points that lie on a
real trajectory of the field $\omega\chi(s,\mu,\epsilon)$ staying for ever
inside the annulus (32). In general there are four of these regions: a
symmetric pair of inner regions, denote them
$\mathsf{R}_{I,\omega}(\check{\mu},\check{\epsilon})$,
$\mathsf{R}_{I,\omega}^{\mathsf{P}}\\!(\check{\mu},\check{\epsilon})$, and a
symmetric pair of outer regions, denote them
$\mathsf{R}_{O,\omega}(\check{\mu},\check{\epsilon})$,
$\mathsf{R}_{O,\omega}^{\mathsf{P}}\\!(\check{\mu},\check{\epsilon})$, see
Figure 4 (a). Corresponding to these regions, there are four domains
$\Omega(\check{\mu},\check{\epsilon})$: a symmetric pair of inner domains
$\Omega_{I}$, $\Omega_{I}^{\mathsf{P}}\\!$ (resp. outer domains $\Omega_{O}$,
$\Omega_{O}^{\mathsf{P}}\\!\,$), obtained as ramified unions of the regions
$\mathsf{R}_{I,\omega}$, $\mathsf{R}_{I,\omega}^{\mathsf{P}}\\!$ (resp.
$\mathsf{R}_{O,\omega}$, $\mathsf{R}_{O,\omega}^{\mathsf{P}}\\!\,$) over
varying $\omega$.
To understand how do they behave in dependence on the parameters, let’s first
look at the regions $\mathsf{R}_{\bullet,\omega}$. First of all, it follows
from Remark 18 that if $|\omega|=1$ then
$\mathsf{R}_{\bullet,\omega}(\check{\mu},\check{\epsilon})=\omega^{-1}\mathsf{R}_{\bullet,1}(\omega^{2}\check{\mu},\omega^{4}\check{\epsilon}),\quad\bullet=O,I.$
Now, when $\mu\in\Sigma_{I}(\epsilon)\cup\\{\pm\sqrt{\epsilon}\\}$, the inner
regions $\mathsf{R}_{I,1}(\check{\mu},\check{\epsilon})$ become empty, and
they split in two components when $\epsilon=0$ ($\mu\neq 0$). The outer
regions $\mathsf{R}_{O,\omega}(\check{\mu},\check{\epsilon})$ are empty
whenever a separatrix of the origin of the field $\omega\chi(s,\mu,\epsilon)$
leaves the disc of radius $\delta_{s}$ (32): this happens for values of
$(\mu,\epsilon)$ close to the bifurcation set $\Sigma_{O}$ (40), see Figure 4
(b). Therefore a bifurcation of the region $\mathsf{R}_{O,\omega}$ occurs when
a separatrix touches the boundary of the disc of radius $\delta_{s}$ from
inside for the first time: at that moment the region ceases to exist as no
outer points can be joined to both $s_{i}$ and $-s_{j}$ inside the annulus.
Figure 4: The outer and inner regions
$\mathsf{R}_{O,1}(\check{\mu},\check{\epsilon})$ and
$\mathsf{R}_{I,1}(\check{\mu},\check{\epsilon})$ (with $\omega=1$) inside the
annulus (32) for (a) $\epsilon\in i\mathbb{R}^{+},\ \mu=0$, (b) $\epsilon\in
i\mathbb{R}^{+}$, $\mu$ close to $\Sigma_{I}(\epsilon)$:
$\mathsf{R}_{O,1}=\mathsf{R}_{O,1}^{\mathsf{P}}\\!=\emptyset$. – Compare with
the corresponding vector fields in Figure 3 (ii).
The same kind of bifurcation happens to the domains
$\Omega_{I}(\check{\mu},\check{\epsilon})$ and
$\Omega_{O}(\check{\mu},\check{\epsilon})$, but this time it is delayed by the
effect of variation of $\omega$. We will consider these domains as being
parametrized by $(\check{\mu},\check{\epsilon})$ from a ramified space
$\check{\mathcal{M}}$ defined below, whose boundary is composed by the
bifurcation sets.
Let $\eta$, $\delta_{s}$ be as in Proposition 14 and let
$\delta_{\mu},\,\delta_{\epsilon}$ be small enough,
$\mathsf{S}=\\{|s|<\delta_{s}\\},\quad\mathsf{E}=\\{|\epsilon|<\delta_{\epsilon}\\},\quad\mathsf{M}=\\{|\mu|<\delta_{\mu}\\},$
(41)
and define a ramified sectorial cover $\check{\mathsf{E}}$ of $\mathsf{E}$ as
$\check{\mathsf{E}}=\\{\check{\epsilon}\mid|\check{\epsilon}|<\delta_{\epsilon}\
\&\ |\arg\check{\epsilon}|<2\pi+\eta\\},$
with each $\check{\epsilon}$ being projected to $\epsilon\in\mathsf{E}$. For
each value of $\omega$ and $\check{\epsilon}\in\check{\mathsf{E}}$, let
$\mathsf{M}_{\omega}(\check{\epsilon})$ denote the connected component of the
set
$\\{\mu\in\mathsf{M}\mid\mathsf{R}_{O,\omega}(\mu,\epsilon)\neq\emptyset\\}\,\\!\smallsetminus\\!\,\Sigma_{\text{I}}(\epsilon)$
that contains the point $\mu\\!=\\!\sqrt{\check{\epsilon}}$. By Remark 18, for
$|\omega|=1$ we can write
$\mathsf{M}_{\omega}(\check{\epsilon})=\omega^{-2}\mathsf{M}_{1}(\omega^{4}\check{\epsilon})$.
Define the domain $\check{\mathsf{M}}(\check{\epsilon})$ of ramified parameter
$\check{\mu}$ as a ramified union
$\check{\mathsf{M}}(\check{\epsilon})=\bigcup_{\begin{subarray}{c}|\omega|=1\\\
|\arg\omega|<\eta\end{subarray}}\\!\
\\!\mathsf{M}_{\omega}(\check{\epsilon})=\bigcup_{\begin{subarray}{c}|\omega|=1\\\
|\arg\omega|<\eta\end{subarray}}\\!\
\\!\omega^{-2}\mathsf{M}_{1}(\omega^{4}\check{\epsilon}),$
with $\check{\mu}\\!=\\!\sqrt{\check{\epsilon}}$ as the ramification point
included in $\check{\mathsf{M}}(\check{\epsilon})$. See Figure 5.
Figure 5: The ramified domains $\check{\mathsf{M}}(\check{\epsilon})$ for the
parameter $\check{\mu}$ depending on $\check{\epsilon}\in\check{\mathsf{E}}$.
Also define their union $\check{\mathsf{M}}$ in the
$(\check{\mu},\check{\epsilon})$-space fibered over $\check{\mathsf{E}}$ as
$\check{\mathsf{M}}:=\bigcup_{\check{\epsilon}\in\check{\mathsf{E}}}\check{\mathsf{M}}(\check{\epsilon})\times\\{\check{\epsilon}\\}.$
(42)
If $\,\bar{\epsilon}\in\check{\mathsf{E}}\cap e^{-2\pi i}\check{\mathsf{E}}\,$
and $\tilde{\epsilon}=e^{2\pi i}\bar{\epsilon}\in\check{\mathsf{E}}\cap
e^{2\pi i}\check{\mathsf{E}}$ are two ramified parameters in
$\check{\mathsf{E}}$ that correspond to the same $\epsilon$, then the two
domains $\check{\mathcal{M}}(\bar{\epsilon})$,
$\check{\mathcal{M}}(\tilde{\epsilon})$ form together a ramified cover of the
$\mu$-space $\mathsf{M}$ (if $\delta_{\mu}$, $\delta_{\epsilon}$ in (41) are
sufficiently small), see Figure 5. _The space $\check{\mathcal{M}}$ is a
single simply connected ramified cover of the whole space
$\mathsf{M}\\!\times\\!\mathsf{E}$._
From now on, let
$s_{1}(\check{\mu},\check{\epsilon}):=\sqrt{\check{\mu}+\sqrt{\check{\epsilon}}},\quad
s_{2}(\check{\mu},\check{\epsilon}):=\sqrt{\check{\mu}-\sqrt{\check{\epsilon}}}$
(43)
with the $\sqrt{\ }$ being such that for $\check{\mu},\check{\epsilon}>0$ and
$\arg\epsilon=0,$ it coincides with the standard square root. Le’s agree that
out of the two inner regions,
$\mathsf{R}_{I,\omega}(\check{\mu},\check{\epsilon})$ is the one consisting of
trajectories from $s_{1}(\check{\mu},\check{\epsilon})$ to
$s_{2}(\check{\mu},\check{\epsilon})$, and that out of the two outer regions
(both consisting of trajectories from $s_{1}(\check{\mu},\check{\epsilon})$ to
$-s_{1}(\check{\mu},\check{\epsilon})$),
$\mathsf{R}_{I,\omega}(\check{\mu},\check{\epsilon})$ is the upper one.
Now for each $(\check{\mu},\check{\epsilon})\in\check{\mathcal{M}}$ let
$\mathring{\Omega}_{\bullet}(\check{\mu},\check{\epsilon})=\bigcup_{\begin{subarray}{c}\omega\text{
such that}\\\
\mu\in\mathsf{M}_{\omega}(\check{\epsilon})\end{subarray}}\mathsf{R}_{\bullet,\omega}(\check{\mu},\check{\epsilon}),\quad\bullet=O,I$
(44)
be a ramified union of the regions in the $s$-plane, and let
$\Omega_{I}(\check{\mu},\check{\epsilon})\quad(\text{resp. }\
\Omega_{O}(\check{\mu},\check{\epsilon})\,)$
be $\mathring{\Omega}_{I}(\check{\mu},\check{\epsilon})$ (resp.
$\mathring{\Omega}_{O}(\check{\mu},\check{\epsilon})$) with the corresponding
zero points
$s_{1}(\check{\mu},\check{\epsilon}),s_{2}(\check{\mu},\check{\epsilon})$
(resp.
$s_{1}(\check{\mu},\check{\epsilon}),-s_{1}(\check{\mu},\check{\epsilon})$) of
$(s^{2}\\!-\mu)^{2}\\!-\epsilon$ from its adherence added as in Proposition
14. Again there is a natural projection
$\Omega_{\bullet}(\check{\mu},\check{\epsilon})\ni\check{s}\mapsto
s\in\mathbb{C}.$
(a) $0<\epsilon<\mu^{2},\ \mu>0$
(b) $\epsilon=0<\mu$
(c) $\epsilon=\mu=0$
Figure 6: The domains $\Omega_{O}(\check{\mu},\check{\epsilon})$ and
$\Omega_{I}(\check{\mu},\check{\epsilon})$ for selected values of
$\mu,\epsilon$.
Finally let $\Omega_{\bullet}$ be the union of all
$\Omega_{\bullet}(\check{\mu},\check{\epsilon})$ in the space
$(\check{s},\check{\mu},\check{\epsilon})$ fibered over $\check{\mathcal{M}}$
$\Omega_{\bullet}:=\bigcup_{(\check{\mu},\check{\epsilon})\in\check{\mathcal{M}}}\Omega_{\bullet}(\check{\mu},\check{\epsilon})\times\\{(\check{\mu},\check{\epsilon})\\}.$
While the outer domain $\Omega_{O}(\check{\mu},\check{\epsilon})$ is connected
nonempty for all $(\check{\mu},\check{\epsilon})\in\check{\mathcal{M}}$,
following from its construction, the inner domain becomes empty whenever
$\mu^{2}=\epsilon$, and splits into two components if $\epsilon=0$:
$\Omega_{I}(\check{\mu},0)=\Omega_{I\\!+\\!}(\check{\mu},0)\cup\Omega_{I\\!-\\!}(\check{\mu},0)$,
with
$\,\Omega_{I\\!+\\!}(\check{\mu},0)\cap\Omega_{I\\!-\\!}(\check{\mu},0)\\!=\\!\\{\\!\sqrt{\check{\mu}}\\}$
(see Figure 6).
###### Proposition 14 revisited.
Let the systems $\Delta^{\\!s}(s,\mu,\epsilon)$ be as in (23) and $\mkern
2.0mu\overline{\Delta}^{s}(s,\mu,\epsilon)$ be as in (24). On the domains
$\Omega_{\bullet}$ and $\Omega_{\bullet}^{\mathsf{P}}\\!=e^{\pi
i}\Omega_{\bullet}$, $\bullet=O,I$, defined above, there exist unique
diagonalising transformations
$\,F_{\bullet}\in\mathrm{GL}_{2}(\mathcal{B}(\Omega_{\bullet})),$ | $\,F_{\bullet}{}^{*}\Delta^{\\!s}=\mkern 2.0mu\overline{\Delta}^{s}$
---|---
$F_{\bullet}^{\mathsf{P}}\\!\in\mathrm{GL}_{2}(\mathcal{B}(\Omega_{\bullet}^{\mathsf{P}}\\!)),$ | $F_{\bullet}^{\mathsf{P}}\\!\,{}^{*}\Delta^{\\!s}=\mkern 2.0mu\overline{\Delta}^{s}$
(see Definition 11 and Notation 13), determined by
$\,F_{I}(s_{1},\check{\mu},\check{\epsilon})=\left(\begin{smallmatrix}1&0\\\\[3.0pt] 0&\kappa_{I}(\check{\mu},\check{\epsilon})\end{smallmatrix}\right)$, | $\,F_{I}(s_{2},\check{\mu},\check{\epsilon})=\left(\begin{smallmatrix}\kappa_{I}(\check{\mu},\check{\epsilon})&0\\\\[3.0pt] 0&1\end{smallmatrix}\right),$
---|---
$F_{O}(s_{1},\check{\mu},\check{\epsilon})=\left(\begin{smallmatrix}1&0\\\\[3.0pt] 0&\kappa_{O}(\check{\mu},\check{\epsilon})\end{smallmatrix}\right)$, | $F_{O}(s_{2},\check{\mu},\check{\epsilon})=\left(\begin{smallmatrix}\kappa_{O}(\check{\mu},\check{\epsilon})&0\\\\[3.0pt] 0&1\end{smallmatrix}\right),$
(45)
with unique functions
$\,\kappa_{O},\,\kappa_{I}\in\mathcal{C}(\check{\mathcal{M}})\cap\mathcal{O}(\,\mathrm{int}\,\check{\mathcal{M}})$.
For the following discussion we will want to fix a branch $\Psi_{\bullet}$ of
the fundamental solution $\Psi$ (29) of the diagonal system $\mkern
2.0mu\overline{\Delta}^{s}$ on the domains $\Omega_{\bullet}$. In order to do
so, we need to split the inner domain
$\Omega_{I}(\check{\mu},\check{\epsilon})$ in two parts:
$\Omega_{I\\!+\\!}(\check{\mu},\check{\epsilon})$ and
$\Omega_{I\\!-\\!}(\check{\mu},\check{\epsilon})$, corresponding to the two
components of $\mathring{\Omega}_{I}(\check{\mu},0)$ when
$\check{\epsilon}=0$.
First we divide each inner region
$\mathsf{R}_{I,\omega}(\check{\mu},\check{\epsilon})$ of the field
$\omega\chi$ into two parts by cutting it along a chosen trajectory going from
the repelling point $s_{1}(\check{\mu},\check{\epsilon})$ to the attracting
point $s_{2}(\check{\mu},\check{\epsilon})$. If $\theta$ is the function (28),
one knows that the imaginary part of $\omega^{-1}\theta$ stays constant along
each trajectory. Using (43), we can write $\theta$ as
$\theta(\check{s},\check{\mu},\check{\epsilon})=\tfrac{s_{1}(\check{\mu},\check{\epsilon})}{2\sqrt{\check{\epsilon}}}\log\tfrac{s-s_{1}(\check{\mu},\check{\epsilon})}{s+s_{1}(\check{\mu},\check{\epsilon})}-\tfrac{s_{2}(\check{\mu},\check{\epsilon})}{2\sqrt{\check{\epsilon}}}\log\tfrac{s-s_{2}(\check{\mu},\check{\epsilon})}{s+s_{2}(\check{\mu},\check{\epsilon})},$
and calculate (by setting $s=0$) that points on the unique separatrix
connecting 0 to $s_{2}$ satisfy
$\mathrm{Im}\,(\tfrac{\theta(s,\check{\mu},\check{\epsilon})}{\omega})\in\mathrm{Im}\,(\tfrac{s_{1}(\check{\mu},\check{\epsilon})-s_{2}(\check{\mu},\check{\epsilon})}{2\,\omega\sqrt{\check{\epsilon}}})+\mathrm{Im}\,(\tfrac{s_{2}(\check{\mu},\check{\epsilon})}{\omega\sqrt{\check{\epsilon}}}\pi
i)\,\mathbb{Z}.$
To cut $\mathsf{R}_{I,\omega}$ we will use the “opposite” incoming trajectory
to $s_{2}$, that is the trajectory, whose points $s$ satisfy
$\mathrm{Im}\,(\tfrac{\theta(s,\check{\mu},\check{\epsilon})}{\omega})\in\mathrm{Im}\,(\tfrac{s_{1}(\check{\mu},\check{\epsilon})+s_{2}(\check{\mu},\check{\epsilon})}{2\,\omega\sqrt{\check{\epsilon}}})+\mathrm{Im}\,(\tfrac{s_{2}(\check{\mu},\check{\epsilon})}{\omega\sqrt{\check{\epsilon}}}\pi
i)\,\mathbb{Z}$
(the imaginary part of $\omega^{-1}\theta$ is shifted by the half-phase
$\mathrm{Im}\,(\tfrac{s_{2}(\check{\mu},\check{\epsilon})}{2\omega\sqrt{\check{\epsilon}}}\pi
i)$).
For each $\omega$ this trajectory divides $\mathsf{R}_{I,\omega}$ into:
$\mathsf{R}_{I\\!+\\!,\omega}$ and $\mathsf{R}_{I\\!-\\!,\omega}$, see Figure
7 (a).
We define $\Omega_{I\\!+\\!}(\check{\mu},\check{\epsilon})$ and
$\Omega_{I\\!-\\!}(\check{\mu},\check{\epsilon})$ in the same way as in (44)
for each
$(\check{\mu},\check{\epsilon})\in\check{\mathcal{M}}\\!\smallsetminus\\!\\{\check{\mu}^{2}\\!=\\!\check{\epsilon}\\}$.
Hence
$\Omega_{I}=\Omega_{I\\!+\\!}\cup\Omega_{I\\!-\\!},\qquad\text{(see Figure
\ref{fig:Omega1}\,(b))}$
and we set $F_{I\\!+\\!}=F_{I}\\!\mid_{I\\!+\\!}$ and
$F_{I\\!-\\!}=F_{I}\\!\mid_{I\\!-\\!}$.
Figure 7: (a) The regions
$\mathsf{R}_{I\\!\pm,1}(\check{\mu},\check{\epsilon})$ of the vector field
$\chi$ with the dividing trajectory from $s_{1}$ to $s_{2}$ dotted. (Picture
with $\epsilon,\mu\in\mathbb{R}^{+},\,\mu^{2}>\epsilon$, compare to Figure 3
(i).) (b) The inner domains $\Omega_{I\\!+\\!}(\check{\mu},\check{\epsilon})$
and $\Omega_{I\\!-\\!}(\check{\mu},\check{\epsilon})$, see Figure 6 (a) for
the whole
$\Omega_{I}(\check{\mu},\check{\epsilon})=\Omega_{I\\!+\\!}(\check{\mu},\check{\epsilon})\cup\Omega_{I\\!-\\!}(\check{\mu},\check{\epsilon})$.
###### Definition 19.
Let $\Phi_{1}$, $\Phi_{2}$ be two fundamental solutions of a linear system on
two domains $U_{1}$, $U_{2}$ with connected non-empty intersection $U_{1}\cap
U_{2}$. We call the matrix $C=\Phi_{1}^{-1}\Phi_{2}$ _a connection matrix_
between $\Phi_{1}$ and $\Phi_{2}$ and represent it schematically as
$\Phi_{1}\xrightarrow{\ C\ }\Phi_{2}.$
Remark that if $M=\Phi(e^{2\pi i}x)\Phi(x)^{-1}$ is a matrix of monodromy in
the positive direction, then the corresponding arrow is drawn in the opposite
direction: see the figure below.
Choice of fundamental solutions $\Psi_{\bullet}$. On interior
$\mathring{\Omega}_{\bullet}$ of each of the domains $\Omega_{\bullet}$,
$\bullet=O,{I\\!+\\!},{I\\!-\\!}$, we fix a branch
$\Psi_{\bullet}(\check{s},\check{\mu},\check{\epsilon})$ of the fundamental
solution $\Psi(\check{s},\check{\mu},\check{\epsilon})$ (29) of the diagonal
system $\mkern 2.0mu\overline{\Delta}^{s}$ so that the connection matrices
between them are as in Figure 8.
Figure 8: The connection matrices between the fundamental solutions
$\Psi_{\bullet}$ for each fixed parameter $(\check{\mu},\check{\epsilon})$,
with $\check{\mu}^{2}\neq\check{\epsilon}\neq 0$, where $N_{1}$ and $N$ are as
in (46) and (47). If $\check{\epsilon}=0$ then
$s_{1}(\check{\mu},0)=s_{2}(\check{\mu},0)$ and the matrices $N_{1}$ are
missing from the picture. If $\check{\mu}^{2}\\!=\\!\check{\epsilon}$ then
only the fundamental solutions $\Psi_{O}$ and $\Psi_{O}^{\mathsf{P}}\\!$
persist together with the two connection matrices $-iI$. (Picture for
$(\check{\mu},\check{\epsilon})$ as in Figure 6(a)).
The matrices of monodromy of $\Psi(s,\check{\mu},\check{\epsilon})$ around the
points $s_{1}(\check{\mu},\check{\epsilon})$, resp.
$s_{2}(\check{\mu},\check{\epsilon})$, in the positive direction are are
independent of the choice of the branch, and are given by
$N_{1}(\check{\mu},\check{\epsilon})=\begin{pmatrix}e^{\frac{s_{1}(\check{\mu},\check{\epsilon})}{\sqrt{\check{\epsilon}}}\pi
i}\\!\\!\\!&0\\\
0&\\!\\!\\!e^{-\frac{s_{1}(\check{\mu},\check{\epsilon})}{\sqrt{\check{\epsilon}}}\pi
i}\end{pmatrix},\quad
N_{2}(\check{\mu},\check{\epsilon})=\begin{pmatrix}e^{-\frac{s_{2}(\check{\mu},\check{\epsilon})}{\sqrt{\check{\epsilon}}}\pi
i}\\!\\!\\!&0\\\
0&\\!\\!\\!e^{\frac{s_{2}(\check{\mu},\check{\epsilon})}{\sqrt{\check{\epsilon}}}\pi
i}\end{pmatrix},$ (46)
They satisfy
$N_{i}^{\mathsf{P}}\\!=N_{i}^{-1},\quad i=1,2.$
The matrix of monodromy of $\Psi(s,\check{\mu},\check{\epsilon})$ around both
of the points
$s_{1}(\check{\mu},\check{\epsilon}),\,s_{2}(\check{\mu},\check{\epsilon})$ is
equal to
$N(\check{\mu},\check{\epsilon})=N_{1}(\check{\mu},\check{\epsilon})N_{2}(\check{\mu},\check{\epsilon})=\begin{pmatrix}e^{\frac{s_{1}-s_{2}}{\sqrt{\check{\epsilon}}}\pi
i}\\!\\!\\!&0\\\ 0&\\!\\!\\!e^{-\frac{s_{1}-s_{2}}{\sqrt{\check{\epsilon}}}\pi
i}\end{pmatrix}.$ (47)
At the limit when $\check{\epsilon}\to 0$ we get
$\,N(\check{\mu},0)=\begin{pmatrix}e^{\frac{1}{\sqrt{\check{\mu}}}\pi
i}\\!\\!&0\\\ 0&\\!\\!e^{-\frac{1}{\sqrt{\check{\mu}}}\pi i}\end{pmatrix}$, is
for $\mu\neq 0$ the matrix of monodromy around the double zero
$s_{1}(\check{\mu},0)=s_{2}(\check{\mu},0)$, while none of the matrices
$N_{1}(\check{\mu},\check{\epsilon}),\ N_{2}(\check{\mu},\check{\epsilon})$
has a limit, which is the reason for splitting the domain $\Omega_{I}$ into
$\Omega_{I\\!+}$, $\Omega_{I\\!-}$ and choosing $\Psi_{I\\!+}$, $\Psi_{I\\!-}$
in the way we did. Therefore the fundamental solution$\Psi_{I\\!\pm}$ are well
defined on the whole $\mathring{\Omega}_{I\\!\pm}$ and $\Psi_{O}$ is well
defined on the whole $\mathring{\Omega}_{O}$.
Considering now the fundamental solutions $F_{\bullet}\Psi_{\bullet}$ of
$\Delta^{\\!s}$ (23) on the ramified
$(\check{s},\check{\mu},\check{\epsilon})$-space, there is a connection matrix
whenever for a point
$(s,\mu,\epsilon)\in\mathsf{S}\times\mathsf{M}\times\mathsf{E}$ there are two
domains $\Omega(\check{\mu},\check{\epsilon})$ that cover $s$, either both for
the same value of the ramified parameter
$(\check{\mu},\check{\epsilon})\in\check{\mathcal{M}}$, or for two different
ramified parameters corresponding to the same $(\mu,\epsilon)$. The collection
of all these connection matrices carries all information about the analytic
equivalence class of the system $\Delta$ (20).
###### Proposition 20.
Let $\Delta(x,\mu,\epsilon)$, $\Delta^{\prime}(x,\mu,\epsilon)$ be two
families of parametric systems (20) and let $\Delta^{\\!s}(s,\mu,\epsilon)$,
$\Delta^{\prime s}(s,\mu,\epsilon)$ be their transforms as in (23). Let
$F_{\bullet}$, $F^{\prime}_{\bullet}$ be normalizing transformations for
$\Delta^{\\!s}$, $\Delta^{\prime s}$:
$F_{\bullet}{}^{*}\Delta^{\\!s}=\mkern
2.0mu\overline{\Delta}^{s}=F_{\bullet}^{\prime}{}^{*}\Delta^{\\!s}$
on the domains $\bullet=O,{I\\!+\\!},{I\\!-\\!}$ defined above. If all the
connection matrices associated to the fundamental solutions
$F_{\bullet}\Psi_{\bullet}$ of $\Delta^{\\!s}$ agree with those associated to
the fundamental solutions $F^{\prime}_{\bullet}\Psi_{\bullet}$ of
$\Delta^{\prime s}$, then the two parametric families of systems $\Delta$,
$\Delta^{\prime}$ are analytically equivalent.
###### Proof.
Let
$H(s,\mu,\epsilon):=F^{\prime}_{\bullet}(\check{s},\check{\mu},\check{\epsilon})\,F_{\bullet}^{-1}(\check{s},\check{\mu},\check{\epsilon})$.
Since all the connection matrices are equal, this is a well defined non-
ramified invertible matrix function defined on the union of the projections of
the domains $\Omega_{\bullet}$ to $(s,\mu,\epsilon)$-space,
$\bullet=O,{I\\!+\\!},{I\\!-\\!}$. It is bounded on a neighborhood of each
singularity $s_{i}$, hence $H$ can be analytically extended on
$(\mathsf{S}\\!\smallsetminus\\!\\{0\\})\times\\!\mathsf{M}\\!\times\\!\mathsf{E}$,
where $\mathsf{S},\mathsf{M},\mathsf{E}$ are as in (33). It satisfies
$H=H^{\mathsf{P}}\\!$: if $s$ is in the projection of
$\Omega_{\bullet}(\check{\mu},\check{\epsilon})\,$ and
$\,H(s,\check{\mu},\epsilon)=F^{\prime}_{\bullet}(\check{s},\check{\mu},\check{\epsilon})\,F_{\bullet}^{-1}(\check{s},\check{\mu},\check{\epsilon})\,$
then $-s$ is in the projection of $\Omega_{\bullet}^{\mathsf{P}}\\!\,$ and
$\displaystyle H(-s,\check{\mu},\epsilon)$
$\displaystyle={F^{\prime}_{\bullet}}^{\mathsf{P}}\\!(e^{\pi
i}\check{s},\check{\mu},\check{\epsilon})\,(F_{\bullet}^{\mathsf{P}}\\!(e^{\pi
i}\check{s},\check{\mu},\check{\epsilon}))^{-1}=$
$\displaystyle=\left(\begin{smallmatrix}0&1\\\
1&0\end{smallmatrix}\right)F^{\prime}_{\bullet}(\check{s},\check{\mu},\check{\epsilon})\,F_{\bullet}^{-1}(\check{s},\check{\mu},\check{\epsilon})\left(\begin{smallmatrix}0&1\\\
1&0\end{smallmatrix}\right)=$ $\displaystyle=\left(\begin{smallmatrix}0&1\\\
1&0\end{smallmatrix}\right)H(\check{s},\check{\mu},\epsilon)\left(\begin{smallmatrix}0&1\\\
1&0\end{smallmatrix}\right).$
Hence the function
$G(x,\mu,\epsilon):=S(s,\mu,\epsilon)\,VH(s,\mu,\epsilon)\,V^{-1}S^{-1}(s,\mu,\epsilon),$
with $S,V$ as in (22), is well defined. The fundamental solutions
$\Phi_{\bullet}(\check{x},\check{\mu},\check{\epsilon})=S(s)\,VF_{\bullet}(\check{s},\check{\mu},\check{\epsilon})\,\Psi_{\bullet}(\check{s},\check{\mu},\check{\epsilon})$
(resp.
$\Phi^{\prime}_{\bullet}(\check{x},\check{\mu},\check{\epsilon})=S(s)\,VF^{\prime}_{\bullet}(\check{s},\check{\mu},\check{\epsilon})\,\Psi_{\bullet}(\check{s},\check{\mu},\check{\epsilon})$)
of the systems $\Delta(x,\mu,\epsilon)$ (resp.
$\Delta^{\prime}(x,\mu,\epsilon)$) can for $(\check{\mu},\check{\epsilon})\neq
0$ be analytically extended on a neighborhood of the point $x\\!=\\!-\mu$
(i.e. $s\\!=\\!0$) which is non-singular for these systems. As
$G=\Phi^{\prime}_{\bullet}\Phi_{\bullet}^{-1}$, it means that
$\,G^{*}\Delta^{\prime}=\Delta$ and that $G$ is an invertible analytic matrix
function which is well defined and continuous on
$(\mathsf{X}\\!\times\\!\mathsf{M}\\!\times\\!\mathsf{E})\,\\!\smallsetminus\\!\\{0\\}$,
where
$\mathsf{X}:=\\{|x|\leq\delta_{s}^{2}\\!-\delta_{\mu}\\},$
and analytic for $x\neq-\mu$, $\mu^{2}\neq\epsilon$, $\epsilon\neq 0$.
Therefore it is analytic on the whole neighborhood
$\mathsf{X}\\!\times\\!\mathsf{M}\\!\times\\!\mathsf{E}$ of 0. ∎
Figure 9: The connection matrices between the fundamental solutions
$F_{\bullet}\Psi_{\bullet}$ for a fixed parameter
$(\check{\mu},\check{\epsilon})$, $\check{\mu}^{2}\neq\check{\epsilon}$. For
$\check{\mu}^{2}=\check{\epsilon}$, only the fundamental solutions
$F_{O}\Psi_{O}$ and $F_{O}^{\mathsf{P}}\\!\Psi_{O}^{\mathsf{P}}\\!$ persist,
with the two corresponding connection matrices $-iC_{0}$,
$-iC_{0}^{\mathsf{P}}\\!$. (Picture with $(\check{\mu},\check{\epsilon})$ as
in Figures 8 and 6(a)).
###### Lemma 21.
Let $F_{\bullet}$ be the normalizing transformations from Proposition 14
satisfying (36) with the uniquely determined functions $\kappa_{\bullet}$, and
let $\Psi_{\bullet}$ be as Figure 8. Then for each fixed
$(\check{\mu},\check{\epsilon})\in\check{\mathcal{M}}$ the connection matrices
between the solutions $F_{\bullet}\Psi_{\bullet}$ on the domains
$\Omega_{\bullet}(\check{\mu},\check{\epsilon})$ are given in Figure 9 with
the matrices
$C_{0}(\check{\mu},\check{\epsilon}),\ldots,C_{4}(\check{\mu},\check{\epsilon})$
equal to
$C_{0}=\begin{pmatrix}1&i\gamma\\\ 0&1\end{pmatrix}$, | $C_{3}=\begin{pmatrix}1&i\kappa^{-1}e^{\\!-2a\pi i}\\\ 0&\kappa^{-1}\end{pmatrix},$
---|---
$C_{1}=\begin{pmatrix}1&i\kappa^{-1}(\gamma-e^{\,2a\pi i}\\!\\!-e^{\\!-2a\pi i})\\\ 0&1\end{pmatrix}$, | $C_{4}=\begin{pmatrix}1&-i\kappa^{-1}e^{\,2a\pi i}\\\ 0&\kappa^{-1}\end{pmatrix},$
$C_{2}=\begin{pmatrix}1&0\\\ -i\kappa e^{\,2a\pi i}&1\end{pmatrix}$, |
(48)
where
$\displaystyle a(\check{\mu},\check{\epsilon})$
$\displaystyle:=\left\\{\hskip-6.0pt\begin{array}[]{ll}\frac{s_{1}(\check{\mu},\check{\epsilon})-s_{2}(\check{\mu},\check{\epsilon})}{2\sqrt{\check{\epsilon}}}&\text{
if }\ \check{\epsilon}\neq 0,\\\\[6.0pt] \frac{1}{2\sqrt{\check{\mu}}}&\text{
if }\ \check{\epsilon}=0\ \text{ and }\ \check{\mu}\neq 0,\end{array}\right.$
(51) $\displaystyle\kappa(\check{\mu},\check{\epsilon})$
$\displaystyle:=\frac{\kappa_{O}(\check{\mu},\check{\epsilon})}{\kappa_{I}(\check{\mu},\check{\epsilon})}$
(52)
and $\gamma(\mu,\epsilon)$ is the analytic invariant of the system
$\Delta(x,\mu,\epsilon)$ (20).
###### Proof.
From Lemma 16 we know that a connection matrix on an intersection domain that
is adjacent to point $s_{1}(\check{\mu},\check{\epsilon})$ (resp.
$s_{2}(\check{\mu},\check{\epsilon})$) must be upper triangular (resp. lower
triangular), with the diagonal terms determined by the values of the
corresponding pair of transformations
$F_{\bullet}(s_{1}(\check{\mu},\check{\epsilon}),(\check{\mu},\check{\epsilon}))$
(resp.
$F_{\bullet}(s_{2}(\check{\mu},\check{\epsilon}),(\check{\mu},\check{\epsilon}))$).
Hence we have
$C_{0}=\left(\begin{smallmatrix}1&c_{0}\\!\\\\[3.0pt]
0&1\end{smallmatrix}\right),\quad
C_{1}=\left(\begin{smallmatrix}1&c_{1}\\!\\\\[3.0pt]
0&1\end{smallmatrix}\right),\quad
C_{2}=\left(\begin{smallmatrix}1&0\\\\[3.0pt]
c_{2}&1\end{smallmatrix}\right),\quad
C_{3}=\left(\begin{smallmatrix}1&c_{3}\\\\[3.0pt]
0&\kappa^{-1}\\!\end{smallmatrix}\right),\quad
C_{4}=\left(\begin{smallmatrix}1&c_{4}\\\\[3.0pt]
0&\kappa^{-1}\\!\end{smallmatrix}\right),$
for some
$c_{0}(\check{\mu},\check{\epsilon}),\ldots,c_{4}(\check{\mu},\check{\epsilon})$.
Let $M(\check{\mu},\check{\epsilon})$ be the monodromy matrix of the
fundamental solution
$\,\Phi_{O}(\check{x},\check{\mu},\check{\epsilon})=S(s)\,VF_{O}(\check{s},\check{\mu},\check{\epsilon})\,\Psi_{O}(\check{s},\check{\mu},\check{\epsilon})\,$
of the system $\Delta$ around the two singular points
$x=\pm\sqrt{\check{\epsilon}}$ in the positive direction. On one hand we have
$\displaystyle M$
$\displaystyle=\Psi_{O}^{-1}(\check{s})F_{O}^{-1}(\check{s})V^{-1}S^{-1}(s)\cdot
S(-s)\,VF_{O}(e^{\pi i}\check{s})\,\Psi_{O}(e^{\pi i}\check{s})$
$\displaystyle=\Psi_{O}^{-1}(\check{s})F_{O}^{-1}(\check{s})V^{-1}S^{-1}(s)\cdot
S(s)\,VF_{O}^{\mathsf{P}}\\!(\check{s})\,\Psi_{0}^{\mathsf{P}}\\!(\check{s})\left(\begin{smallmatrix}0&1\\\\[3.0pt]
1&0\end{smallmatrix}\right)=-i\,C_{0}\\!\left(\begin{smallmatrix}0&1\\\\[3.0pt]
1&0\end{smallmatrix}\right),$
on the other hand, as apparent from Figure 9,
$M=C_{3}C_{1}NC_{2}C_{3}^{-1},$
where $\,N=\left(\begin{smallmatrix}e^{\,2a\pi i}&0\\\\[3.0pt] 0&e^{\\!-2a\pi
i}\end{smallmatrix}\right).$ Therefore
$\displaystyle M=-i\,C_{0}\\!\left(\begin{smallmatrix}0&1\\\\[3.0pt]
1&0\end{smallmatrix}\right)$ $\displaystyle=\,C_{3}C_{1}NC_{2}C_{3}^{-1},$
$\displaystyle\begin{pmatrix}\\!-ic_{0}\\!&\\!\\!-i\\\
-i\\!&\\!0\end{pmatrix}$ $\displaystyle=\begin{pmatrix}\\!e^{\,2a\pi
i}\\!+e^{\\!-2a\pi i}c_{2}(c_{1}\\!+\\!c_{3})&\kappa e^{\\!-2a\pi
i}(c_{1}\\!+\\!c_{3})(1\\!-\\!c_{2}c_{3})-\kappa e^{\,2a\pi i}c_{3}\\!\\\
\kappa^{-1}e^{\\!-2a\pi i}c_{2}&e^{\\!-2a\pi
i}(1\\!-\\!c_{2}c_{3})\end{pmatrix},$
which implies that
$\gamma=\mathrm{tr}M=-ic_{0}=e^{\,2a\pi i}+\,e^{\\!-2a\pi i}+\,e^{\\!-2a\pi
i}c_{1}c_{2},$ (53) $c_{2}c_{3}=1,\quad\text{and}\quad c_{2}=-i\kappa
e^{\,2a\pi i},\quad c_{3}=i\kappa^{-1}e^{\\!-2a\pi i}.$
From Figure 9 one also sees that
$C_{3}C_{1}=C_{0}C_{4},$
which gives the matrix $C_{4}$. ∎
The matrices
$C_{0}(\check{\mu},\check{\epsilon}),\ldots,C_{4}(\check{\mu},\check{\epsilon})$
of Lemma 21 determine for each fixed
$(\check{\mu},\check{\epsilon})\in\check{\mathcal{M}}$ all the relations
between the set of fundamental solutions
$F_{\bullet}(\cdot,\check{\mu},\check{\epsilon})$, $\bullet=O,I\\!+,I\\!-$ and
their symmetric counterparts
$F_{\bullet}^{\mathsf{P}}\\!(\cdot,\check{\mu},\check{\epsilon})$. We will now
look at the situation of two different
$(\check{\mu},\check{\epsilon})\in\check{\mathcal{M}}$ corresponding to the
same value of $(\mu,\epsilon)$. One finds that the corresponding connection
matrices can be expressed in terms of the values of $C_{0},\ldots,C_{4}$ for
the two ramified parameters, while certain cocycle relations must be
satisfied.
###### Lemma 22.
Let $F_{\bullet},\ \Psi_{\bullet}$ be as in Lemma 21.
* (a)
For each $\check{\epsilon}$ fixed, let
$\bar{\mu},\tilde{\mu}\in\check{\mathcal{M}}(\check{\epsilon})$ be two points
that project to the same $\mu$, so that
$\tilde{\mu}-\sqrt{\check{\epsilon}}=e^{2\pi
i}(\bar{\mu}-\sqrt{\check{\epsilon}})$, i.e. $\tilde{\mu}$ is $\bar{\mu}$ plus
one positive turn around the ramification point $\sqrt{\check{\epsilon}}$ in
$\check{\mathcal{M}}(\check{\epsilon})$. So
$s_{1}(\tilde{\mu},\check{\epsilon})=s_{1}(\bar{\mu},\check{\epsilon}),\quad
s_{2}(\tilde{\mu},\check{\epsilon})=e^{\pi
i}s_{2}(\bar{\mu},\check{\epsilon}).$
Then
$\,\Omega_{O}(\tilde{\mu},\check{\epsilon})=\Omega_{O}(\bar{\mu},\check{\epsilon}),$
and
$F_{O}(\check{s},\tilde{\mu},\check{\epsilon})=F_{O}(\check{s},\bar{\mu},\check{\epsilon}),\quad\Psi_{O}(\check{s},\tilde{\mu},\check{\epsilon})=\Psi_{O}(\check{s},\bar{\mu},\check{\epsilon}).$
We have
$\,\kappa_{O}(\tilde{\mu},\check{\epsilon})=\kappa_{O}(\bar{\mu},\check{\epsilon})=:\kappa_{O}(\mu,\check{\epsilon}),\,$
$\kappa_{O}(\mu,\check{\epsilon})=\frac{\kappa(\tilde{\mu},\check{\epsilon})\,\kappa(\bar{\mu},\check{\epsilon})\,e^{\frac{s_{2}(\bar{\mu},\check{\epsilon})}{\sqrt{\check{\epsilon}}}\pi
i}}{e^{\frac{s_{2}(\bar{\mu},\check{\epsilon})}{\sqrt{\check{\epsilon}}}\pi
i}-e^{-\frac{s_{2}(\bar{\mu},\check{\epsilon})}{\sqrt{\check{\epsilon}}}\pi
i}}.$ (54)
* (b)
Let
$(\bar{\mu},\bar{\epsilon}),\,(\tilde{\mu},\tilde{\epsilon})\in\check{\mathcal{M}}$
be two values of the ramified parameter corresponding to the same
$(\mu,\epsilon)$ so that $(\tilde{\mu},\tilde{\epsilon})=e^{2\pi
i}(\bar{\mu},\bar{\epsilon})$, more precisely, for
$|\mu|\gg\sqrt{|\epsilon|}$, $(\tilde{\mu},\tilde{\epsilon})$ is obtained from
$(\bar{\mu},\bar{\epsilon})$ by simultaneously turning both $\check{\epsilon}$
and $\check{\mu}$. So
$s_{1}(\tilde{\mu},\tilde{\epsilon})=e^{\pi
i}s_{2}(\bar{\mu},\bar{\epsilon}),\quad
s_{2}(\tilde{\mu},\tilde{\epsilon})=e^{\pi i}s_{1}(\bar{\mu},\bar{\epsilon}),$
and
$N(\tilde{\mu},\tilde{\epsilon})=N(\bar{\mu},\bar{\epsilon})^{-1}.$
Then
$\,\Omega_{I\\!+}(\tilde{\mu},\check{\epsilon})=\Omega_{I\\!-}^{\mathsf{P}}\\!(\bar{\mu},\check{\epsilon}),\
\Omega_{I\\!-}(\tilde{\mu},\check{\epsilon})=\Omega_{I\\!+}^{\mathsf{P}}\\!(\bar{\mu},\check{\epsilon}),\,$
and
$F_{I\\!+}(\check{s},\tilde{\mu},\check{\epsilon})=F_{I\\!-}^{\mathsf{P}}\\!(\check{s},\bar{\mu},\check{\epsilon}),\quad\Psi_{I\\!+}(\check{s},\tilde{\mu},\check{\epsilon})=-i\,\Psi_{I\\!-}^{\mathsf{P}}\\!(\check{s},\bar{\mu},\check{\epsilon}),$
---
$F_{I\\!-}(\check{s},\tilde{\mu},\check{\epsilon})=F_{I\\!+}^{\mathsf{P}}\\!(\check{s},\bar{\mu},\check{\epsilon}),\quad\Psi_{I\\!-}(\check{s},\tilde{\mu},\check{\epsilon})=-i\,\Psi_{I\\!+}^{\mathsf{P}}\\!(\check{s},\bar{\mu},\check{\epsilon})\,N(\bar{\mu},\bar{\epsilon})^{-1}.$
Therefore
$C_{1}(\tilde{\mu},\tilde{\epsilon})=N(\bar{\mu},\bar{\epsilon})^{-1}C_{2}(\bar{\mu},\bar{\epsilon})^{\mathsf{P}}\\!N(\bar{\mu},\bar{\epsilon}),\quad
C_{2}(\tilde{\mu},\tilde{\epsilon})=C_{1}(\bar{\mu},\bar{\epsilon})^{\mathsf{P}}\\!,$
(55)
and we have
$\displaystyle\kappa_{I}(\tilde{\mu},\tilde{\epsilon})$
$\displaystyle=\kappa_{I}(\bar{\mu},\bar{\epsilon}),$ (56)
$\displaystyle\gamma(\mu,\epsilon)$ $\displaystyle=e^{2\bar{a}\pi
i}\,+\,e^{-2\bar{a}\pi i}\,+\,\bar{\kappa}\,\tilde{\kappa}\,e^{-2\bar{a}\pi
i},$ (57)
where $\,\check{a}:=a(\check{\mu},\check{\epsilon})\,\text{ and
}\,\check{\kappa}:=\kappa(\check{\mu},\check{\epsilon})\,\text{ are defined in
\eqref{eq:a} and \eqref{eq:kappa}.}$
###### Proof.
(a) For each $\check{\epsilon}\in\check{\mathsf{E}}$ the ramification of the
$\check{\mu}$-parameter domain $\check{\mathcal{M}}(\check{\epsilon})$
corresponds to the bifurcation $\Sigma_{I}(\epsilon)$: the difference between
$(\bar{\mu},\check{\epsilon})$ and $(\tilde{\mu},\check{\epsilon})$ is that of
crossing the line $\Sigma_{I}(\epsilon)$. Since this bifurcation affects only
the inner regions of the field $\chi$, it therefore affects only the internal
domains $\Omega_{I\\!\pm}$, $\Omega_{I\\!\pm}^{\mathsf{P}}\\!$, while the
outer domains are not affected. Therefore
$\Omega_{O}(\tilde{\mu},\check{\epsilon})=\Omega_{O}(\bar{\mu},\check{\epsilon})$
and consequently
$F_{O}(\check{s},\tilde{\mu},\check{\epsilon})=F_{O}(\check{s},\bar{\mu},\check{\epsilon})$.
To obtain the assertion (54), it is enough to prove it for generic values of
$(\mu,\epsilon)$, and extend it to the other values by continuity. So we can
assume that $\epsilon\neq 0$, $\mu^{2}\neq\epsilon$ and moreover that both of
the points $s_{1}(\check{\mu},\check{\epsilon})$,
$s_{2}(\check{\mu},\check{\epsilon})$ are non-resonant – see Remark 12 (a).
Therefore, aside from the transformations
$F_{\bullet}(\check{s},\check{\mu},\check{\epsilon})$,
$\bullet=O,I\\!+,I\\!-$, we have also the transformations
$F_{i}(\check{s},\check{\mu},\check{\epsilon})$ on
$\Omega_{i}(\check{\mu},\check{\epsilon})$, $i=1,2$, from Remark 12 (a),
satisfying
$F_{1}(\cdot,\tilde{\mu},\check{\epsilon})=F_{1}(\cdot,\bar{\mu},\check{\epsilon}),\quad
F_{2}(\cdot,\tilde{\mu},\check{\epsilon})=F_{2}^{\mathsf{P}}\\!(\cdot,\bar{\mu},\check{\epsilon}).$
Let $A_{i}$ be the connection matrix between $F_{i}\Psi_{I\\!+}$ and
$F_{I\\!+}\Psi_{I\\!+}$:
$F_{I\\!+}\Psi_{I\\!+}=F_{i}\Psi_{I\\!+}A_{i}.$
See Figure 10.
It is easy to see that the monodromy of $F_{I\\!+}\Psi_{I\\!+}$ around the
point $s_{1}$ (resp. $s_{2}$) is equal to
$C_{1}N_{1}=A_{1}^{-1}N_{1}A_{1},\quad(\,\text{resp.
}N_{2}C_{2}=A_{2}^{-1}N_{2}A_{2}\,),$
from which one can calculate using Lemma 21 that
$A_{1}=\begin{pmatrix}1&\frac{1}{e^{2\frac{s_{1}}{\sqrt{\check{\epsilon}}}\pi
i}-1}c_{1}\\\ 0&\kappa_{I}\end{pmatrix},\quad
A_{2}=\begin{pmatrix}\kappa_{I}&0\\\
\frac{1}{1-e^{-2\frac{s_{2}}{\sqrt{\check{\epsilon}}}\pi
i}}c_{2}&1\end{pmatrix},$
with
$c_{1}=i\kappa^{-1}(\gamma-e^{\frac{s_{1}-s_{2}}{\sqrt{\check{\epsilon}}}\pi
i}\\!\\!-e^{-\frac{s_{1}-s_{2}}{\sqrt{\check{\epsilon}}}\pi i})$, and
$c_{2}=-i\kappa e^{\frac{s_{1}-s_{2}}{\sqrt{\check{\epsilon}}}\pi i}$.
Figure 10: Connection matrices between fundamental solutions
$F_{\bullet}\Psi_{\bullet}$ of $\Delta^{\\!s}$ for fixed $\check{\epsilon}$ in
Lemma 22(a), using the following kind of notation: $\,\mkern
2.0mu\overline{X}=X(\bar{\mu},\check{\epsilon})$,
$\,\widetilde{X}=X(\tilde{\mu},\check{\epsilon})$. (Picture with
$\arg\check{\epsilon}=0$.) The corresponding diagram for the diagonal
solutions $\Psi_{\bullet}$ of $\bar{\Delta}^{s}$ is obtained by erasing all
the $F$’s and replacing thematrices $A_{i},\,C_{i}$ by identities. The top
arrow in the diagram here $\mkern 2.0mu\overline{F}_{2}^{\mathsf{P}}\\!\mkern
2.0mu\overline{\Psi}_{I\\!+}^{\mathsf{P}}\\!\xrightarrow{\ -i\mkern
2.0mu\overline{N}_{1}\ }\widetilde{F}_{2}\widetilde{\Psi}_{I\\!+}$ follows
from the corresponding arrow $\mkern
2.0mu\overline{\Psi}_{I\\!+}^{\mathsf{P}}\\!\xrightarrow{\ -i\mkern
2.0mu\overline{N}_{1}\ }\widetilde{\Psi}_{I\\!+}$ which one can easily read in
the corresponding diagram for the diagonal solutions.
Knowing that
$F_{2}(\cdot,\tilde{\mu},\check{\epsilon})=F_{2}^{\mathsf{P}}\\!(\cdot,\bar{\mu},\check{\epsilon})$
one can see from Figure 10 that
$\displaystyle
A_{2}(\tilde{\mu},\check{\epsilon})\,C_{3}^{-1}(\tilde{\mu},\check{\epsilon})$
$\displaystyle=N_{1}^{-1}(\bar{\mu},\check{\epsilon})\,A_{2}^{\mathsf{P}}\\!(\bar{\mu},\check{\epsilon})\,N_{1}(\bar{\mu},\check{\epsilon})\,(C_{4}^{\mathsf{P}}\\!)^{-1}(\bar{\mu},\check{\epsilon}),$
$\displaystyle\begin{pmatrix}\kappa_{I}(\tilde{\mu},\check{\epsilon})&-i\kappa_{I}(\tilde{\mu},\check{\epsilon})\frac{1}{e_{1}e_{2}}\\\\[6.0pt]
i\kappa(\tilde{\mu},\check{\epsilon})\frac{e_{1}}{e_{2}-e_{2}^{-1}}&\kappa(\tilde{\mu},\check{\epsilon})\frac{e_{2}}{e_{2}-e_{2}^{-1}}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\kappa(\bar{\mu},\check{\epsilon})\frac{e_{2}}{e_{2}-e_{2}^{-1}}&-i\kappa(\bar{\mu},\check{\epsilon})\frac{e_{1}^{-1}}{e_{2}-e_{2}^{-1}}\\\\[6.0pt]
i\kappa_{I}(\bar{\mu},\check{\epsilon})\frac{e_{1}}{e_{2}}&\kappa_{I}(\bar{\mu},\check{\epsilon})\end{pmatrix}$
with
$e_{1}:=e^{\frac{s_{1}(\bar{\mu},\check{\epsilon})}{\sqrt{\check{\epsilon}}}\pi
i}=e^{\frac{s_{1}(\tilde{\mu},\check{\epsilon})}{\sqrt{\check{\epsilon}}}\pi
i},\quad
e_{2}:=e^{\frac{s_{2}(\bar{\mu},\check{\epsilon})}{\sqrt{\check{\epsilon}}}\pi
i}=e^{-\frac{s_{2}(\tilde{\mu},\check{\epsilon})}{\sqrt{\check{\epsilon}}}\pi
i}.$
This is satisfied if and only if
$\kappa_{I}(\tilde{\mu},\check{\epsilon})\kappa_{I}(\bar{\mu},\check{\epsilon})=\kappa_{O}(\mu,\check{\epsilon})\tfrac{e_{2}}{e_{2}-e_{2}^{-1}},$
which is equivalent to (54).
Similarly, one would find that
$A_{1}(\tilde{\mu},\check{\epsilon})C_{3}^{-1}(\tilde{\mu},\check{\epsilon})=A_{1}(\bar{\mu},\check{\epsilon})C_{3}^{-1}(\bar{\mu},\check{\epsilon}),$
which is satisfied without imposing any new condition, since
$A_{1}C_{3}^{-1}=\begin{pmatrix}1&\frac{i\gamma
e^{-\frac{s_{1}}{\sqrt{\check{\epsilon}}}\pi
i}-ie^{\frac{s_{2}}{\sqrt{\check{\epsilon}}}\pi
i}-ie^{-\frac{s_{2}}{\sqrt{\check{\epsilon}}}\pi
i}}{e^{\frac{s_{1}}{\sqrt{\check{\epsilon}}}\pi
i}-e^{-\frac{s_{1}}{\sqrt{\check{\epsilon}}}\pi i}}\\\
0&\kappa_{O}\end{pmatrix}.$
(b) Similarly to (a), the passage between $(\bar{\mu},\bar{\epsilon})$,
$\bar{\mu}\in\check{\mathcal{M}}(\bar{\epsilon}),$ and
$(\tilde{\mu},\tilde{\epsilon})=e^{2\pi i}(\bar{\mu},\bar{\epsilon})$,
$\tilde{\mu}\in\check{\mathcal{M}}(\tilde{\epsilon}),$ is that of crossing the
curve $\Sigma_{O}(\epsilon)$, which affects only the outer regions, and hence
the outer domains. The inner domains rotate together with their vertices
$s_{1}(\check{\mu},\check{\epsilon}),\ s_{2}(\check{\mu},\check{\epsilon})$,
therefore
$\Omega_{I\\!+}(\tilde{\mu},\tilde{\epsilon})=\Omega_{I\\!-}^{\mathsf{P}}\\!(\bar{\mu},\bar{\epsilon})$
and
$\Omega_{I\\!-}(\tilde{\mu},\tilde{\epsilon})=\Omega_{I\\!+}^{\mathsf{P}}\\!(\bar{\mu},\bar{\epsilon})$.
So we have
$F_{I\\!+}(\cdot,\tilde{\mu},\tilde{\epsilon})=F_{I\\!-}^{\mathsf{P}}\\!(\cdot,\bar{\mu},\bar{\epsilon}),\quad
F_{I\\!-}(\cdot,\tilde{\mu},\tilde{\epsilon})=F_{I\\!+}^{\mathsf{P}}\\!(\cdot,\bar{\mu},\bar{\epsilon}).$
One can see from Figure 11 that the fundamental solutions $\Psi_{I\\!\pm}$ of
the diagonal system $\mkern 2.0mu\overline{\Delta}^{\,s}$ satisfy
$\Psi_{I\\!+}(\cdot,\tilde{\mu},\tilde{\epsilon})=-i\,\Psi_{I\\!-}^{\mathsf{P}}\\!(\cdot,\bar{\mu},\bar{\epsilon}),\quad\Psi_{I\\!-}(\cdot,\tilde{\mu},\tilde{\epsilon})=-i\,\Psi_{I\\!+}^{\mathsf{P}}\\!(\cdot,\bar{\mu},\bar{\epsilon})\,N(\bar{\mu},\bar{\epsilon}).$
This then implies (55), i.e.
$C_{1}(\bar{\mu},\bar{\epsilon})=\left(\begin{smallmatrix}1&-i\widetilde{\kappa}e^{\\!-2\bar{a}\pi i}\\\\[3.0pt] 0&1\end{smallmatrix}\right),\quad$ | $C_{1}(\tilde{\mu},\tilde{\epsilon})=\left(\begin{smallmatrix}1&-i\bar{\kappa}e^{\\!-2\tilde{a}\pi i}\\\\[3.0pt] 0&1\end{smallmatrix}\right)$,
---|---
$C_{2}(\bar{\mu},\bar{\epsilon})=\left(\begin{smallmatrix}1&0\\\\[3.0pt] -i\bar{\kappa}e^{\,2\bar{a}\pi i}&1\end{smallmatrix}\right),$ | $C_{2}(\tilde{\mu},\tilde{\epsilon})=\left(\begin{smallmatrix}1&0\\\\[3.0pt] -i\widetilde{\kappa}e^{\,2\tilde{a}\pi i}&1\end{smallmatrix}\right),$
where $\tilde{a}=-\bar{a}$, then (57) follows from (53).
Figure 11: Connection matrices between fundamental solutions $\Psi_{\bullet}$
for $(\tilde{\mu},\tilde{\epsilon})=e^{2\pi i}(\bar{\mu},\bar{\epsilon})$,
using the following kind of notation: $\,\mkern
2.0mu\overline{X}=X(\bar{\mu},\bar{\epsilon})$,
$\,\widetilde{X}=X(\tilde{\mu},\tilde{\epsilon})$.
∎
All the connection matrices between the fundamental solutions
$F_{\bullet}\Psi_{\bullet}$ can now be determined from Lemmas 21 and 22.
###### Proposition 23.
(a) Let $\Delta$ be a system (20), $\Delta^{\\!s}$ its transform (23), let
$F_{\bullet}$ be the normalizing transformations from Proposition 14
determined by the condition (45) and let $\Psi_{\bullet}$ be as in Figure 8.
The collection of all the connection matrices between the fundamental
solutions $F_{\bullet}\Psi_{\bullet}$ is uniquely determined by
$\kappa=\frac{\kappa_{O}}{\kappa_{I}}$ and by the invariant $\gamma$,
satisfying the relation (57).
(b) Let $\gamma(\mu,\epsilon)$ be a germ of analytic function and assume that
there exists an analytic germ $Q(\mu,\epsilon)$ such that $\gamma=2\cos\,2\pi
Q$. Let $a(\check{\mu},\check{\epsilon})$ be as in (51) and
$\,b(\check{\mu},\check{\epsilon}):=\frac{s_{1}+s_{2}}{2\sqrt{\check{\epsilon}}}.$
Then any triple of functions
$\kappa_{I},\,\kappa_{O},\,\kappa=\frac{\kappa_{O}}{\kappa_{I}}\in\mathcal{B}(\check{\mathcal{M}})$
with $\kappa_{O}(0,0)=1$ and $\kappa_{I}(\check{\mu},0)=1,\
\check{\mu}\in\check{\mathcal{M}}(0),$ satisfying the relations (54), (56) and
(57) of Lemma 22 are equal to
$\displaystyle\kappa_{I}\\!$
$\displaystyle=\frac{\sqrt{\frac{s_{1}s_{2}}{\check{\epsilon}}}\,\Gamma(\frac{s_{1}}{\sqrt{\check{\epsilon}}})\,\Gamma(\frac{s_{2}}{\sqrt{\check{\epsilon}}})}{\Gamma(1\\!+\\!b\\!-\\!Q)\,\Gamma(b\\!+\\!Q)}\,e^{\,2b\log
b\,-\,\frac{s_{1}}{\sqrt{\check{\epsilon}}}\log\frac{s_{1}}{\sqrt{\check{\epsilon}}}\,-\,\frac{s_{2}}{\sqrt{\check{\epsilon}}}\log\frac{s_{2}}{\sqrt{\check{\epsilon}}}\,+\,f_{I}},$
$\displaystyle\kappa_{O}\\!$
$\displaystyle=2\pi\frac{\Gamma(\frac{s_{1}}{\sqrt{\check{\epsilon}}})\,\Gamma(1+\frac{s_{1}}{\sqrt{\check{\epsilon}}})}{\Gamma(1\\!+\\!a\\!-\\!Q)\,\Gamma(1\\!+\\!b\\!-\\!Q)\,\Gamma(a\\!+\\!Q)\,\Gamma(b\\!+\\!Q)}\,e^{\,2a\log
a\,+\,2b\log
b\,-\,2\frac{s_{1}}{\sqrt{\check{\epsilon}}}\log\frac{s_{1}}{\sqrt{\check{\epsilon}}}\,+\,f_{O}},$
$\displaystyle\kappa\,$
$\displaystyle=2\pi\frac{\sqrt{\frac{s_{1}}{s_{2}}}\,\Gamma(\frac{s_{1}}{\sqrt{\check{\epsilon}}})\,\Gamma(\frac{s_{2}}{\sqrt{\check{\epsilon}}})^{-1}}{\Gamma(1\\!+\\!a\\!-\\!Q)\,\Gamma(a\\!+\\!Q)}\,e^{\,2a\log
a\,-\,\frac{s_{1}}{\sqrt{\check{\epsilon}}}\log\frac{s_{1}}{\sqrt{\check{\epsilon}}}\,+\,\frac{s_{2}}{\sqrt{\check{\epsilon}}}\log\frac{s_{2}}{\sqrt{\check{\epsilon}}}\,+\,f},$
where $\Gamma$ is the gamma function and
$f=(s_{1}\\!+\\!s_{2})\,g(s_{1}s_{2},\,s_{1}^{2}\\!+\\!s_{2}^{2}),\quad
f_{I}=(s_{1}\\!-\\!s_{2})\,g(\\!-s_{1}s_{2},\,s_{1}^{2}\\!+\\!s_{2}^{2}),\quad
f_{O}=f+f_{I},$
for a unique analytic germ $g$.
###### Proof.
(a) All the connection matrices between the fundamental solutions
$F_{\bullet}\Psi_{\bullet}$ can be determined from Lemmas 21 and 22.
(b) Let $\kappa^{\prime}_{I},\,\kappa^{\prime}_{O},\,\kappa^{\prime}$ be the
functions given above for $g=0$. Denote
$\sigma:(\bar{\mu},\check{\epsilon})\mapsto(\tilde{\mu},\check{\epsilon})$ the
continuation map from Lemma 22 (a), and
$\rho:(\bar{\mu},\bar{\epsilon})\mapsto(\tilde{\mu},\tilde{\epsilon})$ the
continuation map from Lemma 22 (b). Hence,
$s_{1}\\!\circ\\!\sigma=s_{1},\quad s_{2}\\!\circ\\!\sigma=e^{\pi
i}s_{2},\quad a\\!\circ\\!\sigma=b,\quad b\\!\circ\\!\sigma=a,$
$s_{1}\\!\circ\\!\rho=e^{\pi i}s_{2},\quad s_{2}\\!\circ\\!\rho=e^{\pi
i}s_{1},\quad a\\!\circ\\!\rho=e^{-\pi i}a,\quad b\\!\circ\\!\rho=b,$
and the identities (52):
$\kappa^{\prime}=\frac{\kappa^{\prime}_{O}}{\kappa^{\prime}_{I}}$,
$\eqref{eq:11}:\
\kappa^{\prime}_{O}\\!\circ\\!\sigma=\kappa^{\prime}_{O}=\frac{\kappa^{\prime}\,(\kappa^{\prime}\\!\circ\\!\sigma)\,e^{\frac{s_{2}}{\sqrt{\check{\epsilon}}}\pi
i}}{2i\,\sin\frac{s_{2}}{\sqrt{\check{\epsilon}}}\pi},$
(56): $\kappa^{\prime}_{I}\\!\circ\\!\rho=\kappa^{\prime}_{I}$, and
$\eqref{eq:12}:\ -e^{-\\!2a\pi
i}\kappa^{\prime}\,(\kappa^{\prime}\\!\circ\\!\rho)+2\cos\,2\pi a=2\cos\,2\pi
Q=\gamma$
are easily verified using the standard reflection formula
$\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin\pi z}$. It follows from the Stirling
formula:
$\begin{split}\Gamma(1+z)\sim\sqrt{2\pi
z}\,(\frac{z}{e})^{z}\left(1+O(\tfrac{1}{z})\right)\ &\mbox{ in the sector at
infinity where }\\\\[-6.0pt] &\quad|\arg z|<\eta\,\text{ for any
}\,0<\eta<\pi,\end{split}$
that $\,\kappa^{\prime}_{I}(\check{\mu},0):=\lim_{\check{\epsilon}\to
0}\kappa^{\prime}_{I}(\check{\mu},\check{\epsilon})=1\,$ and
$\,\kappa^{\prime}_{O}(0,0):=\lim_{(\check{\mu},\check{\epsilon})\to
0}\kappa^{\prime}_{O}(\check{\mu},\check{\epsilon})=1\,$ with the limits being
inside $\check{\mathcal{M}}$.
Now if $\kappa_{I},\,\kappa_{O},\,\kappa$ are another functions satisfying the
assumptions of the lemma, let
$f_{I}:=\log\frac{\kappa_{I}}{\kappa^{\prime}_{I}},\quad
f_{O}:=\log\frac{\kappa_{O}}{\kappa^{\prime}_{O}},\quad
f:=\log\frac{\kappa}{\kappa^{\prime}},$
then it follows that
$f_{I}=f\\!\circ\\!\sigma,\quad f_{O}=f+f_{I},\quad\text{and that}\quad
f_{O}\\!\circ\\!\sigma=f_{O},\quad f_{I}\\!\circ\\!\rho=f_{I},\quad
f\\!\circ\\!\rho=-f.$
One may deduce that all three $f_{O}$,
$f_{I}=\frac{1}{2}(f_{O}+f_{O}\\!\circ\\!\rho)$,
$f=\frac{1}{2}(f_{O}-f_{O}\\!\circ\\!\rho)$ are non-ramified, and hence
analytic, functions of $(s_{1},s_{2})$. Since $0=\lim_{\check{\epsilon}\to
0}f_{I}=\lim_{s_{1}-s_{2}\to 0}f_{I}$, we can write $f_{I}=(s_{1}-s_{2})\,g$
with $g$ that is both $\sigma$\- and $\rho$-invariant, thus an analytic
function of $s_{1}s_{2}=\sqrt{\check{\mu}^{2}-\check{\epsilon}}$ and of
$s_{1}^{2}+s_{2}^{2}=2\check{\mu}$ (which are algebraically independent and
form a Hilbert basis of the space of polynomials of $(s_{1},s_{2})$ that are
invariant to the action of $\sigma$ and $\rho$). ∎
###### Corollary 24.
Two germs of parametric families of systems $\Delta(x,\mu,\epsilon)$ and
$\Delta^{\prime}(x,\mu,\epsilon)$ (20) are analytically equivalent by means of
a germ of analytic transformation $G(x,\mu,\epsilon)$ satisfying
$G(x,\mu,\epsilon)=I\quad\text{whenever}\quad x^{2}-\epsilon=0,$ (58)
if and only if they have the same $\kappa$.
###### Proof.
If $G$ satisfying (58) is such that $\,G^{*}\Delta^{\prime}=\Delta$, and if
$F_{\bullet}$, $\bullet=O,I$, are the uniquely determined diagonalizing
transformations of Proposition 14 for $\Delta^{\\!s}$ (23):
$F_{\bullet}{}^{*}\\!\Delta^{\\!s}=\mkern 2.0mu\overline{\Delta}^{s}$, then
$\,F^{\prime}_{\bullet}=V^{-1}S^{-1}TSVF_{\bullet}\,$ must be the uniquely
determined transformations for $\Delta^{\prime s}$, and therefore
$\kappa_{\bullet}=\det F_{\bullet}=\det
F_{\bullet}^{\prime}=\kappa^{\prime}_{\bullet}.$
Conversely, if $\kappa=\kappa^{\prime}$, than also
$\kappa_{O}=\kappa^{\prime}_{O}$, $\kappa_{I}=\kappa^{\prime}_{I}$ and
$\gamma=\gamma^{\prime}$, which are determined by (54), (52) and (57).
Therefore the collections of connection matrices are the same for the two
systems (Proposition 23), and the transformation $G(x,\mu,\epsilon)$ of the
proof of Proposition 20 has the property (58). ∎
Now everything is ready to finish the proof of the Theorem I.
###### Proof of Theorem I (a).
Let $\Delta(x,\mu,\epsilon),\,\Delta^{\prime}(x,\mu,\epsilon)$ be two families
of systems (20), $\Delta^{\\!s}(s,\mu,\epsilon)$, $\Delta^{\prime
s}(s,\mu,\epsilon)$ their transforms 23 and $F_{\bullet}$,
$F^{\prime}_{\bullet}$ be the normalizing transformations from Proposition
(14) determined by the condition (45) with $\kappa_{\bullet}$,
$\kappa^{\prime}_{\bullet}$. Suppose that their invariants
$\gamma=\gamma^{\prime}$ are the same. We want to show that the two families
of systems $\Delta,\,\Delta^{\prime}$ are then analytically equivalent. We
know that
$\kappa_{I}(\check{\mu},0)=1=\kappa^{\prime}_{I}(\check{\mu},0)\quad\text{and}\quad\kappa_{O}(0,0)=1=\kappa^{\prime}_{O}(0,0).$
Let $\delta(\check{\mu},\check{\epsilon})$ depending continuously on the
parameters $(\check{\mu},\check{\epsilon})\in\check{\mathcal{M}}$ be such that
$\frac{\kappa^{\prime}_{O}}{\kappa^{\prime}_{I}}=\delta^{2}\frac{\kappa_{O}}{\kappa_{I}},\quad\delta(0,0)=1.$
The relation (57) implies that
$\delta(\tilde{\mu},\tilde{\epsilon})\,\delta(\bar{\mu},\bar{\epsilon})=1.$
Put
$F^{\prime\prime}_{O}=\delta^{-1}\,F^{\prime}_{O},\quad
F^{\prime\prime}_{I}=F^{\prime}_{I}\left(\begin{smallmatrix}\delta^{-1}&0\\\\[3.0pt]
0\ &\delta\,\end{smallmatrix}\right).$
They are also normalizing transformations for the system $\Delta^{\prime s}$:
$F^{\prime\prime*}\Delta^{\prime s}=\Delta^{\prime s}$. One can easily verify
that the connection matrices between the fundamental solutions
$F^{\prime\prime}_{\bullet}\Psi_{\bullet}$ are exactly the same as those
between the fundamental solutions $F_{\bullet}\Psi_{\bullet}\,$ (with
$\Psi_{\bullet}$ as in Figure 9), hence one concludes by Proposition 20.
The general case of two parametric families of systems
$\Delta(z,m),\,\Delta^{\prime}(z,m)$ in Theorem I that have the same formal
invariants $h(z,m),\,\lambda(z,m),\,\alpha(z,m)$ is reduced to the former case
using Lemma 8 and Proposition 9. ∎
To conclude, we shall summarize in Theorem III the results of the preceding
section on existence of normalizing transformations (Proposition 14), which we
formulate this time in general for all parametric systems $\Delta(z,m)$
unfolding $\Delta_{0}$ (3). The main interest of this theorem is that it shows
a way of generalizing the concept of formal equivalence of systems to
parametric systems that is particularly suited for treating multiple
singularities and their confluence.
Let
$\check{x}(\check{s},\check{\mu})=\check{s}^{2}-\check{\mu},$
a one-to-one map from the ramified coordinate $\check{s}$ to the ramified
coordinate $\check{x}$ projecting on $x$, be lifting of the map
$x(s,\mu)=s^{2}-\mu$ (21). Let
$\mathsf{X}:=\\{|x|<\delta_{s}^{2}-\delta_{\mu}\\},\quad\check{\mathsf{X}}:=\\{|\check{x}|<\delta_{s}^{2}-\delta_{\mu}\\}.$
Then the ramified images of $\Omega_{O},\,\Omega_{I}$ in the
$\check{x}$-coordinate:
$\check{x}(\Omega_{0}(\check{\mu},\check{\epsilon}))\cap\check{\mathsf{X}}$,
$\,\check{x}(\Omega_{I}(\check{\mu},\check{\epsilon}))\cap\check{\mathsf{X}}$,
cover for each $(\check{\mu},\check{\epsilon})$ a full neighborhood of each
singular point $x=\pm\sqrt{\epsilon}$. Define
$\check{X}_{O}(\check{\mu},\check{\epsilon}),\quad\check{X}_{I}(\check{\mu},\check{\epsilon}),$
depending continuously on
$(\check{\mu},\check{\epsilon})\in\check{\mathcal{M}}$, as simply connected
ramified extensions of these images, in such a way that they agree with them
near each singularity, are open away of the singularities, and the union of
their projections covers for each $(\check{\mu},\check{\epsilon})$ all
$\mathsf{X}$. The transformations (26)
$T_{\bullet}(\check{x},\check{\mu},\check{\epsilon})=S(s)VF_{\bullet}(\check{s},\check{\mu},\check{\epsilon})E_{\bullet}^{-1}(\check{s},\check{\mu},\check{\epsilon})V^{-1}S^{-1}(s),\quad\bullet=O,I$
extend then on $\check{X}_{\bullet}$ as normalizing transformation for the
parametric system $\Delta(x,\mu,\epsilon)$:
$T_{\bullet}\in\mathrm{GL}_{2}(\mathcal{B}(\check{X}_{\bullet})),\quad
T_{\bullet}^{*}\Delta=\widehat{\Delta}\ \eqref{eq:model2}.$
In the general case, when $\Delta(z,m)$ is a germ of a parametric system with
formal invariants $h(z,m)$, $\lambda(x,m)$, $\alpha(z,m),\,$ let
$\,x(z,m),\,\mu(m),\,\epsilon(m)\,$ be as in (16), (17). Define a ramified
cover $\check{M}$ of a neighborhood $M$ of 0 in the $m$-space as the lifting
of the cover $\check{\mathcal{M}}$ of $\mathsf{M}\\!\times\\!\mathsf{E}$
induced by the map $(\mu(m),\epsilon(m)):M\to\mathsf{M}\\!\times\\!\mathsf{E}$
$\textstyle{\mathllap{\check{M}\ni\,}\check{m}\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\
(\check{\mu}(\check{m}),\check{\epsilon}(\check{m}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathrlap{\,\in\check{\mathcal{M}}}}$$\textstyle{\mathllap{M\ni\,}m\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\
(\mu(m),\epsilon(m))\mathrlap{\,\in\mathsf{M}\\!\times\\!\mathsf{E}}}$
Define the ramified domains $O,I$ in $(\check{z},\check{m})$-space as
preimages of $\check{X}_{O}$, $\check{X}_{I}$ in the
$(\check{x},\check{\mu},\check{\epsilon})$-map
$O(\check{m})=\\{\check{z}\mid\check{x}(\check{z},\check{m})\in\check{X}_{O}(\check{\mu}(\check{m}),\check{\epsilon}(\check{m}))\\},\quad
I(\check{m})=\\{\check{z}\mid\check{x}(\check{z},\check{m})\in\check{X}_{I}(\check{\mu}(\check{m}),\check{\epsilon}(\check{m}))\\},$
(where $\check{x}(\check{z},\check{m})$ is lifting of the map $x(z,m)$ (16)).
###### Theorem III.
Let $\Delta(z,m)$ be a germ of a parametric family of systems (1) unfolding a
system (3) of generic type (4), and let $\widehat{\Delta}(z,m)$ be the
corresponding formal model system (8). On the ramified parametric domains
$\,O(\check{m}),I(\check{m}),\ \check{m}\in\check{M},\,$ defined above, whose
union covers a full neighborhood of 0 in the $(z,m)$-space, there exist
bounded normalizing transformations $T_{O}(\check{z},\check{m}),\
T_{I}(\check{z},\check{m})$:
$T_{O}\in\mathrm{GL}_{2}(\mathcal{B}(O)),\quad
T_{I}\in\mathrm{GL}_{2}(\mathcal{B}(I)),$
such that
$T_{O}^{*}\Delta=T_{I}^{*}\Delta=\widehat{\Delta}.$
###### Proof.
Follows from Lemma 8, Propositions 9 and 14 and the preceding considerations.
∎
## 3 Concluding remarks
The formulation of Theorems I and II is connected to the particular case
studied here: unfolding of a system $\Delta_{0}$ of Poincaré rank $k=1$ in
dimension $n=2$. It is benefiting from the fact that, in this case, one does
not need more than a single analytic invariant besides of the formal ones,
which then can be easily calculated from the monodromy. One cannot hope for
such a nice easy formulation in more general situations when $k>1$ or $n>2$.
On the other hand, most of the methods used in the previous section in order
to prove these theorems can be successfully generalized and used in other
situations. In particular, one should be able to obtain an analog of Theorem
III, which is one of the key steps in the analytic classification.
When dealing with a case of a single singularity of a (non-parametric) system,
the standard procedure is to pass first through the formal classification
(formal power series equivalence) and then, using (multi-)summability, to
obtain a chain of sectorial normalizations asymptotic to the formal
transformation, hence a stronger variant of Theorem III. However, we suggest
that in problems involving parameters and/or multiple singular points, one
should relax the asymptoticity of the normalizing transformations by demanding
instead that the transformations are just bounded near each singularity, while
at the same time allowing coverings by more general kind of domains (sector-
like or spiraling sector-like near each singularity), as in Theorem III here,
or as previously in [LR] (Theorem 4.21) in the case of unfolding of non-
resonant singularity of Poincaré rank 1. This would define a new kind of
equivalence, more general then the standard formal equivalence. In this
approach, each such chain of normalizing transformations then determines a
“cocycle” of bounded sectorial automorphisms of some chosen model system on
the intersections of the domains of the covering. This should be seen as an
element of a certain “cohomology” space (generalizing the Stokes cohomology)
whose elements would be identified with analytic equivalence classes of
systems. A suitable cohomology theory that would allow to deal with such
cocycles needs yet to be formulated.
Another interesting question that comes to one’s mind is that of whether
Corollary 6 stays valid also in more general situations: considering two
families of systems depending on the same parameter, if the systems restricted
to each parameter value are analytically equivalent on some open set
independent of the parameter, does that imply that they are also equivalent as
parametric families on that set?
## Acknowledgment
I would like to express my most sincere thanks to my advisor, Christiane
Rousseau, for her support and encouragement.
## References
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* [Ba] W. Balser, Formal power series and linear systems of meromorphic ordinary differential equations, Springer, 2000.
* [BJL] W. Balser, W.B. Jurkat, D.A. Lutz, Birkhoff Invariants and Stokes’ Multipliers for Meromorphic Linear Differential Equations, J. Math. Anal. Appl. 71 (1979), 48–94.
* [Glu] A.A. Glutsyuk, Resonant Confluence of Singular Points and Stokes Phenomena, J. Dynam. Contr. Syst. 10 (2004), 253–302.
* [IY] Y. Ilyashenko, S. Yakovenko, Lectures on Analytic Differential Equations, Graduate Studies in Mathematics 86, American Mathematical Society, Providence, 2008.
* [JLP1] W.B. Jurkat, D.A. Lutz, A. Peyerimhoff, Birkhoff invariants and effective calculations for meromorphic linear differential equations I, J. Math. Anal. Appl. 53 (1976), 438–470.
* [JLP2] W.B. Jurkat, D.A. Lutz, A. Peyerimhoff, Birkhoff invariants and effective calculations for meromorphic linear differential equations II, Houston J. Math. 2 (1976), 207–238.
* [LR] C. Lambert, C. Rousseau, Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank 1, Moscow Math. Journal 12 (2012), 77–138.
* [Si] Y. Sibuya, Linear differential equations in the complex domain : problems of analytic continuation, Translations of Mathematical Monographs 82, American Mathematical Society, Providence, 1990.
|
arxiv-papers
| 2013-01-22T16:30:46 |
2024-09-04T02:49:40.576999
|
{
"license": "Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/",
"authors": "Martin Klime\\v{s}",
"submitter": "Martin Klimes",
"url": "https://arxiv.org/abs/1301.5228"
}
|
1301.5286
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-006 LHCb-PAPER-2012-037 22 January 2013
Measurement of the fragmentation fraction ratio $f_{s}/f_{d}$ and its
dependence on $B$ meson kinematics
The LHCb collaboration†††Authors are listed on the following pages.
The relative production rate of $B^{0}_{s}$ and $B^{0}$ mesons is determined
with the hadronic decays $B^{0}_{s}\rightarrow D^{-}_{s}\pi^{+}$ and
$B^{0}\rightarrow D^{-}K^{+}$. The measurement uses data corresponding to 1.0
fb-1 of $pp$ collisions at a centre-of-mass energy of $\sqrt{s}=7$ TeV
recorded in the forward region with the LHCb experiment. The ratio of
production rates, $f_{s}/f_{d}$, is measured to be $0.238\pm 0.004\pm 0.015\pm
0.021$, where the first uncertainty is statistical, the second systematic, and
the third theoretical. This is combined with a previous LHCb measurement to
obtain $f_{s}/f_{d}~{}=0.256\pm 0.020$. The dependence of $f_{s}/f_{d}$ on the
transverse momentum and pseudorapidity of the $B$ meson is determined using
the decays $B^{0}_{s}\rightarrow D^{-}_{s}\pi^{+}$ and $B^{0}\rightarrow
D^{-}\pi^{+}$. There is evidence for a decrease with increasing transverse
momentum, whereas the ratio remains constant as a function of pseudorapidity.
In addition, the ratio of branching fractions of the decays $B^{0}\rightarrow
D^{-}K^{+}$ and $B^{0}\rightarrow D^{-}\pi^{+}$ is measured to be $0.0822\pm
0.0011\,(\textrm{stat})\pm 0.0025\,(\textrm{syst})$.
Submitted to Journal of High Energy Physics
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves
Jr22,35, S. Amato2, Y. Amhis7, L. Anderlini17,f, J. Anderson37, R.
Andreassen57, R.B. Appleby51, O. Aquines Gutierrez10, F. Archilli18, A.
Artamonov 32, M. Artuso53, E. Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J.
Back45, C. Baesso54, V. Balagura28, W. Baldini16, R.J. Barlow51, C.
Barschel35, S. Barsuk7, W. Barter44, A. Bates48, Th. Bauer38, A. Bay36, J.
Beddow48, I. Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28, E. Ben-
Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, A.
Berezhnoy29, R. Bernet37, M.-O. Bettler44, M. van Beuzekom38, A. Bien11, S.
Bifani12, T. Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36,
C. Blanks50, J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N.
Bondar27, W. Bonivento15, S. Borghi51, A. Borgia53, T.J.V. Bowcock49, E.
Bowen37, C. Bozzi16, T. Brambach9, J. van den Brand39, J. Bressieux36, D.
Brett51, M. Britsch10, T. Britton53, N.H. Brook43, H. Brown49, A. Büchler-
Germann37, I. Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O.
Callot7, M. Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A.
Carbone14,c, G. Carboni21,k, R. Cardinale19,i, A. Cardini15, H. Carranza-
Mejia47, L. Carson50, K. Carvalho Akiba2, G. Casse49, M. Cattaneo35, Ch.
Cauet9, M. Charles52, Ph. Charpentier35, P. Chen3,36, N. Chiapolini37, M.
Chrzaszcz 23, K. Ciba35, X. Cid Vidal34, G. Ciezarek50, P.E.L. Clarke47, M.
Clemencic35, H.V. Cliff44, J. Closier35, C. Coca26, V. Coco38, J. Cogan6, E.
Cogneras5, P. Collins35, A. Comerma-Montells33, A. Contu15, A. Cook43, M.
Coombes43, G. Corti35, B. Couturier35, G.A. Cowan36, D. Craik45, S.
Cunliffe50, R. Currie47, C. D’Ambrosio35, P. David8, P.N.Y. David38, I. De
Bonis4, K. De Bruyn38, S. De Capua51, M. De Cian37, J.M. De Miranda1, L. De
Paula2, W. De Silva57, P. De Simone18, D. Decamp4, M. Deckenhoff9, H.
Degaudenzi36,35, L. Del Buono8, C. Deplano15, D. Derkach14, O. Deschamps5, F.
Dettori39, A. Di Canto11, J. Dickens44, H. Dijkstra35, P. Diniz Batista1, M.
Dogaru26, F. Domingo Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil Suárez34,
D. Dossett45, A. Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A. Dziurda23, A.
Dzyuba27, S. Easo46,35, U. Egede50, V. Egorychev28, S. Eidelman31, D. van
Eijk38, S. Eisenhardt47, U. Eitschberger9, R. Ekelhof9, L. Eklund48, I. El
Rifai5, Ch. Elsasser37, D. Elsby42, A. Falabella14,e, C. Färber11, G.
Fardell47, C. Farinelli38, S. Farry12, V. Fave36, D. Ferguson47, V. Fernandez
Albor34, F. Ferreira Rodrigues1, M. Ferro-Luzzi35, S. Filippov30, C.
Fitzpatrick35, M. Fontana10, F. Fontanelli19,i, R. Forty35, O. Francisco2, M.
Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, E. Furfaro21, A. Gallas
Torreira34, D. Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J. Garofoli53,
P. Garosi51, J. Garra Tico44, L. Garrido33, C. Gaspar35, R. Gauld52, E.
Gersabeck11, M. Gersabeck51, T. Gershon45,35, Ph. Ghez4, V. Gibson44, V.V.
Gligorov35, C. Göbel54, D. Golubkov28, A. Golutvin50,28,35, A. Gomes2, H.
Gordon52, M. Grabalosa Gándara5, R. Graciani Diaz33, L.A. Granado Cardoso35,
E. Graugés33, G. Graziani17, A. Grecu26, E. Greening52, S. Gregson44, O.
Grünberg55, B. Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53,
G. Haefeli36, C. Haen35, S.C. Haines44, S. Hall50, T. Hampson43, S. Hansmann-
Menzemer11, N. Harnew52, S.T. Harnew43, J. Harrison51, P.F. Harrison45, T.
Hartmann55, J. He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando
Morata34, E. van Herwijnen35, E. Hicks49, D. Hill52, M. Hoballah5, C.
Hombach51, P. Hopchev4, W. Hulsbergen38, P. Hunt52, T. Huse49, N. Hussain52,
D. Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12, J. Imong43, R.
Jacobsson35, A. Jaeger11, E. Jans38, F. Jansen38, P. Jaton36, F. Jing3, M.
John52, D. Johnson52, C.R. Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M.
Karacson35, T.M. Karbach35, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A.
Keune36, B. Khanji20, O. Kochebina7, I. Komarov36,29, R.F. Koopman39, P.
Koppenburg38, M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M.
Kreps45, G. Krocker11, P. Krokovny31, F. Kruse9, M. Kucharczyk20,23,j, V.
Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G.
Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E. Lanciotti35, G.
Lanfranchi18,35, C. Langenbruch35, T. Latham45, C. Lazzeroni42, R. Le Gac6, J.
van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O.
Leroy6, Y. Li3, L. Li Gioi5, M. Liles49, R. Lindner35, C. Linn11, B. Liu3, G.
Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33, N. Lopez-March36, H.
Lu3, J. Luisier36, H. Luo47, A. Mac Raighne48, F. Machefert7, I.V.
Machikhiliyan4,28, F. Maciuc26, O. Maev27,35, S. Malde52, G. Manca15,d, G.
Mancinelli6, N. Mangiafave44, U. Marconi14, R. Märki36, J. Marks11, G.
Martellotti22, A. Martens8, L. Martin52, A. Martín Sánchez7, M. Martinelli38,
D. Martinez Santos39, D. Martins Tostes2, A. Massafferri1, R. Matev35, Z.
Mathe35, C. Matteuzzi20, M. Matveev27, E. Maurice6, A. Mazurov16,30,35,e, J.
McCarthy42, R. McNulty12, B. Meadows57,52, F. Meier9, M. Meissner11, M.
Merk38, D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S. Monteil5, D.
Moran51, P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K. Müller37, R.
Muresan26, B. Muryn24, B. Muster36, P. Naik43, T. Nakada36, R. Nandakumar46,
I. Nasteva1, M. Needham47, N. Neufeld35, A.D. Nguyen36, T.D. Nguyen36, C.
Nguyen-Mau36,o, M. Nicol7, V. Niess5, R. Niet9, N. Nikitin29, T. Nikodem11, S.
Nisar56, A. Nomerotski52, A. Novoselov32, A. Oblakowska-Mucha24, V.
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Palutan18, J. Panman35, A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J.
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A. Zhelezov11, A. Zhokhov28, L. Zhong3, A. Zvyagin35.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25National Center for Nuclear Research (NCBJ), Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
56Institute of Information Technology, COMSATS, Lahore, Pakistan, associated
to 53
57University of Cincinnati, Cincinnati, OH, United States, associated to 53
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pMassachusetts Institute of Technology, Cambridge, MA, United States
## 1 Introduction
The ratio of fragmentation fractions $f_{s}/f_{d}$ quantifies the relative
production rate of $B^{0}_{s}$ mesons with respect to $B^{0}$ mesons.
Knowledge of this quantity is essential when determining any $B^{0}_{s}$
branching fraction at the LHC. The measurement of the branching fraction of
the rare decay $B^{0}_{s}\\!\rightarrow\mu^{+}\mu^{-}$ [1] is the prime
example where a precise measurement of $f_{s}/f_{d}$ is crucial for reaching
the highest sensitivity in the search for physics beyond the Standard Model.
The branching fractions of a large number of $B^{0}$ and $B^{+}$ decays have
been measured to high precision at the $B$ factories [2], but no $B^{0}_{s}$
branching fraction is yet known with sufficiently high precision to be used as
a normalisation channel.
The relative production rates of $b$ hadrons are determined by the
fragmentation fractions $f_{u}$, $f_{d}$, $f_{s}$, $f_{c}$ and $f_{\mathchar
28931\relax}$, which describe the probability that a $b$ quark will hadronize
into a $B_{q}$ meson (where $q=u,d,s,c$), or a $b$ baryon,
respectively111Charge conjugation is implied throughout this paper.. The ratio
of fragmentation fractions $f_{s}/f_{d}$ has been previously measured at LHCb
with hadronic [3] and semileptonic decays [4], and the resulting values were
combined [4].
In this paper, the ratio of fragmentation fractions $f_{s}/f_{d}$ is
determined using $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ and
$B^{0}\\!\rightarrow D^{-}K^{+}$ decays collected in $pp$ collisions at a
centre-of-mass energy of $\sqrt{s}=7$ TeV, with data corresponding to an
integrated luminosity of $1.0~{}\mbox{\,fb}^{-1}$ recorded with the LHCb
detector. Since the ratio of branching fractions of the two decay channels is
theoretically well understood [5], their relative decay rates can be used to
determine the ratio of fragmentation fractions for $B^{0}_{s}$ and $B^{0}$
mesons through
$\displaystyle\frac{f_{s}}{f_{d}}$ $\displaystyle=$ $\displaystyle\frac{{\cal
B}(B^{0}\\!\rightarrow D^{-}K^{+})}{{\cal B}(B^{0}_{s}\\!\rightarrow
D^{-}_{s}\pi^{+})}\frac{\epsilon_{DK}}{\epsilon_{D_{s}\pi}}\frac{N_{D_{s}\pi}}{N_{DK}}$
(1) $\displaystyle=$
$\displaystyle\Phi_{\mathrm{PS}}\left|\frac{V_{us}}{V_{ud}}\right|^{2}\left(\frac{f_{K}}{f_{\pi}}\right)^{2}\frac{\tau_{B^{0}}}{\tau_{B^{0}_{s}}}\frac{1}{{\cal
N}_{a}{\cal N}_{F}}\frac{{\cal B}(D^{-}\rightarrow K^{+}\pi^{-}\pi^{-})}{{\cal
B}(D_{s}^{-}\rightarrow
K^{+}K^{-}\pi^{-})}\frac{\epsilon_{DK}}{\epsilon_{D_{s}\pi}}\frac{N_{D_{s}\pi}}{N_{DK}},$
where $N$ corresponds to a signal yield, $\epsilon$ corresponds to a total
efficiency, $\tau_{B_{s}^{0}}/\tau_{B^{0}}=0.984\pm 0.011$ [6] corresponds to
the ratio of lifetimes and ${\cal B}(D^{-}\rightarrow
K^{+}\pi^{-}\pi^{-})=(9.14\pm 0.20)\%$ [7] and ${\cal B}(D_{s}^{-}\rightarrow
K^{+}K^{-}\pi^{-})=(5.50\pm 0.27)\%$ [8] correspond to the $D^{-}_{(s)}$ meson
branching fractions. The factor ${\cal N}_{a}=1.00\pm 0.02$ accounts for the
ratio of non-factorizable corrections [9], ${\cal N}_{F}=1.092\pm 0.093$ for
the ratio of $B^{0}_{(s)}\rightarrow D^{-}_{(s)}$ form factors [10], and
$\Phi_{\mathrm{PS}}=0.971$ for the difference in phase space due to the mass
differences of the initial and final state particles. The numerical values
used for the CKM matrix elements are $|V_{us}|=0.2252$, $|V_{ud}|=0.97425$,
and for the decay constants are $f_{\pi}=130.41$ MeV, $f_{K}=156.1$ MeV, with
negligible uncertainties, below 1% [2]. The measurement is not statistically
limited by the size of the $B^{0}\\!\rightarrow D^{-}K^{+}$ sample , and
therefore the theoretically less clean $B^{0}\\!\rightarrow D^{-}\pi^{+}$
decays, where exchange diagrams contribute to the total amplitude, do not
contribute to the knowledge of $f_{s}/f_{d}$ .
The ratio of fragmentation fractions can depend on the centre-of-mass energy,
as well as on the kinematics of the $B^{0}_{(s)}$ meson, as was studied
previously at LHCb with partially reconstructed $B$ decays [4]. The dependence
of the ratio of fragmentation fractions on the transverse momentum $p_{\rm T}$
and pseudorapidity $\eta$ of the $B^{0}_{(s)}$ meson is determined using fully
reconstructed $B^{0}\\!\rightarrow D^{-}\pi^{+}$ and $B^{0}_{s}\\!\rightarrow
D^{-}_{s}\pi^{+}$ decays. Since it is only the dependence that is of interest
here, the more abundant $B^{0}\\!\rightarrow D^{-}\pi^{+}$ decay is used
rather than the $B^{0}\\!\rightarrow D^{-}K^{+}$ decay. The
$B^{0}\\!\rightarrow D^{-}K^{+}$ and $B^{0}\\!\rightarrow D^{-}\pi^{+}$ decays
are also used to determine their ratio of branching fractions, which can be
used to quantify non-factorizable effects in such heavy-to-light decays [9].
The paper is organised as follows: the detector is described in Sec. 2,
followed by the event selection and the relative selection efficiencies in
Sec. 3. The fit to the mass distributions and the determination of the signal
yields are discussed in Sec. 4. The systematic uncertainties are presented in
Sec. 5, and the final results are given in Sec. 6.
## 2 Detector and software
The LHCb detector [11] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift tubes placed downstream. Data are
taken with both magnet polarities. The combined tracking system has momentum
resolution $\Delta p/p$ that varies from 0.4% at
5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter222Impact
parameter (IP) is defined as the transverse distance of closest approach
between the track and a primary interaction. resolution of 20$\,\upmu\rm m$
for tracks with high transverse momentum. Charged hadrons are identified using
two ring-imaging Cherenkov detectors.
The trigger [12] consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage which applies a
full event reconstruction. The events used in this analysis are selected at
the hardware stage by requiring a cluster in the calorimeters with transverse
energy larger than 3.6 GeV. The software stage requires a two-, three- or
four-track secondary vertex with a high sum of the $p_{\rm T}$ of the tracks
and a significant displacement from the primary $pp$ interaction vertices
(PVs). At least one track should have $p_{\rm T}$ greater than
$1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, track fit $\chi^{2}$ over the
number of degrees of freedom less than two, and IP $\chi^{2}$ with respect to
the associated primary interaction greater than sixteen. The IP $\chi^{2}$ is
defined as the difference between the $\chi^{2}$ from the vertex fit of the
associated PV reconstructed with and without the considered track. A
multivariate algorithm is used for the identification of the secondary
vertices consistent with the decay of a $b$ hadron.
In the simulation, $pp$ collisions are generated using Pythia 6.4 [13] with a
specific LHCb configuration [14]. Decays of hadronic particles are described
by EvtGen [15], whilst final state radiation is generated using Photos [16].
The interaction of the generated particles with the detector and its response
are implemented using the Geant4 toolkit [17, *Agostinelli:2002hh] as
described in Ref. [19].
## 3 Event selection
The three decay modes, $B^{0}\rightarrow D^{-}\pi^{+}$, $B^{0}\rightarrow
D^{-}K^{+}$ and $B^{0}_{s}\rightarrow D_{s}^{-}\pi^{+}$, are topologically
very similar and can therefore be selected using the same event selection
criteria, thus minimizing efficiency differences between the modes. The
$B^{0}_{(s)}$ candidates are reconstructed from a $D_{(s)}^{-}$ candidate and
an additional pion or kaon (the “bachelor” particle), with the $D_{(s)}^{-}$
meson decaying to $K^{+}\pi^{-}\pi^{-}$ ($K^{+}K^{-}\pi^{-}$).
After the trigger selection, a loose preselection is made using the
$B^{0}_{(s)}$ and $D_{(s)}^{-}$ masses, lifetimes and vertex qualities. A
boosted decision tree (BDT) [20] is used to further separate signal from
background. The BDT is trained on half the $B^{0}_{s}\\!\rightarrow
D^{-}_{s}\pi^{+}$ data sample. The most discriminating variables are the
$B^{0}_{(s)}$ impact parameter $\chi^{2}$, the pointing angle of the
$B^{0}_{(s)}$ candidate to the primary vertex and the $p_{\rm T}$ of the
tracks. A cut value for the BDT output variable was chosen to optimally reduce
the number of combinatorial background events, retaining approximately 84% of
the signal events.
The $D_{(s)}^{-}$ candidates are identified by requiring the invariant mass
under the $K^{+}\pi^{-}\pi^{-}$ ($K^{+}K^{-}\pi^{-}$) hypothesis to fall
within the selection window 1844 – 1890 (1944 – 1990)
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The relative efficiency of the
selection procedure is evaluated for all decay modes using simulated events,
generated with the appropriate Dalitz plot structures [21, 22]. Since the
analysis is only sensitive to relative efficiencies, the impact of any
discrepancy between data and simulation is small.
The final $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ and $B^{0}\\!\rightarrow
D^{-}\pi^{+}$ event samples are obtained after a particle identification (PID)
criterion, based on the difference in log-likelihood between the kaon and pion
hypotheses (DLL). A cut on the bachelor particle is placed at
DLL($K-\pi$)$<0$. The $B^{0}\\!\rightarrow D^{-}K^{+}$ sample is selected by
requiring DLL($K-\pi$)$>5$ for the bachelor particle. The
$D_{s}^{-}\rightarrow K^{+}K^{-}\pi^{-}$ decay is distinguished from
$D^{-}\rightarrow K^{+}\pi^{-}\pi^{-}$ decays by imposing DLL($K-\pi$)$>5$ on
the kaon candidate with the same charge as the $D$ meson, whilst the DLL
criteria for the $\pi^{-}$ and $K^{+}$ are identical between $D^{-}$ and
$D_{s}^{-}$ and are used to discriminate $D^{-}_{(s)}$ decays from background.
The PID performance as a function of $p_{\rm T}$ and $\eta$ of the track is
estimated from data using a calibration sample of approximately 27 million
$D^{*-}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}(K^{+}\pi^{-})\pi^{-}$ decays, which
are selected using kinematic criteria only.
## 4 Event yields
(a)(b)(c) Figure 1: Invariant mass distributions of (a) $B^{0}\\!\rightarrow
D^{-}\pi^{+}$ (b) $B^{0}\\!\rightarrow D^{-}K^{+}$ and (c)
$B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ candidates. The solid line is the
result of the fit and the dotted line indicates the signal. The stacked
background shapes follow the same top-to-bottom order in the legend and the
plot. The $B^{0}_{s}$ and $\kern 1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}^{0}_{b}$ backgrounds in the $B^{0}\\!\rightarrow D^{-}\pi^{+}$
mass distribution are invisibly small. The resulting signal yields are listed
in Table 1. For illustration purposes the figures include events from both
magnet polarities, although they are fitted separately as described in the
text.
The relative yields of the three decay modes are determined from unbinned
extended maximum likelihood fits to the mass distributions of the
reconstructed $B^{0}_{(s)}$ candidates as shown in Fig. 1. In order to achieve
the highest sensitivity, the sample is separated according to the two magnet
polarities, allowing for possible differences in PID performance and in
running conditions. A simultaneous fit to the two magnet polarities is
performed for each decay mode, with the peak position and width of each signal
shared between the two.
The signal mass shape is described by a Gaussian distribution with power-law
tails on either side to model the radiative tail and non-Gaussian detector
effects. It consists of a Crystal Ball function [23]
$\displaystyle
f_{\mathrm{left}}(m,\alpha,n,\mu,\sigma)=N\cdot\left\\{\begin{array}[]{ll}e^{-\frac{(m-\mu)^{2}}{2\sigma^{2}}},&\mathrm{for}\;\frac{m-\mu}{\sigma}>-\alpha\\\
\left(\frac{n}{|\alpha|}\right)^{n}\cdot
e^{-|\alpha|^{2}/2}\cdot\left(\frac{n}{|\alpha|}-|\alpha|-\frac{m-\mu}{\sigma}\right)^{-n},&\mathrm{for}\;\frac{m-\mu}{\sigma}\leq-\alpha\\\
\end{array}\right.$ (4)
and a second, similar but mirrored, function to describe the right tail,
resulting in the signal mass shape
$f_{\mathrm{2CB}}(m)=f_{\mathrm{left}}(m)+f_{\mathrm{right}}(m)$. The
parameters of the tails are obtained from simulated events. The mean $\mu$ and
the width $\sigma$ of the Gaussian distribution are equal in both Crystal Ball
functions, and are allowed to vary in the fit. The parameter $N$ is a
normalisation factor.
Three classes of background are considered in the fit: fully reconstructed
decays where at least one track is misidentified, partially reconstructed
decays with or without misidentified tracks and combinatorial background. The
shapes of the invariant mass distributions for the partially reconstructed
decays are taken from large samples of simulated events. The main sources are
$B^{0}\rightarrow D^{-}\rho^{+}\,$and $B^{0}\\!\rightarrow
D^{*-}\pi^{+}(K^{+})$ for the $B^{0}\\!\rightarrow D^{-}\pi^{+}(K^{+})$
sample, and $B^{0}_{s}\rightarrow D^{-}_{s}\rho^{+}\,$and
$B^{0}_{s}\rightarrow D^{*-}_{s}\pi^{+}\,$for the $B^{0}_{s}\\!\rightarrow
D^{-}_{s}\pi^{+}$ sample.
The invariant mass distributions of the misidentified decays are affected by
the PID criteria. The shapes are obtained from simulated events, with the
appropriate mass hypothesis applied. The distribution is then reweighted in a
data-driven way, according to the particle identification cut efficiency
obtained from the calibration sample, which is strongly dependent on the
momentum of the particle.
Despite the small $\pi\rightarrow K$ misidentification probability of 2.8%,
the largest misidentified background in the $B^{0}\\!\rightarrow D^{-}K^{+}$
sample originates from Cabibbo-favoured $B^{0}\\!\rightarrow D^{-}\pi^{+}$
decays where the bachelor pion is misidentified as a kaon. The shape of this
particular misidentified decay is determined from data using a high purity
sample of $B^{0}\\!\rightarrow D^{-}\pi^{+}$ decays (see Fig. 1(a)), obtained
by selecting events in a narrow mass window 5200–5340
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The yield of this prominent
peaking background is allowed to vary in the fit and is found to be consistent
with the expected yield based on the $B^{0}\\!\rightarrow D^{-}\pi^{+}$ signal
yield and the misidentification probability. The contamination of
$B^{0}\\!\rightarrow D^{-}\pi^{+}$ events in the $B^{0}_{s}\\!\rightarrow
D^{-}_{s}\pi^{+}$ sample can be caused by the misidentification of either pion
from the $D^{-}$ decay. The misidentification probability is 2.0% (3.2%) for
the higher (lower) $p_{\rm T}$ pion. After selecting the $D^{-}_{s}$ candidate
within the mass window around the nominal $D^{-}_{s}$ mass [2], the number of
misidentified pions is reduced to 0.75% (0.02%). The yield of this background
is constrained in the fit, based on the $B^{0}\\!\rightarrow D^{-}\pi^{+}$
signal yield, the misidentification probability and their associated
uncertainties.
The yield of $\kern 1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}^{0}_{b}\rightarrow\mathchar 28931\relax^{-}_{c}\pi^{+}$decays is
allowed to vary in the fit. The cross-feeds from $B^{0}\\!\rightarrow
D^{-}K^{+}$ and $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ events in the
$B^{0}\\!\rightarrow D^{-}\pi^{+}$ signal is small, and are constrained to
their respective predicted yields. In addition, a contribution from the rare
$B^{0}\rightarrow D^{-}_{s}\pi^{+}\,$decay is expected with a yield of 3.3%
compared to the $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ signal, and is
accounted for accordingly.
Table 1: Yields obtained from the fits to the invariant mass distributions. Signal | Yield
---|---
$B^{0}\\!\rightarrow D^{-}\pi^{+}$ | $106\,197\pm 344$
$B^{0}\\!\rightarrow D^{-}K^{+}$ | $7\,664\pm\,\,\,99$
$B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ | $17\,419\pm 155$
The combinatorial background consists of events with random pions and kaons,
forming a fake $D^{-}$ or $D^{-}_{s}$ candidate, as well as real $D^{-}$ or
$D^{-}_{s}$ mesons, that combine with a random pion or kaon. The combinatorial
background is modelled with an exponential shape.
The results of the fits are presented in Fig. 1, and the corresponding signal
yields are listed in Table 1. The total yields of the decays
$B^{0}\\!\rightarrow D^{-}\pi^{+}$ and $B^{0}\\!\rightarrow D^{-}K^{+}$ are
used to determine the ratio of their branching fractions, while the event
yields of the decays $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ and
$B^{0}\\!\rightarrow D^{-}K^{+}$ are used to measure the average ratio of
fragmentation fractions.
The dependence of the relative $b$-hadron production fractions as a function
of the transverse momentum and pseudorapidity of the $B^{0}_{(s)}$ meson is
studied in the ranges $2.0<\eta<5.0$ and $1.5<\mbox{$p_{\rm T}$}<40$
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, using $B^{0}\\!\rightarrow
D^{-}\pi^{+}$ and $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ decays. The event
sample is subdivided in 20 bins in $p_{\rm T}$ and 10 bins in $\eta$, with the
bin sizes chosen to obtain approximately equal number of events per bin. The
fitting model for each bin is the same as that for the integrated samples,
apart from the treatment of the exponent of the combinatorial background
distribution, which is fixed to the value obtained from the fits to the
integrated sample.
## 5 Systematic uncertainties
The systematic uncertainties on the measurement of the relative event yields
of the $B^{0}\\!\rightarrow D^{-}\pi^{+}$, $B^{0}\\!\rightarrow D^{-}K^{+}$
and $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ decay modes are related to
trigger and offline selection efficiency corrections, particle identification
calibration and the fit model.
Table 2: Systematic uncertainties for the measurement of the corrected ratio of event yields used for the measurements of $f_{s}/f_{d}$ and the relative branching fraction of $B^{0}\\!\rightarrow D^{-}K^{+}$. The systematic uncertainty in $p_{\rm T}$ and $\eta$ bins is shown as a range in the last column, and the total systematic uncertainty is the quadratic sum of the uncorrelated uncertainties. The systematic uncertainties on the ratio of $B^{0}\\!\rightarrow D^{-}\pi^{+}$ and $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ yields that are correlated among the bins do not affect the dependence on $p_{\rm T}$ or $\eta$, and are not accounted for in the total systematic uncertainty. Source | $\frac{B^{0}\\!\rightarrow D^{-}\pi^{+}}{B^{0}\\!\rightarrow D^{-}K^{+}}$ (%) | $\frac{B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}}{B^{0}\\!\rightarrow D^{-}K^{+}}$ (%) | $\frac{B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}}{B^{0}\\!\rightarrow D^{-}\pi^{+}}$ (%)
---|---|---|---
Detector acceptance | | |
and reconstruction | $0.7$ | $0.7$ | $2.0-2.9$
Hardware trigger efficiency | $2.0$ | $2.0$ | $0.8$
Offline selection | $1.2$ | $1.1$ | $1.2$
BDT cut | $1.0$ | $1.0$ | $1.5$
PID selection | $1.0$ | $1.5$ | $1.1$
Comb. background | $0.7$ | $1.0$ | $0.8$
Signal shape (tails) | $0.5$ | $0.6$ | [correl.]
Signal shape (core) | $0.8$ | $1.0$ | [correl.]
Charmless background | $0.4$ | – | [correl.]
Total | $3.1$ | $3.4$ | $3.2-3.8$
The response to charged pions and kaons of the hadronic calorimeter used at
the hardware trigger level has been investigated. As the hardware trigger
mostly triggers on the high-$p_{\rm T}$ bachelor, a systematic uncertainty of
2% is assigned to the ratio of trigger efficiencies for the decays
$B^{0}\\!\rightarrow D^{-}K^{+}$ and $B^{0}\\!\rightarrow D^{-}\pi^{+}$,
estimated from dedicated studies with $D^{*-}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}(K^{+}\pi^{-})\pi^{-}$ decays. This
uncertainty is assumed to be uncorrelated between the individual bins in the
binned analysis.
The relative selection efficiencies from simulation are studied by varying the
BDT criterion, changing the signal yields by about $\pm 25\%$. The variation
of the relative efficiency is 1.0% which is assigned as systematic
uncertainty.
The uncertainty on the PID efficiencies is estimated by comparing, in
simulated events, the results obtained using the $D^{*-}$ calibration sample
to the true simulated PID performance on the signal decays. The corresponding
uncertainty ranges from 1.0% to 1.5% for the different measurements.
The exponent of the combinatorial background distribution is allowed to vary
in the fits to the $B^{0}\\!\rightarrow D^{-}\pi^{+}$ and
$B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ mass distributions. By studying
$D^{-}\pi^{-}$ and $D^{-}K^{-}$ combinations, it is suggested that the value
of the exponent is smaller for the $B^{0}\\!\rightarrow D^{-}K^{+}$ decays
than for the $B^{0}\\!\rightarrow D^{-}\pi^{+}$ decays, and therefore in the
fit to the $B^{0}\\!\rightarrow D^{-}K^{+}$ candidates the exponent is fixed
to half the value found in the fit to the $B^{0}\\!\rightarrow D^{-}\pi^{+}$
sample. The uncertainty on the signal yields due to the shape of the
combinatorial background is estimated by reducing the exponent to half its
value in the fits to the $B^{0}\\!\rightarrow D^{-}\pi^{+}$ and
$B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ mass distributions, and by taking a
flat background for the fit to the $B^{0}\\!\rightarrow D^{-}K^{+}$ mass
distribution. An uncertainty of 1.0% (0.7%) is assigned to the relative
$B^{0}\\!\rightarrow D^{-}K^{+}$ and $B^{0}_{s}\\!\rightarrow
D^{-}_{s}\pi^{+}$ ($B^{0}\\!\rightarrow D^{-}\pi^{+}$) yields.
The tails of the signal distributions are fixed from simulation due to the
presence of large amounts of partially reconstructed decays in the lower
sidebands. The uncertainty on the signal yield is estimated by varying the
parameters that describe the tails by 10%. The uncertainty from the shape of
the central peak is taken from a fit allowing for two different widths for the
Crystal Ball functions in Eq. 4, leading to a 1.0% (0.8%) uncertainty on the
relative $B^{0}\\!\rightarrow D^{-}K^{+}$ and $B^{0}_{s}\\!\rightarrow
D^{-}_{s}\pi^{+}$ ($B^{0}\\!\rightarrow D^{-}\pi^{+}$) yields.
The contribution of _charmless_ $B$ decays without an intermediate $D$ meson
is ignored in the fit. To evaluate the systematic uncertainty due to these
decays, the $B$ mass spectra for candidates in the sidebands of the $D$ mass
distribution are examined. A contribution of 0.4% relative to the signal yield
is found in the $B^{0}\\!\rightarrow D^{-}\pi^{+}$ decay mode, and no
contribution is seen in the other modes. For the $B^{0}\\!\rightarrow
D^{-}\pi^{+}$ decay mode no correction is applied and the full size is taken
as an uncertainty. No systematic uncertainty is assigned for the other decay
modes.
The various sources of the systematic uncertainty that contribute to the
uncertainties on the ratios of signal yields are listed in Table 2. No
uncertainty is associated to the $\kern
1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}^{0}_{b}\rightarrow\mathchar
28931\relax^{-}_{c}\pi^{+}$background, as the yield is allowed to vary in the
fit. Other cross checks, like varying the $B^{0}\rightarrow
D^{-}_{s}\pi^{+}\,$yield in the $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ fit
or including $\kern 1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}^{0}_{b}\rightarrow\mathchar 28931\relax^{-}_{c}K^{+}\,$in the
$B^{0}\\!\rightarrow D^{-}K^{+}$ fit, show a negligible effect on the signal
yields.
All systematic variations are also performed in bins, and the corresponding
relative changes in the ratio of yields have been quantified. Variations
showing correlated behaviour do not affect the slope and are therefore not
considered further.
## 6 Results
The relative signal yields of the decays $B^{0}\\!\rightarrow D^{-}\pi^{+}$,
$B^{0}\\!\rightarrow D^{-}K^{+}$ and $B^{0}_{s}\\!\rightarrow
D^{-}_{s}\pi^{+}$ are used to determine the branching fraction of the decay
$B^{0}\\!\rightarrow D^{-}K^{+}$, and the ratio of fragmentation fractions
$f_{s}/f_{d}$ .
The efficiency corrected ratio of $B^{0}\\!\rightarrow D^{-}K^{+}$ and
$B^{0}\\!\rightarrow D^{-}\pi^{+}$ signal yields results in the ratio of
branching fractions
$\frac{{\cal B}\left(B^{0}\\!\rightarrow D^{-}K^{+}\right)}{{\cal
B}\left(B^{0}\\!\rightarrow D^{-}\pi^{+}\right)}=0.0822\pm
0.0011\,(\textrm{stat})\pm 0.0025\,(\textrm{syst}).$
This is combined with the world average branching fraction ${\cal
B}\left(B^{0}\\!\rightarrow D^{-}\pi^{+}\right)=(26.8\pm 1.3)\times 10^{-4}$
[2], to give
${\cal B}\left(B^{0}\\!\rightarrow D^{-}K^{+}\right)=(2.20\pm 0.03\pm 0.07\pm
0.11)\times 10^{-4},$
where the first uncertainty is statistical, the second is systematic and the
last is due to the uncertainty on the $B^{0}\\!\rightarrow D^{-}\pi^{+}$
branching fraction.
The ratio of fragmentation fractions is determined from the efficiency
corrected event yields. The ratio of efficiencies is $0.913\pm 0.027$. This
results in
$\displaystyle\frac{f_{s}}{f_{d}}$ $\displaystyle=$ $\displaystyle(0.261\pm
0.004\pm 0.017)\times\frac{1}{{\cal N}_{a}{\cal N}_{F}}$ $\displaystyle=$
$\displaystyle 0.238\pm 0.004\pm 0.015\pm 0.021\,,$
where the first uncertainty is statistical, the second is systematic
containing the sources listed in Table 2 as well as errors from external
measurements, and the third is theoretical, due to the knowledge of ${\cal
N}_{a}$ and ${\cal N}_{F}$. The last source is dominated by the uncertainty on
the form factor ratio.
This measurement supersedes and is in agreement with the previous
determination with hadronic decays [3]. It also agrees with the previous
measurement based on semileptonic decays [4]. The two independent results are
combined taking into account the various sources of correlated systematic
uncertainties, notably the $D^{-}_{(s)}$ branching fractions and $B^{0}_{(s)}$
lifetimes, to give
$\frac{f_{s}}{f_{d}}=0.256\pm 0.020,$ (6)
which supersedes the previous measurement from LHCb.
(a)(b) Figure 2: Ratio of fragmentation fractions $f_{s}/f_{d}$ as functions
of (a) $p_{\rm T}$ and (b) $\eta$. The errors on the data points are the
statistical and uncorrelated systematic uncertainties added in quadrature. The
solid line is the result of a linear fit, and the dashed line corresponds to
the fit for the no-dependence hypothesis. The average value of $p_{\rm T}$ or
$\eta$ is determined for each bin and used as the center of the bin. The
horizontal error bars indicate the bin size. Note that the scale is zero
suppressed.
The value of $f_{s}/f_{d}$ in bins of $p_{\rm T}$ or $\eta$ is determined
using the $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ and $B^{0}\\!\rightarrow
D^{-}\pi^{+}$ decay modes and is presented in Fig. 2. A linear $\chi^{2}$ fit
gives
$\displaystyle f_{s}/f_{d}~{}(\mbox{$p_{\rm T}$})$ $\displaystyle=$
$\displaystyle(0.256\pm 0.020)+(-2.0\pm 0.6)\times
10^{-3}/\mathrm{\,Ge\kern-1.00006ptV}/c\times(\mbox{$p_{\rm
T}$}-\langle\mbox{$p_{\rm T}$}\rangle)$ $\displaystyle f_{s}/f_{d}~{}(\eta)$
$\displaystyle=$ $\displaystyle(0.256\pm 0.020)+(0.005\pm
0.006)\times(\eta-\langle\eta\rangle),$
with $\langle\mbox{$p_{\rm T}$}\rangle=10.4$ GeV/$c$ and
$\langle\eta\rangle=3.28$. The data points are normalised with a scale factor
to match the average value of 0.256. The uncertainty associated to this
parameter is taken from Eq. 6, whilst the error from the fit is 0.003 for both
$p_{\rm T}$ and $\eta$.
The p-value for this linear fit is found to be 0.16 (0.87) for $p_{\rm T}$
($\eta$). The observed slope for the dependence on the transverse momentum of
the $B^{0}_{(s)}$ meson deviates from zero with a significance of three
standard deviations. No indication of a dependence on $\eta(B)$ is found.
## 7 Conclusions
The relative production rate of $B^{0}_{s}$ and $B^{0}$ mesons is determined
using the hadronic decays $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ and
$B^{0}\\!\rightarrow D^{-}K^{+}$ resulting in $f_{s}/f_{d}~{}=0.238\pm
0.004(\textrm{stat})\pm 0.015(\textrm{syst})\pm 0.021(\textrm{theo})$. This
value is consistent with a previous LHCb measurement based on semileptonic
decays, with which it is averaged to obtain $f_{s}/f_{d}~{}=0.256\pm 0.020$.
The ratio of fragmentation fractions $f_{s}/f_{d}$ is determined as a function
of the transverse momentum and pseudorapidity of the $B^{0}_{(s)}$ meson, and
a variation consistent with a linear dependence on the transverse momentum of
the the $B^{0}_{(s)}$ meson is observed, with a significance of three standard
deviations. In addition, the ratio of branching fractions of the decays
$B^{0}\\!\rightarrow D^{-}K^{+}$ and $B^{0}\\!\rightarrow D^{-}\pi^{+}$ is
measured to be $0.0822\pm 0.0011\,(\textrm{stat})\pm 0.0025\,(\textrm{syst})$.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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|
arxiv-papers
| 2013-01-22T19:16:00 |
2024-09-04T02:49:40.594254
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, A. Adametz, B. Adeva,\n M. Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio,\n M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S.\n Amato, Y. Amhis, L. Anderlini, J. Anderson, R. Andreassen, R. B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, S. Bachmann, J. J. Back, C. Baesso, V. Balagura, W. Baldini, R. J.\n Barlow, C. Barschel, S. Barsuk, W. Barter, A. Bates, Th. Bauer, A. Bay, J.\n Beddow, I. Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M.\n Benayoun, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O.\n Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P. M.\n Bj{\\o}rnstad, T. Blake, F. Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov,\n V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T. J. V.\n Bowcock, E. Bowen, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D.\n Brett, M. Britsch, T. Britton, N. H. Brook, H. Brown, A. B\\\"uchler-Germann,\n I. Burducea, A. Bursche, J. Buytaert, S. Cadeddu, O. Callot, M. Calvi, M.\n Calvo Gomez, A. Camboni, P. Campana, A. Carbone, G. Carboni, R. Cardinale, A.\n Cardini, H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, M.\n Cattaneo, Ch. Cauet, M. Charles, Ph. Charpentier, P. Chen, N. Chiapolini, M.\n Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P. E. L. Clarke, M. Clemencic,\n H. V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, E. Cogneras, P. Collins,\n A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, G. Corti, B. Couturier,\n G. A. Cowan, D. Craik, S. Cunliffe, R. Currie, C. D'Ambrosio, P. David, P. N.\n Y. David, I. De Bonis, K. De Bruyn, S. De Capua, M. De Cian, J. M. De\n Miranda, L. De Paula, W. De Silva, P. De Simone, D. Decamp, M. Deckenhoff, H.\n Degaudenzi, L. Del Buono, C. Deplano, D. Derkach, O. Deschamps, F. Dettori,\n A. Di Canto, J. Dickens, H. Dijkstra, P. Diniz Batista, M. Dogaru, F. Domingo\n Bonal, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F.\n Dupertuis, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V.\n Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, U. Eitschberger, R.\n Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, D. Elsby, A. Falabella, C.\n F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, D. Ferguson, V.\n Fernandez Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, C.\n Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C.\n Frei, M. Frosini, S. Furcas, E. Furfaro, A. Gallas Torreira, D. Galli, M.\n Gandelman, P. Gandini, Y. Gao, J. Garofoli, P. Garosi, J. Garra Tico, L.\n Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph.\n Ghez, V. Gibson, V. V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A.\n Gomes, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L. A. Granado\n Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S. Gregson, O.\n Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G.\n Haefeli, C. Haen, S. C. Haines, S. Hall, T. Hampson, S. Hansmann-Menzemer, N.\n Harnew, S. T. Harnew, J. Harrison, P. F. Harrison, T. Hartmann, J. He, V.\n Heijne, K. Hennessy, P. Henrard, J. A. Hernando Morata, E. van Herwijnen, E.\n Hicks, D. Hill, M. Hoballah, C. Hombach, P. Hopchev, W. Hulsbergen, P. Hunt,\n T. Huse, N. Hussain, D. Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J.\n Imong, R. Jacobsson, A. Jaeger, E. Jans, F. Jansen, P. Jaton, F. Jing, M.\n John, D. Johnson, C. R. Jones, B. Jost, M. Kaballo, S. Kandybei, M. Karacson,\n T. M. Karbach, I. R. Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji, O.\n Kochebina, I. Komarov, R. F. Koopman, P. Koppenburg, M. Korolev, A.\n Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F.\n Kruse, M. Kucharczyk, V. Kudryavtsev, T. Kvaratskheliya, V. N. La Thi, D.\n Lacarrere, G. Lafferty, A. Lai, D. Lambert, R. W. Lambert, E. Lanciotti, G.\n Lanfranchi, C. Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van\n Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, O. Leroy, Y.\n Li, L. Li Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben,\n J. H. Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, J. Luisier, H. Luo, A.\n Mac Raighne, F. Machefert, I. V. Machikhiliyan, F. Maciuc, O. Maev, S. Malde,\n G. Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, M. Martinelli, D.\n Martinez Santos, D. Martins Tostes, A. Massafferri, R. Matev, Z. Mathe, C.\n Matteuzzi, M. Matveev, E. Maurice, A. Mazurov, J. McCarthy, R. McNulty, B.\n Meadows, F. Meier, M. Meissner, M. Merk, D. A. Milanes, M.-N. Minard, J.\n Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F.\n Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, P. Naik, T. Nakada, R.\n Nandakumar, I. Nasteva, M. Needham, N. Neufeld, A. D. Nguyen, T. D. Nguyen,\n C. Nguyen-Mau, M. Nicol, V. Niess, R. Niet, N. Nikitin, T. Nikodem, S. Nisar,\n A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S.\n Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J. M. Otalora Goicochea, P.\n Owen, B. K. Pal, A. Palano, M. Palutan, J. Panman, A. Papanestis, M.\n Pappagallo, C. Parkes, C. J. Parkinson, G. Passaleva, G. D. Patel, M. Patel,\n G. N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A.\n Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D. L. Perego, E. Perez\n Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina,\n K. Petridis, A. Petrolini, A. Phan, E. Picatoste Olloqui, B. Pietrzyk, T.\n Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F. Polci, G. Polok, A.\n Poluektov, E. Polycarpo, D. Popov, B. Popovici, C. Potterat, A. Powell, J.\n Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian, J. H. Rademacker, B.\n Rakotomiaramanana, M. S. Rangel, I. Raniuk, N. Rauschmayr, G. Raven, S.\n Redford, M. M. Reid, A. C. dos Reis, S. Ricciardi, A. Richards, K. Rinnert,\n V. Rives Molina, D. A. Roa Romero, P. Robbe, E. Rodrigues, P. Rodriguez\n Perez, G. J. Rogers, S. Roiser, V. Romanovsky, A. Romero Vidal, J. Rouvinet,\n T. Ruf, H. Ruiz, G. Sabatino, J. J. Saborido Silva, N. Sagidova, P. Sail, B.\n Saitta, C. Salzmann, B. Sanmartin Sedes, M. Sannino, R. Santacesaria, C.\n Santamarina Rios, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta,\n M. Savrie, D. Savrina, P. Schaack, M. Schiller, H. Schindler, S. Schleich, M.\n Schlupp, M. Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune,\n R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K.\n Senderowska, I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I.\n Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko,\n V. Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, N. A. Smith, E.\n Smith, M. Smith, K. Sobczak, M. D. Sokoloff, F. J. P. Soler, F. Soomro, D.\n Souza, B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S.\n Stahl, O. Steinkamp, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U.\n Straumann, V. K. Subbiah, S. Swientek, V. Syropoulos, M. Szczekowski, P.\n Szczypka, T. Szumlak, S. T'Jampens, M. Teklishyn, E. Teodorescu, F. Teubert,\n C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S. Tolk, D.\n Tonelli, S. Topp-Joergensen, N. Torr, E. Tournefier, S. Tourneur, M. T. Tran,\n M. Tresch, A. Tsaregorodtsev, P. Tsopelas, N. Tuning, M. Ubeda Garcia, A.\n Ukleja, D. Urner, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P.\n Vazquez Regueiro, S. Vecchi, J. J. Velthuis, M. Veltri, G. Veneziano, M.\n Vesterinen, B. Viaud, D. Vieira, X. Vilasis-Cardona, A. Vollhardt, D.\n Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo{\\ss}, H. Voss, R.\n Waldi, R. Wallace, S. Wandernoth, J. Wang, D. R. Ward, N. K. Watson, A. D.\n Webber, D. Websdale, M. Whitehead, J. Wicht, D. Wiedner, L. Wiggers, G.\n Wilkinson, M. P. Williams, M. Williams, F. F. Wilson, J. Wishahi, M. Witek,\n W. Witzeling, S. A. Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, F. Xing, Z.\n Xing, Z. Yang, R. Young, X. Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev,\n F. Zhang, L. Zhang, W. C. Zhang, Y. Zhang, A. Zhelezov, A. Zhokhov, L. Zhong,\n A. Zvyagin",
"submitter": "Rose Koopman",
"url": "https://arxiv.org/abs/1301.5286"
}
|
1301.5347
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-004 LHCb-PAPER-2012-045 January 24, 2013
Analysis of the resonant components in $\kern
3.73305pt\overline{\kern-3.73305ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$
The LHCb collaboration†††Authors are listed on the following pages.
Interpretation of $C\\!P$ violation measurements using charmonium decays, in
both the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ systems, can be subject to
changes due to “penguin” type diagrams. These effects can be investigated
using measurements of the Cabibbo-suppressed $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ decays. The final state composition of this channel is
investigated using a 1.0 fb-1 sample of data produced in 7 TeV $pp$ collisions
at the LHC and collected by the LHCb experiment. A modified Dalitz plot
analysis is performed using both the invariant mass spectra and the decay
angular distributions. An improved measurement of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ branching fraction of $(3.97\pm 0.09\pm 0.11\pm
0.16)\times 10^{-5}$ is reported where the first uncertainty is statistical,
the second is systematic and the third is due to the uncertainty of the
branching fraction of the decay
$B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-}$ used as a
normalization channel. Significant production of $f_{0}(500)$ and $\rho(770)$
resonances is found in the substructure of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ final state, and
this indicates that they are viable final states for $C\\!P$ violation
studies. In contrast evidence for the $f_{0}(980)$ resonance is not found.
This allows us to establish the first upper limit on the branching fraction
product ${\cal{B}}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)\right)\times{\cal{B}}\left(f_{0}(980)\rightarrow\pi^{+}\pi^{-}\right)<1.1\times
10^{-6}$, leading to an upper limit on the absolute value of the mixing angle
of the $f_{0}(980)$ with the $f_{0}(500)$ of less than $31^{\circ}$, both at
90% confidence level.
Submitted to Physical Review D
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht9, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves
Jr22,35, S. Amato2, Y. Amhis7, L. Anderlini17,f, J. Anderson37, R.
Andreassen57, R.B. Appleby51, O. Aquines Gutierrez10, F. Archilli18, A.
Artamonov 32, M. Artuso53, E. Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J.
Back45, C. Baesso54, V. Balagura28, W. Baldini16, R.J. Barlow51, C.
Barschel35, S. Barsuk7, W. Barter44, Th. Bauer38, A. Bay36, J. Beddow48, I.
Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28, E. Ben-Haim8, M.
Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, A. Berezhnoy29, R.
Bernet37, M.-O. Bettler44, M. van Beuzekom38, A. Bien11, S. Bifani12, T.
Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36, C. Blanks50,
J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N. Bondar27, W.
Bonivento15, S. Borghi51, A. Borgia53, T.J.V. Bowcock49, E. Bowen37, C.
Bozzi16, T. Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M.
Britsch10, T. Britton53, N.H. Brook43, H. Brown49, I. Burducea26, A.
Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo
Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14,c, G. Carboni21,k, R.
Cardinale19,i, A. Cardini15, H. Carranza-Mejia47, L. Carson50, K. Carvalho
Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, P. Chen3,36, N. Chiapolini37, M. Chrzaszcz 23, K. Ciba35, X.
Cid Vidal34, G. Ciezarek50, P.E.L. Clarke47, M. Clemencic35, H.V. Cliff44, J.
Closier35, C. Coca26, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins35, A.
Comerma-Montells33, A. Contu15, A. Cook43, M. Coombes43, G. Corti35, B.
Couturier35, G.A. Cowan36, D. Craik45, S. Cunliffe50, R. Currie47, C.
D’Ambrosio35, P. David8, P.N.Y. David38, I. De Bonis4, K. De Bruyn38, S. De
Capua51, M. De Cian37, J.M. De Miranda1, L. De Paula2, W. De Silva57, P. De
Simone18, D. Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C.
Deplano15, D. Derkach14, O. Deschamps5, F. Dettori39, A. Di Canto11, J.
Dickens44, H. Dijkstra35, P. Diniz Batista1, M. Dogaru26, F. Domingo
Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil Suárez34, D. Dossett45, A.
Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A. Dziurda23, A. Dzyuba27, S.
Easo46,35, U. Egede50, V. Egorychev28, S. Eidelman31, D. van Eijk38, S.
Eisenhardt47, U. Eitschberger9, R. Ekelhof9, L. Eklund48, I. El Rifai5, Ch.
Elsasser37, D. Elsby42, A. Falabella14,e, C. Färber11, G. Fardell47, C.
Farinelli38, S. Farry12, V. Fave36, D. Ferguson47, V. Fernandez Albor34, F.
Ferreira Rodrigues1, M. Ferro-Luzzi35, S. Filippov30, C. Fitzpatrick35, M.
Fontana10, F. Fontanelli19,i, R. Forty35, O. Francisco2, M. Frank35, C.
Frei35, M. Frosini17,f, S. Furcas20, E. Furfaro21, A. Gallas Torreira34, D.
Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J. Garofoli53, P. Garosi51,
J. Garra Tico44, L. Garrido33, C. Gaspar35, R. Gauld52, E. Gersabeck11, M.
Gersabeck51, T. Gershon45,35, Ph. Ghez4, V. Gibson44, V.V. Gligorov35, C.
Göbel54, D. Golubkov28, A. Golutvin50,28,35, A. Gomes2, H. Gordon52, M.
Grabalosa Gándara5, R. Graciani Diaz33, L.A. Granado Cardoso35, E. Graugés33,
G. Graziani17, A. Grecu26, E. Greening52, S. Gregson44, O. Grünberg55, B.
Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53, G. Haefeli36,
C. Haen35, S.C. Haines44, S. Hall50, T. Hampson43, S. Hansmann-Menzemer11, N.
Harnew52, S.T. Harnew43, J. Harrison51, P.F. Harrison45, T. Hartmann55, J.
He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van
Herwijnen35, E. Hicks49, D. Hill52, M. Hoballah5, C. Hombach51, P. Hopchev4,
W. Hulsbergen38, P. Hunt52, T. Huse49, N. Hussain52, D. Hutchcroft49, D.
Hynds48, V. Iakovenko41, P. Ilten12, R. Jacobsson35, A. Jaeger11, E. Jans38,
F. Jansen38, P. Jaton36, F. Jing3, M. John52, D. Johnson52, C.R. Jones44, B.
Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35, T.M. Karbach35, I.R.
Kenyon42, U. Kerzel35, T. Ketel39, A. Keune36, B. Khanji20, O. Kochebina7, I.
Komarov36,29, R.F. Koopman39, P. Koppenburg38, M. Korolev29, A. Kozlinskiy38,
L. Kravchuk30, K. Kreplin11, M. Kreps45, G. Krocker11, P. Krokovny31, F.
Kruse9, M. Kucharczyk20,23,j, V. Kudryavtsev31, T. Kvaratskheliya28,35, V.N.
La Thi36, D. Lacarrere35, G. Lafferty51, A. Lai15, D. Lambert47, R.W.
Lambert39, E. Lanciotti35, G. Lanfranchi18,35, C. Langenbruch35, T. Latham45,
C. Lazzeroni42, R. Le Gac6, J. van Leerdam38, J.-P. Lees4, R. Lefèvre5, A.
Leflat29,35, J. Lefrançois7, O. Leroy6, Y. Li3, L. Li Gioi5, M. Liles49, R.
Lindner35, C. Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E.
Lopez Asamar33, N. Lopez-March36, H. Lu3, J. Luisier36, H. Luo47, F.
Machefert7, I.V. Machikhiliyan4,28, F. Maciuc26, O. Maev27,35, S. Malde52, G.
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Marks11, G. Martellotti22, A. Martens8, L. Martin52, A. Martín Sánchez7, M.
Martinelli38, D. Martinez Santos39, D. Martins Tostes2, A. Massafferri1, R.
Matev35, Z. Mathe35, C. Matteuzzi20, M. Matveev27, E. Maurice6, A.
Mazurov16,30,35,e, J. McCarthy42, R. McNulty12, B. Meadows57,52, F. Meier9, M.
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Monteil5, D. Moran51, P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K.
Müller37, R. Muresan26, B. Muryn24, B. Muster36, P. Naik43, T. Nakada36, R.
Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35, A.D. Nguyen36, T.D.
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Nikodem11, S. Nisar56, A. Nomerotski52, A. Novoselov32, A. Oblakowska-Mucha24,
V. Obraztsov32, S. Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35,
M. Orlandea26, J.M. Otalora Goicochea2, P. Owen50, B.K. Pal53, A. Palano13,b,
M. Palutan18, J. Panman35, A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J.
Parkinson50, G. Passaleva17, G.D. Patel49, M. Patel50, G.N. Patrick46, C.
Patrignani19,i, C. Pavel-Nicorescu26, A. Pazos Alvarez34, A. Pellegrino38, G.
Penso22,l, M. Pepe Altarelli35, S. Perazzini14,c, D.L. Perego20,j, E. Perez
Trigo34, A. Pérez-Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G.
Pessina20, K. Petridis50, A. Petrolini19,i, A. Phan53, E. Picatoste Olloqui33,
B. Pietrzyk4, T. Pilař45, D. Pinci22, S. Playfer47, M. Plo Casasus34, F.
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Popovici26, C. Potterat33, A. Powell52, J. Prisciandaro36, V. Pugatch41, A.
Puig Navarro36, W. Qian4, J.H. Rademacker43, B. Rakotomiaramanana36, M.S.
Rangel2, I. Raniuk40, N. Rauschmayr35, G. Raven39, S. Redford52, M.M. Reid45,
A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, V. Rives
Molina33, D.A. Roa Romero5, P. Robbe7, E. Rodrigues51, P. Rodriguez Perez34,
G.J. Rogers44, S. Roiser35, V. Romanovsky32, A. Romero Vidal34, J. Rouvinet36,
T. Ruf35, H. Ruiz33, G. Sabatino22,k, J.J. Saborido Silva34, N. Sagidova27, P.
Sail48, B. Saitta15,d, C. Salzmann37, B. Sanmartin Sedes34, M. Sannino19,i, R.
Santacesaria22, C. Santamarina Rios34, E. Santovetti21,k, M. Sapunov6, A.
Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D. Savrina28,29, P.
Schaack50, M. Schiller39, H. Schindler35, S. Schleich9, M. Schlupp9, M.
Schmelling10, B. Schmidt35, O. Schneider36, A. Schopper35, M.-H. Schune7, R.
Schwemmer35, B. Sciascia18, A. Sciubba18,l, M. Seco34, A. Semennikov28, K.
Senderowska24, I. Sepp50, N. Serra37, J. Serrano6, P. Seyfert11, M. Shapkin32,
I. Shapoval40,35, P. Shatalov28, Y. Shcheglov27, T. Shears49,35, L.
Shekhtman31, O. Shevchenko40, V. Shevchenko28, A. Shires50, R. Silva
Coutinho45, T. Skwarnicki53, N.A. Smith49, E. Smith52,46, M. Smith51, K.
Sobczak5, M.D. Sokoloff57, F.J.P. Soler48, F. Soomro18,35, D. Souza43, B.
Souza De Paula2, B. Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S.
Stahl11, O. Steinkamp37, S. Stoica26, S. Stone53, B. Storaci37, M.
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M. Szczekowski25, P. Szczypka36,35, T. Szumlak24, S. T’Jampens4, M.
Teklishyn7, E. Teodorescu26, F. Teubert35, C. Thomas52, E. Thomas35, J. van
Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39, D. Tonelli35, S. Topp-
Joergensen52, N. Torr52, E. Tournefier4,50, S. Tourneur36, M.T. Tran36, M.
Tresch37, A. Tsaregorodtsev6, P. Tsopelas38, N. Tuning38, M. Ubeda Garcia35,
A. Ukleja25, D. Urner51, U. Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez
Gomez33, P. Vazquez Regueiro34, S. Vecchi16, J.J. Velthuis43, M. Veltri17,g,
G. Veneziano36, M. Vesterinen35, B. Viaud7, D. Vieira2, X. Vilasis-
Cardona33,n, A. Vollhardt37, D. Volyanskyy10, D. Voong43, A. Vorobyev27, V.
Vorobyev31, C. Voß55, H. Voss10, R. Waldi55, R. Wallace12, S. Wandernoth11, J.
Wang53, D.R. Ward44, N.K. Watson42, A.D. Webber51, D. Websdale50, M.
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Wilkinson52, M.P. Williams45,46, M. Williams50,p, F.F. Wilson46, J. Wishahi9,
M. Witek23, S.A. Wotton44, S. Wright44, S. Wu3, K. Wyllie35, Y. Xie47,35, F.
Xing52, Z. Xing53, Z. Yang3, R. Young47, X. Yuan3, O. Yushchenko32, M.
Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang53, W.C. Zhang12, Y. Zhang3,
A. Zhelezov11, A. Zhokhov28, L. Zhong3, A. Zvyagin35.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25National Center for Nuclear Research (NCBJ), Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
56Institute of Information Technology, COMSATS, Lahore, Pakistan, associated
to 53
57University of Cincinnati, Cincinnati, OH, United States, associated to 53
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pMassachusetts Institute of Technology, Cambridge, MA, United States
## 1 Introduction
$C\\!P$ violation measurements using neutral $B$ meson decays into
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons are of prime importance
both for determinations of Standard Model (SM) parameters and searching for
physics beyond the SM. In the case of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays, the final state
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}_{\rm\scriptscriptstyle S}$
is the most important for measuring $\sin 2\beta$ [1, *Adachi:2012et,
*:2012ke], while in the case of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays, used to measure
$\phi_{s}$, only the final states ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ [4, 5, 6, *Abazov:2011ry, *:2012fu], and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ [9] have been
used so far, where the largest component of the latter is
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}(980)$[10, *Stone:2008ak].
The decay rate for these ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ modes
is dominated by the color-suppressed tree level diagram, an example of which
is shown for $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays in Fig.
1(a), while penguin processes, an example of which is shown in Fig. 1(b), are
expected to be suppressed. Theoretical predictions on the effects of such
“penguin pollution” vary widely for both $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays [12, *Li:2006vq,
*Boos:2004xp, *Ciuchini:2005mg, *Bhattacharya:2012ph, *Fleischer:1999nz,
*Faller:2008gt], so it is incumbent upon experimentalists to limit possible
changes in the value of the $C\\!P$ violating angles measured using other
decay modes.
Figure 1: (a) Tree level and (b) penguin diagram examples for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays into
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}_{\rm\scriptscriptstyle S}$.
The decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ can occur via a Cabibbo suppressed tree level diagram,
shown in Fig. 2(a), or via several penguin diagrams. An example is shown in
Fig. 2(b), while others are illustrated in Ref. [19]. These decays are
interesting because they can also be used to measure or limit the amount of
penguin pollution. The advantage in using the decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ arises because the relative amount of pollution is
larger. In the allowed decays, e.g. $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$, the penguin amplitude is multiplied by
a factor of $\lambda^{2}Re^{i\phi}$, where $\lambda$ is the sine of the
Cabibbo angle $(\approx 0.22)$, while in the suppressed decays the factor
becomes $R^{\prime}e^{i\phi^{\prime}}$, where $R$ and $R^{\prime}$, and $\phi$
and $\phi^{\prime}$ are expected to be similar in size [19]. A similar study
uses the decay $B_{s}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ [20, *Aaij:2012di, *DeBruyn:2010hh].
Figure 2: (a) Tree level and (b) penguin diagram for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays into
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$.
$C\\!P$ violation measurements in the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ mode utilizing $B^{0}-\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mixing determine $\sin 2\beta^{\rm
eff}$ which can be compared to the well measured $\sin 2\beta$. Differences
can be used to estimate the magnitude of penguin effects. Knowledge of the
final state structure is the first step in this program. Such measurements on
$\sin 2\beta^{\rm eff}$ have been attempted in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ system by using the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{0}$ final state [23,
*Aubert:2008bs, *Jung:2012pz].
In order to ascertain the viability of such $C\\!P$ violation measurements we
perform a full “Dalitz like” analysis of the final state. Regions in
$\pi^{+}\pi^{-}$ mass that correspond to spin-0 final states would be $C\\!P$
eigenstates. Final states containing vector resonances, such as the
$\rho(770)$ can be analyzed in a similar manner as was done for the decay
$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ [4, 5, 6, *Abazov:2011ry, *:2012fu].
It is also of interest to search for the $f_{0}(980)$ contribution and to
obtain information concerning the mixing angle between the $f_{0}(980)$ and
the $f_{0}(500)$111This particle has been identified previously as the
$f_{0}(600)$ or $\sigma$ resonance. partners in the scalar nonet, as the
latter should couple strongly to the $d\bar{d}$ system. Branching fractions
for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ and ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rho^{0}$ have previously been measured by the BaBar collaboration [26,
*Aubert:2007xw].
In this paper the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$ and
$\pi^{+}\pi^{-}$ mass spectra and decay angular distributions are used to
determine the resonant and non-resonant components. This differs from a
classical Dalitz plot analysis [28] because one of the particles in the final
state, the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson, has spin-1
and its three decay amplitudes must be considered. We first show that there
are no evident structures in the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}$ invariant mass, and then model the $\pi^{+}\pi^{-}$ invariant
mass with a series of resonant and non-resonant amplitudes. The data are then
fitted with the coherent sum of these amplitudes. We report on the resonant
structure and the $C\\!P$ content of the final state.
## 2 Data sample and selection requirements
The data sample consists of $1.0~{}\rm fb^{-1}$ of integrated luminosity
collected with the LHCb detector [29] using $pp$ collisions at a center-of-
mass energy of 7 TeV. The detector is a single-arm forward spectrometer
covering the pseudorapidity range $2<\eta<5$, designed for the study of
particles containing $b$ or $c$ quarks. Components include a high precision
tracking system consisting of a silicon-strip vertex detector surrounding the
$pp$ interaction region, a large-area silicon-strip detector located upstream
of a dipole magnet with a bending power of about $4{\rm\,Tm}$, and three
stations of silicon-strip detectors and straw drift-tubes placed downstream.
The combined tracking system has a momentum222We work in units where $c=1$.
resolution $\Delta p/p$ that varies from 0.4% at
5$\mathrm{\,Ge\kern-1.00006ptV}$ to 0.6% at
100$\mathrm{\,Ge\kern-1.00006ptV}$, and an impact parameter resolution of
20$\,\upmu\rm m$ for tracks with large transverse momentum ($p_{\rm T}$) with
respect to the proton beam direction. Charged hadrons are identified using two
ring-imaging Cherenkov (RICH) detectors. Photon, electron and hadron
candidates are identified by a calorimeter system consisting of scintillating-
pad and preshower detectors, an electromagnetic calorimeter and a hadronic
calorimeter. Muons are identified by a system composed of alternating layers
of iron and multiwire proportional chambers. The trigger consists of a
hardware stage, based on information from the calorimeter and muon systems,
followed by a software stage that applies a full event reconstruction [30].
Events are triggered by a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\\!\rightarrow\mu^{+}\mu^{-}$ decay, requiring two identified muons with
opposite charge, $\mbox{$p_{\rm T}$}(\mu^{\pm})$ greater than
500$\mathrm{\,Me\kern-1.00006ptV}$, an invariant mass within
120$\mathrm{\,Me\kern-1.00006ptV}$ of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass [31], and form a vertex
with a fit $\chi^{2}$ less than 16. After applying these requirements, there
is a large ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ signal over a small
background [32]. Only candidates with dimuon invariant mass between
$-$48$\mathrm{\,Me\kern-1.00006ptV}$ and +43$\mathrm{\,Me\kern-1.00006ptV}$
relative to the observed ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass
peak are selected, corresponding a window of about $\pm 3\sigma$. The
requirement is asymmetric because of final state electromagnetic radiation.
The two muons subsequently are kinematically constrained to the known
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass.
Other requirements are imposed to isolate $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ candidates with high signal yield
and minimum background. This is accomplished by combining the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$
candidate with a pair of pion candidates of opposite charge, and then testing
if all four tracks form a common decay vertex. Pion candidates are each
required to have $p_{\rm T}$ greater than 250$\mathrm{\,Me\kern-1.00006ptV}$,
and the scalar sum of the two transverse momenta, $\mbox{$p_{\rm
T}$}(\pi^{+})+\mbox{$p_{\rm T}$}(\pi^{-})$, must be larger than
900$\mathrm{\,Me\kern-1.00006ptV}$. The impact parameter (IP) is the distance
of closest approach of a track to the primary vertex (PV). To test for
inconsistency with production at the PV, the IP $\chi^{2}$ is computed as the
difference between the $\chi^{2}$ of the PV reconstructed with and without the
considered track. Each pion must have an IP $\chi^{2}$ greater than 9. Both
pions must also come from a common vertex with an acceptable $\chi^{2}$ and
form a vertex with the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ with a
$\chi^{2}$ per number of degrees of freedom (ndf) less than 10 (here ndf
equals five). Pion and kaon candidates are positively identified using the
RICH system. Cherenkov photons are matched to tracks, the emission angles of
the photons compared with those expected if the particle is an electron, pion,
kaon or proton, and a likelihood is then computed. The particle identification
makes use of the logarithm of the likelihood ratio comparing two particle
hypotheses (DLL). For pion selection we require DLL$(\pi-K)>-10$.
The four-track $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ candidate
must have a flight distance of more than 1.5 mm, where the average decay
length resolution is 0.17 mm. The angle between the combined momentum vector
of the decay products and the vector formed from the positions of the PV and
the decay vertex (pointing angle) is required to be less than $2.5^{\circ}$.
Events satisfying this preselection are then further filtered using a
multivariate analyzer based on a Boosted Decision Tree (BDT) technique [33].
The BDT uses six variable that are chosen in a manner that does not introduce
an asymmetry between either the two muons or the two pions. They are the
minimum DLL($\mu-\pi$) of the $\mu^{+}$ and $\mu^{-}$, the minimum $p_{\rm T}$
of the $\pi^{+}$ and $\pi^{-}$, the minimum of the IP $\chi^{2}$ of the
$\pi^{+}$ and $\pi^{-}$, the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ vertex $\chi^{2}$, the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ pointing angle, and the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ flight distance. There is
discrimination power between signal and background in all of these variables,
especially the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ vertex
$\chi^{2}$.
The background sample used to train the BDT consists of the events in the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mass sideband having
$5566<m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-})<5616$$\mathrm{\,Me\kern-1.00006ptV}$. The signal sample
consists of two million $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(\rightarrow\mu^{+}\mu^{-})\pi^{+}\pi^{-}$ Monte Carlo simulated events
that are generated uniformly in phase space, using Pythia [34] with a special
LHCb parameter tune [35], and the LHCb detector simulation based on Geant4
[36] described in Ref. [37]. Separate samples are used to train and test the
BDT. The distributions of the BDT classifier for signal and background are
shown in Fig. 3. To minimize a possible bias on the signal acceptance due to
the BDT, we choose a relatively loose requirement of the BDT classifier
$>0.05$ which has a 96% signal efficiency and a 92% background rejection rate.
Figure 3: Distributions of the BDT classifier for both training and test
samples of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ signal
and background events. The signal samples are from simulation and the
background samples are from data. Figure 4: Invariant mass of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ combinations. The
data are fitted with a double-Gaussian signal and several background
functions. The (red) solid double-Gaussian function centered at
5280$\mathrm{\,Me\kern-1.00006ptV}$ is the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ signal, the (brown) dotted line
shows the combinatorial background, the (green) short-dashed shows the $B^{-}$
background, the (purple) dot-dashed line shows the contribution of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ decays, the (black) dot-long dashed is the sum of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\eta^{\prime}(\rightarrow\rho\gamma)$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi(\rightarrow\pi^{+}\pi^{-}\pi^{0})$ backgrounds, the (light blue)
long-dashed is the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-}\pi^{+}$ reflection, and the (blue) solid line is the total.
The invariant mass of the selected ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ combinations, where the dimuon pair is constrained to
have the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass, is shown in Fig.
4. There are signal peaks at both the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ masses on top of the background.
Double-Gaussian functions are used to fit both signal peaks. They differ only
in their mean values, which are determined by the data. The core Gaussian
width is also allowed to vary, while the fraction and width ratio of the
second Gaussian is fixed to that obtained in the fit of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ events. (The details of the fit are given in Ref. [10].) Other
components in the fit model take into account background contributions. One
source is from $B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-}$decays, which contributes when the $K^{-}$ is misidentifed as a
$\pi^{-}$ and then combined with a random $\pi^{+}$; the smaller
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{-}$ mode contributes when it
is combined with a random $\pi^{+}$. The next source contains $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\eta^{\prime}(\rightarrow\rho\gamma)$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi(\rightarrow\pi^{+}\pi^{-}\pi^{0})$ decays where the $\gamma$ and
the $\pi^{0}$ are ignored respectively. Finally there is a $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-}\pi^{+}$ reflection where the $K^{-}$ is misidentified as
$\pi^{-}$. Here and elsewhere charged conjugated modes are included when
appropriate. The exponential combinatorial background shape is taken from
same-sign combinations, that are the sum of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{+}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{-}\pi^{-}$ candidates. The
shapes of the other components are taken from the simulation with their
normalizations allowed to vary. The fit gives $5287\pm 112$ signal and
$3212\pm 80$ background candidates within $\pm
20$$\mathrm{\,Me\kern-1.00006ptV}$ of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mass peak, where a
$K^{0}_{\rm\scriptscriptstyle S}$ veto, discussed later, is applied.
We use the well measured
$B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-}$ mode as a
normalization channel to determine the branching fractions. To minimize the
systematic uncertainty from the BDT selection, we employ a similar selection
on $B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-}$ decays
after requiring the same pre-selection except for particle identification
criteria on the $K^{-}$ candidates. Similar variables are used for the BDT
except that the variables describing the combination of $\pi^{+}$ and
$\pi^{-}$ in the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$
final state are replaced by ones describing the $K^{-}$ meson. For BDT
training, the signal sample uses simulated events and the background sample
consists of the data events in the region
$5400<m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-})<5450$$\mathrm{\,Me\kern-1.00006ptV}$. The resulting invariant
mass distribution of the candidates satisfying BDT classifier $>0.05$ is shown
in Fig. 5. Fitting the distribution with a double-Gaussian function for the
signal and linear function for the background gives $350{,}727\pm 633$ signal
and $4756\pm 103$ background candidates within $\pm
20$$\mathrm{\,Me\kern-1.00006ptV}$ of the $B^{-}$ mass peak.
Figure 5: Invariant mass of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-}$ combinations. The data points are fitted with a double-Gaussian
function for signal and a linear function for background. The dotted line
shows the background, and the (blue) solid line is the total.
## 3 Analysis formalism
We apply a formalism similar to that used in Belle’s analysis [38] of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow K^{-}\pi^{+}\chi_{c1}$
decays and later used in LHCb’s analysis of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ decays [10]. The decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$, with ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-}$, can be described by four variables. These
are taken to be the invariant mass squared of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$ ($s_{12}\equiv
m^{2}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+})$), the invariant
mass squared of $\pi^{+}\pi^{-}$ ($s_{23}\equiv m^{2}(\pi^{+}\pi^{-})$), where
we use label 1 for ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$, 2 for
$\pi^{+}$ and 3 for $\pi^{-}$, the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ helicity angle ($\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$), which is the angle of the $\mu^{+}$ in the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ rest frame with respect to the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ direction in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ rest frame, and the angle between
the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\pi^{+}\pi^{-}$ decay
planes ($\chi$) in the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ rest
frame. To improve the resolution of these variables we perform a kinematic fit
constraining the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ masses to their nominal values
[31], and recompute the final state momenta. To simplify the probability
density function, we analyze the decay process after integrating over $\chi$,
which eliminates several interference terms.
### 3.1 The decay model for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$
The overall probability density function (PDF) given by the sum of signal,
$S$, and background, $B$, functions is
$F(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})=\frac{f_{\rm sig}}{{\cal{N}}_{\rm
sig}}\varepsilon(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})S(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})+\frac{(1-f_{\rm sig})}{{\cal{N}}_{\rm
bkg}}B(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}),$
(1)
where $f_{\rm sig}$ is the fraction of the signal in the fitted region and
$\varepsilon$ is the detection efficiency. The fraction of the signal is
obtained from the mass fit and is fixed for the subsequent analysis. The
normalization factors are given by
$\displaystyle{\cal{N}}_{\rm sig}$ $\displaystyle=$
$\displaystyle\int\\!\varepsilon(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})S(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})\,ds_{12}ds_{23}d\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}},$ $\displaystyle{\cal{N}}_{\rm bkg}$ $\displaystyle=$
$\displaystyle\int\\!B(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})\,ds_{12}ds_{23}d\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}.$ (2)
The event distribution for $m^{2}(\pi^{+}\pi^{-})$ versus
$m^{2}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+})$ in Fig. 6 shows
obvious structure in $m^{2}(\pi^{+}\pi^{-})$. To investigate if there are
visible exotic structures in the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}$ system as claimed in similar decays [39], we examine the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$ mass distribution shown
in Fig. 7 (a). No resonant effects are evident. Figure 7 (b) shows the
$\pi^{+}\pi^{-}$ mass distribution. There is a clear peak at the $\rho(770)$
region, a small bump around 1250$\mathrm{\,Me\kern-1.00006ptV}$, but no
evidence for the $f_{0}(980)$ resonance. The favored $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ decay is mostly rejected by the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ vertex $\chi^{2}$ selection, but
about 150 such events remain. We eliminate them by excluding the candidates
that have $|m(\pi^{+}\pi^{-})-m_{K^{0}_{\rm\scriptscriptstyle S}}|<$
25$\mathrm{\,Me\kern-1.00006ptV}$, where $m_{K^{0}_{\rm\scriptscriptstyle S}}$
is the $K^{0}_{\rm\scriptscriptstyle S}$ mass [31].
Figure 6: Distribution of $m^{2}(\pi^{+}\pi^{-})$ versus
$m^{2}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+})$ for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ candidate decays within $\pm
20$$\mathrm{\,Me\kern-1.00006ptV}$ of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mass.
Figure 7: Distribution of (a) $m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+})$ and (b) $m(\pi^{+}\pi^{-})$ for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ candidate decays within $\pm
20$$\mathrm{\,Me\kern-1.00006ptV}$ of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mass shown with the solid line.
The (red) points with error bars show the background contribution determined
from $m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-})$ fits
performed in each bin.
#### 3.1.1 The signal function
The signal function for $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ is
taken to be the coherent sum over resonant states that can decay into
$\pi^{+}\pi^{-}$, plus a possible non-resonant S-wave contribution333The
interference terms between different helicities are zero because we integrate
over the angular variable $\chi$.
$S(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})=\sum_{\lambda=0,\pm
1}\left|\sum_{i}a^{R_{i}}_{\lambda}e^{i\phi^{R_{i}}_{\lambda}}\mathcal{A}_{\lambda}^{R_{i}}(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})\right|^{2},$ (3)
where
$\mathcal{A}_{\lambda}^{R_{i}}(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})$ is the amplitude of the decay via an intermediate resonance $R_{i}$
with helicity $\lambda$. Each $R_{i}$ has an associated amplitude strength
$a_{\lambda}^{R_{i}}$ for each helicity state $\lambda$ and a phase
$\phi_{\lambda}^{R_{i}}$. Note that the spin-0 component can only have a
$\lambda=0$ term. The amplitudes for each $i$ are defined as
$\mathcal{A}_{\lambda}^{R}(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})=F_{B}^{(L_{B})}\;F_{R}^{(L_{R})}\;A_{R}(s_{23})\;\Big{(}\frac{P_{B}}{m_{B}}\Big{)}^{L_{B}}\;\Big{(}\frac{P_{R}}{\sqrt{s_{23}}}\Big{)}^{L_{R}}\;T_{\lambda}\;\Theta_{\lambda}(\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}),$ (4)
where $P_{B}$ is the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ momentum
in the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ rest frame and
$P_{R}$ is the momentum of either of the two pions in the dipion rest frame,
$m_{B}$ is the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mass,
$F_{B}^{(L_{B})}$ and $F_{R}^{(L_{R})}$ are the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ meson and $R$ resonance Blatt-
Weisskopf barrier factors [40], $L_{B}$ is the orbital angular momentum
between the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$\pi^{+}\pi^{-}$ system, and $L_{R}$ is the orbital angular momentum in the
$\pi^{+}\pi^{-}$ decay and is equal to the spin of resonance $R$ because pions
have spin-0. Since the parent $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ has spin-0 and the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ is a vector, when the
$\pi^{+}\pi^{-}$ system forms a spin-0 resonance, $L_{B}=1$ and $L_{R}=0$. For
$\pi^{+}\pi^{-}$ resonances with non-zero spin, $L_{B}$ can be 0, 1 or 2 (1, 2
or 3) for $L_{R}=1(2)$ and so on. We take the lowest $L_{B}$ as the default
and consider the other possibilities in the systematic uncertainty.
The Blatt-Weisskopf barrier factors $F_{B}^{(L_{B})}$ and $F_{R}^{(L_{R})}$
are
$\displaystyle F^{(0)}$ $\displaystyle=$ $\displaystyle 1,$ $\displaystyle
F^{(1)}$ $\displaystyle=$ $\displaystyle\frac{\sqrt{1+z_{0}}}{\sqrt{1+z}},$
(5) $\displaystyle F^{(2)}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{z_{0}^{2}+3z_{0}+9}}{\sqrt{z^{2}+3z+9}}.$
For the $B$ meson $z=r^{2}P_{B}^{2}$, where the hadron scale $r$ is taken as
$5.0\mathrm{\,Ge\kern-1.00006ptV}^{-1}$, and for the $R$ resonance
$z=r^{2}P_{R}^{2}$ with $r$ taken as $1.5\mathrm{\,Ge\kern-1.00006ptV}^{-1}$
[41]. In both cases $z_{0}=r^{2}P_{0}^{2}$ where $P_{0}$ is the decay daughter
momentum calculated at the resonance pole mass.
The angular term, $T_{\lambda}$, is obtained using the helicity formalism and
is defined as
$T_{\lambda}=d^{J}_{\lambda 0}(\theta_{\pi\pi}),$ (6)
where $d$ is the Wigner $d$-function, $J$ is the resonance spin,
$\theta_{\pi\pi}$ is the $\pi^{+}\pi^{-}$ resonance helicity angle which is
defined as the angle of the $\pi^{+}$ in the $\pi^{+}\pi^{-}$ rest frame with
respect to the $\pi^{+}\pi^{-}$direction in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ rest frame and calculated from the
other variables as
$\cos\theta_{\pi\pi}=\frac{\left[m^{2}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+})-m^{2}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{-})\right]m(\pi^{+}\pi^{-})}{4P_{R}P_{B}m_{B}}.$ (7)
The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ helicity dependent term
$\Theta_{\lambda}(\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}})$ is
defined as
$\displaystyle\Theta_{\lambda}(\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})$ $\displaystyle=$
$\displaystyle\sqrt{\sin^{2}\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}\;\;\;\;\;\;\;\;\;\text{for}\;\;\text{helicity}=0$ (8)
$\displaystyle=$
$\displaystyle\sqrt{\frac{1+\cos^{2}\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}{2}}\;\;\text{for}\;\;\text{helicity}=\pm 1.$
The function $A_{R}(s_{23})$ describes the mass squared shape of the resonance
$R$, that in most cases is a Breit-Wigner (BW) amplitude. Complications arise,
however, when a new decay channel opens close to the resonant mass. The
proximity of a second threshold distorts the line shape of the amplitude. This
happens for the $f_{0}(980)$ resonance because the $K^{+}K^{-}$ decay channel
opens. Here we use a Flatté model [42] which is described below.
The BW amplitude for a resonance decaying into two spin-0 particles, labeled
as 2 and 3, is
$A_{R}(s_{23})=\frac{1}{m^{2}_{R}-s_{23}-im_{R}\Gamma(s_{23})}~{},$ (9)
where $m_{R}$ is the resonance pole mass, $\Gamma(s_{23})$ is its energy-
dependent width that is parametrized as
$\Gamma(s_{23})=\Gamma_{0}\left(\frac{P_{R}}{P_{0}}\right)^{2L_{R}+1}\left(\frac{m_{R}}{\sqrt{s_{23}}}\right)F^{2}_{R}~{}.$
(10)
Here $\Gamma_{0}$ is the decay width when the invariant mass of the daughter
combinations is equal to $m_{R}$.
The Flatté model is parametrized as
$A_{R}(s_{23})=\frac{1}{m_{R}^{2}-s_{23}-im_{R}(g_{\pi\pi}\rho_{\pi\pi}+g_{KK}\rho_{KK})}.$
(11)
The constants $g_{\pi\pi}$ and $g_{KK}$ are the $f_{0}(980)$ couplings to
$\pi\pi$ and $K\kern 1.99997pt\overline{\kern-1.99997ptK}{}$ final states
respectively. The $\rho$ factors account for the Lorentz-invariant phase space
and are given as
$\displaystyle\rho_{\pi\pi}$ $\displaystyle=$
$\displaystyle\frac{2}{3}\sqrt{1-\frac{4m^{2}_{\pi^{\pm}}}{m^{2}(\pi^{+}\pi^{-})}}+\frac{1}{3}\sqrt{1-\frac{4m^{2}_{\pi^{0}}}{m^{2}(\pi^{+}\pi^{-})}},$
(12) $\displaystyle\rho_{KK}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{1-\frac{4m^{2}_{K^{\pm}}}{m^{2}(\pi^{+}\pi^{-})}}+\frac{1}{2}\sqrt{1-\frac{4m^{2}_{K^{0}}}{m^{2}(\pi^{+}\pi^{-})}}.$
(13)
For non-resonant processes, the amplitude
$\mathcal{A}(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})$ is derived from Eq. 4, considering that the $\pi^{+}\pi^{-}$ system
is S-wave (i.e. $L_{R}=0$, $L_{B}=1$) and $A_{R}(s_{23})$ is constant over the
phase space $s_{12}$ and $s_{23}$. Thus, it is parametrized as
$\mathcal{A}(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})=\frac{P_{B}}{m_{B}}\sqrt{\sin^{2}\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}.$ (14)
#### 3.1.2 Detection efficiency
Figure 8: Distributions of $\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$ for the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$
simulated sample in (a) the entire dipion mass region and (b) $\rho(770)$
region.
The detection efficiency is determined from a sample of two million $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(\rightarrow\mu^{+}\mu^{-})\pi^{+}\pi^{-}$ simulated events that are
generated uniformly in phase space. Both $s_{12}$ and $s_{13}$ are centered at
about $18.4\mathrm{\,Ge\kern-1.00006ptV}^{2}$. We model the detection
efficiency using the symmetric dimensionless Dalitz plot observables
$x=s_{12}/{\rm\mathrm{\,Ge\kern-1.00006ptV}}^{2}-18.4,~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}y=s_{13}/{\rm\mathrm{\,Ge\kern-1.00006ptV}}^{2}-18.4.$ (15)
These variables are related to $s_{23}$ since
$s_{12}+s_{13}+s_{23}=m^{2}_{B}+m^{2}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}+m^{2}_{\pi^{+}}+m^{2}_{\pi^{-}}.$ (16)
The acceptance in $\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$
is not uniform, but depends on $s_{23}$, as shown in Fig. 8. If the efficiency
was independent of $s_{23}$, then the curves would have the same shape. On the
other hand, no clear dependence on $s_{12}$ is seen. Thus the efficiency model
can be expressed as
$\varepsilon(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})=\varepsilon_{1}(x,y)\times\varepsilon_{2}(s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}).$ (17)
To study the $\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$
acceptance, we fit the $\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$ distributions from simulation in 24 bins of $m^{2}(\pi^{+}\pi^{-})$
with the function
$\varepsilon_{2}(s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})=\frac{1+a\cos^{2}\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}{2+2a/3},$ (18)
giving 24 values of $a$ as a function of $m^{2}(\pi^{+}\pi^{-})$. The
resultant distribution shown in Fig. 9 can be described by an exponential
function
$a(s_{23})=\exp(a_{1}+a_{2}s_{23}),$ (19)
with $a_{1}=-1.48\pm 0.20$ and $a_{2}=(-1.45\pm
0.33)\mathrm{\,Ge\kern-1.00006ptV}^{-2}$.
Figure 9: Exponential fit to the acceptance parameter $a(s_{12})$ used in Eq.
18.
Equation 18 is normalized with respect to
$\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$. Thus, after
integrating over $\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$,
Eq. 17 becomes
$\int_{-1}^{+1}\varepsilon(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})d\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}=\varepsilon_{1}(x,y).$ (20)
This term of the efficiency is parametrized as a symmetric fourth order
polynomial function given by
$\displaystyle\varepsilon_{1}(x,y)$ $\displaystyle=$ $\displaystyle
1+\epsilon^{\prime}_{1}(x+y)+\epsilon^{\prime}_{2}(x+y)^{2}+\epsilon^{\prime}_{3}xy+\epsilon^{\prime}_{4}(x+y)^{3}+\epsilon^{\prime}_{5}xy(x+y)$
(21)
$\displaystyle+\epsilon^{\prime}_{6}(x+y)^{4}+\epsilon^{\prime}_{7}xy(x+y)^{2}+\epsilon^{\prime}_{8}x^{2}y^{2},$
where the $\epsilon^{\prime}_{i}$ are the fit parameters.
Table 1: Efficiency parameters to describe the acceptance on the signal Dalitz-plot. | Parameter | Value
---|---|---
| $\epsilon^{\prime}_{1}$ | $0.142\pm 0.010$
| $\epsilon^{\prime}_{2}$ | $0.101\pm 0.014$
| $\epsilon^{\prime}_{3}$ | $0.0082\pm 0.0005$
| $\epsilon^{\prime}_{4}$ | $0.027\pm 0.007$
| $\epsilon^{\prime}_{5}$ | $0.0052\pm 0.0003$
| $\epsilon^{\prime}_{6}$ | $0.0028\pm 0.0010$
| $\epsilon^{\prime}_{7}$ | $0.00074\pm 0.00017$
| $\epsilon^{\prime}_{8}$ | $-0.000105\pm 0.000008$
| $\chi^{2}/\rm ndf$ | 308/298
Figure 10 shows the polynomial function
Figure 10: Parametrized detection efficiency as a function of
$m^{2}(\pi^{+}\pi^{-})$ versus $m^{2}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+})$ determined from simulation. The $z$-axis scale is arbitrary.
obtained from a fit to the Dalitz-plot distributions of simulated events. The
projections of the fit are shown in Fig. 11 and the resulting parameters are
given in Table 1.
Figure 11: Projections onto (a) $m^{2}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+})$ and (b) $m^{2}(\pi^{+}\pi^{-})$ of the simulated Dalitz plot
used to determine the efficiency parameters. The points represent the
simulated event distributions and the curves the projections of the polynomial
fits.
#### 3.1.3 Background composition
Backgrounds from $B$ decays into ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ final states have already been discussed in Section 2. The main
background source is combinatorial and its shape can be determined from the
same-sign $\pi^{\pm}\pi^{\pm}$ combinations within $\pm
20$$\mathrm{\,Me\kern-1.00006ptV}$ of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mass peak; this region also
contains the small $B^{-}$ background. In addition, there is background
arising from partially reconstructed $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays including $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\eta^{\prime}(\rightarrow\rho\gamma)$, $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi(\rightarrow\pi^{+}\pi^{-}\pi^{0})$, and a $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-}\pi^{+}$ reflection, which cannot be present in same-sign
combinations. We use simulated samples of inclusive $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays, and exclusive $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}(892)$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}_{2}(1430)$ decays to
model the additional backgrounds. The background fraction of each source is
studied by fitting the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ candidate invariant mass distributions in bins of
$m^{2}(\pi^{+}\pi^{-})$. The resulting background distribution in the $\pm
20$$\mathrm{\,Me\kern-1.00006ptV}$ $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ signal region is shown in Fig. 12.
It is fit by histograms from the same-sign combinations and two additional
simulations, giving a partially reconstructed $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ background of 12.8%, and a
reflection background that is 5.2% of the total background.
Figure 12: The $m^{2}(\pi\pi)$ distribution of background. The (black)
histogram with error bars shows the same-sign data combinations with
additional background from simulation, the (blue) points with error bars show
the background obtained from the mass fits, the (black) dashed line is the
partially reconstructed $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
background, and the (red) dotted is the misidentified $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-}\pi^{+}$ contribution.
The background is parametrized as
$B(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})=\frac{m(\pi^{+}\pi^{-})}{2P_{R}P_{B}m_{B}}B_{1}(s_{23},\cos\theta_{\pi\pi})\times\left(1+\alpha\cos^{2}\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\right),$ (22)
where the first part $\frac{m(\pi^{+}\pi^{-})}{2P_{R}P_{B}m_{B}}$ converts
phase space from $s_{12}$ to $\cos\theta_{\pi\pi}$, and
$\displaystyle B_{1}(s_{23},\cos\theta_{\pi\pi})=$
$\displaystyle\left[B_{2}(\zeta)\frac{p_{B}}{m_{B}}+\frac{b_{0}}{(m^{2}_{0}-s_{23})^{2}+m_{0}^{2}\Gamma_{0}^{2}}\right]$
(23)
$\displaystyle\times\frac{1+c_{1}q(\zeta)|\cos\theta_{\pi\pi}|+c_{4}p(\zeta)\cos^{2}\theta_{\pi\pi}}{2[1+c_{1}q(\zeta)/2+c_{4}p(\zeta)/3]}.$
The variable $\zeta=2(s_{23}-s_{\rm min})/(s_{\rm max}-s_{\rm min})-1$, where
$s_{\rm min}$ and $s_{\rm max}$ give the fit boundaries, $B_{2}(\zeta)$ is a
fifth-order Chebychev polynomial with parameters $b_{i}$ ($i=1$–5), and
$q(\zeta)$ and $p(\zeta)$ are both second-order Chebychev polynomials with
parameters $c_{i}$ ($i$=2, 3, 5, 6), and $c_{1}$, and $c_{4}$ are free
parameters. In order to better approximate the real background in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ signal region, the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{\pm}\pi^{\mp}$ candidates
are kinematically constrained to the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass. A fit to the same-sign
sample, with additional background from simulation, determines $b_{i}$,
$c_{i}$, $m_{0}$ and $\Gamma_{0}$. Figure 13 shows the mass squared
projections from the fit. The fitted background parameters are shown in Table
2.
The $\left(1+\alpha\cos^{2}\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\right)$ term is a function of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ helicity angle. The
$\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ distribution of
background is shown in Fig. 14, and is fit with the function
$1+\alpha\cos^{2}\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ that
determines the parameter $\alpha=-0.38\pm 0.04$. We have verified that
$\alpha$ is independent of $s_{23}$.
Figure 13: Projections of invariant mass squared of (a) $m^{2}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+})$ and (b) $m^{2}(\pi^{+}\pi^{-})$ of the background Dalitz plot. The points with error bars show the same-sign combinations with additional background from simulation. Figure 14: distribution of the background in $\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ resulting from ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ candidate mass fits in each bin of $\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$. The curve represents the fitted function $1+\alpha\cos^{2}\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$. Table 2: Parameters for the background model used in Eq. 23. | Parameter | Value
---|---|---
| $b_{0}$ | $(4.4\pm 1.2)\times 10^{-3}\mathrm{\,Ge\kern-1.00006ptV}^{4}$
| $m_{0}$ | $0.767\pm 0.005\mathrm{\,Ge\kern-1.00006ptV}$
| $\Gamma_{0}$ | $0.101\pm 0.015\mathrm{\,Ge\kern-1.00006ptV}$
| $b_{1}$ | $-0.52\pm 0.07$
| $b_{2}$ | $0.22\pm 0.05$
| $b_{3}$ | $-0.14\pm 0.06$
| $b_{4}$ | $0.11\pm 0.04$
| $b_{5}$ | $-0.06\pm 0.04$
| $c_{1}$ | $-0.70\pm 0.04$
| $c_{2}$ | $-0.4\pm 0.3$
| $c_{3}$ | $1.9\pm 0.2$
| $c_{4}$ | $0.42\pm 0.03$
| $c_{5}$ | $1.7\pm 0.8$
| $c_{6}$ | $2.5\pm 0.8$
| $\chi^{2}/{\rm ndf}$ | 252/284
### 3.2 Fit fractions
While a complete description of the decay is given in terms of the fitted
amplitudes and phases, the knowledge of the contribution of each component can
be summarized by defining a fit fraction, ${\cal{F}}^{R}_{\lambda}$, as the
integration of the squared amplitude of $R$ over the Dalitz plot divided by
the integration of the entire signal function,
${\cal{F}}^{R}_{\lambda}=\frac{{\int}\left|a^{R}_{\lambda}e^{i\phi^{R}_{\lambda}}\mathcal{A}_{\lambda}^{R}(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})\right|^{2}ds_{12}\;ds_{23}\;d\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}{{\int}S(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})~{}ds_{12}\;ds_{23}\;d\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}.$ (24)
Note that the sum of the fit fractions over all $\lambda$ and $R$ is not
necessarily unity due to the potential presence of interference between two
resonances. If the Dalitz plot has more destructive interference than
constructive interference, the total fit fraction will be greater than one.
Interference term fractions are given by
${\cal{F}}^{RR^{\prime}}_{\lambda}=\mathcal{R}e\left(\frac{{\int}a^{R}_{\lambda}\;a^{R^{\prime}}_{\lambda}e^{i(\phi^{R}_{\lambda}-\phi^{R^{\prime}}_{\lambda})}\mathcal{A}_{\lambda}^{R}(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}){\mathcal{A}_{\lambda}^{R^{\prime}}}^{*}(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})ds_{12}\;ds_{23}\;d\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}{{\int}S(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})~{}ds_{12}\;ds_{23}\;d\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}\right),$ (25)
and the sum of the two is
$\sum_{\lambda}\left(\sum_{R}{\cal{F}}^{R}_{\lambda}+\sum_{RR^{\prime}}^{R\neq
R^{\prime}}{\cal{F}}^{RR^{\prime}}_{\lambda}\right)=1.$ (26)
Note that interference terms between different spin-$J$ states vanish, because
the $d^{J}_{\lambda 0}$ angular functions in $\mathcal{A}^{R}_{\lambda}$ are
orthogonal.
The statistical errors of the fit fractions depend on the statistical errors
of every fitted magnitude and phase, and their correlations. Therefore, to
determine the uncertainties the covariance matrix and parameter values from
the fit are used to generate 500 sample parameter sets. For each set, the fit
fractions are calculated. The distributions of the obtained fit fractions are
described by bifurcated Gaussian functions. The widths of the Gaussians are
taken as the statistical errors on the corresponding parameters. The
correlations of fitted parameters are also taken into account.
## 4 Final state composition
### 4.1 Resonance models
To study the resonant structures of the decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ we use those combinations with an invariant mass within
$\pm 20$$\mathrm{\,Me\kern-1.00006ptV}$ of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mass peak and apply a
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}_{\rm\scriptscriptstyle S}$
veto. The total number of remaining candidates is 8483, of which $3212\pm 80$
are attributed to background. Possible resonances in the decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ are listed in Table 3. In addition, there could be some
contribution from non-resonant $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ decays.
Table 3: Possible resonances in the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ decay mode. Resonance | Spin | Helicity | Resonance
---|---|---|---
| | | formalism
$f_{0}(500)$ | 0 | 0 | BW
$\rho(770)$ | 1 | $0,\pm 1$ | BW
$\omega(782)$ | 1 | $0,\pm 1$ | BW
$f_{0}(980)$ | 0 | 0 | Flatté
$f_{2}(1270)$ | 2 | $0,\pm 1$ | BW
$f_{0}(1370)$ | 0 | 0 | BW
$\rho(1450)$ | 1 | $0,\pm 1$ | BW
$f_{0}(1500)$ | 0 | 0 | BW
$\rho(1700)$ | 1 | $0,\pm 1$ | BW
$f_{0}(1710)$ | 0 | 0 | BW
Table 4: Breit-Wigner resonance parameters. Resonance | Mass ($\mathrm{\,Me\kern-1.00006ptV}$) | Width ($\mathrm{\,Me\kern-1.00006ptV}$) | Source
---|---|---|---
$f_{0}(500)$ | $513\pm 32$ | $335\pm 67$ | CLEO [43]
$\rho(770)$ | $775.49\pm 0.34$ | $149.1\pm 0.8$ | PDG [31]
$\omega(782)$ | $782.65\pm 0.12$ | $8.49\pm 0.08$ | PDG [31]
$f_{2}(1270)$ | $1275.1\pm 1.2$ | $185.1_{-2.4}^{+2.9}$ | PDG [31]
$f_{0}(1370)$ | $1475\pm 6$ | $113\pm 11$ | LHCb [10]
$\rho(1450)$ | $1465\pm 25$ | $400\pm 60$ | PDG [31]
$f_{0}(1500)$ | $1505\pm 6$ | $109\pm 7$ | PDG [31]
$\rho(1700)$ | $1700\pm 20$ | $250\pm 100$ | PDG [31]
$f_{0}(1710)$ | $1720\pm 6$ | $135\pm 8$ | PDG [31]
The masses and widths of the BW resonances are listed in Table 4. When used in
the fit they are fixed to these values except for the parameters of the
$f_{0}(500)$ resonance which are constrained by their uncertainties. Besides
the mass and width, the Flatté resonance shape has two additional parameters
$g_{\pi\pi}$ and $g_{KK}$, which are also fixed in the fit to values obtained
in our previous Dalitz analysis of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ [10], where a large fraction of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays are to
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}(980)$. The parameters are
taken to be $m_{0}=939.9\pm 6.3$$\mathrm{\,Me\kern-1.00006ptV}$,
$g_{\pi\pi}=199\pm 30$$\mathrm{\,Me\kern-1.00006ptV}$ and
$g_{KK}/g_{\pi\pi}=3.0\pm 0.3$. All background and efficiency parameters are
fixed in the fit.
To determine the complex amplitudes in a specific model, the data are fitted
maximizing the unbinned likelihood given as
$\mathcal{L}=\prod_{i=1}^{N}F(s^{i}_{12},s^{i}_{23},\theta^{i}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}),$ (27)
where $N$ is the total number of candidates, and $F$ is the total PDF defined
in Eq. 1. The PDF is constructed from the signal fraction $f_{\rm sig}$, the
efficiency model
$\varepsilon(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})$, the background model
$B(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}})$, and
the signal model
$S(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}})$. In
order to ensure proper convergence using the maximum likelihood method, the
PDF needs to be normalized. This is accomplished by first normalizing the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ helicity dependent part
$\varepsilon(s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})\Theta_{\lambda}(\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})$ over $\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ by
analytical integration. This integration results in additional factors as a
function of $s_{23}$. We then normalize the mass dependent part multiplied by
the additional factors using numerical integration over 500$\times$500 bins.
The fit determines the relative amplitude magnitudes $a_{\lambda}^{R_{i}}$ and
phases $\phi_{\lambda}^{R_{i}}$ defined in Eq. 3; we choose to fix
$a_{0}^{\rho(770)}$ to 1. As only relative phases are physically meaningful,
one phase in each helicity grouping has to be fixed; we choose to fix those of
the $f_{0}(500)$ and the $\rho(770)$ ($|\lambda|=1$) to 0. In addition, since
the final state ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$
is a self-charge-conjugate mode and as we do not determine the $B$ flavor, the
signal function is an average of $B^{0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays. If we do not consider
$\pi^{+}\pi^{-}$ partial waves of a higher order than D-wave, then we can
express the differential decay rate derived from Eqs. 3, 4 and 8 in terms of
S-, P-, and D-waves including helicity 0 and $\pm 1$
$\displaystyle\frac{d\overline{\Gamma}}{dm_{\pi\pi}d\cos\theta_{\pi\pi}d\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}$ (28) $\displaystyle=$ $\displaystyle\left|{\cal
A}^{s}_{S_{0}}e^{i\phi^{s}_{S_{0}}}+{\cal
A}^{s}_{P_{0}}e^{i\phi^{s}_{P_{0}}}\cos\theta_{\pi\pi}+{\cal
A}^{s}_{D_{0}}e^{i\phi^{s}_{D_{0}}}\left(\frac{3}{2}\cos^{2}\theta_{\pi\pi}-\frac{1}{2}\right)\right|^{2}\sin^{2}\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$ $\displaystyle+$ $\displaystyle\left|{\cal A}^{s}_{P_{\pm
1}}e^{i\phi^{s}_{P_{\pm 1}}}\frac{1}{2}\sin\theta_{\pi\pi}+{\cal
A}^{s}_{D_{\pm 1}}e^{i\phi^{s}_{D_{\pm
1}}}\sqrt{\frac{3}{2}}\sin\theta_{\pi\pi}\cos\theta_{\pi\pi}\right|^{2}\frac{1+\cos^{2}\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}{2}$
for $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays, where ${\cal
A}^{s}_{k_{\lambda}}$ and $\phi^{s}_{k_{\lambda}}$ are the sum of amplitudes
and reference phase for the spin-$k$ resonance group, respectively. The
$B^{0}$ function for decays is similar, but $\theta_{\pi^{+}\pi^{-}}$ and
$\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ are changed to
$\pi-\theta_{\pi^{+}\pi^{-}}$ and
$\pi-\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ respectively, as a
result of using $\pi^{-}$ and $\mu^{-}$ to define the helicity angles,
yielding
$\displaystyle\frac{d\Gamma}{dm_{\pi\pi}d\cos\theta_{\pi\pi}d\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}$ (29) $\displaystyle=$ $\displaystyle\left|{\cal
A}^{s}_{S_{0}}e^{i\phi^{s}_{S_{0}}}-{\cal
A}^{s}_{P_{0}}e^{i\phi^{s}_{P_{0}}}\cos\theta_{\pi\pi}+{\cal
A}^{s}_{D_{0}}e^{i\phi^{s}_{D_{0}}}\left(\frac{3}{2}\cos^{2}\theta_{\pi\pi}-\frac{1}{2}\right)\right|^{2}\sin^{2}\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$ $\displaystyle+$ $\displaystyle\left|{\cal A}^{s}_{P_{\pm
1}}e^{i\phi^{s}_{P_{\pm 1}}}\frac{1}{2}\sin\theta_{\pi\pi}-{\cal
A}^{s}_{D_{\pm 1}}e^{i\phi^{s}_{D_{\pm
1}}}\sqrt{\frac{3}{2}}\sin\theta_{\pi\pi}\cos\theta_{\pi\pi}\right|^{2}\frac{1+\cos^{2}\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}{2}.$
Summing Eqs. 28 and 29 results in cancellation of the interference involving
the $\lambda=0$ terms for spin-1, and the $\lambda=\pm 1$ terms for spin-2, as
they appear with opposite signs for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and $B^{0}$ decays. Therefore, we
have to fix one phase in spin-1 ($\lambda=0$) group ($\phi^{s}_{P_{0}}$) and
one in spin-2 ($\lambda=\pm 1$) group ($\phi^{s}_{D_{\pm 1}}$); the phases of
$\rho(770)$ ($\lambda=0$) and $f_{2}(1270)$ ($\lambda=\pm 1$) are fixed to
zero. The other phases in each corresponding group are relative to that of the
fixed resonance.
### 4.2 Fit results
To find the best model, we proceed by fitting with all the possible resonances
and a non-resonance (NR) component, then subsequently remove the most
insignificant component one at a time. We repeat this procedure until each
remaining contribution has more than 3 statistical standard deviation
($\sigma$) significance. The significance is estimated from the fit fraction
divided by its statistical uncertainty. The best fit model contains six
resonances, the $f_{0}(500)$, $f_{0}(980)$, $f_{2}(1270)$, $\rho(770)$,
$\rho(1450)$, and $\omega(782)$.
In order to compare the different models quantitatively an estimate of the
goodness of fit is calculated from three-dimensional partitions of the one
angular and two mass squared variables. We use the Poisson likelihood
$\chi^{2}$ [44] defined as
$\chi^{2}=2\sum_{i=1}^{N_{\rm
bin}}\left[x_{i}-n_{i}+n_{i}\text{ln}\left(\frac{n_{i}}{x_{i}}\right)\right],$
(30)
where $n_{i}$ is the number of events in the three dimensional bin $i$ and
$x_{i}$ is the expected number of events in that bin according to the fitted
likelihood function. A total of 1021 bins ($N_{\rm bin}$) are used to
calculate the $\chi^{2}$, based on the variables
$m^{2}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+})$,
$m^{2}(\pi^{+}\pi^{-})$, and
$\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$. The
$\chi^{2}/\text{ndf}$ and the negative of the logarithm of the likelihood,
$\rm-ln\mathcal{L}$, of the fits are given in Table 5; ndf is equal to $N_{\rm
bin}-1-N_{\rm par}$, where $N_{\rm par}$ is the number of fitting parameters.
The difference between the best fit results and fits with one additional
component is taken as a systematic uncertainty. Figure 15 shows the best fit
model projections of $m^{2}(\pi^{+}\pi^{-})$,
$m^{2}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+})$,
$\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ and
$m(\pi^{+}\pi^{-})$. We calculate the fit fraction of each component using Eq.
24. For a P- or D-wave resonance, we report its total fit fraction by summing
all the helicity components, and the fraction of the helicity $\lambda=0$
component. The results are listed in Table 6. Systematic uncertainties will be
discussed in Section 6. Two interesting ratios of fit fractions are
($0.93_{-0.22-0.23}^{+0.37+0.47}$)% for $\omega(782)$ to $\rho(770)$, and
($9.5_{-3.4}^{+6.7}\pm 3.0$)% for $f_{0}(980)$ to $f_{0}(500)$.
The fit fractions of the interference terms are computed using Eq. 25 and
listed in Table 7. Table 8 shows the resonant phases from the best fit. For
the systematic uncertainty study, Table 9 shows the fit fractions of
components for the best model with one additional resonance.
Table 5: Values of $\chi^{2}/\text{ndf}$ and $\rm-ln\mathcal{L}$ of different resonance models. Resonance model | $\rm-ln\mathcal{L}$ | $\chi^{2}/\text{ndf}$ | Probability (%)
---|---|---|---
Best Model | 35292 | 1058/1003 | 11.1
Best Model + $\rho(1700)$ | 35284 | 1045/ 999 | 15.0
Best Model + NR | 35284 | 1058/1001 | 10.3
Best Model + $f_{0}(1370)$ | 35285 | 1047/1001 | 15.2
Best Model + $f_{0}(1500)$ | 35287 | 1049/1001 | 14.4
Best Model + $f_{0}(1710)$ | 35289 | 1052/1001 | 12.6
Figure 15: Dalitz fit projections of (a) $m^{2}(\pi^{+}\pi^{-})$, (b) $m^{2}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+})$, (c) $\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ and (d) $m(\pi^{+}\pi^{-})$ for the best model. The points with error bars are data, the signal fit is shown with a (red) dashed line, the background with a (black) dotted line, and the (blue) solid line represents the total. In (a) and (d), the shape variations near the $\rho(770)$ mass is due to $\rho(770)-\omega(782)$ interference, and the dip at the $K^{0}_{\rm\scriptscriptstyle S}$ mass [31] is due to the $K^{0}_{\rm\scriptscriptstyle S}$ veto. Table 6: Fit fractions and significances of contributing components for the best model, as well as the fractions of the helicity $\lambda=0$ part. The significance takes into account both statistical and systematic uncertainties. Components | Fit fraction (%) | $\lambda=0$ fraction | Significance ($\sigma$)
---|---|---|---
$\rho(770)$ | $62.8_{-2.9-4.8}^{+4.8+2.8}$ | $0.63\pm 0.04_{-0.03}^{+0.06}$ | 11.2
$\omega(782)$ | $0.59_{-0.13-0.14}^{+0.23+0.27}$ | $0.30^{+0.26}_{-0.18}\pm 0.05$ | 3.1
$f_{0}(980)$ | $1.53^{+0.77+0.43}_{-0.50-0.35}$ | 1 | 2.5
$f_{2}(1270)$ | $8.9\pm 1.1\pm 1.0$ | $0.76\pm 0.06\pm 0.05$ | 5.9
$\rho(1450)$ | $5.3^{+2.5+5.6}_{-1.4-0.9}$ | $0.28^{+0.17+0.08}_{-0.13-0.12}$ | 3.2
$f_{0}(500)$ | $16.2\pm 2.0^{+6.0}_{-2.0}$ | 1 | 5.7
Sum | 95.2 | |
Table 7: Interference fractions ${\cal{F}}_{\lambda}^{RR^{\prime}}$ (%) computed using Eq 25. Note that the diagonal elements are fit fractions defined in Eq 24. | | $\rho$ | $\omega$ | $\rho$ | $f_{0}$ | $f_{0}$ | $f_{2}$ | $\rho$ | $\omega$ | $\rho$ | $f_{2}$
---|---|---|---|---|---|---|---|---|---|---|---
| | 770 | 782 | 1450 | 980 | 500 | 1270 | 770 | 782 | 1450 | 1270
| $|\lambda|$ | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1
$\rho(770)$ | 0 | 39.44 | $-0.02$ | $-0.89$ | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\omega(782)$ | 0 | | 0.18 | $-0.05$ | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\rho(1450)$ | 0 | | | 1.47 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$f_{0}(980)$ | 0 | | | | 1.53 | 2.08 | 0 | 0 | 0 | 0 | 0
$f_{0}(500)$ | 0 | | | | | 16.15 | 0 | 0 | 0 | 0 | 0
$f_{2}(1270)$ | 0 | | | | | | 6.72 | 0 | 0 | 0 | 0
$\rho(770)$ | 1 | | | | | | | 23.32 | 0.29 | 0 | 0
$\omega(782)$ | 1 | | | | | | | | 0.41 | $-0.07$ | 0
$\rho(1450)$ | 1 | | | | | | | | | 3.80 | 0
$f_{2}(1270)$ | 1 | | | | | | | | | | 2.14
Table 8: Resonant phases from the best fit. Components | Phase (deg)
---|---
$\rho(770)$, $\lambda=0$ | 0 (fixed)
$\rho(770)$, $|\lambda|=1$ | 0 (fixed)
$\omega(782)$, $\lambda=0$ | $-84\pm 31$
$\omega(782)$, $|\lambda|=1$ | $-70\pm 16$
$f_{0}(980)$ | $103\pm 17$
$f_{2}(1270)$, $\lambda=0$ | $-87\pm 12$
$f_{2}(1270)$, $|\lambda|=1$ | 0 (fixed)
$\rho(1450)$, $\lambda=0$ | $-162\pm 22$
$\rho(1450)$, $|\lambda|=1$ | $160\pm 48$
$f_{0}(500)$ | 0 (fixed)
Table 9: Fit fractions (%) of contributing components for the best model with adding one additional resonance. | Best | $+\rho(1700)$ | $+f_{0}(1370)$ | $+f_{0}(1500)$ | $+f_{0}(1710)$ | +NR
---|---|---|---|---|---|---
$\rho(770)$ | $62.8_{-2.9}^{+4.8}$ | $59.5_{-2.8}^{+3.1}$ | $62.6_{-2.5}^{+3.9}$ | $62.4_{-2.5}^{+4.1}$ | $63.3_{-3.0}^{+5.6}$ | $63.4_{-2.7}^{+3.8}$
$\omega(782)$ | $0.59_{-0.13}^{+0.23}$ | $0.58_{-0.15}^{+0.22}$ | $0.60_{-0.15}^{+0.26}$ | $0.60_{-0.15}^{+0.25}$ | $0.59_{-0.15}^{+0.25}$ | $0.59_{-0.14}^{+0.25}$
$f_{0}(980)$ | $1.53_{-0.50}^{+0.77}$ | $1.54_{-0.53}^{+0.75}$ | $1.54_{-0.55}^{+0.76}$ | $1.50_{-0.54}^{+0.78}$ | $1.55_{-0.51}^{+0.76}$ | $1.74_{-0.55}^{+0.80}$
$f_{2}(1270)$ | $8.9\pm 1.1$ | $8.1\pm 1.2$ | $8.8\pm 1.1$ | $8.8\pm 1.1$ | $8.9\pm 1.1$ | $8.8\pm 1.1$
$\rho(1450)$ | $5.3_{-1.4}^{+2.5}$ | $10.8_{-3.6}^{+5.4}$ | $4.7_{-1.1}^{+1.6}$ | $4.9_{-1.2}^{+1.9}$ | $5.7_{-2.3}^{+4.0}$ | $4.6_{-1.3}^{+2.1}$
$f_{0}(500)$ | $16.2\pm 2.0$ | $15.6\pm 1.9$ | $16.6\pm 2.0$ | $16.9\pm 2.1$ | $16.3\pm 2.1$ | $21.9\pm 3.8$
$\rho(1700)$ | - | $3.4_{-1.5}^{+2.7}$ | - | - | - | -
$f_{0}(1370)$ | - | - | $1.3_{-0.5}^{+0.8}$ | - | - | -
$f_{0}(1500)$ | - | - | - | $1.0_{-0.4}^{+0.7}$ | - | -
$f_{0}(1710)$ | - | - | - | - | $0.4_{-0.2}^{+0.4}$ | -
NR | - | - | - | - | - | $4.5_{-1.7}^{+2.7}$
Sum | 95.2 | 99.6 | 96.2 | 96.0 | 96.7 | 105.5
### 4.3 Helicity angle distributions
We show the helicity angle distributions in the $\rho(770)$ mass region
defined within one full width of the $\rho(770)$ resonance (the width values
are given in Table 4) in Fig. 16. The
$\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ and
$\cos\theta_{\pi\pi}$ background subtracted and efficiency corrected
distributions for this mass region are presented in Fig. 17. The distributions
are in good agreement with the best fit model.
Figure 16: Helicity angle distributions of (a)
$\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ ($\chi^{2}$/ndf
=15/20) and (b) $\cos\theta_{\pi\pi}$ ($\chi^{2}$/ndf =14/20) in the
$\rho(770)$ mass region defined within one full width of the $\rho(770)$ mass.
The points with error bars are data, the signal fit to the best model is shown
with a (red) dashed line, the background with a (black) dotted line, and the
(blue) solid line represents the total.
Figure 17: Background subtracted and efficiency corrected helicity
distributions of (a) $\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$ ($\chi^{2}$/ndf =20/20) and (b) $\cos\theta_{\pi\pi}$ ($\chi^{2}$/ndf
=13/20) in the $\rho(770)$ mass region defined within one full width of the
$\rho(770)$ mass. The points with error bars are data and the solid blue lines
show the fit to the best model.
## 5 Branching fractions
Branching fractions are measured by normalizing to the well measured decay
mode $B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-}$,
which has two muons in the final state and has the same triggers as the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ decays. Assuming equal production of charged and neutral
$B$ mesons at the LHC due to isospin symmetry, the branching fraction is
calculated as
${\cal B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-})=\frac{N_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}}/\epsilon_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}}}{N_{B^{-}}/\epsilon_{B^{-}}}\times{\cal
B}(B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-}),$ (31)
where $N$ and $\epsilon$ denote the yield and total efficiency of the decay of
interest. The branching fraction ${\cal
B}(B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-})=(10.18\pm 0.42)\times 10^{-4}$ is determined from an average of
recent Belle [45] and BaBar [46] measurements that are corrected with respect
to the reported values, which assume equal production of charged and neutral
$B$ mesons at the $\Upsilon(4S)$, using the measured value of
$\frac{\Gamma(B^{+}B^{-})}{\Gamma(B^{0}\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0})}=1.055\pm 0.025$ [47].
Signal efficiencies are derived from simulations including trigger,
reconstruction, and event selection components. Since the efficiency to detect
the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ final state
is not uniform across the Dalitz plane, the efficiency is averaged according
to the Dalitz model, where the best fit model is used. The
$K^{0}_{\rm\scriptscriptstyle S}$ veto efficiency is also taken into account.
Small corrections are applied to account for differences between the
simulation and the data. We measure the kaon and pion identification
efficiencies with respect to the simulation using
$D^{*+}\rightarrow\pi^{+}D^{0}(\rightarrow K^{-}\pi^{+})$ events selected from
data. The efficiencies are measured in bins of $p_{\rm T}$ and $\eta$ and the
averages are weighted using the signal event distributions in the data.
Furthermore, to ensure that the $p$ and $p_{\rm T}$ distributions of the
generated $B$ mesons are correct we weight the $B^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ simulation samples using
$B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ data, respectively.
Finally, the simulation samples are weighted with the charged tracking
efficiency ratio between data and simulation in bins of $p$ and $p_{\rm T}$ of
the track. The average of the weights is the correction factor. The total
correction factors are below 1.04 and largely cancel between the signal and
normalization channels. Multiplying the simulation efficiencies and correction
factors gives the total efficiency ($1.163\pm 0.003\pm 0.017$)% for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ and ($3.092\pm 0.012\pm 0.038$)% for
$B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-}$, where the
first uncertainty is statistical and the second is systematic.
Using $N_{B^{-}}=350{,}727\pm 633$ and $N_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}}=5287\pm 112$, we measure
${\cal B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-})=(3.97\pm 0.09\pm 0.11\pm 0.16)\times 10^{-5},$
where the first uncertainty is statistical, the second is systematic and the
third is due to the uncertainty of ${\cal
B}(B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-})$. The
systematic uncertainties are discussed in Section 6. Our measured value is
consistent with and more precise than the previous BaBar measurement of
$(4.6\pm 0.7\pm 0.6)\times 10^{-5}$ [26].
Table 10 shows the branching fractions of resonant modes calculated by
multiplying the fit fraction and the total branching fraction of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$. Since the $f_{0}(980)$ contribution has a significance
of less than 3$\sigma$ we quote also an upper limit of ${\cal{B}}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)\right)\times{\cal{B}}\left(f_{0}(980)\rightarrow\pi^{+}\pi^{-}\right)<1.1\times
10^{-6}$ at 90% confidence level (CL); this is the first such limit. The limit
is calculated assuming a Gaussian distribution as the central value plus 1.28
times the addition in quadrature of the statistical and systematic
uncertainties. This branching ratio is predicted to be in the range
$(1-3)\times 10^{-6}$ if the $f_{0}(980)$ resonance is formed of tetra-quarks,
but can be much smaller if the $f_{0}(980)$ is a standard quark anti-quark
resonance [19]. Our limit is at the lower boundary of the tetra-quark
prediction, and is consistent with a quark anti-quark resonance with a small
mixing angle. In Section 7.2, we show that the mixing angle, describing the
admixture of $s\bar{s}$ and light quarks, is less than 31∘ at 90% CL.
The other branching fractions are consistent with and more precise than the
previous measurements from BaBar [26, *Aubert:2007xw]. Using ${\cal
B}(\omega\rightarrow\pi^{+}\pi^{-})=(1.53^{+0.11}_{-0.13})\%$ [31], we measure
$\frac{{\cal B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\omega)}{{\cal B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rho^{0})}=0.61^{+0.24+0.31}_{-0.14-0.16},$
and
${\cal B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\omega)=(1.5_{-0.3-0.4}^{+0.6+0.7})\times 10^{-5}.$
This is consistent with the LHCb measurement $\frac{{\cal B}(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\omega)}{{\cal B}(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rho^{0})}=0.89\pm 0.19^{+0.07}_{-0.13}$, using the
$\omega\rightarrow\pi^{+}\pi^{-}\pi^{0}$ mode [48].
Table 10: Branching fractions for each channel. The upper limit at 90% CL is also quoted for the $f_{0}(980)$ resonance which has a significance smaller than 3$\sigma$. The first uncertainty is statistical and the second the total systematic. Channel | ${\cal B}(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}R,R\rightarrow\pi^{+}\pi^{-})$ | Upper limit of $\cal{B}$
---|---|---
| | (at 90% CL)
$\rho(770)$ | $(2.49_{-0.13-0.23}^{+0.20+0.16})\times 10^{-5}$ | -
$\omega(782)$ | $(2.3_{-0.5-0.6}^{+0.9+1.1})\times 10^{-7}$ | -
$f_{0}(980)$ | $(6.1^{+3.1+1.7}_{-2.0-1.4})\times 10^{-7}$ | $<1.1\times 10^{-6}$
$f_{2}(1270)$ | $(3.5\pm 0.4\pm 0.4)\times 10^{-6}$ | -
$\rho(1450)$ | $(2.1^{+1.0+2.2}_{-0.6-0.4})\times 10^{-6}$ | -
$f_{0}(500)$ | $(6.4\pm 0.8^{+2.4}_{-0.8})\times 10^{-6}$ | -
## 6 Systematic uncertainties
Table 11: Relative systematic uncertainties on branching fractions (%). Source | Uncertainty (%)
---|---
Tracking efficiency | 1.0
Material and physical effects | 2.0
Particle identification efficiency | 1.0
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ $p$ and $p_{\rm T}$ distributions | 0.5
$B^{-}$ $p$ and $p_{\rm T}$ distributions | 0.5
Dalitz modeling | 0.6
Background modeling | 0.5
Sum of above sources | 2.7
$\mathcal{B}(B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-})$ | 4.1
Total | 4.9
The contributions to the systematic uncertainties on the branching fractions
are listed in Table 11. Since the branching fractions are measured with
respect to the $B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-}$ mode, which has a different number of charged tracks than the
decays of interest, a 1% systematic uncertainty is assigned due to differences
in the tracking performance between data and simulation. Another 2%
uncertainty is assigned because of the difference between two pions and one
kaon in the final states, due to decay in flight, multiple scattering, and
hadronic interactions. Small uncertainties are introduced if the simulation
does not have the correct $B$ meson kinematic distributions. We are relatively
insensitive to any differences in the $B$ meson $p$ and $p_{\rm T}$
distributions since we are measuring the relative rates. By varying the $p$
and $p_{\rm T}$ distributions we see at most a change of 0.5%. There is a 1.0%
systematic uncertainty assigned for the relative particle identification
efficiencies (0.5% per particle). These efficiencies have been corrected from
those predicted in the simulation by using the data from
$D^{*+}\rightarrow\pi^{+}D^{0}(\rightarrow K^{-}\pi^{+})$. A 0.6% uncertainty
is included for the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{-}\pi^{+}$ efficiency, estimated by changing the best model to that
including all possible resonances. The $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ signal yield is changed by 0.5%
when the shape of the combinatorial background is changed from an exponential
to a linear function. The total systematic uncertainty is obtained by adding
each source of systematic uncertainty in quadrature as they are uncorrelated.
In addition, the largest source is $4.1\%$ due to the uncertainty of
$\mathcal{B}(B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-})$ which is quoted separately.
Table 12: Absolute systematic uncertainties on the results of the Dalitz analysis. Item | Acceptance | Background | Fit model | Resonance parameters | Total
---|---|---|---|---|---
Fit fractions (%)
$\rho(770)$ | $\pm 0.9$ | ${}^{+2.0}_{-3.1}$ | ${}^{+0.6}_{-3.2}$ | $\pm 1.6$ | ${}^{+2.8}_{-4.8}$
$\omega(782)$ | $\pm 0.08$ | ${}^{+0.23}_{-0.06}$ | ${}^{+0.11}_{-0.10}$ | ${}^{+0.028}_{-0.014}$ | ${}_{-0.14}^{+0.27}$
$f_{0}(980)$ | $\pm 0.03$ | ${}_{-0.17}^{+0.24}$ | ${}^{+0.21}_{-0.18}$ | ${}_{-0.24}^{+0.29}$ | ${}^{+0.43}_{-0.35}$
$f_{2}(1270)$ | $\pm 0.06$ | ${}_{-0.59}^{+0.45}$ | ${}^{+0.85}_{-0.76}$ | $\pm 0.36$ | $\pm 1.0$
$\rho(1450)$ | $\pm 0.10$ | ${}^{+0.5}_{-0.6}$ | ${}^{+5.6}_{-0.7}$ | ${}^{+0.4}_{-0.3}$ | ${}^{+5.6}_{-0.9}$
$f_{0}(500)$ | $\pm 0.4$ | ${}_{-0.9}^{+1.6}$ | ${}^{+5.7}_{-1.6}$ | $\pm 0.6$ | ${}^{+6.0}_{-2.0}$
$\lambda=0$ fractions (%)
$\rho(770)$ | $\pm 1.0$ | ${}_{-2.0}^{+1.7}$ | ${}^{+4.9}_{-1.5}$ | $\pm 2.1$ | ${}^{+5.7}_{-3.4}$
$\omega(782)$ | $\pm 1.5$ | ${}_{-1.8}^{+3.5}$ | ${}^{+2.8}_{-3.5}$ | ${}_{-1.7}^{+1.2}$ | ${}^{+4.9}_{-4.5}$
$f_{2}(1270)$ | $\pm 0.3$ | $\pm 2.4$ | ${}^{+3.7}_{-3.4}$ | $\pm 1.5$ | $\pm 4.5$
$\rho(1450)$ | $\pm 0.9$ | ${}_{-8.4}^{+4.8}$ | ${}^{+5.5}_{-5.1}$ | ${}^{+4.2}_{-6.1}$ | ${}^{+8.4}_{-11.6}$
Ratio of fit fractions (%)
$\omega(782)/\rho(770)$ | $\pm 0.13$ | ${}_{-0.11}^{+0.41}$ | ${}_{-0.16}^{+0.18}$ | ${}_{-0.022}^{+0.034}$ | ${}^{+0.47}_{-0.23}$
$f_{0}(980)/f_{0}(500)$ | $\pm 0.3$ | ${}^{+1.5}_{-1.1}$ | ${}^{+1.0}_{-2.1}$ | ${}_{-1.8}^{+2.2}$ | $\pm 3.0$
The sources of the systematic uncertainties on the results of the Dalitz plot
analysis are summarized in Table 12. For the uncertainties due to the
acceptance or background modeling, we repeat the data fit 100 times where the
parameters of acceptance or background modeling are generated according to the
corresponding covariance matrix. We also study the acceptance function by
changing the minimum IP $\chi^{2}$ requirement from 9 to 12.5 on both of the
pion candidates. As shown previously [10], this increases the $\chi^{2}$ of
the fit to the angular distributions by one unit. The acceptance function is
then applied to the data with the original minimum IP $\chi^{2}$ selection of
9, and the likelihood fit is redone and the uncertainties are estimated by
comparing the results with the best fit model. The larger of the two
variations is taken as uncertainty due to the acceptance.
We study the effect of ignoring the experimental mass resolution in the fit by
comparing fits between different pseudo-experiments with and without the
resolution included. As the widths of the resonances we consider are much
larger than the mass resolution, we find that the effects are negligible
except for the $\omega(782)$ resonance whose fit fraction is underestimated by
($0.09\pm 0.08$)%. Thus, we apply a $0.09\%$ correction to the $\omega(782)$
fraction and assign an additional $\pm 0.08\%$ in the acceptance systematic
uncertainty. The results shown in the previous sections already include this
correction.
In the default fit, the signal fraction $f_{\rm sig}=0.621\pm 0.009$, defined
in Eq. 1 is fixed; we vary its value within its error to estimate the
systematic uncertainty. The change is added in quadrature with the background
modeling uncertainties.
The uncertainties due to the fit model include adding each resonance that is
listed in Table 4 but not used in the best model, changing the default values
of $L_{B}$ in P- and D-wave cases, varying the hadron scale $r$ parameters for
the $B$ meson and $R$ resonance to $3.0\mathrm{\,Ge\kern-1.00006ptV}^{-1}$ for
both, replacing the $f_{0}(500)$ model by a Zhou and Bugg function [49, 50]
and using the alternate Gounaris and Sakurai model [51] for $\rho$ resonances.
Then the largest variations among those changes are assigned as the systematic
uncertainties for modeling (see Table 12).
Finally, we repeat the data fit by varying the mass and width of resonances
(see Table 4) within their errors one at a time, and add the changes in
quadrature.
## 7 Further results and implications
### 7.1 Resonant structure
The largest intermediate state in $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ decays is the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rho(770)$ mode. Beside the $\rho(770)$, significant $f_{2}(1270)$ and
$f_{0}(500)$ contributions are also seen. The smaller $\omega(782)$ and
$\rho(1450)$ resonances have 3.1$\sigma$ and 3.2$\sigma$ significances
respectively, including systematic uncertainties. The systematic uncertainties
reduce the significance of the $f_{0}(980)$ to below $3\sigma$. Replacing the
$f_{0}(500)$ by a non-resonant component increases $-\ln{\cal L}$ by 117, and
worsens the $\chi^{2}$ by 192 with the same ndf resulting in a fit confidence
level of $1.8\times 10^{-7}$. Thus the $f_{0}(500)$ state is firmly
established in $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ decays.
As discussed in the introduction, a region with only S- and P-waves is
preferred for measuring $\sin 2\beta^{\rm eff}$. The best fit model
demonstrates that the mass region within $\pm
149$$\mathrm{\,Me\kern-1.00006ptV}$ (one full width) of the $\rho(770)$ mass
contains only $(0.72\pm 0.09)\%$ D-wave contribution, thus this region can be
used for a clean $C\\!P$ measurement. The S-wave in this region is
(11.9$\pm$1.7)%, where the fraction is the sum of individual fit fractions and
the interference.
### 7.2 Mixing angle between $f_{0}(980)$ and $f_{0}(500)$
The scalar nonet is quite an enigma. The mysteries are summarized in Ref.
[52], and in the “Note on scalar mesons” in the PDG [31]. Let us contrast the
masses of the lightest vector mesons with those of the scalars, listed in
Table 13.
Table 13: Masses of light vector and scalar resonances. All values are taken from [31], except for the $f_{0}(500)$ [43]. Isospin | Vector particle | Vector mass ($\mathrm{\,Me\kern-1.00006ptV}$) | Scalar particle | Scalar mass ($\mathrm{\,Me\kern-1.00006ptV}$)
---|---|---|---|---
0 | $\omega$ | 783 | $f_{0}(500)$ | 513
1 | $\rho$ | 776 | $a_{0}$ | 980
1/2 | $K^{*}$ | 980 | $\kappa$ | 800
0 | $\phi$ | 1020 | $f_{0}$ | 980
For the vector particles, the $\omega$ and $\rho$ masses are nearly degenerate
and the masses increase as the $s$-quark content increases. For the scalar
particles, however, the mass dependence differs in several ways which requires
an explanation. Some authors introduce the concept of $q\bar{q}q\bar{q}$
states or superpositions of the four-quark state with the $q\bar{q}$ state. In
either case, the $I=0$ $f_{0}(500)$ and the $f_{0}(980)$ are thought to be
mixtures of the underlying states whose mixing angle has been estimated
previously (see Ref. [19] and references contained therein).
The mixing is parameterized by a 2$\times$2 rotation matrix characterized by
the angle $\varphi_{m}$, giving in our case
$\displaystyle|f_{0}(980)\rangle$ $\displaystyle=$
$\displaystyle\;\;\;\cos\varphi_{m}|s\bar{s}\rangle+\sin\varphi_{m}|n\bar{n}\rangle$
$\displaystyle|f_{0}(500)\rangle$ $\displaystyle=$
$\displaystyle-\sin\varphi_{m}|s\bar{s}\rangle+\cos\varphi_{m}|n\bar{n}\rangle,$
$\displaystyle{\rm where~{}}|n\bar{n}\rangle$ $\displaystyle\equiv$
$\displaystyle\frac{1}{\sqrt{2}}\left(|u\bar{u}\rangle+|d\bar{d}\rangle\right).$
(32)
In this case only the $|d\bar{d}\rangle$ part of the $|n\bar{n}\rangle$ wave
function contributes (see Fig. 2). Thus we have
$\tan^{2}\varphi_{m}=\frac{{\cal{B}}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)\right)}{{\cal{B}}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(500)\right)}\frac{\Phi(500)}{\Phi(980)},$ (33)
where the $\Phi$ terms denote the phase space factors. The phase space in this
pseudoscalar to vector-pseudoscalar decay is proportional to the cube of the
$f_{0}$ three-momentum. Taking the average of the momentum dependent phase
space over the resonant line shapes results in the ratio of phase space
factors $\frac{\Phi(500)}{\Phi(980)}$ being equal to 1.25.
Using the data shown in Table 10 we determine the ratio of branching fractions
for both resonances resulting in the $\pi^{+}\pi^{-}$ final state as
$\frac{{\cal{B}}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)\right)\times{\cal{B}}\left(f_{0}(980)\rightarrow\pi^{+}\pi^{-}\right)}{{\cal{B}}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(500)\right)\times{\cal{B}}\left(f_{0}(500)\rightarrow\pi^{+}\pi^{-}\right)}=(9.5^{+6.7}_{-3.4}\pm
3.0)\%.$
This value must be corrected for the individual branching fractions of the
$f_{0}$ resonances into the $\pi^{+}\pi^{-}$ final state.
BaBar has measured $\frac{{\cal{B}}\left(f_{0}(980)\rightarrow
K^{+}K^{-}\right)}{{\cal{B}}\left(f_{0}(980)\rightarrow\pi^{+}\pi^{-}\right)}=0.69\pm
0.32$ using $B\rightarrow KKK$ and $B\rightarrow K\pi\pi$ decays [53]. BES
obtained relative branching ratios using $\psi(2S)\rightarrow\gamma\chi_{c0}$
decays where the $\chi_{c0}\rightarrow f_{0}(980)f_{0}(980)$, and either both
$f_{0}(980)$ candidates decay into $\pi^{+}\pi^{-}$ or one into
$\pi^{+}\pi^{-}$ and the other into $K^{+}K^{-}$ pairs [54, *Ablikim:2005kp].
From their results we obtain $\frac{{\cal{B}}\left(f_{0}(980)\rightarrow
K^{+}K^{-}\right)}{{\cal{B}}\left(f_{0}(980)\rightarrow\pi^{+}\pi^{-}\right)}=0.25^{+0.17}_{-0.11}$[56].
Averaging the two measurements gives
$\frac{{\cal{B}}\left(f_{0}(980)\rightarrow
K^{+}K^{-}\right)}{{\cal{B}}\left(f_{0}(980)\rightarrow\pi^{+}\pi^{-}\right)}=0.35_{-0.14}^{+0.15}\,.$
(34)
Assuming that the $\pi\pi$ and $KK$ decays are dominant we obtain
${\cal{B}}\left(f_{0}(980)\rightarrow\pi^{+}\pi^{-}\right)=\left(46\pm
6\right)\%,$ (35)
where we have assumed that the only other decays are to $\pi^{0}\pi^{0}$, half
of the $\pi^{+}\pi^{-}$ rate, and to neutral kaons, taken equal to charged
kaons. We use
${\cal{B}}\left(f_{0}(500)\rightarrow\pi^{+}\pi^{-}\right)=\frac{2}{3}$, which
results from isospin Clebsch-Gordon coefficients, and assuming that the only
decays are into two pions. Since we have only an upper limit on the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}(980)$ final state, we will
only find an upper limit on the mixing angle, so if any other decay modes of
the $f_{0}(500)$ ($f_{0}(980)$) exist, they would make the limit more (less)
stringent. Our limit then is
$\tan^{2}\varphi_{m}=\frac{{\cal{B}}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)\right)}{{\cal{B}}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(500)\right)}\frac{\Phi(500)}{\Phi(980)}<0.35~{}{\rm
at~{}90\%~{}confidence~{}level},$
which translates into a limit
$|\varphi_{m}|<31^{\circ}~{}{\rm at~{}90\%~{}confidence~{}level}.$
Various mixing angle measurements have been derived in the literature and
summarized in Ref. [19]. There are a wide range of values including: (a) using
$D_{s}^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ transitions which give a range
$35^{\circ}<|\varphi_{m}|<55^{\circ}$, (b) using radiative decays where two
solutions were found either $\varphi_{m}=4^{\circ}\pm 3^{\circ}$ or
$136^{\circ}\pm 6^{\circ}$, (c) using resonance decays from both
$\phi\rightarrow\gamma\pi^{0}\pi^{0}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\omega\pi\pi$ where a
value of $\varphi_{m}\simeq 20^{\circ}$ was found, (d) using the $D^{\pm}$ and
$D_{s}^{\pm}$ decays into $f_{0}(980)\pi^{\pm}$ and $f_{0}(980)K^{\pm}$ where
values of $\varphi_{m}=31^{\circ}\pm 5^{\circ}$ or $42^{\circ}\pm 7^{\circ}$
were found.
## 8 Conclusions
We have studied the resonance structure of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ using a modified Dalitz plot analysis where we also
include the decay angle of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
meson. The decay distributions are formed from a series of final states
described by individual $\pi^{+}\pi^{-}$ interfering decay amplitudes. The
largest component is the $\rho(770)$ resonance. The data are best described by
adding the $f_{2}(1270)$, $f_{0}(500)$, $\omega(782)$, $\rho(1450)$ and
$f_{0}(980)$ resonances, where the $f_{0}(980)$ resonance contributes less
than $3\sigma$ significance. The results are listed in Table 6.
We set an upper limit ${\cal{B}}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)\right)\times{\cal{B}}\left(f_{0}(980)\rightarrow\pi^{+}\pi^{-}\right)<1.1\times
10^{-6}$ at 90% confidence level that favors somewhat a quark anti-quark
interpretation of the $f_{0}(980)$ resonance. We also have firmly established
the existence of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}(500)$
intermediate resonant state in $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays, and limit the absolute
value of the mixing angle between the two lightest scalar states to be less
than $31^{\circ}$ at 90% confidence level.
Our six-resonance best fit shows that the mass region within one full width of
the $\rho(770)$ contains mostly P-wave, $(11.9\pm 1.7)$% S-wave, and only
$(0.72\pm 0.09)$% D-wave. Thus this region can be used to perform $C\\!P$
violation measurements, as the S- and P-wave components can be treated in the
same manner as in the analysis of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ [4, 5, 6, *Abazov:2011ry, *:2012fu]. The measured value of the
asymmetry can be compared to that found in other modes such as $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{0}$ in order to ascertain
the possible effects due to penguin amplitudes.
The measured branching ratio is
${\cal B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-})=(3.97\pm 0.09\pm 0.11\pm 0.16)\times 10^{-5},$
where the first uncertainty is statistical, the second is systematic and the
third is due to the uncertainty of ${\cal
B}(B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-})$. The
largest contribution is the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rho(770)$ mode with a branching fraction of
$(2.49_{-0.13-0.23}^{+0.20+0.16})\times 10^{-5}$.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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arxiv-papers
| 2013-01-22T21:51:08 |
2024-09-04T02:49:40.609268
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, A. Adametz, B. Adeva,\n M. Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio,\n M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S.\n Amato, Y. Amhis, L. Anderlini, J. Anderson, R. Andreassen, R. B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, S. Bachmann, J. J. Back, C. Baesso, V. Balagura, W. Baldini, R. J.\n Barlow, C. Barschel, S. Barsuk, W. Barter, Th. Bauer, A. Bay, J. Beddow, I.\n Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G.\n Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M.\n van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P. M. Bj{\\o}rnstad, T.\n Blake, F. Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov, V. Bocci, A.\n Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T. J. V. Bowcock, E.\n Bowen, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M.\n Britsch, T. Britton, N. H. Brook, H. Brown, I. Burducea, A. Bursche, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, H. Carranza-Mejia,\n L. Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo, Ch. Cauet, M. Charles,\n Ph. Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal,\n G. Ciezarek, P. E. L. Clarke, M. Clemencic, H. V. Cliff, J. Closier, C. Coca,\n V. Coco, J. Cogan, E. Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A.\n Cook, M. Coombes, G. Corti, B. Couturier, G. A. Cowan, D. Craik, S. Cunliffe,\n R. Currie, C. D'Ambrosio, P. David, P. N. Y. David, I. De Bonis, K. De Bruyn,\n S. De Capua, M. De Cian, J. M. De Miranda, L. De Paula, W. De Silva, P. De\n Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi, L. Del Buono, C. Deplano, D.\n Derkach, O. Deschamps, F. Dettori, A. Di Canto, J. Dickens, H. Dijkstra, P.\n Diniz Batista, M. Dogaru, F. Domingo Bonal, S. Donleavy, F. Dordei, A. Dosil\n Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, A.\n Dzyuba, S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, S.\n Eisenhardt, U. Eitschberger, R. Ekelhof, L. Eklund, I. El Rifai, Ch.\n Elsasser, D. Elsby, A. Falabella, C. F\\\"arber, G. Fardell, C. Farinelli, S.\n Farry, V. Fave, D. Ferguson, V. Fernandez Albor, F. Ferreira Rodrigues, M.\n Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M. Fontana, F. Fontanelli, R.\n Forty, O. Francisco, M. Frank, C. Frei, M. Frosini, S. Furcas, E. Furfaro, A.\n Gallas Torreira, D. Galli, M. Gandelman, P. Gandini, Y. Gao, J. Garofoli, P.\n Garosi, J. Garra Tico, L. Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M.\n Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V. V. Gligorov, C. G\\\"obel, D.\n Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa G\\'andara, R.\n Graciani Diaz, L. A. Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E.\n Greening, S. Gregson, O. Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C.\n Hadjivasiliou, G. Haefeli, C. Haen, S. C. Haines, S. Hall, T. Hampson, S.\n Hansmann-Menzemer, N. Harnew, S. T. Harnew, J. Harrison, P. F. Harrison, T.\n Hartmann, J. He, V. Heijne, K. Hennessy, P. Henrard, J. A. Hernando Morata,\n E. van Herwijnen, E. Hicks, D. Hill, M. Hoballah, C. Hombach, P. Hopchev, W.\n Hulsbergen, P. Hunt, T. Huse, N. Hussain, D. Hutchcroft, D. Hynds, V.\n Iakovenko, P. Ilten, R. Jacobsson, A. Jaeger, E. Jans, F. Jansen, P. Jaton,\n F. Jing, M. John, D. Johnson, C. R. Jones, B. Jost, M. Kaballo, S. Kandybei,\n M. Karacson, T. M. Karbach, I. R. Kenyon, U. Kerzel, T. Ketel, A. Keune, B.\n Khanji, O. Kochebina, I. Komarov, R. F. Koopman, P. Koppenburg, M. Korolev,\n A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F.\n Kruse, M. Kucharczyk, V. Kudryavtsev, T. Kvaratskheliya, V. N. La Thi, D.\n Lacarrere, G. Lafferty, A. Lai, D. Lambert, R. W. Lambert, E. Lanciotti, G.\n Lanfranchi, C. Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van\n Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, O. Leroy, Y.\n Li, L. Li Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben,\n J. H. Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, J. Luisier, H. Luo, F.\n Machefert, I. V. Machikhiliyan, F. Maciuc, O. Maev, S. Malde, G. Manca, G.\n Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G. Martellotti,\n A. Martens, L. Martin, A. Mart\\'in S\\'anchez, M. Martinelli, D. Martinez\n Santos, D. Martins Tostes, A. Massafferri, R. Matev, Z. Mathe, C. Matteuzzi,\n M. Matveev, E. Maurice, A. Mazurov, J. McCarthy, R. McNulty, B. Meadows, F.\n Meier, M. Meissner, M. Merk, D. A. Milanes, M.-N. Minard, J. Molina\n Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F.\n Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, P. Naik, T. Nakada, R.\n Nandakumar, I. Nasteva, M. Needham, N. Neufeld, A. D. Nguyen, T. D. Nguyen,\n C. Nguyen-Mau, M. Nicol, V. Niess, R. Niet, N. Nikitin, T. Nikodem, S. Nisar,\n A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S.\n Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J. M. Otalora Goicochea, P.\n Owen, B. K. Pal, A. Palano, M. Palutan, J. Panman, A. Papanestis, M.\n Pappagallo, C. Parkes, C. J. Parkinson, G. Passaleva, G. D. Patel, M. Patel,\n G. N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A.\n Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D. L. Perego, E. Perez\n Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina,\n K. Petridis, A. Petrolini, A. Phan, E. Picatoste Olloqui, B. Pietrzyk, T.\n Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F. Polci, G. Polok, A.\n Poluektov, E. Polycarpo, D. Popov, B. Popovici, C. Potterat, A. Powell, J.\n Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian, J. H. Rademacker, B.\n Rakotomiaramanana, M. S. Rangel, I. Raniuk, N. Rauschmayr, G. Raven, S.\n Redford, M. M. Reid, A. C. dos Reis, S. Ricciardi, A. Richards, K. Rinnert,\n V. Rives Molina, D. A. Roa Romero, P. Robbe, E. Rodrigues, P. Rodriguez\n Perez, G. J. Rogers, S. Roiser, V. Romanovsky, A. Romero Vidal, J. Rouvinet,\n T. Ruf, H. Ruiz, G. Sabatino, J. J. Saborido Silva, N. Sagidova, P. Sail, B.\n Saitta, C. Salzmann, B. Sanmartin Sedes, M. Sannino, R. Santacesaria, C.\n Santamarina Rios, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta,\n M. Savrie, D. Savrina, P. Schaack, M. Schiller, H. Schindler, S. Schleich, M.\n Schlupp, M. Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune,\n R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K.\n Senderowska, I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I.\n Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko,\n V. Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, N. A. Smith, E.\n Smith, M. Smith, K. Sobczak, M. D. Sokoloff, F. J. P. Soler, F. Soomro, D.\n Souza, B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S.\n Stahl, O. Steinkamp, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U.\n Straumann, V. K. Subbiah, S. Swientek, V. Syropoulos, M. Szczekowski, P.\n Szczypka, T. Szumlak, S. T'Jampens, M. Teklishyn, E. Teodorescu, F. Teubert,\n C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S. Tolk, D.\n Tonelli, S. Topp-Joergensen, N. Torr, E. Tournefier, S. Tourneur, M. T. Tran,\n M. Tresch, A. Tsaregorodtsev, P. Tsopelas, N. Tuning, M. Ubeda Garcia, A.\n Ukleja, D. Urner, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P.\n Vazquez Regueiro, S. Vecchi, J. J. Velthuis, M. Veltri, G. Veneziano, M.\n Vesterinen, B. Viaud, D. Vieira, X. Vilasis-Cardona, A. Vollhardt, D.\n Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo{\\ss}, H. Voss, R.\n Waldi, R. Wallace, S. Wandernoth, J. Wang, D. R. Ward, N. K. Watson, A. D.\n Webber, D. Websdale, M. Whitehead, J. Wicht, J. Wiechczynski, D. Wiedner, L.\n Wiggers, G. Wilkinson, M. P. Williams, M. Williams, F. F. Wilson, J. Wishahi,\n M. Witek, S. A. Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, F. Xing, Z.\n Xing, Z. Yang, R. Young, X. Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev,\n F. Zhang, L. Zhang, W. C. Zhang, Y. Zhang, A. Zhelezov, A. Zhokhov, L. Zhong,\n A. Zvyagin",
"submitter": "Sheldon Stone",
"url": "https://arxiv.org/abs/1301.5347"
}
|
1301.5433
|
# Far-ultraviolet Observations of the Taurus-Perseus-Auriga Complex
Tae-Ho Lim11affiliation: Korea Advanced Institute of Science and Technology
(KAIST), 373-1 Guseong-dong, Yuseong-gu, Daejeon, Korea 305-701, Republic of
Korea Korea Advanced Institute of Science and Technology (KAIST)
[email protected] Kyoung-Wook Min11affiliation: Korea Advanced Institute of
Science and Technology (KAIST), 373-1 Guseong-dong, Yuseong-gu, Daejeon, Korea
305-701, Republic of Korea Korea Advanced Institute of Science and Technology
(KAIST) Kwang-Il Seon22affiliation: Korea Astronomy and Space Science
Institute (KASI), 61-1 Hwaam-dong, Yuseong-gu, Daejeon, Korea 305-348,
Republic of Korea Korea Astronomy and Space Science Institute (KASI)
###### Abstract
We have constructed a far-ultraviolet (FUV) continuum map of the Taurus-
Auriga-Perseus complex, one of the largest local associations of dark clouds,
by merging the two data sets of GALEX and FIMS, which made observations at
similar wavelengths. The FUV intensity varies significantly across the whole
region, but the diffuse FUV continuum is dominated by dust scattering of
stellar photons. A diffuse FUV background of $\sim$1000 CU is observed, part
of which may be attributable to the scattered photons of foreground FUV light,
located in front of the thick clouds. The fluorescent emission of molecular
hydrogen constitutes $\sim$10 % of the total FUV intensity throughout the
region, generally proportional to the local continuum level. We have developed
a Monte Carlo radiative transfer code and applied it to the present clouds
complex to obtain the optical properties of dust grains and the geometrical
structures of the clouds. The albedo and the phase function asymmetry factor
were estimated to be $0.42^{+0.05}_{-0.05}$, and $0.47^{+0.11}_{-0.27}$,
respectively, in accordance with theoretical estimations as well as recent
observations. The distance and thickness of the four prominent clouds in this
complex were estimated using a single slab model applied individually to each
cloud. The results obtained were in good agreement with those from other
observations in the case of the Taurus cloud, as its geometrical structure is
rather simple. For other clouds, which were observed to have multiple
components, the results gave distances and thicknesses encompassing all
components of each cloud. The distance and thickness estimations were not
crucially sensitive to the exact values of the albedo and the phase function
asymmetry factor, while the locations of the bright field stars relative to
the clouds as initial photon sources seem to be the most important factor in
the process of fitting.
###### Subject headings:
ISM: clouds — ISM: dust, extinction — ISM: individual (Taurus, Auriga and
Perseus complex) — ISM: structure — ultraviolet: ISM
## 1\. Introduction
The immense complex of the Taurus, Perseus, and Auriga (TPA) clouds is known
to be one of the largest local associations of dark nebula. According to
Ungerechts et al. (1987), the total masses of the whole complex is $\sim
2\times 10^{5}$ solar mass. The TPA complex contains star-forming cloudlets
(TMC1, TMC2, NGC 1333, and IC 348, for example), a bright emission nebula (NGC
1499, also known as California Nebula), and stellar associations (Per OB2, for
example). Low-mass star formation processes are believed to be taking place in
the eastern section of the complex (Taurus and Auriga), where mostly low-mass
T Tauri type stars are observed, while both low- and high-mass stars are being
formed in the western section of the Perseus complex (the Per OB2 association,
the open cluster NGC 1333, and so on). The whole complex may be physically
associated with the Per OB2 stellar association, centered at (l, b)=(160.5,
-14.7) with a diameter of $\sim 20^{\circ}$, which is the second closest OB
association to the Sun with a distance of $\sim$300 pc on average (Belikov et
al., 2002). Recent supernova explosions in the Per OB2 association are
believed to have created an expanding HI supershell that is moving into the
surrounding interstellar medium (Sancisi et al., 1974; Hartmann & Burton,
1997).
The TPA complex may consist of multiple layers of molecular gas located at
various distances. The Taurus cloud is believed to spread between 100 pc and
200 pc (Ungerechts et al., 1987), with a large part of it blocked by the cloud
itself (Elias, 1978; Kenyon et al., 1994). The most distant part of the Taurus
molecular cloud complex may be interacting with a supershell blown by the Per
OB2 association (Olano et al., 1987). Cernis (1990) argued that the cloud in
the region of NGC 1333, one of the star forming regions located in and around
the Perseus cloud, consists of two layers, and estimated their distances to be
$\sim$170 pc and $\sim$230 pc based on an extinction analysis. The Auriga dark
cloud, located at a distance similar to that of the Taurus cloud, is usually
regarded as part of the Taurus-Auriga complex, and was suggested to overlap
with the California cloud at its farthest side (Ungerechts et al., 1987). The
California cloud seems to be the most distant cloud in this region: Lada et
al. (2009) argued that the cloud is located at $\sim$450 pc from the Sun.
The interstellar radiation field (ISRF), composed of radiations originating
directly from stellar sources as well as the diffuse background, plays an
important role in the physical and chemical processes of the interstellar
medium (ISM). For example, far-ultraviolet (FUV) ISRF photons may ionize
atoms, dissociate molecules, or heat gases by ejecting electrons from dust
grains or by directly exciting atoms and molecules. The ISRF itself is also
affected by interactions with the ISM. In fact, most of the diffuse FUV
background radiation is believed to originate from the scattering of starlight
by interstellar dust (Bowyer, 1991; Seon, 2011). The effect of dust scattering
is well observed in reflection nebulae and dark clouds as well as in the
diffuse galactic light (DGL). Hence, the observations of these objects have
been used not only to trace interstellar dust, but also to obtain the optical
properties of dust itself, such as the albedo (a) and the phase function
asymmetry factor (g). In the case of the DGL, dust scattering was observed to
be moderately reflective in the forward direction (Lillie et al., 1976;
Joubert et al., 1983; Hurwitz et al., 1991; Murthy et al., 1991, 1994, 1995;
Witt et al., 1997), with the estimated values of a and g in general agreement
with those of the theoretical model for carbonaceous-silicate grains, i.e.,
0.40-0.60 and 0.55-0.65, respectively, at UV wavelengths (Weingartner &
Draine, 2001). For reflection nebulae (Murthy et al., 1993; Witt, 1992; Gordon
et al., 1994; Gibson et al., 2003) and individual clouds (Witt et al., 1990;
Mattila, 1970; Hurwitz, 1994), however, the results were varied and dependent
on the individual targets, probably due to uncertainties in the scattering
geometry of the target clouds as well as the star systems that were the
dominant sources of photons (Draine, 2003; Bowyer, 1991). Calzetti et al.
(1995) determined a and g to be 0.7-0.8 and $\sim$0.75 around 1600 Å,
respectively, for dust toward IC 43, while Gibson et al. (2003) estimated a
and g to be $\sim$0.22 and $\sim$0.74, respectively, for wavelengths below
2200 Å for the Pleiades nebula. For the Ophiuchus cloud, Sujatha et al. (2005)
found a and g to be $\sim$0.40 and $\sim$0.55, respectively, at $\sim$1100 Å,
while Lee et al. (2008) obtained $\sim$0.36 and $\sim$0.52 for a and g,
respectively, for 1370 Å–1670 Å.
|
---|---
Figure 1.— The FIMS continuum image (a) and the GALEX FUV image (b).
The main goal in this line of research has been the estimation of the optical
properties of dust grains. The geometry of the clouds and the distribution of
stellar photon sources involved, however, were also assumed to influence the
intensity of light scattered by dust in the clouds, thereby providing
constraints on the geometrical structures of the clouds themselves. For
example, Sujatha et al. (2005) assumed two extended sheet-like structures for
their study on the Ophiuchus cloud. Shalima et al. (2006) also used the
morphology of a plane-parallel dust sheet in their scattering simulation of
the Orion nebula. Although these assumed clouds geometries were highly
idealistic, the estimated distances were more or less reasonable when compared
to results based on other observations. In this paper, we examine the results
of the FUV observations made for the TPA region and study the clouds
geometries, though the results might still be very simple. We perform Monte
Carlo simulations of dust scattering and compare the results with the observed
diffuse FUV emission map. In fact, it was revealed in the study of the Taurus
region that the continuum emission is closely related to dust extinction (Lee
et al., 2006). There have been several reports on the structures of the clouds
in this region, mostly based on star counting methods, extinction studies, and
comparisons with the distances of the associated stars and nebulae. We show
that the present method of dust scattering simulations gives consistent
estimates regarding the clouds’ geometries in this region. We also demonstrate
that the average albedo value obtained for this region is in good agreement
with a theoretical estimation.
## 2\. Data Reduction
We employed two data sets for the present study. The first is from
observations made by the FUV Imaging Spectrograph (FIMS) flown aboard the
Korean microsatellite STSAT-1, which was launched on September 27, 2003 and
operated for a year and a half. The second consists of FUV archival data of
the Galaxy Evolution Explorer (GALEX). FIMS is an imaging spectrograph
optimized for the observation of diffuse emissions with two wavelength
channels, a short wavelength channel covering $\lambda\lambda$ 900–1150 Å, and
a long wavelength channel covering $\lambda\lambda$ 1350–1700 Å. Both channels
have a resolving power of $\lambda/\Delta\lambda\sim$ 550\. We analyzed only
the long wavelength channel data in the present study since the short
wavelength channel was contaminated by many airglow lines. The field of view
of the long wavelength channel is $7.4^{\circ}\times 4.3^{\prime}$, with a
5–10′ imaging resolution along the slit, and the spectral half-energy line
width, averaged over the angular field, is 3.2 Å. The instrument, its in-orbit
performance, and the data analysis procedures are described in Edelstein et
al. (2006a, 2006b).
The FIMS data set that covers the TPA area of 154∘ – 180∘ in Galactic
longitude and -28∘ – -2∘ in Galactic latitude is composed of 38 observations
made during the sky survey toward this region. The total exposure time is
around 3.6 s per pixel, on average. Only the data from 1360 Å to 1680 Å,
excluding the intense O I airglow line at 1356 Å, was used for the present
analysis. The FIMS continuum image of the diffuse emission was constructed by
spatially binning the data with a square bin of 0.4∘ x 0.4∘ size, after
removing pixels that had more than 9,000 Continuum Unit (photons s-1 cm-1 sr-1
Å-1, hereafter CU) as these were suspected to be contaminated by bright stars.
The 9000 CU criterion was chosen as the data of GALEX, which intentionally
avoided the observation of bright stars, did not exceed 9000 CU in this
region. Figure 1(a) shows the resulting FIMS image, where the regions of
missing data are shown in black. While the map shows the expected general
features of the TPA region, including the dark Taurus cloud and the bright Per
OB2 association, we can also see that the bright regions of the Per OB2
association are very extended across the scanning direction. In fact, this
extended appearance is due to instrumental scattering along the slit.
Furthermore, some areas of the FIMS image were made with extremely short
exposures. Hence, we merged the GALEX FUV data with the FIMS data to remedy
these shortcomings, as the wavelength range of the GALEX data is similar to
that of the FIMS, i.e., 1350 – 1750 Å.
From the GALEX data release website, we extracted the All Sky Surveys imaging
data (AIS of GR4/GR5 Data Release) of the TPA region, and plotted the image
with 0.4∘ x 0.4∘ bins in Figure 1(b). We compared the intensities of the two
observations pixel by pixel to check for possible biases that might exist
between them. We limited our comparison to only the regions in which the FIMS
intensity was less than 6000 CU, to exclude the bright pixels affected by the
instrumental scattering and the pixels where FIMS exposure time was greater
than 2.5 s. In Figure 2, we show the resulting scatter plot, in which we can
see a remarkable correlation between the two data sets. The linear fitting
between the two intensities is given by $I_{GALEX}=1.01\times I_{FIMS}+185.50$
in CU. The standard deviation of the data about the linear fit is $\sim$ 550
CU. Since we believe GALEX data are more reliable than FIMS data in the region
where the GALEX data are available because of instrumental scattering and
short exposure times of FIMS, we used GALEX data as the basis of the merged
map and filled the regions where GALEX did not observe with the FIMS data
scaled according to the above linear fitting function. The resulting merged
map is shown in Figure 3(a).
Figure 2.— A pixel-to-pixel scatter plot for the intensities of the TPA region
observed by FIMS and GALEX showing a remarkable agreement. The red solid line
is a linear fit of the data points.
## 3\. Results
### 3.1. Examination of the FUV continuum map
|
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|
Figure 3.— (a) FUV intensity (in CU) overlaid by the dust extinction contours,
with thicker contours representing higher dust extinction levels, (b)
H$\alpha$ emission map shown in Rayleighs, (c) SFD dust extinction map,
E(B-V), and (d) CO emission map, shown in K km/s. The locations A, B, and C in
panel (a) are those with distinct physical characteristics and discussed in
the text.
Figure 3 shows images of the TPA region constructed at different wavelengths:
(a) the FUV map obtained by merging the FIMS and GALEX data, (b) H$\alpha$
emission map (Finkbeiner, 2003), (c) Schlegel, Finkbeiner & Davis (SFD) Dust
Survey map (Schlegel et al., 1998), and (d) the CO emission map (Dame et al.,
2001). The FUV continuum map of Figure 3(a) shows bright enhancement in the
regions of the Per OB2 association and the Pleiades cluster, as well as dark
regions coincident with the Taurus and Auriga clouds. Hence, the FUV flux is
bright on the low-longitude half where H$\alpha$ emission is also bright,
while FUV flux seems to be heavily absorbed by the dense cloud of the Taurus-
Auriga complex in the high-longitude half, where the dust extinction level is
high. The existence of a dense cloud, however, does not necessarily imply that
FUV intensity will be low there. For example, the California cloud near the
Per OB2 association is rich in dust but FUV intensity is still bright there, a
feature related to their respective distances, to be discussed later. In the
dust extinction and CO emission maps, shown in Figure 3(c) and Figure 3(d),
respectively, a common void or cavity is seen surrounded by the dark clouds of
the TPA complex. The void is coincident with the brightest region in the
H$\alpha$ emission map of Figure 3(b) in the vicinity of the center of the Per
OB2 association. Though not noticeable here, a dust emission ring is seen
around the cavity according to sub-millimeter and far-infrared maps (Watson et
al., 2005). In fact, the void region was identified to be an HI supershell
with a diameter of $\sim$100 pc, which is believed to have been formed by the
interaction of the Per OB2 association with the ambient interstellar medium
(Sancisi et al., 1974; Heiles, 1984). The bright spot south of the Taurus
cloud is the reflection nebula around the Pleiades cluster.
We have selected three distinct subregions as shown in Figure 3(a) and made a
pixel-to-pixel plot of the FUV intensity against dust extinction, with
different colors according to the subregions. Subregion A includes the Taurus
cloud which is optically thick with low FUV intensity, and subregion B is
optically thin with a little higher FUV intensity. Subregion C is a region of
high FUV intensity that includes both optically thin and thick areas. The
result is shown in Figure 4. First, we note that region A, which includes the
thick Taurus cloud, indeed shows a general anti-correlation between the FUV
intensity and the dust extinction level, implying that FUV is heavily absorbed
by the dense Taurus cloud. On the other hand, the result for region B shows a
positive correlation, in agreement with the general notion that FUV intensity
increases with dust extinction level in the optically thin region (Hurwitz,
1994; Luhman et al., 1996). Indeed, most of the data points in region B
correspond to small values of color excess, below $\sim$0.5. On the other
hand, region C shows severe fluctuations, obviously due to the non-uniform
strong radiation fields of the Per OB2 association over a wide range of the
color excess values. Another interesting feature seen in Figure 4 is that FUV
intensity does not decrease below 1000 CU even at a very high extinction
level, part of which we may attribute to the scattered photons of the
foreground FUV light, located in front of the thick clouds (Lee et al., 2006).
Hurwitz et al. (1991) also observed a similar diffuse background for the
Galactic latitudes $10^{\circ}<|b|<20^{\circ}$.
Figure 4.— A scatter plot of the FUV intensity against dust extinction: the
blue, orange, and black colors correspond to the regions A, B, and C in Figure
3(a), respectively.
### 3.2. Monte Carlo simulation of dust scattering
As shown in Figure 4, the scatter plot of FUV intensity against the dust
extinction level shows complicated behaviors that depend on local extinction
levels, fluctuations of local radiation fields as well as mixing of the
foreground and background FUV photons. On the other hand, such complicated
behaviors might also make it possible to extract information regarding the
structure of the clouds, especially their relative locations, by comparing the
observed FUV intensities with those of dust-scattering simulations that
incorporate local photon sources and an assumed distribution of clouds. In
fact, similar methods have been previously applied to several cloud systems
(Gordon et al., 2001; Shalima et al., 2006; Lee et al., 2008; Sujatha et al.,
2005, 2007).
|
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|
Figure 5.— Spectra of the whole TPA region and subregions A, B, and C, in which the fluorescent emission lines of molecular hydrogen are indicated with vertical red bars. The dashed blue lines represent simple models for the continuum emissions. |
---|---
Figure 6.— (a) Dust scattered FUV map constructed from the observations and
(b) the Monte Carlo simulation results. The California nebula region is
excluded from our dust scattering simulations as its H$\alpha$ intensity is
extremely high and the associated two-photon effect is difficult to estimate.
Our main purpose in the present study is to understand through simulations the
dust scattering properties and the morphological structures in this broad
region. More specifically, we seek to understand the distances and thicknesses
of the TPA clouds, which includes both thick clouds and strong radiation
fields of OB associations, based on the observed FUV emission features. Hence,
it is necessary to identify and remove contributions other than the dust-
scattered emission from the observed FUV image. In addition to dust-scattering
of stellar photons, diffuse interstellar FUV emissions can have three
different origins including: i) ion line emission coming from hot gas of
10${}^{4.5}-$ 105 K; ii) H2 fluorescent emission originating from the outer
regions of molecular clouds irradiated by interstellar UV photons; and iii)
two-photon effects which are mostly relevant to ionized regions. Of the other
FUV sources, ion line emissions should be negligible in the present region of
thick clouds. On the other hand, we expect significant H2 fluorescent
emissions in this region where both thick clouds and strong radiation fields
exist. Based on the spectral observations of FIMS, Figure 5 indeed shows that
H2 fluorescent emission is detectable throughout the whole TPA region. The
contribution from the H2 fluorescent emission lines is estimated to be
$\sim$10.8 %, $\sim$11.4 %, and $\sim$9.5 % of the total FUV intensity for the
regions A, B, and C, respectively, and $\sim$10% for the whole region. We have
accordingly subtracted 10 % of the total intensity from the map shown in
Figure 3(a) to account for the contribution from H2 fluorescent emission.
Estimation of the two-photon effect can be a little tricky since H$\alpha$
emission can also originate from regions other than ionized regions, and
H$\alpha$ photons may also be scattered in a manner similar to the FUV
continuum photons. Furthermore, Reynolds (1990) suggested a two-photon
continuum emission of $\sim$60 CU for 1 Rayleigh of H$\alpha$ line in the case
of a diffuse ionized gas, while Seon (2011) estimated it to be $\sim$26 CU for
1 Rayleigh of H$\alpha$ emission of a cooling ionized gas. Hence, we have
excluded the California nebula region, where H$\alpha$ intensity is especially
high, in our simulation and applied the formula of the FUV continuum intensity
of 60 CU for 1 Rayleigh intensity of H$\alpha$ throughout the region as a
simple approximation. We will discuss later the dependence of the optical
properties of dust on the models of the two-photon effects. Finally, we have
subtracted uniformly an isotropic FUV emission of $\sim$300 CU (Bowyer, 1991)
although this value was obtained for the high Galactic latitude region. The
resulting FUV map is shown in Figure 6(a).
We employed a Monte Carlo Radiative Transfer technique, in which photons were
allowed to have multiple scatterings (Witt, 1977; Wood et al., 1999; Seon,
2009, 2012). At each scattering, after moving a mean free path distance
determined by the optical depth, the photons were randomly scattered into an
angle with a probability given by the Henyey-Greenstein function (Henyey &
Greenstein, 1941). We adopted the peeling-off technique to achieve simulation
efficiency, i.e., regardless of the actual scattering angle determined
randomly, the probability that the photon would be scattered in the direction
toward the Sun was calculated for each scattering, and the corresponding
fraction of the intensity was added up for the final image (Yusef-Zadeh et
al., 1984). We did not consider any wavelength changes upon scattering. The
free parameters to be determined by the simulation were the albedo, the
asymmetry factor, and the distribution of dust, while the initial photon
sources were chosen to be the bright stars in the TPA region. There are,
however, infinite numbers of possible choices for the three-dimensional
spatial distribution of dust under the given information of integrated
extinction along the lines of sight, which we adopted from the SFD dust
extinction map (Schlegel et al., 1998). Therefore, we assumed, for convenience
that the dust was distributed uniformly inside a slab whose thickness and
distance from the Sun were to be determined. The E(B-V) values of the SFD map
were converted to optical depths at 1565Å, a representative FUV wavelength of
the present study, with an assumed RV value of 3.1.
Table 1Distance to the front face and the thickness of the four selected clouds, obtained by treating each cloud individually. clouds | Taurus | Perseus | Auriga | California
---|---|---|---|---
Distance to the front face (pc) | 120 ${}^{+20}_{-80}$ | 70 ${}^{+30}_{-20}$ | 140 ${}^{+20}_{-90}$ | 170 ${}^{+10}_{-10}$
Thickness (pc) | 110 ${}^{+20}_{-50}$ | 230 ${}^{+30}_{-40}$ aaThe large thicknesses of the Perseus, California, and Auriga clouds may be due to multiple layers. Refer to Section 4 for their discussions. | 190 ${}^{+70}_{-20}$ aaThe large thicknesses of the Perseus, California, and Auriga clouds may be due to multiple layers. Refer to Section 4 for their discussions. | 320 ${}^{+50}_{-30}$ aaThe large thicknesses of the Perseus, California, and Auriga clouds may be due to multiple layers. Refer to Section 4 for their discussions.
Distance to the center (pc) | $\sim$175 | $\sim$185 | $\sim$235 | $\sim$330
The simulation box consisted of a 400 $\times$ 400 $\times$ 400 pixel, with
each pixel equivalent to 1 pc. The Sun was located at the center of the front
face of the simulation box. A total of 88307 stars was employed as the
radiation sources, consisting of 18212 TD-1 stars and 70095 Hipparcos stars.
The luminosities of the stars were calculated using the observed FUV stellar
fluxes for the TD-1 stars and Castelli’s stellar models for the Hipparcos
stars, based on the listed spectral types, with extinction effects taken into
account (Thompson et al., 1978; Perryman et al., 1997; Castelli et al., 2003).
As a first step, we assumed a single dust slab model for the entire TPA
region, which was to be improved at the second step by considering individual
clouds separately, using the values of the albedo and the asymmetry factor
obtained in the first step. Simulations were carried out for the following
parameter ranges: the albedo values varied from 0.20 to 0.60 with 0.01 steps,
the asymmetry factor varied from 0.0 to 0.80 with 0.01 steps, the distance to
the front face varied from 10 pc to 390 pc with 10 pc steps, and the thickness
of the slab varied from 10 pc to 390 pc with 10 pc steps. The best-fit
parameters were determined by comparing the simulation results with the
observed FUV map using the chi-square minimization.
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Figure 7.— A scatter plot of the simulated FUV intensity against the observed FUV intensity for the regions of the Taurus, Perseus, Auriga, and California clouds. Panel a) shows the result obtained using a single dust slab model and panel b) shows the result obtained with the individual clouds treated separately. The navy, sky blue, yellow, and orange colors correspond to the regions of the Taurus, Perseus, Auriga, and California clouds, respectively. |
---|---
|
Figure 8.— 1-$\sigma$ confidence contours of the distance to the front face
and the thickness, plotted for the four clouds identified in Figure 3(d). The
cross symbol indicates the best-fit values. For this simulation, fixed values
of the albedo (a = 0.42) and the asymmetry factor (g = 0.47) were used.
Figure 6(b) shows the simulated map with the best-fit parameters: 0.42 for the
albedo, 0.47 for the asymmetry factor, 50 pc for the distance to the front
face, and 220 pc for the thickness. The error ranges of the parameter values
are mainly due to the statistical fluctuations of the FIMS data with low
exposure time: the 1-$\sigma$ ranges of the albedo and the asymmetry factor
are 0.37 to 0.47 and 0.20 to 0.58, respectively. Nevertheless, the estimated
albedo of 0.42 is close to the theoretical model value of $\sim$0.40, while
the asymmetry factor is rather small compared to the theoretical expectation
of $\sim$0.65 (Weingartner & Draine, 2001). Jo et al. (2012) obtained 0.43 and
0.45 for the albedo and the asymmetry factor, respectively, in their
simulation for the Orion-Eridanus Superbubble. The simulated map shown in
Figure 6(b) looks similar to the observed FUV map of Figure 6(a), but we note
that the region around the California nebula is less bright and the Taurus
region is less dark in the simulated map than in the observed map. Indeed, the
intensity varies less over the whole TPA region in the simulated map than in
the observed map. The reason for the reduced variability could be the
assumption of a single slab model for the entire region. For example, the
reduction of intensity in the region around the California nebula was
compensated by an increase in intensity in the Taurus region as the reduced
chi-square was to be minimized with a single dust slab model. Hence, it should
be meaningful for the individual clouds to be treated separately with their
own distances and thicknesses. With the values of the albedo (a = 0.42) and
the asymmetry factor (g = 0.47) determined from the single slab model, we
carried out simulations separately for the individual clouds of Taurus,
Perseus, Auriga, and California. For the California cloud, we enlarged the
simulation box to 400 pc $\times$ 400 pc $\times$ 800 pc as the cloud was
found to extend beyond 400 pc. Relative to the single slab model, the pixel-
to-pixel scatter plot of the simulated FUV intensity against the observed FUV
intensity shows significant improvement, as can be seen in Figure 7. Table 1
shows the improved distances and thicknesses of the TPA clouds, and Figure 8
shows contour plots for 1-$\sigma$ error ranges. These best-fit values are in
good agreement with the previous estimations based on other methods, as will
be discussed in the following section. In the cases of the Taurus and Auriga
clouds, the distances to the front faces seem to have rather large error
ranges in the lower limit. Unfortunately, very few bright stars exist in front
of these clouds that can constrain well the distances to their front faces.
|
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Figure 9.— A scatter plot of the simulated FUV intensity against the observed
FUV intensity for the two subregions A and B of the Perseus cloud, marked in
Figure 3(d): (a) when the whole system of the Perseus cloud is treated as a
single layer, and (b) when the two sub-regions are treated individually. The
green and purple data points correspond to the regions of A, where NGC 1333 is
located, and B, where IC 348 is located, respectively.
## 4\. Discussion
We have estimated the locations of the four prominent clouds in the TPA region
by comparing the observed FUV images with the results of the Monte Carlo
simulations of dust scattering. For the Taurus cloud, we obtained a distance
to the front face and a thickness of the cloud of $120^{+20}_{-80}$ pc and
$110^{+20}_{-50}$ pc, respectively. There have been many reports on the
physical structure of the Taurus cloud. Elias (1978) estimated the mean
distance to the Taurus complex to be $\sim$135 pc, based on the distances to
the stars associated with the Taurus cloud. Ungerechts et al. (1987) suggested
that the cloud spreads between 100 pc and 200 pc, based on the results of star
counting, the distances of the reflection nebulae, and the color excess of the
field stars. Straizys (2001) also reported that extinction begins to increase
sharply at 130-180 pc. The present result agrees well with all these
observations. For the Perseus cloud, the present simulation estimates a
distance to the front face and a thickness of $70^{+30}_{-20}$ pc and
$230^{+30}_{-40}$ pc, respectively. The scatter plot of the simulated vs. the
observed FUV intensity in Figure 7(b) shows severely scattered data points for
the Perseus cloud, indicating that the model cloud does not adequately
describe its actual structure. In fact, the data points corresponding to the
observation are spread over a wide range of intensity, which we believe comes
from the assumption of a single layer for the cloud. Hence, we have divided
the Perseus cloud into two sub-regions and fit them separately. We name Per A
as the western part of the cloud and Per B as the eastern part, as shown in
Figure 3(d), and these include the two regions NGC 1333 and IC 348,
respectively. With the same fixed values of the albedo (a = 0.42) and the
asymmetry factor (g = 0.47) as used in the previous simulations, we find a
distance to the front face and a thickness of the Per A region of
$100^{+30}_{-30}$ pc and $180^{+30}_{-50}$ pc, respectively, and a distance to
the front face and a thickness of the Per B region of $\sim$320 pc and
$\sim$30 pc within ${}^{+10}_{-10}$ pc error range, respectively. The scatter
plot is now improved considerably, as shown in Figure 9. For the western
portion of the Perseus cloud, Cernis (1990) obtained $\sim$170 pc and
$\sim$230 pc for the distances of the two layers in this region, while Hirota
et al. (2008), using the parallax observation of the 22 GHz H2O maser,
directly measured the distance of NGC 1333 to be $\sim$240 pc. The rather
thick cloud obtained in the present simulation for Per A may reflect the
multi-layer nature of the cloud suggested by Cernis (1990). The distance to
the eastern portion of the Perseus cloud containing IC 348 was estimated to be
larger than 300 pc (Herbig & Jones, 1983). It is interesting that the
thickness was estimated to be rather small for Per B in the present
simulation. It was seen that the intensity of the scattered emission was very
high when the cloud was placed at a distance below 300 pc. We believe the
emission observed in the Per B region comes from backward scattering reflected
by the cloud itself.
Figure 10.— Distance estimations of the four clouds based on the present Monte
Carlo simulations of diffuse FUV emissions, compared with previous
observations.
For the Auriga cloud, the distance to its front face and its thickness are
estimated in the present study to be $140^{+20}_{-90}$ pc and
$190^{+70}_{-20}$ pc, respectively, and for the California cloud, they are
$170^{+10}_{-10}$ pc and $320^{+50}_{-30}$ pc, respectively. Eklof (1959)
suggested that the Auriga cloud has a two-layered structure near the eastern
end of the California cloud, with the layers located at $\sim$115-135 pc and
$\sim$270-380 pc, based on an extinction study. Ungerechts et al. (1987)
determined a distance to the California cloud of $\sim$350 pc while Lada et
al. (2009) suggested that the cloud is located at $\sim$450 pc from the Sun.
Straizys (2001) argued that the California cloud region is two-layered, with
the first layer at $\sim$160 pc and the second layer at $\sim$350 pc. With the
present single-slab model, it is impossible to reproduce such multi-layer
structures. Instead, we obtained single thick clouds, encompassing two layers
of the clouds. We note that the estimated value of $\sim$330 pc to the center
of the California cloud is larger than the distance to the Per OB2 association
($\sim$300 pc on average, Belikov et al. (2002)). Hence, the California cloud,
being located behind the Per OB2 association which is a candidate source for
the bright FUV emission seen in this region, does not block the emission from
the association. Figure 10 shows a summary of our distance estimations for the
four clouds, along with previous observations for the corresponding clouds.
We have plotted in Figure 11 the velocity-clipped CO emission maps for the
ranges given in the Local Standard of Rest (LSR) frame: (a) from -21.8 km/s to
+30.2 km/s, (b) from -10.7 km/s to -0.3 km/s, (c) from +0.3 km/s to +2.9 km/s,
and (d) from +4.2 km/s to +10.7 km/s. These maps are reproduced from the CO
sky survey data by Dame et al. (2001). While the differences in the LSR
velocities do not necessarily imply their relative distances, they may give
insightful clues to the physical associations of the corresponding clouds. For
example, the California cloud is seen in Figure 11(b) and no other clouds are
seen in that velocity range. In Figure 11(d), however, we can see indications
of the Taurus, Auriga, and Perseus clouds, but not of the California cloud.
Such a grouping may indicate that the California cloud is physically not
connected to other clouds, while the rest of the clouds system may be more or
less associated. Furthermore, parts of the Auriga and Perseus clouds are also
seen in Figure 11(c). Hence, the fact that these clouds are seen in a wide
range of the LSR velocities may be consistent with them having multi-layered
structures. Bally (2008) also noted that the large velocity gradient observed
in the Perseus cloud may be due to the nature of multi-components superimposed
along the line of sight.
Figure 11.— Velocity-clipped CO emission maps for the ranges (a) from -21.8
km/s to +30.2 km/s, (b) from -10.7 km/s to -0.3 km/s, (c) from +0.3 km/s to
+2.9 km/s, and (d) from +4.2 km/s to +10.7 km/s.
The error ranges of the estimated optical properties are a little large,
especially in the value of the asymmetry factor, where the reason was
attributed earlier to the statistical fluctuations of the FIMS data with low
exposure time. Nevertheless, the best fit values of the distances to the
respective front face and the thicknesses of the clouds do not seem to change
much when even the upper or lower limits of these error ranges are used for
fitting. For example, the changes are mostly $\sim$10 pc for the distances to
the front faces and $\sim$30 pc for the thicknesses of the clouds. The most
sensitive parameters for fitting are instead the locations of the field stars
as the photon sources relative to the clouds, not the optical properties of
dust. We have also made a number of assumptions in the present study to obtain
the optical properties of dust and the geometrical structures of the clouds in
the TPA region. First, we used a conversion formula for the FUV continuum
intensity of 60 CU for 1 Rayleigh intensity of H$\alpha$ to estimate the two-
photon effects, which holds only for the case of a hot diffuse ionized gas
whose temperature is $\sim$8000 K (Seon, 2011). To see the effect of the
choice of the conversion factor, we ran the same Monte Carlo simulation by but
instead subtracting the two-photon effect with a conversion factor of 30 CU
for 1 Rayleigh of H$\alpha$ in one case and even with no subtraction in
another case. The results were a = 0.43, g = 0.49 for the former case, and a =
0.44, g = 0.52 for the latter case. All these values are within the 1-sigma
range of the nominal case, a = $0.42^{+0.05}_{-0.05}$ and g =
$0.47^{+0.11}_{-0.27}$, implying that the choice of the conversion factor does
not alter the conclusions made in the present study. Next, we subtracted 300
CU as an isotropic background of the FUV emission, which was obtained for high
Galactic regions. When we subtracted 500 CU instead, only the thicknesses of
the clouds changed while the distances to the respective front face remained
the same. The thickness decreased by $\sim$10 % for the Auriga and the
California clouds as well as Per A, and $\sim$30 % for the Taurus cloud
region, which is darkest of all and thus most affected. The reason for the
decrease in thickness is that reduction in FUV intensity was accomplished by
increasing the shielding effect in the case of these clouds. On the other
hand, the thickness of Per B increased by $\sim$20 pc, reflecting the nature
of backward scattering in the case of Per B. Nevertheless, these new values
are all within the 1-sigma range of the results obtained for the nominal case.
## 5\. Conclusions
We have constructed a FUV continuum map of the TPA complex using FIMS and
GALEX data. The morphological features seen in the complex are well understood
in terms of dust scattering, as demonstrated by Monte Carlo simulations. The
following are the main findings of the present study:
1. 1.
The diffuse FUV emission seen in the complex originates mostly from scattering
of stellar photons by dust grains.
2. 2.
The FUV intensity of 1000 CU is observed at a very high extinction level,
which we regard as diffuse background and attribute to scattered FUV photons
located in the foreground to the thick clouds.
3. 3.
Molecular hydrogen fluorescent emission constitutes $\sim$10% of the total FUV
intensity throughout the region.
4. 4.
We have derived the following scattering parameters for this region based on
the Monte Carlo Radiative Transfer (MCRT) simulations: albedo (a) =
$0.42^{+0.05}_{-0.05}$ and asymmetry factor (g) = $0.47^{+0.11}_{-0.27}$.
These values agree well with those obtained previously for the Orion-Eridanus
Superbubble region as well as theoretical estimations.
5. 5.
We have estimated the distances to the four prominent clouds in this complex,
which are in good agreement with those estimated observationally using other
methods. The thickness of each cloud was also reasonably determined when it
has a simple structure such as the Taurus cloud. The present single slab
model, however, was not able to reproduce multi-layered clouds, but gives
rather thick clouds, instead.
6. 6.
The geometrical structures of the clouds are less sensitive to the exact
values of the albedo and the asymmetry factor. Instead, the locations of the
field stars relative to the clouds are the main factors that constrain the
distance and thickness of the clouds.
FIMS/SPEAR is a joint project of KAIST and KASI (Korea) and UC Berkeley (USA),
funded by the Korea MOST and NASA grant NAG5-5355. This research was supported
by Basic Science Research Program (2010-0023909) and National Space Laboratory
Program (2008-2003226) through the National Research Foundation of Korea (NRF)
funded by the Ministry of Education, Science and Technology.
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|
arxiv-papers
| 2013-01-23T08:48:35 |
2024-09-04T02:49:40.625493
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tae-Ho Lim, Kyoung-Wook Min, and Kwang-Il Seon",
"submitter": "Taeho Lim",
"url": "https://arxiv.org/abs/1301.5433"
}
|
1301.5620
|
# Stratified Graphene-Noble Metal Systems for Low-Loss Plasmonics Applications
L. Rast [email protected] Applied Chemicals and Materials Division,
National Institute of Standards and Technology, Boulder, CO 80305 T. J.
Sullivan Mathematics Institute, University of Warwick, Coventry, CV4 7AL,
United Kingdom V. K. Tewary Applied Chemicals and Materials Division,
National Institute of Standards and Technology, Boulder, CO 80305
###### Abstract
We propose a composite layered structure for tunable, low-loss plasmon
resonances, which consists of a noble-metal thin film coated in graphene and
supported on a hexagonal boron nitride (hBN) substrate. We calculate electron
energy loss spectra (EELS) for these structures, and numerically demonstrate
that bulk plasmon losses in noble-metal films can be significantly reduced,
and surface coupling enhanced, through the addition of a graphene coating and
the wide-bandgap hBN substrate. Silver films with a trilayer graphene coating
and hBN substrate demonstrated surface plasmon-dominant spectral profiles for
metallic layers as thick as $34\,$nm. A continued-fraction expression for the
effective dielectric function, based on a specular reflection model which
includes boundary interactions, is used to systematically demonstrate plasmon
peak tunability for a variety of configurations. Variations include substrate,
plasmonic metal, and individual layer thickness for each material. Mesoscale
calculation of EELS is performed with individual layer dielectric functions as
input to the effective dielectric function calculation, from which the loss
spectra are directly determined.
## I Introduction
Plasmonic devices hold promise for a host of metamaterials applications, due
to their ability to control light propagation on a subwavelength scale. Noble
metal nanomaterials are in some ways ideal components in such devices – they
possess tunable, large amplitude plasmon resonances, which can be excited at
optical wavelengths. Johnson and Christy (1972) Nevertheless, metallic
plasmonic devices at infrared and visible wavelengths present significant
challenges due to bulk plasmon losses. These large losses severely limit the
practicality of these materials for a wide variety of applications,
particularly in telecommunications and photovoltaics. West _et al._ (2010)
Bulk losses may be mitigated through the use of very thin noble-metal films;
however, fabrication of uniform thin metal films is experimentally
challenging. Thin films tend to form islands and often require adhesion layers
that alter the electronic structure of the device. Boragno _et al._ (2009);
Rha and Park (1997) Additionally, thin noble-metal films still suffer
significant resistive losses in the visible regime. Atwater and Polman (2010)
Composite materials designed to shift plasmon resonances to a low-loss regime,
while increasing coupling to surface plasmons and diminishing bulk resonances,
provide clear advantages from a fabrication standpoint. Systems that combine
the strong and tunable optical-wavelength plasmon resonances of noble metals
with materials possessing improved transport properties could mitigate
surface-plasmon losses in the visible regime.
Graphene is an ideal candidate for such a composite structure due to its
unparalleled carrier mobility. This allows for extremely enhanced and tunable
electromagnetic response spectra when doped with other plasmonic materials, or
fabricated as a component of a multilayer structure. Monolayer graphene has a
response spectrum dominated by absorption peaks at $\approx 4.5\,$eV and
$\approx 15\,$eV, the $\pi$ and $\pi$ \+ $\sigma$ surface plasmons. Graphene
plasmon resonance provides low losses in the frequency regime below the
optical phonon frequency of $0.2\,$eV, where large losses are typically
present for metallic plasmonic materials. Geim and Novoselov (2007)
Hexagonal boron nitride (hBN) provides several advantages as a substrate for
graphene-based plasmonic materials, as it is an ultra-flat wide bandgap
insulator with excellent thermal transport properties. Xue _et al._ (2011)
The flat morphology and uniformity provided by hBN are are desirable as key
factors in preserving the transport properties of graphene. Xue _et al._
(2011) Results of our numerical study clearly elucidate the advantages of the
hBN spectral response for low-loss plasmonics applications. We demonstrate in
this work that through the combined advantages of graphene and noble-metal
films, along with careful choice of substrate material and layer thickness,
multilayer systems can be tuned for low-loss surface plasmon resonance (SPR).
Metamaterials derive their exotic properties in part from advantageous
behaviors of their constituent materials, as well as collective behaviors that
emerge due to interactions within the system. The complex task of designing
metamaterials tailored for applications as diverse as photovoltaics,
biosensing, and microscopy gives rise to a need to develop efficient methods
for predicting metamaterial properties. These methods must realistically treat
both the individual material properties and the interactions among the
constituent materials. We detail and employ such a method for the calculation
of electron energy loss spectra (EELS) of multilayer structures consisting of
graphene layers on noble-metal (silver, gold, and copper) films with silicon
and hBN substrates.
The effective dielectric function is based on a specular reflection model,
first derived by Lambin et al. Lambin _et al._ (1985), and takes into account
the boundary conditions across each layer in the stratified structure. The use
of the efficient continued-fraction expression along with pre-prepared
libraries of dielectric functions for the individual materials opens up the
possibility of multilayer graphene composites by design.
## II EELS Calculation Details
### II.1 General Procedure
Figure 1: Multilayer structure: EELS are calculated for a structure consisting
of graphene top layer(s), noble-metal middle layer, and semiconductor
substrate.
EELS are calculated for a variety of multilayer sandwich structures as
depicted in Fig. 1. In general, the sandwich structure consists of a noble-
metal middle layer, a graphene coating of an integer number of layers, and a
semiconducting substrate. Graphene layer numbers vary from 1 to 20. Noble
metals employed include silver, gold, and copper, and are varied in thickness.
The effect of two different substrates is considered: (1) a more traditional
substrate, Si with a $4\,$nm top layer of SiO2 (Fig. 1 (a)), and (2) hBN (Fig.
1 (b)). $4\,$nm is a typical thickness for the SiO2 layer, which arises from
the thermal processing of the Si substrate. Seah and White (2002) The Si layer
is assumed to be semi-infinite. In the case of the hBN substrate, an integer
number of hBN layers is considered, with a semi-infinite vacuum layer below
the hBN layer. The single-layer thickness for both graphene and hBN are taken
to be $3.4\,$Å, the natural inter-layer spacing for graphite. Berashevich and
Chakraborty (2011) In all cases, a semi-infinite vacuum layer exists above the
top ($z=0$) layer.
Individual complex dielectric functions are obtained for each layer. We
calculate the values through ab-initio methods for graphene and hBN. Empirical
values from the literature are used for the metal and silicon substrate
layers. These values are then stored for use as input to a continued-fraction
algorithm, which yields the effective dielectric function. This algorithm is
outlined in Section II.2.
### II.2 The Effective Dielectric Function
The effective dielectric function, $\xi(\omega,k,z)$, of the stratified
structure in Fig. 1 is that of Lambin et al. Lambin _et al._ (1985) The
expression for $\xi$ was derived from EELS theory in a reflection geometry.
The expression has been shown to be applicable to both phonons Lambin _et
al._ (1985) and polaritons Dereux _et al._ (1988) in stratified structures
with histogram-like dielectric functions (continuous within each layer) and
interacting interfaces. Though Lambin’s initial paper containing the
derivation focused on semiconducting materials, the expression and the
formalism from which it is derived also apply to surface plasmon structure of
alternating of metal-insulator layers. Dereux _et al._ (1988); Economou
(1969) It is worth noting that the specular reflection model agrees well with
the spectroscopic predictions of the Bloch hydrodynamic model in the small
wave vector regime considered in our work. Ritchie and Marusak (1966)
The $z$ coordinate is in the direction perpendicular to the free surface of
the sample, extending from the $z=0$ surface to $-\infty$. $\mathbf{k}$
denotes the surface excitation (plasmon or phonon) wave vector and $\omega$ is
the frequency of excitation.
$\xi(\mathbf{k},\omega,z)=\frac{i\mathbf{D}(\mathbf{k},\omega,z)\cdot\mathbf{n}}{\mathbf{E}(\mathbf{k},\omega,z)\cdot\mathbf{k}/k},$
(1)
where
$\mathbf{D}(\mathbf{k},\omega,z)=\epsilon(\omega,z)\mathbf{E}(\mathbf{k},\omega,z)$,
and $\epsilon(\omega,z$) is the long wavelength dielectric function (tensor)
of the material at $z$. $\xi$ remains continuous even in the case of sharp
interfaces parallel to the $x$-$y$ directions below the surface (as is the
case in our multilayer system). This is due to the interface boundary
conditions: continuity of $D_{\perp}$ and $E_{\parallel}$.
The effective dielectric function $\xi_{0}(k,\omega)$ (Eq. 3) is a solution to
the Riccati equation (Eq. 2), in the long-wavelength approximation $k\approx
0$, at the $z=0$ surface. Lambin _et al._ (1985) We fix $k$ as $k=0.05\,$Å-1
for both the ab-initio calculations and the composite calculation. Eq. 2 was
derived for heterogeneous materials made of a succession of layers (with
homogeneous dielectric functions within each layer), the layers having
parallel interfaces. $\epsilon(z)$ are complex functions, with positive
imaginary parts at $z=0$. Lambin _et al._ (1985)
$\displaystyle\frac{1}{k}\frac{\mathrm{d}\xi(z)}{\mathrm{d}z}+\frac{\xi^{2}(z)}{\epsilon(z)}=\epsilon(z)$
(2)
$\xi_{0}=a_{1}-\frac{b_{1}^{2}}{a_{1}+a_{2}-\frac{b_{2}^{2}}{a_{2}+a_{3}-\frac{b_{3}^{2}}{a_{3}+a_{4}-\cdots}}}$
(3)
where
$\displaystyle a_{i}=\epsilon_{i}\coth(kd_{i})$ (4)
and
$\displaystyle b_{i}=\epsilon_{i}/\sinh(kd_{i}).$ (5)
Once individual dielectric functions are obtained, this procedure allows for
the performance of mesoscale EELS calculations of a wide variety of
structures. Layer thickness and materials can easily be substituted in the
calculation, with each EELS calculation running in a fraction of a second
(nearly independent of the spectral range). EELS are calculated directly from
the effective dielectric function as
$\mathrm{EELS}=\mathrm{Im}\left[\frac{-1}{\xi(\omega,k)+1}\right].$ (6)
Inspection of Eq. 3 reveals that for $\mathrm{Im}[\epsilon_{i}]>0$,
$\mathrm{Im}[\xi_{0}]>0$. The EELS spectra given by Eq. 6 are then positive.
### II.3 Noble-Metal Dielectric Functions
The copper, silver, and gold complex dielectric functions are empirical values
by Johnson and Christy Johnson and Christy (1972) obtained by reflection and
transmission spectroscopy on vacuum-evaporated films at room temperature.
Film-thickness in the Johnson and Christy study ranged from $185\,$Å–
$500\,$Å. Dielectric functions in the film-thickness range $250\,$Å – $500\,$Å
did not vary significantly. In our work, the intermediate value $340\,$Å was
chosen to represent bulk mode dominant (yet still nanoscale) metallic thin
films.
### II.4 SiO2 and Si Dielectric Constants
Relative static permittivities of 3.9 and 11.68 are used for the SiO2 and Si
dielectric constants, respectively. These are reasonable and widely-used
values from the literature. Murarka (2003); Yi (2012)
### II.5 Graphene and hBN Individual Layer Dielectric Functions
Complex dielectric functions for graphene and hBN are displayed in Fig. 2 (a)
and Fig. 2 (b), respectively. These ab-initio calculations use the time-
dependent density functional theory in the local density approximation (LDA),
and are implemented in the Python code GPAW, a real-space electronic structure
code using the projector augmented wave method. 111Certain commercial
equipment, instruments, or materials are identified in this paper in order to
specify the experimental procedure adequately. Such identification is not
intended to imply recommendation or endorsement by the National Institute of
Standards and Technology, nor is it intended to imply that the materials or
equipment identified are necessarily the best available for the purpose.,
Mortensen _et al._ (2005); Enkovaara _et al._ (2010); Walter _et al._
(2008); Yan _et al._ (2011) Both graphene and hBN dielectric functions are
calculated in the armchair configuration for a momentum transfer value of
$0.05\,$Å-1, along the $\bar{\Gamma}$-$\bar{M}$ direction of the surface
Brillouin zone. The armchair configuration and value of $k$ are selected for
comparison of EELS with existing data in the literature obtained via density
functional theory (DFT) methods. The $k$-point sampling with $30\times
30\times 1$ Monkhorst–Pack grid was chosen for the band-structure and EELS
calculations for both graphene and hBN.
Our model utilizes dielectric functions due to surface parallel excitations
only, as the effective dielectric function is derived in a specular reflection
geometry. This is a reasonable approximation as out-of-plane excitations are
minimal in graphene at energies less than $\approx 10\,$eV, and extreme UV
radiation is outside of the regime of interest for this study. Gass _et al._
(2008) A lattice constant of $2.46\,$Å is used for graphene, and $2.50\,$Å for
hBN, as hBN is nearly isomorphic to the graphene, except for the slightly
larger lattice constant. Song _et al._ (2010)
The dielectric function we have obtained for graphene (see Fig. 2 (a)) is
nearly identical to those of Yan et al. Yan _et al._ (2011), where the
authors used the projector augmented wave methodology implemented in GPAW in
terms of linear combinations of atomic orbitals, with a momentum transfer of
$0.046\,$Å-1, along the $\bar{\Gamma}$-$\bar{M}$ direction of the surface
Brillouin zone. Both our results and that of Yan et al. display collective
peaks for free-standing single-layer graphene at $\approx 5\,$ eV and $\approx
15\,$ eV. These values also agree with experimental EELS results for single-
layer graphene plasmons (with in-plane excitation), for example Eberlein et
al. Eberlein _et al._ (2008) find the $\pi$ plasmon at 4.7 eV and $\pi$ \+
$\sigma$ at 14.6 eV.
A low-energy peak (below 1 eV) is also apparent in Fig. 2 (a). This feature,
which has been observed in ab-intio calculations by others Trevisanutto _et
al._ (2008); Yan _et al._ (2011), corresponds to a broad shoulder in the
EELS, and is due to the low-energy $\pi$ $\rightarrow$ $\pi^{\star}$ single-
particle excitation. Low energy graphene features, including intraband
transitions are generally quite dependent upon the value of momentum transfer.
Hwang and Das Sarma (2007) These features are outside the energy regime of
interest for the surface and bulk plasmon peaks in this paper, work in which
very low-energy excitations are a focus should not neglect momentum transfer
dependence. In the context of these calculations, increasing momentum transfer
values lead to a shift towards higher energies of the $\pi$ plasmons for
intrinsic graphene. Yan _et al._ (2011); Gao (2011) Others, for example Gao
et al. Gao (2011) have found (via the time-dependent local density
approximation) linear dispersion for the $\pi$ plasmon (the key feature in our
energy regime of interest) in single-layer graphene for both
$\bar{\Gamma}$-$\bar{M}$ and $\bar{\Gamma}$-$\bar{K}$ directions. The
calculations of Gao et al. were found to agree well with experiment.
Our hBN dielectric function (see Fig. 2 (b)) compares well with that of Yan et
al. Yan _et al._ (2012), who obtained their results in the long-wavelength
limit, using the LDA-adiabatic local density approximation method. Both our
results for hBN spectra and that of Yan et al. display a broad absorbtion peak
with onset at $\approx 4.5\,$ eV and maximum at $\approx 5.75\,$ eV, which is
in good agreement with previous literature. Wirtz _et al._ (2006) The
dielectric function obtained for hBN is also similar to the experimental
results of Tarrio and Schnatterly Tarrio and Schnatterly (1989), where the
authors measured a peak in the imaginary part of the dielectric function at
$\approx 6\,$ eV. Differences between the complex dielectric function used in
this study and that of Tarrio and Schnatterly (1989) are attributable to the
larger momentum transfer value of $0.13\,$Å-1 in the Tarrio and Schnatterly
study.
Figure 2: Complex relative dielectric function $\epsilon(\omega)$ for
graphene (a) and hBN (b). Real and imaginary parts
($\epsilon^{\prime}(\omega)$ and $\epsilon^{\prime\prime}(\omega)$) are
represented by solid and dotted lines, respectively.
## III RESULTS
### III.1 Decreasing metallic film thickness and the begrenzung effect
Figure 3: EELS: (a), (b), and (c) demonstrate the effect of differing
thickness for the Ag, Au, and noble-metal layer, respectively, with a SiO2/Si
substrate, and without a graphene coating. Film thicknesses are $34\,$nm
(solid line), $20\,$nm (long dashes), $10\,$nm (short dashes), and $4\,$nm
(dotted line).
Figures 3 (a)–(c) (respectively, Ag, Au, and Cu on SiO2/Si, without graphene
top coating) demonstrate the effect of decreasing noble metal layer thickness.
These data should serve as a basis for comparison with Fig. 4 (a)–(c), which
are discussed in detail in the next section.
As the noble metal film thickness is reduced in Fig. 3 (a), the so called
_begrenzung effect_ is apparent. An increase in the surface-to-volume ratio in
the metal causes enhanced coupling to the surface resonance and diminished
coupling to the bulk modes. Ritchie (1957); Osma and Garcia de Abajo (1997)
In the case of a thin metallic slab, empirical models have been quite
thoroughly explored. Upon the introduction of a boundary to an infinite
metallic slab, a negative (begrenzung) peak is introduced at the same energy
as the bulk peak, and a trailing surface peak appears. Ritchie (1957) The
surface peak becomes more pronounced with decreasing thickness, as does the
negative begrenzung peak, decreasing the net bulk-plasmon amplitude. A sharp
transition can be observed, between $34\,$nm and $10\,$nm Ag film thickness,
in which surface modes become dominant. This transition can also be observed
for Au and Cu films, between $34\,$nm and $20\,$nm. This is consistent with
observations in the well-validated and widely-used empirical data by Johnson
and Christy. Johnson and Christy (1972)
### III.2 SPR Enhancement due to the graphene coating
#### III.2.1 SiO2/Si Substrate
Figure 4: EELS for varying numbers of layers for the graphene film: (a)
$34\,$nm Ag layer and SiO2/Si substrate, (b) the surface peak in (a), and (c)
$34\,$nm Ag layer and monolayer hBN substrate. In both cases the graphene
layer numbers are 0 (solid line), 1 (long dashes), 3 (intermediate-length
dashes), 10 (short dashes), and 20 (dotted line).
Figures 4 (a) and (b) demonstrate the effect of the addition of graphene
coatings of varyious thicknesses to the Ag surface, in the case of a SiO2/Si
substrate. For up to three layers of graphene, the surface peak remains
virtually unchanged. However, at three layers, the bulk peak amplitude is
reduced by approximately $50\%$. This effect is attributed to the introduction
of a thin boundary layer, decreasing the bulk resonance amplitude. Imposition
of a boundary leads to simultaneous diminished coupling to bulk modes and
enhanced coupling to surface modes, physically similar to the aforementioned
begrenzung effect. Ritchie (1957); Osma and Garcia de Abajo (1997); Egerton
(2011) At 20 layers, as we would expect, the bulk peak broadens significantly
(indicating increased losses) and the system has spectral properties
(broadening and peak position) resembling those of graphite on silver. Peak
positions calculated for the 20-layer graphene coating ($\approx 2.6\,$eV and
$\approx 3.5\,$eV) are very close to those measured in EELS of thin films of
Ag nanoparticles evaporated on graphitic surfaces, for example $2.2\,$eV and
$3.4\,$eV. Palmer _et al._ (2002) Differences in peak position (particularly
for the lower-energy peak) are attributable to differences in morphology
between the Ag slab in our calculations and the film of Ag nanoparticles in
the experimental samples.
#### III.2.2 hBN Substrate
Figure 4 (c) demonstrates the effect of the addition of graphene coatings of
varying thickness to the silver surface, in the case of a hBN substrate. The
use of the hBN substrate dramatically enhances the SPR peak, as well as
shifting the surface peak to an energy very close to that of the bulk peak,
even the case where no graphene coating is used. This blue-shift of the SPR is
due to the much smaller real part of the hBN dielectric function in comparison
to Si. Decreasing substrate dielectric function is known to dramatically blue-
shift SPR. G. Xu and Yoshimura (2003)
Addition of a single-layer graphene coating serves to further diminish the
bulk peak without significant degradation of the surface peak. For up to 3
layers of graphene, the surface peak does not broaden significantly. As in the
case of the SiO2/Si substrate, broadening of the surface peak for more than 3
layers of graphene corresponds to increased losses. Indeed, at 10 graphene
layers the response is what one would expect from a graphite coating. A
similar transition in plasmonic behavior at 10 layers has been observed
experimentally for multilayer graphene EELS. Eberlein _et al._ (2008)
#### III.2.3 Ideal configurations for Ag: enhancement of surface modes
Figure 5: EELS:(a) Comparison of the use of a SiO2/Si substrate (solid line),
a mono-, tri-, 10-, and 20-layer hBN substrate (long dashes, intermediate-
length dashes, short dashes, and a dotted line, respectively). No graphene
coating is used and the Ag film is $34\,$nm thick. (b) Close-up of EELS peaks
for the hBN substrates. The suspended sample (“vacuum substrate”) is
represented by the solid line in this case. Mono-, tri-, 10-, and 20-layer hBN
substrates are represented by long dashes, intermediate-length dashes, short
dashes, and a dotted line, respectively.
Figure 5 depicts EELS for a $34\,$nm Ag film (with no graphene coating) on
various substrates: SiO2/Si, mono-layer, tri-layer, and 10-layer hBN, and
semi-infinite vacuum (as would occur for a suspended sample). Introduction of
the high band-gap hBN substrate dramatically enhances surface modes as
compared to SiO2/Si. The result is striking — the sample with a monolayer hBN
substrate has a spectral profile nearly identical to that of a suspended
sample. The EELS remain very nearly identical to the suspended case for up to
10 layers. This is in accordance with experimental results that have found the
electron mobility of graphene on hBN to be nearly that of suspended graphene.
Xue _et al._ (2011) The similarity of the hBN substrate to the vacuum is
expected due to its ultra-wide bandgap, and in the case of our model partly
due to the atomic layer thickness. Upon inspection of Fig. 2 (b), it is clear
that both real and imaginary parts of hBN’s dielectric function are small;
therefore at a thickness of $3.4\,$Å, the properties of this substrate
approach those of the vacuum. The hBN substrate appears to dramatically
enhance coupling to the surface plasmon resonance when compared to the SiO2/Si
substrate, as well as reducing the peak strength of the lossy bulk plasmon.
Figure 6: (a)EELS: trilayer graphene on Ag layers (of varying thickness) with
a mono-layer hBN substrate. (b) Free-standing Ag films of various thicknesses.
In both (a) and (b) Ag layer thicknesses are $50\,$nm (solid line), $34\,$nm
(long dashes), $20\,$nm (intermediate-length dashes), and $10\,$nm (short
dashes), and $4\,$nm (dotted line).
Fig. 6 (a) demonstrates the combined effects of graphene coating and hBN
substrate for various Ag film thicknesses. Comparison of Fig. 6 (a) with Fig.
3 (a) reveals that the introduction of a tri-layer graphene coating and hBN
substrate to the Ag film results in a surface plasmon intensity and relative
surface to bulk intensity similar to a significantly thinner Ag film on the
SiO2/Si substrate. For example, the surface plasmon intensity and relative
surface to bulk intensity of the $34\,$nm Ag case in Fig. 6 (a) is comparable
to the $20\,$nm case in Fig. 3 (a). Additionally, the surface plasmon
intensity and relative surface to bulk intensity of the $20\,$nm Ag case in
Fig. 6 (a) is comparable to the $10\,$nm case in Fig. 3(a).
Comparison of Fig. 6 (a) with Fig. 6 (b) (which shows spectra for free-
standing Ag of various thicknesses) further elucidates the effect of a few
graphene layers. For thicker Ag films, a few-layer graphene coating diminishes
the bulk peak without significant broadening of the SPR. However, for very
thin Ag films, of 4nm, for example, the effect of diminishing the bulk peak is
negligible and the addition of the graphene coating of three layers only
slightly broadens the SPR. This is in keeping with the idea that for bulk Ag,
the addition of a graphene layer enhances surface coupling, through the
begrenzung effect. This is further demonstrated by the lack of significant
broadening of the SPR peak for a single-layer graphene coating (see Fig. 4
(c)).
### III.3 Au and Cu Noble-Metal Film Composites
#### III.3.1 Spectral changes of Au and Cu films due to the graphene coating
One may wish to employ a noble metal other than Ag in multilayer structures
such as depicted in this paper. Cu has the advantage of lower cost and is
therefore attractive for industrial applications. Both Cu and Au may also be
of interest due to inherent surface plasmons that occur at longer wavelengths
than those of Ag. In this section we demonstrate the effect of graphene
coatings and the hBN substrate on Au and Cu films. For reference, Fig. 3 (b)
and (c) demonstrate the effect of reduction of noble-metal film thickness for
Au and Cu films (respectively) on a SiO2/Si substrate. The begrenzung effect
is again increasingly apparent in both Au and Cu film plasmonic response as
film thickness decreases. At $20\,$nm, both Au and Cu display strong SPR and
broad absorption for higher energies, rather than a bulk plasmon peak.
Inspection of Fig. 7 (a) and (b) reveals that the introduction of a graphene
coating to an Au or Cu surface does measurably strengthen surface modes while
reducing bulk plasmon intensity, although the effect is not as dramatic as in
the Ag case (see Fig. 4). Coatings as thick as 20 layers further enhance and
do not significantly broaden the surface peak. A jump in broad absorption
(corresponding to increased losses) at higher energies is apparent in the case
of both Au and Cu for more than three graphene layers.
Figure 7: EELS: (a) The effect of various numbers of layers for the graphene
film, in the case of a $34\,$nm Au layer and SiO2/Si substrate, and (b) The
effect of varyious numbers of layers for the graphene film, in the case of a
$34\,$nm Cu layer and SiO2/Si substrate. Graphene layer numbers are 0 (solid
line), 1 (long dashes), 3 (intermediate-length dashes), 10 (short dashes), and
20 (dotted line).
#### III.3.2 Metallic films with hBN substrate: noble-metal comparison
Figure 8: EELS: $34\,$nm Au (a) and Cu (b) films, with monolayer graphene
coatings and various substrates including SiO2/Si (solid line), hBN monolayer
(dashes), and vacuum (dotted line) substrates.
Figure 8 (a) and (b) are the EELS for Au and Cu films of thickness $34\,$nm
(respectively), with monolayer graphene coating, comparing the SiO2/Si
substrate, hBN substrate, and suspended samples. Surface peaks at $\approx
2\,$eV for both Au and Cu are surface peaks due to change in dielectric
function across the metal/SiO2/Si substrate interface. The addition of the hBN
layer in these cases appears to sharpen and enhance Au and Cu inherent
plasmons. Notably, in the Au and Cu cases, the hBN substrate appears to
produce a similar spectral profile to the suspended “vacuum substrate”, as in
the case of the Ag film. Rigorous optimization of both the surface coating and
substrate for both Au and Cu films is the subject of further investigation.
## IV DISCUSSION
In this study we systematically investigated the effect of the use of
different plasmonic materials, different semiconducting substrates, and
different layer thicknesses in a multilayer graphene-based plasmonic composite
structure. For mono-, bi-, and tri-layer graphene, in the case of $34\,$nm Ag
layers (where bulk modes would dominate in the absence of a graphene coating),
bulk plasmon modes are significantly diminished, while maintaining the
strength of surface modes.
Reduced plasmon losses for graphene coated plasmonic metals on hBN substrates
likely originates within the context of our mesoscopic model from two primary
physical effects: (1) The addition of a graphene boundary layer on the
metallic surface reduces coupling of excitations to bulk plasmons through the
begrenzung effect. The origin of the begrenzung effect is a reduction of the
degrees of freedom for excitations, and thus further surface confinement comes
at the expense of bulk oscillations, leading to reduced losses. (2) The
strongly insulating hBN substrate diminishes bulk losses and enhances surface
confinement particularly through a reduction in scattering.
Traditional substrates such as Si are known to degrade the electron transport
properties of graphene as compared to suspended samples. Xue _et al._ (2011)
Reduced transport properties are due to various scattering mechanisms. There
is strong evidence that scattering in these systems is due in large part to
various substrate interactions including interfacial phonons (which are taken
into account in our work), surface charge traps, and substrate stabilized
ripples. Bolotin _et al._ (2008)
It is therefore reasonable to mitigate scattering in graphene-based systems
for plasmonics (and devices for plasmonics in general) by suspending samples,
or through the use of alternative substrates. Bolotin _et al._ (2008) In this
work, for Ag films, the use of a hBN substrate (rather than the more
traditional silicon) is found to shift plasmonic coupling towards surface
modes, both diminishing bulk losses and enhancing the SPR peak amplitude. This
effect is particularly dramatic in the case of the $34\,$nm Ag film (including
those with no graphene top-coating), as an Ag film at this thickness on an
SiO2/Si substrate produces EELS that are extremely dominated by the bulk
resonance. Johnson and Christy (1972) This is interesting in light of recent
experimental results, showing electron mobility of graphene on hBN to be
similar to that of suspended graphene. Xue _et al._ (2011) We expect SPR
enhancement for other wide-bandgap substrates due to enhanced surface plasmon
field confinement. Further comparison of alternate wide-bandgap substrate
materials, such as SiC, are the subject of future work.
As expected, bulk modes were quenched for $4\,$nm Au and Cu films. For Ag,
bulk modes very nearly vanished at this thickness, and are nearly undetectable
when coupled with the hBN substrate. However, the morphology of very thin
metallic films deposited on graphene is difficult to control, often forming
island-like structures of various sizes. Boragno _et al._ (2009) This is also
the case for graphene deposited on a metallic substrate, as agglomeration
below a critical thickness is a general property of thin films. Rha and Park
(1997) Results of this study indicate that Ag films as thick as $34\,$nm, when
coated with a graphene film and placed on a wide-bandgap substrate such as
hBN, may also be employed for low-loss plasmon resonance applications. An hBN
substrate, due to its ultra-flat morphology, would also lend additional
uniformity to the structure.
The mesoscopic model used in these calculations has several limitations that
are worth discussing. Results of this study are valid in the long-wavelength
limit for which the continued fraction expression by Lambin et al. was
derived. Additionally, coupling between layers is classical (via boundary
conditions), and as a result inter-layer hopping is neglected. This tunnelling
has been determined by angle-resolved photoemission spectroscopy to be
important for graphene band-structure, leading to $\pi$ band splitting, which
increases with layer number ($\approx 0.7\,$eV at 4 layers). Ohta _et al._
(2007) Multilayer graphene excitation spectra have been calculated by Ohta et
al. with the inclusion of inter-layer hopping in the kinetic Hamiltonian.
Tunneling is found to be primarily important in accurately producing the low
energy region of the excitation spectrum ($\omega\sim v_{F}k$), where $v_{F}$
is the Fermi velocity). Yuan _et al._ (2011) Though tunnelling is not as
critical in the regime of interest for this work, calculations for graphene-
based multilayer plasmonic systems geared towards lower-energy applications
should take care to include inter-layer hopping.
More general EELS, accurate for a wider range of $k$ values and angles of
incidence will require first principles calculations for the entire structure,
and incorporate higher level quasiparticle interactions (including the effect
of excitons). Using guidance from the results of this study, future work
consisting of 1-3 layers of graphene on silver, with the hBN substrate are the
subject of future work. The band-gap for hBN is underestimated by about 33% in
the LDA Giovannetti _et al._ (2007) — the effect of hBN on surface
confinement of plasmons may be even more dramatic than is demonstrated in this
work. A quasiparticle GW correction to the LDA calculation will be employed,
as it has been demonstrated to bring the hBN bandgap into close agreement with
experimental results. Arnaud _et al._ (2006)
As the individual layers in these structures are nearly isomorphic, rather
than exactly isomorphic, future work will incorporate the effect of lattice
strain on the optical properties of the composite. Strain engineering is
expected to provide a further means of plasmon resonance tunability.
Sciammarella _et al._ (2010) Additionally the effect of configurations other
than the flat-armchair morphology for graphene is the subject of ongoing work.
The effect of rippling and defects in graphene-based multilayer structures is
of particular interest, as is a rigorous quantification of the uncertainties
involved in the modelling and manufacturing stages, and optimal design against
those uncertainties.
###### Acknowledgements.
The authors would like to thank Katie Rice, Ann Chiaramonti Debay, and Alex
Smolyanitsky for helpful discussions. This research was performed while the
first author held a National Research Council Research Associateship Award at
the National Institute of Standards and Technology. This work represents an
official contribution of the National Institute of Standards and Technology
and is not subject to copyright in the USA.
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|
arxiv-papers
| 2013-01-23T20:13:28 |
2024-09-04T02:49:40.645097
|
{
"license": "Public Domain",
"authors": "L. Rast, T. J. Sullivan, V. K. Tewary",
"submitter": "Lauren Rast",
"url": "https://arxiv.org/abs/1301.5620"
}
|
1301.5627
|
# Exoplanet Clouds Chapter v2.1.docx
Clouds and Hazes in Exoplanet
Atmospheres
Mark S. Marley
NASA Ames Research Center
Andrew S. Ackerman
NASA Goddard Institute for Space Studies
Jeffrey N. Cuzzi
NASA Ames Research Center
Daniel Kitzmann
Zentrum für Astronomie und Astrophysik,
Technische Universität Berlin
## Abstract
Clouds and hazes are commonplace in the atmospheres of solar system planets
and are likely ubiquitous in the atmospheres of extrasolar planets as well.
Clouds affect every aspect of a planetary atmosphere, from the transport of
radiation, to atmospheric chemistry, to dynamics and they influence - if not
control - aspects such as surface temperature and habitability. In this review
we aim to provide an introduction to the role and properties of clouds in
exoplanetary atmospheres. We consider the role clouds play in influencing the
spectra of planets as well as their habitability and detectability. We briefly
summarize how clouds are treated in terrestrial climate models and consider
the far simpler approaches that have been taken so far to model exoplanet
clouds, the evidence for which we also review. Since clouds play a major role
in the atmospheres of certain classes of brown dwarfs we briefly discuss brown
dwarf cloud modeling as well. We also review how the scattering and extinction
efficiencies of cloud particles may be approximated in certain limiting cases
of small and large particles in order to facilitate physical understanding.
Since clouds play such important roles in planetary atmospheres, cloud
modeling may well prove to be the limiting factor in our ability to interpret
future observations of extrasolar planets.
## 1 Introduction
Clouds and hazes are found in every substantial solar system atmosphere and
are likely ubiquitous in extrasolar planetary atmospheres as well. They
provide sinks for volatile compounds and influence both the deposition of
incident flux and the propagation of emitted thermal radiation. Consequently
they affect the atmospheric thermal profile, the global climate, the spectra
of scattered and emitted radiation, and the detectability by direct imaging of
a planet. As other chapters in this book attest, clouds and hazes are a
complex and deep subject. In this chapter we will broadly discuss the roles of
clouds and hazes as they relate to the study of exoplanet atmospheres. In
particular we will focus on the challenge of exoplanet cloud modelling and
discuss the impact of condensates on planetary climates.
The terms “clouds” and “hazes” are sometimes used interchangeably. Here we use
the term “cloud” to refer to condensates that grow from an atmospheric
constituent when the partial pressure of the vapor exceeds its saturation
vapor pressure. Such supersaturation is typically produced by atmospheric
cooling, and cloud particles will generally evaporate or sublimate in
unsaturated conditions. A general framework for such clouds in planetary
atmospheres is provided by Sánchez-Lavega et al. (2004). By “haze” we refer to
condensates of vapor produced by photochemistry or other non-equilibrium
chemical processes. This usage is quite different from that of the terrestrial
water cloud microphysics literature where the distinction depends on water
droplet size and atmospheric conditions.
Because exoplanetary atmospheres can plausibly span such a wide range of
compositions as well as temperature and pressure conditions, a large number of
species may form clouds. Depending on conditions, clouds in a solar
composition atmosphere can include exotic refractory species such as Al2O3,
CaTiO3, Mg2SiO4 and Fe at high temperature and Na2S, MnS, and of course H2O at
lower temperatures. Many other species condense as well including CO2 in cold,
Mars-like atmospheres and NH3 in the atmospheres of cool giants, like Jupiter
and Saturn. In Earth-like atmospheres water clouds are likely important
although Venus-like conditions and sulfuric-acid or other clouds are
possibilities as well. Depending on atmospheric temperature, pressure, and
composition the range of possibilities is very large. Furthermore not all
clouds condense directly from the gas to a solid or liquid phase as the same
species. For example in the atmosphere of a gas giant exoplanet solid MnS
cloud particles are expected to form around 1400 K from the net reaction H2S +
Mn$\rightarrow$MnS(s) + H2 (Visscher et al. 2006).
Clouds strongly interact with incident and emitted radiative fluxes. The
clouds of Earth and Venus increase the planetary Bond albedo (the fraction of
all incident flux that is scattered back to space) and consequently decrease
the equilibrium temperature. Clouds can also “trap” infrared radiation and
heat the atmosphere. Hazes, in contrast, because of their usually smaller
particle sizes can scatter incident light away from a planet but not strongly
affect emergent thermal radiation, and thus predominantly result in a net
cooling. The hazes of Titan play such an “anti-greenhouse effect” role in the
energy balance of the atmosphere. For these reasons global atmospheric models
of exoplanets, including those aiming to define the habitable zone given
various assumptions, must consider the effects of clouds. However clouds are
just one ingredient in such planetary atmosphere models. Bulk atmospheric
composition, incident flux, gravity, chemistry, molecular and atomic
opacities, and more must all be integrated along with the effect of clouds in
order to construct realistic models. Introductory reviews by Burrows & Orton
(2010) and Seager & Deming (2010) cover the important fundamentals atmospheric
modeling and place cloud models in their broader context.
In the remainder of this chapter we discuss the importance of clouds to
exoplanet atmospheres, particularly considering their impact on habitability,
discuss cloud modeling in general and the types of models developed for
exoplanet studies, and finally briefly review observations of exoplanet
clouds. Because clouds have played such a large role in efforts to understand
the atmospheres of brown dwarfs, we also briefly review the findings of this
field. We conclude with an overview of how the Mie opacity of particles
behaves in various limits and consider the case of fluffy particles.
## 2 Importance of Clouds and Hazes in Exoplanet Atmospheres
### 2.1 Albedo, Detectability, and Characterization
Before discussing the role on clouds on the spectra of extrasolar planets it
is worthwhile to review the various albedos which enter the discussion. The
Bond or bolometric albedo is the fraction of all incident light, integrated
over the entire stellar spectrum, which is scattered back to space by a
planet. This albedo is a single number and enters into the computation of a
planet’s equilibrium temperature (the temperature an airless planet would have
if its thermal emission were equal to the incident radiation which it
absorbs). It is also useful to know the monochromatic ratio of all scattered
to incident light as a function of wavelength, which is the spherical albedo.
For historical reasons it is more common to discuss the geometric albedo,
which is specifically the wavelength-dependent ratio of the light scattered by
the entire planet in the direction directly back towards its star compared to
that which would be so scattered by a perfectly reflecting Lambert disk of the
same radius as the planet. Care must be taken to distinguish all these albedos
as they can differ markedly from one another even for the same planet and the
literature is rife with confusion between them.
Perhaps the single most important effect of clouds is to brighten the
reflected light spectra of exoplanets, particularly at optical wavelengths (
$0.38<\lambda<1$ μm). For planets with appreciable atmospheres, Rayleigh
scattering is most efficient in the blue. At longer wavelengths however
absorption by either the planetary solid surface or oceans (for terrestrial
planets) or by atmospheric gaseous absorbers becomes important in the red.
This is because for common molecules at planetary temperatures, such as H2O,
CO2, or CH4, vibrational-rotational transitions become important at
wavenumbers below about 15,000 cm-1 or wavelength $\lambda>0.6$ μm. Except for
diatomic species (notably O2) planetary atmospheres are generally not warm
enough to exhibit strong electronic absorption features in the optical. Thus
the reflected light or geometric albedo spectrum of a generic cloudless planet
with an atmosphere would be bright at blue wavelengths from Rayleigh
scattering and dark in the red and at longer wavelengths from gaseous
molecular or surface absorption.
Clouds, however, tend to be bright with a fairly gray opacity through the
optical. Thus a thick, scattering cloud can brighten a planet in the far red
by scattering more light back to space than a cloudless planet. As a result
two similar planets, one with and one without cloud cover, will have very
different geometric albedos in the red, and consequently differing brightness
contrasts with their parent stars. The mean Earth water clouds increase the
contrast between the reflection spectrum of an Earth-like planet and its host
star by one order of magnitude (Kitzmann et al. 2011b). This effect, first
noted for giant planets by Marley et al. (1999) and Sudarsky et al. (2000) and
further explored in Cahoy et al. (2010) is illustrated in Figure 1 which plots
geometric albedo as a function of $\lambda$ for giant planets with and without
clouds. A warm, cloudless atmosphere is dark in reflected light while a cooler
atmosphere, sporting water clouds, is much brighter. In this case clouds
affect the model spectra far more than a factor of three difference in the
atmospheric abundance of heavy elements, which is also shown.
In the case of searches for planets by direct coronagraphic imaging in
reflected light, clouds may even control whether or not a planet is
discovered. Depending on the spectral bandpass used for planet discovery at a
given distance from its primary star a cloudy planet may be brighter and more
detectable than a cloudless one. Discussions of the influence of clouds and
albedo on detectability include those by Tinetti et al. (2006a) and Kitzmann
et al. (2011a).
Figure 1: Model geometric albedo spectra for Jupiter-like planets at 2 AU
(green) and 0.8 AU (red) from their parent star. Solid and dashed lines lines
show models with a solar abundance and a 3-times enhanced abundance of heavy
elements respectively. Prominent absorption features are labeled. The 2 AU
model planets possess a thick water cloud while the 0.8 AU models are warmer
and cloudless and consequently darker in reflected light. Figure modified from
Cahoy et al. (2010).
Once a planet is detected, spectra are needed to characterize atmospheric
abundances of important molecules. For terrestrial extrasolar planets this is
best done by the analysis of the thermal emission spectrum (Selsis, 2004;
Tinetti et al., 2012). Clouds, however, may conceal the thermal emission from
the surface and dampen spectral features of molecules (e.g., the bio-
indicators N2O or O3). Indeed, clouds on Earth have a larger impact on the
emitted infrared flux than the differences between night and day (Hearty et
al., 2009; Tinetti et al., 2006a). Thermal emission spectra are therefore very
sensitive to the types and fractional coverages of clouds present in the
atmosphere. At high spectral resolution the most important terrestrial
molecular spectral features in the mid-infrared, such as O3, remain detectable
even for cloud covered conditions in many cases. Figure 2 shows thermal
emission spectra affected by low-level water droplet and high-level water ice
clouds of an Earth-like planet orbiting different main-sequence dwarf stars
(adopted from Vasquez et al., 2013). With increasing cloud cover of either
cloud type the important 9.6 μm absorption band of ozone is strongly dampened,
along with an overall decrease in the thermal radiation flux. For some cases
presented in Fig. 2 (e.g. for the F-type star and 100% high-level cloud cover)
the ozone band seems to be completely absent or even appears in emission
rather than absorption (F-type star, 100% low-level clouds). At lower spectral
resolution, such as could be obtained for terrestrial exoplanets in the near
future, clouds render the molecular features even less detectable; thus clouds
will strongly affect the determination of their atmospheric composition. For
example, as shown by Kitzmann et al. (2011a) a substantial amount of water
clouds in an Earth-like atmosphere can completely hide the spectral signature
of the bio-indicator ozone in low-resolution thermal emission spectra.
Figure 2: Planetary thermal emission spectra influenced by low-level water
droplet (left panel) and high-level ice clouds (right panel) for an Earth-like
planet orbiting different kinds of main-sequence host stars (adopted from
Vasquez et al. 2013). For each central star the spectra are shown for
different cloud coverages. Note especially the strong impact of the cloud
layers on the 9.6 μm absorption band of ozone.
Clouds can also obscure spectral features originating from the surface of a
planet. In principle, signals of surface vegetation (“vegetation red edge”)
are present in the reflected light spectra of Earth-like planets. However, as
often pointed out (e.g., Arnold et al. (2002) and Hamdani et al. (2006)) this
spectral feature can easily be concealed by clouds. For the detectability of
possible vegetation signatures of terrestrial extrasolar planets Montañés-
Rodríguez et al. (2006) and Tinetti et al. (2006b) concluded that clouds play
a crucial role for these signatures in the reflection spectra. Thus, apart
from the scattering characteristics of different planetary surface types, the
presence of clouds has been found to be one of the most important factors
determining reflection spectra.
Transmission spectra of transiting planets can be used to obtain many
atmospheric properties, such as atmospheric composition and temperature
profiles. Transmission spectroscopy of extrasolar giant planets has already
proven to be a successful method for the characterization of giant exoplanets.
As shown by Pallé et al. (2009) the major atmospheric constituents of a
terrestrial planet remain detectable in transmission spectra even at very low
signal-to-noise ratios. Thus, transmission spectra can provide more
information about the atmospheres of exoplanets than reflection spectra.
Theoretical transmission spectra of Earth-sized transiting planets have been
studied by Ehrenreich et al. (2006) including the effects of optically thick
cloud layers. Their results show that the transmission spectra only contain
information about the atmosphere above the cloud layer and that clouds can
effectively increase the apparent radius of the planet. The impact of clouds
on the transmission spectra therefore depends strongly on their atmospheric
height. If cloud layers are only located in the lower atmosphere, which is
already opaque due to absorption and scattering by gas species, their overall
effect on the spectrum will be small (Kaltenegger & Traub, 2009).
### 2.2 Habitability of Extrasolar Terrestrial Planets
From the terrestrial bodies with an atmosphere in the solar system we know
that clouds are a common phenomenon and should also be expected to occur in
atmospheres of terrestrial extrasolar planets. Apart from the usual well-known
greenhouse gases, clouds have the most important climatic impact in the
atmospheres of terrestrial planets by affecting the energy budget in several
ways. Firstly, clouds can scatter incident stellar radiation back to space
resulting in atmospheric cooling (albedo effect). On the other hand, clouds
can trap thermal radiation within the lower atmosphere by either absorption
and re-emission at their local temperature (classical greenhouse effect) or by
scattering thermal radiation back towards the surface (scattering greenhouse
effect), which heats the lower atmosphere and planetary surface. All these
effects are determined by the wavelength-dependent optical properties of the
cloud particles (absorption and scattering cross-sections, single scattering
albedo, asymmetry parameter, scattering phase function). These properties can
differ considerably for different cloud forming species (owing to their
refractive indices) and atmospheric conditions (composition and temperature
structure). Note that the single scattering albedo, yet a fourth type of
albedo (see Section 2.1 for the others), measures the fraction of all incident
light scattered by a single cloud particle.
Life as we know it requires the presence of liquid water to form and survive.
In the context of terrestrial exoplanets we are primarily concerned with
habitable conditions on the planetary surface. Lifeforms may of course also
exist in other environments, such as deep under the planetary surface or
within a subsurface ocean (Lammer et al., 2009). Since there is no possibility
of detecting the presence of such habitats by remote observations our current
definition of a habitable terrestrial planet assumes a reservoir of liquid
water somewhere on its surface. Liquid surface water implies that these
planets would also have abundant water vapor in their atmospheres. We can
therefore safely assume that water (and water ice) clouds will naturally occur
in the atmospheres of these exoplanets throughout the classical habitable
zone.
As such, H2O and CO2 clouds are the prime focus for habitable exoplanets.
Other possible condensing species also found in our solar system include C2H6
and CH4 (e.g., in the atmosphere of Titan), N2 (Triton), or H2SO4 (Venus,
Earth). Any chemical species in liquid form on a planetary surface can in
principle be considered as a potential cloud-forming species in the atmosphere
if its atmospheric partial pressure is high enough and the temperature low
enough.
Given these considerations it is clear that clouds are important for the
determination of the boundaries of the classical habitable zone around
different kinds of stars. In the next two subsections we discuss some of the
ways that clouds influence the habitable zone boundaries.
#### 2.2.1 Inner Edge of the Habitable Zone
The closer a terrestrial planet orbits its host star the greater stellar
insolation and consequently surface temperature. This leads to enhanced
evaporation of surface water and increases the amount of water vapor in the
atmosphere. The strong greenhouse effect by this additional water vapor
further increases the surface temperature and therefore the evaporation of
surface water, resulting in a positive feedback cycle (see Covey et al., this
volume).
The inner boundary of the classical habitable zone is usually defined either
as the distance from a host star at which an Earth-like planet completely
loses its liquid surface water by evaporation (runaway greenhouse limit), or
as the distance from the host star when water vapor can reach the upper
atmosphere (water loss limit). The latter is the definition favored by Kasting
et al. (1993); the former definition is used by Hart (1979). In the present
Earth atmosphere water escape is limited by the cold trap at the tropopause
which loses its efficiency in a sufficiently warm, moist atmosphere (Kasting
et al., 1993).
Using a one-dimensional atmospheric model, Kasting et al. (1993) estimated
that the inner boundary of the habitable zone for the runaway greenhouse
scenario for an Earth around the Sun would be located at 0.84 AU. This
distance, however, was calculated with the clouds treated as a surface albedo
effect and neglecting any feedbacks. For example once formed clouds increase
the planetary albedo which in turn partially reduces the temperature increase
due to the higher stellar insolation and the cloud greenhouse effect. Kasting
(1988) investigated some feedback effects and concluded that with a single
layer of thick water clouds Earth could be moved as close as 0.5 AU (for a
cloud coverage of 100%) or 0.67 AU (50% cloud coverage) from the Sun before
all liquid surface water would be lost. Goldblatt & Zahnle (2011) explored the
various feedback issues in detail and concluded that more sophisticated
modeling approaches are necessary to explore habitability.
Water clouds also contribute to the greenhouse effect. Depending on the
temperature at which the cloud emits thermal radiation, its greenhouse effect
can match or exceed the albedo effect. Cold clouds, such as cirrus, are net
greenhouse warmers. Low clouds with temperatures near the effective radiating
temperature affect climate almost entirely by their albedo, which depends on
such factors as patchiness, cloud thickness, and the size distributions and
composition of cloud particles. The exact climatic impact of clouds therefore
crucially depends on the balance between their greenhouse and albedo effects.
Whether the greenhouse or albedo effect dominates for clouds forming under
runaway greenhouse conditions cannot be easily determined a priori. A better
understanding of the cloud microphysics and convection processes in moist
atmospheres during a runaway greenhouse process is needed to determine the
cloud properties and their fractional coverage near the inner habitable zone
(HZ) boundary. The effect of clouds under runaway greenhouse conditions
represents one of the most important unresolved issues in planetary climate.
Note that this discussion is relevant to wet planets with moist atmospheres
fed by extensive seas. Dry, land or desert, planets will have unsaturated
atmospheres in the tropics and thus radiate at a higher temperature and cool
more efficiently than planets with a water saturated atmosphere. As a result
the habitable zone for such planets may be larger (Abe et al., 2011).
#### 2.2.2 Outer Edge of the Habitable Zone
The outer edge of the habitable zone is set by the point at which there is no
longer liquid water available at the surface as it is locked in ice. With
falling insolation planets found progressively farther away from their central
star have lower atmospheric and surface temperatures. With lower surface
temperatures, the removal of CO2 from the atmosphere due to the carbonate-
silicate cycle, which controls the amount of CO2 in Earth’s atmosphere on time
scales of order a million years, becomes less efficient (see Covey et al.,
this volume). Thus, if the terrestrial planet is still geologically active,
CO2 can accumulate in the atmosphere by volcanic outgassing. With decreasing
atmospheric temperatures, CO2 will condense at some point to form clouds
composed of CO2 ice crystals. Because condensation nuclei can be expected to
be available in atmospheres of terrestrial planets, the most dominant
nucleation process for the formation of CO2 clouds is usually assumed to be
heterogeneous nucleation (Glandorf et al 2002), in which cloud particles form
on existing seed particles.
Just like water clouds, the presence of CO2 clouds will result in an increase
of the planetary albedo by scattering incident stellar radiation back to
space. However, in contrast to water, CO2 ice is almost transparent in the
infrared (Hansen, 1997, 2005) except within some strong absorption bands. Thus
CO2 clouds are unlikely to trigger a substantial classical greenhouse effect
by absorption and re-emission of thermal radiation.
On the other hand, CO2 ice particles can efficiently scatter thermal
radiation. This allows for a scattering greenhouse effect in which a fraction
of the outgoing thermal radiation is scattered back towards the planetary
surface (Forget & Pierrehumbert, 1997). Depending on the cloud properties this
scattering greenhouse effect can outweigh the albedo effect and can in
principle increase the surface temperature above the freezing point of water.
However, the scattering greenhouse effect is a complex function of the cloud
optical depth and particle size (Colaprete and Toon, 2003). Furthermore,
because the greenhouse effect of CO2 clouds depends on the scattering
properties of the ice particles, the particle shape (which cannot be expected
as spherical) or particle surface roughness may play an important role.
However, these effects cannot be easily quantified because neither the
particle shapes nor their surface properties are known. These and other
uncertainties in the cloud microphysics of CO2 ice in cool CO2 dominated
atmospheres makes the calculation of the position of the outer HZ boundary
complicated. More details on the formation of CO2 clouds in the present and
early Martian atmosphere can be found in Glandorf et al. (2002), Colaprete &
Toon (2003), and Määttänen et al. (2005), as well as in Esposito et al. (this
volume). Inferences about CO2 cloud particle sizes in the current martian
atmosphere as constrained by a variety of datasets are presented by Hu et al.
(2012).
For a fully cloud-covered early Mars with thick a CO2 dominated atmosphere and
CO2 clouds composed of spherical ice particles Forget & Pierrehumbert (1997)
estimated that the outer boundary of the HZ is located at 2.4 AU, in contrast
to the cloud-free boundary of 1.67 AU by Kasting et al. (1993). This greater
value has been further used by Selsis et al. (2007) to extrapolate the effects
of CO2 clouds on the outer HZ boundary towards other main sequence central
stars (see also Kaltenegger & Sasselov, 2011).
For terrestrial super-Earths it has been suggested by Pierrehumbert & Gaidos
(2011) that these planets could have retained much of their primordial H2
atmosphere owing to their greater mass. According to their model study, the
classical habitable zone might be far larger than expected for a CO2 dominated
atmosphere although this study did not explore the impact of clouds.
## 3 Clouds and Radiation
Clouds are important for planetary atmospheres because they interact with both
incident short wavelength radiation from the parent star and emergent thermal
radiation emitted by the planetary surface, if present, and the atmosphere
itself. Perhaps the simplest example of such interaction occurs for spherical
cloud particles composed of a single constituent. In this case the wavelength
dependent optical properties can be computed from Mie calculations that depend
only on particle size and the wavelength-dependent complex refractive index of
the bulk condensate. In this section we briefly review the basics of the
interaction of cloud particles with radiation and summarize the use of Mie
theory for modeling this interaction as well as point out some useful
simplifications that can be applied in certain limiting cases.
### 3.1 Basic Radiative Properties of Cloud Particles
Cloud opacity ultimately depends upon the radiative properties of the
constituent particles. A particle with radius r has a cross section to scatter
losslessly or to absorb incident radiation at some wavelength λ, given by Cs
or Ca respectively. These cross sections are defined as $C_{s}=Q_{s}\pi r^{2}$
and $C_{a}=Q_{a}\pi r^{2}$, and their sum is the extinction cross section Ce.
The scattering and absorption efficiencies Qs and Qa, and, from them, the
extinction efficiency Qe = Qs \+ Qa is thus defined. The particle single
scattering albedo is $\omega=Q_{s}/Q_{e}$, and the scattering phase function
is $P\left(\theta\right)$. These quantities can be computed by a Mie code,
which computes the electromagnetic wave propagation through spherical
particles, given a tabulation of the optical properties of the cloud material.
Good physically-based introductions to radiative properties of particles can
be found in van de Hulst (1957), Hansen & Travis (1974), and Liou (2002).
Modeling cloud particle radiative properties can seem forbidding; workers
often think a Mie scattering code, with all its exotic predictions, is needed.
However, in many applications, much simpler approaches not only can provide
very good quantitative fidelity - and greater physical insight - but also can
be applied directly to more complex kinds of particles than the spheres for
which Mie theory is derived.
The direct beam of radiant flux vertically traversing a cloud layer of
particles composed of some material q is reduced by a factor
$\exp\left(-\tau_{q}\right)$, where the layer optical depth is
$\tau_{q}=nQ_{e}\pi r^{2}H_{q}=nC_{e}H_{q}$, and n is the particle number
density, r the particle radius, and Hq the vertical thickness of the layer
(assuming a uniform vertical distribution for simplicity). Qe and thus
$\tau_{q}$ are functions of the wavelength λ, through the λ–dependent real and
imaginary refractive indices of the material in question (nr , ni; see below,
and Draine & Lee, 1984 or Pollack et al., 1994 or http://www.astro.uni-
jena.de/Laboratory/Database/databases.html for typical values). The particle
opacity ϰ (in units of length-squared per mass) is the effective particle
cross section per unit mass of either solids or gas.
If the phase function is strongly forward scattering, as can be the case for
particles with $r/\lambda>10$ or so (Hansen & Travis 1974), much of the
“scattering” can be approximated as unattenuated radiation by reducing the
layer optical depth and renormalizing the particle albedo and phase function.
This is done by applying simple corrections to both ω and $\tau_{q}$ known as
similarity relations. Irvine, (1975), van de Hulst (1980), and Liou (2002)
review these, and present a number of ways to solve the radiative transfer
problem in general.
### 3.2 Refractive Indices and Opacity
All calculations of particle radiative properties (Qe, Qs, Qa, $\omega$ and
P(ϑ)) begin with the refractive indices of the material and the particle size.
While Mie codes are available for download (e.g.,
ftp://climate1.gsfc.nasa.gov/wiscombe/Single_Scatt/), analytical
approximations that are valid in all the relevant limits can often be of great
use. Such approximations rely on the fact that realistic clouds have size and
shape distributions that average away the exotic fluctuations shown by Mie
theory for particles of specific sizes (see Hansen & Travis 1974 for
examples). With current computational capabilities, Mie calculations are not
onerous, but if many wavelengths and/or grids of numerous models are of
interest, the burden is compounded, so one needs to understand whether such
detailed predictions are actually needed.
One simple approximation to scattering by equidimensional, but still
irregular, particles was developed by Pollack & Cuzzi (1980). This approach
uses Mie theory for particles with small-to-moderate $r/\lambda$, where shape
irregularities are indiscernible to the waves involved and Mie calculations
converge rapidly. For larger $r/\lambda$ a combination of diffraction,
external reflection, and internal transmission is used as adjusted for shape
and parametrized by laboratory experiments. The approach is easy to apply in
cases where the angular distribution of scattered radiation is important, as
long as a particle size distribution smears away the significant oscillations
in scattering properties which Mie theory predicts for monodisperse spheres
(Hansen & Travis 1974). This method saves on computational effort for
particles with large $r/\lambda$ but does involve a Mie code for the smaller
particles.
Even simpler approaches are feasible. In many cases of interest, only globally
averaged emergent fluxes or reflectivities, or perhaps heating calculations,
are needed; here, the details of the phase function are of less interest than
the optical depth and particle albedo, which are based only on the
efficiencies. For these cases one can do fairly well using asymptotic
expressions for efficiencies in the limiting regimes $r/\lambda{\ll}1$ and
$r/\lambda{\gg}1$, connecting them with a suitable bridging function. These
expressions depend on the refractive indices of the material in question.
The simplest limit is when $r/\lambda{\gg}1$ (geometrical optics); in fact,
many of the cloud models discussed in this chapter fall into this regime at
1-10 μm wavelengths. In this range it is convenient to neglect diffraction,
which is strongly concentrated in the forward direction and can be lumped with
the direct beam as noted above. Then, assuming the particle has density ρ and
is itself opaque, $Q_{e}{\leq}1$ and Ce reduces to the physical cross section.
Thus the solids-based opacity is simply $\kappa=3\pi r^{2}/4\rho\pi
r^{3}=3/4r\rho$. In this regime, particle growth drastically reduces the
opacity because the surface to mass ratio decreases linearly with radius
(Miyake et al., 1993; Pollack et al., 1994).
At the opposite extreme, when $r/\lambda{\ll}1$(the Rayleigh limit), simple
analytical expressions also apply (Draine & Lee 1984; Bohren & Huffman 1983).
For simplicity below we give the expressions for $n_{i}{\ll}n_{r}{\sim}1$
(appropriate for silicates, oxides, water, but not iron), but the general
expressions are not much more complicated (Draine & Lee 1984). Thus
$Q_{a}=24xn_{r}n_{i}/\left(n_{r}^{2}+2\right)^{2}$ and
$Q_{s}=8x^{4}\left(n_{r}^{2}-1\right)^{2}/3\left(n_{r}^{2}+2\right)^{2}$,
where $x=2\pi r/\lambda$(see also van de Hulst, 1957, p. 70). Because Qs
decreases much faster than Qa with decreasing $r/\lambda$ in this regime,
small particles become not only less effective at interacting with radiation,
but increasingly absorbers/emitters rather than scatterers. In this limit,
their cross section $C_{a}=Q_{a}\pi r^{2}$ per unit particle mass becomes
constant since Qa is proportional to r.
### 3.3 Shape and Porosity
In the case of terrestrial cloud droplets, in which condensation dominates
growth, and raindrops, in which coagulation dominates growth, spherical
particles of constant density are assumed, provided they fall slowly (though
drag-induced deformation is taken into account when computing terminal fall-
speeds for large raindrops). However, if particles condense from their vapor
phase as solids, tiny initial monomers may instead coagulate by sticking into
porous aggregates, perhaps having some fractal properties where the density
may depend on the size. Non-spherical particle shape adds complexity to the
computation of optical properties, which is commonly the case for solid
particles. For instance, the single-scattering properties of ice particles in
the terrestrial atmosphere depend on not only the geometric shape of the
crystals (which depend on the temperature), but also the microscopic surface
roughness. As noted by Fu et al. (1998), the extinction and absorption cross
sections depend mainly on projected areas and particle volumes (note that
random orientation is typically assumed), while the scattering phase function
are largely determined by the aspect ratio of the crystal components and their
small-scale roughness (Fu, 2007).
In the general exoplanet case for tiny particles in gas of typical densities,
plausible inter-particle collision velocities are very small and lead to
sticking but only minor compaction (see e.g., Cuzzi and Hogan 2003, Dominik
and Tielens 1997) until particles become large enough (10-100 μm) where
sedimentation velocity can exceed several meters per second and compaction or
bouncing arise (section 6.3; see Dominik et al. 2007, Zsom et al. 2010). The
thermodynamic properties of condensates and the temperature-pressure (T-P)
structure of the atmosphere in question determine whether the condensate
appears as a liquid or a solid. Figure 3 shows condensation curves of a number
of important cloud-forming compounds, along with the T-P profiles of a range
of exoplanetary and substellar objects.
Models of growing aggregates have been fairly well developed in the literature
of protoplanetary disks (e.g., Weidenschilling, 1988; Dominik & Tielens, 1997;
Beckwith et al., 2000; Dominik et al., 2007; Blum, 2010). The properties of
clouds made of this rather different kind of particle will differ in their
opacity and vertical distribution from predictions of the simplest models;
these properties in turn will impact, and can in principle be constrained by,
remote observations of reflected and emitted radiation.
Figure 3: Model atmospheric temperature-pressure profiles for brown dwarfs
(solid) and condensation curves (dashed) for a variety of compounds. For each
model effective temperature three curves are shown, corresponding to a
cloudless calculation (blue) and two cloudy models with varying sedimentation
efficiency, fsed = 2 (red) and 4 (orange). Figure modified from Morley et al.
(2012).
### 3.4 Porous Particles and Effective Medium Theory
The most straightforward way of modeling porous aggregates is to model their
effective refractive indices based on their constituent materials and
porosity. If the monomers from which the particles are made are smaller than
the wavelength in question, they act as independent dipoles immersed in an
enveloping medium (the medium can be another material; here we assume it is
vacuum). The aggregate as a whole can then be modeled as having effective
refractive indices which depend only on the porosity of the aggregate and the
refractive indices (but not the size) of the monomers. This is the so-called
effective medium theory (EMT); several variants are discussed by Bohren &
Huffman (1983), Ossenkopf (1991), Stognienko et al. (1995), and Voschinnikov
et al. (2006).
For most combinations of materials, EMTs can be even further approximated by
simple volume averages such that the refractive indices of the particle as a
whole can be written (for a simple two-component particle with component 2
being vacuum) as $n_{i}=fn_{\mathit{i1}}+\left(1-f\right)n_{\mathit{i2}}$ and
$\left(n_{r}-1\right)=f\left(n_{\mathit{r1}}-1\right)+\left(1-f\right)\left(n_{\mathit{r2}}-1\right)$,
where nr and ni are the effective real and imaginary indices of the aggregate
as a whole, and component 1 has volume fraction f. If a material which has
large refractive indices, such as iron, is involved, the full expressions are
needed (see Wright, 1987; Bohren & Huffman, 1983; or other basic references).
We will assume the porous aggregates are roughly equidimensional, not needle-
like structures, but even that added complication can often be tractable in
semi-analytical ways for most materials (Wright 1987; Bohren & Huffman 1983).
In the $r/\lambda{\ll}1$ regime, porosity plays little role, both because the
opacity is simply proportional to the total mass regardless of how it is
distributed (see above), and because ever-smaller particles are unlikely to be
aggregates of ever-smaller monomer constituents, but are more likely to be
monomers themselves. Ferguson et al. (2007) present numerical calculations
which show that, for the tiny particles they modeled (0.35 nm - 0.17 μm)
indeed this expectation is fulfilled to first order, also showing that mixing
of materials within these tiny aggregates has little effect (see also Allard
et al. 2001, Helling et al. 2008, Witte et al. 2009 and references therein).
Particles this small are likely to be well-mixed at all levels of typical
exoplanet atmospheres (see below). Some targets of interest may have high haze
layers where this regime is of interest.
However, in the $r/\lambda{\gg}1$ regime, which covers typical planetary
clouds observed in the optical or near-infrared, porosity does affect opacity.
Specifically, for a particle of mass M and porosity $\varphi$, the opacity
$\kappa=C_{e}/M=3/4\rho r=3/4\rho_{s}\left(1-\varphi\right)r$ where
$r=\left(3M/4\rho_{s}\left(1-\varphi\right)\right)^{1/3}$ is the actual
radius. Comparing the opacity of this particle to that of a “solid” particle
with the same mass having radius rs , gives
$\kappa/\kappa_{s}=\left(1-\varphi\right)^{-2/3}$. The porous particle thus
has a larger opacity for the same mass, and indeed reaches the geometrical
optics (λ independent) limit at lower mass than a nonporous particle. In this
regime, effective medium theory should be a convenient and valid way of
modeling porous and/or aggregate grain properties (section 3.2; see also
Helling et al. 2008 and references therein).
Variable composition introduces further complexity to computing single-
scattering properties, requiring a mixing rule to compute the refractive index
for the composite particle. Different mixing rules can introduce uncertainties
of order 1 and 10% respectively in the real and imaginary refractive indices
for black carbon inclusions in liquid water droplets (Lesins et al., 2002).
## 4 Cloud Models
The cloud properties required to compute radiative fluxes are rather
extensive. Thermal emission requires the emissivity and temperature of the
cloud particles. Vertical fluxes can be computed from knowledge of the
vertical distribution of particle cross-section (for each species of particle)
together with the wavelength dependent scattering phase function and the
single-scattering albedo. For a basic two-stream approach these quantities can
be boiled down to a vertical profile of extinction coefficient, asymmetry
parameter, and single-scattering albedo.
Complexities can arise if particle temperature differs from the local
atmospheric temperature by virtue of latent heat exchange and radiative
heating of the particles, but this should be rare in exoplanet applications
(Woitke & Helling 2003) and typically the particles are assumed to be at
ambient atmospheric air temperature. Clouds that are not horizontally uniform
are another possible complication that can be considered.
The required scattering and absorption total cross sections and overall
asymmetry parameter needed for input to a radiative transfer model are found
by integration of the single-scattering properties and emissivities over the
particle size distributions. The size distributions in turn can be computed by
a so-called bin model which tracks the number of particles in multiple
different bin sizes as the particles interact with the atmosphere. Such an
approach is computationally expensive, and a more efficient treatment is to
assume a particular functional form with a small number of free parameters.
For terrestrial applications log-normal distributions, with three parameters
(total number, geometric mean radius, and geometric standard deviation) and
gamma distributions, also with three parameters (total number, slope
parameter, and shape parameter) are commonly used. Separate size distributions
are used for particles of different phases and bulk densities (e.g., raindrops
and snowflakes) and for each mode of a multimodal size distribution. For
example, the parameterized size distribution of cloud droplets, which grow
principally through condensation in the terrestrial atmosphere, are treated
distinctly from raindrops, which grow principally through collisions. Although
a poor approximation for cloud droplets, it is often assumed that the shape
parameter is zero for other cloud species, which allows the gamma distribution
to collapse to an exponential distribution, thereby reducing the free
parameters from three to two.
For the computation of heating rates from the divergence of radiative fluxes,
the vertical distribution of any absorbing and emitting species is obviously
critical. Also, any vertical redistribution by scatterers in bands with
emission or absorption requires that their vertical distribution is accurate.
If horizontal photon transport is unimportant, it is feasible to represent the
global atmosphere–or the atmosphere within any model column of finite
horizontal extent, for that matter–with two columns, one clear and one cloudy
(for example see the discussion in the context of brown dwarfs in Marley et
al. (2010)). Assumptions about the vertical overlap of clouds are critical and
can induce large errors in top-of-atmosphere fluxes as well as heating rates
(e.g., Barker et al., 1999).
In the context of extrasolar planet atmosphere modeling we must connect a
simple, usually 1D model of the atmosphere with what is potentially a complex
brew of cloud properties which we would ideally need to know. However it is
clear that the number of variables can quickly grow to unmanageable extent. In
this chapter we discuss various approaches that have been taken to address
this problem.
### 4.1 Conceptual Framework for Cloud Modeling
Given a profile of atmospheric temperature as a function of elevation or
pressure we can ask where clouds form and what are their radiative properties.
Here we give a simplified discussion of the problem following Sanchez-Lavega
et al. (2004).
The abundance of a given atmospheric species a in the vapor can be given by
its mass mixing ratio $m_{a}=\rho_{a}/\rho=\epsilon P_{a}/P$ where $\rho$ and
P are the density and pressure of the “background” atmosphere (that is, not
including species a) and $\rho_{a}$ and Pa are the density and partial
pressure of vapor species a. Here $\epsilon=\mu_{a}/\mu$ or the ratio of the
molecular weight of species a, $\mu_{a}$, to the background gas mean
molecular weight $\mu$. A not unreasonable assumption is that any vapor in
excess of the saturation vapor pressure of a, $P_{v,a}\left(T\right)$,
condenses out. We can define the saturation ratio of a as
$f_{a}=P_{a}/P_{v,a}\left(T\right)$.
Deep in the atmosphere, below cloud base, $P_{a}{\ll}P_{v,a}$ so the species
is found in the gas phase. In a Lagrangian framework one can imagine a rising
parcel of gas that cools adiabatically as it expands, and at some point it may
reach $P_{a}=P_{v,a}$ and a saturation ratio $f_{a}=1$. In an Eulerian
framework this condition requires that the thermal profile intersects with the
vapor pressure curve for a given constituent, for instance as the 550 K model
intersects the H2O vapor pressure curve in Figure 3. However the thermal
profile for the 1300 K model does not intersect the H2O vapor pressure curve.
Thus H2O would be expected to condense in the cooler, but not the warmer,
atmosphere. To this point the problem is relatively straightforward for those
species that condense directly from the gas phase, such as water. For other
species, however, condensation instead proceeds by a net chemical reaction
when the condensed phase has a lower Gibb’s free energy than the vapor phase.
One such example is H2S + 2Na $\rightleftharpoons$ Na2S(s) + H2.
Above the condensation level a number of issues arise. First, the assumption
that all vapor in excess of saturation condenses may be faulty. Condensation
may require a supersaturation $\left(f_{a}>1\right)$ before it proceeds, owing
to the thermodynamic energy barrier of forming new particles. If so, what
degree of supersaturation is required? Above cloud base, which is the lowest
level at which condensation occurs, abundance of the condensed phase will
depend on a balance between the convective mixing of particles from below and
the downward sedimentation of the condensate particles. If the sedimentation
rate of some portion of the size distribution of condensate is greater than
the vertical mixing velocity scale, the scale height for the condensed phase,
Ha, will be smaller than the atmospheric scale height H evaluated at cloud
base. In the solar system typical values of Ha/H range from 0.05 to 0.20
(Sanchez-Lavega et al. 2004).
Nucleation is the starting process for the formation of a condensed phase
(either liquid or solid) from a gaseous state, or the formation of solid from
a liquid state. It creates an initial distribution of nuclei (embryos) which,
if large enough, are stable with respect to the higher-entropy phase and tend
to grow into larger particles by condensation or freezing. Such a phase
transition can only occur spontaneously under thermodynamically favorable
conditions (for the following discussion we first focus on the process of
condensate nucleation from the vapor phase). Such conditions require the
saturation ratio $f_{a}$ to exceed unity ($f_{a}=1$ characterizes phase
equilibrium).
The simplest nucleation mechanism is homogeneous nucleation, where the initial
nuclei are formed by random spontaneous collisions of monomers in the vapor
phase (e.g. single H2O molecules) into a molecular cluster. This process is,
however, connected to an energy barrier that can prevent the formation of a
stable new phase. A supersaturated gas phase possesses a high Gibb’s free
energy, whereas at the same time molecules in the condensed phase would be at
a lower potential energy. Thus, removing molecules from the vapour phase and
adding them into a condensed phase would in principle lower the total free
energy of the entire thermodynamical system. The corresponding net change in
free energy depends mainly on the volume of an embryo (as a function of the
particle radius $r$) and the supersaturation of the gas.
However, the creation and growth of nuclei also implies that a new surface is
generated which additionally contributes to the free energy owing to surface
tension. It is only thermodynamically favourable to form a condensed phase
when the net reduction of free energy in the system $\left({\sim}r^{3}\right)$
is greater than (or equal to) the free energy required from the surface
tension $\left({\sim}r^{2}\right)$. Therefore, there exists a critical radius
for which these two contributions balance each other. Embryos smaller than the
critical size are unstable and will tend to evaporate again quickly whereas
those with sizes larger than the critical radius will tend to grow freely. The
critical radius is, in particular, a function of the saturation ratio. Low
supersaturations yield very large critical radii while, on the other hand, for
increasing values of $f_{a}$ the size of the critical embryos decreases.
Homogeneous nucleation usually requires a very high supersaturation and is
therefore unlikely to occur in atmospheres of terrestrial planets on a large
scale (observed supersaturations for water vapor in the atmosphere of Earth
are normally in the range of a few percent, for example), because
heterogeneous nucleation occurs at much lower supersaturations and thus
quenches the supersaturation well before homogeneous nucleation occurs.
The predominant nucleation process from the vapor for terrestrial planetary
atmospheres is thought to be heterogeneous nucleation. Here, the initial
distribution of nuclei is formed on pre-existing surfaces. The presence of
such surfaces substantially lowers the supersaturation required to form the
critical clusters. Depending on the properties of these surfaces (e.g. whether
they are soluble with respect to the condensed phase) nucleation is already
possible for saturation ratios close to unity. Such surfaces can be provided
by dust, sea salt, pollen, or even bacteria. In case of terrestrial planets
these particles (condensation nuclei) are usually largely available because of
mechanical process associated with the planetary surface–such as wave
breaking, bubble bursting, and dust saltation–and from the formation of haze
particles (such as sulfuric acid or sulfate droplets) that result from
photochemical processes. Therefore, heterogeneous nucleation can reasonably be
expected as the dominant nucleation mechanism for terrestrial extrasolar
planets. This assumption, however, complicates the treatment of cloud
formation because details on such condensation nuclei (composition, number
density, size distribution) are not available in case of exoplanets.
Additionally, the formation of a solid phase (e.g. ice crystals) can in
principle occur by different pathways. It can either form directly from the
gas phase by homogeneous or heterogeneous nucleation. On the other hand, it is
also possible to form the liquid phase as an intermediate step, followed by
freezing of the supercooled liquid into the solid phase afterwards. Whether
this indirect or the direct pathway occurs depends on the properties of the
condensing species and on the local atmospheric conditions. Water ice cloud
crystals in the Earth’s atmosphere form by homogeneous and heterogeneous
freezing of liquid in mixed-phase clouds (such as cumulonimbus and Arctic
stratiform clouds) and heterogeneous nucleation in cirrus clouds. In the
Martian atmosphere CO2 clouds form directly by heterogeneous nucleation from
the gas phase.
Given a composition of the cloud the task for any cloud model then becomes one
of computing cloud particle sizes and their number distribution through the
atmosphere above cloud base. A very simple solution would be to simply assume
a mean particle size and a cloud scale height and this is effectively the
approach many investigations have take to explore the effect of exoplanet
clouds. For the remainder of this section we consider efforts to more
rigorously derive expected particle sizes and the vertical distribution of
cloud particles. We begin by considering the most thoroughly modeled clouds,
terrestrial clouds of liquid water and water ice. We then move on to cloud
models that have been constructed for terrestrial and giant extrasolar planets
and conclude by considering the lessons learned from efforts to model clouds
expected in brown dwarf atmospheres.
### 4.2 Perspective from Earth Science
Cloud modeling of terrestrial clouds comes in a wide assortment of classes.
For detailed cloud studies of limited spatial extent, dynamical frameworks
range from 0-D parcel models, to 1-D column models, to 2-D eddy-resolving
models, to 3-D large-eddy simulations and cloud-resolving models. The
difference between such models is the number of spatial dimensions represented
in the governing equations. Another varying aspect of cloud models is the
degree of detail in describing cloud microphysics. The simplest models simply
assume that all vapor in excess of saturation condenses and assume a fixed
size for the cloud particles. Others assume a functional shape for the cloud
particle size distributions and parameterize the microphysical processes,
prognosing one, two, or three moments of the size distribution for each
condensate species (such as cloud water, rain, cloud ice, snow, graupel, and
hail). The most detailed approach is to resolve the cloud particle size
distributions without making any assumptions about the functional shape of the
size distributions.
Global climate model frameworks similarly range from one-dimensional radiative
convective models (e.g., Manabe & Wetherald, 1967), to two-dimensional,
zonally averaged models (e.g., Schneider, 1972), to modern three-dimensional
general circulation models (GCMs). (See Schneider & Dickinson, 1974 for an
early, comprehensive review of approaches to global climate modeling.) With
respect to clouds in global models, one-dimensional radiative convective
models suffer from a major deficiency, namely predicting plausible estimates
of horizontal cloud coverage (see Ramanathan & Coakley, 1978). A two-
dimensional framework is an intermediate step, though modern climate studies
rely principally on three-dimensional GCMs. The rest of this section will
focus on clouds in modern GCMs.
The representation of clouds in GCMs remains a major challenge, as cloud
feedbacks constitute a leading source of uncertainty in current model-derived
estimates of overall climate sensitivity, which are typically cast in terms of
the sensitivity of globally averaged surface temperature to changes in
radiative forcings (see Hansen et al., 1984). The response of tropical cirrus
clouds to increasing sea surface temperatures has been a topic of great debate
in the last two decades. At one extreme is the thermostat hypothesis of
Ramanathan & Collins (1991), which suggests that increased water vapor leads
to more extensive, thicker anvils that will on net cool the planet through
increased albedo, while at the other extreme is the argument of Lindzen et al.
(2001) that greater condensate loading in a warmer climate precipitate more
efficiently and lead to a dryer upper atmosphere that traps less infrared
energy, driving a negative water vapor feedback. Both hypotheses produce a
negative climate feedback for tropical clouds, one employing increased solar
reflection, the other relying on increased infrared emission. While these are
provocative ideas that have spawned countless evaluations of the hypothesis
(with neither withstanding scrutiny), the predominant concept currently is
that the tropical climate feedback for modern GCMs is determined primarily by
the response of clouds in the marine boundary layer (Bony & Dufresne, 2005),
of which the transition from overcast stratocumulus to broken cloud fields of
trade cumulus is a leading primary candidate responsible for that response.
With respect to cloud feedbacks generally, attention on the climate feedback
of cirrus clouds formed from the detrainment of deep tropical convection has
been supplanted to a large degree by more recent focus on the climate feedback
of shallow clouds.
A fundamental problem in representing clouds in the GCMs is that the native
GCM grid cells are very coarse, of order 100 km horizontally and 1 km
vertically. Although model resolution steadily improves as computing power
advances, the problem of convection and clouds being unresolved persists in
models designed to span the globe and simulate climate over time scales of
decades to millennia. The basic assumption in a conventional treatment of
convection and clouds in a GCM is that cloud properties and precipitation
rates in a model grid cell, which is much larger horizontally than any cloud,
can be computed based on the mean properties of the grid cell. The purpose of
a cloud parameterization is to compute cloud properties and precipitation
rates from those mean properties.
A somewhat recent version of the NASA GISS (Goddard Institute for Space
Studies) GCM (Schmidt et al., 2006) includes a conventional treatment of
clouds and convection. The atmospheric model grid spacing is roughly 2 × 2.5°
with 20 or 23 vertical layers, and thus all convection and cloud physics are
necessarily highly parameterized. Deep convection is parameterized based on
the convective instability of model columns using idealized updrafts and
downdrafts, which detrain air into stratiform cloud layers. The stratiform
cloud cover is computed as a diagnostic function of grid-scale relative
humidity, and the relative humidity thresholds used to compute cloud cover in
that diagnostic function serve as the principle tuning knobs for the GCM, with
the dual tuning targets of top-of-atmosphere radiative balance and an overall
albedo reasonably close to satellite-based estimates (note that such tuning
can easily result in exaggerated cloudiness in some regions that make up for
insufficient cloudiness in others). In the GISS GCM the only prognostic cloud
variable is the mass mixing ratio of cloud condensate, which is a fundamental
component of the cloud parameterization. Any precipitation is assumed to
evaporate or fall out in one time step (30 minutes) and the phase of the
condensate is probabilistically determined from temperature to allow for a
modest amount of supercooling (liquid colder than the melting point) on
average. The standard version of the GCM assumes different cloud droplet
concentrations over ocean and land and also assumes a fixed number
concentration for ice particles, and a fixed effective variance of the
condensate size distributions is assumed for computing cloud optical
thickness.
A more complex approach for stratiform cloud microphysics is used in the most
recent version of the NCAR (National Center for Atmospheric Research) GCM.
That scheme uses two moments (mass and number) for two prognostic (cloud water
and cloud ice) and two diagnostic (rain and snow) hydrometeor species
(Morrison & Gettelman, 2009). The rapidly sedimenting species are treated
diagnostically with a tridiagonal solver to allow for long time steps (20
minutes with 2 microphysics substeps) in a manner that avoids numerical
instability associated with falling through more than one layer during a time
step. A novel aspect of this microphysics scheme is that by assuming a
particular subgrid-scale distribution of cloud water, microphysical processes
that involve cloud water take into account the problem of grid-averaging over
nonlinear process rates (Pincus & Klein, 2000). Taking into account joint
subgrid-scale distributions of just two moments for two species complicates
the math considerably (Larson & Griffin, 2012).
An alternative to parameterizing convection within grid scales O(100 km) is
the multiscale modeling framework (Randall et al., 2003), in which two-
dimensional cloud-resolving models are embedded within each GCM column. While
some convection-related aspects of the global circulation are treated well by
such an approach, like the more traditional approach to GCM cloud
parameterization, the pervasive and climatologically important regime of
shallow marine convection is poorly represented in such models (e.g., Marchand
& Ackerman, 2010). (Explicit resolution of such clouds requires horizontal
resolution O(100 m) and vertical resolution O(10 m).) Avoiding issues related
to embedding 2-D slices within GCM columns (problems including how to orient
the slices and shortcomings including the use of periodic lateral boundary
conditions for each embedded 2-D sub-model), the most expensive, yet perhaps
straightforward approach to climate modeling is the Earth Simulator, a global
cloud-resolving model with simulations run on a 3.5-km horizontal grid (Satoh
et al., 2008). The computational demands of such an approach are vast, with
order 1019 grid cells in such a model. Even such a brute-force approach falls
far short of the grid resolution required to explicitly simulate shallow
marine convection, however. It is safe to say that even on the most powerful
computing platforms that global simulations of Earth will be saddled with
cloud and convective parameterizations for the foreseeable future.
### 4.3 Exoplanet Clouds
In extreme contrast to the situation for Earth outlined above, there are
currently no observational constraints for atmospheres of terrestrial
exoplanets that would provide information about what kind of clouds may have
to be considered for a particular exoplanet. Available observables are
confined to basic planetary parameters, like radius and orbital inclination
(if a transit event can be observed), planetary mass (from the radial velocity
method), orbital eccentricity and distance, and additionally the type of the
central star. Consequently, we are faced with the difficult problem of
modeling clouds in an environment without actually knowing any further
details.
Cloud formation can only be treated theoretically in compliance with the
chemical composition of the atmosphere because it determines the condensing
species forming a cloud. Considering how diverse atmospheres of terrestrial
exoplanets can be expected to be, the self-consistent modelling of cloud
formation in such atmospheres without observational constraints or theoretical
predictions is somewhat ambiguous. The composition of a terrestrial planet
cannot be easily deduced from simple theoretical arguments. In the case of a
planet which has lost its primordial atmosphere the atmospheric composition is
determined by the combination of the outgassed chemical species from the
planet’s interior and the volatiles delivered by impacts of asteroids and
comets and will therefore depend on the planet’s mantle composition, physical
processes in the planetary interior, and the composition and sizes of
impactors. Another factor with a huge impact on the atmospheric composition is
also the possible existence of a biosphere that interacts chemically with the
atmosphere.
The long-term evolution of the atmospheric composition, such as that resulting
from the carbonate-silicate cycle as on Earth, depends also on the occurrence
of plate tectonics. To date, the only known planet with plate tectonics is
Earth. It is currently not quite well understood under which conditions a
planetary crust will start plate tectonics and how this process is maintained
over an extended period of time. This is especially true for more massive
terrestrial planets like super-Earths, where there is much controversy
regarding plate tectonics (see e.g. Valencia et al. 2007, O’Neill et al.
2007).
Other important processes determining the atmospheric composition are the
escape mechanisms of atmospheric gas to space. While thermal escape is a
function of the atmospheric temperature, planetary mass, and the molecular
weight, the non-thermal escape processes (erosion of the atmosphere by a
stellar wind, for example) are much more complicated (e.g. Lammer et al. 2008
or Tian, this volume). They not only depend on the activity of the central
star–which itself is a function of the stellar type and its particular stellar
evolution–but also on the possible existence of a planetary magnetic field
(linked to the rotation rate of the planet), which can protect the planetary
atmosphere against loss processes.
In contrast to a low-mass planet like Earth, a more massive terrestrial super-
Earth might retain a part of its primordial hydrogen dominated atmosphere,
partly enriched by volcanic outgassing or additional external delivery. Such
atmospheres may be vastly different from those known within our solar system.
A discussion of possible atmospheric compositions can be found in Seager &
Deming (2010).
The secondary atmosphere of an Earth-like planet (Earth-like with respect to
the chemical composition of the planetary mantle) would most likely be rich in
H2O and CO2 (Schaefer et al. 2012). Therefore, clouds composed of these
species are of prime interest for habitable Earth-like planets. Cooler
atmospheres can also contain significant amounts of CH4 and CH3, or SO2 in
case of high temperatures (Schaefer et al. 2012).
In this environment effective cloud models must match the problem at hand. For
example highly sophisticated terrestrial cloud microphysics and dynamics
models are not required in order to ascertain the range of plausible albedos
for a hypothetical Earth-like terrestrial planet. However more sophisticated
approaches than simple ad-hoc models may be needed to interpret the colors of
a directly imaged planet.
### 4.4 Cloud Models for Terrestrial Exoplanets
In principle the basic mathematical description of cloud microphysics in
atmospheres of terrestrial exoplanets do not deviate from their solar system
counterparts. The temporal and spatial evolution of the cloud particle size
distribution can be described by means of a master equation (“general dynamic
equation”) incorporating all relevant gain and loss processes. This includes
nucleation, evaporation, sedimentation, coagulation/coalescence, diffusion, or
hydrodynamical transport (see Pruppacher & Klett, 1996 or Lamb & Verlinde
2011). While the numerical solution of the master equation is still quite
challenging it can be efficiently performed by the methods summarised in
Williams & Loyalka (1991) or by applying more advanced methods (e.g.
continuous and discontinuous Galerkin methods by Sandu & Borden, 2003).
Self-consistent modeling of the formation and temporal and spatial evolution
of clouds is an unsolved problem. An ideal treatment would require a thorough
knowledge of the state of the atmosphere, including its composition, the
spatial distribution of chemical species, atmospheric temperature and
dynamics, and the size distribution and density of potential cloud
condensation nuclei and heterogeneous freezing nuclei. However, even in the
terrestrial atmosphere the formation of ice at temperatures warmer than the
homogeneous freezing temperature for water (about 233 K for typical drop
sizes) is not well understood (e.g., Fridlind et al., 2007, 2012); far less,
if anything, is known about heterogeneous freezing and condensation nuclei in
other atmospheres and would hardly be detectable by remote observations of
terrestrial extrasolar planets. Lack of laboratory data to derive the
necessary microphysical rates under atmospheric conditions more “exotic” than
found in the solar system further complicates the problem of describing cloud
formation in atmospheres of exoplanets.
Additionally, many atmospheric models for exoplanets are restricted to one
spatial dimension and are often considered to be stationary, which makes a
detailed description of cloud microphysics very difficult. While the climatic
effects of clouds can be approximately treated in a one-dimensional model
atmosphere, a consistent modelling of cloud formation would, in principle,
require a three-dimensional dynamical atmospheric model as described in
Section 4.2. In comparison to one-dimensional models, however, three-
dimensional general circulation models contain many more free parameters and
are very computationally intensive. Properties such as the surface orography,
distribution of surface types (fractions of oceans and land mass), and the
local distribution of chemical species (affected by volcanism for example)
play a major role for the dynamics and chemistry of the atmosphere (Joshi,
2003), and affect directly the formation and evolution of clouds. Since none
of these detailed properties are known for a terrestrial exoplanet many
additional assumptions about the planet and its atmosphere have to enter the
calculations. On the other hand, three-dimensional models are the only
opportunity to obtain information about the possible temporal variation and
fractional coverage of clouds and their distribution throughout the
atmosphere. Such results might be required to analyze transmission spectra of
terrestrial exoplanetary atmospheres containing patchy clouds.
Given the aforementioned challenges due to the lack of observational
constraints, clouds in atmospheres of terrestrial exoplanets are usually
treated in a simplified way. The simplest method to account for the effects of
clouds in an atmospheric model is a modified surface albedo. This approach has
been widely used in the past (e.g., Kasting et al. (1993), Segura et al.
(2003, 2005), or Grenfell et al. (2007)). The surface albedo of these kind of
models is modified to yield a specified surface temperature for a given
reference scenario. For example, a common reference scenario is an Earth-like
planet around the Sun at a known orbital distance of 1 AU. The surface albedo
is then adjusted to obtain the mean surface temperature of Earth (288 K),
thereby mimicking the climatic effects of clouds. This adjusted surface albedo
value is then used in all subsequent model calculations, assuming that the net
effect of the clouds is invariant from changes of the atmospheric conditions,
or type of central star. This approach makes no assumptions about the physical
nature of the clouds, their composition, size, or optical properties but
instead estimates their effects based on the original tuning to surface
temperature. While a modified surface albedo can crudely describe the climatic
effect of clouds, it cannot simulate their impact on the planetary spectra.
A more detailed approach is to consider model scenarios where the properties
of clouds can be assumed to be approximately known. For a completely Earth-
like planet, one could expect that the properties of clouds in such an
atmosphere would closely resemble those found on Earth. This approach was used
by Kitzmann et al. (2010) to study the impact of mean Earth water clouds in
the atmospheres of Earth-like extrasolar planets. In contrast to a modified
surface albedo, the wavelength-dependent optical properties of the cloud
particles are explicitly taken into account to study their effects for
different incident stellar spectra and other parameters. Thus, in addition to
the influence on the atmospheric and surface temperatures, their impact on the
planetary spectra can also be studied in detail using this modelling approach
(e.g., Kaltenegger et al., 2007). If the properties of the clouds are not
known (e.g., for a CO2 cloud in a thick CO2 dominated atmosphere of a super-
Earth) parameter studies can be performed, varying the cloud properties over a
wide range to estimate the possible effect of clouds. This approach has been
used by Forget & Pierrehumbert (1997) for CO2 ice clouds in an early Martian
atmosphere, for example.
More detailed treatments of cloud formation in exoplanetary atmospheres
include simplified air parcel and vertically resolved one-dimensional cloud
models based on those originally developed for the Earth atmosphere. Such
models can be used to determine mean cloud properties under different
atmospheric conditions (see Neubauer et al. (2011, 2012) for several cloud
species (e.g. H2O and H2SO4) or Zsom et al. (2012) for water (droplet and ice)
clouds, for example). However, these more detailed descriptions of cloud
formation already need to include many additional assumptions, such as the
distribution of cloud condensation nuclei, which strongly influence the
resulting cloud properties.
While cloud models for terrestrial exoplanets so far lack many of the
sophisticated and detailed cloud microphysics needed to reproduce the
complicated cloud structures known from Earth observations they nonetheless
offer an important first-order estimate of cloud effects in exoplanetary
atmospheres. One of the largest uncertainties for one-dimensional models is
the treatment of fractional clouds. Unless the atmosphere is globally
supersaturated, thus resulting in a completely cloud-covered planet, the
fraction of the atmosphere where clouds are present has to be introduced as a
free parameter. Thus, one-dimensional models are incapable of organically
describing planets with fractional cloud-cover.
Although one-dimensional models are commonly employed for many terrestrial
exoplanet applications, three-dimensional models have also been used. For the
terrestrial super-Earth Gliese 581d, Wordsworth et al. (2011) adapted a Mars
global circulation model that included a simplified treatment of CO2 ice cloud
formation. This microphysical model assumes a certain size and number density
of cloud condensation nuclei and equally distributes the condensable material
among them, accounting also for the sedimentation and hydrodynamical transport
of the cloud particles within the atmosphere. The corresponding formation of
CO2 clouds lead to an increase of the surface temperature of Gliese 581d in
their model calculations. The same approach was also used by Wordsworth et al.
(2010) in a one-dimensional atmospheric model for the same planet.
### 4.5 Giant Planet Cloud Models
Unlike the case for Earthlike planets where water clouds are the greatest
concern, a wide variety of species may condense in the hydrogen-helium
dominated atmospheres of giant planets. Sánchez-Lavega et al. (2004) review
the standard framework for cloud formation in giant planets. Homogeneous
condensation occurs when the partial pressure of a species in the gas phase
exceeds its saturation vapor pressure at a given temperature in the
atmosphere. Sánchez-Lavega et al. tabulate the vapor pressures for many
relevant species. Other expressions for additional species can be found in
Ackerman & Marley (2001) and Morley et al. (2012). Curves tracing the set of
pressure and temperature conditions at which a given species condenses
assuming equilibrium chemistry (“condensation equilibrium curves”) are shown
for many species in Figure 3 and a schematic illustration of the resulting
cloud decks is shown in Figure 4.
Once a cloud layer forms the condensate is removed from the overlying
atmosphere and thus is no longer available to react at lower temperatures
higher in the atmosphere. Thus the calculation of the chemical equilibrium
state for the atmosphere must account for the presence of the cloud. Such a
“condensation chemistry” is distinct from equilibrium chemistry calculations
in which the condensate remains in communication with the gas phase and is
available for reaction at lower temperatures. Condensation chemistry is
discussed in detail by Fegley & Lodders (1996), Lodders & Fegley (2002), and
Visscher et al. (2006) as well as by Burrows & Sharp (1999) in the context of
brown dwarf models and Sudarsky et al. (2003) for extrasolar giant planets.
The schematic Figure 4 accounts for condensation chemistry. If iron were not
sequestered into a deep cloud layer in Jupiter’s atmosphere the Fe would react
with gaseous H2S to form FeS, thus removing H2S from the observable
atmosphere, in contradiction to observations (Fegley & Lodders, 1996).
Once the cloud base pressure is found the challenge is to describe the
variation in cloud particle sizes and number densities above this level. Early
attempts to develop cloud models for use in giant solar system atmospheres
included the work of Lewis (1969), Rossow (1978), and Carlson et al. (1988).
These and other early works are reviewed by Ackerman & Marley (2001). In this
subsection we focus on more recent modeling approaches that are in use today,
in particular the cloud models of Ackerman & Marley and of Helling and
collaborators.
Figure 4: Schematic illustration (modified from Lodders 2004) of cloud layers
expected in extrasolar planet atmospheres based on consideration of
equilibrium chemistry in the presence of precipitation. The three panels
correspond roughly to effective temperatures Teff of approximately 120 K
(Jupiter-like, left), to 600 K (middle) to 1300 K (right). Note that with
falling atmospheric temperature the more refractory clouds form at
progressively greater depth in the atmosphere and new clouds composed of more
volatile species form near the top of the atmosphere.
#### 4.5.1 Ackerman & Marley
Iron and silicates condense in the atmospheres of warm brown dwarfs (Figure 3)
and these clouds must be accounted for in models of brown dwarf emergent
spectra. Early modeling attempts for such clouds simply computed the mass of
dust that would be found in chemical equilibrium for a given assumed initial
abundance of gas (as if the gas were isolated at a given pressure and
temperature from the rest of the atmosphere. The lower the atmospheric
temperature the more dust that would be present. Atmospheric models following
such a prescription (such as the “DUSTY” models of Allard et al. 2001)
adequately reproduced the near-infrared colors of the warmest brown dwarfs but
predicted far too great of a dust load in cooler objects. Thus it was apparent
that an accounting for sedimentation of grain particles was required. One
approach used in the literature was to set a variable “critical” temperature
for a given cloud such that cloud particles would only be found between cloud
base and the specified temperature (Tsuji, 2002). Another approach was to
limit the cloud to be confined within an arbitrary distance, usually one
scale-height, of cloud base. Both such approaches required the choice of an
arbitrary particle size for the grains. The advantage of such approaches is
that they are computationally very tractable for modeling and thus allow the
exploration of a large parameter space. One disadvantage is that it is
difficult to consider particle size effects and other complexities.
In order to allow for vertically-varying particle number densities and sizes a
second approach was suggested by Ackerman & Marley (2001). In their
formulation downward transport of particles by sedimentation is balanced by
upwards mixing of vapor and condensate (either solid grains or liquid drops),
$-K_{\mathrm{zz}}\frac{{\partial}q_{t}}{{\partial}z}-f_{\mathrm{sed}}w^{*}q_{c}=0$
(1)
where Kzz is the vertical eddy diffusion coefficient, qt is the mixing ratio
of condensate and vapor, qc is the mixing ratio of condensate, w* is the
convective velocity scale, and fsed is a dimensionless parameter that
describes the efficiency of sedimentation. The solution of this equation
allows computation of a self-consistent variation in condensate number density
and particle size with altitude above an arbitrary cloud base.
In their model the cloud base is found by determining at which point in the
atmosphere the local gas abundance exceeds the local condensate saturation
vapor pressure $P_{v,a}$ at which point the atmosphere becomes saturated. In
cases where the formation of condensates does not proceed by homogeneous
condensation an equivalent vapor pressure curve is computed, as described by
Morley et al. (2012).
The Ackerman & Marley cloud model has the advantage of not requiring knowledge
of microphysical processes to compute particle sizes. For a given
sedimentation efficiency clouds are simply assumed to have grown large enough
to provide the required downward mass flux that balances Equation (1). Since
the solution is numerically rapid a large number of models can be computed and
compared with data within a tractable amount of time. Sample model
temperature-pressure profiles along with equilibrium condensation curves are
illustrated in Figure 3.
Considering the simplicity of this approach, the model has fared fairly well
in comparisons with data. Stephens et al. (2009) for example compared model
spectra for L and T dwarfs to a large database of near- to mid-infrared
spectra. They found that cloudy L dwarfs can generally be well fit by clouds
computed with fsed = 1 to 2 while early T dwarfs, which exhibit thinner
clouds, are better fit with fsed = 3 to 4. The model thus provides a framework
for describing mean global clouds in a 1-dimensional sense, but the model
lacks the ability to explain why the sedimentation efficiency might change at
effective temperatures around 1200 K, where the near-infrared spectra of L
dwarfs evolves over a small temperature range.
Marley et al. (2010) considered the effect of partial cloudiness on L dwarf
spectra computed with the Ackerman & Marley (2001) cloud model. Their method
assumed that clear and cloudy columns of atmosphere had the same temperature
profile and together emitted the flux corresponding to a specified effective
temperature. They found that partially cloudy L dwarfs would have emergent
spectra comparable to standard models with homogeneous cloud cover but with
larger values of fsed. Thus a dwarf with 50% clear skies and 50% cloudy skies
with fsed = 2 ends up with a model spectrum comparable to that of a
homogeneous cloud cover with fsed = 4.
#### 4.5.2 Helling and Collaborators
The most extensive body of work on cloud formation in giant exoplanet and
brown dwarf atmospheres has been undertaken by Helling and her collaborators
(Helling & Woitke, 2006; Helling et al., 2008; Witte et al., 2009, 2011; de
Kok et al., 2011) who follow the trajectory of seed particles from the top of
their model atmospheres as they sink downwards. The seeds grow and accrete
condensate material as they fall, resulting in “dirty” or compositionally
layered grains. This work extends the dust moment method from Gail et al.
(1984) and Gail & Sedlmayr (1988). It accounts for the microphysics of grain
growth given these conditions and available relevant laboratory data. Particle
nucleation is explicitly computed, taking into account barriers to grain
formation.
Because condensation is envisioned in this framework to proceed downwards from
the top of the atmosphere rather than upwards from the deep atmosphere and
because grains are allowed to interact with the surrounding gas, the cloud
composition predicted by the Helling approach differs substantially from that
employed in the Ackerman & Marley framework. An example is shown in Figure 5
for a model brown dwarf (log g = 5 in CGS units, Teff = 1600 K). Here Helling
et al. (2008) predict that in addition to the usual Fe, MgSiO3 and Mg2SiO4
cloud layers, additional condensates including SiO2 and MgO will form. These
latter species are not predicted by equilibrium condensation for a cooling gas
mixed upwards from higher temperature and pressure conditions The TiO2 cloud
seeds arising at the model at the top of the atmosphere in this approach is
also evident.
In fact the presence of these initial TiO2 seed particles at the top of the
model atmosphere in the Helling framework deserves some discussion. In the
equilibrium chemistry condensation framework, Ti-bearing condensates (e.g.,
CaTiO3) would form much deeper in the atmosphere as a gas parcel rises
vertically and cools. Precipitation of such particles would remove the
condensate from the gas phase and notable amounts of refractory TiO2 seed
particles would not be expected to arrive at the top of the atmosphere.
Conversely in the Helling conceptual framework the formation of CaTiO3 at the
expected equilibrium cloud base is thought to be kinetically inhibited (since
multiple collisions of molecules would be required) resulting in the
refractory TiO2 clusters being mixed further upwards to ultimately seed
condensation in the cooler upper reaches of the atmosphere. Thus the Helling
approach fundamentally assumes that vertical mixing timescales, at least in
localized columns, are faster than condensation timescales. As the seeds
eventually fall from the top of the atmosphere they then encounter other
condensable molecules which are likewise assumed not to have not been cold
trapped below and the seeds then accrete these species and grow. The model
iterates to find a self-consistent solution. This top-down approach thus
differs from most of the other cloud modeling approaches discussed in the
literature which generally conceive of a condensation sequence operating from
the depths of the atmosphere upwards with species sequentially condensing, as
conceptually shown in Figure 4. Atmospheric mixing by breaking gravity waves
(Freytag et al. 2010) might provide a mechanism to stir the atmosphere
sufficiently to deliver the seed particles.
Because of the complexity of the computational approach required to compute
cloud properties in this framework there have been fewer comparisons between
model spectra computed with the Helling clouds and data than has been the case
with the Ackerman & Marley cloud model. Some direct comparisons between
different cloud models are shown by Helling et al. (2008). Ultimately only a
thorough comparison of the predictions of all cloud modeling schools and data
will be required to establish which conceptual framework is a better
approximation over which ranges of conditions. An application of the Helling
framework to the clouds of Jupiter would be enlightening.
Figure 5: Composition of atmospheric cloud layers for a Teff = 1600 K,
log(g)=5 brown dwarf as computed by the dust model of Helling and
collaborators (Helling et al. 2008). The vertical axis is atmospheric
temperature with the top of the atmosphere to the top of the figure. The
horizontal axis gives the relative volumes Vs of each dust species indicated
by line labels as a ratio to the total dust volume Vtot. Unlike the Ackerman &
Marley condensation equilibrium approach, this model predicts that MgO and
SiO2 are important condensates along with the TiO2 seed particles that are
formed at the top of the model.
#### 4.5.3 Other Approaches
More simplified approaches have also been taken for cloud models, such as
specifying particle sizes and cloud heights. Sudarsky et al. (2000; 2003)
computed model exoplanet albedo spectra given various particle size and cloud
height assumptions. In particular they utilized the Deirmendjian (1964) size
distribution and explored the effects of changing mean cloud particle sizes
and widths of the size distribution. Sudarsky et al. (2000) also considered
the effects of various plausible photochemical hazes on giant planet albedo
spectra.
Cooper et al. (2003) employed the timescale comparison framework pioneered by
Rossow (1978) to compute cloud models for brown dwarfs and extrasolar giant
planets. In this approach various timescales for particle nucleation, growth,
and sedimentation are compared to derive expected condensate particle sizes.
As discussed by Ackerman & Marley (2001), a difficulty with the Rossow method
is that some of the critical timescales depend upon unknown factors,
particularly the assumed supersaturation. Nevertheless the Cooper et al. model
provides a useful survey of likely particle sizes for species of interest
expected under various combinations of gravity and effective temperature. For
example, in agreement with most of the other cloud models Cooper et al.
predict typical silicate grain sizes in the range of 10 to 200 μm.
Tsuji and collaborators (Tsuji 2002, 2005; Tsuji et al. 2004) have computed
brown dwarf models by specifying cloud-top and cloud-base temperatures. For
the directly imaged planets Currie et al. (2011), Madhusudhan et al. (2011),
and Bowler et al. (2010) employ a variety of approaches to specify cloud
properties and explore parameter space. Approaches such as these offer the
advantage of quickly exploring the phase space of possible models and
establishing the effect of plausible cloud models on spectra. The lack of
physical complexity in such models is offset by their useful ability to offer
qualitative understanding of the effect of various condensate properties.
### 4.6 Lessons Learned from Cloudy Brown Dwarfs
Brown dwarfs–hydrogen-helium rich objects with masses between about 12 and 80
times that of Jupiter (MJ)–have been a proving ground for understanding the
role of clouds in exotic atmospheres. This is because the class of brown
dwarfs known as the L dwarfs have atmospheric temperatures in the regime in
which iron and silicate grains condense from the gas phase. It is apparent
from the available data that these refractory condensates do not form a pall
of particles mixed through the entire atmosphere, but rather form discrete
cloud layers. As such L dwarfs were the first objects outside the solar system
for which a detailed description of clouds was required in order to interpret
their emitted spectra.
There have been several comparisons of cloudy brown dwarf atmosphere models to
observations. The most extensive to date are presented by Cushing et al.
(2008) and Stephens et al. (2009). These authors compared atmosphere models of
Saumon & Marley (2008) computed using the Ackerman & Marley (2001) cloud model
to a variety of L- and T-type brown dwarf spectra from 0.8 to 15 μm. For the L
dwarfs and early T-dwarfs the cloudy models clearly did a better job
reproducing the data than cloudless models. The tunable fsed parameter, with
typical values between 1 and 3, allowed sufficient dynamic range to generally
reproduce most of the observed spectra. Witte et al. (2011) meanwhile compare
model spectra computed with the Helling et al. (2008) cloud formulation to
spectra of L-type brown dwarfs. Likewise their cloud model shows much better
agreement with data than either cloudless or very cloudy models with no dust
sedimentation. In all of these studies the matches between models and data are
very good in many cases, but nevertheless there remain notable spectral
mismatches and it is clear that a more sophisticated cloud model would be
required to fit all objects.
The most important lesson learned from the campaign to model brown dwarfs may
be that large grids of atmosphere models–including a variety of cloud
descriptions–are required. Models should not be so complex that the creation
of such grids are a challenge. In the next section we review the available
exoplanet data relevant to clouds. As we will see the exoplanet data do not
yet require large systematic model grids, but such models will undoubtedly be
required as more data become available. Ideally more models will be available
than has been the case for brown dwarfs, thereby permitting more systematic
comparisons of various cloud modeling frameworks to large exoplanet datasets.
## 5 Observations of exoplanet clouds
### 5.1 Transiting Planets
#### 5.1.1 Transmission Spectra
Perhaps the most convincing evidence of high altitude clouds or hazes on any
extrasolar planet to date is found in the case of the transiting planet HD
189733b. This 1.1-Jupiter mass planet orbits a bright nearby K star and is
thus an excellent target for detailed studies of its atmosphere. This planet
is notable because its transit radius–the apparent size of the planet as a
function of wavelength–follows a smooth power law as first measured by Pont et
al. (2008). Signatures of molecular or atomic absorption expected from a
clear, solar composition atmosphere are almost absent, although Na is detected
at high spectral resolution (Huitson et al., 2012). Figure 6 illustrates the
smooth variation in atmospheric transmission as measured from 0.3 through 1 μm
(Pont et al., 2008; Sing et al., 2009; Gibson et al., 2011). The red curve
presents a pure Rayleigh scattering model while a gas opacity only model is
also shown. With the possible exception of a spectral feature at 1.5 μm the
smooth variation of planet radius with wavelength extends to at least 2.5 μm
(Gibson et al., 2011).
Figure 6: Observed transmission spectrum (points) for transiting hot Jupiter
planet HD 189733b (Sing et al. 2011). The width of the wavelength bin for each
measurement is indicated by the x-axis error bars. The y-axis error bars
denote the 1-ς error. The smooth curve denotes the prediction of a simple haze
Rayleigh scattering-only model while the lower curve is a model for gaseous
absorption only from Fortney et al. (2010). Further details in Sing et al.
(2011). Figure courtesy J. Fortney.
The most natural explanation for the HD 189733b transmission spectrum is that
a population of small, high albedo particles is present in the atmosphere at
low pressures (Lecavelier des Etangs et al., 2008). In the terminology of Mie
scattering this requires that the scattering efficiency is large compared to
the absorption efficiency or $Q_{\mathit{abs}}{\ll}Q_{\mathit{scat}}$.
Lecavelier des Etangs et al. demonstrate that this in turn requires a material
with an imaginary index of refraction that is small compared to the real index
and suggest MgSiO3 as a possible candidate. In the condensation chemistry
framework (Section 3.4) however, silicates are expected to condense much
deeper in the atmosphere and the mechanism by which small grains could be
transported to the upper atmosphere has yet to be fully explored, although
vigorous mixing is a likely explanation.
Another transiting planet for which particulate opacity may be important is GJ
1214b. This 6.5-Earth mass planet orbits an M star. Planetary structure models
that fit the observed mass and radius of the planet can be found with either a
thick hydrogen atmosphere comprising less than 3% of the mass of the planet or
a water rich planet surrounded by a steam atmosphere. Transmission spectra of
a large scale height, hydrogen-dominated atmosphere (Figure 7) are predicted
to exhibit well defined absorption bands while a water dominated atmosphere
would have a much smaller scale height and consequently a very smooth
transmission spectrum (Miller-Ricci & Fortney, 2010).
The observed flat transmission spectrum from the optical to perhaps 5 μm (Bean
et al., 2010, 2011; Berta et al., 2012; Desert et al., 2011) is consistent
with a small scale-height atmosphere, thus apparently favoring the water-rich
alternative. However high altitude clouds or hazes, as with HD 189733 b, could
also be concealing atmospheric absorption bands if they lie at altitudes above
200 mbar (Bean et al., 2010 and Figure 7). Since the nominal model pressure-
temperature profile does not cross the condensation curve of any major species
at solar abundance (although ZnS and KCl do condense around 1 bar), Bean et
al. proposed photochemical products as the likely source of the haze rather
than clouds. However as shown in Figure 7, Morley et al. (2013) find that very
extended (fsed=0.1) equilibrium sulphide clouds, including Na2S, in an
atmosphere enriched in heavy elements can also attenuate the flux sufficiently
to reproduce the data. Such small values of fsed are seen in some terrestrial
regimes (Ackerman & Marley 2001) but whether or not this would be plausible in
the atmosphere of GJ 1214b remains to be investigated.
Miller-Ricci Kempton et al. (2012) explored possible photochemical pathways in
the atmosphere of this planet to explore possible mechanisms for haze
production and found that second order hydrocarbons, including C2H2, C2H4, and
C2H6 are efficiently produced by UV photolysis of methane in the atmosphere.
While their model did not explore the chemistry to hydrocarbons of order
higher than C2H6, they argue that polymerization of the initial photochemical
products are likely and that complex hydrocarbons including tholins and soots
are likely to form. Morley et al. (2013) demonstrate that such a hydrocarbon
haze could well explain the flat transmission spectrum observed for this
planet if the atmosphere is indeed hydrogen rich with efficient methane
photolysis (with results very similar to the cloudy case shown in Figure 7).
In an unpublished manuscript Zahnle et al. (2009) argue that for solar
composition atmospheres in general there is a range of atmospheric
temperatures near 1000 K that would be expected to favor methane photolysis
and the formation of higher order hydrocarbon soots or hazes. A definitive
exploration of this point would require new generations of computer codes that
can follow the fate of hydrocarbon species produced by photochemistry.
Figure 7: Observed (points) and model (lines) transit radius of GJ 1214b.
Datapoints are from multiple observations as described in Morley et al.
(2013). Models are for a cloudless H2-He rich atmosphere with 50 times the
solar abundance of heavy elements (grey) and the same atmosphere with the
equilibrium abundance of clouds computed with fsed = 0.1 (red). Figure
modified from Morley et al. (2013).
While it is still too soon to definitely characterize the atmospheres of
either HD 189733b or GJ 1214b, it is apparent that cloud or haze opacity is
likely important in at least some transiting planet atmospheres. The James
Webb Space Telescope will obtain transit spectra of dozens of planets and
characterize atmospheric absorbers as a function of planet mass and
composition and the degree of stellar insolation. Such comprehensive studies
will help to map out the conditions under which clouds and hazes are found.
#### 5.1.2 Transiting Planets in Reflected Light
In addition to the light transmitted through the atmosphere of transiting
planets, measurements have also been made of the the apparent brightness of
the day side of transiting planets. By comparing the brightness of a
transiting system immediately before and after a planet is occulted by its
primary star the combination of light reflected and emitted by the planet can
be measured. In the limit in which the planet does not emit but rather shines
only by reflected light within the passband such a measurement provides the
mean geometric albedo of the planet. However since such a measurement is most
tractable for large planets orbiting close to their stars, in other words the
hot Jupiter class of planets, thermal emission cannot generally be
disregarded.
Upper limits on the geometric albedo have been placed on many planets (see the
summary in Demory et al. (2011)), generally finding the passband averaged
geometric albedo to be less than about 30% and in some cases much less
(Kipping & Spiegel, 2011). Such a finding is not surprising for hot giant
planet atmospheres (Marley et al., 1999; Sudarsky et al., 2000, 2003) since in
the absence of a cloud layer most of the incident flux is absorbed rather than
scattered. Despite these predictions as constraints became available from
studies of transiting planets the preponderance of low albedos was often
treated as surprising when compared to the bright albedo of Jupiter.
Persuasive evidence has been found, however, indicating that at least one hot
Jupiter has a large geometric albedo that is likely attributable to a haze or
cloud layer. Kepler-7b (Latham et al., 2010) has a geometric albedo averaged
over the Kepler bandpass (423 to 897 nm, Koch et al., 2010) of $0.32\pm 0.03$
(Demory et al., 2011). Such a high albedo is more typical of the cloudy solar
system giants (e.g., Karkoschka, 1994) than a deep scattering atmosphere.
Demory et al. thus argue that the most likely explanation is that this
particular planet has a bright photochemical haze or cloud layer, perhaps
similar to that seen in the transit spectra of HD 189733b. More detailed
modeling to constrain the properties of the scattering layer has not yet been
done and to date transit spectra have not been obtained for Kepler 7b.
As albedos are measured for more planets with appropriate corrections for
thermal emission, it may become apparent which particular combinations of
planet mass, composition, and stellar incident flux (particularly including UV
flux) conspire to produce high albedo planets.
### 5.2 Directly Imaged Planets
#### 5.2.1 Young Jupiters
Giant planets start their lives in a warm, extended state with luminosities
much greater than at later times after they have cooled. For this reason giant
planets are easier to detect at young ages of a few hundred million years or
less and several have already been detected, including 2M 1207b and the
planets orbiting the A star HR 8799 (Marois et al., 2008; 2010). The near-
infrared thermal emission of all of these objects (which have effective
temperatures near 1000 K) points to the presence of substantial refractory
cloud decks, most likely comprised of iron and silicate grains (e.g., Marois
et al., 2008; Barman et al., 2011). Much of the literature on these objects
has focused on constraining the properties of these clouds because in field
brown dwarfs clouds have largely dissipated by 1000 K. Currie et al. (2011),
Madhusudhan et al. (2011), and Bowler et al. (2010) also all explored models
for the directly imaged planets and agreed that clouds played a critical role
in shaping their thermal emission at a lower effective temperature than is
typical for more massive field brown dwarfs.
Marley et al. (2012) also constructed model atmospheres for these planets and
concurred that clouds are present in the atmospheres of HR 8799 b, c, and d.
They applied the cloud model of Ackerman & Marley (2001) and found that they
could reproduce much of the available data by using cloud parameters typically
seen in warmer L dwarfs. They argue from mass balance considerations that mean
cloud particle size likely varies inversely with $\sqrt{g}$ where g is the
gravitational acceleration and that all else being equal the column optical
depth of a cloud varies proportionately to $\sqrt{g}$. The net result of these
scaling relationships is that clouds in lower gravity objects tend to be
similar to clouds found at warmer effective temperatures in higher gravity,
more massive brown dwarfs. They find that as objects cool their spectra can be
modeled by increasing the sedimentation efficiency, fsed in Equation (1). The
effective temperature at which fsed begins to increase varies with gravity
such that lower gravity objects begin to lose their refractory clouds at
warmer effective temperatures than low mass objects.
At cooler effective temperatures near 600 K another set of condensates become
important in field dwarfs. Morley et al. (2012) have demonstrated that Na2S
and other clouds (Figure 3) moderately redden the near-IR colors of late T
type brown dwarfs. Once young giant planets in this effective temperature
range are discovered it will be possible to test if the scaling relationships
seen at higher effective temperature persist.
Ultimately understanding the nature of clouds in the warm, young giant planets
hinges on understanding why the cloud clearing effective temperature varies as
it does. In the next few years many more young giant planets are expected to
be discovered by the upcoming GPI and SPHERE exoplanet surveys.
#### 5.2.2 Giants in Reflected Light
No giant planet has yet been imaged in reflected light although such an
observation is less technically difficult than imaging a terrestrial planet in
reflected starlight and a number of space telescope missions have been
proposed. As with terrestrial planets, clouds play a large role in affecting
the reflection spectra of a giant planet (Marley et al., 1999; Sudarsksy et
al., 2002; Cahoy et al., 2010).
One way of visualizing the effect of clouds in a giant planet atmosphere is to
imagine a Jupiter twin lying at progressively closer distances to the sun
(Figure 4). The optical geometric albedo spectrum of Jupiter generally
consists of a bright continuum punctuated by methane absorption bands. The
bright continuum is formed by scattering from gas and stratospheric hazes in
the blue and the bright ammonia clouds at longer wavelengths. The methane
bands are formed from the column of gas overlying the clouds. Below the
ammonia cloud-tops lie cloud decks of NH4SH and H2O.
A Jupiter twin (with the same internal heat flux) lying closer to its star at
2AU would have a warmer atmosphere in which the ammonia and ammonium
hydrosulphide clouds would not condense. Instead the planet would be covered
by a bright global layer of water clouds which would give a very high albedo
(Figure 1). At even closer distances to its star the atmosphere would be too
warm for water clouds to form. Gas absorption thus overwhelms gaseous Rayleigh
scattering and the planet becomes much darker (Figure 1) than the cloudless
case.
## 6 Additional Topics
### 6.1 Clouds of Low Porosity Aggregates
Most of the discussion in this chapter has focused on cloud or haze particles
as fully dense spheres. However fluffy or porous aggregate particles may very
well form and such particles behave differently both as they interact with the
atmosphere and with radiation. In this section we simplify the Ackerman &
Marley (2001) approach to illustrate how low-porosity aggregates would behave.
In this simplified version, we neglect the fraction of the condensible species
present as vapor above the cloud base, so $q_{t}$= $q_{c}$. We stress that
this treatment is not a replacement for the complete model.
When the particles are too small to settle (see below) the condensate mixing
ratio $q=q_{c}=\rho_{c}/\rho_{g}$ is constant and so has an infinite scale
height $H_{q}$ (the condensate has the same scale height as the atmosphere).
However, when the cloud is significantly settled such that the condensate
mixing ratio scale height $H_{q}{\ll}H,$ and if it is assumed that the vapor
mixing ratio is negligible compared to the condensate ratio then Equation (1)
governing the vertical cloud distribution can be approximated as
$K_{\mathit{zz}}\mathit{dq}/\mathit{dz}-v_{f}q=0$, where Kzz = Lw* is the
vertical eddy diffusion coefficient, a property of macroscopic turbulence with
typical lengthscale L and large (energy-containing) eddy velocity w*; L is
usually taken to be the atmospheric scale height H, although this might not be
valid if the associated large eddy timescale L/w* is much longer than the
rotation period of the object (Schubert & Zhang, 2000) or the temperature
profile is stable (Ackerman & Marley use an ad hoc stability correction).
Then, for a settled cloud with uniform mass mixing ratio q and effective
thickness Hq, the above equation is approximated by
$K_{\mathit{zz}}q/H_{q}-v_{f}q=0$, thus the cloud thickness is roughly
$H_{q}=Hw^{*}/v_{f}$. The scale height in the simplified exponential solution
for condensate mass mixing ratio of Ackerman & Marley (2001) is essentially
the same. That is, when the convective eddy velocity is much larger than the
settling velocity, the cloud particles are not able to settle (
$f_{\mathit{sed}}=0$).
Particle settling velocities depend upon the dynamical regime. The settling
velocity vf is the product of the local gravity g and the gas drag stopping
time of the particle, ts. Ackerman & Marley (2001), equation B2, contains a
bridging expression β that covers both the so-called Stokes and Epstein drag
regimes, in which r is respectively greater than, or less than, the gas
molecule mean free path $l_{m}=m_{H_{2}}/\sigma_{H_{2}}\rho_{g}$, where
$m_{H_{2}}$ and $\sigma_{H_{2}}$ are the mass and cross section of a hydrogen
molecule and $\rho_{g}$ is the gas density. In equations B1 and B3 of Ackerman
& Marley (2001) the density contrast is essentially the particle density
because $\rho{\gg}\rho_{g}$ for all applications of interest. So, vf is
proportional to ρ, and for particles with porosity $\varphi$,
$\rho=\rho_{s}\left(1-\varphi\right)=\rho_{s}f$. In the Epstein regime (r/lm <
1) the expression for ts is very simple: $t_{s}=r\rho/c\rho_{g}$ where c is
the sound speed. Cuzzi & Weidenschilling (2006) show how the stopping time in
the Stokes regime (r/lm > 1) is essentially larger by a factor of r/lm.
We can thus show that for porous particles under local gravity g, having the
same mass as a particle with no porosity,
$v_{f}=\mathit{gr}\rho/c\rho_{g}=gr_{s}\rho_{s}\left(1-\varphi\right)^{2/3}/c\rho_{g}$,
and thus
$H_{q}=Hw^{*}/v_{f}=H_{q\left(\mathit{solid}\right)}\left(1-\varphi\right)^{-2/3}$.
Porous particles are lofted to greater heights than non-porous particles of
the same mass, because of their slower settling velocity. In this sense they
behave like smaller particles; however, their radiative behavior is not that
of smaller particles, as noted above; porous particles in the regime
$r/\lambda{\gg}1$ indeed have large and wavelength-independent opacity.
### 6.2 Polarization
Polarization may provide an additional avenue for characterizing cloudy
planets. Unlike the stellar radiation emitted by the central star, the light
scattered by clouds will in general be polarized. Thus, investigating the
polarized radiation scattered by clouds in contrast to the non-polarized
stellar radiation may be an opportunity for characterizing cloudy exoplanetary
atmospheres at short wavelengths in the future (see Stam 2008).
Giant planets can also be polarized in the thermal infrared. In this case the
required asymmetry in radiation emitted across the apparent planetary disk
must be provided either by rotational flattening or irregularities in the
global cloud cover. Marley & Sengupta (2011) investigated the former mechanism
and de Kok et al. (2011) the latter. Both sets of authors found that in the
most favorable circumstances polarization fractions of a few percent were
plausible and that in such cases polarization confirms the presence of a
scattering condensate layer.
While polarization undoubtedly provides additional constraints on cloud
particle sizes (e.g., Bréon & Colzy, 2000), condensate phase (e.g., van
Diedenhoven et al., 2012a), and even asymmetry parameter (van Diedenhoven et
al., 2012b) and atmospheric structure, it may in practice be of limited value
for studies of exoplanets. Within the solar system the most well known
discovery attributable to a polarization measurement is the particle size of
the clouds of Venus (Hansen & Hovenier, 1974). An imaging photopolarimeter
carried on the Pioneer 10 and 11 spacecraft also constrained the vertical
structure of Jupiter’s clouds (Gehrels et al., 1974). However in general other
techniques have proven more valuable. Especially given the low signal to noise
and difficulty inherent in any measurement of an extrasolar planet, the value
of further dividing the light into polarization channels must be weighed
against other potential measurements (e.g., obtaining higher resolution
spectroscopy).
## 7 Conclusions
As with the planetary atmospheres of solar system planets, clouds are expected
to play major roles in the vertical structure, chemistry, and reflected and
emitted spectra of extrasolar planets. That said, the most extensive
experience to date with clouds outside of the solar system has been with the
brown dwarfs. The presence of refractory clouds in L dwarfs and sulphide and
salt clouds in late T dwarfs has been well established and a number of methods
have been developed to model these clouds.
Compared to cloud modeling approaches within the solar system or particularly
on Earth, exoplanet cloud models are still in their infancy. In many cases
arbitrary clouds are employed that specify a range of plausible cloud
properties that are usually sufficient to explore parameter space. More
sophisticated efforts attempt to derive particle sizes, composition, and
number density as a function of height through the atmosphere from a given set
of assumptions will be needed once higher resolution, broad wavelength
spectral data become available.
As of the time of this chapter’s writing the best evidence for clouds in
extrasolar planet atmospheres lies in the spectra of the planets orbiting the
nearby A star HR 8799. Planets b, c, and d each have red near-infrared colors
that are best explained by global refractory cloud decks that have persisted
to lower effective temperatures than in higher mass field brown dwarfs. This
persistence of clouds to lower effective temperatures for lower gravity
objects continues a trend that has already been recognized among brown dwarfs.
Among the transiting planets there is convincing evidence for high altitude
clouds–or perhaps a photochemical haze–in the atmosphere of HD 189733b. The
transit spectra of this planet lacks the deep absorption bands expected for a
clear atmosphere of absorbing gas but rather exhibits a smoothly varying
absorption profile likely caused by small grain scattering. A second planet,
GJ 1214b also has a transit spectrum lacking absorption features. In this case
the spectrum may be attributable either to a small atmospheric scale height
resulting from a high mean molecular weight composition or from clouds.
These early detections are like distant clouds seen at sunset presaging a
coming storm. The GPI, SPHERE, and many other direct imaging planet searches
are expected to discover dozens of young, self-luminous extrasolar giant
planets over the coming decade (Oppenheimer & Hinkley, 2009; Traub &
Oppenheimer, 2010). Many of these planets will exist in the effective
temperature range in which clouds shape their emergent spectra. Meanwhile
continued studies of transiting planets, particularly by the upcoming James
Webb Space Telescope will probe the atmospheres of the transiting planets and
test the hypothesis that hazes of photochemical origin are important in some
regimes. Finally future space based coronagraphic telescopes may eventually
image extrasolar giant and terrestrial planets in reflected light. All such
efforts to discover and characterize extrasolar planets will hinge upon an
understanding of the role clouds play in shaping the climate and atmospheres
of planets.
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|
arxiv-papers
| 2013-01-23T20:44:22 |
2024-09-04T02:49:40.657339
|
{
"license": "Public Domain",
"authors": "Mark S. Marley, Andrew S. Ackerman, Jeffrey N. Cuzzi, Daniel Kitzmann",
"submitter": "Daniel Kitzmann",
"url": "https://arxiv.org/abs/1301.5627"
}
|
1301.5666
|
# Characterizations of the Quaternionic Mannheim curves in Euclidean space
$E^{4}$
O. Zekİ Okuyucu Department of Mathematics, University of Bilecik Şeyh
Edebali, Turkey [email protected]
###### Abstract.
In [5], Matsuda and Yorozo obtained that Mannheim curves in 4-dimensional
Euclidean space. In this study, we define a quaternionic Mannheim curve and we
give some characterizations of them in Euclidean $3-$space and $4-$space.
###### Key words and phrases:
Quaternion algebra, Mannheim curve, Euclidean space
###### 2010 Mathematics Subject Classification:
11R52; 53A04.
## 1\. Introduction
The geometry of curves in a Euclidean space have been developed a long time
ago and we have a deep knowledge about it. In the theory of curves in
Euclidean space, one of the important and interesting problem is
characterizations of a regular curve. We can characterize some curves via
their relations between the Frenet vectors of them. For instance Mannheim
curve is a special curve and it is characterized using the Frenet vectors of
its Mannheim curve couple.
In 2007, the definition of Mannheim curves in Euclidean 3-space is given by H.
Liu and F. Wang [4] with the following:
###### Definition 1.1.
Let $\alpha$ and $\beta$ be two curves in Euclidean 3-space If there exists a
corresponding relationship between the space curves $\alpha$ and $\beta$ such
that, at the corresponding points of the curves, the principal normal lines of
$\alpha$ coincides with the binormal lines of $\beta$, then $\alpha$ is called
a Mannheim curve, and $\beta$ is called a Mannheim partner curve of $\alpha$.
In their paper, they proved that a given curve is a Mannheim curve if and only
if then for $\lambda\in\mathbb{R}$ it has
$\lambda\left(\varkappa^{2}+\tau^{2}\right)=\varkappa$ curve Also in 2009,
Matsuda and Yorozo, in [5], defined generalized Mannheim curves in Euclidean
4-space. If the first Frenet vector at each point of $\alpha$ is included in
the plane generated by the second Frenet vector and the third Frenet vector of
$\beta$ at corresponding point under a bijection, which is from $\alpha$ to
$\beta$. Then the curve $\alpha$ is called generalized Mannheim curve and the
curve $\beta$ is called generalized Mannheim mate curve of $\alpha$. And they
gave a theorem such that if the curve $\alpha$ is a generalized Mannheim curve
in Euclidean 4-space, then the first curvature function $k_{1}$ and second
curvature functions $k_{2}$ of the curve $\alpha$ satisfy the equality:
$k_{1}(s)=\mu\left\\{\left(k_{1}(s)\right)^{2}+\left(k_{2}(s)\right)^{2}\right\\}$
where $\mu$ is a positive constant number.
The quaternion was introduced by Hamilton. His initial attempt to generalize
the complex numbers by introducing a three-dimensional object failed in the
sense that the algebra he constructed for these three-dimensional object did
not have the desired proporties. On the 16th October 1843 Hamilton discovered
that the appropriate generalization is one in which the scalar(real) axis is
left unchanged whereas the vector(imaginary) axis is supplemented by adding
two further vectors axes.
In 1987, The Serret-Frenet formulas for a quaternionic curve in $E^{3}$ and
$E^{4}$ was defined by Bharathi and Nagaraj [7] and then in 2004, Serret-
Frenet formulas for quaternionic curves and quaternionic inclined curves have
been defined in Semi-Euclidean space by Çöken and Tuna in 2004 [1].
In 2011 Güngör and Tosun studied quaternionic rectifying curves [8]. Also, Gök
et.al [6, 3] defined a new kind of slant helix in Euclidean space $E^{4}$ and
semi-Euclidean space $E_{2}^{4}.$ It called quaternionic $B_{2}$-slant helix
in Euclidean space $E^{4}$ and semi-real quaternionic $B_{2}$-slant helix in
semi-Euclidean space $E_{2}^{4},$ respectively. Recently, Sağlam, in [2], has
studied on the osculating spheres of a real quaternionic curve in Euclidean
$4-$space.
In this study, we define quaternionic Mannheim curves and we give some
characterizations of them in Euclidean $3$ and $4$ space.
## 2\. Preliminaries
Let $Q_{H}$ denotes a four dimensional vector space over the field $H$ of
characteristic grater than 2. Let $e_{i}$ $(1\leq i\leq 4)$denote a basis for
the vector space. Let the rule of multiplication on $Q_{H}$ be defined on
$e_{i}$ $(1\leq i\leq 4)$ and extended to the whole of the vector space by
distributivity as follows:
A real quaternion is defined with
$q=a\overrightarrow{e_{1}}+b\overrightarrow{e_{2}}+c\overrightarrow{e_{3}}+de_{4}$
where $a,b,c,d$ are ordinary numbers. Such that
$\begin{array}[c]{cc}e_{4}=1,&e_{1}^{2}=e_{2}^{2}=e_{3}^{2}=-1,\\\
e_{1}e_{2}=e_{3},&e_{2}e_{3}=e_{1},\text{ }e_{3}e_{1}=e_{2},\\\
e_{2}e_{1}=-e_{3},&e_{3}e_{2}=-e_{1},\text{ }e_{1}e_{3}=-e_{2}.\end{array}$
(2.1)
If we denote $S_{q}=d$ and
$\overrightarrow{V_{q}}=a\overrightarrow{e_{1}}+b\overrightarrow{e_{2}}+c\overrightarrow{e_{3}}$,
we can rewrite real quaternions the basic algebraic form
$q=S_{q}+\overrightarrow{V_{q}}$ where $S_{q}$ is scalar part of $q$ and
$\overrightarrow{V_{q}}$ is vectorial part. Using these basic products we can
now expand the product of two quaternions to give
$p\times q=S_{p}S_{q}-\langle\overrightarrow{V_{p}},\text{
}\overrightarrow{V_{q}}\rangle+S_{p}\overrightarrow{V_{q}}+S_{q}\overrightarrow{V_{p}}+\overrightarrow{V_{p}}\wedge\overrightarrow{V_{q}}\text{
for every }p,q\in Q_{H},$ (2.2)
where we have use the inner and cross products in Euclidean space $E^{3}$ [7].
There is a unique involutory antiautomorphism of the quaternion algebra,
denoted by the symbol $\gamma$ and defined as follows:
$\gamma
q=-a\overrightarrow{e_{1}}-b\overrightarrow{e_{2}}-c\overrightarrow{e_{3}}+de_{4}\text{
for every
}q=a\overrightarrow{e_{1}}+b\overrightarrow{e_{2}}+c\overrightarrow{e_{3}}+de_{4}\in
Q_{H}$
which is called the “Hamiltonian conjugation”. This defines the symmetric,
real valued, non-degenerate, bilinear form $h$ are follows:
$h(p,\text{ }q)=\frac{1}{2}\left[\text{ }p\times\gamma q+q\times\gamma
p\right]\text{ for }p,q\in Q_{H}.$
And then, the norm of any $q$ real quaternion denotes
$\left\|q\right\|^{2}=h(q,q)=q\times\gamma q.$ (2.3)
The concept of a spatial quaternion will be used of throughout our work. $q$
is called a spatial quaternion whenever $q+\gamma q=0.$ $\left[2\right].$
The Serret-Frenet formulae for quaternionic curves in $E^{3}$ and $E^{4}$ are
follows:
###### Theorem 2.1.
The three-dimensional Euclidean space $E^{3}$ is identified with the space of
spatial quaternions $\left\\{p\in Q_{H}\text{ }\left|\text{ }p+\gamma
p=0\right.\right\\}$ in an obvious manner. Let $I=\left[0,1\right]$ denotes
the unit interval of the real line $\mathbb{R}.$ Let
$\alpha:I\subset\mathbb{R}\longrightarrow Q_{H}$
$s\longrightarrow\alpha(s)=\underset{i=1}{\overset{3}{\sum}}\alpha_{i}(s)\overrightarrow{e_{i}},\text{
}1\leq i\leq 3.$
be an arc-lenghted curve with nonzero curvatures $\left\\{k,\text{
}r\right\\}$ and $\left\\{t(s),\text{ }n(s),\text{ }b(s)\right\\}$ denotes the
Frenet frame of the curve $\alpha$. Then Frenet formulas are given by
$\left[\begin{array}[c]{c}t^{{}^{\prime}}\\\ n^{{}^{\prime}}\\\
b^{\prime}\end{array}\right]=\left[\begin{array}[c]{ccc}0&k&0\\\ -k&0&r\\\
0&-r&0\end{array}\right]\left[\begin{array}[c]{c}t\\\ n\\\
b\end{array}\right]$ (2.4)
where $k$ is the principal curvature, $r$ is torsion of $\alpha$ [7].
###### Theorem 2.2.
The four-dimensional Euclidean spaces $E^{4}$ is identified with the space of
quaternions. Let $I=\left[0,1\right]$ denotes the unit interval of the real
line $\mathbb{R}.$ Let
$\alpha^{\left(4\right)}:I\subset\mathbb{R}\longrightarrow Q_{H}$
$s\longrightarrow\alpha^{\left(4\right)}(s)=\underset{i=1}{\overset{4}{\sum}}\alpha_{i}(s)\overrightarrow{e_{i}},\text{
}1\leq i\leq 4,\text{ }\overrightarrow{e_{4}}=1.$
be a smooth curve in $E^{4}$ with nonzero curvatures $\left\\{K,\text{
}k,\text{ }r-K\right\\}$ and $\left\\{T(s),\text{ }N(s),\text{
}B_{1}(s),\text{ }B_{2}(s)\right\\}$ denotes the Frenet frame of the curve
$\alpha$. Then Frenet formulas are given by
$\left[\begin{array}[c]{c}T^{{}^{\prime}}\\\ N^{{}^{\prime}}\\\
B_{1}^{\prime}\\\
B_{2}^{\prime}\end{array}\right]=\left[\begin{array}[c]{cccc}0&K&0&0\\\
-K&0&k&0\\\ 0&-k&0&(r-K)\\\
0&0&-(r-K)&0\end{array}\right]\left[\begin{array}[c]{c}T\\\ N\\\ B_{1}\\\
B_{2}\end{array}\right]$ (2.5)
where $K$ is the principal curvature, $k$ is the torsion and $(r-K)$ is
bitorsion of $\alpha^{\left(4\right)}$ [7].
## 3\. Characterizations of the Quaternionic Mannheim curve
In this section, we define a quaternionic Mannheim curve and we give some
characterizations of them.in Euclidean $3$ and $4$ space.
###### Definition 3.1.
Let $\alpha\left(s\right)$ and $\beta\left(s^{\ast}\right)$ be two spatial
quaternionic curves in $\mathbb{E}^{3}.$ $\left\\{t(s),\text{ }n(s),\text{
}b(s)\right\\}$ and $\left\\{t^{\ast}(s^{\ast}),\text{
}n^{\ast}(s^{\ast}),\text{ }b^{\ast}(s^{\ast})\right\\}$ are Frenet
frames,respectively, on these curves. $\alpha\left(s\right)$ and
$\beta\left(s^{\ast}\right)$ are called spatial quaternionic Mannheim curves
if $n(s)$ and $b^{\ast}(s^{\ast})$ are linearly dependent.
###### Theorem 3.2.
Let $\alpha:I\subset\mathbb{R}\rightarrow$ $\mathbb{E}^{3}$ be a spatial
quaternionic Mannheim curve with the arc lenght parameter $s$ and
$\beta:I\subset\mathbb{R}\rightarrow\mathbb{E}^{3}$ be spatial quaternionic
Mannheim partner curve of $\alpha$ with the arc lenght parameter $s^{\ast}$.
Then
$d\left(\alpha\left(s\right),\beta\left(s^{\ast}\right)\right)=\text{constant,
\ \ for all }s\in I$
###### Proof.
From Definition $\left(\text{\ref{tan 3.1}}\right)$, we can write
$\alpha\left(s\right)=\beta\left(s^{\ast}\right)+\lambda\left(s^{\ast}\right)b\left(s^{\ast}\right)$
(3.1)
Differentiating the Eq. (3.1) with respect to $s^{\ast}$ and by using the
Frenet equation, we get
$\frac{d\alpha\left(s\right)}{ds}\frac{ds}{ds^{\ast}}=t^{\ast}(s^{\ast})+\lambda^{\shortmid}(s^{\ast})b^{\ast}(s^{\ast})-\lambda^{\ast}(s^{\ast})r^{\ast}(s^{\ast})n^{\ast}(s^{\ast})$
If we denote $\frac{d\alpha\left(s\right)}{ds}=t(s)$
$t(s)=\frac{ds^{\ast}}{ds}\left[t^{\ast}(s^{\ast})+\lambda^{\shortmid}(s^{\ast})b^{\ast}(s^{\ast})-\lambda^{\ast}(s^{\ast})r^{\ast}(s^{\ast})n^{\ast}(s^{\ast})\right]$
and
$h\left(t(s),n\left(s\right)\right)=\frac{ds^{\ast}}{ds}\left[\begin{array}[c]{c}h\left(t^{\ast}(s^{\ast}),n\left(s\right)\right)+\lambda^{\ast\prime}\left(s\right)h\left(b^{\ast}(s^{\ast}),n\left(s\right)\right)\\\
-\lambda^{\ast}(s^{\ast})r^{\ast}(s^{\ast})h\left(n^{\ast}(s^{\ast}),n\left(s\right)\right)\end{array}\right]$
Since $\left\\{n\left(s\right),b^{\ast}(s^{\ast})\right\\}$ is a linearly
dependent set, we get
$\lambda^{\ast\prime}\left(s\right)=0$
that is, $\lambda^{\ast}$ is constant function on $I.$ This completes the
proof. ∎
###### Theorem 3.3.
Let $\left\\{\alpha,\beta\right\\}$ be a Mannheim curve couple in
$\mathbb{E}^{3}.$Then measure of the angle between the tangent vector fields
of spatial quaternionic curves $\alpha(s)$ and $\beta(s^{\ast})$ is constant.
###### Proof.
Let $\alpha:I\subset\mathbb{R}\rightarrow$ $\mathbb{E}^{3}$ and
$\beta:I\subset\mathbb{R}\rightarrow$ $\mathbb{E}^{3}$ be spatial quaternionic
curves with arc-length $s$ and $s^{\ast}$ respectively. We show that
$h\left(t(s),t^{\ast}(s^{\ast})\right)=\cos\theta=\text{constant}$ (3.2)
Differentiating Eq. (3.2) with respect to $s,$ we get
$\displaystyle\frac{d}{ds}h\left(t(s),t^{\ast}(s^{\ast})\right)$
$\displaystyle=h\left(\frac{dt(s)}{ds},t^{\ast}(s^{\ast})\right)+h\left(t(s),\frac{dt^{\ast}(s^{\ast})}{ds^{\ast}}\frac{ds^{\ast}}{ds}\right)$
$\displaystyle=h\left(k(s)n(s),t^{\ast}(s^{\ast})\right)+h\left(t(s),k^{\ast}(s^{\ast})n^{\ast}(s^{\ast})\frac{ds^{\ast}}{ds}\right)$
$\displaystyle=0$
Thus,
$h\left(t(s),t^{\ast}(s^{\ast})\right)=\text{constant}$
∎
###### Theorem 3.4.
Let $\alpha:I\subset\mathbb{R}\rightarrow$ $\mathbb{E}^{3}$ be spatial
quaternionic curves with the arc-length parameter $s$. Then $\alpha$ is
spatial quaternionic Mannheim curve if and only if
$k(s)=\lambda\left(k^{2}(s)+r^{2}(s)\right)$
where $\lambda_{1},$ $\lambda_{2}$ are constants.
###### Proof.
If $\alpha$ is spatial quaternionic Mannheim curve, we can write
$\beta\left(s\right)=\alpha\left(s\right)+\lambda(s)n\left(s\right)$
Differentiating the above equality and by using the Frenet equations, we get
$\frac{d\beta(s)}{ds}=\left[\left(1-\lambda(s)k\left(s\right)\right)t(s)+\lambda^{\shortmid}(s)n(s)+\lambda(s)r\left(s\right)b\left(s\right)\right]$
as $\left\\{n\left(s\right),b^{\ast}(s^{\ast})\right\\}$ is a linearly
dependent set, we get
$\lambda^{\shortmid}(s)=0.$
This means that $\lambda$ is constant. Thus we have
$\frac{d\beta(s)}{ds}=\left(1-\lambda k\left(s\right)\right)t(s)+\lambda
r\left(s\right)b\left(s\right).$
On the other hand, we have
$t^{\ast}=\frac{d\beta}{ds}\frac{ds}{ds^{\ast}}=\left[\left(1-\lambda
k\left(s\right)\right)t(s)+\lambda
r\left(s\right)b\left(s\right)\right]\frac{ds}{ds^{\ast}}.$
By taking the derivative of this equation with respect to $s^{\ast}$ and
appliying the Frenet formulas we obtain
$\displaystyle\frac{dt^{\ast}}{ds}\frac{ds}{ds^{\ast}}$
$\displaystyle=\left[-\lambda k^{\shortmid}(s)t(s)+\left(k(s)-\lambda
k^{2}(s)-\lambda r^{2}(s)\right)n(s)+\lambda
r^{\shortmid}(s)b(s)\right]\left(\frac{ds}{ds^{\ast}}\right)^{2}$
$\displaystyle+\left[\left(1-\lambda k\left(s\right)\right)t(s)+\lambda
r\left(s\right)b\left(s\right)\right]\frac{d^{2}s}{ds^{\ast^{2}}}$
From this equation we get
$k(s)=\lambda\left(k^{2}(s)+r^{2}(s)\right).$
Conversely, if $k(s)=\lambda\left(k^{2}(s)+r^{2}(s)\right),$ then we can
easily see that $\alpha$ is a Mannheim curve. ∎
###### Theorem 3.5.
Let $\alpha:I\subset\mathbb{R}\rightarrow$ $\mathbb{E}^{3}$ be spatial
quaternionic Mannheim curve with arc-length parameter $s$. Then $\beta$ is the
spatial quaternionic Mannheim partner curve of $\alpha$ if and only if the
curvature functions $k^{\ast}\left(s^{\ast}\right)$ and $r^{\ast}(s^{\ast})$
of $\beta$ satisfy the following equation
$\frac{dr^{\ast}}{ds^{\ast}}=\frac{k^{\ast}}{\mu}\left(1+\mu^{2}r^{\ast^{2}}\right),$
where $\mu$ is constant.
###### Proof.
Let $\alpha:I\subset\mathbb{R}\rightarrow$ $\mathbb{E}^{3}$ be spatial
quaternionic Mannheim curve. Then, we can write
$\alpha\left(s^{\ast}\right)=\beta\left(s^{\ast}\right)+\mu\left(s^{\ast}\right)b^{\ast}\left(s^{\ast}\right)$
(3.3)
for some function $\mu\left(s^{\ast}\right)$. By taking the derivative of this
equation with respect to $s^{\ast}$ and using the Frenet equations we obtain
$t\frac{ds}{ds^{\ast}}=t^{\ast}\left(s^{\ast}\right)+\mu^{\shortmid}\left(s^{\ast}\right)b^{\ast}\left(s^{\ast}\right)-\mu\left(s^{\ast}\right)r^{\ast}\left(s^{\ast}\right)n^{\ast}\left(s^{\ast}\right).$
And then, we know that
$\left\\{n\left(s\right),b^{\ast}\left(s^{\ast}\right)\right\\}$ is a linearly
dependent set, so we have
$\mu^{\shortmid}\left(s^{\ast}\right)=0.$
This means that $\mu\left(s^{\ast}\right)$ is a constant function. Thus we
have
$t\frac{ds}{ds^{\ast}}=t^{\ast}\left(s^{\ast}\right)-\mu
r^{\ast}\left(s^{\ast}\right)n^{\ast}\left(s^{\ast}\right).$ (3.4)
On the other hand, we have
$t=t^{\ast}\cos\theta+n^{\ast}\sin\theta$ (3.5)
where $\theta$ is the angle between $t$ and $t^{\ast}$ at the corresponding
points of $\alpha$ and $\beta.$ By taking the derivative of this equation with
respect to $\overline{s}$ and using the Frenet equations we obtain
$kn\frac{ds}{ds^{\ast}}=-\left(k^{\ast}+\frac{d\theta}{ds^{\ast}}\right)\sin\theta
t^{\ast}+\left(k^{\ast}+\frac{d\theta}{ds^{\ast}}\right)\cos\theta
n^{\ast}+r^{\ast}\sin\theta b^{\ast}.$
From this equation and the fact that the
$\left\\{n\left(s\right),b^{\ast}\left(s^{\ast}\right)\right\\}$ is a linearly
dependent set, we get
$\left\\{\begin{array}[c]{c}\left(k^{\ast}+\frac{d\theta}{ds^{\ast}}\right)\sin\theta=0\\\
\left(k^{\ast}+\frac{d\theta}{ds^{\ast}}\right)\cos\theta=0.\end{array}\right.$
For this reason we have
$\frac{d\theta}{ds^{\ast}}=-k^{\ast}.$ (3.6)
From the Eq. (3.4) and Eq. (3.5) and notice that $t^{\ast}$ is orthogonal to
$b^{\ast}$, we find that
$\frac{ds}{ds^{\ast}}=\frac{1}{\cos\theta}=-\frac{\mu r^{\ast}}{\sin\theta}.$
Then we have
$\mu r^{\ast}=-\tan\theta.$
By taking the derivative of this equation and applying Eq. $(3.6)$, we get
$\mu\frac{dr^{\ast}}{ds^{\ast}}=k^{\ast}\left(1+\mu^{2}r^{\ast^{2}}\right)$
that is
$\frac{dr^{\ast}}{ds^{\ast}}=\frac{k^{\ast}}{\mu}\left(1+\mu^{2}r^{\ast^{2}}\right).$
Conversely, if the curvature $k^{\ast}$ and torsion $r^{\ast}$ of the curve
$\beta$ satisfy
$\frac{dr^{\ast}}{ds^{\ast}}=\frac{k^{\ast}}{\mu}\left(1+\mu^{2}r^{\ast^{2}}\right)$
for constant $\mu,$ then we define a curve a curve by
$\alpha\left(s^{\ast}\right)=\beta\left(s^{\ast}\right)+\mu
b^{\ast}\left(s^{\ast}\right)$ (3.7)
and we will show that $\alpha$ is a spatial quaternionic Mannheim curve and
$\beta$ is the spatial quaternionic partner curve of $\alpha$. By taking the
derivative of Eq. $(3.7)$ with respect to $\overline{s}$ twice, we get
$t\frac{ds}{ds^{\ast}}=t^{\ast}-\mu r^{\ast}n^{\ast},$ (3.8)
$kn\left(\frac{ds}{ds^{\ast}}\right)^{2}+t\frac{d^{2}s}{ds^{\ast^{2}}}=\mu
k^{\ast}r^{\ast}t^{\ast}+\left(k^{\ast}-\mu\frac{dr^{\ast}}{ds^{\ast}}\right)n^{\ast}-\mu
r^{\ast^{2}}b^{\ast}$ (3.9)
respectively. Taking the cross product of Eq. (3.8) with Eq. (3.9) and
noticing that
$k^{\ast}-\mu\frac{dr^{\ast}}{ds^{\ast}}+\mu^{2}k^{\ast}r^{\ast^{2}}=0,$
we have
$kb\left(\frac{ds}{ds^{\ast}}\right)^{3}=\mu^{2}r^{\ast^{3}}t^{\ast}+\mu
r^{\ast^{2}}n^{\ast}.$ (3.10)
By taking the cross product of Eq. (3.8) with Eq. (3.10), we get
$kn\left(\frac{ds}{ds^{\ast}}\right)^{4}=-\mu
r^{\ast^{2}}\left(1+\mu^{2}r^{\ast^{2}}\right)b^{\ast}.$
This means that the principal normal vector field of the spatial quaternionic
curve $\alpha$ and binormal vector field of the spatial quaternionic curve
$\beta$ are linearly dependent set. And so $\alpha$ is a spatial quaternionic
Mannheim curve and $\beta$ is spatial quaternionic Mannheim partner curve of
$\alpha$. ∎
###### Definition 3.6.
A quaternionic curve $\alpha^{\left(4\right)}:I\subset\mathbb{R}\rightarrow$
$\mathbb{E}^{4}$ is a quaternionic Mannheim curve if there exists a
quaternionic curve
$\beta^{\left(4\right)}:\overline{I}\subset\mathbb{R}\rightarrow$
$\mathbb{E}^{4}$ such that the second Frenet vector at each point of
$\alpha^{\left(4\right)}$ is included the plane generated by the third Frenet
vector and the fourth Frenet vector of $\beta^{\left(4\right)}$ at
corresponding point under $\varphi,$ where $\varphi$ is a bijection from
$\alpha^{\left(4\right)}$ to $\beta^{\left(4\right)}$. The curve
$\beta^{\left(4\right)}$ is called the quaternionic Mannheim partner curve of
$\alpha^{\left(4\right)}.$
###### Theorem 3.7.
Let $\alpha^{\left(4\right)}:I\subset\mathbb{R}\rightarrow$ $\mathbb{E}^{4}$
and $\beta^{\left(4\right)}:\overline{I}\subset\mathbb{R}\rightarrow$
$\mathbb{E}^{4}$ be quaternionic Mannheim curve couple with arc-length $s$ and
$\overline{s},$ respectively. Then
$d\left(\alpha^{\left(4\right)}\left(s\right),\beta^{\left(4\right)}\left(\overline{s}\right)\right)=\lambda(s)=\text{constant,
\ for all }s\in I$ (3.11)
###### Proof.
From the Definition $\left(\text{\ref{tan 3.2}}\right)$, quaternionic Mannheim
partner curve $\beta^{\left(4\right)}$ of $\alpha^{\left(4\right)}$ is given
by the following equation
$\beta^{\left(4\right)}\left(s\right)=\alpha^{\left(4\right)}(s)+\lambda(s)N\left(s\right).$
where $\lambda\left(s\right)$ is a smooth function. A smooth function
$\psi:s\in I\rightarrow\psi\left(s\right)=\overline{s}\in\overline{I}$ is
defined by
$\psi\left(s\right)=\overset{s}{\underset{0}{\int}}\left\|\frac{d\alpha^{\left(4\right)}\left(s\right)}{ds}\right\|ds=\overline{s}.$
The bijection
$\varphi$:$\alpha^{\left(4\right)}\rightarrow\beta^{\left(4\right)}$ is
defined by
$\varphi\left(\alpha^{\left(4\right)}(s)\right)=\beta^{\left(4\right)}(\psi\left(s\right)).$
Since the second Frenet vector at each point of $\alpha^{\left(4\right)}$ is
included the plane generated by the third Frenet vector and the fourth Frenet
vector of $\beta^{\left(4\right)}$ at corresponding point under $\varphi,$ for
each $s\in I,$ the Frenet vector $N(s)$ is given by the linear combination of
Frenet vectors $\overline{B}_{1}\left(\psi\left(s\right)\right)$ and
$\overline{B}_{2}\left(\psi\left(s\right)\right)$ of $\beta^{\left(4\right)},$
that is, we can write
$N(s)=g(s)\overline{B}_{1}\left(\psi\left(s\right)\right)+h(s)\overline{B}_{2}\left(\psi\left(s\right)\right),$
where $g(s)$ and $h(s)$ are smooth functions on $I.$ So we can write
$\beta^{\left(4\right)}\left(\psi(s)\right)=\alpha^{\left(4\right)}\left(s\right)+\lambda(s)N\left(s\right).$
(3.12)
Differentiating Eq. (3.12) with respect to $s$ and by using the Frenet
equations, we get
$\overline{T}\left(\psi\left(s\right)\right)\psi^{\shortmid}(s)=\left[\left(1-\lambda
K\left(s\right)\right)T(s)+\lambda^{\prime}\left(s\right)N\left(s\right)+\lambda\left(s\right)k\left(s\right)B_{1}\left(s\right)\right].$
By the fact that:
$h\left(\overline{T}\left(\varphi\left(s\right)\right),g(s)\overline{B}_{1}\left(\psi\left(s\right)\right)+h(s)\overline{B}_{2}\left(\psi\left(s\right)\right)\right)=0,$
we have
$\lambda^{\prime}\left(s\right)=0$
that is, $\lambda\left(s\right)$ is constant function on $I.$ This completes
the proof. ∎
###### Theorem 3.8.
If the quaternionic curve
$\alpha^{\left(4\right)}:I\subset\mathbb{R}\rightarrow$ $\mathbb{E}^{4}$ is a
quaternionic Mannheim curve, then the first and second curvature functions of
$\alpha^{\left(4\right)}$ satisfy the equality:
$K(s)=\lambda\left\\{K^{2}(s)+k^{2}(s)\right\\}$
where $\lambda$ is constant.
###### Proof.
Let $\beta^{\left(4\right)}$ be a quaternionic Mannheim partner curve of
$\alpha^{\left(4\right)}.$ Then we can write
$\beta^{\left(4\right)}\left(\psi\left(s\right)\right)=\alpha^{\left(4\right)}\left(s\right)+\lambda
N\left(s\right)$
Differentiating, we get
$\overline{T}\left(\psi\left(s\right)\right)\psi^{\shortmid}(s)=\left[\left(1-\lambda
K\left(s\right)\right)T(s)+\lambda k\left(s\right)B_{1}\left(s\right)\right],$
taht is,
$\overline{T}\left(\psi\left(s\right)\right)=\frac{1-\lambda
K\left(s\right)}{\psi^{\shortmid}(s)}T(s)+\frac{\lambda
k\left(s\right)}{\psi^{\shortmid}(s)}B_{1}\left(s\right)$
where $\psi^{\shortmid}(s)=\sqrt{\left(1-\lambda K(s)\right)^{2}+\left(\lambda
k(s)\right)^{2}}$ for $s\in I.$ By differentiation of both sides of the above
equality with respect to $s$, we have
$\displaystyle\psi^{\shortmid}(s)\overline{K}\left(\overline{s}\right)\overline{N}\left(\overline{s}\right)$
$\displaystyle=\left(\frac{1-\lambda
K\left(s\right)}{\psi^{\shortmid}(s)}\right)^{\shortmid}T(s)$
$\displaystyle+\left(\frac{\left(1-\lambda K\left(s\right)\right)K(s)-\lambda
k\left(s\right)^{2}}{\psi^{\shortmid}(s)}\right)N(s)$
$\displaystyle+\left(\frac{\lambda
k\left(s\right)}{\psi^{\shortmid}(s)}\right)^{\shortmid}B_{1}\left(s\right)-\frac{\lambda
k\left(s\right)\left(r(s)-K(s)\right)}{\psi^{\shortmid}(s)}B_{2}\left(s\right).$
By the fact:
$h\left(\overline{N}\left(\varphi\left(s\right)\right),g(s)\overline{B}_{1}\left(\psi\left(s\right)\right)+h(s)\overline{B}_{2}\left(\psi\left(s\right)\right)\right)=0,$
we have that coefficient of $N$ in the above equation is zero, that is,
$\left(1-\lambda K\left(s\right)\right)K(s)-\lambda k\left(s\right)^{2}=0.$
Thus, we have
$K(s)=\lambda\left\\{K^{2}(s)+k^{2}(s)\right\\}$
for $s\in I$. This completes the proof. ∎
###### Theorem 3.9.
Let $\alpha^{\left(4\right)}:I\subset\mathbb{R}\rightarrow$ $\mathbb{E}^{4}$
be quaternionic curven with arc-length $s$ whose curvature functions $K(s)$
and $k(s)$ are non-constant functions and satisfy the equality:
$K(s)=\lambda\left\\{K^{2}(s)+k^{2}(s)\right\\},$ where $\lambda$ is constant.
If the quaternıonic curve $\beta^{\left(4\right)}$ is given by
$\beta^{\left(4\right)}\left(\overline{s}\right)=\alpha^{\left(4\right)}\left(s\right)+\lambda
N\left(s\right),$ then $\alpha^{\left(4\right)}$ is a quaternionic Mannheim
curve and $\beta^{\left(4\right)}$ is the quaternionic Mannheim partner curve
of $\alpha^{\left(4\right)}$.
###### Proof.
Let $\overline{s}$ be the arc-length of the quaternionic curve
$\beta^{\left(4\right)}$. That is, $\overline{s}$ is defined by
$\overline{s}=\overset{s}{\underset{0}{\int}}\left\|\frac{d\alpha^{\left(4\right)}\left(s\right)}{ds}\right\|ds$
for $s\in I$. We can write a smooth function $\psi:s\in
I\rightarrow\psi\left(s\right)=\overline{s}\in\overline{I}$. By the assumption
of the curvature functions $K(s)$ and $k(s)$, we have
$\displaystyle\psi^{\shortmid}(s)$ $\displaystyle=\sqrt{\left(1-\lambda
K(s)\right)^{2}+\left(\lambda k(s)\right)^{2}},$
$\displaystyle\psi^{\shortmid}(s)$ $\displaystyle=\sqrt{1-\lambda K(s)}$
for $s\in I$. Then we can easily write
$\displaystyle\beta^{\left(4\right)}\left(\overline{s}\right)$
$\displaystyle=\beta^{\left(4\right)}\left(\psi\left(s\right)\right)$
$\displaystyle=\alpha^{\left(4\right)}\left(s\right)+\lambda N\left(s\right)$
for the quaternionic curve $\beta^{\left(4\right)}$. If we differentiate both
sides of the above equality with respect to $s,$ we get
$\psi^{\shortmid}(s)\overline{T}\left(\psi(s)\right)=T(s)+\lambda\left\\{-K(s)T(s)+k(s)B_{1}(s)\right\\}.$
And so we have,
$\overline{T}\left(\psi(s)\right)=\sqrt{1-\lambda K(s)}T(s)+\frac{\lambda
k(s)}{\sqrt{1-\lambda K(s)}}B_{1}(s).$ (3.13)
Differentiating the above equality with respect to $s$ and by using the Frenet
equations, we get
$\displaystyle\psi^{\shortmid}(s)\overline{K}\left(\psi(s)\right)\overline{N}\left(\psi(s)\right)$
$\displaystyle=\left(\sqrt{1-\lambda
K(s)}\right)^{\shortmid}T(s)+\left(\frac{K(s)\left(1-\lambda
K(s)\right)-\lambda k^{2}(s)}{\sqrt{1-\lambda K(s)}}\right)N(s)$
$\displaystyle+\left(\frac{\lambda k(s)}{\sqrt{1-\lambda
K(s)}}\right)^{\shortmid}B_{1}(s)+\frac{\lambda
k(s)\left(r(s)-K(s)\right)}{\sqrt{1-\lambda K(s)}}B_{2}(s)$
From our assumption, it holds
$\frac{K(s)\left(1-\lambda K(s)\right)-\lambda k^{2}(s)}{\sqrt{1-\lambda
K(s)}}=0.$
We find the coefficient of $N(s)$ in the above equality vanishes. Thus the
vector $\overline{N}\left(\psi(s)\right)$ is given by linear combination of
$T(s),$ $B_{1}(s)$ and $B_{2}(s)$ for each $s\in I.$ And the vector
$\overline{T}\left(\psi(s)\right)$ is given by linear combination of $T(s)$
and $B_{1}(s)$ for each $s\in I$ in the Eq. (3.13). As the curve
$\beta^{\left(4\right)}$ is quaternionic curve in $\mathbb{E}^{4}$, the vector
$N(s)$ is given by linear combination of
$\overline{B}_{1}\left(\overline{s}\right)$ and
$\overline{B}_{2}\left(\overline{s}\right)$. For this reason, the second
Frenet curve at each point of $\alpha^{(4)}$ is included in the plane
generated the third Frenet vector and the fourth Frenet vector of
$\beta^{\left(4\right)}$ at corresponding point under $\varphi$. Here the
bijection $\varphi:\alpha^{\left(4\right)}\rightarrow\beta^{\left(4\right)}$
is defined by
$\varphi\left(\alpha^{\left(4\right)}(s)\right)=\beta^{\left(4\right)}(\psi\left(s\right))$.
This completes the proof. ∎
## References
* [1] A. C. Çöken, A. Tuna, On the quaternionic inclined curves in the semi-Euclidean space $E_{2}^{4}$, Appl. Math. Comput., 155 (2004), 373-389.
* [2] D. Sağlam, On the Osculating Spheres of a Real Quaternionic Curve in the Euclidean Space $E^{4}$, International J.Math. Combin., Vol.3(2012), 46-53.
* [3] F. Kahraman, İ. Gök, H. H. Hacısalihoğlu, On the Quaternionic $B_{2}$ Slant helices In the semi-Euclidean Space $E_{2}^{4},$ Appl. Math. Comput., 218 (2012) 6391–6400.
* [4] H., Liu, and F., Wang, Mannheim partner curves in 3-space, J. Geom., 88 (2008) 120 – 126.
* [5] H. Matsuda and S. Yorozu, On Generalized Mannheim Curves in Euclidean 4-Space, Nihonkai Math. J., Vol.20 (2009), 33–56.
* [6] İ. Gök, O.Z. Okuyucu, F. Kahraman, H. H. Hacısalihoğlu. On the Quaternionic $B_{2}-$ Slant Helices in the Euclidean Space $E^{4}.$ Adv. Appl. Cliff ord Algebras, DOI 10.1007/s00006-011-0284-6.
* [7] K. Bharathi, M. Nagaraj, Quaternion valued function of a real Serret-Frenet formulae, Indian J. Pure Appl. Math. 16 (1985) 741-756.
* [8] M. A. Güngör, M. Tosun, Some characterizations of quaternionic rectifying curves, Diff. Geom. Dyn. Syst.,Vol.13, (2011), 89-100.
|
arxiv-papers
| 2013-01-23T22:45:14 |
2024-09-04T02:49:40.678910
|
{
"license": "Public Domain",
"authors": "O. Zeki Okuyucu",
"submitter": "Osman Zeki Okuyucu",
"url": "https://arxiv.org/abs/1301.5666"
}
|
1301.5850
|
# The neat embedding problem for algebras other than cylindric algebras and
for infinite dimensions
Robin Hirsch and Tarek Sayed Ahmed
###### Abstract
Hirsch and Hodkinson proved, for $3\leq m<\omega$ and any $k<\omega$, that the
class $S{\mathfrak{Nr}}_{m}{\bf CA}_{m+k+1}$ is strictly contained in
$S{\mathfrak{Nr}}_{m}{\bf CA}_{m+k}$ and if $k\geq 1$ then the former class
cannot be defined by any finite set of first order formulas, within the latter
class. We generalise this result to the following algebras of $m$-ary
relations for which the neat reduct operator ${\mathfrak{Nr}}_{m}$ is
meaningful: polyadic algebra with or without equality and substitution
algebra. We also generalise this result to allow the case where $m$ is an
infinite ordinal, using quasipolyadic algebra in place of polyadic algebra
(with or without equality). 111 Mathematics Subject Classification: 03G15,
03C10. Key words: algebraic logic, cylindric algebras, quasi-polyadic
algebras, substitution algebras, neat reducts, neat embeddings.
Cylindric algebra is an algebraic correspondent of first-order logic with no
constants or functions, more specifically $n$-dimensional cylindric algebra,
${\bf CA}_{n}$, is an algebraic correspondent of first-order logic restricted
to $n$ indexed variables, for finite $n$. An algebra in ${\bf CA}_{n}$ is a
boolean algebra together with a cylindrfier ${\sf c}_{i}$, which acts as a
unary operator and corresponds to existential quantification of the $i$’th
variable, and a diagonal ${\sf d}_{ij}$ element corresponding to the equaltify
of the $i$th and $j$’ variable, for $i,j<n$. For $m<n$, the _neat reduct_
${\mathfrak{Nr}}_{m}{\mathcal{C}}$ of a ${\mathcal{C}}\in{\bf CA}_{n}$ is the
$m$-dimensional cylindric algebra obtained by restricting to those elements
$c\in{\mathcal{C}}$ such that ${\sf c}_{i}c=c$, for $m\leq i<n$ and
restricting to those cylindrifiers and diagonals indexed by $m$. It is not the
case that every algebra in ${\bf CA}_{m}$ is the neat reduct of an algebra in
${\bf CA}_{n}$, nor need it even be a subalgebra of a neat reduct of an
algebra in ${\bf CA}_{n}$. Furthermore, $S{\mathfrak{Nr}}_{m}{\bf
CA}_{m+k+1}\neq S{\mathfrak{Nr}}_{m}{\bf CA}_{m}$, whenever $3\leq m<\omega$
and $k<\omega$ [6]. A consequence of this is that there are $m$ variable
formulas that can be proved with $m+k+1$ variables, but not with $m+k$
variables, in a certain, fairly typical, proof system.
Other algebras may be defined corresponding to restrictions or extensions of
the $n$ variable first order logic described above. Because our focus is on
neat reducts, we will only consider $n$-dimensional algebras where the
cylindrifiers ${\sf c}_{i}$ are included, or at least are definable, within
the set of operators of the algebra. Without that restriction it would not be
possible to define a neat reduct and our algebras would correspond to first
order logic without quantifiers, we do not consider that case here. But we
might choose to drop the diagonals from our signature (corresponding to first
order logic without equality), or we may add permutation operators,
corresponding to permutations of the variables in first order logic. By
generalising the results about neat reducts of $m$ dimensional cylindric
algebras to other $m$ dimensional algebras, such as polyadic algebras with or
without equality, diagonal free cylindric algebras, substitution algebras, one
may deduce that there are $m$ variable formulas provable with $m+k+1$
variables but not with $m+k$ variables in logical proof systems similar to the
one in [7], but with additional inference rules (corresponding to polyadic
equality algebras $({\bf PEA}))$ or fewer weaker inference rules
(corresponding to substitution algebras $({\bf Sc}))$.
## Preliminaries
For cardinals $m,n$ we write ${}^{m}n$ for the cardinal of the set of maps
from $m$ to $n$. If $U$ is an ultrafilter over $\wp(I)$ and if
${\mathcal{A}}_{i}$ is some structure (for $i\in I$) we write either
$\Pi_{i\in I}{\mathcal{A}}_{i}/U$ or $\Pi_{i/U}{\mathcal{A}}_{i}$ for the
ultraproduct of the ${\mathcal{A}}_{i}$ over $U$. Fix some ordinal $n\geq 2$.
For $i,j<n$ the replacement $[i/j]$ is the map that is like the identity on
$n$, except that $i$ is mapped to $j$ and the transposition $[i,j]$ is the
like the identity on $n$, except that $i$ is swapped with $j$. A map
$\tau:n\rightarrow n$ is finitary if the set $\\{i<n:\tau(i)\neq i\\}$ is
finite, so if $n$ is finite then all maps $n\rightarrow n$ are finitary. It is
known, and indeed not hard to show, that any finitary permutation is a product
of transpositions and any finitary non-injective map is a product of
replacements.
The standard reference for all the classes of relations mentioned previously
is [4]. Each class in $\\{{\bf Df}_{n},{\bf Sc}_{n},{\bf CA}_{n},{\bf
PA}_{n},{\bf PEA}_{n},{\bf QPA}_{n},{\bf QPEA}_{n}\\}$ consists of boolean
algebras with extra operators, as shown in figure 1, where ${\sf d}_{ij}$ is a
nullary operator (constant), ${\sf c}_{i},{\sf s}_{\tau},{\sf s}_{[{i}/{j}]}$
and ${\sf s}_{[i,j]}$ are unary operators, for $i,j<n,\;\tau:n\rightarrow n$.
For finite $n$, polyadic algebras are the same as quasi-polyadic algebra and
for the infinite dimensional case we restrict our attention to quasi-polyadic
algebras in ${\bf QPA}_{n},{\bf QPEA}_{n}$. Each class is defined by a finite
axiom schema of equations. Existing in a somewhat scattered form in the
literature, equations defining ${\bf Sc}_{n},{\bf QPA}_{n}$ and ${\bf
QPEA}_{n}$ are given in the appendix, definition 15. For ${\bf CA}_{n}$ we
follow the standard axiomatization given in definition 1.1.1 in [3]. For any
operator $o$ of any of these signatures, we write ${\sf
dim}(o)\;(\subseteq\alpha)$ for the set of dimension ordinals used by $o$,
e.g. ${\sf dim}({\sf c}_{i})=\\{i\\},\;{\sf dim}({\sf
s}_{[{i}/{j}]})=\\{i,j\\}$. An algebra ${\mathcal{A}}$ in ${\bf QPEA}_{n}$ has
operators that can define any operator of ${\bf QPA}_{n},{\bf CA}_{n},\;{\bf
Sc}_{n}$ and ${\bf Df}_{n}$, thus we may obtain the reducts
${\mathfrak{Rd}}_{\bf K}({\mathcal{A}})$ for ${\bf K}\in\\{{\bf QPEA}_{n},{\bf
QPA}_{n},{\bf CA}_{n},{\bf Sc}_{n},{\bf Df}_{n}\\}$ and it turns out that the
reduct always satisfies the equations defining the relevant class so
${\mathfrak{Rd}}_{\bf K}({\mathcal{A}})\in{\bf K}$, similarly from any algebra
${\mathcal{A}}$ in any of the classes ${\bf QPEA}_{n},{\bf QPA}_{n},{\bf
CA}_{n},{\bf Sc}_{n}$ we may obtain the reduct ${\mathfrak{Rd}}_{\bf
Sc}({\mathcal{A}})\in{\bf Sc}_{n}$ [1].
$\begin{array}[]{l|l}\mbox{class}&\mbox{extra operators}\\\ \hline\cr{\bf
Df}_{n}&{\sf c}_{i}:i<n\\\ {\bf Sc}_{n}&{\sf c}_{i},{\sf s}_{[i/j]}:i,j<n\\\
{\bf CA}_{n}&{\sf c}_{i},{\sf d}_{ij}:i,j<n\\\ {\bf PA}_{n}&{\sf c}_{i},{\sf
s}_{\tau}:i<n,\;\tau\in\;^{n}n\\\ {\bf PEA}_{n}&{\sf c}_{i},{\sf d}_{ij},{\sf
s}_{\tau}:i,j<n,\;\tau\in\;^{n}n\\\ {\bf QPA}_{n}&{\sf c}_{i},{\sf
s}_{[i/j]},{\sf s}_{[i,j]}:i,j<n\\\ {\bf QPEA}_{n}&{\sf c}_{i},{\sf
d}_{ij},{\sf s}_{[i/j]},{\sf s}_{[i,j]}:i,j<n\end{array}$ Figure 1: Non-
boolean operators for the classes
Let ${\bf K}\in\\{{\bf QPEA},{\bf QPA},{\bf CA},{\bf Sc},{\bf Df}\\}$, let
${\mathcal{A}}\in{\bf K}_{n}$ and let $2\leq m\leq n$ (possibly infinite
ordinals). The _reduct to $m$ dimensions_
${\mathfrak{Rd}}_{m}({\mathcal{A}})\in{\bf K}_{m}$ is obtained from
${\mathcal{A}}$ by discarding all operators with indices $m\leq i<n$. The
_neat reduct to $m$ dimensions_ is the algebra
${\mathfrak{Nr}}_{m}({\mathcal{A}})\in{\bf K}_{m}$ with base
$\\{a\in{\mathcal{A}}:m\leq i<n\rightarrow{\sf c}_{i}a=a\\}$ where all the
operators are induced from ${\mathcal{A}}$ (see [3, definition 2.6.28] for the
${\bf CA}$ case). More generally, for $\Gamma\subseteq n$ we write
${\mathfrak{Nr}}_{\Gamma}{\cal{A}}$ for the algebra whose base is
$\\{a\in{\mathcal{A}}:i\in n\setminus\Gamma\rightarrow{\sf c}_{i}a=a\\}$ with
all the operators $o$ of ${\mathcal{A}}$ where ${\sf dim}(o)\subseteq\Gamma$.
Let ${\mathcal{A}}\in{\bf K}_{m},\;{\mathcal{B}}\in{\bf K}_{n}$. An injective
homomorphism $f:{\mathcal{A}}\rightarrow{\mathcal{B}}$ is a _neat embedding_
if the range of $f$ is a subalgebra of ${\mathfrak{Nr}}_{m}({\mathcal{B}})$.
The notions of neat reducts and neat embeddings have proved useful in
analyzing the number of variables needed in proofs, as well as for proving
representability results, via the so-called neat embedding theorems [16, 11,
12, 17]. For ${\bf K}$ as above and $\alpha$ an ordinal, let ${\bf
RK}_{\alpha}$ denote the set of representable algebras in ${\bf K}_{\alpha}$.
Then a typical neat embedding theorem, essentially due to Henkin [4], states
that ${\bf RK}_{\alpha}=S{\mathfrak{Nr}}_{\alpha}{\bf
K}_{\alpha+\omega}=\bigcap_{k\in\omega}S{\mathfrak{Nr}}_{\alpha}{\bf
K}_{\alpha+k}$; this is known to hold for all such ${\bf K}$ except for ${\bf
Df}$.
Let $\alpha\leq\beta$ be ordinals and let $\rho:\alpha\rightarrow\beta$ be an
injection. For any $\beta$-dimensional algebra ${\mathcal{B}}$ (substution,
cylindric or quasi-polyadic algebra with or without equality) we define an
$\alpha$-dimensional algebra ${\mathfrak{Rd}}^{\rho}({\mathcal{B}})$, with the
same base and boolean structure as ${\mathcal{B}}$, where the $(ij)$th
diagonal of ${\mathfrak{Rd}}^{\rho}({\mathcal{B}})$ is ${\sf
d}_{\rho(i)\rho(j)}\in{\mathcal{B}}$ (if diagonals are included in the
signature of the algebra), the $i$th cylindrifier is ${\sf c}_{\rho(i)}$, the
$i$ for $j$ replacement operator is the operator ${\sf s}^{\rho(i)}_{\rho(j)}$
of ${\mathcal{A}}$, the $ij$ transposition operator is ${\sf
s}_{\rho(i)\rho(j)}$ (if included in the signature), for $i,j<\alpha$. It is
easy to check, for ${\bf K}\in\\{{\bf Df},{\bf Sc},{\bf CA},{\bf QPA},{\bf
QPEA}\\}$, that if ${\mathcal{B}}\in{\bf K}_{\beta}$ then
${\mathfrak{Rd}}^{\rho}({\mathcal{B}})\in{\bf K}_{\alpha}$. Also, for
${\mathcal{B}}\in{\bf K}_{\beta}$ and $x\in{\mathcal{B}}$, we define
${\mathfrak{Rl}}_{x}({\mathcal{B}})$ by ‘restriction to $x$’, so the base is
the set of elements of ${\mathcal{B}}$ below $x$, where the boolean unit is
$x$, boolean zero and sum are not changed, boolean complementation is relative
to $x$, and the result of applying any non-boolean operator is obtained by
using the operator for ${\mathcal{B}}$ and intersecting with $x$. It is not
always the case that ${\mathfrak{Rl}}_{x}({\mathcal{B}})$ is a ${\bf
K}_{\beta}$ (we can lose commutativity of cylindrifiers).
The main question we address in this paper is whether
$S{\mathfrak{Nr}}_{m}{\bf K}_{n}={\bf K}_{m}$, where $m<n$ are possibly
infinite ordinals and ${\bf K}\in\\{{\bf Df},{\bf Sc},{\bf CA},{\bf QPA},{\bf
QPEA}\\}$ and, if not, whether $S{\mathfrak{Nr}}_{m}{\bf K}_{n}$ may be
defined within ${\bf K}_{m}$ using only finitely many axioms (or finitely many
axiom schemas, when $m$ is infinite). The case ${\bf K}={\bf Df}$ of diagonal
free algebra is easily answered: $S{\mathfrak{Nr}}_{m}{\bf Df}_{n}={\bf
Df}_{m}$, for $3\leq m\leq n$, see [4, theorem 5.1.31]. We show that in all
the other cases, the answers are negative. In order to generalise the results
of [6] to these other classes of algebra, we define an $m$-dimensional
polyadic equality type algebra ${\mathfrak{C}}(m,n,r)$ where $3\leq m\leq
n,\;r<\omega$ (see definition 4 below). These algebras are based on a relation
algebra construction that first appeared in [9] or see [8, section 15.2],
modified here so that the elements become $n$-dimensional rather than two
dimensional. Still, although they are $n$-dimensional, all of their elements
are generated by two dimensional elements. We will then prove the following
theorem.
###### Theorem 1.
1. I.
${\mathfrak{C}}(m,n,r)\in{\mathfrak{Nr}}_{m}{\bf PEA}_{n}$,
2. II.
${\mathfrak{Rd}}_{{\bf Sc}}{\mathfrak{C}}(m,n,r)\not\in
S{\mathfrak{Nr}}_{m}{\bf Sc}_{n+1}$,
3. III.
$\Pi_{r/U}{\mathfrak{C}}(m,n,r)$ is elementarily equivalent to a countable
polyadic equality algebra ${\mathfrak{C}}\in{\mathfrak{Nr}}_{m}{\bf
PEA}_{n+1}$.
The proof of this theorem is the substantial part of this paper. The proofs
are similar to proofs of corresponding results in [9] but modified for the
signatures considered here and with some further modifications. To prove the
first two parts, the algebras we consider will only have elements generated by
two dimensional elements, however for the third part we will also use elements
that are essentially three dimensional (hence we will introduce three
dimensional hypernetworks, for this part only). From this theorem we deduce
the following.
###### Corollary 2.
Let ${\bf K}\in\\{{\bf PEA},{\bf PA},{\bf CA},{\bf Sc}\\}$, let $3\leq
m<n<\omega$. Then $S{\mathfrak{Nr}}_{m}{\bf K}_{n+1}$ is a proper subclass of
$S{\mathfrak{Nr}}_{m}{\bf K}_{n}$ which cannot be defined, within
$S{\mathfrak{Nr}}_{m}{\bf K}_{n}$ by any finite set of first order formulas.
###### Proof.
We remarked earlier that for each choice of ${\bf K}$ and each $n$, the
operators of ${\bf Sc}_{n}$ are definable by the operators of ${\bf K}_{n}$
which are themselves definable by the operators of ${\bf PEA}_{n}$. Hence, it
follows from (I) that ${\mathfrak{Rd}}_{\bf
K}{\mathfrak{C}}(m,n,r)\in{\mathfrak{Nr}}_{m}{\bf K}_{n}$, from (II)
${\mathfrak{Rd}}_{\bf K}{\mathfrak{C}}(m,n,r)\not\in S{\mathfrak{Nr}}_{m}{\bf
K}_{n+1}$, for $r<\omega$ and from (III) ${\mathfrak{Rd}}_{\bf
K}({\mathfrak{C}})\in{\mathfrak{Nr}}_{m}{\bf K}_{n+1}$. Now suppose for
contradiction that $\phi$ is a formula defining $S{\mathfrak{Nr}}_{m}{\bf
K}_{n}$ within $S{\mathfrak{Nr}}_{m}{\bf K}_{n+1}$. Let $U$ be any non-
principal ultrafilter over $\omega$. Since ${\mathfrak{Rd}}_{\bf
K}{\mathfrak{C}}(m,n,r)\in{\mathfrak{Nr}}_{m}{\bf K}_{n}\setminus
S{\mathfrak{Nr}}_{m}{\bf K}_{n+1}$, ${\mathfrak{Rd}}_{\bf
K}({\mathfrak{C}}(m,n,r))\not\models\phi$, for each $r<\omega$. By Łoś’s
theorem, $\Pi_{r/U}\;{\mathfrak{Rd}}_{\bf
K}{\mathfrak{C}}(m,n,r)\models\neg\phi$. By elementary equivalence
${\mathfrak{C}}\models\neg\phi$, contradicting (III). ∎
We will prove (I), (II), (III) below, after we have defined the algebras
${\mathfrak{C}}(m,n,r)$. For some time to come we restrict our attention to
finite ordinals, which we denote by $m,n\ldots,$ etc. Now we define our
algebras:
###### Definition 3.
Define a function $\kappa:\omega\times\omega\rightarrow\omega$ by
$\kappa(x,0)=0$ (all $x<\omega$) and $\kappa(x,y+1)=1+x\times\kappa(x,y))$
(all $x,y<\omega$). For $n,r<\omega$ let
$\psi(n,r)=\kappa((n-1)r,(n-1)r)+1$
and let $\psi(n,J)=\omega$, for any infinite linear order $J$. [All of this is
simply to ensure that $\psi(n,r)$ is sufficiently big compared to $n,r$ for
the proof of non-embedability to work.] For any $n<\omega$ and any linear
order $J$, let
$Bin(n,J)=\\{Id\\}\cup\\{a^{k}(i,j):i<n-1,\;j\in J,\;k<\psi(n,J)\\}$
For $i<n-1,\;j\in J,\;k<\psi(n,J)$ let
$\displaystyle a(i,\\_)$ $\displaystyle=\\{a^{k}(i,j):j\in
J,\;k<\psi(n,J))\\},$ $\displaystyle a(\\_,j)$
$\displaystyle=\\{a^{k}(i,j):i<n-1,\;k<\psi(n,J)\\},$ $\displaystyle a^{k}$
$\displaystyle=\\{a^{k}(i,j):i<n-1,\;j\in J\\},$ $\displaystyle a(i,j)$
$\displaystyle=\\{a^{k}(i,j):k<\psi(n,J)\\},$ $\displaystyle a(\\_,>j)$
$\displaystyle=\bigcup_{j<j^{\prime}\in J}a(\\_,j^{\prime})$ $\displaystyle
a(\\_,\leq j)$ $\displaystyle=\bigcup_{j^{\prime}\leq j\in
J}a(\\_,j^{\prime})\mbox{ and }$ $\displaystyle a$
$\displaystyle=\bigcup_{i<n-1}a(i,\\_).$
Let $3\leq m\leq n<\omega$ and let $J$ be any linear order. Let $F(m,n,J)$ be
the set of all functions $f:m\times m\to Bin(n,J)$ such that for all $x,y,z<m$
we have $f(x,x)=Id,\;f(x,y)=f(y,x)$, and $(f(x,y),f(y,z),f(x,z))\not\in Forb$,
where $Forb$ is the following set of triples
$\begin{array}[]{c}\\{(Id,b,c):b\neq c\in Bin(n,J)\\}\\\ \cup\\\
\\{(a^{k}(i,j),a^{k^{\prime}}(i,j),a^{k^{*}}(i,j^{\prime})):k,k^{\prime},k^{*}<\psi(n,r),\;i<n-1,\;j^{\prime}\leq
j\in J\\}.\end{array}$
For any $f,g\in F(m,n,J)$ and $x<m$ we write $f\equiv_{x}g$ if for all $y,z\in
m\setminus\\{x\\}$ we have $f(y,z)=g(y,z)$, and for $\tau:m\to m$ we write
$(f\tau)$ for the function defined by
$(f\tau)(x,y)=f(\tau(x),\tau(y)).$ (1)
Clearly $(f\tau)\in F(m,n,J)$. For the next couple of sections we will
consider cases where $J$ is a finite linear order, i.e. some finite ordinal
$r<\omega$.
###### Definition 4.
The base of ${\mathfrak{C}}(m,n,r)$ is the power set of $F(m,n,r)$ and the
operators are
* •
the boolean operators $+,-$ are union and set complement,
* •
the diagonal ${\sf d}_{xy}=\\{f\in F(m,n,r):f(x,y)=Id\\}$,
* •
the cylindrifier ${\sf c}_{x}(X)=\\{f\in F(m,n,r):\exists g\in
X\;f\equiv_{x}g\\}$ and
* •
the polyadic ${\sf s}_{\tau}(X)=\\{f\in F(m,n,r):f\tau\in X\\}$,
for $x,y<m,\;X\subseteq F(m,n,r)$ and $\tau:m\to m$.
Let $x,y<m$ and let $b\in Bin(n,r)$. Define
$b_{x,y}=\\{f\in F(m,n,r):f(x,y)=b\\}\;\;\in{\mathfrak{C}}(m,n,r)$ (2)
Observe, for any $x,y,z<m$ and $\lambda,\mu,\rho\in Bin(n,r)$, that
$(\lambda,\mu,\rho)\in Forb\iff\lambda_{x,y}\cdot\mu_{y,z}\cdot\rho_{x,z}=0,$
(3)
in particular we will use the case $(x,y,z)=(0,1,2)$, later.
###### Lemma 5.
For $3\leq m,\;2\leq n$ and $r<\omega$ the algebra ${\mathfrak{C}}(m,n,r)$
satisfies all of the axioms defining ${\bf PEA}_{m}$ (see definition 15,
noting that for finite $m$, ${\bf PEA}_{m}$ is the same as ${\bf QPEA}_{m}$)
except, perhaps, the commutativity of cylindrifiers ${\sf c}_{x}{\sf
c}_{y}(X)={\sf c}_{y}{\sf c}_{x}(X)$.
###### Proof.
Routine. ∎
###### Lemma 6.
If $3\leq m\leq m^{\prime}$ then
${\mathfrak{C}}(m,n,r)\cong{\mathfrak{Nr}}_{m}{\mathfrak{C}}(m^{\prime},n,r)$.
###### Proof.
The isomorphism maps $X\subseteq F(m,n,r)$ to $\\{f\in
F(m^{\prime},n,r):f{\restriction_{m\times m}}\in X\\}$. ∎
###### Lemma 7.
For $3\leq m\leq n$ and $r<\omega$, ${\mathfrak{C}}(m,n,r)\in{\bf PEA}_{m}$.
###### Proof.
If $r=0$ then $Bin(n,r)=\\{Id\\}$ so ${\mathfrak{C}}(m,n,0)$ is the trivial
algebra hence ${\mathfrak{C}}(m,n,0)\in{\bf PEA}_{m}$. Now assume $r>0$. In
view of lemma 5 we only have to check the commutativity of cylindrifiers:
${\sf c}_{x}{\sf c}_{y}X={\sf c}_{y}{\sf c}_{x}X$, for $x,y<m$. This equation
is trivial if $x=y$ so assume not. By additivity, it suffices to check the
case where $X$ is an atom, $X=\\{f\\}$ for some $f\in F(m,n,r)$, that is we
must show that $g\in{\sf c}_{x}{\sf c}_{y}\\{f\\}\iff g\in{\sf c}_{y}{\sf
c}_{x}\\{f\\}$. Thus given $g\equiv_{xy}f$, it suffices to find $h\in
F(m,n,r)$ such that $f\equiv_{x}h\equiv_{y}g$. If there is $z<m,\;z\neq x,y$
and $f(y,z)=Id$ then the required $h$ is $g[y/z]$, or if $g(z,x)=Id$ the
required $h$ is $f[x/z]$. Suppose there is no such $z$, so for each
$z<m,\;z\neq x,y$ we have $f(y,z),g(x,z)\in a$. Let $h:m\times m\to Bin(n,r)$
be identical to $f$ on pairs not involving $x$, be identical to $g$ on pairs
not involving $y$ (this is well-defined, since $f\equiv_{xy}g$) and let
$h(x,y)=h(y,x)=a^{0}(i,0)$, where $i$ is the least number below $n-1$ such
that it is not the case that there is $z\neq x,y<m$ and $f(y,z),g(x,z)\in
a(i,\\_)$. Since $m\leq n$ and there are only $m-2$ possible values of $z$ in
$m\setminus\\{x,y\\}$ and $n-1$ possible values of $i$, such an $i$ must
exist. This defines $h$, it is now easy to check that $h\in F(m,n,r)$.
∎
We can now prove theorem 1 (I): if $3\leq m\leq n$ and $r<\omega$ then
${\mathfrak{C}}(m,n,r)\cong{\mathfrak{Nr}}_{m}({\mathfrak{C}}(n,n,r))$ by
lemma 6 and ${\mathfrak{C}}(n,n,r)\in{\bf PEA}_{n}$ by lemma 7, so
${\mathfrak{C}}(m,n,r)\in{\mathfrak{Nr}}_{m}{\bf PEA}_{n}$. Next, we prove
theorem 1 (II).
###### Lemma 8.
Let $3\leq m<\omega,\;2\leq n<\omega,\;r<\omega$. ${\mathfrak{Rd}}_{\bf
Sc}{\mathfrak{C}}(m,n,r)\not\in S{\mathfrak{Nr}}_{m}{\bf Sc}_{n+1}$.
###### Proof.
Suppose, for contradiction, that ${\mathcal{X}}\in{\bf Sc}_{n+1}$ and
$\prime:{\mathfrak{Rd}}_{\bf
Sc}{\mathfrak{C}}(m,n,r)\rightarrow{\mathfrak{Nr}}_{m}{\mathcal{X}}$ is an
isomorphism. Let $B\subseteq Bin(n,r)$ and let $i<j<n+1$. Define
$\alpha(B,i,j)=\sum\\{{\sf s}_{[{0}/{i}]}{\sf s}_{[{1}/{j}]}f^{\prime}:f\in
F(m,n,r),\;f(0,1)\in B\\}\;\;\in{\mathcal{X}}$
For $b\in Bin(n,r)$ we may write $\alpha(b,i,j)$ instead of
$\alpha(\\{b\\},i,j)$. (Though we do not need it, it can be checked that
$\alpha(b,i,j)=b_{i,j}^{\prime}$, for $i,j<m$.) By additivity of the
substitutions
$\sum_{b\in Bin(n,r)}\alpha(b,i,j)=1.$ (4)
Further, if $i<j<k<n+1$ and $(u,v,w)\in Forb$ then
$0=\alpha(u,i,j)\cdot\alpha(v,j,k)\cdot\alpha(w,i,k)\;\;(=\delta\mbox{ say}),$
(5)
(if this were not the case then $0\neq{\sf s}_{[{0}/{i}]}{\sf
s}_{[{1}/{j}]}{\sf s}_{[{2}/{k}]}\delta\leq u_{01}\cdot v_{12}\cdot w_{02}$,
contrary to (3)). Let
$\beta^{k}=\alpha(a^{k}(0,0),0,n)\cdot\prod_{i<n}\alpha(Id,0,i)\;\in{\mathcal{X}}.$
Since there is $f\in F(m,n,r)$ with $f(0,1)=a^{k}(0,0)$, letting
$\sigma:n+1\rightarrow n+1$ be the function
$\sigma(1)=n,\;\sigma(i)=0\;(i<n+1,\;i\neq 1$), we have $0\neq{\sf
s}_{\sigma}f^{\prime}\leq\beta^{k}$, and clearly for $k\neq
k^{\prime}<\psi(n,r)$ we have
$\beta^{k}\cdot\beta^{k^{\prime}}\leq\alpha(a^{k}(0,0),0,n)\cdot\alpha(a^{k^{\prime}}(0,0),0,n)=0$.
Let $S_{0}=\\{\beta^{k}:k<\psi(n,r)\\}$. We will prove by induction over $t$
that if $t\leq(n-1)r$ there is a set $S_{t}$ with
$|S_{t}|>\kappa((n-1)r,(n-1)r-t)$ and functions
$I_{t}:\\{0,\ldots,n-1\\}\to\\{0,\ldots,n-2\\},\;J_{t}:\\{0,\ldots,n-1\\}\to\\{0,\ldots,r-1\\}$,
such that for all $\beta,\beta^{\prime}\in S_{t}$
1. 1.
if $l<n$ then $\beta\leq\alpha(a(I_{t}(l),J_{t}(l)),l,n)$,
2. 2.
there is $k<\psi(n,r)$ unique to $\beta\in S_{t}$ such that
$\beta\leq\alpha(a^{k},0,n)$,
3. 3.
${\sf c}_{n}\beta={\sf c}_{n}\beta^{\prime}$,
To see that the case $t=0$ holds: let $I_{0}(i)=0,\;J_{0}(i)=0$ (all $i<n$).
Given functions $I_{t},J_{t}$ as above and $i<n-1$ let the _index of $i$ with
respect to $I_{t},J_{t}$_ be
$ind(i,I_{t},J_{t})=max(\\{J_{t}(l):l<n-1,\;I_{t}(l)=i\\}\cup\\{-1\\}).$
Define the _rank_ $\rho(I_{t},J_{t})=\sum_{i<n-1}ind(i,I_{t},J_{t})$. Observe
that $ind(0,I_{0},J_{0})=0$ and $ind(i,I_{0},J_{0})=-1$ for $0<i<n-1$, so
$\rho(I_{0},J_{0})=0+(n-2)\times(-1)=2-n$. We also assume, inductively,
1. 1.
$\rho(I_{t},J_{t})\geq 2-n+t$.
We have seen that this last inductive condition also holds for $t=0$.
Let $0\leq t<(n-1)r$ and assume these properties hold. Since $|{\sf
dom}(I_{t})|=n$ and $|{\sf rng}(I_{t})|\leq n-1$ there must be $u<v<n$ with
$I_{t}(u)=I_{t}(v)$. Pick such a pair $(u,v)$ and let $l=u$ if $J_{t}(u)\leq
J_{t}(v)$, else let $l=v$. Note, by choice of $l$, that if
$I^{\prime},J^{\prime}$ are functions identical to $I,J$, respectively, except
perhaps on $l$, then $\rho(I^{\prime},J^{\prime})\geq\rho(I,J)$.
Since $t<(n-1)r$ we have
$|S_{t}|>\kappa((n-1)r,(n-1)r-t)\geq\kappa((n-1)r,1)=1$. Fix some
$\beta_{0}\in S_{t}$. For each $\beta\in S_{t}\setminus\\{\beta_{0}\\}$, since
${\sf c}_{n}\beta={\sf c}_{n}\beta_{0}$, we have ${\sf c}_{n}{\sf c}_{l}{\sf
s}_{[{n}/{l}]}{\sf c}_{l}(\beta)={\sf c}_{n}{\sf c}_{l}(\beta_{0})$, hence
${\sf c}_{l}\beta_{0}\;\cdot\;{\sf s}_{[{n}/{l}]}{\sf c}_{l}\beta\neq 0$. By
(4), there is $b\in Bin(n,r)$ such that
$\delta={\sf c}_{l}\beta_{0}\;\cdot\;{\sf s}_{[{n}/{l}]}{\sf
c}_{l}\beta\;\cdot\;\alpha(b,l,n)\neq 0.$
We know that
$\beta_{0}\leq\alpha(a^{k_{0}}(0,0),0,n),\;\beta\leq\alpha(a^{k}(0,0),0,n)$
for some $k_{0}\neq k<\psi(n,r)$, so
$\delta\leq\alpha(a^{k_{0}}(0,0),0,l)\cdot\alpha(a^{k}(0,0)(0,n))\cdot\alpha(b,l,n)$.
By (5), $(a^{k_{0}}(0,0),a^{k}(0,0),b)\not\in Forb$ and we cannot have $b=Id$.
Hence $b=a^{k^{\prime}}(i,j)$ for some $i<n-1,\;j<r,\;k^{\prime}<\psi(n,r)$.
For $i<n,\;j<r$ let
$S(i,j)=\\{{\sf c}_{l}\beta_{0}\;\cdot\;{\sf s}_{[{n}/{l}]}{\sf
c}_{l}\beta\cdot\alpha(a^{k}(i,j),l,n):k<\psi(n,r),\;\beta\in
S_{t}\setminus\\{\beta_{0}\\}\\}\setminus\\{0\\}.$
By cardinalities, there is a fixed $i_{0}<n-1$ and $j_{0}<r$ such that
$|S(i_{0},j_{0})|\geq\frac{|S_{t}|-1}{(n-1)r}>\frac{\kappa((n-1)r,(n-1)r-t)-1}{(n-1)r}=\kappa((n-1)r,(n-1)r-(t+1))$.
Let $S_{t+1}=S(i_{0},j_{0})$, let $I_{t+1}$ be identical to $I_{t}$ except
that $l\mapsto i_{0}$ and let $J_{t+1}$ be identical to $J_{t}$ except that
$l\mapsto j_{0}$. If $i_{0}\not\in{\sf rng}(I_{t})$ then
$Ind(i_{0},I_{t+1},J_{t+1})=j_{0}\geq 0>-1=Ind(i_{0},I_{t},J_{t})$, otherwise
for any $p<n+1$ if $I_{t}(p)=i_{0}$ then $j_{0}>J_{t}(p)$, by (5) applied to
$(p,l,n)$, so $j_{0}=Ind(i_{0},I_{t+1},J_{t+1})>Ind(i_{0},I_{t},J_{t})$.
Either way, $\rho(I_{t+1},J_{t+1})>\rho(I_{t},J_{t})$. Hence
$S_{t+1},I_{t+1},J_{t+1}$ satisfies induction hypothesis 3. The other
induction hypotheses are straightforward.
By induction, the properties hold for all $t\leq(n-1)r$. Letting $t=(n-1)r$,
we have a set $S_{(n-1)r}$ of size strictly greater than
$\kappa((n-1)r,(n-1)r-(n-1)r)=\kappa((n-1)r,0)=0$, i.e. non-empty, and there
are functions $I_{t},J_{t}$ of rank at least $(2-n)+((n-1)r)=(n-1)(r-1)+1$, an
impossibility since for each $i<n-1$ the maximum index $i$ can have is $r-1$,
hence the maximum possible rank is $(n-1)(r-1)$. We conclude that
${\mathfrak{Rd}}_{\bf Sc}{\mathfrak{C}}(m,n,r)\not\in S{\mathfrak{Nr}}_{m}{\bf
Sc}_{n+1}$. ∎
We now concentrate on proving (III), that $\Pi_{r/U}\;{\mathfrak{C}}(m,n,r)\in
S{\mathfrak{Nr}}_{m}{\bf PEA}_{n+1}$, for any non-principal ultrafilter $U$. A
standard ultraproduct argument shows that
$\Pi_{r/U}\;{\mathfrak{C}}(m,n,r)\cong{\mathfrak{C}}(m,n,\Pi_{r/U}\;r)$ so we
have to prove that ${\mathfrak{C}}(m,n,J)\in S{\mathfrak{Nr}}_{m}{\bf
PEA}_{n+1}$, where $J=\Pi_{r/U}\;r$. Note that $J$ is a linear order
containing an infinite ascending chain. First we define a game.
###### Definition 9.
Let $m,n<\omega$, let $J$ be a linear order and let $\Lambda=(n+1)^{3}$. An
_$m$ -hypernetwork_ $h=(f,g)$ consists of some $f\in F(m,n,J)$ and a ternary
function $g:\;^{3}m\rightarrow\Lambda$ such that for all
$x,y,z,x^{\prime},y^{\prime},z^{\prime}<m$, if
$f(x,x^{\prime})=f(y,y^{\prime})=f(z,z^{\prime})=Id$ then
$g(x,y,z)=g(x^{\prime},y^{\prime},z^{\prime})$. For $X\subseteq m$ we say that
$h=(f,g)$ is _strict over $X$_ if $x\neq y\in X\Rightarrow f(x,y)\neq Id$. If
$m^{\prime}\leq m$ and $h=(f,g)$ is an $m$-hypernetwork then
$h{\restriction_{m^{\prime}}}$ denotes the $m^{\prime}$-hypernetwork obtained
from $h$ by restriction to $m^{\prime}$.
We define a game $G=G(m,n,J)$ as follows. A _play_ of $G$ is a sequence
$h_{0},h_{1},\ldots,h_{t},\;\;(t<\omega)$ of $(n+1)$-hypernetworks. In round
$t<\omega$ of the game, $\forall$ plays either an $m$-dimensional move
$\theta$ by choosing any $m$-hypernetwork $\theta$ or an amalgamation move
$(u,v,\sigma,\tau,x,y)$ where $u,v<t,\;\sigma,\tau:n+1\rightarrow
n+1,\;x,y<n+1$ and $h_{u}\sigma\equiv_{xy}h_{v}\tau$. In response to an
$m$-dimensional move $\theta$, $\exists$ must play a (n+1)-hypernetwork
$h_{t}$ such that $h_{t}{\restriction_{m}}=\theta$. In response to an
amalgamation move $(u,v,\sigma,\tau,x,y)$, $\exists$ must play a
$(n+1)$-hypernetwork $h_{t}$ such that
$h_{u}\sigma\equiv_{x}h_{t}\equiv_{y}h_{v}\tau$. If she fails to provide such
a response to either kind of move then she loses the play in that round. If
$\exists$ does not lose in any of the $\omega$ rounds of $G$ then she wins the
play.
###### Lemma 10.
Let $3\leq m<n<\omega$ and let $J$ be a linear order containing an infinite
ascending sequence. $\exists$ has a winning strategy in $G(m,n,J)$.
###### Proof.
Let $k_{0}<k_{1}<k_{2}\ldots\;\;\in J$ be an infinite ascending sequence, let
$K=\\{k_{0},k_{1},\ldots\\}\subseteq J$. We describe $\exists$’s strategy.
Consider round $t$ of a play of the game. Suppose, inductively, that $\exists$
has successfully implemented her strategy in all previous rounds $s<t$, the
play so far is $h_{0},h_{1},\ldots h_{t-1}$. Suppose $\forall$ plays an
$m$-dimensional move $\theta$. Let $\sigma:(n+1)\rightarrow m$ be the function
defined by
$\sigma(i)=\left\\{\begin{array}[]{ll}i&(i<m)\\\ 0&(m\leq
i<n+1)\end{array}\right.$
$\exists$ plays the hypernetwork $\theta\sigma$. Observe that if
$X\subseteq(n+1)$ and $|X|>m$ then $\theta\sigma$ is not strict over $X$.
Now suppose $\forall$ plays an amalgamation move $(u,v,\sigma,\tau,x,y)$ where
$h_{u}\sigma\equiv_{xy}h_{v}\tau$. To avoid trivialities assume $x\neq y$.
$\exists$ is required to find $h_{t}=(f_{t},g_{t})$ such that
$h_{u}\sigma\equiv_{x}h_{t}\equiv_{y}h_{v}\tau$. These equivalences uniquely
determine the value of $f_{t}$ on any pair from $n+1$ except $(x,y)$ and
$(y,x)$ and they determine the value of $g_{t}$ on any triple from $n+1$
except those involving both $x$ and $y$. If there is $w<t$ and
$\rho:n+1\rightarrow n+1$ such that
$h_{u}\sigma\equiv_{x}h_{w}\rho\equiv_{y}h_{v}\tau$ then $\exists$ lets
$h_{t}=h_{w}\rho$ (if there is more than one possible solution, then any will
do). Since such a move by $\forall$ is clearly superflous we will assume
henceforth that $\forall$ never makes such a move. Furthermore, we will assume
that if $\forall$ plays the amalgamation move $(u,v,\sigma,\tau,x,y)$ then
there is no $u^{\prime}<u$ and $\sigma^{\prime}:n+1\rightarrow n+1$ such that
$h_{u}\sigma\equiv_{x}h_{u^{\prime}}\sigma^{\prime}$ (if such a $u^{\prime}<u$
and $\sigma^{\prime}$ existed then $\forall$ could instead play
$(u^{\prime},v,\sigma^{\prime},\tau,x,y)$) and there is no $v^{\prime}<v$ and
$\tau^{\prime}:n+1\rightarrow n+1$ such that
$h_{v}\tau\equiv_{y}h_{v^{\prime}}\tau^{\prime}$.
Now, although we have not yet entirely defined $f_{t}$, for each
$x^{\prime},y^{\prime}<n+1$ we already know whether
$f_{t}(x^{\prime},y^{\prime})=Id$ or not (we cannot have $f_{t}(x,y)=Id$, by
our assumption about $\forall$-moves). For any
$x_{1},y_{1},z_{1},x_{2},y_{2},z_{2}<n+1$ we write
$(x_{1},y_{1},z_{1})\sim(x_{2},y_{2},z_{2})$ iff
$f_{t}(x_{1},x_{2})=f_{t}(y_{1},y_{2})=f_{t}(z_{1},z_{2})=Id$.
First $\exists$ defines $g_{t}:^{3}\\!\\!(n+1)\rightarrow\Lambda$ by defining
$g_{t}$ on all triples involving both $x$ and $y$ in such a way that if
$\overline{x}$ is any triple involving $x$ and $y$ and if $\overline{y}$ is
any triple of elements of $(n+1)$ then
$g_{t}(\overline{x})=g_{t}(\overline{y})\iff(\overline{x}\sim\overline{y})$.
Since $\sim$ is clearly an equivalence relation and since $\Lambda=(n+1)^{3}$,
the range of $g_{t}$ is large enough to allow this.
Secondly, $\exists$ defines $f_{t}\in F(n+1,n,J)$ by letting
$f_{t}(x,y)=a^{0}(i,k)$ where
* •
$k\in K$ is greater than each element of the finite set $\\{k^{\prime}\in
K:\exists s<t,\;x^{\prime},y^{\prime}<n+1\;f_{u}(x^{\prime},y^{\prime})\in
a(\\_,k^{\prime})\\}$, least possible subject to that.
* •
$i<n-1$ is least such that there is no $w<n+1$ and $j\in J\setminus K$ with
$f_{u}\sigma(y,w),f_{v}\tau(w,x)\in a(i,j)$.
We will prove that the strategy may be implemented, in particular the $i<n-1$
required in the second part may always be found. To prove our claim, suppose
for contradiction that there are $w_{0},w_{1},\ldots,w_{n-2}<n+1$ such that
for each $i<n-1$ there is $j\in J\setminus K$ and
$f_{u}\sigma(y,w_{i}),(f_{v}\tau)(w_{i},x)\in a(i,j)$. Observe that $f_{u}$ is
strict over $\\{\sigma(y),\sigma(w_{i}):i<n-1\\}$, so $f_{u}$ was itself
played in response to an amalgamation move, say
$(u^{\prime},v^{\prime},\sigma^{\prime},\tau^{\prime},x^{\prime},y^{\prime})$.
By our assumption that there is no $u^{*}<u$ and $\sigma^{*}$ such that
$f_{u}\sigma\equiv_{x}f_{u^{*}}\sigma^{*}$, we see that
$\\{\sigma(y^{\prime}),\sigma(x^{\prime})\\}\subseteq\\{y,w_{0},w_{1},\ldots,w_{n-2}\\}$.
Inductively, $\exists$ chose $f_{u}(\sigma(y^{\prime}),\sigma(x^{\prime}))\in
a(\\_,k^{\prime})$ for some $k^{\prime}\in K$, hence
$\\{\sigma(y^{\prime}),\sigma(x^{\prime})\\}\subseteq\\{w_{0},\ldots,w_{n-2}\\}$.
Similarly, $f_{v}$ was played in response to an amalgamation move
$(u^{*},v^{*},\sigma^{*},\tau^{*},x^{*},y^{*}),\;f_{v}(\tau(y^{*}),\tau(x^{*}))\in
a(\\_,k^{*})$ (some $k^{*}\in K$) and
$\\{\tau(y^{*}),\tau(x^{*})\\}\subseteq\\{w_{0},\ldots,w_{n-2}\\}$. By
uniqueness of $k^{\prime}$ and $k^{*}$ we deduce that $k^{\prime}=k^{*},\;u=v$
and
$\\{\sigma(y^{\prime}),\sigma(x^{\prime})\\}=\\{\tau(y^{*}),\tau(x^{*})\\}$.
When $\exists$ played $f_{u}$ she ensured that for each $w_{h}\;(h<n-1)$ the
label $g_{u}(\sigma(y^{\prime}),\sigma(x^{\prime}),\sigma(w_{h}))$ is unique
but it is equal to $g_{v}(\tau(y^{\prime}),\tau(x^{\prime}),\tau(w_{h}))$
(since $g_{u}\sigma\equiv_{xy}g_{v}\tau$), hence $\sigma(w_{h})=\tau(w_{h})$.
But then, define $\rho:(n+1)\rightarrow(n+1)$ by $\rho(v)=\sigma(v)$, for
$v\in(n+1)\setminus\\{x\\}$, and $\rho(x)=\tau(x)$. Then
$h_{u}\sigma\equiv_{x}h_{u}\rho\equiv_{y}h_{v}\sigma$, contrary to our
assumption. This proves the claim and proves that $\exists$’s strategy can
always be implemented.
By choice of $i,k$ it is clear that $f_{t}$ avoids all forbidden triples so
indeed $f_{t}\in F(n+1,n,J)$. ∎
###### Lemma 11.
Let $3\leq m<n<\omega$ and let $J$ be a countable linear order containing an
infinite ascending sequence. Then
${\mathfrak{C}}(m,n,J)\in{\mathfrak{Nr}}_{m}{\bf PEA}_{n+1}$.
###### Proof.
Consider a play of $G(m,n,J)$ in which $\exists$ plays her winning strategy
and $\forall$ plays all possible $m$-dimensional moves and all possible
amalgamation moves. Since $J$ is countable, this can be scheduled. Let $H$ be
the set of all hypernetworks occuring in the play. As in definition 4, the
power set $\wp(H)$ is the base of a ${\bf PEA}_{n+1}$-type algebra
${\mathfrak{C}}$, where ${\sf d}_{ij=}\\{(f,g)\in H:f(i,j)=Id\\},\;{\sf
c}_{i}(X)=\\{h\in H:\exists h^{\prime}\in X,\;h^{\prime}\equiv_{i}h\\}$ and
$s_{\tau}(X)=\\{h\in H:h\tau\in X\\}$, for
$i,j<n+1,\;\tau:(n+1)\rightarrow(n+1)$. As with lemma 5 it is easy to see that
${\mathfrak{C}}$ satisfies all the ${\bf PEA}_{n+1}$ axioms other than
commutativity of cylindrifiers, and since $H$ is closed under amalgamation,
commutativity holds too, so ${\mathfrak{C}}\in{\bf PEA}_{n+1}$. The map
$\lambda:{\mathfrak{C}}(m,n,J)\rightarrow{\mathfrak{Nr}}_{m}{\mathfrak{C}}$
defined by $\iota(f)=\\{(f^{\prime},g^{\prime})\in
H:f^{\prime}{\restriction_{m}}=f\\}$ is easily shown to be an isomorphism. ∎
Since $\Pi_{r/U}{\mathfrak{C}}(m,n,r)\cong{\mathfrak{C}}(m,n,\Pi_{r/U}r)$ and
$\Pi_{r/U}r$ contains an infinite ascending sequence, this proves theorem
1(III) and completes the proof of theorem 1.
## Infinite dimensional case
Now we prove the infinite dimensional case, by lifting the dimensions for the
finite case to the transfinite; a trick due to Monk. Our proof has the same
structure as the finite dimensional case, but naturally we need an infinite
dimensional quasi- polyadic equality algebra. Let $\alpha$ be an infinite
ordinal. For each finite subset $\Gamma\subseteq\alpha$ let
$\rho_{\Gamma}\mbox{ be the unique order preserving bijection from
}|\Gamma|\mbox{ onto }\Gamma.$
Let $I=\\{\Gamma:\Gamma\subseteq\alpha,|\Gamma|<\omega\\}$. For each
$\Gamma\in I$, let $M_{\Gamma}=\\{\Delta\in I:\Gamma\subseteq\Delta\\}$, and
let $F$ be an ultrafilter on $I$ such that $\forall\Gamma\in I,\;M_{\Gamma}\in
F$ (such an ultrafilter exists because $M_{\Gamma_{1}}\cap
M_{\Gamma_{2}}=M_{\Gamma_{1}\cup\Gamma_{2}}$). Let $r<\omega$, $1\leq
k<\omega$, and let ${{\mathfrak{C}}}_{\Gamma}^{r}$ be an algebra similar to
${\bf QPEA}_{\alpha}$ such that
${\mathfrak{Rd}}^{\rho_{\Gamma}}{{\mathfrak{C}}}_{\Gamma}^{r}={{\mathfrak{C}}}(|\Gamma|,|\Gamma|+k,r).$
Let
${\mathfrak{B}}^{r}=\prod_{\Gamma/F}{\mathfrak{C}}_{\Gamma}^{r}.$
###### Theorem 12.
Let $U$ be any non-principal ultraproduct over $\omega$.
1. 1.
${\mathfrak{B}}^{r}\in S{\mathfrak{Nr}}_{\alpha}{\bf QPEA}_{\alpha+k}$,
2. 2.
${\mathfrak{Rd}}_{\bf Sc}{\mathfrak{B}}^{r}\not\in
S{\mathfrak{Nr}}_{\alpha}{\bf Sc}_{\alpha+k+1}$ and
3. 3.
$\Pi_{r/U}{\mathfrak{B}}^{r}\in S{\mathfrak{Nr}}_{\alpha}{\bf
QPEA}_{\alpha+k+1}$.
But first a lemma.
###### Lemma 13.
Let $\alpha$ be an infinite ordinal, let $X$ be any finite subset of $\alpha$,
let $I=\\{\Gamma:X\subseteq\Gamma\subseteq\alpha,\;|\Gamma|<\omega\\}$. For
each $\Gamma\in I$ let $M_{\Gamma}=\\{\Delta\in I:\Delta\supseteq\Gamma\\}$
and let $F$ be any ultrafilter over $I$ such that for all $\Gamma\in I$ we
have $M_{\Gamma}\in F.$ Let ${\mathcal{A}}_{\Gamma},{\mathcal{B}}_{\Gamma}$ be
${\bf QPEA}_{\alpha}$-type algebras. If for each $\Gamma\in I$ we have
${\mathfrak{Rd}}^{\rho_{\Gamma}}{\mathcal{A}}_{\Gamma}={\mathfrak{Rd}}^{\rho_{\Gamma}}{\mathcal{B}}_{\Gamma}$
then
$\Pi_{\Gamma/F}{\mathcal{A}}_{\Gamma}=\Pi_{\Gamma/F}{\mathcal{B}}_{\Gamma}$.
Furthermore, if ${\mathfrak{Rd}}^{\rho_{\Gamma}}{\mathcal{A}}_{\Gamma}\in{\bf
QPEA}_{|\Gamma|}$, for each $\Gamma\in I$ then
$\Pi_{\Gamma/F}{\mathcal{A}}_{\Gamma}\in{\bf QPEA}_{\alpha}$.
###### Proof.
Standard proof, by Łoś’ theorem. For the first part, note that the base of
$\Pi_{\Gamma/F}{\mathcal{A}}_{\Gamma}$ is identical to the base of
$\Pi_{\Gamma/F}{\mathfrak{Rd}}^{\rho_{\Gamma}}{\mathcal{A}}_{\rho}$ which is
identical to the base of $\Pi_{\Gamma/F}{\mathcal{B}}_{\Gamma}$, by the
assumption in the first part of the lemma. Each operator $o$ of ${\bf
QPEA}_{\alpha}$ is the same for both ultraproducts because $\\{\Gamma\in
I:{\sf dim}(o)\subseteq{\sf rng}(\rho_{\Gamma})\\}\in F$.
For the second part, it suffices to prove that each of the defining axioms for
${\bf QPEA}_{\alpha}$ holds for $\Pi_{\Gamma/F}{\mathcal{A}}_{\Gamma}$. Let
$\sigma=\tau$ be one of the defining equations for ${\bf QPEA}_{\alpha}$, the
number of dimension variables involved is certainly finite, indeed it can be
at most four (see definition 15.8). Take any $i,j,k,l\in\alpha$, we must prove
that
$\Pi_{\Gamma/F}{\mathcal{A}}_{\Gamma}\models\sigma(i,j,k,l)=\tau(i,j,k,l)$. If
$i,j,k,l\in{\sf rng}(\rho_{\Gamma})$, say
$i=\rho_{\Gamma}(i_{0}),\;j=\rho_{\Gamma}(j_{0}),\;k=\rho_{\Gamma}(k_{0}),\;l=\rho_{\Gamma}(l_{0})$,
then
${\mathfrak{Rd}}^{\rho_{\Gamma}}{\mathcal{A}}_{\Gamma}\models\sigma(i_{0},j_{0},k_{0},l_{0})=\tau(i_{0},j_{0},k_{0},l_{0})$,
since ${\mathfrak{Rd}}^{\rho_{\Gamma}}{\mathcal{A}}_{\Gamma}\in{\bf
QPEA}_{|\Gamma|}$, so
${\mathcal{A}}_{\Gamma}\models\sigma(i,j,k,l)=\tau(i,j,k,l)$. Hence
$\\{\Gamma\in
I:{\mathcal{A}}_{\Gamma}\models\sigma(i,j,k,l)=\tau(i,j,k,l)\\}\supseteq\\{\Gamma\in
I:i,j,k,l\in{\sf rng}(\rho_{\Gamma}\\}\in F$, hence
$\Pi_{\Gamma/F}{\mathcal{A}}_{\Gamma}\models\sigma(i,j,k,l)=\tau(i,j,k,l)$.
Thus $\Pi_{\Gamma/F}{\mathcal{A}}_{\Gamma}\in{\bf QPEA}_{\alpha}$. ∎
###### Proof of theorem 12.
For the first part, for each $\Gamma\in I$ we know that
${\mathfrak{C}}(|\Gamma|+k,|\Gamma|+k,r)\in{\bf QPEA}_{|\Gamma|+k}$ and
${\mathfrak{Nr}}_{|\Gamma|}{\mathfrak{C}}(|\Gamma|+k,|\Gamma|+k,r)\cong{\mathfrak{C}}(|\Gamma|,|\Gamma|+k,r)$
(by lemmas 6 and 7). Let $\sigma_{\Gamma}$ be the one to one function
$(|\Gamma|+k)\rightarrow(\alpha+k)$ where
$\rho_{\Gamma}\subseteq\sigma_{\Gamma}$ and
$\sigma_{\Gamma}(|\Gamma|+i)=\alpha+i$ for each $i<k$. Let
${\mathcal{A}}_{\Gamma}$ be an algebra similar to a ${\bf QPEA}_{\alpha+k}$
such that
${\mathfrak{Rd}}^{\sigma_{\Gamma}}{\mathcal{A}}_{\Gamma}={\mathfrak{C}}(|\Gamma|+k,|\Gamma|+k,r)$.
By the second part of lemma 13, with $\alpha+k$ in place of $\alpha$,
$m\cup\\{\alpha+i:i<k\\}$ in place of $X$,
$\\{\Gamma\subseteq\alpha+k:|\Gamma|<\omega,\;X\subseteq\Gamma\\}$ in place of
$I$, and with $\sigma_{\Gamma}$ in place of $\rho_{\Gamma}$, we know that
$\Pi_{\Gamma/F}{\cal{A}}_{\Gamma}\in{\bf QPEA}_{\alpha+k}$.
We prove that
${\mathfrak{B}}^{r}\subseteq{\mathfrak{Nr}}_{\alpha}\Pi_{\Gamma/F}{\mathcal{A}}_{\Gamma}$.
Recall that ${\mathfrak{B}}^{r}=\Pi_{\Gamma/F}{\mathfrak{C}}^{r}_{\Gamma}$ and
note that $C^{r}_{\Gamma}\subseteq A_{\Gamma}$ (the base of
${\mathfrak{C}}^{r}_{\Gamma}$ is ${\mathfrak{C}}(|\Gamma|,|\Gamma|+k,r)$, the
base of ${\mathcal{A}}_{\Gamma}$ is
${\mathfrak{C}}(|\Gamma|+k,|\Gamma|+k,r)$). So, for each $\Gamma\in I$,
$\displaystyle{\mathfrak{Rd}}^{\rho_{\Gamma}}{\mathfrak{C}}_{\Gamma}^{r}$
$\displaystyle={\mathfrak{C}}((|\Gamma|,|\Gamma|+k,r)$
$\displaystyle\cong{\mathfrak{Nr}}_{|\Gamma|}{\mathfrak{C}}(|\Gamma|+k,|\Gamma|+k,r)$
$\displaystyle={\mathfrak{Nr}}_{|\Gamma|}{\mathfrak{Rd}}^{\sigma_{\Gamma}}{\cal{A}}_{\Gamma}$
$\displaystyle={\mathfrak{Rd}}^{\sigma_{\Gamma}}{\mathfrak{Nr}}_{\Gamma}{\cal{A}}_{\Gamma}$
$\displaystyle={\mathfrak{Rd}}^{\rho_{\Gamma}}{\mathfrak{Nr}}_{\Gamma}{\cal{A}}_{\Gamma}$
By the first part of lemma 13 we deduce that
$\Pi_{\Gamma/F}{\mathfrak{C}}^{r}_{\Gamma}\cong\Pi_{\Gamma/F}{\mathfrak{Nr}}_{\Gamma}{\cal{A}}_{\Gamma}\subseteq{\mathfrak{Nr}}_{\alpha}\Pi_{\Gamma/F}{\cal{A}}_{\Gamma}$,
proving (1).
Now we prove (2), ${\mathfrak{Rd}}_{\bf
Sc}{\mathfrak{B}}^{r}\not\in{\mathfrak{Nr}}_{\alpha}{\bf Sc}_{\alpha+k+1}$.
For this assume, seeking a contradiction, that ${\mathfrak{Rd}}_{{\bf
Sc}}{\mathfrak{B}}^{r}\in S{\mathfrak{Nr}}_{\alpha}{\bf Sc}_{\alpha+k+1}$,
i.e. ${\mathfrak{Rd}}_{\bf
Sc}{\mathfrak{B}}^{r}\subseteq{\mathfrak{Nr}}_{\alpha}{\mathfrak{C}}$, where
${\mathfrak{C}}\in{\bf Sc}_{\alpha+k+1}$. Let $3\leq m<\omega$ and
$\lambda:m+k+1\rightarrow\alpha+k+1$ be the function defined by $\lambda(i)=i$
for $i<m$ and $\lambda(m+i)=\alpha+i$ for $i<k+1$. Then
${\mathfrak{Rd}}^{\lambda}({\mathfrak{C}})\in{\bf Sc}_{m+k+1}$ and
${\mathfrak{Rd}}_{m}{\mathfrak{Rd}}_{{\bf
Sc}}{\mathfrak{B}}^{r}\subseteq{\mathfrak{Nr}}_{m}{\mathfrak{Rd}}^{\lambda}({\mathfrak{C}})$.
Let ${\cal{A}}={\mathfrak{Rd}}_{m}{\mathfrak{Rd}}_{\bf Sc}{\mathfrak{B}}^{r}$.
We have just shown that
${\cal{A}}\in S{\mathfrak{Nr}}_{m}{\bf Sc}_{m+k+1}.$ (6)
For $n>m$, let
$x_{n}=\\{f\in F(n,n+k,r):m\leq j<n\to\exists i<m\;f(i,j)=Id\\}.$
Then $x_{n}\in{\mathfrak{C}}(n,n+k,r)$ and ${\sf c}_{i}x_{n}\cdot{\sf
c}_{j}x_{n}=x_{n}$ for distinct $i,j<m$. Furthermore
${I_{n}:{\mathfrak{C}}}(m,m+k,r)\cong{\mathfrak{Rl}}_{x_{n}}{\mathfrak{Rd}}_{m}{{\mathfrak{C}}}(n,n+k,r).$
via
$I_{n}(S)=\\{f\in F(n,n+k,r):f\upharpoonright m\times m\in S,\forall j(m\leq
j<n\to\exists i<m\;f(i,j)=Id)\\}.$
So for each $\Gamma\in I$, $I_{|\Gamma|}$ is an isomorphism
${{\mathfrak{C}}}(m,m+k,r)\cong{\mathfrak{Rl}}_{x_{|\Gamma|}}{\mathfrak{Rd}}_{m}{{\mathfrak{C}}}(|\Gamma|,|\Gamma|+k,r).$
Let $x=(x_{|\Gamma|}:\Gamma\in I)/F\;(\in{\mathfrak{B}}^{r})$ and let
$\iota(b)=(I_{|\Gamma|}b:\Gamma\in I)/F$ for $b\in{\mathfrak{C}}(m,m+k,r)$.
Then $\iota$ is an isomorphism from ${\mathcal{{\mathfrak{Rd}}}}_{{\bf
Sc}}C(m,m+k,r)$ into ${\mathfrak{Rd}}_{{\bf
Sc}}{\mathfrak{Rl}}_{x}{\mathfrak{Rd}}_{m}{\mathfrak{B}}^{r}={\mathfrak{Rl}}_{x}{\mathfrak{Rd}}_{m}{\mathfrak{Rd}}_{{\bf
Sc}}{\mathfrak{B}}^{r}={\mathfrak{Rl}}_{x}{\cal{A}}$. Now ${\cal{A}}\in
S{\mathfrak{Nr}}_{m}{\bf Sc}_{m+k+1}$, by (6), and ${\sf c}_{i}x\cdot{\sf
c}_{j}x=x\leq{\sf s}_{[j/i]}x,$ so that ${\sf s}_{[j/i]}x\cdot{\sf
s}_{[i/j]}x=x$ for any distinct $i,j<m$.
We will now prove that
${\mathfrak{Rl}}_{x}{\cal{A}}\in S{\mathfrak{Nr}}_{m}{\bf Sc}_{m+k+1}.$ (7)
This will yield a contradiction, because it follows from this that
${\mathfrak{Rd}}_{\bf
Sc}{\mathfrak{C}}(m,m+k,r)\subseteq{\mathfrak{Rl}}_{x}{\cal{A}}\in
S{\mathfrak{Nr}}_{m}{\bf Sc}_{m+k+1}$, contrary to theorem 1(II).
The proof of this is similar to the proof of [3, theorem 2.6.38]. Let $n=k+1.$
We know that ${\cal{A}}\in S{\mathfrak{Nr}}_{m}{\bf Sc}_{m+n}$, so let
${\cal{A}}\subseteq{\mathfrak{Nr}}_{m}{\mathfrak{C}}$, where
${\mathfrak{C}}\in{\bf Sc}_{m+n}$. Let $b=x\cdot\prod_{i<n}{\sf
s}_{[0/{m+i}]}x$; then $b\in{\mathfrak{C}}.$ We proceed by induction on $n$ to
show that
${\mathfrak{Rl}}_{x}{\cal{A}}\subseteq{\mathfrak{Nr}}_{m}{\mathfrak{Rl}}_{b}{\mathfrak{C}}$
and simultaneously that ${\sf s}_{[k/l]}b\cdot{\sf s}_{[l/k]}b=b$ for all
$k,l<m+n$. The latter condition will guarantee that
${\mathfrak{Rl}}_{b}{\mathfrak{C}}\in{\bf Sc}_{m+n}$. Indeed, commutativity of
cylindrifiers follows from the fact for any $d\in C$, and distinct $k,l<m+n$,
we have ${\sf c}_{l}({\sf c}_{k}d\cdot b)\cdot b\leq{\sf c}_{l}({\sf
c}_{k}d\cdot b)$. For the base of the induction, assume that $n=1$. Then for
any $k,l<m$, $k\neq l,$ we have
${\sf s}_{[k/m]}x={\sf s}_{[k/m]}({\sf s}_{[k/l]}x\cdot{\sf s}_{[l/k]}x)={\sf
s}_{[k/l]}x\cdot{\sf s}_{[k/m]}{\sf s}_{[l/k]}x.$
By $x\leq{\sf s}_{[k/l]}x$, we get
$x\cdot{\sf s}_{[k/m]}x=x\cdot{\sf s}_{[k/m]}{\sf s}_{[l/k]}x=x\cdot{\sf
s}_{[k/m]}{\sf s}_{[l/m]}x.$
And similarly,
$x\cdot{\sf s}_{[l/m]}x=x\cdot{\sf s}_{[l/m]}{\sf s}_{[k/m]}x=x\cdot{\sf
s}_{[k/m]}{\sf s}_{[l/m]}x.$
Therefore $x\cdot{\sf s}_{[k/m]}x=x\cdot{\sf s}_{[l/m]}x$ for all $k,l<m$, and
so it readily follows that for any non-empty finite $\Gamma\subseteq m,$ we
have (*)
$b=x\cdot\prod_{i\in\Gamma}{\sf s}_{[i/m]}x.$
Now consider $k,l<m$. Then:
$\begin{split}{\sf s}_{[l/k]}b\cdot{\sf s}_{[k/l]}b&={\sf
s}_{[l/k]}(x\cdot{\sf s}_{[k/m]}x)\cdot{\sf s}_{[l/k]}(x\cdot{\sf
s}_{[l/m]}x)\\\ &={\sf s}_{[l/k]}x\cdot{\sf s}_{[k/m]}x\cdot{\sf
s}_{[l/k]}x\cdot{\sf s}_{[l/m]}x\\\ &=x\cdot{\sf s}_{[k/m]}x\cdot{\sf
s}_{[l/m]}x=b.\\\ \end{split}$
And similarly,
${\sf s}_{[k/m]}b\cdot{\sf s}_{[m/k]}b=b.$
We have proved that ${\sf s}_{[k/l]}b\cdot{\sf s}_{[l/k]}b=b$ for all $k,l\leq
m$. Now ${\mathfrak{Rl}}_{x}{\cal{A}}$ can be embedded into
${\mathfrak{Nr}}_{m}{\mathfrak{Rl}}_{b}{\mathfrak{C}}$ via
$f(a)=a\cdot{\sf s}_{[0/m]}x.$
Indeed, for any $a\in{\mathfrak{Rl}}_{x}{\cal{A}}$, we have $f(a)\leq b$. That
$f$ preserves the boolean operations is obvious. We show that $f$ is one to
one. Assume that $f(c)=f(d)$, then $c\cdot{\sf s}_{[0/m]}x=d\cdot{\sf
s}_{[0/m]}x.$ Applying ${\sf c}_{m}$ to both sides of the equation, we get
that $c\cdot{\sf c}_{m}{\sf s}_{[0/m]}x=d\cdot{\sf c}_{m}{\sf s}_{[0/m]}x$.
Hence
$c=c\cdot{\sf c}_{0}x=d\cdot{\sf c}_{0}x=d.$
We show that $f$ is a neat embedding. Let
${\cal{A}}^{\prime}={\mathfrak{Rl}}_{x}{\cal{A}}$ and
${\mathfrak{B}}^{\prime}={\mathfrak{Rl}}_{b}{\mathfrak{B}}$. Let
$a\in{\cal{A}}^{\prime}$. Then we have
$\begin{split}{\sf c}_{m}^{{\mathfrak{B}}^{\prime}}f(a)&={\sf
c}_{m}^{{\mathfrak{B}}^{\prime}}(a\cdot{\sf s}_{[0/{m}]}x)\\\ &={\sf
c}_{m}(a\cdot{\sf s}_{[0/m]}x)\cdot b\\\ &=a\cdot{\sf c}_{m}{\sf
s}_{[0/m]}x\cdot b\\\ &=a\cdot{\sf c}_{0}x\cdot b=f(a).\\\ \end{split}$
To show that $f$ preserves the non-boolean operations, we note that from (*)
we get for any $i,j\in m$, $b={\sf s}_{[i/m]}{\sf s}_{[j/m]}x\cdot b$. This
will be used in our derivations. First we check that $f$ preserves
cylindrifiers. For $a\in{\cal{A}}^{\prime}$ and $j<m$, we have
$\begin{split}f({\sf c}_{j}^{{\cal{A}}^{\prime}}a)&=f({\sf c}_{j}a\cdot x)\\\
&={\sf c}_{j}a\cdot x\cdot{\sf s}_{[0/m]}a\\\ &={\sf c}_{j}a\cdot b\\\ &={\sf
c}_{j}a\cdot{\sf s}_{[j/m]}x\cdot b\\\ &={\sf c}_{j}(a\cdot{\sf
s}_{[j/m]}x)\cdot b\\\ &={\sf c}_{j}f(a)\cdot b\\\ &={\sf
c}_{j}^{{\mathfrak{B}}^{\prime}}f(a).\\\ \end{split}$
Next we show that $f$ preserves substitutions corresponding to replacements.
For $a\in{\cal{A}}^{\prime}$ and distinct $i,j\in m$, we have
$\begin{split}f({\sf s}_{[j/i]}^{{\cal{A}}^{\prime}}a)&=f({\sf
s}_{[j/i]}a\cdot x)\\\ &={\sf s}_{[j/i]}a\cdot x\cdot{\sf s}_{[0/m]}x\\\
&={\sf s}_{[j/i]}a\cdot b\\\ &={\sf s}_{[j/i]}a\cdot{\sf s}_{[i/m]}{\sf
s}_{[j/m]}x\cdot b\\\ &={\sf s}_{[j/i]}a\cdot{\sf s}_{[j/i]}{\sf
s}_{[i/m]}{\sf s}_{[j/m]}x\cdot b\\\ &={\sf s}_{[j/i]}(a\cdot{\sf
s}_{[i/m]}{\sf s}_{[j/m]}x)\cdot b\\\ &={\sf s}_{[j/i]}(a\cdot{\sf
s}_{[i/m]}{\sf s}_{[j/m]}x\cdot b)\cdot b\\\ &={\sf s}_{[j/i]}(a\cdot{\sf
s}_{[0/m]}x\cdot b)\cdot b\\\ &={\sf s}_{[j/i]}(a\cdot{\sf s}_{[0/m]}x)\cdot
b\\\ &={\sf s}_{[i/j]}f(a)\cdot b\\\ &={\sf
s}_{[i/j]}^{{\mathfrak{B}}^{\prime}}f(a).\\\ \end{split}$
Now assume that $n\geq 2$. Let
$b^{\prime}=x\cdot\prod_{i<n-1}{\sf s}_{[0/m+i]}x.$
We have
${\cal{A}}\subseteq{\mathfrak{Nr}}_{m}{\mathfrak{Nr}}_{m+n-1}{\mathfrak{C}}.$
By the induction hypothesis, we get (**)
${\mathfrak{Rl}}_{a}{\cal{A}}\subseteq{\mathfrak{Nr}}_{m}{\mathfrak{Rl}}_{b^{\prime}}{\mathfrak{Nr}}_{m+n-1}{\mathfrak{C}},$
and
${\sf s}_{[k/l]}b^{\prime}\cdot{\sf s}_{[l/k]}b^{\prime}=b^{\prime},$
for all $k,l<m+n-1$. But
$b^{\prime}\cdot{\sf s}_{[0/m+n-1]}b^{\prime}=b^{\prime}\cdot{\sf
s}_{[0/m+n-1]}x\cdot\prod_{i<n-1}{\sf s}_{[0/m+i]}x,$
and so
$b=b^{\prime}\cdot{\sf s}_{[0/m+n-1]}b^{\prime}.$
It follows that
${\mathfrak{Rl}}_{b^{\prime}}{\mathfrak{Nr}}_{m+n-1}{\mathfrak{C}}\subseteq{\mathfrak{Nr}}_{m+n-1}{\mathfrak{Rl}}_{b}{\mathfrak{C}},$
and we are done by (**).
Thus we have proved (7), contradicting theorem 1(II). Hence
${\mathfrak{Rd}}_{\bf Sc}{\mathfrak{B}}^{r}\not\in{\mathfrak{Nr}}_{\alpha}{\bf
Sc}_{\alpha+k+1}$, as required.
Now we prove the third part of the theorem, putting the superscript $r$ to
use. Recall that
${\mathfrak{B}}^{r}=\Pi_{\Gamma/F}{\mathfrak{C}}^{r}_{\Gamma}$, where
${\mathfrak{C}}^{r}_{\Gamma}$ has the type of ${\bf QPEA}_{\alpha}$ and
${\mathfrak{Rd}}^{\rho_{\Gamma}}{\mathfrak{C}}^{r}_{\Gamma}={\mathfrak{C}}(|\Gamma|,|\Gamma|+k,r)$.
We know that
$\Pi_{r/U}{\mathfrak{Rd}}^{\rho_{\Gamma}}{\mathfrak{C}}^{r}_{\Gamma}=\Pi_{r/U}{\mathfrak{C}}(|\Gamma|,|\Gamma|+k,r)\subseteq{\mathfrak{Nr}}_{|\Gamma|}{\mathcal{A}}_{\Gamma}$,
for some ${\mathcal{A}}_{\Gamma}\in{\bf QPEA}_{|\Gamma|+k+1}$.
Let $\lambda_{\Gamma}:|\Gamma|+k+1\rightarrow\alpha+k+1$ extend
$\rho_{\Gamma}:|\Gamma|\rightarrow\Gamma\;(\subseteq\alpha)$ and satisfy
$\lambda_{\Gamma}(|\Gamma|+i)=\alpha+i$
for $i<k+1$. Let ${\mathfrak{F}}_{\Gamma}$ be a ${\bf QPEA}_{\alpha+k+1}$ type
algebra such that
${\mathfrak{Rd}}^{\lambda_{\Gamma}}{\mathfrak{F}}_{\Gamma}={\mathcal{A}}_{\Gamma}$.
As before, by the second part of lemma 13,
$\Pi_{\Gamma/F}{\mathfrak{F}}_{\Gamma}\in{\bf QPEA}_{\alpha+k+1}$. And
$\displaystyle\Pi_{r/U}{\mathfrak{B}}^{r}$
$\displaystyle=\Pi_{r/U}\Pi_{\Gamma/F}{\mathfrak{C}}^{r}_{\Gamma}$
$\displaystyle\cong\Pi_{\Gamma/F}\Pi_{r/U}{\mathfrak{C}}^{r}_{\Gamma}$
$\displaystyle\subseteq\Pi_{\Gamma/F}{\mathfrak{Nr}}_{|\Gamma|}{\mathcal{A}}_{\Gamma}$
$\displaystyle=\Pi_{\Gamma/F}{\mathfrak{Nr}}_{|\Gamma|}{\mathfrak{Rd}}^{\lambda_{\Gamma}}{\mathfrak{F}}_{\Gamma}$
$\displaystyle\subseteq{\mathfrak{Nr}}_{\alpha}\Pi_{\Gamma/F}{\mathfrak{F}}_{\Gamma},$
proving the lemma. ∎
###### Corollary 14.
Let $\alpha$ be an infinite ordinal, let $k\in\omega$. Let ${\bf K}$ be any
class between ${\bf Sc}$ and ${\bf QPEA}$. Then $S{\mathfrak{Nr}}_{\alpha}{\bf
K}_{\alpha+k+1}\subset S{\mathfrak{Nr}}_{\alpha}{\bf K}_{\alpha+k}.$
Furthermore, $S{\mathfrak{Nr}}_{\alpha}{\bf K}_{\alpha+k+1}$ is not finite
schema axiomatisable over $S{\mathfrak{Nr}}_{\alpha}{\bf K}_{\alpha+k}$.
The first part of the corollary is credited to Pigozzi in [3], for cylindric
algebras; however it seems that Pigozzi did not publish his proof, and we have
not found a published proof elsewhere. See [4, definition 4.1.4] for the
precise definition of finitely schema axiomatisability and see [4, theorem
4.1.7] to see how non finite schema axiomatisability follows from theorem 12.
We summarize the status of the neat embedding problem in the following two
tables. The first table adresses the case when $3\leq\alpha<\omega$ and the
second table addresses the case when $\alpha\geq\omega.$ The first row in the
first table is proved in [4, theorem 5.1.3], the third row of the first table
is in [9], while the last two rows in the second table are proved in [2,
theorem 3.3], [4, theorem 5.4.17]. The rest is proved in the present paper.
Algebras | Status of the Neat Embedding Problem for $3\leq\alpha<\omega,\;m<\omega$
---|---
${\bf Df}_{\alpha}$ | $S{\mathfrak{Nr}}_{\alpha}{\bf Df}_{\beta}={\bf Df}_{\alpha}$ for all $\beta>\alpha.$
${\bf Sc}_{\alpha}$ | $S{\mathfrak{Nr}}_{\alpha}{\bf Sc}_{\alpha+m+1}$ is n.f.a. over $S{\mathfrak{Nr}}_{\alpha}{\bf Sc}_{\alpha+m}$
${\bf CA}_{\alpha}$ | $S{\mathfrak{Nr}}_{\alpha}{\bf CA}_{\alpha+m+1}$ is n.f.a. over $S{\mathfrak{Nr}}_{\alpha}{\bf CA}_{\alpha+m}$
${\bf PA}_{\alpha}$ | $S{\mathfrak{Nr}}_{\alpha}{\bf PA}_{\alpha+m+1}$ is n.f.a. over $S{\mathfrak{Nr}}_{\alpha}{\bf PA}_{\alpha+m}$
${\bf PEA}_{\alpha}$ | $S{\mathfrak{Nr}}_{\alpha}{\bf PEA}_{\alpha+m+1}$ is n.f.a. over $S{\mathfrak{Nr}}_{\alpha}{\bf PEA}_{\alpha+m}$
Algebras | Status of the Neat Embedding Problem for $\alpha\geq\omega,\;m<\omega$
---|---
${\bf Df}_{\alpha}$ | $S{\mathfrak{Nr}}_{\alpha}{\bf Df}_{\beta}={\bf Df}_{\alpha}$ for all $\beta>\alpha.$
${\bf Sc}_{\alpha}$ | $S{\mathfrak{Nr}}_{\alpha}{\bf Sc}_{\alpha+m+1}$ is n.f.a. over $S{\mathfrak{Nr}}_{\alpha}{\bf Sc}_{\alpha+m}$
${\bf CA}_{\alpha}$ | $S{\mathfrak{Nr}}_{\alpha}{\bf CA}_{\alpha+m+1}$ is n.f.a. over $S{\mathfrak{Nr}}_{\alpha}{\bf CA}_{\alpha+m}$
${\bf QPA}_{\alpha}$ | $S{\mathfrak{Nr}}_{\alpha}{\bf QPA}_{\alpha+m+1}$ is n.f.a. over $S{\mathfrak{Nr}}_{\alpha}{\bf QPA}_{\alpha+m}$
${\bf QPEA}_{\alpha}$ | $S{\mathfrak{Nr}}_{\alpha}{\bf QPEA}_{\alpha+m+1}$ is n.f.a. over $S{\mathfrak{Nr}}_{\alpha}{\bf QPEA}_{\alpha+m}$
${\bf PA}_{\alpha}$ | $S{\mathfrak{Nr}}_{\alpha}{\bf PA}_{\beta}={\bf PA}_{\alpha}$ for all $\beta>\alpha$
${\bf PEA}_{\alpha}$ | $S{\mathfrak{Nr}}_{\alpha}{\bf PA}_{\beta}={\bf PEA}_{\alpha}$ for all $\beta>\alpha$
For $\alpha=0,1$, the problem is trivial. For $\alpha=2,$ we have for ${\bf
K}\in\\{{\bf Df},{\bf Sc},{\bf QPA}\\}$ and $\beta>2$,
$S{\mathfrak{Nr}}_{2}{\bf K}_{\beta}={\bf K}_{2}$. The ${\bf Df}$ case is
trivial, the ${\bf Sc}$ and ${\bf QPA}$ cases follow from [4, theorem 5.4.33]
without much ado. On the other hand, for ${\bf K}\in\\{{\bf CA},{\bf QPEA}\\}$
(where diagonal elements are present) and $\beta>2$, ${\bf
S}{\mathfrak{Nr}}_{2}{\bf K}_{\beta}={\bf RK}_{2}$ with ${\bf RK}_{2}$
denoting representable algebras in ${\bf K}_{2}.$ This follows from [4,
theorems 3.2.65, 5.4.34].
## Appendix A Appendix
###### Definition 15.
Substitution Algebra, ${\bf Sc}$
[13].
Let $\alpha$ be an ordinal. By a substitution algebra of dimension $\alpha$,
briefly an ${\bf Sc}_{\alpha},$ we mean an algebra
${{\cal{A}}}=(A,+,-,{\sf c}_{i},{\sf s}_{[{i}/{j}]}:i,j<\alpha)$
where $(A,+,-)$ is a boolean algebra, ${\sf c}_{i},{\sf s}_{[{i}/{j}]}$ are
unary operations on ${\cal{A}}$ (for $i,j<\alpha$) satisfying the following
equations for all $i,j,k,l<\alpha$:
1. 1.
${\sf c}_{i}0=0,\;x\leq{\sf c}_{i}x,\;{\sf c}_{i}(x\cdot{\sf c}_{i}y)={\sf
c}_{i}x\cdot{\sf c}_{i}y$, and ${\sf c}_{i}{\sf c}_{j}x={\sf c}_{j}{\sf
c}_{i}x,$
2. 2.
${\sf s}_{[{i}/{i}]}x=x,$
3. 3.
${\sf s}_{[{i}/{j}]}$ are boolean endomorphisms,
4. 4.
${\sf s}_{[{i}/{j}]}{\sf c}_{i}x={\sf c}_{i}x,$
5. 5.
${\sf c}_{i}{\sf s}_{[{i}/{j}]}x={\sf s}_{[{i}/{j}]}x$ whenever $i\neq j,$
6. 6.
${\sf s}_{[{i}/{j}]}{\sf c}_{k}x={\sf c}_{k}{\sf s}_{[{i}/{j}]}x$, whenever
$k\notin\\{i,j\\},$
7. 7.
${\sf c}_{i}{\sf s}_{[{j}/{i}]}x={\sf c}_{j}{\sf s}_{[{i}/{j}]}x,$
8. 8.
${\sf s}_{[{j}/{i}]}{\sf s}_{[{i}/{k}]}x={\sf s}_{[{i}/{k}]}{\sf
s}_{[{j}/{i}]}x$, whenever $|\\{i,j,k,l\\}|=4,$
9. 9.
${\sf s}_{[{l}/{i}]}{\sf s}_{[{j}/{l}]}x={\sf s}_{[{l}/{i}]}{\sf
s}_{[{j}/{i}]}x$.
Quasipolyadic algebra, ${\bf QPA}$
[18].
A quasipolyadic algebra of dimension $\alpha$, briefly a ${\bf QPA}_{\alpha}$,
is an algebra
${\cal{A}}=(A,+,-,{\sf c}_{i},{\sf s}_{[{i}/{j}]},{\sf s}_{[i,j]}:i,j<n)$
where the reduct to ${\bf Sc}_{\alpha}$ is a substitution algebra (it
satisfies (1)–(9) above) and additionally it satisfies the following equations
for all $i,j,k<\alpha$:
1. 2’
${\sf s}_{[{i}/{i}]}x={\sf s}_{[i,i]}x=x,\text{ and }{\sf s}_{[i,j]}x={\sf
s}_{[j,i]}x,$
2. 3’
${\sf s}_{[i,j]}$ are boolean endomorphisms
3. 1.
${\sf s}_{[i,j]}{\sf s}_{[i,j]}x=x$,
4. 2.
${\sf s}_{[i,j]}{\sf s}_{[i,k]}x={\sf s}_{[j,i]}{\sf s}_{[i,j]}x~{}~{}\text{
if }~{}~{}|\\{i,j,k\\}|=3$,
5. 3.
${\sf s}_{[i,j]}{\sf s}_{[{j}/{i}]}x={\sf s}_{[{i}/{j}]}x$.
Quasipolyadic equality algebra, ${\bf QPEA}$
[18].
A quasipolyadic equality algebra of dimension $\alpha$, briefly a ${\bf
QPEA}_{\alpha}$ is an algebra
${\mathfrak{B}}=({\cal{A}},{\sf d}_{ij})_{i,j<\alpha}$
where ${\cal{A}}$ is a ${\bf QPA}_{\alpha}$ (i.e. it satisfies all the
equations above), ${\sf d}_{ij}$ is a constant and the following equations
hold, for all $i,j,k<\alpha$:
1. 1.
${\sf s}_{[{i}/{j}]}{\sf d}_{ij}=1$,
2. 2.
$x\cdot{\sf d}_{ij}\leq{\sf s}_{[{i}/{j}]}x$.
## References
* [1] Andréka, H., Givant, S., Mikulas, S. Németi, I., Simon A., Notions of density that imply representability in algebraic logic. Annals of Pure and Applied logic, 91(1998), p. 93 –190.
* [2] Daigneault, A., and Monk, J.D., Representation Theory for Polyadic algebras. Fund. Math. 52(1963), p.151-176.
* [3] Henkin,L., Monk,J.D., and Tarski, A., Cylindric Algebras Part I. North Holland, 1971.
* [4] Henkin,L., Monk,J.D., and Tarski,A., Cylindric Algebras Part II. North Holland, 1985.
* [5] Hirsch R. Relation algebra reducts of cylindric algebras and complete representations. Journal of Symbolic Logic, 72(2) (2007), p.673-703.
* [6] Hirsch R., Hodkinson I, Maddux R., Relation algebra reducts of cylindric algebras and an application to proof theory. Journal of Symbolic Logic, 67 (2002), p.197-213.
* [7] Hirsch, R., Hodkinson, I. and Maddux, R., On provability with finitely many variables. Bulletin of Symbolic Logic, 8(3)(2002), p.329-347.
* [8] Hirsch R., Hodkinson I., Relation algebras by games. Studies in Logic and the Foundations of Mathematics. Volume 147. (2002)
* [9] Hirsch R., Hodkinson I., Relation algebras from cylindric algebras, II. Annals of Pure and Applied Logic, 112(2001), p. 267–297.
* [10] Monk., J.D., Non-finitizability of classes of representable cylindric algebras. Journal of Symbolic Logic, 34 (1969), p.331-343.
* [11] Németi, I., The Class of Neat Reducts of Cylindric Algebras is Not a Variety But is closed w.r.t. HP. Notre Dame Journal of Formal logic, 24(3) (1983), p 399-409.
* [12] Németi,I., Algebraisation of quantifier logics, an introductory overview. Math.Inst. Budapest, Preprint, No 13 (1996). A shortened version appeared in Studia Logica, 50(4) (1991), p.465-569.
* [13] Pinter, C. A simple algebra of first order logic. Notre Dame Journal of Formal Logic, 1(1973), p.361-366.
* [14] Pinter, C. Cylindric algebras and algebras of substitutions. Transactions of the American Mathematical Society, 175(1973), p.167-179.
* [15] Sayed Ahmed, T. and Németi I, On neat reducts of algebras of logic. Studia Logica, 62 (2) (2001), p.229-262.
* [16] Sayed Ahmed, T., and Samir B., A Neat embedding theorem for expansions of cylindric algebras. Logic journal of IGPL, 15 (2007), p. 41-51.
* [17] Sayed Ahmed, T., Neat reducts and neat embeddings in cylindric algebras In ’Cylindric-like algebras and Algebraic Logic’, Bolyai Society of Mathematical Studies, vol 22, Editors: Andréka H, Ferenzci, M. and Németi. I (2013).
* [18] Sain, I. Thompson, R. Strictly finite schema axiomatization of quasi-polyadic algebras. In ‘Algebraic Logic’ North Holland, Editors: Andréka H, Monk D,. and Németi. (1990) I. p.539-572.
|
arxiv-papers
| 2013-01-24T17:58:47 |
2024-09-04T02:49:40.692042
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tarek Sayed Ahmed",
"submitter": "Tarek Sayed Ahmed",
"url": "https://arxiv.org/abs/1301.5850"
}
|
1301.5855
|
MILC Collaboration
# Leptonic decay-constant ratio $f_{K^{+}}/f_{\pi^{+}}$ from lattice QCD with
physical light quarks
A. Bazavov Physics Department, Brookhaven National Laboratory, Upton, NY
11973, USA C. Bernard [email protected] Department of Physics, Washington
University, St. Louis, MO 63130, USA C. DeTar Department of Physics and
Astronomy, University of Utah, Salt Lake City, UT 84112, USA J. Foley
Department of Physics and Astronomy, University of Utah, Salt Lake City, UT
84112, USA W. Freeman Department of Physics, The George Washington
University, Washington, DC 20052, USA Steven Gottlieb Department of Physics,
Indiana University, Bloomington, IN 47405, USA U.M. Heller American Physical
Society, One Research Road, Ridge, NY 11961, USA J.E. Hetrick Physics
Department, University of the Pacific, Stockton, CA 95211, USA J. Kim
Department of Physics, University of Arizona, Tucson, AZ 85721, USA J. Laiho
SUPA Department of Physics and Astronomy, University of Glasgow, Glasgow G12
8QQ, Scotland, UK Department of Physics, Syracuse University, Syracuse, NY
13244, USA L. Levkova Department of Physics and Astronomy, University of
Utah, Salt Lake City, UT 84112, USA Department of Physics, University of
Arizona, Tucson, AZ 85721, USA M. Lightman Department of Physics, Washington
University, St. Louis, MO 63130, USA J. Osborn Argonne Leadership Computing
Facility, Argonne National Laboratory, Argonne, IL 60439, USA S. Qiu
Department of Physics and Astronomy, University of Utah, Salt Lake City, UT
84112, USA R.L. Sugar Department of Physics, University of California, Santa
Barbara, CA 93106, USA D. Toussaint [email protected] Department of
Physics, University of Arizona, Tucson, AZ 85721, USA R.S. Van de Water
[email protected] Theoretical Physics Department, Fermi National Accelerator
Laboratory, Batavia 60510, USA R. Zhou Department of Physics, Indiana
University, Bloomington, IN 47405, USA
###### Abstract
A calculation of the ratio of leptonic decay constants $f_{K^{+}}/f_{\pi^{+}}$
makes possible a precise determination of the ratio of CKM matrix elements
$|V_{us}|/|V_{ud}|$ in the Standard Model, and places a stringent constraint
on the scale of new physics that would lead to deviations from unitarity in
the first row of the CKM matrix. We compute $f_{K^{+}}/f_{\pi^{+}}$
numerically in unquenched lattice QCD using gauge-field ensembles recently
generated that include four flavors of dynamical quarks: up, down, strange,
and charm. We analyze data at four lattice spacings $a\approx 0.06,0.09,0.12$,
and 0.15 fm with simulated pion masses down to the physical value 135 MeV. We
obtain $f_{K^{+}}/f_{\pi^{+}}=1.1947(26)(37)$, where the errors are
statistical and total systematic, respectively. This is our first physics
result from our $N_{f}=2+1+1$ ensembles, and the first calculation of
$f_{K^{+}}/f_{\pi^{+}}$ from lattice-QCD simulations at the physical point.
Our result is the most precise lattice-QCD determination of
$f_{K^{+}}/f_{\pi^{+}}$, with an error comparable to the current world
average. When combined with experimental measurements of the leptonic
branching fractions, it leads to a precise determination of
$|V_{us}|/|V_{ud}|=0.2309(9)(4)$ where the errors are theoretical and
experimental, respectively.
###### pacs:
13.20.-v, 12.38.Gc, 12.15.Hh
Motivation. — Leptonic decays of charged pseudoscalar mesons are sensitive
probes of quark flavor-changing interactions. Experimental measurements of the
leptonic decay widths, when combined with precise theoretical calculations of
the leptonic decay constants, enable the determination of elements of the
Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing matrix. Further, when combined
with independent determinations of the CKM matrix elements from other
processes such as semileptonic meson decays, they make possible precise tests
of the Standard-Model CKM framework. Here we present a lattice-QCD calculation
of the decay-constant ratio $f_{K^{+}}/f_{\pi^{+}}$, which may be used to
determine $|V_{us}|/|V_{ud}|$ via Marciano:2004uf ; Aubin:2004fs
$\frac{\Gamma(K\to l\bar{\nu}_{l})}{\Gamma(\pi\to
l\bar{\nu}_{l})}=\frac{|V_{us}|^{2}}{|V_{ud}|^{2}}\frac{f_{K}^{2}}{f_{\pi}^{2}}\frac{m_{K}}{m_{\pi}}\frac{\left(1-\frac{m_{l}^{2}}{m_{K}^{2}}\right)^{2}}{\left(1-\frac{m_{l}^{2}}{m_{\pi}^{2}}\right)^{2}}\left[1+\delta_{\rm
EM}\right]\,,$ (1)
where $l=e,\mu$ and $\delta_{\rm EM}$ is a known 1-loop QED correction
Decker:1994ea ; Finkemeier:1995gi . The ratio $f_{K^{+}}/f_{\pi^{+}}$ can be
calculated to subpercent precision using numerical lattice-QCD simulations
Laiho:2009eu ; Colangelo:2010et because the Monte Carlo statistical errors
are correlated between the numerator and denominator and the explicit
dependence on the lattice scale drops out.
Lattice QCD enables ab initio calculations of nonperturbative hadronic matrix
elements. The QCD Lagrangian is formulated in discrete Euclidean spacetime,
after which the now finite-dimensional path integral can be solved numerically
with Monte Carlo methods. Once the $N_{f}+1$ parameters of the QCD action
(quark masses and gauge coupling) are fixed by matching to an equal number of
experimental measurements, all other outputs of the lattice simulation are
predictions of the theory. In practice, currently available computing
resources limit the lattice spacing in most recent simulations to $a\gtrsim
0.06$ fm, the spacetime volume of the lattice to $L^{3}\sim(4~{}{\rm
fm})^{3}$, and the pion mass to $m_{\pi}\gtrsim 200$ MeV. These choices
introduce systematic errors that must be quantified, but the good agreement
between lattice-QCD calculations and experiment for a wide range of
observables Davies:2003ik ; Kronfeld:2012uk demonstrates that the errors are
controlled.
Many precise lattice-QCD weak-matrix element calculations relevant for quark-
flavor physics have been obtained using our $N_{f}=2+1$ flavor gauge-field
ensembles generated using the asqtad-improved staggered quark action for the
dynamical $u$, $d$, and $s$ quarks Bernard:2001av ; Aubin:2004wf ;
Bazavov:2009bb . Our calculations of $f_{K+}/f_{\pi^{+}}$ on those ensembles
Aubin:2004fs ; Bazavov:2009bb ; Bazavov:2010hj ultimately reached a precision
of $\sim\\!0.6\%$. We are now embarking on a new program of configuration
generation using the highly-improved staggered quark (HISQ) action
Follana:2006rc , which has two primary advantages. The most important
staggered discretization errors are approximately 3 times smaller for the HISQ
action than for the asqtad action at the same lattice spacing Bazavov:2010ru ;
this makes it possible to undertake simulations at the physical pion mass,
which require a box of approximately $(5~{}{\rm fm})^{3}$. The improved quark
dispersion relation for the HISQ action also allows the inclusion of dynamical
charm quarks in the simulations.
One of the largest sources of uncertainty in lattice-QCD determinations of
quantities such as $f_{K^{+}}/f_{\pi^{+}}$ is from the extrapolation of the
simulation results to the physical up- and down-quark masses. In this work, we
replace this error with a negligible interpolation error by simulating at
light-quark masses very close to or even below their physical values. The
calculation presented here, using the configurations with four flavors of
dynamical HISQ quarks, lays the foundation for a larger quark flavor-physics
program to compute pion, kaon, and charmed weak matrix elements, and
eventually even those with bottom quarks, on these ensembles.
Lattice-QCD Calculation. — This calculation is based on a subset of our
(2+1+1)-flavor ensembles described in Refs. Bazavov:2010ru ; Bazavov:2012uw .
These ensembles use a one-loop Symanzik-improved gauge action for the gluons
Luscher:1985zq ; Hart:2008sq , and the HISQ action for the dynamical $u$, $d$,
$s$, and $c$ quarks. The simulated strange and charm sea-quark masses ($m_{s}$
and $m_{c}$) are fixed at approximately their physical values. The simulated
up and down sea-quark masses are taken to be degenerate with a common mass
$m_{l}\approx m_{s}/27$, such that the simulated pion mass is approximately
the physical value 135 MeV Beringer:1900zz . Additional ensembles with
slightly heavier pions ($m_{l}=m_{s}/10$) allow us to correct
$f_{K^{+}}/f_{\pi^{+}}$ a posteriori for the slight mistuning of the simulated
sea-quark masses. The spatial volumes are sufficiently large ($m_{\pi}L>3.5$)
that finite-volume corrections are expected to be at the subpercent level for
light pseudoscalar meson masses and decay constants, and we confirm this with
an explicit finite-volume study. We use four lattice spacings $a\approx 0.15$,
0.12, 0.09, and 0.06 fm to enable a controlled extrapolation to the continuum
($a\to 0$) limit. The specific numerical simulation parameters are given in
Table 1.
Table 1: $N_{f}=2+1+1$ gauge-field ensembles used in this work. The lattice spacing in fm is approximate. Ensembles for which the number of equilibrated lattices ${N}_{\rm lats}$ has reached 1000 are considered complete. At $a\approx 0.12$ fm, we have three ensembles with identical simulation parameters except for the spatial volume. For these ensembles, the given pion masses and ${N}_{\rm lats}$ are approximate, since they differ slightly for the three volumes. $a$(fm) | $L$(fm) | $M_{\pi,5}$(MeV) | $M_{\pi,{\rm RMS}}$(MeV) | ${N}_{\rm lats}$
---|---|---|---|---
0.15 | 3.66 | 214 | 352 | 1000
0.15 | 4.82 | 133 | 311 | 1000
0.12 | {2.84,3.79,4.73} | 215 | 294 | 1000
0.12 | 5.83 | 133 | 241 | 1000
0.09 | 4.33 | 215 | 244 | 1000
0.09 | 5.63 | 128 | 173 | 707
0.06 | 3.79 | 223 | 229 | 678
0.06 | 5.45 | 134 | 143 | 310
Figure 1: Pion mass (left) and decay constant (right) versus lattice spatial
extent. The open circles show the numerical data with statistical errors. The
staggered NLO $\chi$PT prediction is shown as a solid red line. The continuum
NLO (NNLO) $\chi$PT predictions are shown as dashed (dot-dashed) blue lines.
The close agreement of the staggered and continuum NLO $\chi$PT fits for
$f_{\pi}$ is a numerical coincidence resulting from the cancellation of
effects from pion taste splittings and quark-disconnected hairpin
contributions, and is not seen at other lattice spacings.
Naive lattice discretizations of the quark action suffer from the problem of
fermion doubling. The staggered quark action only partly eliminates fermion
doublers, reducing the number of species from 16 to 4. The remaining species
are referred to as “tastes” to distinguish them from physical flavors. Quarks
of different taste can interact by exchanging gluons with momentum components
close to the lattice cutoff $p_{\mu}\approx\pi/a$. Taste-changing interactions
split the mass degeneracy of pions composed of different quark tastes. The
taste splittings are of ${\mathcal{O}}(\alpha_{S}^{2}a^{2})$ for the HISQ
action, and decrease rapidly with lattice spacing. The taste-Goldstone and
root-mean-squared (RMS) sea-pion masses are given in Table 1; the difference
between the two is only about 10 MeV for the finest $a\approx 0.06$ fm
ensembles used in this analysis. More details can be found in Ref.
Bazavov:2012uw .
The dynamical HISQ simulations use the fourth-root procedure to remove the
unwanted taste degrees of freedom. Due to taste-changing interactions, the
rooted Dirac operator is nonlocal at nonzero lattice spacing, and leads to
violations of unitarity Prelovsek:2005rf ; Bernard:2006zw ; Bernard:2006ee ;
Bernard:2007qf . Nevertheless, there are strong theoretical arguments and
numerical evidence that the continuum limit of the rooted staggered lattice
theory is indeed QCD Shamir:2004zc ; Shamir:2006nj ; Follana:2004sz ;
Durr:2004as ; Durr:2004ta ; Wong:2004nk ; Durr:2006ze ; Donald:2011if .
We obtain the meson masses and decay constants from fits of two-point
correlation functions with a pseudoscalar interpolating operator at both the
source and sink. For each valence-quark mass combination, we compute the decay
constant using a partially-conserved axial current relation:
$F_{xy}=(m_{x}+m_{y})\langle
0|\bar{x}\gamma_{5}y|P_{x\bar{y}}\rangle/M_{xy}^{2}\,,$ (2)
where $M_{xy}$ is the mass of the pseudoscalar meson $P_{x\bar{y}}$ with bare
valence-quark masses $m_{x}$ and $m_{y}$. We choose a fitting range for the
correlators such that, in the case of degenerate valence-quark masses, a
single-exponential fit gives a good fit as measured by the correlated
$\chi^{2}/{\rm dof}$ and $p$-value. We include both the ground state and an
opposite-parity excited state for the nondegenerate correlators. Statistical
errors are estimated by jackknifing. The ground-state mass and amplitudes are
stable with respect to a reasonable reduction of the minimum time separation
of the source and sink, and with the inclusion of additional excited states;
therefore the error due to excited-state contamination can be neglected
compared with the statistical errors. Additional details of these fits can be
found in Ref. Bazavov:2012us .
Chiral perturbation theory ($\chi$PT) predicts that the finite-volume
corrections to pion and kaon masses and decay constants are at the per mill
level in our simulations, and are therefore comparable to the size of the
statistical errors in our numerical data. We check this expectation with an
explicit comparison of these quantities measured on three $a\approx 0.12$ fm
ensembles with identical simulation parameters except for the spatial lattice
volume (see Table 1). We fit the data at the three volumes to the functional
form:
$aX(L)\\!=\\!A_{X}\left(1+B_{X}(m_{\pi},L)\right)$ (3)
where $X=\\{m_{\pi},f_{\pi},m_{K},f_{K}\\}$, $A_{X}$ is a free parameter, and
the function $B_{X}$ is determined at a given order in $\chi$PT. We try both
the NLO expression for $B_{X}$ in staggered $\chi$PT Aubin:2003mg ;
Aubin:2003uc , and the continuum NLO and resummed NNLO expressions
Colangelo:2005gd . Figure 1 compares the numerical lattice data for $m_{\pi}$
and $f_{\pi}$ with these expressions. NLO staggered $\chi$PT describes the
$f_{\pi}$ data very well at all three volumes, and describes the $m_{\pi}$,
$m_{K}$, and $f_{K}$ data adequately (to $\sim\\!0.4\%$ or better). We
therefore use NLO staggered finite-volume $\chi$PT to correct our simulation
data in our central fit, and use the continuum NLO and NNLO finite-volume
$\chi$PT corrections to estimate the systematic uncertainty.
Because our lattice-QCD simulations are isospin-symmetric (the up and down
sea-quark masses are equal), we adjust the experimental inputs to what they
would be in a world without electromagnetism or isospin violation before
matching the simulation data to experiment to find the strange quark mass
$m_{s}$ and the average light quark mass $\hat{m}=(m_{u}+m_{d})/2$. We follow
the approach of Ref. Aubin:2004fs , but with updated values for the EM
correction to Dashen’s theorem. For the kaon, we consider the isospin-averaged
mass $M_{\widehat{K}}^{2}=(M_{K^{+}}^{2}+M_{K^{0}}^{2})_{\rm QCD}/2$, where
the subscript “QCD” indicates that the leading EM effects in the masses were
removed from the experimental masses Beringer:1900zz ; we take the parameter
$\Delta_{\rm EM}=0.65(7)_{\rm stat}(14)_{\rm syst}$ that characterizes
violations of Dashen’s theorem from our ongoing lattice QED+QCD simulations
using asqtad sea quarks Basak:2012zx .
In this work, we tune the quark masses and lattice scale and determine
$f_{K^{+}}/f_{\pi^{+}}$ in a single, self-contained analysis as follows. We
begin with numerical lattice data for meson masses and decay constants for a
selection of valence-quark masses that allow us to adjust for slight
mistunings and interpolate or extrapolate to the physical values. In order to
reduce autocorrelations below measurable levels, the single-elimination
jackknife results are blocked by 20 lattices (100 or 120 molecular dynamics
time units) before the final averaging. The physical-quark-mass $a\approx
0.06$ fm ensemble has an insufficient number of configurations for blocks that
large. Instead, we block by 5 trajectories on that ensemble and rescale the
resulting errors by a factor of 1.17, which is determined by comparing the
errors at block sizes around 20 to those at block size 5 on the
$m_{l}=0.1m_{s}$, $a\approx 0.06$ fm ensemble, as well as an additional
$m_{l}=0.2m_{s}$, $a\approx 0.06$ fm ensemble not otherwise discussed here. We
then construct the squared ratio of pseudoscalar-meson mass to decay constant,
$M_{xx}^{2}/F_{xx}^{2}$, which $\chi$PT predicts to be approximately
proportional to the valence-quark mass $m_{x}$. Using the lightest valence-
quark mass on each ensemble, we interpolate or extrapolate data for this ratio
linearly in $m_{x}$ to where it equals the experimental value for
$M_{\pi^{0}}^{2}/f_{\pi^{+}}^{2}$. This fixes the physical value of $\hat{m}$.
We obtain the lattice spacing from requiring that the decay constant at this
mass equal the experimental value for $f_{\pi^{+}}$ Rosner:2010ak , where we
use a quadratic interpolation/extrapolation through the lightest three
valence-quark masses. We then take nondegenerate “kaons” in which the strange
valence-quark mass ($m_{y}$) is 1.0 or 0.8 times the strange sea-quark mass
and the lighter valence-quark mass ($m_{x}$) is the lightest available, and
linearly interpolate in the valence strange-quark mass to where
$2M_{xy}^{2}-M_{xx}^{2}=2M_{\widehat{K}}^{2}-M_{\pi^{0}}^{2}$; this fixes the
physical strange-quark mass $m_{s}$. Once we know $m_{s}$, we obtain the
$u$-$d$ mass-splitting from the difference between the EM-subtracted neutral
and charged kaon masses: $m_{d}-m_{u}=(M_{K^{0}}^{2}-M_{\hat{K}}^{2})_{\rm
QCD}/(\partial M^{2}_{sx}/\partial m_{x})$. Finally, we obtain $f_{K^{+}}$ by
linearly interpolating the decay constant $F_{xy}$ in the light valence-quark
mass to $m_{u}$ and in the strange valence-quark mass to $m_{s}$. In
$f_{K^{+}}$, we neglect isospin violations from the sea quarks, and we neglect
all isospin violations in $f_{\pi^{+}}$, because these effects are of NNLO in
$\chi$PT Colangelo:2010et . We also slightly correct $f_{K}$ a posteriori for
sea-quark mass mistuning using the slope with respect to the light sea-quark
mass, $\partial f_{K}/\partial m_{l}$, measured by comparing the value of
$f_{K}$ from the physical-mass ensembles with that from ensembles with
$m_{l}\approx 0.1m_{s}$. The results for $f_{K^{+}}/f_{\pi^{+}}$ are shown
versus $a^{2}$ in Fig. 2.
Figure 2: Continuum extrapolation of $f_{K^{+}}/f_{\pi^{+}}$. Our central
value is obtained from a quadratic-in-$a^{2}$ fit to the physical-quark-mass
data points (red circles), adjusted for tuning errors to the physical sea
quark masses using the $0.1m_{s}$ data points (blue squares). The size of the
adjustments is too small to be visible on this plot. The black star shows
$f_{K^{+}}/f_{\pi^{+}}$ at the physical quark masses in the continuum, where
the inner solid error bar is statistical and the outer dotted error bar
includes the continuum-extrapolation error added in quadrature. An
alternative, constant-in-$a^{2}$ fit to the finest two physical-mass data
points, is shown by the dashed horizontal line. Its deviation from the central
value gives our estimate of the continuum extrapolation error (see text). The
dotted error bars on the $0.1m_{s}$ data show the systematic error from the
valence-quark mass tuning added in quadrature to the statistical error (see
text); the tuning uncertainty on the physical-mass ensembles is negligible.
We extrapolate $f_{K^{+}}/f_{\pi^{+}}$ at physical quark masses to the
continuum limit quadratically in $a^{2}$. We estimate the uncertainty in the
continuum extrapolation from alternative fits with different ansätze for the
$a^{2}$ dependence and excluding the coarsest ensemble(s). Two reasonable
choices are a linear extrapolation of the three physical-mass data points with
$a\leq 0.12$ fm, and a constant fit to the two physical-mass data points with
$a\leq 0.09$ fm. Of these, the constant fit, shown in Fig. 2, gives the larger
difference from the central extrapolation, $0.28\%$. Additional extrapolations
have also been tried, including a quadratic-in-$a^{2}$, linear in $m_{l}$ fit
to all the $m_{l}\approx m_{s}/27$ and $m_{l}=0.1m_{s}$ data points, and a
similar fit restricted to ensembles with $a\leq 0.12$ fm. These fits give
differences from the central value that are comparable to or smaller than the
difference seen above, as does a simple linear-in-$a^{2}$ extrapolation of the
two finest $a\leq 0.09$ fm physical-mass points. We thus take $0.28\%$ as our
estimate of the discretization error. Note that, when the scale-setting
procedure is changed by repeating the analysis using other quantities to fix
the lattice scale instead of $f_{\pi}$ (see Ref. Bazavov:2012uw for further
details), we find a $0.17\%$ difference with the central value, well within
the estimate for the continuum extrapolation error. To estimate the finite-
volume error, we repeat the entire analysis using finite-volume corrections
for the pion (kaon) masses and decay constants calculated at NNLO (NLO) in
continuum $\chi$PT. We propagate the statistical uncertainties in the tuned
quark masses and lattice scale throughout the analysis via the jackknife
method, so they are already folded into the statistical error. We estimate the
systematic uncertainty from the quark-mass tuning and scale setting by taking
the difference of our central tuning procedure with one where we
interpolate/extrapolate the ratio $M^{2}_{xx}/F^{2}_{xx}$ quadratically in
$m_{x}$. For the physical quark-mass ensembles (and hence for the continuum
extrapolated result), the valence-quark masses are very close to the tuned
physical values, so the choice of interpolation fit function makes a
negligible difference. On the 0.1 $m_{s}$ ensembles, however, valence masses
as light as physical are not available, so the choice of fit function for the
valence-quark extrapolation has a greater impact. The systematic errors from
the tuning procedure are plotted as dotted error bars in Fig. 2. We estimate
the uncertainty due to EM effects by varying the values of the EM-subtracted
meson masses used in the quark-mass tuning; this primarily affects $m_{u}$. We
vary the parameter $\Delta_{EM}$ by its total error Basak:2012zx . We also
repeat the tuning including a previously-neglected EM correction to the
neutral kaon mass, $\delta M^{2}_{K^{0},{\rm EM}}=901(8)_{\rm stat}~{}{\rm
MeV}^{2}$ Bernard:CD12 . Note that direct EM effects on the weak matrix
elements are by definition removed from $f_{\pi^{+}}$ and $f_{K^{+}}$
Beringer:1900zz .
Results and Conclusions. — We obtain the following determination of the ratio
$f_{K^{+}}/f_{\pi^{+}}$:
$f_{K^{+}}/f_{\pi^{+}}=1.1947(26)_{\rm stat}(33)_{a^{2}}(17)_{\rm FV}(2)_{\rm
EM},$ (4)
with an approximately 0.4% total uncertainty. Our result agrees with, but is
more precise than, other independent lattice-QCD calculations Follana:2007uv ;
Durr:2010hr ; Aoki:2010dy ; Bazavov:2010hj ; Laiho:2011np , and its error is
competitive with that of the current lattice-QCD world average Laiho:2009eu ;
Colangelo:2010et . Because we use $f_{\pi}$ to set the lattice scale, we also
obtain a result for $f_{K^{+}}=155.80(34)_{\rm
stat}(48)_{\mathrm{sys}}(24)_{f_{\pi}}$ where the errors are due to
statistics, systematics in $f_{K}/f_{\pi}$, and the uncertainty in $f_{\pi}$.
This agrees with the experimental determination of $f_{K^{+}}$ assuming CKM
unitarity Rosner:2010ak .
By combining $f_{K^{+}}/f_{\pi^{+}}$ from Eq. (4) with recent experimental
results for the leptonic branching fractions Antonelli:2010yf , we obtain
$|V_{us}|/|V_{ud}|=0.2309(9)_{\rm theo.}(4)_{\rm exp.}$. Taking $|V_{ud}|$
from nuclear $\beta$ decay Hardy:2008gy , we also obtain
$|V_{us}|=0.2249(8)_{\rm theo.}(4)_{\rm exp.}(1)_{V_{ud}}$, which agrees with
and has comparable errors to the determination from $K\to\pi\ell\nu$
semileptonic decay Lubicz:2009ht ; Boyle:2010bh ; Antonelli:2010yf ;
Bazavov:2012cd . Further, our result places stringent constraints on new-
physics scenarios that would lead to deviations from unitarity in the first
row of the CKM matrix. We find
$1-|V_{ud}|^{2}-|V_{us}|^{2}-|V_{ub}|^{2}=0.0003(6)$, which is in excellent
agreement with the Standard-Model value of zero.
We are currently extending the ensemble with the physical pion mass at
$a\approx 0.06$ fm as well as the other ensembles with fewer than 1000
equilibrated lattices. We will include this additional data in a longer work
that updates this result and presents determinations of the charmed
pseudoscalar decay constants. We will also perform a more sophisticated
analysis using staggered chiral perturbation theory Aubin:2003mg ;
Aubin:2003uc ; Komijani:2012fq to obtain the low-energy constants of
$\chi$PT. Our result for $f_{K^{+}}/f_{\pi^{+}}$ lays the foundation for a new
large-scale lattice-QCD physics program using the $N_{f}=2+1+1$ flavor HISQ
gauge-field ensembles with physical pion masses. These ensembles will enable
calculations of light and heavy-light hadronic weak matrix elements with
unprecedented precision, and will ultimately strengthen tests of the Standard
Model and improve the reach of searches for new physics in the quark-flavor
sector.
NB: After this work was submitted for peer review, another determination of
$f_{K}/f_{\pi}$ using our HISQ ensembles and consistent with our result was
posted by HPQCD Dowdall:2013rya .
Acknowledgments. — We thank Christine Davies and Andreas Kronfeld for useful
discussions and comments on the manuscript. We thank Maarten Golterman for
pointing out a critical typo in the abstract. Computations for this work were
carried out with resources provided by the USQCD Collaboration, the Argonne
Leadership Computing Facility and the National Energy Research Scientific
Computing Center, which are funded by the Office of Science of the United
States Department of Energy; and with resources provided by the National
Center for Atmospheric Research, the National Center for Supercomputing
Applications, the National Institute for Computational Science, and the Texas
Advanced Computing Center, which are funded through the National Science
Foundation’s Teragrid/XSEDE and Blue Waters Programs. We thank the staffs of
NICS, ALCF and NCSA for their assistance with block time grants and Early
Science usage. This work was supported in part by the U.S. Department of
Energy under Grants No. DE-FG02-91ER40628 (C.B., M.L.), No. DE-FC02-06ER41446
(C.D., L.L., J.F.), No. DE-FG02-91ER40661 (S.G., R.Z.), No. DE-FG02-85ER40237
(J.L.), No. DE-FG02-04ER-41298 (D.T.); No. DE-FC02-06ER-41439 (J.K.,A.B.,L.L),
by the National Science Foundation under Grants No. PHY-1067881, No.
PHY-0757333, No. PHY-0703296 (C.D., L.L., J.F., S.Q.), No. PHY-0555397,
(A.B.), No. PHY-0903536, (A.B.,J.K.), No. PHY-0757035 (R.S.), and by the
Science and Technology Facilities Council and the Scottish Universities
Physics Alliance (J.L.). This manuscript has been co-authored by employees of
Brookhaven Science Associates, LLC, under Contract No. DE-AC02-98CH10886 with
the U.S. Department of Energy. Fermilab is operated by Fermi Research
Alliance, LLC, under Contract No. DE-AC02-07CH11359 with the U.S. Department
of Energy.
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|
arxiv-papers
| 2013-01-24T18:18:27 |
2024-09-04T02:49:40.702857
|
{
"license": "Public Domain",
"authors": "A. Bazavov, C. Bernard, C. DeTar, J. Foley, W. Freeman, Steven\n Gottlieb, U. M. Heller, J. E. Hetrick, J. Kim, J. Laiho, L. Levkova, M.\n Lightman, J. Osborn, S. Qiu, R. L. Sugar, D. Toussaint, R. S. Van de Water,\n R. Zhou",
"submitter": "Ruth Van de Water",
"url": "https://arxiv.org/abs/1301.5855"
}
|
1301.5867
|
CALT 68-2902
# Global CKM Fits with the Scan Method
Gerald Eigen University of Bergen, 5007 Bergen, Norway Gregory Dubois-
Felsmann SLAC National Accelerator Laboratory, Stanford, California 94309,
USA David G. Hitlin Frank C. Porter California Institute of Technology,
Pasadena, California 91125, USA
###### Abstract
We present results of unitary triangle fits based on the scan method. This
frequentist approach employs Gaussian uncertainties for experimental
quantities, but avoids assumptions about the distribution of theoretical
errors. Instead, we perform a large number of fits, scanning over regions of
plausible theory errors for each quantity. We retain those fits meeting a
specific confidence level criterion, thereby constructing a region in the
$\bar{\rho}-\bar{\eta}$ plane using the “standard” measurements (CKM matrix
elements, $\sin 2\beta,B^{0}_{d,s}$ mixing, $\epsilon_{K}$). In addition we
use branching fraction and $C\\!P$ asymmetry measurements of $B$ decays to
pseudoscalar pseudoscalar, pseudoscalar vector, vector vector and $a_{1}$
pseudoscalar final states to determine $\alpha$, $D^{(*)}K^{(*)}$ modes to
determine $\gamma$, and $D^{(*)}\pi$ and $D\rho$ modes to determine
$2\beta+\gamma$. We parameterize individual decay amplitudes in terms of
color-allowed tree, color-suppressed tree, penguin, singlet penguin,
electroweak penguin, as well as $W$-exchange and $W$-annihilation amplitudes.
With this parameterization, we obtain a good fit to the measured branching
fractions and $C\\!P$ asymmetries within the Standard Model ansatz, with no
new physics contributions. This simultaneous fit allows us to determine, for
the first time in a global fit, the correlation between $\alpha$ and $\beta$,
as well as between $\gamma$ and $\beta$.
###### pacs:
12.15.Hh, 13.25.Hw, 14.65.Fy
## I Introduction
The phase of the Cabibbo-Kobayashi-Maskawa (CKM) matrix ckm is responsible
for $C\\!P$ violation in the Standard Model (SM). The unitarity relations
within the CKM matrix provide an excellent laboratory to test this prediction,
the relation ${V_{ub}}^{*}V_{ud}+{V_{cb}}^{*}V_{cd}+{V_{tb}}^{*}V_{td}=0$
being particularly useful, since many measurements and theory inputs in the
$B$ and $K$ systems can be combined for this test. Tests (including
simultaneous estimation of the fundamental CKM parameters) such as those by
the CKMfitter ckmfitter and UT${}_{\it fit}\ $utfit groups are in common
circulation. The former is a frequentist technique, the latter Bayesian. The
scan method scan presented herein is a frequentist-based fitting technique to
determine the parameters of the CKM matrix and test consistency with the
standard model. The scan method takes a different approach in its treatment of
theoretical uncertainties, as well as in the construction of confidence sets.
We first describe the scan method in section II. The results for the global
fits are presented in section III. We begin with a comparison with CKMfitter
and UTfit, employing the scan method to extract CKM parameters using inputs
standardized for the book Physics of the $B$ Factories pbf . Then we look at
the current situation using inputs from PDG12 pdg12 and HFAG HFAG ,
investigating the question of consistency of the results with SM expectations.
We call such fits “baseline” fits. Their characteristic is that the $\chi^{2}$
function has of the order of ${\cal O}(20)$ terms and ${\cal O}(10)$ fit
parameters including explicit inputs for $\alpha$ and $\gamma$. To take into
account the correlations between $\alpha$ and $\beta$ or $\gamma$ and $\beta$
in the extraction of the parameters of the unitarity triangle, we perform a
fit in which we replace the inputs for $\alpha$ and $\gamma$ by branching
fractions and $C\\!P$ asymmetries of $B$ decays to pseudoscalar pseudoscalar
($PP$), pseudoscalar vector ($PV$), vector vector ($VV$) and $a_{1}$
pseudoscalar ($a_{1}P$) final states and $B$ decays to $D^{(*)}K^{(*)}$,
$D^{(*)}\pi$, and $D\rho$ final states. We refer to the latter as “full” fits.
Thus, we are able for the first time to include the correlations between
$\alpha$ and $\beta$ and $\gamma$ and $\beta$ in the extraction of the
parameters of the unitarity triangle.
In section IV, we perform stand-alone determinations of the angles $\alpha$
and $\gamma$, which are then used in the baseline fits. First, we determine
the correlations between $\alpha$ and $\beta$ by performing fits to
measurements of $B$ decays to $PP,PV,VV$, and $a_{1}P$ final states. Next, we
determine the correlations between $\gamma$ and $\beta$ by performing fits to
measurements of $B$ decays to $D^{(*)}K^{(*)}$, $D^{(*)}\pi$ and $D\rho$ final
states. Finally, the results are summarized in section V.
## II Fit Methodology
The scan method accounts for the theoretical uncertainties in the QCD
parameters $f_{B_{s}}$, $\xi_{f}=f_{B_{s}}/f_{B_{d}}$, $B_{B_{s}}$,
$\xi_{b}=B_{B_{s}}/B_{B_{d}}$, and $B_{K}$ laiho10 ; laiho2 and the CKM
parameters $|V_{ub}|$ and $|V_{cb}|$ by scanning over the range allowed by
theory uncertainties using fixed grid or Monte Carlo (MC) methods. In the
baseline fit, we combine measurements of $\Delta m_{B_{d}},\Delta
m_{B_{s}},\epsilon_{K},|V_{cb}|,|V_{ub}|,|V_{ud}|$, $|V_{us}|$, $|V_{cd}|$,
$|V_{cs}|$, $|V_{tb}|$, $\sin 2\beta,\alpha$ and $\gamma$ in the $\chi^{2}$
function, Eq. II. The angle brackets ($<>$) here indicate the experimental
averages.
$\displaystyle\chi^{2}(\bar{\rho},\bar{\eta},p_{i},t_{j})=$
$\displaystyle\Bigl{(}\frac{\langle\Delta m_{B}{{}_{d,s}}\rangle-\Delta
m_{B_{d,s}}(\bar{\rho},\bar{\eta},p_{i},t_{j})}{\sigma_{\Delta
m_{B_{d,s}}}}\Bigr{)}^{2}+\Bigl{(}\frac{\langle|V_{cb,ub,ud,us}|\rangle-|V_{cb,ub,ud,us}|(\bar{\rho},\bar{\eta},p_{i},t_{j})}{\sigma_{|V_{cb,ub,ud,us}|}}\Bigr{)}^{2}$
$\displaystyle+$
$\displaystyle\Bigl{(}\frac{\langle|\epsilon_{K}|\rangle-\epsilon_{K}(\bar{\rho},\bar{\eta},p_{i},t_{j})}{\sigma_{\epsilon_{K}}}\Bigr{)}^{2}+\Bigl{(}\frac{\langle
S_{\psi K^{0}}\rangle-\sin
2\beta(\bar{\rho},\bar{\eta},p_{i})}{\sigma_{S_{\psi
K^{0}}}}\Bigr{)}^{2}+\Bigl{(}\frac{\langle\alpha\rangle-\alpha(\bar{\rho},\bar{\eta},p_{i})}{\sigma_{\alpha}}\Bigr{)}^{2}$
$\displaystyle+$
$\displaystyle\Bigl{(}\frac{\langle\gamma\rangle-\gamma(\bar{\rho},\bar{\eta},p_{i})}{\sigma_{\gamma}}\Bigr{)}^{2}+\sum_{k}\Bigl{(}\frac{\langle{\cal
M}_{k}\rangle-{\cal M}_{k}(p_{i})}{\sigma_{{\cal
M}_{k}}}\Bigr{)}^{2}+\sum_{n}\Bigl{(}\frac{\langle{\cal T}_{n}\rangle-{\cal
T}_{n}(p_{i},t_{j})}{\sigma_{{\cal T}_{n}}}\Bigr{)}^{2}.$ (1)
Table 1: Observables used in the baseline fits and full fits. Fit type I are baseline fits with inputs specified by CKMfitter and UTfit (July 2012) to compare the fit results among the different fit methodologies. The values of $|V_{cb}|$ and $|V_{ub}|$ have only a total uncertainty. We list a second set in which experimental and theory uncertainties are separated allowing scans over the theoretical component of the uncertainties in $|V_{cb}|$ and $|V_{ub}|$. Fit type II are the baseline fits using the most recent input values to test the SM with and without the inclusion of ${\cal B}(B^{+}\rightarrow\tau^{+}\nu)$. The first set of $\alpha$ and $\gamma$ values has been fixed by CKMfitter and UTfit, while the second set results from our global fits to branching fractions and $C\\!P$ asymmetries of $B$ decay modes. Fit type III represents the full fits in which $\alpha$ and $\gamma$ are replaced by branching fraction and $C\\!P$ asymmetry measurements of the $B$ decay modes. fit type | $m_{t}^{\rm pole}~{}\rm[GeV/c^{2}]$ | $\overline{m}_{c}(m_{c})~{}\rm[GeV/c^{2}]$ | $\Delta m_{B_{d}}~{}\rm[ps^{-1}]$ | $\Delta m_{B_{s}}~{}\rm[ps^{-1}]$
---|---|---|---|---
I | $173.2\pm 0.9$ pbf | $1.275\pm 0.025$ pdg12 | $0.508\pm 0.004$ pdg12 | $17.719\pm 0.042$ pdg12
II, III | $173.5\pm 0.9$ pdg12 | same | same | same
fit type | $|V_{cb}|$ | $|V_{ub}|$ | $|V_{ud}|$ | $|V_{us}|$
I | $(4.16\pm 0.038\pm 0.05)\times 10^{-2}$ pbf | $(3.95\pm 0.38\pm 0.39)\times 10^{-3}$ pbf | $\ 0.97425\pm 0.0002\ $ colangelo | $\ 0.2208\pm 0.0039$ colangelo
II, III | $(4.09\pm 0.07\pm 0.09)\times 10^{-2}$ pdg12 | $(4.15\pm 0.31\pm 0.39)\times 10^{-3}$ pdg12 | same | same
fit type | $\epsilon_{K}$ | $\sin 2\beta$ | $\alpha$ | $\gamma$
I | $(2.228\pm 0.0011)\times 10^{-3}$ pdg12 | $0.0677\pm 0.020$ HFAG | $(88.0\pm 5.0)^{\circ}$ pbf | $(67.0\pm 11)^{\circ}$ pbf
II, III | same | same | $(84.6\pm 2.1)^{\circ}$ 111Input from global fit to Unitarity Triangle angles (see below); not used in the full fit (III). | $(79.7\pm 4.2)^{\circ}$${\rm\ {}^{a}}$
fit type | $|V_{cd}|$ | $|V_{cs}|$ | $|V_{tb}|$ | ${\cal B}(B^{+}\rightarrow\tau^{+}\nu)$
I | not used | not used | not used | $(1.15\pm 0.23)\times 10^{-4}$
II, III | $0.23\pm 0.011$ pdg12 | $1.023\pm 0.036$ pdg12 | $0.97\pm 0.08$ pdg12 | same
Table 2: QCD parameters used in the baseline fits. The first row shows the inputs for fit type I that were the averages listed by the Lattice group in July 2012. The second row shows the values with separate “statistical” and theory uncertainties used in the baseline fit II and the full fit (III). fit type | $f_{B_{s}}$ [MeV] | $\xi_{f}$ | $B_{B_{s}}$ | $\xi_{b}$ | $B_{K}$ | Ref.
---|---|---|---|---|---|---
I | $\ 227.6\pm 2.2\pm 4.5$ | $\ 1.201\pm 0.012\pm 0.012$ | $\ 1.33\pm 0.06$ | $1.05\pm 0.07$ | $\ 0.7643\pm 0.0034\pm 0.0091$ | laiho2 ; laiho10
II, III | same | same | $\ 1.33\pm 0.018\pm 0.06$ | $1.05\pm 0.025\pm 0.07$ | same | gamiz
fit type | $\eta_{cc}$ | $\eta_{ct}$ | $\eta_{tt}$ | $\eta_{b}$ | |
I, II, III | $\ 1.39\pm 0.35$ | $\ 0.47\pm 0.04$ | $\ 0.5765\pm 0.0065$ | $\ 0.551\pm 0.007$ | | nierste
The dependence of the predicted values on the quantities
$\bar{\rho},\bar{\eta},p_{i}$, and $t_{j}$ is described in detail in the
Appendix. The $p_{i}$ are measured inputs to these predicted values, including
the Wolfenstein parameters wolf $A$ and $\lambda$ and the quark masses and
$B$ meson masses. We add terms in the $\chi^{2}$, denoted generically by
${\cal M}_{k}$, accounting for the contributions from the uncertainties in the
$p_{i}$. Note that the dependence on these terms introduces correlations in
the $\chi^{2}$ expression. The $t_{j}$ represent parameters having a
theoretical uncertainty, e.g. the QCD parameters as well as $|V_{ub}|$ and
$|V_{cb}|$.
We are careful to distinguish among different kinds of uncertainties.
Observables with experimental errors only (statistical and systematic) are
assumed to be Gaussian-distributed. Theoretical quantities such as lattice-
derived QCD parameters, and inputs to $|V_{ub}|$ and $|V_{cb}|$, typically
have two types of uncertainties. The first type of error is of a “statistical”
nature, resulting from an input with statistical uncertainties or from Monte
Carlo statistics in lattice calculations. We assume this error to be Gaussian-
distributed and add corresponding terms to the $\chi^{2}$, denoted by ${\cal
T}_{n}$. The second type of uncertainty is a theory error with no known
underlying statistical distribution. We therefore make no assumption as to the
distribution of these errors, and instead perform a scan over a large range of
plausible values, doing a $\chi^{2}$ minimization at each point. Our use of
such fit results is described in the next section.
The scan includes the QCD parameters ($f_{B_{s}}$, $\xi_{f}$, $B_{B_{s}}$,
$\xi_{b}$ and $B_{K}$) and the CKM matrix elements $|V_{ub}|$ and $|V_{cb}|$.
The QCD corrections $\eta_{cc},\ \eta_{ct}$ and $\eta_{tt}$ used in the
determination of $\epsilon_{K}$ and $\eta_{b}$ that appear in the prediction
of $\Delta m_{B_{d}}$ also have theory errors. Although the fit methodology is
able to scan over them, we do not do so, since this is unnecessary at the
current level of precision. We parametrize $\eta_{cc}$ in terms of
$\overline{m}_{c}(m_{c})$ and $\alpha_{s}$ nierste . Tables 1 and 2 summarize
the input parameters for our baseline fits. We are preparing a more detailed
article that includes a fuller discussion of all input values, provides a
study of the correlations among the theory uncertainties and shows further
results unfit .
## III Results of the Global Fits
We present in this section the results for the global fits according to the
scan method, beginning with a comparison with the results for CKMfitter and
UT${}_{\it fit}\ $ for a common set of inputs. Then we look at the scan method
results for a more current set of inputs, including both a stand-alone
determination of the angles $\alpha$ and $\gamma$ and a fit explicitly
incorporating the measurements that enter into the $\alpha$ and $\gamma$
determination.
### III.1 Comparison to CKMfitter and UT${}_{\it fit}\ $
Figure 1: Overlay of $68\%~{}{\rm CL}$ contours in the $\bar{\rho}-\bar{\eta}$
plane for the inputs of Physics of the $B$ Factories with scanning over
$f_{B_{s}}$, $\xi_{f}$ and $B_{K}$. The black points show the central values
of each accepted fit.
Figure 2: Overlay of $68\%~{}{\rm CL}$ contours in the $\bar{\rho}-\bar{\eta}$
plane for the inputs of Physics of $B$ the Factories with scanning over
$f_{B_{s}},\xi_{f},B_{K},|V_{ub}|$ and $|V_{cb}|$. The green ellipses show
selected contours of accepted fits.
Table 3: Comparison of unitarity triangle parameters for different fitting techniques using inputs for the book Physics of the $B$ Factories. The second and third columns show the fit results from CKM fitter and UTfit. The fourth column shows our fit result if no scanning over theory parameters is performed and experimental uncertainties and theory uncertainties are added in quadrature. The fifth column shows the fit results if we scan over the QCD parameters $f_{B_{s}},\xi_{f}{\rm and\ }B_{K}$. The sixth column shows the fit results if we scan, in addition, over the theory errors associated with $|V_{ub}|$ and $|V_{cb}|$. | | | Our fit | Scan method
---|---|---|---|---
Parameter | CKMfitter | UT${}_{\it fit}\ $ | no scan | scan over | scan over
| pbf | pbf | | $\ f_{B_{s}},\xi_{f},B_{K}\ $ | $\ f_{B_{s}},\xi_{f},B_{K}$, $|V_{cb}|$, $|V_{ub}|\ $
$\ \bar{\rho}$ | $\ 0.129^{+0.027}_{-0.022}$ | $\ 0.132\pm 0.020\ $ | $\ 0.134\pm 0.041\ $ | $0.132^{+0.048}_{-0.042}$ | $0.139^{+0.048}_{-0.052}$
$\ \bar{\eta}$ | $\ 0.345\pm 0.014\ $ | $\ 0.348\pm 0.013$ | $\ 0.348^{+0.024}_{-0.023}$ | $0.348^{+0.026}_{-0.025}$ | $0.341^{+0.034}_{-0.025}$
$\ \beta~{}[^{\circ}]$ | $21.6^{+0.8}_{-0.7}$ | $21.8^{+0.8}_{-0.7}$ | $\ 21.9^{+2.4}_{-2.2}$ | $21.8^{+2.5}_{-2.1}$ | $21.6^{+2.6}_{-2.2}$
$\ \alpha~{}[^{\circ}]$ | $88.8^{+4.2}_{-3.6}$ | $88.6\pm 2.9$ | $\ 89.2^{+1.7}_{-2.9}$ | $89.0^{+2.9}_{-3.6}$ | $90.6^{+2.8}_{-5.9}$
$\ \gamma~{}[^{\circ}]$ | $68.96^{+3.5}_{-4.2}$ | $69.4\pm 3.1$ | $68.9^{+5.1}_{-4.1}$ | $69.2^{+5.5}_{-5.2}$ | $67.8^{+7.4}_{-5.1}$
To begin, we compare the performance of the scan method with that of CKMfitter
and UT${}_{\it fit}\ $using 19 input measurements ($|V_{ud}|$, $|V_{us}|$,
$|V_{cb}|$, $|V_{ub}|$, $\epsilon_{K}$, $\Delta m_{B_{d}}$, $\Delta
m_{B_{s}}$, $\sin 2\beta$, $\alpha$, $\gamma$, $f_{B_{s}}$, $B_{B_{s}}$,
$\xi_{f}$, $\xi_{b}$, $B_{K}$, $m_{t}^{\rm pole}$, $\overline{m}_{c}(m_{c})$,
$m_{B_{d}}$, $m_{B_{s}}$), choosing values specified for the book Physics of
the $B$ Factories pbf to fit 13 parameters ($\bar{\rho}$, $\bar{\eta}$, $A$,
$\lambda$, $f_{B_{s}}$, $B_{B_{s}}$, $\xi_{f}$, $\xi_{b}$, $B_{K}$,
$m_{t}^{\rm pole}$, $\overline{m}_{c}(m_{c})$, $m_{B_{d}}$, $m_{B_{s}}$). In
these fits, which we call fit type I in Tables 1 and 2, we use the central
values and measurement errors for $\alpha$ and $\gamma$ in the $\chi^{2}$
function. We compute $\overline{m}_{t}(m_{t})$, which enters into the Inami-
Lim functions inami for $\Delta m_{B_{d}}$, $\Delta m_{B_{s}}$ and
$\epsilon_{K}$, in the $\overline{MS}$ scheme from the pole mass $m_{t}^{\rm
pole}$ at three loop level for six quarks melnikov00 ; gray90 ; broadhurst91 .
We plot $1\sigma$ contours in the $\bar{\rho}-\bar{\eta}$ plane according to
the prescription described below (see Eq. 2). For the central value, we select
the fit with the highest $P(\chi^{2})$. We take the $\pm 1\sigma$
uncertainties from the maximum and minimum values of the envelope of all
contours. We perform three different fits. In the first, we combine theory and
experimental uncertainties, treating these as Gaussian. In the second, we scan
over the theory uncertainties in $f_{B_{s}},B_{B_{s}},B_{K}$. In the third, we
separate theory uncertainties from experimental uncertainties in $|V_{ub}|$
and $|V_{cb}|$ (see Table 1) thus scanning over theory uncertainties in
$|V_{ub}|,|V_{cb}|$, as well as those for $f_{B_{s}},\xi_{f}$ and $B_{K}$. We
performed the separation according to the ratio of experimental to theory
uncertainties listed in PDG10 pdg10 keeping the total uncertainty unchanged.
Figure 2 shows the overlay of $68\%$ confidence level contours in the
$\bar{\rho}-\bar{\eta}$ plane for the second fit. Since values of $|V_{ub}|$
and $|V_{cb}|$, which have substantial theoretical uncertainties, are not
scanned the accepted contours are similar. Figure 2 shows the comparable
results for the third fit, i.e, the most complete scan. Table 3 lists our
results in comparison to those from CKMfitter and UTfit. The three methods
yield broadly similar results. Scanning over theoretical errors increases the
uncertainties, often substantially. With this methodology the resulting
uncertainties are typically around a factor of two larger than those from
CKMfitter and UTfit. Large differences may be expected with UTfit, as UTfit is
a Bayesian approach and includes prior distributions for the theoretically
uncertain quantities.
The differences with CKMfitter are more subtle, as both are frequentist
approaches. The essential difference between the two interval estimation
methods is the following: CKMfitter uses the conventional change in $\chi^{2}$
from its minimum value to determine confidence regions. The motivation for the
scan method is to test the hypothesis that the standard model is correct,
against the alternative that it is incorrect. The statistical test adopted is
the $\chi^{2}$ test. Hence, this motivation is reflected in our confidence
region determination, in which we use the method of inverting a test
acceptance region (as described in standard statistics texts, for example
Shao2003 ). That is, the confidence regions are determined by comparing
$\chi^{2}$ values with a critical value instead of looking for a change in
$\chi^{2}$.
Specifically, the algorithm for a $1-\alpha$ confidence region in $d$
dimensions of a $p$ dimensional parameter space with $n$ measurements is as
follows. First, determine the acceptance region, at the $\alpha$ significance
level, by determining the critical value $\chi_{c}^{2}$ such that
$P(\chi^{2}\geq\chi_{c}^{2};n-p+d|H_{0})\geq\alpha,$ (2)
where $H_{0}$ is the hypothesis of the standard model, and $n-p+d$ is the
number of degrees of freedom. The $d$ is added back here because the critical
region for the test is constructed for each point in the $d$-dimensional
subspace of the full parameter space. The confidence region is then given by
all those points in the $d$ dimensional parameter subspace for which
$\chi^{2}\leq\chi^{2}_{c}$, under $H_{0}$.
For the theoretical uncertainties, CKMfitter makes an implicit scan by
incorporating a term in the likelihood function for each theoretically
uncertain quantity. The term equals 1 or 0 depending on whether the theory
parameter is in the theoretically “allowed” region or not. A modified
algortihm is also available, which makes a smooth transition between 1 and 0.
In the scan method, the scan over theoretical parameter space is explicit. No
theoretical term is included in the likelihood function. Instead, each point
in the theoretical parameter space of interest is treated as a possible value,
and the fit procedure is performed at each such point.
Depending on the value of the minimum $\chi^{2}$, the regions determined by
either method may be larger or smaller. If the best fit gives a high
$p$-value, then the scan method regions will be larger. On the other hand, if
the best fit gives a low $p$-value, the CKMfitter regions will be larger. In
the present instance, the best fit $p$-value is 0.76, hence the scan method
confidence intervals will tend to be larger than those from CKMfitter.
Reflecting its origins in goodness-of-fit testing, the scan method region goes
to null in the limit where the fit fails at the specified confidence level. In
the limit of no theoretical uncertainties, both methods give valid frequentist
confidence regions, with different properties. However, this comparison points
out the importance of methodology in forming conclusions, and hence of
examining the problem with multiple approaches.
Table 4: The $95\%~{}{\rm CL}$ ranges for unitarity triangle parameters from our baseline fit scans without the inclusion of ${\cal B}(B^{+}\rightarrow\tau^{+}\nu)$, from our baseline fit scans with the inclusion of ${\cal B}(B^{+}\rightarrow\tau^{+}\nu)$ and our full fit scans without the inclusion of ${\cal B}(B^{+}\rightarrow\tau^{+}\nu)$. The values of $\alpha$ and $\gamma$ are, in these cases, computed from the values of $\bar{\rho}$ and $\bar{\eta}$. Parameter | $\bar{\rho}$ | $\bar{\eta}$ | $\beta~{}[^{\circ}]$ | $\alpha~{}[^{\circ}]$ | $\gamma~{}[^{\circ}]$
---|---|---|---|---|---
baseline fit: scan without $B^{+}\rightarrow\tau^{+}\nu\ $ | $\ 0.069-0.147\ $ | $\ 0.319-0.395\ $ | $\ 19.0-24.7\ $ | $\ 82.7-88.5\ $ | $\ 68.8-77.9\ $
baseline fit: scan with $B^{+}\rightarrow\tau^{+}\nu$ | $0.073-0.147$ | $0.324-0.396$ | $19.3-24.8$ | $82.8-88.4$ | $68.8-77.4$
full fit: scan without $B^{+}\rightarrow\tau^{+}\nu$ | $0.070-0.151$ | $0.318-0.395$ | $18.8-24.8$ | $82.4-89.0$ | $67.9-77.9$
Figure 3: Overlay of $95\%~{}{\rm CL}$ contours in the $\bar{\rho}-\bar{\eta}$
plane for the baseline fit scan with 22 measurements without including ${\cal
B}(B^{+}\rightarrow\tau^{+}\nu)$. The green ellipses show selected contours of
accepted fits.
Figure 4: Overlay of $95\%~{}{\rm CL}$ contours in the $\bar{\rho}-\bar{\eta}$
plane for the baseline fit scan with 23 measurements with inclusion of ${\cal
B}(B^{+}\rightarrow\tau^{+}\nu)$. The green ellipses show selected contours of
accepted fits.
### III.2 Fit results with direct inputs of $\alpha$ and $\gamma$
To test the SM with the scan method, we perform the same baseline fits with
updated input parameters also listed in Tables 1 and 2. We refer to these fits
as type II fits. We use PDG12 $|V_{ub}|$ and $|V_{cb}|$ averages pdg12 . Since
the $|V_{ub}|$ and $|V_{cb}|$ results from exclusive modes are significantly
lower than those from inclusive modes, the PDG uses scaling factors on the
total errors of 2.6 and 2.0, respectively. If we have any fit satisfying
$P(\chi^{2})>5\%$, the SM is deemed to be compatible with the data at the
present level of theoretical uncertainties. We compute 95% CL contours
according to the inversion of a test acceptance region prescription described
above. That is, we compute contours based on Eq. 2, with d=1, such that taking
the extrema of a contour along a parameter axis provides a 95% CL interval for
the parameter. In the limit of no theoretical uncertainties, the procedure has
the stated coverage. Figure 4 shows the overlay of $95\%~{}{\rm CL}$ contours
in the $\bar{\rho}-\bar{\eta}$ plane for all accepted baseline fits using 22
measurements to fit 13 parameters, as in Section III. To obtain a range of
values for a given parameter, we take the extrema of the union of the contours
attached to accepted fits. Table 4 shows the range of the unitarity triangle
parameters thus obtained (hereinafter referred to as the $95\%~{}{\rm CL}$
range).
The $B^{+}\rightarrow\tau^{+}\nu$ branching fraction conjugate , measured by B
AB AR babar11 and Belle belle11 , is very sensitive to contributions from a
charged Higgs boson. The PDG average ${\cal
B}(B^{+}\rightarrow\tau^{+}\nu)=(1.65\pm 0.34)\times 10^{-4}$ pdg12 ; HFAG is
larger than the SM prediction of ${\cal
B}(B^{+}\rightarrow\tau^{+}\nu)=(1.2\pm 0.25)\times 10^{-4}$ silverman . Even
with these high values of ${\cal B}(B^{+}\rightarrow\tau^{+}\nu)$, we obtain a
sizeable allowed $\bar{\rho}-\bar{\eta}$ region; there is no conflict with the
SM. Belle has now presented a new measurement of ${\cal
B}(B^{+}\rightarrow\tau^{+}\nu)=(0.72^{+0.27}_{-0.25}({\rm stat})\pm 0.11\
({\rm sys}))\times 10^{-4}$ belle12 that reduces the world average to ${\cal
B}(B^{+}\rightarrow\tau^{+}\nu)=(1.14\pm 0.22)\times 10^{-4}$. Figure 4 shows
our results in the $\bar{\rho}-\bar{\eta}$ plane for this world average and
Table 4 summarizes the $95\%~{}{\rm CL}$ ranges of unitarity parameters. The
inclusion of the present ${\cal B}(B^{+}\rightarrow\tau^{+}\nu)$ world average
has hardly any impact on the $\bar{\rho}-\bar{\eta}$ plane.
Figure 5: Overlay of $95\%~{}{\rm CL}$ contours in the $\bar{\rho}-\bar{\eta}$
plane for fits with 257 measurements without ${\cal
B}(B^{+}\rightarrow\tau^{+}\nu)$. The green ellipses show selected contours of
accepted fits.
Figure 6: The $95\%~{}{\rm CL}$ contour in the $\alpha-\beta$ plane from a fit
including the $B\rightarrow PP$, $B\rightarrow PV$, $B\rightarrow VV$ and
$B\rightarrow Pa_{1}$ modes. The black dot shows the central value from the
fit. The vertical band shows the $68\%~{}{\rm CL}$ range for $\beta$ obtained
from $\sin 2\beta$ measurements HFAG .
### III.3 Fit Results using Individual Measurements of Branching Fractions
and $C\\!P$ Asymmetries of Various Decays
We also perform fits, called type III fits, in which, instead of treating the
values of $\alpha$ and $\gamma$ as inputs, we directly include the
measurements that determine them. This allows us to determine the correlations
in the extraction of the various unitarity triangle angles from the fit. We
omit ${\cal B}(B^{+}\rightarrow\tau^{+}\nu)$ in these fits; this has little
effect on the results.
We replace the direct $\alpha$ measurement term in the $\chi^{2}$ function by
all measured branching fractions and $C\\!P$ asymmetries in $B\rightarrow PP$,
$B\rightarrow PV$, $B\rightarrow VV$ and $B\rightarrow a_{1}P$ modes in the
fit. Following the Gronau-Rosner approach gronau , we parametrize amplitudes
in terms of tree, color-suppressed tree, penguin, singlet penguin,
$W$-annihilation/$W$-exchange, electroweak and color-suppressed electroweak
diagrams (up to $\lambda^{2}$ beyond leading order). Thus for order
$\lambda^{2}$, we consider SU(3) corrections for the tree, color-suppressed
tree and penguin diagrams. Since calculations of branching fractions use the
$B^{+}$ and $B^{0}_{d}$ lifetimes, we add $\chi^{2}$ terms for these lifetimes
in the fit.
We replace the direct $\gamma$ measurement term in the $\chi^{2}$ function, by
all measured branching fractions and $C\\!P$ asymmetries for $B^{+}\rightarrow
D^{(*)}K^{+}$ and $B^{+}\rightarrow DK^{*+}$ decays analyzed in the Giri-
Grossman-Soffer-Zupan (GGSZ) ggsz , Gronau-London-Wyler (GLW) glw and Atwood-
Dunietz-Soni (ADS) methods ads . We also include branching fractions and
$C\\!P$ asymmetries for $B^{+}\rightarrow D^{(*)}\pi^{+}$ decays analyzed in
the ADS method. Here, the observables are calculated in terms of $b\rightarrow
c\bar{u}s(d)$ and $b\rightarrow u\bar{c}s(d)$ amplitudes. We also include
time-dependent $C\\!P$ asymmetries in $B^{0}\rightarrow D^{(*)+}\pi^{-}$ and
$B^{0}\rightarrow D^{(*)+}\rho^{-}$ decays that determine
$\sin(2\beta+\gamma)$. Thus, the total number of measurements in the global
fits increases to 257; the fit has 120 parameters. Figure 5 shows the
$95\%~{}{\rm CL}$ contours of all accepted fits in the $\bar{\rho}-\bar{\eta}$
plane. Table 4 summarizes the $95\%~{}{\rm CL}$ ranges for unitarity triangle
parameters. This result shows that the 257 measurements are in good agreement
with the SM. This procedure accounts for possible correlations between
$\alpha,\gamma$ and the other Wolfenstein parameters (e.g., $\beta$).
## IV Determination of $\alpha$ and $\gamma$
In addition to the global fits of the CKM matrix, we perform separate fits
that determine the unitarity triangle angles $\alpha$ and $\gamma$. These
results are then used as inputs to the baseline fits. They are also used to
investigate the correlation with the angle $\beta$.
### IV.1 Determination of $\alpha$
For the $\alpha$ determination we combine all measured branching fractions and
$C\\!P$ asymmetries in $B\rightarrow PP$, $B\rightarrow PV$ and $B\rightarrow
VV$ modes. We first perform separate fits for each class of decays. As in
Section IV we parameterize the observables in terms of amplitudes, following
the Gronau-Rosner method. We include $f_{B_{s}}$ and $\xi_{f}$ in the fit, but
we do not scan over them, since these parameters appear only in the
$W$-exchange and $W$-annihilation diagrams that are at order $\lambda^{2}$
with respect to the tree diagram; any variation in these parameters is
absorbed by adjusting the magnitude of the $W$-annihilation/$W$-exchange
diagrams, leaving $\alpha$ and the remaining parameters unchanged. Thus, we
perform a single fit for each class of decays and plot $95\%$ CL
$\alpha-\beta$ contours. Table 5 shows the central values for $\alpha$ and
$\beta$ with uncertainties obtained by changing $\Delta\chi^{2}$ by one for
all fits. In addition, we list the correlation between $\alpha$ and $\beta$
and the fit probability. The correlation coefficients vary between -18% and
5%.
We next perform a combined fit of all measurements in $B\rightarrow
PP,B\rightarrow PV$ and $B\rightarrow VV$ and $B\rightarrow Pa_{1}$ modes to
extract $\alpha$. We use 185 measurements HFAG to determine 96 parameters.
Figure 6 shows the $95\%~{}{\rm CL}$ contour in the $\alpha-\beta$ plane,
which encompasses the world average $\beta=(21.4\pm 0.8)^{o}$ measured using
$b\rightarrow c\bar{c}s$ modes. The fit probability is $P(\chi^{2})=38.6\%$.
The correlation coefficient is about -4%. These results show that all
measurements in $B\rightarrow PP,B\rightarrow PV$ and $B\rightarrow VV$ and
$B\rightarrow Pa_{1}$ modes are consistent with the SM description and no new
physics amplitudes are required.
### IV.2 Determination of $\gamma$
Table 5: Measurements of $\alpha$, $\beta$ and $\gamma$ from fits of branching fractions and $C\\!P$ asymmetries in $B\rightarrow PP$, $B\rightarrow PV$, $B\rightarrow VV$ decays and in a combination of all modes combined plus $B\rightarrow Pa_{1}$ decays ($\alpha$) and $B\rightarrow D^{(*)}K(\pi)+B\rightarrow DK^{*}(\rho)$ decays ($\gamma$). | $B\rightarrow PP$ | $B\rightarrow PV$ | $B\rightarrow VV$ | $B$ modes combined | $B\rightarrow D^{(*)}K(\pi)+B\rightarrow DK^{*}(\rho)$
---|---|---|---|---|---
$\ \alpha~{}[^{\circ}]$ | $85.9^{+3.0}_{-2.7}$ | $82.4^{+4.1}_{-4.3}$ | $83.8^{+5.5}_{-5.6}$ | $84.7^{+2.1}_{-2.1}$ | \-
$\ \gamma~{}[^{\circ}]$ | - | - | - | - | $79.6^{+4.1}_{-4.2}$
$\ \beta~{}[^{\circ}]$ | $20.8^{+2.1}_{-1.9}$ | $20.5^{+3.6}_{-3.4}$ | $24.3^{+6.4}_{-4.9}$ | $21.1^{+1.6}_{-1.6}$ | $22.8^{+7.7}_{-2.1}$
$\alpha-\beta$ correlation | 0.052 | -0.182 | -0.151 | -0.035 | \-
$\gamma-\beta$ correlation | - | - | - | - | -0.194
p-value | 0.50 | 0.369 | 0.248 | 0.386 | 0.051-0.071
For the $\gamma$ determination, we use branching fraction and $C\\!P$
asymmetries of $B^{+}\rightarrow D^{(*)}K^{+}$ and $B^{+}\rightarrow DK^{*+}$
decays analyzed in the GGSZ ggsz , GLW glw and ADS ads methods. We also
include branching fractions and $C\\!P$ asymmetries of $B^{+}\rightarrow
D^{(*)}\pi^{+}$ decays analyzed in the ADS method and time-dependent $C\\!P$
asymmetries in $B^{0}\rightarrow D^{(*)+}\pi^{-}$ and $B^{0}\rightarrow
D^{(*)+}\rho^{-}$ decays that determine $\sin(2\beta+\gamma)$. We separate the
CKM factors, $|V_{us}V_{ub}^{*}|/|V_{cs}V_{cb}^{*}|$ and
$|V_{ud}V_{ub}^{*}|/|V_{cd}V_{cb}^{*}|$ from the ratio of $b\rightarrow u$ to
$b\rightarrow c$ amplitudes. We include the ratios $|V_{us}/V_{ud}|$ and
$|V_{ub}/V_{cb}|$ in the fit, scanning over $|V_{ub}|$ and $|V_{cb}|$. Since
the predictions contain a product of CKM factors and ratios of amplitudes, it
is necessary to constrain the CKM factors in the fit to obtain sensible values
for the amplitude ratios and CKM factors.
We use 56 measurements HFAG to extract 19 fit parameters, scanning over the
constraint $|V_{ub}|/|V_{cb}|$. The probabilities for these fits range between
$P(\chi^{2})=5.1\%$ and $P(\chi^{2})=7.1\%$. Figure 7 shows the resulting
contours at $95\%~{}{\rm CL}$ in the $\gamma-\beta$ plane. Table 5 lists the
fit results. Again, these 56 modes are well-described within the SM and no new
physics amplitudes are required. The $\gamma$–$\beta$ correlation coefficient
is $-19$%.
Figure 7: Overlay of $95\%~{}{\rm CL}$ contours in the $\gamma-\beta$ plane
for fits including the $B\rightarrow DK^{(*)},DK^{*},D^{(*)}\pi,D\rho$ modes.
The vertical band shows the $68\%~{}{\rm CL}~{}\beta$ region obtained from
$\sin 2\beta$ measurements HFAG .
## V Conclusion
The three fitting approaches: CKMfitter, UT${}_{\it fit}\ $and the scan method
yield similar central values for $\bar{\rho}$ and $\bar{\eta}$ when presented
with identical inputs. However, the allowed region in the
$\bar{\rho}-\bar{\eta}$ plane is substantially larger in the scan method. This
may be expected from the difference with the Bayesian methodology of
UT${}_{\it fit}\ $, in which prior distributions are assigned for the
theoretical uncertainties. The differences with CKMfitter are more subtle, as
both are frequentist approaches. In fact, the larger scan method intervals is
not a given; with different measurement values the situation could reverse in
the comparison with CKMfitter. This comparison points out the importance of
methodology in forming conclusions, and hence of examining the problem with
multiple approaches.
Using the scan method, we find no tension with the SM even when we include the
current PDG value of ${\cal B}(B^{+}\rightarrow\tau^{+}\nu)=(1.65\pm
0.34)\times 10^{-4}$; the scan yields global fits consistent with the SM at
$95\%~{}{\rm CL}$. When we include the recent Belle result, the ${\cal
B}(B^{+}\rightarrow\tau^{+}\nu)$ branching fraction has hardly any impact on
the $\bar{\rho}-\bar{\eta}$ plane.
Our global fit allows us to determine and incorporate the correlation between
$\alpha$ and $\beta$, as well as between $\gamma$ and $\beta$. Using all
measured branching fractions and $C\\!P$ asymmetries of $B\rightarrow PP$,
$B\rightarrow PV$, $B\rightarrow VV$, $B\rightarrow a_{1}P$ modes and
$B^{+}\rightarrow D^{(*)}K^{+},B\rightarrow DK^{*+}$ modes that are sensitive
to $\alpha$ and $\gamma$, respectively, we observe small changes in the
allowed region in the $\bar{\rho}-\bar{\eta}$ plane. From separate fits of
branching fractions and $C\\!P$ asymmetries in these modes, we determine
$\alpha-\beta$ and $\gamma-\beta$ contours. Though $\alpha$ measurements agree
with each other and $\beta$ results are consistent with $\sin 2\beta$ from
$b\rightarrow c\bar{c}s$ modes, some correlation among Wolfenstein parameters
in the different measurements is observed. The values of $\alpha$, determined
from a fit to all measured branching fractions and $C\\!P$ asymmetries of
$B\rightarrow PP$, $B\rightarrow PV$, $B\rightarrow VV$ and $B\rightarrow
Pa_{1}$ modes, and $\gamma$, extracted from a fit to $B^{+}\rightarrow
D^{(*)}K^{+},B\rightarrow DK^{*+}$, $B\rightarrow D^{(*)}\pi$ and
$B^{0}\rightarrow D^{+}\rho^{-}$ modes, agree with the SM expectations.
## Acknowledgments
This work is supported in part by the U. S. Department of Energy under Grant
DE-FG02-92-ER40701 and by the NFR (Norway).
## Appendix
In the baseline fits, the $\chi^{2}$ function (Eq. II) includes 23 or 22
terms, depending on whether ${\cal B}(B\to\tau\nu)$ is included or not. In
this Appendix, we describe the dependence of the predicted values used in the
$\chi^{2}$ expression on the quantities $\bar{\rho},\bar{\eta},p_{i}$, and
$t_{j}$.
The CKM matrix elements are parameterized in terms of Wolfenstein parameters
$\bar{\rho}$, $\bar{\eta}$, $A$ and $\lambda$ up to order ${\cal
O}(\lambda^{9})$ wolf ; buras94 ; ahn :
$\displaystyle V_{ud}=$ $\displaystyle
1-\frac{1}{2}\lambda^{2}-\frac{1}{8}\lambda^{4}-\frac{1}{16}\lambda^{6}\bigl{(}1+8A^{2}(\rho^{2}+\eta^{2})\bigr{)}-\frac{1}{128}\lambda^{8}\bigl{(}5-32A^{2}(\rho^{2}+\eta^{2})\bigr{)},$
$\displaystyle V_{us}=$
$\displaystyle\lambda\bigl{(}1-\frac{1}{2}A^{2}\lambda^{6}(\rho^{2}+\eta^{2})\bigr{)},$
$\displaystyle V_{ub}=$ $\displaystyle A\lambda^{3}(\rho-i\eta),$
$\displaystyle V_{cd}=$
$\displaystyle-\lambda\Bigl{(}1-\frac{1}{2}A^{2}\lambda^{4}\bigl{(}1-2(\rho+i\eta)\bigr{)}-\frac{1}{2}A^{2}\lambda^{6}(\rho+i\eta)\Bigr{)},$
$\displaystyle V_{cs}=$ $\displaystyle
1-\frac{1}{2}\lambda^{2}-\frac{1}{8}\lambda^{4}\bigl{(}1+4A^{2}\bigr{)}-\frac{1}{16}\lambda^{6}\bigl{(}1-4A^{2}+16A^{2}(\rho+i\eta)\bigr{)}-\frac{1}{128}\lambda^{8}\bigl{(}5-8A^{2}+16A^{4}\bigr{)},$
$\displaystyle V_{cb}=$ $\displaystyle
A\lambda^{2}(1-\frac{1}{2}A^{2}\lambda^{6}\bigl{(}\rho^{2}+\eta^{2}\bigr{)}),$
$\displaystyle V_{td}=$ $\displaystyle
A\lambda^{3}\bigl{(}1-\rho-i\eta\bigr{)}+\frac{1}{2}A\lambda^{5}\bigl{(}\rho+i\eta\bigr{)}+\frac{1}{8}A\lambda^{7}\bigl{(}1+4A^{2}\bigr{)}\bigl{(}\rho+i\eta\bigr{)},$
$\displaystyle V_{ts}=$
$\displaystyle-A\lambda^{2}\Bigl{(}1-\frac{1}{2}\lambda^{2}\bigl{(}1-2(\rho+i\eta)\bigr{)}-\frac{1}{8}\lambda^{4}-\frac{1}{16}\lambda^{6}\bigl{(}1+8A^{2}(\rho+i\eta)\bigr{)}\Bigr{)},$
$\displaystyle V_{tb}=$ $\displaystyle
1-\frac{1}{2}A^{2}\lambda^{4}-\frac{1}{2}A^{2}\lambda^{6}(\rho^{2}+\eta^{2})-\frac{1}{8}A^{4}\lambda^{8}.$
(3)
The Unitarity Triangle angles $\beta$, $\alpha$ and $\gamma$ are implemented
by
$\displaystyle\sin 2\beta$ $\displaystyle=$
$\displaystyle\frac{2\bar{\eta}(1-\bar{\rho})}{(1-\bar{\rho})^{2}+\bar{\eta}^{2}},$
$\displaystyle\tan\alpha$ $\displaystyle=$
$\displaystyle\frac{\bar{\eta}}{\bar{\eta}^{2}+\bar{\rho}(\bar{\rho}-1)},$
$\displaystyle\tan\gamma$ $\displaystyle=$
$\displaystyle\frac{\bar{\eta}}{\bar{\rho}}.$ (4)
The oscillation frequencies for $B^{0}_{d}\bar{B}^{0}_{d}$ and
$B^{0}_{s}\bar{B}^{0}_{s}$ mixing are computed according to:
$\displaystyle\Delta m_{B_{d}}$ $\displaystyle=$
$\displaystyle\frac{G^{2}_{F}}{6\pi^{2}}\eta_{B}m_{B_{d}}\frac{f^{2}_{B_{s}}}{\xi_{f}^{2}}\frac{B_{B_{s}}}{\xi_{b}}m_{W}^{2}S(x_{t})|V_{td}V^{*}_{tb}|^{2},$
$\displaystyle\Delta m_{B_{s}}$ $\displaystyle=$
$\displaystyle\frac{G^{2}_{F}}{6\pi^{2}}\eta_{B}m_{B_{s}}f^{2}_{B_{s}}B_{B_{s}}m_{W}^{2}S(x_{t})|V_{ts}V^{*}_{tb}|^{2},$
(5)
where $G_{F}$ is the Fermi constant, $\eta_{B}$ is a QCD correction, $m_{W}$
is the $W$ mass, $S(x_{t})$ is the Inami-Lim function inami and
$x_{t}=\overline{m}_{t}^{2}/m^{2}_{W}$ where the top quark mass is calculated
in the $\rm\overline{MS}$ scheme. We have expressed the $B^{0}_{d}$ decay
constant and bag parameters in terms of the $B^{0}_{s}$ decay constant and bag
parameters and their ratios $\xi_{f}$ and $\xi_{b}$ since the latter have
smaller uncertainties. The explicit relation between $\overline{m}_{t}(m_{t})$
and $m_{t}^{\rm pole}$ is given by
$\overline{m}_{t}(m_{t})=m_{t}^{\rm
pole}\bigl{(}1-\frac{4}{3}\bigl{(}\frac{\alpha_{s}}{\pi}\bigr{)}-9.1253\bigl{(}\frac{\alpha_{s}}{\pi}\bigr{)}^{2}-80.4045\bigl{(}\frac{\alpha_{s}}{\pi}\bigr{)}^{3}\bigr{)},$
(6)
where $\alpha_{s}(\overline{m}_{t})=0.1068\pm 0.0018$ is calculated in the
$\overline{MS}$ scheme for six quark flavors at the scale of the pole mass
buras97 ; ckmfitter .
$C\\!P$ violation in the $K^{0}\bar{K}^{0}$ system is represented by the
parameter $\epsilon_{K}$. In the SM, this is proportional to the off-diagonal
matrix element of the mixing matrix divided by the $K^{0}_{L}-K^{0}_{S}$ mass
difference $\Delta m_{K}$ yielding buchalla95
$\epsilon_{K}=C_{\epsilon}\kappa_{\epsilon}\exp{i\phi_{\epsilon}}B_{K}\Bigl{(}{\rm
Im}[(V_{cs}V^{*}_{cd})^{2}]\eta_{cc}S(x_{c})+{\rm
Im}[(V_{ts}V^{*}_{td})^{2}]\eta_{tt}S(x_{t})+2{\rm
Im}[V_{cs}V^{*}_{cd}V_{ts}V^{*}_{td}]\eta_{ct}S(x_{c},x_{t})\Bigr{)},$ (7)
where $\kappa_{\epsilon}=0.94\pm 0.02$ buras10 , $\phi_{\epsilon}=(43.5\pm
0.7)^{\circ}$, $x_{c}=\overline{m}_{c}^{2}/m^{2}_{W}$, $\overline{m}_{c}$ is
the charm quark mass in the $\rm\overline{MS}$ scheme. The constant is given
by
$C_{\epsilon}=\frac{G^{2}_{F}f^{2}_{K}m_{K}m^{2}_{W}}{12\pi^{2}\sqrt{2}\Delta
m_{K}},$ (8)
where $m_{K}$ and $f_{K}$ are the kaon mass and kaon decay constant,
respectively. The contribution from the decay rate difference has been
neglected. We use NLO calculations of the QCD parameters; $\eta_{cc}$ and
$\eta_{ct}$ calculations at NNLO brod10 ; brod12 have now been done.
The $B^{+}\rightarrow\tau^{+}\nu$ branching fraction is given by
${\cal
B}(B^{+}\rightarrow\tau^{+}\nu)=\frac{G^{2}_{F}}{8\pi}m_{B^{+}}m^{2}_{\tau}\Bigl{(}1-\frac{m^{2}_{\tau}}{m^{2}_{B^{+}}}\Bigr{)}\frac{f^{2}_{B_{s}}}{\xi_{f}^{2}}|V_{ub}|^{2}\tau_{B^{+}},$
(9)
where $\tau_{B^{+}}$ is the $B^{+}$ lifetime, $m_{B^{+}}$ is the $B^{+}$ mass
and $m_{\tau}$ is the $\tau^{+}$ mass,
We also add Gaussian terms in the $\chi^{2}$ function for the quark masses
$m_{t}^{\rm pole}$ and $\overline{m}_{c}(m_{c})$, meson messes $m_{B^{0}_{d}}$
and $m_{B^{0}_{s}}$, and the Gaussian parts of the QCD parameters $B_{K}$,
$f_{B_{s}}$, $\xi_{f}$, $B_{B_{s}}$, and $\xi_{b}$.
## References
* (1) N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963); M. Kobayashi and T. Maskawa, Prog. Th. Phys. 49, 652 (1973).
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* (4) G. P. Dubois-Felsmann, G. Eigen, D. G. Hitlin and F. C. Porter, arXiv:hep-ph/0308262v2 (2003).
* (5) Physics of the $B$ Factories, eds. A. Bevan et al., Springer-Verlag March, in preparation.
* (6) J. Beringer et al. (Particle Data Group), Phys. Rev. D86, 010001 (2012).
* (7) D. Asner et al., arXiv:1010.1589v3 (2011); www.slac.stanford.edu/xorg/hfag/triangle/index.html.
* (8) J. Laiho, E. Lunghi and R. S. Van de Water, Phys. Rev. D81, 034503 (2010).
* (9) J. Laiho, E. Lunghi and R. S. Van de Water, End of 2012 update: http://mypage.iu.edu/$\sim$elunghi/webpage/ LatAves/page7/page7.html.
* (10) L. Wolfenstein, Phys. Rev. Lett. 51, 1945 (1983).
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* (12) G. Colangelo et al., Eur. Phys. J. C71, 1695 (2011).
* (13) E. Gamiz et al., Phys. Rev. D80, 014503 (2009).
* (14) G. Eigen, G. P. Dubois-Felsmann, D. G. Hitlin and F. C. Porter, detailed article in preparation.
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* (20) J. Shao, Mathematical Statistics, 2nd ed., (Springer-Verlag, New York, 2003).
* (21) $C\\!P$ conjugate states are implied.
* (22) P. del Amo Sanchez et al. (B AB AR Collaboration), Phys. Rev. D81, 051101 (2010); J.P. Lees et al. (B AB AR Collaboration), Phys. Rev. D88, 031102 (2013).
* (23) K. Hara et al., Phys. Rev. D82, 071101 (2010).
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* (27) A. Giri, Y. Grossman, A. Soffer and J. Zupan, Phys. Rev. D68, 054018 (2003).
* (28) M. Gronau and D. London, Phys. Lett. B253, 483 (1991); M. Gronau and D. Wyler, Phys. Lett. B265, 172 (1991).
* (29) D. Atwood, I. Dunietz, A. Soni, Phys. Rev. Lett. 78, 3257 (1997); Phys. Rev. D63, 036005 (2001).
* (30) A.J. Buras, M.E. Lautenbacher and G. Ostermaier, Phys Rev. D50, 3433 (1994).
* (31) Y.H. Ahn, H.Y. Cheng and S. Oh, Phys. Lett. B703, 571 (2011).
* (32) A.J. Buras and R. Fleischer, Adv.Ser.Direct.High Energy Phys. 15, 65 (1998).
* (33) G. Buchalla, A.J. Buras and M.E. Lautenbacher, Rev. Mod. Phys. 68 1125 (1996).
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|
arxiv-papers
| 2013-01-24T19:15:10 |
2024-09-04T02:49:40.712838
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Gerald Eigen, Gregory Dubois-Felsmann, David G. Hitlin, Frank C.\n Porter",
"submitter": "Gerald Eigen",
"url": "https://arxiv.org/abs/1301.5867"
}
|
1301.5950
|
# Reply to ”Comment on ’Detecting Non-Abelian Geometric Phases with Three-
Level $\Lambda$ Atoms’ ”
Yan-Xiong Du Zheng-Yuan Xue Xin-Ding Zhang Hui Yan [email protected]
Laboratory of Quantum Information Technology, School of Physics and
Telecommunication Engineering, South China Normal University, Guangzhou
510006, China
###### Abstract
In this reply, we address the comment by Ericsson and Sjoqvist on our paper
[Phys. Rev. A 84, 034103 (2011)]. We point out that the zero gauge field is
not the evidence of trivial geometric phase for a non-Abelian SU(2) gauge
field. Furthermore, the recalculation shows that the non-Abelian geometric
phase we proposed in the three-level $\Lambda$ system is indeed experimentally
detectable.
###### pacs:
03.65.Vf, 03.67.Lx, 32.90.+a
There are three points in the preceding comment Ericsson : (i) the sign in
Eq.(7) in the original paperdu is wrong; (ii) non-Abelian geometrical
phase(GP) of a large-detuned $\Lambda$ system ( $\sin\gamma\approx 0$ with
$\gamma$ defined below) in the adiabatic approximation is trivial since the
corresponding gauge field vanishes; (iii) the GP is still small in the general
case with relatively large $\gamma$ and cannot be separated from the dynamical
phase .
In this reply, we address the above comments. We agree with point (i).
However, this mistake doesn’t influence the validity of the main result. We
re-calculate the population difference induced by the geometric phase after
correcting this sign error and find that the maximum population difference can
still reach $20\%$. Such big population difference can be easily detected in a
typical experiment.
We focus on the large-detuning case, where
$\tan\gamma=(\sqrt{\Delta^{2}+\Omega^{2}}-\Delta)/\Omega\approx 0$ with
$\Delta$ $(\Omega)$ the detuning (effective Rabi frequency). The large
detuning can be easily realized through choosing proper parameters, i.e.
$\Delta=1$ GHz and $\Omega=1$ MHz. As corrected in the commentEricsson , the
gauge potentials in the large detuning case can be written as
$\displaystyle A_{\theta}$ $\displaystyle\approx$ $\displaystyle
i\cos{\varphi}\sigma_{y}+i\sin{\varphi}\sigma_{x},$ (1) $\displaystyle
A_{\varphi}$ $\displaystyle\approx$
$\displaystyle-i\sin^{2}{\theta}\sigma_{z}+i\sin{\theta}\cos{\theta}\cos{\varphi}\sigma_{x}$
$\displaystyle-i\sin{\theta}\cos{\theta}\sin{\varphi}\sigma_{y}.$
Then we also agree that the corresponding gauge field $F_{\theta\varphi}$ must
vanish.
One will think of such potentials like Eq.(1) and (2) may not bring any
physical observable effects because of the vanished field strength. In the
following we will illustrate with some examples that this granted judgement is
wrong for non-Abelian gauge fields. The first example is about the vacuum
solution in a Yang-Mills theory demonstrated in detail in Ref.Jackiw . Jackiw
and Rebbi demonstrated thatJackiw , a type of gauge potentials in the Yang-
Mills theory are gauge equivalent to the potential $A=0$ and thus the field
strengths are vanished; however, the potentials should not be removed from the
integrations over the field configurations by the gauge fixing procedure, and
they argued that physical effects are associated with them. The second example
we will address in detail is the well-known case that spin $1/2$ behavior in a
time-dependent magnetic field $\mathbf{B}=(B_{x}(t),B_{y}(t),B_{z}(t))$Berry ;
Bliokh . The interacting Hamiltonian is
$H=\hbar\chi\mathbf{\sigma}\mathbf{B},$ (3)
here $\sigma$ is the Pauli Matrixes and $\chi$ is the coupling constant. This
Hamiltonian can be diagonalized through $H_{dg}=U^{-1}HU$ with the unitary
matrix
$U=\frac{1}{\sqrt{2B}}\left(\begin{array}[]{cc}\sqrt{B+B_{z}}&\sqrt{B-B_{z}}\\\
\frac{B_{x}+iB_{y}}{\sqrt{B+B_{z}}}&-\frac{B_{x}+iB_{y}}{\sqrt{B-B_{z}}}\\\
\end{array}\right),$ (4)
here the corresponding eigenvalues are $E_{\pm}=\pm\hbar\chi B$,
$B=|\mathbf{B}|$. For a pure gauge potential with the unitary matrix (4) as
$\mathbf{A}(\mathbf{m})=iU^{-1}\frac{\partial
U}{\partial\mathbf{m}},\mathbf{m}=(B_{x},B_{y},B_{z}),$ (5)
the non-Abelian gauge field $\mathbf{F}$ is vanishing Bliokh . To recover the
non-trivialness of such potential, one should choose a specific closed loop in
this parameter space, i.e. , a loop with large enough $B$ to maintain the
adiabatic approximation. In this case, the off-diagonal elements of
$\mathbf{A}(\mathbf{m})$ are rejected and the adiabatic gauge potentials
$\mathbf{A}^{(ad)}$ are given by
$\mathbf{A}^{(ad)}=(\frac{B_{y}}{2B(B\pm B_{z})},-\frac{B_{x}}{2B(B\pm
B_{z})},0),$ (6)
where different signs correspond to two eigenvalue $E_{\pm}$. The
corresponding field tensors are given by
$F_{ij}^{(ad)}=\frac{\partial A_{j}^{(ad)}}{\partial m_{i}}-\frac{\partial
A_{i}^{(ad)}}{\partial m_{j}}=\mp\varepsilon_{ijk}\frac{B_{k}}{2B^{3}},$ (7)
$\varepsilon_{ijk}$ is the three order antisymmetric tensor. It can be seen
that the closed path integral $\oint\mathbf{A}^{(ad)}d\mathbf{l}$ in these
monopole field strength is just the case of Berry phase. This integral is non-
zero and will bring physical observable effects. Furthermore, the integral
should hold when we shrink the curve by decreasing $B$ linearly, then we
return to the case of full matrix $\mathbf{A}(\mathbf{m})$. Therefore,
$\oint\mathbf{A}(\mathbf{m})d\mathbf{l}$ also contains a flux and will bring
physical observable effects. Namely, the non-vanishing
$\oint\mathbf{A}(\mathbf{m})d\mathbf{l}$ is guaranteed by
$\oint\mathbf{A}^{(ad)}d\mathbf{l}$ in the Abelian case. And hence, we show
with the second exampe that the gauge potential with a zero gauge field will
also have observable effects for its non-vanishing closed path integral in the
non-Abelian case.
Figure 1: (color online).(a) The new result of population difference after two
composed path with different orders. (b) The population difference $P_{d}$
versus parameter $\alpha$ when $\beta=0.5$. The parameters $\alpha$ and
$\beta$ represent the relative amplitude and the time delay of Raman pulses in
the stimulated Raman adiabatic transitions as the same in the origin paper du
.
The situation of three-level $\Lambda$ atoms interacting with lasers is
similar with the above analyzation. The unitary matrix diagonalize Hamiltonian
(1) in du reads as
$\Gamma=\left(\begin{array}[]{ccc}\cos{\theta}&-\sin{\theta}e^{-i\varphi}&0\\\
\sin{\theta}\cos{\gamma}e^{i\varphi}&\cos{\theta}\cos{}\gamma&-\sin{\gamma}\\\
\sin{\theta}\sin{\gamma}e^{i\varphi}&\cos{\theta}\sin{\gamma}&\cos{\gamma}\end{array}\right),$
(8)
and $\gamma$ is given by
$\tan{\gamma}=(\sqrt{\Delta^{2}+\Omega^{2}}-\Delta)/\Omega$. One can set large
detuning $\Delta$ to achieve the non-Abelian geometric phase Eqs. (1,2). The
corresponding gauge field is zero. To recover the non-trivialness of Eqs.
(1,2), suitable $\Delta$ and $\Omega$ should be chose to reject the non-
diagonal elements. Then we return to the Abelian case of which the gauge
potential is $A^{\pm}_{\varphi}=\mp\sin^{2}{\theta}$ is observable Zhu . The
non-zero closed path integral of $A^{\pm}_{\varphi}$ can be extended to the
non-Abelian case just as the above analyzation. For example, we choose a
closed loop of
$\mathbf{A}=(A^{\prime}_{\theta},A^{\prime}_{\varphi})=i(A_{\theta},A_{\varphi})$
along $C$ with $\theta=\theta_{0}$, $\varphi\in[0,2\pi]$, and then the
integral can be derived as
$\oint\mathbf{A}d\mathbf{R}=\int_{0}^{2\pi}A^{\prime}_{\varphi}d\varphi=\pi(1-\cos
2\theta_{0})\sigma_{z}=\frac{1}{2}\Omega(C)\sigma_{z},$ (9)
where $\Omega(C)$ is the solid angle spanned by $C$ which shows the geometric
feature of the evolution. Clearly this integral is generally non-vanishing and
then the corresponding geometric phase is non-trivial.
The result of a vanishing gauge field brings physical effects is
counterintuitive. Indeed, this comes from the fact that the Stroke theorem in
the non-Abelian case is not a direct generalization of the Abelian case. It
has been realized that the surface integral in the non-Abelian case should
depend on the gauge field $F_{\theta\varphi}$ as well as the gauge potential
$\mathbf{A}$niu ; Barrett . Actually, the phase factor given as
$U=\hat{P}\exp\left(i\oint\mathbf{A}\cdot d\mathbf{l}\right)$ (10)
with $\hat{P}$ being the operator of chronological ordering, which is called
the Wilson loopWilson , is of particular important for a non-Abelian field.
This phase factor is transformed covariantly and may results in the observable
physical effects even with zero field strengthBliokh ; Yang .
We re-calculate the possibly observed effects based on Eqs.(1) and (2). The
new results are show in Fig.1, which should replace the results of Fig.2 in du
. The procedure and parameters are the same with those in our original paperdu
, but the gauge potentials are replaced with those in Eqs(1) and (2). We can
find from Fig.1 that the maximum population difference can still reach almost
$20\%$. Such big population difference can be easily detected in a typical
experiment. Since $\gamma\approx 0.001$ for $\Gamma/\Delta=0.001$, we know
that the effects induced by the dynamical phase could be neglected in the
above calculation, as shown in the commentEricsson . Therefore, the population
differences in Fig.1 are indeed induced by the non-Abelian gauge potential.
Moreover, we directly calculate the Schrodinger equation
$i\hbar\partial_{t}|\psi\rangle=H|\psi\rangle$ with the Hamiltonian given by
Eq.(1) in Ref.Ericsson . We find that the results are the same with those in
Fig.1 when the parameters are the same for the two different methods. It
further confirms that the population difference is indeed caused by the non-
vanished phase factor defined by the Wilson loop.
In summary, the main point that the authors used to defend in the comment is
that the induced gauge field is zero and thus the GP is trivial. However, we
have shown that this causality doesn’t hold for an SU(2) gauge field.
Furthermore, we have shown that the non-Abelian GP in our proposal is non-
trivial for its non-vanishing closed loop integral and can be detected through
the induced significant population difference.
Acknowledge: The authors thank Z. B. Li, D. W. Zhang and S. L. Zhu for helpful
discussions. This work was supported by the NSF of China(No. 11104085, No.
11125417, and No. 91121023 ), the SKPBR of China(No. 2011CB922104) and PCSIRT.
Du was aslo supported by the SRFGS of SCNU.
## References
* (1) M. Ericsson and E. Sjoqvist, comment (to be published in Phys. Rev. A).
* (2) Y. X. Du, Z. Y. Xue, X. D. Zhang and H. Yan, Phys. Rev. A 84, 034103 (2011).
* (3) R. Jackiw and C. Rebbi, Phys. Rev. Lett. 37, 172 (1976).
* (4) M. V. Berry, Proc. R. Soc. Lond. A 392, 45 (1984).
* (5) K. Yu. Bliokh and Yu. P. Bliokh, Anna. of Phys. 319, 13 (2005).
* (6) S. L. Zhu, H. Fu, C. J. Wu, S. C. Zhang and L. M. Duan, Phys. Rev. Lett. 97, 240401 (2006).
* (7) A. Bohm, A. Mostafazadeh, H. Koizumi, Q. Niu, J. Zwanziger, The Geometric Phase in Quantum Systems (Springer, New York, 2003).
* (8) T. Barrett and D. Grimes, Advanced Electromagnetism: Foundations, Theory, and application (World Scientific Publishing Co. Pte. Ltd., Singapore, 1995).
* (9) K.G. Wilson, Phys. Rev. D 10 (1974) 2445.
* (10) T. T. Wu and C. N. Yang, Phys. Rev. D 12, 3845 (1975); ibid. 12, 3843 (1975).
|
arxiv-papers
| 2013-01-25T02:18:22 |
2024-09-04T02:49:40.722716
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Y. X. Du, Z. Y. Xue, X. D. Zhang, H. Yan",
"submitter": "Yanxiong Du Mr.",
"url": "https://arxiv.org/abs/1301.5950"
}
|
1301.6077
|
11institutetext: Institute of Nuclear Physics, ul. Radzikowskiego 152, 31–342
Cracow, Poland
# TiC Lattice Dynamics from ab initio Calculations
P.T. Jochym K. Parlinski M. Sternik [email protected]
###### Abstract
Ab initio calculations and a direct method have been applied to derive the
phonon dispersion curves and phonon density of states for the TiC crystal. The
results are compared and found to be in a good agreement with the experimental
neutron scattering data. The force constants have been determined from the
Hellmann-Feynman forces induced by atomic displacements in the
$2$$\times$$2$$\times$$2$ supercell. The calculated phonon density of states
suggests that vibrations of Ti atoms form acoustic branches, whereas the
motion of C atoms is confined to optic branches. The elastic constants have
been found using the deformation method and compared with the results obtained
from acoustic phonon slopes.
###### pacs:
63.20.–ePhonons and vibrations in crystal lattices and 71.15.NTotal energy
calculations
The transition-metal carbide compounds, of which TiC is a representative, are
of big scientific and technological interest because of their striking
mechanical properties, extreme hardness combined additionally with metallic
electrical and thermal conductivity. They have usually a rock-salt crystal
structure – like some ionic crystals – while having properties typical for
materials with covalent bonding. It suggests that the bonding have some mixed
nature, which makes these materials very interesting as an object of study.
Over a large range of fractional carbon content from TiC to TiC0.5 toth the
TiC is stable in the NaCl structure, with space group Fm${\bar{3}}$m. In
practice, it is usually nonstoichiometric and contains carbon vacancy defects.
The phonon dispersion curves of nearly stoichiometric TiC0.95 have been
measured along main symmetry directions by Pintschovius et al pint using the
inelastic neutron scattering technique. In the same paper, the influence of
the carbon content on the phonon dispersion curves has been additionally
checked by measuring a sample with lower carbon concentration TiC0.89. The
results shown that the optic modes could differ by 3.5%, while the acoustic
modes are less sensitive to carbon content. The elastic constants derived from
the measured acoustic dispersion curves agree very well with those obtained by
ultrasonic measurements for TiC0.91 crystal exp_dataA .
Very interesting properties and simple structure of TiC caused that efforts to
understand its properties using first-principle calculations have been
undertaken several times neckel ; zhukov ; price ; haglund . The electronic
structure, bulk modulus, and elastic constants have been found TiC_GGA by
means of accurate first-principles total-energy calculations using the full
potential linear muffin-tin orbital method. The calculated values are
generally in very good agreement with experiment.
In this paper we intend to extend first-principle calculations to describe the
phonon dispersion curves and phonon density of TiC. The method which was is
based on the total energy calculation and Hellmann–Feynman (HF) forces.
Dispersion relations are calculated by direct method frank ; kresse ;
parlinski ; ackland ; sluiter , in which the force constants of the dynamical
matrix are calculated from HF forces. Elastic constants and bulk modulus have
been estimated by straight evaluation of energy derivatives with respect to
deformation.
The energies and forces of the TiC crystal were calculated by the method of
total energy minimization, using a norm–conserving pseudopotentials as an
approximation of the atomic core – valence electron interaction lin ;
goniakowskiA . For an excellent review of ab initio total–energy calculations
see Ref.payne and further references given there. For the lattice dynamics
calculations the $2$$\times$$2$$\times$$2$ supercell with periodic boundary
conditions and 64 atoms has been used. For optimizing the structure and for
direct calculation of energy deformations $1$$\times$$1$$\times$$1$ supercell
has been utilized. The ab initio total energy calculations were done with the
CASTEP package castep and standard pseudopotentials provided within this
package. These pseudopotentials were constructed within LDA approximation. The
Ti pseudopotential treats 3p electrons as belonging to the core. We have
employed non–local variant of those potentials parametrized in the reciprocal
space with 900 eV and 600 eV cut–off energy for $1$$\times$$1$$\times$$1$ and
$2$$\times$$2$$\times$$2$ supercell, respectively. Tests which had been made
with the cut-off energy of 600 eV applied to the $1$$\times$$1$$\times$$1$
supercell, showed that in this case the cohesive energy is only 0.2 eV higher,
and the lattice constant is longer by 0.0006Å(0.01%), comparing to the 900 eV
cut-off. Therefore, we have used 600 eV cut-off energy for a larger supercell
calculation.
A generalized gradient approximation (GGA) has been used for the exchange
energy term of the valence states of the Hamiltonian perdew . Although this
procedure is not quite consistent, there are indications goniakowskiB that
LDA generated pseudopotentials used with GGA exchange term in crystals are
sufficiently accurate for energies and structures. The integration over the
Brillouin zone has been performed with weighted summation over wave vectors
generated by Monkhorst-Pack scheme pack using 0.1 Å-1 and 0.06 Å-1 grid sizes
which leads to 4 and 32 wave vectors for $1$$\times$$1$$\times$$1$ supercell
and to 1 and 4 wave vectors for $2$$\times$$2$$\times$$2$ supercell,
respectively. No significant difference due to different number of wave
vectors, 1 or 4, has been found in calculated phonon frequencies. The results
of dispersion curve presented here are generated from the 0.06 Å-1 data set
with 4 wave vectors.
Quantity | Exper. | Non-metallic | | Metallic |
---|---|---|---|---|---
# k | | 4 | 4 | 14 | 32 | 63
$a_{0}$ | $4.327$ | $4.345$ | $4.35$ | $4.342$ | $4.345$ | $4.342$
$B$ | $2.4$ | $2.42$ | $2.41$ | $3.527$ | $2.418$ | $2.427$
$\omega_{\Gamma}$ | $16.3$ | $16.5$ | $19.0$ | $11.8$ | $15.6$ | $15.6$
$\omega_{X}^{1}$ | $8.5$ | $8.5$ | $10.2$ | $5.1$ | $9.2$ | $8.5$
$\omega_{X}^{2}$ | $10.8$ | $10.0$ | $11.1$ | $10.3$ | $10.7$ | $10.7$
$\omega_{X}^{3}$ | $16.5$ | $17.0$ | $18.2$ | $12.8$ | $15.7$ | $15.3$
$\omega_{X}^{4}$ | $18.5$ | $17.8$ | $19.3$ | $16.7$ | $18.1$ | $17.7$
Table 1: Comparison of results from metallic and non-metallic approach
Since TiC is a weakly metallic compound it could have been appropriate to use
a smearing of electron levels in the Brillouin zone integration. This
procedure is computationally more expensive. Thus we have performed a series
of tests to compare lattice constants, bulk moduli and phonons in $\Gamma$ and
X points obtained using gaussian smearing in the range from 4 to 0.1 eV and
denser k-space grids (4, 14, 32, 63 k-points) for one unit cell configuration.
The comparison of results shown in the Table 1 indicate that, for sufficiently
dense k-space grid used, these quantities are not very sensitive (specially
the lattice constant and bulk modulus) to the choice of method of Brillouin
zone integration. Thus we have decided to use much simpler and faster non-
metallic approach in the final $2$$\times$$2$$\times$$2$ supercell
calculation. Note also that results obtained in the metallic regime lead to
larger divergency of phonon frequencies from experimental data.
The minimization of the total energy with respect to the lattice constant was
performed within the CASTEP minimizer module with $1$$\times$$1$$\times$$1$
supercell and cross-checked with $2$$\times$$2$$\times$$2$ supercell. The
equilibrium lattice constant is $a=4.3448$Å which could be compared with the
experimental value $a=4.3269$Ånaray . Such a good agreament may be
coincidental, since inclusion of 3p electrons in the valence band as well as
gradient corrections in core states tends to change the equilibrium lattice
constant garcia ; juan ; seifert .
The HF forces are defined as
${\bf F}_{i}({\bf n},\nu)=-\partial E_{tot}/\partial{\bf R}_{i}({\bf n},\nu)$
(1)
where ${\bf n},\nu$ are the indices of the unit cell and atom within the unit
cell, respectively, and $i$ is the Cartesian component. At extremum all HF
forces vanish. Non-zeroth HF forces arise, when a single atom $({\bf m},\mu)$
is displaced by $u_{j}({\bf m},\mu)$ from its equilibrium position. The
arising forces are related with the cummulant force constants $\Phi_{ij}$ by a
relation parlinski ; ackland
${\bf F}_{i}({\bf n},\nu)=-\sum_{{\bf m},\mu,j}\Phi_{ij}({\bf n},\nu;{\bf
m},\mu)u_{j}({\bf m},\mu)$ (2)
Cummulant force constants appear as a result of periodic boundary conditions
imposed on the supercell. To calculate HF forces an atomic configuration with
a single displaced atom must be minimized with respect to the electronic part
only. Each of such runs provides $3n$ HF forces, where $n=8$ or $64$ is the
number of atoms in the $1$$\times$$1$$\times$$1$ or $2$$\times$$2$$\times$$2$
supercell, respectively. Two runs with displacements along $z$ direction, one
for Ti and second for C were performed. To minimize the systematic errors two
other runs with negative displacements were carried on. The displacement
amplitudes were set to $1\%$ of the lattice constant. This amplitude has been
selected after number of careful tests in which displacements ware varied from
$0.1\%$ to $1.6\%$. It has been proven that the anharmonic contributions are
negligable up to $1\%$ of displacement. At smaller displacements, close to
$0.1\%$ the HF forces became too small and the uncertainty of force constant
evaluation is not acceptable.
Figure 1: Absolute eigenvalues of the force constant matrices of the TiC
crystal as a function of interatomic distance.
The data of $4\times 3n$ HF forces ${\bf F}_{i}({\bf n},\nu)$ and four
displacements $u_{j}({\bf m},\mu)$ form an overdetermined set of equations,
Eq.(2), for the force constants. This system was solved by the singular value
decomposition algorithm parlinski ; numrec , which automatically provides the
least-squares solution. For $2$$\times$$2$$\times$$2$ supercell the 768
components of the HF forces lead to 33 non-zero independent parameters of the
force constants. Knowledge of these force constants allows one to test the
range of interaction potential. For that, similarly to procedure used in
sluiter , we have calculated the absolute values of the eigenvalues of the
$3\times 3$ $\Phi_{ij}({\bf n},\nu;{\bf m},\mu)$ matrices and plotted them
against a distance between atoms $({\bf n},\nu)$ and $({\bf m},\mu)$. The
result for $2$$\times$$2$$\times$$2$ supercell is depicted in Fig. 1. It shows
that the interaction drops down at least by two orders of magnitudes within
the size of the supercell. The scatter of points is too large to assign any
rule which could govern the decaying of the force constants parameters with
distance. From Fig. 1 it is clear that the $1$$\times$$1$$\times$$1$ supercell
is too small to be used for calculation of dispersion curves since the
considered force constants terminate at radius of $a\sqrt{3}/2=0.866a$.
The knowledge of the cummulant force constants allows one to define an
approximate dynamical matrix, in which the summation over all atoms is
confined to those atoms which reside within the volume of the supercell. The
approximate dynamical matrix becomes equal to the conventional one at discrete
wave vectors ${\bf k}_{L}$ given be the equation $\exp(2\pi i{\bf
k}_{L}\cdot{\bf L})=1$ parlinski , where ${\bf L}=({\bf L}_{1},{\bf
L}_{2},{\bf L}_{3})$ are the lattice vectors of the supercell. At the wave
vectors ${\bf k}_{L}$ the phonon frequencies $\omega^{2}({\bf k}_{L})$,
calculated by diagonalization of the approximate dynamical matrix, are the
same as those calculated from the exact dynamical matrix. For
$2$$\times$$2$$\times$$2$ supercell the exact wave vectors are at $\Gamma$, X,
L, W, and two other wave vector points, namely, the midpoint between $\Gamma$
and X along $\langle 1,0,0\rangle$ and $\langle 1,1,1\rangle$ directions. The
advantage of the above described direct method is that it does not impose any
limit to the range of interaction. When the supercell size is smaller than the
range of interaction, the direct method interpolates the dispersion curves
between the exact points.
Figure 2: Phonon dispersion relations of TiC crystal calculated from
$2$$\times$$2$$\times$$2$ supercell (full line). The same dispersion curves
divided by a factor $1.05$ (dashed line). Experimental points taken from Ref.
pint .
The dispersion curves calculated on the basis of $2$$\times$$2$$\times$$2$
supercell are shown in Fig. 2. They correspond to temperature T=0. At the same
Figure 2 we compare the calculated dispersion curves with the experimental
phonon frequencies measured by the inelastic neutron scattering pint .
Different symbols of experimental points correspond to different phonon
polarizations established experimentally.
The experimental points are about $\Delta\omega=0.05\omega$ lower then the
calculated values. One of the reasons for such situation could be that
experimental values are measured at room temperature and anharmonic effects
might deminish the phonon frequencies. Another reason may be related with a
non-stoichiometric concentration of carbon in measured samples, which diminish
in average phonon frequencies as well. The efect that experimental phonon
points are at lower frequencies then the calculated ones appears as well in
other ab-initio calculations frank ; parlinski .
The theory of lattice dynamics provides summation rules which follow from
translational and rotational invariances of the crystal. For TiC these
invariances lead to two equations, which can be used to set the values of on-
site force constant parameters. If these conditions are satisfied the acoustic
phonon branches at k=0 would point to $\omega=0$. Our dispersion curves,
without imposed translational-rotational invariances point at k$=0$ to
$\omega=0.31$THz. Since the invariance conditions should be satisfied within
the ab initio and direct methods, we have not corrected our dispersion curves
displayed in Fig. 2 according to this effect.
Figure 3: Phonon density spectrum $g(\omega)$ of the TiC crystal and partial
phonon density spectra $g_{i,\alpha}(\omega)$, where $\alpha=$ C or Ti and
$i=x,$ $y$ or $z$ are the displacement directions. Experimental points taken
from Ref. pint .
By sampling the dynamical matrix for many wave vectors, one can calculate the
phonon density function $g(\omega)$, and the partial phonon density functions
$g_{x,Ti}(\omega)$ and $g_{x,C}(\omega)$. The $g(\omega)$ describes the number
of phonon frequencies in an interval around $\omega$, while $g_{x,Ti}$ and
$g_{x,C}$ specify the number of phonon frequencies in the interval around
$\omega$ but modified by the population of the $x$ displacement component of
either Ti or C atom. The density of states functions are shown in Fig. 3. They
are conventionally normalized to $\int g(\omega)d\omega=1$ and $\int
g_{x,\alpha}(\omega)d\omega=1/6$, where $\alpha=$ Ti or C. The curves show
that motions within acoustic dispersion curves are almost entirely due to Ti
atoms. The sum of Titanium density functions
($g_{x,Ti}(\omega)+g_{y,Ti}(\omega)+g_{z,Ti}(\omega)$) fits to the total
density function $g(\omega)$ below 14 THz. Part of $g(\omega)$ above the gap
is mainly due to C atoms. Thus, TiC form a rather special crystal, in which
the Ti atoms, as heavier, form a frame for elastic motion, and the C atoms
vibrate within the optical modes. We add, however, that the magnitude of the
force constants between Ti - C and between Ti - Ti and C - C at the same
distances remains of the same order. Thus, TiC does not consist of two weakly
bounded subsystems. In Fig. 3 the experimentally determined phonon density of
states pint ; gompf is also shown. The agreement within acoustic region is
quite qood. In optic part the experimental resolution of $\approx 2.5$THz and
the nonsoichiometry of carbon in the sample lead to broadening of the optic
band.
The elastic constants can be evaluated from the slopes of acoustic branches of
phonon dispersion curves. Here, we have used the invariance conditions to set
the acoustic modes to $\omega=0$ at k$=0$. For the fcc structure, the set of
seven equations for three unknown variables $c_{ij}$ has to be solved: $\rho
v^{2}_{\alpha,L}=c_{11}$; $\rho v^{2}_{\alpha,T}=c_{44}$; $\rho
v^{2}_{\beta,L}=c_{11}+4c_{44}+2c_{12}$; $\rho
v^{2}_{\beta,T}=c_{11}+c_{44}-c_{12}$; $\rho
v^{2}_{\gamma,L}=c_{11}+2c_{44}+c_{12}$; $\rho
v^{2}_{\gamma,T_{h}}=c_{11}-c_{12}$; $\rho v^{2}_{\gamma,T_{3}}=c_{44}$, where
the velocities $v_{i}$ are the slopes of appropriate acoustic branches. Here,
$\alpha,\beta$ and $\gamma$ indexes describe three main directions in the
reciprocal space $\langle 0,0,1\rangle$, $\langle 1,1,1\rangle$ and $\langle
0,1,1\rangle$, respectively. The density of TiC $\rho=4849.6$ kg/m3 was found
using the calculated lattice constant. The solution of this set of equations,
by means of the least-squares fit, results in $c_{11}=5.7$, $c_{12}=2.5$ and
$c_{44}=1.51$ MBar. Comparing these values with experimental data (Table 2)
one notices considerable discrepancies, especially for $c_{12}$. Closer look
to dispersion curves, Fig. 2, assures us that the transverse acoustic mode in
$\langle 1,1,0\rangle$ direction and the longitudinal in $\langle
1,1,1\rangle$ direction do not follow exactly the experimental points. We have
checked, that such discrepancy of slope is sufficient to produce the observed
deviations of the calculated elastic constants, specially of $c_{12}$. The
discrepancy in slopes is so small that it could be assigned to the missing
force constants describing mainly the long range Coulomb interaction. Indeed,
the direct method did not handle well interactions exceeding the size of the
supercell, except for the special points $\Gamma$, X, L, W mentioned earlier.
Result | $B$ | $c_{11}$ | $c_{12}$ | $c_{44}$
---|---|---|---|---
Exp. TiC0.91 exp_dataA | $(2.422)$ | $5.145$ | $1.060$ | $1.788$
Exp. TiC exp_dataB | $2.4$ | $5.00$ | $1.13$ | $1.75$
Exp. TiC0.95 pint | $(2.5)$ | $5.4$ | $1.1$ | $1.8$
Exp. TiC0.89 pint | $(2.5)$ | $5.2$ | $1.1$ | $1.8$
Calculations TiC_GGA | $2.2$ | $4.7$ | $0.97$ | $1.67$
Ours, acoustic modes | $(3.57)$ | $5.7$ | $2.5$ | $1.51$
Ours, deform. energy | $2.42$ | $5.39$ | $0.94$ | $1.92$
Ours, stress–strain | $(2.46)$ | $5.21$ | $1.09$ | $1.93$
Table 2: Experimental and calculated bulk modulus $B$ and elastic constants
$c_{ij}$ of the TiC crystal, in units of $10^{11}{\rm Nm}^{-2}=1{\rm MBar}$.
Values in parenthesis are calculated from $B=\frac{1}{3}(c_{11}+2c_{12})$.
To rule out that the discrepancy for $c_{12}$ does not follow from ab-initio
calculations, we have derived the bulk modulus and the elastic constants by
two conventional methods TiC_GGA . The bulk modulus was calculated in the
$1$$\times$$1$$\times$$1$ supercell from the total energy as a function of the
lattice constant. This data have been fitted to a third order polynomial.
Hence, the derivative $\frac{\partial E}{\partial V}$, and the bulk modulus
$B$ were calculated from the relation: $B=-V\frac{\partial E}{\partial V}$,
where $V$ is the volume of TiC crystallographic unit cell. The $B$ values are
given in Table 2 and they agree quite well with experimental data.
A calculation of $c_{11}$, $c_{12}$, $c_{44}$ elastic constants involved a
similar procedure, in which the crystal was deformed by lengthening and sheer
deformations. Deformations from $0.5\%$ to $3\%$ in length and from $1$ to $5$
degrees in angle have been used. The results of small and large deformations
are consistent. The deformed lattices have space groups Fm$\bar{3}$m, I4/mmm
and I/mmm, for bulk expansion, elongation along z, and sheer modes. In all
these deformed lattices all atoms remain in the high-symmetry sites which
guaranties an extremum of potential energy at those positions. Thus, in the
case of cubic TiC, one is entitled not to carry on the relaxation of internal
degrees of freedom during supercell deformations. The calculated elastic
constants are compared to experimental data in Table 2. They agree with
calculated values given in Ref.TiC_GGA and are in good agreement with
experiment.
We have also checked the above results using stresses calculated by CASTEP.
The elastic constants were found as first derivatives of the stress–strain
relationship. For that, we have used the same runs of electronic calculations
as for the elastic constant derivation given above using deformation
dependence of energy. The results are included in Table 2 and they show
slightly better agreement with the experimental data.
In summary, we have shown that the ab initio calculations of HF forces,
together with the direct method lead to satisfactory description of phonons in
the TiC crystal. As usual the frequency of the calculated dispersion curves
are slightly too high. The acoustic modes are not sufficiently accurate, thus
the elastic constants calculated from them differ from those found through the
supercell deformations. The acoustic part of density of states fits well to
neutron experimental data.
###### Acknowledgements.
The computational advices of J. Oleszkiewicz and the use of facilities of the
Computer and Network Center of the Jagellonian University, Cracow, where the
calculations have been done, are kindly acknowledged. This work was partially
supported by the Polish State Committee of Scientific Research (KBN), grant No
2 PO3B 145 10.
## References
* (1) L.E. Toth, Transition Metal Carbides and Nitrides (Academic Press, New York 1971).
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* (4) A. Neckel, R. Eibler, P.Weinberger and K. Schwartz, J. Phys C 9, 579 (1976).
* (5) V.P. Zhukov, V.A. Gubanov, O. Jepsen, N.E. Christensen and O.K. Andersen, J. Phys. Chem. Solids 49, 841 (1988).
* (6) D.L. Price and B.R. Cooper, Phys. Rev. B 39, 4945 (1989).
* (7) J. Häglund, A.F. Guillerment, G. Grimvall and M. Körling, Phys. Rev. B 48, 11685 (1993).
* (8) R. Ahuja, O. Eriksson, J. M. Wills, B. Johansson, Phys. Rev. B 53, 3073 (1996).
* (9) W.Frank, C.Elsässer, and M,Fähnle, Phys.Rev.Lett. 74, 1791 (1995).
* (10) G.Kresse, J.Furthmüller, and J.Hafner, Europhys.Lett. 32, 729 (1995).
* (11) K. Parlinski, Z. Q. Li and Y. Kawazoe, Phys. Rev. Lett. 78, 4063 (1997).
* (12) G. J. Ackland, M. C. Warren and S. J. Clark, J. Phys.: Condens. Matter 9, 7861–7872 (1997).
* (13) M.H.F. Sluiter, M. Weinert, Y. Kawazoe, Europhys. Lett. 43, 183 (1998).
* (14) J.S. Lin, A. Qteish, M.C. Payne and V. Heine, Phys. Rev. B 47, 4174 (1993).
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* (16) M.C. Payene, M.P. Teter, D.C. Allan, T.A. Arias, J.D. Joannopoulos, Rev. of Mod. Phys. 64, 1045 (1992)
* (17) CASTEP is a part of $Cerius^{2}$ system developed by Molecular Simulations.
* (18) J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Singh, C. Fiolhais, Phys. Rev. B 46, 6671 (1992).
* (19) J. Goniakowski, J. M. Holender, L. N. Kantorovich, M. J. Gillan, J. A. White, Phys. Rev. B 53, 957 (1996)
* (20) H.J. Monkhorst and J.D. Pack, Phys. Rev. B 13, 5188 (1976).
* (21) I. Náray-Szabó, Inorganic Crystal Chemistry (Akadémiai kiadó, Budapest, 1969), p.301.
* (22) A. García, C. Elsässer, J. Zhu, S.G. Louie, M.L. Cohen, Phys. Rev. B 46, 9829 (1992)
* (23) Y.M. Juan and E. Kaxiras, Phys. Rev. B 48, 14 994 (1993)
* (24) K. Seifert, J. Hafner, J. Furthmüller, G. Kresse, J. Phys. Condens. Matter 7, 3683 (1995)
* (25) W. H. Press, B.P. Flamery, S.A. Tenkolsky, and W.T. Vetterling, Numerical Recipes (Cambridge Univ. Press, 1988), p.301
* (26) F. Gompf and W. Reichardt, 1977, unpublished.
* (27) M. M. Choy at. al., Numerical Data and Functional Relationships in Science and Technology (Springer, Berlin, 1979), vol.11.
|
arxiv-papers
| 2013-01-25T15:58:00 |
2024-09-04T02:49:40.730554
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Pawe{\\l} T. Jochym, Ma{\\l}gorzata Sternik, Krzysztof Parlinski",
"submitter": "Pawe{\\l} T. Jochym",
"url": "https://arxiv.org/abs/1301.6077"
}
|
1301.6083
|
11institutetext: Institute of Nuclear Physics, ul. Radzikowskiego 152, 31–342
Cracow, Poland
# Ab Initio Lattice Dynamics and Elastic Constants of ZrC
P.T. Jochym K. Parlinski [email protected]
###### Abstract
Ab initio calculations and a direct method are applied to derive the phonon
dispersion relations and phonon density of states for the ZrC crystal. The
results are in good agreement with neutron scattering data. The force
constants are determined from the Hellmann-Feynman forces induced by atomic
displacements in a $2$$\times$$2$$\times$$2$ supercell. The elastic constants
are found using the deformation method and successfully compare with
experimental data.
###### pacs:
63.20.–ePhonons in crystal lattices and 71.15.MbDensity functional theory
Studies of transition-metal carbides have been carried out several times.
Phenomenological model calculations weber ; gupta ; yadav managed to
reproduce the phonon dispersion relations quite well, but the interactions
were quite sophisticated , some of them including three-body interactions and
double shell models. Recently, there have been several studies based on ab
initio methodsneckel ; zhukov ; price ; haglund . The electronic structure,
bulk modulus, and elastic constants, as well as the phonon dispersion
relations have been found for TiC, TiN and TiO compounds by means of first-
principles total-energy calculations mytic ; TiC_GGA . Generally, the
calculated values show good agreement with experiments. This provides a
motivation to extend the investigation to heavier transition metals and to
treat zirconium carbide, ZrC. This crystal is an important material in nuclear
energy technology, since it provides a filling medium for fuel particles. It
is also used for surface hardening and coverage of cathodes of X-ray sources.
In this paper we continue to study the lattice dynamics and elastic properties
of transition-metal compounds. See mytic and references given there for a
more detailed description of the method.
The transition-metal carbide compounds, of which ZrC is a representative, are
of considerable scientific and technological interest because of their
striking mechanical properties, extreme hardness combined with metallic
electrical and thermal conductivities. The ZrC crystal has NaCl structure and
it is usually non-stoichiometric, mainly owing to carbon-vacancy defects. The
phonon dispersion relations of ZrC have been measured along main symmetry
directions by Smith et al smithA ; smithB . Elastic constants, in turn, have
been measured for two concentrations of carbon (ZrC0.89 and ZrC0.94) by Chang
and Graham chang .
In this paper we extend the first-principle calculations to describe the
phonon dispersion curves, phonon density and elastic constants of ZrC. The
method which is used, is based on the total energy calculation and
Hellmann–Feynman (HF) forces. The phonon dispersion relations are calculated
by the direct method and the Phonon program frank ; kresse ; parlinski ;
ackland ; sluiter ; phonon , in which the force constants of the dynamical
matrix are derived from HF forces. Alternatively, one could use the
linear–response method gonze ; giannozzi for the evaluation of ab initio
phonon frequencies at a predetermined set of Brillouin zone points. Elastic
constants and bulk modulus are estimated by straightforward evaluation of
energy derivatives with respect to the deformations.
The energies and HF forces of ZrC crystal are calculated by the method of
total energy minimization, using norm-conserving pseudopotentials as an
approximation for the atomic core–valence electron interaction lin ;
goniakowskiA ; payne . This method allows to include the pressure in
calculations as well as anharmonic effects. For the lattice dynamics
calculations a $2$$\times$$2$$\times$$2$ supercell with periodic boundary
conditions and 64 atoms was used. For optimizing the structure and for the
direct calculation of stress-strain relations a $1$$\times$$1$$\times$$1$
supercell was utilized. The ab initio total energy calculations were done with
the CASTEP package castep and standard pseudopotentials constructed within
LDA approximation and provided within this package. The Zr pseudopotential
treats 4d electrons as belonging to the valence band. All pseudopotentials
were parametrized in the reciprocal space with 680 eV cut–off energy.
Tests, which were made with the $1$$\times$$1$$\times$$1$ supercell and with
cut-off energies of 680 eV and 900 eV, showed that in the first case the
cohesive energy is only 0.01 eV higher, and the lattice constant changes by
only 0.0007 Å(0.01%). Therefore, we used the 680 eV cut-off energy for all
remaining calculations.
The Local Density Approximation (LDA) as well as the Generalized Gradient
Approximation (GGA) were used for the exchange energy term of the valence
states of the Hamiltonian perdew . The integration over the Brillouin zone was
performed with weighted summation over wave vectors generated by the
Monkhorst-Pack scheme pack using grid spacings from 0.1 Å-1 to 0.04 Å-1 which
lead to sets of k-vectors containing from 4 to 118 wave vectors for the
$1$$\times$$1$$\times$$1$ supercell and from 1 to 14 wave vectors for the
$2$$\times$$2$$\times$$2$ supercell. The convergence of the force constants
was achieved for grid spacing below 0.05 Å-1. Hence, we carried on the
optimization for a 0.05 Å-1 grid spacing, which leads to 32 and 4 k-points for
the $1$$\times$$1$$\times$$1$ and the $2$$\times$$2$$\times$$2$ supercells,
respectively. However, we used a 64 wave-vector set for the LDA calculation of
the elastic constants. The metallic character of the TiC compound, studied in
mytic , indicated that the use of smearing does not substantially improve the
quality of phonon dispersion relations. Thus, we did not use smearing in the
case of ZrC. The use of non–metallic summation of the Brillouin zone is
further justified by the fact that the electron density of states at the Fermi
level is small for ZrC and similar compounds: TiC and HfC weber ; biltz , and
the related errors are of the same order as the errors of the DFT approach
itself mytic . Namely, they are less then 0.05% for the lattice constant, less
then 0.3% for the bulk modulus, and less then 5% for phonon frequencies.
Figure 1: Phonon dispersion relations of ZrC crystal calculated with a
$2$$\times$$2$$\times$$2$ supercell within GGA (full line) and LDA (dashed
line) approximations. The experimental points are taken from Ref. smithA ;
smithB .
The minimization of the total energy leads to the equilibrium lattice constant
of the stoichiometric ZrC $a=4.691$Å for LDA and $a=4.695$Å for GGA. These
values could be compared with the experimental values of $a=4.6994$Å for
ZrC0.94 and $a=4.7004$Å for ZrC0.89chang . The phonon dispersion relations,
which correspond to temperature T=0, are shown in Fig. 1. These are compared
with experimental phonon frequencies measured by inelastic neutron scattering
smithA ; smithB . For GGA, the calculated force constants for all 10
coordination spheres of the $2$$\times$$2$$\times$$2$ supercell are presented
in Table 1.
Distance (Å) | From | To | $xx$ | $yy$ | $zz$ | $yz$ | $xz$ | $xy$
---|---|---|---|---|---|---|---|---
0.0 | C(0,0,0) | C(0,0,0) | $219.36$ | $219.36$ | $219.36$ | | |
0.0 | Zr${}_{(\frac{1}{2},\frac{1}{2},\frac{1}{2})}$ | Zr${}_{(\frac{1}{2},\frac{1}{2},\frac{1}{2})}$ | $284.67$ | $284.67$ | $284.67$ | | |
2.3447 | C(0,0,0) | Zr${}_{(\frac{1}{2},0,0)}$ | $-24.20$ | $-23.82$ | $-23.82$ | | |
3.3159 | C(0,0,0) | C${}_{(\frac{1}{2},\frac{1}{2},0)}$ | $-5.77$ | $-5.77$ | $0.64$ | | | $-10.41$
3.3159 | Zr${}_{(\frac{1}{2},\frac{1}{2},\frac{1}{2})}$ | Zr${}_{(1,1,\frac{1}{2})}$ | $-15.67$ | $-15.67$ | $-3.68$ | | | $-31.66$
4.0611 | C(0,0,0) | Zr${}_{(\frac{1}{2},\frac{1}{2},\frac{1}{2})}$ | $-1.90$ | $-1.90$ | $-1.90$ | $-1.20$ | $-1.20$ | $-1.20$
4.6894 | C(0,0,0) | C(1,0,0) | $-21.97$ | $1.02$ | $1.02$ | | |
4.6894 | Zr${}_{(\frac{1}{2},\frac{1}{2},\frac{1}{2})}$ | Zr${}_{(1\frac{1}{2},\frac{1}{2},\frac{1}{2})}$ | $-17.47$ | $3.82$ | $3.82$ | | |
5.2429 | C(0,0,0) | Zr${}_{(1,\frac{1}{2},0)}$ | $2.63$ | $-0.01$ | $0.53$ | | |
5.7433 | C(0,0,0) | C${}_{(1,\frac{1}{2},\frac{1}{2})}$ | $0.37$ | $0.36$ | $0.36$ | $-0.15$ | $-0.02$ | $-0.02$
5.7433 | Zr${}_{(\frac{1}{2},\frac{1}{2},\frac{1}{2})}$ | Zr${}_{(1\frac{1}{2},1,1)}$ | $0.39$ | $0.28$ | $0.28$ | $0.41$ | |
6.6318 | C(0,0,0) | C(1,1,0) | $-0.94$ | $-0.94$ | $-0.05$ | | |
6.6318 | Zr${}_{(\frac{1}{2},\frac{1}{2},\frac{1}{2})}$ | Zr${}_{(1\frac{1}{2},1\frac{1}{2},\frac{1}{2})}$ | $0.52$ | $0.52$ | $-0.06$ | | |
7.0341 | C(0,0,0) | Zr${}_{(1,1,\frac{1}{2})}$ | $-0.13$ | $-0.13$ | $-0.09$ | | |
8.1223 | C(0,0,0) | C(1,1,1) | $-0.03$ | $-0.03$ | $-0.03$ | | |
8.1223 | Zr${}_{(\frac{1}{2},\frac{1}{2},\frac{1}{2})}$ | Zr${}_{(1\frac{1}{2},1\frac{1}{2},1\frac{1}{2})}$ | $-0.01$ | $-0.01$ | $-0.01$ | | |
Table 1: Values of the non-zero elements of the force-constant matrices
derived from a $2$$\times$$2$$\times$$2$ supercell in GGA (in N/m). All force-
constant matrices are symmetric.
From Fig. 1 one sees that the experimental data at the $\Gamma$ point are
about $\Delta\omega=0.05\omega$ lower than the calculated values. A reason for
this could be that the experimental values are measured at room temperature
and that anharmonic effects diminish the phonon frequencies. Another reason
may relate to the non-stoichiometric concentration of carbon, which diminishes
the average phonon frequencies as well. In other ab-initio calculations for
alkali metalsfrank , graphite and diamondkresse the experimental phonon
points also are at frequencies lower than calculated ones. On the same Fig. 1
we see the influence of the different exchange-correlation energy
approximations on phonon frequencies. In this case, the LDA appears to be far
less accurate than the GGA.
Figure 2: Phonon density of states $g(\omega)$ of the ZrC crystal and partial
phonon density of states $g_{i,\alpha}(\omega)$, where $\alpha=$ C or Zr and
$i=x,$ $y$ or $z$ are the displacement directions. These are the GGA results.
Experimental data taken from Ref. gompf .
By sampling the dynamical matrix at many wave vectors, one can calculate the
phonon density of states $g(\omega)$, and the partial phonon density of states
$g_{x,Zr}(\omega)$ and $g_{x,C}(\omega)$. The $g(\omega)$ describes the number
of phonon frequencies in an interval around $\omega$, while $g_{x,Zr}$ and
$g_{x,C}$ specify the number of phonon frequencies in an interval around
$\omega$, but only taking into account vibrations of Zr and C atoms,
respectively. The density-of-state functions calculated from the GGA data are
shown in Fig. 2. They are conventionally normalized to $\int
g(\omega)d\omega=1$ and $\int g_{x,\alpha}(\omega)d\omega=1/6$, where
$\alpha=$ Zr or C. The curves show that motions within acoustic dispersion
curves are almost entirely due to Zr atoms. The sum of the zirconium density
of states ($g_{x,Zr}(\omega)+g_{y,Zr}(\omega)+g_{z,Zr}(\omega)$) fits the
total density of states $g(\omega)$ below 10 THz. The part of $g(\omega)$
above the gap is mainly due to vibrations of C atoms. Thus, ZrC forms quite a
special crystal, in which the heavy Zr atoms, form a frame for the elastic
motion, and the C atoms vibrate within the optical modes. We see in Table 1,
that the magnitudes of the force constants between Zr–C and between Zr–Zr and
C–C at a similar distance remain of the same order. Thus, ZrC does not consist
of two weakly bounded subsystems. The situation is similar in TiC mytic . In
Fig. 2 the experimentally determined phonon density of states taken from
Ref.gompf is also shown. The agreement within the acoustic region is quite
good. In the optical region the experimental resolution of $\approx 2.5$ THz
and the non-stoichiometry of carbon lead to considerable broadening.
Result | $B$ | $c_{11}$ | $c_{12}$ | $c_{44}$
---|---|---|---|---
Expt ZrC0.94 smithA | $(2.231)$ | $4.720$ | $0.987$ | $1.593$
Expt ZrC0.89 smithA | $(2.225)$ | $4.682$ | $0.997$ | $1.573$
ZrC extrapolated | $(2.238)$ | $4.766$ | $0.975$ | $1.617$
Calc. LDA | $(2.270)$ | $4.716$ | $1.047$ | $1.348$
Calc. LDA (energ.) | $2.295$ | | |
Calc. GGA | $(2.279)$ | $4.802$ | $1.018$ | $1.697$
Calc. GGA (energ.) | $2.32$ | | |
Table 2: Experimental and calculated bulk modulus $B$ and elastic constants
$c_{ij}$ of the ZrC crystal, in units of $10^{11}{\rm Nm}^{-2}=1{\rm MBar}$.
Values in parentheses are calculated from $B=\frac{1}{3}(c_{11}+2c_{12})$.
The bulk modulus was calculated in the $1$$\times$$1$$\times$$1$ supercell
from the relation of pressure to the lattice constant. These data were fitted
to a fourth order polynomial. Hence, the derivative ${\partial P}/{\partial
V}$, and the bulk modulus $B$ were calculated from the relation:
$B=-V({\partial P}/{\partial V})$, where $V$ is the volume of the ZrC
crystallographic unit cell. For LDA and GGA, the $B$ values are given in Table
2 and they agree quite well with experimental data. We also derived the bulk
modulus from the energy-deformation relationship and the stress–strain
relations, and the last proved to be more accurate than differentiation of the
total energy curve (see Table 2).
We used stress–strain relations to obtain the elastic constants. The
calculations of $c_{11}$, $c_{12}$, $c_{44}$ elastic constants involved
elongation and shear deformations. Deformations from $0.5\%$ to $3\%$ in
length and from $1$ to $5$ degrees in angle are used. The results of small and
large deformations are consistent. The deformed lattices have space groups
Fm$\bar{3}$m, I4/mmm and I/mmm, for bulk expansion, elongation along z, and
shear modes, respectively. In all these deformed lattices all atoms remain in
high-symmetry sites so that the relaxation of the internal degrees of freedom
during the supercell deformations are not necessary. The calculated elastic
constants are compared with the experimental data in Table 2. Generally they
are in good agreement with experiments. The $c_{12}$, $c_{44}$ elastic
constants are better reproduced by the GGA, but also $c_{11}$ fits better with
GGA, when experimental data are extrapolated to the stoichiometric ZrC (third
row in Table 2).
In summary, we have shown that the ab initio calculations of HF forces
together with the direct method lead to a satisfactory description of phonons
and elastic constants in the ZrC crystal. The phonon dispersion relations and
the density of states fit well to neutron experimental data. The GGA offered
better phonon frequencies and elastic constants.
###### Acknowledgements.
The use of facilities of the ACC “Cyfronet”, Cracow, where the calculations
have been done, are kindly acknowledged. This work was partially supported by
the Polish State Committee of Scientific Research (KBN), grant No 2 PO3B 004
14, and computational grant No. KBN/SGI_ORIGIN_2000/IFJ/128/1998.
## References
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* (24) CASTEP is a part of $Cerius^{2}$ system developed by Molecular Simulations.
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|
arxiv-papers
| 2013-01-25T16:32:43 |
2024-09-04T02:49:40.737175
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Pawe{\\l} T. Jochym, Krzysztof Parlinski",
"submitter": "Pawe{\\l} T. Jochym",
"url": "https://arxiv.org/abs/1301.6083"
}
|
1301.6090
|
# Construction of two-dimensional quantum field models through Longo-Witten
endomorphisms
Yoh Tanimoto 111Supported by Deutscher Akademischer Austauschdienst, Hausdorff
Institut für Mathematik and Alexander von Humboldt Stiftung, and in part by
Courant Research Centre “Higher Order Structures in Mathematics”, Göttingen.
e-mail: [email protected]
Institut für Theoretische Physik, Universität Göttingen
Friedrich-Hund-Platz 1, 37077 Göttingen, Germany.
###### Abstract
We present a procedure to construct families of local, massive and interacting
Haag-Kastler nets on the two-dimensional spacetime through an operator-
algebraic method. An existence proof of local observable is given without
relying on modular nuclearity.
By a similar technique, another family of wedge-local nets is constructed
using certain endomorphisms of conformal nets recently studied by Longo and
Witten.
## 1 Introduction
In a series of papers [18, 33, 6] we have investigated operator-algebraic
methods based on conformal field theory to construct quantum field models on
two-dimensional spacetime with a weak localization property. Although we
succeeded to obtain various examples and general structural results, there was
missing one important property: strict localization. In the present paper we
construct further new families of quantum field models and prove their strict
locality. In other words, we construct interacting two-dimensional Haag-
Kastler nets, using techniques from conformal nets.
In recent years the operator-algebraic approach (algebraic QFT) to quantum
field theory has seen many developments. A fundamental idea is that the whole
quantum field model can be recovered from the set of observables localized in
an unbounded wedge-shaped region and the spacetime translations [7].
Conversely, a strategy to construct models is first to obtain models localized
in wedges then to prove the existence of fully localized observables. The most
successful application of this strategy is the construction of scalar
factorizing S-matrix models in two dimensions [22].
Construction of interacting quantum field theory in four dimensions remains
one of the most important open problems in mathematical physics. Further steps
toward higher dimensions within the above operator-algebraic approach have
been obtained [13, 23], however, the second step to find localized observables
has so far turned out to be unsuccessful [13]. Furthermore, we found that
conformal covariance implies triviality of scattering in two dimensions [34]
yet it is possible to construct weakly localized, conformally covariant models
[33]. Thus construction of weakly localized models should be considered as an
important but intermediate step.
We have constructed weakly localized massless models using chiral components
of two-dimensional conformal field theory [18, 33, 6], however, strict
locality is disproved in some of them and open in the others. The only known
operator-algebraic technique to find local observables is wedge-splitting [12]
(see Section 5). This property is apparently difficult to hold in massless
models, especially known to be invalid in conformal case. Actually, one
encounters a similar problem in the form factor bootstrap program to
integrable models [31]: one starts with a given S-matrix and matrix
coefficients of local operators are given as solutions of the so-called form
factor equations. One can easily observe that the convergence of these matrix
coefficients is much worse in massless case. Hence, in order to avoid such
technical difficulty, one should try to construct massive models (besides, the
convergence of form factors is a difficult problem even in massive models
[1]).
Although our previous constructions are fundamentally based on conformal field
theory, one can still find a connection with massive models. We will
investigate this issue more systematically in a separate paper [5]. Here we
use again Longo-Witten endomorphisms [27] to construct weakly localized
massive models. Then we show that some of them are actually strictly local by
examining wedge-splitting. Note that in [22] wedge-splitting has been proved
through modular nuclearity [12]. We provide both a direct proof of wedge-
splitting and a simple proof of modular nuclearity of certain models. Once
strict locality is demonstrated, one can apply the standard scattering theory
and check that S-matrix is nontrivial. As we will see, this construction can
be considered as a generalization of the Federbush model. The Federbush model
is an integrable model with S-matrix independent from rapidity [28]. Local
observables in this model have been found in [29] for certain range of
coupling constant. We construct Haag-Kastler nets which have the same S-matrix
as those Federbush models with the coupling constant of arbitrary size. The
model has multiple one-particle spectrum, hence the strict locality has not
been treated in [22]. We construct another family of wedge-local nets
(Borchers triples) using Longo-Witten endomorphisms [27]. Although strict
locality is open for this latter family, two constructions look quite similar.
The present construction uses as a starting point the free field or any
quantum field which gives a wedge-split net. The new models are constructed on
the tensor product of the Hilbert space of the input field (net). One could
call our construction a “deformation” of the trivial combination of the input
fields. The term “deformation” had a specific sense in [13] and [23], and in
massless cases it has been revealed that the connection with free (trivial
scattering) models is generic [33]. In addition, one notes that such a
“deformation” is always related to a certain structure of the input field. The
deformation in [13] exploits the translation covariance, while the chiral
decomposition plays a crucial role in [33, 6]. If one desires further
different deformations, it is a natural idea to introduce a further structure
to the “undeformed” model. In this paper this idea will be realized using
tensor product and inner symmetry. Such structure is easily found in examples.
Indeed, our procedure in Section 5 can be applied to the tensor product of
Lechner’s models [22] and immediately gives new strict local nets.
This paper is organized as follows: after introductory Sections 1, 2, we first
exhibit ways to construct Borchers triples. Section 3 is of general nature:
given a Borchers triple with certain symmetry, we show a way to construct new
Borchers triples. Section 4 demonstrates how to construct massive Borchers
triples out of Longo-Witten endomorphisms of the ${\rm U}(1)$-current net.
Then we turn to strict locality in Section 5. We collect arguments to prove
strict locality of the models from Section 3. Then we apply them to the
concrete cases including the complex massive free field to obtain interacting
Haag-Kastler nets (Section 6). The two-particle S-matrix of the models from
Section 4 will be also calculated.
## 2 Preliminaries
### 2.1 Two dimensional Haag-Kastler net and Borchers triples
In algebraic QFT, models of quantum field theory are realized as nets of von
Neumann algebras [20]. A Haag-Kastler net (of von Neumann algebras)
${\mathcal{A}}$ on ${\mathbb{R}}^{2}$ is a map $O\mapsto{\mathcal{A}}(O)$ from
the set of open regions $\\{O\\}$ in two dimensional Minkowski space
${\mathbb{R}}^{2}$ to the set of von Neumann algebras on a fixed Hilbert space
${\mathcal{H}}$ such that
1. (1)
(Isotony) If $O_{1}\subset O_{2}$, then
${\mathcal{A}}(O_{1})\subset{\mathcal{A}}(O_{2})$.
2. (2)
(Locality) If $O_{1}$ and $O_{2}$ are spacelike separated, then
${\mathcal{A}}(O_{1})$ and ${\mathcal{A}}(O_{2})$ commute.
3. (3)
(Poincaré covariance) There is a (strongly continuous) unitary representation
$U$ of the (proper orthochronous) Poincaré group
${{\mathcal{P}}^{\uparrow}_{+}}$ such that
$U(g){\mathcal{A}}(O)U(g)^{*}={\mathcal{A}}(gO)$ for
$g\in{{\mathcal{P}}^{\uparrow}_{+}}$.
4. (4)
(Positivity of energy) The joint spectrum of the restriction of $U$ to the
translation subgroup ($\cong{\mathbb{R}}^{2}$) is contained in the closed
forward lightcone
$V_{+}:=\\{(p_{0},p_{1})\in{\mathbb{R}}^{2}:p_{0}\geq|p_{1}|\\}$.
5. (5)
(Vacuum vector) There is a (up to scalar) unique vector $\Omega$ which is
invariant under $U(g)$ and cyclic for ${\mathcal{A}}(O)$.
Note that the condition on the vacuum $\Omega$ contains the Reeh-Schlieder
property, which can be derived if one assumes the additivity of the net. We
take this as an axiom for net for simplicity (see the discussion in [35,
Section 2]). From these assumptions, the following is automatic [3].
1. (6)
(Irreducibility) The von Neumann algebra
$\bigvee_{O\subset{\mathbb{R}}^{2}}{\mathcal{A}}(O)$ is equal to
$B({\mathcal{H}})$.
Precisely, the triple $({\mathcal{A}},U,\Omega)$ should be called a Haag-
Kastler net, however we say ${\mathcal{A}}$ is a net if no confusion arises.
If one has a net ${\mathcal{A}}$, under certain conditions one can define
S-matrix (see Section 6). The main objective in this paper is to construct
examples of two-dimensional Haag-Kastler nets with nontrivial S-matrix.
Yet, it appears very difficult to construct such nets from a scratch.
Fortunately, Borchers showed that it suffices to have a triple of a single von
Neumann algebra associated with the wedge-shaped region
$W_{\mathrm{R}}:=\\{a\in{\mathbb{R}}^{2}:a_{1}>|a_{0}|\\}$, the spacetime
translations and the vacuum. More precisely, a Borchers triple is a triple
$({\mathcal{M}},T,\Omega)$ of a von Neumann algebra ${\mathcal{M}}$, a unitary
representation $U$ of ${\mathbb{R}}^{2}$ and a vector $\Omega$ such that
1. (1)
If $a\in W_{\mathrm{R}}$, then ${\hbox{\rm
Ad\,}}T(a)({\mathcal{M}})\subset{\mathcal{M}}$.
2. (2)
The joint spectrum of $T$ is contained in $V_{+}$.
3. (3)
$\Omega$ is cyclic and separating for ${\mathcal{M}}$.
The correspondence between nets and Borchers triples is as follows (see [7,
22]): If $({\mathcal{A}},U,\Omega)$ is a (two-dimensional) Haag-Kastler net,
then $({\mathcal{A}}(W_{\mathrm{R}}),U|_{{\mathbb{R}}^{2}},\Omega)$ is a
Borchers triple, where $U$ is restricted to the translation subgroup
$\cong{\mathbb{R}}^{2}$.
On the other hand, if $({\mathcal{M}},T,\Omega)$ is a Borchers triple, then
one can define a net as follows: in the two-dimensional Minkowski space any
double cone $D$ is represented as the intersection of two wedges
$D=(W_{\mathrm{R}}+a)\cap(W_{\mathrm{L}}+b)$. Then we define for double cones
${\mathcal{A}}(D):={\hbox{\rm Ad\,}}T(a)({\mathcal{M}})\cap{\hbox{\rm
Ad\,}}T(b)({\mathcal{M}}^{\prime})$. For a general region $O$ we take
${\mathcal{A}}(O):=\bigvee_{D\subset O}{\mathcal{A}}(D)$, where the union runs
all the double cones contained in $O$. Then one can observe that
${\mathcal{A}}$ satisfies isotony and locality. Borchers further proved that
the representation $T$ of ${\mathbb{R}}^{2}$ extends to a representation $U$
of ${{\mathcal{P}}^{\uparrow}_{+}}$ through Tomita-Takesaki theory [7] such
that ${\mathcal{A}}$ is covariant and $\Omega$ is invariant under $U$ (the
representation of boosts is given by the modular group). Positivity of energy
is inherited from $T$ of the Borchers triple. The only missing property is
cyclicity of $\Omega$ for local algebras ${\mathcal{A}}(O)$. Indeed, there are
examples of Borchers triples which fail to satisfy this cyclicity [33, Theorem
4.16].
Hence, a general strategy to construct Haag-Kastler nets is first to construct
Borchers triples and then to check cyclicity of the vacuum. This program has
been first completed in [22]. Note that the latter condition is actually very
hard to check directly. In [22], Lechner proved instead the so-called modular
nuclearity [12], which is a sufficient condition for the cyclicity of the
vacuum. In this paper, we will provide new examples of Borchers triples for
which the cyclicity of the vacuum can be proved without relying on modular
nuclearity (see Section 5.2). We say that a Borchers triple is strictly local
if $\Omega$ is cyclic for ${\mathcal{A}}(O)$ constructed as above. A strictly
local Borchers triple corresponds to a Haag-Kastler net.
If there is a unitary operator $V$ which commutes with $U(g)$ and if
${\hbox{\rm Ad\,}}V$ preserves each local algebra ${\mathcal{A}}(O)$, then we
call ${\hbox{\rm Ad\,}}V$ an inner symmetry of the net ${\mathcal{A}}$. By the
uniqueness of $\Omega$, ${\hbox{\rm Ad\,}}V$ preserves the vacuum state
$\langle\Omega,\cdot\,\Omega\rangle$. By the irreducibility, the automorphism
${\hbox{\rm Ad\,}}V$ is implemented uniquely by $V$ up to a scalar. We require
always that $V\Omega=\Omega$. By this requirement, the implementation is
unique. A general strategy to use them in order to construct new Borchers
triples will be explained in Section 3 and concrete examples of inner symmetry
will be discussed in Section 6.1. A collection of inner symmetries may form a
group $G$. In such a case we say that $G$ acts on the net by inner symmetry.
Similarly one can consider an action of $V$ on Borchers triples. In this case,
one says that $V$ implements an inner symmetry if ${\hbox{\rm Ad\,}}V$
preserves ${\mathcal{M}}$ and commutes with $T$. We say also that $G$ acts by
inner symmetry when such $V$’s form a group.
### 2.2 Conformal nets and Longo-Witten endomorphisms
In some of our constructions, the main ingredients come from chiral conformal
field theory. Let us summarize here its operator-algebraic treatment (see also
[19, 26]). An open, connected, non dense and non empty subset $I$ of $S^{1}$
is called an interval in $S^{1}$. We identify ${\mathbb{R}}$ as a dense subset
in $S^{1}$ by the stereographic projection. The Möbius group ${\rm
PSL}(2,{\mathbb{R}})$ acts on $S^{1}={\mathbb{R}}\cup\\{\infty\\}$ and under
this identification it contains translations and dilations of ${\mathbb{R}}$.
A (Möbius covariant) net (of von Neumann algebras) on $S^{1}$ is an assignment
of von Neumann algebras ${\mathcal{A}}_{0}(I)$ on a common Hilbert space
${\mathcal{H}}_{0}$ to intervals $I$ such that
1. (1)
(Isotony) If $I_{1}\subset I_{2}$, then
${\mathcal{A}}_{0}(I_{1})\subset{\mathcal{A}}_{0}(I_{2})$.
2. (2)
(Locality) If $I_{1}$ and $I_{2}$ are disjoint, then
${\mathcal{A}}_{0}(I_{1})$ and ${\mathcal{A}}_{0}(I_{2})$ commute.
3. (3)
(Möbius covariance) There is a (strongly continuous) unitary representation
$U_{0}$ of ${\rm PSL}(2,{\mathbb{R}})$ such that
$U_{0}(g){\mathcal{A}}_{0}(I)U_{0}(g)^{*}={\mathcal{A}}_{0}(gI)$ for $g\in{\rm
PSL}(2,{\mathbb{R}})$.
4. (4)
(Positivity of energy) The generator of the subgroup of translation in the
representation $U_{0}$ has positive spectrum.
5. (5)
(Vacuum vector) There is a (up to scalar) unique vector $\Omega_{0}$ which is
invariant under $U_{0}(g)$ and cyclic for ${\mathcal{A}}_{0}(I)$.
As in two dimensions, we call ${\mathcal{A}}_{0}$ a net, however the actual
object of interest is the triple $({\mathcal{A}}_{0},U_{0},\Omega_{0})$. From
these assumptions many properties automatically follow, among which of
importance in our application are
1. (6)
(Irreducibility) The von Neumann algebra $\bigvee_{I\subset
S^{1}}{\mathcal{A}}_{0}(I)$ is equal to $B({\mathcal{H}}_{0})$.
2. (7)
(Haag duality on $S^{1}$) It holds that
${\mathcal{A}}_{0}(I)^{\prime}={\mathcal{A}}_{0}(I^{\prime})$, where
$I^{\prime}$ denotes the interior of the complement of $I$ in $S^{1}$.
3. (8)
(Bisognano-Wichmann property) The modular group $\Delta_{0}^{it}$ of
${\mathcal{A}}_{0}({\mathbb{R}}_{+})$ associated with $\Omega_{0}$ is equal to
$U_{0}(\delta(-2\pi t))$, where $\delta$ is the dilation in ${\rm
PSL}(2,{\mathbb{R}})$.
An inner symmetry of a conformal net ${\mathcal{A}}_{0}$ is a collection of
automorphisms of each local algebra ${\mathcal{A}}_{0}(O)$ implemented by a
common unitary operator $V_{0}$ which preserves the vacuum state
$\langle\Omega_{0},\cdot\,\Omega_{0}\rangle$. In conformal (Möbius covariant)
case, it automatically follows that $V_{0}$ commutes with $U_{0}(g)$ thanks to
Bisognano-Wichmann property.
A Longo-Witten endomorphism of ${\mathcal{A}}_{0}$ is an endomorphism of
${\mathcal{A}}_{0}({\mathbb{R}}_{+})$, implemented by a unitary operator
$V_{0}$ which commutes with $U_{0}(g)$, where $g$ is a translation. An inner
symmetry restricted to ${\mathcal{A}}_{0}({\mathbb{R}}_{+})$ is a Longo-Witten
endomorphism.
We will give a concrete example of conformal net and of Longo-Witten
endomorphisms in Section 4.
### 2.3 The massive scalar free field
Our main construction strategy is based on simpler examples with certain
properties. Let us quickly review the simplest quantum field.
The free field is constructed from an irreducible representation of the
Poincaré group through second quantization. We use the notation which is to
some extent consistent with [25]. The one-particle Hilbert space is
${\mathcal{H}}_{1}:=L^{2}({\mathbb{R}},d\theta)$ and for the mass $m>0$ the
(proper orthochronous) Poincaré group acts by
$U_{1}(a,\lambda)\psi(\theta)=e^{ip(\theta)\cdot a}\psi(\theta-\lambda)$,
where $p(\theta):=(m\cosh(\theta),m\sinh(\theta))$ parametrizes the mass
shell. We introduce the (auxiliary) unsymmetrized Hilbert space
${\mathcal{H}}^{\Sigma}:=\bigoplus{\mathcal{H}}_{1}^{\otimes n}$ and the
(physical) symmetrized Hilbert space $\mathcal{H}_{\mathrm{r}}:=\bigoplus
Q_{n}{\mathcal{H}}_{1}^{\otimes n}$, where $Q_{n}$ is the projection onto the
symmetric subspace.
Let us denote the $n$-th component of a vector
$\Psi\in\mathcal{H}_{\mathrm{r}}$ by $(\Psi)_{n}$. For a vector
$\psi\in{\mathcal{H}}_{1}$ and $\Psi$ which has only finitely many components,
the creation operator $a^{\dagger}$ is defined by
$(a^{\dagger}(\psi)\Psi)_{n}=Q_{n}(\psi\otimes\Psi_{n-1})$. The annihilation
operator is the adjoint $a(\psi)=a^{\dagger}(\psi)^{*}$. With this notation,
$a^{\dagger}$ is linear and $a$ is antilinear with respect to $\psi$. The free
quantum field $\phi$ is now given by
$\phi(f):=a^{\dagger}(f^{+})+a(J_{1}f^{-}),\;\;\;\;\;\;\;\;\;\;f^{\pm}(\theta)=\frac{1}{2\pi}\int
d^{2}af(a)e^{\pm ip(\theta)\cdot a},$
where $f$ is a test function $\mathscr{S}({\mathbb{R}}^{2})$ and
$J_{1}\psi(\theta)=\overline{\psi(\theta)}$. This field is local, in the sense
that if $f$ and $g$ have spacelike separated supports, then $\phi(f)$ and
$\phi(g)$ commute on an appropriate domain.
Finally we introduce the free field net. For each open region
$O\subset{\mathbb{R}}^{2}$, we set
$\mathcal{A}_{\mathrm{r}}(O):=\\{e^{i\phi(f)}:{\rm supp}f\subset
O\\}^{\prime\prime}.$
We have the second quantized representation $U_{\mathrm{r}}:=\Gamma(U_{1})$
and the Fock vacuum vector $\Omega_{\mathrm{r}}\in\mathcal{H}_{\mathrm{r}}$.
This triple $(\mathcal{A}_{\mathrm{r}},U_{\mathrm{r}},\Omega_{\mathrm{r}})$ is
a two-dimensional Haag-Kastler net and referred to as the free massive scalar
net. The subscript r is intended for the real scalar field. Furthermore, this
net satisfies the modular nuclearity [12, Section 4], a property which we will
explain in more detail in Section 5.
### 2.4 Scalar factorizing S-matrix models
The free net has many important features, but is not interacting. Lechner has
constructed a large family of interacting nets [22]. Here we only briefly
summarize the construction and their fundamental properties.
Let $S_{2}(\theta)$ be a bounded analytic function on the strip
$0<\Im\theta<\pi$, continuous on the boundary, with the property
$S_{2}(\theta)^{-1}=\overline{S_{2}(\theta)}=S_{2}(-\theta)=S_{2}(\theta+i\pi)$
for $\theta\in{\mathbb{R}}$. This is called the two-particle scattering
function.
This time again the construction of the (wedge-local) field and the net is
based on the one-particle space ${\mathcal{H}}_{1}$ above. For
$\Psi_{n}\in{\mathcal{H}}_{1}^{\otimes n}$, the $S_{2}$-permutation is given
by
$(D_{S_{2},n}(\tau_{j})\Psi)(\theta_{1},\cdots,\theta_{n})=S_{2}(\theta_{j+1}-\theta_{j})\Psi(\theta_{1},\cdots,\theta_{j+1},\theta_{j},\cdots,\theta_{n}).$
Let $Q_{S_{2},n}$ be the orthogonal projection onto the subspace of
${\mathcal{H}}_{1}^{\otimes n}$ invariant under
$\\{D_{S_{2},n}(\tau_{j}):1\leq j\leq n-1\\}$. The full Hilbert space is
${\mathcal{H}}_{S_{2}}:=\bigoplus Q_{S_{2},n}{\mathcal{H}}_{1}^{\otimes n}$
and the representation $U_{1}$ promotes to $U_{S_{2}}$ on
${\mathcal{H}}_{S_{2}}$ by the second quantization. We define similarly the
creation and annihilation operators
$(z_{S_{2}}^{\dagger}(\psi)\Phi)_{n}=Q_{S_{2},n}(\psi\otimes\Phi_{n-1})$ and
$z_{S_{2}}(\psi)=z_{S_{2}}^{\dagger}(\psi)^{*}$. The quantum field is defined
accordingly by
$\phi_{S_{2}}(f):=z_{S_{2}}^{\dagger}(f^{+})+z_{S_{2}}(J_{1}f^{-}),\;\;\;\;\;\;\;\;\;\;f^{\pm}=\frac{1}{2\pi}\int
d^{2}af(a)e^{\pm ip(\theta)\cdot a},$
but this time $\phi_{S_{2}}$ is only wedge-local, hence the net is defined
through Borchers triple.
The von Neumann algebra ${\mathcal{M}}_{S_{2}}$ is defined by
${\mathcal{M}}_{S_{2}}:=\\{e^{i\phi_{S_{2}}(f)}:{\rm supp}f\subset
W_{\mathrm{L}}\\}^{\prime}$
(note that here we take only the single commutant and $f$ has support in
$W_{\mathrm{L}}$, while ${\mathcal{M}}_{S_{2}}$ corresponds to
$W_{\mathrm{R}}$. By the wedge-duality, this is just a matter of convention
(see [25])). The triple $({\mathcal{M}}_{S_{2}},U_{S_{2}},\Omega_{S_{2}})$ is
a Borchers triple [21]. Furthermore, the modular nuclearity holds if $S_{2}$
is regular and fermionic [22], hence the triple is wedge-split in those cases.
## 3 Borchers triples through inner symmetries
In this Section we make the first step in our main construction. We start with
a Borchers triple with inner symmetry and construct new triples. Note that,
differently from our previous results [33, 6], the new triples are defined on
a different Hilbert space although the formulae look very similar.
First we state a key lemma, which can be obtained as a special case of [33,
Lemma 4.1].
###### Lemma 3.1.
Let $\mathcal{M}_{\mathrm{c}}$ be a von Neumann algebra and $Q_{\mathrm{c}}$ a
self-adjoint operator such that ${\rm sp}\,Q_{\mathrm{c}}\subset{\mathbb{Z}}$
and ${\hbox{\rm Ad\,}}e^{i2\pi\kappa
Q_{\mathrm{c}}}(\mathcal{M}_{\mathrm{c}})=\mathcal{M}_{\mathrm{c}}$ for any
$\kappa\in{\mathbb{R}}$. Then, for $\kappa\in{\mathbb{R}}$,
$x\otimes{\mathbbm{1}}$ commutes with ${\hbox{\rm Ad\,}}e^{i2\pi\kappa
Q_{\mathrm{c}}\otimes Q_{\mathrm{c}}}(x^{\prime}\otimes{\mathbbm{1}})$ where
$x\in\mathcal{M}_{\mathrm{c}}$ and
$x^{\prime}\in\mathcal{M}_{\mathrm{c}}^{\prime}$.
###### Proof.
For any $\kappa\in{\mathbb{R}}$, $x$ commutes with ${\hbox{\rm
Ad\,}}e^{i2\pi\kappa Q_{\mathrm{c}}}(x^{\prime})$ by assumption, hence the
commutativity lemma [33, Lemma 4.1] applies. ∎
Analogously one can prove the following.
###### Lemma 3.2.
Let $N$ be an integer, $\mathcal{M}_{\mathrm{c}}$ a von Neumann algebra and
$Q_{\mathrm{c}}$ a self-adjoint operator such that ${\rm
sp}\,Q_{\mathrm{c}}\subset{\mathbb{Z}}$ and ${\hbox{\rm Ad\,}}e^{i\frac{2\pi
k}{N}Q_{\mathrm{c}}}(\mathcal{M}_{\mathrm{c}})=\mathcal{M}_{\mathrm{c}}$ for
any $k\in{\mathbb{Z}}$. Then, for $k\in{\mathbb{Z}}$, $x\otimes{\mathbbm{1}}$
commutes with ${\hbox{\rm Ad\,}}e^{i\frac{2\pi k}{N}Q_{\mathrm{c}}\otimes
Q_{\mathrm{c}}}(x^{\prime}\otimes{\mathbbm{1}})$ where
$x\in\mathcal{M}_{\mathrm{c}}$ and
$x^{\prime}\in\mathcal{M}_{\mathrm{c}}^{\prime}$.
Now let $(\mathcal{M}_{\mathrm{c}},T_{\mathrm{c}},\Omega_{\mathrm{c}})$ be a
Borchers triple with an action of $S^{1}$ or the finite cyclic group
${\mathbb{Z}}_{N}$ by inner symmetry. The argument of this Section goes
parallelly for both groups. There is a unique unitary operator
$V_{\mathrm{c}}(\kappa)$ which implements the inner symmetry by $S^{1}$. For a
${\mathbb{Z}}_{N}$-action we take $V_{\mathrm{c}}$ such that
$V_{\mathrm{c}}^{k}$ implements the inner symmetry for $k\in{\mathbb{Z}}_{N}$.
One can choose $Q_{\mathrm{c}}$ such that
$V_{\mathrm{c}}(\kappa)=e^{i2\pi\kappa Q_{\mathrm{c}}}$ or
$V_{\mathrm{c}}=e^{i\frac{2\pi}{N}Q_{\mathrm{c}}}$, respectively. For an
action of $S^{1}$, $Q_{\mathrm{c}}$ is unique. For ${\mathbb{Z}}_{N}$, we
remark that we can concretely choose $Q_{\mathrm{c}}$ as follows: for each
integer $j\in[0,N-1]$ we put
$\hat{V}_{\mathrm{c}}(j):=\frac{1}{N}\sum_{k}e^{-i\frac{2\pi
jk}{N}}V_{\mathrm{c}}(k)$. It is clear that $\hat{V}_{\mathrm{c}}(j)$ is an
orthogonal projection and $\hat{V}_{\mathrm{c}}(j)\hat{V}_{\mathrm{c}}(l)=0$
if $j\not\equiv l\mod N$. In other words, $k\mapsto V_{\mathrm{c}}^{k}$ is a
representation of ${\mathbb{Z}}_{N}$, whose dual group is also
${\mathbb{Z}}_{N}$ and we extract the spectral projections. Then we can take
$Q_{\mathrm{c}}:=\sum_{j}j\hat{V}_{\mathrm{c}}(j)$. By this construction, it
is clear that $Q_{\mathrm{c}}$ commutes with the translation $T_{\mathrm{c}}$.
An operator $Q_{\mathrm{c}}$ with the properties specified above is not unique
since one can amplify each spectral component by $N$, but if we consider the
operator $e^{i\frac{2\pi k}{N}Q_{\mathrm{c}}\otimes Q_{\mathrm{c}}}$, it does
not depend on the choice of $Q_{\mathrm{c}}$ (see also the remark after [33,
Theorem 4.17]). One sees that ${\rm sp}\,Q_{\mathrm{c}}\subset{\mathbb{Z}}$
since ${\mathbbm{1}}=V_{\mathrm{c}}(1)=e^{i2\pi Q_{\mathrm{c}}}$ for an
$S^{1}$-action or ${\mathbbm{1}}=V_{\mathrm{c}}^{N}=e^{i2\pi Q_{\mathrm{c}}}$
for a ${\mathbb{Z}}_{N}$-action (or directly by our choice).
Now we turn to the construction of Borchers triples. Our objects act on the
tensor product Hilbert space
$\widetilde{\mathcal{H}}_{\mathrm{c}}:=\mathcal{H}_{\mathrm{c}}\otimes\mathcal{H}_{\mathrm{c}}$.
Let us denote $\widetilde{V}_{\mathrm{c},\kappa}=e^{i2\pi\kappa
Q_{\mathrm{c}}\otimes Q_{\mathrm{c}}}$ for an $S^{1}$-action and
$\widetilde{V}_{\mathrm{c},k}=e^{i\frac{2\pi k}{N}Q_{\mathrm{c}}\otimes
Q_{\mathrm{c}}}$ for a ${\mathbb{Z}}_{N}$-action. The tensor product
representation $\widetilde{T}_{\mathrm{c}}(a):=T_{\mathrm{c}}(a)\otimes
T_{\mathrm{c}}(a)$ has positive spectrum and preserves the new vacuum
$\widetilde{\Omega}_{\mathrm{c}}:=\Omega_{\mathrm{c}}\otimes\Omega_{\mathrm{c}}$.
The von Neumann algebra is given respectively by
$\displaystyle\widetilde{\mathcal{M}}_{\mathrm{c},\kappa}$ $\displaystyle:=$
$\displaystyle\\{x\otimes{\mathbbm{1}},{\hbox{\rm
Ad\,}}\widetilde{V}_{\mathrm{c},\kappa}({\mathbbm{1}}\otimes
y):x,y\in\mathcal{M}_{\mathrm{c}}\\}^{\prime\prime},$
$\displaystyle\widetilde{\mathcal{M}}_{\mathrm{c},k}$ $\displaystyle:=$
$\displaystyle\\{x\otimes{\mathbbm{1}},{\hbox{\rm
Ad\,}}\widetilde{V}_{\mathrm{c},k}({\mathbbm{1}}\otimes
y):x,y\in\mathcal{M}_{\mathrm{c}}\\}^{\prime\prime}.$
Note the difference from our previous construction [33]: first of all, this
time the input is the two-dimensional Borchers triple, while we used chiral
components in [33]. Accordingly, the representation
$\widetilde{T}_{\mathrm{c}}$ is now just the two copies of the given one. The
formula for $\widetilde{\mathcal{M}}_{\mathrm{c},\kappa}$ or
$\widetilde{\mathcal{M}}_{\mathrm{c},k}$ is also similar, but this time we
take both $x$ and $y$ from the same algebra $\mathcal{M}_{\mathrm{c}}$.
###### Theorem 3.3.
The triples
$(\widetilde{\mathcal{M}}_{\mathrm{c},\kappa},\widetilde{T}_{\mathrm{c}},\widetilde{\Omega}_{\mathrm{c}})$
and
$(\widetilde{\mathcal{M}}_{\mathrm{c},k},\widetilde{T}_{\mathrm{c}},\widetilde{\Omega}_{\mathrm{c}})$
are Borchers triples for $\kappa\in{\mathbb{R}}$ and $k\in{\mathbb{Z}}_{N}$,
respectively.
###### Proof.
We give a proof only for
$(\widetilde{\mathcal{M}}_{\mathrm{c},\kappa},\widetilde{T}_{\mathrm{c}},\widetilde{\Omega}_{\mathrm{c}})$
constructed from an $S^{1}$-action. The other case is completely parallel.
As in [33, Theorem 4.17], the properties of $\widetilde{T}_{\mathrm{c}}$ and
$\widetilde{\Omega}_{\mathrm{c}}$ are readily checked. As for the relation
${\hbox{\rm
Ad\,}}\widetilde{T}_{\mathrm{c}}(a)(\widetilde{\mathcal{M}}_{\mathrm{c},\kappa})\subset\widetilde{\mathcal{M}}_{\mathrm{c},\kappa}$
for $a\in W_{\mathrm{R}}$, one can prove this by noting that
$\widetilde{V}_{\mathrm{c},\kappa}$ and $\widetilde{T}_{\mathrm{c}}(a)$
commute and the assumptions that $\mathcal{M}_{\mathrm{c}}$ and
$T_{\mathrm{c}}$ have the relation.
The cyclicity of $\widetilde{\Omega}_{\mathrm{c}}$ for
$\widetilde{\mathcal{M}}_{\mathrm{c},\kappa}$ follows from the cyclicity of
$\Omega_{\mathrm{c}}$ for $\mathcal{M}_{\mathrm{c}}$ and the fact that
$\widetilde{V}_{\mathrm{c},\kappa}({\mathbbm{1}}\otimes
y)\widetilde{\Omega}_{\mathrm{c}}=({\mathbbm{1}}\otimes
y)\widetilde{\Omega}_{\mathrm{c}}$. To show the separating property, we need
again to prepare a sufficiently big algebra in the commutant:
$\widetilde{\mathcal{M}}_{\mathrm{c},\kappa}^{1}:=\\{{\hbox{\rm
Ad\,}}\widetilde{V}_{\mathrm{c},k}(x^{\prime}\otimes{\mathbbm{1}}),{\mathbbm{1}}\otimes
y^{\prime}:x^{\prime},y^{\prime}\in\mathcal{M}_{\mathrm{c}}^{\prime}\\}^{\prime\prime}.$
It is easy to see that $\widetilde{\Omega}_{\mathrm{c}}$ is cyclic for
$\widetilde{\mathcal{M}}_{\mathrm{c},\kappa}^{1}$. In order to see that
$\widetilde{\Omega}_{\mathrm{c}}$ is separating for
$\widetilde{\mathcal{M}}_{\mathrm{c},\kappa}$, it is enough to prove that
$\widetilde{\mathcal{M}}_{\mathrm{c},\kappa}$ and
$\widetilde{\mathcal{M}}_{\mathrm{c},\kappa}^{1}$ commute. This follows from
Lemma 3.1, as in [33, Theorem 4.2]. ∎
At the end of this Section, we remark that the existence of inner symmetry is
not at all exceptional. Indeed, if
$(\mathcal{M}_{\mathrm{c}},T_{\mathrm{c}},\Omega_{\mathrm{c}})$ is a (wedge-
split) Borchers triple, then the tensor product
$(\mathcal{M}_{\mathrm{c}}\otimes\mathcal{M}_{\mathrm{c}},T_{\mathrm{c}}\otimes
T_{\mathrm{c}},\Omega_{\mathrm{c}}\otimes\Omega_{\mathrm{c}})$ has the flip
automorphism which commutes with $T_{\mathrm{c}}\otimes T_{\mathrm{c}}$ and
preserves $\Omega_{\mathrm{c}}\otimes\Omega_{\mathrm{c}}$, hence it is an
inner symmetry ${\mathbb{Z}}_{2}$. We present more examples in Section 6.1.
## 4 Borchers triples through Longo-Witten endomorphisms on the
$U(1)$-current net
Here we exhibit another procedure to produce Borchers triples in a more
concrete way. We take the free massive net as the starting point (Section
2.3). The formulae are quite similar to those in [33], but should not be
confused. Strict locality of the models constructed here is not investigated
in the present paper.
### 4.1 Reduction to lightray
Let $(\mathcal{A}_{\mathrm{r}},U_{\mathrm{r}},\Omega_{\mathrm{r}})$ be the
free massive net. The representation $U_{\mathrm{r}}$ can be restricted to the
positive lightray, which we denote by $U_{\mathrm{r}}^{+}$ and we obtain a
one-dimensional Borchers triple
$(\mathcal{M}_{\mathrm{r}},U_{\mathrm{r}}^{+},\Omega_{\mathrm{r}})$, namely a
von Neumann algebra
$\mathcal{M}_{\mathrm{r}}:=\mathcal{A}_{\mathrm{r}}(W_{\mathrm{R}})$, a
positive energy representation $U_{\mathrm{r}}^{+}$ of ${\mathbb{R}}$ and a
cyclic separating vector $\Omega_{\mathrm{r}}$ for $\mathcal{A}_{\mathrm{r}}$
invariant under $U_{\mathrm{r}}^{+}$ such that ${\hbox{\rm
Ad\,}}U_{\mathrm{r}}^{+}(t)(\mathcal{M}_{\mathrm{r}})\subset\mathcal{M}_{\mathrm{r}}$
for $t\in{\mathbb{R}}_{+}$. We denote the negative lightlike translation by
$U_{\mathrm{r}}^{-}$.
Let us recall the ${\rm U}(1)$-current net
$({{\mathcal{A}}^{(0)}},U_{0},\Omega_{0})$, which is a conformal net. For its
definition, see our previous discussion [33, Section 5]. The point is that the
Hilbert space is naturally isomorphic to the Fock space
$\mathcal{H}_{\mathrm{r}}$ of the massive free net, whose one-particle space
is $L^{2}({\mathbb{R}},d\theta)$ and one considers the second quantization
operators. The above one-dimensional triple
$(\mathcal{M}_{\mathrm{r}},U_{\mathrm{r}}^{+},\Omega_{\mathrm{r}})$ is
actually unitarily equivalent to the triple
$({{\mathcal{A}}^{(0)}}({\mathbb{R}}_{+}),T_{0},\Omega_{0})$, where
$({{\mathcal{A}}^{(0)}},U_{0},\Omega_{0})$ is the ${\rm U}(1)$-current net and
$T_{0}$ is the restriction of $U_{0}$ to the translation subgroup. This will
be explained in more detail in [5].
In particular, we can exploit the Longo-Witten endomorphisms found in [27].
Recall that, for an inner symmetric function $\varphi(z)$, namely the boundary
value on ${\mathbb{R}}$ of a bounded analytic function on $0<\Im z<\pi$, one
considers the operator $V_{\varphi}:=\Gamma(\varphi(P_{1}))$, where $P_{1}$ is
the generator of the restriction of $U_{\mathrm{r}}^{+}$ on the one-particle
space ${\mathcal{H}}_{1}$, $\varphi(P_{1})$ denotes a functional calculus and
$\Gamma$ is the second quantization (note that in general the order of second
quantization and functional calculus cannot be exchanged:
$\Gamma(\varphi(P_{1}))\neq\varphi(\Gamma(P_{1}))$. Then ${\hbox{\rm
Ad\,}}V_{\varphi}$ preserves $\mathcal{M}_{\mathrm{r}}$ and $V_{\varphi}$
commutes with $U_{\mathrm{r}}^{+}$. Furthermore, $V_{\varphi}$ commutes with
$U_{\mathrm{r}}^{-}$ since
$U_{\mathrm{r}}^{-}(a)=\Gamma\left(\exp\left(\frac{it}{P_{1}}\right)\right)$,
as we see in [5].
### 4.2 Construction of Borchers triples
We work on the tensor product Hilbert space
$\widetilde{\mathcal{H}}_{\mathrm{r}}:=\mathcal{H}_{\mathrm{r}}\otimes\mathcal{H}_{\mathrm{r}}$.
We fix an inner symmetric function $\varphi$. As above, $P_{1}$ is the one-
particle lightlike translation. Let us recall our argument [33, Section 5].
The physical Hilbert space $\mathcal{H}_{\mathrm{r}}$ is included in the
unsymmetrized Fock space ${\mathcal{H}}^{\Sigma}$. We consider $m$ commuting
operators on ${\mathcal{H}}_{1}^{\otimes m}$:
$\\{{\mathbbm{1}}\otimes\cdots\otimes\underset{j\mbox{-th}}{P_{1}}\otimes\cdots\otimes{\mathbbm{1}}:1\leq
j\leq m\\}.$
For $1\leq j\leq m$ and $1\leq k\leq n$, let us define operators on
${\mathcal{H}}^{m}\otimes{\mathcal{H}}^{n}$:
$\displaystyle P_{j,k}^{m,n}$
$\displaystyle:=\underset{j\mbox{-th}}{\left({\mathbbm{1}}\otimes\cdots\otimes\frac{1}{P_{1}}\otimes\cdots\otimes{\mathbbm{1}}\right)}\otimes{\left({\mathbbm{1}}\otimes\cdots\otimes\underset{k\mbox{-th}}{P_{1}}\otimes\cdots\otimes{\mathbbm{1}}\right)}$
$\displaystyle R^{m,n}_{\varphi}$
$\displaystyle:=\prod_{j,k}\varphi(P_{j,k}^{m,n}),$
where $\varphi(P_{j,k}^{m,n})$ is the functional calculus on
${\mathcal{H}}_{1}^{\otimes m}\otimes{\mathcal{H}}_{1}^{\otimes n}$. Now, our
key operator on the unsymmetrized space
${\mathcal{H}}^{\Sigma}\otimes{\mathcal{H}}^{\Sigma}$ is
$\widetilde{R}_{\varphi}:=\bigoplus_{m,n}R_{\varphi}^{m,n}=\bigoplus_{m,n}\prod_{j,k}\varphi(P^{m,n}_{j,k}),$
where for $m=0$ or $n=0$ we set $R_{j,k}^{m,n}={\mathbbm{1}}$ as a convention.
It is easy to see that $\widetilde{R}_{\varphi}$ naturally restricts to
partially symmetrized subspaces
$\mathcal{H}_{\mathrm{r}}\otimes{\mathcal{H}}^{\Sigma}$ and
${\mathcal{H}}^{\Sigma}\otimes\mathcal{H}_{\mathrm{r}}$ and to the totally
symmetrized space $\mathcal{H}_{\mathrm{r}}\otimes\mathcal{H}_{\mathrm{r}}$.
Let $E_{1}\otimes E_{1}\otimes\cdots\otimes E_{1}$ be the joint spectral
measure of
$\\{{\mathbbm{1}}\otimes\cdots\otimes\underset{k\mbox{-th}}{P_{1}}\otimes\cdots\otimes{\mathbbm{1}}:1\leq
k\leq n\\}$. One obtains the following expression:
$\displaystyle\varphi(P_{j,k}^{m,n})$ $\displaystyle=$
$\displaystyle\int\left({\mathbbm{1}}\otimes\cdots\otimes\underset{j\mbox{-th}}{\varphi\left(\frac{p_{k}}{P_{1}}\right)}\otimes\cdots{\mathbbm{1}}\right)\otimes\left({\mathbbm{1}}\otimes\cdots\underset{k\mbox{-th}}{dE_{1}(p_{k})}\otimes\cdots{\mathbbm{1}}\right)$
$\displaystyle=$
$\displaystyle\int\left({\mathbbm{1}}\otimes\cdots\underset{j\mbox{-th}}{dE_{1}(p_{j})}\otimes\cdots{\mathbbm{1}}\right)\otimes\left({\mathbbm{1}}\otimes\cdots\otimes\underset{k\mbox{-th}}{\varphi\left(\frac{P_{1}}{p_{j}}\right)}\otimes\cdots{\mathbbm{1}}\right)$
Similarly to our previous case [33, Section 5.2], we decompose
$\widetilde{R}_{\varphi}$ with respect only to the right or left component:
$\displaystyle\widetilde{R}_{\varphi}$ $\displaystyle=$
$\displaystyle\bigoplus_{n}\int\prod_{k}\Gamma\left(\varphi\left(\frac{p_{k}}{P_{1}}\right)\right)\otimes
dE_{1}(p_{1})\otimes\cdots\otimes dE_{1}(p_{n})$ $\displaystyle=$
$\displaystyle\bigoplus_{m}\int dE_{1}(p_{1})\otimes\cdots\otimes
dE_{1}(p_{m})\otimes\prod_{j}\Gamma\left(\varphi\left(\frac{P_{1}}{p_{j}}\right)\right)$
For the proof, we refer to [33, Section 5.2]. The expression in the first line
naturally restricts to the partially symmetrized space
$\mathcal{H}_{\mathrm{r}}\otimes{\mathcal{H}}^{\Sigma}$ and the second
expression to ${\mathcal{H}}^{\Sigma}\otimes\mathcal{H}_{\mathrm{r}}$.
We have a variant of [33, Lemma 5.2].
###### Lemma 4.1.
It holds for $x\in\mathcal{A}_{\mathrm{r}}(W_{\mathrm{R}})$ and
$x^{\prime}\in\mathcal{A}_{\mathrm{r}}(W_{\mathrm{R}})^{\prime}$ that
$[x\otimes{\mathbbm{1}},{\hbox{\rm
Ad\,}}\widetilde{R}_{\varphi}(x^{\prime}\otimes{\mathbbm{1}})]=0$
on the Hilbert space
$\mathcal{H}_{\mathrm{r}}\otimes\mathcal{H}_{\mathrm{r}}$. Similarly, for
$y\in\mathcal{A}_{\mathrm{r}}(W_{\mathrm{R}})$ and
$y^{\prime}\in\mathcal{A}_{\mathrm{r}}(W_{\mathrm{R}})^{\prime}$ one has
$[{\hbox{\rm Ad\,}}\widetilde{R}_{\varphi}({\mathbbm{1}}\otimes
y),{\mathbbm{1}}\otimes y^{\prime}]=0.$
###### Proof.
The operator $\widetilde{R}_{\varphi}$ is disintegrated into second
quantization operators as we saw above. First we consider the first of the
commutators above. The operator $\widetilde{R}_{\varphi}$ restricts naturally
to $\mathcal{H}_{\mathrm{r}}\otimes{\mathcal{H}}^{\Sigma}$ and
$x\otimes{\mathbbm{1}}$ and $x^{\prime}\otimes{\mathbbm{1}}$ extend naturally
to $\mathcal{H}_{\mathrm{r}}\otimes{\mathcal{H}}^{\Sigma}$. The statement is
equivalent to
$[{\hbox{\rm
Ad\,}}\widetilde{R}_{\overline{\varphi}}(x\otimes{\mathbbm{1}}),x^{\prime}\otimes{\mathbbm{1}}]=0.$
It is easy to see that if $\varphi(z)$ is an inner symmetric function, then so
are $\overline{\varphi(1/\overline{z})}$ and hence
$\overline{\varphi(p_{k}/\overline{z})}$ for $p_{k}\geq 0$. Let us calculate
the commutation relation on
$\mathcal{H}_{\mathrm{r}}\otimes{\mathcal{H}}^{\Sigma}$.
${\hbox{\rm
Ad\,}}\widetilde{R}_{\overline{\varphi}}(x\otimes{\mathbbm{1}})=\bigoplus_{n}\int{\hbox{\rm
Ad\,}}\left(\prod_{k}\Gamma\left(\overline{\varphi}\left(\frac{p_{k}}{P_{1}}\right)\right)\right)(x)\otimes
dE_{1}(p_{1})\otimes\cdots\otimes dE_{1}(p_{n}).$
And this commutes with $x\otimes{\mathbbm{1}}$. Indeed, since
$x\in\mathcal{A}_{\mathrm{r}}(W_{\mathrm{R}})={\mathcal{A}}_{0}({\mathbb{R}}_{+})$
and
$x^{\prime}\in\mathcal{A}_{\mathrm{r}}(W_{\mathrm{R}})^{\prime}={\mathcal{A}}_{0}({\mathbb{R}}_{-})$,
it follows that ${\hbox{\rm
Ad\,}}\Gamma\left(\overline{\varphi}\left(\frac{p_{k}}{P_{1}}\right)\right)(x)\in{\mathcal{A}}_{0}({\mathbb{R}}_{+})$
for any $p_{k}\geq 0$ by the result of Longo and Witten [27] (see also the
beginning of this Section), and by the fact that the spectral support of
$E_{1}$ is positive. Then the equality naturally restricts to
$\mathcal{H}_{\mathrm{r}}\otimes\mathcal{H}_{\mathrm{r}}$.
The proof of the second commutator goes more simply. We only have to consider
the expression
${\hbox{\rm Ad\,}}\widetilde{R}_{\varphi}({\mathbbm{1}}\otimes
y)=\bigoplus_{m}\int dE_{1}(p_{1})\otimes\cdots\otimes
dE_{1}(p_{m})\otimes{\hbox{\rm
Ad\,}}\left(\prod_{j}\Gamma\left(\varphi\left(\frac{P_{1}}{p_{j}}\right)\right)\right)(y).$
The rest of the argument is parallel as above. ∎
Let $T_{\mathrm{r}}$ be the restriction of $U_{\mathrm{r}}$ to translation
(not to be confused with trace). Our Borchers triple is given as follows.
###### Theorem 4.2.
The triple
* •
$\widetilde{\mathcal{M}}_{\mathrm{r},\varphi}:=\\{x\otimes{\mathbbm{1}},{\hbox{\rm
Ad\,}}\widetilde{R}_{\varphi}({\mathbbm{1}}\otimes
y):x,y\in\mathcal{A}_{\mathrm{r}}(W_{\mathrm{R}})\\}^{\prime\prime}$
* •
$\widetilde{T}_{\mathrm{r}}=T_{\mathrm{r}}\otimes T_{\mathrm{r}}$
* •
$\widetilde{\Omega}_{\mathrm{r}}=\Omega_{\mathrm{r}}\otimes\Omega_{\mathrm{r}}$
is a Borchers triple.
###### Proof.
The conditions on $\widetilde{T}_{\mathrm{r}}$ and
$\widetilde{\Omega}_{\mathrm{r}}$ are readily satisfied since they are same as
the tensor product net. The operators $\widetilde{R}_{\varphi}$ and
$\widetilde{T}_{\mathrm{r}}$ commute since both are the functional calculus of
the same spectral measure (recall that
$T_{\mathrm{r}}(t_{+},t_{-})=U_{\mathrm{r}}^{+}(t_{+})U_{\mathrm{r}}^{-}(t_{-})$),
hence $\widetilde{T}_{\mathrm{r}}(t_{+},t_{-})$ sends
$\widetilde{\mathcal{M}}_{\mathrm{r},\varphi}$ into itself for
$(t_{+},t_{-})\in W_{\mathrm{R}}$. The vector
$\widetilde{\Omega}_{\mathrm{r}}$ is cyclic for
$\widetilde{\mathcal{M}}_{\mathrm{r},\varphi}$ since
$\widetilde{\mathcal{M}}_{\mathrm{r},\varphi}\widetilde{\Omega}_{\mathrm{r}}\supset\\{(x\otimes{\mathbbm{1}})\cdot\widetilde{R}_{\varphi}\cdot({\mathbbm{1}}\otimes
y)\cdot\widetilde{\Omega}_{\mathrm{r}}\\}=\\{(x\otimes{\mathbbm{1}})\cdot({\mathbbm{1}}\otimes
y)\cdot\widetilde{\Omega}_{\mathrm{r}}\\}$ and the latter is total by the
Reeh-Schlieder property of the tensor product net.
We see the separating property of $\widetilde{\Omega}_{\mathrm{r}}$ as
follows. Consider a von Neumann algebra
$\widetilde{\mathcal{M}}_{\mathrm{r},\varphi}^{1}:=\\{{\hbox{\rm
Ad\,}}\widetilde{R}_{\varphi}(x^{\prime}\otimes{\mathbbm{1}}),{\mathbbm{1}}\otimes
y^{\prime}:x^{\prime},y^{\prime}\in\mathcal{A}_{\mathrm{r}}(W_{\mathrm{R}})^{\prime}\\}^{\prime\prime}.$
One verifies that $\widetilde{\Omega}_{\mathrm{r}}$ is cyclic for
$\widetilde{\mathcal{M}}_{\mathrm{r},\varphi}^{1}$ as above, hence we only
have to show that $\widetilde{\mathcal{M}}_{\mathrm{r},\varphi}$ and
$\widetilde{\mathcal{M}}_{\mathrm{r},\varphi}^{1}$ commute. This has been done
by Lemma 4.1. ∎
One can actually show that
$\widetilde{\mathcal{M}}_{\mathrm{r},\varphi}^{1}=(\widetilde{\mathcal{M}}_{\mathrm{r},\varphi})^{\prime}$,
so this confusing notation is justified. Indeed, one has only to check the
modular group of $\widetilde{\mathcal{M}}_{\mathrm{r},\varphi}$ with respect
to $\widetilde{\Omega}_{\mathrm{r}}$ is the same for the tensor product, which
follows from the field representation in the next Section and the argument of
[23].
In this presentation we took $\mathcal{A}_{\mathrm{r}}$ as the starting point.
It is also possible to take the models in [22] or more general models with
more particle spectrum [25]. We will discuss this (slight) generalization in
[5]. In this paper we do not consider strict locality of these Borchers
triples, although the author expects that a similar proof to [22] should work.
## 5 General arguments for strict locality
We are now concerned with the main problem in the construction of nets through
Borchers triples: the strict locality of the models in Section 3. The key
argument is the wedge-split property.
In general, if one has an inclusion of von Neumann algebras
${\mathcal{N}}\subset{\mathcal{M}}$, this is said to be split if there is a
type I factor ${\mathcal{R}}$ such that
${\mathcal{N}}\subset{\mathcal{R}}\subset{\mathcal{M}}$. Furthermore, if
${\mathcal{N}}\subset{\mathcal{M}}$ is an inclusion of factors and $\Omega$ is
cyclic and separating for ${\mathcal{M}}$, then the nuclearity of the map
${\mathcal{N}}\ni x\longmapsto\Delta^{\frac{1}{4}}x\Omega$ implies the split
property, where $\Delta$ is the modular operator for ${\mathcal{M}}$ with
respect to $\Omega$ [11, Proposition 2.3].
We say that a Borchers triple $({\mathcal{M}},T,\Omega)$ is wedge-split if
${\hbox{\rm Ad\,}}T(a)({\mathcal{M}})\subset{\mathcal{M}}$ is split for any
$a\in W_{\mathrm{R}}$. Note that $W_{\mathrm{R}}$ is defined as the open wedge
and split inclusion for lightlike translation is not required. It is easy to
see already from wedge-split property that the intersection
${\mathcal{M}}\cap{\hbox{\rm Ad\,}}T(a)({\mathcal{M}})$ for $a\in
W_{\mathrm{R}}$ is nontrivial. Let $\Delta$ be the modular operator for
${\mathcal{M}}$ with respect to $\Omega$. One says that an inclusion
${\mathcal{N}}\subset{\mathcal{M}}$ satisfies modular nuclearity if the map
${\mathcal{N}}\ni x\longmapsto\Delta^{\frac{1}{4}}x\Omega$ is nuclear. A
Borchers triple $({\mathcal{M}},T,\Omega)$ is said to satisfy modular
nuclearity if the inclusion ${\hbox{\rm
Ad\,}}T({\mathcal{M}})\subset{\mathcal{M}}$ has modular nuclearity for any
$a\in W_{\mathrm{R}}$. From the above remark, modular nuclearity implies
wedge-split property. The strict locality of the models in [22] was proved
through modular nuclearity.
Actually, wedge-split inclusion is sufficient for the strict locality [22,
Theorem 2.5].
###### Theorem 5.1 (Lechner).
If a Borchers triple $({\mathcal{M}},T,\Omega)$ is wedge-split, then it is
strictly local.
This theorem is stated with the assumption of modular nuclearity, however, the
actual proof depends only on wedge-split property.
Let us recall our main construction strategy (Section 3): starting with a
given Borchers triple
$(\mathcal{M}_{\mathrm{c}},T_{\mathrm{c}},\Omega_{\mathrm{c}})$ with inner
symmetry ${\hbox{\rm Ad\,}}V_{\mathrm{c}}(\kappa)$ of $S^{1}$, we constructed
a new Borchers triple
$(\widetilde{\mathcal{M}}_{\mathrm{c},\kappa},\widetilde{T}_{\mathrm{c}},\widetilde{\Omega}_{\mathrm{c}})$
on the tensor product Hilbert space. As we will see later (Section 6.1), this
gives a nontrivial scattering even if the initial triple comes from the free
field. This can be generalized to the following program: take a Borchers
triple with a good property (either wedge-split property or modular
nuclearity) and prove that the construction of Section 3 leads again to
Borchers triples with the same property.
We carry out this program in two ways. First we present a proof through wedge-
split property and then we use modular nuclearity. Note that strict locality
is not “good enough” for this program. We will exhibit examples of strictly
local Borchers triples for which the new triples constructed as in Section 3
violate strict locality. Another remark is that our arguments in Section 5.2
through wedge-split property are valid for both $S^{1}$\- and
${\mathbb{Z}}_{N}$-actions, but those in Section 5.3 through modular
nuclearity apply so far only to finite cyclic group actions. Already for the
simplest compact group $S^{1}$ the proofs break down, as we will see.
### 5.1 Common arguments
Let us collect some facts which will be commonly used in the following (see
also [33, Section 4.4.1]). In this Subsection, we do not use the properties of
Borchers triple.
In the following, we consider only an action of $S^{1}$. The case of
${\mathbb{Z}}_{N}$ is analogous. Let $\mathcal{M}_{\mathrm{c}}$ be a von
Neumann algebra, $\Omega_{\mathrm{c}}$ a cyclic separating vector for
$\mathcal{M}_{\mathrm{c}}$ and $V_{\mathrm{c}}(\kappa)$ be a unitary
representation of $S^{1}$ which preserves $\Omega_{\mathrm{c}}$ such that
$\alpha_{\mathrm{c},\kappa}:={\hbox{\rm Ad\,}}V_{\mathrm{c}}(\kappa)$ is an
automorphism of $\mathcal{M}_{\mathrm{c}}$ and
$V_{\mathrm{c}}(1)={\mathbbm{1}}$. In other words, there is an action
$\alpha_{\mathrm{c}}$ of $S^{1}$ on $\mathcal{M}_{\mathrm{c}}$ which preserves
the state $\langle\Omega_{\mathrm{c}},\cdot\,\Omega_{\mathrm{c}}\rangle$.
#### Discrete Fourier expansion on von Neumann algebras
For any element $x\in\mathcal{M}_{\mathrm{c}}$ and $l\in{\mathbb{Z}}$, we
define
$x_{l}:=\int_{0}^{1}d\kappa\,{\hbox{\rm
Ad\,}}V_{\mathrm{c}}(\kappa)(x)e^{-i2\pi l\kappa},$
It holds that $x=\sum_{l\in{\mathbb{Z}}}x_{l}$ and ${\hbox{\rm
Ad\,}}V_{\mathrm{c}}(\kappa)(x_{l})=\alpha_{\mathrm{c},\kappa}(x_{l})=e^{i2\pi
l\kappa}x_{l}$. The sum is $*$-strongly convergent. Let us denote by
$\mathcal{M}_{\mathrm{c}}^{\alpha_{\mathrm{c}}}(l)$ the set of elements $x$ of
$\mathcal{M}_{\mathrm{c}}$ such that $\alpha_{\mathrm{c},\kappa}(x)=e^{i2\pi
l\kappa}x$. Then it holds that
$x_{l}\in\mathcal{M}_{\mathrm{c}}^{\alpha_{\mathrm{c}}}(l)$ and
$\mathcal{M}_{\mathrm{c}}^{\alpha_{\mathrm{c}}}(l_{1})\cdot\mathcal{M}_{\mathrm{c}}^{\alpha_{\mathrm{c}}}(l_{2})\subset\mathcal{M}_{\mathrm{c}}^{\alpha_{\mathrm{c}}}(l_{1}+l_{2})$.
These are well-known facts, however, if necessary the reader is referred to
[33, Proposition 4.9] for a proof (for the case of ${\mathbb{Z}}_{N}$, the
proof is easy to adapt).
Next we consider the twisting. As in Section 3, let $Q_{\mathrm{c}}$ be a
self-adjoint operator such that $V_{\mathrm{c}}(\kappa)=e^{i2\pi\kappa
Q_{\mathrm{c}}}$. Then for
$y\in\mathcal{M}_{\mathrm{c}}^{\alpha_{\mathrm{c}}}(m)$, it holds that
${\hbox{\rm Ad\,}}\widetilde{V}_{\mathrm{c},\kappa}({\mathbbm{1}}\otimes
y)={\hbox{\rm Ad\,}}e^{i2\pi\kappa Q_{\mathrm{c}}\otimes
Q_{\mathrm{c}}}({\mathbbm{1}}\otimes y)=e^{i2\pi m\kappa
Q_{\mathrm{c}}}\otimes y=V_{\mathrm{c}}(m\kappa)\otimes y,$
where $\widetilde{V}_{\mathrm{c},\kappa}=e^{i2\pi\kappa Q_{\mathrm{c}}\otimes
Q_{\mathrm{c}}}$ as in Section 3. For the proof, see [33, Lemma 4.10]. The
proof is written again for an action of $S^{1}$, however, the adaptation is
easy for ${\mathbb{Z}}_{N}$. Note that one has to consider spectral subspaces
of $V_{\mathrm{c}}$ parametrized by ${\mathbb{Z}}_{N}$. The identification
${\mathbb{Z}}_{N}={\mathbb{Z}}/N{\mathbb{Z}}$ should be always kept in mind
and numbers, e.g., $e^{i\frac{2\pi k}{N}}$ are well-defined for
$k\in{\mathbb{Z}}_{N}$.
Finally we remark that $\widetilde{\mathcal{M}}_{\mathrm{c},\kappa}$ is the
$*$-strong closure of the linear span of the elements of the form
$x_{l}V_{\mathrm{c}}(m\kappa)\otimes y_{m}$, where
$x_{l}\in\mathcal{M}_{\mathrm{c}}^{\alpha_{\mathrm{c}}}(l)$ and
$y_{m}\in\mathcal{M}_{\mathrm{c}}^{\alpha_{\mathrm{c}}}(m)$. Furthermore, any
element $\widetilde{x}\in\widetilde{\mathcal{M}}_{\mathrm{c},\kappa}$ can be
decomposed as follows:
$\widetilde{x}=\sum_{l,m\in{\mathbb{Z}}}\widetilde{x}_{l,m}(V_{\mathrm{c}}(m\kappa)\otimes{\mathbbm{1}}),\,\,\,\,\,\,\,\widetilde{x}_{l,m}\in\mathcal{M}_{\mathrm{c}}\otimes\mathcal{M}_{\mathrm{c}},$
and the sum is $*$-strongly convergent. This decomposition corresponds to the
discrete Fourier expansion with respect to the action $({\hbox{\rm
Ad\,}}V_{\mathrm{c}})\otimes({\hbox{\rm Ad\,}}V_{\mathrm{c}})$ of the group
$S^{1}\times S^{1}$. The group $S^{1}\times S^{1}$ acts also on
$\mathcal{M}_{\mathrm{c}}\otimes\mathcal{M}_{\mathrm{c}}$ and there is a
decomposition into components as well. For a corresponding proof, see again
[33, Lemma 4.14].
#### Modular operator and inner symmetry
Let $\Delta_{\mathrm{c}}$ and $J_{\mathrm{c}}$ be the modular operator and the
modular conjugation for $\mathcal{M}_{\mathrm{c}}$ with respect to
$\Omega_{\mathrm{c}}$. The implementing unitary $V_{\mathrm{c}}$ of
$\alpha_{\mathrm{c}}$ and $\Delta_{\mathrm{c}},J_{\mathrm{c}}$ commute. The
modular operator and the modular conjugation for
$\mathcal{M}_{\mathrm{c}}\otimes\mathcal{M}_{\mathrm{c}}$ with respect to
$\widetilde{\Omega}_{\mathrm{c}}:=\Omega_{\mathrm{c}}\otimes\Omega_{\mathrm{c}}$
are $\Delta_{\mathrm{c}}\otimes\Delta_{\mathrm{c}}$ and $J_{\mathrm{c}}\otimes
J_{\mathrm{c}}$, respectively. Now we can determine the modular objects for
$\widetilde{\mathcal{M}}_{\mathrm{c},\kappa}=\mathcal{M}_{\mathrm{c}}\otimes{\mathbbm{1}}\vee{\hbox{\rm
Ad\,}}\widetilde{V}_{\mathrm{c},\kappa}({\mathbbm{1}}\otimes\mathcal{M}_{\mathrm{c}})$.
###### Proposition 5.2.
The operators $\Delta_{\mathrm{c}}\otimes\Delta_{\mathrm{c}}$ is the modular
operator for $\widetilde{\mathcal{M}}_{\mathrm{c},k}$ with respect to
$\widetilde{\Omega}_{\mathrm{c}}$.
###### Proof.
We will give two proofs. The first one is a direct calculation and the second
one uses the so-called KMS condition.
A direct calculation goes as follows. We will see that actually the modular
conjugation is $\widetilde{V}_{\mathrm{c},\kappa}(J_{\mathrm{c}}\otimes
J_{\mathrm{c}})$. Since we have already the candidates for the modular
objects, we only have to check their actions. First we take
$x_{l}\in\mathcal{M}_{\mathrm{c}}^{\alpha_{\mathrm{c}}}(l)$ and
$y_{m}\in\mathcal{M}_{\mathrm{c}}^{\alpha_{\mathrm{c}}}(m)$. The operator
$x_{l}V_{\mathrm{c}}(m\kappa)\otimes y_{m}$ belongs to
$\widetilde{\mathcal{M}}_{\mathrm{c},\kappa}$ and it holds that
$\displaystyle\widetilde{V}_{\mathrm{c},\kappa}(J_{\mathrm{c}}\otimes
J_{\mathrm{c}})(\Delta_{\mathrm{c}}\otimes\Delta_{\mathrm{c}})(x_{l}V_{\mathrm{c}}(m\kappa)\otimes
y_{m})\widetilde{\Omega}_{\mathrm{c}}$ $\displaystyle=$
$\displaystyle\widetilde{V}_{\mathrm{c},\kappa}(S_{c}\otimes
S_{c})(x_{l}\Omega_{\mathrm{c}}\otimes y_{m}\Omega_{\mathrm{c}})$
$\displaystyle=$
$\displaystyle\widetilde{V}_{\mathrm{c},\kappa}(x_{l}^{*}\Omega_{\mathrm{c}}\otimes
y_{m}^{*}\Omega_{\mathrm{c}})$ $\displaystyle=$ $\displaystyle e^{i2\pi
lm\kappa}(x_{l}^{*}\Omega_{\mathrm{c}}\otimes y_{m}^{*}\Omega_{\mathrm{c}})$
$\displaystyle=$
$\displaystyle(\alpha_{\mathrm{c},-m\kappa}(x_{l}^{*})\Omega_{\mathrm{c}})\otimes(y_{m}^{*}\Omega_{\mathrm{c}})$
$\displaystyle=$
$\displaystyle(V_{\mathrm{c}}(-m\kappa)x_{l}^{*}\Omega_{\mathrm{c}})\otimes(y_{m}^{*}\Omega_{\mathrm{c}})$
$\displaystyle=$ $\displaystyle(x_{l}V_{\mathrm{c}}(m\kappa)\otimes
y_{m})^{*}(\Omega_{\mathrm{c}}\otimes\Omega_{\mathrm{c}}),$
namely, the operator $\widetilde{V}_{\mathrm{c},\kappa}(J_{\mathrm{c}}\otimes
J_{\mathrm{c}})(\Delta_{\mathrm{c}}\otimes\Delta_{\mathrm{c}})$ acts correctly
as the modular involution. Since $\widetilde{\mathcal{M}}_{\mathrm{c},\kappa}$
is the $*$-strong closure of the linear span of such elements, the modular
involution actually coincides with
$\widetilde{V}_{\mathrm{c},\kappa}(J_{\mathrm{c}}\otimes
J_{\mathrm{c}})(\Delta_{\mathrm{c}}\otimes\Delta_{\mathrm{c}})$. Then the
conclusion follows immediately from the uniqueness of the polar decomposition.
In the second proof, we use the uniqueness of the modular automorphism group
with respect to the KMS condition. In general, for a von Neumann algebra
${\mathcal{M}}$ and a state $\psi$, if there is a one-parameter automorphisms
$\sigma_{t}$ which satisfies the KMS condition, namely if for any pair
$x,y\in{\mathcal{M}}$ there is an analytic function $f(t)$ on the strip $0<\Im
t<1$ and continuous on the boundary such that $f(t)=\psi(\sigma_{t}(x)y)$ and
$f(t+i)=\psi(y\sigma_{t}(x))$ for $t\in{\mathbb{R}}$, then $\sigma_{t}$ is the
modular automorphism [32, Theorem VIII.1.2]. In our case, the state is the
vacuum
$\langle\widetilde{\Omega}_{\mathrm{c}},\cdot\,\widetilde{\Omega}_{\mathrm{c}}\rangle$.
This is the KMS state on the von Neumann algebra
$\mathcal{M}_{\mathrm{c}}\otimes\mathcal{M}_{\mathrm{c}}$ with respect to
$\sigma^{\widetilde{\Omega}_{\mathrm{c}}}_{t}={\hbox{\rm
Ad\,}}(\Delta_{\mathrm{c}}^{it}\otimes\Delta_{\mathrm{c}}^{it})$. Hence for
any pair
$\widetilde{x},\widetilde{y}\in\mathcal{M}_{\mathrm{c}}\otimes\mathcal{M}_{\mathrm{c}}$
of elements there is an analytic function as above. Actually the decomposition
with respect to $\alpha_{\mathrm{c}}\otimes\alpha_{\mathrm{c}}$ commutes with
$\sigma^{\widetilde{\Omega}_{\mathrm{c}}}$ (see the remark about discrete
Fourier expansion) and it holds that
$\displaystyle\langle\widetilde{\Omega}_{\mathrm{c}},\sigma^{\widetilde{\Omega}_{\mathrm{c}}}_{t}(\widetilde{x})\widetilde{y}\widetilde{\Omega}_{\mathrm{c}}\rangle$
$\displaystyle=$
$\displaystyle\sum_{l,m}\langle\widetilde{\Omega}_{\mathrm{c}},\sigma^{\widetilde{\Omega}_{\mathrm{c}}}_{t}(\widetilde{x}_{l,m})\widetilde{y}_{-l,-m}\widetilde{\Omega}_{\mathrm{c}}\rangle,$
$\displaystyle\langle\widetilde{\Omega}_{\mathrm{c}},\widetilde{y}\sigma^{\widetilde{\Omega}_{\mathrm{c}}}_{t}(\widetilde{x})\widetilde{\Omega}_{\mathrm{c}}\rangle$
$\displaystyle=$
$\displaystyle\sum_{l,m}\langle\widetilde{\Omega}_{\mathrm{c}},\widetilde{y}_{-l,-m}\sigma^{\widetilde{\Omega}_{\mathrm{c}}}_{t}(\widetilde{x}_{l,m})\widetilde{\Omega}_{\mathrm{c}}\rangle$
by the orthogonality of the vacuum acted on by the component
$\widetilde{x}_{l,m}$ etc. Next, by considering the pair
$\widetilde{x}_{l,m},\widetilde{y}_{-l,-m}$ and the KMS condition for
$\mathcal{M}_{\mathrm{c}}\otimes\mathcal{M}_{\mathrm{c}}$ there is an analytic
function
$\widetilde{f}_{l,m}(t)=\langle\widetilde{\Omega}_{\mathrm{c}},\sigma^{\widetilde{\Omega}_{\mathrm{c}}}_{t}(\widetilde{x}_{l,m})\widetilde{y}_{-l,-m}\widetilde{\Omega}_{\mathrm{c}}\rangle$,
$\widetilde{f}_{l,m}(t+i)=\langle\widetilde{\Omega}_{\mathrm{c}},\widetilde{y}_{-l,-m}\sigma^{\widetilde{\Omega}_{\mathrm{c}}}_{t}(\widetilde{x}_{l,m})\widetilde{\Omega}_{\mathrm{c}}\rangle$
(of course the dependence of $\widetilde{f}_{l,m}$ on $\widetilde{x}$ and
$\widetilde{y}$ is implicit). Let us turn to
$\widetilde{\mathcal{M}}_{\mathrm{c},\kappa}$. By the same argument as above,
an inner product decomposes into suitable combination and the decomposition is
compatible with the action of
$\sigma^{\widetilde{\Omega}_{\mathrm{c}}}_{t}={\hbox{\rm
Ad\,}}(\Delta_{\mathrm{c}}^{it}\otimes\Delta_{\mathrm{c}}^{it})$. For a pair
of elements $\widetilde{x}_{l,m}(e^{i2\pi m\kappa
Q_{\mathrm{c}}}\otimes{\mathbbm{1}})$ and $\widetilde{y}_{-l,-m}(e^{-i2\pi
m\kappa Q_{\mathrm{c}}}\otimes{\mathbbm{1}})$, where
$\widetilde{x}_{l,m},\widetilde{y}_{-l-m}\in\mathcal{M}_{\mathrm{c}}\otimes\mathcal{M}_{\mathrm{c}}$,
we have
$\displaystyle\langle\widetilde{\Omega}_{\mathrm{c}},\sigma^{\widetilde{\Omega}_{\mathrm{c}}}_{t}(\widetilde{x}_{l,m}(e^{i2\pi
m\kappa Q_{\mathrm{c}}}\otimes{\mathbbm{1}}))\widetilde{y}_{-l,-m}(e^{-i2\pi
m\kappa
Q_{\mathrm{c}}}\otimes{\mathbbm{1}})\widetilde{\Omega}_{\mathrm{c}}\rangle$
$\displaystyle=$ $\displaystyle e^{-i2\pi
lm\kappa}\langle\widetilde{\Omega}_{\mathrm{c}},\sigma^{\widetilde{\Omega}_{\mathrm{c}}}_{t}(\widetilde{x}_{l,m})\widetilde{y}_{-l,-m}\widetilde{\Omega}_{\mathrm{c}}\rangle$
$\displaystyle\langle\widetilde{\Omega}_{\mathrm{c}},\widetilde{y}_{-l,-m}(e^{-i2\pi
m\kappa
Q_{\mathrm{c}}}\otimes{\mathbbm{1}})\sigma^{\widetilde{\Omega}_{\mathrm{c}}}_{t}(\widetilde{x}_{l,m}(e^{i2\pi
m\kappa
Q_{\mathrm{c}}}\otimes{\mathbbm{1}}))\widetilde{\Omega}_{\mathrm{c}}\rangle$
$\displaystyle=$ $\displaystyle e^{-i2\pi
lm\kappa}\langle\widetilde{\Omega}_{\mathrm{c}},\widetilde{y}_{-l,-m}\sigma^{\widetilde{\Omega}_{\mathrm{c}}}_{t}(\widetilde{x}_{l,m})\widetilde{\Omega}_{\mathrm{c}}\rangle$
One observes that the right hand sides are equal to $\widetilde{f}_{l,m}$ up
to the constant $e^{-i2\pi lm\kappa}$. In other words, the KMS condition is
satisfied for the pair with respect to
$\sigma^{\widetilde{\Omega}_{\mathrm{c}}}$. Therefore for an arbitrary linear
combination of such components the KMS condition holds as well, thanks to the
decomposition of the inner product into $(l,m)$-components. Such linear
combinations is $*$-strongly dense in
$\widetilde{\mathcal{M}}_{\mathrm{c},\kappa}$, hence the above-cited
uniqueness theorem applies to see that
$\sigma^{\widetilde{\Omega}_{\mathrm{c}}}_{t}={\hbox{\rm
Ad\,}}(\Delta_{\mathrm{c}}^{it}\otimes\Delta_{\mathrm{c}}^{it})$ is the
modular automorphism of $\widetilde{\mathcal{M}}_{\mathrm{c},\kappa}$ with
respect to
$\langle\widetilde{\Omega}_{\mathrm{c}},\cdot\,\widetilde{\Omega}_{\mathrm{c}}\rangle$
(indeed, the KMS condition on a $*$-strongly dense subalgebra is enough by [9,
Proposition 5.7]. One should note that in [9] the KMS condition is defined on
a dense set of analytic elements). Since
$\Delta_{\mathrm{c}}\otimes\Delta_{\mathrm{c}}$ preserves the vacuum vector
$\Omega_{\mathrm{c}}\otimes\Omega_{\mathrm{c}}$, it must coincide with the
modular operator. ∎
It is interesting to compare our proof with [12, 22] where the modular objects
were calculated through unbounded operators affiliated to the von Neumann
algebras. One recalls also that in our previous work the modular objects were
indirectly determined by scattering theory [33, Section 3].
###### Proposition 5.3.
It holds that
$(\widetilde{\mathcal{M}}_{\mathrm{c},\kappa})^{\prime}=\left({\hbox{\rm
Ad\,}}\widetilde{V}_{\mathrm{c},k}(\mathcal{M}_{\mathrm{c}}^{\prime}\otimes{\mathbbm{1}})\vee{\mathbbm{1}}\otimes\mathcal{M}_{\mathrm{c}}^{\prime}\right)=:\widetilde{\mathcal{M}}_{\mathrm{c},\kappa}^{1}$.
###### Proof.
It can be observed that the right hand side commute with
$\widetilde{\mathcal{M}}_{\mathrm{c},\kappa}$ by the same argument as in
Theorem 3.3, so we have
$\widetilde{\mathcal{M}}_{\mathrm{c},\kappa}^{1}\subset(\widetilde{\mathcal{M}}_{\mathrm{c},\kappa})^{\prime}$.
Since we already know that the modular operator for
$(\widetilde{\mathcal{M}}_{\mathrm{c},\kappa})^{\prime}$ with respect to
$\Omega_{\mathrm{c}}\otimes\Omega_{\mathrm{c}}$ is
$\Delta_{\mathrm{c}}^{-1}\otimes\Delta_{\mathrm{c}}^{-1}$ from Proposition
5.2, it is immediate to see that
$\widetilde{\mathcal{M}}_{\mathrm{c},\kappa}^{1}$ is globally invariant under
the modular group of $(\widetilde{\mathcal{M}}_{\mathrm{c},\kappa})^{\prime}$
with respect to $\Omega_{\mathrm{c}}\otimes\Omega_{\mathrm{c}}$. Then the two
algebras coincide since $\Omega_{\mathrm{c}}\otimes\Omega_{\mathrm{c}}$ is
cyclic for $\widetilde{\mathcal{M}}_{\mathrm{c},\kappa}^{1}$ by a standard
application of Takesaki’s theorem [32, Theorem IX.4.2] (see [34, Theorem
A.1]). ∎
Variations of Propositions 5.2 and 5.3 hold for an action of
${\mathbb{Z}}_{N}$ with trivial changes.
### 5.2 Proof through wedge-split property
Here we take as the starting point a Borchers triple
$(\mathcal{M}_{\mathrm{c}},T_{\mathrm{c}},\Omega_{\mathrm{c}})$ which is
wedge-split. As before, we assume that there is an action of $S^{1}$
implemented by $V_{\mathrm{c}}(\kappa)$. With this action and for
$\kappa\in{\mathbb{R}}$ we can construct a Borchers triple
$(\widetilde{\mathcal{M}}_{\mathrm{c},\kappa},\widetilde{T}_{\mathrm{c}},\widetilde{\Omega}_{\mathrm{c}})$
as in Section 3. We are going to prove that this is again wedge-split, hence
strictly local.
Recall that [15] for a split inclusion
$(\mathcal{N}_{\mathrm{c}}\subset\mathcal{M}_{\mathrm{c}},\Omega_{\mathrm{c}})$
equipped with a cyclic separating vector $\Omega_{\mathrm{c}}$ for
$\mathcal{M}_{\mathrm{c}},\mathcal{N}_{\mathrm{c}}$ and
$\mathcal{M}_{\mathrm{c}}\cap\mathcal{N}_{\mathrm{c}}$ (where
$\mathcal{N}_{\mathrm{c}}$ will be ${\hbox{\rm
Ad\,}}T_{\mathrm{c}}(a)(\mathcal{M}_{\mathrm{c}})$ for some $a\in
W_{\mathrm{R}}$), there is a canonical type I factor
$\mathcal{R}_{\mathrm{c}}$ such that
$\mathcal{N}_{\mathrm{c}}\subset\mathcal{R}_{\mathrm{c}}\subset\mathcal{M}_{\mathrm{c}}$.
Moreover, $\mathcal{R}_{\mathrm{c}}$ is given by the formula
$\mathcal{R}_{\mathrm{c}}=\mathcal{N}_{\mathrm{c}}\vee
J\mathcal{N}_{\mathrm{c}}J=\mathcal{M}_{\mathrm{c}}\cap
J\mathcal{M}_{\mathrm{c}}J$, where $J$ is the modular conjugation for
$\mathcal{M}_{\mathrm{c}}\cap\mathcal{N}_{\mathrm{c}}$ with respect to
$\Omega_{\mathrm{c}}$. If $V_{\mathrm{c}}$ is a unitary operator which
preserves $\Omega_{\mathrm{c}}$ and ${\hbox{\rm Ad\,}}V_{\mathrm{c}}$
preserves both $\mathcal{N}_{\mathrm{c}}$ and $\mathcal{M}_{\mathrm{c}}$, then
$\mathcal{R}_{\mathrm{c}}$ is preserved under ${\hbox{\rm
Ad\,}}V_{\mathrm{c}}$ as well.
###### Lemma 5.4.
The von Neumann algebra
$\widetilde{\mathcal{R}}_{\mathrm{c},\kappa}:=\mathcal{R}_{\mathrm{c}}\otimes{\mathbbm{1}}\vee{\hbox{\rm
Ad\,}}\widetilde{V}_{\mathrm{c},\kappa}({\mathbbm{1}}\otimes\mathcal{R}_{\mathrm{c}})$
is a factor.
###### Proof.
Since we know that
$(\widetilde{\mathcal{R}}_{\mathrm{c},\kappa})^{\prime}={\hbox{\rm
Ad\,}}\widetilde{V}_{\mathrm{c},\kappa}(\mathcal{R}_{\mathrm{c}}^{\prime}\otimes{\mathbbm{1}})\vee{\mathbbm{1}}\otimes\mathcal{R}_{\mathrm{c}}^{\prime}$
by Proposition 5.3 (in Section 5.1 we did not assume that the von Neumann
algebra comes from a Borchers triple), we only have to show that
$\widetilde{\mathcal{R}}_{\mathrm{c},\kappa}\vee(\widetilde{\mathcal{R}}_{\mathrm{c},k})^{\prime}=\mathcal{R}_{\mathrm{c}}\otimes\mathcal{R}_{\mathrm{c}}^{\prime}\vee{\hbox{\rm
Ad\,}}\widetilde{V}_{\mathrm{c},\kappa}(\mathcal{R}_{\mathrm{c}}^{\prime}\otimes\mathcal{R}_{\mathrm{c}})$
is equal to $B(\mathcal{H}_{\mathrm{c}}\otimes\mathcal{H}_{\mathrm{c}})$.
We show that $V_{\mathrm{c}}(\kappa)\otimes{\mathbbm{1}}$ and
${\mathbbm{1}}\otimes V_{\mathrm{c}}(\kappa)$ are contained in
$\widetilde{\mathcal{R}}_{\mathrm{c},\kappa}\vee(\widetilde{\mathcal{R}}_{\mathrm{c},\kappa})^{\prime}$
for each $\kappa\in{\mathbb{R}}$. Indeed,
$V_{\mathrm{c}}(\kappa)\otimes{\mathbbm{1}}$ implements an automorphism on
$\mathcal{R}_{\mathrm{c}}\otimes{\mathbbm{1}}$ and since the latter is a type
I factor, there is a unitary $u\in\mathcal{R}_{\mathrm{c}}$ which implements
the same automorphism. At the same time
$V_{\mathrm{c}}(\kappa)\otimes{\mathbbm{1}}$ implements an automorphism of
$\mathcal{R}_{\mathrm{c}}^{\prime}\otimes{\mathbbm{1}}$, and there is an
implementing unitary $u^{\prime}\in\mathcal{R}_{\mathrm{c}}^{\prime}$. Then
$uu^{\prime}$ implements ${\hbox{\rm Ad\,}}V_{\mathrm{c}}(\kappa)$ on
${\mathcal{B}}(\mathcal{H}_{\mathrm{c}})=\mathcal{R}_{\mathrm{c}}\vee\mathcal{R}_{\mathrm{c}}^{\prime}$,
hence $uu^{\prime}V_{\mathrm{c}}(\kappa)^{*}$ must be a scalar. We may assume
$uu^{\prime}=V_{\mathrm{c}}(\kappa)$. Furthermore, obviously ${\hbox{\rm
Ad\,}}V_{\mathrm{c}}(\kappa)(u^{\prime})={\hbox{\rm
Ad\,}}u^{\prime}(u^{\prime})=u^{\prime}$, namely $u^{\prime}$ is fixed under
the automorphism ${\hbox{\rm Ad\,}}V_{\mathrm{c}}(\kappa)$, hence ${\hbox{\rm
Ad\,}}\widetilde{V}_{\mathrm{c},\kappa}(u^{\prime}\otimes{\mathbbm{1}})=u^{\prime}\otimes{\mathbbm{1}}$.
This implies that
$u^{\prime}\otimes{\mathbbm{1}}\in(\widetilde{\mathcal{R}}_{\mathrm{c},\kappa})^{\prime}$
hence
$uu^{\prime}\otimes{\mathbbm{1}}=V_{\mathrm{c}}(\kappa)\otimes{\mathbbm{1}}$
is in
$\widetilde{\mathcal{R}}_{\mathrm{c},\kappa}\vee(\widetilde{\mathcal{R}}_{\mathrm{c},\kappa})^{\prime}$.
An analogous proof works for ${\mathbbm{1}}\otimes V_{\mathrm{c}}(\kappa)$.
The rest is easy since $\widetilde{V}_{\mathrm{c},\kappa}$ is obtained from
the functional calculus of $V_{\mathrm{c}}(\kappa)\otimes{\mathbbm{1}}$ and
${\mathbbm{1}}\otimes V_{\mathrm{c}}(\kappa)$. ∎
This Lemma actually works even for an action of ${\mathbb{R}}$. In contrast,
we need the periodicity of the action in the following.
###### Theorem 5.5.
The von Neumann algebra
$\widetilde{\mathcal{R}}_{\mathrm{c},\kappa}:=\mathcal{R}_{\mathrm{c}}\otimes{\mathbbm{1}}\vee{\hbox{\rm
Ad\,}}\widetilde{V}_{\mathrm{c},\kappa}({\mathbbm{1}}\otimes\mathcal{R}_{\mathrm{c}})$
is a type I factor.
###### Proof.
We have seen that $\widetilde{\mathcal{R}}_{\mathrm{c},\kappa}$ is a factor.
What remains is to show that $\widetilde{\mathcal{R}}_{\mathrm{c},\kappa}$
contains a minimal projection. As in Lemma 5.4 we take implementing unitary
$u(\kappa)\in\mathcal{R}_{\mathrm{c}}$ for ${\hbox{\rm
Ad\,}}V_{\mathrm{c}}(\kappa)$, this time indicating the dependence on
$\kappa$. By a classical result by Bargmann, we may assume that $u(\kappa)$ is
a one-parameter group of unitaries [2, Section 6a], where our group is $S^{1}$
hence there is no nontrivial antisymmetric bilinear form on its Lie algebra
${\mathbb{R}}$.
Thus the spectrum of the representation $u$ is discrete and contained in
${\mathbb{Z}}$, hence the fixed point
$\mathcal{R}_{\mathrm{c}}^{\alpha_{\mathrm{c}}}=\mathcal{R}_{\mathrm{c}}\cap\\{u(\kappa):\kappa\in{\mathbb{R}}\\}^{\prime}$
is the relative commutant in $\mathcal{R}_{\mathrm{c}}$ of the spectral
projections of the representation $u$. Hence it is the direct sum of type I
factors and admits a minimal projection.
Let us take such a minimal projection $p$ in
$\mathcal{R}_{\mathrm{c}}^{\alpha_{\mathrm{c}}}$. In general, if
$y\in\mathcal{R}_{\mathrm{c}}^{\alpha_{\mathrm{c}}}$, then ${\hbox{\rm
Ad\,}}\widetilde{V}_{\mathrm{c},\kappa}({\mathbbm{1}}\otimes
y)={\mathbbm{1}}\otimes y$ and hence the inclusion
$\mathcal{R}_{\mathrm{c}}\otimes\mathcal{R}_{\mathrm{c}}^{\alpha_{\mathrm{c}}}\subset\widetilde{\mathcal{R}}_{\mathrm{c},\kappa}$
holds. Obviously $p$ must be a subprojection of a discrete point spectrum of
$u$. Furthermore, it is immediate that $p\otimes p$ is a minimal projection in
$\mathcal{R}_{\mathrm{c}}\otimes\mathcal{R}_{\mathrm{c}}^{\alpha_{\mathrm{c}}}$.
We claim that it is a minimal projection in
$\widetilde{\mathcal{R}}_{\mathrm{c},\kappa}$. In fact, suppose that
$\widetilde{q}\leq p\otimes p$ in
$\widetilde{\mathcal{R}}_{\mathrm{c},\kappa}$. Then $\widetilde{q}$ and
${\mathbbm{1}}\otimes u(\kappa)$ commute since the spectral projections of the
reprensentation ${\mathbbm{1}}\otimes u$ either contains or disjoint from
$p\otimes p$, hence so does it hold with $\widetilde{q}$. This implies that
$\widetilde{q}$ belongs to
$\mathcal{R}_{\mathrm{c}}\otimes\mathcal{R}_{\mathrm{c}}^{\alpha_{\mathrm{c}}}$
(consider the discrete Fourier expansion
$\widetilde{q}=\sum\widetilde{q}_{lm}(V_{\mathrm{c}}(m\kappa)\otimes{\mathbbm{1}})$,
then it is possible that ${\hbox{\rm Ad\,}}(1\otimes
u(\kappa))(\widetilde{q})=\widetilde{q}$ only if $\widetilde{q}_{lm}$ vanish
except $m=0$), then $\widetilde{q}=p\otimes p$ by the minimality of $p\otimes
p$ in this restricted algebra. This is the minimality of $p\otimes p$ in
$\widetilde{\mathcal{R}}_{\mathrm{c},\kappa}$. ∎
We remark that the intermediate type I factor constructed here is not the
canonical one of [15]. An explicit formula for it involves the modular
conjugation of the relative commutant, which is only abstractly determined.
The proof of Theorem 5.5 can be easily adapted to the case of an action of
${\mathbb{Z}}_{N}$.
In terms of wedge-split Borchers triple, we apply Theorem 5.5 with
$\mathcal{N}_{\mathrm{c}}={\hbox{\rm
Ad\,}}T_{\mathrm{c}}(a)(\mathcal{M}_{\mathrm{c}})$ to see the split inclusion
$\mathcal{N}_{\mathrm{c}}\otimes{\mathbbm{1}}\vee{\hbox{\rm
Ad\,}}\widetilde{V}_{\mathrm{c},\kappa}({\mathbbm{1}}\otimes\mathcal{N}_{\mathrm{c}})\subset\mathcal{R}_{\mathrm{c}}\otimes{\mathbbm{1}}\vee{\hbox{\rm
Ad\,}}\widetilde{V}_{\mathrm{c},\kappa}({\mathbbm{1}}\otimes\mathcal{R}_{\mathrm{c}})\subset\mathcal{M}_{\mathrm{c}}\otimes{\mathbbm{1}}\vee{\hbox{\rm
Ad\,}}\widetilde{V}_{\mathrm{c},\kappa}({\mathbbm{1}}\otimes\mathcal{M}_{\mathrm{c}}),$
where the middle element is $\widetilde{\mathcal{R}}_{\mathrm{c},\kappa}$ and
the last element is $\widetilde{\mathcal{M}}_{\mathrm{c},\kappa}$. Since
$\widetilde{T}_{\mathrm{c}}$ and $\widetilde{V}_{\mathrm{c},\kappa}$ commute,
${\hbox{\rm
Ad\,}}\widetilde{T}_{\mathrm{c}}(a)(\widetilde{\mathcal{M}}_{\mathrm{c},\kappa})\subset\widetilde{\mathcal{M}}_{\mathrm{c},\kappa}$
is split with an intermediate type I factor
$\widetilde{\mathcal{R}}_{\mathrm{c},\kappa}$ (which implicitly depends on
$a$). Then we have the following with the help of Theorem 5.1.
###### Theorem 5.6.
If a given Borchers triple
$(\mathcal{M}_{\mathrm{c}},T_{\mathrm{c}},\Omega_{\mathrm{c}})$ with an action
of $S^{1}$ by inner symmetry is wedge-split, then the Borchers triple
$(\widetilde{\mathcal{M}}_{\mathrm{c},\kappa},\widetilde{T}_{\mathrm{c}},\widetilde{\Omega}_{\mathrm{c}})$
is wedge-split, hence strictly local.
One can repeat parallel proofs for an action of ${\mathbb{Z}}_{N}$ to obtain
the same result.
### 5.3 Proof through modular nuclearity
Let us give another proof of strict locality, based on modular nuclearity.
Here we have to restrict ourselves to the case of an action of
${\mathbb{Z}}_{N}$. Of course one can take an arbitrary finite subgroup
${\mathbb{Z}}_{N}$ in $S^{1}$, hence physically this should not be considered
as an essential restriction. Let
$(\widetilde{\mathcal{M}}_{\mathrm{c},k},\widetilde{T}_{\mathrm{c}},\widetilde{\Omega}_{\mathrm{c}})$
again be constructed as in Section 3 for a fixed $k\in{\mathbb{Z}}_{N}$.
Again, thanks to Theorem 5.1, it is enough to prove the modular nuclearity of
the new triple.
Let us start with a trivial observation.
###### Lemma 5.7.
Let $(\mathcal{M}_{\mathrm{c}},T_{\mathrm{c}},\Omega_{\mathrm{c}})$ be a
Borchers triple with modular nuclearity. Then the triple
$(\mathcal{M}_{\mathrm{c}}\otimes\mathcal{M}_{\mathrm{c}},\widetilde{T}_{\mathrm{c}},\widetilde{\Omega}_{\mathrm{c}})$
has modular nuclearity.
###### Proof.
The modular objects are the tensor products as well, hence the nuclearity norm
simply gets squared. ∎
###### Theorem 5.8.
Let $(\mathcal{M}_{\mathrm{c}},T_{\mathrm{c}},\Omega_{\mathrm{c}})$ be a
Borchers triple with modular nuclearity. Then the triple
$(\widetilde{\mathcal{M}}_{\mathrm{c},k},\widetilde{T}_{\mathrm{c}},\widetilde{\Omega}_{\mathrm{c}})$
has modular nuclearity.
###### Proof.
For a fixed $a\in W_{\mathrm{R}}$, we have to show that the inclusion
${\hbox{\rm
Ad\,}}\widetilde{T}_{\mathrm{c}}(a)(\widetilde{\mathcal{M}}_{\mathrm{c},k})\subset\widetilde{\mathcal{M}}_{\mathrm{c},k}$
has modular nuclearity with respect to $\widetilde{\Omega}_{\mathrm{c}}$. By
assumption and Lemma 5.7 we know that $({\hbox{\rm
Ad\,}}\widetilde{T}_{\mathrm{c}}(a)(\mathcal{M}_{\mathrm{c}}\otimes\mathcal{M}_{\mathrm{c}})\subset\mathcal{M}_{\mathrm{c}}\otimes\mathcal{M}_{\mathrm{c}})$
has modular nuclearity. This means that the map
$\mathcal{M}_{\mathrm{c}}\otimes\mathcal{M}_{\mathrm{c}}\ni\widetilde{x}\longmapsto\left(\Delta_{\mathrm{c}}^{\frac{1}{4}}\otimes\Delta_{\mathrm{c}}^{\frac{1}{4}}\right)\widetilde{x}(\Omega_{\mathrm{c}}\otimes\Omega_{\mathrm{c}})$
is a nuclear map.
Let us consider an element $\widetilde{x}$ of
$\mathcal{M}_{\mathrm{c}}\otimes\mathcal{M}_{\mathrm{c}}$. As we have seen in
the end of Section 5.1, it can be decomposed as
$\widetilde{x}=\sum_{l,m}\widetilde{x}_{l,m}$, and $\widetilde{x}_{l,m}$ are
the matrix components in
$\mathcal{M}_{\mathrm{c}}\otimes\mathcal{M}_{\mathrm{c}}$. For such
$\widetilde{x}$, we define an element
$\tau_{k}(\widetilde{x})\in\widetilde{\mathcal{M}}_{\mathrm{c},k}$ as follows:
$\tau_{k}(\widetilde{x})=\sum_{l,m}\widetilde{x}_{l,m}(V_{\mathrm{c}}^{km}\otimes{\mathbbm{1}}).$
This is a finite sum, thus there is no problem of convergence. This map
$\tau_{k}$ is onto, as any element in $\widetilde{\mathcal{M}}_{\mathrm{c},k}$
in decomposed in the form above. It is important to observe that the action on
the vector $\widetilde{\Omega}_{\mathrm{c}}$ is unchanged. In other words, it
holds that
$\widetilde{x}\widetilde{\Omega}_{\mathrm{c}}=\tau_{k}(\widetilde{x})\widetilde{\Omega}_{\mathrm{c}}$.
Now, we know that the modular operator of
$\widetilde{\mathcal{M}}_{\mathrm{c},k}$ with respect to
$\widetilde{\Omega}_{\mathrm{c}}$ is
$\Delta_{\mathrm{c}}\otimes\Delta_{\mathrm{c}}$, hence the map in question is
$\tau_{k}(\widetilde{x})\longmapsto(\Delta_{\mathrm{c}}^{\frac{1}{4}}\otimes\Delta_{\mathrm{c}}^{\frac{1}{4}})\tau_{k}(\widetilde{x})\widetilde{\Omega}_{\mathrm{c}}=(\Delta_{\mathrm{c}}^{\frac{1}{4}}\otimes\Delta_{\mathrm{c}}^{\frac{1}{4}})\widetilde{x}\widetilde{\Omega}_{\mathrm{c}}$.
By assumption and Lemma 5.7 we know that the map
$\widetilde{x}\longmapsto(\Delta_{\mathrm{c}}^{\frac{1}{4}}\otimes\Delta_{\mathrm{c}}^{\frac{1}{4}})\tau_{k}(\widetilde{x})\widetilde{\Omega}_{\mathrm{c}}$
is nuclear.
${\widetilde{x}}$${\widetilde{x}\widetilde{\Omega}_{\mathrm{c}}=\tau_{k}(\widetilde{x})\widetilde{\Omega}_{\mathrm{c}}}$${(\Delta_{\mathrm{c}}^{\frac{1}{4}}\otimes\Delta_{\mathrm{c}}^{\frac{1}{4}})\widetilde{x}\widetilde{\Omega}_{\mathrm{c}}}$${\tau_{k}(\widetilde{x})}$
In the diagram above, the left triangle commutes and the straight line above
is nuclear. Since a composition of a bounded linear map and a nuclear map is
again nuclear, we only have to show that $\tau_{k}^{-1}$ is bounded. For this
purpose, let us recall how explicitly $\\{\widetilde{x}_{l,m}\\}$ are obtained
(see [33, Proposition 4.9] for a corresponding formula for $S^{1}$-action):
$\widetilde{x}_{l,m}=\frac{1}{N^{2}}\sum_{j_{1},j_{2}}e^{-i\frac{2\pi(j_{1}l+j_{2}m)}{N}}{\hbox{\rm
Ad\,}}(V_{\mathrm{c}}^{j_{1}}\otimes V_{\mathrm{c}}^{j_{2}})(\widetilde{x}).$
Correspondingly for the decomposition of $\tau_{k}(\widetilde{x})$, we have
$\widetilde{x}_{l,m}(V_{\mathrm{c}}^{km}\otimes{\mathbbm{1}})=\frac{1}{N^{2}}\sum_{j_{1},j_{2}}e^{-i\frac{2\pi(j_{1}l+j_{2}m)}{N}}{\hbox{\rm
Ad\,}}(V_{\mathrm{c}}^{j_{1}}\otimes
V_{\mathrm{c}}^{j_{2}})(\tau_{k}(\widetilde{x})).$
In particular, we see $\|\widetilde{x}_{l,m}\|\leq\|\tau_{k}(\widetilde{x})\|$
and hence $\|\widetilde{x}\|=\|\sum_{l,m}\widetilde{x}_{l,m}\|\leq
N^{2}\|\tau_{k}(\widetilde{x})\|$. In other words, $\tau_{k}^{-1}$ is bounded
by $N^{2}$. ∎
Note that here the finiteness of $N$ is crucial. The author does not know if
the same holds for the action of e.g. $S^{1}$, although one can always take an
arbitrary finite subgroup ${\mathbb{Z}}_{N}$ of $S^{1}$.
### 5.4 Counterexamples from massless case
In previous Sections we started with a strictly local Borchers triple,
constructed new triples and proved strict locality. Although our method may
seem quite simple, it is neither trivial nor purely group-theoretic. In order
to see this, we take the massless models, for which one can construct Borchers
triples but strict locality fails. See also [33, Section 4.4].
Let $({\mathcal{A}}_{0},T_{0},\Omega_{0})$ be any strongly additive conformal
(diffeomorphism covariant) net with an action of ${\mathbb{Z}}_{N}$ by inner
symmetry (there are many such nets, e.g. the tensor product of the ${\rm
U}(1)$-current or the loop group nets). A two-dimensional massless net can be
constructed by tensor product: ${\mathcal{A}}(I_{+}\times
I_{-}):={\mathcal{A}}_{0}(I_{+})\otimes{\mathcal{A}}_{0}(I_{-}),T(t_{+},t_{-}):=T_{0}(t_{+})\otimes
T_{0}(t_{-}),\Omega:=\Omega_{0}\otimes\Omega_{0}$. On the net ${\mathcal{A}}$
there is an action of ${\mathbb{Z}}_{N}$ by inner symmetry which acts only on
the left component of
${\mathcal{A}}_{0}(I_{+})\otimes{\mathcal{A}}_{0}(I_{-})$. Let $Q$ be the
generator with ${\rm sp}\,Q\subset{\mathbb{Z}}$, which is of the form
$Q_{0}\otimes{\mathbbm{1}}$. With the wedge algebra
${\mathcal{M}}:={\mathcal{A}}_{0}({\mathbb{R}}_{-})\otimes{\mathcal{A}}_{0}({\mathbb{R}}_{+})$,
the twisted Borchers triple is given by
$\displaystyle\widetilde{\mathcal{M}}_{\mathrm{c},k}$ $\displaystyle:=$
$\displaystyle{\mathcal{M}}\otimes{\mathbbm{1}}\vee{\hbox{\rm
Ad\,}}e^{i\frac{2\pi k}{N}Q\otimes Q}({\mathbbm{1}}\otimes{\mathcal{M}})$
$\displaystyle=$
$\displaystyle{\mathcal{A}}_{0}({\mathbb{R}}_{-})\otimes{\mathcal{A}}_{0}({\mathbb{R}}_{+})\otimes{\mathbbm{1}}\vee{\hbox{\rm
Ad\,}}e^{i\frac{2\pi k}{N}Q\otimes Q}({\mathbbm{1}}\otimes{\mathcal{M}})$
$\displaystyle\cong$
$\displaystyle\left({\mathcal{A}}_{0}({\mathbb{R}}_{-})\otimes{\mathbbm{1}}\vee{\hbox{\rm
Ad\,}}e^{i\frac{2\pi k}{N}Q_{0}\otimes
Q_{0}}({\mathbbm{1}}\otimes{\mathcal{A}}_{0}({\mathbb{R}}_{-}))\right)\otimes\left({\mathcal{A}}_{0}({\mathbb{R}}_{+})\otimes{\mathcal{A}}_{0}({\mathbb{R}}_{+})\right),$
where in the last line we interchanged the second and the third components for
brevity. We consider the relative commutant of the wedge shifted by
$a=(a_{+},a_{-}),a_{+}<0,a_{-}>0$. With the above interchanged notation, it is
clear that the $-$-component (the 3rd and 4th tensor components) of the
intersection is simply
${\mathcal{A}}_{0}((0,a_{-}))\otimes{\mathcal{A}}_{0}((0,a_{-}))$. As for the
$+$-component (the 1st and 2nd tensor components), one observes that it is
almost same as the intersection calculated in [33, Theorem 4.16]. The only
change is that the direction of the 2nd component is reversed. This does not
affect the proof, indeed, the inner symmetry commutes with translation and the
positivity of energy is used only through Reeh-Schlieder property hence is not
essential. Thus we have
$\widetilde{\mathcal{M}}_{\mathrm{c},k}\cap{\hbox{\rm
Ad\,}}\widetilde{T}_{\mathrm{c}}(a)(\widetilde{\mathcal{M}}_{\mathrm{c},k})=\left({\mathcal{A}}_{0}^{\alpha_{0}}((a_{+},0))\otimes{\mathcal{A}}_{0}^{\alpha_{0}}((a_{+},0))\right)\otimes\left({\mathcal{A}}_{0}((0,a_{-}))\otimes{\mathcal{A}}_{0}((0,a_{-}))\right),$
where ${\mathcal{A}}_{0}^{\alpha_{0}}$ denotes the fixed point with respect to
${\hbox{\rm Ad\,}}e^{i\frac{2\pi k}{N}Q_{0}}$. In particular with $k=1$, this
does not satisfies the Reeh-Schlieder property and strict locality fails if
${\mathbb{Z}}_{N}$ acts nontrivially.
This counterexample shows that our proof of locality is by no means purely
group-theoretic. Namely, in order to obtain strict locality, it is necessary
to assume stronger property than strict locality itself of the original net
(wedge-split property or modular nuclearity as above). The above massless
counterexample appears to be related to the subtlety in massless bootstrap
program: the convergence of form factors is typically worse in massless models
and even the fundamental commutativity theorem relies on the behavior of form
factors, which is also worse (c.f. [31]). Hence if one aims at constructing
Wightman fields or Haag-Kastler net out of form factors, the problem of
convergence is inevitable.
## 6 Realization as deformed fields and scattering theory
In Sections 3 and 4 we constructed families of Borchers triples more or less
operator-algebraically. Such a construction was also useful for the argument
of strict locality for one case (Section 5). However, we still have to show
that the resulting nets of von Neumann algebras are really new, or more
desirably have nontrivial interaction and for this purpose the previous
presentation is not very convenient. Fortunately, it turns out that the nets
are accompanied by (wedge-local) quantum fields which create one-particle
states and the scattering process can be calculated [8]. We follow the
notations of [25]. The author expects that this field presentation would help
to prove strict locality for the construction in Section 4.
### 6.1 The complex free field and inner symmetry
#### The Zamolodchikov-Fadeev algebra
Here we consider the construction in Section 3 applied to the complex free
massive field net. The complex massive free net is given simply by the tensor
product
$\mathcal{M}_{\mathrm{c}}:=\mathcal{M}_{\mathrm{r}}\otimes\mathcal{M}_{\mathrm{r}},T_{\mathrm{c}}:=T_{\mathrm{r}}\otimes
T_{\mathrm{r}},\Omega_{\mathrm{c}}:=\Omega_{\mathrm{r}}\otimes\Omega_{\mathrm{r}}$.
The Hilbert space
$\mathcal{H}_{\mathrm{c}}:=\mathcal{H}_{\mathrm{r}}\otimes\mathcal{H}_{\mathrm{r}}$
is canonically isomorphic to the Fock space
${\mathcal{F}}({\mathcal{H}}_{1}\oplus{\mathcal{H}}_{1})$. The U(1) symmetry
transformation of the complex field is constructed as follows: On the “one-
particle space” ${\mathcal{H}}_{1}\oplus{\mathcal{H}}_{1}$ we consider the
following operator
$V_{1}(\kappa)\left(\begin{array}[]{c}\xi\\\
\eta\end{array}\right)=\left(\begin{array}[]{cc}\cos 2\pi\kappa&-\sin
2\pi\kappa\\\ \sin 2\pi\kappa&\cos
2\pi\kappa\end{array}\right)\left(\begin{array}[]{c}\xi\\\
\eta\end{array}\right),\,\,\,\kappa\in{\mathbb{R}}$
The second quantized promotion to $\mathcal{H}_{\mathrm{c}}$ is denoted by
$V_{\mathrm{c}}(\kappa):=\Gamma(V_{1}(\kappa))$. The operator
$V_{\mathrm{c}}(\kappa)$ obviously commutes with $T_{\mathrm{c}}$ and
preserves $\Omega_{\mathrm{c}}$. Moreover, for field operators one has
$\displaystyle{\hbox{\rm
Ad\,}}V_{\mathrm{c}}(\kappa)(\phi(f)\otimes{\mathbbm{1}})$ $\displaystyle=$
$\displaystyle\cos 2\pi\kappa(\phi(f)\otimes{\mathbbm{1}})+\sin
2\pi\kappa({\mathbbm{1}}\otimes\phi(f)),$ $\displaystyle{\hbox{\rm
Ad\,}}V_{\mathrm{c}}(\kappa)({\mathbbm{1}}\otimes\phi(g))$ $\displaystyle=$
$\displaystyle-\sin 2\pi\kappa(\phi(g)\otimes{\mathbbm{1}})+\cos
2\pi\kappa({\mathbbm{1}}\otimes\phi(g)),$
and it holds that ${\hbox{\rm
Ad\,}}V_{\mathrm{c}}(\kappa)(e^{i(\phi(f)\otimes{\mathbbm{1}})})=e^{i\cos
2\pi\kappa\phi(f)}\otimes e^{i\sin 2\pi\kappa\phi(f)}$ and ${\hbox{\rm
Ad\,}}V_{\mathrm{c}}(\kappa)(e^{i({\mathbbm{1}}\otimes\phi(g)})=e^{-i\sin
2\pi\kappa\phi(g)}\otimes e^{i\cos 2\pi\kappa\phi(g)}$. By considering $f$ and
$g$ supported in $W_{\mathrm{R}}$, we conclude that ${\hbox{\rm
Ad\,}}V_{\mathrm{c}}(\kappa)(\mathcal{M}_{\mathrm{c}})=\mathcal{M}_{\mathrm{c}}$.
In other words, $V_{\mathrm{c}}(\kappa)$ implements an inner symmetry of the
group ${\mathbb{R}}/{\mathbb{Z}}\cong S^{1}$.
As we have seen in Section 5, our proof of strict locality works for an action
of $S^{1}$. Hence in the following we consider only that case.
Since $(\mathcal{M}_{\mathrm{r}},T_{\mathrm{r}},\Omega_{\mathrm{r}})$ is
wedge-split, so is the tensor product
$(\mathcal{M}_{\mathrm{c}},T_{\mathrm{c}},\Omega_{\mathrm{c}})$. By
considering the above action of $S^{1}$ by inner symmetry, one can construct
Borchers triples
$(\widetilde{\mathcal{M}}_{\mathrm{c},\kappa},\widetilde{T}_{\mathrm{c}},\widetilde{\Omega}_{\mathrm{c}})$
as in Section 3, which we know to be strictly local by Section 5.
We first take a closer look at the action of $V_{\mathrm{c}}(\kappa)$. The
matrix $V_{1}(\kappa)$ expressed above can be diagonalized by
$\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}{\mathbbm{1}}&i{\mathbbm{1}}\\\
{\mathbbm{1}}&-i{\mathbbm{1}}\end{array}\right)$ into
$\left(\begin{array}[]{cc}e^{i2\pi\kappa}&0\\\
0&e^{-i2\pi\kappa}\end{array}\right)$. Correspondingly we define
${\mathcal{H}}_{1,\pm}:=\\{\psi\oplus\pm i\psi:\psi\in{\mathcal{H}}_{1}\\}$.
Then the full Fock space $\mathcal{H}_{\mathrm{c}}$ can be decomposed into
${\mathbb{Z}}$-graded subspaces
$\mathcal{H}_{\mathrm{c}}=\bigoplus_{l\in{\mathbb{Z}}}\mathcal{H}_{\mathrm{c}}^{l}$
and we may assume that the generator $Q_{\mathrm{c}}$ of
$V_{\mathrm{c}}(\kappa)$ acts by $l{\mathbbm{1}}$ on
$\mathcal{H}_{\mathrm{c}}^{l}$ (by definition of the grading), and
$V_{\mathrm{c}}(\kappa)=e^{i2\pi\kappa Q_{\mathrm{c}}}$. Hence on the Hilbert
space of our interest
$\widetilde{\mathcal{H}}_{\mathrm{c}}=\mathcal{H}_{\mathrm{c}}\otimes\mathcal{H}_{\mathrm{c}}=\bigoplus\mathcal{H}_{\mathrm{c}}^{l}\otimes\mathcal{H}_{\mathrm{c}}^{m}$,
it is clear that $Q_{\mathrm{c}}\otimes Q_{\mathrm{c}}$ acts by
$lm{\mathbbm{1}}$ on
$\mathcal{H}_{\mathrm{c}}^{l}\otimes\mathcal{H}_{\mathrm{c}}^{m}$.
Now the operator $a^{\dagger}_{+}(\psi):=a^{\dagger}(\psi\oplus(i\psi))$ on
$\mathcal{H}_{\mathrm{c}}$ increments the grading and so does
$a_{-}(\psi):=a(\psi\oplus(-i\psi))$. On the other hand,
$a^{\dagger}_{-}(\psi):=a^{\dagger}(\psi\oplus(-i\psi))$ and
$a_{+}(\psi):=a(\psi\oplus(i\psi))$ decrement the grading.
Now it is easy to see the following twisted commutation relations:
$\displaystyle a_{\pm}^{\dagger}(\psi_{1})\otimes{\mathbbm{1}}\cdot{\hbox{\rm
Ad\,}}\widetilde{V}_{\mathrm{c},\kappa}({\mathbbm{1}}\otimes
a_{\pm}^{\dagger}(\psi_{2}))-e^{\pm(\mp i2\pi\kappa)}\cdot{\hbox{\rm
Ad\,}}\widetilde{V}_{\mathrm{c},\kappa}({\mathbbm{1}}\otimes
a_{\pm}^{\dagger}(\psi_{2}))\cdot
a_{\pm}^{\dagger}(\psi_{1})\otimes{\mathbbm{1}}$ $\displaystyle=$
$\displaystyle 0,$ $\displaystyle
a_{\pm}(\psi_{1})\otimes{\mathbbm{1}}\cdot{\hbox{\rm
Ad\,}}\widetilde{V}_{\mathrm{c},\kappa}({\mathbbm{1}}\otimes
a_{\pm}^{\dagger}(\psi_{2}))-e^{\mp(\mp i2\pi\kappa)}\cdot{\hbox{\rm
Ad\,}}\widetilde{V}_{\mathrm{c},\kappa}({\mathbbm{1}}\otimes
a_{\pm}^{\dagger}(\psi_{2}))\cdot a_{\pm}(\psi_{1})\otimes{\mathbbm{1}}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle
a_{\pm}^{\dagger}(\psi_{1})\otimes{\mathbbm{1}}\cdot{\hbox{\rm
Ad\,}}\widetilde{V}_{\mathrm{c},\kappa}({\mathbbm{1}}\otimes
a_{\pm}(\psi_{2}))-e^{\pm(\pm i2\pi\kappa)}\cdot{\hbox{\rm
Ad\,}}\widetilde{V}_{\mathrm{c},\kappa}({\mathbbm{1}}\otimes
a_{\pm}(\psi_{2}))\cdot a_{\pm}^{\dagger}(\psi_{1})\otimes{\mathbbm{1}}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle
a_{\pm}(\psi_{1})\otimes{\mathbbm{1}}\cdot{\hbox{\rm
Ad\,}}\widetilde{V}_{\mathrm{c},\kappa}({\mathbbm{1}}\otimes
a_{\pm}(\psi_{2}))-e^{\mp(\pm i2\pi\kappa)}\cdot{\hbox{\rm
Ad\,}}\widetilde{V}_{\mathrm{c},\kappa}({\mathbbm{1}}\otimes
a_{\pm}(\psi_{2}))\cdot a_{\pm}(\psi_{1})\otimes{\mathbbm{1}}$
$\displaystyle=$ $\displaystyle 0,$
where the signs $\pm$ etc. in the first term correspond to respectively to
$\pm$ etc. in the constant factor in the second term. The commutation relation
between objects with or without ${\hbox{\rm
Ad\,}}\widetilde{V}_{\mathrm{c},\kappa}$ follows trivially from the usual
ones. Namely, we have
$[a_{\pm}^{\dagger}(\psi_{1}),a_{\pm}^{\dagger}(\psi_{2})]=0,\;\;\;[a_{\pm}^{\dagger}(\psi_{1}),a_{\pm}(\psi_{2})]=\langle\psi_{2},\psi_{1}\rangle{\mathbbm{1}},\;\;\;$
and all other combinations commute (note that
$\langle\,\cdot\,,\cdot\,\rangle$ is linear in the second argument and
$a^{\dagger}(\cdot)$ is linear and $a(\cdot)$ is antilinear). In other words,
these operator-valued distributions satisfy the Zamolodchikov-Fadeev algebra,
with the S-matrix given by the phase factors. This two-particle scattering
matrix (see below) does not depend on the rapidity $\theta=\log p$.
Note that this set of commutation relations can be summarized in the form of
matrix. We take a basis $\\{e_{1,+},e_{1,-},e_{2,+},e_{2,-}\\}$ on
${\mathbb{C}}^{2}\otimes{\mathbb{C}}^{2}$ and accordingly $\\{e_{1,+}\otimes
e_{1,+},e_{1,+}\otimes e_{1,-},e_{1,+}\otimes e_{2,+},e_{1,+}\otimes
e_{2,-},\cdots\\}$ on
$({\mathbb{C}}^{2}\otimes{\mathbb{C}}^{2})\otimes({\mathbb{C}}^{2}\otimes{\mathbb{C}}^{2})$,
where
$(\mathcal{H}_{\mathrm{r}}\oplus\mathcal{H}_{\mathrm{r}})\oplus(\mathcal{H}_{\mathrm{r}}\oplus\mathcal{H}_{\mathrm{r}})=(\mathcal{H}_{\mathrm{r}}\otimes{\mathbb{C}}^{2})\otimes{\mathbb{C}}^{2}$
is understood. The signs $\pm$ refer to the structure of the complex free
field, while indices $1,2$ are the first and the second copies of the field.
The two-particle S-matrix $\widetilde{S}_{\mathrm{c},\kappa}(\theta)$ is given
on this basis by (note that this is constant with respect to $\theta$ and the
dependence on the choice of $N$ is always implicitly understood):
$\left(\begin{array}[]{cccc|cccc|cccc|cccc}1&&&&&&&&&&&&&&&\\\
&&&&1&&&&&&&&&&&\\\ &&&&&&&&e^{i2\pi\kappa}&&&&&&&\\\
&&&&&&&&&&&&e^{-i2\pi\kappa}&&&\\\ \hline\cr&1&&&&&&&&&&&&&&\\\
&&&&&1&&&&&&&&&&\\\ &&&&&&&&&e^{-i2\pi\kappa}&&&&&&\\\
&&&&&&&&&&&&&e^{i2\pi\kappa}&&\\\ \hline\cr&&e^{-i2\pi\kappa}&&&&&&&&&&&&&\\\
&&&&&&e^{i2\pi\kappa}&&&&&&&&&\\\ &&&&&&&&&&1&&&&&\\\ &&&&&&&&&&&&&&1&\\\
\hline\cr&&&e^{i2\pi\kappa}&&&&&&&&&&&&\\\ &&&&&&&e^{-i2\pi\kappa}&&&&&&&&\\\
&&&&&&&&&&&1&&&&\\\ &&&&&&&&&&&&&&&1\end{array}\right)$
And it is straightforward that this complies the conditions of [25, Definition
2.1] if the charge conjugation $J_{\mathrm{c}}$ is introduced which exchanges
$+$ and $-$ fields in each component. This is said to be diagonal in the sense
of [25, Section 6]. The advantage of our methods is that the strict locality
can be seen as an immediate consequence of Section 5.
This two-particle S-matrix is nontrivial only between different components.
One could say that the interaction occurs only between particles of different
species but there is no self-interaction. This is clear also from the
construction: one component remains unchanged and the other component is just
shifted by a unitary equivalence, thus the twisting exists only between
different components.
With the help of the analysis [25], we have the following.
###### Theorem 6.1.
The triple
$(\widetilde{\mathcal{M}}_{\mathrm{c},\kappa},\widetilde{T}_{\mathrm{c}},\widetilde{\Omega}_{\mathrm{c}})$
is strictly local and the corresponding Haag-Kastler net is asymptotically
complete and interacting and the S-matrix is factorizing and its two-particle
S-matrix is given as above.
#### Comparison of von Neumann algebras
By definition we have
$\widetilde{\mathcal{M}}_{\mathrm{c},\kappa}=\mathcal{M}_{\mathrm{c}}\otimes{\mathbbm{1}}\vee{\hbox{\rm
Ad\,}}\widetilde{V}_{\mathrm{c},\kappa}({\mathbbm{1}}\otimes\mathcal{M}_{\mathrm{c}})$
and $\mathcal{M}_{\mathrm{c}}$ is generated by the exponential of fields
$\phi(f)\otimes{\mathbbm{1}},{\mathbbm{1}}\otimes\phi(g)$, where $f,g$ are
real test functions with ${\rm supp}f,{\rm supp}g\subset W_{\mathrm{R}}$ and
$\mathcal{H}_{\mathrm{c}}=\mathcal{H}_{\mathrm{r}}\otimes\mathcal{H}_{\mathrm{r}}$
is understood. First let us consider the
$\mathcal{M}_{\mathrm{c}}\otimes{\mathbbm{1}}$ component. Following [25], we
consider pairs of complex valued test functions $f,g$ such that
$\overline{f}=g$. The complex field which generates the wedge algebra in [25]
is given in our notation by
$\displaystyle\phi_{\mathrm{c}}(f\oplus g)$ $\displaystyle=$ $\displaystyle
a_{\mathrm{c}}^{\dagger}(f^{+}\oplus g^{+})+a_{\mathrm{c}}(J_{1}(f^{-}\oplus
g^{-}))$ $\displaystyle:=$ $\displaystyle
a_{+}^{\dagger}(f^{+})+a_{-}^{\dagger}(g^{+})+a_{+}(\overline{g^{-}})+a_{-}(\overline{f^{-}})$
$\displaystyle=$
$\displaystyle\phi(f)\otimes{\mathbbm{1}}+{\mathbbm{1}}\otimes
i\phi(f)+\phi(g)\otimes{\mathbbm{1}}+{\mathbbm{1}}\otimes i\phi(g)$
$\displaystyle=$
$\displaystyle\phi(f+\overline{f})\otimes{\mathbbm{1}}+{\mathbbm{1}}\otimes
i\phi(f-\overline{f})$ $\displaystyle=$ $\displaystyle\phi(2\Re
f)\otimes{\mathbbm{1}}-{\mathbbm{1}}\otimes\phi(2\Im f),$
where $J_{1}(\xi\oplus\eta)=\overline{\eta}\oplus\overline{\xi}$ and
$\xi,\eta\in L^{2}({\mathbb{R}},d\theta)$ and
$f^{\pm}(\theta),g^{\pm}(\theta)$ are defined as before. Namely, the fields in
complex and real basis are just the linear combination of each other. Note
that $\phi$ and $a^{\dagger}$ are linear but $a$ is antilinear.
Similarly, one has ${\hbox{\rm
Ad\,}}\widetilde{V}_{\mathrm{c},\kappa}(\phi_{\mathrm{c}}(f\oplus
g))={\hbox{\rm Ad\,}}\widetilde{V}_{\mathrm{c},\kappa}(\phi_{\mathrm{c}}(2\Re
f)\otimes{\mathbbm{1}}-{\mathbbm{1}}\otimes\phi_{\mathrm{c}}(2\Im f))$ From
this one easily shows that the wedge algebra generated by the Zamolodchikov-
Fadeev fields in the sense of [25] is equal to
$\widetilde{\mathcal{M}}_{\mathrm{c},\kappa}=\mathcal{M}_{\mathrm{c}}\otimes{\mathbbm{1}}\vee{\hbox{\rm
Ad\,}}\widetilde{V}_{\mathrm{c},\kappa}({\mathbbm{1}}\otimes\mathcal{M}_{\mathrm{c}})$.
#### Relation to the Federbush model
One notices that this S-matrix is very similar to the one of the Federbush
model [28, 14, 30], although here the fields are bosonic. However, our
procedure can be easily adapted to fermionic nets. Moreover, in the
traditional approach there were technical problems: One can construct local
fields only for small coupling constant [29], or if one takes the bootstrap
approach, the convergence of form factors are not clear [1]. Here this problem
is completely solved. We can prove the existence of local operators for any
value of $\kappa$ if we consider the action of $S^{1}$. By comparing the
S-matrix, this corresponds to an arbitrary value of the coupling constant.
More importantly, our construction is not restricted to the Federbush models.
One can take any wedge-split net with inner symmetry. This contains, for
example, the tensor product of one of Lechner’s models [22], instead of the
real free field. One can consider $n$ copies of the real free field, which
have $\mathrm{O}(n)$ symmetry, then take any subgroup of $\mathrm{O}(n)$
isomorphic to $S^{1}$ or ${\mathbb{Z}}_{N}$. This should correspond to the
Lie-algebraic generalization of the Federbush models, whose form factors were
proposed in [14]. It works also with $n$ copies of one of Lechner’s models.
Furthermore, the constructed net admits again inner symmetry and is wedge-
split, hence one can repeat the construction to obtain further new models (on
a bigger Hilbert space).
#### Some technical remarks on inner symmetry
The wedge-algebra of our two-dimensional nets is given by the tensor product
twisted by the inner symmetry. One realizes that the whole net still admits an
action of $S^{1}\times S^{1}$ by inner symmetry and the fixed point is the
same for the untwisted tensor product. In other words, the fixed point net has
different (free and interacting) extension. This does not contradict the DHR
reconstruction theorem [16, 17], which does not work with a compact group in
two dimensions.
Our models violate the parity symmetry, which is clear from the S-matrix.
However, the extended parity which interchanges the two components is
preserved (see [30, Section 6.3.3]). As noted in [10, Theorem 3.3], the parity
symmetry is essential for the Bisognano-Wichmann property in two dimensions.
Although this is not necessarily related to our models, we present a simple
counterexample. One takes the complex free field, which admits $S^{1}$-inner
symmetry with Bisognano-Wichmann property. Then one can simply replace the
Lorentz boosts by the composition of Lorentz boosts and the inner symmetry.
This still satisfies all the axioms of net but violates the parity symmetry
which must have the appropriate commutation relation with the boosts.
Accordingly, the inner symmetry and Poincaré symmetry do not necessarily
commute. For example, we can take a net with a noncommutative Lie group
symmetry and replace the boosts as above. The proof in four dimensions [15,
Theorem 10.4] does not work in two dimensions since the Lorentz group is
abelian, and hence the Poincaré group has finite dimensional unitary
representations.
### 6.2 Real free field and Longo-Witten endomorphisms
Let us see that the Borchers triples constructed in Section 4 admit a wedge-
local field interpretation. Since the twisting operator
$\widetilde{R}_{\varphi}$ is given on each particle number space, the
calculation is straightforward. We only need the following commutation
relations:
$a^{\dagger}(\psi_{1})\otimes{\mathbbm{1}}\cdot{\hbox{\rm
Ad\,}}\widetilde{R}_{\varphi}({\mathbbm{1}}\otimes a^{\dagger}(\psi_{2}))-\int
dpdp^{\prime}\psi_{1}(p)\psi_{2}(p^{\prime})\varphi\left(\frac{p}{p^{\prime}}\right){\hbox{\rm
Ad\,}}\widetilde{R}_{\varphi}({\mathbbm{1}}\otimes
a^{\dagger}(p^{\prime}))\cdot a^{\dagger}(p)\otimes{\mathbbm{1}}=0,$
or with rapidity $\theta=\log p$ and in terms of operator-valued distributions
one has
$a^{\dagger}(\theta)\otimes{\mathbbm{1}}\cdot{\hbox{\rm
Ad\,}}\widetilde{R}_{\varphi}({\mathbbm{1}}\otimes
a^{\dagger}(\theta^{\prime}))-\varphi\left(e^{\theta-\theta^{\prime}}\right){\hbox{\rm
Ad\,}}\widetilde{R}_{\varphi}({\mathbbm{1}}\otimes
a^{\dagger}(\theta^{\prime}))\cdot\alpha^{\dagger}(\theta)\otimes{\mathbbm{1}}=0.$
Note that if $\varphi$ is an inner symmetric function, then
$\varphi(e^{\theta})$ is a bounded analytic function in the strip
$0<\Im\theta<\pi$ and $\varphi(e^{i\pi-\theta})=\varphi(-e^{-\theta})$ has the
same property. In the matrix form, it can be written as
$\widetilde{S}_{\mathrm{r},\varphi}(\theta)=\left(\begin{array}[]{cccc}1&&&\\\
&&\varphi(e^{\theta})&\\\ &\varphi(-e^{-\theta})&&\\\ &&&1\end{array}\right)$
on the basis $\\{e_{1}\otimes e_{1},e_{1}\otimes e_{2},e_{2}\otimes
e_{1},e_{2}\otimes e_{2}\\}$. This is again a diagonal S-matrix. From its
simple form, it is expected that the proof of modular nuclearity is similar to
the one in [22].
It is clear that also in these models the interaction occurs only between
different components.
## 7 Summary and outlook
In this paper, we presented a novel procedure to obtain interacting quantum
field models realized as nets of observables. One can view this procedure as
first preparing a pair of models then making them couple.
The method is thoroughly operator-algebraic and in the most abstract setting
of Section 5 no field picture (wedge-local or not) is required. On the other
hand, the interaction is fairly weak. When the field description is available,
a pair of particles of different species obtains a phase during the
interaction and no momentum transfer occurs. A connection with wedge-local
field approach and our previously constructed Borchers triples has been found
in [24]. It is worth investigating how to obtain more general integrable
models (e.g. [22, 25]) purely operator-algebraically, where a “self-
interaction” occurs.
Furthermore, an interesting variant has been obtained in [6]. Strict locality
of the models therein is not known, but S-matrix shows a phenomenon which
resembles particle production. It is desired to establish strict locality of
these models or to find massive correspondents.
More ambitiously, certain relations are claimed between integrable models and
higher- dimensional gauge theories (e.g. [4]). The author wishes to study
such connections with analytical approach.
#### Acknowledgment.
I would like to thank Marcel Bischoff and Karl-Henning Rehren for helpful
discussions.
## References
* [1] H. Babujian and M. Karowski. Towards the construction of Wightman functions of integrable quantum field theories. In Proceedings of 6th International Workshop on Conformal Field Theory and Integrable Models, volume 19, pages 34–49, 2004.
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* [3] Hellmut Baumgärtel. Operator algebraic methods in quantum field theory. Akademie Verlag, Berlin, 1995.
* [4] Niklas Beisert, Changrim Ahn, Luis F. Alday, and et al. Review of AdS/CFT integrability: An overview. Lett. Math. Phys., 99(1-3):3–32, 2012.
* [5] Marcel Bischoff and Yoh Tanimoto. Integrable QFT and Longo-Witten endomorphisms. in preparation., 2012.
* [6] Marcel Bischoff and Yoh Tanimoto. Construction of Wedge-Local Nets of Observables through Longo-Witten Endomorphisms. II. Comm. Math. Phys., 317(3):667–695, 2013.
* [7] H.-J. Borchers. The CPT-theorem in two-dimensional theories of local observables. Comm. Math. Phys., 143(2):315–332, 1992.
* [8] Hans-Jürgen Borchers, Detlev Buchholz, and Bert Schroer. Polarization-free generators and the $S$-matrix. Comm. Math. Phys., 219(1):125–140, 2001.
* [9] Ola Bratteli and Derek W. Robinson. Operator algebras and quantum statistical mechanics. 2. Equilibrium states. Models in quantum statistical mechanics. Texts and Monographs in Physics. Springer-Verlag, Berlin, second edition, 1997.
* [10] R. Brunetti, D. Guido, and R. Longo. Modular structure and duality in conformal quantum field theory. Comm. Math. Phys., 156(1):201–219, 1993.
* [11] Detlev Buchholz, Claudio D’Antoni, and Roberto Longo. Nuclear maps and modular structures. I. General properties. J. Funct. Anal., 88(2):233–250, 1990.
* [12] Detlev Buchholz and Gandalf Lechner. Modular nuclearity and localization. Ann. Henri Poincaré, 5(6):1065–1080, 2004.
* [13] Detlev Buchholz, Gandalf Lechner, and Stephen J. Summers. Warped convolutions, Rieffel deformations and the construction of quantum field theories. Comm. Math. Phys., 304(1):95–123, 2011.
* [14] O. A. Castro-Alvaredo and A. Fring. Form factors from free fermionic Fock fields, the Federbush model. Nuclear Phys. B, 618(3):437–464, 2001.
* [15] S. Doplicher and R. Longo. Standard and split inclusions of von Neumann algebras. Invent. Math., 75(3):493–536, 1984.
* [16] Sergio Doplicher and John E. Roberts. A new duality theory for compact groups. Invent. Math., 98(1):157–218, 1989.
* [17] Sergio Doplicher and John E. Roberts. Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics. Comm. Math. Phys., 131(1):51–107, 1990.
* [18] Wojciech Dybalski and Yoh Tanimoto. Asymptotic completeness in a class of massless relativistic quantum field theories. Comm. Math. Phys., 305(2):427–440, 2011.
* [19] Fabrizio Gabbiani and Jürg Fröhlich. Operator algebras and conformal field theory. Comm. Math. Phys., 155(3):569–640, 1993.
* [20] Rudolf Haag. Local quantum physics. Texts and Monographs in Physics. Springer-Verlag, Berlin, second edition, 1996. Fields, particles, algebras.
* [21] Gandalf Lechner. Polarization-free quantum fields and interaction. Lett. Math. Phys., 64(2):137–154, 2003.
* [22] Gandalf Lechner. Construction of quantum field theories with factorizing $S$-matrices. Comm. Math. Phys., 277(3):821–860, 2008.
* [23] Gandalf Lechner. Deformations of quantum field theories and integrable models. Comm. Math. Phys., 312(1):265–302, 2011.
* [24] Gandalf Lechner, Jan Schlemmer, and Yoh Tanimoto. On the equivalence of two deformation schemes in quantum field theory. Lett. Math. Phys., to appear.
* [25] Gandalf Lechner and Christian Schützenhofer. Towards an operator-algebraic construction of integrable global gauge theories. arXiv:1208.2366, 2012.
* [26] Roberto Longo. Real Hilbert subspaces, modular theory, ${\rm SL}(2,{\bf R})$ and CFT. In Von Neumann algebas in Sibiu: Conference Proceedings, pages 33–91. Theta, Bucharest, 2008.
* [27] Roberto Longo and Edward Witten. An algebraic construction of boundary quantum field theory. Comm. Math. Phys., 303(1):213–232, 2011.
* [28] S. N. M. Ruijsenaars. Scattering theory for the Federbush, massless Thirring and continuum Ising models. J. Funct. Anal., 48(2):135–171, 1982.
* [29] S. N. M. Ruijsenaars. The Wightman axioms for the fermionic Federbush model. Comm. Math. Phys., 87(2):181–228, 1982/83.
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* [32] M. Takesaki. Theory of operator algebras. II, volume 125 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 2003. Operator Algebras and Non-commutative Geometry, 6.
* [33] Yoh Tanimoto. Construction of Wedge-Local Nets of Observables Through Longo-Witten Endomorphisms. Comm. Math. Phys., 314(2):443–469, 2012.
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* [35] Mihály Weiner. An algebraic version of Haag’s theorem. Comm. Math. Phys., 305(2):469–485, 2011.
|
arxiv-papers
| 2013-01-25T16:58:17 |
2024-09-04T02:49:40.744232
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yoh Tanimoto",
"submitter": "Yoh Tanimoto",
"url": "https://arxiv.org/abs/1301.6090"
}
|
1301.6099
|
# CG-content log-ratio distributions of Caenorhabditis elegans and Drosophila
melanogaster mirtrons
Denise Fagundes-Lima [email protected] Departamento de Ciências
Biológicas, Universidade Federal de Ouro Preto, 35400-000 Ouro Preto, MG,
Brazil
and Departamento de Física, Universidade Federal de Minas Gerais,31270-901
Belo Horizonte, MG, Brazil Gerald [email protected] Departamento de
Física, Universidade Federal de Minas Gerais,31270-901 Belo Horizonte, MG,
Brazil
###### Abstract
Mirtrons are a special type of pre-miRNA which originate from intronic regions
and are spliced directly from the transcript instead of being processed by
Drosha. The splicing mechanism is better understood for the processing of mRNA
for which was established that there is a characteristic CG content around
splice sites. Here we analyse the CG-content ratio of pre-miRNAs and mirtrons
and compare them with their genomic neighbourhood in an attempt to establish
key properties which are easy to evaluate and to understand their biogenesis.
We propose a simple log-ratio of the CG-content comparing the precursor
sequence and is flanking region. We discovered that Caenorhabditis elegans and
Drosophila melanogaster mirtrons, so far without exception, have smaller CG-
content than their genomic neighbourhood. This is markedly different from
usual pre-miRNAs which mostly have larger CG-content when compared to their
genomic neighbourhood. We also analysed some mammalian and primate mirtrons
which, in contrast the invertebrate mirtrons, have higher CG-content ratio.
## Introduction
During the last decade, a wealth of small RNAs were discovered and with them
new classes of biological regulators emerged. Among those, microRNAs (or
miRNAs) due to their crucial role in genomic regulation are perhaps the most
intensively studied. miRNAs are involved in the regulation of numerous
cellular processes including differentiation, development, apoptosis,
proliferation, the stress response and they change the expression of genes in
several human diseases such as diabetes, cancer and neuromuscular dystrophy
zhang07 ; iorio12 ; hussain12 .
miRNAs are non-coding RNAs first identified in 1993 in the nematode
Caenorhabditis elegans lee93 . Canonical miRNAs are derived from primary miRNA
transcripts (pri-miRNA), usually long nucleotide sequences that form specific
hairpin-shaped stem–loop secondary structures. Pri-miRNA may originate one or
more hairpins typically with 55–70 nucleotide (nt) in length. In animals, pri-
miRNAs are cleaved by the nuclear Drosha RNase III enzyme to release precursor
miRNA (pre-miRNA) hairpins. These are then transported to the cytoplasm by
Exportin-5 (Exp5) and cleaved by the Dicer RNase III enzyme to generate a very
short miRNA/miRNA* duplex kim09 . One of the strands, called mature miRNA
(22–25nt), is incorporated into a RISC complex (RNA induced silencing complex)
and guides the complex to the target mRNA to regulate gene expression while
the other strand seems to take on other biological functions kim09 ; okamura08
; bhayani11 . In animals, most of the miRNA functions are related to down-
regulation of genes.
Ruby _et al._ ruby07 showed the existence of intronic pre-miRNAs in
Drosophila melanogaster and C. elegans that bypass Drosha processing providing
an alternative pathway for miRNA biogenesis okamura07 . These pre-miRNAs were
called ‘mirtrons’ and the main difference between them and canonical miRNAs is
that intronic sequences form lariats and the mirtrons are originated by
splicing okamura07 ; ruby07 ; berezikov07 . Flynt _et al._ flynt10
reclassified a subset of mirtrons in D. melanogaster as “tailed mirtrons”,
which have substantial $3^{\prime}$ overhangs and are targets of exosome-
mediated $3^{\prime}-5^{\prime}$ trimming, which allows functional pre-miRNA
to be generated. The existence of mirtrons in mammalians (human, macaque,
chimpanzee, rat and/or mouse) was reported by Berezikov _et al._ berezikov07
where they identified, using computational and experimental strategies, 3 well
conserved mirtrons expressed in diverse mammals, 16 primate specific mirtrons,
and 46 candidate mirtrons in primates.
For mRNA, which is processed by splicing, Zhang _et al._ zhang11 determined
that there is a characteristic CG content around splice sites. Also, it was
shown that alternative splicing is promoted by the secondary RNA structure
shepard08 which is strongly determined by CG content weber06 . MicroRNAs are
co-expressed with mRNAs morlando08 ; shomron09 and, in particular, mirtrons
are seemingly not processed by the Drosha microprocessor but by splicing only.
With splicing being dependent on thermodynamic stability could there be some
characteristic CG content which would set aside mirtrons from ordinary pre-
miRNAs? Of special interest would be properties which would help to understand
the splicing mechanism proposed for mirtrons okamura07 .
Here we set out to characterise precursor sequences of miRNAs and mirtrons in
terms of CG-content and also Gibbs free energies for D. melanogaster and C.
elegans. We found that the CG-content shows marked differences for both types
of small RNA. Also, we performed the same analysis for mammalian mirtrons
reported by Berezikov _et al._ berezikov07 , again our results show important
differences albeit opposite of those for the two invertebrates.
## Methods
To characterise the small RNAs we compare the CG-content of the precursor
sequences which originate the pre-miRNAs and mirtrons to the CG-content of
their neighbouring regions. The rationale for this approach is that if the
neighbouring DNA sequence has an important difference in thermodynamic
stability, when compared to the precursor sequence there should be tell tale
signs of it in the CG-content fractions. We define the CG-content fraction as
$f=\frac{\text{number of C and G nucleotides}}{\text{total number of
nucleotides}}$ (1)
Two types of CG-content fractions are used, one $f_{P}$ is related to either
the precursor miRNA or the precursor mirtron. The other $f_{N}$ accounts for
the total CG-contents of the 150 base pairs downstream and upstream of the
precursor sequence which forms the neighbourhood of the precursor. The
flanking sequence length was chosen to be of the same order of magnitude of
the length of canonical pre-miRNAs. We perfomed the same analysis with longer
flanking sequences (up to 250 nt, not presented), but found no difference from
the results reported in this work. Both CG-contents are combined to form a
log-ratio between the precursor and its neighbourhood
$R=\log_{2}\left(\frac{f_{P}}{f_{N}}\right)$ (2)
A positive ratio means that the CG-content $f_{P}$ of the precursor sequence
is larger than that of its neighbours. Since CG-content is related to
thermodynamic stability we may infer that $R>0$ generally means that the
flanking DNA region is less stable that the precursor region. To ease the
notation we use
$\begin{array}[]{cl}R^{+}\rightarrow R>0&\text{\emph{precursor} region has
larger CG-content},f_{P}>f_{N}\\\ R^{-}\rightarrow R<0&\text{\emph{flanking}
region has larger CG-content},f_{N}>f_{P}\end{array}$
To evaluate the statistical significance of our findings we use the
Kolmogorov-Smirnov test press92 . Even though this statistical test is well
established, given the question which is posed in this work it is perhaps more
intuitive and simpler to quantify the significance by using simple
combinatorial probabilities. Therefore, we also calculate the probability
$p^{-}$ of drawing $k$ pre-miRNAs, all with $R^{-}$, purely by chance
$p^{-}=\left(\frac{n^{-}}{n^{-}+n^{+}}\right)^{k}$ (3)
where $n^{-}$ and $n^{+}$ are number of known pre-miRNAs with $R^{-}$ and
$R^{+}$, respectively.
The database used to obtain the precursor miRNA and mirtrons of D.
melanogaster and C. elegans was from mirBASE version 16 griffiths-jones06 ;
kozomara11 , which is one of the main on-line repositories for microRNA
sequences. For extracting the flanking sequences we used the complete genome
file of D. melanogaster version r5.34 drysdale05 and version WS223 for C.
elegans stein01 . We retrieved the precursor miRNA/mirtron sequences by
searching for an exact match within the complete genome files. For each
sequence four types of matches were performed: the original sequence, the
reversed sequence, the complementary sequence and the reversed-complementary
sequence.
The mirtrons reported by Berezikov _et al._ berezikov07 were collected from
the supplemental data, the neighbouring sequences for each these mirtrons were
obtained from Ensemble API and databases hubbard02 .
To complete our analysis we also calculated the average Gibbs free energies of
mirtrons and ordinary pre-miRNA. In this work we use the RNAfold program from
the Vienna package hofacker03 with default parameters to obtain the Gibbs
free energies, $\Delta G$.
## Results and Discussion
Figure 1: CG-content ratio $R$ distribution for a) 151 canonical miRNAs and b)
18 mirtrons of D. melanogaster and c) 170 canonical miRNAs and d) 5 mirtrons
of C. elegans. Table 1: CG-content ration $R$ and free energy $\Delta G$
characteristics and of canonical pre-miRNAs and mirtrons of invertebrates.
Also shown are number of sequences $n^{\pm}$ with $R^{\pm}$, average CG-
content ratio $\langle R\rangle$, average free energy $\langle\Delta G\rangle$
and average precursor length $\langle N\rangle$.
Organism | RNA type | total | $n^{+}$ | $n^{-}$ | $n^{+}:n^{-}$ | $\langle R\rangle$ | $\langle\Delta G\rangle$ (kcal/mol) | $\langle N\rangle$ (nt)
---|---|---|---|---|---|---|---|---
C. elegans | mirtrons | 5 | 0 | 5 | | $0.81\pm 0.08$ | $-20.28\pm 4.21$ | $62.22\pm 6.72$
D. melanogaster | mirtrons | 18 | 0 | 18 | | $0.76\pm 0.10$ | $-22.03\pm 6.55$ | $69.05\pm 16.68$
D. melanogaster | tailed mirtrons | 7 | 2 | 5 | $1:1.8$ | $0.84\pm 0.24$ | $-20.36\pm 10.19$ | $93.00\pm 42.24$
C. elegans | canonical miRNAs11footnotemark: 1 | 170 | 131 | 39 | $3.3:1$ | $1.18\pm 0.23$ | $-35.10\pm 9.09$ | $91.78\pm 14.26$
D. melanogaster | canonical miRNAs11footnotemark: 1 | 151 | 97 | 54 | $1.8:1$ | $1.08\pm 0.21$ | $-33.89\pm 8.25$ | $94.00\pm 18.50$
11footnotemark: 1
mirtrons excluded.
Table 2: CG-content ratio $R$ and free energy $\Delta G$ characteristics and
of canonical miRNAs considering only those with $R^{-}$.
Organism | RNA type | $n^{-}$ | $\langle R^{-}\rangle$ | $\langle\Delta G\rangle$ (kcal/mol) | $\langle N\rangle$ (nt)
---|---|---|---|---|---
C. elegans | canonical miRNAs | 39 | $0.90\pm 0.08$ | $-30.17\pm 8.72$ | $90.95\pm 17.50$
D. melanogaster | canonical miRNAs | 54 | $0.87\pm 0.10$ | $-30.0\pm 5.40$ | $90.40\pm 11.78$
In Fig. 1 we show the distribution of CG-content log-ratio $R$ for canonical
pre-miRNAs and mirtrons, defined in Methods, of D. melanogaster and C.
elegans. The content log-ratio $R$ for canonical pre-miRNAs, Fig. 1a, is
roughly gaussian with a peak around $R=0$. This means that for this type pre-
miRNA there appears to be no strong preferential ratio for CG-content within
the precursor sequence and its neighbours, although a bias towards $R^{+}$ is
clearly noticeable. In stark contrast, all 18 mirtrons of D. melanogaster have
$R<0$ ($R^{-}$) as shown in Fig. 1b. Even though the number of reported
mirtrons is still small, the probability of picking 18 small RNAs with $R^{-}$
by chance alone, considering the distribution for canonical pre-miRNA, is
$p^{-}=9.1\times 10^{-9}$, see Eq. (3). The Kolmogorov-Smirnov distibution
test yields $p^{-}=7.8\times 10^{-9}$ which essentially confirms the simple
combinatorial probability. Therefore, the occurrence of 18 $R^{-}$ pre-miRNA
entirely by chance is very unlikely.
Some authors describe mirtrons as tightly packed between exons hussain12 , but
in our analysis we have found that this is not the case. Most mirtrons are
surrounded by intronic sequences not exons. This seems consistent if one
considers that intronic regions of D. melanogaster are about 750 to 1000 nt in
length on average presgraves06 and that mirtrons are typically 60 nt in
length. Therefore, $R^{-}$ means that the immediate flanking region which is
also intronic is more stable than the precursor region. One possible
explanation for the predominance of $R^{-}$ would be if intronic regions were
of highest CG-content. However, the intronic regions of D. melanogaster have
one of the smallest CG-content in this genome: 0.4 as compared to 0.52 for
coding regions. The fact that the surrounding region of mirtrons has a higher
CG content, which is unusual for intronic regions, suggest a role in the
processing of these special types of miRNAs. Therefore, we may speculate that
$R^{-}$ may play a role in the mirtron splicing mechanism in a similar fashion
to what happens for messenger RNA zhang11 .
For canonical C. elegans pre-miRNAs we observe a similar gaussian shaped
distribution of the ratio $R$ (Fig. 1c) but with strong bias towards $R^{+}$.
Tab. 2 shows that the $n^{+}:n^{-}$ ratio is of three $R^{+}$ pre-miRNAs for
each $R^{-}$ pre-miRNA. To date there are only five mirtrons reported and they
all show $R^{-}$ (Fig. 1d), similar to the mirtrons of D. melanogaster. Even
though this number is very small, it is still intriguing given the strong bias
toward $R^{+}$ in canonical pre-miRNAs. Indeed, the probability of picking 5
pre-miRNAs all with $R^{-}$ is small, the combinatorial probability being
$p^{-}=6.3\times 10^{-4}$. Again, the Kolmogorov-Smirnov test provides
$5.5\times 10^{-4}$ in agreement with the combinatorial probability.
To complete our comparative analysis of mirtrons and canonical pre-miRNAs, we
also calculated the Gibbs free energies. Clearly, given the $R^{-}$ nature of
the mirtons, one would expect these to be generally less stable than the
average pre-miRNAs. In Fig. 2 we show the distribution of free energy $\Delta
G$ for both invertebrates, and detailed quantities are also given in Tab. 2.
Except for one notable exception, all mitrons show $\Delta G$ larger than
$-30$ kcal/mol, confirming their instability. In contrast, canonical pre-
miRNAs are distributed over a much larger range of energies. Certainly, the
fact that mirtrons are much shorter than canonical pre-miRNAs, 60 nt compared
to 90 nt on average, largely accounts for this. But is a free energy larger
than $-30$ kcal/mol sufficient to result in $R^{-}$? To answer this, we
isolated all canonical pre-miRNAs with $R^{-}$ and recalculated the their
$\Delta G$ distribution, which are shown as red bars in Figs. 2a and 2c and
summarised in Tab. 2. Essentially, we find a considerable number of $R^{+}$
pre-miRNAs with $\Delta G>-30$ kcal/mol. In other words, a pre-miRNAs with
$\Delta G>-30$ kcal/mol does not imply in $R^{-}$. Therefore, the free energy
distribution alone does not explain why all mirtrons of D. melanogaster and C.
elegans are $R^{-}$.
Figure 2: Average free energy $\Delta G$ distribution for a) 151 canonical
pre-miRNAs and b) 18 mirtrons of D. melanogaster. and c) 170 canonical miRNAs
and d) 5 mirtrons of C. elegans. Red bars are for $R^{-}$ miRNAs.
The next question is whether other types of reported mirtrons, such as primate
and mammalian mirtrons show the same $R$ distribution as D. melanogaster and
C. elegans? As shown in Tab. 3, in terms of CG-content ration $R$ and average
free energy $\Delta G$ these mirtrons appear not to be biased to any
particular value. Berezikov _et al._ berezikov07 found that the GC content of
mammalian mirtrons was much higher than that of invertebrate miRNAs but, in
comparison with their neighbours regions, we found that they tend generally to
$R^{+}$ (_precursor_ region has larger CG-content), see Fig. 3. We have not
attempted to generate the distribution of mammalian pre-miRNAs due to the
number of large genomes which would have to be processed.
Table 3: CG-content ratio $R$ and free energy $\Delta G$ characteristics of
specific vertebrate mirtrons.
Organism | RNA type | total | $n^{+}$ | $n^{-}$ | $n^{+}:n^{-}$ | $\langle R\rangle$ | $\Delta G$ (kcal/mol) | $\langle N\rangle$ (nt)
---|---|---|---|---|---|---|---|---
mammalians | putative mirtrons | 13 | 8 | 5 | $1.5:1$ | $0.98\pm 0.16$ | $-30.02\pm 13.20$ | $87.92\pm 9.52$
primates | specific mirtrons | 16 | 13 | 3 | $4.3:1$ | $1.12\pm 0.11$ | $-30.32\pm 11.69$ | $83.62\pm 24.36$
primates | candidate mirtrons | 4522footnotemark: 2 | 40 | 5 | $8:1$ | $1.11\pm 0.09$ | $-41.18\pm 12.39$ | $90.44\pm 21.94$
22footnotemark: 2
Ref. berezikov07, reports 46 candidates mirtrons for primates, yet
supplementary tables only show 45 sequences.
Figure 3: CG-content ratio $R$ distribution for a) 13 putative mammalian
mirtrons, b) 16 specific primate mirtrons and c) 46 candidate primate
mirtrons.
## Conclusions
We have introduced the concept of CG-content log-ratio of precursor sequences
and flanking regions and discovered that all D. melanogaster and C. elegans
mirtrons are $R^{-}$. This cannot be explained by the CG-content of the
intronic region and neither by the fact that mirtrons are generally shorter
and less stable than pre-miRNAs. Usual pre-miRNAs of these organisms only show
a moderate bias towards $R^{+}$. This finding appears to support the notion
that mirtrons are spliced in a similar fashion to mRNA instead of being
processed by Drosha. For mammalian mirtrons we have found no such bias, but we
noticed that these also display several important differences when compared to
the vertebrate mirtrons which were considered in this work, such as
differences in length and free energy.
## Acknowledgements
We are grateful to J. M. Ortega for helpful suggestions. Funding: CNPq,
Fapemig and National Institute of Science and Technology for Complex Systems.
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|
arxiv-papers
| 2013-01-25T17:29:39 |
2024-09-04T02:49:40.758371
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Denise Fagundes-Lima and Gerald Weber",
"submitter": "Gerald Weber",
"url": "https://arxiv.org/abs/1301.6099"
}
|
1301.6191
|
# Reuse, Temporal Dynamics, Interest Sharing, and Collaboration in Social
Tagging Systems
ELIZEU SANTOS-NETO DAVID CONDON NAZARENO ANDRADE ADRIANA IAMNITCHI MATEI
RIPEANU University of British Columbia University of South Florida
Universidade Federal de Campina Grande University of South Florida
University of British Columbia
###### Abstract
User-generated content is shaping the dynamics of the World Wide Web. Indeed,
an increasingly large number of systems provide mechanisms to support the
growing demand for content creation, sharing, and management. Tagging systems
are a particular class of these systems where users share and collaboratively
annotate content such as photos and URLs. This collaborative behavior and the
pool of user-generated metadata create opportunities to improve existing
systems and to design new mechanisms. However, to realize this potential, it
is necessary to first understand the usage characteristics of current systems.
This work addresses this issue characterizing three tagging systems
(CiteULike, Connotea and del.icio.us) while focusing on three aspects: i) the
patterns of information (tags and items) production; ii) the temporal dynamics
of users’ tag vocabularies; and, iii) the social aspects of tagging systems.
The analysis of the patterns of information production shows that users
publish new content more often than they annotate already existing content in
the system. The opposite, however, occurs for tags; the level of tag reuse is
much higher. This observation provides evidence that tags are indeed used for
categorization. The relative difference between the rate of item publication
and tag reuse suggests that tags are potentially useful as an additional
source of information for item recommendation techniques.
The study of the temporal dynamics of user vocabularies shows that the growth
rate of tag vocabularies across the user population over time decreases at
early ages, stabilizes, and returns to increase for older users. Moreover, a
closer look into the change of vocabulary contents over time shows that
despite the fact that tag vocabularies are slowly growing in size with user
age, the relative frequency in which each tag is used converges relatively
quickly in a user’s lifetime. Mechanisms that rely on tag-based user
similarity offer opportunities to harness the above observation by attempting
to strike a balance between the accuracy of vocabulary similarity estimates,
the data volume required for estimation, and the freshness of the data used.
Finally, the characterization of social aspects of tagging unveils the
relationship between the implicit user ties, as inferred from the similarity
between users’ activity, and their explicit social ties, as represented by co-
membership in discussion groups or semantic similarity between tag
vocabularies. In particular, the results show that mechanisms that aim to
harness the social ties between users can exploit the fact that implicit
social ties complement their explicit counterparts with finer-grained strength
information.
## 1 Introduction
Tagging systems [Mathes (2004), Hammond et al. (2005), Marlow et al. (2006),
Macgregor and McCulloch (2006), Farooq et al. (2007)] are a ubiquitous
manifestation of online peer-production of information [Benkler (2006)], a
production mode commonplace in today’s World Wide Web [Ramakrishnan and
Tomkins (2007)]. The annotation feature, often referred to as simply tagging,
has been originally designed to support personal content management. However,
as this feature exposes user preferences and their temporal dynamics,
similarities between users, and the aggregated characteristics of the user
population, annotations have been recognized for their potential to support a
wider range of mechanisms such as social search [Yahia et al. (2008)],
recommendation [Sigurbjörnsson and van Zwol (2008)], and search optimization
[Yanbe et al. (2007), Heymann et al. (2008), Huang et al. (2008)].
Moreover, tagging is increasingly important in online social systems and, more
recently, motivates new initiatives such as OpenAnnotation
111http://openannotation.org that aims to enable users to annotate content on
the web without depending on specific systems. Therefore, understanding social
tagging through characterization and modeling of usage patterns is important,
as understanding the current systems can better inform the design of future
annotation platforms such as Hypothes.is 222http://hypothes.is. Finally,
characterizing social tagging systems can both unveil new opportunities and
improve existing mechanisms.
This work extends our previous study [Santos-Neto et al. (2009)] and addresses
this need for characterization by investigating unexplored aspects of social
tagging behaviour as well as complementing previous characterization studies
(presented in Section 2). In particular, it focuses on three major aspects of
the tagging activity that have attracted relatively little attention in the
past: i) the dynamics of tag and items produced via collaborative annotation;
ii) the temporal dynamics of users’ tag vocabularies; and, iii) the
characteristics of the social ties between users in these systems. A marked
difference from this work to previously similar characterization studies is
that, this study takes one step further by offering observations across
multiple social tagging systems, which allows for a richer analysis of tagging
behaviour.
To study the productions of tags and items, Section 4 concentrates on two
metrics: i) item re-tagging, a measure of the degree to which items are
repeatedly tagged; and ii) tag reuse, a measure of the degree to which users
reuse a tag to perform new annotations.
The analysis of the evolution of the users tag vocabularies (i.e., the set of
tags a user assigns to her items) in Section 5 focuses on the evolution of the
user vocabularies over time.
The investigation of social ties between pairs of users focuses first on
unveiling the characteristics of the implicit ties between users based on the
similarity between their tagging activities (Section 6). Additionally, this
work explores the relationship between the strength of such implicit ties and
those of more explicit social ties such as co-membership in discussion groups
and semantic similarity of tag vocabularies (Section 7). Studying the
relationship among the implicit and explicit ties is relevant as we test
whether the implicit ties based on usage similarity provide information about
the potential creation of explicit social ties and ultimately for
collaboration.
This study uses activity traces from three distinct tagging systems -
CiteULike, Connotea, and del.icio.us (detailed in Section 3). We believe that
this selection of systems samples the diversity of the tagging ecosystem, as
they are three emblematic tagging systems for the type of content they target,
with CiteULike and Connotea concentrating in bookmarking of academic
citations, and del.icio.us focusing on general URLs. The in-depth analysis of
these three systems reveals regularities and relevant variations in tagging
behavior.
The main findings of this work are:
* $\blacksquare$
The characteristics of peer production of information are qualitatively
similar across systems but differ quantitatively, as suggested by the observed
rates of item re-tagging and tag reuse. In all three systems investigated,
users produce new items at higher rate than they produce new tags. However,
the observed rates in CiteULike and Connotea are different from del.icio.us.
As the three systems provide essentially similar annotation features, these
findings suggest that the target audience and the type of annotated content
play an important role in the users’ tagging behavior (Section 4).
* $\blacksquare$
User tag vocabularies are constantly growing, but at different rates depending
on the age of the user. However, despite the constant increase in size, the
relative usage frequency of tags in a vocabulary converges to a stable ranking
at early stages of a user’s lifetime in the system. These observations have
implications for applications that rely on tag-vocabulary similarity (e.g.,
recommender systems): these applications can use only a subsample of the
entire user activity to estimate vocabulary similarity between users.
Moreover, applications can aim to strike a balance between the accuracy of
similarity estimates, the data volume used for estimation, and the freshness
of the data. (Section 5)
* $\blacksquare$
The observed levels of activity similarity between pairs of users are the
result of shared interested as opposed to generated by chance. The
distributions of activity similarity strength deviate significantly from those
produced by a Random Null Model (RNM) [Reichardt and Bornholdt (2008)]. This
suggests that the implicit ties between users, as defined by their activity
similarity levels, capture latent information about user relationships that
may offer support for optimizing system mechanisms. (Section 6)
* $\blacksquare$
The implicit social ties are related to explicit indicators of collaboration.
We show that user pairs that share interests over items (i.e., annotate the
same items) have higher similarity regarding the groups they participate
together and higher semantic similarity of their tag vocabularies (even after
eliminating the portions of tagging activity that is related to the items they
tag in common). (Section 7).
These characteristics have practical implications for the design of mechanisms
that rely on implicit user interactions such as collaborative search [Evans
and Chi (2008), Yahia et al. (2008)], spam detection [Koutrika et al. (2008),
Neubauer et al. (2009)], recommendation [Santos-Neto et al. (2007), Jäschke et
al. (2007), Sigurbjörnsson and van Zwol (2008), Song et al. (2008)] and the
desig of incentives [Santos-Neto et al. (2010)] as outlined in Section 8.
## 2 Related Work
This section contextualizes this work along four main topics: i) general
characterization studies of peer production of information in tagging systems;
ii) characterization of the evolution of tag vocabularies; iii) graph-based
approaches to study activity similarity among users; and, iv) design of tag-
based support mechanisms.
### 2.1 General Characterization Studies
Previous characterization studies focusing on tagging systems vary along three
main aspects: i) the system analyzed from social bookmarking systems such as
del.icio.us, CiteULike, and Bibsonomy to content sharing systems like Flickr
and YouTube; and, ii) the focus of the characterization system-, tag-, item-
or user-centric analysis; and, iii) the method of investigation - qualitative
or quantitative research methods.
Nevertheless these works share the same intent: they address the high level
set of questions that relate to characterizing the usage patterns observed and
gaining insight into the underlying processes that generate them. These works
propose models that can be used to explain the observed characteristics of
tagging activity such as the incentives behind tagging, the relative frequency
of tags over time for a given item, the interval between tag assignments
performed by users and the distributions of activity volume.
Hammond et al. [Hammond et al. (2005)] is, perhaps, the first work to perform
an initial stuy and to discuss the characteristics of social tagging, its
potential, and the incentives behind tagging itself. The study comments on the
features provided by different social tagging systems and discusses
preliminary reasons that incentivize users to annotate and share content
online. Following on the question of incentives, Ames et al. [Ames and Naaman
(2007)] study tagging in online social media websites by interviewing 13 users
on the fundamental question of why do people tag? Based on user answers, the
authors suggest that tagging serves to support content organization or to
communicate aspects about the content. These actions can be either socially-
or personally-driven. More recent studies have followed the analysis of
incentives at a larger scale [Strohmaier et al. (2012)]. Our study supports
and, more importantly, extends these result by performing a large-scale user
behavior analysis (covering more than 700,000 users) in three tagging systems.
Although, we do not focus on the question of incentives particularly, the
quantitative analysis we present highlight and provide stronger evidence of
existing incentives hypothesized by previous works.
One of the first works on the quantitative characterization of tagging systems
is an item-centric characterization of del.icio.us that proposes the
Eggenberger-Polya’s urn model [Eggenberger and Polya (1923)] as an explanation
to the observed relative frequencies of tags applied to an item [Golder and
Huberman (2006)]. Cattuto et al. [Cattuto et al. (2007)] show in a tag-centric
characterization that the observed tag co-occurrence patterns in del.icio.us
is well modeled by the Yule-Simon’s stochastic process [Simon (1955)].
Similarly, Capocci et al. [Capocci et al. (2009)] show that the tag
interarrival time distribution follows a power-law. Using a different
approach, Chi and Mytkowicz [Chi and Mytkowicz (2008)] study the impact of
user population growth in the efficiency of tags to retrieve items in
del.icio.us. More recent works, focus on a characterization of social tagging
systems that analyzes the impact of using tagging on external applications
such as information retrieval and expert-generated content [Gu et al. (2011),
Li et al. (2011), Lu et al. (2010), Seki et al. (2010)].
Another stream of characterization studies focuses on user-centric analysis.
Nov et al. [Nov et al. (2008)] present a user-centric qualitative study on the
motivations behind content tagging in Flickr, where they suggest that users
tag content due to a mixture of individual like personal content organization,
and social motivation such as to help others in finding photos from a
particular place. In a previous study, we characterize the user-centric
properties of tagging activity from two social bookmarking systems designed
for academic citation management: CiteULike and Bibsonomy. The observations
suggest that user activity across the system follows the Hoerl model [Santos-
Neto et al. (2007)].
Our work complements and extends these previous studies as it investigates a
combination of user-, item- and tag-centric characteristics. Moreover, it
explores different aspects of tagging activity, such as the levels of item re-
tagging and tag reuse over time and the relationship between implicit and
explicit user ties in tagging systems. By applying a quantitative approach on
a broad population of users and multiple tagging systems, this study also
offers new insights on user behavior that complement previous qualitative work
by Ames and Naaman [Ames and Naaman (2007)].
### 2.2 Evolution of Users’ Tag Vocabularies
Tags represent to a certain extent the user perception or intended use of an
item. It is natural, therefore, to assume that the set of tags (i.e, tag
vocabulary) of a given user provides information about her topics of interest,
which is useful to design other mechanisms that support efficient content
usage such as recommender systems. Naturally, if tag vocabularies are stable
over time, that is, if inclusion of new tags and shifts in the tag usage
frequency observed in a vocabulary are rare, a mechanism can delay updates on
the vocabulary snapshot used to base its predictions. Indeed, this study shows
that this is the case (Section 5).
Previous studies on the characterization of the evolution of tag vocabulary
can be divided in two categories: first, studies that aim to quantify and
model the growth of tag vocabularies at both the system- and user-level
[Cattuto et al. (2007), Cattuto et al. (2009)]; and, second, studies that
estimate shifts in the tag vocabularies over time such as evolution of the tag
popularity distribution of item-level tag vocabularies [Halpin et al. (2007)],
and the variation of tag usage frequency across predefined tag classes [Golder
and Huberman (2006)] (i.e., factual tags, subjective tags and personal tags)
[Sen et al. (2006)].
In summary, these previous studies show that: i) the system-level and user-
level tag vocabulary growth is sublinear; ii) item-level tag popularity
distribution converges to a power-law; and, iii) the usage frequency of tag
categories shifts over time.
This study extends previous works by evaluating different facets of the
vocabulary evolution. First, this work goes beyond the estimation of
vocabulary growth, focusing on the evolution of tag usage frequency. Second,
it concentrates on individual, user-level tag vocabularies, as opposed to the
item-level vocabularies as in the previous studies. Finally, it uses a
different methodology to estimate the difference between tag vocabularies from
different points in time. Finally, we note that we use a different approach
that does not make assumptions about the categories of tags that appear in the
user tag vocabularies, an approach used by previous works.
### 2.3 Interest Sharing Analysis
An alternative way to characterize tagging systems is a graph-centric
approach. Two users are connected by a weighted edge with strength
proportional to the similarity between the tagging activities of these two
users. In this study, this similarity is referred to as an implicit social tie
between users. Note that other types of connections between users are
possible. In particular, we refer to explicit social ties as explicit
indicators of user collaboration, such as co-membership in discussion groups.
This approach has been used by Iamnitchi et al. [Iamnitchi et al. (2011),
Iamnitchi et al. (2004)] to characterize scientific collaborations, the web,
and peer-to-peer networks. The same model has been used by Li et al. [Li et
al. (2011)] to target the problem of finding users with similar interests in
online social networking sites. The authors use a del.icio.us data set and
define links between users based on the similarity of their tags. Their
conclusions support the intuition that tags accurately represent the content
by showing that tags assigned to a URL match to a great extent the keywords
that summarize that URL. Additionally, they design and evaluate a system that
clusters users based on similar interests and identifies topics of interests
in a tagging community.
Another focus of graph-centric characterizations is to determine structural
features in the graph formed by connecting users, items and tags based on
similarity. Hotho et al. [Hotho et al. (2006)] models a collaborative tagging
system as a tripartite network (the network connects users, items and tags in
a hypergraph) and design a ranking algorithm to enable search in social
tagging systems. Using the same tripartite network model, Cattuto et al.
[Cattuto et al. (2007)] study Bibsonomy and show the existence of small-world
patterns in such networks representing social tagging systems. Krause et al.
[Krause et al. (2008)] also explore the topology of a tagging system, but the
one formed by item similarity, to compare the folksonomy inferred from search
logs and tagging systems. Their results suggest that search keywords can be
considered as tags to URLs. More recently, Kashoob et al. [Kashoob and
Caverlee (2012)] characterizes and model the temporal evolution of sub-
communities in social tagging systems by looking into the similarity between
users vocabularies.
Our study differs from these previous investigations in three aspects: first,
the characterization of tagging activity similarity between users focuses on
the system-wide concentration and intensity of pairwise similarities, as
opposed to the topological characteristics. Second, our methodology provide a
principled way to test whether the user similarity observed in social tagging
systems is the product of interest sharing among users or chance. Finally, we
investigate possible correlations between the observed levels of activity
similarity between users (i.e., the implicit social ties) and the external
indicators of explicit collaboration (i.e., the explicit social ties) as co-
membership to discussion groups and semantic similarity of tag vocabularies
(Sections 6 and 7). We note that our methodology is inspired by a previous
work by Reichardt and Bornholdt that studies the patterns of similarity of
product preferences among buyers and sellers on eBay [Reichardt and Bornholdt
(2008)].
### 2.4 System Design
System characterization work is primarily motivated by it potential impact on
system design. Thus, several studies propose to exploit characteristics of
tagging systems to improve mechanisms such as recommendation [Jäschke et al.
(2007), Sigurbjörnsson and van Zwol (2008), Song et al. (2008)], spam
detection [Koutrika et al. (2008), Krause et al. (2008), Neubauer et al.
(2009), Noll et al. (2009)], top-k querying techniques [Schenkel et al.
(2008), Yahia et al. (2008)], and search and ranking [Hotho et al. (2006),
Yanbe et al. (2007), Heymann et al. (2008)].
The present work adds to these studies by providing evidence that tagging
activity can be useful to support such mechanisms. For example, the
characteristics of vocabulary evolution, as presented in Section 5, can be
used in the design of tagging systems in distributed platforms to adjust the
frequency in which the user profiles are updated across nodes/users.
## 3 Data Collection and Notation
This section describes the tagging systems analyzed as well as their
respective activity traces collected and analyzed in this study, and
introduces the basic notation used in the rest of this article.
We choose to analyze three tagging systems: CiteULike, Connotea and
del.icio.us. The first two are designed to help users organize references to
scientific publications, while the third is a social bookmarking tool for any
type of URL.
The main reason to focus on these systems is their popularity. Additionally,
studying systems that target different audiences enables a broader comparison
between tagging systems that target a niche of web users such as the
scientific community (i.e., CiteULike and Connotea) and a system where any web
user is a potential client (i.e., del.icio.us). Furthermore, the
characterizations of multiple classes of systems are complementary. Our
intuition is that a study of more specialized tagging systems – in this case,
for managing academic publications – may reveal social structures that are
harder to identify in generic systems such as del.icio.us.
CiteULike, Connotea, and del.icio.us target different types of content and
users, though all three systems can be described in terms of the same abstract
entities. In these systems each user maintains a library: a collection of
bookmarked items that, for the systems we study, are either citation records
linked to online articles or URLs to generic web pages. A user may assign tags
to items in her library. Additionally, a user may also tag items in other
user’s public library. Tags may serve to group items, as a form of
categorization, or to help find items in the future [Golder and Huberman
(2006), Nov et al. (2008)]. The tagging activity can be private (i.e., only
the user who generated the tags and items can access these annotations) or
public. The analysis presented in the next sections concentrates on the public
portion of the activity. A user can see what (public) tags other users
assigned to an item when she is tagging it, thus the user is able to reinforce
the choice of tags as appropriate by repeating the tags previously assigned to
that item.
In the case of CiteULike and Connotea, an item can be added to a user’s
library (an action often referred to as item posting) in three ways: i) browse
popular scientific literature portals (e.g., ACM Portal, IEEE Explorer,
arXiv.org) and use their features that automate item posting; ii) search for
items already present in other users’ libraries and add them to her own
library; and, iii) post a new item manually. In del.icio.us, users can use
automatic bookmarking features or manually bookmark URLs.
Table 3 presents a summary of the data sets used in this investigation. The
CiteULike and Connotea data sets consist of all tag assignments since the
creation of each system in late 2004 until January 2009. The CiteULike dataset
is available directly from its website. For Connotea, we built a crawler that
leverages Connotea’s API to collect tagging activity since December 2004 (no
earlier activity was retrieved). Finally, the del.icio.us dataset is available
at the website of a previous study by Görlitz et al. [Görlitz et al.
(2008)]333http://www.tagora-project.eu/.
Note that we do not have access to browsing or click traces. The traces
analyzed in this work contain records that indicate when items are annotated
with a given tag and who was the user, but the traces do not inform whether a
tag is subsequently used by a user to navigate through the system, for
example. The data sets are ’cleaned’ to reduce sources of noise, such as the
default tag ’no-tag’ in CiteULike, tags composed only of symbols and other
tags like the automatically generated ’bibtex-import’, which are clear
outliers in the popularity distribution.
Summary of data sets used in this study CiteULike Connotea del.icio.us
Activity Period 11/2004 – 01/2009 12/2004 – 01/2009 01/2003 – 12/2006 # Users
40,327 34,742 659,470 # Items 1,325,565 509,311 18,778,597 # Tags (distinct)
274,982 209,759 2,370,234 # Tag Assignments 4,835,488 1,671,194 140,126,555
#### Notation.
The rest of this paper uses the following notation to formally refer to the
entities that comprise tagging systems. A tagging system is composed of a set
of users, items and tags, respectively denoted by $U$, $I$, $T$. The tagging
activity in the system is a set of tuples $(u,i,w,t)$, where $u\in U$ is a
user who tagged item $i\in I$ with tag $w\in T$ at time $t$. The activity of a
user $u\in U$ can be characterized by $A_{u}$, $I_{u}$ and $T_{u}$, which are
respectively the set of tag assignments performed by $u$, the set of items
annotated, and the vocabulary or set of tags used by u. The user’s activity
from the beginning of the trace up to a particular point in time is denoted by
$A_{u}(t_{0},t)$, $I_{u}(t_{0},t)$ and $T_{u}(t_{0},t)$, where $t_{0}$ and $t$
are timestamps, $t_{0}$ represents the begin of the trace, and $t_{0}\leq t$.
The next sections focus on the analysis of the traces described above,
starting with the characteristics of peer production of information in these
systems.
## 4 Tag Reuse and Item Re-Tagging
Let a new item (or tag) be an item (or tag) that has never been used in an
annotation in the tagging system. If users introduce new items and tags
frequently, efficiently harnessing information based on collective action is
difficult, if not impossible. This is so because in this case information
about future user actions towards the annotation of an item or use of a tag is
then hard to predict: prediction relies on the historical use of items and
tags; new items or tags have no history in the system. Understanding the
degree to which items are repeatedly tagged and tags reused can therefore help
estimating the potential efficiency of techniques that rely on similarity of
past user activity (e.g., recommender systems). To this end, this section
addresses the following questions:
* Q1.1.
What is the rate of repeated item annotation and tag reuse? (Section 4.1)
* Q1.2.
Is the flow of new incoming users a major factor in the observed low rates of
repeated item annotation? (Section 4.2)
* Q1.3.
Are the reuse patterns we observe the result of different usage
characteristics of a group of high-volume power users, or are they pervasive
through the entire user population? (Section 4.3)
The rest of this section first formalizes the metrics item re-tagging and tag
reuse used to address these questions. Second, it characterizes the levels of
item re-tagging and tag reuse as well as the level of activity generated by
returning users. Finally, it discusses the implications of the usage
characteristics discovered.
### 4.1 Levels of Item Re-tagging and Tag Reuse
An item is re-tagged (repeatedly tagged) if one or more users tag it again
(with the same or different tags) after it was tagged for the first time.
Similarly, a tag is reused if it appears in the trace more than once (for the
same or different items) with different timestamps. We aim to determine which
portion of the activity falls in these categories.
###### Definition 4.1.
The level of item re-tagging during a time interval $[t_{f-1},t_{f})$ is the
ratio between the number of items tagged during that interval that have also
been tagged in the past $[t_{0},t_{f})$ to the total number of items tagged
during the interval $[t_{f-1},t_{f})$, as expressed by Equation 1. Tag reuse,
denoted by $tr(t_{f-1},t_{f})$, is similarly defined.
$ir(t_{f-1},t_{f})=\frac{|I(t_{0},t_{f-1})\cap
I(t_{f-1},t_{f})|}{|I(t_{f-1},t_{f})|}$ (1)
We use this definition to determine the aggregate level of item re-tagging and
tag reuse in CiteULike, Connotea and del.icio.us. Table 4.1 presents the
median daily item re-tagging and tag reuse over the entire traces (i.e., the
time interval $[t_{f-1},t_{f})$ encompasses a day). The results show that
CiteULike and Connotea have relatively low levels of item re-tagging while
del.icio.us has a higher level of item re-tagging, yet all three systems
present similarly high levels of tag reuse. We hypothesize that the observed
difference in item re-tagging between del.icio.us and their counterparts in
CiteULike and Connotea is due to the type of content users bookmark in each
system (with URLs of any type in the former, and academic literature in the
latter).
A summary of daily item re-tagging and tag reuse Re-Tagged Items Reuse Tags
Median Std. Dev. Median Std. Dev. CiteULike 0.15 0.07 0.84 0.12 Connotea 0.07
0.06 0.77 0.21 del.icio.us 0.45 0.17 0.86 0.07
To test whether these aggregate levels are a result of stable behavior over
time, Figure 1 presents the moving average (with a window size of 30 days) of
daily item re-tagging and tag reuse. Overall, these results show that all
three systems go through a bootstrapping period, after which they stabilize,
with the levels of item re-tagging and tag reuse stabilizing much sooner for
CiteULike and Connotea than that for del.icio.us. However, the tag reuse
levels have a similar evolution pattern in all three systems.
On the one hand, from the perspective of personal content management, the
observed levels of item re-tagging and tag reuse, together with the much
larger number of items than tags in these systems, suggest that users exploit
tags as an instrument to categorize items according to, for example, topics of
interest or intent of usage (’toread’, ’towatch’). On the other hand, from the
social (or collaborative) perspective, the relatively high level of tag reuse
taken together with the low level of item reuse suggests that users may have
common interest over some topics, but not necessarily over specific items.
These quantitative results suggest that tags are used in the way previous
exploratory qualitative study Ames and Naaman discusses [Ames and Naaman
(2007)].
Figure 1: Daily item re-tagging (left) and tag reuse (right). The curves are
smoothed by a moving average with window size $n=30$
A question that arises from the above observations is whether the levels of
item re-tagging and tag reuse are generated by the same user or by different
users. We observe that virtually none of the item re-tagging events are
produced by the user who originally introduced the item to the system:
generally, users do not add new tags to describe the items they collected and
annotated once.
As illustrated by Figure 2 (left), about $50\%$ of tag reuse is self-reuse
(i.e., the reuse of a tag by a user who already used it first). This level of
tag self-reuse indicates that users will often tag multiple items with the
same tag, a behavior consistent with the use of tagging for item
categorization and personal content management, as discussed above.
Additionally, the fact that half of the tag reuse is not self-reuse reinforces
the notion that users do share tags, which indicates potentially similar
interests. In Section 6, we further investigate this social aspect of tag
reuse by defining and evaluating interest sharing among users, as implied by
the similarity between users’ activity (i.e., tags and items).
### 4.2 New Incoming Users
To understand whether the observed low level of item re-tagging is due to a
high rate of new users joining the community, we estimate the levels of
activity generated by returning users (as opposed to new users that join the
community). Figure 2 (right) shows that, after a short bootstrap period, the
level of tagging activity generated by returning users remains stable at about
80% over the rest of the trace for both CiteULike and Connotea. In
del.icio.us, the percentage of activity represented by returning users is even
higher, with above 95% of daily activity performed by returning users.
Thus, the low levels of item re-tagging are the outcome of expanding interests
of returning users, instead of a constant stream of new users joining the
community and introducing new items.
Figure 2: Self-tag reuse (left) and daily activity generated by returning
users (right). The curves are smoothed by a moving average with window size
$n=30$
### 4.3 The Influence of Power Users
Finally, we investigate the influence of highly active users in the observed
item re tagging and tag reuse levels. To this end, we compare the observed
item re-tagging and tag reuse with and without the activity produced by such
power users. In this experiment, we define power users as the top-1% most
active users according to the number of annotations produced, and calculate
item re-tagging and tag reuse as before.
The experiments test the hypothesis that the levels of item re-tagging and tag
reuse are the same with and without the activity produced by these power
users. To this end, we apply the Kolmogorov-Smirnov test (KS-test) on the two
samples of activity (i.e., with and without the power users) with the null
hypothesis that the item re-tagging and tag reuse observed in the two samples
come from the same distribution (i.e, $H_{0}=$ the item re-tagging and tag
reuse levels are equally distributed with and without the power users). Using
the KS-test is appropriate as it does not require that the samples are drawn
from a normal distribution.
At a confidence level of $99\%(\alpha=0.01,p=1-\alpha)$, we can reject the
null hypothesis for all the systems, except the item re-tagging levels for
del.icio.us (see the p-values in Table 4.3). This means that removing the
activity produced by the power users leads to statistically different levels
of item re-tagging and tag reuse as indicated by the D-statistic in Table 4.3
(i.e., the maximum difference between the two distributions).
We hypothesize as the del.icio.us is a system that focuses on social
bookmarking of URLs of any type (as opposed to be restricted to scientific
articles in CiteULike and Connotea), removing the top 1% most active users do
not affect the observed levels of item re-tagging because some items will
attract the attention of many other less active users. These users contribute,
therefore, in large part for the observed levels of item re-tagging in
del.icio.us.
The statistical test results reject the hypothesis that the item re-tagging
and tag reuse observations with and without the power users are equal Re-
Tagged Items D-Statistic $p$-value $<$ CiteULike 0.03516 $2.2\times 10^{-16}$
Connotea 0.1889 $2.2\times 10^{-16}$ del.icio.us 0.0475 0.0768 Reuse Tags
D-Statistic $p$-value $<$ CiteULike 0.2858 $2.2\times 10^{-16}$ Connotea
0.2132 $2.2\times 10^{-16}$ del.icio.us 0.1371 $3.23\times 10^{-16}$
### 4.4 Summary and Implications
The observed user behavior impacts the efficiency of systems that rely on the
inferred similarity among items, such as recommender systems. On the one hand,
the relatively low level of item re-tagging suggests a highly sparse data set
(i.e., attempting to connect users based on similar items will connect only
few user pairs). A sparse data set poses challenges when designing recommender
systems as they typically rely on the similarity of users based on their past
activity to make recommendations.
On the other hand, the higher level of tag reuse confirms that analyzing tags
has the potential to circumvent, or at least alleviate, the sparsity problem
described above. The tags and users that relate to each item could not only
serve to link items and build an item-to-item structure, but could also
potentially provide semantic information about items. This information may
help, for instance, to design better bibliography and citation management
tools for the research community.
The results on analyzing the impact of power users in the observed levels of
item re-tagging and tag reuse support two ideas: first, the notion that some
users are instrumental on reducing the sparsity on tagging data sets (i.e.,
without power users, tags and items would be reused less, therefore
potentially lesser items would be connected through tags and users). In fact,
recommender systems benefit directly from the activity produced by such power
users, as they can connect more items via repeated tag usage. Second, the role
of power users differs from system to system, potentially due to effects of
population size and diversity of interests. In the largest and most diverse
system, we consider, reuse is a result of the activity of less active users
rather than only power users.
Finally, despite the sparse data set problem, the fact that users tend to
permanently add fresh content, as indicated by the low level of item re-
tagging, implies that the approach proposed by Yanbe et al. [Yanbe et al.
(2007)] would be useful in a search portal for academic content. They suggest
that content updated often in tagging systems can be used to improve the
freshness and relevance of search results produced by a search engine. Portals
for academic publications, such as Google Scholar, could exploit this fact to
improve the freshness and relevance of their search results by using a
combination of the PageRank ranking algorithm [Brin and Page (1998)] and
annotations from systems like CiteULike, Connotea and del.icio.us.
## 5 Temporal Dynamics of Users’ Tag Vocabularies
The item re-tagging and tag reuse analysis presented in the previous section
shows that users constantly produce new information in the system, by adding
both new items to their libraries and tags to their vocabularies, though at
different rates.
Although user tag vocabularies are constantly growing, it is unclear whether
the growth rate is uniform over time. More importantly, vocabulary growth may
or may not imply changes in the relative tag usage frequency by a given user.
Changes in these frequency can indicate shifts in user interests over time.
To better understand these aspects of tagging activity, this section
characterizes the temporal dynamics of user tag vocabularies. In particular,
we study the rate of change of user vocabularies over time, as it quantifies
the growth rate and changes in tag usage frequency for each user vocabulary.
Overall, the objective is to answer the following question:
* Q2.
How do users’ vocabularies change over time?
To address this question, this section quantifies the evolution of user tag
vocabularies by considering both their vocabulary growth and the tag usage
frequency at different points in time. More specifically, the experiments
first characterize the growth of user vocabularies, and, second, estimate the
distance between tag vocabularies as expressed by the distance between
snapshots of a user’s vocabulary at various points in time and her final
vocabulary. To take into account tag usage frequency the tags are ordered
according to their frequency (i.e., the number of times the user annotated an
item with the tag).
We note that this investigation is different from, but complements, previous
work [Kashoob and Caverlee (2012), Cattuto et al. (2007), Halpin et al.
(2007), Sen et al. (2006)]: first, it performs a user-centric vocabulary
analysis as opposed to a system-centric characterization; second, it studies
both growth and change in vocabulary content in contrast to only one of the
dimensions; and, finally, our characterization concentrates on the entire user
population, as opposed to subcommunities of interests (as indicated by tags)
or the evolution of such communities. Yet, the methodology we introduce in
this study can be applied in addition to those proposed in previous works for
a richer understanding of tag vocabulary evolution. The rest of this section
describes the methodology applied to identify the vocabulary evolution,
presents the results, and discusses its implications on system design.
### 5.1 Methodology
We introduce time in the definition of a user vocabulary by defining the tag
vocabulary of a user $T_{u}(s,f)$ as the set of tags used within the tag
assignment interval $[s,f]$. A particular case is $T_{u}(1,n)$ when $1$ and
$n$ indicate the timestamps of the first and the last observed tagging
assignment by user $u$, respectively. Thus $T_{u}(1,n)=T_{u}$ and represents
the user’s entire vocabulary.
Vocabulary growth. To analyze the vocabulary growth, we track the distribution
of growth rates across the user population for the duration of the traces. The
goal is to understand whether the growth rate changes according to the user
age. Therefore, we measure the following ratio:
$\frac{|T_{u}(1,k+1)|-|T_{u}(1,k)|}{|T_{u}(1,k+1)|}$ (2)
where $k\in[1,n]$ for all users in the system (i.e., $1$ and $n$ represent the
timestamp of the first and last tag assignments of a particular users,
respectively).
Vocabulary change. To measure the rate of change in the content of the
vocabularies, we consider vocabularies as sets of tags ordered in decreasing
order of usage frequency (i.e., number of times the tag was used to annotate
any item), and apply a distance metric as follows.
In this context, the final tag vocabulary, $T_{u}(1,n)$ is taken as a
reference point to study the evolution of tag vocabularies in terms of the
usage frequency of individual tags. The rationale behind the choice of this
reference is that according to the tag reuse results in Section 4, user tag
vocabularies are constantly growing. Therefore, it is unlikely that splitting
the activity trace into disjoint windows could help identifying meaningful
evolution patterns. Instead, we trace the evolution of a user’s tag vocabulary
by comparing the distance of incremental snapshots to her final vocabulary.
This way, it is possible to understand the rate of convergence of user
vocabularies over time. The experiment consists of calculating the distance
from the tag vocabularies $T_{u}(1,k)$ ($k\in[2,n]$), to the reference tag
vocabulary $T_{u}(1,n)$.
A traditional metric to calculate the distance between two lists of ordered
elements is the Kendall’s $\tau$ distance [Kendall (1938)], which considers
the number of pairwise swaps of adjacent elements necessary to make the lists
similarly ordered. However, Kendall’s $\tau$ distance assumes that both lists
are composed of the same elements. Since we are interested in the evolution of
tag vocabularies over time, this assumption is not valid in our case: tag
vocabularies are likely to contain different tags at different times due to
the constant inclusion of new tags.
Therefore, we apply the generalized Kendall’s $\tau$ distance, as defined by
Fagin et al. [Fagin et al. (2003)], which relaxes the restriction mentioned
above and accounts for elements that are present in one permutation, but are
missing in the other. Similar to the original Kendall’s $\tau$ distance, the
generalized version of the metric counts the number of pairwise swaps of items
necessary to make the lists similarly ordered. Additionally, the generalized
version counts the absence of items via a parameter $p$. This parameter can be
set between 0 and 1, which allows various levels of certainty about the order
of absent items. For example, in the case that two items are missing from one
list, but present on the other, setting $p=0$ indicates that there are not
enough information to decide whether the two items are in the same other or
not. Conversely, setting $p=1$ indicates that there is full information
available to consider the absence as an increase in the distance between the
lists. In the experiments that follow we use $p=1$.
### 5.2 Results and Implications
Our analysis filters out users that had negligible activity considering only
users with at least 10 annotations. This sample is responsible for
approximately $93\%$, $61\%$, and $90\%$ of the total system activity in terms
of tag assignments in CiteULike, Connotea, and del.icio.us, respectively.
#### V
ocabulary growth rate. Figure 3 illustrates vocabulary growth rate across the
user population in the three systems studied. The x-axis indicates categories
of users according to their age (i.e., number of days since their first
recorded tag assignment), while the y-axis indicates the growth rate relative
to each user vocabulary. For each of the systems studied we present two plots:
labeled ’median’ and ’90th percentile’. A point in the median plot indicates
that 50
The results show that, for the duration of the traces analyzed, the median
growth rate (Figure 3 – left) is relatively larger for older users. On the
other hand, if we take the 90th percentile growth rate (Figure 3 – right),
except the very young users, we observe that the rate is relatively the same
for all age groups with a slightly smaller rate for users in the middle of the
age spectrum. An important observation is that except for the growth rate of
young vocabularies, the $90^{th}$ percentile reaches a maximum rate of 0.1.
This means that for $90\%$ of users, their vocabularies growth rate upper
bound is $10\%$.
Figure 3: The vocabulary growth pattern in the systems studied: CiteULike
(left), Connotea (center), and delicious (left)
#### V
ocabulary change. Figure 4 changes the focus from growth rate to the rate of
change in users’ vocabularies. The figure presents the rate of change in the
contents of user vocabularies by taking into account the frequency of tags and
calculating the distance between vocabulary snapshots. The results show that
the distance from the vocabulary at earlier ages to its final state (i.e.,
Kendall-tau distance $t(T_{u}(1,k),T_{u}(1,n))$, where $k\in[2,n]$) decreases
rapidly in the first 100 days for 50% of users.
Figure 4: Rate of change in the tag usage frequency in the user vocabularies:
CiteULike (left), Connotea (center), and del.icio.us (right)
## 6 Interest Sharing
The analysis of item re-tagging and tag reuse in Section 4 suggests that the
observed level of re-tagging is the result of different users interested in
the same item and annotating it. We dub this similarity in item related
activity item-based interest sharing. Similarly, we dub the similarity in tag
related activity tag-based interest sharing. This section defines and
characterizes pairwise interest sharing between users as implied by their
annotation activity in CiteULike, Connotea and del.icio.us.
Analyzing interest sharing is relevant for information retrieval mechanisms
such as search engines tailored for tagging systems [Yahia et al. (2008), Zhou
et al. (2008)], which can exploit pairwise user similarity to estimate the
relevance of query results. This section focuses in particular on
characterizing interest sharing distributions across the user-pairs in the
system and addresses the following question:
* Q3.
How is interest sharing distributed across the pairs of users in the system?
However, this section goes one step further and studies the system-wide
characteristics of interest sharing and the implicit social structure that can
be inferred from it. Moreover, the next section investigates the relationship
between interest sharing (as inferred from activity similarity) and explicit
indicators of collaboration such as co-membership in discussion groups and
semantic similarity between tag vocabularies (Section 7).
### 6.1 Quantifying Activity Similarity
We use the Asymmetric Jaccard Similarity Index [Jaccard (1912)] to quantity
similarity between the item (or tag-) sets of two users. We note that previous
work (including ours) has used the Jaccard Index to quantify interest sharing:
Stoyanovich et al. [Stoyanovich et al. (2008)] used this index to model shared
user interest in del.icio.us and to evaluate its efficiency in predicting
future user behavior. Chi, Pirolli and Lam [Chi et al. (2007)] applied the
symmetric index to determine the diversity of users and its impact in a social
search setting. Our analysis considerably extends that performed in previous
work (as discussed in Section 2).
The formal definition of item-based interest-sharing metric is as follows (the
tag-based version is defined similarly and denoted by $w_{T}$):
###### Definition 6.1.
The level of item-based interest sharing between two users, $k$ and $j$, as
perceived by $k$, is the ratio between the size of the intersection of the two
item sets and the size of the item set of that user.
$\centering w_{I}(k,j)=\frac{|I_{k}\cap I_{j}|}{|I_{k}|}\@add@centering$ (3)
Equation 3 captures how much the interests of a user $u_{k}$ match those of
another user $u_{j}$, from the perspective of $u_{k}$. We opt for the
asymmetric similarity index rather than the symmetric version (which uses the
size of the union of the two sets as the denominator in Equation 3) to account
for the observation that the distribution of item set sizes in our data is
heavily skewed. As a result, the situation where a user has a small item set
contained in another user’s much larger item set happens often. In such cases,
the symmetric index would define that there is little similarity between
interests, while the asymmetric index accurately reflects that, from the
standpoint of the user with smaller item set, there is a large overlap of
interests. From the perspective of the user with a large item set, however,
only a small part of his interests intersect with those of the other user.
### 6.2 How is Interest Sharing Distributed across the System?
This section presents the distribution of pairwise interest sharing in
CiteULike, Connotea and del.icio.us. We first find that approximately 99.9% of
user pairs in CiteULike and del.icio.us share no interest over items (i.e.,
$w_{I}(k,j)=0$). In Connotea, the percentage is virtually the same: 99.8%. For
the tag-based interest sharing, the percentage of user pairs with no tag-based
shared interest (i.e., $w_{T}(k,j)=0$) is slightly lower: 83.8%, 95.8% and
99.7% for CiteULike, Connotea and del.icio.us, respectively. Such sparsity in
the pairwise user similarity supports the conjecture that users are drawn to
tagging systems primarily by their personal content management needs, as
opposed to the desire of collaborating with others.
The rest of this section focuses on the remaining user pairs, that is, those
user pairs that have shared interest either over items or tags. To
characterize these user pairs, we determine the cumulative probability
distribution (CDF) of item- and tag-based interest sharing for these sets of
user pairs in all three systems.
Figure 5: Distributions for item- and tag-based interest sharing (for pairs of
users with non-zero sharing) in CiteULike, Connotea and del.icio.us
Figure 5 shows that, in all three systems, the typical intensity of tag-based
interest sharing is higher than its item-based counterpart. This is not
surprising: after all, all three systems include two to three times more items
than tags. However, there is qualitative difference across systems with
respect the concentration of item-based and tag-based interest sharing levels,
with del.icio.us showing a much wider gap between the distributions.
The difference between the levels of item- and tag-based interest sharing
suggests the existence of latent organization among users as reflected by
their fields of interest. We hypothesize that this observation is due to a
large number of user pairs that have similar tag vocabularies regarding high-
level topics (e.g., computer networks), but have diverging interests in
specific sub-topics (e.g., internet routing versus firewall traversal
techniques), which could explain the relatively lower item-based interest
sharing compared to the observed tag-based interest sharing.
Finally, to provide a better perspective in the tag-based interest sharing
levels, we compare the observed values to that of controlled studies on the
vocabulary of users describing computer commands [16]. The tag-based interest
sharing level, as observed in Figure 6, is approximately 0.2 (or less) for 80%
of the user pairs that have some interest sharing, while Furnas et al. [16]
show that in an experiment where participants are instructed to provide a word
to name a command based on its description such that it is an intuitive name
and more likely to be understood by other people, the ratio of agreement
between two participants is in the interval [0.1, 0.2] (i.e., number of times
two participants use the same word divided by the total number of participant
pairs).
These observations suggest that observed tag-based interest sharing is due to
conscious choice of terms from vocabularies that are shared among users,
rather than by chance. We look more closely into this aspect in the next
section by constructing a baseline to compare the observed interest sharing
levels to that of a random null model.
### 6.3 Comparing to a Baseline
The goal of this section is to better understand the interest sharing levels
we observe. In particular, we focus on the following high-level question:
* Q4.
Do the interest sharing distributions we observe differ significantly from
those produced by random tagging behavior?
For this investigation, we compare the observed interest sharing distribution
to that obtained in a system with users that have an identical volume of
activity and the same user-level popularity distributions for items or tags,
but do not act according to their personal interests. Instead, in the random
null model (RNM) [Reichardt and Bornholdt (2008)], the chance that a user is
interested in an item or tag is simply that item or tag’s popularity in the
user’s vocabulary.
The reason to perform this experiment is the following: we aim to validate our
intuition that the interest sharing metric distils useful user behavior
information. If the interest-sharing levels we observe in the three real
systems at hand are more concentrated than those generated by the RNM, then
interest sharing metric captures relevant information about similarity of user
preferences, rather than simply coincidence in the tagging activity.
To reiterate, the random null model (RNM) is produced by emulating a tagging
system activity that preserves the main macro-characteristics of the real
systems we explore (such as the number of items, tags, and users, as well as
item and tag popularity, and user activity distributions), but where users
make random tag assignments. As such, random assignments are used here as the
opposite of interest-driven assignments.
To test our hypothesis, we compare the two sets of data (real and RNM-
generated) in terms of the numbers of user pairs with non-zero interest
sharing and the interest-sharing intensity distribution. Because of its
probabilistic nature, we use the RNM to generate five synthetic traces
corresponding to each of the real systems we analyze. For the rest of this
section, the RNM results represent averages over the five RNM traces for each
system. We confirmed that the five synthetic traces represent a large enough
sample to guarantee a narrow 95% confidence interval for the average interest
sharing observed from the RNM simulations.
Our data analysis shows that the observed interest sharing deviates
significantly from that generated by random behavior in two important
respects.
First, interest sharing (and, consequently, the similarity between users) is
more concentrated in the real systems than in the corresponding simulated RNM.
More specifically, the number of user pairs that share some item-based
interest (i.e., $w_{I}(k,j)>0$) is approximately three times smaller in the
real systems than in the RNM-generated ones. Tag-based interest sharing
follows a similar trend.
Second, interest sharing distribution deviates significantly from that
produced by a RNM. We compare the cumulative distribution function (CDF) for
the interest sharing intensity for the user-pairs that have some shared
interest (i.e., $w(k,j)>0$). Figure 6 presents the Q-Q plots that directly
compare the quantiles of the distributions of interest-sharing levels derived
from the actual trace and those derived from the simulated RNM. A deviation
from the diagonal indicates a difference between these distributions: The
higher the points are above the diagonal, the larger the difference between
the observed interest-sharing levels and those generated by the RNM.
Figure 6: Q-Q plots that compare the interest sharing distributions for the
observed vs. simulated (i.e., the RNM model) for CiteULike (left) and Connotea
(right)
We note that the only interest-sharing distribution that is close to the one
produced by the RNM is for Connotea’s tag-based interest sharing (Figure 6).
However, there is still a significant deviation from randomness: the real
activity trace leads to three times fewer user-pairs that share interest than
the corresponding RNM.
### 6.4 Summary and Implications
This section provides a metric to estimate pairwise interest sharing between
users, offers a characterization of interest-sharing levels in CiteULike and
Connotea; and investigates whether the observed interest sharing in these
systems deviates from that produced by chance, given the amount of activity
users had. Such reference is given by a random null model (RNM) that preserves
the macro characteristics of the systems we investigate, but uses random tag
assignments.
The comparison highlights two main characteristics of the interest sharing:
first, interest sharing is significantly more concentrated in the real traces
than in the RNM-generated activity: in quantitative terms, three times fewer
user pairs share interests in the real traces. Second, most of the time, for
the user pairs that have non-zero interest sharing the observed interest-
sharing intensity is significantly higher in each real system than in its RNM
equivalent.
We conjecture that a possible explanation for these observations is as
follows. Let us consider that the set of tags that can be assigned to an item
is largely limited by the set of topics that item is related to. In this case,
intuitively, the probability of choosing a tag is conditional to the set of
topics the item is related to. At one extreme, the maximum diversity of topics
occurs when there is a one-to-one mapping between topics and tags, that is,
when each tag introduces a different topic. The RMN simulates the other
extreme, a single topic that encompasses all tags in the system.
However, in real systems, the interests for each individual user are limited
to a finite set of topics, which is likely to determine their tag vocabulary.
This leads to a concentration of interest sharing, as implied by the tag
similarity, on few user pairs, yet at higher intensity than that produced by
the RNM.
Finally, and most importantly, the divergence between the observed and the
RNM-generated interest sharing distributions shows that activity similarity,
our metric to quantify interest sharing intensity, embeds information about
user self-organization according to their preferences. This information, in
turn, could be exploited by mechanisms that rely on implicit relationships
between users. The next section seeks evidence about the existence of such
information by analyzing the relationship implicit user ties, as inferred from
the similarity between users’ activity, and their explicit social ties, as
represented by co membership in discussion groups or semantic similarity
between tag vocabularies.
## 7 Shared Interest and Indicators of Collaboration
The previous section characterizes interest sharing across all user pairs in
each system and suggests that it encodes information about user behavior, as
its distribution deviates significantly from that produced by a random null
model.
This section complements this characterization and evaluates whether the
implicit user relationships that can be derived from high levels of interest
sharing correlate with explicit online social behavior. More specifically,
this section addresses the following question:
* Q5.
Are there correlations between interest sharing and explicit indicators of
social behavior?
Before starting the analysis, it is important to mention that the number of
externally observable elements of user behavior to which we have access is
limited by the design of the tagging systems themselves (e.g., the tagging
systems collect limited information on user attributes) and by our limited
access to data (e.g., we do not have access to browsing traces or search
logs).
One CiteULike feature, however, is useful for this analysis: CiteULike allows
users to explicitly declare membership to groups and to share items among a
selected subset of co-members – an explicit indicator of user collaboration in
the system. Thus, this feature enables an investigation about the relationship
between interest sharing and group co-membership (which we assume to indicate
collaboration). We note that a similar experiment could be performed using the
explicit friendship links in del.icio.us, for example. However, this data is
not available to our study.
Along the same lines, we use a second external signal: semantic similarity
between tag vocabularies. More specifically, we test the hypothesis that item-
based interest sharing relates to semantic similarity between user
vocabularies. The underlying assumption here is that users who (have the
potential to) collaborate employ semantically similar vocabularies.
This section presents the methodology and the results of these two experiments
that mine the relationship between interest sharing and indicators of
collaboration. In brief, our conclusions are:
* $\blacksquare$
User pairs with positive item-based interest sharing have a much higher
similarity in terms of group co-membership and semantic tag vocabulary, than
users who have no interest sharing.
* $\blacksquare$
On the other side, we find no correlation between the intensity of the
interest sharing and the collaboration levels as implied by group co-
membership or vocabulary similarity.
### 7.1 Group Membership
In CiteULike, approximately 11% of users declare membership to one or more
groups. While the percentage may seem small, they are the most active users:
these users generate 65% of tag assignments, and introduce 51% of items and
50% of tags. For this section we limit our analysis to the user pairs for
which both users are members of at least one group. Also, the analysis focuses
on groups that have two or more users (about 50% of all groups) as groups with
only one user are obviously not representative of potential collaboration.
The goal is to explore the possible relationship between item-based interest
sharing and co-membership in one or more groups. Let $H_{u}$ be the set of
groups in which the user $u$ participates. We determine the group-based
similarity $w_{H}(u,v)$ between two users $u$ and $v$ using the asymmetric
Jaccard index, similar to the item-based definition in Eq. 3, but considering
the sets of groups users participate in. Based on this similarity definition,
we study whether the intensity of item-based interest sharing between two
users with non-zero interest sharing (i.e., $w_{I}(u,v)>0$) correlates with
group membership similarity.
We find no correlation between $w_{I}(u,v)$ – the item-based similarity – and
$w_{H}(u,v)$ – the group-based similarity. More precisely, Pearson’s
correlation coefficient is approximately $0.12$, and Kendall’s $\tau$ is about
$0.05$. This is surprising as one would expect that being part of the same
discussion groups is a good predictor to the intensity in which users share
interest over items. Therefore, we look into these correlations in more
detail.
To put these correlation results in perspective, we look at group similarity
for two distinct groups of user pairs: those with no item-based interest
sharing ($w_{I}(u,v)=0$) and those with some interest sharing
($w_{I}(u,v)>0$). We observe that, although the group information is
relatively sparse, pairs of users with positive interest sharing are more
likely to be members of the same group than the user pairs where
$w_{I}(u,v)=0$. In particular, 4% of the user pairs with $w_{I}(u,v)>0$ have
$w_{H}(u,v)>0.2$, while twenty times fewer user pairs with $w_{I}(u,v)=0$ have
$w_{H}(u,v)>0.2$.
These observations suggest that activity similarity is a necessary, but not
sufficient condition for higher-level collaboration, such as participation in
the same discussion groups. Although users share interest over items, and may
implicitly benefit from each other tagging activity (e.g., using one another’s
tags to navigate the system), this may not directly lead to users actively
engaging in explicit collaborative behavior. Conversely, the lack of interest
sharing strongly suggests a lack of collaborative behavior.
### 7.2 Semantic Similarity of Tag Vocabularies
This section complements the previous analysis on the relationship between
item-based interest sharing and collaboration indicators via group co-
membership. It investigates the potential relation between item-based interest
sharing of a pair of users and the semantic similarity between their tag
vocabularies, that is, the set of tags each has applied to items in its
library. Since, through this experiment we aim to understand the potential for
user collaboration through similar vocabularies, when comparing vocabularies
for a user pair, we exclude the tags applied to the items the two users have
tagged in common – a these tags have a likely high similarity.
The rest of this section is organized as follows: it presents the metric used
to estimate the semantic similarity of two tag vocabularies; discusses
methodological issues; and, finally, presents the evaluation results.
Estimating semantic similarity: We use the lexical database WordNet to
estimate the semantic similarity between individual tags. WordNet consists of
a set of hierarchical trees representing semantic relations between word
senses such as synonymy (the same or similar meaning) and hypernymy/hyponymy
(one term is a more general sense of the other). Different methods have been
implemented to quantify semantic similarity using WordNet. In particular,
WordNet::Similarity – a Perl module – provides a set of semantic similarity
measures [Pedersen et al. (2004)].
For our experiments, we use the Leacock-Chodorow similarity metric [Budanitsky
and Hirst (2006)], as previous experiments, based on human judgments, suggest
that it best captures the human perception of semantic similarity. The metric
is derived from the negative log of the path length between two word senses in
the WordNet ”is-a” hierarchy, and is only usable between word pairs where each
has at least one noun sense.
Additionally, we explore a method to extend coverage to a larger subset of
users’ tag vocabularies, with an approach that builds on the yago ontology,
developed and described by Suchanek et al. [Suchanek et al. (2007), Suchanek
et al. (2008)]. yago (”Yet Another Great Ontology”) is built from the entries
in Wikipedia 444http://wikipedia.org, a collaborative online encyclopedia. The
standardized formatting of Wikipedia makes it possible for information to be
automatically extracted from the work of thousands of individual contributors
and used as the raw material of a generalized ontology. The primary content of
the yago ontology is a set of fact tables consisting of bilateral relations
between entities, such as ”bornIn”, a table of relations between persons and
their birthplaces. Five of the relations are of particular interest to us
because they contain links between entities mentioned in Wikipedia and terms
found in WordNet. In this way, we are able to identify some tags as probable
personal, collective, or place names, and use the WordNet links from yago to
map these on to a set of corresponding WordNet terms.
We used a merged tag vocabulary from CiteULike and Connotea datasets. A little
over 13% of the tags in the merged repertoire had direct matches in WordNet.
By adding the tags matched through comparison with yago’s WordNet links, this
was increased to 28.6% of unique tags applied by users of both systems. Note,
however, that these tags cover up to 75% of the tagging activity in the two
systems, as shown in Table 7.2.
The share of tagging activity captured by the tag vocabularies in CiteULike
and Connotea that is found in WordNet and WorldNet combined with yago lexical
databases. As we use an anonymous version of the del.icio.us dataset, with all
the users, items, and tags identified by numbers, this precluded us to perform
the same analysis using WordNet and yago for del.icio.us. WordNet only
WordNet \+ yago CiteULike 62.1% 79.5% Connotea 51.3% 65.3% Combined 57.4%
73.4%
In order to match tags gathered from the two systems with corresponding
entities in yago, we replaced all non-ASCII characters, such as accented
letters, with their nearest ASCII equivalents; removed all characters other
than letters and numerals, and reduced all the yago entities to lower case
(tags from both systems being already reduced to lower case). We allowed
partial matches, but required that the end of a tag correspond to a word
boundary in the yago entity or vice-versa. Thus, we were able to develop a
mapping between about 58,600 tags from the merged vocabulary and 57,900
distinct WordNet senses, with most tags matching multiple WordNet senses.
Given that the addition of WordNet terms identified by mapping through yago
effectively increases the total depth of the tree being considered, the
Leacock-Chodorow algorithm required that we adjust all tag pair similarity
scores accordingly in order to fairly compare the WordNet-only and
WordNet+yago scores. The maximum possible similarity with WordNet alone is
$\log(1/40)$ or $3.689$; whereas with WordNet \+ yago it is $\log(1/42)$ or
$3.738$.
We define the similarity $sim(t_{1},t_{2})$ between two tags $(t_{1},t_{2})$
as the maximum Leacock-Chodorow similarity between every available noun sense
of $t_{1}$ and $t_{2}$. Thus, we define the semantic similarity between the
tag vocabularies $T_{u}$ and $T_{v}$ of two users, $u$ and $v$, as perceived
by $u$, is denoted by $s(u,v)$, and determined by the ratio between the sum
over the pairwise tag similarities and the size of $u$’s vocabulary, as
expressed by Eq. 4 below.
$\centering sim(u,v)=\frac{\sum_{t_{1}\in T_{u},t_{2}\in
T_{v}}sim(t_{1},t_{2})}{|T_{u}|}\@add@centering$ (4)
We then calculate the corresponding value of $s(v,u)$ by reversing the $u$ and
$v$ terms in Eq. 4 and record the smaller of the two – i.e.
$min(s(u,v),s(v,u))$ – as the undirected tag vocabulary similarity between the
two users $u$ and $v$. We note that this metric is based on the Modified
Hausdorff Distance (MHD) [Dubuisson and Jain (1994)].
#### Methodological issues.
There are three practical issues regarding our experimental design that
deserve a note. First, to avoid bias, if two users assigned the same tags to
the same item, we omit these tags from their vocabularies, before determining
the aggregate similarity. By eliminating from vocabularies the tags that have
been used on exactly the same items, we eliminate the tags on which the two
users have most likely already converged. We look only at the remaining parts
of the vocabularies where convergence is not apparent. Second, the Leacock-
Chodorow similarity metric only considers words that have noun senses in
WordNet, because it is calculated from paths through the ”is-a” hierarchy,
only defined for nouns. Tags in both systems considered may include words or
phrases from any language, abbreviations, or even arbitrary strings invented
by the user, while WordNet consists mainly of common English words. A third
methodological issue was that matching tags to yago entries, in some cases,
returned an unmanageably large set of distinct WordNet senses. We accordingly
eliminated those tags that were above the 99th percentile in distinct WordNet
senses matched, which were those returning more than 167 distinct senses.
Figure 7: CDFs of tag vocabulary similarity for user pairs with positive
(bottom curve) and zero (top curve) activity similarity. CiteULike (left);
Connotea (right)
#### Results.
We use sampling to test, in both CiteULike and Connotea, whether there is a
significant difference in tag vocabulary similarity between two sets of user
pairs: one where all users have no item-based interest sharing and one with
positive item-based interest sharing (we sample each group with $n=4000$
pairs). This analysis shows that the vocabularies of user pairs with interest
sharing are significantly more similar than those of user pairs with no
interest sharing (Figure 7). The median vocabulary similarity for user pairs
with positive interest sharing $\mu_{c}=2.112$ ($\pm 0.02$, $99\%$ c.i.) is
about 1.6 times that of user pairs with no interest sharing $\mu_{u}=1.308$
($\pm 0.04$, $99\%$ c.i.). This salient difference in the vocabulary
similarity suggests that the item-based interest sharing embeds information
about the ”language” shared by the users to describe the items they are
interested in.
### 7.3 Summary and Implications
This section takes a first step towards understanding the relationship between
the implicit user ties, as inferred from pairwise interest sharing, and their
explicit social ties. First, we look at correlations between the item-based
interest sharing and the group-based similarity. The observations indicate
that although the intensity of item-based activity interest sharing does not
correlate with explicit collaborative behavior, as implied by group co-
membership, user pairs with some interest sharing are more than one order of
magnitude more likely to participate in similar groups.
Second, we evaluate the relationship between item-based interest similarity
and the semantic similarity of tag vocabularies. We discover that, although
the two do not yield a Pearson’s correlation, item-based interest similarity
does embed information about the expected semantic similarity between user
vocabularies.
These results have implications on the design of mechanisms that aim to
predict collaborative behavior, as these mechanisms could exploit item-based
similarity to set expectations about group-based and vocabulary-based
similarity. Moreover, assuming that the tagging activity characteristics of
spammers differ from legitimate users, one could use deviations from observed
relationship between item-based similarity and the two indicators of
collaborative behavior presented here to detect malicious user behavior.
## 8 Conclusions
Tagging systems have been widely adopted by today’s World Wide Web. These
systems provide users with the ability to annotate and share content. The peer
produced annotations (or tags) and shared items create a valuable pool of
metadata for mechanisms that aim to harness or predict user preferences such
as recommendation systems and social search. However, to efficiently use this
information, it is first necessary to understand the usage characteristics of
tagging systems.
To this end, this work studies two major aspects of usage characteristics in
tagging systems: i) the dynamics of peer production of information; and, ii)
the relationship between implicit and explicit social ties between users.
To address the first aspect, this work analyzes the user behavior
characteristics at the individual and aggregate level in three tagging systems
that focus on distinct applications: CiteULike and Connotea – personal
management of academic citation records; and, del.icio.us – a popular social
bookmarking system.
In particular, the characterization of peer production of information focuses
on three user activity indicators: i) item re-tagging, a measure for the
degree to which users re-tag the items already existing in the system; ii) tag
reuse, a measure for the degree to which users reuse a tag perform new
annotations; and, iii) the temporal dynamics of user tag vocabularies, a user-
centric analysis of the tag vocabulary evolution over time.
To address the second aspect, we define interest sharing, a metric the
activity similarity between a pair of users. Through experiments that compare
with a random null model, we show that interest sharing metric captures
relevant information about similarity of user preferences, rather than simply
coincidence in the tagging activity. Additionally, we present an analysis of
the relationship between the implicit ties, as represented by activity
similarity between users with respect to their tagging activity, and more
explicit ties, such as co-membership in discussion groups and semantic
similarity of tag vocabularies.
A summary of the main findings of this study is the following:
1. 1.
The qualitative characteristics of peer production of information are similar
across different systems, but they differ quantitatively, as indicated by the
relative levels of item re-tagging and tag reuse.
2. 2.
Interest sharing (the metric that quantifies the similarity between pairs of
user’s tagging activity) is significantly concentrated on a small fraction of
user pairs. This is a characteristic of intelligent choices made by users in
tagging systems, and not an implicit result of tagging activity volumes and
tag/item popularity distributions as indicated by a comparison of the observed
interest sharing distribution to that of a system with the same macro
characteristics yet where random tag assignments are used.
3. 3.
User tag vocabularies are constantly growing, but at different rates depending
on the age of the user. However, despite the constant increase in size, the
relative usage frequency of tags in a vocabulary tends to converge to a stable
ranking at early stages of a user’s lifetime in the system, as observed in the
analysis of the distance between tag vocabularies at different points in time
and the user tag vocabulary considering the entire trace.
4. 4.
The implicit and explicit social ties are related, as suggested by the
observed higher intensity in group co-membership and tag vocabulary semantic
similarity for those user pairs that share interest over items.
The implications of these results have ramifications along multiple fronts of
system design including: i) recommender systems – as the concentration of
interest sharing on a small fraction of user pairs indicate a highly sparse
dataset, it demands more sophisticated techniques to achieve better precision
and recall results; ii) malicious user detection – as spam detection
mechanisms, specially tailored for tagging systems, could use deviations from
the characteristics of interest sharing of a non-malicious user population to
detect malicious users; iii) design of distributed infrastructure for tagging
systems – the characteristics of the evolution of user tag vocabularies
together with the ’sparse’ interest sharing support the intuition that it is
possible to design and implement a distributed infrastructure to support
tagging features, as it would imply in low communication cost among the parts.
## 9 Acknowledgments
The authors would like thank the Research Computing Center at the University
of South Florida for allowing us to use their infrastructure in part of our
experiments. Elizeu Santos-Neto was partially supported by the BC Innovation
Council Fellowship and AUCC LACREG Exchange Grant. Finally, thanks to Lauro
Beltrão Costa and Abdullah Gharaibeh for insightful discussions.
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|
arxiv-papers
| 2013-01-25T23:31:28 |
2024-09-04T02:49:40.771724
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Elizeu Santos-Neto, David Condon, Nazareno Andrade, Adriana Iamnitchi,\n Matei Ripeanu",
"submitter": "Elizeu Santos-Neto",
"url": "https://arxiv.org/abs/1301.6191"
}
|
1301.6254
|
# Elastic properties and phase stability of AgBr under pressure.
P.T. Jochym [email protected] K. Parlinski Institute of Nuclear
Physics,
ul. Radzikowskiego 152, 31–342 Cracow, Poland
###### Abstract
Ab initio calculations have been used to derive the elastic constants,
equation of state and free energy of NaCl, KOH and CsCl-type structures of
AgBr. The elastic constants have been derived by the stress–strain relation.
The shear elastic constants C44 has proved to be exceptionally soft. We have
found that at temperature T=0 the phase transition from the low pressure NaCl-
type to the high-pressure CsCl-type structure goes over a monoclinic KOH-type
structure. The free energy comparison indicates that the monoclinic phase
ranges from 8 to 35 GPa.
###### pacs:
05.70.F, 64.60, 64.70, 71.15.Mb
Silver halide crystals are of considerable interest, since they exhibit
deviation from the ideal ionic character due to the presence of the
d-electrons in Ag+ ions. Numerous studies of these crystals have been
undertaken in the past years using various forms of interaction potentials
including three-body interactions and van der Waals forces,singh ; gupta as
well as the ab initio total energy pseudopotential calculations.chelikowsky
There are also many experimental works studying various aspects of these
solids.loje ; bridgmanA ; bridgmanB ; slykhouse ; vaidya ; fjeldly ; mellander
; ves The results of studies using phenomenological potentials generally show
good agreement with experiment.gupta This studies use sophisticated schemes
involving three body interactions and van der Waals potentials. The ab initio
results for transition pressure seems to be less accuratechelikowsky . There
is one ab initio study of phase stability of silver bromide AgBr.nunes
Nevertheless, the transition mechanism is still unclear for this compound.
Several models have been proposed, bridgmanA ; bridgmanB ; slykhouse ; nunes ;
hull ; kusaba ; schock but the transition mechanism remains still an open
problem. The ab initio study by Nunes, Allen and Martins,nunes which
investigated the energy of several candidate structures, proposed an
intermediate trigonal phase with space group P$3_{1}21$. In contrary recent
experimental work of Hull and Keenhull suggested an existence of an
intermediate phase with a monoclinic, KOH-type structure and shear instability
as the transition mechanism. The silver bromide AgBr is stable at ambient
temperature and pressure in the rocksalt structure Fm$\bar{3}$m. In ambient
temperature and high pressures it has a second stable phase with CsCl
structure Pm$\bar{3}$m. The phase transition in between these phases has been
reportedbridgmanB ; slykhouse ; bassett ; riggleman to occur at Pt=8.3 GPa.
In this paper we are studying elastic properties and phase stability of the
AgBr as a function of pressure. For NaCl and CsCl structures we have scanned
pressures in the interval P=0–200 GPa, and calculated the free energies and
elastic constants. In the range of P=0–40 GPa we have also studied the phase
stability of the free, non-constrained by any symmetry, AgBr crystal ,
searching for any intermediate phases as well as general transition mechanism.
The non-constrained crystal was treated as having P1 symmetry, however, its
unit cell content was not allowed to change. Hence, phase transitions to
phases with multiplication unit cells were not taken into account.
## I Methodology
The calculations have been performed using Density Functional Theory (DFT)
approach with ultrasoft pseudopotentials and Generalised Gradient
Approximation (GGA).lin ; goniakowski ; payene We have used CASTEPcastep
implementation of this method and pseudopotentials provided with this package.
The elastic constants of the NaCl phase have been also verified by the
independent calculation using VASPvasp . The electronic minimisation used
density mixing scheme and pseudopotentials parametrised in the reciprocal
space. The summation over the Brillouin zone has been performed with weighted
summation over wave vectors generated by Monkhorst-Pack scheme.pack All
computations were done in $1$$\times$$1$$\times$$1$ supercell with 8 or 2 or 8
atoms for NaCl or CsCl or KOH structures, respectively. Some tests for NaCl
has been carried out on the primitive, others on the face–centered cubic unit
cell.
Due to high demands of the accuracy of the free energy, we have conducted
extensive tests of convergence of the energy and pressure with respect to
energy cutoff and Brillouin zone grid spacing parameters. Thus, we have
performed a series of calculations with cutoff energies in the range of
$E_{cut}$=300 – 450eV and k-space grid spacing in the range $\Delta$k=0.02 –
0.1 Å-1. In the light of obtained results we have decided to use in our
computations a 0.4 Å-1 grid spacing and 340eV energy cutoff. The chosen grid
corresponds to 32 wave-vectors for NaCl structure and 256 wave-vectors for
CsCl structure. Such a large number of wave-vectors made calculations rather
time-consuming. But since we are comparing energies of different structures
with different unit cell volumes and Brillouin zone shapes, sizes and
k-sampling, we need a very good convergence of energy. At zero pressure the
lattice constants of the optimised NaCl structure is a= 5.7859 Å, while the
experimental lattice constants at P=0 GPa and T=300 K is A=5.7745 Å.
## II Free Energy and Equation of State
Figure 1: Lower part: Free energy of AgBr in NaCl (+, solid line) and CsCl
($\times$, dashed line) structures. Upper part: Free energy difference
F${}_{\mathrm{C}sCl}-$GNaCl between CsCl and NaCl structures as a function of
pressure. The vertical arrow denotes predicted phase transition pressure.
To investigate the phase stability of the crystal and to estimate the phase
transition pressure, one must evaluate the free energy of all involved phases
and choose the one with the lowest energy as stable. Since in ab initio
calculations the temperature T=0, the free energy is: $F=E+PV$. We have
performed calculations of the crystal free energy $F$ for several pressures
from 0 to 50 GPa for both NaCl and CsCl structures. Fig 1 shows the results of
these computations. In the lower part the absolute values of free energies per
atom are plotted together with cubic polynomials fitted to these data. In the
upper part the difference of $F_{CsCl}-F_{NaCl}$ is plotted against pressure.
The difference changes sign at the pressure Pt=27 GPa (arrow). We see that our
calculations correctly predict that the rocksalt structure is more stable at
low pressure and that at higher pressures the stability passes to CsCl
structure . The large difference between experimental Pt=8.3 GPa and
calculated Pt=27 GPa transition pressure could be attributed to a few factors.
The main reason could be related to the appearance of the intermediate KOH-
type structure, discussed below. Here, we mention other possible factors.
First, the experimental data bridgmanA ; bridgmanB are measured only at
ambient temperature, while the calculations were performed at T=0. At finite
temperature the entropy contribution to the Gibbs free energy $G=E+PV-TS$ can
also be different for the two structures, since the closest coordination
shells contain different number of ions. Another reason could be related to
the fact that the van der Waals interaction is neglected in DFT calculations,
and that leads to under-binding of the crystal. This effect has already been
noticed for silver iodide AgI crystal, where DFT predicts the phase transition
pressure to be 10 GPa,gupta ; chelikowsky while experimentally it is 4 GPa.
Figure 2: Variation of relative volumes with pressure for NaCl(+) and CsCl
($\times$) structure. Experimental points ($\bullet,\blacktriangle$) are taken
from BridgmanbridgmanA ; bridgmanB and Vaidyavaidya respectively. The dashed
line shows results of theoretical study of Gupta and Singh gupta . The lines
are present calculations.
The data used to calculate free energy provide also the equation of state of
the crystals. Fig. 2 shows the calculated equation of state for NaCl and CsCl
structures as well as the experimental data measured by Bridgman and
Vaidya.bridgmanA ; bridgmanB ; vaidya We have also included in figure 2 the
results of phenomenological study of Gupta and Singh.gupta This work was able
to predict the transition pressure much better, but it fails to reproduce the
volume compression of the phase transition. Present calculations show
excellent agreement with experimental data for the low pressure range up to 6
GPa. In this interval the relative error in V(P)/V(0) ratio is below 0.5%. It
should be noted that the Bridgman measurements, are limited to 10 GPa and it
would be interesting to increase the pressures to at least 15 GPa. We have
also fitted the Murnaghan equation of state to rocksalt structure data below
15 GPa:
$V=V_{0}\exp\left[-\frac{\ln\left(\frac{PA_{2}}{A_{1}}+1\right)}{A_{2}}\right]$
(1)
achieving good fit for $A_{1}$=41.01 GPa and $A_{2}$=4.35.
As seen from Fig. 2 the calculated relative volume change from NaCl to CsCl
structure at the transition pressure Pt=8.3 GPa is a 6% compression. We are
not aware of any experimental measurements of this value, however, we can
estimate it from the Bridgman databridgmanB to be at least 3%.
## III Elastic constants
Figure 3: Bulk modulus of AgBr calculated for NaCl structure from $P(V)$
function (solid line) and $c_{ij}$ data (+) from Fig. 4. The bulk modulus for
CsCl structure is calculated from $P(V)$ function (dashed line) and from
$c_{ij}$ data ($\times$) as well. Experimental data for T=195 K ($\circ$) and
T=300 K ($\Box$) are taken from Lojeloje . The dotted lines denote
experimental pressure derivatives of B at zero pressure.
The behaviour of elastic properties of the AgBr is specially interesting in
the vicinity of the phase transition. Fig. 3 shows pressure dependence of the
bulk modulus as calculated from the equation of state, as well as from the
following relationship with elastic constants:
$B=-V\frac{\partial P(V)}{\partial V}=\frac{1}{3}(c_{11}+2c_{12})$ (2)
The difference in the bulk modulus B between NaCl and CsCl structures is quite
small. On the same figure 3 we have included two dotted lines showing
experimental pressure derivatives of the bulk modulus at P=0 and T=195 K and
T=300 K.loje From these data we see that our calculations reproduce the
experimental bulk modulus quite well and that the results from different
methods are consistent within 10% .
We have used the standard stress–strain relationship to derive all three
elastic constants as a function of pressure. The method is based on
constructing a set of linear equations from stress–strain relationships for
several deformations of the unit cell. This set of equations represents a
general form of the Hook’s law and can be solved with respect to the elastic
constants. Since in practice this set of equations is over-determined, to
solve it we have used a singular value decomposition algorithm which
automatically provides a least squares solution of the setnumrec . For each
pressure three negative and three positive deformations of each kind, namely,
bulk, shear and tetragonal elongation, have been set up.
Figure 4: Elastic constants $c_{11}$, $c_{12}$, $c_{44}$ of AgBr in NaCl (solid lines and $\circ$, $\Box$, $\diamond$) and CsCl (dashed lines and $\vartriangle,\vartriangleleft,\triangledown$) structures respectively. The lines are cubic fits to the calculation data. The vertical solid arrow denotes predicted shear instability pressure. Table 1: Experimental and calculated bulk modulus $B$ and elastic constants $c_{ij}$ of the AgBr crystal, in GPa. Values in parenthesis are calculated from $B=\frac{1}{3}(c_{11}+2c_{12})$. Experimental data for T=195 K and T=300 K are taken from Lojeloje . Result | $B$ | $c_{11}$ | $c_{12}$ | $c_{44}$
---|---|---|---|---
Expt T=300 K | $40.5$ | $56.1$ | $32.7$ | $7.2$
Expt T=195 K | $43.81$ | $63.13$ | $34.14$ | $7.65$
Present | $(44.84)$ | $64.09$ | $35.21$ | $8.39$
$B=-V\partial P(V)/\partial V$ | $42.56$ | | |
Table 1 compares the calculated and experimental elastic constants at ambient
pressure and two temperatures: T=195 K and T=300 K. Present results show quite
good agreement with the measurements, although the calculations are performed
at T=0. The temperature data points allow to extrapolate the experimental
values to T=0. Thus, we have estimated that from T=195 K to T=0 K the elastic
constants might increase by about 5%. The mechanism of the phase transition
from NaCl to CsCl structure involves variation of the $c_{44}$ elastic
constant. We have calculated $c_{44}$ for both NaCl and CsCl structures and
for several pressures in the interval from 0 to 40 GPa . The results are shown
at Fig. 4. We can clearly see that the two lines representing the $c_{44}$
elastic constants , intersect at pressure P=21 GPa, and moreover at value
close to zero. The pressure P=21 GPa of the shear instability and P=27 GPa of
the free energy equality are not exactly the same. However, we shall see in
the next section that in this pressure interval another phase could be more
stable than the cubic NaCl or CsCl structures.
Figure 5: Relation between cubic NaCl and monoclinic KOH (left) and between
cubic CsCl and monoclinic KOH (right) unit cells. The KOH unit cell is drawn
with dashed lines.
In previous sections the calculation were performed for NaCl and CsCl
structures under constrains that both phases remain always cubic. Now, we
release this restrictions and allow the unit cell to deform to a structure of
P1 symmetry being intermediate configuration between the NaCl and CsCl ones.
## IV Intermediate phase
Figure 6: Transition from NaCl to CsCl structure. Solid lines are guide to the
eye. Figure 7: Cell parameters in NaCl structure to CsCl structure transition
of AgBr. Solid lines are fourth degree fits to the data points.
When the intermediate configuration becomes monoclinic it resembles the KOH-
type structure. The relationship between NaCl, KOH and CsCl structures is
shown on Fig.5. Setting the monoclinic angle $\beta$=90∘ or $\beta$=125,27∘
one reestablishes the NaCl or CsCl structures, respectively. Hull and and
Keenhull pointed out that this monoclinic phase, may provide a proper
transition path between NaCl and CsCl structures. A detailed study of the
above phase transition mechanism has been performed and we refer the results
below. We have prepared a model of the AgBr crystal in the KOH-type structure
with cell volume corresponding to the NaCl structure at P=0. The $\beta$ angle
of the monoclinic unit cell was set to 110∘. Then the crystal geometry
inducing atomic relaxations was optimised with respect to free energy for
various increasing pressures from 0 to 45 GPa.
Figure 8: Free energy difference in the transition of AgBr from NaCl (squares)
to CsCl structure (circles) through KOH structure (crosses). The polynomial
fit to the CsCl data points is taken as the reference line (zero level). The
lines are guide to the eye.
At each pressure the minimisation procedure was reiterated several times since
large changes in the unit cell size involved changes of the Brillouin zone
grid. The results are plotted in Fig 6. One notices that at about 8 GPa, there
is a clear tendency to depart from the pure NaCl structure to an intermediate
monoclinic structure. There is also a clear jump of lattice parameters at the
P=25–30 GPa range. Simultaneously, the $\alpha$, and $\gamma$ angles do not
change by more then 0.05∘ for the scanned pressure range. To verify the last
findings we have carried out two series of calculations each starting from the
same either low or high pressure configurations.
Table 2: Calculated elastic constants (in GPa) of the monoclinic, KOH-type structure of the AgBr under pressure of P=20 GPa. Pressure [GPa] | $c_{11}$ | $c_{22}$ | $c_{33}$ | $c_{12}$ | $c_{13}$ | $c_{23}$
---|---|---|---|---|---|---
$20$ | $156$ | $175$ | $229$ | $152$ | $84$ | $98$
The starting low and high pressure configurations were taken from optimised
monoclinic configurations at Plo=25 GPa or Phi=30 GPa, respectively. The
results are shown in Fig. 7. It is clear from this figure that both monoclinic
structures may show weak metastable behaviour. However, the energy difference
is small and it is possible that this effect will manifest itself only in low
temperatures.
Table 3: Calculated structural parameters of the intermediate KOH-type structure of the AgBr crystal. $P$ [GPa] | $a$ [Å] | $b$ [Å] | $c$ [Å] | $\beta$ [∘] | $b/a$ | $c/a$
---|---|---|---|---|---|---
$0$ | $4.0803$ | $4.1064$ | $5.7601$ | $90.0$ | $1.006$ | $1.412$
$5$ | $3.9521$ | $3.9819$ | $5.5982$ | $89.9$ | $1.008$ | $1.417$
$10$ | $3.8226$ | $3.9411$ | $5.4886$ | $91.6$ | $1.031$ | $1.436$
$15$ | $3.7368$ | $3.9091$ | $5.4100$ | $96.1$ | $1.046$ | $1.448$
$20$ | $3.5445$ | $4.0156$ | $5.3589$ | $101.4$ | $1.133$ | $1.512$
$25$ | $3.4033$ | $4.1011$ | $5.3403$ | $107.0$ | $1.205$ | $1.569$
$30$ | $3.2993$ | $4.1976$ | $6.7106$ | $132.7$ | $1.272$ | $2.034$
$35$ | $3.2316$ | $4.2612$ | $6.2892$ | $130.4$ | $1.319$ | $1.946$
$40$ | $3.1725$ | $4.4554$ | $5.5465$ | $125.6$ | $1.404$ | $1.748$
$45$ | $3.1489$ | $4.4375$ | $5.4855$ | $125.5$ | $1.409$ | $1.742$
Furthermore, the energy of the high-pressure configuration is always above the
low-pressure one. Consequently, the high-pressure structure is probably never
present in the real crystal, and we will omit this structure from now on.
Difference of the free energy of the three optimised candidate structures
(NaCl, KOH, CsCl) are shown in Fig. 8. The polynomial fit to the CsCl free
energy data is taken as the reference line (“0” level). At zero pressure the
NaCl structure is the most stable one. When we increase the pressure above
about 8 GPa, the cubic symmetry is broken, the monoclinic angle $\beta$ starts
to increase and the monoclinic phase is favoured. At the special point at P=21
GPa, at which the shear elastic constants $c_{44}$ of NaCl and CsCl structure
vanish on the Fig. 4, the monoclinic phase is still the most stable one. Only
above P=35 GPa the CsCl structure becomes the most stable phase. However, for
the finite temperature case this might change, since the NaCl and CsCl
structures have different number of atoms in coordination shells and hence
different entropy terms in the Gibbs free energies. Since there are no low
temperature measurements of pressure dependent behaviour of the AgBr we cannot
compare the present results with experimental findings.
We have also, for completeness, calculated some elastic constants of the
intermediate KOH phase of the crystal. The monoclinic phase has 13 independent
elastic constants. However, due to the small energy difference between phases
connected by the shear deformation, only 6 of them can be calculated
accurately. The remaining 7 constants are too small and could be only
estimated to be below 10 GPa. This is quite expected result. The 6 large
elastic constants, calculated under pressure of P=20 GPa are presented in the
Table 2. The values are consistent with the values of elastic constants of
NaCl and CsCl structures depicted on the Fig. 4.
Summarising, our results support the hypothesis formulated by Hull and
Keen,hull that the phase transition from NaCl to CsCl phase, goes through the
intermediate KOH-type, monoclinic structure.
## V Conclusions
The present DFT calculations show that at temperature T=0 K the cubic NaCl and
CsCl-type structures of AgBr are stable at pressures 0–8 GPa and above 35 GPa,
respectively. Moreover, the calculation performed in constrains of these two
cubic structures permit to find the equation of state and the elastic
constants in the whole interesting range of pressures from 0 to 45 GPa. It is
interesting to see that the shear elastic constants $c_{44}$ for both phases
NaCl and CsCl vanish close to 21 GPa. The derived equation of state and the
elastic constants characterise properly the stable cubic phase, and
additionally can provide an useful estimations of similar quantities within
the intermediate monoclinic phase. The calculation of the elastic constants of
the monoclinic KOH phase showed that, due to the small energy difference
between phases, the KOH phase is very soft to shear deformations and its
elastic constants are consistent with the elastic constants of the other two
phases.
The study of the intermediate phase has been done without point symmetry
elements constraints. This means that any deformation of the simulated
supercell could have taken place. The only imposed limitation was the number
of atoms in the supercell. Due to this approach the supercell remained always
monoclinic (or cubic for special values of monoclinic angle $\beta$). The
values of the free energy of the three candidate phases indicate that the
monoclinic phase becomes the most stable one in the pressure interval from 8
to 35 GPa. This structure has previously been proposed to be the
transformation path between NaCl and CsCl-type phases. And indeed, the free
energy of the intermediate monoclinic phase seems to be lower then these of
the cubic ones. However the difference in energy is quite small, which means
that defects and impurities present in the crystal could show a large impact
on the monoclinic phase.
###### Acknowledgements.
The use of facilities of the ACC “Cyfronet”, Cracow, where the calculations
have been done, are kindly acknowledged. This work was partially supported by
the Polish State Committee of Scientific Research (KBN), grant No 2 PO3B 004
14, and computational grant No. KBN/SGI_ORIGIN_2000/IFJ/128/1998.
## References
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* (5) P.W. Bridgman, Proc. Am. Acad. Arts Sci., 74, 21 (1940)
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* (8) W.A. Bassett and T. Takahashi, Am. Mineral. 50, 1576 (1965)
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* (10) S.N. Vaidya and G.C. Kennedy, J. Phys. Chem. Solids, 32 951 (1971)
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* (12) B.E. Mellander, Phys. Rev. B 26 5886 (1982)
* (13) S. Ves, D. Glötzel, M. Cardona and H. Overhof, Phys. Rev. B 24 3073 (1981)
* (14) G.S. Nunes, P.B. Allen, J.L. Martins, Phys. Rev. B 57, 5098 (1998)
* (15) S. Hull, D.A. Keen, Phys. Rev. B 59, 750 (1999)
* (16) K. Kusaba, Y. Syono, T. Kikegawa, O. Shimomura, J. Phys. Chem. Solids 56, 751 (1995)
* (17) R.N. Schock and J.C. Jamieson, J. Phys. Chem. Solids 30, 1527 (1969)
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|
arxiv-papers
| 2013-01-26T13:35:32 |
2024-09-04T02:49:40.787503
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Pawe{\\l} T. Jochym, Krzysztof Parlinski",
"submitter": "Pawe{\\l} T. Jochym",
"url": "https://arxiv.org/abs/1301.6254"
}
|
1301.6349
|
# Construction of Nilpotent Jordan Algebras Over any Arbitrary Fields
A.Hegazi , H.Abdelwahab
Mathematics department, Faculty of Science, Mansoura University, Egypt
[email protected]
[email protected]
###### Abstract
A cohomological approach for classifying nilpotent Jordan algebras, every
nilpotent Jordan algebras can be constructed by the second cohomolgy of
nilpotent Jordan algebras of low dimension. Also we describe the analogue of
Skjelbred-Sund method for classifying nilpotent lie algebras to classify
nilpotent Jordan algebras and carry out a procudure for constructing nilpotent
Jordan algebras over arbitrary fields. By using this method, we construct all
non-isomorphic nilpotent Jordan algebras up to dimension three over any field,
and all non-isomorphic nilpotent Jordan algebras of dimension four over an
algebraic closed field of characteristic $\neq 2$ and over the real field
$\mathbb{R}$. Also commutative nilpotent associative algebras are classified,
since any commutative nilpotent associative algebras are nilpotent Jordan
algebras. We show that there are up to isomorphism $13$ nilpotent Jordan
algebras of dimension $4$ over an algebraic closed field of characteristic
$\neq 2,$ and $4$ of those are not associative, yielding $9$ commutative
nilpotent associative algebras. Also up to isomorphism there are $17$
nilpotent Jordan algebras of dimension $4$ over the real field $\mathbb{R}$,
and $5$ of those are not associative, yielding $12$ commutative nilpotent
associative algebras.
Keywords : Jordan algebras, Nilpotent, Centeral extension, Cohomology,
Isomorphism, Automorphism group.
2010 Mathematics Subject Classification : 17C10, 17C55, 17-08.
## 1 Introduction
Jordan algebras were introduced in the early 1930’s by a physicist, P. Jordan,
in an attempt to generalize the formalism of quantum mechanics. Little appears
to have resulted in this direction, but un-anticipated relationships between
these algebras and Lie groups and the foundations of geometry have been
discovered.
The structure of this paper, in section $2$ we define the notion of Extension
of Jordan algebra which is similar to the concept of Extension of Groups, let
$N,F$ and $G$ be groups and let $G$ have a normal subgroup $\bar{N}$ which is
isomorphic to $N$, that is, $\bar{N}$ $\cong N$. Recall that a subgroup is
called normal or invariant if one has $g\bar{N}g^{-1}=\bar{N}$ for all $g$
$\in$ $G$. (At the Jordan algebras level a normal subgroup yields an ideal.)
The group $G$ is called an extension of $F$ by $N$ if the factor group $G/N$
is isomorphic with the group $F$, i.e. $G/\bar{N}$ $\cong F$. The relationship
between the groups $N,F$ and $G$ can be represented by the sequence
$N\overset{\Gamma}{\longrightarrow}G\overset{\Lambda}{\longrightarrow}F$
where $\Gamma$ is an injective group homomorphism with im$\Gamma=\bar{N}$ and
where $\Lambda$ is a surjective group homomorphism with ker$\Lambda=\bar{N}$.
A systematic study of group extensions has been performed by the German
mathematician O. Schreier in $1926$.
In section $3,$ we focus on the Centeral Extension and in sections $4,5$ and
$6$ we descripe the analouge of of The Skjelbred-Sund Method, and carry out a
procudure for constructing nilpotent Jordan algebras over any arbitrary field.
Finally, classify nilpotent Jordan algebras of dimension $\leq 4.$
###### Definition 1
A Jordan algebra $J$ is a vector space over a field $K$ equipped with a
symmetric bilinear map $B:J\times J\longrightarrow J$ such that :
$B(B(x,x),B(x,y)=B(x,B(B(x,x),y))\text{ },\text{all }x,y\in J.$
which is called the Jordan identity.
Let $J$ be a Jordan algebra over a field $K.$For any integer $m\in\mathbb{N}$
we define a lower centeral series as the descending chain of ideals
$c^{1}(J)=J\supseteq c^{2}(J)=B(J,J)\supseteq
c^{3}(J)=B(c^{2}(J),J)\supseteq.....\supseteq c^{m}(J)=B(c^{m-1}(J),J)$
###### Definition 2
A Jordan algebra $J$ over a field $K$ is said to be nilpotent of nilindex $n$
if :
$c^{n}(J)=\\{0\\}\text{ \ \ and \ \ }c^{n-1}(J)\neq\\{0\\}.$
Also we call $n$ the nilpotency class of $J$ and $J$ is called $n$-step
nilpotent.
###### Definition 3
Let $J$ be a Jordan algebra over a field $K$ then the ideal :
$Z(J)=\\{x\in J:B(x,y)=0\ \forall x\in J\\}$
is called the centre of $J.$
So, any nilpotent Jordan algebra have non-trivial centre $Z(J)\neq\\{0\\}.$In
the Jordan algebra $J,$ we will denote the product $B(x,y)$ by $x\circ y$ and
$x^{2}$ for $B(x,x).$Then the jordan identity is written as follow :
$x^{2}\circ(x\circ y)=x\circ(x^{2}\circ y)\text{ \ , all }x,y\in J.$
## 2 Extensions of Jordan algebras
In this section we are going to focus on extensions of Jordan algebras.
Loosely speaking an extension of a Jordan algebra $J$ is an enlargement of $J$
by some other Jordan algebra. To be somewhat more concrete we consider the
following construction. Starting with two Jordan algebras $L$ and $M$ over the
same field $K$ we consider the Cartesian product $J=L\times M$. Elements of
this set are ordered pairs $(l,m)$ with $l\in L$ and $m\in M$. Defining
addition of such pairs by $(l,m)+(l^{{}^{\prime}},m^{{}^{\prime}}):=$
$(l+l^{{}^{\prime}},m+m^{{}^{\prime}})$ and multiplication by scalars
$\alpha\in K$ as $\alpha(l,m):=$ $(\alpha l,\alpha m)$ the set $J$ becomes a
vector space which will also be denoted by $J$. Using the Jordan
multiplication on $L$ and $M$ we define on $J$ the multiplication
$(l,m)\circ(l^{{}^{\prime}},m^{{}^{\prime}})=(l\circ_{L}l^{{}^{\prime}},m\circ_{M}m^{{}^{\prime}})$
One easily sees that this is a Jordan multipliction on $J$. Defining next the
map $\epsilon$ $:m\in M\longrightarrow$ $(0,m)$ $\in J$ and the map $\lambda:$
$(l,m)\in J\longrightarrow$ $m\in M$ one readily verifies that $\epsilon$ is
an injective Jordan algebra homomorphism, while $\lambda$ is a surjective
Jordan algebra homomorphism. Moreover, im $\epsilon$ $=$ ker $\lambda$. The
Jordan algebra $J$ with these properties is called a trivial extension of of
$L$ by $M$ ( or of $M$ by $L$ ) .
Instead of denoting the elements of $L$ by ordered pairs we will frequently
use the notation $(l,m)=l+m$ . The vector space $J$ is then written as
$J=L\oplus M$
and the Jordan multiplication is in this notation given by
$(l+m)\circ(l^{{}^{\prime}}+m^{{}^{\prime}})=l\circ_{L}l^{{}^{\prime}}+m\circ_{M}m^{{}^{\prime}}$
This example of a trivial extension is generalized in the following concept.
Let $L,J$ and $M$ be Jordan algebras and let these algebras be related in the
following way.
* •
There exists an injective Jordan algebra homomorphism
$\epsilon:L\longrightarrow J\text{ .}$
* •
There exists an surjective Jordan algebra homomorphism
$\lambda:J\longrightarrow M\text{ .}$
* •
The Jordan algebra homomorphisms $\epsilon$ and $\lambda$ are related by
$\text{im }\epsilon=\text{ker }\lambda\text{ }.$
Then the Jordan algebra $J$ is called an extension of $M$ by $L$. The
relationship is summarized by the sequence
$L\overset{\epsilon}{\longrightarrow}J\overset{\lambda}{\longrightarrow}M$
We elaborate a little on this concept. Since $\lambda$ is a jordan algebra
homomorphism one obtains that ker $\lambda$ is an ideal in $J$ and since
$\lambda$ is surjective First Isomorphism Theorem entails a Jordan algebra
isomorphism between the quotient algebra $J/$ker $\lambda$ and $L$ ,
$J/\text{ker }\lambda\cong M$
Using im $\epsilon=$ ker $\lambda$ this relation can be written as
$J/\text{im }\epsilon\cong M$
Since $\epsilon$ is injective the Jordan algebras $L$ and im $\epsilon$ are
Jordan isomorphic, i.e $J/L\cong M$. From these properties one sees that it
makes sense to call the Jordan algebra $J$ is an extension of $M$ by $L$.
###### Definition 4
Let $L$, $J$ and $M$ be Jordan algebras over the field $K$. Let
$\lambda:J\longrightarrow M$
be a surjective Jordan algebra homomorphism and
$\epsilon:L\longrightarrow J$
an injective Jordan algebra homomorphism. Then the sequence
$L\overset{\epsilon}{\longrightarrow}J\overset{\lambda}{\longrightarrow}M$ (1)
is called an extension of $M$ by $L$ if $\epsilon$ maps $L$ onto the kernel
ker $\lambda$ $\subset$ $J$ of the map $\lambda$. That is if
$\text{im }\epsilon=\text{ker }\lambda\text{ }.$
The kernel ker $\lambda$ is called the kernel of the extension.
Loosely speaking $J$ itself, instead of the sequence (1), will be called an
extension of $M$ by $L$.
It may happen that there exist several extensions of $M$ by $L$. To classify
extensions we define the notion of equivalent extensions.
###### Definition 5
Two sequences
$L\overset{\epsilon}{\longrightarrow}J\overset{\lambda}{\longrightarrow}M$
and
$L\overset{\epsilon^{{}^{\prime}}}{\longrightarrow}J^{{}^{\prime}}\overset{\lambda^{{}^{\prime}}}{\longrightarrow}M$
are called equivalent extensions if there exists a Jordan algebra isomorphism
$\phi:J\longrightarrow J^{{}^{\prime}}$ such that
$\phi\circ\epsilon=\epsilon^{{}^{\prime}},\text{ \
}\lambda^{{}^{\prime}}\circ\phi=\lambda\text{ .}$
One easily sees that equivalence of extensions is an equivalence relation.
The concept of a Jordan algebra extension can be formulated more succinctly
using the concept of an exact sequence.Let $\\{J_{i}\\}$ be a sequence of
Jordan algebras and $\\{\phi_{i}\\}$ a sequence of Jordan algebra
homomorphisms $\phi_{i}:$ $J_{i}\longrightarrow J_{i+1}$, then the sequence
$.....\longrightarrow
J_{i-1}\overset{\phi_{i-1}}{\longrightarrow}J_{i}\overset{\phi_{i}}{\longrightarrow}J_{i+1}\overset{\phi_{i+1}}{\longrightarrow}.....$
is called exact if one has for each $i$
$\text{im }\phi_{i-1}=\text{ker }\phi_{i}\text{ .}$
Let $L,J$ and $M$ be Jordan algebras and let us denote by $0$ the Jordan
algebra consisting of the zero element only. Then the sequence
$0\overset{f}{\longrightarrow}L\overset{g}{\longrightarrow}J$ is exact if and
only if $g$ is an injective Jordan algebra homomorphism. Indeed, since im
$f=0$ $\in L$ the requirement ker $g$ $=$ im $f$ forces $g$ to be injective.
One usually omits the mapping $f$ i.e., one writes $0\longrightarrow
L\overset{g}{\longrightarrow}J$ . Likewise the sequence
$J\overset{f}{\longrightarrow}M\longrightarrow 0$ is exact if and only if $f$
is a surjective Jordan algebra homomorphism. Consequently we have the
following proposition.
###### Proposition 6
The sequence $L\longrightarrow J\longrightarrow M$ of Jordan algebras is an
extension of $M$ by $L$ if and only if the sequence
$0\longrightarrow
L\overset{\epsilon}{\longrightarrow}J\overset{\lambda}{\longrightarrow}M\longrightarrow
0$
is exact .
Next we introduce some particular types of extensions. For this we need the
following concept from linear algebra. Two subspaces $V_{1}$ and $V_{2}$ of a
vector space $V$ are called complementary if $V$ is the direct sum of $V_{1}$
and $V_{2}$.
###### Definition 7
An extension
$L\overset{\epsilon}{\longrightarrow}J\overset{\lambda}{\longrightarrow}M$
is called:
trivial if there exists an ideal $I$ $\subset J$ complementary to ker
$\lambda$, i.e.
$J=ker\text{ }\lambda\oplus I\text{ \ \ \ \ \ }(\text{ Jordan algebra direct
sum }),$
split if there exists a Jordan subalgebra $S\subset J$ complementary to ker
$\lambda$,
i.e.
$J=ker\text{ }\lambda\oplus S\text{\ \ \ \ \ }(\text{ vector space direct sum
}),$
centeral if the kernel ker $\lambda$ is contained in the center $Z(J)$ of $J$,
i.e.
$ker\text{ }\lambda\subset Z(J)\text{ .}$
## 3 2-Cocycles on Jordan algebra
In this section we take a closer look at central extensions of Jordan
algebras. The Jordan algebras in this section will be over the field $K$ .
Recall that the sequence of Jordan algebras
$L\overset{\epsilon}{\longrightarrow}J\overset{\lambda}{\longrightarrow}M$
is a central extension (of $M$ by $L$) if one has for the Jordan algebra
homomorphisms $\epsilon$ and $\lambda$ the following properties: $\epsilon$ is
injective, $\lambda$ is surjective, im $\epsilon=$ ker $\lambda$ and the
kernel ker $\lambda$ is contained in the center $Z(J)$ of $J$. Hence im
$\epsilon$ = ker $\lambda$ is a Jordan algebra with trivial multiplicatin.
Since $Z(J)$ is a Jordan subalgebra with trivial multiplicatin and since $L$
and im $\epsilon$ are isomorphic, $L$ is a Jordan algebra with trivial
multiplicatin too. A central extension of a Jordan algebra $M$ by a Jordan
algebra with trivial multiplicatin( a vector space ) $L$ can be obtained with
the help of a so called $2$-cocycle on $M$.
###### Definition 8
Let $J$ be a Jordan algebra and $V$ be avector space over $K$. A bilinear map
$\theta:J\times J\longrightarrow V$
is called a Jordan $2$-cocycle from $J$ to $V$ if it satisfies for all $x,y\in
J$ the following conditions :
* •
$\theta(x,y)=\theta(y,x)$ ($\theta$ is symmetric ) .
* •
$\theta(x^{2},x\circ y)=\theta(x,x^{2}\circ y)$ ( Jordan identity for
$2$-cocycles ) .
The set of $2$-cocycles from $J$ to $V$ is denoted by $Z^{2}(J,V)$ . One
easily sees that $Z^{2}(J,V)$ is a vector space if one defines the vector
space operations as follows. Let $\theta_{1}$ and $\theta_{2}$ be $2$-cocycles
rom $J$ to $V$, then their linear combination $\lambda_{1}$ $\theta_{1}+$
$\lambda_{2}$ $\theta_{2}$ ( $\lambda_{1},\lambda_{2}\in K$ ) is defined by
$(\lambda_{1}\theta_{1}+\lambda_{2}\theta_{2})(x,y):=\lambda_{1}\theta_{1}(x,y)+\lambda_{2}\theta_{2}(x,y)\text{
.}$
This linear combination is again a Jordan $2$-cocycle.
Let $\theta\in$ $Z^{2}(J,V)$ ,and set $J_{\theta}=J\oplus V.$ Using Jordan
multiplication on $J$ and the cocycle $\theta$ from $J$ to $V$ e define on
$J_{\theta}$ the multiplication
$(x+v)\circ(y+w)=x\circ_{J}y+\theta(x,y)$
For $x,y\in J,v,w\in V$ .
###### Lemma 9
$J_{\theta}$ is a Jordan algebra if and only if $\ \theta\in$ $Z^{2}(J,V)$ .
Proof. For $x+v,y+w\in J_{\theta},$we have that :
$\displaystyle(x+v)^{2}\circ((x+v)\circ(y+w))$ $\displaystyle=$
$\displaystyle(x\circ_{J}x+\theta(x,x))\circ(x\circ_{J}y+\theta(x,y))$
$\displaystyle=$ $\displaystyle
x^{2}\circ_{J}(x\circ_{J}y)+\theta(x^{2},x\circ_{J}y)$
$\displaystyle(x+v)\circ((x+v)^{2}\circ(y+w))$ $\displaystyle=$
$\displaystyle(x+v)\circ((x^{2}+\theta(x,x))\circ(y+w))$ $\displaystyle=$
$\displaystyle(x+v)\circ(x^{2}\circ_{J}y+\theta(x\circ_{J}x,y))$
$\displaystyle=$ $\displaystyle
x\circ_{J}(x^{2}\circ_{J}y)+\theta(x,x^{2}\circ_{J}y)$
From Jordan identity it follows that
$(x+v)^{2}\circ((x+v)\circ(y+w))=(x+v)\circ((x+v)^{2}\circ(y+w)$ if and only
if $\theta(x^{2},x\circ y)=\theta(x,x^{2}\circ y)$ then $J_{\theta}$ is a
Jordan algebra if and only if $\theta\in$ $Z^{2}(J,V)$ .
###### Lemma 10
Let $\theta\in$ $Z^{2}(J,V)$ then $J_{\theta}$ is a centeral extension of $J$
by $V$ .
Proof. From the previous lemma $J_{\theta}$ is a Jordan algebra and $V$ is a
Jordan algebra with the trivial multiplication . The sequence
$V\overset{i}{\longrightarrow}J\oplus V\overset{\pi}{\longrightarrow}J$ is
exact( $i$ be the injection map and $\pi$ be the projection map ) and $V=$ker
$\pi\subset Z(J_{\theta})$ . Hence $J_{\theta}$ is a centeral extension of $J$
by $V$ .
Hence, a Jordan $2$-cocycle $\theta$ from $J$ to $V$ ( dim$V=k$ ) allows for
the construction of a $k$-dimensional central extension $J_{\theta}=J\oplus V$
where the Jordan multiplication on $J_{\theta}$ is given by
$(x+v)\circ(y+w)=x\circ_{J}y+\theta(x,y)$
For $x,y\in J,v,w\in V$ .
Now we will show that a Jordan $2$-cocycles can be obtained from the”abstract
form” of a central extension. Considering the extension
$L\overset{\epsilon}{\longrightarrow}J\overset{\lambda}{\longrightarrow}M$
with im $\epsilon=$ker $\lambda\subset Z(J)$ we show that one can find
$2$-cocycles from $M$ to $L$. Consider a linear map $s:M\longrightarrow J$
satisfying
$\lambda\circ s=id_{M}$
A map with this property is called a section of $J$. With the help of a
section one can define a bilinear map $\theta^{{}^{\prime}}:M\times
M\longrightarrow J$ by taking for all $x,y\in M$
$\theta^{{}^{\prime}}(x,y)=s(x\circ y)-s(x)\circ s(y)$ (2)
Notice that $\theta^{{}^{\prime}}$ is identically zero if $s$ is a Jordan
algebra homomorphism. Notice also that $\theta^{{}^{\prime}}$ is symmetric.
From equation $\lambda\circ s=id_{M}$ and fact that $\lambda$ is a Jordan
algebra homomorphism one sees that
$\lambda(\theta^{{}^{\prime}}(x,y))=0$
Hence, we have for all $x,y$ $\in M$
$\theta^{{}^{\prime}}(x,y)\in\text{ker }\lambda\subset Z(J)\text{ .}$ (3)
Using the injectivity of the map $\epsilon:L\longrightarrow J$ to define the
map
$\theta:M\times M\longrightarrow J\text{ .}$
Given by
$\theta:=\epsilon^{-1}\circ\theta^{{}^{\prime}}.$
$\theta$ is bilinear and symmetric . It remains to show that $\theta$ satisfy
the second condition in Defenition 8 . Let $x,y\in M$ then by using Jordan
identity, equation (2) and property (3) we have
$\displaystyle\theta^{{}^{\prime}}(x^{2},x\circ y)$ $\displaystyle=$
$\displaystyle s(x^{2}\circ(x\circ y))-s(x^{2})\circ s(x\circ y)$
$\displaystyle=$ $\displaystyle s(x\circ(x^{2}\circ
y))-s(x^{2})\circ(\theta^{{}^{\prime}}(x,y)+s(x)\circ s(y))$ $\displaystyle=$
$\displaystyle s(x\circ(x^{2}\circ y))-s(x^{2})\circ(s(x)\circ s(y))$
$\displaystyle=$ $\displaystyle s(x\circ(x^{2}\circ
y))-s(x)\circ(s(x^{2})\circ s(y))$ $\displaystyle=$ $\displaystyle
s(x\circ(x^{2}\circ y))-s(x)\circ(\theta^{{}^{\prime}}(x^{2},y)+s(x^{2}\circ
y))$ $\displaystyle=$ $\displaystyle s(x\circ(x^{2}\circ y))--s(x)\circ
s(x^{2}\circ y)$ $\displaystyle=$
$\displaystyle\theta^{{}^{\prime}}(x,x^{2}\circ y)\text{ .}$
Hence, $\theta(x^{2},x\circ y)=\epsilon^{-1}(\theta^{{}^{\prime}}(x^{2},x\circ
y))=\epsilon^{-1}(\theta^{{}^{\prime}}(x,x^{2}\circ y))=\theta(x,x^{2}\circ
y)$ . Then $\theta$ is a $2$-cocycles from $M$ to $L$. We have thus shown that
$2$-cocycles are quite natural objects in central extensions.
###### Corollary 11
$J_{\theta}$ is a centeral extension of $J$ by $V$ if and only if $\
\theta\in$ $Z^{2}(J,V)$.
We now proceed with the properties of cocycles. A special type of $2$-cocycles
is given by so-called $2$-coboundaries.
###### Definition 12
Let $J$ be a Jordan algebra and $V$ be avector space over $K$. Then a linear
map
$f:J\longrightarrow V.$
is called a $1$-cochain from $J$ to $V$. The set of all $1$-cochains from $J$
to $V$ is denoted by $C^{1}(J,V).$
Notice that $C^{1}(J,V)$ be the vector space $Hom(J,V)$ and hence it has, just
as$Z^{2}(J,V)$, an obvious vector space structure. Using cochains one defines
$2$-coboundaries as follows.
###### Definition 13
Let $J$ be a Jordan algebra,$V$ be avector space over $K$ and let
$f:J\longrightarrow V$ be a $1$-cochain from $J$ to $V$. Then the bilinear map
$\delta f:J\times J\longrightarrow V.$
defined by
$(\delta f)(x,y):=f(x\circ y).$
is called a $2-$coboundary from $J$ to $V$.
Next we show that $\delta f$ is a $2$-cocycle from $J$ to $V$. The symmetry of
$\delta f$ follows from
$(\delta f)(x,y):=f(x\circ y)=f(y\circ x)=(\delta f)(y,x).$
Furthermore the Jordan identity of $J$ imply
$(\delta f)(x^{2},x\circ y)=f(x^{2}\circ(x\circ y))=f(x\circ(x^{2}\circ
y))=(\delta f)(x,x^{2}\circ y).$
for all $x,y\in J.$ Hence a $2$-coboundary from $J$ to $V$ is a $2$-cocycle
from $J$ to $V$. Consequently, the map $\delta:f\longrightarrow\delta f$ is a
map from $C^{1}(J,V)$ to $Z^{2}(J,V)$.
###### Definition 14
The map
$\delta:f\in C^{1}(J,V)\longrightarrow\delta f\in Z^{2}(J,V).$
where $\delta f$ is defined by $(\delta f)(x,y):=f(x\circ y)$ for all $x,y\in
J,$is called the coboundary operator .
Using the notions of $2$-cocycles and $1$-cochains, more precisely the vector
spaces $Z^{2}(J,V)$ and $C^{1}(J,V)$, one defines the second cohomology group
of a Jordan algebra $J$ by$V$ .
###### Definition 15
Let $J$ be a Jordan algebra,$V$ be avector space over $K.$ Then the quotient
vector space
$H^{2}(J,V):=Z^{2}(J,V)\diagup\delta C^{1}(J,V)$
is called the second cohomology group of $J$ by $V$.
Elements of $H^{2}(J,V)$ are equivalence classes of $2$-cocycles and two
$2$-cocycles $\theta_{1}$ and $\theta_{2}$ are called equivalent cocycles if
they differ by a $2$-coboundary, i.e. if $\theta_{1}$ $=$ $\theta_{2}+$
$\delta f$ for some $f$ $\in$ $C^{1}(J,V)$. Equivalent $2$-cocycles are called
cohomologous.
###### Lemma 16
Let $\theta_{1}$ and $\theta_{2}$ be equivalent Jordan cocycles and let
$J_{\theta_{1}},J_{\theta 2}$ be respectively the centeral extensions
constructed with these Jordan cocycles .Then the centeral extensions
$J_{\theta_{1}}$ and $J_{\theta 2}$ are are equivalent extensions.
Proof. According to the definition of equivalent Jordan cocycles we have
$\theta_{1}=\theta_{2}+\delta f$ with $f\in C^{1}(J,V)$ a 1-cochain. Define
$\sigma:J_{\theta_{2}}\longrightarrow J_{\theta_{2}+\delta f}$ by
$\sigma(x+v)=x+f(x)+v$ . Let $x+v\in\ker\sigma$ then
$\sigma(x+v)=x+f(x)+v=0,$hence $x=v=0.$ Thus $\ker\sigma=\\{0\\},$ this shows
that $\sigma$ is an invertible linear transformation. Morever,for all
$x+v,y+w\in J_{\theta_{2}}$
$\displaystyle\sigma((x+v)\circ_{J_{\theta_{2}}}(y+w))$ $\displaystyle=$
$\displaystyle\sigma(x\circ_{J}y+\theta_{2}(x,y))$ $\displaystyle=$
$\displaystyle x\circ_{J}y+f(x\circ_{J}y)+\theta_{2}(x,y)$ $\displaystyle=$
$\displaystyle x\circ_{J}y+\delta f(x,y)+\theta_{2}(x,y)$ $\displaystyle=$
$\displaystyle x\circ_{J}y+(\theta_{2}+\delta f)(x,y)$ $\displaystyle=$
$\displaystyle x\circ_{J}y+\theta_{1}(x,y)$ $\displaystyle=$
$\displaystyle(x+f(x)+v)\circ_{J_{\theta_{1}}}(y+f(y)+w)$ $\displaystyle=$
$\displaystyle\sigma(x+v)\circ_{J_{\theta_{1}}}\sigma(y+w).$
Then $J_{\theta_{1}}$ and $J_{\theta_{2}}$ are isomorphic. Hence, cohomologous
2-cocycles yield equivalent (central) extensions.
###### Corollary 17
A cohomology class $\theta\in H^{2}(J,V)$ defines a central extension of the
Jordan algebra $J$ by $V$ which is unique up to equivalence.
The cocycle $\theta=0$ $\in H^{2}(J,V)$ gives a trivial central extension of a
Jordan algebra $J$ by $V$. Namely
$J_{0}=J\oplus V$ ( direct sum of Jordan algebras )
and the Jordan multiplication is given by
$(x+V)\circ_{J_{0}}(y+w)=x\circ_{J}y.$
for all $x,y\in J$ and $v,w\in V.$
For any $f\in C^{1}(J,V)$ a $1$-cochains, $\delta f=0+\delta f.$ Hence,
$2$-cocycles which are obtained from $1$-cochains are cohomologous with the
trivial $2$-cocycle $\theta=0$ we obtain the following corollary.
###### Corollary 18
A central extension defined by a $2$-coboundary is equivalent with a trivial
central extension.
Let $V$ be $m$-dimensional vector space then
$Z^{2}(J,V)=Z^{2}(J,K^{m})=Z^{2}(J,K)^{m}$ and
$H^{2}(J,V)=H^{2}(J,K^{m})=H^{2}(J,K)^{m}.$ So for any $\theta\in Z^{2}(J,V)$
we may write $\theta=(\theta_{1},\theta_{2},.....,\theta_{m})\in
Z^{2}(J,K)^{m}.$ Let $e_{1},e_{2},..........,e_{m}$ be a basis of $V$ then
$\theta(x,y)=\overset{m}{\underset{i=1}{\sum}}\theta_{i}(x,y)e_{i\text{ }}$
where $\theta_{i}\in Z^{2}(J,K)$ , and $\theta$ is a $2-$coboundary if and
only if all $\theta_{i}\in\delta C^{1}(J,K).$
###### Lemma 19
dim $\delta C^{1}(J,K)=$dim $J^{2}$
Proof. Let $J$ be n-dimensional Jordan algebra with basis
$<x_{1},x_{2},...........,x_{n}>$ and $J^{\ast}$ generated by the dual basis
$<x_{1}^{\ast},x_{2}^{\ast},...........,x_{n}^{\ast}>$ defined by
$x_{i}^{\ast}(x_{i})=1$ and $x_{i}^{\ast}(x_{j})=0$ if $i\neq j.$Let
$<x_{r},x_{r+1},...........,x_{r+s}>$ be a basis of $J^{2}.$Notice that
$C^{1}(J,K)$ is the dual vector space of $J.$ The coboundary operator
$\delta:J^{\ast}\longrightarrow Z^{2}(J,K)$
defined by
$(\delta x_{k}^{\ast})(x_{i},x_{j})=x_{k}^{\ast}(x_{i}\circ x_{j})$
for $x_{k}^{\ast}\in J$ ∗ and $x_{i},x_{j}\in J$. Let $x_{k}\notin J^{2}$ then
$\delta$ $x_{k}^{\ast}=0,$ so $\delta C^{1}(J,K)$ is spanned by
$<x_{r}^{\ast},x_{r+1}^{\ast},...........,x_{r+s}^{\ast}>$ which have a
dimension equal to the dimension of $J^{2}.$
## 4 Analouge of The Skjelbred-Sund Method
Let $J$ be a Jordan algebra with non-trivial centre $Z(J)$. We will show that
such algebra are centeral extensions of smaller Jordan algebras .
###### Lemma 20
Every $n$-dimensional Jordan algebra with non-trivial centre is a centeral
extension of a lower dimensional Jordan algebra .
Proof. Let $J$ be $n-$dimensional Jordan algebra with non trivial centre$Z(J)$
, then $J/Z(J)$ is a Jordan algebra of smaller dimension . Let
$\pi:J\longrightarrow J/Z(J)$ be a projection map, choose an injective linear
map $\eta:J/Z(J)\longrightarrow J$ such that $\pi(\eta(\bar{x}))=\bar{x}$ for
all $\bar{x}\in J/Z(J).$ For $\bar{x},\bar{y}\in J/Z(J)$ we have
$\eta(\bar{x})\circ\eta(\bar{y})-\eta(\bar{x}\circ\bar{y})\in Z(J)$. This
define a symmetric bilinear map
$\theta:J/Z(J)\times J/Z(J)\longrightarrow Z(J)$
defined by
$\theta(\bar{x},\bar{y})=\eta(\bar{x})\circ\eta(\bar{y})-\eta(\bar{x}\circ\bar{y}).$
Then
$\displaystyle\theta(\bar{x}^{2},\bar{x}\circ\bar{y})$ $\displaystyle=$
$\displaystyle\eta(\bar{x}^{2})\circ\eta(\bar{x}\circ\bar{y})-\eta(\bar{x}^{2}\circ(\bar{x}\circ\bar{y}))$
$\displaystyle=$
$\displaystyle(\eta(\bar{x})\circ\eta(\bar{x})-\theta(\bar{x},\bar{x}))\circ(\eta(\bar{x})\circ\eta(\bar{y})-\theta(\bar{x},\bar{y}))-\eta(\bar{x}\circ(\bar{x}^{2}\circ\bar{y})$
$\displaystyle=$
$\displaystyle\eta(\bar{x})\circ(\eta(\bar{x})^{2}\circ\eta(\bar{y}))-\eta(\bar{x}\circ(\bar{x}^{2}\circ\bar{y})$
$\displaystyle=$
$\displaystyle\eta(\bar{x})\circ(\eta(\bar{x}^{2})\circ\eta(\bar{y}))-\eta(\bar{x}\circ(\bar{x}^{2}\circ\bar{y})$
$\displaystyle=$ $\displaystyle\theta(\bar{x},\bar{x}^{2}\circ\bar{y}).$
Hence, $\theta$ is $2-$cocycle. It remains to show that
$(J/Z(J))_{\theta}=J/Z(J)\oplus Z(J)$ is isomorphic to $J$. Let $x\in J$, then
x can uniquely written as $x=\eta(\bar{y})+z$, where $\bar{y}\in J/Z(J)$ and
$z\in Z(J).$ Define $\phi:J\longrightarrow(J/Z(J))_{\theta}$ by
$\phi(x)=\bar{y}+z.$ Then $\phi$ is bijective and
$\displaystyle\phi(x_{1}\circ_{J}x_{2})$ $\displaystyle=$
$\displaystyle\phi((\eta(\bar{y}_{1})+z_{1})\circ_{J}(\eta(\bar{y}_{2})+z_{2}))$
$\displaystyle=$
$\displaystyle\phi(\eta(\bar{y}_{1})\circ_{J}\eta(\bar{y}_{2}))$
$\displaystyle=$
$\displaystyle\phi(\eta(\bar{y}_{1}\circ_{J/Z(J)}\bar{y}_{2})+\theta(\bar{y}_{1},\bar{y}_{2}))$
$\displaystyle=$
$\displaystyle\bar{y}_{1}\circ\bar{y}_{2}+\theta(\bar{y}_{1},\bar{y}_{2})$
$\displaystyle=$
$\displaystyle(\bar{y}_{1}+z_{1})\circ_{(J/Z(J))_{\theta}}(\bar{y}_{2}+z_{2})$
$\displaystyle=$
$\displaystyle\phi(x_{1})\circ_{{}_{(J/Z(J))_{\theta}}}\phi(x_{2})$
Then $\phi$ is an isomorhism.
So in particular, Every $n$-dimensional nilpotent Jordan algebras is a
centeral extension of a lower dimensional nilpotent Jordan algebra .
###### Definition 21
Let $\ \theta\in Z^{2}(J,V)$ then
$\theta^{\bot}=\\{x\in J:\theta(x,y)=0\text{ for all }y\in J\\}$
is called the radical of $\theta$ .
Let us now fix a basis $\\{e_{1},...,e_{r}\\}$ of $V$ . A cocycle $\theta$
$\in Z_{2}(J,V)$ such that
$\theta(x,y)=\overset{r}{\underset{i=1}{\sum}}\theta_{i}(x,y)e_{i\text{ }}$
where $\theta_{i}\in Z^{2}(J,K).$ Then
$\theta^{\perp}=\theta_{1}^{\perp}\cap\theta_{2}^{\perp}\cap....\cap\theta_{r}^{\perp}$
###### Lemma 22
Let $\ \theta\in Z^{2}(J,V)$ then $Z(J_{\theta})=(\theta^{\bot}\cap
Z(J))\oplus V.$
Proof. Let $x+v\in Z(J_{\theta})$ then ($x+v)\circ_{J_{\theta}}(y+w)=0$ for
all $y+w\in J_{\theta}.$Then $x\circ_{J}y+\theta(x,y)=0$ for all $y+w\in
J_{\theta}.$It follows that $x\in\theta^{\bot}\cap Z(J)$ then
$x+v\in(\theta^{\bot}\cap Z(J))\oplus V.$On the other hand , suppose that
$x+v\in(\theta^{\bot}\cap Z(J))\oplus V$ then for all $y+w\in J_{\theta}$ we
have ($x+v)\circ_{J_{\theta}}(y+w)=x\circ_{J}y+\theta(x,y)=0$ then $x+v\in
Z(J_{\theta}).$ Then $Z(J_{\theta})=(\theta^{\bot}\cap Z(J))\oplus V.$
A subspace $W$ of $H^{2}(J,K)$ is said to be allowable if
$\underset{{}_{\theta\in W}}{\cap}\theta^{\perp}\cap Z(J)=0.$
###### Corollary 23
$\theta^{\bot}\cap Z(J)=0$ if and only if $Z(J_{\theta})=V.$
If $\theta^{\bot}\cap Z(J)\neq 0$ then $J_{\theta}$ can be obtaines as a
centeral extension of another Jordan algebra $\tilde{J}$ by
$(\theta^{\bot}\cap Z(J))\oplus V.$ Then to avoid constructing the same Jordan
algebra as a centeral extension of different Jordan algebra we want to
restrict to $\theta$ such that $\theta^{\bot}\cap Z(J)=0$ .
Let $J$ be a Jordan over a field $K$ . For each $\theta\in Z^{2}(J,V)$ and
$\phi\in Aut(J)$ ,the automorphism group of $J$ , we define
$\phi\theta(x,y)=\theta(\phi x,\phi y)$ for any $x,y\in J.$ So $Aut(J)$ acts
on $Z^{2}(J,V)$ , and $\phi\theta\in\delta C^{1}(J,V)$ if and only if
$\theta\in\delta C^{1}(J,V).$ Then $Z^{2}(J,V)$ and $\delta C^{1}(J,V)$ are
invariant under the action of $Aut(J).$ So $Aut(J)$ acts on $H^{2}(J,V)$ .
Let $\theta_{1},\theta_{2}\in H^{2}(J,V)$ and $\theta_{1}^{\bot}\cap
Z(J)=\theta_{2}^{\bot}\cap Z(J)=0$ , i.e
$Z(J_{\theta_{1}})=Z(J_{\theta_{2}})=V.$ Assume that $J_{\theta_{1}}$ and
$J_{\theta_{2}}$ are isomorphic, let $\alpha:J_{\theta_{1}}\longrightarrow
J_{\theta_{2}}$ be an isomorphism .Dividing with the common center $V$ we
obtain an automorphism $\alpha_{0}:J\longrightarrow J.$ We can realize
$\alpha$ as a matrix to a suitable basis for $J\oplus V$ which assumed to
contain a basis for $J$ and a basis for $V$ :
$\alpha=\left(\begin{array}[]{cc}\begin{array}[]{c}\alpha_{0}\\\
\end{array}&\begin{array}[]{c}0\\\ \end{array}\\\
\varphi&\psi\end{array}\right)$
Where $\alpha_{0}\in Aut(J)$ , $\psi=\alpha\mid_{V}\in Gl(v)$ and $\varphi\in
Hom(J,V).$
Now $\alpha$ preserves the Jordan products and writing $\circ_{1}$and
$\circ_{2}$ for the Jordan products of $J_{\theta_{1}}$ and $J_{\theta_{2}}$
respectively, we have
$\alpha((x+v)\circ_{1}(y+w))=\alpha(x+v)\circ_{2}\alpha(y+w)\text{ \ ; \
}x,y\in J\text{ and }v,w\in V.$
Where
$\displaystyle\alpha(x+v)$ $\displaystyle=$
$\displaystyle\alpha_{0}(x)+\varphi(x)+\psi(v)$ $\displaystyle\alpha(y+w)$
$\displaystyle=$ $\displaystyle\alpha_{0}(y)+\varphi(y)+\psi(w).$
Then
$\displaystyle\alpha((x+v)\circ_{1}(y+w))$ $\displaystyle=$
$\displaystyle\alpha(x\circ y+\theta_{1}(x,y))$ $\displaystyle=$
$\displaystyle\alpha_{0}(x\circ y)+\varphi(x\circ y)+\psi(\theta_{1}(x,y)).$
$\displaystyle\alpha((x+v)\circ_{2}\alpha(y+w))$ $\displaystyle=$
$\displaystyle\alpha_{0}((x)+\varphi(x)+\psi(v))\circ_{2}(\alpha_{0}(y)+\varphi(y)+\psi(w))$
$\displaystyle=$
$\displaystyle\alpha_{0}(x)\circ\alpha_{0}(y)+\theta_{2}(\alpha_{0}(x),\alpha_{0}(y))$
This yields
$\displaystyle\theta_{2}(\alpha_{0}(x),\alpha_{0}(y))$ $\displaystyle=$
$\displaystyle\varphi(x\circ y)+\psi(\theta_{1}(x,y))\ \ ,\ x,y\in J$
$\displaystyle\theta_{2}(\alpha_{0}(x),\alpha_{0}(y))$ $\displaystyle=$
$\displaystyle\psi(\theta_{1}(x,y))\text{ \ mod }\delta C^{1}(J,V)$
In case of $\theta_{1}=\theta_{2}=\theta$ we obtain the following description
of $Aut(J_{\theta})$
###### Lemma 24
Let J be a nilpotent Jordan algebra .Let $\theta\in H^{2}(J,V)$ and
$\theta^{\bot}\cap Z(J)=0.$ Then the automorphism group $Aut(J_{\theta})$ of
the extension algebra $J_{\theta}$ consists of all linear operators of the
matrix form
$\alpha=\left(\begin{array}[]{cc}\begin{array}[]{c}\alpha_{0}\\\
\end{array}&\begin{array}[]{c}0\\\ \end{array}\\\
\varphi&\psi\end{array}\right)$
Where $\alpha_{0}\in Aut(J)$ , $\psi=\alpha\mid_{V}\in Gl(V)$ and $\varphi\in
Hom(J,V).$ Such that :
$\theta(\alpha_{0}(x),\alpha_{0}(y))=\varphi(x\circ y)+\psi\theta(x,y)\
,\forall\ x,y\in J.$
Thus $J_{\theta_{1}}$ and $J_{\theta_{2}}$ are isomorphic if and only if there
exist $\alpha_{0}\in Aut(J)$ and $\psi\in Gl(V)$ such that
$\alpha_{0}\theta_{2}=\psi\theta_{1}$ mod $\delta C^{1}(J,V)$, i.e
$\alpha_{0}\theta_{2}$ and $\psi\theta_{1}$ are cohomologous. Let
$\theta_{1}=(\theta_{11},\theta_{12},.....,\theta_{1s})\in Z^{2}(J,V)$ ,
$\theta_{2}=(\theta_{21},\theta_{22},.....,\theta_{2s})\in Z^{2}(J,V)$ and
$e_{1},e_{2},..........,e_{m}$ be a basis of $V$ then
$\theta_{1}(x,y)=\overset{s}{\underset{i=1}{\sum}}\theta_{1i}(x,y)e_{i\text{
}}$ and
$\theta_{2}(x,y)=\overset{s}{\underset{i=1}{\sum}}\theta_{2i}(x,y)e_{i\text{
}}.$Suppose that $J_{\theta_{1}}$ and $J_{\theta_{2}}$ are isomorphic , then
$\overset{s}{\underset{i=1}{\sum}}\alpha_{0}\theta_{2i}(x,y)e_{i\text{
}}=\overset{s}{\underset{i=1}{\sum}}\theta_{1i}(x,y)\psi(e_{i\text{ }}).$ Let
$\psi(e_{i\text{ }})=\overset{m}{\underset{j=1}{\sum}}a_{ji}e_{j}$ then
$\overset{s}{\underset{i=1}{\sum}}\alpha_{0}\theta_{2i}(x,y)e_{i\text{
}}=\overset{s}{\underset{i=1}{\sum}}\overset{m}{\underset{j=1}{\sum}}\theta_{1i}(x,y)a_{ji}e_{j}$
hence :
$\alpha_{0}\theta_{2i}(x,y)=\overset{s}{\underset{i=1}{\sum}}\theta_{1i}(x,y)a_{ji}\text{
\ \ mod }\delta C^{1}(J,V)$ (4)
It follows that $\alpha_{0}\theta_{2i}$ span the same subspace of $H^{2}(J,K)$
as the $\theta_{1i}$ . We have proved
###### Lemma 25
Let $\theta=(\theta_{1},\theta_{2},.....,\theta_{m})$ and
$\eta=(\eta_{1},\eta_{2},.....,\eta_{m})\in H^{2}(J,V)$ and $\theta^{\bot}\cap
Z(J)=\eta^{\bot}\cap Z(J)=0$ . Then $J_{\theta}$ and $J_{\eta}$ are isomorphic
if and only if there exist $\phi\in Aut(J)$ such that $\phi\theta_{i}$ span
the same subspace of $H^{2}(J,K)$ as the $\eta_{i}$ .
i.e Two allowable subspaces give isomorphic Jordan algebras if and only if
they are in the same $Aut(J)$-orbit.
Let $J=I_{1}\oplus I_{2}$ be the direct sum of two ideals. Suppose that
$I_{2}$ is contained in the centre of $J$. Then $I_{2}$ is called a centeral
component of $J$.
Jordan algebra $J=I_{1}\oplus I_{2}$ with centeral component $I_{2}$ is a
trivial centeral extension of $I_{1}$ by $I_{2}.$ i.e Jordan algebras with
centeral components are simply obtained by taking direct sums of Jordan
algebras of smaller dimension with trivial Jordan algebras ( Vector spaces ).
Therefore when constructing Jordan algebra with non trivial centre as centeral
extension we want to avoid constructing those with centeral components. The
folowing Lemma will help us to exclude Jordan algebras wit centeral components
.
###### Lemma 26
Let $\theta(x,y)=\underset{i=1}{\overset{r}{\sum}}$ $\theta_{i}(x,y)e_{i}\in
H^{2}(J,V)$ and $\theta^{\bot}\cap Z(J)=0$ . Then $J_{\theta}$ has a centeral
component if and only if $\theta_{1},\theta_{2},.....,\theta_{r}$ are linearly
dependent .
Proof. Suppose that $\theta_{1},\theta_{2},.....,\theta_{r}$ are linearly
dependent, then there exist a proper subset of
$\theta_{1},\theta_{2},.....,\theta_{r}$ span the same as $\theta_{i}.$ So we
may assume that that some of $\theta_{i}$ are zero , hence $J_{\theta}$ has a
centeral component .
On the other hand, Let $\theta(x,y)=\underset{i=1}{\overset{r=s+t}{\sum}}$
$\theta_{i}(x,y)e_{i}\in H^{2}(J,V)$ and $\theta^{\bot}\cap Z(J)=0.$Suppose
that $J_{\theta}=J\oplus V$ has a centeral component $B$ , and $B$ is
contained in $V$ . Write $J_{\theta}=J\oplus W\oplus B$ , then $J_{\theta}$
can be considered as a trivial centeral extension of $\tilde{J}=J\oplus W$ by
$B$ .The centre of $J_{\theta}$ is equal to $V=W\oplus B$ .Consider
$\\{e_{1},e_{2},.....,e_{s}\\}$ and $\\{e_{s+1},e_{s+2},.....,e_{t}\\}$ are
basis of $W\ $and $B$ respectively. Also $\tilde{J}=J\oplus W$ can be
considered as a centeral extension of $J$ by$W$ , then there exist a Jordan
cocycle $\psi(x,y)=\underset{i=1}{\overset{s}{\sum}}$ $\psi_{i}(x,y)e_{i}$
such that $\psi^{\bot}\cap Z(J)=0\ $and $J_{\psi}=\tilde{J}=J\oplus W.$ Then
$J_{\theta}$ is a centeral extension of $J_{\psi}$by $B$ with trivial Jordan
cocycle,, then
$\theta(x,y)=\underset{i=1}{\overset{r=s+t}{\sum}}\theta_{i}(x,y)e_{i}=\psi(x,y)=\underset{i=1}{\overset{s}{\sum}}\psi_{i}(x,y)e_{i}$
Hence $\theta_{1},\theta_{2},.....,\theta_{r}$ are lineary dependent .
So to exclude centeral components, we must have $\theta_{i}$ to be linearly
independent .
A Jordan algebra $\tilde{J}$ is said to be a descendant of the Jordan algebra
$J$ if $\tilde{J}\diagup Z(\tilde{J})$ $\cong J$ and $Z(\tilde{J})\leq$’
$\tilde{J}^{2}$. If dim $Z(\tilde{J})$ $=r$ then $\tilde{J}$ is also referred
to as a step$-r$ descendant. A descendant of a nilpotent Jordan algebra is
nilpotent. Conversely, if $\tilde{J}$ is a finite-dimensional nilpotent Jordan
algebra over a field $K$, then by Lemma (20) $\tilde{J}$ is either a
descendant of a smaller-dimensional nilpotent Jordan algebra when $\tilde{J}$
has no centeral components, or $\tilde{J}$ $=J\oplus Kx$ where $J$ is an ideal
of $\tilde{J}$ and $Kx$ is a centeral component of $\tilde{J}$ ($Kx$ is a
$1$-dimensional Vector space viewed as trivial Jordan algebra).
All our previous observations can be summarized as follows:
###### Theorem 27
Let $J$ be a Lie algebra, let $V$be a vector space with fixed basis
$\\{e_{1},...,e_{r}\\}$ over a field $K$, and let $\theta$, $\eta$ be elements
of $Z^{2}(J,V).$
1. 1.
The Jordan algebra $J_{\theta}$ is a step-$r$ descendant of $J$ if and only if
$\theta^{\perp}$ $\cap Z(J)$ $=0$ and the image of the subspace
$<\theta_{1},...,\theta_{r}>$ in $H^{2}(J,K)$ is $r$-dimensional.
2. 2.
Suppose that $\eta$ is an other element of $Z^{2}(J,V)$ and that
$J_{\theta},J_{\eta}$ are descendants of $J$. Then $J_{\theta}\cong J_{\eta}$
if and only if images of the subspaces $<\theta_{1},...,\theta_{r}>$and
$<\eta_{1},...,\eta_{r}>$ in $H^{2}(J,K)$ are in the same orbit under the
action of $Aut(J)$.
It follows that there is a one-to-one correspondence between the set of
isomorphism types of step-$r$ descendants of $J$ and the Aut($J$)-orbits on
the $r$-dimensional allowable subspaces of $H^{2}(J,K)$. Hence the
classification of $n$-dimensional nilpotent Jordan algebras requires that we
determine these orbits for all nilpotent Jordan algebras of dimension at most
$n-1$.
Let $G_{r}(H^{2}(J,K))$ be the Grassmanian of subspaces of dimension $r$ in
$H^{2}(J,K)$ . There is anatural action of $Aut(J)$ on $G_{r}(H^{2}(J,K))$
defined by :
$\phi<\theta_{1},\theta_{2},.....,\theta_{r}>=<\phi\theta_{1},\phi\theta_{2},.....,\phi\theta_{r}>$
for $V=<$ $\theta_{1},\theta_{2},.....,\theta_{r}>\in G_{r}(H^{2}(J,K))$ and
$\phi\in Aut(J)$ .
Note that if $\ \\{\theta_{1},\theta_{2},.....,\theta_{r}\\}$ is linear
independent so
is$\\{\phi\theta_{1},\phi\theta_{2},.....,\phi\theta_{r}\\}$.Define
$U_{r}(J)=\\{V=<\theta_{1},\theta_{2},.....,\theta_{r}>\in
G_{r}(H^{2}(J,K)):\theta_{i}^{\bot}\cap Z(J)=0;i=1,2,...,r\\}.$
###### Lemma 28
$U_{r}(J)$ is stable under the action of $Aut(J).$
Proof. Let $\phi\in Aut(J)$ and $V=<\theta_{1},\theta_{2},.....,\theta_{r}>\in
U_{r}(J).$ Let $x\in(\phi\theta_{i})^{\perp}=\\{y\in
J:\phi\theta_{i}(y,J)=0\\}$ then $\phi(x)\in\theta_{i}^{\perp}$ hence
$x\in\phi^{-1}\theta_{i}^{\perp}=\\{\phi^{-1}(y):y\in\theta_{i}^{\perp}\\}$ .
Also let $\phi^{-1}(x)\in\phi^{-1}\theta_{i}^{\perp}$ then
$\phi\theta_{i}(\phi^{-1}(x),J)=\theta_{i}(x,J)=0$ hence
$\phi^{-1}(x)\in(\phi\theta_{i})^{\perp}.$ Then
$(\phi\theta_{i})^{\perp}=\phi^{-1}\theta_{i}^{\perp}$ and
$\phi^{-1}(Z(J))=Z(J).$ Therefore $(\phi\theta_{i})^{\bot}\cap
Z(J)=\phi^{-1}(\theta_{i}^{\bot}\cap Z(J))=0$ then $\phi V\in U_{r}(J).$
Let $U_{r}(J)\diagup Aut(J)$ be the set of $Aut(J)$-orbits of $U_{r}(J).$ Then
there exists a canonical one-to-one correspondense from $U_{r}(J)\diagup
Aut(J)$ onto the set of isomorphism classes of Jordan algebras without
centeral components which are central extensions of $J$ by $V$ and have
$r$-dimensional center where $r=\dim V$.
All our previous observations can be summarized to have an analogue of the
Skejelbred-Sund theorem for Jordan algebras as follows:
###### Theorem 29
Let $J$ be a Jordan algebra over a field $K$.The isomorphism clases of Jordan
algebras $\tilde{J}$ with centre $V$ of dimension $r$ ,$\tilde{J}\diagup
V\cong J$ and without centeral component are in bijective correspondence with
the elements in $U_{r}(J)\diagup Aut(J).$
By this theorem, we may construct all nilpotent Jordan algebras of dimension
$n$, given those algebras of dimension less than $n$, by centeral extension.
## 5 Constructing nilpotent Jordan algebras
In this section we carry out a procudure for constructing nilpotent Jordan
algebras over any arbitrary field.
The classification procedure :
Let $J$ be a Jordan algebras with basis $e_{1},e_{2},....,e_{n-r}$ then the
dual basis of the space of all symmetric bilinear forms $\theta:J\times
J\longrightarrow K$ are $\sum_{e_{i},e_{j}}$ $:n-r\geq i\geq j=1$ with
$\sum_{e_{i},e_{j}}(e_{l},e_{m})=\sum_{e_{i},e_{j}}(e_{m},e_{l})=1$ if $i=l$
and $j=m$ ,and takes the value $0$ otherwise, then any $2$-cocycle $\theta\in$
$Z^{2}(J,K)$ can be represented by $\theta=\underset{i\geq
j=1}{\overset{n-r}{\sum}}c_{ij}\sum_{e_{i},e_{j}}$ such that the elements
$c_{i,j}$ satisfy Jordan identity for $2-$cocycles . By Theorem (29) we have a
procdure that takes as input a Jordan algebra $J$ of dimension $n-r$ it
outputs all nilpotent Jordan algebras $\tilde{J}$ of dimension $n$ such that
$\tilde{J}\diagup Z($ $\tilde{J})\cong J$, and $\tilde{J}$ has no centeral
components. It runs as follows :
* •
For a given nilpotent Jordan algebra $J$ of dimension $n-r$, we list at first
its center to help us identify the $2$-cocycles satisfying $\theta^{\perp}$
$\cap Z(J)$ $=0.$
* •
Compute $Z^{2}(J,K)$: When computing the $2$-cocycles just list all the
constraints on the elements $c_{ij}.$
* •
Compute $\delta C^{1}(J,K):$ For $e_{l}\in J^{2}$ define
$\theta_{l}=\underset{i,j}{\sum}\left(\sum_{e_{i},e_{j}}\right)$ such that
$e_{i}\circ e_{j}=e_{l}.$ Then $\delta C^{1}(J,K)$ is spanned by $\theta_{l}$
for all $e_{l}\in J^{2},$
* •
Compute $H^{2}(J,K):$ The complement of $\delta C^{1}(J,K)$ in $Z^{2}(J,K)$ be
$H^{2}(J,K).$ For $\theta=\underset{i\geq
j=1}{\overset{n-r}{\sum}}c_{i,j}\sum_{e_{i},e_{j}}\in$ $Z^{2}(J,K)$ put
$c_{i,j}=0$ for all $\sum_{e_{i},e_{j}}=\theta_{l}\in$ $\delta C^{1}(J,K).$
* •
Consider $\theta\in H^{2}(J,V)$ with
$\theta(x,y)=\overset{r}{\underset{i=1}{\sum}}\theta_{i}(x,y)e_{n-r+i}$ where
the $\theta_{i}\in H^{2}(J,K)$ are linearly indpendent, and $\theta^{\perp}$
$\cap Z(J)$ $=0.$ Find a list of representatives of the orbits of $Aut(J)$
acting on the $\theta$.
* •
For each $\theta$ found, construct $J_{\theta}$. Discard the isomorphic ones.
Remarks :
1. 1.
If charcteristic $K\neq 2,$ one can linearize $\theta(x^{2},x\circ
y)=\theta(x,x^{2}\circ y)$ to obtain :
$\theta(x,v\circ(y\circ z))+\theta(y,v\circ(x\circ z))+\theta(z,v\circ(x\circ
y))=\theta(x\circ y,z\circ v)+\theta(y\circ z,x\circ v)+\theta(x\circ z,y\circ
v)$
for all $x,y,z,v\in J.$
2. 2.
One can use Lemma $\left(\ref{n}\right)$ to compute the automorphism group,and
the system of equations $\left(\ref{k}\right)$for testing if two cocycles are
in the same $Aut(J)$-orbit by checking solvalbality. However in the next
section we construct an algorithm for testing isomorphism of any two Jordan
algebra, by this algorithm we can also compute the automorphism group.
3. 3.
The procedure only gives those nilpotent Jordan algebras without cetral
components. So we have to add the nilpotent Jordan algebras obtained by by
taking direct sum of a smaller dimensional nilpotent Jordan algebra with
trivial Jordan algebra (that has trivial multiplication).
## 6 Deciding isomorphism of Jordan algebras
A classical problem is to know how many different (up to isomorphisms) finite
dimensional Jordan algebras exist for each dimension. Let $J$ be a Jordan
algebra of dimension $n$, then it has a basis $e_{1},e_{2},....,e_{n}.$ It
follows that there are constants $c_{ij}^{k}$ for $n\geq i\geq j=1$ such that
$e_{i}\circ e_{j}=\underset{k=1}{\overset{n}{\sum}}c_{ij}^{k}e_{k}.$
Let $J_{1}$ and $J_{2}$ be two Jordan algebras for which we want to decide
whether they are isomorphic or not. If the dimension of $J_{1}$ and $J_{2}$
are not equal, then $J_{1}$ and $J_{2}$ can never be isomorphic. Furthermore,
suppose that $\phi:J_{1}\longrightarrow J_{2}$ is an isomorphism of Jordan
algebras, then $\phi(Z(J_{1}))=Z(J_{2})$ and
$\phi(C^{k}(J_{1}))=C^{k}(J_{2}).$ So if $\dim Z(J_{1})\neq\dim Z(J_{2})$ then
$J_{1}$ and $J_{2}$ can never be isomorphic. Also if $\dim
C^{k}(J_{1})\neq\dim C^{k}(J_{2})$ hen $J_{1}$ and $J_{2}$ can never be
isomorphic. By this way we may able to decide that $J_{1}$ and $J_{2}$ are not
isomorphic. There is a direct method for testing isomorphism of Jordan
algebras by using Grobner bases. Let $e_{1},e_{2},....,e_{n}$ be a basis of
$J_{1}$ and let $\tilde{e}_{1},\tilde{e}_{2},....,\tilde{e}_{n}$ be a basis of
$J_{2}.$ Let $(c_{ij}^{k})$ and $(\gamma_{ij}^{k})$ be the structure constants
of $J_{1}$ and $J_{2}$, respectively. A map $\phi:J_{1}\longrightarrow J_{2}$
is an isomorphism of Jordan algebras if and only if it satisfies the following
requirements :
* •
$\phi(e_{i}\circ e_{j})=\phi(e_{i})\circ\phi(e_{j})$ for $n\geq i\geq j=1,$
* •
$\phi$ is non-singular.
Let $\phi:J_{1}\longrightarrow J_{2}$ is an isomorphism of Jordan algebras
given by $\phi(e_{i})=\overset{n}{\underset{j=1}{\sum}}a_{ij}\tilde{e}_{j}$
then
$\phi(e_{i}\circ
e_{j})=\phi\left(\underset{k=1}{\overset{n}{\sum}}c_{ij}^{k}e_{k}\right)=\underset{k=1}{\overset{n}{\sum}}c_{ij}^{k}\phi(e_{k})=\underset{k,m=1}{\overset{n}{\sum}}c_{ij}^{k}a_{km}\tilde{e}_{m},$
and
$\displaystyle\phi(e_{i})\circ\phi(e_{j})$ $\displaystyle=$
$\displaystyle\left(\overset{n}{\underset{k=1}{\sum}}a_{ik}\tilde{e}_{k}\right)\circ\left(\overset{n}{\underset{l=1}{\sum}}a_{jl}\tilde{e}_{l}\right)$
$\displaystyle=$
$\displaystyle\overset{n}{\underset{k,l=1}{\sum}}a_{ik}a_{jl}(\tilde{e}_{k}\circ\tilde{e}_{l})=\overset{n}{\underset{k,l,m=1}{\sum}}\gamma_{kl}^{m}a_{ik}a_{jl}\tilde{e}_{m}.$
Hence the first requirement amounts to the following $\frac{n(n+1)}{2}$
equations in the variables $a_{ij}$ :
$\underset{k=1}{\overset{n}{\sum}}c_{ij}^{k}a_{km}-\overset{n}{\underset{k,l=1}{\sum}}\gamma_{kl}^{m}a_{ik}a_{jl}=0\text{
for }n\geq i\geq j=1\text{ and }1\leq m\leq n.$
By the second erquirement the determinant of the matrix $(a_{ij})$ not equal
to $0$, $\det(a_{ij})\neq 0.$
###### Theorem 30
(Generalized Weak Nullstellensatz). Let $k$ be any field. A system of
polynomials $f_{1},...,f_{m}$ $\in$ $k[x1,....,xn]$has no common zero over the
algebraic closure of $k$, if and only if, $1\in I$ where $I$ be the ideal
generated by the polynomials $f_{1},...,f_{m}.$
So we consider the ideal $I$ generated by the polynomials
$\underset{k=1}{\overset{n}{\sum}}c_{ij}^{k}a_{km}-\overset{n}{\underset{k,l=1}{\sum}}\gamma_{kl}^{m}a_{ik}a_{jl}=0\text{
for }n\geq i\geq j=1\text{ and }1\leq m\leq n.$
together with the polynomial $b\det(a_{ij})-1$ in the polynomial ring
$k[a_{ij},b].$ Then there is a solution of this system if and only if $I\neq
k[a_{ij},b].$ The algorithm for calculating Grobener bases yields a method for
deciding whether or not $1\in I.$ Then by Grobener bases calculations we can
decide isomorphism of Jordan algebras.
###### Algorithm 31
Testing isomorphism for Jordan algebras.
Input : Two Jordan algebras $(J_{1},\circ_{1})$ with basis
$e_{1},e_{2},....,e_{n}$ and $(J_{2},\circ_{2})$ with basis
$\tilde{e}_{1},\tilde{e}_{2},....,\tilde{e}_{n}.$
Output ; True if $J_{1}\cong J_{2}$ and false in other case.
1. 1.
Compute the following system of equations :
$\phi(e_{i}\circ_{1}e_{j})-\phi(e_{i})\circ_{2}\phi(e_{j})=0\ \text{for }n\geq
i\geq j=1.$
where $\phi=(a_{ij})_{n\times n}$
2. 2.
To ensure that $\phi$ is going to be non-singular, add the following relation
with a new variable $b$ :
$b\det\phi-1=0.$
3. 3.
Compute a grobener bases $G$ of the ideal
$I=<\\{\phi(e_{i}\circ_{1}e_{j})-\phi(e_{i})\circ_{2}\phi(e_{j})\\}_{n\geq
i\geq j=1}\cup\\{b\det\phi-1\\}>$ in the polynomial ring $k[a_{ij},b].$
4. 4.
$G=\\{1\\}?$
1. (a)
Yes.
Return False.
2. (b)
No.
Output : True ; Return $G$.
If the output of the algorithm is true, then it provides the equations of an
algebraic variety whose points are all possiple values for $\phi,$ hence we
have a descripiton of the automorphism group in the case $J_{1}=J_{2}$ .
## 7 Nilpotent Jordan algebras of dimension$\leq 4$
In this section we use the analouge Skjelbred-Sund method to classify
nilpotent Jordan algebras of dimension $\leq 3$ over any field, and four
dimensional nilpotent Jordan algebras in the following cases :
* •
Over algebraic closed field $K$ and $ch(K)\neq 2.$
* •
Over the real field $\mathbb{R}.$
Here we denote the $j$-th algebra of dimension $i$ by $J_{i,j}$ .
### 7.1 Nilpotent Jordan algebras of dimension $1$
Let $J$ be a Jordan algebra of dimension $1$ spanned by a. If $J$ is nilpotent
then $Z(J)$ is non-trivial. Hence $Z(J)$ is spanned by $a$. It follows that
there is only one nilpotent Jordan algebra of dimension $1$ over any arbitrary
field,$J_{1,1},$ it spanned by $a$ and $a^{2}=0$.
$J_{1,1}$ | All multiplications are zero.
---|---
### 7.2 Nilpotent Jordan algebras of dimension $2$
To construct all nilpotent Jordan algebras of dimension $2,$ firstly we
consider the trivial $1$-dimensional Central extension of the nilpotent Jordan
algebras of dimension $1$ corresponding to $\theta=0$ ( Algebras that are
direct sum of an algebra of dimension $1$ and a $1$-dimensional trivial Jordan
algebra, isomorphic to $J_{1,1}$) to get nilpotent Jordan algebras with
centeral component . Finally consider non trivial $1$\- dimensional Central
extension of the nilpotent Jordan algebras of dimension $1$ to get nilpotent
Jordan algebras without centeral component.
#### 7.2.1 Nilpotent Jordan algebras with Centeral component
1. 1.
1- dimensional Trivial Central extension of $J_{1,1}$ corresponding to
$\theta=0.$
1. (a)
We get the Jordan algebra :
$J_{2,1}=J_{1,1}\oplus J_{1,1}$ | All multiplications are zero.
---|---
#### 7.2.2 Nilpotent Jordan algebras without Centeral component
1. 1.
1\- dimensional Central extension of $J_{1,1}$
Here we get that $Z^{2}(J_{1,1},K)$ is spanned by $\sum_{a,a}$ and $\delta
C^{1}(J_{1,1},K)=0$, then $H^{2}(J_{1,1},K)$ is spanned by $\sum_{a,a}.$ So we
get only one cocycle $\theta=\sum_{a,a}$, yielding the algebra :
$J_{2,2}$ | $a^{2}=b.$
---|---
###### Theorem 32
Up to isomorphism there exactly two nilpotent Jordan algebras of dimension $2$
over any field $K$ which are isomorphic to the following nilpotent Jordan
algebras :
Nilpotent Jordan algebras with Centeral component
---
$J_{2,1}=J_{1,1}\oplus J_{1,1}$.
Nilpotent Jordan algebras without Centeral component
$J_{2,2}$ $:$ $a^{2}=b.$
### 7.3 Nilpotent Jordan algebras of dimension $3$
From the previous subsection we get that the $2$-dimensional nilpotent Jordan
algebras over any field $K$ are : $J_{2,1}$ and $J_{2,2}.$ Using those algebra
to construct $3$-dimensional nilpotent Jordan algebras as follow :
#### 7.3.1 Nilpotent Jordan algebras with Centeral component
1. 1.
1- dimensional Trivial Central extension of $J_{2,1}$ corresponding to
$\theta=0.$
We get the Jordan algebra :
$J_{3,1}=J_{2,1}\oplus J_{1,1}$ | All multiplications are zero.
---|---
2. 2.
1\- dimensional Trivial Central extension of $J_{2,2}$ corresponding to
$\theta=0.$
We get the Jordan algebra :
$J_{3,2}=J_{2,2}\oplus J_{1,1}$ | $a^{2}=b$.
---|---
#### 7.3.2 Nilpotent Jordan algebras without Centeral component
1. 1.
1\- dimensional Central extension of $J_{2,1}$
Here we get that $Z^{2}(J_{2,1},K)$ is spanned by $\sum_{a,a}$,$\sum_{a,b}$
and $\sum_{b,b},$ also we get that $\delta C^{1}(J_{2,1},K)=0$. Then
$H^{2}(J_{2,1},K)$ is consists of
$\theta=\alpha_{1}\sum_{a,a}+\alpha_{2}\sum_{a,b}+\alpha_{3}\sum_{b,b}.$ The
centre of $J_{2,1},Z(J_{2,1}),$ spanned by $a$ and $b$, it follows that
$\theta:J_{2,1}\times J_{2,1}\longrightarrow K$ such that
$\theta^{\perp}\cap Z(J_{2,1})=0$
is nondegenerate since $\theta^{\perp}=0.$ The autommorphism group consists of
$:$
$\phi=\left(\begin{array}[]{cc}a_{11}&a_{12}\\\
a_{21}&a_{22}\end{array}\right),\det\phi\neq 0.$
Characteristic $K\neq 2$
Every symmetric bilinear form is diagonalizable, so we may assume that
$\theta=\sum_{a,a}+\alpha\sum_{b,b}$ and $\alpha\neq 0.$ Write
$\phi\theta=\alpha_{1}^{\prime}\sum_{a,a}+\alpha_{2}^{\prime}\sum_{a,b}+\beta\sum_{b,b},$
then
$\displaystyle\alpha_{1}^{\prime}$ $\displaystyle=$ $\displaystyle
a_{11}^{2}+\alpha a_{21}^{2}$ $\displaystyle\alpha_{2}^{\prime}$
$\displaystyle=$ $\displaystyle a_{11}a_{12}+\alpha a_{21}a_{22}$
$\displaystyle\beta$ $\displaystyle=$ $\displaystyle a_{12}^{2}+\alpha
a_{22}^{2}$
To fix $\alpha_{1}^{\prime}=1$ and $\alpha_{2}^{\prime}=0,$ choose
$a_{21}=a_{12}=0$ and $a_{11}=1.$ Then
$\phi\left(\sum_{a,a}+\alpha\sum_{b,b}\right)=\sum_{a,a}+\alpha
a_{22}^{2}\sum_{b,b}.$ This show that for any $\alpha,\beta,a_{22}\in
K^{\ast}$ such that $\beta=\alpha a_{22}^{2},$ then $\theta_{1,\alpha}$ and
$\theta_{1,\beta}$ are in the same $Aut(J_{2,1})$-orbit. On the other hand ,
suppose that $\theta_{1,\alpha}$ and $\theta_{1,\beta}$ are in the same
$Aut(J_{2,1})$-orbit. Then there exist $\phi\in Aut(J_{2,1})$ and $\lambda\in
K^{\ast}$ such that $\phi\theta_{1,\alpha}=\lambda\theta_{1,\beta},$ so we
have
$\phi^{t}\left(\begin{array}[]{cc}1&0\\\
0&\alpha\end{array}\right)\phi=\allowbreak\lambda\left(\begin{array}[]{cc}1&0\\\
0&\beta\end{array}\right)\allowbreak$
it follows that $\beta=\left(\frac{\det\phi}{\lambda}\right)^{2}\alpha.$ Hence
$\theta_{1,\alpha}$ and $\theta_{1,\beta}$ are in the same
$Aut(J_{2,1})$-orbit if and only if there is a $\delta\in K^{\ast}$ with
$\beta=\delta^{2}\alpha.$ So we get the algebra
$J_{3,3}^{\alpha}$ | $a^{2}=c$ , $b^{2}=\alpha c$ , $\alpha\in K^{\ast}\diagup\left(K^{\ast}\right)^{2}.$
---|---
Over algebraic closed field $K$,
$K^{\ast}\diagup\left(K^{\ast}\right)^{2}=\\{1\\}.$ So we get only one
$J_{3,3}$ | $a^{2}=c$ , $b^{2}=c.$
---|---
Over real field $\mathbb{R},$
$\mathbb{R}^{\ast}\diagup\left(\mathbb{R}^{\ast}\right)^{2}=\\{\pm 1\\}.$ So
we get two algebras
$J_{3,3}$ | $a^{2}=c$ , $b^{2}=c.$
---|---
$J_{3,3}^{-1}$ | $a^{2}=c$ , $b^{2}=-c.$
Over finite field $K,K^{\ast}\diagup\left(K^{\ast}\right)^{2}=\\{\pm 1\\}.$ So
we get two algebras
$J_{3,3}$ | $a^{2}=c$ , $b^{2}=c.$
---|---
$J_{3,3}^{-1}$ | $a^{2}=c$ , $b^{2}=-c.$
Over rational field
$\mathbb{Q},\mathbb{Q}^{\ast}\diagup\left(\mathbb{Q}^{\ast}\right)^{2}$ is
infinite. Then there is an infinite number of them.
Characteristic $K=2$
If $\theta$ is alternate we obtain only one cocycle $\theta_{1}=\sum_{a,b}$.
On the other hand if $\theta$ is not alternate, then it is diagonalizable. In
this case $K^{\ast}\diagup\left(K^{\ast}\right)^{2}=\\{1\\}$. so we obtain
only one cocycle $\theta_{2}=\sum_{a,a}+\sum_{b,b}.$ Clearly
$\theta_{1},\theta_{2}$ are inequivalent bilinear form, hence they not lie in
the same $Aut(J_{2,1})$-orbit,[ $\phi\theta_{1}=(\det\phi)\theta_{1}.$ i.e
$\phi\theta_{1}$ is a multiple of $\theta_{1},$hence it is not conjugate to
$\theta_{2}.$ Then$\left(J_{2,1}\right)_{\theta_{1}}$ and
$\left(J_{2,1}\right)_{\theta_{2}}$ are not isomorphic]. We get two algebras
$J_{3,3}$ | $a^{2}=c$ , $b^{2}=c.$
---|---
$J_{3,4}$ | $a\circ b=c.$
2. 2.
1\- dimensional Central extension of $J_{2,2}$
Here we get that $Z^{2}(J_{2,1},K)$ is spanned by $\sum_{a,a}$ and
$\sum_{a,b}.$ Moreover $\delta C^{1}(J_{1,1},K)$ is spanned by $\sum_{a,a}$,
then $H^{2}(J_{1,1},K)$ is spanned by $\sum_{a,b}.$ So we get only one cocycle
$\theta=\sum_{a,b}$, yielding the algebra
$J_{3,4}$ | $a^{2}=b$ , $a\circ b=c$.
---|---
3. 3.
2\- dimensional Central extension of $J_{1,1}$
We have that $H^{2}(J_{1,1},K)$ is 1-dimensional spanned by $\sum_{a,a},$ then
there is no 2- dimensional Central extension of $J_{1,1}.$
We can summarize some of our results in the following theorems :
###### Theorem 33
Up to isomorphism there exist $4$ nilpotent Jordan algebras of dimension $3$
over algebraic closed field $K$ and $ch(K)\neq 2$ which are isomorphic to one
of the following pairwise non-isomorphic nilpotent Jordan algebras :
Nilpotent Jordan algebras with Centeral component
---
$J_{3,1}=J_{2,1}\oplus J_{1,1}.$
$J_{3,2}=J_{2,2}\oplus J_{1,1}.$
Nilpotent Jordan algebras without Centeral component
$J_{3,3}$ $:$ $a^{2}=c$ , $b^{2}=c.$
$J_{3,4}$ $:$ $a^{2}=b$ , $a\circ b=c$.
###### Theorem 34
Up to isomorphism there exist $5$ nilpotent Jordan algebras of dimension $3$
over field $K$ and $ch(K)=2$ which are isomorphic to one of the following
pairwise non-isomorphic nilpotent Jordan algebras :
Nilpotent Jordan algebras with Centeral component
---
$J_{3,1}=J_{2,1}\oplus J_{1,1}.$
$J_{3,2}=J_{2,2}\oplus J_{1,1}.$
Nilpotent Jordan algebras without Centeral component
$J_{3,3}$ $:$ $a\circ b=c.$
$J_{3,4}$ $:$ $a^{2}=c$ , $b^{2}=c.$
$J_{3,5}$ $:$ $a^{2}=b$ , $a\circ b=c$.
###### Theorem 35
Up to isomorphism there exist $5$ nilpotent Jordan algebras of dimension $3$
over $\mathbb{R}$ which are isomorphic to one of the following pairwise non-
isomorphic nilpotent Jordan algebras :
Nilpotent Jordan algebras with Centeral component
---
$J_{3,1}=J_{2,1}\oplus J_{1,1}.$
$J_{3,2}=J_{2,2}\oplus J_{1,1}.$
Nilpotent Jordan algebras without Centeral component
$J_{3,3}^{\alpha=\pm 1}$ $:$ $a^{2}=c$ , $b^{2}=\alpha c.$ $\left(\alpha=\pm
1\right)$
$J_{3,4}$ $:$ $a^{2}=b$ , $a\circ b=c.$
We are showed that also the number of nilpotent Jordan algebras of dimension
$3$ over finite field is five for any characteristic.
###### Theorem 36
Every nilpotent Jordan algebra of dimension $\leq 3$ is a commutative
nilpotent associative algebra.
It follows that all commutative nilpotent associative algebras of dimension
$\leq 3$ are classified, and all previous theorems are valid for commutative
nilpotent associative algebras.
### 7.4 Nilpotent Jordan algebras of dimension $4$
In this subsection we construct all nonisomorphic nilpotent algebras of
dimension four :
* •
Over algebraic closed field $K$ and characteristic $K\neq 2.$
* •
Over real field $\mathbb{R}.$
Here we denote the $j$-th algebra of dimension $i$ over $K$ and $\mathbb{R}$
by $J_{i,j}(K)$ and $J_{i,j}(\mathbb{R})$ respectively.
#### 7.4.1 Nilpotent Jordan algebras with Centeral component
1. 1.
1- dimensional Trivial Central extension of $J_{3,1}$ corresponding to
$\theta=0.$
We get the Jordan algebra :
$J_{4,1}(K)=J_{3,1}\oplus J_{1,1}$ | All multiplications are zero.
---|---
$J_{4,1}(\mathbb{R})=J_{3,1}\oplus J_{1,1}$ | All multiplications are zero.
---|---
2. 2.
1- dimensional Trivial Central extension of $J_{3,2}$ corresponding to
$\theta=0.$
We get the Jordan algebra :
$J_{4,2}(K)=J_{3,2}\oplus J_{1,1}$ | $a^{2}=b$.
---|---
$J_{4,2}(\mathbb{R})=J_{3,2}\oplus J_{1,1}$ | $a^{2}=b$.
---|---
3. 3.
1- dimensional Trivial Central extension of $J_{3,3}^{\alpha}$,$\alpha\in
K^{\ast}\diagup\left(K^{\ast}\right)^{2},$corresponding to $\theta=0.$
We get the Jordan algebra :
$J_{4,3}(K)=J_{3,3}\oplus J_{1,1}$ | $a^{2}=c$ , $b^{2}=c$ .
---|---
$J_{4,3}^{\alpha=\pm 1}(\mathbb{R})=J_{3,3}^{\alpha=\pm 1}\oplus J_{1,1}$ | $a^{2}=c$ , $b^{2}=\alpha c$ . $\left(\alpha=\pm 1\right)$
---|---
4. 4.
1- dimensional Trivial Central extension of $J_{3,4}$ corresponding to
$\theta=0.$
We get the Jordan algebra :
$J_{4,4}(K)=J_{3,4}\oplus J_{1,1}$ | $a^{2}=b$ , $a\circ b=c$.
---|---
$J_{4,4}(\mathbb{R})=J_{3,4}\oplus J_{1,1}$ | $a^{2}=b$ , $a\circ b=c$.
---|---
#### 7.4.2 Nilpotent Jordan algebras without Centeral component
1. 1.
1\- dimensional Central extension of $J_{3,1}$
Here we get that $Z^{2}(J_{3,1},K)$ is spanned by
$\sum_{a,a},\sum_{b,b},\sum_{c,c},\sum_{a,b},\sum_{a,c}$ and $\sum_{b,c},$
also we get that $\delta C^{1}(J_{3,1},K)=0$. Then $H^{2}(J_{3,1},K)$ consists
of
$\theta=\alpha_{1}\sum_{a,a}+\alpha_{2}\sum_{b,b}+\alpha_{3}\sum_{c,c}+\alpha_{4}\sum_{a,b}+\alpha_{5}\sum_{a,c}+\alpha_{6}\sum_{b,c}.$
The centre of $J_{3,1},Z(J_{3,1}),$ spanned by $a,b$ and $c$, it follows that
$\theta:J_{3,1}\times J_{3,1}\longrightarrow K$ such that
$\theta^{\perp}\cap Z(J_{3,1})=0$
is nondegenerate since $\theta^{\perp}=0.$ The autommorphism
group,$Aut(J_{3,1}),$ consists of $:$
$\phi=\left(\begin{array}[]{ccc}a_{11}&a_{12}&a_{13}\\\
a_{21}&a_{22}&a_{23}\\\ a_{31}&a_{32}&a_{33}\end{array}\right),\det\phi\neq
0.$
Over algebraic closed field $K$ and characteristic $K\neq 2:$
Up to equivalence there is only one nondegenerate symmetric bilinear form,
$\sum_{a,a}+\sum_{b,b}+\sum_{c,c}.$ So there is only one $Aut(J_{3,1})$-orbit
$\theta=\sum_{a,a}+\sum_{b,b}+\sum_{c,c},$ yielding the algebra
$J_{4,5}(K)$ | $a^{2}=d$ , $b^{2}=d$ $,c^{2}=d$.
---|---
Over real field $\mathbb{R}:$
By Sylvester’s law of inertia and Signature, up to equivalence there are four
nondegenerate symmetric bilinear form
$\theta_{1}=\sum_{a,a}+\sum_{b,b}+\sum_{c,c}$
$\theta_{2}=\sum_{a,a}+\sum_{b,b}-\sum_{c,c}$
$\theta_{3}=-\sum_{a,a}-\sum_{b,b}+\sum_{c,c}$
$\theta_{4}=-\sum_{a,a}-\sum_{b,b}-\sum_{c,c}.$
We see that $\theta_{1}$ and $\theta_{4}$ are in the same $Aut(J_{3,1})$-orbit
(since $\theta_{1}=-\theta_{4}$), also $\theta_{2}$ and $\theta_{3}$ are in
the same $Aut(J_{3,1})$-orbit (since $\theta_{2}=-\theta_{3}$). It remains to
check that if $\theta_{1}$ and $\theta_{2}$ are in the same
$Aut(J_{3,1})$-orbit or not. We claim that $\theta_{1}$ and $\theta_{2}$ are
not in the same $Aut(J_{3,1})$-orbit. Let $\phi\in Aut(J_{3,1})$ and
$\lambda\in\mathbb{R}^{\ast}$ such that $\phi\theta_{1}=\lambda\theta_{2}.$
Then
$\displaystyle a_{11}^{2}+a_{21}^{2}+a_{31}^{2}$ $\displaystyle=$
$\displaystyle\lambda$ (5) $\displaystyle a_{12}^{2}+a_{22}^{2}+a_{32}^{2}$
$\displaystyle=$ $\displaystyle\lambda$ (6) $\displaystyle
a_{13}^{2}+a_{23}^{2}+a_{33}^{2}$ $\displaystyle=$ $\displaystyle-\lambda$ (7)
$\displaystyle a_{11}a_{12}+a_{21}a_{22}+a_{31}a_{32}$ $\displaystyle=$
$\displaystyle 0$ $\displaystyle a_{11}a_{13}+a_{21}a_{23}+a_{31}a_{33}$
$\displaystyle=$ $\displaystyle 0$ $\displaystyle
a_{12}a_{13}+a_{22}a_{23}+a_{32}a_{33}$ $\displaystyle=$ $\displaystyle 0$
Equations $\left(\ref{l1}\right),\left(\ref{l2}\right)$ with
$\left(\ref{l3}\right)$ lead to $\lambda=0$ and $\phi=0.$ So $\theta_{1}$ and
$\theta_{2}$ are not in the same $Aut(J_{3,1})$-orbit, hence we get only two
algebras
$J_{4,5}^{\alpha=\pm 1}(\mathbb{R})$ | $a^{2}=d$ , $b^{2}=d$ $,c^{2}=\alpha d$.
---|---
2. 2.
1\- dimensional Central extension of $J_{3,2}$
Here we get that $Z^{2}(J_{3,2},K)$ is spanned by
$\sum_{a,a},\sum_{a,b},\sum_{a,c},\sum_{b,c}$ and $\sum_{c,c}.$ Moreover,
$\delta C^{1}(J_{3,2},K)$ is spanned by $\sum_{a,a}$. Then $H^{2}(J_{3,2},K)$
consists of
$\theta:=\alpha_{1}\sum_{a,b}+\alpha_{2}\sum_{a,c}+\alpha_{3}\sum_{b,c}+\alpha_{4}\sum_{c,c}.$
The centre of $J_{3,2},Z(J_{3,2}),$ is spanned by $b$ and $c.$ Furthermore the
automorphism group,$Aut(J_{3,2}),$ consists of :
$\phi=\begin{pmatrix}a_{11}&0&0\\\ a_{21}&a_{11}^{2}&a_{23}\\\
a_{31}&0&a_{33}\end{pmatrix}\text{ , }\det\phi\neq 0.$
The automorphism group acts as follows :
$\displaystyle\alpha_{1}$ $\displaystyle\longrightarrow$ $\displaystyle
a_{11}^{3}\alpha_{1}+a_{31}a_{11}^{2}\alpha_{3}$ $\displaystyle\alpha_{2}$
$\displaystyle\longrightarrow$ $\displaystyle
a_{11}a_{23}\alpha_{1}+a_{11}a_{33}\alpha_{2}+(a_{21}a_{33}+a_{31}a_{23})\alpha_{3}+a_{31}a_{33}\alpha_{4}$
$\displaystyle\alpha_{3}$ $\displaystyle\longrightarrow$ $\displaystyle
a_{11}^{2}a_{33}\alpha_{3}$ $\displaystyle\alpha_{4}$
$\displaystyle\longrightarrow$ $\displaystyle
2a_{23}a_{33}\alpha_{3}+a_{33}^{2}\alpha_{4}.$
We distinguish two cases.
Case 1 : First suppose that $\alpha_{3}\neq 0,$ then we can divide to get
$\alpha_{3}=1.$ So we may assume that $\alpha_{3}=1,$ choose
$a_{11}=a_{33}=1$, this leads to :
$\displaystyle\alpha_{1}$ $\displaystyle\longrightarrow$
$\displaystyle\alpha_{1}+a_{31}$ $\displaystyle\alpha_{2}$
$\displaystyle\longrightarrow$ $\displaystyle
a_{23}\alpha_{1}+\alpha_{2}+(a_{21}+a_{31}a_{23})+a_{31}\alpha_{4}$
$\displaystyle\alpha_{3}$ $\displaystyle\longrightarrow$ $\displaystyle 1$
$\displaystyle\alpha_{4}$ $\displaystyle\longrightarrow$ $\displaystyle
2a_{23}+\alpha_{4}.$
By taking $a_{31}=-\alpha_{1}$ and $a_{23}=-\frac{1}{2}\alpha_{4},$ we get
$\alpha_{1}\longrightarrow 0$ and $\alpha_{4}\longrightarrow 0.$ Then we can
assume that $\alpha_{1}=\alpha_{4}=0$ and $\alpha_{3}=1.$ To conserve this we
set $a_{31}=a_{23}=0$ and $a_{11}=a_{33}=1,$ in this case we get
$\displaystyle\alpha_{1}$ $\displaystyle\longrightarrow$ $\displaystyle 0$
$\displaystyle\alpha_{2}$ $\displaystyle\longrightarrow$
$\displaystyle\alpha_{2}+a_{21}$ $\displaystyle\alpha_{3}$
$\displaystyle\longrightarrow$ $\displaystyle 1$ $\displaystyle\alpha_{4}$
$\displaystyle\longrightarrow$ $\displaystyle 0.$
By taking $a_{21}=-\alpha_{2},$ then $\alpha_{2}\longrightarrow 0.$ Then we
may assume that $\theta=\sum_{b,c},$ so we get the algebra :
$J_{4,6}(K)$ | $a^{2}=b\text{ },\text{ }b\circ c=d.$
---|---
$J_{4,6}(\mathbb{R})$ | $a^{2}=b\text{ },\text{ }b\circ c=d.$
---|---
Case 2 : If $\alpha_{3}=0,$ this ledas to
$\displaystyle\alpha_{1}$ $\displaystyle\longrightarrow$ $\displaystyle
a_{11}^{3}\alpha_{1}$ $\displaystyle\alpha_{2}$ $\displaystyle\longrightarrow$
$\displaystyle
a_{11}a_{23}\alpha_{1}+a_{11}a_{33}\alpha_{2}+a_{31}a_{33}\alpha_{4}$
$\displaystyle\alpha_{3}$ $\displaystyle\longrightarrow$ $\displaystyle 0$
$\displaystyle\alpha_{4}$ $\displaystyle\longrightarrow$ $\displaystyle
a_{33}^{2}\alpha_{4}.$
In order to have $\theta^{\perp}\cap Z(J_{3,2})=0,$ we need $\alpha_{1}\neq 0$
and $(\alpha_{2},\alpha_{4})\neq(0,0).$ So after dividing we may assume
$\alpha_{1}=1.$ Choose $a_{11}=a_{33}=1,a_{31}=0$ and $a_{23}=-\alpha_{2},$
this leads to :
$\displaystyle\alpha_{1}$ $\displaystyle\longrightarrow$ $\displaystyle 1$
$\displaystyle\alpha_{2}$ $\displaystyle\longrightarrow$ $\displaystyle 0$
$\displaystyle\alpha_{3}$ $\displaystyle\longrightarrow$ $\displaystyle 0$
$\displaystyle\alpha_{4}$ $\displaystyle\longrightarrow$
$\displaystyle\alpha_{4}\neq 0.$
then we get the cocycle $\theta=\sum_{a,b}+\alpha_{4}\sum_{c,c}.$ Now the
problem is to describe exactly when there exist $\phi\in Aut(J_{3,2})$ with
$\phi(\sum_{a,b}+\alpha_{4}\sum_{c,c})=\lambda(\sum_{a,b}+\alpha_{4}^{{}^{\prime}}\sum_{c,c})$
for some $\lambda\in K^{\ast}.$ Then we get
$\phi(\sum_{a,b}+\alpha_{4}\sum_{c,c})=\lambda(\sum_{a,b}+\alpha_{4}^{{}^{\prime}}\sum_{c,c})$
if and only if there exist $a_{11},a_{33}\in K^{\ast}$ such that
$a_{33}^{2}\alpha_{4}=a_{11}^{3}\alpha_{4}^{{}^{\prime}}.$ Then there is only
one $Aut(J_{3,2})$-orbit, with representative $\theta=\sum_{a,b}+\sum_{c,c}.$
So we get the algebra
$J_{4,7}(K)$ | $a^{2}=b\text{ },\text{ }a\circ b=d\text{ },\text{ }c^{2}=d.$
---|---
$J_{4,7}(\mathbb{R})$ | $a^{2}=b\text{ },\text{ }a\circ b=d\text{ },\text{ }c^{2}=d.$
---|---
Remarks :
* •
For any $\alpha,\beta\in K^{\ast}$ there exist $x,y\in K^{\ast}$ such that
$x^{3}\alpha=y^{2}\beta,$ by taking $x=\alpha\beta$ and $y=\alpha^{2}\beta.$
* •
Let $J_{4,7}^{\alpha}:a^{2}=b$ , $a\circ b=d$ , $c^{2}=\alpha d.$ Now putting
$a^{{}^{\prime}}=\alpha a,b^{{}^{\prime}}=\alpha^{2}b,c^{{}^{\prime}}=\gamma
c,d^{{}^{\prime}}=\alpha^{3}d$ we get the same multiplication table, but the
parameter has changed to $1$.So we may assume $\alpha=1,$ and we get only one
algebra $J_{4,7}$.
* •
Cocycles $\sum_{a,b}+\sum_{c,c}$ and $\sum_{b,c}$ not lie in the same
$Aut(J_{3,2})$-orbit,
$\phi\left(\sum_{a,b}+\sum_{c,c}\right)=a_{11}^{3}\sum_{a,b}+(a_{11}a_{23}+a_{31}a_{33})\sum_{a,c}+a_{33}^{2}\sum_{c,c}.$
Clearly $J_{4,6}$ and $J_{4,7}$ are not isomorphic, $J_{4,6}$ not associative
but $J_{4,7}$ is a commutative nilpotent associative algebra.
3. 3.
1\- dimensional Central extension of $J_{3,3}^{\alpha},\alpha\in
K^{\ast}\diagup\left(K^{\ast}\right)^{2}$
Here we get that $Z^{2}(J_{3,3}^{\alpha},K)$ consists of :
$\theta:=\alpha_{1}\sum_{a,a}+\alpha_{2}\sum_{b,b}+\alpha_{3}\sum_{a,b}+\alpha_{4}\sum_{a,c}+\alpha_{5}\sum_{b,c}.$
Moreover, $\delta C^{1}(J_{3,3}^{\alpha},K)$ is spanned by
$\sum_{a,a}+\alpha\sum_{b,b},$ then $H^{2}(J_{3,3}^{\alpha},K)$ is consists of
$\theta=\alpha_{1}\sum_{a,a}+\alpha_{2}\sum_{b,b}+\alpha_{3}\sum_{a,b}+\alpha_{4}\sum_{a,c}+\alpha_{5}\sum_{b,c}$
modulo $\sum_{a,a}+\alpha\sum_{b,b}.$ Furthermore the automorphism
group,$Aut(J_{3,3}^{\alpha})$, consists of :
$\phi=\left(\begin{array}[]{ccc}a_{11}&a_{12}&0\\\ a_{21}&a_{22}&0\\\
a_{31}&a_{32}&a_{33}\end{array}\right)$
such that
$\displaystyle a_{11}^{2}+\alpha a_{21}^{2}$ $\displaystyle=$ $\displaystyle
a_{33}\text{ },\text{ }a_{12}^{2}+\alpha a_{22}^{2}=a_{33}$ $\displaystyle
a_{11}a_{12}+\alpha a_{21}a_{22}$ $\displaystyle=$ $\displaystyle 0\text{
},\text{ }a_{33}\delta=a_{33}(a_{11}a_{22}-a_{21}a_{12})\neq 0.$
Write
$\phi\theta=\alpha_{1}^{\prime}\sum_{a,a}+\alpha_{2}^{\prime}\sum_{b,b}+\alpha_{3}^{\prime}\sum_{a,b}+\alpha_{4}^{\prime}\sum_{a,c}+\alpha_{5}^{\prime}\sum_{b,c}.$
Then
$\displaystyle\alpha_{1}^{{}^{\prime}}$ $\displaystyle=$ $\displaystyle
a_{11}^{2}\alpha_{1}+a_{21}^{2}\alpha_{2}+2a_{11}a_{21}\alpha_{2}+a_{11}a_{31}\alpha_{4}+a_{21}a_{31}\alpha_{5}$
$\displaystyle\alpha_{2}^{{}^{\prime}}$ $\displaystyle=$ $\displaystyle
a_{12}^{2}\alpha_{1}+a_{22}^{2}\alpha_{2}+2a_{11}a_{21}\alpha_{2}+a_{12}a_{32}\alpha_{4}+a_{22}a_{32}\alpha_{5}$
$\displaystyle\alpha_{3}^{{}^{\prime}}$ $\displaystyle=$ $\displaystyle
a_{11}a_{12}\alpha_{1}+a_{22}a_{21}\alpha_{2}+(a_{11}a_{22}+a_{21}a_{12})\alpha_{3}$
$\displaystyle+(a_{11}a_{32}+a_{31}a_{12})\alpha_{4}+(a_{21}a_{32}+a_{31}a_{22})\alpha_{5}$
$\displaystyle\alpha_{4}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\delta\left(a_{22}\alpha\alpha_{3}-a_{12}\alpha_{4}\right)$
$\displaystyle\alpha_{5}^{{}^{\prime}}$ $\displaystyle=$ $\displaystyle
a_{33}\left(a_{12}\alpha_{3}+a_{22}\alpha_{4}\right).$
The centre of $J_{3,3}^{\alpha},Z(J_{3,3}^{\alpha}),$ is spanned by $c$, so in
order to have $\theta^{\perp}\cap Z(J_{3,3}^{\alpha})=0$ we need one of
$\alpha_{4},\alpha_{5}$ nonzero. Without loss of generality, we can assume
that $\alpha_{4}=1$. Then
$\theta:=\alpha_{1}\sum_{a,a}+\alpha_{2}\sum_{b,b}+\alpha_{3}\sum_{a,b}+\sum_{a,c}+\alpha_{5}\sum_{b,c}$
modulo $\sum_{a,a}+\alpha\sum_{b,b}.$
Case $\alpha=1$ :
We get
$\displaystyle\alpha_{1}^{{}^{\prime}}$ $\displaystyle=$ $\displaystyle
a_{11}^{2}\alpha_{1}+a_{21}^{2}\alpha_{2}+2a_{11}a_{21}\alpha_{2}+a_{11}a_{31}+a_{21}a_{31}\alpha_{5}$
$\displaystyle\alpha_{2}^{{}^{\prime}}$ $\displaystyle=$ $\displaystyle
a_{12}^{2}\alpha_{1}+a_{22}^{2}\alpha_{2}+2a_{11}a_{21}\alpha_{2}+a_{12}a_{32}+a_{22}a_{32}\alpha_{5}$
$\displaystyle\alpha_{3}^{{}^{\prime}}$ $\displaystyle=$ $\displaystyle
a_{11}a_{12}\alpha_{1}+a_{22}a_{21}\alpha_{2}+(a_{11}a_{22}+a_{21}a_{12})\alpha_{3}$
$\displaystyle+a_{11}a_{32}+a_{31}a_{12}+(a_{21}a_{32}+a_{31}a_{22})\alpha_{5}$
$\displaystyle\alpha_{4}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\delta\left(a_{22}-a_{12}\alpha_{5}\right)$
$\displaystyle\alpha_{5}^{{}^{\prime}}$ $\displaystyle=$ $\displaystyle
a_{33}\left(a_{12}+a_{22}\alpha_{5}\right).$
If $1+\alpha_{5}^{2}\neq 0:$
We can choose $a_{22}=\frac{1}{\delta\left(1+\alpha_{5}^{2}\right)}$ and
$a_{12}=\frac{-\alpha_{5}}{\delta\left(1+\alpha_{5}^{2}\right)}$ to get
$\alpha_{4}^{{}^{\prime}}=1$ and $\alpha_{5}^{{}^{\prime}}=0.$ So may assume
that $\alpha_{4}=1$ and $\alpha_{5}=0,$ to fix $\theta$, choose
$a_{12}=a_{21}=0$ and $a_{11}=a_{22}=1.$ It follows that
$\displaystyle\alpha_{1}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\alpha_{1}+a_{31}$ $\displaystyle\alpha_{2}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle\alpha_{2}$
$\displaystyle\alpha_{3}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\alpha_{3}+a_{32}$ $\displaystyle\alpha_{4}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle 1$ $\displaystyle\alpha_{5}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle 0.$
Choose $a_{31}=\alpha_{2}-\alpha_{1}$ and $a_{32}=-\alpha_{3}.$
So we can assume that
$\theta=\alpha_{2}\left(\sum_{a,a}+\sum_{b,b}\right)+\sum_{a,c}$ modulo
$\sum_{a,a}+\sum_{b,b},$ then $\theta=\sum_{a,c}.$ We get algebra
$J_{4,8}(K)$ | $a^{2}=c\text{ },\text{ }b^{2}=c\text{ },\text{ }a\circ c=d.$
---|---
$J_{4,8}(\mathbb{R})$ | $a^{2}=c\text{ },\text{ }b^{2}=c\text{ },\text{ }a\circ c=d.$
---|---
If $1+\alpha_{5}^{2}=0:$
Over algebraic closed field $K$ and characteristic $K\neq 2:$
We have
$\displaystyle\alpha_{1}^{{}^{\prime}}$ $\displaystyle=$ $\displaystyle
a_{11}^{2}\alpha_{1}+a_{21}^{2}\alpha_{2}+2a_{11}a_{21}\alpha_{2}+a_{11}a_{31}+a_{21}a_{31}\alpha_{5}$
$\displaystyle\alpha_{2}^{{}^{\prime}}$ $\displaystyle=$ $\displaystyle
a_{12}^{2}\alpha_{1}+a_{22}^{2}\alpha_{2}+2a_{11}a_{21}\alpha_{2}+a_{12}a_{32}+a_{22}a_{32}\alpha_{5}$
$\displaystyle\alpha_{3}^{{}^{\prime}}$ $\displaystyle=$ $\displaystyle
a_{11}a_{12}\alpha_{1}+a_{22}a_{21}\alpha_{2}+(a_{11}a_{22}+a_{21}a_{12})\alpha_{3}$
$\displaystyle+a_{11}a_{32}+a_{31}a_{12}+(a_{21}a_{32}+a_{31}a_{22})\alpha_{5}$
$\displaystyle\alpha_{4}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\delta\left(a_{22}-a_{12}\alpha_{5}\right)$
$\displaystyle\alpha_{5}^{{}^{\prime}}$ $\displaystyle=$ $\displaystyle
a_{33}\left(a_{12}+a_{22}\alpha_{5}\right).$
Choose $a_{12}=a_{21}=0,a_{11}=a_{22}=1,a_{31}=-\alpha_{1}$ and
$a_{32}=\frac{-\alpha_{2}}{\alpha_{5}}$ (note that $\alpha_{5}\neq 0$ since
$1+\alpha_{5}^{2}=0$). It follows that
$\alpha_{1}^{{}^{\prime}}=\alpha_{2}^{{}^{\prime}}=0,\alpha_{4}^{{}^{\prime}}=1,\alpha_{5}^{{}^{\prime}}=\alpha_{5}$
and $1+\alpha_{5}^{2}=1+\alpha_{5}^{\prime 2}=0.$
So we assume that
$\theta:=\alpha_{3}\sum_{a,b}+\sum_{a,c}+\alpha_{5}\sum_{b,c}$ such that
$1+\alpha_{5}^{2}=0.$ If $\alpha_{3}=0$ we get the algebra :
$a^{2}=c\text{ },\text{ }b^{2}=c\text{ },\text{ }a\circ c=d,b\circ
c=\alpha_{5}d,1+\alpha_{5}^{2}=0.$
---
The change of basis $a\longrightarrow
a,b\longrightarrow-\alpha_{5}b,c\longrightarrow c$ and $d\longrightarrow d,$
yields to
$J_{4,9}(K)$ | $a^{2}=c\text{ },\text{ }b^{2}=-c\text{ },\text{ }a\circ c=d$ , $b\circ c=d$ .
---|---
If $\alpha_{3}\neq 0$ , by consider the previous change of the basis we get
the algebra :
$a^{2}=c\text{ },\text{ }b^{2}=-c\text{ },\text{ }a\circ c=d$ $,b\circ c=d$ ,
$a\circ b=\alpha_{3}d.$
---
Now putting
$a^{\prime}=\alpha_{3}a,b^{\prime}=\alpha_{3}b,c^{\prime}=\alpha_{3}^{2}c,d^{\prime}=\alpha_{3}^{3}d$
we get the same multiplication table, but the parameter has changed to 1. So
we may assume $\alpha_{3}=1$, and we get only algebra
$J_{4,10}(K)$ | $a^{2}=c\text{ },\text{ }b^{2}=-c\text{ },\text{ }a\circ c=d$ , $b\circ c=d$ , $a\circ b=d.$
---|---
Over real field $\mathbb{R}:$
There is no $\alpha_{5}\in\mathbb{R}$ such that $1+\alpha_{5}^{2}=0.$
We are completed 1\- dimensional Central extension of $J_{3,3}^{\alpha}$
$\alpha\in K^{\ast}\diagup\left(K^{\ast}\right)^{2}$over algebraic closed
field $K$ and characteristic $K\neq 2$ since
$K^{\ast}\diagup\left(K^{\ast}\right)^{2}=\\{1\\}.$ To complete classification
over $\mathbb{R},$ it remains to consider case $\alpha=-1.$
Case $\alpha=-1$ :
We get :
$\displaystyle\alpha_{1}^{{}^{\prime}}$ $\displaystyle=$ $\displaystyle
a_{11}^{2}\alpha_{1}+a_{21}^{2}\alpha_{2}+2a_{11}a_{21}\alpha_{2}+a_{11}a_{31}+a_{21}a_{31}\alpha_{5}$
$\displaystyle\alpha_{2}^{{}^{\prime}}$ $\displaystyle=$ $\displaystyle
a_{12}^{2}\alpha_{1}+a_{22}^{2}\alpha_{2}+2a_{11}a_{21}\alpha_{2}+a_{12}a_{32}+a_{22}a_{32}\alpha_{5}$
$\displaystyle\alpha_{3}^{{}^{\prime}}$ $\displaystyle=$ $\displaystyle
a_{11}a_{12}\alpha_{1}+a_{22}a_{21}\alpha_{2}+(a_{11}a_{22}+a_{21}a_{12})\alpha_{3}$
$\displaystyle+a_{11}a_{32}+a_{31}a_{12}+(a_{21}a_{32}+a_{31}a_{22})\alpha_{5}$
$\displaystyle\alpha_{4}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle-\delta\left(a_{22}+a_{12}\alpha_{5}\right)$
$\displaystyle\alpha_{5}^{{}^{\prime}}$ $\displaystyle=$ $\displaystyle
a_{33}\left(a_{12}+a_{22}\alpha_{5}\right).$
As we do before
If $1-\alpha_{5}^{2}\neq 0:$
We can choose $a_{22}=\frac{-1}{\delta\left(1-\alpha_{5}^{2}\right)}$ and
$a_{12}=\frac{\alpha 5}{\delta\left(1-\alpha_{5}^{2}\right)}$ to get
$\alpha_{4}^{{}^{\prime}}=1$ and $\alpha_{5}^{{}^{\prime}}=0.$ So may assume
that $\alpha_{4}=1$ and $\alpha_{5}=0,$ to fix $\theta$, choose
$a_{12}=a_{21}=0$ and $a_{11}=a_{22}=1.$ It follows that
$\displaystyle\alpha_{1}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\alpha_{1}+a_{31}$ $\displaystyle\alpha_{2}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle\alpha_{2}$
$\displaystyle\alpha_{3}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\alpha_{3}+a_{32}$ $\displaystyle\alpha_{4}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle 1$ $\displaystyle\alpha_{5}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle 0.$
Choose $a_{31}=-\alpha_{2}-\alpha_{1}$ and $a_{32}=-\alpha_{3}.$
So we can assume that
$\theta=-\alpha_{2}\left(\sum_{a,a}-\sum_{b,b}\right)+\sum_{a,c}$ modulo
$\sum_{a,a}-\sum_{b,b},$ then $\theta=\sum_{a,c}.$ We get algebra
$J_{4,9}(\mathbb{R})$ | $a^{2}=c\text{ },\text{ }b^{2}=-c\text{ },\text{ }a\circ c=d.$
---|---
If $1-\alpha_{5}^{2}=0:$
We have
$\displaystyle\alpha_{1}^{{}^{\prime}}$ $\displaystyle=$ $\displaystyle
a_{11}^{2}\alpha_{1}+a_{21}^{2}\alpha_{2}+2a_{11}a_{21}\alpha_{2}+a_{11}a_{31}+a_{21}a_{31}\alpha_{5}$
$\displaystyle\alpha_{2}^{{}^{\prime}}$ $\displaystyle=$ $\displaystyle
a_{12}^{2}\alpha_{1}+a_{22}^{2}\alpha_{2}+2a_{11}a_{21}\alpha_{2}+a_{12}a_{32}+a_{22}a_{32}\alpha_{5}$
$\displaystyle\alpha_{3}^{{}^{\prime}}$ $\displaystyle=$ $\displaystyle
a_{11}a_{12}\alpha_{1}+a_{22}a_{21}\alpha_{2}+(a_{11}a_{22}+a_{21}a_{12})\alpha_{3}$
$\displaystyle+a_{11}a_{32}+a_{31}a_{12}+(a_{21}a_{32}+a_{31}a_{22})\alpha_{5}$
$\displaystyle\alpha_{4}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle-\delta\left(a_{22}+a_{12}\alpha_{5}\right)$
$\displaystyle\alpha_{5}^{{}^{\prime}}$ $\displaystyle=$ $\displaystyle
a_{33}\left(a_{12}+a_{22}\alpha_{5}\right).$
Choose $a_{12}=a_{21}=0,a_{11}=a_{22}=1,a_{31}=-\alpha_{1}$ and
$a_{32}=\frac{-\alpha_{2}}{\alpha_{5}}$ (note that $\alpha_{5}\neq 0$ since
$1-\alpha_{5}^{2}=0$). It follows that
$\alpha_{1}^{{}^{\prime}}=\alpha_{2}^{{}^{\prime}}=0,\alpha_{4}^{{}^{\prime}}=1,\alpha_{5}^{{}^{\prime}}=\alpha_{5}$
and $1-\alpha_{5}^{2}=1-\alpha_{5}^{\prime 2}=0.$
So we assume that
$\theta:=\alpha_{3}\sum_{a,b}+\sum_{a,c}+\alpha_{5}\sum_{b,c}$ such that
$1-\alpha_{5}^{2}=0.$ If $\alpha_{3}=0$ we get the algebra :
$a^{2}=c\text{ },\text{ }b^{2}=-c\text{ },\text{ }a\circ c=d,b\circ
c=\alpha_{5}d,1-\alpha_{5}^{2}=0.$
---
The change of basis $a\longrightarrow
a,b\longrightarrow\alpha_{5}b,c\longrightarrow c$ and $d\longrightarrow d,$
yields to
$J_{4,10}(\mathbb{R})$ | $a^{2}=c\text{ },\text{ }b^{2}=-c\text{ },\text{ }a\circ c=d$ , $b\circ c=d$ .
---|---
If $\alpha_{3}\neq 0$ , by consider the previous change of the basis we get
the algebra :
$a^{2}=c\text{ },\text{ }b^{2}=-c\text{ },\text{ }a\circ c=d$ $,b\circ c=d$ ,
$a\circ b=\alpha_{3}d.$
---
Now putting
$a^{\prime}=\alpha_{3}a,b^{\prime}=\alpha_{3}b,c^{\prime}=\alpha_{3}^{2}c,d^{\prime}=\alpha_{3}^{3}d$
we get the same multiplication table, but the parameter has changed to 1. So
we may assume $\alpha_{3}=1$, and we get only algebra
$J_{4,11}(\mathbb{R})$ | $a^{2}=c\text{ },\text{ }b^{2}=-c\text{ },\text{ }a\circ c=d$ , $b\circ c=d$ , $a\circ b=d.$
---|---
Remark :
One can consider the algebra $J:a\circ b=c$ instead of $J_{3,3}^{\alpha}$ over
algebaic closed field $K$ and $ch(K)\neq 2$, by change of basis
$a\longrightarrow a+\sigma b,b\longrightarrow a-\sigma b,c\longrightarrow 2c$
and $\sigma^{2}+1=0.$ Also for $J_{3,3}^{\alpha},\alpha=-1$ in the case of the
Real field $\mathbb{R},$ by change of basis $a\longrightarrow
a+b,b\longrightarrow a-b,c\longrightarrow 2c.$ Calculations by this
consideration may by some simple. Here we get that $H^{2}(J,K)$ consists of
$\theta=\alpha_{1}\sum_{a,a}+\alpha_{2}\sum_{b,b}+\alpha_{3}\sum_{a,c}+\alpha_{4}\sum_{b,c}$
, and the automorphism group consists of
$\phi=\left(\begin{array}[]{ccc}a_{11}&a_{12}&0\\\ a_{21}&a_{22}&0\\\
a_{31}&a_{32}&a_{11}a_{22}+a_{21}a_{12}\end{array}\right),a_{11}^{2}a_{22}^{2}-a_{12}^{2}a_{21}^{2}\neq
0\text{ and \ }a_{11}a_{21}=a_{12}a_{22}=0\text{ }$
Then$\ \
\phi\theta=\alpha_{1}^{{}^{\prime}}\sum_{a,a}+\alpha_{2}^{{}^{\prime}}\sum_{b,b}+\alpha_{3}^{{}^{\prime}}\sum_{a,c}+\alpha_{4}^{{}^{\prime}}\sum_{b,c}$
$\displaystyle\alpha_{1}^{{}^{\prime}}$ $\displaystyle=$ $\displaystyle
a_{11}^{2}\alpha_{1}+a_{21}^{2}\alpha_{2}+2a_{11}a_{31}\alpha_{3}+2a_{21}a_{31}\alpha_{4}$
$\displaystyle\alpha_{2}^{{}^{\prime}}$ $\displaystyle=$ $\displaystyle
a_{12}^{2}\alpha_{1}+a_{22}^{2}\alpha_{2}+2a_{12}a_{32\alpha_{3}}+2a_{22}a_{32}\alpha_{4}$
$\displaystyle\alpha_{3}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\left(a_{11}a_{22}+a_{21}a_{12}\right)\left(a_{11}\alpha_{3}+a_{21}\alpha_{4}\right)$
$\displaystyle\alpha_{4}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\left(a_{11}a_{22}+a_{21}a_{12}\right)\left(a_{12}\alpha_{3}+a_{22}\alpha_{4}\right)$
We find that there exist only three orbits,
$\theta_{1}=\sum_{a,c},\theta_{2}=\sum_{a,c}+\sum_{b,b}$ and
$\theta_{3}=\sum_{a,c}+\sum_{b,c},$ which be the same number of orbits we get
for these algebras.
4. 4.
1\- dimensional Central extension of $J_{3,4}$
Here $Z^{2}(J_{3,4},K)$ is spanned by $\sum_{a,a},\sum_{a,b}$ and
$\sum_{a,c}+\sum_{b,b}.$ Moreover, $\delta C^{1}(J_{3,4},K)$ is spanned by
$\sum_{a,a}$ and $\sum_{a,b}$. Then $H^{2}(J_{3,4},K)$ is spanned by
$\theta:=\sum_{a,c}+\sum_{b,b},$ so we get only one algebra
$J_{4,11}(K)$ | $a^{2}=b\text{ },\text{ }a\circ b=c$ , $a\circ c=d$ , $b^{2}=d.$
---|---
$J_{4,12}(\mathbb{R})$ | $a^{2}=b\text{ },\text{ }a\circ b=c$ , $a\circ c=d$ , $b^{2}=d.$
---|---
5. 5.
2\- dimensional Central extension of $J_{2,1}$
Here $H^{2}(J_{2,1},K)$ is spanned by $\sum_{a,a},\sum_{a,b}$ and
$\sum_{b,b},$ then $G_{2}(H^{2}(J_{2,1},K))$ consists of :
$\theta:=\alpha_{1}$
$\sum_{a,a}\wedge\sum_{a,b}+\alpha_{2}\sum_{a,a}\wedge\sum_{b,b}+\alpha_{3}\sum_{a,b}\wedge\sum_{b,b}.$
The autommorphism group,$Aut(J_{2,1}),$ consists of $:$
$\phi=\left(\begin{array}[]{cc}a_{11}&a_{12}\\\
a_{21}&a_{22}\end{array}\right),\det\phi\neq 0.$
Now we need to find representatives of $U_{2}(J_{2,1})\diagup Aut(J_{2,1}),$
write
$\phi\theta:=\alpha_{1}^{\prime}$
$\sum_{a,a}\wedge\sum_{a,b}+\alpha_{2}^{\prime}\sum_{a,a}\wedge\sum_{b,b}+\alpha_{3}^{\prime}\sum_{a,b}\wedge\sum_{b,b}.$
Then
$\displaystyle\alpha_{1}^{\prime}$ $\displaystyle=$
$\displaystyle\left(a_{11}^{2}\alpha_{1}+a_{11}a_{21}\alpha_{2}+a_{21}^{2}\alpha_{3}\right)\det\phi$
$\displaystyle\alpha_{2}^{\prime}$ $\displaystyle=$
$\displaystyle\left(2a_{11}a_{12}\alpha_{1}+(a_{11}a_{22}+a_{12}a_{21})\alpha_{2}+2a_{21}a_{22}\alpha_{3}\right)\det\phi$
$\displaystyle\alpha_{3}^{\prime}$ $\displaystyle=$
$\displaystyle\left(a_{12}^{2}\alpha_{1}+a_{12}a_{22}\alpha_{2}+a_{22}^{2}\alpha_{3}\right)\det\phi.$
Without loss of generality, we can assume that $\alpha_{1}=1$ since
$\left(\alpha_{1},\alpha_{2},\alpha_{3}\right)$ $\neq\left(0,0,0\right).$
Choose $a_{11}=a_{22}=1,a_{21}=0,a_{12}=\frac{-\alpha_{2}}{2}$ to get
$\alpha_{1}^{\prime}=1$ and $\alpha_{2}^{\prime}=0.$
So we may assume $\theta=$
$\sum_{a,a}\wedge\sum_{a,b}+\alpha_{3}\sum_{a,b}\wedge\sum_{b,b}.$
If $\alpha_{3}=0:$
Then we get the cocycle $\theta_{1}=\sum_{a,a}\wedge\sum_{a,b}.$
If $\alpha_{3}\neq 0:$
$\displaystyle\alpha_{1}^{\prime}$ $\displaystyle=$
$\displaystyle\left(a_{11}^{2}\alpha_{1}+a_{21}^{2}\alpha_{3}\right)\det\phi$
$\displaystyle\alpha_{2}^{\prime}$ $\displaystyle=$
$\displaystyle\left(2a_{11}a_{12}\alpha_{1}+2a_{21}a_{22}\alpha_{3}\right)\det\phi$
$\displaystyle\alpha_{3}^{\prime}$ $\displaystyle=$
$\displaystyle\left(a_{12}^{2}\alpha_{1}+a_{22}^{2}\alpha_{3}\right)\det\phi.$
To fix $\alpha_{2}^{\prime}=0$ choose $a_{21}=a_{12}=0,$ then
$\displaystyle\alpha_{1}^{\prime}$ $\displaystyle=$
$\displaystyle\det\phi\left(a_{11}^{2}\right)$
$\displaystyle\alpha_{2}^{\prime}$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle\alpha_{3}^{\prime}$ $\displaystyle=$
$\displaystyle\det\phi\left(a_{22}^{2}\alpha_{3}\right).$
Over algebraic closed field $K$ and characteristic $K\neq 2:$
In this case every element is a square, then we choose
$a_{11}=1,a_{22}=\frac{1}{\sqrt{-\alpha_{3}}}$ and then dividing by
$\det\phi.$ We get $\theta_{2}=$
$\sum_{a,a}\wedge\sum_{a,b}-\sum_{a,b}\wedge\sum_{b,b}=\left(\sum_{a,a}+\sum_{b,b}\right)\wedge\sum_{a,b}.$
We claim that $\theta_{1}$ and $\theta_{2}$ are not in the same orbit. Let
$\phi\in Aut(J_{2,1})$ and $\lambda\in K^{\ast}$ such that $\phi$
$\theta_{1}=\lambda\theta_{2},$then
$\displaystyle\lambda$ $\displaystyle=$
$\displaystyle\left(a_{11}^{2}\right)\det\phi$ $\displaystyle 0$
$\displaystyle=$ $\displaystyle\left(2a_{11}a_{12}\right)\det\phi$ (8)
$\displaystyle-\lambda$ $\displaystyle=$
$\displaystyle\left(a_{12}^{2}\right)\det\phi.$
Equation $\left(\ref{m}\right)$ ensures that $\lambda=0.$ So we get only two
$Aut(J_{2,1})$-orbits over $K,$ yielding the two algebras
$J_{4,12}(K)$ | $a^{2}=c\text{ },\text{ }a\circ b=d$ .
---|---
$J_{4,13}(K)$ | $a^{2}=c\text{ },\text{ }b^{2}=c$ , $a\circ b=d$ .
---|---
Over real field $\mathbb{R}:$
If $\alpha_{3}<0,$ choose $a_{11}=1,a_{22}=\frac{1}{\sqrt{-\alpha_{3}}}$ and
then dividing by $\det\phi.$We get :
$\theta_{2}=$
$\sum_{a,a}\wedge\sum_{a,b}-\sum_{a,b}\wedge\sum_{b,b}=\left(\sum_{a,a}+\sum_{b,b}\right)\wedge\sum_{a,b}.$
If $\alpha_{3}>0,$ choose $a_{11}=1,a_{22}=\frac{1}{\sqrt{\alpha_{3}}}$ and
then dividing by $\det\phi.$ We get :
$\theta_{3}=$
$\sum_{a,a}\wedge\sum_{a,b}+\sum_{a,b}\wedge\sum_{b,b}=\left(\sum_{a,a}-\sum_{b,b}\right)\wedge\sum_{a,b}.$
Clearly $\theta_{1}$ is not conjugate to any of $\theta_{2}$ and $\theta_{3}.$
It remains to check if $\theta_{2}$ and $\theta_{3}$ are in the same
$Aut(J_{2,1})$-orbit or not. Let $\phi\in Aut(J_{2,1})$ and
$\lambda\in\mathbb{R}^{\ast}$ such that $\phi$
$\theta_{3}=\lambda\theta_{2},$then
$\displaystyle\lambda$ $\displaystyle=$
$\displaystyle\left(a_{11}^{2}+a_{21}^{2}\right)\det\phi$ (9) $\displaystyle
0$ $\displaystyle=$
$\displaystyle\left(2a_{11}a_{12}+2a_{21}a_{22}\right)\det\phi$
$\displaystyle-\lambda$ $\displaystyle=$
$\displaystyle\left(a_{12}^{2}+a_{22}^{2}\right)\det\phi.$ (10)
Equations $\left(\ref{m1}\right)$ and $\left(\ref{m2}\right)$ Lead to
$\lambda=0$ and $\phi=0.$ So we get only three $Aut(J_{2,1})$-orbits over
$\mathbb{R},$ yielding the three algebras
$J_{4,13}(\mathbb{R})$ | $a^{2}=c\text{ },\text{ }a\circ b=d$ .
---|---
$J_{4,14}(\mathbb{R})$ | $a^{2}=c\text{ },\text{ }b^{2}=c$ , $a\circ b=d$ .
---|---
$J_{4,15}(\mathbb{R})$ | $a^{2}=c\text{ },\text{ }b^{2}=-c$ , $a\circ b=d$ .
---|---
Summarize our results in the following theorems :
###### Theorem 37
Up to isomorphism there exist $13$ nilpotent Jordan algebras of dimension $4$
over algebraic closed field $K$ and $ch(K)\neq 2$ which are isomorphic to one
of the following pairwise non-isomorphic nilpotent Jordan algebras :
Nilpotent Jordan algebras with Centeral component
---
$J_{4,1}=J_{3,1}\oplus J_{1,1}.$ $:$All multiplications are zero.
$J_{4,2}=J_{3,2}\oplus J_{1,1}.$ $:$ $a^{2}=b.$
$J_{4,3}=J_{3,3}\oplus J_{1,1}.$ $:$ $a^{2}=c$ , $b^{2}=c.$
$J_{4,4}=J_{3,4}\oplus J_{1,1}.$ $:$ $a^{2}=b$ , $a\circ b=c.$
Nilpotent Jordan algebras without Centeral component
$J_{4,5}$ $:$ $a^{2}=d$ , $b^{2}=d$ $,c^{2}=d$..
$J_{4,6}$ $:$ $a^{2}=b\text{ },\text{ }b\circ c=d.$
$J_{4,7}$ $:$ $a^{2}=b\text{ },\text{ }a\circ b=d\text{ },\text{ }c^{2}=d.$
$J_{4,8}$ $:$ $a^{2}=c\text{ },\text{ }b^{2}=c\text{ },\text{ }a\circ c=d.$
$J_{4,9}$ $:a^{2}=c\text{ },\text{ }b^{2}=-c\text{ },\text{ }a\circ c=d$ ,
$b\circ c=d$ .
$J_{4,10}$ $:$ $a^{2}=c\text{ },\text{ }b^{2}=-c\text{ },\text{ }a\circ c=d$ ,
$b\circ c=d$ , $a\circ b=d.$
$J_{4,11}$ $:$ $a^{2}=b\text{ },\text{ }a\circ b=c$ , $a\circ c=d$ ,
$b^{2}=d.$
$J_{4,12}$ $:$ $a^{2}=c\text{ },\text{ }a\circ b=d$ .
$J_{4,13}$ $:$ $a^{2}=c\text{ },\text{ }b^{2}=c$ , $a\circ b=d$ .
We see that all nilpotent Jordan algebras of dimension $4$ over algebraic
closed field $K$ and $ch(K)\neq 2$ are associative except
$J_{4,6},J_{4,8},J_{4,9}$ and $J_{4,10}.$
###### Theorem 38
Up to isomorphism there exist $9$ commutative nilpotent associative algebras
of dimension $4$ over algebraic closed field $K$ and $ch(K)\neq 2$ which are
isomorphic to one of the following pairwise non-isomorphic commutative
nilpotent associative algebras :
Nilpotent Jordan algebras with Centeral component
---
$J_{4,1}=J_{3,1}\oplus J_{1,1}.$ $:$All multiplications are zero.
$J_{4,2}=J_{3,2}\oplus J_{1,1}.$ $:$ $a^{2}=b.$
$J_{4,3}=J_{3,3}\oplus J_{1,1}.$ $:$ $a^{2}=c$ , $b^{2}=c.$
$J_{4,4}=J_{3,4}\oplus J_{1,1}.$ $:$ $a^{2}=b$ , $a\circ b=c.$
Nilpotent Jordan algebras without Centeral component
$J_{4,5}$ $:$ $a^{2}=d$ , $b^{2}=d$ $,c^{2}=d$..
$J_{4,7}$ $:$ $a^{2}=b\text{ },\text{ }a\circ b=d\text{ },\text{ }c^{2}=d.$
$J_{4,11}$ $:$ $a^{2}=b\text{ },\text{ }a\circ b=c$ , $a\circ c=d$ ,
$b^{2}=d.$
$J_{4,12}$ $:$ $a^{2}=c\text{ },\text{ }a\circ b=d$ .
$J_{4,13}$ $:$ $a^{2}=c\text{ },\text{ }b^{2}=c$ , $a\circ b=d$ .
###### Theorem 39
Up to isomorphism there exist $17$ nilpotent Jordan algebras of dimension $4$
over $\mathbb{R}$ which are isomorphic to one of the following pairwise non-
isomorphic nilpotent Jordan algebras :
Nilpotent Jordan algebras with Centeral component
---
$J_{4,1}=J_{3,1}\oplus J_{1,1}.$ $:$All multiplications are zero.
$J_{4,2}=J_{3,2}\oplus J_{1,1}.$ $:$ $a^{2}=b.$
$J_{4,3}^{\alpha=\pm 1}=J_{3,3}^{\alpha=\pm 1}\oplus J_{1,1}.$ $:$ $a^{2}=c$ ,
$b^{2}=\alpha c$ . $\ \ \ \ \ \ \ \ \left(\alpha=\pm 1\right)$
$J_{4,4}=J_{3,4}\oplus J_{1,1}.$ $:$ $a^{2}=b$ , $a\circ b=c.$
Nilpotent Jordan algebras without Centeral component
$J_{4,5}^{\alpha=\pm 1}$ $:$ $a^{2}=d$ , $b^{2}=d$ $,c^{2}=\alpha d$ . $\ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(\alpha=\pm 1\right)$
$J_{4,6}$ $:$ $a^{2}=b\text{ },\text{ }b\circ c=d.$
$J_{4,7}$ $:$ $a^{2}=b\text{ },\text{ }a\circ b=d\text{ },\text{ }c^{2}=d.$
$J_{4,8}^{\alpha=\pm 1}$ $:$ $a^{2}=c\text{ },\text{ }b^{2}=\alpha c\text{
},\text{ }a\circ c=d.$ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left(\alpha=\pm 1\right)$
$J_{4,9}$ $:a^{2}=c\text{ },\text{ }b^{2}=-c\text{ },\text{ }a\circ c=d$ ,
$b\circ c=d$ .
$J_{4,10}$ $:$ $a^{2}=c\text{ },\text{ }b^{2}=-c\text{ },\text{ }a\circ c=d$ ,
$b\circ c=d$ , $a\circ b=d.$
$J_{4,11}$ $:$ $a^{2}=b\text{ },\text{ }a\circ b=c$ , $a\circ c=d$ ,
$b^{2}=d.$
$J_{4,12}$ $:$ $a^{2}=c\text{ },\text{ }a\circ b=d$ .
$J_{4,13}^{,\alpha=\pm 1.}$ $:$ $a^{2}=c\text{ },\text{ }b^{2}=\alpha c$ ,
$a\circ b=d$ . $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left(\alpha=\pm 1\right)$
Also, all nilpotent Jordan algebras of dimension $4$ over $\mathbb{R}$ are
associative except $J_{4,6},J_{4,8}^{\alpha=\pm 1},J_{4,9}$ and $J_{4,10}.$
###### Theorem 40
Up to isomorphism there exist $12$ commutative nilpotent associative algebras
of dimension $4$ over $\mathbb{R}$ which are isomorphic to one of the
following pairwise non-isomorphic commutative nilpotent associative algebras :
Nilpotent Jordan algebras with Centeral component
---
$J_{4,1}=J_{3,1}\oplus J_{1,1}.$ $:$All multiplications are zero.
$J_{4,2}=J_{3,2}\oplus J_{1,1}.$ $:$ $a^{2}=b.$
$J_{4,3}^{\alpha=\pm 1}=J_{3,3}^{\alpha=\pm 1}\oplus J_{1,1}.$ $:$ $a^{2}=c$ ,
$b^{2}=\alpha c$ . $\ \ \ \ \ \ \ \ \left(\alpha=\pm 1\right)$
$J_{4,4}=J_{3,4}\oplus J_{1,1}.$ $:$ $a^{2}=b$ , $a\circ b=c.$
Nilpotent Jordan algebras without Centeral component
$J_{4,5}^{\alpha=\pm 1}$ $:$ $a^{2}=d$ , $b^{2}=d$ $,c^{2}=\alpha d$ . $\ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(\alpha=\pm 1\right)$
$J_{4,7}$ $:$ $a^{2}=b\text{ },\text{ }a\circ b=d\text{ },\text{ }c^{2}=d.$
$J_{4,11}$ $:$ $a^{2}=b\text{ },\text{ }a\circ b=c$ , $a\circ c=d$ ,
$b^{2}=d.$
$J_{4,12}$ $:$ $a^{2}=c\text{ },\text{ }a\circ b=d$ .
$J_{4,13}^{,\alpha=\pm 1.}$ $:$ $a^{2}=c\text{ },\text{ }b^{2}=\alpha c$ ,
$a\circ b=d$ . $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left(\alpha=\pm 1\right)$
## References
* [1] ST. Skjelbred and T. Sund. On the classification of nilpotent Lie algebras. Technical report Mathematisk Institutt, Universitetet i Oslo, (1977).
* [2] Gong M. P. Classification of Nilpotent Lie Algebras of Dimension 7\. PhD thesis, University of Waterloo, Canada, 1998.
* [3] R.E. Beck, B. Kolman, Construction of nilpotent Lie algebras over arbitrary fields, in: Paul S.Wang (Ed.), Proceedings of the 1981 ACM Symposium on Symbolic and Algebraic Computation, ACM, New York, 1981, pp. 169–174.
* [4] de Graaf W. A. Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2, J. Algebra. 309(2)(2007), 640-653.
* [5] D. Cox, J. Little, D. O’Shea, Ideals, Varieties and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Springer, New York, 1992.
* [6] A. A. Albert. A structure theory for Jordan algebras, Ann.Math. 48 (1946),546-567.
* [7] W. A. de Graaf. Classification of nilpotent associative algebras of small dimension. arXiv:1009.5339v1 [math.RA], 2010.
* [8] O. Timothy O’Meara. Introduction to quadratic forms. Classics in Mathematics. Springer Verlag, Berlin, 2000. Reprint of the 1973 edition.
* [9] Nathan Jacobson. Lectures in abstract algebra. Springer-Verlag, New York, 1975. Volume II: Linear algebra, Reprint of the 1953 edition [Van Nostrand, Toronto, Ont.], Graduate Texts in Mathematics, No. 31.
* [10] Milnor, J.; Husemoller, D. Symmetric Bilinear Forms. Springer-Verlag New York Heidelberg Berlin 1973,ISBN 0-387-06009-X .
* [11] Jean-Pierre Serre. A Course in Arithmetic, Springer-Verlag, 1973.
* [12] Schafer, Richard D. (1995) [1966], An Introduction to Nonassociative Algebras, Dover, ISBN 0-486-68813-5
* [13] E. A. Dekerf , A. P. E Ten Kroode. Lie Algebras, Finite and Infinite Dimensional Lie Algebras and Applications in Physics part 2, Elsevier, ISBN 0444828362 .
* [14] Karlheinz Spindler. Abstract Algebra With Applications, Volumes I, Marcel Dekker 1994. ISBN 0-8247-9144-4
|
arxiv-papers
| 2013-01-27T13:37:14 |
2024-09-04T02:49:40.799860
|
{
"license": "Public Domain",
"authors": "A.S.Hegazi, H.Abdelwahab",
"submitter": "Ahmed Hegazi Prof",
"url": "https://arxiv.org/abs/1301.6349"
}
|
1301.6365
|
# Fixed effects Selection in high dimensional Linear Mixed Models
Florian Rohart1,2, Magali San-Cristobal 2 and Béatrice Laurent 1
1 UMR 5219, Institut de Mathématiques de Toulouse,
INSA de Toulouse, 135 Avenue de Rangueil, 31077 Toulouse cedex 4, France
2 UMR 444 Laboratoire de Génétique Cellulaire,
INRA Toulouse, 31320 Castanet Tolosan cedex, France
(2012)
###### Abstract
We consider linear mixed models in which the observations are grouped. A
$\ell^{1}$-penalization on the fixed effects coefficients of the log-
likelihood obtained by considering the random effects as missing values is
proposed. A multicycle ECM algorithm is used to solve the optimization
problem; it can be combined with any variable selection method developed for
linear models. The algorithm allows the number of parameters $p$ to be larger
than the total number of observations $n$; it is faster than the lmmLasso
(Schelldorfer et al.,, 2011) since no $n\times n$ matrix has to be inverted.
We show that the theoretical results of Schelldorfer et al., (2011) apply for
our method when the variances of both the random effects and the residuals are
known. The combination of the algorithm with a variable selection method
(Rohart,, 2011) shows good results in estimating the set of relevant fixed
effects coefficients as well as estimating the variances; it outperforms the
lmmLasso both in the common case $(p<n)$ and in the high-dimensional case
$(p\geq n)$.
## 1 Introduction
More and more real data sets are high-dimensional data because of the widely-
used new technologies such as high-thoughput DNA/RNA chips or RNA seq in
biology. The high-dimensional setting -in which the number of parameters $p$
is greater than the number of observations $n$\- generally implies that the
problem can not be solved. In order to address this problem, some conditions
are usually added such as a sparsity condition -which means that a lot of
parameters are equal to zero- or a well-conditioning of the variance matrix of
the observations, among others. A lot of work has been done to address the
problem of variable selection, mainly in a linear model $Y=X\beta+\epsilon$,
where $X$ is an $n\times p$ matrix containing the observations and $\epsilon$
is a n-vector of i.i.d random variables, usually Gaussian. One of the oldest
method is the Akaike Information Criterion (AIC), which is a penalization of
the log-likelihood by a function of the number of parameters included in the
model. More recently, the Lasso (Least Absolute Shrinkage and Selection
Operator) (Tibshirani,, 1996) revolutionized the field with both a simple and
powerful method: $\ell^{1}$-penalization of the least squares estimate which
exactly shrinks to zero some coefficients. The Lasso has some extensions, a
group Lasso (Yuan and Lin,, 2007), an adaptive Lasso (Huang et al.,, 2008) and
a more stable version known as BoLasso (Bach,, 2009), for example. A
penalization on the likelihood is not the only way to perform variable
selection. Indeed statistical testing has also been used recently (Rohart,,
2011) and it appears to give good results.
In all methods cited above, the observations are supposed to be independent
and identically distributed. When a structure information is available, such
as family relationships or common environmental effects, these methods are no
longer adapted. In a linear mixed model, the observations are assumed to be
clustered, hence the variance-covariance matrix $V$ of the observations is no
longer diagonal but could be assumed to be block diagonal in some cases. A lot
of literature about linear mixed models concerns the estimation of the
variance components, either with a maximum likelihood estimation (ML)
(Henderson,, 1973, 1953) or a restricted maximum likelihood estimation (REML)
which accounts for the loss in degrees of freedom due to fitting fixed effects
(Patterson and Thompson,, 1971; Harville,, 1977; Henderson,, 1984; Foulley et
al.,, 2006). However, both methods assume that each fixed effect and each
random effect is relevant. This assumption might be wrong and leads to false
estimation of the parameters, especially in a high-dimensional analysis.
Contrary to the linear model, there is little literature about selection of
fixed effects coefficients in a linear mixed model in a high-dimensional
setting.
Both Bondell et al., (2010) and Ibrahim et al., (2011) used a penalized
likelihood to perform selection of both the fixed and the random effects.
However, their simulation studies were only designed in a low dimensional
context. Bondell et al., (2010) introduced a constrained EM algorithm to solve
the optimization problem, however the algorithm does not really cope with the
problem of high dimension. To our knowledge, only Schelldorfer et al., (2011)
studied the topic in a high dimensional setting. Their paper introduced an
algorithm based on a $\ell^{1}$-penalization of the maximum likelihood
estimator in order to select the relevant fixed effects coefficients. As
highlighted in their paper, their algorithm relies on the inversion of the
variance matrix of the observations $V$, which can be time-consuming. Finally,
their method depends on a regularization parameter that has to be tuned, as
for the original Lasso. As this question remains an open problem, they
proposed the use of the Bayesian Information Criterion (BIC) to choose the
penalty.
All methods are usually considered with one grouping factor -meaning one
partition of the observations-, which can be sometimes misappropriate when the
observations are divided w.r.t two factors or more; for instance when a family
relationship and a common environmental effect are considered.
We present in this paper another way to perform selection of the fixed effects
in a linear mixed model. We propose to consider the random effects as missing
data, as done in Bondell et al., (2010) or in Foulley, (1997), and to add a
$\ell^{1}$-penalization on the log-likelihood of the complete data. Our method
allows the use of several different grouping factors. We propose a multicycle
ECM algorithm (Foulley,, 1997; McLachlan and Krishnan,, 2008; Meng and Rubin,,
1993) to solve the optimization problem; this algorithm possesses convergence
properties. In addition, we show that the use of BIC in order to tune the
regularization parameter as proposed by Schelldorfer et al., (2011) could
sometimes turn out to be misappropriate.
We give theoretical results when the variances of the observations are known.
Due to the design of the algorithm that is decomposed into steps, the
algorithm can be combined with any variable selection method built for linear
models. Nevertheless, the performance of the combination strongly depends on
the variable selection method that is used. As there is little literature on
the selection of the fixed effects in a high-dimensional linear mixed model,
we will mainly compare our results to those of Schelldorfer et al., (2011).
This paper extends the analysis on a real data-set coming from a project in
which hundreds of pigs have been studied. The aim is to enlighten
relationships between some phenotypes of interest and metabolomic data (Rohart
et al.,, 2012). Linear mixed models are appropriate since the observations are
repeated data from different environments (groups of animals are reared
together in the same conditions). Some individuals are also genetically
related, in a family effect. The data set consists in $506$ individuals from
$3$ breeds, $8$ environments and $157$ families. The metabolomic data contains
$p=375$ variables. We will investigate the Daily Feed Intake (DFI) phenotype.
This paper is organized as follows: we will first describe the linear mixed
model and the objective function, then we will present the multicycle ECM
algorithm that is used to solve the optimization problem of the objective
function. Section 3 gives a generalization of the algorithm of Section 2 that
can be used with any variable selection method developed for linear models.
Finally, we will present results from a simulation study showing that the
combination of this new algorithm with a good variable selection method
performs well, in terms of selection of both the fixed and random effects
coefficients (Section 4), before applying the method on a real data set in
Section 5.
## 2 The method
Let us introduce some notations that will be used throughout the paper.
$Var({a})$ denotes the variance-covariance matrix of the vector ${a}$. For all
$a>0$, set $I_{a}$ to be the identity matrix of $\mathbb{R}^{a}$. For
$A\in\mathbb{R}^{n\times p}$, let $A_{I,J}$ $A_{.,J}$ and $A_{I,.}$ denote
respectively the submatrix of $A$ composed of elements of $A$ whose rows are
in $I$ and columns are in $J$, whose columns are in $J$ with all rows, and
whose rows are in $I$ with all columns. Moreover, we set for all $a>0,b>0$,
$0_{a}$ to be the vector of size $a$ with all its coordinates equal to $0$ and
$0_{a\times b}$ to be the null matrix of size $a\times b$. Let us denote $|A|$
the determinant of matrix $A$.
### 2.1 The linear mixed model setup
We consider the linear mixed model in which the observations are grouped and
we suppose that only a small subset of the fixed effects coefficients are non-
zero. The aim of this paper is to recover this subset through an algorithm
that will be presented in the next section. In the present section we explicit
the linear mixed model and our objective function.
Mixed models are often considered with a single grouping factor, meaning that
each observation belongs to one single group. In this paper we allow several
grouping factors. Assume there are $q$ random effects and $q$ grouping factors
$(q\geq 1)$, where some grouping factors may be identical. The levels of the
factor $k$ are denoted $\\{1,2,\dots,N_{k}\\}$. The $i^{th}$-observation
belongs to the groups $(i_{1},\dots,i_{q})$, where for all $l=1,\dots,q$,
$i_{l}\in\\{1,2,\dots,N_{l}\\}$. We precise that two observations can belong
to the same group of one grouping factor whereas they can belong to different
groups of another grouping factor.
Let $n$ be the total number of observations with
$n=\sum_{i=1}^{N_{k}}n_{i,k},\forall k\leq q$, where $n_{i,k}$ is the number
of observations within group $i$ from the grouping factor $k$. Denote
$N=\sum_{k=1}^{q}N_{k}$.
The linear mixed model can be written as
$y=X\beta+\sum_{k=1}^{q}Z_{k}u_{k}+\epsilon,$ (1)
where
* •
$y$ is the set of observed data of length $n$,
* •
$\beta$ is an unknown vector of $\mathbb{R}^{p}$;
$\beta=(\beta_{1},\dots,\beta_{p})$,
* •
$X$ is the $n\times p$ matrix of fixed effects; $X=(X_{1},\dots,X_{p})$,
* •
For $k=1,\dots,q$, $u_{k}$ is a $N_{k}$-vector of the random effect
corresponding to the grouping factor $k$, ,
* •
For $k=1,\dots,q$, $Z_{k}$ is a $n\times N_{k}$ incidence matrix corresponding
to the grouping factor $k$,
* •
$\epsilon=(\epsilon_{1},\dots,\epsilon_{n})^{\prime}$ is a Gaussian vector
with i.i.d. components $\epsilon\sim\mathcal{N}_{n}(0,\sigma_{e}^{2}I_{n})$,
where $\sigma_{e}$ is an unknown positive quantity. We denote by $R$ the
variance-covariance matrix of $\epsilon$, $R=\sigma_{e}^{2}I_{n}$.
To fix ideas, let us give a example of matrices $Z_{k}$ for $n=6$ and two
random effects.
Let $Z_{1}=\begin{pmatrix}1&0&0\\\ 1&0&0\\\ 0&1&0\\\ 0&1&0\\\ 0&0&1\\\
0&0&1\end{pmatrix}$ and $Z_{2}=\begin{pmatrix}x_{1}&0&0\\\ x_{2}&0&0\\\
0&x_{3}&0\\\ 0&x_{4}&0\\\ 0&0&x_{5}\\\ 0&0&x_{6}\end{pmatrix}$. The grouping
factors 1 and 2 are the same for the two random effects $u_{1}$ and $u_{2}$,
and $Z_{2}$ is the incidence matrix of the interaction of the variable
$x=(x_{1},\dots,x_{6})$ and the grouping factor.
Throughout the paper, we assume that
$u_{k}\sim\mathcal{N}_{N_{k}}(0,\sigma_{k}^{2}I_{N_{k}})$, where $\sigma_{k}$
is an unknown positive quantity. We denote
$u=(u_{1}^{\prime},\dots,u_{k}^{\prime})^{\prime}$, $Z$ the concatenation of
$(Z_{1},\dots,Z_{q})$, $G$ the block diagonal matrix of
$\sigma_{1}^{2}I_{N_{1}},\dots,\sigma_{q}^{2}I_{N_{q}}$ and $\Gamma$ the block
diagonal matrix of $\gamma_{1}I_{N_{1}},\dots,\gamma_{q}I_{N_{q}}$, where
$\gamma_{k}=\sigma_{e}^{2}/\sigma_{k}^{2}$.
Remark that with these notations, Model (1) can also be written as:
$y=X\beta+Zu+\epsilon$.
In the following, we assume that $\epsilon,{u}_{1},\dots,{u}_{q}$ are mutually
independent. Thus $Var({u}_{1},\dots,{u}_{q},\epsilon)=\begin{pmatrix}G&{0}\\\
{0}&R\end{pmatrix}$. We consider the matrices $X$ and
$\\{Z_{k}\\}_{1,\dots,q}$ to be fixed design.
Note that our model (1) and the one in Schelldorfer et al., (2011) are almost
identical when all the grouping factors are identical, except that we supposed
$u_{1}\dots,u_{q}$ to be independent while they did not make this assumption.
Nevertheless, for their simulation study, they considered i.i.d. random
effects.
Let us denote by $J$ the set of the indices of the relevant fixed effects of
Model (1); $J=\\{j,\beta_{j}\neq 0\\}$. The aim of this paper is to estimate
$J$, $\beta$, $G$ and $R$. In the whole paper, the number of fixed effects $p$
can be larger than the total number of observations $n$. However, we focus on
the case where only a few fixed-effects are relevant. We also assume that only
a few grouping factors are included in the model since this paper was
motivated by such a case on a real data set, see Section 5. Hence we assume
$N+|J|<n$.
### 2.2 A $\ell^{1}$ penalization of the complete log-likelihood
In the following, we consider the fixed effects coefficients $\beta$ and the
variances $\sigma_{1}^{2},\dots,\sigma_{q}^{2},\sigma_{e}^{2}$ as parameters
and $\\{u_{k}\\}_{k\in\\{1,\dots,q\\}}$ as missing data. We denote
$\Phi=(\beta,\sigma_{1}^{2},\dots,\sigma_{q}^{2},\sigma_{e}^{2})$.
The log-likelihood of the complete data ${x}=({y},{u})$ is
$L(\Phi;{x})=L_{0}(\beta,\sigma_{e}^{2},\sigma_{1}^{2},\dots,\sigma_{q}^{2};\epsilon)+\sum_{k=1}^{q}L_{k}(\sigma_{k}^{2};u_{k}),$
(2)
where
$\displaystyle-2L_{0}(\beta,\sigma_{e}^{2},\sigma_{1}^{2},\dots,\sigma_{q}^{2};\epsilon)=n\log(2\pi)+n\log(\sigma_{e}^{2})+\left|\left|y-X\beta-\sum_{k=1}^{q}Z_{k}u_{k}\right|\right|^{2}/\sigma_{e}^{2},$
(3a) $\displaystyle\forall
k\in\\{1,\dots,q\\},-2L_{k}(\sigma_{k}^{2};u_{k})=N_{k}\log(2\pi)+N_{k}\log(\sigma_{k}^{2})+\left|\left|u_{k}\right|\right|^{2}/\sigma_{k}^{2}.$
(3b)
Indeed, (2) comes from
$p(x|\Phi)=p(y|\beta,u_{1},\dots,u_{q},\sigma_{e}^{2})\Pi_{k=1}^{q}p(u|\sigma_{k}^{2})$;
(3a) comes from
$L_{0}(\beta,\sigma_{e}^{2},\sigma_{1}^{2},\dots,\sigma_{q}^{2};\epsilon)=L_{0}(\sigma_{e}^{2};\epsilon)=n\log(2\pi)+n\log(\sigma_{e}^{2})+\epsilon^{\prime}\epsilon/\sigma_{e}^{2}$
because $\epsilon|\sigma_{e}^{2}\sim\mathcal{N}_{n}(0,\sigma_{e}^{2}I_{n})$
and (3b) from
$u_{k}|\sigma_{k}^{2}\sim\mathcal{N}_{N_{k}}(0,\sigma_{k}^{2}I_{N_{k}})$.
Since we allow the number of fixed-effects $p$ to be larger than the total
number of observations $n$, the usual maximum likelihood (ML) or restricted
maximum likelihood (REML) approaches do not apply. As we assumed that $\beta$
is sparse -many coefficients are assumed to be null- and since we want to
recover that sparsity, we add a $\ell^{1}$ penalty on $\beta$ to the log-
likelihood of the complete data (2). Indeed a $\ell^{1}$ penalization is known
to induce sparsity in the solution, as in the Lasso method (Tibshirani,, 1996)
or the lmmLasso method (Schelldorfer et al.,, 2011). Thus we consider the
following objective function to be minimized:
$g(\Phi;{x})=-2L(\Phi;{x})+\lambda|\beta|_{1},$ (4)
where $\lambda$ is a positive regularization parameter. Remark that the
function $g$ could have been obtained from a Bayesian setting considering a
Laplace prior on $\beta$.
It is interesting to note that finding a minimum of the objective function
$\eqref{gtheta}$ is a non-linear, non-differentiable and non convex problem.
But more importantly, one thing that strikes out -especially from (3b)- is
that the function $g$ is not lower-bounded. Indeed, $L(\Phi;{x})$ tends to
infinity when both $u_{k}$ and $\sigma_{k}$ tends toward $0$. It is a well-
known problem of degeneracy of the likelihood, especially studied in Gaussian
mixture model (Biernacki and Chrétien,, 2003) but not much concerning mixed
models. In linear mixed models, some authors focus on the log-likelihood of
the marginal model in which the random effects are integrated out in the
matrix of variance of the observations $Y$, such as in Schelldorfer et al.,
(2011):
$y=X\beta+\epsilon,\text{ where }\epsilon\sim\mathcal{N}(0,V).$
Note that $V=ZGZ^{\prime}+R$. The degeneracy of the likelihood can also appear
in the marginal model when the determinant of $V$ tends toward zero. This
phenomenon is likely to happen in a high dimensional context when too much
fixed-effects enter the model, that is to say when the amount of
regularization chosen by the penalty of the lmmLasso (Schelldorfer et al.,,
2011) or by $\lambda$ in (4) is not large enough.
Because of the non lower-boundness of the likelihood, the problem of
minimizing the function $g$ is ill-posed: we are not interested in the
minimization of $g$ on the parameter space
$\\{\beta\in\mathbb{R}^{p},\sigma_{1}^{2}\geq 0,\dots,\sigma_{q}^{2}\geq
0,\sigma_{e}^{2}\geq 0\\}$ but more interested in minimizing $g$ inside the
parameter space
$\Lambda=\\{\beta\in\mathbb{R}^{p},\sigma_{1}^{2}>0,\dots,\sigma_{q}^{2}>0,\sigma_{e}^{2}>0\\}.$
Instead of adding a $\ell^{1}$ penalty on the random effect as Bondell et al.,
(2010), we will use the degeneracy of the likelihood at the frontier of the
parameter space $\Lambda$ to perform selection of the random effects. Indeed,
if it exists $1\leq k\leq q$ such that the minimization process of the
function $g$, defined by (4), takes place at the frontier $\sigma_{k}^{2}=0$
of the parameter space $\Lambda$, then the grouping factor $k$ is deleted from
the model (1). Nevertheless, our method is more restrictive than the one of
Bondell et al., (2010) since we assume $N+|J|<n$.
The minimization process of the function $g$ can coincide with the deletion of
the random effect $k$, for $1\leq k\leq q$, for two reasons: either the true
underlying model was different from the fitted one -some grouping factors are
included in the model although there is no need to-, or because the
initialization of the minimization process was to close to an attraction
domain of $(u_{k},\sigma_{k}^{2})=(0_{N_{k}},0)$ (Biernacki and Chrétien,,
2003).
When selection of the random effects is performed in the linear mixed model
(1) with $q$ random effects, a new model is fitted with $q-1$ grouping factor
and the objective function is modified accordingly. The selection of the
random effects can be performed until no grouping factor remains, then a
linear model is considered.
In the next section we will use a multicycle ECM algorithm in order to solve
the minimization of (4); it performs selection of both the fixed and the
random effects.
### 2.3 A multicycle ECM algorithm
The multicycle ECM algorithm (Meng and Rubin,, 1993; Foulley,, 1997; McLachlan
and Krishnan,, 2008) used to solve the minimization problem of (4) contains
four steps -two E steps interlaced with two M steps-; each will be described
in this section.
Recall that $\Phi=(\beta,\sigma_{1}^{2},\dots,\sigma_{q}^{2},\sigma_{e}^{2})$
is the vector of the parameters to estimate and that
$u=(u_{1}^{\prime},\dots,u_{k}^{\prime})^{\prime}$ is a vector of missing
values. For the sake of simplicity, we denote $\mathcal{K}=\\{1,\dots,q\\}$
and $\sigma_{\mathcal{K}}^{2}=\\{\sigma_{k}^{2}\\}_{k\in\mathcal{K}}$.
The multicyle ECM algorithm is an iterative algorithm. We will index the
iterations by $t\in\mathbb{N}$. $\Theta^{[t]}$ will denote the current
estimation of the parameter $\Theta$ at iteration $t$.
Let $E_{u|y,\Phi=\Phi^{[t]}}$ denote the conditional expectation under the
distribution of $u$ given the vector of observations $y$ and the current
estimation of the set of parameters $\Phi$ at iteration $t$.
#### 2.3.1 First E-step
Let denote
$Q(\Phi;\Phi^{[t]})=E_{u|y,\Phi=\Phi^{[t]}}[g(\Phi;x)].$
We can decompose $Q$ as follows:
$Q(\Phi;\Phi^{[t]})=Q_{0}(\beta,\sigma_{\mathcal{K}}^{2},\sigma_{e}^{2};\Phi^{[t]})+\sum_{k=1}^{q}Q_{k}(\sigma_{k}^{2};\Phi^{[t]}),$
where
$Q_{0}(\Phi;\Phi^{[t]})=n\ \log(2\pi)+n\
\log(\sigma_{e}^{2[t]})+E_{u|y,\Phi=\Phi^{[t]}}(\epsilon^{\prime}\epsilon)/\sigma_{e}^{2[t]}+\lambda|\beta^{[t]}|_{1}\\\
$
and
$\forall
k\in{\mathcal{K}},Q_{k}(\sigma_{k}^{2};\Phi^{[t]})=N\log(2\pi)+N\log(\sigma_{k}^{2[t]})+E_{u|y,\Phi=\Phi^{[t]}}({u_{k}}^{\prime}u_{k})/\sigma_{k}^{2[t]}.$
By definition, we have for all $1\leq i\leq
n,Var_{u|y,\Phi=\Phi^{(t)}}\left(\epsilon_{i}\right)=E_{u|y,\Phi=\Phi^{[t]}}(\epsilon_{i}^{2})-\left|\left|E_{u|y,\Phi=\Phi^{[t]}}\left(\epsilon_{i}\right)\right|\right|^{2}$.
Hence
$E_{u|y,\Phi=\Phi^{[t]}}(\epsilon^{\prime}\epsilon)=\left|\left|E_{u|y,\Phi=\Phi^{[t]}}\left(\epsilon\right)\right|\right|^{2}+tr\left(Var_{u|y,\Phi=\Phi^{(t)}}\left(\epsilon\right)\right).$
We can then explicit
$E_{u|y,\Phi=\Phi^{[t]}}(\epsilon^{\prime}\epsilon)=\left|\left|y-X\beta^{[t]}-ZE\left(u|y,\Phi=\Phi^{[t]}\right)\right|\right|^{2}+tr\left(ZVar\left(u|y,\Phi^{[t]}\right)Z^{\prime}\right).$
(5)
According to the denomination of Henderson, (1973),
$E\left(u|y,\Phi=\Phi^{[t]}\right)$ is the BLUP (Best Linear Unbiased
Prediction) of $u$ for the vector of parameters $\Phi$ equal to $\Phi^{[t]}$.
Let us denote $u^{[t+1/2]}=E\left(u|y,\Phi=\Phi^{[t]}\right)$, we have that
$u^{[t+1/2]}=(Z^{\prime}Z+\Gamma^{[t]})^{-1}Z^{\prime}\left(y-X\beta^{[t]}\right).$
#### 2.3.2 M-Step for $\beta$
The next step performs a minimization of
$Q_{0}(\beta,\sigma_{\mathcal{K}}^{2},\sigma_{e}^{2};\Phi^{[t]})$ with respect
to $\beta$:
$\beta^{[t+1]}=\underset{\beta}{Argmin}\left(\dfrac{1}{\sigma_{e}^{2[t]}}\left|\left|\left(y-Zu^{[t+1/2]}\right)-X\beta\right|\right|^{2}+\lambda\left|\beta\right|_{1}\right).$
(6)
Remark that (6) is a Lasso on $\beta$ with the vector of “observed” data
$\left(y-Zu^{[t+1/2]}\right)$ and the penalty $\lambda\sigma_{e}^{2[t]}$.
#### 2.3.3 Second E-Step
A second E-step is performed with the actualization of the vector of missing
values $u$:
$u^{[t+1]}=E\left(u|y,\beta=\beta^{[t+1]},\sigma_{1}^{2}=\sigma_{1}^{2[t]},\dots,\sigma_{q}^{2}=\sigma_{q}^{2[t]},\sigma_{e}^{2}=\sigma_{e}^{2[t]}\right)$,
thus
$u^{[t+1]}=(Z^{\prime}Z+\Gamma^{[t]})^{-1}Z^{\prime}\left(y-X\beta^{[t+1]}\right).$
We define $\forall k\in\mathcal{K}$, $u_{k}^{[t+1]}$ to be the element of size
$N_{k}$ that corresponds to the grouping factor $k$ in $u^{[t+1]}$.
#### 2.3.4 M-step for $(\sigma_{1}^{2},\dots,\sigma_{q}^{2},\sigma_{e}^{2})$
The actualization of the variances $\\{\sigma_{k}^{2}\\}_{1\leq k\leq q}$ and
$\sigma_{e}^{2}$ are performed with the minimization of $\\{Q_{k}\\}_{1\leq
k\leq q}$ and $Q_{0}$ respectively.
Let $k\in\mathcal{K}$, the minimization of $Q_{k}$ with respect to
$\sigma_{k}^{2}$ gives:
$\sigma_{k}^{2[t+1]}=E\left(u_{k}^{\prime}u_{k}|y,\sigma_{k}^{2[t]},\sigma_{e}^{2[t]},\beta^{[t+1]}\right)/N_{k}$.
Besides,
$E\left(u_{k}^{\prime}u_{k}|y,\sigma_{k}^{2[t]},\sigma_{e}^{2[t]},\beta^{[t+1]}\right)=\left|\left|E\left(u_{k}|y,\sigma_{k}^{2[t]},\sigma_{e}^{2[t]},\beta^{[t+1]}\right)\right|\right|^{2}+tr\left(Var\left(u_{k}|y,\sigma_{k}^{2[t]},\sigma_{e}^{2[t]},\beta^{[t+1]}\right)\right).$
Moreover we have, thanks to Henderson, (1973),
$Var\left(u_{k}|y,\sigma_{k}^{2[t]},\sigma_{e}^{2[t]},\beta^{[t+1]}\right)=T_{k,k}\sigma_{e}^{2[t]},$
where $T_{k,k}$ is defined as follows:
$\displaystyle\left(Z^{\prime}Z+\Gamma^{[t]}\right)^{-1}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}Z_{1}^{\prime}Z_{1}+\gamma_{1}^{[t]}I_{N_{1}}&Z_{1}^{\prime}Z_{2}&\ldots&Z_{1}^{\prime}Z_{q}\\\
Z_{2}^{\prime}Z_{1}&Z_{2}^{\prime}Z_{2}+\gamma_{2}^{[t]}I_{N_{2}}&\ldots&Z_{2}^{\prime}Z_{q}\\\
\vdots&\vdots&\ddots&\vdots\\\
Z_{q}^{\prime}Z_{1}&Z_{q}^{\prime}Z_{2}&\ldots&Z_{q}^{\prime}Z_{q}+\gamma_{q}^{[t]}I_{N_{q}}\end{pmatrix}^{-1}$
$\displaystyle=$
$\displaystyle\begin{pmatrix}T_{1,1}&T_{1,2}&\ldots&T_{1,q}\\\
T_{1,2}^{\prime}&T_{2,2}&\ldots&T_{2,q}\\\ \vdots&\vdots&\ddots&\vdots\\\
T_{1,q}^{\prime}&T_{2,q}^{\prime}&\ldots&T_{q,q}\end{pmatrix}.$
Thus, for all $k\in\mathcal{K}$:
$\sigma_{k}^{2[t+1]}=\dfrac{1}{N_{k}}\left[\left|\left|u_{k}^{[t+1]}\right|\right|^{2}+tr\left(T_{k,k}\right)\sigma_{e}^{2[t]}\right].$
The minimization of $Q_{0}$ with respect to $\sigma_{e}^{2}$ gives:
$\sigma_{e}^{2[t+1]}=E_{u|y,\Phi=\Phi^{[t]}}(\epsilon^{\prime}\epsilon)/n$.
From (5), we have
$\sigma_{e}^{2[t+1]}=\dfrac{1}{n}\left[\left|\left|y-X\beta^{[t+1]}-Zu^{[t+1]}\right|\right|^{2}+tr\left(Z(Z^{\prime}Z+\Gamma^{[t]})^{-1}Z^{\prime}\right)\sigma_{e}^{2[t]}\right].$
Since
$\displaystyle
tr\left(Z\left(Z^{\prime}Z+\Gamma^{(t)}\right)^{-1}Z^{\prime}\right)$
$\displaystyle=$ $\displaystyle
tr\left(\left(Z^{\prime}Z+\Gamma^{(t)}\right)^{-1}Z^{\prime}Z\right)$
$\displaystyle=$ $\displaystyle
N-tr\left[\left(Z^{\prime}Z+\Gamma^{(t)}\right)^{-1}\Gamma^{[t]}\right]$
$\displaystyle=$ $\displaystyle
N-\sum_{k=1}^{q}\gamma_{k}^{[t]}tr\left(T_{k,k}\right)$
we have
$\sigma_{e}^{2[t+1]}=\dfrac{1}{n}\left[\left|\left|y-X\beta^{[t+1]}-Zu^{[t+1]}\right|\right|^{2}+\left(N-\sum_{k=1}^{q}\gamma_{k}^{[t]}tr\left(T_{k,k}\right)\right)\sigma_{e}^{2[t]}\right].$
In summary, the algorithm is the following:
###### Algorithm 2.1 (Lasso+).
Initialization:
Set $\mathcal{K}=\\{1,\dots,q\\}$. Initialize the set of parameters
$\Phi^{[0]}=(\sigma_{\mathcal{K}}^{2[0]},\sigma_{e}^{2[0]},\beta^{[0]})$.
Define $\Gamma^{[0]}$ as the block diagonal matrix of
$\gamma_{1}^{[0]}I_{N_{1}},\dots,\gamma_{q}^{[0]}I_{N_{q}}$, where
$\gamma_{k}^{[0]}=\sigma_{e}^{2[0]}/\sigma_{k}^{2[0]}$. Define $Z$ as the
concatenation of $Z_{1},\dots,Z_{q}$ and
$u=(u_{1}^{\prime},\dots,u_{q}^{\prime})^{\prime}$.
Until convergence:
1\. E-step
$u^{[t+1/2]}=(Z^{\prime}Z+\Gamma^{[t]})^{-1}Z^{\prime}(y-X\beta^{[t]})$
2\. M-step
$\beta^{[t+1]}=\underset{\beta}{Argmin}\left(\left|\left|\left(y-Zu^{[t+1/2]}\right)-X\beta\right|\right|^{2}+\lambda\sigma_{e}^{2[t]}\left|\beta\right|_{1}\right)$
3\. E-step
$u^{[t+1]}=(Z^{\prime}Z+\Gamma^{[t]})^{-1}Z^{\prime}(y-X\beta^{[t+1]})$
4\. M-step
(a) For k in $\mathcal{K}$, set
${\sigma^{2}_{k}}^{[t+1]}=\left|\left|u^{[t+1]}_{k}\right|\right|^{2}/N_{k}+tr\left(T_{k,k}\right)\sigma_{e}^{2[t]}/N_{k}$
(b) Set
$\sigma_{e}^{2[t+1]}=\dfrac{1}{n}\left[\left|\left|y-X\beta^{[t+1]}-Zu^{[t+1]}\right|\right|^{2}+\sum_{k\in\mathcal{K}}\left(N_{k}-\gamma_{k}^{[t]}tr\left(T_{k,k}\right)\right)\sigma_{e}^{2[t]}\right]$
(c) For k in $\mathcal{K}$, if
$\left(\left|\left|u^{[t+1]}_{k}\right|\right|^{2}/N_{k}<10^{-4}\sigma_{e}^{2[t]}\right)$
then $\mathcal{K}=\mathcal{K}\backslash\\{k\\}$
Define $Z$ as the concatenation of $\\{Z_{k}\\}_{k\in\mathcal{K}}$ and $u$ as
the transpose of the concatenation of
$\\{u_{k}^{\prime}\\}_{k\in\mathcal{K}}$.
Set $\Gamma^{[t+1]}$ as the block diagonal matrix of
$\\{\gamma_{k}^{[t+1]}I_{N_{k}}\\}_{k\in\mathcal{K}}$, where for all
$k\in\mathcal{K}$,
$\gamma_{k}^{[t+1]}=\sigma_{e}^{2[t+1]}/\sigma_{k}^{2[t+1]}$.
end
The convergence of Algorithm 2.1 is ensured since it is a multicycle ECM
algorithm (Meng and Rubin,, 1993).
Three stopping criteria are used to stop the convergence process of the
algorithm: a condition on $||\beta^{[t+1]}-\beta^{[t]}||^{2}$, a condition on
$||u_{k}^{[t+1]}-u_{k}^{[t]}||^{2}$ for each random effect $u_{k}$ and a
condition on $||L(\Phi^{[t+1]},x)-L(\Phi^{[t]},x)||^{2}$ where $L(\Phi,x)$ is
the log-likelihood defined by (2). The convergence takes place when all the
criteria are fulfilled. We also add a fourth condition that controls the
number of iterations. We choose to initialize the algorithm 2.1 as follows:
for all $1\leq k\leq q,\sigma_{k}^{2[0]}=\frac{0.4}{q}\
\sigma_{e}^{2[-1]},\sigma_{e}^{2[0]}=0.6\ \sigma_{e}^{2[-1]}$, and
$(\sigma_{e}^{2[-1]},\beta^{[0]})$ is estimated from a linear estimation
(without the random effects) of the Lasso at the given penalty $\lambda$. We
will study in Section 4.4 the influence of the initialization of the algorithm
on simulated data.
Note that Step 4(c) performs the selection on the random effects; we decide to
delete a random effect when its variance became lower that
$10^{-4}\sigma_{e}^{2[t]}$.
The estimation of the set of parameters $\Phi$ is biased (Zhang and Hunag,,
2008). One last step can be added in order to address this problem once both
Algorithm 2.1 has converged and the penalization parameter $\lambda$ has be
tuned. Indeed, one should prefer to use Algorithm 2.1 in order to estimate
both the support of $\beta$ and the support of the random effects, and then to
estimate the set $\Phi$ with a classical mixed model estimation on the model:
$y=X\beta_{\hat{J}}+\sum_{k\in S}Z_{k}u_{k}+\epsilon,$
where $\hat{J}$ and $S$ are the estimated set of indices of the relevant fixed
effects and the estimated set of indices of the relevant random effects
respectively.
###### Proposition 2.2.
When the variances are known, the minimization of our objective function (4)
is the same as the minimization of
$Q(\beta)=(y-X\beta)^{\prime}V^{-1}(y-X\beta)+\lambda|\beta|_{1}$, which is
the objective function of Schelldorfer et al., (2011) at known variances.
Let us recall that Schelldorfer et al., (2011) obtained theoretical results on
the consistency of their method. According to Proposition 2.2, these results
apply to our method in the case of known variances. The proof of Proposition
2.2 is given in Appendix C.
Note that when individuals are genetically related through a known
relationship matrix $A$, we have $u\sim\mathcal{N}_{n}(0,\sigma_{s}^{2}A)$,
with $\sigma_{s}>0$. Thanks to Henderson, (1973), $A^{-1}$ can be directly
computed. In all that precede, the changes are the following : the matrix
$\Gamma$ becomes the matrix $\sigma_{e}^{2}/\sigma_{s}^{2}A^{-1}$ and
$||u||^{2}$ becomes $u^{\prime}A^{-1}u$.
### 2.4 The tuning parameter
Algorithms 2.1 involves a regularization parameter $\lambda$; the solution
depends on this parameter. This amount of shrinkage has to be tuned. We choose
the use of the Bayesian Information Criterion (BIC) (Schwarz,, 1978):
$\lambda_{BIC}=\underset{\lambda}{Argmin}\\{\log|V_{\lambda}|+(y-X\hat{\beta}_{\lambda})^{\prime}V_{\lambda}^{-1}(y-X\hat{\beta}_{\lambda})+d_{\lambda}.\log(n)\\},$
where
$V_{\lambda}=\sum_{k\in\mathcal{K}}\hat{\sigma}_{k}^{2}Z_{k}Z^{\prime}_{k}+\hat{\sigma}_{e}^{2}I_{n}$
and $\hat{\sigma}_{k}^{2},\hat{\sigma}_{e}^{2},\hat{\beta}_{\lambda}$ are
obtained from the minimization of the objective function $g$ defined by (4).
Moreover, $d_{\lambda}:=\sum_{k=1}^{p}1_{\sigma_{k}\neq
0}+|\hat{J}_{\lambda}|$ is the sum of the number of non-zero variance-
covariance parameters and the number of non-zero fixed effects coefficients
included in the model which has been selected with the regularization
parameter $\lambda$.
Other methods can be used to choose $\lambda$ such as AIC or cross-validation,
among others. An advantage of BIC over cross-validation is mainly the gain of
computational time.
In the next section, we propose a generalization of Algorithm 2.1 which allows
the use of any variable selection methods developed for linear models.
## 3 A generalized algorithm
Algorithm 2.1 gives good results, as it can be seen in the simulation study of
Section 4. Nevertheless, since Step 2 of Algorithm 2.1 aims at selecting the
relevant coefficients of $\beta$ in a linear model, the Lasso method can be
replaced with any variable selection method built for linear models. If the
chosen variable selection method optimizes a criterion, such as the adaptive
Lasso (Zou,, 2006) or the elastic net (Zou and Hastie,, 2005), the algorithm
thus obtained remains a multicycle ECM algorithm and the convergence property
still applies. However, the convergence property does not hold for methods
that do not optimize a criterion.
Algorithm 2.1 can be reshaped for a generalized algorithm as follows:
###### Algorithm 3.1.
Initialization:
Initialize the set of parameters
$\Phi^{[0]}=(\sigma_{\mathcal{K}}^{2[0]},\sigma_{e}^{2[0]},\beta^{[0]})$. Set
$\mathcal{K}=\\{1,\dots,q\\}$.
Define $\Gamma^{[0]}$ as the block diagonal matrix of
$\gamma_{1}^{[0]}I_{N_{1}},\dots,\gamma_{q}^{[0]}I_{N_{q}}$, where
$\gamma_{k}^{[0]}=\sigma_{e}^{2[0]}/\sigma_{k}^{2[0]}$. Define $Z$ as the
concatenation of $Z_{1},\dots,Z_{q}$ and
$u=(u_{1}^{\prime},\dots,u_{q}^{\prime})^{\prime}$.
Until convergence:
1\. $u^{[t+1/2]}=(Z^{\prime}Z+\Gamma^{[t]})^{-1}Z^{\prime}(y-X\beta^{[t]})$
2\. Variable selection and estimation of $\beta$ in the linear model
$y-Zu^{[t+1/2]}=X\beta+\epsilon^{[t]}$, where
$\epsilon^{[t]}\sim\mathcal{N}(0,\sigma_{e}^{2[t]}I_{n})$.
3\. $u^{[t+1]}=(Z^{\prime}Z+\Gamma^{[t]})^{-1}Z^{\prime}(y-X\beta^{[t+1]})$
4\. (a) For k in $\mathcal{K}$, set
${\sigma^{2}_{k}}^{[t+1]}=\left|\left|u^{[t+1]}_{k}\right|\right|^{2}/N_{k}+tr\left(T_{k,k}\right)\sigma_{e}^{2[t]}/N_{k}$
(b) Set
$\sigma_{e}^{2(t+1)}=\dfrac{1}{n}\left[\left|\left|y-X\beta^{[t+1]}-Zu^{[t+1]}\right|\right|^{2}+\sum_{k\in\mathcal{K}}\left(N_{k}-\gamma_{k}^{[t]}tr\left(T_{k,k}\right)\right)\sigma_{e}^{2[t]}\right]$
(c) For k in $\mathcal{K}$, if
$\left(\left|\left|u^{[t+1]}_{k}\right|\right|^{2}/N_{k}<10^{-4}\sigma_{e}^{2[t]}\right)$
then $\mathcal{K}=\mathcal{K}\backslash\\{k\\}$
Define $Z$ as the concatenation of $\\{Z_{k}\\}_{k\in\mathcal{K}}$ and $u$ as
the transpose of the concatenation of
$\\{u_{k}^{\prime}\\}_{k\in\mathcal{K}}$.
Set $\Gamma^{[t+1]}$ as the block diagonal matrix of
$\\{\gamma_{k}^{[t+1]}I_{N_{k}}\\}_{k\in\mathcal{K}}$, where for all
$k\in\mathcal{K}$,
$\gamma_{k}^{[t+1]}=\sigma_{e}^{2[t+1]}/\sigma_{k}^{2[t+1]}$.
end
We choose to initialize Algorithm 3.1 as follows: for all $1\leq k\leq
q,\sigma_{k}^{2[0]}=\frac{0.4}{q}\ \sigma_{e}^{2[-1]},\sigma_{e}^{2[0]}=0.6\
\sigma_{e}^{2[-1]}$, and $(\sigma_{e}^{2[-1]},\beta^{[0]})$ is estimated from
a linear estimation (without the random effects) of the method used at Step 2.
In the following we propose to combine Algorithm 2.1 with a method that does
not need a tuning parameter, namely the procbol method (Rohart,, 2011). The
procbol method is a sequential multiple hypotheses testing which statistically
determines the set of relevant variables in a linear model $y=X\beta+\epsilon$
where $\epsilon$ is an i.i.d Gaussian noise. This method is a two-step
procedure: the first step orders the variables taking into account the
observations $y$ and the second step uses multiple hypotheses testing to
separate the relevant variables from the irrelevant ones. The procbol method
is proved to be powerful under some conditions on the signal in Rohart,
(2011).
In Section 4, we show that the combination of Algorithm 3.1 and the procbol
method performs well on simulated data.
## 4 Simulation study
The purpose of this section is to compare different methods that aim at
selecting both the correct fixed effects coefficients and the relevant random
effects in a linear mixed model (1), but also to look at the improvement
obtained from including random effects in the model.
### 4.1 Presentation of the methods
We compare several methods, some of them are designed to work in a linear
model: Lasso (Tibshirani,, 1996), adLasso (Zou,, 2006) and procbol (Rohart,,
2011), while others are designed to work in a linear mixed model: lmmLasso
(Schelldorfer et al.,, 2011), Algorithm 2.1 (labelled as Lasso+),
adLasso+Algorithm 3.1 (labelled as adLasso+) and procbol+Algorithm 3.1
(labelled as pbol+).
The initial weights of the adLasso and adLasso+ are set to be equal to
$1/|\hat{\beta}_{i}|$ where for all $i\in\\{1,\dots,p\\}$, $\hat{\beta}_{i}$
is the Ordinary Least Squares (OLS) estimate of $\beta_{i}$ in the model
$y=X_{i}\beta_{i}+\epsilon_{i}$.
The second step of the procbol method performs multiple hypotheses testing
with an estimation of unknown quantiles related to the matrix $X$. The
calculation of these quantiles at each iteration of the convergence process
would make the combination of the procbol method and Algorithm 3.1 almost
impossible to run; however, since the data matrix $X$ stays the same
throughout the algorithm, the quantiles also do. Thus the procbol method was
adapted to be run several times on the same data set by keeping the calculated
quantiles, which led to a enormous gain of computational time. Some parameters
of the procbol method were changed in order to limit the time of one iteration
of the convergence process, as follows. The parameter $m$ which stands for the
number of bootstrapped samples used to sort the variables (first step of the
procbol method) was set to $10$. The number of variables ordered at the first
step of the procbol method was set to $40$. Note that when the procbol method
was used in a linear model, we set $m=100$ as advised in Rohart, (2011). Both
the procbol method and the pbol+ method were set with a user-level of
$\alpha\in\\{0.1,0.05\\}$, which stands for the level of the testing
procedure.
Concerning all methods that needed a tuning parameter, we set it using the
Bayesian Information Criterion described in Section 2.4. A particular
attention has to be drawn on the tuning of the regularization parameter of
some methods that could be tricky in some cases due to the degeneracy of the
likelihood, especially Lasso and adLasso, see Appendix B.
### 4.2 Design of our simulation study
Concerning the design of our simulations, we set $X_{1}$ to be the vector of
$\mathbb{R}^{n}$ whose coordinates are all equal to $1$ and we considered four
models. For each model, the response variable $y$ is computed via
$y=\sum_{j=1}^{5}X_{i_{j}}\beta_{i_{j}}+\sum_{k=1}^{q}Z_{k}u_{k}+\epsilon$,
where $J=\\{i_{1},\dots,i_{5}\\}\subset\\{1,\dots,p\\}$, with two random
effects ($q=2$) being standard Gaussian ($\sigma_{1}^{2}=\sigma_{2}^{2}=1$)
and $\epsilon$ being a vector of independent standard Gaussian variables. The
models used to fit the data differ in the number of parameters $p$, the number
of random effects $q$ and the dependence structure of the $X_{i}$’s. For each
model, we have that for all $j=2,\dots,p$: $\sum_{i=1}^{n}X_{j,i}=0$ and
$\frac{1}{n}\sum_{i=1}^{n}X_{j,i}^{2}=1$. For $k=1,\dots,q$, the random
effects regression matrix $Z_{k}$ corresponds to the design matrix of the
interaction between the $k^{th}$ column of $X$ and the grouping factor $k$,
which gives a $n\times N_{k}$ matrix. The design of the matrices $Z_{k}$’s
means that the first $q$ grouping variables generates both a fixed effect
(corresponds to $\beta_{k}$’s) and a random effect (corresponds to $u_{k}$’s).
As advised in Schelldorfer et al., (2011), the variables that generate both a
fixed and a random effect do not undergo feature selection; otherwise the
fixed effect coefficients of those variables tends to be shrunken towards 0.
The set of variables that do not undergo feature selection can change at each
step of the convergence process of our algorithms. Indeed, as soon as a
variable does not generate a random effect anymore, the fixed effect
corresponding to that variable undergoes feature selection again.
The models are defined as follows:
* •
$M_{1}$: $n=120$, $p=80$, $\beta_{J}=2/3$. For all
$j=2,\dots,p,X_{j}\sim\mathcal{N}_{n}(0,I_{n})$.The division of the
observations for the two random effects are the same; for all $k\leq
2:N_{k}=20,\forall i\in\\{1,..,20\\}\ n_{i,k}=6$. This model is fitted
assuming $q=3$.
* •
$M_{2}$: $n=120$, $p=300$, $\beta_{J}=3/4$. The covariates are generated from
a multivariate normal distribution with mean zero and covariance matrix
$\Sigma$ with the pairwise correlation
$\Sigma_{kk^{\prime}}=\rho^{|k-k^{\prime}|}$ and $\rho=0.5$. The division of
the observations for the two random effects are the same; for all $k\leq
2:N_{k}=20,\forall i\in\\{1,..,20\\}\ n_{i,k}=6$.
* •
$M_{3}$: $n=120$, $p=300$, $\beta_{J}=2/3$. For all
$j=2,\dots,p,X_{j}\sim\mathcal{N}_{n}(0,I_{n})$. The division of the
observations for the two random effects are different: $N_{1}=20,\forall
i\in\\{1,..,20\\}\ n_{i,1}=6$ and $N_{2}=15,\forall i\in\\{1,..,15\\}\
n_{i,2}=8$
* •
$M_{4}$: $n=120$, $p=600$, $\beta_{J}=2/3$. For all
$j=2,\dots,p,X_{j}\sim\mathcal{N}_{n}(0,I_{n})$. The division of the
observations for the two random effects are the same; for all $k\leq
2:N_{k}=20,\forall i\in\\{1,..,20\\}\ n_{i,k}=6$.
For models $M_{1},M_{3},M_{4}$, we set $J=\\{1,\dots,5\\}$. For model $M_{2}$,
we set $J=\\{1,2,i_{3},i_{4},i_{5}\\}$ where
$\\{i_{3},i_{4},i_{5}\\}\subset\\{3,\dots,p\\}$.
In each model, the aim is to recover both the set of relevant fixed effects
coefficients $J$ and the set of relevant random effects; but also to estimate
the variance of both the random effects and the residuals. To judge the
quality of the methods, we use several criterion: the percentage of true model
recovered under the label ‘Truth’ (both $J$ and the set of relevant random
effects), the percentage of times the true set of fixed effects is recovered
‘$\hat{J}=J$’, the cardinal of the estimated set of fixed effects coefficients
$|\hat{J}|$, the number of true positive $TP$, the estimated variance
$\hat{\sigma}_{e}^{2}$ of the residuals, the estimated variances
$\hat{\sigma}_{1}^{2},\dots,\hat{\sigma}_{q}^{2}$ of the random effects and
the mean squared error $mse$ calculated as an $\ell^{2}$ error rate between
the reality -$X\beta$\- and the estimation -$X\hat{\beta}$-. We also
calculated the Signal-to-Noise Ratio (SNR) as
$||X\beta||_{2}^{2}/||\sum_{k=1}^{q}Z_{k}u_{k}+\epsilon||_{2}^{2}$ for each of
the replications.
### 4.3 Comments on the results
The detailed results of the simulation study are available in Appendix A. A
summary of the main results is shown in Figure 1 ($\alpha=0.1$ for the procbol
method and the pbol+ method). No results are given for the lmmLasso of
Schelldorfer et al., (2011) in Model $M_{3}$ since two different grouping
factors are considered and the R-package lmmLasso does not include that
setting.
Figure 1: Summary of the results of the simulation study for models
$M_{1}-M_{4}$ ($X$ axis). Results of ‘Truth’ (a), ‘$\hat{J}=J$’ (b) and Mean
Squared Error (c) for each model.
In all models, there is an improvement of the results when we switch from a
simple linear model to a linear mixed model; indeed there is a significant
difference between Lasso and Lasso+ or procbol and pbol+, especially with
model $M_{4}$.
On all models, lmmLasso and Lasso+ give very similar results; this is not
surprising since both are a $\ell^{1}$-penalization of the log likelihood,
except for model $M_{1}$ where lmmLasso seems to give better results. This
difference comes from the coding of the R-package that contains the lmmLasso
method. Indeed, a variable that generates both a fixed and a random effect
does not undergo feature selection in the lmmLasso method when the random
effect tends towards zero, whereas the Lasso+ method would allow it.
We observed on our simulation study that both lmmLasso and Lasso+ are very
sensitive to the choice of the regularization parameter. On most simulations
of model $M_{4}$ in which $p=600$, we observed an edge effect between a
regularization parameter that selects few fixed effects (fewer than $15$) and
a regularization parameter that selects too much fixed-effects ($|\hat{J}|>n$)
and thus stops the algorithm because we assumed that the number of relevant
fixed-effects is lower than $\text{min}(n-1,p)$, see Figure 2. Nevertheless,
the weights included in the adLasso+ seems to smooth this phenomenon, see
Figure 2 for the same simulation as Figure 2. Remark that for the run of model
$M_{4}$ which is on Figure 2, Lasso+ could select the true model for a
regularization parameter around $0.22$ whereas adLasso+ could not as a noisy
variable enters the set of selected variables before all the relevant fixed-
effects do.
Figure 2: Number of selected fixed effects coefficients depending on the value
of the regularization parameter for one run of model M4, for the method (a)
Lasso+ and (b) adLasso+. The grid of the penalty is as thin as $10^{-7}$ next
to the area $|\hat{J}|>n$ in (a) and $10^{-3}$ in (b).
Concerning the adLasso+ method, it appears to improve the Lasso+ method,
except for model $M_{4}$ where the true model is only selected once over the
$100$ replications. On this particular model $M_{4}$, adLasso+ selects more
fixed effects but less relevant ones than Lasso+. This could mean that the
initial weights are not adapted to this case. Despite the result of ‘Truth’,
the mse is lower for adLasso+ than for Lasso+.
Algorithm 3.1 combined with the procbol method (pbol+) gives the best results
over all tested methods for all models. Indeed the percentage of true model
recovered is the largest over all methods, the estimation of the fixed effects
is really close to the reality and the $mse$ is the lowest among the tested
methods. Nevertheless, due to the bias of the Lasso, the results in term of
mse for Lasso+ and lmmLasso could easily be improved with a linear mixed model
estimation as said in Section 2.3 (see Appendix). Yet, the results of pbol+
are mitigated for model $M_{1}$. Indeed, the percentage of true model
recovered is lower than in the other models because of the selection of the
random effects that lacks efficiency (the results concerning the selection of
the fixed-effects are equivalent as in the other models, as shown in Figure
1). Nonetheless, the results are still better than for the others methods.
Moreover, a relevant random effect was never falsely deleted in all models and
for all methods. It is interesting to note that the pbol+ method always
converged on our simulations.
A R-package “MMS” is available on CRAN (http://cran.r-project.org). This
package contains tools to perform fixed effects selection in linear mixed
models; it contains the previous methods denoted as Lasso+, adLasso+, pbol+,
among others.
All the results presented in this section were obtained with a specific
initialization of the algorithms. The next paragraph is dedicated to the
analysis of the influence of that specific initialization.
### 4.4 Influence of the initialization of our algorithms
Both Algorithm 2.1 and Algorithm 3.1 start with an initialization of the
parameter $\Phi=(\sigma_{1}^{2},\dots,\sigma_{q}^{2},\sigma_{e}^{2},\beta)$.
We choose to initialize each algorithm with the following setting: for all
$1\leq k\leq q,\sigma_{k}^{2[0]}=\frac{0.4}{q}\
\sigma_{e}^{2[-1]},\sigma_{e}^{2[0]}=0.6\ \sigma_{e}^{2[-1]}$, and
$(\sigma_{e}^{2[-1]},\beta^{[0]})$ is estimated from a linear estimation
(without the random effects) of the method used at Step 2.
In the current Section, we choose different initializations of Algorithm 2.1
and Algorithm 3.1, both on Model $M_{4}$ (see Section 4). The initial values
of the variances were set from $0.1$ to $10$ and of the fixed effects
coefficients from $-100$ to $100$. Each algorithm always converged towards the
same point, whatever the initialization of $\Phi$, not shown. However, the
farther $\Phi^{[0]}$ is set from the true estimation of $\Phi$, the higher is
the number of iterations of the algorithms.
## 5 Application on a real data-set
In this section we analyze a real data set which comes from Rohart et al.,
(2012). The aim of this analysis is to pinpoint metabolomic data that
describes a phenotype taking into account all the available information such
as the breed, the batch effect and the relationship between individuals. Here
we will study the Daily Feed Intake phenotype (DFI). We model the data as
follows:
$y=X_{B}\beta_{B}+X_{M}\beta_{M}+Z_{E}u_{E}+Z_{F}u_{F}+\epsilon,$ (7)
where $y$ is the DFI phenotype, $X_{B},X_{M},Z_{E},Z_{F}$ are the design
matrices of the breed effect, the metabolomic data, the batch effect and the
family effect, respectively. We consider two random effects: the batch and the
family, considering that each level of these factors is a random sample drawn
from a much larger population of batches and families, contrary to the breed
factor. Note that the coefficients $\beta_{B}$ do not undergo feature
selection.
We compare several methods on this model: Lasso, adLasso, procbol, Lasso+,
adLasso+ and pbol+ (see Section 4). The model which is considered for the
first three methods is $y=X_{B}\beta_{B}+X_{M}\beta_{M}+\epsilon$. Both
methods procbol and pbol+ were set with a user-level of $\alpha=0.1$. The
results are presented in Table 1.
| $|\hat{J}|$ | $\hat{\sigma}_{e}^{2}$ | $\hat{\sigma}_{E}^{2}$ | $\hat{\sigma}_{F}^{2}$
---|---|---|---|---
Lasso | 14 | $3.8\times 10^{-2}$ | - | -
adLasso | 21 | $3.4\times 10^{-2}$ | - | -
procbol | 11 | $4.1\times 10^{-2}$ | - | -
Lasso+ | $11$ | $3.2\times 10^{-2}$ | $3.2\times 10^{-3}$ | $6.4\times 10^{-3}$
adLasso+ | $10$ | $3.3\times 10^{-2}$ | $2.5\times 10^{-3}$ | $6.5\times 10^{-3}$
pbol+ | 5 | $3.4\times 10^{-2}$ | $5.9\times 10^{-3}$ | $6.5\times 10^{-3}$
Table 1: Results for the real data set
We observe that considering random effects leads to a decrease of both the
residual variance and the number of selected metabolomic variables. This
behavior is in accordance with the simulation study. The question that arises
from this analysis is to know whether the variables which are selected in the
linear mixed models are more relevant than in the linear model. Biological
analyses remain to be done to answer that question.
Table 2 gives the computational time of one run when we only consider the
batch effect -in order to be able to compute the lmmLasso-, showing that the
Lasso+ method is much faster than the lmmLasso method for a large number of
observations (due to the inversion of the matrix of variance $V$ at each step
of the convergence process). The simulation was performed at a regularization
parameter that selects the same model for the two methods, on a $2.80$GHz CPU
with $8.00$Go of RAM.
Methods | CPU Time
---|---
Lasso+ | 0.80
lmmLasso | 24.28
Table 2: CPU Time on a single run that selects the same model
## 6 Conclusion
In this paper, we proposed to add a $\ell^{1}$-penalization of the complete
log-likelihood in order to perform selection of the fixed effects in a linear
mixed model. The multicycle ECM algorithm used to minimize the objective
function also performs random effects selection. This algorithm gives the same
results as the lmmLasso of Schelldorfer et al., (2011) when the random effects
are assumed to be independent, but faster. Theoretical results are identical
to those of Schelldorfer et al., (2011) when the variances are known. The
structure of our algorithm gives the possibility to combine it with any
variable selection method built for linear models, but at the price of
possibly loosing the convergence property. Nonetheless, the combined procbol
method appears to give good results on simulated data and outperforms other
approaches.
We applied all these methods to a real data set showing that the residual
variance can be reduced, even with a small set of selected variables.
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## Appendix A - Results of the simulation study
Table 3: Results of model $M_{1}$. The percentage of true model recovered was
recorded -‘Truth’- as well as $\hat{J}=J$. $|J|$ is the number of fixed
effects selected and $TP$ the number of relevant fixed effects selected. The
signal to noise ratio is equal to $SNR=0.78(0.13)$. Standard errors are given
in parentheses, for $100$ runs.
| Truth | $\hat{J}=J$ | $|\hat{J}|$ | TP | $\hat{\sigma}_{e}^{2}$ | $\hat{\sigma}_{1}^{2}$ | $\hat{\sigma}_{2}^{2}$ | $\hat{\sigma}_{3}^{3}$
---|---|---|---|---|---|---|---|---
Ideal | 1 | 5 | 5 | 5 | 1 | 1 | 1 | 0
Lasso | - | 0.15 | 4.95 | 4.13 | 3.27 | - | - | -
| | | (1.90) | (1.12) | (0.62) | - | - | -
adLasso | - | 0.16 | 5.25 | 4.26 | 2.91 | - | - | -
| | | (1.84) | (0.89) | (0.59) | - | - | -
procbol | - | 0.59 | 4.70 | 4.58 | 2.83 | - | - | -
$\alpha=0.1$ | | | (0.78) | (0.61) | (0.57) | - | - | -
procbol | - | 0.45 | 4.47 | 4.40 | 2.89 | - | - | -
$\alpha=0.05$ | | | (0.67) | (0.62) | (0.58) | - | - | -
Lasso+ | 0.21 | 0.34 | 6.42 | 5.00 | 1.04 | 0.88 | 0.98 | 0.02
| | | (1.64) | (0.00) | (0.21) | (0.37) | (0.44) | (0.06)
adLasso+ | 0.21 | 0.35 | 6.34 | 4.99 | 0.94 | 0.86 | 0.95 | 0.02
| | | (1.41) | (0.10) | (0.18) | (0.36) | (0.41) | (0.06)
lmmLasso | 0.29 | 0.39 | 6.15 | 5.00 | 1.01 | 0.89 | 0.96 | 0.02
| | | (1.29) | (0.00) | (0.19) | (0.38) | (0.42) | (0.06)
pbol+ | 0.55 | 0.89 | 5.18 | 5.00 | 0.92 | 0.87 | 0.97 | 0.03
$\alpha=0.1$ | | | (0.50) | (0.00) | (0.18) | (0.37) | (0.41) | (0.06)
pbol+ | 0.59 | 0.93 | 5.08 | 5.00 | 0.93 | 0.88 | 0.97 | 0.03
$\alpha=0.05$ | | | (0.30) | (0.00) | (0.17) | (0.37) | (0.41) | (0.06)
| $\hat{\beta}_{1}$ | $\hat{\beta}_{2}$ | $\hat{\beta}_{3}$ | $\hat{\beta}_{4}$ | $\hat{\beta}_{5}$ | MSE
---|---|---|---|---|---|---
Ideal | 0.67 | 0.67 | 0.67 | 0.67 | 0.67 | 0.00
Lasso | 0.67 | 0.29 | 0.31 | 0.41 | 0.17 | 0.79
| (0.27) | (0.26) | (0.20) | (0.19) | (0.16) | (0.42)
adLasso | 0.69 | 0.42 | 0.46 | 0.58 | 0.27 | 0.60
| (0.27) | (0.33) | (0.25) | (0.23) | (0.22) | (0.37)
procbol | 0.69 | 0.63 | 0.68 | 0.65 | 0.49 | 0.44
$\alpha=0.1$ | (0.27) | (0.32) | (0.17) | (0.30) | (0.33) | (0.31)
procbol | 0.69 | 0.63 | 0.68 | 0.62 | 0.43 | 0.51
$\alpha=0.05$ | (0.27) | (0.32) | (0.17) | (0.33) | (0.36) | (0.30)
Lasso+ | 0.69 | 0.65 | 0.49 | 0.41 | 0.43 | 0.35
| (0.25) | (0.28) | (0.17) | (0.11) | (0.11) | (0.17)
adLasso+ | 0.69 | 0.64 | 0.59 | 0.57 | 0.48 | 0.26
| (0.25) | (0.27) | (0.15) | (0.12) | (0.14) | (0.15)
lmmLasso | 0.69 | 0.65 | 0.66 | 0.41 | 0.43 | 0.30
| (0.25) | (0.28) | (0.11) | (0.11) | (0.10) | (0.15)
pbol+ | 0.69 | 0.67 | 0.67 | 0.66 | 0.65 | 0.19
$\alpha=0.1$ | (0.25) | (0.28) | (0.12) | (0.10) | (0.10) | (0.14)
pbol+ | 0.69 | 0.67 | 0.67 | 0.66 | 0.65 | 0.18
$\alpha=0.05$ | (0.25) | (0.28) | (0.11) | (0.10) | (0.10) | (0.13)
Table 4: Results of model $M_{2}$. The percentage of true model recovered was
recorded -‘Truth’- as well as $\hat{J}=J$. $|J|$ is the number of fixed
effects selected and $TP$ the number of relevant fixed effects selected. The
signal to noise ratio is equal to $SNR=1.02(0.21)$. Standard errors are given
in parentheses, for $100$ runs.
Results | Truth | $\hat{J}=J$ | $|\hat{J}|$ | TP | $\hat{\sigma}_{e}^{2}$ | $\hat{\sigma}_{1}^{2}$ | $\hat{\sigma}_{2}^{2}$
---|---|---|---|---|---|---|---
Ideal | 1 | 5 | 5 | 5 | 1 | 1 | 1
Lasso | - | 0.11 | 5.02 | 3.86 | 3.62 | - | -
| | | (2.69) | (1.35) | (0.96) | - | -
adLasso | - | 0.09 | 6.06 | 4.24 | 3.05 | - | -
| | | (2.66) | (1.16) | (0.87) | - | -
procbol | - | 0.24 | 3.95 | 3.76 | 3.62 | - | -
$\alpha=0.1$ | | | (1.22) | (1.06) | (0.95) | - | -
procbol | - | 0.21 | 3.60 | 3.47 | 3.53 | - | -
$\alpha=0.05$ | | | (1.25) | (1.14) | (0.87) | - | -
Lasso+ | 0.17 | 0.17 | 7.60 | 4.92 | 1.25 | 0.91 | 0.93
| | | (2.64) | (0.37) | (0.28) | (0.40) | (0.48)
adLasso+ | 0.08 | 0.08 | 8.26 | 5.00 | 0.99 | 0.90 | 0.85
| | | (3.15) | (0.00) | (0.21) | (0.38) | (0.41)
lmmLasso | 0.17 | 0.17 | 7.65 | 4.93 | 1.24 | 0.91 | (0.93)
| | | (2.49) | (0.36) | (0.26) | (0.40) | (0.48)
pbol+ | 0.91 | 0.91 | 4.86 | 4.85 | 1.01 | 0.95 | 0.88
$\alpha=0.1$ | | | (0.59) | (0.58) | (0.28) | (0.38) | (0.41)
pbol+ | 0.80 | 0.80 | 4.57 | 4.57 | 1.11 | 0.93 | 0.88
$\alpha=0.05$ | | | (0.93) | (0.93) | (0.39) | (0.38) | (0.39)
| $\hat{\beta}_{i_{1}}$ | $\hat{\beta}_{i_{2}}$ | $\hat{\beta}_{i_{3}}$ | $\hat{\beta}_{i_{4}}$ | $\hat{\beta}_{i_{5}}$ | MSE
---|---|---|---|---|---|---
Ideal | 0.75 | 0.75 | 0.75 | 0.75 | 0.75 | 0.00
Lasso | 0.79 | 0.47 | 0.21 | 0.19 | 0.17 | 1.19
| (0.27) | (0.31) | (0.19) | (0.17) | (0.16) | (0.57)
adLasso | 0.79 | 0.64 | 0.36 | 0.35 | 0.29 | 0.84
| (0.27) | (0.38) | (0.24) | (0.24) | (0.22) | (0.55)
procbol | 0.79 | 0.72 | 0.50 | 0.57 | 0.52 | 0.82
$\alpha=0.1$ | (0.27) | (0.49) | (0.40) | (0.38) | (0.38) | (0.55)
procbol | 0.79 | 0.75 | 0.44 | 0.50 | 0.45 | 0.93
$\alpha=0.05$ | (0.27) | (0.50) | (0.42) | (0.41) | (0.40) | (0.56)
Lasso+ | 0.82 | 0.91 | 0.35 | 0.35 | 0.33 | 0.54
| (0.26) | (0.26) | (0.13) | (0.11) | (0.13) | (0.24)
adLasso+ | 0.81 | 0.82 | 0.51 | 0.52 | 0.49 | 0.33
| (0.25) | (0.25) | (0.14) | (0.13) | (0.14) | (0.17)
lmmLasso | 0.82 | 0.91 | 0.35 | 0.35 | 0.33 | 0.53
| (0.26) | (0.26) | (0.13) | (0.11) | (0.13) | (0.23)
pbol+ | 0.79 | 0.76 | 0.70 | 0.73 | 0.72 | 0.23
$\alpha=0.1$ | (0.25) | (0.26) | (0.22) | (0.17) | (0.18) | (0.28)
pbol+ | 0.80 | 0.79 | 0.64 | 0.66 | 0.66 | 0.35
$\alpha=0.05$ | (0.25) | (0.28) | (0.29) | (0.28) | (0.28) | (0.43)
Table 5: Results of model $M_{3}$. The percentage of true model recovered was
recorded -‘Truth’- as well as $\hat{J}=J$. $|J|$ is the number of fixed
effects selected and $TP$ the number of relevant fixed effects selected. The
signal to noise ratio is equal to $SNR=0.83(0.16)$. Standard errors are given
in parentheses, for $100$ runs.
Results | Truth | $\hat{J}=J$ | $|\hat{J}|$ | TP | $\hat{\sigma}_{e}^{2}$ | $\hat{\sigma}_{1}^{2}$ | $\hat{\sigma}_{2}^{2}$
---|---|---|---|---|---|---|---
Ideal | 1 | 5 | 5 | 5 | 1 | 1 | 1
Lasso | - | 0.22 | 4.96 | 4.13 | 3.32 | - | -
| | | (2.18) | (1.10) | (0.80) | - | -
adLasso | - | 0.20 | 6.10 | 4.58 | 2.85 | - | -
| | | (2.19) | (0.70) | (0.72) | - | -
procbol | - | 0.28 | 4.37 | 4.12 | 2.90 | - | -
$\alpha=0.1$ | | | (1.08) | (0.77) | (0.79) | - | -
procbol | - | 0.26 | 4.17 | 3.97 | 2.97 | - | -
$\alpha=0.05$ | | | (1.12) | (0.83) | (0.82) | - | -
Lasso+ | 0.20 | 0.20 | 7.07 | 4.99 | 1.11 | 0.91 | 0.92
| | | (2.01) | (0.10) | (0.22) | (0.36) | (0.46)
adLasso+ | 0.24 | 0.24 | 6.70 | 4.97 | 0.97 | 0.88 | 0.88
| | | (1.51) | (0.17) | (0.19) | (0.34) | (0.45)
lmmLasso | - | - | - | - | - | - | -
| | | - | - | - | - | -
pbol+ | 0.93 | 0.93 | 5.09 | 5.00 | 0.95 | 0.91 | 0.89
$\alpha=0.1$ | | | (0.38) | (0.00) | (0.17) | (0.33) | (0.44)
pbol+ | 0.95 | 0.95 | 5.08 | 5.00 | 0.95 | 0.91 | 0.89
$\alpha=0.05$ | | | (0.44) | (0.00) | (0.17) | (0.33) | (0.44)
| $\hat{\beta}_{1}$ | $\hat{\beta}_{2}$ | $\hat{\beta}_{3}$ | $\hat{\beta}_{4}$ | $\hat{\beta}_{5}$ | MSE
---|---|---|---|---|---|---
Ideal | 0.67 | 0.67 | 0.67 | 0.67 | 0.67 | 0.00
Lasso | 0.69 | 0.69 | 0.18 | 0.20 | 0.27 | 0.90
| (0.25) | (0.32) | (0.17) | (0.17) | (0.17) | (0.40)
adLasso | 0.69 | 0.68 | 0.32 | 0.36 | 0.46 | 0.60
| (0.25) | (0.32) | (0.21) | (0.21) | (0.22) | (0.32)
procbol | 0.73 | 0.65 | 0.48 | 0.51 | 0.57 | 0.63
$\alpha=0.1$ | (0.34) | (0.13) | (0.36) | (0.36) | (0.35) | (0.42)
procbol | 0.73 | 0.65 | 0.44 | 0.49 | 0.56 | 0.68
$\alpha=0.05$ | (0.34) | (0.13) | (0.38) | (0.38) | (0.36) | (0.43)
Lasso+ | 0.71 | 0.71 | 0.40 | 0.38 | 0.43 | 0.41
| (0.24) | (0.29) | (0.12) | (0.11) | (0.11) | (0.19)
adLasso+ | 0.71 | 0.69 | 0.50 | 0.48 | 0.56 | 0.30
| (0.24) | (0.29) | (0.16) | (0.14) | (0.13) | (0.18)
lmmLasso | - | - | - | - | - | -
| - | - | - | - | - | -
pbol+ | 0.71 | 0.69 | 0.67 | 0.65 | 0.68 | 0.19
$\alpha=0.1$ | (0.24) | (0.29) | (0.12) | (0.10) | (0.10) | (0.16)
pbol+ | 0.71 | 0.69 | 0.67 | 0.65 | 0.68 | 0.19
$\alpha=0.05$ | (0.24) | (0.29) | (0.12) | (0.10) | (0.10) | (0.16)
Table 6: Results of model $M_{4}$. The percentage of true model recovered was
recorded -‘Truth’- as well as $\hat{J}=J$. $|J|$ is the number of fixed
effects selected and $TP$ the number of relevant fixed effects selected. The
signal to noise ratio is equal to $SNR=0.63(0.11)$. Standard errors are given
in parentheses, for $100$ runs.
Results | Truth | $\hat{J}=J$ | $|\hat{J}|$ | TP | $\hat{\sigma}_{e}^{2}$ | $\hat{\sigma}_{1}^{2}$ | $\hat{\sigma}_{2}^{2}$
---|---|---|---|---|---|---|---
Ideal | 1 | 5 | 5 | 5 | 1 | 1 | 1
Lasso | - | 0.00 | 2.81 | 2.06 | 4.08 | - | -
| | | (2.80) | (1.30) | (0.84) | - | -
adLasso | - | 0.00 | 5.64 | 3.03 | 3.38 | - | -
| | | (4.10) | (1.22) | (0.88) | - | -
procbol | - | 0.15 | 3.85 | 3.61 | 3.23 | - | -
$\alpha=0.1$ | | | (1.00) | (0.95) | (0.73) | - | -
procbol | - | 0.15 | 3.48 | 3.34 | 3.39 | - | -
$\alpha=0.05$ | | | (1.00) | (0.99) | (0.80) | - | -
Lasso+ | 0.25 | 0.25 | 7.13 | 4.99 | 1.21 | 0.93 | 1.03
| | | (1.84) | (0.10) | (0.27) | (0.41) | (0.40)
adLasso+ | 0.01 | 0.01 | 9.56 | 4.87 | 0.94 | 0.89 | 0.98
| | | (4.01) | (0.37) | (0.26) | (0.37) | (0.37)
lmmLasso | 0.25 | 0.25 | 7.22 | 4.99 | 1.19 | 0.93 | 1.03
| | | (1.95) | (0.10) | (0.25) | (0.40) | (0.40)
pbol+ | 0.82 | 0.82 | 5.21 | 4.99 | 0.92 | 0.97 | 1.00
$\alpha=0.1$ | | | (0.56) | (0.10) | (0.17) | (0.39) | (0.34)
pbol+ | 0.88 | 0.88 | 5.10 | 4.98 | 0.93 | 0.97 | 1.00
$\alpha=0.05$ | | | (0.41) | (0.14) | (0.16) | (0.39) | (0.34)
| $\hat{\beta}_{1}$ | $\hat{\beta}_{2}$ | $\hat{\beta}_{3}$ | $\hat{\beta}_{4}$ | $\hat{\beta}_{5}$ | MSE
---|---|---|---|---|---|---
Ideal | 0.67 | 0.67 | 0.67 | 0.67 | 0.67 | 0.00
Lasso | 0.60 | 0.06 | 0.06 | 0.06 | 0.11 | 1.27
| (0.25) | (0.15) | (0.11) | (0.11) | (0.15) | (0.32)
adLasso | 0.60 | 0.15 | 0.18 | 0.17 | 0.26 | 0.99
| (0.25) | (0.26) | (0.19) | (0.19) | (0.23) | (0.33)
procbol | 0.60 | 0.55 | 0.38 | 0.44 | 0.43 | 0.83
$\alpha=0.1$ | (0.25) | (0.31) | (0.38) | (0.39) | (0.40) | (0.43)
procbol | 0.60 | 0.53 | 0.32 | 0.38 | 0.35 | 0.91
$\alpha=0.05$ | (0.25) | (0.32) | (0.38) | (0.40) | (0.41) | (0.39)
Lasso+ | 0.62 | 0.55 | 0.31 | 0.35 | 0.37 | 0.46
| (0.25) | (0.27) | (0.11) | (0.12) | (0.12) | (0.20)
adLasso+ | 0.61 | 0.56 | 0.41 | 0.43 | 0.48 | 0.39
| (0.25) | (0.26) | (0.16) | (0.18) | (0.16) | (0.19)
lmmLasso | 0.62 | 0.55 | 0.31 | 0.35 | 0.38 | 0.45
| (0.25) | (0.27) | (0.11) | (0.12) | (0.12) | (0.19)
pbol+ | 0.60 | 0.64 | 0.67 | 0.67 | 0.67 | 0.21
$\alpha=0.1$ | (0.25) | (0.28) | (0.10) | (0.11) | (0.13) | (0.15)
pbol+ | 0.60 | 0.64 | 0.67 | 0.67 | 0.67 | 0.20
$\alpha=0.05$ | (0.25) | (0.27) | (0.10) | (0.13) | (0.13) | (0.15)
Table 7: Results of model $M_{4}$ when a ML linear regression is added after the convergence of the algorithm. The percentage of true model recovered was recorded -‘Truth’- as well as $\hat{J}=J$. $|J|$ is the number of fixed effects selected and $TP$ the number of relevant fixed effects selected. The signal to noise ratio is equal to $SNR=0.63(0.11)$. Standard errors are given in parentheses, for $100$ runs. | Ideal | lmmLasso | Lasso+
---|---|---|---
Truth | 1 | 0.25 | 0.25
$\hat{J}=J$ | 1 | 0.25 | 0.25
$|\hat{J}|$ | 5 | 7.22(1.95) | 7.13(1.84)
$TP$ | 5 | 4.99(0.10) | 4.99(0.10)
$\hat{\sigma}_{e}^{2}$ | 1 | 1.19(0.25) | 1.21(0.27)
$\hat{\sigma}_{1}^{2}$ | 1 | 0.96(0.39) | 0.96(0.40)
$\hat{\sigma}_{2}^{2}$ | 1 | 1.01(0.36) | 1.01(0.36)
$\hat{\beta}_{1}$ | 0.67 | 0.61(0.25) | 0.61(0.25)
$\hat{\beta}_{2}$ | 0.67 | 0.62(0.28) | 0.62(0.28)
$\hat{\beta}_{3}$ | 0.67 | 0.61(0.12) | 0.61(0.12)
$\hat{\beta}_{4}$ | 0.67 | 0.63(0.12) | 0.63(0.12)
$\hat{\beta}_{5}$ | 0.67 | 0.62(0.14) | 0.62(0.14)
$mse$ | 0 | 0.40(0.17) | 0.40(0.17)
## Appendix B - Remark on the tuning parameter
The tuning of the regularization parameter could be tricky for some methods,
especially the Lasso method and the adLasso method. In this section, we look
at the causes.
We shall begin to consider the classical linear model before studying the
linear mixed model. Let us first look at the Lasso method when only applied in
a classical linear model. We compare two penalizations of the likelihood: BIC
and the Extended BIC (EBIC) (Chen and Chen,, 2008). The EBIC penalizes a space
of dimension $k$ with a term that depends on the number of spaces that have
the same dimension, which is $\frac{p!}{k!(p-k)!}$; thus EBIC penalizes more
the complex spaces than BIC. Figure 3 shows the behavior of the BIC and EBIC
criteria, the log-likelihood and the residual variance for several values of
the regularization parameter of the Lasso in a low dimensional case ($p=80$).
We observe that tuning the regularization parameter in this case raises no
problem.
(a) BIC or EBIC depending on the value of the regularization parameter of the
Lasso method
(b) $-2\times$log-Likelihood depending on the regularization parameter of the
Lasso method
(c) Residual variance depending on the regularization parameter of the Lasso
method
Figure 3: One simulation of linear model for the Lasso method with
$n=120,p=80$ and $\beta_{J}=1$.
Let us now consider a simulation in a high dimensional context in which we
have $n=120$ observations and $p=600$ explanatory variables. Results of the
two methods for choosing the regularization parameter of Lasso are presented
in Figure 4.
(a) BIC or EBIC depending on the value of the regularization parameter of the
Lasso
(b) $-2\times$log-Likelihood depending on the regularization parameter of the
Lasso method
(c) Residual variance depending on the regularization parameter of the Lasso
method
Figure 4: One simulation of linear model for the Lasso method with
$n=120,p=600$ and $\beta_{J}=1$.
Firstly, we confirm that EBIC is more conservative than BIC and penalizes more
the complex spaces. On the far left of Figure 4(a), we observe that both the
BIC and the EBIC curves decrease when the regularization parameter is close to
zero. This phenomenon is due to the degeneracy of the likelihood that can be
seen in Figure 4(b) (stated in Section 2 for mixed models, it can also happen
in linear models). Figure 4(c) shows that the degeneracy of the likelihood
comes from the residual variance that drops to zero when the regularization
parameter is close to zero, and thus when too much variables enter the model.
To conclude, we see that both BIC and EBIC penalties are not sufficiently
strong to completely balance the degeneracy of the likelihood; however, EBIC
penalty leads to select a more parsimonious model while BIC penalty selects a
more complex model. Nonetheless, the EBIC penalty is usually too much
conservative in practice, that is why the simulation study used the BIC
penalty. When the degeneracy happens -as it is likely to occur as $p$ grows-,
the regularization parameter should be optimized over an area that does not
contain the explosion of the likelihood, that means that the area should not
contain the far left part of Figure 4(a) where the criterion decreases.
We now look at the Lasso+ method. As mentioned in the paper, the maximal
number of fixed-effects that can be selected with the Lasso+ method is small
compared to $n$ or $p$. Thus, the degeneracy of the likelihood never occurred
in our simulations (Figure 5). However, if this phenomenon happens, the choice
of the grid of the regularization parameter should follow the same advice as
the one given above for the classical linear model.
(a) BIC or EBIC depending on the value of the regularization parameter of the
Lasso+ method
(b) $-2\times$log-Likelihood depending on the regularization parameter of the
Lasso+ method
(c) Residual variance depending on the regularization parameter of the Lasso+
method
(d) Residual variance depending on the regularization parameter of the Lasso+
method
Figure 5: One simulation of linear mixed model with $n=120,p=600,\beta_{J}=1$
and two i.i.d. random effects.
## Appendix C - Proof of Proposition 2.2
$G$ and $R$ are supposed to be known. Thus the minimization of our objective
function $g$ reduces to the minimization of the following function in
$(\beta,u)$:
$h(u,\beta)=(y-X\beta-Zu)^{\prime}R^{-1}(y-X\beta-
Zu)+u^{\prime}G^{-1}u+\lambda|\beta|_{1}$.
Let denote $(\hat{u},\hat{\beta})=\underset{(u,\beta)}{\text{argmin
}}h(u,\beta)$. Since the function $h$ is convex, we have:
$(\hat{u},\hat{\beta})=\\{\begin{array}[]{l
l}u(\beta)=\underset{u}{\text{argmin }}h(u,\beta)\\\
\hat{\beta}=\underset{\beta}{\text{argmin }}h(u(\beta),\beta)\\\
\hat{u}=u(\hat{\beta})\\\ \end{array}\right.$.
Since $\dfrac{\partial h(u,\beta)}{\partial u}$ exists, we can explicit the
minimum of $h$ in $u$:
$(\hat{u},\hat{\beta})=\\{\begin{array}[]{l
l}u(\beta)=(Z^{\prime}R^{-1}Z+G^{-1})^{-1}Z^{\prime}R^{-1}(y-X\beta)\\\
\hat{\beta}=\underset{\beta}{\text{argmin }}h(u(\beta),\beta)\\\
\hat{u}=u(\hat{\beta})\\\ \end{array}\right.$
Thus, we obtain:
$\displaystyle h(u(\beta),\beta)$ $\displaystyle=$ $\displaystyle(y-X\beta-
Zu(\beta))^{\prime}R^{-1}(y-X\beta-
Zu(\beta))+u^{\prime}G^{-1}u+\lambda|\beta|_{1}$ $\displaystyle=$
$\displaystyle(y-X\beta)^{\prime}R^{-1}(y-X\beta)-(y-X\beta)R^{-1}Zu(\beta)-(Zu(\beta))^{\prime}R^{-1}(y-X\beta)$
$\displaystyle+(Z\hat{u})^{\prime}R^{-1}Zu(\beta)+u(\beta)^{\prime}G^{-1}u(\beta)+\lambda|\beta|_{1}$
$\displaystyle=$
$\displaystyle(y-X\beta)^{\prime}$$R^{-1}-R^{-1}Z(Z^{\prime}R^{-1}Z+G^{-1})^{-1}Z^{\prime}R^{-1}$$(y-X\beta)+\lambda|\beta|_{1}$
Denote $W=R^{-1}-R^{-1}Z(Z^{\prime}R^{-1}Z+G^{-1})^{-1}Z^{\prime}R^{-1}$. We
can show that $W=(Z^{\prime}GZ+R^{-1})^{-1}=V^{-1}$. This result comes from
the equivalence between the resolution of Henderson’s equations (Henderson,,
1973) and the generalized least squares.
To conclude, we have that
$(\hat{u},\hat{\beta})=\left((Z^{\prime}R^{-1}Z+G^{-1})^{-1}Z^{\prime}R^{-1}(y-X\hat{\beta}),\underset{\beta}{\text{argmin
}}(y-X\beta)^{\prime}V^{-1}(y-X\beta)+\lambda|\beta|_{1}\right)$.
|
arxiv-papers
| 2013-01-27T15:39:44 |
2024-09-04T02:49:40.812758
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Florian Rohart, Magali San-Cristobal and B\\'eatrice Laurent",
"submitter": "Florian Rohart",
"url": "https://arxiv.org/abs/1301.6365"
}
|
1301.6427
|
# Fundamental Inequalities and Identities Involving Mutual and Directed
Informations in Closed-Loop Systems
Milan S. Derpich, Eduardo I. Silva and Jan Østergaard M.S. Derpich and E.I
Silva are with the Department of Electronic Engineering, Universidad Técnica
Federico Santa María, Casilla 110-V, Valparaíso, Chile (email:
[email protected], [email protected]). Their work was supported in part
by CONICYT through grants FONDECYT Nr. 1120468, Nr. 1130459, and Anillo
ACT-53. J. Østergaard is with the Department of Electronic Systems, Aalborg
University, Niels Jernes Vej 12, DK-9220, Aalborg, Denmark (email:
[email protected]).
###### Abstract
We present several novel identities and inequalities relating the mutual
information and the directed information in systems with feedback. The
internal blocks within such systems are restricted only to be causal mappings,
but are allowed to be non-linear, stochastic and time varying. Moreover, the
involved signals can be arbitrarily distributed. We bound the directed
information between signals inside the feedback loop by the mutual information
between signals inside and outside the feedback loop. This fundamental result
has an interesting interpretation as a law of conservation of information
flow. Building upon it, we derive several novel identities and inequalities,
which allow us to prove some existing information inequalities under less
restrictive assumptions. Finally, we establish new relationships between
nested directed informations inside a feedback loop. This yields a new and
general data-processing inequality for systems with feedback.
## I Introduction
The notion of directed information introduced by Massey in [1] assesses the
amount of information that causally “flows” from a given random and ordered
sequence to another. For this reason, it has increasingly found use in diverse
applications, from characterizing the capacity of channels with feedback [1,
2, 3, 4], the rate distortion function under causality constraints [5],
establishing some of the fundamental limitations in networked control [6, 7,
8, 9, 10, 11], determining causal relationships in neural networks [12], to
portfolio theory and hypothesis testing [13], to name a few.
The directed information from a random111 Hereafter we use non-italic letters
(such as $\operatorname{x}$) for random variables, denoting a particular
realization by the corresponding italic character, $x$. sequence
$\operatorname{x}^{k}$ to a random sequence $\operatorname{y}^{k}$ is defined
as
$\displaystyle
I(\operatorname{x}^{k}\to\operatorname{y}^{k})\triangleq\sum\limits_{i=1}^{k}I(\operatorname{y}(i);\operatorname{x}^{i}|\operatorname{y}^{i-1}),$
(1)
where the notation $\operatorname{x}^{i}$ represents the sequence
$\operatorname{x}(1),\operatorname{x}(2),\ldots,\operatorname{x}(i)$. The
causality inherent in this definition becomes evident when comparing it with
the mutual information between $\operatorname{x}^{k}$ and
$\operatorname{y}^{k}$, given by
$I(\operatorname{x}^{k},\operatorname{y}^{k})=\sum\nolimits_{i=1}^{k}I(\operatorname{y}(i);\operatorname{x}^{k}|\operatorname{y}^{i-1})$.
In the latter sum, what matters is the amount of information about the entire
sequence $\operatorname{x}^{k}$ present in $\operatorname{y}(i)$, given the
past values $\operatorname{y}^{i-1}$. By contrast, in the conditional mutual
informations in the sum of (1), only the past and current values of
$\operatorname{x}^{k}$ are considered, that is, $\operatorname{x}^{i}$. Thus,
$I(\operatorname{x}^{k}\to\operatorname{y}^{k})$ represents the amount of
information causally conveyed from $\operatorname{x}^{k}$ to
$\operatorname{y}^{k}$.
There exist several results characterizing the relationship between
$I(\operatorname{x}^{k}\to\operatorname{y}^{k})$ and
$I(\operatorname{x}^{k};\operatorname{y}^{k})$. First, it is well known that
$I(\operatorname{x}^{k}\to\operatorname{y}^{k})\leq
I(\operatorname{x}^{k};\operatorname{y}^{k})$, with equality if and only if
$\operatorname{y}^{k}$ is causally related to $\operatorname{x}^{k}$ [1]. A
conservation law of mutual and directed information has been found in [14],
which asserts that
$I(\operatorname{x}^{k}\to\operatorname{y}^{k})+I(0*\operatorname{y}^{k-1}\to\operatorname{x}^{k})=I(\operatorname{x}^{k};\operatorname{y}^{k})$,
where $0*\operatorname{y}^{k-1}$ denotes the concatenation
$0,\operatorname{y}(1),\ldots,\operatorname{y}^{k-1}$.
Given its prominence in settings involving feedback, it is perhaps in these
scenarios where the directed information becomes most important. For instance,
the directed information has been instrumental in characterizing the capacity
of channels with feedback (see, e.g., [3, 15, 4] and the references therein),
as well as the rate-distortion function in setups involving feedback [16, 9,
11, 10, 5].
In this paper, our focus is on the relationships (inequalities and identities)
involving directed and mutual informations within feedback systems, as well as
between directed informations involving different signals within the
corresponding feedback loop. In order to discuss some of the existing results
related to this problem, it is convenient to consider the general feedback
system shown in Fig. 1-(a). In this diagram, the blocks
$\mathcal{S}_{1},\ldots,\mathcal{S}_{4}$ represent possibly non-linear and
time-varying causal systems such that the total delay of the loop is at least
one sample. In the same figure,
$\operatorname{r},\operatorname{p},\operatorname{s},\operatorname{q}$ are
exogenous random signals (scalars, vectors or sequences), which could
represent, for example, any combination of disturbances, noises, random
initial states or side informations. We note that any of these exogenous
signals, in combination with its corresponding deterministic mapping
$\mathcal{S}_{i}$, can also yield any desired stochastic causal mapping.
For the simple case in which all the systems $\\{\mathcal{S}_{i}\\}_{i=1}^{4}$
are linear time invariant (LTI) and stable, and assuming
$\operatorname{p},\operatorname{x},\operatorname{q}=0$ (deterministically), it
was shown in [17] that $I(\operatorname{r}^{k}\to\operatorname{e}^{k})$ does
not depend on whether there is feedback from $\operatorname{e}$ to
$\operatorname{u}$ or not. Inequalities between mutual and directed
informations in a less restricted setup, shown in Fig. 1-(b), have been found
in [7, 8]. In that setting (a networked-control system), $G$ is a strictly
causal LTI dynamic system having (vector) state sequence
$\\{\mathbf{x}(i)\\}_{i=0}^{\infty}$, with
$\mathbf{x}_{0}\triangleq\mathbf{x}(0)$ being the random initial state in its
state-space representation. The external signal $\operatorname{r}$ (which
could correspond to a disturbance) is statistically independent of
$\operatorname{s}$, the latter corresponding to, for example, side information
or channel noise. Both are also statistically independent of $\mathbf{x}_{0}$.
$\operatorname{x}$$\operatorname{y}$$\operatorname{s}$$\operatorname{p}$$\operatorname{r}$$\operatorname{q}$$\mathcal{S}_{1}$$\mathcal{S}_{3}$$\mathcal{S}_{4}$$\mathcal{S}_{2}$$\operatorname{u}$$G$$\mathbf{x}_{0}$$D$$E$$\operatorname{x}$$\operatorname{e}$$\operatorname{u}$$f$$\operatorname{w}$$\operatorname{r}$$\operatorname{e}$$\operatorname{y}$$\operatorname{s}$(a)(b)
Figure 1: (a): The general system considered in this work. (b): A special
case, corresponding to the closed-loop system studied in [7].
The blocks labeled $E$, $D$ and $f$ correspond to an encoder, a decoder and a
channel, respectively, all of which are causal. The channel $f$ maps
$\operatorname{s}^{k}$ and $\operatorname{x}^{k}$ to $\operatorname{y}(k)$ in
a possibly time-varying manner, i.e.,
$\operatorname{y}(k)=f(k,\operatorname{x}^{k},\operatorname{s}^{k}).$
Similarly, the concatenation of the encoder, the channel and the decoder, maps
$\operatorname{s}^{k}$ and $\operatorname{w}^{k}$ to $\operatorname{u}(k)$ as
a possibly time-dependent function
$\operatorname{u}(k)=\psi(k,\operatorname{w}^{k},\operatorname{s}^{k}).$ Under
these assumptions, the following fundamental result was shown in [8, Lemma
5.1]:
$\displaystyle
I(\mathbf{x}_{0},\operatorname{r}^{k}\,;\,\operatorname{u}^{k})-I(\operatorname{r}^{k};\operatorname{u}^{k})\geq
I(\mathbf{x}_{0};\operatorname{e}^{k}).$ (2)
By further assuming in [8] that the decoder $D$ in Fig. 1-(b) is
deterministic, the following Markov chain naturally holds,
$\displaystyle(\mathbf{x}_{0},\operatorname{r}^{k})\longleftrightarrow\operatorname{y}^{k}\longleftrightarrow\operatorname{u}^{k},$
(3)
leading directly to
$\displaystyle
I(\mathbf{x}_{0},\operatorname{r}^{k}\,;\,\operatorname{y}^{k})-I(\operatorname{r}^{k};\operatorname{u}^{k})\geq
I(\mathbf{x}_{0};\operatorname{e}^{k}),$ (4)
which is found in the proof of [8, Corollary 5.3]. The deterministic nature of
the decoder $D$ played a crucial role in the proof of this result, since
otherwise the Markov chain (3) does not hold, in general, due to the feedback
from $\operatorname{u}$ to $\operatorname{y}$.
Notice that both (2) and (4) provide lower bounds to the difference between
two mutual informations, each of them relating a signal external to the loop
(such as $\mathbf{x}_{0},\operatorname{r}^{k}$) to a signal internal to the
loop (such as $\operatorname{u}^{k}$ or $\operatorname{y}^{k}$). Instead, the
inequality
$\displaystyle I(\operatorname{x}^{k}\to\operatorname{y}^{k})\geq
I(\operatorname{r}^{k};\operatorname{y}^{k}),$ (5)
which holds for the system in Fig. 1-(a) and appears in [1, Theorem 3] (and
rediscovered later in [6, Lemma 4.8.1]), involves the directed information
between two internal signals and the mutual information between the second of
these and an external sequence. A related bound, similar to (4) but involving
information rates and with the leftmost mutual information replaced by the
directed information from $\operatorname{x}^{k}$ to $\operatorname{y}^{k}$
(which are two signals internal to the loop), has been obtained in [7, Lemma
4.1]:
$\displaystyle\bar{I}(\operatorname{x}\to\operatorname{y})-\bar{I}(\operatorname{r};\operatorname{u})\geq\lim_{k\to\infty}\frac{I(\mathbf{x}(0);\operatorname{e}^{k})}{k},$
(6)
with
$\bar{I}(\operatorname{x}\to\operatorname{y})\triangleq\lim_{k\to\infty}\frac{1}{k}I(\operatorname{x}^{k}\to\operatorname{y}^{k})$
and
$\bar{I}(\operatorname{r};\operatorname{u})\triangleq\lim_{k\to\infty}\frac{1}{k}I(\operatorname{r}^{k};\operatorname{u}^{k})$,
provided $\sup_{i\geq
0}\operatorname{E}\left[\mathbf{x}(i)^{T}\mathbf{x}(i)\right]<\infty$. This
result relies on three assumptions: a) that the channel $f$ is memory-less and
satisfies a “conditional invertibility” property, b) a finite-memory
condition, and c) a fading-memory condition, these two related to the decoder
$D$ (see Fig. 1). It is worth noting that, as defined in [7], these
assumptions upon $D$ exclude the use of side information by the decoder and/or
the possibility of $D$ being affected by random noise or having a random
internal state which is non-observable (please see [7] for a detailed
description of these assumptions).
The inequality (5) has recently been extended in [4, Theorem 1], for the case
of discrete-valued random variables and assuming
$\operatorname{s}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}(\operatorname{r},\operatorname{p},\operatorname{q})$,
as the following identity (written in terms of the signals and setup shown in
Fig. 1-(a)):
$\displaystyle
I(\operatorname{x}^{k}\to\operatorname{y}^{k})=I(\operatorname{p}^{k},\operatorname{y}^{k})+I(\operatorname{x}^{k}\to\operatorname{y}^{k}|\operatorname{p}^{k}).$
(7)
Letting $\operatorname{q}=\operatorname{s}$ in Fig. 1-(a) and with the
additional assumption that
$(\operatorname{p},\operatorname{s})\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\operatorname{q}$, it was also shown in [4,
Theorem 1] that
$\displaystyle
I(\operatorname{x}^{k}\to\operatorname{y}^{k})=I(\operatorname{p}^{k};\operatorname{y}^{k})+I(\operatorname{q}^{k-1};\operatorname{y}^{k})+I(\operatorname{p}^{k};\operatorname{q}^{k-1}|\operatorname{y}^{k}),$
(8)
for the cases in which
$\operatorname{u}(i)=\operatorname{y}(i)+\operatorname{q}(i)$ (i.e., when the
concatenation of $\mathcal{S}_{4}$ and $\mathcal{S}_{1}$ corresponds to a
summing node). In [4], (7) and (8) play important roles in characterizing the
capacity of channels with noisy feedback.
To the best of our knowledge, (2), (4), (5) (6), (7) and (8) are the only
results available in the literature which lower bound the difference between
an internal-to-internal directed information and an external-to-internal
mutual information. There exist even fewer published results in relation to
inequalities between two directed informations involving only signals internal
to the loop. To the best of our knowledge, the only inequality of this type in
the literature is the one found in the proof of Theorem 4.1 of [9]. The latter
takes the form of a (quasi) data-processing inequality for directed
informations in closed-loop systems, and states that
$\displaystyle
I(\operatorname{x}^{k}\to\operatorname{y}^{k}\parallel\operatorname{q}^{k})\geq
I(\operatorname{x}^{k}\to\operatorname{u}^{k}),$ (9)
provided222Here, and in the sequel, we use the notation
$\operatorname{x}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\operatorname{y}$ to mean “$\operatorname{x}$
is independent of $\operatorname{y}$”.
$\operatorname{q}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}(\operatorname{r},\operatorname{p})$ and if
$\mathcal{S}_{4}$ is such that $\operatorname{y}^{i}$ is a function of
$(\operatorname{u}^{i},\operatorname{q}^{i})$ (i.e., if $\mathcal{S}_{4}$ is
conditionally invertible) $\forall i$. In (9),
$\displaystyle
I(\operatorname{x}^{k}\to\operatorname{y}^{k}\parallel\operatorname{q}^{k})\triangleq\sum\limits_{i=1}^{k}I(\operatorname{y}(i);\operatorname{x}^{i}|\operatorname{y}^{i-1},\operatorname{q}^{i})$
(10)
corresponds to the causally conditioned directed information defined in [2].
Inequality (9) plays a crucial role [9], since it allowed lower bounding the
average data rate across a digital error-free channel by a directed
information. (In [9], $\operatorname{q}$ corresponded to a random dither
signal in an entropy-coded dithered quantizer.)
In this paper, we derive a set of information identities and inequalities
involving pairs of sequences (internal or external to the loop) in feedback
systems. The first of these is an identity which, under an independence
condition, can be interpreted as a law of conservation of information flows.
The latter identity is the starting point for most of the results which follow
it. Among other things, we extend (4) and (6) to the general setup depicted in
Fig. 1-(a), where none of the assumptions made in [7, 8, 9] (except causality)
needs to hold. Moreover, we will prove the validity of (9) without assuming
the conditional invertibility of $\mathcal{S}_{4}$ nor that
$\operatorname{q}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}(\operatorname{r},\operatorname{p})$. The
latter result is one of four novel data-processing inequalities derived in
Section III-B, each involving two nested directed informations valid for the
system depicted in Fig. 1-(a). The last of these is a complete closed-loop
counterpart of the traditional open-loop data-processing inequality.
The remainder of this paper begins with a description of the systems under
study and the extension of Massey’s directed information to the case in which
each of the blocks in the loop may introduce an arbitrary, non-negative delay
(i.e., we do not allow for anticipation). The information identities and
inequalities are presented in Section III. For clarity of the exposition, all
the proofs are deferred to Section IV. A brief discussion of potential
applications of our results is presented in Section V, which is followed by
the conclusions in Section VI.
## II Preliminaries
### II-A System Description
We begin by providing a formal description of the systems labeled
$\mathcal{S}_{1}\ldots\mathcal{S}_{4}$ in Fig. 1-(a). Their input-output
relationships are given by the possibly-varying deterministic mappings333For
notational simplicity, we omit writing their time dependency explicitly.
$\displaystyle\operatorname{e}(i)$
$\displaystyle=\mathcal{S}_{1}(\operatorname{u}^{i-d_{1}(i)},\operatorname{r}^{i}),$
(11a) $\displaystyle\operatorname{x}(i)$
$\displaystyle=\mathcal{S}_{2}(\operatorname{e}^{i-d_{2}(i)},\operatorname{p}^{i}),$
(11b) $\displaystyle\operatorname{y}(i)$
$\displaystyle=\mathcal{S}_{3}(\operatorname{x}^{i-d_{3}(i)},\operatorname{s}^{i}),$
(11c) $\displaystyle\operatorname{u}(i)$
$\displaystyle=\mathcal{S}_{4}(\operatorname{y}^{i-d_{4}(i)},\operatorname{q}^{i}),$
(11d)
where $\operatorname{r},\operatorname{p},\operatorname{s},\operatorname{q}$
are exogenous random signals and the (possibly time-varying) delays
$d_{1},d_{2},d_{3},d_{4}\in\\{0,1,\ldots\\}$ are such that
$d_{1}(k)+d_{2}(k)+d_{3}(k)+d_{4}(k)\geq 1,\;\;\;\;\forall k\in\mathbb{N}.$
That is, the concatenation of $\mathcal{S}_{1},\ldots,\mathcal{S}_{4}$ has a
delay of at least one sample. For every $i\in\\{1,\ldots,k\\}$,
$\operatorname{r}(i)\in\mathbb{R}^{n_{\operatorname{r}}(i)}$, i.e.,
$\operatorname{r}(i)$ is a real random vector whose dimension is given by some
function $n_{\operatorname{r}}:\\{1,\ldots,k\\}\to\mathbb{N}$. The other
sequences
($\operatorname{q},\operatorname{p},\operatorname{s},\operatorname{x},\operatorname{y},\operatorname{u}$)
are defined likewise.
### II-B A Necessary Modification of the Definition of Directed Information
As stated in [1], the directed information (as defined in (1)) is a more
meaningful measure of the flow of information between $\operatorname{x}^{k}$
and $\operatorname{y}^{k}$ than the conventional mutual information
$I(\operatorname{x}^{k};\operatorname{y}^{k})=\sum\nolimits_{i=1}^{k}I(\operatorname{y}(i);\operatorname{x}^{k}|\operatorname{y}^{i-1})$
when there exists causal feedback from $\operatorname{y}$ to
$\operatorname{x}$. In particular, if $\operatorname{x}^{k}$ and
$\operatorname{y}^{k}$ are discrete-valued sequences, input and output,
respectively, of a forward channel, and if there exists strictly causal,
perfect feedback, so that $\operatorname{x}(i)=\operatorname{y}(i-1)$ (a
scenario utilized in [1] as part of an argument in favor of the directed
information), then the mutual information becomes
$\displaystyle I(\operatorname{x}^{k};\operatorname{y}^{k})$
$\displaystyle=H(\operatorname{y}^{k})-H(\operatorname{y}^{k}|\operatorname{x}^{k})=H(\operatorname{y}^{k})-H(\operatorname{y}^{k}|\operatorname{y}^{k-1})=H(\operatorname{y}^{k})-H(\operatorname{y}(k)|\operatorname{y}^{k-1})=H(\operatorname{y}^{k-1}).$
Thus, when strictly causal feedback is present,
$I(\operatorname{x}^{k};\operatorname{y}^{k})$ fails to account for how much
information about $\operatorname{x}^{k}$ has been conveyed to
$\operatorname{y}^{k}$ through the forward channel that lies between them.
It is important to note that, in [1] (as well as in many works concerned with
communications), the forward channel is instantaneous, i.e., it has no delay.
Therefore, if a feedback channel is utilized, then this feedback channel must
have a delay of at least one sample, as in the example above. However, when
studying the system in Fig. 1-(a), we may need to evaluate the directed
information between signals $\operatorname{x}^{k}$ and $\operatorname{y}^{k}$
which are, respectively, input and output of a strictly casual forward channel
(i.e., with a delay of at least one sample), whose output is instantaneously
fed back to its input. In such case, if one further assumes perfect feedback
and sets $\operatorname{x}(i)=\operatorname{y}(i)$, then, in the same spirit
as before,
$\displaystyle I(\operatorname{x}^{k}\to\operatorname{y}^{k})$
$\displaystyle=\sum\limits_{i=1}^{k}I(\operatorname{y}(i);\operatorname{x}^{i}|\operatorname{y}^{i-1})=\sum\limits_{i=1}^{k}\left[H(\operatorname{y}(i)|\operatorname{y}^{i-1})-H(\operatorname{y}(i)|\operatorname{x}^{i},\operatorname{y}^{i-1})\right]=H(\operatorname{y}^{k}).$
As one can see, Massey’s definition of directed information ceases to be
meaningful if instantaneous feedback is utilized.
It is natural to solve this problem by recalling that, in the latter example,
the forward channel had a delay, say $d$, greater than one sample. Therefore,
if we are interested in measuring how much of the information in
$\operatorname{y}(k)$, not present in $\operatorname{y}^{i-1}$, was conveyed
from $\operatorname{x}^{i}$ through the forward channel, we should look at the
mutual information
$I(\operatorname{y}(i);\operatorname{x}^{i-d}|\operatorname{y}^{i-1})$,
because only the input samples $\operatorname{x}^{i-d}$ can have an influence
on $\operatorname{y}(i)$. For this reason, we introduce the following,
modified notion of directed information
###### Definition 1 (Directed Information with Forward Delay)
In this paper, the directed information from $\operatorname{x}^{k}$ to
$\operatorname{y}^{k}$ through a forward channel with a non-negative time
varying delay of $d(i)$ samples is defined as
$\displaystyle
I(\operatorname{x}^{k}\to\operatorname{y}^{k})\triangleq\sum\limits_{i=1}^{k}I(\operatorname{y}(i);\operatorname{x}^{i-d(i)}|\operatorname{y}^{i-1}).$
(12)
For a zero-delay forward channel, the latter definition coincides with
Massey’s.
Likewise, we adapt the definition of causally-conditioned directed information
to the definition
$\displaystyle
I(\operatorname{x}^{k}\to\operatorname{y}^{k}\parallel\operatorname{e}^{k})\triangleq\sum\limits_{i=1}^{k}I(\operatorname{y}(i);\operatorname{x}^{i-d_{3}(i)}|\operatorname{y}^{i-1},\operatorname{e}^{i-d_{2}(i)}).$
when the signals $\operatorname{e}$, $\operatorname{x}$ and $\operatorname{y}$
are related according to (11).
Before finishing this section, it is convenient to recall the following
identity (a particular case of the chain rule of conditional mutual
information [18]), which will be extensively utilized in the proofs of our
results:
$\displaystyle
I(\operatorname{a},\operatorname{b};\operatorname{c}|\operatorname{d})=I(\operatorname{b};\operatorname{c}|\operatorname{d})+I(\operatorname{a};\operatorname{c}|\operatorname{b},\operatorname{d}).$
(13)
## III Information Identities and Inequalities
### III-A Relationships Between Mutual and Directed Informations
We begin by stating a fundamental result, which relates the directed
information between two signals within a feedback loop, say $\operatorname{x}$
and $\operatorname{y}$, to the mutual information between an external set of
signals and $\operatorname{y}$:
###### Theorem 1
In the system shown in Fig. 1-(a), it holds that
$\displaystyle
I(\operatorname{x}^{k}\to\operatorname{y}^{k})=I(\operatorname{q}^{k},\operatorname{r}^{k},\operatorname{p}^{k}\to\operatorname{y}^{k})-I(\operatorname{q}^{k},\operatorname{r}^{k},\operatorname{p}^{k}\to\operatorname{y}^{k}\parallel\operatorname{x}^{k})\leq
I(\operatorname{p}^{k},\operatorname{q}^{k},\operatorname{r}^{k}\,;\operatorname{y}^{k}),\;\;\;\;\forall
k\in\mathbb{N},$ (14)
with equality achieved if $\operatorname{s}$ is independent of
$(\operatorname{p},\operatorname{q},\operatorname{r})$. $\blacktriangle$
This fundamental result, which for the cases in which
$\operatorname{s}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}(\operatorname{p},\operatorname{q},\operatorname{r})$
can be understood as a law of conservation of information flow, is illustrated
in Fig. 2. For such cases, the information causally conveyed from
$\operatorname{x}$ to $\operatorname{y}$ equals the information flow from
$(\operatorname{q},\operatorname{r},\operatorname{p})$ to $\operatorname{y}$.
When $(\operatorname{p},\operatorname{q},\operatorname{r})$ are not
independent of $\operatorname{s}$, part of the mutual information between
$(\operatorname{p},\operatorname{q},\operatorname{r})$ and $\operatorname{y}$
(corresponding to the term
$I(\operatorname{q}^{k},\operatorname{r}^{k},\operatorname{p}^{k}\to\operatorname{y}^{k}\parallel\operatorname{x}^{k})$)
can be thought of as being “leaked” through $\operatorname{s}$, thus bypassing
the forward link from $\operatorname{x}$ to $\operatorname{y}$. This provides
an intuitive interpretation for (14).
$\operatorname{r}$$\mathcal{S}_{1}$$\operatorname{u}$$\operatorname{p}$$\mathcal{S}_{2}$$\operatorname{e}$$\operatorname{q}$$\mathcal{S}_{4}$$\operatorname{x}$$\operatorname{s}$$\mathcal{S}_{3}$$\operatorname{y}$
Figure 2: The flow of information between exogenous signals
$(\operatorname{p},\operatorname{q},\operatorname{r})$ and the internal signal
$\operatorname{y}$ equals the directed information from $\operatorname{x}^{k}$
to $\operatorname{y}^{k}$ when $\operatorname{s}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}(\operatorname{p},\operatorname{q},\operatorname{r})$.
###### Remark 1
Theorem 1 implies that $I(\operatorname{x}^{k}\to\operatorname{y}^{k})$ is
only a part of (or at most equal to) the information “flow” between all the
exogenous signals entering the loop outside the link
$\operatorname{x}\to\operatorname{y}$ (namely
$(\operatorname{q},\operatorname{r},\operatorname{p})$), and
$\operatorname{y}$. In particular, if
$(\operatorname{p},\operatorname{q},\operatorname{r})$ were deterministic,
then $I(\operatorname{x}^{k}\to\operatorname{y}^{k})=0$, regardless of the
blocks $\mathcal{S}_{1},\ldots,\mathcal{S}_{4}$ and irrespective of the nature
of $\operatorname{s}$. $\blacktriangle$
###### Remark 2
By using (13),
$I(\operatorname{p}^{k},\operatorname{q}^{k},\operatorname{r}^{k};\operatorname{y}^{k})=I(\operatorname{r}^{k};\operatorname{y}^{k})+I(\operatorname{p}^{k},\operatorname{q}^{k};\operatorname{y}^{k}|\operatorname{r}^{k})$.
Then, applying Theorem 1, we recover (5), whenever
$\operatorname{s}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}(\operatorname{q},\operatorname{r},\operatorname{p})$.
Thus, [1, Theorem 3] and [6, Lemma 4.8.1]) can be obtained as a corollary of
Theorem 1. $\blacktriangle$
The following result provides an inequality relating
$I(\operatorname{x}^{k}\to\operatorname{y}^{k})$ with the separate flows of
information $I(\operatorname{r}^{k};\operatorname{y}^{k})$ and
$I(\operatorname{p}^{k},\operatorname{q}^{k}\,;\,\operatorname{y}^{k})$.
###### Theorem 2
For the system shown in Fig. 1-(a), if
$\operatorname{s}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}(\operatorname{p},\operatorname{q},\operatorname{r})$
and $\operatorname{r}^{k}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}(\operatorname{p}^{k},\operatorname{q}^{k})$,
then
$\displaystyle I(\operatorname{x}^{k}\to\operatorname{y}^{k})$
$\displaystyle\geq
I(\operatorname{r}^{k};\operatorname{y}^{k})+I(\operatorname{p}^{k},\operatorname{q}^{k}\,;\,\operatorname{y}^{k}).$
(15)
with equality if and only if the Markov chain
$(\operatorname{p}^{k},\operatorname{q}^{k})\leftrightarrow\operatorname{y}^{k}\leftrightarrow\operatorname{r}^{k}$
holds.
Theorem 2 shows that, provided
$(\operatorname{p},\operatorname{q},\operatorname{r})\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\operatorname{s}$,
$I(\operatorname{x}^{k}\to\operatorname{y}^{k})$ is lower bounded by the sum
of the individual flows from all the subsets in any given partition of
$(\operatorname{p}^{k},\operatorname{q}^{k},\operatorname{r}^{k})$, to
$\operatorname{y}^{k}$, provided these subsets are mutually independent.
Indeed, both theorems 1 and 2 can be generalized for any appropriate choice of
external and internal signals. More precisely, let $\Theta$ be the set of all
external signals in a feedback system. Let $\alpha$ and $\beta$ be two
internal signals in the loop. Define $\Theta_{\alpha,\beta}\subset\Theta$ as
the set of exogenous signals which are introduced to the loop at every
subsystem $\mathcal{S}_{i}$ that lies in the path going from $\alpha$ to
$\beta$. Thus, for any $\rho\in\Theta\setminus\Theta_{\alpha,\beta}$, if
$\Theta_{\alpha,\beta}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\Theta\setminus\Theta_{\alpha,\beta}$, we have
that (14) and (15) become
$\displaystyle I(\alpha\to\beta)$
$\displaystyle=I(\Theta\setminus\\{\Theta_{\alpha,\beta}\\};\beta),$ (16)
$\displaystyle I(\alpha\to\beta)-I(\rho;\beta)$ $\displaystyle\geq
I(\Theta\setminus\\{\rho\cup\Theta_{\alpha,\beta}\\};\beta),$ (17)
respectively.
To finish this section, we present a stronger, non-asymptotic version of
inequality (6):
###### Theorem 3
In the system shown in Fig. 1-(a), if
$(\operatorname{r},\operatorname{p},\operatorname{q},\operatorname{s})$ are
mutually independent, then
$\displaystyle I(\operatorname{x}^{k}\to\operatorname{y}^{k})$
$\displaystyle=I(\operatorname{r}^{k};\operatorname{u}^{k})+I(\operatorname{p}^{k};\operatorname{e}^{k})+I(\operatorname{q}^{k};\operatorname{y}^{k})+I(\operatorname{p}^{k};\operatorname{u}^{k}|\operatorname{e}^{k})+I(\operatorname{r}^{k},\operatorname{p}^{k};\operatorname{y}^{k}|\operatorname{u}^{k}).$
(18)
$\blacktriangle$
As anticipated, Theorem 3 can be seen as an extension of (6) to the more
general setup shown in Fig. 1-(a), where the assumptions made in [7, Lemma
4.1] do not need to hold. In particular, letting the decoder $D$ and
$\operatorname{x}_{0}$ in Fig. 1-(b) correspond to $\mathcal{S}_{4}$ and
$\operatorname{p}^{k}$ in Fig. 1-(a), respectively, we see that inequality
(15) holds even if $D$ and $E$ have dependent initial states, or if the
internal state of $D$ is not observable [19].
Theorem 3 also admits an interpretation in terms of information flows. This
can be appreciated in the diagram shown in Fig. 3, which depicts the
individual full-turn flows (around the entire feedback loop) stemming from
$\operatorname{q}$, $\operatorname{r}$ and $\operatorname{p}$. Theorem 3
states that the sum of these individual flows is a lower bound for the
directed information from $\operatorname{x}$ to $\operatorname{y}$, provided
$\operatorname{q},\operatorname{r},\operatorname{p},\operatorname{s}$ are
independent.
$\operatorname{r}$$\mathcal{S}_{1}$$\operatorname{u}$$\operatorname{p}$$\mathcal{S}_{2}$$\operatorname{e}$$\operatorname{q}$$\mathcal{S}_{4}$$\operatorname{x}$$\operatorname{s}$$\mathcal{S}_{3}$$\operatorname{y}$
Figure 3: A representation of the three first information flows on the right-
hand-side of (18).
### III-B Relationships Between Nested Directed Informations
This section presents three closed-loop versions of the data processing
inequality relating two directed informations, both between pairs of signals
internal to the loop. As already mentioned in Section I, to the best of our
knowledge, the first inequality of this type to appear in the literature is
the one in Theorem 4.1 in [9] (see (9)). Recall that the latter result stated
that
$I(\operatorname{x}^{k}\to\operatorname{y}^{k}\parallel\operatorname{q}^{k})\geq
I(\operatorname{x}^{k}\to\operatorname{u}^{k})$, requiring $\mathcal{S}_{4}$
to be such that $\operatorname{y}^{i}$ is a deterministic function of
$(\operatorname{u}^{i},\operatorname{q}^{i})$ and that
$\operatorname{q}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}(\operatorname{r},\operatorname{p})$. The
following result presents another inequality which also relates two nested
directed informations, namely,
$I(\operatorname{x}^{k}\to\operatorname{y}^{k})$ and
$I(\operatorname{e}^{k}\to\operatorname{y}^{k})$, but requiring only that
$\operatorname{s}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}(\operatorname{q},\operatorname{r},\operatorname{p})$.
###### Theorem 4
For the closed-loop system in Fig. 1-(b), if
$(\operatorname{q},\operatorname{r},\operatorname{p})\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\operatorname{s}$, then
$\displaystyle I(\operatorname{x}^{k}\to\operatorname{y}^{k})$
$\displaystyle\geq I(\operatorname{e}^{k}\to\operatorname{y}^{k}).$ (19)
$\blacktriangle$
Notice that Theorem 4 does not require $\operatorname{p}$ to be independent of
$\operatorname{r}$ or $\operatorname{q}$. This may seem counter-intuitive upon
noting that $\operatorname{p}$ enters the loop between the link from
$\operatorname{e}$ to $\operatorname{x}$.
The following theorem is an identity between two directed informations
involving only internal signals. It can also be seen as a complement to
Theorem 4, since it can be directly applied to establish the relationship
between $I(\operatorname{e}^{k}\to\operatorname{y}^{k})$ and
$I(\operatorname{e}^{k}\to\operatorname{u}^{k})$.
###### Theorem 5
For the system shown in Fig. 1-(a), if
$(\operatorname{q},\operatorname{s})\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}(\operatorname{r},\operatorname{p})$, then
$\displaystyle I(\operatorname{x}^{k}\to\operatorname{y}^{k})\leq
I(\operatorname{x}^{k}\to\operatorname{u}^{k})+I(\operatorname{q}^{k}\,;\,\operatorname{y}^{k})+I(\operatorname{r}^{k},\operatorname{p}^{k};\operatorname{y}^{k}|\operatorname{u}^{k})+I(\operatorname{q}^{k};\operatorname{r}^{k}|\operatorname{u}^{k},\operatorname{y}^{k}).$
(20)
with equality if, in addition, $\operatorname{q}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\operatorname{s}$. In the latter case, it
holds that
$\displaystyle
I(\operatorname{x}^{k}\to\operatorname{y}^{k})=I(\operatorname{x}^{k}\to\operatorname{u}^{k})+I(\operatorname{q}^{k}\,;\,\operatorname{y}^{k})+I(\operatorname{r}^{k},\operatorname{p}^{k};\operatorname{y}^{k}|\operatorname{u}^{k}).$
(21)
$\blacktriangle$
Notice that, by requiring additional independence conditions upon the
exogenous signals (specifically, $\operatorname{q}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\operatorname{s}$), Theorem 5 (and, in
particular, (21)) yields
$\displaystyle I(\operatorname{x}^{k}\to\operatorname{y}^{k})\geq
I(\operatorname{x}^{k}\to\operatorname{u}^{k}),$ (22)
which strengthens the inequality in [9, Theorem 4.1] (stated above in (9)).
More precisely, (22) does not require conditioning one of the directed
informations and holds irrespective of the invertibility of the mappings in
the loop.
A closer counterpart of (9) (i.e., of [9, Theorem 4.1]), involving
$I(\operatorname{x}^{k}\to\operatorname{y}^{k}\parallel\operatorname{q}^{k})$,
is presented next.
###### Theorem 6
For the system shown in Fig. 1-(a), if
$(\operatorname{q},\operatorname{s})\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}(\operatorname{r},\operatorname{p})$, then
$\displaystyle
I(\operatorname{x}^{k}\to\operatorname{y}^{k}|\operatorname{q}^{k})=I(\operatorname{x}^{k}\to\operatorname{u}^{k})+I(\operatorname{r}^{k},\operatorname{p}^{k};\operatorname{y}^{k}|\operatorname{u}^{k})+I(\operatorname{q}^{k};\operatorname{r}^{k}|\operatorname{u}^{k},\operatorname{y}^{k})\overset{(\dagger)}{=}I(\operatorname{x}^{k}\to\operatorname{y}^{k}\parallel\operatorname{q}^{k}).$
(23)
where the equality labeled $(\dagger)$ hods if, in addition, the Markov chain
$\displaystyle\operatorname{q}_{i+1}^{k}\longleftrightarrow\operatorname{q}^{i}\longleftrightarrow\operatorname{s}^{i}$
(24)
is satisfied for all $i\in\\{1,\ldots,k\\}$. $\blacktriangle$
Thus, provided $(\operatorname{q},\operatorname{s})\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}(\operatorname{r},\operatorname{p})$, (23)
yields that (9) holds regardless of the invertibility of $\mathcal{S}_{4}$,
requiring instead that, for all $i\in\\{1,\ldots,k\\}$, any statistical
dependence between $\operatorname{q}^{k}$ and $\operatorname{s}^{i}$ resides
only in $\operatorname{q}^{i}$ (i.e., that Markov chain (24) holds).
The results derived so far relate directed informations having either the same
“starting” sequence or the same “destination” sequence. We finish this section
with the following corollary, which follows directly by combining theorems 4
and 5 and relates directed informations involving four different sequences
internal to the loop.
###### Corollary 1 (Full Closed-Loop Directed Data Processing Inequality)
For the system shown in Fig. 1-(a), if
$(\operatorname{q},\operatorname{s})\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}(\operatorname{r},\operatorname{p})$ and
$\operatorname{q}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\operatorname{s}$, then
$\displaystyle
I(\operatorname{x}^{k}\to\operatorname{y}^{k})\overset{(a)}{\geq}I(\operatorname{e}^{k}\to\operatorname{u}^{k})+I(\operatorname{q}^{k}\,;\,\operatorname{y}^{k})+I(\operatorname{r}^{k};\operatorname{y}^{k}|\operatorname{u}^{k})\geq
I(\operatorname{e}^{k}\to\operatorname{u}^{k}).$ (25)
Equality holds in $(a)$ if, in addition,
$\operatorname{r}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\operatorname{p}$ (i.e., if
$(\operatorname{q},\operatorname{r},\operatorname{p},\operatorname{s})$ are
mutually independent). $\blacktriangle$
To the best of our knowledge, Corollary 1 is the first result available in the
literature providing a lower bound to the gap between two nested directed
informations, involving four different signals inside the feedback loop. This
result can be seen as the first full extension of the open-loop (traditional)
data-processing inequality, to arbitrary closed-loop scenarios. (Notice that
there is no need to consider systems with more than four mappings, since all
external signals entering the loop between a given pair of internal signals
can be regarded as exogenous inputs to a single equivalent deterministic
mapping.)
## IV Proofs
We start with the proof of Theorem 1.
###### Proof:
It is clear from Fig. 1-(a) and from (11) that the relationship between
$\operatorname{r}$, $\operatorname{p}$, $\operatorname{q}$,
$\operatorname{s}$, $\operatorname{x}$ and $\operatorname{y}$ can be
represented by the diagram shown in Fig. 4.
$q^{i}$$p^{i}$$r^{i}$$\theta^{i}$$s^{i}$$y(i)$$x_{i+1-d_{3}}^{i-1+d_{1}+d_{2}+d_{4}}$$x^{i-d_{3}}$$y^{i-1}$
Figure 4: Representation of the system of Fig. 1-(b) highlighting the
dependency between $p$, $q$, $r$, $s$, $x$ and $y$. The dependency on $i$ of
the delays $d_{1}(i),\ldots,d_{4}(i)$ is omitted for clarity.
From this diagram and Lemma 1 (in the appendix) it follows that if
$\operatorname{s}$ is independent of
$(\operatorname{r},\operatorname{p},\operatorname{q})$, then the following
Markov chain holds:
$\displaystyle\operatorname{y}(i)$
$\displaystyle\longleftrightarrow(\operatorname{x}^{i-d_{3}(i)},\operatorname{y}^{i-1})\longleftrightarrow(\operatorname{p}^{i},\operatorname{q}^{i},\operatorname{r}^{i}).$
(26)
Denoting the triad of exogenous signals
$\operatorname{p}^{k},\operatorname{q}^{k},\operatorname{r}^{k}$ by
$\displaystyle\theta^{k}\triangleq(\operatorname{p}^{k},\operatorname{q}^{k},\operatorname{r}^{k}),$
(27)
we have the following
$\displaystyle I(\operatorname{x}^{k}\to\operatorname{y}^{i})$
$\displaystyle=\sum\limits_{i=1}^{k}I(\operatorname{y}(i);\operatorname{x}^{i-d_{3}(i)}|\operatorname{y}^{i-1})$
$\displaystyle\overset{\eqref{eq:chainrule_I}}{=}\sum\limits_{i=1}^{k}\left[I(\theta^{i},\operatorname{x}^{i-d_{3}(i)};\operatorname{y}(i)|\operatorname{y}^{i-1})-I(\theta^{i};\operatorname{y}(i)|\operatorname{x}^{i-d_{3}(i)},\operatorname{y}^{i-1})\right]$
$\displaystyle\overset{(a)}{=}\sum\limits_{i=1}^{k}\left[I(\theta^{i};\operatorname{y}(i)|\operatorname{y}^{i-1})-I(\theta^{i};\operatorname{y}(i)|\operatorname{x}^{i-d_{3}(i)},\operatorname{y}^{i-1})\right]$
(28a)
$\displaystyle\overset{(b)}{\leq}\sum\limits_{i=1}^{k}I(\theta^{i};\operatorname{y}(i)|\operatorname{y}^{i-1})\overset{(c)}{\leq}\sum\limits_{i=1}^{k}I(\theta^{k};\operatorname{y}(i)|\operatorname{y}^{i-1})$
(28b) $\displaystyle=I(\theta^{k};\operatorname{y}^{k}).$ (28c)
In the above, $(a)$ follows from the fact that, if $\operatorname{y}^{i-1}$ is
known, then $\operatorname{x}^{i-d_{3}(i)}$ is a deterministic function of
$\theta^{i}$. The resulting sums on the right-hand side of (28a) correspond to
$I(\operatorname{q}^{k},\operatorname{r}^{k},\operatorname{p}^{k}\to\operatorname{y}^{k})-I(\operatorname{q}^{k},\operatorname{r}^{k},\operatorname{p}^{k}\to\operatorname{y}^{k}\parallel\operatorname{x}^{k})$,
and thereby proving the first part of the theorem, i.e., the equality in (14).
In turn, $(b)$ stems from the non-negativity of mutual informations, turning
into equality if $\operatorname{s}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}(\operatorname{r},\operatorname{p},\operatorname{q})$,
as a direct consequence of the Markov chain in (26). Finally, equality holds
in $(c)$ if $\operatorname{s}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}(\operatorname{q},\operatorname{r},\operatorname{p})$,
since $\operatorname{y}$ depends causally upon $\theta$. This shows that
equality in (14) is achieved if $\operatorname{s}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}(\operatorname{q},\operatorname{r},\operatorname{p})$,
completing the proof. ∎
###### Proof:
Apply the chain-rule identity (13) to the RHS of (14) to obtain
$\displaystyle
I(\theta^{k};\operatorname{y}^{k})=I(\operatorname{p}^{k},\operatorname{q}^{k},\operatorname{r}^{k};\operatorname{y}^{k})=I(\operatorname{p}^{k},\operatorname{q}^{k};\operatorname{y}^{k}|\operatorname{r}^{k})+I(\operatorname{r}^{k};\operatorname{y}^{k}).$
(29)
Now, applying (13) twice, one can express the term
$I(\operatorname{p}^{k},\operatorname{q}^{k};\operatorname{y}^{k}|\operatorname{r}^{k})$
as follows:
$\begin{split}I(\operatorname{p}^{k},\operatorname{q}^{k};\operatorname{y}^{k}|\operatorname{r}^{k})&=I(\operatorname{p}^{k},\operatorname{q}^{k}\,;\,\operatorname{y}^{k},\operatorname{r}^{k})-I(\operatorname{p}^{k},\operatorname{q}^{k};\operatorname{r}^{k})=I(\operatorname{p}^{k},\operatorname{q}^{k}\,;\,\operatorname{y}^{k},\operatorname{r}^{k})\\\
&=I(\operatorname{p}^{k},\operatorname{q}^{k};\operatorname{y}^{k})+I(\operatorname{p}^{k},\operatorname{q}^{k};\operatorname{r}^{k}|\operatorname{y}^{k}),\end{split}$
(30)
where the second equality follows since
$(\operatorname{p}^{k},\operatorname{q}^{k})\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\operatorname{r}^{k}$. The result then follows
directly by combining (30) with (29) and (14). ∎
###### Proof:
Since $\operatorname{q}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}(\operatorname{r},\operatorname{p},\operatorname{s})$,
$\displaystyle I(\operatorname{x}^{k}\to\operatorname{y}^{k})$
$\displaystyle\overset{(a)}{=}I(\operatorname{x}^{k}\to\operatorname{u}^{k})+I(\operatorname{q}^{k};\operatorname{y}^{k})+I(\operatorname{r}^{k},\operatorname{p}^{k};\operatorname{y}^{k}|\operatorname{u}^{k})$
(31)
$\displaystyle\overset{(b)}{=}I(\operatorname{r}^{k},\operatorname{p}^{k};\operatorname{u}^{k})+I(\operatorname{q}^{k};\operatorname{y}^{k})+I(\operatorname{r}^{k},\operatorname{p}^{k};\operatorname{y}^{k}|\operatorname{u}^{k})$
(32)
$\displaystyle\overset{(c)}{=}I(\operatorname{r}^{k};\operatorname{u}^{k})+I(\operatorname{p}^{k};\operatorname{u}^{k}|\operatorname{r}^{k})+I(\operatorname{q}^{k};\operatorname{y}^{k})+I(\operatorname{r}^{k},\operatorname{p}^{k};\operatorname{y}^{k}|\operatorname{u}^{k}),$
(33)
where $(a)$ is due to Theorem 5, $(b)$ follows from Theorem 1 and the fact
that $(\operatorname{s},\operatorname{q})\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}(\operatorname{r},\operatorname{p})$ and $(c)$
from the chain rule of mutual information. For the second term on the RHS of
the last equation, we have
$\displaystyle
I(\operatorname{p}^{k};\operatorname{u}^{k}|\operatorname{r}^{k})$
$\displaystyle\overset{(a)}{=}I(\operatorname{p}^{k};\operatorname{u}^{k}|\operatorname{r}^{k})+I(\operatorname{p}^{k};\operatorname{r}^{k})=I(\operatorname{p}^{k};\operatorname{r}^{k},\operatorname{u}^{k})$
(34)
$\displaystyle\overset{(b)}{=}I(\operatorname{p}^{k};\operatorname{r}^{k},\operatorname{u}^{k},\operatorname{e}^{k})-I(\operatorname{p}^{k};\operatorname{e}^{k}|\operatorname{r}^{k},\operatorname{u}^{k})$
(35)
$\displaystyle\overset{(c)}{=}I(\operatorname{p}^{k};\operatorname{r}^{k},\operatorname{u}^{k},\operatorname{e}^{k})$
(36)
$\displaystyle\overset{(d)}{=}I(\operatorname{p}^{k};\operatorname{e}^{k})+I(\operatorname{p}^{k};\operatorname{r}^{k},\operatorname{u}^{k}|\operatorname{e}^{k})$
(37)
$\displaystyle\overset{(e)}{=}I(\operatorname{p}^{k};\operatorname{e}^{k})+I(\operatorname{p}^{k};\operatorname{u}^{k}|\operatorname{e}^{k})+I(\operatorname{p}^{k};\operatorname{r}^{k}|\operatorname{u}^{k},\operatorname{e}^{k})$
(38)
$\displaystyle\overset{(f)}{=}I(\operatorname{p}^{k};\operatorname{e}^{k})+I(\operatorname{p}^{k};\operatorname{u}^{k}|\operatorname{e}^{k}),$
(39)
where $(a)$ holds since $\operatorname{r}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\operatorname{p}$, $(b)$, $(d)$ and $(e)$ stem
from the chain rule of mutual information (13), and $(c)$ is a consequence of
the Markov chain
$\operatorname{e}^{k}\leftrightarrow(\operatorname{u}^{k},\operatorname{r}^{k})\leftrightarrow\operatorname{p}^{k}$
which is due to the fact that
$\operatorname{e}^{k}=\mathcal{S}_{1}(\operatorname{u}^{k-d_{1}(k)},\operatorname{r}^{k})$.
Finally, $(f)$ is due to the Markov chain
$\operatorname{r}^{k}\leftrightarrow(\operatorname{u}^{k},\operatorname{e}^{k})\leftrightarrow\operatorname{p}^{k}$,
which holds because $\operatorname{r}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}(\operatorname{p},\operatorname{s},\operatorname{q})$
as a consequence of Lemma 1 in the appendix (see also Fig. 1-(a)).
Substitution of (39) into (33) yields (18), thereby completing the proof. ∎
###### Proof:
Since
$(\operatorname{p},\operatorname{q},\operatorname{r})\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\operatorname{s}$, we can apply (5) (where now
$(\operatorname{q},\operatorname{r})$ plays the role of $\operatorname{r}$),
and obtain
$\displaystyle I(\operatorname{x}^{k}\to\operatorname{y}^{k})\geq
I(\operatorname{q}^{k},\operatorname{r}^{k};\operatorname{y}^{k}).$ (40)
Now, we apply Theorem 1, which gives
$\displaystyle
I(\operatorname{q}^{k},\operatorname{r}^{k};\operatorname{y}^{k})\geq
I(\operatorname{e}^{k}\to\operatorname{y}^{k}),$ (41)
completing the proof. ∎
###### Proof:
Applying Theorem 1, since
$(\operatorname{r},\operatorname{p})\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}(\operatorname{s},\operatorname{q})$,
$\displaystyle
I(\operatorname{x}^{k}\to\operatorname{u}^{k})=I(\operatorname{r}^{k},\operatorname{p}^{k}\,;\,\operatorname{u}^{k}).$
(42)
For the other directed information, we have that
$\displaystyle I(\operatorname{x}^{k}\to\operatorname{y}^{k})$
$\displaystyle\overset{(a)}{\leq}I(\operatorname{r}^{k},\operatorname{p}^{k},\operatorname{q}^{k}\,;\,\operatorname{y}^{k})$
$\displaystyle\overset{\eqref{eq:chainrule_I}}{=}I(\operatorname{q}^{k}\,;\,\operatorname{y}^{k})+I(\operatorname{r}^{k},\operatorname{p}^{k}\,;\,\operatorname{y}^{k}|\operatorname{q}^{k})$
(43)
$\displaystyle\overset{\eqref{eq:chainrule_I}}{=}I(\operatorname{q}^{k}\,;\,\operatorname{y}^{k})+I(\operatorname{r}^{k},\operatorname{p}^{k}\,;\,\operatorname{u}^{k},\operatorname{y}^{k}|\operatorname{q}^{k})-I(\operatorname{r}^{k},\operatorname{p}^{k};\operatorname{u}^{k}|\operatorname{q}^{k},\operatorname{y}^{k})$
$\displaystyle\overset{(b)}{=}I(\operatorname{q}^{k}\,;\,\operatorname{y}^{k})+I(\operatorname{r}^{k},\operatorname{p}^{k}\,;\,\operatorname{u}^{k},\operatorname{y}^{k}|\operatorname{q}^{k})$
$\displaystyle\overset{\eqref{eq:chainrule_I}}{=}I(\operatorname{q}^{k}\,;\,\operatorname{y}^{k})+I(\operatorname{r}^{k},\operatorname{p}^{k}\,;\,\operatorname{u}^{k},\operatorname{y}^{k},\operatorname{q}^{k})-I(\operatorname{r}^{k},\operatorname{p}^{k};\operatorname{q}^{k})$
$\displaystyle\overset{\eqref{eq:chainrule_I}}{=}I(\operatorname{q}^{k}\,;\,\operatorname{y}^{k})+I(\operatorname{r}^{k},\operatorname{p}^{k}\,;\,\operatorname{u}^{k})+I(\operatorname{r}^{k},\operatorname{p}^{k};\operatorname{y}^{k},\operatorname{q}^{k}|\operatorname{u}^{k})-I(\operatorname{r}^{k},\operatorname{p}^{k};\operatorname{q}^{k})$
$\displaystyle\overset{(c)}{\leq}I(\operatorname{q}^{k}\,;\,\operatorname{y}^{k})+I(\operatorname{r}^{k},\operatorname{p}^{k}\,;\,\operatorname{u}^{k})+I(\operatorname{r}^{k},\operatorname{p}^{k};\operatorname{y}^{k},\operatorname{q}^{k}|\operatorname{u}^{k})$
$\displaystyle\overset{\eqref{eq:chainrule_I}}{=}I(\operatorname{q}^{k}\,;\,\operatorname{y}^{k})+I(\operatorname{r}^{k},\operatorname{p}^{k}\,;\,\operatorname{u}^{k})+I(\operatorname{r}^{k},\operatorname{p}^{k};\operatorname{y}^{k}|\operatorname{u}^{k})+I(\operatorname{q}^{k};\operatorname{r}^{k}|\operatorname{u}^{k},\operatorname{y}^{k})$
(44)
$\displaystyle\overset{(d)}{\leq}I(\operatorname{q}^{k}\,;\,\operatorname{y}^{k})+I(\operatorname{r}^{k},\operatorname{p}^{k}\,;\,\operatorname{u}^{k})+I(\operatorname{r}^{k},\operatorname{p}^{k};\operatorname{y}^{k}|\operatorname{u}^{k}),$
(45)
where $(a)$ follows from Theorem 1, which also states that equality is reached
if and only if
$(\operatorname{r},\operatorname{p},\operatorname{q})\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\operatorname{s}$. In turn, $(b)$ is due to
the fact that $\operatorname{u}^{k}$ is a deterministic function of
$\operatorname{q}^{k},\operatorname{y}^{k}$. Equality $(c)$ holds if and only
if $(r,\operatorname{p})\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\operatorname{q}$. Finally, from Lemma 1 (in
the appendix), $(d)$ turns into equality if
$\operatorname{q}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}(\operatorname{r},\operatorname{p},\operatorname{s})$.
Substitution of (42) into (45) yields (21), completing the proof. ∎
###### Proof:
We begin with the second part of the theorem, proving the validity of the
equality $(\dagger)$ in (23). We have the following:
$\displaystyle
I(\operatorname{x}^{k}\to\operatorname{y}^{k}\parallel\operatorname{q}^{k})$
$\displaystyle=\sum\limits_{i=1}^{k}I(\operatorname{y}(i);\operatorname{x}^{i-d_{3}(i)}|\operatorname{y}^{i-1},\operatorname{q}^{i})$
(46)
$\displaystyle\overset{\eqref{eq:chainrule_I}}{=}\sum\limits_{i=1}^{k}\left[I(\operatorname{r}^{i},\operatorname{p}^{i},\operatorname{x}^{i-d_{3}(i)};\operatorname{y}(i)|\operatorname{y}^{i-1},\operatorname{q}^{i})-I(\operatorname{r}^{i},\operatorname{p}^{i};\operatorname{y}(i)|\operatorname{x}^{i-d_{3}(i)},\operatorname{y}^{i-1},\operatorname{q}^{i})\right]$
(47)
$\displaystyle\overset{(a)}{\leq}\sum\limits_{i=1}^{k}I(\operatorname{r}^{i},\operatorname{p}^{i},\operatorname{x}^{i-d_{3}(i)};\operatorname{y}(i)|\operatorname{y}^{i-1},\operatorname{q}^{i})$
(48)
$\displaystyle\overset{(b)}{=}\sum\limits_{i=1}^{k}I(\operatorname{r}^{i},\operatorname{p}^{i};\operatorname{y}(i)|\operatorname{y}^{i-1},\operatorname{q}^{i})$
(49)
$\displaystyle\overset{\eqref{eq:chainrule_I}}{=}\sum\limits_{i=1}^{k}\left[I(\operatorname{r}^{i},\operatorname{p}^{i},\operatorname{q}_{i+1}^{k};\operatorname{y}(i)|\operatorname{y}^{i-1},\operatorname{q}^{i})-I(\operatorname{q}_{i+1}^{k};\operatorname{y}(i)|\operatorname{y}^{i-1},\operatorname{q}^{i},\operatorname{r}^{i},\operatorname{p}^{i})\right]$
(50)
$\displaystyle\overset{(c)}{=}\sum\limits_{i=1}^{k}\left[I(\operatorname{r}^{i},\operatorname{p}^{i},\operatorname{q}_{i+1}^{k};\operatorname{y}(i)|\operatorname{y}^{i-1},\operatorname{q}^{i})\right]$
(51)
$\displaystyle\overset{\eqref{eq:chainrule_I}}{=}\sum\limits_{i=1}^{k}\left[I(\operatorname{r}^{i},\operatorname{p}^{i};\operatorname{y}(i)|\operatorname{y}^{i-1},\operatorname{q}^{k})+I(\operatorname{q}_{i+1}^{k};\operatorname{y}(i)|\operatorname{y}^{i-1},\operatorname{q}^{i})\right]$
(52)
$\displaystyle\overset{(d)}{=}\sum\limits_{i=1}^{k}I(\operatorname{r}^{i},\operatorname{p}^{i};\operatorname{y}(i)|\operatorname{y}^{i-1},\operatorname{q}^{k})$
(53)
$\displaystyle\overset{(e)}{\leq}\sum\limits_{i=1}^{k}I(\operatorname{r}^{k},\operatorname{p}^{k};\operatorname{y}(i)|\operatorname{y}^{i-1},\operatorname{q}^{k})=I(\operatorname{r}^{k},\operatorname{p}^{k};\operatorname{y}^{i}|\operatorname{q}^{k})$
(54)
where equality holds in $(a)$ if and only if the Markov chain
$\operatorname{s}^{i}\leftrightarrow\operatorname{q}^{i}\leftrightarrow(\operatorname{r}^{i},\operatorname{p}^{i})$
holds for all $i\in\\{1,\ldots,k\\}$ (as a straightforward extension of Lemma
1). In our case, the latter Markov chain holds since we are assuming
$(\operatorname{q}^{k},\operatorname{s}^{k})\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}(\operatorname{r}^{k},\operatorname{p}^{k})$.
In turn, $(b)$ stems from the fact that, for all $i\in\\{1,\ldots,k\\}$,
$\operatorname{x}^{i-d_{3}(i)}$ is a function of
$\operatorname{y}^{i-1},\operatorname{q}^{i},\operatorname{r}^{i},\operatorname{p}^{i}$.
To prove $(c)$, we resort to (13) and write
$\displaystyle
I(\operatorname{q}_{i+1}^{k};\operatorname{y}(i)|\operatorname{y}^{i-1},\operatorname{q}^{i},\operatorname{r}^{i},\operatorname{p}^{i})$
$\displaystyle=I(\operatorname{q}_{i+1}^{k};\operatorname{y}^{i},\operatorname{r}^{i},\operatorname{p}^{i}|\operatorname{q}^{i})-I(\operatorname{q}_{i+1}^{k};\operatorname{y}^{i-1},\operatorname{r}^{i},\operatorname{p}^{i}|\operatorname{q}^{i})$
(55)
From the definitions of the blocks (in (11)), it can be seen that, given
$\operatorname{q}^{i}$, the triad of random sequences
$(\operatorname{y}^{i},\operatorname{r}^{i},\operatorname{p}^{i})$ is a
deterministic function of (at most)
$(\operatorname{s}^{i},\operatorname{r}^{i},\operatorname{p}^{i})$. Recalling
that $(\operatorname{q}^{k},\operatorname{s}^{k})\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}(\operatorname{r},\operatorname{p}^{k})$ and
that
$\operatorname{q}_{i+1}^{k}\leftrightarrow\operatorname{q}^{i}\leftrightarrow\operatorname{s}^{i}$
(see (24)), it readily follows that
$\operatorname{q}_{i+1}^{k}\leftrightarrow\operatorname{q}^{i}\leftrightarrow(\operatorname{r}^{i},\operatorname{p}^{i},\operatorname{s}^{i})$,
and thus each of the mutual informations on the right-hand-side of (55) is
zero. To verify the validity of $(d)$, we use (13) and obtain
$\displaystyle
I(\operatorname{q}_{i+1}^{k};\operatorname{y}(i)|\operatorname{y}^{i-1},\operatorname{q}^{i})=I(\operatorname{q}_{i+1}^{k};\operatorname{y}^{i}|\operatorname{q}^{i})-I(\operatorname{q}_{i+1}^{k};\operatorname{y}^{i-1}|\operatorname{q}^{i}),$
(56)
where $(d)$ now follows since $0\leq
I(\operatorname{q}_{i+1}^{k};\operatorname{y}^{i-1}|\operatorname{q}^{i})\leq
I(\operatorname{q}_{i+1}^{k};\operatorname{y}^{i}|\operatorname{q}^{i})\leq
I(\operatorname{q}_{i+1}^{k};\operatorname{y}^{i},\operatorname{r}^{i},\operatorname{p}^{i}|\operatorname{q}^{i})$,
where the last term in this chain of inequalities was shown to be zero in the
proof of $(d)$. Equality holds in $(e)$ if and only if
$(\operatorname{r}^{k},\operatorname{p}^{k})\leftrightarrow(\operatorname{r}^{i},\operatorname{p}^{i},\operatorname{q}^{i},\operatorname{y}^{i-1})\leftrightarrow\operatorname{y}(i)$,
a Markov chain which is satisfied in our case from the fact that
$(\operatorname{q},\operatorname{s})\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}(\operatorname{r},\operatorname{p})$ and from
Lemma 1.
Finally, since
$(\operatorname{r}^{k},\operatorname{p}^{k})\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}(\operatorname{q}^{k},\operatorname{s}^{k})$,
we have that the chain of equalities from (43) to (44) holds, from which we
conclude that
$\displaystyle
I(\operatorname{r}^{k},\operatorname{p}^{k};\operatorname{y}^{i}|\operatorname{q}^{k})=I(\operatorname{r}^{k},\operatorname{p}^{k}\,;\,\operatorname{u}^{k})+I(\operatorname{r}^{k},\operatorname{p}^{k};\operatorname{y}^{k}|\operatorname{u}^{k})+I(\operatorname{q}^{k};\operatorname{r}^{k}|\operatorname{u}^{k},\operatorname{y}^{k}).$
(57)
Inserting this result into (54) and invoking Theorem 1 we arrive at equality
$(\dagger)$ in (23).
To prove the first equality the (23), it suffices to notice that
$I(\operatorname{x}^{k}\to\operatorname{y}^{k}|\operatorname{q}^{k})$
corresponds to the sum on the right-hand-side of (53), from where we proceed
as with the first part. This completes the proof of the theorem. ∎
## V Potential Applications
Information inequalities and, in particular, the data-processing inequality,
have played a fundamental role in Information Theory and its applications [20,
21, 22, 23, 24, 25, 26, 27]. It is perhaps the lack of a similar body of
results associated with the directed information (and with non-asymptotic,
causal information transmission) which has limited the extension of many
important information-theoretic ideas and insights to situations involving
feedback or causality constraints [28, 5]. Two such areas, already mentioned
in this paper, are the understanding of the fundamental limitations arising in
networked control systems over noiseless digital channels, and causal rate
distortion problems. In those contexts, causality is of paramount relevance an
thus the directed information appears, naturally, as the appropriate measure
of information flow (see, for example, [9, 5, 11, 29, 30] and [7]). We believe
that our results might help gaining insights into the fundamental trade-offs
underpinning those problems, and might also allow for the solution of open
problems such as, for instance, characterizing the minimal average data-rate
that guarantees a given performance level [10] (an improved version of the
latter paper, which extensively uses the results derived here, is currently
under preparation by the authors). On a different vein, directed mutual
information plays a role akin to that of (standard) mutual information when
characterizing channel feedback capacity (see, e.g., [3, 4] and the references
therein). Our results may also play a role in expanding the understanding of
communication problems over channels used with feedback, particularly when
including in the analysis additional exogenous signals such as a random
channel state, interference and, in general, any form of side information.
Thus, we hope that the inequalities and identities presented in Section III
may help in extending results such as dirty-paper coding [31], watermarking
[32], distributed source coding [33, 34, 25, 26], multi-terminal coding [35,
36], and data encryption [37], to scenarios involving causal feedback.
## VI Conclusions
In this paper, we have derived fundamental relations between mutual and
directed informations in general discrete-time systems with feedback. The
first of these is an inequality between the directed information between to
signals inside the feedback loop and the mutual information involving a subset
of all the exogenous incoming signals. The latter result can be interpreted as
a law of conservation of information flows for closed-loop systems. Crucial to
establishing these bounds was the repeated use of chain rules for conditional
mutual information as well as the development of new Markov chains. The proof
techniques do not rely upon properties of entropies or distributions, and the
results hold in very general cases including non-linear, time-varying and
stochastic systems with arbitrarily distributed signals. Indeed, the only
restriction is that all blocks within the system must be causal mappings, and
that their combined delay must be at least one sample. A new generalized data
processing inequality was also proved, which is valid for nested directed
informations within the loop. A key insight to be gained from this inequality
was that the further apart the signals are in the loop, the lower is the
directed information between them. This closely resembles the behavior of
mutual information in open loop systems, where it is well known that any
independent processing of the signals can only reduce their mutual
information.
## VII Appendix
###### Lemma 1
In the system shown in Fig. 5, the exogenous signals
$\operatorname{r},\operatorname{q}$ are mutually independent and
$\mathcal{S}_{1},\mathcal{S}_{2}$ are deterministic (possibly time-varying)
causal maps characterized by
$\operatorname{y}^{i}=\mathcal{S}_{1}(\operatorname{r}^{i},\operatorname{u}^{i})$,
$\operatorname{u}^{i}=\mathcal{S}_{2}(\operatorname{q}^{i},\operatorname{y}^{i})$,
$\forall i\in\\{1,\ldots,k\\}$, for some $k\subset\mathbb{N}$.
$\mathcal{S}_{1}$$r$$\mathcal{S}_{2}$$q$$u$$y$
Figure 5: Two arbitrary causal systems $\mathcal{S}_{1},\mathcal{S}_{2}$
interconnected in a feedback loop. The exogenous signals
$\operatorname{r},\operatorname{q}$ are mutually independent.
For this system, the following Markov chain holds
$\displaystyle\operatorname{r}^{k}\longleftrightarrow(\operatorname{u}^{k},\operatorname{y}^{k})\longleftrightarrow\operatorname{q}^{k},\;\;\;\;\forall
k\in\mathbb{K}.$ (58)
###### Proof:
Since
$\operatorname{y}^{k}=\mathcal{S}_{1}(\operatorname{r}^{k},\operatorname{u}^{k})$
and
$\operatorname{u}^{k}=\mathcal{S}_{2}(\operatorname{q}^{k},\operatorname{y}^{k})$
are deterministic functions, it follows that for every possible pair of
sequences $y^{k},u^{k}$, the sets
$\rho_{y^{k},u^{k}}\triangleq\\{r^{k}:y^{k}=\mathcal{S}_{1}(r^{k},u^{k})\\}$
and
$\phi_{y^{k},u^{k}}\triangleq\\{q^{k}:u^{k}=\mathcal{S}_{2}(q^{k},y^{k})\\}$
are also deterministic. Thus,
$(\operatorname{u}^{k},\operatorname{y}^{k})=(u^{k},y^{k})\iff\operatorname{r}^{k}\in\rho_{y^{k},u^{k}}$
and
$(\operatorname{u}^{k},\operatorname{y}^{k})=(u^{k},y^{k})\iff\operatorname{q}^{k}\in\phi_{y^{k},u^{k}}$.
This means that for every pair of Borel sets $(R,Q)$ of appropriate
dimensions,
$\displaystyle\Pr\\{\operatorname{r}^{k}\in R$
$\displaystyle,\operatorname{q}^{k}\in
Q|\operatorname{y}^{k}=y^{k},\operatorname{u}^{k}=u^{k}\\}$
$\displaystyle\overset{\hphantom{(a)}}{=}\Pr\\{\operatorname{r}^{k}\in
R,\operatorname{q}^{k}\in
Q|\operatorname{r}^{k}\in\rho_{y^{k},u^{k}}\,,\,\operatorname{q}^{k}\in\phi_{y^{k},u^{k}}\\}$
$\displaystyle\overset{\hphantom{(a)}}{=}\Pr\\{\operatorname{r}^{k}\in
R|\operatorname{r}^{k}\in\rho_{y^{k},u^{k}}\,,\,\operatorname{q}^{k}\in\phi_{y^{k},u^{k}}\\}\Pr\\{\operatorname{q}^{k}\in
Q|\operatorname{r}^{k}\in(\rho_{y^{k},u^{k}}\cap
R)\,,\,\operatorname{q}^{k}\in\phi_{y^{k},u^{k}}\\}$
$\displaystyle\overset{(a)}{=}\Pr\\{\operatorname{r}^{k}\in
R|\operatorname{r}^{k}\in\rho_{y^{k},u^{k}}\\}\Pr\\{\operatorname{q}^{k}\in
Q|\operatorname{q}^{k}\in\phi_{y^{k},u^{k}}\\}$
$\displaystyle\overset{\hphantom{(a)}}{=}\Pr\\{\operatorname{r}^{k}\in
R|\operatorname{y}^{k}=y^{k},\operatorname{u}^{k}=u^{k}\\}\Pr\\{\operatorname{q}^{k}\in
Q|\operatorname{y}^{k}=y^{k},\operatorname{u}^{k}=u^{k}\\},$
where $(a)$ follows from the fact that
$\operatorname{r}^{k}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\operatorname{q}^{k}$. This completes the
proof. ∎
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|
arxiv-papers
| 2013-01-28T02:00:32 |
2024-09-04T02:49:40.825738
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Milan S. Derpich, Eduardo I. Silva and Jan {\\O}stergaard",
"submitter": "Milan Derpich",
"url": "https://arxiv.org/abs/1301.6427"
}
|
1301.6477
|
# Statistical features of freely decaying two-dimensional hydrodynamic
turbulence
A.N. Kudryavtseva,b, E.A. Kuznetsova,c,d, E.V. Sereshchenkoa,b,e
[email protected] a Novosibirsk State University, 630090 Novosibirsk, Russia
b Khristianovich Institute of Theoretical and Applied Mechanics, SB RAS,
630090 Novosibirsk, Russia
c Lebedev Physical Institute, RAS, 119991 Moscow, Russia
d Landau Institute for Theoretical Physics, RAS, 119334 Moscow, Russia
e Far-Eastern Federal University, 690091 Vladivostok, Russia
###### Abstract
Statistical characteristics of freely decaying two-dimensional hydrodynamic
turbulence at high Reynolds numbers are numerically studied. In particular,
numerical experiments (with resolution up to $8192\times 8192$) provide a
Kraichnan-type turbulence spectrum $E_{k}\sim k^{-3}$. By means of spatial
filtration, it is found that the main contribution to the spectrum comes from
the sharp vorticity gradients in the form of quasi-shocks. Such quasi-
singularities are responsible for a strong angular dependence of the spectrum
owing to well-localized (in terms of the angle) jets with minor and/or large
overlapping. In each jet, the spectrum decreases as $k^{-3}$. The behavior of
the third-order structure function accurately agrees with Kraichnan direct
cascade concept corresponding to a constant enstrophy flux. It is shown that
the power law exponents $\zeta_{n}$ for higher structure functions grow more
slowly than the linear dependence of $n$, which testifies to turbulence
intermittency.
PACS: 52.30.Cv, 47.65.+a, 52.35.Ra
1\. Introduction. It is known that developed two-dimensional hydrodynamic
turbulence, in contrast to three-dimensional turbulence, in the inertial
interval of scales possesses an additional integral of motion — enstrophy.
Enstrophy is an integral of the squared vorticity
$\int_{s}\Omega^{2}d\mathbf{r}$. As Kraichnan demonstrated in 1967 [1], the
existence of this integral generates in the inertial interval its own
Kolmogorov spectrum of turbulence:
$E(k)~{}\sim~{}k^{-3},$ (1)
(now called the Kraichnan spectrum). This spectrum corresponds to a constant
enstrophy flux toward the region of small scales. Simultaneously, according to
[1], a conventional Kolmogorov spectrum $E(k)~{}\sim~{}k^{-5/3}$ with a
constant energy flux directed toward the region of small values of $k$
(inverse cascade) is formed in the region of large scales. After that paper
was published, many numerical experiments were performed (we note only some of
them: [2]-[9]; a more complete list can be found in [10]), which testified in
favor of existence of the Kraichnan spectrum. At the same time, already in the
first numerical experiments there was observed (see, e.g., [2]) the emergence
of sharp vorticity gradients (consistent with the high Reynolds number). It
corresponds to the formation of jumps similar to shock waves which thickness
is small as compared with their length. Based on this observation, Saffman
[11] proposed another spectrum, which differs from the Kraichnan distribution:
$E(k)~{}\sim~{}k^{-4}$. The Saffman spectrum was obtained under the assumption
that the main contribution to the spectrum comes from isotropically
distributed vorticity shocks. On the other hand, it follows from simple
considerations that the spectrum with such shocks is expected to have the
Kraichnan type behavior rather than the Saffman distribution. Indeed, if one
considers the Fourier transform $\Omega_{k}$ of a vorticity jump
$\Omega\propto\theta(x)$, where $\theta(x)$ is the Heaviside function, then
one can easily obtain $\Omega_{k}\propto k^{-1}$, that immediately yields the
spectrum $E(k)~{}\sim~{}k^{-3}$. However, the situation is not too simple. If
one assumes that the characteristic length of the step $L\gg k^{-1}$, then the
energy distribution from one such step in the $k$-space has the form of a jet
with an apex angle of the order of $\left(kL\right)^{-1}$. The maximum value
of the energy distribution along the jet decreases as $\propto k^{-3}$ [10,
12, 13]. As was demonstrated in [10, 12], such an estimate at $kL\gg 1$
follows from the rigorous analysis.
The main attention in this work is paid to the numerical study of statistical
characteristics of the direct cascade of freely decaying two-dimensional
turbulence and the influence of vorticity quasi-shocks on these
characteristics. As compared to previous works,
(i) the spatial resolution is significantly increased (up to $K\times
K=8192^{2}$ points). High-resolution numerical simulations confirmed all
previous results [10, 13];
(ii) it is proved numerically that the spectrum (1) arises owing to vorticity
quasi-shocks, which form a system of jets with weak and strong overlapping in
the $k$-space;
(iii) in each of these jets, the decrease in $E(k)$ at high values of $k$ is
proportional to $~{}k^{-3}$, which yields the spectrum (1) after averaging
over the angle;
(iv) the third-order structure function $\langle(\delta\Omega)^{2}\delta
v_{\|}\rangle$ depends linearly on $R=|\mathbf{r}-\mathbf{r}^{\prime}|$ (where
$\delta\Omega=\Omega(\mathbf{r})-\Omega(\mathbf{r}^{\prime})$, $\delta v_{\|}$
is the projection of the velocity difference onto the vector $\mathbf{R}$), in
complete agreement with the Kraichnan theory. The higher structure functions
of velocity have the exponents $\zeta_{n}$, which grow more slowly than the
first power of $n$, testifying to turbulence intermittency.
2\. Main equations and numerical scheme. As was noted in Introduction, the
emergence of sharp gradients of vorticity was observed in the majority of
numerical experiments (see, e.g., [2]-[9]) modeling two-dimensional turbulence
at high Reynolds numbers, i.e., in the regime where the Euler equation can be
considered in the zero approximation instead of the Navier–Stokes equation.
In terms of vorticity $\Omega={\partial v_{y}}/{\partial x}-{\partial
v_{x}}/{\partial y}$, the Euler equation is written as
$\frac{\partial\Omega}{\partial
t}+(\mathbf{v}\nabla)\Omega=0\quad\quad\mbox{\rm
with}\quad\quad\mbox{div}~{}\mathbf{v}=0,$ (2)
i.e., $\Omega$ is a Lagrangian invariant. As was demonstrated in [10, 12], the
reason for the emergence of strong gradients of vorticity in two-dimensional
turbulence is connected with compressibility of the field
$\mathbf{B}=\mbox{rot}\,(\Omega\hat{z}$), which was called divorticity. This
vector is tangent to the level lines $\Omega(\mathbf{r})=\mbox{\rm const}$,
and the field $\mathbf{B}$ is frozen into the fluid, i.e., it satisfies the
equation [6, 14]
$\frac{\partial\mathbf{B}}{\partial t}=\mbox{\rm
rot}~{}[\mathbf{v}\times\mathbf{B}].$ (3)
It follows from (3) that $\mathbf{B}$ changes owing to the velocity component
$\mathbf{v}_{n}$ normal to $\mathbf{B}$, which is the velocity of the lines of
the field $\mathbf{B}$ due to its frozenness. The Lagrangian trajectories
determined by $\mathbf{v}_{n}$,
$\frac{d\mathbf{r}}{dt}=\mathbf{v}_{n}(\mathbf{r},t);\quad\mathbf{r}|_{t=0}=\mathbf{a},$
(4)
define the mapping
$\mathbf{r}=\mathbf{r}(\mathbf{a},t).$ (5)
In this case Eq. (3) admits integration [10, 12]:
$\mathbf{B}(\mathbf{r},t)=\frac{(\mathbf{B}_{0}(\mathbf{a})\cdot\nabla_{a})\mathbf{r}(\mathbf{a},t)}{J},$
(6)
where $\mathbf{B_{0}}(\mathbf{a})$ is the initial value of the field
$\mathbf{B}$, and $J$ is the Jacobian of mapping (5). As the trajectories (5)
are defined by the normal component of velocity rather than by velocity
proper, the Jacobian $J$ is not fixed. It can take arbitrary values, in
particular, it can tend to zero. This implies compressibility of the field
$\mathbf{B}$. It should be noted that Eq. (6) is an analog of the vortex lines
representation, which was introduced in [15, 16] for the three-dimensional
Euler equation.
It is known that the emergence of discontinuities in gas dynamics is
associated with vanishing of the Jacobian of the corresponding compressible
mapping, i.e., the transformation from the Eulerian to the Lagrangian
description. Just the decrease in $J$ is responsible for the occurrence of
sharp gradients of vorticity resulting from compressibility $\mathbf{v}_{n}$:
$\mbox{div}\,\mathbf{v}_{n}\neq 0$. In the two-dimensional incompressible
Euler hydrodynamics, however, singularity formation is impossible in a finite
time [17]; possibly, it can occur in an infinite time. Numerical experiments
[10, 13] yield an exponential increase in vorticity gradients with the
characteristic increment of the order of the maximum value of vorticity rather
than double-exponential growth, as it follows from the estimates made in [18].
The same behavior is observed in our numerical experiments, which ensure a
better resolution (see the next paragraph).
Let us briefly describe the numerical scheme used. The equation for vorticity
(2) was solved numerically in a square box with periodic boundary conditions
along both coordinates. The box size was chosen to be equal to unity. To
eliminate bottleneck instability and ensure a sink of energy at small scales,
we introduced dissipation into the right hand side of Eq. (2), which was taken
in the form of hyperviscosity
$(-1)^{n+1}\mu_{n}\nabla^{2n}\omega,\,\,\mu_{n}=10^{-20}\left(\frac{2048}{K}\right)^{2n},\quad
n=3.$
The use of hyperviscosity allows appreciable reduction of the range of scales
where the influence of dissipation is significant and, therefore, expansion of
the inertial region of the spectrum [19]. When the spatial resolution was
changed, the hyperviscosity coefficient was scaled in such a manner that the
value of dissipation for the shortest wave perturbations that could be
presented on the computational grid remained constant. The effect of
hyperviscosity on the flow as a whole always remained extremely small: in the
numerical experiments, the total energy was preserved within 0.01 % .
Equation (2) with hyperviscosity was solved by the pseudo-spectral Fourier
method [20], i.e., the derivatives were calculated and the Poisson equation
(for finding the stream function from vorticity) was solved in the spectral
space, whereas the nonlinear terms were calculated on a computational grid in
the physical space. The transformations from the physical to the spectral
space and back were performed through the Fast Fourier Transform (FFT) with
the FFTW library used for that purpose [21].
Figure 1: Initial distribution of vorticity, $N=20$
Figure 2: Distribution of vorticity at $t=100$, $N=20$
The use of hyperviscosity gives rise to a stiff term in the equation; because
of this term, the time step of integration is restricted for any explicit
scheme by a prohibitively small value ($\sim 1/K^{2n}$). To avoid it, time
integration was performed by a hybrid third-order Runge–Kutta /
Crank–Nicholson method [22]. The convective term is approximated explicitly,
and an implicit scheme is used for the dissipative (linear) term.
The computations were performed both on a conventional multiprocessor cluster
(for which the code was parallelized with the help of the MPI library; up to
128 processors were used) and on a GPU cluster (with the use of the NVIDIA
CUDA technology) at the Computer Center of the Novosibirsk State University.
The spatial resolution was up to 8192$\times$8192 points.
3\. Numerical experiments. The initial condition was chosen in the form of two
sets of vortices with positive and negative vorticity. All vortices had a
Gaussian shape with the maximum value of $|\Omega|$ equal to unity and with
random sizes (the vortex radii were uniformly distributed in a certain
interval); the vortex centers were also randomly distributed over the entire
domain. The total vorticity was equal to zero. The number of positive vortices
was equal to the number of negative vortices; this number varied depending on
the spatial resolution from $N=10$ at 2048$\times$2048 to $N=40$ at the
maximum spatial resolution. Variations of the initial conditions did not
change the qualitative behavior of the vortices and their statistical
characteristics. A typical initial condition is shown in Fig. 1.
Figure 2 shows the vorticity distribution at $t=100$. This time is close to
the maximum value $t_{max}$ at which the computation was terminated. The time
$t=t_{max}$ is determined from the condition that the edge of the energy
spectrum $k_{bound}$ reaches $2/3$ of the maximum value $k_{max}$ in the box.
At $t>t_{max}$, the aliasing phenomenon arises [20], where the spectrum is
distorted because harmonics located at a distance of $2k_{max}$ from each
other in the spectral space cannot be distinguished on the computational grid.
As is seen from Fig. 2 (resolution $8192^{2}$), the structure of vortices at
$t=100$ is substantially changed owing to interaction of vortices, emergence
of shear flows generated by vortices, and their stretching. Figure 3 shows the
angularly averaged compensated spectrum of turbulence $k^{3}E(k)$ at different
instants of time. At times close to $t_{max}$, a plateau is formed in the
compensated spectrum, i.e., a Kraichnan-type spectrum $E(k)\sim k^{-3}$ is
formed almost on two decades.
As was noted previously (see also [10, 13]), such a spectrum is formed owing
to the sharp vorticity gradients. Figure 4 shows the distribution of
$|\mathbf{B}|\equiv|\nabla\Omega|$ at $t=100$, which demonstrates a
significant increase ($\sim 100$) in the initial gradients in the vicinity of
lines corresponding to the positions of vorticity shocks. Between the lines,
$|\mathbf{B}|$ practically does not increase. At greater times ($t>100$), the
lines become considerably longer; correspondingly, the pattern contains more
lines, and the distribution of these lines becomes rather twisted (turbulent).
At the initial stage, at several turnover times $n$
($T_{\ast}=n2\pi/\Omega_{max}$), $|\mathbf{B}|$ grows almost exponentially,
and then the saturation is observed. Thus, for Fig. 5, the number of turnovers
is $n\approx 4$, and $|\mathbf{B}|$ increases almost by a factor of $300$.
Figure 3: Compensated spectrum $k^{3}E(k)$ at different times for the initial
distribution shown in Fig. 1
A significant increase in vorticity gradients and in the density of lines
corresponding to the positions of vorticity shocks testifies that their role
in spectrum formation is rather important. This (principal) issue was not
completely clarified in [10, 13]. To verify numerically that the Kraichnan-
type spectrum $E(k)\sim k^{-3}$ is generated by sharp gradients of vorticity,
we performed spatial filtration, in contrast to [10, 13], where filtration in
the $k$-space was performed. In the distribution of the field
$\mathbf{B}(\mathbf{r})$, we retained only those regions where $|\mathbf{B}|$
exceeded some threshold value $B_{0}>0$. In regions with $|\mathbf{B}|<B_{0}$,
the field $\mathbf{B}$ was set to zero. Using the “filtered” field
$\widetilde{\mathbf{B}}\left(\mathbf{r}\right)$, we found the Fourier
transform $\widetilde{\mathbf{B}}_{k}$. Further the Fourier transform of the
filtered velocity was determined:
$\mathbf{v}_{k}=-\frac{1}{k^{4}}\left[\mathbf{k\times}\left[\mathbf{k\times}\widetilde{\mathbf{B}}_{k}\right]\right],$
and then the “filtered” spectrum $\widetilde{E}(k)$. (It is evident that this
procedure introduces its own contribution to the spectrum $\widetilde{E}(k)$;
this contribution is ensured by the shocks $|\widetilde{\mathbf{B}}|$
generated by power-law tails decreasing at least as $k^{-4}$. In other words,
filtration alters only the long-wave part of the spectrum.)
Figure 4: Distribution of $|\mathbf{B}|$ at $t=100$ for the initial condition
shown in Fig. 1
Figure 5: Maximum value of $|\mathbf{B}|$ versus time for the initial
condition shown in Fig. 1 (logarithmic scale, the straight line corresponds to
the exponential growth)
Figure 6 shows the “filtered” spectra $k^{3}\widetilde{E}(k)$ for different
threshold values $B_{0}$. As a result of such filtration, the (compensated)
spectrum $k^{3}\widetilde{E}(k)$ experienced changes only in the region of
small values of $k$ and remained unchanged in the region of large values of
$k$, which is a numerical consequence of the fact that the Kraichnan-type
spectra are induced by vorticity quasi-singularities.
The reason for the emergence of the dependence $k^{-3}$ for the spectrum is
closely connected with its angular distribution. Figure 7 shows the
distribution of the two-dimensional spectrum $\epsilon(k_{x},k_{y})$. This
distribution considerably varies with the angle. It is a number of jets with
weak and/or strong overlapping. This behavior is also clearly visible in the
angular dependence of the spectra with different values of $k$ (Fig. 8). Along
each jet, the spectrum $\epsilon_{k}$ changes as $k^{-4}$, which yields (after
angular averaging) a Kraichnan-type distribution $E(k)$ (1).
Apart from the spectrum, important characteristics of turbulence are structure
functions. As was noted above, at times $t$ close to $t_{max}$, the
distribution of the lines of the maximum values of $|\mathbf{B}|$ becomes
substantially more complicated; we can say that it becomes turbulized. In
particular, the angular overlapping of jets becomes more noticeable, and the
spectrum becomes more isotropic. To elucidate whether the turbulence
considered here is close to the Kraichnan-type turbulence, we performed
numerical experiments at $t$ close to $t_{max}$ and obtained the dependence of
the correlation function
$D(R)=\langle(\Omega(\mathbf{r})-\Omega(\mathbf{r^{\prime}}))^{2}\left([\mathbf{v}(\mathbf{r})-\mathbf{v}(\mathbf{r^{\prime}})]\cdot[\mathbf{R}]\right)/R\rangle$
on $R=|\mathbf{r}-\mathbf{r^{\prime}}|$. For the direct cascade, this
dependence on $R$, as was predicted by Kraichnan (see, e.g., review [23]),
should be linear with a proportionality coefficient $\eta$, which is the flux
of enstrophy toward the region of small scales. Figure 9 shows that this
dependence on $R$ is indeed close to linear, especially at small values of
$R$.
Figure 6: Filtered compensated spectra $k^{3}\widetilde{E}(k)$ for different
threshold values $B_{0}$
Figure 7: Distribution of the two-dimensional spectrum
$k^{4}\epsilon(k_{x},k_{y})$.
Figure 8: Compensated spectrum $k^{4}\epsilon$ versus the angle $\phi$ for
$k_{0}=500$, $k_{0}=1000$, and $k_{0}=1500$
At large values of $R$, there is a small inflection, which, in our opinion, is
associated with deviation of the spectrum from the isotropic one, i.e., with
the jet-like structure of the spectrum.
Figure 9: Correlation function $D(R)$
In addition to the correlation function $D(R)$, we also calculated the
structural functions of velocity
$S_{n}(R)=\left\langle\left[\left(\mathbf{v}(\mathbf{r^{\prime}})-\mathbf{v}(\mathbf{r})\right)\cdot\frac{\mathbf{r}^{\prime}-\mathbf{r}}{R}\right]^{n}\right\rangle$.
It is known that the dependence of the power of the structure functions
$\zeta_{n}$ ($S_{n}(R)\sim R^{\zeta_{n}}$) on $n$ characterizes the degree of
turbulence intermittency. For the Kolmogorov–Kraichnan regime, $\zeta_{n}$ is
a linear dependence of $n$. Our numerical experiment shows that the local
dependence $\zeta_{n}$ (Fig. 10) deviates from a linear dependence and has a
tendency to saturation, especially at large values of $R$, which testifies to
intermittency of this type of turbulence. It should be noted that all
$\zeta_{n}$ in the absence of the dissipative scale depend linearly on $n$ as
$R\to 0$ (see Fig. 10). It is caused by the analytical dependence of velocity
difference at small scales.
4\. Conclusion. The main resume of this work is the fact that the Kraichnan-
type power-law spectrum is formed due to quasi-singularities, which appear in
solving the Cauchy problem for the two-dimensional Euler equation. Though the
collapse as the process of singularity formation in a finite time is forbidden
in two-dimensional hydrodynamics, but there is a tendency to the emergence of
such singularities. As it follows from our numerical experiments, vorticity
gradients increase by more than two orders of magnitude. In our opinion, this
process can be enhanced in the presence of forcing. This assumption is based
on the presence of a logarithmic correction in terms of $k$ in the Kraichnan
spectrum, which is associated with a weak interaction nonlocality. It is also
of interest that the third-order correlation function $D(R)$ demonstrates the
Kraichnan-type behavior at the stage of turbulence decay, when a plateau is
formed in the compensated spectrum $k^{3}E(k)$, which does not contradict the
conclusion that spectrum (1) is generated by vorticity quasi-shocks. At a
sufficiently late stage of turbulence development, however, the turbulence
spectrum has a rather anisotropic distribution over the angle owing to jets.
The authors thank G.E. Falkovich and J.J. Rasmussen for useful discussions.
This work was supported by the grant of the Government of the Russian
Federation for state support of academic research performed under supervision
of leading scientists in Russian educational institutions of higher
professional education (contract No. 11.G34.31.0035 dated November 25, 2010
between the Ministry of Education and Sciences of the Russian Federation,
Novosibirsk State University, and leading scientist), by the RFBR (grant No.
12-01-00943), by the program ”Fundamental problems of nonlinear dynamics in
mathematical and physical sciences” of the RAS Presidium, and by the grant No.
NSh 6170.2012.2 for state support of leading scientific schools of the Russian
Federation.
Figure 10: Exponents $\zeta_{n}$ (local) as functions of $R$
E.K. would like to acknowledge his gratitude for hospitality of the
Observatoire de la Côte d’Azur (Nice, France), where this paper was finalized.
## References
* [1] R. Kraichnan, Phys. Fluids, 11, 1417 (1967).
* [2] D.K. Lilly, J. Fluid Mech., 45, 395 (1971).
* [3] J.C. McWilliams, J. Fluid Mech., 146, 21 (1984).
* [4] S. Kida, J. Phys. Soc. Japan, 54, 2840 (1985).
* [5] M.E. Brachet, M. Meneguzzi, P.L. Sulem, Phys. Rev. Lett., 57, 683 (1986).
* [6] M.E. Brachet, M. Meneguzzi, P.L. Sulem, J. Fluid Mech. 194, 333-349 (1988).
* [7] R. Benzi, S. Patarnello, P. Santangelo, Europhys. Lett., 3, 811 (1986).
* [8] B. Legras, B. Santangelo, R. Benzi, Europhys. Lett., 5, 37 (1988); B. Santangelo, R. Benzi, and B. Legras, Phys. Fluids A, 1, 1027 (1989).
* [9] K. Okhitani, Phys. Fluids A, 3, 1598 (1991).
* [10] E.A. Kuznetsov, V. Naulin, A.H. Nielsen, and J. Juul Rasmussen, Phys. Fluids 19, 105110 (2007).
* [11] P.G. Saffman, Stud. Appl. Maths, 50, 49 (1971).
* [12] E.A. Kuznetsov, JETP Letters, 80, 83 (2004).
* [13] E.A. Kuznetsov, V. Naulin, A.H. Nielsen, and J.J. Rasmussen, Theor. Comput. Fluid Dyn. 24, 253-258 (2010).
* [14] J. Weiss, Physica D, 48, 273 (1991).
* [15] E.A. Kuznetsov, V.P. Ruban, JETP Letters, 67, 1076 (1998); Phys. Rev. E, 61, 831 (2000).
* [16] E.A. Kuznetsov, JETP Letters, 76, 346 (2002).
* [17] W. Wolibner, Math. Z. 37, 698 (1933); V. I. Yudovich, Comput. Math. Math. Phys. 3, 1407 1456 (1963); T. Kato, Arch. Rat. Mech. Anal. 25, 189 (1967).
* [18] H.A. Rose and P.L. Sulem, Journal de Physique 39, 441-484 (1978).
* [19] V. Borue and S.A. Orszag, Europhys. Lett 20, 687-692 (1995).
* [20] C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods, Springer, 2006.
* [21] www.fftw.org.
* [22] P. R. Spalart, R. D. Moser, and M. M. Rogers, J. Comput. Phys., 96, 297 (1991).
* [23] G. Boffetta and R. E. Ecke, Annu. Rev. Fluid Mech. 44, 427 (2012).
|
arxiv-papers
| 2013-01-28T08:44:25 |
2024-09-04T02:49:40.836963
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A.N. Kudryavtsev, E.A. Kuznetsov, E.V. Sereshchenko",
"submitter": "Evgenii A. Kuznetsov",
"url": "https://arxiv.org/abs/1301.6477"
}
|
1301.6502
|
# A variational approach to complex Hessian equations in $\mathbb{C}^{n}$
Lu Hoang Chinh Lu Hoang Chinh
Chalmers University of Technology
Mathematical Sciences
SE- 412 96 Gothenburg,
Sweden [email protected]
###### Abstract.
Let $\Omega$ be a $m$-hyperconvex domain of $\mathbb{C}^{n}$ and $\beta$ be
the standard Kähler form in $\mathbb{C}^{n}$. We introduce finite energy
classes of $m$-subharmonic functions of Cegrell type,
$\mathcal{E}_{m}^{p}(\Omega),p>0$ and $\mathcal{F}_{m}(\Omega)$. Using a
variational method we show that the degenerate complex Hessian equation
$(dd^{c}\varphi)^{m}\wedge\beta^{n-m}=\mu$ has a unique solution in
$\mathcal{E}^{1}_{m}(\Omega)$ if and only if every function in
$\mathcal{E}^{1}_{m}(\Omega)$ is integrable with respect to $\mu$. If $\mu$
has finite total mass and does not charge $m$-polar sets, then the equation
has a unique solution in $\mathcal{F}_{m}(\Omega)$.
## 1\. Introduction
Let $\Omega$ be a bounded domain of $\mathbb{C}^{n}$ and $m$ be an integer
such that $1\leq m\leq n.$ We consider complex $m$-Hessian equations of the
form
(1.1) $(dd^{c}\varphi)^{m}\wedge\beta^{n-m}=\mu,$
where $\beta:=dd^{c}|z|^{2}$ is the standard Kähler form in $\mathbb{C}^{n}$
and $\mu$ is a positive Radon measure.
The border cases $m=1$ and $m=n$ correspond to the Laplace equation which is a
classical subject and the complex Monge-Ampère equation which was studied
intensively in the recent years by many authors.
The complex $m$-Hessian equation was first studied by Li [21]. He used the
well-known continuity method to solve the non-degenerate Dirichlet problem for
equation (1.1) (where the data is smooth and we seek for smooth solutions) in
strongly $m$-pseudoconvex domains. One of its degenerate counterparts was
studied by Błocki [4]. More precisely, he solved the homogeneous equation with
continuous boundary data and initiated a potential theory for this equation.
Recently, Abdullaev and Sadullaev [30] also considered $m$-polar sets and
$m$-capacity for $m$-subharmonic functions. When the right-hand side $\mu$ has
density in $L^{p}(\Omega)$ ($p>n/m$) Dinew and Kołodziej proved in [9] that
given a continuous boundary data, the Dirichlet problem of equation (1.1) has
a unique continuous solution. The Hölder regularity of the solution has been
recently studied by Nguyen Ngoc Cuong [27]. He also showed how to construct
solutions from subsolutions [26]. A viscosity approach to this equation has
been developed in [24] which generalize results in [34] and [11].
The real Hessian equation is a classical subject which was studied intensively
in the recent years. The reader can find a survey for this in [33]. It was
explained in [9] that real and complex Hessian equations are very different
and direct adaptations of the real methods to the complex setting often fails.
The corresponding complex $m$-Hessian equation on compact Kähler manifolds has
been studied by many authors. It has the following form
(1.2) $(\omega+dd^{c}\varphi)^{m}\wedge\omega^{n-m}=\mu,$
where $(X,\omega)$ is a compact Kähler manifold of dimension $n$ and $1\leq
m\leq n$ and $\mu$ is a positive Radon measure.
When $\mu=f\omega^{n}$, $f>0$ is a smooth function satisfying the
compatibility condition $\int_{X}f\omega^{n}=\int_{X}\omega^{n}$, this is a
generalization of the well-known Calabi-Yau equation [31]. In [19], Kokarev
gave some conditions on the measure $\mu$ and on the holomorphic sectional
curvature of the metric so that equation (1.2) has a $\omega$-plurisubharmonic
solution. In general, if $\varphi$ solves equation (1.2) the form
$\omega+dd^{c}\varphi$ is not positive. This lack of positivity prevents one
from copying the proof of Yau’s Theorem without assuming a positivity
condition on the holomorphic bisectional curvature. Hou, Ma, Wu [15], and
Jbilou [17] independently proved that equation (1.2) has a smooth solution
provided this positivity condition. Another effort from Hou, Ma and Wu [16]
showed that one can obtain a $\mathcal{C}^{2}$ estimate if a gradient estimate
holds. As suggested by these authors, this estimate can be used in some blow-
up analysis. This blow-up analysis reduces the problem of solving equation
(1.2) to a Liouville-type theorem for $m$-subharmonic functions in
$\mathbb{C}^{n}$ which was recently proved by Dinew and Kołodziej [10] and the
solvability of equation (1.2) is thus confirmed on any compact Kähler
manifold.
When $0\leq f\in L^{p}(X,\omega^{n})$ for some $p>n/m,$ Dinew and Kołodziej
recently proved that (1.2) admits a unique continuous weak solution. The
result also holds when the right-hand side $f=f(x,\varphi)$ depends on
$\varphi$ (see [23]).
To deal with more singular measures (measures of finite energy), the
variational method developed in [5] is a powerful method. However, applying
this method to the complex Hessian equation (1.2) need further studies on the
local Dirichlet problem and on the regularizing process which are not yet
available and seem to be very difficult.
As a matter of fact, it is interesting to first develop this approach for the
complex Hessian equation in the flat case, i.e the case when the metric is
$\beta$. This is the aim of this paper.
The paper is organized as follows. In section 2, we recall basic facts about
$m$-subharmonic functions and the complex $m$-Hessian operators. At the end of
section 2 we give a connection between the m-polarity and the Hausdorff
measure of a set. Using this one can find examples of $m$-polar sets ($m<n$)
which are not pluripolar. In section 3, we study finite energy classes of
$m$-subharmonic functions inspired by [7, 8]. An $m$-subharmonic function
$\varphi$ belongs to the class $\mathcal{E}^{1}_{m}(\Omega)$ if the Hessian
measure $H_{m}(\varphi)=(dd^{c}\varphi)^{m}\wedge\beta^{n-m}$ is well-defined
and with respect to which $\varphi$ is integrable. The class
$\mathcal{F}_{m}^{a}(\Omega)$ consists of non-positive $m$-subharmonic
functions whose Hessian measures are well-defined, of finite total mass and do
not charge $m$-polar sets. In section 4, we develop a variational approach
inspired by [5] (see also [2]) to solve equation (1.1) with a "finite energy"
right-hand side.
The main results are the followings.
Theorem 1. Let $\mu$ be a positive Radon measure in $\Omega$, an
$m$-hyperconvex domain. Then $\mathcal{E}^{1}_{m}(\Omega)\subset
L^{1}(\Omega,\mu)$ if and only if there exists a unique
$\varphi\in\mathcal{E}^{1}_{m}(\Omega)$ such that
$(dd^{c}\varphi)^{m}\wedge\beta^{n-m}=\mu.$
To prove this result we use a variational method introduced in [5]. Our result
generalizes the result in [2]. Using this and following [8] we also get:
Theorem 2. Let $\mu$ be a positive Radon measure in an $m$-hyperconvex domain
$\Omega$ such that $\mu(\Omega)<+\infty$ and $\mu$ does not charge $m$-polar
sets. Then there exists a unique $\varphi\in\mathcal{F}^{a}_{m}(\Omega)$ such
that $(dd^{c}\varphi)^{m}\wedge\beta^{n-m}=\mu.$
## 2\. Preliminaries
### 2.1. m-subharmonic functions and the Hessian operator
In the whole paper, $\beta$ denotes the standard Kähler form in
$\mathbb{C}^{n}.$ In this section we summarize basic facts about
$m$-subharmonic functions and the Hessian operator which will be used in the
next sections. Most of these results can be found in [26, 27], [30] or can be
proved similarly as in the case of plurisubharmonic functions (see for example
[18], [20]).
###### Definition 2.1.
Let $\alpha$ be a real $(1,1)$-form in $\Omega$, a domain of $\mathbb{C}^{n}$.
We say that $\alpha$ is $m$-positive in $\Omega$ if the following inequalities
hold
$\alpha^{j}\wedge\beta^{n-j}\geq 0,\ \ \forall j=1,...,m.$
Let $T$ be a current of bidegree $(n-k,n-k)$, with $k\leq m$. Then $T$ is
called $m$-positive if for all $m$-positive $(1,1)$-forms
$\alpha_{1},...,\alpha_{k},$ we have
$\alpha_{1}\wedge...\wedge\alpha_{k}\wedge T\geq 0.$
###### Definition 2.2.
A function $u:\Omega\rightarrow\mathbb{R}\cup\\{-\infty\\}$ is called
$m$-subharmonic if it is subharmonic and the current $dd^{c}u$ is
$m$-positive. The class of all $m$-subharmonic functions in $\Omega$ will be
denoted by $\mathcal{SH}_{m}(\Omega).$
###### Definition 2.3.
Let $\Omega$ be a bounded domain in $\mathbb{C}^{n}$. Then $\Omega$ is called
$m$-hyperconvex if there exists a continuous $m$-subharmonic function
$\varphi:\Omega\rightarrow\mathbb{R}^{-}$ such that
$\\{\varphi<c\\}\Subset\Omega$, for every $c<0.$
From now on, we always assume that $\Omega$ is $m$-hyperconvex.
We list in the following proposition some elementary facts on
$m$-subharmonicity.
###### Proposition 2.4.
(i) If $u$ is $\mathcal{C}^{2}$ smooth then $u$ is $m$-subharmonic if and only
if the form $dd^{c}u$ is $m$-positive in $\Omega.$
(ii) If $u,v\in\mathcal{SH}_{m}(\Omega)$ then $\lambda u+\mu
v\in\mathcal{SH}_{m}(\Omega),\forall\lambda,\mu>0.$
(iii) If $u$ is $m$-subharmonic in $\Omega$ then the standard regularization
$u*\chi_{\epsilon}$ are also $m$-subharmonic in
$\Omega_{\epsilon}:=\\{x\in\Omega\ /\ d(x,\partial\Omega)>\epsilon\\}$.
(iv) If $(u_{l})\subset\mathcal{SH}_{m}(\Omega)$ is locally uniformly bounded
from above then $(\sup u_{l})^{*}\in\mathcal{SH}_{m}(\Omega)$, where $v^{*}$
is the upper semi continuous regularization of $v$.
(v)
$PSH(\Omega)=\mathcal{SH}_{n}(\Omega)\subset...\subset\mathcal{SH}_{m}(\Omega)\subset...\subset\mathcal{SH}_{1}(\Omega)=\mathcal{SH}(\Omega).$
(vi) Let $\emptyset\neq U\subset\Omega$ be a proper open subset such that
$\partial U\cap\Omega$ is relatively compact in $\Omega$. If
$u\in\mathcal{SH}_{m}(\Omega)$, $v\in\mathcal{SH}_{m}(U)$ and $\limsup_{x\to
y}v(x)\leq u(y)$ for each $y\in\partial U\cap\Omega$ then the function $w$,
defined by
$w(z)=\begin{cases}u(z)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\rm if}\
z\in\Omega\setminus U,\\\ \max(u(z),v(z))\ \ {\rm if}\ z\in U\end{cases}$
is $m$-subharmonic in $\Omega.$
For locally bounded $m$-subharmonic functions $u_{1},...,u_{p}$ (with $p\leq
m$) we can inductively define a closed $m$-positive current (following Bedford
and Taylor [3]).
###### Lemma 2.5.
Let $u_{1},...,u_{k}$ (with $k\leq m$) be locally bounded $m$-subharmonic
functions in $\Omega$ and let $T$ be a closed $m$-positive current of bidegree
$(n-p,n-p)$ (with $p\geq k$). Then we can define inductively a closed
$m$-positive current
$dd^{c}u_{1}\wedge dd^{c}u_{2}\wedge...\wedge dd^{c}u_{k}\wedge T,$
and the product is symmetric, i.e.
$dd^{c}u_{1}\wedge dd^{c}u_{2}\wedge...\wedge dd^{c}u_{p}\wedge
T=dd^{c}u_{\sigma(1)}\wedge dd^{c}u_{\sigma(2)}\wedge...\wedge
dd^{c}u_{\sigma(k)}\wedge T,$
for every permutation $\sigma:\\{1,...,k\\}\to\\{1,...,k\\}.$
In particular, the Hessian measure of $u\in\mathcal{SH}_{m}(\Omega)\cap
L^{\infty}_{\rm loc}$ is defined to be
$H_{m}(u)=(dd^{c}u)^{m}\wedge\beta^{n-m}.$
###### Proof.
See [30]. ∎
###### Theorem 2.6.
Let $(u_{0}^{j}),...(u_{k}^{j})$ be decreasing sequences of $m$-subharmonic
functions in $\Omega$ converging to
$u_{0},...,u_{k}\in\mathcal{SH}_{m}(\Omega)\cap L^{\infty}_{\rm loc}(\Omega)$
respectively. Let $T$ be a closed $m$-positive current of bidegree $(n-p,n-p)$
(with $p\geq k$) on $\Omega$. Then
$u_{0}^{j}.dd^{c}u_{1}^{j}\wedge...\wedge dd^{c}u_{k}^{j}\wedge
T\rightharpoonup u_{0}.dd^{c}u_{1}\wedge...\wedge dd^{c}u_{k}\wedge T,$
weakly in the sense of currents.
###### Proof.
See [30]. ∎
One of the most important properties of $m$-subharmonic functions is the
quasicontinuity. Every $m$-subharmonic function is continuous outside an
arbitrarily small open subset. The $m$-Capacity is used to measure the
smallness of these sets.
###### Definition 2.7.
Let $E\subset\Omega$ be a Borel subset. The $m$-capacity of $E$ with respect
to $\Omega$ is defined to be
${\rm Cap}_{m}(E,\Omega):=\sup\Big{\\{}\int_{E}H_{m}(\varphi)\ /\
\varphi\in\mathcal{SH}_{m}(\Omega),0\leq\varphi\leq 1\Big{\\}}.$
The $m$-Capacity shares the same elementary properties as the Capacity
introduced by Bedford and Taylor.
###### Proposition 2.8.
i) ${\rm Cap}_{m}(E_{1},\Omega)\leq{\rm Cap}_{m}(E_{2},\Omega)$ if
$E_{1}\subset E_{2},$
ii) ${\rm Cap}_{m}(E,\Omega)=\lim_{j\to\infty}{\rm Cap}_{m}(E_{j},\Omega)$ if
$E_{j}\uparrow E$,
iii) ${\rm Cap}_{m}(E,\Omega)\leq\sum{\rm Cap}_{m}(E_{j},\Omega)$ for $E=\cup
E_{j}.$
The following results can be proved by repeating the arguments in [20].
###### Theorem 2.9.
Every $m$-subharmonic function $u$ defined in $\Omega$ is quasi-continuous.
This means that for any positive number $\epsilon$ one can find an open set
$U\subset\Omega$ with ${\rm Cap}_{m}(U,\Omega)<\epsilon$ and such that $u$
restricted to $\Omega\setminus U$ is continuous.
###### Theorem 2.10.
Let $\\{u_{k}^{j}\\}_{j=1}^{\infty}$ be a locally uniformly bounded sequence
of $m$-subharmonic functions in $\Omega$ for $k=1,2,...,N\leq m$ and let
$u_{k}^{j}\uparrow u_{k}\in\mathcal{SH}_{m}(\Omega)\cap L^{\infty}_{\rm loc}$
almost everywhere as $j\to\infty$ for $k=1,2,...,N.$ Then
$dd^{c}u_{1}^{j}\wedge...\wedge
dd^{c}u_{N}^{j}\wedge\beta^{n-m}\rightharpoonup dd^{c}u_{1}\wedge...\wedge
dd^{c}u_{N}\wedge\beta^{n-m}.$
###### Theorem 2.11 (Integration by parts).
Let $u,v\in\mathcal{SH}_{m}(\Omega)\cap L^{\infty}_{\rm loc}(\Omega)$ such
that $\lim_{z\to\partial\Omega}u(z)=\lim_{z\to\partial\Omega}v(z)=0.$ Then
$\int_{\Omega}udd^{c}v\wedge T=\int_{\Omega}vdd^{c}u\wedge T,$
where $T=dd^{c}\varphi_{1}\wedge...\wedge
dd^{c}\varphi_{m-1}\wedge\beta^{n-m}$ with
$\varphi_{1},...,\varphi_{m-1}\in\mathcal{SH}_{m}(\Omega)\cap L^{\infty}_{{\rm
loc}}(\Omega)$. The equality is understood in the sense that if one of the two
terms is finite then so is the other, and they are equal.
###### Theorem 2.12 (Maximum principle).
If $u,v\in\mathcal{SH}_{m}(\Omega)\cap L^{\infty}_{\rm loc}(\Omega)$ then
$1{\hskip-2.5pt}\hbox{{I}}_{\\{u>v\\}}H_{m}(\max(u,v))=1{\hskip-2.5pt}\hbox{{I}}_{\\{u>v\\}}H_{m}(u).$
###### Theorem 2.13 (Comparison principle).
Let $u,v\in\mathcal{SH}_{m}(\Omega)\cap L^{\infty}_{\rm loc}(\Omega)$ such
that
$\liminf_{z\to\partial\Omega}(u(z)-v(z))\geq 0.$
Then
$\int_{\\{u<v\\}}(dd^{c}v)^{m}\wedge\beta^{n-m}\leq\int_{\\{u<v\\}}(dd^{c}u)^{m}\wedge\beta^{n-m}.$
###### Definition 2.14.
For a subset $E$ of a domain $\Omega\subset\mathbb{C}^{n}$ we define the
relative $m$-extremal function of $E$ by
$u_{m,E,\Omega}:=\sup\\{u\in\mathcal{SH}_{m}(\Omega)\ /\ u<0,\ \text{and}\
u\leq-1\ \text{on}\ E\\}.$
It is easy to see that $u_{m,E,\Omega}^{*}$ is $m$-subharmonic in $\Omega.$
###### Proposition 2.15.
i) If $E_{1}\subset E_{2}$ then $u_{E_{2}}\leq u_{E_{1}}.$
ii) If $E\subset\Omega_{1}\subset\Omega_{2}$ then $u_{m,E,\Omega_{2}}\leq
u_{m,E,\Omega_{1}}.$
iii) If $K_{j}\downarrow K,$ with $K_{j}$ compact in $\Omega$ then $(\lim
u_{m,K_{j},\Omega}^{*})^{*}=u_{m,K,\Omega}^{*}.$
As in [18] we have the following formula for the $m$-extremal functions of
concentric balls.
###### Lemma 2.16.
Let $0<r<R$ and set $a=\dfrac{n}{m}>1$. The $m$-extremal function of $B(r)$
with respect to $B(R)$ is given by
$u_{m,B(r),B(R)}=\max\Big{(}\dfrac{R^{2-2a}-\|z\|^{2-2a}}{r^{2-2a}-R^{2-2a}},-1\Big{)}.$
###### Proposition 2.17.
If $E\Subset\Omega$ then one has $\lim_{z\to w}u_{m,E,\Omega}(z)=0$ for any
$w\in\partial\Omega$.
###### Proposition 2.18.
Let $K\subset\Omega$ be a compact subset which is the union of closed balls,
then $u_{K}^{*}=u_{K}$ is continuous. In particular, if $K\subset\Omega$ is an
arbitrary compact set and $\epsilon<{\rm dist}(K,\partial\Omega)$, then
$u_{K_{\epsilon}}$ is continuous, where $K_{\epsilon}=\\{z\in\Omega\ /\ {\rm
dist}(z,K)\leq\epsilon\\}.$
Sketch of proof: From Lemma 2.16 we know that the $m$-extremal function of
balls is continuous. Let $K$ be a compact set which is a union of closed
balls, $K=\cup B_{j}$ and let $u$ be its $m$-extremal function. Since $u\leq
u_{m,B_{j},\Omega}$, it is easy to see that $u^{*}\equiv-1$ on $K$. The same
arguments as in [18, Proposition 4.5.3] show that $u$ is continuous in
$\Omega.$
###### Definition 2.19.
The outer $m$-capacity of a Borel set $E\subset\Omega$ is defined by
${\rm Cap}_{m}^{*}(E,G):=\inf\left\\{{\rm Cap}_{m}(G,\Omega)\ /\ E\subset
G,G{\ \rm is\ an\ open\ subset\ of\ }\Omega\right\\}.$
###### Theorem 2.20.
If $E\Subset\Omega$ is a Borel subset then
${\rm Cap}^{*}_{m}(E,\Omega)=\int_{\Omega}H_{m}(u_{m,E,\Omega}^{*}),\ \ {\rm
and}\ {\rm Cap}^{*}_{m}(E,\Omega)={\rm Cap}_{m}(E,\Omega)\ \ {\rm if}\ E\ {\rm
is\ compact}.$
We compute the $m$-Capacity of the concentric balls.
###### Example 2.21.
For every $0<r<R$ we have
${\rm Cap}_{m}(B(r),B(R))=\dfrac{2^{n}(n-m)}{m.n!(r^{2-2a}-R^{2-2a})^{m}}.$
###### Definition 2.22.
Let $\Omega$ be an open set in $\mathbb{C}^{n}$, and let
$\mathcal{U}\subset\mathcal{SH}_{m}(\Omega)$ be a family of functions which is
locally bounded from above. Define
$u(z)=\sup\\{v(z)\ /\ v\in\mathcal{U}\\}.$
Sets of the form $\mathcal{N}=\\{z\in\Omega\ /\ u(z)<u^{*}(z)\\}$ and all
their subsets are called $m$-negligible.
###### Definition 2.23.
A set $E\subset\mathbb{C}^{n}$ is called $m$-polar if
$E\subset\\{v=-\infty\\}$ for some $v\in\mathcal{SH}_{m}(\mathbb{C}^{n})$ and
$v\not\equiv-\infty.$
###### Theorem 2.24.
Let $E\Subset\Omega$. Then $E$ is $m$-negligible $\Leftrightarrow$ $E$ is
$m$-polar $\Leftrightarrow$ ${\rm Cap}^{*}_{m}(E,\Omega)=0$.
### 2.2. $m$-polarity of sets with small Hausdorff measure
In this section, following [22] we give a sufficient condition for a set being
$m$-polar using Hausdorff measure. We then give examples of $m$-polar sets
($m<n$) which are not pluripolar.
###### Definition 2.25.
A function $h:[0,1)\rightarrow\mathbb{R}^{+}$ is called measuring function if
it is increasing and $\lim_{r\to 0}h(r)=h(0)=0.$
For a measuring function $h$, the $h$-Hausdorff measure of
$E\subset\mathbb{C}^{n}$ is defined by (see [22], [25], [32])
$\Lambda_{h}(E):=\lim_{\delta\to 0}\left(\inf\sum_{j}h(r_{j})\right),$
where the infimum is taken over all coverings of $E$ by balls $B_{j}$ of radii
$r_{j}\leq\delta.$
###### Theorem 2.26.
Let $H(r)=r^{2n-2m}$ ($n>m\geq 1$). Then every subset $E\subset\mathbb{C}^{n}$
satisfying $\Lambda_{H}(E)<+\infty$ is $m$-polar.
###### Proof.
We can assume that $E\Subset B=B(0,R).$ Suppose that $E$ is not $m$-polar.
Then ${\rm Cap}^{*}_{m,B}(E)>0$ and $u:=u^{*}_{m,E,B}\not\equiv 0.$ Set
$E_{1}:=\\{z\in E\ /\ u(z)=-1\\}.$ Then by Theorem 2.24, $E\setminus E_{1}$ is
$m$-polar and hence
$\int_{E\setminus E_{1}}H_{m}(u)=0.$
Therefore, as $H_{m}(u)$ is a regular Borel measure and $E_{1}$ is a Borel
set, we can find a compact set $K\subset E_{1}$ such that
$\int_{K}H_{m}(u)>0.$
We claim that for every bounded open set $\Omega\supset K$, ${\rm
Cap}_{m}(K,\Omega)\leq C.\Lambda_{H}(K)$, where $C>0$ is a constant
independent of $\Omega.$ Indeed, let $\delta:={\rm dist}(K,\Omega)$ and fix
$\epsilon\in(0,1)$ such that $\epsilon<\delta/4.$ We cover $K$ by open balls
$B(z_{j},r_{j})$ such that $r_{j}<\epsilon.$ We may assume that
$B(z_{j},r_{j})\subset\Omega.$ From (2.21) and after simple computations we
get
${\rm Cap}_{m}(B(z_{j},r_{j}),B(z_{j},\delta/2))\leq C.r_{j}^{2n-2m},$
where $C>0$ is a constant depending only on $n,m.$ Using this and the
monotonicity and subadditivity of the outer $m$-capacity we get
$\displaystyle{\rm Cap}^{*}_{m}(K,\Omega)$ $\displaystyle\leq$
$\displaystyle\sum_{j}{\rm Cap}_{m}^{*}(B(z_{j},r_{j}),\Omega)$
$\displaystyle\leq$ $\displaystyle\sum_{j}{\rm
Cap}^{*}_{m}(B(z_{j},r_{j}),B(z_{j},\delta/2))\leq C\sum_{j}H(r_{j}).$
Now, the claim follows by taking the infimum over all such coverings and
letting $\epsilon\to 0.$
For each $\delta>0$ set $\Omega_{\delta}:=\\{z\in B\ /\ {\rm
dist}(z,K)<\delta\\}.$ Since $u$ is continuous at every point on $K$ and
$u(x)=-1$ for every $x\in K$, we get
$c(\delta):={\rm osc}_{\Omega_{\delta}}(u)\rightarrow 0\ \ {\rm as}\ \delta\to
0.$
For any $z\in\Omega_{\delta}$ we have $0\leq\frac{u(z)+1}{c(\delta)}\leq 1.$
Thus,
$\displaystyle{\rm Cap}^{*}_{m}(K,\Omega_{\delta})$ $\displaystyle\geq$
$\displaystyle{\rm
Cap}_{m}(K,\Omega_{\delta})\geq\int_{K}H_{m}\left(\frac{u+1}{c(\delta)}\right)=\frac{1}{c(\delta)^{m}}\int_{K}H_{m}(u).$
We then get ${\rm Cap}^{*}_{m}(K,\Omega_{\delta})\to+\infty$ as $\delta\to 0$.
The Claim yields $\Lambda_{H}(E)=+\infty.$ ∎
###### Example 2.27.
Assume that $1\leq m<n$ and let $E$ be the Cantor set constructed in [22] (see
also [1], [6], [25]) and $E^{n}=E\times E\times...\times E$ ($n$ times). Since
$\lim_{r\to 0}H(r)/[\log(1/r)]^{-n}=0$, we can choose the sequence
$(\ell_{j})$ defining $E$ such that $E^{n}$ is not pluripolar but
$\Lambda_{H}(E^{n})=0$. This implies that $E^{n}$ is $m$-polar in view of
Theorem 2.26.
## 3\. Finite energy classes
In this section we study finite energy classes of $m$-subharmonic functions in
$m$-hyperconvex domains. They are generalizations of Cegrell’s classes [7, 8]
for plurisubharmonic functions.
### 3.1. Definitions and properties
In pluripotential theory one of the most important steps is to regularize
singular plurisubharmonic functions. It can be easily done locally by
convolution with a smooth kernel. The following theorem explains how to do it
globally in a $m$-hyperconvex domain. Let $\mathcal{SH}_{m}^{-}(\Omega)$
denote the class of non-positive functions in $\mathcal{SH}_{m}(\Omega).$
###### Theorem 3.1.
For each $\varphi\in\mathcal{SH}_{m}^{-}(\Omega)$ there exists a sequence
$(\varphi_{j})$ of $m$-sh functions verifying the following conditions:
(i) $\varphi_{j}$ is continuous on $\bar{\Omega}$ and $\varphi_{j}\equiv 0$ on
$\partial\Omega;$
(ii) each $H_{m}(\varphi_{j})$ has compact support,
(iii) $\varphi_{j}\downarrow\varphi$ on $\Omega.$
###### Proof.
If $B$ is a closed ball in $\Omega$ then by Proposition 2.18, the $m$-extremal
function $u_{m,B,\Omega}$ is continuous on $\bar{\Omega}$ and ${\rm
supp}(H_{m}(u))\Subset\Omega.$ We can follow the lines in [8, Theorem 2.1]. ∎
###### Definition 3.2.
We let $\mathcal{E}_{m}^{0}(\Omega)$ denote the class of bounded functions in
$\mathcal{SH}_{m}^{-}(\Omega)$ such that
$\underset{z\to\partial\Omega}{\lim}\varphi(z)=0$ and
$\int_{\Omega}H_{m}(\varphi)<+\infty.$
For each $p>0$, $\mathcal{E}_{m}^{p}(\Omega)$ denote the class of functions
$\varphi\in\mathcal{SH}_{m}(\Omega)$ such that there exists a decreasing
sequence $(\varphi_{j})\subset\mathcal{E}_{m}^{0}(\Omega)$ satisfying
(i) $\lim_{j}\varphi_{j}=\varphi,$ in $\Omega$ and
(ii) $\sup_{j}\int_{\Omega}(-\varphi_{j})^{p}H_{m}(\varphi_{j})<+\infty.$
If we require moreover that $\sup_{j}\int_{\Omega}H_{m}(\varphi_{j})<+\infty$
then, by definition, $\varphi$ belongs to $\mathcal{F}_{m}^{p}(\Omega).$
###### Definition 3.3.
We let $\mathcal{F}_{m}(\Omega)$ denote the class of functions
$u\in\mathcal{SH}_{m}^{-}(\Omega)$ such that there exists a sequence
$(u_{j})\subset\mathcal{E}_{m}^{0}(\Omega)$ decreasing to $u$ in $\Omega$ and
$\sup_{j}\int_{\Omega}H_{m}(u_{j})<+\infty.$
###### Definition 3.4.
We define the $p$-energy ($p>0$) of $\varphi\in\mathcal{E}_{m}^{0}(\Omega)$ by
${\rm E}_{p}(\varphi):=\int_{\Omega}(-\varphi)^{p}H_{m}(\varphi).$
If $p=1$ we drop the index and denote by ${\rm E}(\varphi)={\rm
E}_{1}(\varphi)$.
We generalize Hölder inequality in the following lemma. When $m=n$ it is a
result of Persson [28]. Our proof uses the same idea.
###### Lemma 3.5.
Let $u,v_{1},...,v_{m}\in\mathcal{E}_{m}^{0}(\Omega)$ and $p\geq 1.$ We have
$\int_{\Omega}(-u)^{p}dd^{c}v_{1}\wedge...\wedge
dd^{c}v_{m}\wedge\beta^{n-m}\leq D_{p}({\rm E}_{p}(u))^{\frac{p}{m+p}}{\rm
E}_{p}(v_{1})^{\frac{1}{m+p}}...{\rm E}_{p}(v_{m})^{\frac{1}{m+p}},$
here $D_{1}=1$ and for each $p>1$, $D_{p}:=p^{p\alpha(p,m)/(p-1)}$, where
$\alpha(p,m)=(p+2)\Big{(}\frac{p+1}{p}\Big{)}^{m-2}-p-1.$
###### Proof.
Let
$F(u,v_{1},...,v_{m})=\int_{\Omega}(-u)^{p}dd^{c}v_{1}\wedge...\wedge
dd^{c}v_{m}\wedge\beta^{n-m},\
u,v_{1},...,v_{m}\in\mathcal{E}_{m}^{0}(\Omega).$
Thanks to [28, Theorem 4.1] it suffices to prove that
(3.1) $F(u,v,v_{1},...,v_{m-1})\leq
a(p)F(u,u,v_{1},...,v_{m-1})^{\frac{p}{p+1}}F(v,v,v_{1},...,v_{m-1})^{\frac{1}{p+1}},$
where $a(p)=1$ if $p=1$ and $a(p)=p^{\frac{p}{p-1}}$ if $p>1.$ Set
$T=dd^{c}v_{1}\wedge...\wedge dd^{c}v_{m-1}\wedge\beta^{n-m}.$ When $p=1,$
(3.1) becomes
$\int_{\Omega}(-u)dd^{c}v\wedge T\leq\Big{(}\int_{\Omega}(-u)dd^{c}u\wedge
T\Big{)}^{\frac{1}{2}}\Big{(}\int_{\Omega}(-v)dd^{c}v\wedge
T\Big{)}^{\frac{1}{2}},$
which is the Cauchy-Schwarz inequality. In the case $p>1,$ integrating by
parts we get
$\int_{\Omega}(-u)^{p}dd^{c}v\wedge T\leq
p\int_{\Omega}(-u)^{p-1}(-v)dd^{c}u\wedge T.$
By using Hölder inequality we obtain
$\int_{\Omega}(-u)^{p}dd^{c}v\wedge T\leq
p\Big{(}\int_{\Omega}(-u)^{p}dd^{c}u\wedge
T\Big{)}^{\frac{p-1}{p}}\Big{(}\int_{\Omega}(-v)^{p}dd^{c}u\wedge
T\Big{)}^{\frac{1}{p}}.$
Now, interchanging $u$ and $v$ we get
$\int_{\Omega}(-v)^{p}dd^{c}u\wedge T\leq
p\Big{(}\int_{\Omega}(-u)^{p}dd^{c}v\wedge
T\Big{)}^{\frac{1}{p}}\Big{(}\int_{\Omega}(-v)^{p}dd^{c}v\wedge
T\Big{)}^{\frac{p-1}{p}}.$
Combining the above two inequalities we obtain the result.
∎
Thanks to Lemma 3.5 we can bound
$\int_{\Omega}(u_{0})^{p}dd^{c}u_{1}\wedge...\wedge
dd^{c}u_{m}\wedge\beta^{n-m}$ by ${\rm E}_{p}(u_{j}),j=0,...,m$ if $p\geq 1.$
To get similar estimates when $p\in(0,1)$ we can follow the lines in [14]:
###### Lemma 3.6.
Let $u,v\in\mathcal{E}_{m}^{0}(\Omega)$ and $0<p<1$. If T is a closed
$m$-positive current of type $T=dd^{c}v_{1}\wedge...\wedge
dd^{c}v_{m-k}\wedge\beta^{n-m},$ where $u_{j}\in\mathcal{SH}_{m}(\Omega)\cap
L^{\infty}_{\rm loc}$, then
$\int_{\Omega}(-u)^{p}(dd^{c}v)^{k}\wedge T\leq
2\int_{\Omega}(-u)^{p}(dd^{c}u)^{k}\wedge
T+2\int_{\Omega}(-v)^{p}(dd^{c}v)^{k}\wedge T.$
###### Proof.
The same as in the proof of [14, Proposition 2.5]. ∎
###### Proposition 3.7.
Let $0<p<1$. There exists $C_{p}>0$ such that
$0\leq\int_{\Omega}(-\varphi_{0})^{p}dd^{c}\varphi_{1}\wedge\cdots\wedge
dd^{c}\varphi_{m}\wedge\beta^{n-m}\leq C_{p}\max_{0\leq j\leq m}{\rm
E}_{p}(\varphi_{j}),$
for all $0\geq\varphi_{0},\ldots,\varphi_{m}\in\mathcal{E}_{m}^{0}(\Omega).$
###### Proof.
See [14, Proposition 2.10] ∎
From Lemma 3.5 and Proposition 3.7 we easily get the following result.
###### Corollary 3.8.
Let $(u_{j})$ be a sequence in $\mathcal{E}_{m}^{0}(\Omega)$ and $p>0.$ Assume
also that $\sup_{j}{\rm E}_{p}(u_{j})<+\infty.$ Then
$u=\sum_{j=1}^{\infty}2^{-j}u_{j}\ \text{belongs to }\
\mathcal{E}_{m}^{p}(\Omega).$
From the above facts, we can prove the convexity the classes
$\mathcal{E}_{m}^{p}(\Omega),\mathcal{E}_{m}(\Omega)$ by the same way as in
[7, 8].
###### Theorem 3.9.
By $\mathcal{E}$ we denote one of the classes $\mathcal{E}_{m}^{0}(\Omega),$
$\mathcal{F}_{m}(\Omega)$, $\mathcal{E}_{m}^{p}(\Omega)$,
$\mathcal{F}_{m}^{p}(\Omega)$, $p>0.$ They are convex and moreover, if
$v\in\mathcal{E}$, $u\in\mathcal{SH}_{m}^{-}(\Omega)$, $u\geq v$, then
$u\in\mathcal{E}.$
### 3.2. Definition of the complex Hessian operator and basic properties
In this section we prove that the complex Hessian operator $H_{m}$ is well-
defined for functions in $\mathcal{F}_{m}(\Omega)$ and in
$\mathcal{E}_{m}^{p}(\Omega),p>0.$ We follow the arguments in [8].
As in [8], continuous functions in $\mathcal{E}_{m}^{0}(\Omega)$ can be
considered as test functions.
###### Lemma 3.10.
$\mathcal{C}_{0}^{\infty}(\Omega)\subset\mathcal{E}_{m}^{0}(\Omega)\cap\mathcal{C}(\Omega)-\mathcal{E}_{m}^{0}(\Omega)\cap\mathcal{C}(\Omega).$
###### Theorem 3.11.
Let $u^{p}\in\mathcal{F}_{m}(\Omega),p=1,...,m$ and
$(g^{p}_{j})_{j}\subset\mathcal{E}_{m}^{0}(\Omega)$ such that
$g^{p}_{j}\downarrow u^{p},\forall p$. Then the sequence of measures
$dd^{c}g^{1}_{j}\wedge dd^{c}g^{2}_{j}\wedge...\wedge
dd^{c}g^{m}_{j}\wedge\beta^{n-m}$
converges weakly to a positive Radon measure which does not depend on the
choice of the sequences $(g^{p}_{j}).$ We then define
$dd^{c}u^{1}\wedge...\wedge dd^{c}u^{m}\wedge\beta^{n-m}$ to be this weak
limit.
###### Proof.
See [8, Theorem 4.2]. ∎
It is convenient to use the notation
$H_{m}(u_{1},...,u_{m}):=dd^{c}u^{1}\wedge...\wedge
dd^{c}u^{m}\wedge\beta^{n-m}$. When $u_{1}=...=u_{m}=u$ we simply write
$H_{m}(u)$ instead of $H_{m}(u,...,u).$
###### Definition 3.12.
A function $u$ belongs to the class $\mathcal{F}_{m}^{a}(\Omega)$ if
$u\in\mathcal{F}_{m}(\Omega)$ and $H_{m}(u)$ vanishes on $m$-polar sets.
###### Corollary 3.13.
Let $u_{1},...,u_{m}\in\mathcal{F}_{m}(\Omega)$ and $u_{1}^{j},...,u_{m}^{j}$
be sequences of functions in
$\mathcal{E}_{m}^{0}(\Omega)\cap\mathcal{C}(\Omega)$ decreasing to
$u_{1},...,u_{m}$ respectively such that
$\sup_{j,p}\int_{\Omega}H_{m}(u_{j}^{p})<+\infty.$
Then for each $\varphi\in\mathcal{E}_{m}^{0}(\Omega)\cap\mathcal{C}(\Omega)$
we have
$\lim_{j\to+\infty}\int_{\Omega}\varphi dd^{c}u_{1}^{j}\wedge...\wedge
dd^{c}u_{m}^{j}\wedge\beta^{n-m}=\int_{\Omega}\varphi
dd^{c}u_{1}\wedge...\wedge dd^{c}u_{m}\wedge\beta^{n-m}.$
###### Proof.
It is clear that
(3.2) $\sup_{j}\int_{\Omega}dd^{c}u_{1}^{j}\wedge...\wedge
dd^{c}u_{m}^{j}\wedge\beta^{n-m}<+\infty.$
Fix $\epsilon>0$ small enough and consider
$\varphi_{\epsilon}=\max(\varphi,-\epsilon).$ The function
$\varphi-\varphi_{\epsilon}$ is continuous and compactly supported in
$\Omega.$ It follows from Theorem 3.11 that
$\lim_{j\to+\infty}\int_{\Omega}(\varphi-\varphi_{\epsilon})dd^{c}u_{1}^{j}\wedge...\wedge
dd^{c}u_{m}^{j}\wedge\beta^{n-m}=\int_{\Omega}(\varphi-\varphi_{\epsilon})dd^{c}u_{1}\wedge...\wedge
dd^{c}u_{m}\wedge\beta^{n-m}.$
Observe also that $|\varphi_{\epsilon}|\leq\epsilon$. By using (3.2), we get
the result. ∎
###### Corollary 3.14.
Assume that $(u_{j})\subset\mathcal{E}_{m}^{0}(\Omega)$ decreases to $u$ such
that
$\sup_{j}\int_{\Omega}H_{m}(u_{j})<+\infty$
Then for every $h\in\mathcal{E}_{m}^{0}(\Omega)$ we have the weak convergence
$hH_{m}(u_{j})\rightharpoonup hH_{m}(u).$
###### Proof.
For every test function $\chi$ the function $h\chi$ is upper semicontinuous.
Thus,
$\liminf_{j\to+\infty}\int_{\Omega}(-h)\chi
H_{m}(u_{j})\geq\int_{\Omega}(-h)\chi H_{m}(u).$
Let $\Theta$ be any cluster point of this the sequence $(-h)H_{m}(u_{j})$.
From the above inequality we infer that $\Theta\geq(-h)H_{m}(u).$ Moreover, it
follows from Corollary 3.13 that the sequence $\int_{\Omega}(-h)H_{m}(u_{j})$
increases to $\int_{\Omega}(-h)H_{m}(u).$ This implies that the total mass of
$\Theta$ is less than or equal to the total mass of $(-h)H_{m}(u)$ and hence
these measures are equal. ∎
###### Theorem 3.15.
Let $u_{1},...,u_{m}\in\mathcal{E}_{m}^{p}(\Omega),\ p>0$ and
$(g^{j}_{1}),(g^{j}_{2}),...,(g^{j}_{m})\subset\mathcal{E}_{m}^{0}(\Omega)$ be
such that $g^{j}_{k}\downarrow u_{k},\forall k=1,...,m$ and
$\sup_{j,k}{\rm E}_{p}(g^{j}_{k})<+\infty.$
Then the sequence of measures $dd^{c}g^{j}_{1}\wedge
dd^{c}g^{j}_{2}\wedge...\wedge dd^{c}g^{j}_{m}\wedge\beta^{n-m}$ converges
weakly to a positive Radon measure which does not depend on the choice of the
sequences $(g^{j}_{k}).$ We then define $dd^{c}u_{1}\wedge...\wedge
dd^{c}u_{m}\wedge\beta^{n-m}$ to be this weak limit.
###### Proof.
Since the problem is local we can assume that
$u_{1},...,u_{m}\in\mathcal{F}_{m}^{p}(\Omega).$ Indeed, let $K$ be a compact
subset of $\Omega.$ For each $j\in\mathbb{N},k=1,...,m$ consider
$h_{k}^{j}:=\sup\\{u\in\mathcal{SH}_{m}(\Omega)\ /\ u\leq g_{k}^{j}\ {\rm on}\
K\\}.$
Then by using a standard balayage argument we see that $H_{m}(h_{k}^{j})$ is
supported in $K$. It follows that $h_{k}^{j}$ decreases to
$v_{k}\in\mathcal{F}_{m}^{p}(\Omega)$. Observe also that $v_{k}=u_{k}$ on $K.$
Now, fix $h\in\mathcal{E}_{m}^{0}(\Omega).$ Then
$\int_{\Omega}hdd^{c}g^{j}_{1}\wedge dd^{c}g^{j}_{2}\wedge...\wedge
dd^{c}g^{j}_{m}\wedge\beta^{n-m}$
is decreasing to a finite number. Thus the limit
$\lim_{j}\int_{\Omega}hdd^{c}g^{j}_{1}\wedge dd^{c}g^{j}_{2}\wedge...\wedge
dd^{c}g^{j}_{m}\wedge\beta^{n-m}$ exists for every
$h\in\mathcal{E}_{m}^{0}(\Omega).$ In view of Lemma 3.10, this implies the
weak convergence of the sequence
$dd^{c}g^{j}_{1}\wedge dd^{c}g^{j}_{2}\wedge...\wedge
dd^{c}g^{j}_{m}\wedge\beta^{n-m}.$
To prove the last statement it suffices to follow the lines in [8, Theorem
4.2]. ∎
### 3.3. Integration by parts and the comparison principle
In this section we prove that integration by parts is valid in the classes
$\mathcal{E}_{m}^{p}(\Omega),p>0$ and $\mathcal{F}_{m}(\Omega).$ Following a
classical argument of Cegrell in [8] we prove that the comparison principle
holds in $\mathcal{E}_{m}^{p}(\Omega)$ for $0<p\leq 1$. To prove the
comparison principle in $\mathcal{E}_{m}^{p}(\Omega),p>1$ and in
$\mathcal{F}_{m}(\Omega)$ we need strong convergence results which will be
established in the next section.
From Theorem 3.11 and Corollary 3.13 we prove the integration by parts formula
for functions in $\mathcal{F}_{m}(\Omega)$.
###### Theorem 3.16.
Integration by parts is allowed in $\mathcal{F}_{m}(\Omega)$, more precisely
$\int_{\Omega}udd^{c}v\wedge T=\int_{\Omega}vdd^{c}u\wedge T,$
where $u,v,\varphi_{1},...,\varphi_{m-1}\in\mathcal{F}_{m}(\Omega)$ and
$T=dd^{c}\varphi_{1}\wedge...\wedge dd^{c}\varphi_{m-1}\wedge\beta^{n-m}$ and
the equality means that if one of the two terms are finite then they are
equal.
###### Proof.
Let $u_{j},v_{j},\varphi_{1}^{j},..,\varphi_{m-1}^{j}$ be sequences in
$\mathcal{E}_{m}^{0}(\Omega)\cap\mathcal{C}(\Omega)$ decreasing to
$u,v,\varphi_{1},...,\varphi_{m-1}$ respectively such that their total mass
are uniformly bounded:
$\sup_{j}\int_{\Omega}dd^{c}v_{j}\wedge T_{j}<+\infty,\ \
\sup_{j}\int_{\Omega}dd^{c}u_{j}\wedge T_{j}<+\infty,$
where $T_{j}=dd^{c}\varphi_{1}^{j}\wedge...\wedge
dd^{c}\varphi_{m-1}^{j}\wedge\beta^{n-m}.$ Theorem 3.11 gives us that
$dd^{c}u_{j}\wedge T_{j}\rightharpoonup dd^{c}u\wedge T.$ For each fixed
$k\in\mathbb{N}$ and any $j>k$ we have
$\int_{\Omega}v_{k}dd^{c}u_{k}\wedge
T_{k}\geq\int_{\Omega}v_{k}dd^{c}u_{j}\wedge
T_{j}\geq\int_{\Omega}v_{j}dd^{c}u_{j}\wedge T_{j}.$
We then infer that the sequence of real numbers
$\int_{\Omega}v_{j}dd^{c}u_{j}\wedge T$ decreases to some
$a\in\mathbb{R}\cup\\{-\infty\\}$. By letting $j\to+\infty$ and using
Corollary 3.13 we get
$\int_{\Omega}v_{k}dd^{c}u\wedge T\geq a,$
from which we obtain $\int_{\Omega}vdd^{c}u\wedge T\geq a.$ For each fixed $k$
we also have
$\displaystyle\int_{\Omega}vdd^{c}u\wedge T$ $\displaystyle\leq$
$\displaystyle\int_{\Omega}v_{k}dd^{c}u\wedge
T=\lim_{j\to+\infty}\int_{\Omega}v_{k}dd^{c}u_{j}\wedge T_{j}$
$\displaystyle\leq$ $\displaystyle\int_{\Omega}v_{k}dd^{c}u_{k}\wedge T_{k}.$
This implies that $\int_{\Omega}vdd^{c}u\wedge T=a,$ from which the result
follows. ∎
Let $u\in\mathcal{E}_{m}^{p}(\Omega),p>0.$ It is clear form the definition
that the $H_{m}(u)$ does not charge $m$-polar sets. One expects that
$H_{m}(u)$ is dominated by the Capacity. The following results tell us more
about that.
###### Lemma 3.17.
Let $U$ be an open subset of $\Omega$ and
$\varphi\in\mathcal{E}_{m}^{0}(\Omega),\ p\geq 1.$ Then
$\int_{U}H_{m}(\varphi)\leq{\rm Cap}_{m}(U)^{\frac{p}{p+m}}{\rm
E}_{p}(\varphi)^{\frac{m}{p+m}}.$
###### Proof.
We can suppose that $U$ is relatively compact in $\Omega.$ Denote by
$u=u_{m,U,\Omega}$ the $m$-extremal function of $U$ in $\Omega.$ Then
$u\in\mathcal{E}_{m}^{0}(\Omega)$ and $u=-1$ in $U.$ From Lemma 3.5 we have
$\displaystyle\int_{U}H_{m}(\varphi)$ $\displaystyle\leq$
$\displaystyle\int_{\Omega}(-u)^{p}H_{m}(\varphi)\leq{\rm
E}_{p}(u)^{\frac{p}{m+p}}{\rm E}_{p}(\varphi)^{\frac{m}{m+p}}$
$\displaystyle\leq$
$\displaystyle\Big{(}\int_{\Omega}H_{m}(u)\Big{)}^{\frac{p}{m+p}}{\rm
E}_{p}(\varphi)^{\frac{m}{p+m}}={\rm Cap}_{m}(U)^{\frac{p}{p+m}}{\rm
E}_{p}(\varphi)^{\frac{m}{p+m}}.$
∎
###### Lemma 3.18.
Let $U\subset\Omega$ be an open subset and
$\varphi\in\mathcal{E}_{m}^{0}(\Omega),\ 0<p\leq 1.$ Then for each
$\epsilon>0$ small enough we have
$\int_{U}H_{m}(\varphi)\leq 2({\rm Cap}_{m}(U))^{1-m\epsilon}+2{\rm
Cap}_{m}(U)^{p\epsilon}.{\rm E}_{p}(\varphi).$
###### Proof.
Without loss of generality we can assume that $U\Subset\Omega.$ Let $u$ be the
$m$-extremal function of $U$ with respect to $\Omega.$ Put $a={\rm
Cap}_{m}(U)=\int_{\Omega}H_{m}(u).$ If $a=0$, we are done. Thus, we can assume
that $a>0.$ By applying Lemma 3.6 we obtain
$\displaystyle\int_{U}H_{m}(\varphi)$ $\displaystyle\leq$ $\displaystyle
a^{p\epsilon}\int_{\Omega}(-u/a^{\epsilon})^{p}H_{m}(\varphi)$
$\displaystyle\leq$ $\displaystyle 2a^{p\epsilon}{\rm
E}_{p}(u/a^{\epsilon})+2a^{p\epsilon}{\rm E}_{p}(\varphi)$ $\displaystyle\leq$
$\displaystyle 2a^{1-m\epsilon}+2a^{p\epsilon}{\rm E}_{p}(\varphi).$
∎
The following result is the so-called maximum principle.
###### Theorem 3.19.
Let $u_{1},...,u_{m}\in\mathcal{E}_{m}^{p}(\Omega),\ p>0$ and
$v\in\mathcal{SH}_{m}(\Omega)$. Then
$1{\hskip-2.5pt}\hbox{{I}}_{A}H_{m}(u_{1},...,u_{m})=1{\hskip-2.5pt}\hbox{{I}}_{A}H_{m}(\max(u_{1},v),...,\max(u_{m},v)),$
where $A=\cap_{j=1}^{m}\\{u_{j}>v\\}$ and
$H_{m}(u_{1},...,u_{m})=dd^{c}u_{1}\wedge...\wedge
dd^{c}u_{m}\wedge\beta^{n-m}.$
###### Proof.
Let $(u_{1}^{j},...u_{m}^{j})$ be sequences in $\mathcal{E}_{m}^{0}(\Omega)$
decreasing to $u_{1},...,u_{m}$ respectively as in the definition of
$\mathcal{E}_{m}^{p}(\Omega).$ We can assume that they are continuous in
$\Omega.$ Set $v_{k}^{j}:=\max(u_{k}^{j},v),k=1,...,m.$ Then since the set
$A_{j}:=\cap_{k=1}^{m}\\{u_{k}^{j}>v\\}$ is open, we get
(3.3)
$1{\hskip-2.5pt}\hbox{{I}}_{A_{j}}H_{m}(u_{1}^{j},...,u_{m}^{j})=1{\hskip-2.5pt}\hbox{{I}}_{A_{j}}H_{m}(v_{1}^{j},...,v_{m}^{j}).$
Set $u^{j}:=\min(u_{1}^{j},...,u_{m}^{j})$ and $u:=\min(u_{1},...,u_{m}).$
Consider $\psi_{j}:=\max(u^{j}-v,0)$. Then
$\psi_{j}\downarrow\psi:=\max(u-v,0)$, all of them are quasi-continuous.
Fix $\delta>0$ and set $g_{j}:=\frac{\psi_{j}}{\psi_{j}+\delta},\
g=\frac{\psi}{\psi+\delta}.$ By multiplying (3.3) with $g_{j}$ we obtain
(3.4)
$g_{j}H_{m}(u_{1}^{j},...,u_{m}^{j})=g_{j}H_{m}(v_{1}^{j},...,v_{m}^{j}).$
Now, let $\chi\in\mathcal{C}_{0}^{\infty}(\Omega)$ be a test function and fix
$\epsilon>0.$ By Theorem 2.9, there exists an open subset $U\subset\Omega$
such that ${\rm Cap}_{m}(U)<\epsilon,$ and there exist $\varphi_{j},\varphi$
continuous functions in $\Omega$ which coincide with $\psi_{j},\psi$
respectively on $K:=\Omega\setminus U.$ The monotone convergence
$\psi_{j}\downarrow\psi$ implies that $\varphi_{j}$ converges uniformly to
$\varphi$ on $K\cap{\rm Supp}\chi,$ which in turn implies the uniform
convergence of $h_{j}=\frac{\varphi_{j}}{\varphi_{j}+\delta}$ on $K\cap{\rm
Supp}\chi$ to $h=\frac{\varphi}{\varphi+\delta}.$
In the next arguments, we let $C$ denote a positive constant which does not
depend on $j,\epsilon.$ Since $g_{j},h_{j}$ are uniformly bounded , Lemma 3.17
and Lemma 3.18 give us
(3.5) $\Big{|}\int_{\Omega}\chi
g_{j}H_{m}(u_{1}^{j},...,u_{m}^{j})-\int_{\Omega}\chi
h_{j}H_{m}(u_{1}^{j},...,u_{m}^{j})\Big{|}\leq
C.\int_{U}H_{m}(u_{1}^{j},...,u_{m}^{j})\leq C.\epsilon^{q},$
where $q$ is some positive constant. The last inequality follows since
$H_{m}(u_{1},...,u_{m})\leq H_{m}(u_{1}+...+u_{m})$ and since
$\mathcal{E}_{m}^{p}(\Omega)$ is convex. We also obtain
(3.6) $\displaystyle\Big{|}\int_{\Omega}\chi
g\,H_{m}(u_{1},...,u_{m})-\int_{\Omega}\chi h\,H_{m}(u_{1},...,u_{m})\Big{|}$
$\displaystyle\leq$ $\displaystyle C.\int_{U}H_{m}(u_{1}+...+u_{m})$
$\displaystyle\leq$ $\displaystyle C.\epsilon^{q}.$
Moreover, since $h$ is continuous on $\Omega$ and
$H_{m}(u_{1}^{j},...,u_{m}^{j})\rightharpoonup H_{m}(u_{1},...,u_{m})$, we get
$\lim_{j\to+\infty}\int_{\Omega}\chi.h(H_{m}(u_{1}^{j},...,u_{m}^{j})-H_{m}(u_{1},...,u_{m}))=0.$
Hence, we obtain
$\displaystyle\limsup_{j\to+\infty}\Big{|}\int_{\Omega}\chi
h_{j}\,H_{m}(u_{1}^{j},...,u_{m}^{j})-\int_{\Omega}\chi
h\,H_{m}(u_{1},...,u_{m})\Big{|}$ $\displaystyle\leq$
$\displaystyle\limsup_{j\to+\infty}\int_{\Omega}\chi.|h_{j}-h|H_{m}(u_{1}^{j},...,u_{m}^{j}).$
Since $h_{j}$ converges uniformly to $h$ on $K\cap\rm{supp}\chi$, we have
$\displaystyle\int_{\Omega}\chi.|h_{j}-h|H_{m}(u_{1}^{j},...,u_{m}^{j})$
$\displaystyle=$
$\displaystyle\int_{U}\chi.|h_{j}-h|H_{m}(u_{1}^{j},...,u_{m}^{j})$
$\displaystyle+$
$\displaystyle\int_{K}\chi.|h_{j}-h|H_{m}(u_{1}^{j},...,u_{m}^{j})$
$\displaystyle\leq$ $\displaystyle C.\int_{U}H_{m}(u_{1}^{j},...,u_{m}^{j})$
$\displaystyle+$
$\displaystyle\|h_{j}-h\|_{L^{\infty}(K\cap\rm{supp}\chi)}\int_{\Omega}\chi
H_{m}(u_{1}^{j},...,u_{m}^{j}).$
From the two inequalities above we get
(3.7) $\displaystyle\limsup_{j\to+\infty}\Big{|}\int_{\Omega}\chi
h_{j}\,H_{m}(u_{1}^{j},...,u_{m}^{j})-\int_{\Omega}\chi
h\,H_{m}(u_{1},...,u_{m})\Big{|}$ $\displaystyle\leq$ $\displaystyle
C.\epsilon^{q},$
where $q$ is some positive constant. From (3.5), (3.6) and (3.7), we see that
$\limsup_{j}\Big{|}\int_{\Omega}\chi
g_{j}H_{m}(u_{1}^{j},...,u_{m}^{j})-\int_{\Omega}\chi
gH_{m}(u_{1},...,u_{m})\Big{|}\leq C.\epsilon^{q}.$
We then see that $g_{j}H_{m}(u_{1}^{j},...,u_{m}^{j})\rightharpoonup
gH_{m}(u_{1},...,u_{m})$. In the same way, we get
$g_{j}H_{m}(v_{1}^{j},...,v_{m}^{j})\rightharpoonup gH_{m}(v_{1},...,v_{m}),$
and hence $gH_{m}(u_{1},...,u_{m})=gH_{m}(v_{1},...,v_{m}).$ The result
follows by letting $\delta$ go to zero. ∎
We now return to the integration by parts formula in the class
$\mathcal{E}_{m}^{p}(\Omega)$, $p>0$. We first need the following convergence
result.
###### Corollary 3.20.
Let $u_{1},...,u_{m}\in\mathcal{E}_{m}^{p}(\Omega),p>0$ and
$u_{1}^{j},...,u_{m}^{j}$ be sequences of functions in
$\mathcal{E}_{m}^{0}(\Omega)$ decreasing to $u_{1},...,u_{m}$ respectively
such that
$\sup_{j,k}\int_{\Omega}(-u_{k}^{j})^{p}H_{m}(u_{k}^{j})<+\infty.$
Then for each $\varphi\in\mathcal{E}_{m}^{0}(\Omega)$ we have
$\lim_{j\to+\infty}\int_{\Omega}\varphi dd^{c}u_{1}^{j}\wedge...\wedge
dd^{c}u_{m}^{j}\wedge\beta^{n-m}=\int_{\Omega}\varphi
dd^{c}u_{1}\wedge...\wedge dd^{c}u_{m}\wedge\beta^{n-m}.$
###### Proof.
Without loss of generality we can assume that the right hand side is finite.
Since if it is $-\infty$ then the equality is obvious. We can also assume that
$-1\leq\varphi\leq 0.$ We will use a truncation argument. For each
$k,j\in\mathbb{N}$ set $u_{l,k}^{j}:=\max(u_{l}^{j},k\psi)$, and
$u_{l,k}:=\max(u_{l},k\psi)$, $l=1,...,m.$ Here we set
$\psi:=-(-\varphi)^{q}$, where $q=\min(1,1/p).$
We claim that, for any $k$,
$\lim_{j\to+\infty}\int_{\Omega}(-\varphi)H_{m}(u_{1,k}^{j},...,u_{m,k}^{j})=\int_{\Omega}(-\varphi)H_{m}(u_{1,k},...,u_{m,k}).$
Indeed, the inequality "$\geq$" follows from the fact that the sequence of
Hessian measures converges and $-\varphi$ is lower semi continuous. Moreover,
it follows from Theorem 2.11 that we can integrate by parts in the right hand
side, which implies the inequality $"\leq"$. Thus, the Claim is proved.
Thus, it is enough to prove that
$\Big{|}\int_{\Omega}(-\varphi)H_{m}(u_{1,k}^{j},...,u_{m,k}^{j})-\int_{\Omega}(-\varphi)H_{m}(u_{1}^{j},...,u_{m}^{j})\Big{|}\leq\epsilon(k),$
where $0<\epsilon(k)\to 0$ as $k\to+\infty$ and (of course) $\epsilon(k)$ does
not depend on $j.$ If we can prove it then the same estimate holds for the
limit functions $u_{1},...,u_{m}$ and we are done.
In the following arguments we use $C_{1},C_{2},...$ to denote positive
constants that do not depend on $j,k.$ By Theorem 3.19 and since $\varphi$ is
bounded it suffices to estimate
$\int_{\\{u_{l}^{j}\leq
k\psi\\}}(-\varphi)H_{m}(u_{1,k}^{j},...,u_{m,k}^{j}),\ \ l=1,...,m.$
But we can bound this term by using Lemma 3.5 (for $p\geq 1$), Proposition 3.7
(for $0<p<1$) and the fact that the $p$-energy of these functions are
uniformly bounded:
$\displaystyle\int_{\\{u_{l}^{j}\leq
k\psi\\}}(-\varphi)H_{m}(u_{1,k}^{j},...,u_{m,k}^{j})$ $\displaystyle\leq$
$\displaystyle\frac{1}{k^{p}}\int_{\Omega}(-u_{l,k}^{j})^{p}H_{m}(u_{1,k}^{j},...,u_{m,k}^{j})\leq\frac{C_{1}}{k^{p}}.$
In the last step the constant $C_{1}$ depends on the $p$-energy of
$u_{l,k}^{j},l=1,...,m$. Note also that $u_{l,k}^{j}\geq u_{l}^{j}$. Now, for
functions $u,v\in\mathcal{E}_{m}^{0}(\Omega)$ such that $u\leq v$ we always
have $E_{p}(v)\leq C_{2}.E_{p}(u)$ (where $C_{2}$ does not depend on $u,v$).
To see this we can use integration by parts (if $p<1$) or use Lemma 3.5. This
explains why the constant $C_{1}$ in the above estimate does not depend on
$k.$ Thus, the proof is complete. ∎
###### Theorem 3.21.
Integration by parts is allowed in $\mathcal{E}_{m}^{p}(\Omega),p>0.$ More
precisely, assume that $u,v\in\mathcal{E}_{m}^{p}(\Omega)$ and $T$ is a closed
$m$-positive current of type
$T=dd^{c}\varphi_{1}\wedge...dd^{c}\varphi_{m-1}\wedge\beta^{n-m}$, where
$\varphi_{j}\in\mathcal{E}_{m}^{p}(\Omega),\forall j.$ Then
$\int_{\Omega}udd^{c}v\wedge T=\int_{\Omega}vdd^{c}u\wedge T,$
where the equality means that if one of the two terms is finite then so is the
other and they are equal.
###### Proof.
Thanks to Corollary 3.20 the same arguments as in the proof of Theorem 3.16
can be used here. ∎
###### Theorem 3.22.
Let $u,v\in\mathcal{E}_{m}^{p}(\Omega),p>0$ (or $\mathcal{F}_{m}(\Omega)$)
such that $u\leq v$ on $\Omega.$ Then
$\int_{\Omega}H_{m}(u)\geq\int_{\Omega}H_{m}(v).$
###### Proof.
Let $(u_{j}),(v_{j})$ be two sequences in $\mathcal{E}_{m}^{0}(\Omega)$
decreasing to $u,v$ as in the definition of $\mathcal{E}_{m}^{p}(\Omega).$ Fix
$h\in\mathcal{E}_{m}^{0}(\Omega)\cap\mathcal{C}(\Omega).$ We can suppose that
$u_{j}\leq v_{j},\ \forall j.$ Integrating by parts we get
$\int_{\Omega}(-h)H_{m}(v_{j})\leq\int_{\Omega}(-h)H_{m}(u_{j}).$
Corollary 3.13 and Corollary 3.20 then yield
$\lim_{j\to\infty}\int_{\Omega}(-h)H_{m}(v_{j})=\int_{\Omega}(-h)H_{m}(v),\ \
\ {\rm and}\ \ \
\lim_{j\to\infty}\int_{\Omega}(-h)H_{m}(u_{j})=\int_{\Omega}(-h)H_{m}(u).$
Combining them we obtain
$\int_{\Omega}(-h)H_{m}(v)\leq\int_{\Omega}(-h)H_{m}(u).$
The result follows by letting $h$ decrease to $-1.$ ∎
###### Theorem 3.23.
If $u\in\mathcal{E}_{m}^{p}(\Omega)$, $p>0$, then ${\rm
E}_{p}(u):=\int_{\Omega}(-u)^{p}H_{m}(u)<+\infty.$ If
$(u_{0}^{j}),...,(u_{m}^{j})$ are sequences in $\mathcal{E}_{m}^{0}(\Omega)$
decreasing to $u_{0},...,u_{m}\in\mathcal{E}_{m}^{p}(\Omega)$ respectively
then
$\int_{\Omega}(-u_{0}^{j})dd^{c}u_{1}^{j}\wedge...\wedge
dd^{c}u_{m}^{j}\wedge\beta^{n-m}\nearrow\int_{\Omega}(-u_{0})dd^{c}u_{1}\wedge...\wedge
dd^{c}u_{m}\wedge\beta^{n-m}.$
###### Proof.
Let $(u_{j})$ be a sequence in $\mathcal{E}_{m}^{0}(\Omega)$ decreasing to $u$
and having uniformly bounded $p$-energy. Then
$\int_{\Omega}(-u)H_{m}(u)\leq\liminf_{j\to+\infty}\int_{\Omega}(-u_{j})H_{m}(u_{j})<+\infty.$
We now prove the second statement. We can assume that the sequences have
uniformly bounded $p$-energy. It follows from Theorem 3.15 that
$T_{j}:=dd^{c}u_{1}^{j}\wedge...\wedge
dd^{c}u_{m}^{j}\wedge\beta^{n-m}\rightharpoonup T:=dd^{c}u_{1}\wedge...\wedge
dd^{c}u_{m}\wedge\beta^{n-m}.$
Furthermore since $(-u_{0}^{j})\uparrow(-u_{0})$ and since all of them are
lower semicontinuous, we have
$\liminf_{j}\int_{\Omega}(-u_{0}^{j})T_{j}\geq\int_{\Omega}(-u_{0}^{j})T.$
Thus, it suffices to prove that
$\int_{\Omega}(-u_{0})T_{j}\leq\int_{\Omega}(-u_{0})T,\forall j.$
But it can be easily seen by integrating by parts thanks to Theorem 3.21. The
proof is thus complete. ∎
###### Theorem 3.24.
If $0<p\leq 1$ and $u,v\in\mathcal{E}_{m}^{p}(\Omega)$ then
$\int_{\\{u>v\\}}H_{m}(u)\leq\int_{\\{u>v\\}}H_{m}(v).$
###### Proof.
Fix $h\in\mathcal{E}_{m}^{0}(\Omega)\cap\mathcal{C}(\Omega).$ The measure
$H_{m}(v)$ does not charge $m$-polar sets. We can easily show that for almost
every $r,$
$\int_{\\{v=ru\\}}(-h)H_{m}(v)=0.$
This allows us to restrict ourself to the case
$\int_{\\{u=v\\}}(-h)H_{m}(v)=0.$ From Theorem 3.19, we get
$1{\hskip-2.5pt}\hbox{{I}}_{\\{u>v\\}}H_{m}(u)=1{\hskip-2.5pt}\hbox{{I}}_{\\{u>v\\}}H_{m}(\max(u,v)),\
\text{and}\
1{\hskip-2.5pt}\hbox{{I}}_{\\{u<v\\}}H_{m}(v)=1{\hskip-2.5pt}\hbox{{I}}_{\\{u<v\\}}H_{m}(\max(u,v)).$
Furthermore, as in the proof of Theorem 3.22, we can prove that
$\int_{\Omega}(-h)H_{m}(\max(u,v))\leq\int_{\Omega}(-h)H_{m}(u).$
From this we get
$\displaystyle\int_{\\{u>v\\}}(-h)H_{m}(u)$ $\displaystyle=$
$\displaystyle\int_{\\{u>v\\}}(-h)H_{m}(\max(u,v))$ $\displaystyle\leq$
$\displaystyle\int_{\Omega}(-h)H_{m}(\max(u,v))+\int_{\\{u<v\\}}hH_{m}(\max(u,v))$
$\displaystyle\leq$
$\displaystyle\int_{\Omega}(-h)H_{m}(v)+\int_{\\{u<v\\}}hH_{m}(v)=\int_{\\{u>v\\}}(-h)H_{m}(v).$
The above arguments hold since all terms are finite. This is no longer true if
$p>1.$ Now, letting $h\downarrow-1$ we obtain the result. ∎
###### Remark 3.25.
We proved in the above arguments that
$\int_{\\{u>v\\}}(-h)H_{m}(v)\leq\int_{\\{u>v\\}}(-h)H_{m}(u)$
if $u,v\in\mathcal{E}_{m}^{p}(\Omega),0<p\leq 1$ and
$h\in\mathcal{E}_{m}^{0}(\Omega)\cap\mathcal{C}(\Omega).$ Thanks to the
regularization theorem (Theorem 3.1) it also holds for every
$h\in\mathcal{SH}_{m}^{-}(\Omega).$
###### Theorem 3.26.
Let $u,v\in\mathcal{E}_{m}^{p}(\Omega)$, $0<p\leq 1$, such that $H_{m}(u)\geq
H_{m}(v).$ Then $u\leq v$ in $\Omega$.
###### Proof.
See [7, Theorem 4.5]. ∎
## 4\. The variational approach
In this section we use a variational method to solve the equation
$H_{m}(u)=\mu,$ where $\mu$ is a positive Radon measure. We characterize the
range of $H_{m}(u)$ when $u$ runs in $\mathcal{E}_{m}^{p}(\Omega).$
Our results are direct generalizations of the classical case of
plurisubharmonic functions (see [2], [7, 8]). The variational approach for the
complex Monge-Ampère equation was first introduced in [5].
### 4.1. The energy functional
We recall some useful results obtained from previous sections. For
$\varphi\in\mathcal{E}^{1}_{m}(\Omega)$, we define its energy by ${\rm
E}(\varphi)=\int_{\Omega}(-\varphi)H_{m}(\varphi).$
* •
If $0\geq u_{j}\downarrow u$ and $u\in\mathcal{E}^{1}_{m}(\Omega)$, then by
Theorem 3.23, we have ${\rm E}(u_{j})\uparrow{\rm E}(u).$
* •
If $u,v\in\mathcal{E}^{1}_{m}(\Omega)$ and $u\leq v$ then ${\rm E}(u)\geq{\rm
E}(v).$
###### Lemma 4.1.
(i) If $(u_{j})\subset\mathcal{E}^{1}_{m}(\Omega)$ then
$(\sup_{j}u_{j})^{*}\in\mathcal{E}^{1}_{m}(\Omega).$
(ii) If $(u_{j})\in\mathcal{E}^{1}_{m}(\Omega)$ such that $\sup_{j}{\rm
E}(u_{j})<+\infty$ and $u_{j}\downarrow u,$ then
$u\in\mathcal{E}^{1}_{m}(\Omega).$
(iii) For each $C>0$,
$\mathcal{E}_{m}^{1,C}:=\\{u\in\mathcal{E}^{1}_{m}(\Omega)\ /\ {\rm E}(u)\leq
C\\}$ is convex and compact in $\mathcal{SH}_{m}(\Omega).$
###### Proof.
(i) Let $(\varphi_{j})$ be a sequence of continuous functions in
$\mathcal{E}_{m}^{0}(\Omega)$ decreasing to $\varphi:=(\sup_{j}u_{j})^{*}.$
Since $u_{j}\leq\varphi_{j}$, we have $\sup_{j}{\rm E}(\varphi_{j})<+\infty,$
which implies that $\varphi\in\mathcal{E}^{1}_{m}(\Omega).$
(ii) Let $(\varphi_{j})$ be a sequence in
$\mathcal{E}_{m}^{0}(\Omega)\cap\mathcal{C}(\Omega)$ decreasing to $u.$ Set
$\psi_{j}:=\max(u_{j},\varphi_{j})$. Then
$\psi_{j}\in\mathcal{E}^{1}_{m}(\Omega),\forall j$ and ${\rm
E}(\psi_{j})\leq{\rm E}(u_{j})$. Thus, $u\in\mathcal{E}^{1}_{m}(\Omega).$
(iii) Let $(u_{j})$ be a sequence in $\mathcal{E}_{m}^{1,C}$. Since
$\sup_{j}{\rm E}(u_{j})<+\infty,$ $(u_{j})$ can not go uniformly to $-\infty$
in $\Omega.$ Thus, there exists a subsequence (still denoted by $(u_{j})$)
converging to $u\in\mathcal{SH}_{m}(\Omega)$ in $L^{1}_{\rm loc}(\Omega)$. Set
$\varphi_{j}:=(\sup_{k\geq j}u_{k})^{*}\in\mathcal{E}^{1}_{m}(\Omega),\forall
j.$
Then $\varphi_{j}\downarrow u$ and $\sup_{j}{\rm E}(\varphi_{j})\leq C.$ In
view of (ii), we have $u\in\mathcal{E}^{1}_{m}(\Omega),$ and since
$(-\varphi_{j})\uparrow(-u)$, all of them being lower semicontinuous we get,
for each fixed $k\in\mathbb{N}$,
$C\geq\liminf_{j\to\infty}\int_{\Omega}(-\varphi_{j})H_{m}(\varphi_{j})\geq\liminf_{j\to\infty}\int_{\Omega}(-\varphi_{k})H_{m}(\varphi_{j})\geq\int_{\Omega}(-\varphi_{k})H_{m}(u).$
By monotone convergence Theorem we see that $E_{1}(u)\leq C.$ This means
$u\in\mathcal{E}_{m}^{1,C}.$ ∎
###### Lemma 4.2.
Let $\mu$ be a positive Radon measure in $\Omega$ such that
$\mu(\Omega)<+\infty$ and $\mu$ does not charge $m$-polar sets. Let $(u_{j})$
be a sequence in $\mathcal{SH}_{m}^{-}(\Omega)$ which converges in $L^{1}_{\rm
loc}$ to $u\in\mathcal{SH}_{m}^{-}(\Omega).$ If
$\sup_{j}\int_{\Omega}(-u_{j})^{2}d\mu<+\infty$ then
$\int_{\Omega}u_{j}d\mu\to\int_{\Omega}ud\mu.$
###### Proof.
Since $\int_{\Omega}u_{j}d\mu$ is bounded it suffices to prove that every
cluster point is $\int_{\Omega}ud\mu.$ Without loss of generality we can
assume that $\int_{\Omega}u_{j}d\mu$ converges. Since the sequence $u_{j}$ is
bounded in $L^{2}(\mu)$, one can apply Banach-Saks theorem to extract a
subsequence (still denoted by $u_{j}$) such that
$\varphi_{N}:=\frac{1}{N}\sum_{j=1}^{N}u_{j}$
converges in $L^{2}(\mu)$ and $\mu$-almost everywhere to $\varphi.$ Observe
also that $\varphi_{N}\to u$ in $L^{1}_{\rm loc}$. For each $j\in\mathbb{N}$
set
$\psi_{j}:=(\sup_{k\geq j}\varphi_{k})^{*}.$
Then $\psi_{j}\downarrow u$ in $\Omega.$ But $\mu$ does not charge the
$m$-polar set $\\{(\sup_{k\geq j}\varphi_{k})^{*}>\sup_{k\geq
j}\varphi_{k}\\}.$ We thus get $\psi_{j}=\sup_{k\geq j}\varphi_{k}$
$\mu$-almost everywhere. Therefore, $\psi_{j}$ converges to $\varphi$
$\mu$-almost everywhere hence $u=\varphi$ $\mu$-almost everywhere. This yields
$\lim_{j}\int_{\Omega}u_{j}d\mu=\lim_{j}\int_{\Omega}\varphi_{j}d\mu=\int_{\Omega}ud\mu.$
∎
###### Lemma 4.3.
The functional ${\rm E}:\mathcal{E}^{1}_{m}(\Omega)\rightarrow\mathbb{R}$ is
lower semicontinuous.
###### Proof.
Suppose that $u,u_{j}\in\mathcal{E}^{1}_{m}(\Omega)$ and $u_{j}$ converges to
$u$ in $L^{1}_{\rm loc}(\Omega).$ We are to prove that $\liminf_{j}{\rm
E}(u_{j})\geq{\rm E}(u).$ For each $j\in\mathbb{N}$, the function
$\varphi_{j}:=(\sup_{k\geq j}u_{k})^{*}$
belongs to $\mathcal{E}^{1}_{m}(\Omega)$ and $\varphi_{j}\downarrow u.$ Hence
${\rm E}(\varphi_{j})\uparrow{\rm E}(u)$. We also have ${\rm E}(u_{j})\geq{\rm
E}(\varphi_{j})$ from which the result follows. ∎
###### Definition 4.4.
Let $\mu$ be a positive Radon measure in $\Omega.$ The functional
$\mathcal{F}_{\mu}:\mathcal{E}^{1}_{m}(\Omega)\rightarrow\mathbb{R}$ is
defined by
$\mathcal{F}_{\mu}(u)=\frac{1}{m+1}{\rm E}(u)+\mathcal{L}_{\mu}(u),$
where $\mathcal{L}_{\mu}(u)=\int_{\Omega}ud\mu.$ We say that
$\mathcal{F}_{\mu}$ is proper (with respect to ${\rm E}$) if
$\mathcal{F}_{\mu}\to+\infty$ whenever ${\rm E}\to+\infty.$
###### Definition 4.5.
For each $p>0$, let $\mathcal{M}_{p}$ denote the set of all positive Radon
measures $\mu$ in $\Omega$ such that $\mathcal{E}_{m}^{p}(\Omega)\subset
L^{p}(\Omega,\mu).$
###### Proposition 4.6.
Let $\mu$ be a positive Radon measure on $\Omega$ and $p>0.$ Then
$\mu\in\mathcal{M}_{p}$ if and only if there exists a positive constant
$C=C(p)>0$ such that
$\int_{\Omega}(-u)^{p}d\mu\leq C.{\rm E}_{p}(u)^{\frac{p}{m+p}},\ \forall
u\in\mathcal{E}_{m}^{p}(\Omega).$
###### Proof.
The "if" statement is evident. To prove the "only if", suppose by
contradiction that $\mu\in\mathcal{M}_{p}$ and there exists a sequence
$(u_{j})\subset\mathcal{E}_{m}^{p}(\Omega)$ such that
$\int_{\Omega}(-u_{j})^{p}d\mu\geq 4^{jp}{\rm E}_{p}(u_{j})^{\frac{p}{m+p}}.$
For simplicity we can assume that ${\rm E}_{p}(u_{j})=1,\forall j.$ By
Corollary 3.8, $v=\sum_{j=1}^{\infty}2^{-j}v_{j}$ belongs to
$\mathcal{E}_{m}^{p}(\Omega).$ But
$\int_{\Omega}(-v)^{p}d\mu\geq\int_{\Omega}(-2^{-j}v_{j})^{p}d\mu\geq
2^{jp}\to+\infty,$
which contradicts $\mathcal{E}_{m}^{p}(\Omega)\subset L^{p}(\mu).$ ∎
###### Remark 4.7.
If $u,v\in\mathcal{E}_{m}^{p}(\Omega)$ and
$(u_{j}),(v_{j})\subset\mathcal{E}_{m}^{0}(\Omega)$ decrease to $u,v$
respectively, then by Lemma 3.5 and Proposition 3.7 we have
$\int_{\Omega}(-u)^{p}H_{m}(v)\leq\liminf_{j}\int_{\Omega}(-u_{j})^{p}H_{m}(v_{j})<+\infty.$
Thus, $\mathcal{E}_{m}^{p}(\Omega)\subset L^{p}(H_{m}(v))$ and by Proposition
4.6 there exists $C_{v}>0$ such that
$\int_{\Omega}(-u)^{p}H_{m}(v)\leq C_{v}{\rm E}_{p}(u)^{\frac{p}{p+m}},\
\forall u\in\mathcal{E}_{m}^{p}(\Omega).$
It is not clear how to obtain this inequality directly by using Hölder
inequality.
###### Lemma 4.8.
If $u,v\in\mathcal{E}^{1}_{m}(\Omega)$ then ${\rm
E}(u+v)^{\frac{1}{m+1}}\leq\,{\rm E}(u)^{\frac{1}{m+1}}+\,{\rm
E}(v)^{\frac{1}{m+1}}\,.$ Moreover, if $\mu\in\mathcal{M}_{1}$ then
$\mathcal{F}_{\mu}$ is convex and proper.
###### Proof.
It follows from Lemma 3.5 that
${\rm E}(u+v)\leq{\rm E}(u)^{\frac{1}{m+1}}\,{\rm E}(u+v)^{\frac{m}{m+1}}+{\rm
E}(v)^{\frac{1}{m+1}}\,{\rm E}(u+v)^{\frac{m}{m+1}}\ ,$
which implies that ${\rm E}^{\frac{1}{m+1}}$ is convex since it is homogeneous
of degree $1$. So, ${\rm E}$ is also convex. If $\mu$ belongs to
$\mathcal{M}_{1}$, there exists $A>0$ such that
$\|u\|_{L^{1}(\mu)}\leq A.{\rm E}(u)^{\frac{1}{1+m}}\,,\qquad\text{for every
}u\in\mathcal{E}^{1}_{m}(\Omega)\,.$
We thus obtain
$\mathcal{F}_{\mu}(u_{j})=\frac{1}{m+1}{\rm
E}(u_{j})-\|u_{j}\|_{L^{1}(\mu)}\geq\frac{1}{m+1}{\rm E}(u_{j})-A.{\rm
E}(u_{j})^{\frac{1}{m+1}}\to\infty.$
∎
Let $u:\Omega\rightarrow\mathbb{R}\cup\\{-\infty\\}$ be an upper
semicontinuous function. Suppose that there exists
$w\in\mathcal{E}^{1}_{m}(\Omega)$ such that $w\leq u.$ We define the
projection of $u$ on $\mathcal{E}^{1}_{m}(\Omega)$ by
$P(u):=\sup\\{v\in\mathcal{E}^{1}_{m}(\Omega)\ /\ v\leq u\\}.$
Using a standard balayage argument, we obtain the following result.
###### Lemma 4.9.
Let $u:\Omega\rightarrow\mathbb{R}$ be a continuous function. Suppose that
there exists $w\in\mathcal{E}^{1}_{m}(\Omega)$ such that $w\leq u.$ Then
$\int_{\\{P(u)<u\\}}H_{m}(P(u))=0.$
###### Proof.
Without loss of generality we can assume that $w$ is bounded. From Choquet’s
lemma, there exists an increasing sequence
$(u_{j})\subset\mathcal{E}^{1}_{m}(\Omega)\cap L^{\infty}(\Omega)$ such that
$(\lim_{j}u_{j})^{*}=P(u).$
Let $x_{0}\in\\{P(u)<u\\}.$ Since $u$ is continuous, there exists
$\epsilon>0,r>0$ such that
$P(u)(x)<u(x_{0})-\epsilon<u(x),\ \ \forall x\in B=B(x_{0},r).$
For each fixed $j$, by approximating $u_{j}|_{\partial B}$ from above by a
sequence of continuous functions on $\partial B$ and by using [9, Theorem
2.10], we can find a function $\varphi_{j}\in\mathcal{SH}_{m}(B)$ such that
$\varphi_{j}=u_{j}$ on $\partial B$ and $H_{m}(\varphi_{j})=0$ in $B.$ The
comparison principle gives us that $\varphi_{j}\geq u_{j}$ in $B$. The
function $\psi_{j}$, defined by $\psi_{j}=\varphi_{j}$ in $B$ and
$\psi_{j}=u_{j}$ in $\Omega\setminus B,$ belongs to
$\mathcal{E}^{1}_{m}(\Omega)\cap L^{\infty}(\Omega).$ For each $x\in\partial
B$ we have $\varphi_{j}(x)=u_{j}(x)\leq P(u)(x)\leq u(x_{0})-\epsilon.$ It
then follows that $\varphi_{j}\leq u(x_{0})-\epsilon$ in $B$ since
$u(x_{0})-\epsilon$ is a constant and $\varphi_{j}$ is $m$-sh. Hence,
$u_{j}\leq\psi_{j}\leq u$ in $\Omega.$ This implies that
$(\lim\psi_{j})^{*}=P(u).$
It follows from Theorem 2.10 that $H_{m}(\psi_{j})\rightharpoonup
H_{m}(P(u)).$ Therefore,
$H_{m}(P(u))(B)\leq\liminf_{j\to+\infty}H_{m}(\psi_{j})(B)=0,$
from which the result follows. ∎
###### Lemma 4.10.
Let $u,v\in\mathcal{E}^{1}_{m}(\Omega)$ and suppose that $v$ is continuous.
For each $t<0,$ we define
$h_{t}=\frac{P(u+tv)-tv-u}{t}.$
Then for each $0\leq k\leq m,$
(4.1) $\lim_{t\nearrow
0}\int_{\Omega}h_{t}(dd^{c}u)^{k}\wedge(dd^{c}P(u+tv))^{m-k}\wedge\beta^{n-m}=0\,.$
In particular,
(4.2) $\lim_{t\nearrow
0}\int_{\Omega}\frac{P(u+tv)-u}{t}(dd^{c}u)^{k}\wedge(dd^{c}P(u+tv))^{m-k}\wedge\beta^{n-m}=\int_{\Omega}vH_{m}(u)\,.$
###### Proof.
An easy computation shows that $h_{t}$ is decreasing in $t$ and $0\leq
h_{t}\leq-v.$ For each fixed $s<0$ we have
$\displaystyle\lim_{t\nearrow
0}\int_{\Omega}h_{t}(dd^{c}u)^{k}\wedge(dd^{c}P(u+tv))^{m-k}\wedge\beta^{n-m}$
$\displaystyle\leq\lim_{t\nearrow
0}\int_{\Omega}h_{s}(dd^{c}u)^{k}\wedge(dd^{c}P(u+tv))^{m-k}\wedge\beta^{n-m}$
$\displaystyle=\int_{\Omega}h_{s}(dd^{c}u)^{m}\wedge\beta^{n-m}\leq\int\limits_{\\{P(u+sv)-sv<u\\}}(-v)(dd^{c}u)^{m}\wedge\beta^{n-m}.$
Let $u_{k}\in\mathcal{E}_{m}^{0}(\Omega)\cap C(\Omega)$ be a sequence
decreasing to $u$ such that
$\int\limits_{\\{P(u+sv)-sv<u\\}}(-v)(dd^{c}u)^{m}\wedge\beta^{n-m}\leq
2\int\limits_{\\{P(u_{k}+sv)-sv<u\\}}(-v)(dd^{c}u)^{m}\wedge\beta^{n-m}\,.$
Taking into account Remark 3.25 and Lemma 4.9 we can conclude that
$\displaystyle\int\limits_{\\{P(u_{k}+sv)-sv<u\\}}(-v)(dd^{c}u)^{m}\wedge\beta^{n-m}$
$\displaystyle\leq\int\limits_{\\{P(u_{k}+sv)-sv<u_{k}\\}}(-v)(dd^{c}(P(u_{k}+sv)-sv))^{m}\wedge\beta^{n-m}$
$\displaystyle\leq-sM\to 0\,,\qquad\text{as }s\to 0.$
Here, $M$ is a positive constant which depends only on $m$, $\|v\|$, and
$\int_{\Omega}v\,H_{m}(u+v)$. Equality (4.2) follows from equality (4.1). The
proof is thus complete. ∎
###### Lemma 4.11.
Let $u,v\in\mathcal{E}^{1}_{m}(\Omega)$, and assume that $v$ is continuous.
Then
$\frac{d}{dt}\Big{|}_{t=0}{\rm E}(P(u+tv))=\int_{\Omega}(-v)H_{m}(u).$
###### Proof.
If $t>0$, $P(u+tv)=u+tv.$ It is easy to see that
$\frac{d}{dt}\Big{|}_{t=0^{+}}{\rm E}(P(u+tv))=\int_{\Omega}(-v)H_{m}(u).$
To compute the left-derivative observe that
$\displaystyle\frac{1}{t}\left(\int_{\Omega}(-P(u+tv))(dd^{c}P(u+tv))^{m}\wedge\beta^{n-m}-\int_{\Omega}(-u)(dd^{c}u)^{m}\wedge\beta^{n-m}\right)$
$\displaystyle=\sum_{k=0}^{m}\int_{\Omega}\frac{u-P(u+tv)}{t}(dd^{c}u)^{k}\wedge(dd^{c}P(u+tv))^{m-k}\wedge\beta^{n-m}.$
It suffices to apply Lemma 4.10. ∎
### 4.2. Resolution
In this section we use the variational formula established above to solve the
equation $H_{m}(u)=\mu$ in finite energy calsses of Cegrell type, where $\mu$
is a positive Radon measure. Our main results represented in the introduction
follow from these theorems. The following lemma is important for the sequel.
###### Lemma 4.12.
Let $\mu$ be a positive Radon measure such that $\mathcal{F}_{\mu}$ is proper
and lower semicontinuous on $\mathcal{E}^{1}_{m}(\Omega)$. Then there exists
$\varphi\in\mathcal{E}^{1}_{m}(\Omega)$ such that
$\mathcal{F}_{\mu}(\varphi)=\inf_{\psi\in\mathcal{E}^{1}_{m}(\Omega)}\mathcal{F}_{\mu}(\psi).$
###### Proof.
Let $(\varphi_{j})\subset\mathcal{E}^{1}_{m}(\Omega)$ be such that
$\lim_{j}\mathcal{F}_{\mu}(\varphi_{j})=\inf_{\psi\in\mathcal{E}^{1}_{m}(\Omega)}\mathcal{F}_{\mu}(\psi)\leq
0.$
From the properness of the functional $\mathcal{F}_{\mu}$, we obtain
$\sup_{j}{\rm E}(\varphi_{j})<+\infty.$ It follows that the sequence
$(\varphi_{j})$ forms a compact subset of $\mathcal{E}^{1}_{m}(\Omega).$ Hence
there exists a subsequence (still denoted by $(\varphi_{j})$) such that
$\varphi_{j}$ converges to $\varphi$ in $L^{1}_{\rm loc}(\Omega)$. Since
$\mathcal{F}_{\mu}$ is lower semicontinuous we have
$\liminf_{j\to\infty}\mathcal{F}_{\mu}(\varphi_{j})\geq\mathcal{F}_{\mu}(\varphi).$
We then deduce that $\varphi$ is a minimum point of $\mathcal{F}_{\mu}$ on
$\mathcal{E}^{1}_{m}(\Omega).$ ∎
We now prove a Dirichlet principle.
###### Theorem 4.13.
Let $\varphi\in\mathcal{E}^{1}_{m}(\Omega)$ and $\mu\in\mathcal{M}_{1}$. Then
$H_{m}(\varphi)=\mu\Leftrightarrow\mathcal{F}_{\mu}(\varphi)=\inf_{\psi\in\mathcal{E}^{1}_{m}(\Omega)}\mathcal{F}_{\mu}(\psi).$
###### Proof.
Assume first that $H_{m}(\varphi)=\mu.$ Let
$\psi\in\mathcal{E}^{1}_{m}(\Omega).$ By Lemma 3.5 and Hölder inequality we
get
$\int_{\Omega}(-\psi)H_{m}(\varphi)\leq{\rm E}(\psi)^{1/(1+m)}.{\rm
E}(\varphi)^{m/(1+m)}\leq\frac{1}{m+1}{\rm E}(\psi)+\frac{m}{m+1}{\rm
E}(\varphi).$
We then easily obtain $\mathcal{F}_{\mu}(\psi)\geq\mathcal{F}_{\mu}(\varphi).$
Now, assume that $\varphi$ minimizes $\mathcal{F}_{\mu}$ on
$\mathcal{E}^{1}_{m}(\Omega).$ Let $\psi$ be a continuous function in
$\mathcal{E}^{1}_{m}(\Omega)$ and consider the function $g(t)={\rm
E}(P(\varphi+t.\psi))+\mathcal{L}_{\mu}(\varphi+t.\psi),\ \ t\in\mathbb{R}.$
Since $P(\varphi+t\psi)\leq\varphi+t\psi$, we have that
$g(t)\geq\mathcal{F}_{\mu}(P(\varphi+t\psi))\geq\mathcal{F}_{\mu}(\varphi)=g(0),\
\forall t.$
It follows that $g$ attains its minimum at $t=0$, hence $g^{\prime}(0)=0$.
Since $\mu\in\mathcal{M}_{1}$, $\mathcal{L}_{\mu}$ is finite on
$\mathcal{E}^{1}_{m}(\Omega)$ which implies that
$\frac{d}{dt}\mathcal{L}_{\mu}(\varphi+t.\psi)=\mathcal{L}_{\mu}(\varphi).$
This coupled with Lemma 4.11 yields
$\int_{\Omega}\psi H_{m}(\varphi)=\int_{\Omega}\psi d\mu.$
The test function $\psi$ is taken arbitrarily, so it follows that
$\mu=H_{m}(\varphi).$ ∎
###### Theorem 4.14.
Let $\mu$ be a positive Radon measure such that
$\mathcal{E}^{1}_{m}(\Omega)\subset L^{1}(\Omega,\mu).$ Then there exists a
unique $u\in\mathcal{E}^{1}_{m}(\Omega)$ such that $H_{m}(u)=\mu.$
###### Proof.
The uniqueness follows from the comparison principle.
We prove the existence. Suppose first that $\mu$ has compact support
$K\Subset\Omega$, and let $h_{K}:=h_{m,K,\Omega}^{*}$ denote the $m$-extremal
function of $K$ with respect to $\Omega.$ Set
$\mathcal{M}=\left\\{\nu\geq 0\ /\ {\rm supp}(\nu)\subset
K,\;\int_{\Omega}(-\varphi)^{2}d\nu\leq C.{\rm
E}(\varphi)^{\frac{2}{m+1}}\text{ for every
}\varphi\in\mathcal{E}^{1}_{m}(\Omega)\right\\},$
where $C$ is a fixed constant such that $C>2{\rm E}(h_{K})^{\frac{m-1}{m+1}}.$
For each compact $L\subset K$ , we have $h_{K}\leq h_{L}$. We deduce that
${\rm E}(h_{L})\leq{\rm E}(h_{K}).$ Therefore, for every
$\varphi\in\mathcal{E}^{1}_{m}(\Omega)$, we have
$\displaystyle\int_{\Omega}(-\varphi)^{2}H_{m}(h_{L})$ $\displaystyle\leq$
$\displaystyle
2\|h_{L}\|\int_{\Omega}(-\varphi)(dd^{c}\varphi)\wedge(dd^{c}h_{L})^{m-1}\wedge\omega^{n-m}$
$\displaystyle\leq$ $\displaystyle
2\left(\int_{\Omega}(-\varphi)H_{m}(\varphi)\right)^{\frac{2}{m+1}}\left(\int_{\Omega}(-h_{L})H_{m}(h_{L})\right)^{\frac{m-1}{m+1}}$
$\displaystyle\leq$ $\displaystyle C.{\rm E}(\varphi)^{\frac{2}{m+1}}.$
This implies that $H_{m}(h_{L})\in\mathcal{M}$ for every compact $L\subset K$.
Put $T=\sup\\{\nu(\Omega)\ /\ \nu\in\mathcal{M}\\}.$ We claim that
$T<+\infty.$ In fact, since $\Omega$ is $m$-hyperconvex, there exists
$h\in\mathcal{SH}_{m}^{-}(\Omega)\cap\mathcal{C}(\bar{\Omega})$ such that
$K\Subset\\{h<-1\\}\Subset\Omega.$ For each $\nu\in\mathcal{M}$, we have
$\nu(K)\leq\int_{K}(-h)d\nu\leq C.{\rm E}(h)^{\frac{2}{m+1}},$
from which the claim follows.
Fix $\nu_{0}\in\mathcal{M}$ such that $\nu_{0}(\Omega)>0$. Let
$\mathcal{M}^{\prime}$ denote the set of all probability measures $\nu$ in
$\Omega$ supported in $K$ such that
$\int_{\Omega}(-\varphi)^{2}d\nu\leq\left(\frac{C}{T}+\frac{C}{\nu_{0}(\Omega)}\right){\rm
E}(\varphi)^{\frac{2}{m+1}},\ \
\forall\varphi\in\mathcal{E}^{1}_{m}(\Omega)\,.$
Then, for each $\nu\in\mathcal{M}$ and
$\varphi\in\mathcal{E}^{1}_{m}(\Omega),$
$\displaystyle\int_{\Omega}(-\varphi)^{2}\frac{(T-\nu(\Omega))d\nu_{0}+\nu_{0}(\Omega)d\nu}{T\nu_{0}(\Omega)}$
$\displaystyle\leq$
$\displaystyle\frac{T-\nu(\Omega)}{T\nu_{0}(\Omega)}\int_{\Omega}(-\varphi)^{2}d\nu_{0}+\frac{1}{T}\int_{\Omega}(-\varphi)^{2}d\nu$
$\displaystyle\leq$
$\displaystyle\left(C\frac{T-\nu(\Omega)}{T\nu_{0}(\Omega)}+\frac{C}{T}\right){\rm
E}(\varphi)^{\frac{2}{m+1}}$ $\displaystyle\leq$
$\displaystyle\left(\frac{C}{\nu_{0}(\Omega)}+\frac{C}{T}\right){\rm
E}(\varphi)^{\frac{2}{m+1}}.$
From this we infer that
$\frac{(T-\nu(\Omega))\nu_{0}+\nu_{0}(\Omega)\nu}{T\nu_{0}(\Omega)}\in\mathcal{M}^{\prime},\text{
for every }\nu\in\mathcal{M}\,.$
We conclude that $\mathcal{M}^{\prime}$ is (non empty) convex and weakly
compact in the space of probability measures. It follows from a generalized
Radon-Nykodim Theorem [29] that there exists a positive measure
$\nu\in\mathcal{M}^{\prime}$ and a positive function $f\in L^{1}(\nu)$ such
that $\mu=f\,d\nu+\nu_{s}$, where $\nu_{s}$ is orthogonal to
$\mathcal{M}^{\prime}$. Observe also that every measures orthogonal to
$\mathcal{M}^{\prime}$ is supported in some $m$-polar set since
$H_{m}(h_{L})\in\mathcal{M}$ for each $L\Subset K.$ We then deduce that
$\nu_{s}\equiv 0$ since $\mu$ does not charge $m$-polar sets.
From Lemma 4.2, Lemma 4.8 we see that for each
$\lambda\in\mathcal{M}^{\prime},$ the functional $\mathcal{F}_{\lambda}$ is
proper and lower semicontinuous. For each $j\in\mathbb{N}$ set
$\mu_{j}=\min(f,j)\nu$. Then $\mathcal{L}_{\mu_{j}}$ is also continuous on
$\mathcal{E}^{1}_{m}(\Omega)$ and $\mathcal{F}_{\mu_{j}}$ is proper since
$\mu_{j}\leq j.\nu.$ Therefore, by Lemma 4.12 and Theorem 4.13, there exists
$u_{j}\in\mathcal{E}^{1}_{m}(\Omega)$ such that $H_{m}(u_{j})=\mu_{j}$. It is
clear from the comparison principle that $\\{u_{j}\\}$ decreases to a function
$u\in\mathcal{E}^{1}_{m}(\Omega)$ which solves $H_{m}(u)=\mu$.
It remains to treat the case when $\mu$ does not have compact support. Let
$\\{K_{j}\\}$ be an exhaustive sequence of compact subsets of $\Omega$ and
consider $\mu_{j}=\chi_{K_{j}}d\mu.$ Let $u_{j}\in\mathcal{E}^{1}_{m}(\Omega)$
solve $H_{m}(u_{j})=\mu_{j}.$ Observe also that $(u_{j})$ decreases to
$u\in\mathcal{SH}_{m}^{-}(\Omega).$ It suffices to prove that $\sup_{j}{\rm
E}(u_{j})<+\infty$. Since $\mu\in\mathcal{M}_{1}$, we have
${\rm
E}(u_{j})=\int_{\Omega}(-u_{j})H_{m}(u_{j})=\int_{K_{j}}(-u_{j})d\mu\leq\int_{\Omega}(-u_{j})d\mu\leq
A.{\rm E}(u_{j})^{\frac{1}{m+1}}.$
This implies that ${\rm E}(u_{j})$ is uniformly bounded, hence $u$ belongs to
$\mathcal{E}^{1}_{m}(\Omega)$ and the result follows. ∎
## 5\. Some applications
###### Lemma 5.1.
Let $\mu$ be a positive Radon measure having finite mass
$\mu(\Omega)<+\infty.$ Assume that $\mu\leq H_{m}(\psi)$, where $\psi$ is a
bounded $m$-sh function in $\Omega.$ Then there exists a unique function
$\varphi\in\mathcal{E}_{m}^{0}(\Omega)$ such that $\mu=H_{m}(\varphi).$
###### Proof.
Without loss of generality, we can assume that $-1\leq\psi\leq 0.$ Consider
$h_{j}=\max(\psi,jh)$, where $h\in\mathcal{E}_{m}^{0}(\Omega)$ is an
exhaustion function of $\Omega.$ Let $A_{j}:=\\{z\in\Omega\ /\ jh<-1\\}.$ From
Theorem 4.14, there exists
$(\varphi_{j})_{j}\subset\mathcal{E}_{m}^{0}(\Omega)$ such that
$H_{m}(\varphi_{j})=1{\hskip-2.5pt}\hbox{{I}}_{A_{j}}\mu,\ \forall j.$ Thus,
$0\geq\varphi_{j}\geq h_{j}\geq\psi,\ \text{and}\
\varphi_{j}\downarrow\varphi\in\mathcal{E}_{m}^{0}(\Omega).$
∎
Using this lemma we can prove the comparison principle for the classes
$\mathcal{E}_{m}^{p}(\Omega)$ with $p>1.$
###### Theorem 5.2.
If $p>1$ and $u,v\in\mathcal{E}_{m}^{p}(\Omega)$ then
$\int_{\\{u>v\\}}H_{m}(u)\leq\int_{\\{u>v\\}}H_{m}(v).$
###### Proof.
Fix $0\not\equiv h\in\mathcal{E}_{m}^{0}(\Omega)\cap\mathcal{C}(\Omega)$.
Assume first that $v$ is bounded and vanishes on the boundary. Let $K_{j}$ be
an exhaustion sequence of compact subsets of $\Omega.$ Use Lemma 5.1 to solve
$H_{m}(v_{j})=1{\hskip-2.5pt}\hbox{{I}}_{K_{j}}H_{m}(v).$ Then
$v_{j}\downarrow v.$ Now, the arguments of the proof of Theorem 5.2 can be
applied to yield
$\int_{\\{u>v_{j}\\}}(-h)H_{m}(u)\leq\int_{\\{u>v_{j}\\}}(-h)H_{m}(v_{j})=\int_{\\{u>v_{j}\\}\cap
K_{j}}(-h)H_{m}(v).$
Letting $j\to+\infty$ we get
$\int_{\\{u>v\\}}(-h)H_{m}(u)\leq\int_{\\{u>v\\}}(-h)H_{m}(v).$
It remains to remove the assumption on $v.$ For each $k\in\mathbb{N}$ set
$\varphi_{k}:=\max\left(v,-k(-h)^{1/p}\right).$
Since $\varphi_{k}$ is bounded and vanishes on $\partial\Omega$, by the above
arguments we get
$\int_{\\{u>\varphi_{k}\\}}(-h)H_{m}(u)\leq\int_{\\{u>\varphi_{k}\\}}(-h)H_{m}(\varphi_{k}).$
Set $A_{k}:=\\{v>-k(-h)^{1/p}\\}$ and $B_{k}:=\\{v\leq-k(-h)^{1/p}\\}.$ On
$B_{k}$ we have $(-h)\leq\frac{(-\varphi_{k})^{p}}{k^{p}}.$ It then follows
that
$\int_{B_{k}}(-h)H_{m}(\varphi_{k})\leq\frac{1}{k^{p}}E_{p}(\varphi_{k})\leq\frac{C}{k^{p}}E_{p}(v),$
where $C>0$ does not depend on $k.$
It follows from Theorem 3.19 that $H_{m}(\varphi_{k})=H_{m}(v)$ on $A_{k}$. We
thus get
$\int_{\\{u>\varphi_{k}\\}}(-h)H_{m}(u)\leq\int_{\\{u>\varphi_{k}\\}\cap
A_{k}}(-h)H_{m}(v)+\frac{C}{k^{p}}E_{p}(v).$
It suffices now to let $k\to+\infty.$ ∎
Now we prove a decomposition theorem of Cegrell type.
###### Theorem 5.3.
Let $\mu$ be a positive measure in $\Omega$ which does not charge $m$-polar
sets. Then there exists $\varphi\in\mathcal{E}_{m}^{0}(\Omega)$ and $0\leq
f\in L^{1}_{\rm loc}(H_{m}(\varphi))$ such that $\mu=f.H_{m}(\varphi).$
###### Proof.
We first assume that $\mu$ has compact support. By applying Theorem 4.14 we
can find $u\in\mathcal{E}^{1}_{m}(\Omega)$ and $0\leq f\in L^{1}(H_{m}(u))$
such that $\mu=f.H_{m}(u),$ and $\text{supp}(H_{m}(u))\Subset\Omega.$ Consider
$\psi=(-u)^{-1}\in\mathcal{SH}_{m}(\Omega)\cap L^{\infty}_{\rm loc}(\Omega).$
Then $(-u)^{-2m}H_{m}(u)\leq H_{m}(\psi).$ Since $H_{m}(u)$ has compact
support in $\Omega,$ we can modify $\psi$ in a neighborhood of
$\partial\Omega$ such that $\psi\in\mathcal{E}_{m}^{0}(\Omega).$ It follows
from Lemma 5.1 that
$(-u)^{-2m}H_{m}(u)=H_{m}(\varphi),\ \varphi\in\mathcal{E}_{m}^{0}(\Omega).$
This gives us $\mu=f(-u)^{2m}.H_{m}(\varphi).$
It remains to consider the case $\mu$ does not have compact support. Let
$(K_{j})$ be an exhaustive sequence of compact subsets of $\Omega.$ From
previous arguments there exists $u_{j}\in\mathcal{E}_{m}^{0}(\Omega)$ and
$f_{j}\in L^{1}(H_{m}(u_{j}))$ such that
$1{\hskip-2.5pt}\hbox{{I}}_{K_{j}}\mu=f_{j}H_{m}(u_{j}).$ Take a sequence of
positive numbers $(a_{j})$ satisfying
$\varphi:=\sum_{j=1}^{\infty}a_{j}u_{j}\in\mathcal{E}_{m}^{0}(\Omega).$ The
measure $\mu$ is absolutely continuous with respect to $H_{m}(\varphi).$ Thus,
$\mu=gH_{m}(\varphi)\ \text{and}\ g\in L^{1}_{\rm loc}(H_{m}(\varphi)).$
∎
###### Theorem 5.4.
Let $\mu$ be a positive Radon measure on $\Omega$ such that
$\mathcal{E}_{m}^{p}(\Omega)\subset L^{p}(\mu),p>0.$ Then there exists a
unique $\varphi\in\mathcal{E}_{m}^{p}(\Omega)$ such that $H_{m}(\varphi)=\mu.$
###### Proof.
The uniqueness follows from the comparison principle. Let us prove the
existence result. Since $\mu$ does not charge $m$-polar sets, applying the
decomposition theorem (Theorem 5.3) we get
$\mu=fH_{m}(u),\ \ u\in\mathcal{E}_{m}^{0}(\Omega),\ 0\leq f\in L^{1}_{\rm
loc}(H_{m}(u)).$
For each $j$, use Lemma 5.1 to find
$\varphi_{j}\in\mathcal{E}_{m}^{0}(\Omega)$ such that
$H_{m}(\varphi_{j})=\min(f,j)H_{m}(u).$
By Proposition 4.6, $\sup_{j}{\rm E}_{p}(\varphi_{j})<+\infty.$ Thus, the
comparison principle gives us that
$\varphi_{j}\downarrow\varphi\in\mathcal{E}_{m}^{p}(\Omega)$ which solves
$H_{m}(\varphi)=\mu.$ ∎
## 6\. Proof of the main results
### 6.1. Proof of Theorem 1
###### Proposition 6.1.
Let $u,v\in\mathcal{E}_{m}^{p}(\Omega),\ p>0.$ There exist two sequences
$(u_{j}),(v_{j})\subset\mathcal{E}_{m}^{0}(\Omega)$ decreasing to $u,v$
respectively such that
$\lim_{j\to+\infty}\int_{\Omega}(-u_{j})^{p}H_{m}(v_{j})=\int_{\Omega}(-u)^{p}H_{m}(v).$
In particular, if $\varphi\in\mathcal{E}_{m}^{p}(\Omega)$ then there exists
$(\varphi_{j})\subset\mathcal{E}_{m}^{0}(\Omega)$ decreasing to $\varphi$ such
that
${\rm E}_{p}(\varphi_{j})\to{\rm E}_{p}(\varphi).$
###### Proof.
Let $(u_{j})$ be a sequence in $\mathcal{E}_{m}^{0}(\Omega)$ decreasing to $u$
such that $\sup_{j}\int_{\Omega}(-u_{j})^{p}H_{m}(u_{j})<+\infty.$ Since
$H_{m}(v)$ vanishes on $m$-polar sets Theorem 5.3 gives
$H_{m}(v)=fH_{m}(\psi),\ \ \psi\in\mathcal{E}_{m}^{0}(\Omega),\ 0\leq f\in
L^{1}_{\rm loc}(H_{m}(\psi)).$
For each $j$, use Lemma 5.1 to find $v_{j}\in\mathcal{E}_{m}^{0}(\Omega)$ such
that
$H_{m}(v_{j})=\min(f,j)H_{m}(\psi).$
By the comparison principle
$v_{j}\downarrow\varphi\in\mathcal{E}_{m}^{p}(\Omega)$ which solves
$H_{m}(\varphi)=H_{m}(v).$ It implies that $\varphi\equiv v.$ We then have
$\int_{\Omega}(-u)^{p}H_{m}(v)=\lim_{j}\int_{\Omega}(-u_{j})^{p}\min(f,j)H_{m}(\psi)=\lim_{j}\int_{\Omega}(-u_{j})^{p}H_{m}(v_{j}).$
∎
Theorem 1 is a consequence of the following result.
###### Theorem 6.2.
Let $\mu$ be a positive Radon measure in $\Omega$ and $p>0.$ Then we have
$\mu=H_{m}(\varphi)\ {\rm with}\
\varphi\in\mathcal{E}_{m}^{p}(\Omega)\Leftrightarrow\mathcal{E}_{m}^{p}(\Omega)\subset
L^{p}(\Omega,\mu).$
###### Proof.
The implication $\Leftarrow$ has been proved in Theorem 4.14 and Theorem 5.4.
Now, assume that $\mu=H_{m}(\varphi)$ with
$\varphi\in\mathcal{E}_{m}^{p}(\Omega)$. Let $\psi$ be another function in
$\mathcal{E}_{m}^{p}(\Omega)$. By Proposition 6.1 there exist sequences
$(\varphi_{j}),(\psi_{j})$ in $\mathcal{E}_{m}^{0}(\Omega)$ having uniformly
bounded energy such that
$\int_{\Omega}(-\psi)^{p}H_{m}(\varphi)=\lim_{j}\int_{\Omega}(-\psi_{j})^{p}H_{m}(\varphi_{j}),\
\ \lim_{j}{\rm E}_{p}(\varphi_{j})={\rm E}_{p}(\varphi),\ \lim_{j}{\rm
E}_{p}(\psi_{j})={\rm E}_{p}(\psi).$
It suffices now to apply Lemma 3.5. ∎
### 6.2. Proof of Theorem 2
###### Proof.
We first prove the existence result. Since $\mu$ does not charge $m$-polar
sets the decomposition theorem yields
$\mu=fH_{m}(u),\ \ u\in\mathcal{E}_{m}^{0}(\Omega),\ 0\leq f\in L^{1}_{\rm
loc}(H_{m}(u)).$
For each $j$ use Lemma 5.1 to find $\varphi_{j}\in\mathcal{E}_{m}^{0}(\Omega)$
such that
$H_{m}(\varphi_{j})=\min(f,j)H_{m}(u).$
Besides, $\sup_{j}\int_{\Omega}H_{m}(\varphi_{j})\leq\mu(\Omega)<+\infty.$
Thus, $\varphi_{j}\downarrow\varphi\in\mathcal{F}_{m}(\Omega)$ in view of the
comparison principle. The limit function $\varphi$ solves $H_{m}(\varphi)=\mu$
as required.
The uniqueness can be proved by the same ways as in [8, Lemma 5.14]. Assume
that $\psi\in\mathcal{F}_{m}(\Omega)$ solves $H_{m}(\psi)=\mu.$ We are to
prove that $\varphi=\psi.$ Let $(K_{j})$ be an exhaustive sequence of compact
subsets of $\Omega$ such that $h_{j}=h_{m,K_{j},\Omega}$ is continuous. For
each $j$, the function $\psi_{j}:=\max(\psi,j.h_{j})$ belongs to
$\mathcal{E}_{m}^{0}(\Omega)$, and $\psi_{j}\downarrow\psi.$ Set
$d_{j}:=\frac{\psi_{j}}{j}-h_{j}=\max(\frac{\psi}{j}-h_{j},0).$ Then
$d_{j}\leq 1{\hskip-2.5pt}\hbox{{I}}_{\\{\psi>j.h_{j}\\}}$ and
$1-d_{j}\downarrow 0.$ For $s>j$, by the comparison principle we get
$\displaystyle 0\leq d_{j}H_{m}(\max(\psi,s.h_{j}))$ $\displaystyle\leq$
$\displaystyle
1{\hskip-2.5pt}\hbox{{I}}_{\\{\psi>j.h_{j}\\}}H_{m}(\max(\psi,s.h_{j}))$
$\displaystyle=$ $\displaystyle
1{\hskip-2.5pt}\hbox{{I}}_{\\{\psi>j.h_{j}\\}}H_{m}(\max(\psi,j.h_{j}))$
$\displaystyle=$ $\displaystyle H_{m}(\max(\psi,s.h_{j})).$
Letting $s$ tend to $+\infty$ and using Corollary 3.14 we get
(6.1) $d_{j}.H_{m}(\psi)\leq
1{\hskip-2.5pt}\hbox{{I}}_{\\{\psi>j.h_{j}\\}}H_{m}(\max(\psi,j.h_{j}))\leq
H_{m}(\psi).$
Recall that from the first part we have
$\mu=fH_{m}(u),\ \ u\in\mathcal{E}_{m}^{0}(\Omega),\ 0\leq f\in L^{1}_{\rm
loc}(H_{m}(u)).$
and $H_{m}(\varphi_{p})=\min(f,p)H_{m}(u)$ for each $p\in\mathbb{N}.$ For each
$p,j$ we can find $v_{j}^{p}\in\mathcal{E}_{m}^{0}(\Omega)$ such that
$H_{m}(v_{j}^{p})=(1-d_{j})H_{m}(\varphi_{p}).$
Using (6.1) we get
$\displaystyle H_{m}(\varphi_{p})$ $\displaystyle=$ $\displaystyle
d_{j}.H_{m}(\varphi_{p})+(1-d_{j})H_{m}(\varphi_{p})$ $\displaystyle\leq$
$\displaystyle d_{j}H_{m}(\psi)+(1-d_{j})H_{m}(\varphi_{p})$
$\displaystyle\leq$ $\displaystyle
1{\hskip-2.5pt}\hbox{{I}}_{\\{\psi>jh_{j}\\}}H_{m}(\psi_{j})+H_{m}(v_{j}^{p})$
$\displaystyle\leq$ $\displaystyle H_{m}(\psi_{j})+H_{m}(v_{j}^{p}).$
This couped with the comparison principle yield $\varphi_{p}\geq
v_{j}^{p}+\psi_{j}$. Letting $p\to+\infty$ we obtain $\varphi\geq
v_{j}+\psi_{j}$, where $v_{j}\in\mathcal{F}_{m}(\Omega)$ solves
$H_{m}(v_{j})=(1-d_{j})H_{m}(\varphi).$ Since $H_{m}(\varphi)$ does not charge
$m$-polar sets, by monotone convergence theorem the total mass of
$H_{m}(v_{j})$ goes to $0$ as $j\to+\infty.$ This implies that $v_{j}$
increases to $0$ and hence $\varphi\geq\psi.$
Now, we prove that $\varphi\leq\psi.$ Let
$\psi_{j},t_{j}\in\mathcal{E}_{m}^{0}(\Omega)$ such that
$H_{m}(w_{j})=d_{j}H_{m}(\psi_{j})$ and
$H_{m}(t_{j})=(1-d_{j})H_{m}(\psi_{j}).$ Since $H_{m}(\varphi_{p})$ increases
to $H_{m}(\varphi)$, the comparison principle can be applied for $\varphi$ and
$w_{j}$ which implies that $w_{j}\geq\varphi.$ But, applying again the
comparison principle for $t_{j}+w_{j}$ and $\psi_{j}$ we get
$t_{j}+w_{j}\leq\psi_{j}.$ Furthermore, the total mass of $H_{m}(t_{j})$ can
be estimated as follows
$\displaystyle\int_{\Omega}H_{m}(t_{j})$ $\displaystyle=$
$\displaystyle\int_{\Omega}H_{m}(\psi_{j})-\int_{\Omega}d_{j}H_{m}(\psi_{j})$
$\displaystyle\leq$
$\displaystyle\int_{\Omega}H_{m}(\psi)-\int_{\Omega}d_{j}^{2}H_{m}(\psi)$
$\displaystyle\leq$ $\displaystyle 2\int_{\Omega}(1-d_{j})H_{m}(\psi)\to 0.$
This implies that $t_{j}$ converges in $m$-capacity to $0.$ Indeed, for every
$\epsilon>0$ and $m$-subharmonic function $-1\leq\theta\leq 0$, by the
comparison principle we have
$\displaystyle\epsilon^{m}\int_{\\{t_{j}<-\epsilon\\}}H_{m}(\theta)$
$\displaystyle\leq$
$\displaystyle\int_{\\{t_{j}<-\epsilon\theta\\}}H_{m}(\epsilon\theta)$
$\displaystyle\leq$
$\displaystyle\int_{\\{t_{j}<-\epsilon\theta\\}}H_{m}(t_{j})\leq\int_{\Omega}H_{m}(t_{j})\to
0.$
Thus, we can deduce that $\varphi\leq\psi$ which implies the equality. ∎
We prove in the above uniqueness theorem that every
$\psi\in\mathcal{F}^{a}_{m}(\Omega)$ can be approximated from above by a
sequence $\psi_{j}\in\mathcal{E}_{m}^{0}(\Omega)$ such that $H_{m}(\psi_{j})$
increases to $H_{m}(\psi).$ This type of convergence is strong enough to prove
the comparison principle for the class $\mathcal{F}^{a}_{m}(\Omega).$ We thus
get
###### Theorem 6.3.
The comparison principle is valid for functions in
$\mathcal{F}^{a}_{m}(\Omega).$
## 7\. Examples
###### Lemma 7.1.
If $\varphi\in\mathcal{E}_{m}^{p}(\Omega)$, $p>0$ then ${\rm
Cap}_{m}(\varphi<-t)\leq C.{\rm E}_{p}(\varphi).\frac{1}{t^{m+p}},$ where
$C>0$ is a constant depending only on $m.$
###### Proof.
Without loss of generality we can assume that
$\varphi\in\mathcal{E}_{m}^{0}(\Omega).$ Fix
$u\in\mathcal{SH}^{-}_{m}(\Omega)$ such that $-1\leq u\leq 0.$ Observe that,
for any $t>0,$
$(\varphi<-2t)\subset(\varphi<tu-t)\subset(\varphi<-t).$
Thus, by the comparison principle (Theorem 2.13) we have
$\displaystyle\int_{\\{\varphi<-2t\\}}H_{m}(u)\leq\frac{1}{t^{m}}\int_{\\{\varphi<tu-t\\}}H_{m}(tu-t)\leq\frac{1}{t^{m}}\int_{\\{\varphi<tu-t\\}}H_{m}(\varphi)$
$\displaystyle\leq\frac{1}{t^{m}}\int_{\\{\varphi<-t\\}}H_{m}(\varphi)\leq\frac{1}{t^{m+p}}\int_{\Omega}(-\varphi)^{p}H_{m}(\varphi).$
∎
###### Proposition 7.2.
Let $\mu=fdV,$ where $0\leq f\in L^{p}(\Omega,dV),\frac{n}{m}>p>1.$ Then
$\mu=H_{m}(\varphi),\ \varphi\in\mathcal{F}_{m}^{q}(\Omega),\ \forall
q<\frac{nm(p-1)}{n-mp}.$
###### Proof.
Fix $0<r<n/(n-m).$ By Hölder’s inequality and [9, Proposition 2.1], there
exists $C>0$ depending only on $p,r,\|f\|_{p}$ such that
(7.1) $\mu(K)\leq C.{\rm Vol}(K)^{\frac{p-1}{p}}\leq C.{\rm
Cap}_{m}(K)^{\frac{r(p-1)}{p}}.$
Take $0<q<\frac{nm(p-1)}{n-mp}$ and $u\in\mathcal{E}_{m}^{q}(\Omega).$ By
Theorem 5.4 it suffices to show that $u\in L^{q}(\mu)$ which is, in turn,
equivalent to showing that
$\int_{1}^{+\infty}\mu(u<-t^{1/q})dt<+\infty.$
The latter follows easily from (7.1) and Lemma 7.1, which completes the proof.
∎
The exponent $q(p)=\frac{nm(p-1)}{n-mp}$ is sharp in view of the following
example.
###### Example 7.3.
Consider $\varphi_{\alpha}=1-\|z\|^{-2\alpha},$ where $\alpha$ is a constant
in $(0,\frac{n-m}{m}).$ An easy computation shows that
$\varphi_{\alpha}\in\mathcal{F}_{m}(\Omega)$ and
$H_{m}(\varphi_{\alpha})=C.\|z\|^{-2m(\alpha+1)}dV=f_{\alpha}dV.$
Then
$\varphi_{\alpha}\in\mathcal{F}_{m}^{q}(\Omega)\Longleftrightarrow
q<\frac{n-m}{\alpha}-m,$
while
$f_{\alpha}\in L^{p}(\Omega,dV)\Longleftrightarrow p<\frac{n}{m(\alpha+1)}.$
Acknowledgements. The paper is taken from my Ph.D Thesis defended on 30th
November 2012. It is a great pleasure to express my deep gratitude to my
advisor Ahmed Zeriahi for inspirational discussions and enlightening
suggestions. I am also indebted to Vincent Guedj for constant help and
encouragements. I also would like to thank Urban Cegrell for very useful
discussions on the decomposition theorem. I also thank the reviewer for
his/her thorough review and highly appreciate the comments and suggestions,
which significantly contributed to improving the quality of the paper.
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|
arxiv-papers
| 2013-01-28T10:16:02 |
2024-09-04T02:49:40.844742
|
{
"license": "Public Domain",
"authors": "Lu Hoang Chinh",
"submitter": "Chinh Lu Hoang",
"url": "https://arxiv.org/abs/1301.6502"
}
|
1301.6554
|
How (not) to assess the importance of correlations for the matching of
spontaneous and evoked activity
József Fiser1, Máté Lengyel2, Cristina Savin2, Gergő Orbán2,3, Pietro Berkes4
1Department of Cognitive Science, Central European University, 2Computational
and Biological Learning Lab, Department of Engineering, University of
Cambridge, 3Wigner Research Institute for Physics, Hungarian Academy of
Sciences, 4Volen National Center for Complex Systems, Brandeis University
Contents
February 2013: Comment on Okun et al. (2012) p.1 March 2013: Response to the
reply of Okun et al. (2013) to our comments p.1
Comment on okun12
In a recent publication, okun12 criticise our earlier paper (berkes11) in
which we analysed multi-neuron firing patterns in V1 of awake ferrets. We
described a remarkably close match between the distribution of multi-neuron
firing patterns during spontaneous activity (recorded in the absence of visual
stimulation) and during evoked activity recorded while stimulating with
naturalistic visual stimuli. We argued that this match confirmed predictions
about the cortex implementing a statistically optimal internal model of the
environment. In contrast, okun12 claim that this match can be a consequence of
trivial statistical properties of multi-neuron firing patterns, specifically
fluctuations in overall population firing rates, and as such these
epiphenomenal results are not indicative of optimal internal models. Below we
explain that the analyses conducted by okun12 suffer from both conceptual and
statistical flaws which invalidate their interpretation of their own data. We
also show that if the correct analyses are performed on our original data set,
the claims of okun12 regarding our findings do not hold and in fact these
analyses provide additional support for our main conclusion.
Following the logic of berkes11, central to okun12’s analysis is the
construction of a surrogate data set that respects as constraints some simple
statistical properties of the experimentally measured distribution of multi-
neuron firing patterns, but is otherwise as random as possible (it has maximum
entropy given the constraints). One can then test whether there is a
difference between the original results obtained with the real data and the
results obtained using surrogate data. If no significant difference can be
found, then the original results are deemed epiphenomenal. berkes11 used the
simplest possible such surrogate: a distribution under which all neurons fire
independently from each other at their true observed firing rates – this has
been a standard test for the role of correlations (of 2nd and higher order) in
shaping the distribution of neural responses since the seminal paper of
schneidman06. berkes11 reported that this surrogate data set did not capture
some essential aspects of the real neural responses (berkes11, Fig. 3A-B) and
thus that their results could not be trivially explained by single neuron
firing rate dynamics.
Building on recent advances in the field of maximum entropy analyses of neural
data (tkacik12), okun12 go one step further and construct a more sophisticated
surrogate, which preserves not only single neuron firing rates, but also the
distribution of the number of co-active units at a time. This is an important
control because, in general, this second constraint introduces correlations at
all orders, without any functionally relevant pairwise coupling between
neurons. Indeed, the number of degrees of freedom of this surrogate (a measure
of its complexity) is still only linear in the number of units, as it is in
the simple surrogate that only cares about single neuron firing rates, while
in a true second-order surrogate (schneidman06), that is able to capture the
detailed pairwise (functional) connections between neurons, this measure
scales quadratically with the number of units. The surrogate used by okun12 is
also appealing because it allows a simple biological interpretation: a network
in which units are functionally disconnected (or randomly connected) but
undergo synchronized fluctuations of activity would produce data whose
statistics would match those of this surrogate (see Fig. 8 in okun12).
okun12 then perform the following key analyses using this surrogate data:
1. 1.
They show that the surrogate data is very close to the true data recorded in
A1 (okun12, Fig. 3).
2. 2.
They replace both the true spontaneous and evoked activity distributions with
their surrogate counterparts and show that some of the results of berkes11
(berkes11, Figs. 2B, 3A-B) can be replicated: the statistical dissimilarity
between these surrogates can be minimal in adult animals both in A1 and V1
(okun12, Figs. 4B and 5) and substantially decrease over (simulated)
development (okun12, Figs. 7 and 8).
These analyses have questionable relevance for the earlier results of berkes11
on both counts:
1. 1.
Assessing match between surrogate and true data.
1. a)
okun12 never demonstrate directly a match between surrogate and true data for
their V1 recordings. Given that all the rest of their analyses pertaining
directly to the results of berkes11 (obtained in V1) are performed on the V1
data set (okun12, Figs. 7-8), demonstrating this match would have been a
necessary prerequisite for validating their approach.
2. b)
In fact, even for their A1 data, okun12’s own analyses do indicate a
divergence between the surrogate and true data that appears to be significant
(okun12, Fig. 3B, difference between gray and red bars). This significance is
never tested statistically, even though if there was a significant divergence
that would invalidate their claims about their surrogate reproducing the
relevant statistics of the true data (as is the case in their Fig. 4C, see
below in point 2b).
3. c)
Repeating their analysis with their surrogate on the data set used in
berkes11, we find that the divergence between real and surrogate data
significantly increases over development (Spearman’s $\rho=0.68,p=0.005$ for
spontaneous, and Spearman’s $\rho=0.57,p=0.027$ for movie-evoked activities)
and reaches highly significant levels in adult age groups (Fig. 1A).
2. 2.
Using surrogate-to-surrogate divergences.
1. a)
Measuring the divergence between surrogate spontaneous and surrogate evoked
activities, as okun12 did, is prone to trivial outcomes. Imagine that the
original spontaneous and evoked activities are truly being matched by all
statistical measures. Then it follows, by definition, that their corresponding
surrogate versions, which respect only a subset of these statistical measures,
will also be automatically matched to each other.***As an example, imagine
that there are two data sets, each with complicated non-Gaussian looking
histograms. We create a Gaussian surrogate for each, matching their mean and
variance. If the original histograms looked the same in all their gory detail,
their Gaussian surrogates will also look the same – but this does not mean
that the original histograms didn’t match in ways that went beyond their mean
and variance (Fig. 2A). (This is strictly true only if the divergence between
the true distributions is zero, but will usually hold for small but non-zero
divergences as well.) As expected, this effect can be demonstrated in the data
set of berkes11 using okun12’s surrogate: the divergence between surrogate
spontaneous and surrogate movie-evoked activity significantly decreases over
development (Spearman’s $\rho=-0.75,p=0.0014$) and reaches non-significant
levels in adult animals (Fig. 1B, purple bars). To illustrate that this effect
is indeed trivial, we computed the same divergence with the simple surrogate
used by berkes11 respecting only single-neuron firing rates, and found
qualitatively identical results (Spearman’s $\rho=-0.71,p=0.003$, Fig. 1B,
pink bars).
2. b)
The appropriate way to establish the importance of higher-order statistics for
matching two histograms is to perform analyses of the kind shown in Figure 3B
of berkes11: surrogate spontaneous must be compared to true evoked activities
(or vice versa). If it is only the lower order moments reproduced by the
surrogate that matter for the original match between the true distributions,
then this divergence using the surrogate should be similar to that computed
between the true distributions. By inversion, a substantially larger
surrogate-to-true divergence than true-to-true divergence would indicate that
higher order correlations were important for the original true-to-true
divergence. okun12 show only a single example of such an analysis using their
A1 (but not V1) data (okun12, Fig. 4C) and even this analysis is inconclusive
at best. According to their conclusion from this analysis, not supported by
any statistical test, there is no substantial difference between surrogate-to-
true and true-to-true divergences – even though their figure suggests that
there is (the green dots are visibly above the diagonal). In fact, okun12
themselves acknowledge that the difference they found in their divergences is
due to the inappropriate fit of their surrogate to their real data in the
first place (okun12, Fig. 3B, see point 1b above). This assessment seems to be
correct, but it also points to a crucial flaw undermining their ensuing claims
as explained above: when surrogate-to-true divergences are larger than true-
to-true divergences then higher-order correlations do play a major role.
3. c)
Most importantly, when the correct surrogate-to-true analysis is performed on
our data, we find that in contrast to true-to-true divergences (Fig. 1C, red
bars), surrogate-to-true divergences do not decrease significantly over time
(Spearman’s $\rho=-0.32,p=0.246$) and remain significantly above baseline even
in adult animals (Fig. 1C, purple bars), further strengthening berkes11’s
earlier results obtained using the simpler surrogate analysis (Fig. 1C, pink
bars).
In summary, we concur with the spirit of okun12’s work: showing a low (and
developmentally decreasing) divergence between spontaneous and evoked
activities is necessary but not sufficient for claiming statistical
optimality. In fact, this is in line with the approach already taken by
berkes11 who provided a set of controls specifically to address this issue,
including controls for the effects of spatial correlations, temporal
correlations, and stimulus statistics. okun12 elaborate on one of these
controls, concerning spatial correlations. Indeed, the surrogate they proposed
as an improved control does allow for a more stringent test of trivial factors
affecting spatial correlations than the simple surrogate normally used
(schneidman06; berkes11). In that respect, their paper represents an important
contribution which will be a useful addition to the toolbox of maximum entropy
models in systems neuroscience (schneidman06; tkacik12).
Unfortunately, however, there are two reasons why the conclusions okun12 drew
from their otherwise useful surrogate are questionable. First, the way they
ended up using this surrogate in their analyses was mostly inappropriate to
the main question they tried to address, and thus provided trivial results.
(And in those cases in which their analysis was appropriate, in their Fig. 4C,
their results seem to agree with those reported in berkes11) Second, we
performed the correct analyses with their surrogate on our data, and found
that these provide additional support for the main conclusion of berkes11: the
match between spontaneous and movie-evoked activities in the visual cortex is
remarkable, cannot be explained by trivial statistical properties and it is,
therefore, indicative of statistically optimal internal models in the cortex.
## References
Figure 1:
A. Divergences between true activity distributions and their surrogate
counterparts. As a reference, the dashed and dotted lines show the average of
the within-condition baselines (dotted for S, dashed for M), computed with
within-condition data split into two halves.
B. Divergences between movie-evoked (M) and spontaneous (S) activities using
the true data (red bars), the surrogate used by okun12 respecting single
neuron firing rates and population rate distributions (purple bars), and the
original surrogate used by berkes11 respecting only single neuron firing rates
(pink bars). Red bars and dashed line are reproduced from berkes11
C. Divergences between true movie evoked and the true (red bars) surrogate
spontaneous activities (purple bars: okun12’s surrogate, pink bars: berkes11’s
surrogate). Red and pink bars and dashed line (computed as in panel A) are
reproduced from berkes11
Asterisks denote significant differences from respective baselines (A, B) or
between bars (C), ${}^{*}p<0.05,^{**}p<0.01,^{***}p<0.001$, m-test (for
details see berkes11).
An abbreviated version of this text has also been posted as a Letter to the
Editor of the _Journal of Neuroscience_ as a reply to okun12.
Response to the reply of okun13 to our comments
Following our comment (above) on their original paper (okun12), okun13
recently published a reply. Our summary for where this discussion stands is
the following. There are two general points that okun12 (2012, 2013) made
about how the learning of internal models could and should be tested in
experiments, which we agree with in principle, although with some important
qualifications. There are also two important specific points about their own
results and analyses of data, and the way they interpreted those results, that
we disagree with and still find technically flawed. In our view, these flaws
will need to be corrected and some of the main conclusions of okun12 will need
to be revisited in light of these analyses. We detail these points below.
Agreement on general points
Looking at KL divergences (or any other similar measure of divergence) between
spontaneous and evoked activities alone cannot be taken as conclusive evidence
that an internal model of the environment has been brought about by learning.
We could not agree more and have never in fact said otherwise. In our original
paper, we wrote that our findings “do not address the degree to which
statistical adaptation in the cortex is driven by visual experience or by
developmental programs” (berkes11), and thus interpreted our results as a sign
of adaptation (learning-driven or otherwise), not learning _per se_.†††A
simple content analysis of berkes11 shows 13 occurrences of words with a root
“adapt” and 0 occurrence of words starting with “learn”. Thus, okun12’s
statement that our result alone “does not indicate that an environmental model
has been learned” echoes our earlier views. However, we now also have results
from newer experiments using lid-sutured animals that had no normal vision
until the moment of the tests (fiser12; savin13). These results, to which
okun13 also refer and which we will present in detail elsewhere, have since
provided additional evidence clearly indicating, at least in the oldest age
group, that visual experience does have a significant role in the specificity
of the match between evoked and spontaneous activities. This is compatible
with stimulus specificity only emerging in this age group in control animals
(berkes11, Fig. 4). Strangely, okun13 interpret these results of ours as
evidence for specifically non-learning related changes driving adaptation in
all but the last age group. However, it is important to note that even in the
first three age groups our results do not exclude forms of learning that our
manipulation (lid suture) did not affect (such as cross-modal learning,
learning from low spatial frequencies, etc), or our data recording and
analyses techniques were unable to pick up. Thus, okun13’s interpretation is
not quite warranted.
It is important to control for the effects of populations rate fluctuations in
maximum entropy model-based analyses of the role of correlations. We agree
with this statement, too. To make the historical context clear: at the time of
the publication of berkes11, the “standard” way of demonstrating the
importance of correlations in maximum entropy analyses was to follow
schneidman06 and compare the true data to the simple surrogate that only
controls for single neuron firing rates but not for the population rate
distribution. This is precisely the practice we followed in berkes11. The
contribution of okun12 and others (tkacik12) is that they suggested a more
stringent test, controlling for the effects of population rate fluctuations,
which warrants the revisiting of _all_ earlier results that used just the
simpler surrogate. We have now revisited our results published in berkes11,
and confirmed them (see above, Fig. 1), and it would be interesting to see
which other earlier results (such as schneidman06, and, importantly, the
results of okun12 themselves, as we argue below) survive this new, more
stringent test.
Disagreement on specific results and interpretations in okun12
Population rate fluctuations account for most of the seemingly correlation-
driven effects in the data of okun12 which therefore implies that they offer
an alternative explanation of the results of berkes11. In our comment on their
paper (see above) we stated that this conclusion by okun12 rests on performing
the wrong comparison of divergences that were computed with their otherwise
valuable new surrogate. In their reply, okun13 dismiss our criticism by
stating that “the point [of their paper, okun12] was not that $\mathrm{P}$ and
$\hat{\mathrm{P}}$‡‡‡Where their $\hat{\mathrm{P}}$ is
$\tilde{\mathrm{P}}_{\mathrm{Okun}}$ in our notation, and $\mathrm{P}$ and
$\mathrm{Q}$ can stand for movie-evoked (M) or spontaneous (S) activities. are
identical” and that “the relationship of
$\mathrm{D}[\mathrm{P}\|\hat{\mathrm{Q}}]$ and
$\mathrm{D}[\mathrm{P}\|\mathrm{Q}]$ was not central [to their analysis]”.
However, as we said before and explain in more detail below in a separate
section, this is precisely the shortcoming of their approach that should be
mended: analyzing those relationships would have been crucial to support their
claims about the (un)importance of true pairwise correlations, and the
analysis they chose instead to perform (comparing
$\mathrm{D}[\mathrm{P}\|\mathrm{Q}]$ with
$\mathrm{D}[\tilde{\mathrm{P}}_{\mathrm{Okun}}\|\tilde{\mathrm{Q}}_{\mathrm{Okun}}]$)
is not only much more indirect in general than the analyses we suggested and
performed on our data (computing
$\mathrm{D}[\mathrm{P}\|\tilde{\mathrm{P}}_{\mathrm{Okun}}]$ and also
comparing $\mathrm{D}[\mathrm{P}\|\mathrm{Q}]$ with
$\mathrm{D}[\mathrm{P}\|\tilde{\mathrm{Q}}_{\mathrm{Okun}}]$), but in this
case it also leads to wrong conclusions. Importantly, while we have taken
seriously the issues they had raised and re-analyzed our data using the
surrogate they suggested, thus confirming the crucial (and developmentally
increasing) importance of true pairwise (and higher order) correlations for
the matching of spontaneous and evoked activities in our data set (see above),
Okun et al. did not perform the analyses we pointed out were necessary on
their data set. As a result, judging from the analyses they have conducted so
far, there is no indication that the important effects of true pairwise
correlations shown in our data set are not present in theirs – although the
final word can only be said once they perform the correct analyses on their
data set.
A random network, such as that shown in okun12, could reproduce the key
results of berkes11 When making this claim, okun13 continue to ignore one of
the most important controls we already presented in berkes11. In fact,
precisely to address this issue, we performed in our original paper (berkes11)
additional tests using evoked activities obtained using different kinds of
artificial stimulus ensembles (block noise or drifting gratings), and showed
that the divergence of these evoked activities from spontaneous activity was
significantly larger than the original divergence obtained using natural image
movie-evoked activities (see Figure 4 therein). This shows that the cortex is
indeed specifically adapted for natural stimuli. Given the importance of this
control, it is puzzling why nowhere in okun12’s original paper (2012) or in
their latest reply (2013) there is any reflection on how their random network
fares on it. Therefore, in lack of evidence otherwise, we are led to believe
that this control clearly rules out the random-network account of our results
proposed by okun12
As a final conclusion, before we present our detailed argument about the right
comparison of divergences below, we want to stress again that in our view the
challenge is now for okun12 to correct these technical flaws by using the
right divergences and comparisons on the data they recorded, and by performing
all necessary tests on their simulations, before they draw conclusions about
their own results and those presented in berkes11.
Detailed comments on the right comparison of divergences
In their reply to our comment, okun13 argue that instead of performing the
analyses we suggested and performed on our data (computing
$\mathrm{D}[\mathrm{P}\|\tilde{\mathrm{P}}_{\mathrm{Okun}}]$ and also
comparing $\mathrm{D}[\mathrm{P}\|\mathrm{Q}]$ with
$\mathrm{D}[\mathrm{P}\|\tilde{\mathrm{Q}}_{\mathrm{Okun}}]$), the comparison
of $\mathrm{D}[\mathrm{P}\|\mathrm{Q}]$ with
$\mathrm{D}[\tilde{\mathrm{P}}_{\mathrm{Okun}}\|\tilde{\mathrm{Q}}_{\mathrm{Okun}}]$
is the relevant analysis to perform.§§§To be precise: for their A1 data they
did actually perform the tests we propose, but still drew conclusions from the
other test, which we think is misleading, for their V1 data they did not even
perform the tests that would have been necessary. In their eyes, the fact that
the difference between these divergences is small supports their claim that
“word distributions primarily reflect changes in population rate dynamic”.
This is despite the fact that they agree with us that a finding that both
divergences are zero (or close to it) is uninformative as to the importance of
correlations (or in general, statistics not controlled by maximum entropy
surrogates, Fig. 2A). In contrast, note that the divergence we suggest should
be used, $\mathrm{D}[\mathrm{P}\|\tilde{\mathrm{Q}}_{\mathrm{Okun}}]$, is able
to dissect the role of higher-order statistical structure in causing a match
between two distributions in this case (Fig. 2A).
Interestingly, okun13 insist that when the individual divergences,
$\mathrm{D}[\mathrm{P}\|\mathrm{Q}]$ and
$\mathrm{D}[\tilde{\mathrm{P}}_{\mathrm{Okun}}\|\tilde{\mathrm{Q}}_{\mathrm{Okun}}]$,
are significantly larger than zero, comparing them is still meaningful.
Unfortunately, this is simply not the case: in general, comparing
$\mathrm{D}[\mathrm{P}\|\mathrm{Q}]$ with
$\mathrm{D}[\tilde{\mathrm{P}}\|\tilde{\mathrm{Q}}]$ (where
$\tilde{\mathrm{X}}$ is some maximum entropy surrogate of $\mathrm{X}$) can be
very misleading for two reasons. First, because the way differences between
different kinds of statistics (e.g. moments) of two distributions jointly
determine the total Kullback-Leibler (KL) divergence between two distributions
is highly non-linear. Second, because the KL divergence is not a proper
metric. Fig. 2B shows two examples in which both
$\mathrm{D}[\mathrm{P}\|\mathrm{Q}]$ and
$\mathrm{D}[\tilde{\mathrm{P}}\|\tilde{\mathrm{Q}}]$ are large and are very
close to each other. However, in one case (Fig. 2B, top) they only differ in
their lower order moments (which are being controlled by the maximum entropy
surrogates), while in the other case (Fig. 2B, bottom) they differ both in
these lower order moments and in higher order moments (that are not controlled
by the maximum entropy surrogates). In general, it can also be seen (Fig. 2C)
that $\mathrm{D}[\mathrm{P}\|\mathrm{Q}]$ varies non-monotonically as the
differences between $\mathrm{P}$ and $\mathrm{Q}$ in higher-order statistics
change, such that any arrangement of $\mathrm{D}[\mathrm{P}\|\mathrm{Q}]$
being greater or smaller than
$\mathrm{D}[\tilde{\mathrm{P}}\|\tilde{\mathrm{Q}}]$ is possible. Therefore,
it is not at all necessarily true that the difference of these two divergences
are zero when their higher-order statistics match. This demonstrates that the
fact that $\mathrm{D}[\mathrm{P}\|\mathrm{Q}]$ and
$\mathrm{D}[\tilde{\mathrm{P}}\|\tilde{\mathrm{Q}}]$ are large and nearly
identical, may be a reflection of “a property specific to word distributions
recorded in the sensory cortex” (okun13), but it tells us nothing about the
role of higher-order statistical structure in causing a mismatch between
$\mathrm{P}$ and $\mathrm{Q}$ – which is what the original claim of okun12 was
about. Note that our suggested divergence cannot resolve this issue either
(Fig. 2B), and consequently, we never made any conjectures from the large-
divergence case. As a side note, it would be interesting to see whether there
are other divergence-comparisons that may be useful in this case.
In summary, while the comparison okun12 advocate is unable to resolve, in
general, the role of correlations in the case of either a match or a mismatch
between evoked and spontaneous activities, the comparison we proposed and used
throughout can resolve this role in a match (but not in a mismatch). Thus,
while using the correct comparison we have convincingly demonstrated that in
our data set, correlations do play a major role in shaping neural activities
and causing a match between evoked and spontaneous activities, okun12 still
need to perform these analyses, especially on their V1 data set. Until these
analyses are performed, their claims about the (un)importance of correlations
in their data set, including the schematic of the relative distances between
distributions they presented (Fig.1 in okun13) remains unjustified and
potentially misleading. In fact, looking at the correct divergences in our
data set, the version of their schematic would look rather different in adult
animals: $\mathrm{P}$ and $\mathrm{Q}$ would be close
($\mathrm{D}[\mathrm{M}\|\mathrm{S}]~{}=100$, Fig. 1B and C red), so would be
$\tilde{\mathrm{P}}_{\mathrm{Okun}}$ and $\tilde{\mathrm{Q}}$
($\mathrm{D}[\tilde{\mathrm{M}}_{\mathrm{Okun}}\|\tilde{\mathrm{S}}_{\mathrm{Okun}}]~{}=100$,
Fig. 1B purple), but the cross-distances would be much larger
($\mathrm{D}[\mathrm{M}\|\tilde{\mathrm{M}}_{\mathrm{Okun}}]~{}=200$, Fig. 1A
orange; $\mathrm{D}[\mathrm{S}\|\tilde{\mathrm{S}}_{\mathrm{Okun}}]~{}=200$,
Fig. 1C purple;
$\mathrm{D}[\mathrm{M}||\tilde{\mathrm{S}}_{\mathrm{Okun}}]~{}=300$, Fig. 1C
purple).
Figure 2:
A. When both $\mathrm{D}[\mathrm{P}\|\mathrm{Q}]$ (right panels, black) and
$\mathrm{D}[\tilde{\mathrm{P}}\|\tilde{\mathrm{Q}}]$ (right panels, dark grey)
are small and they are nearly identical, this does not distinguish between
whether the match between $\mathrm{P}$ and $\mathrm{Q}$ is only governed by
their lower order statistics (top) or also by their higher order statistics
(bottom). Note that using $\mathrm{D}[\mathrm{P}\|\tilde{\mathrm{Q}}]$(right
panels, light grey) it is possible to distinguish between these two cases. In
the examples shown, $\mathrm{P}$ and $\mathrm{Q}$ (left panels, red and green
solid lines) are Gaussian scale mixtures (with two components), and their
surrogates ($\tilde{\mathrm{P}}$ and $\tilde{\mathrm{Q}}$, left panels, red
and green dotted lines) are moment-matched normal distributions. Panels on the
right show respective divergences. True distributions for the top panels are
almost exactly Gaussians, hence they are near-identical to their surrogates.
True distributions for the bottom panels are leptokurtotic and hence are not
identical to their surrogates. $\mathrm{P}$ and $\mathrm{Q}$ are identical in
both panels except for a very small shift in their means.
B. When both $\mathrm{D}[\mathrm{P}\|\mathrm{Q}]$ and
$\mathrm{D}[\tilde{\mathrm{P}}\|\tilde{\mathrm{Q}}]$ are large and they are
nearly identical, this still does not distinguish between whether the mismatch
between $\mathrm{P}$ and $\mathrm{Q}$ is only governed by their lower order
statistics (top) or also by their higher order statistics (bottom). Note that
$\mathrm{D}[\mathrm{P}\|\tilde{\mathrm{Q}}]$ also does not resolve this issue.
Lines in left panels and bars in right panels as in A. For the top panels,
$\mathrm{P}$ and $\mathrm{Q}$ differ only in their means. For the bottom
panels, $\mathrm{P}$ and $\mathrm{Q}$ also differ in their kurtosis (but not
their variance or skewness) which is not controlled by their surrogates.
C. Lines show $\mathrm{D}[\mathrm{P}\|\mathrm{Q}]$ (black) and
$\mathrm{D}[\tilde{\mathrm{P}}\|\tilde{\mathrm{Q}}]$ (dark grey) as the
kurtosis of $\mathrm{Q}$ is varied relative to that of $\mathrm{P}$, while
keeping its mean, variance, and skewness fixed. Bottom and top panels as in B,
green points show the kurtoses of the $\mathrm{Q}$ distributions used there.
Note that in the case shown in the bottom panel a substantial difference in
kurtosis between $\mathrm{P}$ and $\mathrm{Q}$ is accompanied by zero
difference in the two divergences.
## References
|
arxiv-papers
| 2013-01-28T14:22:50 |
2024-09-04T02:49:40.859377
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J\\'ozsef Fiser, M\\'at\\'e Lengyel, Cristina Savin, Gerg\\\"o Orb\\'an,\n Pietro Berkes",
"submitter": "Cristina Savin",
"url": "https://arxiv.org/abs/1301.6554"
}
|
1301.6814
|
A Sampling Theorem for Computational Diffraction
Daniel J. Merthe1,∗
1University of Southern California,
Los Angeles, CA 90089, USA
∗Corresponding author: [email protected]
###### Abstract
A major challenge of many diffraction calculations, using some form of the
Rayleigh-Sommerfeld formulas, is the integration of a highly oscillatory
integrand. Here we derive a potentially useful alternative form of solution to
the Helmholtz equation, which implies a sampling theorem for the evaluation of
a diffracted scalar field. This alternative solution bears close resemblance
to the Rayleigh-Sommerfeld diffraction formulas, but instead incorporates the
boundary conditions digitally. Hence, the integration is replaced by a simple
summation. This formulation may be more efficient for accurate computer-based
calculation of the diffracted scalar field.
## 1 Introduction
The diffraction of monochromatic electromagnetic radiation by obstacles much
larger than the wavelength is often treated satisfactorily by the Scalar
Diffraction Theory. That is, one solves the Helmholtz equation with certain
boundary conditions. We let $U$ be any component of either the electric or
magnetic fields, defined in the half-space $z\geq 0$, free of charges or
currents. If the scalar field $U$ is simple harmonic in time with frequency
$\omega$, then it satisfies the Helmholtz equation,
$\nabla^{2}U+k^{2}U=0$ (1)
where $\nabla^{2}$ is the Laplacian differential operator,
$k=\omega/c=2\pi/\lambda$ is the wavenumber, $c$ is the speed of light and
$\lambda$ is the wavelength. The boundary conditions are usually given as
follows [1]. For some known function $U_{0}(x,y)$, the scalar field $U$ on the
boundary plane $z=0$ is $U(x,y,0)=U_{0}(x,y)$. If $r$ is the distance from the
origin to any point in the half-space, we assume the Sommerfeld radiation
condition, $\lim_{r\rightarrow\infty}U=O(1/r)$, such that the total energy is
bounded. Additionally, we assume that all radiation is propagating in the
positive $z$-direction.
One solution to this boundary value problem is the Rayleigh-Sommerfeld
diffraction formula of the first kind, which can be written as [1],
$U(x,y,z)=\frac{-1}{2\pi}\iint
U_{0}(x^{\prime},y^{\prime})\frac{\partial}{\partial
z}\frac{\exp(ikR)}{R}dx^{\prime}dy^{\prime}$ (2)
where $R$ is the distance between the points $P$ and $Q$ with coordinates
$(x,y,z)$ and $(x^{\prime},y^{\prime},0)$, respectively, and the integration
is taken over the entire plane $z=0$. This integral can be evaluated exactly
only for a small class of boundary value functions $U_{0}(x,y)$. Under certain
additional constraints, it can be approximated by the Fresnel or Fraunhofer
diffraction integrals [2, 3, 4], which essentially reduces the problem to
Fourier analysis. However, the diffracted field is often found from Eq. (2),
given some general incident field $U_{0}(x,y)$, by evaluating the integral
numerically using a computer.
The principal challenge of evaluating the right side of Eq. (2) numerically is
creating a sampling mesh fine enough to accurately represent the highly
oscillatory integrand. The exponential term oscillates spatially with a
mimimum wavelength of $\lambda$. Thus, it seems necessary that the spacing
between mesh points be much smaller than $\lambda$. Here we show that a
sampling mesh with spacing $\lambda/2$ between points is sufficient to obtain
high accuracy for most practical applications. Moreover, if the field is
composed only of homogeneous plane waves, then this produces the exact result.
We propose a modified version of the Rayleigh-Sommerfeld solution, which
incorporates the boundary conditions discretely. To arrive at this
formulation, we employ the angular spectrum method [5] of field analysis. Both
the two-dimensional and three-dimensional versions of the result are
discussed.
## 2 The Angular Spectrum: Homogeneous and Inhomogeneous Wave Components
The Rayleigh-Sommerfeld solution to the Helmholtz equation is obtained by
using the Kirchhoff integral formula in conjunction with the Green’s function
solution [1]. Yet, there is an alternative method for solving Eq. (1),
referred to as the angular spectrum method. It can be shown[5, 1] that the
general solution to Eq. (2) in the half-space $z\geq 0$ with the given
boundary conditions can be written as
$U(x,y,z)=\iint_{-\infty}^{\infty}A(u,v)\exp(ik[ux+vy+wz])\ du\ dv$ (3)
where $w=+\sqrt{1-u^{2}-v^{2}}$, and
$A(u,v)=\lambda^{-2}\iint_{-\infty}^{\infty}U_{0}(x,y)\exp(-ik[ux+vy])\ dx\
dy$ (4)
is the so-called angular spectrum of the field $U$, as a function of the
direction cosines $u$ and $v$. Evidently, the angular spectrum function is
just the Fourier transform of the boundary value function $U_{0}(x,y)$. For
each plane wave component,
$U_{u,v}=A(u,v)\exp(ik[ux+vy+wz])$
in the integrand of Eq. (3), there are two physically distinct possibilities:
$u^{2}+v^{2}\leq 1$ and $u^{2}+v^{2}>1$. In the former, $w$ is real and the
plane wave is called homogeneous. In the latter case, $w$ is imaginary and the
plane wave is called inhomogeneous,
$\exp(ik[ux+vy+wz])=\exp(-k|w|z)\exp(ik[ux+vy]).$ (5)
This latter kind of wave is also known as an evanescent wave, because it
propagates in the plane $z=0$ and vanishes rapidly in the positive $z$
direction. Based on this consideration, we can express the field $U$ as the
sum of its homogeneous and inhomogeneous plane wave components,
$U(x,y,z)=U_{H}(x,y,z)+U_{I}(x,y,z)$ (6)
where the homogeneous component is
$U_{H}(x,y,z)=\iint_{u^{2}+v^{2}\leq 1}A(u,v)\exp(ik[ux+vy+wz])\ du\ dv,$ (7)
and the inhomogeneous component is
$U_{I}(x,y,z)=\iint_{u^{2}+v^{2}>1}A(u,v)\exp(-k|w|z)\exp(ik[ux+vy])\ du\ dv.$
(8)
Let us place an upper bound on $|U_{I}|$ for any (real) value of $kz$.
Applying Schwarz’ inequality to the right side of Eq. (8) gives
$\begin{split}|\iint_{u^{2}+v^{2}>1}A(u,v)\exp(-k|w|z)&\exp(ik[ux+vy])\ du\
dv|^{2}\\\ &\leq\iint_{u^{2}+v^{2}>1}|A(u,v)|^{2}\ du\ dv\ \cdot\
\iint_{u^{2}+v^{2}>1}\exp(-2k|w|z)\ du\ dv\end{split}$ (9)
The second integral on the right side is easily evaluated. Switching from
Cartesian to polar coordinates in the space of the direction cosines, with
radial coordinate $\rho=\sqrt{u^{2}+v^{2}}$, this integral takes the form,
$\iint_{u^{2}+v^{2}>1}\exp(-2k|w|z)\ du\
dv=2\pi\int_{1}^{\infty}\exp(-2kz\sqrt{\rho^{2}-1})\ \rho\ d\rho$ (10)
Then, using the substitution $t^{2}=\rho^{2}-1$ yields
$2\pi\int_{1}^{\infty}\exp(-2kz\sqrt{\rho^{2}-1})\ \rho\
d\rho=2\pi\int_{0}^{\infty}\exp(-2kzt)\ t\ dt$ (11)
Finally, applying integration by parts to the right side has the result,
$\iint_{u^{2}+v^{2}>1}\exp(-2k|w|z)\ du\ dv=\frac{\pi}{2k^{2}z^{2}}$ (12)
Combining this with Eq. (10), we obtain
$|U_{I}(x,y,z)|\leq\frac{1}{kz}\sqrt{\frac{\pi}{2}\iint_{u^{2}+v^{2}>1}|A(u,v)|^{2}\
du\ dv}.$ (13)
In other words, for any given angular spectrum function $A(u,v)$, the optical
field is given asymptically by
$U(x,y,z)=U_{H}(x,y,z)+O\left(1/kz\right).$ (14)
as $kz\rightarrow\infty$. In most applications, the value of $kz$ is very
large, e.g. $\sim 10^{6}$ for visible light. It follows that the inhomogeneous
plane wave component $U_{I}$ is usually nearly zero in almost all of the half-
space $z>0$. This contribution will be neglected in the following analysis.
In dealing only with the homogenous plane wave component, it is desirable to
express $U_{H}$ in terms of its values at the boundary $z=0$. The function
$U_{H}(x,y,0)$ can be expressed in simple terms of the known function
$U_{0}(x,y)$. It follows from elementary Fourier analysis and the convolution
theorem that
$U_{H}(x,y,0)=U_{0}(x,y)\ast\frac{J_{1}(k\sqrt{x^{2}+y^{2}})}{\lambda\sqrt{x^{2}+y^{2}}}$
(15)
where $\ast$ is the convolution operator and $J_{1}$ is the first order Bessel
function of the first kind. It is easy to see that as the wavelength $\lambda$
approaches zero, the second term in the convolution becomes the two
dimensional Dirac delta function $\delta^{2}(x,y)$, with the property
$U_{0}\ast\delta^{2}=U_{0}$. In regions where $U_{0}(x,y)$ varies with length
scales much larger than $\lambda$, this implies $U_{H}(x,y,0)\approx
U_{0}(x,y)$. This approximation will fail at the boundaries of the obstructing
obstacle where the field has a discontinuity. However, if the size of the
obstacle is much larger than the wavelength, then the field near this boundary
will contribute negligibly to the diffracted field at points far away from the
obstacle.
## 3 Diffraction in Two Dimensions
Many configurations of diffraction are two-dimensional in form, e.g.
diffraction by an infinitely long slit. Then as a first case, let us assume
that the boundary field $U_{0}(x,y)=U_{0}(x)$ on the plane $z=0$ has no
dependence on the $y$-coordinate. Then, from Eq. (4) the angular spectrum can
be written as
$A(u,v)=\delta(v)A^{\prime}(u)$ (16)
where $\delta$ is the (one dimensional) Dirac delta function. Inserting this
into Eq. (7) then yields
$U_{H}(x,y,z)=U_{H}(x,z)=\int_{-1}^{1}A(u)\exp(ik[ux+wz])\ du$ (17)
where here $w=+\sqrt{1-u^{2}}$ and the dash is dropped from the angular
spectrum function $A(u)$. As expected, the homogeneous component in $z\geq 0$
also has no $y$-dependence. The same argument obviously applies to $U_{I}$.
To evaluate the integral in Eq. (17), the function $A(u)$ need only be defined
on the interval $u\in[-1,1]$. Within this domain, let us expand $A(u)$ as the
Fourier series,
$A(u)=\sum_{m=-\infty}^{\infty}c_{m}\exp(-i\pi mu)$ (18)
which is always possible for any sufficiently well-behaved function $A(u)$
[6]. The expansion coefficients $c_{m}$ are found by the inversion formula,
$c_{m}=\frac{1}{2}\int_{-1}^{1}A(u)\exp(i\pi mu)\ du.$ (19)
Comparing Eq. (19) to Eq. (17), it is immediately apparent that
$c_{m}=\frac{1}{2}U_{H}\left(\frac{m\lambda}{2},0\right).$ (20)
Combining this relation with Eqs. (16) and (17) therefore has the result
$U_{H}(x,z)=\sum_{m=-\infty}^{\infty}U_{H}\left(\frac{m\lambda}{2},0\right)G\left(x-\frac{m\lambda}{2},z\right)$
(21)
where
$G(x,z)=\frac{1}{2}\int_{-1}^{1}\exp(ik[ux+wz])\ du.$ (22)
We can compare Eq. (21) to Eq. (2). For the homogeneous component $U_{H}$ of
the optical field $U$, we replace the integral of Eq. (2) with the simple sum
in Eq. (21). That is, instead of requiring knowledge of the whole function
$U_{H}$ on the boundary $z=0$, we only need to know its values at uniformly
spaced points $x=m\lambda/2$ for all integer $m$. Moreover, if the field is
only non-zero for a finite region on $z=0$, then one only needs a finite set
of known values of $U_{H}$ on $z=0$ to perfectly compute $U_{H}$ anywhere else
in the half-space $z\geq 0$.
Let us now evaluate the integral on the right side of Eq. (22) for a clearer
expression of the kernel function $G(x,z)$. Firstly, consider the two
important limiting cases: $z=0$ and $z>>\lambda$. The first case is simple,
$G(x,0)=\frac{1}{2}\int_{-1}^{1}\exp(ikux)\ du=\sin(kx)/kx,$ (23)
which reduces the right side of Eq. (21) to Whittaker’s cardinal series
expansion [7, 8, 9, 10] of the function $U_{H}(x,0)$. For the second case
($z>>\lambda$, or equivalently $kz>>1$), we make the trigonometric
substitution, $u=\sin\eta$, such that $w=\cos\eta$ and using basic
trigonometric identities Eq. (22) becomes
$G(x,z)=\frac{1}{2}\int_{-\pi/2}^{\pi/2}\exp(ikr\cos[\eta-\theta])\ \cos\eta\
d\eta,$ (24)
switching to polar coordinates: $x=r\sin\theta$ and $y=r\cos\theta$. The
restriction $z\geq 0$ implies that the angle $\theta$ is confined to
$\theta\in[-\pi/2,\pi/2]$. Because we have $kz>>1$, the exponential term
oscillates rapidly, except near the stationary points of
${\cos[\eta-\theta]}$. These stationary points are $\eta=\theta+l\pi$ for any
integer $l$. However, within the integration interval $[-\pi/2,\pi/2]$ there
can only be one stationary point, namely $\eta=\theta$. Furthermore, near this
point $\cos[\eta-\theta]\approx 1-[\eta-\theta]^{2}/2$. Invoking the principle
of stationary phase [11], it follows that
$\frac{1}{2}\int_{-\pi/2}^{\pi/2}\exp(ikr\cos[\eta-\theta])\ \cos\eta\
d\eta\approx\frac{\exp(ikr)}{2}\cos\theta\int_{-\infty}^{\infty}\exp\left(-ikr\frac{[\eta-\theta]^{2}}{2}\right)\
d\eta.$ (25)
This well known integral is easily evaluated, resulting in the approximation
$G(x,z>>\lambda)\approx\sqrt{\frac{\pi}{2}}\frac{\exp(ikr)}{\sqrt{ikr}}\cos\theta.$
(26)
This function is just the asymptotic form (for large $kr$) of the two
dimensional field established by an oscillating dipole at the origin, with the
dipole moment aligned with the $x$-axis. Combining Eqs. (21) and (26), we
obtain
$U_{H}(x,z>>\lambda)\approx\sqrt{\frac{\pi}{2ik}}\sum_{m=-\infty}^{\infty}U_{H}\left(\frac{m\lambda}{2},0\right)\frac{\exp(ikR_{m})}{\sqrt{R_{m}}}\cos\theta_{m}$
(27)
where $R_{m}$ is the distance between the points at $(x,z)$ and
$(m\lambda/2,0)$, and $\theta_{m}$ is the angle between the ray connecting
these two points and the $z$-axis as illustrated in Fig. 1.
Fig. 1: A two-dimensional electromagnetic disturbance, composed solely of
homogeneous plane waves and satisfying the given boundary conditions, is
uniquely determined at any point $(x,z)$ in the half-space $z\geq 0$ by its
values on the boundary $z=0$ at points separated by $\lambda$/2. The field is
constructed by placing individual oscillating dipoles at each of these
boundary points and adding up all contributions at the point of interest.
To the knowledge of the author, there is no exact closed-form expression for
the kernel function $G(x,z)$ in terms of elementary functions for the two-
dimensional case.
## 4 Diffraction in Three Dimensions
We now examine the more general case of diffraction in three dimensions,
dealing only with the homogeneous component $U_{H}$ of the field $U$. Starting
from Eq. (7), the evaluation of the integral only requires that the angular
spectrum function $A(u,v)$ be defined on the unit disc $u^{2}+v^{2}\leq 1$.
Let the function $f(u,v)$ be defined on the square domain, in which both $u$
and $v$ are bound to the interval $[-1,1]$, such that
$f(u,v)=\left\\{\begin{array}[]{rl}A(u,v)&u^{2}+v^{2}\leq 1\\\
0&u^{2}+v^{2}>1\end{array}\right\\}.$ (28)
Within this square domain, the function $f(u,v)$ can be expanded as the two-
dimensional Fourier series,
$f(u,v)=\sum_{m,n=-\infty}^{\infty}c_{m,n}\exp(-i\pi[mu+nv]).$ (29)
As before, the expansion coefficients are
$\begin{array}[]{rl}c_{m,n}&=\frac{1}{4}\int_{-1}^{1}\int_{-1}^{1}f(u,v)\exp(i\pi[mu+nv])\
du\ dv\\\ \\\ &=\frac{1}{4}\iint_{u^{2}+v^{2}\leq 1}A(u,v)\exp(i\pi[mu+nv])\
du\ dv\\\ \\\
&=\frac{1}{4}U_{H}\left(\frac{m\lambda}{2},\frac{n\lambda}{2},0\right)\end{array}$
(30)
where the third equality follows from the second upon comparing with Eq. (7).
Because $f(u,v)=A(u,v)$ within the unit disc, the function $A(u,v)$ can be
replaced by $f(u,v)$ in Eq. (7). Combining this with Eqs. (29) and (30) then
yields
$U_{H}(x,y,z)=\sum_{m,n=-\infty}^{\infty}U_{H}\left(\frac{m\lambda}{2},\frac{n\lambda}{2},0\right)G\left(x-\frac{m\lambda}{2},y-\frac{n\lambda}{2},z\right)$
(31)
where
$G(x,y,z)=\frac{1}{4}\iint_{u^{2}+v^{2}\leq 1}\exp(ik[ux+vy+wz])\ du\ dv.$
(32)
is the three dimensional kernel function.
To evaluate the integral in Eq. (32), we make use of the coordinate
transformations illustrated by Fig. 2 and described as follows. The approach
is similar to that in Ref. [12], but here a more careful transformation of the
limits of integration is required. We first make the transformation
$(x,y,z)\rightarrow(x^{\prime},y^{\prime},z^{\prime})$, consisting of a
rotation about the origin, such that the point $P$ with coordinates $(x,y,z)$
lies on the $z^{\prime}$-axis in the new coordinate system. That is,
$x^{\prime}=y^{\prime}=0$ for point $P$. The same transformation is also
applied to the direction cosines,
$(u,v,w)\rightarrow(u^{\prime},v^{\prime},w^{\prime})$. This change of
variables preserves the dot product,
$ux+vy+wz=u^{\prime}x^{\prime}+v^{\prime}y^{\prime}+w^{\prime}z^{\prime}=w^{\prime}r$
(33)
where $r=\sqrt{x^{2}+y^{2}+z^{2}}$. The ratio of the differential areas $du\
dv$ and $du^{\prime}\ dv^{\prime}$ (given by the determinant of the Jacobian)
is $\cos\theta$, where $\theta=\cos^{-1}(z/r)$ is the angle between point $P$
and the $z$-axis. With this,
$\frac{1}{4}\iint\exp(ik[ux+vy+wz])\ du\
dv=\frac{\cos\theta}{4}\iint\exp(ikw^{\prime}r)\ du^{\prime}\ dv^{\prime}.$
(34)
This form suggests a switch to spherical coordinates in the space of the
direction cosines,
$\begin{array}[]{rl}u^{\prime}&=\ \sin\eta\ \cos\psi\\\ v^{\prime}&=\
\sin\eta\ \sin\psi\\\ w^{\prime}&=\ \cos\eta\end{array}$ (35)
where $\eta$ and $\psi$ are the azimuthal and polar angles, respectively, of
the point with coordinates $(u^{\prime},v^{\prime},w^{\prime})$. The
differential area transforms once more as
$du^{\prime}\ dv^{\prime}=\sin\eta\ \cos\eta\ d\eta\ d\psi.$ (36)
The limits of integration, defined by $u^{2}+v^{2}\leq 1$ in the initial
coordinate system, transform accordingly. The polar angle $\psi$ is integrated
from 0 to $2\pi$. The azimuthal angle $\eta$ is integrated from 0 to
$\pi/2+\Delta\eta$, where $\Delta\eta$ is the angle between the planes $z=0$
and $z^{\prime}=0$ as seen from the origin at angle $\psi$. Applying the
spherical law of sines [13] to the geometry shown in the inset of Fig. 2, it
can be easily verified that $\Delta\eta=\sin^{-1}(\cos\psi\sin\theta)$.
Therefore, the integral in Eq. (32) becomes
$\begin{array}[]{rl}\frac{1}{4}\iint_{u^{2}+v^{2}\leq 1}\exp(ik[ux+vy+wz])\
du\
dv&=\frac{\cos\theta}{4ik}\frac{d}{dr}\int_{0}^{2\pi}\int_{0}^{\frac{\pi}{2}+\sin^{-1}(\cos\psi\sin\theta)}\exp(ikr\cos\eta)\sin\eta\
d\eta\ d\psi\\\ \\\
&=\frac{1}{4ik}\frac{d}{dz}\int_{0}^{2\pi}\int_{0}^{\frac{\pi}{2}+\sin^{-1}(\cos\psi\sin\theta)}\exp(ikr\cos\eta)\sin\eta\
d\eta\ d\psi.\end{array}$ (37)
The chain rule is applied, $\cos\theta\ (d/dr)=(dr/dz)(d/dr)=d/dz$, to obtain
the second equality.
Fig. 2: Geometry of the coordinate transformation. The point $P$ in the new
coordinate system lies on the $z^{\prime}$-axis. The point given by
coordinates $(u,v,w)$ lies on the unit half-sphere $w\geq 0$. The shaded
region is the plane $z=0$. In the inset, right angles of the spherical
triangle are denoted by small squares.
The integral on the right side of Eq. (37) can now be evaluated with the
substitution $\tau=\cos\eta$, yielding
$\begin{array}[]{rl}\frac{1}{4}\iint_{u^{2}+v^{2}\leq 1}\exp(ik[ux+vy+wz])\
du\
dv&=\frac{1}{4ik}\frac{d}{dz}\int_{0}^{2\pi}\int_{-\sin\theta\cos\psi}^{1}\exp(ikr\tau)\
d\tau\ d\psi\\\ \\\
&=\frac{1}{4ik}\frac{d}{dz}\int_{0}^{2\pi}\frac{\exp(ikr)-\exp(-ikr\sin\theta\cos\psi)}{ikr}\
d\psi\\\ \\\ \end{array}$ (38)
and finally,
$G(x,y,z)=-\frac{1}{2\pi}\left(\frac{\lambda}{2}\right)^{2}\frac{d}{dz}\frac{\exp(ikr)-J_{0}(kr\sin\theta)}{r}.$
(39)
where $J_{0}$ is the zeroth order Bessel function of the first find, which
admits the identity [14],
$J_{0}(s)=\frac{1}{2\pi}\int_{0}^{2\pi}\exp(is\cos\phi)\ d\phi,$ (40)
used to obtain Eq. (39) from Eq. (38). Note that, whereas we were only able to
obtain an approximate simplification of the kernel function $G$ in the two-
dimensional case, Eq. (39) is an exact expression of this function for three
dimensions. Applying this to Eq. (31) gives the discretized diffraction
formula,
$U_{H}(x,y,z)=-\frac{1}{2\pi}\left(\frac{\lambda}{2}\right)^{2}\sum_{m,n=-\infty}^{\infty}U_{H}\left(\frac{m\lambda}{2},\frac{n\lambda}{2},0\right)\frac{d}{dz}\frac{\exp(ikR_{m,n})-J_{0}(kR_{m,n}\sin\theta_{m,n})}{R_{m,n}}$
(41)
where
$R_{m,n}=\sqrt{\left(x-m\lambda/2\right)^{2}+\left(y-n\lambda/2\right)^{2}+z^{2}}$
is the distance between the point at $(m\lambda/2,n\lambda/2,0)$ and point
$P$, and $\theta_{m,n}=\cos^{-1}(z/R_{m,n})$ is the angle between the ray
connecting these two points and the $z$-axis, as illustrated in Fig. 3.
Equation (41) bears much resemblance to the Rayleigh-Sommerfeld integral
formula of Eq. (2), with the exception of the Bessel function term, and more
importantly the integral is replaced by a sum.
Fig. 3: The homogeneous component $U_{H}$ of the diffracted field at the point
$(x,y,z)$ in the half-space $z\geq 1$ is completely determined by its values
on the rectangular lattice of points at $(m\lambda/2,n\lambda/2,0)$, where $m$
and $n$ are integers. It is found by placing secondary sources at each lattice
point and adding up each contribution at the point of interest.
## 5 Discussion and Conclusions
We present a solution to the scalar Helmholtz equation, the cornerstone of
Scalar Diffraction Theory, which inherently implies a potentially useful
sampling theorem regarding the propagation of electromagnetic radiation in
free space. Summarily, if the scalar field is composed only of homogeneous
plane waves, then it can be perfectly reconstructed from the knowledge of its
values on a lattice of points on the boundary $z=0$. The lattice of points has
uniform spacing of $\lambda/2$ between collinear points. If the field is
composed of inhomogeneous plane waves, then one can always choose some point
$z$ such that this contribution to the total field becomes negligibly small.
Going further, the solution presented in Eq. (41) can be straightforwardly
interpreted as a matrix equation, if the field were to be evaluated on a
secondary lattice of points for some $z>0$. Posing the diffraction calculation
in this form would be well-suited for numerical algorithms performed by a
computer. The operation could possibly be optimized for efficiency, analogous
to the Fast Fourier Transform. Such a development would significantly reduce
the time needed to calculate the diffracted field in many applications.
## 6 Acknowledgements
I would like to thank Wayne R. McKinney of Lawrence Berkeley National
Laboratory for several valuable discussions on this topic.
## References
* [1] J. W. Goodman, _Introduction to Fourier Optics_ (McGraw-Hill, 1988), chap. 3, 2nd ed.
* [2] M. Born and E. Wolf, _Principles of Optics_ (Pergamom Press, 1980), chap. 8, 6th ed.
* [3] J. W. Goodman, _Introduction to Fourier Optics_ (McGraw-Hill, 1988), chap. 4, 2nd ed.
* [4] E. Hecht, _Optics_ (Addison Wesley, 2002), chap. 10, 4th ed.
* [5] L. Mandel and E. Wolf, _Optical Coherence and Quantum Optics_ (Cambridge University Press, 1995), chap. 3.
* [6] M. L. Boas, _Mathematical Methods in the Physical Sciences_ (John Wiley and Sons, 2006), chap. 7.
* [7] E. T. Whittaker, “On the functions which are represented by the expansions of the interpolation theory,” Proc. Royal Soc. Edinburgh 35, 181–194 (1915).
* [8] J. M. Whittaker, “On the cardinal function of interpolation theory,” Proc. Edinburgh Math. Soc. 1, 41–46 (1927).
* [9] C. E. Shannon, “A mathematical theory of communication,” The Bell System Technical Journal 27, 379–423 (1948).
* [10] J. W. Goodman, _Introduction to Fourier Optics_ (McGraw-Hill, 1988), chap. 2, 2nd ed.
* [11] A. Erdélyi, _Asymptotic Expansions_ (Dover, reprint, 1956), chap. 2.
* [12] H. Weyl, “Ausbreitung elektromagnetischer wellen uber einem ebenem leiter,” Ann. Phys. 60, 481–500 (1919).
* [13] D. Zwillinger, _Standard Mathematical Tables and Formulae_ (CRC Press, 2003), chap. 6.
* [14] M. Abramowitz and I. A. Stegun, _Handbook of Mathematical Functions_ (National Bureau of Standards, 1964), chap. 9, 10th ed.
|
arxiv-papers
| 2013-01-29T01:32:33 |
2024-09-04T02:49:40.876701
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Daniel J. Merthe",
"submitter": "Daniel Merthe",
"url": "https://arxiv.org/abs/1301.6814"
}
|
1301.6970
|
# Non-classicality of molecular vibrations activating electronic dynamics at
room temperature
Edward J. O’Reilly Alexandra Olaya-Castro [email protected] Department of
Physics and Astronomy, University College London, Gower Street, London, WC1E
6BT United Kingdom.
###### Abstract
Advancing the debate on the possible advantages of quantum effects in
biomolecular systems requires illustrations of biomolecular prototypes that
can exhibit and use non-classical phenomena to aid performance. Here we
investigate non-classical fluctuations in molecular dimers present in
photosynthetic light-harvesting antennae. We show that exciton-vibrational
resonances enable non-classical occupation fluctuations of the driven
collective high-energy vibration at room temperature. This phenomenon is a
manifestation of strong modulation of low-phonon occupation levels through the
transient formation of hybrid exciton-vibration states. For the prototype
considered, direct quantitative relations seem to hold between non-
classicality and optimal exciton population transfer.
The experimental investigation of quantum effects in photosynthetic energy
transfer Engel07 ; Panitchayangkoon07 ; Calhoun09 ; Collini10 ; Harel12 has
inspired exciting research towards asserting the relevance of non-classical
behaviour in the biological function of the molecular components of living
organisms. Theoretical works have made emphasis on the quantumness exhibited
by electronic degrees of freedom Whaley11 and on the key role of
environmental fluctuations Plenio08 ; Rebentrost09 . In order to understand
the full range of possible non-classical phenomena present during room-
temperature dynamics of electronic excitations in light-harvesting antennae it
is important to expand the analysis to the molecular motions that play a
fundamental role in driving exciton transport Renger01 . Research in this
direction is timely given that experiments Richards12 ; Turner12 and
theoretical works Womick11 ; Chin12a ; Chin12b ; Kreisbeck12 ; Kolli12 ;
Christensson12 ; Tiwari12 are highlighting the importance of concerted
electronic and vibrational dynamics in energy transfer. Moreover, recent
experimental techniques able to manipulate vibrational states Biggs09 ;
Biggs12 and probe their quantum properties may indeed provide experimental
access to the non-classicality of vibrational modes in biomolecular systems.
Here we consider a prototype molecular dimer with an structured exciton-phonon
interaction and investigate non-classical behaviour of molecular vibrations
that steer energy dynamics and exciton coherences in this molecular unit.
While previous studies have considered coherent exciton-vibration dynamics in
molecular dimersKuhn96 ; Chin12b ; Christensson12 , it is unclear how this
deviates from classical (wave-like) coherent behaviour and if so whether there
are quantitative relationships between non-classicality and effective energy
transfer.
The prototype of interest consists of two chromophores coupled by weak
electronic interactions and subject to the influence of both a low-energy
thermal background and a discrete high-frequency intramolecular vibration. The
key aspect of this molecular vibration is that its energy matches closely the
exciton gap but it is much larger than the thermal energy scale. Dimers of
this type naturally occur in several light-harvesting proteins Womick11 ;
Doust05 and are of particular interest as they represent systems where,
although counter-intuitive, exciton coherence beating has been probed Wong12 .
Insights into the importance of resonances between exciton energy mismatch and
vibrational frequencies for efficient energy transfer can be gained from
Förster theory Forster59 . However, the wider implications of such resonances
for optimal spatio-temporal distribution of energy Kolli12 ; Perdomo10 , for
manipulation of exciton coherence Chin12a ; Kolli12 ; Chin12b and for non-
adiabatic excitation dynamics Tiwari12 have only started to be clarified. An
issue that remains to be tackled, is the extent to which such resonances can
expose the non-classical features of the molecular motions activating
transport. In the dimer investigated here, commensurate energies of exciton
gap and discrete vibrations imply that after input of exciton population,
effective energy distribution proceeds by driving the collective relative
displacement mode out of thermal equilibrium towards a non-classical state.
Non-classicality of the state of this relative motion is unambiguously
indicated by sub-Poissonian occupation statistics Mandel79 or alternatively
by strong modulation of low-phonon occupations Klyshko96 and is a
manifestation of transient collective coherent behaviour between excitons and
vibrations. Low-energy thermal fluctuations stabilize energy distribution to
the lowest exciton state before destroying non-classicality of the collective
mode assisting the process. Meanwhile, coherent interaction between electronic
degrees of freedom and such a mode manifests itself in the beating of
excitonic coherences with frequency components that vary depending on the
coupling to the thermal bath. When time-averaging over a relevant time scale,
we find direct quantitative relations between population transfer and the
degree of non-classicality as indicated by the average of negative values of
the Mandel $Q-$parameter Mandel79 . This relationship holds even when the
electronic system is initialized in a statistical mixture of exciton states.
Hence, efficient photosynthetic energy transport could take advantage of non-
classicality.
The field of quantum optics has developed a solid framework to quantify the
non-classicality of bosonic fields Davidovich96 . It therefore provides useful
conceptual and quantitative tools to investigate non-classicality of the
harmonic vibrational degrees of freedom of interest in this work. From the
perspective of quantum optics, non-classicality arises if the state of the
system of interest cannot be expressed as a statistical mixture of coherent
states defining a valid probability measure, this then leads to non-positive
or highly singular values of the Glauber-Sudarshan $P$-function or the Wigner
functionScully97 . However, experimental characterization of these
distributions is not always a straightforward task. Alternatively, it is of
interest to investigate signatures of non-classicality. For a single-mode
field, a sub-Poissonian occupation number distribution quantified by the
Mandel’s $Q-$parameter Mandel79 is a signature of deviations from classical
behaviour. The Mandel $Q-$parameter characterizes the departure of the
occupation number distribution $P(n)$ from Poissonian statistics through the
inequality
$\displaystyle
Q=\frac{\langle\hat{n}^{2}\rangle-\langle\hat{n}\rangle^{2}}{\langle\hat{n}\rangle}-1<0$
(1)
where $\langle\hat{n}\rangle$ and $\langle\hat{n}^{2}\rangle$ denote the first
and second moments of the bosonic number operator $\hat{n}$ respectively.
Vanishing $Q$ indicates Poissonian number statistics where the mean of
$\hat{n}$ equals its variance (such is the case of a coherent state of light).
For a chaotic thermal state one finds that $Q=\langle\hat{n}\rangle>0$
indicating that particles are ‘bunched’. The departure from classical (wave-
like) behaviour is characterised by $Q<0$ or particle number distributions
where the variance is less than the mean, as is the case for Fock states. The
occupation fluctuations associated with a sub-Poissonian distribution
originate from the quantized nature of the field and cannot be described in
terms of an underlying _classical_ stochastic process. A criteria that
directly connects non-classical occupation fluctuations and the negativity of
a quasiprobability distribution is an inequality indicating modulation of
nearest number occupationsKlyshko96 . In this work we use the above framework
to quantify non-classicality of the undamped vibrational modes supporting
exciton transport in the molecular dimer of interest. We consider each
chromophore with an excited electronic state of energy $\varepsilon_{i}$
strongly coupled to a quantized vibrational mode of high-frequency
($\omega_{\textrm{vib}}>K_{B}T$) described by the bare Hamiltonians
$H_{\textrm{el}}=~{}\sum_{i=1,2}\varepsilon_{i}\sigma^{+}_{i}\sigma^{-}_{i}$
and
$H_{\textrm{vib}}=~{}\omega_{\textrm{vib}}(b_{1}^{\dagger}b_{1}+~{}b_{2}^{\dagger}b_{2})$
respectively. Inter-chromophore coupling is generated by dipole-dipole
interactions of the form
$H_{\textrm{d-d}}=V(\sigma^{+}_{1}\sigma^{-}_{2}+\sigma^{+}_{2}\sigma^{-}_{1})$.
The electronic excited states interact with their local vibrational
environments with strength $g$, linearly displacing the corresponding mode
coordinate, $H_{\textrm{el-
vib}}=g\sum_{i=1,2}\sigma^{+}_{i}\sigma^{-}_{i}(b^{\dagger}_{i}+b_{i})$. In
the above $b^{\dagger}_{i}$($b_{i}$) creates (annihilates) a phonon of the
vibrational mode of chromophore $i$ while $\sigma^{\pm}_{i}$ creates or
destroys an electronic excitation at site $i$. The eigenstates $|X\rangle$ and
$|Y\rangle$ of $H_{\textrm{el}}+H_{\textrm{d-d}}$ denote exciton states with
energy splitting given by $\Delta
E=\sqrt{(\varepsilon_{1}-\varepsilon_{2})^{2}+4V^{2}}$. Transformation into
collective mode coordinates shows that only the mode corresponding to the
relative displacement of the modes with the bosonic operators
$b^{({\dagger})}=(b^{({\dagger})}_{1}-b^{({\dagger})}_{2})/\sqrt{2}$, couples
to the excitonic system. The centre of mass mode decouples from the electronic
degrees of freedom. The occupation of the relative displacement mode of
frequency $\omega_{\textrm{vib}}$ is denoted by $n$ and the non-classical
properties of such mode is the subject of interest in this article.
Our prototype system corresponds to the central dimer of the phycoerythrin 545
(PE545) antennae found in cryptophyte marine algae Doust04 . Exciton coherence
beating involving this dimer has been probed Collini10 and a structured
spectral density has been phenomenologically determined via linear
spectroscopy Novoderezhkin10 . The electronic parameters for the dimer taken
from reference Novoderezhkin10 are $\varepsilon_{1}=19574\textrm{cm}^{-1}$,
$\varepsilon_{2}=18532\textrm{cm}^{-1}$ and $V=92.2\textrm{cm}^{-1}$. The
intramolecular mode that is separated from the set of vibrational degrees of
freedom has frequency $\omega_{\textrm{vib}}=1111\textrm{cm}^{-1}$ being
quasi-resonant with the exciton energy splitting $\Delta
E=1058.2\textrm{cm}^{-1}$. The coupling strength of each chromophore to the
this mode is $g=\omega_{\textrm{vib}}\sqrt{0.0578}=267.1\textrm{cm}^{-1}$
obtained from Doust04 ; Novoderezhkin10 , and the thermal energy scale at room
temperature is $k_{\rm B}T\approx 200\textrm{cm}^{-1}$. We consider the
influence of the low energy thermal fluctuations via a thermal bath $H_{B}$
described by a continuous distribution of harmonic modes, each linearly and
independently coupled to the excited electronic states $H_{\textrm{el}}$ with
coupling strength characterised by a Drude spectral density
$J(\omega)=(\lambda\Omega_{c}\omega/\pi)/{(\omega^{2}+{\Omega_{c}}^{2}})^{2}$
with reorganization energy $\lambda=110\textrm{cm}^{-1}$ and cut-off frequency
$\Omega_{c}=100\textrm{cm}^{-1}$ Hoda12 , hence the strength of the system-
bath interaction is parametrized by $\sqrt{\lambda\Omega_{c}}$ Prior10 . Since
$J(\omega_{\textrm{vib}})\ll g^{2}$, this does not over-account for the
influence of the resonant mode on the electronic system. Importantly, although
the electronic coupling $V$ is the smallest energy scale in the prototype
dimer, the exciton dynamics and phonon fluctuations here investigated will not
be properly described by Förster theory. In order to accurately account for
the effects of the thermal bath we have extended the hierarchical expansion of
the dynamics Tanimura89 ; Ishizaki09 ; Shi09 to explicitly include the
quantum interactions between electronic excitations and the selected mode.
This approach allows us to account accurately for the effects of the low-
energy thermal bath. In calculating dynamics we consider an initial state, in
which the upper energy exciton eigenstate is rapidly excited while both high-
energy quantized vibrations and low-energy thermal modes are initially in
thermal equilibrium at room temperature. Truncating the quantized modes at an
occupation Fock level $M=6$ adequately describes both reduced dynamics of the
collective quantized mode and the electronic dynamics. A description of the
numerical methodology can be found in the appendix.
Figure 1: a, Representation of the pigments and protein environment of the
PE545 complex (Protein Data Bank ID code 1XG0Doust04 ). The central dimer is
highlighted. Also shown are the energy levels of the exciton-vibration states
used to describe energy transfer $\left|{X,n}\right\rangle$. b, Coherent
dynamics of the exciton-vibration dimer. Oscillatory population of the lowest
exciton population $\rho_{YY}(t)$ (thick blue curve) is accompanied by
oscillations of the inter-exciton coherence in the ground-state of the
vibrational mode $|\rho_{X0-Y0}(t)|$ (thin red curve). c, Mandel $Q$-parameter
of the vibrational mode for biological electronic coupling and d for zero
electronic coupling.
Hamiltonian dynamics and non-classicality.- We first illustrate the quantum
coherent dynamics of excitation when coupled to the undamped quasiresonant
high-energy vibrational modes in the absence of the low-energy thermal bath.
This allows us to identify relevant time scales in the evolution of a state
with no initial excitonic superpositions:
$\rho(t_{0})=\left|{X}\right\rangle\langle
X|\otimes\varrho^{\textrm{th}}_{\textrm{vib}}$ which written in the basis of
exciton-vibration states of the form $|X,n\rangle$ becomes
$\rho(t_{0})=\sum_{n}P_{th}(n)\left|{X,n}\right\rangle\langle X,n|$. Here $n$
denotes the phonon occupation number of the collective mode while $P_{th}(n)$
denotes the thermal occupation of such level. We investigate the population of
the lowest excitonic state $\rho_{YY}(t)=\sum_{n}\langle
Y,n|\rho(t)|Y,n\rangle$, the absolute value of the coherence
$\rho_{X0-Y0}(t)=\langle X,0|\rho(t)|Y,0\rangle$ which denotes the inter-
exciton coherence in the ground-state of the collective vibrational mode, and
the non-classicality given by negative values of $Q(t)$. Hamiltonian evolution
generates coherent transitions from states $|X,n\rangle$ to $|Y,n+1\rangle$
(see Figure 1(a)) with a rate $f$ that depends on the exciton delocalization
($|V|/\Delta\epsilon$), the coupling to the mode $g$, and the phonon
occupation $n$ i.e. $f\simeq g(2|V|/\Delta\varepsilon)\sqrt{(n+1)/2}$. Since
$\omega_{\textrm{vib}}\gg K_{\rm B}T$ the ground state of the effective mode
is largely populated, such that the Hamiltonian evolution of the initial state
is largely dominated by the evolution of the state $|X,0\rangle$. This implies
that the energetically close exciton-vibration state $|Y,1\rangle$ becomes
coherently populated at a rate $f\simeq g(2|V|/\Delta\epsilon)$, leading to
the oscillatory pattern observed in the probability of occupation
$\rho_{YY}(t)$ as illustrated in Figure 1 (b). The low-frequency oscillations
of the dynamics of $\rho_{YY}(t)$ cannot be assigned to the exciton or the
vibrational degrees of freedom alone as expected from quantum-coherent
evolution of the exciton-plus-effective mode system. For instance, if the mode
occupation is restricted to at most $n=1$, the period of the amplitude of
$\rho_{YY}(t)$ is given approximately by the inverse of $(1/2)(\sqrt{(\Delta
E-\alpha)^{2}+(2g^{2}4V^{2}/\alpha\Delta E)}-(\Delta E+\alpha))$ with
$\alpha^{2}=2g^{2}+\omega_{\textrm{vib}}^{2}$ and
$(2g^{2}4V^{2}/\alpha^{2}\Delta E^{2})\ll 1$. Coherent exciton population
transfer is then accompanied by coherent dynamics of the inter-exciton
coherence $|\rho_{X0-Y0}(t)|$ with the main amplitude modulated by the same
low-frequency oscillations of $\rho_{YY}(t)$ and a superimposed fast
oscillatory component of frequency close to $\omega_{\textrm{vib}}$ as can be
seen in Figure 1 (b). This fast driving component arises from local
oscillatory displacements: when $V\simeq 0$ the time evolution of each local
mode is determined by the displacement operator with amplitude
$\alpha(t)={2g}(1-\exp(-i\omega_{\textrm{vib}}t))/\omega_{\textrm{vib}}$
Palma96 . As the state $|Y,1\rangle\langle Y,1|$ is coherently populated, the
collective quantized mode is driven out of equilibrium towards a non-classical
state in which selective occupation of the first vibrational level takes
place. This manifests itself in sub-Poissonian phonon statistics as indicated
by negative values of $Q(t)$ shown in Figure 1(c). Similar phenomena have been
described in the context of electron transport in a nanoelectromechanical
system Merlo08 ; Cavaliere08 . Importantly, such sub-Poissonian statistics
arises only when the electronic interaction is finite. For comparison, Figure
1(d) shows that if $V=0$, a local excitation drives the mode towards a
classical thermal displaced state with super-Poissonian statistics ($Q(t)>0$).
In short, non-classicality of the relative displacement mode quasiresonant
with the excitonic transition arises through the formation of exciton-
vibration states.
Energy and coherence dynamics - Interplay between vibration-activated dynamics
and thermal fluctuations leads to two distinct regimes of energy transport.
For our consideration of weak electronic coupling, the coherent transport
regime is determined approximately by $\sqrt{\lambda\Omega_{c}}\leq
2g|V|/\Delta\epsilon$. Here population of the low-lying exciton state is
dominated by coherent transitions between exciton-vibration states and the
rapid short-time growth of $\rho_{YY}(t)$ in this regime can be traced back to
coherent transfer of population from $|X,0\rangle\langle X,0|$ to
$|Y,1\rangle\langle Y,1|$. At longer time scales thermal fluctuations induce
incoherent transitions from $|X,0\rangle\langle X,0|$ to $|Y,0\rangle\langle
Y,0|$ with a rate proportional to $\sqrt{\lambda\Omega_{c}}$, thereby
stabilizing population of $|Y,1\rangle\langle Y,1|$ to a particular value (see
Figure 2(a)). In contrast, for $\sqrt{\lambda\Omega_{c}}>2g|V|/\Delta\epsilon$
population transfer to $\rho_{YY}(t)$ is incoherent although the dynamics is
still accompanied by generation and coherent evolution of excitonic coherences
(see Figures 2(g) and 2(h)). In this incoherent transport regime
$\rho_{YY}(t)$ has a slow but continuous rise reflecting the fact that the
incoherent transitions from $|X,0\rangle\langle X,0|$ to $|Y,0\rangle\langle
Y,0|$ now contribute significantly to the transport. However, transfer to
$\rho_{YY}(t)$ is always more efficient with the quasiresonant mode than in
the situations where only thermal-bath induced transitions are considered (see
dashed lines in Figures 2(d) and 2(g)). The underlying reason is that the
system is transiently evolving towards a thermal configuration of exciton-
vibration states. Hence, in both regimes excitation transfer to the lowest
exciton state is achieved by transiently sharing energy with the relative
displacement mode. The transition from coherent to incoherent exciton
population dynamics is also marked by the onset of energy dissipation of the
exciton-vibration system as shown in figures 2(c),2(f) and 2(i). While
population transfer is coherent, energy dissipation into the thermal bath is
prevented (Fig. 2(c) and 2(f)).
The beating patterns of the coherence $\rho_{X0-Y0}(t)$ reveals the structured
nature of the exciton-phonon interaction and witnesses whether there is
coherent exciton-vibration evolution. Because of this the frequency components
of such oscillatory dynamics vary depending of the coupling to the thermal
bath as can be seen in Figures 2(b), (e) and (h) . In the coherent regime
$\rho_{X0-Y0}(t)$ follow exciton populations with the main amplitude modulated
by the same relevant energy difference between exciton-vibration states (see
Figures 2(b) and (e)). In contrast, in the regime of incoherent transport the
short-time oscillations of $\rho_{X0-Y0}(t)$ (between $t=0$ and $t=0.1$ps)
arise from electronic correlations due to bath-induced renormalization of the
electronic Hamiltonian Silbey84 and as such, the frequency of these short-
time oscillations correspond to transitions of renormalized excitons. As the
main amplitude is damped, this exciton coherence retains the superimposed
driving at a frequency $\omega_{\textrm{vib}}$. The dynamical features
presented in Figures 2(g) and 2(h) confirm previous findings based on
perturbative calculations Kolli12 and agrees with the time-scales of the
exciton coherence beating reported in the two-dimensional spectroscopy of
cryptophyte algae Wong12 .
Figure 2: The dynamics of $\rho_{YY}(t)$ (top row), $|\rho_{X0,Y0}(t)|$
(middle row) and Energy of the exciton vibration system (bottom row) for three
interaction strengths to the low-energy thermal bath: a-c:
$\lambda=6\textrm{cm}^{-1}$, d-f $\lambda=20\textrm{cm}^{-1}$ and g-i:
$\lambda=110\textrm{cm}^{-1}$
Non-classicality - Interaction with a large thermal environment leads to the
emergence of classicality. Indeed, as the coupling of the electronic degrees
of freedom to the thermal background is increased, the maximum non-
classicality of the collective mode as indicated by the most negative value of
$Q(t)$ is reduced. However, the time interval for which the mode remains in a
non-classical state is not a monotonic function of $\lambda$. For moderate
values of $\lambda$, the relative displacement mode spends longer periods in
states with non-classical fluctuations i.e. periods for which $Q(t)<0$ as seen
in Figures 3(a)-(b). Hence relatively medium coupling to a thermal background
assists selective population of state $|Y,1\rangle$ thereby stabilizing non-
classicality at a particular level. An important feature of the results
presented in Figures 3(a)-3(c) is therefore that non-classical features
surviving the picosecond time scale are present across the whole range of
thermal bath couplings. This is consistent with a slow decay of the exciton-
vibration coherence $\rho_{X0,Y1}$ (not shown). Interestingly, the non-
classical properties of the collective mode resemble non-classicality of
thermal states (completely incoherent states) that are excited by a single
quanta Zavatta07 . The non-classical properties predicted by $Q(t)$ also agree
with the non-classicality predicted by a parameter proposed by Klyshko
Klyshko96 as is described in appendix B.
Figure 3: a-c Dynamics of the Mandel $Q$-parameter for
$\lambda=6,20,110\textrm{cm}^{-1}$. Figure 4: Time integrated averages of
exciton population $\rho_{YY}(t)$, inter-exciton coherent in the lowest
vibrational level $|\rho_{X0-Y0}(t)|$ and non-classicality as quantified by
$Q(t)\Theta[-Q(t)]$ (blue, red and green respectively) as functions of
coupling to the thermal background by fixing environment cut-off frequency
$\Omega_{c}=100\textrm{cm}^{-1}$ and varying reorganization energy $\lambda$.
Non-classicality is witnessed by exciton population dynamics. In order to
illustrate this, we now investigate quantitative relations between non-
classicality and energy transport by considering relevant integrated averages
in the time scale of coherent exciton-vibration dynamics denoted by $\tau$.
For the parameters chosen, this time scale is about half a picosecond and
corresponds to the time in which excitation energy would be distributed away
to other chromophores Kolli12 . The time integrated averages over $\tau$ are
defined as: $\langle
F[\rho(t)]\rangle_{\tau}=\frac{1}{\tau}\int_{0}^{\tau}\textrm{d}t~{}F[\rho(t)]~{},$
where $F[\rho(t)]$ corresponds to the exciton population $\rho_{YY}(t)$ and
the non-classicality of the collective quantized mode through periods of sub-
Poissonian statistics $Q(t)\Theta[-Q(t)]$ as functions of the coupling to the
bath $\lambda$. As shown in Figure 4 the average exciton population and non-
classicality follow a similar non-monotonic trend as a function of the
coupling to the thermal bath, indicating a direct quantitative relation
between efficient energy transfer in the time scale $\tau$ and the degree of
non-classicality. The appearance of maximal point in the average non-
classicality as a function of the system-bath coupling indicates that the
average quantum response of the relative displacement mode to the impulsive
electronic excitation, is optimal for a small amount of thermal noise.
The above results indicate that the degree of delocalization of the initial
exciton state is key to enabling non-classical fluctuations of the collective
quantized vibration. This suggests that statistical mixtures of excitons with
finite purity can still trigger such non-classical response. To illustrate
this we now consider mixed initial states of the form
$\rho(t_{0})=\varrho_{\textrm{ex}}\otimes\varrho^{\textrm{th}}_{\textrm{vib}}$
where $\varrho_{\textrm{ex}}=r\left|{X}\right\rangle\langle
X|-(1-r)\left|{Y}\right\rangle\langle Y|$ with $1/2\leq r\leq 1$. The
associated linear entropy quantifying the mixedness of the initial exciton
states is given by $S_{L}=2r(r-1)$. The time-averaged non-classicality (Figure
5 (a)) and average population transfer (Figure 5 (b)) follow similar
decreasing monotonic trends for mixed states. These findings then suggest
direct quantitative relationships between non-classicality and exciton
population transport in the relevant time scale.
Figure 5: The time integrated averages of exciton population
$\rho_{YY}(t)-\rho_{YY}(0$) and non-classicality as quantified by
$Q(t)\Theta[-Q(t)]$ as functions of the linear entropy $S_{L}=2r(r-1)$ for
$0<r<1/2$. Curves are for different system-bath couplings.
Outlook.- Our work shows that in molecular dimers operating in the weak
electronic coupling regime, exciton-vibrational resonances can trigger a non-
classical response of a quasiresonant collective high-energy vibration thereby
taking advantage of quantum fluctuations to efficiently populate a target
exciton state. These non-classical fluctuations are observed for a variety of
initial exciton conditions including statistical mixtures suggesting that such
non-classicality may be triggered even under incoherent input of electronic
population. These quantum phenomena could be experimentally accessible by
transient coherent ultrafast phonon spectroscopy Waldermann08 which is
sensitive to low-phonon populations of high-energy vibrations Lee11 . The
framework here discussed could also be used to give more insight into the
possible non-classical response of vibrational modes in other chemical sensors
Turin96 which are conjectured to operate through weak electronic
interactionsBrookes07 to sense vibrational frequencies of the order of a
thousand wavenumbers Franco11 .
From a more general perspective, our work illustrates how exciton-vibrational
resonances allow for rich dynamics of energy and coherences thereby providing
a mechanism for adjusting the rates of entropy production and output energy.
In this context, the associated vibrational dynamics could be understood as an
internal quantum mechanism for thermodynamic control Erez08 . Understanding
how quantum phenomena affect the thermodynamics of energy conversion in
biomolecules could provide an unifying framework to quantify the advantages of
quantum phenomena in a variety of biomolecular transport processes
Novoderezhkin04 ; Hay12 .
Acknowledgements.- We thank Greg Scholes and Rienk van Grondelle for
discussions, and Avinash Kolli for support on numerical calculations in the
initial stages of this project. Financial support from the Engineering and
Physical Sciences Research Council (EPSRC UK) is gratefully acknowledged.
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## Appendix A Hierarchical expansion of exciton-vibration dynamics
Here we provide details of the model for a system consisting of $N$ electronic
degrees of freedom each interacting with a quantized vibrational mode and the
method used to accurately calculate the dynamics of these exciton-vibration
systems. The system Hamiltonian
$H_{S}=H_{\textrm{el}}\otimes\mathbbm{1}_{\textrm{vib}}+\mathbbm{1}_{\textrm{el}}\otimes
H_{\textrm{vib}}+H_{\textrm{el-vib}}$, is defined by
$H_{\textrm{el}}=\sum_{i=1}^{N}\varepsilon_{i}\sigma^{+}_{i}\sigma^{-}_{i}+\sum_{i,j\leq
i}^{N}V_{ij}(\sigma^{+}_{i}\sigma^{-}_{j}+\sigma^{+}_{j}\sigma^{-}_{i})$ (2)
$H_{\textrm{vib}}=\sum_{i=1}^{N}\omega_{\textrm{vib}}^{i}b^{\dagger}_{i}b_{i}$
(3)
and
$H_{\textrm{el-
vib}}=g_{i}\sigma^{+}_{i}\sigma^{-}_{i}\otimes(b^{\dagger}_{i}+b_{i})~{}.$ (4)
Here, the operators $\sigma^{\pm}_{i}$ create/annihilate electronic
excitations at site $i$, while $b^{\dagger}_{i}$ ($b_{i}$) create (annihilate)
phonons associated with the vibrational mode of site $i$. $\varepsilon_{i}$
denotes the excitation energies and $V_{ij}$ the strength of interactions
between sites. $\omega_{\textrm{vib}}^{i}$ is the frequency of the vibrational
mode at site $i$ and $g_{i}=\sqrt{S_{i}}\omega_{\textrm{vib}}^{i}$ the
coupling of the mode to its site. $S_{i}$ is the Huang-Rhys factor or mean
number of phonons in the ‘polaron cloud’ formed when the mode is fully
displaced. The the high energy modes considered here, we expand the quantized
vibrational modes of the system in the basis of single phonon (Fock) states
(e.g. $b_{i}=\sum_{n}^{M}(n+1)^{1/2}\left|{n}\right\rangle\langle n+1|$)
truncating at $M$ phonons. This system is then coupled to a continuum of
bosonic modes (the thermal background)
$H_{B}=\sum_{\textbf{k}}\omega_{\textbf{k}}b_{\textbf{k}}^{\dagger}b_{\textbf{k}}$,
with an interaction
$H_{I}=\sum_{\textbf{k}}\sum_{i}g_{\textbf{k},i}(\sigma^{+}_{i}\sigma^{-}_{i}\otimes\mathbbm{1}_{\textrm{vib}})(b_{\textbf{k}}^{\dagger}+b_{\textbf{k}})~{},$
(5)
such that a mode with frequency $\omega_{\textbf{k}}$ is linearly coupled to
the electronic excitation at site $i$ with a strength $g_{\textbf{k,i}}$ as
specified by a spectral density
$J_{i}(\omega)=\sum_{\textbf{k}}|g_{\textbf{k,i}}|^{2}\delta(\omega-\omega_{\textbf{k}})$.
We assume the bath is spatially uncorrelated and identical for each site.
An infinite hierarchy of coupled differential equations[34, 35, 36] is used to
express the dynamics of the exciton-vibration system density matrix
$\rho_{S}(t)$. These read as,
$\displaystyle\partial_{t}\tilde{\rho}_{\boldsymbol{n}}(t)$
$\displaystyle=-i/{\hbar}[H_{S},\tilde{\rho}_{\boldsymbol{n}}(t)]-\sum_{i=1}^{N}\sum_{k=0}^{K}n_{ik}v_{ik}\tilde{\rho}_{\boldsymbol{n}}(t)$
$\displaystyle{-i\sum_{i=1}^{N}\left(\sum_{k=1}^{\infty}\frac{c_{ik}}{\nu_{ik}}-\sum_{k=1}^{K}\frac{c_{ik}}{\nu_{ik}}\right)\left[Q_{i},\left[Q_{i},\tilde{\rho}_{\boldsymbol{n}}(t)\right]\right]}$
$\displaystyle-i\sum_{i=1}^{N}\sum_{k=0}^{K}\sqrt{(n_{ik}+1)|c_{ik}|}\left[Q_{i},\;\tilde{\rho}_{\boldsymbol{n_{ik}^{+}}}\right]$
$\displaystyle-i\sum_{i=1}^{N}\sum_{k=0}^{K}\sqrt{n_{ik}/|c_{ik}|}\left(c_{ik}\,Q_{i}\tilde{\rho}_{\boldsymbol{n_{ik}^{-}}}-c_{ik}^{*}\,\tilde{\rho}_{\boldsymbol{n_{ik}^{-}}}Q_{i}\right)~{}.$
(6)
for the case of a Drude spectral density and in a rescaled formulation[36]
which admits their efficient adaptive time-step numerical integration.
Operators $\tilde{\rho}_{\boldsymbol{n}}$ are indexed by the multi-index
$\boldsymbol{n}$ which has entries $n_{ik}$ and $n=\sum_{ik}n_{ik}$.
$\boldsymbol{n_{ik}^{\pm}}$ is $\boldsymbol{n}$ with entry $n_{jk}\rightarrow
n_{jk}\pm 1$. The reduced density matrix of the system $\rho_{S}$ is given by
the operator $\tilde{\rho}_{\boldsymbol{0}}$. The coefficients $\nu_{ik}$ and
$c_{ik}$ are the Matsubara frequencies and coefficients appearing in the
(truncated) exponential decomposition $\sum_{k=0}^{K}c_{ik}e^{-\nu_{ik}t}$ of
the bath correlation function applied in Ref [35]. In the above terms $k>K$
are truncated with a Markovian approximation scheme [35]. This results in
accurate dynamics for $K=1$ at high temperatures ($\beta\Omega_{c}<1$). In the
present exciton-vibration case the system-bath coupling operator takes the
form $Q_{i}=\sigma_{i}^{+}\sigma_{i}^{-}\otimes\mathbbm{1}_{\textrm{vib}}$.
## Appendix B $B_{0}$ parameter
Modulations of adjacent phonon number occupation as quantified by negative
$B_{n}$, guarantees negative regions of a quasi-probability distribution[27]
as is a potentially an experimentally accessible quantity. The non-
classicality indicated by sub-Poissonian statistics is confirmed by negative
$B_{0}=2P(0)P(2)-P(1)$. This arises from selective population of the $P(1)$
level of the relative displacement vibrational mode.
Figure 6: $B_{0}$ for a) $\lambda=0\textrm{cm}^{-1}$, b)
$\lambda=6\textrm{cm}^{-1}$, c) $\lambda=20\textrm{cm}^{-1}$ and d)
$\lambda=110\textrm{cm}^{-1}$
|
arxiv-papers
| 2013-01-29T16:25:49 |
2024-09-04T02:49:40.888611
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Edward J. O'Reilly, Alexandra Olaya-Castro",
"submitter": "Edward O'Reilly",
"url": "https://arxiv.org/abs/1301.6970"
}
|
1301.7006
|
††thanks:
# A Unified Community Detection, Visualization and Analysis method
Michel Crampes [email protected] http://www.lgi2p.ema.fr/~crampes
Michel Plantié [email protected] http://www.lgi2p.ema.fr/~plantie
Ecole des Mines d’Ales, Parc Georges Besse, 30035 Nîmes Cedex
###### Abstract
Community detection in social graphs has been attracting researchers’
attention for a long time. With social networks now so widespread on the
Internet, community detection has recently become a major field of research.
Most contributions focus on defining algorithms that optimize the so-called
”modularity function”. Interest was initially limited to unipartite graph
inputs and partitioned community outputs. More recently, bipartite graphs,
directed graphs and overlapping communities have all been investigated. Few
contributions however have encompassed all three types of graphs
simultaneously. In this paper, we present a method that unifies community
detection for these three types of graphs while at the same time merges
partitioned and overlapping communities. Moreover, the results are visualized
in a way that allows for analysis and semantic interpretation. For validation
purposes, this method is experimented on some well-known simple benchmarks and
then applied to real data in three cases. In two examples of photo sets with
tagged people, social networks are revealed. A second type of application is
of particular interest herein. After applying our method to Human Brain
Tractography Data provided by a team of neurologists, clusters of white fibers
are produced in accordance with other well-known clustering techniques. Our
approach to visualizing overlapping clusters also provides a better
understanding of the neurological team’s results, which lead to the
possibility of applying community detection methods to other fields, such as
data analysis with original enhanced performances.
## I introduction
Social group analysis is a long-standing field of research, particularly in
the social sciences. Newman2003 reveals that some studies data back as early
as the 19th century; moreover, this review identified 21 results relative to
the unique case of the Southern Women Dataset. Thanks to the Internet and
growth in online social networks, community detection has become a major field
of research in computer sciences. Many algorithms have been proposed (see
several surveys on this topic: Fortunato2009 ; Papadopoulos2011 ; Yang2010 ;
Porter2009 . These algorithms may be classified into four dimensions:
* •
The first dimension is dedicated to input data types, which may be:
* –
unipartite graphs (nodes belonging to a single class),
* –
bipartite graphs (nodes belonging to two classes, with no edges between nodes
of the same class),
* –
multipartite graphs (more than two classes, with no edges between nodes of the
same class),
* –
directed graphs (whereby the edges are oriented) - moreover, the edges may be
weighted or unweighted.
* •
The second dimension is important since it pertains to the capacity of
algorithms to comply with the experimenter’s objectives, which may consist of
a predefined number of communities (vs. a preference to determine this number
after performing the math), a final partitioning (with every element belonging
to a unique community), or overlapping communities (with some elements
potentially belonging to several communities).
* •
In the third dimension, the algorithms are classified by performance, e.g.
execution speed, data volume, stability, determinism vs. heuristics.
* •
The fourth and final dimension is scarcely mentioned in the various
contributions. Since group detection is called upon by other scientists or for
purposes of industrial data analysis, it is of utmost importance that the
results be validated by the ’customer’ and eventually analyzed and
interpreted. This last objective requires both the algorithm and the entire
methodology, from inputs to results presentation, to be easy to understand and
implement from a variety of perspectives.
For this paper, the last requirement cited above will serve as a guide for
proposing a unifying method applicable to community detection, visualization
and analysis. Our first contribution will unify the three types of graphs
listed above. This step will entail reducing bipartite graphs and directed
graphs into unipartite graphs. The notion of building bridges from one context
to another (e.g. from directed graphs to bipartite graphs) has already been
explored by other authors. To the best of our best knowledge however, these
attempts have only been piecemeal and, moreover, have not yielded any
visualization or analysis tools. With this specific goal in mind, we will
introduce a simple model of unified modularity for bipartite and directed
graphs that differs from other authors’ models. Ours will be formally derived
from the typical GN modularity model for unipartite graphs Girvan2002 ,
whereas other authors have used GN modularity for the a priori building of a
specific model for bipartite graphs. These differences will be discussed
below. We will then be able to extract partitioned communities from
unipartite, bipartite or directed graphs with any algorithm that accepts
unipartite graphs as inputs. In this paper, we will apply the Louvain
algorithm Blondel2008 , which is known for its efficiency in producing
partitioned communities from extensive datasets. This algorithm is also
applicable to weighted and unweighted graphs.
Our second contribution relates to the merging of partitioned and overlapping
communities. Most authors propose models that extract either partitioned or
overlapping communities, with the former being more frequent. According to our
approach, when inputs are bipartite graphs, both node classes share the same
resulting communities. Though the fact of having the same number of
communities for both classes has been criticized by some authors Suzuki2009 ,
we will show herein that such a condition is relevant and especially useful
for validating and analyzing the resulting communities. It is indeed possible
to define overlapping communities in both classes using different belonging
functions. From this result, we will present a model that merges partitioned
and quantified overlapping communities, in a way that offers a unique view, as
supported by a matrix visualization method that is both easy to produce and
capable of facilitating analysis and interpretation.
In a third contribution, we will demonstrate how this composite view is of
special interest in analyzing the detected communities. Other authors
typically analyze their results through a comparison with other authors’
results. While this is an option with our approach, it is also possible to
observe different features of the extracted communities, then analyze them and
create the potential for knowledge extraction.
For validation and comparison with other authors, our entire method has been
experimented on small traditional unipartite and bipartite benchmarks. We have
generated some interesting insight, which extends beyond previously reported
results. For instance, when applying the Louvain algorithm Blondel2008 to
detect communities in either bipartite or unipartite graphs, it is possible to
observe local modularity optima and propose hypotheses that explain their
origins. We can then apply our method to real medium-sized bipartite graphs,
in a step that reveals significant properties such as community compactness
and the role of inter-community objects. These results are valuable when
observed in data like the people-photo datasets targeted by our experiments.
Beyond community detection, our method has been applied to brain data
extracted through ’tractography’ by a team of neurologists and
psychoneurologists seeking to extract macro connections between different
brain areas. Our results were compared with other results they obtained when
applying traditional data analysis methods, and our community detection
analysis tools led them to question their datasets and consider new hypotheses
on brain connections. These two experiments are described in the paper in
noting the following consideration. When working with people who are not
specialists in the field (e.g. neurologists), it is of utmost importance that
we justify our results and, above all else, explain any unexpected
observations. It may also be necessary to improve results when ’greedy
algorithms’, such as Louvain’s, might classify some people too quickly or more
generally place objects into communities. We will show in this paper that our
method is especially well suited to address such issues and may contribute to
a better detection and understanding of communities or clusters in complex
networks.
The next section will present a state-of-the-art on community detection
techniques using different types of graphs. Section III will follow by
focusing on a new method to unify all types of graphs. Section IV will then
display how our unifying method is particularly valuable in computing and
analyzing overlapping communities. In section V will discuss the visualization
issue as applied to overlapping communities, while section VI will provide
several practical results on various types of graph datasets.
## II BIPARTITE GRAPHS AND SOCIAL NETWORK DETECTION, STATE OF THE ART
### II.1 _Unipartite graph partitioning_.
_As stated above, several state-of-the-art assessments have already addressed
the community detection problemPapadopoulos2011 ; Porter2009 ; Yang2010 ;
Fortunato2009 . These have mainly focused on unipartite graph partitioning,
i.e. each individual belonging to just one community. The calculation
performed is based on maximizing a mathematical criterion, in most cases
modularity _Newman2004 , representing the maximum number of connections within
each community and the minimum number of links with external communities.
Various methods have been developed to identify the optimum, e.g. greedy
algorithms Newman2004a ; Noack2008 , spectral analysis Newman2006 , or a
search for the most centric edges Newman2004 .One of the most efficient greedy
algorithms for extracting partitioned communities from large (and possibly
weighted) graphs is Louvain Blondel2008 . In a very comprehensive state-of-
the-art reportFortunato2009 other newer partitioned community detection
methods are described.
### II.2 _Extracting overlapping communities from unipartite graphs_.
The partitioning of communities, despite being mathematically attractive, is
not satisfactory to describe reality. Each individual has ’several lives’ and
usually belongs to several communities based on family, professional and other
activities. For example, researchers may be affiliated with several partially
overlapping communities when considering their scientific discipline, priority
interests and scientific monitoring activities. As such, a greater number of
methods take into account the possibility for overlapping communities. The so-
called “k-clique percolation method” Palla2005 detects overlapping
communities by allowing nodes to belong to multiple k-cliques. A more recent
method adapted to bipartite networks, based on an extension of the k-clique
community detection algorithm, is presented in
SuneLehmannMartinSchwartzLarsKaiHansen2008 . Several methods find overlapping
communities through local fitness optimization Lee2010c Lancichinetti2009 .
The ”Label Propagation Algorithms” (LPA) are efficient methods for detecting
overlapping communities, e.g. Gregory2009 . Lee2010c uses a greedy clique
expansion method to determine overlapping communities, via a two-step process:
identify separated cliques and expand them for overlapping by means of
optimizing a local fitness criterion. Some research has provided results in
the form of hypergraph communities, such as in Estrada2005 ; Chakraborty2012 .
Other methods are found in scientific papers, yet most of these are prone to
major problems due to high computational complexity. More recently, Wu Wu2012a
proposed a fast overlapping community detection method for large, real-world
unipartite networks. Evans2010 derives n order clique graphs from unipartite
graphs to produce partitionned and overlapping communities using Louvain
algorithm. Interestingly with this method the number of communities is
parameterized by the order n.
### II.3 From unipartite to bipartite graphs.
The vast majority of community detection techniques are not really inspired by
the meaning (semantics) of the relationship between nodes. When considering
”semantics”, it becomes necessary to focus on bipartite or ”multi-partite”
graphs, i.e. graphs whose nodes are divided into several separate subsets and
whose edges only link nodes from different subsets. One example of this type
of graph is the set of photos from a Facebook account along with their ’tags’
MichelPlantie2011 or else the tripartite network of epistemic graphs Roth2005
linking researchers, their publications and keywords in these publications.
Mining communities are often formed by converting a multipartite graph into a
monopartite graph, through assigning a link between two nodes should they
share a common property. Guimera Guimera2007 offered a modularity measurement
for bipartite graphs, even those using a weighting parameter based on the
number of shared properties, and then reduced the problem to a classical graph
partitioning. In doing so however, semantics are lost; hence, many researchers
retain the multiparty graph properties by extending the notion of modularity
to these types of graphs Newman2004 or else adapting the algorithms
originally designed for unipartite graphs Suzuki2009 ; NeubauerNicolas ;
Barber2007 ; Murata2009 ; Evans2009 . In LiuXin2010 Liu and Murata presented
a new and efficient algorithm based on LPA for bipartite networks.
### II.4 Analysis and interpretation tools.
An important feature of the bipartite graph is its semantic value. A number of
authors have developed methods to analyze and interpret the community results.
Suzuki Suzuki2009 compared modularity measurements stemming from various
detection methods on well-known examples. This analysis proved limited however
because it was exclusively quantitative and did not take semantics into
account. Moreover, the analyses in LiuXin2010 were limited to modularity
optimization, wherein detected communities from several authors were compared
on the basis of their inner stability on standard benchmarks. Wu Wu2012a also
focused on a quantitative evaluation in terms of both modularity measurement
and computation time. In all these contributions, the quality analysis has
been limited to a modularity assessment, while a semantic analysis has hardly
ever been considered. In contrast, we will show in the following section that
our method provides for a deep interpretation of community membership, even
though quantitative considerations (namely computational performance and
modularity optimization) are still being included.
## III Unifying bipartite, directed and unipartite graphs
### III.1 From unipartite graph modularity to bipartite graph bimodularity
Most authors introduce modularity into bipartite graphs using a probabilistic
analogy with the modularity for unipartite graphs. Conversely, we have
formally derived bipartite graph modularity from unipartite graph modularity
and refer to it as ”bimodularity” since it involves both types of nodes in the
communities.
In formal terms, a bipartite graph $G=(U,V,E)$ is a graph $G^{\prime}=(N,E)$
where node set $N$ is the union of two independent sets $U$ and $V$ and
moreover the edges only connect pairs of vertices $(u,v)$ where $u$ belongs to
$U$ and $v$ belongs to $V$.
$N=U\cup V$,
$U\cap V=\emptyset$,
$E\subseteq U\times V$
$Let\>r=|U|\>and\>s=|V|,\>then\>|N|=n=r+s$
The unweighted biadjacency matrix of a bipartite graph $G=(U,V,E)$ is a
$r\times s$ matrix $B$ in which $B_{i,j}=1\>iff(u_{i},v_{j})\in
E\>and\>B_{i,j}=0\>iff(u_{i},v_{j})\notin E$.
It must be pointed out that the row margins in $B$ represent the degrees of
nodes $u_{i}$ while the columns’ margins represent the degrees of nodes
$v_{j}$. Conversely, in $B^{t}$ , the transpose of $B$, row’s margins
represent the degrees of nodes $v_{j}$ and columns’ margins represent the
degrees of nodes $u_{i}$. Let’s now define a new matrix as the adjacency
matrix $A^{\prime}$ of $G^{\prime}$ which is the off-diagonal block square
matrix:
$A^{\prime}=\left(\begin{array}[]{cc}0_{r}&B\\\ B^{t}&0_{s}\end{array}\right)$
where $0_{r}$ is an all zero square matrix of order $r$ and $0_{s}$ is an all
zero square matrix of order $s$.
Community detection in graphs consists of identifying subsets of densely
connected nodes with sparse connections between subsets. Each subset becomes a
candidate community. Two main strategies are possible: 1) in the partitions,
all communities are distinct; and 2) conversely, overlapping communities may
share nodes. As a first step, we will look to identify non-overlapping
communities, i.e. graph partitions; then, the second step will involve a
generalization to overlapping communities.
Modularity is an indicator often used to measure the quality of graph
partitions Newman2004 . First defined for a unipartite graph, several
modularity variants have been proposed for bipartite graph partitioning and
overlapping communities. These variants will be summarized in a subsequent
section. In Equation (2) we propose a new formal definition of modularity for
bipartite graphs (which we call herein bimodularity) using the biadjacency
matrix. This new definition differs from all modularity definitions previously
known for bipartite graphs, which are usually defined a priori through
probabilistic reasoning. In our specific case, bimodularity is a posteriori
defined since it has been formally derived from an expression of modularity
for unipartite graphs. The corresponding demonstration is detailed in Appendix
1 (Section IX). In the following subsection, we will introduce the notion of
bimodularity for bipartite graphs and justify its application for both
directed graphs and unipartite graphs, with the aim of unifying the approach
for all three types of graphs.
#### III.1.1 _Bimodularity for bipartite graphs_
Let $G=(U,V,E)$ be a bipartite graph with its biadjancecy matrix $B$ and
adjacency off-diagonal block matrix $A^{\prime}$. Since $A^{\prime}$ is
symmetric, a unipartite graph $G^{\prime}=(N,E)$ defined by this matrix
actually exists. Let’s consider Newman’s modularity Newman2004 for this graph
$G^{\prime}$. It is a function $Q$ of both matrix $A^{\prime}$ and the
communities detected in $G^{\prime}$ :
$Q=\frac{1}{2m}\sum_{i,j}\left[A^{\prime}_{ij}-\frac{k_{i}k_{j}}{2m}\right]\delta(c_{i},c_{j})$
(1)
where $A^{\prime}_{ij}$ represents the weight of the edge between $i$ and $j$,
$k_{i}=\sum_{j}A^{\prime}_{ij}$ is the sum of the weights of the edges
attached to vertex $i$, $c_{i}$ denotes the community to which vertex $i$ is
assigned, the Kronecker’s function $\delta(u,v)$ equals $1$ if $u=v$ and $0$
otherwise and $m=1/2\sum_{ij}A^{\prime}_{ij}$. Hereafter we only consider
binary graphs and weights are equal to 0 or 1.
After several transformations we show (see Appendix 1, Section IX) that this
modularity can also be written using the biadjacency matrix $B$ of the
bipartite graph $G=(U,V,E)$:
$Q=\frac{1}{m}\sum_{ij}[B_{ij}-\frac{(k_{i}+k_{j})\text{\texttwosuperior}}{4m}]\delta(c_{i},c_{j})$
(2)
where $k_{i}$ is the margin of row $i$ in $B$, $k_{j}$ the margin of column
$j$ in $B$ and $m=\sum_{ij}B_{ij}=\frac{1}{2}\times\sum_{ij}A^{\prime}_{ij}=m$
in (1).
Another interesting formulation to be used is the following (Appendix 1,
Section IX):
$Q=\sum_{c}[\frac{|e_{c}|}{m}\text{\textendash}(\frac{(d_{u|c}+d_{v|c})}{2\times
m})\text{\texttwosuperior}]$ (3)
where $|e_{c}|$ is the number of edges in community $c$, and $d_{w|c}$ is the
degree of node $w$ belonging to $c$.
Since in the general case $B$ is not symmetric, this definition thus
characterizes modularity for bipartite graphs after their extension into new
unipartite graphs. It then becomes possible to apply any partitioning
algorithm for unipartite graphs to matrix $A^{\prime}$ and obtain a result
where both types of nodes are bound in the same communities, except in the
case of singletons (i.e. nodes without edges). To distinguish this definition
from unipartite graph modularity and given that it is able to bind both types
of nodes, we have called it bimodularity and in Section III.3 will compare it
with other authors’ modularity models for bipartite graphs.
#### III.1.2 _Bimodularity for_ directed _graphs_
A directed graphs is a of the form $G^{d}=(N,E^{d})$ where $N$ is a set of
nodes and $E^{d}$ is a set of ordered pairs of nodes belonging to
$N:\;E^{d}\subseteq N\times N$. From the model in (1) Leicht Leicht2007 use
probabilistic reasoning ’insights’ to derive the following modularity for
directed network:
$Q=\frac{1}{m}\sum_{ij}\left[A_{ij}-\frac{k_{i}^{in}k_{j}^{out}}{m}\right]\delta(c_{i},c_{j})$
(4)
where $k_{i}^{in}$ and $k_{j}^{out}$ are the in - and out- degrees of vertices
$i$ and $j$, $A$ is the asymmetric adjacency matrix, and $m=\sum_{ij}A_{ij}$ =
$\sum_{i}k_{i}^{in}=\sum_{i}k_{j}^{out}$. Symmetry is then restored and
spectral optimization applied to extract non-overlapping communities. This
model leads to a node partition that does not distinguish between the in and
out roles; the nodes are simply clustered within the various communities.
To compare these authors’ method to ours, we transformed directed graphs into
bipartite graphs (this transformation was also suggested in Guimera’s work
Guimera2007 when applying their method for bipartite networks to directed
graphs, as will be seen below). At this point, let’s differentiate the nodes’
roles into $N\times N$. Along these lines, we duplicate $N$ and consider two
identical sets $N^{out}$ and $N^{in}$. The original directed graph $G^{d}$ is
transformed into a bipartite graph $G=(N^{out},N^{in},E)$ in which nodes
appear twice depending on their ’out’ or ’in’ role and moreover the asymmetric
adjacency matrix $A$ plays the role of biadjacency matrix $B$ in bipartite
graphs. We can now define bimodularity for directed graphs as follows:
$Q=\frac{1}{m}\sum_{ij}[A_{ij}-\frac{(k_{i}^{in}+k_{j}^{out})\text{\texttwosuperior}}{4m}]\delta(c_{i},c_{j})$
(5)
After applying any algorithm for a unipartite graph on the corresponding
adjacency matrix $A^{\prime}$ we obtain a partition where some nodes may
belong to the same community twice or instead may appear in two different
communities. Each model has its pros and cons. Leicht’s model Leicht2007 is
preferable when seeking a single partition with no role distinction. Our model
is attractive when seeking to distinguish between ’in’ and ’out’ roles, e.g.
between producers and customers where anyone can play either role. The brain
data example that follows will demonstrate that our model is particularly well
suited for analyzing real data.
#### III.1.3 _Bimodularity for unipartite graphs_
In the above presentation, we introduced bimodularity for bipartite graphs as
a formal derivative of unipartite graph modularity. It is dually possible to
consider unipartite graphs as bipartite graphs, in defining unipartite graph
bimodularity and in extracting communities as if unipartite graphs were
bipartite graphs. To proceed, we must consider the original symmetric
adjacency matrix $A$ as an asymmetric biadjacency matrix $B$ B (with the same
nodes on both dimensions) and build a new adjacency matrix $A^{\prime}$ using
the original adjacency matrix $A$ twice on the off-diagonal, as if the nodes
had been cloned. When applying a unipartite graph partitioning algorithm, we
then obtain communities in which all nodes appear twice. This method only
works if we add to $A$ the unity matrix $I$ (with the same dimensions as $A)$
before building $A^{\prime}$. The first diagonal in $A$ in fact only contains
0s since no loops are generally present in a unipartite graph adjacency
matrix. Semantically adding $I$ to $A$ means that all objects will be linked
to their respective clones in $A^{\prime}$ .This is a necessary step in that
when extracting communities, the objects must drag their clones into the same
communities in order to maintain connectivity. In practice therefore, for
unipartite graphs, we build $A$’ with $A+I$.
It may seem futile to perform such a transformation from a unipartite graph to
a bipartite one in order to find communities in unipartite graphs given that
for computing bipartite graph partitioning, we have already made the extension
into unipartite graphs using their (symmetric) adjacency matrix. This
transformation is nonetheless worthwhile for several reasons. First, when
appearing twice, nodes should be associated with their clones. If the
resulting communities do not display this property, i.e. a node’s clone lies
in another community, then the original matrix is not symmetric and can be
considered as the adjacency matrix of a directed graph. This conclusion has
been applied to the human brain tractography data clustering, which will be
described in the experimental section below.
Conversely, if we are sure that the original adjacency matrix is symmetric,
then a result where all nodes are associated with their clones in the same
communities would be a good indicator of the quality of the clustering
algorithm and moreover provides the opportunity to compare our bipartite graph
approach with other unipartite graph strategies. This is also a method we
introduced into our experiment (see the karate application below) for the
purpose of verifying the validity of results.
Lastly, the most important benefit consists of building overlapping
communities and ownership functions for unipartite graphs using the method
explained in Section IV below. Although transforming unipartite graphs into
bipartite graphs requires more computation, it also provides considerable
information, which justifies its application in a variety of contexts.
### III.2 Unifying bipartite and unipartite graph partitioning
The modularity in Equation (1) merely considers unipartite graphs. The
bimodularity in Equation (2) considers two types of nodes, with both types in
the same community set. We can now show that it is feasible to easily compute
partitioned communities in bipartite graphs by using their adjacency off-
diagonal matrices and applying unipartite graph partitioning algorithms.
$\mathit{G^{\prime}}$ is a unipartite graph; consequently, it is possible to
apply its adjacency matrix $A^{\prime}$ to any algorithm for extracting
communities from unipartite graphs. $A^{\prime}$ is also the off-diagonal
adjacency matrix of bipartite graph $\mathit{G}$. Since bimodularity in
Equation (2) has been formally derived from modularity in Equation (1), then
computing communities in the unipartite graph $\mathit{G^{\prime}}$ using
symmetric matrix $A^{\prime}$ and the modularity of Equation (1) is equivalent
to computing communities in $\mathit{G}$ using matrix $B$ and bimodularity in
Equation 2. As a result, we obtain communities for the corresponding bipartite
graph $\mathit{G}$ where both types of nodes are bound.
### III.3 Comparing bimodularity with other modularity models and
partitioning algorithms
Several modularity models have been proposed in the literature for unipartite,
bipartite, directed weighted and unweighted graphs. This section will compare
our bimodularity model with the main proposals from other researchers.
#### III.3.1 Standard probabilistic modularity models for bipartite graphs
Most modularity models for bipartite graphs are derived from Newman’s
modularity for unipartite graphs. Authors typically use “$A\textrm{ }$” to
label their biadjacency matrix, i.e. the matrix whose rows represent one set
and columns the other set. We have chosen herein to call this matrix $B$ in
order to avoid confusion with the symmetric matrix $A$ for unipartite graphs;
hence, we used $A^{\prime}$ to refer to the corresponding adjacency matrix,
which is the off-diagonal block square matrix.
Using such definitions, some authors have probabilistically defined modularity
as follows Barber2007 LiuXin2010 :
$Q=\frac{1}{m}\sum_{i,j}\left[A_{ij}-\frac{k_{i}k_{j}}{m}\right]\delta(c_{i},c_{j})$
(6)
where $m=\sum_{ij}A_{ij}$, $k_{i}$ is the margin of row$i$ in $A$ and $k_{j}$
the margin of column $j$ in $A$
Differences may exist in the nomenclature from one author to the next, but
their definitions remain equivalent. Let’s call this model the standard
probabilistic modularity model for bipartite graphs since it is the most
common reference, though we will see below that some authors have slightly
adapted this reference.
To compare our bimodularity definition for bipartite graphs with Equation (6)
we must first rename the other biadjacency matrix $A$ with our equivalent
biadjacency matrix $B$ and replace $m=\sum_{ij}A_{ij}$ by $m=\sum_{ij}B_{ij}$.
Once these changes have been introduced, their new definition becomes:
$Q=\frac{1}{m}\sum_{i,j}[B_{ij}-\frac{k_{i}k_{j}}{m}]\delta(c_{i,}c_{j})$ (7)
which is to be compared with our bimodularity formula:
$Q=\frac{1}{m}\sum_{ij}[B_{ij}-\frac{(k_{i}+k_{j})\text{\texttwosuperior}}{4m}]\delta(c_{i},c_{j})$
(8)
The effects of using either this formula or the other can be observed in the
number of communities in each set, as well as in the node distribution by type
through modularity optimization. According to our definition, both types of
nodes are explicitly bound. If we were to consider the unipartite graph
represented by the off-diagonal matrix $A^{\prime}$, we can apply any
unipartite graph algorithm for detecting communities, and both types of nodes
are then regrouped into the same communities (except for singletons). This
side effect is not explicit in Equation (5). However since
$\delta(c_{i},c_{j})$ in Equation (5) specifies that the summation is applied
to both types of objects belonging to the same community, the side effect is
the same: optimizing the standard bipartite graph modularity should yield a
partitioning of both types of nodes in the same communities (this analysis is
also found in MURATA2010 : i.e.: “This definition implicitly indicates that
the numbers of communities of both types are equal”). Both modularities should
then produce the same results in terms of node type distribution.
As far as the number of communities and node ownership are concerned, it is
more difficult to compare the results of both these models, in particular if
various algorithms are applied depending on the selected model. For instance,
in the Southern Women experiment described below, we found 3 communities when
applying Louvain, while LiuXin2010 found four communities using their
original LPAb+ algorithm. These authors however only provided a quantitative
evaluation via comparison with other algorithms on computation performance and
modularity optimization; in contrast, we have provided qualitative analysis as
well, which allows for semantics justification on the partitioning.
#### III.3.2 Bimodularity versus other bipartite graph modularity models
All bipartite graph modularity models analyzed in this section have been
derived from the original Newman-Girvan model given in Equation (1).
Guimera Guimera2007 introduced an original probabilistic model after
considering node co-occurrences of one set (’actors’) in the other set
(’teams’). Partitioning using simulated annealing is only applied to one set
at a time, introducing an important difference with respect to our model,
which binds both types of nodes in the same communities. Another valuable
contribution by the authors consists of modeling directed graphs as bipartite
graphs, in the same way we did in the previous section.
Barber’s model Barber2007 is particularly instructive since it starts from
the off-diagonal adjacency matrix $A$ to define modularity for bipartite
graphs, just like our efforts. This approach however yields a different result
when defining modularity. Barber considered the above standard probabilistic
model for bipartite graphs and, consequently, did not apply any algorithm to
the unipartite graph; instead he applied a complex spectral model, whereby
optimization is recursively applied to one set then the other set, thus
leading to partitioning where the same number of communities exists for both
sets. In our symmetric method, both partitions are computed at the same time
using a simple unipartite graph algorithm. In Barber’s result, communities of
one node type are not directly linked to communities of the other node type;
in our case, both node types are bound in communities, with the benefit being
that a given node type introduces semantics into regrouping the other node
type and vice versa.
Murata MURATA2010 introduced a new bipartite graph modularity model after
critiquing Barber’s and Guimera’s models for producing equal numbers of
communities in both sets. The authors in MURATA2010 also compared various
modularities and algorithms regarding the issue of modularity optimization
applied to a bipartite graph benchmark. In their comparison, three bipartite
modularities were found to be the best candidates when at least four
communities are present. These three models compared were Barber’s, Murata’s
and Suzuki’s, all of which will be analyzed below. It is worthwhile to observe
that Barber’s symmetric model yields as good results as its two asymmetric
competitors. In LiuXin2010 Murata et al. reintroduced Barber’s model along
with the implicit aggregation of both sets in the same communities.
With the aim of avoiding equal numbers of communities (symmetric clustering
according to the authors), Suzuki2009 introduced another ’shared clustering’
modularity measurement for bipartite graphs and compared it to Newman’s and
Murata’s model. Experiments were limited to the Southern Women’s benchmark,
and it will be shown in our experimental section that even with such a
benchmark, the superiority of this model has not been proven. Moreover, as
explained above, Murata MURATA2010 compared this model with other models and
could not confirm the fact that asymmetric models are better than symmetric
models.
#### III.3.3 Conclusion on bimodularity versus other bipartite graph
modularity models
After several proposals, it appears that the most widely accepted model for
bipartite graph modularity is what we refer to as the ’standard probabilistic
model’, which while not proven has been intuitively proposed by Barber, as
inspired by Newman’s model for unipartite graphs. This model implicitly
aggregates both sets of nodes in the communities. In our model, we have
transformed bipartite graphs into unipartite graphs, and bimodularity has been
formally derived from unipartite graph modularity. As a result, we can apply
any unipartite graph algorithm and expect partitioning wherever both types of
nodes are associated. Far from being a drawback, we will now show that this
association is especially attractive in the field of overlapping detection and
community analysis.
## IV Detection and analysis of community overlapping
### IV.1 Adding semantics to communities
The fact that both types of nodes are bound in their communities yields
several important results. First, in considering one type of nodes, a
community can be defined by associating a subset of nodes from the other type.
In other words, nodes from one set provides semantics for the grouping of
nodes from the other set and moreover may qualitatively explain regroupings,
as will be seen below. This semantic perspective has not been considered by
any of the other authors, a situation due to the fact that in other
contributions, either the number of communities differs for both types of
nodes (e.g. Murata2009 , oor else when both types of nodes contain the same
number of communities they are not bound in each community Guimera2007 ;
Barber2007 .
Binding both types of nodes into the same communities yields other pertinent
results. For one thing, it is possible to define belonging functions and
consequently obtain quantified overlapping communities. In the following
discussion, we will consider three possible belonging functions, which may
expose community overlapping in a different light.
### IV.2 Probabilistic function
Let’s adopt the Southern Women’s benchmark, which will be more thoroughly
described in Section V below. Applying the Louvain community detection
algorithm for unipartite graphs yields a partition where Women and Events are
regrouped into three exclusive communities. Let’s call these communities
$c_{1}$ , $c_{2}$ and $c_{3}$ . . Now, let’s suppose the fictitious case in
which woman $w_{1}$ participated in events $e_{1}$, $e{}_{2}$, $e_{3}$ and
$e_{4}$ . furthermore, $w_{1}$, $e_{1}$ and $e{}_{2}$ are classified in
$c_{1}$, while$e_{3}$ is classified in $c_{2}$ and $e_{4}$ is classified in
$c_{3}$. We can then define a probability function as follows:
$P(u_{i}\in c)=\frac{1}{k_{i}}{\textstyle\sum_{j}B_{ij}\delta(c_{j})}$ (9)
where $c$ is a community, $k_{i}=\sum_{j}B_{ij}$ and $\delta(c_{j})=1$ if
$v_{j}\in c$ or $\delta(c_{j})=0$ if $v_{j}\notin c$
In $P(u_{i}\in c)$ the numerator includes all edges linking $u_{i}$ to
properties $v_{j}\in c$ and the denominator contains all edges linking $u_{i}$
to all other nodes. With this function in the present example the probability
of $w_{1}$ being classified in community $e_{1}$ equals $\frac{2}{4}$, and her
probabilities of being classified in $c_{2}$ and in $c_{3}$ are $\frac{1}{4}$
each. The probability a node belongs to a given community is the percentage of
its links to this community as a proportion of the total number of links to
all communities. In other words, the greater the proportion of links to a
given community, the higher the expectation of belonging to this community.
### IV.3 Legitimacy function
It is possible to add more semantics in order to decide which community a
given node should join. The legitimacy function serves to measure the node
involvement in a community from the community’s perspective. The more strongly
a node is linked to other nodes in a community, the greater its legitimacy to
belong to the particular community. In the Southern Women’s example, let’s
assume that after partitioning, $c_{1}$ contains 7 events, $c_{2}$ 5 events
and $c_{3}$ 2 events (which is actually the case in the experiment presented
below). Then, $w_{1}$ would have a $\frac{2}{7}$ legitimacy for $c_{1}$,
$\frac{1}{5}$ for $c_{2}$ and $\frac{1}{2}$ for $c_{3}$. The legitimacy
function can thus be formalized as follows:
$L(u_{i}\in c)=\frac{\sum{}_{j}B_{ij}\delta(c_{j})}{|\\{v\in c\\}|}$ (10)
where $c$ is a community,$\delta(c_{j})=1$ if $v_{j}\in c$ or
$\delta(c_{j})=0$ if $v_{j}\notin c$
The numerator in this expression is the same as the probabilistic function
numerator. Only the denominator is different.
### IV.4 Reassignment Modularity function
Reassigning node $w$ from $C_{1}$ to $C_{2}$ eitherincreases or decreases the
modularity defined in Equation(1). Such a change is referred to as
Reassignment Modularity ($RM_{w:C_{1}\rightarrow C_{2}}$).
Let $w$ be a node $u$ or $v$. If w is withdrawn from $C_{1}$ and reassigned
to$C_{2}$, then we can define $RM_{w:C_{1}\rightarrow C_{2}}$ =$Q_{w\in
C_{2}}$-$Q_{w\in C_{1}}$
where Q is the modularity value in
$Q=\sum_{c}[\frac{|e_{c}|}{m}\text{\textendash}(\frac{(d_{u|c}+d_{v|c})}{2\times
m})\text{\texttwosuperior}]$ (11)
Let $l_{w|i}=l_{w,w^{\prime}|w^{\prime}\in C_{i}}$ be the number of edges
between $w$ and $w^{\prime}$ where $w^{\prime}\in C_{i}$,
Let $d_{w}$ be the degree of $w$, |$e_{i}|$ the number of edges in $C_{i}$ and
$d_{C_{i}}$= $d_{u|c_{i}}+d_{v|c_{i}}$
then
$Q_{w\in C_{2}}$-$Q_{w\in C_{1}}$ =
$[\frac{1}{m}(|e_{1}|-l_{w|1})+\frac{1}{m}(|e_{2}|+l_{w|2})-(\frac{(d_{C_{1}}-d_{w})^{2}}{(2m)^{2}}+\frac{(d_{C_{2}}+d_{w})^{2}}{(2m)^{2}})]-[\frac{1}{m}|e_{1}|-\frac{(d_{C_{1}})\text{\texttwosuperior}}{(2m)^{2}}+\frac{1}{m}|e_{2}|-(\frac{(d_{C_{2}})\text{\texttwosuperior}}{(2m)^{2}})]$
and
$RM_{w:C_{1}\rightarrow
C_{2}}=\frac{1}{m}(l_{w|2}-l_{w|1})-\frac{2}{(2m)^{2}}[d_{w}^{2}+d_{w}(d_{C_{2}}-d_{C_{1}})]$
(12)
This equation can be partly validated if after withdrawing $w$ from $C_{1}$ we
put it back into $C_{1}$: considering that $C_{2}$ is in fact $C_{1}$ without
$w$, we get $d_{C_{2}}=d_{C_{1}}-d_{w}$. Then replacing $d_{C_{2}}$ in
equation (20) by its value yields $RM_{w:C_{1}\rightarrow C_{1}}=0$.
A second validation can be performed with Equation (IX.2) in Wu2012a .Although
the authors’ demonstration is limited, it can still be noticed that their
final formula resembles ours with a slight difference (i.e. division by 2 in
their case) due to their definition of modularity for overlapping communities.
Moreover, in arguing that the right part of their equation is not meaningful
for large graphs, the authors only considered $dEQ=\frac{l_{2}-l_{1}}{2m}$
which is the equivalent of $\frac{1}{m}(l_{w|2}-l_{w|1})$ in our Reassignment
Modularity definition. In our case, we did not limit reassignment to large
graphs and moreover kept the whole value in Equation (20).
### IV.5 Other functions
Other assignment criteria may be explored, such as cardinality of communities:
when hesitating in a tie, an individual may prefer joining a small community
or, conversely, a large one. These criteria may be considered in practical
applications but will not be taken into account within the scope of the
present paper. Among the measures presented in this section, two will be used
to display our results, namely the Legitimacy function and the Reassignment
Modularity function. These two functions are of tremendous value in inferring
semantic analyses on overlapping results.
## V Visualization
Visualization is of utmost importance since it provides a means for analyzing
and interpreting the communities extracted from graphs. In most cases,
unipartite graphs are displayed using vertices and edges. Though these graphs
in a large and non-planar form quickly become entangled, it is still possible
to observe a certain structure by the use of graph drawing techniques
BattistaG.1999 . Visualizing partitioning is simple on unipartite graphs
through vertex coloring, provided the number of communities remains limited,
such as in Blondel2008 . With regard to community overlapping, the
visualization step is much more complicated (even for unipartite graphs) since
vertices may belong to several communities and, consequently, cannot be easily
assigned several colors. One possible solution would be to consider the
community assignment as a bipartite graph with one set of nodes being the
graph vertices and the other set being the communities and edges linking
vertices to their communities. Bipartite graphs can be represented like any
other graphs or as two-layer graphs, with both sets being organized in lines
either horizontally or vertically. When community assignment is represented
with edges however, the original edges between vertices cannot be shown on the
same graph without creating confusion. It is therefore more appropriate to
separately visualize the overlap through a biadjacency matrix, whose rows
represent communities and columns the original vertices. From this
perspective, each community is depicted with its core members and each vertex
with the various communities where it belongs. Other visualization techniques
have been studied, particularly when considering hypergraphs on which
communities are the hyperedges and vertices the nodes. For example, Euler
diagrams have been explored in Riche2010a and Simonetto2009 , Galois lattices
in Brandes2010 ; Johnson2006 , and linesets in Alper2011 , they are limited
however by the hypergraph size, with no clear representation emerging for
medium-sized hypergraphs. For this reason, we have only considered the
biadjacency visualization strategy in the Karate club experiment conducted
below, which proves to be valuable for our visual analysis of community
overlapping. Moreover, this representation technique offers the possibility of
indicating ownership functions, such as legitimacy (see Section IV.3), which
is not possible with other representations. When unipartite graphs are not
excessively large, it becomes important to retain all original information; it
is also possible to show the original unipartite adjacency matrices and then
add community overlapping. This possibility will not be discussed within the
scope of the present paper, which is solely dedicated to studying the
community overlapping visualization and does not consider the details of
vertex relations. Conversely, when the graph is too large, it is no longer
possible to even show community details, in which case it is better to
consider a square stochastic block matrix that only displays communities with
their relationship degrees (see Faust19925 and Guimera2009 ).
The visualization of community overlapping is even more complicated for
bipartite graphs since two of these graphs need to be represented: the
original graph, and the computed community overlap structure. As explained
above, two main strategies are available to visualize bipartite graphs: a
bilayer graph, and the biadjacency matrix. In limiting our objective to
representing partitioning, any symbolic representation, such as node shape or
coloring in the bilayer graph or even in the original graph, proves sufficient
(see for instance Figure 2 in Barber2007 or Figure 6 in Suzuki2009 ).
Visualizing community overlap requires alternative strategies. Along these
lines, we have explored two matrix representations. A detailed representation,
not presented within the scope of this paper, is based on the biadjacency
matrix of the original bipartite graph. The other bipartite representation, in
which rows are communities and columns people, will be provided below. Two
dual representations may coexist for a given situation, one for the first set
the other for the second, with both sets sharing the same communities. In the
Southern Women’s experiment described below, we analyze such a dual
representation while retaining the original bilayer bipartite graph. The two
other experiments introduced in the next section will be limited to
representing one set of vertices according to the semantic focus.
## VI Experimentation
This section will consider several benchmarks from various sources. We begin
by applying our method to two simple graphs: the so-called ”karate club”
unipartite graph from Zachary1977 shows friendship relations between members
of a karate sport club; and the ”Southern Women” bipartite graph depicts
relations between southern American women participating in several events. Our
method is then applied to a medium-sized dataset extracted from a real-world
situation. For this purpose, we consider a bipartite graph (people tagged on
photos) drawn from a student’s ”Facebook” account containing an average number
of photos and people. Lastly, this same method will be applied to human brain
data in order to derive dependencies between several areas in the brain.
### VI.1 Karate club
The karate club graph Zachary1977 is a well-known benchmark showing
friendship relations between members of a karate club; it is a unipartite
graph on which many partitioning algorithms have been experimented.
Consequently, this set-up makes it possible not only to verify that our method
for bipartite graphs when applied to unipartite graphs meets expected results,
but also to assess the additional knowledge extracted from overlapping.
We began by directly applying the Louvain algorithm to the original unipartite
graph, represented by its adjacency matrix $A$. which yielded four separate
communities (as shown in 2). These are the same communities extracted by other
authors, e.g. Newman2004 . During a second experiment, we considered that the
adjacency matrix $A$ is in fact a biadjacency matrix $B$ which is
representative of a bipartite graph whose corresponding objects are the club
members and whose properties are also club members. An edge exists in the
bipartite graph between a club member-object and a club member-property
provided an edge is present between the two club members in the original
unipartite graph. The new $A^{\prime}$ adjacency matrix is
$A^{\prime}=\left[\begin{array}[]{cc}O_{r}&B\\\
B^{t}&O_{s}\end{array}\right]$, where $B=A+I$. and where $I$ is the identity
matrix (as explained in section III.1.3). We once again apply the Louvain
algorithm to $A^{\prime}$.
_Results._ As expected, these same four communities identified in the
unipartite graph have been extracted from the bipartite graph, with the same
individuals appearing twice in each community (see Figure 2). This initial
result confirms the absence of bias when transforming a unipartite graph into
a bipartite one. The second result is more pertinent because it reveals an
overlap between communities when considering legitimacy values. If we were to
consider just the cell colorings in the figure, an overlap would be observable
whenever at least one node from a community is linked to other nodes in
another community. The legitimacy values that indicate the involvement of each
node in each community offer an effective tool for identifying and analyzing
new features. Some slight differences have been noted in works by other
authors: for example, in page 2, Porter Porter2009 placed node number 10 in
the second community. In our case, this node has been placed in the first
community, though the legitimacy value suggests that it should have been
placed in the second one, in which case the situation would be reversed in the
second community and node 10 would have a legitimacy value that alters its
placement in the first community. Node 10 is thus in a hesitation mode between
the two communities.
To the best of our knowledge, this experiment represents the first time Karate
communities are shown as separate and overlapping. Partitioning provides a
practical way to observe communities; however, overlapping reveals the extent
to which partitioning reduces the amount of initial information. With our
method for example, it can be seen that some nodes actually straddle several
communities, e.g. node 10 in our experiment.
Figure 1: Karate club graph with partitionned communities
(see a color version at this web adress :
www.lgi2p.ema.fr/plantie/PR/FIGURE-1.jpg) Figure 2: Karate club communities
and modularity measures
(see a color version at this web adress :
www.lgi2p.ema.fr/plantie/PR/FIGURE-2.jpg)
### VI.2 Southern Women
This benchmark has been studied by most authors interested in checking their
partitioning algorithm for bipartite graphs. The goal here is to partition,
into various groups, 18 women who attended 14 social events according to their
level of participation in these events. In his well-known cross-sectional
study, Freeman compared results from 21 authors, most of whom identified two
groups.
_Results._ In Figure 3, the bipartite graph is depicted as a bilayer graph in
the middle with women at the top and events below; moreover, the edges between
women and events represent woman-event participations. Three clusters with
associated women and events have been found and eventually shown with black,
dark grey and light grey colorings. This result is more accurate than the
majority of results presented in Freeman ]; only one author found three female
communities. Beyond mere partitioning, Figure 3 presents overlapping
communities using two overlapping functions, namely legitimacy and
reassignment modularity (RM). Legitimacy overlapping and RM for women are
placed just above female partitioning; for events, both are symmetrically
shown below event partitioning. As expected, reassignment in the same
community produces a zero RM value. The best values for legitimacy and RM have
been underscored. Only the values of woman 8 and event 8 indicate that they
could have been in another community. This is the outcome of early assignment
during the first Louvain phase for entities with equal or nearly equal
probabilities across several communities. It can be observed in Freeman that
woman 8’s community is also debated by several authors; our results appear to
be particularly pertinent in terms of both partitioning and overlapping.
One possible criticism of our result may stem from the fact that women and
events are correlated, which may cause bias, such as in the number of
communities. When comparing our results to those of other authors however, the
merging of our blue and yellow communities produces their corresponding second
community. In their trial designed to obtain a varying number of communities
in both sets, Suzuki Suzuki2009 found a large number of singletons. Their
results were far from those presented in Freeman , while ours were compatible
and more highly detailed.
In conclusion, results on the Southern Women’s benchmark are particularly
relevant. Moreover, our visualization enables observing community
partitioning, overlapping and possible assignment contradictions. The
application of reassignment for better modularity optimization will be tested
in a subsequent work.
Figure 3: Women Events communities and modularity measures
(see a color version at this web adress :
www.lgi2p.ema.fr/plantie/PR/FIGURE-3.jpg)
### VI.3 Facebook account
Three Facebook photo files were downloaded from various Facebook (FB)
accounts. All these files were extracted with the consent of their owners,
none of whom were members of the research team. A person was considered to be
linked to a photo if he/she had been tagged in the photo producing a bipartite
graph. We evaluated data from FB photo tags and not friendship relations.
Community extraction using our method reveals some common features among the
datasets. These features are shown in Figure _4 for one FB photo file, in
which 274 people could be identified in a total of 644 photos._
_Results._ Communities are seldom overlapping, which supports the notion that
the photos were taken at different times in the owner’s life (this is to be
confirmed in a forthcoming study). When the owner was asked to comment on the
communities, two main observations were submitted. The various groups of
people were indeed consistent, yet with one exception. The owner was
associated in the partition with a group she had met on only a few occasions
and not associated with other groups of close friends. An analysis of the
results provided a good explanation, which is partially displayed in Figure 4.
From this view, the FB account owner is in the first community on the left,
yet she is also present in most of the other communities (see grey color
levels in the first column). Although at first glance it might be assumed that
she is not part of other communities, our visualization indicates that such is
not the case. She is present in most communities, even though she is mainly
identified in the first one. Three types of photos can be distinguished in
this first community. More than 200 photos only contain the owner’s tag, plus
a few photos with unique tags of another community member; for every other
person, at least one photo tags him/her with the owner. This first community
has in fact been built from the first group with photos of unique owner’s tags
associated with the owner. The owner’s tag thus encompasses photos containing
two people, one of whom is the owner. It turns out that this group is
predominantly the owner’s group.
In conclusion, partitioning only the bipartite graph would have produced a
major pitfall: the owner would have been isolated in a community that is not
his/her top preference. With our method, merging partitioning and overlapping
exposes better multiple regroupings with broader affinities. Other communities
also showed high consistency when considering the photos: each community was
associated with some particular event responsible for gathering a group of the
FB account owner’s friends.
Figure 4: Facebook account communities and modularity measures
(see a color version at this web adress :
www.lgi2p.ema.fr/plantie/PR/FIGURE-4.jpg)
### VI.4 Brain Data
Our method was initially designed for human community detection and analysis.
In this experiment, we have demonstrated how it can be applied to other data
analysis techniques as well. The brain dataset was collected on a single
patient by a research team affiliated with the ”Human Connectome” project
working on brain tractography techniques Catani;2012 . These techniques use
Magnetic Resonance Imaging (MRI) and Diffusion Tensor Imaging (DTI) to explore
white matter tracks between brain regions. Probabilistic tractography produces
’connectivity’ matrices between Regions Of Interest (ROI) in the brain. For
the case we studied, ’seed’ ROIs were located in the occipital lobe and
’target’ ROIs throughout the entire brain. The goal here was to detect
possible brain areas in the occipital lobe through ROI clustering on the basis
of similar track behavior. In Catani;2012 ], the research team used Spectral
Clustering (SC) to combine ROIs. It is interesting to note that SC is one of
numerous techniques that have traditionally been applied in social community
detection, e.g. by Bonacich on the Southern Women’s benchmark Freeman . SC
results are limited to community partitioning (though in theory overlapping
could also be computed). The goal was to experiment with our method and
produce both partitioning and overlapping analyses of brain areas.
The original matrix contained 1,914 rows and 374 columns, with cells denoting
the probabilities of linkage between ROIs. We considered this matrix as a
bipartite graph biadjacency matrix with weighted values and then applied our
community detection method. Figure 6 presents the results of ROI community
partitioning and overlapping. Each grey level color in the first row is
associated with a community that gathers several ROIs. Each ROI is represented
by a column that indicates its belonging to the other communities. When a cell
is highlighted with a grey color, a nonzero overlapping value exists for both
this ROI and the corresponding community (with community numbers being plotted
on the left-hand side of the figure). This value has been computed with the
legitimacy function, which has been extended to the weighted edges, i.e. the
weighted sum of values from cerebral hemisphere zones (ELF) within the
selected community. Each community is associated with a threshold value
corresponding to the maximum weighted legitimacy above which the community
would lose a full member. For each community, this threshold value is
automatically computed in order to include all ROI members of the community.
_Results._ We found 7 communities when neurologists selected 8 clusters with
SP and after choosing the most significant eigenvectors on a scree test. Let’s
observe that two communities overlap heavily on all others, which thus overlap
to a lesser extent. If we were to lower all thresholds to a zero value, then
the overlapping would be maximized (see Figure 5). Figure 6 confirms the
strong interest in this set-up that simultaneously exhibits overlapping and
non-overlapping data. These results have been taken into account by a team of
neurological researchers as different observations recorded on brain
parcellation.
Figure 5: Brain data communities and modularity measures
(see a color version at this web adress :
www.lgi2p.ema.fr/plantie/PR/FIGURE-5.jpg) Figure 6: Brain data communities and
modularity measures
(see a color version at this web adress :
www.lgi2p.ema.fr/plantie/PR/FIGURE-6.jpg)
## VII Discussion et Conclusion
In this paper, we have demonstrated the feasibility of unifying bipartite
graphs, directed graphs and unipartite graphs after: considering them as
bipartite graphs with their biadjacency matrix, then building the
corresponding unipartite graph with the off-diagonal adjacency matrix, and
finally preparing for community building through a unipartite graph
partitioning algorithm. Along these lines, we formally derived a bipartite
graph modularity model from the standard unipartite modularity model. It was
then proven that any unipartite graph partitioning algorithm aimed at
optimizing the standard unipartite modularity model will lead to a bipartite
graph partitioning, wherein both types of nodes are bound in the communities.
In the special case of directed graphs, nodes appear twice in potentially
different communities depending on their roles; for unipartite graphs, nodes
are cloned and appear with their clones in the same communities.
Any approach that produces communities from bipartite graphs associating both
types of nodes has been qualified as a ’symmetric’ method in Suzuki2009 . Such
approaches have been criticized by some authors, who argue that the number of
communities in both types of nodes is often skewed Murata2009 ; Suzuki2009 .
Though this point of view remains defensible, we have shown that most authors
ultimately introduced a standard probabilistic model for bipartite graphs that
implicitly associates both types of nodes. Moreover, all the experiments we
identified in other papers actually present results that appear to be more
accurate with symmetric models.
Our approach is not limited to unifying different types of graphs; we also
introduced the possibility of unifying, into just a single view, the
partitioning and overlapping communities. This development is possible thanks
to associating both types of nodes in the communities. Moreover, overlapping
can be characterized through several functions presenting different semantic
meanings. For instance, it is possible to identify those nodes that define the
community cores, i.e. those who belong exclusively to just one community and,
conversely, those who serve as bridges between different communities. Some of
these functions even create the possibility to compute reassignment values,
which may then be used for fine-tuning the greedy partitioning algorithms.
Practically speaking, when applying our method to various benchmarks and
datasets, we are able to extract meaningful communities and display surprising
overlapping properties. Other authors’ models limit the goal to identifying
communities. We extend far beyond this point and provide tools for analyzing
and interpreting results. We can understand how some entities may be
hesitating between various community assignments, and moreover it is possible
to measure this ’fuzziness’. Some entities may even be reassigned in the
aftermath in pursuit of improved modularity optimization.
Lastly, we introduced an essential result after experimenting on real brain
datasets, supplied by a research team from the Connectome project. Many
traditional data analysis techniques have often been implemented by authors
for the purpose of community detection in unipartite or bipartite graphs, e.g.
Spectral Clustering or hierarchical clustering. These methods however require
specifying a number of clusters or setting a threshold. Recent community
detection algorithms, based on modularity detection, do not require any such
subjective orientation. We have applied algorithms of this type (e.g. Louvain)
so as to work with clustering in data analysis and compare it to traditional
data analysis techniques. The results are very similar for both approaches,
yet two main differences are noteworthy. First, community grouping is solely
dependent on the original data and does not require the a priori choice of
eigenvalues. Second, we were easily able to provide both partitioning and
overlapping communities with ownership functions. This result is of particular
interest when dealing with brain data in which community borders are not clear
cut.
### Acknowledgements
The authors would like to thank the Connectome research team, as part of the
“CAFO” project (ANR-09-RPDOC-004-01 project) as well as the CRICM UPMC
U975/UMRS 975/UMR 7225 research group for providing the original brain
dataset.
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* [42] W W Zachary. An information flow model for conflict and fission in small groups. Journal of Anthropological Research, 33(4):452–473, 1977.
## IX Annex 1
### IX.1 From modularity to bimodularity
In this Appendix, we will provide full details of the demonstration that
yielded Equation (2)
For the sake of convenience, let’s use the definition of unipartite graph
modularity offered in Newman [21]. It is a function $Q$ of matrix
$A^{{}^{\prime}}$ and the communities detected in $G$ [29]:
$Q=\frac{1}{2m}\sum_{i,j}\left[A_{ij}^{{}^{\prime}}-\frac{k_{i}k_{j}}{2m}\right]\delta(c_{i},c_{j})$
(13)
where $A_{ij}^{{}^{\prime}}$ denotes the weight of the edge between $i$ and
$j$, $k_{i}=\sum_{j}A_{ij}^{{}^{\prime}}$ is the sum of the weights of edges
attached to vertex $i$, $c_{i}$ is the community to which vertex $i$ has been
assigned, the Kronecker’s function $\delta(u,v)$ equals $1$ if $u=v$ and $0$
otherwise and $m=1/2\sum_{ij}A_{ij}^{{}^{\prime}}$. Hereafter, we will only
consider binary graphs and weights that are equal to 0 or 1.
In our particular case (i.e. where $A^{\prime}$ is the off-diagonal block
adjacency matrix of a bipartite graph), we apply the following
transformations:
Let’s rename $i_{1}$ as index $i$ when $1\leq i\leq r$ and $i_{2}$ when
$r<i\leq r+s$. Conversely, let’s rename $j_{1}$ the index $j$ when $1\leq
j\leq r$ and $j_{2}$ when $r<j\leq r+s$.
To avoid confusion between the $A^{{}^{\prime}}$’s indices and $B$’s indices
let’s rename $B$ indices $i_{b}$ and $j_{b}$ : $1\leq i_{b}\leq r$ and $1\leq
j_{b}\leq s$ (see a representation of $A$ matrix below (Equation 14))
$A^{\prime}=\begin{array}[]{c|c|c|c|c}A^{{}^{\prime}}\>indexes&&&\\\
\downarrow\rightarrow&....j_{1}....&....j_{2}....&\\\ \hline\cr...&&&...\\\
i_{1}&O_{r}&B&i_{b}&r\>rows\\\ ...&&&...\\\ \hline\cr...&&&...\\\
i_{2}&B^{t}&O_{s}&j_{b}&s\>rows\\\ ...&&&...\\\
\hline\cr&....i{}_{b}....&....j_{b}....&\leftarrow\uparrow\\\ &&&B\>indexes\\\
\hline\cr&r\>columns&s\>columns&\end{array}$ (14)
Let’s call $k_{i_{b}}$ the margin of row $i_{b}$ in $B$ and $k_{j_{b}}$ the
margin of column $j_{b}$ in $B$.
$k_{i_{b}}=\sum_{j_{b}}B_{i_{b}j_{b}}=\sum_{j_{2}}A_{i_{1}j_{2}}^{{}^{\prime}}=\sum_{i_{2}}A_{i_{2}j_{1}}^{{}^{\prime}},\>where\>i_{b}=i_{1}=j_{1}$
(15)
$k_{j_{b}}=\sum_{i_{b}}B_{i_{b}j_{b}}=\sum_{i_{1}}A_{i_{1}j_{2}}^{{}^{\prime}}=\sum_{j_{1}}A_{i_{2}j_{1}}^{{}^{\prime}},\>where\>j_{b}=i_{2}\text{\textendash}r=j_{2}\text{\textendash}r$
(16)
$k_{i_{b}}$is the degree of node $u_{i_{b}}$
$k_{j_{b}}$ is the degree of node $v_{j_{b}}$
Let’s define
$k_{i/j_{1}}=\sum{}_{j_{1}}A_{ij_{1}}^{{}^{\prime}}\>and\>k_{i/j_{2}}=\sum_{j_{2}}A_{ij_{2}}^{{}^{\prime}}$
Conversely :
$k_{j/i_{1}}=\sum_{i_{1}}A_{ji_{1}}^{{}^{\prime}}\>and\>k_{j/i_{2}}=\sum_{i_{2}}A_{ji_{2}}^{{}^{\prime}}$
Hence :
$k_{i}=\sum_{j}A_{ij}^{{}^{\prime}}=k_{i/j_{1}}+k_{i/j_{2}}$
$k_{j}=\sum_{i}A_{ij}^{{}^{\prime}}=k_{j/i_{1}}+k_{j/i_{2}}$
By taking into account the structure and properties of $A$ in (15) and (16)
for the indices we derive the following properties :
$k_{i/j_{1}}$ has non-zero values only for $i=i_{2}$, with $k_{j_{b}}$ the
degree of node $v_{j_{b}}$:
$k_{i/j_{1}}=k_{i_{2}/j_{1}}=\sum_{j_{1}}A_{i_{2}j_{1}}^{{}^{\prime}}=\sum_{i_{1}}A_{i_{1}j_{2}}^{{}^{\prime}}=k_{j_{2}/i_{1}}=k_{j_{b}}$
(17)
$k_{i/j_{2}}$ has non-zero values only for $i=i_{1},$ with $k_{i{}_{b}}$ the
degree of node $u_{i{}_{b}}$:
$k_{i/j_{2}}=k_{i_{1}/j_{2}}=\sum_{j_{2}}A_{i_{1}j_{2}}^{{}^{\prime}}=\sum_{i_{2}}A_{i_{2}j_{1}}^{{}^{\prime}}=k_{j_{1}/i_{2}}=k_{i_{b}}$
(18)
Moreover and more directly:
$k_{j/i_{1}}$ offers values only for $j=j_{2}$:
$k_{j/i_{1}}=k_{j_{2}/i_{1}}=k_{i_{2}/j_{1}}=k_{j_{b}}$, the degree of node
$v_{j_{b}}$.
$k_{j/i_{2}}$ offers values only for $j=j_{1}$:
$k_{j/i_{2}}=k_{j_{1}/i_{2}}=k_{i_{1}/j_{2}}=k_{i_{b}}$, the degree of node
$u_{i_{b}}$.
### IX.2 Analysing second part of $Q$ in (13)
Using these properties of matrix $A^{{}^{\prime}}$, it is now possible to
analyse $\sum_{ij}k_{i}k_{j}$. in equation (1).
Next, by developing $k_{i}$ and $k_{j}$ in $A^{{}^{\prime}}$ we obtain:
$\sum_{ij}k_{i}k_{j}=\sum_{ij}(k_{i/j_{1}}+k_{i/j_{2}})(k_{j/i_{1}}+k_{j/i_{2}})$
$=\sum_{ij}k_{i/j_{1}}k_{j/i_{1}}+\sum_{ij}k_{i/j_{2}}k_{j/i_{2}}+\sum_{ij}k_{i/j_{1}}k_{j/i_{2}}+\sum_{ij}k_{i/j_{2}}k_{j/i_{1}}$
$=\sum_{i_{2}j_{2}}k_{i_{2}/j_{1}}k_{j_{2}/i_{1}}+\sum_{i_{1}j_{1}}k_{i_{1}/j_{2}}k_{j_{1}/i_{2}}+\sum_{i_{2}j_{1}}k_{i_{2}/j_{1}}k_{j_{1}/i_{2}}+\sum_{i_{1}j_{2}}k_{i_{1}/j_{2}}k_{j_{2}/i_{1}}$
(19)
Let’s note that $\sum_{ij}k_{i/.}k_{j/.}=\sum_{i}k_{i/.}\sum_{j}k_{j/.}$ where
the dot may take any value in ${i_{1},i_{2},j_{1},j_{2}}$
Let $c$ be a community, in equation (1) summations $\sum_{ij}k_{i}k_{j}$ on
indices $i$ and $j$ may only be applied under the condition
$\delta(c_{i},c_{j})=1$. Where an edge is present between two nodes $u$ and
$v$ belonging to $c$: $\delta(c_{i},c_{j})=1$ and $\delta(c_{j},c_{i})=1$.
Consequently for each row $i$ representing a node belonging to $c$, a
corresponding column $j$ represents this same node belonging to $c$ and vice
versa.
From (17), (18), property (5) and the above observation:
$\sum_{ij}k_{i/j_{1}}k_{j/i_{1}}\delta(c_{i},c_{j})=\sum_{i}k_{i/j_{1}}\sum_{j}k_{j/i_{1}}\delta(c_{i},c_{j})=\sum_{i_{2}}k_{i_{2}/j_{1}}\sum_{j_{2}}k_{j_{2}/i_{1}}\delta(c_{i_{2}},c_{j_{2}})=\sum_{j_{b}}k_{j_{b}}\sum_{j_{b}}k_{j_{b}}=[\sum_{j_{b}}k_{j_{b}}]{{}^{2}}$
$\sum_{ij}k_{i/j_{2}}k_{j/i_{2}}\delta(c_{i},c_{j})=\sum_{i}k_{i/j_{2}}\sum_{j}k_{j/i_{2}}\delta(c_{i},c_{j})=\sum_{i_{1}}k_{i_{1}/j_{2}}\sum_{j_{1}}k_{j_{1}/i_{2}}\delta(c_{i_{2}},c_{j_{2}})=\sum_{i_{b}}k_{i_{b}}\sum_{i_{b}}k_{i_{b}}=[\sum_{i_{b}}k_{i_{b}}]{{}^{2}}$
$\sum_{ij}k_{i/j_{1}}k_{j/i_{2}}\delta(c_{i},c_{j})=\sum_{i}k_{i/j_{1}}\sum_{j}k_{j/i_{2}}\delta(c_{i},c_{j})=\sum_{i_{2}}k_{i_{2}/j_{1}}\sum_{j_{1}}k_{j_{1}/i_{2}}\delta(c_{i_{2}},c_{j_{1}})=\sum_{j_{b}}k_{j_{b}}\sum_{i_{b}}k_{i_{b}}$
$\sum_{ij}k_{i/j_{2}}k_{j/i_{1}}\delta(c_{i},c_{j})=\sum_{i}k_{i/j_{2}}\sum_{j}k_{j/i_{1}}\delta(c_{i},c_{j})=\sum_{i_{1}}k_{i_{1}/j_{2}}\sum_{j_{2}}k_{j_{2}/i_{1}}\delta(c_{i_{2}},c_{j_{1}})=\sum_{i_{b}}k_{i_{b}}\sum_{j_{b}}k_{j_{b}}$
where $j_{b}=i_{2}\text{\textendash}r=j_{2}\text{\textendash}r$ ,
$i_{b}=i_{1}=j_{1}$, $u_{i_{b}}\in c$ and $v_{i_{b}}\in c$ these last two
conditions can also be formalized with $\delta(c_{i_{b}},c_{j_{b}})=1$ if
$u_{i_{b}}$ and $v_{i_{b}}$ belong to the same community $c$ and
$\delta(c_{i_{b}},c_{j_{b}})=0$ otherwise.
This development yields :
$\sum_{ij}k_{i}k_{j}=[\sum_{j_{b}}k_{j_{b}}]{{}^{2}}+[\sum_{i_{b}}k_{i_{b}}]{{}^{2}}+2[\sum_{j_{b}}k_{j_{b}}][\sum_{i_{b}}k_{i_{b}}]=\sum_{i_{b}j_{b}}(k_{i_{b}}+k_{j_{b}}){{}^{2}}$
and:
$\sum_{ij}k_{i}k_{j}\delta(c_{i},c_{j})=\sum_{i_{b}j_{b}}(k_{i_{b}}+k_{j_{b}})^{2}\delta(c_{i_{b}},c_{j_{b}})$
(20)
Equation (20) can be rewritten using the degrees of nodes:
$\sum_{i_{b}}k_{i_{b}}$ is the sum of the degrees of nodes $u_{i_{b}}$
belonging to $c$ under the condition $\delta$ in equation (20). We denote this
$d_{u|c}$ .
$\sum_{j_{b}}k_{j_{b}}$is the sum of the degrees of nodes $v_{j_{b}}$
belonging to $c$ under the condition $\delta$ in equation (20) and has been
called $d_{v|c}$.
$Then\>\sum_{ij}k_{i}k_{j}\delta(c_{i},c_{j})=(d_{u|c}+d_{v|c})^{2}$ (21)
### IX.3 Analysing first part in (13)
First part in $Q$ is $\sum_{ij}A_{ij}^{{}^{\prime}}$ . Let’s examine what it
represents in terms of $B$. It is possible to identify matrix $B$ in $A$ using
indices $i_{1}$ and $j_{2}$. Conversely $B^{t}$ can be identified with indices
$i_{2}$ and $j_{1}$:
For $i=i_{1}\>A_{ij}$s only produce values for $j=j_{2}$, moreover for
$i=i_{2}$ $A_{ij}^{{}^{\prime}}$s only produce values for $j=j_{1}$ with
$A_{i_{1}j_{2}}^{{}^{\prime}}=B_{i_{b}j_{b}}$ and
$A_{i_{2}j_{1}}^{{}^{\prime}}=B_{i_{b}j_{b}}^{t}$ under typical conditions
regarding indices.
Then
$\sum_{ij}A_{ij}^{{}^{\prime}}=\sum_{i_{1}j_{2}}A_{i_{1}j_{2}}^{{}^{\prime}}+\sum_{i_{2}j_{1}}A_{i_{2}j_{1}}^{{}^{\prime}}$
And
$\sum_{ij}A_{ij}^{{}^{\prime}}\delta(c_{i},c_{j})=\sum_{i_{1}j_{2}}A_{i_{1}j_{2}}^{{}^{\prime}}\delta(c_{i_{1}},c_{j_{2}})+\sum_{i_{2}j_{1}}A_{i_{2}j_{1}}^{{}^{\prime}}\delta(c_{i_{2}},c_{j_{1}})$
The left-hand side of the sum equals the number of edges from nodes $u$ to
nodes $v$ inside $c$.
The right-hand side is the number of edges from these same nodes $v$ and $u$
inside $c$.
This set-up then leads to:
$\sum_{i_{1}j_{2}}A_{i_{1}j_{2}}^{{}^{\prime}}\delta(c_{i_{1}},c_{j_{2}})=\sum_{i_{2}j_{1}}A_{i_{2}j_{1}}^{{}^{\prime}}\delta(c_{i_{2}},c_{j_{1}})\>with\>i_{1}=j_{2}\>and\>i_{2}=j_{1}$
$Then\>\sum_{ij}A_{ij}^{{}^{\prime}}\delta(c_{i},c_{j})=2\sum_{i_{1}j_{2}}A_{i_{1}j_{2}}^{{}^{\prime}}\delta(c_{i_{1}},c_{j_{2}})=2\sum_{i_{b}j_{b}}B_{i_{b}j_{b}}\delta(c_{i_{b}},c_{j_{b}})$
(22)
This value can also be formalized using the number of edges:
$\sum_{i_{b}j_{b}}B_{i_{b}j_{b}}\delta(c_{i_{b}},c_{j_{b}})=|(u_{i_{b|c}},v_{j_{b|c}})|=|e_{i_{b|c},j_{b|c}}|\>where\>e_{i_{b|c},j_{b|c}}\in
E\>\&\>u_{i_{b|c}},v_{j_{b|c}}\in c$ (23)
For the entire matrix
$A^{{}^{\prime}}:\sum_{ij}A_{ij}^{{}^{\prime}}=2\sum_{i_{b}j_{b}}B_{i_{b}j_{b}}$
From equation (1), $m=1/2\sum_{ij}A_{ij}^{{}^{\prime}}$
Let’s now define $m_{b}=\sum_{i_{b}j_{b}}B_{i_{b}j_{b}}=|e_{i_{b}j_{b}}|$
where $e_{i_{b}j_{b}}\in E$
Then $m=\frac{1}{2}\times\sum_{ij}A_{ij}^{{}^{\prime}}=\frac{1}{2}\times
2\times\sum_{i_{b}j_{b}}B_{i_{b}j_{b}}=m_{b}$
### IX.4 Bimodularity
Lastly, by removing sub-index $b$, which had only been introduced to
distinguish indices $i$ and $j$ when applied to $A^{{}^{\prime}}$ or $B$, we
can redefine the $A^{{}^{\prime}}$ modularity in terms of $B$:
$Q=\frac{1}{m}\sum_{ij}[B_{ij}-\frac{(k_{i}+k_{j})\text{\texttwosuperior}}{4m}]\delta(c_{i},c_{j})$
(24)
In terms of edges, by simplifying $e_{i_{b|c},j_{b|c}}$ as $e_{c}$
(where$e_{c}$ has both ends in $c$) and by dropping sub-index $b$ Equation
(24) becomes:
$Q=\sum_{c}[\frac{|e_{c}|}{m}\text{\textendash}(\frac{(d_{u|c}+d_{v|c})}{2\times
m})\text{\texttwosuperior}]$ (25)
The term bimodularity refers to this definition of modularity for bipartite
graphs since both types of nodes are bound. In previous sections, we have
validated the above results on the basis of another author’s graph modularity
models. It can thus be concluded that (24) offers a good candidate for
bipartite graph modularity that takes some specific characteristics into
account.
|
arxiv-papers
| 2013-01-29T18:02:44 |
2024-09-04T02:49:40.899548
|
{
"license": "Public Domain",
"authors": "Michel Crampes, Michel Planti\\'e",
"submitter": "Michel Plantie",
"url": "https://arxiv.org/abs/1301.7006"
}
|
1301.7009
|
# Amending the Vafa-Witten Theorem
Chuan Li2111Email:[email protected]. and Qing Wang1,2222Email:
[email protected] author
1Center for High Energy Physics, Tsinghua University, Beijing 100084, People’s
Republic of China
2Department of Physics, Tsinghua University, Beijing 100084, People’s Republic
of China444mailing address
###### Abstract
The strong version of the Vafa-Witten theorem is shown may not to hold because
the zero condensate from a direct computation of the order parameter is found
to be a result on the symmetric vacuum. The validity of the Vafa-Witten
theorem relies then on its weak version, that the Goldstone boson is absent in
vector-like gauge theories with vanishing $\theta$ angle. The existence of a
charged $\rho$ meson condensate, which violates electromagnetic gauge
symmetry, is consistent with this weak version of the Vafa-Witten theorem when
applied to strong magnetic fields in QCD.
###### pacs:
11.30.Qc, 13.40-f, 12.38.-t, 74.90.+n
††preprint: TUHEP-TH-13178
The behavior of the vacuum is a very important problem in quantum field
theory. The Vafa-Witten theorem VW offers a strong constraint on the vacuum
of vector-like gauge theories. Recently, It was shown that in a sufficiently
strong magnetic field, the QCD vacuum can undergo a transition to a new phase
when charged $\rho$ mesons condenseEFTSC . In this phase the vacuum behaves as
an anisotropic inhomogeneous superconductor that supports superconductivity
along the axis of the magnetic field and generates $\rho$ vortices. As these
initial results are either from an effective theory for $\rho$ mesonsEFTrho
or from a phenomenological NJL model NJLSC , an investigation of the
underlying fundamental QCD is needed. For QCD, the results seem to be
controversial: paper QCDnegative claims that in terms of the Vafa-Witten
theorem, charged vector mesons cannot condense out in a magnetic field,
whereas a later paper QCDpositive states that this is not the case. Indeed,
both papers agree that there is no Goldstone boson which originally is taken
as a typical signature of global $U(1)_{I_{3}}$ isospin symmetry breaking in
QCD for strong magnetic fields. The work QCDnegative treats it as indicating
an absence of condensation, whereas the work QCDpositive explains it as the
result of the Higgs mechanism. That is, due to the lock of the $U(1)_{I_{3}}$
with electro-magnetic $U(1)_{\mathrm{em}}$ gauge symmetry, charged
$\rho$-meson condensation will induce spontaneous $U(1)_{\mathrm{em}}$
breaking; the photon will then obtain mass by ’eating’ a Goldstone boson as
its longitudinal component. Considering that the absence of a Goldstone boson
can have different interpretations, the direct computation of a nonzero
condensate becomes ever more important. Paper QCDnegative explicitly performs
this computation and shows that the condensate is zero, whereas paper
QCDpositive avoids directly discussing this issue. Nevertheless, nonzero
charged $\rho$ meson condensation is preferred in effective theories and
phenomenological model computations, and more importantly, intuition based on
the important formula for the $\rho$ meson mass in an external magnetic field
$\displaystyle
m^{2}_{\rho^{\pm}}(B_{\mathrm{ext}})=m^{2}_{\rho^{\pm}}-eB_{\mathrm{ext}}$ (1)
tells us that when the external magnetic field exceeds its critical value
$m^{2}_{\rho^{\pm}}/e$, the effective $\rho$ mass
$m_{\rho^{\pm}}(B_{\mathrm{ext}})$ becomes imaginary. This is a sign of an
unstable vacuum similar to the famous situation for the electroweak Higgs
potential; specifically, the curvature of the effective potential for a
charged $\rho$ meson field near the origin (symmetric vacuum) is to be changed
from concave to convex if the effective $\rho$ mass becomes imaginary. As long
as the effective potential is bounded from below, convex behavior of the
effective potential near origin demands that there should then exist at least
two new non-symmetric vacuums characterized by nonzero condensates. One might
guess that, in QCDnegative , a no-condensate result is due to the fact that
there are some unexpected infinities not counted in the computation that
multiply the obtained zero creating a finite nonzero condensate.
Unfortunately, up to now, this kind of infinity has not been found. One is
then lead to question the reliability of the computation. The purpose of this
paper is to show that this is really the case, i.e., the isospin violation
condensation in QCD with $\theta=0$ might not be zero. There exists a loop-
hole in the direct computation of the condensate either in the original Vafa-
Witten’s work or in the paper QCDnegative .
The Vafa-Witten theorem shows that the vector-like global symmetries cannot be
spontaneously broken in vector-like theories with zero theta angle. The result
for continuous global symmetries is supported by two facts: direct computation
shows a zero condensate for the order parameter, and a combination of various
inequalities indicate no Goldstone boson. If both of these two facts are true,
we call that the strong version of Vafa-Witten theorem holds. As discussed
above, the absence of a Goldstone boson cannot be treated as a criterion for
the spontaneous isospin symmetry breaking in QCD with strong external magnetic
fields. In this special case, the validity of the strong version of Vafa-
Witten theorem relies solely on the direct computation of the condensate in
QCD and in the rest of the paper, we shall mainly focus on this issue.
In 2-flavor QCD with an external magnetic field, the charged $\rho$ meson
condensate in terms of quark field $\psi$ is555Here, we perform calculations
in Minkowski space, the rotation to Euclidian space being unnecessary for the
present discussions.
$\displaystyle\langle
0|\overline{\psi}(x)\tau_{\pm}\gamma_{\pm}\psi(x)|0\rangle_{B}\sim\overline{\mathrm{tr}\bigg{[}\gamma_{\pm}\tau_{\pm}~{}\frac{1}{i\not{D}-m}\bigg{]}(x,x)}\;,$
(2)
where $\tau$ and $\gamma$ are the Pauli and gamma matrices defined in the
isospin and spinor spaces, respectively, $\tau_{\pm}=(\tau_{1}\pm\tau_{2})/2$
and $\gamma_{\pm}=(\gamma_{1}\pm i\gamma_{2})/\sqrt{2}$. Subscript ’$B$’
denotes the background external magnetic field, ’tr’ signifies the trace for
isospin and spinor indices,
$D_{\mu}=\partial_{\mu}-igT^{a}A^{a}_{\mu}-iqA^{\mathrm{em}}$ is the QCD
covariant derivative for the gluon fields $A^{a}_{\mu}$ in the presence of an
external $U(1)_{\mathrm{em}}$ field $A^{\mathrm{em}}_{\mu}$ with $g$ the
strong coupling constant, and $q=e(\tau_{3}+1/3)/2$ the electric charge for
quarks. We take the u and d quarks with the same current mass $m$. The average
in (2) is computed for the gluon field with the standard QCD path integral
measure; $\sim$ is to note that we have ignored the multiplication of some
irrelevant finite nonzero constant. Because term $\frac{1}{i\not{D}-m}$ only
has $1$ and $\tau_{3}$ components and no $\tau_{\pm}$ components in the
isospin space, then
$\mathrm{tr}\big{[}\gamma_{\pm}\tau_{\pm}~{}\frac{1}{i\not{D}-m}\big{]}(x,x)=0$
due to fact that
$\mathrm{tr}_{I_{3}}(\tau_{\pm})=\mathrm{tr}_{I_{3}}(\tau_{\pm}\tau_{3})=0$,
where $\mathrm{tr}_{I_{3}}$ is the trace of the isospin component. This
situation is similar to isospin violation discussed in the original Vafa-
Witten paper, where the condensate is
$\displaystyle\langle
0|\overline{\psi}(x)\tau_{3}\psi(x)|0\rangle\sim\overline{\mathrm{tr}\bigg{[}\tau_{3}~{}\frac{1}{i\not{\tilde{D}}-m}\bigg{]}(x,x)}\;,$
(3)
where $\tilde{D}_{\mu}=\partial_{\mu}-igT^{a}A^{a}_{\mu}$ is the pure QCD
covariant derivative corresponding to the gluon fields. In (3), the term
$\frac{1}{i\not{\tilde{D}}-m}$ has no $\tau_{3}$ component in isospin space,
hence
$\mathrm{tr}\big{[}\tau_{3}~{}\frac{1}{i\not{\tilde{D}}-m}\big{]}(x,x)=0$ due
to the fact that $\mathrm{tr}_{I_{3}}(\tau_{3})=0$. To avoid trivial zeroes,
QCDnegative adds to the Lagrangian an explicit infinitesimal isospin breaking
perturbation term $\lim_{\epsilon\rightarrow
0}\epsilon\overline{\psi}\Gamma\psi$ with $\Gamma$ depending on isospins,
spinors and space-time coordinate. Vafa-Witten does a similar thing by
separating $m_{u}\neq m_{d}$ infinitesimally. Both studies show that the
result is stable under infinitesimal isospin-violated perturbations. Thus,
from direct computation giving vanishing condensates, QCDnegative is at the
same level as the original Vafa-Witten discussion.
To examine whether it is possible to avoid the above no-condenste results for
(2) and (3), notice that in the effective theory and the NJL model, the
covariant derivative $D_{\mu}$ includes the $\rho$ field as
$\rho_{\mu}^{(a)}\tau_{a}$. This $\rho$ field dependence of the covariant
derivative will generate $\tau_{\pm}$ and $\tau_{3}$ components in the
$\frac{1}{i\not{D}-m}$ and $\frac{1}{i\not{\tilde{D}}-m}$, thereby prohibiting
the appearance of a final zero. If one further argues that those effective
theories and phenomenological models are not QCD, then we can do a similar
thing in QCD by applying the technique we developed in Ref.WQ2000 to
integrate out quark and gluon fields and integrate in bilocal colorless meson
fields $\Phi(x,x^{\prime})$ and $\Pi(x,x^{\prime})$ exactly, modifying (2) to
$\displaystyle\overline{\mathrm{tr}\bigg{[}\gamma_{\pm}\tau_{\pm}~{}\frac{1}{i\not{D}-m}\bigg{]}(x,x)}$
$\displaystyle\sim$
$\displaystyle\int\mathcal{D}\Phi\mathcal{D}\Pi~{}\mathrm{tr}\bigg{[}\gamma_{\pm}\tau_{\pm}~{}\frac{1}{i\not{\partial}+q\not{A}^{\mathrm{em}}-m-\Pi}\bigg{]}(x,x)~{}\mathrm{exp}\bigg{\\{}\mathrm{Trln}(i\not{\partial}+q\not{A}^{\mathrm{em}}-m-\Pi)$
(4) $\displaystyle+~{}i\int
d^{4}x_{1}d^{4}x_{1}^{\prime}N_{c}\Phi^{\sigma\rho}(x_{1},x_{1}^{\prime})\Pi^{\sigma\rho}(x_{1},x_{1}^{\prime})+N_{c}{\displaystyle\sum_{n=2}}\int
d^{4}x_{1}\cdots d^{4}x_{n}d^{4}x_{1}^{\prime}\cdots
d^{4}x_{n}^{\prime}~{}\frac{(-i)^{n}(N_{c}g^{2})^{n-1}}{n!}$
$\displaystyle\times~{}\overline{G}^{\sigma_{1}\cdots\sigma_{n}}_{\rho_{1}\cdots\rho_{n}}(x_{1},x_{1}^{\prime},\cdots,x_{n},x_{n}^{\prime})\Phi^{\sigma_{1}\rho_{1}}(x_{1},x_{1}^{\prime})\cdots\Phi^{\sigma_{n}\rho_{n}}(x_{n},x_{n}^{\prime})\bigg{\\}}\;,$
where $\sigma$ and $\rho$ are isospin and spinor index sets, Tr is the trace
over color, isospin, spinor, and space-time indices, and
$\overline{G}^{\sigma_{1}\cdots\sigma_{n}}_{\rho_{1}\cdots\rho_{n}}(x_{1},x_{1}^{\prime},\cdots,x_{n},x_{n}^{\prime})$
is the n-point gluon Green’s function. For (3), the corresponding result is
just to replace $\gamma_{\pm}\tau_{\pm}$ in (4) by $\tau_{3}$ and to ignore
the $q\not{A}^{\mathrm{em}}$ term. We see due to the appearance of isospin
field $\Pi$ in $\frac{1}{i\not{\partial}+q\not{A}^{\mathrm{em}}-m-\Pi}$ that
the original argument for a vanishing result now does not hold.
Suppose (4) does yield a nonzero result; one could still question whether (2)
and (3) give zero results. Why does this no-condensate result, once the path
integral is changed to some other form, become nonzero? To answer these
questions, notice that once spontaneous symmetry breaking occurs, the order
parameter should be a multi-valued quantity. However, (2) and (3) themselves
only give a single valued zero result for the order parameters $\langle
0|\overline{\psi}(x)\tau_{\pm}\gamma_{\pm}\psi(x)|0\rangle_{B}$ and $\langle
0|\overline{\psi}(x)\tau_{3}\psi(x)|0\rangle$; the expected multi-value result
does not show up, while (4) does implies the existence of multi-value result.
Here the multivaluedness is associated with possible multiple local minima of
the effective energy functional. These suggest that (2) and (3) are just
results pertaining to the symmetric vacuum. Without the knowledge of other
possible vacuums, it is dangerous to jump to conclusion only with the
information on the symmetric vacuum. To search for multi-valued results, one
needs to find some alternative expression which can include in the non-
symmetric vacuum effect, like that given by (4). The standard way to deal with
this situation is to introduce an effective action to search for different
vacuums and assess which is physical according to its vacuum energy. In more
detail, suppose $\phi(x)$ is an order parameter such as $\langle
0|\overline{\psi}(x)\tau_{\pm}\gamma_{\pm}\psi(x)|0\rangle_{B}$ or $\langle
0|\overline{\psi}(x)\tau_{3}\psi(x)|0\rangle$; introduce an external source
term $J(x)\phi(x)$ into the Lagrangian of the path integral and construct the
connected generating functional $W[J]$ on $J$ and classical field
$\phi_{c}(x)=\frac{\delta W[J]}{\delta J(x)}$. The condensate is then the
value of $\phi_{c}(x)$ when we switch off the external source $J=0$. This
$\langle 0|\phi(x)|0\rangle=\phi_{c}\big{|}_{J=0}$ simply corresponds to the
present direct computation of the condensate given in (2) and (3).
In practice, usually we do not know whether $W[J]$ is a single-valued
functional on $J$ or not. We usually start computations by assuming a single-
valued functional, hence $\phi_{c}(x)$ should be single valued. To retain
multi-valued condensate solutions, instead of directly calculating $\phi_{c}$,
one introduces an effective action $\Gamma[\phi_{c}]=W[J]-\int
d^{4}x\phi_{c}(x)J(x)$, and expresses the effective action $\Gamma$ in terms
of the classical field $\phi_{c}(x)$, not the original external source $J$. It
is then easy to show $\frac{\delta\Gamma[\phi_{c}]}{\delta\phi_{c}(x)}=-J(x)$,
which implies that $\phi_{c}(x)$ is an extremum of $\Gamma$. If $J=0$, one
only needs to search for a real minimum of the effective potential (multiplied
by the space-time volume and a minus sign, it is just the translational
invariant part of the effective action) to obtain the physical condensation.
For spontaneous symmetry breaking, with the exception of the conventional
symmetric solution $\phi_{c}(x)\big{|}_{J=0}=0$, there usually exist other
nonzero solutions and these correspond to lower vacuum energies. In this
analysis to evaluate physical condensate, the key is that the external source
should not vanish before obtaining the final physical condensate. For (2), the
corresponding generating functional is
$\displaystyle e^{iW[J_{\pm}]}$ $\displaystyle=$
$\displaystyle\int\mathcal{D}A^{a}_{\mu}\mathcal{D}\overline{\psi}\mathcal{D}\psi~{}\mathrm{exp}\bigg{\\{}i\int
d^{4}x~{}\bigg{[}-\frac{1}{4}G_{\mu\nu}^{a}G^{a\mu\nu}+\overline{\psi}(i\not{D}\\!-\\!m\\!+\\!\gamma_{\pm}\tau_{\pm}J_{\pm})\psi\bigg{]}\bigg{\\}}$
$\displaystyle=$
$\displaystyle\int\mathcal{D}A^{a}_{\mu}~{}\mathrm{exp}\bigg{\\{}\mathrm{Trln}[i\not{D}\\!-\\!m\\!+\\!\gamma_{\pm}\tau_{\pm}J_{\pm}]+i\int
d^{4}x~{}[-\frac{1}{4}G_{\mu\nu}^{a}G^{a\mu\nu}]\bigg{\\}}\;,$
where $G_{\mu\nu}^{a}$ is the gluon field strength and in the second equality
we have integrated out the quark fields. The classical field $\rho_{\pm}(x)$
is defined as
$\displaystyle\rho_{\pm}(x)=\frac{\delta W[J_{\pm}]}{\delta
J_{\pm}(x)}=\frac{\int\\!\mathcal{D}A^{a}_{\mu}\mathrm{tr}\bigg{[}\gamma_{\pm}\tau_{\pm}\frac{-i}{i\not{D}-m+\gamma_{\pm}\tau_{\pm}J_{\pm}}\bigg{]}(x,x)\mathrm{exp}\bigg{\\{}\mathrm{Trln}[i\not{D}\\!-\\!m\\!+\\!\gamma_{\pm}\tau_{\pm}J_{\pm}]+i\int\\!d^{4}x[-\frac{1}{4}G_{\mu\nu}^{a}G^{a\mu\nu}]\bigg{\\}}}{\int\mathcal{D}A^{a}_{\mu}~{}\mathrm{exp}\bigg{\\{}\mathrm{Trln}[i\not{D}\\!-\\!m\\!+\\!\gamma_{\pm}\tau_{\pm}J_{\pm}]+i\int
d^{4}x~{}[-\frac{1}{4}G_{\mu\nu}^{a}G^{a\mu\nu}]\bigg{\\}}}\;.$ (6)
Note here the external-source-dependent isospin violation term
$\gamma_{\pm}\tau_{\pm}J_{\pm}$ appearing in the denominator of the integrand
in addition to the covariant derivative $\not{D}$. This makes the result
nonzero as long as we do not let the external source $J_{\pm}$ vanish. The
effective action is
$\displaystyle\Gamma[\rho_{\pm}]=W[J_{\pm}]-\int
d^{4}x~{}\rho_{\pm}(x)J_{\pm}(x)\;,$ (7)
which satisfies
$\frac{\delta\Gamma[\rho_{\pm}]}{\delta\rho_{\pm}(x)}=-J_{\pm}(x)$. Here we
emphasize that once we take $J_{\pm}=0$ at the very beginning in (Amending the
Vafa-Witten Theorem) and (6), as has been done in (2) and (3), we will just
get the symmetric solution $\rho_{\pm}(x)\big{|}_{J_{\pm}=0}=0$. The nonzero
solution can be obtained only when we come to calculate $\Gamma[\rho_{\pm}]$,
which demands that one must not let the external source $J_{\pm}$ vanish in
the process of computation. Instead, the external source must be replaced with
its function on the classical field $J_{\pm}=J_{\pm}[\rho_{\pm}]$. Is this
constraint on vanishing external source late so important? What is the effect
if we replace it with its function on $\rho_{\pm}$? Note that to prove a
theorem, we need to investigate all possibilities, while to show the theorem
is not true, just a counter example is enough. Here we exhibit an explicit
example that zero condensate from directly calculation of generating
functional, once considering its effective action, does lead physical nonzero
condensations. This just contradicts with the logic of Vafa-Witten theorem
that zero condensate from directly calculation of generating functional will
lead physical zero condensation. To avoid QCD complexities and see clearly the
core of the problem, we take this counter example as simple as possible, which
will describe what happens and show strong version of the Vafa-Witten theorem
may not to hold.
We consider a 0-dimension field model with generating functional $W[J]$ given
as
$\displaystyle W[J]=12\lambda
r^{\frac{4}{3}}\cos(\Theta+\frac{4}{3}\pi)[\cos(\Theta+\frac{4}{3}\pi)+\cos(3\Theta)]\;,$
(8)
where $\lambda$ and $r$ are two real positive numbers of the model, quantity
$\Theta$ relates these and the external source $J$ through
$\displaystyle 3\Theta=\mathrm{arc~{}cos}\left(\frac{J}{8r\lambda}\right)\;.$
(9)
The classical field $\phi$ is defined as
$\displaystyle\phi=\frac{\delta W[J]}{\delta
J}=2r^{\frac{1}{3}}\cos(\Theta+\frac{4}{3}\pi)\;,$ (10)
where we have used the expression for $W[J]$ given in (8) and relation (9) to
obtain an explicit expression for the classical field $\phi$ in terms of
$\Theta$. Notice that switching off the external source $J=0$ corresponds to
$\displaystyle\Theta\bigg{|}_{J=0}=\frac{\pi}{6}\;,$ (11)
which allows the classical field to vanish, $\phi\big{|}_{J=0}=0$. This is a
single-valued vanishing result, just as in (2) or (3). To generate nonzero
$\phi\big{|}_{J=0}$, we first solve (10) and express the external source $J$
in terms of the classical field $\phi$ to obtain
$\displaystyle J=4\lambda(\phi^{3}-3r^{\frac{2}{3}}\phi)\;.$ (12)
Noting that as our model is 0-dimensional, i.e., there is no space-time
integration and the functional differential reverts to an ordinary derivative,
we introduce the effective action as
$\displaystyle\Gamma=W-J\phi=6\lambda
r^{\frac{2}{3}}\phi^{2}-\lambda\phi^{4}\;,$ (13)
where, with the help of (12), we can perform some algebraic computations to
express $\Gamma$ in terms of the classical field $\phi$. The result $\Gamma$
given by (13) is simply the 0-dimensional $\lambda\phi^{4}$ model with the
wrong-sign mass-squared, $12\lambda r^{\frac{2}{3}}$. The vacuum is determined
by $\frac{\delta\Gamma}{\delta\phi}=0$ and it leads to
$\displaystyle[\phi^{2}-3r^{\frac{2}{3}}]\phi=0\;.$ (14)
$\phi=0$ is the solution for the symmetrical vacuum, which is already
predicted from (10) and (11) originally by directly computing the generating
functional (8). The other two nonzero solutions
$\phi=\pm\sqrt{3}r^{\frac{1}{3}}$ are new ones which cannot be obtained from
(10) and (11). One can easily check that these two solutions have equal lower
vacuum energies $V=-\Gamma|_{J=0}$ in comparison with the zero solution, hence
correspond to physical condensates. This toy model thus produces nonzero
condensates from a single-valued generating functional for the symmetrical
vacuum, as long as we use the effective action formalism. The reason we call
(8) and (9) the generating functional on the symmetrical vacuum is that there
exist two other generating functionals on non-symmetric vacuums by changing
(9) to
$\displaystyle
3\Theta=\left\\{\begin{array}[]{l}2\pi-\mathrm{arc~{}cos}\left(\frac{J}{8r\lambda}\right)\\\
{}\\\
4\pi+\mathrm{arc~{}cos}\left(\frac{J}{8r\lambda}\right)\end{array}\right.\;,$
(18)
which leads exactly to the same classical field expression (10) and the
effective action (13). When we switch off the external source, $J=0$, instead
of (11), these two generating functionals on the non-symmetric vacuums give
$\displaystyle\Theta\big{|}_{J=0}=\left\\{\begin{array}[]{l}\frac{\pi}{2}\\\
{}\\\ \frac{3\pi}{2}\end{array}\right.\;,$ (22)
which combined with (10) just give the nonzero solutions
$\phi=\pm\sqrt{3}r^{\frac{1}{3}}$. Indeed, the three expressions from (9) and
(18) just cover the three different solutions of equation (12), if we want
solutions expressing $\phi$ in terms of the external source $J$. $\phi$ as
function of $J$ is a three-valued function. In other words, it has three
branches: the one given by (9) corresponds to the symmetric branch, the other
two given by (18) to the non-symmetric branches. We refer to these results as
those on symmetric and non-symmetric vacuums, respectively. Our original
generating functional (8) just chooses the symmetric branch. This toy model
indicates that due to the choice of a generating functional on the symmetrical
vacuum, (2) and (3) produces a no-condensate result does not imply that the
physical result must vanish.
One may criticize the simplicity of this toy model that first it is in
0-dimension space and then there is no motion for the particle; second there
is no path integral and then may be no quantum effect. To overcome these
shortcomings, we generalize above 0-dimensional $\lambda\phi^{4}$ model to
D-dimensional $O(N)$ $\lambda\phi^{4}$ model by considering following
generating functional
$\displaystyle
e^{i\tilde{W}_{D}[J]}=\int[{\mathcal{D}}\phi_{i}]~{}\exp\bigg{\\{}i\int
d^{D}x~{}\bigg{[}\phi_{i}(x)[-\frac{1}{2}\partial_{x}^{2}+6\tilde{\lambda}\tilde{r}^{\frac{2}{3}}]\phi_{i}(x)-\frac{\tilde{\lambda}}{N}[\phi_{i}(x)\phi_{i}(x)]^{2}+J_{i}(x)\phi_{i}(x)\bigg{]}\bigg{\\}}\;,$
(23)
where $\tilde{\lambda}$ and $\tilde{r}$ are two real positive numbers of the
model. $\phi_{i}(x)$ has $N$ components, $J_{i}(x)$ is corresponding external
source, $i=1,2,\ldots,N$. We discuss the large N limit of the model. Under
this limit, there is an overall factor $N$ appears in the exponential of the
integrand of the path integral due to summation of different components of
$\phi(x)$ fields, which will suppress all loop contributions Jackiw
$\displaystyle\tilde{W}_{D}[J]\stackrel{{\scriptstyle
N\rightarrow\infty}}{{====\Rightarrow}}\int
d^{D}x~{}\bigg{[}\phi_{i,c}(x)[-\frac{1}{2}\partial_{x}^{2}+6\tilde{\lambda}\tilde{r}^{\frac{2}{3}}]\phi_{i,c}(x)-\frac{\tilde{\lambda}}{N}[\phi_{i,c}(x)\phi_{i,c}(x)]^{2}+J_{i}(x)\phi_{i,c}(x)\bigg{]}$
(24)
where $\phi_{i,c}(x)$ is the solution of equation
$\displaystyle\frac{\delta}{\delta\phi_{i,c}(x)}\int
d^{D}y~{}\bigg{[}\phi_{i,c}(y)[-\frac{1}{2}\partial_{y}^{2}+6\tilde{\lambda}\tilde{r}^{\frac{2}{3}}]\phi_{i,c}(y)-\frac{\tilde{\lambda}}{N}[\phi_{i,c}(y)\phi_{i,c}(y)]^{2}+J_{i}(y)\phi_{i,c}(y)\bigg{]}=0\;,$
(25)
which leads,
$\displaystyle
J_{i}(x)=\bigg{[}\frac{4\tilde{\lambda}}{N}\phi_{j,c}(x)\phi_{j,c}(x)+(\partial_{x}^{2}-12\tilde{\lambda}\tilde{r}^{\frac{2}{3}})\bigg{]}\phi_{i,c}(x)\;.$
(26)
Above equation is difficult to solve, we consider a translational invariant
situation that external source $J_{i}$ and $\phi_{i,c}$ are all independent of
space-time coordinates. Then above equation is reduced to
$\displaystyle\tilde{J}=\bigg{[}\frac{4\tilde{\lambda}}{N}\tilde{\phi}^{2}-12\tilde{\lambda}\tilde{r}^{\frac{2}{3}}\bigg{]}\tilde{\phi}\;,$
(27)
where $\tilde{J}=\sqrt{J_{i}J_{i}}$ and
$\tilde{\phi}=\sqrt{\phi_{i,c}\phi_{i,c}}$ are modulus of $J_{i}$ and
$\phi_{i,c}$ in $O(N)$ space respectively. Above equation is the same as
Eq.(12) if we identify $\tilde{J}$, $\tilde{\phi}$, $\tilde{\lambda}/N$,
$N\tilde{r}^{2/3}$ with $J$, $\phi$, $\lambda$, $r^{2/3}$, respectively. This
equation has three solutions if we want to express the classical field
$\tilde{\phi}$ in terms of the external source $\tilde{J}$. Among them, one
solution cannot satisfy positivity requirement for $\tilde{J}$ and
$\tilde{\phi}$, the left two solutions are
$\displaystyle\tilde{\phi}=2N^{\frac{1}{2}}r^{\frac{1}{3}}\cos(\tilde{\Theta}+\frac{4}{3}\pi)\;,$
(28)
and
$\displaystyle
3\tilde{\Theta}=\left\\{\begin{array}[]{lll}\mathrm{arc~{}cos}\left(\frac{\tilde{J}}{8N^{\frac{1}{2}}\tilde{r}\tilde{\lambda}}\right)&&\mbox{symmetric
branch}\\\ {}\\\
2\pi-\mathrm{arc~{}cos}\left(\frac{\tilde{J}}{8N^{\frac{1}{2}}\tilde{r}\tilde{\lambda}}\right)&{}{}{}{}&\mbox{non-
symmetric branch}\end{array}\right.\;,$ (32)
The translational invariant generating functional of large N limit
D-dimensional $O(N)$ $\lambda\phi^{4}$ model becomes
$\displaystyle\tilde{W}_{D}[J]\stackrel{{\scriptstyle
N\rightarrow\infty;~{}J_{i}(x)=J_{i}}}{{=========\Rightarrow}}12N\tilde{\lambda}\tilde{r}^{\frac{4}{3}}\cos(\tilde{\Theta}+\frac{4}{3}\pi)[\cos(\tilde{\Theta}+\frac{4}{3}\pi)+\cos(3\tilde{\Theta})]\int
d^{D}x\;.$ (33)
We see that path integral now generates two different branch generating
functionals, one is for symmetric vacuum, another is for non-symmetric vacuum.
If one chooses the symmetric one as QCDnegative and VW at the very beginning
of the computation for condensate, it will lead vanishing result. The
difference with QCDnegative and VW is that now instead of single value
generating functional, path integral gives multi-value result. The reason
present direct computation for this D-dimensional $O(N)$ $\lambda\phi^{4}$
model resulting in multi-value result is that path integral involves the
degree of freedom of order parameter $\phi_{i}$, just like (4). While in
QCDnegative and VW , due to choice of fundamental quark as dynamical variable
and the lack of explicitly extracting out the degree of freedom of order
parameter, the generating functional unfortunately choose the symmetric
branch.
In more general situations, if we assume the effective action $\Gamma[\phi]$
be a single-valued function, then for spontaneous symmetry breaking, solving
equation $\frac{\delta\Gamma[\phi]}{\delta\phi(x)}=-J(x)$ to express $\phi(x)$
in terms of $J$ will usually lead to a multi-valued function, or a function
which has several branches, because spontaneous symmetry breaking demands
multi-vacuum solutions. Among these branches, one is the symmetric branch
corresponding to the generating functional on the symmetric vacuum. The other
branches lead to generating functionals on non-symmetric vacuums. The no-
condensate results obtained in QCDnegative and Vafa-Witten’s paper VW is due
to choosing the generating functional on the symmetric vacuum from the very
beginning. The proofs in VW and QCDnegative do not deny the other
nonsymmetric vacuum (or other branch of the generating functional), while our
expression (4) explicitly show the possibility of existence of other
nonsymmetric vacuum. Then just the discussion on the symmetric vacuum may not
represent the correct result of the whole system. To avoid that misleading
result, the safest way is to calculate the effective action, or changing over
to the generating functional on the non-symmetric vacuum as done in an
effective theory or the NJL model or (6), where the path integral is already
improved involving degree of freedom relate to order parameter as our
D-dimensional $O(N)$ $\lambda\phi^{4}$ model. Taking just an infinitesimal
perturbation, as Vafa-Witten and Ref.QCDnegative had done, is not enough to
cross-over from the symmetric vacuum to the non-symmetric vacuum.
With this discussion on direct computation of condensate, we find that the no-
condensate result in the Vafa-Witten theorem and QCDnegative may not be true
due to its prejudiced choice of a generating functional on symmetric vacuum
and lack the information for other possible nonsymmetric vacuum. Ignoring this
possible defect, the Vafa-Witten theorem has to reduce to its weaker form,
i.e., it only proves that in vector-like gauge theories with $\theta=0$, there
is no Goldstone boson. The charged $\rho$ meson condensate in QCD for strong
magnetic fields happen to have no Goldstone boson due to existence of Higgs
mechanism.
The explanation of generating functional on symmetric vacuum may also help to
understand the dispute over the parity violation problem in QCD Parity . Once
we accept the fact that the generating functional Vafa-Witten used in their
proof is the one associated with the symmetric vacuum, then the absence of a
Goldstone boson obtained from this generating functional is also questionable.
Considering that up to now, unlike the vanishing condensate case, there is no
exceptions to the result of no Goldstone boson in various effective field
theories and phenomenological models, this partial result may still be true.
In fact, one can check that original Vafa-Witten’s discussion on two-point
correlation function of isospin current for QCD in a strong magnetic field is
still true in presence non-vanishing isospin source introduced as in
Eq.(Amending the Vafa-Witten Theorem) and the exponential fall-off behavior
for the correlation function remains, thus insuring the correctness of the
weak version of the Vafa-Witten theorem.
It should be emphasized that our discussions are not only valid for the Vafa-
Witten theorem, but also apply to all kinds of direct computations of various
physical quantities and all generating functional-based results. That is, once
the system has spontaneous symmetry breaking or multi-vacuums, one must
carefully check the results obtained from the direct computation of path
integrals to see that these are not performed just on the non-physical vacuum.
## ACKNOWLEDGMENTS
We thank Professor Xiaoyuan Li for recommending the present topic to us. This
work is supported by the National Science Foundation of China (NSFC) under
Grants No.11075085, Specialized Research Fund Grants No.20110002110010 for the
Doctoral Program of High Education of China, and Tsinghua University
Initiative Scientific Research Program.
## References
* (1) C.Vafa and E.Witten, Nucl.Phys. B234, 173(1984).
* (2) M.N.Chernodub, Phys.Rev.D82,085011(2010); M.N.Chernodub, J.Van Doorsselaere and H.Verschelde, Phys.Rev. D85, 045002(2012).
* (3) D.Djukanovic, M.R.Schindler, J.Gegelia, and S.Scherer, Phys.Rev.Lett. 95, 012001(2005).
* (4) M. N. Chernodub, Phys.Rev.Lett, 106, 142003(2011).
* (5) Y. Hidaka and A. Yamamoto, arXiv:1209.0007[hep-ph].
* (6) M.N.Chernodub, Phys.Rev. D86, 107703(2012).
* (7) Q.Wang, Y.P.Kuang, X.L.Wang and M.Xiao, Phys.Rev. D61, 054011(2000).
* (8) R.Jackiw, Phys.Rev. D9, 1686(1974).
* (9) X.D.Ji, Phys.Lett. B554, 33(2003).
|
arxiv-papers
| 2013-01-29T18:19:03 |
2024-09-04T02:49:40.913831
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chuan Li and Qing Wang",
"submitter": "Wang Qing",
"url": "https://arxiv.org/abs/1301.7009"
}
|
1301.7064
|
# How to Scale a Code in the Human Dimension
Matthew J. Turk ([email protected])
Columbia University
This is a re-telling of a talk given at Scientific Software Days in December,
2012, at the Texas Advanced Computing Center. The slides and video of this
talk can be found at http://scisoftdays.org/meetings/2012/. Abstract: As
scientists’ needs for computational techniques and tools grow, they cease to
be supportable by software developed in isolation. In many cases, these needs
are being met by communities of practice, where software is developed by
domain scientists to reach pragmatic goals and satisfy distinct and enumerable
scientific goals. We present techniques that have been successful in growing
and engaging communities of practice, specifically in the yt and Enzo
communities.
## 1 Why “Community?”
Astrophysics, and particularly computational astrophysics, is dominated by
vertically-integrated, small-population research collaborations. The concept
of a community of researchers – sharing physics modules, improvements to
simulation codes, analysis techniques, technology – is somewhat foreign. In
fact, this idea of “community” is often viewed as a detriment to the
individual researcher, rather than as a benefit to the field. I participate in
two vibrant, active communities in computational astrophysics, those
surrounding the simulation code Enzo and the analysis code yt. Within these
two communities, we have found a somewhat surprising result: the empowerment
of the community does not come at the expense of individual success. In fact,
with actively shepherded and cultivated community participation and processes,
the opposite is true: the betterment of the community comes at the enrichment
of the individual.
In addition to the concrete, measurable benefits we receive from community-
focused development, we have also realized that community development is
essential to the continued health of the field of computational astrophysics.
By focusing on developing a community of practice, where the goals and
technology are driven by active participants, we have been able to expand the
class of astrophysical problems to which the technologies have been applied.
This has led not to consolidation of research interests, but rather a
broadening, with improvements, technology, and techniques being directly
shared between working domain scientists.
In this paper, I outline the infrastructure and the techniques with which we
have cultivated these two complementary, but distinct, communities and the
various conscious decisions we have made to ensure their growth and
sustainability.
## 2 Introduction: yt and Enzo
The yt Project (http://yt-project.org/) is an open source, community-developed
analysis code for simulated astrophysical data. Largely written in Python,
Cython and C, it is parallelized using MPI and OpenMP and has been used to
analyze datasets whose size range from small (tens of megabytes) to large
(tens of terabytes). yt is designed to abstract out underlying technical
aspects of simulation data such as file format, units, geometric conventions,
and parameter storage in such a way that is neutral to the underlying
simulation platforms. The flagship simulation platforms, where we conducted
detailed testing and support all functionality, are are Enzo, FLASH, Orion,
Castro, Nyx, Piernik and NMSU-ART. In the current development branch
(described briefly below) this has been expanded to include limited support
for SPH and N-body codes such as Gadget, as well as full support for Octree
codes such as RAMSES and ART. yt provides a language for describing physical
regions and applying data processing techniques to those regions: rather than
focusing on selecting grid patches, particles or octs and then masking out
overlapping regions, it develops concepts of geometric regions, regions
defined by fluid quantities, and processes that transform those regions into
quantitative values. yt is best thought of not as an application, but as a
library for asking and answering questions about data. This aspect of yt
naturally encourages technical contributions, as every user of the software
package typically writes analysis code that builds upon its underlying
machinery. Although its mission has expanded in recent years to include the
application of microphysical solvers, standardized input/output data formats,
creating initial conditions, and even studying data from earthquake
simulations, at its core yt remains a tool for interrogating astrophysical
data and answering questions about the underlying physics of that data.
The defining characteristic of yt is not that it is written in Python, or the
operations it does, but rather that it is supported by a participatory
community of scientists interested in both using and developing it. The yt
community draws its members primarily from computational astrophysicists:
individuals who conduct and analyze simulations. The first version was written
in late 2006 by and for a single scientist, but over the last several years
has attracted 170 subscribers to the general mailing list and 50 for the
development mailing list; in 2012 it was identified as one of the most highly
used codes on the NSF NICS analysis machine Nautilus [Szczepanski et al.,
2012]. Over the course of its history, nearly 40 people have contributed
changesets, ranging from tiny to very large, and the mailing lists are
relatively active, averaging between one and four messages a day. In 2012 we
held our first workshop at the FLASH center in Chicago, and we are holding a
second workshop in March of 2013 at UC Santa Cruz. Despite this relatively
high level of activity for a project in computational astrophysics, the
community has faced several challenges as well as taken steps to directly
address these challenges.
I am the original author of yt, having created it during my graduate school
career to analyze and visualize data created by the simulation code Enzo
(http://enzo-project.org/), an adaptive mesh refinement simulation code. Enzo
itself has undergone a number of changes over its years, both in the manner in
which it is developed and the problems to which it can be applied. Enzo began
as a codebase largely developed by a single individual, but through the
stewardship of the Laboratory for Computational Astrophysics (LCA) it grew
into a large, open source (but mostly not community-based) development
project. A direct consequence of both the technology and the prevailing
mindsets led to divergent lines of “public” and “internal” development,
resulting in a highly fragmented community of users and developers and many
different branches of the code itself. Over the course of a few years in 2009
and 2010, through concerted efforts from the Enzo community, it transitioned
from a closed development model with periodic open releases into a fully-open,
transparent development model based around contributions from community
members. This included resolving many divergent code bases and produced the
Enzo 2.0 release. The code is now developed using many best practices of
software development including testing, version control, peer review,
infrastructure design and investment of stakeholders in the development
roadmap. This has resulted in contributions of both physics modules and
infrastructure improvements, and has even brought to light bugs that have
subsequently been fixed by community members. This transition, from semi-open
to community-driven, has brought an energy and excitement to the development
of Enzo that seems likely to sustain it for years to come. Many of the items
discussed below, about how to shape and foster community, have been applied to
the Enzo community as well as yt.
## 3 What is “Community”?
Open Source is often used as a synonym for community development, but in
practice they are better thought of as two different, but overlapping, modes
of development. The growth of a community, where individuals participate in
discussions, report problems, provide enhancements, and support other
community members, is neither necessitated by software being open source, nor
is it a foregone conclusion for open source software.
For scientific software, this situation is slightly more complex, as the
adoption of open source methods are often met with resistance. In addition to
this, members of a given community related to software are just as likely – if
not more likely – to be working on similar projects, competing for mindshare
among the academic public, and even competing for funding or jobs. Building
communities that are able to thrive despite these barriers can result in
considerable, non-local benefits: the aphorism “a rising tide lifts all ships”
is nowhere more true than in community-driven scientific codes. Even the
presence of other, engaged community members means that there are more people
able to answer questions from newcomers and provide assistance and energy
toward solving problems.
The concept of openness in science amongst academics is something of a
paradox. Often, utilizing commodity tools is viewed as a very positive trait,
whereas releasing software is occasionally viewed with skepticism. While this
is changing, and in some ways quite rapidly, the idea of source code being
shared between potential competitors is still often seen as dangerous or even
anathema to scientific progress. This typically breaks along three primary
objections:111For a broader and more quantitative study, see Stodden [2010]
1. 1.
Why should I give up my competitive advantage?
2. 2.
How can I manage supporting a code?
3. 3.
What if someone finds a bug in my work?
Answering these questions in detail is beyond the scope of this document. But,
a few comments can be made. The first two objections to releasing code are
directly addressed by a community of practice. As noted above, the benefits
from collaborative, participatory communities can alleviate the burdens of
support as well as provide advantages to scientific inquiry that would
otherwise not be present. In both the Enzo and yt communities, as the
community has scaled in size, with it has scaled the number of eager, helpful
participants in contributing modules (which are then shared, collectively) as
well as provide support for problems and issues on the mailing list. When
discussing these aspects of competitive advantage and support, it is also
worthwhile to frame the discussion in terms of generalization and
specialization. As an example, in the early stages of developing a simulation
code, it’s entirely reasonable that a single individual can manage every
aspect of the code: the IO framework, parallelism, hydrodynamics, gravity, and
so on. But as the simulation code becomes larger, more general, and more
mature, it becomes unwieldy for a single person to manage and develop all
aspect of a code. Within Enzo we have seen this as the code has grown to
include many different hydrodynamic modules (including magnetohydrodynamics),
chemistry modules, radiative transport, star particle implementations, and
parallelism strategies. The overall generalization of the code base has
resulted in a deep specialization in some areas of the code. By investing in a
community of individuals, the overall management costs are reduced as
individuals specialize in different subsections of the code. This pipeline for
conducting research using a code such as Enzo shares many characteristics with
observational research; the data is constructed using the simulation platform,
relies on many pieces of diverse infrastructure such as IO libraries,
libraries for parallelism and libraries for solving differential equations,
and is then passed on to the analysis platform which processes the data to
produce results.
The third objection enumerated above is somewhat more subtle. The implicit
question is not, “What if someone finds a bug?” but really “What if someone
sees my work and is able to show that it is flawed?” This objection is the
most challenging, as it neatly aligns with the conflict between personal
advancement and the collective advancement of science. The glib, simple
solution to this would be to prioritize an incremental, layered approach to
science rather than preserving the professional benefits of previous,
potentially incorrect results. In this approach, “bugs” and flaws in results
are not hidden from view and defended from exposure, but rather found and
corrected. Unfortunately, such prioritization is a subtle form of the
prisoner’s dilemma: it is maximally successful when all members of the
community participate in revealing shortcomings and problems. The personal
solution is somewhat more subtle, and not entirely clear. From a pragmatic
perspective, identification of bugs enables higher-quality, longer-lasting
results, less likely to be overturned by subsequent investigations by
competitors.
As computational science matures, and as computational techniques permeate
every aspect of scientific inquiry, it is natural that the software utilized
in scientific inquiry grows more complex. Scientific software in many
respects, particularly in astrophysics, is thought of as something of a
second-class citizen – in years past, the concept of scientific software being
developed in isolation, placed on a website and largely abandoned became
pervasive. This methodology simply will not scale with the complexity of
projects necessary for modern scientific inquiry; we have reached the age of
advanced algorithms being applied in non-trivial ways to complex, physically-
rich datasets. The cyberinfrastructure necessary to address problems in
computational science is no longer tractably solved by individuals working in
isolation – community projects must become the new norm.
Part of the reason that communities developed around scientific codes can be
beneficial is also a component of the challenges within these communities. An
active community with participants sharing enhancements, features, and
assistance relies on the participants developing those enhancements and
understanding the code base well enough to provide assistance to less
experienced individuals. Scientific code development is strongly driven by
pragmatic, short-term needs of scientific inquiry; as a result, most
improvements are motivated by the next paper, or the next talk, or advancing a
particular line of inquiry likely to result in a publication and the
affiliated respect from peers for solving a challenging scientific problem.
The common argument against sharing advancements, techniques and tools is that
it undercuts competitive advantage. This form of individualism results not
only in researchers refraining from sharing, but more importantly, prevents
them from benefiting from the work of others. A strong community of sharing
results in a stronger technical code base, even if it has the side effect of a
moderate reduction in perceived intellectual priority for individual
researchers. If I share a technique that can be applied to Population III star
formation simulations, I can no longer be the only one to apply that technique
– and thus the only person who can answer that type of problem. I lose a small
amount of intellectual priority. However, I am now able to receive
improvements to the technique, suggestions for further enhancements, and am
thus able to extend my scientific inquiry further than before. The most
important aspect of this is that altruistic goals (sharing, reproducibility,
openness) align exactly with non-altruistic goals. It does not matter if I
share because I hope to gain something in return or because I believe in
openness; the end result is that the health of the community has been improved
by participation.
The strongest bias we have seen in yt is not against releasing of code or
contributing back, but against the notion of conducting any development at all
– whether that be contributing code, documentation or even reporting bugs.
This cognitive block comes from a fundamental misunderstanding of how
scientific codes are developed, one that we have been attempting to remedy
within our own ranks. The traditional view of open source scientific code
development has been that of two groups – developers and users. This
segregation of individuals results in an implicit, but difficult to overcome,
feeling of boundaries between responsibilities. Often responsibilities such as
verification of results, inspecting results, and tracking (and understanding)
modifications to the code base are left to “developers.” Unfortunately, all of
these responsibilities are essential components of the scientific method!
Segregating responsibilities in this way leads to misunderstandings about the
nature of scientific codes, many of which are constantly under development and
improvement, and the nature of the results that they can produce. This is even
evidenced in simple things such as using the word “user” to describe community
members, further emphasizing a distinction between people who “use” the code
and people who “make” the code. In Pawlik et al. [2012], the authors
thoughtfully describe the role of scientists in software not as a black and
white, user or developer distinction, but rather as a continuum of greys.
Often, the self-application of a term like “user,” with its connotations,
results from self-assigned roles, or perceptions of individual abilities. As
peers in a global scientific community, the distinction between “users” and
“developers” is actively harmful, when in reality scientists are tasked with
occupying that grey area in the middle.
In both the yt and Enzo communities, we have seen this behavior play out
numerous times. Those members of the community who have been the most giving
of enhancements, assistance, and technology (scripts, modules, bug fixes) are
typically those who are able to best take advantage of the technology to ask
complex questions about the physical world. Furthermore, the process of
opening up a module and contributing it to an open source software community
provides the opportunity for receiving enhancements in functionality, bug
reports, and synergistic applications of that module.
In the yt project, there are a number of individuals who are the primary
contributors and a larger number of individuals who have provided occasional
bug fixes or individual, isolated modules. The influence of non-developing
community members on code changes, roadmap efforts, and development
methodology is not only allowed, but encouraged and solicited. Furthermore, we
actively solicit contributions in the form of scripts, modules, and
enhancements or modifications to the yt codebase. This initiative has resulted
in several positive effects, all of which have strengthened yt as a community,
independent of their effects on yt as software. Individuals who do not feel
motivated to or capable of contributing in a technical sense are more likely
to contribute through helping other users, providing documentation, and even
networking and advertising the project. But the more prominent effect is a
sense of investment in the project. This sense of investment results in
quantifiable results (such as more mailing list activity, a larger number of
commits) as well as results that are more difficult to quantify, such as
positive feelings, friendly discourse, excitement, word of mouth, and so on.
## 4 Challenges
Even beyond these concerns, community building can be challenging within the
confines of the traditional realm of academia. The reward structure in
astrophysical academia, for funding, jobs and mindshare, is highly-correlated
with influence, high-impact papers, and citation counts[see also Howison and
Herbsleb, 2011]. Often developing software, or participating in academic
communities, is seen as orthogonal to these goals, even though this is not
necessarily true. In light of this, soliciting development contributions is
particularly difficult. Developing a tool in support of a publication (for
instance, a module for yt) is seen as a worthwhile use of time. Despite the
benefits that come from making that tool available, the time needed to “clean
it up” and provide a modicum of support is not as immediately beneficial, as
it does not result in an additional publication, additional citations, and so
on. Often, those contributions that would be the most beneficial to the
project as a whole are the most difficult to motivate. These secondary, yet
important, goals include improvements to documentation (particularly
narrative, instructive documentation), testing of individual components (as
opposed to large-scale integration testing), and infrastructure improvements
such as optimization and maintainability refactoring. As a result, tool
builders are not always favored by the academic reward structure. This is not
unique to computational astrophysics, but is also seen frequently within the
instrumentation community. A common pattern seen within yt is a spirited
discussion of idealistic goals for the project, but then pressing deadlines
and other local concerns typically temper enthusiasm and execution. I myself
am not only guilty of this, but perhaps the most common instigator!
The challenge to community building I find the most worrisome is that
presented by the so-called “citation economy” in astrophysics. The influence
of a given piece of code is often measured by citations to a method paper,
which is by definition a fixed and archived document. In 2011, a method paper
for yt was published. Enzo’s method paper is still under preparation, but
papers in 1997 and 2004 are often cited as references for Enzo. Because ADS
provides reverse-indexing capabilities, the number of citations to these
papers provides an immediate method of gauging the influence of the software
that they describe. In the intervening time between publication of the method
papers, new contributors have joined the collaborations and contributed
substantial and non-trivial enhancements to the code base. In fact, this is
not only what has already happened, but an explicit, named goal of the future
of these two code bases! This results in an unfortunate conflict of interests
between new contributors and existing contributors. It is in the best
interests of existing contributors, who contributed to the code base prior to
the publication of a method paper, to consolidate citations in the canonical
paper; unfortunately, it is in the best interests of new contributors to
publish a new method paper, so that they can begin to “collect” citations. In
whose interest is it to contribute to a code base without rewards mediated
through citations, one of the most common metrics of success in academia? I
believe this is among the greatest challenges to community developed codes in
astrophysics, one that will continue to grow in importance as software
projects inevitably scale beyond a handful of contributors222Alternate metrics
of software citation are being explored by, e.g. the ASCL [Allen et al.,
2012], but they primarily address discoverability of codes and do not attempt
to address the issue of distributing credit to contributors..
## 5 Strategies
Not all of the challenges enumerated above can be addressed directly. However,
in both the yt and Enzo communities, we have addressed the technical and
social challenges through conscious development of infrastructure and
techniques. These have been designed to foster good will, encourage
contributions, and build social capital between contributors. Below, I list
several steps we have taken over the last several years toward the goal of
growing and engaging communities of practice.333A related work is Prlić and
Procter [2012] where the authors have compiled a set of very useful and
helpful guidelines for scientific software development.
Most important, however, these need to be viewed as the investment they are.
These strategies, particularly the social strategies, require not only focused
energy and time, but an emotional investment. You must design the community
you want. This design extends far beyond designing software and algorithms; it
includes thinking about the diversity of contributors and community members,
the tone of discourse within the community, the projected enthusiasm within
the project, and even the congeniality with which feedback – especially
critical feedback – is received. The culture seeded in a community will self-
propagate; whether this is a culture of neglect, a culture of homogeneity, a
culture of kindness, a culture of brash arrogance or even a culture of
openness, this will flow outward from the core[Bacon, 2009, Trapani, 2011,
Allsopp, 2012]. Both Enzo and yt have attempted to build cultures that promote
respect and excitement.
### 5.1 Technical Infrastructure
Within both yt and Enzo, we utilize several pieces of technical infrastructure
that ease the process of growing communities [see also Fogel, 2005]. These
include open and freely-joinable mailing lists, a completely open development
process based on the distributed version control system (DVCS) mercurial, and
a code review and mentoring process to ensure contributions are high-quality,
with a minimum of friction. Furthermore, we utilize methods of communication
that enable participation and that are tuned for low- to high-latency
interaction.
Both yt and Enzo have seen an enormous growth in contributions following
migration to DVCS; this is not unique to these projects, but is a hallmark of
the lowered barrier to entry presented by DVCS. In contrast to centralized
version control systems such as CVS and Subversion, DVCS systems are fully-
distributed. Every clone, or checkout, of a repository is a peer with every
other clone; this enables anyone who checks out a repository to track their
own changes to the repository, and provides the (technical) ability to much
more easily contribute changes upstream. The unique versioning of local copies
of a code base also allow unique revision specifiers to be applied globally;
rather than “version 4.2 with modifications (unspecified)” code that generates
a given result can be uniquely identified with a globally-unique hash. Both yt
and Enzo are hosted on BitBucket, a code hosting provider (roughly
functionally equivalent with others such as RhodeCode, GitHub, Gitorious and
Savannah). It provides technical infrastructure for code review, accepting
changes, tracking contributions and identifying specific revisions of the
code. By using this system we lower the barrier to entry for contributors,
enabling submission of locally-developed changes. By using DVCS, we encourage
scientists using yt or Enzo to track their own changes to it. Even if this
code is never submitted for inclusion in the primary repository, this enables
provenance tracking and debugging.
Development of scientific software also requires several difference degrees of
communication; as the software itself is not the only project contributors are
working on, we have found that communication is naturally divided into three
categories, based on latency and urgency. Immediate communication, such as in-
person meetings (or “code sprints”) or Google Hangouts, are useful for low-
latency planning, discussion, and collaboration. Tools such as Internet Relay
Chat (IRC) provide a means of medium-latency communication; often in the #yt
channel on FreeNode, an IRC network, between 10 and 20 people can be found at
any time. Messages left here are usually responded to relatively quickly,
which leads to a low-latency discussion where problems can be resolved much
more quickly than through email. Finally, nearly all planning and detailed
discussion happens through high-latency mediums such as comments on code
changes and mailing lists. The mailing lists for yt (which are titled the
unfortunately-chosen names yt-users and yt-dev) are open, indexed by Google,
and freely joinable. We have often (successfully) encouraged discussions here
to turn into collaborations and code sharing, and in fact we have had a number
of successful opportunities for long-term planning and invasive code changes
here. They serve not only as broadcast media, but as venues for soliciting
input and contributions.
Nearly as important as providing mechanisms for communication is reducing the
barrier to entry for new contributors. This includes ensuring a smooth process
to finding the appropriate location in the source code, making changes,
submitting those changes for review, and a kind, thoughtful mentoring process
for new code contributors. The most common path we have seen for new code
contributors is relatively straightforward:
1. 1.
Individual applies software package to meet own goals
2. 2.
Individual develops a modification to the code, for either an enhancement, a
bug fix, or a contribution of an example
3. 3.
This change is submitted for inclusion in the main yt repository
4. 4.
The change is read, reviewed and tested by community members to ensure that
standards for coding, documentation and testing are all met.
5. 5.
The individual begins to participate in the community
The main codebase is not suitable for contributions of all types; we have
found individuals eager to contribute scripts that exist as supplemental
tools, or that were used in the writing of a paper. To support this desire, we
provide a location (https://hub.yt-project.org/) to submit these scripts as
well, where links to external source code repositories are collected alongside
descriptions of the individual projects. In this way, by providing mechanisms
and structures, as well as an open and active solicitation for technical
contributions, we have found that we can foster a sense of community and
accomplishment among individuals.
We have found that even minor barriers to contribution (or even software
deployment!) can build up a “technical friction” to participation that results
in losing contributions and community members. In an effort to alleviate this,
we provide easy access to the source code, detailed instructions for
contributing code, a mechanism for communication, and scripts that assist with
the entire process. We have found that by ensuring that we have a uniform
review process (where even code written by founding members of the community
is reviewed) we ensure that the code is of acceptable quality, tests are
included, and that undocumented code is discouraged.
### 5.2 Social Techniques
In Fitzpatrick and Collins-Sussman [2012] the authors describe a strategy for
social interactions among so-called “geek” teams. I will not provide a
detailed retelling of this, but despite focus on applying the techniques
described in the book to software engineers, in many ways they can be applied
directly to collaborations around scientific software projects. The authors
enumerate three characteristics which they suggest applying to all
communication and interaction:
* •
Humility
* •
Respect
* •
Trust
Communication by text, in particular about technical topics, is often stripped
of the nuances and inflections that convey emotion. In textual media,
therefore, it is even more important to guide discussion in a way that
encourages participation. Within the yt community, we are very careful to
ensure that the tones on the mailing list, code review and in IRC should be
conducted in this way: the health of a community can be judged by how it
treats newcomers and novices as well as how it treats experienced, advanced
participants. Even in framing analysis interactions between peers in this way,
one can see how communication can serve the growth of scientific communities.
In scientific software communities, an aspect that often goes unmentioned is
that we arguably want to foster discussions between peers, not discussions
between an elite class and a subservient class. By guiding the discussion with
a focus on humility, respect and trust (HRT as abbreviated by Fitzpatrick and
Collins-Sussman), discussions can become more congenial, more productive, and
can lead to a greater spirit of collective focus.
Humility can be well-characterized in how a community responds to negative
feedback. In a community in which I participate, I witnessed a discussion
between an experienced community member and a newcomer. The newcomer reported
what they thought was a bug in a particular routine. The experienced community
member, who had made the change that introduced the potential bug, responded
with a very abrupt, dismissive response. “It’s like that for a very good
reason. Don’t touch it.” By terminating discussion at that point, it sent the
clear signal that a discussion of the reasoning behind a code change was
simply not necessary; the message was not only that the reasoning was beyond
reproach, but that a discussion was not worth the time it took to have. The
better solution, of an open and humble discussion of the background and
reasoning behind the decision, would have been to engage the newcomer and
explain the reasoning behind a decision. This takes an investment of time, but
certainly no more than the investment of time it would take to justify such a
decision to a referee or journal reviewer, and the potential gains in this
case would be a new contributor and an increase in social capital amongst the
community.
Respect can be seen in how peers contribute to the development of the code. In
principle, when developing a scientific code, we are engaging in scientific
discourse, attempting to push the field of astrophysics forward in both our
understanding of and our ability to ask questions of the natural world. One of
the most common ways I’ve seen this ignored is when a person writes to a
mailing list or appears on IRC and says some variation of “I’ve noticed
something is acting strangely with …” The all-too-common response is simply,
“You’re doing it wrong,” or even worse, a dismissive instruction to “Just read
the documentation.” This is rarely even prefaced with the obvious question of,
“How did you expect it to act?” It terminates discussion and discourages
individuals from returning. This has the direct effect of reducing
participation, and disproportionately impacts new contributors. A lack of
respect can infect and transform a community into an unfriendly, unwelcoming
environment; the most valuable attribute a community of scientific software
developers has is its ability to engage in scientific practice, and that is
obstructed by a lack of respect.
Trust within scientific software communities is perhaps more subtle. What we
have seen with yt and Enzo is not that trust is an unwavering faith in someone
to produce high-quality, scientifically correct code, but a trust in the
process and the stewardship of a code. Within communities of scientific
software, we have seen this evidenced when contributors let go of a project.
As a community matures, individuals cannot contribute to the breadth of sub-
projects that they did when it was young; furthermore, as scientific interests
shift, the leader of an individual project or aspect of the scientific code
may move on to other things. Trusting that the community will be able to
steward and shepherd that project is a crucial ingredient in addressing
sustainability and burnout, and an essential ingredient in expressing trust
for peers and other community members.
The motion of projects between individuals is a difficult social situation. We
attempt to address this in yt by emphasizing pride over ownership. By changing
the conversation from dominion to stewardship, this changes how individuals
approach projects, and helps enable them to regard external contributions in a
positive way, rather than as threats to their accomplishments.
HRT are not always easy attributes to focus on. Particularly in a competitive
field such as astrophysics, it can be difficult to spend time thinking about
how words or conversations will be perceived, particularly when these
conversations are between potential competitors. But by remaining focused on
engaging community members as peers, treating them with HRT and spending time
and energy on thoughtful communication, the community will be more likely to
grow and flourish.
## 6 Conclusions
As computational science has grown both more complex and more pervasive,
communities of practice surrounding scientific software projects have become
essential to the health and vibrancy of computational science projects.
Unfortunately, the academic reward system does not always align with the
development of software projects, which can lead to stagnation or corner
cutting in important tasks such as infrastructure, documentation and testing.
This largely results from the overlap between the traditional roles of
developers and users in software development, a distinction that can in fact
actively harm the scientific process.
To grow communities around any software project, the structure and character
of the community itself must be carefully considered; you must design this
community. This includes setting the tone of discourse, consciously fostering
a diverse set of participants, and being open and enthusiastic. This process
can be assisted with technical infrastructure. This includes using distributed
version control systems, conducting open communication in media that suit the
style of communication, and streamlining the participation process and
providing a clear method of contributing. Technical infrastructure alone
cannot “solve” the development of a community; social infrastructure and
standards of conduct must also be developed. Conducting business while
remaining focused on humility, respect and trust can help to ensure that
newcomers feel welcomed, existing contributors and peers feel validated and
respected, and that ultimately as interests change over time projects do not
stagnate under the weight of code or project ownership.
A vibrant, active community brings sustainable development, synergistic
applications and development of code, and considerable rewards in potential
collaborations and discourse. I am grateful for the opportunity to have
participated in both the yt and Enzo communities, and grateful for the
concrete rewards – scientific, social and technical – that my participation
has allowed me.
As a closing note, despite all of the successes we have seen in the yt and
Enzo communities, we still have considerable room to grow from both the social
and technical standpoints. I hope that as communities driven both by long-
reaching goals and the pragmatic needs of scientific inquiry, we can continue
to remain healthy, vibrant and growing. I believe that the single biggest
problem within scientific software communities is ensuring credit for
community participation can be shared with new contributors. This affects
motivations at essentially every level, and because of its very nature can
dramatically affect the health and stunt the growth of scientific software
communities.
## Acknowledgments
I would like to thank the organizers of the Texas Scientific Software Days
conference, Victor Eijkhout, Sergey Fomel, Andy R. Terrel, and Michael Tobis
for a wonderful conference full of excellent talks. I’d also like to thank the
other attendees and speakers for fascinating and enlightening conversations
about scientific software. I am supported by an NSF CI TraCS fellowship
(OCI-1048505). Any opinions, findings, and conclusions or recommendations
expressed in this publication are my own and do not necessarily reflect the
views of the National Science Foundation.
I would also like to thank the members of both the yt and Enzo communities.
Much of this talk – and approach to community building – was developed in
collaboration with them. I’m grateful for feedback provided by Nathan Goldbaum
and Brian O’Shea on a draft of this manuscript. In addition, I would like to
thank Britton Smith and Greg Bryan both for extremely valuable feedback in
preparing this paper and for leading by example in community development for
scientific codes.
## References
* Allen et al. [2012] A. Allen, P. Teuben, R. J. Nemiroff, and L. Shamir. Practices in Code Discoverability: Astrophysics Source Code Library. In P. Ballester, D. Egret, and N. P. F. Lorente, editors, _Astronomical Data Analysis Software and Systems XXI_ , volume 461 of _Astronomical Society of the Pacific Conference Series_ , page 627, September 2012.
* Allsopp [2012] John Allsopp. The proof of the pudding, November 2012. URL http://www.webdirections.org/blog/the-proof-of-the-pudding/.
* Bacon [2009] Jono Bacon. _The Art of Community. Building the New Age of Participation._ 2009\.
* Fitzpatrick and Collins-Sussman [2012] Brian W. Fitzpatrick and Ben Collins-Sussman. _Team geek : a software developer’s guide to working well with others_. O’Reilly, Sebastopol. CA, 2012. ISBN 978-1-449-30244-3.
* Fogel [2005] Karl Fogel. _Producing Open Source Software: How to Run a Successful Free Software Project_. O’Reilly Media, Inc., October 2005. ISBN 0596007590.
* Howison and Herbsleb [2011] James Howison and James D. Herbsleb. Scientific software production: incentives and collaboration. In Pamela J. Hinds, John C. Tang, Jian Wang, Jakob E. Bardram, and Nicolas Ducheneaut, editors, _CSCW_ , pages 513–522. ACM, 2011. ISBN 978-1-4503-0556-3.
* Pawlik et al. [2012] A. Pawlik, J. Segal, and M. Petre. Documentation practices in scientific software development. In _Cooperative and Human Aspects of Software Engineering (CHASE), 2012 5th International Workshop on_ , pages 113 –119, june 2012. doi: 10.1109/CHASE.2012.6223004.
* Prlić and Procter [2012] Andreas Prlić and James B. Procter. Ten simple rules for the open development of scientific software. _PLoS Computational Biology_ , 8(12):e1002802+, December 2012. ISSN 1553-7358. doi: 10.1371/journal.pcbi.1002802. URL http://dx.doi.org/10.1371/journal.pcbi.1002802.
* Stodden [2010] Victoria Stodden. The scientific method in practice: Reproducibility in the computational sciences. Technical Report Working Paper 4773-10, MIT Sloan School of Management, February 2010. URL http://ssrn.com/abstract=1550193.
* Szczepanski et al. [2012] A. Szczepanski, T. Baer, Y. Mack, J. Huang, and S. Ahern. Usage of a hpc data analysis and visualization system. _Computer_ , PP(99):1, 2012. ISSN 0018-9162. doi: 10.1109/MC.2012.192.
* Trapani [2011] Gina Trapani. Your community is your best feature, April 2011. URL http://smarterware.org/7819/my-codeconf-talk-your-community-is-your-bes%t-feature.
|
arxiv-papers
| 2013-01-29T21:00:01 |
2024-09-04T02:49:40.923103
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Matthew J. Turk",
"submitter": "Matthew Turk",
"url": "https://arxiv.org/abs/1301.7064"
}
|
1301.7084
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-005 LHCb-PAPER-2012-044 29 January 2013
Exclusive ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$
production in pp collisions at $\sqrt{s}=7$ $\mathrm{\,Te\kern-2.07413ptV}$
The LHCb collaboration†††Authors are listed on the following pages.
Exclusive ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$
vector meson production has been observed in the dimuon channel using the LHCb
detector. The cross-section times branching fractions to two muons with
pseudorapidities between 2.0 and 4.5 are measured to be
$\sigma_{pp\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(\rightarrow\mu^{+}\mu^{-})}(2.0<\eta_{\mu^{\pm}}<4.5)=307\pm 21\pm
36~{}\text{pb},$
$\sigma_{pp\rightarrow\psi{(2S)}(\rightarrow\mu^{+}\mu^{-})}(2.0<\eta_{\mu^{\pm}}<4.5)=7.8\pm
1.3\pm 1.0~{}\text{pb},$
where the first uncertainties are statistical and the second are systematic.
The measurements are found to be in good agreement with results from previous
experiments and theoretical predictions. The
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ photoproduction cross-section
has been measured as a function of the photon-proton centre-of-mass energy.
The results are consistent with measurements obtained at HERA and confirm a
similar power law behaviour for the photoproduction cross-section.
To be submitted to Journal of Physics G
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht9, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves
Jr22,35, S. Amato2, Y. Amhis7, L. Anderlini17,f, J. Anderson37, R.
Andreassen57, R.B. Appleby51, O. Aquines Gutierrez10, F. Archilli18, A.
Artamonov 32, M. Artuso53, E. Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J.
Back45, C. Baesso54, V. Balagura28, W. Baldini16, R.J. Barlow51, C.
Barschel35, S. Barsuk7, W. Barter44, Th. Bauer38, A. Bay36, J. Beddow48, I.
Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28, E. Ben-Haim8, M.
Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, A. Berezhnoy29, R.
Bernet37, M.-O. Bettler44, M. van Beuzekom38, A. Bien11, S. Bifani12, T.
Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36, C. Blanks50,
J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N. Bondar27, W.
Bonivento15, S. Borghi51, A. Borgia53, T.J.V. Bowcock49, E. Bowen37, C.
Bozzi16, T. Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M.
Britsch10, T. Britton53, N.H. Brook43, H. Brown49, I. Burducea26, A.
Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo
Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14,c, G. Carboni21,k, R.
Cardinale19,i, A. Cardini15, H. Carranza-Mejia47, L. Carson50, K. Carvalho
Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, P. Chen3,36, N. Chiapolini37, M. Chrzaszcz 23, K. Ciba35, X.
Cid Vidal34, G. Ciezarek50, P.E.L. Clarke47, M. Clemencic35, H.V. Cliff44, J.
Closier35, C. Coca26, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins35, A.
Comerma-Montells33, A. Contu15, A. Cook43, M. Coombes43, G. Corti35, B.
Couturier35, G.A. Cowan36, D. Craik45, S. Cunliffe50, R. Currie47, C.
D’Ambrosio35, P. David8, P.N.Y. David38, I. De Bonis4, K. De Bruyn38, S. De
Capua51, M. De Cian37, J.M. De Miranda1, L. De Paula2, W. De Silva57, P. De
Simone18, D. Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C.
Deplano15, D. Derkach14, O. Deschamps5, F. Dettori39, A. Di Canto11, J.
Dickens44, H. Dijkstra35, P. Diniz Batista1, M. Dogaru26, F. Domingo
Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil Suárez34, D. Dossett45, A.
Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A. Dziurda23, A. Dzyuba27, S.
Easo46,35, U. Egede50, V. Egorychev28, S. Eidelman31, D. van Eijk38, S.
Eisenhardt47, U. Eitschberger9, R. Ekelhof9, L. Eklund48, I. El Rifai5, Ch.
Elsasser37, D. Elsby42, A. Falabella14,e, C. Färber11, G. Fardell47, C.
Farinelli38, S. Farry12, V. Fave36, D. Ferguson47, V. Fernandez Albor34, F.
Ferreira Rodrigues1, M. Ferro-Luzzi35, S. Filippov30, C. Fitzpatrick35, M.
Fontana10, F. Fontanelli19,i, R. Forty35, O. Francisco2, M. Frank35, C.
Frei35, M. Frosini17,f, S. Furcas20, E. Furfaro21, A. Gallas Torreira34, D.
Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J. Garofoli53, P. Garosi51,
J. Garra Tico44, L. Garrido33, C. Gaspar35, R. Gauld52, E. Gersabeck11, M.
Gersabeck51, T. Gershon45,35, Ph. Ghez4, V. Gibson44, V.V. Gligorov35, C.
Göbel54, D. Golubkov28, A. Golutvin50,28,35, A. Gomes2, H. Gordon52, M.
Grabalosa Gándara5, R. Graciani Diaz33, L.A. Granado Cardoso35, E. Graugés33,
G. Graziani17, A. Grecu26, E. Greening52, S. Gregson44, O. Grünberg55, B.
Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53, G. Haefeli36,
C. Haen35, S.C. Haines44, S. Hall50, T. Hampson43, S. Hansmann-Menzemer11, N.
Harnew52, S.T. Harnew43, J. Harrison51, P.F. Harrison45, T. Hartmann55, J.
He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van
Herwijnen35, E. Hicks49, D. Hill52, M. Hoballah5, C. Hombach51, P. Hopchev4,
W. Hulsbergen38, P. Hunt52, T. Huse49, N. Hussain52, D. Hutchcroft49, D.
Hynds48, V. Iakovenko41, P. Ilten12, R. Jacobsson35, A. Jaeger11, E. Jans38,
F. Jansen38, P. Jaton36, F. Jing3, M. John52, D. Johnson52, C.R. Jones44, B.
Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35, T.M. Karbach35, I.R.
Kenyon42, U. Kerzel35, T. Ketel39, A. Keune36, B. Khanji20, O. Kochebina7, I.
Komarov36,29, R.F. Koopman39, P. Koppenburg38, M. Korolev29, A. Kozlinskiy38,
L. Kravchuk30, K. Kreplin11, M. Kreps45, G. Krocker11, P. Krokovny31, F.
Kruse9, M. Kucharczyk20,23,j, V. Kudryavtsev31, T. Kvaratskheliya28,35, V.N.
La Thi36, D. Lacarrere35, G. Lafferty51, A. Lai15, D. Lambert47, R.W.
Lambert39, E. Lanciotti35, G. Lanfranchi18,35, C. Langenbruch35, T. Latham45,
C. Lazzeroni42, R. Le Gac6, J. van Leerdam38, J.-P. Lees4, R. Lefèvre5, A.
Leflat29,35, J. Lefrançois7, O. Leroy6, Y. Li3, L. Li Gioi5, M. Liles49, R.
Lindner35, C. Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E.
Lopez Asamar33, N. Lopez-March36, H. Lu3, J. Luisier36, H. Luo47, F.
Machefert7, I.V. Machikhiliyan4,28, F. Maciuc26, O. Maev27,35, S. Malde52, G.
Manca15,d, G. Mancinelli6, N. Mangiafave44, U. Marconi14, R. Märki36, J.
Marks11, G. Martellotti22, A. Martens8, L. Martin52, A. Martín Sánchez7, M.
Martinelli38, D. Martinez Santos39, D. Martins Tostes2, A. Massafferri1, R.
Matev35, Z. Mathe35, C. Matteuzzi20, M. Matveev27, E. Maurice6, A.
Mazurov16,30,35,e, J. McCarthy42, R. McNulty12, B. Meadows57,52, F. Meier9, M.
Meissner11, M. Merk38, D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54,
S. Monteil5, D. Moran51, P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47,
K. Müller37, R. Muresan26, B. Muryn24, B. Muster36, P. Naik43, T. Nakada36, R.
Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35, A.D. Nguyen36, T.D.
Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, R. Niet9, N. Nikitin29, T.
Nikodem11, S. Nisar56, A. Nomerotski52, A. Novoselov32, A. Oblakowska-Mucha24,
V. Obraztsov32, S. Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35,
M. Orlandea26, J.M. Otalora Goicochea2, P. Owen50, B.K. Pal53, A. Palano13,b,
M. Palutan18, J. Panman35, A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J.
Parkinson50, G. Passaleva17, G.D. Patel49, M. Patel50, G.N. Patrick46, C.
Patrignani19,i, C. Pavel-Nicorescu26, A. Pazos Alvarez34, A. Pellegrino38, G.
Penso22,l, M. Pepe Altarelli35, S. Perazzini14,c, D.L. Perego20,j, E. Perez
Trigo34, A. Pérez-Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G.
Pessina20, K. Petridis50, A. Petrolini19,i, A. Phan53, E. Picatoste Olloqui33,
B. Pietrzyk4, T. Pilař45, D. Pinci22, S. Playfer47, M. Plo Casasus34, F.
Polci8, G. Polok23, A. Poluektov45,31, E. Polycarpo2, D. Popov10, B.
Popovici26, C. Potterat33, A. Powell52, J. Prisciandaro36, V. Pugatch41, A.
Puig Navarro36, W. Qian4, J.H. Rademacker43, B. Rakotomiaramanana36, M.S.
Rangel2, I. Raniuk40, N. Rauschmayr35, G. Raven39, S. Redford52, M.M. Reid45,
A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, V. Rives
Molina33, D.A. Roa Romero5, P. Robbe7, E. Rodrigues51, P. Rodriguez Perez34,
G.J. Rogers44, S. Roiser35, V. Romanovsky32, A. Romero Vidal34, J. Rouvinet36,
T. Ruf35, H. Ruiz33, G. Sabatino22,k, J.J. Saborido Silva34, N. Sagidova27, P.
Sail48, B. Saitta15,d, C. Salzmann37, B. Sanmartin Sedes34, M. Sannino19,i, R.
Santacesaria22, C. Santamarina Rios34, E. Santovetti21,k, M. Sapunov6, A.
Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D. Savrina28,29, P.
Schaack50, M. Schiller39, H. Schindler35, S. Schleich9, M. Schlupp9, M.
Schmelling10, B. Schmidt35, O. Schneider36, A. Schopper35, M.-H. Schune7, R.
Schwemmer35, B. Sciascia18, A. Sciubba18,l, M. Seco34, A. Semennikov28, K.
Senderowska24, I. Sepp50, N. Serra37, J. Serrano6, P. Seyfert11, M. Shapkin32,
I. Shapoval40,35, P. Shatalov28, Y. Shcheglov27, T. Shears49,35, L.
Shekhtman31, O. Shevchenko40, V. Shevchenko28, A. Shires50, R. Silva
Coutinho45, T. Skwarnicki53, N.A. Smith49, E. Smith52,46, M. Smith51, K.
Sobczak5, M.D. Sokoloff57, F.J.P. Soler48, F. Soomro18,35, D. Souza43, B.
Souza De Paula2, B. Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S.
Stahl11, O. Steinkamp37, S. Stoica26, S. Stone53, B. Storaci37, M.
Straticiuc26, U. Straumann37, V.K. Subbiah35, S. Swientek9, V. Syropoulos39,
M. Szczekowski25, P. Szczypka36,35, T. Szumlak24, S. T’Jampens4, M.
Teklishyn7, E. Teodorescu26, F. Teubert35, C. Thomas52, E. Thomas35, J. van
Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39, D. Tonelli35, S. Topp-
Joergensen52, N. Torr52, E. Tournefier4,50, S. Tourneur36, M.T. Tran36, M.
Tresch37, A. Tsaregorodtsev6, P. Tsopelas38, N. Tuning38, M. Ubeda Garcia35,
A. Ukleja25, D. Urner51, U. Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez
Gomez33, P. Vazquez Regueiro34, S. Vecchi16, J.J. Velthuis43, M. Veltri17,g,
G. Veneziano36, M. Vesterinen35, B. Viaud7, D. Vieira2, X. Vilasis-
Cardona33,n, A. Vollhardt37, D. Volyanskyy10, D. Voong43, A. Vorobyev27, V.
Vorobyev31, C. Voß55, H. Voss10, R. Waldi55, R. Wallace12, S. Wandernoth11, J.
Wang53, D.R. Ward44, N.K. Watson42, A.D. Webber51, D. Websdale50, M.
Whitehead45, J. Wicht35, J. Wiechczynski23, D. Wiedner11, L. Wiggers38, G.
Wilkinson52, M.P. Williams45,46, M. Williams50,p, F.F. Wilson46, J. Wishahi9,
M. Witek23, S.A. Wotton44, S. Wright44, S. Wu3, K. Wyllie35, Y. Xie47,35, F.
Xing52, Z. Xing53, Z. Yang3, R. Young47, X. Yuan3, O. Yushchenko32, M.
Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang53, W.C. Zhang12, Y. Zhang3,
A. Zhelezov11, A. Zhokhov28, L. Zhong3, A. Zvyagin35.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25National Center for Nuclear Research (NCBJ), Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
56Institute of Information Technology, COMSATS, Lahore, Pakistan, associated
to 53
57University of Cincinnati, Cincinnati, OH, United States, associated to 53
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pMassachusetts Institute of Technology, Cambridge, MA, United States
## 1 Introduction
Exclusive vector meson production through photoproduction, $\gamma
p\rightarrow Vp$, has attracted much interest both experimentally and
theoretically as it provides a rich testing ground for QCD. At sufficiently
high meson masses, perturbative QCD (pQCD) can be used to predict the
production cross-section [1], but as the masses decrease this approach ceases
to work due to non-perturbative effects. Light vector meson production is best
described by Regge theory [2], which models the process using soft pomeron
exchange. This theory predicts a cross-section which is almost flat with
respect to $W$, the photon-proton centre-of-mass energy. In contrast, the
cross-section for exclusive ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
production has been observed to have a strong power law dependence on $W$ [3,
4]. This feature is described in pQCD by a hard pomeron (or two gluon
exchange) and the fact that the gluon density in the proton increases rapidly
with decreasing $x$, the fractional momentum of the proton carried by the
parton [1]. A consistent description of the transition regime between
perturbative and non-perturbative QCD is a challenge.
The cross-section for exclusive ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
and $\psi{(2S)}$ production is calculable within pQCD though with large
uncertainties. The simplest leading-order calculation is proportional to
$[\alpha_{S}(m_{V}^{2}/4)xg(x,m_{V}^{2}/4)]^{2}$, where $m_{V}$ is the vector
meson mass, $\alpha_{S}(m_{V}^{2}/4)$ is the strong coupling constant and
$g(x,m_{V}^{2}/4)$ is the gluon parton density function (PDF) evaluated at the
relevant scale, $m_{V}^{2}/4$. Hence a measurement of these processes at the
LHC can constrain the gluon PDF [1].
More exotic effects can also be searched for in this process. The forward
acceptance of the LHCb detector is sensitive to $W$ values between 10 and 2000
GeV and allows the gluon density to be probed down to $x=5\times 10^{-6}$ [5],
lower than any previous experiment at a scale of a few
$\mathrm{\,Ge\kern-1.00006ptV}$. At a sufficiently low value of $x$, the
strong growth of the gluon density with decreasing $x$ is expected to be
restrained by gluon recombination [6]. This saturation effect could be visible
as a reduction in the photoproduction cross-section with respect to the power
law behaviour at $W$ values accessible at the LHC [7]. Exclusive vector meson
production also provides a promising channel to investigate the existence of
the odderon [8].
The HERA experiments measured ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
photoproduction using electron-proton collisions in the range 20 $\leq W\leq$
305 GeV [3, 4]. The data are consistent with a power law dependence of the
form $\sigma=a(W/1\mathrm{\,Ge\kern-1.00006ptV})^{\delta}$ with $a=3$ nb and
$\delta=0.72$ [7]. The CDF experiment measured exclusive
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production in proton-antiproton
collisions in a similar kinematic range as HERA and found consistent results
[9].
This paper presents measurements of the cross-section times branching
fractions for exclusive ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$\psi{(2S)}$ mesons to produce two muons in the pseudorapidity range
$2.0<\eta_{\mu^{\pm}}<4.5$ at a proton-proton centre-of-mass energy of
$\sqrt{s}=7$ $\mathrm{\,Te\kern-1.00006ptV}$. The results are compared with
previous experiments and a number of theoretical models. The SuperChic [10]
and Starlight [11] models use a parameterisation of the HERA results to
predict the photoproduction cross-section at the LHC. Motyka and Watt [7] use
an equivalent photon approximation combined with a dipole model, which is
found to reproduce the main features of the HERA data. Gonçalves and Machado
[12] use a colour dipole approach and the Colour Glass Condensate formalism
which also agrees with the HERA data. Schäfer and Szczurek [13] use an
explicit dynamical model of the photoproduction amplitude, which is evaluated
in terms of an unintegrated gluon distribution. This model predicts a cross-
section which is higher than that observed at HERA.
## 2 Detector and data samples
The LHCb detector [14] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$ (forward region), designed for the study of
particles containing $b$ or $c$ quarks. The detector includes a high precision
tracking system consisting of a silicon-strip vertex detector (VELO)
surrounding the proton-proton interaction region, a large-area silicon-strip
detector located upstream of a dipole magnet with a bending power of about
$4{\rm\,Tm}$, and three stations of silicon-strip detectors and straw drift-
tubes placed downstream. The combined tracking system has a momentum
resolution $\Delta p/p$ that varies from 0.4$\%$ at
5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6$\%$ at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an impact parameter resolution
of 20$\,\upmu\rm m$ for tracks with high transverse momentum. Charged hadrons
are identified using two ring-imaging Cherenkov detectors. Photon, electron
and hadron candidates are identified by a calorimeter system consisting of
scintillating-pad (SPD) and pre-shower detectors, an electromagnetic
calorimeter and a hadronic calorimeter. The SPD also provides a measurement of
the charged particle multiplicity in an event. Muons are identified by a muon
system composed of alternating layers of iron and multiwire proportional
chambers. The trigger consists of a hardware stage, based on information from
the calorimeter and muon systems, followed by a software stage which applies a
full event reconstruction. The VELO also has sensitivity to charged particles
with momenta above $\sim$100${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ in the
pseudorapidity range $-3.5<\eta<-1.5$ (backward region) and extends the
sensitivity of the forward region to $1.5<\eta<5$.
The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ mesons are
identified through their decay to two muons. The protons are only marginally
deflected and remain undetected inside the beam pipe. Therefore the signature
for exclusive vector meson production is an event containing two muons and no
other activity. The analysis is performed using 36 $\mbox{\,pb}^{-1}$ of
proton-proton collision data collected at LHCb in 2010. In this data taking
period, the average number of interactions per bunch crossing varied up to a
maximum of about 2.5, with most luminosity accumulated at larger values. In
order to identify exclusive candidates, the analysis is restricted to events
with a single interaction.
Dedicated Monte Carlo (MC) generators have been used to produce signal and
background events which are passed through a GEANT4 [15] based detector
simulation, the trigger emulation and the event reconstruction chain of the
LHCb experiment. Two generators have been used to produce samples of exclusive
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$: Starlight
[11] and SuperChic [10]. A sample of $\chi_{c}$ production by double pomeron
fusion, which forms a background for the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ analysis, has been produced
with SuperChic.
## 3 Event selection
The hardware trigger used in this analysis requires a single muon track with
transverse momentum $p_{\rm T}$ $>$ 400${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$,
or two muon tracks both with $p_{\rm T}$ $>$
80${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$, in coincidence with a low SPD
multiplicity ($<$ 20 hits). The software trigger used requires either a dimuon
with invariant mass greater than
2.9${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, or a dimuon with invariant mass
greater than 1${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ with a distance of
closest approach of the two muons to each other less than 150$\,\upmu\rm m$
and a dimuon $p_{\rm T}$ less than 900${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$.
The selection of exclusive events begins with the requirement of two
reconstructed muons in the forward region. It is also required that there are
no other tracks and no photons in the detector. Thus rapidity gaps, regions
devoid of reconstructed charged and neutral objects, are defined. Requiring
just two tracks, produced from a single particle decay, ensures two rapidity
gaps which sum to 3.5 units in the forward region. An additional rapidity gap
is obtained by requiring that there are no tracks in the backward region. The
VELO is able to exclude tracks within a certain rapidity gap depending on the
$z$ position from which the tracks originate and the event topology. The mean
backward rapidity gap where tracks are excluded is 1.7 with a root mean square
of about 0.5. Figure 1 shows that requiring a backward rapidity gap affects
the distribution of tracks in the forward region; with a backward rapidity gap
required, there is a clear peak at precisely two forward tracks. These are
candidates for exclusive production.
$\begin{array}[]{cc}\includegraphics[width=216.81pt]{Fig1a.pdf}&\includegraphics[width=216.81pt]{Fig1b.pdf}\\\
\vspace*{-1.0cm}\hfil\end{array}$
Figure 1: Number of tracks in the forward region for dimuon triggered events
which in the backward region have (left) one or more tracks or (right) no
tracks.
Both muons are required to be in the pseudorapidity range
$2.0<\eta_{\mu^{\pm}}<4.5$. Muon pair candidates, having invariant masses
within 65${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ mass values
[16], are selected.
### 3.1 Non-resonant background determination
A number of background processes have been considered including a non-resonant
contribution, inclusive prompt charmonium production, inelastic
photoproduction and exclusive $\chi_{c}$ and
$\psi{(2S)}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}+X$
productions. The non-resonant background is evaluated by fitting the dimuon
invariant-mass distribution, parameterizing the resonances with a Crystal Ball
function [17] and the continuum with an exponential function. Figure 2
displays the fit results. The non-resonant background is estimated to account
for (0.8 $\pm$ 0.1)$\%$ and (16 $\pm$ 3)$\%$ of the events within
65${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ mass values,
respectively.
$\begin{array}[]{cc}\includegraphics[width=216.81pt]{Fig2a.pdf}&\includegraphics[width=216.81pt]{Fig2b.pdf}\\\
\vspace*{-1.0cm}\hfil\end{array}$
Figure 2: Invariant mass distributions in the regions of (left) the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and (right) $\psi{(2S)}$ mass
peaks for events with exactly two tracks, no photons and a dimuon with $p_{\rm
T}$ below 900${\mathrm{\,Me\kern-0.90005ptV\\!/}c}$. The overall fits to the
data are shown by the full curves while the dashed curves show the background
contributions. Figure 3: Feynman diagrams displaying (a) exclusive
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ photoproduction and (b)
inelastic ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ photoproduction where
a small number of additional particles are produced due to gluon radiation and
(c,d) proton dissociation.
### 3.2 Inelastic background determination
The requirement of two tracks and no other visible activity enriches the
sample in exclusive events. However, this does not guarantee that there is no
other activity in the regions outside the LHCb acceptance. The contributions
from two non-exclusive processes have been considered: inclusive prompt
charmonium produced through colour strings, which leads to large numbers of
additional particles; and inelastic ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ photoproduction (as shown in Fig. 3), where gluon radiation or proton
dissociation lead to a small number of additional particles. The former has
been evaluated using simulated samples of prompt charmonium, generated using
PYTHIA [18] and normalised using a data sample where the zero backward tracks
requirement has been removed; it is found to be negligible.
The inelastic photoproduction of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ mesons is the dominant background in this analysis. Most of the
additional particles produced in these events escape down the beam pipe and
thus the inelastic production cross-section as a function of track
multiplicity is expected to peak at two tracks. An extrapolation of the
inelastic production cross-section from higher track multiplicities is not
possible as no reliable simulation is available. Instead, this background is
determined from a fit to the $p_{\rm T}$ spectrum of the exclusive candidates.
The signal shape is taken from simulation while the background distribution is
estimated from data as described below. The result is shown in Fig. 4 for
events selected by the software trigger, which has no restriction on the
dimuon $p_{\rm T}$. The agreement between the predictions and the data is
good. To reduce the inelastic background contribution the dimuon $p_{\rm T}$
is required to be less than 900${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$. Below
this value the fit estimates that (70 $\pm$ 4)$\%$ of the events are
exclusive.
Figure 4: Transverse momentum distribution for exclusive
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates with exactly two
tracks and no photons. The points represent the data. The fit contains an
exclusive signal component (short-dashed green curve) as estimated by
SuperChic and an inelastic background component (long-dashed red curve) as
estimated from data.
The signal shape is taken from the dimuon $p_{\rm T}$ distribution of the
SuperChic simulation which assumes a $\mbox{$p_{\rm T}$}^{2}$ distribution of
the form $\exp(-b\mbox{$p_{\rm T}$}^{2})$. Regge theory predicts [2] that the
slope, $b$, of the $\mbox{$p_{\rm T}$}^{2}$ distribution increases with $W$
according to
$b=b_{0}+4\alpha^{\prime}\ln{W\over W_{0}},$ (1)
where fits to H1 and ZEUS data [3, 4] give values of $b_{0}$ = 5
GeV${}^{-2}c^{2}$, $\alpha^{\prime}=0.2$ GeV${}^{-2}c^{2}$ at a scale
$W_{0}=90$ GeV. Producing an object of mass $m_{V}$ at a rapidity $y$ accesses
$W$ values of
$(W_{\pm})^{2}=m_{V}\sqrt{s}\exp(\pm|y|),$ (2)
where the two solutions, $W_{+}$ and $W_{-}$, correspond to the photon being
either an emitter or a target. The total cross-section has contributions
coming from both $W_{+}$ and $W_{-}$, the relative amounts of which can be
determined from the photon energy spectrum of the proton given in Ref. [19].
To determine an appropriate $b$ value to describe the $p_{\rm T}$ spectrum in
LHCb, the rapidity range $2.0<y<4.5$ is split into ten equally sized bins and
$W_{\pm}$ values are calculated for each bin using Eq.(2). Two corresponding
$b$ values are found using Eq.(1) and then weighted according to the photon
energy spectrum. The mean and root mean square of $b$ in the rapidity bins are
6.1 GeV${}^{-2}c^{2}$ and 0.3 GeV${}^{-2}c^{2}$, respectively. These values
are consistent with the results of the fit described below where $b$ is left
as a free parameter.
$\begin{array}[]{cc}\begin{@subfigure}\includegraphics[width=195.12767pt]{Fig5a.pdf}\end{@subfigure}&\begin{@subfigure}\includegraphics[width=195.12767pt]{Fig5b.pdf}\end{@subfigure}\\\
\begin{@subfigure}\includegraphics[width=195.12767pt]{Fig5c.pdf}\end{@subfigure}&\begin{@subfigure}\includegraphics[width=195.12767pt]{Fig5d.pdf}\end{@subfigure}\\\
\vspace*{-1.0cm}\hfil\end{array}$
Figure 5: (a) Transverse momentum distribution in
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ events with no backward tracks
and no photons when there are three (black squares) and eight (red open
circles) forward tracks. The results of fitting a Novosibirsk function to the
three and eight forward track data are represented by the dashed black and the
full red curves respectively. The values for (b) $P_{0}$, (c) $\sigma$ and (d)
$\alpha$ of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ $p_{\rm T}$
distribution are shown as a function of the number of forward tracks.
The shape for the inelastic background is taken from data. Non-exclusive
candidates are selected by requiring events to contain more than two tracks in
the forward region. The $p_{\rm T}$ distributions for events with 3, 4, 5, 6,
7 and 8 forward tracks are each fitted separately with a Novosibirsk
function111 The Novosibirsk function is defined as
$N(P;P_{0},\sigma,\alpha)=A\exp(-0.5(\ln^{2}(1+\Lambda\alpha(P-P_{0}))/\alpha^{2}+\alpha^{2}))$
where $\Lambda=\sinh(\alpha\sqrt{\ln 4})/(\sigma\alpha\sqrt{\ln 4})$, $P_{0}$
is the peak position, $\sigma$ is the width and $\alpha$ is the tail
parameter. , which is a skewed Gaussian function with three parameters for the
peak position ($P_{0}$), width ($\sigma$) and tail ($\alpha$).
Figure 5(a) displays two of the fitted $p_{\rm T}$ spectra for the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson in events with no
backward tracks and no photons when there are 3 and 8 forward tracks. The
values for $P_{0}$, $\sigma$ and $\alpha$ are extracted from each fit and
plotted as a function of the number of forward tracks in Fig. 5(b), (c) and
(d), respectively. A linear extrapolation of each parameter is made to predict
the inelastic background shape for events with two forward tracks giving
$P_{0}=0.55\pm 0.09{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, $\sigma=0.56\pm
0.06{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $\alpha=0.72\pm 0.11$.
Although a full theoretical prediction is not possible [20] a dependence of
the $p_{\rm T}$ peak position on the number of tracks is expected from general
kinematic considerations; additional gluon radiation will impart a greater
$p_{\rm T}$ to the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ as well as
giving extra tracks inside LHCb, while proton disintegration, leading to
additional tracks, is also more likely to occur at higher $p_{\rm T}$. This is
found to be the case for a number of related processes using simulated events:
single and central diffractive events in PYTHIA; the diphoton production of
muon pairs with similar masses to the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ in which one or both of the protons disintegrate, as described by
LPAIR [21]; and minimum bias events in PYTHIA. A linear dependence is also
observed in $Z\rightarrow\mu^{+}\mu^{-}$ events in data although the average
$p_{\rm T}$ is one order of magnitude larger than in exclusive
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production.
Systematic uncertainties on the inelastic background determination due to both
the signal and background shapes have been assessed. To determine the former,
the fit to the data is repeated leaving $b$ as a free parameter, which returns
a value of 5.8 $\pm$ 0.8 GeV${}^{-2}c^{2}$. A variation in the estimated
percentage of exclusive events of 4$\%$ is observed when $b$ is changed by
$\pm 1$ GeV${}^{-2}c^{2}$. The uncertainty due to the background shape is
found by varying the parameters of the phenomenological shape within their
statistical uncertainties resulting in a change of 4$\%$ in the estimated
fraction of exclusive events. Therefore, it is estimated that ($70\pm 4\pm
6)\%$ of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ sample consists of
exclusive events, where the first uncertainty is statistical and the second is
systematic due to out understanding of the signal and background shapes. It is
assumed that the $\psi{(2S)}$ sample has the same proportion of exclusive
events.
### 3.3 Exclusive feed-down background determination
Exclusive $\chi_{c}$ production can feed down to give a fake exclusive
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$, via the
$\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma$ decay,
when the photon is very soft or is outside the detector acceptance. There is
no corresponding resonance above the $\psi{(2S)}$. Exclusive $\chi_{c}$
candidate events are identified in the data as those containing a
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and a single photon. The
background from $\chi_{c}$ feed-down is then estimated by scaling the number
of observed $\chi_{c}$ candidates by the ratio of fake exclusive
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ to exclusive $\chi_{c}$
candidates in simulated $\chi_{c}$ events. The feed-down from $\chi_{c}$ decay
is estimated to account for (9.0 $\pm$ 0.8)$\%$ of the exclusive
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates, where the
uncertainty includes a contribution due to the uncertainty on the photon
reconstruction efficiency in simulation.
Exclusive $\psi{(2S)}$, in particular
$\psi{(2S)}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}+X$, can
also feed down to give a fake exclusive ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ if the additional particles are undetected. This is estimated from a
simulated sample of $\psi{(2S)}$ events which has been normalised using the
number of observed $\psi{(2S)}\rightarrow\mu^{+}\mu^{-}$ events in data. The
amount of feed-down to the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ from
the $\psi{(2S)}$ is estimated to be (1.8 $\pm$ 0.3)$\%$.
### 3.4 Selection summary
Table 1: Summary of selection requirements. Quantity | Requirement
---|---
Dimuon mass | within 65${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of known value
Dimuon $p_{\rm T}$ | $\mbox{$p_{\rm T}$}<$ 900${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$
Muon $\eta$ | 2.0 $<\eta_{\mu^{\pm}}<$ 4.5
Number of backward tracks | 0
Number of forward tracks | 2
Number of photons | 0
The requirements for the selection of exclusive
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ events are
summarised in Table 1. In total, 1492 exclusive
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and 40 exclusive $\psi{(2S)}$
candidate events pass the selection requirements. The overall purities
(including inelastic, non-resonant, and feed-down backgrounds where
appropriate) are estimated to be $(62\pm 4\pm 5)$% for the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ sample and $(59\pm 4\pm 5)$%
for the $\psi{(2S)}$ sample.
A cross-section times branching fraction,
$\sigma_{V\rightarrow\mu^{+}\mu^{-}}$, is calculated for the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and the $\psi{(2S)}$ using the
number of selected events, $N$, and the equation
$\sigma_{V\rightarrow\mu^{+}\mu^{-}}=pN/(\epsilon L)$ where $\epsilon$
represents the efficiency for selecting the events, $p$ is the purity of the
sample, and $L$ is the luminosity which has been determined with an
uncertainty of 3.5% [22].
## 4 Efficiency determination
The efficiency $\epsilon$ is the product of five components,
$\epsilon_{\mathrm{trigger}}\times\epsilon^{2}_{\mathrm{track}}\times\epsilon^{2}_{\mathrm{muon}}\times\epsilon_{\mathrm{sel}}\times\epsilon_{\mathrm{single}}$
where: $\epsilon_{\mathrm{trigger}}$ is the efficiency for triggering on
events that pass the offline selection; $\epsilon_{\mathrm{track}}$ is the
efficiency for reconstructing a track within the fiducial region of the
measurement; $\epsilon_{\mathrm{muon}}$ is the efficiency for identifying a
track as a muon; $\epsilon_{\mathrm{sel}}$ is the efficiency of the selection
requirements in the kinematic range of the measurement; and
$\epsilon_{\mathrm{single}}$ is the efficiency for selecting single
interaction events. The first four components have been determined from
simulation. The fifth component accounts for the fact that the selection
requirements reject signal events that are accompanied by a visible proton-
proton interaction in the same beam crossing.
The number of visible proton-proton interactions per beam crossing, $n$, is
assumed to follow a Poisson distribution, $P(n)=\mu^{n}\exp(-\mu)/n!$, where
$\mu$ is the average number of visible interactions. The probability that a
signal event is not rejected due to the presence of another visible
interaction is given by $P(0)$ and, therefore,
$\epsilon_{\mathrm{single}}=\exp(-\mu)$. This has been calculated throughout
the data-taking period in roughly hour-long intervals. Variations in $\mu$
during this interval have been studied and found to have a negligible effect.
The spread in the value of $\mu$ for different crossing bunch-pairs is small
and its effect is neglected. The impact of detector noise has also been
investigated using data taken when either one beam or no beam circulated and a
systematic uncertainty of 0.7$\%$ is evaluated leading to a determination for
$\epsilon_{\mathrm{single}}$ of $(21.1\pm 0.1)\%$.
A systematic uncertainty of 4$\%$ is assigned to
$\epsilon_{\mathrm{trigger}}$. This is based on the difference between the
value measured in data and in simulation as determined in Ref. [23].
A tag-and-probe technique with ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-}$ decays [24] has been used to study
$\epsilon_{\mathrm{track}}$. The estimated value from this method is found to
agree within 1$\%$ with the simulation. This value is taken as a systematic
uncertainty per track.
A systematic uncertainty of 2.5$\%$ per muon is assigned to the determination
of $\epsilon_{\mathrm{muon}}$. This is based on the results from Ref. [23]
using a tag-and-probe technique.
The fraction of exclusive ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
mesons below the $p_{\rm T}$ selection threshold of
900${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ is found to vary by 1$\%$ when $b$,
the $p_{\rm T}$ shape parameter, is changed by $\pm$ 1 GeV${}^{-2}c^{2}$. Thus
a systematic uncertainty of 1$\%$, due to the uncertainty in the $p_{\rm T}$
shape, is assigned to the determination of $\epsilon_{\mathrm{sel}}$. No
systematic uncertainty is assigned due to the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ polarisation; it is assumed to
be transversely polarised due to s-channel helicity conservation. A summary of
the systematic uncertainties of the analysis is shown in Table 2.
Table 2: Relative systematic uncertainties on the measurement. Source | Uncertainty ($\%$)
---|---
Luminosity | 3.5
Trigger efficiency | 4
Tracking efficiency | 2
Identification efficiency | 5
Selection efficiency | 1
Single interaction efficiency | 0.7
$\psi{(2S)}$ background (${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ analysis) | 0.3
$\chi_{c}$ background (${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ analysis) | 0.8
Signal shape of dimuon $p_{\rm T}$ fit | 6
Background shape of dimuon $p_{\rm T}$fit | 6
## 5 Results
The cross-section times branching fraction to two muons with pseudorapidities
between 2.0 and 4.5 is measured for exclusive
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ to be
$\sigma_{pp\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(\rightarrow\mu^{+}\mu^{-})}(2.0<\eta_{\mu^{\pm}}<4.5)=307\pm 21\pm
36~{}\text{pb},$
$\sigma_{pp\rightarrow\psi{(2S)}(\rightarrow\mu^{+}\mu^{-})}(2.0<\eta_{\mu^{\pm}}<4.5)=7.8\pm
1.3\pm 1.0~{}\text{pb},$
where the first uncertainty is statistical and the second is systematic.
These results are compared to a number of predictions for exclusive production
in Table 3. The predictions for Starlight and SuperChic have been determined
using samples of generated events with a full LHCb simulation. The other
predictions are obtained by scaling the differential cross-section in rapidity
for each model by an acceptance factor corresponding to the fraction of mesons
at a given rapidity that have both muons in the fiducial volume, as determined
using Starlight. For the models of Motyka and Watt, and Gonçalves and Machado,
a rescattering correction of 0.8 has been assumed [13]. The prediction of
Schäfer and Szczurek is significantly higher than the data; good agreement is
observed with all other predictions.
Table 3: Comparison of cross-section times branching fraction measurements (pb) with theoretical predictions. Predictions | $\sigma_{pp\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}(\rightarrow\mu^{+}\mu^{-})}$ | $\sigma_{pp\rightarrow\psi{(2S)}(\rightarrow\mu^{+}\mu^{-})}$
---|---|---
Gonçalves and Machado | 275 |
Starlight | 292 | 6.1
Motyka and Watt | 334 |
SuperChic | 396 |
Schäfer and Szczurek | 710 | 17
LHCb measured value | $307\pm 21\pm 36$ | $7.8\pm 1.3\pm 1.0$
Table 4: Cross-section measurements (nb) as a function of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ rapidity. Rapidity | 2.00-2.25 | 2.25-2.50 | 2.50-2.75 | 2.75-3.00
---|---|---|---|---
${\mathrm{d}\sigma\over\mathrm{d}y}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$ | 3.2 $\pm$ 0.8 $\pm$ 0.9 | 4.5 $\pm$ 0.5 $\pm$ 0.8 | 5.3 $\pm$ 0.4 $\pm$ 0.9 | 4.4 $\pm$ 0.3 $\pm$ 0.7
Rapidity | 3.00-3.25 | 3.25-3.50 | 3.50-3.75 | 3.75-4.00
${\mathrm{d}\sigma\over\mathrm{d}y}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$ | 5.5 $\pm$ 0.3 $\pm$ 0.8 | 4.8 $\pm$ 0.3 $\pm$ 0.7 | 5.2 $\pm$ 0.3 $\pm$ 0.8 | 4.8 $\pm$ 0.4 $\pm$ 0.8
Rapidity | 4.00-4.25 | 4.25-4.50 | |
${\mathrm{d}\sigma\over\mathrm{d}y}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$ | 4.7 $\pm$ 0.5 $\pm$ 0.9 | 4.1 $\pm$ 0.9 $\pm$ 1.3 | |
Combining the vector meson branching fractions to two muons with acceptance
factors determined from Starlight gives a ratio of $\psi{(2S)}$ to
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production of 0.19 $\pm$ 0.04.
This can be compared to a value of 0.16 according to Starlight and about 0.2
according to Schäfer and Szczurek. CDF measured this ratio to be 0.14 $\pm$
0.05 [9] and at HERA it was measured to be 0.166 $\pm$ 0.012 [3, 4] although
these were at different values of $W$.
The differential ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ cross-section
is also measured in ten bins of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
rapidity. The trigger and selection efficiencies are calculated from the
simulation in bins of rapidity. The systematic uncertainties are dominated by
the purity and the statistical uncertainty coming from the size of the
simulation sample. The purity within each bin is assumed to be the same as in
the integrated sample. An acceptance factor in each rapidity bin has been
calculated using the Starlight simulation. Table 4 summarises the differential
cross-section result.
The present results can be compared to H1 and ZEUS results [3, 4] for the
photoproduction of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$. This is
possible as the underlying production mechanism is the same: at HERA the
photon radiates from an electron, while at the LHC the photon radiates from a
proton. The differential cross-section for proton-proton exclusive
photoproduction of a vector meson with mass $m_{V}$ can be obtained by
weighting the photon-proton exclusive production cross-section by the photon
flux, ${d}n/{d}k$, for a photon of energy $k$ [11, 7]
${{d}\sigma\over{d}y}_{pp\rightarrow pVp}=r(y)\biggl{[}k_{+}{dn\over
dk_{+}}\sigma_{\gamma p\rightarrow Vp}(W_{+})+k_{-}{dn\over
dk_{-}}\sigma_{\gamma p\rightarrow Vp}(W_{-})\biggr{]},$ (3)
$k_{\pm}\approx(m_{V}/2)\exp(\pm|y|),$ (4)
where $W_{\pm}$ is defined as in Eq.(2) and $r(y)$ is an absorptive correction
which depends on $y$ [13].
Assuming the validity of a power law dependence of the form $aW^{\delta}$ to
describe $\sigma_{\gamma p\rightarrow Vp}$, the proton-proton differential
cross-section can be written as
${d\sigma\over dy}_{pp\rightarrow
pVp}=a(2\sqrt{s})^{\delta/2}r(y)\biggl{[}{dn\over
dk_{+}}{k_{+}}^{1+\delta/2}+{dn\over dk_{-}}{k_{-}}^{1+\delta/2}\biggr{]}.$
(5)
The parameters for the power law dependence of the photoproduction cross-
section are found by fitting the differential cross-section data from Table 4
with the functional form given in Eq.(5). The uncertainties between bins are
taken to be uncorrelated for all sources except for the purity, which is fully
correlated between bins. For the description of ${dn\over dk}$, the photon
energy spectrum given in Ref. [19] is used. The absorptive corrections have
been calculated in Ref. [13] for proton-proton collisions and have a value of
0.85 at $y=0$ and 0.75 at $y=3$ with a rather flat dependence on $y$. This
analysis assumes a shape $r(y)=0.85-0.1|y|/3$. The fit to the data in Table 4
gives values of $a=0.8^{+1.2}_{-0.5}$ nb and $\delta=0.92\pm 0.15$ with a
$\chi^{2}$ of 4.3 for 8 degrees of freedom, indicating the results are
consistent with the hypothesis of a power law dependence. The values obtained
are also consistent with the results from HERA, albeit with much larger
uncertainties.
### 5.1 Evaluation of the photon-proton cross-section
$\begin{array}[]{c}\includegraphics[width=368.57964pt]{Fig6.pdf}\vspace*{-1.0cm}\end{array}$
Figure 6: Dependence of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
photoproduction cross-section on the centre-of-mass energy of the photon-
proton system. The blue (red) triangles represent the data from H1 (ZEUS) [3,
4]. The black dots and squares are derived from the LHCb differential cross-
section as a function of rapidity. The dashed and full lines are the power law
dependences determined from the HERA and LHCb data, respectively. The
uncertainty on the LHCb power law determination is shown by the shaded band.
The differential cross-sections for the process $pp\rightarrow
p{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}p$ given in Table 4 are
transformed into cross-sections for the process $\gamma
p\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}p$ using a re-
arrangement of Eq.(3)
$\sigma_{\gamma p\rightarrow Vp}(W_{\pm})={{1/r(y)}{d\sigma\over
dy}_{pp\rightarrow pVp}-k_{\mp}{dn\over dk_{\mp}}\sigma_{\gamma p\rightarrow
Vp}(W_{\mp})\over k_{\pm}{dn\over dk_{\pm}}}.$ (6)
The photoproduction cross-sections at $W_{+}$ and $W_{-}$ are determined
independently using Eq.(6) and substituting into the right-hand side the
expected cross-section for the alternative $W$ solution from the power law
determined above.
The LHCb data are plotted together with the H1 and ZEUS data in Fig. 6. The
power law dependences determined from LHCb data and from HERA data are also
indicated. The uncertainty on the LHCb power law determination is shown by the
shaded band. Only experimental uncertainties are shown on the LHCb data, which
have significant bin-to-bin correlations due to the purity determination.
Theoretical uncertainties that account for the power law assumption and the
absorptive correction are not included. Both are smaller than the current
experimental uncertainties; using the HERA power law in place of the LHCb
power law in Eq.(6) changes the estimated values of $\sigma_{\gamma
p\rightarrow Vp}$ by about half the experimental uncertainty, while changing
the absorptive correction by 5$\%$ changes $\sigma_{\gamma p\rightarrow Vp}$
by about one quarter of the experimental uncertainty. With the precision of
the current data, the LHCb results are consistent with HERA and confirm a
similar power law behaviour for the photoproduction cross-section.
## 6 Conclusion
The first observations of exclusive ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ and $\psi{(2S)}$ production in proton-proton collisions have been
made. The cross-sections times branching fraction to two muons with
pseudorapidities between 2.0 and 4.5 are measured to be $307\pm 21\pm
36~{}\text{pb}$ and $7.8\pm 1.3\pm 1.0~{}\text{pb}$ for exclusive
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$, respectively
The measured cross-sections are in agreement with the theoretical predictions
of Starlight, SuperChic, Gonçalves and Machado, and Motyka and Watt. The
differential cross-section for ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
production as a function of rapidity has also been measured. This has allowed
the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ photoproduction cross-
section as a function of the photon-proton centre-of-mass energy to be
determined. The data are consistent with a power law dependence and the
parametric form is in broad agreement with previous results from H1 and ZEUS.
## Acknowledgements
We thank Lucian Harland-Lang, Valery Khoze, James Stirling and Graeme Watt for
many helpful discussions. We express our gratitude to our colleagues in the
CERN accelerator departments for the excellent performance of the LHC. We
thank the technical and administrative staff at the LHCb institutes. We
acknowledge support from CERN and from the national agencies: CAPES, CNPq,
FAPERJ and FINEP (Brazil); NSFC (China); CNRS/IN2P3 and Region Auvergne
(France); BMBF, DFG, HGF and MPG (Germany); SFI (Ireland); INFN (Italy); FOM
and NWO (The Netherlands); SCSR (Poland); ANCS/IFA (Romania); MinES, Rosatom,
RFBR and NRC “Kurchatov Institute” (Russia); MinECo, XuntaGal and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7. The Tier1 computing centres are supported by IN2P3 (France), KIT
and BMBF (Germany), INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain),
GridPP (United Kingdom). We are thankful for the computing resources put at
our disposal by Yandex LLC (Russia), as well as to the communities behind the
multiple open source software packages that we depend on.
## References
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|
arxiv-papers
| 2013-01-29T22:12:17 |
2024-09-04T02:49:40.934124
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, A. Adametz, B. Adeva,\n M. Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio,\n M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S.\n Amato, Y. Amhis, L. Anderlini, J. Anderson, R. Andreassen, R.B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W. Baldini, R.J.\n Barlow, C. Barschel, S. Barsuk, W. Barter, Th. Bauer, A. Bay, J. Beddow, I.\n Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G.\n Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M.\n van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T.\n Blake, F. Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov, V. Bocci, A.\n Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, E.\n Bowen, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M.\n Britsch, T. Britton, N.H. Brook, H. Brown, I. Burducea, A. Bursche, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, H. Carranza-Mejia,\n L. Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo, Ch. Cauet, M. Charles,\n Ph. Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal,\n G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V.\n Coco, J. Cogan, E. Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A.\n Cook, M. Coombes, G. Corti, B. Couturier, G.A. Cowan, D. Craik, S. Cunliffe,\n R. Currie, C. D'Ambrosio, P. David, P.N.Y. David, I. De Bonis, K. De Bruyn,\n S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, W. De Silva, P. De\n Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi, L. Del Buono, C. Deplano, D.\n Derkach, O. Deschamps, F. Dettori, A. Di Canto, J. Dickens, H. Dijkstra, P.\n Diniz Batista, M. Dogaru, F. Domingo Bonal, S. Donleavy, F. Dordei, A. Dosil\n Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, A.\n Dzyuba, S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, S.\n Eisenhardt, U. Eitschberger, R. Ekelhof, L. Eklund, I. El Rifai, Ch.\n Elsasser, D. Elsby, A. Falabella, C. F\\\"arber, G. Fardell, C. Farinelli, S.\n Farry, V. Fave, D. Ferguson, V. Fernandez Albor, F. Ferreira Rodrigues, M.\n Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M. Fontana, F. Fontanelli, R.\n Forty, O. Francisco, M. Frank, C. Frei, M. Frosini, S. Furcas, E. Furfaro, A.\n Gallas Torreira, D. Galli, M. Gandelman, P. Gandini, Y. Gao, J. Garofoli, P.\n Garosi, J. Garra Tico, L. Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M.\n Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V.V. Gligorov, C. G\\\"obel, D.\n Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa G\\'andara, R.\n Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E.\n Greening, S. Gregson, O. Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C.\n Hadjivasiliou, G. Haefeli, C. Haen, S.C. Haines, S. Hall, T. Hampson, S.\n Hansmann-Menzemer, N. Harnew, S.T. Harnew, J. Harrison, P.F. Harrison, T.\n Hartmann, J. He, V. Heijne, K. Hennessy, P. Henrard, J.A. Hernando Morata, E.\n van Herwijnen, E. Hicks, D. Hill, M. Hoballah, C. Hombach, P. Hopchev, W.\n Hulsbergen, P. Hunt, T. Huse, N. Hussain, D. Hutchcroft, D. Hynds, V.\n Iakovenko, P. Ilten, R. Jacobsson, A. Jaeger, E. Jans, F. Jansen, P. Jaton,\n F. Jing, M. John, D. Johnson, C.R. Jones, B. Jost, M. Kaballo, S. Kandybei,\n M. Karacson, T.M. Karbach, I.R. Kenyon, U. Kerzel, T. Ketel, A. Keune, B.\n Khanji, O. Kochebina, I. Komarov, R.F. Koopman, P. Koppenburg, M. Korolev, A.\n Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F.\n Kruse, M. Kucharczyk, V. Kudryavtsev, T. Kvaratskheliya, V.N. La Thi, D.\n Lacarrere, G. Lafferty, A. Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G.\n Lanfranchi, C. Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van\n Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, O. Leroy, Y.\n Li, L. Li Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben,\n J.H. Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, J. Luisier, H. Luo, F.\n Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, S. Malde, G. Manca, G.\n Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G. Martellotti,\n A. Martens, L. Martin, A. Mart\\'in S\\'anchez, M. Martinelli, D. Martinez\n Santos, D. Martins Tostes, A. Massafferri, R. Matev, Z. Mathe, C. Matteuzzi,\n M. Matveev, E. Maurice, A. Mazurov, J. McCarthy, R. McNulty, B. Meadows, F.\n Meier, M. Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez,\n S. Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F. Muheim, K.\n M\\\"uller, R. Muresan, B. Muryn, B. Muster, P. Naik, T. Nakada, R. Nandakumar,\n I. Nasteva, M. Needham, N. Neufeld, A.D. Nguyen, T.D. Nguyen, C. Nguyen-Mau,\n M. Nicol, V. Niess, R. Niet, N. Nikitin, T. Nikodem, S. Nisar, A. Nomerotski,\n A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O.\n Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora Goicochea, P. Owen, B.K.\n Pal, A. Palano, M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, C.\n Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel, G.N. Patrick, C.\n Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A. Pellegrino, G. Penso, M.\n Pepe Altarelli, S. Perazzini, D.L. Perego, E. Perez Trigo, A. P\\'erez-Calero\n Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina, K. Petridis, A.\n Petrolini, A. Phan, E. Picatoste Olloqui, B. Pietrzyk, T. Pila\\v{r}, D.\n Pinci, S. Playfer, M. Plo Casasus, F. Polci, G. Polok, A. Poluektov, E.\n Polycarpo, D. Popov, B. Popovici, C. Potterat, A. Powell, J. Prisciandaro, V.\n Pugatch, A. Puig Navarro, W. Qian, J.H. Rademacker, B. Rakotomiaramanana,\n M.S. Rangel, I. Raniuk, N. Rauschmayr, G. Raven, S. Redford, M.M. Reid, A.C.\n dos Reis, S. Ricciardi, A. Richards, K. Rinnert, V. Rives Molina, D.A. Roa\n Romero, P. Robbe, E. Rodrigues, P. Rodriguez Perez, G.J. Rogers, S. Roiser,\n V. Romanovsky, A. Romero Vidal, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino,\n J.J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann, B.\n Sanmartin Sedes, M. Sannino, R. Santacesaria, C. Santamarina Rios, E.\n Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D.\n Savrina, P. Schaack, M. Schiller, H. Schindler, S. Schleich, M. Schlupp, M.\n Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R.\n Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska,\n I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P.\n Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V.\n Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, N.A. Smith, E.\n Smith, M. Smith, K. Sobczak, M.D. Sokoloff, F.J.P. Soler, F. Soomro, D.\n Souza, B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S.\n Stahl, O. Steinkamp, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U.\n Straumann, V.K. Subbiah, S. Swientek, V. Syropoulos, M. Szczekowski, P.\n Szczypka, T. Szumlak, S. T'Jampens, M. Teklishyn, E. Teodorescu, F. Teubert,\n C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S. Tolk, D.\n Tonelli, S. Topp-Joergensen, N. Torr, E. Tournefier, S. Tourneur, M.T. Tran,\n M. Tresch, A. Tsaregorodtsev, P. Tsopelas, N. Tuning, M. Ubeda Garcia, A.\n Ukleja, D. Urner, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P.\n Vazquez Regueiro, S. Vecchi, J.J. Velthuis, M. Veltri, G. Veneziano, M.\n Vesterinen, B. Viaud, D. Vieira, X. Vilasis-Cardona, A. Vollhardt, D.\n Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo\\ss, H. Voss, R. Waldi,\n R. Wallace, S. Wandernoth, J. Wang, D.R. Ward, N.K. Watson, A.D. Webber, D.\n Websdale, M. Whitehead, J. Wicht, J. Wiechczynski, D. Wiedner, L. Wiggers, G.\n Wilkinson, M.P. Williams, M. Williams, F.F. Wilson, J. Wishahi, M. Witek,\n S.A. Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z. Yang,\n R. Young, X. Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L.\n Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, A. Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Ronan McNulty",
"url": "https://arxiv.org/abs/1301.7084"
}
|
1301.7139
|
# Solution to the one-dimensional telegrapher’s equation subject to a
backreaction boundary condition
Thorsten Prüstel Laboratory of Systems Biology
National Institute of Allergy and Infectious Diseases
National Institutes of Health Martin Meier-Schellersheim Laboratory of
Systems Biology
National Institute of Allergy and Infectious Diseases
National Institutes of Health
###### Abstract
We discuss solutions of the one-dimensional telegrapher’s equation in the
presence of boundary conditions. We revisit the case of a radiation boundary
condition and obtain an alternative expression for the already known Green’s
function. Furthermore, we formulate a backreaction boundary condition, which
has been widely used in the context of diffusion-controlled reversible
reactions, for a one-dimensional telegrapher’s equation and derive the
corresponding Green’s function.
11footnotetext: Email: [email protected], [email protected]
## 1 Introduction
Although the diffusion equation [6, 4]
$\frac{\partial p(x,t)}{\partial t}=D\frac{\partial^{2}p(x,t)}{\partial
x^{2}}$ (1.1)
can be successfully applied to a broad range of different phenomena, it
suffers from a pathological defect: The implied signal propagation speed is
infinite in the sense that any perturbation $\delta p(x,t)$ localized at one
point in space instantaneously influences $p(x,t)$ at any other point. Still,
the diffusion equation provides an excellent approximation, as long as the
time scale on which inertial effects must not be neglected is small compared
to the relevant time scale of the considered problem. This is not always the
case, though. For instance, the movement of animals and biological cells
proceeds with a rather well-defined and finite velocity [5] and the diffusion
equation approximation becomes invalid.
The telegrapher’s equation [9, 16, 19, 15] may be considered as a
generalization of the diffusion equation that overcomes the described
deficiencies. Being a hyperbolic differential equation, it exhibits wave-like
features on short time scales while diffusion-like features dominate on longer
time scales. In particular, the signal propagation speed is bounded and well-
defined. Like the diffusion equation, the telegrapher’s equation may be viewed
from a macroscopic, phenomenological and, at least in one dimension, from a
microscopic point of view.
The phenomenological approach starts with Fick’s first law that asserts that
the probability flux is linearly related to the gradient of the probability
density
$j(x,t)=-D\frac{\partial p(x,t)}{\partial x}.$ (1.2)
Combining Eq. (1.2) with the conservation law
$\frac{\partial p}{\partial t}+\frac{\partial j}{\partial x}=0$ (1.3)
immediately yields the diffusion equation. Consequently, Ficks’ first law and
the telegrapher’s equation are not compatible with each other and one has to
seek a modification of Eq. (1.2). A suitable generalization was suggested by
Cattaneo in an attempt to replace Fourier’s law in the theory of heat waves.
In the present context Cattaneo’s equation [10] reads as
$T\frac{\partial j}{\partial t}=-\bigg{(}D\frac{\partial p}{\partial
x}+j\bigg{)},$ (1.4)
where $T$ sets the relaxation time scale. Obviously, for $T\rightarrow 0$ one
recovers Eq. (1.2). We note for later use that
$j(x,t)=-c^{2}\int^{t}_{0}e^{-(t-\tau)/T}\frac{\partial p(x,\tau)}{\partial
x}\,d\tau$ (1.5)
satisfies Eq. (1.4), given that one identifies the diffusion constant with the
speed $c$ and the relaxation time $T$ according to $D=c^{2}T$. Cattaneo’s
equation Eq. (1.4) combined with the conservation equation Eq. (1.3) leads to
the telegrapher’s equation
$\frac{\partial^{2}p}{\partial t^{2}}+\frac{1}{T}\frac{\partial p}{\partial
t}=c^{2}\frac{\partial^{2}p}{\partial x^{2}}.$ (1.6)
Note that $T$ interpolates between the ballistic and diffusion regime in the
sense that Eq. (1.6) reduces to the wave equation in the limit
$T\rightarrow\infty$, while one arrives at the diffusion equation Eq. (1.1) in
the limit $T\rightarrow 0,c\rightarrow\infty,c^{2}T\rightarrow
D=\text{finite}$.
Turning to a microscopic point of view, we recall the well-known fact that the
diffusion equation can be obtained as the continuum limit of the simple
uncorrelated random walk, also referred to as Brownian motion. Similarly, at
least in one dimension, the telegrapher’s equation emerges from a correlated
random walk that takes into account correlations in the direction of movement
[7, 18, 9]. In this context Brownian motion appears as a contracted
description of the more general correlated random walk, being valid only if
the characteristic correlation time decays sufficiently fast. The persistence
of the correlated walker, i.e. the tendency that each step points in the same
direction as the previous one, necessitates to consider two separate
probability density functions (PDF). Let $a(x,t|x_{0})$ denote the PDF to find
the particle at $x$, given that it was at $x_{0}$ for $t=0$ and that it is
moving in the positive $x$ direction at time $t$, and let $b(x,t|x_{0})$ the
corresponding PDF for moving in the negative $x$ direction. It follows that
the PDF $p(x,t|x_{0})$ independent of the direction in which the particle is
moving at $t$ is given by the superposition
$p(x,t|x_{0})=a(x,t|x_{0})+b(x,t|x_{0}).$ (1.7)
It can be shown that the time evolution of $p(x,t|x_{0})$ is governed by the
telegrapher’s equation Eq. (1.6) [14].
The broad range of potential applications of the telegrapher’s equation [19],
its blend of wave and diffusion-like features and relation to other important
equations like the diffusion, wave and Dirac equation [8], motivate a thorough
study of its solutions.
In this paper, we are interested in solutions to Eq. (1.6) in the presence of
a single boundary [14, 12, 13]. Previously, a radiation boundary condition
(BC), the purely absorbing and reflecting limiting cases and the corresponding
Green’s functions (GF) have been derived [14]. Here, we will revisit the
absorbing and radiation boundary cases. We will obtain an alternative form of
the GFs by directly inverting the Laplace transform via the Bromwich contour
integral and we will discuss how the alternative forms relate to the
expressions obtained earlier. Then, we will discuss and derive a backreaction
BC suitable for a one-dimensional telegrapher’s equation. The backreaction BC
generalizes the radiation BC, which only describes irreversible reactions
between the particle and boundary: Once the particle gets absorbed, it stays
trapped forever. In contrast, the backreaction BC incorporates desorption or
return of population at the boundary. Consequently, the backreaction BC
figures prominently in the theory of reversible diffusion-influenced reactions
[3] and the GF of the diffusion equation subject to that BC have been obtained
in one, two, and three dimensions [2, 17, 11]. Finally, we will focus on the
purely absorption-desorption case and calculate the associated GF by make use
of the Bromwich contour integral again. In his way, we can derive and discuss
all obtained GFs in a coherent way.
## 2 Radiation boundary condition
We will start by summarizing some results obtained earlier [14]. Eq. (1.6) has
to be supplemented by initial conditions. A choice that can be motivated from
an underlying random walk picture is
$\displaystyle p(x,t=0|x_{0})$ $\displaystyle=$
$\displaystyle\delta(x-x_{0}),$ (2.1) $\displaystyle\frac{\partial p}{\partial
t}\bigg{|}_{t=0}$ $\displaystyle=$ $\displaystyle 0.$ (2.2)
The solution to the telegrapher’s equation that satisfy these initial
conditions is known as the free-space GF [14]
$\displaystyle p_{\text{free}}(x,t|x_{0})=e^{-t/(2T)}f_{0}(t,|x-x_{0}|),$
(2.3)
where
$\displaystyle f_{0}(t,x)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\delta(ct-x)+\frac{\Theta(ct-x)}{4cT}\bigg{[}I_{0}(u)+\frac{t}{2uT}I_{1}(u)\bigg{]},$
(2.4) $\displaystyle u$ $\displaystyle=$
$\displaystyle\frac{\sqrt{c^{2}t^{2}-x^{2}}}{2cT}.$ (2.5)
Here $\Theta(ct-x)$ denotes the Heaviside step function and $I_{0},I_{1}$
refer to the modified Bessel functions of first kind [1, Ch. (9.6.)]. For
later reference we note that the Laplace transform of $f_{0}(t,x)$ is
$\tilde{f}_{0}(s,x)=\frac{1}{2c}\frac{s+\frac{1}{2T}}{\sqrt{s^{2}-\frac{1}{4T^{2}}}}e^{-x/c\sqrt{s^{2}-\frac{1}{4T^{2}}}}$
(2.6)
The free-space GF can be used to obtain GFs that satisfy the telegrapher’s
equation in the presence of boundaries. Typically, one seeks a solution of the
form
$p(x,t|x_{0})=p_{\text{free}}(x,t|x_{0})+h(x,t),$ (2.7)
where $h(x,t)$ has to be a solution of the telegrapher’s equation Eq. (1.6).
Furthermore, it is chosen in such a way that Eq. (2.7) satisfy the specified
BC. Henceforth, we will assume without loss of generality that the boundary is
located at $x=0$ and that $x\geq 0,x_{0}\geq 0$.
In Ref. [14] it has been shown that the appropriate radiation BC is quite
different to the one in the diffusion case and that it takes the form
$c\frac{\partial p}{\partial
x}\bigg{|}_{x=0}=\frac{\beta}{2-\beta}\bigg{(}\frac{\partial p}{\partial
t}+\frac{1}{T}p\bigg{)}\bigg{|}_{x=0}.$ (2.8)
Here, $\beta$ denotes the probability that a particle reaching the boundary
gets actually absorbed
$a(x=0,t|x_{0})=(1-\beta)b(x=0,t|x_{0}).$ (2.9)
Thus, $\beta=1$ corresponds to the purely absorbing limit, while $\beta=0$
corresponds to a purely reflecting boundary. Furthermore, seeking a solution
of the form
$p(x,t|x_{0})=e^{-t/(2T)}P(x,t|x_{0}),$ (2.10)
one finds for the Laplace transform of $P(x,t|x_{0})$ [14]
$\tilde{P}(x,s|x_{0})=\frac{s+\frac{1}{2T}}{2c\rho}\bigg{[}e^{-\rho|x-x_{0}|/c}-\Omega
e^{-\rho(x+x_{0})/c}\bigg{]},$ (2.11)
where
$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\sqrt{s^{2}-\frac{1}{4T^{2}}},$ (2.12) $\displaystyle\Omega$
$\displaystyle=$
$\displaystyle\frac{\beta(s+\frac{1}{2T})-(2-\beta)\rho}{\beta(s+\frac{1}{2T})+(2-\beta)\rho}.$
(2.13)
It is easy to show that for purely absorbing and reflecting BCs the expression
for $\Omega$ reduces to
$\displaystyle\Omega$ $\displaystyle=$ $\displaystyle
2T(s-\rho),\,\,\quad\text{if}\,\,\beta=1,$ (2.14) $\displaystyle\Omega$
$\displaystyle=$ $\displaystyle 1,\quad\qquad\qquad\text{if}\,\,\beta=0,$
(2.15)
respectively. Taking into account these identities, Eqs. (2.6), (2.11) and
elementary properties of the Laplace transform, one arrives at the following
form for the GFs corresponding to purely absorbing and reflecting BC
$\displaystyle p_{\text{abs}}(x,t|x_{0})$ $\displaystyle=$ $\displaystyle
e^{-t/2T}\big{[}f_{0}(t,|x-x_{0}|)-f_{1}(t,x+x_{0})\big{]},$ (2.16)
$\displaystyle p_{\text{ref}}(x,t|x_{0})$ $\displaystyle=$ $\displaystyle
e^{-t/2T}\big{[}f_{0}(t,|x-x_{0}|)+f_{0}(t,x+x_{0})\big{]},$ (2.17)
where the function $f_{1}$ is given by
$\displaystyle f_{1}(t,x)$ $\displaystyle=$
$\displaystyle\frac{\Theta(ct-x)}{8cT}\bigg{[}I_{0}(u)+2\bigg{(}\frac{ct-x}{ct+x}\bigg{)}^{1/2}I_{1}(u)+$
(2.18) $\displaystyle+$
$\displaystyle\bigg{(}\frac{ct-x}{ct+x}\bigg{)}I_{2}(u)\bigg{]}.$
Finally, to obtain the inverse Laplace transform in the case of a radiation BC
one can proceed by expanding the denominator in a power series to find the
following expression for the GF satisfying the BC Eq. (2.8) [14]
$\displaystyle p_{\text{rad}}(x,t|x_{0})$ $\displaystyle=$ $\displaystyle\beta
p_{\text{abs}}(x,t|x_{0})+(1-\beta)p_{\text{ref}}(x,t|x_{0})+$ (2.19)
$\displaystyle-$
$\displaystyle\frac{1}{8cT}\beta(1-\beta)e^{-t/2T}\sum^{\infty}_{n=0}c_{n}g_{n}(t,x+x_{0}),$
The coefficients are defined by
$\displaystyle c_{0}$ $\displaystyle=$ $\displaystyle 1,$ $\displaystyle
c_{1}$ $\displaystyle=$ $\displaystyle 4-\beta,$ $\displaystyle c_{2}$
$\displaystyle=$ $\displaystyle(2-\beta)(3-\beta)+1,$ $\displaystyle c_{n}$
$\displaystyle=$ $\displaystyle(2-\beta)^{3}(1-\beta)^{n-3},\,\,\,\,n\geq 3,$
$\displaystyle g_{n}(t,x)$ $\displaystyle=$
$\displaystyle\Theta(ct-x)\bigg{(}\frac{ct-x}{ct+x}\bigg{)}^{n/2}I_{n}(u).$
(2.20)
### 2.1 Alternative expression for the Green’s function
In what follows, we will take a different route to derive alternative
expressions for the GFs subject to purely absorbing and radiation BC. By
making use of the Bromwich contour integral, we will derive an alternative
form of the GFs as an important preparatory step. In fact, by employing for
all discussed BC the same technique, hardly any extra work will be necessary
to establish the expression for the GF that satisfies the backreaction BC.
Focusing first on the purely absorbing case, we start again from the Laplace
domain and write
$\tilde{p}_{\text{abs}}(x,s|x_{0})=\tilde{p}_{\text{free}}(x,s|x_{0})+\tilde{h}_{\text{abs}}(x,s).$
(2.21)
Now, our strategy will be to invert $\tilde{h}_{\text{abs}}$ directly by using
the Bromwich contour integral. Henceforth, to streamline the notation, we
define
$\displaystyle a$ $\displaystyle=$ $\displaystyle T^{-1},$ $\displaystyle k$
$\displaystyle=$ $\displaystyle\frac{x+x_{0}}{c}.$
Then, taking into account Eqs. (2.11), (2.12) and (2.13), the Bromwich contour
integral takes the form
$\displaystyle h_{\text{abs}}(x,t|x_{0})=\frac{1}{2\pi
i}\int^{\gamma+i\infty}_{\gamma-i\infty}e^{st}\tilde{h}_{\text{abs}}(x,s)ds=$
$\displaystyle-\frac{1}{4\pi
ic}\int^{\gamma+i\infty}_{\gamma-i\infty}e^{ts-k\sqrt{s(s+a)}}\frac{s+a}{\sqrt{s(s+a)}}\frac{\sqrt{s+a}-\sqrt{s}}{\sqrt{s+a}+\sqrt{s}}ds.\quad$
(2.22)
To proceed we have to discuss the analytical structure of the integrand, see
Fig. 1. First, we note that it has two branch points, at $s=-1/T$ and $s=0$,
necessitating the specification of two branch cuts. We choose both branch cuts
along the negative real axis, leading to a potentially discontinuous integrand
in the region $s<0$. Therefore, we employ the keyhole contour depicted in Fig.
1. It turns out that the branch cuts’ effects cancel each other along the
overlapping region $s<-1/T$, i.e. along $\mathcal{C}^{-1/T}_{-\infty}$,
$\mathcal{C}^{-\infty}_{-1/T}$, and the integrand is only discontinuous for
$-1/T<s<0$. Therefore, the sole contributions coming from the integration on
each sides of the branch cut are due to the integrals along
$\mathcal{C}^{0}_{-1/T}$ and $\mathcal{C}^{-1/T}_{0}$. Along
$\mathcal{C}^{0}_{-1/T}$ we choose $s=re^{i\pi}$ and it follows
$\sqrt{s+a}=\sqrt{a-r}$ and $\sqrt{s}=i\sqrt{r}$. We get
$\int_{\mathcal{C}^{0}_{-1/T}}e^{st}\tilde{h}_{\text{abs}}(x,s)ds=\int^{a}_{0}\frac{e^{-tr-
ik\sqrt{r(a-r)}}}{i\sqrt{r(a-r)}}(a-r)\frac{\sqrt{a-r}-i\sqrt{r}}{\sqrt{a-r}+i\sqrt{r}}dr.$
(2.23)
Similarly, to evaluate the integral along $\mathcal{C}^{-1/T}_{0}$ , we choose
$s=re^{-i\pi}$ and note that the ensuing integral is the negative complex
conjugate of the integral given in Eq. (2.23)
$\int_{\mathcal{C}^{0}_{-1/T}}e^{st}\tilde{h}_{\text{abs}}(x,s)ds=-\bigg{(}\int_{\mathcal{C}^{-1/T}_{0}}e^{st}\tilde{h}_{\text{abs}}(x,s)ds\bigg{)}^{\ast},$
(2.24)
where $\ast$ denotes complex conjugation. Furthermore, the integrand has no
poles neither inside nor on the contour and the contributions from the small
circles
$\mathcal{C}_{\epsilon_{2}},\mathcal{C}_{\epsilon_{1}},\mathcal{C}_{\epsilon_{3}}$
around the origin and $-1/T$, respectively, vanish. Thus, using Eqs. (2.24),
(2.23), (2.1) and making the substitutions
$r\rightarrow\varphi+1/2a,\,\varphi\rightarrow 1/2a\xi$, we finally arrive at
$\displaystyle h_{\text{abs}}(x,t)=\frac{a\Theta(ct-k)}{4\pi
c}e^{-1/2at}\int^{1}_{-1}e^{-1/2at\xi}(1-\xi)\times$
$\displaystyle\bigg{[}\frac{\cos(1/2ka\sqrt{1-\xi^{2}})}{\sqrt{1-\xi^{2}}}\xi+\sin(1/2ka\sqrt{1-\xi^{2}})\bigg{]}d\xi.$
(2.25)
The question arises how the found expression relates to the expression
obtained in Ref. [14], cp. also Eqs. (2.16) and (2.18). Actually, one can
explicitly show that Eq. (2.1) can be cast in an integral free form. To this
end we first note the identity
$\frac{1}{\pi}\int^{1}_{-1}e^{-1/2at\xi}\frac{\cos(1/2ak\sqrt{1-\xi^{2}})}{\sqrt{1-\xi^{2}}}d\xi=I_{0}(1/2a\sqrt{t^{2}-k^{2}}).$
(2.26)
Then, it follows that the terms involving the cosine in Eq. (2.1) can be
rewritten by taking the first and second order derivative of the expression on
the rhs of Eq. (2.26) with respect to $t$. Also, upon integration by parts we
can bring the term involving the sinus in a form that can be expressed
analogously.
The general case of a radiation boundary conditions can now be treated in a
similar way. Again, the part of the GF that takes into account the BC can be
expressed by a Bromwich integral
$\displaystyle h_{\text{rad}}(x,t)=-\frac{1}{4\pi
ic}\int^{\gamma+i\infty}_{\gamma-i\infty}e^{ts-k\sqrt{s(s+a)}}\times$
$\displaystyle\frac{s+a}{\sqrt{s(s+a)}}\frac{(1-\eta)\sqrt{s+a}-(1+\eta)\sqrt{s}}{(1-\eta)\sqrt{s+a}+(1+\eta)\sqrt{s}}ds,$
(2.27)
where $\eta=1-\beta$. Because the analytical properties of the integrand in
Eq. (2.1) are analogous to the case of purely absorbing BC, we employ the same
keyhole contour and proceed in the same way to find
$\displaystyle h_{\text{rad}}(x,t)=\frac{a\Theta(ct-k)}{4\pi
c}e^{-1/2at}\int^{1}_{-1}\frac{e^{-1/2at\xi}}{1+\eta^{2}+2\eta\xi}(1-\xi)\times\qquad\qquad\qquad$
$\displaystyle\bigg{[}\frac{\cos(\frac{ka}{2}\sqrt{1-\xi^{2}})}{\sqrt{1-\xi^{2}}}[2\eta+(1+\eta^{2})\xi]+[1-\eta^{2}]\sin(\frac{ka}{2}\sqrt{1-\xi^{2}})\bigg{]}d\xi.$
(2.28)
Here, two remarks are in order. First, as $\eta\rightarrow 0$, one recovers
the expression obtained for $h_{\text{abs}}$ Eq. (2.1). Second, we may expand
the denumerator $1+\eta^{2}+2\eta\xi$ in powers of $2\eta\xi/(1+\eta^{2})$,
because $2\eta/(1+\eta^{2})<1$ for $\eta<1$. As explained for
$h_{\text{abs}}$, the resulting integrals can be expressed as suitable time
derivatives of the expression on the rhs of Eq. (2.26), and in this way one
may recover the series expansion given in Eqs. (2.19), (2).
## 3 Backreaction boundary condition
Next, we will turn to reversible interactions with the boundary: when the
particle gets trapped this does not mean necessarily that it stays trapped
forever. First, we will derive a backreaction BC that is suitable for the one-
dimensional telegrapher’s equation. The derivation will follow the line of
reasoning in Ref. [14] and it will turn out that the backreaction BC is
somewhat more complicated than the corresponding BC in the case of diffusion,
resembling the situation for the radiation BC.
Conventionally, the salient assumption underlying the backreaction BC is that
the rate of desorption is proportional to the total absorbed population.
Hence, generalizing Eq. (2.9), we postulate that at the boundary the PDFs that
take into account the directions are related by
$a(x,t|x_{0})|_{x=0}=(1-\beta)b(x,t|x_{0})|_{x=0}+\frac{\beta\kappa}{2}[1-S(t|x_{0})].$
(3.1)
Here, $S(t|x_{0})$ denotes the survival probability, defined by
$S(t|x_{0})=\int^{\infty}_{0}p(x,t|x_{0})\,dx,$ (3.2)
and therefore $1-S(t|x_{0})$ is the total absorbed population. The total
probability density becomes
$\displaystyle p(0,t|x_{0})$ $\displaystyle=$ $\displaystyle
a(0,t|x_{0})+b(0,t|x_{0})$ (3.3) $\displaystyle=$
$\displaystyle(2-\beta)b(0,t|x_{0})+\frac{\beta\kappa}{2}[1-S(t|x_{0})].$
Additionally, we can conclude that
$a(0,t|x_{0})-b(0,t|x_{0})=-\frac{\beta}{2-\beta}\bigg{[}p(0,t|x_{0})-\kappa\big{[}1-S(t|x_{0})\big{]}\bigg{]}.$
(3.4)
Moreover, one has the dynamic equation [14]
$c\frac{\partial p}{\partial x}=-\frac{\partial}{\partial
t}(a-b)-\frac{1}{T}(a-b),$ (3.5)
which is valid in particular at the boundary $x=0$, and hence we arrive at the
backreaction BC for the one-dimensional telegrapher’s equation
$c\frac{\partial p}{\partial
x}\bigg{|}_{x=0}=\frac{\beta}{2-\beta}\bigg{\\{}\frac{\partial p}{\partial
t}\bigg{|}_{x=0}+\kappa\frac{\partial S(t|x_{0})}{\partial
t}+\frac{1}{T}\bigg{[}p|_{x=0}-\kappa\big{(}1-S(t|x_{0})\big{)}\bigg{]}\bigg{\\}}$
(3.6)
To shed more light on the nature of the parameter $\kappa$ it is instructive
to consider the limiting case of the BC Eq. (3.6). Taking the limit
$c\rightarrow\infty$, $T\rightarrow 0$ and $\beta/(2-\beta)\rightarrow 0$ such
that $c^{2}T=:D$ and $\beta/(2-\beta)c=:\kappa_{a}$ are kept finite, we
recover the radiation-backreaction BC [2] for the diffusion equation
$D\frac{\partial p}{\partial
x}\bigg{|}_{x=0}=\kappa_{a}p|_{x=0}-\kappa_{d}\big{(}1-S(t|x_{0})\big{)},$
(3.7)
where $\kappa_{d}:=\kappa_{a}\kappa$. On the other hand, when we take the same
limit as before with the exception that $\beta/(2-\beta)c\rightarrow\infty$,
we arrive at the absorption-backreaction BC [2]
$p|_{x=0}=\kappa\big{[}1-S(t|x_{0})\big{]}.$ (3.8)
To derive a GF of the telegrapher’s equation subject to the BC specified by
Eq. (3.6), one has to find a suitable expression for the survival probability.
To this end, we briefly recall how one proceeds in the case of diffusion. From
the equation of continuity Eq. (1.3) and the definition of the survival
probability Eq. (3.2) it follows
$\frac{\partial S(t|x_{0})}{\partial t}=j(x,t|x_{0})|_{x=0}.$ (3.9)
Assuming that the current vanishes at infinity and that Fick’s first law Eq.
(1.2) is valid one obtains
$S(t|x_{0})=1-\int^{t}_{0}\frac{\partial p(x,t|x_{0})}{\partial
x}\bigg{|}_{x=0}\,dt^{\prime}.$ (3.10)
Obviously, Eq. (3.10) cannot be used in the context of the telegrapher’s
equation, because we already discussed that Fick’s law has to be replaced by
Cattaneo’s equation Eq. (1.4). Hence, we combine Eq. (1.5) and Eq. (3.9),
leading to
$S(t|x_{0})=1-c^{2}\int^{t}_{0}\int^{t^{\prime}}_{0}e^{-(t^{\prime}-\tau)/T}\frac{\partial
p(x,\tau|x_{0})}{\partial x}\,d\tau\,dt^{\prime}.$ (3.11)
The derived expression for the survival probability can be introduced in the
BC Eq. (3.6). Henceforth, we will focus on the case $\beta=1.$ Then, in the
Laplace domain the BC becomes
$\bigg{(}c+\frac{\kappa c^{2}}{s}\bigg{)}\frac{\partial\tilde{p}}{\partial
x}\bigg{|}_{x=0}=\bigg{(}s+\frac{1}{T}\bigg{)}\tilde{p}\bigg{|}_{x=0}.$ (3.12)
As before, we make the ansatz
$\tilde{p}(x,s|x_{0})=\tilde{p}_{\text{free}}(x,s|x_{0})+\tilde{h}_{\text{back}}(x,s).$
(3.13)
The function $h_{\text{back}}(x,t)$ has to be a solution of the telegrapher’s
equation and hence takes in the Laplace domain the form
$\tilde{h}_{\text{back}}(x,s)=A(s)e^{-x/c\sqrt{s(s+\frac{1}{T})}}.$ (3.14)
The constant $A(s)$ must be chosen in such a way that Eq. (3.13) satisfies the
BC. We obtain
$A(s)=-\frac{1}{2c}\frac{s+a}{\sqrt{s(s+a)}}\frac{s+a-\big{(}1+\frac{c\kappa}{s}\big{)}\sqrt{s(s+a)}}{s+a+\big{(}1+\frac{c\kappa}{s}\big{)}\sqrt{s(s+a)}}e^{-\frac{x_{0}}{c}\sqrt{s(s+a)}}.$
(3.15)
As before, we can calculate $h_{\text{back}}(x,t)$ via the Bromwich integral
$\displaystyle h_{\text{back}}(x,t)=-\frac{1}{4\pi
ic}\int^{\gamma+i\infty}_{\gamma-i\infty}e^{ts-k\sqrt{s(s+a)}}\times$
$\displaystyle\frac{s+a}{\sqrt{s(s+a)}}\bigg{[}\frac{\sqrt{s+a}-\big{(}1+\frac{c\kappa}{s}\big{)}\sqrt{s}}{\sqrt{s+a}+\big{(}1+\frac{c\kappa}{s}\big{)}\sqrt{s}}\bigg{]}ds.$
(3.16)
We can closely follow the line of reasoning presented for the case of
absorbing and radiation BC. We may specify the same branch cut structure,
employ the same integration contour and make the same substitutions. Thus, we
arrive at
$\displaystyle h_{\text{back}}(x,t|x_{0})=\frac{a\Theta(ct-k)}{4\pi
c}e^{-1/2at}\int^{1}_{-1}\frac{e^{-1/2at\xi}(1-\xi)}{c^{2}\kappa^{2}+a/2(a-2c\kappa)(1+\xi)}\times\qquad\qquad$
$\displaystyle\bigg{[}\frac{\cos(1/2ka\sqrt{1-\xi^{2}})}{\sqrt{1-\xi^{2}}}\Pi(\xi)+\sin(1/2ka\sqrt{1-\xi^{2}})\Gamma(\xi)\bigg{]}d\xi.$
(3.17)
Here, we introduced the functions
$\displaystyle\Pi(\xi)$ $\displaystyle=$
$\displaystyle\frac{1}{2}a^{2}\xi^{2}+\frac{1}{2}a(a-2c\kappa)\xi-c\kappa+c^{2}\kappa^{2},$
(3.18) $\displaystyle\Gamma(\xi)$ $\displaystyle=$
$\displaystyle\frac{1}{2}a^{2}\xi+\frac{1}{2}a(a-2c\kappa).$ (3.19)
In the limit $\kappa\rightarrow 0$, we recover the result for the purely
absorbing BC Eq. (2.1). Furthermore, we note that
$\bigg{|}\frac{\frac{a}{2}(a-2c\kappa)}{\frac{a}{2}(a-2c\kappa)+c^{2}\kappa^{2}}\bigg{|}<1,$
(3.20)
and therefore in this case also it is possible to expand the denumerator in a
power expansion in
$\frac{\frac{a}{2}(a-2c\kappa)}{\frac{a}{2}(a-2c\kappa)+c^{2}\kappa^{2}}\xi.$
Using Eq. (2.26) this means that we can dispense with the integral
representation altogether in the case of the backreaction BC also.
Figure 1: Integration contour used in Eq. (2.1).
### Acknowledgments
This research was supported by the Intramural Research Program of the NIH,
National Institute of Allergy and Infectious Diseases.
We would like to thank Bastian R. Angermann and Frederick Klauschen for
stimulating discussions.
## References
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|
arxiv-papers
| 2013-01-30T04:30:33 |
2024-09-04T02:49:40.945471
|
{
"license": "Public Domain",
"authors": "Thorsten Pr\\\"ustel and Martin Meier-Schellersheim",
"submitter": "Thorsten Pr\\\"ustel",
"url": "https://arxiv.org/abs/1301.7139"
}
|
1301.7153
|
# Weak Concurrent Kleene Algebra with Application to Algebraic Verification
Annabelle McIver1 Tahiry Rabehaja1 Georg Struth3 1 Department of Computing
Macquarie University,
Sydney, Australia
Email: {annabelle.mciver,tahiry.rabehaja}@mq.edu.au
2 Department of Computer Science
University of Sheffield,
United Kingdom,
Email: [email protected]
###### Abstract
We propose a generalisation of concurrent Kleene algebra Hoa09 that can take
account of probabilistic effects in the presence of concurrency. The algebra
is proved sound with respect to a model of automata modulo a variant of rooted
$\eta$-simulation equivalence. Applicability is demonstrated by algebraic
treatments of two examples: algebraic may testing and Rabin’s solution to the
choice coordination problem.
This work has been supported by the iMQRES grant from Macquarie University.
## 1 Introduction
Kleene algebra generalises the language of regular expressions and, as a basis
for reasoning about programs and computing systems, it has been used in
applications ranging from compiler optimisation, program refinement,
combinatorial optimisation and algorithm design Con71 ; Koz94 ; Koz00a ;
Koz00b ; Mci06 . A number of variants of the original axiom system and
language of Kleene algebra have extended its range of applicability to include
probability Mci05 with the most recent being the introduction of a
concurrency operator Hoa09 . Main benefits of the algebraic approach are that
it captures some essential aspects of computing systems in a simple and
concise way and that the calculational style of reasoning it supports is very
suitable for automated theorem proving.
In this paper we continue this line of work and propose _weak concurrent
Kleene algebra_ , which extends the abstract probabilistic Kleene algebra
Mci05 with the concurrency operator of concurrent Kleene algebra Hoa09 and
thus supports reasoning about concurrency in a context of probabilistic
effects. This extension calls for a careful evaluation of the axiom system so
that it accurately accounts for the interactions of probabilistic choice,
nondeterministic choice and the treatment of concurrency. For example
probabilistic Kleene algebra accounts for the presence of probability in the
_failure_ of the original distributive law $x(y+z)=xy+xz$ which is also absent
in most process algebras. That is because when the terms $x,y,z$ are
interpreted as probabilistic programs, with $xy$ meaning “first execute $x$
and then $y$” and $+$ interpreted as a nondeterministic choice, the expression
on the left hand side exhibits a greater range of nondeterminism than the
right in the case that $x$ includes probabilistic behaviours. For example if
$x$ is interpreted as a program which flips a bit with probability $1/2$ then
the following nondeterministic choice in $y+z$ can always be resolved so that
$y$ is executed _if and only if_ the bit was indeed flipped. This is not a
behaviour amongst those described by $xy+xz$, where the nondeterminism is
resolved before the bit is flipped and therefore its resolution is unavoidably
independent of the flipping. Instead, in contexts such as these,
distributivity be replaced by a weaker law:
$\textrm{Sub-distributivity:}\qquad xy+xz~{}\leq~{}x(y+z)~{}.$ (1)
Elsewhere Rab11 we show that this weakening of the original axioms of Kleene
algebra results in a complete system relative to a model of nondeterministic
automata modulo simulation equivalence.
The behaviour of the concurrency operator of concurrent Kleene algebra Hoa09
is captured in particular by the _Interchange law_ :
$(x\|y)(u\|v)~{}\leq~{}(xu)\|(yv)$
which expresses that there is a lesser range of nondeterministic executions on
the left where, for example, the execution of $u$ is constrained to follow a
complete execution of $x$ run concurrently with $y$ but on the right it is
not.
Our first contribution is the construction of a concrete model of abstract
probabilistic automata (where the probability is at the action level) over
which to interpret terms composed of traditional Kleene algebra together with
concurrent composition. In this interpretation, each term represents an
automaton. For example in Equation (1), $x,y$ and $z$ are automata and so is
$xy+xz$. We show that the axiom system of concurrent Kleene algebra weakened
to allow for the presence of probability is sound with respect to those
probabilistic automata. Our use of probabilistic automata is similar to models
where the resolution of probability and nondeterminism can be interleaved;
concurrent composition of automata models CSP synchronisation Hoa78 in that
context. Finally we use a notion of rooted $\eta$-simulation to interpret the
inequality $\leq$ used in algebraic inequations.
Our second contribution is to explore some applications of our axiomatisation
of weak concurrent Kleene algebra, to explain our definition of rooted
$\eta$-simulation in terms of may testing Nic83 , and to demonstrate the proof
system on Rabin’s distributed consensus protocol Rab82 .
One of the outcomes of this study is to expose the tensions between the
various aspects of system execution. Some of the original concurrent Kleene
algebra axioms Hoa09 required for the concurrency operator now fail to be
satisfiable in the presence of probabilistic effects and synchronisation
supported by the interchange law. For example, the term $1$ from Kleene
algebra (interpreted as “do nothing”) can no longer be a neutral element for
the concurrency operator $\|$ — we only have the specific equality $1\|1=1$
and not the more general $1\|x=x$. In fact we chose to preserve the full
interchange law in our choice of axioms because it captures so many notions of
concurrency already including exact parallel and synchronisation, suggesting
that it is a property about general concurrent interactions.
A feature of our approach is to concentrate on broad algebraic structures in
order to understand how various behaviours interact rather than to study
precise quantitative behaviours. Thus we do not include an explicit
probabilistic choice operator in the signature of the algebra — probability
occurs explicitly only in the concrete model as a special kind of asynchronous
probabilistic action combined with internal events (events that the
environment cannot access). This allows the specification of complex
concurrent behaviour to be simplified using applications of weak
distributivity embodied by Equation (1) and/or the interchange law as
illustrated by our case study. Finally we note that the axiomatisation we give
is entirely in terms of first-order expressions and therefore is supported by
first-order reasoning. Thus all of our algebraic proofs has been implemented
within the Isabelle/HOL theorem proving environment. These proof can be found
in a repository of formalised algebraic theorems.
111http://staffwww.dcs.shef.ac.uk/people/G.Struth/isa/
In Section 2 we explore the axiomatisation of the new algebra. It is
essentially a mixture of probabilistic and concurrent Kleene algebras.
Sections 3 and 4 are devoted to showing the consistency of our approach. A
concrete model based on automata and $\eta$-simulation is constructed. In
section 5, we compare our approach with probabilistic automata (automata that
exhibit explicit probability) and probabilistic simulation. We conclude that,
up to some constraint, the concrete model is a very special case of that more
general model. In sections 6 and 7, we present some applications, in
particular an algebraic version of may testing is studied and variations of
the specification of Rabin’s protocol are explored.
In this paper $x,y,$ etc represent algebraic expressions or variables. Terms
are denoted $s,t,$ etc. Letters $a,b,$ etc stand for actions and $\tau$
represents an internal action. An automaton associated with a term or an
expression is usually denoted by the same letter. Other notation is introduced
as we need it.
In this extended abstract we can only explain the main properties of weak
concurrent Kleene algebra and sketch the construction of the automaton model.
Detailed constructions and proofs of all statements in this paper can be found
in an appendix.
## 2 Axiomatisation
A Kleene algebra is a structure that encodes algebraically the sequential
behaviour of a system. It is generally presented in the form of an idempotent
222Idempotence refers to the operation $+$ i.e. $x+x=x$. semiring structure
$(K,+,\cdot,0,1)$ where $x\cdot y$ (sequential composition) is sometimes
written using juxtaposition $xy$ in expressions. The term $0$ is the neutral
element of $+$ and $1$ is the neutral element of $\cdot$. The semiring is then
endowed with a unary Kleene star $*$ representing finite iteration to form a
Kleene algebra. This operator is restricted by the following axioms:
$\displaystyle\textrm{Left unfold:}\qquad 1+xx^{*}$ $\displaystyle=$
$\displaystyle x^{*},$ (2) $\displaystyle\textrm{Left induction:}\hskip
4.2679pt\qquad xy\leq y$ $\displaystyle\Rightarrow$ $\displaystyle x^{*}y\leq
y,$ (3)
where $x\leq y$ if and only if $x+y=y$. In the sequel our interpretations will
be over a version of probabilistic automata. In particular we will interpret
$\leq$ and $=$ as $\eta$-simulations.
Often, the dual of (2-3) i.e. $1+x^{*}x=x^{*}$ and $yx\leq y\Rightarrow
yx^{*}\leq y$ are also required. However, (2) and (3) are sufficient here and
the dual laws follow from continuity of sequential composition for finite
automata.
In a Kleene algebra, the semiring structure supports two distributivity laws:
$\displaystyle\textrm{Left distributivity:}\qquad\hskip 2.27621ptxy+xz$
$\displaystyle=$ $\displaystyle x(y+z),$ (4) $\displaystyle\textrm{Right
distributivity:}\qquad(x+y)z$ $\displaystyle=$ $\displaystyle xz+yz.$ (5)
Equation (4) however is not valid in the presence of probability. For example,
compare the behaviour of probabilistic choice in the diagrams below. Here,
$\mathtt{flip}_{p}$ denotes the process that flips a $p$-biased coin, which we
can represent by a probabilistic automaton (details are given in Section 3).
$\scriptstyle{\mathtt{flip}_{p}}$$\scriptstyle{\mathtt{flip}_{p}}$
$\scriptstyle{\mathtt{flip}_{p}}$$\scriptstyle{x}$$\scriptstyle{y}$$\scriptstyle{x}$$\scriptstyle{y}$
In the right diagram, the choice between $a$ and $b$ can be based on the
outcome of the coin flip but such resolution is not possible in the left-hand
diagram. We express the greater range of possible outcomes by the general
inequation (1), specifically here it becomes
$(\mathtt{flip}_{p})y+(\mathtt{flip}_{p})z\leq(\mathtt{flip}_{p})(y+z).~{}$
(6)
As mentioned above, the zero of a Kleene algebra satisfies:
$\displaystyle\textrm{Left annihilation:}\qquad 0x$ $\displaystyle=$
$\displaystyle 0,$ (7) $\displaystyle\textrm{Right annihilation:}\qquad x0$
$\displaystyle=$ $\displaystyle 0.$ (8)
In our interpretation that includes concurrency, we assume that $0$ captures
_deadlock_. However, axiom (8) is no longer appropriate because we should be
able to differentiate between the process doing an action and deadlocking from
a process that is just deadlocked.
###### Definition 1.
A weak probabilistic Kleene algebra is a structure $(K,+,\cdot,*,0,1)$ that
satisfies the axioms of Kleene algebra except there is no left distributivity
(it is replaced by (1)) and Equation (8) does not hold generally.
A concurrency operator was added to Kleene algebra by Hoare et al Hoa09 . Our
concurrency operator $\|$ satisfies the following standard axioms:
$\displaystyle\textrm{Associativity:}\qquad x\|(y\|z)$ $\displaystyle=$
$\displaystyle(x\|y)\|z,$ (9) $\displaystyle\textrm{Commutativity:}\hskip
36.98866ptx\|y$ $\displaystyle=$ $\displaystyle y\|x,$ (10)
$\displaystyle\textrm{One-idempotence:}\hskip 38.41139pt1\|1$ $\displaystyle=$
$\displaystyle 1.$ (11)
In Hoa09 , $\|$ satisfies the identity $1\|x=x$ which we do not have here
because in the concrete model, we will interpret $\|$ as the synchronisation
operator found in CSP Hoa78 . However, we still maintain the instance of that
law in the special case $x=1$ (see axiom (11)) where $1$ is interpreted as “do
nothing”.
Next we have the axioms dealing the interaction of $\|,+$ and $\cdot$.
$\displaystyle\textrm{Monotonicity :}\qquad x\|y+x\|z$ $\displaystyle\leq$
$\displaystyle x\|(y+z)$ (12) $\displaystyle\textrm{Interchange-law:}\hskip
8.5359pt(x\|y)(u\|v)$ $\displaystyle\leq$ $\displaystyle(xu)\|(yv)$ (13)
The interchange law is the most interesting axiom of concurrent Kleene
algebra. In fact it allows the derivation of many properties involving $\|$.
To illustrate this in the probabilistic context, consider a probabilistic
vending machine $\mathtt{VM}$ which we describe as the expression
$\mathtt{VM}\ =\
\mathtt{coin}\cdot\mathtt{flip}_{p}\cdot(\tau_{h}\cdot(\mathtt{tea}+1)+\tau_{t}\cdot(\mathtt{coffee}+1))$
where $\mathtt{coin},\mathtt{tea},\mathtt{coffee},\tau_{h},\tau_{t}$ and
$\mathtt{flip}_{p}$ are all represented by automata. That is the vending
machine accepts a coin and then decides internally whether it will enable the
button coffee or tea. The decision is determined by the action
$\mathtt{flip}_{p}$ 444i.e. the automaton that performs a $\mathtt{flip}_{p}$
action. which (as explained later) enables either $\tau_{h}$ or $\tau_{t}$.
The actions $\tau_{t}$ and $\tau_{h}$ are internal and the user cannot access
them. Now, a user who wants to drink tea is specified as
$\mathtt{U}\ =\ \mathtt{coin}\cdot(\mathtt{tea}+1).$
The system becomes $\mathtt{U}\|\mathtt{VM}$ where the concurrent operation is
CSP like and synchronises on $\mathtt{coin},\mathtt{tea}$ and
$\mathtt{coffee}$. The interchange law together with the other axioms and some
system assumptions imply the following inequation:
$\mathtt{U}\|\mathtt{VM}\ \geq\
\mathtt{coin}\cdot\mathtt{flip}_{p}\cdot(\tau_{h}\cdot(\mathtt{tea}+1)+\tau_{t})$
(14)
which is proved automatically in our repository. In other words, the user will
only be satisfied with probability at least $p$ since the right-hand side
equation says that the tea action can only be enabled provided that $\tau_{h}$
is enabled, and in turn that is determined by the result of the
$\mathtt{flip}_{p}$ action.
Now we are ready to define our algebra.
###### Definition 2.
A weak concurrent Kleene algebra is a weak probabilistic Kleene algebra
$(K,+,\cdot,*,0,1)$ with a concurrency operator $\|$ satisfying (9-13)
We assume the operators precedence $*<\cdot<\|<+$.
###### Proposition 3.
Let $s,t$ be terms, the following equations holds in weak concurrent Kleene
algebra.
1. 1.
All the operators are monotonic.
2. 2.
$(s^{*}\|t^{*})^{*}=s^{*}\|t^{*}$.
3. 3.
$(s\|t)^{*}\leq s^{*}\|t^{*}$.
4. 4.
$(s+t)^{*}=(s^{*}t^{*})^{*}$.
## 3 Concrete Model
### 3.1 Semantic Space
We use nondeterministic automata to construct a concrete model. An automaton
is denoted by a tuple
$(P,\longrightarrow,i,F)$
where $P$ is a set of states. The set $\longrightarrow\subseteq
P\times\Sigma\times P$ is a transition relation and we write
$x\stackrel{{\scriptstyle a}}{{\longrightarrow}}y$ when there is a transition,
labelled by $a$, from state $x$ to state $y$. The alphabet $\Sigma$ is left
implicit and considered to be fixed for every automaton. The state $i\in P$ is
the initial state and $F\subseteq P$ is the set of final states of the
automaton. In the sequel, we will denote an automaton
$(P,\longrightarrow,i,F)$ by its set of states $P$ when no confusion is
possible.
The actions in the alphabet $\Sigma$ are categorised into three kinds:
* •
internal: actions that will be “ignored” by the simulation relation (as in
$\tau_{h}$ and $\tau_{t}$). Internal actions are never synchronised by $\|$.
* •
external: actions that _can_ be synchronised. Probabilistic actions are
external (as in $\mathtt{flip}_{p}$) but they are _never_ synchronised.
* •
synchronised: external actions that will be synchronised when applying $\|$
(as in $\mathtt{coin},\mathtt{tea}$ and $\mathtt{coffee}$). These actions are
determined by a set of external actions $A$. More specifically, $\|$ refers to
${}_{A}\\!\|$ which we assume is fixed and given beforehand.
The special case of probabilistic choice is modelled by combining
probabilistic and internal actions. That is a process that does $a$ with
probability $p$ and does $b$ with probability $1-p$ is interpreted as the
following automaton
$\scriptstyle{\mathtt{flip}_{p}}$$\scriptstyle{\tau_{h}}$$\scriptstyle{\tau_{l}}$$\scriptstyle{a}$$\scriptstyle{b}$
where $\mathtt{flip}_{p}\in\Sigma$ represents the action of flipping a
$p$-biased coin which produces head with probability $p$ and tail with
probability $1-p$. The internal actions $\tau_{t}$ and $\tau_{h}$ are enabled
according to the result of $\mathtt{flip}_{p}$. Hence only one of $\tau_{h}$
and $\tau_{t}$ will be enabled just after the coin flip. Since $\tau_{t}$ and
$\tau_{h}$ are internal actions, the choice is internal and based upon the
outcome of $\mathtt{flip}_{p}$. The important facts here are that the choice
after $\mathtt{flip}_{p}$ is internal so could be based on the probabilistic
outcome of $\mathtt{flip}_{p}$ and that the environment cannot interfere with
that choice. These two behavioural characteristics are what we consider to be
the most general features of probability in a concurrent setting and they are
those which we axiomatise and record in our concrete model.
Next, we impose some conditions on the automata to ensure soundness.
* -
reachability: every state of the automaton is reachable by following a finite
path from the initial state.
* -
initiality: there is no transition that leads to the initial state. This means
that $a^{*}$ corresponds to the automata associated to $1+aa^{*}$ rather than
a self loop labeled by $a\in\Sigma$.
We denote by $\mathbf{Aut}$ the set of automata satisfying these two
conditions. The next step is to define the operators that act on
$\mathbf{Aut}$. We use the standard inductive construction found in Coh09 ;
Gla90 ; Rab11 and the diagrams illustrating the constructions are given in
the appendix.
* Deadlock: $0$
This is the automaton that has only one state, namely the initial state, and
no transition at all. It is the tuple $(\\{i\\},\emptyset,i,\emptyset)$.
* Skip: $1$
This is the automaton that has only one state $i$ which is both initial and
final. This automaton has no transition i.e. is denoted by
$(\\{i\\},\emptyset,i,\\{i\\})$.
* Single action:
The automata associated with $a$ is $i\stackrel{{\scriptstyle
a}}{{\longrightarrow}}\circ$ where $i$ is the initial state and $\circ$ is a
final state. It is the tuple $(\\{i,\circ\\},\\{i\stackrel{{\scriptstyle
a}}{{\longrightarrow}}\circ\\},i,\\{\circ\\})$.
* Addition: $P+Q$
This automaton is obtained using the standard construction of identifying the
initial states of $P$ and $Q$. (This is possible due to the initiality
property.)
* Multiplication: $PQ$ (or $P\cdot Q$)
This automaton is constructed in the standard way of identifying copies of the
initial state of $Q$ with final states of $P$.
* Concurrency: $P_{A}\\!\|Q$
This automaton is constructed as in CSP Hoa78 . It is a sub-automaton of the
Cartesian product of $P$ and $Q$. The initial state is $(i_{P},i_{Q})$ and
final states are reachable elements of $F_{P}\times F_{Q}$. Notice that the
set $A$ never contains probabilistic actions. Further explanation about
${}_{A}\\!\|$ is given below.
* Kleene star: $P^{*}$
This automaton is the result of repeating $P$ allowing a successful
termination after each (possibly empty) full execution of $P$. The initial
state of $P^{*}$ is final and copies of the initial state of $P$ are
identified with the final states of $P$.
All automata begin with an initial state and end in some final or deadlock
state. Our main use of final states is in the construction of sequential
composition and Kleene star.
The concurrency operator ${}_{A}\\!\|$ synchronises transitions labeled by an
action in $A$ and interleaves the others (including internal transitions). As
in CSP, a synchronised transition waits for a corresponding synchronisation
action from the other argument of ${}_{A}\\!\|$. This is another reason we do
not have $1_{\\{a\\}}\\!\|P=P$ because if $P=i_{P}\stackrel{{\scriptstyle
a}}{{\longrightarrow}}\circ$ and $i_{P}$ is not a final state, then
$1_{\\{a\\}}\\!\|P=(\\{(i,i_{P})\\},\emptyset,(i,i_{P}),\emptyset)=0.$
###### Proposition 4.
These operations of weak concurrent Kleene algebra are well defined on
$\mathbf{Aut}$ that is if $P,Q\in\mathbf{Aut}$ then $P+Q,PQ,P_{A}\\!\|Q$ and
$P^{*}$ are elements of $\mathbf{Aut}$.
The proof consists of checking that $P+Q,PQ,P\|Q$ and $P^{*}$ satisfy the
reachability and initiality conditions whenever $P$ and $Q$ satisfy the same
conditions. (See Proposition 20 in the appendix).
In the sequel, whenever we use an unframed concurrency operator $\|$, we mean
that the frame $A$ has been given and remains fixed.
### 3.2 Equivalence
The previous subsection has given us the objects and operators needed to
construct our concrete model. Next we turn to the interpretation of equality
for our concrete interpretation.
Following the works found in Coh09 ; Rab11 ; Mil71 , we again use a
simulation-like relation to define valid equations in the concrete model. More
precisely, due to the presence of internal actions, we will use an
$\eta$-simulation as the basis for our equivalence.
Before we give the definition of simulation, we need the following notation.
Given the state $x$ and $y$, we write $x\Rightarrow y$ if there exists a path,
possibly empty, from $x$ to $y$ such that it is labelled by internal actions
only. This notation is also used in Gla90 with the same meaning.
###### Definition 5.
Let $P,Q$ be automata, a relation $S\subseteq P\times Q$ (or $S:P\rightarrow
Q)$ is called $\eta$-simulation if
* –
$(i_{P},i_{Q})\in S$,
* –
if $(x,y)\in S$ and $x\stackrel{{\scriptstyle
a}}{{\longrightarrow}}x^{\prime}$ then
* a)
if $a$ is internal then there exits $y^{\prime}$ such that $y\Rightarrow
y^{\prime}$ and $(x^{\prime},y^{\prime})\in S$,
* b)
if $a$ is external then there exists $y_{1}$ and $y^{\prime}$ in $Q$ such that
$y\Rightarrow y_{1}\stackrel{{\scriptstyle a}}{{\longrightarrow}}y^{\prime}$
and $(x,y_{1})\in S$ and $(x^{\prime},y^{\prime})\in S$.
* –
if $(x,y)\in S$ and $x\in F_{P}$ then $y\in F_{Q}$.
A simulation $S$ is rooted if $(i_{P},y)\in S$ implies $y=i_{Q}$. If there is
a rooted simulation from $P$ to $Q$ then we say that $P$ is simulated by $Q$
and we write $P\leq Q$. Two processes $P$ and $Q$ are simulation equivalent if
$P\leq Q$ and $Q\leq P$, and we write $P\equiv Q$. In the sequel, rooted any
$\eta$-simulation will be referred simply as a simulation.
Relations satisfying Definition 5 are also $\eta$-simulation in the sense of
Gla90 where property (a) is replaced by:
$\textrm{if }a\textrm{ is internal then }(x^{\prime},y)\in S.$ (15)
The identity relation (drawn as dotted arrow) in the following diagram is a
simulation relation satisfying Definition 5, but it is not a simulation in the
sense of Gla90 .
$\scriptstyle{\tau}$$\scriptstyle{\tau}$$\textstyle{\circ\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\circ}$
We need the identity relation to be a simulation here because in our proof of
soundness, more complex simulations are constructed from identity relations.
###### Proposition 6.
The following statements hold.
1. 1.
The relational composition of two rooted $\eta$-simulations is again a rooted
$\eta$-simulation. That is, if $S,T$ are rooted $\eta$-simulations then
$S\circ T$ is also a rooted $\eta$-simulation, where $\circ$ denotes
relational composition.
2. 2.
The simulation relation $\leq$ is a preorder on $\mathbf{Aut}$.
Proposition 6 is proven in Proposition 21 of the appendix.
Therefore, $\equiv$ as determined by Definition 5 is an equivalence. In fact,
we prove in the following proposition that it is a congruence with respect to
$+$.
###### Proposition 7.
The equivalence relation $\equiv$ is a congruence with respect to $+$ and
$P\leq Q$ iff $P+Q\equiv Q$.
The proof adapts and extends the one found in Gla90 and the specialised
version for our case is Proposition 22 in the appendix.
It is well documented that $\eta$-simulation is not a congruence without the
rootedness condition Gla90 . A typical example is given by the expressions
$\tau a+\tau b$ and $\tau(a+b)$. The automata associated to these expressions
are equivalent under non-rooted $\eta$-simulation.
The manipulation of probabilistic actions is also an important facet of our
model. We assume that probabilistic actions are not synchronised and in that
respect they are similar to internal actions. However probabilistic actions
cannot be treated as internal as the following examples illustrates. Consider
the action $\mathtt{flip}_{1/2}$ which flips a fair coin. If $\mathtt{flip}$
is an internal action then the inequality
$(\mathtt{flip}_{1/2})(\tau a+\tau b)\leq(\mathtt{flip}_{1/2})\tau
a+(\mathtt{flip}_{1/2})\tau b$
would be valid when interpreted in the concrete model. In other words, we
would have the following simulation:
$\scriptstyle{\mathtt{flip}_{1/2}}$$\scriptstyle{\mathtt{flip}_{1/2}}$$\scriptstyle{\mathtt{flip}_{1/2}}$$\scriptstyle{\tau}$$\scriptstyle{\tau}$$\scriptstyle{\tau}$$\scriptstyle{\tau}$$\scriptstyle{a}$$\scriptstyle{b}$$\scriptstyle{a}$$\scriptstyle{b}$
But this relationship (which implies distributivity of $\mathtt{flip}_{p}$
through $+$) does not respect the desired behaviour of probability which, as
we explained earlier, satisfies only a weaker form of distributivity. Whence,
we assume that probabilistic actions such as $\mathtt{flip}_{1/2}$ are among
the external actions which will never be synchronised.
## 4 Soundness
In this section, we prove that the set $\mathbf{Aut}$ endowed with the
operators defined in Subsection 3.1 modulo rooted $\eta$-simulation
equivalence (Subsection 3.2) forms a weak concurrent Kleene algebra.
The first part is to prove that $\mathbf{Aut}$ is a weak probabilistic Kleene
algebra.
###### Proposition 8.
$(\mathbf{Aut},+,\cdot,*,0,1)$ is a weak probabilistic Kleene algebra.
The proof consists of detailed verifications of the axioms for weak
probabilistic Kleene algebra (see Proposition 23 in the appendix).
The second part consists of proving that $\|$ satisfies the equations (9-13).
Associativity depends heavily on the fact that both concurrent compositions
involved in $x\|y\|z$ have the same frame set. For instance, let
$\Sigma=\\{a,b,c\\}$. The identities
$(a_{\\{a\\}}\\!\|b)_{\\{c\\}}\\!\|a=ab0+ba0$
and
$a_{\\{a\\}}\\!\|(b_{\\{c\\}}\\!\|a)=ab+ba$
are valid in the concrete model. Hence, the first process will always go into
a deadlock state though the second one will always terminate successfully.
Therefore, to have associativity, the concurrency operator must have a fixed
frame.
###### Proposition 9.
$(\mathbf{Aut},+,\cdot,_{A}\\!\|,1)$ satisfies equations (9\- 13) modulo
rooted $\eta$-simulation equivalence for any set of synchronisable actions
$A\subseteq\Sigma$ (i.e. no probabilistic actions).
Associativity is mainly a consequence of the fact that there is only one frame
for $\|$. The other axioms need to be checked thoroughly (see Proposition 24).
Our soundness result directly follows from these two propositions.
###### Theorem 10.
$(\mathbf{Aut},+,\cdot,_{A}\\!\|,*,0,1)$ is a weak concurrent Kleene algebra
for any set of synchronisable actions $A\subseteq\Sigma$.
In this theorem, the frame $A$ is fixed beforehand. In other words, a model of
weak concurrent Kleene algebra is constructed for each possible choice of $A$.
In particular, if $A$ is empty then the concurrency operator is interleaving
all actions i.e. no actions are synchronised. This particular model satisfies
the identity $1_{\emptyset}\\!\|x=x$ of the original concurrent Kleene algebra
found in Hoa09 .
The sequential and concurrent composition actually have stronger properties in
the concrete model. If we consider finite automata only — automata with
finitely many states and transitions— then we show that these two operators
are conditionally Scott continuous in the sense of Rab11 (see Proposition 25
and 27 in the appendix).
## 5 Relationship to Probabilistic Processes
Firstly, it is shown in Mci04 that a probabilistic choice $a_{p}\\!\\!\oplus
b$ simulates the nondeterministic choice $a+b$. A similar result also holds in
our setting. In the absence of internal transitions, simulation has been also
defined elsewhere Coh09 ; Gla90 ; Rab11 which we will refer to as strong
simulation. Recall that
$(\mathtt{flip}_{p})a+(\mathtt{flip}_{p})b\leq(\mathtt{flip}_{p})(a+b)$ is a
general property of probabilistic Kleene algebra Mci05 so it is valid under
strong simulation equivalence Coh09 ; Rab11 . Due to the absence of internal
actions, the middle part of the diagram of Figure 1 does not exist with
respect to strong simulation equivalence.
In the context of Definition 5, the right-hand simulation of Figure 1 is the
refinement of probabilistic choice by nondeterminism. This example gives an
explicit distinction between $(\mathtt{flip}_{p})(a+b)$ and
$(\mathtt{flip}_{p})a+(\mathtt{flip}_{p})b$ by considering the fact that the
choice in $(\mathtt{flip}_{p})(a+b)$ can depend on the probabilistic outcome
of $(\mathtt{flip}_{p})$, but this is not the case for
$(\mathtt{flip}_{p})a+(\mathtt{flip}_{p})b$.
$\scriptstyle{\mathtt{flip}_{p}}$$\scriptstyle{\mathtt{flip}_{p}}$$\scriptstyle{\mathtt{flip}_{p}}$$\scriptstyle{\mathtt{flip}_{p}}$$\scriptstyle{\tau}$$\scriptstyle{\tau}$$\scriptstyle{a}$$\scriptstyle{b}$$\scriptstyle{a}$$\scriptstyle{b}$$\scriptstyle{a}$$\scriptstyle{b}$
Figure 1: Refinements between probabilistic choice and nondeterminism.
Secondly, we discuss about the relationship between our concrete model and
probabilisitic automata. Remind that our interpretation of probability lies in
the use of actions that implicitly contain probabilistic information. In its
most general form, a probabilistic choice between $n$ possibilities can be
written as
$\mathtt{flip}_{p_{1},\dots,p_{n}}\cdot(\tau_{1}\cdot
a_{1}+\dots+\tau_{n}\cdot a_{n})$
where $\sum_{i}p_{i}=1$. In this algebraic expression, we implicitly ensure
that each guard $\tau_{i}$ is enabled with a corresponding probability
$p_{i}$. Therefore, if these $\tau_{i}$’s are not found directly after the
execution of the probabilistic action then matching them with the
corresponding $p_{i}$ becomes a difficult task. We call $p$-automaton 555The
name $p$-automata describes probabilistic automata and as we will see later
on, there is a relationship between the two of them. a transition system as
per the definition of Subsection 3.1 such that if a probabilistic action has
associated $\tau$ transitions then all of them follow that action directly.
Another complication also arises from the use of these $\tau_{i}$’s. Consider
the following two processes
$\mathtt{flip}_{p_{1},p_{2}}\cdot(\tau_{1}\cdot a+\tau_{2}\cdot b)$
and
$\mathtt{flip}_{p_{1},p_{2}}\cdot(\tau_{1}\cdot b+\tau_{2}\cdot a)$
where $p_{1}+p_{2}=1$. We can construct a (bi)simulation relation between the
corresponding automata though the probabilities of doing an $a$ are different.
Hence we need to modify the definition of $\eta$-simulation (Definition 5) to
account for these particular structure.
###### Definition 11.
A $p$-simulation $S$ between two $p$-automata $P,Q$ is a $\eta$-simulation
such that if
* -
$x\stackrel{{\scriptstyle\mathtt{flip}_{p_{1},\dots,p_{n}}}}{{\longrightarrow}}x^{\prime}\stackrel{{\scriptstyle\tau_{i}}}{{\longrightarrow}}x_{i}^{\prime\prime}$
is a transition in $P$,
* -
$y\stackrel{{\scriptstyle\mathtt{flip}_{p_{1},\dots,p_{n}}}}{{\longrightarrow}}y^{\prime}\stackrel{{\scriptstyle\tau_{i}}}{{\longrightarrow}}y_{i}^{\prime\prime}$
is a transition in $Q$,
* -
and $(x,y)\in S$
then $(x_{i}^{\prime\prime},y_{i}^{\prime\prime})\in S$, for each
$i=1,\dots,n$.
This definition ensures that the probability of doing a certain action from
$y$ is greater than doing that action from $x$. With similar proofs as in the
previous Sections, we can show that the set of $p$-automata modulo
$p$-simulation forms again a weak concurrent Kleene algebra. We denote
$p$-$\mathbf{Aut}$ the set of $p$-automata modulo $p$-simulation.
We will now show that this definition is a very special case of probabilistic
simulation on probabilistic automata. To simplify the comparison, we assume
that $\tau$ transitions occur only as part of these probabilistic choices in
$p$-automata.
###### Definition 12.
A probabilistic automaton is defined as a tuple $(P,\longrightarrow,\Delta,F)$
where $P$ is a set of states, $\longrightarrow$ is a set of labelled
transitions from state to distributions 666We assume that all distributions
are finitely supported. of states i.e. $\longrightarrow\subseteq
P\times\Sigma\times\mathcal{D}P$, $\Delta$ is the initial distribution and
$F\subseteq P$ is a set of final states.
The notion of simulation also exists for probabilistic automata Seg94 and, in
particular, simulation and failure simulation is discussed in Den07 where
they are proven to be equivalent to may and must testing respectively.
To give a proper definition of probabilistic simulation, we need the following
notations which are borrowed from Den07 and Gla90 . Given a relation
$R\subseteq P\times\mathcal{D}Q$, the lifting of $R$ is a relation
$\hat{R}\subseteq\mathcal{D}P\times\mathcal{D}Q$ such that $\phi\hat{R}\psi$
iff:
* -
$\phi=\sum_{x}p_{x}\delta_{x}$, 777We denote by $\delta_{x}$ the point
distribution concentrated on $x$.
* -
for each $x\in\mathrm{supp}(\phi)$ (the support of $\phi$) there exists
$\psi_{x}\in\mathcal{D}Q$ such that $xR\psi_{x}$,
* -
$\psi=\sum_{x}p_{x}\psi_{x}$.
Similarly, the lifting of a transition relation
$\stackrel{{\scriptstyle\tau}}{{\longrightarrow}}$ is denote
$\stackrel{{\scriptstyle\hat{\tau}}}{{\longrightarrow}}$ whose reflexive
transitive closure is denote
$\stackrel{{\scriptstyle\hat{\tau}}}{{\Longrightarrow}}$. For each external
action $a$, we write $\stackrel{{\scriptstyle\hat{a}}}{{\Longrightarrow}}$ for
the sequence
$\stackrel{{\scriptstyle\hat{\tau}}}{{\Longrightarrow}}\stackrel{{\scriptstyle
a}}{{\longrightarrow}}$.
###### Definition 13.
A probabilistic simulation $S$ between two probabilistic automata $P$ and $Q$
is a relation $S\subseteq R\times\mathcal{D}Q$ such that:
* -
$(\Delta_{P},\Delta_{Q})\in\hat{S}$,
* -
if $(x,\psi)\in S$ and $x\stackrel{{\scriptstyle a}}{{\longrightarrow}}\phi$
then there exists $\psi^{\prime}\in\mathcal{D}Q$ such that
$\psi\stackrel{{\scriptstyle\hat{a}}}{{\Longrightarrow}}\psi^{\prime}$ and
$(\phi,\psi^{\prime})\in\hat{S}$ (for every $a\in\Sigma\cup\\{\tau\\}$).
* -
if $x\in F_{P}$ and $(x,\psi)\in S$ then $\mathrm{supp}(\psi)\subseteq F_{Q}$.
we denote by $\mathbf{ProbAut}$ the set of probabilistic automata modulo
simulation equivalence.
We can now construct a mapping
$\epsilon:p\textrm{-}\mathbf{Aut}\rightarrow\mathbf{ProbAut}$ such that each
instance of structure similar to
$\mathtt{flip}_{p_{1},\dots,p_{n}}\cdot(\tau_{1}\cdot
a_{1}+\dots+\tau_{n}\cdot a_{n})$ is collapsed into probabilistic transitions.
More precisely, let $P\in p\textrm{-}\mathbf{Aut}$ and $\longrightarrow$ be
its transition relation. The automaton $\epsilon(P)$ has the same state space
as $P$ (up to accessibility with respect to the transitions of $\epsilon(P)$).
The initial distribution of $\epsilon(P)$ is $\delta_{i_{P}}$ and the set of
final states of $\epsilon(P)$ is $F_{P}$ again 888Notice that by assuming the
structure $\mathtt{flip}_{p_{1},\dots,p_{n}}\cdot(\tau_{1}\cdot
a_{1}+\dots+\tau_{n}\cdot a_{n}$, the state between the flip action the
corresponding $\tau$ transitions is never a final state. Hence we are safe to
use $F_{P}$ as the final state of $\epsilon(P)$ .
The set of transitions $\longrightarrow_{\epsilon(P)}$ is constructed as
follow. Let $x\stackrel{{\scriptstyle a}}{{\longrightarrow}}x^{\prime}$ be a
transition of $P$, there are two possible cases:
* a)
if $a$ is probabilistic i.e. of the form $\mathtt{flip}_{p_{1},\dots,p_{n}}$
and is followed by the $\tau_{i}$’s, then the transition
$x\stackrel{{\scriptstyle\tau}}{{\longrightarrow}}p_{1}\delta_{x_{1}^{\prime}}+\dots+p_{n}\delta_{x_{n}^{\prime}}$
is in $\longrightarrow_{\epsilon(P)}$ where
$x^{\prime}\stackrel{{\scriptstyle\tau_{i}}}{{\longrightarrow}}x_{i}^{\prime}$
is a transition in $P$.
* b)
else the transition $x\stackrel{{\scriptstyle
a}}{{\longrightarrow}}x^{\prime}$ is in $\longrightarrow_{\epsilon(P)}$.
We now prove that $\epsilon$ is a monotonic function from
$p\textrm{-}\mathbf{Aut}$ to $\mathbf{ProbAut}$.
###### Proposition 14.
If $P\leq Q$ then $\epsilon(P)\leq\epsilon(Q)$.
###### Proof.
Assume that $S$ is a $p$-simulation from $P$ to $Q$. Consider the exact same
relation but restricted to the state space of $\epsilon(P)$ and $\epsilon(Q)$.
We show that this restriction is a probabilistic simulation.
* -
Obviously, $(\delta_{i_{P}},\delta_{i_{Q}})\in\hat{S}$.
* -
Let $x\stackrel{{\scriptstyle a}}{{\longrightarrow}}\phi$ and
$(x,\psi)\in\hat{S}$. Since $\tau$ transitions only occur as part of
probabilistic choices, we have two possibilities:
* –
$x\stackrel{{\scriptstyle\tau}}{{\longrightarrow}}p_{1}\delta_{x_{1}^{\prime}}+\dots+p_{n}\delta_{x_{n}^{\prime}}$
is a transition of $\epsilon(P)$ and $(x,\psi)\in S$ where $\psi=\delta_{y}$.
Since $(x,y)$ belongs to the original $S$. In this case,
$y\stackrel{{\scriptstyle\tau}}{{\longrightarrow}}p_{1}\delta_{y_{1}^{\prime}}+\dots+p_{n}\delta_{y_{n}^{\prime}}$
is a transition of $\epsilon(Q)$ and each $(x_{i}^{\prime},y_{i}^{\prime})$
belongs to the original $S$ (Definition of $p$-simulation).
* –
$x\stackrel{{\scriptstyle a}}{{\longrightarrow}}x^{\prime}$ and $a$ is an
external action. Therefore there are two possibilities again,
$y\stackrel{{\scriptstyle\tau_{i}}}{{\longrightarrow}}y_{i}\stackrel{{\scriptstyle
a}}{{\longrightarrow}}y^{\prime}$ or $y\stackrel{{\scriptstyle
a}}{{\longrightarrow}}y^{\prime}$. In both cases, we have
$(x^{\prime},y^{\prime})\in S$.
* -
Conservation of final states follows easily from the fact that $S$ is a
$p$-simulation.∎
Since our Definition (13) implies the definition of probabilistic simulation
in Den07 , we conclude that maximal probability of doing a particular action
in $p$-automata is increased by $p$-simulation. This remark provides a formal
justification of our earlier example. That is, Equation (14) ensures that the
maximal probability that a buyer will be satisfied when using the
probabilistic vending machine is at least $1/2$ because the maximal
probability of a trace containing $\mathtt{tea}$ in the automata described by
$\mathtt{coin}\cdot\mathtt{flip}\cdot(\tau_{h}\cdot(\mathtt{tea}+1)+\tau_{t}$
is $1/2$.
In the proof of proposition 14, the simulation constructed is a very
particular case of probabilistic simulation so it is too weak to establish
certain relationships between $p$-automata. For instance, the automaton
represented by $a_{p}\\!\\!\oplus(a_{q}\\!\\!\oplus b)$ should be equivalent
to $a_{p+q-pq}\\!\\!\oplus b$ but Definition 11 will not provide such
equality. This line of research is part of our future work where we will study
proper probabilistic automata and simulations against weak concurrent Kleene
algebra.
## 6 Algebraic Testing
In this section, we describe an algebraic treatment of _testing_. Testing is a
natural ordering for processes that was studied first in Nic83 . The idea is
to “measure” the behaviour of the process with respect to the environment. In
other words, given two processes $x$ and $y$ and a set of test processes $T$,
the goal is to compare the processes $x\|t$ and $y\|t$ for every $t\in T$. In
our case, the set $T$ will contain all processes.
We consider a function $o$ from the set of terms to the set of internal
expressions $I=\\{x\ |\ x\leq 1\\}$. The function $o:T_{\Sigma}\rightarrow I$
is defined by
$\begin{array}[]{lll}o(x)=x\textrm{ if }x\in I&&o(st)=o(s)o(t)\\\
o(a)=\tau\textrm{ for any a }\in\Sigma-I&&o(s^{*})=1\\\
o(s+t)=o(s)+o(t)&&o(s\|t)\leq o(s)o(t)\end{array}$
In the model, the function $o$ is interpreted by substituting each external
action with the internal action $\tau$ ($o(a)=\tau$ for any $a\in\Sigma-I$).
Then any final state is labelled by $1$ and deadlock states are labelled by
$0$. Inductively, we label a state that leads to some final state by $1$, else
it is labelled by $0$. This is motivated by the fact that $x0=0$ for any $x\in
I$ so each transition leading to deadlock states only will be removed.
Therefore, only states labelled by $1$ will remain and the transitions between
them. Hence, $o(s)\neq 0$ iff the resulting automaton contains at least one
state labelled by $1$. In other words, $o(s)=0$ iff $x$ must not terminate
successfully.
Without loss of generality (by considering automata modulo simulation), we
assume that $\tau$ is the only internal action in $\Sigma$ and it satisfies
$\tau\tau=\tau$. This equation is valid in the concrete model.
The existence of a well-defined function $o$ satisfying these conditions
depends on our definition of simulation. That is, we can show that if $P\leq
Q$ then $o(P)\leq o(Q)$ where we have abused notation by writing $o(P)$ as the
application of $o$ on the term associated to $P$. A detailed discussion about
this can be found in the appendix under Remark 28.
###### Definition 15.
The may testing order is given by
$x\sqsubseteq_{\texttt{\tiny{may}}}y\quad\textrm{ iff }\quad\forall t\in
T_{\Sigma}.\left[o(y\|t)=0\Rightarrow o(x\|t)=0\right].~{}$
We now provide some results about algebraic may testing. It follows from
monotonicity of $\|$ with respect to $\leq$ (Proposition 3) that may ordering
$\sqsubseteq_{\texttt{\tiny{may}}}$ is weaker than the rooted
$\eta$-simulation order.
###### Proposition 16.
$x\leq y$ implies $x\sqsubseteq_{\texttt{\tiny{may}}}y$.
In fact, $\sqsubseteq_{\texttt{\tiny{may}}}$ is too weak compared to $\leq$:
may testing is equivalent to language equivalence. Given a term $s$, the
language $Tr(s)$ of $s$ is the set of finite words formed by external actions
and are accepted by the automata represented by $s$. In other word, it is the
set of finite traces in the sense of CSP which lead to final states. The
precise definition of this language equivalence can be found in the appendix
and so is the proof of the following proposition (Proposition 29 of the
appendix).
###### Proposition 17.
In $\mathbf{Aut}$, $\sqsubseteq_{\texttt{\tiny{may}}}$ reduces to language
equivalence.
We have shown that $\sqsubseteq_{\texttt{\tiny{may}}}$ is equivalent to
language equivalence and hence it is weaker than our simulation order. This is
also a consequence of the fact that our study of may testing is done in a
qualitative way because the probabilities are found implicitly within actions.
A quantitative study of probabilistic testing orders can be found in Den07 .
## 7 Case Study: Rabin’s Choice Coordination
The problem of choice coordination is well known in the area of distributed
systems. It usually appears in the form of processes voting for a common goal
among some possibilities. Rabin has proposed a probabilistic protocol which
solves the problem Rab82 and a sequential specification can be found in Mci04
.
We specify the protocol in our algebra and prove that a fully concurrent
specification is equivalent to a sequential one. Once this has been done, the
full verification can proceed by reusing the techniques for sequential
reasoning Mci04 .
The protocol consists of a set of tourists and two places: a church $C$ and a
museum $M$. Each tourist has a notepad where he keeps track of an integer $k$.
Each place has a board where tourists can read and write. We denote by $L$
(resp. $R$) the value on the church board (resp. museum board).
In this section, we use $\cdot$ again for the sequential composition to make
the specifications clearer.
* •
The church is specified as $C=(c!L)^{*}\cdot(c?L)$ where the channel $c$
represents the church’s door. $c!L$ means that the value of $L$ is available
to be read in the channel $c$ and $c?L$ waits for an input which is used as
value for $L$ in the subsequent process.
In other words, each tourist can read as many times as they want from the
church board but write on it only once. Repeated writing will be considered in
the specification of the protocol.
Similarly, the museum is specified as $M=(m!R)^{*}\cdot(m?R)$.
* •
Each tourist is specified as $P(\alpha,k)$ where $\alpha\in\\{c,m\\}$ is the
door before which the tourist currently stands and $k$ is the actual value
written on his notepad. A detailed description of $P$ can be found in the
appendix but roughly, we have
$P(\alpha,k)=(\alpha?K)\cdot\mathtt{rabin}\cdot[\alpha:=\underline{\alpha}]~{}$
where $\underline{c}=m$ and $\underline{m}=c$. In other words, the tourist
reads the value on the place specified by $\alpha$, executes Rabin’s protocol
rabin and then goes to the other place. Notice that the process rabin contains
the probabilistic component of Rabin’s protocol. Essentially, it describes the
rules that are used by each tourist to update their actual value for $k$ with
respect to the value on the board and vice versa.
The whole specification of the protocol executed by each tourist is described
by the automata of Figure 2
$\textstyle{P(\alpha,k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha?K}$$\scriptstyle{[K=here]}$$\scriptstyle{[K\neq
here]}$$\textstyle{\bullet}$$\scriptstyle{\alpha!here}$$\scriptstyle{[k>K]}$$\scriptstyle{[k<K]}$$\scriptstyle{[k=K]}$$\scriptstyle{[k:=K]}$$\scriptstyle{\mathtt{flip}_{1/2}}$$\scriptstyle{\alpha!k}$$\scriptstyle{\tau_{h}}$$\scriptstyle{\tau_{t}}$$\scriptstyle{[\alpha:=\underline{\alpha}]}$$\scriptstyle{[k:=K+2]}$$\scriptstyle{[k:=\overline{K+2}]}$$\textstyle{\circ}$$\scriptstyle{\alpha!k}$$\scriptstyle{[\alpha:=\underline{\alpha}]}$$\textstyle{\circ}$
Figure 2: $p$-automaton that describes the protocol $P(\alpha,k)$ executed by
each tourist.
We are ready to specify the whole system. Assume we have two tourists $P$ and
$Q$ (our result generalises easily to $n$ tourists). The tourists’ joint
action is specified as $(P+Q)^{*}$. This ensures that when a tourist has
started his turn by reading the board, he will not be interrupted by any other
tourist until he is done and goes inside the current place or to the other
place. This condition is crucial for the protocol to work properly.
The actions of the locations process are specified by $(M+C)^{*}$ which
ensures that each tourist can be at one place at a time only — this is a
physical constraint. Now, the whole system is specified by
$\mathtt{init}\cdot\left([P(\alpha,u)+Q(\beta,v)]^{*}_{\\{c,m\\}}\\!\|(M+C)^{*}\right)$
(16)
where $\mathtt{init}$ is the initialisation of the values on the boards,
notepads and initial locations. Specification 16 describes the most arbitrary
behaviour of the tourists compatible with visiting and interacting with the
locations in the manner described above. Rabin’s design of the protocol means
that this behaviour is equivalent to a serialised execution where first one
location is visited, followed by the other. We can write that behaviour
behaviour as $[((P+Q)\|M)^{*}((P+Q)\|C)^{*}]^{*}$, where (for this section
only) we denote the concurrency operator by $\|$ instead of
${}_{\\{c,m\\}}\\!\|$ to make the notation lighter. The next theorem says that
this more uniform execution is included in
$S=[P(\alpha,u)+Q(\beta,v)]^{*}\|(M+C)^{*}$, described by Specification 16.
###### Theorem 18.
We have
$S\geq[((P+Q)\|M)^{*}((P+Q)\|C)^{*}]^{*}$
The proof is a simple application of Proposition 3. Theorem 18 means $S$ could
execute all possible actions related to door $M$, and then those at door $C$,
and then back to door $M$ and so one. In fact, we can also prove the converse
i.e. Proposition 18 could be strengthen to equality. But for that, we need the
continuity of the operators $\cdot$ and $\|$.
###### Theorem 19.
In the concrete model, the specification of Rabin’s protocol satisfies
$S=[((P+Q)\|M)^{*}((P+Q)\|C)^{*}]^{*}$
The proof of this theorem depends heavily on the fact that the concurrent and
sequential compositions are continuous in the the concrete model. The complete
proof can be found in the appendix.
In the proof, if we stopped at the distribution over $\|$, we obtain the
equivalent specification
$S=[(P+Q)\|M+(P+Q)\|C]^{*}$
which describes a simpler situation where P or Q interacts at the Museum or at
the Church. This is similar to the sequential version found in Mci04 , which
can be treated by standard probabilistic invariants to complete a full
probabilistic analysis of the protocol.
## 8 Conclusion
An algebraic account of probabilistic and concurrent system has been presented
in this paper. The idea was to combine probabilistic and concurrent Kleene
algebra. A soundness result with respect to automata and rooted
$\eta$-simulation has been provided. The concrete model ensures not only the
consistency of the axioms but provides also a semantic space for systems
exhibiting probabilistic, nondeterministic and concurrent behaviour. We also
showed that the model has stronger properties than just the algebraic
axiomatisation. For instance, sequential and concurrent compositions are both
continuous in the case of finite automata.
We provided some applications of the framework. An algebraic account of may
testing has been discussed in Section 6. It was shown that may ordering
reduces to language equivalence.
We also provided a case study of Rabin’s solution to the choice coordination
problem. A concurrent specification was provided and it was shown to be
structurally equivalent to the sequential one given in Mci04 .
Though the algebra was proven to be powerful enough to derive non-trivial
properties for concrete protocols, the concrete model still needs to be
refined. For instance, the inclusion of tests is important especially for the
construction of probabilistic choices. Tests need to be introduced carefully
because their algebraic characterisation are subtle due to presence of
probability. We also need to improve and refine the manipulation of
quantitative properties in the model as part of our future work.
Finally, it is customary to motivate automated support for algebraic
approaches. The axioms system for weak concurrent Kleene algebra is entirely
first-order, therefore proof automation is supported and automatised version
of our algebraic proofs can be found in our repository.
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## Appendix
The following proofs, diagrams, remarks and other results are only included to
add further clarification of the contents of the present paper. It is left to
the discression of the reviewers to choose whether they will read these proofs
or not.
## Appendix A Diagrams, Theorems and Proofs
Diagram of the Operators: The construction are done inductively from $0,1$ and
elements of the alphabet $\Sigma$.
* -
Deadlock: $0$.
This is the automaton that has only one state, no transition and no final
state.
* -
Skip: $1$
This is the automaton $\circ$ which has only one state which is both initial
and final and has no transition.
* -
Single action:
The automaton associated to $a\in\Sigma$ is $i\stackrel{{\scriptstyle
a}}{{\longrightarrow}}\circ$ where $i$ is the initial state and $\circ$ is a
final state.
* -
Addition: $P+Q$.
This is constructed by identifying the initial states of $P$ and $Q$. This
construction is allowed because of the initiality condition (Figure 3).
$\textstyle{i_{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{b}$$\textstyle{+}$$\textstyle{i_{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\scriptstyle{d}$$\textstyle{=}$$\textstyle{P^{\prime}_{1}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
i_{P+Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{b}$$\scriptstyle{d}$$\scriptstyle{c}$$\textstyle{Q^{\prime}_{2}}$$\textstyle{P^{\prime}_{1}}$$\textstyle{P^{\prime}_{2}}$$\textstyle{Q^{\prime}_{1}}$$\textstyle{Q^{\prime}_{2}}$$\textstyle{P^{\prime}_{2}}$$\textstyle{Q^{\prime}_{1}}$
Figure 3: Automaton for $P+Q$.
* -
Multiplication: $PQ$.
This is constructed by identifying each final state of $P$ with the initial
state of $Q$ (Figure 4).
$\textstyle{P^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{\cdot}$$\textstyle{i_{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\scriptstyle{d}$$\textstyle{=}$$\textstyle{P^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{\dots}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\circ}$$\scriptstyle{\dots}$$\textstyle{Q^{\prime}_{1}}$$\textstyle{Q^{\prime}_{2}}$$\textstyle{\dots}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\dots}$$\scriptstyle{c}$$\scriptstyle{d}$$\textstyle{Q^{\prime}_{1}}$$\textstyle{Q^{\prime}_{2}}$
Figure 4: Automaton for $PQ$. The symbol $\circ$ denotes a final state and the
symbol $\bullet$ is final if and only if $i_{Q}$ is final in $Q$. Notice that
this construction is done for each final state of $P$.
* -
Concurrency: $P_{A}\\!\|Q$
This is constructed as a sub-automaton of the Cartesian product of $P$ and $Q$
following CSP [4]. Assuming $a\in A$ and $b,d\notin A$, the concurrent
composition $P_{A}\\!\|Q$ is inductively constructed as in Figure 5
$\textstyle{i_{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{b}$$\textstyle{{}_{A}\\!\|}$$\textstyle{i_{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{d}$$\textstyle{=}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces(i_{P},i_{Q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{b}$$\scriptstyle{d}$$\textstyle{P^{\prime}_{1}}$$\textstyle{P^{\prime}_{2}}$$\textstyle{Q^{\prime}_{1}}$$\textstyle{Q^{\prime}_{2}}$$\textstyle{P^{\prime}_{1}\
\\!\\!{}_{A}\\!\|Q_{1}}$$\textstyle{P^{\prime}_{2}\
\\!\\!{}_{A}\\!\|Q}$$\textstyle{P_{A}\\!\|Q^{\prime}_{2}}$ Figure 5: Automaton
for $P_{A}\\!\|Q$. The action $a$ has been synchronised and $b,d$ were
interleaved. Notice that $b$ or $d$ could be internal. The initial state of
the automata is the pair $(i_{P},i_{Q})$ and the final states are the elements
of $F_{P}\times F_{Q}$.
Notice that $A\subseteq\Sigma$ is a set of synchronised action and does not
contain any (strictly) probabilistic actions such as $\mathtt{flip}(p)$, for
$p\in]0,1[$.
* -
Kleene star: $P^{*}$
This is the result of repeating $P$ allowing a successful termination after
each (possibly empty) full execution of $P$. In the diagram of Figure 6, we
just picture one transition from the initial state and one final state. The
construction needs to be performedfor each initial transition and final state.
$\textstyle{(i_{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{P^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{\circ)^{*}}$$\textstyle{=}$$\textstyle{\circ\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{P^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{\circ\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$
Figure 6: Automaton for $P^{*}$.
Notice the initial state of $P^{*}$ is a final state too.
###### Proposition 20.
These operations are well defined on $\mathbf{Aut}$ that is if
$P,Q\in\mathbf{Aut}$ then $P+Q,PQ,P_{A}\\!\|Q$ and $P^{*}$ are elements of
$\mathbf{Aut}$.
###### Proof.
The proof is by induction on the structure of the automata $P$ and $Q$. For
the base case, it is obvious that $0,1$ and $i\stackrel{{\scriptstyle
a}}{{\longrightarrow}}\circ$ satisfy the reachability and initiality
conditions.
Let $P,Q\in\mathbf{Aut}$. It is easy to see from the diagrams that $P+Q,P\|Q$
and $P^{*}$ belongs to $\mathbf{Aut}$ too. $PQ$ satisfies the initiality
condition because the initial state is $i_{P}$. For reachability, let $x\in
Q$. Then $x$ is reachable from $i_{Q}$ which in turn is reachable from $i_{P}$
by the definition of sequential composition. ∎
###### Proposition 21.
The following statements hold.
1. 1.
The relational composition of two rooted $\eta$-simulations is again a rooted
$\eta$-simulation. That is, if $S,T$ are rooted $\eta$-simulations then
$S\circ T$ is also a rooted $\eta$-simulation, where $\circ$ denotes
relational composition.
2. 2.
The simulation relation $\leq$ is a preorder on $\mathbf{Aut}$.
###### Proof.
1. 1.
Let $S:P\rightarrow Q$ and $T:Q\rightarrow R$ be simulations and let us show
that $ST:P\rightarrow R$ is a simulation.
* –
Evidently, $(i,i)\in ST$.
* –
Let $(x,z)\in ST$ and $x\stackrel{{\scriptstyle
a}}{{\longrightarrow}}x^{\prime}$. By definition of the relational composition
there exists $y\in Q$ such that $(x,y)\in S$ and $(y,z)\in T$.
* a)
if $a$ is internal, there exists $y^{\prime}\in Q$ such that $y\Rightarrow
y^{\prime}$ and $(x^{\prime},y^{\prime})\in S$. Since $y\Rightarrow
y^{\prime}$ consists of a sequence of finite internal transition, there exists
$z^{\prime}\in T$ such that $(y^{\prime},z^{\prime})\in T$ and $z\Rightarrow
z^{\prime}$. Hence $(x^{\prime},z^{\prime})\in ST$ and $z\Rightarrow
z^{\prime}$.
* b)
If $a$ is external, there exists $y_{1},y^{\prime}\in Q$ such that
$y\Rightarrow y_{1}\stackrel{{\scriptstyle a}}{{\longrightarrow}}y^{\prime}$
and $(x,y_{1})\in S$ and $(x^{\prime},y^{\prime})\in S$. Since $(y,z)\in T$
and $y\Rightarrow y_{1}$, there exists $z_{1}\in R$ such that
$(y_{1},z_{1})\in T$ and $z\Rightarrow z_{1}$. Again, since $T$ is a
simulation and $y_{1}\stackrel{{\scriptstyle
a}}{{\longrightarrow}}y^{\prime}$, there exists $z_{2},z^{\prime}\in R$ such
that $z_{1}\Rightarrow z_{2}\stackrel{{\scriptstyle
a}}{{\longrightarrow}}z^{\prime}$ and $(y_{1},z_{2})\in T$ and
$(y^{\prime},z^{\prime})\in T$. Hence, by transitivity of $\Rightarrow$, we
have $z\Rightarrow z_{2}\stackrel{{\scriptstyle
a}}{{\longrightarrow}}z^{\prime}$ and $(x,z_{2})\in ST$ and
$(x^{\prime},z^{\prime})\in ST$.
* –
Let $(x,z)\in ST$ and $x\in F_{P}$, there exists $y\in Q$ such that $(x,y)\in
S$ and $(y,z)\in T$. So $y\in F_{Q}$ and hence $z\in F_{R}$.
* –
Let $(i,z)\in ST$, there exists $y\in Q$ such that $(i,y)\in S$ and $(y,z)\in
T$. So $y=i$ and hence $z=i$.
2. 2.
For reflexivity, the identity relation is a rooted $\eta$-simulation and
transitivity follows from 1.
∎
###### Proposition 22.
The equivalence relation $\equiv$ is a congruence with respect to $+$ and
$P\leq Q$ iff $P+Q\equiv Q$.
###### Proof.
Let $S:P\rightarrow Q$ and $S^{\prime}:P^{\prime}\rightarrow Q^{\prime}$ be
routed $\eta$-simulations. We show that $S\cup
S^{\prime}:P+P^{\prime}\rightarrow Q+Q^{\prime}$ is again a routed
$\eta$-simulation.
* -
Since initial states are identified in the construction of $+$, we have
$(i_{P+P^{\prime}},i_{Q+Q^{\prime}})=(i_{P},i_{Q})\in S\cup S^{\prime}$.
* -
Let $(x,y)\in S\cup S^{\prime}$ and $x\stackrel{{\scriptstyle
a}}{{\longrightarrow}}x^{\prime}$ be a transition of $P$ (the case where this
transition belongs to $P^{\prime}$ is dealt with the exact same way). We have
two cases:
1. 1.
if $(x,y)\in S$, then either $a$ is internal and hence $(x^{\prime},y)\in S$
(so in $S\cup S^{\prime}$ too) or there exists $y_{1},y^{\prime}\in Q$ such
that $y\Rightarrow y_{1}\stackrel{{\scriptstyle
a}}{{\longrightarrow}}y^{\prime}$ is a path in $Q$ and $(x,y_{1})\in S$ and
$(x^{\prime},y^{\prime})\in S$. By definition of $+$, $y\Rightarrow
y_{1}\rightarrow y^{\prime}$ is again a path in $Q+Q^{\prime}$ such that
$(x,y_{1})\in S\cup S^{\prime}$ and $(x^{\prime},y^{\prime})\in S\cup
S^{\prime}$.
2. 2.
if $(x,y)\in S^{\prime}$, then $x=i_{P}$ because $x\stackrel{{\scriptstyle
a}}{{\longrightarrow}}x^{\prime}$ is assumed to be a transition in $P$. Since
the initial states are merged, $(i_{P^{\prime}},y)\in S^{\prime}$ and
therefore $y=i_{Q^{\prime}}=i_{Q+Q^{\prime}}=i_{Q}$. Therefore $(x,y)\in S$
and we are back to Case 1.
* -
Let $(x,y)\in S\cup S^{\prime}$ and $x\in F_{P}$ (the case $x\in
F_{P^{\prime}}$ is similar). We have two cases again, $(x,y)\in S$ and $y\in
F_{P^{\prime}}$. Or $(x,y)\in S^{\prime}$ and then $x\in P\cap P^{\prime}$.
Hence $x=i$ and we are back to the first case again.
* -
$S\cup S^{\prime}$ is rooted because $S$ and $S^{\prime}$ are both rooted.
Now assume $P\leq Q$. Then $P+Q\leq Q+Q\equiv Q$ follows from the fact that
$\leq$ is a congruence and the idempotence of $+$ in Proposition 8. Moreover,
since $id_{Q}:Q\rightarrow Q$ is a simulation, we have $P+Q\equiv Q$.
Conversely, assume $P+Q\equiv Q$, since $id_{P}:P\rightarrow P+Q$ is a
simulation we have $P\leq Q$ by transitivity of $\leq$. ∎
###### Proposition 23.
$(\mathbf{Aut},+,\cdot,*,0,1)$ is a weak probabilistic Kleene algebra.
###### Proof.
Associativity and commutativity of $+$ and $0+x=x$ follows easily from the
fact that $+$ is base on $\cup$.
* •
Idempotence of $+$: since the union is made disjoint, we assume $P_{c}$ is a
copy of $P$ where every states is indexed by $c$. Then $id_{P}:P\rightarrow
P+P_{c}$ is a simulation and $\\{(y,x)\ |\ y=x\textrm{ or }y=x_{c}\\}$ is a
simulation from $P+P_{c}$ to $P$.
* •
Associativity of $\cdot$: associativity follows from the same proof found in
[9] because identity relations are simulation and our multiplication here is
exactly the $\varepsilon$-free version of the multiplication there.
* •
$1$ is neutral for $\cdot$: it follows easily from the construction that
$1P=P$ and $P1=P$.
* •
Subdistributivity 3: to show that $PQ+PR\leq P(Q+R)$, it suffices to show that
$PR\leq P(Q+R)$ and derive the result from idempotence of $+$. Remind that
$id_{P}$ and $id_{Q}$ are simulation so it suffices to show that $id_{P}\cup
id_{Q}:PQ\rightarrow P(Q+R)$ is again a simulation. Obviously, $(i,i)\in S$
and it is rooted and conserves final states. Moreover
$\longrightarrow_{P(Q+R)}\supseteq\longrightarrow_{PQ}$. Hence $id_{P}\cup
id_{Q}$ is a simulation.
* •
Right distributivity 5: let $P_{c}$ be a disjoint copy of $P$, then the
relation
$S=\\{(x,y)\ |\ y=x\textrm{ or }y=x_{c}\\}$
from $(Q+R)P$ to $QP_{c}+RP$ is rooted and preserves final states. Let
$(x,y)\in S$ and $x\stackrel{{\scriptstyle
a}}{{\longrightarrow}}x^{\prime}\in\longrightarrow_{(Q+R)P}$. Remind that
$\displaystyle\longrightarrow_{(Q+R)P}=\longrightarrow_{Q}\cup\longrightarrow_{R}\cup\longrightarrow_{P}-\\{i\stackrel{{\scriptstyle
a}}{{\longrightarrow}}z\in\longrightarrow_{P}\\}$
$\displaystyle\cup\\{z\stackrel{{\scriptstyle
a}}{{\longrightarrow}}z^{\prime}\ |\ z\in F_{Q}\cup F_{R}\textrm{ and
}i\stackrel{{\scriptstyle
a}}{{\longrightarrow}}z^{\prime}\in\longrightarrow_{P}\\}$
If the transition belongs to the first three sets then we are done, else we
can assume $x\in F_{Q}$ and $i\stackrel{{\scriptstyle
a}}{{\longrightarrow}}x^{\prime}\in\longrightarrow_{P}$ i.e. $y=x$ and
$x^{\prime}\in P$. We have
$\longrightarrow_{QP_{c}+RP}\supseteq\\{z\stackrel{{\scriptstyle
a}}{{\longrightarrow}}z^{\prime}_{c}\ |\ z\in F_{Q}\textrm{ and
}i\stackrel{{\scriptstyle
a}}{{\longrightarrow}}z^{\prime}_{c}\in\longrightarrow_{P_{c}}\\}$
so $x\stackrel{{\scriptstyle
a}}{{\longrightarrow}}x^{\prime}_{c}\in\longrightarrow_{QP+RP}$ and
$(x^{\prime},x^{\prime}_{c})\in S$. Similarly, we can prove that if $(x,y)\in
S$ and $y\stackrel{{\scriptstyle a}}{{\longrightarrow}}y^{\prime}$ then there
exists $x^{\prime}$ such that $(x^{\prime}y^{\prime})\in S$ and
$x\stackrel{{\scriptstyle a}}{{\longrightarrow}}x^{\prime}$. Hence $S$ is a
bisimulation.
* •
Left unfold 2: Let $x_{*}$ be a state in $P^{*}$ and $x$ the corresponding
state in the unfolded version $(1+PP^{*})$ i.e. $x$ is considered as a state
of $P$. The rooted version of relation $S=id_{P^{*}}\cup\\{(x_{*},x)\\}$ is a
rooted $\eta$-bisimulation from $P^{*}$ to $1+PP^{*}$.
* •
Left induction 3: as in (10), the proof is again similar to [9] because rooted
$\eta$-simulation are stable by union.
∎
###### Proposition 24.
$(\mathbf{Aut},+,\cdot,_{A}\\!\|,1)$ satisfies equations (9\- 13) modulo
rooted $\eta$-simulation equivalence for any set of synchronisable actions
$A\subseteq\Sigma$ (i.e. no probabilistic actions).
###### Proof.
$1_{A}\\!\|1=1$ follows directly from the definition of ${}_{A}\\!\|$ and the
simulation used for the commutativity is $\\{((x,y),(y,x))\ |\ x\in P\textrm{
and }y\in Q\\}$.
* (9)
For associativity, we show that if $(x,(y,z))\stackrel{{\scriptstyle
a}}{{\longrightarrow}}(x^{\prime},(y^{\prime},z^{\prime}))\in\longrightarrow_{P_{A}\\!\|(Q_{A}\\!\|R)}$
then $((x,y),z)\stackrel{{\scriptstyle
a}}{{\longrightarrow}}((x^{\prime},y^{\prime}),z^{\prime})\in\longrightarrow_{(P_{A}\\!\|Q)_{A}\\!\|R}$.
* –
If $a\notin A$, then
* *
$x\stackrel{{\scriptstyle a}}{{\longrightarrow}}x^{\prime}$ and
$y=y^{\prime},z=z^{\prime}$. So $(x,y)\stackrel{{\scriptstyle
a}}{{\longrightarrow}}(x^{\prime},y)\in\longrightarrow_{P_{A}\\!\|Q}$ and
hence $((x,y),z)\stackrel{{\scriptstyle
a}}{{\longrightarrow}}((x^{\prime},y),z)\in\longrightarrow_{(P_{A}\\!\|Q)_{A}\\!\|R}$
because $a\notin A$.
* *
or $x=x^{\prime}$ and $(y,z)\stackrel{{\scriptstyle
a}}{{\longrightarrow}}(y^{\prime},z^{\prime})$. Since $a\notin A$:
* ·
$y\stackrel{{\scriptstyle a}}{{\longrightarrow}}y^{\prime}$ and
$z=z^{\prime}$, hence $((x,y),z)\stackrel{{\scriptstyle
a}}{{\longrightarrow}}((x,y^{\prime}),z)\in\longrightarrow_{(P_{A}\\!\|Q)_{A}\\!\|R}$,
* ·
or $y=y^{\prime}$ and $z\stackrel{{\scriptstyle
a}}{{\longrightarrow}}z^{\prime}$ and hence $((x,y),z)\stackrel{{\scriptstyle
a}}{{\longrightarrow}}((x,y),z^{\prime})\in\longrightarrow_{(P_{A}\\!\|Q)_{A}\\!\|R}$.
* –
If $a\in A$, then $x\stackrel{{\scriptstyle a}}{{\longrightarrow}}x^{\prime}$
and $(y,z)\stackrel{{\scriptstyle
a}}{{\longrightarrow}}(y^{\prime},z^{\prime})$. Since $a$ is again
synchronised in $Q_{A}\\!\|R$, $y\stackrel{{\scriptstyle
a}}{{\longrightarrow}}y^{\prime}$ and $z\stackrel{{\scriptstyle
a}}{{\longrightarrow}}z^{\prime}$. So $(x,y)\stackrel{{\scriptstyle
a}}{{\longrightarrow}}(x^{\prime},y^{\prime})\in\longrightarrow_{P_{A}\\!\|Q}$
and hence $((x,y),z)\stackrel{{\scriptstyle
a}}{{\longrightarrow}}((x^{\prime},y^{\prime}),z^{\prime})\in\longrightarrow_{(P_{A}\\!\|Q)_{A}\\!\|R}$.
Since ${}_{A}\\!\|$ is commutative, we deduce that
$\longrightarrow_{(P_{A}\\!\|Q)_{A}\\!\|R}=\longrightarrow_{P_{A}\\!\|(Q_{A}\\!\|R)}$
so the identity relations could again be used for the simulation.
* (12)
To prove monotonicity, we consider the relation
$S:P_{A}\\!\|Q+{P_{c}}_{A}\\!\|R\to P_{A}\\!\|(Q+R)$ as in the case of
multiplication i.e. $S=\\{((x,y),(x,y)),((x_{c},y),(x,y))\ |\ x\in P\wedge
y\in Q\cup R\\}$ and $x_{c}$ is the copy of the state $x\in P$ in $P_{C}$. Let
$((x_{c},y),(x,y))\in S$ (the case $((x,y),(x,y))\in S$ is easier and can be
handled in the same way) and $(x_{c},y)\stackrel{{\scriptstyle
a}}{{\longrightarrow}}(x^{\prime}_{c},y^{\prime})\in\longrightarrow_{P_{A}\\!\|Q+P_{A}\\!\|R}$.
By definition of $+$, that transition belongs to
$\longrightarrow_{P_{A}\\!\|Q}$ or $\longrightarrow_{P_{A}\\!\|R}$. Since the
first component is a copy of $x$, we have $(x_{c},y)\stackrel{{\scriptstyle
a}}{{\longrightarrow}}(x^{\prime}_{c},y^{\prime})\in\longrightarrow_{P_{A}\\!\|R}$
that is $y,y^{\prime}\in R$.
* –
if $a\notin A$, then
* *
$x_{c}\stackrel{{\scriptstyle a}}{{\longrightarrow}}x^{\prime}_{c}$ and
$y=y^{\prime}$, so $x\stackrel{{\scriptstyle
a}}{{\longrightarrow}}x^{\prime}\in\longrightarrow_{P}$ and hence
$(x,y)\stackrel{{\scriptstyle
a}}{{\longrightarrow}}(x^{\prime},y)\in\longrightarrow_{P_{A}\\!\|(Q+R)}$ and
$((x^{\prime}_{c},y),(x^{\prime},y))\in S$ by definition of $S$.
* *
or $x_{c}=x^{\prime}_{c}$ and $y\stackrel{{\scriptstyle
a}}{{\longrightarrow}}y^{\prime}$, so $(x,y)\stackrel{{\scriptstyle
a}}{{\longrightarrow}}(x,y^{\prime})\in\longrightarrow_{P_{A}\\!\|(Q+R)}$ and
$((x,y^{\prime}),(x_{c},y^{\prime}))\in S$.
* –
if $a\in A$, then $x_{c}\stackrel{{\scriptstyle
a}}{{\longrightarrow}}x^{\prime}_{c}$ and $y\stackrel{{\scriptstyle
a}}{{\longrightarrow}}y\in\longrightarrow_{R}^{\prime}$. So
$x\stackrel{{\scriptstyle
a}}{{\longrightarrow}}x^{\prime}\in\longrightarrow_{P}$ and hence
$(x,y)\stackrel{{\scriptstyle
a}}{{\longrightarrow}}(x^{\prime},y^{\prime})\in\longrightarrow_{P_{A}\\!\|(Q+R)}$
and $((x^{\prime}_{c},y^{\prime}),(x^{\prime},y^{\prime}))\in S$.
* (13)
Firstly notice that the set of states of $(P\|Q)(P^{\prime}\|Q^{\prime})$
(where the frame $A$ of the concurrency operator is left implicit) is a subset
of $(P\times Q)\cup(P^{\prime}\times Q^{\prime})$ which is in turn a subset of
$(P\cup P^{\prime})\times(Q\cup Q^{\prime})$. Hence we consider the injection
$id$ of the former set to the later one and the relation defined in Figure 7
$\displaystyle S=id$ $\displaystyle\cup$
$\displaystyle\\{((x^{\prime},i),(x^{\prime},y))\ |\ y\in F_{Q}\textrm{ and
}i\stackrel{{\scriptstyle
a}}{{\longrightarrow}}x^{\prime}\in\longrightarrow_{P^{\prime}}\textrm{ and
}i\in Q^{\prime}\\}$ $\displaystyle\cup$
$\displaystyle\\{((i,y^{\prime}),(x,y^{\prime}))\ |\ x\in F_{P}\textrm{ and
}i\stackrel{{\scriptstyle
a}}{{\longrightarrow}}y^{\prime}\in\longrightarrow_{Q^{\prime}}\textrm{ and
}i\in P^{\prime}\\}$ Figure 7: Construction of the simulation to prove the
interchange law.
We show that $S$ is a simulation in our sense.
* -
Since $(i,i)=i$ is related to itself. In particular, $S$ is rooted because
$x^{\prime}\neq i$ in the second set in the definition of $S$ (resp. for the
third set) and HCI.
* -
Let $(x,y)\in(P\|Q)(P^{\prime}\|Q^{\prime})$ such that
$(x,y)\stackrel{{\scriptstyle a}}{{\longrightarrow}}(x^{\prime},y^{\prime})$.
We have the following cases.
* *
The transition is in $\longrightarrow_{P\|Q}$, in which case
$(x,y),(x^{\prime},y^{\prime})\in P\times Q$.
* ·
if $a\notin A$ then $x\stackrel{{\scriptstyle
a}}{{\longrightarrow}}x^{\prime}\in\longrightarrow_{P}$ and $y=y^{\prime}$ or
$x=x^{\prime}$ and $y\stackrel{{\scriptstyle
a}}{{\longrightarrow}}y^{\prime}\in\longrightarrow_{Q}$. By definition of the
sequential composition again, these transitions belong to
$\longrightarrow_{PP^{\prime}}$ or $\longrightarrow_{QQ^{\prime}}$
respectively. Hence $(x,y)\stackrel{{\scriptstyle
a}}{{\longrightarrow}}(x^{\prime},y^{\prime})\in\longrightarrow_{PP^{\prime}\|QQ^{\prime}}$.
* ·
if $a\in A$ then $x\stackrel{{\scriptstyle
a}}{{\longrightarrow}}x^{\prime}\in\longrightarrow_{P}$ and
$y\stackrel{{\scriptstyle
a}}{{\longrightarrow}}y^{\prime}\in\longrightarrow_{Q}$. As in the previous
case, the considered transition exists in $PP^{\prime}\|QQ^{\prime}$
* *
The transition is in $\longrightarrow_{P^{\prime}\|Q^{\prime}-\\{(i,i)\\}}$.
This case is similar to the previous one because because $x\neq i$ and $y\neq
i$ as states of $P^{\prime}$ and $Q^{\prime}$ respectively.
* *
It is a linking transition i.e. $(x,y)\in F_{P\|Q}$ and
$(i,i)\stackrel{{\scriptstyle
a}}{{\longrightarrow}}(x^{\prime},y^{\prime})\in\longrightarrow_{P^{\prime}\|Q^{\prime}}$.
Then $x\in F_{P}$ and $y\in F_{Q}$ and we have two cases:
* ·
if $a\notin A$, then $i\stackrel{{\scriptstyle
a}}{{\longrightarrow}}x^{\prime}\in\longrightarrow_{P^{\prime}}$ and
$y^{\prime}=i$ or $x^{\prime}=i$ and $i\stackrel{{\scriptstyle
a}}{{\longrightarrow}}y^{\prime}\in\longrightarrow_{Q^{\prime}}$. In the first
case, the definition of $S$ implies that
$((x^{\prime},y^{\prime}),(x^{\prime},y))\in S$ and since $a\notin A$, we have
$(x,y)\stackrel{{\scriptstyle
a}}{{\longrightarrow}}(x^{\prime},y)\in\longrightarrow_{PP^{\prime}\|QQ^{\prime}}$.
Similarly for the other case.
* ·
if $a\in A$, then $i\stackrel{{\scriptstyle
a}}{{\longrightarrow}}x^{\prime}\in\longrightarrow_{P^{\prime}}$ and
$i\stackrel{{\scriptstyle
a}}{{\longrightarrow}}y^{\prime}\in\longrightarrow_{Q^{\prime}}$. Then
$x\stackrel{{\scriptstyle
a}}{{\longrightarrow}}x^{\prime}\in\longrightarrow_{PP^{\prime}}$ and
$y\stackrel{{\scriptstyle
a}}{{\longrightarrow}}y^{\prime}\in\longrightarrow_{QQ^{\prime}}$. Hence
$(x,y)\stackrel{{\scriptstyle
a}}{{\longrightarrow}}(x^{\prime},y^{\prime})\in\longrightarrow_{PP^{\prime}\|QQ^{\prime}}$.
* *
Let $((x^{\prime},i),(x^{\prime},y))\in S$ as in the above definition of $S$
and $(x^{\prime},i)\stackrel{{\scriptstyle
a}}{{\longrightarrow}}(x^{\prime\prime},y^{\prime\prime})\in\longrightarrow_{P^{\prime}\|Q^{\prime}}$.
* ·
If $a\notin A$, then $x^{\prime}\stackrel{{\scriptstyle
a}}{{\longrightarrow}}x^{\prime\prime}\in\longrightarrow_{P^{\prime}}$ and
$y^{\prime\prime}=i$ or $x^{\prime}=x^{\prime\prime}$ and
$i\stackrel{{\scriptstyle
a}}{{\longrightarrow}}y^{\prime\prime}\in\longrightarrow_{Q^{\prime}}$. In the
first case, $(x^{\prime},y)\stackrel{{\scriptstyle
a}}{{\longrightarrow}}(x^{\prime\prime},y)\in\longrightarrow_{PP^{\prime}\|QQ^{\prime}}$
because $a\notin A$ and $(x^{\prime\prime},y)\in S$ because
$y^{\prime\prime}=i$. In the second case, $y\stackrel{{\scriptstyle
a}}{{\longrightarrow}}y^{\prime\prime}\in\longrightarrow_{QQ^{\prime}}$ and
hence $(x^{\prime},y)\stackrel{{\scriptstyle
a}}{{\longrightarrow}}(x^{\prime\prime},y^{\prime\prime})\in\longrightarrow_{PP^{\prime}\|QQ^{\prime}}$
because $x^{\prime}=x^{\prime\prime}$.
* ·
If $a\in A$, then $x^{\prime}\stackrel{{\scriptstyle
a}}{{\longrightarrow}}x^{\prime\prime}\in\longrightarrow_{P^{\prime}}$ and
$i\stackrel{{\scriptstyle
a}}{{\longrightarrow}}y^{\prime\prime}\in\longrightarrow_{Q^{\prime}}$. By
definition of sequential composition, $y\stackrel{{\scriptstyle
a}}{{\longrightarrow}}y^{\prime\prime}\in\longrightarrow{QQ^{\prime}}$ and
since $a\in A$, $(x^{\prime},y)\stackrel{{\scriptstyle
a}}{{\longrightarrow}}(x^{\prime\prime},y^{\prime\prime})\in\longrightarrow_{PP^{\prime}\|QQ^{\prime}}$.
* *
The case $((i,y^{\prime}),(x,y^{\prime}))\in S$ is similar.
* -
Let $((x,y),(u,v))\in S$ such that $(x,y)\in
F_{(P\|Q)(P^{\prime}\|Q^{\prime})}$. That is,
$x\in(F_{P^{\prime}}-\\{i\\})\cup
o_{P^{\prime}\|Q^{\prime}}((i,i))F_{P}\subseteq(F_{P^{\prime}}-\\{i\\})\cup
o_{P^{\prime}}(i)F_{P}$
and similarly for $y$. Hence, if tuple belongs to $id$ then we are done.
Assume $i\stackrel{{\scriptstyle
a}}{{\longrightarrow}}x\in\longrightarrow_{P^{\prime}}$ and $y=i$ (the other
case is proved in exactly the same way), then $u=x$ and $v\in F_{Q}$. Since
$(x,i)$ is a final state, we have
$F_{QQ^{\prime}}=(F_{Q^{\prime}}-\\{i\\})\cup F_{Q}$ and $x^{\prime}\in
F_{P^{\prime}}-\\{i\\}$. Hence $(u,v)\in F_{PP^{\prime}}\times
F_{QQ^{\prime}}$.
Finally, since simulation preserves reachability, the reachable part of
$(P\|Q)(P^{\prime}\|Q^{\prime})$ is simulated by the reachable part of
$PP^{\prime}\|QQ^{\prime}$.
∎
###### Proposition 25.
The sequential composition is (conditionally) continuous from the left and the
right in $\mathbf{Aut}_{f}$. That is, if $(P_{i})_{i}$ is a $\leq$-directed
set of finite automata with limit $P$ then $\sup_{i}P_{i}K=PK$ and
$\sup_{i}KP_{i}=KP$.
We denote $\mathbf{Aut}_{f}$ the set of finite automata satisfying the
reachability and initiality conditions. $\mathbf{Aut}_{f}$ is a subalgebra of
$\mathbf{Aut}$. The proof is similar to our proof in [9]. The only difference
is from the manipulation of $0$ (because we do not have $x0=0$ in this
setting) and hence Proposition 25 is a generalised version of the continuity
in [9].
###### Proof.
We first define a notion of residuation on $\mathbf{Aut}_{f}$. For automata
$P$ and $Q$ we define the automaton $P/Q$ with initial state $i_{P/Q}=i_{P}$,
final states $F_{P/Q}=\\{x\in P\ |\ Q\leq P_{x}\\}$, where $P_{x}$ is
constructed from $P$ by making its initial state into $x$. We make the
resulting automaton reachable by discarding all states not reachable from $x$.
Notice that $P_{x}$ does not necessarily satisfy HCI. In this case, we unfold
each transition from $x$ once and isolate $x$ but keeping a disjoint copy of
it to make sure that the resulting automata is bisimulation equivalent to the
non-rooted version.
We now show that $RQ\leq P\textrm{ iff }R\leq P/Q$. Assume $S$ is a simulation
from $RQ$ to $P$. That means $S$ generates a simulation from $Q$ to $P_{x}$
for some $x$. It follows from the definition of $P/Q$ that $S$ generates a
simulation from $R$ to $P/Q$, since the state $x$ become final state of $G/H$
and they are images of the final states of $K$ under the simulation generated
by $S$.
For the converse direction, suppose that $S$ is a simulation from $R$ to
$P/Q$. By Theorem 8, multiplication is isotone, hence $RQ\leq(P/Q)Q$, and it
remains to show that $(P/Q)Q\leq P$.
First, if $F_{P/Q}$ is empty and then $R$ has no final state either and $RQ=R$
by definition of sequential composition. Hence $RQ=R\leq P/Q\leq P$.
Assume $F_{P/Q}$ is not empty and let $S^{\prime}$ be a simulation from $RQ$
to $(P/Q)Q$. By construction of $P/Q$, we know that there exists a simulation
$S_{x}$ from $Q$ to $P_{x}$ for all final states $x\in F_{P/Q}$. Moreover,
there is a relation $T:P/Q\rightarrow P$ satisfying all properties of
simulation except the final state property, namely a restriction of the
identity relation $id_{P}$. We can show that $T^{\prime}=(\cup_{x}S_{x})\cup
T$ is indeed a simulation from $(P/Q)Q$ to $P$ and $S^{\prime}\circ
T^{\prime}$ is a simulation from $RQ$ to $P$.
It then follows from general properties of Galois connections that $(\cdot H)$
is (conditionally) completely additive, hence right continuous.
It remains to show left continuity. Let $(Q_{i})_{i}$ be a directed set of
automata such that $\sup_{i}Q_{i}=Q$ and let $P$ be any automaton. Then
$\sup_{i}(PQ_{i})\leq PQ$ because multiplication is monotone and it remains to
show $PQ\leq\sup_{i}(PQ_{i})$. Let us assume that $\sup_{i}(PQ_{i})\leq R$. We
will show that $PQ\leq R$.
By definition of supremum, $PQ_{i}\leq R$ for all $i$, hence there is a set of
states $X_{i}=\\{x\in R\ |\ Q_{i}\leq R_{x}\\}$, that is, the set of all those
states in $R$ from which $Q_{i}$ is simulated. Obviously, $X_{i}\subseteq
X_{j}$ if $Q_{j}\leq Q_{i}$ in the directed collection. But since
$R\in\mathbf{Aut}_{f}$ has only finitely many states, there must be a minimal
set $X$ in that directed set such that all $Q_{i}$ are simulated by $R_{x}$
for some $x\in X$. Therefore $Q=\sup_{i}Q_{i}\leq R_{x}$ for all $x\in X$.
There exists a simulation $S_{X}:PQ_{i}\rightarrow R$ for some $i$ such that
the residual automaton $R/Q_{i}$ has precisely $X$ as its set of final states.
We can thus take the union of $S_{X}$ restricted to $P$ with all simulations
yielding $Q\leq R_{x}$ for all $x\in X$ and verify that this is indeed a
simulation of $PQ$ to $R$. ∎
We denote $L(P)=\\{t\ |\ t\textrm{ is a tree and }t\leq P\\}$ the tree
language associated to $P$. We have
###### Lemma 26.
$P\leq Q$ iff $L(P)\subseteq L(Q)$.
A specialized version of this theorem could be found in [9]. In this paper, we
prove it for our rooted $\eta$-simulation.
###### Proof.
By transitivity of simulation, we have $P\leq Q$ implies $L(P)\subseteq L(Q)$
so it suffices to show the converse.
Let $L(P)\subseteq L(Q)$ and consider the relation $S:P\rightarrow Q$ such
that $(x,y)\in S$ iff $L(P_{x})\subseteq L(Q_{y})$, where $P_{x}$ is the
automata constructed from $P$ with initial state $x$ as in the previous proof.
We now show that the rooted version of $S$ is a simulation.
* •
Since $L(P)\subseteq L(Q)$, we have $(i_{P},i_{Q})\in S$.
* •
Let $(x,y)\in S$ and $x\in F_{P}$, then $1\in L(P_{x})\subseteq L(Q_{y})$.
Hence $y\in F_{Q}$.
* •
Let $(x,y)\in S$, $L(P_{x})\subseteq L(Q_{y})$ and $x\stackrel{{\scriptstyle
a}}{{\longrightarrow}}x^{\prime}$ be a transition of $P$. There are two cases:
* a)
$a$ is internal: for any tree $t$, $at\leq t$. Hence
$L(P_{x^{\prime}})\subseteq L(Q_{y})$ i.e. $(x^{\prime},y)\in S$.
* b)
$a$ is external: assume for a contradiction that for each $y^{\prime}_{i}\in
Q$ such that $y\Rightarrow y_{1}\stackrel{{\scriptstyle
a}}{{\longrightarrow}}y^{\prime}_{i}$, there exists $t_{i}\in
L(P_{x^{\prime}})$ such that $t_{i}\notin L(Q_{y^{\prime}_{i}})$. Since
$Q\in\mathbf{Aut}_{f}$, there are only finitely many such $y^{\prime}_{i}$. By
definition of $\eta$-simulation, $a(\sum_{i}t_{i})\in L(P_{x})$ and from it
follows from the hypothesis that $a(\sum_{i}t_{i})\in L(Q_{y})$ i.e.
$a(\sum_{i}t_{i})\leq Q_{y}$. It follows from the definition of
$\eta$-simulation that there exists $y^{\prime}_{j}$ such that $y\Rightarrow
y_{1}\stackrel{{\scriptstyle a}}{{\longrightarrow}}y^{\prime}_{j}$ and
$\sum_{i}t_{i}\leq y^{\prime}_{j}$ which implies $t_{j}\leq\sum_{i}t_{i}\leq
y_{j}$, a contradiction.
* •
Making the relation $S$ rooted does not affect the well-definedness of $S$ as
a simulation because the automata $P,Q$ are rooted.
∎
###### Proposition 27.
The concurrency operator $\|$ is (conditionally) continuous in
$\mathbf{Aut}_{f}$.
###### Proof.
We need to show that for any $\leq$-directed sequence
$(Q_{i})_{i}\subseteq\mathbf{Aut}_{f}$ such that $\sup_{i}Q_{i}=Q$, we have
$\sup_{i}(P\|Q_{i})=P\|Q$, where the frame is left implicit.
Firstly, we show that
$L(P\|Q)=\downarrow(L(P)\|L(Q))=\downarrow\\{t\|t^{\prime}\ |\ t\in L(P)\wedge
t^{\prime}\in L(Q)\\}$
where $\downarrow X$ is the down closure of $X$. Since $\|$ is monotone,
$t\|t^{\prime}\in L(P\|Q)$. Conversely, let $t\in L(P_{A}\\!\|Q)$. By
unfolding $P$ and $Q$ up to the depth of $t$, we can find two tree
$t_{P},t_{Q}$ such that $t\leq{t_{P}}\|t_{Q}$ and hence
$t\in\downarrow(L(P)\|L(Q))$.
Secondly, we have
$\displaystyle L(P\|Q)$ $\displaystyle=$
$\displaystyle\downarrow\\{t\|t^{\prime}\ |\ t\in P\wedge
t^{\prime}\in\cup_{i}L(Q_{i})\\}$ $\displaystyle=$
$\displaystyle\downarrow\cup_{i}\\{t\|t^{\prime}\ |\ t\in P\wedge
t^{\prime}\in L(Q_{i})\\}$ $\displaystyle=$
$\displaystyle\cup_{i}\downarrow\\{t\|t^{\prime}\ |\ t\in P\wedge
t^{\prime}\in L(Q_{i})\\}$ $\displaystyle=$ $\displaystyle\cup_{i}L(P\|Q_{i})$
and directedness ensures that $L(Q)=\cup_{i}L(Q_{i})$
111111$\cup_{i}L_{i}\subseteq Q$ is obvious and the converse could be proven
by showing that if $t\in L(Q)$ and $t\notin L(Q_{i})$ for every $i$, then
there exists $Q^{\prime}$ constructed from $Q$ “minus” some part of $t$ such
that $Q_{i}\leq Q^{\prime}<Q$ for any $i$.. Therefore, Lemma 26 ensures that
$P\|Q=\sup_{i}(P\|Q_{i})$. ∎
###### Remark 28.
The following remarks ensures the existence of an $o$ function satisfying the
properties listed in Section 6
1. 1.
The axiom $\tau\tau=\tau$ ensures that $o(x)\in\\{0,\tau,1\\}$ for any $x\in
T_{\Sigma}$. If $o(x)=0$ then $x$ will never terminate successfully. If
$o(x)=1$, then $x$ may terminate successfully without the execution of any
action 121212A rigorous proof of this fact could be done by induction on the
structure of $x$., and if $o(x)=\tau$ then $x$ may terminate successfully
after the execution of some action.
2. 2.
The interpretation of $o$ in the concrete model respects simulation. In fact,
let $P,Q$ be the automata representing some terms in $T_{\Sigma}$ and
$S:P\rightarrow Q$ be a simulation. After replacing each action in $P,Q$ by
$\tau$, $S$ remains a simulation by Propriety (a) of Definition 5. Therefore
* -
if $o(P)=1$ then the initial state of $P$ is final and so is the initial state
of $Q$,
* -
if $o(P)=\tau$ then the initial state of $P$ leads to some final state and so
is the initial state of $Q$ i.e. $1\leq o(Q)$,
* -
if $o(P)=0$ then we are done,
and in all three cases $o(P)\leq o(Q)$. Hence, it is safe to assume that $o$
is well defined on $T_{\Sigma}$ modulo the axioms of weak concurrent Kleene
algebra. In particular, $o$ is monotonic with respect to the restriction of
the natural order of the algebra on $I$.
3. 3.
The last property $o(x\|y)\leq o(x)o(y)$ is in general a strict inequality.
For instance, if $a,b$ are synchronised actions then $o(a\|b)=o(0)=0$ but
$o(a)o(b)=\tau\tau=\tau$.
###### Proposition 29.
In $\mathbf{Aut}$, $\sqsubseteq_{\texttt{\tiny{may}}}$ reduces to language
equivalence.
Remind that we assume there is only one non-trivial internal action, namely
$\tau$, and it satisfies $\tau\tau=\tau$.
###### Proof.
Firstly, the language of the automata associated to $x$ is given by
$\displaystyle Tr(x)$ $\displaystyle=$ $\displaystyle\\{t\ |\ t\textrm{ is
linear, loop-free, has only $\tau$ as}$ $\displaystyle\quad\textrm{non-
synchronised action and }t\leq_{\tau}x\\}$
where $x\leq_{\tau}y$ if there is a simulation between the automata
represented by $x$ and $y$ such that all non-synchronised actions are replaced
by $\tau$. This ensures for instance that $Tr(\mathtt{flip})=\\{\tau\\}$.
Remind that $Tr(x\|y)=Tr(x)\cap Tr(y)$ 131313A small difference from CSP is
that we consider only words terminating to final states but since
$F_{x\|y}=F_{x}\times F_{y}$, we are safe to use most of the general
properties found in CSP such as $Tr(x\|y)=\\{t\ |\ t_{|x}\in Tr(x)\wedge
t_{|y}\in Tr(y)\\}$ . because elements of $Tr(x)$ are of the form $w\tau$ or
$w$ (modulo the equivalence from $\leq_{\tau}$) where $w$ is a word formed of
synchronised actions only.
For the direct implication, assume $x\sqsubseteq_{\texttt{\tiny{may}}}y$ and
let $t\in Tr(x)$. Then $o(x\|t)\neq 0$ and since $x\|t\leq y\|t$, we have
$o(y\|t)\neq 0$. Since $t$ has synchronised actions only (or possibly ends
with $\tau$) and $o(y\|t)\neq 0$, then $t\in Tr(y)$ that is $Tr(x)\subseteq
Tr(y)$.
Conversely, let $Tr(x)\subseteq Tr(y)$ and $z\in T_{\Sigma}$.
$Tr(x\|z)=Tr(x)\cap Tr(z)\subseteq Tr(y)\cap Tr(z)=Tr(y\|z)$. So if
$o(y\|z)=0$ then $y\|z$ has no final state and hence $Tr(y\|z)=\\{0\\}$. Hence
$Tr(x\|z)=\\{0\\}$ i.e. $x\|z$ has no final state that is $o(x\|z)=0$. ∎
## Appendix B Specification of Rabin’s Protocol.
Remind that $P(\alpha,k)$ is the specification of a tourist in from of the
door $\alpha\in\\{m,c\\}$ and has $k$ written on his notepad (Figure 8).
$\textstyle{P(\alpha,k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha?K}$$\scriptstyle{[K=here]}$$\scriptstyle{[K\neq
here]}$$\textstyle{\bullet}$$\scriptstyle{\alpha!here}$$\scriptstyle{[k>K]}$$\scriptstyle{[k<K]}$$\scriptstyle{[k=K]}$$\scriptstyle{[k:=K]}$$\scriptstyle{\mathtt{flip}_{1/2}}$$\scriptstyle{\alpha!k}$$\scriptstyle{\tau_{h}}$$\scriptstyle{\tau_{t}}$$\scriptstyle{[\alpha:=\underline{\alpha}]}$$\scriptstyle{[k:=K+2]}$$\scriptstyle{[k:=\overline{K+2}]}$$\textstyle{\circ}$$\scriptstyle{\alpha!k}$$\scriptstyle{[\alpha:=\underline{\alpha}]}$$\textstyle{\circ}$
Figure 8: Interpretation of $P(\alpha,k)$ in term of automata with imiplicit
probability.
Any action of the form $[a]$ are considered internal. The symbol $\circ$
denotes final states and $\bullet$ is a deadlock state. In this protocol,
deadlock state is used to specify that the tourist has come to a decision and
the common place would be the value of $\alpha$ when the deadlock state is
reached.
###### Theorem 30.
In the concrete model, the specification of Rabin’s protocol satisfies
$S=[((P+Q)\|M)^{*}((P+Q)\|C)^{*}]^{*}$
Firstly, notice that if $\cdot$ is (conditionally) continuous then
$x^{*}=\sup_{n\in\mathbb{N}}(1+x)^{n}$. The proof relies on the fact that
$f_{x}^{n}(0)=(1+x)^{n}$ where $f_{x}(y)=1+x\cdot y$ and the result follows by
taking the limit.
###### Proof.
The above property allows us to express $x^{*}$ as the limit of finite
iterations of $x$ interleaved with successful termination. We have
$\displaystyle(P+Q)^{*}\|(M+C)^{*}$ $\displaystyle=$
$\displaystyle\sup_{m}(1+P+Q)^{m}\|\sup_{n}(1+M+C)^{n}$ $\displaystyle=$
$\displaystyle\sup_{m}\sup_{n}\left[(1+P+Q)^{m}\|(1+M+C)^{n}\right]$
The processes $P$ and $Q$ are essentially delimited by $\alpha?K$ and
$\alpha!K$ which ensures the following properties of the system
$\displaystyle X\cdot A\|Y\cdot B$ $\displaystyle=$
$\displaystyle[X\|Y]\cdot[A\|B]$ (17) $\displaystyle X\cdot A\|1$
$\displaystyle=$ $\displaystyle 0$ (18) $\displaystyle Y\cdot B\|1$
$\displaystyle=$ $\displaystyle 0$ (19)
for every processes $A,B$ and where $X=P+Q$ is the collection of tourists and
$Y=M+C$ is the collection of places.
In particular,
$1\|(1+X)^{n}=1\|(1+X)^{n-1}+1\|X\cdot(1+X)^{n}=1\|(1+X)^{n-1}$
and by induction, since $1\|1=1$,
$1\|(1+X)^{n}=1$ (20)
for every $n\in\mathbb{N}$. Similarly, $1\|(1+Y)^{n}=1$.
On the other hand, let us denote $T_{m,n}=(1+X)^{m}\|(1+Y)^{n}$, then
$\displaystyle T_{m,n}$ $\displaystyle=$
$\displaystyle\left[(1+X)^{m-1}+X\cdot(1+X)^{m-1}\right]\|$
$\displaystyle\qquad\left[(1+Y)^{n-1}+Y\cdot(1+Y)^{n-1}\right]$
$\displaystyle=$ $\displaystyle T_{m-1,n-1}+X\cdot(1+X)^{m-1}\|(1+Y)^{n-1}+$
$\displaystyle\qquad(1+X)^{m-1}\|Y\cdot(1+Y)^{m-1}+$
$\displaystyle\qquad[X\|Y]\cdot T_{m-1,n-1}$ $\displaystyle=$
$\displaystyle(1+X\|Y)\cdot T_{m-1,n-1}+U_{m-1,n-1}+$ $\displaystyle\qquad
V_{m-1,n-1}$
where
$\displaystyle U_{m-1,n-1}$ $\displaystyle=$ $\displaystyle
X\cdot(1+X)^{m-1}\|[(1+Y)^{n-1}$ $\displaystyle=$ $\displaystyle
U_{m-1,n-2}+[X\|Y]\cdot T_{m-1,n-2}$ $\displaystyle=$ $\displaystyle
U_{m-1,n-3}+[X\|Y]\cdot T_{m-1,n-3}+$ $\displaystyle\qquad[X\|Y]\cdot
T_{m-1,n-3}$
Since the sequence $(1+Y)^{n}$ is monotone, $T_{m,n}\leq T_{m,n^{\prime}}$ for
every $n\leq n^{\prime}$ and therefore $U_{m-1,n-1}\leq U_{m-1,0}+[X\|Y]\cdot
T_{m-1,n-1}$. But Property 18 implies that $U_{m-1,0}=0$.
Similarly, $V_{m-1,n-1}\leq[X\|Y]\cdot T_{m-1,n-1}$. Hence
$T_{m,n}=(1+[X\|Y])\cdot T_{m-1,n-1}.$
By induction, we show that
$T_{m,n}=(1+[X\|Y])^{\inf(m,n)}$
because $T_{0,n}=T_{m,0}=1$ by Equation 20.
Finally, we have
$\displaystyle X^{*}\|Y^{*}$ $\displaystyle=$
$\displaystyle\sup_{m}\sup_{n}(1+X)^{m}\|(1+Y)^{n}$ $\displaystyle=$
$\displaystyle\sup_{n}(1+[X\|Y])^{n}$ $\displaystyle=$
$\displaystyle(X\|Y)^{*}$
∎
|
arxiv-papers
| 2013-01-30T07:53:48 |
2024-09-04T02:49:40.953473
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Annabelle McIver and Tahiry Rabehaja and Georg Struth",
"submitter": "Tahiry Rabehaja",
"url": "https://arxiv.org/abs/1301.7153"
}
|
1301.7228
|
# Roles of icosahedral and crystal-like order
in hard spheres glass transition
Mathieu Leocmach Hajime Tanaka [email protected] Institute of
Industrial Science, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo
153-8505, Japan
(Received September 27, 2011)
###### Abstract
A link between structural ordering and slow dynamics has recently attracted
much attention from the context of the origin of glassy slow dynamics.
Candidates for such structural order are icosahedral, exotic amorphous, and
crystal-like. Each type of order is linked to a different scenario of glass
transition. Here we experimentally access local structural order in
polydisperse hard spheres by particle-level confocal microscopy. We identify
the key structures as icosahedral and fcc-like order, both statistically
associated with slow particles. However, when approaching the glass
transition, the icosahedral order does not grow in size whereas crystal-like
order grows. It is the latter that governs the dynamics and is linked to
dynamic heterogeneity. This questions the direct role of the local icosahedral
ordering in glassy slow dynamics and suggests that the growing lengthscale of
structural order is essential for the slowing down of dynamics and the
nonlocal cooperativity in particle motion.
##
Upon cooling, a liquid usually undergoes a first order transition to an
ordered ground state (crystal or quasi-crystal). However, it is generally
possible to avoid the transition and to form a metastable supercooled liquid.
Upon further cooling the dynamics slows down dramatically by many orders of
magnitude, leading to the impossibility to equilibrate the system in the
experimental time scale below a temperature $T_{g}$ named the glass transition
temperature. The nature of the glass transition (thermodynamic or kinetic,
structural or purely dynamical) is still a matter of debate after decades of
intensive research (see (Cavagna, 2009; Berthier & Biroli, 2011) for
comprehensive reviews).
A part of the answer is believed to come from the dynamics in itself: a
supercooled liquid is dynamically heterogeneous (see (Berthier & Biroli, 2011)
for a review) and the characteristic size of the dynamical heterogeneity is
growing when approaching the glass transition (Yamamoto & Onuki, 1998; Donati
et al., 1999) (although non-monotonic behaviour has been observed recently Kob
et al. (2011)). The dynamical arrest may then be the analogue of the slowing
down observed near a critical point when the characteristic size of the
fluctuations diverges, although slowing down at a particle level is absent in
ordinary critical phenomena. Here we note that the lengthscale defined by the
dynamical heterogeneity is not static (one-time spatial correlation) but
dynamic (two-time spatial correlation).
Modern spin-glass-type theories of the structural glass transition Lubchenko &
Wolynes (2007); Biroli et al. (2008); Parisi & Zamponi (2010) introduce a
possible static length called ‘point-to-set’ length that makes neither
assumption nor description of the ‘amorphous order’ underlying the glass
transition. It was demonstrated theoretically Montanari & Semerjian (2006) for
lattice models that the growth of a dynamical length scale imposes a lower
bound on the point-to-set length. However this constraint is expected to
become relevant only in a low temperature regime inaccessible to present-day
simulations Kob et al. (2011). Indeed static lengths have been measured
uncorrelated to the dynamic one above this regime Charbonneau et al. (2012).
The relation between the dynamical and the static length scales is a corner
stone of the debate between geometrical frustration-based descriptions,
facilitation-based descriptions, and the random first order transition (RFOT)
theory.
In order to unveil static quantities, it is tempting to link dynamical
heterogeneity to structural heterogeneity. Stable structures should move less
than unstable ones. Widmer-Cooper & Harrowell (2005) demonstrated that the
initial configuration of a simulation, independently of the initial dynamics,
has a role in the heterogeneity of the dynamics. Berthier & Jack (2007)
pointed out later that this influence does not exist at the particle level,
but on larger scales: medium-range particle configuration can be statistically
linked to medium-range dynamical heterogeneity. However these arguments are
based on the iso-configurational ensemble (starting several simulations with
different dynamics but with the same initial configuration), a method that has
no experimental equivalent.
Given that a glass shows no long-range periodic order, structures different
from a crystal, icosahedral Steinhardt et al. (1983); Sadoc & Mosseri (1999);
Tarjus et al. (2005) or amorphous order Lubchenko & Wolynes (2007), are most
often pointed as responsible for the dynamical arrest. Since the pioneering
work of Frank Frank (1952), icosahedral order is the archetypal model of
amorphous structures (Spaepen, 2000). Locally, an icosahedron maximises the
density of packing, but its five-fold symmetry is incompatible with long-range
periodicity. Geometrical frustration associated with icosahedral ordering is
proposed to be crucial for vitrification Steinhardt et al. (1983); Sadoc &
Mosseri (1999); Tarjus et al. (2005). It was also proposed that the
percolation of icosahedral order is responsible for glass transition Tomida &
Egami (1995). Evidence of icosahedral ordering has indeed been found in very
dense hard sphere packings (Bernal & Mason, 1960; Clarke & Jónsson, 1993;
Anikeenko & Medvedev, 2007; Karayiannis et al., 2011; Charbonneau et al.,
2012), in various simulation models (Steinhardt et al., 1983; Tomida & Egami,
1995; Doye et al., 2003; Pedersen et al., 2010; Coslovich, 2011) and more
recently in experimental dense metallic liquids (Reichert et al., 2000; Celino
et al., 2007) and glasses (Luo et al., 2004; Wang et al., 2011).
Geometrical frustration in icosahedral order (or more generally locally
favoured structures of the liquid) plays a central role in original versions
of the spin-glass-type theory Steinhardt et al. (1983); Sadoc & Mosseri (1999)
and the frustration-limited domain theory of the glass transition (see (Tarjus
et al., 2005) for a review). The former theory considers the icosahedral
ordering under frustration. On the other hand, the latter theory takes as a
reference state an unfrustrated icosahedral order existing in a curved space,
even if (and because) this reference state is impossible for the real system
to reach. If the space were curved, the locally preferred order of the liquid
(icosahedral) would form a continuous ordered phase upon cooling. In the
euclidean space, however, this transition is avoided, yet the avoided
transition temperature still acts as a critical point giving rise to diverging
lengthscales (icosahedral domain size and defect size) that would explain the
dynamical arrest. This theory suggests that the frustration-limited domains
are the origin of the dynamical heterogeneity. Both theories are still very
popular and icosahedral order is often cited as the reference amorphous
structure particularly in metallic glasses (Reichert et al., 2000; Celino et
al., 2007; Luo et al., 2004; Wang et al., 2011). Yet, to our knowledge,
numerical systems where a link between icosahedral order and dynamical
heterogeneity has been observed (Doye et al., 2003; Pedersen et al., 2010;
Coslovich, 2011), always have a (quasi-)crystalline ground state including
icosahedra. Thus, icosahedra could be regarded as a structural motif linked to
the crystalline ground state in these systems. From this point of view, these
results support neither the spin-glass nor frustration-limited domain
approaches based on frustration in icosahedral order, but rather the crystal-
based approach (see below).
An alternative way of explaining the phenomenology of the glass transition is
based on the structure of the crystal. Crystallisation, even if avoided, may
have an influence on the supercooled fluid or the glass (Tanaka, 1998; Cavagna
et al., 2003; van Megen et al., 2009) and is taken as a reference state.
Simulations Tanaka et al. (2010); Pedersen et al. (2010); Coslovich (2011) and
experiments (Tanaka et al., 2010) show that a local bond orientational order
compatible with the ground state (crystal) symmetry does extend to medium
range in several model supercooled systems at accessible temperatures. In this
class of systems, these transient but relatively stable structures seem to be
correlated with the slow regions because they are low free-energy
configurations. Furthermore, they lack extended positional order and thus
could not be detected by macroscopic diffraction experiments. We emphasize
that high crystal-like bond orientation regions are not crystal nuclei with
solid nature, but transient medium-range structural order formed in a liquid
as thermal fluctuations.
In the present work, we focus on the colloidal (hard spheres) supercooled
liquid Pusey & van Megen (1987) for two reasons. First, hard sphere-like
colloidal particles can be tracked by confocal microscopy, giving access to
the positions and dynamics of individual particles Kegel & van Blaaderen
(2000); Weeks et al. (2000), thus allowing microscopic analysis of the
dynamical and structural heterogeneity. Second, hard spheres have a well
defined crystalline ground state of face-centred-cubic (fcc) or hexagonal-
close-packed (hcp) symmetry and locally favoured structures of icosahedral
symmetry. This enables us to distinguish the different scenarios of glass
transition, in particular, the frustration-limited domain scenario, the
amorphous order scenario, and the medium-range crystalline order scenario. We
will show that both kinds of order are statistically associated with slower
than average particles. However icosahedral order remains local, whereas the
crystal-like order correlation length grows when approaching the glass
transition. Moreover, the medium-range crystalline order is associated with
the slow regions of the dynamic heterogeneities. We confirm that both
structural and dynamical length scale grow in the same (critical-like) way.
Our result suggest that, when not compatible with an avoided crystallisation,
local ordering seems to play a minor role in the dynamical aspects of the
glass transition.
## Results
### Dynamics
We were able to follow the volume fraction dependence of the dynamics over
nearly three orders of magnitude (see Fig. 1), confirming the basic
phenomenology observed in experiments (Pusey & van Megen, 1987; Kegel & van
Blaaderen, 2000; Weeks et al., 2000; Berthier & Biroli, 2011) as well as
simulations (Tanaka et al., 2010) of similar systems. In Fig. 1a we fit the
self-intermediate scattering function by a stretched exponential $\propto
e^{-(t/\tau_{\alpha})^{\beta}}$ that defines the structural relaxation time
$\tau_{\alpha}$. Here $\beta$ is the stretching exponent.
The non-Gaussianity of the dynamics is monitored by the kurtosis of
displacements distribution (Fig. 1b)
$\alpha_{2}(t)=\frac{3\left\langle{(\Delta
r(t))^{4}}\right\rangle}{5{\left\langle{(\Delta
r(t))^{2}}\right\rangle}^{2}}-1,$ (1)
where $\Delta r(t)=|\mbox{\boldmath$r$}(t_{o}+t)-\mbox{\boldmath$r$}(t_{0})|$
is the displacement of a particle after a time interval $t$. At a given volume
fraction, $\alpha_{2}(t)$ peaks at the characteristic time scale of the
dynamic heterogeneity $t^{\text{dh}}(\phi)$.
The volume fraction dependence of $\tau_{\alpha}$ is super-Arrhenius and we
fit it in Fig. 1d by the Vogel-Fulcher-Tammann (VFT) law
$\tau_{\alpha}=\tau_{\alpha}^{0}\exp(D\phi/(\phi_{0}-\phi))$, where $D$ is the
fragility index, $\phi_{0}$ is the hypothetical ideal glass transition volume
fraction, and $\tau_{\alpha}^{0}$ is the microscopic timescale. We notice that
$t^{\text{dh}}$ corresponds to the end of the plateau, always slightly shorter
than $\tau_{\alpha}$.
The spatial heterogeneity of the dynamics is characterised by a four-point
susceptibility $\chi_{4}(t)$ and a four-point structure factor $S_{4}(q,t)$
Flenner et al. (2011). To the reader unfamiliar with these terms (defined in
Methods), we note that $S_{4}(q,t)\times N/N_{s}$ is the structure factor of a
system containing only the slow particles.
As shown in Fig. 1c, the small wave-vector behaviour of $S_{4}(q,t)$ is well
described by the asymptotic Ornstein-Zernike function in Fourier space:
$S_{4}(q\rightarrow 0,t)\approx\frac{\chi_{4}(t)}{1+\xi_{4}(t)^{2}q^{2}},$ (2)
where $\xi_{4}(t)$ is the four-point correlation length (see Methods for the
details of the fit). We found that at each volume fraction the maximum of
$\chi_{4}(t)$ is reached at times longer than the relaxation time
$\tau_{\alpha}$. In summary, we have $t^{\text{dh}}\leq\tau_{\alpha}\leq
t_{\chi_{4}}$. For simplicity, we omit time dependence in $S_{4}$, $\xi_{4}$
and $\chi_{4}$ when $t=t_{\chi_{4}}$. The volume fraction dependence of the
resulting correlation length is plotted in the inset of Fig. 1. In the volume
fraction range that we were able to study, it follows a power law
$\xi_{4}=\xi_{4,0}((\phi_{0}-\phi)/\phi_{0})^{-\nu},$ (3)
where $\nu=2/3$ and $\phi_{0}$ is independently determined by the above VFT
fit; thus the bare correlation length $\xi_{4,0}$ is the only fitting
parameter (Tanaka et al., 2010). Note that the lengths are in units of the
mean diameter $\sigma$.
### Bond orientational ordering
The local structures of our suspensions are identified by a detailed analysis
of Steinhardt’s bond orientational order (boo) (Steinhardt et al., 1983),
including original improvements (see Methods). Figure 2 summarizes our
structure identification results. We find the distinctive signature of medium-
ranged crystalline order (mrco) of fcc type, without excluding some hcp. We
find no trace of bcc-like structure. As for the aperiodic structures,
icosahedral order stands out without any other obvious rivals (dodecahedra are
present but can be considered as twisted icosahedra).
We establish that in our system the tendency towards icosahedral order is best
monitored by the order parameter called $w_{6}$, whereas $Q_{6}$ is the
natural crystal-like bond orientational order parameter (see Methods for the
definitions of $w_{6}$ and $Q_{6}$). The population of local structures in the
$(w_{6},Q_{6})$-plane is shown in Fig. 2c-d, which confirms that the two
tendencies are present in the supercooled liquid of hard spheres. In
particular, icosahedra are present even at a relatively low volume fraction
(Fig. 2d). Both tendencies become more pronounced for deeper supercooling
(Fig. 2c). Furthermore, Fig. 2c shows clearly that icosahedral order and
crystalline order are incompatible and frustrate one another.
### Bond order mobility
In the introduction we mentioned the dynamic propensity invented by Widmer-
Cooper & Harrowell (2005) as a trick only accessible via simulations. In our
system, since we know the relevant structural order parameters ($w_{6}$ and
$Q_{6}$), we can compute the square displacement of all the particles with the
same initial local structure (iso-bond order ensemble) and call it bond order
mobility. We define the $w_{6}$-mobility as
$\Delta
r^{2}(w_{6},t)\equiv\left\langle\frac{\sum\limits_{i}{\left\|\mathbf{r}_{i}(t)-\mathbf{r}_{i}(0)\right\|^{2}\delta(w_{6}^{i}-w_{6})}}{\sum\limits_{i}{\delta(w_{6}^{i}-w_{6})}}\right\rangle,$
(4)
where the brackets denote ensemble averaging. $\Delta r^{2}(Q_{6},t)$ and
$\Delta r^{2}(w_{6},Q_{6},t)$ can be defined in the same way. The iso-
configurational propensity is very powerful since it does not require any a
priori information on structures and thus can probe any static order if it
exists. This feature is absent in the bond order mobility, which instead has
the merit of accessibility even in a system that cannot be restarted ab
libitum from exactly the same overall configuration. In general the two
quantities are not equivalent.
Figure 3 shows the bond order mobility (calculated at the characteristic time
of the dynamical heterogeneity, $t^{\text{dh}}$). Both structural order
parameters have a clear correlation with the dynamics: the more structured is
the environment, the more likely the particle will be slow. For all volume
fractions, the mobility displays the same linear dependence upon $w_{6}$ (Fig.
3b). Once scaled by the mean square displacement all points collapse on the
same line. This is not the case for the $Q_{6}$ dependence (Fig. 3c) which is
more and more pronounced with increasing the degree of supercooling. In our
most deeply supercooled sample, the crystal-like environments can be in
average ten times slower than the bulk, whereas even the almost perfect
icosahedra are in average only $40\%$ slower than the bulk. A complete
screening of all the possible boo mobilities excludes a link of slow dynamics
to any other local symmetries (including exotic amorphous order), at least in
our system.
To explain our mobility data, we can refer to the cage picture, where a
particle is considered as trapped in the cage formed by its neighbours. Once
the particle has escaped from its cage, it is free to diffuse. In general,
this picture is not correct due to dynamical heterogeneity, or nonlocal
cooperativity of motion. In the case of the icosahedral environment, however,
it seems to hold: the normalisation by the bulk msd takes into account the
diffusion once out of the cage, and the remaining universal dependence upon
$w_{6}$ holds the information about the “quality” of the cage. Efficient
packing makes the 13 particles icosahedra more stable than a disordered
structure. Thus the central particle has a low probability to escape from its
cage and start diffusing. After cage escaping, however, it diffuses in average
like any other particle. This suggests that the influence of icosahedral order
on the dynamics is only local because of its isolated character. By contrast
the non-trivial $\phi$ dependence of the $Q_{6}$-mobility calls for nonlocal
explanations.
### Real space patterns and length scale
To observe the ordered regions in real space, we define the thresholds
$w_{6}^{*}$ and $Q_{6}^{*}$ so that the bond order mobility at the threshold
is half between the bulk and the (extrapolated) perfect structure. In our most
deeply supercooled sample, this criterion yields $Q_{6}^{*}=0.25$ for crystal-
like structures (see Fig. 3d) and $w_{6}^{*}=-0.023$ for icosahedral
structures (see Fig. 3e). These thresholds are arbitrary but coherent with
each other. We display in Fig. 4b typical configurations of the ordered
neighbourhoods in our deepest supercooled sample. With our thresholds, we see
only small patches of icosahedral order that are not reaching medium range,
whereas clusters of crystal-like order are much larger than icosahedral
clusters and their sizes reach medium range.
Here it may be worth stressing that mrco does not correspond to crystal
nucleus (compare Fig. 4a and b). However sub-critical nuclei are not unrelated
to crystal-like boo. Nuclei are systematically embedded into much larger mrco.
One can explain this phenomenon as perfect wetting of the nuclei by coherently
bond-ordered regions Kawasaki & Tanaka (2010). We note that the size of the
crystal nuclei is smaller than the critical nucleus size and thus they appear
only transiently. Furthermore, examples of crystal-like structures without
embedded crystal nucleus, in addition to the continuity of $Q_{6}$ values
(Fig. 2), suggest that the crystal nuclei are the extreme part of the bond
order fluctuations. Indeed it has been shown in hard spheres O’Malley & Snook
(2005); Schilling et al. (2010); Kawasaki & Tanaka (2010), Gaussian-core model
Russo & Tanaka (2012) and probably Wahnström binary Lennard-Jones Pedersen et
al. (2010) that parts of mrco, fluid regions with a structure reminiscent of
the crystal, act as precursors to crystallisation if the local density there
is high enough. However in these systems the presence of precursors to
crystallisation does not imply the formation of a critical nucleus and growth.
Furthermore, mrco is always present (transiently) in the supercooled liquid:
mrco are usual fluctuations of the system, as common as the fast and slow
regions of the dynamic heterogeneity Tanaka et al. (2010). We also note that
the average density within mrco is the same as within disordered liquid
regions and thus different from within crystal nuclei Kawasaki & Tanaka
(2010).
Now we consider a link between static spatial structures and dynamics.
Reflecting the short-range nature of icosahedral order, we were not able to
extract the associated lengthscale that would grow when approaching the glass
transition. By contrast, the spatial extent of crystal-like boo is well
described by $G_{6}(r)$, the spatial correlation function of $Q_{6m}$ (see,
e.g., Tanaka et al. (2010)). Both dynamical and structural lengthscales
(defined in Methods) are increasing while approaching the glass transition in
a coherent manner (the inset of Fig. 1). We can see that near $\phi_{0}$ the
growth of both dynamic and static correlation length can be well expressed by
the same power law (Eq. (3)), although the range is rather limited. In
agreement with (Tanaka et al., 2010) it suggests that the dynamical
heterogeneity in hard spheres is a manifestation of critical-like fluctuations
of the crystal-like bond orientational order parameter, and neither those of
icosahedral nor amorphous order. In relation to this, we stress that
$G_{6}(r)$ is not sensitive to aperiodic structures such as icosahedral and
amorphous order (see Methods).
Of course, one can set a more permissive threshold on $w_{6}$ and observe more
icosahedra. A percolating network of icosahedral neighbourhood can even be
found at a high volume fraction if the threshold encompasses enough particles
(see Fig. 5c). However, the imperfect icosahedra particles that must be
included to form this network have a mobility comparable to the bulk (Fig.
3b). Furthermore, we checked that this percolation is of the 3D random
percolation class (cluster size distribution follows a power-law of exponent
$\approx 2.1$, see Fig. 3a), meaning that a comparable network could be
obtained by picking randomly the same number of particles (see Fig. 5d). We
conclude that the concept of icosahedral network is not physically meaningful
at least in our system.
As a final but an intuitive check, we compare the spatial repartition of
stable crystal-like order (Fig. 4c) and slow particles (Fig. 4d). As shown in
(Berthier & Jack, 2007), the correlation between structure and dynamics is not
a one-to-one correspondence at the particle level but a statistical
relationship on larger length scales. We were able to show visually this
correlation by averaging out short time ($t^{\text{dh}}/2$) fluctuations of
$Q_{6}$; by calculating the square displacement of each particle over
$2\tau_{\alpha}\approx 4.5t^{\text{dh}}$ which allow to accumulate many
rearrangement events; and by coarse-graining this value over the particle’s
neighbours (Berthier & Jack, 2007) (see Eq. (16) in Methods). Figure 4d
displays the $10\%$ slowest particles according to this criterion and their
neighbours. The correlation of shape and size is clearly seen between Fig. 4c
and Fig. 4d, although it is not perfect. The slight disagreement comes from
the following reasons: some clusters of fcc-like order disappear soon after
the initiation of the displacement measurement and some particle groups, which
come back to their original positions after the period of $2\tau_{\alpha}$,
can be accounted as slow particles. We could find no such correlation between
icosahedral particles (Fig. 4b) and slow regions.
Note that we recover the correlation between crystal-like order and slowness
by two independent methods: directly in instantaneous boo space via mobility
on moderate times; and indirectly via positions and shapes of (short time
averaged) ordered regions coherent with those of slowest regions (on long
times). In the former method, quality of statistics comes from ensemble
averaging over many realisations of each structure. In the latter method, it
comes from both coarse graining and the length of trajectories that allow to
accumulate many rearrangement events.
## Discussion
We have demonstrated that supercooled hard-sphere liquids display local
icosahedral ordering as well as structural ordering linked to the avoided
crystalline (fcc) ground state. Both types of packing have low local free
energy and thus are locally favoured. Yet the icosahedral order remains local
even at deep supercooling and (consistently with the arguments of (Berthier &
Jack, 2007)) is not able to influence significantly the global dynamics. By
contrast, the slowing down associated with the crystal-like boo is nonlocal
and long-lived. The characteristic length scale of the crystal-like order
grows when approaching the glass transition in the same way as that of the
dynamical heterogeneity. This suggests that the structural origin of the
dynamical arrest is linked to the avoided crystallisation and neither to the
condensation of the local (icosahedral) order of the liquid nor to the exotic
amorphous order at least in our system. We note that the development of
structural order and its link to dynamic heterogeneity are at odds with purely
kinetic scenarios of glass transition.
How general is our crystal-like order scenario? Not all glass formers
crystallise, indeed the crystal structure of many glass formers is unknown.
However in a system where the crystallisation is possible but unlikely (a
definition that include many practical glass formers), we expect structures
locally reminiscent of the symmetry of the crystal to be present as common
fluctuations of the supercooled liquid. Their relative stability and ability
to reach medium-range sizes makes them a good candidate to explain the slow
regions of the dynamic heterogeneity. In that class of systems static and
dynamical lengths would grow together. Yet the generality and limitation of
this scenario need to be checked carefully in the future.
We note that Charbonneau et al. (2012) are not finding growing 6-fold bond
order length scale in a binary hard sphere mixture. Hopkins et al. (2011,
2012) showed that for this size ratio if the two components de-mix they
crystallise independently in fcc or hcp; however the symmetry-breaking phase
of the mixed system is not known and could possibly be not 6-fold symmetric.
This de-mixing required for crystallization may destroy a link between
configurations of low local free energy in the liquid state and the crystal
structure. Indeed, the study of boo in binary systems with the concept of bond
orientational order is not promising, as reported in (Tanaka et al., 2010;
Kawasaki & Tanaka, 2011) for 2D binary soft disk simulations.
For some systems, the first order transition avoided upon supercooling leads
to a symmetry compatible with icosahedral order. The symmetry-breaking phase
can be a quasi-crystal (Doye et al., 2003) or a Frank-Kasper crystal, as
recently demonstrated for the Wahnström mixture (Pedersen et al., 2010). In
that case, the icosahedral order can be regarded as part of the defective
crystalline bond orientational order that slows down the system, so the
icosahedral order can play a role in the slowing down. However, we showed here
that when the icosahedral order is not compatible with the crystal symmetry,
its role in the slowing down is actually minor.
We do not say that the local (icosahedral) order of the liquid is unimportant
in the glass transition. Actually it plays a role in preventing the first
order transition (crystallisation) to happen, thus allowing supercooling and
the glass transition Tanaka (2003). Furthermore, this frustration against
crystallisation may be linked to the fragility of the glass former (Tanaka,
1998; Tanaka et al., 2010). Competing crystal-like and non-crystal-like
orderings may be important in many glass-forming systems covering colloidal,
molecular, polymeric, oxide, chalcogenide, and metallic glasses. Indeed, our
competing ordering scenario seems very consistent with a recent finding in
metallic liquids Liu et al. (2010) and may explain the high glass-forming
ability of bulk metallic glassformers Wang et al. (2004). An intimate link
between the liquid and crystal structure may also provide a mechanism of fast
phase change materials Wuttig & Yamada (2007).
## Methods
### Experimental
We used pmma (poly(methyl methacrylate)) colloids sterically stabilized with
methacryloxypropyl terminated pdms(poly(dimethyl siloxane)) and fluorescently
labelled with rhodamine isothiocyanate chemically bonded to the pmma. The size
distribution, as characterised by scanning electron microscopy imaging (see
Supplementary Figure S1), showed a long tail towards small sizes, leading to
the polydispersity larger than 6%. This amount of polydispersity allows us to
avoid or at least delay crystallisation Zaccarelli et al. (2009) (see below
the crystallization monitoring method) but is too low for fractionation to
happen (Fasolo & Sollich, 2003). The colloids were suspended in a solvent
mixture of cis-decalin and cyclohexyl-bromide for both optical index and
density matching. To screen any (weak) electrostatic interactions, we
dissolved tetrabutylammonium bromide salt, to a concentration of
$300\text{\,}\mathrm{nmol}\text{\,}{\mathrm{L}}^{-1}$ (Royall et al., 2005).
The estimated Debye screening length is $13\text{\,}\mathrm{nm}$, well below
the length scale of the colloids. To avoid errors due to the swelling of the
particles, their size was measured in the same solvent by the following
technique: short-ranged depletion attraction was induced by adding non-
adsorbing polystyrene to a dilute suspension to drive particles into contact;
the position of the first peak of the $g(r)$ gave the diameter
$\sigma=$3.356\pm 0.03\text{\,}\mathrm{\SIUnitSymbolMicro m}$$.
After careful shear melting, the samples are filled into square
$500\text{\,}\mathrm{\SIUnitSymbolMicro m}$ capillaries (Vitrocom). The most
deeply supercooled sample showed a slight indication of sedimentation even
after careful density matching. To avoid this effect we used a thinner
$$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$\times$1\text{\,}\mathrm{mm}$$
cell. The data were collected on a Leica SP5 confocal microscope, using
$532\text{\,}\mathrm{nm}$ laser excitation. The temperature was controlled on
both stage and objective lens, allowing a more precise density matching. Our
polydisperse colloids were detected using a novel multi-scale algorithm (see
Supplementary Methods on the details), which allows us to retrieve both
position and size of each particle with about $1\%$ precision, and this for
any arbitrary size distribution. We stress that standard particle tracking
methods Kegel & van Blaaderen (2000); Weeks et al. (2000) are not able to
track all the particles in our polydisperse samples (see Supplementary Figure
S2). The large size of our colloids allows a comparable precision in position
for both in-plane and out-of-plane coordinates (see Supplementary Figure S1).
The particles are tracked in time to extract dynamical informations. Typically
$\approx 5000$ trajectories can be followed over a few structural relaxation
times.
### Dynamics
To characterise the dynamic heterogeneity, we define a microscopic overlap
function $w_{i}(t)=\Theta[b-|\mathbf{r}_{i}(t)-\mathbf{r}_{i}(0)|]$, where
$\Theta(x)$ is Heaviside’s step function, $\mathbf{r}_{i}(t)$ is the position
of particle $i$ at a time $t$ and $b=0.3\langle\sigma\rangle$. The number of
overlapping particles is $N_{s}(t)$ and in the thermodynamic limit its
fluctuations define the four-point susceptibility:
$\chi_{4}=(\langle N_{s}^{2}\rangle-\langle N_{s}\rangle^{2})/\langle
N\rangle.$ (5)
In a simulation with a fixed number of particles $N$ and a limited size, some
fluctuations are forbidden, thus Eq. (5) do not give the right value of
$\chi_{4}$ (see (Flenner et al., 2011)). However in our experiments we are
observing a small portion of the whole colloidal suspension, thus we can use
time average to sample all the fluctuations of this large system, and Eq. (5)
becomes exact and ensemble-independent.
The four-point structure factor Flenner et al. (2011) is
$S_{4}(q,t)=N^{-1}(\left\langle
W(\mathbf{q},t)W(-\mathbf{q},t)\right\rangle-|\left\langle
W(\mathbf{q},t)\right\rangle^{2}|),$ (6)
where $W(\mathbf{q},t)$ is the Fourier transform of $w_{i}(t)$:
$W(\mathbf{q},t)=\sum_{i}w_{i}(t)\exp(-\imath\mathbf{q}\cdot\mathbf{r}_{i}(0)).$
(7)
Our experimental data do not have periodic boundary conditions, so we must use
a window function to ensure the correct correlation, especially at small $q$.
Here we use the Hamming window, but we checked that our results are not
affected by other reasonable choices of the window function. The results are
shown in Fig. 1c.
As stressed in (Flenner et al., 2011), the value $S_{4}(q=0)$ is crucial for
the quality of the fit in Eq. (2). Here we use the value of $\chi_{4}$ given
by Eq. (5) and we fit $S_{4}(q)$ for $q<1.5/\xi_{4}$.
### Structure identification
Steinhardt et al. (1983) defined the boo of the $\ell$-fold symmetry as a
$2\ell+1$ vector:
$q_{\ell m}(i)=\frac{1}{N_{i}}\sum_{0}^{N_{i}}Y_{\ell
m}(\theta(\mathbf{r}_{ij}),\phi(\mathbf{r}_{ij})),$ (8)
where the $Y_{\ell m}$ are spherical harmonics and $N_{i}$ is the number of
bonds of particle $i$. In the analysis, one uses the rotational invariants
defined as:
$\displaystyle q_{\ell}=$
$\displaystyle\sqrt{\frac{4\pi}{2l+1}\sum_{m=-\ell}^{\ell}|q_{\ell m}|^{2}},$
(9) $\displaystyle w_{\ell}=$
$\displaystyle\sum_{m_{1}+m_{2}+m_{3}=0}\left(\begin{array}[]{ccc}\ell&\ell&\ell\\\
m_{1}&m_{2}&m_{3}\end{array}\right)q_{\ell m_{1}}q_{\ell m_{2}}q_{\ell
m_{3}},$ (12) $\displaystyle\hat{w}_{\ell}=$ $\displaystyle
w_{\ell}{\left(\sum_{m=-\ell}^{\ell}|q_{\ell m}|^{2}\right)}^{-\frac{3}{2}},$
(13)
where the term in brackets in Eq. (12) is the Wigner 3-j symbol. Note that the
popular (Steinhardt et al., 1983; Lechner & Dellago, 2008) $\hat{w}_{\ell}$
can show a large norm if its denominator (or $q_{\ell}$) has small values
(particle with low $\ell$-fold symmetry). Therefore, we have used here
$w_{\ell}$ that pulls clearly apart highly symmetric particles from the fluid
distribution.
Following Lechner & Dellago (2008), we coarse-grain the tensorial boo over the
neighbours:
$Q_{\ell m}(i)=\frac{1}{N_{i}+1}\left(q_{\ell m}(i)+\sum_{j=0}^{N_{i}}q_{\ell
m}(j)\right),$ (14)
and define coarse-grained invariants $Q_{\ell}$ and $W_{\ell}$ in the same way
as Eqs. (9) and (12). Structures with and without spatial extendability are
then much easier to tell apart (Lechner & Dellago, 2008). For non-extendable
local structures like icosahedra, their $Q_{\ell}$ and $W_{\ell}$ are buried
into the liquid distribution. With this in mind, we looked for extendable
structures in the $\\{Q_{\ell},W_{\ell}\\}$ space, and for non-extendable
(aperiodic) structures in the $\\{q_{\ell},w_{\ell}\\}$ space, with
$\ell=4,6,8,10$.
In both Eqs. (8) and (14), the sum runs over the “neighbours” (labelled with
$j$) of particle $i$. The definition of the neighbours is crucial for the
consistency of the analysis. The most popular criterion in literature is that
particles closer than $R_{\text{max}}$ are neighbours, with $R_{\text{max}}$
the first minimum of $g(r)$. To account for the polydispersity, we used a
threshold in the scaled distance
$\hat{r}_{ij}=r_{ij}/(\sigma_{i}+\sigma_{j})<\hat{R}_{\text{max}}$, with
$\hat{R}_{\text{max}}$ corresponding to the first minimum of $g(\hat{r})$. We
made a first round of analysis using this definition, constructing maps
similar to Fig. 2 but noisier. These maps allowed us to discard the
possibility of bcc (high $Q_{6}$, low $Q_{4}$, high $Q_{10}$) and generally
any structure except fcc (high $Q_{6}$, high $Q_{4}$ and $w_{4}<0$), hcp (high
$Q_{6}$ and $w_{4}>0$) or icosahedron (very negative $w_{6}$, high $q_{6}$ and
$q_{10}$). Dodecahedron can be found for
$w_{6}>w_{6}^{\text{dod}}\approx-0.00782$, i.e., between the liquid and the
icosahedral order. We are thus left with structures where the particles
naturally have 12 neighbours.
We then took a new boo analysis, this time taking the first 12 closest
particles (in $\hat{r}$) as neighbours. This criterion reduces the noise in
the distribution of the invariants and makes detected structures more easily
comparable with crystallographic archetypes. It also avoids artefacts near
$R_{\text{max}}$ in the correlation functions and removes the negative
correlation between the number of neighbours and the value of the $q_{\ell}$.
The figures presented here were constructed from this second round of
analysis.
### Crystal nuclei
To avoid any confusion between our imperfect crystal-like order and well-
formed but small transient crystals, we detect crystal nuclei via the most
standard method in the literature ten Wolde et al. (1996); Zaccarelli et al.
(2009). For each neighbouring particle, we compute the normalized scalar
product of the (non coarse-grained) $q_{6m}$:
$s(i,j)=\frac{\sum_{m=-6}^{6}q_{6m}(i)q_{6m}^{*}(j)}{\sqrt{\sum_{m=-6}^{6}|q_{6m}(i)|^{2}}\sqrt{\sum_{m=-6}^{6}|q_{6m}(j)|^{2}}}.$
(15)
The bond between particles $i$ and $j$ is considered crystalline if
$s(i,j)>0.7$ Zaccarelli et al. (2009). A particle is crystalline if more than
half of its bonds ($\geq 7$) are crystalline.
Polydispersity is an important controlling factor which determines the ease of
crystallization of colloidal suspensions (see, e.g., Zaccarelli et al.
(2009)). Polydispersity of our sample was high enough to prevent
crystallisation in bulk over our experimental time scale. However,
heterogeneous nucleation from walls sometimes interfered our experiments, thus
we had to discard such samples. We find no large crystal nucleus in the
samples analysed in the text. Even at deep supercooling a crystal nucleus
never gathers more than 8 particles. This is smaller than the critical nucleus
size (2-3 particle diameters according to Auer & Frenkel (2001) and classical
nucleation theory). In agreement, crystal-like solid regions appear only
transiently without growth, i.e. we never observed crystal nucleation in bulk
for our samples. Truly crystalline particles account for less than $1\%$ of
the system. Figure 4 allows to compare the spatial extent of crystal nuclei
with that of the crystal-like clusters or of the slow regions. It demonstrates
that trivial crystallisation can be responsible for neither the spatial extent
of the dynamical heterogeneities nor glassy slow dynamics.
### Slow particles patterns
The above definition of neighbours is used consistently in the coarse graining
of square displacements:
$\mathrm{bar}{\Delta r^{2}}_{i}(t)=\frac{1}{N(i)+1}\left({\Delta
r^{2}}_{i}(t)+\sum_{j=0}^{N(i)}{\Delta r^{2}}_{j}(t)\right),$ (16)
where ${\Delta
r^{2}}_{i}(t)=\left\|\mathbf{r}_{i}(t)-\mathbf{r}_{i}(0)\right\|^{2}$ and the
neighbours are defined at initial time.
### Icosahedral percolation
We studied clusters formed by icosahedral particles as a bidirectional graph.
The nodes of the graph are all the particles, and at the beginning we have no
edge. When a particle is ‘activated’ the bonds between this particle and its
12 neighbours are added to the graph. It is often meaningful to activate the
particles as a function of an order parameter, for example sorted by
increasing $w_{6}$; the most icosahedral particle connects first and we
activate particles that are less and less icosahedral until percolation and
beyond. For our most supercooled sample, the percolation threshold is
identified near $w_{6}\approx-0.011$ (see Fig. 5c) which corresponds to
$7.5\%$ of activated particles. We stress that this level of icosahedral order
corresponds to an average mobility very close ($\gtrsim 90\%$) to the bulk
(see Fig. 3b).
In this way, the cluster size distribution and fractal dimension can be
studied systematically at each value of the threshold on the order parameter.
In Fig. 5a-b we report characteristics of icosahedral clusters when the
threshold is set just below percolation (all particles with $w_{6}<-0.012$ are
activated). The cluster size distribution shows an exponent of $2.1$,
characteristic of the random percolation universality class. The fractal
dimension is close to $2$.
One can also activate particles in a random fashion and measure the fraction
of activated particles needed for percolation to occurs. We performed 100 try
of this Metropolis algorithm on 160 configurations, in order to estimate the
percolation probability function of the fraction of activated particles.
Results are reported in Fig. 5d and indicate that the peculiar case of
icosahedral order clusters has nothing special.
### Bond order spatial correlation
We defined the coarse-grained bond order correlation function
$G_{6}(r)=\frac{4\pi}{13}\frac{\sum_{i,j}\sum_{m=-6}^{6}Q_{6m}(i)Q_{6m}^{*}(j)\delta(r_{ij}-r)}{\sum_{i,j}\delta(r_{ij}-r)},$
(17)
which is not sensible to aperiodic structures. After baseline subtraction, the
envelope of $G_{6}(r)$ is fitted by the Ornstein-Zernike form to extract the
crystal-like bond orientational order correlation length $\xi_{6}$. We checked
that $g_{6}(r)$, defined from the non-coarse-grained bond order in the same
way as Eq. (17), yield similar results. Insensibility against aperiodic
structures comes from the scalar product in Eq. (17) Tomida & Egami (1995),
not from coarse-graining.
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##
#### Acknowledgments
The authors thank Paddy Royall and John Russo for fruitful discussions. This
work was partially supported by a grant-in-aid from the Ministry of Education,
Culture, Sports, Science and Technology, Japan and Aihara Project, the FIRST
program from JSPS, initiated by CSTP.
#### Author Contributions
H.T. conceived the research, M.L. performed experiments and analysis, M.L. and
H.T. discussed and wrote the manuscript.
#### Author Information
The authors declare that they have no competing financial interests.
Correspondence and requests for materials should be addressed to H.T.
Figure 1: Dynamics of the system. a, Decay of the self-intermediate scattering
(ISF) function computed at wave vector
$q=2\pi/\left\langle\sigma\right\rangle$. Lines are stretched exponential fits
from which we extract the structural relaxation time $\tau_{\alpha}$. b, Non-
Gaussian parameter, whose maximum defines the characteristic time of the
dynamic heterogeneity $t^{\text{dh}}$. c, Collapse of the small-$q$ regimes of
the four-point structure factor on the Orstein-Zernike function (solid curve).
d, $\phi$-dependence of $\tau_{\alpha}$ (circles) and of $t^{\text{dh}}$
(squares). The solid curve is the VFT fit for $\tau_{\alpha}$
($\phi_{0}=0.600$, $D=0.328$). Inset: Dynamical ($\xi_{4}$, squares) and
structural ($\xi_{6}$, circles) correlation lengths scaled by their respective
bare correlation length, respectively $\xi_{4,0}=0.206\sigma$ and
$\xi_{6,0}=0.129\sigma$. The curve is the power-law given in the text. All
times are scaled by the Brownian time $\tau_{B}$.
Figure 2: Population of local particle configurations specified by the boo
parameters. The value of boo parameter for a perfect structure is indicated by
its name’s position on each map. a in the $(Q_{4},Q_{6})$-plane, b
$(w_{4},Q_{6})$-plane and c-d $(w_{6},Q_{6})$-plane. a-c are the populations
at deep supercooling ($\phi=0.575\pm 0.03$); d is the same as c but for a
liquid near the freezing point ($\phi=0.497\pm 0.03$). The colour at a given
couple of BOO parameters indicate how often (log scale) this couple is
detected (per units of area in this plane). The arrows stress the ordering
tendencies: the tendency towards fcc is always visible, a weak tendency
towards hcp can also be distinguished in b, and the tendency towards
icosahedral order (ico) is visible in c. Conflicting tendencies towards fcc
and ico in c is a manifestation of competing orderings, or frustration,
between them. Straight lines are drawn at important thresholds: $Q_{6}^{*}$,
$w_{6}^{*}$, $w_{6}^{\text{dod}}$ and $w_{4}=0$ that differentiates between
fcc and hcp.
Figure 3: Bond order mobility. a, Normalised mobility in the
$(w_{6},Q_{6})$-plane for our most deeply supercooled sample ($\phi=0.575\pm
0.03$). The colour scale is saturated at $1.5$ times the bulk mean square
displacement. b-c, Normalised mobility for icosahedral and crystalline order
parameters respectively at volume fraction $0.535$ (squares), $0.555$
(triangles) and $0.575$ (diamonds), all $\pm 0.03$. Bulk mean square
displacement is scaled to be at 1 (horizontal line). Perfect structures are on
the edge of each plot. The lines are a guide for the eye, stressing the
collapse of the $w_{6}$-mobility at all volume fractions in b and the absence
of such collapse in c. The scattering at low volume fractions is due to poor
averaging of rare structures. Straight lines in a-c corresponds to the
important thresholds: $Q_{6}^{*}$, $w_{6}^{*}$ and $w_{6}^{\text{dod}}$.
Examples of crystal-like cluster and distorted icosahedron at the respective
threshold values are shown in d and e, respectively.
Figure 4: Computer reconstruction from confocal microscopy coordinates in our
most deeply supercooled sample. Depth is $\sim 12\sigma$. Only particles of
interest and their neighbours are displayed. Each particle is plotted with its
real radius. a, Particles with more than $7$ crystalline bonds. b, A typical
configuration of bond ordered particles. Icosahedral particles are shown in
the same colour if they belong to the same cluster. If a particle is
neighbouring both crystal-like and icosahedral structures, it is displayed as
icosahedral. c, Crystal-like particles alone (the order parameter was averaged
over $t^{\text{dh}}/2$). d, Slow particles with respect to the coarse-grained
displacement. Due to particles going in and out of the field of view,
assignment of particles located very near the edges of c and d were not
accurate and have been removed.
Figure 5: Imperfect icosahedral clusters. a, Cluster size distribution
(compatible with random percolation) for nearly percolating threshold
($w_{6}<-0.012$). b, Radius of gyration (indicating a fractal dimension
$d_{f}\approx 2$ given by the straight line) for the case of a. c, Size of the
largest non-percolating cluster as a function of the threshold on $w_{6}$.
Percolation takes place near $w_{6}\approx-0.011$. Vertical lines indicate the
position of $w_{6}^{*}$ and $w_{6}^{\text{dod}}$ respectively. d, Translation
of c in terms of the fraction of activated particles (dots) together with the
probability of the onset of percolation when particles are randomly activated
(gray shading). We can see that activating particles as a function of
icosahedral order and those activated randomly have almost the same
percolation threshold ($\approx 7.5\%$ of the particles activated).
|
arxiv-papers
| 2013-01-30T13:52:20 |
2024-09-04T02:49:40.968444
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mathieu Leocmach and Hajime Tanaka",
"submitter": "Mathieu Leocmach",
"url": "https://arxiv.org/abs/1301.7228"
}
|
1301.7237
|
Present address: ]Laboratoire de Physique, CNRS UMR 5672, Ecole Normale
Supérieure de Lyon, 46 allée d’Italie 69364 Lyon cedex 07, France.
# A novel particle tracking method with individual particle size measurement
and its application to ordering in glassy hard sphere colloids
Mathieu Leocmach [email protected] [ Hajime Tanaka
[email protected] Institute of Industrial Science, University of Tokyo,
4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan
###### Abstract
Particle tracking is a key to single-particle-level confocal microscopy
observation of colloidal suspensions, emulsions, and granular matter. The
conventional tracking method has not been able to provide accurate information
on the size of individual particle. Here we propose a novel method to localise
spherical particles of arbitrary relative sizes from either 2D or 3D
(confocal) images either in dilute or crowded environment. Moreover this
method allows us to estimate the size of each particle reliably. We use this
method to analyse local bond orientational ordering in a supercooled
polydisperse colloidal suspension as well as the heterogeneous crystallisation
induced by a substrate. For the former, we reveal non-trivial couplings of
crystal-like bond orientational order and local icosahedral order with the
spatial distribution of particle sizes: Crystal-like order tends to form in
regions where very small particles are depleted and the slightly smaller size
of the central particle stabilizes icosahedral order. For the latter, on the
other hand, we found that very small particles are expelled from crystals and
accumulated on the growth front of crystals. We emphasize that such
information has not been accessible by conventional tracking methods.
## I Introduction
Experimental physicists often need to recognize objects, count them, follow
them and characterise them. Perrin (1913) had to count colloids by hand to
establish the sedimentation-diffusion equilibrium. Nowadays computer vision
algorithms are used routinely in the lab to track hundreds of thousands of
objects as diverse as stars in a galaxy Bertin and Arnouts (1996), tracers in
a microfluidic device Wereley and Meinhart (2010), pattern formation in
polymer systems Tanaka _et al._ (1986, 1989), dust in a plasma, bacteria
Zhang _et al._ (2010); Gibiansky _et al._ (2010) or viruses in a living cell
Brandenburg and Zhuang (2007). In all these cases, tracking is possible if the
particles are either point-like and far apart, or several pixel wide and
almost monodisperse in size (at least the smallest dimension for anisotropic
objects Zhang _et al._ (2010); Gibiansky _et al._ (2010)). To our knowledge,
algorithm that allow tracking of polydisperse particles in crowded
environments have not reached the soft matter community.
Overall particle size distribution in colloidal suspensions and emulsions
influences crystallisation Pusey (1987); Henderson _et al._ (1996); Fasolo
and Sollich (2003); Schöpe _et al._ (2007); Pusey _et al._ (2009), glass
forming ability Pusey (1987); Henderson _et al._ (1996); Senkov and Miracle
(2001); Schöpe _et al._ (2007); Pusey _et al._ (2009), sedimentation Binks
(1998); Leocmach _et al._ (2010) and emulsion stability Biben _et al._
(1993); Binks (1998) among other physical phenomena. It can be characterized
by various methods that rely on measurements done in well-controlled
environments Lange (1995); Provder (1997); Finder _et al._ (2004). However
the local size distribution is not accessible experimentally _in situ_ and has
thus not been studied so far.
Particle-level microscopy experiments usually access the coordinates of the
particles via the algorithm proposed by Crocker and Grier (1996). The original
noisy image is blurred by convolution with a Gaussian kernel of width $\sigma$
to yield a soft peak per particle. Local intensity maxima within this blurred
image give the coordinates of the particles with pixel resolution. Sub-pixel
resolution ($0.1\sim 0.3$ pixels error) can be achieved by taking the centre
of mass of a neighbourhood around the local maxima. The extension of this
algorithm to localize particles in three-dimensional (3D) confocal microscopy
images has been done in two ways: either tracking particles in each confocal
plane and reconstructing the results (2D-flavour) (van Blaaderen and Wiltzius,
1995; Lu _et al._ , 2007), or full image analysis on three dimensional
pictures (3D-flavour) (Dinsmore _et al._ , 2001).
The choice of the width $\sigma$ of the blurring kernel is critical: if it is
too small, then the intensity profile is flat near the centre of a particle,
leading to multiple and ill-localized maxima per particle; if it is too large,
then the peaks of nearby particles overlap, leading to shifts in the detected
positions (Baumgartl and Bechinger, 2005; Jenkins and Egelhaaf, 2008), or even
fusion of the particles (only one particle detected instead of two). If the
colloids are fairly monodisperse one can argue (at least in the 3D-flavour)
that there exists a range of possible width where the choice of $\sigma$ has
almost no effect on the number of particles detected. Choosing $\sigma$ within
this range gives confidence in the localisation results.
Figure 1: Visualisation of the results of various tracking methods for the
same portion of image. (a) Multiscale 3D tracking. (b) Reconstruction from
multiscale 2D tracking. (c-h) Crocker and Grier method in 3D with blurring
radius increasing from $2\text{\,}\mathrm{p}\mathrm{x}$ to
$4.5\text{\,}\mathrm{p}\mathrm{x}$ by steps of
$0.5\text{\,}\mathrm{p}\mathrm{x}$. The circles on each picture are the result
of 2D multiscale tracking of each XY slice of the 3D pictures. Sphere are
displayed with radii determined by the tracking methods in (a) and (b), and
equal to the blurring radius for (c)-(h).
However, we found that no such “good blur width” exists in a sample of
moderate ($6-7\%$) polydispersity (see Fig. 1c-h). The detection of smaller
particles with small blurring is incompatible with the detection of the larger
particles, and conversely. This unacceptable failure of the Crocker and Grier
(1996) algorithm, as well as the want of the particles’ radii data, triggered
our design of a novel localisation algorithm that would be robust even for a
system of any finite polydispersity, which is unavoidable and sometime desired
in real experiments. Recently Kurita and Weeks (2011a, b) have designed a
sizing method using particle coordinates from confocal experiments. However
their method do not work at the image processing level and relies on
coordinates extracted via the Crocker and Grier (1996) algorithm. If these
coordinates are wrong or if some particles are missed (as shown in Fig. 1),
the output of their method could not be exact.
The key notion to detect objects of unknown and possibly diverse sizes in an
image is the _scale space_ Lindeberg (1993). A popular implementation for
isotropic objects (or “blobs”) is the Scale Invariant Feature Transform (sift)
of Lowe (2004). It is often used to match between different images from
complex objects consisting of many rigidly linked blobs (_e.g._ , to create a
large scale image from overlapping pictures) (Lowe, 2004; Urschler _et al._ ,
2006; Cheung and Hamarneh, 2009). To our knowledge, this method has never been
used for the quantitative localisation and sizing of independent single-blob
objects like spherical colloids, droplets in an emulsion, or crystal nuclei.
Here we apply this new particle tracking method to study how the sizes of
particles affect local structural ordering in a supercooled colloidal
suspension and the process of heterogeneous nucleation from a substrate. We
reveal non-trivial local couplings between such orderings and the spatial
distribution of particle sizes, which may provide crucial information for our
understanding on how the polydispersity influences liquid dynamics and
ordering phenomena.
The organization of our paper is as follows. In Section II, we will describe
our localisation method and its results on synthetic but more and more
realistic data. In Section III of the paper, we will focus on the specific
case of 3D confocal data. In Section IV we will apply our method to a
crystallizing glass of polydisperse hard spheres observed by confocal
microscopy and discuss the influence of size distribution on the ordering
process. We conclude in Section V.
## II Localisation and sizing method
In this section, we will start by recalling the principle of sift, then we
will explain our method in the ideal case of an isolated binary ball, to add
successively finite dilution and difference in brightness between the
particles.
### II.1 Scale invariant feature transform
The sift consists in convolving the original image $I$ by Gaussian kernels
$g_{\sigma_{s}}$ of logarithmically increasing widths $\sigma_{s}$ to obtain a
series of blurred images $G_{s}$
$\displaystyle\forall s>0,\quad G_{s}=I\star g_{\sigma_{s}},$ (1) where
$\star$ is the convolution operator and $\displaystyle\forall
s>0,\quad\sigma_{s}=2^{s/n}\sigma_{0},$ (2)
with $n$ a fixed integer. Following Ref. Lowe (2004) we use $\sigma_{0}=1.6$
and $n=3$. Bright objects in the original image appear as bright blobs in the
blurred images, and the blobs fuse together as the kernel width increases (see
Fig. 1c-h). This can be seen as a series of low-pass filters in the frequency
domain. If we take the difference between consecutive blurred images we obtain
a comb of band pass filtered versions of $I$:
$\forall s>1,\quad DoG_{s}=G_{s}-G_{s-1}$ (3)
The difference of Gaussians ($DoG$) response function defined in this way
depends on the position in space $\vec{r}$ and on the scale $s$. Bright
objects in the original image are detected as local minima in $DoG$. With this
procedure any feature with a radius as small as
$2\text{\,}\mathrm{p}\mathrm{x}$ and as large as its distance to the edges of
the image can be detected. Furthermore, the intensity of the response is
optimal at a $\sigma$ that can be related to the size of the object (see
below). Thus a local minima in both space and scale in the response function
$DoG$ indicates both localisation and _size_ of an object, without any
assumption on the target size.
Figure 2: Application of sift in 2D. (a) Fluorescent droplets in a
microfluidic device. The wall of the device is visible in the top-left corner.
(b) Nucleation under phase contrast microscope. The blue circles are the
result of our algorithm. For any tracking algorithm some detection failure
near the edge of the image are unavoidable and indeed visible here since we
show the whole pictures used for tracking.
Because of the inherent polydispersity of many soft matter systems, the
possible applications of sift-based localisation are countless. For example,
droplets of an emulsion can be followed through a microfluidic device. Fig. 2a
shows the result of our version of the sift on a very polydispered and dense
emulsion observed by wide-field fluorescence microscopy Montagne _et al._
(2011). The sizes extracted are obviously correct except when the system
cannot be considered as 2D. Another possible application is nucleation rate
measurement. When a phase transition proceeds via nucleation growth, the sizes
of the nuclei are very diverse, making their automatic counting difficult by
other methods. As shown in Fig. 2b the sift allows to count and to measure the
size of nuclei during a liquid-liquid transition in a water-glycerol mixture
Murata and Tanaka (2012).
However, to the extent of our knowledge, the sift has not been used in a
physics context to obtain reliable measurements of particles numbers and sizes
from well calibrated images. We will address this reliability in the
following.
We will mainly cope with the three-dimensional extension of the sift. As in
the case of the Crocker and Grier (1996) algorithm, the sift can be extended
in three dimensions (_e.g._ , confocal microscopy images) in two ways: either
by extracting 2D blobs independently on each slice and then reconstructing 3D
objects (Fig. 1b) or by working directly in three dimensions Urschler _et
al._ (2006); Cheung and Hamarneh (2009) (Fig. 1a). We found that the former
method is prone to errors, missing about a tenth of the particles in our best
implementation (Fig. 1b). The 3D results presented in this paper are obtained
solely by the later method.
We stress that a volumetric implementation of sift implies a large amount of
data (typically more than $1\text{\,}\mathrm{G}\mathrm{b}$ for a
$($256\text{\,}\mathrm{p}\mathrm{x}$)^{3}$ picture) and thus requires careful
memory management. Our best implementation (C++) on a 4 cores i7 computer
takes less that $10\text{\,}\mathrm{s}$ to extract the positions and scales of
$\sim 10^{4}$ particles in such a volumetric picture. A much slower
implementation (Python+Scipy, single core) takes $1\text{\,}\mathrm{s}$ to
deal with $($1600\text{\,}\mathrm{p}\mathrm{x}$)^{2}$ 2D images like Fig. 2b.
We would expect real-time processing for 2D images on GPU-enabled
implementations.
### II.2 Sub-pixel and sub-scale resolution
Here we assume perfect noiseless, distortion-free, images. The objects to
localise are thus (pixellised) balls of uniform intensity. To mimic the low
resolution of experimental images in our test images (see Fig. 3), we draw
uniformly white balls on a 4 to 16 times larger image and then we reduce
accordingly the resolution using area resampling.
Figure 3: Results from perfect images. (a-b) Sizing of an isolated sphere.
Left of the vertical line our algorithm uses a doubled image. (c) Localisation
error and (d) sizing error function of the distance between two particles.
Oscillations are due to off-lattice centre position. (e) Size distribution
extracted from digitized configuration of 4000 monodisperse hard spheres at
$0.50$ volume fraction (the vertical line indicates the input radius). The
tail to the right is due to particles on the edges of the image who have fewer
neighbours and thus are more ‘dilute’. (d) and (e) also show the effect of
finite dilution correction up to convergence.
The $DoG$ constructed above is defined on a $d+1$ dimensional grid, with $d$
the spatial dimensionality. One can interpolate it as a continuous function of
the position $\vec{r}$ and the scale $s$ and thus localise the minima with a
precision below the grid size, which corresponds to sub-pixel resolution on
the position and sub-scale sizing (Lowe, 2004). We found that second order
estimate of the spatial derivatives and first order estimates of the scale
derivatives gave the best precision.
The object-by-object optimal scale determination allows us to perform the
spatial sub-pixel resolution step for each object on an image that is blurred
just enough to have neither a flat intensity profile nor an influence of the
overlap with a nearby object’s blob (an effect that plagues the Crocker and
Grier (1996) algorithm Baumgartl and Bechinger (2005); Jenkins and Egelhaaf
(2008)). We found that if for a given object the $DoG$ is minimum at
$\sigma_{s}$, the best image to use is $G_{s-1}$. This leads to a spatial
resolution below $0.1$ pixels in the worst case (when particles are at hard-
core contact), which is the same as the average precision claimed by Crocker
and Grier (1996). Moreover, when particle’s surfaces are further than $1$
pixel, the error on the positions is less than $0.02$ pixels ($0.3\%$ of the
diameter) (see Fig. 3c).
### II.3 Sizing at infinite dilution
The analytical response of a binary ball to a Gaussian blur, at the centre of
the ball is a function of dimensionless ratio $x=R/\sqrt{2}\sigma$: $G(x)$
(see Appendix for an exact expression). The $DoG$ response function is the
difference between the two such functions. However, the choice of the width of
the two functions is not arbitrary: we make the difference between two
consecutive blurred images, the image blurred by $\sigma_{s+1}$ and the image
blurred by $\sigma_{s}$. Therefore, in each of our discrete $DoG$ images the
value at the centre of the particle can be expressed as
$\displaystyle DoG(R,\sigma_{s},\alpha)=DoG(x_{s},\alpha)$
$\displaystyle=G(x_{s}/\alpha)-G(x_{s}),$ (4) with $\alpha=2^{1/n}$. Given
sub-scale refinement, this can be written as a continuous function of
$\sigma$: $\displaystyle DoG(R,\sigma,\alpha)=DoG(x,\alpha)$
$\displaystyle=G(x/\alpha)-G(x).$ (5)
Here it is clear that minimizing $DoG(x,\alpha)$ with respect to $x$ yields a
value $x^{*}$ that depends only on $\alpha$. Exact calculation yields (see
Appendix):
$x^{*}=R/\sqrt{2}\sigma^{*}=\sqrt{\frac{d\ln\alpha}{1-\alpha^{-2}}},$ (6)
where $d$ is the spatial dimensionality. Practically one obtains $\sigma^{*}$,
the value of $\sigma$ that minimises $DoG(R,\sigma,\alpha)$, by a polynomial
fit for discrete data $\sigma_{j}$. Eq. (6) allows to translate the
$n$-dependent $\sigma^{*}$ to the parameter-free real radius of the particle,
$R$. The error on $R$ does depend on the number of subdivisions. We found that
with $n=3$ the radius of an isolated pixelated ball can be indeed measured
within $0.3\%$ relative error with this method (see Fig. 3b).
The scale $s=0.5$ corresponds (via Eq. (2) and Eq. (6)) to the smaller
detectable radius $R_{min}\approx$3.5\text{\,}\mathrm{p}\mathrm{x}$$
($\sigma_{0}=1.6$, $n=3$). In order to detect small objects, Lowe (2004)
recommends to double the size of the input image using linear interpolation
prior to building the first level of the pyramid. This method fares relatively
well in noiseless images (see Fig. 3b) despite larger errors, but we found
that it has to be used with care in confocal microscopy images due to noise
and deconvolution artefacts. In addition doubling an image size implies a
8-fold increase in memory consumption (reaching $60\text{\,}\mathrm{GB}$ in
the case of a $($512\text{\,}\mathrm{p}\mathrm{x}$)^{3}$ original image). Our
implementation allows this on $64\text{\,}\mathrm{bit}$ computers by relying
on memory mapped files, nevertheless it is mostly impractical. All following
tests results are obtained without relying on doubled images.
### II.4 Edge and overlap removal
The difference-of-Gaussian function has a strong response not only at the
centre of bright objects but also along their edges. To eliminate these
spurious detections, Lowe (2004) suggests to construct the local Hessian
matrix around each minimum of the $DoG$ and then compare its eigenvalues to
identify elongated objects.
We found that this was often not enough, especially in crowded environments
where an isolated void induces local minima of the $DoG$ response with rather
isotropic signatures on the edge of nearby particles. An other case not
covered by the Hessian technique is the physically hierarchical structure of
many soft matter systems, _e.g._ , particles forming clusters. In this
situation our algorithm detects a blob for each particle and a much larger
blob for the cluster. In both cases the $DoG$ response of the spurious feature
is smaller (less negative) than the one of the valid particles covering the
same portion of space.
Our method to remove the spurious feature is as follow: we looked for pairs of
particles closer to each other than the larger of their radii (assuming
infinite dilution). This means the centre of one of the particles is situated
inside the other. In the physical systems we study, this cannot be correct due
to excluded volume effects, thus we remove the feature of lesser $DoG$
response. We name this method ‘half overlap removal’. One may be tempted to
implement a full overlap removal (no pair of particle closer than the sum of
their radii) to eliminate even more spurious detection. We found that the gain
was extremely limited ($0.01\%$ of the detected features suppressed) and that
imprecision in positioning or in sizing caused valid particles to be
discarded.
We note that the orientation of non-spherical particles or the deformation of
soft objects can be monitored in principle by using the local Hessian matrix.
### II.5 Finite dilution
The Gaussian (and $DoG$) response of a particle decays rapidly away from its
surface (see Eq. (21). We found that even at contact the position shift
induced by a nearby particle was less than a tenth of a pixel (see Fig. 3c).
However when two particles are closer than a few times the blurring width,
their influence on each other cannot be neglected: the minima of the $DoG$ is
effectively shifted toward smaller scales, leading to smaller radii if one
uses Eq. (6) out of the dilute context (see Fig. 3d).
The response of $N$ particles of radii $\\{R_{i}\\}$ can be superimposed at
any scale $\sigma$ and at any point of space, in particular we define
$DoG_{i}$ as the response at the centre of particle $i$.
$DoG_{i}(\\{A_{j}\\},\\{r_{ij}\\},\\{R_{j}\\},\sigma)=\sum_{j}A_{j}DoG(r=r_{ij},R=R_{j},\sigma),$
(7)
where the function $DoG(r,R,\sigma)$ is the response of a particle of radius
$R$ at distance $r$ from its centre; $r_{ij}$ is the distances between
particles $i$ and $j$ and $\\{A_{i}\\}$ are the respective brightness of the
particles. The sift algorithm yields $\\{\sigma_{i}^{*}\\}$ so that for all
$i$, $DoG_{i}$ is minimum with respect to $\sigma$ at $\sigma_{i}^{*}$, thus
differentiating Eq. (7) with respect to $\sigma$:
$\forall i,\quad\frac{\partial
DoG_{i}}{\partial\sigma}(\\{A_{j}\\},\\{r_{ij}\\},\\{R_{j}\\},\sigma_{i}^{*})=0.$
(8)
The system defined by Eq. (8) is non-linear with respect to $\\{R_{j}\\}$ but
can be solved iteratively by Newton’s method:
$\left[\frac{\partial^{2}DoG_{i}}{\partial
R_{j}\partial\sigma}\right]\times\left(\\{R_{j}\\}^{(k+1)}-\\{R_{j}\\}^{(k)}\right)=-\frac{\partial
DoG_{i}}{\partial\sigma},$ (9)
where the matrix and the right hand side are computed given
$(\\{A_{j}\\},\\{r_{ij}\\},\sigma_{i}^{*})$ and iteratively
$\\{R_{j}\\}^{(k)}$, with the upper parenthesised index indicating the
iteration rank. The results of Eq. (6) are good starting values for the radii.
Using Eq. (7), the elements of the matrix simplify to
$\frac{\partial^{2}DoG_{i}}{\partial
R_{j}\partial\sigma}=A_{j}\frac{\partial^{2}DoG}{\partial
R\partial\sigma}(r=r_{ij},R=R_{j}^{(k)},\sigma=\sigma_{i}^{*}).$ (10)
In principle, Eq. (9) is a $N\times N$ system of equations. However the $DoG$
functions and its derivatives are rapidly decaying functions, thus the
matrices are actually very sparse (about as many non-zero coefficients as
particles in the first coordination shell), alleviating dramatically the
computational burden when using sparse system solvers. Fig. 3d shows the
result of such correction for two identical particles, where Eq. (9) converges
in a single iteration. In a many body case (see Fig. 3e) the convergence is
reached in two iterations; however, the extremely low error of the dilute case
is not totally recovered ($\approx 3\%$ relative error rather than $0.3\%$).
### II.6 Brightnesses
To solve Eq. (9), one needs knowledge of the brightnesses $\\{A_{i}\\}$. In a
first approximation, they can be assumed equal to a constant, which allows to
simplify them out. This is often a sensible approximation. Nevertheless, the
particles in an experimental image are not uniformly bright due to synthesis
imperfection (quantity of dye fixed by each particle) and photo bleaching. If
one does not take into account the relative brightness of the particles, less
bright particles will appear smaller.
A better approximation is to measure during the sift process the value of the
$DoG$ response at the position and scale of each particle, _i.e._
$DoG_{i}(\\{A_{j}\\},\\{r_{ij}\\},\\{R_{j}\\},\sigma_{i}^{*})$. Given the
(iterative) values of the $\\{R_{j}\\}^{(k)}$, one can solve Eq. (7) to get an
iterative value of $\\{A_{i}\\}^{(k)}$. With respect to the brightnesses, Eq.
(7) is a linear system of $N$ equations with $N$ unknowns, thus directly
solvable. It is also as sparse as Eq. (9).
To sum up, the coefficients $\\{A_{i}\\}$ can be computed along with the radii
in a joint iterative process:
$\\{R_{i}\\}^{(0)}\xleftarrow{\text{Eq.~{}(\ref{eq:scale_dil})}}\\{\sigma_{i}^{*}\\}$
repeat
$\\{A_{i}\\}^{(k+1)}\xleftarrow{\text{Eq.~{}(\ref{eq:superpos})}}\\{DoG_{i}\\},\\{R_{i}\\}^{(k)},\\{\sigma_{i}^{*}\\},\\{r_{ij}\\}$
$\\{R_{i}\\}^{(k+1)}\xleftarrow{\text{Eq.~{}(\ref{eq:Newton})}}\\{R_{i}\\}^{(k)},\\{A_{i}\\}^{(k+1)},\\{\sigma_{i}^{*}\\},\\{r_{ij}\\}$
until convergence
In our tests, we found that for both cases with and without the brightness
determination this algorithm converges quickly in one or two iterations.
## III Application to 3D confocal microscopy images
### III.1 Effect of a point spread function
Figure 4: Deconvolution. Detail of the same $YZ$ slice of (a) original
confocal image, (b) previous blurred by $\sigma_{0}=1.6$, (c) previous
deconvolved by measured kernel. Circles indicate the tracked particles
position and size when using whether (b) or (c) as first Gaussian layer
$G_{0}$. All three centres are in the slice
$\pm$0.5\text{\,}\mathrm{p}\mathrm{x}$$. (d) Radial distribution function of
almost monodisperse sticky spheres localised (blue) without and (red dash)
with deconvolution.
Real images suffer from optical limitations, _e.g._ , the point-spread
function (psf) of the microscope. In particular, spherical particles observed
by confocal microscopy appear elongated along $Z$. The influence of such
anisotropic distortion on the Gaussian and $DoG$ responses is not trivial and
most of the equations given in Appendix do not have analytical equivalents. In
particular, the minimum of the $DoG$ response is found at larger values of
$R/\sigma$, thus the naive use of the methods detailed in the previous section
lead to overestimate particle sizes.
In addition, the overlap of neighbouring particle images is no longer
negligible when the particles are aligned along $Z$. This leads to large
imprecision in particle positions, especially in the case of anisotropic
environment (_e.g._ , isolated pair of particles, interface, colloidal gel).
In Fig. 4d the radial distribution function of almost monodisperse colloids
with short range attraction displays a spurious shoulder before its first
peak, indicating that particles centres are often found closer than the sum of
their hard core radii, as illustrated in Fig. 4b.
We found that these issues can be better addressed by pre-processing the
images (as detailed below) rather than post processing the sift output _via_
analytical methods.
### III.2 Deconvolution
The image $y$ acquired by a microscope can be expressed as
$y=x\star h+\epsilon,$ (11)
where $x$ is the perfect image, $h$ is the psf of the microscope and
$\epsilon$ is the noise independent of both $x$ and $h$. The process of
estimating $x$ from $y$ and some theoretical or measured expression of $h$ is
called deconvolution. Deconvolution in the presence of noise is a difficult
problem Riad (1986). Hopefully here we do not need to reconstruct the original
image, but only our first Gaussian blurred version of it, _i.e._ ,
$G_{0}=x\star g_{\sigma_{0}}$ starting from $y_{0}=y\star g_{\sigma_{0}}$.
Indeed, after a reasonable amount of blur in three dimensions, the noise can
be neglected and we thus obtain:
$y_{0}\approx G_{0}\star h,$ (12)
or in Fourier space
$\mathcal{F}[y_{0}]=\mathcal{F}[G_{0}]\times\mathcal{F}[h].$ (13)
Once $\mathcal{F}[h]$ is known the deconvolution reduces to a simple division
in Fourier space. Let us measure $\mathcal{F}[h]$ in an isotropic system where
we can write
$\displaystyle\left\langle\left|\mathcal{F}_{X}[x]\right|^{2}\right\rangle$
$\displaystyle=\left\langle\left|\mathcal{F}_{Z}[x]\right|^{2}\right\rangle.$
(14)
Using Eq. (11) we obtain
$\displaystyle\frac{\left\langle\left|\mathcal{F}_{X}[y]\right|^{2}\right\rangle}{\left|\mathcal{F}_{X}[h]\right|^{2}}=\frac{\left\langle\left|\mathcal{F}_{Z}[y]\right|^{2}\right\rangle}{\left|\mathcal{F}_{Z}[h]\right|^{2}}.$
(15)
Here $\mathcal{F}_{X}$ indicates the Fourier transform along axis $X$. In
point scanning confocal imaging, the psf has negligible lobes along $X$ and
$Y$ ($X$ only for line scanning), thus we have
$\displaystyle\left|\mathcal{F}_{Z}[h]\right|^{2}=\frac{\left\langle\left|\mathcal{F}_{Z}[y]\right|^{2}\right\rangle}{\left\langle\left|\mathcal{F}_{X}[y]\right|^{2}\right\rangle}.$
(16)
For example an Hermitian kernel (real valued spectrum) is
$\displaystyle\mathcal{F}_{Z}[h]=\sqrt{\frac{\left\langle\left|\mathcal{F}_{Z}[y]\right|^{2}\right\rangle}{\left\langle\left|\mathcal{F}_{X}[y]\right|^{2}\right\rangle}}.$
(17)
Fig. 4b-c shows the particles localized from the original image and from the
deconvolved image. Deconvolution mends both size overestimation and
imprecision in $z$ coordinate. This also translates in the radial distribution
function (Fig. 4d), where the spurious shoulder before the first peak
disappears.
### III.3 From optical to hard-core radius
The method exposed above relies on the uniformity of the brightness within
each particle to extract optical radii that is a good approximation to the
physical (hard core) radii. However a colloidal particle often shows a smooth
intensity profile in itself under the microscope, less bright at the edge than
at the centre of the particle. This can be due to inhomogeneous fixation of
the dye during the colloid synthesis, but the bottom line is the in-plane psf
not corrected for in our above deconvolution method.
Under such conditions, the particles are detected smaller than their expected
sizes. The real to optical size ratio has to be set using our knowledge of the
sample. For example one may use the position of the first peak of the $g(r)$
to measure the average hard-core diameter. In general the real to optical size
ratio may depend on the size of the particles and may evolve in time (_i.e._ ,
because of photobleaching). We present below a detailed analysis of these
issues for a case system.
We also note that darker edges lead to smaller influence of a particle on the
$DoG$ response of its neighbours. A smaller optical radius makes the particles
optically further from each other. Accordingly, we found that finite dilution
corrections were less important in real images than in our synthetic test
images. For the same reason, the scaling factor must be measured and applied
after — _not_ before — finite dilution corrections.
## IV Revealing polydispersity effects on structural orderings in a glassy
colloidal hard sphere liquid
Here we apply our tracking method to access couplings between the spatial
particle size distribution and local structural ordering in a supercooled
colloidal liquid with size polydispersity.
### IV.1 Experimental
We used pmma (poly(methyl methacrylate)) colloids sterically stabilized with
methacryloxypropyl terminated pdms (poly(dimethyl siloxane)) and fluorescently
labelled with rhodamine isothiocyanate chemically bonded to the pmma. The
colloids were suspended in a solvent mixture of cis-decalin and cyclohexyl-
bromide for both optical index and density matching. To screen any (weak)
electrostatic interactions, we dissolved tetrabutylammonium bromide salt, to a
concentration of $300\text{\,}\mathrm{nmol}\text{\,}{\mathrm{L}}^{-1}$ (Royall
_et al._ , 2005). The estimated Debye screening length is
$13\text{\,}\mathrm{nm}$, well below the length scale of the colloids that can
be considered as hard spheres. The $8\text{\,}\mathrm{b}\mathrm{i}\mathrm{t}$
graylevel data was collected on a Leica SP5 confocal microscope, using
$532\text{\,}\mathrm{nm}$ laser excitation and voxel size of
$($283\text{\,}\mathrm{nm}$)^{2}\times$293\text{\,}\mathrm{nm}$$.
To realize a more precise density matching, the temperature was controlled on
both stage and objective lens. This setup allows us to alter the buoyancy
simply by a temperature change, thus to set the effective gravity upward,
downward or almost null. After careful shear melting, the sample was filled
into a $$100\text{\,}\mathrm{\SIUnitSymbolMicro
m}$\times$1\text{\,}\mathrm{mm}$$ capillary (Vitrocom) and set on the
microscope stage. We then spent a few days to find the temperature
corresponding to exact density matching (within $0.1\text{\,}\mathrm{K}$).
This waiting time was enough for crystallites to form on both top and and
bottom walls. Then, we heat up our sample by a few degrees compared to the
density-matching temperature in order to make the colloids heavier than the
solvent. At the top of the sample (close to the objective lens and thus
allowing much clear imaging) the volume fraction drops and all crystallites
melt. Finally, we set back the thermostat to the density-matching temperature,
allowing the top of the sample to slowly return to its supercooled state. We
could then observe the heterogeneous nucleation from the beginning. We tracked
the sample from top to bottom and we are thus sure that the crystallite
actually form at the wall and do not come from the rest of the sample. Namely,
crystallisation in this sample is caused only by heterogeneous nucleation on
the substrates, presumably due to the wall-induced enhancement of crystal-like
bond orientational ordering Watanabe _et al._ (2011).
### IV.2 Global size distribution
We localise the particles using our method with a preblur of only
$\sigma_{0}=1.0$ (see Section II.1) to be able to detect the smaller particles
($R_{min}\approx$2\text{\,}\mathrm{p}\mathrm{x}$$) without expensive
oversampling. We checked that choosing higher $\sigma_{0}$ truncates the size
distribution but has no other effect on its shape. After removing half-
overlapping features, we applied a single iteration of finite dilution
corrections for both intensities and radii to obtain the optical radius of
each particle.
We then estimate the hard-core diameter function of the optical radius $R$ by
locating the first peak of the partial radial distribution function $g_{R}(r)$
of the particles having $R_{i}$ close to $R$. As shown in Fig. 5a, the real to
optical size ratio is around $1.5$ and is rather constant respective to $R$,
thus a single overall scaling factor is enough. It can be determined by a
single partial $g_{R}(r)$ with $R$ near the peak of the size distribution, or
by locating the peak of $g(\hat{r})$, with
$\hat{r}_{ij}=r_{ij}/(R_{i}+R_{j})$. We found that the real-to-optical size
ratio was increasing with time due to photobleaching (see Fig. 5b). We fit
this increase by a linear relation and applied the resulting time-dependent
ratio to obtain the real size of each particles.
The consistency of our method can be checked by constructing the radial
distribution function $g(r)$. In monodisperse hard spheres, the $g(r)$ should
have a sharp first peak at $r=2R$ corresponding to hard core contact.
Polydispersity implies hard core contacts at various $r$ and thus broadens the
peak. One can recover a sharp peak by constructing $g(\hat{r})$ (this time
$\hat{r}$ is scaled by the real radii, not the optical ones). In Fig. 5d we
successfully used the sizes measured by our method to rectify the first peak.
Fig. 5c shows the size distribution only $140$ dry colloids measured on high
resolution scanning electron microscopy (sem) images. We also show the size
distribution obtained by our _in situ_ measurements ($\sim 1.7\times 10^{6}$
instantaneous sizing). The main peak of the later compares well with the
former once a solvent swelling of $25\%$ in radius is taken into account.
However the small sampling of the SEM measurements completely misses the tail
toward small sizes featuring two low peaks that probably correspond to
secondary and ternary nucleation during the synthesis of the colloids Bosma
_et al._ (2002); Poon _et al._ (2012).
From our data we compute an overall volume fraction of $0.60$, almost constant
during the experiment. The polydispersity estimated from sem data is $6.2\%$.
This is coherent with a fit of the main peak of our data ($6.9\%$), however
the polydispersity of the whole distribution including the tail toward small
sizes is $14.8\%$. We will see below how such a complex size distribution
affects the physical behaviour of the system.
Figure 5: _In situ_ sizing of colloids in a glass. (a) Position of the first
peak of the partial $g_{R}(r)$ function of the optical radius $R$. Solid line
corresponds to a real-to-optical size ratio of $1.5$. (b) Time dependence of
the real-to-optical size ratio. Solid line is a linear fit. (c) Size
distribution estimated by our algorithm (dashed line). Comparison with the
estimation from sem of only $140$ dry particles (steps) is possible once
$25\%$ of swelling is taken into account (full line). (d) First peak of the
radial distribution function with (full line) and without (dashed) the
individual sizes data.
### IV.3 Polydispersity and structural heterogeneity in a supercooled
colloidal liquid
Hard sphere supercooled liquids and glasses contain both medium ranged
crystal-like bond orientational ordering (mrco) and local icosahedral ordering
Leocmach and Tanaka (2012). The later acts as a frustration against the
expansion of the former and thus against crystallisation. Both kind of
structures can be detected using bond orientational order (boo) introduced by
Steinhardt _et al._ (1983) (see Fig. 6). As we have described elsewhere
Leocmach and Tanaka (2012), crystal-like bond ordering is well described by
the scalar bond order parameter $Q_{6}$, and the icosahedral bond ordering by
$w_{6}$. fcc or hcp crystals under thermal vibrations typically have
$Q_{6}>0.4$ Lechner and Dellago (2008). We distinguish mrco from liquid
structures by a threshold at $Q_{6}=0.25$. The perfect 13 particles
icosahedron is the minimum of $w_{6}$, at negative values. We consider that a
neighbourhood is icosahedral when $w_{6}<-0.023$.
Figure 6: Structure visualisation at $t=0$ (left) and
$t=$40\text{\,}\mathrm{h}$$ (right). Small particles
($R<$1.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$) are shown in green, crystal-
like ordered particles ($Q_{6}>0.25$) in red and icosahedral particles and
their neighbours in purple. Other particles are not shown.
In Fig. 7, we compare the overall size distribution to the size distribution
within each remarkable structure. The roles of the particle at the centre of
an icosahedron and of the particles surrounding it are asymmetric. The size
distribution of the particles at the surface of icosahedra is almost identical
to the overall size distribution, however the size distribution of the
particles at the center of icosahedra features a second peak at radii about
$80\%$ smaller than the main peak. A centre to surface size ratio near $0.8$
seems to stabilize icosahedral order. This is consistent with a recent
simulation study by Shimono and Onodera Shimono and Onodera (2012).
Figure 7: Size distribution within local structures in a supercooled hard
sphere colloidal liquid. (a) With increasing local crystalline bond
orientational order: all particles, MRCO particles ($0.4>Q_{6}>0.25$) and
crystalline particles ($Q_{6}>0.4$). Crystalline ordered regions are less
polydisperse. (b) Particles at the centre or on the surface of icosahedra
($w_{6}<-0.023$). A small central particle with large surface particles tend
to stabilize icosahedra.
We note that the tail of small particles characteristic of our overall size
distribution is less pronounced in mrco and almost disappears in well-formed
crystals (Fig. 7a). The width of the main peak of the distribution does not
change significantly upon crystal-like ordering. This means that a small
amount of polydispersity is acceptable inside mrco, coherently with the
results for the (defective) crystalline lattice by Fasolo and Sollich (2003).
However mrco cannot appear where markedly smaller particles are present. We
also checked the difference between fcc ($w_{4}<0$) and hcp ($w_{4}>0$) to
find that they have both exactly the same size distribution (not shown). This
suggests that the size distribution controls the formation of mrco but has no
influence on its symmetry (probably, the crystal polymorph). We stress that
this coupling of mrco and icosahedral ordering with the spatial distribution
of particle sizes does not imply fractionation, but rather suggests that a
rather homogeneous size distribution tends to help or stabilize mrco and a
slightly smaller size of a particle also stabilizes the formation of
icosahedral order around it. We will see below that the fractionation appends
later in the crystallisation process.
### IV.4 Polydispersity and heterogeneous crystallisation in a supercooled
colloidal liquid
As explained above, we triggered heterogeneous crystal nucleation on the top
wall of our container. At each time step we can construct a $z$-dependent
density profile for any species of interest. In Fig. 8 we show at two time
steps the density profiles of large particles
($R>$1.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$), small particles
($R<$1.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$) and particles with medium to
high crystalline order ($Q_{6}>0.25$). Icosahedral particles (not shown) are
an order of magnitude fewer than small particles (see Fig. 6) and are
basically irrelevant. Near the walls, the density profiles show oscillations
characteristic of layering (see, _e.g._ , Refs. Watanabe _et al._ (2011);
Scheidler _et al._ (2003)), whose wavelength corresponds to the peak of the
size distribution (the ‘majority species’). We note that the small particles
also shows layering on the same wavelength, although with a smaller amplitude.
Figure 8: Instantaneous density profiles at $t=0$ (left) and
$t=$40\text{\,}\mathrm{h}$$ (right). Top: large particles
($R>$1.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$) without smoothing (black
line) and with a Gaussian smoothing of width $2R_{\text{peak}}$ (gray area).
Bottom: small particles ($R<$1.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$) with
the same color code. For all figures, the red dashed curve is the smoothed
density of ordered particles ($Q_{6}>0.25$) irrespective of their size. Figure
9: Heterogeneous nucleation. Successive computer reconstruction of the first
three layers from the wall, every $12\text{\,}\mathrm{h}$, seen from the fluid
side. Particles sizes are according to our method. Small
($R<$1.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$) particles are shown in
green. Other particles are coloured according to their crystal-like order.
Note how the small particles are expelled from the forming crystallites to
concentrate in the grain boundaries.
Once the short oscillations removed via a Gaussian smoothing of width
$2R_{\text{peak}}$, one can see that the density of large particles is
constant from top to bottom of the sample (the slight dip on the edges is due
to a small misalignment between our $z$ axis and the normal of the wall). It
confirms the accuracy of our density matching but also stresses that
crystallisation takes place without an increase in density. The smoothed
density profile of small particles starts almost flat. However the small
particles are expelled from the forming crystal, and as the crystalline front
propagates it pushes in front of itself a growing ridge of small particles,
very apparent after $40\text{\,}\mathrm{h}$ in both Fig. 8 and Fig. 6.
We confirmed this view by looking at the behaviour of small particles and
crystalline order in a constant plain (Fig. 9 and Supplementary Movie 1).
Before crystallisation, small particles are scattered randomly and mrco
develop where few very small particles are present. Then, crystallisation
proceeds from the mrco, pushing away the small particles that accumulate in
the grain boundaries. As a consequence, heterogeneous crystallisation do not
proceed via a spatially homogeneous layer-by-layer growth, as observed at low
polydispersity Sandomirski _et al._ (2011), but by many nucleation-growth
events reflecting the initial fluctuations of the mrco. This dynamics is
reminiscent of the homogeneous nucleation case Kawasaki and Tanaka (2010);
Russo and Tanaka (2012a, b) but enhanced by the wetting of the wall by
crystalline bond orientational order Watanabe _et al._ (2011).
To sum up, the small particles affect the growth of the crystal by coupling
ordering to diffusion. Without the small particles, moving grain boundaries
implies moving particles only locally and thus propagation is easy. With very
small particles, moving grain boundaries implies moving them by the same
amount by diffusion. In the same way, the crystallisation front has to push
the small particles in front of it, effectively increasing the polydispersity
in the melt and thus slowing its propagation. Indeed, weeks after preparation
our samples was still only partially crystallised and highly polycrystalline.
We note that this coupling is due to the conserved nature of the species
‘small particles’. Non-conserved species defined by local structural ordering
such as icosahedra cannot produce such a coupling.
## V Conclusion
We have presented a new method to extract the coordinates of particles of very
diverse sizes even when they are very close (at contact) to each other. This
opens new experimental possibilities in soft matter systems where the
particles are polydisperse (_e.g._ , colloids, emulsions, and granular
particles) or even changing size (_e.g._ , phase separation dynamics). Our
method also reliably extracts the size of individual particle, which allows a
_in situ_ analysis of the consequences of naturally occurring size
distributions (beyond monodisperse, bidisperse or Gaussian distribution).
We showed that in the case of colloidal hard sphere, a size distribution with
a main peak and a tail toward small sizes induces different behaviours than
what would be expected from a Gaussian size distribution. Our method allowed
us to measure this distribution properly and to look into the interactions
between size distribution, local ordering and heterogeneous crystal nucleation
in a glass. We found that the presence of a long tail toward small sizes in
the size distribution, was leading to highly polycrystalline materials that
may be of interest for engineering purposes. We hope that this novel particle
tracking method with a capability of particle-size determination will be
applied to a wide field in soft matter physics.
## Acknowledgements
We are grateful to Koshi Hasatani, Anthony Genot and Yannick Rondelez for
providing the image of Fig. 2a; to Ken-ichiro Murata for providing the image
of Fig. 2b; to John Russo for providing us with simulation data of dense hard
spheres, which are used in Fig. 3e; to Hideyo Tsurusawa for the synthesis of
the colloid and the confocal data of Fig. 4; and to Rebecca Rice and C. Paddy
Royall for sem measurements of the size distribution of dried particles. This
work was partially supported by a grant-in-aid from the Ministry of Education,
Culture, Sports, Science and Technology, Japan and also by Aihara Project, the
FIRST program from JSPS, initiated by CSTP.
## Appendix: Gaussian response of a binary ball
A binary ball $\mathcal{B}_{R}$ of radius $R$ is convolved by a Gaussian
kernel of width $\sigma$. The response at a distance $r$ (along z-axis without
loss of generality) is
$\displaystyle G(r,R,\sigma)=$
$\displaystyle\frac{1}{(2\pi)^{3/2}\sigma^{3}}\int_{\mathcal{B}_{R}}{e^{-\frac{(z-r)^{2}+y^{2}+x^{2}}{2\sigma^{2}}}dxdydz}$
(18) or in spherical coordinates $(\rho,\theta,\phi)$ after integration along
$\phi$ $\displaystyle=$
$\displaystyle\frac{e^{-\frac{r^{2}}{2\sigma^{2}}}}{\sqrt{2\pi}\sigma^{3}}\int_{0}^{R}\rho^{2}e^{-\frac{\rho^{2}}{2\sigma^{2}}}d\rho\int_{0}^{\pi}\sin\theta
e^{\frac{\rho r}{\sigma^{2}}\cos\theta}d\theta$ (19) $\displaystyle=$
$\displaystyle\frac{1}{r\sigma\sqrt{2\pi}}\left[\int_{0}^{R}\rho
e^{-\frac{(\rho-r)^{2}}{2\sigma^{2}}}d\rho-\int_{0}^{R}\rho
e^{-\frac{(\rho+r)^{2}}{2\sigma^{2}}}d\rho\right]$ (20) this can be integrated
using the error function $\operatorname{erf}$ $\displaystyle G(r,R,\sigma)$
$\displaystyle=\frac{1}{2}\left(g(r,R,\sigma)+g(-r,R,\sigma)\right)$ (21) with
$\displaystyle g(r,R,\sigma)$
$\displaystyle=\operatorname{erf}\left(\frac{R+r}{\sigma\sqrt{2}}\right)+\sqrt{\frac{2}{\pi}}\frac{\sigma}{r}e^{-\frac{(R+r)^{2}}{2\sigma^{2}}}$
(22) At the centre of the ball, Eq. (21) reduces to $\displaystyle
G(r=0,R,\sigma)$
$\displaystyle=\operatorname{erf}\left(\frac{R}{\sigma\sqrt{2}}\right)-\sqrt{\frac{2}{\pi}}\frac{R}{\sigma}e^{-\frac{R^{2}}{2\sigma^{2}}}$
(23)
In addition, we compute the following useful partial derivatives
$\displaystyle\frac{\partial g}{\partial\sigma}(r,R,\sigma)$
$\displaystyle=\frac{\left(R^{2}+rR+\sigma^{2}\right)}{r\sigma^{2}\sqrt{2\pi}}e^{-\frac{(R+r)^{2}}{2\sigma^{2}}}$
(24) $\displaystyle\frac{\partial^{2}g}{\partial\sigma\partial R}(r,R,\sigma)$
$\displaystyle=-R\frac{(R+r)^{2}-\sigma^{2}}{r\sigma^{4}\sqrt{2\pi}}e^{-\frac{(R+r)^{2}}{2\sigma^{2}}}$
(25) $\displaystyle\frac{\partial G}{\partial\sigma}(r=0,R,\sigma)$
$\displaystyle=-\sqrt{\frac{2}{\pi}}\frac{R^{3}}{\sigma^{4}}e^{-\frac{R^{2}}{2\sigma^{2}}}$
(26) $\displaystyle\frac{\partial^{2}G}{\partial\sigma\partial
R}(r=0,R,\sigma)$
$\displaystyle=\sqrt{\frac{2}{\pi}}R^{2}\frac{R^{2}-3\sigma^{2}}{\sigma^{6}}e^{-\frac{R^{2}}{2\sigma^{2}}}$
(27)
The difference of Gaussians response is
$\displaystyle DoG(r,R,\sigma,\alpha)$
$\displaystyle=G(r,R,\alpha\sigma)-G(r,R,\sigma)$ (28) and has the partial
derivative relative to the scale $\displaystyle\frac{\partial
DoG}{\partial\sigma}(r,R,\sigma,\alpha)$ $\displaystyle=\alpha\frac{\partial
G}{\partial\sigma}(r,R,\alpha\sigma)-\frac{\partial
G}{\partial\sigma}(r,R,\sigma)$ (29)
When the difference of Gaussians is minimum at the centre of the ball, we
combine Eq. (26) and Eq. (29) to get
$\frac{R}{\sigma\sqrt{2}}=\sqrt{\frac{3\ln\alpha}{1-\alpha^{-2}}}$ (30)
One can show that the factor $3$ in the radical is the dimensionality of the
space, thus Eq. (30) can be generalised to other dimensions.
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|
arxiv-papers
| 2013-01-30T14:24:05 |
2024-09-04T02:49:40.978997
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mathieu Leocmach and Hajime Tanaka",
"submitter": "Mathieu Leocmach",
"url": "https://arxiv.org/abs/1301.7237"
}
|
1301.7247
|
# Uniqueness for an inviscid stochastic dyadic model on a tree
Luigi Amedeo Bianchi
###### Abstract
In this paper we prove that the lack of uniqueness for solutions of the tree
dyadic model of turbulence is overcome with the introduction of a suitable
noise. The uniqueness is a weak probabilistic uniqueness for all
$l^{2}$-initial conditions and is proven using a technique relying on the
properties of the $q$-matrix associated to a continuous time Markov chain.
## 1 Introduction
inline,caption=Stress similarities with [5]]The author should stress out in
the introduction that the results have already been proven in [5] putting in
evidence the new formulation of Section6. He has to clarify what has been done
in the previous literature and would be the novelty in his paper. The
deterministic dyadic model on a tree
$\begin{cases}dX_{j}=(c_{j}X_{\overline{\jmath}}^{2}-\sum_{k\in{\mathcal{O}}_{j}}c_{k}X_{j}X_{k})dt\\\
X_{0}(t)=0,\end{cases}$ (1)
was introduced as a wavelet description of Euler equations in [15] and studied
in [4] as a model for energy cascade in turbulence. It can be seen as a
generalization with more structure of the so called dyadic model of
turbulence, studied in [7]. As we show in section 2 this deterministic model
(1) does not have uniqueness in $l^{2}$. The aim of this paper is to prove
that we can restore uniqueness with the introduction of a suitable random
noise:
$dX_{j}=(c_{j}X_{\overline{\jmath}}^{2}-\sum_{k\in\mathcal{O}_{j}}c_{k}X_{j}X_{k})dt+\sigma
c_{j}X_{\overline{\jmath}}\circ
dW_{j}-\sigma\sum_{k\in\mathcal{O}_{j}}c_{k}X_{k}\circ dW_{k},$ (2)
with $\left(W_{j}\right)_{j\in J}$ a sequence of independent Brownian motions.
Let’s also assume deterministic initial conditions for (2):
$X\left(0\right)=x=\left(x_{j}\right)_{j\in J}\in l^{2}$. The main result of
this paper is the weak uniqueness of solution for (2), proven in theorem 7.2.
This paper can be seen as a generalization to the dyadic tree model of the
results proven for the classic dyadic model in [5], but the proof of
uniqueness given here relies of a new, different approach (see also [8]) based
on a general abstract property instead of a trick (see Section 6). The
$q-$matrix we rely on is closely related to an infinitesimal generator, so the
technique is valid for a larger class of models.
inline,caption=Sets $J$ and $\mathcal{O}_{j}$]I think that the introduction of
the countable sets $J$ and $\mathcal{O}_{j}$ is not a good idea. I don’t see
the difference between the two. The author should clarify or change the
notation. inline,color=green,nolist]As stated below one is a subset of the
other: $J$ is the countable set of all nodes of the tree, while
$\mathcal{O}_{j}$ is the finite subset of all the children of node $j$ The set
$J$ is a countable set and its elements are called nodes. We assume for the
nodes a tree-like structure, where given $j\in J$, $\overline{\jmath}$ is the
(unique) father of the node $j$, and $\mathcal{O}_{j}\subset J$ is the finite
set of offsprings of $j$. In $J$ we identify a special node, called root and
denoted by $0$. It has no parent inside $J$, but with slight notation abuse we
will nevertheless use the symbol $\bar{0}$ when needed.
We see the nodes as eddies of different sizes, that split and transfer their
kinetic energies to smaller eddies along the tree. To formalize this idea we
consider the eddies as belonging to discrete levels, called _generations_ ,
defined as follows. For all $j\in J$ we define the generation number
$|j|\in{\mathbb{N}}$ such that $|0|=0$ and $|k|=|j|+1$ for all
$k\in\mathcal{O}_{j}$.
To every eddy $j\in J$ we associate an _intensity_ $X_{j}(t)$ at time $t$,
such that $X_{j}^{2}(t)$ is the kinetic energy of the eddy $j$ at time $t$.
The relations among intensities are those given in (1) for the deterministic
model and (2) for the perturbed stochastic model. The coefficients $c_{j}$ are
positive real numbers that represent the speed of the energy flow on the tree.
The idea of a stochastic perturbation of a deterministic model is well
established in the literature, see [6] for the classical dyadic model, but
also [11], [9] for different models. This stochastic dyadic model falls in the
family of shell stochastic models. Deterministic shell models have been
studied extensively in [12] while stochastic versions have been investigated
for example in [10] and [16].
When dealing with uniqueness of solutions in stochastic shell models, the
inviscid case we study is more difficult than the viscous one, since the more
regular the space is, the simpler the proof and the operator associated to the
viscous system regularizes, see for example [3] about GOY models, where the
results are proven only in the viscous case.
In (2) the parameter $\sigma\neq 0$ is inserted just to stress the open
problem of the zero noise limit, for $\sigma\to 0$. This has provided an
interesting selection result for simple examples of linear transport equations
(see [2]), but it is nontrivial in our nonlinear setting, due to the
singularity that arises with the Girsanov transform, for example in (6).
It is worth noting that the form of the noise is unexpected: one could think
that the stochastic part would mirror the deterministic one, which is not the
case here, since there is a $j$-indexed Brownian component where we’d expect a
$\overline{\jmath}$ one, and there is a $k$-indexed one instead of a $j$ one.
One could argue that this is not the only possible choice for the random
perturbation. On one hand we chose a multiplicative noise, instead of an
additive one, but this is due to technical reasons (see [14]). On the other
hand, there are other possible choices, for example the Brownian motion could
depend on the father and not on the node itself, so that brothers would share
the same Brownian motion. But the choice we made is dictated by the fact that
we’d like to have a formal conservation of the energy, as we have in the
deterministic case (see [4]). If we use Itô formula to calculate
$\displaystyle\frac{1}{2}dX_{j}^{2}$ $\displaystyle=X_{j}\circ dX_{j}$
$\displaystyle=(c_{j}X_{\bar{\jmath}}^{2}X_{j}-\sum_{k\in{\mathcal{O}}_{j}}c_{k}X_{j}^{2}X_{k})dt+\sigma
c_{j}X_{\bar{\jmath}}X_{j}\circ
dW_{j}-\sigma\sum_{k\in{\mathcal{O}}_{j}}c_{k}X_{j}X_{k}\circ dW_{k},$
we can sum _formally_ on the first $n+1$ generations, taking $X_{0}(t)=0$:
$\displaystyle\sum_{|j|=0}^{n}\frac{1}{2}dX_{j}^{2}$
$\displaystyle=-\sum_{|j|=n}\sum_{k\in\mathcal{O}_{j}}[c_{k}X_{j}^{2}X_{k}dt+\sigma
c_{k}X_{j}X_{k}\circ dW_{k}]$
$\displaystyle=-\sum_{|j|=n}\sum_{k\in\mathcal{O}_{j}}c_{k}X_{j}X_{k}(X_{j}dt+\sigma\circ
dW_{k}),$
since the series is telescoping in both the drift and the diffusion parts
independently. That means we have P-a.s. the formal conservation of energy, if
we define the energy as
${\mathcal{E}}_{n}(t)=\sum_{|j|\leq
n}X_{j}^{2}(t)\qquad{\mathcal{E}}(t)=\sum_{j\in
J}X_{j}^{2}(t)=\lim_{n\to\infty}{\mathcal{E}}_{n}(t).$
## 2 Non-uniqueness in the deterministic case
In [7] it has been proven that there exists examples of non uniqueness of
$l^{2}$ solutions for the dyadic model if we consider solutions of the form
$Y_{n}(t)=\dfrac{a_{n}}{t-t_{0}}$, called self-similar solutions, with
$(a_{n})_{n}\in l^{2}$. Thanks to the lifting result (Proposition 4.2 in [4])
that is enough to obtain two different solutions of the dyadic tree model,
with the same initial conditions.
inline,caption=reference introducing self similar solutions]The author should
point out to reference [5] istead of [4] where these self-similar solutions
have been introduced first. Moreover, [5] was dealing exactly with the same
inviscid equation (2). inline, color=green,nolist]It’s good to point out that
self similar solutions have been introduced in [5] for the dyadic model, but
in [4] for the dyadic tree model. Moreover it isn’t completely correct to say
that the equation is the same, since the noise is different and there are many
more interactions in the tree model. Following the same idea of self-similar
solutions, introduced in [7] and [5] for the classic dyadic model and in
section 5.1 in [4] for the tree dyadic model, we can construct a direct
counterexample to uniqueness of solutions. In order to do this we need an
existence result stronger than the one proven in [4].
###### Theorem 2.1.
For every $x\in l^{2}$ there exists at least one finite energy solution of
(1), with initial conditions $X(0)=x$ and such that
$\sum_{j\in J}X_{j}^{2}(t)\leq\sum_{j\in J}X_{j}^{2}(s)\qquad\forall\,0\leq
s\leq t.$
The proof of this theorem is classical, via Galerkin approximations, and
follows that of theorem 3.3 in [4].
Now we recall the time reversing technique. We may consider the system (1) for
$t\leq 0$: given a solution $X(t)$ of this system for $t\geq 0$, we can define
$\widehat{X}(t)=-X(-t)$, which is a solution for $t\leq 0$, since
$\displaystyle\frac{d}{dt}\widehat{X}_{j}(t)=\frac{d}{dt}X_{j}(-t)$
$\displaystyle=c_{j}X_{\overline{\jmath}}^{2}(-t)-\sum_{k\in\mathcal{O}_{j}}c_{k}X_{j}(-t)X_{k}(-t)$
$\displaystyle=c_{j}\widehat{X}_{\overline{\jmath}}^{2}(t)-\sum_{k\in\mathcal{O}_{j}}c_{k}\widehat{X}_{j}(t)\widehat{X}_{k}(t).$
We can now consider the self similar solutions for the tree dyadic model, as
introduced in [4], $X_{j}(t)=\dfrac{a_{j}}{t-t_{0}}$, defined for $t>t_{0}$,
with $t_{0}<0$ and with $(a_{j})_{j\in J}\in l^{2}$ such that
$\begin{cases}a_{\bar{0}}=0\\\
a_{j}+c_{j}a_{\bar{\jmath}}^{2}=\displaystyle{\sum_{k\in\mathcal{O}_{j}}c_{k}a_{j}a_{k}},\qquad\forall
j\in J.\end{cases}$
We time-reverse them and we define
$\widehat{X}_{j}(t)=-X_{j}(-t)\qquad\forall j\in J\qquad t<-t_{0},$
which, as we pointed out earlier, is a solution of (1) in $(-\infty,-t_{0})$,
with $-t_{0}>0$. Since
$\lim_{t\to+\infty}|X_{j}(t)|=0\quad\textrm{and}\quad\lim_{t\to
t_{0}^{+}}|X_{j}(t)|=+\infty,\ \forall j\in J$
we have
$\lim_{t\to-\infty}|\widehat{X}_{j}(t)|=0\quad\textrm{and}\quad\lim_{t\to-
t_{0}^{-}}|\widehat{X}_{j}(t)|=+\infty,\ \forall j\in J.$
Thanks to theorem 2.1 there is a solution $\widetilde{X}$, with initial
conditions $x=\widehat{X}(0)$, and this solution is a finite energy one, so,
in particular, doesn’t blow up in $-t_{0}$. Yet it has the same initial
conditions of $\widehat{X}$, so we can conclude that there is no uniqueness of
solutions in the deterministic case.
## 3 Itô formulation
Let’s write the infinite dimensional system (2) in Itô formulation:
$dX_{j}=(c_{j}X_{\overline{\jmath}}^{2}-\sum_{k\in\mathcal{O}_{j}}c_{k}X_{j}X_{k})dt+\sigma
c_{j}X_{\overline{\jmath}}dW_{j}\\\
-\sigma\sum_{k\in\mathcal{O}_{j}}c_{k}X_{k}dW_{k}-\frac{\sigma^{2}}{2}(c_{j}^{2}+\sum_{k\in\mathcal{O}_{j}}c_{k}^{2})X_{j}dt.$
(3)
We will use this formulation since it’s easier to handle the calculations,
while all results can also be stated in the Stratonovich formulation.
So let’s now introduce the definition of weak solution. A filtered probability
space $(\Omega,\mathcal{F}_{t},P)$ is a probability space
$(\Omega,\mathcal{F}_{\infty},P)$ together with a right-continuous filtration
$(\mathcal{F}_{t})_{t\geq 0}$ such that $\mathcal{F}_{\infty}$ is the
$\sigma$-algebra generated by $\bigcup_{t\geq 0}\mathcal{F}_{t}$.
###### Definition 1.
Given $x\in l^{2}$, a weak solution of (2) in $l^{2}$ is a filtered
probability space $(\Omega,\mathcal{F}_{t},P)$, a $J$-indexed sequence of
independent Brownian motions $(W_{j})_{j\in J}$ on
$(\Omega,\mathcal{F}_{t},P)$ and an $l^{2}$-valued process $(X_{j})_{j\in J}$
on $(\Omega,\mathcal{F}_{t},P)$ with continuous adapted components $X_{j}$
such that
$X_{j}=x_{j}+\int_{0}^{t}[c_{j}X_{\overline{\jmath}}^{2}(s)-\sum_{k\in\mathcal{O}_{j}}c_{k}X_{j}(s)X_{k}(s)]ds\\\
+\int_{0}^{t}\sigma
c_{j}X_{\overline{\jmath}}(s)dW_{j}(s)-\sum_{k\in\mathcal{O}_{j}}\int_{0}^{t}\sigma
c_{k}X_{k}(s)dW_{k}(s)\\\
-\frac{\sigma^{2}}{2}\int_{0}^{t}(c_{j}^{2}+\sum_{k\in\mathcal{O}_{j}}c_{k}^{2})X_{j}(s)ds,$
(4)
for every $j\in J$, with $c_{0}=0$ and $X_{0}(t)=0$. We will denote this
solution by
$(\Omega,\mathcal{F}_{t},P,W,X),$
or simply by $X$.
###### Definition 2.
A weak solution is an energy controlled solution if it is a solution as in
Definition 1 and it satisfies
$P(\sum_{j\in J}X_{j}^{2}(t)\leq\sum_{j\in J}x_{j}^{2})=1,$
for all $t\geq 0$.
###### Theorem 3.1.
There exists an energy controlled solution to (3) in
$L^{\infty}(\Omega\times[0,T],l^{2})$ for $(x_{j})\in l^{2}$.
We will give a proof of this Theorem at the end of Section 7. It is a weak
existence result and uses the Girsanov transform.
We’ll prove in the following result that a process satisfying (3) satisfies
(2) too.
###### Proposition 3.2.
If $X$ is a weak solution, for every $j\in J$ the process $(X_{j}(t))_{t\geq
0}$ is a continuous semimartingale, so the following equalities hold:
$\displaystyle\int_{0}^{t}\sigma c_{j}X_{\overline{\jmath}}(s)\circ dW_{j}(s)$
$\displaystyle=\int_{0}^{t}\sigma
c_{j}X_{\overline{\jmath}}(s)dW_{j}(s)-\frac{\sigma^{2}}{2}\int_{0}^{t}c_{j}^{2}X_{j}(s)ds$
$\displaystyle\int_{0}^{t}\sigma\sum_{k\in\mathcal{O}_{j}}c_{k}X_{k}(s)\circ
dW_{k}(s)$ $\displaystyle=\sum_{k\in\mathcal{O}_{j}}\int_{0}^{t}\sigma
c_{k}X_{k}(s)dW_{k}(s)+\frac{\sigma^{2}}{2}\int_{0}^{t}\sum_{k\in\mathcal{O}_{j}}c_{k}^{2}X_{j}(s)ds,$
where the Stratonovich integrals are well defined. So $X$ satisfies the
Stratonovich formulation of the problem (2).
###### Proof.
We know that
$\int_{0}^{t}\sigma c_{j}X_{\overline{\jmath}}(s)\circ
dW_{j}(s)=\int_{0}^{t}\sigma
c_{j}X_{\overline{\jmath}}(s)dW_{j}(s)+\frac{\sigma
c_{j}}{2}[{X_{\overline{\jmath}}},W_{j}]_{t},$
but from (2) we have that the only contribution to
$[{X_{\overline{\jmath}}},W_{j}]$ is given by the $-\sigma c_{j}X_{j}\circ
dW_{j}$ term, so
$\frac{\sigma c_{j}}{2}[{X_{\overline{\jmath}}},W_{j}]_{t}=\frac{\sigma
c_{j}}{2}[-\int_{0}^{t}\sigma c_{j}X_{j}\circ
dW_{j},W_{j}]_{t}=-\frac{\sigma^{2}c_{j}^{2}}{2}\int_{0}^{t}X_{j}ds.$
Now if we consider the other integral, we have
$\int_{0}^{t}\sigma\sum_{k\in\mathcal{O}_{j}}c_{k}X_{k}(s)\circ
dW_{k}(s)=\sum_{k\in\mathcal{O}_{j}}\int_{0}^{t}\sigma
c_{k}X_{k}(s)dW_{k}(s)+\sum_{k\in\mathcal{O}_{j}}\frac{\sigma
c_{k}}{2}[X_{k},W_{k}]_{t}.$
For each $X_{k}$ we get, with the same computations, that the only
contribution to $[X_{k},W_{k}]_{t}$ comes from the term $\sigma
c_{k}X_{j}\circ dW_{k}$, so that we get
$\frac{\sigma c_{k}}{2}[X_{k},W_{k}]_{t}=\frac{\sigma
c_{k}}{2}[\int_{0}^{t}\sigma c_{k}X_{j}\circ
dW_{k},W_{k}]_{t}=\frac{\sigma^{2}c_{k}^{2}}{2}\int_{0}^{t}X_{j}ds.\qed$
## 4 Girsanov transform
Let’s consider (3) and rewrite it as
$dX_{j}=c_{j}X_{\overline{\jmath}}(X_{\overline{\jmath}}dt+\sigma dW_{j})-\\\
\sum_{k\in\mathcal{O}_{j}}c_{k}X_{k}(X_{j}dt+\sigma
dW_{k})-\frac{\sigma^{2}}{2}(c_{j}^{2}+\sum_{k\in\mathcal{O}_{j}}c_{k}^{2})X_{j}dt.$
(5)
The idea is to isolate $X_{\overline{\jmath}}dt+\sigma dW_{j}$ and prove
through Girsanov’s theorem that they are Brownian motions with respect to a
new measure $\widehat{P}$ in $(\Omega,\mathcal{F}_{\infty})$, simultaneously
for every $j\in J$. This way (3) becomes a system of linear SDEs under the new
measure $\widehat{P}$. The infinite dimensional version of Girsanov’s theorem
can be found in [17] and [13].
###### Remark 1.
We can obtain the same result under Stratonovich formulation.
Let $X$ be an energy controlled solution: its energy ${\mathcal{E}}(t)$ is
bounded, so we can define the process
$M_{t}=-\frac{1}{\sigma}\sum_{j\in
J}\int_{0}^{t}X_{\overline{\jmath}}(s)dW_{j}(s)$ (6)
which is a martingale. Its quadratic variation is
$[M]_{t}=\frac{1}{\sigma^{2}}\int_{0}^{t}\sum_{j\in
J}X_{\overline{\jmath}}^{2}(s)ds.$
Because of the same boundedness of ${\mathcal{E}}(t)$ stated above, by the
Novikov criterion $\exp(M_{t}-\frac{1}{2}[M]_{t})$ is a (strictly) positive
martingale. We now define $\widehat{P}$ on $(\Omega,\mathcal{F}_{t})$ as
$\displaystyle\frac{d\widehat{P}}{dP}\Big{|}_{\mathcal{F}_{t}}$
$\displaystyle=\exp(M_{t}-\frac{1}{2}[M]_{t})$
$\displaystyle=\exp(-\frac{1}{\sigma}\sum_{j\in
J}\int_{0}^{t}X_{\overline{\jmath}}(s)dW_{j}(s)-\frac{1}{2\sigma^{2}}\int_{0}^{t}\sum_{j\in
J}X_{\overline{\jmath}}^{2}(s)ds),$ (7)
for every $t\geq 0$. $P$ and $\widehat{P}$ are equivalent on each
$\mathcal{F}_{t}$, because of the strict positivity of the exponential.
We can now prove the following:
###### Theorem 4.1.
If $(\Omega,\mathcal{F}_{t},P,W,X)$ is an energy controlled solution of the
nonlinear equation (3), then $(\Omega,\mathcal{F}_{t},\widehat{P},B,X)$
satisfies the linear equation
$dX_{j}=\sigma
c_{j}X_{\overline{\jmath}}dB_{j}(t)-\sigma\sum_{k\in\mathcal{O}_{j}}c_{k}X_{k}dB_{k}(t)-\frac{\sigma^{2}}{2}(c_{j}^{2}+\sum_{k\in\mathcal{O}_{j}}c_{k}^{2})X_{j}dt,$
(8)
where the processes
$B_{j}(t)=W_{j}(t)+\int_{0}^{t}\frac{1}{\sigma}X_{\overline{\jmath}}(s)ds$
are a sequence of independent Brownian motions on
$(\Omega,\mathcal{F}_{t},\widehat{P})$, with $\widehat{P}$ defined by (7).
###### Proof.
Now let’s define
$B_{j}(t)=W_{j}(t)+\int_{0}^{t}\frac{1}{\sigma}X_{\overline{\jmath}}(s)ds.$
Under $\widehat{P}$, $(B_{j}(t))_{j\in J,t\in[0,T]}$ is a sequence of
independent Brownian motions.
Since
$\displaystyle\sigma\int_{0}^{t}c_{j}{X_{\overline{\jmath}}}(s)dB_{j}(s)$
$\displaystyle=\sigma\int_{0}^{t}c_{j}{X_{\overline{\jmath}}}(s)dW_{j}(s)+\int_{0}^{t}c_{j}{X_{\overline{\jmath}}}^{2}(s)ds$
$\displaystyle\sigma\int_{0}^{t}c_{k}X_{k}(s)dB_{k}(s)$
$\displaystyle=\sigma\int_{0}^{t}c_{k}X_{k}(s)dW_{k}(s)+\int_{0}^{t}c_{k}X_{j}(s)X_{k}(s)ds\quad
k\in\mathcal{O}_{j}.$
Then (5) can be rewritten in integral form as
$X_{j}(t)=x_{j}+\sigma\int_{0}^{t}c_{j}{X_{\overline{\jmath}}}(s)dB_{j}(s)-\sigma\sum_{k\in\mathcal{O}_{j}}\int_{0}^{t}c_{k}X_{k}(s)dB_{k}(s)\\\
-\frac{\sigma^{2}}{2}\int_{0}^{t}(c_{j}^{2}+\sum_{k\in\mathcal{O}_{j}}c_{k}^{2})X_{j}(s)ds,$
(9)
which is a linear stochastic equation. ∎
###### Remark 2.
We can write our linear equation (8) also in Stratonovich form:
$dX_{j}=\sigma c_{j}{X_{\overline{\jmath}}}\circ
dB_{j}(t)-\sum_{k\in\mathcal{O}_{j}}\sigma c_{k}X_{k}\circ dB_{k}(t).$
###### Remark 3.
If we look at (8) we can see that it is possible to drop the $\sigma$,
considering it a part of the coefficients $c_{j}$.
inline,caption=quadratic variation]In equation (10) the author writes
$[X_{j}]_{t}$ while he uses another notation elsewhere. I suggest that he
sticks with only one formulation for consistency. inline,
color=green,nolist]Changed notation in equation (7) and surroundings We can
use Itô formula to calculate
$\displaystyle\frac{1}{2}dX_{j}^{2}$
$\displaystyle=X_{j}dX_{j}+\frac{1}{2}d[X_{j}]_{t}$ (10) $\displaystyle=\sigma
c_{j}{X_{\overline{\jmath}}}X_{j}dB_{j}-\sigma\sum_{k\in\mathcal{O}_{j}}c_{k}X_{j}X_{k}dB_{k}$
$\displaystyle\quad-\frac{\sigma^{2}}{2}(c_{j}^{2}+\sum_{k\in\mathcal{O}_{j}}c_{k}^{2})X_{j}^{2}dt+\frac{\sigma^{2}}{2}(c_{j}^{2}X_{\overline{\jmath}}^{2}+\sum_{k\in\mathcal{O}_{j}}c_{k}^{2}X_{k}^{2})dt$
$\displaystyle=-\frac{\sigma^{2}}{2}(c_{j}^{2}+\sum_{k\in\mathcal{O}_{j}}c_{k}^{2})X_{j}^{2}dt+dN_{j}+\frac{\sigma^{2}}{2}(c_{j}^{2}X_{\overline{\jmath}}^{2}+\sum_{k\in\mathcal{O}_{j}}c_{k}^{2}X_{k}^{2})dt,$
with
$N_{j}(t)=\sigma\int_{0}^{t}c_{j}X_{\overline{\jmath}}X_{j}dB_{j}-\sigma\sum_{k\in\mathcal{O}_{j}}\int_{0}^{t}c_{k}X_{j}X_{k}dB_{k}.$
(11)
This equality will be useful in the following.
We now present an existence result also for system (8).
###### Proposition 4.2.
There exists a solution of (8) in $L^{\infty}(\Omega\times[0,T],l^{2})$ with
continuous components, with initial conditions $x\in l^{2}$.
###### Proof.
Fix $N\geq 1$ and consider the finite dimensional stochastic linear system
$\begin{cases}dX_{j}^{N}=\sigma
c_{j}X_{\overline{\jmath}}^{N}dB_{j}(t)-\sigma\sum_{k\in\mathcal{O}_{j}}c_{k}X_{k}^{N}dB_{k}(t)\\\
\qquad\quad-\frac{\sigma^{2}}{2}(c_{j}^{2}+\sum_{k\in\mathcal{O}_{j}}c_{k}^{2})X_{j}^{N}dt&j\in
J,\ 0\leq|j|\leq N\\\ X_{k}^{N}(t)\equiv 0&k\in J,\ |k|=N+1\\\
X_{j}^{N}(0)=x_{j}&j\in J,\ 0\leq|j|\leq N.\end{cases}$ (12)
This system has a unique global strong solution $(X_{j}^{N})_{j\in J}$. We can
compute, using (10) and the definition of $N_{j}$ in (11),
$\displaystyle\frac{1}{2}d(\sum_{|j|\leq N}(X_{j}^{N}(t))^{2})$
$\displaystyle=\sum_{|j|\leq
N}(-\frac{\sigma^{2}}{2}(c_{j}^{2}+\sum_{k\in\mathcal{O}_{j}}c_{k}^{2})(X_{j}^{N})^{2}dt+dN_{j}^{N}$
$\displaystyle\quad+\frac{\sigma^{2}}{2}(c_{j}^{2}(X_{\overline{\jmath}}^{N})^{2}+\sum_{k\in\mathcal{O}_{j}}c_{k}^{2}(X_{k}^{N})^{2})dt)$
$\displaystyle=-\sum_{|j|=N}\frac{\sigma^{2}}{2}\sum_{k\in\mathcal{O}_{j}}c_{k}^{2}(X_{j}^{N})^{2}dt$
$\displaystyle=-\frac{\sigma^{2}}{2}\sum_{|k|=N+1}c_{k}^{2}(X_{\overline{k}}^{N})^{2}\leq
0.$
Hence
$\sum_{|j|\leq N}(X_{j}^{N}(t))^{2}\leq\sum_{|j|\leq N}x_{j}^{2}\leq\sum_{j\in
J}x_{j}^{2}\qquad\widehat{P}-\textrm{a.s.}\quad\forall t\geq 0.$
This implies that there exists a sequence $N_{m}\uparrow\infty$ such that
$(X_{j}^{N_{m}})_{j\in J}$ converges weakly to some $(X_{j})_{j\in J}$ in
$L^{2}(\Omega\times[0,T],l^{2})$ and also weakly star in
$L^{\infty}(\Omega\times[0,T],l^{2})$, so $(X_{j})_{j\in J}$ is in
$L^{\infty}(\Omega\times[0,T],l^{2})$.
Now for every $N\in{\mathbb{N}}$, $(X_{j}^{N})_{j\in J}$ is
in${\mathrm{Prog}}$, the subspace of progressively measurable processes in
$L^{2}(\Omega\times[0,T],l^{2})$. But ${\mathrm{Prog}}$ is strongly closed,
hence weakly closed, so $(X_{j})_{j\in J}\in{\mathrm{Prog}}$.
We just have to prove that $(X_{j})_{j\in J}$ solves (8). All the one
dimensional stochastic integrals that appear in each equation in (9) are
linear strongly continuous operators ${\mathrm{Prog}}\to L^{2}(\Omega)$, hence
weakly continuous. Then we can pass to the weak limit in (12). Moreover from
the integral equations (9) we have that there is a modification of the
solution which is continuous in all the components. ∎
## 5 Closed equation for $E_{\widehat{P}}[X_{j}^{2}(t)]$
###### Proposition 5.1.
For every energy controlled solution $X$ of the nonlinear equation (3),
$E_{\widehat{P}}[X_{j}^{2}(t)]$ is finite for every $j\in J$ and satisfies
$\frac{d}{dt}E_{\widehat{P}}[X_{j}^{2}(t)]=-\sigma^{2}(c_{j}^{2}+\sum_{k\in\mathcal{O}_{j}}c_{k}^{2})E_{\widehat{P}}[X_{j}^{2}(t)]\\\
+\sigma^{2}c_{j}^{2}E_{\widehat{P}}[X_{\overline{\jmath}}^{2}(t)]+\sigma^{2}\sum_{k\in\mathcal{O}_{j}}c_{k}^{2}E_{\widehat{P}}[X_{k}^{2}(t)].$
(13)
###### Proof.
Let $(\Omega,\mathcal{F}_{t},P,W,X)$ be an energy controlled solution of the
nonlinear equation (3), with initial condition $X\in l^{2}$ and let
$\widehat{P}$ be the measure given by Theorem 4.1. Denote by $E_{\widehat{P}}$
the expectation with respect to $\widehat{P}$ in $(\Omega,\mathcal{F}_{t})$.
Notice that
$E_{\widehat{P}}[\int_{0}^{T}X_{j}^{4}(t)dt]<\infty\qquad\forall j\in J.$ (14)
For energy controlled solutions from the definition we have that $P$-a.s.
$\sum_{j\in J}X_{j}^{4}(t)\leq\max_{j\in J}X_{j}^{2}(t)\sum_{j\in
J}X_{j}^{2}(t)\leq(\sum_{j\in J}x_{j}^{2})^{2},$
because of the behavior of the energy we showed. But on every
$\mathcal{F}_{t}$, $P\sim\widehat{P}$, so
$\widehat{P}(\sum_{j\in J}X_{j}^{4}(t)\leq(\sum_{j\in J}x_{j}^{2})^{2})=1,$
and (14) holds.
From (14) it follows that $M_{j}(t)$ is a martingale for every $j\in J$.
Moreover
$E_{\widehat{P}}[\sum_{j\in J}X_{j}^{2}(t)]<\infty,$
since $X_{j}(t)$ is an energy controlled solution and the condition is
invariant under the change of measure $P\leftrightarrow\widehat{P}$ on
$\mathcal{F}_{t}$ and, in particular,
$E_{\widehat{P}}[X_{j}^{2}(t)]<\infty\qquad\forall j\in J.$
Now let’s write (10) in integral form:
$X_{j}^{2}(t)-x_{j}^{2}=-\sigma^{2}\int_{0}^{t}(c_{j}^{2}+\sum_{k\in\mathcal{O}_{j}}c_{k}^{2})X_{j}^{2}(s)ds\\\
+2\int_{0}^{t}dN_{j}(s)+\sigma^{2}\int_{0}^{t}(c_{j}^{2}X_{\overline{\jmath}}^{2}(s)+\sum_{k\in\mathcal{O}_{j}}c_{k}^{2}X_{k}^{2}(s))ds.$
We can take the $\widehat{P}$ expectation,
$E_{\widehat{P}}[X_{j}^{2}(t)]-x_{j}^{2}=-\sigma^{2}\int_{0}^{t}(c_{j}^{2}+\sum_{k\in\mathcal{O}_{j}}c_{k}^{2})E_{\widehat{P}}[X_{j}^{2}(s)]ds\\\
+\sigma^{2}\int_{0}^{t}c_{j}^{2}E_{\widehat{P}}[X_{\overline{\jmath}}^{2}(s)]ds+\sigma^{2}\sum_{k\in\mathcal{O}_{j}}\int_{0}^{t}c_{k}^{2}E_{\widehat{P}}[X_{k}^{2}(s)]ds,$
where the $N_{j}$ term vanishes, since it’s a $\widehat{P}$-martingale. Now we
can derive and the proposition is established. ∎
It’s worth stressing that $E_{\widehat{P}}[X_{j}^{2}(t)]$ satisfies a closed
equation. Even more interesting is the fact that this is the forward equation
of a continuous-time Markov chain, as we will see in the following section.
## 6 Associated Markov chain
We want to show and use this characterization of the second moments equation
as the forward equation of a Markov chain, taking advantage of some known
results in the Markov chains theory. We follow the transition functions
approach to continuous times Markov chains; we don’t assume any knowledge of
this theory, so we will provide the basic definitions and results we need.
More results can be found in the literature, see for example [1].
###### Definition 3.
A non-negative function $f_{j,l}(t)$ with $j,l\in J$ and $t\geq 0$ is a
_transition function_ on $J$ if $f_{jl}(0)=\delta_{jl}$,
$\sum_{l\in J}f_{jl}(t)\leq 1\qquad\forall j\in J,\ \forall t\geq 0,$
and it satisfies the semigroup property (or Chapman-Kolmogorov equation)
$f_{jl}(t+s)=\sum_{h\in J}f_{jh}(t)f_{hl}(s)\qquad\forall j,l\in J,\ \forall
t,s\geq 0.$
###### Definition 4.
A _$q$ -matrix_ $Q=(q_{jl})_{j,l\in J}$ is a square matrix such that
$\displaystyle 0\leq q_{jl}<+\infty\qquad\forall j\neq l\in J,$
$\displaystyle\sum_{l\neq j}q_{jl}\leq-q_{jj}\eqqcolon
q_{j}\leq+\infty\qquad\forall j\in J.$
A $q$-matrix is called _stable_ if all $q_{j}$’s are finite, and
_conservative_ if
$q_{j}=\sum_{l\neq j}q_{jl}\qquad\forall j\in J.$
If $Q$ is a $q$-matrix, a $Q$-function is a transition function $f_{jl}(t)$
such that $f^{\prime}_{jl}(0)=Q$.
The $q$-matrix shows a close resemblance to the infinitesimal generator of the
transition function, but they differ, since the former doesn’t determine a
unique transition function, while the latter does. Still this approach can be
seen as a generator approach to Markov chains in continuous times.
Now let’s see these objects in our framework: let’s write (13) in matrix form.
Let $Q$ be the infinite dimensional matrix which entries are defined as
$q_{j,j}=-\sigma^{2}(c_{j}^{2}+\sum_{k\in\mathcal{O}_{j}}c_{k}^{2})\quad
q_{j,\bar{\jmath}}=\sigma^{2}c_{j}^{2}\quad
q_{j,k}=\mathbbm{1}_{\\{k\in\mathcal{O}_{j}\\}}\sigma^{2}c_{k}^{2}\
\textrm{for }k\neq j,\bar{\jmath}$
###### Proposition 6.1.
The infinite matrix $Q$ defined above is the stable and conservative
$q$-matrix. Moreover $Q$ is symmetric.
###### Proof.
It’s easy to check that $Q$ is a stable and conservative $q$-matrix. First of
all $q_{j,j}<0$ for all $j\in J$ and $q_{j,l}\geq 0$ for all $j\neq l$. Then
$q_{j}=\sum_{l\neq
j}q_{j,l}=q_{j,\overline{\jmath}}+\sum_{k\in\mathcal{O}_{j}}q_{j,k}=\sigma^{2}c_{j}^{2}+\sum_{k\in\mathcal{O}_{j}}\sigma^{2}c_{k}^{2}=-q_{j,j}.$
Moreover it is very easy to check that the matrix is symmetric:
$q_{ij}=\left\\{\begin{aligned} \sigma^{2}c_{j}^{2}\quad
l=\overline{\jmath}\quad&\Leftrightarrow\quad
j\in\mathcal{O}_{l}\quad\sigma^{2}c_{j}^{2}\\\ \sigma^{2}c_{l}^{2}\quad
l\in\mathcal{O}_{j}\quad&\Leftrightarrow\quad
j=\overline{l}\quad\sigma^{2}c_{l}^{2}\end{aligned}\right\\}=q_{lj}$
∎
Since $Q$ is a $q$-matrix we can construct the process associated, as a jump
and hold process on the space state, which in our case is the tree of the
dyadic model. The process will wait in node $j$ for an exponential time of
parameter $q_{j}$, and then will jump to $\bar{\jmath}$ or
$k\in\mathcal{O}_{j}$ with probabilities $q_{j,\bar{\jmath}}/q_{j}$ and
$q_{j,k}/q_{j}$ respectively. This process is a continuous time Markov chain
that has $J$ as a state space and also has the same skeleton as the dyadic
tree model, meaning that the transition probabilities are non-zero only if one
of the nodes is the father of the other one.
Given a $q$-matrix $Q$, it is naturally associated with two (systems of)
differential equations:
$\displaystyle y^{\prime}_{jl}(t)=\sum_{h\in J}y_{jh}(t)q_{hl}$ (15)
$\displaystyle y_{jl}^{\prime}(t)=\sum_{h\in J}q_{jh}y_{hl}(t),$
called forward and backwards Kolmogorov equations, respectively.
###### Lemma 6.2.
Given a stable, symmetric and conservative $q$-matrix $Q$, then the unique
nonnegative solution of the forward equations (15) in
$L^{\infty}([0,\infty),l^{1})$, given a null initial condition $y(0)=0$, is
$y(t)=0$.
###### Proof.
Let $y$ be a generic solution, then
$\left\\{\begin{aligned} &\frac{d}{dt}y_{j}(t)=\sum_{i\in J}y_{i}(t)q_{ij}\\\
&y_{j}(t)\geq 0\quad j\in J\\\ &y_{j}(0)=0\quad j\in J\\\ &\sum_{j\in
J}y_{j}(t)<+\infty.\end{aligned}\right.$ (16)
We can consider for every node
$\hat{y}_{j}=\int_{0}^{+\infty}e^{-t}y_{j}(t)dt$, the Laplace transform in 1\.
From the last equation of the system above, we have $\sum_{j}\hat{y}_{j}\leq
M$, for some constant $M>0$, so in particular we can consider $k\in J$ such
that $\hat{y}_{k}\geq\hat{y}_{j}$, for all $j\in J$.
Now we want to show that $y_{k}^{\prime}(t)$ is bounded: thanks to the
symmetry and stability of $Q$ we have
$|y_{k}^{\prime}(t)|\leq|-q_{k}y_{k}(t)|+|\sum_{l\neq k}y_{l}(t)q_{lk}|\leq
q_{k}M+q_{k}M<+\infty.$
We can integrate by parts
$\hat{y}_{k}=\int_{0}^{+\infty}e^{-t}y_{k}^{\prime}(t)dt=\int_{0}^{+\infty}e^{-t}\sum_{l\in
J}y_{l}(t)q_{lk}dt=\sum_{l\in J}\hat{y}_{l}q_{lk}\\\
=-\hat{y}_{k}q_{k}+\sum_{l\neq
k}\hat{y}_{l}q_{lk}\leq\hat{y}_{k}(-q_{k}+\sum_{l\neq k}q_{kl})=0,$ (17)
where the last equality follows from the conservativeness of $Q$, and we used
the stability and symmetry. Now we have $\hat{y}_{k}=0$ and so all
$\hat{y}_{j}=0$, hence $y_{j}(t)=0$ for all $j\in J$, for all $t\geq 0$. ∎
## 7 Uniqueness
Now we can use the results of the previous section to prove the main results
of this paper.
###### Theorem 7.1.
There is strong uniqueness for the linear system (8) in the class of energy
controlled $L^{\infty}(\Omega\times[0,T],l^{2})$ solutions.
###### Proof.
By linearity of (8) it is enough to prove that for null initial conditions
there is no nontrivial solution. Since we have (13), proposition 6.1 and lemma
6.2, then $E_{\widehat{P}}[X_{j}^{2}(t)]=0$ for all $j$ and $t$, hence $X=0$
a.s. ∎
Let’s recall that we already proved an existence result for (8) with
proposition 4.2.
###### Theorem 7.2.
There is uniqueness in law for the nonlinear system (3) in the class of energy
controlled $L^{\infty}(\Omega\times[0,T],l^{2})$ solutions.
###### Proof.
Assume that $(\Omega^{(i)},\mathcal{F}_{t}^{(i)},P^{(i)},W^{(i)},X^{(i)})$,
$i=1,2$, are two solutions of (3) with the same initial conditions $x\in
l^{2}$. Given $n\in{\mathbb{N}}$, $t_{1},\ldots,t_{n}\in[0,T]$ and a
measurable and bounded function $f:(l^{2})^{n}\to{\mathbb{R}}$, we want to
prove that
$E_{P^{(1)}}[f(X^{(1)}(t_{1}),\ldots,X^{(1)}(t_{n}))]=E_{P^{(2)}}[f(X^{(2)}(t_{1}),\ldots,X^{(2)}(t_{n}))].$
(18)
By theorem 4.1 and the definition of $\widehat{P}$ given in (7) we have that,
for $i=1,2$,
$E_{P^{(i)}}[f(X^{(i)}(t_{1}),\ldots,X^{(i)}(t_{n}))]=\\\
E_{\widehat{P}^{(i)}}[\exp\\{-M_{T}^{(i)}+\frac{1}{2}[M^{i},M^{(i)}]_{T}\\}f(X^{(i)}(t_{1}),\ldots,X^{(i)}(t_{n}))],$
(19)
where $M^{(i)}$is defined as in (6). We have proven in proposition 4.2 and
theorem 7.1 that the linear system (8) has a unique strong solution. Thus it
has uniqueness in law on $\mathcal{C}([0,T],{\mathbb{R}})^{\mathbb{N}}$ by
Yamada-Watanabe theorem, that is under the measures $\widehat{P}^{(i)}$, the
processes $X^{(i)}$ have the same laws. For a detailed proof of this theorem
in infinite dimension see [17].
Now we can also include $M^{(i)}$ in the system and conclude that
$(X^{(i)},M^{(i)})$ under $\widehat{P}^{(i)}$ have laws independent of
$i=1,2$, hence, through (19), we have (18). ∎
We can now conclude with the proof of Theorem 3.1.
###### Proof of Theorem 3.1.
Let $(\Omega,\mathcal{F}_{t},\widehat{P},B,X)$ be the solution of (8) in
$L^{\infty}(\Omega\times[0,T],l^{2})$ provided by theorem 7.1. We follow the
same argument as in Section 4, only from $\widehat{P}$ to $P$. We construct
$P$ as a measure on $(\Omega,\mathcal{F}_{T})$ satisfying
$\frac{dP}{d\widehat{P}}\Big{|}_{\mathcal{F}_{T}}=\exp(\widehat{M}_{T}-\frac{1}{2}[\widehat{M},\widehat{M}]_{T}),$
where $\widehat{M}_{t}=\frac{1}{\sigma}\sum_{j\in
J}\int_{0}^{t}X_{\overline{\jmath}}(s)dB_{j}(s)$. Under $P$ the processes
$W_{j}(t)=B_{j}(t)-\int_{0}^{t}\frac{1}{\sigma}X_{\overline{\jmath}}(s)ds,$
are a sequence of independent Brownian motions. Hence
$(\Omega,\mathcal{F}_{t},P,W,X)$ is a solution of (3) and it is in
$L^{\infty}$, since $P$ and $\widehat{P}$ are equivalent on $\mathcal{F}_{T}$.
∎
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{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Luigi Amedeo Bianchi",
"submitter": "Luigi Amedeo Bianchi",
"url": "https://arxiv.org/abs/1301.7247"
}
|
1301.7253
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KOBE-TH-13-1
HRI-P-13-01-001
RECAPP-HRI-2013-001
$CP$ phase from twisted Higgs
vacuum expectation value in extra dimension
Yukihiro Fujimoto, aaa E-mail: [email protected] Kenji Nishiwaki,
bbb E-mail: [email protected] and Makoto Sakamoto ccc E-mail:
[email protected]
a,cDepartment of Physics, Kobe University, Kobe 657-8501, Japan
bRegional Centre for Accelerator-based Particle Physics,
Harish-Chandra Research Institute, Allahabad 211 019, India
2 September 2024
We propose a new mechanism for generating a $CP$ phase via Higgs a vacuum
expectation value originating from geometry of an extra dimension. A twisted
boundary condition is the key to produce an extra-dimension coordinate-
dependent vacuum expectation value, which contains a $CP$ phase degree of
freedom and can be a new source of a $CP$ phase in higher-dimensional gauge
theories. As an illustrative example, we apply our mechanism to a five-
dimensional gauge theory with point interactions and show that our mechanism
can dynamically produce a nontrivial $CP$-violating phase with electroweak
symmetry breaking, even though the five-dimensional model does not include any
$CP$-violating phases of Yukawa couplings in the five-dimensional Lagrangian
because of a single generation of five-dimensional fermions. We apply our
mechanism to a model with point interactions, which has no source of
$CP$-violating phases in the couplings of the higher-dimensional action, and
show that a nontrivial $CP$ phase dynamically appears.
## 1 Introduction
Pursuing the origin of the generations of the fermions is one of the important
themes in particle physics. The three generations (or more) are necessary to
produce the Kobayashi–Maskawa $CP$ phase, which causes $CP$-violating effects
and was proposed in Ref. [1]. If the number of generations were less than $3$,
any complex phases in the Cabbibo–Kobayashi–Maskawa (CKM) matrix would be
absorbed into phases of quark fields and then the Kobayashi-Maskawa
$CP$-violating mechanism would not work.
Extra-dimensional field theory is one of the appealing candidates beyond the
standard model (SM). Many studies have been done up to today based on many
ideas for pursuing the origin of fermion flavor [2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18]. Especially, we can find several attractive
models to solve the generation problem, in which the three generations of the
four-dimensional chiral fermions are dynamically realized from a single
generation of higher-dimensional fermions. However, such models have a common
problem: The number of higher-dimensional Yukawa couplings is not enough to
produce a $CP$-violating phase because of a single generation of fermions in
higher-dimensions, so we could not obtain a $CP$-violating phase à la
Kobayashi-Maskawa. Therefore, in models to solve the generation problem, we
need some new sources of $CP$-violating phases other than higher-dimensional
Yukawa couplings. Otherwise, those models without a $CP$-violating phase
should be discarded as phenomenological ones.111 In the gauge-Higgs
unification model, a similar problem arises because of lack of degree of
freedom in the Yukawa sector of an original five-dimensional action. The ways
to overcome this point have been studied [19, 20, 21].
In this paper, we propose a new mechanism to produce a $CP$ phase in the
context of five-dimensional gauge theories. Allowing a twisted boundary
condition (BC) for the Higgs doublet leads to a Higgs vacuum expectation value
(VEV) with an extra-dimension coordinate-dependent phase, which contains a
$CP$ phase degree of freedom. The properties of such kinds of scalar VEVs have
been studied in Refs. [22, 23, 24, 25, 26, 27, 28, 29, 30, 31].222 Point
interactions on $S^{1}$, which are additional boundary points (on $S^{1}$),
have been studied in Refs [32, 33, 34, 35, 36]. We can consider another
possibility that some terms are localized in boundary points at tree level
[37, 38, 39, 40]. We note that the electroweak symmetry is dynamically broken
at that time.
As a demonstration of our mechanism, we apply the mechanism to a five-
dimensional gauge theory with point interactions in which three generations in
four dimensions are produced from a single generation in five dimensions. We
show that a nontrivial CP phase dynamically appears in the CKM matrix in four
dimensions, even though any coupling constants in the five-dimensional
Lagrangian have no $CP$ phases. Our purpose of this paper is to show that our
mechanism does work as a new source of the $CP$ violation.
This paper is organized as follows. In Sec. 2, we discuss and verify a
possibility of the Higgs doublet with a twisted BC to explain the origin of
the $CP$ phase in the CKM matrix. In Sec. 3, we construct a model with point
interactions and a scalar singlets for which the VEV depends on the extra
coordinate exponentially. In Sec. 4, we check that the CP phase originating
from our mechanism can explain the CKM properties in the above model. Here, we
also discuss the properties of the realized quark masses and other mixings
briefly. In Sec. 5, we summarize our results and discuss some aspects of our
model. In the Appendix, details of choosing parameters is explored.
## 2 Position-dependent VEV (also as $CP$ phase) with twisted boundary
condition
In this section, we propose a new mechanism for generating $CP$ phase with
twisted boundary condition of a five-dimensional scalar $H$ on $S^{1}$.
Hereafter, we use a coordinate $y$ to indicate the position in the extra
space. A key aspect is that broken phase can be realized with the scalar, and
at the same time, the VEV profile itself turns out to be $y$-position
dependent and complex, which means that the scalar VEV possibly triggers the
$CP$ violation. Interestingly, the $y$-position dependence disappears in the
gauge boson masses, even though the VEV of $H$ depends on $y$. This is because
the $y$ dependence of the VEV of $H$ is cancelled out in the squared form
$H^{\dagger}H$. This property is very important and it works as an usual four-
dimensional Higgs mechanism without violating electroweak precision
measurements at the tree level. When we consider the situation that $H$ is the
$SU(2)_{W}$ Higgs doublet, we can dynamically generate both the suitable
electroweak symmetry breaking (EWSB) and the $CP$-violating phase
simultaneously. We note that its $SU(N)$ extension is possible and
straightforward.
The action we consider is
$\displaystyle S_{H}$ $\displaystyle=\int
d^{4}x\int_{0}^{L}dy\bigg{\\{}{H^{\dagger}(\partial_{M}\partial^{M}+{M}^{2})H-\frac{\lambda}{2}(H^{\dagger}H)^{2}}\bigg{\\}},$
(2.1)
where $M$ and $\lambda$ are the bulk mass and quartic coupling, respectively.
Since $S^{1}$ is a multiply connected space, we can impose the twisted
boundary condition on $H$ as [22, 23, 24, 25, 26]
$\displaystyle H(y+L)=e^{i\theta}H(y).$ (2.2)
Here, we take the range of $\theta$ as $-\pi<\theta\leq\pi$. $L$ shows the
circumference of $S^{1}$, and we choose the metric convention as
$\eta_{MN}=\eta^{MN}=\text{diag}(-1,1,1,1,1).$ The Latin indices run from $0$
to $3,\,5\ (\text{or\ }y)$, and Greek ones run from $0$ to $3$, respectively.
We note that the VEV of $\langle H(y)\rangle$ should be determined by
minimizing the functional
$\displaystyle\mathcal{E}[H]=\int_{0}^{L}dy\bigg{\\{}{\left|\partial_{y}H\right|^{2}-{M}^{2}\left|H\right|^{2}+\frac{\lambda}{2}\left|H\right|^{4}}\bigg{\\}}{,}$
(2.3)
because the VEV can possess the $y$ dependence to minimize the energy. Here,
we assume that the four-dimensional (4D) Lorentz invariance is unbroken.
After introducing $\mathcal{H}(y)$ by
$\displaystyle{H(y)=e^{i{\frac{\theta}{L}y}}\mathcal{H}(y),\quad\mathcal{H}(y+L)=\mathcal{H}(y),}$
(2.4)
the functional $\mathcal{E}$ can be rewritten as
$\displaystyle\mathcal{E}[H]$
$\displaystyle=\mathcal{E}_{1}[\mathcal{H}]+\mathcal{E}_{2}[\mathcal{H}],$
(2.5) $\displaystyle\mathcal{E}_{1}[\mathcal{H}]$
$\displaystyle=\int_{0}^{L}dy\bigg{\\{}{\left|\partial_{y}\mathcal{H}\right|^{2}}+i\frac{\theta}{L}\Big{(}(\partial_{y}\mathcal{H})^{\dagger}\mathcal{H}-\mathcal{H}^{\dagger}\partial_{y}\mathcal{H}\Big{)}\bigg{\\}},$
(2.6) $\displaystyle\mathcal{E}_{2}[\mathcal{H}]$
$\displaystyle=\int_{0}^{L}dy\bigg{\\{}{\frac{\lambda}{2}\left(\left|\mathcal{H}\right|^{2}-\frac{1}{\lambda}\left[{M}^{2}-\left(\frac{\theta}{L}\right)^{2}\right]\right)^{2}-\frac{1}{2\lambda}\left[{M}^{2}-\left(\frac{\theta}{L}\right)^{2}\right]^{2}}\bigg{\\}},$
(2.7)
where ${\cal E}_{1}$ corresponds to the contribution from the $y$-kinetic term
of ${\cal H}$.
Since $\mathcal{H}(y)$ satisfies the periodic boundary condition,
$\mathcal{H}(y)$ can be decomposed as
$\displaystyle{\mathcal{H}(y)=\sum_{n=-\infty}^{\infty}\frac{\vec{a}_{n}}{\sqrt{L}}e^{i2\pi
n\frac{y}{L}},}$ (2.8)
where $\vec{a}_{n}$ is a two-component $SU(2)_{W}$ constant vector.
Substituting Eq. (2.8) into $\mathcal{E}_{1}$, we obtain the expression
$\displaystyle\mathcal{E}_{1}=\sum_{n=-\infty}^{\infty}\left[\left(\frac{2\pi
n+\theta}{L}\right)^{2}-\left(\frac{\theta}{L}\right)^{2}\right]{\left|\vec{a}_{n}\right|^{2}\geq
0}$ (2.9)
and we can conclude that the minimum of $\mathcal{E}_{1}$ is given by
$\mathcal{E}_{1}=0$ when the values of $\theta$ and $\vec{a}_{n}$ satisfy one
of the conditions
(i) $\displaystyle{-\pi<\theta<\pi\text{\ and\ }{\vec{a}_{n}}=0\,(n\not=0):}$
$\displaystyle\mathcal{H}=\frac{{\vec{a}_{0}}}{\sqrt{L}},$ (2.10) (ii)
$\displaystyle\theta=\pi\text{\ and\ }{\vec{a}_{n}}=0\,(n\not=0,-1):$
$\displaystyle{\mathcal{H}=\frac{\vec{a}_{0}}{\sqrt{L}}\ \ \text{or}\ \
\mathcal{H}=\frac{{\vec{a}_{-1}}}{\sqrt{L}}e^{-i2\pi\frac{y}{L}},}$ (2.11)
where $\vec{a}_{0}$ in Eq. (2.10) and $\vec{a}_{0}$ or $\vec{a}_{-1}$ in Eq.
(2.11) are still undetermined. The functional $\mathcal{E}_{2}$ takes the
minimum value if the following condition is fulfilled:
$\displaystyle\left|\mathcal{H}\right|^{2}=\begin{cases}{\frac{1}{\lambda}\left({M}^{2}-\left(\frac{\theta}{L}\right)^{2}\right)}&\text{for\
}{{M}^{2}}-\left(\frac{\theta}{L}\right)^{2}>0\\\ 0&\text{for\
}{{M}^{2}}-\left(\frac{\theta}{L}\right)^{2}\leq 0\end{cases}.$ (2.12)
Combining the above two results and using the $SU(2)_{W}$ global symmetry, we
can show that the VEV $\langle H(y)\rangle$ is given, without loss of
generality, as
(I)
${M}^{2}-\left(\frac{\theta}{L}\right)^{2}>0$
$\displaystyle\left\langle
H(y)\right\rangle=\begin{cases}\frac{v}{\sqrt{2}}\,e^{i{\frac{\theta}{L}y}}\begin{pmatrix}0\\\
1\end{pmatrix}&\text{for\ }{-\pi<\theta<\pi},\\\
\frac{v}{\sqrt{2}}\,e^{i{\frac{{\pi}}{L}y}}\begin{pmatrix}0\\\
1\end{pmatrix}\text{\ \ or\ \
}\frac{v}{\sqrt{2}}\,e^{{-i\frac{{\pi}}{L}y}}\begin{pmatrix}0\\\
1\end{pmatrix}&\text{for\ }{\theta=\pi},\\\ \end{cases}$ (2.13)
(II)
${M}^{2}-\left(\frac{\theta}{L}\right)^{2}\leq 0$
$\displaystyle\left\langle H(y)\right\rangle=\begin{pmatrix}0\\\
0\end{pmatrix}{,}$ (2.14)
where $v$ is given by
$\displaystyle\left(\frac{v}{\sqrt{2}}\right)^{2}:={\left|\left\langle
H(y)\right\rangle\right|^{2}}={\frac{1}{\lambda}\left({M}^{2}-\left(\frac{\theta}{L}\right)^{2}\right)}.$
(2.15)
From now on, we will assume the case of (I)
${M}^{2}-\left(\frac{\theta}{L}\right)^{2}>0$.
Now we discuss some properties of the derived VEV in Eq. (2.13). Differently
from the SM, the VEV possesses $y$-position dependence, and its broken phase
is realized only in the case of ${M}^{2}-\left(\frac{\theta}{L}\right)^{2}>0$.
But like the SM, the squared VEV (2.15) is still constant even though $\langle
H(y)\rangle$ depends on $y$. This means that after $v\sqrt{L}$ is set as
$246\,\text{GeV}$, where the mass dimension of $v$ is $3/2$, the same
situation as the SM occurs in the EWSB sector. On the other hand, the $y$
dependence of the Higgs VEV in Eq. (2.13) is an important consequence for the
Yukawa sector. Since the VEV of the Higgs doublet appears linearly in each
Yukawa term, the overlap integrals which lead to effective 4D Yukawa couplings
will produce a nontrivial $CP$ phase in the CKM matrix.
In terms of the VEV and physical Higgs modes $h^{(n)}(x)$, $H$ can be expanded
as
$\displaystyle H(x,y)\rightarrow\sum_{n=-\infty}^{\infty}\begin{pmatrix}0\\\
\frac{1}{\sqrt{2}}\left(ve^{i\frac{\theta}{L}y}\delta_{n,0}+h^{(n)}(x)\frac{1}{\sqrt{L}}e^{i\left(\frac{2\pi
n+\theta}{L}\right)y}\right)\end{pmatrix},$ (2.16)
which obeys the boundary condition (2.2). The physical masses $\mu_{h^{(n)}}$
of the zero mode $(n=0)$ and the Kaluza–Klein (KK) modes $(n\not=0)$ are
easily calculated from Eq. (2.1) as
$\displaystyle\mu_{h^{(n)}}^{2}=\begin{cases}2\left(M^{2}-\left(\frac{\theta}{L}\right)^{2}\right)=\lambda
v^{2}&\text{for\ }n=0\\\ {2{M}^{2}+\frac{(\theta+2\pi
n)^{2}}{2L^{2}}+\frac{(\theta-2\pi
n)^{2}}{2L^{2}}-3\left(\frac{\theta}{L}\right)^{2}={\lambda v^{2}}+\frac{(2\pi
n)^{2}}{L^{2}}}&\text{for\ }{n\geq 1}\end{cases},$ (2.17)
with the hermiticity condition for a real field on $S^{1}$:
$h^{(n)\dagger}=h^{(-n)}$.
We mention that the relation between $\mu_{h^{(n)}}$ and $\lambda$ for $n=0$
in Eq. (2.17) is totally the same as that of the standard model. We also
comment on the Higgs-quarks couplings in our model. As shown in Eq. (2.16),
the profiles of the VEV and the Higgs physical zero mode are the same as
$e^{i\frac{\theta}{L}y}$ up to the coefficients. This means that the strengths
of the couplings are equivalent to those of the SM even though the mode
function gets to be $y$-position dependent. As a result, the decay branching
ratios of the Higgs boson are the same as those of the standard model.333Being
different from the universal extra dimension case [41, 42, 43, 44, 45], the
“low” KK mass less than a TeV scale is not allowed after considering the level
mixing in the top sector [46]. Then, the significant deviations do not occur
in the loop-induced single Higgs production via gluon fusion and Higgs decay
processes to a pair of photons and gluons in our model.
## 3 Model with point interactions on $S^{1}$
Figure 1: The wave function profiles of the quarks and the VEV of $\Phi(y)$
are schematically depicted. Here we take $L_{0}^{(q)}=L_{0}^{(\Phi)}=0$. Note
that all the profiles have the periodicity along $y$ with the same period $L$.
Differently from the model on an interval in Ref. [46], we can find the
$(1,3)$ elements of the mass matrices due to the periodicity along the $y$
direction. Figure 2: This is an overview of our system with point
interactions. The red (blue) circular spots show the Dirichlet BC for left-
(right-)handed part at the corresponding boundary points, respectively. The
green, purple, and black circular spots represent the ordinary periodic in Eq.
(3.19), the Robin BCs in Eq. (3.16), and the twisted BCs in Eq. (2.2),
respectively. It is noted that we adopt the assumption in Eq. (4.3).
In the previous section, we introduced the twisted BC for the $SU(2)_{W}$
doublet $H$ and generated the EWSB by the $y$-position-dependent complex VEV
in Eq. (2.13) in the case of $M^{2}-(\theta/L)^{2}>0$. We expect that this VEV
also works as the source of the $CP$ phase of the CKM matrix, but here an
important issue, which we should think about carefully, exists.
If all the profiles of the three-generation quarks are flat, an effective
phase appears after integration over $y$ just as an overall factor, which can
be removed by $U(1)$ rephasing and never works as a physical $CP$ phase. To
circumvent this difficulty, profiles of the quarks are required to be
localized. On the other hand, field localization (in extra dimensions) is
known as an effective way of explaining the quark mass hierarchy and pattern
of flavor mixing. In this section, we consider a model with point interactions
as an illustrative example. Point interaction can be considered as zero-
thickness brane and we can arrange it anywhere in the bulk space of $S^{1}$.
At the location of a point interaction, we can consider five-dimensional (5D)
gauge-invariant boundary conditions, for which the variety is rich compared
with the case of $Z_{2}$ orbifolding. After we introduce three point
interactions for a 5D fermion, its zero-mode profile gets to be chiral, split
and localized. This situation is just what we want.444 Another interesting
idea for generating three-generation structure and field localization is
introducing magnetic flux on the torus [47, 48, 49]. We emphasize that flavor
mixing is naturally realized as overlapping of localized quark profiles. In
the model, an additional gauge singlet scalar is required for generating the
large mass hierarchy of the quarks. Its (almost) exponential shape in the VEV
is also generated by a suitable boundary condition at the corresponding point
interactions.
This basic idea is found in Ref. [46]. Nevertheless, there are two different
points between the models in this paper and in Ref. [46]:
* •
In the previous model [46], the Higgs VEV cannot possess a nontrivial complex
phase, and a $CP$ phase in the CKM matrix has not been realized. On the other
hand, the VEV in our present model has a $y$-position-dependent complex phase,
which will produce a $CP$ phase of the CKM matrix.
* •
In the previous model [46], the extra dimension has been taken to be an
interval, where the twisted BC in Eq. (2.2) cannot be realized. In the present
model, we set the extra dimension to be a circle $S^{1}$, for which the
geometry is compatible with the twisted BC (2.2).
In the following part, we briefly explain how to construct our model. The 5D
action for fermions is given by555 We adopt the representations of the gamma
matrices as $\Gamma_{\mu}=\gamma_{\mu},\
\Gamma_{y}=\Gamma^{y}=-i\gamma^{5}=\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$
and the Clifford algebra is defined as
$\left\\{\Gamma_{M},\Gamma_{N}\right\\}=-2\eta_{MN}.$
$\displaystyle S$ $\displaystyle=\int
d^{4}x\int_{0}^{L}dy\Bigg{\\{}\Big{[}\overline{Q}\left(i\partial_{M}\Gamma^{M}+M_{Q}\right)Q+\overline{\mathcal{U}}\left(i\partial_{M}\Gamma^{M}+M_{\mathcal{U}}\right)\mathcal{U}$
$\displaystyle\phantom{=\int d^{4}x\int_{0}^{L_{3}}\Bigg{\\{}\
\,}+\overline{\mathcal{D}}\left(i\partial_{M}\Gamma^{M}+M_{\mathcal{D}}\right)\mathcal{D}\Big{]}\Bigg{\\}},$
(3.1)
where we introduce an $SU(2)_{W}$ doublet ($Q$), an up-quark singlet
($\mathcal{U}$), and a down-type singlet ($\mathcal{D}$) with the
corresponding bulk masses ($M_{Q},M_{\mathcal{U}},M_{\mathcal{D}}$). We note
that our model contains only one generation for 5D quarks but each 5D quark
produces three generations of the 4D quarks, as we will see below.
We adopt the following BCs for $Q,\mathcal{U},\mathcal{D}$ with an
infinitesimal positive constant $\varepsilon$ [46]:
$\displaystyle Q_{R}$ $\displaystyle=0\qquad\text{at}\quad
y=L_{0}^{(q)}+\varepsilon,\,L_{1}^{(q)}\pm\varepsilon,\,L_{2}^{(q)}\pm\varepsilon,\,L_{3}^{(q)}-\varepsilon,$
(3.2) $\displaystyle\mathcal{U}_{L}$ $\displaystyle=0\qquad\text{at}\quad
y=L_{0}^{(u)}+\varepsilon,\,L_{1}^{(u)}\pm\varepsilon,\,L_{2}^{(u)}\pm\varepsilon,\,L_{3}^{(u)}-\varepsilon,$
(3.3) $\displaystyle\mathcal{D}_{L}$ $\displaystyle=0\qquad\text{at}\quad
y=L_{0}^{(d)}+\varepsilon,\,L_{1}^{(d)}\pm\varepsilon,\,L_{2}^{(d)}\pm\varepsilon,\,L_{3}^{(d)}-\varepsilon,$
(3.4)
where $\Psi_{R}$ and $\Psi_{L}$ denote the eigenstates of $\gamma^{5}$, i.e.,
$\Psi_{R}\equiv\frac{1+\gamma^{5}}{2}\Psi$ and
$\Psi_{L}\equiv\frac{1-\gamma^{5}}{2}\Psi$. Here $L^{(i)}_{j}$ for $i=q,u,d$
and $j=0,1,2,3$ means the positions of point interactions for the 5D fermions.
See Figs. 1 and 2 for details. A crucial consequence of the above BCs is that
there appear threefold degenerated left- (right-)handed zero modes in the mode
expansions of $Q$ ($\mathcal{U},\mathcal{D}$) and that they form the three
generations of the quarks. The details have been given in Ref. [46]. We will
not repeat the discussions here.
The fields $Q,\mathcal{U},\mathcal{D}$ with the BCs in Eqs (3.2)–(3.4) are KK
decomposed as follows:
$\displaystyle Q(x,y)=\begin{pmatrix}U(x,y)\\\ D(x,y)\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\sum_{i=1}^{3}u^{(0)}_{iL}(x)f_{q^{(0)}_{iL}}(y)\\\
\sum_{i=1}^{3}d^{(0)}_{iL}(x)f_{q^{(0)}_{iL}}(y)\end{pmatrix}+(\text{KK
modes}),$ (3.5) $\displaystyle\mathcal{U}(x,y)$
$\displaystyle=\sum_{i=1}^{3}u^{(0)}_{iR}(x)f_{u^{(0)}_{iR}}(y)+(\text{KK
modes}),$ (3.6) $\displaystyle\mathcal{D}(x,y)$
$\displaystyle=\sum_{i=1}^{3}d^{(0)}_{iR}(x)f_{d^{(0)}_{iR}}(y)+(\text{KK
modes}).$ (3.7)
Here the zero-mode functions are obtained in the following forms:
$\displaystyle f_{q^{(0)}_{iL}}(y)$
$\displaystyle=\mathcal{N}_{i}^{(q)}e^{M_{Q}(y-L_{i-1}^{(q)})}\Big{[}\theta(y-L_{i-1}^{(q)})\theta(L_{i}^{(q)}-y)\Big{]}$
$\displaystyle\text{in\ }[L_{0}^{(q)},L_{3}^{(q)}],$ (3.8) $\displaystyle
f_{u^{(0)}_{iR}}(y)$
$\displaystyle=\mathcal{N}_{i}^{(u)}e^{-M_{\mathcal{U}}(y-L_{i-1}^{(u)})}\Big{[}\theta(y-L_{i-1}^{(u)})\theta(L_{i}^{(u)}-y)\Big{]}$
$\displaystyle\text{in\ }[L_{0}^{(u)},L_{3}^{(u)}],$ (3.9) $\displaystyle
f_{d^{(0)}_{iR}}(y)$
$\displaystyle=\mathcal{N}_{i}^{(d)}e^{-M_{\mathcal{D}}(y-L_{i-1}^{(d)})}\Big{[}\theta(y-L_{i-1}^{(d)})\theta(L_{i}^{(d)}-y)\Big{]}$
$\displaystyle\text{in\ }[L_{0}^{(d)},L_{3}^{(d)}],$ (3.10)
where
$\displaystyle\Delta L^{(l)}_{i}$
$\displaystyle=L^{(l)}_{i}-L^{(l)}_{i-1}\qquad(\text{for }i=1,2,3;\ l=q,u,d),$
(3.11) $\displaystyle\mathcal{N}_{i}^{(q)}=\sqrt{\frac{2M_{Q}}{e^{2M_{Q}\Delta
L_{i}^{(q)}}-1}},\quad\mathcal{N}_{i}^{(u)}=\sqrt{\frac{2M_{\mathcal{U}}}{1-e^{-2M_{\mathcal{U}}\Delta
L_{i}^{(u)}}}},\quad\mathcal{N}_{i}^{(d)}=\sqrt{\frac{2M_{\mathcal{D}}}{1-e^{-2M_{\mathcal{D}}\Delta
L_{i}^{(d)}}}}.$ (3.12)
$\mathcal{N}_{i}^{(q)},\mathcal{N}_{i}^{(u)},\mathcal{N}_{i}^{(d)}$ are the
wave function normalization factors for
$f_{q^{(0)}_{iL}},f_{u^{(0)}_{iL}},f_{d^{(0)}_{iL}}$, respectively.
Since the length of the total system is universal, $L^{(l)}_{3}-L^{(l)}_{0}\
(l=q,u,d)$ should be equal to the circumference of $S^{1}$, i.e.
$\displaystyle
L:=L^{(q)}_{3}-L^{(q)}_{0}=L^{(u)}_{3}-L^{(u)}_{0}=L^{(d)}_{3}-L^{(d)}_{0}.$
(3.13)
Note that all the mode functions in Eqs. (3.8)–(3.10) (and a form of a singlet
VEV in Eq. (3.17)) are periodic with the common period $L$, whereas we do not
indicate that thing explicitly in Eqs. (3.8)–(3.10).
In this model, the large mass hierarchy is naturally explained with the Yukawa
sector
$\displaystyle S_{\mathcal{Y}}=\int
d^{4}x\int_{0}^{{L}}dy\bigg{\\{}\Phi\Big{[}-\mathcal{Y}^{(u)}\overline{Q}(i\sigma_{2}H^{\ast})\mathcal{U}-\mathcal{Y}^{(d)}\overline{Q}H\mathcal{D}\Big{]}+\text{h.c.}\bigg{\\}},$
(3.14)
where $\mathcal{Y}^{(u)}/\mathcal{Y}^{(d)}$ is the Yukawa coupling for
up-/down-type quark; $H$ and $\Phi$ are an $SU(2)_{W}$ scalar doublet and a
singlet. It should be noted that although the Yukawa couplings
$\mathcal{Y}^{(u)}$ and $\mathcal{Y}^{(d)}$ can be complex, they cannot be an
origin of the $CP$ phase of the CKM matrix because our model contains only a
single quark generation so that the number of the 5D Yukawa couplings is not
enough to produce a $CP$ phase in the CKM matrix. An outline of our system is
depicted in Fig. 1. Note that the five terms of
$\overline{Q}(i\sigma_{2}H^{\ast})\mathcal{U},\overline{Q}H\mathcal{D},\Phi\overline{Q}Q,\Phi\overline{\mathcal{U}}\mathcal{U},\Phi\overline{\mathcal{D}}\mathcal{D}$
with the Pauli matrix $\sigma_{2}$ are excluded by introducing a discrete
symmetry $H\rightarrow-H,\Phi\rightarrow-\Phi$. $\Phi$ is a gauge singlet and
there is no problem with gauge universality violation.666 If there exists the
doublet-singlet mixing term $-CH^{\dagger}H\Phi^{\dagger}\Phi$ with a
coefficient $C$, which cannot be prohibited by the discrete symmetry
$H\to-H,\Phi\to-\Phi$ in our theory, gauge universality violation should be
revisited. A bound from the universality in $Z$ boson gauge couplings was
already calculated as $CL\lesssim 0.003$ (when a KK scale is around a few TeV)
in a model on an interval [46]. In this paper, we simply ignore this term.
The 5D action and the BCs for $\Phi$ are assumed to be of the form [46, 50]
$\displaystyle S_{\Phi}=\int
d^{4}x\int_{0}^{{L}}dy\bigg{\\{}{\Phi^{\dagger}\left(\partial_{M}\partial^{M}-{M_{\Phi}}^{2}\right)\Phi-\frac{\lambda_{\Phi}}{2}\left(\Phi^{\dagger}\Phi\right)^{2}}\bigg{\\}},$
(3.15) $\displaystyle\Phi+L_{+}\partial_{y}\Phi$
$\displaystyle=0\qquad\text{at}\quad y=L_{0}^{(\Phi)}+\varepsilon,$
$\displaystyle\Phi-L_{-}\partial_{y}\Phi$ $\displaystyle=0\qquad\text{at}\quad
y=L_{3}^{(\Phi)}-\varepsilon,$ (3.16)
where $M_{\Phi}$ ($\lambda_{\Phi}$) is the bulk mass (quartic coupling) of the
scalar singlet $\Phi$ and $L_{\pm}$ can take values in the range of
$-\infty\leq L_{\pm}\leq\infty$ and $L_{0}^{(\Phi)}$ and $L_{3}^{(\Phi)}$
indicate the locations of the two “end points” of the singlet.
The VEV of $\Phi$ with the BCs, named Robin BCs, in Eq. (3.16) is expressed in
terms of Jacobi’s elliptic functions in general and its phase structure has
been discussed in Ref [50]. We adopt a specific form in the region
$[L_{0}^{(\Phi)}+\varepsilon,L_{3}^{(\Phi)}-\varepsilon]$ [46],
$\displaystyle\left\langle\Phi(y)\right\rangle=\left[{\frac{M_{\Phi}}{\sqrt{\lambda_{\Phi}}}}\left\\{\sqrt{1+X}-1\right\\}^{1/2}\right]\times\frac{1}{\text{cn}\left({M_{\Phi}}\left\\{1+X\right\\}^{1/4}(y-y_{0}),\sqrt{\frac{1}{2}\left(1+\frac{1}{\sqrt{1+X}}\right)}\right)},$
(3.17)
with
$\displaystyle X:={\frac{4\lambda_{\Phi}|Q|}{M^{4}_{\Phi}}}.$ (3.18)
Here $y_{0}$ and $Q$ are parameters which appear after integration on $y$ and
we focus on the choice of $Q<0$. We note that the values of $y_{0}$ and $Q$
are automatically determined after choosing those of $L_{\pm}$. As shown in
Ref. [46], we get the form of $\left\langle\Phi(y)\right\rangle$ to be an
(almost) exponential function of $y$ by choosing suitable parameter
configurations. Although there is a discontinuity in the wave function profile
of $\langle\Phi\rangle$ between $y=L_{0}^{(\Phi)}+\varepsilon$ and
$y=L_{3}^{(\Phi)}-\varepsilon$ in Eqs. (3.16), this type of BC is derived from
the variational principle on $S^{1}$ and leads to no inconsistency [50].
The BCs for the 5D $SU(3)_{C},SU(2)_{W},U(1)_{Y}$ gauge bosons
$G_{M},W_{M},B_{M}$ are selected as
$\displaystyle{G_{M}|_{y=0}=G_{M}|_{y=L},\quad\partial_{y}G_{M}|_{y=0}=\partial_{y}G_{M}|_{y=L},}$
(3.19)
where we only show the $G_{M}$’s case. In this configuration, we obtain the
standard model gauge bosons in zero modes. Based on the discussion in Sec. 2,
we conclude that the W and Z bosons become massive and their masses are
suitably created through “our” Higgs mechanism as $m_{W}\simeq
81\,\text{GeV},m_{Z}\simeq 90\,\text{GeV}$. The overview of the BCs is
summarized in Fig. 2. We mention that, on $S^{1}$ geometry, $G^{(0)}_{y}$,
$W^{(0)}_{y}$, and $B^{(0)}_{y}$ would exist as massless 4D scalars at the
tree level, but they will become massive via quantum corrections and are
expected to be uplifted to near KK states. We will discuss those modes in
another paper. We should note that in our model on $S^{1}$ with point
interactions, the 5D gauge symmetries are intact under the BCs summarized in
Fig. 2.777 In Refs. [51, 52, 53], the 5D gauge invariance has been discussed
from a quantum mechanical supersymmetry point of view. Hence the unitarity in
the scattering processes of massive particles are ensured in our model.888
Some related works are found in Refs. [54, 55, 56, 57, 58, 59, 60, 61].
## 4 $CP$ phase in the CKM matrix
In this section, we verify that our mechanism can actually produce a
nontrivial $CP$ phase in the CKM matrix. We further would like to find a set
of parameter configurations in which the quark mass hierarchy and the
structure of the CKM matrix are derived naturally. In the following analysis,
we rescale all the dimensional valuables by the $S^{1}$ circumference $L$ to
make them dimensionless and the rescaled valuables are indicated with the
tilde $\tilde{{}}$ .
We set the parameters concerning the scalar singlet $\Phi$ as
$\displaystyle{\tilde{M}_{\Phi}}=8.67,\quad\tilde{y}_{0}=-0.1,\quad{\tilde{\lambda}_{\Phi}}=0.001,\quad|\tilde{Q}|=0.001,$
(4.1)
where the VEV profile becomes an (almost) exponential function of $y$, which
is suitable for generating the large mass hierarchy.999 The smallness of $Q$
is not an unnatural thing because they are resultant values derived from the
two input parameters $L_{\pm}$, for which the dimensionless values are within
$\mathcal{O}(10)$ as in Eq. (4.2). We note that $\lambda_{\Phi}$ always
appears in the form of the singlet VEV in Eq. (3.17) as the combination
$|Q|\lambda_{\Phi}$. $\lambda_{\Phi}$ in itself only affects the overall
normalization. Therefore some room might remain for a more “natural” choice of
$\lambda_{\Phi}$. In this case, the values of $L_{\pm}$ in Eq. (3.16)
correspond to
$\displaystyle\frac{1}{\tilde{L}_{+}}=-6.07,\quad\frac{1}{\tilde{L}_{-}}=8.69,$
(4.2)
where the broken phase is realized [46].
$M_{ij}^{(u)}$ | $a$ | $b$ | $M_{ij}^{(d)}$ | $a$ | $b$
---|---|---|---|---|---
$M_{11}^{(u)}$ | $L_{0}^{(u)}$ | $L_{1}^{(u)}$ | $M_{11}^{(d)}$ | $L_{0}^{(d)}$ | $L_{1}^{(d)}$
$M_{22}^{(u)}$ | $L_{1}^{(q)}$ | $L_{2}^{(u)}$ | $M_{22}^{(d)}$ | $L_{1}^{(q)}$ | $L_{2}^{(d)}$
$M_{33}^{(u)}$ | $L_{2}^{(q)}$ | $L$ | $M_{33}^{(d)}$ | $L_{2}^{(q)}$ | $L$
$M_{12}^{(u)}$ | $L_{1}^{(u)}$ | $L_{1}^{(q)}$ | $M_{12}^{(d)}$ | $L_{1}^{(d)}$ | $L_{1}^{(q)}$
$M_{23}^{(u)}$ | $L_{2}^{(u)}$ | $L_{2}^{(q)}$ | $M_{23}^{(d)}$ | $L_{2}^{(d)}$ | $L_{2}^{(q)}$
$M_{31}^{(u)}$ | $L$ | $L+L_{0}^{(u)}$ | $M_{31}^{(d)}$ | $L$ | $L+L_{0}^{(d)}$
Table 1: The summary table for the overlap integrals in Eq. (4.6).
As in the previous analysis [46], the signs of the fermion bulk masses are
assigned as $M_{Q}>0,M_{\mathcal{U}}<0,M_{\mathcal{D}}>0$ to make much larger
overlapping in the up-quark sector than in down ones for top mass. Here we
assume the positions of the two end points of both the quark doublet and the
scalar singlet are the same
$\displaystyle L_{0}^{(q)}=L_{0}^{(\Phi)}=0,\quad
L_{3}^{(q)}=L_{3}^{(\Phi)}=L,$ (4.3)
where we set $L_{0}^{(q)}$ and $L_{0}^{(\Phi)}$ as zero. In addition, we also
assume that the orders of the positions of point interactions are settled as
$\displaystyle
0<L_{0}^{(u)}<L_{1}^{(u)}<L_{1}^{(q)}<L_{2}^{(u)}<L_{2}^{(q)}<L<L_{3}^{(u)},$
$\displaystyle
0<L_{0}^{(d)}<L_{1}^{(d)}<L_{1}^{(q)}<L_{2}^{(d)}<L_{2}^{(q)}<L<L_{3}^{(d)}.$
(4.4)
Here our up-quark mass matrix ${\mathcal{M}}^{(u)}$ and that of down ones
${\mathcal{M}}^{(d)}$ take the forms
$\displaystyle\mathcal{M}^{(u)}=\begin{bmatrix}M^{(u)}_{11}&M^{(u)}_{12}&M^{(u)}_{13}\\\
0&M^{(u)}_{22}&M^{(u)}_{21}\\\
0&0&M^{(u)}_{33}\end{bmatrix},\quad\mathcal{M}^{(d)}=\begin{bmatrix}M^{(d)}_{11}&M^{(d)}_{12}&M^{(d)}_{13}\\\
0&M^{(d)}_{22}&M^{(d)}_{21}\\\ 0&0&M^{(d)}_{33}\end{bmatrix},$ (4.5)
where the row (column) index of the mass matrices shows the generations of the
left- (right-)handed fermions, respectively. Differently from the model on an
interval in Ref. [46], the $(1,3)$ elements of the mass matrices are allowed
geometrically due to the periodicity along the $y$ direction. The general form
of the nonzero matrix elements of ${\mathcal{M}}^{(u)}$ and
${\mathcal{M}}^{(d)}$ can be expressed as
$\displaystyle
M^{(\kappa)}_{ij}={\mathcal{Y}}^{(\kappa)}\int_{a}^{b}\\!\\!dyf_{q^{(0)}_{iL}}(y)f_{\kappa^{(0)}_{jR}}(y)\langle\Phi(y)\rangle{\langle
H(y)\rangle},$ (4.6)
where $\kappa$ indicates the up/down type of quark and the concrete
information is stored in Table 1.
### 4.1 Quark masses and mixing parameters
The parameters which we use for calculation are
$\begin{array}[]{llll}\tilde{L}_{0}^{(q)}=0,&\tilde{L}_{1}^{(q)}={0.30},&\tilde{L}_{2}^{(q)}={0.660},&\tilde{L}_{3}^{(q)}=1,\\\
\tilde{L}_{0}^{(u)}={0.024},&\tilde{L}_{1}^{(u)}={0.026},&\tilde{L}_{2}^{(u)}={0.52},&\tilde{L}_{3}^{(u)}={1.024},\\\
\tilde{L}_{0}^{(d)}={0.07},&\tilde{L}_{1}^{(d)}={0.18},&\tilde{L}_{2}^{(d)}={0.646},&\tilde{L}_{3}^{(d)}={1.07},\\\
\tilde{M}_{Q}={6},&\tilde{M}_{\mathcal{U}}={-6},&\tilde{M}_{\mathcal{D}}={5},&\theta={3},\end{array}$
(4.7)
where the twist angle $\theta$ is a dimensionless value and should be within
the range $-\pi<\theta\leq\pi$. We note that $\tilde{L}_{3}^{(q)}$,
$\tilde{L}_{3}^{(u)}$, and $\tilde{L}_{3}^{(d)}$ are considered to be not
independent degree of freedom, for which the values are automatically
determined after we choose the other positions of the point interactions. In
Appendix A, we will comment on the orders of significant digits of the input
parameters in Eq. (4.7). We should note that in our system, the EWSB is only
realized on the condition of ${M}^{2}-\left(\frac{\theta}{L}\right)^{2}>0$ as
in Eqs. (2.13). Recently, the ATLAS and CMS experiments have announced that
the physical Higgs mass is around $126\,\text{GeV}$ over $5\sigma$ confidence
level [62, 63]. $\tilde{\lambda}$ is $0.262$ irrespective of the value of $L$,
while $\tilde{M}$ is slightly dependent on the value of $L$ as $3.01303$
$(3.00052)$ in the case of $M_{\text{KK}}=2\,\text{TeV}$
($M_{\text{KK}}=10\,\text{TeV}$), where $M_{\text{KK}}$ is a typical scale of
the KK mode and defined as $2\pi/L$. Here some tuning is required to obtain
the suitable values realizing the EWSB.
After the diagonalization of the two mass matrices, the quark masses are
evaluated as
$\begin{array}[]{lll}m_{\text{up}}={2.5}\,\text{MeV},&m_{\text{charm}}={1.339}\,\text{GeV},&m_{\text{top}}={173.3}\,\text{GeV},\\\
m_{\text{down}}={4.8}\,\text{MeV},&m_{\text{strange}}={104}\,\text{MeV},&m_{\text{bottom}}={4.183}\,\text{GeV},\\\
\vspace{-4mm}\\\
\displaystyle\frac{m_{\text{up}}}{m_{\text{up}}|_{\text{exp.}}}={1.07},&\displaystyle\frac{m_{\text{charm}}}{m_{\text{charm}}|_{\text{exp.}}}={1.05},&\displaystyle\frac{m_{\text{top}}}{m_{\text{top}}|_{\text{exp.}}}=1.00,\\\
\vspace{-4mm}\\\
\displaystyle\frac{m_{\text{down}}}{m_{\text{down}}|_{\text{exp.}}}={0.993},&\displaystyle\frac{m_{\text{strange}}}{m_{\text{strange}}|_{\text{exp.}}}={1.10},&\displaystyle\frac{m_{\text{bottom}}}{m_{\text{bottom}}|_{\text{exp.}}}=1.00,\end{array}$
(4.8)
and the absolute values of the CKM matrix elements are given as101010 The
values of $\tilde{\mathcal{Y}}^{(u)}$ and $\tilde{\mathcal{Y}}^{(d)}$ are also
chosen as $\tilde{\mathcal{Y}}^{(u)}=-0.0532+0.0156\,i$ and
$\tilde{\mathcal{Y}}^{(d)}=-0.00335-0.00146\,i$ by setting the initial
conditions $M_{33}^{(u)}=m_{t}$ and $M_{33}^{(d)}=m_{b}$ with the top mass
$m_{t}$ and the bottom mass $m_{b}$.
$\displaystyle|V_{\text{CKM}}|=\begin{bmatrix}{0.971}&{0.238}&{0.00377}\\\
{0.237}&{0.971}&{0.0403}\\\
{0.00887}&{0.0395}&0.999\end{bmatrix},\quad\left|\frac{V_{\text{CKM}}}{V_{\text{CKM}}|_{\text{exp.}}}\right|=\begin{bmatrix}0.997&1.06&{1.07}\\\
1.06&0.998&{0.978}\\\ {1.02}&{0.978}&1.00\end{bmatrix}.$ (4.9)
### 4.2 $CP$ phase
The Jarlskog parameter $J$ containing information about the $CP$ phase is
defined by
$\displaystyle\text{Im}\left[(V_{\text{CKM}})_{ij}(V_{\text{CKM}})_{kl}(V_{\text{CKM}}^{\ast})_{il}(V_{\text{CKM}}^{\ast})_{kj}\right]=J\sum_{m,n=1}^{3}\epsilon_{ikm}\epsilon_{jln}$
(4.10)
with the completely antisymmetric tensor $\epsilon$, and is invariant under
the $U(1)$ unphysical rephasing operations of six types of quarks [64, 65].
This value is easily estimated as
$\displaystyle J={3.23}\times 10^{-5},\quad\frac{J}{J|_{\text{exp.}}}={1.09},$
(4.11)
where we also provide the differences from the latest experimental values in
Ref. [66]. All the deviations from the latest experimental values are within
about $10\%$, and we can conclude that the situation of the SM is suitably
generated. In Appendix A, we discuss distribution patterns of quark mass-
matrix elements and required orders in tuning the input parameters with the
results for realizing the accuracy.
## 5 Summary and discussion
In this paper, we have proposed a new mechanism for generating a $CP$ phase
via a Higgs VEV originating from the geometry of an extra dimension. A twisted
BC for the Higgs doublet has been found to lead to an extra-dimension
coordinate-dependent phase in the Higgs VEV, which contains a nontrivial $CP$
phase degree of freedom. This mechanism is useful for generating a $CP$ phase
to a single generation extra-dimensional field theory incorporating with a
generation production mechanism. The electroweak symmetry breaking is also
generated dynamically due to the twisted boundary condition with the suitable
W- and Z-boson masses.
As an illustrative example, we applied our mechanism to a five-dimensional
gauge theory on a circle with point interactions [46]. Point interactions,
which are additional boundary points with respect to the extra dimension, are
responsible for producing the three generations while the model consists of a
single-generation fermion and make the quark profiles be localized. Since each
element of the mass matrices picks up a different phase through the overlap
integrals, there is some possibility of realizing a nontrivial $CP$ phase.
After numerical calculations, we found that a nontrivial $CP$ phase appears
with good precision, maintaining the property of the original model in which
the generations, the quark mass hierarchy and the CKM matrix appear from the
geometry of the extra dimension. Certainly, our new mechanism for generating a
$CP$ phase via Higgs VEV works.
A key point of our mechanism is that we can generate both the EWSB and a $CP$
phase simultaneously as a complex Higgs VEV. To make our $CP$-violation
mechanism work correctly, quark profiles should be split and localized. In
this situation, flavor mixing and mass hierarchy of the quarks are also
naturally activated. We would like to emphasize that in the model adopting our
mechanism, all the concepts of quark flavor in the SM, namely EWSB, the number
of generations, flavor mixing, mass hierarchy, and $CP$ violation, are
interlinked closely.
One of the most important remaining tasks is to construct a model which brings
both the quarks and the leptons into perspective. Using our mechanism, not
only the quark sector but also the lepton sector can acquire a nontrivial $CP$
phase. Since the origin of the $CP$ phase is common, we can predict the value
of the $CP$ phase of the lepton sector after fitting the value of the $CP$
phase in the quark sector. The result will be reported elsewhere.
Accommodation of our mechanism to another single generation model is also an
important task.
Another crucial topic is the stability of the system. Our system is possibly
threatened with instability. Some mechanisms will be required to stabilize the
moduli representing the positions of point interactions (branes).111111 Moduli
stabilization via Casimir energy in the system where a scalar takes the Robin
BCs (but no point interaction in the bulk) has been studied in Refs.[67, 68,
69]. In a multiply connected space of $S^{1}$, there is another origin of
gauge symmetry breaking, i.e., the Hosotani mechanism [70, 71]. Since further
gauge symmetry breaking causes a problem in the model, we need to insure that
the Hosotani mechanism does not occur. To this end, we might introduce
additional 5D matter to prevent zero modes of $y$ components of gauge fields
from acquiring nonvanishing VEVs. We will leave those issues in future work.
## Acknowledgments
The authors would like to thank T.Kugo for valuable discussions. K.N. is
partially supported by funding available from the Department of Atomic Energy,
Government of India for the Regional Centre for Accelerator-based Particle
Physics (RECAPP), Harish-Chandra Research Institute. This work is supported in
part by a Grant-in-Aid for Scientific Research [Grants No. 22540281 and No.
20540274 (M.S.)] from the Japanese Ministry of Education, Science, Sports and
Culture.
## Appendix
## Appendix A Considering input-parameter dependence
Figure 3: The left scatter plot shows the distribution of
$M_{33}^{(u)}-M_{13}^{(u)}$ when we choose 100,000 points randomly around the
configuration in Eq.(4.7) within $\pm 10\%$ being consistent with the order in
Eq. (4.4). The right one represents the same thing when we pick up 100,000
points randomly only with following the order in Eq. (4.4).
Figure 4: From left to right, distributions of
$m_{\text{up}}-m_{\text{down}}$, $m_{\text{charm}}-m_{\text{strange}}$ and
$m_{\text{top}}-m_{\text{bottom}}$ with 10,000 random points within $\pm 20\%$
parameter deviations from the central values in Eq. (4.7 following the order
in Eq. (4.4). The green (yellow) band in the left and center plots represents
$\pm 30\%$ range from the central values in Eq. (4.7). In the right plot, we
skip depicting the bands because all the shown ranges are covered by them.
Figure 5: From left to right, distributions of
$\sin^{2}{\theta_{12}}-\sin^{2}{\theta_{23}}$ and $\sin^{2}{\theta_{13}}-J$
with 10,000 random points within $\pm 20\%$ parameter deviations from the
central values in Eq. (4.7) following the order in Eq. (4.4). The allowed
region is just near the point $(0,0)$ in both the plots.
In this appendix, we discuss distribution patterns of quark mass-matrix
elements and required orders in tuning the input parameters with the results
given in Sec. 4.1 and 4.2.
We first focus on the matrices in Eq. (4.5). In our model, the geometry of the
extra dimension strongly restricts the form of the matrices. In fact, we
cannot fill all the elements of the mass matrices and at least three of the
nine elements for each mass matrix have to be zero, as shown in Eq. (4.5).
This property is contrasted with that of the standard model, where all the
mass matrix elements are free parameters. This fact means that possible
patterns of mass matrices are constrained by the shape of the geometry of our
model.
Furthermore, it turns out that the values of the nonzero elements in the mass
matrices (4.5) cannot be controlled freely. To see this, we investigated
correlations of matrix elements. In the left figure of Fig. 3, we chose
100,000 points randomly around the configuration in Eq.(4.7) within $\pm 10\%$
being consistent with the order in Eq. (4.4) and depicted the resultant values
as two scatter plots ($M_{33}^{(u)}-M_{13}^{(u)}$). The right figure of Fig. 3
shows the same thing when we pick up 100,000 points randomly only with
following the order in Eq. (4.4).
It follows from Fig. 3 that we find no random distribution in the
$M^{(u)}_{33}-M^{(u)}_{13}$ plane and a strong correlation between
$M^{(u)}_{13}$ and $M^{(u)}_{33}$. We further see the property that the
typical value of $M^{(u)}_{13}$ is much smaller than that of $M^{(u)}_{33}$.
In the quark sector, the mixing angles are known to be small, so that off-
diagonal elements of the mass matrices will be preferred to be subleading
compared with the diagonal ones with suitable magnitudes. Our geometry
realizes this point naturally via its geometry.
From the above observations, we may conclude that the quark mass matrices in
our model are considerably restricted from the geometry of the extra
dimension, and hence that it is nontrivial to reproduce the quark-related
properties of the standard model, although we have 16 input parameters for
quark profiles, where 13 parameters are independent, to explain the 10
standard model parameters (6 quarks masses and 4 CKM parameters).
As another consideration, we investigate the resultant quark masses and CKM
matrix elements when we change the input parameters around the central values
in Eq. (4.7) and then find that the quark masses and CKM matrix elements are
very sensitive to some of the input parameters.
To show the sensitivity of the input parameters, we first alter all the
parameters randomly within $10\%$, $1\%$, $0.1\%$, and $0.01\%$, respectively,
obeying the order in Eq. (4.4), and calculate the masses and the elements.
When we proceed with the above procedure 100,000 times in each case, 11,
11,039, 81,955, and 100,000 points survive after putting the cut where all the
resultants are within $15\%$. These results indicate that parameter tuning
less than $1\%$ is required as our inputs are so in Eq. (4.7). The scatter
plots in Figs. 4 and 5 represent the distributions of the physical parameters
with 10,000 random points within $\pm 20\%$ around the central values, where
the CKM angles and the Jarlskog parameter are apt to getting away from the
required range easily, for which the experimental central values are
$\sin^{2}{\theta_{12}}\sim 0.05$, $\sin^{2}{\theta_{23}}\sim 0.002$,
$\sin^{2}{\theta_{13}}\sim 0.00001$, and $J\sim 0.00003$, respectively. This
issue is explained by the fact that off-diagonal elements of the CKM matrix
are closely related to those of the up- and down-quark mass matrices, which at
least parts of them are, sensitive to perturbation of the input parameters. We
can also find the tendency that bottom and top masses are not away from the
central values by the perturbation.
On the other hand, we try to alter an input parameter separately. In each
case, points of the numbers in Table 2 pass the cut which rejects the
possibilities that at least one resultant value is out of the $\pm 15\%$
deviation range from the central value in Eq. (4.7). According to the result,
we understand that quark masses and mixing angles are sensitive to the
positions of point interactions, while those are insensitive to (absolute
values of) the bulk masses and the twisted angle. Then we can conclude that
parameter tuning in $M_{Q}$, $M_{\mathcal{U}}$, $M_{\mathcal{D}}$, and
$\theta$ in Eq. (4.7) is not always necessary.
Finally, we briefly comment on the required orders of significant digits in
the input parameters. As we have discussed before based on Table 2, the system
is insensitive to $M_{Q}$, $M_{\mathcal{U}}$, $M_{\mathcal{D}}$, and $\theta$
around the central region of the parameters, and then single digits are
sufficient for them. On the other hand, for $L_{2}^{(q)}$ and $L_{2}^{(d)}$,
as also expressed in Table 2, triple digits are required because of their
great sensitivity. For the other values, tuning up to double digits is enough
for our purpose since they are less sensitive than $L_{2}^{(q)}$ and
$L_{2}^{(d)}$ as shown in Table 2.
parameter | $L_{1}^{(q)}$ | $L_{2}^{(q)}$ | $L_{0}^{(u)}$ | $L_{1}^{(u)}$ | $L_{2}^{(u)}$ | $L_{0}^{(d)}$ | $L_{1}^{(d)}$ | $L_{2}^{(d)}$ | $M_{Q}$ | $M_{\mathcal{U}}$ | $M_{\mathcal{D}}$ | $\theta$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\\#$ of surviving points | 105 | 11 | 46 | 85 | 42 | 270 | 172 | 10 | 679 | 730 | 372 | 971
Table 2: Numbers of surviving points out of 1,000 ones after the cut where
all the physical resultants are within $\pm 15\%$ with changing an input
parameter within $\pm 10\%$ from the central values in Eq. (4.7) individually.
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|
arxiv-papers
| 2013-01-30T15:18:13 |
2024-09-04T02:49:40.998108
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yukihiro Fujimoto, Kenji Nishiwaki, and Makoto Sakamoto",
"submitter": "Kenji Nishiwaki",
"url": "https://arxiv.org/abs/1301.7253"
}
|
1301.7294
|
# Dismantling Hubble’s Legacy?
Michael J. Waya,b aNASA/Goddard Institute for Space Studies, New York, NY,
10029 USA bDepartment of Astronomy, Uppsala University, Uppsala, Sweden
###### Abstract
Edwin Hubble is famous for a number of discoveries that are well known to
amateur and professional astronomers, students and the general public. The
origins of these discoveries are examined and it is demonstrated that, in each
case, a great deal of supporting evidence was already in place. In some cases
the discoveries had either already been made, or competing versions were not
adopted for complex scientific and sociological reasons.
## 1 Introduction
Edwin Hubble is considered one of the titans of early 20th century
observational cosmology. He is credited in most textbooks111See Smith (2009,
p. 98). and the popular literature for a series of important discoveries made
between 1920 and 1930:
* •
The confirmation of the Island Universe hypothesis
* •
The classification of extragalactic nebulae
* •
The discovery of a linear relationship between distance and velocity for
extragalactic nebulae, providing the first evidence for the expanding universe
* •
The brightness profile of galaxies
The discoveries above are well-known to the astronomy community and most
astronomers would associate them solely with Edwin Hubble; yet this is a gross
oversimplification. Astronomers and historians are beginning to revise that
standard story and bring a more nuanced version to the public’s attention.
This paper is adding to this burgeoning reappraisal.222Some examples include
Nussbaumer & Bieri (2009); Kragh & Smith (2003); Bartusiak (2010).
As a (small) counter-narrative, William Hoyt (1980, p. 411), in his
biographical memoir of V. M. Slipher exclaims that “[Slipher] probably made
more fundamental discoveries than any other observational astronomer of the
twentieth century.”333Also see Hall (1970). Clearly some historians in the
1970s and 1980s thought that Slipher made more fundamental discoveries than
Hubble.444See contributions in this book by John Peacock, Joseph S. Tenn,
Robert Smith, Laird Thompson and Kevin Schindler for more on Slipher’s
discoveries. Yet how can that be true given all we _know_ today? In this paper
we re-examine Hubble’s discoveries in some detail in order to see if they are
better understood in a broader context. Given the focus on V. M. Slipher at
this conference we will also explicitly discuss his contributions in two of
the cases above.
## 2 Discovery of the Island Universe
The hypothesis of Island Universes has a long history going back at least to
the 18th Century with contributions by: Swedenborg (1734); Wright (1750); Kant
(1755)555Kant actually cited the work of Thomas Wright (1750)., and Lambert
(1761).666See contribution by Ayala in this book. William Herschel (1785)
initially believed that the spiral nebulae were external to the Milky Way, but
later changed his mind. Knut Lundmark (1927e, Chapter 1) does an excellent job
of explaining the origins of the Island Universe that I do not believe has
been much bettered by time.777The essence of the hypothesis, in an early 20th
century context, was that the universe was populated by many Milky Way
galaxies known then as spiral nebulae. This was opposed to the belief that the
universe consisted of a single Milky Way object with satellites such as spiral
nebulae, globular clusters and Magellanic cloud-like objects.
To get from philosophical speculation to modern quantification one must fast-
forward to the late 19th and early 20th century to find a large number of
investigations of objects termed “Nebulae” with the new art of photography and
ever larger telescopes.888See Gingerich (1987) for more on this early period.
For example, Huggins & Miller (1864) were deeply interested in the spectra of
nebulae, while astronomers such as Isaac Roberts (1903) built photographic
catalogs.999Only later was Roberts’ catalog compiled and completed by his wife
Mrs. Isaac Roberts. Using the catalogs of nebulae like that of Roberts’ and
others Lundmark (1925, pg.869) claimed that nearly 1200 spiral nebulae proper
motions had been measured at that time. We now know this claim was incorrect –
most likely it was an incorrect assessment of observational errors.101010Only
in 2012 was the proper motion of M31 possibly measured by Sohn et al. (2012)
using optical observations. The field was clearly in its infancy, but progress
on distance estimates to objects like globular clusters and spiral nebulae was
moving rapidly forward.
Table 1 lists all of the main distance estimates to spiral nebulae (known to
this author) from the late 1800s until 1930 when standard candles began to be
found in spiral nebulae.
Table 1.: Early distance estimates to Spiral Nebulae Reference | Object | Distancea | Method
---|---|---|---
Herschel (1786) | M31 | $<$17,200 b | color/magnitude
Nichol (1850) | “cluster” | 154,800c | magnitude comparison
– | | 302,505 | –
Clark (1890) | M31 | 564? | nova of 1885
Clark (1903) | M31 | $<$1000 | Size
Bohlin (1907) | M31 | 19 | parallax
Very (1911) | M31 | 4,000 | diameters
Very (1911) | M31 | 1,600 | S Andromedae
Wolf (1912) | M31d | 32,000 | diameters
Curtis (1915b) | spirals | 10,000 | astrometry/radial velocity
Pease (1916) | NGC 4594 | 25,000 | astrometry/radial velocity
Curtis (1917) | M31 | 20,000,000 | novae
– | – | 100,000 | novaee
Shapley (1917) | M31 | 1,000,000 | “bright stars”
van Maanen (1918) | M31 | 250 | parallax
Lundmark (1919) | M31 | 650,000 | novae
Curtis (1920) | misc | 4,000,000 | novae
– | misc | 1,000,000 | novae
– | misc | 500,000 | novae
Lundmark (1921h) | M33 | 1,000,000 | “bright stars”
Luplau-Janssen & Haarh (1922) | M31 | 326,000 | novaef
Öpik (1922) | M31 | 1,500,000 | luminosity/mass
Hubble (1922d) | M33 | 100,000 | “stars”
Shapley (1923) | NGC 6822 | 1,000,000 | diameters/“bright stars”
Hubble (1925a) | M31/33 | 930,000 | Cepheids
Hubble (1925c) | NGC 6822 | 700,000 | Cepheids,“bright-stars”
Lundmark (1925) | M31,M87 | 1,400,000 | novae
– | – | 8,000,000 | novae
Lundmark (1925) | M104 | 56,000,000 | Öpik (1922) method
Hubble (1926a) | M33 | 850,000 | Cepheids,Blue-Giants
Hubble (1929c) | M31 | 900,000 | Cepheids,novae
M31 value (Dec. 2012)g | M31 | 2,588,440 | 19 Methods
The ability to estimate _accurate_ distances of objects beyond the reach of
parallax only came into being with the publishing of the period–luminosity
relationship for Cepheid Variable stars by Henrietta Leavitt & Edward
Pickering (1912) and its later calibration by Ejnar Hertzsprung (1913); Henry
Norris Russell (1913)111111Using 13 Cepheids and later Harlow Shapley
(1918)121212Using 11 Cepheids of Hertzsprung’s original 13. all utilizing the
Lewis Boss (1910) catalog of proper motions.131313Although Shapley’s distances
were strongly contested by a number of people including Curtis (1921). Before
Cepheids were discovered in spiral nebulae there were a number of attempts to
use novae as standard candles to measure the distances to spiral nebulae
(Curtis 1917; Shapley 1917; Lundmark 1919; Luplau-Janssen & Haarh 1922). For
example, Heber Curtis (1917) calculated an average distance to the spiral
nebulae of 20,000,000 light years in one case and found them to be around 100
times as distant as the galactic novae in another. Lundmark obtained a
distance to Andromeda of 650,000 light years. Shapley (1918) attempted to
compare the “brightest stars” in our own galaxy to that of Andromeda and
stated:
> … the minimum distance of the Andromeda Nebula must be of order a million
> light years. At that remote distance the diameter of this largest of spirals
> would be about 50,000 light years a value that now appears most probable as
> a minimum for our galactic system.141414This is ironic given his later
> disavowals of his larger distance estimates and those related to novae in
> spiral nebulae (Shapley 1919b).
Initially the novae studies in Andromeda were difficult to reconcile with a
supernovae observed in Andromeda $\sim$35 years previously (Krueger et al.
1885; Hartwig 1885a, b; de Vaucouleurs & Corwin 1985)151515This supernova was
later denoted ‘S Andromedae’. At that time supernovae were unknown so they
were easily confused with normal novae., but given the multiple observations
of fainter novae observed in spiral nebulae Curtis (and later others) was
persuaded to drop S Andromedae and Z Centauri161616Z Centauri (in NGC 5253)
was another bright nova observed by Pickering (1895). as anomalies.
Unfortunately for the novae derived distance measurements, other observations
at that time called their accuracy into question. In the mid–1910s Curtis
(1915a) and then Lampland (1916) detected rotation in spiral nebulae, but the
former did not believe his own detection while the latter’s results were not
influential (Smith 1982, p. 31). However, additional observations of this sort
by van Maanen et al. (1916) in Messier 101 (M101) and later in Messier 33
(M33) and other nebulae (van Maanen 1923) along with the support of James
Jeans (1917) convinced many astronomers like Shapley (1919a) that novae
derived distances to spiral nebulae were impossible to reconcile without
superluminal speeds of spiral nebulae rotation.171717This was also the time
when Shapley came up with his 300,000 light year diameter Milky Way galaxy,
much larger ($>$ 30 times) than any other estimate at that time.
However, in spite of the confusing novae observations and (incorrectly)
observed rotation of spirals the evidence continued to mount that the spiral
nebulae were indeed very distant objects. The first evidence of this was a
fascinating paper by Öpik (1922) where “an expression is derived for the
absolute distance in terms of the linear speed $v_{o}$ at an angular distance
$\rho$ from the center, the apparent luminosity $i$, and $E$, the energy
radiated per unit mass.” He calculated a distance of 1.5 million light years
to M31. One year later Shapley (1923) used diameters of galaxies and the
brightness of super-giant stars in NGC 6822 to state:
> The above considerations all indicate that the distance of N.G.C. 6822 is of
> the order of a million light years. It appears to be a great star cloud that
> is at least three or four times as far away as the most distant of known
> globular clusters and probably quite beyond the limits of the galactic
> system.
This quote is particularly interesting in light of Shapley’s long standing
opposition to the Island Universe hypothesis and his super-galaxy model
(Shapley 1921), but he still would not let go of his super-galaxy model just
yet.
The issue was effectively settled by two papers from Hubble in 1925 in which
he derived distances from Cepheid variables found in M31 and M33 (Hubble
1925a) of 930,000 light years181818The original Cepheid that Hubble discovered
in M31 now has a modern ephemeris and light curve published by Templeton et
al. (2011). and in NGC 6822 (Hubble 1925c) of 700,000 light years. Note that
there were no citations to previous distance estimates in the former paper and
only a reference to Shapley (1918) for his calibration of the Cepheid variable
light curves in the latter.
Still, there was some confusion about van Maanen’s spiral nebulae observations
among his contemporaries including Knut Lundmark. Lundmark (1922c) was at
first dismissive of van Maanen’s measurements because of a preponderance of
conflicting data. However, by 1922 he had changed his mind after having
measured some of van Maanen’s plates himself (Lundmark 1927e).191919See page
17 where he states that [in 1922?] “When remeasuring Messier 33 during my stay
at Mount Wilson the situation seemed to be rather hopeless for the followers
of the island–universe theory.” Upon re-measuring the motions in M33 a couple
of years later, Lundmark concluded that van Maanen’s measurements were flawed
(Lundmark 1926c, 1927e). However, it would not be until 1935 that van Maanen
(1935) would nearly admit that his measurements of the rotation of spiral
nebulae were false. In the same issue of _The Astrophysical Journal_ Hubble
(1935) published his own measurements showing any measured rotation to be
within the measurement errors. Clearly Hubble wanted to make sure that the
persistent observations of van Maanen were dismissed by publishing his own
measurements given his (now) elevated status as one of the more highly
respected astronomers of his day.202020Christianson (1996, Chapter 11)
explains what happened in more detail. Thus it took nearly a decade after
Hubble’s 1925 paper for van Maanen’s measurements to be completely disposed of
and the Island Universe theory to be confirmed. Of course many astronomers
felt the matter had been settled all the way back in 1926.212121As mentioned
above (Lundmark 1926c) but also Luyten (1926): “It is now universally accepted
that the spiral nebula are millions of light years distant.”
Still, as mentioned by Robert Smith (2008, p. 114) “…what was missing to
settle the dispute on the spirals was a method of calculating their distances
that a great majority of astronomers could agree was accurate.” Clearly Hubble
(1925a) provided that method with his observations of Cepheids in spirals, but
a great many people before him made his observations possible. As Smith (2009,
p. 74) points out “… it is appropriate to view Hubble as confirming rather
than discovering the extragalactic nature of spirals. But, following the
dictum of John Herschel that he who proves discovers, Hubble was given the
credit.” Many important contributions to this story have been forgotten and
most textbooks in astronomy today, if they discuss the “Island Universe”
confirmation at all, bestow 100% of the credit on Hubble with scant attention
to the earlier observations that clearly supported his measurements.
At that time the use of Cepheids as standard candles was considered a very
reliable method of estimating distance. However, by the mid-1950s it had
become clear that Hubble’s distances measurements contained significant
systematic errors. Recalibration of the Cepheids by Walter Baade (1956) later
helped to show that Andromeda was twice as far away, and was actually larger
than our own Milky Way. This would have serious implications for the Big Bang
theory in the 1930s and 1940s.
### 2.1 Slipher’s Contribution to the Island Universe story
The anniversary date for this conference was intended to overlap with the
published date of Slipher’s first observation of a doppler shift in a spiral
nebula (Andromeda) on 17 September 1912 (Slipher 1913). He obtained an
astounding value of –384 km s-1. This was surprising because it was nearly an
order-of-magnitude higher than any other measured doppler shift in the heavens
at that time. By 1917 Slipher had observed 25 spiral nebulae, the largest
having a redshift of 1100 km s-1 (Slipher 1917). As we have seen above the
debate on whether spiral nebulae were Island Universes went on until Hubble
discovered Cepheids in Andromeda and other spiral nebulae. Given the 25 spiral
nebulae with radial velocities discovered by Slipher in 1917 (21 of which were
redshifts) why didn’t the astronomical community realize these objects could
not be bound to the Milky Way and must be Island Universes? In fact a number
of people did reach this conclusion including Campbell (1917) and Hertzsprung
(see Robert Smith’s chapter in this book). Still, it took several years for
Slipher to convince the community that what he was observing was real. As well
the reticence to push this interpretation by Slipher himself was related to
his modest personality.222222See Section 7 and Robert Smith’s chapter in this
book. In reality what made it difficult (and would make it difficult for _all_
proponents of an Island Universe theory) were the observations of internal
motion in Spiral Nebulae by Adrian van Maanen that began with his first
publication on the subject in July of 1916 (van Maanen 1916).232323See David
DeVorkin’s contribution to this proceedings and the first chapter in Smith
(1982). Had it not been for the erroneous observations of van Maanen it is
likely that Slipher’s observations would have provided strong support for the
Island Universe theory.
## 3 Classification of Extra-galactic Nebulae
The first systematic classification of nebulae was probably attempted by
Herschel (1786) in his paper titled “Catalogue of One Thousand New Nebulae and
Clusters of Stars.” Therein he described eight different classes of objects:
1. 1.
Bright nebulae [93 examples]
2. 2.
Faint nebulae [402]
3. 3.
Very faint nebulae [376]
4. 4.
Planetary nebulae [29]
5. 5.
Very large nebulae [24]
6. 6.
Very compressed and rich clusters of stars [19]
7. 7.
Pretty much compressed clusters of large or small stars [17]
8. 8.
Coarsely scattered clusters of stars [40]
His descriptions of the nebulae were extremely detailed using terms with
single letter abbreviations, for example: B. Bright, S. Small, v. very, e.
extremely, R. Round, M. in the middle, l. a little, g. gradually, r.
resolvable, m. milky. These were used in combinations, one of his own examples
being vgmbM: (v)ery (g)radually (m)uch (b)righter in the (M)iddle.
Later Lord Rosse (1850) gave the term spiral to some of Herschel’s nebulae by
using his new 1.8m telescope “Leviathon of Parsonstown,” but he described it
first via a drawing of M51 presented to the 15th meeting of the British
Association for the Advancement of Science (Rosse 1845; Hoskin 1982; Dewhirst
& Hoskin 1991).
Table 2.: Early Classification schemes for Extragalactic–Nebulae Reference | Notes
---|---
Herschel (1786) | “first comprehensive scheme?”
Rosse (1850) | terminology “Spirals” used
Wolf (1908) | “widely cited scheme”
Bailey (1908) | –
Pahlen (1911) | –
Bigourdan (1914) | –
Shaw (1915) | –
Curtis et al. (1918) | “bars”
Curtis (1919) | –
Jeans (1919) | Theoretical
Reynolds (1920) | Classification of spirals like Hubble (1922a)
Hubble (1922a) | Preliminary scheme
Lundmark (1926a) | Preliminary scheme
Hubble (1926b) | More complete scheme
Lundmark (1927e) | Full scheme
Shapley (1927) | –
Jeans (1928) | Tuning-fork diagram suggestion
(Hubble 1936) | Tuning fork diagram added to create complete scheme
The classification scheme of Herschel (1786) (with later modifications by son
John Herschel) was considered unwieldy and complicated, but was probably the
only one referred to consistently until new schemes in the early 20th century
such as that of Wolf (1908).242424The classification scheme of Wolf (1908) is
mentioned in a number of articles, but without a proper citation. The UGC
catalog of Nilson (1973, p. 452) cites the year 1909: “The well-known
classification system for nebulae, the Wolf code a-w with 23 standard objects,
was presented for the first time in 1909.” It was actually 1908. Wolf’s
classification scheme worked for all types of nebulae. He not only lists
specific examples of each but also includes a table of images. He labeled them
with letters ‘a–w’ (there is no letter ‘j’, but rather an ‘h’ and ‘ho’).
Another interesting scheme was developed by Bigourdan (1914).
Today Hubble (1926b) is generally given credit for coming up with the first
usable classification scheme of “Galaxies”, or as they came to be known
“Extra-galactic nebulae”. In fact the Table in his 1926 paper is titled
“Classification of Nebulae” which included both “Galactic nebulae” and “Extra-
galactic nebulae” which is an extension of his earlier work (Hubble 1922a).
Using his 1922 work as a basis, he had tried to build his Extra-galactic
nebulae classification scheme in-line with the nebular evolutionary model of
Jeans (1919).
One of the great strengths of the Hubble (1926b) paper was his formula that
described the spiral divisions:
$m_{t}=C-5log(d)$ (1)
where $m_{t}$$=$total magnitude, $d=$diameter of the nebulae, $C=$Constant
describing each object in his sequence (1–3):
Sa(1),Sb(2),Sc(3),SBa(1),SBb(2),SBc(3) where 1$=$Early, 2$=$Intermediate,
3$=$Late. Hence as a nebulae aged from “Early” to “Late” the diameter and
luminosity would change accordingly and (again) in-line with the theoretical
work of Jeans (1919). However, other models at that time (Lindblad 1927)
contradicted some aspects of Jean’s evolutionary sequence (Jeans
1919).252525In particular Lindblad (1927) states, “We do not assume a general
development from less flattened to more flattened system of higher angular
speed of rotation [like that of Jeans].” Lindblad also believed that the
centers of the the nebulae were simply unresolved faint stars – contrary to
Jeans.
David Block & Ken Freeman (2008) appear to most recently describe how Hubble’s
entire “Extra-galactic nebulae” classification scheme was remarkably similar
to one developed by John Reynolds (1920). In fact they present clear evidence
that Hubble (at this time) was aware of the Reynolds (1920) paper via an
unpublished memo written to Reynolds, which they reproduce in their book.
However, Hubble (1922a, 1926b) does not cite Reynolds (1920) in these papers,
although he does give credit to Curtis et al. (1918) for the recognition of
bars in spiral nebulae.
A year after he introduced his 1926 classification scheme, Hubble (1927f)
mentioned a paper from earlier in 1927 in which Reynolds (1927) criticizes
Hubble’s published classification scheme of 1926. Hubble again omits mention
of Reynolds (1920) while Reynolds not only mentions Hubble’s work, but an even
earlier classification scheme by Shaw (1915).
What makes Hubble’s omission of Reynolds (1920) particularly troubling is that
he accused Lundmark of plagarism not only in personal correspondence262626He
asked Lundmark to explain himself and threatened to publish his accusation
(Holmberg 1999, p. 103)., but also on page 3 of his 53-page classification
scheme article (Hubble 1926b):272727It was published a few months after
(Lundmark 1926a).
> Meanwhile K. Lundmark, who was present at the Cambridge meeting and has
> since been appointed a member of the Commission, has recently published
> (Arkiv för Matematik, Astronomi och Fysik, Band 19B, No.8, 1926) a
> classification, which, except for nomenclature, is practically identical
> with that submitted by me. Dr. Lundmark makes no acknowledgments or
> references to the discussions of the Commission other than those for the use
> of the term ‘galactic’.
This is rather remarkable because in no paper published by Hubble between 1920
and 1930 is the classification scheme of Reynolds (1920) mentioned. Lundmark
(1927e) strongly rebutted Hubble on page 24 of his 127-page paper titled
“Studies of Anagalactic Nebulae.”282828See Appendix C for Lundmark’s full
reply. The latter denotes the classification scheme of Wolf in one of the
columns292929No citation is provided, but he must be referring to Wolf (1908).
next to his own, but also mentions other work that preceded his (page 23):
> Classifications of nebulae based on photographic material have been made by
> Bailey, Curtis, Mrs. Isaac Roberts, Max Wolf, Hubble and others.
We do not have the space here to delve deeply into the personalities of Hubble
or Lundmark, yet we may get some feeling for what their contemporaries felt
about them and how they felt about their contemporaries via the limited notes
placed in papers and in their personal correspondence. Some of the latter can
be found in Smith (1982), while some specific examples in the case of Hubble
are described in Christianson (1996, Chapter 11).303030A particular quote from
Walter Adams, then the director at Mt. Wilson, is worth repeating in reference
to the conflict between Hubble and van Maanen over the distances to the
spirals: “This is not the first case in which Hubble has seriously injured
himself in the opinion of scientific men by the intemperate and intolerant way
in which he has expressed himself.” This was in reference to the conflict
between Hubble and van Maanen that Adams had to negotiate as director.
On the other hand Lundmark has been called enigmatic by Smith (1982), but at
least some of his contemporaries appreciated his general attitude. Take this
quote from Ludwik Silberstein (1925):
> … I should like to express my deep gratitude to Dr. Lundmark for having
> devoted so much attention to the discussion of this problem from a perfectly
> impartial attitude.
This was in reply to a paper by Lundmark (1924b) that critized Silberstein
(1924d) for his use of globular clusters (GCs) to determine the curvature
radius of the Universe. Lundmark felt that GCs were not distant
enough.313131It may be amusing to note that six years later Lundmark (1930d)
asked whether the GCs and Extragalactic (he used the word Anagalactic) nebulae
were related. Holmberg (1999) in his Chapter titled “Lundmark and the Lund
Observatory” also paints a picture of a complex character, but whose
bitterness towards some of his Swedish colleagues appeared to surface later in
his career when his scientific productivity was waning.
Finally, it is clear that Lundmark had been thinking of a classification
scheme for nebulae at least since 1922 (Teerikorpi 1989) and even discussed a
simplified “class of objects” in Lundmark (1925). Lundmark’s scheme for
“Anagalactic nebulae”323232His name for “Extra-galactic nebulae.” was broken
into 4 groups (see page 22 of Lundmark (1927e)): 1.) Anomalous nebulae (Aa),
2.) Globular, elliptical, elongated, ovate or lenticular nebulae (Ae), 3.)
Magellanic (“irregular”) nebulae (Am), 4.) Spiral nebulae (As), where the
degree of condensation toward the center was his main criteria. On the other
hand Hubble (1926b) separated his ellipticals by eccentricity. Spirals were
separated based on form and degree of arm development. In fact the Lundmark
and Hubble schemes were not considered the same by their
contemporaries.333333See Smith (1982, p. 152).
A year later Shapley (1927) joined in the classification attempts with a model
that incorporated aspects of the work of both Lundmark and Hubble, but his
model was not adopted. Contrary to Hubble, Shapley carefully cited his
predecessors including Bailey (1908); Reynolds (1920); Wolf (1908); Hubble
(1922a); Lundmark (1926a); Hubble (1926b).
Hubble’s classification scheme is also noted for its later tuning fork design
to separate the barred spirals from non–barred ones (Hubble 1936). Block et
al. (2004) have pointed out that Hubble was not the first to describe the
tuning fork diagram – that was originally proposed by Jeans (1928).
It is generally acknowledged that Hubble’s classification scheme became
standard because it had an evolutionary component and mathematical description
(see Equation 1) that previous schemes did not. But as should be clear from
Table 2 there were a great many classification schemes leading up to that of
Hubble’s which surely influenced him. Is it troubling that Hubble does not
readily cite two of the most important and influential schemes (before his own
was published) of Reynolds (1920) and Wolf (1908) and yet accuses a
contemporary (Lundmark) of plagarism on the basis of scant evidence? This lack
of citation by Hubble will be further discussed in Section 6.
## 4 Discovery of the “Hubble Constant”
A great deal has been written in recent years on the topic of the discovery of
the expanding universe (Nussbaumer & Bieri 2009; Shaviv 2011; Kragh & Smith
2003; Smith 1982). A number of accusations have been levelled against Hubble
(Block 2011), some of which have been discredited (Livio 2011). Several
chapters in this book contain discussions on the discovery of the expanding
universe (see chapters by Cormac O’Raifeartaigh, Ari Belenkiy, Harry
Nussbaumer, John Peacock, and Robert Smith). For that reason there is no need
to go into a great deal of detail here, but suffice it to say that this
“discovery” is even more complicated than the other stories described above.
In Table 3 one can see a steady progression of three related measures: 1.) the
solar motion with respect to the nebulae, 2.) the radius of curvature of the
universe and 3.) the linear relation of velocity and distance for the spiral
nebulae that lent support to an expanding universe model over a static one in
the first half of the 20th century. Note that papers that do not explicitly
discuss observational data are not included in the table. Some further details
on selected publications from Table 3 are worth mentioning in detail:
Table 3.: Early estimates of solar motion, curvature radius and Ho via spiral nebulae Reference | Date | Notes
---|---|---
Truman (1916)a | 1915/12/30 | RA=20h, Dec=-20∘, V=–670km s-1 (14 spirals)
Young & Harper (1916)a | 1916/02/01 | RA=20h24m, Dec=-12∘10′, V=–598$\pm$234km s-1
| | (15 spirals + MC)
Paddock (1916)b | 1916/05/00 | V=–295$\pm$202 km s-1, K=+248$\pm$88 km s-1
| | Using Young & Harper (1916) data.
Wirtz (1916) | 1916/08/00 | Various values of RA, Dec, V, but no K term
Wirtz (1917) | 1916/12/06 | Various values of RA, Dec, V, but no K term
Slipher (1917) | 1917/04/13 | RA=22h, Dec=-22∘, V=–700 km s-1
de Sitter (1917) | 1917/07/00 | First estimates of R for Models A and B
Wirtz (1918) | 1917/12/00 | K=+656 km s-1, V=–830 km s-1
Shapley & Shapley (1919) | 1918/11/00 | Magnitude vs. Velocity
Lundmark (1920) | 1920/01/26 | K=+587 km s-1 (29 spirals, page 75)
Wirtz (1922) | 1921/10/00 | K=+656 km s-1, V=–820 km s-1
Friedmann (1922) | 1922/05/29 | Set M=5$\times 10^{21}M_{\hbox{$\odot$}}$, $\lambda$=0 and found
| | “world period”=1010 years
Wirtz (1924) | 1924/03/00 | Distance vs. velocity: $v$(km)=2200–1200$\times$log($Dm$)c
Silberstein (1924d) | 1924/03/00 | Distance vs. velocity Relation and calculation of
| | curvature radius (R)
Silberstein (1924e) | 1924/03/08 | Calculates R for 11 GCd
Silberstein (1924a) | 1924/04/26 | First distance vs. velocity plot for _GC and LMC/SMC_
Lundmark (1924b) | 1924/06/00 | First distance vs. velocity plot for _spiral nebulae_ ,
| | Kspirals= +800 km s-1
Silberstein (1924b) | 1924/06/07 | Same method and data as before with new estimate of R
Silberstein (1924c) | 1924/10/00 |
Silberstein (1924f) | 1924/Late | Distance vs. velocity plot
Silberstein (1925) | 1925/01/00 | R=7.2$\times 10^{12}$ A.U. and updated plot of distance
| | vs. velocity for GC+MC+M33
Strömberg (1925) | 1925/06/00 | Estimates of K, but no relation found for
| | distance vs. velocity
Lundmark (1925) | 1925/06/00 | Re-defines: K=k+lr+mr2, First time for a
| | variable K–term
Dose (1927) | 1926/11/00 | K=+765$\pm$111 km s-1 for spirals (no variable K–term)
Lemaître (1927) | 1927/04/00 | Discovers that K is linearly dependent on distance
Robertson (1928) | 1928/00/00 | Expects a correlation between velocity and distance
Hubble (1929b) | 1929/01/17 | Distance vs. Velocity plot using Cepheid distances
| | yields a linear fit
de Sitter (1930) | 1930/05/26 | Using observational data calculates R and estimates
| | slope of velocity vs. distance
Hubble & Humason (1931) | 1931/03/00 | Updated list of distances and velocities yields
| | 558 km s-1 Mpc-1
Oort (1931) | 1931/11/30 | Finds Ho=290 km s-1 Mpc-1 after finding
| | some distance inaccuracies
* •
O.H. Truman (1916): was the first to measure the solar motion relative to the
spiral nebulae like that of Campbell (1913) and Airy (1860):
V = X cos$\alpha$ cos$\delta$ \+ Y sin$\alpha$ cos$\delta$ \+ Z sin$\delta$
343434$\alpha$=Right Ascenson, $\delta$=Declination, X,Y,Z = velocity
components of our sun through space.
* •
George Paddock (1916): realized there may be a non-random spiral nebulae
recessional component (denoted as K). He didn’t believe it was real, but
others quickly thought otherwise. This paper contains the first appearance of
the “K correction”353535Not to be confused with the modern usage of “K
correction” which refers to resetting the colors of a galaxy to the rest
frame. in the formula for solar motion:
V = X cos$\alpha$ cos$\delta$ \+ Y sin$\alpha$ cos$\delta$ \+ Z sin$\delta$ \+
K.363636K is the average recessional velocity of spiral nebulae observed in km
s-1.
* •
Willem de Sitter (1917): attempted to measure the radius of curvature of the
universe (R) for Einstein’s model A and his own model B in a number of ways.
In model A he made an estimate of the mass and total volume of the universe to
obtain R$\leq$5$\times$1013 Astronomical Units (A.U.)373737de Sitter (1917, p.
25) (790$\times 10^{6}$ light years). In model B he made a number of estimates
(all found in Section 6 of his paper), but perhaps the most relevant are:
1. 1.
“If we accept the existence of a number of galactic systems whose average
mutual distances are of the order 1010 all we can say is that $\pi$R must be
several times 1010 or roughly $R>10^{11}$[A.U.].” (1.6$\times 10^{6}$ light
years)
2. 2.
“For the lesser Magellanic cloud Hertzsprung found the distance
r$>$6$\times$109. The radial velocity is about +150 km sec-1. This gives
$R>2\times 10^{11}$[A.U.].” (3$\times 10^{6}$ light years)
3. 3.
By averaging the velocities of 3 spiral nebulae (+600 km s-1) and their
distances (326,000 light years) he obtained $R=3\times 10^{11}$A.U.
(4.7$\times 10^{6}$ light years)
* •
Shapley & Shapley (1919): “The speed of spiral nebulae is dependent to some
extent upon apparent brightness, indicating a relation of speed to distance
or, possibly, to mass.” [our emphasis]
* •
Alexander Friedmann (1922): derived the first non-static solutions in addition
to the solutions of Einstein (1917) (model A) and de Sitter (1917) (model B).
He estimated the age of the universe, but we do not know where he obtained his
numbers from. He set the mass of the universe M=5$\times
10^{21}M_{\hbox{$\odot$}}$, set $\lambda$=0 and found a “world period”=1010
years. However see Ari Belenkiy’s chapter in this book. Belenkiy believes this
number should have been written 109 years.
* •
Silberstein (1924d): estimated a distance vs. velocity relation for 7 GC, LMC
and SMC383838GC=Globular Clusters, LMC=Large Magellanic Could, SMC=Small
Magellanic Cloud), R$=$94$\times$106 light years (R=6$\times$1012 A.U.). His
estimate was later criticized by Lundmark (1924b) (among others) for only
using 7 of 16 known GC. Lundmark (1924b) states: “… there is no good reason
for selecting such an arbitrary limit for excluding objects which do not give
a rather constant value of R.” Lundmark (1924b) also complained that the low
value of K (+31 for GCs vs. +800 km s-1 for spiral nebulae) “suggests that the
former are comparatively near as compared with the latter”, implying that they
were inappropriate for calculating R. [Published March, 1924]
* •
Silberstein (1924e): for 11 GC R$<$94$\times$106 light years
(R$<$6$\times$1012 A.U.) [Published March 8, 1924]
* •
Silberstein (1924a): first distance vs. velocity plot with 2 fits for 11
_Globular Clusters_ including the LMC and SMC. (one fit only used 8 GC).
R$<$158$\times$106 light years (R$<$1$\times$1013 A.U.). [Published April 26,
1924]
* •
Silberstein (1924b): used the same objects as in Silberstein (1924a) and found
R=110–126$\times$106 light years (R=7–8$\times$1012 a.u.). [Published June 7,
1924]
* •
Lundmark (1924b): first distance vs. velocity plot for _spiral nebulae_.
Included separate plots for GC+LMC+SMC, and a variety of stellar types, but
without any lines fit to any samples since he did not feel it was warranted.
He used the novae distances to Andromeda and then used spiral nebulae
diameters (assuming constant nebular diameters and luminosities) for the other
spirals in his sample in comparison with Andromeda – his x-axis units were in
‘Distances of Andromeda Nebula.’ He also calculated K=+800 km s-1 (for
spirals), R=19.7$\times$1012 A.U. (3$\times$108 light years) for GC (but with
a very large dispersion of 26.1$\times$1012 A.U.)393939Lundmark stated: “As
the dispersion in R is 26.1$\times$1012 km and thus considerably higher than
what could be expected from the dispersions in V and r, it does not seem that
the curvature of space-time, at least for the present, can be determined with
any accuracy by using the displacements in the spectra of globular clusters.”
and 2.4–6.6$\times$1012 A.U. (3.8$\times$107–1$\times$108 light years) for
spirals.404040Note: Lundmark quoted R in units of kilometers, but it is clear
in comparisons with Silberstein’s papers and with his own distance estimates
to GCs and spiral nebulae that he must have meant to write A.U. rather than
km. Duerbeck & Seitter (1999) fit Lundmark’s data in the plot for spirals
using Lundmark’s smaller value of the distance to Andromeda of 0.2 Mpc to
yield a slope with the origin through zero of 90 km s-1 Mpc-1. If one uses an
Andromeda distance of 0.5 Mpc (derived by Lundmark using novae) one obtains 36
km s-1 Mpc-1. Also see Peacock’s contribution in this volume. [Published June,
1924]
* •
Silberstein (1924c): R=7.2$\times$1012 A.U. (1.1$\times$108 light years).
[Published September 6, 1924]
* •
Silberstein (1924f): distance vs. velocity plot for 11 GC, plus the LMC, SMC
and M33. Values for R were nearly the same as in Silberstein (1924d).
[Published late 1924]
* •
Silberstein (1925): newly updated distance vs. velocity plot using 18 GC, the
LMC, SMC, and M33. Obtained R=7.2$\times 10^{12}$ A.U. (1.1$\times$108 light
years). [Published January 1925]
* •
Strömberg (1925): after an extensive investigation into spiral nebulae (and
separately GCs) he considered correlations between distance vs. velocity and
positions on the sky (cos $\lambda$) vs. velocity. He ended his article by
stating: “In conclusion we may say that we have found no sufficient reason to
believe that there exists any dependence of radial motion upon distance from
the sun. The only dependence fairly well established is one that is a function
of position in the sky.” He plotted _two_ equivalent correlations for the
globular clusters on the same plot with wildly different slopes and stated:
“It is significant, however, that the regression-line for the clusters does
not go through the origin as expected from the theory.” [Published June 1925]
* •
Lundmark (1925): initially believed the K-term was a constant for spirals, but
decided it was given by $K=k+lr+mr^{2}$. Here k,l,m are constants, and the r
is relative distance via the apparent diameter. Solving with 44 velocities
gave k=513,l=10.365,m=0.047.
* •
Georges Lemaître (1927): derived a non–static solution to Einstein’s equations
and coupled it to observations to reveal a linear distance vs. redshift
relation with a slope of 670 or 575 km s-1 Mpc-1 (depending on how the data is
grouped). Radial velocities were from Strömberg (1925), distances from
apparent magnitudes given in Hubble (1926b) that were taken from Hopmann
(1921) and Holetschek (1907).
* •
Howard Percy Robertson (1928): “… we should nevertheless expect a correlation
v$\approxeq$cl/R between assigned velocity v, distance l, and radius of the
observable world R.” [equation 17] Using the data of Hubble (1926b) for
distances and Slipher (Eddington 1923) for velocities he obtained R=2$\times
10^{27}$cm (1.3$\times 10^{14}$ A.U., 2.1$\times 10^{9}$ light years). Hilmar
Duerbeck & Waltraut Seitter (1999) have estimated his distance vs. velocity
slope as 460 km s-1 Mpc-1. Robertson also says that a similar relation to that
of equation 17 was deduced by Weyl (1923).
* •
Hubble (1929b): used Cepheids and bright stars for distances and spiral
nebulae Doppler shifts mostly from Slipher (Eddington 1923). He found a linear
relation between distance and velocity using the data available (grouping them
two ways) and an updated solar motion equation:
V=Xcos$\alpha$cos$\delta$+Ysin$\alpha$cos$\delta$+Zsin$\delta$+kr, where the
old K is now a function linearly dependent upon distance (K=kr). He quoted a
slope of $\sim$465 $\pm$50 km s-1 Mpc-1 for 24 objects, and $\sim$513 $\pm$60
km s-1 Mpc-1 for 9 groups. He stated: “The outstanding feature, however, is
the possibility that the velocity-distance relation may represent the de
Sitter effect, and hence that numerical data may be introduced into
discussions of the general curvature of space.”
* •
de Sitter (1930): used observational data from nebulae and calculated the
slope of the velocity vs. distance linear fit: V/cr=0.5$\times 10^{-27}$ c.g.s
(V/r$\sim$450 km s-1 Mpc-1) which for model A yields R${}_{A}=2.3\times
10^{27}$cm = 1.5$\times 10^{14}$A.U. and for model B yields R${}_{B}=2\times
10^{27}$cm = 1.3$\times 10^{14}$A.U.
Perhaps the most notable name that readers will find missing from the table
and individual descriptions is Arthur Eddington. Eddington took part in this
project in important ways that did not include actual “discoveries”:
1. 1.
He participated in a number of important discussions with most of the authors
listed in Table 3.414141See Smith (1982)
2. 2.
He was responsible for the re-publication and translation of Lemaître’s 1927
paper in _Monthly Notices of the Royal Astronomical Society_ (Lemaître 1931).
He initially brought Lemaître’s work to the attention of the world in his May
1930 paper (Eddington 1930).
3. 3.
He published the final list of Slipher’s radial velocities (Eddington 1923).
4. 4.
By some he is even considered to be the transition figure who triggered the
major paradigm change from the “static or stationary” model of the universe to
an “evolving geometry” (Ellis 1990).424242See page 98, Table 6.1.
Overall we find that Lemaître was the first to seek and find a linear relation
between distance and velocity _in the context of an expanding universe_ , but
that a number of other actors (e.g. Carl Wirtz, Ludwik Silberstein, Knut
Lundmark, Edwin Hubble, Willem de Sitter) were looking for a relation that fit
into the context of de Sitter’s Model B world with its spurious radial
velocities. This is discussed in a number of other papers in this book (see
contributions by Harry Nussbaumer, Cormac O’Raifeartaigh, and Ari Belenkiy).
### 4.1 Slipher’s Contribution to the Expanding Universe Story
Slipher’s radial velocities played a critical role in _all_ of the
publications listed in Table 3 above. Lets look at Slipher’s data in several
of the most important papers in this table.
Lundmark (1925) used Slipher’s radial velocity data of spirals to look for a
relation between distance and velocity. While he did not cite Slipher’s work,
he did state on page 866, “Mainly on account of the enthusiastic and skilful
work of V. M. Slipher we have now knowledge of 44 radial velocities of spiral
nebulae.” Why wouldn’t Lundmark cite Slipher’s work containing the 44 radial
velocities he used? In fact Slipher never published his final list of radial
velocities, the final list was found in Arthur Eddington’s book of 1923
(Eddington 1923).434343See the contribution by Robert Smith in this book, and
Slipher’s table of velocities in Eddington (1923) reproduced in Ari Belenkiy’s
contribution.
Lemaître (1927) also used the radial velocities of Slipher, but Slipher’s name
did not appear in this paper. Rather he cited the work of Strömberg (1925) as
his source. Strömberg (1925) listed 56 velocities obtained from Slipher (it
included some globular clusters in addition to spiral nebulae), but stated
“Slipher’s determinations are given without references….” Strömberg otherwise
praised Slipher in his Introduction stating “..but through the perseverance of
Professor V. M. Slipher, a fairly large number of such velocities has been
derived.” Perhaps Lemaitre could be forgiven as he was mainly a theorist, but
it’s troubling that he didn’t take the time to cite the original sources of
his data.
Hubble (1929b) used Slipher’s radial velocities for 20 out of 24 objects
444444The numbers come from Peacock’s chapter in this book. listed in his
famous Figure 1 showing a “Velocity–Distance Relation among Extra-Galactic
Nebulae”. Hubble gave no attribution to Slipher in this paper, only stating
that “Radial velocities of 46 extra-galactic nebulae are now available….”
However, he did give credit to a few people, mentioning that two of the
distances listed in his Table 1 were those of Shapley (he gave no citation),
three velocities were those of Humason, and with the exception of three
measured by himself, the rest of the visual magnitudes listed were
“Holetschek’s visual magnitude as corrected by Hopmann.”
### 4.2 Hubble Finds his Expanding Universe?
It is commonly believed that Hubble not only discovered an expanding universe,
but that he was also looking for it. The former is credible, but the latter is
not. Thus far historians have unearthed no evidence that Hubble was searching
for the clues to an expanding universe when he published his 1929 paper
(Hubble 1929b). Given the timing of events it is difficult to reconcile. There
were only a few people with knowledge of a non-static solution to Einstein’s
equations in 1928-29:
1. 1.
Alexander Friedman: passed away in 1925.
2. 2.
Yuri Krutkov & Paul Eherenfest who worked to get Friedman’s papers published
and negotiated with Einstein over their validity (see Belenkiy in this book).
3. 3.
Georges Lemaître: his 1927 paper was published in French in an obscure Belgian
journal. He sent his paper to at least Einstein, de Sitter and Eddington.
4. 4.
Einstein: discussed Lemaître’s paper with him at the 1927 Solvay conference,
454545See Belenkiy’s chapter in this book. but told him he did not believe in
his solution. For the first time Lemaître also learned from Einstein of the
Friedman 1922 and 1924 papers.
5. 5.
De Sitter: it is not clear that he ever read Lemaître’s 1927 paper prior to
1930.
6. 6.
Eddington: appears to have forgotten about Lemaître’s 1927 paper until he was
reminded of it in early 1930.464646See Nussbaumer & Bieri (2009), Chapter 11.
It was not until May 1930 when the papers of Eddington (1930) and de Sitter
(1930) were published that the rest of the world became aware of the non-
static solutions of Lemaître and later the earlier solutions of Friedman.
## 5 Brightness Profile of Galaxies
Reynolds (1913) was perhaps the first to attempt the measurement of the light
profile of the Andromeda Galaxy, but only across the bulge, not out to the
spiral arms. His careful measurements yielded:
$Luminosity=\frac{Constant}{(x+I)^{2}}$ (2)
where $x=$ distance from the center of the nucleus/bulge along the major axis
(out to a diameter of 7′).
Seven years later Reynolds (1927) went after more nuclei in a number of spiral
nebulae (M65, M99, M100, M94, and M64) but with mixed success at applying Eq.
2.
Three years later Hubble (1930c) describes a “Distribution of Luminosity in
Elliptical Nebulae.” In this particular case Hubble does cite the Reynolds
(1913, 1927) papers, but perhaps since Reynolds was focused on the bulges of
spiral nebulae rather than Ellipticals Hubble didn’t feel the need to ignore
his competitor. Regardless, Hubble later generalized the relation for
Ellipticals as:
$I=\frac{I_{o}}{\left(\frac{r}{a}+I\right)^{2}}$ (3)
Hubble (1930c, p. 133) also gives credit where credit is due:
> The pioneer investigations along this line are due to J. H. Reynolds, who,
> in 1913 found that the luminosity along the major axis of M 3I, out to 7′
> from the nucleus, could be represented by the formula L=Constant/(x+I)2.
Here we have a (unique?) case where Hubble has properly cited and praised his
predecessor. This relation is now referred to as the Hubble Luminosity
Profile, but perhaps it would be more properly named the Reynolds–Hubble
Luminosity Profile.
In the end it is perhaps not so relevant as there are a number of other far
more popular profiles in use today including the de Vaucouleurs (1948), King
(1962) and Sérsic (1963) profiles.
## 6 Combing Through the Literature
It may be possible to better quantify Hubble’s unwillingness to cite his
predecessors by examining the literature more closely. Most bibliographies in
the 1920s were not compiled at the back of each article as it is done today. A
work was cited by the person’s name and the citation would be contained in a
footnote at the bottom of the same page. For this reason the SAO/NASA
Astrophysical Data Service (ADS) does not have complete bibliographies for the
papers of the period of interest. This author took all of the scientific
publications of Hubble from 1920 through 1930 and attempted to put together a
pseudo-bibliography for each paper. Thankfully Hubble did not write that many
papers in comparison with Lundmark (see Appendices A and B). A pseudo-
bibliography here means that if someone’s name was mentioned in the context of
a previous publication but no bibliographic information was included it was
included as a citation. Full citations are also included in the counting.
Knut Lundmark was chosen as a comparative figure to Hubble as he was probably
considered by his peers to be a figure of equal stature during the 1920s
474747This was more so in the beginning of the 1920s, less so in the latter
after Hubble’s many discoveries.. Lundmark wrote many more papers than Hubble,
but ADS does not have a complete set of the papers he wrote during this period
of time. A handwritten book was obtained 484848The author is unknown, hence
there is no citation for it. from the Uppsala University Library that
contained a listing of all of Lundmark’s papers in his career (even it is not
complete, but more so than ADS). Some of the papers missing from ADS found in
this book are included in Appendix B. Papers were not included that were
considered popular science.494949Lundmark wrote at least 35 articles for the
Swedish magazine _Populär Astronomisk Tidskrift_ , none of which are presently
in ADS (There is an effort to make it so). Some of the articles are on
historical figures such as Tycho Brahe, obituaries, and discoveries or reports
from meetings. He also wrote several popular science books and encyclopedia
articles that are not included. The recording of citations in Lundmark’s
papers were approached in the same manner as for Hubble.
To ensure these are relatively comparable figures ADS was used to see how
often other authors mentioned Lundmark and Hubble in their articles. Using the
ADSLabs Fulltext Service505050Currently residing at
http://labs.adsabs.harvard.edu/fulltext Statistics were compiled on how many
times authors mentioned Lundmark and Hubble by name from 1920 through 1930.
Astronomers did not consistently include a full citation to other authors’
works, but often only referenced the author’s name. All papers have been
eliminated where Lundmark or Hubble have cited/mentioned
themselves.515151There are certain to be a number of complaints about this
methodology. For example, perhaps Lundmark simply likes to includes lots of
citations, or Hubble only cites “big shots”, etc. Those and others are
certainly valid complaints, but they should not distract one from making an
attempt. From Appendices A and B, it is apparent that Lundmark did not simply
fill up his papers with citations for the sake of doing so, but because he had
a broad knowledge of the scientific literature in his field of study. This is
confirmed by Holmberg (1999, p. 130):
> He had read widely and perhaps knew the astronomical literature better than
> most astronomers…
While Hubble was inconsistent in his citations, this inconsistency was not
necessarily reflected by the status of the individual he did or did not cite.
To attempt to properly quantify these kinds of tendencies would require a much
larger and sophisticated effort than that provided herein; nonetheless, it is
clear from Figure 1 that in the early half of the 1920s it is Lundmark who is
“cited” more frequently, while in the latter half it is Hubble. This is not
surprising given not only the facilities with which Hubble was able to conduct
his research (including the largest aperture telescope in the world at Mt.
Wilson), but also Hubble’s success at promoting himself and Mt. Wilson as
described above.
Figure 1.: Top: Black are the number of publications 1920 through 1930 that
cite Knut Lundmark per year, white are the total number of citations per year.
Bottom: The same for Hubble.
Figure 2 shows how often Lundmark and Hubble cited other authors.
Unfortunately there is no way to accomplish this within ADS at
present.525252The author is currently working with leading text data mining
researchers to make this possible in the future. To obtain these numbers was
difficult and required reading through every paper and counting the number of
actual citations, not just names, to authors.535353By citation we are being
flexible in that a citation can simply refer to an author’s work by name
(without a journal reference), but they can only be “cited” once per paper
(this is sometimes difficult to discern). Figure 2 shows a clear trend in that
Lundmark cites authors almost twice as frequently as Hubble. If one allows all
of Lundmark’s publications (Hubble’s longest was 65 pages, while Lundmark’s
has 3 over 100 including one over 250) Lundmark is still over a factor of two
higher in citation rate per page.
Figure 2.: Top Left: Citations per page in Lundmark 1920–1930 publications of
less than 70 pages. Top right: Citation per page in all Lundmark 1920–1930
publications. Bottom left: Hubble citations per page in all publications
1920–1930.
A more specific comparison can be made by examining two papers by Hubble and
one by Lundmark (Figure 3). The first one of Hubble was 51 pages in length
(Hubble 1926a); not including pages with tables or photos taking up a full
page it is only 41. A second Hubble paper (Hubble 1929c) is 63 pages in
length; without full page tables/photos only 41. For Lundmark a paper 195
pages in length was chosen (Lundmark 1927e) (91 eliminating full pages with
tables/photos). In both Hubble and Lundmark’s papers the number of author
names per paper was counted (not including their own names).545454This is a
looser criterion from that above where only _citation_ related names were
compiled. In the two Hubble papers there were a total of 66 and 75 names
mentioned (many repeated of course), which comes to 1.61 and 1.83 names per
page. For Lundmark’s paper there were 423 names mentioned which comes to 4.65
names per page.
Figure 3.: Top Left: Number of author names (including duplicates) found in
(Hubble 1926a). Top Right: same for (Hubble 1929c). Lower Left: Same for
(Lundmark 1927e). Note that all of the scales on the y-axes are distinct, as
are the x-axes between the top two and bottom plot.
Of course the raw numbers as presented in Figure 3 should be renormalized to
the number of pages, but one can see that even dividing Lundmark’s numbers by
a little over a factor of two would reveal that he still mentioned his
colleagues much more frequently than Hubble. It should also be obvious to the
knowledgeable reader that all of the more highly-cited names shown are the
expected authors in this particular domain.
One could also extend this type of study in numerous directions to better
quantify these effects. For example, it could be interesting to compare how
often authors cite others who their work explicitly relies upon. Do the works
on classification of Herschel, Reynolds, or Wolf show up in the classification
works of Lundmark or Hubble? We know from above that Reynolds (1920) was never
found in any of Hubble’s work, but what about the others who would have
influenced Hubble?
## 7 The Making of Mythic Heroes
It would be inappropriate to suggest that Hubble was irrelevant to the history
astronomy, but is clear that many of the advances discussed above would have
happened within a short period of time of their original discovery even if
Hubble had never worked as an astronomer.555555As a (distant) analogy lets
consider the accomplishments of Alexander the Great. Were his accomplishments
inevitable because the Macedonian state was ready for a strong ruler after the
death of his father Philip? Wasn’t the Macedonian version of the Greek phalanx
with its longer spear and the high level of training required to master this
technology in place well before Alexander? If one answers yes to these things,
then perhaps another “strong man” could have come to the helm of the state and
accomplished much the same as Alexander. Or if Philipp had lived he surely
would have attempted many of the same things that Alexander attempted. This
may be contrary to the assertion by Smith (2009, p. 72) that technological
determinism alone does not explain Hubble’s discoveries. Indeed, technological
determinism, the belief that such discoveries were inevitable given facilities
such as the large telescopes at Mt. Wilson, is not needed. Instead one may
consider historical determinism “lite.” Clearly there were many astronomers
working in these topics and steady progress was being made in each field. If
Hubble had not found Cepheids in spiral nebulae in 1924/25, then someone else
surely would have within a few years. Lundmark’s classification scheme, or the
earlier scheme of Reynolds for spirals, could have easily replaced Hubble’s at
some level and the linear distance-velocity relation had already been
postulated by Lemaître in 1927. Without Hubble it is clear that within a short
period of time someone else would have been given credit for each of these
initial discoveries, Stigler’s Law of Eponymy notwithstanding (Merton 1965;
Stigler 1980).565656The most famous quote from Stigler’s paper is: “No
scientific discovery is named after its discoverer.” Hubble’s success in
gaining credit for his classification scheme and linear distance-velocity
relation may be related to his verification of the Island Universe hypothesis
– after the latter, his prominence as a major player in astronomy was
affirmed. As pointed out by Merton (1968) credit for simultaneous (or nearly
so) discoveries is usually given to eminent scientists over lesser-known
ones.575757This has been termed “The Matthew Effect.”
One may also consider the competition between the Lick, Lowell and Mt. Wilson
Observatories in early observational cosmology. Because Lowell Observatory did
not enjoy a high level of esteem, 585858The poor reputation was initially due
to Percival Lowell’s claim to have discovered canals engineered by intelligent
beings on Mars. it may have taken Slipher more effort than other professional
astronomers to convince the community of the validity of his initial
discoveries (see contributions by Joseph S. Tenn, Robert Smith and others in
this book). In addition, Slipher had a modest personality and was not given to
boasting or promoting his accomplishments in public. On the other hand,
consider Lundmark or Wirtz who did not have regular access to telescopes,
instrumentation and support facilities like that of Lick (Clark 36-in [1888]
and Crossley 36-in [1905]), Mt. Wilson (60-in [1908] and Hooker 100-in
[1917]), or Yerkes (40-in [1895]).595959Lundmark did visit and observe at Mt.
Wilson before his falling out with Hubble over their classification schemes.
He also spent time at Lick and Mt. Wilson over a period of two years (1921-22)
thanks to the Sweden-America foundation (Holmberg 1999, p. 94). At that time
he was able to borrow plates from Lick Observatory. In 1929 and in 1932 he
again visited several American observatories (including Mt. Wilson) to utilize
their plate collections for his Lund General Catalogue project (Ibid., pp.
109-116). Did Lundmark and Wirtz have to write more cogent and highly
interesting/readable papers to get the community to follow their research in a
“competition for attention” (Collins 1975, p. 480)? If so, then we know that
Lundmark succeeded at some level because his 1920 thesis (Lundmark 1920) was
read by astronomers such as Shapley and van Maanen before he had even arrived
at Mt. Wilson and Lick Observatories (Holmberg 1999, p. 94). In fact,
Lundmark’s work was so highly regarded that a “Memorial Volume” was edited by
Martin Johnson (1961) and published three years after Lundmark’s death. It
included contributions by a number of highly regarded astronomers such as
Milton Humason (p. 26), Harlow Shapley (p. 37), Boris Vorontsov-Velyaminov (p.
43), Fritz Zwicky (p. 55) and Gustaf Strömberg (p. 95).
Of course language may have also played a role. None of Wirtz’s major works
were published in English,606060Much of observational cosmology research was
being produced at American observatories, while much of the theory was being
promoted by Eddington, de Sitter and others in English language publications
in Europe during the trying economic times after World War I. whereas Lundmark
wrote mainly in English language journals from around 1920.616161Lundmark also
published in Swedish in popular science publications and in English and
Swedish in journals like _Arxiv för Matematik, Astronomi och Fysik_ , and
_Lund or Uppsala Observatory Reports_. Is it possible that an astronomer like
Hubble coming from the premier observatory didn’t need to be as explicit in
his citing of previous work because people had to read and utilize his results
regardless of the quality of the background scholarship?
There is also the issue of the relative decline of research activities at
smaller institutions that took place in the U.S.A. (and Europe) around the
turn of the century given that the latest observational astronomy research
required large amounts of capital (Lankford 1997, Chapter 7). As well, the
First World War did not do anything positive for the facilities at European
institutions in this sense. Simply consider Shapley’s move from Mt. Wilson to
Harvard around 1925. Clearly he knew he was giving up access to one of the
finest observatories in the world to work at an institution with inferior
equipment, although as director of Harvard Observatory he was to obtain
sufficient research funding for otherwise large-scale projects.
One should take account of these changes in order to better quantify how the
attitudes of scientists working at the premier institutions could have changed
and how credit for discoveries was subsequently awarded. While it may be a
stretch to quote from William Pitt that “Unlimited power is apt to corrupt the
minds of those who possess it.” One should not rule out such effects on the
minds of successful scientists. Aggression and competition surely play some
kind of role – a role which sociologists have already explored in a number of
scientific contexts, including the “necessity” to defend one’s position and
ideas (Lankford 1997, p. 187). Collins (1975, p. 482) has also suggested that
as scientists move into higher status positions they may take on more
aggressive roles. Certainly we have seen aspects of this behavior in Hubble in
his footnote dispute with Lundmark (also see Chapter 11 in Christianson
(1996)), and his persistent tendency to defend his discoveries as sole
achievements of himself and Mt. Wilson.626262Hubble wrote to de Sitter in 1930
(Hubble 1930b) in response to de Sitter’s recent publication (de Sitter 1930)
of a velocity-distance relation and his lack of sufficient credit to Hubble:
“I consider the velocity-distance relation, its formulation, testing and
confirmation, as a Mount Wilson contribution and I am deeply concerned in its
recognition as such.”
It has been argued (see Kragh & Smith 2003) that much of Hubble’s fame at-
large came after his death in 1953.636363Although in 1934 Lundmark was already
lamenting a wave of “Hubbleianism” (Holmberg 1999, p. 100) This retrospective
view of Hubble’s accomplishments would certainly fit in with the well-known
hypothesis of Thomas Kuhn (1962) that:
> There is a persistent tendency to make the history of science look linear or
> cumulative, a tendency that even affects scientists looking back at their
> own research.
Of course, a similar analysis could apply to many scientific discoveries of
importance in the early 20th century. An outstanding example is Henrietta
Levitt’s discovery of a period–luminosity relation for Cepheid variable stars
and the number of scientists it took to place it on a reliable and accurate
footing.
## 8 Conclusion
Can one say anything definitive about the credit that Hubble has received for
the seminal discoveries discussed herein, and his lack of acknowledgement of
the work of others? At the present time it does not seem possible to quantify
these observations. The line of research presented in this work is only an
initial attempt. With new text data mining technologies growing in strength
one should be able to better quantify some of the assertions above in the near
future. Given what is known today it would not be fair to suggest that Hubble
was a Lavoisier-like figure who regularly claimed the discoveries of others as
his own (Butterfield 1959, pp. 206-9), but that he was inconsistent in
awarding credit.
Future researchers are certain to mine the literature in detail to examine how
major scientists cited (or not) their colleagues, but will this influence the
writers of today? Perhaps this could be accomplished by demonstrating that
over the long-term one may be discredited for neglecting to cite relevant work
that one relied upon.
Take two recent books from the 2000s that were highly read in the scientific
community: Stephen Wolfram’s book (Wolfram 2002) and that of Roger Penrose
(Penrose 2005). Wolfram’s book contains almost no citations to other work,
while Penrose makes a valiant attempt to cite others for a book even broader
in scope than Wolfram’s. Needless to say Wolfram was pilloried in the
scientific and popular press for this lack of attribution and general belief
that his ideas have broader application than possible (e.g. Casti 2002; Hayes
2002; Economist 2002). Perhaps the general public agrees as well: Wolfram has
received 3 stars out of 5 (from 344 reviews) on Amazon.com as of 2012/12/05
(with a large number of 1 stars (102) and 2 stars (61). Penrose received 4 out
of 5 (from 204 reviews) with very few 1 (12) or 2 (12) stars. Penrose’s
bibliography is 30 pages in length (pp. 1050–1080) and has received rather
more favorable reviews (e.g. Johnson 2005; Blank 2006).
How credit is awarded for a discovery is often a complex issue and should not
be oversimplified – yet this happens time and again. Another well-known
example in this field is the discovery of the Cosmic Microwave Background (see
Alpher & Herman (1988); Gribbin (1998); Kragh (1999)).
The problem is larger than awarding credit within a given field as outsiders
pick anecdotal stories from astronomy, and not only get them wrong, but
oversimplify them.646464A recent example would be Taleb (2010) in his
_Inadvertent Discoveries_ subsection of Chapter 11 (on page 168 of the
paperback edition) where he discussed the discovery of the Cosmic Microwave
Background Radiation. This can also happen in the case of professional
astronomers/physicists (see Greene 2011).656565To take one of many mis-
statements in this piece: “In 1929, the American astronomer Edwin Hubble
discovered that distant galaxies are all rushing away from us.” One only needs
to read a smattering of the contributions in this book to understand how
misguided Greene was in his assumptions. Perhaps, as a community, astronomers
can learn to do better and this book could be the beginning, at least in this
particular domain.
#### Acknowledgments
This research has made use of NASA’s Astrophysics Data System Bibliographic
Services.
Thanks are due to the ever-efficient librarians at The Goddard Institute for
Space Studies, Zoe Wai and Josefina Mora, without whom many of the references
in this paper would have taken much more time to locate. I also wish to thank
Samuel Regandell, Bengt Edvardsson and Kjell Lundgren of the Astronomy
department at Uppsala University for their help in locating (and scanning)
some hard-to-obtain manuscripts by Knut Lundmark in their basement storage. My
wife Elisabeth also deserves praise for patiently listening to my many
complaints about the literature and soothing my stress from co-organizing this
conference and Proceedings.
This work has also benefited from comments by Bengt Edvardsson, John Peacock,
and Jeffrey Scargle and my excellent co-Editor Deidre Hunter.
## Appendix A Hubble’s published papers 1 Jan. 1920 through 31 Dec. 1930
No. | Reference | Pages | Citations | Notesa
---|---|---|---|---
1 | Seares & Hubble (1920a) | 15.0 | 15 | +
2 | Hubble (1920) | 18.0 | 15 | +
3 | Seares & Hubble (1920b) | – | – | See ref. 1
4 | Hubble (1921) | – | – | N/A not Hubble’s work?
5 | Hubble (1922b) | 39.0 | 44 | +
6 | Hubble (1922c) | 38.5 | 17 | +
7 | Hubble (1922a) | – | – | See ref. 5
8 | Hubble & Lundmark (1922a) | 2.0 | 4 | +
9 | Hubble (1922d) | – | – | See ref. 6
10 | Hubble (1923a) | 1.0 | 3 | *
11 | Hubble (1923b) | 2.5 | 2 | –
12 | Hubble (1925d) | 26.0 | 37 | !
13 | Hubble (1925a) | 3.0 | 2 | –
14 | Hubble (1925b) | – | – | See ref. 13
15 | Hubble (1925c) | – | – | See ref. 12
16 | Hubble (1926c) | 39.5 | 56 | +
17 | Hubble (1926b) | – | – | See ref. 17
18 | Hubble (1926d) | 52.0 | 28 | –
19 | Hubble (1926a) | – | – | See ref. 16
20 | Hubble (1926e) | 2.0 | 5 | *
21 | Hubble (1927e) | – | – | N/A Popular Article
22 | Hubble & Duncan (1927a) | 5.5 | 7 | –
23 | Hubble (1927a) | – | – | See ref. 34
24 | Hubble & Humason (1927) | – | – | N/A written in 1934?
25 | Hubble (1927d) | – | – | N/A written in 1936?
26 | Hubble (1927c) | – | – | See ref. 10
27 | Hubble (1927b) | 3.0 | 2 |
28 | Hubble & Duncan (1927b) | – | – | See ref. 22
29 | Hubble (1927f) | 5.5 | 5 | +
30 | Hubble (1928) | 4.0 | 6 | *
31 | Hubble (1929a) | – | – | N/A Article “excerpt”
32 | Hubble (1929d) | – | – | See ref. 33
33 | Hubble (1929c) | 62.0 | 63 | !
34 | Hubble (1929b) | 6.0 | 8 | +
35 | Hubble (1930c) | – | – | See ref. 37
36 | Hubble & Duncan (1930) | 1.0 | 2 | +
37 | Hubble (1930a) | 17.0 | 19 | +
## Appendix B Lundmark’s published papers 1 Jan. 1920 through 31 Dec. 1930
No. | Reference | Pages | Citations | Notesa
---|---|---|---|---
1 | Lundmark (1920) | 78.0 | 256 | –
2 | Lundmark (1921a) | 2.0 | 10 | –
3 | Lundmark (1921b) | 2.0 | 10 | –
4 | Lundmark (1921e) | 1.5 | 1 | –
5 | Lundmark (1921d) | 15.0 | 27 | !
6 | Lundmark (1921g) | 1.5 | 7 | *
7 | Lundmark (1921c) | 3.0 | 11 | !
8 | Lundmark (1921f) | 3.0 | 5 | *
9 | Lundmark (1921h) | 4.0 | 15 | !
10 | Lundmark (1922i) | 3.5 | 13 | –
11 | Lundmark & Luyten (1922d) | 4.0 | 13 | +
12 | Lundmark (1922h) | 2.0 | 1 | *
13 | Lundmark (1922l) | 1.0 | 2 | *
14 | Lundmark (1922b) | 1.0 | 7 | *
15 | Lundmark (1922j) | 10.0 | 39 | !
16 | Lundmark (1922f) | 0.5 | 2 | *
17 | Lundmark (1922c) | 8.0 | 18 | +!
18 | Lundmark & Luyten (1922b) | 2.0 | 13 | +!
19 | Lundmark & Luyten (1922a) | 16.0 | 44 | –
20 | Lundmark (1922k) | 9.0 | 22 | +
21 | Lundmark (1922a) | 9.0 | 17 | +
22 | Lundmark (1922g) | 5.0 | 28 | +
23 | Lundmark (1922d) | 1.5 | 14 | *+!
24 | Lundmark & Luyten (1922c) | 0.3 | 3 | –
25 | Hubble & Lundmark (1922b) | 2.0 | 4 | +
26 | Lundmark (1922e) | 8.0 | 3 | –
27 | Lundmark & Luyten (1923b) | 2.0 | 2 | –
28 | Lundmark & Luyten (1923e) | 2.0 | 9 | +!
29 | Lundmark & Luyten (1923d) | 3.0 | 6 | +!
30 | Lundmark (1923b) | 25.0 | 28 | –
31 | Lundmark & Luyten (1923a) | 4.5 | 7 | +
32 | Lundmark & Luyten (1923c) | 2.5 | 5 | +
33 | Lundmark (1923a) | 2.0 | 4 | +
34 | Lundmark (1924b) | 24.0 | 62 | +
35 | Lundmark (1924c) | 4.0 | 7 | +
36 | Lundmark (1924a) | 2.5 | 7 | +
37 | Lundmark (1925) | 31.0 | 110 | +!
38 | Lundmark (1926d) | 6.0 | 6 | +
39 | Lundmark (1926e) | – | – | ?
40 | Lundmark (1926c) | – | – | See ref. 38
41 | Lundmark (1926a) | 2.0 | 5 | *
42 | Lundmark (1926b) | 6.0 | 13 | +
43 | Lundmark (1927a) | 6.0 | 5 | –
44 | Lundmark (1927d) | – | – | ?
45 | Lundmark (1927b) | – | – | ?
No. | Reference | Pages | Citations | Notesa
---|---|---|---|---
46 | Lundmark & Luyten (1927b) | – | – | ?
47 | Lundmark (1927c) | 15.0 | 63 | +
48 | Lundmark & Luyten (1927a) | 2.0 | 3 | +
49 | Lundmark & Luyten (1927c) | 2.0 | 5 | +
50 | Lundmark (1927e) | 133.0 | 195 | !
51 | Lundmark (1928c) | – | – | ?
52 | Lundmark (1928a) | 6.0 | 10 | +
53 | Lundmark & Ark (1928) | 8.0 | 14 | +
54 | Lundmark (1928b) | – | – | ?
55 | Lundmark (1930d) | 1.5 | 4 | *
56 | Lundmark (1930b) | 1.0 | 1 | *
57 | Lundmark (1930c) | 30.0 | 16 | *
58 | Lundmark (1930a) | 3.0 | 5 | *
## Appendix C Lundmark’s reply to Hubble May 1927
Lundmark’s reply to Hubble’s accusation of plagiarism was presented as a paper
to the Royal Society of Science of Upsala on May 6, 1927 and later published
in the _New Proceedings of the Royal Society of Science of Upsala_ 666666Nova
Acta Regiae Societatis Scientiarum Upsaliensis, Volumen Extra Ordinem Editum
1927 and provides a unique look at how at least one of Hubble’s peers viewed
him at that time. I produce this quote (verbatim) here since this volume was,
until recently, very difficult to obtain.
> “In his paper, Extragalactic nebulae, Aph. J. 64:321, 1926, E. P. Hubble
> makes an attack on me which is written in such a tone that I hesitate to
> give any answer at all. Still, I may take the occasion to state a few facts.
>
> I was present at the Cambridge meeting of the Astronomical Union.
>
> I was not then a member of the Commission of Nebulae.
>
> I did not have any, access whatsoever to the memorandum or to other writings
> of E. P. Hubble, neither did I have access to the report of nebulae (which
> does not give details of Hubble’s classification) until at the end of the
> meeting, neither did I recognize until I obtained a letter from Hubble at
> the end of 1926 that he had made another classification of nebulae than the
> one published in his paper, A general study of the Diffuse Galactic Nebulae,
> Mt Wils. Contr. No. 241, 1922.
>
> As much as I heard of the discussion in the committee of nebulae the only
> question was if the terms $\gg$galactic$\gg$ and $\gg$extragalactic
> nebulae$\gg$ should be accepted, From the discussion I got the impression
> that the intention of Hubble was to force through his nomenclature, One of
> the members told me outside the discussion that Hubble had suggested the
> subdivision $\gg$logarithmic spirals$\gg$ but I did not understand that this
> suggestion was given in any memorandum to the Union, Now when reading
> Hubble’s paper I am glad to note that he seems not to have carried out the
> unhappy idea introducing the term $\gg$logarithmic spirals$\gg$ Slight
> changes in his classification might have been introduced since the Cambridge
> meeting.
>
> Hubble’s statement that my classification except for nomenclature is
> practically identical with the one submitted by him is not correct. Hubble
> classifies his subgroups according to eccentricity or form of the spirals or
> degree of development while I use the degree of concentration towards the
> centre. As to the three main groups, elliptical, spiral and magellanic
> nebulae it may be of interest to note that the two first are slightly older
> than Hubble and myself. The term elliptical nebulae thus is used by
> Alexander in 1852 and the term spiral by Rosse in 1845; The importance of
> the magellanic group has been pointed out by myself Observatory 47, 277,
> 1924 earlier than by Hubble. _As to Hubble’s way of acknowledging his
> predecessors I have no reason to enter upon this question here._ ” (the
> latter is our emphasis).
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|
arxiv-papers
| 2013-01-30T17:13:16 |
2024-09-04T02:49:41.010141
|
{
"license": "Public Domain",
"authors": "Michael J. Way (NASA Goddard Institute for Space Studies)",
"submitter": "Michael Way",
"url": "https://arxiv.org/abs/1301.7294"
}
|
1301.7473
|
# Information driven self-organization of complex robotic behaviors
Georg Martius1, Ralf Der1, Nihat Ay1,2
1Max Planck Institute for Mathematics, Leipzig, Germany
2Santa Fe Institute, Santa Fe, USA
{martius|ralfder|nay}@mis.mpg.de
###### Abstract
Information theory is a powerful tool to express principles to drive
autonomous systems because it is domain invariant and allows for an intuitive
interpretation. This paper studies the use of the predictive information (PI),
also called excess entropy or effective measure complexity, of the
sensorimotor process as a driving force to generate behavior. We study
nonlinear and nonstationary systems and introduce the time-local predicting
information (TiPI) which allows us to derive exact results together with
explicit update rules for the parameters of the controller in the dynamical
systems framework. In this way the information principle, formulated at the
level of behavior, is translated to the dynamics of the synapses. We underpin
our results with a number of case studies with high-dimensional robotic
systems. We show the spontaneous cooperativity in a complex physical system
with decentralized control. Moreover, a jointly controlled humanoid robot
develops a high behavioral variety depending on its physics and the
environment it is dynamically embedded into. The behavior can be decomposed
into a succession of low-dimensional modes that increasingly explore the
behavior space. This is a promising way to avoid the curse of dimensionality
which hinders learning systems to scale well.
## 1 Introduction
Autonomy is a puzzling phenomenon in nature and a major challenge in the world
of artifacts. A key feature of autonomy in both natural and artificial systems
is seen in the ability for independent exploration [9]. In animals and humans,
the ability to modify its own pattern of activity is not only an indispensable
trait for adaptation and survival in new situations, it also provides a
learning system with novel information for improving its cognitive
capabilities, and it is essential for development. Efficient exploration in
high-dimensional spaces is a major challenge in building learning systems. The
famous exploration-exploitation trade-off was extensively studied in the area
of reinforcement learning [60]. In a Bayesian formulation this trade-off can
be optimally solved [22], however it is computationally intractable. A more
conceptual solution is to provide the agent with an intrinsic motivation [51,
55] for focusing on certain things and thus constraining the exploration to a
smaller space. To approach this problem in a more fundamental way we consider
mechanisms for goal-free exploration of the dynamical properties of a physical
system, e. g. a robot. If the exploration is rooted in the agent in a self-
determined way, i. e. as a deterministic function of internal state variables
and not via a pseudo-random generator it has the chance to escape the curse of
dimensionality. Why? Because specific features of the system such as
constrains and other embodiment effects can be exploited to reduce the search
space. Thus an exploration strategy taking the particular body and environment
into account is vital for building efficient learning algorithms for high-
dimensional robotic systems. But how can goal-free exploration be useful to
actually pursue goals? We show that a variety of coordinated sensorimotor
patterns are formed that may be used to quickly construct more complex
behaviors using a second level of learning. It may also be used more directly
in combination with reinforcement learning where the typical random
exploration is substituted or augmented by the goal-free exploration leading
presumably to a large speedup.
The solution for such a general problem needs a core paradigm in order to be
relevant for a large class of systems. In recent years, information theory has
come into the focus of researchers interested in a number of related issues
ranging from quantifying and better understanding autonomous systems [7, 33,
23, 56, 24, 64, 53] to questions of spontaneity in biology and technical
systems [10] to the self-organization of robot behavior [2, 65].
A systematic approach requires both a convenient definition of the information
measure and a robust, real time algorithm for the maximization of that
measure. This paper studies in detail the use of the predictive information
(PI) of a robot’s sensorimotor process. The predictive information of a
process quantifies the total information of past experience that can be used
for predicting future events. Technically, it is defined as the mutual
information between the past and the future of the time series. It has been
argued [8] that predictive information, also termed excess entropy [13] and
effective measure complexity [26], is the most natural complexity measure for
time series. By definition, predictive information of the sensor process is
high if the robot manages to produce a stream of sensor values with high
information content (in the Shannon sense) by using actions that lead to
predictable consequences. A robot maximizing PI therefore is expected to show
a high variety of behavior without becoming chaotic or purely random. In this
working regime, somewhere between order and chaos, the robot will explore its
behavioral spectrum in a self-determined way in the sense discussed above.
This paper studies the control of robots by simple neural networks whose
parameters (synaptic strengths and threshold values) are adapted on-line to
maximize (a modified) PI of the sensor process. These rules define a mechanism
for behavioral variability as a deterministic function formulated at the
synaptic level. For linear systems a number of features of the PI maximization
method have been demonstrated [2]. In particular, it could be shown that the
principle makes the system to explore its behavior space in a systematic
manner. In a specific case, the PI maximization caused the controller of a
stochastic oscillator system to sweep through the space of available
frequencies. More importantly, if the world is hosting a latent oscillation,
the controller will learn by PI maximization to go into resonance with this
inherent mode of the world. This is encouraging, since maximizing the PI means
(at least in this simple example) to recognize and amplify the latent modes of
the robotic system.
The present paper is devoted to the extension of the above mentioned method to
nonlinear systems with nonstationary dynamics. This leads to a number of novel
elements in the present approach. Commonly information theoretic measures are
optimized in the stationary state. This is not adequate for a robot in a self-
determined process of behavioral development. This paper develops a more
appropriate measure for this purpose called the time-local predictive
information (TiPI) for general nonstationary processes by using a specific
windowing technique and conditioning. Moreover, the application of information
theoretic measures in robotics is often restricted to the case of a finite
state-action space with discrete actions and sensor values. Also these
restrictions are overcome in this paper so that it can be used immediately in
physical robots with high dimensional state-action space. This will be
demonstrated by examples with two robots in a physically realistic simulation.
The approach is seen to work from scratch, i. e. without any knowledge about
the robot, so that everything has to be inferred from the sensor values alone.
In contrast to the linear case the nonlinearities and the nonstationarity
introduce a number of new phenomena, for instance the self-switching dynamics
in a simple hysteresis system and the spontaneous cooperation of physically
coupled systems. In high-dimensional systems we observe behavioral patterns of
reduced dimensionality that are dependent on the body and the environment of
the robot.
### 1.1 Relation to other work
Finding general mechanisms that help robots and other systems to more
autonomy, is the topic of intensive recent research. The approaches are widely
scattered and follow many different routes so that we give in the following
just a few examples.
#### 1.1.1 Information theoretic measures
Information theory has been used recently in a number of approaches in
robotics in order (i) to understand how input information is structured by the
behavior [34, 33] and (ii) to quantify the nature of information flows inside
the brain [23, 56, 24] and in behaving robots [53, 64]. An interesting
information measure is the empowerment, quantifying the amount of Shannon
information that an agent can “inject into” its sensor through the
environment, affecting future actions and future perceptions. Recently,
empowerment has been demonstrated to be a viable objective for the self-
determined development of behavior in the pole balancer problem and other
agents in continuous domains [28].
Driving exploration by maximizing PI can also be considered as an alternative
to the principle of homeokinesis as introduced in Der and Liebscher [15], Der
[14] that has been applied successfully to a large number of complex robotic
systems, see Der et al. [19, 18, 20], Der and Martius [16] and the recent book
Der and Martius [17]. Moreover, this principle has also been extended to form
a basis for a guided self-organization of behavior [39, 17].
#### 1.1.2 Intrinsic motivation
As mentioned above, the self-determined and self-directed exploration for
embodied autonomous agents is closely related to many recent efforts to equip
the robot with a motivation system producing internal reward signals for
reinforcement learning in pre-specified tasks. Pioneering work has been done
by Schmidhuber using the prediction progress as a reward signal in order to
make the robot curious for new experiences [50, 58, 52]. Related ideas have
been put forward in the so called play ground experiment [29, 43]. There have
been also a few proposals to autonomously form a hierarchy of competencies
using the prediction error of skill models [3] or more abstractly to balance
skills and challenges [57]. Predictive information can also be used as an
intrinsic motivation in reinforcement learning [66] or additional fitness in
evolutionary robotics [46].
#### 1.1.3 Embodiment
The past two decades in robotics have seen the emergence of a new trend of
control in robotics which is rooted more deeply in the dynamical systems
approach to robotics using continuous sensor and action variables. This
approach yields more natural movements of the robots and allows to exploit
embodiment effects in an effective way, see Pfeifer and Bongard [44], Pfeifer
et al. [45] for an excellent survey. The approach described in the present
paper is tightly coupled to the ideas of exploiting the embodiment, since the
development of behavioral modes is entire dependent on the dynamical coupling
of the body, brain, and its environment.
#### 1.1.4 Spontaneity
We would like to briefly discuss the implications of using a self-
determined111Self-determined is understood here has “only based its own
internal laws” and deterministic mechanism of exploration to the understanding
of variability in animal behavior. In the animal kingdom, there is increasing
evidence showing that animals from invertebrates to fish, birds, and mammals
are equipped with a surprising degree of variety in response to external
stimulation [4, 25, 59, 6]. So far, it is not clear how this behavioral
variability is created. Ideas cover the whole range from the quantum effects
[30] (pure and inexorable randomness) to thermal fluctuations at the molecular
level to the assumption of pure spontaneity [42], rooting the variability in
the existence of intrinsic, purely deterministic processes.
This paper shows that a pure spontaneity is enough to produce behavioral
variations, and as in animals, their exact source appears “indecipherable”
from an observer point of view. If the variation of behavior in animals is
produced in a similar way, this would bring new insights into the free will
conundrum [10].
## 2 Methods
We start with the general expressions for the predictive information (PI) and
introduce a derived quantity called time-local predictive information (TiPI)
more suitable for the intended treatment of nonstationary systems. Based on
the specific choice of the time windows we derive estimates of the TiPI for
general stochastic dynamical systems and give explicit expressions for the
special case of a Gaussian noise. The explicit expressions are used for the
derivation of the parameter dynamics of the controller (exploration dynamics)
obtained by gradient ascending the TiPI. Besides giving the exploration
dynamics as a batch rule we also derive, in the sense of a stochastic gradient
rule, the one-shot gradient. The resulting combined dynamics (system plus
exploration dynamics) is a deterministic dynamical system, where the self-
exploration of the system becomes a part of the strategy. These general
results are then applied to the case of the sensorimotor loop and we discuss
their Hebbian nature.
### 2.1 Predictive information
The PI of a time discrete process $\left\\{S_{t}\right\\}_{t=a}^{b}$ with
values in $\mathbb{R}^{n}$ is defined [8] as the mutual information between
the past and the future, relative to some instant of time $a\leq t_{0}<b$
$I\left(S_{\text{future
}};S_{\text{past}}\right)=\left\langle\ln\frac{p\left(s_{\text{future
}},s_{\text{past}}\right)}{p\left(s_{\text{past}}\right)p\left(s_{\text{future}}\right)}\right\rangle=H\left(S_{\text{future
}}\right)-H\left(S_{\text{future }}|S_{\text{past}}\right)$ (1)
where the averaging is over the joint probability density distribution
$p\left(s_{\text{past}},s_{\text{future }}\right)$ with
$\text{past}:=\\{a,\dots,t_{0}\\}$ and $\text{future}:=\\{t_{0}+1,\dots,b\\}$.
In more detail, we use the (differential) entropy $H\left(S\right)$ of a
random variable $S$ given by
$H\left(S\right)=-\int p\left(s\right)\ln p\left(s\right)\,\mathrm{d}s$
where $p\left(s\right)$ is the probability density distribution of the random
variable $S$. The conditional entropy $H\left(S_{t}|S_{t-1}\right)$ is defined
accordingly
$H\left(S_{t}|S_{t-1}\right)=-\int\int p\left(s_{t}|s_{t-1}\right)\ln
p\left(s_{t}|s_{t-1}\right)\,\mathrm{d}s_{t}\,p(s_{t-1})\,\mathrm{d}s_{t-1}$
$p\left(s_{t}|s_{t-1}\right)$ being the conditional probability density
distribution of $s_{t}$ given $s_{t-1}$. As is well known, in the case of
continuous variables, the individual entropy components
$H\left(S_{\text{future }}\right)$, $H\left(S_{\text{future
}}|S_{\text{past}}\right)$ may well be negative whereas the PI is always
positive and will exist even in cases where the individual entropies diverge.
This is a very favorable property deriving from the explicit scale invariance
of the PI [2].
The usefulness of the PI for the development of explorative behaviors of
autonomous robots has been discussed earlier222In experiments with a coupled
chain of robots [21] it was observed that the PI of just a single sensor, one
of the wheel counters of an individual robot, already yields essential
information on the behavior of the robot chain. The PI turned out to be
maximal if the individual robots managed to cooperate so that the chain as a
whole could navigate effectively. This is remarkable in that a one-dimensional
sensor process can already give essential information on the behavior of a
very complex physical object under real world conditions. These results give
us some encouragement to study the role of PI and other information measures
for specific sensor processes as is done in the present paper., see Ay et al.
[1], Der et al. [21], Ay et al. [2]. This paper continues these investigations
for the case of more general situations than those discussed before. In order
to do so, we have to introduce some specifications necessary for the
development of a versatile and stable algorithm realizing the increase of PI
in the sensor process at least approximately.
Let us start with simplifying eq. (1). If $\left\\{S_{t}\right\\}_{t=a}^{b}$
is a Markov process, see Ay et al. [1], the PI is given by the mutual
information (MI) between two successive time steps, i. e. instead of eq. (1)
we have
$I\left(S_{t};S_{t-1}\right)=\left\langle\ln\frac{p\left(s_{t},s_{t-1}\right)}{p\left(s_{t}\right)p\left(s_{t-1}\right)}\right\rangle=H\left(S_{t}\right)-H\left(S_{t}|S_{t-1}\right)$
(2)
the averaging being done over the joint probability density
$p\left(s_{t},s_{t-1}\right)$. Actually, any realistic sensor process will
only be in exceptional cases purely Markovian. However, we can use the
simplified expression (2)—let us call it the one-step PI—also for general
sensor processes taking it as the definition of the objective function driving
the autonomous exploration dynamics to be derived.
#### 2.1.1 Nonstationarity and time-local predictive information (TiPI)
Most applications done so far were striving for the evaluation of the PI in a
stationary state of the system. With our robotic applications, this is neither
necessary nor adequate. The robot is to develop a variety of behavioral modes
ideally in a open-ended fashion, which will certainly not lead to a stationary
distribution of sensor values. The PI would change on the timescale of the
behavior. How can one obtain in this case the probability distributions of
$p(s_{t})$? The solution we suggest is to introduce a conditioning on an
initial state in a moving time window and thus obtain the distributions from
our local model as introduced below. More formally, let us consider the
following setting. Let $t$ be the current instant of time and $\tau$ be the
length of a time window $\tau$ steps into the past. We study the process in
that window with a fixed starting state $s_{t-\tau}$ so that all distributions
in eq. (2) are conditioned on state $s_{t-\tau}$. For instance, instead of
$p\left(s_{t}\right)$ in eq. (2), we have to use333In the Markovian case this
boils down to $p\left(s_{t}|s_{t-\tau}\right)=\int\cdots\int
p\left(s_{t}|s_{t-1}\right)\cdots
p\left(s_{t-\tau+1}|s_{t-\tau}\right)ds_{t-1}\cdots\,\mathrm{d}s_{t-\tau+1}$
$p\left(s_{t}|s_{t-\tau}\right)=\int\cdots\int
p\left(s_{t},\ldots,s_{t-\tau+1}|s_{t-\tau}\right)\,\mathrm{d}s_{t-1}\cdots\,\mathrm{d}s_{t-\tau+1}$
(3)
and the related expression for $p\left(s_{t},s_{t-1}|s_{t-\tau}\right)$, where
$p\left(s_{t},\ldots,s_{t-\tau+1}|s_{t-\tau}\right)$ is the joint probability
distribution for the process in the time window, conditioned on $s_{t-\tau}$.
As to notation, the conditional probabilities depend explicitly on time so
that $p\left(s_{t^{\prime}}|s_{t^{\prime}-1}\right)$ is different from
$p\left(s_{t^{\prime\prime}}|s_{t^{\prime\prime}-1}\right)$ in general if
$t^{\prime}\neq t^{\prime\prime}$, with equality only in the stationary state.
As a result we obtain the new quantity, written in a short-hand notation as
$I^{\tau}\left(S_{t};S_{t-1}\right):=I\left(S_{t};S_{t-1}|S_{t-\tau}=s_{t-\tau}\right)$
(4)
which we call _time-local predictive information_ (TiPI). Note the difference
to the conditional mutual information where an averaging over $s_{t-\tau}$
would take place. Analogously we define the time local entropy as
$H^{\tau}\left(S_{t}\right):=H\left(S_{t}|S_{t-\tau}=s_{t-\tau}\right)$ (5)
### 2.2 Estimating the TiPI
To evaluate the TiPI only the kernels have to be known which can be sampled by
the agent on the basis of the measured sensor values. However, in order to get
explicit update rules driving the increase of the TiPI, these kernels have to
be known as a function of the parameters of the system, in particular those of
the controller. This can be done by learning the kernels as a function of the
parameters. A related approach, followed in this paper, is to learn a model of
the time series, i.e. learning a function
$\psi:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ acting as a time series
predictor $S_{t}=\psi\left(S_{t-1}\right)+\Xi$ with realization
$s_{t}=\psi\left(s_{t-1}\right)+\xi_{t}$ (6)
for any time $t$, $\xi_{t}$ being the prediction error, also called the noise
in the following. $\psi$ can be realized for instance by a neural network that
can be trained with any of the standard supervised learning techniques. A
concrete example will be considered below, see eq. (25). The relation to the
kernel notation is obtained by observing that
$p\left(s_{t}|s_{t-1}\right)=\int\delta\left(s_{t}-\psi\left(s_{t-1}\right)-\xi_{t}\right)p_{\Xi}\left(\xi_{t}\right)\,\mathrm{d}\xi_{t}=p_{\Xi}\left(s_{t}-\psi\left(s_{t-1}\right)\right)$
(7)
where $\delta\left(x\right)$ is the Dirac delta distribution and
$p_{\Xi}\left(\xi\right)$ is the probability density of the random variable
$\Xi$ (prediction error) which may depend on the state $s$ itself
(multiplicative noise).
The case of linear systems, where $\psi\left(s\right)=Ls+b$ with a constant
matrix $L$, has been treated in Ay et al. [2] revealing many interesting
properties of the PI. How can we translate the findings of the linear systems
to the case of nonlinear systems? As it turns out, the nonlinearities
introduce many difficulties into the evaluation of the PI as it becomes clear
already in a one-dimensional bistable system as treated in Ay et al. [1].
Higher dimensional systems bring even more of such difficulties so that we
propose to consider the information quantity on a new basis. The idea is to
study the TiPI of the error propagation dynamics in the stochastic dynamical
system instead of the process $S_{t}$ itself.
#### 2.2.1 Error propagation dynamics
Let us introduce a new variable describing the deviation of the actual
dynamics, eq. (6), from the deterministic prediction in a certain time window.
We define for a time window starting at time $t-\tau$
$\delta
s_{t^{\prime}}=s_{t^{\prime}}-\psi^{t^{\prime}-\left(t-\tau\right)}\left(s_{t-\tau}\right)$
(8)
for any time $t^{\prime}$ with $t-\tau\leq t^{\prime}\leq t$ and
$\psi^{\left(0\right)}\left(s\right)=s$. As to notation, $\delta s$ denotes a
single variable not to be confused with the Dirac function. Intuitively
$\delta s_{t^{\prime}}$ captures how the prediction errors occurred since the
start of the time window are propagated up to time $t^{\prime}$.
Figure 1: The time window and the error propagation dynamics used for
calculating the TiPI, eq. (41). In principle, the process is considered many
times with always the same starting value but different realizations of the
noise $\xi$. Note that, when using the one-shot gradients, only one
realization is needed, see section 2.3.
Figure 1 illustrates the transformed state and the relevant distributions of
the belonging process $\delta S_{t}$. Interestingly the TiPI on the process
$\delta S$ is equivalent to the one on the original process $S$, see eq. (41).
The dynamics of the $\delta s$ can be approximated by linearization as
$\delta s_{t^{\prime}}=L\left(s_{t^{\prime}-1}\right)\delta
s_{t^{\prime}-1}+\xi_{t^{\prime}}+O(\|\xi_{t}\|^{2})$ (9)
using the Jacobian
$L_{ij}(s)=\frac{\partial\psi_{i}(s)}{\partial s_{j}}$ (10)
Assuming the prediction errors (noise) $\xi$ to be both small and Gaussian we
obtain an explicit expression for the TiPI on $\delta S$
$I^{\tau}\left(\delta S_{t}:\delta
S_{t-1}\right)=\frac{1}{2}\ln\left|\Sigma_{t}\right|-\frac{1}{2}\ln\left|D_{t}\right|$
(11)
where $\Sigma=\left\langle\delta s\delta s^{\top}\right\rangle$ is the
covariance matrix of $\delta S$ and $D=\left\langle\xi\xi^{\top}\right\rangle$
is the covariance matrix of the noise. The derivation and further details are
in the Appendix section A. The results for linear systems in Ay et al. [2] can
be obtained from the general case considered here by $\tau\rightarrow\infty$.
When looking at eq. (11) one sees that the entropies are expressed in terms of
covariance matrices. This is exact in the case of Gaussian distributions. In
the general case this may be considered as an approximation to the true TiPI.
Alternatively, we can also consider eq. (11) as the definition of a new
objective function for any process if we agree to measure variability not in
terms of entropies but more directly in terms of the covariance matrices.
### 2.3 The exploration dynamics
Our aim is the derivation of an algorithm driving the behavior of the agent
toward increasing TiPI. Let us assume that the function
$\psi:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ depends on a set of parameters
$\theta$ so that we may write the dynamics as
$s_{t}=\psi\left(s_{t-1},\theta_{t}\right)+\xi_{t+1}$ (12)
For instance, if $\psi$ is a neural network as introduced further below, the
parameter set $\theta$ comprises just the synaptic weights and threshold
values of the neurons.
#### 2.3.1 Gradient ascending the TiPI
Based on the TiPI, eq. (11), a rule for the parameter dynamics is given by the
gradient step to be executed at each time $t$
$\Delta\theta_{t}=\varepsilon\frac{\partial
I^{\tau}}{\partial\theta}=\varepsilon\frac{\partial}{\partial\theta}\ln\left|\Sigma_{t}\right|$
(13)
where $\varepsilon$ is the update rate and
$\theta_{t+1}=\theta_{t}+\Delta\theta_{t}$. The term $\ln|D|$ from eq. (11)
has been omitted assuming that $\xi$ is essentially noise which is not
depending on the parameters of the controller. This is justifiable in the case
of parsimonious control as realized by the low-complexity controller networks.
These generate typically well predictable (low noise) behaviors as shown in
the applications studied below.
In order to get more explicit expressions, let us consider the case of very
short time windows. With $\tau=1$ there is no learning signal since $\Sigma=D$
meaning that $I^{\tau}=0$. So, $\tau=2$ is the most simple nontrivial case.
The parameter dynamics is given by
$\Delta\theta_{t}=\varepsilon\delta u_{t}^{\top}\frac{\partial
L\left(t-1\right)}{\partial\theta}\delta s_{t-1}$ (14)
where $\delta s$ and the auxiliary vector $\delta u$ are given as
$\displaystyle\delta s_{t-1}$ $\displaystyle=s_{t-1}-\psi\left(s_{t-2}\right)$
(15) $\displaystyle\text{ }\delta s_{t}$
$\displaystyle=s_{t}-\psi\left(\psi\left(s_{t-2}\right)\right)\text{ }$ (16)
$\displaystyle\delta u_{t}$ $\displaystyle=\Sigma_{t}^{-1}\delta s_{t}$ (17)
$\displaystyle\Sigma_{t}$ $\displaystyle=\left\langle\delta s_{t}\delta
s_{t}^{\top}\right\rangle$ (18)
stipulating the noise is different from zero (though possibly infinitesimal)
and employing the self-averaging property of a stochastic gradient, see below.
The general parameter dynamics for arbitrary $\tau$ is derived in Appendix
section B. However, in the applications described below, already the simple
parameter dynamics with $\tau=2$ will be seen to create most complex behaviors
of the considered physical robots.
In a nutshell, eq. (13) reveals already the main effect of TiPI maximization:
increasing $|\Sigma|$ means increasing the norm of $\delta s$ (in the
$\Sigma$-metric see eq. (56)). This is achieved by increasing the
amplification of small fluctuations in the sensorimotor dynamics which is
equivalent to increasing the instability of the system dynamics, see also the
more elaborate discussion in Der et al. [21].
#### 2.3.2 Learning vs. exploration dynamics
Usually, updating the parameters of a system according to a given objective is
called learning. In that sense, the gradient ascent on the TiPI defines a
learning dynamics. However, we would like to avoid this notion here, since
actually nothing is learnt. Instead by the interplay between the system and
the parameter dynamics, the combined system never reaches a final behavior
corresponding to the goal of a learning process. Therefore we prefer the
notion _exploration dynamics_ for the dynamics in the parameter space that is
driven by the TiPI maximization.
#### 2.3.3 One-shot gradients
The formulas for the gradient (eqs. (14) and (56)) were obtained by tacitly
invoking the self-averaging properties of the gradient, i. e. by simply
replacing $\left\langle\delta s\delta s^{\top}\right\rangle$ with $\delta
s\delta s^{\top}$ in eq. (51). This still needs a little discussion. Actually,
the self-averaging is exactly valid only in the limit of sufficiently small
$\varepsilon$, with $\varepsilon$ eventually being driven to zero in a
convenient way. However, our scenario is different. What we are aiming at is
the derivation of an intrinsic mechanism for the self-determined and self-
directed exploration using the TiPI and related objectives. The essential
point is that self-exploration is driven by a deterministic function of the
states (sensor values) of the system itself.
Equation (14) obtained from the gradient of the TiPI fulfills these aims very
well—any change of the system parameters and hence of the behavior is given in
terms of the predecessor states in the short time window. With finite (and
often quite large) $\varepsilon$ eqs. (14)-(18) are just a rough approximation
of the original TiPI but, in view of our goal, the one-shot nature of the
gradient is favorable as it supports the explorative nature of the exploration
dynamics generating interesting synergy effects.
#### 2.3.4 Synergy of system and exploration dynamics
A further central aspect of our approach is the interplay between the system
and the parameter dynamics driven by the TiPI maximization process. In
specific cases, the latter may show convergence as in conventional approaches
based on stationary states. An example is given by the one-parameter system
studied in Ay et al. [1] realizing convergence to the so called effective
bifurcation point. However, with a richer parametrization and/or more complex
systems, instead of convergence, the combined system (state + parameter
dynamics) never comes to a steady state due to the intensive interplay between
the two dynamical components if $\varepsilon$ is kept finite. An example will
be given in the Results section.
Typically, the TiPI landscape permanently changes its shape due to the fact
that increasing the TiPI means in general a destabilization of the system
dynamics. If the latter is in an attractor, increasing the TiPI destabilizes
the attractor until it may disappear altogether with a complete restructuring
of the TiPI landscape. This is but one of the possible scenarios where the
exploration dynamics engages into an intensive and persistent interplay with
the system dynamics. This interplay leads to many synergistic effects between
system and exploration dynamics and makes the actual flavor of the method.
#### 2.3.5 Self-directed search
The common approach to solve the exploration–exploitation dilemma in learning
problems is to use some randomization of actions in order to get the necessary
exploration and then decrease the randomness to exploit the skills acquired so
far. This is prone to the curse of dimensionality if the systems are gaining
some complexity. Randomness can also be introduced by using a deterministic
policy with a random component in the parameters, as quite successfully
applied to evolution strategies and reinforcement learning [54, 27].
Our approach is also to use deterministic policies (given by the function $K$)
but aims at making exploration part of the policy. So, instead of relegating
exploration to the obscure activities of a random number generator, variation
of actions should be generated by the responses of the system itself. This
replaces randomness with spontaneity and is hoped (and will be demonstrated)
to restrict the search space automatically to the physically relevant
dimensions defined by the embodiment of the system.
Formally, we call a search self-directed if there exists a function $\alpha$
so that the change in the parameters
$\Delta\theta_{t}=\alpha\left(s_{t},\ldots,s_{t-\tau},\theta_{t}\right)$ (19)
is given as a deterministic function of the states in a certain time window
(of length $\tau$) and the parameter set $\theta$ itself. In this paper,
$\alpha$ is given by the gradient of the predictive information in the one-
shot formulation, see section 2.3.3.
In more general terms, we believe that randomization of actions makes the
agent heteronomous, its fate being determined by an obscure (to him) procedure
(the pseudo-random number generator) alien to the nature of its dynamics. The
agent is autonomous in the ’genuine’ sense only if it varies its actions
exclusively by its own internal laws [49]. In our approach, according to eq.
(19), exploration is driven entirely by the dynamics of the system itself so
that exploration is coupled in an intimate way to the pattern of behavior the
robot is currently in. The danger might be that in this way the exploration is
restricted too much. As our experiments show, this is not so for active motion
patterns in high dimensional systems. This fact can be attributed to the
destabilization effect incurred by the TiPI maximization, see above and [21].
For stabilizing behaviors, however, the exploration may be too restrictive.
### 2.4 The sensorimotor loop
Let us now specify the above expressions to the case of a sensorimotor loop,
in particular a neurally controlled robotic system. The dynamical systems
formulation is obtained now by writing our predictor for the next sensor
values as a function of both the sensors and the actions so that
$s_{t}=\phi\left(s_{t-1},a_{t-1}\right)+\xi_{t}$ (20)
where $\phi$ represents the so-called forward model and $\xi_{t}$ is the
prediction error as before. As the next step, we consider the controller also
as a deterministic function $K:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$
generating actions (motor values) $a_{t}\in\mathbb{R}^{m}$ as a function of
the sensor values $s_{t}\in\mathbb{R}^{n}$ so that
$a_{t}=K\left(s_{t}\right)$ (21)
In the applications, $K$ will be realized as a (feed-forward) neural network.
Using eq. (21) in eq. (20) we obtain the map $\psi$ modeling our sensor
process as
$\psi\left(s_{t-1}\right)=\phi\left(s_{t-1},K\left(s_{t-1}\right)\right)$ (22)
In Ay et al. [2] a standard linear control system was studied where
$K\left(s\right)=Cs\text{, \ }\phi\left(s,a\right)=Ts+Va$ and
$\psi\left(s\right)=\left(T+VC\right)s$. This paper will consider a nonlinear
generalization of that case in specific robotic applications.
#### 2.4.1 Exploration dynamics for neural control systems
In the present setting, we assume that both the controller $K$ and the forward
model $\phi$ of our robot are realized by neural networks, the controller
being given by a single-layer neural network as
$K\left(s\right)=g\left(Cs+h\right)$ (23)
the set of parameters $\theta$ now given by $C$ and $h$. In the concrete
applications to be given below, we specifically use
$g_{i}\left(z\right)=\tanh\left(z_{i}\right)$ (to be understood as a vector
function so that $g:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$).
Moreover, the forward model $\phi$ is given by a layer of linear neurons, so
that
$\phi\left(s,a\right)=Va+Ts+b$ (24)
The matrices $V$, $T$ and the vector $b$ represent the parametrization of the
forward model that is adapted on-line by a supervised gradient procedure to
minimize the prediction error $\xi^{\top}\xi$ as
$\Delta V=\eta_{\phi}\xi a^{\top}\text{,\ \ }\Delta T=\eta_{\phi}\xi
s^{\top}\text{,\ \ }\Delta b=\eta_{\phi}\xi$ (25)
In the applications, the learning rate $\eta_{\phi}$ is large such that the
low complexity of the model is compensated by a very fast adaptation process.
In contrast to the forward model parameters, the controller parameters are to
be adapted to maximize the TiPI. For that the map $\psi$ (eq. (22)) is
required which becomes $\psi\left(s\right)=Vg\left(Cs+h\right)+Ts+b$ with
Jacobian matrix
$L=VG^{\prime}\left(z\right)C+T$ (26)
where $z=Cs+h$ is the postsynaptic potential and
$G^{\prime}(z)=\textrm{diag}[g^{\prime}(z_{1}),\dots,g^{\prime}(z_{m})]$ (27)
is the diagonal matrix of the derivatives of the activation functions for each
control neuron.
In the applications given below, we are using the short-time window, with the
general exploration dynamics given by eq. (14). The explicit exploration
dynamics for this neural setting with $g(z)=\tanh(z)$ are given as
$\displaystyle\frac{1}{\varepsilon}\Delta C_{ij}$
$\displaystyle=\delta\mu_{i}\delta s_{j}-\gamma_{i}a_{i}s_{j}$ (28)
$\displaystyle\frac{1}{\varepsilon}\Delta h_{i}$
$\displaystyle=-\gamma_{i}a_{i}$ (29)
where all variables are time dependent and are at time $t$, except $\delta s$
which is at time $t-1$. The vector $\delta\mu\in\mathbb{R}^{m}$ is defined as
$\delta\mu_{t}=G^{\prime}V^{\top}\delta
u_{t}=G^{\prime}V^{\top}\Sigma^{-1}\delta s_{t}$ (30)
(see eq. (17)), and the channel specific learning rates $\gamma_{i}$ are
$\gamma_{i}=2\left(C\delta s_{t-1}\right)_{i}\delta\mu_{i}$ (31)
The derivation and generalization to aribrary activation functions are
provided in the Appendix section C. The update rules for $\tau>2$ are given by
a sum of such terms, with appropriate redefinitions of the vector $\delta\mu$,
see eq. (55).
#### 2.4.2 The Hebbian nature of the update rules
In order to interpret these rules in more neural terms, we at first note that
the last term in eq. (28) is of an anti-Hebbian structure. In fact, it is
given by the product of the output value $a_{i}$ of neuron $i$ times the input
$s_{j}$ into the $j$-th synapse of that neuron, the $\gamma_{i}$ (which are
positive, as a rule) being interpreted as a neuron specific learning rate.
Moreover, we may also consider the term $\delta\mu_{i}\delta s_{j}$ as a kind
of Hebbian since it is again given by a product of values that are present at
the ports of the synapse $j$ of neuron $i$. The factor $\delta s_{j}$ can be
considered as a signal directly feeding into the input side of the synapse
$C_{ij}$. Moreover, $\delta\mu$ given as $\delta\mu=G^{\prime}V^{\top}\delta
u$ is obtained by using $\delta u$ as the vector of output errors in the
$\psi$ network and propagating this error back to the layer of the motor
neurons by means of the standard backpropagation algorithm.
These results make the generalization to more complicated, multi-layer
networks straightforward. However already the simple setting produces an
overwhelming behavioral variety, see the case studies below.
More intuitively the Hebbian term acts as a self-amplification and increases
the Lyapunov exponents. In the linear case [2] this leads eventually to the
divergence of the dynamics such that the PI does not exist any longer. With
the nonlinearities, the latter effect is avoided, but the system is driven
into the saturation region of the motor neurons. However, the second term in
eq. (28), by its anti-Hebbian nature, is seen to counteract this tendency. The
net effect of both terms is to drive the motor neurons towards a working
regime where the reaction of the motors to the changes in sensor values is
maximal. This is understandable, given that maximum entropy in the sensor
values requires a high sensorial variety that can be achieved by that
strategy.
## 3 Results
We apply our theory to three case studies to illuminate the main features.
First a hysteresis systems is considered to exemplify the consequences of
nonstationarity and the resulting interplay between the exploration dynamics
and the system dynamics in a nutshell. In section 3.2 a physical system of
many degrees of freedom is controlled by independent controllers that
spontaneously cooperate. Finally in section 3.3 we apply the method to a
jointly controlled humanoid robot in various situations to illustrate the
exploration process in a high-dimensional embodied system.
### 3.1 Hysteresis systems
Nonstationary processes are the main target of our theory, made accessible by
the special windowing and averaging technique presented in this paper for the
first time. In order to work out the consequences, let us consider an
idealized situation where the above derivations, in particular eqs. (14)-(18),
are the exact update rules for increasing the TiPI.
Let us consider a single neuron in an idealized sensorimotor loop, where the
sensor values are $s_{t}=a_{t-1}+\xi_{t}$ (the white Gaussian noise $\xi$ is
added explicitly). This case corresponds to the dynamical system
$s_{t}=\tanh\left(Cs_{t-1}+h\right)+\xi_{t}$ (32)
where now $s_{t}\in\mathbb{R}^{1}$. The system was studied earlier [21] in the
special case of $h=0$ and it was shown that the maximization of the PI self-
regulates the system parameter $C$ towards a slightly supercritical value
$(1<C\ll 2)$. There, the system is at the so called effective bifurcation
point where it is bistable but still sensitive to the noise.
Let us start with keeping $C$ fixed at some supercritical value (e. g.
$C=1.1$) and concentrating on the behavior of the bistable system as a
function of the threshold value $h$. The interesting point is that the system
shows hysteresis. This can be demonstrated best by rewriting the dynamics in
state space as a gradient descent. Let us introduce the postsynaptic potential
$z_{t}=Cs_{t}+h$ and rewrite eq. (32) in terms of $z_{t}$ as
$\Delta z_{t}=-\frac{\partial}{\partial z_{t}}U\left(z_{t}\right)+\xi_{t+1}$
(33)
where $\Delta z_{t}=z_{t+1}-z_{t}$ and the potential is
$U\left(z\right)=-C\ln\cosh z-hz+\frac{z^{2}}{2}$ (using
$\frac{\partial}{\partial z}\ln\cosh z=\tanh z$). In that picture, the
hysteresis properties of the system are most easily demonstrated by Fig. 2.
Figure 2: The hysteresis cycle in the gradient picture. The diagrams show the
stages of one hysteresis cycle starting from $h=0$ (A) with the state at $z>0$
as represented by the sphere. Decreasing $h$ creates the asymmetric situation
(B). If $h=-h_{c}$ the saddle-node bifurcation happens, i. e. both the maximum
at z=0 and the right minimum disappear so that the system shifts to the left
minimum of the potential (C). Increasing $h$ until $h=0$ brings us back to the
initial situation with the state shifted to the other well see (D,E). The
diagrams (F) and (G) depict the switching from the minimum at $z<0$ to the
minimum at $z>0$ by increasing $h$. By decreasing $h$ until $h=0$ the
hysteresis cycle is finished, see (H,I).
This phenomenon can be related directly to the destabilization effect of the
exploration dynamics. In the potential picture, stability is increasing with
the well depth. Hence, the exploration dynamics, aiming at the destabilization
of the system, is decreasing the depth of the well more and more until the
well disappears altogether, see Fig. 2, and the state switches to the other
well where the procedure restarts.
#### 3.1.1 Deterministic self-induced hysteresis oscillation
Now we show that in the one-dimensional case the parameter dynamics is
independent of white noise. This implies we can in the state dynamics make the
limit of vanishing noise strength and obtain a fully deterministic system.
Again we only consider the two-step window ($\tau=2$). Using $\delta
s_{t}=\xi_{t}+L\delta s_{t-1}=\xi_{t}+L\xi_{t-1}$ (eq. (9)) we find that the
TiPI, according to eq. (50)
$I^{\tau}=\frac{1}{2}\ln\left(1+L^{2}\right)$
is independent of the noise. Analogously to eqs. (28)-(31) we obtain the
update rules for $C$ and $h$ as the gradient ascent on $I^{\tau}$ and thus the
full state-parameter dynamics (with $|\xi|\to 0$) is given by
$\displaystyle s_{t}$ $\displaystyle=g\left(C_{t-1}s_{t-1}+h_{t-1}\right)$
(34) $\displaystyle C_{t}$
$\displaystyle=C_{t-1}+\gamma\left(1/(2C_{t-1})-s_{t-1}a_{t-1}\right)$ (35)
$\displaystyle h_{t}$ $\displaystyle=h_{t-1}-\gamma s_{t}$ (36)
with $\gamma=2L^{2}/(1+L^{2})$.
Apart from the definition of $\gamma$ (that just modulates the speed of the
parameter dynamics), the extended dynamical system agrees in the one-
dimensional case with that derived from the principle of homeokinesis,
discussed in detail in Der and Martius [17]. Let us therefore only briefly
sketch the most salient features of the dynamics. Keeping $C$ fixed at some
supercritical value, as above the most important point is that, instead of
converging towards a state of maximum TiPI, the $h$ dynamics drives the neuron
through its hysteresis cycle as shown in Fig. 2, which we call a self-induced
hysteresis oscillation, see Fig. 3(A).
A $C=1.2$ | B full dynamics
---|---
|
Figure 3: State and parameter dynamics in the one-dimensional system. (A) Only
$h$ dynamics (fixed $C=1.2$); the bias $h$ oscillates around zero and causes
the state $s$ to jump between the positive and negative fixed points. The TiPI
is seen to increase steadily until it eventually drops back when the state is
jumping. (B) With full dynamics ($C,h$). $C$ increases until it oscillates
around its average at $C\approx 1.2$ where the hysteresis cycle starts.
Parameters: $h_{0}=0.1$, $s_{0}=0.8$, $C_{0}=0$ in (B), $\varepsilon=0.002$
For the full dynamics (with eq. (35)) the results are given in Fig. 3(B)
showing that the feedback strength $C$ in the loop converges indeed toward the
regime with the hysteresis oscillation. This demonstrates that the latter is
not an artifact present only under the specific parametrization. In fact, we
encounter this phenomenon in many applications with complex high-dimensional
robotic systems, see the experiments with the Armband below and many examples
treated in Der and Martius [17].
Interestingly this behavior is not restricted to simple hysteresis systems but
is of more general relevance. For instance, in two-dimensional systems a
second order hysteresis was observed, corresponding to a sweep through the
frequency space of the self-induced oscillations [17]. It would be interesting
to relate this fast synaptic dynamics to the spike-timing-dependent plasticity
[38] or other plasticity rules [63] found in the brain.
#### 3.1.2 About time windows
Before giving the applications to embodied systems, let us have a few remarks
on the special nature of the time windowing technique as compared to the
common settings. Let us consider again the bistable system with the bias $h$
as the only parameter and with finite noise. Figure 4 depicts a typical
situation with $h\neq 0$ so that the wells are of different depth. The figure
depicts the qualitative difference between the classical attitude of
considering information measures in very large time windows, large enough for
the process to reach total equilibrium, as compared to our nonstationarity
approach where the TiPI is estimated on the basis of a comparatively short
window444Note that the time to stay in a well is exponentially increasing with
the depth of the well and decreasing exponentially with the strength of the
noise [48]. Mean first passage times can readily exceed physical times (on the
time scale of the behavior) by orders of magnitudes..
Figure 4: The probability density distributions with different time windows of
the stochastic process in an asymmetric double well potential. The mean first
passage time $T_{f}$ of switching between wells is one characteristic time
constant of the process [48], $T_{f}$ increasing exponentially with the
barrier height. If observing the process in a window of length $T\gg T_{f}$,
the distribution of (A) will be observed. In that situation, the TiPI is
maximal if the wells are of equal depth ($h=0$). However, with windows of
length $T\ll T_{f}$, the system state will be predominantly in one of the
wells generating the distributions shown in (B), (C). Gradient ascending the
TiPI will decrease the well depth as long as the probability mass is still
concentrated in that well. This is what drives the hysteresis cycle depicted
in Fig. 2.
While in the former case convergence of the hysteresis parameter $h$ towards
the equilibrium condition $h=0$ is reached, there is no convergence in the
nonstationary case. Instead, one obtains a self-induced hysteresis
oscillation. This is generic for a large class of phenomena based on the
synergy effects between system and exploration dynamics, see section 2.3.4,
which open new horizons for the explorative capabilities of the agent. In the
context of homeokinesis, this phenomenon has already been investigated in many
applications, see Der and Martius [17]. This paper provides a new, information
theoretic basis and opens new horizons for applications as the matrix
inversions inherent to the homeokinesis approach are avoided.
### 3.2 Spontaneous cooperation with decentralized control
Let us now give examples illustrating the specific properties of the present
approach. We start with an example of strongly decentralized control where the
TiPI driven parameter dynamics leads to the emergence of collective modes.
Earlier papers have already demonstrated this phenomenon for a chain of
passively coupled mobile robots [1, 21, 65]. In the setting of Ay et al. [1],
Der et al. [21], each wheel was being controlled by a single neuron with a
synapse of strength $C$ defining the feedback strength in each of the
sensorimotor loops. There was no bias. As it turned out, the TiPI in the
sensorimotor loop is maximal if the synaptic strength $C$ is at its critical
value where the system is bistable but still reacts to the external
perturbations, i. e. the system is at its so-called effective bifurcation
point [17]. As compared to the present setting, these results correspond to
using a time window of infinite length, stipulating the presence of a
stationary state.
The situation is entirely different when using the short time window and large
update rates allowing for the synergy effects. In experiments with the robot
chain, we observe better cooperativity with the hysteresis oscillations and
better exploration capabilities. The reason can be seen in the fact that the
self-regulated bias oscillations help the chain to better get out of impasse
situations. We do not give details here, since we will study in the following
an example that demonstrates the synergy effects even more convincingly.
#### 3.2.1 The Armband
The Armband considered here is a complicated physical object with $18$ degrees
of freedom, see Fig. 5. The physics of the robot is simulated realistically in
the LpzRobots simulator [40]. The program source code for this and the next
simulation is available from [41]. Each joint is controlled by an individual
controller, a single neuron driven by TiPI maximization, as with the robot
chain treated in Der et al. [21]. The controller receives the measured joint
angle or slider position and the output of the controller defines the target
joint angle or target slider position to be realized by the motor. The motors
are implemented as simulated servomotors in order to be as close to reality as
possible. Moreover, the forces are limited so that, due to the interaction
with obstacles or the entanglement of the system’s different degrees of
freedom, the true joint angle may differ substantially from the target angle.
These deviations drive the interplay between system and exploration dynamics.
Figure 5: The Armband. The robot has 12 hinge and 6 slider joints, each
actuated by a servo motor and equipped with a proprioceptive sensor measuring
the joint angle or slider length. The robot is strongly underactuated so that
it can not take on a wheel like form where locomotion were trivial.
In the experiments, we use the controller given by eq. (23) and the update
rules for the parameter dynamics as given by eqs. (35) and (36). The adaptive
forward model is given by eq. (24) with $T=0$ and the appropriate learning
rules eq. (25). In order to demonstrate the constitutive role of the synergy
effect, we started by studying the system with fixed $C$ and $h=0$. In
contrast to the chain of mobile robots, with fixed parameters there is no
parameter regime where the Armband shows substantial locomotion. This result
suggests that, as compared to the chain of mobile robots, the specific
embodiment of the Armband is more demanding for the emergence of the
collective effect.
In order to assess the effects appropriately, note that potential locomotion
depends on the forces the motors are able to realize. For instance, if the
robot is strongly actuated, the command $a=0$ for each of the motors drives
each joint to its center position so that the shape of the robot is nearly
circular, locomotion readily taking place under the influence of very weak
external influences. In order to avoid such trivial effects, we use an
underactuated setting so that gravitational or environmental forces are
deforming the robot substantially, see Fig. 5.
Figure 6: Role of the fast synaptic dynamics: depending on the speed of the
synaptic dynamics defined by $\varepsilon$, the locomotion properties are
changing drastically. Depicted is the distance traveled by the robot in 10 min
simulated time on an empty plane. The inset gives a close up view for low
$\varepsilon$, demonstrating that the locomotion starts only if $\varepsilon$
exceeds a certain threshold value. Shown is the mean and standard deviation of
10 runs each.
The situation changes drastically if the $h$ dynamics is included. As
demonstrated by Fig. 6, substantial locomotion sets in only if $\varepsilon$
is large enough so that the exploration dynamics is sufficiently fast for the
synergy effect to unfold. Also, as the experiments show, the effect is stable
for a very wide range of $\varepsilon$ and under varying external conditions.
It is also notable, that the Armband robot shows a definite reaction to
external influences. For instance, obstacles in its path are either surmounted
or cause the robot to invert its velocity, see Fig. 7. The latter effect is
observed in particular in the underactuated regime defined above, so that the
reflection is not the result of the elastic collision but it is actively
controlled by the involvement of the exploration dynamics. The role of the
latter is also demonstrated by the fact that locomotion stops as soon as the
update rate $\varepsilon$ is put to zero, see Fig. 8 and the corresponding
video S1 on [41].
Figure 7: Regular locomotion pattern and interaction with the environment.
Plotted are the center positions of the 6 rigid segments in space for an
interval of 40 sec. One line is highlighted for visibility. The trajectory
starts while the robot is moving to the left (A) and is hitting the wall (B)
(black box) and locomotes to the right (C) showing a very regular pattern.
Then it overcomes an obstacle (D) and hits the wall (E) and moves back (F).
The behavior is cyclic. Parameter: $\varepsilon=0.5$. Figure 8: Armband robot
surmounting an obstacle and inverting speed at a wall. Screen shots from the
simulation for Fig. 7. The order is row-wise from left to right. The last two
pictures show the situation after switching off the parameter dynamics
$\varepsilon=0$ for a few seconds (the robots stops) and enabling it again
(starts moving).
The Armband has also been investigated recently using artificial evolution for
the controller [47], demonstrating convincingly the usefulness of the
evolution strategy for obtaining recurrent neural networks that make the
Armband roll into a given direction. There are several differences to our
approach, both conceptually and in the results. While in the evolution
strategy the fitness function was designed for the specific task and many
generations were necessary to get the performance, in our approach the rolling
modes are emerging right away by themselves. Moreover, the modes are sensitive
to the environment, for instance by inverting velocity upon collisions with a
wall, they are flexible (changing to a jumping behavior on several occasions)
and resilient under widely differing physical conditions. Interestingly, these
behaviors are achieved with an extremely simple neural controller, the
functionality of a recurrent network being substituted by the fast synaptic
dynamics.
### 3.3 High dimensional case – the Humanoid
Let us now study the properties of the exploration dynamics in a general (not
decentralized) control task. We consider a humanoid robot with $17$ degrees of
freedom. Each joint is driven by a simulated servo motor, the motor values
$a\in\mathbb{R}^{17}$ sent by the controller are the target angles of the
joints and sensor values $s\in\mathbb{R}^{17}$ are the true, observed angles.
This is the only knowledge the robot has about its physical state.
The aim of this experiment is to investigate in how far the robot develops
behaviors with high variability so that it explores its sensorimotor
contingencies. Given that there is no externally defined goal for the behavior
development, will the robot develop a high behavioral variety depending on its
physics and the environment it is dynamically embedded into?
A normal | B bungee | C high bar | D pit
---|---|---|---
| | |
Figure 9: The Humanoid robot in four different scenarios. (A) Normal
environment with flat ground. (B) The robot is hanging at a bungee like
spring. (C) The robot is attached to a high bar. (D) Robot is fallen into a
narrow pit.
That this happens indeed is demonstrated by the videos S2, S3, S4 and S5 on
[41]. However, we want a more objective quantity to assess the relation
between body and behavior. We provide two different measures for that purpose.
One idea is to use the parameter constellation of the controller itself for
characterizing the behavior—different behaviors should reflect in
characteristic parameter configurations of the controller. In order to study
this idea, we place the robot in different scenarios, see Fig. 9, always
starting with the same initial parameter configuration (using the result of a
preparatory learning phase in the bungee setting), letting the robot move
independently for 40 min physical time. Without any additional noise, the
dynamics is deterministic so that variations are introduced by starting the
robot in different poses, i. e. in a straight upright position and in slightly
tilted poses ($0.5^{\circ}$ and $5^{\circ}$ slanted to the front). We then
compared the parameter values of the controller matrix $C$ at each second (1
s) for all simulations and calculated a hierarchical clustering reflecting the
differences between the $C$ matrices. Figure 10 shows the resulting
dendrogram.
Figure 10: Parameter similarity for the behavior in different environments
(Fig. 9). Plotted is the results of a hierarchical clustering based on the
difference between the parameters in each of the simulations (averaged over
time). For each of the four environments there are three initial poses:
$0^{\circ}$ (straight upright), $0.5^{\circ}$ and $5^{\circ}$ slanted to the
front. The parameters for runs in the same environment are clustered together.
This supports the observation that the embodiment plays an essential role in
the generation of behavior. More importantly the physical conditions are
reflected in the parameters and are thus internalized. We used the squared
norm of the difference of the absolute values of the matrix elements. The
absolute values were used because a common structure in the parameters are
rotation matrices and there the same qualitative behavior is obtained with
inverted signs. Parameters: $\varepsilon=0.001$ ($\eta=0.005$)
Obviously, there is a distinct grouping of the $C$ matrices according to the
environment the robot is in and the behaviors developing in the respective
situation. Distances between the groups are different, the most pronounced
group corresponding to the behavior in the pit situation. This seems plausible
since the constraints are most distinctive here, driving the robot to
behaviors that are markedly different from the situation with the bungee
setting, say, where all joints (extremities, hip, back) can move much more
freely. There is a second pronounced group—the robot clinging to the high
bar—whereas the distances between the $C$ matrices controlling the robot lying
on the ground and hanging at the bungee rope is less pronounced. However, by
visual inspection the emerging behaviors in the two latter situations appear
quite different (compare videos S2 and S3 on [41])—a finding that is not so
clear in the matrix distance method.
Figure 11: Dimensionality of behavior on different time scales. Humanoid robot
in bungee setup running 40 min with different control settings. The sensor
data is partitioned into chunks of a fixed length, the graph depicting the
effective dimension over the length of the chunks for different settings. In
order to test the method we start with a uniformly distributed noise signal
for motor commands (“noise signal”). As expected the observed dimension is
maximal. The sensor values produced by that random controller show a lower
dimension (“noise ctrl.”) as is expected due to the low pass filtering
property of the mechanical system. All other cases are with the TiPI
maximization controller with different update rates $\varepsilon$. In
particular, the comparison with the $\varepsilon=0$ case demonstrates that the
exploration dynamics produces more complex behaviors than any fixed
controller.
Figure 12: Behavioral changes with time. Pairwise distances of chunks with
length 10 s. Distance is defined as the length of the vector of maximal
projections of the first 6 principal components.
In order to get an additional measure we start from the idea that the TiPI
maximization method produces a series of behaviors that are qualified by a
high dynamical complexity generated in a controlled way. The latter point
means that the dimensionality of the time series of the sensor values is much
less than that of the mechanical system – if the behavior of the robot is well
controlled (think of a walking pattern) a few master observables will be
sufficient to describe the dynamics of the mechanical system. We have tried
different methods from dynamical system theory for finding the effective
dimension of that time series without much success. The reason was found to be
in the strongly nonstationary nature of the compound dynamics (system plus
exploration dynamics) making low dimensional behaviors to emerge and disappear
in a rapid sequence. So, in the long run the full space of the dynamical
system is visited so that globally a seemingly high dimensional behavior is
observed.
In order to cope with this nonstationary characteristic, we developed a
different method, splitting the whole time series into chunks and using an
elementary principal component analysis (PCA) in order to define the effective
dimension in each chunk: on each chunk a PCA is performed and the number of
principal components required to capture $95\%$ of the data’s variance is
plotted (mean and standard deviation for all chunks of the same length). In
order to avoid discretization artifacts we linearly interpolate the required
number of components to obtain a real number.
The results presented in Fig. 11 corroborate the above hypothesis on the
dimensionality of the behaviors. In particular, we observe the increase of the
effective dimension if the chunk length is increasing, mixing different low
dimensional behaviors. The latter point is made even more obvious in Fig. 12
depicting the overlap between the behaviors in chunks at different times. This
overlap is large if the behaviors are essentially the same and small if the
behavior has changed in the time span between the chunks. As the figure
demonstrates, the overlap is indeed large for short time spans, but behaviors
can reemerge after some time. Altogether, the results demonstrate that our
TiPI maximization method effectively explores the behavior space of high-
dimensional robotic systems by exciting their low-dimensional modes, avoiding
in this way the curse of dimensionality.
## 4 Discussion
Can a robot develop its skills completely on its own, driven by the sole
objective to gain more and more information about its body and its interaction
with the world? This question raises immediately further issues such as (i)
what is the relevant information for the robot and (ii) how can one find a
convenient update rule that realizes the gradient ascent on this information
measure. We have studied the predictive information of the stream of sensor
values as a tentative answer to the first question and, based on that, could
give exact answers to the second question for simple cases. Earlier work was
restricted to linear systems [2]. In order to be applicable to actual robotic
systems we extend it to the case of nonlinear controllers and to nonstationary
processes leading to a new measure called TiPI (time-local predictive
information). Using several approximations we have been still able to obtained
analytical results. In this way we derived an explicit exploration dynamics
for the controller parameters based on an information maximization principle,
namely by maximizing the TiPI using gradient ascent. For neural networks the
gradient yields a fast synaptic dynamics which is essentially local in nature.
Interestingly the TiPI landscape (on which the gradient is calculated)
continuously changes its shape due to the general destabilization of the
system dynamics inherent in maximizing the TiPI. For instance if the system
dynamics is in an attractor, increasing the TiPI destabilizes the attractor
until it may disappear altogether with a complete restructuring of the TiPI
landscape. This is another reason why nonstationary processes have to be
handled and why no convergence of the parameter dynamics is desired.
We studied a one-dimensional hysteresis system in order to work out the
consequences of the nonstationary. The parameter dynamics leads to a slightly
supercritical regime and additionally a self-induced hysteresis oscillation
emerges. This is a useful new property as shown in the experiment with the
Armband robot, a high-dimensional robot with a complicated dynamics. Despite
the highly decentralized control—each joint is controlled individually—the
robot develops coherent and global pattern of behavior. This is enabled by the
continuous adaptation and spontaneous mutual cooperation of the individual
controllers (hysteresis elements). We find the effect to be very robust
against the speed of the exploration dynamics. Interestingly in the one-
dimensional case the update formulas are independent of white noise and we can
obtain an exploration dynamics in a fully deterministic system.
The new theoretical basis also allows for controlling complex high-dimensional
robotic systems. This is demonstrated by a series of experiments with the
Humanoid robot, now jointly controlled by a single high-dimensional
controller. Given that there is no externally defined goal for the behavior
development, will the robot develop a high behavioral variety depending on its
physics and the environment it is dynamically embedded into? Our results
support a positive answer to this question. We quantify the dimensionality and
temporal structure of the behavior and find a succession of low-dimensional
modes that increasingly explore the behavior space. Furthermore we show that
environmental factors influence the internal as well as behavioral
development. Without additional noise, the deterministic dynamics leads to an
individual development which depends decisively on the particular experiences
made during the lifetime.
The exploration dynamics can be viewed as a self-directed search process,
where the directions to explore are created from the dynamics of the system
itself. Without a random component the changes of the parameters are
deterministically given as a function of the sensor values and internal
parameters in a certain time window. For an embodied system this means in
particular that constraints, responses and current knowledge of the dynamical
interaction with the environment can directly be used to advance further
exploration. Randomness is replaced with spontaneity which we demonstrate to
restrict the search space automatically to the physically relevant dimensions.
Its effectiveness is shown in the Humanoid experiments and we argue that this
is a promising way to avoid the curse of dimensionality.
What is the relation of the parameter dynamics described here to other work on
maximizing information quantities in neural systems? Maximizing the mutual
information between input and output of a neuron, known as InfoMax, yields a
very similar parameter dynamics [5]. Interestingly, when applied to a feed-
forward network an independent component analysis can be performed. Also
similar rules have been obtained in Triesch [61] where the entropy of the
output of a neuron was maximized under the condition of a fixed average output
firing-rate Triesch [61]. The resulting dynamics is called intrinsic
plasticity as it acts on the membrane instead of on the synaptic level and it
was shown to result in the emergence of complex dynamical phenomena [11, 31,
62, 32]. In Markovic and Gros [36, 37] a related dynamics is obtained at the
synaptic level of a feedback circuit realized by an autaptic (self)
connection. In a recurrent network of such neurons it was shown that any
finite update rate ($\varepsilon$ in our case) destroys all attractors,
leading to intermittently bursting behavior and self-organized chaos.
Our work differs in two aspects. On the one hand, we use the information
theoretical principle at the behavioral level of the whole system by
maximizing the TiPI on the full sensorimotor loop, whereas they use it at the
neuronal level. Nevertheless we manage to root the information paradigm back
to the level of the synaptic dynamics of the involved neurons. On the other
hand, as a direct consequence of that approach, there is no need to specify
the average output activity of the neurons. Instead the latter is self-
regulating by the closed loop setting. Independent of the specific
realization, the general message is that these self-regulating neurons realize
a specific working regime where they are both active and sensitive to
influences of their environment. If embedded into a feedback setting many
interesting phenomena are produced. Instead of studying them in internal
(inside the “brain”) recurrences, we embed such neurons into a feedback loop
with complex physical systems where the self-active and highly responsive
nature of these neurons produces similar phenomena at the behavioral level.
In the current form, our approach is limited to the control of robots where
the sensorimotor dynamics can be, in its essence, modeled by a simple feed-
forward neural network. The parameter dynamics can also be calculated for more
complex controllers, such as recurrent networks, which remains for future
work. In this study only proprioceptive sensors measuring joint angles have
been used. However, our newest experiences have shown that also other sensors
e. g. current sensors, acceleration sensor or velocity sensors can be
successfully integrated.
To conclude, information theory is a powerful tool to express principles to
drive autonomous systems because it is domain invariant and allows for an
intuitive interpretation. We present for the first time, to our knowledge, a
method linking information theoretic quantities on the behavioral level
(sensor values) to explicit dynamical rules on the internal level (synaptic
weights) in a systematic way. This opens new horizons for the applicability of
information theory to the sensorimotor loop and autonomous systems.
## Acknowledgments
The project was supported by the DFG (SPP 1527).
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## Appendix A Estimating the TiPI
As stated above we consider the TiPI on the process of error propagations
because it allows us to derive explicit expressions. Thus we start with the
definition of the error propagation to derive eq. (11) and provide further
insights.
As a first step, using the notion of an orbit of the dynamical system defined
by the map $\psi:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ we define a sequence
of states $\hat{s}_{t^{\prime}}\in\mathbb{R}^{n}$
$\hat{s}_{t^{\prime}}=\psi^{t^{\prime}-\left(t-\tau\right)}\left(s_{t-\tau}\right)$
(37)
for any time $t^{\prime}$ within the time window $t-\tau\leq t^{\prime}\leq t$
starting from state $\hat{s}_{t-\tau}=s_{t-\tau}$. $\psi^{k}(s)$ denotes the
$k$-fold iteration of the map $\psi$ with $\psi^{(0)}\left(s\right)=s$. We can
consider $\hat{s}_{t^{\prime}}$ as the predicted state over
$t^{\prime}-\left(t-\tau\right)$ time steps. In particular, the prediction
over $\tau$ steps is $\hat{s}_{t}=\psi^{\tau}\left(s_{t-\tau}\right)$.
The error propagation can now be defined as the difference
$\delta s_{t^{\prime}}=s_{t^{\prime}}-\hat{s}_{t^{\prime}}$ (38)
between the true state $s_{t^{\prime}}$, eq. (6), and the state
$\hat{s}_{t^{\prime}}$ obtained by the deterministic dynamics ($\psi$), see
Fig. 1. The dynamics of the $\delta s_{t^{\prime}}$ obeys the rule555Proof:
Using $\hat{s}_{t^{\prime}}=\psi\left(\hat{s}_{t^{\prime}-1}\right)$ we write
$\displaystyle\delta s_{t^{\prime}}$
$\displaystyle=s_{t^{\prime}}-\hat{s}_{t^{\prime}}=\psi\left(s_{t^{\prime}-1}\right)+\xi_{t^{\prime}}-\psi\left(\hat{s}_{t^{\prime}-1}\right)$
$\displaystyle=\psi\left(\hat{s}_{t^{\prime}-1}+\delta
s_{t^{\prime}-1}\right)-\psi\left(\hat{s}_{t^{\prime}-1}\right)+\xi_{t^{\prime}}$
$\displaystyle=L\left(s_{t^{\prime}-1}\right)\delta
s_{t^{\prime}-1}+\xi_{t^{\prime}}+O(\|\xi\|^{2})$
$\delta s_{t^{\prime}}=L\left(s_{t^{\prime}-1}\right)\delta
s_{t^{\prime}-1}+\xi_{t^{\prime}}+O(\|\xi_{t}\|^{2})$ (39)
with starting state $\delta s_{t-\tau}=0$ and $L(s)$ denoting the Jacobian
matrix of $\psi$. In the following we will use this approximation which is
arbitrary good for infinitesimally small noise. Note that this dynamics
corresponds to that of a linear system666In a linear system, $L$ is
independent of the state. In this case
$\hat{s}_{t^{\prime}}=L\hat{s}_{t^{\prime}-1}$ such that the dynamical
evolution of $\delta s$ and $s$ are the same., however with state dependent
dynamical operator $L$.
As a remark, in the case of finite noise, we can obtain a related exact rule
by using the mean value theorem of differential calculus stating that under
mild restrictions one can find a state
$\tilde{s}_{t^{\prime}}\in[\hat{s}_{t^{\prime}},s_{t^{\prime}}]$ so that
$\delta s_{t^{\prime}}=L\left(\tilde{s}_{t^{\prime}-1}\right)\delta
s_{t^{\prime}-1}+\xi_{t^{\prime}}$ (40)
yields the exact dynamics of the multi-step prediction error $\delta s_{t}$.
The interesting point now is that $I^{\tau}\left(S_{t}:S_{t-1}\right)$ (eq.
(4)) is equal to that of the process defined by the error propagation
dynamics777Consider two random vectors $S$ and $S^{\prime}$ together with the
shifted vectors $U=S+a$ and $U^{\prime}=S^{\prime}+a^{\prime}$. Using that the
probability distribution functions (pdf) $p_{S}\left(s\right)$ and
$p_{U}\left(u\right)$ obey
$p_{U}\left(u\right)=p_{U}\left(s+a\right)=p_{S}\left(s\right)$ one obtains
$H\left(S\right)=H\left(U\right)$. Analogously, the joint pdf’s obey
$p_{UU^{\prime}}\left(u,u^{\prime}\right)=p_{UU}\left(s+a,s^{\prime}+a^{\prime}\right)=p_{SS^{\prime}}\left(s,s^{\prime}\right)$
so that $H\left(S^{\prime}|S\right)=H\left(U^{\prime}|U\right)$., i. e.
$I^{\tau}\left(S_{t}:S_{t-1}\right)=I^{\tau}\left(\delta S_{t}:\delta
S_{t-1}\right)$ (41)
This result is central for the following arguments—we will make use of the
fact that the dynamics eq. (39) is more easily treated to obtain explicit
estimates for the TiPI and its gradient.
#### Explicit expressions
By iterating eq. (39) we obtain an explicit expression for $\delta s_{t}$
(using here and in the following $L\left(t^{\prime}\right)$ for
$L\left(s\,_{t^{\prime}}\right)$)
$\delta s_{t}=\sum_{k=0}^{\tau-1}L^{\left(k\right)}\left(t-1\right)\xi_{t-k}$
(42)
with
$L^{\left(k\right)}\left(t-1\right)=L\left(t-1\right)\cdots
L\left(t-k\right)\text{, and }L^{\left(0\right)}=\mathbf{I}\text{ }$ (43)
for any $t$. In general it is very complicated to obtain the entropy of
$\delta S_{t}$ in realistic situations with high dimensional physical systems.
Therefore we will base the further considerations on a convenient estimate of
the latter. With white Gaussian noise, the process $\delta S_{t}$ is Gaussian
as well, i. e. $\delta S_{t}\sim\mathcal{N}\left(0,\Sigma_{t}\right)$ (it is a
linear combination of independent Gaussians), so that the entropy is given in
terms of the covariance matrix $\Sigma_{t}$ of the random vector $\delta
S_{t}$ as [12]
$H^{\tau}\left(\delta
S_{t}\right)=\frac{1}{2}\ln\left|\Sigma_{t}\right|+\frac{n}{2}\ln 2\pi e$ (44)
$\left|A\right|$ denoting the determinant of a square matrix $A$ and
$\Sigma_{t}=\left\langle\delta S_{t}\delta S_{t}^{\top}\right\rangle=\int
p\left(\delta s_{t}\right)\delta s_{t}\delta s_{t}^{\top}\,\mathrm{d}\delta
s_{t}$ (45)
is the covariance matrix of $\delta S_{t}$ and $p\left(\delta s_{t}\right)$ is
the probability density distribution of the random variable $\delta S_{t}$.
Using eq. (42), explicit expressions for $\Sigma$ can readily be obtained, see
eq. (49) below.
By the same arguments, the conditional entropy is defined, using eq. (7), as
$H^{\tau}\left(\delta S_{t}|\delta
S_{t-1}\right)=H^{\tau}\left(\Xi_{t}\right)=\frac{1}{2}\ln\left|D_{t}\right|+\frac{n}{2}\ln
2\pi e$ (46)
with
$D_{t}=\left\langle\Xi_{t}\Xi_{t}^{\top}\right\rangle=\int
p\left(\xi_{t}\right)\xi_{t}\xi_{t}^{\top}\,\mathrm{d}\xi_{t}$ (47)
where $\Xi$ denotes the process of the noise with $p\left(\xi\right)$ being
the probability density function of $\Xi\sim\mathcal{N}\left(0,D_{t}\right)$.
Thus we obtain the estimate of the TiPI as
$I^{\tau}\left(\delta S_{t}:\delta
S_{t-1}\right)=\frac{1}{2}\ln\left|\Sigma_{t}\right|-\frac{1}{2}\ln\left|D_{t}\right|$
(48)
which is the entropy of the state $\delta s$ minus that of the noise.
#### White noise
Explicit expressions revealing more details of the theory are obtained for the
case of white noise, meaning
$\left\langle\xi_{t}\xi_{t^{\prime}}^{\top}\right\rangle=\mathbf{0}$ if $t\neq
t^{\prime}$, so that using eq. (42) in eq. (45) yields
$\Sigma=\sum_{k=0}^{\tau-1}L^{\left(k\right)}D\left(L^{\left(k\right)}\right)^{\top}$
(49)
In particular, in the case of $\tau=2$, the shortest nontrivial time window,
we find
$\Sigma=D+LDL^{\top}.$
It is also useful to introduce the transformed dynamical operator888This
corresponds to using a so-called whitening transformation on the state
dynamics, replacing in eq. (40) the state vector $\delta s$ by a new vector
$\delta x=\sqrt{D^{-1}}\delta s$ so that the covariance matrix of the noise in
the $\delta x$ dynamics is just the unit matrix.
$\hat{L}=\sqrt{D^{-1}}L\sqrt{D}$ which leads to
$\Sigma=\sum_{k=0}^{\tau-1}\sqrt{D}\hat{L}^{\left(k\right)}\left(\hat{L}^{\left(k\right)}\right)^{\top}\sqrt{D}$
and (using $\left|\sqrt{D}M\sqrt{D}\right|=\left|MD\right|=$
$\left|M\right|\left|D\right|$)
$I^{\tau}\left(\delta S_{t}:\delta
S_{t-1}\right)=\frac{1}{2}\ln\left|\sum_{k=0}^{\tau-1}\hat{L}^{\left(k\right)}\left(\hat{L}^{\left(k\right)}\right)^{\top}\right|$
(50)
Interestingly, the $\hat{L}$ operators also exist if the overall noise
strength $\lambda=\left\|\xi\right\|$ goes to zero, so that $I^{\tau}$ stays
finite999Introducing $\hat{D}=\lambda^{-2}D$ where $\hat{D}$ stays finite with
$\lambda\rightarrow 0$, we have
$\hat{L}=\sqrt{\hat{D}^{-1}}L\sqrt{\hat{D}}=\sqrt{D}L\sqrt{D}$ since $\lambda$
cancels out. although the defining entropies, conditioned on the state
$s_{t-\tau}$, are equal to zero in the deterministic system.
#### The linear case
For linear systems explicit expressions for the PI were obtained in Ay et al.
[2]. In this case $L$ is not dependent on the state $s_{t}$ of the system so
that $L^{\left(k\right)}=L^{k}$ in eq. (43). Using eq. (49), with
$\tau\rightarrow\infty$, we reobtain the results101010Note that all
eigenvalues of the Jacobi matrix $L$ must be less than one by absolute value
so that the limes will exist. This requirement also guarantees that the
conditioning on $s_{t-\tau}$ looses its influence for $\tau\rightarrow\infty$.
of Ay et al. [2]. Under the additional assumption that $L$ is a normal matrix
and the noise is isotropic the explicit expression
$\Sigma=\left(\mathbb{I}-LL^{\top}\right)$ was obtained.
## Appendix B Explicit gradient step
In order to derive the general gradient step on the TiPI based on eq. (13) we
need to calculate the derivative
$\frac{\partial}{\partial\theta}\ln\left|\Sigma_{t}\right|$. Considering any
(square) matrix $M$ depending on a single parameter $\theta_{k}$ of the set
$\theta$ we have111111We write $\frac{1}{M}$ for $M^{-1}$ here and in the
following. (see for example Magnus and Neudecker [35])
$\frac{\partial}{\partial M}\ln\left|M\right|=\frac{1}{M^{\top}}$
and
$\frac{\partial}{\partial\theta_{k}}\ln\left|M\right|=\sum_{ij}M_{ji}^{-1}\frac{\partial
M_{ij}}{\partial\theta_{k}}=\mathrm{Tr}\left(\left(M^{-1}\right)^{\top}\frac{\partial
M}{\partial\theta_{k}}\right)$
so that, using $\Sigma=\Sigma^{\top}=\left\langle\delta s\delta
s^{\top}\right\rangle$ and omitting the time index
$\frac{\partial}{\partial\theta_{k}}\ln\left|\Sigma\right|=\mathrm{Tr}\left(\frac{1}{\Sigma}\frac{\partial}{\partial\theta_{k}}\left\langle\delta
s\delta s^{\top}\right\rangle\right)$ (51)
By using the cyclic invariance of the trace we obtain from eq. (51)
$\frac{\partial}{\partial\theta}\ln\left|\Sigma_{t}\right|=\left\langle\delta
s_{t}^{\top}\Sigma^{-1}\frac{\partial}{\partial\theta}\delta
s_{t}\right\rangle$ (52)
now valid for the entire set of parameters $\theta$. By eq. (40) we obtain
(ignoring the dependence of $\xi$ on the parameter)
$\frac{\partial}{\partial\theta}\delta s_{t^{\prime}}=\frac{\partial
L\left(t^{\prime}-1\right)}{\partial\theta}\delta
s_{t^{\prime}-1}+L\left(t^{\prime}-1\right)\frac{\partial}{\partial\theta}\delta
s_{t^{\prime}-1}$
so that by iteration
$\frac{\partial}{\partial\theta}\delta
s_{t}=\sum_{l=1}^{\tau-1}L^{\left(l-1\right)}\left(t-1\right)\frac{\partial
L\left(t-l\right)}{\partial\theta}\delta s_{t-l}$
where $L^{\left(k\right)}\left(t-1\right)$ is given in eq. (43). Using
$a^{\top}Wb=\left(W^{\top}a\right)^{\top}b$, we write
$\frac{\partial}{\partial\theta}\ln\left|\Sigma_{t}\right|=\sum_{l=1}^{\tau-1}\left\langle\delta
u_{t-l+1}^{\top}\frac{\partial L\left(t-l\right)}{\partial\theta}\delta
s_{t-l}\right\rangle$ (53)
where ($\Sigma$ is symmetric)
$\delta
u_{t-l+1}=\left(L^{\left(l-1\right)}\left(t-1\right)\right)^{\top}\Sigma_{t}^{-1}\delta
s_{t}$ (54)
Stipulating the self-averaging property of the stochastic gradient, see
section One-shot gradients for details, we realize the update rule as
$\Delta\theta=\varepsilon\sum_{l=1}^{\tau-1}\delta
u_{t-l+1}^{\top}\frac{\partial L\left(t-l\right)}{\partial\theta}\delta
s_{t-l}$ (55)
Here we see again that $\tau=2$ is the simplest non-trivial case where the sum
consists of a single term.
#### Characterizing the parameter dynamics
In order to better characterize the parameter dynamics, let us consider for
the moment $\Sigma$ at the r. h. s. of eq. (51) to be some fixed, positive
matrix (not depending on the parameters $\theta_{k}$). Then, we can write
$\mathrm{Tr}\left(\frac{1}{\Sigma}\frac{\partial}{\partial\theta_{k}}\left\langle\delta
s\delta
s^{\top}\right\rangle\right)=\frac{\partial}{\partial\theta_{k}}\left\langle\mathrm{Tr}\left(\frac{1}{\Sigma}\delta
s\delta
s^{\top}\right)\right\rangle=\frac{\partial}{\partial\theta_{k}}\left\langle\delta
s^{\top}\frac{1}{\Sigma}\delta s\right\rangle$
(using the cyclic invariance of the trace in the last step). The update rule
eq. (13) becomes using again the self-averaging
$\Delta\theta=\varepsilon\frac{\partial}{\partial\theta}\left\|\delta
s\right\|_{\Sigma}^{2}$ (56)
where $\left\|a\right\|_{M}^{2}=a^{\top}M^{-1}a$ defines the length of a
vector $a$ in the metric given by $M$ (considered fixed in the current
gradient step). From eq. (56) it becomes obvious that following the gradient
is to increase the norm of $\delta s$ in the $\Sigma$ metric.
## Appendix C Neural networks—derivation of the update rule
We derive the parameter dynamics for neural networks eq. (28) from the general
parameter dynamics for the two-step time window given by eq. (14). According
to eq. (26) we have $L=VG^{\prime}\left(z\right)C+T$ with $z=Cs+h$ and
$G^{\prime}(z)=\mathrm{diag}[g_{1}^{\prime}(z),\dots,g_{m}^{\prime}(z)]$.
Putting this into eq. (14) yields (omitting the time indices)
$\displaystyle\frac{1}{\varepsilon}\Delta C_{ij}$ $\displaystyle=\delta
u^{\top}\frac{\partial L}{\partial C_{ij}}\delta s$ $\displaystyle=\delta
u^{\top}VG^{\prime}\frac{\partial C}{\partial C_{ij}}\delta s+\delta
u^{\top}V\frac{\partial G^{\prime}}{\partial C_{ij}}C\delta s$
$\displaystyle=\left(G^{\prime}V^{\top}\delta u\right)_{i}\delta s_{j}+\delta
u^{\top}V\frac{\partial G^{\prime}}{\partial C_{ij}}C\delta s$ (57)
The second term remains to be calculated. Because $G^{\prime}$ is a diagonal
matrix the vectors on both sides of the derivative carry the index $i$ such
that we get
$\delta u^{\top}V\frac{\partial G^{\prime}}{\partial C_{ij}}C\delta
s=\left(\delta u^{\top}V\right)_{i}g_{i}^{\prime\prime}(z)s_{j}\left(C\delta
s\right)_{i}$ (58)
In the case of $g\left(z\right)=\tanh\left(z\right)$ we find, using
$g_{i}^{\prime\prime}\left(z\right)=-2g_{i}^{\prime}\left(z\right)g_{i}\left(z\right)$
and $a=g(z)$
$\delta u^{\top}V\frac{\partial G^{\prime}}{\partial C_{ij}}C\delta
s=-2\left(\delta u^{\top}VG^{\prime}\right)_{i}\left(C\delta
s\right)_{i}a_{i}s_{j}=-\gamma_{i}a_{i}s_{j}$ (59)
with
$\displaystyle\gamma_{i}$ $\displaystyle=2\left(C\delta
s\right)_{i}\delta\mu_{i}$ (60) $\displaystyle\delta\mu_{i}$
$\displaystyle=\left(G^{\prime}V^{\top}\delta u\right)_{i}$ (61)
The final update rule follows by putting eq. (59) and eq. (61) into eq. (57)
$\Delta C_{ij}=\varepsilon\delta\mu_{i}\delta
s_{j}-\varepsilon\gamma_{i}a_{i}s_{j}$ (62)
Analogously we obtain the parameter dynamics of $h$ as
$\displaystyle\frac{1}{\varepsilon}\Delta h_{i}$ $\displaystyle=\delta
u^{\top}\frac{\partial L}{\partial h_{i}}\delta s=\delta
u^{\top}V\frac{\partial G^{\prime}}{\partial h_{i}}C\delta s$
$\displaystyle=\left(\delta
u^{\top}V\right)_{i}g_{i}^{\prime\prime}(z)\left(C\delta
s\right)_{i}=-\gamma_{i}a_{i}$ (63)
A more compact matrix notation can be obtained by introducing the diagonal
matrix $\Gamma$
$\Gamma=\mathrm{diag}[\gamma_{1},\cdots,\gamma_{i}]$
and thus (reintroducing the time indices)
$\displaystyle\frac{1}{\varepsilon}\Delta C_{t}$
$\displaystyle=\delta\mu_{t}\delta s_{t-1}^{\top}-\Gamma_{t}a_{t}s_{t}^{\top}$
(64) $\displaystyle\frac{1}{\varepsilon}\Delta h_{t}$
$\displaystyle=-\Gamma_{t}a_{t}$ (65)
In the case of arbitrary neuron activation functions $g$ we obtain equivalent
formula by defining
$\gamma_{i}=-\frac{g_{i}^{\prime\prime}}{g_{i}^{\prime}g_{i}}\left(C\delta
s\right)_{i}\delta\mu_{i}$ (66)
Note the factor $-\frac{g_{i}^{\prime\prime}}{g_{i}^{\prime}g_{i}}$ is $2$ in
the case of $g=\tanh$.
In the derivation of eqs. (59) and (63) we ignored the dependence of the state
$s$ in $g^{\prime}\left(Cs+h\right)$ on the parameters $C$ and $h$. This
dependence can be considered explicitly if the state is at a fixed point. In
that case, a more detailed discussion in Der and Martius [17] (section 6.2)
shows that the effect of the derivative can be condensed into the so-called
_sense_ parameter $\alpha$ multiplying $\gamma$. Thus we replace $\gamma$ as
$\gamma_{i}\leftarrow\alpha\gamma_{i}$ (67)
where $\alpha$ is an empirical constant, typically $\alpha\geq 1$, by which
the sensitivity of the sensorimotor dynamics to external perturbations can be
regulated. This works also in more general cases like a limit cycle dynamics,
see Der and Martius [17].
## Appendix D Learning the inverse covariance matrix
Note that the covariance matrix given in eq. (15) can be easily obtained by
the on-line update rule
$\Delta\Sigma_{t}=\eta\left(\delta s_{t}\delta s_{t}^{\top}-\Sigma_{t}\right)$
(68)
or
$\Sigma_{t+1}=\left(1-\eta\right)\Sigma_{t}+\eta\delta s_{t}\delta
s_{t}^{\top}$ (69)
realizing a sampling over a restricted period of time. The update rate $\eta$
defines the time horizon $t_{H}\propto\eta^{-1}$ for the averaging. The only
remaining nontrivial operation in that setting is the inversion of the
covariance matrix $\Sigma$. However, this can also be reduced to elementary
operations by using the Sherman-Morrison formula as given by
$\left(A+uv^{\top}\right)^{-1}=A^{-1}-\frac{1}{1+v^{\top}A^{-1}u}A^{-1}uv^{\top}A^{-1}$
Putting $A=\left(1-\eta\right)\Sigma$ and $uv^{\top}=\eta\delta s\delta
s^{\top}$ we get
$\left(\left(1-\eta\right)\Sigma_{t}+\eta\delta s_{t}\delta
s_{t}^{\top}\right)^{-1}\\!\\!=\frac{1}{1-\eta}\Sigma_{t}^{-1}-\frac{\eta}{\left(1-\eta\right)^{2}\left(1+\frac{\eta}{1-\eta}\delta
s_{t}^{\top}\Sigma_{t}^{-1}\delta s_{t}\right)}\Sigma_{t}^{-1}\delta
s_{t}\delta s_{t}^{\top}\Sigma_{t}^{-1}$
and thus
$\Sigma_{t+1}^{-1}=\frac{1}{1-\eta}\Sigma_{t}^{-1}-\frac{\beta}{1-\eta}\Sigma_{t}^{-1}\delta
s_{t}\delta s_{t}^{\top}\Sigma_{t}^{-1}$
where $\beta\in\mathbb{R}$ is given by
$\beta=\frac{\eta}{\left(1-\eta+\eta\delta s_{t}^{\top}\Sigma_{t}^{-1}\delta
s_{t}\right)}$
Note that $\delta s_{t}^{\top}\Sigma_{t}^{-1}\delta s_{t}$ featuring in the
denominator of $\beta$ is a scalar so that with $\Sigma_{t}^{-1}$ given there
is no matrix inversion to be done.
If $\Sigma_{t}$ is an $n\times n$ matrix, the cost of getting $\Sigma_{t+1}$
is $O\left(n^{2}\right)$. This is very favorable if the dimension of the
sensor space is large. Using the above formula, the only true inversion (of
order $O\left(n^{3}\right)$) has to be done just once, when starting the
process (with a convenient initialization of $\Sigma$).
|
arxiv-papers
| 2013-01-30T23:44:25 |
2024-09-04T02:49:41.030277
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Georg Martius, Ralf Der, Nihat Ay",
"submitter": "Georg Martius",
"url": "https://arxiv.org/abs/1301.7473"
}
|
1301.7484
|
# Perturbation analysis for Moore-Penrose inverse of closed operators on
Hilbert spaces
FAPENG DU
School of Mathematical and Physical Sciences, Xuzhou Institute of Technology
Xuzhou 221008, Jiangsu Province, P.R. China
E-mail: [email protected] YIFENG XUE
Department of mathematics, East China Normal University
Shanghai 200241, P.R. China
###### Abstract
In this paper, we investigate the perturbation for the Moore-Penrose inverse
of closed operators on Hilbert spaces. By virtue of a new inner product
defined on $H$, we give the expression of the Moore-Penrose inverse
$\bar{T}^{\dagger}$ and the upper bounds of $\|\bar{T}^{\dagger}\|$ and
$\|\bar{T}^{\dagger}-T^{\dagger}\|$. These results obtained in this paper
extend and improve many related results in this area.
2000 Mathematics Subject Classification: 15A09, 47A55
Key words: generalized inverse, Moore-Penrose inverse, stable perturbation,
closed operators
## 1 Introduction
An operator $\bar{T}=T+\delta T$ is called the stable perturbation of $T$ if
$R(\bar{T})\cap N(T^{+})=\\{0\\}$. This notation is introduced by Chen and the
second author in [2, 3]. Later it is generalized to the Banach algebra by the
second author in [15] and to Hilbert $C^{*}$–module by Xu, Wei and Gu in [17].
Using this notation the upper bounds for generalized inverse or Moore–Penrose
inverse of bounded linear operators are discussed(See all references). A
classical result about upper bounds is
$\|\bar{T}^{\dagger}\|\leq\frac{\|T^{\dagger}\|}{1-\|T^{\dagger}\|\|\delta
T\|},\quad\frac{\|\bar{T}^{\dagger}-T^{\dagger}\|}{\|T^{\dagger}\|}\leq\frac{1+\sqrt{5}}{2}\frac{\|T^{\dagger}\|}{1-\|T^{\dagger}\|\|\delta
T\|}.$
In recent years, the perturbation analysis for generalized inverses of closed
operators has been appeared. Some results similar to the perturbation analysis
of bounded linear operators are obtained when $\delta T$ is a T–bounded linear
operator(see [9],[10],[13]).
But there are some unsolved questions. What is the result of the perturbation
for closed operators $T\in C(X,Y)$ when $\delta T$ is a linear operators? What
is the expression of the Moore-Penrose inverse $(T+\delta T)^{\dagger}$ and
how to estimate the upper bounds of $\|\bar{T}^{\dagger}\|$ and
$\|\bar{T}^{\dagger}-T^{\dagger}\|$ when $X,\,Y$ are Hilbert spaces ? The
first question has been solved in [7]. Now we discuss the second question in
this paper.
Let $H,K$ be Hilbert spaces, $T\in C(H,K)$ defined on $D(T)$, $\delta T\in
L(H,K)$ be a linear operators. We introduce a new norm $\|\cdot\|_{T}$ on
$D(T)$ such that $(D(T),\|\cdot\|_{T})$ be a Hilbert spaces and give the
expression of $(T+\delta T)^{\dagger}$ and the upper bounds of
$\|\bar{T}^{\dagger}\|$ and $\|\bar{T}^{\dagger}-T^{\dagger}\|$ when $\delta
T$ is a bounded linear operators on $(D(T),\|\cdot\|_{T})$.
## 2 Preliminaries
Let $X,Y$ be Banach spaces, $L(X,Y),\,C(X,Y)$ and $B(X,Y)$ denote the set of
linear operators, densely-defined closed operators and bounded linear
operators from $X$ to $Y$, respectively. For an operator $T\in L(X,Y)$,
$D(T),\,R(T),\,\ker T$ denoted by the domain, the range and the null spaces of
$T$, respectively.
Let $V$ be a closed subspace of $X$. Recall that $V$ is complemented in $X$ if
there is a closed subspace $U$ in $X$ such that $V\cap U=\\{0\\}$ and $X=V+U$.
In this case, we set $X=V\dotplus U$ and $U=V^{c}$.
###### Definition 2.1
[7] Let $T\in C(X,Y)$. If there is $S\in C(Y,X)$ with $D(S)\supset R(T)$ and
$R(S)\subset D(T)$ such that
$TST=T\;\text{on}\ D(T),\quad STS=S\ \text{on}\;D(S),$
then $S$ is called a generalized inverse of $T$, which is also denoted by
$T^{+}$.
Clearly, $P=I-ST$ (resp. $Q=TS$) are idempotent operators on $D(T)$ (resp.
$D(S)$) with $R(P)=\ker T$ (resp. $R(Q)=R(T)$).
###### Proposition 2.1
Let $T\in C(X,Y)$. Then $T^{+}\in C(Y,X)$ exists if and only if
$X=\ker T\oplus\overline{R(T^{+})},\quad Y=\overline{R(T)}\oplus\ker T^{+}.$
In addition, $T^{+}$ is bounded if $R(T)$ closed.
Proof. $(\Rightarrow).$ If $T^{+}\in C(Y,X)$, then we have
$D(T)=R(S)+\ker T,\quad D(S)=R(T)+\ker S.$
So the assertion follows since $D(T)$ (rsep. $D(S)$) are densely in $X$(rsep.
$Y$).
$(\Leftarrow).$ See Proposition 2.2 in [7].
###### Lemma 2.1
[7] Let $T\in C(X,Y)$ such that $T^{+}$ exists. Let $\delta T\colon D(\delta
T)\rightarrow D(T^{+})$ be a linear operators. Assume that $I+\delta
TT^{+}\colon D(T^{+})\rightarrow D(T^{+})$ is bijective. Put $\bar{T}=T+\delta
T$ and $G=T^{+}(I+\delta TT^{+})^{-1}$. Then the following statements are
equivalent:
1. $(1)$
$R(\bar{T})\cap\ker T^{+}=\\{0\\};$
2. $(2)$
$\bar{T}G\bar{T}=\bar{T},\;G\bar{T}G=G$ and $R(\bar{T}^{+})=R(T^{+})$,
$\ker\bar{T}^{+}=\ker T^{+}$.
3. $(3)$
$(I+\delta TT^{+})^{-1}\bar{T}$ maps $\ker T$ into $R(T);$
4. $(4)$
$(I+\delta TT^{+})^{-1}R(\bar{T})=R(T);$
5. $(5)$
$(I+T^{+}\delta T)^{-1}\ker T=\ker\bar{T}$.
Let $H$ and $K$ be Hilbert spaces. For $T\in C(H,K)$, let
$P_{\overline{R(T)}}$ (resp. $P_{\ker T}$) denote the orthogonal projection
from $K$ (resp. $H$) to $\overline{R(T)}$ (resp. $\ker T$).
###### Definition 2.2
Let $T\in C(H,K)$. Then there is a unique $S\in C(K,H)$ with
$D(S)=R(T)+R(T)^{\perp}$ and $R(S)=\ker T^{\perp}\cap D(T)$ such that
$\displaystyle TST$ $\displaystyle=T\ \text{on}\ D(T),\,\ $ $\displaystyle\,\
STS$ $\displaystyle=S\ \text{on}\ D(S),$ $\displaystyle TS$
$\displaystyle=P_{\overline{R(T)}}\ \text{on}\ D(S),\,\ $ $\displaystyle\,\
ST$ $\displaystyle=I-P_{\ker T}\ \text{on}\ D(T).$
The operator $S$ is called the Moore–Penrose inverse of $T$, denoted by
$T^{\dagger}$. Clearly, $\ker T^{\dagger}=R(T)^{\perp}$ and
$R(T^{\dagger})=\ker T^{\perp}\cap D(T)$. In addition, if $R(T)$ is closed,
then $S$ is bounded.
## 3 Perturbation analysis of M-P inverse on Hilbert spaces
In this section, we investigate the expression of M-P inverse
$\bar{T}^{\dagger}$ and the upper bound of $\|\bar{T}^{\dagger}\|$ and
$\|\bar{T}^{\dagger}-T^{\dagger}\|$.
$\forall x\in H$, let
$\|x\|_{G}=\|x\|+\|Tx\|,$
then we know $T$ is closed if and only if $(D(T),\|\cdot\|_{G})$ is a Banach
space([11, P191]). Clearly $T$ is a bounded linear operators on
$(D(T),\|\cdot\|_{G})$ since $\|Tx\|\leq\|x\|_{G}$.
Denote $(\cdot,\cdot)_{H}$ be a inner product on $H$. $\forall x,y\in D(T)$,
let
$(x,y)_{T}=(x,y)_{H}+(Tx,Ty)_{K}.$
It is easy to check that $(x,y)_{T}$ is a inner product on $D(T)$.
Let
$\|x\|^{2}_{T}=(x,x)_{T},$
then
$\|x\|^{2}_{T}=(x,x)_{T}=(x,x)_{H}+(Tx,Tx)_{K}=\|x\|^{2}+\|Tx\|^{2},$
that is,
$\|x\|_{T}=(\|x\|^{2}+\|Tx\|^{2})^{\frac{1}{2}}.$
Since
$\frac{\sqrt{2}}{2}\|x\|_{G}\leq\|x\|_{T}\leq\|x\|_{G},$
we know $\|\cdot\|_{G}$ equivalence to $\|\cdot\|_{T}$. So $T$ is closed if
and only if $(D(T),\|\cdot\|_{T})$ is a Hilbert space. For convenience, we
denote $(D(T),\|\cdot\|_{T})$ by $D_{T}$ in the context.
Consider a mapping as following:
$\displaystyle\tau:D(T)\subset H\rightarrow D_{T}$ $\displaystyle\tau
x=x,\quad\forall x\in D(T)$
Clearly, $\tau$ is defined on $D(T)$ and $R(\tau)=D_{T}$.
Let $x_{n}\subset D(T)$ and $x_{n}\xrightarrow{\|\cdot\|}x,\;\tau
x_{n}\xrightarrow{\|\cdot\|_{T}}y$, then
$0\leftarrow\|\tau x_{n}-y\|^{2}_{T}=\|x_{n}-y\|^{2}+\|T(x_{n}-y)\|^{2}.$
So $\|x_{n}-y\|\rightarrow 0$. This indicate $y=\tau x=x\in D(T)$. Hence,
$\tau\in C(H,D_{T})$.
Clearly,
$\displaystyle\tau^{\dagger}$ $\displaystyle=\rho\in B(D_{T},H);$
$\displaystyle\rho x$ $\displaystyle=x,\;x\in D_{T}.$
###### Lemma 3.1
[6] Let $A\in C(L,K),B\in C(H,L)$ with $R(A),R(B),R(AB)$ closed and
$R(B)\subseteq D(A)$. Assume that $AB\in C(H,K)$. Then
$\displaystyle(AB)^{\dagger}$
$\displaystyle=P_{\ker(AB)^{\perp}}(B^{\dagger}(A^{\dagger}ABB^{\dagger})^{\dagger}A^{\dagger})\times$
$\displaystyle\\{A(A^{\dagger}ABB^{\dagger})(A^{\dagger}ABB^{\dagger})^{\dagger}A^{\dagger}+(A^{\dagger})^{*}(A^{\dagger}ABB^{\dagger})(A^{\dagger}ABB^{\dagger})^{\dagger}A^{*}-I\\}^{-1}.$
###### Lemma 3.2
Let $T\in C(H,K)$, then $T^{+}\in B(K,H)$ if and only if $T^{+}\in
B(K,D_{T})$, and in this case
$\|T^{+}\|^{2}\leq\|T^{+}\|^{2}_{T}\leq\|T^{+}\|^{2}+\|TT^{+}\|^{2}.$
Proof.
If $T^{+}\in B(K,H)$, then $TT^{+}\in B(K)$. $\forall x\in K$,
$\|T^{+}x\|^{2}_{T}=\|T^{+}x\|^{2}+\|T(T^{+}x)\|^{2}\leq(\|T^{+}\|^{2}+\|TT^{+}\|^{2})\|x\|^{2}.$
Hence, $T^{+}\in B(K,D_{T})$ and
$\|T^{+}\|^{2}_{T}\leq\|T^{+}\|^{2}+\|TT^{+}\|^{2}$.
Conversely, if $T^{+}\in B(K,D_{T})$, then $\forall x\in K$,
$\|T^{+}x\|^{2}=\|T^{+}x\|^{2}_{T}-\|TT^{+}x\|^{2}\leq\|T^{+}x\|^{2}_{T}.$
Hence, $T^{+}\in B(K,H)$ and $\|T^{+}\|\leq\|T^{+}\|_{T}$.
From the above, we have
$\|T^{+}\|^{2}\leq\|T^{+}\|^{2}_{T}\leq\|T^{+}\|^{2}+\|TT^{+}\|^{2}.$
###### Lemma 3.3
Let $T\in C(H,K)$ with $R(T)$ closed. If $T$ has generalized inverse $T^{+}$,
then $T^{\dagger}\in B(K,H)$ and
$T^{\dagger}=-P_{\ker T^{\perp}}(I+P(I-P-P^{*})^{-1})T^{+}(I-Q-Q^{*})^{-1}.$
Proof. Since $R(T)$ closed, we have $T^{+}\in B(K,H)$. So $T^{+}\in
B(K,D_{T})$ by Lemma 3.2. Thus, $Q=TT^{+}\in B(K),\;P=I-T^{+}T\in B(D_{T})$
are idempotent operators. Now we consider the Moore-Penrose inverse
$T^{\dagger}$ of $T$ on $D_{T}$. From [4], we have $T^{\dagger}\in B(K,D_{T})$
and
$T^{\dagger}=-(I+P(I-P-P^{*})^{-1})T^{+}(I-Q-Q^{*})^{-1}.$
Since $T^{\dagger}\in B(K,D_{T})$, we have $T^{\dagger}\in B(K,H)$ by Lemma
3.2. Noting that $T\in C(H,K)$ is a compound operator by $T\in B(D_{T},K)$ and
$\tau\in C(H,D_{T})$. Therefore, by Lemma 3.1, we have
$T^{\dagger}=-P_{\ker T^{\perp}}(I+P(I-P-P^{*})^{-1})T^{+}(I-Q-Q^{*})^{-1}.$
###### Theorem 3.1
Let $T\in C(H,K)$ with $T^{\dagger}\in B(K,H)$, $\delta T\in B(D_{T},K)$ such
that $\bar{T}=T+\delta T$ closed, $D(T)\subseteq D(\delta T)$. If $I+\delta
TT^{\dagger}$ is invertible and $R(\bar{T})\cap N(T^{\dagger})=\\{0\\}$, then
$\bar{T}^{\dagger}\in B(K,H)$ and
$\bar{T}^{\dagger}=-P_{\ker\bar{T}^{\perp}}(I+\bar{P}(I-\bar{P}-\bar{P}^{*})^{-1})G(I-\bar{T}G-(\bar{T}G)^{*})^{-1},$
where $G=T^{\dagger}(I+\delta TT^{\dagger})^{-1},\;\bar{P}=I-G\bar{T}$.
Proof. $\forall x\in D(T)$, there is an $M$ such that $\|\delta Tx\|\leq
M\|x\|_{T}$ since $\delta T\in B(D_{T},K)$. Thus, $\forall y\in K$,
$\|\delta TT^{\dagger}y\|^{2}\leq\|\delta
T\|^{2}_{T}\|T^{\dagger}y\|^{2}_{T}\leq\|\delta
T\|^{2}_{T}(\|T^{\dagger}\|^{2}+1)\|y\|^{2}.$
Hence $G=T^{\dagger}(I+\delta TT^{\dagger})^{-1}\in B(K,H)$ be the generalized
inverse of $\bar{T}$ by Lemma 2.1. By Lemma 3.3, $\bar{T}^{\dagger}\in B(K,H)$
and
$\bar{T}^{\dagger}=-P_{\ker\bar{T}^{\perp}}(I+\bar{P}(I-\bar{P}-\bar{P}^{*})^{-1})G(I-\bar{T}G-(\bar{T}G)^{*})^{-1}.$
###### Remark 3.1
If $\delta T$ is T–bounded, i.e., there are constants $a,\,b>0$ such that
$\|\delta Tx\|\leq a\|x\|+b\|Tx\|,\quad\forall\,x\in D(T),$
then $\delta T\in B(D_{T},K)$. Indeed,
$\|\delta Tx\|^{2}\leq(a\|x\|+b\|Tx\|)^{2}\leq
2(\max(a,b))^{2}(\|x\|^{2}+\|Tx\|^{2})=2(\max(a,b))^{2}\|x\|^{2}_{T}.$
Let $M,N$ are two closed subspaces of $H$. Set
$\delta(M,N)=\sup\\{dist(\mu,N)|\|\mu\|=1,\mu\in M\\}.$
We call $\hat{\delta}(M,N)=\max\\{\delta(M,N),\delta(N,M)\\}$ the gap between
subspaces $M$ and $N$.
###### Proposition 3.1
[11]
(1)
$\delta(M,N)=0$ if and only if $M\subset N$
(2)
$\hat{\delta}(M,N)=0$ if and only if $M=N$
(3)
$\hat{\delta}(M,N)=\hat{\delta}(N,M)$
(4)
$0\leq\delta(M,N)\leq 1$, $0\leq\hat{\delta}(M,N)\leq 1$
(5)
$\hat{\delta}(M,N)=\|P-Q\|$, where $P,Q$ are orthogonal projection on $M,N$,
respectively.
For convenience, we set $\|\delta T\|_{T}=\underset{\|x\|_{T}\leq
1}{\sup}\dfrac{\|\delta Tx\|}{\|x\|_{T}}$ if $\delta T\in B(D_{T},K)$.
###### Lemma 3.4
Under the assumptions of Theorem 3.1, we have
1. 1.
$\delta(R(T),R(\bar{T}))\leq\|\delta
T\|_{T}(\|T^{\dagger}\|^{2}+1)^{\frac{1}{2}}.$
2. 2.
$\delta(\ker T,\ker(\bar{T}))\leq\|\bar{T}^{\dagger}\|\|\delta T\|_{T}.$
Proof. Noting that $\|\delta Tx\|\leq\|\delta T\|_{T}\|x\|_{T}$ and
$\bar{T}^{\dagger}\in B(K,H)$.
$(1).$ Let $u\in R(T)$ with $\|u\|=1$, then there is a $x\in D(T)$ such that
$u=Tx$.
$\displaystyle dist(u,R(\bar{T}))$
$\displaystyle\leq\|u-\bar{T}(T^{\dagger}Tx)\|=\|\delta TT^{\dagger}Tx\|$
$\displaystyle\leq\|\delta T\|_{T}\|T^{\dagger}Tx\|_{T}$
$\displaystyle\leq\|\delta T\|_{T}(\|T^{\dagger}\|^{2}+1)^{\frac{1}{2}}\|u\|.$
Hence, $\delta(R(T),R(\bar{T}))\leq\|\delta
T\|_{T}(\|T^{\dagger}\|^{2}+1)^{\frac{1}{2}}$.
$(2).$ Let $x\in\ker T$ with $\|x\|=1$, then $Tx=0$
$\displaystyle dist(x,\ker(\bar{T}))$
$\displaystyle\leq\|x-(I-\bar{T}^{\dagger}\bar{T})x\|=\|\bar{T}^{\dagger}\delta
Tx\|$ $\displaystyle\leq\|\bar{T}^{\dagger}\|\|\delta Tx\|$
$\displaystyle\leq\|\bar{T}^{\dagger}\|_{T}\|\delta T\|_{T}\|x\|.$
Hence, $\delta(\ker T,\ker(\bar{T}))\leq\|\bar{T}^{\dagger}\|\|\delta
T\|_{T}$.
###### Lemma 3.5
Let M and N be closed subspaces of H. Suppose that $M\cap
N^{\perp}=\\{0\\}$.Then
$\delta(M,N)=\|P_{M}-P_{N}\|.$
Proof. If $\delta(M,N)=1$, then $1=\hat{\delta}(M,N)=\|P_{M}-Q_{N}\|$, Thus
$\delta(M,N)=\|P_{M}-P_{N}\|$.
Assume that $\delta(M,N)=\delta<1$, then $\forall x\in M$,
$\|(I-P_{N})P_{M}x\|=dist(P_{M}x,N)\leq\|P_{M}x\|\delta(M,N)\leq\delta\|x\|.$
So by Lemma 3 of [14], we know $\delta(M,N)=\|P_{M}-P_{N}\|$.
###### Lemma 3.6
Under the assumptions of Theorem 3.1, we have
$\|TT^{\dagger}-\bar{T}\bar{T}^{\dagger}\|=\delta(R(T),R(\bar{T})).$
Proof. Since $T^{\dagger}\in B(K,H)$, we have
$\ker(T^{\dagger})=R(T)^{\perp}$. So
$\ker(T^{\dagger})=\ker(\bar{T}^{\dagger})$ implies
$R(T)\cap\ker(\bar{T}^{\dagger})=\\{0\\}$. By Lemma 3.5, we know
$\|TT^{\dagger}-\bar{T}\bar{T}^{\dagger}\|=\delta(R(T),R(\bar{T})).$
###### Theorem 3.2
Under the assumptions of Theorem 3.1, we have
(1)
$\|\bar{T}^{\dagger}\|\leq\|(1+\delta
TT^{\dagger})^{-1}\|(\|T^{\dagger}\|^{2}+1)^{\frac{1}{2}}$.
(2)
$\|\bar{T}^{\dagger}-T^{\dagger}\|\leq\dfrac{1+\sqrt{5}}{2}\|\bar{T}^{\dagger}\|\|\delta
T\|_{T}(\|T^{\dagger}\|^{2}+1)^{\frac{1}{2}}$.
Proof. $(1).$ Since $\delta T\in B(D_{T},K)$, we have
$\|\delta TT^{\dagger}x\|\leq\|\delta
T\|_{T}\|\|T^{\dagger}x\|_{T}\leq\|\delta
T\|_{T}(\|T^{\dagger}\|^{2}+1)^{\frac{1}{2}}\|x\|.$
Hence, $\delta TT^{\dagger}\in B(K)$. By Lemma 2.1 ,we have
$G=T^{+}(I+\delta TT^{+})^{-1}\in B(K,H),\;P=I-G\bar{T}\in
B(D_{T}),\;Q=\bar{T}G\in B(K)$
and $\|(I+P(I-P-P^{*})^{-1})x\|_{T}\leq\|x\|_{T},\;x\in D(T).$ So,
$\displaystyle\|\bar{T}^{\dagger}x\|^{2}$
$\displaystyle\leq\|\bar{T}^{\dagger}x\|^{2}_{T}=\|-(I+P(I-P-P^{*})^{-1})G(I-Q-Q^{*})^{-1}x\|^{2}_{T}$
$\displaystyle\leq\|G(I-Q-Q^{*})^{-1}x\|^{2}_{T}$
$\displaystyle\leq\|T^{\dagger}(1+\delta
TT^{\dagger})^{-1}(I-Q-Q^{*})^{-1}x\|^{2}+\|TT^{\dagger}(1+\delta
TT^{\dagger})^{-1}(I-Q-Q^{*})^{-1}x\|^{2}$
$\displaystyle\leq(\|T^{\dagger}\|^{2}+1)\|(1+\delta
TT^{\dagger})^{-1}\|^{2}\|x\|^{2}.$
$(2).$ Since $D(T)$ dense in $H$, we can extend $I-T^{\dagger}T$ to the whole
spaces $H$ such that $P_{\ker T}|_{D(T)}=I-T^{\dagger}T$ and $P_{\ker T}$ is
an orthogonal projection form $H$ onto $\ker T$. Similarly, we extend
$I-\bar{T}^{\dagger}\bar{T}$ to the whole spaces $H$ such that
$P_{\ker(\bar{T})}|_{D(\bar{T})}=I-\bar{T}^{\dagger}\bar{T}$ and
$P_{\ker(\bar{T})}$ is an orthogonal projection form $H$ onto $\ker(\bar{T})$.
Clearly, $\forall x\in D(T)$,
$(T^{\dagger}T-\bar{T}^{\dagger}\bar{T})x=(P_{\ker(\bar{T})}-P_{\ker T})x.$
Noting that $\ker T\cap\ker(\bar{T})^{\perp}=\\{0\\}$, we have $\|P_{\ker
T}-P_{\ker(\bar{T})}\|=\delta(\ker T,\ker(\bar{T}))$ by Lemma 3.5.
$\forall y\in K$ and $\|y\|=1$,
$\displaystyle\bar{T}^{\dagger}y-T^{\dagger}y$
$\displaystyle=-\bar{T}^{\dagger}\delta
TT^{\dagger}TT^{\dagger}y+\bar{T}^{\dagger}(\bar{T}\bar{T}^{\dagger}-TT^{\dagger})(I-TT^{\dagger})y$
$\displaystyle+(P_{\ker(\bar{T})}-P_{\ker T})T^{\dagger}y.$
Using the proof of Proposition 7 in [14], we have
$\|\bar{T}^{\dagger}y-T^{\dagger}y\|^{2}_{T}\leq\frac{3+\sqrt{5}}{2}\|\bar{T}^{\dagger}\|^{2}\|\delta
T\|^{2}_{T}(\|T^{\dagger}\|^{2}+1).$
Since
$\|\bar{T}^{\dagger}y-T^{\dagger}y\|\leq\|\bar{T}^{\dagger}y-T^{\dagger}y\|_{T}$,
we have
$\|\bar{T}^{\dagger}-T^{\dagger}\|\leq\frac{1+\sqrt{5}}{2}\|\bar{T}^{\dagger}\|\|\delta
T\|_{T}(\|T^{\dagger}\|^{2}+1)^{\frac{1}{2}}.$
## 4 Perturbation analysis for $Tx=b$ in Hilbert spaces
In this section, we consider the perturbation of the least square solution of
the following two equations
(1)
$Tx=b,$
(2)
$\bar{T}\bar{x}=\bar{b},$ $(\bar{b}=b+\delta b)$
As we know the solutions of
$\|Tx-b\|=\min_{z\in D(T)}\|Tz-b\|$
are $x=T^{\dagger}b+(I-T^{\dagger}T)z,\forall z\in D(T)$, denoted by $S(T,b)$,
i.e.
$S(T,b)=\\{x:x=T^{\dagger}b+(I-T^{\dagger}T)z,\forall z\in D(T)\\}$
Similarly,
$S(\bar{T},\bar{b})=\\{\bar{x}:\bar{x}=\bar{T}^{\dagger}\bar{b}+(I-\bar{T}^{\dagger}\bar{T})z,\forall
z\in D(\bar{T})\\}$
###### Theorem 4.1
Under the assumptions of Theorem 3.1, we have
$(1)$ For any solution $x=T^{\dagger}b+(I-T^{\dagger}T)z$ in $S(T,b)$, there
exist $\bar{x}\in S(\bar{T},\bar{b})$ such that
$\|\bar{x}-x\|\leq\|\bar{T}^{\dagger}\|\|(b-Tx)+(\delta b-\delta Tx)\|.$
$(2)$ For any solution
$\bar{x}=\bar{T}^{\dagger}\bar{b}+(I-\bar{T}^{\dagger}\bar{T})z$ in
$S(\bar{T},\bar{b})$, there exist $x\in S(T,b)$ such that
$\|\bar{x}-x\|\leq\|T^{\dagger}\|\|(\bar{b}-\bar{T}\bar{x})-(\delta b-\delta
T\bar{x})\|.$
Proof. $(1)$ Taking
$\bar{x}=\bar{T}^{\dagger}\bar{b}+(I-\bar{T}^{\dagger}\bar{T})(T^{\dagger}b+(I-T^{\dagger}T)z).$
Then
$\displaystyle\|\bar{x}-x\|$
$\displaystyle=\|\bar{T}^{\dagger}\bar{b}+(I-\bar{T}^{\dagger}\bar{T})(T^{\dagger}b+(I-T^{\dagger}T)z)-(T^{\dagger}b+(I-T^{\dagger}T)z)\|$
$\displaystyle=\|\bar{T}^{\dagger}\bar{b}-(\bar{T}^{\dagger}\bar{T})(T^{\dagger}b+(I-T^{\dagger}T)z)\|$
$\displaystyle=\|\bar{T}^{\dagger}\delta
b+(I-\bar{T}^{\dagger}\bar{T})T^{\dagger}b-\bar{T}^{\dagger}\bar{T}(I-T^{\dagger}T)z+(\bar{T}^{\dagger}-T^{\dagger})b\|$
$\displaystyle=\|\bar{T}^{\dagger}\delta
b+(I-\bar{T}^{\dagger}\bar{T})T^{\dagger}b-\bar{T}^{\dagger}\bar{T}(I-T^{\dagger}T)z$
$\displaystyle+(-\bar{T}^{\dagger}\delta
TT^{\dagger}b+\bar{T}^{\dagger}(I-TT^{\dagger})b-(I-\bar{T}^{\dagger}\bar{T})T^{\dagger}b)\|$
$\displaystyle=\|\bar{T}^{\dagger}(\bar{b}-\bar{T}x)\|$
$\displaystyle\leq\|\bar{T}^{\dagger}\|\|(b-Tx)+(\delta b-\delta Tx)\|.$
$(2)$ Taking
$x=T^{\dagger}b+(I-T^{\dagger}T)(\bar{T}^{\dagger}\bar{b}+(I-\bar{T}^{\dagger}\bar{T})z).$
By similarly computation to $(1)$, we have
$\|\bar{x}-x\|\leq\|T^{\dagger}\|\|(\bar{b}-\bar{T}\bar{x})-(\delta b-\delta
T\bar{x})\|.$
###### Theorem 4.2
Under the assumptions of Theorem 3.1, we have
$\displaystyle\|\bar{T}^{\dagger}\bar{b}-T^{\dagger}b\|$
$\displaystyle\leq\|(1+\delta
TT^{\dagger})^{-1}\|(\|T^{\dagger}\|^{2}+1)^{\frac{1}{2}}\|\delta b\|$
$\displaystyle+\frac{1+\sqrt{5}}{2}\|(1+\delta TT^{\dagger})^{-1}\|\|\delta
T\|_{T}(1+\|T^{\dagger}\|^{2})^{\frac{1}{2}}\|b\|.$
Proof. By Theorem 3.2, we have
$\displaystyle\|\bar{T}^{\dagger}\bar{b}-T^{\dagger}b\|$
$\displaystyle=\|\bar{T}^{\dagger}\delta b+(\bar{T}^{\dagger}-T^{\dagger})b\|$
$\displaystyle\leq\|\bar{T}^{\dagger}\delta
b\|+\|\bar{T}^{\dagger}-T^{\dagger}\|\|b\|$ $\displaystyle\leq\|(1+\delta
TT^{\dagger})^{-1}\|(\|T^{\dagger}\|^{2}+1)^{\frac{1}{2}}\|\delta b\|$
$\displaystyle+\frac{1+\sqrt{5}}{2}\|(1+\delta TT^{\dagger})^{-1}\|\|\delta
T\|_{T}(1+\|T^{\dagger}\|^{2})^{\frac{1}{2}}\|b\|.$
## 5 Conclusion
In this paper, we extend the perturbation analysis of Moore-Penrose inverse
for bounded linear operators to closed operators. By virtue of a new inner
product, we give the expression of the Moore-Penrose inverse
$\bar{T}^{\dagger}$ and the upper bounds of $\|\bar{T}^{\dagger}\|$ and
$\|\bar{T}^{\dagger}-T^{\dagger}\|$. As an application, we study the
perturbation of the least square solution. These results enrich and improve
the perturbation theory of Moore-Penrose inverse described in [16].
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* [6] F. Du, Y. Xue, The reverse order law for generalized inverse of closed operators. Chinese Quart. J. Math. (2013).
* [7] F. Du, Y. Xue, The characterizations of the stable perturbation of a closed operator by a linear operator in Banach spaces. Linear Algebra Appl. 438(2013), 2046-2053.
* [8] C.W. Groetsch, Representations of generalized inverse. J. Math. Anal. Appl., 49(1975), 154-157.
* [9] Q. Huang, W. Zhai, Perturbation and expressions for generalized inverses in Banach spaces and Moore-penrose inverses in Hilbert spaces of closed operators. Linear Algebra Appl. 435(2011), 117-127.
* [10] Q. Huang, On perturbations for oblique projection generalized inverses of closed linear operators in Banach spaces. Linear Algebra Appl. 434(12)(2011), 2368-2474.
* [11] T. Kato, Perturbation Theory for Linear Operators. Springer-Verlag, New York, 1984.
* [12] M.Z. Nashed, Generalized inverse and Applications. Academic Press, New York, 1976.
* [13] Y. Wang, H. Zhang, Perturbation analysis for oblique projection generalized inverses of closed operators in Banach spaces. Linear Algebra Appl. 426(2007), 1-11.
* [14] Y. Xue, G. Chen, Some equivalent conditions of stable perturbation of operators in Hilbert spaces. Applied Math. comput. 147(2004), 65-772.
* [15] Y. Xue, Stable perturbation in Banach algebras. J. Aust. Math. soc. 83(2007), 1-14.
* [16] Y. Xue, Stable Perturbations of Operators and Related Topics, World Scientific, 2012.
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|
arxiv-papers
| 2013-01-31T00:35:07 |
2024-09-04T02:49:41.046095
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fapeng Du and Yifeng Xue",
"submitter": "Yifeng Xue",
"url": "https://arxiv.org/abs/1301.7484"
}
|
1301.7521
|
HOMOLOGY GROUPS OF PIPELINE PETRI NETS 111This work was performed as a part of
the Strategic Development Program at the National Educational Institutions of
the Higher Education, N 2011-PR-054
A.A. Husainov, E.S. Bushmeleva, T.A. Trishina
###### Abstract
We study homology groups of elementary Petri nets for the pipeline systems. We
show that the integral homology groups of these nets equal the group of
integers in dimensions $0$ and $1$, and they are zero in other dimensions. We
prove that directed homology groups of elementary Petri nets are zero in all
dimensions.
2000 Mathematics Subject Classification 18G10, 18G35, 55U10, 68Q10, 68Q85
Key words: homology of small categories, trace monoid, asynchronous transition
system, elementary Petri net, pipeline, semicubical set.
## Introduction
The homology groups of elementary Petri nets were introduced in [1]. In [1],
it was built also an algorithm to compute the first homology group of
asynchronous systems, based on which we calculated the homology of Petri net
of pipeline ${\mathfrak{P}}_{3}$, consisting of three transitions. In [2], an
algorithm for computing the homology groups of elementary Petri net was
developed. An example of calculating the homology groups of elementary Petri
net for pipeline ${\mathfrak{P}}_{4}$
$\textstyle{t_{1}}$ $\scriptstyle{p_{1}}$$\textstyle{t_{2}}$
$\scriptstyle{p_{2}}$$\textstyle{t_{3}}$
$\scriptstyle{p_{3}}$$\textstyle{t_{4}}$
was considered.
We calculated the homology group for ${\mathfrak{P}}_{n}$, with $n=2,3,4,5$ by
the software described in [3]. In these cases, the homology group for the
pipelines are
$H_{0}({\mathfrak{P}}_{n})=H_{1}({\mathfrak{P}}_{n})={\,\mathbb{Z}}$, and
$H_{k}({\mathfrak{P}}_{n})=0$ for $k\geqslant 2$. There was a conjecture that
this is true for all $n\geqslant 2$. In this paper, we prove this conjecture.
In addition, the second co-author of this paper has developed the software
designed to calculate the directed homology groups of the state spaces.
Calculations showed that the directed homology groups of elementary Petri nets
${\mathfrak{P}}_{n}$ for $n=2,3,4,5$ are equal to zero in all dimensions. In
this paper, we show that this is true for all $n\geqslant 2$.
## 1 Preliminaries
For any category ${\mathscr{C}}$ denote by ${\rm Ob\,}{\mathscr{C}}$ the class
of its objects, ${\rm Mor\,}{\mathscr{C}}$ the class of all morphisms, and
${\mathscr{C}}^{op}$ the opposite category. For any $A,B\in{\rm
Ob\,}{\mathscr{C}}$ the set of all morphisms $A\to B$ in a category
${\mathscr{C}}$ is denoted by ${\mathscr{C}}(A,B)$.
### 1.1 Partial maps
For any sets $S$ and $T$ a partial map $f:S\rightharpoonup T$ is a relation
$f\subseteq S\times T$ with the following property:
$(\forall s\in S)(\forall t_{1},t_{2}\in T)~{}(s,t_{1})\in
f~{}\&~{}(s,t_{2})\in f~{}\Rightarrow~{}t_{1}=t_{2}\,.$
We denote $PSet$ category of sets and partial maps.
For any set $S$ denote $S_{*}=S\sqcup\\{*\\}$. Let ${\rm Set}_{*}$ be a
category of sets of the form $S_{*}$ where $S$ are arbitrary sets. For any of
its objects $S_{*}$ and $T_{*}$, let the set of morphisms ${\rm
Set}_{*}(S_{*},T_{*})$ consists of maps $f:S_{*}\to T_{*}$ satisfying the
equality $f(*)=*$. It is easy to see that functor $\Phi:PSet\to{\rm Set}_{*}$,
defined on objects as $\Phi(S)=S_{*}$, and on morphisms
$\Phi(f)(x)=\left\\{\begin{array}[]{cl}f(x),&\mbox{ if }f(x)\mbox{
defined},\\\ ,&\mbox{ if }f(x)\mbox{ is not defined or }x=*,\\\
\end{array}\right.$
will provide an isomorphism of categories $PSet$ and ${\rm Set}_{*}$. This
isomorphism allows us to identify category $PSet$ with ${\rm Set}_{*}$ and we
can consider a partial map as the (total) function preserving point $*$.
### 1.2 Partial action of a monoid
Suppose $M$ be a monoid. The operation in $M$ is denoted by $\cdot_{M}$. Each
monoid $M$ will be considered as a small category that has single object
${\,\rm pt}_{M}$. The set of morphisms of this category is the set $M$. In
particular, the opposite monoid $M^{op}$ is defined. Product of the elements
$\mu_{1},\mu_{2}\in M$ in this monoid is given by
$\mu_{1}\cdot_{M^{op}}\mu_{2}=\mu_{2}\cdot_{M}\mu_{1}$. Partial right action
of a monoid $M$ on the set $S$ is an arbitrary homomorphism $M^{op}\to
PSet(S,S)$. In accordance with the above agreement, a partial action will be
viewed as a homomorphism $M^{op}\to{\rm Set}_{*}(S_{*},S_{*})$. It follows
that it can be regarded as a functor $M^{op}\to{\rm Set}_{*}$.
### 1.3 Trace monoids and state space
Let $E$ be a set, and let $I\subseteq E\times E$ be a irreflective symmetric
relation on $E$. Let $M(E,I)$ be a monoid generated by the set $E$ and given
by the equalities $ab=ba$ for all $(a,b)\in I$. The monoid $M(E,I)$ is called
to be a free partially commutative monoid or trace monoid. $M(E,I)$ is equal
to the quotient monoid $E^{*}/\equiv$ where $E^{*}$ is the monoid of words and
$\equiv$ is the smallest congruence relation containing a pair of $(ab,ba)$
for all pairs $(a,b)\in I$. The space of states is a pair $(S,M(E,I))$, where
$M(E,I)$ is a trace monoid with action on a set $S$.
### 1.4 The state space of elementary Petri nets
For any elementary Petri net ${\,\mathcal{N}}=(P,E,pre,post,s_{0})$, we
consider a trace monoid $M(E,I)$, generated by the set $E$ with the
independence relation
$(a,b)\in I~{}\Leftrightarrow~{}(pre(a)\cup post(a))\cap(pre(b)\cup
post(b))=\emptyset.$
Suppose $a\in E$. For any subset $s\subseteq P$, with properties
* •
$pre(a)\subseteq s$,
* •
$(s\setminus pre(a))\cap post(a)=\emptyset$,
we denote $s\cdot a=(s\setminus pre(a))\cup post(a)$. This defines for each
$a\in E$ a partial map $s\mapsto s\cdot a$, of the set $\\{0,1\\}^{P}$ in
$\\{0,1\\}^{P}$. Here the subsets $s\subseteq P$ are considered as their
characteristic functions.
The obtained partial map (corresponding to the relation of the transition from
[4]), can be defined as the set of triples $(s,a,s^{\prime})$ with the
property
$pre(a)\subseteq s~{}\&~{}post(a)\subseteq s^{\prime}~{}\&~{}s\setminus
pre(a)=s^{\prime}\setminus post(a).$
Using the induction on the length of the trace $\mu=a_{1}a_{2}\cdots a_{n}$ by
the formula $s\cdot a_{1}a_{2}\cdots a_{n}=(s\cdot a_{1}a_{2}\cdots
a_{n-1})\cdot a_{n}$, we can define a partial right action of a monoid
$M(E,I)$ on $\\{0,1\\}^{P}$. The resulting state space
$(\\{0,1\\}^{P},M(E,I))$ will be called a state space of elementary Petri
nets.
If we consider only the set of reachable states instead of $\\{0,1\\}^{P}$,
then this partial action will define the space of reachable states of the
elementary Petri net.
## 2 Homology of semicubical sets and state spaces
Recall the definition of semicubical set and its homology groups [5]. We
describe semicubical set corresponding to a state category with the same
homology groups.
### 2.1 Semicubical sets
A semicubical set $X=(X_{n},\partial_{i}^{n,\varepsilon})$ is given as a
sequence sets $X_{n}$, $n\geqslant 0$ with a family of maps
$\partial_{i}^{n,\varepsilon}:X_{n}\to X_{n-1}$, defined by $1\leqslant
i\leqslant n$, $\varepsilon\in\\{0,1\\}$, for which the following diagrams
$\textstyle{X_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{j}^{n,\beta}}$$\scriptstyle{\partial_{i}^{n,\alpha}}$$\textstyle{X_{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{j-1}^{n-1,\beta}}$$\textstyle{X_{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{i}^{n-1,\alpha}}$$\textstyle{X_{n-2}}$
are commutative for all $n\geqslant 2,~{}1\leqslant i<j\leqslant n$.
Morphism of semicubical sets $f:X\to\tilde{X}$ is a sequence of maps
$f_{n}:X_{n}\to\tilde{X}_{n}$, $n\geqslant 0$, for which the diagrams
$\textstyle{X_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{n}}$$\scriptstyle{\partial_{i}^{n,\varepsilon}}$$\textstyle{\tilde{X}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\partial}_{i}^{n,\varepsilon}}$$\textstyle{X_{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{n-1}}$$\textstyle{\tilde{X}_{n-1}}$
are commutative for all $1\leqslant i\leqslant n$ and
$\varepsilon\in\\{0,1\\}$. If the morphism semicubical sets consist of
inclusions $X_{n}\subseteq\tilde{X}_{n}$ for all $n\geqslant 0$, then $X$ is
called to be a semicubical subset of $\tilde{X}$.
### 2.2 Homology of semicubical sets
Let $X=(X_{n},\partial_{i}^{n,\varepsilon})$ be a semicubical set. Consider
the complex
$0\leftarrow LX_{0}\stackrel{{\scriptstyle
d_{1}}}{{\leftarrow}}LX_{1}\stackrel{{\scriptstyle
d_{2}}}{{\leftarrow}}LX_{2}\stackrel{{\scriptstyle
d_{1}}}{{\leftarrow}}\cdots,$
consisting of free Abelian groups $LX_{n}$, $n\geqslant 0$, generated by the
sets $X_{n}$, and differentials, acting on the basis elements by the formula
$d_{n}(\sigma)=\sum\limits_{i=1}^{n}(-1)^{i}(\partial_{i}^{n,1}(\sigma)-\partial_{i}^{n,0}(\sigma)).$
The homology groups of this complex are called to be homology groups
$H_{n}(X)$ of semicubical set $X$, $n\geqslant 0$.
###### Proposition 2.1
Suppose that $X=X_{1}\cup X_{2}$ is a union of semicubical subsets. Then there
is a long exact sequence of groups
$0\leftarrow H_{0}(X)\leftarrow H_{0}(X_{1})\oplus H_{0}(X_{2})\leftarrow
H_{0}(X_{1}\cap X_{2})\leftarrow\cdots\\\ \cdots\leftarrow H_{n}(X)\leftarrow
H_{n}(X_{1})\oplus H_{n}(X_{2})\leftarrow H_{n}(X_{1}\cap
X_{2})\leftarrow\cdots\\\ $
Proof. Consider the complex $C_{n}(X)=LX_{n}$ with differentials
$d_{n}(\sigma)=\sum\limits_{i=1}^{n}(-1)^{i}(\partial_{i}^{n,1}(\sigma)-\partial_{i}^{n,0}(\sigma))$.
Consider the exact sequence of complexes associated with the homomorphism
$\sigma_{1}\oplus\sigma_{2}\mapsto\sigma_{1}-\sigma_{2}$,
$0\rightarrow C_{n}(X_{1}\cap
X_{2})\stackrel{{\scriptstyle\theta_{n}}}{{\rightarrow}}C_{n}(X_{1})\oplus
C_{n}(X_{2})\stackrel{{\scriptstyle-}}{{\rightarrow}}C_{n}(X_{1}\cup X_{2})\to
0,$
where $\theta_{n}(\sigma)=\sigma\oplus\sigma$ for each $\sigma\in(X_{1}\cap
X_{2})_{n}$. A long exact sequence corresponding to this short exact sequence,
will be required. $\Box$
### 2.3 Homology of state category and semicubical sets
Let $(S,M(E,I))$ be a state space. Consider an arbitrary linear order relation
on $E$. It defines a semicubical set
$Q_{n}(S,E,I)=\\{(s,a_{1},\ldots,a_{n})\in S\times
E^{n}~{}|~{}a_{1}<\cdots<a_{n}~{}\&~{}s\cdot a_{1}\cdots a_{n}\in S\\\
~{}\&~{}(a_{i},a_{j})\in I\mbox{ for all }1\leqslant i<j\leqslant
n\\},~{}~{}n\geqslant 0,$
with boundary operators
$\partial_{i}^{n,\varepsilon}(s,a_{1},\ldots,a_{n})=(s\cdot
a_{i}^{\varepsilon},a_{1},\ldots,a_{i-1},a_{i+1},\ldots,a_{n}),$
for $1\leqslant i\leqslant n$, $\varepsilon\in\\{0,1\\}$. There $a_{i}^{0}=1$
and $a_{i}^{1}=a_{i}$.
Let $(S,M(E,I))$ be a state space. Its state category $K(S)$ is defined as a
small category with objects $s\in S$. Morphisms
$s\stackrel{{\scriptstyle\mu}}{{\to}}t$ are defined as triples of elements
$s,t\in S$ and $\mu\in M(E,I)$ satisfying $s\cdot\mu=t$. Composition is given
by the formula
$(t\stackrel{{\scriptstyle\nu}}{{\to}}u)\circ(s\stackrel{{\scriptstyle\mu}}{{\to}}t)=(s\stackrel{{\scriptstyle\mu\nu}}{{\to}}u)$.
For any elementary Petri net ${\,\mathcal{N}}$, we consider the state space
and its state category. Let $K(S)$ be a full subcategory of the state category
consisting of reachable states. Homology groups $H_{n}({\,\mathcal{N}})$ is
defined as homology groups $H_{n}(K(S))$ of the nerve of the category $K(S)$.
According to [2, Corollary 4], $H_{n}(K(S))\cong H_{n}(Q(S,E,I))$, for all
$n\geqslant 0$. Therefore, the homology groups of elementary Petri nets
$H_{n}({\,\mathcal{N}})$ are isomorphic to the homology of semicubical sets
corresponding to its space of reachable states.
## 3 State category of the pipeline Petri net and its homology groups
In this section, we will explore the state category of pipeline elementary
Petri net and calculate the integral homology groups of this category.
### 3.1 State category of pipeline elementary Petri net
We consider pipeline Petri net ${\mathfrak{P}}_{n}$:
$\textstyle{t_{1}}$ $\scriptstyle{p_{1}}$$\textstyle{t_{2}}$
$\scriptstyle{p_{2}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{t_{n-1}}$
$\scriptstyle{p_{n-1}}$$\textstyle{t_{n}}$
Represent the state category of this net as union of two partially ordered
sets, each of which has the least and the greatest element.
Let ${\,\mathcal{N}}_{n}$ be the following Petri net
$\scriptstyle{p_{1}}$$\textstyle{t_{2}}$
$\scriptstyle{p_{2}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{t_{n-1}}$
$\scriptstyle{p_{n-1}}$$\textstyle{t_{n}}$
Denote by ${\mathscr{C}}_{n}$ its the state category. Every partially ordered
set can be considered as a small category ${\mathscr{C}}$ such that for any
objects $x,y\in{\rm Ob\,}{\mathscr{C}}$, the set ${\mathscr{C}}(x,y)$ contains
no more than one element, and if the conjunction of
${\mathscr{C}}(x,y)\not=\emptyset$ and ${\mathscr{C}}(y,x)\not=\emptyset$
implies $x=y$.
Any state of elementary Petri net ${\,\mathcal{N}}_{n}$ is given by an
arbitrary function $f:\\{p_{1},p_{2},\cdots,p_{n-1}\\}\to\\{0,1\\}$. It
follows that the state can be given by a sequence of values
$\varepsilon_{1}\varepsilon_{2}\cdots\varepsilon_{n-1}$, where
$\varepsilon_{i}=f(p_{i})$ of $f$ for $1\leqslant i\leqslant n-1$. Let $S_{1}$
the set of states beginning with $\varepsilon_{1}=1$, and $S_{0}$ the set of
states, beginning with $\varepsilon_{1}=0$. The set of all states will be
equal to $S=S_{1}\sqcup S_{0}$. In general, the elements of $S_{1}$ and
$S_{0}$ are connected by transitions
$10\varepsilon_{3}\cdots\varepsilon_{n-1}\stackrel{{\scriptstyle
t_{2}}}{{\to}}01\varepsilon_{3}\cdots\varepsilon_{n-1}$.
For example, if $n=4$, then $S_{1}$ and $S_{0}$ are linearly ordered sets
corresponding columns of the diagram:
$\textstyle{111\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t_{4}}$$\textstyle{011\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t_{4}}$$\textstyle{110\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t_{3}}$$\textstyle{010\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t_{3}}$$\textstyle{101\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t_{4}}$$\scriptstyle{t_{2}}$$\textstyle{001\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t_{4}}$$\textstyle{100\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t_{2}}$$\textstyle{000}$
Transitions between $S_{1}$ and $S_{0}$ is shown by the dashed arrows.
For any state $x=\varepsilon_{1}\cdots\varepsilon_{n-1}$, we introduce a
number $|x|=\varepsilon_{1}\cdot 2^{n-2}+\varepsilon_{2}\cdot
2^{n-3}+\cdots+\varepsilon_{n-1}\cdot 2^{0}$. For example, $|11\cdots
1|=2^{n-1}-1$ and $|00\cdots 0|=0$. If there is a transition
$x\stackrel{{\scriptstyle t_{k}}}{{\to}}y$, then $|x|<|y|$
###### Lemma 3.1
Category ${\mathscr{C}}_{n}$ is a partially ordered set having the greatest
and least elements.
Proof. By induction, we prove that ${\mathscr{C}}_{n}$ is a partially ordered
set with the least element of $11\cdots 1$ and the greatest element $00\cdots
0$. Let it proved for $n-1$. Then ${\mathscr{C}}_{n-1}$ is the partially
ordered set with the least and greatest elements. It is easy to see that the
full subcategory of ${\mathscr{C}}_{n}$ with set of objects $S_{1}$ and
$S_{0}$ are isomorphic to the category ${\mathscr{C}}_{n-1}$. So, they will be
partially ordered sets. We denote these posets by $S_{1}$ and $S_{0}$. In
$S_{1}$, a least element of the induction assumption is $11\cdots 1$, and the
greatest element equals $10\cdots 0$. In $S_{0}$, a least element is $01\cdots
1$, and the greatest equals $00\cdots 0$.
Let $x\mu=y$, $x\in S_{1}$, $y\in S_{0}$. Then $\mu$ contains the letter
$t_{2}$. Therefore, there exist $\mu_{1}$ and $\mu_{2}$, such that
$\mu=\mu_{1}t_{2}\mu_{2}$, and $\mu_{1}$ does not contain $t_{2}$. Now, we
will rearrange the letters contained in $mu_{1}$ with the letter $t_{2}$,
until we do not meet $t_{3}$. For some $\mu_{1}$ and $\mu_{2}$, we obtain the
decomposition $\mu=\mu_{1}t_{3}t_{2}\mu_{2}$.
The following step we will perform until we get the decomposition of the form
$\mu=t_{k}\cdots t_{2}\mu^{\prime}$.
Suppose that $\mu=\mu_{1}tt_{k-1}\cdots t_{2}$, for some $k\geqslant 3$ and
$t\in E$. We describe the action taken by us in each of the following possible
cases:
1. (i)
$t=t_{k}$ $\Rightarrow$ increase $k$ by $1$,
2. (ii)
$t=t_{i}$, for some $i>k$ $\Rightarrow$ swap $t$ with $t_{k-1}$, $t_{k-2}$,
$\cdots$, $t_{2}$.
For $i\leqslant k-1$, a decomposition of the form
$\mu=\mu_{1}t_{i}t_{k-1}\cdots t_{2}$ impossible, since an element
$x^{\prime}t_{k-1}\cdots t_{2}$ is defined if and only if $x^{\prime}=11\cdots
10\varepsilon_{k}\cdots\varepsilon_{n-1}$. Such element $x^{\prime}$ can not
be obtained after the action of $t_{i}$ for $2\leqslant i\leqslant k-1$. It
follows that the iteration of the described action will lead to the
decomposition $\mu=t_{k}\cdots t_{2}\mu^{\prime}$.
Now we will prove that $x\mu=x\nu=y\in S$ implies $\mu=\nu$, which implies
that ${\mathscr{C}}_{n}$ is a preordered set.
If $x\mu=x\nu=y\in S_{0}$ and $y\in S_{0}$, then $\mu=\nu$ because of $S_{0}$
is a partially ordered set. The same is true if $x\mu=x\nu=y\in S_{1}$ and
$y\in S_{1}$. There are no $\mu\in M(E,I)$ and $x\in S_{0}$, for which
$x\mu\in S_{1}$.
Now consider the case $x\mu=x\nu=y\in S_{0}$, $x\in S_{1}$. In this case,
there exist decompositions $\mu=t_{k}\cdots t_{2}\mu^{\prime}$ and
$\nu=t_{m}\cdots t_{2}\nu^{\prime}$. Since $xt_{k}t_{k-1}\cdots
t_{2}\mu^{\prime}\in S_{0}$, then $x=11\cdots
10\varepsilon_{k+1}\cdots\varepsilon_{n-1}$, for some $\varepsilon_{k+1}$,
$\cdots,$ $\varepsilon_{n-1}\in\\{0,1\\}$. Hence $m=k$. We get the equality
$xt_{k}\cdots t_{2}\mu^{\prime}=xt_{k}\cdots t_{2}\nu^{\prime}$
Since $xt_{k}\cdots t_{2}\in S_{0}$ and $S_{0}$ – partially ordered set it
follows from this equation $\mu^{\prime}=\nu^{\prime}$. Therefore, $\mu=\nu$
and ${\mathscr{C}}_{n}$ – preordered set. If for some
$x=\varepsilon_{1}\cdots\varepsilon_{n-1}$, $y=\delta_{1}\cdots\delta_{n-1}$,
there exists $\mu\not=1$, which $x\mu=y$, then $|x|>|y|$. Therefore, from
$x\mu=y$ and $y\nu=x$ will follow the $\mu=\nu=1$. Hence otnshenie preorder
will antisymmetric and ${\mathscr{C}}_{n}$ is partially ordered set. $\Box$
### 3.2 The homology groups of pipeline
Let ${\mathfrak{P}}_{n}$ – elementary net pipeline. Delete event $t_{1}$, we
have a network ${\,\mathcal{N}}_{n}$. Remove from ${\mathfrak{P}}_{n}$ event
$t_{2}$, we get the following elementary network:
$\textstyle{t_{1}}$$\scriptstyle{p_{1}}$
$\scriptstyle{p_{2}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{t_{n-1}}$
$\scriptstyle{p_{n-1}}$$\textstyle{t_{n}}$
We denote it by ${\,\mathcal{N}}^{\prime}_{n}$.
###### Proposition 3.2
Semicubical set $Q({\mathfrak{P}}_{n})$ is the union of subsets semicubical
$Q({\,\mathcal{N}}_{n})\cup Q({\,\mathcal{N}}^{\prime}_{n})$. Intersection of
$Q({\,\mathcal{N}}_{n})\cap Q({\,\mathcal{N}}^{\prime}_{n})$ semicubical be
set with a two components, each of which is isomorphic
$Q({\,\mathcal{N}}_{n-1})$.
Proof. There are $Q_{0}({\mathfrak{P}}_{n})=Q_{0}({\,\mathcal{N}}_{n})\cup
Q_{0}({\,\mathcal{N}}^{\prime}_{n})$, because all of these sets are
$S=\\{0,1\\}^{n-1}$. If $k\geqslant 1$, then since $t_{1}$ and $t_{2}$
dependent, they can not belong a set of mutually independent events
$(e_{1},\cdots,e_{m})$. This means that for every $(s,e_{1},\cdots,e_{m})\in
Q_{m}({\mathfrak{P}}_{n})$ will be a $t_{1}\notin\\{e_{1},\cdots,e_{m}\\}$ or
$t_{2}\notin\\{e_{1},\cdots,e_{m}\\}$, where $(s,e_{1},\cdots,e_{m})\in
Q_{m}({\,\mathcal{N}}_{n})\cup Q_{m}({\,\mathcal{N}}^{\prime}_{n})$. Because
of $(s,e_{1},\cdots,e_{m})\in Q_{m}({\,\mathcal{N}}_{n})$ follows $(s\cdot
e_{i}^{\varepsilon},e_{1},\cdots,e_{i-1},e_{i+1},\cdots,e_{m})\in
Q_{m-1}({\,\mathcal{N}}_{n})$ and from $(s,e_{1},\cdots,e_{m})\in
Q_{m}({\,\mathcal{N}}^{\prime}_{n})$ follows $(s\cdot
e_{i}^{\varepsilon},e_{1},\cdots,e_{i-1},e_{i+1},\cdots,e_{m})\in
Q_{m-1}({\,\mathcal{N}}^{\prime}_{n})$, attachments commute with the boundary
operators. So, $Q({\mathfrak{P}}_{n})$ equals the union its cubic subsets of
$Q({\,\mathcal{N}}_{n})$ and $Q({\,\mathcal{N}}^{\prime}_{n})$. $\Box$
###### Theorem 3.3
$H_{0}({\mathfrak{P}}_{n})=H_{1}({\mathfrak{P}}_{n})={\,\mathbb{Z}}$, and
$H_{k}({\mathfrak{P}}_{n})=0$ at $k>1$.
Proof. By Proposition 3.2, $Q({\mathfrak{P}}_{n})=Q({\,\mathcal{N}}_{n})\cup
Q({\,\mathcal{N}}^{\prime}_{n})$. Apply Proposition 2.1. By Lemma 3.1 category
network states ${\,\mathcal{N}}_{n}$ has initial and terminal object. Let
${\,\mathbb{I}}=\\{0,1\\}$ – partially ordered set consisting of the elements
of $0$ and $1$, with the usual order relation $0<1$. It is easy to see that
the category of network conditions ${\,\mathcal{N}}^{\prime}_{n}$ is
isomorphic to the product of ${\,\mathbb{I}}\times{\mathscr{C}}_{n-2}$.
Therefore it has initial and terminal facilities too. From this follow
isomorphisms
$H_{k}(Q({\,\mathcal{N}}_{n}))=H_{k}(Q({\,\mathcal{N}}^{\prime}_{n}))=0$, for
$k>0$. The exact sequence 2.1 leads to the isomorphisms
$H_{k}(Q({\mathfrak{P}}_{n}))\cong H_{k-1}(Q({\,\mathcal{N}}_{n})\cap
Q({\,\mathcal{N}}^{\prime}_{n}))$, for $k\geqslant 2$. Of Proposition 3.2
follows $Q({\,\mathcal{N}}_{n})\cap Q({\,\mathcal{N}}^{\prime}_{n})\cong
Q({\,\mathcal{N}}_{n-1})\sqcup Q({\,\mathcal{N}}_{n-1})$. We get
$H_{k}(Q({\mathfrak{P}}_{n}))\cong H_{k-1}(Q({\,\mathcal{N}}_{n-1}))\oplus
H_{k-1}(Q({\,\mathcal{N}}_{n-1}))=0$, where $k\geqslant 2$. Consequently,
$H_{k}({\mathfrak{P}}_{n})=0$ for $k\geqslant 2$. Of Proposition 2.1 also
obtain accurate sequence
$0\leftarrow H_{0}(Q({\mathfrak{P}}_{n}))\leftarrow
H_{0}(Q({\,\mathcal{N}}_{n}))\oplus H_{0}(Q({\,\mathcal{N}}^{\prime}_{n}))\\\
\leftarrow H_{0}(Q({\,\mathcal{N}}_{n})\cap
Q({\,\mathcal{N}}^{\prime}_{n}))\leftarrow
H_{1}(Q({\mathfrak{P}}_{n}))\leftarrow 0.$
Group $H_{0}$ freely generated connected components semicubical sets.
Homomorphism
$H_{0}(\theta):H_{0}(Q({\,\mathcal{N}}_{n})\cap
Q({\,\mathcal{N}}^{\prime}_{n}))\to H_{0}(Q({\,\mathcal{N}}_{n}))\oplus
H_{0}(Q({\,\mathcal{N}}^{\prime}_{n}))$
induced chain homomorphism
$\theta_{k}:C_{k}(Q({\,\mathcal{N}}_{n})\cap
Q({\,\mathcal{N}}^{\prime}_{n}))\to C_{k}(Q({\,\mathcal{N}}_{n}))\oplus
C_{k}(Q({\,\mathcal{N}}^{\prime}_{n})),$
defined by $\sigma\in(Q({\,\mathcal{N}}_{n})\cap
Q({\,\mathcal{N}}^{\prime}_{n}))_{k}$ by the formula
$\theta_{k}(\sigma)=\sigma\oplus\sigma$. This implies that $H_{0}(\theta)$
acts on the homology classes of the formula
$H_{0}(\theta)(cls(\sigma))=cls(\sigma)\oplus cls(\sigma)$. Because these
classes homology are connected components semicubical sets, then
$H_{0}(\theta)$ is a homomorphism
${\,\mathbb{Z}}\oplus{\,\mathbb{Z}}\to{\,\mathbb{Z}}\oplus{\,\mathbb{Z}}$,
given by the matrix $\left(\begin{array}[]{cc}1&1\\\ 1&1\end{array}\right)$.
But $H_{1}({\mathfrak{P}}_{n})$ is isomorphic to kernel of the homomorphism
$H_{0}(\theta)$, and $H_{0}({\mathfrak{P}}_{n})$ – cokernel. Hence, for
example, with a cast of this matrix to Smith normal form, we obtain
$H_{0}({\mathfrak{P}}_{n})=H_{1}({\mathfrak{P}}_{n})={\,\mathbb{Z}}$. $\Box$
## 4 Directed homology
We present supporting information on the homology groups Goubault semicubical
sets. We study the properties of directed homology category states and their
interpretation in dimension $0$. Calculate the direction of homology Petri net
pipeline.
### 4.1 Homology groups of Goubault
Let $X=(X_{n},\partial^{n,\varepsilon}_{i})$ semicubical set. For all
$\varepsilon\in\\{0,1\\}$, consider the chain complex of abelian groups
$C_{n}(X)=L(X_{n})$ with the differentials
$0\leftarrow C_{0}(X)\stackrel{{\scriptstyle
d_{1}^{\varepsilon}}}{{\leftarrow}}C_{1}(X)\leftarrow\cdots
C_{n-1}(X)\stackrel{{\scriptstyle
d_{n}^{\varepsilon}}}{{\leftarrow}}C_{n}(X)\leftarrow\cdots,$
where
$d_{n}^{\varepsilon}(\sigma)=\sum\limits_{i=1}^{n}(-1)^{i}\partial_{i}^{n,\varepsilon}(\sigma)$.
Homology groups $H_{n}^{\varepsilon}(X)$ of the complex called homology groups
Goubault semicubical set $X$. They were studied in [6].
Groups $H_{n}^{0}(X)$ are called initial, and $H_{n}^{1}(X)$ – final homology
groups Goubault.
###### Proposition 4.1
Let $X=X_{1}\cup X_{2}$ – union semicubical subsets $X_{1}\subseteq X$ and
$X_{2}\subseteq X$. Then for every $\varepsilon\in\\{0,1\\}$ there is a long
exact sequence
$0\leftarrow H_{0}^{\varepsilon}(X)\leftarrow H_{0}^{\varepsilon}(X_{1})\oplus
H_{0}^{\varepsilon}(X_{2})\leftarrow H_{0}^{\varepsilon}(X_{1}\cap
X_{2})\leftarrow\cdots\\\ \cdots\leftarrow H_{n}^{\varepsilon}(X)\leftarrow
H_{n}^{\varepsilon}(X_{1})\oplus H_{n}^{\varepsilon}(X_{2})\leftarrow
H_{n}^{\varepsilon}(X_{1}\cap X_{2})\leftarrow\cdots\\\ $
Proof repeats almost verbatim the proof of Proposition 2.1.
### 4.2 Directed homology groups of the state category
Homology groups of a small category ${\mathscr{C}}$ with coefficients in the
functor $F:{\mathscr{C}}\rightarrow{\rm Ab}$ is defined as values
$\underrightarrow{\lim}_{n}^{{\mathscr{C}}}F$ of left derived functors of
$\underrightarrow{\lim}^{{\mathscr{C}}}:{\rm Ab}^{{\mathscr{C}}}\to{\rm Ab}$
on $F$.
Let $(S,M(E,I))$ be a state space, and $K(S)$ be its the state category.
Consider the functors $\Delta^{0}{\,\mathbb{Z}}:K(S)\to{\rm Ab}$ and
$\Delta^{1}{\,\mathbb{Z}}:K(S)^{op}\to{\rm Ab}$, taking on the objects the
constant values
$\Delta^{0}{\,\mathbb{Z}}(s)=\Delta^{1}{\,\mathbb{Z}}(s)={\,\mathbb{Z}}$. And
on the morphisms defined by the formulas
$\Delta^{\varepsilon}{\,\mathbb{Z}}(s\stackrel{{\scriptstyle\mu}}{{\to}}s^{\prime})=\left\\{\begin{array}[]{cl}1_{s},&\mbox{
}\mu=1,\\\ 0,&\mbox{ }\mu\not=1,\\\ \end{array}\right.$
###### Definition 4.1
Directed homology groups of the state space defined by the formulas
$H_{n}^{0}(S,M(E,I))=_{def}\underrightarrow{\lim}_{n}^{K(S)}\Delta^{0}{\,\mathbb{Z}},\quad
H_{n}^{1}(S,M(E,I))=_{def}\underrightarrow{\lim}_{n}^{K(S)^{op}}\Delta^{1}{\,\mathbb{Z}}.$
For any elementary Petri nets ${\,\mathcal{N}}$ its homology groups
$H_{n}^{\varepsilon}({\,\mathcal{N}})$ defined as the homology groups of its
state space.
###### Proposition 4.2
For elementary Petri nets ${\,\mathcal{N}}_{n}$ obtained from the pipeline
Petri nets ${\mathfrak{P}}_{n}$ by deleting the transition $t_{1}$, homology
groups equal
$H_{k}^{\varepsilon}({\,\mathcal{N}}_{n})=\left\\{\begin{array}[]{cc}{\,\mathbb{Z}},&\mbox{
if }k=0\\\ 0,&\mbox{ if }k>0.\end{array}\right.$
Proof. By Lemma 3.1, the state category ${\mathscr{C}}_{n}$ of Petri net
${\,\mathcal{N}}_{n}$ has an initial and terminal objects. If a small category
has a terminal object, then the colimit functor on the category equals to the
value of this functor on the terminal object. Hence
$H_{k}^{\varepsilon}({\,\mathcal{N}}_{n})=\underrightarrow{\lim}_{k}\Delta^{\varepsilon}{\,\mathbb{Z}}=0$
for $k>0$ and
$H_{0}^{\varepsilon}({\,\mathcal{N}}_{n})=\underrightarrow{\lim}\Delta^{\varepsilon}{\,\mathbb{Z}}={\,\mathbb{Z}}$.
$\Box$
According to [2, Corollary 5] for any linear order on $E$, there are
isomorphisms
$\underrightarrow{\lim}_{n}^{K(S)}\Delta^{0}{\,\mathbb{Z}}\cong
H_{n}^{0}(Q(S,E,I)),\quad\underrightarrow{\lim}_{n}^{K(S)^{op}}\Delta^{1}{\,\mathbb{Z}}\cong
H_{n}^{1}(Q(S,E,I)).$ (1)
Hence we get the following interpretation of directed homology for a state
space in dimension $0$. State $s$ is called to be a deadlock, if there is no
$a\in E$ satisfying $s\cdot a\in S$. It is called a sender, if there is no
such a pair $s^{\prime}\in S$ and $a\in E$ for which $s^{\prime}\cdot a=s$. In
the state category, deadlock is an object $s$ which has not a morphisms
$\alpha\not=1_{s}$ with $dom\alpha=s$. A sender $s$ has not $\alpha\not=1_{s}$
such that $cod\alpha=s$.
###### Proposition 4.3
The group $H_{0}^{0}(S,M(E,I))$ is isomorphic to the free abelian group
generated by deadlocks, and $H_{0}^{1}(S,M(E,I))$ generated by senders.
Proof. It follows from [2, Corollary 5] that $H_{0}^{0}(S,M(E,I))$ is
isomorphic to the cokernel of $d_{1}^{0}:LQ_{1}(S,E,I)\to LQ_{0}(S,E,I)$
defined as $d_{1}^{0}(s,a)=s$. Cokernel is generated by the set obtained by
the identification with $0$ of all $s\in S$ for which there are $a\in E$
satisfying $s\cdot a\in S$. It is removed from the set $S$ all objects which
are no deadlocks. The remaining set generates the cokernel. It is proved
similarly that $H_{0}^{1}(S,M(E,I))$ generated by senders. $\Box$
###### Example 4.2
State category of pipeline elementary Petri nets has no senders or deadlocks,
hence $H_{0}^{0}({\mathfrak{P}}_{n})=H_{0}^{1}({\mathfrak{P}}_{n})=0$.
### 4.3 Directed homology groups of pipeline
According to the above example, groups $H_{0}^{0}({\mathfrak{P}}_{n})$ and
$H_{0}^{1}({\mathfrak{P}}_{n})$ equal zero. The following statement shows that
$H_{k}^{0}({\mathfrak{P}}_{n})$ and $H_{k}^{1}({\mathfrak{P}}_{n})$ equal $0$
for all $k\geqslant 0$.
###### Theorem 4.4
$H_{k}^{\varepsilon}({\mathfrak{P}}_{n})=0$, for all $n\geqslant 2$,
$k\geqslant 0$ and $\varepsilon\in\\{0,1\\}$.
Proof. Above, we found that the semicubical set $Q({\mathfrak{P}}_{n})$ is
equal to union of semicubical sets $Q({\,\mathcal{N}}_{n})$ and
$Q({\,\mathcal{N}}^{\prime}_{n})$. The state categories of
${\,\mathcal{N}}_{n}$ and ${\,\mathcal{N}}^{\prime}_{n}$ have the greatest and
the least elements. This means that
$H_{k}^{\varepsilon}({\,\mathcal{N}}_{n})=H_{k}^{\varepsilon}({\,\mathcal{N}}^{\prime}_{n})=0$
for $k>0$, and
$H_{0}^{\varepsilon}({\,\mathcal{N}}_{n})=H_{k}^{\varepsilon}({\,\mathcal{N}}^{\prime}_{n})={\,\mathbb{Z}}$.
By the suggestion of 3.2, $Q({\,\mathcal{N}}_{n})\cap
Q({\,\mathcal{N}}^{\prime}_{n})\cong Q({\,\mathcal{N}}_{n-1})\sqcup
Q({\,\mathcal{N}}_{n-1})$. It follows that the exact sequence of the fragment
suggests 4.1
$\leftarrow H_{k-1}^{\varepsilon}(Q({\,\mathcal{N}}_{n})\cap
Q({\,\mathcal{N}}^{\prime}_{n}))\leftarrow
H_{k}^{\varepsilon}(Q({\mathfrak{P}}_{n}))\leftarrow
H_{k}^{\varepsilon}(Q({\,\mathcal{N}}_{n}))\oplus
H_{k}^{\varepsilon}(Q({\,\mathcal{N}}^{\prime}_{n}))\leftarrow$
leads to the equations $H_{k}^{\varepsilon}(Q({\mathfrak{P}}_{n}))=0$ for
$k\geqslant 2$. In addition, there is an exact sequence
$0\leftarrow H_{0}^{\varepsilon}(Q({\mathfrak{P}}_{n}))\leftarrow
H_{0}^{\varepsilon}(Q({\,\mathcal{N}}_{n}))\oplus
H_{0}^{\varepsilon}(Q({\,\mathcal{N}}^{\prime}_{n}))\stackrel{{\scriptstyle
H_{0}^{\varepsilon}(\theta)}}{{\leftarrow}}\\\
H_{0}^{\varepsilon}(Q({\,\mathcal{N}}_{n})\cap
Q({\,\mathcal{N}}^{\prime}_{n}))\leftarrow
H_{1}^{\varepsilon}(Q({\mathfrak{P}}_{n}))\leftarrow 0$
By example 4.2 and isomorphisms (1), we have
$H_{0}^{\varepsilon}(Q({\mathfrak{P}}_{n}))=0$. We obtain an epimorphism
${\,\mathbb{Z}}\oplus{\,\mathbb{Z}}\stackrel{{\scriptstyle
H^{\varepsilon}_{0}(\theta)}}{{\to}}{\,\mathbb{Z}}\oplus{\,\mathbb{Z}}$. But
${\,\mathbb{Z}}\oplus{\,\mathbb{Z}}$ is projective object in the category of
Abelian groups. It follows that there exists a homomorphism $\gamma$ such that
$H_{0}^{\varepsilon}(\theta)\circ\gamma=1_{{\,\mathbb{Z}}\oplus{\,\mathbb{Z}}}$.
This means that the determinant of $H^{\varepsilon}_{0}(\theta)$ is invertible
where the kernel $H_{0}^{\varepsilon}(\theta)$ equally zero. Consequently,
$H_{1}^{\varepsilon}(Q({\mathfrak{P}}_{n}))=0$. We have proved that
$H_{k}^{\varepsilon}(Q({\mathfrak{P}}_{n}))=0$ for all $k\geqslant 0$.
Isomorphisms (1) give $H_{k}^{\varepsilon}({\mathfrak{P}}_{n})\cong
H_{k}^{\varepsilon}(Q({\mathfrak{P}}_{n}))$. $\Box$
## References
* [1] Husainov A., “On the homology of small categories and asynchronous transition systems”, Homology Homotopy Appl., 6:1 (2004), 439–471; http://www.rmi.acnet.ge/hha
* [2] Husainov A.A., “The Homology of Partial Monoid Actions and Petri Nets”, Appl. Categor. Struct., 2012. DOI: 10.1007/s10485-012-9280-9
* [3] Husainov A.A., Bushmeleva E.S., “Homology of Asynchronous Systems”, Actual Problems of Mathematics, Physics, and Computer Science in University and School: Materials International Scientific Conference March 23, 2012, Komsomolsk-on-Amur, Humanities and Education Amur State University, 2012. 24–31.
* [4] Winskel G., Nielsen M. Models for Concurrency. Handbook of Logic in Computer Science, Vol.IV, ed. Abramsky, Gabbay and Maibaum. Oxford University Press, 1995\. P.1–148.
* [5] Husainov A.A., “Homology groups of semicubical sets”, Sib. Math. Journal., 49:1 (2008), 224–237.
* [6] Goubault E., The Geometry of Concurrency, Thesis Doct. Phylosophy (Mathematics), Ecole Normale Supérieure 1995, 349 p.
* [7] Gaucher P., “About the globular homology of higher dimensional automata”, Topol. Geom. Differ., 43:2 (2002), 107–156.
|
arxiv-papers
| 2013-01-31T06:23:16 |
2024-09-04T02:49:41.053533
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ahmet A. Husainov, E.S. Bushmeleva, T.A. Trishina",
"submitter": "Ahmet Husainov A.",
"url": "https://arxiv.org/abs/1301.7521"
}
|
1301.7551
|
# Quantum measurements and maps preserving strict convex combinations and pure
states
Lihua Yang, Jinchuan Hou Department of Mathematics, Taiyuan University of
Technology , Taiyuan 030024, P. R. China [email protected];
[email protected]; [email protected]
###### Abstract.
In this paper, a characterization of maps between quantum states that preserve
pure states and strict convex combinations is obtained. Based on this
characterization, a structural theorem for maps between multipartite quantum
states that preserve separable pure states and strict convex combinations is
established. Then these results are applied to characterize injective (local)
quantum measurements and answer some conjectures proposed in
[J.Phys.A:Math.Theor. 45 (2012) 205305].
PACS. 03.65.Aa, 03.65.Ud, 03.65.Db, 03.67.-a
Key words and phrases. quantum measurement, pure states, separable pure
states, strict convex combination preserving maps
This work is partially supported by Natural Science Foundation of China
(11171249, 11271217), and a grant from International Cooperation Program in
Sciences and Technology of Shanxi (2011081039).
## 1\. Introduction
In the theory of quantum mechanics, a state is a positive operator of trace 1
acting on a complex Hilbert space $H$. Denote by ${\mathcal{S}}(H)$ and
$\mathcal{P}ur(H)$ respectively the set of all states and the set of all pure
states ($i.e.$ rank-1 projections) on $H$. In quantum information theory we
deal, in general, with multipartite systems. The underlying space $H$ of a
multipartite composite quantum system is a tensor product of underlying spaces
$H_{i}$ of its subsystems, that is $H=H_{1}\otimes H_{2}\otimes\cdots\otimes
H_{n}$. If $n=2$, the system is called a bipartite system. The definition of
multipartite separability was introduced in [13] as a natural extension of the
notion of separability in bipartite case [14]. Let us denote the set of all
states in an $n$-partite system by ${\mathcal{S}}(H_{1}\otimes\cdots\otimes
H_{n})$. In the case $\dim H<\infty$, a state
$\rho\in{\mathcal{S}}(H_{1}\otimes\cdots\otimes H_{n})$ is said to be (fully)
separable if it admits a representation of the form
$\rho=\sum_{i}p_{i}\rho^{(1)}_{i}\otimes\cdots\otimes\rho^{(n)}_{i},$
where $p_{i}>0$ with $\sum_{i}p_{i}=1$ and
$\rho_{i}^{(k)}\in{\mathcal{S}}(H_{k})$. Otherwise, $\rho$ is said to be
entangled. Denote by respectively ${\mathcal{S}}_{\rm sep}(H_{1}\otimes
H_{2}\otimes\cdots\otimes H_{n})$ and ${\mathcal{P}ur}_{\rm sep}(H_{1}\otimes
H_{2}\otimes\cdots\otimes H_{n})$ the set of all separable states and the set
of all separable pure states on $H_{1}\otimes H_{2}\otimes\cdots\otimes
H_{n}$. It is obvious that
$\begin{array}[]{rl}&{\mathcal{P}ur}_{\rm sep}(H_{1}\otimes
H_{2}\otimes\cdots\otimes
H_{n})={\mathcal{P}ur}(H_{1})\otimes{\mathcal{P}ur}(H_{2})\otimes\cdots\otimes{\mathcal{P}ur}(H_{n})\\\
=&\\{P_{1}\otimes P_{2}\otimes\cdots\otimes
P_{n}:P_{i}\in{\mathcal{P}ur}(H_{i}),i=1,2,\ldots,n\\}.\end{array}$
The theory of maps on the set of states plays an important role in quantum
computation and quantum information science. It is important to understand,
characterize, and construct different classes of maps on states. For instance,
all quantum channels and quantum operations are completely positive linear
maps; in quantum error correction, one has to construct the recovery map for a
given channel; to study the entanglement of states, one constructs NCP (non
completely positive) positive maps and entanglement witnesses. Many
researchers pay their attention to the problem of characterizing the maps on
the states; ref. [1, 2, 3, 6].
The present paper is motivated by some conjectures proposed in [8] where the
bijective maps $\Phi:{\mathcal{S}}_{\rm sep}(H_{1}\otimes
H_{2})\rightarrow{\mathcal{S}}_{\rm sep}(H_{1}\otimes H_{2})$ that preserve
strict convex combinations were studied.
Recall that a map $\phi$ between convex sets is said to be (strict) convex
combination preserving if, for any $\rho,\sigma\in{\mathcal{S}}(H)$ and
$t\in[0,1]$ ($t\in(0,1)$), there is some $s$ with $0\leq s\leq 1$ ($0<s<1$)
such that $\phi(t\rho+(1-t)\sigma)=s\phi(\rho)+(1-s)\phi(\sigma)$. It is
obvious that $\phi$ preserves (strict) convex combination if and only if
$\phi([\rho,\sigma])\subseteq[\phi(\rho),\phi(\sigma)]$
($\phi((\rho,\sigma))\subseteq(\phi(\rho),\phi(\sigma))$), where $[A,B]$
stands for the closed (open) line segment joint $A$ and $B$, that is,
$[A,B]=\\{tA+(1-t)B:0\leq t\leq 1\\}$ ($(A,B)=[A,B]\setminus\\{A,B\\}$, here
we define $(\rho,\rho)=\\{\rho\\}$). The maps preserve strict convex
combinations are closely related to quantum measurements. In quantum mechanics
a fine-grained quantum measurement is described by a collection $\\{M_{m}\\}$
of measurement operators acting on the Hilbert space $H$ corresponding to the
system satisfying $\sum_{m}M_{m}^{*}M_{m}=I$; ref. for example, [4]. Let
$M_{j}$ be a measurement operator. If the state of the quantum system is
$\rho\in{\mathcal{S}}(H)$ before the measurement, then the state after the
measurement is $\frac{M_{j}\rho M_{j}^{*}}{{\rm Tr}(M_{j}\rho M_{j}^{*})}$
whenever $M_{j}\rho M_{j}^{*}\not=0$. If we fix an $M_{j}=M$ is fixed, we get
a measurement map $\phi$ defined by $\phi(\rho)=\frac{M\rho M^{*}}{{\rm
Tr}(M\rho M^{*})}$ from the convex subset ${\mathcal{S}}_{M}(H)=\\{\rho:M\rho
M^{*}\not=0\\}$ of the (convex) set ${\mathcal{S}}(H)$ into
${\mathcal{S}}(H)$, which is a map that preserves the strict convex
combinations and sends pure states to pure states. If $M$ is invertible
(injective), then $\phi:{\mathcal{S}}(H)\rightarrow{\mathcal{S}}(H)$ is
bijective (injective) and will be called an invertible (injective) measurement
map.
The problem of characterizing the strict convex combination maps between
quantum states was firstly attacked by [5]. It is shown in [5] that a
bijective map $\phi:{\mathcal{S}}(H)\rightarrow{\mathcal{S}}(H)$, $\dim H\geq
2$, is (strict) convex combination preserving if and only if $\phi$ is an
invertible quantum measurement map or the composition of transpose and an
invertible quantum measurement map. Note that
$\phi:{\mathcal{S}}(H)\rightarrow{\mathcal{S}}(H)$ is bijective and (strict)
convex combination preserving imply that $\phi$ preserves pure states in both
directions, that is, $\phi({\mathcal{P}ur}(H))={\mathcal{P}ur}(H)$. Let
$\psi:{\mathcal{S}}_{\rm sep}(H_{1}\otimes H_{2})\rightarrow{\mathcal{S}}_{\rm
sep}(H_{1}\otimes H_{2})$ be a bijective map. In [8], based on the work of
[5], it is shown that, if $\Phi$ is (strict) convex combination preserving and
if
$None$ $\begin{array}[]{rl}&\psi(P_{1}\otimes P_{2})\ \mbox{ is a product
state for any}\ P_{i}\in{\mathcal{S}}(H_{i})\ \mbox{\rm with}\\\ &{\rm
rank}P_{i}=1\ \mbox{ and }{\rm rank}P_{j}=2,\ \ (1\leq i\not=j\leq
2),\end{array}$
then $\psi$ is a composition of an invertible local quantum measurement (i.e.,
the map of the form $\rho\mapsto\frac{(S\otimes T)\rho(S\otimes T)^{*}}{{\rm
Tr}((S\otimes T)\rho(S\otimes T)^{*})}$ with $S,T$ invertible) and some of the
following maps: the transpose, the partial transpose and the swap. In the
sequel we say a map is essentially a (local) quantum measurement if it is a
(local) quantum measurement or a composition of a (local) quantum measurement
with any one of the following maps: the transpose, the partial transpose, and
the swap.
It is conjectured in [8] that if a bijective map $\psi:{\mathcal{S}}_{\rm
sep}(H_{1}\otimes H_{2})\rightarrow{\mathcal{S}}_{\rm sep}(H_{1}\otimes
H_{2})$ is (strict) convex combination preserving, then it sends in fact
product states to product states, and thus the additional assumption Eq.(1.1)
in the result just mentioned above is superfluous.
The purpose of the present paper is to answer these conjectures affirmatively
for finite dimensional systems. To do this, we need first to characterize
general maps between states that preserve the strict convex combinations and
send pure states to pure states. Let $H$ and $K$ be complex Hilbert spaces of
dimension $\geq 2$ with $\dim H<\infty$. We show in Section 2 that a map
$\psi:{\mathcal{S}}(H)\rightarrow{\mathcal{S}}(K)$ preserves strict convex
combinations and pure states if and only if it has one of the following three
forms: (1) $\psi$ is contractive to a pure state, i.e, there exists a pure
state $Q\in{\mathcal{P}ur}(K)$ such that $\psi(\rho)=Q$ for all
$\rho\in{\mathcal{S}}(H)$; (2) there exist distinct pure states
$Q_{i}\in{\mathcal{P}ur}(K)$, $i=1,2$ such that
$\psi({\mathcal{P}ur}(H))=\\{Q_{1},Q_{2}\\}$, and
$\psi({\mathcal{S}}(H))\subseteq[Q_{1},Q_{2}]$; (3) $\dim H\leq\dim K$ and
there exists an injective operator $M\in\mathcal{B}(H,K)$ such that
$\psi(\rho)=\frac{M\rho M^{*}}{{\rm Tr}(M\rho M^{*})}$ for all
$\rho\in{\mathcal{S}}(H)$, or $\psi(\rho)=\frac{M\rho^{t}M^{*}}{{\rm
Tr}(M\rho^{t}M^{*})}$ for all $\rho\in{\mathcal{S}}(H)$, where $A^{t}$ is the
transpose of $A$ with respect to an arbitrarily fixed orthonormal basis of
$H$, that is, $\psi$ is essentially an injective quantum measurement (see
Theorem 2.5). Note that, by our result, if $\psi$ is strict convex combination
preserving and pure state preserving and if $\psi$ is not continuous, then
$\psi$ must have the form (2), and an example of such map is given (Remark
2.7). Based on the results obtained in Section 2, we are able to give a
structure theorem of maps $\psi:{\mathcal{S}}_{\rm sep}(H_{1}\otimes
H_{2})\rightarrow{\mathcal{S}}_{\rm sep}(K_{1}\otimes K_{2})$ that preserve
strict convex combinations and separable pure states in Section 3, where
$2\leq\dim H_{i}<\infty$ and $\dim K_{i}\geq 2$, $i=1,2$. We show that such
maps can have ten possible forms (Theorem 3.2). Consequently, if the range of
$\psi$ is non-collinear or a singleton, then $\psi$ sends product states to
product states (Corollary 3.3); and moreover, if the range of $\psi$ also
contains a state $\sigma$ so that both of its reductions ${\rm
Tr}_{1}(\sigma)$ and ${\rm Tr}_{2}(\sigma)$ have rank $\geq 2$, then $\psi$ is
essentially an injective local quantum measurement (Corollary 3.4). These
results particularly answer the conjectures in [8] mentioned above. Section 4
is a brief discussion of the same topic for multipartite systems. The similar
structure theorem is valid for maps $\psi:{\mathcal{S}}_{\rm sep}(H_{1}\otimes
H_{2}\otimes\cdots\otimes H_{n})\rightarrow{\mathcal{S}}_{\rm
sep}(K_{1}\otimes K_{2}\otimes\cdots\otimes K_{n})$ that preserve strict
convex combinations and separable pure states, but with more complicated
expressions. Particularly, if the range of $\psi$ is non-collinear or a
singleton, then $\psi$ sends product states to product states; and moreover,
if the range of $\psi$ contains a state $\sigma$ so that each reduction ${\rm
Tr}^{i}(\sigma)$ has rank $\geq 2$, then $\psi$ is essentially a local
injective quantum measurement (Theorem 4.1, Corollary 4.2). Section 5 is a
short conclusion.
## 2\. Maps preserving pure states and strict convex combinations
In this section we characterize the maps between the convex sets of quantum
states that send pure states to pure states and preserve the strict convex
combinations.
We start by giving a simple lemma which is easily checked.
Lemma 2.1. Let $\\{Q_{1},Q_{2},\cdots,Q_{r}\\}$ be a linearly independent set
of rank one projections acting on a Hilbert space $H$. If
$\sum_{i=1}^{r}t_{i}Q_{i}=P_{0}$ is a projection for some
$t_{i}>0$,$i=1,2,\ldots,r$, then $\\{Q_{i}\\}_{i=1}^{r}$ are orthogonal and
$t_{i}=1$ for all $i$.
Let ${\bf H}_{m}$ be the real linear space of all $m\times m$ Hermitian
matrices and let ${\mathcal{P}}_{m}$ be the set of all rank-1 $m\times m$
projection matrices. The next lemma comes from [3] which can be viewed as a
characterization of linear preservers of pure states. Also, ref. [9] for
infinite dimensional case.
Lemma 2.2. Suppose $\phi:{\bf H}_{m}\rightarrow{\bf H}_{n}$ is a linear map
satisfying $\phi({\mathcal{P}}_{m})\subseteq{\mathcal{P}}_{n}$. Then one of
the following holds:
(i) There is $Q\in{\mathcal{P}}_{n}$ such that $\phi(A)={\rm Tr}(A)Q$ for all
$A\in{\bf H}_{m}$.
(ii) $m\leq n$ and there is a $U\in M_{n\times m}$ with $U^{*}U=I_{m}$ such
that $\phi(A)=UAU^{*}$ for all $A\in{\bf H}_{m}$, or $\phi(A)=UA^{t}U^{*}$ for
all $A\in{\bf H}_{m}$.
The following lemma is the main result in [12], which gives a characterization
of strict convex combination preserving maps in terms of linear ones.
Lemma 2.3. Let $X$ and $Y$ be real linear spaces and $D\subseteq X$ a nonempty
convex subset. Assume that $\phi:D\rightarrow Y$ is a strict convex
combination preserving map such that $\phi(D)$ is non-collinear (i.e.,
$\phi(D)$ contains a nondegenerate triangle). Then, there exist a linear
transformation $A:X\rightarrow Y$, a linear functional
$f:X\rightarrow{\mathbb{R}}$, a vector $y_{0}\in Y$, and a scalar
$b\in{\mathbb{R}}$ such that
$f(x)+b>0\quad\mbox{for all }\quad x\in D$
and
$\phi(x)=\frac{Ax+y_{0}}{f(x)+b}\quad\mbox{for all }\quad x\in D.$
By using of Lemma 2.1 and Lemma 2.3 we can prove the following lemma, which is
also crucial for proving our main result.
Lemma 2.4. Let $H$ be a complex Hilbert space with $2\leq\dim H=r<\infty$ and
$\phi:{\mathcal{S}}(H)\rightarrow{\mathcal{S}}(H)$ be a map preserving pure
state and strict convex combinations. If $\phi(\frac{1}{r}I)=\frac{1}{r}I$ and
${\rm ran}\phi$ is non-collinear, then $\phi$ is affine.
Proof. As $\phi$ preserves pure states and strict convex combinations, by
Lemma 2.1, $\phi(\frac{1}{r}I)=\frac{1}{r}I$ implies that $\phi$ maps
orthogonal pure states to orthogonal pure states. Also, by Lemma 2.3, $\phi$
is strict convex combination preserving and ${\rm ran}\phi$ is non-collinear
together imply that $\phi$ has the form
$\phi(\rho)=\frac{\Gamma(\rho)+D}{f(\rho)+d}$ for any
$\rho\in{\mathcal{S}}(H)$, where
$\Gamma:\mathcal{B}_{sa}(H)\rightarrow\mathcal{B}_{sa}(H)$ is a linear
transformation, $f:\mathcal{B}_{\rm sa}(H)\rightarrow\mathbb{R}$ is a linear
functional, $D\in\mathcal{B}_{\rm sa}(H)$ and $d\in\mathbb{R}$ with
$f(\rho)+d>0$ for all $\rho\in{\mathcal{S}}(H)$, ${\mathcal{B}}_{\rm sa}(H)$
is the real linear space of all self-adjoint operators in ${\mathcal{B}}(H)$.
Since $\dim H<\infty$, $\Gamma$ and $f$ are continuous. It follows that $\phi$
is continuous. To prove the lemma, we consider two cases of $\dim H>2$ and
$\dim H=2$ respectively.
Case 1. $\dim H>2$.
We will show that $f$ is a constant on ${\mathcal{S}}(H)$, that is, there is a
real number $a$ such that $f(\rho)=a$ for all $\rho\in{\mathcal{S}}(H)$.
For any normalized orthogonal basis $\\{e_{i}\\}_{i=1}^{r}$ of $H$, let
$P_{i}=e_{i}\otimes e_{i}$. We first claim that $f(e_{i}\otimes
e_{i})=f(e_{j}\otimes e_{j})$ for any $i$ and $j$. Since $\phi$ preserves pure
states, there is a pure state $Q_{i}=x_{i}\otimes x_{i}$ such that
$x_{i}\otimes x_{i}=Q_{i}=\phi(P_{i})=\frac{\Gamma(e_{i}\otimes
e_{i})+D}{f(e_{i}\otimes e_{i})+d}.$
So
$\Gamma(e_{i}\otimes e_{i})+D=(f(e_{i}\otimes e_{i})+d)(x_{i}\otimes x_{i}).$
As $\phi(\frac{I}{r})=\frac{I}{r}$ and
$\frac{I}{r}=\frac{1}{r}\sum_{i=1}^{r}e_{i}\otimes e_{i}$, we have
$\frac{I}{r}=\phi(\frac{1}{r}\sum_{i=1}^{r}e_{i}\otimes
e_{i})=\frac{\Gamma(\sum_{i=1}^{r}\frac{1}{r}e_{i}\otimes
e_{i})+D}{f(\sum_{i=1}^{r}\frac{1}{r}e_{i}\otimes
e_{i})+d}=\frac{\sum_{i=1}^{r}\frac{1}{r}\Gamma(e_{i}\otimes
e_{i})+r\frac{1}{r}D}{\sum_{i=1}^{r}\frac{1}{r}f(e_{i}\otimes
e_{i})+r\frac{1}{r}d},$
that is,
$\frac{I}{r}=\frac{\frac{1}{r}(\sum_{i=1}^{r}(\Gamma(e_{i}\otimes
e_{i})+D))}{\frac{1}{r}(\sum_{i=1}^{r}(f(e_{i}\otimes
e_{i})+d))}=\frac{\sum_{i=1}^{r}(\Gamma(e_{i}\otimes
e_{i})+D)}{\sum_{i=1}^{r}(f(e_{i}\otimes e_{i})+d)}.$
Let $A_{i}=\Gamma(e_{i}\otimes e_{i})+D$ and $a_{i}=f(e_{i}\otimes e_{i})+d$.
Then $A_{i}=a_{i}Q_{i}$ and the above equation becomes to
$I=r(\frac{A_{1}+A_{2}+\cdots+A_{r}}{a_{1}+a_{2}+\cdots+a_{r}})=r(\frac{a_{1}Q_{1}+a_{2}Q_{2}+\cdots+a_{r}Q_{r}}{a_{1}+a_{2}+\cdots+a_{r}}).$
Applying Lemma 2.1, we see that $\frac{ra_{i}}{a_{1}+a_{2}+\cdots+a_{r}}=1$
for each $i=1,2,\ldots,r$ and hence
$a_{1}=a_{2}=\cdots=a_{r}=\frac{a_{1}+a_{2}+\cdots+a_{r}}{r}.$
This implies that there is some scalar $a$ such that $f(e_{i}\otimes e_{i})=a$
holds for all $i$. Now for arbitrary unit vectors $x$, $y\in H$, as $\dim
H>2$, there is a unit vector $z\in H$ such that $z\in[x,y]^{\bot}$. It follows
from what proved above that $f(x\otimes x)=f(z\otimes z)=f(y\otimes y)$. So
$f(x\otimes x)=a$ for all unit vectors $x\in H$. Since each state is a convex
combination of pure states, by the linearity of $f$, we get that $f(\rho)=a$
holds for every state $\rho$. Therefore, we have
$\phi(\rho)=\frac{\Gamma(\rho)+D}{a+d}$
holds for all $\rho$. Then by the linearity of $\Gamma$, it is clear that
$\phi$ is affine, i.e., for any states $\rho$, $\sigma$ and scalar $\lambda$
with $0\leq\lambda\leq 1$,
$\phi(\lambda\rho+(1-\lambda)\sigma)=\lambda\phi(\rho)+(1-\lambda)\phi(\sigma)$.
Case 2. $\dim H=2$.
By fixing an orthonormal basis of $H$ we may identify $\mathcal{S}(H)$ with
$\mathcal{S}_{2}$, the convex set of $2\times 2$ positive matrices with the
trace 1. Then $\phi:\mathcal{S}_{2}\rightarrow\mathcal{S}_{2}$ is a map
preserving pure states and strict convex combinations satisfying
$\phi(\frac{1}{2}I_{2})=\frac{1}{2}I_{2}$. Let us identify $\mathcal{S}_{2}$
with the Bloch ball representation
$(\mathbb{R}^{3})_{1}=\\{(x,y,z)^{t}\in\mathbb{R}^{3}:x^{2}+y^{2}+z^{2}\leq
1\\}$ by the following way. Let
$\pi:(\mathbb{R}^{3})_{1}\rightarrow\mathcal{S}_{2}$ be the map defined by
$(x,y,z)^{t}\longmapsto\frac{1}{2}I_{2}+\frac{1}{2}\left(\begin{array}[]{cc}z&x-iy\\\
x+iy&-z\end{array}\right).$
$\pi$ is a bijective affine isomorphism. Note that $v=(x,y,z)^{t}$ satisfies
$x^{2}+y^{2}+z^{2}=1$ if and only if the corresponding matrix $\pi(v)$ is a
pure state, and $0=(0,0,0)^{t}$ if and only if the corresponding matrix is
$\pi(0)=\frac{1}{2}I$. The map
$\phi:\mathcal{S}_{2}\rightarrow\mathcal{S}_{2}$ induces a map
$\hat{\phi}:(\mathbb{R}^{3})_{1}\rightarrow(\mathbb{R}^{3})_{1}$ by the
following equation
$\phi(\rho)=\pi(\hat{\phi}(\pi^{-1}(\rho))).$
Since $\phi$ is pure state and strict convex combination preserving and
continuous, and $\pi$ is an affine isomorphism, it is easily checked that the
map $\hat{\phi}$ is strict convex combination preserving and maps the surface
of $(\mathbb{R}^{3})_{1}$ into the surface of $(\mathbb{R}^{3})_{1}$. Since
$\phi(\frac{1}{2}I)=\frac{1}{2}I$, we have that
$\hat{\phi}((0,0,0)^{t})=(0,0,0)^{t}$. It is also clear that the range of
$\hat{\phi}$ is non-collinear.
Now applying Lemma 2.3 to $\hat{\phi}$, there exists a linear transformation
$L:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$, a linear functional
$f:\mathbb{R}^{3}\rightarrow\mathbb{R}$, a vector $u_{0}\in\mathbb{R}^{3}$ and
a scalar $s\in\mathbb{R}$ such that $f((x,y,z)^{t})+s>0$ and
$\hat{\phi}((x,y,z)^{t})=\frac{L((x,y,z)^{t})+u_{0}}{f((x,y,z)^{T})+s}$
for each $(x,y,z)^{T}\in(\mathbb{R}^{3})_{1}$. Since
$\hat{\phi}((0,0,0)^{T})=(0,0,0)^{t}$, we have $u_{0}=0$ and $s>0$.
Furthermore, the linearity of $f$ implies that there are real scalars $r_{1}$,
$r_{2}$, $r_{3}$ such that $f((x,y,z)^{t})=r_{1}x+r_{2}y+r_{3}z$. We claim
that $r_{1}=r_{2}=r_{3}=0$ and hence $f=0$. If not, then there is a vector
$(x_{0},y_{0},z_{0})^{T}$ satisfying $x_{0}^{2}+y_{0}^{2}+z_{0}^{2}=1$ such
that $f((x_{0},y_{0},z_{0})^{T})=r_{1}x_{0}+r_{2}y_{0}+r_{3}z_{0}\neq 0$. It
follows that
$1=\|\hat{\phi}((x_{0},y_{0},z_{0})^{t})\|=\|\frac{L((x_{0},y_{0},z_{0})^{t})}{r_{1}x_{0}+r_{2}y_{0}+r_{3}z_{0}+s}\|,$
and thus
$\|L((x_{0},y_{0},z_{0})^{t})\|=r_{1}x_{0}+r_{2}y_{0}+r_{3}z_{0}+s.$
Similarly
$\|L((-x_{0},-y_{0},-z_{0})^{t})\|=-r_{1}x_{0}-r_{2}y_{0}-r_{3}z_{0}+s.$
By the linearity of $L$ we have
$r_{1}x_{0}+r_{2}y_{0}+r_{3}z_{0}+s=-r_{1}x_{0}-r_{2}y_{0}-r_{3}z_{0}+s$.
Hence $r_{1}x_{0}+r_{2}y_{0}+r_{3}z_{0}=0$, a contradiction. So, we have
$f=0$, and thus $\hat{\phi}=\frac{L}{s}$ is affine. Now it is clear that
$\phi$ is affine as $\pi$ is an affine isomorphism. $\Box$
The following is the main result of this section.
Theorem 2.5. Let $H,K$ be complex Hilbert spaces with $2\leq\dim H<\infty$ and
${\mathcal{S}}(H),{\mathcal{S}}(K)$ the convex sets of all states on $H,K$,
respectively. Let $\psi:{\mathcal{S}}(H)\rightarrow{\mathcal{S}}(K)$ be a map.
Then $\psi$ preserves pure states and strict convex combinations (that is,
$\psi({\mathcal{P}ur}(H))\subseteq{\mathcal{P}ur}(K)$ and
$\psi((\rho,\sigma))\subseteq(\psi(\rho),\psi(\sigma))$ for any
$\rho,\sigma\in{\mathcal{S}}(H)$) if and only if one of the following holds:
(1) There exists $\sigma_{0}\in{\mathcal{P}ur}(K)$ such that
$\psi(\rho)=\sigma_{0}$ for all $\rho\in{\mathcal{S}}(H)$.
(2) There exist distinct pure states $Q_{i}\in{\mathcal{P}ur}(K)$, $i=1,2$
such that $\psi({\mathcal{P}ur}(H))=\\{Q_{1},Q_{2}\\}$, and a map
$h:{\mathcal{S}}(H)\rightarrow[0,1]$ such that, for any
$\rho_{1},\rho_{2}\in{\mathcal{S}}(H)$ and any $t\in(0,1)$,
$h(t\rho_{1}+(1-t)\rho_{2})=sh(\rho_{1})+(1-s)h(\rho_{2})$ for some
$s\in(0,1)$, and $\psi(\rho)=h(\rho)Q_{1}+(1-h(\rho))Q_{2}$ for all
$\rho\in{\mathcal{S}}(H)$.
(3) $\dim H\leq\dim K$ and there exists an injective operator
$M\in\mathcal{B}(H,K)$ such that $\psi(\rho)=\frac{M\rho M^{*}}{{\rm Tr}(M\rho
M^{*})}$ for all $\rho\in{\mathcal{S}}(H)$, or
$\psi(\rho)=\frac{M\rho^{t}M^{*}}{{\rm Tr}(M\rho^{t}M^{*})}$ for all
$\rho\in{\mathcal{S}}(H)$, where $A^{t}$ is the transpose of $A$ with respect
to an arbitrarily fixed orthonormal basis of $H$.
We remark here that the form (3) can be restated as:
(3′) $\dim H\leq\dim K$ and there exists an injective linear or conjugate
linear operator $M:H\rightarrow K$ such that $\psi(\rho)=\frac{M\rho
M^{*}}{{\rm Tr}(M\rho M^{*})}$ for all $\rho\in{\mathcal{S}}(H)$.
This statement is more convenient some times.
The following corollary is immediate, which essentially gives a
characterization of injective quantum measurement maps or the transpose of an
injective quantum measurement. We say that a map $\psi$ is open line segment
preserving if $\psi((\rho,\sigma))=(\psi(\rho),\psi(\sigma))$ for any
$\rho,\sigma$.
Corollary 2.6. Let $H,K$ be complex Hilbert spaces with $2\leq\dim H<\infty$
and $\psi:{\mathcal{S}}(H)\rightarrow{\mathcal{S}}(K)$ be a map. Then the
following statements are equivalent.
(1) $\psi$ is strict convex combination preserving with
$\psi({\mathcal{P}ur}(H))\subseteq{\mathcal{P}ur}(K)$ and non-collinear range.
(2) $\psi$ is open line segment preserving with
$\psi({\mathcal{P}ur}(H))\subseteq{\mathcal{P}ur}(K)$ and non-collinear range.
(3) $\dim H\leq\dim K$ and there exists an injective operator
$M\in\mathcal{B}(H,K)$ such that $\psi(\rho)=\frac{M\rho M^{*}}{{\rm Tr}(M\rho
M^{*})}$ for all $\rho\in{\mathcal{S}}(H)$, or
$\psi(\rho)=\frac{M\rho^{t}M^{*}}{{\rm Tr}(M\rho^{t}M^{*})}$ for all
$\rho\in{\mathcal{S}}(H)$, where $A^{t}$ is the transpose of $A$ with respect
to an arbitrarily fixed orthonormal basis of $H$.
Particularly, if $\psi$ is bijective, then, by [5, Lemma 2.1], we have
$\psi(\mathcal{P}ur(H))={\mathcal{P}ur}(K)$. Also the surjectivity of $\psi$
implies the surjectivity of $M$. Thus the above corollary is a generalization
of the main result in [5] for finite dimensional case.
Remark 2.7. If the map $\psi$ has the form (1) or (3) of Theorem 2.5, then
$\psi$ is continuous. However, if $\psi$ has the form (2), $\psi$ is not
continuous and may have erratic behavior. For example, Assume $H$ is of
dimension 2. Let $Q_{1},Q_{2}$ be two distinct pure states on $K$. Divide
${\mathcal{P}ur}(H)$ into two disjoint parts
${\mathcal{P}ur}(H)={\mathcal{P}}_{1}\cup{\mathcal{P}}_{2}$ with the property
$P\in{\mathcal{P}}_{1}\Leftrightarrow P^{\bot}\in{\mathcal{P}}_{2}$ and define
$\psi(P)=Q_{1}$ if $P\in{\mathcal{P}}_{1}$; $\psi(P)=Q_{2}$ if
$P\in{\mathcal{P}}_{2}$; $\psi(tP_{1}+(1-t)P_{2})=\frac{1}{2}(Q_{1}+Q_{2})$ if
$P_{i}\in{\mathcal{P}}_{i}$, $i=1,2$, and $t\in(0,1)$, where $Q_{1},Q_{2}$ are
any distinct pure states on $K$. Then,
$\psi:{\mathcal{S}}(H)\rightarrow{\mathcal{S}}(K)$ is strict convex
combination preserving and
$\psi({\mathcal{P}ur}(H))\subset{\mathcal{P}ur}(K)$. $\psi$ has the form (2)
in Theorem 2.5.
Now let us start to prove the main result of this section.
Proof of Theorem 2.5.
If $\psi$ has the form (1) or (2) or (3), it is clear that $\psi$ is pure
states and strict convex combination preserving. Conversely, assume that
$\psi$ is pure states and strict convex combination preserving. We will show
that $\psi$ has one of the forms stated in (1), (2) and (3).
Assume $\dim H=m<\infty$.
As $\frac{1}{m}I\in{\mathcal{S}}(H)$, where $I$ is the identity on $H$,
$\psi(\frac{1}{m}I)$ is positive with trace 1. So
$\psi(\frac{1}{m}I)=\frac{RR^{*}}{{\rm Tr}(RR^{*})}$ for some bounded linear
operator $R$ from $H$ into $K$.
Claim 1. $\dim{\rm ran}R\leq m$ and ${\rm ran}\psi(\rho)\subseteq{\rm ran}R$
holds for all $\rho\in\mathcal{S}(H)$.
Firstly we will show that ${\rm ran}\psi(P)\subseteq{\rm ran}R$ holds for any
pure state $P\in\mathcal{S}(H)$. Let $P=P_{1}$. There exist pure states
$\\{P_{2},\cdots,P_{m}\\}$ such that $\\{P_{1},P_{2},\cdots,P_{m}\\}$ is an
orthogonal set satisfying $P_{1}+P_{2}+\cdots+P_{m}=I$. Since $\psi$ is strict
convex combination preserving, there are $p_{i}\in(0,1)$ $(i=1,2,\cdots,m)$
with $\sum_{i=1}^{m}p_{i}=1$ such that
$\frac{RR^{*}}{{\rm
Tr}(RR^{*})}=\psi(\frac{1}{m}I)=\psi(\frac{1}{m}\sum_{i=1}^{m}P_{i})=\sum_{i=1}^{m}p_{i}\psi(P_{i}).$
Note that $\psi(P_{i})$s are rank-1 projections by the assumption. It follows
that $\dim{\rm ran}R\leq\sum_{i=1}^{m}\dim{\rm ran}\psi(P_{i})\leq m$ and, for
any $i$, we have $0\leq\psi(P_{i})\leq p_{i}^{-1}\frac{RR^{*}}{{\rm
Tr}(RR^{*})}$. Hence ${\rm ran}\psi(P_{i})\subseteq{\rm ran}R$ for all $i$.
Particularly, ${\rm ran}\psi(P)={\rm ran}\psi(P_{1})\subseteq{\rm ran}R$.
For any $\rho\in{\mathcal{S}}(H)$, let $\rho=\sum_{i=1}^{m}t_{i}P_{i}$ be its
spectral resolution. As $t_{i}\geq 0$, $\sum_{i}t_{i}=1$ and $\psi$ is strict
convex combination preserving, there are $s_{i}\in[0,1]$ with
$\sum_{i}s_{i}=1$ and $s_{i}=0$ if $t_{i}=0$ such that
$\psi(\rho)=\sum_{i=1}^{m}s_{i}\psi(P_{i})$. Now it is clear that ${\rm
ran}\psi(\rho)\subseteq{\rm ran}R$ because ${\rm ran}\psi(P_{i})\subseteq{\rm
ran}R$ for all $i=1,2,\ldots,m$.
As a result, if $R$ is of rank-1, then it is clear that $\psi$ has the form
(1) in Theorem 2.5.
So, in the sequel, we assume that rank$(R)=r\geq 2$. Thus we can define a map
$\phi:\mathcal{S}(H)\rightarrow\mathcal{S}(H)$ by
$None$ $\phi(\rho)=\frac{R^{[-1]}\psi(\rho)R^{[-1]*}}{{\rm
Tr}(R^{[-1]}\psi(\rho)R^{[-1]*})},$
where $R^{[-1]}$ is the Moore-Penrose generalized inverse of $R$. It is clear
that $\phi(\mathcal{P}ur(H))\subseteq\mathcal{P}ur(H)$ and $\phi$ preserves
strict convex combination. Write $\phi(\frac{1}{m}I_{m})=\frac{Q_{0}}{{\rm
Tr}(Q_{0})}$, where $Q_{0}=R^{[-1]}R$. This implies that $Q_{0}$ is a
projection with ${\rm rank}Q_{0}=r\leq m$. It follows that, there exists an
orthonormal set $\\{e_{1},\ldots,e_{r}\\}\subset H$ such that
$\sum_{i=1}^{r}\phi(P_{i})=Q_{0}$, where $P_{i}=e_{i}\otimes e_{i}$. Let
$H_{1}={\rm span}\\{e_{1},\ldots,e_{r}\\}$ and $K_{1}=Q_{0}(H)$. Then $\dim
H_{1}=\dim K_{1}=r$ and
$\widetilde{\phi}=\phi|_{{\mathcal{S}}(H_{1})}:{\mathcal{S}}(H_{1})\rightarrow{\mathcal{S}}(K_{1})$
is a strict convex combination preserver sending pure states to pure states.
As
$\widetilde{{\phi}}(\frac{1}{r}I_{H_{1}})=\sum_{i=1}^{r}q_{i}\phi(P_{i})=SS^{*}$
is an invertible state on $K_{1}$, it induces a strict convex combination
preserver
$\widehat{{\phi}}:{\mathcal{S}}(H_{1})\rightarrow{\mathcal{S}}(H_{1})$
satisfying
$\widehat{\phi}({\mathcal{P}ur}(H_{1}))\subseteq{\mathcal{P}ur}(H_{1})$ and
$\widehat{\phi}(\frac{1}{r}I_{H_{1}})=\frac{1}{r}I_{H_{1}}$, where
$\widehat{\phi}$ is defined by
$None$
$\widehat{\phi}(\rho)=\frac{S^{-1}\widetilde{\phi}(\rho){S^{-1}}^{*}}{{\rm
Tr}(S^{-1}\widetilde{\phi}(\rho){S^{-1}}^{*})}.$
It follows that $\widehat{\phi}$ maps orthogonal pure states to orthogonal
ones.
Claim 2. If $r\geq 3$, then there is a unitary operator $U:H_{1}\rightarrow
H_{1}$ such that either $\widehat{\phi}(\rho)=U\rho U^{*}$ for every
$\rho\in{\mathcal{S}}(H_{1})$ or $\widehat{\phi}(\rho)=U\rho^{t}U^{*}$ for
every $\rho\in{\mathcal{S}}(H_{1})$.
If $r\geq 3$, then $\widehat{\phi}({\mathcal{S}}(H_{1}))$ contains at least 3
rank one projections which are orthogonal to each other and hence non-
collinear. So, by Lemma 2.4, $\widehat{\phi}$ is affine, that is
$\widehat{\phi}(t\rho+(1-t)\sigma)=t\widehat{\phi}(\rho)+(1-t)\widehat{\phi}(\sigma)$
holds for any $\rho,\sigma\in{\mathcal{S}}(H_{1})$ and $t\in[0,1]$. Thus
$\widehat{\phi}$ is injective and can be extended to an injective linear map
from ${\mathcal{B}}_{\rm sa}\cong{\bf H}_{r}$ into ${\mathcal{B}}_{\rm
sa}\cong{\bf H}_{r}$.
Now by Lemma 2.2, there is a unitary operator $U:H_{1}\rightarrow H_{1}$ such
that either $\widehat{\phi}(\rho)=U\rho U^{*}$ for every
$\rho\in{\mathcal{S}}(H_{1})$ or $\widehat{\phi}(\rho)=U\rho^{t}U^{*}$ for
every $\rho\in{\mathcal{S}}(H_{1})$.
If $P$ and $Q$ are projections and $PQ=0$, we say that $P$ and $Q$ are
orthogonal, denoted by $P\bot Q$. $P^{\bot}$ stands for $I-P$.
Claim 3. If $r=2$, then either
(i) there exist
${\mathcal{P}}_{1},{\mathcal{P}}_{2}\subseteq{\mathcal{P}ur}(H_{1})$
satisfying ${\mathcal{P}}_{1}\cup{\mathcal{P}}_{2}={\mathcal{P}ur}(H_{1})$ and
$P,I_{2}-P$ can not be in the same ${\mathcal{P}}_{i}$; and there exist
$Q_{1},Q_{2}\in{\mathcal{P}ur}(H_{1})$ with $Q_{1}\bot Q_{2}$ such that
$\widehat{\phi}({\mathcal{P}}_{i})=\\{Q_{i}\\}$, $i=1,2$, and
$\widehat{\phi}({\mathcal{S}}(H_{1})\subseteq[Q_{1},Q_{2}]$.
(ii) there exists a unitary operator $U:H_{1}\rightarrow H_{1}$ such that
either $\widehat{\phi}(\rho)=U\rho U^{*}$ for every
$\rho\in{\mathcal{S}}(H_{1})$ or $\widehat{\phi}(\rho)=U\rho^{t}U^{*}$ for
every $\rho\in{\mathcal{S}}(H_{1})$.
If there exist $Q_{1},Q_{2},Q_{3}\in\widehat{\phi}({\mathcal{S}}(H_{1}))$ such
that they are non-collinear, then Lemma 2.4 is applicable. It follows that
$\widehat{\phi}$ is affine and can be extended to a linear or conjugate linear
map, still denoted by $\widehat{\phi}$ from ${\bf H}_{2}$ into ${\bf H}_{2}$,
which is unital and rank-1 projection preserving. Therefore, use Lemma 2.2 we
see that (ii) is true.
Assume that $\widehat{\phi}({\mathcal{S}}(H_{1})$ is collinear; then there
exist $P_{1},P_{2}\in{\mathcal{S}}(H_{1})$ such that
$\widehat{\phi}(P_{i})=Q_{i}$, $Q_{1}+Q_{2}=I_{2}$ and
$\widehat{\phi}({\mathcal{S}}(H_{1})\subseteq[Q_{1},Q_{2}]$. It entails that
$\widehat{\phi}({\mathcal{P}ur}(H_{1}))=\\{Q_{1},Q_{2}\\}$. Let
${\mathcal{P}}_{i}=\widehat{\phi}^{-1}(\\{Q_{i}\\})$, $i=1,2$. Then,
${\mathcal{P}}_{1}\cap{\mathcal{P}}_{2}=\emptyset$ and
${\mathcal{P}}_{1}\cup{\mathcal{P}}_{2}={\mathcal{P}ur}(H_{1})$. If
$P\in{\mathcal{P}}_{1}$, then
$\frac{1}{2}\widehat{\phi}(P)+\frac{1}{2}\widehat{\phi}(P^{\bot})=\widehat{\phi}(\frac{1}{2}(P+P^{\bot}))=\frac{1}{2}I_{2}.$
So $\widehat{\phi}(P^{\bot})=Q_{2}$, and $P^{\bot}\in{\mathcal{P}}_{2}$. Now,
it is clear that (i) holds.
Claim 4. If $r=m\geq 3$, then $\psi(\rho)=\frac{M\rho M^{*}}{{\rm Tr}(M\rho
M^{*})}$ for all $\rho\in{\mathcal{S}}(H)$; or
$\psi(\rho)=\frac{M\rho^{t}M^{*}}{{\rm Tr}(M\rho^{t}M^{*})}$ for all
$\rho\in{\mathcal{S}}(H)$, where $M:H\rightarrow K$ is an injective linear
operator. So, $\psi$ has the form (3) of Theorem 2.5.
In fact, in this case we have $\widehat{\phi}=\widetilde{\phi}=\phi$. By Claim
2, there is a unitary $U:H\rightarrow H$ such that $\phi(\rho)=U\rho U^{*}$
for all $\rho$ or $\phi(\rho)=U\rho^{t}U^{*}$ for all $\rho$. Let $M=RU$. Then
$M:H\rightarrow K$ is injective and the claim holds.
By use of Claim 4, the following claim is obvious.
Claim 5. If $r=m=2$, then either $\psi$ has the form (3) or has the form (2)
of Theorem 2.5.
Claim 6. If $2=r<m$, then $\psi$ has the form (2).
As $r=2$, $\widehat{\phi}$ has two possible forms (i) and (ii) stated in Claim
3.
If $\widehat{\phi}$ has the form (i), then there exist distinct pure states
$Q_{1},Q_{2}$ on $K$ such that $\psi({\mathcal{P}ur}(H))=\\{Q_{1},Q_{2}\\}$.
It is clear that, in this case, we have $\psi(\rho)\in[Q_{1},Q_{2}]$ for every
$\rho\in{\mathcal{S}}(H)$, that is, $\psi$ is of the form (2) stated in
Theorem 2.5.
We assert that the case (ii) does not occur. If $\widehat{\phi}$ takes form
(ii), then there exists an orthogonal set of pure states
$\\{P_{1}=e_{1}\otimes e_{1},\ldots,P_{m}=e_{m}\otimes e_{m}\\}$ such that
ran$(\psi)={\mathcal{S}}(K_{1})$, where $K_{1}={\rm span}\\{u_{1},u_{2}\\}$
with $u_{i}\otimes u_{i}=\psi(e_{i}\otimes e_{i})$, $i=1,2$. Thus $\psi$ is
continuous when restricted on ${\mathcal{S}}(H_{1})$ with $H_{1}={\rm
span}\\{e_{1},e_{2}\\}$. Note that $\psi(P_{3})=\psi(P_{1})$ or $\psi(P_{2})$,
say $\psi(P_{3})=\psi(P_{1})=Q_{1}$. Let $P(\alpha,\beta)=(\alpha e_{1}+\beta
e_{2})\otimes(\alpha e_{1}+\beta e_{2})\in{\mathcal{P}ur}(H_{1})$, where
$\alpha,\beta\in{\mathbb{C}}$ satisfying $|\alpha|^{2}+|\beta|^{2}=1$. Then,
$P(\alpha,\beta)$ is continuous and hence $\psi(P(\alpha,\beta))$ is
continuous in $\alpha,\beta$. Since
$\\{P(\alpha,\beta),P(\bar{\beta},-\bar{\alpha}),P_{3},\ldots,P_{m}\\}$ is
still a complete orthogonal set of rank-1 projections, one of them must be
$\psi(P_{3})=Q_{1}$ and another be $Q_{2}$. It follows that the range of
$\psi(P(\alpha,\beta))$ can take at most two distinct value and thus must be a
constant function. However, $\psi(P(1,0))=\psi(P_{1})=Q_{1}$ and
$\psi(P(0,1))=\psi(P_{2})=Q_{2}$, a contradiction. So, this case can not
occur, finishing the proof of Claim 6.
Claim 7. The case of $3\leq r<m$ can not occur.
On the contrary, suppose $m>r\geq 3$. Then by Lemma 2.3, $\phi$ is continuous.
Choose any orthogonal set of rank one projections $\\{P_{i}=x_{i}\otimes
x_{i}\\}_{i=1}^{m}$ satisfying $\sum_{i=1}^{m}P_{i}=I$. Then there exists
$p_{i}>0$ with $\sum_{i=1}^{m}p_{i}=1$ such that
$\phi(\frac{1}{m}I)=\phi(\frac{1}{m}\sum_{i=1}^{m}P_{i})=\sum_{i=1}^{m}p_{i}\phi(P_{i})=\frac{1}{r}Q_{0}$.
It follows that $m$ rank-1 projections $\phi(P_{1}),\cdots,\phi(P_{m})$ are
linearly dependent. Without loss of generality, assume
$\\{\phi(P_{1}),\cdots,\phi(P_{r})\\}$ is linearly independent. Then for any
$j>r$, there exists $i_{j}\leq r$ such that $\phi(P_{j})=\phi(P_{i_{j}})$. So
there exist $q_{i}>0$, $(1\leq i\leq r)$, such that
$\phi(\frac{1}{m}I)=\sum_{i=1}^{r}q_{i}\phi(P_{i})$. By Lemma 2.1, we obtain
that $q_{i}=\frac{1}{r}$ and $\\{\phi(P_{i})\\}_{i=1}^{r}$ is an orthogonal
set of rank-1 projections. Consequently we obtain that, for any two orthogonal
rank-1 projections $Q_{1}=y_{1}\otimes y_{1}$, $Q_{2}=y_{2}\otimes y_{2}$ on
$H$, either $\phi(Q_{1})=\phi(Q_{2})$ or $\phi(Q_{1})\bot\phi(Q_{2})$. Let
$Q^{\prime}_{1}=(\alpha y_{1}+\beta y_{2})\otimes(\alpha y_{1}+\beta y_{2})$,
$Q^{\prime}_{2}=(\bar{\beta}y_{1}-\bar{\alpha}y_{2})\otimes(\bar{\beta}y_{1}-\bar{\alpha}y_{2})$,
where $|\alpha|^{2}+|\beta|^{2}=1$. Clearly $Q^{\prime}_{1}\bot
Q^{\prime}_{2}$. We assert that:
$None$
$\phi(Q_{1})=\phi(Q_{2})\Rightarrow\phi(Q^{\prime}_{1})=\phi(Q^{\prime}_{2})\quad\mbox{\rm
and}\quad\phi(Q_{1})\bot\phi(Q_{2})\Rightarrow\phi(Q^{\prime}_{1})\bot\phi(Q^{\prime}_{2}).$
To see this, let $f(\alpha,\beta)=\phi(Q^{\prime}_{1})$,
$g(\alpha,\beta)=\phi(Q^{\prime}_{2})$ and let
$h(\alpha,\beta)=\|f(\alpha,\beta)-g(\alpha,\beta)\|$. As $\phi$ is
continuous, we see that $f,g$ and $h$ are continuous in $\alpha,\beta$. Also
note that $h(\alpha,\beta)\in\\{0,1\\}$ for any $(\alpha,\beta)$. Hence, if
$\phi(Q_{1})=\phi(Q_{2})$, then $h(1,0)=0$, which forces
$h(\alpha,\beta)\equiv 0$ and consequently, $f(\alpha,\beta)=g(\alpha,\beta)$
for all $\alpha,\beta$; if $\phi(Q_{1})\bot\phi(Q_{2})$, then
$h(\alpha,\beta)\equiv h(0,1)=1$, which implies that $f(\alpha,\beta)\bot
g(\alpha,\beta)$ for all $\alpha,\beta$. So the assertion (2.4) is true.
Now for the chosen orthogonal set $\\{P_{i}\\}_{i=1}^{m}$, as $3\leq r<m$, by
what proved above, we can rearrange the order of $\\{\phi(P_{i})\\}_{i=1}^{m}$
so that $\\{\phi(P_{1}),\ldots,\phi(P_{r})\\}$ is an orthogonal set. Then
$\phi(P_{1})+\cdots+\phi(P_{r})=Q_{0}$, $\phi(P_{r+j})$ equals to
$\phi(P_{i})$ for some $i$ with $1\leq i\leq r$.
Assume that there exist two distinguished projections in
$\\{\phi(P_{r+1}),\ldots,\phi(P_{m})\\}$, say
$\phi(P_{r+1})\not=\phi(P_{r+2})$. Let $P^{\prime}_{r+1}=(\alpha x_{r+1}+\beta
x_{r+2})\otimes(\alpha x_{r+1}+\beta x_{r+2})$ and
$P^{\prime}_{r+2}=(\bar{\beta}x_{r+1}-\bar{\alpha}x_{r+2})\otimes(\bar{\beta}x_{r+1}-\bar{\alpha}x_{r+2})$
with $|\alpha|^{2}+|\beta|^{2}=1$. By Eq.(2.4), $f(\alpha,\beta)\bot
g(\alpha,\beta)$, where $f(\alpha,\beta)=\phi(P^{\prime}_{r+1})$ and
$g(\alpha,\beta)=\phi(P^{\prime}_{r+2})$. Since
$\\{P_{1},\ldots,P_{r},P^{\prime}_{r+1},P^{\prime}_{r+2},\ldots,P_{m}\\}$ is
still orthogonal, we see that
$f(\alpha,\beta)\in\\{\phi(P_{1}),\ldots,\phi(P_{r})\\}$. The continuity of
$f$ then implies that $f(\alpha,\beta)\equiv\phi(P_{i_{0}})$ for some $1\leq
i_{0}\leq r$. Similarly, $g(\alpha,\beta)\equiv\phi(P_{i_{1}})$ for some
$1\leq i_{1}\leq r$. Since $f$ and $g$ has the same range, we must have
$i_{0}=i_{1}$, but this contradicts to $f(1,0)\bot g(1,0)$.
Therefore, we may assume that $\phi(P_{r+1})=\cdots=\phi(P_{m})=\phi(P_{1})$.
Now Let $P^{\prime}_{1}=(\alpha x_{1}+\beta x_{2})\otimes(\alpha x_{1}+\beta
x_{2})$ and
$P^{\prime}_{2}=(\bar{\beta}x_{1}-\bar{\alpha}x_{2})\otimes(\bar{\beta}x_{1}-\bar{\alpha}x_{2})$
with $|\alpha|^{2}+|\beta|^{2}=1$. Denote
$f(\alpha,\beta)=\phi(P^{\prime}_{1})$ and
$g(\alpha,\beta)=\phi(P^{\prime}_{2})$. Then by Eq.(2.4) again we have
$f(\alpha,\beta)\bot g(\alpha,\beta)$. Note that
$f(1,0)=\phi(P_{1})=\phi(P_{r+1})\in\\{f(\alpha,\beta),g(\alpha,\beta)\\}$ for
any $(\alpha,\beta)$. This entails
$\\{f(\alpha,\beta),g(\alpha,\beta)\\}=\\{\phi(P_{1}),\phi(P_{2})\\}$ for any
$(\alpha,\beta)$. So it follows from the continuity of $f$ and $g$ that
$f(\alpha,\beta)\equiv\phi(P_{1})$ and $g(\alpha,\beta)\equiv\phi(P_{2})$,
contradicting to the fact that $f$ and $g$ has the same range. So the claim is
true.
Combining Claims 1-7, we see that $\psi$ preserves pure state and strict
convex combinations will imply that $\psi$ takes one of the form (1), (2) and
(3), completing the proof of Theorem 2.5. $\Box$
## 3\. Maps preserving separable pure states and strict convex combinations:
bipartite systems
This section is devoted to giving a structure theorem of maps preserve
separable pure states and strict convex combinations for bipartite systems.
Using this structure theorem we are able to answer the conjectures for finite
dimensional case mentioned in the introduction section that were proposed in
[8] .
The following simple lemma may be found in [10].
Lemma 3.1 Let $H$ be a complex Hilbert space of any dimension and
$T\in{\mathcal{B}}(H)$ a finite rank operator. Then $\frac{1}{{\rm
rank}T}\|T\|_{\rm Tr}^{2}\leq\|T\|_{2}^{2}\leq\|T\|_{\rm Tr}^{2}$, where
$\|T\|_{\rm Tr}$ and $\|T\|_{2}$ are respectively the trace-norm and the
Hilbert-Schmidt norm of $T$.
Recall that ${\mathcal{S}}_{\rm sep}(H_{1}\otimes H_{2})$ and
$\mathcal{P}ur(H_{1})\otimes\mathcal{P}ur(H_{2})$ stand respectively for the
convex set of all separable states and the set of all separable pure states in
bipartite system $H_{1}\otimes H_{2}$. Now let us present the main result of
this section.
Theorem 3.2 Let $H_{1},H_{2},K_{1},K_{2}$ be complex Hilbert spaces with
$2\leq\dim H_{i}<\infty$, $i=1,2$. Let $\psi:{\mathcal{S}}_{\rm
sep}(H_{1}\otimes H_{2})\rightarrow{\mathcal{S}}_{\rm sep}(K_{1}\otimes
K_{2})$ be a map. If $\psi$ preserves separable pure states and strict convex
combinations, then one of the following statements holds.
(1) There exists $R_{1}\otimes
R_{2}\in\mathcal{P}ur(K_{1})\otimes\mathcal{P}ur(K_{2})$ such that
$\psi(A\otimes B)=R_{1}\otimes R_{2}$
for all $A\in{\mathcal{S}}(H_{1})$ and $B\in{\mathcal{S}}(H_{2})$.
(2) $\dim H_{1}\leq\dim K_{1}$, there exist $R_{2}\in\mathcal{P}ur(K_{2})$ and
injective $M_{1}\in\mathcal{B}(H_{1},K_{1})$ such that
$\psi(A\otimes B)=\frac{M_{1}AM_{1}^{*}}{{\rm Tr}(M_{1}AM_{1}^{*})}\otimes
R_{2}\ \mbox{\rm for all }A\in{\mathcal{S}}(H_{1})\ \mbox{\rm and}\
B\in{\mathcal{S}}(H_{2})$
or
$\psi(A\otimes B)=\frac{M_{1}A^{t}M_{1}^{*}}{{\rm
Tr}(M_{1}A^{t}M_{1}^{*})}\otimes R_{2}\ \mbox{\rm for all
}A\in{\mathcal{S}}(H_{1})\ \mbox{\rm and}\ B\in{\mathcal{S}}(H_{2}).$
(3) $\dim H_{2}\leq\dim K_{2}$, there exist $R_{1}\in\mathcal{P}ur(K_{1})$ and
injective $M_{2}\in\mathcal{B}(H_{2},K_{2})$ such that
$\psi(A\otimes B)=R_{1}\otimes\frac{M_{2}BM_{2}^{*}}{{\rm
Tr}(M_{2}BM_{2}^{*})}\ \mbox{\rm for all }A\in{\mathcal{S}}(H_{1})\ \mbox{\rm
and}\ B\in{\mathcal{S}}(H_{2})$
or
$\psi(A\otimes B)=R_{1}\otimes\frac{M_{2}B^{t}M_{2}^{*}}{{\rm
Tr}(M_{2}B^{t}M_{2}^{*})}\ \mbox{\rm for all }A\in{\mathcal{S}}(H_{1})\
\mbox{\rm and}\ B\in{\mathcal{S}}(H_{2}).$
(4) $\dim H_{2}\leq\dim K_{1}$, there exist $R_{2}\in\mathcal{P}ur(K_{2})$ and
injective $M_{1}\in\mathcal{B}(H_{2},K_{1})$ such that
$\psi(A\otimes B)=\frac{M_{1}BM_{1}^{*}}{{\rm Tr}(M_{1}BM_{1}^{*})}\otimes
R_{2}\ \mbox{\rm for all }A\in{\mathcal{S}}(H_{1})\ \mbox{\rm and}\
B\in{\mathcal{S}}(H_{2})$
or
$\psi(A\otimes B)=\frac{M_{1}B^{t}M_{1}^{*}}{{\rm
Tr}(M_{1}B^{t}M_{1}^{*})}\otimes R_{2}\ \mbox{\rm for all
}A\in{\mathcal{S}}(H_{1})\ \mbox{\rm and}\ B\in{\mathcal{S}}(H_{2}).$
(5) $\dim H_{1}\leq\dim K_{2}$, there exist $R_{1}\in\mathcal{P}ur(K_{1})$ and
injective $M_{2}\in\mathcal{B}(H_{1},K_{2})$ such that
$\psi(A\otimes B)=R_{1}\otimes\frac{M_{2}AM_{2}^{*}}{{\rm
Tr}(M_{2}AM_{2}^{*})}\ \mbox{\rm for all }A\in{\mathcal{S}}(H_{1})\ \mbox{\rm
and}\ B\in{\mathcal{S}}(H_{2})$
or
$\psi(A\otimes B)=R_{1}\otimes\frac{M_{2}A^{t}M_{2}^{*}}{{\rm
Tr}(M_{2}A^{t}M_{2}^{*})}\ \mbox{\rm for all }A\in{\mathcal{S}}(H_{1})\
\mbox{\rm and}\ B\in{\mathcal{S}}(H_{2}).$
(6) $\dim H_{i}\leq\dim K_{i}$, $i=1,2$, there exist injective
$M_{1}\in\mathcal{B}(H_{1},K_{1})$ and $M_{2}\in\mathcal{B}(H_{2},K_{2})$ such
that
$\psi(A\otimes B)=\frac{M_{1}\Psi_{1}(A)M_{1}^{*}}{{\rm
Tr}(M_{1}\Psi_{1}(A)M_{1}^{*})}\otimes\frac{M_{2}\Psi_{2}(B)M_{2}^{*}}{{\rm
Tr}(M_{2}\Psi_{2}(B)M_{2}^{*})}$
for all $A\in{\mathcal{S}}(H_{1})\ \mbox{\rm and}\ B\in{\mathcal{S}}(H_{2})$,
where $\Psi_{i}:{\mathcal{B}}(H_{i})\rightarrow{\mathcal{B}}(H_{i})$, $i=1,2$,
is the identity, or the transpose.
(7) $\dim H_{1}\leq\dim K_{2}$ and $\dim H_{2}\leq\dim K_{1}$, there exist
injective $M_{1}\in\mathcal{B}(H_{2},K_{1})$ and
$M_{2}\in\mathcal{B}(H_{1},K_{2})$ such that
$\psi(A\otimes B)=\frac{M_{1}\Psi_{2}(B)M_{1}^{*}}{{\rm
Tr}(M_{1}\Psi_{2}(B)M_{1}^{*})}\otimes\frac{M_{2}\Psi_{1}(A)M_{2}^{*}}{{\rm
Tr}(M_{2}\Psi_{1}(A)M_{2}^{*})}$
for all $A\in{\mathcal{S}}(H_{1})\ \mbox{\rm and}\ B\in{\mathcal{S}}(H_{2})$,
where $\Psi_{i}:{\mathcal{B}}(H_{i})\rightarrow{\mathcal{B}}(H_{i})$, $i=1,2$,
is the identity, or the transpose.
(8) $\min\\{\dim H_{1},\dim H_{2}\\}\leq\dim K_{1}$, there exist
$R_{2}\in{\mathcal{P}ur}(K_{2})$ and a strict convex combination preserving
map $\varphi_{1}:{\mathcal{S}}_{\rm sep}(H_{1}\otimes
H_{2})\rightarrow{\mathcal{S}}(K_{1})$ such that
$\psi(A\otimes B)=\varphi_{1}(A\otimes B)\otimes R_{2}$
for all $A\in\mathcal{S}(H_{1})$ and $B\in\mathcal{S}(H_{2})$. Moreover,
$\varphi_{1}$ satisfies that for each $P\otimes
Q\in\mathcal{P}ur(H_{1})\otimes\mathcal{P}ur(H_{2})$, $\varphi_{1}(P\otimes
Q)=\frac{M_{P}QM_{P}^{*}}{{\rm
Tr}(M_{P}QM_{P}^{*})}=\frac{N_{Q}PN_{Q}^{*}}{{\rm Tr}(N_{Q}PN_{Q}^{*})}$ for
some injective linear or conjugate linear (may not synchronously) operators
$M_{P}:H_{2}\rightarrow K_{1}$ and $N_{Q}:H_{1}\rightarrow K_{1}$.
(9) $\min\\{\dim H_{1},\dim H_{2}\\}\leq\dim K_{2}$, there exist
$R_{1}\in\mathcal{P}ur(K_{1})$ and a strict convex combination preserving map
$\varphi_{2}:{\mathcal{S}}_{\rm sep}(H_{1}\otimes
H_{2})\rightarrow{\mathcal{S}}(K_{2})$ such that
$\psi(A\otimes B)=R_{1}\otimes\varphi_{2}(A\otimes B)$
for all $A\in\mathcal{S}(H_{1})$ and $B\in\mathcal{S}(H_{2})$. Moreover,
$\varphi_{2}$ satisfies that, for each $P\otimes
Q\in\mathcal{P}ur(H_{1})\otimes\mathcal{P}ur(H_{2})$, $\varphi_{2}(P\otimes
Q)=\frac{M_{P}QM_{P}^{*}}{{\rm
Tr}(M_{P}QM_{P}^{*})}=\frac{N_{Q}PN_{Q}^{*}}{{\rm Tr}(N_{Q}PN_{Q}^{*})}$, for
some injective linear or conjugate linear (may not synchronously) operators
$M_{P}:H_{2}\rightarrow K_{2}$ and $N_{Q}:H_{1}\rightarrow K_{2}$.
(10) There exist $P_{i}^{\prime}\in{\mathcal{P}ur}(K_{1})$ and
$Q_{i}^{\prime}\in{\mathcal{P}ur}(K_{2})$, $i=1,2$ such that
$\psi({\mathcal{P}ur}(H_{1})\otimes{\mathcal{P}ur}(H_{2}))=\\{P_{1}^{\prime}\otimes
Q_{1}^{\prime},P_{2}^{\prime}\otimes Q_{2}^{\prime}\\}$, and
ran$(\psi)\subseteq[P_{1}^{\prime}\otimes Q_{1}^{\prime},P_{2}^{\prime}\otimes
Q_{2}^{\prime}]$.
Where the transpose is taken with respect to an arbitrarily fixed orthonormal
basis.
Proof. Suppose $\dim H_{1}=m$, $\dim H_{2}=n$. If the range of $\psi$ is
collinear, then it is clear that either (1) holds or (10) holds. So, in the
sequel, we assume that the range of $\psi$ is non-collinear. Denote by
$\mathcal{B}_{\rm sa}(H)$ the real linear space of all self-adjoint operators
on the Hilbert space $H$. Consider the partial traces ${\rm
Tr_{1}}:\mathcal{B}_{\rm sa}(K_{1}\otimes K_{2})\rightarrow\mathcal{B}_{\rm
sa}(K_{2})$ and ${\rm Tr_{2}}:\mathcal{B}_{\rm sa}(K_{1}\otimes
K_{2})\rightarrow\mathcal{B}_{\rm sa}(K_{1})$ on $\mathcal{B}_{\rm
sa}(K_{1}\otimes K_{2})\equiv\mathcal{B}_{\rm
sa}(K_{1})\otimes\mathcal{B}_{\rm sa}(K_{2})$ defined by ${\rm
Tr_{1}}(A\otimes B)=({\rm Tr}A)B$ and ${\rm Tr_{2}}(A\otimes B)=({\rm Tr}B)A$.
Clearly, ${\rm Tr_{1}}$ and ${\rm Tr_{2}}$ are linear maps. Define two maps
$\phi_{1}:(\mathcal{S}(H_{1}),\mathcal{S}(H_{2}))\rightarrow\mathcal{S}(K_{1})$
and
$\phi_{2}:(\mathcal{S}(H_{1}),\mathcal{S}(H_{2}))\rightarrow\mathcal{S}(K_{2})$
by
$\phi_{1}(A,B)={\rm Tr_{2}}(\psi(A\otimes B))\ {\rm and}\ \phi_{2}(A,B)={\rm
Tr_{1}}(\psi(A\otimes B)).$
Notice that
$\psi(P\otimes Q)=\phi_{1}(P,Q)\otimes\phi_{2}(P,Q)\ {\rm for\ all}\
P\in\mathcal{P}ur(H_{1})\ {\rm and}\ Q\in\mathcal{P}ur(H_{2}).$
Fix a $Q\in\mathcal{P}ur(H_{2})$; then the maps
$\phi_{1}(\cdot,Q):\mathcal{S}(H_{1})\rightarrow\mathcal{S}(K_{1})$ and
$\phi_{2}(\cdot,Q):\mathcal{S}(H_{1})\rightarrow\mathcal{S}(K_{2})$ are both
strict convex combination preserving and
$\phi_{1}(\mathcal{P}ur(H_{1}),Q)\subseteq\mathcal{P}ur(K_{1})$ while
$\phi_{2}(\mathcal{P}ur(H_{1}),Q)\subseteq\mathcal{P}ur(K_{2})$. Therefore,
applying Theorem 2.5 to $\phi_{1}(\cdot,Q)$ and $\phi_{2}(\cdot,Q)$,
respectively, we get that, for $i=1,2$, either
(i) there exists pure state $R_{iQ}\in{\mathcal{P}ur}(K_{i})$ such that
$\phi_{i}(A,Q)=R_{iQ}$ for all $A\in\mathcal{S}(H_{1})$;
or
(ii) there are pure states
$R_{1Q}^{(i)},R_{2Q}^{(i)}\in{\mathcal{P}ur}(K_{i})$ and a strict convex
combination preserving map $h_{iQ}:{\mathcal{S}}(H_{1})\rightarrow[0,1]$ such
that $\phi_{i}({\mathcal{P}ur}(H_{1}),Q)=\\{R_{1Q}^{(i)},R_{2Q}^{(i)}\\}$ and
$\phi_{i}(A,Q)=h_{iQ}(A)R_{1Q}^{(i)}+(1-h_{iQ}(A))R_{2Q}^{(i)}$ for all
$A\in\mathcal{S}(H_{1})$;
or
(iii) there exists an injective linear or conjugate linear operator
$M_{iQ}:H_{1}\rightarrow K_{i}$ such that
$\phi_{i}(A,Q)=\frac{M_{iQ}AM_{iQ}^{*}}{{\rm Tr}(M_{iQ}AM_{iQ}^{*})}$ for all
$A\in\mathcal{S}(H_{1})$.
As $\psi$ is strict convex combination preserving and ${\rm ran}\psi$ is non-
collinear, Lemma 2.3 is applicable. Thus we have
$\psi(\rho)=\frac{\Gamma(\rho)+D}{f(\rho)+d}$, where $\Gamma:\mathcal{B}_{\rm
sa}(H_{1}\otimes H_{2})\rightarrow\mathcal{B}_{\rm sa}(K_{1}\otimes K_{2})$ is
a linear map, $D\in\mathcal{B}_{\rm sa}(K_{1}\otimes K_{2})$ is an operator,
$f:\mathcal{B}_{\rm sa}(H_{1}\otimes H_{2})\rightarrow\mathbb{R}$ is a linear
functional and $d\in\mathbb{R}$ with $f(\rho)+d>0$ for all
$\rho\in\mathcal{S}_{\rm sep}(H_{1}\otimes H_{2})$. As both $H_{1}$ and
$H_{2}$ are finite dimensional, $\Gamma$, $f$ are continuous. It follows that
$\psi$ is continuous. So both $\phi_{1}$ and $\phi_{2}$ are continuous on
$\mathcal{S}(H_{1})$. These facts will be used frequently.
Let us first consider the map $\phi_{1}$.
Claim 1. Either $\phi_{1}(\cdot,Q)$ has the form (i) for all
$Q\in\mathcal{P}ur(H_{2})$ or $\phi_{1}(\cdot,Q)$ has the form (iii) for all
$Q\in\mathcal{P}ur(H_{2})$.
As mentioned above, for any fixed pure state $Q\in{\mathcal{P}ur}(H_{2})$,
$\phi_{1}(\cdot,Q)$ takes one of the forms (i)-(iii). As $\phi_{1}$ is
continuous, by Remark 2.7 we see that $\phi_{1}(\cdot,Q)$ can not have the
form (ii) for any $Q\in\mathcal{P}ur(H_{2})$. Thus, for any $Q$,
$\phi_{1}(\cdot,Q)$ takes the form (i) or the form (iii).
Furthermore, we will show that either $\phi_{1}(\cdot,Q)$ has the form (i) for
all $Q\in\mathcal{P}ur(H_{2})$ or $\phi_{1}(\cdot,Q)$ has the form (iii) for
all $Q\in\mathcal{P}ur(H_{2})$.
To do this, for any $A\in{\mathcal{S}}(H_{1})$ so that ${\rm rank}(A)\geq 2$,
define $F_{A}:\mathcal{P}ur(H_{2})\rightarrow\mathbb{R}$ by
$F_{A}(Q)=\|\phi_{1}(A,Q)\|_{2}$, where $\|\cdot\|_{2}$ is the Hilbert-Schmidt
norm. Notice that when $\phi_{1}(\cdot,Q)$ takes the form (i), then
$F_{A}(Q)=\|\phi_{1}(A,Q)\|_{2}=1$; when $\phi_{1}(\cdot,Q)$ takes the form
(iii), then $\phi_{1}(A,Q)=\frac{M_{1Q}AM_{1Q}^{*}}{{\rm
Tr}(M_{1Q}AM_{1Q}^{*})}$ and
$F_{A}(Q)=\|\phi_{1}(A,Q)\|_{2}=\frac{\|M_{1Q}AM_{1Q}^{*}\|_{2}}{\|M_{1Q}A^{\frac{1}{2}}\|_{2}^{2}}<1$
as ${\rm rank}A\geq 2$ and $M_{1Q}$ is injective.
If there exist two distinct $Q_{1},Q_{2}\in\mathcal{P}ur(H_{2})$, such that
$\phi_{1}(\cdot,Q_{1})$ has the form (i) while $\phi_{1}(\cdot,Q_{2})$ has the
form (iii), that is, $\phi_{1}(\cdot,Q_{1})=R_{1Q_{1}}$,
$\phi_{1}(\cdot,Q_{2})=\frac{M_{1Q_{2}}(\cdot)M_{1Q_{2}}^{*}}{{\rm
Tr}(M_{1Q_{2}}(\cdot)M_{1Q_{2}}^{*})}$. Let $Q_{1}=x\otimes x$ and
$Q_{2}=y\otimes y$ with unit vectors $x,y\in H_{2}\cong\mathbb{C}^{n}$. Note
that $x$ and $y$ are linearly independent. For any $t\in[0,1]$, define
$Q(t)=\frac{1}{\|x+t(y-x)\|^{2}}(x+t(y-x))\otimes(x+t(y-x))\in\mathcal{P}ur(H_{2}).$
Then, $Q(t)$ is continuous in $t$, $Q(0)=Q_{1}$, $Q(1)=Q_{2}$, and
$\phi_{1}(\cdot,Q(t))$ has the form (i) or (iii) for any $t$. Fix an
$A\in{\mathcal{S}}(H_{1})$ with rank$A\geq 2$ and let
$t_{0}=\max\\{t\in[0,1]:F_{A}(Q(t))=1\\}$. Then $F_{A}(Q(t_{0}))=1$ and so
$\phi_{1}(\cdot,Q(t_{0}))$ has the form (i). Thus
$\|\phi_{1}(A,Q(t_{0}))\|_{2}=1$ for any $A\in{\mathcal{S}}(H_{1})$. For any
$1\geq t>t_{0}$, $\phi_{1}(\cdot,Q(t))$ has the form (iii). Thus there exist
$\\{t_{n}\\}$, $t_{n}>t_{0}$, such that $\phi_{1}(\cdot,Q(t_{n}))$ has the
form (iii) and $t_{n}\rightarrow t_{0}$. Then by Lemma 3.1, for any given
sufficient small $\varepsilon>0$, there exist
$\\{A_{t_{n}}\\}\subseteq{\mathcal{S}}(H_{1})$ with ${\rm rank}A_{t_{n}}=2$,
such that
$\frac{1}{2}\leq\|\phi_{1}(A_{t_{n}},Q(t_{n}))\|_{2}^{2}\leq\frac{1}{2}+\varepsilon<1$.
The reason of the existence of $\\{A_{t_{n}}\\}$ for each $n$ is that, we can
find rank-2 operator $B_{t_{n}}\in$ran$\phi_{1}(\cdot,Q_{t_{n}})$ such that
$\frac{1}{2}\leq\|B_{t_{n}}\|_{2}^{2}\leq\frac{1}{2}+\varepsilon<1$, thus
there exists $A_{t_{n}}\in{\mathcal{S}}(H_{1})$ such that
$B_{t_{n}}=\frac{M_{t_{n}}A_{t_{n}}M_{t_{n}}^{*}}{{\rm
Tr}(M_{t_{n}}A_{t_{n}}M_{t_{n}}^{*})}$. As $M_{t_{n}}$ is injective, we see
that rank$A_{t_{n}}=2$. Now, since ${\mathcal{S}}(H_{1})$ is a compact set,
$\\{A_{t_{n}}\\}\subseteq{\mathcal{S}}(H_{1})$ has a convergent subsequence
$\\{A_{t_{n_{i}}}\\}\subseteq\\{A_{t_{n}}\\}$, say $A_{t_{n_{i}}}\rightarrow
A_{0}$ as $t_{n_{i}}\rightarrow t_{0}$. Then the continuity of $\psi$ entails
that
$\|\phi_{1}(A_{t_{n_{i}}},Q(t_{n_{i}}))\|_{2}^{2}\rightarrow\|\phi_{1}(A_{0},Q(t_{0}))\|_{2}^{2}$.
But this is a contradiction because $\|\phi_{1}(A_{0},Q(t_{0}))\|_{2}^{2}=1$
and
$\|\phi_{1}(A_{t_{n_{i}}},Q(t_{n_{i}}))\|_{2}^{2}\leq\frac{1}{2}+\varepsilon<1$.
Thus either $\phi_{1}(\cdot,Q)$ has the form (i) for all
$Q\in\mathcal{P}ur(H_{2})$ or $\phi_{1}(\cdot,Q)$ has the form (iii) for all
$Q\in\mathcal{P}ur(H_{2})$.
Similarly, we have
Claim 1′. Either $\phi_{2}(\cdot,Q)$ has the form (i) for all
$Q\in\mathcal{P}ur(H_{2})$ or $\phi_{2}(\cdot,Q)$ has the form (iii) for all
$Q\in\mathcal{P}ur(H_{2})$.
Claim 2. One of the following holds:
(a) For all $Q\in\mathcal{P}ur(H_{2})$, both $\phi_{1}(\cdot,Q)$ and
$\phi_{2}(\cdot,Q)$ have the form (i).
(b) For all $Q\in\mathcal{P}ur(H_{2})$, $\phi_{1}(\cdot,Q)$ has the form (i)
and $\phi_{2}(\cdot,Q)$ has the form (iii).
(c) For all $Q\in\mathcal{P}ur(H_{2})$, $\phi_{1}(\cdot,Q)$ has the form (iii)
and $\phi_{2}(\cdot,Q)$ has the form (i).
We need only to check that, for all $Q\in\mathcal{P}ur(H_{2})$,
$\phi_{1}(\cdot,Q)$ and $\phi_{2}(\cdot,Q)$ can not have the form (iii)
simultaneously. Suppose, on the contrary, there exists
$Q_{0}\in\mathcal{P}ur(H_{2})$ such that both $\phi_{1}(\cdot,Q_{0})$ and
$\phi_{2}(\cdot,Q_{0})$ are of the form (iii). Then there exist injective
linear or conjugate linear operators $M_{1Q_{0}}:H_{1}\rightarrow K_{1}$ and
$M_{2Q_{0}}:H_{1}\rightarrow K_{2}$ such that
$\phi_{i}(A,Q_{0})=\frac{M_{iQ_{0}}AM_{iQ_{0}}^{*}}{{\rm
Tr}(M_{iQ_{0}}AM_{iQ_{0}}^{*})}$ for all $A\in{\mathcal{S}}(H_{1})$, $i=1,2$.
Thus, we must have $\dim H_{1}\leq\min\\{\dim K_{1},\dim K_{2}\\}$ and
$\begin{array}[]{rl}\psi(P\otimes
Q_{0})=&\phi_{1}(P,Q_{0})\otimes\phi_{2}(P,Q_{0})=(\frac{M_{1Q_{0}}PM_{1Q_{0}}^{*}}{{\rm
Tr}(M_{1Q_{0}}PM_{1Q_{0}}^{*})})\otimes(\frac{M_{2Q_{0}}PM_{2Q_{0}}^{*}}{{\rm
Tr}(M_{2Q_{0}}PM_{2Q_{0}}^{*})})\\\ =&\frac{(M_{1Q_{0}}\otimes
M_{2Q_{0}})(P\otimes P)(M_{1Q_{0}}\otimes M_{2Q_{0}})^{*}}{{\rm
Tr}((M_{1Q_{0}}\otimes M_{2Q_{0}})(P\otimes P)(M_{1Q_{0}}\otimes
M_{2Q_{0}})^{*})}\end{array}$
for all $P\in\mathcal{P}ur(H_{1})$. Particularly, take $P_{1}=e_{1}\otimes
e_{1}$, $P_{2}=e_{2}\otimes e_{2}$, $P_{3}=\frac{1}{2}(e_{1}\otimes
e_{1}+e_{1}\otimes e_{2}+e_{2}\otimes e_{1}+e_{2}\otimes e_{2})$ and
$P_{4}=\frac{1}{2}(e_{1}\otimes e_{1}-e_{1}\otimes e_{2}-e_{2}\otimes
e_{1}+e_{2}\otimes e_{2})$, where $e_{1},e_{2}\in H_{1}$ are unit vectors with
$e_{1}\perp e_{2}$. Then $P_{1}+P_{2}=P_{3}+P_{4}$ and so $P_{1}\otimes
Q_{0}+P_{2}\otimes Q_{0}=P_{3}\otimes Q_{0}+P_{4}\otimes Q_{0}$. As $\psi$ is
strict convex combination preserving, there exist $s_{1}\in(0,1)$ and
$s_{2}\in(0,1)$, such that
$\begin{array}[]{rl}&\psi(\frac{1}{2}P_{1}\otimes
Q_{0}+\frac{1}{2}P_{2}\otimes Q_{0})=s_{1}\psi(P_{1}\otimes
Q_{0})+(1-s_{1})\psi(P_{2}\otimes Q_{0})\\\ =&s_{1}\frac{(M_{1Q_{0}}\otimes
M_{2Q_{0}})(P_{1}\otimes P_{1})(M_{1Q_{0}}\otimes M_{2Q_{0}})^{*}}{{\rm
Tr}((M_{1Q_{0}}\otimes M_{2Q_{0}})(P_{1}\otimes P_{1})(M_{1Q_{0}}\otimes
M_{2Q_{0}})^{*})}+(1-s_{1})\frac{(M_{1Q_{0}}\otimes M_{2Q_{0}})(P_{2}\otimes
P_{2})(M_{1Q_{0}}\otimes M_{2Q_{0}})^{*}}{{\rm Tr}((M_{1Q_{0}}\otimes
M_{2Q_{0}})(P_{2}\otimes P_{2})(M_{1Q_{0}}\otimes
M_{2Q_{0}})^{*})}\end{array}$
and
$\begin{array}[]{rl}&\psi(\frac{1}{2}P_{3}\otimes
Q_{0}+\frac{1}{2}P_{4}\otimes Q_{0})=s_{2}\psi(P_{3}\otimes
Q_{0})+(1-s_{2})\psi(P_{4}\otimes Q_{0})\\\ =&s_{2}\frac{(M_{1Q_{0}}\otimes
M_{2Q_{0}})(P_{3}\otimes P_{3})(M_{1Q_{0}}\otimes M_{2Q_{0}})^{*}}{{\rm
Tr}((M_{1Q_{0}}\otimes M_{2Q_{0}})(P_{3}\otimes P_{3})(M_{1Q_{0}}\otimes
M_{2Q_{0}})^{*})}+(1-s_{2})\frac{(M_{1Q_{0}}\otimes M_{2Q_{0}})(P_{4}\otimes
P_{4})(M_{1Q_{0}}\otimes M_{2Q_{0}})^{*}}{{\rm Tr}((M_{1Q_{0}}\otimes
M_{2Q_{0}})(P_{4}\otimes P_{4})(M_{1Q_{0}}\otimes
M_{2Q_{0}})^{*})}.\end{array}$
It follows that there exist $a_{i}\neq 0$, $i=1,2,3,4$, such that
$a_{1}P_{1}\otimes P_{1}+a_{2}P_{2}\otimes P_{2}=a_{3}P_{3}\otimes
P_{3}+a_{4}P_{4}\otimes P_{4}.$
As $P_{4}=P_{1}+P_{2}-P_{3}$, then $0=a_{1}P_{1}\otimes
P_{1}+a_{2}P_{2}\otimes P_{2}-a_{3}P_{3}\otimes P_{3}-a_{4}P_{4}\otimes
P_{4}=P_{1}\otimes a_{1}P_{1}+P_{2}\otimes a_{2}P_{2}-P_{3}\otimes
a_{3}P_{3}-P_{1}\otimes a_{4}P_{4}-P_{2}\otimes a_{4}P_{4}+P_{3}\otimes
a_{4}P_{4}=P_{1}\otimes(a_{1}P_{1}-a_{4}P_{4})+P_{2}\otimes(a_{2}P_{2}-a_{4}P_{4})-P_{3}\otimes(a_{3}P_{3}-a_{4}P_{4})$.
As $P_{1}$, $P_{2}$ and $P_{3}$ are linearly independent, by [7], we have
$a_{1}P_{1}-a_{4}P_{4}=0$, $a_{2}P_{2}-a_{4}P_{4}=0$ and
$a_{3}P_{3}-a_{4}P_{4}=0$, which is a contradiction. So the claim is true.
Similarly, one can check that
Claim 3. One of the following holds:
(a′) For all $P\in\mathcal{P}ur(H_{1})$, both $\phi_{1}(P,\cdot)$ and
$\phi_{2}(P,\cdot)$ have the form (i).
(b′) For all $P\in\mathcal{P}ur(H_{1})$, $\phi_{1}(P,\cdot)$ has the form (i)
and $\phi_{2}(P,\cdot)$ has the form (iii).
(c′) For all $P\in\mathcal{P}ur(H_{1})$, $\phi_{1}(P,\cdot)$ has the form
(iii) and $\phi_{2}(P,\cdot)$ has the form (i).
Claim 4. (a) and (a′) can not hold simultaneously.
In fact, if (a) and (a′) hold, that is, for all $Q\in\mathcal{P}ur(H_{2})$, we
have $\phi_{i}(A,Q)=R_{iQ}$, and, for all $P\in\mathcal{P}ur(H_{1})$, we have
$\phi_{i}(P,B)=R_{iP}$. Fix $P_{0}\in\mathcal{P}ur(H_{1})$ and
$Q_{0}\in\mathcal{P}ur(H_{2})$. Then we get
$\phi_{i}(P,Q)=\phi_{i}(P,Q_{0})=\phi_{i}(P_{0},Q_{0})=R_{i}.$
Therefore, $\psi(P\otimes Q)=\phi_{1}(P,Q)\otimes\phi_{2}(P,Q)=R_{1}\otimes
R_{2}$ for all $P\otimes Q\in\mathcal{P}ur(H_{1})\otimes\mathcal{P}ur(H_{2})$.
We know that for any $A\otimes
B\in{\mathcal{S}}(H_{1})\otimes{\mathcal{S}}(H_{2})$, there exist
$\\{A_{i}\\}_{i=1}^{m}\subseteq\mathcal{P}ur(H_{1})$,
$\\{B_{j}\\}_{j=1}^{n}\subseteq\mathcal{P}ur(H_{2})$,
$\\{a_{i}\\}_{i=1}^{m}\subseteq[0,1]$, $\\{b_{j}\\}_{j=1}^{n}\subseteq[0,1]$
satisfying $\sum_{i=1}^{m}a_{i}=1$, $\sum_{j=1}^{n}b_{j}=1$, such that
$A=\sum_{i=1}^{m}a_{i}A_{i}$, $B=\sum_{j=1}^{n}b_{j}B_{j}$. Then, there exist
$c_{ij}\geq 0$ with $\sum_{i,j}c_{ij}=1$ such that $\psi(A\otimes
B)=\sum_{i,j}c_{ij}\psi(A_{i}\otimes B_{j})=R_{1}\otimes R_{2}$, this
contradicts to the assumption that the range of $\psi$ is non-collinear. So
the Claim 4 is true.
Claim 5. If (a) and (b′) hold, then $\psi$ has the form (3), that is, there
exist $R_{1}\in\mathcal{P}ur(K_{1})$ and injective linear or conjugate linear
operator $M_{2}:H_{2}\rightarrow K_{2}$ such that
$\psi(A\otimes B)=R_{1}\otimes\frac{M_{2}BM_{2}^{*}}{{\rm
Tr}(M_{2}BM_{2}^{*})}$
for all $A\in{\mathcal{S}}(H_{1})$ and $B\in{\mathcal{S}}(H_{2})$. It is clear
that $\dim H_{2}\leq\dim K_{2}$.
In this case, for any $P\otimes
Q\in\mathcal{P}ur(H_{1})\otimes\mathcal{P}ur(H_{2})$ we have
$\psi(P\otimes Q)=\phi_{1}(P,Q)\otimes\phi_{2}(P,Q)=R_{1Q}\otimes
R_{2Q}=R_{1P}\otimes\frac{M_{2P}QM_{2P}^{*}}{{\rm Tr}(M_{2P}QM_{2P}^{*})},$
which implies that $R_{1Q}=R_{1P}$ is independent of $P,Q$, and
$\frac{M_{2P}QM_{2P}^{*}}{{\rm Tr}(M_{2P}QM_{2P}^{*})}=R_{2Q}$. Thus for any
fixed $Q\in\mathcal{P}ur(H_{2})$, $\frac{M_{2P_{1}}QM_{2P_{1}}^{*}}{{\rm
Tr}(M_{2P_{1}}QM_{2P_{1}}^{*})}=\frac{M_{2P_{2}}QM_{2P_{2}}^{*}}{{\rm
Tr}(M_{2P_{2}}QM_{2P_{2}}^{*})}=R_{2Q}$ holds for any distinct
$P_{1}\in\mathcal{P}ur(H_{1})$ and $P_{2}\in\mathcal{P}ur(H_{1})$. Thus for
all $Q\in\mathcal{P}ur(H_{2})$, we have $\frac{M_{2P_{1}}QM_{2P_{1}}^{*}}{{\rm
Tr}(M_{2P_{1}}QM_{2P_{1}}^{*})}=\frac{M_{2P_{2}}QM_{2P_{2}}^{*}}{{\rm
Tr}(M_{2P_{2}}QM_{2P_{2}}^{*})}$, that is, $\frac{M_{2P}QM_{2P}^{*}}{{\rm
Tr}(M_{2P}QM_{2P}^{*})}$ is independent of $P$. So there exist
$R_{1}\in\mathcal{P}ur(K_{1})$ and injective linear or conjugate linear
operator $M_{2}:H_{2}\rightarrow K_{2}$ such that $\psi(P\otimes
Q)=R_{1}\otimes\frac{M_{2}QM_{2}^{*}}{{\rm Tr}(M_{2}QM_{2}^{*})}$ for all
separable pure states $P\otimes Q$.
Now, for any $B\in{\mathcal{S}}(H_{2})$ and $P\in{\mathcal{P}ur}(H_{1})$,
writing $B=\sum_{j=1}^{n}b_{j}B_{j}$ as in the proof of Claim 4, we have
$\psi(P\otimes
B)=\psi(P\otimes\sum_{j=1}^{n}b_{j}B_{j})=\sum_{j=1}^{n}b^{\prime}_{j}\psi(P\otimes
B_{j})=\sum_{j=1}^{n}b^{\prime}_{j}(R_{1}\otimes\frac{M_{2}B_{j}M_{2}^{*}}{{\rm
Tr}(M_{2}B_{j}M_{2}^{*})})=R_{1}\otimes\sum_{j=1}^{n}b^{\prime}_{j}\frac{M_{2}B_{j}M_{2}^{*}}{{\rm
Tr}(M_{2}B_{j}M_{2}^{*})}$ is a product state. But we already know that
$\phi_{2}(P,\cdot)=\frac{M_{2}(\cdot)M_{2}^{*}}{{\rm
Tr}(M_{2}(\cdot)M_{2}^{*})}$, so ${\rm Tr}_{1}(\psi(P\otimes
B))=\phi_{2}(P,B)=\frac{M_{2}BM_{2}^{*}}{{\rm Tr}(M_{2}BM_{2}^{*})}$. Thus
$\psi(P\otimes B)=R_{1}\otimes\frac{M_{2}BM_{2}^{*}}{{\rm
Tr}(M_{2}BM_{2}^{*})}$. Then, for any $A\in{\mathcal{S}}(H_{1})$, writing
$A=\sum_{i=1}^{m}a_{i}A_{i}$ as in the proof of Claim 4, we obtain
$\psi(A\otimes B)=\psi(\sum_{i=1}^{m}a_{i}A_{i}\otimes
B)=\sum_{i=1}^{m}a^{\prime}_{i}\psi(A_{i}\otimes
B)=\sum_{i=1}^{m}a^{\prime}_{i}(R_{1}\otimes\frac{M_{2}BM_{2}^{*}}{{\rm
Tr}(M_{2}BM_{2}^{*})})=R_{1}\otimes(\sum_{i=1}^{m}a^{\prime}_{i}\frac{M_{2}BM_{2}^{*}}{{\rm
Tr}(M_{2}BM_{2}^{*})}))$, which is a product states. Therefore, we must have
$\psi(A\otimes B)=R_{1}\otimes\frac{M_{2}BM_{2}^{*}}{{\rm
Tr}(M_{2}BM_{2}^{*})}$, where $M_{2}$ is linear or conjugate linear. In the
case that $M_{2}$ is a conjugate linear operator, it is well known that, there
exists a linear operator $N_{2}$ such that $\frac{M_{2}BM_{2}^{*}}{{\rm
Tr}(M_{2}BM_{2}^{*})}=\frac{N_{2}B^{t}N_{2}^{*}}{{\rm
Tr}(N_{2}B^{t}N_{2}^{*})}$ for all $B$, where the transpose is taken with
respect to an arbitrarily fixed orthonormal basis of $H_{2}$. So the claim is
true.
Similarly, one can show the following Claims 6-8.
Claim 6. If (a) and (c′) hold, then (4) holds.
Claim 7. If (b) and (a′) hold, then $\psi$ has the form (5).
Claim 8. If (c) and (a′) hold, then $\psi$ takes the form (2).
Claim 9. If (b) and (c′) hold, then $\psi$ has the form (7).
Suppose that (b) and (c′) hold; then
$\phi_{1}(P,Q)=\phi_{1}(P_{0},Q)=\frac{M_{1P_{0}}QM_{1P_{0}}^{*}}{{\rm
Tr}(M_{1P_{0}}QM_{1P_{0}}^{*})}$ and
$\phi_{2}(P,Q)=\phi_{2}(P,Q_{0})=\frac{M_{2Q_{0}}PM_{2Q_{0}}^{*}}{{\rm
Tr}(M_{2Q_{0}}PM_{2Q_{0}}^{*})}$. Thus, we obtain
$\psi(P\otimes
Q)=\phi_{1}(P,Q)\otimes\phi_{2}(P,Q)=\frac{M_{1P_{0}}QM_{1P_{0}}^{*}}{{\rm
Tr}(M_{1P_{0}}QM_{1P_{0}}^{*})}\otimes\frac{M_{2Q_{0}}PM_{2Q_{0}}^{*}}{{\rm
Tr}(M_{2Q_{0}}PM_{2Q_{0}}^{*})}$
for all $P\in\mathcal{P}ur(H_{1})$ and $Q\in\mathcal{P}ur(H_{2})$, where
$M_{1P_{0}}:H_{2}\rightarrow K_{1}$ and $M_{2Q_{0}}:H_{1}\rightarrow K_{2}$
are injective linear or conjugate linear operators. It follows that $\dim
H_{1}\leq\dim K_{2}$ and $\dim H_{2}\leq\dim K_{1}$. Let $M_{1}=M_{1P_{0}}$
and $M_{2}=M_{2Q_{0}}$. Then $\psi(P\otimes Q)=\frac{M_{1}QM_{1}^{*}}{{\rm
Tr}(M_{1}QM_{1}^{*})}\otimes\frac{M_{2}PM_{2}^{*}}{{\rm Tr}(M_{2}PM_{2}^{*})}$
for all separable pure states $P\otimes Q$. For any $A\otimes
B\in{\mathcal{S}}(H_{1}\otimes H_{2})$, write $A=\sum_{i=1}^{m}a_{i}A_{i}$ and
$B=\sum_{j=1}^{n}b_{j}B_{j}$ by the spectral theorem, where $a_{i},b_{j}\geq
0$ with $\sum_{i=1}^{m}a_{i}=1,\sum_{j=1}^{n}b_{j}=1$ and $A_{i}$s, $B_{j}$s
pure states. Thus, for any $Q\in{\mathcal{P}ur}(H_{2})$, $\psi(A\otimes
Q)=\psi(\sum_{i=1}^{m}a_{i}A_{i}\otimes
Q)=\sum_{i=1}^{m}a^{\prime}_{i}\psi(A_{i}\otimes
Q)=\sum_{i=1}^{m}a^{\prime}_{i}(\frac{M_{1}QM_{1}^{*}}{{\rm
Tr}(M_{1}QM_{1}^{*})}\otimes\frac{M_{2}A_{i}M_{2}^{*}}{{\rm
Tr}(M_{2}A_{i}M_{2}^{*})})=\frac{M_{1}QM_{1}^{*}}{{\rm
Tr}(M_{1}QM_{1}^{*})}\otimes(\sum_{i=1}^{m}a^{\prime}_{i}\frac{M_{2}A_{i}M_{2}^{*}}{{\rm
Tr}(M_{2}A_{i}M_{2}^{*})})$ is a product state. As
$\phi_{2}(\cdot,Q)=\frac{M_{2}(\cdot)M_{2}^{*}}{{\rm
Tr}(M_{2}(\cdot)M_{2}^{*})}$, we obtain that ${\rm Tr}_{1}(\psi(A\otimes
Q))=\phi_{2}(A,Q)=\frac{M_{2}AM_{2}^{*}}{{\rm Tr}(M_{2}AM_{2}^{*})}$. So we
must have $\psi(A\otimes Q)=\frac{M_{1}QM_{1}^{*}}{{\rm
Tr}(M_{1}QM_{1}^{*})}\otimes\frac{M_{2}AM_{2}^{*}}{{\rm
Tr}(M_{2}AM_{2}^{*})}$. It follows that $\psi(A\otimes
B)=\psi(A\otimes\sum_{j=1}^{n}b_{j}B_{j})=\sum_{j=1}^{n}b^{\prime}_{j}\psi(A\otimes
B_{j})=\sum_{j=1}^{n}b^{\prime}_{j}(\frac{M_{1}B_{j}M_{1}^{*}}{{\rm
Tr}(M_{1}B_{j}M_{1}^{*})}\otimes\frac{M_{2}AM_{2}^{*}}{{\rm
Tr}(M_{2}AM_{2}^{*})})=(\sum_{j=1}^{n}b^{\prime}_{j}\frac{M_{1}B_{j}M_{1}^{*}}{{\rm
Tr}(M_{1}B_{j}M_{1}^{*})})\otimes\frac{M_{2}AM_{2}^{*}}{{\rm
Tr}(M_{2}AM_{2}^{*})}$. On the other hand, we also have $\psi(P\otimes
B)=\psi(P\otimes\sum_{j=1}^{n}b_{j}B_{j})=\sum_{j=1}^{n}b^{\prime}_{j}\psi(P\otimes
B_{j})=\sum_{j=1}^{n}b^{\prime}_{j}(\frac{M_{1}B_{j}M_{1}^{*}}{{\rm
Tr}(M_{1}B_{j}M_{1}^{*})}\otimes\frac{M_{2}PM_{2}^{*}}{{\rm
Tr}(M_{2}PM_{2}^{*})})=(\sum_{j=1}^{n}b^{\prime}_{j}\frac{M_{1}B_{j}M_{1}^{*}}{{\rm
Tr}(M_{1}B_{j}M_{1}^{*})})\otimes\frac{M_{2}PM_{2}^{*}}{{\rm
Tr}(M_{2}PM_{2}^{*})}$. As
$\phi_{1}(P,\cdot)=\frac{M_{1}(\cdot)M_{1}^{*}}{{\rm
Tr}(M_{1}(\cdot)M_{1}^{*})}$, then ${\rm Tr}_{2}(\psi(P\otimes
B))=\phi_{1}(P,B)=\frac{M_{1}BM_{1}^{*}}{{\rm Tr}(M_{1}BM_{1}^{*})}$. Thus we
get $\psi(P\otimes B)=\frac{M_{1}BM_{1}^{*}}{{\rm
Tr}(M_{1}BM_{1}^{*})}\otimes\frac{M_{2}PM_{2}^{*}}{{\rm Tr}(M_{2}PM_{2}^{*})}$
and then $\psi(A\otimes B)=\psi(\sum_{i=1}^{m}a_{i}A_{i}\otimes
B)=\sum_{i=1}^{m}a_{i}\psi(A_{i}\otimes
B)=\sum_{i=1}^{m}a^{\prime}_{i}(\frac{M_{1}BM_{1}^{*}}{{\rm
Tr}(M_{1}BM_{1}^{*})}\otimes\frac{M_{2}A_{i}M_{2}^{*}}{{\rm
Tr}(M_{2}A_{i}M_{2}^{*})})=\frac{M_{1}BM_{1}^{*}}{{\rm
Tr}(M_{1}BM_{1}^{*})}\otimes(\sum_{i=1}^{m}a^{\prime}_{i}\frac{M_{2}A_{i}M_{2}^{*}}{{\rm
Tr}(M_{2}A_{i}M_{2}^{*})})$. Now it is clear that $\psi(A\otimes
B)=\frac{M_{1}BM_{1}^{*}}{{\rm
Tr}(M_{1}BM_{1}^{*})}\otimes\frac{M_{2}AM_{2}^{*}}{{\rm Tr}(M_{2}AM_{2}^{*})}$
for any $A\in{\mathcal{S}}(H_{1})$ and $B\in{\mathcal{S}}(H_{2})$. Hence the
claim is true.
Similarly, we have
Claim 10. If (c) and (b′) hold, then $\psi$ has the form (6).
Claim 11. If (b) and (b′) hold, then $\psi$ has the form (9).
Assume (b) and (b′) hold synchronously. Then for any $P\otimes
Q\in\mathcal{P}ur(H_{1})\otimes\mathcal{P}ur(H_{2})$ we have $\psi(P\otimes
Q)=\phi_{1}(P,Q)\otimes\phi_{2}(P,Q)=R_{1Q}\otimes\frac{M_{2Q}PM_{2Q}^{*}}{{\rm
Tr}(M_{2Q}PM_{2Q}^{*})}=R_{1P}\otimes\frac{M_{2P}QM_{2P}^{*}}{{\rm
Tr}(M_{2P}QM_{2P}^{*})}$. It follows that there exist
$R_{1}\in\mathcal{P}ur(K_{1})$ such that $R_{1Q}=R_{1P}=R_{1}$ and
$\frac{M_{2Q}PM_{2Q}^{*}}{{\rm
Tr}(M_{2Q}PM_{2Q}^{*})}=\frac{M_{2P}QM_{2P}^{*}}{{\rm Tr}(M_{2P}QM_{2P}^{*})}$
for all $P$, $Q$. Thus there exists a strict convex combination preserving map
$\varphi_{2}:{\mathcal{S}}_{\rm sep}(H_{1}\otimes
H_{2})\rightarrow{\mathcal{S}}(K_{2})$ such that, for each $P\otimes
Q\in{\mathcal{P}ur}(H_{1})\otimes{\mathcal{P}ur}(H_{2})$,
$\varphi_{2}(P\otimes Q)=\frac{M_{P}QM_{P}^{*}}{{\rm
Tr}(M_{P}QM_{P}^{*})}=\frac{N_{Q}PN_{Q}^{*}}{{\rm Tr}(N_{Q}PN_{Q}^{*})}$ for
some injective, may not synchronously, linear or conjugate linear operators
$M_{P}:H_{2}\rightarrow K_{2}$, $N_{Q}:H_{1}\rightarrow K_{2}$, and
$\psi(\rho)=R_{1}\otimes\varphi_{2}(\rho)$
for all $\rho\in{\mathcal{S}}_{\rm sep}(H_{1}\otimes H_{2})$. In this case
$\max\\{\dim H_{1},\dim H_{2}\\}\leq\dim K_{2}$. So $\psi$ has the form (9)
and Claim 11 is true.
Similarly,
Claim 12. If (c) and (c′) hold, then $\psi$ takes the form (8).
Combining the claims 4-12, we complete the proof of Theorem 3.2. $\Box$
By Theorem 3.2, the following corollary is immediate, which gives an
affirmative answer to a conjecture in [8] without the injectivity assumption.
Corollary 3.3 Let $\psi:{\mathcal{S}}_{\rm sep}(H_{1}\otimes
H_{2})\rightarrow{\mathcal{S}}_{\rm sep}(K_{1}\otimes K_{2})$ be a map with
$2\leq\dim H_{i}<\infty$, $i=1,2$, and ${\rm ran}\psi$ non-collinear or a
singleton (i.e., contains only one element). If $\psi$ preserves separable
pure states and strict convex combinations, then it sends product states to
product states.
Now we give a characterization of injective local quantum measurements, which
reveals that, in almost all situations the maps preserving separable pure
states and strict convex combinations are essentially the injective local
quantum measurements.
Corollary 3.4 Let $H_{1},H_{2},K_{1},K_{2}$ be Hilbert spaces with $2\leq\dim
H_{i}<\infty$, $i=1,2$ and let $\psi:{\mathcal{S}}_{\rm sep}(H_{1}\otimes
H_{2})\rightarrow{\mathcal{S}}_{\rm sep}(K_{1}\otimes K_{2})$ be a map. Then
the following statements are equivalent.
(1) $\psi$ is strict convex combination preserving,
$\psi({\mathcal{P}ur}(H_{1})\otimes{\mathcal{P}ur}(H_{2}))\subseteq{\mathcal{P}ur}(K_{1})\otimes{\mathcal{P}ur}(K_{2})$
and the range of $\psi$ is non-collinear containing a state $\sigma$ so that
both reductions ${\rm Tr}_{1}(\sigma)$ and ${\rm Tr}_{2}(\sigma)$ have rank
$\geq 2$.
(2) $\psi$ is open line segment preserving,
$\psi({\mathcal{P}ur}(H_{1})\otimes{\mathcal{P}ur}(H_{2}))\subseteq{\mathcal{P}ur}(K_{1})\otimes{\mathcal{P}ur}(K_{2})$
and the range of $\psi$ is non-collinear containing a state $\sigma$ so that
both reductions ${\rm Tr}_{1}(\sigma)$ and ${\rm Tr}_{2}(\sigma)$ have rank
$\geq 2$.
(3) Either
(1∘) there exist injective operators $M_{1}\in\mathcal{B}(H_{1},K_{1})$ and
$M_{2}\in\mathcal{B}(H_{2},K_{2})$ such that
$\psi(\rho)=\frac{(M_{1}\otimes M_{2})\Phi(\rho)(M_{1}\otimes M_{2})^{*}}{{\rm
Tr}((M_{1}\otimes M_{2})\Phi(\rho)(M_{1}\otimes M_{2})^{*})}$
for all $\rho\in{\mathcal{S}}_{\rm sep}(H_{1}\otimes H_{2})$;
or
(2∘) there exist injective operators $M_{1}\in\mathcal{B}(H_{2},K_{1})$ and
$M_{2}\in\mathcal{B}(H_{1},K_{2})$ such that
$\psi(\rho)=\frac{(M_{1}\otimes M_{2})\Phi(\Theta(\rho))(M_{1}\otimes
M_{2})^{*}}{{\rm Tr}((M_{1}\otimes M_{2})\Phi(\Theta(\rho))(M_{1}\otimes
M_{2})^{*})}$
for all $\rho\in{\mathcal{S}}_{\rm sep}(H_{1}\otimes H_{2})$.
Here $\Phi$ is the identity, or the transpose, or the partial transpose of the
first system or the partial transpose of the second system with respect to an
arbitrarily fixed product basis, $\Theta$ is the swap.
Proof. (1)$\Leftrightarrow$(2)$\Leftarrow$(3) is obvious, we need to check the
(1)$\Rightarrow$(3).
Assume (1). It is clear that $\psi$ has one of the forms Theorem 3.2.(1)-(10)
as $\psi$ satisfies all the conditions of Theorem 3.2. Furthermore, the
assumption that there exists $\sigma$ in ${\rm ran}\psi$ so that rank${\rm
Tr}_{i}(\sigma)\geq 2$, $i=1,2$ forces that $\psi$ can only take the form (6)
or (7), that is, $\psi(A\otimes B)=\frac{M_{1}\Psi_{1}(A)M_{1}^{*}}{{\rm
Tr}(M_{1}\Psi_{1}(A)M_{1}^{*})}\otimes\frac{M_{2}\Psi_{2}(B)M_{2}^{*}}{{\rm
Tr}(M_{2}\Psi_{2}(B)M_{2}^{*})}$ for all $A\otimes B\in{\mathcal{S}}_{\rm
sep}(H_{1}\otimes H_{2})$, or $\psi(A\otimes
B)=\frac{M_{1}\Psi_{2}(B)M_{1}^{*}}{{\rm
Tr}(M_{1}\Psi_{2}(B)M_{1}^{*})}\otimes\frac{M_{2}\Psi_{1}(A)M_{2}^{*}}{{\rm
Tr}(M_{2}\Psi_{1}(A)M_{2}^{*})}$ for all $A\otimes B\in{\mathcal{S}}_{\rm
sep}(H_{1}\otimes H_{2})$, where $\Psi_{i}$ is the identity or the transpose
with respect to an arbitrarily fixed orthonormal basis, $i=1,2$. Thus either
(i) $\psi(A\otimes B)=\frac{(M_{1}\otimes M_{2})\Phi(A\otimes B)(M_{1}\otimes
M_{2})^{*}}{{\rm Tr}(M_{1}\otimes M_{2})\Phi(A\otimes B)(M_{1}\otimes
M_{2})^{*})}$ for all $A\otimes B\in{\mathcal{S}}_{\rm sep}(H_{1}\otimes
H_{2})$; or
(ii) $\psi(A\otimes B)=\frac{(M_{1}\otimes M_{2})\Phi(\Theta(A\otimes
B))(M_{1}\otimes M_{2})^{*}}{{\rm Tr}(M_{1}\otimes M_{2})\Phi(\Theta(A\otimes
B))(M_{1}\otimes M_{2})^{*})}$ for all $A\otimes B\in{\mathcal{S}}_{\rm
sep}(H_{1}\otimes H_{2})$,
where $\Phi$ is the identity, or the transpose, or the partial transpose of
the first system or the partial transpose of the second system with respect to
an arbitrarily fixed product basis, $\Theta$ is the swap.
Let
$\Delta(\rho)=\frac{(M_{1}^{[-1]}\otimes
M_{2}^{[-1]})\psi(\Phi(\rho))(M_{1}^{[-1]}\otimes M_{2}^{[-1]})^{*}}{{\rm
Tr}((M_{1}^{[-1]}\otimes M_{2}^{[-1]})\psi(\Phi(\rho))(M_{1}^{[-1]}\otimes
M_{2}^{[-1]})^{*})}$
if $\psi$ has form (i), and let
$\Delta(\rho)=\frac{\Theta[(M_{1}^{[-1]}\otimes
M_{2}^{[-1]})\psi(\Phi(\rho))(M_{1}^{[-1]}\otimes M_{2}^{[-1]})^{*}]}{{\rm
Tr}((M_{1}^{[-1]}\otimes M_{2}^{[-1]})\psi(\Phi(\rho))(M_{1}^{[-1]}\otimes
M_{2}^{[-1]})^{*})}$
if $\psi$ takes the form (ii). Then $\Delta:{\mathcal{S}}_{\rm
sep}(H_{1}\otimes H_{2})\rightarrow{\mathcal{S}}_{\rm sep}(H_{1}\otimes
H_{2})$ is a bijective map preserving separable pure states and strict convex
combinations. Furthermore $\Delta(A\otimes B)=A\otimes B$ for all $A\otimes
B\in{\mathcal{S}}_{\rm sep}(H_{1}\otimes H_{2})$. Then by [8, Lemma 2.7], we
get $\Delta(\rho)=\rho$ for any $\rho\in{\mathcal{S}}_{\rm sep}(H_{1}\otimes
H_{2})$. Now it is clear that $\psi$ has the form (1∘) or (2∘), finishing the
proof. $\Box$
The following result is a generalization of the main result in [8] by omitting
the additional assumption in Eq.(1.1) for finite dimensional case, and thus,
answer affirmatively a conjecture proposed in [8] , as mentioned in the
introduction section.
Theorem 3.5 Let $H_{i},K_{i},$ be complex Hilbert spaces with $2\leq\dim
H_{i}<\infty$, $i=1,2$, and $\psi:{\mathcal{S}}_{\rm sep}(H_{1}\otimes
H_{2})\rightarrow{\mathcal{S}}_{\rm sep}(K_{1}\otimes K_{2})$ an bijective
map. Then $\Phi$ is convex combination preserving if and only if either
(1) there exist invertible operators $S\in\mathcal{B}(H_{1},K_{1})$ and
$T\in\mathcal{B}(H_{2},K_{2})$ such that
$\psi(\rho)=\frac{(S\otimes T)\Psi(\rho)(S\otimes T)^{*}}{{\rm Tr}((S\otimes
T)\Psi(\rho)(S\otimes T)^{*})}$
for all $\rho\in{\mathcal{S}}_{\rm sep}(H_{1}\otimes H_{2})$;
or
(2) there exist invertible operators $S\in\mathcal{B}(H_{2},K_{1})$ and
$T\in\mathcal{B}(H_{1},K_{2})$ such that
$\psi(\rho)=\frac{(S\otimes T)\Psi(\Theta(\rho))(S\otimes T)^{*}}{{\rm
Tr}((S\otimes T)\Psi(\Theta(\rho))(S\otimes T)^{*})}$
for all $\rho\in{\mathcal{S}}_{\rm sep}(H_{1}\otimes H_{2})$.
Here $\Psi$ is the identity, or the transpose, or the partial transpose of the
first system or the partial transpose of the second system with respect to an
arbitrarily fixed product basis, $\Theta$ is the swap.
Proof. We need only check the “only if” part. By the assumption, $\psi$ is
bijective and strict convex combination preserving from ${\mathcal{S}}_{\rm
sep}(H_{1}\otimes H_{2})$ onto ${\mathcal{S}}_{\rm sep}(K_{1}\otimes K_{2})$.
Particularly, the range of $\psi$ is non-collinear. By [8], we have
$\psi({\mathcal{P}ur}_{\rm sep}(H_{1}\otimes H_{2}))={\mathcal{P}ur}_{\rm
sep}(K_{1}\otimes K_{2})$, that is, $\psi$ preserves separable pure states in
both directions.
Then by Corollary 3.4, $\psi$ has either the form (1) or the form (2) with
$T,S$ injective. Since $\psi({\mathcal{P}ur}_{\rm sep}(H_{1}\otimes
H_{2}))={\mathcal{P}ur}_{\rm sep}(K_{1}\otimes K_{2})$, it is clear that both
$T$ and $S$ are invertible. Hence the theorem is true. $\Box$
## 4\. Maps preserving separable pure states and strict convex combinations:
Multipartite systems
The results similar to that in Section 3 for bipartite case are valid for
multipartite cases, of course, with more complicated expressions. The proofs
are also similar. In this section we only list some of them, which have
relatively simple expressions and may have more applications. The meanings of
the notations used here are also similar to that in Section 3.
Suppose $\dim H_{i}=n_{i}$. For $1\leq r_{1}<\cdots<r_{p}\leq n$, define the
partial trace which is a linear map ${\rm
Tr}^{r_{1},\cdots,r_{p}}:{\mathcal{B}}_{\rm
sa}(\otimes_{i=1}^{n}H_{i})\rightarrow{\mathcal{B}}_{\rm
sa}(\otimes_{j=1}^{p}H_{r_{j}})$ as follows:
$\otimes_{i=1}^{n}A_{i}\longmapsto(\prod_{i\neq{r_{1},\cdots,r_{p}}}{\rm
Tr}A_{i})\otimes_{j=1}^{p}A_{r_{j}}.$
In particular, the linear map ${\rm Tr}^{r}:{\mathcal{B}}_{\rm
sa}(\otimes_{i=1}^{n}H_{i})\rightarrow{\mathcal{B}}_{\rm sa}(H_{r})$ is given
by ${\rm Tr}^{r}(\otimes_{i=1}^{n}A_{i})=(\prod_{i\neq r}{\rm
Tr}(A_{i}))A_{r}$. We call ${\rm Tr}^{r}(\rho)$ the reduction state of
$\rho\in{\mathcal{S}}(H_{1}\otimes H_{2}\otimes\cdots\otimes H_{n})$ in the
subsystem ${\mathcal{S}}(H_{r})$. A multipartite state
$\rho\in{\mathcal{S}}(\otimes_{i=1}^{n}H_{i})$ is called a product state if
$\rho=\otimes_{i=1}^{n}\rho_{i}$ for some $\rho_{i}\in{\mathcal{S}}(H_{i})$.
The following result corresponds to Corollary 3.3 and Corollary 3.4.
Theorem 4.1 Let $\psi:{\mathcal{S}}_{\rm sep}(H_{1}\otimes
H_{2}\otimes\cdots\otimes H_{n})\rightarrow{\mathcal{S}}_{\rm
sep}(K_{1}\otimes K_{2}\otimes\cdots\otimes K_{n})$ be a strict convex
combination preserving map, with $2\leq\dim H_{i}<\infty$, $i=1,2,\cdots,n$,
and
$\psi(\mathcal{P}ur(H_{1})\otimes\mathcal{P}ur(H_{2})\otimes\cdots\otimes\mathcal{P}ur(H_{n}))\subseteq\mathcal{P}ur(K_{1})\otimes\mathcal{P}ur(K_{2})\otimes\cdots\otimes\mathcal{P}ur(K_{n})$.
(1) If the range of $\psi$ is non-collinear or a singleton, then $\psi$ maps
product states to product states.
(2) If the range of $\psi$ is non-collinear and contains a state $\sigma$ so
that its reduction state ${\rm Tr}^{i}(\sigma)$ has rank $\geq 2$ for each
$i=1,2,\ldots,n$, then there exist a permutation
$\pi:(1,\cdots,n)\mapsto(\pi(1),\cdots,\pi(n))$ of $(1,\cdots,n)$ and
injective linear or conjugate linear (may not simultaneously) operators
$M_{j}:H_{\pi(j)}\rightarrow K_{j}$, $j=1,\cdots,n$, such that
$\psi(\rho)=\frac{(M_{1}\otimes\cdots\otimes
M_{n})\Theta_{\pi}(\rho)(M_{1}^{*}\otimes\cdots\otimes M_{n}^{*})}{{\rm
Tr}((M_{1}\otimes\cdots\otimes
M_{n})\Theta_{\pi}(\rho)(M_{1}^{*}\otimes\cdots\otimes M_{n}^{*}))}$
for all $\rho\in{\mathcal{S}}_{\rm sep}(H_{1}\otimes\cdots\otimes H_{n})$.
Here $\Theta_{\pi}:{\mathcal{B}}_{\rm sa}(H_{1}\otimes
H_{2}\otimes\cdots\otimes H_{n})\rightarrow{\mathcal{B}}_{\rm
sa}(H_{\pi(1)}\otimes H_{\pi(2)}\otimes\cdots\otimes H_{\pi(n)})$ is a linear
map determined by $\Theta_{\pi}(A_{1}\otimes A_{2}\otimes\cdots\otimes
A_{n})=A_{\pi(1)}\otimes A_{\pi(2)}\otimes\cdots\otimes A_{\pi(n)}$. It is
clear that $\dim H_{\pi(j)}\leq\dim K_{j}$.
The following result is corresponding to Theorem 3.5.
Corollary 4.2 Let $\psi:{\mathcal{S}}_{\rm sep}(H_{1}\otimes
H_{2}\otimes\cdots\otimes H_{n})\rightarrow{\mathcal{S}}_{\rm
sep}(K_{1}\otimes K_{2}\otimes\cdots\otimes K_{n})$ be a bijective map with
$2\leq\dim H_{i}<\infty$, $i=1,2,\cdots,n$. Then $\psi$ is strict convex
combination preserving if and only if there exist a permutation $\pi$ of
$(1,2,\cdots,n)$ and invertible linear or conjugate linear (may not
simultaneously) operators $M_{j}:H_{\pi(j)}\rightarrow K_{j}$, $j=1,\cdots,n$,
such that
$None$ $\psi(\rho)=\frac{(M_{1}\otimes\cdots\otimes
M_{n})\Theta_{\pi}(\rho)(M_{1}^{*}\otimes\cdots\otimes M_{n}^{*})}{{\rm
Tr}((M_{1}\otimes\cdots\otimes
M_{n})\Theta_{\pi}(\rho)(M_{1}^{*}\otimes\cdots\otimes M_{n}^{*}))}$
holds for all $\rho\in{\mathcal{S}}_{\rm sep}(H_{1}\otimes\cdots\otimes
H_{n})$. Here $\Theta_{\pi}:{\mathcal{B}}_{\rm sa}(H_{1}\otimes
H_{2}\otimes\cdots\otimes H_{n})\rightarrow{\mathcal{B}}_{\rm
sa}(H_{\pi(1)}\otimes H_{\pi(2)}\otimes\cdots\otimes H_{\pi(n)})$ is the
linear map determined by $\Theta_{\pi}(A_{1}\otimes A_{2}\otimes\cdots\otimes
A_{n})=A_{\pi(1)}\otimes A_{\pi(2)}\otimes\cdots\otimes A_{\pi(n)}$. It is
clear that $\dim H_{\pi(j)}=\dim K_{j}$.
## 5\. Conclusion
The quantum measurement map preserve strict convex combinations and sends pure
states to pure states. Similarly, each local quantum measurement map preserves
separable pure states and strict convex combinations. These facts make it
interesting to study the problem of characterizing the maps between states in
(multipartite) quantum systems that are (separable) pure state preserving and
strict convex combination preserving. These problems are basic and interesting
both in quantum information science and mathematics science, and their
solutions will present a geometric characterization of (local) quantum
measurements and help us to understand better the quantum measurement.
In the present paper, we give a characterization of the maps
$\phi:{\mathcal{S}}(H)\rightarrow{\mathcal{S}}(K)$ with $2\leq\dim H<\infty$
that preserve pure states and strict convex combinations, and give a structure
theorem of the maps $\psi:{\mathcal{S}}(H_{1}\otimes H_{2}\otimes\cdots\otimes
H_{n})\rightarrow{\mathcal{S}}(K_{1}\otimes K_{2}\otimes\cdots\otimes K_{n})$
with $2\leq\dim H_{i}<\infty$ that preserve separable pure states and strict
convex combinations. From these results we get a characterization of injective
(local) quantum measurements. In almost all situations, for example, in the
case that the range of $\psi$ is non-collinear or a singleton, $\psi$ sends
product states to product states. In particular, if the range of $\psi$ is
non-collinear and contains an element with each reduction having rank $\geq
2$, then $\phi$ is essentially an injective local quantum measurement. Thus we
answer affirmatively two conjectures proposed in [8].
Finally we remark that, the main results obtained in [5, 8] hold for both
finite dimensional systems and infinite dimensional systems. However, in the
present paper we only deal with finite dimensional systems. We conjecture that
the results in this paper are also valid for infinite dimensional case but new
tools are needed to prove this.
Acknowledgement. The authors wish to give their thanks to the referees for
helpful comments and suggestions to improve the expression of this paper.
## References
* [1] Alfsen, E., Shultz, F., Unique decompositions, faces, and automorphisms of separable states, Journal of Mathematical Physics 51(2010), 052201.
* [2] Bengtsson, I., Zyczkowski, K., Geometry of Quantum States, An introduction to quantum entangument, Cambridge University Press, Cambridge, 2006.
* [3] Friedland, S., Li C.-K., Poon Y.-T. and Sze N.-S. The automorphisms group of separable states in quantum information theory J. Math. Phys. 52 (2011) 042203.
* [4] Gudder, S., A structure for quantum measurements, Reports on Mathematical Physics, 55 (2005) 2, 249-267.
* [5] He, K. Hou, J. and Li C.-K., A geometric characteristic of invertible quantum measurement maps, J. Funct. Anal. 264 (2013), 464-478.
* [6] Hou, J., A characterization of positive linear maps and criteria for entangled quantum states, J. Phys. A: Math. Theor. 43 (2010) 385201.
* [7] Hou, J., On the tensor products of operators, Acta Math. Sinica (New Ser.), 9 (1993), 195-202.
* [8] Hou, J. and Liu, L., Quantum measurement and maps preserving convex combinations of separable states, J. Phys. A: Math. Theor. 45 (2012) 205305.
* [9] Hou J. and Qi X., Linear maps preserving separability of pure states, arXiv:1210.2155
* [10] Mendonca, P. E. M. F., Napolitano, R. d. J., etc., Alternative fidelity measure between quantum states, Phy. Rev. A 78 (2008) 052330.
* [11] Nielsen, M. A., Chuang, I. L., Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000.
* [12] Z. P$\breve{a}$les, Characterization of segment and convexity preserving maps, arXiv:1212.1268v1.
* [13] Vedral, V., Plenio, M. B., Jacobs, K., Knight, P. L., Phys. Rev. Lett. 78 (1997) 2275; Phys. Rev. Lett. 57 (1997) 4452
* [14] Werner, R. F., Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model, Phys. Rev. A 40 (1989) 4277.
|
arxiv-papers
| 2013-01-31T08:59:53 |
2024-09-04T02:49:41.064984
|
{
"license": "Public Domain",
"authors": "Lihua Yang, Jinchuan Hou",
"submitter": "Jinchuan Hou",
"url": "https://arxiv.org/abs/1301.7551"
}
|
1301.7553
|
# Lie ring isomorphisms between nest algebras on Banach spaces
Juan Deng , Jinchuan Hou Department of Mathematics, Taiyuan University of
Technology, Taiyuan 030024, P. R. China [email protected];
[email protected] and Xiaofei Qi Department of Mathematics, Shanxi
University, Taiyuan, 030006, P. R. China [email protected]
###### Abstract.
Let ${\mathcal{N}}$ and ${\mathcal{M}}$ be nests on Banach spaces $X$ and $Y$
over the (real or complex) field $\mathbb{F}$ and let $\mbox{\rm
Alg}{\mathcal{N}}$ and $\mbox{\rm Alg}{\mathcal{M}}$ be the associated nest
algebras, respectively. It is shown that a map $\Phi:{\rm
Alg}{\mathcal{N}}\rightarrow{\rm Alg}{\mathcal{M}}$ is a Lie ring isomorphism
(i.e., $\Phi$ is additive, Lie multiplicative and bijective) if and only if
$\Phi$ has the form $\Phi(A)=TAT^{-1}+h(A)I$ for all $A\in\mbox{\rm
Alg}{\mathcal{N}}$ or $\Phi(A)=-TA^{*}T^{-1}+h(A)I$ for all $A\in\mbox{\rm
Alg}{\mathcal{N}}$, where $h$ is an additive functional vanishing on all
commutators and $T$ is an invertible bounded linear or conjugate linear
operator when $\dim X=\infty$; $T$ is a bijective $\tau$-linear transformation
for some field automorphism $\tau$ of $\mathbb{F}$ when $\dim X<\infty$.
2010 Mathematical Subject Classification. Primary 47L35, 47B49
Key words and phrases. Banach spaces, nest algebras, Lie ring isomorphisms
This work is supported by National Natural Science Foundation of China
(11171249,11101250, 11271217) and Youth Foundation of Shanxi Province
(2012021004).
## 1\. Introduction and main results
Let $\mathcal{R}$ and $\mathcal{R}^{\prime}$ be two associative rings. Recall
that a map $\phi:\mathcal{R}\rightarrow\mathcal{R}^{\prime}$ is called a
multiplicative map if $\phi(AB)=\phi(A)\phi(B)$ for any $A,B\in\mathcal{R}$;
is called a Lie multiplicative map if $\phi([A,B])=[\phi(A),\phi(B)]$ for any
$A,B\in\mathcal{R}$, where $[A,B]=AB-BA$ is the Lie product of $A$ and $B$
which is also called a commutator. In addition, a map
$\phi:\mathcal{R}\rightarrow\mathcal{R}^{\prime}$ is called a Lie
multiplicative isomorphism if $\phi$ is bijective and Lie multiplicative; is
called a Lie ring isomorphism if $\phi$ is bijective, additive and Lie
multiplicative. If $\mathcal{R}$ and $\mathcal{R}^{\prime}$ are algebras over
a field $\mathbb{F}$, $\phi:{\mathcal{R}}\rightarrow{\mathcal{R}}^{\prime}$ is
called a Lie algebraic isomorphism if $\phi$ is bijective, $\mathbb{F}$-linear
and Lie multiplicative. For the study of Lie ring isomorphisms between rings,
see [3, 5, 10] and the references therein. In this paper we focus our
attention on Lie ring isomorphisms between nest algebras on general Banach
spaces.
Let $X$ be a Banach space over the (real or complex) field ${\mathbb{F}}$ with
topological dual $X^{*}$. $\mathcal{B}(X)$ stands for the algebra of all
bounded linear operators on $X$. A nest $\mathcal{N}$ on $X$ is a complete
totally ordered subspace lattice, that is, a chain of closed (under norm
topology) subspaces of $X$ which is closed under the formation of arbitrary
closed linear span (denote by $\bigvee$) and intersection (denote by
$\bigwedge$), and which includes $\\{0\\}$ and $X$. The nest algebra
associated with a nest $\mathcal{N}$, denoted by ${\rm Alg}\mathcal{N}$, is
the weakly closed operator algebra consisting of all operators that leave
every subspace $N\in\mathcal{N}$ invariant. For $N\in\mathcal{N}$, let
$N_{-}=\bigvee\\{M\in\mathcal{N}\mid M\subset N\\}$ and
$N_{-}^{\perp}=(N_{-})^{\perp}$, where $N^{\perp}=\\{f\in X^{*}\mid
N\subseteq\ker(f)\\}$. If $\mathcal{N}$ is a nest on $X$, then
${\mathcal{N}}^{\perp}=\\{N^{\perp}\mid N\in{\mathcal{N}}\\}$ is a nest on
$X^{*}$ and $({\rm Alg}{\mathcal{N}})^{*}\subseteq{\rm
Alg}{\mathcal{N}}^{\perp}$. If ${\mathcal{N}}=\\{(0),X\\}$, we say that
$\mathcal{N}$ is a trivial nest, in this case,
Alg${\mathcal{N}}={\mathcal{B}}(X)$. Non-trivial nest algebras are very
important reflexive operator algebras that are not semi-simple, not semi-prime
and not self-adjoint. If $\dim X<\infty$, a nest algebra on $X$ is isomorphic
to an algebra of upper triangular block matrices. Nest algebras are studied
intensively by a lot of literatures. For more details on basic theory of nest
algebras, the readers can refer to [6, 8].
In [9], Marcoux and Sourour proved that every Lie algebraic isomorphism
between nest algebras on separable complex Hilbert spaces is a sum
$\alpha+\beta$, where $\alpha$ is an algebraic isomorphism or the negative of
an algebraic anti-isomorphism and $\beta:{\rm
Alg}{\mathcal{N}}\rightarrow{\mathbb{C}}I$ is a linear map vanishing on all
commutators, that is, satisfying $\beta([A,B])=0$ for all $A,B\in{\rm
Alg}{\mathcal{N}}$.
Qi and Hou in [11] generalized the result of Marcoux and Sourour by
classifying certain Lie multiplicative isomorphisms. Note that, a Lie
multiplicative isomorphism needs not be additive. Let ${\mathcal{N}}$ and
${\mathcal{M}}$ be nests on Banach spaces $X$ and $Y$ over the (real or
complex) field $\mathbb{F}$, respectively, with the property that if
$M\in{\mathcal{M}}$ such that $M_{-}=M$, then $M$ is complemented in $Y$
(Obviously, this assumption is not needed if $Y$ is a Hilbert space or if
$\dim Y<\infty$). Let $\mbox{\rm Alg}{\mathcal{N}}$ and $\mbox{\rm
Alg}{\mathcal{M}}$ be respectively the associated nest algebras, and let
$\Phi:{\rm Alg}{\mathcal{N}}\rightarrow{\rm Alg}{\mathcal{M}}$ be a bijective
map. Qi and Hou in [11] proved that, if $\dim X=\infty$ and if there is a
nontrivial element in $\mathcal{N}$ which is complemented in $X$, then $\Phi$
is a Lie multiplicative isomorphism if and only if there exists a map
$h:\mbox{\rm Alg}{\mathcal{N}}\rightarrow{\mathbb{F}}I$ with $h([A,B])=0$ for
all $A,B\in\mbox{\rm Alg}{\mathcal{N}}$ such that $\Phi$ has the form
$\Phi(A)=TAT^{-1}+h(A)$ for all $A\in\mbox{\rm Alg}{\mathcal{N}}$ or
$\Phi(A)=-TA^{*}T^{-1}+h(A)$ for all $A\in\mbox{\rm Alg}{\mathcal{N}}$, where,
in the first form, $T:X\rightarrow Y$ is an invertible bounded linear or
conjugate-linear operator so that $N\mapsto T(N)$ is an order isomorphism from
$\mathcal{N}$ onto $\mathcal{M}$, while in the second form, $X$ and $Y$ are
reflexive, $T:X^{*}\rightarrow Y$ is an invertible bounded linear or
conjugate-linear operator so that $N^{\bot}\mapsto T(N^{\bot})$ is an order
isomorphism from ${\mathcal{N}}^{\bot}$ onto $\mathcal{M}$. If $\dim
X=n<\infty$, identifying nest algebras with upper triangular block matrix
algebras, then $\Phi$ is a Lie multiplicative isomorphism if and only if there
exist a field automorphism $\tau:{\mathbb{F}}\rightarrow{\mathbb{F}}$ and
certain invertible matrix $T$ such that either $\Phi(A)=TA_{\tau}T^{-1}+h(A)$
for all $A$, or $\Phi(A)=-T(A_{\tau})^{\rm tr}T^{-1}+h(A)$ for all $A$, where
$A_{\tau}=(\tau(a_{ij}))$ for $A=(a_{ij})$ and $A^{\rm tr}$ is the transpose
of $A$. Particularly, above results give a characterization of Lie ring
isomorphisms between nest algebras for finite-dimensional case, and for
infinite-dimensional case under the mentioned assumptions on $\mathcal{N}$ and
$\mathcal{M}$.
Recently, Wang and Lu in [12] generalized Marcoux and Sourour’s result from
another direction, and proved that every Lie algebraic isomorphism between
nest algebras ${\rm Alg}{\mathcal{N}}$ and ${\rm Alg}{\mathcal{M}}$ for any
nests $\mathcal{N}$ and $\mathcal{M}$ on Banach spaces $X$ and $Y$
respectively can be decomposed as $\alpha+\beta$, where $\alpha$ is an
algebraic isomorphism or the negative of an algebraic anti-isomorphism and
$\beta:{\rm Alg}{\mathcal{N}}\rightarrow{\mathbb{F}}I$ is a linear map
vanishing on each commutator. Because Lie algebraic isomorphisms were
characterized in [4] for the case that the nest $\mathcal{N}$ has a nontrivial
complemented element, Wang and Lu in [12] mainly dealt with the case that all
nontrivial elements of $\mathcal{N}$ are not complemented.
The purpose of the present paper is to characterize all Lie ring isomorphisms
between nest algebras of Banach space operators for any nests. Note that, the
Lie ring isomorphisms are very different from algebraic ones. For example, the
method used in [4] to characterize Lie algebraic isomorphisms for the case
that the nest $\mathcal{N}$ has a nontrivial complemented element is not valid
for characterizing Lie ring isomorphisms. Algebraic isomorphisms between nest
algebras are continuous, however ring isomorphisms are not necessarily
continuous for finite-dimensional case.
The following are the main results of this paper.
Theorem 1.1. Let ${\mathcal{N}}$ and ${\mathcal{M}}$ be nests on Banach spaces
$X$ and $Y$ over the (real or complex) field $\mathbb{F}$, and, ${\rm
Alg}{\mathcal{N}}$ and ${\rm Alg}{\mathcal{M}}$ be the associated nest
algebras, respectively. Then a map $\Phi:{\rm Alg}{\mathcal{N}}\rightarrow{\rm
Alg}{\mathcal{M}}$ is a Lie ring isomorphism, that is, $\Phi$ is additive,
bijective and satisfies $\Phi([A,B])=[\Phi(A),\Phi(B)]$ for all $A,B\in{\rm
Alg}{\mathcal{N}}$, if and only if $\Phi$ has the form $\Phi(A)=\Psi(A)+h(A)I$
for all $A\in{\rm Alg}{\mathcal{N}}$, where $\Psi$ is a ring isomorphism or
the negative of a ring anti-isomorphism between the nest algebras and
$h:\mbox{\rm Alg}{\mathcal{N}}\rightarrow{\mathbb{F}}$ is an additive
functional satisfying $h([A,B])=0$ for all $A,B\in\mbox{\rm
Alg}{\mathcal{N}}$.
The ring isomorphisms and the ring anti-isomorphisms between nest algebras of
Banach space operators were characterized in [7, Theorem 2.2, Theorem 2.7 and
Remark 2.6]. Using these results and Theorem 1.1, we can get more concrete
characterization of Lie ring isomorphisms. Recall that a map $S:W\rightarrow
V$ with $W,V$ linear spaces over a field $\mathbb{F}$ is called $\tau$-linear
if $S$ is additive and $S(\lambda x)=\tau(\lambda)Sx$ for all $x\in W$ and
$\lambda\in{\mathbb{F}}$, where $\tau$ is a field automorphism of
$\mathbb{F}$.
Theorem 1.2. Let ${\mathcal{N}}$ and ${\mathcal{M}}$ be nests on Banach spaces
$X$ and $Y$ over the (real or complex) field $\mathbb{F}$, and let ${\rm
Alg}{\mathcal{N}}$ and ${\rm Alg}{\mathcal{M}}$ be the associated nest
algebras, respectively. Then a map $\Phi:{\rm Alg}{\mathcal{N}}\rightarrow{\rm
Alg}{\mathcal{M}}$ is a Lie ring isomorphism if and only if there exist an
additive functional $h:\mbox{\rm Alg}{\mathcal{N}}\rightarrow{\mathbb{F}}$
satisfying $h([A,B])=0$ for all $A,B\in\mbox{\rm Alg}{\mathcal{N}}$ and a
field automorphism $\tau:{\mathbb{F}}\rightarrow{\mathbb{F}}$ such that one of
the following holds.
(1) There exists a $\tau$-linear transformation $T:X\rightarrow Y$ such that
the map $N\mapsto T(N)$ is an order isomorphism from $\mathcal{N}$ onto
$\mathcal{M}$ and
$\Phi(A)=TAT^{-1}+h(A)I\ \ {for\ all}\ \ A\in{\mbox{\rm Alg}}\mathcal{N}.$
(2) $X$ and $Y$ are reflexive, there exists a $\tau$-linear transformation
$T:X^{*}\rightarrow Y$ such that the map $N_{-}^{\perp}\mapsto
T(N_{-}^{\perp})$ is an order isomorphism from ${\mathcal{N}}^{\perp}$ onto
${\mathcal{M}}$ and
$\Phi(A)=-TA^{*}T^{-1}+h(A)I\ \ {for\ all}\ \ A\in{\mbox{\rm
Alg}}\mathcal{N}.$
Moreover, if $\dim X=\infty$, the above $T$ is in fact an invertible bounded
linear or conjugate-linear operator; if ${\mathbb{F}}={\mathbb{R}}$, $T$ is
linear.
For the finite dimensional case, it is clear that every nest algebra on a
finite dimensional space is isomorphic to an upper triangular block matrix
algebra. Let ${\mathcal{M}}_{n}({\mathbb{F}})$ denote the algebra of all
$n\times n$ matrices over $\mathbb{F}$. Recall that an upper triangular block
matrix algebra ${\mathcal{T}}={\mathcal{T}}(n_{1},n_{2},\ldots,n_{k})$ is a
subalgebra of ${\mathcal{M}}_{n}({\mathbb{F}})$ consisting of all $n\times n$
matrices of the form
$A=\left(\begin{array}[]{cccc}A_{11}&A_{12}&\ldots&A_{1k}\\\
0&A_{22}&\ldots&A_{2k}\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\ldots&A_{kk}\end{array}\right),$
where $n_{1},n_{2},\cdot\cdot\cdot,n_{k}$ are finite sequence of positive
integers satisfying $n_{1}+n_{2}+\cdot\cdot\cdot+n_{k}=n$ and
$A_{ij}\in{\mathcal{M}}_{n_{i}\times n_{j}}({\mathbb{F}})$, the space of all
$n_{i}\times n_{j}$ matrices over $\mathbb{F}$. Thus by Theorem 1.2, we get a
characterization of Lie ring isomorphisms between upper triangular block
matrix algebras.
Corollary 1.3. Let ${\mathbb{F}}$ be the real or complex field, and $m,n$ be
positive integers greater than 1. Let
${\mathcal{T}}={\mathcal{T}}(n_{1},n_{2},\ldots,n_{k})\subseteq{\mathcal{M}}_{n}({\mathbb{F}})$
and
${\mathcal{S}}={\mathcal{T}}(m_{1},m_{2},\ldots,m_{r})\subseteq{\mathcal{M}}_{m}({\mathbb{F}})$
be upper triangular block matrix algebras, and
$\Phi:{\mathcal{T}}\rightarrow{\mathcal{S}}$ be a map. Then $\Phi$ is a Lie
ring isomorphism if and only if $m=n$, and there exist an additive functional
$\phi:{\mathcal{T}}\rightarrow{\mathbb{F}}$ satisfying $\phi([A,B])=0$ for all
$A,B\in{\mathcal{T}}$, a field automorphism
$\tau:{\mathbb{F}}\rightarrow{\mathbb{F}}$ such that either
(1) ${\mathcal{T}}={\mathcal{S}}$, there exists an invertible matrix
$T\in{\mathcal{T}}$ such that
$\Phi(A)=TA_{\tau}T^{-1}+\phi(A)I\ \ for\ \ all\ \ A\in{\mathcal{T}};$
or
(2) $(n_{1},n_{2},\ldots,n_{k})=(m_{r},m_{r-1},\ldots,m_{1})$, there exists an
invertible block matrix $T=(T_{ij})_{k\times k}$ with
$T_{ij}\in{\mathcal{M}}_{n_{i},n_{j}}({\mathbb{F}})$ and $T_{ij}=0$ whenever
$i+j>k+1$, such that
$\Phi(A)=-TA_{\tau}^{\rm tr}T^{-1}+\phi(A)I\ \ for\ \ all\ \
A\in{\mathcal{T}}.$
Where $A_{\tau}=(\tau(a_{ij}))_{n\times n}$ for $A=(a_{ij})_{n\times
n}\in{\mathcal{M}}_{n}({\mathbb{F}})$ and $A^{\rm tr}$ is the transpose of
$A$. If ${\mathbb{F}}={\mathbb{R}}$, then $\Phi$ is a Lie algebraic
isomorphism.
Corollary 1.3 is also a consequence of [11, Corollary 2.2].
Since the Lie ring isomorphisms between nest algebras on finite-dimensional
Banach spaces were already characterized in [11, Corollary 2.2], to give a
classification of all Lie ring isomorphisms between nest algebras of Banach
space operators, it suffices to prove Theorem 1.1 for the infinite-dimensional
cases without any additional assumption on the nests. It is clear that $\dim
X=\infty\Leftrightarrow\dim Y=\infty$.
The remain part of the paper is to prove the main result Theorem 1.1 under the
assumption that both $X$ and $Y$ are infinite-dimensional. Our approach borrow
and combine some ideas developed in [11] and [12]. In Section 2 we give
preliminary lemmas, some of them are also parts of the proof of the main
result. Section 3 deals with the case that both $(0)$ and $X$ are limit points
of the nest $\mathcal{N}$, that is, $(0)=(0)_{+}$ and $X_{-}=X$. The case that
$X_{-}\not=X$ and $X_{-}$ is complemented or $(0)\not=(0)_{+}$ and $(0)_{+}$
is complemented is discussed in Section 4. And finally, the case that
$X_{-}\not=X$ and $X_{-}$ is not complemented or $(0)\not=(0)_{+}$ and
$(0)_{+}$ is not complemented is considered in Section 5.
## 2\. Preliminary lemmas
In this section, we give some preliminary lemmas, definitions and symbols
which are needed in other sections to prove the main result.
Let $X$ and $Y$ be Banach spaces over $\mathbb{F}$, and let $\mathcal{N}$ and
$\mathcal{M}$ be nests on $X$ and $Y$. Let Alg$\mathcal{N}$ and
Alg$\mathcal{M}$ be associated nest algebras, respectively. It is well known
that the commutant of a nest algebra is trivial, i.e., if
$T\in{\mathcal{B}}(X)$ and $TA=AT$ for every operator $A\in$Alg$\mathcal{N}$,
then $T=\lambda I$ for some scalar $\lambda\in\mathbb{F}.$ This fact will be
used in this paper without any specific explanation. In addition, the symbols
${\rm ran}T$, $\ker T$ and ${\rm rank}T$ stand for the range, the kernel and
the rank (i.e., the dimension of ran$T$) of an operator $T$, respectively. For
$x\in X$ and $f\in X^{*}$, $x\otimes f$ stands for the operator on $X$ with
rank not greater than 1 defined by $(x\otimes f)y=f(y)x$ for every $y$. Some
times we use $\langle x,f\rangle$ to present the value $f(x)$ of $f$ at $x$.
The following lemma is also well known which gives a characterization of rank
one operators in nest algebras.
Lemma 2.1. ([8]) Let ${\mathcal{N}}$ be a nest on a (real or complex) Banach
space $X$ and $x\in X$, $f\in X^{*}$. Then $x\otimes f\in{\rm Alg}\mathcal{N}$
if and only if there exists a subspace $N\in{\mathcal{N}}$ such that $x\in N$
and $f\in N_{-}^{\perp}$.
For any non-trivial element $E\in\mathcal{N}$, define
$None$ $\mathcal{J}({\mathcal{N}},E)=\\{A\in{\rm Alg}{\mathcal{N}}:AE=0\ \
{\rm and}\ \ A^{*}E^{\perp}=0\\}.$
In [12], Wang and Lu proved that $\mathcal{L}$ is a proper maximal commutative
Lie algebra ideal in ${\rm Alg}\mathcal{N}$ if and only if
${\mathcal{L}}=\mathbb{F}I+{\mathcal{J}}({\mathcal{N}},E)$ for some unique
$E\in{\mathcal{N}}$. The following lemma shows that any maximal commutative
Lie ring ideal also rises in this way.
Lemma 2.2. ${\mathcal{J}}$ is a proper maximal commutative Lie ring ideal in
${\rm Alg}{\mathcal{N}}$ if and only if it is a proper maximal commutative Lie
algebra ideal.
Proof. Assume that $\mathcal{J}$ is a maximal commutative Lie ring ideal. Then
for any $A\in{\rm Alg}{\mathcal{N}}$, any $C\in{\mathcal{J}}$ and any
$\lambda\in\mathbb{F}$, we have $[A,\lambda
C]=\lambda[A,C]\in\mathbb{F}\mathcal{J}$, which implies that
$\mathbb{F}\mathcal{J}$ is a Lie ring ideal. It is obvious that
$\mathbb{F}\mathcal{J}$ is also commutative. So
$\mathbb{F}\mathcal{J}\subseteq\mathcal{J}$ as $\mathcal{J}$ is maximal. Note
that $\mathbb{F}{\mathcal{J}}\supseteq{\mathcal{J}}$. Thus we get
$\mathbb{F}\mathcal{J}=\mathcal{J}$. It follows that $\mathcal{J}$ is also a
Lie algebra ideal. The converse is obvious. $\Box$
In the rest part of this paper, we assume that $\Phi:{\rm
Alg}{\mathcal{N}}\rightarrow{\rm Alg}{\mathcal{M}}$ is a Lie ring isomorphism.
If ${\mathcal{N}}$ contains at least one nontrivial element, by Lemma 2.2, for
any nontrivial element $E\in\mathcal{N}$,
$\Phi(\mathbb{F}I+\mathcal{J}(\mathcal{N},E))$ is a maximal commutative Lie
ring ideal in ${\rm Alg}\mathcal{M}$. Hence there is a unique nontrivial
element $F\in{\mathcal{M}}$ such that
$\Phi(\mathbb{F}I+\mathcal{J}(\mathcal{N},E))=\mathbb{F}I+\mathcal{J}(\mathcal{M},F).$
Define a map
$None$
$\hat{\Phi}:{\mathcal{N}}\setminus{\\{(0),X\\}}\rightarrow{\mathcal{M}}\setminus{\\{(0),Y\\}}$
by
$\Phi(\mathbb{F}I+\mathcal{J}(\mathcal{N},E))=\mathbb{F}I+\mathcal{J}(\mathcal{M},\hat{\Phi}(E))$.
With the symbols introduced above and by an argument similar to [12, Lemmas
4.1, 4.3, 4.4], one can show that the following lemma is still true for the
Lie ring isomorphism $\Phi$.
Lemma 2.3. $\hat{\Phi}$ in Eq.(2.2) is bijective and is either order-
preserving or order-reversing, that is, $\hat{\Phi}$ is an order isomorphism
or a reverse-order isomorphism from ${\mathcal{N}}$ onto ${\mathcal{M}}$ if we
extend the definition of $\hat{\Phi}$ so that $\hat{\Phi}((0))=(0)$ or $Y$ and
$\hat{\Phi}(X)=Y$ or $(0)$ accordingly.
By Lemma 2.3, for any $A\in{\mathcal{J}}({\mathcal{N}},E)$ with nontrivial
$E\in{\mathcal{N}}$, there exists a unique operator
$B\in{\mathcal{J}}({\mathcal{M}},\hat{\Phi}(E))$ such that
$\Phi(A)-B\in{\mathbb{F}}I$. Thus we can define another map
$None$ $\bar{\Phi}:\bigcup\\{{\mathcal{J}}({\mathcal{N}},E):E\in{\mathcal{N}}\
{\rm is\
nontrivial}\\}\rightarrow\bigcup\\{{\mathcal{J}}({\mathcal{M}},F):F\in{\mathcal{M}}\
{\rm is\ nontrivial}\\}$
with the property that
$\bar{\Phi}(A)\in{\mathcal{J}}({\mathcal{M}},\hat{\Phi}(E))$ and
$\Phi(A)-\bar{\Phi}(A)\in{\mathbb{F}}I$ for any
$A\in{\mathcal{J}}({\mathcal{N}},E)$.
Similar to [12, Lemma 4.2], we have
Lemma 2.4. $\bar{\Phi}$ is a bijective map and
$\bar{\Phi}({\mathcal{J}}({\mathcal{N}},E))={\mathcal{J}}({\mathcal{M}},\hat{\Phi}(E))$
for every non-trivial $E\in{\mathcal{N}}$.
Next we discuss the idempotents in nest algebras. Denote by
$\mathcal{E}(\mathcal{N})$ the set of all idempotents in ${\rm
Alg}{\mathcal{N}}$.
Lemma 2.5. ([11, Lemma 2.2]) Let ${\mathcal{N}}$ be a nest on a (real or
complex) Banach space $X$ and $A\in{\rm Alg}\mathcal{N}$.
(1) $A\in{\mathbb{F}}I+\mathcal{E}({\mathcal{N}})$ if and only if
$[A,[A,[A,T]]]=[A,T]$ for all $T\in{\rm Alg}\mathcal{N}$.
(2) $A$ is the sum of a scalar and an idempotent operator with range in
$\mathcal{N}$ if and only if $[A,[A,T]]=[A,T]$ for all $T\in{\rm
Alg}\mathcal{N}$.
By Lemma 2.5, if $P$ is an idempotent operator in ${\rm Alg}{\mathcal{N}}$,
then $\Phi(P)=Q+\lambda_{P}I$, where $\lambda_{P}\in{\mathbb{F}}$ and $Q$ is
an idempotent operator in ${\rm Alg}{\mathcal{M}}$. Furthermore, if ${\rm
ran}P\in{\mathcal{N}}$, then ${\rm ran}Q\in{\mathcal{M}}$. So we can define a
map
$None$
$\tilde{\Phi}:\mathcal{E}(\mathcal{N})\rightarrow\mathcal{E}(\mathcal{M})$
by $\tilde{\Phi}(P)=\Phi(P)-\lambda_{P}I$. It is easily seen that
$\tilde{\Phi}$ is a bijective map from ${\mathcal{E}}({\mathcal{N}})$ onto
${\mathcal{E}}({\mathcal{M}})$; see [11].
Now, for any nontrivial element $E\in{\mathcal{N}}$, define two sets
$None$
$\Omega_{1}(\mathcal{N},E)=\\{P\in\mathcal{E}(\mathcal{N}):PE=0\\}\quad{\rm
and}\quad\Omega_{2}(\mathcal{N},E)=\\{P\in\mathcal{E}(\mathcal{N}):P^{*}E^{\bot}=0\\}.$
For any nontrivial element $F\in{\mathcal{M}}$, the sets
$\Omega_{1}(\mathcal{M},F)$ and $\Omega_{2}(\mathcal{M},F)$ can be analogously
defined. Note that, if $P\in\Omega_{1}({\mathcal{N}},E)$, then one can easily
check $P^{*}E^{\perp}\not=0$; if $E$ is not complemented, then
$PE=0\Rightarrow(I-P)^{*}E^{\perp}\not=0$ and
$P^{*}E^{\perp}=0\Rightarrow(I-P)E\not=0$. These facts are needed in the proof
of Lemma 2.6.
Still, by an argument similar to [12, Lemmas 5.2-5.5], one can show that the
following lemma is true for Lie ring isomorphisms $\Phi$, with $\hat{\Phi}$
and $\tilde{\Phi}$ defined in Eq.(2.2) and Eq.(2.4) respectively.
Lemma 2.6. Assume that $E\in{\mathcal{N}}$ is nontrivial and not complemented
in $X$.
(1) Either
$\tilde{\Phi}(\Omega_{1}({\mathcal{N}},E))\subseteq\Omega_{1}({\mathcal{N}},\hat{\Phi}(E))$
or
$I-\tilde{\Phi}(\Omega_{1}({\mathcal{N}},E))\subseteq\Omega_{2}({\mathcal{N}},\hat{\Phi}(E))$,
and either
$\tilde{\Phi}(\Omega_{2}({\mathcal{N}},E))\subseteq\Omega_{2}({\mathcal{N}},\hat{\Phi}(E))$
or
$I-\tilde{\Phi}(\Omega_{2}({\mathcal{N}},E))\subseteq\Omega_{1}({\mathcal{N}},\hat{\Phi}(E))$.
(2) If $\Omega_{1}({\mathcal{N}},\hat{\Phi}(E))$ and
$\Omega_{2}({\mathcal{N}},\hat{\Phi}(E))$ are not empty, then
$\tilde{\Phi}(\Omega_{1}({\mathcal{N}},E))\subseteq\Omega_{1}({\mathcal{N}},\hat{\Phi}(E))$
if and only if
$\tilde{\Phi}(\Omega_{2}({\mathcal{N}},E))\subseteq\Omega_{2}({\mathcal{N}},\hat{\Phi}(E))$.
(3) If $F\in{\mathcal{N}}$ is also nontrivial and not complemented in $X$ with
$F<E$, then $\Omega_{1}({\mathcal{N}},E)\not=\emptyset$ and
$\tilde{\Phi}(\Omega_{1}({\mathcal{N}},E))\subseteq\Omega_{1}({\mathcal{N}},\hat{\Phi}(E))$
together imply that
$\tilde{\Phi}(\Omega_{1}({\mathcal{N}},F))\subseteq\Omega_{1}({\mathcal{N}},\hat{\Phi}(F))$.
The following lemma gives a characterization of complemented elements
$E\in{\mathcal{N}}$ by the operators in ${\mathcal{J}}({\mathcal{N}},E)$ and
${\mathcal{E}}({\mathcal{N}})$, which is needed to prove that $\hat{\Phi}$
preserves the complementarity.
Lemma 2.7. Assume that $E\in{\mathcal{N}}$ is a nontrivial element. The
following statements are equivalent.
(1) $E\in{\mathcal{N}}$ is complemented in $X$.
(2) There exists some idempotent $P\in{\rm Alg}{\mathcal{N}}$ such that
$[P,A]=A$ for any $A\in{\mathcal{J}}({\mathcal{N}},E)$.
(3) There exists some idempotent $P\in{\rm Alg}{\mathcal{N}}$ such that
$[P,A]=A$ for any rank one operators $A\in{\mathcal{J}}({\mathcal{N}},E)$.
Furthermore, we have $E={\rm ran}P$ for $P$ in (2) and (3).
Proof. (1)$\Rightarrow$(2). If $E\in{\mathcal{N}}$ is complemented in $X$,
there exists an idempotent $P\in{\rm Alg}{\mathcal{N}}$ such that ${\rm
ran}P=E$. For any $A\in{\mathcal{J}}({\mathcal{N}},E)$, we have $PA=A$ and
$AP=0$. Hence $[P,A]=A$.
(2)$\Rightarrow$(3) is obvious.
(3)$\Rightarrow$(1). Assume that there exists some idempotent $P\in{\rm
Alg}{\mathcal{N}}$ such that $[P,A]=A$ for any rank-1 operator
$A\in{\mathcal{J}}({\mathcal{N}},E)$. According to the space decomposition
$X={\rm ran}P+\ker P$, for any $A\in{\mathcal{J}}({\mathcal{N}},E)$, we have
$P=\left(\begin{array}[]{cc}I&0\\\ 0&0\end{array}\right)\quad{\rm and}\quad
A=\left(\begin{array}[]{cc}A_{11}&A_{12}\\\ A_{21}&A_{22}\end{array}\right).$
Then $[P,A]=A$ implies that $A=\left(\begin{array}[]{cc}0&A_{12}\\\
0&0\end{array}\right)$. So $PA=A$ and $AP=0$ hold for all
$A\in{\mathcal{J}}({\mathcal{N}},E)$. Take any $y\in E$ and any $g\in
E^{\perp}$. It is easy to check that $y\otimes
g\in{\mathcal{J}}({\mathcal{N}},E)$. So $Py\otimes g=y\otimes g$ and $y\otimes
gP=0$. It follows that $Py=y$ and $P^{*}g=0$ for all $y\in E$ and all $g\in
E^{\perp}$. Thus we get $E\subseteq{\rm ran}P$ and $E^{\bot}\subseteq\ker
P^{*}$. If $E\not={\rm ran}P$, then there exist $x\in{\rm ran}P$ and $g\in
E^{\bot}$ such that $\langle x,g\rangle=1$. This leads to a contraction that
$0=\langle x,P^{*}g\rangle=\langle Px,g\rangle=\langle x,g\rangle=1$. Hence we
must have ${\rm ran}P=E$ and $E$ is complemented in $X$. $\Box$
Lemma 2.8. Non-trivial element $E\in{\mathcal{N}}$ is complemented in $X$ with
${\rm ran}P=E$ if and only if $\hat{\Phi}(E)$ is complemented in $Y$ with
${\rm ran}\tilde{\Phi}(P)=\hat{\Phi}(E)$. Here $P\in{\rm Alg}{\mathcal{N}}$ is
an idempotent.
Proof. It is clear that $\widehat{\Phi^{-1}}=\hat{\Phi}^{-1}$. So we need only
to show that $\hat{\Phi}(E)$ is complemented in $Y$ whenever
$E\in{\mathcal{N}}$ is complemented in $X$. Indeed, if $E\in{\mathcal{N}}$ is
complemented, by Lemma 2.7, there exists some idempotent $P\in{\rm
Alg}{\mathcal{N}}$ such that $[P,A]=A$ for any
$A\in{\mathcal{J}}({\mathcal{N}},E)$. By definitions of $\hat{\Phi}$,
$\bar{\Phi}$ and $\tilde{\Phi}$ (ref. (2.2)-(2.4)), there exists some scalar
$\gamma$ such that
$\bar{\Phi}(A)+\gamma
I=\Phi(A)=\Phi([P,A])=[\Phi(P),\Phi(A)]=[\tilde{\Phi}(P),\bar{\Phi}(A)].$
Since $\bar{\Phi}$ maps ${\mathcal{J}}({\mathcal{N}},E)$ onto
${\mathcal{J}}({\mathcal{M}},\hat{\Phi}(E))$ by Lemma 2.4, we see that
$B+\gamma I=[\tilde{\Phi}(P),B]$ holds for any
$B\in{\mathcal{J}}({\mathcal{M}},\hat{\Phi}(E))$. Assume that $B$ is of
rank-1. If $\gamma\not=0$, then $[\tilde{\Phi}(P),B]=B+\gamma I$ is a sum of
nonzero scalar and a rank-1 operator, which is impossible since a commutator
can not be the sum of a nonzero scalar and a compact operator. Hence
$\gamma=0$, and $B=[\tilde{\Phi}(P),B]$ for all rank-1 operators
$B\in{\mathcal{J}}({\mathcal{M}},\hat{\Phi}(E))$. By Lemma 2.7 again,
$\hat{\Phi}(E)$ is complemented in $Y$ with ${\rm
ran}\tilde{\Phi}(P)=\hat{\Phi}(E)$. $\Box$
The following lemma is obvious.
Lemma 2.9. $\Phi({\mathbb{F}}I)={\mathbb{F}}I$.
Finally, we give a lemma, which is needed to prove our main result. Let $E$
and $F$ be subspaces of $X$ and $X^{*}$, respectively. Denote by $E\otimes F$
the set $\\{x\otimes f:x\in E,f\in F\\}$.
Lemma 2.10. Let $X_{i}$ be an infinite-dimensional Banach space, $i=1,2$. Let
$E_{i}$ and $F_{i}$ be closed subspaces with dimensions $>2$ of $X_{i}$ and
$X_{i}^{*}$ respectively. Let $\mathcal{A}_{i}$ be a unital subalgebra of
${\mathcal{B}}(X_{i})$ containing $E_{i}\otimes F_{i}$. Suppose that
$\Psi:{\mathcal{A}}_{1}\rightarrow{\mathcal{A}}_{2}$ is an additive bijective
map satisfying $\Psi(\mathbb{F}I)=\mathbb{F}I$ and
$\Phi({\mathbb{F}}I+E_{1}\otimes F_{1})={\mathbb{F}}I+E_{2}\otimes F_{2}$.
Then there is a map $\gamma:E_{1}\otimes F_{1}\rightarrow\mathbb{F}$ and a
field automorphism $\tau:\mathbb{F}\rightarrow\mathbb{F}$ such that either
(1) $\Psi(x\otimes f)=\gamma(x,f)I+Cx\otimes Df$ for all $x\in E_{1}$ and
$f\in F_{1},$ where $C:E_{1}\rightarrow E_{2}$ and $D:F_{1}\rightarrow F_{2}$
are two $\tau$-linear bijective maps; or
(2) $\Psi(x\otimes f)=\gamma(x,f)I+Df\otimes Cx$ for all $x\in E_{1}$ and
$f\in F_{1},$ where $C:E_{1}\rightarrow F_{2}$ and $D:F_{1}\rightarrow E_{2}$
are two $\tau$-linear bijective maps.
Lemma 2.10 can be proved by a similar approach as that in [2] and we omit its
proof here.
Since the “if” part of Theorem 1.1 is obvious, we need only to check the “only
if” part. If $\mathcal{N}$ is a trivial nest, that is,
${\mathcal{N}}=\\{(0),X\\}$, then ${\mathcal{M}}=\\{(0),Y\\}$ by Lemma 2.3. So
$\Phi$ is a Lie ring isomorphism from ${\mathcal{B}}(X)$ onto
${\mathcal{B}}(Y)$. Bai, Du and Hou showed in [1] that every Lie
multiplicative isomorphism between prime rings with a non-trivial idempotent
element is of the form $\psi+\beta$ with $\psi$ a ring isomorphism or the
negative of a ring anti-isomorphism and $\beta$ a central valued map vanishing
on each commutator. Note that ${\mathcal{B}}(X)$ is prime and contains non-
trivial idempotents if $\dim X\geq 2$. Hence, for the case that $\mathcal{N}$
is trivial, Theorem 1.1 follows from [1].
So, in the rest sections we always assume that ${\mathcal{N}}$ is nontrivial.
Thus ${\mathcal{M}}$ is also nontrivial by Lemma 2.3. We shall complete the
proof of Theorem 1.1 by considering several cases according to the situations
of $(0)_{+}$ and $X_{-}$, which will be dealt with separably in Sections 3-5.
## 3\. The case that $(0)_{+}=(0)$ and $X_{-}=X$
In this section, we deal with the case that both $(0)$ and $X$ are limit
points of $\mathcal{N}$, i.e., $(0)_{+}=(0)$ and $X_{-}=X$.
Keep the definitions of $\hat{\Phi}$, $\bar{\Phi}$ in mind; ref. Eqs.(2.2) and
(2.3).
Lemma 3.1. Assume that $(0)_{+}=(0)$ and $X_{-}=X$. Let $E\in{\mathcal{N}}$ be
a nontrivial element. Then for any $x\in E$ and $f\in E^{\bot}$,
$\bar{\Phi}(x\otimes f)$ is a rank one operator.
Proof. Since, by Lemma 2.3, the bijective map
$\hat{\Phi}:{\mathcal{N}}\rightarrow{\mathcal{M}}$ is either order-preserving
or order-reversing, we have $(0)_{+}=(0)$ and $Y_{-}=Y$ in $\mathcal{M}$.
Write $R=\bar{\Phi}(x\otimes f)$ with $\bar{\Phi}$ defined in Eq.(2.3). Then
$R\in\mathcal{J}(\mathcal{M},\hat{\Phi}(E))$. If, on the contrary, ${\rm
rank}R\geq 2$, then, since $\bigcup\\{M:M\in{\mathcal{M}}\\}$ is a dense
linear manifold of $Y$, there exists a nontrivial element
$M_{0}\in{\mathcal{M}}$ and two vectors $u,v\in M$ such that $Ru$ and $Rv$ are
linearly independent. As $\bigcap\\{M:(0)<M\in{\mathcal{M}}\\}=\\{0\\}$, there
exists some nontrivial $L\in{\mathcal{M}}$ such that $Ru,Rv\not\in L$. Let
$Y_{L}={\rm span}\\{Ru,L\\}$. By Hahn-Banach theorem, there exists some $g\in
Y_{L}^{\bot}$ such that $g(Rv)\not=0$. Then $g\in L^{\bot}$ satisfying
$g(Ru)=0$ and $g(Rv)\neq 0.$ It is easily checked that
$L<\hat{\Phi}(E)<M_{0}$. Take $z\in L$, $h\in M_{0}^{\bot}$, and let
$A=\bar{\Phi}^{-1}(z\otimes g)$, $B=\bar{\Phi}^{-1}(u\otimes h)$. By Lemma
2.4, $A\in\mathcal{J}(\mathcal{N},\hat{\Phi}^{-1}(L))$ and
$B\in\mathcal{J}(\mathcal{N},\hat{\Phi}^{-1}(M_{0})).$
If $\hat{\Phi}$ is order-preserving, we have
$\hat{\Phi}^{-1}(L)<E<\hat{\Phi}^{-1}(M_{0})$. So
$\Phi(A(x\otimes f)B)=\Phi([A,[x\otimes f,B]])=[z\otimes g,[R,u\otimes
h]]=(z\otimes g)R(u\otimes h)=0.$
It follows from the injectivity of $\Phi$ that $A(x\otimes f)B=0$, which
implies $A(x\otimes f)=0$ or $(x\otimes f)B=0.$ If $A(x\otimes f)=0,$ then
$0=\Phi([A,x\otimes f])=[z\otimes g,R]=(z\otimes g)R\not=0$, a contradiction;
if $(x\otimes f)B=0,$ then $0=\Phi([x\otimes f,B])=[R,u\otimes h]=R(u\otimes
h)\not=0,$ a contradiction.
If $\hat{\Phi}$ is order-reversing, we have
$\hat{\Phi}^{-1}(M_{0})<E<\hat{\Phi}^{-1}(L).$ So
$\Phi(B(x\otimes f)A)=\Phi([B,[x\otimes f,A]])=[u\otimes h,[R,z\otimes
g]]=(z\otimes g)R(u\otimes h)=0,$
which yields that either $B(x\otimes f)=0$ or $(x\otimes f)A=0$. By a similar
argument to that of the above, one can obtain a contradiction.
Therefore, we must have ${\rm rank}\bar{\Phi}(x\otimes f)={\rm rank}R=1$.
$\Box$
Proof of Theorem 1.1 for the case that $(0)=(0)_{+}$ and $X_{-}=X$.
By Lemma 3.1, Lemma 2.9 and the bijectivity of $\bar{\Phi}$, we have proved
that, for every nontrivial element $E\in{\mathcal{N}}$,
$\Phi(\mathbb{F}I+E\otimes
E^{\bot})=\mathbb{F}I+\hat{\Phi}(E)\otimes\hat{\Phi}(E)^{\bot}$. So, by Lemma
2.10, there exists a ring automorphism
$\tau_{E}:\mathbb{F}\rightarrow\mathbb{F}$ and a map $\gamma_{E}:E\otimes
E^{\bot}\rightarrow\mathbb{F}$ such that either
$None$ $\Phi(x\otimes f)=\gamma_{E}(x,f)I+C_{E}x\otimes D_{E^{\perp}}f$
holds for all $x\in E$ and $f\in E^{\bot}$, where
$C_{E}:E\rightarrow\hat{\Phi}(E)$ and
$D_{E^{\perp}}:E^{\bot}\rightarrow\hat{\Phi}(E)^{\bot}$ are two $\tau$-linear
bijective maps; or
$None$ $\Phi(x\otimes f)=\gamma_{E}(x,f)I+D_{E^{\perp}}f\otimes C_{E}x$
holds for all $x\in E$ and $f\in E^{\bot}$, where
$C_{E}:E\rightarrow\hat{\Phi}(E)^{\perp}$ and
$D_{E^{\perp}}:E^{\bot}\rightarrow\hat{\Phi}(E)$ are two $\tau$-linear
bijective maps.
It is easily checked that, if there is a nontrivial $E_{0}\in{\mathcal{N}}$
such that Eq.(3.1) holds, then Eq.(3.1) holds for any nontrivial
$E\in{\mathcal{N}}$; If there is a nontrivial $E_{0}\in{\mathcal{N}}$ such
that Eq.(3.2) holds, then Eq.(3.1) holds for any nontrivial
$E\in{\mathcal{N}}$.
Assume that Eq.(3.1) holds for a nontrivial $E\in{\mathcal{N}}$. Then, for any
$N\in{\mathcal{N}}$, any $x\in E\cap N$ and any $f\in N^{\perp}\cap
E^{\perp}$, we have
$\Phi(x\otimes f)=\gamma_{E}(x,f)I+C_{E}x\otimes
D_{E^{\perp}}f=\gamma_{N}(x,f)I+C_{N}x\otimes D_{N^{\perp}}f.$
Since ${\rm rank}I=\infty$, the above equation yields
$\gamma_{N}(x,f)=\gamma_{E}(x,f)$ and $C_{E}x\otimes
D_{E^{\perp}}f=C_{N}x\otimes D_{N^{\perp}}f$ for any $x\in N\cap E$. This
entails that there exists an automorphism
$\tau:{\mathbb{F}}\rightarrow{\mathbb{F}}$ so that $\tau_{N}=\tau_{E}=\tau$
for any $N,E\in{\mathcal{N}}$, and if $N\subset E$, then there exists a scalar
$\alpha_{EN}$ such that $C_{E}|_{N}=\alpha_{EN}C_{N}$ and
$D_{N^{\perp}}|_{E^{\perp}}=\alpha_{EN}D_{E^{\perp}}$. Now fix
$E\in{\mathcal{N}}$. For any $N\in{\mathcal{N}}$, we define
$\left\\{\begin{array}[]{lll}\tilde{C}_{N}=C_{N},&\tilde{D}_{N_{-}^{\perp}}=D_{N^{\perp}},&\mbox{\rm
if}\ \ N=E;\\\
\tilde{C}_{N}=\alpha_{EN}C_{N},&\tilde{D}_{N_{-}^{\perp}}=\frac{1}{\alpha_{EN}}D_{N^{\perp}},&\mbox{\rm
if}\ \ N\subset E;\\\
\tilde{C}_{N}=\frac{1}{\alpha_{EN}}C_{N},&\tilde{D}_{N_{-}^{\perp}}=\alpha_{EN}D_{N^{\perp}},&\mbox{\rm
if}\ \ N\supset E.\end{array}\right.$
It is obvious that $\\{\tilde{C_{N}}:N\not=(0),X\\}$ and
$\\{\tilde{D}_{N_{-}^{\perp}}:N\not=(0),X\\}$ are well defined with
$\tilde{C}_{N}|_{E}=\tilde{C}_{E}$ and
$\tilde{D}_{E^{\perp}}|_{N^{\perp}}=\tilde{D}_{N^{\perp}}$ whenever
$E\subseteq N$. Thus there exist bijective $\tau$-linear maps
$C:\bigcup\\{N\in{\mathcal{N}}:N\not=(0),X\\}\rightarrow\bigcup\\{M\in{\mathcal{M}}:M\not=(0),Y\\}$
and
$D:\bigcup\\{N^{\perp}:N\in{\mathcal{N}},N\not=(0),X\\}\rightarrow\bigcup\\{M^{\perp}:M\in{\mathcal{M}},M\not=(0),Y\\}$
such that $C|_{N}=\tilde{C}_{N}$ and
$D|_{N^{\perp}}=\tilde{D}_{N_{-}^{\perp}}$ for any
$N\in{\mathcal{N}}\backslash\\{(0),X\\}$.
By now, we have shown that, there exist bijective $\tau$-linear maps
$C:\bigcup\\{N\in{\mathcal{N}}:N\not=(0),X\\}\rightarrow\bigcup\\{M\in{\mathcal{M}}:M\not=(0),Y\\}$
and
$D:\bigcup\\{N^{\perp}:N\in{\mathcal{N}},N\not=(0),X\\}\rightarrow\bigcup\\{M^{\perp}:M\in{\mathcal{M}},M\not=(0),Y\\}$,
and a map $\gamma:\bigcup\\{E\times
E^{\perp}:E\in{\mathcal{N}}\setminus\\{(0),X\\}\\}\rightarrow{\mathbb{F}}$,
such that for any $x\in N$ and $f\in N^{\perp}$ with
$N\in{\mathcal{N}}\setminus\\{(0),X\\}$, we have
$None$ $\Phi(x\otimes f)=\gamma(x,f)I+Cx\otimes Df.$
Therefore, for any $A\in{\rm Alg}\mathcal{N}$, any $x\in N$ and any $f\in
N^{\perp}$, by Eq.(3.3), we have
$\Phi([A,x\otimes f])=[\Phi(A),\Phi(x\otimes f)]=\Phi(A)Cx\otimes Df-
Cx\otimes\Phi(A)^{*}Df$
and
$\begin{array}[]{rl}\Phi([A,x\otimes f])=&\Phi(Ax\otimes f-x\otimes A^{*}f)\\\
=&(\gamma(Ax,f)-\gamma(x,A^{*}f))I+CAx\otimes Df-Cx\otimes
DA^{*}f.\end{array}$
Combining the above two equations and noting that $I$ is of infinite-rank, one
obtains that
$Cx\otimes\Phi(A)^{*}Df-Cx\otimes DA^{*}f=\Phi(A)Cx\otimes Df-CAx\otimes Df$
holds for any $x\in N$, $f\in N^{\bot}$ and any nontrivial
$N\in{\mathcal{N}}$. Note that $D$ is bijective. So there exists a scalar
$h(A)$ such that
$None$ $\Phi(A)Cx=CAx+h(A)Cx$
for all $x\in\bigcup\\{N\in{\mathcal{N}}:N\not=(0),X\\}$. It is clear that $h$
is additive as a functional of Alg$\mathcal{N}$. Define
$\Psi(A)=\Phi(A)-h(A)I$ for all $A\in{\rm Alg}{\mathcal{N}}$. Then, by
Eq.(3.4), for any $A,B\in{\rm Alg}{\mathcal{N}}$ and any
$x\in\bigcup\\{N\in{\mathcal{N}}:N\not=(0),X\\}$, we have
$\Psi(AB)Cx=CABx=\Psi(A)CBx=\Psi(A)\Psi(B)Cx.$
Since $\bigcup\\{N\in{\mathcal{N}}:N\not=(0),X\\}$ is dense in $X$ and $C$ is
bijective, it follows that $\Psi(AB)=\Psi(A)\Psi(B)$ for all $A,B\in{\rm
Alg}\mathcal{N}$, that is, $\Psi$ is a ring isomorphism and
$\Phi(A)=\Psi(A)+h(A)I$ for all $A\in{\rm Alg}\mathcal{N}$.
Similarly, if Eq.(3.2) holds, one can check that $\Phi(A)=-\Psi(A)+h(A)I$ for
all $A\in{\rm Alg}\mathcal{N}$, where $\Psi$ is a ring anti-isomorphism and
$h$ is an additive functional.
This completes the proof of Theorem 1.1 for the case that $(0)_{+}=(0)$ and
$X_{-}=X$. $\Box$
## 4\. The case $X_{-}\not=X$ and $X_{-}$ is complemented or $(0)_{+}\not=0$
and $(0)_{+}$ is complemented
We give only the proof in detail for the case that $X_{-}\not=X$ and $X_{-}$
is complemented. The case that $(0)\not=(0)_{+}$ and $(0)_{+}$ is complemented
in $X$ can be dealt with similarly.
Proof of Theorem 1.1 for the case that $X_{-}\not=X$ and $X_{-}$ is
complemented in $X$.
Assume that $X_{-}\not=X$ and $X_{-}$ is complemented in $X$.
Since $X_{-}$ is complemented, there exists an idempotent $P_{0}\in{\rm
Alg}\mathcal{N}$ such that ${\rm ran}P_{0}=X_{-}$.
With $\tilde{\Phi}$ as in Eq.(2.4) and by a similar argument to that in the
proof of [11, Theorem 3.1], one can show that there exist an idempotent
operator $Q_{0}$ and a scalar $\lambda_{P_{0}}$ such that
$\Phi(P_{0})=Q_{0}+\lambda_{P_{0}}I$ with $Q_{0}=\tilde{\Phi}(P_{0})$, ${\rm
ran}Q_{0}\in\mathcal{M}$ and the following statements hold:
(a) If there is an idempotent $P_{1}\in{\rm Alg}{\mathcal{N}}$ such that
$P_{1}<P_{0}$ and $\tilde{\Phi}(P_{1})<Q_{0}$ (or $P_{1}>P_{0}$ and
$\tilde{\Phi}(P_{1})>Q_{0}$), then for any $P\in{\rm Alg}{\mathcal{N}}$,
$P<P_{0}\Rightarrow\tilde{\Phi}(P)<Q_{0}$ and
$P>P_{0}\Rightarrow\tilde{\Phi}(P)>Q_{0}$.
(b) If there is an idempotent $P_{1}\in{\rm Alg}{\mathcal{N}}$ such that
$P_{1}<P_{0}$ and $\tilde{\Phi}(P_{1})>Q_{0}$ (or $P_{1}>P_{0}$ and
$\tilde{\Phi}(P_{1})<Q_{0}$), then for any $P\in{\rm Alg}{\mathcal{N}}$,
$P<P_{0}\Rightarrow\tilde{\Phi}(P)>Q_{0}$ and
$P>P_{0}\Rightarrow\tilde{\Phi}(P)<Q_{0}$.
Claim 4.1. If (a) occurs, then $\Phi=\Psi+h$, where $\Psi:{\rm
Alg}{\mathcal{N}}\rightarrow{\rm Alg}{\mathcal{M}}$ is a ring isomorphism and
$h:{\rm Alg}{\mathcal{N}}\rightarrow{\mathbb{F}}I$ is an additive map
vanishing all commutators.
(a) implies that ${\rm ran}Q_{0}=Y_{-}$ by Lemmas 2.3 and 2.8. For the
convenience, let ${\mathcal{A}}_{11}=P_{0}({\rm Alg}{\mathcal{N}})P_{0}$,
${\mathcal{A}}_{12}=P_{0}({\rm Alg}{\mathcal{N}})(I-P_{0})$,
${\mathcal{A}}_{22}=(I-P_{0})({\rm Alg}{\mathcal{N}})(I-P_{0})$,
${\mathcal{B}}_{11}=Q_{0}({\rm Alg}{\mathcal{M}})Q_{0}$,
${\mathcal{B}}_{12}=Q_{0}({\rm Alg}{\mathcal{M}})(I-Q_{0})$ and
${\mathcal{B}}_{22}=(I-Q_{0})({\rm Alg}{\mathcal{M}})(I-Q_{0})$. Then ${\rm
Alg}{\mathcal{N}}={\mathcal{A}}_{11}\dot{+}{\mathcal{A}}_{12}\dot{+}{\mathcal{A}}_{22}$
and ${\rm
Alg}{\mathcal{M}}={\mathcal{B}}_{11}\dot{+}{\mathcal{B}}_{12}\dot{+}{\mathcal{B}}_{22}$.
We will prove Claim 4.1 by several steps.
Step 1. $\Phi({\mathcal{A}}_{12})={\mathcal{B}}_{12}$.
The proof is the same as that of [11, Lemma 2.8].
Step 2. $\Phi({\mathcal{A}}_{ii})\subseteq{\mathcal{B}}_{ii}+{\mathbb{F}}I$,
$i=1,2$.
For any $A_{11}\in{\mathcal{A}}_{11}$, denote
$\Phi(A_{11})=S_{11}+S_{12}+S_{22}$, where $S_{ij}\in{\mathcal{B}}_{ij}$. Then
$0=\Phi([A_{11},P_{0}])=[\Phi(A_{11}),\Phi(P_{0})]=[\Phi(A_{11}),Q_{0}],$
which implies that $S_{12}=0$. Let $P\in{\mathcal{A}}_{22}$ be any idempotent
with $P\not=I-P_{0}$. It is clear that $P<(I-P_{0})$. Then $I-P>P_{0}$. As
$\Phi$ meets (a), we have $\tilde{\Phi}(I-P)>Q_{0}$, that is,
$\tilde{\Phi}(P)<I-Q_{0}$. It follows from $[A_{11},P]=0$ that
$[\Phi(A_{11}),\tilde{\Phi}(P)]=[S_{22},\tilde{\Phi}(P)]=0$. By the
arbitrariness of $P$ and the bijectivity of $\tilde{\Phi}$, we see that
$S_{22}$ commutes with every idempotent in ${\mathcal{B}}_{22}$. Note that
${\rm ran}Q_{0}=Y_{-}$. So $\mathcal{B}_{22}=\mathcal{B}(\ker Q_{0})$, which
implies $S_{22}\in{\mathbb{F}}(I-Q_{0})$. It follows that
$\Phi(A_{11})=S_{11}+\lambda(I-Q_{0})=(S_{11}-\lambda Q_{0})+\lambda I$ and
hence, $\Phi(\mathcal{A}_{11})\subseteq\mathcal{B}_{11}+\mathbb{F}I$.
Assume that $A_{22}\in\mathcal{A}_{22}$. In the same way as above, one can
show that, $\Phi(A_{22})=T_{11}+T_{22}$ for some
$T_{ii}\in{\mathcal{B}}_{ii}$, $i=1,2$, with $T_{11}$ commuting with every
idempotent in ${\mathcal{B}}_{11}$. For any $B_{11}\in\mathcal{B}_{11}$, by
the surjectivity of $\Phi$, there exists some $A_{0}\in{\rm Alg}\mathcal{N}$
such that $\Phi(A_{0})=B_{11}$. Furthermore, $A_{0}=A_{11}+\lambda I$ for some
$A_{11}\in{\mathcal{A}}_{11}$ and some scalar $\lambda$ by Step 1. Thus we
have
$[B_{11},T_{11}]=[B_{11},T_{22}+T_{11}]=[\Phi(A_{0}),\Phi(A_{22})]=\Phi([A_{0},A_{22}])=\Phi([A_{11}+\lambda
I,A_{22}])=0$
for all $B_{11}\in\mathcal{B}_{11}$, which implies $T_{11}\in\mathbb{F}I$ as
${\mathcal{A}}_{11}$ is a nest algeba. So
$\Phi(\mathcal{A}_{22})\subseteq\mathcal{B}_{22}+\mathbb{F}I$.
By Steps 1-2, for each $A_{12}\in{\mathcal{A}}_{12}$, there exists
$B_{12}\in{\mathcal{B}}_{12}$ such that $\Phi(A_{12})=B_{12}$; for each
$A_{ii}\in{\mathcal{A}}_{ii}$, $i=1,2$, there exist
$B_{ii}\in{\mathcal{B}}_{ii}$ and $\lambda_{ii}\in{\mathbb{F}}$ such that
$\Phi(A_{ii})=B_{ii}+\lambda_{ii}I$. We claim that $B_{ii}$ and $\lambda_{ii}$
are uniquely determined. In fact, if
$\Phi(A_{ii})=B_{ii}+\lambda_{ii}I=B_{ii}^{{}^{\prime}}+\lambda^{{}^{\prime}}_{ii}I$,
then $B_{ii}-B_{ii}^{{}^{\prime}}\in{\mathbb{F}}I$, which implies that
$B_{ii}=B_{ii}^{{}^{\prime}}$ and $\lambda_{ii}=\lambda^{{}^{\prime}}_{ii}$.
Let $\Psi(A_{ij})=B_{ij}$ and
$\Psi(A)=\Psi(A_{11})+\Psi(A_{12})+\Psi(A_{22})$. Then we define a map
$\Psi:{\rm Alg}{\mathcal{N}}\rightarrow{\rm Alg}{\mathcal{M}}$ and a map
$h:{\rm Alg}{\mathcal{N}}\rightarrow{\mathbb{F}}I$ with
$h(A)=\Phi(A)-\Psi(A)\in{\mathbb{F}}I$. Now, imitating the proof of [11,
Lemmas 2.10-2.13], one can show that $\Psi:{\rm
Alg}{\mathcal{N}}\rightarrow{\rm Alg}{\mathcal{M}}$ is a ring isomorphism and
$h:{\rm Alg}{\mathcal{N}}\rightarrow{\mathbb{F}}I$ is an additive map
satisfying $h([A,B])=0$ for all $A,B$. Hence Claim 4.1 is true.
Claim 4.2. If $\Phi$ satisfies (b), then $\Phi=-\Psi+h$, where $\Psi:{\rm
Alg}{\mathcal{N}}\rightarrow{\rm Alg}{\mathcal{M}}$ is a ring anti-isomorphism
and $h:{\rm Alg}{\mathcal{N}}\rightarrow{\mathbb{F}}I$ is an additive map
vanishing all commutators.
If (b) holds, then ${\rm ran}Q_{0}=(0)_{+}\in{\mathcal{M}}$ and $(0)_{+}$ is
complemented in $\mathcal{M}$ by Lemma 2.3 and 2.8. Consider the map
$\Phi^{\prime}:{\rm Alg}{\mathcal{N}}\rightarrow({\rm Alg}{\mathcal{M}})^{*}$
defined by $\Phi^{\prime}(A)=-\Phi(A)^{*}$ for all $A\in{\rm
Alg}{\mathcal{N}}$.
By imitating the proof of [11, Lemmas 2.14], one can show that
$\Phi^{\prime}:{\rm Alg}{\mathcal{N}}\rightarrow({\rm Alg}{\mathcal{M}})^{*}$
is a Lie multiplicative bijective map. Since, for any nontrivial idempotent
operator $P\in{\rm Alg}{\mathcal{N}}$, $\Phi(P)=\tilde{\Phi}(P)+\lambda_{P}I$
for some $\lambda_{P}\in{\mathbb{C}}$, we have
${\Phi}^{\prime}(P)=-\tilde{\Phi}(P)^{*}-\lambda_{P}I$. Now define a map
$\tilde{{\Phi}^{\prime}}:{\mathcal{E}}{(\mathcal{N})}\rightarrow{\mathcal{E}}{(\mathcal{M})}^{*}$
by $\tilde{{\Phi}^{\prime}}(P)=I-\tilde{\Phi}(P)^{*}$ for all idempotents $P$.
Since $\Phi$ satisfies (b), for any nonzero idempotent $P_{1}\in{\rm
Alg}{\mathcal{N}}$, if $P_{1}<P_{0}$, we have
$\tilde{{\Phi}^{\prime}}(P_{1})=I-\tilde{\Phi}(P_{1})^{*}<I-\tilde{\Phi}(P_{0})^{*}=\tilde{{\Phi}^{\prime}}(P_{0})$;
if $P_{1}>P_{0}$, we have
$\tilde{{\Phi}^{\prime}}(P_{1})=I-\tilde{\Phi}(P_{1})^{*}>I-\tilde{\Phi}(P_{0})^{*}=\tilde{{\Phi}^{\prime}}(P_{0})$.
Hence ${\Phi}^{\prime}$ satisfies (a).
Note that ${\mathcal{M}^{\perp}}=\\{M^{\perp}:M\in{\mathcal{M}}\\}$ is a nest
on $Y^{*}$. Since ${\rm ran}Q_{0}\in{\mathcal{M}}$, we have $\ker
Q_{0}^{*}=({\rm ran}Q_{0})^{\perp}\in{\mathcal{M}}^{\perp}$, and so ${\rm
ran}(I-Q_{0}^{*})\in{\mathcal{M}}^{\perp}$. With respect to the decomposition
$Y^{*}={\rm ran}(I-Q_{0}^{*})\dot{+}{\rm ran}Q_{0}^{*}$, we have ${\rm
Alg}{\mathcal{M}}^{\perp}={\mathcal{D}}_{11}\dot{+}{\mathcal{D}}_{12}\dot{+}{\mathcal{D}}_{22}$,
where ${\mathcal{D}}_{11}=(I-Q_{0}^{*})({\rm
Alg}{\mathcal{M}}^{\perp})(I-Q_{0}^{*})$,
${\mathcal{D}}_{12}=(I-Q_{0}^{*})({\rm Alg}{\mathcal{M}}^{\perp})Q_{0}^{*}$,
${\mathcal{D}}_{22}=Q_{0}^{*}({\rm Alg}{\mathcal{M}}^{\perp})Q_{0}^{*}$. Since
${\rm Alg}{\mathcal{M}}=Q_{0}({\rm Alg}{\mathcal{M}})Q_{0}\dot{+}Q_{0}({\rm
Alg}{\mathcal{M}})(I-Q_{0})\dot{+}(I-Q_{0})({\rm
Alg}{\mathcal{M}})(I-Q_{0})={\mathcal{B}}_{11}\dot{+}{\mathcal{B}}_{12}\dot{+}{\mathcal{B}}_{22}$,
we have ${\mathcal{B}}_{11}^{*}\subseteq{\mathcal{D}}_{22}$,
${\mathcal{B}}_{12}^{*}\subseteq{\mathcal{D}}_{12}$ and
${\mathcal{B}}_{22}^{*}\subseteq{\mathcal{D}}_{11}$. Hence $({\rm
Alg}{\mathcal{M}})^{*}={\mathcal{B}}_{22}^{*}\dot{+}{\mathcal{B}}_{12}^{*}\dot{+}{\mathcal{B}}_{11}^{*}\subseteq{\rm
Alg}{\mathcal{M}}^{\perp}$.
Step 1. $\Phi^{\prime}({\mathcal{A}}_{12})={\mathcal{B}}_{12}^{*}$.
The proof is the same as that of [11, Lemma 2.15].
Step 2.
$\Phi^{\prime}({\mathcal{A}}_{ii})\subseteq{\mathcal{B}}_{jj}^{*}+{\mathbb{F}}I$,
$i,j=1,2$ and $i\not=j$.
For any $A_{11}\in{\mathcal{A}}_{11}$, write
$\Phi^{\prime}(A_{11})=S_{22}^{*}+S_{12}^{*}+S_{11}^{*}$, where
$S_{ij}\in{\mathcal{B}}_{ij}$. Then
$0=\Phi^{\prime}([A_{11},P_{0}])=[\Phi^{\prime}(A_{11}),\Phi^{\prime}(P_{0})]=[\Phi^{\prime}(A_{11}),-Q_{0}^{*}]=[Q_{0}^{*},\Phi^{\prime}(A_{11})],$
which implies that $S_{12}^{*}=0$.
Let $P\in{\mathcal{A}}_{22}$ be any idempotent with $P<I-P_{0}$. As
$\Phi^{\prime}$ satisfies (a), we have
$\widetilde{\Phi^{\prime}}(P)<\widetilde{\Phi^{\prime}}(I-P_{0})=I-\tilde{\Phi}(P_{0})^{*}=Q_{0}^{*}$.
It follows from $[A_{11},P]=0$ that
$0=[\Phi^{\prime}(A_{11}),{\Phi}^{\prime}(P)]=[S_{11}^{*},\Phi^{\prime}(P)]$.
Since $P$ is arbitrary, we see that $S_{11}^{*}$ commutes with every
idempotent in ${\mathcal{B}}_{11}^{*}$, which implies that $S_{11}$ commutes
with every idempotent in ${\mathcal{B}}_{11}$. Noting that
$\mathcal{B}_{11}=\mathcal{B}(\ker Q_{0})$ in this case, so we must have
$S_{11}\in{\mathbb{F}}I_{\ker Q_{0}}$. By the arbitrariness of $A_{11}$ we
obtain that
$\Phi^{\prime}(\mathcal{A}_{11})\subseteq\mathcal{B}_{22}^{*}+\mathbb{F}I$.
Similarly, one can show that, for any $A_{22}\in{\mathcal{A}}_{22}$,
$\Phi^{\prime}(A_{22})=T_{22}^{*}+T_{11}^{*}$, where $T_{22}$ commutes with
every idempotent in ${\mathcal{B}}_{22}$. Taking any
$B_{22}\in\mathcal{B}_{22}$, by the surjectivity of $\Phi^{\prime}$, there
exists some $A_{0}\in{\rm Alg}\mathcal{N}$ such that
$\Phi^{\prime}(A_{0})=B_{22}^{*}$. Furthermore, $A_{0}=A_{11}+\lambda I$ for
some $A_{11}\in{\mathcal{A}}_{11}$ and some scalar $\lambda$. Then
$[B_{22}^{*},T_{22}^{*}]=[B_{22}^{*},T_{22}^{*}+T_{11}^{*}]=[\Phi^{\prime}(A_{0}),\Phi^{\prime}(A_{22})]=\Phi^{\prime}([A_{0},A_{22}])=0$
for all $B_{22}\in\mathcal{B}_{22}$. Thus $[B_{22},T_{22}]=0$ for all
$B_{22}\in{\mathcal{B}}_{22}$, which implies $T_{22}\in\mathbb{F}I_{{\rm
ran}Q_{0}}$ as ${\mathcal{B}}_{22}$ is a nest algebra. Therefore
$\Phi^{\prime}({\mathcal{A}}_{22})\subseteq{\mathcal{B}}_{11}^{*}+{\mathbb{F}}I$.
By Steps 1-2, for each $A_{12}\in{\mathcal{A}}_{12}$, there exists
$B_{12}\in{\mathcal{B}}_{12}$ such that $\Phi^{\prime}(A_{12})=B_{12}^{*}$;
for each $A_{ii}\in{\mathcal{A}}_{ii}$ ($i=1,2$), there exist
$B_{jj}\in{\mathcal{B}}_{jj}$ and $\lambda_{jj}\in{\mathbb{F}}$ such that
$\Phi^{\prime}(A_{ii})=B_{jj}^{*}+\lambda_{jj}I$, $i,j=1,2$ and $i\not=j$. It
is easily seen that $B_{jj}$ and $\lambda_{jj}$ are uniquely determined.
Define a map $\Psi^{\prime}:{\rm Alg}{\mathcal{N}}\rightarrow({\rm
Alg}{\mathcal{M}})^{*}$ by $\Psi^{\prime}(A_{12})=B_{12}^{*}$,
$\Psi^{\prime}(A_{11})=B_{22}^{*}$, $\Psi^{\prime}(A_{22})=B_{11}^{*}$ and
$\Psi^{\prime}(A)=\Psi^{\prime}(A_{11})+\Psi^{\prime}(A_{12})+\Psi^{\prime}(A_{22})$
for $A=A_{11}+A_{12}+A_{22}$. Set
$h^{\prime}(A)=\Phi^{\prime}(A)-\Psi^{\prime}(A)$. It is clear that
$h^{\prime}$ maps ${\rm Alg}{\mathcal{N}}$ into ${\mathbb{F}}I$. By [11, Lemma
2.17], one can show that $\Psi^{\prime}$ is a ring isomorphism. Since, for any
$A\in{\rm Alg}{\mathcal{N}}$, there exists a unique element $S\in{\rm
Alg}{\mathcal{M}}$ such that $\Psi^{\prime}(A)=S^{*}$, we can define a map
$\Psi:{\rm Alg}{\mathcal{N}}\rightarrow{\rm Alg}{\mathcal{M}}$ by $\Psi(A)=S$.
Thus $\Psi(A)^{*}=\Psi^{\prime}(A)$ for every $A$ and hence $\Psi$ is a ring
anti-isomorphism. Let $h:{\rm Alg}{\mathcal{N}}\rightarrow{\mathbb{F}}I$ be
the map defined by $h^{\prime}(A)=-h(A)^{*}$ for all $A\in{\rm
Alg}{\mathcal{N}}$. Clearly, $h([A,B])=0$ for all $A,B\in{\rm
Alg}{\mathcal{N}}$. Furthermore, we have
$-\Phi(A)^{*}=\Phi^{\prime}(A)=\Psi^{\prime}(A)+h^{\prime}(A)=\Psi(A)^{*}+h^{\prime}(A)=(\Psi(A)-h(A))^{*},$
which yields $\Phi(A)=-\Psi(A)+h(A)$ for every $A\in{\rm Alg}{\mathcal{N}}$.
This completes the proof of Claim 4.2.
By Claim 4.1 and Claim 4.2, Theorem 1.1 holds for the case that $X_{-}\not=X$
and $X_{-}$ is complemented in $X$. $\Box$
## 5\. The case that $X_{-}\not=X$ and $X_{-}$ is not complemented or
$(0)\not=(0)_{+}$ and $(0)_{+}$ is not complemented
In this section, we deal with the case that $X_{-}\not=X$ and $X_{-}$ is not
complemented or $(0)\not=(0)_{+}$ and $(0)_{+}$ is not complemented. Here we
borrow some ideas developed in [12]. Note that, not like [12], we do not
assume that all non-trivial elements in the nests are not complemented. Also,
we give only the detail of our proof for the case that $X_{-}\not=X$ and
$X_{-}$ is not complemented in $X$. The other case can be checked similarly.
Assume that $X_{-}\not=X$ and $X_{-}$ is not complemented in $X$.
Note that, there are three possible situations that $(0)$ may have, that is,
(1∘) $(0)_{+}\not=(0)$ and $(0)_{+}$ is complemented in $X$, (2∘)
$(0)_{+}=(0)$ and (3∘) $(0)_{+}\not=(0)$ and $(0)_{+}$ is not complemented in
$X$. By Section 4, Theorem 1.1 is true if the situation (1∘) occurs. So what
we need to deal with is either (2∘) or (3∘).
Recall that $\hat{\Phi}$, $\bar{\Phi}$ and $\tilde{\Phi}$ are maps defined in
Eqs.(2.2)-(2.4), and $\Omega_{1}({\mathcal{N}},E)$ and
$\Omega_{2}({\mathcal{N}},E)$ are defined in Eq.(2.5). By Lemma 2.6, either
$\tilde{\Phi}(\Omega_{1}({\mathcal{N}},X_{-}))\subseteq\Omega_{1}({\mathcal{N}},\hat{\Phi}(X_{-}))$
or $\tilde{\Phi}(\Omega_{1}({\mathcal{N}},X_{-}))\subseteq
I-\Omega_{2}({\mathcal{N}},\hat{\Phi}(X_{-}))$.
The following lemma is crucial for our purpose.
Lemma 5.1. Assume that $(0)<X_{-}<X$ and $X_{-}$ is not complemented. The
following statements are true:
(1) If
$\tilde{\Phi}(\Omega_{1}({\mathcal{N}},X_{-}))\subseteq\Omega_{1}({\mathcal{N}},\hat{\Phi}(X_{-}))$,
then $\Phi(\mathbb{F}I+X\otimes X_{-}^{\bot})=\mathbb{F}I+Y\otimes
Y_{-}^{\bot}$, and there exists a ring automorphism
$\tau:\mathbb{F}\rightarrow\mathbb{F}$, a map $\gamma:X\otimes
X_{-}^{\bot}\rightarrow\mathbb{F}$, bijective $\tau$-linear maps
$C:X\rightarrow Y$ and $D:X_{-}^{\bot}\rightarrow Y_{-}^{\bot}$ such that
$\Phi(x\otimes f)=\gamma(x,f)I+Cx\otimes Df$
holds for all $x\in X$ and $f\in X_{-}^{\bot}$.
(2) If $\tilde{\Phi}(\Omega_{1}({\mathcal{N}},X_{-}))\subseteq
I-\Omega_{2}({\mathcal{N}},\hat{\Phi}(X_{-}))$, then
$\Phi(\mathbb{F}I+X\otimes X_{-}^{\bot})=\mathbb{F}I+(0)_{+}\otimes Y^{*}$,
and there exists a ring automorphism $\tau:\mathbb{F}\rightarrow\mathbb{F}$, a
map $\gamma:X\otimes X_{-}^{\bot}\rightarrow\mathbb{F}$, bijective
$\tau$-linear maps $C:X\rightarrow Y^{*}$ and
$D:X_{-}^{\bot}\rightarrow(0)_{+}$ such that
$\Phi(x\otimes f)=\gamma(x,f)I+Df\otimes Cx$
holds for all $x\in X$ and $f\in X_{-}^{\bot}$.
To prove Lemma 5.1, we consider two cases, that is, the case that
${\mathcal{N}}$ has at least two nontrivial elements and the case that
${\mathcal{N}}$ has only one nontrivial element. These will be done by Lemma
5.3 and Lemma 5.4, respectively.
We first consider the case that ${\mathcal{N}}$ has at least two nontrivial
elements.
The following lemma is crucial for the proof of Lemma 5.3.
Lemma 5.2. Assume that $\mathcal{N}$ has at least two nontrivial elements and
$X_{-}$ is not complemented in $X$. Then the following statements hold.
(i)
$\tilde{\Phi}(\Omega_{1}({\mathcal{N}},X_{-}))\subseteq\Omega_{1}({\mathcal{N}},\hat{\Phi}(X_{-}))$
if and only if $\hat{\Phi}$ is order-preserving.
(ii) $\tilde{\Phi}(\Omega_{1}({\mathcal{N}},X_{-}))\subseteq
I-\Omega_{2}({\mathcal{N}},\hat{\Phi}(X_{-}))$ if and only if $\hat{\Phi}$ is
order-reversing.
Proof. By Lemma 2.6, it suffices to show that
$\tilde{\Phi}(\Omega_{1}({\mathcal{N}},X_{-}))\subseteq\Omega_{1}({\mathcal{N}},\hat{\Phi}(X_{-}))$
implies that $\hat{\Phi}$ is order-preserving and
$\tilde{\Phi}(\Omega_{1}({\mathcal{N}},X_{-}))\subseteq
I-\Omega_{2}({\mathcal{N}},\hat{\Phi}(X_{-}))$ implies that$\hat{\Phi}$ is
order-reversing.
Assume that
$\tilde{\Phi}(\Omega_{1}({\mathcal{N}},X_{-}))\subseteq\Omega_{1}({\mathcal{N}},\hat{\Phi}({X_{-}}))$.
Since $\mathcal{N}$ has at least two nontrivial elements, we may take a
nontrivial element $E\in{\mathcal{N}}$ such that $E<X_{-}$. If, on the
contrary, $\hat{\Phi}$ is order-reversing, then we have
$\hat{\Phi}(E)>\hat{\Phi}(X_{-})$. Fix an idempotent
$P\in\Omega_{1}({\mathcal{N}},X_{-})$. Then we have
$\tilde{\Phi}(P)\in\Omega_{1}({\mathcal{M}},\hat{\Phi}(X_{-}))$. For any
$D_{1}\in{\mathcal{J}}({\mathcal{M}},\hat{\Phi}(X_{-}))$,
$D_{2}\in{\mathcal{J}}({\mathcal{M}},\hat{\Phi}(E))$, let
$C_{1}=\bar{\Phi}^{-1}(D_{1})$, $C_{2}=\bar{\Phi}^{-1}(D_{2})$. By the
definition of $\bar{\Phi}$, we have
$C_{1}\in{\mathcal{J}}({\mathcal{N}},X_{-})$ and
$C_{2}\in{\mathcal{J}}({\mathcal{N}},E)$. Furthermore,
$\begin{array}[]{rl}0=&\Phi([C_{1},[C_{2},P]])=[D_{1},[D_{2},\tilde{\Phi}(P)]]\\\
=&[D_{1},D_{2}\tilde{\Phi}(P)-\tilde{\Phi}(P)D_{2}]=D_{1}D_{2}\tilde{\Phi}(P)-D_{1}\tilde{\Phi}(P)D_{2}.\end{array}$
For any $y_{1}\in\hat{\Phi}(X_{-})$, $y_{2}\in\hat{\Phi}(E)$ and any
$g_{1}\in\hat{\Phi}(X_{-})^{\bot}$, $g_{2}\in\hat{\Phi}(E)^{\bot}$, it is
obvious that $y_{1}\otimes
g_{1}\in{\mathcal{J}}({\mathcal{M}},\hat{\Phi}(X_{-}))$ and $y_{2}\otimes
g_{2}\in{\mathcal{J}}({\mathcal{M}},\hat{\Phi}(E))$. Letting
$D_{i}=y_{i}\otimes g_{i}$, the above equation yields
$None$ $\langle y_{2},g_{1}\rangle
y_{1}\otimes\tilde{\Phi}(P)^{*}g_{2}=\langle\tilde{\Phi}(P)y_{2},g_{1}\rangle
y_{1}\otimes g_{2}.$
Choose $y_{2}\in\hat{\Phi}(E)$ and $g_{1}\in\hat{\Phi}(X_{-})^{\bot}$ such
that $\langle y_{2},g_{1}\rangle\neq 0$. Eq.(5.1) implies that there exists
some scalar $\lambda_{g_{2}}$ such that
$\tilde{\Phi}(P)^{*}g_{2}=\lambda_{g_{2}}g_{2}$ for each
$g_{2}\in\hat{\Phi}(E)^{\bot}$. It follows that there is a scalar $\lambda$
such that $\tilde{\Phi}(P)^{*}g_{2}=\lambda g_{2}$ for all
$g_{2}\in\hat{\Phi}(E)^{\bot}$. Since $\tilde{\Phi}(P)^{*}$ is an idempotent,
either $\lambda=0$ or $\lambda=1$.
If $\lambda=0$, then $\tilde{\Phi}(P)^{*}\hat{\Phi}(E)^{\bot}=\\{0\\}$ and
Eq.(5.1) yields $\langle\tilde{\Phi}(P)y_{2},g_{1}\rangle=0$ for all
$y_{2}\in\hat{\Phi}(E)$ and $g_{1}\in\hat{\Phi}(X_{-})^{\bot}$. It follows
that
$\tilde{\Phi}(P)^{*}\hat{\Phi}(X_{-})^{\bot}\subseteq\hat{\Phi}(E)^{\bot}$. So
$\tilde{\Phi}(P)^{*}\hat{\Phi}(X_{-})^{\bot}=\tilde{\Phi}(P)^{*}(\tilde{\Phi}(P)^{*}\hat{\Phi}(X_{-})^{\bot})\subseteq\tilde{\Phi}(P)^{*}\hat{\Phi}(E)^{\bot}=\\{0\\}$,
which is impossible as
$\tilde{\Phi}(P)\in\Omega_{1}({\mathcal{M}},\hat{\Phi}(X_{-}))$.
If $\lambda=1$, then Eq.(5.1) yields
$\langle\tilde{\Phi}(P)y_{2},g_{1}\rangle=\langle y_{2},g_{1}\rangle$ for all
$y_{2}\in\hat{\Phi}(E)$ and $g_{1}\in\hat{\Phi}(X_{-})^{\bot}$. This implies
that
$(I-\tilde{\Phi}(P))^{*}\hat{\Phi}(X_{-})^{\bot}\subseteq\hat{\Phi}(E)^{\bot}$,
and so
$(I-\tilde{\Phi}(P))^{*}\hat{\Phi}(X_{-})^{\bot}=(I-\tilde{\Phi}(P))^{*}((I-\tilde{\Phi}(P))^{*}\hat{\Phi}(X_{-})^{\bot})\subseteq(I-\tilde{\Phi}(P))^{*}\hat{\Phi}(E)^{\bot}=\\{0\\}.$
This, together with the fact $\tilde{\Phi}(P)\hat{\Phi}(X_{-})=\\{0\\}$
entails ${\rm ran}\tilde{\Phi}(P)=\hat{\Phi}(X_{-})$, a contradiction. Hence
$\hat{\Phi}$ is order-preserving.
Similarly, one can show that $\hat{\Phi}$ is order-reversing if
$\tilde{\Phi}(\Omega_{1}({\mathcal{N}},X_{-}))\subseteq
I-\Omega_{2}({\mathcal{N}},\hat{X_{-}})$. $\Box$
Lemma 5.3. Assume that $\mathcal{N}$ has at least two nontrivial elements and
$(0)<X_{-}<X$ is not complemented in $X$. Then for any $x\in X$ and $f\in
X_{-}^{\bot}$, ${\Phi}(x\otimes f)$ is the sum of a scalar and a rank one
operator. Moreover, the statement (1) and (2) of Lemma 5.1 hold.
Proof. We will complete the proof of the lemma by considering three cases.
Case 1. $x\in X_{-}$ and $f\in X_{-}^{\bot}$.
In this case, let $R=\bar{\Phi}(x\otimes f)$. Then
$R\in\mathcal{J}(\mathcal{M},\hat{\Phi}(X_{-}))$ and $\Phi(x\otimes
f)-\bar{\Phi}(x\otimes f)=\Phi(x\otimes f)-R\in{\mathbb{F}}I$. We show that
$R$ is of rank one. Assume on the contrary that ${\rm rank}R\geq 2$. We will
induce contradiction by considering two subcases.
Subcase 1.1.
$\tilde{\Phi}(\Omega_{1}(\mathcal{N},X_{-}))\subseteq\Omega_{1}(\mathcal{M},\hat{\Phi}(X_{-}))$.
By Lemma 5.2, $\hat{\Phi}$ is order-preserving. It is clear in this case that
we have $(0)<Y_{-}=\hat{\Phi}(X_{-})<Y$.
Since ${\rm rank}R\geq 2$, there are two vectors $u,v\in Y\setminus Y_{-}$
such that $Ru$ and $Rv$ are linearly independent. Take $h\in Y_{-}^{\bot}$
such that $h(u)=1$. Then $u\otimes
h\in\Omega_{1}(\mathcal{M},\hat{\Phi}(X_{-}))$. Let
$B=\tilde{\Phi}^{-1}(u\otimes h)$. By Lemma 5.2(i),
$B\in\Omega_{1}({\mathcal{N}},X_{-})$.
If $(0)_{+}=(0)\in\mathcal{N}$, then $(0)_{+}=(0)\in\mathcal{M}$ by Lemma 2.3.
Thus $(0)=\cap\\{M:(0)\not=M\in{\mathcal{M}}\\}$ and there exists some
nontrivial element $M\in\mathcal{M}$ and $g\in M^{\bot}$ such that $g(Ru)=0$
and $g(Rv)=1.$ Obviously $M<\hat{\Phi}(X_{-})$, and so
$\hat{\Phi}^{-1}(M)<X_{-}$. Take a nonzero vector $z\in M$ and let
$A=\bar{\Phi}^{-1}(z\otimes g).$ Then
$A\in\mathcal{J}(\mathcal{N},\hat{\Phi}^{-1}(M))$, which implies
$A^{*}\hat{\Phi}^{-1}(M)^{\bot}=\\{0\\}$. Thus we have
$\Phi(A(x\otimes f)B)=\Phi([A,[x\otimes f,B]])=[z\otimes g,[R,u\otimes
h]]=(z\otimes g)R(u\otimes h)=0.$
So either $A(x\otimes f)=0$ or $(x\otimes f)B=0.$ If $A(x\otimes f)=0,$ then
$0=\Phi([A,x\otimes f])=[z\otimes g,R]=(z\otimes g)R\not=0,$ which is
impossible; If $(x\otimes f)B=0,$ then $0=\Phi([x\otimes f,B])=[R,u\otimes
h]=R(u\otimes h)\not=0,$ which is also impossible.
If $(0)<(0)_{+}\in\mathcal{N}$, then $(0)<(0)_{+}\in\mathcal{M}$ by Lemma 2.3.
Let $M=\hat{\Phi}((0)_{+}).$ Then
$(0)<M=\hat{\Phi}((0)_{+})<\hat{\Phi}(X_{-})$ as $\mathcal{M}$ has at least
two nontrivial elements. By Lemma 2.8, $M$ is not complemented and thus
infinite-dimensional. So, there is a vector $z\in M$ and a functional $g\in
Y^{*}$ such that $g(Ru)=0$ and $g(z)=g(Rv)=1.$ Let
$A=\tilde{\Phi}^{-1}(z\otimes g).$ As $z\otimes
g\in\Omega_{2}({\mathcal{M}},M)$, we have
$A\in\Omega_{2}(\mathcal{N},(0)_{+})$ by Lemma 2.6(3). It follows that
$A^{*}\hat{\Phi}^{-1}(M)^{\bot}=\\{0\\}$. Now by calculating $\Phi(A(x\otimes
f)B)$ in the same way as the above, one can get a contradiction.
So in the case that $\hat{\Phi}$ is order-preserving, $R$ is of rank one.
Subcase 1.2. $\tilde{\Phi}(\Omega_{1}(\mathcal{N},X_{-}))\subseteq
I-\Omega_{2}(\mathcal{M},\hat{\Phi}(X_{-}))$.
By Lemma 5.1(ii), in this case $\hat{\Phi}$ is order-reversing. So, we have
$(0)<(0)_{+}=\hat{\Phi}(X_{-})<Y$.
If $Y=Y_{-}$, there is a nontrivial element $M\in{\mathcal{M}}$ and vectors
$u,v\in M$ such that $Ru$ and $Rv$ are linearly independent. Let $h\in
M^{\bot}$ and $B=\bar{\Phi}^{-1}(u\otimes h).$ Then
$B\in\mathcal{J}(\mathcal{N},\hat{\Phi}^{-1}(M))$ by Lemma 2.4. Choose $g$ in
$Y^{*}$ such that $g(Ru)=0$ and $g(Rv)=1.$ Let $z\in\hat{\Phi}(X_{-})$ and
$A=\tilde{\Phi}^{-1}(z\otimes g)$. By Lemma 2.6(2),
$I-A\in\Omega_{1}(\mathcal{N},X_{-}).$ It follows from $\hat{\Phi}(X_{-})\leq
M$ that $\hat{\Phi}^{-1}(M)\leq X_{-}$. So we get
$\Phi(B(x\otimes f)(I-A))=\Phi([B,[x\otimes f,I-A]])=[u\otimes h,[R,-z\otimes
g]]=0,$
which implies that either $B(x\otimes f)=0$ or $(x\otimes f)(I-A)=0.$ If
$Bx\otimes f=0$, then we get $0=\Phi([B,x\otimes f])=[u\otimes h,R]=-Ru\otimes
h\not=0$, a contradiction. If $x\otimes f(I-A)=0$, then $0=\Phi([x\otimes
f,I-A])=[R,-z\otimes g]=z\otimes R^{*}g\not=0$, again a contradiction.
If $Y_{-}<Y$, there are two vectors $u,v\in Y$ such that $Ru$ and $Rv$ are
linearly independent. If $u,v\in Y_{-}$, we take $h\in Y_{-}^{\bot}$ and let
$B=\Phi^{-1}(u\otimes h).$ Then
$B\in\mathcal{J}(\mathcal{N},\hat{\Phi}^{-1}(Y_{-}))$. Choose $g\in Y^{*}$
such that $g(Ru)=0$ and $g(Rv)=1.$ Let $z\in\hat{\Phi}(X_{-})$ and
$A=\tilde{\Phi}^{-1}(z\otimes g)$. By Lemma 2.6(2),
$I-A\in\Omega_{1}(\mathcal{N},X_{-}).$ Still, by assumption, we have
$\hat{\Phi}(X_{-})<Y_{-}$, and so $\hat{\Phi}^{-1}(Y_{-})<X_{-}$. Thus by a
similar argument to that in the preceding paragraph, one can get a
contradiction. So we can assume that $u\not\in Y_{-}.$ Take $h\in
Y_{-}^{\bot}$ such that $h(u)=1$ and let $B=\tilde{\Phi}^{-1}(u\otimes h).$
Then $I-B\in\Omega_{2}(\mathcal{N},\hat{\Phi}^{-1}(Y_{-}))$ by Lemma 5.2(ii).
In addition, there exists some $g\in Y^{*}$ such that $g(Ru)=0$ and $g(Rv)=1.$
Let $z=Rv$ and $A=\tilde{\Phi}^{-1}(z\otimes g).$ Then, as
$z\in\hat{\Phi}(X_{-})=(0)_{+}$, we see that $z\otimes
g\in\Omega_{2}({\mathcal{M}},(0)_{+})$ and, by Lemma 2.6 and Lemma 5.2,
$I-A\in\Omega_{1}(\mathcal{N},X_{-}).$ Since $\hat{\Phi}(X_{-})<Y_{-}$, we
have $\hat{\Phi}^{-1}(Y_{-})<X_{-}$. Calculating $\Phi(B(x\otimes f)(I-A))$,
one can get a contradiction.
Hence ${\rm rank}R=1$ and $\Phi(x\otimes f)$ is the sum of a scalar and a rank
one operator, that is, the lemma is true for the case that $x\in X_{-}$ and
$f\in X_{-}^{\bot}$. Moreover, $\bar{\Phi}(X_{-}\otimes
X_{-}^{\bot})=\hat{\Phi}(X_{-})\otimes\hat{\Phi}(X_{-})^{\bot}$.
Case 2. $x\in X\setminus X_{-}$ and $f\in X_{-}^{\perp}$ with $\langle
x,f\rangle=1$.
Let $P=x\otimes f$. Clearly, $P\in\Omega_{1}(\mathcal{N},X_{-})$. By Case 1,
we have $\bar{\Phi}(X_{-}\otimes
X_{-}^{\bot})=\hat{\Phi}(X_{-})\otimes\hat{\Phi}(X_{-})^{\bot}$ and hence
${\Phi}(X_{-}\otimes
X_{-}^{\bot})\subseteq{\mathbb{F}}I+\hat{\Phi}(X_{-})\otimes\hat{\Phi}(X_{-})^{\bot}$.
For the seek of convenience, write $\tilde{P}=\tilde{\Phi}(P)$ and
$\widehat{X_{-}}=\hat{\Phi}(X_{-})$. Then $\Phi(P)-\tilde{P}\in{\mathbb{F}}I$
and it suffices to show that $\tilde{P}$ is the sum of a scalar and a rank-1
idempotent operator.
Subcase 2.1.
$\tilde{\Phi}(\Omega_{1}(\mathcal{N},X_{-}))\subseteq\Omega_{1}(\mathcal{M},\widehat{X_{-}})$.
In this subcase $\hat{\Phi}$ is order-preserving by Lemma 5.2(i), and hence
$\widehat{X_{-}}=Y_{-}$. Note that $\Phi({\mathbb{F}}I)={\mathbb{F}}I$ by
Lemma 2.9. So, applying the fact proved in Case 1, we have
$\Phi(\mathbb{F}I+X_{-}\otimes X_{-}^{\bot})=\mathbb{F}I+Y_{-}\otimes
Y_{-}^{\bot}.$ Thus it follows from Lemma 2.10 that there exists a ring
automorphism $\tau:\mathbb{F}\rightarrow\mathbb{F}$ and a map
$\gamma:X_{-}\times X_{-}^{\bot}\rightarrow\mathbb{F}$ such that either
$None$ $\Phi(y\otimes g)=\gamma(y,g)I+Cy\otimes Dg\ \ {\rm for\ all}\ \ y\in
X_{-}\ \ {\rm and}\ \ g\in X_{-}^{\bot},$
where $C:X_{-}\rightarrow Y_{-}$ and $D:X_{-}^{\bot}\rightarrow Y_{-}^{\bot}$
are two bijective $\tau$-linear maps; or
$None$ $\Phi(y\otimes g)=\gamma(y,g)I+Dg\otimes Cy\ \ {\rm for\ all}\ \ y\in
X_{-}\ \ {\rm and}\ \ g\in X_{-}^{\bot},$
where $C:X_{-}\rightarrow Y_{-}^{\bot}$ and $D:X_{-}^{\bot}\rightarrow Y_{-}$
are two bijective $\tau$-linear maps.
We first show that Eq.(5.3) can not occur. On the contrary, if Eq.(5.3) holds,
for any $y\in X_{-}$ and $g\in X_{-}^{\bot}$, we have
$\Phi([y\otimes g,P])=\Phi(y\otimes P^{*}g)=\gamma(y,P^{*}g)I+DP^{*}g\otimes
Cy$
and
$\Phi([y\otimes g,P])=[Dg\otimes Cy,\tilde{P}]=Dg\otimes\tilde{P}^{*}Cy,$
which imply $Dg\otimes\tilde{P}^{*}Cy=DP^{*}g\otimes Cy$ for all $y\in X_{-}$
and $g\in X_{-}^{\bot}$. Thus there exists some scalar $\lambda$ such that
$D|_{X_{-}^{\bot}}=\lambda DP^{*}|_{X_{-}^{\bot}}$. Since $P$ is of rank one,
we see that $D|_{X_{-}^{\bot}}$ is also of rank one, but this is impossible as
$X_{-}^{\bot}$ is infinite-dimensional.
So Eq.(5.2) holds. Then, for any $y\in X_{-}$ and $g\in X_{-}^{\bot}$, we have
$\Phi([y\otimes g,P])=\Phi(y\otimes P^{*}g)=\gamma(y,P^{*}g)I+Cy\otimes
DP^{*}g$
and
$\Phi([y\otimes g,P])=[Cy\otimes Dg,\tilde{P}]=Cy\otimes\tilde{P}^{*}Dg.$
It follows that $Cy\otimes\tilde{P}^{*}Dg=\gamma(y,P^{*}g)I+Cy\otimes DP^{*}g$
holds for all $y\in X_{-}$ and $g\in X_{-}^{\bot}.$ Since $I$ is of infinite
rank, we have $\gamma(y,P^{*}g)=0$, and $Cy\otimes\tilde{P}^{*}Dg=Cy\otimes
DP^{*}g$. So $\tilde{P}^{*}Dg=DP^{*}g$ for all $g\in X_{-}^{\bot}.$ Since $P$
is of rank one, it follows that the restriction of $\tilde{P}^{*}$ to
$Y_{-}^{\bot}$ is of rank one. Note that $\tilde{P}^{*}Y^{*}\subseteq
Y_{-}^{\bot}$ and
$\tilde{P}^{*}Y^{*}=\tilde{P}^{*}(\tilde{P}^{*}Y^{*})\subseteq\tilde{P}^{*}Y_{-}^{\bot}.$
So $\tilde{P}^{*}$ is of rank one, which implies that $\tilde{P}$ is also of
rank one, as desired.
Subcase 2.2. $\tilde{\Phi}(\Omega_{1}(\mathcal{N},X_{-}))\subseteq
I-\Omega_{2}(\mathcal{M},\widehat{X_{-}})$.
By Lemma 5.2(ii), $\hat{\Phi}$ is order-reversing, and
$\widehat{X_{-}}=(0)_{+}$. Applying the fact proved in Case 1, we have
$\Phi(\mathbb{F}I+X_{-}\otimes
X_{-}^{\bot})=\mathbb{F}I+(0)_{+}\otimes(0)_{+}^{\bot}.$ Thus, by Lemma 2.10
again, there exists a ring automorphism $\tau:\mathbb{F}\rightarrow\mathbb{F}$
and a map $\gamma:X_{-}\times X_{-}^{\bot}\rightarrow\mathbb{F}$ such that
either
$None$ $\Phi(x\otimes f)=\gamma(x,f)I+Cx\otimes Df\ \ {\rm for\ all}\ \ x\in
X_{-}\ \ {\rm and}\ \ f\in X_{-}^{\bot},$
where $C:X_{-}\rightarrow(0)_{+}$ and
$D:X_{-}^{\bot}\rightarrow(0)_{+}^{\bot}$ are two $\tau$-linear bijective
maps; or
$None$ $\Phi(x\otimes f)=\gamma(x,f)I+Df\otimes Cx\ \ {\rm for\ all}\ \ x\in
X_{-}\ \ {\rm and}\ \ f\in X_{-}^{\bot},$
where $C:X_{-}\rightarrow(0)_{+}^{\bot}$ and
$D:X_{-}^{\bot}\rightarrow(0)_{+}$ are two $\tau$-linear bijective maps.
If Eq.(5.4) holds, by calculating $\Phi([y\otimes g,P])$, one obtains
$(I-\tilde{P})Cy\otimes Dg=Cy\otimes DP^{*}g$ for all $y\in X_{-}$ and $g\in
X_{-}^{\bot}.$ Since $P$ is of rank one, we get $D$ is also of rank one, which
is impossible. So we must have that Eq.(5.5) holds. By calculating
$\Phi([y\otimes g,P])$, one can get $(I-\tilde{P})Dg=DP^{*}g$ for all $f\in
X_{-}^{\bot}$, which implies that the restriction of $I-\tilde{P}$ to
$(0)_{+}$ is of rank one. So $I-\tilde{P}$ is of rank one and $\tilde{P}$ is
the sum of a scalar and a rank one operator, as desired.
Summing up, we have proved that $\Phi(x\otimes f)=\Phi(P)$ is the sum of a
scalar and a rank one operator if $x\in X,f\in X_{-}^{\bot}$ with $\langle
x,f\rangle=1$. Moreover,
$\tilde{\Phi}(\Omega_{1}(\mathcal{N},X_{-}))\subseteq\Omega_{1}(\mathcal{M},\widehat{X_{-}})$
implies that Eq.(5.2) holds, while
$\tilde{\Phi}(\Omega_{1}(\mathcal{N},X_{-}))\subseteq
I-\Omega_{2}(\mathcal{M},\widehat{X_{-}})$ implies that Eq.(5.5) holds.
Case 3. $x\in X\setminus X_{-}$ and $f\in X_{-}^{\bot}.$
Now assume that $x\in X\setminus X_{-}$ and $f\in X_{-}^{\bot}.$ We need still
consider two subcases.
Subcase 3.1. $\langle x,f\rangle=\lambda\neq 0$.
Let $P=\lambda^{-1}x\otimes f$. Then the rank-one idempotent
$P\in\Omega_{1}(\mathcal{N},X_{-})$ and $x\otimes f=\lambda P.$ Write
$\tilde{P}=\tilde{\Phi}(P).$ By what proved in Case 2, $\tilde{P}$ is a rank-1
idempotent.
Subcase 3.1.1.
$\tilde{\Phi}(\Omega_{1}(\mathcal{N},X_{-}))\subseteq\Omega_{1}(\mathcal{M},\widehat{X_{-}})$.
By Lemma 5.2(i), $\hat{\Phi}$ is order-preserving, and then
$\widehat{X_{-}}=Y_{-}$. So we still have that either Eq.(5.2) or Eq.(5.3)
holds.
If Eq.(5.2) holds, then for any $y\in X_{-}$ and $g\in X_{-}^{\bot}$, we have
$\Phi([y\otimes g,\lambda P])=[\Phi(y\otimes g),\Phi(\lambda
P)]=Cy\otimes\Phi(\lambda P)^{*}Dg-\Phi(\lambda P)Cy\otimes Dg$
and
$\Phi([y\otimes g,\lambda P])=[\tau(\lambda)Cy\otimes
Dg,\tilde{P}]=\tau(\lambda)Cy\otimes\tilde{P}^{*}Dg.$
It follows that $\tau(\lambda)Cy\otimes\tilde{P}^{*}Dg=Cy\otimes\Phi(\lambda
P)^{*}Dg-\Phi(\lambda P)Cy\otimes Dg$, that is,
$Cy\otimes(\overline{\tau(\lambda)}\tilde{P}^{*}Dg-\Phi(\lambda
P)^{*}Dg)=-\Phi(\lambda P)Cy\otimes Dg\ \ {\rm for\ all}\ y\in X_{-}\ {\rm
and}\ g\in X_{-}^{\perp}.$
Since $C$ is bijective, there exists some scalar $\alpha$ such that
$\tau(\lambda)\tilde{P}^{*}Dg-\Phi(\lambda P)^{*}Dg=\alpha Dg$ for all $g\in
X_{-}^{\bot}$, that is, $(\Phi(\lambda P)^{*}+\alpha
I)|_{Y_{-}^{\bot}}=\tau(\lambda)\tilde{P}^{*}|_{Y_{-}^{\bot}}$ as
$D:X_{-}^{\bot}\rightarrow Y_{-}^{\bot}$ is bijective. Let
$\widetilde{\Phi(\lambda P)}=\Phi(\lambda P)+\overline{\alpha}I.$ Note that
$[\lambda P,A]=[\lambda P,[P,[P,A]]]$ holds for all $A\in{\rm
Alg}\mathcal{N}$. So we get
$None$ $[\widetilde{\Phi(\lambda P)},\Phi(A)]=[\widetilde{\Phi(\lambda
P)},[\tilde{P},[\tilde{P},\Phi(A)]]],\ \ \ \forall A\in{\rm Alg}\mathcal{N}.$
By the bijectivity of $\Phi$, for any $y\in Y$ and $g\in Y_{-}^{\bot}$, there
exists some $A\in{\rm Alg}{\mathcal{N}}$ such that $\Phi(A)=y\otimes g$. Thus
Eq.(5.6) entails
$None$ $(\widetilde{\Phi(\lambda P)}-\widetilde{\Phi(\lambda
P)}\tilde{P})y\otimes g=(\tau(\lambda)\tilde{P}+\widetilde{\Phi(\lambda
P)}-2\widetilde{\Phi(\lambda P)}\tilde{P})y\otimes\tilde{P}^{*}g$
for all $y\in Y$ and $g\in Y_{-}^{\bot}$. By Case 2, we can write
$\tilde{P}=u\otimes h$, where $u\in Y$ and $h\in Y_{-}^{\bot}$ with $\langle
u,h\rangle=1.$ Since $\dim(Y_{-}^{\bot})>2,$ there exists some $g_{1}\in
Y_{-}^{\bot}$ such that $g_{1}$ is linearly independent of $h$. If $\langle
u,g_{1}\rangle\neq 0,$ let $g=g_{1}.$ If $\langle u,g_{1}\rangle=0$, let
$g=h+g_{1}$. Then $g$ and $h$ are linearly independent and $\langle
u,g\rangle\neq 0.$ So $g$ and $\tilde{P}^{*}g$ are also linearly independent.
By Eq.(5.7), we get $(\widetilde{\Phi(\lambda P)}-\widetilde{\Phi(\lambda
P)}\tilde{P})y=0$ for all $y\in Y.$ It follows that $\widetilde{\Phi(\lambda
P)}=\widetilde{\Phi(\lambda P)}\tilde{P}$, which implies
$\widetilde{\Phi(\lambda P)}$ is of rank one. So $\Phi(\lambda
P)=\widetilde{\Phi(\lambda P)}-\bar{\alpha}I$ is the sum of a scalar and a
rank one operator.
We claim that Eq.(5.3) can not occur. If, on the contrary, Eq.(5.3) holds,
then for any $y\in X_{-}$ and $g\in X_{-}^{\bot}$, we have
$\Phi([y\otimes g,\lambda P])=[\Phi(y\otimes g),\Phi(\lambda
P)]=Dg\otimes\Phi(\lambda P)^{*}Cy-\Phi(\lambda P)Dg\otimes Cy$
and
$\Phi([y\otimes g,\lambda P])=\Phi(\lambda y\otimes
P^{*}g)=\tau(\lambda)D(P^{*}g)\otimes Cy.$
It follows that $Dg\otimes\Phi(\lambda P)^{*}Cy=(\Phi(\lambda
P)Dg+\tau(\lambda)D(P^{*}g))\otimes Cy$ for all $y\in X_{-}$ and $g\in
X_{-}^{\perp}.$ So there exists some scalar $\gamma$ such that $\Phi(\lambda
P)^{*}Cy=\gamma Cy$, which implies $\Phi(\lambda P)^{*}=\gamma I$ on
$Y_{-}^{\perp}$ as $C$ is bijective. Now, for any $A\in{\rm Alg}\mathcal{N}$,
by the relation $[\lambda P,A]=[\lambda P,[P,[P,A]]]$, we get $[{\Phi(\lambda
P)},\Phi(A)]=[{\Phi(\lambda P)},[\tilde{P},[\tilde{P},\Phi(A)]]]$.
Particularly, for any $y\in Y$ and $g\in Y_{-}^{\bot}$, there exists some
$A\in{\rm Alg}{\mathcal{N}}$ such that $\Phi(A)=y\otimes g$. Thus we have
$[{\Phi(\lambda P)},y\otimes g]=[{\Phi(\lambda
P)},[\tilde{P},[\tilde{P},y\otimes g]]]$, that is,
$({\Phi(\lambda P)}-{\Phi(\lambda P)}\tilde{P}-\gamma
I+\gamma\tilde{P})y\otimes g=({\Phi(\lambda P)}-2{\Phi(\lambda
P)}\tilde{P}+2\gamma\tilde{P}-\gamma I)y\otimes\tilde{P}^{*}g$
holds for all $y\in Y$ and $g\in Y_{-}^{\bot}$. Still, we can choose $g$ such
that $g$ and $\tilde{P}^{*}g$ are linearly independent. The above equation
yields $({\Phi(\lambda P)}-{\Phi(\lambda P)}\tilde{P}-\gamma
I+\gamma\tilde{P})y=({\Phi(\lambda P)}-2{\Phi(\lambda
P)}\tilde{P}+2\gamma\tilde{P}-\gamma I)y=0$ for all $y\in Y$. This implies
${\Phi(\lambda P)}=\gamma I$, which is contradicting to
$\Phi({\mathbb{F}}I)={\mathbb{F}}I$ and the bijectivity of $\Phi$. So Eq.(5.3)
can not occur.
Subcase 3.1.2. $\tilde{\Phi}(\Omega_{1}(\mathcal{N},X_{-}))\subseteq
I-\Omega_{2}(\mathcal{M},\widehat{X_{-}})$
By Lemma 5.2(ii), $\hat{\Phi}$ is order-reversing and thus
$\widehat{X_{-}}=(0)_{+}$. It follows that either Eq.(5.4) or Eq.(5.5) holds.
By a similar argument to the Subcase 3.1.1 above, one can check that, Eq.(5.4)
can not occur and if Eq.(5.5) holds, then $\Phi(x\otimes f)=\Phi(\lambda P)$
is the sum of a scalar and a rank one operator.
Subcase 3.2. $\langle x,f\rangle=0$.
In this case, take $x_{1}\in X$ such that $\langle x_{1},f\rangle=1$ and let
$x_{2}=x-2x_{1}$, $x_{3}=x-x_{1}.$ Then $\langle x_{i},f\rangle\neq 0$ for
$i=1,2,3$. By Subcase 3.1, $\Phi(x_{i}\otimes f)$ is the sum of a scalar and a
rank one operator. So we can assume $\Phi(x_{i}\otimes
f)=\lambda_{i}I+u_{i}\otimes h_{i}$ for some $\lambda_{i}$, $i=1,2,3.$ Note
that
$(\lambda_{1}+\lambda_{2})I+u_{1}\otimes h_{1}+u_{2}\otimes
h_{2}=\Phi((x_{1}+x_{2})\otimes f)=\Phi(x_{3}\otimes
f)=\lambda_{3}I+u_{3}\otimes h_{3}.$
Since $I$ is of infinite rank, we must have $u_{1}\otimes h_{1}+u_{2}\otimes
h_{2}=u_{3}\otimes h_{3}$. This forces that $\\{u_{1},u_{3}\\}$ or
$\\{h_{1},h_{3}\\}$ is a linearly dependent set. So $u_{1}\otimes
h_{1}+u_{3}\otimes h_{3}$ is of rank one. Thus $\Phi(x\otimes
f)=\Phi((x_{1}+x_{3})\otimes f)=(\lambda_{1}+\lambda_{3})I+u_{1}\otimes
h_{1}+u_{3}\otimes h_{3}$ is the sum of a scalar and a rank one operator.
Combining Cases 1-3 and the bijectivity of $\Phi$, we have shown that
(i) $\Phi({\mathbb{F}}I+X\otimes X_{-}^{\bot})={\mathbb{F}}I+Y\otimes
Y_{-}^{\bot}$ if
$\tilde{\Phi}(\Omega_{1}(\mathcal{N},X_{-}))\subseteq\Omega_{1}(\mathcal{M},\widehat{X_{-}})$;
(ii) $\Phi({\mathbb{F}}I+X\otimes X_{-}^{\bot})={\mathbb{F}}I+(0)_{+}\otimes
Y^{*}$ if $\tilde{\Phi}(\Omega_{1}(\mathcal{N},X_{-}))\subseteq
I-\Omega_{2}(\mathcal{M},\widehat{X_{-}})$.
So Lemma 2.10 is applicable. Observing the arguments in Case 2 and Case 3, it
is easily seen that (i) implies Lemma 5.1(1) and (ii) implies Lemma 5.1(2).
The proof of Lemma 5.3 is finished. $\Box$
For the case that the nest has only one nontrivial element, we have
Lemma 5.4. Assume that $\mathcal{N}=\\{(0),X_{-},X\\}$ and $X_{-}$ is not
complemented in $X$. Then for any $x\in X$ and $f\in X_{-}^{\bot}$,
${\Phi}(x\otimes f)$ is the sum of a scalar and a rank one operator. Moreover,
the statements (1) and (2) of Lemma 5.1 hold.
Proof. Not that $(0)_{+}=X_{-}$ in this situation. By Lemma 2.3,
${\mathcal{M}}=\\{(0),Y_{-},Y\\}$ and $(0)_{+}=Y_{-}$ is not complemented in
$Y$. Let $R\in\mathcal{J}(\mathcal{M},Y_{-})$ so that ${\Phi}(x\otimes
f)-R\in{\mathbb{F}}I$ (Lemma 2.2). We remak here that, since $\mathcal{N}$
only contains one nontrivial element, Lemma 5.2 and Lemma 2.6(3) are not
applicable. Similar to the proof of Lemma 5.3, we consider three cases.
Case 1. $x\in X_{-}$ and $f\in X_{-}^{\bot}$.
In this case, $x\otimes f\in{\mathcal{J}}({\mathcal{N}},X_{-})$. We’ll prove
${\rm rank}R=1$. Assume on the contrary that ${\rm rank}R\geq 2$. Then there
are two vectors $u,v\in Y\setminus Y_{-}$ such that $Ru$ and $Rv$ are linearly
independent. Take $h\in Y_{-}^{\bot}$ such that $h(u)=1$. Then $u\otimes
h\in\Omega_{1}(\mathcal{M},Y_{-})$. Note that $Y_{-}$ is infinite-dimensional.
There is a vector $z\in Y_{-}$ and a functional $g\in Y^{*}$ such that
$g(Ru)=0$ and $g(z)=g(Rv)=1$. Let $A=\tilde{\Phi}^{-1}(z\otimes g)$ and
$B=\tilde{\Phi}^{-1}(u\otimes h)$.
By Lemma 2.6(1), either
$\tilde{\Phi}(\Omega_{1}(\mathcal{N},X_{-}))\subseteq\Omega_{1}(\mathcal{M},Y_{-})$
or
$I-\tilde{\Phi}(\Omega_{1}(\mathcal{N},X_{-}))\subseteq\Omega_{2}(\mathcal{M},Y_{-})$.
If
$\tilde{\Phi}(\Omega_{1}(\mathcal{N},X_{-}))\subseteq\Omega_{1}(\mathcal{M},Y_{-})$,
then $B\in\Omega_{1}({\mathcal{N}},X_{-})$, and by Lemma 2.6(2),
$A\in\Omega_{2}(\mathcal{N},X_{-})$. Thus we get $\Phi(A(x\otimes
f)B)=\Phi([A,[x\otimes f,B]])=[z\otimes g,[R,u\otimes h]]=(z\otimes
g)R(u\otimes h)=0$, which implies either $A(x\otimes f)=0$ or $(x\otimes
f)B=0.$ If $A(x\otimes f)=0,$ then $0=\Phi([A,x\otimes f])=[z\otimes
g,R]=(z\otimes g)R,$ which is impossible; if $(x\otimes f)B=0,$ then
$0=\Phi([x\otimes f,B])=[R,u\otimes h]=R(u\otimes h),$ which is also
impossible. If
$I-\tilde{\Phi}(\Omega_{1}(\mathcal{N},X_{-}))\subseteq\Omega_{2}(\mathcal{M},Y_{-})$,
then $I-B\in\Omega_{2}(\mathcal{N},X_{-})$, and by Lemma 2.6(2),
$I-A\in\Omega_{1}(\mathcal{N},X_{-}).$ Thus $\Phi((I-B)(x\otimes
f)(I-A))=\Phi([I-B,[x\otimes f,I-A]])=[-u\otimes h,[R,-z\otimes g]]=0$. So we
get either $(I-B)(x\otimes f)=0$ or $(x\otimes f)(I-A)=0.$ Still, this is
impossible.
Hence ${\rm rank}R=1$.
Case 2. $x\in X\setminus X_{-}$ and $f\in X_{-}^{\perp}$ with $\langle
x,f\rangle=1$.
In this case $P=x\otimes f\in\Omega_{1}(\mathcal{N},X_{-})$. Write
$\tilde{P}=\tilde{\Phi}(P)$. By Case 1 and Lemma 2.9, we have
$\Phi(\mathbb{F}I+X_{-}\otimes X_{-}^{\bot})=\mathbb{F}I+Y_{-}\otimes
Y_{-}^{\bot}.$ So, by Lemma 2.10, there exists a ring automorphism
$\tau:\mathbb{F}\rightarrow\mathbb{F}$ and a map $\gamma:X_{-}\times
X_{-}^{\bot}\rightarrow\mathbb{F}$ such that either
$None$ $\Phi(y\otimes g)=\gamma(y,g)I+Cy\otimes Dg\ \ {\rm for\ all}\ \ y\in
X_{-}\ \ {\rm and}\ \ g\in X_{-}^{\bot},$
where $C:X_{-}\rightarrow Y_{-}$ and $D:X_{-}^{\bot}\rightarrow Y_{-}^{\bot}$
are two $\tau$-linear bijective maps; or
$None$ $\Phi(y\otimes g)=\gamma(y,g)I+Dg\otimes Cy\ \ {\rm for\ all}\ \ y\in
X_{-}\ \ {\rm and}\ \ g\in X_{-}^{\bot},$
where $C:X_{-}\rightarrow Y_{-}^{\bot}$ and $D:X_{-}^{\bot}\rightarrow Y_{-}$
are two $\tau$-linear bijective maps.
Assume first that Eq.(5.8) holds. If
$\tilde{\Phi}(\Omega_{1}(\mathcal{N},X_{-}))\subseteq\Omega_{1}(\mathcal{M},Y_{-})$,
then $\tilde{P}\in\Omega_{1}({\mathcal{M}},Y_{-})$. Thus, for any $y\in Y$ and
any $g\in Y_{-}^{\perp}$, we have
$\Phi([y\otimes g,P])=\Phi(y\otimes P^{*}g)=\gamma(y,P^{*}g)I+Cy\otimes
DP^{*}g$
and
$\Phi([y\otimes g,P])=[Cy\otimes Dg,\tilde{P}]=Cy\otimes\tilde{P}^{*}Dg.$
Since $I$ is of infinite rank, the above two equations yields
$Cy\otimes\tilde{P}^{*}Dg=Cy\otimes DP^{*}g$, and so $\tilde{P}^{*}Dg=DP^{*}g$
for all $g\in X_{-}^{\bot}.$ Since $P$ is of rank one, the restriction of
$\tilde{P}^{*}$ to $Y_{-}^{\bot}$ is of rank one. Note that
$\tilde{P}^{*}Y^{*}\subseteq Y_{-}^{\bot}$ and
$\tilde{P}^{*}Y^{*}=\tilde{P}^{*}(\tilde{P}^{*}Y^{*})\subseteq\tilde{P}^{*}Y_{-}^{\bot}.$
So $\tilde{P}^{*}$ is of rank one, which implies that $\tilde{P}$ is also of
rank one. Hence $\Phi(P)$ is the sum of a scalar and a rank one operator.
If
$I-\tilde{\Phi}(\Omega_{1}(\mathcal{N},X_{-}))\subseteq\Omega_{2}(\mathcal{M},Y_{-})$,
then $I-\tilde{P}\in\Omega_{2}(\mathcal{M},Y_{-})$. For any $y\in Y$ and any
$g\in Y_{-}^{\perp}$, we have
$\Phi([y\otimes g,P])=\Phi(y\otimes P^{*}g)=\gamma(y,P^{*}g)I+Cy\otimes
DP^{*}g$
and
$\Phi([y\otimes g,P])=[Cy\otimes Dg,\tilde{P}]=-[Cy\otimes
Dg,I-\tilde{P}]=(I-\tilde{P})Cy\otimes Dg.$
The above two equations yield $(I-\tilde{P})Cy\otimes Dg=Cy\otimes DP^{*}g$
for all $y\in X_{-}$ and $g\in X_{-}^{\bot}$, and hence $D$ and $DP^{*}$ are
linearly dependent. Since $P$ is of rank one, $D$ is also of rank one, which
is impossible as $X_{-}^{\bot}$ is infinite-dimensional. Therefore, this case
can not occur.
Similarly, if Eq.(5.9) holds, one can show that
$I-\tilde{\Phi}(\Omega_{1}(\mathcal{N},X_{-}))\subseteq\Omega_{2}(\mathcal{M},Y_{-})$
and $\Phi(P)$ is the sum of a scalar and a rank one operator.
Case 3. $x\in X\setminus X_{-}$ and $f\in X_{-}^{\bot}.$
Note that, we still have that either Eq.(5.8) or Eq.(5.9) holds.
Subcase 3.1. $\langle x,f\rangle=\lambda\neq 0$.
Then there exists a rank-one idempotent $P\in\Omega_{1}(\mathcal{N},X_{-})$
such that $x\otimes f=\lambda P.$ Write $\tilde{P}=\tilde{\Phi}(P)$. By Case
2, $\tilde{P}$ is a rank-1 idempotent.
Assume that Eq.(5.8) holds. If
$\tilde{\Phi}(\Omega_{1}(\mathcal{N},X_{-}))\subseteq\Omega_{1}(\mathcal{M},Y_{-})$,
then $\tilde{P}\in\Omega_{1}({\mathcal{M}},Y_{-})$. So, for any $y\in X_{-}$
and $g\in X_{-}^{\bot}$, we have
$\Phi([y\otimes g,\lambda P])=[\Phi(y\otimes g),\Phi(\lambda
P)]=Cy\otimes\Phi(\lambda P)^{*}Dg-\Phi(\lambda P)Cy\otimes Dg$
and
$\Phi([y\otimes g,\lambda P])=\Phi([\lambda y\otimes
g,P])=[\tau(\lambda)Cy\otimes
Dg,\tilde{P}]=\tau(\lambda)Cy\otimes\tilde{P}^{*}Dg.$
It follows that
$Cy\otimes(\overline{\tau(\lambda)}\tilde{P}^{*}Dg-\Phi(\lambda
P)^{*}Dg)=-\Phi(\lambda P)Cy\otimes Dg\ {\rm for\ all}\ y\in X_{-}\ {\rm and}\
g\in X_{-}^{\perp}.$
Hence there exists some scalar $\alpha$ such that
$\tau(\lambda)\tilde{P}^{*}Dg-\Phi(\lambda P)^{*}Dg=\alpha Dg$ for all $g\in
X_{-}^{\bot}$, which implies that $(\Phi(\lambda P)^{*}+\alpha
I)|_{Y_{-}^{\bot}}=\tau(\lambda)\tilde{P}^{*}|_{Y_{-}^{\bot}}$ as $D$ is
bijective. Let $\widetilde{\Phi(\lambda P)}=\Phi(\lambda
P)+\overline{\alpha}I.$ For any $A\in{\rm Alg}\mathcal{N}$, by the relation
$[\lambda P,A]=[\lambda P,[P,[P,A]]]$, we get $[\widetilde{\Phi(\lambda
P)},\Phi(A)]=[\widetilde{\Phi(\lambda P)},[\tilde{P},[\tilde{P},\Phi(A)]]]$.
Particularly, for any $y\in Y$ and $g\in Y_{-}^{\bot}$, By the bijectivity of
$\Phi$, there exists some $A\in{\rm Alg}{\mathcal{N}}$ such that
$\Phi(A)=y\otimes g$. So we have $[\widetilde{\Phi(\lambda P)},y\otimes
g]=[\widetilde{\Phi(\lambda P)},[\tilde{P},[\tilde{P},y\otimes g]]]$, that is,
$None$ $(\widetilde{\Phi(\lambda P)}-\widetilde{\Phi(\lambda
P)}\tilde{P})y\otimes g=(\tau(\lambda)\tilde{P}+\widetilde{\Phi(\lambda
P)}-2\widetilde{\Phi(\lambda P)}\tilde{P})y\otimes\tilde{P}^{*}g$
holds for all $y\in Y$ and $g\in Y_{-}^{\bot}$. By Case 2, we can write
$\tilde{P}=u\otimes h$, where $u\in Y$ and $h\in Y_{-}^{\bot}$ with $\langle
u,h\rangle=1.$ Since $\dim(Y_{-}^{\bot})>2,$ there exists $g$ with $\langle
u,g\rangle\neq 0$ such that $g$ and $h$ are linearly independent. So $g$ and
$\tilde{P}^{*}g$ are also linearly independent. By Eq.(5.10), we get
$(\widetilde{\Phi(\lambda P)}-\widetilde{\Phi(\lambda P)}\tilde{P})y=0$ for
all $y\in Y$. So $\widetilde{\Phi(\lambda P)}=\widetilde{\Phi(\lambda
P)}\tilde{P}$, which implies $\widetilde{\Phi(\lambda P)}$ is of rank one. So
$\Phi(\lambda P)=\widetilde{\Phi(\lambda P)}-\alpha I$ is the sum of a scalar
and a rank one operator.
We claim that the case
$I-\tilde{\Phi}(\Omega_{1}(\mathcal{N},X_{-}))\subseteq\Omega_{2}(\mathcal{M},Y_{-})$
does not happen. Otherwise, we have
$I-\tilde{P}\in\Omega_{2}(\mathcal{M},Y_{-})$. Then for any $y\in X_{-}$ and
$g\in X_{-}^{\bot}$, by calculating $\Phi([y\otimes g,\lambda P])$, one can
obtain
$Cy\otimes\Phi(\lambda P)^{*}Dg=(\Phi(\lambda
P)-\tau(\lambda)(I-\tilde{P}))Cy\otimes Dg\ {\rm for\ all}\ y\in X_{-}\ {\rm
and}\ g\in X_{-}^{\perp}.$
It follows that there exists some scalar $\gamma$ such that $\Phi(\lambda
P)^{*}Dg=\gamma Dg$, which implies $\Phi(\lambda P)^{*}=\gamma I$ on
$Y_{-}^{\perp}$ as $D$ is bijective. Now, for any $A\in{\rm Alg}\mathcal{N}$,
by the relation $[\lambda P,A]=[\lambda P,[P,[P,A]]]$, we get $[{\Phi(\lambda
P)},\Phi(A)]=[{\Phi(\lambda P)},[\tilde{P},[\tilde{P},\Phi(A)]]]$.
Particularly, for any $y\in Y$ and $g\in Y_{-}^{\bot}$, there exists some
$A\in{\rm Alg}{\mathcal{N}}$ such that $\Phi(A)=y\otimes g$. So we have
$[{\Phi(\lambda P)},y\otimes g]=[{\Phi(\lambda
P)},[\tilde{P},[\tilde{P},y\otimes g]]]$, that is,
$None$ $({\Phi(\lambda P)}-{\Phi(\lambda P)}\tilde{P}-\gamma
I+\gamma\tilde{P})y\otimes g=({\Phi(\lambda P)}-2{\Phi(\lambda
P)}\tilde{P}+2\gamma\tilde{P}-\gamma I)y\otimes\tilde{P}^{*}g$
holds for all $y\in Y$ and $g\in Y_{-}^{\bot}$. Still, we can choose $g$ such
that $g$ and $\tilde{P}^{*}g$ are linearly independent. By Eq.(5.11), we get
$({\Phi(\lambda P)}-{\Phi(\lambda P)}\tilde{P}-\gamma
I+\gamma\tilde{P})y=({\Phi(\lambda P)}-2{\Phi(\lambda
P)}\tilde{P}+2\gamma\tilde{P}-\gamma I)y=0$ for all $y\in Y$. This leads to a
contradiction ${\Phi(\lambda P)}=\gamma I$.
If Eq.(5.9) holds, by a similar argument to that of the above, one can show
that
$I-\tilde{\Phi}(\Omega_{1}(\mathcal{N},X_{-}))\subseteq\Omega_{2}(\mathcal{M},Y_{-})$
and $\Phi(\lambda P)$ is also the sum of a scalar and a rank one operator.
Subcase 3.2. $\langle x,f\rangle=0$.
The proof is the same to that of Subcase 3.2 in Lemma 5.3. We omit it here.
Combining Cases 1-3, we see that the statements (i) and (ii) in the proof of
Lemma 5.3 still hold, and this completes the proof of Lemma 5.4. $\Box$
Proof of Lemma 5.1. It is immediate from Lemma 5.3 and Lemma 5.4. $\Box$
Proof of Theorem 1.1 for the case $X_{-}\not=X$ and $X_{-}$ is not
complemented.
Now let us show that Theorem 1.1 is true for the case that $X_{-}\not=X$ and
$X_{-}$ is not complemented. By Lemma 5.1, (1) or (2) holds.
If Lemma 5.1(1) holds, then $\Phi(\mathbb{F}I+X\otimes
X_{-}^{\bot})=\mathbb{F}I+Y\otimes Y_{-}^{\bot}$, and there exists a ring
automorphism $\tau:\mathbb{F}\rightarrow\mathbb{F}$ and a map $\gamma:X\otimes
X_{-}^{\bot}\rightarrow\mathbb{F}$ such that
$None$ $\Phi(x\otimes f)=\gamma(x,f)I+Cx\otimes Df$
holds for all $x\in X$ and $f\in X_{-}^{\bot}$, where $C:X\rightarrow Y$ and
$D:X_{-}^{\bot}\rightarrow Y_{-}^{\bot}$ are two $\tau$-linear bijective maps.
Thus for any $A\in{\rm Alg}{\mathcal{N}}$, any $x\in X$ and any $f\in
X_{-}^{\perp}$, as $\Phi([A,x\otimes f])=[\Phi(A),\Phi(x\otimes f)]$ and $I$
is of infinite rank, one obtains that
$Cx\otimes(\Phi(A)^{*}Df-DA^{*}f)=(\Phi(A)Cx-CAx)\otimes Df.$
As $D$ is injective, the above equation entails that there exists a scalar
$h(A)$ such that
$None$ $\Phi(A)C=CA+h(A)C.$
It is clear that $h$ is an additive functional on Alg${\mathcal{N}}$. Define
$\Psi:{\rm Alg}{\mathcal{N}}\rightarrow{\rm Alg}{\mathcal{M}}$ by
$\Psi(A)=\Phi(A)-h(A)I$ for all $A\in{\rm Alg}{\mathcal{N}}$. Then, $\Psi$ is
an additive bijection. By Eq.(5.13), for any $A,B\in{\rm Alg}{\mathcal{N}}$,
we have
$\Psi(AB)C=CAB=\Psi(A)CB=\Psi(A)\Psi(B)C.$
Since ran$C=Y$, we see that $\Psi(AB)=\Psi(A)\Psi(B)$ for all $A,B\in{\rm
Alg}\mathcal{N}$, that is, $\Psi$ is a ring isomorphism.
Assume that Lemma 5.1(2) holds. Then $\Phi(\mathbb{F}I+X\otimes
X_{-}^{\bot})=\mathbb{F}I+(0)_{+}\otimes Y^{*}$ and there exists a ring
automorphism $\tau:\mathbb{F}\rightarrow\mathbb{F}$ and a map $\gamma:X\otimes
X_{-}^{\bot}\rightarrow\mathbb{F}$ such that
$None$ $\Phi(x\otimes f)=\gamma(x,f)I+Df\otimes Cx$
holds for all $x\in X$ and $f\in X_{-}^{\bot}$, where $C:X\rightarrow Y^{*}$
and $D:X_{-}^{\bot}\rightarrow(0)_{+}$ are two $\tau$-linear bijective maps. A
similar argument to the above, one can check that there exists an additive
functional $h$ and a ring anti-isomorphism $\Psi$ such that
$\Phi(A)=-\Psi(A)+h(A)I$ for all $A\in{\rm Alg}\mathcal{N}$.
This completes the proof of Theorem 1.1 for the case that $(0)<X_{-}<X$ and
$X_{-}$ is not complemented. $\Box$
The proof of Theorem 1.1 for the case that $(0)<(0)_{+}<X$ and $(0)_{+}$ is
not complemented is similar and we omit it here.
Now, combining Sections 3-5, the proof of Theorem 1.1 is finished.
## References
* [1] Z. F. Bai, S. P. Du, J. C. Hou, Multiplicative Lie isomorphisms between prime rings, Communications in Algebras, 36 (2008), 1626-1633.
* [2] Z. F. Bai, J. C. Hou, S. P. Du, Additive maps preserving rank-one operators on nest algebras, Lin. Multi. Algebra, 58(3) (2010), 269-283.
* [3] K. I. Beidar, W. S. Martindale III and A.V. Mikhalev, Lie isomorphisms in prime rings with involution, J. Algebra, 169 (1994), 304-327.
* [4] D. Benkovi$\check{c}$, D. Eremita, Commuting traces and commutativity preserving maps on triangular algebras, J. Algebra, 280 (2004), 797-824.
* [5] M. Bresar, Commuting traces of biadditive mappings, commutativity preserving mappings, and Lie mappings, Trans. Amer. Math. Soc., 335 (1993), 525-546.
* [6] K. R. Davidson, Nest Algebras, Pitman Research Notes in Mathematics, vol. 191, Longman, London, New York, 1988.
* [7] J. C. Hou, X. L. Zhang, Ring isomorphisms and linear or additive maps preserving zero products on nest algebras, Linear Algebra and its Applications, 387 (2004), 343-360.
* [8] W.E. Longstaff, Operator of rank one in reflexive algebras, Canad. J. Math. 28 (1976) 19-23.
* [9] L. W. Marcoux, Ahmed R. Sourour, Lie isomorphisms of nest algebras, J. Func. Anal., 164 (1999), 163-180.
* [10] W. S. Martindale III, Lie isomorphisms of prime rings, Trans. Amer. Math. Soc., 142 (1969), 437-455.
* [11] X. Qi, J. Hou, Characterization of Lie multiplicative isomorphisms between nest algebras, Sci China Math, 54(11) (2011): 2453-2462.
* [12] T. Wang, F. Lu, Lie isomorphisms of nest algebras on Banaha space, J. Math. Anal. Appl., 391 (2012), 582-594.
|
arxiv-papers
| 2013-01-31T09:07:02 |
2024-09-04T02:49:41.078527
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiaofei Qi, Jinchuan Hou, Juan Deng",
"submitter": "Jinchuan Hou",
"url": "https://arxiv.org/abs/1301.7553"
}
|
1301.7558
|
# Optimality of a class of entanglement witnesses for $3\otimes 3$ systems
Xiaofei Qi Department of Mathematics, Shanxi University , Taiyuan 030006, P.
R. China; [email protected] and Jinchuan Hou Department of
Mathematics
Taiyuan University of Technology
Taiyuan 030024, P. R. China [email protected]
###### Abstract.
Let $\Phi_{t,\pi}:M_{3}({\mathbb{C}})\rightarrow M_{3}({\mathbb{C}})$ be a
linear map defined by
$\Phi_{t,\pi}(A)=(3-t)\sum_{i=1}^{3}E_{ii}AE_{ii}+t\sum_{i=1}^{3}E_{i,\pi(i)}AE_{i,\pi(i)}^{\dagger}-A$,
where $0\leq t\leq 3$ and $\pi$ is a permutation of $(1,2,3)$. We show that
the Hermitian matrix $W_{\Phi_{t,\pi}}$ induced by $\Phi_{t,\pi}$ is an
optimal entanglement witness if and only if $t=1$ and $\pi$ is cyclic.
PACS. 03.67.Mn, 03.65.Ud, 03.65.Db
Key words and phrases. Quantum states, optimal entanglement witness, positive
linear maps, permutations
This work is partially supported by a grant from International Cooperation
Program in Sciences and Technology of Shanxi (2011081039), Natural Science
Foundation of China (11171249, 11101250) and Youth Foundation of Shanxi
Province (2012021004).
## 1\. Introduction
Let $H$ be a separable complex Hilbert space. Recall that a quantum state on
$H$ is a density operator $\rho\in{\mathcal{B}}(H)$ which is positive and has
trace 1. Denote by ${\mathcal{S}}(H)$ the set of all states on $H$. If $H$ and
$K$ are finite dimensional, a state in the bipartite composition system
$\rho\in{\mathcal{S}}(H\otimes K)$ is said to be separable if $\rho$ can be
written as $\rho=\sum_{i=1}^{k}p_{i}\rho_{i}\otimes\sigma_{i},$ where
$\rho_{i}$ and $\sigma_{i}$ are states on $H$ and $K$ respectively, and
$p_{i}$ are positive numbers with $\sum_{i=1}^{k}p_{i}=1$. Otherwise, $\rho$
is entangled.
Entanglement is an important physical resource to realize various quantum
information and quantum communication tasks such as teleportation, dense
coding, quantum cryptography and key distribution [10, 11]. It is very
important but also difficult to determine whether or not a state in a
composite system is separable. One of the most general approaches to
characterize quantum entanglement for bipartite composition systems is based
on the notion of entanglement witnesses (see [4]). A Hermitian matrix $W$
acting on $H\otimes K$ is an entanglement witness (briefly, EW) if $W$ is not
positive and ${\rm Tr}(W\sigma)\geq 0$ holds for all separable states
$\sigma$. Thus, if $W$ is an EW, then there exists an entangled state $\rho$
such that ${\rm Tr}(W\rho)<0$ (that is, the entanglement of $\rho$ can be
detected by $W$). It was shown that, a state is entangled if and only if it is
detected by some entanglement witness [4]. Constructing entanglement witnesses
is a hard task, too. There was a considerable effort in constructing and
analyzing the structure of entanglement witnesses [1, 3, 7, 8, 15]. However,
complete characterization and classification of EWs is far from satisfactory.
Due to the Choi-Jamiołkowski isomorphism [2, 9], a Hermitian matrix
$W\in{\mathcal{B}}(H\otimes K)$ with $\dim H\otimes K<\infty$ is an EW if and
only if there exists a positive linear map which is not completely positive
(NCP) $\Phi:{\mathcal{B}}(H)\rightarrow{\mathcal{B}}(K)$ and a maximally
entangled state $P^{+}\in{\mathcal{B}}(H\otimes H)$ such that
$W=W_{\Phi}=(I_{n}\otimes\Phi)P^{+}$. Recall that a maximally entangled state
is a pure state $P^{+}=|\psi^{+}\rangle\langle\psi^{+}|$ with
$|\psi^{+}\rangle=\frac{1}{\sqrt{n}}(|11\rangle+|22\rangle+\cdots|nn\rangle)$,
where $n=\dim H$ and $\\{|i\rangle\\}_{i=1}^{n}$ is an orthonormal basis of
$H$. Thus, up to a multiple by positive scalar, $W_{\Phi}$ can be written as
the matrix $W_{\Phi}=(\Phi(E_{ij}))_{n\times n}$, where
$E_{ij}=|i\rangle\langle j|$. For a positive linear map
$\Phi:{\mathcal{B}}(H)\rightarrow{\mathcal{B}}(K)$, we always denote
$W_{\Phi}$ the Choi-Jamiołkowski matrix of $\Phi$ with respect to a given
basis of $H$, that is $W_{\Phi}=(\Phi(E_{ij}))_{n\times n}$, and we say that
$W_{\Phi}$ is the witness induced by the positive map $\Phi$. Conversely, for
an EW $W$, we denote $\Phi_{W}$ for the associated positive map so that
$W=W_{\Phi_{W}}$.
For any entanglement witness $W$, let
$\mathcal{D}_{W}=\\{\rho:\rho\in{\mathcal{S}}(H\otimes K),{\rm
Tr}(W\rho)<0\\},$ that is, ${\mathcal{D}}_{W}$ is the set of all entangled
states that detected by $W$. For entanglement witnesses $W_{1},W_{2}$, we say
that $W_{1}$ is finer than $W_{2}$ if
$\mathcal{D}_{W_{2}}\subset\mathcal{D}_{W_{1}}$, denoted by $W_{2}\prec
W_{1}$. While, an entanglement witness $W$ is optimal if there exists no other
witness finer than it. Obviously, a state $\rho$ is entangled if and only if
there is some optimal EW such that ${\rm Tr}(W\rho)<0$. In [10], Lewenstein,
Kraus, Cirac and Horodecki proved that: (1) $W$ is an optimal entanglement
witness if and only if $W-Q$ is no longer an entanglement witness for
arbitrary positive operator $Q$; (2) $W$ is optimal if
${\mathcal{P}}_{W}=\\{|e,f\rangle\in H\otimes K:\langle e,f|W|e,f\rangle=0\\}$
spans the whole $H\otimes K$ (in this case, we say that $W$ has spanning
property). However, the criterion (2) is only a sufficient condition. There
are known optimal witnesses that have no spanning property, for example, the
entanglement witnesses induced by the Choi maps. Recently, Qi and Hou in [12]
gave a necessary and sufficient condition for the optimality of entanglement
witnesses in terms of positive linear maps.
Theorem 1.1. ([12, Theorem 2.2]) Let $H$ and $K$ be finite dimensional complex
Hilbert spaces. Let $\Phi:{\mathcal{B}}(H)\rightarrow{\mathcal{B}}(K)$ be a
positive linear map. Then $W_{\Phi}$ is an optimal entanglement witness if and
only if, for any $C\in{\mathcal{B}}(H,K)$, the map
$X\mapsto\Phi(X)-CXC^{\dagger}$ is not a positive map.
This approach is practical for some situations, especially when the witnesses
have no spanning property. Applying it, Qi and Hou [12] showed that the
entanglement witnesses arising from some positive maps in [13] are
indecomposable optimal witnesses.
If $\dim H=n$, by fixing an orthonormal basis, one may identify
${\mathcal{B}}(H)$ with $M_{n}({\mathbb{C}})$, the $n\times n$ complex matrix
algebra. In this note, we will consider the linear maps $\Phi_{t,\pi}$ defined
by
$None$
$\Phi_{t,\pi}(X)={\small\left(\begin{array}[]{ccc}(2-t)x_{11}+tx_{\pi(1),\pi(1)}&-x_{12}&-x_{13}\\\
-x_{21}&(2-t)x_{22}+tx_{\pi(2),\pi(2)}&-x_{23}\\\
-x_{31}&-x_{32}&(2-t)x_{33}+tx_{\pi(3),\pi(3)}\end{array}\right)},$
where $X=(x_{ij})\in M_{3}({\mathbb{C}})$, $0\leq t\leq 3$ and $\pi$ is any
permutation of $(1,2,3)$. We will show that the necessary and sufficient
condition for the Hermitian matrix $W_{\Phi_{t,\pi}}$ to be an optimal
entanglement witness is that $t=1$ and $\pi$ is cyclic (Theorem 2.2).
## 2\. Main result and proof
In this section, we give the main result and its proof.
Let $\pi$ be a permutation of $(1,2,\ldots,n)$ and $0\leq t\leq n$. For a
subset $F$ of $\\{1,2,\ldots,n\\}$, if $\pi(F)=F$, we say $F$ is an invariant
subset of $\pi$. Let $F$ be an invariant subset of $\pi$. If both $G\subseteq
F$ and $G$ is invariant under $\pi$ imply $G=F$, we say $F$ is a minimal
invariant subset of $\pi$. It is obvious that a minimal invariant subset is a
loop of $\pi$ and $\\{1,2,\ldots,n\\}=\cup_{s=1}^{r}F_{s}$, where
$\\{F_{s}\\}_{s=1}^{r}$ is the set of all minimal invariant subsets of $\pi$.
Denote by $\\#F_{s}$ the cardinal number of $F_{s}$. Then
$\sum_{s=1}^{r}\\#F_{s}=n$. We call $\max\\{\\#F_{s}:s=1,2,\ldots,r\\}$ the
length of $\pi$, denoted by $l(\pi)$. In the case that $l(\pi)=n$, we say that
$\pi$ is cyclic.
The following lemma was shown in [14].
Lemma 2.1. For any permutation $\pi$ of $\\{1,2,3\\}$, let
$\Phi_{t,\pi}:M_{3}({\mathbb{C}})\rightarrow M_{3}({\mathbb{C}})$ be a map
defined by Eq.(1.1). Then $\Phi_{t,\pi}$ is positive if and only if $0\leq
t\leq\frac{3}{l(\pi)}$.
The following is our main result in this note, which states that
$W_{\Phi_{t,\pi}}$ is an optimal EW if and only if $t=1$ and $\pi$ is cyclic.
Theorem 2.2. For any permutation $\pi$ of $\\{1,2,3\\}$, let
$\Phi_{t,\pi}:M_{3}({\mathbb{C}})\rightarrow M_{3}({\mathbb{C}})$ be the map
defined by Eq.(1.1). Then $W_{\Phi_{t,\pi}}$ is an optimal entanglement
witness if and only if $t=1$ and $l(\pi)=3$.
Before stating the main results in this section, let us recall some notions
and give two lemmas that we needed.
Let $l$, $k\in\mathbb{N}$ (the set of all natural numbers), and let
$A_{1},\cdots,A_{k}$, and $C_{1},\cdots,C_{l}\in{\mathcal{B}}(H$, $K$). If,
for each $|\psi\rangle\in H$, there exists an $l\times k$ complex matrix
$(\alpha_{ij}(|\psi\rangle))$ (depending on $|\psi\rangle$) such that
$C_{i}|\psi\rangle=\sum_{j=1}^{k}\alpha_{ij}(|\psi\rangle)A_{j}|\psi\rangle,\qquad
i=1,2,\cdots,l,$
we say that $(C_{1},\cdots,C_{l})$ is a locally linear combination of
$(A_{1},\cdots,A_{k})$, $(\alpha_{ij}(|\psi\rangle))$ is called a local
coefficient matrix at $|\psi\rangle$. Furthermore, if a local coefficient
matrix $(\alpha_{ij}(|\psi\rangle))$ can be chosen for every $|\psi\rangle\in
H$ so that its operator norm
$\|(\alpha_{ij}(|\psi\rangle))\|=\sup\\{\|(\alpha_{ij}(|\psi\rangle)|x\rangle\|:|x\rangle\in{\mathbb{C}}^{\rm
k},\||x\rangle\|\leq 1\\}\leq 1$, we say that $(C_{1},\cdots,C_{l})$ is a
contractive locally linear combination of $(A_{1},\cdots,A_{k})$; if there is
a matrix $(\alpha_{ij})$ such that $C_{i}=\sum_{j=1}^{k}\alpha_{ij}A_{j}$ for
all $i$, we say that $(C_{1},\cdots,C_{l})$ is a linear combination of
$(A_{1},\cdots,A_{k})$ with coefficient matrix $(\alpha_{ij})$.
The following characterization of positive linear maps was obtained in [5],
also, see [6].
Lemma 2.3. Let $H$ and $K$ be complex Hilbert spaces of any dimension,
$\Phi:{\mathcal{B}}(H)\rightarrow{\mathcal{B}}(K)$ be a linear map defined by
$\Phi(X)=\sum_{i=1}^{k}C_{i}XC_{i}^{\dagger}-\sum_{j=1}^{l}D_{j}XD_{j}^{\dagger}$
for all $X$. Then $\Phi$ is positive if and only if $(D_{1},\cdots,D_{l})$ is
a contractive locally linear combination of $(C_{1},\cdots,C_{k})$.
Furthermore, $\Phi$ is completely positive if and only if
$(D_{1},\cdots,D_{l})$ is a linear combination of $(C_{1},\cdots,C_{k})$ with
a contractive coefficient matrix, and in turn, if and only if there exist
$E_{1},E_{2},\ldots,E_{r}$ in ${\rm span}\\{C_{1},\cdots,C_{k}\\}$ such that
$\Phi=\sum_{i=1}^{r}E_{i}(\cdot)E_{i}^{\dagger}.$
Lemma 2.4. Let $t$ be a fixed number with $0<t<1$ and let $x_{1},x_{2},x_{3}$
be any positive numbers with $x_{1}x_{2}x_{3}=1$ and
$(x_{1},x_{2},x_{3})\not=(1,1,1)$. Then we have
$\frac{1-\sum_{i=1}^{3}\frac{1}{(3-t)+tx_{i}}}{\sum_{i=1}^{2}\frac{1}{(3-t)+tx_{i}}-\frac{4}{(3-t+tx_{1})(3-t+tx_{2})}-\frac{1}{(3-t+tx_{1})(3-t+tx_{3})}-\frac{1}{(3-t+tx_{2})(3-t+tx_{3})}}\geq(1-t).$
Proof. Let $f$ be the function in $3$-variables defined by
$\begin{array}[]{rl}&f(x_{1},x_{2},x_{3})\\\
=&\frac{1-\sum_{i=1}^{3}\frac{1}{(3-t)+tx_{i}}}{\sum_{i=1}^{2}\frac{1}{(3-t)+tx_{i}}-\frac{4}{(3-t+tx_{1})(3-t+tx_{2})}-\frac{1}{(3-t+tx_{1})(3-t+tx_{3})}-\frac{1}{(3-t+tx_{2})(3-t+tx_{3})}},\end{array}$
where $t$ is fixed with $0<t<1$ and $x_{1},x_{2},x_{3}$ are any positive
numbers with $x_{1}x_{2}x_{3}=1$ and $(x_{1},x_{2},x_{3})\not=(1,1,1)$. Since
the denominator of $f(x_{1},x_{2},x_{3})$ is not zero whenever
$(x_{1},x_{2},x_{3})\not=(1,1,1)$, a computation shows that
$\begin{array}[]{rl}&f(x_{1},x_{2},x_{3})\geq(1-t)\\\
\Leftrightarrow&1-\sum_{i=1}^{3}\frac{1}{(3-t)+tx_{i}}\geq(\sum_{i=1}^{2}\frac{1}{(3-t)+tx_{i}}-\frac{4}{(3-t+tx_{1})(3-t+tx_{2})}\\\
&-\frac{1}{(3-t+tx_{1})(3-t+tx_{3})}-\frac{1}{(3-t+tx_{2})(3-t+tx_{3})})(1-t)\\\
\Leftrightarrow&g(x_{1},x_{2},x_{3})\geq 0,\end{array}$
where
$\begin{array}[]{rl}g(x_{1},x_{2},x_{3})=&(2t^{2}-2t-3)+(1-t)x_{1}+(1-t)x_{2}+(1-t^{2})x_{3}\\\
&+(2t-t^{2})x_{1}x_{2}+tx_{2}x_{3}+tx_{1}x_{3}.\end{array}$
Thus, to complete the proof of the lemma, we only need to check that the
minimum of the $3$-variable function $g$ is zero on the region $x_{i}>0$ with
$x_{1}x_{2}x_{3}=1$, $i=1,2,3$.
To do this, let
$L(x_{1},x_{2},x_{3},\lambda)=g(x_{1},x_{2},x_{3})+\lambda(x_{1}x_{2}x_{3}-1).$
By the method of Lagrange multipliers, we have the system
$None$ $\begin{cases}L_{x_{1}}^{\prime}=(1-t)+(2t-t^{2})x_{2}+tx_{3}+\lambda
x_{2}x_{3}=0,\\\ L_{x_{2}}^{\prime}=(1-t)+(2t-t^{2})x_{1}+tx_{3}+\lambda
x_{1}x_{3}=0,\\\ L_{x_{3}}^{\prime}=(1-t^{2})+tx_{2}+tx_{1}+\lambda
x_{1}x_{2}=0,\\\ L_{\lambda}^{\prime}=x_{1}x_{2}x_{3}-1=0.\end{cases}$
Solving this system, one obtains
$(x_{2}-x_{1})(2t-t^{2}+\lambda x_{3})=0,$
which implies that
${\rm either}\ \ \ \ x_{1}=x_{2}\ \ \ \ {\rm or}\ \ \ \ 2t-t^{2}+\lambda
x_{3}=0.$
If $2t-t^{2}+\lambda x_{3}=0$, by Eq.(2.1), one gets $x_{3}=\frac{t-1}{t}<0$,
a contradiction. Hence we must have $x_{1}=x_{2}$. Thus, by Eq.(2.1) again, we
have
$(2t-t^{2})x_{1}^{4}+(1-t)x_{1}^{3}-tx_{1}+(t^{2}-1)=0,$
that is,
$None$
$(x_{1}-1)[(2t-t^{2})x_{1}^{3}+(1+t-t^{2})x_{1}^{2}+(1+t-t^{2})x_{1}+(1-t^{2})]=0.$
Note that
$(2t-t^{2})x_{1}^{3}+(1+t-t^{2})x_{1}^{2}+(1+t-t^{2})x_{1}+(1-t^{2})>0$ for
all $x_{1}>0$ and $0<t<1$. So Eq.(2.2) holds if and only if $x_{1}=1$, which
forces $x_{2}=x_{3}=1$. It follows that the function $g(x_{1},x_{2},x_{3})$
takes its extremum at the point $(1,1,1)$. Moreover, it is easy to check that
$(1,1,1)$ is the minimal point of $g(x_{1},x_{2},x_{3})$. Hence
$g(x_{1},x_{2},x_{3})\geq g(1,1,1)=0$ for all $x_{i}>0$ with
$x_{1}x_{2}x_{3}=1$, $i=1,2,3$.
Therefore, the inequality in Lemma 2.4 holds for all $x_{i}>0$, $i=1,2,3$,
with $x_{1}x_{2}x_{3}=1$ and $(x_{1},x_{2},x_{3})\not=(1,1,1)$. The proof is
finished. $\Box$
Now we are in a position to give the proof of Theorem 2.1.
Proof of Theorem 2.1. By Lemma 2.1, $\Phi_{t,\pi}$ is positive whenever $0\leq
t\leq\frac{3}{l(\pi)}$. We will prove the theorem by considering several
cases. Note that, $\Phi_{0,\pi}$ is completely positive; so we may assume that
$t>0$.
Case 1. $l(\pi)=1$.
if $l=1$, then $\pi={\rm id}$ (the identical permutation). In this case,
$\Phi_{t,\pi}$ is a completely positive linear map for all $0<t\leq 3$ (see
[13, Proposition 2.7]), and so $W_{\Phi_{t,\pi}}\geq 0$, which is not an EW.
Case 2. $l(\pi)=2$.
If $l=2$, then $\pi^{2}={\rm id}$. Without loss of generality, assume that
$\pi(1)=2$, $\pi(2)=1$ and $\pi(3)=3$. Since
$\Phi_{t,\pi}(E_{11})=(2-t)E_{11}+tE_{22}$,
$\Phi_{t,\pi}(E_{22})=(2-t)E_{22}+tE_{11}$, $\Phi_{t,\pi}(E_{33})=2E_{33}$ and
$\Phi_{t,\pi}(E_{ij})=-E_{ij}$ with $1\leq i\not=j\leq 3$, the Choi matrix of
$\Phi_{t,\pi}$ is
$\begin{array}[]{rl}W_{\Phi_{t,\pi}}=&\sum_{i=1}^{3}(2-t)E_{ii}\otimes
E_{ii}+tE_{22}\otimes E_{11}+tE_{11}\otimes E_{22}+tE_{33}\otimes
E_{33}-\sum_{i\not=j}E_{ij}\otimes E_{ij}\\\ =&(2-t)E_{11}\otimes
E_{11}+(2-t)E_{22}\otimes E_{22}+2E_{33}\otimes E_{33}\\\ &+tE_{22}\otimes
E_{11}+tE_{11}\otimes E_{22}-\sum_{i\not=j}E_{ij}\otimes E_{ij}.\end{array}$
If $1\leq t\leq\frac{3}{2}$, then let
$C_{1}=(2-t)E_{11}\otimes E_{11}+(2-t)E_{22}\otimes E_{22}+2E_{33}\otimes
E_{33}-\sum_{i\neq j;\pi(i)\neq j}E_{ij}\otimes E_{ij}$
and
$C_{2}=tE_{22}\otimes E_{11}+tE_{11}\otimes E_{22}-E_{12}\otimes
E_{12}-E_{21}\otimes E_{21}.$
It is easily checked that $C_{1}\geq 0$. As $C_{2}^{{\rm
T}_{2}}=tE_{22}\otimes E_{11}+tE_{11}\otimes E_{22}-E_{12}\otimes
E_{21}-E_{21}\otimes E_{12}\geq 0,$ we see that $C_{2}$ is PPT. It is clear
that $C_{1}\not=0$ and $W_{\Phi_{t,\pi}}=C_{1}+C_{2}$. Hence
$W_{\Phi_{t,\pi}}$ is decomposable and not optimal.
If $0<t<1$, then let
$\begin{array}[]{rl}D_{1}=&(2-t)E_{11}\otimes E_{11}+(2-t)E_{22}\otimes
E_{22}+2E_{33}\otimes E_{33}\\\ &-\sum_{i\neq j;\pi(i)\neq j}E_{ij}\otimes
E_{ij}-(1-t)E_{12}\otimes E_{12}-(1-t)E_{21}\otimes E_{21}\end{array}$
and
$D_{2}=tE_{22}\otimes E_{11}+tE_{11}\otimes E_{22}-tE_{12}\otimes
E_{12}-tE_{21}\otimes E_{21}.$
It is also clear that $D_{2}$ is PPT and $D_{1}\geq 0$. We still have
$D_{1}\not=0$ and $W_{\Phi_{t,\pi}}=D_{1}+D_{2}$. Hence $W_{\Phi_{t,\pi}}$ is
decomposable and not optimal.
Case 3. $l(\pi)=3$, i.e., $\pi$ is cyclic.
If $l(\pi)=3$ and $t=1$, then $\pi$ is a cyclic permutation, and by [13,
Theorem 3.2], $W_{\Phi_{1,\pi}}$ is optimal.
In the sequel we always assume that $l(\pi)=3$. Our aim is to prove that
$W_{\Phi_{t,\pi}}$ is not optimal for any $0<t<1$. Without loss of generality,
let $\pi(i)=(i+1)\ {\rm mod}\ 3$, $i=1,2,3$. By Theorem 1.1, to prove that
$W_{\Phi_{t,\pi}}$ is not optimal, we have to prove that there exists a matrix
$C\in M_{3}({\mathbb{C}})$ such that the linear map
$A\mapsto\Phi_{t,\pi}(A)-CAC^{\dagger}$ is positive. Indeed, we will show
that, for any positive number $0<c\leq\sqrt{1-t}$, let $C_{0}={\rm
diag}(c,-c,0)$; then the map $A\mapsto\Phi_{t,\pi}(A)-CAC^{\dagger}$ is
positive.
To do this, let $C_{0}={\rm diag}(c,-c,0)$ with $c>0$ and let $\Psi_{C_{0}}$
be the map defined by
$\begin{array}[]{rl}\Psi_{C_{0}}(A)=&\Phi_{t,\pi}(A)-C_{0}AC_{0}^{\dagger}\\\
=&(3-t)\sum_{i=1}^{3}E_{ii}AE_{ii}^{\dagger}+\sum_{i=1}^{3}E_{i,i+1}AE_{i,i+1}^{\dagger}-A-C_{0}AC_{0}^{\dagger}\end{array}$
for all $A\in M_{3}({\mathbb{C}})$.
If $\Psi_{C_{0}}$ is positive, then by Lemma 2.3, for any unit
$|x\rangle\in{\mathbb{C}}^{3}$, there exist scalars
$\\{\alpha_{i}(|x\rangle)\\}_{i=1}^{3}$,
$\\{\beta_{i}(|x\rangle)\\}_{i=1}^{3}$,
$\\{\delta_{i}(|x\rangle)\\}_{i=1}^{3}$ and
$\\{\gamma_{i}(|x\rangle)\\}_{i=1}^{3}$ such that
$None$
$|x\rangle=I|x\rangle=\sum_{i=1}^{3}\alpha_{i}(|x\rangle)(\sqrt{3-t}E_{ii})|x\rangle+\sum_{i=1}^{3}\beta_{i}(|x\rangle)\sqrt{t}E_{i,i+1}|x\rangle,$
$None$
$C|x\rangle=\sum_{i=1}^{3}\delta_{i}(|x\rangle)(\sqrt{3-t}E_{ii})|x\rangle+\sum_{i=1}^{3}\gamma_{i}(|x\rangle)\sqrt{t}E_{i,i+1}|x\rangle,$
and the matrix
$F_{x}=\left(\begin{array}[]{cccccc}\alpha_{1}(|x\rangle)&\alpha_{2}(|x\rangle)&\alpha_{3}(|x\rangle)&\beta_{1}(|x\rangle)&\beta_{2}(|x\rangle)&\beta_{3}(|x\rangle)\\\
\delta_{1}(|x\rangle)&\delta_{2}(|x\rangle)&\delta_{3}(|x\rangle)&\gamma_{1}(|x\rangle)&\gamma_{2}(|x\rangle)&\gamma_{3}(|x\rangle)\end{array}\right)$
is contractive.
Note that $\|F_{x}\|\leq 1$ if and only if $\|F_{x}F_{x}^{\dagger}\|\leq 1$.
In the sequel, for any unit $|x\rangle\in{\mathbb{C}}^{3}$, we write
$|x\rangle=(|x_{1}|e^{i\theta_{1}},|x_{2}|e^{i\theta_{2}},|x_{3}|e^{i\theta_{3}})^{T}$.
Then $|x_{1}|^{2}+|x_{2}|^{2}+|x_{3}|^{2}=1$.
Subcase 1. $|x_{1}|=|x_{2}|=|x_{3}|=\frac{1}{\sqrt{3}}$.
In Eqs.(2.3)-(2.4), by taking
$(\alpha_{1},\alpha_{2},\alpha_{3})=(\frac{\sqrt{3-t}}{3},\frac{\sqrt{3-t}}{3},\frac{\sqrt{3-t}}{3})\
\ \ {\rm and}\ \
(\delta_{1},\delta_{2},\delta_{3})=(\frac{\sqrt{3-t}c}{3},-\frac{\sqrt{3-t}c}{3},0),$
we get
$(\beta_{1},\beta_{2},\beta_{3})=(\frac{{\sqrt{t}}x_{1}}{3x_{2}},\frac{{\sqrt{t}}x_{2}}{3x_{3}},\frac{{\sqrt{t}}x_{3}}{3x_{1}})\
\ \ {\rm and}\ \
(\gamma_{1},\gamma_{2},\gamma_{3})=(\frac{{\sqrt{t}c}x_{1}}{3x_{2}},-\frac{{\sqrt{t}}cx_{2}}{3x_{3}},0).$
So $\sum_{i=1}^{3}(|\alpha_{i}|^{2}+|\beta_{i}|^{2})=1$,
$\sum_{i=1}^{3}(|\delta_{i}|^{2}+|\gamma_{i}|^{2})=\frac{6c^{2}}{9}$ and
$\sum_{i=1}^{3}(\alpha_{i}\overline{\delta_{i}}+\beta_{i}\overline{\gamma_{i}})=0$.
It follows that
$F_{x}F_{x}^{\dagger}=\left(\begin{array}[]{cc}\sum_{i=1}^{3}(|\alpha_{i}|^{2}+|\beta_{i}|^{2})&\sum_{i=1}^{3}(\alpha_{i}\bar{\delta_{i}}+\beta_{i}\bar{\gamma_{i}})\\\
\sum_{i=1}^{3}(\bar{\alpha_{i}}\delta_{i}+\bar{\beta_{i}}\gamma_{i})&\sum_{i=1}^{3}(|\delta_{i}|^{2}+|\gamma_{i}|^{2})\end{array}\right)=\left(\begin{array}[]{cc}1&0\\\
0&\frac{6c^{2}}{9}\end{array}\right),$
which implies that $\|F_{x}F_{x}^{\dagger}\|\leq 1\Leftrightarrow
c^{2}\leq\frac{9}{6}.$ Hence
$c^{2}\leq 1-t\Rightarrow\|F_{x}F_{x}^{\dagger}\|\leq 1.$
Subcase 2. $x_{i}\not=0$ for all $i=1,2,3$ and
$(|x_{1}|,|x_{2}|,|x_{3}|)\not=(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}})$.
Let $r_{i}=|\frac{x_{i}}{x_{i+1}}|^{2}$ for $i=1,2,3$. Then $r_{i}>0$,
$i=1,2,3$, and $r_{1}r_{2}r_{3}=1$. Take
$(\alpha_{1},\alpha_{2},\alpha_{3})=(\frac{\sqrt{3-t}r_{1}}{t+(3-t)r_{1}},\frac{\sqrt{3-t}r_{2}}{t+(3-t)r_{2}},\frac{\sqrt{3-t}r_{3}}{t+(3-t)r_{3}})$
and
$(\delta_{1},\delta_{2},\delta_{3})=(\frac{\sqrt{3-t}r_{1}c}{t+(3-t)r_{1}},-\frac{\sqrt{3-t}r_{2}c}{t+(3-t)r_{2}},0).$
By Eqs.(2.3)-(2.4), we get
$(\beta_{1},\beta_{2},\beta_{3})=(\frac{\sqrt{r_{1}}}{t+(3-t)r_{1}}e^{i(\theta_{1}-\theta_{2})},\frac{\sqrt{r_{2}}}{t+(3-t)r_{2}}e^{i(\theta_{2}-\theta_{3})},\frac{\sqrt{r_{3}}}{t+(3-t)r_{3}}e^{i(\theta_{3}-\theta_{1})})$
and
$(\gamma_{1},\gamma_{2},\gamma_{3})=(\frac{\sqrt{tr_{1}}c}{t+(3-t)r_{1}}e^{i(\theta_{1}-\theta_{2})},-\frac{\sqrt{tr_{2}}c}{t+(3-t)r_{2}}e^{i(\theta_{2}-\theta_{3})},0).$
So
$f(\alpha_{1},\alpha_{2},\alpha_{3})=\sum_{i=1}^{3}|\alpha_{i}|^{2}+\sum_{i=1}^{3}|\beta_{i}|^{2}=\sum_{i=1}^{3}\frac{r_{i}}{t+(3-t)r_{i}},$
$f_{C_{0}}(\delta_{1},\delta_{2},\delta_{3})=\sum_{i=1}^{3}|\delta_{i}|^{2}+\sum_{i=1}^{3}|\gamma_{i}|^{2}=\frac{r_{1}c^{2}}{t+(3-t)r_{1}}+\frac{r_{2}c^{2}}{t+(3-t)r_{2}}$
and
$\sum_{i=1}^{3}(\alpha_{i}\bar{\delta_{i}}+\beta_{i}\bar{\gamma_{i}})=\frac{r_{1}c}{t+(3-t)r_{1}}-\frac{r_{2}c}{t+(3-t)r_{2}}.$
It follows that
$\begin{array}[]{rl}F_{x}F_{x}^{\dagger}=&\left(\begin{array}[]{cc}\sum_{i=1}^{3}(|\alpha_{i}|^{2}+|\beta_{i}|^{2})&\sum_{i=1}^{3}(\alpha_{i}\bar{\delta_{i}}+\beta_{i}\bar{\gamma_{i}})\\\
\sum_{i=1}^{3}(\bar{\alpha_{i}}\delta_{i}+\bar{\beta_{i}}\gamma_{i})&\sum_{i=1}^{3}(|\delta_{i}|^{2}+|\gamma_{i}|^{2})\end{array}\right)\\\
=&\left(\begin{array}[]{cc}\sum_{i=1}^{3}\frac{r_{i}}{t+(3-t)r_{i}}&\frac{r_{1}c}{t+(3-t)r_{1}}-\frac{r_{2}c}{t+(3-t)r_{2}}\\\
\frac{r_{1}c}{t+(3-t)r_{1}}-\frac{r_{2}c}{t+(3-t)r_{2}}&\frac{r_{1}c^{2}}{t+(3-t)r_{1}}+\frac{r_{2}c^{2}}{t+(3-t)r_{2}}\end{array}\right).\end{array}$
Note that $\|F_{x}F_{x}^{\dagger}\|\leq 1$ if and only if its maximal
eigenvalue $\lambda_{{\rm max}}\leq 1$. By a calculation, it is easily checked
that
$\lambda_{{\rm max}}\leq 1$
holds if and only if
$None$
$c^{2}\leq\frac{1-\sum_{i=1}^{3}\frac{r_{i}}{t+(3-t)r_{i}}}{(1-\sum_{i=1}^{3}\frac{r_{i}}{t+(3-t)r_{i}})(\frac{r_{1}}{t+(3-t)r_{1}}+\frac{r_{2}}{t+(3-t)r_{2}})+(\frac{r_{1}}{t+(3-t)r_{1}}-\frac{r_{2}}{t+(3-t)r_{2}})^{2}},$
where $r_{1},r_{2},r_{3}>0$ with $r_{1}r_{2}r_{3}=1$ and
$(r_{1},r_{2},r_{3})\not=(1,1,1)$. Let
$g(r_{1},r_{2},r_{3})=\frac{1-\sum_{i=1}^{3}\frac{r_{i}}{t+(3-t)r_{i}}}{(1-\sum_{i=1}^{3}\frac{r_{i}}{t+(3-t)r_{i}})(\frac{r_{1}}{t+(3-t)r_{1}}+\frac{r_{2}}{t+(3-t)r_{2}})+(\frac{r_{1}}{t+(3-t)r_{1}}-\frac{r_{2}}{t+(3-t)r_{2}})^{2}}.$
Replacing $r_{i}$ by $\frac{1}{r_{i}}$ in the above function
$g(r_{1},r_{2},r_{3})$, we have
$g(r_{1},r_{2},r_{3})=\frac{1-\sum_{i=1}^{3}\frac{1}{(3-t)+tr_{i}}}{\sum_{i=1}^{2}\frac{1}{(3-t)+tr_{i}}-\frac{4}{(3-t+tr_{1})(3-t+tr_{2})}-\frac{1}{(3-t+tr_{1})(3-t+tr_{3})}-\frac{1}{(3-t+tr_{2})(3-t+tr_{3})}}.$
Now applying Lemma 2.4, we see that
$g(r_{1},r_{2},r_{3})\geq 1-t$
holds for all positive numbers $r_{1},r_{2},r_{3}$ with $r_{1}r_{2}r_{3}=1$
and $(r_{1},r_{2},r_{3})\not=(1,1,1)$. This and Eq.(2.5) imply
$c^{2}\leq(1-t)\Rightarrow\lambda_{{\rm max}}\leq
1\Rightarrow\|F_{x}F_{x}^{\dagger}\|\leq 1.$
Subcase 3. $x_{1}=0$ and $x_{i}\not=0$ for $i=2,3$.
In this case, by Eqs.(2.3)-(2.4), one may choose
$\beta_{1}=\delta_{3}=\gamma_{1}=0$, $\alpha_{3}=\frac{1}{\sqrt{3-t}}$,
$\beta_{2}=\frac{(1-\sqrt{3-t}\alpha_{2})x_{2}}{\sqrt{t}x_{3}}$ and
$\gamma_{2}=\frac{(-c-\sqrt{3-t}\delta_{2})x_{2}}{\sqrt{t}x_{3}}$. Write
$r_{2}=|\frac{x_{2}}{x_{3}}|^{2}=\frac{|x_{2}|^{2}}{1-|x_{2}|^{2}}$. Then by
taking
$\begin{array}[]{rl}&(\alpha_{1},\alpha_{2},\alpha_{3},\beta_{1},\beta_{2},\beta_{3})=(0,\frac{\sqrt{3-t}r_{2}}{t+(3-t)r_{2}},\frac{1}{\sqrt{3-t}},0,\frac{\sqrt{t}\sqrt{r_{2}}}{t+(3-t)r_{2}}e^{i(\theta_{2}-\theta_{3})},0)\end{array}$
and
$(\delta_{1},\delta_{2},\delta_{3},\gamma_{1},\gamma_{2},\gamma_{3})=(0,-\frac{\sqrt{t}\sqrt{r_{2}}c}{t+(3-t)r_{2}}e^{i(\theta_{2}-\theta_{3})},0),$
which meet Eqs.(2.3)-(2.4), we get
$f(\alpha_{1},\alpha_{2},\alpha_{3})=\sum_{i=1}^{3}|\alpha_{i}|^{2}+\sum_{i=1}^{3}|\beta_{i}|^{2}=\frac{r_{2}}{t+(3-t)r_{2}}+\frac{1}{3-t},$
$f_{C_{0}}(\delta_{1},\delta_{2},\delta_{3})=\sum_{i=1}^{3}|\delta_{i}|^{2}+\sum_{i=1}^{3}|\gamma_{i}|^{2}=\frac{r_{2}c^{2}}{t+(3-t)r_{2}}$
and
$\sum_{i=1}^{3}(\alpha_{i}\bar{\delta_{i}}+\beta_{i}\bar{\gamma_{i}})=-\frac{r_{2}c}{t+(3-t)r_{2}}.$
Hence
$F_{x}F_{x}^{\dagger}=\left(\begin{array}[]{cc}\frac{r_{2}}{t+(3-t)r_{2}}+\frac{1}{3-t}&-\frac{r_{2}c}{t+(3-t)r_{2}}\\\
-\frac{r_{2}c}{t+(3-t)r_{2}}&\frac{r_{2}c^{2}}{t+(3-t)r_{2}}\end{array}\right).$
Still, by a calculation, one can easily obtain
$None$ $\|F_{x}F_{x}^{\dagger}\|\leq 1\Leftrightarrow
c^{2}\leq\frac{1-\frac{r_{2}}{t+(3-t)r_{2}}-\frac{1}{3-t}}{\frac{r_{2}}{t+(3-t)r_{2}}-\frac{r_{2}}{(t+(3-t)r_{2})(3-t)}},$
where $r_{2}>0$ is any positive number. Let
$g(r_{2})=\frac{1-\frac{r_{2}}{t+(3-t)r_{2}}-\frac{1}{3-t}}{\frac{r_{2}}{t+(3-t)r_{2}}-\frac{r_{2}}{(t+(3-t)r_{2})(3-t)}}.$
A direct calculation yields that,
$None$ $g(r_{2})\geq 1-t\Leftrightarrow r_{2}\geq\frac{t(2-t)}{t-1}.$
Note that $r_{2}>0$ and $\frac{t(2-t)}{t-1}<0$ as $0<t<1$. So we always have
$r_{2}\geq\frac{t(2-t)}{t-1}$. Thus by Eq.(2.7), we have proved that
$g(r_{2})\geq 1-t$ holds for all positive numbers $r_{2}>0$. It follows from
Eq.(2.6) that
$c^{2}\leq 1-t\Rightarrow\|F_{x}F_{x}^{\dagger}\|\leq 1.$
Subcase 4. $x_{2}=0$ and $x_{i}\not=0$ for $i=1,3$.
Let $r_{3}=|\frac{x_{3}}{x_{1}}|^{2}=\frac{|x_{3}|^{2}}{1-|x_{3}|^{2}}$. By
Eqs.(2.3)-(2.4), we can choose $\beta_{2}=\gamma_{2}=0$,
$\alpha_{1}=\frac{1}{\sqrt{3-t}}$,
$\alpha_{3}=\frac{\sqrt{3-t}r_{3}}{t+(3-t)r_{3}}$,
$\beta_{3}=\frac{(1-\sqrt{3-t}\alpha_{3})x_{3}}{\sqrt{t}x_{1}}$ and
$\gamma_{3}=\frac{-\sqrt{3-t}\delta_{3}x_{3}}{\sqrt{t}x_{1}}$. Now take
$\begin{array}[]{rl}&(\alpha_{1},\alpha_{2},\alpha_{3},\beta_{1},\beta_{2},\beta_{3})=(\frac{1}{\sqrt{3-t}},0,\frac{\sqrt{3-t}r_{3}}{t+(3-t)r_{3}},0,0,\frac{\sqrt{t}\sqrt{r_{3}}}{t+(3-t)r_{3}}e^{i(\theta_{3}-\theta_{1})})\end{array}$
and
$(\delta_{1},\delta_{2},\delta_{3},\gamma_{1},\gamma_{2},\gamma_{3})=(\frac{c}{\sqrt{3-t}},0,0,0,0,0).$
It follows that
$f(\alpha_{1},\alpha_{2},\alpha_{3})=\sum_{i=1}^{3}|\alpha_{i}|^{2}+\sum_{i=1}^{3}|\beta_{i}|^{2}=\frac{r_{3}}{t+(3-t)r_{3}}+\frac{1}{3-t},$
$f_{C_{0}}(\delta_{1},\delta_{2},\delta_{3})=\sum_{i=1}^{3}|\delta_{i}|^{2}+\sum_{i=1}^{3}|\gamma_{i}|^{2}=\frac{c^{2}}{3-t}$
and
$\sum_{i=1}^{3}(\alpha_{i}\bar{\delta_{i}}+\beta_{i}\bar{\gamma_{i}})=\frac{c}{3-t}.$
Hence
$F_{x}F_{x}^{\dagger}=\left(\begin{array}[]{cc}\frac{r_{3}}{t+(3-t)r_{3}}+\frac{1}{3-t}&\frac{c}{3-t}\\\
\frac{c}{3-t}&\frac{c^{2}}{3-t}\end{array}\right).$
Still, one can easily checked that
$\|F_{x}F_{x}^{\dagger}\|\leq 1\Leftrightarrow
c^{2}\leq\frac{1-\frac{r_{3}}{t+(3-t)r_{3}}-\frac{1}{3-t}}{\frac{1}{3-t}-\frac{r_{3}}{(t+(3-t)r_{3})(3-t)}}$
and
$\frac{1-\frac{r_{3}}{t+(3-t)r_{3}}-\frac{1}{3-t}}{\frac{1}{3-t}-\frac{r_{3}}{(t+(3-t)r_{3})(3-t)}}\geq
1-t\Leftrightarrow t\geq(t-1)r_{3},$
where $r_{3}>0$ is any positive number and $0<t<1$. Note that
$t\geq(t-1)r_{3}$ as $0<t<1$ and $r_{3}>0$. Thus we see that we still have
$c^{2}\leq 1-t\Rightarrow\|F_{x}F_{x}^{\dagger}\|\leq 1.$
Subcase 5. $x_{3}=0$ and $x_{i}\not=0$ for $i=1,2$.
Let $r_{1}=|\frac{x_{1}}{x_{2}}|^{2}=\frac{|x_{1}|^{2}}{1-|x_{1}|^{2}}$. By
Eqs.(2.3)-(2.4), one may choose $\beta_{3}=\gamma_{3}=0$,
$\alpha_{2}=\frac{1}{\sqrt{3-t}}$,
$\beta_{1}=\frac{(1-\sqrt{3-t}\alpha_{1})x_{1}}{\sqrt{t}x_{2}}$,
$\delta_{2}=\frac{-c}{\sqrt{3-t}}$ and
$\gamma_{1}=\frac{(c-\sqrt{3-t}\delta_{1})x_{1}}{\sqrt{t}x_{2}}$. Then for
choice
$\begin{array}[]{rl}&(\alpha_{1},\alpha_{2},\alpha_{3},\beta_{1},\beta_{2},\beta_{3})=(\frac{\sqrt{3-t}r_{1}}{t+(3-t)r_{1}},\frac{1}{\sqrt{3-t}},0,\frac{\sqrt{t}\sqrt{r_{1}}}{t+(3-t)r_{1}}e^{i(\theta_{1}-\theta_{2})},0,0)\end{array}$
and
$(\delta_{1},\delta_{2},\delta_{3},\gamma_{1},\gamma_{2},\gamma_{3})=(\frac{\sqrt{3-t}r_{1}c}{t+(3-t)r_{1}},\frac{-c}{\sqrt{3-t}},0,\frac{\sqrt{t}\sqrt{r_{1}}c}{t+(3-t)r_{1}}e^{i(\theta_{1}-\theta_{2})},0,0),$
we get
$f(\alpha_{1},\alpha_{2},\alpha_{3})=\sum_{i=1}^{3}|\alpha_{i}|^{2}+\sum_{i=1}^{3}|\beta_{i}|^{2}=\frac{r_{1}}{t+(3-t)r_{1}}+\frac{1}{3-t},$
$f_{C_{0}}(\delta_{1},\delta_{2},\delta_{3})=\sum_{i=1}^{3}|\delta_{i}|^{2}+\sum_{i=1}^{3}|\gamma_{i}|^{2}=\frac{r_{1}c^{2}}{t+(3-t)r_{1}}+\frac{c^{2}}{3-t}$
and
$\sum_{i=1}^{3}(\alpha_{i}\bar{\delta_{i}}+\beta_{i}\bar{\gamma_{i}})=\frac{r_{1}c}{t+(3-t)r_{1}}-\frac{c}{3-t}.$
So
$F_{x}F_{x}^{\dagger}=\left(\begin{array}[]{cc}\frac{r_{1}}{t+(3-t)r_{1}}+\frac{1}{3-t}&\frac{r_{1}c}{t+(3-t)r_{1}}-\frac{c}{3-t}\\\
\frac{r_{1}c}{t+(3-t)r_{1}}-\frac{c}{3-t}&\frac{r_{1}c^{2}}{t+(3-t)r_{1}}+\frac{c^{2}}{3-t}\end{array}\right).$
It is easily checked that
$None$ $\|F_{x}F_{x}^{\dagger}\|\leq 1\Leftrightarrow
c^{2}\leq\frac{1-\frac{r_{1}}{t+(3-t)r_{1}}-\frac{1}{3-t}}{(1-\frac{r_{1}}{t+(3-t)r_{1}}-\frac{1}{3-t})(\frac{r_{1}}{t+(3-t)r_{1}}+\frac{1}{3-t})+(\frac{r_{1}}{t+(3-t)r_{1}}-\frac{1}{3-t})^{2}},$
where $r_{1}>0$ is any positive number and $0<t<1$. Let
$g(r_{1})=\frac{1-\frac{r_{1}}{t+(3-t)r_{1}}-\frac{1}{3-t}}{(1-\frac{r_{1}}{t+(3-t)r_{1}}-\frac{1}{3-t})(\frac{r_{1}}{t+(3-t)r_{1}}+\frac{1}{3-t})+(\frac{r_{1}}{t+(3-t)r_{1}}-\frac{1}{3-t})^{2}}.$
By a direct calculation, one gets
$g(r_{1})\geq 1-t\Leftrightarrow 1-t^{2}+tr_{1}\geq 0.$
Hence we always have $g(r_{1})\geq 1-t$. This and Eq.(2.8) yield again
$c^{2}\leq 1-t\Rightarrow\|F_{x}F_{x}^{\dagger}\|\leq 1.$
Subcase 6. $x_{1}=x_{2}=0$ and $x_{3}\not=0$.
By Eqs.(2.3)-(2.4), we have $\beta_{2}=\delta_{3}=\gamma_{2}=0$ and
$\alpha_{3}=\frac{1}{\sqrt{3-t}}$. Then take
$(\alpha_{1},\alpha_{2},\alpha_{3},\beta_{1},\beta_{2},\beta_{3})=(0,0,\frac{1}{\sqrt{3-t}},0,0,0)\
{\rm and}\
(\delta_{1},\delta_{2},\delta_{3},\gamma_{1},\gamma_{2},\gamma_{3})=(0,0,0,0,0,0).$
We obtain $F_{x}F_{x}^{\dagger}=\left(\begin{array}[]{cc}\frac{1}{3-t}&0\\\
0&0\end{array}\right),$ which is contractive.
Subcase 7. $x_{1}=x_{3}=0$ and $x_{2}\not=0$.
By Eqs.(2.3)-(2.4), we have $\beta_{1}=\gamma_{1}=0$,
$\alpha_{2}=\frac{1}{\sqrt{3-t}}$ and $\gamma_{2}=\frac{c}{\sqrt{3-t}}$. Then
by taking
$(\alpha_{1},\alpha_{2},\alpha_{3},\beta_{1},\beta_{2},\beta_{3})=(0,\frac{1}{\sqrt{3-t}},0,0,0,0)\
{\rm and}\
(\delta_{1},\delta_{2},\delta_{3},\gamma_{1},\gamma_{2},\gamma_{3})=(0,0,0,0,\frac{c}{\sqrt{3-t}},0),$
we obtain
$F_{x}F_{x}^{\dagger}=\left(\begin{array}[]{cc}\frac{1}{3-t}&\frac{c}{3-t}\\\
\frac{c}{3-t}&\frac{c^{2}}{3-t}\end{array}\right)=\frac{1}{3-t}\left(\begin{array}[]{cc}1&c\\\
c&c^{2}\end{array}\right)$. It is easy to check that
$\|F_{x}F_{x}^{\dagger}\|\leq 1\Leftrightarrow c^{2}\leq(2-t).$
So $c^{2}\leq 1-t$ implies $\|F_{x}F_{x}^{\dagger}\|\leq 1$.
Subcase 8. $x_{2}=x_{3}=0$ and $x_{1}\not=0$.
The case is the same as Case 7.
Thus, by combining Subcases 1-8 and applying Lemma 2.3, we have proved that,
for any matrix $C_{0}={\rm diag}(c,-c,0)$ with $0<c^{2}\leq 1-t$, the map
$A\mapsto\Phi_{t,\pi}(A)-C_{0}AC_{0}^{\dagger}$ is positive. Then, by Theorem
1.1, we see that $W_{\Phi_{t,\pi}}$ is not optimal whenever $l(\pi)=3$ and
$0<t<1$.
The proof is finished. $\Box$
## 3\. Conclusions
Every entangled state can be detected by an optimal entanglement witness. So,
it is important to construct as many as possible optimal EWs. A natural way of
constructing optimal EWs is through NCP positive maps by Choi-Jamiołkowski
isomorphism $\Phi\leftrightarrow W_{\Phi}$. In [14], for $0\leq t\leq n$, a
class of new $D$-type positive maps
$\Phi_{t,\pi}:M_{n}({\mathbb{C}})\rightarrow M_{n}({\mathbb{C}})$ induced by
an arbitrary permutation $\pi$ of $(1,2,\ldots,n)$ was constructed, where
$\Phi_{t,\pi}$ is defined by
$None$
$\Phi_{t,\pi}(A)=(n-t)\sum_{i=1}^{n}E_{ii}AE_{ii}+t\sum_{i=1}^{n}E_{i,\pi(i)}AE_{i,\pi(i)}^{\dagger}-A.$
It was shown in [14] that $\Phi_{t,\pi}$ in NCP positive if and only if
$0<t\leq\frac{n}{l(\pi)}$. In [12], by using Theorem 1.1, we proved that
$W_{\Phi_{1,\pi}}$ is optimal if $l(\pi)=n$ and $\pi^{2}\not={\rm id}$. But it
is not clear that whether or not there exist other optimal
$W_{\Phi_{t,\pi}}$s. We guess there are no.
Conjecture. For $n\geq 3$, $W_{\Phi_{t,\pi}}$ is an optimal entanglement
witness if and only if $t=1$, $l(\pi)=n$ and $\pi^{2}\not={\rm id}$.
The case $n=2$ is simple. It is easily checked that $W_{\Phi_{t,\pi}}$ is
optimal if and only if $t=1$ and $l(\pi)=2$. Note that, $\pi^{2}=$id if $n=2$.
The present note gives an affirmative answer to the above conjecture for the
case $n=3$.
## References
* [1] D. Bru${\ss}$, J. Math. Phys. 43 (2002) 4237.
* [2] M.-D. Choi, Lin. Alg. Appl. 10, 285 (1975); ibid 12, 95 (1975).
* [3] D. Chru$\acute{s}$ci$\acute{n}$ski and A. Kossakowski, Open Systems and Inf. Dynamics 14 (2007) 275.
* [4] M. Horodecki, P. Horodecki, R. Horodecki, Phys. Lett. A 223 (1996) 1.
* [5] J. Hou, J. Operator Theory, 39 (1998), 43-58.
* [6] J. Hou, J. Phys. A: Math. Theor. 43 (2010) 385201; arXiv[quant-ph]: 1007.0560v1.
* [7] J. Hou, X. Qi, Phys. Rev. A 81 (2010) 062351.
* [8] M. A. Jafarizadeh, N. Behzadi, Y. Akbari, Eur. Phys. J. D 55 (2009) 197.
* [9] A. Jamiołkowski, Rep. Math. Phys. 3, 275 (1972).
* [10] M. Lewensetein, B. Kraus, J.I. Cirac, P. Horodecki, Phys. Rev. A 62 (2001) 052310.
* [11] M. A. Nielsen, I. L. Chuang, Cambridge University Press, Cambridge, 2000.
* [12] X. Qi, J. Hou, Phys. Rev. A 85 (2012) 022334.
* [13] X. Qi, J. Hou, J. Phys. A: Math. Theor. 43 (2011) 385201.
* [14] J. C. Hou, Chi-Kwong Li, Yiu-Tung Poon, X. F. Qi, Nung-Sing Sze, Criteria for $k$-positivity of linear maps, arXiv: 1211.036v1.
* [15] G. T$\acute{o}$th, O. G$\ddot{u}$hne, Phys. Rev. Lett. 94 (2005) 060501.
|
arxiv-papers
| 2013-01-31T09:22:35 |
2024-09-04T02:49:41.090837
|
{
"license": "Public Domain",
"authors": "Xiaofei Qi, Jinchuan Hou",
"submitter": "Jinchuan Hou",
"url": "https://arxiv.org/abs/1301.7558"
}
|
1301.7632
|
# Minuscule Schubert varieties and mirror symmetry
Makoto Miura Graduate School of Mathematical Sciences, University of Tokyo,
Komaba Meguro-ku, Tokyo 153-8914, Japan [email protected]
###### Abstract.
We study the mirror symmetry for smooth complete intersection Calabi–Yau
3-folds in minuscule Schubert varieties using their degenerations to Hibi
toric varieties. Listing all these Calabi–Yau 3-folds up to deformation
equivalences, we find a new example of smooth Calabi–Yau 3-folds of Picard
number one, which is a complete intersection in a locally factorial Schubert
variety $\boldsymbol{\Sigma}$ of the Cayley plane $\mathbb{O}\mathbb{P}^{2}$.
We calculate topological invariants and BPS numbers of this Calabi–Yau 3-fold
and conjecture that it has a non-trivial Fourier–Mukai partner.
###### Key words and phrases:
Calabi-Yau, mirror symmetry, minuscule, Schubert variety, toric degeneration.
###### 2010 Mathematics Subject Classification:
Primary 14J32, 14J33; Secondary 14M15, 14M25
## 1\. Introduction
Thousands examples of mirror pairs of Calabi–Yau 3-folds have given a driving
power to the researches of mirror symmetry phenomenon. The earliest examples
of mirror-constructed Calabi–Yau 3-folds include a quintic 3-fold [GP], which
is appeared in relation to a conformal field theory in theoretical physics and
remains a leading example of mirror symmetry, together with complete
intersections in the product of projective spaces and weighted projective
spaces. Further, we have a plenty examples as hypersurfaces [Bat1] and
complete intersections [Bor] in Gorenstein toric Fano varieties. The mirror
construction for these examples is based on the combinatorial duality of
reflexive polytopes and nef-partitions. An extension of this mirror
construction via conifold transition gives other examples as complete
intersections in Grassmannians [BCFKvS1], partial flag manifolds [BCFKvS2] and
smoothing of terminal hypersurfaces in Gorenstein toric Fano 4-folds [BK]. The
mirror construction in these cases is also formulated by [Bat2] as a kind of
monomial-divisor correspondence [AGM].
Studying this mirror construction via conifold transitions, we find a new
example of smooth Calabi–Yau 3-fold in a Schubert variety of the Cayley plane
$\mathbb{O}\mathbb{P}^{2}$, which seems to have a non-trivial Fourier–Mukai
partner. More generally, we study complete intersection Calabi–Yau 3-folds in
the so-called minuscule Schubert varieties in this paper (cf. Definition 2.2)
and obtain the following results:
###### Main results (Proposition 3.1 and Proposition 4.1).
There is a unique deformation equivalent class of smooth complete intersection
Calabi–Yau 3-folds in minuscule Schubert varieties, except for 11 known
examples in homogeneous spaces. This Calabi–Yau 3-fold $X$ is a general
complete intersection of hyperplanes in a Schubert variety
$\boldsymbol{\Sigma}$ in the Cayley plane $\mathbb{O}\mathbb{P}^{2}$ with
$h^{1,1}(X)=1$ and $h^{2,1}(X)=52$, whose topological invariants are
$\deg(X)=33,\ \ \ c_{2}(X)\cdot H=78,\ \ \ \chi(X)=-102,$
where $H$ is the ample generator of the Picard group
$\operatorname{Pic}(X)\cong\mathbb{Z}$.
The Schubert variety $\boldsymbol{\Sigma}\subset\mathbb{O}\mathbb{P}^{2}$
above is a 12 dimensional locally factorial Fano variety of Picard number one
(cf. Definition 2.5 and Proposition 2.9). The Calabi–Yau 3-fold
$X=\boldsymbol{\Sigma}(1^{9})$ in Main results, a general complete
intersection of nine hyperplanes, is our main interest because of its mirror
symmetric property. We study a family of affine complete intersections in
$(\mathbb{C}^{*})^{n}$ which are expected to be birational to the conjectural
mirror manifolds of $X$ [BCFKvS1] and obtain the Picard–Fuchs equation
annihilating the fundamental period. The properties of the monodromy of this
differential equation admit the geometric interpretation around another
maximally unipotent monodromy (MUM) point $x=\infty$ and leads us to
Conjecture 5.6, the existence of a non-trivial Fourier–Mukai partner. The
integrality of the BPS numbers also supports this conjecture.
The paper is organized as follows.
In Section 2 we give some preliminaries. We deal with generalities on posets
and distributive lattices, geometry of minuscule Schubert varieties and Hibi
toric varieties, and toric degenerations. In particular, we bring in the
combinatorial description of singular loci of minuscule Schubert varieties and
Hibi toric varieties, which plays a key role for the proof of Main results.
In Section 3 we make a list of all the deformation equivalent (diffeomorphic)
classes of smooth complete intersection Calabi–Yau 3-folds in minuscule
Schubert varieties. We will see that there is a unique new example of such
Calabi–Yau 3-folds, embedded in a locally factorial Schubert variety
$\boldsymbol{\Sigma}$ in the Cayley plane $\mathbb{O}\mathbb{P}^{2}$
(Proposition 3.1).
In Section 4 we give a computational method of calculating topological
invariants for a kind of Calabi–Yau 3-folds of Picard number one which
degenerate to complete intersection in Gorenstein Hibi toric varieties. We
work on $\boldsymbol{\Sigma}(1^{9})$ as an example, and give a proof of Main
results (Proposition 4.1). The topological Euler number is computed by using a
conifold transition.
In Section 5 we study the mirror symmetry of complete intersection Calabi–Yau
3-folds in minuscule Schubert varieties. We construct special one parameter
families of affine complete intersections in $(\mathbb{C}^{*})^{n}$
conjecturally birational to the mirror families (Conjecture 5.2) and give an
expression for the fundamental periods (Proposition 5.4). We focus on the
example $X=\boldsymbol{\Sigma}(1^{9})$ and obtain the Picard–Fuchs equation
for its mirror, which suggests the existence of a non-trivial Fourier–Mukai
partner of $X$ (Conjecture 5.6). We also perform the monodromy calculation and
the computation of BPS numbers using mirror symmetry. Every result seems very
similar to that happened for the examples of the Pfaffian-Grassmannian [Rød]
[HK] and the Reye congruence Calabi–Yau 3-fold [HT1, HT2].
In Appendix A, we treat other examples in minuscule homogeneous spaces.
In Appendix B, we put the table of BPS numbers calculated for
$X=\boldsymbol{\Sigma}(1^{9})$ and its conjectural Fourier–Mukai partner
$X^{\prime}$.
### Acknowledgments
The author would like to express his deep gratitude to his supervisor
Professor Shinobu Hosono for worthwhile suggestions and warm encouragement. He
greatly appreciates many helpful discussions with Daisuke Inoue, Atsushi
Kanazawa and Fumihiko Sanda at the periodical seminars. He would also like to
thank Professor Yoshinori Gongyo, Professor Takehiko Yasuda, Doctor Atsushi
Ito and Mr. Taro Sano for providing useful comments to improve the work. Part
of this paper was written at Mathematisches Institute Universität Tübingen
during his stay from October 1 to December 25, 2012. He owes a very important
debt to Institutional Program for Young Researcher Overseas Visits by JSPS for
this stay. It is a pleasure to thank Professor Doctor Victor Batyrev for
valuable comments and creating a nice environment for the author.
## 2\. Preliminaries
### 2.1. Posets and distributive lattices
A partially ordered set is called a poset for short. Let $u$ and $v$ be
distinct elements of a poset $P$. There are three possibilities that $u\prec
v$, $v\prec u$ or these elements are incomparable, written as $u\not\sim v$.
We say that $u$ covers $v$ if $u\succ v$ and there is no $w\in P$ with $u\succ
w\succ v$. The Hasse diagram of a poset $P$ is the oriented graph with vertex
set $P$, having an edge $e=\\{u,v\\}$ going down from $u$ to $v$ whenever $u$
covers $v$ in $P$. We can read many informations for a poset by drawing the
Hasse diagram in a plane. For a poset $P$, an order ideal is a subset
$I\subset P$ with the property that
$u\in I\text{ and }v\prec u\text{ imply }v\in I.$
A lattice $L$ is a poset for which each pair of elements $\alpha,\beta\in L$
has the least upper bound $\alpha\vee\beta$ (called the join) and the greatest
lower bound $\alpha\wedge\beta$ (called the meet) in $L$. A distributive
lattice is a lattice on which the following identity holds for all triple
elements $\alpha,\beta,\gamma\in L$
$\displaystyle\alpha\wedge(\beta\vee\gamma)=(\alpha\wedge\beta)\vee(\alpha\wedge\gamma).$
It is easy to see that a finite lattice $L$ has the unique maximal and minimal
element with respect to the partial order on $L$. In a lattice $L$, an element
$\alpha$ is said to be join irreducible if $\alpha$ is neither the minimal
element nor the join of a finite set of other elements.
For a finite poset $P$, the order ideals of $P$ form a distributive lattice
$J(P)$ with the partial order given by set inclusions. The join and the meet
on $J(P)$ correspond to the set union and the set intersection, respectively.
On the other hand, the full subposet of join irreducible elements of $J(P)$
coincides with $P$ by the Birkhoff representation theorem [Bir]. In fact, this
gives a one-to-one correspondence between finite posets and finite
distributive lattices. We give an example of this correspondence in the
following.
###### Example 2.1.
We introduce the example which appears repeatedly throughout this paper. The
Hasse diagrams of $P$ and $J(P)$ are described in Figure 1. The distributive
lattice $J(P)$ is obtained by drawing the Hasse diagrams for each order ideals
of $P$ explicitly. In particular, the maximal element and the minimal element
of $J(P)$ correspond to the whole $P$ and the empty set $\varnothing$,
respectively. A circled vertex in $J(P)$ represents a join irreducible element
of $J(P)$, i.e. the vertex with exactly one edge below. We can easily
reconstruct the poset $P$ as the set of circled vertices with the induced
order in $J(P)$.
Figure 1. The Birkhoff correspondence
Now we introduce further definitions on a finite poset. Denote that the source
$s(e)=u$ and the target $t(e)=v$ for an edge $e=\\{u,v\\}$ of the Hasse
diagram of $P$ if $u$ covers $v$. A chain of length $k$ in $P$ is a sequence
of elements $u_{0}\prec u_{1}\prec\cdots\prec u_{k}\in P$. A chain is called
maximal if there is no $v\prec u_{0}$ or $w\succ u_{k}$ in $P$ and $u_{i}$
covers $u_{i-1}$ for all $1\leq i\leq k$. We call a finite poset $P$ is pure
if every maximal chain has the same length. Let us define the associated
bounded poset $\hat{P}=P\cup\left\\{\hat{0},\hat{1}\right\\}$ for any finite
poset $P$, by extending the partial order on $P$ with $\hat{0}\prec
u\prec\hat{1}$. Every finite poset $P$ has a height function $h$ by defining
$h(u)$ to be the length of the longest chain bounded above by $u$ in
$\hat{P}$. We also define the height $h_{P}$ of $P$ as $h(\hat{1})$. For
example, $h_{P}=9$ for the pure poset $P$ in Example 2.1.
### 2.2. Minuscule Schubert varieties
#### 2.2.1.
We give a brief summary for minuscule homogeneous spaces and minuscule
Schubert varieties. A basic reference of this subject is [LMS]. Let $G$ be a
simply connected simple complex algebraic group, $B$ a Borel subgroup and $T$
a maximal torus in $B$. We denote by $R^{+}$ the set of positive roots and by
$S=\left\\{\alpha_{1},\dots,\alpha_{n}\right\\}$ the set of simple roots. Let
$W$ be the Weyl group of $G$. Denote by $\Lambda$ the character group of $T$,
also called the weight lattice of $G$. The weight lattice $\Lambda$ is
generated by the fundamental weights $\lambda_{1},\dots,\lambda_{n}$ defined
by $(\alpha_{i}^{\vee},\lambda_{j})=\delta_{ij}$ for $1\leq i,j\leq n$, where
$(,)$ is a $W$-invariant inner product and
$\alpha^{\vee}:=2\alpha/(\alpha,\alpha)$. An integral weight $\lambda=\sum
n_{i}\lambda_{i}\in\Lambda$ is said to be dominant if $n_{i}\geq 0$ for all
$i=1,\dots,n$. For an integral dominant weight $\lambda\in\Lambda$, we denote
by $V_{\lambda}$ the irreducible $G$-module of the highest weight $\lambda$.
The associated homogeneous space $G/Q$ of $\lambda$ is the $G$-orbit of the
highest weight vector in the projective space $\mathbb{P}(V_{\lambda})$, where
$Q\supset B$ is the associated parabolic subgroup of $G$. A Schubert variety
in $G/Q$ is the closure of a $B$-orbit in $G/Q$. Recall the following
definition of a minuscule weight, a minuscule homogeneous space and minuscule
Schubert varieties.
###### Definition 2.2 (cf. [LMS, Definition 2.1]).
Let $\lambda\in\Lambda$ be a fundamental weight. We call $\lambda$ minuscule
if it satisfies the following equivalent conditions.
1. (1)
Every weight of $V_{\lambda}$ is in the orbit $W\lambda\subset\Lambda$.
2. (2)
$(\alpha^{\vee},\lambda)\leq 1\text{ for all }\alpha\in R^{+}$.
The homogeneous space $G/Q$ associated with a minuscule weight $\lambda$ is
said to be minuscule. The Schubert varieties in minuscule $G/Q$ are also
called minuscule.
Let us recall that a parabolic subgroup $Q\supset B$ is determined by a subset
$S_{Q}$ of $S$ associated with negative root subgroups. A useful notation for
a homogeneous space $G/Q$ is to cross the nodes in the Dynkin diagram which
correspond to the simple roots in $S\setminus S_{Q}$. With this notation, the
minuscule homogeneous spaces are as shown in Table 1. This contains the
Grassmannians $G(k,n)$, the orthogonal Grassmannians $OG(n,2n)$, even
dimensional quadrics $Q^{2n}$ and, finally, the Cayley plane
$\mathbb{O}\mathbb{P}^{2}=E_{6}/Q_{1}$ and the Freudenthal variety
$E_{7}/Q_{7}$, where we use the Bourbaki labelling for the roots. We omit two
kind of minuscule weights for groups of type $B$ and type $C$, since they give
the isomorphic varieties to those for simply laced groups.
Table 1. Minuscule homogeneous spaces
The Weyl group $W$ is generated by simple reflections $s_{\alpha}\in W$ for
$\alpha\in S$. These generators define the length function $l$ on $W$. Let us
denote by $W_{Q}$ the Weyl group of $Q$, i.e. the subgroup generated by
$\\{s_{\alpha}\in W\mid\alpha\in S_{Q}\\}$, and by $W^{Q}$ the set of minimal
length representatives of the coset $W/W_{Q}$ in $W$. For any $w\in W^{Q}$, we
denote by $X(w)=\overline{BwQ/Q}$ the Schubert variety in $G/Q$ associated
with $w$, which is a $l(w)$-dimensional normal Cohen–Macaulay projective
variety with at worst rational singularities. There is a natural partial order
$\prec$ on $W^{Q}$ called the Bruhat order, defined as $w_{1}\preceq
w_{2}\Leftrightarrow X(w_{1})\subset X(w_{2})$. We recall the following
fundamental fact for minuscule homogeneous spaces.
###### Proposition 2.3 ([Pro, Proposition V.2]).
For a minuscule homogeneous space $G/Q$, the poset $W^{Q}$ is a finite
distributive lattice.
From Proposition 2.3 and the Birkhoff representation theorem, we can define
the minuscule poset $P_{Q}$ for the minuscule $G/Q$ such that $J(P_{Q})=W^{Q}$
as in [Pro]. Moreover, the order ideal $P_{w}\subset P_{Q}$ associated with
$w\in W^{Q}$ is called the minuscule poset for the minuscule Schubert variety
$X(w)\subset G/Q$. For example, the order ideals $P_{Q}$, $\varnothing\subset
P_{Q}$ turn out to be the minuscule posets for the total space $X(w_{Q})=G/Q$
and the $B$-fixed point $X(\mathrm{id})=Q/Q$, respectively, where $w_{Q}$ is
the unique longest element in $W^{Q}$. The minuscule poset for minuscule
Schubert varieties is a generalization of the Young diagram for Grassmann
Schubert varieties.
###### Example 2.4.
An easy method to compute the Hasse diagram of $W^{Q}$ is to trace out the
$W$-orbit of certain dominant weight whose stabilizer coincides with $W_{Q}$
(cf. [BE, $\S$4.3]). Denote by $(ij\cdots k)$ the element
$w=s_{\alpha_{i}}s_{\alpha_{j}}\cdots s_{\alpha_{k}}\in W$, where $s_{\alpha}$
is simple reflection with respect to $\alpha\in S$. The initial part of the
Hasse diagram of $W^{Q}$ for the Cayley plane $\mathbb{O}\mathbb{P}^{2}$ is
the following
where the right covers the left for connected two elements with respect to the
Bruhat order. Thus we obtain the Hasse diagram of the distributive lattice
$W^{Q}$ and hence the minuscule poset for every Schubert variety in
$\mathbb{O}\mathbb{P}^{2}$.
###### Definition 2.5.
In the above notation, let us set $w=(345134265431)\in W^{Q}$. We denote by
$\boldsymbol{\Sigma}$ the associated $12$-dimensional Schubert variety $X(w)$
in the Cayley plane $\mathbb{O}\mathbb{P}^{2}$, which corresponds to the
minuscule poset $P$ in Example 2.1.
###### Remark 2.6.
We remark on another geometric characterization of minuscule homogeneous
spaces in [LMS, Definition 2.1]. A fundamental weight $\lambda$ is minuscule
(Definition 2.2) if and only if the following condition holds.
1. (3)
For the associated homogeneous space $G/Q$, the Chevalley formula
$[H]\cdot[X(w)]=\sum_{w\text{ covers }w^{\prime}}[X(w^{\prime})]$
holds for all $w\in W^{Q}$ in the Chow ring of $G/Q$, where $H$ is the unique
Schubert divisor in $G/Q$ and $W^{Q}$ is the poset with the Bruhat order.
As a corollary, it turns out that the degree of a minuscule Schubert variety
$X(w)$ with respect to $\mathcal{O}_{G/Q}(1)|_{X(w)}$ equals the number of
maximal chains in $J(P_{w})$. For example, we obtain
$\deg\boldsymbol{\Sigma}=33$ by counting the maximal chains in $J(P)$ in
Figure 1.
#### 2.2.2.
Next we introduce further definitions for the description of singularities of
minuscule Schubert varieties. As we expect from the computation in Example
2.4, the Bruhat order on $W^{Q}$ is generated by simple reflections for
minuscule $G/Q$ [LW, Lemma 1.14], that is,
$w_{1}\text{ covers }w_{2}\Leftrightarrow w_{1}=s_{\alpha}\cdot w_{2}\text{
and }l(w_{1})=l(w_{2})+1\text{ for some }\alpha\in S.$
From this fact, a join irreducible element $u\in W^{Q}$ covers the unique
element $s_{\beta_{Q}(u)}\cdot u\in W^{Q}$ where $\beta_{Q}(u)\in S$. Thus we
can define the natural coloration $\beta_{Q}:P_{Q}\rightarrow S$ for a
minuscule poset $P_{Q}$ by simple roots $S$. We also define the coloration
$\beta_{w}$ on each minuscule poset $P_{w}\subset P_{Q}$ by restricting
$\beta_{Q}$ on $P_{w}$. The minuscule poset $P_{w}$ with the coloration
$\beta_{w}:P_{w}\rightarrow S$ has in fact the same information as the
minuscule quiver introduced by Perrin [Per1, Per2], which gives a good
description of geometric properties of minuscule Schubert varieties $X(w)$.
Now we translate the combinatorial notions and useful facts on the geometry of
minuscule Schubert varieties $X(w)$ from [Per1, Per2] in our terminology.
###### Definition 2.7.
Let $P$ be a minuscule poset with the coloration $\beta:P\rightarrow S$.
1. (1)
A peak of $P$ is a maximal element $u$ in $P$.
2. (2)
A hole of $P$ is a maximal element $u$ in $\beta^{-1}(\alpha)$ for some
$\alpha\in S$ such that there are exactly two elements $v_{1},v_{2}\in P$ with
$u\prec v_{i}$ and $(\beta(u)^{\vee},\beta(v_{i}))\neq 0$ ($i=1,2$).
Let us denote by $\mathrm{Peaks}(P)$ and $\mathrm{Holes}(P)$ the set of peaks
and holes of $P$, respectively. A hole $u$ of the poset $P$ is said to be
essential if the order ideal $P^{u}:=\left\\{v\in P\mid v\not\succcurlyeq
u\right\\}$ contains all other holes in $P$.
Let $X(w)$ be a minuscule Schubert variety in $G/Q$ and $P_{w}$ the associated
minuscule poset. Weil and Cartier divisors on $X(w)$ are described in terms of
the poset $P_{w}$. In fact, it is clear that any Schubert divisor coincides
with a Schubert variety $D_{u}$ associated with $P_{w}^{u}$ for some
$u\in\mathrm{Peaks}(P_{w})$. It is well-known that the divisor class group
$\mathrm{Cl}(X(w))$ is the free $\mathbb{Z}$-module generated by the classes
of the Schubert divisors $D_{u}$ for $u\in\mathrm{Peaks}(P_{w})$, and the
Picard group $\mathrm{Pic}(X(w))$ is isomorphic to $\mathbb{Z}$ generated by
$\mathcal{O}_{G/Q}(1)|_{X(w)}$. As we saw in Remark 2.6, the Cartier divisor
corresponding to $\mathcal{O}_{G/Q}(1)|_{X(w)}$ is
$\sum_{u\in\mathrm{Peaks}(P_{w})}D_{u}.$
We use the following results by Perrin about the singularities of $X(w)$.
###### Proposition 2.8 ([Per1, Per2]).
Let $X(w)$ be a minuscule Schubert variety and $P_{w}$ the associated
minuscule poset.
1. (1)
[Per1, Proposition 4.17] An anticanonical Weil divisor of $X(w)$ is
$-K_{X(w)}=\sum_{u\in\mathrm{Peaks}(P_{w})}(h(u)+1)D_{u}.$
In particular, $X(w)$ is Gorenstein if and only if $P_{w}$ is pure. In this
case $X(w)$ is a Fano variety of index $h_{P_{w}}$.
2. (2)
[Per2, Theorem 2.7 (1)] The Schubert subvariety associated with the order
ideal $P_{w}^{u}\subset P_{w}$ for an essential hole $u$ of $P_{w}$ is an
irreducible component of the singular loci of $X(w)$. All the irreducible
components of the singular loci are obtained in this way.
We apply Proposition 2.8 to our main example $\boldsymbol{\Sigma}$ and obtain
the following.
###### Proposition 2.9.
Let $\boldsymbol{\Sigma}$ be the minuscule Schubert variety in
$\mathbb{O}\mathbb{P}^{2}$ defined by Definition 2.5.
1. (1)
$\boldsymbol{\Sigma}$ is a locally factorial Gorenstein Fano variety of index
9.
2. (2)
The singular locus of $\boldsymbol{\Sigma}$ is isomorphic to $\mathbb{P}^{5}$.
###### Proof.
The former holds because the corresponding minuscule poset $P$ (Figure 1) is
pure with $h_{P}=9$ and the unique peak. From the computation of the Hasse
diagram of $W^{Q}$ of $\mathbb{O}\mathbb{P}^{2}$ in Example 2.4, the
coloration $\beta:P\rightarrow S$ is given as the following picture.
Figure 2. The coloration of the minuscule poset $P$ and the singular locus
A unique (essential) hole of $P$ is the circled vertex $u$, whose color is
$\alpha_{2}\in S$. The corresponding Schubert subvariety is described by the
minuscule poset $P^{u}$, which coincides with the singular locus of
$\boldsymbol{\Sigma}$ by Proposition 2.8. It is isomorphic to $\mathbb{P}^{5}$
because the degree equals to one. ∎
Finally, we record the useful vanishing theorems for minuscule Schubert
varieties.
###### Theorem 2.10 ([LMS, Theorem 7.1]).
Let $\lambda$ be a minuscule weight, $G/Q\subset\mathbb{P}(V_{\lambda})$ the
associated homogeneous space and $X(w)\subset G/Q$ a minuscule Schubert
variety.
1. (1)
$H^{0}(\mathbb{P}(V_{\lambda}),\mathcal{O}(m))\rightarrow
H^{0}(X(w),\mathcal{O}(m))$ is surjective for all $m\geq 0$,
2. (2)
$H^{i}(X(w),\mathcal{O}(m))=0\,\text{ for all }m\in\mathbb{Z}\text{ and
}0<i<l(w),$
3. (3)
$H^{l(w)}(X(w),\mathcal{O}(m))=0\,\text{ for all }m\geq 0.$
### 2.3. Hibi toric varieties
#### 2.3.1.
We introduce another kind of projective varieties, called Hibi toric
varieties, deeply related with the minuscule Schubert varieties.
First let us define the polytope $\Delta(P)$ called the order polytope for a
finite poset $P$. These polytopes have been studied in a number of literature,
for example [Sta]. Let $N=\mathbb{Z}P$ and
$M=\operatorname{Hom}_{\mathbb{Z}}(N,\mathbb{Z})$ be the free
$\mathbb{Z}$-modules of rank $\lvert P\rvert$, and $M_{\mathbb{R}}$ the real
scalar extension of $M$. The order polytope $\Delta(P)$ is defined in
$M_{\mathbb{R}}$ as follows.
$\Delta(P):=\left\\{x=\sum_{u\in P}x_{u}\chi_{u}\in[0,1]^{P}\ \Bigg{|}\
x_{u}\leq x_{v}\text{ for all }u\prec v\in P\right\\},$
where $[0,1]^{P}\subset M_{\mathbb{R}}$ is a $\lvert P\rvert$-dimensional unit
cube of all real maps $x:P\rightarrow[0,1]$, and $\left\\{\chi_{u}\mid u\in
P\right\\}\subset M$ is the dual basis of $P\subset N$, i.e.
$\chi_{u}(v)=\delta_{uv}$ for $u,v\in P$. It is easy to see that the order
polytope $\Delta(P)$ is an integral convex polytope of dimension $\lvert
P\rvert$.
###### Definition 2.11.
We call the projective toric variety $\mathbb{P}_{\Delta(P)}$ defined by an
order polytope $\Delta(P)\subset M_{\mathbb{R}}$ the (projective) Hibi toric
variety for the poset $P$.
###### Example 2.12.
One of the simplest examples of order polytopes is the so-called
Gelfand–Tsetlin polytopes for fundamental weights of special linear groups
$SL(n+1,\mathbb{C})$. In general, the Gelfand–Tsetlin polytope for an integral
dominant weight
$\lambda=(\Lambda_{0},\dots,\Lambda_{n})\in\mathbb{Z}^{n+1}/\left<(1,\dots,1)\right>\simeq\Lambda$
is defined by the following inequalities in $\mathbb{R}^{n(n+1)/2}$
$\Lambda_{n}\leq x_{i+1,j+1}\leq x_{i,j}\leq
x_{i,j+1}\leq\Lambda_{i}\quad\text{ for all }0\leq i\leq j\leq n-1.$
These inequalities can be represented in a diagram, like as the left of Figure
3 for $n=3$.
Figure 3. the Gelfand–Tsetlin polytopes for $SL(4,\mathbb{C})$
In fact, the Gelfand–Tsetlin polytope for a fundamental weight
$\lambda=(1,\dots,1,0,\dots,0)$ is the order polytope for a poset of rectangle
shape, like as the right of Figure 3 for example. It coincides with the
minuscule poset for a Grassmannian of type $A$. The corresponding Hibi toric
variety $\mathbb{P}_{\Delta(P)}$ is the toric variety $P(k,n+1)$ defined by
[BCFKvS1].
###### Remark 2.13.
There is the alternative definition of the Hibi toric variety using its
homogeneous coordinate ring, which is standard in the literature. Let $J(P)$
be the distributive lattice of order ideals of a finite poset $P$. Denote by
$\mathbb{C}[J(P)]$ the polynomial ring over $\mathbb{C}$ in $\lvert
J(P)\rvert$ indeterminates $p_{\alpha}\ (\alpha\in J(P))$. Let
$I(J(P))\subset\mathbb{C}[J(P)]$ the homogeneous ideal generated by the
following binomial relations:
$p_{\tau}p_{\phi}-p_{\tau\wedge\phi}p_{\tau\vee\phi}\ (\tau\not\sim\phi).$
One can check that the graded algebra
$A_{J(P)}:=\mathbb{C}\left[J(P)\right]/I(J(P))$ with the standard
$\mathbb{Z}$-grading inherited from $\mathbb{C}\left[J(P)\right]$ coincides
with the homogeneous coordinate ring of the Hibi toric variety
$\mathbb{P}_{\Delta(P)}$ with the embedding defined by the very ample line
bundle associated with $\Delta(P)$. The graded algebra $A_{J(P)}$ is usually
called the Hibi algebra on the distributive lattice $J(P)$ (cf. [Hib]).
#### 2.3.2.
A nice property of Hibi toric varieties is that torus invariant subvarieties
in Hibi toric varieties are also Hibi toric varieties. We need an explicit
description of these subvarieties and their singularities. The subject was
well studied in [Wag] and also in [BL1, BL2] and [LM].
Let us recall the inequalities in the definition of an order polytope
$\Delta(P)$. They are generated by $x_{s(e)}\geq x_{t(e)}$ for all $e\in E$,
where $E$ is the set of edges of the Hasse diagram of
$\hat{P}=P\cup\left\\{\hat{0},\hat{1}\right\\}$ and $x_{\hat{0}}=0$ and
$x_{\hat{1}}=1$. We can get a face of $\Delta(P)$ by replacing some of these
inequalities with equalities. To describe such a replacement, we introduce
some further concepts for posets.
Recall that a full subposet $y\subset\hat{P}$ is a subset of $\hat{P}$ whose
poset structure is that inherited from $\hat{P}$. We call a full subposet
$y\subset\hat{P}$ connected if all the elements in $y$ are connected by edges
in the Hasse diagram of $y$, and convex if $u,v\in y$ and $u\prec w\prec v$
imply $w\in y$.
###### Definition 2.14.
A surjective map $f:\hat{P}\rightarrow\hat{P}^{\prime}$ is called a
contraction of the bounded poset $\hat{P}$ if every fiber $f^{-1}(i)\
(i\in\hat{P}^{\prime})$ is connected full subposet of $\hat{P}$ not containing
both $\hat{0}$ and $\hat{1}$, and the following condition holds for all
$u_{k},v_{k}\in f^{-1}(k)$ and $i\neq j\in\hat{P}^{\prime}$:
$\text{ a relation }u_{i}\prec u_{j}\text{ implies }v_{i}\not\succ v_{j}.$
###### Remark 2.15.
A contraction $f:\hat{P}\rightarrow\hat{P}^{\prime}$ gives a natural partial
order on the image set $\hat{P^{\prime}}$, i.e. the partial order generated by
the following relations:
$i\prec j\Leftrightarrow\text{there exist }u\in f^{-1}(i)\text{ and }v\in
f^{-1}(j)\text{ such that }u\prec v\text{ in }\hat{P}.$
Further, $\hat{P^{\prime}}$ turns out to be a bounded poset by setting
$\hat{1}\in f^{-1}(\hat{1})$ and $\hat{0}\in f^{-1}(\hat{0})$. Hence in fact,
the above definition of contraction coincides with the more abstract
definition in [Wag]; the fiber-connected tight surjective morphism of bounded
posets.
For a contraction $f:\hat{P}\rightarrow\hat{P^{\prime}}$, the corresponding
face of $\Delta(P)$ is given by
$\theta_{f}:=\left\\{x\in\Delta(P)\ \Bigg{|}\ x_{u}=x_{v}\text{ for all
}u,v\in f^{-1}(i)\text{ and }i\in\hat{P^{\prime}}\right\\}.$
Conversely, we can reconstruct the contraction from each face
$\theta_{f}\subset\Delta(P)$ by looking at the coordinates of general point in
$\theta_{f}$. Now we can rephrase the classical fact on the face structure of
order polytopes in our terminology (cf. [Wag, Theorem 1.2]).
###### Proposition 2.16.
Let $P$ be a finite poset, and $\Delta(P)$ the associated order polytope. The
above construction gives a one-to-one correspondence between the faces of
$\Delta(P)$ and the contractions of $\hat{P}$. Moreover, an inclusion of the
faces corresponds to a composition of contractions.
###### Remark 2.17.
It is obvious that the face
$\theta_{\hat{P}\rightarrow\hat{P^{\prime}}}\subset\Delta(P)$ coincides with
the $\lvert P^{\prime}\rvert$-dimensional order polytope $\Delta(P^{\prime})$
under the suitable choice of subspace of $M_{\mathbb{R}}$ and the unimodular
transformation. This means that the torus invariant subvarieties in Hibi toric
varieties are also Hibi toric varieties as noted before.
Similarly to the case of minuscule Schubert varieties, Weil and Cartier
divisors on a Hibi toric variety $\mathbb{P}_{\Delta(P)}$ are naturally
described in terms of the poset $P$. We can show that the divisor class group
$\mathrm{Cl}(\mathbb{P}_{\Delta(P)})$ is the free $\mathbb{Z}$-module of rank
$\lvert E\rvert-\lvert P\rvert$ and the Picard group
$\operatorname{Pic}(\mathbb{P}_{\Delta(P)})$ is the free $\mathbb{Z}$-module
whose rank coincides with the number of connected components of the Hasse
diagram of $P$. We use the following results on singularities of the Hibi
toric varieties.
###### Proposition 2.18 ([HH, Remark 1.6 and Lemma 1.4]).
Let $P$ be a finite poset. The Hibi algebra $A_{J(P)}$ is Gorenstein if and
only if $P$ is pure. In this case the Hibi toric variety
$\mathbb{P}_{\Delta(P)}$ is a Gorenstein Fano variety with at worst terminal
singularities.
###### Theorem 2.19 ([Wag, Theorem 2.3 and Proof of Corollary 2.4]).
Let $\mathbb{P}_{\Delta(P)}$ be a Hibi toric variety for a finite poset $P$. A
face $\theta_{f}\subset\Delta(P)$ corresponds to an irreducible component of
the singular loci of $\mathbb{P}_{\Delta(P)}$ if and only if one of the fiber
$f^{-1}(i)$ of the contraction $f:\hat{P}\rightarrow\hat{P}^{\prime}$ is a
minimal convex cycle and all other fibers $f^{-1}(j)$ $(j\neq i)$ consist of
one element.
###### Proposition 2.20 ([HW, Corollary of Lemma 5]).
Let $P$ be a finite poset. The Poincaré series of the Hibi algebra $A_{J(P)}$
is written as the following form:
$\mathcal{P}(t)=\sum_{i=0}^{\lvert P\rvert}c_{i}(t/(1-t))^{i},$
where $c_{0}=1$ and $c_{i}\ (i\neq 0)$ is the number of the chains of length
$i$ in $J(P)$. In particular, the degree of
$\mathbb{P}_{\Delta(P)}\subset\operatorname{Proj}\mathbb{C}[J(P)]$ equals the
number of maximal chains in $J(P)$.
### 2.4. Toric degenerations
Now we state the key theorem by Gonciulea and Lakshmibai in our terminology,
which gives the relationship between minuscule Schubert varieties and Hibi
toric varieties.
###### Theorem 2.21 ([GL, Theorem 7.34]).
A minuscule Schubert variety $X(w)$ degenerates to the Hibi toric variety
$\mathbb{P}_{\Delta(P_{w})}$, where $P_{w}$ is the minuscule poset for $X(w)$.
More precisely, there exists a flat family $\mathcal{V}\rightarrow\mathbb{C}$
such that $\mathcal{V}_{t}\simeq X(w)$ for all $t\in\mathbb{C}^{*}$ and
$\mathcal{V}_{0}\simeq\mathbb{P}_{\Delta(P_{w})}$.
###### Corollary 2.22.
A Gorenstein minuscule Schubert variety has at worst terminal singularities.
###### Proof.
By [HH] a Gorenstein Hibi toric variety has at worst terminal (and hence
canonical) singularities, Proposition 2.18 (1). It is known that a deformation
of a terminal singularity is again terminal [Nak] as with the case of
canonical singularities [Kaw]. Hence the assertion follows from Theorem 2.21.
∎
## 3\. List of complete intersection Calabi–Yau 3-folds
In this section, we study the smooth complete intersection Calabi–Yau 3-folds
in minuscule Schubert varieties. We show that there is a unique new
deformation equivalent class of such Calabi–Yau 3-folds, that is, the complete
intersection of nine hyperplanes in a locally factorial Schubert variety
$\boldsymbol{\Sigma}$ defined by Definition 2.5.
First we fix some basic terminologies to clarify the meaning of our result.
Let $X(w)$ be a minuscule Schubert variety. We call a subvariety $X\subset
X(w)$ a complete intersection if it is the common zero locus of
$r=\mathrm{codim}X$ global sections of invertible sheaves on $X(w)$. We can
denote by $X=X(w)(d_{1},\dots,d_{r})$ the complete intersection variety of
general $r$ sections of degree $d_{1},\dots,d_{r}$ with respect to
$\mathcal{O}_{G/Q}(1)|_{X(w)}$ since $\operatorname{Pic}X(w)\simeq\mathbb{Z}$.
A Calabi–Yau variety $X$ is a normal projective variety with at worst
Gorenstein canonical singularities and with trivial canonical bundle
$K_{X}\simeq 0$ such that $H^{i}(X,\mathcal{O}_{X})=0$ for all $0<i<\dim X$.
Two smooth varieties $X_{1}$ and $X_{2}$ are called deformation equivalent if
there exist a smooth family $\mathcal{X}\rightarrow U$ over a connected open
base $U\subset\mathbb{C}$ such that $\mathcal{X}_{t_{1}}\simeq X_{1}$ and
$\mathcal{X}_{t_{2}}\simeq X_{2}$ for some $t_{1},t_{2}\in U$. In this case,
$X_{1}$ and $X_{2}$ turn out to be diffeomorphic.
Now we summarize all possible smooth complete intersection Calabi–Yau 3-folds
in minuscule Schubert varieties:
###### Proposition 3.1.
A smooth complete intersection Calabi–Yau 3-fold in a minuscule Schubert
variety is one of that listed in the following table up to deformation
equivalence.
In this table, 10 known examples in Grassmannians of type A include five in
projective spaces;
$\mathbb{P}^{4}(5),\ \mathbb{P}^{5}(2,4),\ \mathbb{P}^{5}(3^{2}),\
\mathbb{P}^{6}(2^{2},3)\text{ and }\mathbb{P}^{7}(2^{4}),$
and five in others, whose mirror symmetry was discussed in [BCFKvS1];
$G(2,5)(1^{2},3),\ G(2,5)(1,2^{2}),\ G(2,6)(1^{4},2),\ G(3,6)(1^{6})\text{ and
}G(2,7)(1^{7}).$
For all these Calabi–Yau 3-folds, the Picard number equals to one.
Before the proof of Proposition 3.1, we show a lemma on the Hibi toric
varieties to handle the complexity of the redundant appearance of deformation
equivalent varieties. The special case of this lemma was already used in
[BCFKvS1] to prove the mirror duality of Hodge numbers for a Calabi–Yau 3-fold
$G(2,5)(1^{2},3)$ and its mirror by reducing the argument to that of a
hypersurface in a Hibi toric variety.
###### Lemma 3.2.
Let $P$ be a finite poset and $P^{*}=P\cup\left\\{\hat{1}\right\\}$,
$P_{*}=P\cup\left\\{\hat{0}\right\\}$ the posets where $u\prec\hat{1}$,
$\hat{0}\prec u$ for all $u\in P$, respectively. The Hibi toric varieties
$\mathbb{P}_{\Delta(P^{*})}$ and $\mathbb{P}_{\Delta(P_{*})}$ are the
projective cones over $\mathbb{P}_{\Delta(P)}$ in
$\operatorname{Proj}\mathbb{C}[J(P^{*})]$ and
$\operatorname{Proj}\mathbb{C}[J(P_{*})]$, respectively.
###### Proof.
Let $1=1_{J(P^{*})}$ be the maximal element in $J(P^{*})$. This $1$ is join
irreducible in $J(P^{*})$ because $J(P)$ has the unique maximal element.
Therefore the variable $p_{1}$ in the homogeneous coordinate ring
$A_{J(P^{*})}$ of $\mathbb{P}_{\Delta(P^{*})}$ does not involve in any
relation $p_{\tau}p_{\phi}-p_{\tau\wedge\phi}p_{\tau\vee\phi}\
(\tau\not\sim\phi)$. Thus the Hibi toric variety $\mathbb{P}_{\Delta(P^{*})}$
is the projective cone over $\mathbb{P}_{\Delta(P)}$. The same argument is
also valid for the minimal element $0=0_{J(P_{*})}$ in $J(P_{*})$ by
considering the order dual of $P$. ∎
###### Proof of Proposition 3.1.
We may assume that the ambient minuscule Schubert variety is Gorenstein. In
fact, from the adjunction formula and the Grothendieck–Lefschetz theorem for
divisor class groups of normal projective varieties [RS], we have an explicit
formula of the canonical divisor as a Cartier divisor,
$K_{X(w)}=-D_{1}-\cdots-D_{r}$ where $D_{j}\subset X(w)$ is a very ample
Cartier divisor of degree $d_{j}$ and $X(w)(d_{1},\dots,d_{r})$ is a general
Calabi–Yau complete intersection.
Let $X(w)$, $X(w^{\prime})$ be minuscule Schubert varieties and $P_{w}$,
$P_{w^{\prime}}$ the corresponding minuscule posets, respectively. Assume that
$P_{w^{\prime}}$ coincides with a $d$-times iterated extension
$(P_{w})^{*\cdots*}_{*\cdots*}$ of $P_{w}$. From Lemma 3.2, it holds that
$\mathbb{P}_{\Delta(P_{w})}$ is isomorphic to a complete intersection of $d$
general hyperplanes in $\mathbb{P}_{\Delta(P_{w^{\prime}})}$. By Theorem 2.21,
there exist the toric degenerations of $X(w)$ and $X(w^{\prime})$ to the Hibi
toric varieties $\mathbb{P}_{\Delta(P_{w})}$ and
$\mathbb{P}_{\Delta(P_{w^{\prime}})}$, respectively. This means that general
complete intersection Calabi–Yau 3-folds $X=X(w)(d_{1},\dots,d_{r})$ and
$X^{\prime}=X(w^{\prime})(1^{d},d_{1},\dots,d_{r})$ can be connected by flat
deformations through a complete intersection
$X_{0}=\mathbb{P}_{\Delta(P_{w})}(d_{1},\dots,d_{r})$. Since $X_{0}$ has at
worst terminal singularities, the Kuranishi space is smooth by [Nam, Theorem
A] and the degenerating loci have a positive complex codimension. Therefore
$X$ and $X^{\prime}$ are connected by smooth deformation. Thus we can
eliminate such redundancy arisen from iterated extensions of minuscule posets.
A Gorenstein minuscule Schubert variety $X(w)$ with minuscule poset $P=P_{w}$
is a $\lvert P\rvert$-dimensional Fano variety of index $h_{P}$ as we stated
in Proposition 2.8 (1). The condition for general complete intersections in
$X(w)$ to be Calabi–Yau 3-folds gives a strong combinatorial restriction for
the poset $P$ as follows,
$h_{P}-1\leq|P|\leq h_{P}+3.$
On the other hand, there is a complete list of the minuscule posets in [Per1].
Hence we can make a list of the complete intersection Calabi–Yau 3-folds by
counting such posets.
We can check that the resulting 3-folds $X\subset X(w)$ with trivial canonical
bundles turn out to be Calabi–Yau varieties after some computation using the
vanishing theorems for $X(w)$, Theorem 2.10. We verify the smoothness of these
3-folds by looking at the codimension of the singular loci of $X(w)$ using
Proposition 2.8 (2). For example, a general linear section
$X=\boldsymbol{\Sigma}(1^{9})$ is smooth since the singular loci of
$\boldsymbol{\Sigma}$ have codimension 7 as we saw in Proposition 2.9. All the
smooth cases are contained in locally factorial minuscule Schubert varieties,
i.e. the minuscule poset $P$ has the unique peak. Thus the Picard number
equals to one again by the Grothendieck–Lefschetz theorem for divisor class
groups [RS]. This completes the proof. ∎
## 4\. Topological invariants
Now we explain our calculation of topological invariants for smooth complete
intersection Calabi–Yau 3-folds in minuscule Schubert varieties by taking
$X=\boldsymbol{\Sigma}(1^{9})$ as an example. The topological invariants mean
the degree $\deg(X)=\int_{X}H^{3}$, the linear form associated with the second
Chern class $c_{2}(X)\cdot H=\int_{X}c_{2}(X)\cup H$ and the Euler number
$\chi(X)=\int_{X}c_{3}(X)$, where $H$ is the ample generator of
$\operatorname{Pic}(X)\simeq\mathbb{Z}$. These three invariants characterize
the diffeomorphic class of smooth simply connected Calabi–Yau 3-folds of
Picard number one [Wal].
###### Proposition 4.1.
The topological invariants of $X=\boldsymbol{\Sigma}(1^{9})$ are
$\mathrm{deg}(X)=33,\ \ \ c_{2}(X)\cdot H=78,\ \ \ \chi(X)=-102.$
###### Proof.
The degree of $X$ coincides with that of the minuscule Schubert variety
$\boldsymbol{\Sigma}\subset\mathbb{OP}^{2}$ since the ample generator
$\mathcal{O}_{\boldsymbol{\Sigma}}(1)$ of
$\operatorname{Pic}\boldsymbol{\Sigma}$ is the restriction of
$\mathcal{O}_{\mathbb{OP}^{2}}(1)$ and $X$ is a linear section. We obtain
$\mathrm{deg}(\boldsymbol{\Sigma})=33$ by using the Chevalley formula of
$\mathbb{O}\mathbb{P}^{2}$ as we already saw in Remark 2.6.
The Schubert variety $V^{0}:=\boldsymbol{\Sigma}$ and its general complete
intersections $V^{j}:=\boldsymbol{\Sigma}(1^{j})$ have at worst rational
singularities (in fact at worst terminal singularities by Corollary 2.22).
Hence the Kawamata–Viehweg vanishing theorem gives
$H^{i}(V^{j},\omega_{V^{j}}\otimes\mathcal{O}_{V^{j}}(k))=H^{i}(V^{j},\mathcal{O}_{V^{j}}(k+j-9))=0\text{\quad
for all }i>0\text{ and }k>0.$
Together with the long cohomology exact sequences of
$0\rightarrow\mathcal{O}_{V^{j}}(k)\rightarrow\mathcal{O}_{V^{j}}(k+1)\rightarrow\mathcal{O}_{V^{j+1}}(k+1)\rightarrow
0,$
the holomorphic Euler number of $X=V^{9}$ becomes
$\chi(X,\mathcal{O}_{X}(1))=\dim H^{0}(X,\mathcal{O}_{X}(1))=\dim
H^{0}(\boldsymbol{\Sigma},\mathcal{O}_{\boldsymbol{\Sigma}}(1))-9=\lvert
J(P)\rvert-9=12.$
On the other hand, it holds that
$\chi(X,\mathcal{O}_{X}(1))=\frac{1}{6}\mathrm{deg}(X)+\frac{1}{12}c_{2}(X)\cdot
H$
from the Hirzebruch–Riemann–Roch theorem of the smooth Calabi–Yau 3-fold $X$.
Thus we obtain $c_{2}(X)\cdot H=78$.
For the topological Euler number $\chi(X)$, we use the toric degeneration of
$\boldsymbol{\Sigma}$ to the Hibi toric variety $\mathbb{P}_{\Delta(P)}$,
Theorem 2.21. A general complete intersection Calabi–Yau 3-fold $X_{0}$ in the
degenerated variety $\mathbb{P}_{\Delta(P)}$ has finitely many nodes by a
Bertini type theorem for toroidal singularities. In fact, we know that three
dimensional Gorenstein terminal toric singularities are at worst nodes. Thus
we obtain a conifold transition $Y$ of $X$, which is a smooth Calabi–Yau
complete intersection in a MPCP resolution $\widehat{\mathbb{P}}_{\Delta(P)}$
of $\mathbb{P}_{\Delta(P)}$ defined by [Bat1]. By Theorem 5.1, the Hodge
numbers of $Y$ can be calculated as $h^{1,1}(Y)=5$ and
$\displaystyle h^{2,1}(Y)$ $\displaystyle=9\left(\lvert
J(P)\rvert-9\right)-\sum_{e\in
E}(l^{*}(9\theta_{e})-9l^{*}(8\theta_{e}))-\lvert P\rvert$
$\displaystyle=96-\sum_{e\in E}(l^{*}(9\theta_{e})-9l^{*}(8\theta_{e})).$
To count the number of interior integral points in each facet, we use
Proposition 2.16 which states a face of the order polytope is also the order
polytope of some poset $P^{\prime}$. For each facet $\theta_{e}$, the
corresponding poset $P^{\prime}$ (or $\hat{P}^{\prime}$) is easily obtained by
replacing the inequality $x_{s(e)}\geq x_{t(e)}$ by the equality
$x_{s(e)}=x_{t(e)}$ and by considering the induced partial order. The Hasse
diagram of resulting posets $P^{\prime}$ are shown in the following table,
where the numbering of edges is chosen from the upper left in a picture of the
Hasse diagram of $\hat{P}$ like as in Figure 1.
As shown in this table, some posets $P^{\prime}$ are pure and others are not
pure. For a pure poset $P^{\prime}$, the face
$\theta_{e}\simeq\Delta(P^{\prime})$ defines Gorenstein Hibi toric variety
$\mathbb{P}_{\Delta(P^{\prime})}$. Then we know
$l^{*}(h_{P^{\prime}}\theta_{e})=1$ and
$l^{*}((h_{P^{\prime}}+1)\theta_{e})=\lvert J(P^{\prime})\rvert$. When
$P^{\prime}$ is not pure, we can also easily obtain the number
$l^{*}(k\theta_{e})$ by counting the points satisfying the inequalities of the
polytope $k\theta_{e}\simeq k\Delta(P^{\prime})$ strictly. For example,
$9\theta_{2}$ contains three internal integral points corresponding to
From the table, we obtain $h^{2,1}(Y)=37$, hence
$\chi(Y)=2(h^{1,1}(Y)-h^{2,1}(Y))=-64$.
Let us recall that the conifold transition is a surgery of Calabi–Yau 3-folds
replacing finite vanishing $S^{3}$ by the same number of exceptional
$\mathbb{P}^{1}\simeq S^{2}$ as in [Cle]. From the inclusion-exclusion
principle of the Euler numbers, $\chi(X)$ and $\chi(Y)$ are related with each
other as:
$\chi(X)=\chi(Y)-2p,$
where $p$ is the number of nodes on $X_{0}$. Then we need to know the total
degree of codimension three singular loci of $\mathbb{P}_{\Delta(P)}$, which
equals the number $p/\prod_{j}d_{j}=p$ in our case. From Proposition 2.18 (2),
an irreducible component of singular loci of the Hibi toric variety
$\mathbb{P}_{\Delta(P)}$ corresponds to a minimal convex cycle in $\hat{P}$.
There are four such cycles (or boxes) $b_{1},\cdots,b_{4}$ and all of them
define the codimension three faces in $\Delta(P)$ as in Proposition 2.16.
Again we can compute the corresponding index poset $P^{\prime}$ of them by the
method used above. The resulting posets $P^{\prime}$ are summarized as
follows.
From Proposition 2.18 (3), we can compute the degree of each irreducible
component of singular loci by counting the maximal chains in $J(P^{\prime})$.
Then we obtain that total degree $p$, that is, the number of nodes on $X_{0}$
is $19$. We conclude $\chi(X)=-102$. ∎
###### Remark 4.2.
1. (1)
The existence of the Calabi–Yau 3-fold with these topological invariants were
previously conjectured by [vEvS] from the monodromy calculations of Calabi–Yau
differential equations. We also perform the similar calculation in the next
section.
2. (2)
It may be possible to calculate the Euler number $\chi(X)$ in another way, by
computing the Chern–Mather class of the Schubert variety
$\boldsymbol{\Sigma}$. For the Grassmann Schubert varieties, this is done by
[Jon] using Zelevinsky’s $IH$-small resolution. In our case, however, it is
known that $\boldsymbol{\Sigma}$ does not admit any $IH$-small resolution
[Per1].
## 5\. Mirror symmetry
In this section, we study the mirror symmetry for Calabi–Yau complete
intersections in Gorenstein Hibi toric varieties and those in minuscule
Schubert varieties. In Subsection 5.2, we focus on the specific example
$X=\boldsymbol{\Sigma}(1^{9})$ and carry out further calculations.
### 5.1. Mirror constructions
Our study of the family of affine complete intersections in
$(\mathbb{C}^{*})^{n}$ which are expected to be birational to the conjectural
mirror family for Calabi–Yau complete intersections in minuscule Schubert
varieties is based on the general method via conifold transition proposed by
[BCFKvS1] [Bat2]. First we apply the Batyrev–Borisov mirror construction
[Bat1] [Bor] to Calabi–Yau complete intersections in Gorenstein Hibi toric
varieties. Second we explain the conjectural mirror construction via conifold
transition and describe a family of affine complete intersections in
$(\mathbb{C}^{*})^{n}$ conjecturally birational to the mirror family for
Calabi–Yau complete intersections in minuscule Schubert varieties. Finally we
establish the closed formula for the fundamental period.
#### 5.1.1. Batyrev–Borisov construction
Let $P$ be a finite pure poset and $J(P)$ the associated distributive lattice
of order ideals. Let $N=\mathbb{Z}P$ and $M=\operatorname{Hom}(N,\mathbb{Z})$
be dual free abelian groups of rank $|P|$, and $N_{\mathbb{R}}$ and
$M_{\mathbb{R}}$ be the real scalar extension of $N$ and $M$, respectively.
Recall that the order polytope $\Delta(P)\subset M_{\mathbb{R}}$ is a
$|P|$-dimensional integral polytope associated with the hyperplane class on
the Gorenstein Hibi toric variety $\mathbb{P}_{\Delta(P)}$. By Proposition
2.18 and Theorem 4.1.9 in [Bat1], a unimodular transformation of
$h_{P}\Delta(P)$ becomes a reflexive polytope corresponding to the
anticanonical sheaf
$-\mathcal{K}_{\mathbb{P}_{\Delta(P)}}=\mathcal{O}(h_{P})$, i.e., it contains
the unique internal integral point $0$ and every facet has integral distance
one to $0$. For instance, we can choose $\Delta:=\sum_{u\in
P}h(u)\chi_{u}-h_{P}\Delta(P)$ as such a reflexive polytope.
The polar dual polytope $\Delta^{*}$ of $\Delta$ also has a good description
[BCFKvS2] [HH]. The set of edges $E$ in the Hasse diagram of $\hat{P}$ plays
an important role. We remark that the abelian groups
$\mathbb{Z}\hat{P}=N\oplus\mathbb{Z}\left\\{\hat{0},\hat{1}\right\\}$ and
$\mathbb{Z}E$ can be viewed as the groups of $0$-chains and $1$-chains of the
natural chain complex associated with the Hasse diagram of $\hat{P}$. The
boundary map in the chain complex is
$\partial:\mathbb{Z}E\longrightarrow\mathbb{Z}\hat{P},\quad e\mapsto
t(e)-s(e).$
We also consider the projection $\mathrm{pr}_{1}:\mathbb{Z}\hat{P}\rightarrow
N$ and the composed map
$\delta:=\mathrm{pr}_{1}\circ\partial:\mathbb{Z}E\longrightarrow N.$
In this notation, the dual polytope $\Delta^{*}$ coincides with the convex
hull of the image $\delta(E)\subset N_{\mathbb{R}}$. Further, the linear map
$\delta$ gives a bijection between $E$ and the set of vertices in
$\Delta^{*}$.
Let $D_{e}$ denote the toric divisor on $\mathbb{P}_{\Delta(P)}$ associated
with an edge $e\in E$ under the above bijection and
$D_{E^{\prime}}:=\sum_{e\in E^{\prime}}D_{e}$ for any subset
$E^{\prime}\subset E$. It is easy to see that there is a linear equivalence
relation
$\sum_{s(e)=u}D_{e}\simeq\sum_{t(e)=u}D_{e},$
for each $u\in P$, and that $\mathcal{O}(D_{E^{k}})$ coincides with a very
ample invertible sheaf $\mathcal{O}(1)$ associated with $\Delta(P)$ for each
$E^{k}:=\left\\{e\in E\mid h(s(e))=k\right\\}$. Let
$X_{0}\subset\mathbb{P}_{\Delta(P)}$ be a general Calabi–Yau complete
intersection of degree $(d_{1},\dots,d_{r})$ with respect to $\mathcal{O}(1)$.
That is, $d_{1},\dots,d_{r}$ satisfies $\sum_{j=1}^{r}d_{j}=h_{P}$. We choose
a nef-partition of $\Delta$, a special kind of Minkowski sum decomposition
$\Delta=\Delta_{1}+\cdots+\Delta_{r}$ of $\Delta$, in the following specific
way. Define subsets $E_{j}$ of edges in $E$ as
$E_{j}=\bigcup_{k=d_{1}+\cdots+d_{j-1}+1}^{d_{1}+\cdots+d_{j}}E^{k}.$
It turns out that $\mathcal{O}(D_{E_{j}})=\mathcal{O}(d_{j})$ and the nef-
partition is arisen from $E=E_{1}\cup E_{2}\cup\cdots\cup E_{r}$. Define
$\nabla_{j}=\operatorname{Conv}(\\{0\\},\delta(E_{j}))$ and the Minkowski sum
$\nabla=\nabla_{1}+\cdots+\nabla_{r}\subset N_{\mathbb{R}}$. From [Bor], it
holds that
$\Delta^{*}=\operatorname{Conv}(\nabla_{1},\dots,\nabla_{r}),\quad\nabla^{*}=\operatorname{Conv}(\Delta_{1},\dots,\Delta_{r})\quad\text{and}\quad\Delta=\Delta_{1}+\cdots+\Delta_{r},$
where $\Delta_{j}$ is the integral polytope in $M_{\mathbb{R}}$ defined by
$\left<\Delta_{i},\nabla_{j}\right>\geq-\delta_{ij}$. One may be also able to
write down more precise combinatorial expressions for $\Delta_{j}$ and
$\nabla^{*}$.
Now we introduce the Batyrev–Borisov mirror of $Y=\hat{X_{0}}$, the strict
transform of $X_{0}$ in a MPCP-resolution $\hat{\mathbb{P}}_{\Delta(P)}$ of
$\mathbb{P}_{\Delta(P)}$. The mirror of $Y\subset\hat{\mathbb{P}}_{\Delta(P)}$
is birational to the set given by the following equations in torus
$(\mathbb{C}^{*})^{|P|}$:
$\tilde{f}_{j}=1-(\sum_{e\in E_{j}}a_{e}t^{\delta(e)})=0\quad\text{ (for all
$1\leq j\leq r$),}$
where each $a_{e}\in\mathbb{C}$ is a parameter. Further, the smooth mirror
manifold $Y^{*}$ of $Y$ is also obtained as the closure of the above set in
MPCP-resolution $\hat{\mathbb{P}}_{\nabla}$ of $\mathbb{P}_{\nabla}$.
In general, the mirror manifold $Y^{*}\subset\hat{\mathbb{P}}_{\nabla}$
actually has the expected stringy (or string-theoretic) Hodge numbers as
proved in [BB1, BB2]. The stringy Hodge numbers of $X_{0}$ coincide with the
usual Hodge numbers of $Y$ if there exists a crepant resolution $Y\rightarrow
X_{0}$. Applying their formula for stringy $(1,*)$-Hodge numbers to the case
of Calabi–Yau complete intersections $X_{0}$ in Gorenstein Hibi toric
varieties $\mathbb{P}_{\Delta(P)}$, we obtain the following convenient
expressions in terms of the poset $P$, which we already used in Proof of
Proposition 4.1.
###### Theorem 5.1 (cf. [BB1, Proposition 8.6]).
The stringy $(1,*)$-Hodge numbers of a general Calabi–Yau complete
intersections $X_{0}$ of degree $(d_{1},\dots,d_{r})$ in a Gorenstein Hibi
toric variety $\mathbb{P}_{\Delta(P)}$ are given by the following formulae
$\displaystyle h_{\mathrm{st}}^{1,1}(X_{0})=$ $\displaystyle\lvert
E\rvert-\lvert P\rvert,\qquad h_{\mathrm{st}}^{1,k}(X_{0})=0\quad(1<k<\lvert
P\rvert-r-1),$ $\displaystyle h_{\mathrm{st}}^{1,\lvert P\rvert-r-1}(X_{0})=$
$\displaystyle\sum_{i\in I}\left[\sum_{J\subset I}(-1)^{\lvert
J\rvert}l\left((d_{i}-d_{J})\Delta(P)\right)\right]-\sum_{J\subset
I}(-1)^{r-\lvert J\rvert}\left[\sum_{e\in
E}l^{*}(d_{J}\theta_{e})\right]-\lvert P\rvert,$
where $I=\\{1,\dots,r\\}$, $d_{J}:=\sum_{j\in J}d_{j}$ and $\theta_{e}$ is the
facet of $P$ corresponding to the edge $e\in E$. The nonzero contribution in
the first term of $h_{\mathrm{st}}^{1,\lvert P\rvert-r-1}(X_{0})$ comes only
from the range of $d_{i}-d_{J}\geq 0$ and in the second term from that of
$d_{J}=h_{P}-1$ or $h_{P}$.
#### 5.1.2. Construction via conifold transition
A conifold transition of a smooth Calabi–Yau 3-fold $X$ is the composite
operation of a flat degeneration of $X$ to $X_{0}$ with finitely many nodes
and a small resolution $Y\rightarrow X_{0}$. The conjecture proposed in
[BCFKvS1] is that the mirror Calabi–Yau 3-folds $Y^{*}$ and $X^{*}$ are again
related in the same way. The construction is depicted as the following
diagram, Figure 4. In the diagram, dashed and solid arrows represent flat
degenerations and small contraction morphisms respectively.
Figure 4. Mirror symmetry and conifold transitions
By an argument in [Bat2] on generalized monomial-divisor correspondence, there
is a natural specialization $Y^{*}_{0}$ of the family of $Y^{*}$ to get the
mirror of $X$. That is, the specialized parameter $(a_{e})_{e\in E}$ should be
$\Sigma(\Delta^{*})$-admissible, i.e. there exists a
$\Sigma(\Delta^{*})$-piecewise linear function
$\phi:N_{\mathbb{R}}\rightarrow\mathbb{R}$ corresponding to a Cartier divisor
on $X$ such that $\phi\circ\delta(e)=\log|a_{e}|$. In all our case,
$\operatorname{Pic}\mathbb{P}_{\Delta(P)}\simeq\operatorname{Pic}X\simeq\mathbb{Z}$
holds. Then we can simply specialize the family to be diagonal, i.e. setting
all the coefficients $a_{e}$ to be a same parameter $a$. Now we repeat the
conjecture of [BCFKvS1] as a specific form in our terminology combining with
the argument on the number of nodes in Proof of Proposition 4.1.
###### Conjecture 5.2 (cf. [BCFKvS1, Conjecture 6.1.2]).
Let $p$ be a number of nodes on a Calabi–Yau 3-fold
$X_{0}\subset\mathbb{P}_{\Delta(P)}$, namely, $p=(\prod_{j=1}^{r}d_{j})\sum
c_{\lvert P^{\prime}\rvert}$ where the sum is over all minimal convex cycles
in $P$ corresponding to the contractions $P\rightarrow P^{\prime}$ with
$\lvert P^{\prime}\rvert=\lvert P\rvert-3$ and $c_{\lvert P^{\prime}\rvert}$
denotes the number of maximal chains in $J(P^{\prime})$. We define a one
parameter family of affine complete intersections in $(\mathbb{C}^{*})^{\lvert
P\rvert}$ defined by the following equations:
$f_{j}=1-a(\sum_{e\in E_{j}}t^{\delta(e)})=0\quad\text{ (for all $1\leq j\leq
r$).}$
The closure $Y_{0}^{*}$ of the above set in a MPCP-resolution
$\hat{\mathbb{P}}_{\nabla}$ has $p$ nodes, satisfying $(|E|-|P|-p-1)$
relations for general $a$. A small resolution $X^{*}\rightarrow Y_{0}^{*}$ is
a mirror manifold of $X$ with the correct Hodge numbers,
$h^{i,j}(X^{*})=h^{3-j,i}(X)$.
###### Remark 5.3.
In the case of smoothing of 3-dimensional Calabi–Yau hypersurfaces in
Gorenstein Hibi toric varieties, we can see Conjecture 5.2 holds by the same
argument as in [BCFKvS1] [BK], i.e., the polar duality of faces gives a one-
to-one correspondence between singular
$\mathbb{P}^{1}\subset\mathbb{P}_{\Delta(P)}$ and torus invariant
$\mathbb{P}^{1}\times\mathbb{P}^{1}\subset\mathbb{P}_{\Delta^{*}}$ which
intersect non-transversally with the closure of the set $\\{f_{1}=0\\}$, and
the MPCP-resolution
$\hat{\mathbb{P}}_{\Delta^{*}}\rightarrow\mathbb{P}_{\Delta^{*}}$ increases
them by $h_{P}$ times.
In general, we can define a one parameter family $Y_{0}^{*}$ for any
dimensional Calabi–Yau variety $X=X(w)(d_{1},\dots,d_{r})$. In the remaining
part, we refer to not only $X^{*}$ but also $Y_{0}^{*}$ as a conjectural
mirror of $X$.
#### 5.1.3. Fundamental period
We derive the explicit form of the fundamental period for the conjectural
mirror family of a Calabi–Yau variety $X(w)(d_{1},\dots,d_{r})$. Obviously,
the coordinate transformation $t_{u}\rightarrow\zeta^{h_{u}}t_{u}$ gives a
$\mathbb{Z}_{h_{P}}$-symmetry $a\rightarrow\zeta a$ in this family, where
$\zeta=e^{2\pi\sqrt{-1}/h_{P}}$. Therefore we should take $x:=a^{h_{P}}$ as a
genuine moduli parameter. The fundamental period $\omega_{0}(x)$ of the mirror
family is defined by integration of the holomorphic $(|P|-r)$-form
$\Omega_{x}$ on a (real) torus cycle $\mathbb{T}$ that vanishes at $x=0$. By
residue theorem, we obtain the following formula up to the constant
multiplication,
$\displaystyle\omega_{0}$
$\displaystyle(x)=\int_{\mathbb{T}}\Omega_{x}=\frac{1}{(2\pi\sqrt{-1})^{|P|}}\int_{|t_{u}|=1}\frac{1}{\prod_{j=1}^{r}f_{j}}\prod_{i=1}^{|P|}\frac{\mathrm{d}t_{i}}{t_{i}}$
$\displaystyle=\sum_{m=0}^{\infty}a^{h_{P}m}\frac{1}{(2\pi\sqrt{-1})^{|P|}}\int_{|t_{u}|=1}\prod_{j=1}^{r}(\sum_{e\in
E_{j}}t^{\delta(e)})^{d_{j}m}\prod_{i=1}^{|P|}\frac{\mathrm{d}t_{i}}{t_{i}}$
$\displaystyle=\sum_{m=0}^{\infty}x^{m}N(J_{j},E_{j})=\sum_{m=0}^{\infty}x^{m}\frac{\prod_{j=1}^{r}(d_{j}m)!}{m!^{h_{P}}}N(J^{k},E^{k}),$
where $J_{j}(m):=\left\\{(j,i)\mid 1\leq i\leq d_{j}m\right\\}$,
$J^{k}(m):=\left\\{(k,i)\mid 1\leq i\leq m\right\\}\subset\mathbb{N}^{2}$ and
$N(J_{j},E_{j}):=\\#{\scriptstyle\left\\{\phi:\bigcup_{j}J_{j}(m)\rightarrow
E\ \Big{|}\ \phi(J_{j}(m))\subset E_{j},\
\sum_{s(e)=u}\phi^{-1}(e)=\sum_{t(e)=u}\phi^{-1}(e)\right\\}}$.
In the case that the Hasse diagram of $P$ (and hence $\hat{P}$) is a plane
graph, we can go further like [BCFKvS2]. This is originally formulated in the
work of Bondal and Galkin [BG] for the Landau–Ginzburg mirror of minuscule
$G/Q$. For the mirror of complete intersections in general minuscule Schubert
varieties, we can also follow the same road because the Hasse diagram of a
minuscule poset is always a plane graph. If the Hasse diagram of $P$ is a
plane graph, we can define the dual graph $B$ of the Hasse diagram of
$\hat{P}$ on a sphere $S^{2}=\mathbb{P}^{1}$ with putting $\hat{1},\hat{0}$ on
$\pm\sqrt{-1}\infty$ respectively. We denote by $b_{L},b_{R}$ the elements
$b\in B$ corresponding to the farthest left and right areas respectively. We
draw the Hasse diagram of $\hat{P}$ and its dual graph $B$ below, Figure 5,
for the minuscule poset of $G(2,6)$ as an example.
Figure 5. An example of $\hat{P}$ and the dual graph $B$
The orientation of an edge $e$ of $B$ is defined as the direction from the
left $l(e)$ to the right $r(e)$. We attain the variable $m_{b}$ for each
element $b\in B$ and set $m_{b_{L}}=0$ and $m_{b_{R}}=m$.
###### Proposition 5.4.
Let $X(w)$ be a minuscule Schubert variety and $P=P_{w}$ its minuscule poset.
Under the above notation, the fundamental period $\omega_{0}(x)$ for the
conjectural mirror family of a Calabi–Yau complete intersection
$X(w)(d_{1},\dots,d_{r})$ is presented in the following:
$\omega_{0}(x)=\sum_{m=0}^{\infty}\frac{\prod_{j=1}^{r}(d_{j}m)!}{m!^{h_{P}}}\sum_{m_{b}\in
B}\prod_{e\in E(B)}\begin{pmatrix}m_{r(e)}\\\ m_{l(e)}\end{pmatrix}x^{m}.$
### 5.2. Mirror symmetry for $\boldsymbol{\Sigma}(1^{9})$
#### 5.2.1. Picard–Fuchs equation
Now we focus on the Calabi–Yau 3-fold $X=\boldsymbol{\Sigma}(1^{9})$. The
fundamental period of the mirror family can be read from the following
diagram.
The vertices of the dual graph $B$ corresponds to the separated areas. The
fundamental period turns out to be
$\displaystyle\omega_{0}(x)$
$\displaystyle=\sum_{m,s,t,u,v}\begin{pmatrix}m\\\
s\end{pmatrix}^{2}\begin{pmatrix}m\\\ v\end{pmatrix}^{2}\begin{pmatrix}m\\\
t\end{pmatrix}\begin{pmatrix}s\\\ t\end{pmatrix}\begin{pmatrix}t\\\
u\end{pmatrix}\begin{pmatrix}v\\\ u\end{pmatrix}x^{m},$
where $x=a^{9}$. With the aid of numerical method, we obtain the Picard–Fuchs
equation for the conjectural mirror of $X$.
###### Proposition 5.5.
Let $\omega_{0}(x)$ be the above power series around $x=0$, which corresponds
to the fundamental period for the conjectural mirror family of the Calabi–Yau
3-fold $X=\boldsymbol{\Sigma}(1^{9})$. This satisfies the Picard–Fuchs
equation $\mathcal{D}_{x}\omega_{0}(x)=0$ with $\theta_{x}=x\partial_{x}$ and
$\displaystyle\mathcal{D}_{x}=$ $\displaystyle
121\theta_{x}^{4}-77x(130\theta_{x}^{4}+266\theta_{x}^{3}+210\theta_{x}^{2}+77\theta_{x}+11)$
$\displaystyle-x^{2}(32126\theta_{x}^{4}+89990\theta_{x}^{3}+103725\theta_{x}^{2}+55253\theta_{x}+11198)$
$\displaystyle-x^{3}(28723\theta_{x}^{4}+74184\theta_{x}^{3}+63474\theta_{x}^{2}+20625\theta_{x}+1716)$
$\displaystyle-7x^{4}(1135\theta_{x}^{4}+2336\theta_{x}^{3}+1881\theta_{x}^{2}+713\theta_{x}+110)-49x^{5}(\theta_{x}+1)^{4}.$
#### 5.2.2. Monodromy calculation
The differential operator $\mathcal{D}_{x}$ in Proposition 5.5 is already
appeared in the lists of [AvEvSZ, vEvSdb] with conjectural topological
invariants, obtained from the monodromy calculation for a particular basis of
the solutions for $\mathcal{D}_{x}$ and some assumptions by mirror symmetry
[vEvS]. The existence of the smooth Calabi–Yau 3-fold
$X=\boldsymbol{\Sigma}(1^{9})$ and our computation of topological invariants
in Proposition 4.1 answer to their conjecture. On the other hand, we observe
that there exists another MUM-point for this equation, not only $x=0$ but
$x=\infty$. In fact, the Riemann scheme of the differential operator
$\mathcal{D}_{x}$ is
$P{\scriptstyle\left\\{\begin{array}[]{cccccc}\zeta_{1}&-11/7&\zeta_{2}&0&\zeta_{3}&\infty\\\
\hline\cr 0&0&0&0&0&1\\\ 1&1&1&0&1&1\\\ 1&3&1&0&1&1\\\ 2&4&2&0&2&1\\\
\end{array}\right\\}}\;\;,$
where $\zeta_{1}<\zeta_{2}<\zeta_{3}$ are the roots of the discriminant
$x^{3}+159x^{2}+84x-1$. The singularities at $x=\zeta_{1}$, $\zeta_{2}$,
$\zeta_{3}$ are called conifold and there is no monodromy around the point
$x=-11/7$, called apparent singularity. We believe that the point $x=\infty$
has also geometric interpretation.
Recall the argument in [CdOGP] based on the mirror symmetry, which involves an
integral symplectic basis of solutions in the original Calabi–Yau geometry.
Let us start from the Frobenius basis of solutions for $\mathcal{D}_{x}$,
namely the unique normalized regular power series solution
$\omega_{0}(x)=1+O(x)$ and the followings:
$\displaystyle\omega_{1}(x)=\omega_{0}(x)\log x+\omega_{1}^{\mathrm{reg}}(x),$
$\displaystyle\omega_{2}(x)=\omega_{0}(x)(\log
x)^{2}+2\omega_{1}^{\mathrm{reg}}(x)\log x+\omega_{2}^{\mathrm{reg}}(x),$
$\displaystyle\omega_{3}(x)=\omega_{0}(x)(\log
x)^{3}+3\omega_{1}^{\mathrm{reg}}(x)(\log
x)^{2}+3\omega_{2}^{\mathrm{reg}}(x)\log x+\omega_{3}^{\mathrm{reg}}(x),$
where $\omega_{k}^{\mathrm{reg}}(x)$ is a regular power series around $x=0$
without constant term. We expect an integral symplectic basis has the
following form:
$\Pi^{X}(x)=\left(\begin{smallmatrix}1&0&0&0\\\ 0&1&0&0\\\
\beta/24&a&\text{{-}}\kappa/2&0\\\
\gamma&\beta/24&0&\kappa/6\end{smallmatrix}\right)\left(\begin{smallmatrix}n_{0}\omega_{0}(x)\\\
n_{1}\omega_{1}(x)\\\ n_{2}\omega_{2}(x)\\\
n_{3}\omega_{3}(x)\end{smallmatrix}\right)\;,$
where $\kappa=-\deg(X)$, $\beta=-c_{2}(X)\cdot H$,
$\gamma=-n_{3}\zeta(3)\chi(X)$, $n_{k}=1/(2\pi i)^{k}$ with the topological
invariants of $X$ in Proposition 4.1 and $a$ is an unknown parameter without
geometric interpretation although it may be consistent to choose
$a\in\deg(X)/2+\mathbb{Z}$.
If there exists a Calabi–Yau geometry around $x=\infty$, we also expect the
existence of similar basis $z\Pi^{Z}(z)$ of solutions for $\mathcal{D}_{z}$
under some appropriate coordinate change $z=c/x$. We denote by $\Pi^{Z}(z)$
the gauge fixed basis, exactly the same form as $\Pi^{X}(x)$ with the
Frobenius basis $\omega^{Z}_{k}(z)$ for
$\mathcal{D}^{Z}_{z}=z\mathcal{D}_{z}z^{-1}$ and unknown parameters $\deg(Z)$,
$c_{2}(Z)\cdot H$, $\chi(Z)$ and $a^{Z}$.
Once passing to a numerical calculation like [vEvS], we obtain the following
results.
1. (1)
There exists the integral symplectic basis $\Pi^{X}(x)$ and $z\Pi^{Z}(z)$ with
the parameters,
$a=a^{Z}=-1/2,\ c=-1,$ $\deg(Z)=21,\ c_{2}(Z)\cdot H=66,\ \chi(Z)=-102.$
2. (2)
The analytic continuation along a path in the upper half plane gives the
following relation of two basis $\Pi^{X}(x)$ and $z\Pi^{Z}(z)$,
$\Pi^{X}(x)=N_{z}S_{xz}z\Pi^{Z}(z),$
with $N_{z}=1$ and the symplectic matrix
$S_{xz}=\left(\begin{smallmatrix}8&4&2&5\\\ 4&0&1&2\\\
10&\text{{-}}25&2&\text{{-}}1\\\
\text{{-}}21&2&\text{{-}}5&\text{{-}}10\end{smallmatrix}\right).$
3. (3)
With respect to the local basis of the analytic continuation of $\Pi^{X}(x)$
and $z\Pi^{Z}(z)$ along a path in the upper half plane, the monodromy matrices
$M^{X}_{p}$ and $M^{Z}_{p}$ at each singular point
$x=-\frac{1}{z}=\zeta_{1},\zeta_{2},0,\zeta_{3},\infty$ have the following
form, respectively:
$M^{X}_{p}$ | $\left(\begin{smallmatrix}169&\text{{-}}80&32&64\\\ 84&\text{{-}}39&16&32\\\ 210&\text{{-}}100&41&80\\\ \text{{-}}441&210&\text{{-}}84&\text{{-}}167\end{smallmatrix}\right)\left(\begin{smallmatrix}13&\text{{-}}8&2&4\\\ 6&\text{{-}}3&1&2\\\ 24&\text{{-}}16&5&8\\\ \text{{-}}36&24&\text{{-}}6&\text{{-}}11\end{smallmatrix}\right)\left(\begin{smallmatrix}1&0&0&0\\\ 1&1&0&0\\\ 16&33&1&0\\\ \text{{-}}12&\text{{-}}17&\text{{-}}1&1\end{smallmatrix}\right)\left(\begin{smallmatrix}1&0&0&1\\\ 0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{smallmatrix}\right)\left(\begin{smallmatrix}286&\text{{-}}130&55&111\\\ 89&\text{{-}}43&17&34\\\ \text{{-}}307&127&\text{{-}}60&\text{{-}}122\\\ \text{{-}}465&218&\text{{-}}89&\text{{-}}179\end{smallmatrix}\right)$
---|---
$M^{Z}_{p}$ | $\left(\begin{smallmatrix}1&0&0&1\\\ 0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{smallmatrix}\right)\left(\begin{smallmatrix}1&3&0&1\\\ 0&1&0&0\\\ 0&\text{{-}}9&1&\text{{-}}3\\\ 0&0&0&1\end{smallmatrix}\right)\left(\begin{smallmatrix}343&\text{{-}}17&83&168\\\ 104&\text{{-}}9&25&50\\\ \text{{-}}496&8&\text{{-}}121&\text{{-}}247\\\ \text{{-}}432&32&\text{{-}}104&\text{{-}}209\end{smallmatrix}\right)\left(\begin{smallmatrix}211&\text{{-}}20&50&100\\\ 105&\text{{-}}9&25&50\\\ 42&\text{{-}}4&11&20\\\ \text{{-}}441&42&\text{{-}}105&\text{{-}}209\end{smallmatrix}\right)\left(\begin{smallmatrix}1&0&0&0\\\ 1&1&0&0\\\ 10&21&1&0\\\ \text{{-}}9&\text{{-}}11&\text{{-}}1&1\end{smallmatrix}\right)$
Table 2. Monodromy matrices
All in the above results strongly indicate the existence of the geometric
interpretation at $x=\infty$. Thus we are led to the following conjecture
based on the homological mirror symmetry similar to the examples of the
Grassmannian–Pfaffian in [Rød] and the Reye congruence Calabi–Yau 3-fold in
[HT1].
###### Conjecture 5.6.
There exists a smooth Calabi–Yau 3-fold $Z$ with $h^{1,1}=1$ whose derived
category of coherent sheaves is equivalent to that of the Calabi–Yau 3-fold
$X=\boldsymbol{\Sigma}(1^{9})$. The topological invariants of $Z$ are
$\mathrm{deg}(Z)=21,\ \ \ c_{2}(Z).H=66,\ \ \ \chi(Z)=-102,$
where $H$ is the ample generator of the Picard group
$\operatorname{Pic}(Z)\simeq\mathbb{Z}$.
The Calabi–Yau 3-fold $Z$ in Conjecture 5.6 can not be birational to $X$
because $h^{1,1}=1$ and $\deg(Z)\neq\deg(X)$, so that it should be a non-
trivial Fourier–Mukai partner of $X$.
#### 5.2.3. BPS numbers
As a further consistency check in Conjecture 5.6 and an application of the
mirror construction, we carry out the computation of BPS numbers by using the
methods proposed by [CdOGP] [BCOV1, BCOV2]. The BPS numbers $n_{g}(d)$ are
related with the Gromov–Witten invariants $N_{g}(d)$ by some closed formula
[GV], hence we obtain the prediction for Gromov–Witten invariants of $X$ and
$Z$ from these computations.
We skip all the details here and only present the results in Appendix B. For
the details, one can get many references in now. Here we have followed [HK],
where a very similar example to ours, the Grassmannian–Pfaffian Calabi–Yau
3-fold, has been analyzed.
## Appendix A Other examples in minuscule $G/Q$
We remark that there are some examples of Calabi–Yau complete intersections in
minuscule homogeneous spaces $G/Q$ that get attention. In these cases one may
be able to know more details because the structure of the small quantum
cohomology ring $QH^{*}(G/Q)$ of $G/Q$ has been revealed intensively by recent
researches [Ber] [KT2] [CMP] for instance.
In particular, the quantum Chevalley formula [CMP, Proposition 4.1] for
minuscule homogeneous spaces $G/Q$ is given in terms of the minuscule poset
$P_{w}$ as follows
$[H]*[X(w)]=\sum_{w\text{ covers
}w^{\prime}}[X(w^{\prime})]+q[X(w)]^{\mathrm{PD}},$
where $q$ is the quantum parameter and $[X(w)]^{\mathrm{PD}}$ denotes the
Poincaré dual of $[X(w)]$, whose minuscule poset corresponds to the order dual
of $P_{Q}\setminus P_{w}$. Of course the limit $q\rightarrow 0$ coincides with
the classical Chevalley formula in Remark 2.6. From this formula, we can
calculate the quantum differential system of minuscule $G/Q$ and reduce this
first order system to a higher order differential system by some computer
aided calculation (see [BCFKvS1] [Gue] for instance). For example, the
orthogonal Grassmannian $OG(5,10)$ and the Cayley plane
$\mathbb{O}\mathbb{P}^{2}$ have the quantum differential operator annihilating
the component associated with the fundamental class as follows:
$\theta_{q}^{11}(\theta_{q}-1)^{5}-q\theta_{q}^{5}(2\theta_{q}+1)(17\theta_{q}^{2}+17\theta_{q}+5)+q^{2},$
$\theta_{q}^{17}(\theta_{q}-1)^{9}-3q\theta_{q}^{9}(2\theta_{q}+1)(3\theta_{q}^{2}+3\theta_{q}+1)(15\theta_{q}^{2}+15\theta_{q}+4)-3q^{2}(3\theta_{q}+2)(3\theta_{q}+4),$
respectively, where $\theta_{q}=q\partial_{q}$. By the quantum hyperplane
section theorem for homogeneous spaces [Kim], we can determine the genus zero
Gromov–Witten potential for each example of Calabi–Yau complete intersections
in minuscule $G/Q$. In fact this procedure also works for general rational
homogeneous spaces $G/Q$ by replacing the above quantum Chevalley formula by
the general one obtained in [FW] though we do not go into the specifics here.
On the other hand, we already know the conjectural mirror family. Hence
comparing both sides, we can prove the classical genus zero ‘mirror theorem’
[Giv] [LLY] for these examples combining Conjecture 5.2. In particular, we can
do this for $OG(5,10)(1^{6},2)$, the remained example of smooth complete
intersection Calabi–Yau 3-folds in minuscule Schubert varieties in Proposition
3.1. The Picard–Fuchs operator for the conjectural mirror family of
$OG(5,10)(1^{6},2)$ coincides with the conjecture in [vEvS],
$\theta_{x}^{4}-2x(2\theta_{x}+1)^{2}(17\theta_{x}^{2}+17\theta_{x}+5)+4x^{2}(\theta_{x}+1)^{2}(2\theta_{x}+1)(2\theta_{x}+3),$
where $\theta_{x}=x\partial_{x}$.
Other dimensional examples also seem meaningful. The $K3$ surface
$OG(5,10)(1^{8})$ is an example which has a nontrivial Fourier–Mukai partner
[Muk], whose mirror symmetry are investigated in [HLOY]. The Picard–Fuchs
operator for the mirror family of $OG(5,10)(1^{8})$ is
$\theta_{x}^{3}-x(2\theta_{x}+1)(17\theta_{x}^{2}+17\theta_{x}+5)+x^{2}(\theta_{x}+1)^{3},$
where $\theta_{x}=x\partial_{x}$. It has exactly the same property as the
Picard–Fuchs operators of conjectural mirror families for $G(2,7)(1^{7})$ and
$\boldsymbol{\Sigma}(1^{9})$, with two MUM-points. Further this operator is
also well-known since it is used essentially in Aṕery’s proof of irrationality
of $\zeta(3)$, with that of the mirror family for $G(2,5)(1^{5})$ in
$\zeta(2)$ case. From this viewpoint, it may be valuable that the conjectural
mirror family of Calabi–Yau 4-fold $\mathbb{O}\mathbb{P}^{2}(1^{12})$ also has
the following 5th-order Picard–Fuchs operator,
$\theta_{x}^{5}-3x(2\theta_{x}+1)(3\theta_{x}^{2}+3\theta_{x}+1)(15\theta_{x}^{2}+15\theta_{x}+4)-3x^{2}(\theta_{x}+1)^{3}(3\theta_{x}+2)(3\theta_{x}+4),$
where $\theta_{x}=x\partial_{x}$. This operator is related to the value
$\zeta(4)$ [Zud] [AZ]. The relationship between the quantum differential
system of rational homogeneous spaces and zeta values are also numerically
confirmed in [Gal] with the definition of the Aṕery characteristic class.
## Appendix B BPS numbers
$d$ | $g=0$ | $g=1$ | $g=2$ | $g=3$ | $g=4$
---|---|---|---|---|---
1 | 252 | 0 | 0 | 0 | 0
2 | 1854 | 0 | 0 | 0 | 0
3 | 27156 | 0 | 0 | 0 | 0
4 | 567063 | 0 | 0 | 0 | 0
5 | 14514039 | 4671 | 0 | 0 | 0
6 | 424256409 | 1029484 | 0 | 0 | 0
7 | 13599543618 | 112256550 | 5058 | 0 | 0
8 | 466563312360 | 9161698059 | 7759089 | 0 | 0
9 | 16861067232735 | 645270182913 | 2496748119 | 151479 | 0
10 | 634912711612848 | 41731465395267 | 438543955881 | 418482990 | -3708
11 | 24717672325914858 | 2557583730349461 | 56118708041940 | 217285861284 | 33975180
Table 3. BPS numbers $n_{g}^{X}(d)$ of $X$ $d$ | $g=0$ | $g=1$ | $g=2$ | $g=3$ | $g=4$
---|---|---|---|---|---
1 | 387 | 0 | 0 | 0 | 0
2 | 4671 | 0 | 0 | 0 | 0
3 | 124323 | 1 | 0 | 0 | 0
4 | 4782996 | 1854 | 0 | 0 | 0
5 | 226411803 | 606294 | 0 | 0 | 0
6 | 12249769449 | 117751416 | 27156 | 0 | 0
7 | 727224033330 | 17516315259 | 33487812 | 252 | 0
8 | 46217599569117 | 2252199216735 | 15885697536 | 7759089 | 0
9 | 3094575464496057 | 265984028638047 | 4690774243470 | 13680891072 | 1127008
10 | 215917815744645750 | 29788858876065588 | 1053460470463461 | 9429360817149 | 12259161360
Table 4. BPS numbers $n_{g}^{Z}(d)$ of $Z$
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|
arxiv-papers
| 2013-01-31T14:53:58 |
2024-09-04T02:49:41.100683
|
{
"license": "Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/",
"authors": "Makoto Miura",
"submitter": "Makoto Miura",
"url": "https://arxiv.org/abs/1301.7632"
}
|
1301.7660
|
Hadron Collider Physics Symposium 2012
11institutetext: Harvard University, Department of Physics, 17 Oxford St.,
Cambridge, MA 02138, USA
# Backgrounds in $H\rightarrow WW^{(\ast)}\rightarrow\ell\nu\ell\nu$ with
ATLAS
Tomo Lazovich On behalf of the ATLAS Collaboration 11
[email protected]
###### Abstract
We present techniques used to estimate the backgrounds in the search for the
Standard Model Higgs boson in the $H\rightarrow
WW^{(\ast)}\rightarrow\ell\nu\ell\nu$ decay channel with the ATLAS experiment
at the LHC. The dataset corresponds to 13 $\text{fb}^{-1}$ of integrated
luminosity taken at a center of mass energy of 8 TeV. Only the final states
with an electron, muon, and zero or one jet are presented here.
## 1 Introduction
In July 2012 the ATLAS ATLASJinst and CMS CMSJinst experiments at the LHC
announced the discovery of a new particle consistent with the long-sought
Higgs boson ATLASDiscovery ; CMSDiscovery . The results presented here
constitute an update of the $H\rightarrow
WW^{(\ast)}\rightarrow\ell\nu\ell\nu$ analysis with a dataset of 13
$\text{fb}^{-1}$ taken at a center of mass energy of 8 TeV HCPConfNote . In
particular, we summarize the methods of background estimation for this search
channel, which focuses on the low mass Higgs signal region.
The $WW^{(\ast)}\rightarrow\ell\nu\ell\nu$ decay channel of the Higgs boson
has a final state defined by two leptons and missing transverse energy
($E_{T}^{\text{miss}}$) from neutrinos which escape detection. The analysis
presented here considers only the final states with one electron, one muon,
and zero or one jets with transverse momentum ($p_{T}$) greater than 25 GeV.
The leptons are required to be isolated and the leading (subleading) lepton
must have $p_{T}>25\text{ }(15)$ GeV. Additionally, the event must have
relative missing transverse energy ($E_{T,\text{rel}}^{\text{miss}}$) greater
than 25 GeV, where
$E_{T,\text{rel}}^{\text{miss}}=E_{T}^{\text{miss}}\sin(\text{min}(\Delta\phi,\frac{\pi}{2}))$
and $\Delta\phi$ is the azimuthal angle between the $E_{T}^{\text{miss}}$ and
the nearest reconstructed lepton or jet. This definition helps to reject
events where a mismeasurement of one of the reconstructed objects is a major
source of the $E_{T}^{\text{miss}}$. After these pre-selection cuts, the
signal region is divided into zero and one jet bins and additional topological
cuts (specific to each bin) are applied to discriminate the Higgs signal from
background contributions.
Figure 1: The jet multiplicity distribution after signal pre-selection cuts
are applied HCPConfNote .
Many processes in the Standard Model (SM) produce final states similar to that
in $H\rightarrow WW^{(\ast)}\rightarrow\ell\nu\ell\nu$. The largest background
contribution is the irreducible SM WW background. The next largest background
contributions come from $t\bar{t}$ and single top production. These
backgrounds are primarily relevant for larger jet multiplicity bins but are
also present in the zero jet bin. Another important background for the
analysis is the W+jets background. Here a single W boson is produced in
association with one or more jets, and one of the jets fakes a final state
lepton. Other backgrounds which will not be discussed in great detail include
the Z+jets and diboson ($WZ$, $ZZ$, $W\gamma$) backgrounds. Figure 1 shows the
jet multiplicity distribution before the selection separates the events into
jet multiplicity bins. After the pre-selection, the WW background is the
dominant background in the zero jet bin while the top background dominates the
higher jet multiplicity bins.
## 2 W+jets and Other Minor Backgrounds
The W+jets background arises from SM W boson production in association with
jets where one jet produces an object reconstructed as a lepton. This can be a
real lepton produced by heavy quark decay or a product of the jet
fragmentation that is incorrectly reconstructed as an electron. The W+jets
background contribution is estimated by a data-driven method called the “fake
factor” method. First, a control region is defined in data by requiring one
lepton with the same identification and isolation criteria as the signal
leptons. The second lepton is required to be anti-identified, satisfying
loosened isolation criteria and failing at least one identification
requirement. These events are then required to pass the full signal selection.
A fake factor, the ratio of the number of lepton candidates passing all
identification requirements and signal selections to the number that are anti-
identified, is derived in an inclusive data dijet sample. This factor is used
to scale the number of events in the control region to the signal region. The
total relative uncertainty on the estimate is 50%, dominated by the systematic
uncertainty on the fake factor.
Figure 2 shows the transverse mass ($m_{T}$) in a same sign validation region
(where two same sign rather than two opposite sign leptons are required). This
region is composed largely of W+jets and $WZ/ZZ/W\gamma$ backgrounds and is
used to validate the modeling of kinematic variables for these samples. This
region shows that the $m_{T}$ is well modeled (within statistics) for these
samples.
Figure 2: $m_{T}$ in the zero jet same sign validation region HCPConfNote .
Here we briefly mention other minor backgrounds which will not be discussed in
further detail. First, the Z+jets background comes from a case where the Z
decays to two leptons and there is fake missing transverse energy in the event
due to the calorimeter resolution. This background is normalized to data in a
control region requiring $m_{\ell\ell}<80$ GeV and
$\Delta\phi_{\ell\ell}>2.9$. Finally, the normalizations for the remaining
backgrounds ($WZ/ZZ/W\gamma$) are taken from Monte Carlo simulation.
Figure 3: $m_{T}$ distribution in the top one jet control region. The rate
predicted by Monte Carlo simulation has not yet been normalized to the data.
HCPConfNote
## 3 $t\bar{t}$ and Single Top Backgrounds
The $t\bar{t}$ and single top backgrounds are normalized together in control
regions (CR) separated by jet multiplicity. The one jet bin is normalized in a
CR defined with the same pre-selection as the signal region (SR) and at least
one b-tagged jet. For the zero jet bin, there are two CRs used for the
background estimate. First, a CR with only the SR pre-selection is used to
estimate the fraction of top events passing a jet veto. A second, b-tagged CR
is then used to estimate the probability of having no other jets reconstructed
in the event and is used as a correction to the fraction estimate from the
first CR.
Figure 3 shows the $m_{T}$ in the one jet CR before any normalization factors
are applied. The normalization factors (NF), or ratio between the data and
Monte Carlo predictions, derived via these methods are $1.04\pm 0.05$ (stat.)
for the zero jet channel and $1.03\pm 0.02$ (stat.) for the one jet channel.
## 4 Standard Model WW Background
The SM WW background is estimated in a CR which uses the SR pre-selection cuts
(two leptons, missing transverse energy) and is separated into jet bins. While
the SR requires $m_{\ell\ell}<50$ GeV, the WW CR requires $m_{\ell\ell}>80$
GeV. This is the largest background in the signal region. The WW modeling in
simulation is done with a tune of Powheg for event generation and Pythia 8 for
parton showering.
Figure 4 shows the $m_{T}$ in the WW zero and one jet CR, before the
application of any WW NF. In both the zero and one jet (but particularly in
the one jet) WW CRs, there is a non-negligible contrbution from the top
backgrounds. Therefore, the top backgrounds are first normalized using the
procedures described in Section 3 before all of the non-WW backgrounds are
subtracted from the event yields in the CR to derive the final normalization.
The ratio of the data (with non WW background subtracted) to the WW simulation
prediction is $1.13\pm 0.04$ (stat.) in the zero jet CR and $0.84\pm 0.08$
(stat.) in the one jet CR. The NF differ between the zero and one jet channels
because these CR are correcting for the over-prediction of the jet
multiplicity distribution by the current Powheg+Pythia 8 tune used by ATLAS.
Figure 4: The $m_{T}$ distribution in the WW zero (one) jet CR is shown in the
top (bottom) plot. The WW rate predicted by Monte Carlo simulation has not yet
been normalized to the data. The top backgrounds have been normalized
according to the procedure described in the text HCPConfNote .
Background | Stat. (%) | Theory (%) | Expt. (%) | Crosstalk (%) | Total (%)
---|---|---|---|---|---
$WW$, $H+\text{0-jet}$ | 3.3 | 7.2 | 1.5 | 6.2 | 13
$WW$, $H+\text{1-jet}$ | 9 | 8 | 12 | 34 | 54
top, $H+\text{1-jet}$ | 2 | 8 | 29 | 1 | 37
Table 1: Total uncertainties on backgrounds normalized using simple NF scaling
in CRs HCPConfNote .
## 5 Background Predictions and Uncertainties
Table 1 shows the total uncertainties on the background normalization for the
backgrounds which use the simple data to simulation scaling in the CR for
normalization. Theoretical uncertainties on the estimates include differences
due to the choice of generator and parton shower/underlying event as well as
other contributions. The experimental uncertainties are dominated the jet
energy scale and resolution and, in the one jet bin, the $b$ tagging
efficiency. The “Crosstalk” column refers to uncertainties on other
backgrounds which must be subtracted from the CR before the normalization of
the desired background can be computed. Notice in particular that the WW one
jet normalization has a large contribution from crosstalk due to the fact that
a top background contribution must be subtracted from the CR before the
normalization is computed.
Table 2 shows the NF derived for all of the backgrounds whose normalizations
are taken from data. In the case of everything except the top zero jet
background, this factor is simply the ratio of the number of Monte Carlo
events to data events in the appropriate CR for that background. We can see
that most of the backgrounds do not require very large corrections to their
normalization (none more than 16%).
Background | 0 jet NF | 1 jet NF
---|---|---
Top | $1.04\pm 0.05$ | $1.03\pm 0.02$
WW | $1.13\pm 0.04$ | $0.84\pm 0.08$
Z+jets | $0.87\pm 0.03$ | $0.85\pm 0.03$
Table 2: Normalization factors (NF) for all backgrounds whose normalizations
are taken from data HCPConfNote .
Figure 5 shows the $m_{T}$ distribution after the signal selection cuts have
been applied for the zero and one jet bin. It can be seen here that the WW
background is dominant in the zero jet bin, while both WW and top are dominant
in the one jet bin. In zero jet, there is a total of $774\pm 9$ (stat.)
expected background events, and $555\pm 5$ (stat.) of those are SM WW events.
In the one jet, out of $386\pm 5$ (stat.) total expected background events,
$118\pm 2$ (stat.) are SM WW events while $134\pm 5$ (stat.) are $t\bar{t}$
events. The difference between the background expectation and the 917 (433)
events observed in data in the zero (one) jet bin are due to the presence of
the Higgs-like signal.
Figure 5: $m_{T}$ distribution in the zero (one) jet bin after all signal
selection cuts in the top (bottom) plot HCPConfNote . The expectation for a
125 GeV Higgs signal is shown in red.
## 6 Conclusion
A wide array of estimation methods can be employed to understand the
complicated background processes that factor into a search for a $H\rightarrow
WW^{(\ast)}\rightarrow\ell\nu\ell\nu$ signal. Simple data to simulation
scaling in control regions is used for backgrounds such as SM WW (or top in
the one jet bin) where the variable shapes are well modeled but their
normalizations may be incorrect. More complicated data-driven methods, such as
the W+jets fake factor method, can also be used when the backgrounds are not
well modeled by simulation alone.
## References
* (1) ATLAS Collaboration, JINST 3 S08003 (2008)
* (2) ATLAS Collaboration, ATLAS-CONF-2012-158, https://cds.cern.ch/record/1493601
* (3) CMS Collaboration, JINST 3 S08004 (2008)
* (4) ATLAS Collaboration, Phys. Lett. B, 716, 1-29 (2012)
* (5) CMS Collaboration, Phys. Lett. B, 716, 30-61 (2012)
|
arxiv-papers
| 2013-01-31T16:04:02 |
2024-09-04T02:49:41.112992
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tomo Lazovich (for the ATLAS Collaboration)",
"submitter": "Tomo Lazovich",
"url": "https://arxiv.org/abs/1301.7660"
}
|
1301.7738
|
# PyPLN: a Distributed Platform for Natural Language Processing
Flávio Codeço Coelho [email protected]
School of Applied Mathematics
Fundação Getulio Vargas
Rio de Janeiro, RJ 22250-900, Brazil Renato Rocha Souza [email protected]
School of Applied Mathematicsl
Fundação Getulio Vargas
Rio de Janeiro, RJ 22250-900, Brazil Álvaro Justen [email protected]
School of Applied Mathematicsl
Fundação Getulio Vargas
Rio de Janeiro, RJ 22250-900, Brazil Flávio Amieiro [email protected]
School of Applied Mathematics
Fundação Getulio Vargas
Rio de Janeiro, RJ 22250-900, Brazil Heliana Mello [email protected]
Faculdade de Letras
Universidade Federal de Minas Gerais
Belo Horizonte, MG 31270-901
###### Abstract
This paper presents PyPLN111pypln.org a distributed platform for Natural
Language Processing. PyPLN leverages a vast array of NLP and text processing
open source tools, managing the distribution of the workload on a variety of
configurations: from a single server to a cluster of linux servers. PyPLN is
developed purely in Python but makes it very easy to incorporate other
softwares for specific tasks, as long as a linux version is available. PyPLN
facilitates analyses both at document and corpus levels, simplifying
management and publication of corpora and analytical results through an easy
to use web interface. In the current (beta) release, it supports English and
Portuguese languages (with support to other languages planned for future
releases). To handle texts in Portuguese language, PyPLN uses the PALAVRAS
parser(Bick, 2000) if the user has it available222PALAVRAS must be purchased
and installed. Currently PyPLN offers the following features: Text extraction
with encoding normalization (to UTF-8), part-of-speech tagging, token
frequency, semantic annotation, n-gram extraction, word and sentence
repertoire, and full-text search across corpora. The platform is licensed as
GPL-v3.
Keywords: Python, data mining, natural language processing, machine learning
## 1 Introduction
The PyPLN platform is a research project in active development in the School
of Applied Mathematics of Fundação Getulio Vargas, Brazil. Its main goal is to
make available a scalable computational platform for a variety of language-
based analyses. Its main target audience is the academic community, where it
can have a powerful impact by making sophisticated computational analyses
doable without the requirement of programming skills on the part of the user.
Among the many features already available we can cite: Simplified access to
corpora with interactive visualization tools, text extraction from HTML and
PDF documents, Encoding detection and conversion to utf-8, Language detection,
Tokenization, full-text search across corpora, part-of-speech tagging, Word
and sentence level statistics, n-gram extraction and word concordance. Many
more features are in development and should become available soon, such as:
semantic annotation, sentiment and text polarity analysis, automatic social
network information monitoring, stylistic analysis and the generation of
Knowledge Organization Systems such as ontologies and thesauri. PyPLN aims for
unrivaled ease of use, and wide availability, through its web interface and
full support to Portuguese language. Besides being a free, uncomplicated
research platform for language scholars capable of handling large corpora,
PyPLN is also a free software platform for distributed text processing, which
can be downloaded and installed by users on their own infrastructure.
The motivations for the development of PyPLN are multifaceted; however, its
major inspiration comes from the current demands spanning from the myriad
possibilities for textual-based analysis inspired by the ever growing number
of unstructured masses of textual data in digital form. A good example of this
type of data are the streams of human communication in social platforms such
as Twitter and Facebook; These social platforms alone have provoked an
extraordinary surge in the development of natural language processing (NLP)
tools. But the applications of NLP are not limited to this area. Traditional
document collections are migrating to digital format and becoming a huge “new”
market for NLP.
Despite the availability of a number of commercial and very good quality free
open source computational solutions for English (e.g GATE, OpenNLP, UIMA), and
even complete programming frameworks for NLP (e.g NLTK for Python, NLP tasks
for R), Portuguese suffers from a scarcity of tools for the automatic
treatment of language. PyPLN tries to fill this niche by providing a solution
which covers both the needs of end-users (via the web interface), and the
computer savvy analyst (via the PyPLN API) while providing the computational
resources in a scalable platform capable of handling very large corpora.
PyPLN is free software licensed under GPLv3. The source code can be downloaded
from pypln.org.
## 2 The Software
### 2.1 Architecture
At the core of any text mining solution is the need to quickly put together
custom analytical workflows. Custom solutions are key, due to the ample
applicability of these tools: for every new data set there is a large set of
original analyses which can be proposed. Though the number of possible
analytical scenarios is very large, they can be framed by the recombination of
relatively simple smaller analytical steps. For example: no textual analysis
can be performed without some form of tokenization, for instance. These
intermediary results are calculated once for each document and cached for
future uses. The modularity of these analytical steps leads to great
performance gains by allowing the distribution of the workload among multiple
cores/computers running in parallel. The Python ecosystem already provides
excellent libraries for this kind of analysis. But merely having a good
library for doing NLP, is not enough. The most interesting problems in NLP
involve large volumes of data which can be neither stored nor processed on a
single PC. PyPLN integrates a number of open-source libraries and applications
into a distributed pipeline for linguistic analysis.
Being a distributed pipeline for analytical text processing, PyPLN applies the
philosophy of Unix pipeline processing to distributed computing on a cluster
environment. It provides a tool capable of handling large volumes of data
while remaining simple to deploy and program for customized analyses. PyPLN
implements distributed processing by relying on ZeroMQ for handling
communication between processes on a single SMP machine or cluster (Coelho et
al., 2012c). Job specifications and data are farmed to worker processes in the
pipeline as JSON messages which are balanced among the cluster. Every step of
the pipeline can be parallelized as long as it applies to more than one
document. Finer grained parallelization is also possible with the
implementation of custom workers. Workers are typically Python scripts which
can, on their turn, call any executable or library available in the system.
Figure 1 depicts the general architecture of PyPLN.
Figure 1: Overview of PyPLN’s architecture.
A common bottleneck in the analysis of large collections of documents is disk
I/O. PyPLN relies on a NoSQL database (Mongodb) to store all the documents and
analytical results. Due to memory mapping, even a single instance of Mongodb
proves much more performant than direct file system storage. For cluster
configurations, the database can be sharded and replicated to maximize
performance and locality of the data with respect to the workers. Any
collection of documents in the PyPLN database is fully searchable as each new
document uploaded is immediately indexed by a high performance search engine.
### 2.2 The Workers
The worker is the basic unit of analysis in the platform. Its basic structure
was designed to be as simple as possible, so that anyone with minimal
knowledge of PyPLN internals can write and contribute a worker to the project.
Figure 2 shows the code of a worker to count the frequency of words in a
document.
⬇
1from pypelinin import Worker
2
3
4class FreqDist(Worker):
5 requires = [’tokens’, ’language’]
6
7 def process(self, document):
8 tokens = [info.lower() for info in document[’tokens’]]
9 frequency_distribution = {token: tokens.count(token) \
10 for token in set(tokens)}
11 fd = frequency_distribution.items()
12 fd.sort(lambda x, y: cmp(y[1], x[1]))
13 return {’freqdist’: fd}
Figure 2: Token counter
Workers are Python modules with a very simple minimal structure but which can
be as complex as their tasks may require. Let’s go through the required lines
of code as can be found in the example of figure 2: It must contain an import
statement like the one shown on line 1; It must declare a Class with an
informative name, which inherits the Worker base class; This class must
declare at least one method named _process_(line 7), and a global variable
named _requires_ (line 5). The _requires_ variable must be assigned a list,
containing analytical results333These may have been calculated before, in
which case they are simply read from the database. on which this worker
depends. In this case, the _FreqDist_ worker depends on the text having been
tokenized before and its language detected. The _process_ method takes one
argument (a Python dictionary that represents the JSON document stored in the
database) and returns a dictionary with at least one key/value pair where the
key corresponds to the name of the result provided.
Analyses which are fundamental and/or pre-requisites for other analyses are
persisted in the document database to be easily accessible for reuse.
Typically most worker implementations wraps code from an existing NLP library
such as NLTK (Bird, 2006) or external executable such as _html2text_. The
wrapping ensures that the calling of the external code conforms with the
requirements of the platform and can be dispatched to other machines of the
cluster to be executed in parallel.
### 2.3 The Pipeline
As mentioned before, distributed processing is implemented in Python using the
metaphor of a pipeline. Pipelines come in two basic flavors in PyPLN. The
first type are built-in ones which consist of basic preprocessing which has to
be applied to every new document added to the system (figure). The second type
are custom pipelines which can be specified by the user and submitted to the
system via the API or, in the future, via the web interface. Figure 3 show the
basic syntax for defining a pipeline. A pipeline is defined as a simple Python
dictionary where every key:value pair represents a pipe.
⬇
1 pipeline = Pipeline({Job(’WorkerName1’): Job(’WorkerName2’),
2 Job(’WorkerName2’): Job(’WorkerName3’)},
3 data={’foo’: ’bar’})
Figure 3: Defining a new pipeline.
Pipelines are created and managed by the pipelinin’
library444github.com/NAMD/pypelinin, which is part of the PyPLN distribution.
By providing means of defining analytical pipelines, PyPLN makes it easy to
store complex analytical workflows in very high-level description language,
which can be parsed and executed by the system with maximal performance. Using
the API programmatically is another way to achieve the same results but with a
little less scalability.
### 2.4 The Web Interface
PyPLN offers a web-based interface for interactive text
analysis555demo.pypln.org in which one can create and manage corpora, perform
full text searches and visualize the analyses performed. Figure 4 shows one
aspect of the web interface displaying the part-of-speech tagging of a
document.
Figure 4: Part-of-speech tagging analysis of PyPLN. Note the various verbal
forms highlighted in the text.
Currently the web interface of PyPLN is the easiest way to interact with the
platform. Either through our free public server or on the user’s own
infrastructure. The installation and deploy of PyPLN’s backend and web front-
end is fully documented 666pypln.org/docs/ and requires no special hardware,
just a single computer running a recent version of Gnu/Linux (for a minimal
install).
When a document is uploaded via the web interface it will pass through the
workers defined in the pipeline. After the analyses are done, the user has
access to interactive visualizations on the web interface, along with links
for the data in machine readable formats. Besides being an easy to use tool
for interactive text processing, the web interface is also meant to be a
showcase of the platform’s capabilities.
All the functionality provided by the web interface is also available as a
REST API, which can be used to interact with PyPLN using your own scripts.
This API is also the only way to communicate with the backend, so all
connections go through the authentication and authorization system.
## 3 Final Remarks
In this paper, we have provided a general description of the PyPLN platform,
developed to make it possible to process large quantities of unstructured
textual data. PyPLN’s goal of integrating a very comprehensive array of NLP
tools into an easy to use web-based platform, is still at least one year away.
But this goal will be approached incrementally and many new features are
already scheduled for the next releases.
Another goal of the PyPLN project is to contribute linguistic knowledge back
to the academic community, in the form of annotated corpora, thesauri, and
other linguistic constructions produced by the platform’s analytical engine.
PyPLN is an open-source project and welcomes the participation of anyone which
would like to see such a platform, as proposed in this paper, grow and remain
available without cost to the community at large.
Despite being on an early stage of development, PyPLN is already very usable
and has been employed to help in computational linguistic studies. (Coelho et
al., 2012a, b)
Acknowledgments
We would like to acknowledge support for this project from Fundação Getulio
Vargas).
## References
* Bick (2000) Eckhard Bick. _The Parsing System ”Palavras”: Automatic Grammatical Analysis of Portuguese in a Constraint Grammar Framework._ Aarhus University Press, 2000.
* Bird (2006) S. Bird. Nltk: the natural language toolkit. In _Proceedings of the COLING/ACL on Interactive presentation sessions_ , pages 69–72. Association for Computational Linguistics, 2006.
* Coelho et al. (2012a) Flávio Codeço Coelho, Cl’audia Torres Codeço, and Renato Rocha Souza. Towards an ontology for mathematical modeling with application to epidemiology. In _Proceedings of the 12th International ISKO Conference - Categories, Relations and Contexts in Knowledge Organization._ , 2012a.
* Coelho et al. (2012b) Flávio Codeço Coelho, Renato Rocha Souza, and Pablo de Camargo Cerdeira. Information mining and visualization of data from the brazilian supreme court (stf): a case study. In _Proceedings of the 12th International ISKO Conference - Categories, Relations and Contexts in Knowledge Organization._ , 2012b.
* Coelho et al. (2012c) Flávio Codeço Coelho, Renato Rocha Souza, and Álvaro Justen. Pypln: a natural language processing pipeline in python. page 6838. Euroscipy, 2012c. URL http://www.euroscipy.org/talk/6838.
|
arxiv-papers
| 2013-01-31T20:21:52 |
2024-09-04T02:49:41.119928
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fl\\'avio Code\\c{c}o Coelho and Renato Rocha Souza and \\'Alvaro Justen\n and Fl\\'avio Amieiro and Heliana Mello",
"submitter": "Flavio Coelho",
"url": "https://arxiv.org/abs/1301.7738"
}
|
1302.0031
|
What We’ve Learned from Heavy Flavour Experiments Since CKM2010
Tim Gershon
Department of Physics, University of Warwick, Coventry, United Kingdom
and
European Organization for Nuclear Research (CERN), Geneva, Switzerland
###### Abstract
A summary of the latest experimental results in heavy flavour physics is
presented.
Proceedings of CKM 2012,
the $7^{\rm th}$ International Workshop on the CKM Unitarity Triangle,
University of Cincinnati, USA, 28 September – 2 October 2012.
## 1 Preliminaries
Heavy flavour physics has seen enormous progress over the last decade or so,
primarily through the successes of the “$B$ factory” experiments, BaBar [1]
and Belle [2], complemented by several notable achievements by the CDF [3] and
D0 [4] experiments based at the Tevatron. Most recently the LHC experiments
have taken on the mantle of producing world leading results in this field (and
others); the LHCb [5] experiment in particular is breaking new ground in
beauty and charm physics, while ATLAS [6] and CMS [7] also have notable $b$
physics components to their programmes.
It must be acknowledged that the successes of the experiments is a direct
consequence of the excellent performance of the accelerators:
* •
PEP-II delivered $433\mbox{\,fb}^{-1}$ of $e^{+}e^{-}$ collisions at the
$\Upsilon(4S)$ resonance to BaBar, plus additional data at other energies;
* •
KEK-B delivered $711\mbox{\,fb}^{-1}$ of $e^{+}e^{-}$ collisions at the
$\Upsilon(4S)$ resonance, and $121\mbox{\,fb}^{-1}$ at the $\Upsilon(5S)$
resonance, to Belle, plus additional data at other energies;
* •
the Tevatron delivered $12\mbox{\,fb}^{-1}$ of $p\bar{p}$ collisions at
$1.96\mathrm{\,Te\kern-1.00006ptV}$ to each of CDF and D0;
* •
during 2011, the LHC delivered $1.2$ ($6$) $\mbox{\,fb}^{-1}$ of $pp$
collisions at $7\mathrm{\,Te\kern-1.00006ptV}$ to LHCb (each of ATLAS and
CMS), and by the time of CKM2012 the 2012 run of the LHC had added $1.5$
($15$) $\mbox{\,fb}^{-1}$ of $8\mathrm{\,Te\kern-1.00006ptV}$ $pp$ collisions
to these respective data samples; additional $pp$ collision data has been
delivered at other energies.
In order to exploit maximally these unprecedented data samples requires novel
detector and innovative analysis techniques. It is impossible to do justice to
the wide range of developments in this area in a single sentence, but as a
brief sample of important breakthroughs: the particle identification
capability provided, e.g., by the BaBar DIRC detector [1], neural network
based event reconstruction as used at Belle [8], trigger algorithms that
enable heavy flavour physics measurements at hadron colliders by exploiting
the displaced vertex signature [9, 10] and silicon detectors to precisely
determine the vertex position, such as the LHCb VELO [5].
## 2 $C\\!P$ violation
$C\\!P$ violation remains one of the deep mysteries of the Standard Model
(SM). Although it is elegantly accommodated within the CKM matrix [11, 12],
and although all measurements to date are consistent with the sole source of
$C\\!P$ violation being the KM phase, continued experimental investigation is
well motivated in order to try to identify additional sources. Indeed, new
sources of $C\\!P$ violation must exist in nature in order to generate the
baryon asymmetry of the Universe. Table 1 lists the channels where $C\\!P$
violation has been observed to date. The list has become longer over the past
few years, and only recently has the first observation of $C\\!P$ violation in
a charged particle decay [13] been achieved. There is still no $5\sigma$
measurement of $C\\!P$ violation in the $B^{0}_{s}$ sector, or in the charm
sector, or in any baryon decay, or in the lepton sector. Clearly, there
remains much to investigate.
Table 1: Observations of $C\\!P$ violation with greater than $5\sigma$
significance, separated by quark-level transition (modified from the review by
D. Kirkby and Y. Nir in Ref. [14], including averages from Ref. [15]). (∗) New
results presented at CKM2012 are not included.
Kaon sector
---
| $\left|\epsilon_{K}\right|=(2.228\pm 0.011)\times 10^{-3}$ [14]
| ${\rm Re}(\epsilon^{\prime}/\epsilon_{K})=(1.65\pm 0.26)\times 10^{-3}$
[16, 17]
$b$ sector
$b\rightarrow c\bar{c}s$ | $S_{\psi K^{0}}=+0.679\pm 0.020$ [18, 19](∗)
$b\rightarrow q\bar{q}s$ | $S_{\eta^{\prime}K^{0}}=+0.59\pm 0.07$ [20, 21], $S_{\phi K^{0}}=+0.74\,^{+0.11}_{-0.13}$ [22, 23],
| $S_{f_{0}(980)K^{0}}=+0.69\,^{+0.10}_{-0.12}$ [22, 23, 24, 25],
$S_{K^{+}K^{-}K^{0}}=+0.68\,^{+0.09}_{-0.10}$ [22, 23]
$b\rightarrow u\bar{u}d$ | $S_{\pi^{+}\pi^{-}}=-0.65\pm 0.07$ [26, 27](∗), $C_{\pi^{+}\pi^{-}}=-0.36\pm 0.06$ [26, 27](∗)
$b\rightarrow c\bar{c}d$ | $S_{\psi\pi^{0}}=-0.93\pm 0.15$ [28, 29], $S_{D^{+}D^{-}}=-0.98\pm 0.17$ [30, 31],
| $S_{D^{*\pm}D^{\mp}}=-0.73\pm 0.11$ [30, 31], $S_{D^{*+}D^{*-}}=-0.71\pm
0.09$ [30, 32, 33]
$b\rightarrow u\bar{u}s$ | $A_{K^{\mp}\pi^{\pm}}=-0.087\pm 0.008$ [26, 34, 35, 36]
$b\rightarrow c\bar{u}s/u\bar{c}s$ | $A_{D_{CP+}K^{\pm}}=+0.19\pm 0.03$ [37, 38, 39, 13]
The determination of $\sin(2\beta)$, where $\beta$ is the angle of the CKM
Unitarity Triangle, provided a spectacular confirmation of the KM scheme [40,
41]. The latest (and, excluding upgrades, most likely final) results from
BaBar [18] and Belle [19] shown in Fig. 1 give a clear visual confirmation of
the large $C\\!P$ violation effect. Further improvement in precision is still
needed, and therefore it is encouraging that LHCb has presented its first
results on this channel [42, 43]. The quantum correlated nature of the $B$
mesons produced in $\Upsilon(4S)\rightarrow B\bar{B}$ decays can in addition
be used to make novel tests of quantum mechanics [44, 45, 46, 47]. Recently,
the BaBar collaboration has extended its $\sin(2\beta)$ analysis to
demonstrate that $T$ is violated, as well as $C\\!P$, in the
$B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}$ system
(but $C\\!PT$ is conserved, as expected) [48].
Figure 1: Results from (left) BaBar [18] and (right) Belle [19] on the
determination of $\sin 2\beta$.
Direct $C\\!P$ violation has been established for several years in both kaon
[49, 50, 51] and $B$ [52, 53] sectors. However, recently new observations have
been made of very large, near maximal, direct $C\\!P$ violation effects. The
LHCb collaboration has reported preliminary studies of the variation of
$C\\!P$ asymmetries across the Dalitz plots of $B^{+}$ decays to the three-
body final states $K^{+}K^{-}K^{+}$, $K^{+}\pi^{+}\pi^{-}$ [54],
$K^{+}K^{-}\pi^{+}$ and $\pi^{+}\pi^{-}\pi^{+}$ [55, 56], revealing regions of
phase space where the direct $C\\!P$ violation effect appears to be in excess
of $50\,\%$. Detailed studies will be necessary to understand the origin of
these effects.
Perhaps of even greater interest is the possibility of $C\\!P$ violation in
the charm system, which is already bounded to be small, but is (at least,
naïvely) expected to be much smaller in the SM. However, several measurements
[57, 58] now point to non-zero direct $C\\!P$ violation effects in the charm
system, specifically in the difference in $C\\!P$ violation between
$D^{0}\rightarrow K^{+}K^{-}$ and $D^{0}\rightarrow\pi^{+}\pi^{-}$ decays,
with the world average being $\Delta a_{C\\!P}^{\rm dir}=(-0.68\pm 0.15)\,\%$
[15]. The emerging consensus (reviewed in Ref. [59]) is that an effect of this
size is unlikely in the SM, but could in principle be generated by QCD
corrections. Improved measurements of the $C\\!P$ violation effects in several
decay channels are needed to resolve the situation. These measurements must
aim at per mille level precision, and will be challenging experimentally.
A similar challenge is faced in the $B$ sector, where the possibility of
anomalous $C\\!P$ violation in mixing has been raised by a measurement by D0
[60] of the inclusive same-sign dimuon asymmetry, which is $3.9\sigma$ away
from the SM prediction (of very close to zero [61]). This measurement is
sensitive to an approximately equal combination of the parameters of $C\\!P$
violation in $B^{0}$ and $B^{0}_{s}$ mixing, $a_{\rm sl}^{d}$ and $a_{\rm
sl}^{s}$, however some sensitivity to the source of the asymmetry can be
obtained by applying additional constraints on the impact parameter to obtain
a sample enriched in either oscillated $B^{0}$ or $B^{0}_{s}$ candidates. In
addition, $a_{\rm sl}^{d}$ and $a_{\rm sl}^{s}$ can be measured individually,
with recent new results from D0 [62, 63], LHCb [64] and BaBar [65]. The latest
world average, shown in Fig. 2, gives $a_{\rm sl}^{d}=-0.0003\pm 0.0021$,
$a_{\rm sl}^{s}=-0.0109\pm 0.0040$ [15], $2.4\sigma$ away from the SM
prediction. Improved measurements of both $a_{\rm sl}^{d}$ and $a_{\rm
sl}^{s}$ are needed.
Figure 2: World average of constraints on the parameters describing $C\\!P$
violation in $B^{0}$ and $B^{0}_{s}$ mixing, $a_{\rm sl}^{d}$ and $a_{\rm
sl}^{s}$. The red point close to $(0,0)$ is the SM prediction [61]. The green
ellipse comes from the D0 inclusive same-sign dimuon analysis [60]; the blue
shaded bands give the world average constraints on $a_{\rm sl}^{d}$ and
$a_{\rm sl}^{s}$ individually; the red ellipse is the world average including
all constraints [15].
## 3 The Unitary Triangle
Many of the most important results in heavy flavour physics can be depicted as
constraints in the so-called CKM Unitarity Triangle, that visualises the
unitarity relation between the first and third columns of the CKM matrix. The
constraints include measurements of the angles and lengths of the sides of the
triangle.111 Here the $\alpha,\beta,\gamma$ notation is used for the angles.
Although other similar triangles can be formed (see, e.g. Ref. [66]), they are
not so intimately related to the experimental programme. Illustrations showing
these constraints, particularly those produced by the CKMfitter [67] and UTfit
[68] collaborations shown in Fig. 3, have become icons of the field.
Figure 3: Constraints on the Unitarity Triangle as compiled by (left)
CKMfitter [67], (right) UTfit [68].
The golden channel measurement of $\sin 2\beta$ from
$B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ decays provides one of the strongest
constraints, giving $\beta=(21.5\,^{+0.8}_{-0.7})^{\circ}$ [15]. BaBar and
Belle have also obtained complementary measurements of $\beta$ using decays
involving the quark-level transitions $b\rightarrow c\bar{c}d$ and
$b\rightarrow q\bar{q}s$. Several updated and improved measurements have
become available recently [31, 32, 33, 22, 69], so that $C\\!P$ violation
effects are now established in these decay processes (see Tab. 1). The
tensions that existed in the data some years ago have been alleviated, and all
measurements of $\beta$ are consistent within their current uncertainties.
Progress on the measurement of the angle $\alpha$ has been somewhat slower in
recent years, although new and improved inputs to the isospin analyses in
$\pi\pi$ [26, 70], $\rho\pi$ [71], $\rho\rho$ [70] and $a_{1}\pi$ [72] systems
have become available. The world average,
$\alpha=(88.5\,^{+4.7}_{-4.4})^{\circ}$ [67] is dominated by the constraints
from the $\rho\rho$ system, where some parameters come from only a single
measurement, and others are in slight tension. Improved measurements, in
particular of $B\rightarrow\rho\rho$ decays, are necessary to ensure that the
determination is under as good control as suggested by the uncertainties.
The precision on the third angle of the Unitarity Triangle, $\gamma$,
determined from $B\rightarrow DK$ and related decay processes, has stubbornly
refused to go below the $10^{\circ}$ level despite great efforts from BaBar
and Belle [73, 74]. The latest combinations from each experiment give (BaBar)
$\gamma=(69\,^{+17}_{-16})^{\circ}$ [75, 76] and (Belle)
$\gamma=(68\,^{+15}_{-14})^{\circ}$ [77]. However, with precision measurements
of some of the relevant observables now available (enabling the first
$5\sigma$ observation of $C\\!P$ violation in $B\rightarrow DK$ decays [13]),
it appears that this barrier may soon be broken. Indeed, the LHCb
collaboration, combining results from $B\rightarrow DK$ processes [13, 78, 79]
obtains $\gamma=(71\,^{+17}_{-16})^{\circ}$ [80, 81]. Further improvement can
be expected as more data and more decay modes are added into the combination.
The use of $B\rightarrow DK$ decays provides a theoretically clean
determination of $\gamma$, since only tree-level diagrams are involved. As
such it can be considered a SM benchmark. Measurements of $\gamma$ from
charmless $B$ decays, however, can be affected by extensions of the SM due to
the loop diagrams involved, but tend to suffer from larger hadronic
uncertainties. A perennial question for CKM workshops is therefore how to
extract clean weak phase information from charmless hadronic $B$ decays,
without losing the ability to probe beyond the SM. Several new ideas have
appeared recently in the literature [82, 83].
Experimentally, new data on direct $C\\!P$ violation in $B$ (including
$B^{0}_{s}$) decays to $K\pi$ are available [36, 26, 84, 85], and the “$K\pi$”
puzzle, namely the difference between the amount of direct $C\\!P$ violation
in $B$ decays to $K^{-}\pi^{+}$ and $K^{-}\pi^{0}$, persists. Improved
measurements of the $B^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{0}$
$C\\!P$ violation parameters are necessary to test the isospin sum-rules that
provide clean tests of the SM [86]. While these must wait for data from Super
$B$ factories to become available, progress on U-spin and flavour SU(3) based
tests of the SM using $B^{0}_{s}\rightarrow K^{+}K^{-}$ can be expected with
promising first results from LHCb already available [87].
As mentioned above, other triangles can be constructed based on the unitarity
of the CKM matrix. One of them is particularly relevant to the $B^{0}_{s}$
sector, and its angle $\beta_{s}$ (in analogy to $\beta$) can be determined
using $b\rightarrow c\bar{c}s$ transitions. Measurements of $\beta_{s}$ are
now available from CDF [88], D0 [89] and ATLAS [90]222 In contrast to the
other analyses, the ATLAS measurement does not use flavour tagging. using
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ decays
and from LHCb using both
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ [91,
92] and ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ [93]
decays. These results, summarised in Fig. 4 (left), in particular those from
LHCb, provide a significant improvement in precision (and have established
$\Delta\Gamma_{s}>0$ [94]; a compilation of results in the
$\Delta\Gamma_{s}$–$1/\Gamma_{s}$ plane is shown in Fig. 4 (right)), while not
confirming the hints of a large anomalous effect that were present in previous
measurements.
Figure 4: (Left) Summary of results in the
$\Delta\Gamma_{s}$–$\phi_{s}^{c\bar{c}s}$ plane [15]. Note that
$\phi_{s}^{c\bar{c}s}\approx-2\beta_{s}$. (Right) Compilation of results in
the $\Delta\Gamma_{s}$–$1/\Gamma_{s}$ plane [90]. In addition to the
constraints from $B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ [88, 89, 90, 92, 93], results of effective lifetime measurements
in $B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ [95, 96], $K^{+}K^{-}$ [97], $D^{-}_{s}\pi^{+}$ [98] and
$D^{-}_{s}\mu^{+}X$ [99] are shown, and compared to the SM prediction for
$\Delta\Gamma_{s}$ [61].
The lengths of the sides of the Unitarity Triangle are determined from
measurements of the magnitudes of CKM matrix elements: $\left|V_{cb}\right|$
and $\left|V_{ub}\right|$ through tree-level decays; $\left|V_{td}\right|$ and
$\left|V_{ts}\right|$ through loop-dominated processes. The latter have to
date been mainly constrained from the neutral $B$ meson oscillation
frequencies $\Delta m_{d}$ and $\Delta m_{s}$ [100, 43, 101], but loop decays
such as $B\rightarrow\rho\gamma$ and $B\rightarrow\pi\mu^{+}\mu^{-}$ can also
be used to probe $\left|V_{td}\right|$ (with corresponding $b\rightarrow s$
transitions probing $\left|V_{ts}\right|$). Since the
$B^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-}$ decay has now been observed [102],
further tests of the SM using these processes become feasible.
Constraints on $\left|V_{ub}\right|$ are obtained from both inclusive and
exclusive processes, which have been in some tension for several years (as
indeed have results on $\left|V_{cb}\right|$). Among the exclusive modes, the
$\pi l\nu$ channel has been studied in detail using a variety of experimental
techniques [103, 104, 105, 106, 107]; a recent compilation is shown in Fig. 5.
Figure 5: Differential branching fraction of $B^{0}\rightarrow\pi^{-}l^{+}\nu$
($l=e,\mu$) [108], using data from BaBar [105] and Belle [106], and lattice
QCD calculations from FNAL/MILC [109].
These results have recently been complemented by new results on
$B\rightarrow\rho l\nu$, $\omega l\nu$, $\eta l\nu$ and $\eta^{\prime}l\nu$
[103, 110, 104, 105, 107]. The latest constraints on $\left|V_{ub}\right|$,
combining BaBar and Belle data, as part of the “Physics of the $B$ factories
legacy book” project (following the procedures developed in Ref. [15])
$\displaystyle\left|V_{ub}\right|_{\rm excl.}$ $\displaystyle=$
$\displaystyle\left[3.23\times\left(1.00\pm 0.05_{\rm exp}\pm 0.08_{\rm
th}\right)\right]\times 10^{-3}\,.$ $\displaystyle\left|V_{ub}\right|_{\rm
incl.}$ $\displaystyle=$ $\displaystyle\left[4.42\times\left(1.000\pm
0.045_{\rm exp}\pm 0.034_{\rm th}\right)\right]\times 10^{-3}\,.$
An average of these numbers has a $\chi^{2}$ (probability) of 3.0 (0.003). The
origin of the discrepancy is not understood, and so the uncertainty is scaled
by $\sqrt{3}$ to give
$\displaystyle\left|V_{ub}\right|_{\rm avg.}$ $\displaystyle=$
$\displaystyle\left[3.95\times\left(1.000\pm 0.096_{\rm exp}\pm 0.099_{\rm
th}\right)\right]\times 10^{-3}\,.$
Better understanding will be needed to reduce the uncertainty.
A complementary determination of $\left|V_{ub}\right|$ can be made from the
leptonic decay $B^{+}\rightarrow\tau^{+}\nu$, using as input the lattice QCD
calculation of the $B^{+}$ decay constant. Such a measurement is sensitive to
extensions to the SM, such as models containing charged Higgs bosons. Previous
results had shown a deviation with the prediction based on the CKM fit, but
the latest result from Belle [111] reduces the tension to below the $2\sigma$
level. It should be noted that the signals for $B^{+}\rightarrow\tau^{+}\nu$
[112, 113, 114, 111] are individually below the $5\sigma$ threshold usually
demanded for discovery, further indicating that claims of a non-SM excess were
premature.
Another channel that is sensitive to potential effects of charged Higgs bosons
is $B\rightarrow D^{(*)}\tau\nu$. These four decays ($B^{+}$ or $B^{0}$ to
final states containing $D$ or $D^{*}$) are all now observed with above
$5\sigma$ significance [115, 116], with the latest results giving an excess
above the SM prediction at $3.4\sigma$ significance [116].333 The Belle
results [115] are not included in this combination since correlations that are
not publicly available need to be taken into account. Moreover the pattern of
the excess in $D\tau\nu$ and $D^{*}\tau\nu$ is inconsistent with that expected
in the simplest model containing charged Higgs bosons, the (type II) two-Higgs
doublet model [117]. This demands further experimental investigation.
## 4 Rare decays
The two most common generic arguments in favour of a continued experimental
programme in quark flavour physics are (i) there must be new sources of
$C\\!P$ violation that may manifest themselves in the quark sector (through
measurements such as those discussed above); (ii) precise measurements of
“rare decays”, i.e. processes that are suppressed in the SM, are sensitive to
physics beyond the SM at scales that can exceed those that can studied at the
energy frontier. Both of these remain strong arguments, although it must be
recognised that neither constitutes a guarantee of discovery in any particular
experiment. In addition, the recent measurement of a large value of the lepton
mixing angle $\theta_{13}$ [118, 119] suggests that the observation of $C\\!P$
violation in neutrino oscillations may become a reality within the next decade
or so. This provides a “best before” date for the $C\\!P$ violation-based
argument. On the other hand, the observation of a Higgs-like particle by ATLAS
[120] and CMS [121], and the non-discovery of any non-SM particle, somewhat
strengthens the argument for precision probes of rare processes (including
$C\\!P$-violating processes) that are sensitive to physics at high scales.
The archetypal flavour-changing neutral-current decay is the $b\rightarrow
s\gamma$ transition. The rate has been measured to be consistent with the SM
prediction [122] within $\sim 10\,\%$ uncertainty, but further studies remain
worthwhile, and new results are becoming available for both inclusive [123,
124] and exclusive decays [125]. In particular, improved measurements of
$C\\!P$ asymmetries and of the polarisation of the emitted photon will help to
constrain, model-independently, the phase space of extensions to the SM.
The $B\rightarrow K^{*}\mu^{+}\mu^{-}$ decay provides an excellent laboratory
to test the SM, via precise measurements of the Wilson coefficients in the
operator product expansion of the effective weak Hamiltonian. Recently new
results have become available from BaBar [126], CDF [127] and LHCb [128, 129].
The latest LHCb analysis has the smallest uncertainties, and allows for the
first time a determination of the zero-crossing point of the forward-backward
asymmetry,
$q^{2}_{0}=(4.9\,^{+1.1}_{-1.3})\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$,
consistent with the SM prediction [130, 131, 132]. A compilation, including
also results from Belle [133] is shown in Fig. 6. However, there is still a
long road to the final goal of a full angular analysis of $B\rightarrow
K^{*}\mu^{+}\mu^{-}$ decays.
Figure 6: Compilation of measurements of (top left) the differential branching
fraction, (top right) the forward-backward asymmetry, (bottom left) the
longitudinal polarisation fraction, (bottom right) the parameter $S_{3}$ in
$B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ decays (from Ref. [129]), as functions
of the dimuon invariant mass squared. The data points are compared to the SM
expectations [134].
The highest profile results in heavy flavour physics must certainly be those
on the $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ decay. This decay is suppressed in
the SM due to three effects: the absence of tree-level flavour-changing-
neutral currents, the CKM factors, and the V-A structure of the weak
interaction (helicity suppression). Many extensions to the SM break at least
one of these factors – of particular interest is that models with extended
Higgs’ sectors (such as SUSY) may break the helicity suppression. This can
potentially lead to large enhancements of the branching fraction over the SM
prediction [135], which is precisely known, in part due to the good control
over the lattice QCD calculations which are used as inputs (as reviewed, e.g.
in Ref. [59]).
The latest results from ATLAS, with $2.4\mbox{\,fb}^{-1}$ [136], CMS, with
$5.0\mbox{\,fb}^{-1}$ [137] and LHCb, with $1.0\mbox{\,fb}^{-1}$ [138] (all
using $7\mathrm{\,Te\kern-1.00006ptV}$ LHC $pp$ collision data recorded in
2011) are shown in Fig. 7. The most restrictive limit from a single experiment
is the LHCb result ${\cal B}(B^{0}_{s}\rightarrow\mu^{+}\mu^{-})<4.5\times
10^{-9}$ at 95 % confidence level [138], while a combination of LHC results
gives ${\cal B}(B^{0}_{s}\rightarrow\mu^{+}\mu^{-})<4.2\times 10^{-9}$ at 95 %
confidence level [139]. The results approach the sensitivity required to probe
the SM expectation, and updates including the 2012 LHC data are hotly
anticipated.444 While these proceedings were in preparation, the LHCb
collaboration announced the first evidence for the
$B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ decay, with branching fraction ${\cal
B}(B^{0}_{s}\rightarrow\mu^{+}\mu^{-})=(3.2\,^{+1.5}_{-1.2})\times 10^{-9}$
[140].
Figure 7: Latest results on the search for
$B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ from (left) ATLAS [136], (middle) CMS
[137], (right) LHCb [138].
## 5 “Bread and butter” measurements
In any summary of a wide range of experimental results, it is natural to focus
attention mainly on “golden channels”. However, by doing so it is possible
that the importance of making improved measurements of other, less glamorous,
observables can be overlooked. It should be stressed that such measurements
are an integral part of the overall programme. Indeed, very often the most
highly cited results depend directly on input from other measurements.
An excellent example comes from the $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$
analysis: in order to measure the absolute branching fraction of any
$B^{0}_{s}$ decay, it is necessary to know the relative production rates of
$B^{0}_{s}$ and $B^{0}$ mesons, denoted as $f_{s}/f_{d}$. Then the branching
fraction can be measured relative to that of a well-known $B^{0}$ (or $B^{+}$,
isospin symmetry is assumed) decay – typically
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ is used .
Measurement of $f_{s}/f_{d}$ [141, 142], however, relies on knowledge of
absolute branching fractions of charmed hadrons, in particular of ${\cal
B}(D^{+}_{s}\rightarrow K^{+}K^{-}\pi^{+})$. This value is currently known to
$5.1\,\%$ precision [143], and future measurements of ${\cal
B}(B^{0}_{s}\rightarrow\mu^{+}\mu^{-})$ will be limited at this level unless
improved results become available. Therefore it is very encouraging that a new
preliminary result on ${\cal B}(D^{+}_{s}\rightarrow K^{+}K^{-}\pi^{+})$ is
available from Belle (as a spin-off from the measurement of ${\cal
B}(D^{+}_{s}\rightarrow\tau^{+}\nu)$ [144]).
This is just one example of the large impact that “bread and butter”
measurements can have. Other examples include lifetime and mass measurements.
It should also be remembered that discoveries may appear in unexpected places.
## 6 Summary
Sometime Cincinnati resident Mark Twain once wrote, “The most permanent
lessons in morals are those which come, not of booky teaching, but of
experience.” This can be interpreted as encouragement for the experimentalist.
To summarise, then, the following (hopefully) permanent lessons have been
learned:
* •
The foundation of accelerator-based physics is, of course, excellent
performance by the accelerators.
* •
Investment in detectors and techniques brings rewards.
* •
Interesting effects could be big …
* –
but they may be very small, so be prepared to be precise,
* –
it seems like there are no ${\cal O}(1)$ deviations from the SM.
* •
Clean theoretical predictions are to be treasured …
* –
but in their absence, data-driven methods to control uncertainties are to be
valued.
* •
$3\sigma$ effects seem to often go away, but $5\sigma$ effects seem to stay …
* –
nevertheless, investigating anomalies is worth the effort, and is sure to
result in something being learned (whether about physics, systematics or
statistics).
* •
Bread and butter measurements provide an essential appetiser to the feast of
interesting observables that heavy flavour experiments can study.
* •
New physics just might be around the corner …
* –
so there is plenty to look forward to in heavy quark physics in the coming
years [145, 146].
Whether or not these lessons have taught us any morals must be left to
CKM2014.
## Acknowledgements
The author wishes to thank the organisers for the invitation, and to
congratulate them for a productive and enjoyable workshop. Vincenzo Chiochia,
Rob Harr, Mikihiko Nakao, Abi Soffer and Mark Williams provided suggestions
for the content of the talk. Rick van Kooten, Marie-Hélène Schune and Karim
Trabelsi commented on drafts of the proceedings. Errors and omissions are the
sole fault of the author. This work is supported by the Science and Technology
Facilities Council (United Kingdom), CERN, and by the European Research
Council under FP7.
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|
arxiv-papers
| 2013-01-31T22:52:55 |
2024-09-04T02:49:41.130870
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tim Gershon",
"submitter": "Tim Gershon",
"url": "https://arxiv.org/abs/1302.0031"
}
|
1302.0079
|
# $\Xi$ baryon strong decays in a chiral quark model
Li-Ye Xiao and Xian-Hui Zhong [email protected] Department of Physics,
Hunan Normal University, and Key Laboratory of Low-Dimensional Quantum
Structures and Quantum Control of Ministry of Education, Changsha 410081,
China
###### Abstract
The strong decays of $\Xi$ baryons up to $N=2$ shell are studied in a chiral
quark model. The strong decay properties of these well-established ground
decuplet baryons are reasonably described. We find that (i) $\Xi(1690)$ and
$\Xi(1820)$ could be assigned to the spin-parity $J^{P}=1/2^{-}$ state
$|70,^{2}{8},1,1,\frac{1}{2}^{-}\rangle$ and the spin-parity $J^{P}=3/2^{-}$
state $|70,^{2}{8},1,1,\frac{3}{2}^{-}\rangle$, respectively. Slight
configuration mixing might exist in these two negative parity states. (ii)
$\Xi(1950)$ might correspond to several different $\Xi$ resonances. The broad
states ($\Gamma\sim 100$ MeV) observed in the $\Xi\pi$ channel could be
classified as the pure $J^{P}=5/2^{-}$ octet state
$|70,^{4}8,1,1,\frac{5}{2}^{-}\rangle$ or the mixed state
$|\Xi\frac{1}{2}^{-}\rangle_{3}$ with $J^{P}=1/2^{-}$. The $\Xi$ resonances
with moderate width ($\Gamma\sim 60$ MeV) observed in the $\Xi\pi$ channel
might correspond to the $J^{P}=1/2^{+}$ excitation
$|56,^{4}10,2,2,\frac{1}{2}^{+}\rangle$. The second orbital excitation
$|56,^{4}10,2,2,\frac{3}{2}^{+}\rangle$ and the mixed state
$|\Xi\frac{1}{2}^{-}\rangle_{1}$ might be candidates for the narrow width
state observed in the $\Lambda\bar{K}$ channel, however, their spin-parity
numbers are incompatible with the moment analysis of the data. (iii)
$\Xi(2030)$ could not be assigned to either any spin-parity $J^{P}=7/2^{+}$
states or any pure $J^{P}=5/2^{+}$ configurations. It seems to favor the
$|70,^{2}8,2,2,\frac{3}{2}^{+}\rangle$ assignment, however, its spin conflicts
with the moment analysis of the data. To find more $\Xi$ resonances, the
observations in the $\Xi(1530)\pi$ and $\Sigma(1385)\bar{K}$ channels are
necessary.
###### pacs:
12.39.Jh, 13.30.-a, 14.20.Lq, 14.20.Mr
## I INTRODUCTION
Understanding the baryon spectrum and the search for the missing baryon
resonances are hot topics in hadron physics. For the scarcity of high quality
antikaon beams and the small production rate for the $\Xi$ resonances via an
electromagnetic probe, the $\Xi$ spectrum is still far from being established.
There are only a few data on the $\Xi$ resonances from the old bubble chamber
experiments with small statistics. Among the eleven $\Xi$ baryons listed in
the review of the Particle Data Group (PDG) Beringer:1900zz , only the ground
octet and decuplet, $\Xi(1320)$ and $\Xi(1530)$, are well established with
four-star ratings. Although the resonances $\Xi(1690)$, $\Xi(1820)$,
$\Xi(1950)$ and $\Xi(2030)$ are three-star states in PDG, only $\Xi(1820)$ has
a determined spin-parity $J^{P}=3/2^{-}$.
A few theoretical studies of the $\Xi$ baryon spectrum can be found in the
literature Isgur:1978xj ; Chao:1980em ; Capstick:1986bm ; Glozman:1995fu ;
Bijker:2000gq ; Pervin:2007wa ; Oh:2007cr ; Melde:2008yr ; Chen:2009de ;
Lee:2002jb ; Schat:2001xr ; Goity:2002pu ; Goity:2003ab . All of the
phenomenological models can well explain the two ground states $\Xi(1320)$ and
$\Xi(1530)$, however, their predictions for the excitations are very
different. For example, there is a lot of controversy about the spin-parity of
$\Xi(1690)$. It might be the first radial excitation of $\Xi$ with
$J^{P}=1/2^{+}$ according to the mass calculations with a nonrelativistic
quark model by Chao _et al._ Chao:1980em . This classification was also
supported by the recent quark model study of Melde _et al._ Melde:2008yr .
However, with a relativized quark model Capstick and Isgur predicted that the
first radial excitation of $\Xi$ should have a larger mass of $\sim 1840$ MeV
Capstick:1986bm . Recently, Pervin and Roberts suggested that $\Xi(1690)$
could be assigned to the first orbital excitation of $\Xi$ with
$J^{P}=1/2^{-}$, although their quark model predicted mass is about 35 MeV
heavier than the mass of $\Xi(1690)$ Pervin:2007wa . They believed that a more
microscopic treatment of spin-orbit interactions, can be expected to drive
this state to slightly lower mass. The calculations from the Skyrme model also
indicated that $\Xi(1690)$ should have $J^{P}=1/2^{-}$ Oh:2007cr .
Furthermore, $\Xi(1690)$ was discussed to be a dynamically generated state
with $J^{P}=1/2^{-}$ Kolomeitsev:2003kt ; Sarkar:2004jh . It should be
mentioned that _BABAR_ collaboration, in a study of
$\Lambda_{c}^{+}\rightarrow\Xi^{-}\pi^{+}K^{+}$, recently found some evidence
that $\Xi(1690)$ has $J^{P}=1/2^{-}$ Aubert:2008ty .
Even for the $\Xi(1820)$ with well determined spin-parity quantum numbers,
$J^{P}=3/2^{-}$, there still exist many puzzles in the explanations of its
nature. Considering $\Xi(1820)$ as a $J^{P}=3/2^{-}$ orbital excitation, its
mass and decay properties could be well explained in the nonrelativistic quark
model Chao:1980em and algebraic model Bijker:2000gq . However, considering
$\Xi(1820)$ as a $J^{P}=3/2^{-}$ excitation, the mass of $\Xi(1820)$ is
obviously underestimated by the relativized quark model Capstick:1986bm and
one-boson exchange model Glozman:1995fu . In the unitary chiral approaches,
$\Xi(1820)$ was even suggested to be a dynamically generated state from the
$\Delta$ decuplet-pion octet chiral interaction Kolomeitsev:2003kt ;
Sarkar:2004jh .
Although $\Xi(1950)$ is a three-star state listed in PDG, not much can be said
about its properties Beringer:1900zz . According to various phenomenological
model predictions Chao:1980em ; Matagne:2011fr ; Faiman:1973br ; Schat:2001xr
; Goity:2002pu ; Goity:2003ab , many states with spin-parity quantum numbers,
$J^{P}=1/2^{\pm}$, $3/2^{\pm}$ and $5/2^{\pm}$, could be candidates for
$\Xi(1950)$, because their predicted masses are close to 1950 MeV. Alitti _et
al._ suggested that the broad $\Xi$ resonance ($\Gamma=80\pm 40$ MeV) with a
mass of $M=1930\pm 20$ MeV observed by them could be classified as the pure
$J^{P}=5/2^{-}$ octet state Alitti:1968zz . With this assignment, the mass and
total and partial decay widths seemed to be reasonably understood. However,
the observations of Biagi _et al._ Biagi:1986vs do not satisfy this
classification. Although they observed a $\Xi$ resonance around 1950 MeV in
the $\Lambda\bar{K}$ invariant mass distribution, its favorable spin-parity
should be $5/2^{+}$ or its spin should be greater than $5/2$ in the natural
spin-parity series $7/2^{-}$, $9/2^{+}$, etc Biagi:1986vs . In fact, there
might be more than one $\Xi$ near 1950 MeV Beringer:1900zz . Recently,
Valderrama, Xie and Nieves proposed the existence of three $\Xi(1950)$ states:
one of these states would be part of a spin-parity $1/2^{+}$ decuplet and the
other two probably would belong to the $5/2^{+}$ and $5/2^{-}$ octets
PavonValderrama:2011gp .
About $\Xi(2030)$, not many discussions can be found in the literature. A
moment analysis of Hemingway:1977uw suggested that the spin of $\Xi(2030)$
should be $J\geq 5/2$. According to the SU(3) symmetry of hadrons, Samios _et
al._ predicted that $\Xi(2030)$ was most likely to be the partner of
$N(1680)$, $\Lambda(1820)$ and $\Sigma(1915)$ with $J^{P}=5/2^{+}$
Samios:1974tw . The quark model mass calculations indicated that the second
$J^{P}=5/2^{+}$ state and $J^{P}=7/2^{+}$ state might be candidates for
$\Xi(2030)$ Chao:1980em ; Pervin:2007wa . However, the recent strong decay
analysis of PavonValderrama:2011gp did not support the classification of
Samios:1974tw .
Fortunately, the situation of the poor knowledge about these hyperon
resonances is to be changed completely with the running of the Japan Proton
Accelerator Research Complex (J-PARC) facility, where high quality antikaon
beams will be available very soon Nagae:2008zz . Then, the $\Xi$ baryons can
be directly produced on the nucleons by the process $\bar{K}N\rightarrow\Xi K$
with large production cross sections. Furthermore, the investigation of $\Xi$
baryons is also one of the major parts of the physics programs at
$\mathrm{\bar{P}ANDA}$ via the process $\bar{p}p\rightarrow\Xi\bar{\Xi}$
Lutz:2009ff , at Jlab via the photo-production processes, $\gamma p\rightarrow
K^{+}K^{+}\Xi^{-}$ and $\gamma p\rightarrow K^{+}K^{+}\Xi^{0}\pi^{-}$
Price:2004xm ; Price:2004hr ; Guo:2007dw , and at BESIII via the process
$\mathrm{Charmonium}\rightarrow\bar{\Xi}\Xi$ Zou:2000wg ; Asner:2008nq , etc.
Thus, these new facilities provide us with good chances to study the $\Xi$
spectrum.
Stimulated by the great progress achieved in experiments, more and more
theoretical interests have begun to focus on the $\Xi$ physics. For example,
some theoretical studies of the production of $\Xi$ via
$\bar{K}N\rightarrow\Xi K$ Sharov:2011xq ; Shyam:2011ys and $\gamma
p\rightarrow K^{+}K^{+}\Xi^{-}$ Nakayama:2006ty ; Man:2011np , and the
determination of the parity of $\Xi$ resonances in these reactions were
started recently Nakayama:2012zp . In this work, we make a systematic study of
the strong decays of the $\Xi$ resonances in a chiral quark model, which has
been developed and successfully used to deal with the strong decays of charmed
baryons and heavy-light mesons Zhong:2007gp ; Liu:2012sj ; Zhong:2010vq ;
Zhong:2009sk ; Zhong:2008kd . Here, we mainly focus on the three-star $\Xi$
resonances $\Xi(1690)$, $\Xi(1820)$, $\Xi(1950)$ and $\Xi(2030)$.
This work is organized as follows. In the subsequent section, baryons in the
constituent quark model are outlined. Then a brief review of the strong decays
described within a chiral quark model approach is given in Sec. III. The
numerical results are presented and discussed in Sec. IV. Finally, a summary
is given in Sec. V.
## II BARYONS IN THE QUARK MODEL
### II.1 Spatial wave functions
A baryon containing three quarks may be described by an oscillator potential
Hamiltonian Bhaduri in a nonrelativistic form:
$\mathcal{H}=\sum_{i=1}^{3}\frac{p_{i}^{2}}{2m_{i}}+\frac{1}{2}K\sum_{i<j}(\mathbf{r}_{j}-\mathbf{r}_{i})^{2},$
(1)
where $\mathbf{r}_{i}$ and $\mathbf{p}_{i}$ are the coordinate and momentum
for the $j$th quark in the baryon rest frame, respectively. The quarks are
confined in an oscillator potential with the potential parameter $K$
independent of the flavor quantum number. One defines the Jacobi coordinates
to eliminate the center mass (c.m.) variables:
$\displaystyle\vec{\rho}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\mathbf{r}_{1}-\mathbf{r}_{2}),$ (2)
$\displaystyle\vec{\lambda}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{6}}\left(\frac{m_{1}\mathbf{r}_{1}+m_{2}\mathbf{r}_{2}}{m_{1}+m_{2}}-2\mathbf{r}_{3}\right),$
(3) $\displaystyle\mathbf{R}_{c.m.}$ $\displaystyle=$
$\displaystyle\frac{m_{1}\mathbf{r}_{1}+m_{2}\mathbf{r}_{2}+m_{3}\mathbf{r}_{3}}{M},$
(4)
where $M\equiv m_{1}+m_{2}+m_{3}$. Using the Jacobi coordinates in Eqs.
(2)-(4), the oscillator Hamiltonian (1) is reduced to
$\displaystyle\mathcal{H}=\frac{P^{2}_{c.m.}}{2M}+\frac{1}{2m_{\rho}}\mathbf{p}^{2}_{\rho}+\frac{1}{2m_{\lambda}}\mathbf{p}^{2}_{\lambda}+\frac{3}{2}K(\rho^{2}+\lambda^{2}),$
(5)
where
$\displaystyle\mathbf{p}_{\rho}=m_{\rho}\dot{\vec{\rho}},\ \
\mathbf{p}_{\lambda}=m_{\lambda}\dot{\vec{\lambda}},\ \
\mathbf{P}_{c.m.}=M\mathbf{\dot{R}}_{c.m.},$ (6)
with
$\displaystyle m_{\lambda}=\frac{3(m_{1}+m_{2})m_{3}}{2M},\ \
m_{\rho}=\frac{2m_{1}m_{2}}{m_{1}+m_{2}}.$ (7)
For a $\Xi$ baryon, it contains two strange ($s$) quarks and a light $u/d$
quark. According to Eqs.(2)-(4), the coordinates, $\mathbf{r}_{1}$,
$\mathbf{r}_{2}$ and $\mathbf{r}_{3}$, can be translated into functions of the
Jacobi coordinates $\lambda$ and $\rho$, which are given by
$\displaystyle\mathbf{r}_{1}$ $\displaystyle=$
$\displaystyle\mathbf{R}_{c.m.}+\frac{1}{\sqrt{6}}\frac{3m^{\prime}}{2m+m^{\prime}}\vec{\lambda}+\frac{1}{\sqrt{2}}\vec{\rho},$
(8) $\displaystyle\mathbf{r}_{2}$ $\displaystyle=$
$\displaystyle\mathbf{R}_{c.m.}+\frac{1}{\sqrt{6}}\frac{3m^{\prime}}{2m+m^{\prime}}\vec{\lambda}-\frac{1}{\sqrt{2}}\vec{\rho},$
(9) $\displaystyle\mathbf{r}_{3}$ $\displaystyle=$
$\displaystyle\mathbf{R}_{c.m.}-\sqrt{\frac{2}{3}}\frac{3m}{2m+m^{\prime}}\vec{\lambda}.$
(10)
Then, the momenta $\mathbf{p}_{1}$, $\mathbf{p}_{2}$ and $\mathbf{p}_{3}$ are
given by
$\displaystyle\mathbf{p}_{1}$ $\displaystyle=$
$\displaystyle\frac{m}{M}\mathbf{P}_{c.m.}+\frac{1}{\sqrt{6}}\mathbf{p}_{\lambda}+\frac{1}{\sqrt{2}}\mathbf{p}_{\rho},$
(11) $\displaystyle\mathbf{p}_{2}$ $\displaystyle=$
$\displaystyle\frac{m}{M}\mathbf{P}_{c.m.}+\frac{1}{\sqrt{6}}\mathbf{p}_{\lambda}-\frac{1}{\sqrt{2}}\mathbf{p}_{\rho},$
(12) $\displaystyle\mathbf{p}_{3}$ $\displaystyle=$
$\displaystyle\frac{m^{\prime}}{M}\mathbf{P}_{c.m.}-\sqrt{\frac{2}{3}}\mathbf{p}_{\lambda},$
(13)
where $m$ and $m^{\prime}$ stand for the masses of strange and $u/d$ quarks,
respectively.
The spatial wave function is a product of the $\rho$-oscillator and the
$\lambda$-oscillator states. With the standard notation, the principal quantum
numbers of the $\rho$-oscillator and $\lambda$-oscillator are
$N_{\rho}=(2n_{\rho}+l_{\rho})$ and $N_{\lambda}=(2n_{\lambda}+l_{\lambda})$,
and the energy of a state is given by
$\displaystyle E_{N}$ $\displaystyle=$
$\displaystyle\left(N_{\rho}+\frac{3}{2}\right)\omega_{\rho}+\left(N_{\lambda}+\frac{3}{2}\right)\omega_{\lambda}\
.$ (14)
The total principal quantum number (i.e. shell number) $N$ is defined as
$\displaystyle N=N_{\rho}+N_{\lambda},$ (15)
and the frequencies of the $\rho$-mode and $\lambda$-mode are
$\displaystyle\omega_{\lambda}=(3K/m_{\lambda})^{1/2},\
\omega_{\rho}=(3K/m_{\rho})^{1/2}.$ (16)
In the constituent quark model, a useful oscillator parameter, i.e. the
potential strength is defined as
$\alpha_{\lambda}=(m_{\lambda}\omega_{\lambda})^{1/2}=\left(\frac{3m^{\prime}}{2m+m^{\prime}}\right)^{1/4}\alpha_{\rho}.$
(17)
For a $\Xi$ baryon, the constituent mass of a strange quark is close to that
of a $u/d$ quark. Thus, the relation
$\alpha_{\lambda}\simeq\alpha_{\rho}\equiv\alpha_{h},$ (18)
is a good approximation.
The total spatial wave function of the oscillator Hamiltonian can be written
as Karl:1969iz ; Zhenping:1991
$\Psi(\mathbf{r}_{1},\mathbf{r}_{2},\mathbf{r}_{3})=\frac{e^{i\mathbf{P}_{c.m.}\cdot\mathbf{R}_{c.m.}}}{\sqrt{(2\pi)^{3}}}\Psi^{\sigma}_{NLL_{Z}}(\vec{\rho},\vec{\lambda}),$
(19)
where $\Psi^{\sigma}_{NLL_{Z}}(\vec{\rho},\vec{\lambda})$ is the harmonic
oscillator wave function, and the $\sigma=s,\rho,\lambda,a$ stands for the
representation of the $S_{3}$ group. The harmonic oscillator wave function is
given by Karl:1969iz ; Zhenping:1991
$\displaystyle\Psi^{\sigma}_{NLL_{Z}}(\vec{\rho},\vec{\lambda})=P^{\sigma}_{NLL_{z}}\frac{\alpha^{3}_{h}}{\pi^{3/2}}e^{-\alpha^{2}_{h}(\vec{\rho}^{2}+\vec{\lambda}^{2})/2},$
(20)
where $P^{\sigma}_{NLL_{z}}$ is a polynomial of $\rho$ and $\lambda$.
Explicitly, the states within $N\leq 2$ can be obtained Zhenping:1991 :
$\displaystyle N=0,~{}~{}L=0,~{}~{}P^{s}_{000}$ $\displaystyle=$
$\displaystyle 1,$ (21) $\displaystyle N=1,~{}~{}L=1,~{}~{}P^{\rho}_{111}$
$\displaystyle=$ $\displaystyle\alpha_{h}\rho_{+},$ (22) $\displaystyle
P^{\lambda}_{111}$ $\displaystyle=$ $\displaystyle\alpha_{h}\lambda_{+},$ (23)
$\displaystyle N=2,~{}~{}L=0,~{}~{}P^{s}_{200}$ $\displaystyle=$
$\displaystyle\frac{\alpha^{2}_{h}}{\sqrt{3}}(\vec{\rho}^{2}+\vec{\lambda}^{2}-3\alpha^{-2}_{h}),$
(24) $\displaystyle~{}~{}P^{\rho}_{200}$ $\displaystyle=$
$\displaystyle\frac{\alpha^{2}_{h}}{\sqrt{3}}2\vec{\rho}\cdot\vec{\lambda},$
(25) $\displaystyle~{}~{}P^{\lambda}_{200}$ $\displaystyle=$
$\displaystyle\frac{\alpha^{2}_{h}}{\sqrt{3}}(\vec{\rho}^{2}-\vec{\lambda}^{2}),$
(26) $\displaystyle~{}~{}L=2,~{}~{}P^{s}_{222}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\alpha^{2}_{h}(\rho^{2}_{+}+\lambda^{2}_{+}),$ (27)
$\displaystyle~{}~{}P^{\rho}_{222}$ $\displaystyle=$
$\displaystyle\alpha^{2}_{h}\rho_{+}\lambda_{+},$ (28)
$\displaystyle~{}~{}P^{\lambda}_{222}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\alpha^{2}_{h}(\rho^{2}_{+}-\lambda^{2}_{+}),$ (29)
$\displaystyle~{}~{}L=1,~{}~{}P^{a}_{211}$ $\displaystyle=$
$\displaystyle\alpha^{2}_{h}(\rho_{+}\lambda_{z}-\lambda_{+}\rho_{z}).$ (30)
### II.2 Flavor and spin wave functions
In the quark model, the SU(3) flavor wave functions of the symmetry decuplet
are Zhenping:1991
$\displaystyle\phi^{s}=\cases{uuu,&$\Delta^{++}$\cr\frac{1}{\sqrt{3}}(uud+udu+duu),&$\Delta^{+}$\cr\frac{1}{\sqrt{3}}(uus+usu+suu),&$\Sigma^{*+}$\cr
ddd,&$\Delta^{-}$\cr\frac{1}{\sqrt{3}}(udd+dud+ddu),&$\Delta^{0}$\cr\frac{1}{\sqrt{3}}(dds+dsd+sdd),&$\Sigma^{*-}$\cr
sss,&$\Omega$\cr\frac{1}{\sqrt{3}}(uss+sus+ssu),&$\Xi^{*0}$\cr\frac{1}{\sqrt{3}}(dss+sds+ssd),&$\Xi^{*-}$\cr\frac{1}{\sqrt{6}}(uds+sud+dsu+sdu+dus+usd),&$\Sigma^{*0}$}$
(31)
the mixed-symmetry octet wave functions are Zhenping:1991
$\displaystyle\phi^{\lambda}=\cases{\frac{1}{\sqrt{6}}(2uud-duu-
udu),&p\cr\frac{1}{\sqrt{6}}(dud+udd-2ddu),&n\cr\frac{1}{\sqrt{6}}(2uus-suu-
usu),&$\Sigma^{+}$\cr\frac{1}{2\sqrt{3}}(sdu+sud+usd+dsu-2uds-2dus),&$\Sigma^{0}$\cr\frac{1}{\sqrt{6}}(2dds-
sdd-dsd),&$\Sigma^{-}$\cr\frac{1}{2}(sud+usd-sdu-
dsu),&$\Lambda$\cr\frac{1}{\sqrt{6}}(sus+uss-2ssu),&$\Xi^{0}$\cr\frac{1}{\sqrt{6}}(dss+sds-2ssd),&$\Xi^{-}$}$
(32)
while the mixed-antisymmetric octet wave functions are Zhenping:1991
$\displaystyle\phi^{\rho}=\cases{\frac{1}{\sqrt{2}}(udu-
duu),&p\cr\frac{1}{\sqrt{2}}(udd-dud),&n\cr\frac{1}{\sqrt{2}}(usu-
suu),&$\Sigma^{+}$\cr\frac{1}{2}(sud+sdu-usd-
dsu),&$\Sigma^{0}$\cr\frac{1}{\sqrt{2}}(dsd-
sdd),&$\Sigma^{-}$\cr\frac{1}{2\sqrt{3}}(usd+sdu-sud-
dsu-2dus+2uds),&$\Lambda$\cr\frac{1}{\sqrt{2}}(uss-
sus),&$\Xi^{0}$\cr\frac{1}{\sqrt{2}}(dss-sds).&$\Xi^{-}$}$ (33)
The total antisymmetric singlet flavor wave function is
$\phi^{a}=\frac{1}{\sqrt{6}}(uds+dsu+sud-dus-usd-sdu).$ (34)
In the quark model, the typical SU(2) spin wave functions for the baryons can
be adopted Koniuk:1979vy ; Copley:1979wj :
$\displaystyle\chi^{s}_{3/2}$ $\displaystyle=$
$\displaystyle\uparrow\uparrow\uparrow,\ \
\chi^{s}_{-3/2}=\downarrow\downarrow\downarrow,$ $\displaystyle\chi^{s}_{1/2}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}(\uparrow\uparrow\downarrow+\uparrow\downarrow\uparrow+\downarrow\uparrow\uparrow),$
$\displaystyle\chi^{s}_{-1/2}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}(\uparrow\downarrow\downarrow+\downarrow\downarrow\uparrow+\downarrow\uparrow\downarrow),$
(35)
for the spin-3/2 states with symmetric spin wave functions,
$\displaystyle\chi^{\rho}_{1/2}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\uparrow\downarrow\uparrow-\downarrow\uparrow\uparrow),$
$\displaystyle\chi^{\rho}_{-1/2}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\uparrow\downarrow\downarrow-\downarrow\uparrow\downarrow),$
(36)
for the spin-1/2 states with mixed antisymmetric spin wave functions, and
$\displaystyle\chi^{\lambda}_{1/2}$ $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{6}}(\uparrow\downarrow\uparrow+\downarrow\uparrow\uparrow-2\uparrow\uparrow\downarrow),$
$\displaystyle\chi^{\lambda}_{-1/2}$ $\displaystyle=$
$\displaystyle+\frac{1}{\sqrt{6}}(\uparrow\downarrow\downarrow+\downarrow\uparrow\downarrow-2\downarrow\downarrow\uparrow),$
(37)
for the spin-1/2 states with mixed symmetric spin wave functions.
The flavor-spin wave functions are representations of SU(6), which are denoted
by $|N_{6},^{2S+1}N_{3}\rangle$, where $N_{6}$ ($N_{3}$) represents the SU(6)
(SU(3)) representation and $S$ stands for the total spin of the wave function.
The symmetric $\mathbf{56}$ representation wave functions are Zhenping:1991
$\displaystyle|56,~{}^{2}8\rangle^{s}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\phi^{\rho}\chi^{\rho}+\phi^{\lambda}\chi^{\lambda}),$
(38) $\displaystyle|56,^{4}10\rangle^{s}$ $\displaystyle=$
$\displaystyle\phi^{s}\chi^{s},$ (39)
the antisymmetric $\mathbf{70}$ representation wave functions are
Zhenping:1991
$\displaystyle|70,~{}^{2}8\rangle^{\rho}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\phi^{\rho}\chi^{\lambda}+\phi^{\lambda}\chi^{\rho}),$
(40) $\displaystyle|70,~{}^{4}8\rangle^{\rho}$ $\displaystyle=$
$\displaystyle\phi^{\rho}\chi^{s},$ (41)
$\displaystyle|70,^{2}10\rangle^{\rho}$ $\displaystyle=$
$\displaystyle\phi^{s}\chi^{\rho},$ (42)
$\displaystyle|70,~{}^{2}1\rangle^{\rho}$ $\displaystyle=$
$\displaystyle\phi^{\alpha}\chi^{\lambda},$ (43)
while the symmetric $\mathbf{70}$ representation wave functions are
Zhenping:1991
$\displaystyle|70,~{}^{2}8\rangle^{\lambda}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\phi^{\rho}\chi^{\rho}-\phi^{\lambda}\chi^{\lambda}),$
(44) $\displaystyle|70,~{}^{4}8\rangle^{\lambda}$ $\displaystyle=$
$\displaystyle\phi^{\lambda}\chi^{s},$ (45)
$\displaystyle|70,^{2}10\rangle^{\lambda}$ $\displaystyle=$
$\displaystyle\phi^{s}\chi^{\lambda},$ (46)
$\displaystyle|70,~{}^{2}1\rangle^{\lambda}$ $\displaystyle=$
$\displaystyle\phi^{\alpha}\chi^{\rho},$ (47)
and the antisymmetric $\mathbf{20}$ representation wave functions are
Zhenping:1991
$\displaystyle|20,^{2}8\rangle^{a}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\phi^{\rho}\chi^{\lambda}-\phi^{\lambda}\chi^{\rho}),$
(48) $\displaystyle|20,^{4}1\rangle^{a}$ $\displaystyle=$
$\displaystyle\phi^{a}\chi^{s}.$ (49)
### II.3 Total wave functions
The total spin-flavor-space wave functions are the SU(6)$\otimes$ O(3)
representations, which are denoted by
$\displaystyle|SU(6)\otimes O(3)\rangle=|N_{6},^{2S+1}N_{3},N,L,J^{P}\rangle.$
(50)
The spin-flavor-space wave functions have to be permutation-symmetric because
the baryon color wave function is totally anti-symmetric. The spin-flavor-
space wave functions up to $N=2$ shell are listed in Tab. 1. So far no
experimental evidence for the existence of the $\mathbf{20}$ SU(6)
representation has been found, thus, we do not consider these excitations in
this work.
Table 1: The spin-flavor-space wave functions of different light baryons, denoted by $|\mathbf{N}_{6},^{2S+1}\mathbf{N}_{3},N,L,J^{P}\rangle$. The Clebsch-Gordan series for the spin and angular-momentum addition $|J,J_{z}\rangle$= $\sum_{L_{z}+S_{z}=J_{z}}\langle LL_{z},SS_{z}|JJ_{z}\rangle\Psi^{\sigma}_{NLL_{z}}\chi_{S_{z}}$ has been omitted. state | N | J L $S$ | wave function
---|---|---|---
$|56,^{2}8,0,0,\frac{1}{2}^{+}\rangle$ | 0 | $\frac{1}{2}$ 0 $\frac{1}{2}$ | $|56,^{2}8\rangle\Psi^{s}_{000}$
$|56,^{4}10,0,0,\frac{3}{2}^{+}\rangle$ | 0 | $\frac{3}{2}$ 0 $\frac{3}{2}$ | $|56,^{4}10\rangle\Psi^{s}_{000}$
$|56,^{2}8,2,0,\frac{1}{2}^{+}\rangle$ | 2 | $\frac{1}{2}$ 0 $\frac{1}{2}$ | $|56,^{2}8\rangle\Psi^{s}_{200}$
$|56,^{4}{10},2,0,\frac{3}{2}^{+}\rangle$ | 2 | $\frac{3}{2}$ 0 $\frac{3}{2}$ | $|56,^{4}10\rangle\Psi^{s}_{200}$
$|56,^{2}{8},2,2,\frac{3}{2}^{+}\rangle$ | 2 | $\frac{5}{2}$ 2 $\frac{1}{2}$ | $|56,^{2}8\rangle\Psi^{s}_{22L_{z}}$
$|56,^{2}{8},2,2,\frac{5}{2}^{+}\rangle$ | 2 | $\frac{5}{2}$ 2 $\frac{1}{2}$ |
$|56,^{4}{10},2,2,\frac{1}{2}^{+}\rangle$ | 2 | $\frac{7}{2}$ 2 $\frac{3}{2}$ |
$|56,^{4}{10},2,2,\frac{3}{2}^{+}\rangle$ | 2 | $\frac{7}{2}$ 2 $\frac{3}{2}$ | $|56,^{4}10\rangle\Psi^{s}_{22L_{z}}$
$|56,^{4}{10},2,2,\frac{5}{2}^{+}\rangle$ | 2 | $\frac{7}{2}$ 2 $\frac{3}{2}$ |
$|56,^{4}{10},2,2,\frac{7}{2}^{+}\rangle$ | 2 | $\frac{7}{2}$ 2 $\frac{3}{2}$ |
$|70,^{2}{8},1,1,\frac{1}{2}^{-}\rangle$ | 1 | $\frac{3}{2}$ 1 $\frac{1}{2}$ | $|70,^{2}8\rangle^{\rho}\Psi^{\rho}_{11L_{z}}+|70,^{2}8\rangle^{\lambda}\Psi^{\lambda}_{11L_{z}}$
$|70,^{2}{8},1,1,\frac{3}{2}^{-}\rangle$ | 1 | $\frac{3}{2}$ 1 $\frac{1}{2}$ |
$|70,^{4}{8},1,1,\frac{1}{2}^{-}\rangle$ | 1 | $\frac{5}{2}$ 1 $\frac{3}{2}$ |
$|70,^{4}{8},1,1,\frac{3}{2}^{-}\rangle$ | 1 | $\frac{5}{2}$ 1 $\frac{3}{2}$ | $|70,^{4}8\rangle^{\rho}\Psi^{\rho}_{11L_{z}}+|70,^{4}8\rangle^{\lambda}\Psi^{\lambda}_{11L_{z}}$
$|70,^{4}{8},1,1,\frac{5}{2}^{-}\rangle$ | 1 | $\frac{5}{2}$ 1 $\frac{3}{2}$ |
$|70,^{2}{10},1,1,\frac{1}{2}^{-}\rangle$ | 1 | $\frac{3}{2}$ 1 $\frac{1}{2}$ | $|70,^{2}{10}\rangle^{\rho}\Psi^{\rho}_{11L_{z}}+|70,^{2}{10}\rangle^{\lambda}\Psi^{\lambda}_{11L_{z}}$
$|70,^{2}{10},1,1,\frac{3}{2}^{-}\rangle$ | 1 | $\frac{3}{2}$ 1 $\frac{1}{2}$ |
$|70,^{2}{8},2,0,\frac{1}{2}^{+}\rangle$ | 2 | $\frac{1}{2}$ 0 $\frac{1}{2}$ | $|70,^{2}{8}\rangle^{\rho}\Psi^{\rho}_{200}+|70,^{2}{8}\rangle^{\lambda}\Psi^{\lambda}_{200}$
$|70,^{4}{8},2,0,\frac{3}{2}^{+}\rangle$ | 2 | $\frac{3}{2}$ 0 $\frac{3}{2}$ | $|70,^{4}{8}\rangle^{\rho}\Psi^{\rho}_{200}+|70,^{4}{8}\rangle^{\lambda}\Psi^{\lambda}_{200}$
$|70,^{2}{10},2,0,\frac{1}{2}^{+}\rangle$ | 2 | $\frac{1}{2}$ 0 $\frac{1}{2}$ | $|70,^{2}{10}\rangle^{\rho}\Psi^{\rho}_{200}+|70,^{2}{10}\rangle^{\lambda}\Psi^{\lambda}_{200}$
$|70,^{2}{8},2,2,\frac{3}{2}^{+}\rangle$ | 2 | $\frac{5}{2}$ 2 $\frac{1}{2}$ | $|70,^{2}{8}\rangle^{\rho}\Psi^{\rho}_{22L_{z}}+|70,^{2}{8}\rangle^{\lambda}\Psi^{\lambda}_{22L_{z}}$
$|70,^{2}{8},2,2,\frac{5}{2}^{+}\rangle$ | 2 | $\frac{5}{2}$ 2 $\frac{1}{2}$ |
$|70,^{4}{8},2,2,\frac{1}{2}^{+}\rangle$ | 2 | $\frac{7}{2}$ 2 $\frac{3}{2}$ |
$|70,^{4}{8},2,2,\frac{3}{2}^{+}\rangle$ | 2 | $\frac{7}{2}$ 2 $\frac{3}{2}$ | $|70,^{4}{8}\rangle^{\rho}\Psi^{\rho}_{22L_{z}}+|70,^{4}{8}\rangle^{\lambda}\Psi^{\lambda}_{22L_{z}}$
$|70,^{4}{8},2,2,\frac{5}{2}^{+}\rangle$ | 2 | $\frac{7}{2}$ 2 $\frac{3}{2}$ |
$|70,^{4}{8},2,2,\frac{7}{2}^{+}\rangle$ | 2 | $\frac{7}{2}$ 2 $\frac{3}{2}$ |
$|70,^{2}{10},2,2,\frac{3}{2}^{+}\rangle$ | 2 | $\frac{5}{2}$ 2 $\frac{1}{2}$ | $|70,^{2}{10}\rangle^{\rho}\Psi^{\rho}_{22L_{z}}+|70,^{2}{10}\rangle^{\lambda}\Psi^{\lambda}_{22L_{z}}$
$|70,^{2}{10},2,2,\frac{5}{2}^{+}\rangle$ | 2 | $\frac{5}{2}$ 2 $\frac{1}{2}$ |
## III strong decay of a baryon in the chiral quark model
In the chiral quark model, the effective low energy quark-meson pseudoscalar
coupling at tree level is given by Li:1995si ; Zhong:2007fx ; Li:1997gda ; qk3
$\displaystyle
H_{m}=\sum_{j}\frac{1}{f_{m}}\bar{\psi}_{j}\gamma^{j}_{\mu}\gamma^{j}_{5}\psi_{j}\vec{\tau}\cdot\partial^{\mu}\vec{\phi}_{m},$
(51)
where $\psi_{j}$ represents the $j$th quark field in a baryon, and $f_{m}$ is
the meson’s decay constant. The pseudoscalar meson octet $\phi_{m}$ is
expressed as
$\displaystyle\phi_{m}=\pmatrix{\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta&\pi^{+}&K^{+}\cr\pi^{-}&-\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta&K^{0}\cr
K^{-}&\bar{K}^{0}&-\sqrt{\frac{2}{3}}\eta}.$ (52)
To match the nonrelativistic harmonic oscillator spatial wave function
${}^{N}\Psi_{LL_{z}}$ in the quark model, we adopt the nonrelativistic form of
Eq. (51) in the calculations, which is given by Li:1995si ; Zhong:2007fx ;
Li:1997gda ; qk3
$\displaystyle H^{nr}_{m}$ $\displaystyle\simeq$
$\displaystyle\sum_{j=1}^{3}\Big{\\{}\frac{\omega_{m}}{E_{f}+M_{f}}\mbox{\boldmath$\sigma$\unboldmath}_{j}\cdot\textbf{P}_{f}+\frac{\omega_{m}}{E_{i}+M_{i}}\mbox{\boldmath$\sigma$\unboldmath}_{j}\cdot\textbf{P}_{i}$
(53)
$\displaystyle-\mbox{\boldmath$\sigma$\unboldmath}_{j}\cdot\textbf{q}+\frac{\omega_{m}}{2\mu_{q}}\mbox{\boldmath$\sigma$\unboldmath}_{j}\cdot\textbf{p}^{\prime}_{j}\Big{\\}}I_{j}\varphi_{m},$
where $\mbox{\boldmath$\sigma$\unboldmath}_{j}$ is the Pauli spin operator on
the the $j$th quark of the baryon, and $\mu_{q}$ is a reduced mass given by
$1/\mu_{q}=1/m_{j}+1/m_{j}^{\prime}$ with $m_{j}$ and $m_{j}^{\prime}$ for the
masses of the $j$th quark in the initial and final baryons, respectively. For
emitting a meson, we have $\varphi_{m}=e^{-i\textbf{q}\cdot\textbf{r}_{j}}$,
and for absorbing a meson we have
$\varphi_{m}=e^{i\textbf{q}\cdot\textbf{r}_{j}}$. In the above nonrelativistic
expansion, $\textbf{p}^{\prime}_{j}$
($=\textbf{p}_{j}-m_{j}/M\textbf{P}_{c.m.}$) is the internal momentum operator
for the $j$th quark in the baryon rest frame. $\omega_{m}$ and q are the
energy and three-vector momentum of the meson, respectively. $\textbf{P}_{i}$
and $\textbf{P}_{f}$ stand for the momenta of the initial final baryons,
respectively. The isospin operator $I_{j}$ in Eq. (53) is expressed as
$\displaystyle I_{j}=\cases{a^{\dagger}_{j}(u)a_{j}(s)&for $K^{+}$,\cr
a^{\dagger}_{j}(s)a_{j}(u)&for $K^{-}$,\cr a^{\dagger}_{j}(d)a_{j}(s)&for
$K^{0}$,\cr a^{\dagger}_{j}(s)a_{j}(d)&for $\bar{K^{0}}$,\cr
a^{\dagger}_{j}(u)a_{j}(d)&for $\pi^{-}$,\cr a^{\dagger}_{j}(d)a_{j}(u)&for
$\pi^{+}$,\cr\frac{1}{\sqrt{2}}[a^{\dagger}_{j}(u)a_{j}(u)-a^{\dagger}_{j}(d)a_{j}(d)]&for
$\pi^{0}$,\cr\frac{1}{\sqrt{2}}[a^{\dagger}_{j}(u)a_{j}(u)+a^{\dagger}_{j}(d)a_{j}(d)]\cos\phi_{P}\cr-a^{\dagger}_{j}(s)a_{j}(s)\sin\phi_{P}&for
$\eta$,}$ (54)
where $a^{\dagger}_{j}(u,d,s)$ and $a_{j}(u,d,s)$ are the creation and
annihilation operators for the $u$, $d$ and $s$ quarks, and $\phi_{P}$ is the
mixing angle of $\eta$ meson in the flavor basis Beringer:1900zz ;
Zhong:2011ht .
In the calculations, we select the initial-baryon-rest system for the decay
processes. The energies and momenta of the initial baryons are denoted by
$(E_{i},\textbf{P}_{i}$), while those of the final state mesons and baryons
are denoted by $(\omega_{f},\textbf{q})$ and $(E_{f},\textbf{P}_{f})$,
respectively. Note that $\textbf{P}_{i}=0$ and $\textbf{P}_{f}=-\textbf{q}$.
For a light pseudoscalar meson emission in the strong decay process of a
baryon, $\mathcal{B}\to\mathcal{B}^{\prime}\mathbb{M}(\mathbf{q})$, the
partial decay amplitudes can be worked out according to
$\displaystyle\mathcal{M}[\mathcal{B}\to\mathcal{B}^{\prime}\mathbb{M}(\mathbf{q})]=3\left\langle\mathcal{B}^{\prime}\left|\left\\{G\mbox{\boldmath$\sigma$\unboldmath}_{3}\cdot\textbf{q}+h\mbox{\boldmath$\sigma$\unboldmath}_{3}\cdot\textbf{p}^{\prime}_{3}\right\\}I_{3}e^{-i\textbf{q}\cdot\textbf{r}_{3}}\right|\mathcal{B}\right\rangle,$
(55)
with
$\displaystyle G\equiv-\frac{\omega_{m}}{E_{f}+M_{f}}-1,\ \
h\equiv\frac{\omega_{m}}{2\mu_{q}},$ (56)
where $\mathcal{B}$ and $\mathcal{B}^{\prime}$ stand for the initial and final
baryon wave functions listed in Tab. 1.
With the derived decay amplitudes, one can calculate the width by Zhong:2007gp
$\Gamma=\left(\frac{\delta}{f_{m}}\right)^{2}\frac{(E_{f}+M_{f})|q|}{4\pi
M_{i}}\frac{1}{2J_{i}+1}\sum_{J_{iz}J_{fz}}|\mathcal{M}_{J_{iz},J_{fz}}|,$
(57)
where the $J_{iz}$ and $J_{fz}$ stand for the the total angular momenta of the
initial and final baryons, respectively. $\delta$ as a global parameter
accounts for the strength of the quark-meson couplings.
In the calculation, the standard quark model parameters are adopted. Namely,
we set $m_{u}=m_{d}=350$ MeV, $m_{s}=450$ MeV, for the constituent quark
masses. The harmonic oscillator parameter $\alpha_{h}$ in the wave function
${}^{N}\Psi_{LL_{z}}$ is taken as $\alpha_{h}=0.40$ GeV. The decay constants
for $\pi$, $K$ and $\eta$ mesons are taken as $f_{\pi}=132$ MeV,
$f_{K}=f_{\eta}=160$ MeV, respectively. The masses of the mesons and baryons
used in the calculations are adopted from the Particle Data Group
Beringer:1900zz .
## IV results AND ANALYSIS
Table 2: The partial and total decay widths (MeV) of the well-established baryons $\Xi(1530)$, $\Sigma(1385)$ and $\Delta(1232)$, which correspond to the same representation $|56,^{4}10,0,0,\frac{3}{2}^{+}\rangle$. The experimental data are obtained from PDG. state | channel | $\Gamma^{th}_{i}$ | $\Gamma^{th}_{total}$ | $\Gamma^{exp}_{total}$
---|---|---|---|---
$\Xi(1530)^{0}$ | $\Xi^{-}\pi^{+}$ | 5.6 | 9.1(input) | $9.1\pm 0.5$
| $\Xi^{0}\pi^{0}$ | 3.5 | |
$\Xi(1530)^{-}$ | $\Xi^{0}\pi^{-}$ | 7.1 | 10.3 | $9.9^{+1.7}_{-1.9}$
| $\Xi^{-}\pi^{0}$ | 3.2 | |
$\Sigma(1385)^{+}$ | $\Sigma^{+}\pi^{0}$ | 1.9 | 27.1 | $36\pm 0.7$
| $\Sigma^{0}\pi^{+}$ | 1.5 | |
| $\Lambda^{0}\pi^{+}$ | 23.7 | |
$\Sigma(1385)^{0}$ | $\Sigma^{+}\pi^{-}$ | 1.8 | 27.9 | $36\pm 5$
| $\Sigma^{-}\pi^{+}$ | 1.3 | |
| $\Lambda^{0}\pi^{0}$ | 24.8 | |
$\Sigma(1385)^{-}$ | $\Sigma^{-}\pi^{0}$ | 1.7 | 28.7 | $39.4\pm 2.1$
| $\Sigma^{0}\pi^{-}$ | 1.8 | |
| $\Lambda^{0}\pi^{-}$ | 25.2 | |
$\Delta(1232)^{++}$ | $p\pi^{+}$ | 63.9 | 63.9 | $114\sim 120$
$\Delta(1232)^{+}$ | $p\pi^{0}$ | 43.7 | 64.7 | $114\sim 120$
| $n\pi^{+}$ | 21.0 | |
$\Delta(1232)^{0}$ | $p\pi^{-}$ | 21.3 | 64.3 | $114\sim 120$
| $n\pi^{0}$ | 43.0 | |
### IV.1 $\Xi$(1530)
$\Xi(1530)$ is the only $\Xi$ resonance whose properties are all reasonably
well known. According to the classification of the quark model, it is assigned
to the $|56,^{4}10,0,0,\frac{3}{2}^{+}\rangle$ representation. In the present
work, the measured width for $\Xi(1530)^{0}\to\Xi\pi$ as an input ( i.e.
$\Gamma=9.1$ MeV Beringer:1900zz ) to determine parameter $\delta$ in Eq.(57),
which gives
$\displaystyle\delta=0.576.$ (58)
With this determined parameter, we can predict the other $\Xi$ resonance
strong decays. First, the strong decays of $\Xi(1530)^{-}$ are calculated. The
results have been listed in Tab. 2, where we find that our predictions are in
agreement with the data. Furthermore, we have noted that $\Sigma(1385)$ and
$\Delta(1232)$ also correspond to the $|56,^{4}10,0,0,\frac{3}{2}^{+}\rangle$
representation. They are the SU(3)-flavor counterparts of $\Xi(1530)$.
Applying the determined value for $\delta$, we calculate the strong decays of
the $\Sigma(1385)$ and $\Delta(1232)$ resonances, the results are listed in
Tab. 2 as well.
It is seen that the predicted widths for $\Sigma(1385)$ are compatible with
the experimental data, and the partial decay width ratio
$\frac{\Gamma[\Sigma(1385)\rightarrow\Sigma\pi]}{\Gamma[\Sigma(1385)\rightarrow\Lambda\pi]}\simeq
0.134\pm 0.09,$ (59)
are in good agreement with the PDG value Beringer:1900zz .
The predicted widths for $\Delta(1232)$ are underestimated by about a factor
of 2 compared with the data. In fact, there are systemic underestimations of
the decay widths for the $\Delta$ and nucleon excitations as well. The reason
is that the parameters used in this work are only fine tuned for the $\Xi$
resonances to make more reliable predictions for the their decay properties.
For simplicity, we do not carry out a global fit to strange and non-strange
baryons. We have noted that the global parameter $\delta=0.576$ used for the
$\Xi$ resonances in this work is slightly smaller than that used for the
$\Delta$ and nucleon resonances in An:2011sb .
As a whole, the predicted strong decay properties for $\Xi(1530)^{-}$,
$\Sigma(1385)$ are compatible with the experimental data. If a global fit to
strange and non-strange baryons is carried out, the predicted strong decay
width of $\Delta(1232)$ can be more close to the data. Thus, the SU(3)-flavor
symmetry approximately holds in these ground decuplet baryons.
### IV.2 $\Xi(1690)$
$\Xi(1690)$ is a three-star state listed in PDG. Its decay width might be less
than $30$ MeV Beringer:1900zz . Three decay modes $\Lambda\bar{K}$,
$\Sigma\bar{K}$ and $\Xi\pi$ were observed in experiments. Recently, _BABAR_
Collaboration found some evidence that the $\Xi^{0}(1690)$ has $J^{P}=1/2^{-}$
in the study of $\Lambda_{c}^{+}\rightarrow\Xi^{-}\pi^{+}K^{+}$ Aubert:2008ty
.
If $\Xi(1690)$ is a $J^{P}=1/2^{-}$ resonance, it should correspond to one of
the first orbital excitation states: $|70,^{2}8,1,1,\frac{1}{2}^{-}\rangle$,
$|70,^{4}8,1,1,\frac{1}{2}^{-}\rangle$ and
$|70,^{2}10,1,1,\frac{1}{2}^{-}\rangle$ or a mixed state between them.
Considering $\Xi(1690)$ as the $|70,^{2}8,1,1,\frac{1}{2}^{-}\rangle$,
$|70,^{4}8,1,1,\frac{1}{2}^{-}\rangle$ and
$|70,^{2}10,1,1,\frac{1}{2}^{-}\rangle$, respectively, we calculate their
decay properties, which are listed in Tab. 3. Comparing the predictions with
the measurements, we find that if $\Xi(1690)$ is assigned to
$|70,^{2}8,1,1,\frac{1}{2}^{-}\rangle$, the calculated decay width
$\displaystyle\Gamma\approx 48~{}\mathrm{MeV},$ (60)
and partial decay width ratios
$\displaystyle\frac{\Gamma(\Xi\pi)}{\Gamma(\Sigma\bar{K})}\simeq
0.2,~{}\frac{\Gamma(\Sigma\bar{K})}{\Gamma(\Lambda\bar{K})}\approx
1.0,~{}\frac{\Gamma(\Xi(1530)\pi)}{\Gamma(\Sigma\bar{K})}\approx 0.0002,$ (61)
are roughly in agreement with the measurements. Neither
$|70,^{4}8,1,1,\frac{1}{2}^{-}\rangle$ nor
$|70,^{2}10,1,1,\frac{1}{2}^{-}\rangle$ could be considered as an assignment
to $\Xi(1690)$ because their partial decay width ratios disagree with the
observations.
Table 3: The total and partial decay widths (MeV) of $\Xi(1690)$ with different $J^{P}=1/2^{-}$ assignments. assignment | $~{}~{}\Xi\pi$ | $\Sigma\bar{K}$ | $\Lambda\bar{K}$ | $\Xi(1530)\pi$ | $\Gamma_{total}$
---|---|---|---|---|---
$|70,^{2}8,1,1,\frac{1}{2}^{-}\rangle$ | 3.69 | 22.30 | 22.15 | 0.005 | 48.14
$|70,^{4}8,1,1,\frac{1}{2}^{-}\rangle$ | 59.05 | 5.58 | 22.15 | 0.001 | 86.78
$|70,^{2}10,1,1,\frac{1}{2}^{-}\rangle$ | 3.69 | 1.39 | 5.54 | 0.005 | 10.63
As we know, configuration mixing between several states with the same $J^{P}$
often occurs via some interactions. Thus, $\Xi(1690)$ might be a mixed state
between $|70,^{2}8,1,1,\frac{1}{2}^{-}\rangle$,
$|70,^{4}8,1,1,\frac{1}{2}^{-}\rangle$ and
$|70,^{2}10,1,1,\frac{1}{2}^{-}\rangle$. Now, we consider the physical states
with $J^{P}=\frac{1}{2}^{-}$ as mixed states between $|70,^{2}8\rangle$,
$|70,^{4}8\rangle$ and $|70,^{2}10\rangle$. According to the standard CKM
matrix method, the physical states can be express as
$\left(\begin{array}[]{c}|\Xi\frac{1}{2}^{-}\rangle_{1}\cr|\Xi\frac{1}{2}^{-}\rangle_{2}\cr|\Xi\frac{1}{2}^{-}\rangle_{3}\end{array}\right)=U\left(\begin{array}[]{c}|70,^{2}8\rangle\cr|70,^{4}8\rangle\cr|70,^{2}10\rangle\end{array}\right),$
(62)
with
$U=\left(\begin{array}[]{ccc}c_{12}c_{13}&s_{12}c_{13}&s_{13}\cr-
s_{12}c_{23}-c_{12}s_{23}s_{13}&c_{12}c_{23}-s_{12}s_{23}s_{13}&s_{23}c_{13}\cr
s_{12}s_{23}-c_{12}c_{23}s_{13}&-c_{12}s_{23}-s_{12}c_{23}s_{13}&c_{23}c_{13}\end{array}\right),$
(63)
where $c_{ij}\equiv\cos\theta_{ij}$ and $s_{ij}\equiv\sin\theta_{ij}$ with
$\theta_{ij}$ the mixing angles to be determined by the experimental data.
In present work, we take $\Xi(1690)$ as $|\Xi\frac{1}{2}^{-}\rangle_{2}$. By
fitting the experimental data of DIONISI 78 Dionisi:1978tg (see Tab. 4), we
have obtained $\theta_{12}\simeq 5^{0}$, $\theta_{13}\simeq 105^{0}$ and
$\theta_{23}\simeq 95^{0}$. The theoretical predictions compared with the data
were listed in Tab. 4. From the table, we find that the decay properties of
$\Xi(1690)$ could be well described with these determined mixing angles. Thus,
$\Xi(1690)$ could be a mixed state.
Table 4: The predicted total and partial decay widths (MeV) and partial decay width ratios (MeV) of $\Xi(1690)\equiv\Psi_{2}$ compared with the experiment data of DIONISI 78 Dionisi:1978tg . The mixing angles are $\theta_{12}=5^{0}$, $\theta_{13}=75^{0}$ and $\theta_{23}=95^{0}$. channel | $\Gamma^{th}_{i}$ | $\Gamma^{th}_{total}$ | $\Gamma^{exp}_{total}$ | channel ratio | $R_{th}$ | $R_{exp}$
---|---|---|---|---|---|---
$\Xi\pi$ | 1.0 | 37 | $44^{+23}_{-23}$ | $\Gamma(\Sigma\bar{K})/\Gamma(\Lambda\bar{K})$ | 2.88 | $2.7^{+0.9}_{-0.9}$
$\Sigma\bar{K}$ | 27.0 | | | $\Gamma(\Xi\pi)/\Gamma(\Sigma\bar{K})$ | 0.04 | $<0.09$
$\Lambda\bar{K}$ | 9.4 | | | $\Gamma(\Xi(1530)\pi)/\Gamma(\Sigma\bar{K})$ | $\sim 10^{-4}$ | $<0.06$
$\Xi(1530)\pi$ | 0.01 | | | | |
With these determined mixing angles, Eq.(62) can be explicitly written as
$\left(\begin{array}[]{c}|\Xi\frac{1}{2}^{-}\rangle_{1}\cr|\Xi(1690)\frac{1}{2}^{-}\rangle\cr|\Xi\frac{1}{2}^{-}\rangle_{3}\end{array}\right)=\left(\begin{array}[]{ccc}-0.26&-0.02&0.97\cr-0.95&-0.17&-0.26\cr
0.17&-0.98&0.02\end{array}\right)\left(\begin{array}[]{c}|70,^{2}8\rangle\cr|70,^{4}8\rangle\cr|70,^{2}10\rangle\end{array}\right),$
(64)
From the above equation, it is obviously seen that the main component of the
physical state $\Xi(1690)$ is $|70,^{2}8,1,1,1/2^{-}\rangle$ ($\sim 90\%$),
which slightly mixes with $|70,^{2}10,1,1,1/2^{-}\rangle$ ($\sim 7\%$) and
$|70,^{4}8,1,1,1/2^{-}\rangle$ ($\sim 3\%$).
In Ref. Aubert:2006ux , the _BABAR_ Collaboration studied the spin of
$\Xi(1690)$ from $\Lambda_{c}^{+}\rightarrow\Lambda K^{+}\bar{K}^{0}$. They
found that the spin of $\Xi(1690)$ is consistent with the value $J=1/2$.
However, they could not determine the parity of $\Xi(1690)$. According to the
mass predictions of quark model, the mass of the first radial ($2S$)
excitation of $\Xi$ with $J^{P}=\frac{1}{2}^{+}$ might be close to $1690$ MeV
Chao:1980em ; Melde:2008yr . Thus, we should consider the possibility of
$\Xi(1690)$ as the positive parity radial ($2S$) excitations with
$J^{P}=\frac{1}{2}^{+}$. In the constituent quark model, there are three
radial ($2S$) excitations with $J^{P}=\frac{1}{2}^{+}$:
$|56,^{2}8,2,0,\frac{1}{2}^{+}\rangle$, $|70,^{2}8,2,0,\frac{1}{2}^{+}\rangle$
and $|70,^{2}10,2,0,\frac{1}{2}^{+}\rangle$. In our calculation, those states’
total decay widths are too small to compare with the experimental data. The
predicted partial decay width ratios are incompatible with the observations as
well. Thus, these states with $J^{P}=\frac{1}{2}^{+}$ as assignments to
$\Xi(1690)$ should be excluded. Furthermore, we also study the strong decay
properties of the other first orbital excitation states with $J^{P}=3/2^{-}$
and $J^{P}=5/2^{-}$. Their decay properties are very different from those of
$\Xi(1690)$. For simplicity, the calculated results are not shown in this
work.
In a word, $\Xi(1690)$ is most likely to be the first orbital excitation of
$\Xi$ with $J^{P}=1/2^{-}$. There might exist configuration mixing in
$\Xi(1690)$. The main component of $\Xi(1690)$ is
$|70,^{2}8,1,1,1/2^{-}\rangle$, which slightly mixes with
$|70,^{2}10,1,1,1/2^{-}\rangle$ and $|70,^{4}8,1,1,1/2^{-}\rangle$. Our
predictions are consistent with the experimental observations. The recent mass
calculations of a constituent quark model support the classification of
$\Xi(1690)$ as a $J^{P}=\frac{1}{2}^{-}$ octet resonance Pervin:2007wa . The
calculations of a Skyrme model Oh:2007cr and unitary chiral approaches
Kolomeitsev:2003kt ; Sarkar:2004jh also indicated that $\Xi(1690)$ has
$J^{P}=\frac{1}{2}^{-}$.
Since $\Xi(1690)$ corresponds to the physical state
$|\Xi\frac{1}{2}^{-}\rangle_{2}$ in Eq.(64), as the counterparts of
$\Xi(1690)$, the other two physical states,
$|\Xi\frac{1}{2}^{-}\rangle_{1}$=-0.26$|70,^{2}8\rangle-0.02|70,^{4}8\rangle+0.97|70,^{2}10\rangle$
and
$|\Xi\frac{1}{2}^{-}\rangle_{3}$=0.17$|70,^{2}8\rangle-0.98|70,^{4}8\rangle+0.02|70,^{2}10\rangle$,
might be observed in experiments. It is easily seen that the main component of
$|\Xi\frac{1}{2}^{-}\rangle_{1}$ is $|70,^{2}10,1,1,1/2^{-}\rangle$ ($\sim
94\%$), while $|\Xi\frac{1}{2}^{-}\rangle_{3}$ is dominated by
$|70,^{4}8,1,1,1/2^{-}\rangle$ ($\sim 96\%$). According to the analysis in
large $N_{c}$ QCD Schat:2001xr ; Goity:2002pu and quark model Isgur:1978xj ,
the masses of $|\Xi\frac{1}{2}^{-}\rangle_{1}$ and
$|\Xi\frac{1}{2}^{-}\rangle_{3}$ are around 1920 MeV. The strong decay
properties of $|\Xi\frac{1}{2}^{-}\rangle_{1}$ and
$|\Xi\frac{1}{2}^{-}\rangle_{3}$ are studied as well. Considering the
uncertainties of the mass predictions, we vary the masses of
$|\Xi\frac{3}{2}^{-}\rangle_{1}$ and $|\Xi\frac{3}{2}^{-}\rangle_{3}$ from
1860 MeV to 1980 MeV. The results are shown in Fig. 1. From the figure, we
find that $|\Xi\frac{1}{2}^{-}\rangle_{1}$ is a narrow state with a width of
$\Gamma\sim 25$ MeV, while $|\Xi\frac{1}{2}^{-}\rangle_{3}$ is a broad state
with a width of $\sim 100$ MeV. The main decay channels of
$|\Xi\frac{1}{2}^{-}\rangle_{1}$ are $\Sigma\bar{K}$ and $\Lambda\bar{K}$,
while the decays of $|\Xi\frac{1}{2}^{-}\rangle_{3}$ are governed by $\Xi\pi$
and $\Lambda\bar{K}$.
Figure 1: The decay properties of $|\Xi\frac{1}{2}^{-}\rangle_{1}$ and
$|\Xi\frac{1}{2}^{-}\rangle_{3}$, respectively, where
$\Psi_{1}\equiv|\Xi\frac{1}{2}^{-}\rangle_{1}$ and
$\Psi_{3}\equiv|\Xi\frac{1}{2}^{-}\rangle_{3}$.
### IV.3 $\Xi(1820)$
In 1987, Biagi _et al._ measured the spin-parity of $\Xi(1820)$ Biagi:1986vs .
They found that its spin-parity is consistent with $J=3/2^{-}$, which is in
good agreement with the quark model predictions Chao:1980em ; Melde:2008yr ;
Pervin:2007wa . In the present work, the study of the well-established
resonance $\Xi(1820)$, on the one hand, can provide an important test of our
model; on the other hand, it can let us obtain more information about the
nature of $\Xi(1820)$.
Assigning $\Xi(1820)$ to the negative states with $J^{P}=1/2^{-}$, $3/2^{-}$
and $5/2^{-}$, respectively, their strong decays are calculated with our
model. The results are listed in Tab. 5, where we find that only the
$|70,^{2}8,1,1,\frac{3}{2}^{-}\rangle$ can be assigned to $\Xi(1820)$. The
detail comparisons of theoretical predictions with measurements are shown in
Tab. 6, from which it is seen that both the decay width and the partial decay
ratios are in agreement with the measurements of ALITTI 69 Alitti:1969rb . Our
predictions about the spin-parity values of $\Xi(1820)$ are consistent with
the other model predictions and experimental determinations.
Table 5: The decay widths (MeV) of $\Xi(1820)^{0}$ with different assignments. assignment | $~{}~{}\Xi\pi$ | $\Sigma\bar{K}$ | $\Lambda\bar{K}$ | $\Xi(1530)\pi$ | $\Gamma_{total}$
---|---|---|---|---|---
$|70,^{2}8,1,1,\frac{3}{2}^{-}\rangle$ | 2.85 | 10.63 | 7.25 | 11.98 | 32.71
$|70,^{4}8,1,1,\frac{1}{2}^{-}\rangle$ | 64.80 | 23.94 | 20.56 | 0.36 | 109.66
$|70,^{4}8,1,1,\frac{3}{2}^{-}\rangle$ | 4.56 | 0.26 | 0.72 | 11.95 | 17.49
$|70,^{4}8,1,1,\frac{5}{2}^{-}\rangle$ | 27.40 | 1.59 | 4.35 | 1.51 | 34.85
$|70,^{2}10,1,1,\frac{1}{2}^{-}\rangle$ | 4.05 | 5.98 | 5.14 | 1.43 | 16.60
$|70,^{2}10,1,1,\frac{3}{2}^{-}\rangle$ | 2.85 | 0.66 | 1.81 | 11.98 | 17.31
Table 6: The predicted total and partial decay width widths (MeV) and partial decay width ratios of $\Xi(1820)$ as the $|70,^{2}8,1,1,\frac{3}{2}^{-}\rangle$ assignment compared with the experimental data of ALITTI 69 Alitti:1969rb . channel | $\Gamma^{th}_{i}$ | $\Gamma^{th}_{total}$ | $\Gamma^{exp}_{total}$ | $\frac{\Gamma_{i}}{\Gamma_{total}}\big{|}_{th}$ | $\frac{\Gamma_{i}}{\Gamma_{total}}\big{|}_{exp}$
---|---|---|---|---|---
$\Xi\pi$ | 2.9 | 32.8 | $24^{+15}_{-10}$ | 0.09 | $0.1\pm 0.1$
$\Sigma\bar{K}$ | 10.6 | | | 0.32 | $0.30\pm 0.15$
$\Lambda\bar{K}$ | 7.3 | | | 0.22 | $0.30\pm 0.15$
$\Xi(1530)\pi$ | 12.0 | | | 0.37 | $0.30\pm 0.15$
Although by considering $\Xi(1820)$ as a pure state
$|70,^{2}8,1,1,\frac{3}{2}^{-}\rangle$, we note that the theoretical
predictions are in good agreement with the experimental observations, there
still exists room for configuration mixing in $\Xi(1820)$ for the
uncertainties of the experimental data. Chao _et al._ employed a quark model
to study the mass spectrum of $\Xi$ resonances Chao:1980em . According to
their study, $\Xi(1820)$ should be a mixed state, which dominates by the octet
$|70,^{2}8,1,1,\frac{3}{2}^{-}\rangle$ components ($\sim 90\%$), while
contains small components of $|70,^{2}10,1,1,\frac{3}{2}^{-}\rangle$ ($\sim
9\%$) and $|70,^{4}8,1,1,\frac{3}{2}^{-}\rangle$ ($\sim 1\%$). Thus, we
consider the physical state $\Xi(1820)$ as an admixture between the octet and
decuplet states with $J^{P}$=$\frac{3}{2}^{-}$. Using the CKM matrix method
discussed in IV.2, and fitting the observed strong decay properties, we obtain
$\left(\begin{array}[]{c}|\Xi\frac{3}{2}^{-}\rangle_{1}\cr|\Xi(1820)\frac{3}{2}^{-}\rangle\cr|\Xi\frac{3}{2}^{-}\rangle_{3}\end{array}\right)=\left(\begin{array}[]{ccc}-0.08&-0.98&0.17\cr
0.96&-0.13&-0.26\cr
0.27&0.14&0.95\end{array}\right)\left(\begin{array}[]{c}|70,^{2}8\rangle\cr|70,^{4}8\rangle\cr|70,^{2}10\rangle\end{array}\right).$
(65)
The theoretical results compared with the data are listed in Tab. 7, where we
find that the predicted partial decay width ratios are in agreement with the
experiment data of ALITTI 69 Alitti:1969rb as well. Considering configuration
mixing effects in $\Xi(1820)$, the decay width is closer to the center values
of observations. From the mixing parameters obtained in Eq. (65), it is seen
that the main component of $\Xi(1820)$ is
$|70,^{2}8,1,1,\frac{3}{2}^{-}\rangle$, which is about $92\%$. As a mixed
state, $\Xi(1820)$ also contains small components of
$|70,^{4}8,1,1,\frac{3}{2}^{-}\rangle$ ($\sim 0.02\%$) and
$|70,^{2}10,1,1,\frac{3}{2}^{-}\rangle$ ($\sim 0.07\%$). Our results are
compatible with the predictions in the quark model Chao:1980em , and the large
$N_{c}$ QCD approach Schat:2001xr ; Goity:2002pu . It should be mentioned that
$\Xi(1820)$ was also suggested to be a dynamically generated state with
$J^{P}=3/2^{-}$ Kolomeitsev:2003kt ; Sarkar:2004jh .
Table 7: The predicted total and partial decay widths (MeV) and partial decay width ratios of $\Xi(1820)$ as the mixed state $|\Xi(1820)\frac{3}{2}^{-}\rangle$ with mixing angles $\theta_{12}=85^{0}$, $\theta_{13}=170^{0}$ and $\theta_{23}=165^{0}$. For a comparison, the experimental data of ALITTI 69 Alitti:1969rb are listed as well. channel | $\Gamma^{th}_{i}$ | $\Gamma^{th}_{total}$ | $\Gamma^{exp}_{total}$ | $\frac{\Gamma_{i}}{\Gamma_{total}}\big{|}_{th}$ | $\frac{\Gamma_{i}}{\Gamma_{total}}\big{|}_{exp}$
---|---|---|---|---|---
$\Xi\pi$ | 2.1 | 23.3 | $24^{+15}_{-10}$ | 0.09 | $0.1\pm 0.1$
$\Sigma\bar{K}$ | 8.0 | | | 0.34 | $0.30\pm 0.15$
$\Lambda\bar{K}$ | 9.2 | | | 0.40 | $0.30\pm 0.15$
$\Xi(1530)\pi$ | 4.0 | | | 0.17 | $0.30\pm 0.15$
In brief, $\Xi(1820)$ could be approximately taken as a pure
$|70,^{2}8,1,1,\frac{3}{2}^{-}\rangle$ state. Slight configuration mixing may
exist in it. All the experimental observations of $\Xi(1820)$ could be well
understood in the constituent quark model.
Figure 2: The decay properties of $|\Xi\frac{3}{2}^{-}\rangle_{1}$ and
$|\Xi\frac{3}{2}^{-}\rangle_{3}$, respectively, where
$\Psi_{1}\equiv|\Xi\frac{3}{2}^{-}\rangle_{1}$ and
$\Psi_{3}\equiv|\Xi\frac{3}{2}^{-}\rangle_{3}$.
If $\Xi(1820)$ is a mixed state indeed, its counterparts
$|\Xi\frac{3}{2}^{-}\rangle_{1}$ and $|\Xi\frac{3}{2}^{-}\rangle_{3}$ in Eq.
(65) might be observed in experiments as well. From Eq. (65) it is seen that
the main components of $|\Xi\frac{3}{2}^{-}\rangle_{1}$ and
$|\Xi\frac{3}{2}^{-}\rangle_{3}$ are $|70,^{4}8,1,1,3/2^{-}\rangle$ ($\sim
96\%$) and $|70,^{2}10,1,1,3/2^{-}\rangle$ ($\sim 90\%$), respectively.
According to the quark model predictions Isgur:1978xj ; Chao:1980em , the
masses of $|\Xi\frac{3}{2}^{-}\rangle_{1}$ and
$|\Xi\frac{3}{2}^{-}\rangle_{3}$ are $\sim 1910$ MeV and $\sim 1970$ MeV,
respectively. The later large QCD calculations Schat:2001xr ; Goity:2002pu
gave similar predictions to those of quark models. The strong decay properties
of $|\Xi\frac{3}{2}^{-}\rangle_{1}$ and $|\Xi\frac{3}{2}^{-}\rangle_{3}$ are
studied as well. The results are shown in Fig. 2. For the uncertainties of the
mass predictions, we vary the masses of $|\Xi\frac{3}{2}^{-}\rangle_{1}$ and
$|\Xi\frac{3}{2}^{-}\rangle_{3}$ from 1850 MeV to 2050 MeV. From the
calculations, we find that if the masses of these two states are larger than
the threshold of $\Sigma(1385)\bar{K}$, the decay channel
$\Sigma(1385)\bar{K}$ dominates their decays. Furthermore, $\Xi(1530)\pi$
contributes significantly to the strong decays of these two states.
Finally, it should be pointed out that although the predicted masses of
$|\Xi\frac{3}{2}^{-}\rangle_{1}$ and $|\Xi\frac{3}{2}^{-}\rangle_{3}$ are
close to that of $\Xi(1950)$, the decay modes and partial decay width ratios
are not consistent with the observations. Thus, $\Xi(1950)$ does not favor any
$J^{P}=3/2^{-}$ assignments.
### IV.4 $\Xi(1950)$
Figure 3: Strong decay properties of the $J^{P}=1/2^{+}$ excitations in the
$N=2$ shell. Figure 4: Strong decay properties of the $J^{P}=3/2^{+}$
excitations in the $N=2$ shell. Figure 5: Strong decay properties of
$J^{P}=3/2^{+}$ and $J^{P}=5/2^{+}$ mixed states suggested by Chao _et
al._Chao:1980em . Figure 6: Strong decay properties of the $J^{P}=5/2^{+}$
excitations in the $N=2$ shell. Figure 7: Strong decay properties of the
$J^{P}=7/2^{+}$ excitations in the $N=2$ shell.
$\Xi(1950)$ was first observed by Badier _et al._ in the invariant mass
distribution $\Xi\pi$ of $K^{-}p\rightarrow\Xi^{-}K\pi$ process Badier:1965zzc
, its observed mass and width are $1933\pm 16$ MeV and $\Gamma\simeq 140\pm
35$ MeV, respectively. Later, several experimental groups also found
structures with a mass of $\sim 1950\pm 50$ MeV in the other processes.
$\Xi(1950)$ was only observed in $\Xi\pi$, $\Xi(1530)\pi$ and $\Lambda\bar{K}$
decay channels Beringer:1900zz . Most of these structures were observed in the
$\Xi\pi$ channel. In the $\Lambda\bar{K}$ channel, only Biagi _et al._ claimed
that they observed one narrow structure ($\Gamma\simeq 25\pm 15$ MeV) with a
mass of $M\simeq 1963$ MeV Biagi:1986vs . They estimated an upper limit on the
ratio of partial widths $\Gamma(\Sigma\bar{K})/\Gamma(\Lambda\bar{K})$ of 2.3,
and also suggested that the spin-parity of this resonance should be $5/2^{+}$
or its spin should be greater than $5/2$ in the natural spin-parity series
$7/2^{-}$, $9/2^{+}$, etc. While in the $\Xi(1530)\pi$ channel, only Briefel
_et al._ reported that they observed a broad structure ($\Gamma\simeq 60\pm
39$ MeV) with a mass of $M\simeq 1964$ MeV Briefel:1977bp . Although
$\Xi(1950)$ is a three-star $\Xi$ resonance listed in PDG, not much can be
said about its properties. According to various model predictions, there are
several $\Xi$ resonances in the $1900-2000$ MeV region Chao:1980em ;
Schat:2001xr ; Goity:2002pu ; Goity:2003ab .
#### IV.4.1 $J^{P}=5/2^{-}$ assignment
$\Xi(1950)$ was first classified as the pure octet $\Xi$ resonance with
$J^{P}=5/2^{-}$ by Alitti _et al._ Alitti:1968zz . In 1968, they observed a
$\Xi$ resonance whose mass and width are $M=1930\pm 20$ MeV and $\Gamma=80\pm
40$ MeV, respectively. The resonance parameters were very close to the
observations of Badier _et al._ in 1965 Badier:1965zzc . The Gell-Mann-Okubo
mass formula indicates this state might be the pure $J^{P}=5/2^{-}$ octet
$\Xi$ resonance Samios:1974tw ; Guzey:2005vz . The detailed SU(3) study of the
total and partial decay widths of the $J^{P}=5/2^{-}$ octet baryons seemed to
give a reasonable and consistent picture Alitti:1968zz . This classification
was also supported by some studies of the mass spectrum of $\Xi$ resonances in
various quark models Chao:1980em ; Pervin:2007wa .
With $\Xi(1950)$ as an assignment to the $J^{P}=5/2^{-}$ octet $\Xi$
resonance, we study its strong decay properties, the results are listed in
Tab. 8. It is seen that the $J^{P}=5/2^{-}$ assignment is a broad state with a
width of $\sim 100$ MeV. Its strong decays are dominated by the $\Xi\pi$
channel. The partial decay widths of $\Xi(1530)\pi$ and $\Lambda\bar{K}$ are
sizeable. Our predictions are in compatible with those in Alitti:1968zz .
Thus, the broad $\Xi$ resonances observed in $\Xi\pi$ channel might be good
candidates for the $J^{P}=5/2^{-}$ octet state.
Table 8: The total and partial decay widths (MeV) of the well-established four-star baryons $N(1675)$, $\Sigma(1775)$ and $\Lambda(1830)$, which correspond to $|70,^{4}8,1,1,\frac{5}{2}^{-}\rangle$. The data are obtained from PDG. assignment | channel | $\Gamma^{th}_{i}$ | $\Gamma^{th}_{total}$ | $\Gamma^{exp}_{total}$ | $\frac{\Gamma_{i}}{\Gamma_{total}}\big{|}_{th}$ | $\frac{\Gamma_{i}}{\Gamma_{total}}\big{|}_{exp}$
---|---|---|---|---|---|---
$N(1675)\frac{5}{2}^{-}$ | $n\pi$ | 25.3 | 81 | $130\sim 165$ | 0.31 | 0.35–0.45
| $\Delta\pi$ | 50.6 | | | 0.62 | 0.50–0.60
| $N\eta$ | 5.8 | | | 0.07 | 0.01
$\Sigma(1775)\frac{5}{2}^{-}$ | $\Sigma\pi$ | 8.0 | 76 | $105\sim 135$ | 0.11 | 0.02–0.05
| $\Lambda\pi^{0}$ | 19.1 | | | 0.25 | 0.14–0.20
| $n\bar{K}$ | 32.1 | | | 0.42 | 0.37–0.43
| $\Sigma(1385)\pi$ | 4.8 | | | 0.06 | 0.08–0.12
| $\Delta(1232)\bar{K}$ | 1.0 | | | 0.01 | …
| $\Lambda(1520)\pi$ | 11.2 | | | 0.14 | 0.17–0.23
$\Lambda(1830)\frac{5}{2}^{-}$ | $\Sigma\pi$ | 52.2 | 94 | $60\sim 110$ | 0.55 | 0.35–0.75
| $\Sigma(1385)\pi$ | 41.7 | | | 0.45 | $>0.15$
$\Xi(1950)\frac{5}{2}^{-}$ | $\Xi\pi$ | 71.33 | 105 | $80\pm 40$ | 0.68 | …
| $\Sigma\bar{K}$ | 7.8 | | | 0.07 | …
| $\Lambda\bar{K}$ | 13.6 | | | 0.13 | …
| $\Xi(1530)\pi$ | 10.9 | | | 0.10 | …
| $\Sigma(1385)\bar{K}$ | 1.2 | | | 0.01 | …
As a by-product, we calculate the strong decays of the other members of
$J^{P}=5/2^{-}$ octet baryons, $N(1675)\frac{5}{2}^{-}$,
$\Sigma(1775)\frac{5}{2}^{-}$ and $\Lambda(1830)\frac{5}{2}^{-}$. The results
are listed in Tab. 8 as well. From the table, it is found that our predictions
of the strong decay properties of $N(1675)\frac{5}{2}^{-}$,
$\Sigma(1775)\frac{5}{2}^{-}$ and $\Lambda(1830)\frac{5}{2}^{-}$ are in
reasonable agreement with the observations.
#### IV.4.2 $J^{P}=1/2^{-}$ assignment
Recently, Valderrama, Xie and Nieves proposed the existence of a spin-parity
state $J^{P}=1/2^{-}$ decuplet belonging to $\Xi(1950)$ PavonValderrama:2011gp
. Now we discuss the possibilities of $\Xi(1950)$ as an assignment to the
$J^{P}=1/2^{-}$ states. In Sec. IV.2, we predicted $\Xi(1690)$ is a mixed
state with $J^{P}=1/2^{-}$. According to the mass calculations of constituent
quark models Chao:1980em ; Chen:2009de and Large $N_{c}$ QCD Schat:2001xr ;
Goity:2002pu , the masses of the counterparts of $\Xi(1690)$,
$|\Xi\frac{1}{2}^{-}\rangle_{1}$ and $|\Xi\frac{1}{2}^{-}\rangle_{3}$, might
be close to 1950 MeV. Thus, they might be candidates for $\Xi(1950)$. The
strong decay properties of $|\Xi\frac{1}{2}^{-}\rangle_{1}$ and
$|\Xi\frac{1}{2}^{-}\rangle_{3}$ had been studied in Sec. IV.2 (see Fig. 1).
Considering $|\Xi\frac{1}{2}^{-}\rangle_{1}$ as an assignment to $\Xi(1950)$,
both the decay width and partial width ratio,
$\Gamma\simeq
27~{}\mathrm{MeV},\frac{\Gamma(\Sigma\bar{K})}{\Gamma(\Lambda\bar{K})}\simeq
2.0,$ (66)
are in agreement with the observations of Biagi 87C Biagi:1986vs ; however,
the spin-parity $J^{P}=1/2^{-}$ disagrees with their suggestion. While
assigning $|\Xi\frac{1}{2}^{-}\rangle_{3}$ to $\Xi(1950)$, we note that its
width and partial width ratios
$\Gamma\simeq
84~{}\mathrm{MeV},\frac{\Gamma(\Sigma\bar{K})}{\Gamma(\Lambda\bar{K})}\simeq
1.6,~{}\frac{\Gamma(\Xi\pi)}{\Gamma(\Sigma\bar{K})}\simeq 2.3,$ (67)
are consistent with those of the broad structures observed in the $\Xi\pi$
channel. Thus, the spin-parity $1/2^{-}$ mixed state
$|\Xi\frac{1}{2}^{-}\rangle_{3}$ could be a good assignment to $\Xi(1950)$.
#### IV.4.3 $J^{P}=1/2^{+}$ assignment
In Ref. Oh:2007cr , Oh predicted that $\Xi(1950)$ might have $J^{P}=1/2^{+}$
in the Skyrme model. We calculate the strong decays of all the excitations of
$\Xi$ with $J^{P}=1/2^{+}$ in the $N=2$ shell. The results are shown in Fig.
3. From the figure, it is found that if the second orbital excitation
$|56,^{4}10,2,2,\frac{1}{2}^{+}\rangle$ is considered as an assignment to
$\Xi(1950)$, the decay width and partial width ratios are
$\Gamma\simeq
53~{}\mathrm{MeV},\frac{\Gamma(\Sigma\bar{K})}{\Gamma(\Lambda\bar{K})}\simeq
0.88,~{}\frac{\Gamma(\Xi\pi)}{\Gamma(\Sigma\bar{K})}\simeq 1.27.$ (68)
The decay width and decay modes are consistent with the observations of
Goldwasser _et al._ Goldwasser:1970fk . It should be pointed out that the
predicted mass of various models for $|56,^{4}10,2,2,\frac{1}{2}^{+}\rangle$
is $\sim 2$ GeV Chao:1980em ; Goity:2003ab ; Chen:2009de , which is slightly
larger than the observation.
#### IV.4.4 $J^{P}=3/2^{+}$ assignment
Furthermore, we calculate the strong decays of all the excitations of $\Xi$
with $J^{P}=3/2^{+}$ in the $N=2$ shell. The results are shown in Fig. 4. It
is found that the second orbital excitation
$|56,^{4}10,2,2,\frac{3}{2}^{+}\rangle$ might be a candidate for $\Xi(1950)$.
The decay width
$\Gamma\simeq 36~{}\mathrm{MeV},$ (69)
and the partial width ratios
$\frac{\Gamma(\Sigma\bar{K})}{\Gamma(\Lambda\bar{K})}\simeq
0.88,~{}\frac{\Gamma(\Xi\pi)}{\Gamma(\Sigma\bar{K})}\simeq
1.27,~{}~{}\frac{\Gamma(\Xi\pi)}{\Gamma(\Xi(1530)\pi)}\simeq 3.16$ (70)
are in agreement with the observations of Biagi 87C Biagi:1986vs and APSELL
70 Apsell:1970uf . However, the spin-parity of
$|56,^{4}10,2,2,\frac{3}{2}^{+}\rangle$ is not consistent with the moment
analysis of Biagi 87C Biagi:1986vs . Furthermore, the predicted mass of
$|56,^{4}10,2,2,\frac{3}{2}^{+}\rangle$ in large $N_{c}$ QCD is obviously
larger than that of $\Xi(1950)$ Goity:2003ab .
Including configuration mixing effects, Chao _et al._ predicted two
$J^{P}=3/2^{+}$ mixed states with masses of $M=1930$ and 1965 MeV,
respectively Chao:1980em . With their mixing scheme, we predict the strong
decays of the two states. The results are shown in Fig. 5. It is obviously
seen that the decay properties of these mixed states are not in agreement with
any observations of $\Xi(1950)$.
Table 9: The total and partial decay widths (MeV) of the well-established four-star baryons $N(1680)$, $\Sigma(1915)$ and $\Lambda(1820)$, which are considered the pure $|56,^{2}8,2,2,\frac{5}{2}^{+}\rangle$ state. The data are obtained from PDG. assignment | channel | $\Gamma^{th}_{i}$ | $\Gamma^{th}_{total}$ | $\Gamma^{exp}_{total}$ | $\frac{\Gamma_{i}}{\Gamma_{total}}\big{|}_{th}$ | $\frac{\Gamma_{i}}{\Gamma_{total}}\big{|}_{exp}$
---|---|---|---|---|---|---
$N(1680)\frac{5}{2}^{+}$ | $n\pi$ | 38.0 | 59 | $120\sim 140$ | 0.64 | 0.65–0.70
| $\Delta\pi$ | 21.0 | | | 0.36 | …
| $N\eta$ | 0.4 | | | 0.01 | 0.0–0.01
| $\Lambda K$ | 0.03 | | | 0 | …
$\Sigma(1915)\frac{5}{2}^{+}$ | $N\bar{K}$ | 1.2 | 63 | $80\sim 160$ | 0.02 | 0.05–0.15
| $\Lambda\pi$ | 10.4 | | | 0.16 | …
| $\Sigma\pi$ | 26.7 | | | 0.42 | …
| $\Sigma(1385)\pi$ | 6.2 | | | 0.10 | …
| $\Delta(1232)\bar{K}$ | 18.4 | | | 0.29 | …
$\Lambda(1820)\frac{5}{2}^{+}$ | $N\bar{K}$ | 14.8 | 30 | $70\sim 90$ | 0.49 | 0.55–0.65
| $\Sigma\pi$ | 5.8 | | | 0.19 | 0.08–0.14
| $\Sigma(1385)\pi$ | 9.8 | | | 0.32 | …
$\Xi(1963)\frac{5}{2}^{+}$ | $\Xi\pi$ | 1.0 | 19 | $25\pm 15$ | 0.05 |
| $\Sigma\bar{K}$ | 9.2 | | | 0.48 |
| $\Lambda\bar{K}$ | 0.8 | | | 0.04 |
| $\Xi(1530)\pi$ | 5.2 | | | 0.27 |
| $\Sigma(1385)\bar{K}$ | 2.5 | | | 0.13 |
#### IV.4.5 $J^{P}=5/2^{+}$ assignment
Recently, Valderrama, Xie and Nieves PavonValderrama:2011gp predicted that
the narrow structure with a mass of $M\simeq 1963$ MeV observed in the
$\Lambda\bar{K}$ channel by Biagi _et al._ Biagi:1986vs (denoted by
$\Xi(1963)$) could be assigned to the partner of the $J^{P}=5/2^{+}$
$N(1680)$, $\Lambda(1820)$ and $\Sigma(1915)$ resonances. According to the
classification of quark model, these resonances could be roughly considered as
the $|56,^{2}8,2,2,\frac{5}{2}^{+}\rangle$ configuration Klempt:2009pi .
Firstly, we calculate the strong decays of the $N(1680)$, $\Lambda(1820)$ and
$\Sigma(1915)$ states, which are listed in Tab. 9. From the table, it is found
that the decay properties of $N(1680)$, $\Lambda(1820)$ and $\Sigma(1915)$
could be roughly understood by taking them as the
$|56,^{2}8,2,2,\frac{5}{2}^{+}\rangle$ configuration. According to the model
predictions, the mass of $\Xi|56,^{2}8,2,2,\frac{5}{2}^{+}\rangle$ is close to
1963 MeV Chao:1980em ; Goity:2003ab . Assigning $\Xi(1963)$ to the
$|56,^{2}8,2,2,\frac{5}{2}^{+}\rangle$ configuration, the predicted strong
decay properties are shown in Tab. 9 as well. It is seen that although the
decay width is in agreement with the data, the partial decay width ratios are
not consistent with the observations of Biagi _et al._ Biagi:1986vs at all.
Furthermore, Chao _et al._ also predicted a mixed states with $J^{P}=5/2^{+}$
having a mass of $M=1935$ MeV Chao:1980em . With their mixing scheme, its
strong decays are studied. The results are shown in Fig. 5 as well. It is seen
that the strong decays are dominated by the $\Sigma\bar{K}$ channel, which
disagrees with any observations of $\Xi(1950)$.
In brief, there might be several $\Xi$ resonances in the $1900-2000$ MeV
region observed in experiments. These states are most likely to correspond to
the pure octet $\Xi$ resonance with $J^{P}=5/2^{-}$, the mixed state
$|\Xi\frac{1}{2}^{-}\rangle_{3}$ with $J^{P}=1/2^{-}$ (its main component is
$|70,^{4}8,1,1,\frac{1}{2}^{-}\rangle$), the second orbital excitation
$|56,^{4}10,2,2,\frac{1}{2}^{+}\rangle$ with $J^{P}=1/2^{+}$, etc. These
states can be easily distinguished by the measurements of their partial decay
width ratios, $\Gamma(\Lambda\bar{K})/\Gamma(\Sigma\bar{K})$ and
$\Gamma(\Sigma\bar{K})/\Gamma(\Xi\pi)$.
Table 10: The decay widths (MeV) of $\Xi(2030)$ with different assignments. assignment | $~{}\Xi\pi$ | $\Sigma\bar{K}$ | $\Lambda\bar{K}$ | $\Xi(1530)\pi$ | $\Sigma(1385)\bar{K}$ | $\Lambda(1405)\bar{K}$ | $\Lambda(1520)\bar{K}$ | $\Gamma_{total}$ | $\Gamma(\Lambda\bar{K}):\Gamma(\Sigma\bar{K})$
---|---|---|---|---|---|---|---|---|---
$|56,^{2}8,2,2,\frac{5}{2}^{+}\rangle$ | 1.6 | 18.5 | 1.3 | 8.2 | 5.7 | 0.4 | 0.01 | 35.8 | 0.07
$|70,^{2}8,2,2,\frac{3}{2}^{+}\rangle$ | 1.4 | 17.9 | 4.5 | 1.4 | 0.6 | 1.5 | 5.4 | 32.7 | 0.25
$|70,^{2}8,2,2,\frac{5}{2}^{+}\rangle$ | 0.8 | 5.8 | 2.7 | 4.1 | 2.8 | 0.05 | 0.07 | 16.3 | 0.46
### IV.5 $\Xi(2030)$
$\Xi(2030)$ is a three-star state listed in PDG. It mainly decays into
$\Sigma\bar{K}$ and $\Lambda\bar{K}$ channels. The decay ratios into the other
channels, such as $\Xi\pi$ and $\Xi(1530)\pi$, are small. The measured decay
width and partial decay width ratio are
$\Gamma\simeq 21\pm
6\mathrm{MeV},~{}\frac{\Gamma(\Lambda\bar{K})}{\Gamma(\Sigma\bar{K})}\simeq
0.22\pm 0.09.$ (71)
A moment analysis of the HEMINGWAY 77 data indicated at a level of three
standard deviations that $J\geq 5/2$ Hemingway:1977uw .
We analyze the strong decay properties for all the configurations in the $N=2$
shell, which are shown in Figs. 3-7. From the figures, we find that only three
excitations $|56,^{2}8,2,2,\frac{5}{2}^{+}\rangle$,
$|70,^{2}8,2,2,\frac{3}{2}^{+}\rangle$ and
$|70,^{2}8,2,2,\frac{5}{2}^{+}\rangle$ have comparable decay widths with that
of $\Xi(2030)$, and mainly decay into $\Sigma\bar{K}$ channel. Considering
them as assignments to $\Xi(2030)$, we collect their decay properties in Tab.
10, where we find that both the decay width and the partial decay ratio
$\Gamma\simeq
33~{}\mathrm{MeV},~{}\frac{\Gamma(\Lambda\bar{K})}{\Gamma(\Sigma\bar{K})}\simeq
0.25.$ (72)
of the $|70,^{2}8,2,2,\frac{3}{2}^{+}\rangle$ configuration are in good
agreement with the observations. However, its spin $J=3/2$ disagrees with the
moment analysis of the HEMINGWAY 77 data Hemingway:1977uw .
As a whole, if we do not care about the moment analysis of the HEMINGWAY 77
data Hemingway:1977uw , $\Xi(2030)$ favors the
$|70,^{2}8,2,2,\frac{3}{2}^{+}\rangle$ assignment. $\Xi(2030)$ could not be
assigned to any pure $J^{P}=7/2^{+}$ states or any admixtures between them. If
the spin-parity of $\Xi(2030)$ is $J^{P}=5/2^{+}$, it is most likely to be a
mixed state, for no pure $J^{P}=5/2^{+}$ configuration could explain the data.
## V Summary
In this work, we have studied the strong decays of the $\Xi$ baryons within
$N\leq 2$ shells in a chiral quark model. The strong decay properties of these
well-established ground decuplet baryons could be reasonably described. We
find that $\Xi(1690)$ should be assigned to the spin-parity $J^{P}=1/2^{-}$
state $|70,^{2}8,1,1,\frac{1}{2}^{-}\rangle$, which might slightly mix with
the other configurations. The $J^{P}=1/2^{-}$ for $\Xi(1690)$ predicted by us
is consistent with the suggestions from the constituent quark model
Pervin:2007wa , Skyrme model Oh:2007cr and unitary chiral approaches
Kolomeitsev:2003kt ; Sarkar:2004jh . The strong decays of the physical
partners of $\Xi(1690)$, $|\Xi\frac{1}{2}^{-}\rangle_{1}$ and
$|\Xi\frac{1}{2}^{-}\rangle_{3}$ are analyzed as well.
$|\Xi\frac{1}{2}^{-}\rangle_{1}$ might be observed in the $\Sigma\bar{K}$ and
$\Lambda\bar{K}$ channels, while $|\Xi\frac{1}{2}^{-}\rangle_{3}$ is possibly
observed in the $\Xi\pi$ and $\Lambda\bar{K}$ channels.
The strong decay properties of $\Xi(1820)$ could be well understood by
assigning it to $|70,^{2}8,1,1,\frac{3}{2}^{-}\rangle$. There might exist
slight configuration mixing in $\Xi(1820)$. Its main component is
$|70,^{2}8,1,1,\frac{3}{2}^{-}\rangle$ ($\sim 92\%$), which is compatible with
the predictions of the quark model Chao:1980em , and large $N_{c}$ QCD
approach Schat:2001xr ; Goity:2002pu . Considering the configuration mixing
effects, we also have studied the strong decay properties of the physical
partners of $\Xi(1820)$, $|\Xi\frac{3}{2}^{-}\rangle_{1}$ and
$|\Xi\frac{3}{2}^{-}\rangle_{3}$. The observations in the
$\Sigma(1385)\bar{K}$ and $\Xi(1530)\pi$ channels are crucial to look for the
other $J^{P}=3/2^{-}$ states $|\Xi\frac{3}{2}^{-}\rangle_{1}$ and
$|\Xi\frac{3}{2}^{-}\rangle_{3}$ in future experiments.
The situation for $\Xi(1950)$ is very complicated. Several $\Xi$ resonances in
the $1900-2000$ MeV region might have been observed in experiments, which is
supported by the mass calculations from various models Chao:1980em ;
Schat:2001xr ; Goity:2002pu ; Goity:2003ab , and the recent strong decay
analysis in PavonValderrama:2011gp . The broad $\Xi$ resonances observed in
the $\Xi\pi$ channel might be good candidates for the $J^{P}=5/2^{-}$ octet
state or the mixed state $|\Xi\frac{1}{2}^{-}\rangle_{3}$ with
$J^{P}=1/2^{-}$. The $\Xi$ resonance with moderate width observed by
Goldwasser _et al._ Goldwasser:1970fk might correspond to the $J^{P}=1/2^{+}$
excitation $|56,^{4}10,2,2,\frac{1}{2}^{+}\rangle$. The existence of a
$J^{P}=1/2^{+}$ $\Xi$ excitation with a mass around 1950 MeV is also suggested
by Oh in a Skyrme model Oh:2007cr . The second orbital excitation
$|56,^{4}10,2,2,\frac{3}{2}^{+}\rangle$ and the mixed state
$|\Xi\frac{1}{2}^{-}\rangle_{1}$ might be candidates for a narrow width state
observed in the $\Lambda\bar{K}$ channel; however, their spin-parity is not
consistent with a moment analysis of the data. The partial decay ratios,
$\Gamma(\Lambda\bar{K})/\Gamma(\Sigma\bar{K})$ and
$\Gamma(\Sigma\bar{K})/\Gamma(\Xi\pi)$, are sensitive to different
assignments. Thus, the measurements of these ratios are crucial to uncover
many puzzles in $\Xi(1950)$.
In present work, $\Xi(2030)$ as any spin-parity $J^{P}=7/2^{+}$ states should
be excluded. The observations of $\Xi(2030)$ do not favor any pure
$J^{P}=5/2^{+}$ configuration as well. If we do not care about the moment
analysis of the HEMINGWAY 77 data, $\Xi(2030)$ favors the
$|70,^{2}8,2,2,\frac{3}{2}^{+}\rangle$ assignment. Further observations in the
$\Xi(1530)\pi$ and $\Sigma(1385)\bar{K}$ channels are necessary.
To provide helpful information for the search for the missing $\Xi$ baryons,
in Figs. 3-7 our predictions of their strong decay properties are shown as
well. From our theoretical results, we find that the strong decays of many
$\Xi$ resonances are dominated by the $\Xi(1530)\pi$ and
$\Sigma(1385)\bar{K}$, thus, in these decay channels, we might find some new
$\Xi$ resonances as well.
## Acknowledgements
This work is supported, in part, by the National Natural Science Foundation of
China (Grant No. 11075051), Program for Changjiang Scholars and Innovative
Research Team in University (Grant No. IRT0964), the Program Excellent Talent
Hunan Normal University, and the Hunan Provincial Natural Science Foundation
(Grants No.11JJ7001 and No. 13JJ1018).
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|
arxiv-papers
| 2013-02-01T05:21:46 |
2024-09-04T02:49:41.140244
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Li-Ye Xiao and Xian-Hui Zhong",
"submitter": "Xianhui Zhong",
"url": "https://arxiv.org/abs/1302.0079"
}
|
1302.0105
|
myctr
# ENTANGLEMENT DYNAMICS IN FINITE QUDIT CHAIN
IN CONSISTENT MAGNETIC FIELD
E. A. IVANCHENKO National Science Center “Institute of Physics and
Technology”, Institute for Theoretical Physics, Akademicheskaya str. 1, 61108
Kharkov, Ukraine
[email protected]
###### Abstract
Based on the Liouville-von Neumann equation, we obtain a closed system of
equations for the description of a qutrit or coupled qutrits in an arbitrary,
time-dependent, external magnetic field. The dependence of the dynamics on the
initial states and the magnetic field modulation is studied analytically and
numerically. We compare the relative entanglement measure’s dynamics in bi-
qudits with permutation particle symmetry. We find the magnetic field
modulation which retains the entanglement in the system of two coupled
qutrits. Analytical formulae for the entanglement measures in finite chains
from 2 to 6 qutrits or 3 quartits are presented.
###### keywords:
Entanglement; qudit; multiqudit chain.
PACS: 03.67.Bg, 03.67.Mg
## 1 Introduction
Multi-level quantum systems are studied extensively, since they have wide
applications. Some of the existing analytical results[1] for spin 1 are
derived in terms of a coherent vector[2]. The class of exact solutions for a
three-level system is given in Ref. AMIshkhanyan. The application of coupled
multi-level systems in quantum devices is actively studied[4]. The study of
these systems is topical in view of possible applications for useful work in
microscopic systems[5]. Exact solutions for two uncoupled qutrits interacting
with the vacuum are obtained in Ref. DerkaczJakobczyk. For the case of qutrits
interacting with a stochastic magnetic field, the exact solutions are obtained
in Ref. MazharAli. The exact solutions for coupled qudits in an alternating
magnetic field, to our knowledge, have not yet been found.
The entanglement in multi-particle coupled systems is an important resource
for many problems in the quantum information science, but its quantitative
value computing is difficult because of different types of entanglement.
Multi-dimensional entangled states are interesting both for the study of the
foundations of quantum mechanics and for the topicality of developing new
protocols for quantum communication. For example, it was shown that for
maximally entangled states of two quantum systems, the qudits break the local
realism stronger than the qubits[8], and the entangled qudits are less
influenced by noise than the entangled qubits. Using entangled qutrits or
qudits instead of qubits is more protective from interception. From a
practical point of view, it is clear that generating and saving the
entanglement in a controlled manner is the primary problem for the realization
of quantum computers. Maximally entangled states are best suited for the
protocols of quantum teleportation and quantum cryptography.
The entanglement and the symmetry are two basic notions of quantum mechanics.
We study the dynamics of multipartite systems, which are invariant at any
subsystem permutation. The aim of this work is finding exact solutions for the
dynamics of coupled qudits interacting with an alternating magnetic field as
well as the comparative analysis of the entanglement measures in a finite
chain of coupled qudits.
The paper is organized as following. The Hamiltonian of the anisotropic qutrit
in an arbitrary alternating magnetic field is described in Sec. II. Then the
system of equations for the description of the qutrit dynamics is derived in
the Bloch vector representation. We introduce a consistent magnetic field,
which describes an entire class of field forms. In section III we find an
analytical solution for the density matrix in the case of isotropic
interaction. Analytical formulae, which describe the entanglement in finite
spin chains of qutrits or quartits, are presented in Sec. IV. The results are
demonstrated graphically in Sec. V at specific parameters. The conclusions are
given in Sec. VI.
## 2 Qutrit
### 2.1 Qutrit Hamiltonian and Liouville-von Neumann equation
We take the qutrit Hamiltonian (for the spin particle with s=1) in the space
of one qutrit $\mathrm{C}^{3}$ in the basis
$|1>=(1,0,0),\;|0>=(0,1,0),\;|-1>=(0,0,1)$, in an external magnetic field
$\overrightarrow{{h}}=(h_{1},h_{2},h_{3})$ with anisotropy, in the form
$\hat{H}(\overrightarrow{h})=h_{1}S_{1}+h_{2}S_{2}+h_{3}S_{3}+Q(S_{3}^{2}-\frac{s(s+1)}{3}E_{2s+1\times
2s+1})+d(S_{1}^{2}-S_{2}^{2}),$ (1)
where $h_{1},\;h_{2},\;h_{3}$ are the Cartesian components of the external
magnetic field in frequency units (we assume $\hbar=1$, Bohr magneton
$\mu_{B}=1$); $S_{1},\;S_{2},\;S_{3}$ are the spin-1 matrices[9];
$E_{2s+1\times 2s+1}$ is the unity matrix; $Q,\;d$ are the anisotropy
constants. When the constants $Q,\;d$ are zeros, then the two Hamiltonian
eigenvalues are symmetrically placed with respect to the zero level.
There exist many useful bases[10]. Allard and Hard[11](AH) formed the
Hermitian basis $C_{\alpha}$ from the linear combinations of the irreducible
tensor operators. This basis is normalized so that
$S_{1}=C_{1,x},S_{2}=C_{1,y},S_{3}=C_{1,z},$ irrespective of the spin quantum
number $s$. It is convenient to construct the spin Hamiltonian for any spin.
Hereinafter we use the Hermitian basis[11]. From the physical point of view,
for important physical applications the basis[11, 12] is preferred. It is not
necessary for the basis to be Hermitian since the results of the calculations
are independent of the choice of base, but there is a significant advantage of
the Hermitian basis. It is useful that the Liouville-von Neuman equation does
not involve complex numbers and can be solved using real algebra. It makes
numerical calculations faster and simplifies the interpretation of the
equation system. The transition matrix determines the coupling between the
generalized Gell-Mann and (AH) Hermitian matrix bases. This coupling for
qutrit is presented in Ref. 9.
The qutrit dynamics in a magnetic field is described in the density matrix
formalism using the Liouville-von Neumann equation
$i\partial_{t}\rho=[\hat{H},\,\rho],~{}\rho(t=0)=\rho_{0}.$ (2)
It is convenient to rewrite Eq. (2) presenting the density matrix $\rho$ in
the decomposition with a full set[11] of orthogonal Hermitian matrices
$C_{\alpha}$ (further the summation over the Greek indices will be from 0 to 8
and over the Latin ones from 1 to 8)
$\rho=\frac{1}{\sqrt{6}}C_{\alpha}R_{\alpha}.$ (3)
Since $\mathrm{Tr\,}C_{i}=0$ for $1\leq i\leq 8$, then from the condition
$\mathrm{Tr\,}\rho=R_{0}$ it follows that $R_{0}=1$. And although the results
are independent of the basis choice, in this basis the functions
$R_{i}=\mathrm{Tr\,}\rho\,C_{i}$ have the concrete physical meaning[11]. The
values $R_{1},R_{2},R_{3}$ are the polarization vector Cartesian components;
$R_{4}$ is the two-quantum coherence contribution in $R_{2}$; $R_{5}$ is the
one-quantum anti-phase coherence contribution in $R_{2}$; $R_{6}$ is the
contribution of the rotation between the phase and anti-phase one-quantum
coherence; $R_{7}$ is the one-quantum anti-phase coherence contribution in
$R_{1}$; $R_{8}$ is the two-quantum coherence contribution in $R_{1}$.
The Liouville-von Neumann equation in terms of the functions $R_{i}$ takes the
form of a closed system of 8 real differential first-order equations. This
system of equations can be written in a compact form as following[12, 13]:
$\partial_{t}R_{l}=e_{ijl}h_{i}R_{j},$ (4)
where $e_{ijl}$ are the structure constants,
$h_{i}=2(h_{1},h_{2},h_{3},0,0,\frac{Q}{\sqrt{3}},0,d)$ are the Hamiltonian
components Eq. (1) in the basis $C_{\alpha}$.
### 2.2 The consistent field
Let us consider the qutrit dynamics in an alternating field of the form
$\vec{h}(t)=\left(\omega_{1}\mathrm{cn}(\omega
t|k),\;\omega_{1}\mathrm{sn}(\omega t|k),\;\omega_{0}\mathrm{dn}(\omega
t|k)\right),$ (5)
where $\mathrm{cn},\mathrm{sn},\mathrm{dn}$ are the Jacobi elliptic
functions[14]. Such field modulation under the changing of the elliptic
modulus $k$ from 0 to 1 describes the whole class of field forms from
trigonometric[15] ($\mathrm{cn}(\omega t|0)=\mathrm{cos}\omega
t,\;\mathrm{sn}(\omega t|0)=\mathrm{sin}\omega t,\;\mathrm{dn}(\omega t|0)=1$
) to the exponentially impulse ones ($\mathrm{cn}(\omega
t|1)=\frac{1}{\mathrm{ch}\omega t},\;\mathrm{sn}(\omega t|1)=\mathrm{th}\omega
t,\;\mathrm{dn}(\omega t|1)=\frac{1}{\mathrm{ch}\omega t}$)[16]. The elliptic
functions $\mathrm{cn}(\omega t|k)$ and$\;\mathrm{sn}(\omega t|k)$ have the
real period $\frac{4K}{\omega}$, while the function $\mathrm{dn}(\omega t|k)$
has a period of half the duration. Here $K$ is the full elliptic integral of
the first kind[14]. In other words, even though the field is periodic with a
common real period $\frac{4K}{\omega}$, but as we can see, the frequency of
the longitudinal field amplitude modulation is twice as high as that of the
transverse field. We call such field consistent.
Let us make use of the substitution $\rho=\alpha_{1}^{-1}r\alpha_{1}$ with the
diagonal matrix $\alpha_{1}=\mathrm{diag\,}(f,1,f^{-1})$, where $f(\omega
t|k)=\mathrm{cn}(\omega t|k)+i\mathrm{sn}(\omega t|k).$ Then we obtain the
equation for the matrix $r$ in the form
$i\partial_{t}r=[\alpha_{1}\hat{H}\alpha_{1}^{-1}-i\alpha_{1}\partial_{t}(\alpha_{1}^{-1}),r].$
(6)
The equation for the matrix $r$ without taking into account the anisotropy can
be written as following
$i\partial_{t}r=[\omega_{1}S_{1}+\delta\,\mathrm{dn}(\omega
t|k)S_{3},r],\;r(t=0)=\rho_{0},\;\delta=\omega_{0}-\omega.$ (7)
At $k=0$ equation (7) describes the dynamics of the qutrit in a circularly
polarized field[15, 17, 18]. The exact solutions of this equation are known,
and under certain initial conditions the explicit formulae are given in Ref.
NathSenGangopadhyay. At the exact resonance, $\omega=\omega_{0}$ it is
straightforward to present (2) in the deformed field ($k\neq 0$) (5) for the
given initial condition $\rho=\rho_{0}$:
$\rho(t)=\alpha_{1}^{-1}e^{-i\omega_{1}tS_{1}}\rho_{0}e^{i\omega_{1}tS_{1}}\alpha_{1}.$
(8)
Explicit solutions for specific initial conditions are given in Ref.
finitequtritchain.
## 3 Bi-qutrit
In the space $\mathrm{C}^{3}\otimes\mathrm{C}^{3}$ the bi-qutrit density
matrix can be written in the Bloch representation
$\varrho=\frac{1}{6}R_{\alpha\beta}C_{\alpha}\otimes
C_{\beta},\;R_{00}=1,\;\varrho(t=0)=\varrho_{0},$ (9)
where $\otimes$ denotes the direct product. The functions $R_{m0},R_{0m}$
characterize the individual qutrits and functions $R_{mn}$ characterize their
correlations.
Let us consider the Hamiltonian of the system of two qutrits with anisotropic
and exchange interaction in a magnetic field in the following form
$H_{2}=\hat{H}(\overrightarrow{h})\otimes E_{2s+1\times 2s+1}+E_{2s+1\times
2s+1}\otimes\hat{H}(\overrightarrow{\bar{h}})+JS_{i}\otimes S_{i},$ (10)
where $\overrightarrow{h}$ and $\overrightarrow{\bar{h}}$ are the magnetic
field vectors in frequency units, which operate on the first and the second
qutrits respectively, and $J$ is the constant of isotropic exchange
interaction. We study the dynamics of two qutrits in the consistent magnetic
field $\overrightarrow{h}=(\omega_{1}\mathrm{cn}(\omega
t|k)),\;\omega_{1}\mathrm{sn}(\omega t|k),\;\omega_{0}\mathrm{dn}(\omega
t|k))$,$\overrightarrow{\bar{h}}=(\varpi_{1}\mathrm{cn}(\varpi
t|k),\;\varpi_{1}\mathrm{sn}(\omega t|k),\;\varpi_{0}\mathrm{dn}(\omega t|k))$
at the anisotropy constants equal to 0. Let us transform the matrix density
$\varrho=\alpha_{2}^{-1}r_{2}\alpha_{2}$ with the matrix
$\alpha_{2}=\alpha_{1}\otimes\alpha_{1}$. The equation for the matrix $r_{2}$
takes the form $i\partial_{t}r_{2}=[\widetilde{H}(\mathrm{dn}(\omega
t|k)),r_{2}]$ with the transformed Hamiltonian
$\widetilde{H}(\mathrm{dn}(\omega t|k))$[9].
Since $\mathrm{dn}(\omega t|k)|_{k=0}=1$, then the transformed Hamiltonian
$\widetilde{H}$ does not depend on time, and the solution for the density
matrix in the circularly polarized field has the form
$\varrho(t)=\alpha_{2}^{-1}e^{-i\widetilde{H}t}\varrho_{0}e^{i\widetilde{H}t}\alpha_{2}|_{k=0}.$
(11)
In the consistent field at resonance $\omega=\varpi_{0}=\omega_{0}=h$ at equal
$\varpi_{1}=\omega_{1}$ the Hamiltonian eigenvalues equal to
$-2J,-J,J,J-2\omega_{1},-J-\omega_{1},J-\omega_{1},-J+\omega_{1},J+\omega_{1},J+2\omega_{1}$.
This allows to find the exact solution in the closed form for any initial
condition since the matrix exponent $e^{i\widetilde{H}t}$ in this case can be
calculated analytically.
For a larger number of the qudits with a pairwise isotropic interaction, the
generalization is evident. In the case of interaction of qudits with a
different dimensionality, the reduction of the original system to the system
with constant coefficients can be done by choosing, for example, the
transformation matrix for spin-3/2 and spin-2 in the form
$\mathrm{diag\,}(f^{3/2},\,f^{1/2},\,f^{-1/2},\,f^{-3/2})\otimes\mathrm{diag\,}(f^{2},\,f,\,1,\,f^{-1},\,f^{-2}).$
(12)
However, the Hamiltonian eigenvalues cannot be found in a simple analytical
form because of the lowering of the system’s symmetry.
## 4 Analytical formulae for entanglement measures
### 4.1 Entanglement in the bi-qutrit
For the initial maximally entangled state which is symmetrical at the particle
permutation
$|\psi>=\frac{1}{\sqrt{3}}\sum_{i=-1}^{1}|i>\otimes|i>,$ (13)
in the consistent field at the resonance $\omega=\varpi_{0}=\omega_{0}=h$ at
equal $\varpi_{1}=\omega_{1},$ the exact solution for the correlation
functions is given in Ref. finitequtritchain. The correlation functions have
the property $R_{\alpha\beta}=R_{\beta\alpha}$, i.e. the symmetry is conserved
during the evolution, since the initial state and the Hamiltonian are
symmetric with respect to the particle permutation.
Given the exact solution, one can find the negative eigenvalues of the partly
transposed matrix $\varrho^{pt}=(T\otimes E)\varrho$ (here $T$ denotes the
transposition): $\epsilon_{1}=\epsilon_{2}=-\frac{1}{27}\sqrt{69+28\cos
3Jt-16\cos 6Jt},\;\epsilon_{3}=-\frac{1}{27}\left(5+4\cos 3Jt\right)$. The
absolute value of the sum of these eigenvalues
$m_{VW}=\sum_{i=1}^{3}|\epsilon_{i}|$ (14)
defines the entanglement measure (negativity) between the qutrits[20].
The entanglement between the qudits can be described quantitatively with the
measure[21]
$m_{SM}=\sqrt{\frac{1}{D-1}(R_{ij}-R_{i0}R_{0j})^{2}},$ (15)
where D is the basis dimension (for qutrit $D=9$). This measure equals to 0
for the separable state and to 1 for the maximally entangled state, and it is
applicable for both pure and mixed states.
That is why for the maximally entangled initial state of two qutrits, the
entanglement in the consistent field is defined by the formulae with the found
solution for the density matrix
$m_{SM}=\frac{1}{81}\sqrt{4457+2776\cos 3Jt-632\cos 6Jt-56\cos 9Jt+16\cos
12Jt}.$ (16)
This measure is numerically equivalent to the measure $m_{VW}$[20, 22] which
is defined by the absolute value of the sum of the negative eigenvalues.
According to the definition for $N$-qudit pure state[23], the entanglement
measure equals to
$\eta_{N}=\frac{1}{N}\sum_{i=1}^{N}S_{i},$ (17)
where $S_{i}=-\mathrm{Tr\,}\rho_{i}\log_{b}\rho_{i}$ is the reduced von
Neumann entropy, the index $i$ numerates the particles, i.e. the other
particles are traced out. We use the logarithm to the base $b$ to ensure that
the maximal measure is normalized to 1. The base $b$ equals to 3 in the qutrit
case.
Since the qutrit reduced matrix eigenvalues equal to
$\lambda_{1}=\lambda_{2}=\frac{1}{27}(5+4\cos
3Jt),\;\lambda_{3}=\frac{1}{27}(17-8\cos 3Jt),$ then the entanglement measure
in the bi-qutrit takes the form
$\eta_{2}=-\sum_{i=1}^{3}\lambda_{i}\log_{3}\lambda_{i}.$ (18)
Normalized to unity the measure I-concurrence which is easy to calculate is
defined by the formulae[24]
$m_{I}=\sqrt{\frac{d}{d-1}}\sqrt{(1-\mathrm{Tr\,}\rho_{1}^{2})}=\frac{1}{9}\sqrt{57+32\cos
3Jt-8\cos 6Jt},$ (19)
where $d=3$ for a qutrit, $\rho_{1}=\frac{1}{\sqrt{6}}C_{\alpha}R_{\alpha 0}$
is the reduced qutrit matrix.
The measures $m_{VW},\;m_{SM},\;\eta_{2},\;m_{I}$ do not depend on the
parameters of the consistent field, the sign of the exchange constant at zero
anisotropy parameters. It should be noted that the Wootters entanglement
measure (the concurrence) in the system of two qubits with an isotropic
interaction in a circularly polarized field at resonance is also independent
of the alternating field amplitude[25], but depends on the exchange constant
$J$ and the initial conditions only.
The numerical solution of the Liouville-von Neumann equation shows that if an
identical external field operates on every qudit, the free Hamiltonian and the
interaction Hamiltonian are commutative operators, the measures considered are
determined only by the symmetric two-body interaction with the interaction
constant of $J$. If a different field operates on every qudit, there arises a
broken permutation symmetry of the total Hamiltonian, which changes the
entanglement dynamics. Thus it is possible to control entanglement by changing
the parameters of an external field.
It is possible to show that the distance measure[26, 27] $\sqrt{{\rm
Tr\,}(\rho(t)-\rho_{0})^{2}}$ depends on the parameters of the consistent
field.
At a zero external field the entanglement measure (15) takes the analytical
form at equal non-zero anisotropy parameters $Q=d=\overline{d}=\overline{Q}$
$m_{SM}(Q)=\frac{1}{\left(9J^{2}+8QJ+16Q^{2}\right)^{2}}\sqrt{\sum_{k=0}^{4}q_{k}\cos\left(k\sqrt{9J^{2}+8QJ+16Q^{2}}\,t\right)},$
(20)
where
$q_{0}=4457J^{8}+11616QJ^{7}+47392Q^{2}J^{6}+85888Q^{3}J^{5}+163072Q^{4}J^{4}+194560Q^{5}J^{3}+221184Q^{6}J^{2}+131072Q^{7}J+65536Q^{8}$;
$q_{1}=8J^{2}(J+2Q)^{2}\left(347J^{4}+518QJ^{3}+1440Q^{2}J^{2}+1504Q^{3}J+1024Q^{4}\right)$;
$q_{2}=-8J^{2}(J+2Q)^{2}\left(79J^{4}+76QJ^{3}+320Q^{2}J^{2}+448Q^{3}J+256Q^{4}\right)$;
$q_{3}=-8J^{3}(7J-4Q)(J+2Q)^{3}(J+4Q)$, $q_{4}=16J^{4}(J+2Q)^{4}$.
### 4.2 Entanglement in the chain of qutrits
We consider the Hamiltonian of the chain of $N$ qutrits with the pairwise
isotropic interaction in the consistent field $\vec{h}(t)$ (5) at resonance in
the following form
$H_{N}=\sum(\vec{h}(t)\overrightarrow{S}\otimes\overbrace{E\otimes\dots\otimes
E}^{N-1}+J\overrightarrow{S}\otimes\overrightarrow{S}\otimes\overbrace{E\otimes\dots\otimes
E}^{N-2}),$ (21)
where the summation is over different possible positions of
$\overrightarrow{S}$ in the direct products. Because the maximally entangled
state of $N$ qutrits
$|\phi>_{N}=\frac{1}{\sqrt{3}}\sum_{i=-1}^{1}|i>^{\otimes N}$ (22)
and the Hamiltonian (21) have a permutation symmetry, it follows that the
density matrix of $N$ qutrits has symmetric correlation functions.
The entanglement measures for many-particle multi-level quantum systems have
not been studied enough and are difficult to calculate in the analytical form,
that is why we will present analytical formulae only for the entropy measure
$\eta_{N}$, which is defined by the eigenvalues of the reduced one-particle
matrices for each qutrit. As the result of the mentioned symmetry the reduced
matrices are equal to each other. Therefore the entanglement measure for $N$
qutrits reads
$\eta_{N}=-\sum_{i=1}^{3}r_{i}\log_{3}{r_{i}}.$ (23)
The eigenvalues of the reduced matrices for 3, 4, 5, and 6 qutrits are
presented in the table below
$\begin{array}[]{ccc}N\setminus r_{i}&r_{1}=r_{2}&r_{3}\\\ &&\\\
3&\frac{29-4\cos 5Jt}{75}&\frac{17+8\cos 5Jt}{75}\\\ 4&\frac{905-98\cos
3Jt-72\cos 7Jt}{2205}&\frac{395+196\cos 3Jt+144\cos 7Jt}{2205}\\\
5&\frac{16919-1944\cos 5Jt-800\cos 9Jt}{42525}&\frac{8687+3888\cos
5Jt+1600\cos 9Jt}{42525}\\\ 6&\frac{21977-1694\cos 3Jt-1936\cos 7Jt-560\cos
11Jt}{53361}&\frac{9407+3388\cos 3Jt+3872\cos 7Jt+1120\cos 11Jt}{53361}.\\\
&&\end{array}$ (24)
The measures $\eta_{3},\;\eta_{4},\;\eta_{5},\;\eta_{6}$ do not depend on the
sign of the exchange constant like the measure $\eta_{2}$.
### 4.3 Entanglement in the bi-quartit
The applied approach for qutrits is translated to qudits. For a spin-3/2
particle or a four-level system, also denoted as a quartit, we take the
Hamiltonian in the space $\mathrm{C}^{4}$ in the basis
$|3/2>=(1,0,0,0),\;|1/2>=(0,1,0,0),\;|-1/2>=(0,0,1,0),\;|-3/2>=(0,0,0,1)$ and
use the matrix representation of a complete set of the Hermitian orthogonal
operators.
We will find the analytical formulae in the bi-quartit and in the 3 quartits
(in bi-pentit, see below) with a pairwise isotropic interaction of the initial
maximally entangled state in a consistent magnetic field at resonance without
taking into account the anisotropy.
The negative eigenvalues of the partly transposed matrix $\varrho^{pt}$ are
equal to $\lambda_{1}=\frac{1}{100}(-13-12\cos
5Jt),\,\lambda_{2}=\lambda_{3}=\lambda_{4}=\lambda_{5}=-\frac{1}{100}\sqrt{409+288\cos
5Jt-72\cos 10Jt},\,\lambda_{6}=\frac{1}{100}(-37+12\cos 5Jt).$ The
entanglement measure in the bi-quartit equals
$m_{VW}^{bi-qrt}=\sum_{i=1}^{6}|\lambda_{i}|.$ (25)
The entanglement between the quartits is described quantitatively with the
measure[21]
$m_{SM}^{bi-qrt}=\frac{\sqrt{1803365+191616\cos 5Jt-35808\cos 10Jt-6912\cos
15Jt+864\cos 20Jt}}{625\sqrt{5}}.$ (26)
Since the quartit reduced matrix eigenvalues equal to
$\lambda_{1}=\lambda_{2}=\frac{1}{100}(13+12\cos
5Jt),\,\lambda_{3}=\lambda_{4}=\frac{1}{100}(37-12\cos 5Jt)$, hence the
measure $\eta_{2}$ reads
$\eta_{2}^{bi-qrt}=-\sum_{i=1}^{4}\lambda_{i}\log_{4}\lambda_{i}.$ (27)
The I-concurrence is equal to
$m_{I}^{bi-qrt}=\frac{1}{25}\sqrt{553+96\cos 5Jt-24\cos 10Jt}.$ (28)
### 4.4 Entanglement in the quartit chain
The eigenvalues of the reduced matrices for 3 quartits are equal to
$r_{1}=r_{2}=0.141+0.068\cos\frac{5Jt}{2}+0.04\cos
8Jt,\,r_{3}=r_{4}=0.359-0.068\cos\frac{5Jt}{2}-0.04\cos 8Jt$. Therefore the
entanglement measure for $3$ quartits is following
$\eta_{3}^{qrt}=-\sum_{i=1}^{4}r_{i}\log_{4}{r_{i}}.$ (29)
### 4.5 Entanglement in the bi-pentit
For the spin-2 particle or a 5-level system, also denoted as a pentit, we take
the Hamiltonian in the space $\mathrm{C}^{5}$ in the basis
$|2>=(1,0,0,0,0),\;|1>=(0,1,0,0,0),\;|0>=(0,0,1,0,0),\;|-1>=(0,0,0,1,0),\;|-2>=(0,0,0,0,1).$
The entanglement between the pentits is described using the measure[21]
$m_{SM}^{bi-pnt}=(0.802+0.106\cos 3Jt-0.019\cos 4Jt+0.242\cos 7Jt-0.098\cos
10Jt-0.088\cos 14Jt+0.067\cos 17Jt-0.014\cos 20Jt)^{1/2}$. The I-concurrence
is determined by the formulae $m_{I}^{bi-pnt}=(0.791+0.114\cos 3Jt-0.018\cos
4Jt-0.005\cos 6Jt+0.230\cos 7Jt-0.079\cos 10Jt-0.079\cos 14Jt+0.060\cos
17Jt-0.015\cos 20Jt)^{1/2}$. We have replaced the exact bulky rational
coefficients by its decimal approximations and the terms less than 0.001 have
removed for inconvenience reduction.
The pentit reduced matrix eigenvalues are equal to
$p_{1}=p_{2}=\frac{1}{6125}(1173-140\cos 3Jt+640\cos 7Jt-448\cos
10Jt),\,p_{3}=p_{4}=\frac{1}{6125}(513+280\cos 3Jt+320\cos 7Jt+112\cos
10Jt),\,p_{5}=\frac{1}{6125}(2753-280\cos 3Jt-1920\cos 7Jt+672\cos 10Jt)$,
hence the measure $\eta_{2}$ reads
$\eta_{2}^{bi-pnt}=-\sum_{i=1}^{5}p_{i}\log_{5}p_{i}.$ (30)
All the measures do not depend on the sign of the exchange constant and the
parameters of the consistent field at zero anisotropy parameters.
Figure 1: Dynamics in the bi-qutrit: in the zero external field at equal
anisotropy constants $Q=0.0250,\,J=-0.1$ (curve 1) and $J=0.1$ (curve 2); in
the consistent field at $J=0.1,Q=0$ the curve 3 shows complete coincidence of
$m_{VW}$ and $m_{SM}$; the curve 4 is the measure $\eta_{2}$; $m_{I}$ is the
curve 5.
Figure 2: Disentanglement of the maximally entangled state in the chain of 2,
3, 4, 5, 6 qutrits with $J=0.1$.
## 5 Numerical results
Although the analytical expressions for the measures in a bi-qutrit
$m_{VW},\;m_{SM}$ are different, but the numerical values are practically
identical. The maximal deviation in the rectangle $(1\geq J\geq
0.01)\times(100\geq t\geq 0)$ equals $0.014$.
Measures $\eta_{2}$ and $m_{I}$ qualitatively coincide with the measures
$m_{VW},\;m_{SM}$.
We have found that the anisotropy of the qutrits disentangles them, namely the
entanglement is decreased down to 0.001 (see graphs 1 and 2 in Fig.2).
In the constant longitudinal field
$\overrightarrow{h}=-\overrightarrow{\overline{h}}=(0,\,0,\,\omega_{0})$ (the
bi-qutrit Hamiltonian eigenvalues are equal to
$J,J,x_{1},x_{2},x_{3},-p,-p,p,p$, where $x_{1},x_{2},x_{3}$ are the roots of
the equation
${x^{3}+2x^{2}J-p^{2}x-2J^{3}=0,\,p=\sqrt{J^{2}+\omega_{0}^{2}}})$ the
Hamiltonian contains the asymmetric part, thus it follows that the density
matrix for the initial symmetric state will not be symmetric because of the
breaking of the symmetry of the particle permutations. The analytical solution
is cumbersome. In the constant longitudinal impulse field
$\overrightarrow{h}=-\overrightarrow{\overline{h}}=(0,\,0,\,2\,(\theta((t-17)(t-60))+\theta((40-t)(57-t)(t-60))))$
the entanglement dynamics is blocked[9] at $\omega_{0}\gg J$ . This points to
the possibility to control the entanglement.
In Fig.2 we present the comparative dynamics of the entropy measure in the
finite qutrit chain. The disentanglement dynamics of the measures
$\eta_{3},\eta_{4},\eta_{5},\eta_{6}$ is similar to the one in the case of two
qutrits, but with smaller oscillation amplitude, i.e. larger number of the
qutrits disentangles less than two qutrits ($0.889\leq\eta_{3}\leq 1$).
The measures in bi-quartit, as shown in Fig.4, qualitative coincide, almost
completely $\eta_{2}^{bi-qrt}$ and $m_{VW}^{bi-qrt}$. The disentanglement 3
quartits is insignificant less than in bi-quartit.
The disentanglement measures in bi-pentit, as shown in Fig.4, qualitative
coincide, almost completely $m_{I}^{bi-pnt}$ and $m_{SM}^{bi-pnt}$.
Figure 3: Disentanglement of the maximally entangled state in the chain of 2,
3 quartits with $J=0.1$. The measures $m_{I}^{bi-qrt}$, $m_{SM}^{bi-qrt}$,
$\eta_{2}^{bi-qrt}$, $m_{VW}^{bi-qrt}$, are presented by the curves 1, 2, 3, 4
respectively; $\eta_{3}^{qrt}$ is the curve 5.
Figure 4: Disentanglement of the maximally entangled state in the chain of 2
pentits with $J=0.1$. The measures $m_{I}^{bi-pnt}$, $m_{SM}^{bi-pnt}$,
$\eta_{2}^{bi-pnt}$, are presented by the curves 1, 2, 3 respectively.
## 6 Conclusion
The comparative analysis of the bi-qutrit entanglement measures on the base of
the analytical solution for the density matrix demonstrates that, in spite of
the different approaches to the derivation of the formulae for the
entanglement, all the formulae yield quite close results (Fig. 2), and the
measures $m_{VW}\ $and$\;m_{SM}$ are practically equal. This is in accordance
with the general results for the entanglement in the systems with a
permutational symmetry[22].
The analytical formulae for the measures $\eta_{3},\eta_{4},\eta_{5},\eta_{6}$
are similar to the measure for two qutrits $\eta_{2}$, but with a numerically
smaller oscillation amplitude, i.e. the larger number of the qutrits
disentangles fewer than two qutrits.
Nevertheless, the comparison of measures in two coupled qutrits, quartits, and
pentits
$\begin{array}[]{cc}bi-qutrit&m_{VW}\cong m_{SM}\\\ bi-quartit&\eta_{2}^{bi-
qrt}\cong m_{VW}^{bi-qrt}\\\ bi-pentit&m_{SM}^{bi-pnt}\cong m_{I}^{bi-pnt}\\\
\end{array}$ (31)
on the base of analytical solutions shows the absence of a full coincidence of
the measures even in a particular case of disentangling a maximally entangled
state. In other words, it is impossible to prefer any measure, there remains
therefore the question concerning the quantitative determination of
entanglement even in case of two multi-level particles.
The author is grateful to A. A. Zippa constant invaluable support.
## References
* [1] F. T. Hioe, Phys. Rev. A 28 (1983) 879.
* [2] F. T. Hioe and J. H. Eberly, Phys. Rev. Lett. 47 (1981) 838.
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* [4] V. E. Zobov et al., JETP Lett. 87 (2008) 334.
* [5] M. O. Scully et al., Science 299 (2003) 862.
* [6] L. Derkacz and L. Jakobczyk Phys. Rev. A 74 (2006) 032313.
* [7] A. Mazhar, Distillability sudden death in qutrit-qutrit systems under global decoherence, quant-ph/0911.0767v1.
* [8] D. Kaszlikowski, et al., Phys. Rev. Lett. 85 (2000) 4418.
* [9] E. A. Ivanchenko, Qutrit: entanglement dynamics in the finite qutrit chain in the consistent magnetic field, quant-ph/1106.2297v1.
* [10] R. A. Bertmann and P. Krammer, Bloch vectors for qudits and geometry of entanglement, quant-ph/0706.1743v1.
* [11] P. Allard and T. Hard, Journal of Mag. Resonance 153 (2001) 15.
* [12] E. A. Ivanchenko J. Math. Phys. 50 (2009) 042704.
* [13] J. N. Elgin Phys. Lett. A 80 (1980) 140.
* [14] M. Abramovitz and I. A. Stegun (ed.), Handbook of Mathematical Functions (Dover, New York, 1964).
* [15] I. I. Rabi, Phys. Rev. 51(1937) 652.
* [16] A. Bambini and P. R. Berman, Phys. Rev. A 23 (1981) 2496.
* [17] J. B. Miller, et al., Journal of Mag. resonance 151 (2001) 228.
* [18] M. Grifoni and P. Hanggi, Driven quantum tunneling, Physics Reports 304 (1998) 229.
* [19] M. R. Nath, et al., Pramana-Journal of Physics 61 (2003) 1089.
* [20] G. Vidal and R. F. Werner, Phys. Rev. A 65 (2002) 32314.
* [21] J. Schlienz and G. Mahler, Phys. Rev. A 52 (1995) 4396.
* [22] G. Toth and O. Gühne, Phys. Rev. Lett. 102 (2009) 170503.
* [23] F. Pan, et al., Int. J. Theor. Phys. 43 (2004) 1241.
* [24] F. Mintert, A. R. R. Carvalho, M. Kus and A. Buchleitner, Measures and dynamics of entangled states. Physics Reports 415 (2005) 207.
* [25] S. X. Zhang, et al., Commun. Theor. Phys. (Beijing, China) 50 (2008) 883\.
* [26] V. Vedral and B. Plenio, Phys. Rev. A 57 (1998) 1619.
* [27] R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Rev. Mod. Phys. 81 (2009) 865; Quantum entanglement, quant-ph/0702225v2.
|
arxiv-papers
| 2013-02-01T08:22:53 |
2024-09-04T02:49:41.150571
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "E. A. Ivanchenko",
"submitter": "Eugene Ivanchenko",
"url": "https://arxiv.org/abs/1302.0105"
}
|
1302.0121
|
An _∞-structure on the cohomology ring of the symmetric group with coefficients in
Stephan Schmid
Let $p$ be a prime. Let $\fp\!\Sp$ be the group algebra of the symmetric group over the finite field $\fp$ with $|\fp|=p$. Let $\fp$ be the trivial $\fp\!\Sp$-module. We present a projective resolution $\pres \fp$ of the module $\fp$ and equip the Yoneda algebra $\Ext^*_{\fp\!\Sp}(\fp,\fp)$ with an $\A_∞$-structure such that $\Ext^*_{\fp\!\Sp}(\fp,\fp)$ becomes a minimal model of the dg-algebra $\Hom^*_{\fp\!\Sp}(\pres \fp, \pres \fp)$.
[0]MSC 2010: 18G15.
§.§ Introduction
Let $R$ be a commutative ring. Let $A$ be a $\Z$-graded $R$-module. Let $m_1:A→A$ be a graded map of degree $1$ with $m_1^2=0$, i.e. a differential on $A$. Let $m_2:A\otimes A→A$ be a graded map of degree $0$ satisfying the Leibniz rule, i.e.
\begin{align*}
m_1\circ m_2 = m_2\circ(m_1\otimes 1 + 1\otimes m_1).
\end{align*}
The map $m_2$ is in general not required to be associative. Instead, we require that for a morphism $m_3:A^{\otimes 3}→A$, the following identity holds.
\[m_2\circ (m_2\otimes 1 - 1\otimes m_2) = m_1\circ m_3 + m_3\circ(m_1\otimes 1^{\otimes 2} + 1\otimes m_1 \otimes 1 + 1^{\otimes 2} \otimes m_1)\]
Following Stasheff, cf. [23], this can be continued in a certain way with higher multiplication maps to obtain a tuple of graded maps $(m_n:A^{\otimes n}→ A)_{n\geq 1}$ of certain degrees satisfying the Stasheff identities, cf. e.g. (<ref>). The tuple $(A, (m_n)_{n\geq 1})$ is then called an $\A_∞$-algebra.
A morphism of $\A_∞$-algebras from $(A',(m'_n)_{n\geq 1})$ to $(A,(m_n)_{n\geq 1})$ is a tuple of graded maps of certain degrees satisfying the identities (<ref>). The first two of these are
\begin{align*}
\eqref{finfrel}[1]: && f_1 \circ m'_1 \eqs & m_1 \circ f_1\\
\eqref{finfrel}[2]: && f_1\circ m'_2 - f_2\circ(m'_1\otimes 1 + 1\otimes m'_1) \eqs& m_1\circ f_2 + m_2\circ (f_1\otimes f_1).
\end{align*}
So a morphism $f=(f_n)_{n\geq 1}$ of $\A_∞$-algebras from $(A',(m'_n)_{n \geq 1})$ to $(A,(m_n)_{n \geq 1})$ contains a morphism of complexes $f_1:(A',m_1')→(A,m_1)$. We say that $f$ is a quasi-isomorphism of $\A_∞$-algebras if $f_1$ is a quasi-isomorphism. Furthermore, there is a concept of homotopy for $\A_∞$-morphisms, cf. e.g. <cit.> and <cit.>.
The history of $\A_∞$-algebras is outlined in [12] and [13].
As already mentioned, Stasheff introduced $\A_∞$-algebras in 1963.
If $R$ is a field, $\mathbb{F}:=R$, we have the following basic results on $\A_∞$-algebras, which are known since the early 1980s.
* Each quasi-isomorphism of $\A_∞$-algebras is a homotopy equivalence, cf. [21], [10], …
* The minimality theorem: Each $\A_∞$-algebra $(A,(m_n)_{n\geq 1})$ is quasi-isomorphic to an $\A_∞$-algebra $(A',\{m'_n\}_{n \geq 1})$ with $m'_1=0$, cf. [9], [8], [21], [5], [7], [18], … . The $\A_∞$-algebra $A'$ is then called a minimal model of $A$.
Suppose given an $\mathbb{F}$-algebra $B$ and suppose given an $B$-module $M$ together with a projective resolution $\pres M$ of $M$.
The homology of the dg-algebra $\Hom^*_B(\pres M,\pres M)$ is the Yoneda algebra $\Ext^*_B(M,M)$. By the minimality theorem, it is possible to construct an $\A_∞$-structure on $\Ext^*_B(M,M)$ such that $\Ext^*_B(M,M)$ becomes a minimal model of the dg-algebra $\Hom^*_B(\pres M,\pres M)$.
For the purpose of this introduction, we will call such an $\A_∞$-structure on $\Ext^*_B(M,M)$ the canonical $\A_∞$-structure on $\Ext^*_B(M,M)$, which is unique up to isomorphisms of $\A_∞$-algebras, cf. <cit.>.
This structure has been calculated or partially calculated in several cases.
Let $p$ be a prime.
For an arbitrary field $\mathbb{F}$, Madsen computed the canonical $\A_∞$-structure on $\Ext^*_{\mathbb{F}[α]/(α^n)}(\mathbb{F},\mathbb{F})$, where $\mathbb{F}$ is the trivial $\mathbb{F}[α]/(α^n)$-module, cf. <cit.>. This can be used to compute the canonical $\A_∞$-structure on the group cohomology $\Ext^*_{\fp {\rm C}_{m}}(\fp,\fp)$, where $m∈\Z_{\geq 1}$ and ${\rm C}_m$ is the cyclic group of order $m$, cf. <cit.>.
Vejdemo-Johansson developed algorithms for the computation of minimal models [24]. He applied these algorithms to compute large enough parts of the canonical $\A_∞$-structures of the group cohomologies $\Ext^*_{\fs {\rm D}_8}(\fs,\fs)$ and $\Ext^*_{\fs {\rm D}_{16}}(\fs,\fs)$ to distinguish them, where ${\rm D}_8$ and ${\rm D}_{16}$ denote dihedral groups. He stated a conjecture on the complete $\A_∞$-structure on $\Ext^*_{\fs {\rm D}_8}(\fs,\fs)$. Furthermore, he computed parts of the canonical $\A_∞$-structure on $\Ext^*_{\fs {\rm Q}_8}(\fs, \fs)$ for the quaternion group ${\rm Q}_8$. He conjecturally stated the minimal complexity of such a structure. Based on this work, there are now built-in algorithms for the Magma computer algebra system. These are capable of computing partial $\A_∞$-structures on the group cohomology of $p$-groups.
In [25], Vejdemo-Johansson examined the canonical $\A_∞$-structure $(m_n)_{n\geq 1}$ on the group cohomology $\Ext^*_{\fp({\rm C}_k\times {\rm C}_l)}(\fp, \fp)$ of the abelian group ${\rm C}_k\times {\rm C}_l$ for $k,l\geq 4$ such that $k,l$ are multiples of $p$. He showed that for infinitely many $n∈\Z_{\geq 1}$, the operation $m_n$ is non-zero.
In [14], Klamt investigated canonical $\A_∞$-structures in the context of the representation theory of Lie-algebras. In particular, given certain direct sums $M$ of parabolic Verma modules, she examined the canonical $\A_∞$-structure $(m'_k)_{k\geq 1}$ on $\Ext^*_{\mathcal{O}^{\mathfrak{p}}}(M, M)$. She proved upper bounds for the maximal $k∈\Z_{\geq 1}$ such that $m'_k$ is non-vanishing and computed the complete $\A_∞$-structure in certain cases.
The result
For $n∈\Z_{\geq 1}$, we denote by $\Sy_n$ the symmetric group on $n$ elements.
The group cohomology $\Ext^*_{\fp\!\Sp}(\fp,\fp)$ is well-known. For example, in <cit.>, it is calculated using group cohomological methods.
Here, we will construct the canonical $\A_∞$-structure on $\Ext^*_{\fp\!\Sp}(\fp, \fp)$.
We obtain homogeneous elements $ι,χ∈\Hom_{\fp\!\Sp}^*(\pres \fp, \pres \fp)=:A$ of degree $|ι|=2(p-1)=: l$ and $|χ|=l-1$ such that $ι^j,χ\circ ι^j=:χι^j$ are cycles for all $j∈\Z_{\geq 0}$ and such that their set of homology classes $\{\overline{ι^j} \mid j∈\Z_{\geq 0}\}\sqcup \{\overline{χι^j} \mid j∈\Z_{\geq 0}\}$ is an $\fp$-basis of $\Ext^*_{\fp\!\Sp}(\fp, \fp)=\Hm^*A$, cf. <ref>.
For all primes $p$, the canonical $\A_∞$-structure $(m'_n:(\Hm^*A)^{\otimes n}→\Hm^*A)_{n\geq 1}$ on $\Hm^*A$ is given as follows.
On the elements $\overline{χ^{a_1}ι^{j_1}}\otimes \cdots \otimes \overline{χ^{a_n}ι^{j_n}}$, $n∈\Z_{\geq 1}$, $a_i∈\{0,1\}$ and $j_i∈\Z_{\geq 0}$ for $i∈\{1,…,n\}$, the maps $m'_n$ are given as follows, cf. <ref>.
If there is an $i∈\{1,…,n\}$ such that $a_i=0$, then
\begin{align*}
m'_n(\overline{χ^{a_1}ι^{j_1}}\otimes \cdots \otimes \overline{χ^{a_n}ι^{j_n}}) \eqs& 0 \hphantom{\overline{χ^{a_1 + a_1}ι^{j_1 + j_2}}} \text{for $n\neq 2$ and}\\
m'_2(\overline{χ^{a_1}ι^{j_1}}\otimes \overline{χ^{a_2}ι^{j_2}}) \eqs & \overline{χ^{a_1 + a_1}ι^{j_1 + j_2}}.% & \text{ and}\\
\end{align*}
If all $a_i$ equal $1$, then
\begin{align*}
m'_n(\overline{χι^{j_1}}\otimes \cdots \otimes \overline{χι^{j_n}}) \eqs&0 \hphantom{(-1)^p\overline{ι^{p-1+j_1+… + j_p}}}\text{for $n\neq p$ and }\\
m'_p(\overline{χι^{j_1}}\otimes \cdots \otimes \overline{χι^{j_p}}) \eqs &%\begin{cases}
(-1)^p\overline{ι^{p-1+j_1+… + j_p}}. %&\text{ and}\\ %&&\text{ if $n=p$} \\
\end{align*}
In particular, we have $m'_n=0$ for all $n∈ \Z_{\geq 1}\setminus\{2,p\}$.
§.§ Outline
Section 1
The goal of <ref> is to obtain a projective resolution of
the trivial $\fp\!\Sp$-Specht module $\fp$. A well-known method for that is "Walking around the Brauer tree", cf. [4].
Instead, we use locally integral methods to obtain a projective resolution in an explicit and straightforward manner.
Over $ℚ$, the Specht modules are absolutely simple. Therefore we have a morphism of $\zp$-algebras
$r:\zp\!\Sp→\prod_{λ\dashv p} \End_{\zp} S^λ_{\zp}=:Γ$ induced by the operation of the elements of $\zp\!\Sp$ on the Specht modules $S^λ$ for partitions $λ$ of $p$, which becomes an Wedderburn isomorphism when tensoring with $ℚ$. So $Γ$ is a product of matrix rings over $\zp$.
There is a well-known description of $\im r=:Λ$,
which we use for $p\geq 3$ to obtain projective $Λ$-modules $\tilde P_k\subseteq Λ$, $k ∈[1,p-1]$, and to construct the indecomposible projective resolution $\pres\zp$ of the trivial $\zp\!\Sp$-Specht module $\zp$. The non-zero parts of $\pres\zp$ are periodic with period length $l=2(p-1)$. In <ref>, we reduce $\pres\zp$ modulo $p$ to obtain a projective resolution $\pres\fp$ of the trivial $\fp\!\Sp$-Specht module $\fp$.
Section 2
The goal of <ref> is to compute a minimal model of the dg-algebra $\Hom_{\fp\!\Sp}^*(\pres\fp, \pres\fp)=: A$ by equipping its homology $\Ext^*_{\fp\!\Sp}(\fp,\fp) = \Hm^*A$ with a suitable $\A_∞$-structure and finding a quasi-isomorphism of $\A_∞$-algebras from $\Hm^* A$ to $A$.
Towards that end, we recall the basic definitions concerning $\A_∞$-algebras and some general results in <ref>.
While there does not seem to be a substantial difference between the cases $p=2$ and $p\geq 3$, we separate them to simplify notation and argumentation.
Consider the case $p\geq 3$. In <ref>, we obtain a set of cycles $\{ι^j \mid j∈\Z_{\geq 0}\}\cup\{χι^j \mid j∈\Z_{\geq 0}\}$ in $A$ such that their homology classes are a graded basis of $\Hm^*A$.
In <ref>, we obtain a suitable $\A_∞$-structure on $\Hm^*A$ and a quasi-isomorphism of $\A_∞$-algebras from $\Hm^*A$ to $A$. For the prime $2$, both steps are combined in the short <ref>.
§.§ Notations and conventions
* For the remainder of this document, $p$ will be a prime with $p\geq 3$.
* Write $l:=2(p-1)$. This will give the period length of the constructed projective resolution of $\fp$ over $\fp\!\Sp$, cf. e.g. (<ref>), <ref>.
* Concerning "$∞$", we assume the set $\Z\cup\{∞\}$ to be ordered in such a way that $∞$ is greater than any integer, i.e. $∞> z$ for all $z∈\Z$, and that the integers are ordered as usual.
* For $a∈\Z$, $b∈\Z\cup \{∞\}$, we denote by $[a,b] := \{z∈\Z \mid a\leq z\leq b\} \subseteq \Z$ the integral interval. In particular, we have $[a,∞] = \{z∈\Z \mid z\geq a\}\subseteq \Z$ for $a∈\Z$.
* For $n∈\Z_{\geq 0}$, $k∈\Z$, let the binomial coefficient $\binom{n}{k}$ be defined by the number of subsets of the set $\{1,…,n\}$ that have cardinality $k$. In particular, if $k<0$ or $k>n$, we have $\binom{n}{k}= 0$. Then the formula $\binom{n}{k-1} + \binom{n}{k} = \binom{n+1}{k}$ holds for all $k∈\Z$.
* For a commutative ring $R$, an $R$-module $M$ and $a,b∈M$, $c∈R$, we write
\begin{align*} b&\equiv_c a &:\Longleftrightarrow & & a-b ∈ cM.\end{align*}
Often we have $M=R$ as module over itself.
* Modules are right-modules unless otherwise specified.
* For sets, we denote by $\sqcup$ the disjoint union of sets.
* $|\cdot|$: For a homogeneous element $x$ of a graded module or a graded map $g$ between graded modules, we denote by $|x|$ resp. $|g|$ their degrees (This is not unique for $x=0$ resp. $g=0$). For $y$ a real number, $|y|$ denotes its absolute value.
Symmetric Groups
Let $n∈\Z_{\geq 1}$. We denote the symmetric group von $n$ elements by $\Sy_n$.
For a partition $λ\dashv n$, we denote the corresponding Specht module by $S^λ$.
Complexes Let $R$ be a commutative ring and $B$ an $R$-algebra.
* For a complex of $B$-modules
\[\cdots \rightarrow C_{k+1}\xrightarrow{d_{k+1}}C_k\xrightarrow{d_k} C_{k-1}\rightarrow\cdots\hphantom{,},\]
its $k$-th boundaries, cycles and homology groups are defined by $\text{B}^k:=\im d_{k+1}$, $\text{Z}^k:=\ker d_{k}$ and $\Hm^k:= \text{Z}^k/\text{B}^k$.
For a cycle $x∈\text{Z}^k$, we denote by $\overline{x}:=x+\text{B}^k∈\Hm^k$ its equivalence class in homology.
* Let
\begin{align*}
C\eqs&(\cdots \rightarrow C_{k+1}\xrightarrow{d_{k+1}}C_k\xrightarrow{d_k} C_{k-1}\rightarrow\cdots)\\
C'\eqs&(\cdots \rightarrow C'_{k+1}\xrightarrow{d'_{k+1}}C'_k\xrightarrow{d'_k} C'_{k-1}\rightarrow\cdots)
\end{align*}
be two complexes of $B$-modules.
Given $z∈\Z$, let
\begin{align*}
\Hom_B^z(C,C') := \prod_{i∈\Z} \Hom_B(C_{i+z},C'_i).
\end{align*}
For an additional complex $C''=(\cdots \rightarrow C''_{k+1}\xrightarrow{d''_{k+1}}C''_k\xrightarrow{d''_k} C''_{k-1}\rightarrow\cdots)$ and maps $h=(h_i)_{i∈\Z}∈\Hom_B^m(C,C')$, $h'=(h'_i)_{i∈\Z}∈\Hom_B^n(C',C'')$, $m,n∈\Z$, we define the composition by component-wise composition as
\begin{align*}
h'\circ h := (h'_{i}\circ h_{i+n})_{i∈\Z} ∈ \Hom_B^{m+n}(C,C'').
\end{align*}
We will assemble elements of $\Hom_B^z(C,C')$ as sums of their non-zero components, which motivates the following notations regarding "extensions by zero" and sums.
For a map $g:C_x→C'_y$ , we define $\ls g\rs_x^y∈\Hom_B^{x-y}(C,C')$ by
\begin{align*}
(\ls g\rs_x^y)_{i} := \begin{cases} g & \text{ for }i=y\\ 0 & \text{ for } i∈\Z\setminus \{y\} \end{cases}\, .
\end{align*}
Let $k∈\Z$. Let $I$ be a (possibly infinite) set. Let $g_i = (g_{i,j})_j∈\Hom_B^k(C,C')$ for $i∈I$ such that $\{i∈I \mid g_{i,j}\neq 0\}$ is finite for all $j∈\Z$.
We define the sum $\sum_{i∈I} g_i∈\Hom_B^k(C,C')$ by
\begin{align*}
\left(\dum_{i∈I} g_i\right)_j := \sum_{i∈I,g_{i,j}\neq 0} g_{i,j}\,.
\end{align*}
The graded $R$-module $\Hom_B^*(C,C') := \bigoplus_{k∈\Z} \Hom_B^k(C,C')$ becomes a complex via the differential $d_{\Hom_B^*(C,C')}$, which is defined on elements $g∈\Hom_B^k(C,C')$, $k∈\Z$ by
\begin{align*}
d_{\Hom_B^*(C,C')}(g) := d' \circ g -(-1)^k g\circ d ∈\Hom_B^{k+1}(C,C'),
\end{align*}
where $d := (d_{i+1})_{i∈\Z}= \sum_{i∈\Z}\ls d_{i+1}\rs^{i}_{i+1}∈\Hom_B^1(C,C)$ and analogously $d' := (d'_{i+1})_{i∈\Z}= \sum_{i∈\Z}\ls d'_{i+1}\rs^{i}_{i+1}∈\Hom_B^1(C',C')$.
An element $h∈\Hom_B^0(C,C')$ is called a complex morphism if it satisfies $d_{\Hom_B^*(C,C')}(h) = 0$, i.e. $d'\circ g = g\circ d$.
§ THE PROJECTIVE RESOLUTION OF FP OVER FPSP
§.§ A description of ZpSp
In this paragraph, we review results found e.g. in <cit.>. We use the notation of [6].
Let $n∈\Z_{\geq 1}$.
A partition of the form $λ^k := (n-k+1,1^{k-1})$, $k∈[1,n]$ is called a hook partition of $n$.
Suppose $λ\dashv n$, i.e. $λ$ is a partition of $n$.
Let $S^λ$ be the corresponding integral Specht module, which is a right $\Z\!\Sy_n$-module, cf. <cit.>.
Then $S^{λ}$ is finitely generated free over $\Z$, cf. <cit.>, having a standard $\Z$-basis consisting of the standard $λ$-polytabloids. We write $n_λ$ for the rank of $S^λ$.
For a tuple $b=(b_2,b_3,…,b_k)$, $k∈[1,n]$, of pairwise distinct elements of $[1,n]$, let $\lpt b\rpt$ be the $λ^k$-polytabloid generated by the $λ^k$-tabloid
execute at end cell=
[line cap=rect,#1] (--.north east) – (--.south east);
execute at end cell=
[line cap=rect,#1] (--.south west) – (--.south east);
execute at end cell=
[line cap=rect,#1] (--.north west) – (--.north east);
inner xsep=-0.2mm, inner ysep=0.4mm,
space/.style=minimum height=1.0em,matrix of nodes,row sep=0.2mm,column sep=1mm]
(m) [matrix of nodes,
outer sep=0pt,
row 2/.style=bottomrule, row 3/.style=bottomrule, row 4/.style=bottomrule, row 5/.style=bottomrule
* ⋯ *
[decorate,thick] (m-1-1.north west) – (m-1-3.north east);
[decorate,thick] (m-1-1.south west) – (m-1-3.south east);
where $*\cdots *$ are the elements of $[1,n]\setminus b$.
Any polytabloid of $S^{λ^k}$ can be expressed this way.
For such a tuple $b$ and distinct elements $y_1,…,y_s∈[1,n]\setminus b$, we denote by $(b,y_1,…,y_s)$ the tuple $(b_2,b_3,…,b_k,y_1,…,y_s)$.
Recall the notations for manipulation of tuples from <ref>.
The $λ^k$-polytabloid $\lpt b\rpt$ is standard iff $2\leq b_2 < b_3 < \cdots < b_k \leq n$, cf. <cit.>. This entails the following lemma.
For $k∈[1,n]$, the rank of $S^{λ^k}$ is given by $n_{λ^k}=\binom{n-1}{k-1}$.
Let $k∈[1,n-1]$. We have the $\Z$-linear box shift morphisms for hooks
\[\begin{array}{rcl}
S^{λ^k} & \overset{f_k}{\longrightarrow} & S^{λ^{k+1}}\\
\lpt b \rpt & \longmapsto & \sum_{s∈[2,n]\setminus b} \lpt (b,s)\rpt.
\end{array}\]
For $x∈S^{λ^k}$ and $ρ∈\Sy_n$, we have
\begin{align}
\label{modn}
f_k(x\cdot ρ) \equiv_n f_k(x)\cdot ρ.
\end{align}
I.e. the composite $(S^{λ^k} \xrightarrow{f_k}S^{λ^{k+1}} \xrightarrow{π} S^{λ^{k+1}}/nS^{λ^{k+1}})$, where $π$ is residue class map, is $\Z\!\Sy_n$-linear.
The following sequence of $\Z$-linear maps is exact.
\begin{align*}
0 \rightarrow S^{λ^1} \xrightarrow{f_1} S^{λ^2} \xrightarrow{f_2} \cdots \xrightarrow{f_{n-1}} S^{λ^n} \rightarrow 0
\end{align*}
We show that $\im f_k\subseteq \ker f_{k+1}$ for $k∈[1,n-2]$, i.e. that $f_{k+1}\circ f_k = 0$. Let $\lpt b\rpt ∈ S^{λ^k}$ be a polytabloid. We obtain
\begin{align*}
f_{k+1}f_k(\lpt b\rpt) \eqs& f_{k+1}\left( \sum_{s∈[2,n]\setminus b} \lpt (b,s)\rpt\right)
%= \sum_{s∈[2,n]\setminus b}\sum_{t∈[2,n]\setminus (b\cup \{s\})} \lpt ((b,s),t)\rpt \\
= \sum_{\substack{s,t∈ [2,n]\setminus b,\\s\neq t}} \lpt (b,s,t)\rpt\\
\eqs& \sum_{\substack{s,t ∈[2,n]\setminus b,\\s<t}} \big(\lpt (b,s,t)\rpt + \lpt(b,t,s)\rpt\big) \overset{\text{cf. \cite[4.3]{Ja78}}}{=} 0.
\end{align*}
Now we show the exactness of the sequence. For convenience, we set and $f_n\colon S^{λ^n}→0$. We define $T^k$ for $k∈[1,n]$ to be the tuple of all tuples $b=(b_2,…,b_k)$ such that $2\leq b_2<b_3 < … < b_k\leq n-1$, where $T^k$ is ordered, say, lexicographically.
Then we set $B^k_{\rm b} := ( \lpt b \rpt \colon b∈T^k)$, which consists of standard $λ^k$-polytabloids.
We set $B^1_{\rm c} := ()$, which is the empty tuple, and for $k∈[2,n]$,
\begin{align*}
B^k_{\rm c} :\eqs& (f_{k-1}(x) \colon x∈B^{k-1}_{\rm b})\\
\eqs& \left(\sum_{s∈[2,n]\setminus b} \lpt (b,s)\rpt \colon b∈T^{k-1}\right)
= \left(\lpt (b,n)\rpt + \sum_{\mathclap{s∈[2,n-1]\setminus b}} \lpt (b,s)\rpt \colon b∈T^{k-1}\right).
\end{align*}
So $B^k_{\rm c} \subseteq \im f_{k-1}$ and thus $f_k(B^k_{\rm c})\subseteq \{0\}$ for $k∈[1,n]$.
By comparing $B^k_{\rm c} \sqcup B^k_{\rm b}$ with the standard basis, we observe that $B^k_{\rm c}\sqcup B^k_{\rm b}$ is a $\Z$-basis of $S^{λ^k}$ for $k∈[1,n]$.
For $k∈[1,n]$, we have
\begin{align*}
n_{\rm b}^k :\eqs& |B^k_{\rm b}| = \binom{n-2}{k-1}\\
n_{\rm c}^k :\eqs& |B^k_{\rm c}| =\left\{\begin{array}{ll} |B^{k-1}_{\rm b}| = \binom{n-2}{k-2} & \text{for }k∈[2,n] \\ 0 = \binom{n-2}{1-2} & \text{for }k=1 \end{array}\right\}
= \binom{n-2}{k-2}.
\end{align*}
For $k∈[1,n-1]$, the morphism $f_k$ maps $\langle B^k_{\rm b}\rangle_{\Z}$ bijectively to $\langle B^{k+1}_{\rm c}\rangle_{\Z}$ and $\langle B^{k}_{\rm c}\rangle_{\Z}$ to zero. So $\ker f_k = \langle B^{k}_{\rm c}\rangle_{\Z}$ and $\im f_k = \langle B^{k+1}_{\rm c}\rangle_{\Z}$. As $B^1_{\rm c} = () = B^n_{\rm b}$, we have also $\im f_0 = \langle B^1_{\rm c}\rangle_{\Z}$ and $\ker f_n = \langle B^n_{\rm c}\rangle_{\Z}$. So the sequence in question is exact.
We equip the Specht modules $S^{λ^k}$ of hook type with the ordered $\Z$-basis $B^k_{\rm c}\sqcup B^k_{\rm b}$. We equip all other Specht modules with the standard $\Z$-basis with an arbitrarily chosen total order. From now on each of these bases will be referred to as the basis of the respective Specht module. We define the $\Z$-algebra
\begin{align*}
Γ^{\Z} := \prod_{λ\dashv n} \Z^{n_λ\times n_λ}.
\end{align*}
Let $λ\dashv n$ and let $B = (b_1,…,b_{n_λ})$ be the basis of $S^{λ}$.
For the multiplication with matrices, we identify $S^λ$ with $\Z^{1\times n_λ}$ via $B$.
Then $S^λ$ becomes a right $Γ^{\Z}$-module via $x\cdot ρ := x\cdot ρ^λ$ for $x∈S^λ$ and $ρ∈Γ^{\Z}$, where $ρ^λ$ is the $λ$-th component of $ρ$.
I.e. $ρ∈Γ^{\Z}$ operates by multiplication with the matrix $ρ^λ$ on the right with respect to the basis $B$.
Similarly, $\bigoplus_{λ\dashv n} S^λ$ becomes a right $Γ^{\Z}$-module. Each $\Z$-endomorphism of $\bigoplus_{λ\dashv n} S^λ$ is represented by the operation of a unique element of $Γ^{\Z}$. As the operation of $\Z\!\Sy_n$ defines such endomorphisms (cf. <cit.>), we obtain a $\Z$-algebra morphism $r^{\Z}:\Z\!\Sy_n → Γ^{\Z}$
such that $y\cdot r^{\Z}(x) = y\cdot x$ for all $λ\dashv n$, $y∈S^λ$, $x∈\Z\Sy_n$.
As the Specht modules give all irreducible ordinary representations of $\Sy_n$, the map $r^{\Z}$ is injective. Because of (<ref>), the image of $r^{\Z}$ is contained in
\begin{align*}
Λ^{\Z} := \{ρ∈Γ^{\Z} \mid f_k(xρ) \equiv_n f_k(x)ρ\, ∀_{k∈[1,n-1]}\, ∀_{x∈S^{λ^k}}\} \subseteq Γ^{\Z}.
\end{align*}
As the basis $B_{\rm c}^k\sqcup B_{\rm b}^k$ of $S^{λ^k}$, $k∈[1,n]$, consists of two parts, we may split each $ρ^{λ^k}$ for a $ρ∈Γ^{\Z}$ into four blocks corresponding to the parts $B^k_{\rm c}$ and $B^k_{\rm b}$:
\begin{equation}
\label{eq:blockmotivation}
ρ^{λ^k} =
\xyprot{
\begin{tikzpicture}[decoration=brace,baseline]
\matrix (m) [matrix of math nodes,left delimiter=(,right delimiter=),
nodes={%rectangle,minimum width=3em,
outer sep=0pt},
row 1/.style={bottomrule},column 1/.style=rightrule] {
ρ^{λ^k}_{\rm cc}\vphantom{ρ^{λ^k}_{\rm bc}} & ρ^{λ^k}_{\rm bc} \\
ρ^{λ^k}_{\rm cb} & ρ^{λ^k}_{\rm bb} \\
\draw[decorate,transform canvas={xshift=1.5em},thick] (m-1-2.north east) -- node[right=2pt] {$n^k_{\rm c}$} (m-1-2.south east);
\draw[decorate,transform canvas={xshift=1.5em},thick] (m-2-2.north east) -- node[right=2pt] {$n^k_{\rm b}$} (m-2-2.south east);
\draw[decorate,transform canvas={yshift=0.5em},thick] (m-1-1.north west) -- node[above=2pt] {$n^k_{\rm c}$} (m-1-1.north east);
\draw[decorate,transform canvas={yshift=0.5em},thick] (m-1-2.north west) -- node[above=2pt] {$n^k_{\rm b}$} (m-1-2.north east);
\end{tikzpicture}
\end{equation}
Suppose given $k∈[1,n-1]$. We represent $f_k$ by a matrix $M_{f_k}$ with respect to the bases of $S^{λ^k}$ and $S^{λ^{k+1}}$, i.e. $f_k(x) = x\cdot M_{f_k}$ for $x∈S^{λ^k}$. As $f_k(B_{\rm b}^k) = B_{\rm c}^{k+1}$ and $f_k(B_{\rm c}^k)\subseteq \{0\}$, the matrix $M_{f_k}$ has the following block form:
\begin{equation*}
M_{f_k} =
\xyprot{
\begin{tikzpicture}[decoration=brace,baseline]
\matrix (m) [matrix of math nodes,left delimiter=(,right delimiter=),
nodes={%minimum width=\widthof{\tikz \node{ABCD};},%rectangle,minimum width=3em,
outer sep=0pt},
row 1/.style={bottomrule},column 1/.style=rightrule] {
{\hphantom{E_{n^k_{\rm b}}}\llap{0}} & 0 \\
E_{n^k_{\rm b}} & 0\vphantom{E_{n^k_{\rm b}}} \\
\draw[decorate,transform canvas={xshift=1.5em},thick] (m-1-2.north east) -- node[right=2pt] {$n^k_{\rm c}$} (m-1-2.south east);
\draw[decorate,transform canvas={xshift=1.5em},thick] (m-2-2.north east) -- node[right=2pt] {$n^k_{\rm b}$} (m-2-2.south east);
\draw[decorate,transform canvas={yshift=0.5em},thick] (m-1-1.north west) -- node[above=2pt] {$n^{k+1}_{\rm c}$} (m-1-1.north east);
\draw[decorate,transform canvas={yshift=0.5em},thick] (m-1-2.north west) -- node[above=2pt] {\rlap{$n^{k+1}_{\rm b}$}\hphantom{$n$}} (m-1-2.north east);
\end{tikzpicture}
\end{equation*}
Here $E_i$ is the $i\times i$-identity matrix for $i∈\Z_{\geq 1}$.
So for $x∈S^{λ^k}$, $ρ∈Γ^{\Z}$ we have
\begin{align*}
f_k(x)\cdot ρ \eqs & x\cdot M_{f_k} \cdot ρ^{λ^{k+1}}
x\cdot \left(\begin{array}{c|c} 0 & 0 \\ \hline E_{n^k_{\rm b}}& 0\end{array}\right)
\cdot \left(\begin{array}{c|c} ρ^{λ^{k+1}}_{\rm cc} & ρ^{λ^{k+1}}_{\rm bc} \\ \hline ρ^{λ^{k+1}}_{\rm cb} \ru{5} & ρ^{λ^{k+1}}_{\rm bb} \end{array}\right)
x\cdot \left(\begin{array}{c|c} 0&0 \\ \hline \ru{5} ρ^{λ^{k+1}}_{\rm cc} & ρ^{λ^{k+1}}_{\rm bc} \end{array}\right)\\
f_k(x\cdot ρ) \eqs & x \cdot ρ^{λ^{k}} \cdot M_{f_k} =
x\cdot \left(\begin{array}{c|c} ρ^{λ^k}_{\rm cc} & ρ^{λ^k}_{\rm bc} \\ \hline ρ^{λ^k}_{\rm cb} \ru{5} & ρ^{λ^k}_{\rm bb} \end{array}\right)
\cdot \left(\begin{array}{c|c} 0 & 0 \\ \hline E_{n^k_{\rm b}}& 0\end{array}\right)
= x\cdot \left(\begin{array}{c|c} ρ^{λ^k}_{\rm bc} & 0 \\ \hline \ru{5} ρ^{λ^k}_{\rm bb} & 0 \end{array}\right).
\end{align*}
This way we have $f_k(x\cdot ρ)\equiv_n f_k(x)\cdot ρ$ for all $x∈S^{λ^k}$ if and only if , $ρ_{\rm bc}^{λ^k} \equiv_n 0$ and $ρ_{\rm bc}^{λ^{k+1}} \equiv_n 0$. So
\begin{align}
\label{lambdazex}
Λ^{\Z} = \{ρ∈Γ^{\Z} \mid (ρ^{λ^k}_{\rm bb} \equiv_n ρ^{λ^{k+1}}_{\rm cc}\text{ for } k∈[1,n-1])\text{ and } (ρ^{λ^k}_{\rm bc} \equiv_n 0\text{ for } k∈[1,n])\}.
\end{align}
We have (cf. e.g. <cit.>)
\[|Γ^{\Z}/Λ^{\Z}|=n^{\frac{1}{2}\sum_{k∈[1,n]} \binom{n-1}{k-1}^2},\]
which is proven by counting the congruences in (<ref>):
\begin{align*}
|Γ^{\Z}/Λ^{\Z}| \eqs& n^{\sum_{k=1}^{n-1} (n_{\rm b}^k)^2 + \sum_{k=1}^{n} n_{\rm b}^k\cdot n_{\rm c}^k}\\
\ovs{n_{\rm b}^n=0}& n^{\sum_{k∈[1,n]} ((n_{\rm b}^k)^2 + n_{\rm b}^k\cdot n_{\rm c}^k)} = n^{\sum_{k∈[1,n]} n_{\rm b}^k(n_{\rm b}^k + n_{\rm c}^k)} \\
\sum_{k∈[1,n]} n_{\rm b}^k(n_{\rm c}^k + n_{\rm b}^k)\eqs& \sum_{k∈[1,n]} \tbinom{n-2}{k-1}\left(\tbinom{n-2}{k-2}+\tbinom{n-2}{k-1}\right)\\
\eqs& \frac{1}{2}\sum_{k∈[1,n]} \left(\tbinom{n-2}{k-1}\tbinom{n-2}{k-2}+\tbinom{n-2}{k-1}^2\right)
+ \frac{1}{2}\sum_{k∈[1,n]} \left(\tbinom{n-2}{k-1}\tbinom{n-2}{k-2}+\tbinom{n-2}{k-2}^2\right)\\
\eqs& \frac{1}{2}\sum_{k∈[1,n]} \tbinom{n-2}{k-1}\left(\tbinom{n-2}{k-2}+\tbinom{n-2}{k-1}\right)
\eqs&\frac{1}{2}\sum_{k∈[1,n]} \tbinom{n-2}{k-1}\tbinom{n-1}{k-1} +\tbinom{n-2}{k-2}\tbinom{n-1}{k-1} =\frac{1}{2}\sum_{k∈[1,n]} \tbinom{n-1}{k-1}^2\\
\end{align*}
Recall that $p\geq 3$ is a prime. Let $n=p$.
We have the commutative diagram of $\Z$-modules
\begin{align}
\begin{aligned}
\label{diagrz}
\xyprot{
\xymatrix@!C{
\Z \Sp\ar@{^{(}->}[d]^{r^{\Z}}\ar@{^{(}->}[r]^{ι^{\Z}\circ r^{\Z}} & Γ^{\Z}\ar@{->>}[r] \ar[d]^{\rotatebox{90}{=}} & Γ^{\Z}/(ι^{\Z}\circ r^{\Z}(\Z \Sp)) \ar@{->>}[d]^{s^{\Z}} \\
Λ^{\Z} \ar@{^{(}->}[r]^{ι^{\Z}} & Γ^{\Z} \ar@{->>}[r] & Γ^{\Z}/Λ^{\Z}
\end{aligned}
\end{align}
The map $ι^{\Z}$ is the inclusion of $Λ^{\Z}$ in $Γ^{\Z}$.
The maps from $Γ^{\Z}$ to $Γ^{\Z}/(ι^{\Z}\circ r^{\Z}(\Z \Sp))$ and to $Γ^{\Z}/Λ^{\Z}$ are the residue class maps. As $r^{\Z}(\Z \Sp)\subseteq Λ^{\Z}$, we have an unique surjective map $s^{\Z}:Γ^{\Z}/(ι^{\Z}\circ r^{\Z}(\Z \Sp))→ Γ^{\Z}/Λ^{\Z}$ such that the right rectangle is commutative. By construction, the rows of the diagram are short exact sequences.
Note that the morphisms of the left rectangle are in fact $\Z$-algebra morphisms.
We will need the following result on the localization of rings.
Let $A$ be a commutative ring. Let $P\subseteq R$ a prime ideal of $A$. Let $A_P$ be the localization of $A$ at $P$. Then
$A_P$ is a flat $A$-module, that is, the functor $\funcblc\underset{A}{\otimes} \leftidx{_A}{(A_P)}{_{A_P}}$ from the category of $A$-modules to the category of $A_P$-modules is exact.
We denote by $\zp$ the localization of $\Z$ at the prime ideal $(p):=p\Z$. We apply the functor $\funcblc\underset{\Z}{\otimes } \zp$ to obtain a commutative diagram (<ref>) of the following form:
\begin{align}
\label{diagr}
\begin{aligned}
\xyprot{
\xymatrix@!C{
\zp\!\Sp\ar@{^{(}->}[d]^{r}\ar@{^{(}->}[r]^{ι \circ r} & Γ\ar@{->>}[r] \ar[d]^{\rotatebox{90}{=}} & Γ/(ι\circ r(\zp\!\Sp)) \ar@{->>}[d]^{s} \\
Λ \ar@{^{(}->}[r]^{ι} & Γ \ar@{->>}[r] & Γ/Λ
\end{aligned}
\end{align}
By <ref>, the functor $\funcblc\underset{\Z}{\otimes } \zp$ is exact, so the short exact sequences are mapped to short exact sequences, monomorphisms to monomorphisms and epimorphisms to epimorphisms. So the rows of diagram (<ref>) are exact and we have mono-/epimorphism as indicated by the arrows. We identify $\Z \Sp \underset{\Z}{\otimes } \zp$ with $\zp\!\Sp$. We identify $Γ^{\Z}\underset{\Z}{\otimes } \zp$ with
\begin{align*}
Γ := \prod_{λ\dashv n} \zp^{n_λ\times n_λ}.
\end{align*}
The map $ι$ realizes $Λ:=Λ^{\Z}\underset{\Z}{\otimes } \zp$ as the following subset of $Γ$, for which we will use notation analogous to (<ref>):
\begin{align*}
Λ = \{ρ∈Γ \mid (ρ^{λ^k}_{\rm bb} \equiv_p ρ^{λ^{k+1}}_{\rm cc}\text{ for } k∈[1,p-1])\text{ and } (ρ^{λ^k}_{\rm bc} \equiv_p 0 \text{ for } k∈[1,p])\}
\end{align*}
As the rows are exact, we identify $(Γ^{\Z}/(ι^{\Z}\circ r^{\Z}(\Z \Sp)) )\underset{\Z}{\otimes } \zp$ with $Γ/(ι\circ r(\zp\!\Sp)$ and $(Γ^{\Z}/Λ^{\Z})\underset{\Z}{\otimes } \zp$ with $Γ/Λ$.
By the classification of finitely generated $\Z$-modules, each finite $\Z$-module $M$ is isomorphic to a finite direct sum of modules of the form $\Z/q^a \Z$, where $q$ is a prime and $a\in \Z_{\geq 0}$. If $q\neq p$ then $(\Z/q^a \Z) \underset{\Z}{\otimes } \zp \cong (0)$. Otherwise $(\Z/p^a \Z) \underset{\Z}{\otimes } \zp \cong \zp/p^a\zp$ and $|(\Z/p^a \Z) \underset{\Z}{\otimes } \zp| = p^a = |\Z/p^a\Z|$.
For $x=p^{a_p} \cdot \prod_{\natop{q \text{ prime}}{q\neq p}} q^{a_q} ∈\Z_{\geq 1}$, we set
\[(x)_p := p^{a_p}.\]
So for finite $M$, we have $|M\underset{\Z}{\otimes } \zp| = (|M|)_p$.
By the total index formula (cf. e.g. <cit.>), we have
\begin{align*}
|Γ^{\Z}/(ι^{\Z}\circ r^{\Z}(\Z \Sp))| = \sqrt{\frac{p!^{p!}}{\prod_{λ \dashv p} n_λ^{n_λ^2}}}\,.
\end{align*}
By the hook formula (cf. <cit.>, <cit.>), we have for $λ\dashv p$
\begin{align*}
(n_λ)_p = \begin{cases} 1 & \text{if $λ$ is a hook-partition} \\ p & \text{ otherwise} \end{cases}\,.
\end{align*}
\begin{align*}
|Γ/(i\circ r(\zp\!\Sp))| \eqs& \left(\sqrt{\frac{p!^{p!}}{\prod_{λ \dashv p} n_λ^{n_λ^2}}}\right)_{\mathllap{p}}
= \sqrt{\frac{p^{p!}}{\prod_{\natop{λ \dashv p}{λ\text{ not a hook}}} (n_λ)_p^{n_λ^2}}}\\
\eqs& \sqrt{\frac{\prod_{λ\dashv p} p^{n_λ^2}}{\prod_{\natop{λ \dashv p}{λ\text{ not a hook}}} p^{n_λ^2}}}
= \sqrt{\prod_{k∈[1,n]} p^{n_{λ^k}^2}} = p^{\frac{1}{2}\sum_{k∈[1,n]}\binom{p-1}{k-1}^2}\\
\eqs& |Γ^{\Z}/Λ^{\Z}| = (|Γ^{\Z}/Λ^{\Z}|)_p= |Γ/Λ|.
\end{align*}
By the pigeon-hole-principle, $s$ is an isomorphism as it is surjective. As (<ref>) has exact rows, $r$ needs to be an isomorphism as well. Note that the functor $\funcblc\underset{\Z}{\otimes } \zp$ transforms morphisms of $\Z$-algebras into morphisms of $\zp$-algebras. In particular, the left rectangle in (<ref>) consists of morphisms of $\zp$-algebras and $r:\zp\!\Sp→Λ$ is an isomorphism of $\zp$-algebras. We have proven the
Recall that $p\geq 3$ is a prime.
For $R$ a ring and $λ\dashv p$ a partition of $p$, the $R\!\Sp$-Specht module $S^λ$ is finitely generatey free over $R$ with dimension independent of $R$, cf. <cit.>. We denote this dimension by $n_λ$.
A partition of the form $λ^k := (p-k+1,1^{k-1})$, $k∈[1,p]$ is called a hook partition of $p$.
Over the valuation ring $\zp$, there is a well-known description of the group algebra $\zp\!\Sp$, cf. e.g. <cit.> (using [19]), cf. also <cit.>:
$n^{k}_{\rm b} = \binom{p-2}{k-1}$, $n^{k}_{\rm c} = \binom{p-2}{k-2}$. Then $n^{k}_{\rm b} + n^{k}_{\rm c} = \binom{p-1}{k-1} = n_{λ^k}$. Set
$ Γ := \prod_{λ \dashv p} \zp^{n_λ\times n_λ} $.
For $ρ∈Γ$, and $λ\dashv p$, we denote by $ρ^λ$ the $λ$-th component of $ρ$. For $λ=λ^k$, $k∈[1,p]$, a hook partition, we name certain subblocks of $ρ^{λ^k}$ as follows.
\begin{equation*}
ρ^{λ^k} =
\xyprot{
\begin{tikzpicture}[decoration=brace,baseline]
\matrix (m) [matrix of math nodes,left delimiter=(,right delimiter=),
nodes={%rectangle,minimum width=3em,
outer sep=0pt},
row 1/.style={bottomrule},column 1/.style=rightrule] {
ρ^{λ^k}_{\rm cc} & ρ^{λ^k}_{\rm bc} \\
ρ^{λ^k}_{\rm cb} & ρ^{λ^k}_{\rm bb} \\
\draw[decorate,transform canvas={xshift=1.5em},thick] (m-1-2.north east) -- node[right=2pt] {$n^k_{\rm c}$ rows} (m-1-2.south east);
\draw[decorate,transform canvas={xshift=1.5em},thick] (m-2-2.north east) -- node[right=2pt] (ru) {$n^k_{\rm b}$ rows} (m-2-2.south east);
\draw[decorate,transform canvas={yshift=0.5em},thick] (m-1-1.north west) -- node[above=2pt] {$n^k_{\rm c}$,} (m-1-1.north east);
\draw[decorate,transform canvas={yshift=0.5em},thick] (m-1-2.north west) -- node[above=2pt] {$n^k_{\rm b}$ \rlap{columns}} (m-1-2.north east);
\node[transform canvas={xshift=1.5em}] at (ru.south east) {.};
\end{tikzpicture}
\end{equation*}
We have the following $\zp$-subalgebra $Λ$ of $Γ$.
\begin{align*}
% \end{align*}
% given by
% \begin{align*}
Λ &:= \{ρ∈Γ \mid ρ^{λ^k}_{\rm bb} \equiv_p ρ^{λ^{k+1}}_{\rm cc}\text{ for }k∈[1,p-1]\text{ and } ρ^{λ^k}_{\rm bc} \equiv_p 0\text{ for }k∈[1,p]\}
\end{align*}
Now there is an isomorphism of $\zp$-algebras
\[\begin{array}{rccc}
r:&\zp\!\Sp & \xrightarrow{\phantom{X}\sim\phantom{X}}& Λ.
\end{array}
\]
such that $ρ∈Λ$ acts on the trivial $\zp\!\Sp$-module $\zp$ by multiplication with its ($1\times 1$ / scalar-)component $ρ^{λ^1}$, i.e. for $x∈\zp\!\Sp$ and for $y∈\zp$, we have $yx = y\cdot r(x)^{λ^1}$.
FIXME: Dieses Bsp. rausmachen.
For $p=3$, the ring $\Z_{(3)}\Sy_3$ is isomorphic to the subring $Λ$ of $Γ=\Z_{(3)}^{1\times 1} \times \Z_{(3)}^{2\times 2} \times \Z_{(3)}^{1\times 1}$
described as
[inner xsep=0.1mm, inner ysep=0.0mm,
space/.style=minimum height=1.0em,matrix of nodes,row sep=0.0mm,column sep=1mm,
(m1) [space, below delimiter=}0$\begin{matrix*}[l]\scriptstyle{\times}\\ \scriptstyle\times \\ \scriptstyle{\times}\end{matrix*}$]
(m2) [right=of m1, space, below delimiter=}0$\begin{matrix*}[l]\scriptstyle\times \\ \scriptstyle{\times}{\times}\end{matrix*}$]
_(3) 3_(3)
_(3) _(3)
(m3) [right=of m2,space, below delimiter=}0$\begin{matrix*}[l] \scriptstyle{\times}{\times}{\times}\end{matrix*}$]
[draw=black, fit=(m1-1-1)](B12) ;
[draw=black, fit=(m2-1-1)](B21) ;
[draw=black, fit=(m2-2-2)](B22) ;
[draw=black, fit=(m3-1-1) ](B31) ;
[black] (B12.east) – node[sloped,above]0.5mm$\scriptscriptstyle 3$ (B21.west);
[black] (B22.east) – node[sloped,above]0.5mm$\scriptscriptstyle 3$ (B31.west);
An entry in this tuple of matrices indicates that an element of $Λ$ must have its corresponding entry in the indicated set. A relation " (n1) ; (n2) [right=of n1] ;
[black] (n1.east) – node[sloped,above]$\scriptscriptstyle p$ (n2.west);
" between (equal sized) subblocks indicates that these subblocks are equivalent modulo $p$, i.e. the difference of corresponding entries is an element of $p\Z_{(p)}$. The blocks are labeled with the diagrams of the corresponding partitions.
Alternatively, $Λ$ is the $\Z_{(3)}$-span of
\begin{align*}
\left( 3, \begin{pmatrix} 0 & 0 \\ 0& 0\end{pmatrix}, 0 \right)=:u, &&
\left( 1, \begin{pmatrix} 1 & 0 \\ 0& 0\end{pmatrix}, 0 \right)=:\tilde e_1, &&
\left( 0, \begin{pmatrix} 0 & 3 \\ 0& 0\end{pmatrix}, 0 \right)=: v_1,\\
\left( 0, \begin{pmatrix} 0 & 0 \\ 1& 0\end{pmatrix}, 0 \right)=: v_2,&&
\left( 0, \begin{pmatrix} 0 & 0 \\ 0& 1\end{pmatrix}, 1 \right)=:\tilde e_2, &&
\left( 0, \begin{pmatrix} 0 & 0 \\ 0& 3\end{pmatrix}, 0\right)=: w.
\end{align*}
We have an orthogonal decomposition $1=\tilde e_1 + \tilde e_2$ into primitive idempotents. Thus we have a decomposition $Λ=\tilde P_1 \oplus \tilde P_2$
into indecomposable projective right modules, where
\begin{align*}
\tilde P_1 :\eqs& \tilde e_1 Λ = \langle u, \tilde e_1, v_1\rangle_{\Z_{(3)}}, &
\tilde P_2 :\eqs& \tilde e_2 Λ = \langle v_2, \tilde e_2, w \rangle_{\Z_{(3)}}.
\end{align*}
A more pictoral description of $\tilde P_i$, $i∈[1,2]$ is that the elements of $\tilde P_i⊂Λ$ are exactly the elements of $Λ$ whose non-zero entries are all in the box in FIXME.
In this case all partitions of $3$ are of hook-type. Thus there appear no full matrix algebras as direct factors of $Λ$.
For $p=5$, the $\zp$-algebra $\Z_{(5)}\Sy_5$ is isomorphic to the subalgebra $Λ$ of $Γ=\Z_{(5)}^{1\times 1}\times \Z_{(5)}^{4\times 4}\times \Z_{(5)}^{6\times 6}\times \Z_{(5)}^{4\times 4}\times \Z_{(5)}^{1\times 1}\times \Z_{(5)}^{5\times 5}\times \Z_{(5)}^{5\times 5}$ described as
[inner xsep=0.1mm, inner ysep=0.0mm,node distance=4mm,
space/.style=minimum height=1.0em,matrix of nodes,row sep=1.0mm,column sep=1mm,
(m1) [space, below delimiter=}0$\begin{matrix*}[l]\scriptstyle{\times}\\ \scriptstyle\times \\ \scriptstyle\times \\ \scriptstyle\times \\ \scriptstyle{\times}\end{matrix*}$]
(m2) [right=of m1, space, below delimiter=}0$\begin{matrix*}[l]\scriptstyle\times \\ \scriptstyle\times \\ \scriptstyle\times \\ \scriptstyle{\times}{\times}\end{matrix*}$]
_(5) 5_(5) 5_(5) 5_(5)
_(5) _(5) _(5) _(5)
_(5) _(5) _(5) _(5)
_(5) _(5) _(5) _(5)
(m3) [right=of m2, space, below delimiter=}0$\begin{matrix*}[l]\scriptstyle\times \\ \scriptstyle\times \\ \scriptstyle{\times}{\times}{\times}\end{matrix*}$]
_(5) _(5) _(5) 5_(5) 5_(5) 5_(5)
_(5) _(5) _(5) 5_(5) 5_(5) 5_(5)
_(5) _(5) _(5) 5_(5) 5_(5) 5_(5)
_(5) _(5) _(5) _(5) _(5) _(5)
_(5) _(5) _(5) _(5) _(5) _(5)
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(m5) [right=of m4,space, below delimiter=}0$\begin{matrix*}[l] \scriptstyle{\times}{\times}{\times}{\times}{\times}\end{matrix*}$]
[draw=black, fit=(m1-1-1)](B12) ;
[draw=black, fit=(m2-1-1)](B21) ;
[draw=black, fit=(m2-2-2.north west) (m2-4-4.south east)](B22) ;
[draw=black, fit=(m3-1-1.north west) (m3-3-3.south east)](B31) ;
[draw=black, fit=(m3-4-4.north west) (m3-6-6.south east)](B32) ;
[draw=black, fit=(m4-1-1.north west) (m4-3-3.south east)](B41) ;
[draw=black, fit=(m4-4-4)] (B42) ;
[draw=black, fit=(m5-1-1)] (B51) ;
[black] (B12.east) – node[sloped,above,pos=0.6]0.5mm$\scriptscriptstyle 5$ (B21.west);
[black] (B22.east) – node[sloped,above,pos=0.6]0.5mm$\scriptscriptstyle 5$ (B31.west);
[black] (B32.east) – node[sloped,above,pos=0.6]0.5mm$\scriptscriptstyle 5$ (B41.west);
[black] (B42.east) – node[sloped,above,pos=0.6]0.5mm$\scriptscriptstyle 5$ (B51.west);
[left=] (C011) at (m1-1-1.south west) ;
[below=] (C11) at (C011) ;
[below=] (C12) at (m1-1-1.south east) ;
[below=] (C13) at (m2-1-1.south west) ;
[below=] (C14) at (m2-1-4.south east) ;
[above=] (C15) at (m2-1-4.north east) ;
[above=] (C16) at (m2-1-1.north west) ;
[above=] (C17) at (m1-1-1.north east) ;
[left=] (C018) at (m1-1-1.north west) ;
[above=] (C18) at (C018) ;
[red,rounded corners=2pt] (C11.center) – (C12.center) – (C13.center) – (C14.center) – (C15.center) –
node[above]1.5mm$\tilde P_1$ (C16.center) – (C17.center) – (C18.center) – cycle;
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[below=] (C22) at (m2-2-4.south east) ;
[below=] (C23) at (m3-1-1.south west) ;
[below=] (C24) at (m3-1-6.south east) ;
[above=] (C25) at (m3-1-6.north east) ;
[above=] (C26) at (m3-1-1.north west) ;
[above=] (C27) at (m2-2-4.north east) ;
[above=] (C28) at (m2-2-1.north west) ;
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[below=] (C32) at (m3-4-6.south east) ;
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[below=] (C34) at (m4-1-4.south east) ;
[above=] (C35) at (m4-1-4.north east) ;
[above=] (C36) at (m4-1-1.north west) ;
[above=] (C37) at (m3-4-6.north east) ;
[above=] (C38) at (m3-4-1.north west) ;
[red,rounded corners=2pt] (C31.center) – (C32.center) – (C33.center) – (C34.center) – (C35.center) – node[above]1.5mm$\tilde P_3$ (C36.center) – (C37.center) – (C38.center) – cycle;
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[below=] (C43) at (m5-1-1.south west) ;
[right=](C044) at (m5-1-1.south east) ;
[below=] (C44) at (C044) ;
[right=](C045) at (m5-1-1.north east) ;
[above=] (C45) at (C045) ;
[above=] (C46) at (m5-1-1.north west) ;
[above=] (C47) at (m4-4-4.north east) ;
[above=] (C48) at (m4-4-1.north west) ;
[red,rounded corners=2pt] (C41.center) – (C42.center) – (C43.center) – (C44.center) – (C45.center) –
node[above]1.5mm$\tilde P_4$ (C46.center) – (C47.center) – (C48.center) – cycle;
[inner xsep=0.1mm, inner ysep=0.0mm,node distance=4mm,
space/.style=minimum height=1.0em,matrix of nodes,row sep=0.0mm,column sep=1mm,
(mx) [space, below delimiter=}0$\begin{matrix*}[l] \scriptstyle{\times}{\times} \\ \scriptstyle{\times}{\times}{\times} \end{matrix*}$]
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(my) [space,right=of mx, below delimiter=}0$\begin{matrix*}[l]\scriptstyle{\times}\\ \scriptstyle{\times}{\times} \\ \scriptstyle{\times}{\times} \end{matrix*}$]
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An entry in this tuple of matrices indicates that an element of $Λ$ must have its corresponding entry in the indicated set. A relation " (n1) ; (n2) [right=of n1] ;
[black] (n1.east) – node[sloped,above]$\scriptscriptstyle 5$ (n2.west);
" between (equal sized) subblocks indicates that these subblocks are equivalent modulo $5$, i.e. the difference of corresponding entries is an element of $5\Z_{(5)}$. The blocks are labeled with the diagrams of the corresponding partitions.
The right ideals $\tilde P_i = \tilde e_iΛ$, $i∈[1,4]=[1,p-1]$ (cf. the definitions below) are framed with red lines.
§.§ A projective resolution of Zp over ZpSp
Recall that $p\geq 3$ is a prime.
Recall from <ref> that $Λ$ is a subring of $Γ=\prod_{λ\dashv p} \zp^{n_λ\times n_λ}$.
For $λ\dashv p$ and $i,j∈[1,n_λ]$, we set $η_{λ,i,j}$ to be the element of $Γ$ such that $(η_{λ,i,j})^{\tilde{λ}} = 0$ for $\tilde{λ}\neq λ$ and $(η_{λ,i,j})^{λ}∈\Z^{n_λ\times n_λ}$ has entry $1$ at position $(i,j)$ and zeros elsewhere.
Let $k∈[1,p-1]$. We obtain the idempotent
\begin{align*}
\tilde e_k := η_{λ^k,n_{\rm c}^{k}+1,n_{\rm c}^{k}+1} + η_{λ^{k+1},1,1} ∈ Λ.
\end{align*}
We define corresponding projective right $Λ$-modules
\begin{align*}
\tilde P_k :\eqs& \tilde e_k Λ \quad \text{ for $k∈[1,p-1]$.}
\end{align*}
Let $A$ be an $R$-algebra and let $e,e'∈A$ be two idempotents. For the right modules $eA$, $e'A$, we have the isomorphism of $R$-Modules
\[ \begin{array}{rcl}
\Hom_A(eA, e'A) &\underset{\sim}{\overset{T_{e',e}}{\longrightarrow}}& e'Ae\\
f &\longmapsto& T_{e',e}(f) := f(e)\\
T_{e',e}^{-1}(e'be):=(ea \mapsto e'bea) &\reflectbox{$\longmapsto$}&
e'be \phantom{:= f(e)X}.
\end{array}\]
Thus given $m∈e'Ae$, the morphism $T_{e',e}^{-1}(m)$ acts on elements $x∈eA$ by the multiplication of $m$ on the left: $(T_{e',e}^{-1}(m))(x) = m\cdot x$.
Given idempotents $e,e',e''∈A$, and elements $f∈\Hom_A(eA,e'A)$, $g∈A(e'A,e''A)$, we have $T_{e'',e}(g\circ f) = g(f(e)) = g(e'f(e)) = g(e')\cdot f(e)= T_{e'',e'}(g)\cdot T_{e',e}(f)$.
We define via <ref>
\begin{equation*}
\begin{array}{lclll}
\hat e_{k} &:\eqs& T_{\tilde e_k, \tilde e_k}^{-1}(\tilde e_k) &∈\Hom_Λ(\tilde P_k,\tilde P_k) & \text{for }k∈[1,p-1]\\
\hat e_{1,1} &:\eqs& T_{\tilde e_1,\tilde e_1}^{-1}(pη_{λ^1,1,1}) &∈\Hom_Λ(\tilde P_1,\tilde P_1) \\
\hat e_{p-1,p-1}&:\eqs& T_{\tilde e_{p-1},\tilde e_{p-1}}^{-1}(pη_{λ^p,1,1}) &∈\Hom_Λ(\tilde P_{p-1},\tilde P_{p-1})\\
\hat e_{k+1,k} &:\eqs& T_{\tilde e_{k+1},\tilde e_k}^{-1}(η_{λ^{k+1},n_{\rm c}^{k+1}+1,1}) &∈\Hom_Λ(\tilde P_k,\tilde P_{k+1}) & \text{for }k∈[1,p-2]\\
\hat e_{k,k+1} &:\eqs& T_{\tilde e_k,\tilde e_{k+1}}^{-1}(pη_{λ^{k+1},1,n_{\rm c}^{k+1}+1}) &∈\Hom_Λ(\tilde P_{k+1},\tilde P_k) & \text{for }k∈[1,p-2].
\end{array}
\end{equation*}
Note that $\hat e_k$ is the identity map on $\tilde P_k$ for $k∈[1,p-1]$.
Moreover, we define the $\zp\!\Sp$-linear map $\hat{ε}:\tilde P_1 → \zp$, $\hat{ε}(ρ) := ρ^{λ^1}$.
It is straightforward to show the following lemma.
We have
\begin{equation*}
\begin{array}{lclr}
\hat e_{1,1} + \hat e_{1,2}\circ \hat e_{2,1} &=& p\hat e_1\\
\hat e_{k,k-1}\circ\hat e_{k-1,k} + \hat e_{k,k+1}\circ\hat e_{k+1,k} &=& p\hat e_k &\text{ for }k∈[2,p-2]\\
\hat e_{p-1,p-2}\circ\hat e_{p-2,p-1} + \hat e_{p-1,p-1} &=& p\hat e_{p-1}\\
\hat{ε}\circ \hat e_{1,1} &=& p\hat{ε}.
\end{array}
\end{equation*}
Furthermore, it is straightforward to check that we obtain a projective resolution of $\zp$ as follows.
We set
\begin{align*}
\tilde{\pr}_i :\eqs& \begin{cases} \tilde P_{ω(i)} & i\geq 0 \\ 0 & i<0 \end{cases},
\end{align*}
where the integer $ω(i)$ is given by the following construction: Recall the stipulation $l:=2(p-1)$. We have $i = jl+r$ for some $j∈\Z$ and $0\leq r\leq l-1$. Then
\begin{align}
\label{omega}
ω(i) := \begin{cases} r+1 & \text{ for }0\leq r \leq p-2\\ l-r = 2(p-1) -r & \text{ for } p-1 \leq r \leq 2(p-1)-1 = l-1 \end{cases}.
\end{align}
So $ω(i)$ increases by steps of one from $1$ to $p-1$ as $i$ runs from $jl$ to $jl+(p-2)$ and $ω(i)$ decreases from $p-1$ to $1$ as $i$ runs from $jl+(p-1)$ to $jl + (l-1)$. Finally we set
\begin{align*}
\hat d_i := \begin{cases} \hat e_{ω(i-1),ω(i)}: \tilde P_{ω(i)} → \tilde P_{ω(i-1)} & i\geq 1\\ 0 & i\leq 0 \end{cases}.
\end{align*}
Now we have the projective resolution of $\zp$
\begin{align}
\label{presform1}
\cdots\xrightarrow{\hat d_3} \tilde{\pr}_2\xrightarrow{\hat d_2}\tilde{\pr}_1\xrightarrow{\hat d_1}\tilde{\pr}_0\xrightarrow{0=\hat d_0} 0\rightarrow\cdots\,,
\end{align}
written more explicitly as
\begin{align*}
\cdots\rightarrow &\tilde P_2\xrightarrow{\hat e_{1,2}} \tilde P_1 \xrightarrow{\hat e_{1,1}} \tilde P_1 \xrightarrow{\hat e_{2,1}} \tilde P_2 \rightarrow …
\rightarrow \tilde P_{p-2} \xrightarrow{\hat e_{p-1,p-2}}\tilde P_{p-1} \nonumber \\
&\xrightarrow{\hat e_{p-1,p-1}} \tilde P_{p-1} \xrightarrow{\hat e_{p-2,p-1}}\tilde P_{p-2}\rightarrow … \rightarrow \tilde P_2\xrightarrow{\hat e_{1,2}} \tilde P_1 \rightarrow 0\rightarrow \cdots\,,
\end{align*}
with augmentation $\hat{ε}:\tilde P_1 → \zp$.
Then let
* $\mathscr{B}^{\Leftrightarrow}:=\{β_{k,x,y}^{\Leftrightarrow} \mid k∈[1,p-1], x,y∈[1,n_{\rm b}^{k}]\}$, where $β_{k,x,y}^{\Leftrightarrow} := η_{λ^k,n_{\rm c}^{k}+x,n_{\rm c}^{k}+y} + η_{λ^{k+1},x,y}$.
* $\mathscr{B}^{\Leftarrow}:=\{β_{k,x,y}^{\Leftarrow} \mid k∈[1,p-1], x,y∈[1,n_{\rm b}^{k}]\}$, where .
* $\mathscr{B}^{\Rightarrow}:=\{β_{k,x,y}^{\Rightarrow} \mid k∈[1,p-1], x,y∈[1,n_{\rm b}^{k}]\}$, where .
* $\mathscr{B}^{\leftarrow} := \{β_{k,x,y}^{\leftarrow} \mid k∈[1,p], x∈[1,n_{\rm b}^{k}],y∈[1,n_{\rm c}^{k}]\}$,
* $\mathscr{B}^{\rightarrow} := \{β_{k,x,y}^{\rightarrow} \mid k∈[1,p], x∈[1,n_{\rm c}^{k}],y∈[1,n_{\rm b}^{k}]\}$,
where .
* $\mathscr{B}^*:=\{η_{λ,x,y} \mid λ\dashv p\text{ not a hook partition}, x,y∈[1,n_λ]\}$.
We have two $\zp$-bases $\mathscr{B}^{\Leftrightarrow} \sqcup \mathscr{B}^{\Leftarrow} \sqcup \mathscr{B}^{\leftarrow} \sqcup \mathscr{B}^{\rightarrow} \sqcup \mathscr{B}^*$ and of $Λ$.
[$p=3$, continuation of <ref>]
The only of the $β_{a,b,c}^d$ that are defined above and that are not shown in <ref> are the following elements.
\begin{align*}
\left( 0, \begin{pmatrix} 3 & 0 \\ 0& 0\end{pmatrix}, 0 \right)= β_{1,1,1}^⇒, &&
\left( 0, \begin{pmatrix} 0 & 0 \\ 0& 0\end{pmatrix}, 3 \right)= β_{2,1,1}^⇒
\end{align*}
$\mathscr{B}^*$ is empty since all partitions are hook partitions.
Once more, see <ref> for an illustration of the case $p=3$.
* $\mathscr{B}_k^{\Leftrightarrow} := (β_{k,1,y}^{\Leftrightarrow}\colon y∈(1,…,n_{\rm b}^{k}))
= (η_{λ^k,n_{\rm c}^{k}+1,n_{\rm c}^{k}+y} + η_{λ^{k+1},1,y}\colon y∈(1,…,n_{\rm b}^{k}))$
* $\mathscr{B}_k^{\Leftarrow} := (β_{k,1,y}^{\Leftarrow}\colon y∈(1,…,n_{\rm b}^{k}))
= (pη_{λ^k,n_{\rm c}^{k}+1,n_{\rm c}^{k}+y}\colon y∈(1,…,n_{\rm b}^{k}))$
* $\mathscr{B}_k^{\Rightarrow} := (β_{k,1,y}^{\Rightarrow}\colon y∈(1,…,n_{\rm b}^{k}))
= (pη_{λ^{k+1},1,y} \colon y∈(1,…,n_{\rm b}^{k}))$
* $\mathscr{B}_k^{\leftarrow} := (β_{k,1,y}^{\leftarrow}\colon y∈(1,…,n_{\rm c}^{{k}}))
=(η_{λ^{k},n_{\rm c}^{k}+1,y} \colon y∈(1,…,n_{\rm c}^k))$
* $\mathscr{B}_k^{\rightarrow} := (β_{k+1,1,y}^{\rightarrow}\colon y∈(1,…,n_{\rm b}^{k+1}))
=(pη_{λ^{k+1},1,n_{\rm c}^{k+1}+y}\colon y∈(1,…, n_{\rm b}^{k+1}))$
Similarly to the bases of $Λ$, the tuples $\mathscr{B}_k^{\Leftrightarrow}\sqcup \mathscr{B}_k^{\Leftarrow} \sqcup \mathscr{B}_k^{\leftarrow} \sqcup \mathscr{B}_k^{\rightarrow}$ and are $\zp$-bases of $\tilde P_k$.
Let $k∈[1,p-1]$. The idempotent $\tilde e_k$ is actually a primitive idempotent and thus $\tilde P_k$ is an indecomposable projective $Λ$-right module: Assume $\tilde e_k=c+c'$ for some idempotents $0\neq c,c'∈Λ$ that are orthogonal, that is $c\cdot c' = c'\cdot c = 0$. Then $\tilde e_k \cdot c = (c+c')c = c^2 = c = c(c+c') = c\cdot \tilde e_k$. Similarly, we have $\tilde e_k \cdot c' = c' = c'\cdot \tilde e_k$. Thus $c,c' ∈\tilde e_kΛ\tilde e_k$. The $\zp$-algebra
\[\tilde e_kΛ\tilde e_k = \langle e_k, β_{k,1,1}^⇐\rangle_{\zp} = \langle e_k, β_{k,1,1}^⇒\rangle_{\zp}\]
is isomorphic to the $\zp$-algebra
\begin{equation*}
\begin{tikzpicture}
\node (n1) {$\zp$};
\node (n2) [right=of n1] {$\zp$};
\draw[black] (n1.east) -- node[above]{$\scriptstyle p$} (n2.west);
\node%[transform canvas={xshift=1.5em}]
at (n1.west) {$\mathllap{J:=}$};
\end{tikzpicture}
\end{equation*}
consisting of elements $\{(a,b)∈\zp\times \zp \mid a\equiv_p b\}$. The only idempotents in $\zp\times \zp$ are $(0,0)∈J$, $(1,1)∈J$, $(1,0)\notin J$ and $(0,1)\notin J$. Thus the identity element $(1,1)$ of $J$ cannot be decomposed into non-trivial idempotents and the same holds for $\tilde e_k$.
We have
\begin{align*}
&{\rm(a)}&&\ker \hat e_{k+1,k} = \langle \mathscr{B}_k^{\leftarrow} \sqcup \mathscr{B}_k^{⇐}\rangle_{\zp},&& \im \hat e_{k+1,k} = \langle \mathscr{B}_{k+1}^{\leftarrow} \sqcup \mathscr{B}_{k+1}^{⇐}\rangle_{\zp} \,\, \text{ for $k∈[1,p-2]$},\\
&{\rm(b)}&&\ker \hat e_{k,k+1} = \langle \mathscr{B}_{k+1}^→\sqcup \mathscr{B}_{k+1}^⇒\rangle_{\zp}, &&\im \hat e_{k,k+1} = \langle \mathscr{B}_k^{→}\sqcup \mathscr{B}_k^⇒ \rangle_{\zp}\,\,\text{ for $k∈[1,p-2]$},\\
&{\rm(c)}&&\ker \hat e_{p-1,p-1} {=} \langle \mathscr{B}_{p-1}^{\leftarrow} \sqcup \mathscr{B}_{p-1}^⇐ \rangle_{\zp}, &&\im \hat e_{p-1,p-1} {=} \langle \mathscr{B}_{p-1}^⇒ \sqcup \mathscr{B}_{p-1}^→\rangle_{\zp},\\
&{\rm(d)}&&\ker \hat e_{1,1} = \langle \mathscr{B}_1^⇒ \sqcup \mathscr{B}_1^→ \rangle_{\zp},&& \im \hat e_{1,1} = \langle \mathscr{B}_1^⇐ \sqcup \mathscr{B}_1^{\leftarrow} \rangle_{\zp}.
\end{align*}
\begin{array}[t]{rcl}
\hat e_{k+1,k}(\mathscr{B}_k^{⇔}) &\svs{\text{R.}\ref{rem:moralg}}& (η_{λ^{k+1},n_{\rm c}^{k+1}+1,1} η_{λ^{k+1},1,y} \colon y∈(1,…,n_{\rm b}^{k}))\\
&\sqs & (η_{λ^{k+1},n_{\rm c}^{{k+1}}+1,y} \colon y∈(1,…,n_{\rm c}^{{k+1}})) = \mathscr{B}_{k+1}^{\leftarrow}\\
\hat e_{k+1,k}(\mathscr{B}_k^{→}) &\svs{\text{R.}\ref{rem:moralg}}& (η_{λ^{k+1},n_{\rm c}^{k+1}+1,1} pη_{λ^{k+1},1,n_{\rm c}^{{k+1}}+y} \colon y∈(1,…,n_{\rm b}^{k+1}))\\
&\sqs& (pη_{λ^{k+1},n_{\rm c}^{k+1}+1,n_{\rm c}^{k+1}+y} \colon y∈(1,…,n_{\rm b}^{k+1})) = \mathscr{B}_{k+1}^{⇐}\\
\hat e_{k+1,k}(\mathscr{B}_k^{\leftarrow}) &\svs[\subseteq]{\text{R.}\ref{rem:moralg}} & \{0\}\\
\hat e_{k+1,k}(\mathscr{B}_k^{⇐}) &\svs[\subseteq]{\text{R.}\ref{rem:moralg}} & \{0\}
\end{array}%\\
% \begin{array}{lrclrcl}
% ⇒& \ker \hat e_{k+1,k} &=& \langle \mathscr{B}_k^{\leftarrow} \sqcup \mathscr{B}_k^{⇐}\rangle,& \im \hat e_{k+1,k} &=& \langle \mathscr{B}_{k+1}^{\leftarrow} \sqcup \mathscr{B}_{k+1}^{⇐}\rangle
% \end{array}
Thus by <ref>, assertion (a) holds.
\begin{array}[t]{rcl}
\hat e_{k,k+1}(\mathscr{B}_{k+1}^{⇔}) &\svs{\text{R.}\ref{rem:moralg}}& (pη_{λ^{k+1},1,n_{\rm c}^{k+1}+1} η_{λ^{k+1},n_{\rm c}^{{k+1}}+1,n_{\rm c}^{{k+1}}+y} \colon y∈(1,…,n_{\rm b}^{{k+1}})) \\
&\sqs& (pη_{λ^{k+1},1,n_{\rm c}^{{k+1}}+y} \colon y∈(1,…,n_{\rm b}^{{k+1}})) = \mathscr{B}_k^{→}\\
\hat e_{k,k+1}(\mathscr{B}_{k+1}^{\leftarrow}) &\svs{\text{R.}\ref{rem:moralg}}& (pη_{λ^{k+1},1,n_{\rm c}^{{k+1}}+1} η_{λ^{k+1},n_{\rm c}^{{k+1}}+1,y} \colon y∈(1,…,n_{\rm c}^{{k+1}})) \\
&\sqs& (pη_{λ^{k+1},1,y} \colon y∈(1,…,n_{\rm b}^{k})) = \mathscr{B}_k^⇒\\
\hat e_{k,k+1}(\mathscr{B}_{k+1}^→) &\svs[\subseteq]{\text{R.}\ref{rem:moralg}} & \{0\}\\
\hat e_{k,k+1}(\mathscr{B}_{k+1}^⇒) &\svs[\subseteq]{\text{R.}\ref{rem:moralg}} & \{0\}
\end{array}
Thus by <ref>, assertion (b) holds.
\begin{array}[t]{rcl}
\hat e_{p-1,p-1}(\mathscr{B}_{p-1}^⇔) &\svs{\text{R.}\ref{rem:moralg}}& (pη_{λ^p,1,1}η_{λ^{(p-1)+1},1,y} \colon y∈(1,…,n_{\rm b}^{{p-1}})) \\
&\sqs& (pη_{λ^{(p-1)+1},1,y} \colon y∈(1,…,n_{\rm b}^{{p-1}})) = \mathscr{B}_{p-1}^⇒\\
\mathscr{B}_{p-1}^→ &\sqs& () \text{ as $n_{\rm b}^{p} = 0$}\\
\hat e_{p-1,p-1}(\mathscr{B}_{p-1}^{\leftarrow}) &\svs[\subseteq]{\text{R.}\ref{rem:moralg}} & \{0\}\\
\hat e_{p-1,p-1}(\mathscr{B}_{p-1}^⇐) &\svs[\subseteq]{\text{R.}\ref{rem:moralg}} & \{0\}
\end{array}
Thus by <ref>, assertion (c) holds.
\begin{array}[t]{rcl}
\hat e_{1,1}(\mathscr{B}_1^⇔) &\svs{\text{R.}\ref{rem:moralg}}& (pη_{λ^1,1,1}η_{λ^1,n_{\rm c}^{1}+1,n_{\rm c}^{1}+y} \colon y∈(1,…,n_{\rm b}^{1}))\\
&\svs{n_{\rm c}^{1}=0}& (pη_{λ^1,n_{\rm c}^{1}+1,n_{\rm c}^{1}+y} \colon y∈(1,…,n_{\rm b}^{1})) = \mathscr{B}_1^⇐\\
\mathscr{B}_1^{\leftarrow} &\sqs& ()\text{ as $n_{\rm c}^{1}=0$}\\
\hat e_{1,1}(\mathscr{B}_1^⇒) &\svs[\subseteq]{\text{R.}\ref{rem:moralg}} & \{0\}\\
\hat e_{1,1}(\mathscr{B}_1^→) &\svs[\subseteq]{\text{R.}\ref{rem:moralg}} & \{0\}
\end{array}
Thus by <ref>, assertion (d) holds.
The trivial $\zp\!\Sp$-module $\zp$ becomes a $Λ$-module via the isomorphism of $\zp$-algebras $r:\zp\!\Sp→Λ$ described in <ref>. We want to construct a projective resolution of $\zp$ over $Λ$.
$Γ$ is a right $Λ$-module as $Λ$ is a subalgebra of $Γ$. The set $Γ^{λ^1}:=\{ρ∈Γ \mid ρ^λ = 0 \text{ for }λ\neq λ^1\}$ is a right $Λ$-submodule of $Γ$. As $n_{\rm c}^{k}= 0$ and $n_{\rm b}^k=1$, $Γ^{λ^1}$ is free over $\zp$ with basis $\{η_{λ^1,1,1}\}$.
Given a partition $λ\dashv p$, the operation of an element $x∈\zp\!\Sp$ on the Specht module corresponding to $λ$ is multiplication with the matrix $r(x)^λ$ with respect to a certain basis of that Specht module, cf. the definition of $r^{\Z}$ in the proof of <ref>.
As $\zp$ is the Specht module corresponding to the trivial partition $λ^1$ of $p$, and as $\zp$ is one-dimensional, the operation of $x∈\zp\!\Sp$ on $\zp$ is multiplication with the scalar $r(x)^{λ^1}$. Thus an element $ρ∈Λ$ operates on $\zp$ via multiplication with the scalar $ρ^{λ^1}$ and we haven an isomorphism of right $Λ$-modules by
\begin{align*}
\begin{array}{rlcl}
\hat{ε}^1 : &Γ^{λ^1} & \longrightarrow & \zp\\
& η_{λ^1,1,1} & \longmapsto & 1.
\end{array}
\end{align*}
We have the morphism of right $Λ$-modules
\begin{align*}
\begin{array}{rlclr}
\hat{ε}^0: &\tilde P_1 &\longrightarrow & Γ^{λ^1}\\
&\tilde e_1x &\longmapsto & η_{λ^1,1,1} \tilde e_1x= η_{λ^1,1,1}x & \text{ for }x∈Λ
\end{array}
\end{align*}
We have $\hat{ε}^0(\tilde e_1) = \hat{ε}^0(η_{λ^1,1,1} + η_{λ^2,1,1}) = η_{λ^1,1,1}$, thus
$\hat{ε}^0$ is surjective as
$\{η_{λ^1,1,1}\}$ is a $\zp$-basis of $Γ^{λ^1}$.
Given $x∈\tilde P_1$, we have
$\hat e_{1,1}(x) = p \hat{ε}^0(x)$ as elements of $Γ$. Thus the maps $\hat e_{1,1}$ and $\hat{ε}^0$ have the same kernel. Concatenation with the isomorphism $\hat{ε}^1$ yields the surjective morphism of right $Λ$-modules
\[ \hat{ε} := \hat{ε}^1 \circ \hat{ε}^0 : \tilde P_1 \longrightarrow \zp,\]
for which we have $\ker \hat{ε} = \ker \hat e_{1,1}$.
With these properties of $\hat{ε}$ and <ref>,
we are able to directly formulate a projective resolution of $\zp$:
We set
\begin{align*}
\tilde{\pr}_i :\eqs& \begin{cases} \tilde P_{ω(i)} & i\geq 0 \\ 0 & i<0 \end{cases},
\end{align*}
where the integer $ω(i)$ is given by the following construction: Recall the stipulation $l:=2(p-1)$. We have $i = jl+r$ for some $j∈\Z$ and $0\leq r\leq l-1$. Then
\begin{align}
\label{omega}
ω(i) := \begin{cases} r+1 & \text{ for }0\leq r \leq p-2\\ l-r = 2(p-1) -r & \text{ for } p-1 \leq r \leq 2(p-1)-1 = l-1 \end{cases}.
\end{align}
So $ω(i)$ increases by steps of one from $1$ to $p-1$ as $i$ runs from $jl$ to $jl+(p-2)$ and $ω(i)$ decreases from $p-1$ to $1$ as $i$ runs from $jl+(p-1)$ to $jl + (l-1)$. Finally we set
\begin{align*}
\hat d_i := \begin{cases} \hat e_{ω(i-1),ω(i)}: \tilde P_{ω(i)} → \tilde P_{ω(i-1)} & i\geq 1\\ 0 & i\leq 0 \end{cases}.
\end{align*}
Now <ref> gives the projective resolution of $\zp$
\begin{align}
\label{presform1}
\cdots\xrightarrow{\hat d_3} \tilde{\pr}_2\xrightarrow{\hat d_2}\tilde{\pr}_1\xrightarrow{\hat d_1}\tilde{\pr}_0\xrightarrow{0=\hat d_0} 0\rightarrow\cdots\,.
\end{align}
More explicitly, we have
\begin{align*}
\cdots\rightarrow &\tilde P_2\xrightarrow{\hat e_{1,2}} \tilde P_1 \xrightarrow{\hat e_{1,1}} \tilde P_1 \xrightarrow{\hat e_{2,1}} \tilde P_2 \rightarrow …
\rightarrow \tilde P_{p-2} \xrightarrow{\hat e_{p-1,p-2}}\tilde P_{p-1} \nonumber \\
&\xrightarrow{\hat e_{p-1,p-1}} \tilde P_{p-1} \xrightarrow{\hat e_{p-2,p-1}}\tilde P_{p-2}\rightarrow … \rightarrow \tilde P_2\xrightarrow{\hat e_{1,2}} \tilde P_1 \rightarrow 0\rightarrow \cdots\,.
\end{align*}
The corresponding extended projective resolution is
\begin{align*}
\cdots \rightarrow &\tilde P_2\xrightarrow{\hat e_{1,2}} \tilde P_1 \xrightarrow{\hat e_{1,1}} \tilde P_1 \xrightarrow{\hat e_{2,1}} \tilde P_2 \rightarrow …
\rightarrow \tilde P_{p-2} \xrightarrow{\hat e_{p-1,p-2}}\tilde P_{p-1} \nonumber \\
&\xrightarrow{\hat e_{p-1,p-1}} \tilde P_{p-1} \xrightarrow{\hat e_{p-2,p-1}}\tilde P_{p-2}\rightarrow … \rightarrow \tilde P_2\xrightarrow{\hat e_{1,2}} \tilde P_1 \xrightarrow{\hat{ε}} \zp \rightarrow 0\rightarrow\cdots\,,
\end{align*}
which is an exact sequence.
We have proven the
tm]tm:przp Recall that $p\geq 3$ is a prime.
The sequence (<ref>) is a projective resolution of $\zp$, with augmentation $\tilde{\pr}_0=\tilde P_1 \xrightarrow{\hat{ε}}\zp$.
Recall that $p\geq 3$ is a prime. We have
\begin{equation*}
\begin{array}{lclr}
\hat e_{1,1} + \hat e_{1,2}\circ \hat e_{2,1} &=& p\hat e_1\\
\hat e_{k,k-1}\circ\hat e_{k-1,k} + \hat e_{k,k+1}\circ\hat e_{k+1,k} &=& p\hat e_k &\text{ for }k∈[2,p-2]\\
\hat e_{p-1,p-2}\circ\hat e_{p-2,p-1} + \hat e_{p-1,p-1} &=& p\hat e_{p-1}\\
\hat{ε}\circ \hat e_{1,1} &=& p\hat{ε}.
\end{array}
\end{equation*}
We have by <ref>
\begin{align*}
T_{\tilde e_1,\tilde e_1}(&\hat e_{1,1} + \hat e_{1,2}\circ\hat e_{2,1}) = T_{\tilde e_1,\tilde e_1}(\hat e_{1,1}) + T_{\tilde e_1,\tilde e_2}(\hat e_{1,2})T_{\tilde e_2,\tilde e_1}(\hat e_{2,1})\\
&= pη_{λ^1,1,1} + pη_{λ^2,1,n_{\rm c}^2+1}η_{λ^2,n_{\rm c}^2+1,1} = p(η_{λ^1,1,1}+η_{λ^2,1,1}) = T_{\tilde e_1,\tilde e_1}(p\hat e_{1})\\
T_{\tilde e_k,\tilde e_k}(&\hat e_{k,k-1}\circ\hat e_{k-1,k} + \hat e_{k,k+1}\circ\hat e_{k+1,k})\\
&= T_{\tilde e_k,\tilde e_{k-1}}(\hat e_{k,k-1})T_{\tilde e_{k-1},\tilde e_k}(\hat e_{k-1,k}) + T_{\tilde e_k,\tilde e_{k+1}}(\hat e_{k,k+1})T_{\tilde e_{k+1},\tilde e_k}(\hat e_{k+1,k})\\
&= η_{λ^k,n_{\rm c}^k+1,1}pη_{λ^k,1,n_{\rm c}^k+1} + pη_{λ^{k+1},1,n_{\rm c}^{k+1}+1}η_{λ^{k+1},n_{\rm c}^{k+1}+1,1}\\
&= p(η_{λ^k,n_{\rm c}^k+1,n_{\rm c}^k+1} + η_{λ^{k+1},1,1}) = T_{\tilde e_k,\tilde e_k}(p\hat e_{k})\\
T_{\tilde e_{p-1},\tilde e_{p-1}}(&\hat e_{p-1,p-2}\circ\hat e_{p-2,p-1} + \hat e_{p-1,p-1})\\
&= T_{\tilde e_{p-1},\tilde e_{p-2}}(\hat e_{p-1,p-2})T_{\tilde e_{p-2},\tilde e_{p-1}}(\hat e_{p-2,p-1}) + T_{\tilde e_{p-1},\tilde e_{p-1}}(\hat e_{p-1,p-1})\\
&= η_{λ^{p-1},n_{\rm c}^{p-1}+1,1}pη_{λ^{p-1},1,n_{\rm c}^{p-1}+1} + pη_{λ^p,1,1}\\
&= p(η_{λ^{p-1},n_{\rm c}^{p-1}+1,n_{\rm c}^{p-1}+1} + η_{λ^p,1,1}) = T_{\tilde e_{p-1},\tilde e_{p-1}}(p\hat e_{p-1}).
\end{align*}
Finally for $x∈\tilde P_1$, we have
\begin{align*}
(\hat{ε}^0\circ \hat e_{1,1})(x) \eqs& η_{λ^1,1,1}\cdot pη_{λ^1,1,1}\cdot x = pη_{λ^1,1,1}\cdot x
= p \hat{ε}^0(x),
\end{align*}
thus $\hat{ε}\circ \hat e_{1,1} = \hat{ε}^1\circ \hat{ε}^0\circ \hat e_{1,1} = p\hat{ε}^1\circ \hat{ε}^0 = p\hat{ε}$.
§.§ A projective resolution of Fp over FpSp
We have $\zp\!\Sp/(p\zp\!\Sp) \simeq \fp\!\Sp$. Since $\zp\!\Sp \simeq Λ$, we identify $\fp\!\Sp = Λ/(pΛ)$.
We obtain the desired projective resolution by reducing the projective resolution of $\zp$ "modulo $p$":
§.§.§ Reduction modulo $I$
Let $R$ be a principal ideal domain.
Let $(A,ρ)$ be an $R$-algebra. Let $I$ be an ideal of $R$. We set $\bar{R}:=R/\!I$.
As $R$ is a principal ideal domain, $ρ(I)A$ is an additive subset of $A$. As $ρ(I)$ is a subset of the center of $A$, $ρ(I)A$ is an ideal of $A$ and $A/(ρ(I)A)=:\bar{A}$ is an $\bar{R}$-algebra.
We regard a right $A$-module $M_A$ as a right $R$-module $M_R$ via $m\cdot r := m\cdot ρ(r)$ for $m∈M$, $r∈R$.
The functors $-\underset{A}{\otimes} \bar{A}$ and $-\underset{R}{\otimes} \bar{R}$ from $\Mod$-$A$ to $\Mod$-$R$ are naturally isomorphic. The natural isomorphism $-\underset{A}{\otimes} \bar{A}→-\underset{R}{\otimes} \bar{R}$ is given at the module $M_A$ by
\begin{equation*}
\begin{array}{rcl}
M_A \underset{A}{\otimes} \leftidx{_A}{\bar{A}}{} &\overset{\sim}{\longrightarrow}& M_R\underset{R}{\otimes} \leftidx{_R}{\bar{R}}{}\\
m\otimes (a+ ρ(I)A) &\longmapsto & ma \otimes (1+I)\\
m\otimes (r+ρ(I)A)&\longmapsfrom & m\otimes (r+I).
\end{array}
\end{equation*}
By the universal property of the tensor product, the two maps given above are well-defined and $R$-linear. Straightforward calculation gives that they invert each other and that we have a natural transformation.
The functor $-\underset{A}{\otimes}\leftidx{_A}{\bar{A}}{_{\bar{A}}}$ from $\Mod$-$A$ to $\Mod$-$\bar A$ maps exact sequences of right $A$-modules that are free and of finite rank as $R$-modules to exact sequences of right $\bar{A}$-modules.
Because $-\underset{A}{\otimes}\leftidx{_A}{\bar{A}}{_{\bar{A}}}$ is an additive functor, it maps complexes to complexes. For considerations of exactness, we may compose our functor with the forgetful functor from $\Mod$-$\bar A$ to $\Mod$-$ R$. This composite is $-\underset{A}{\otimes}\leftidx{_A}{\bar{A}}{}$.
By the natural isomorphism given in <ref>, it suffices to show that $-\underset{R}{\otimes}\leftidx{_R}{\bar{R}}{}$ transforms exact sequences of right $A$-modules that are free and of finite rank as $R$-modules into exact sequences.
Let $\cdots\xrightarrow{d_{i+1}}M_i\xrightarrow{d_i}M_{i-1}\xrightarrow{d_{i-1}}\cdots$ be an exact sequence of right $A$-modules that are free and of finite rank as $R$-modules. Then $\im d_i$ is a submodule of the free $R$-module $M_{i-1}$. As $R$ is a principal ideal domain, $\im d_i$ is free. Hence the short exact sequence $\im d_{i+1} \rightarrow M_i \rightarrow \im d_i$ splits. Now the additive functor $-\underset{R}{\otimes}\leftidx{_R}{\bar{R}}{}$ maps split short exact sequences to (split) short exact sequences and the proof is complete.
§.§.§ Reduction modulo $p$
The isomorphism $r:\zp\!\Sp \rightarrow Λ$ from <ref> induces an isomorphism of $\fp$-algebras $\fp\!\Sp = \zp\!\Sp/(p\zp\!\Sp) \xrightarrow{\bar{r}} Λ/(pΛ)=:\bar{Λ}$.
For the sake of simplicity in the next step, we identify $\bar{Λ}$ and $\fp\!\Sp$ along $\bar{r}$.
Recall that $p\geq 3$ is a prime. Applying the functor $-\underset{Λ}{\otimes} \leftidx{_Λ}{\bar{Λ}}{_{\bar{Λ}}}$ , we obtain
* the projective modules $P_k:= \tilde P_k\underset{Λ}{\otimes} \leftidx{_Λ}{\bar{Λ}}{_{\bar{Λ}}}$
for $k∈[1,p-1]$,
* $\fp:=\zp\underset{Λ}{\otimes} \leftidx{_Λ}{\bar{Λ}}{_{\bar{Λ}}}$ (the trivial $\fp\!\Sp$-module),
\begin{array}[t]{lclll}
e_k&:=&\hat e_k\underset{Λ}{\otimes} \leftidx{_Λ}{\bar{Λ}}{_{\bar{Λ}}}&∈\Hom_{\fp\Sp}(P_k,P_k)&\text{for $k∈[1,p-1]$,} \\
e_{1,1}&:=& \hat e_{1,1}\underset{Λ}{\otimes} \leftidx{_Λ}{\bar{Λ}}{_{\bar{Λ}}}&∈\Hom_{\fp\Sp}(P_1,P_1),\\
e_{p-1,p-1} &:=& \hat e_{p-1,p-1}\underset{Λ}{\otimes} \leftidx{_Λ}{\bar{Λ}}{_{\bar{Λ}}}&∈\Hom_{\fp\Sp}(P_{p-1},P_{p-1}), \\
e_{k+1,k}&:=& \hat e_{k+1,k}\underset{Λ}{\otimes} \leftidx{_Λ}{\bar{Λ}}{_{\bar{Λ}}}&∈\Hom_{\fp\Sp}(P_k,P_{k+1})&\text{for $k∈[1,p-2]$,} \\
e_{k,k+1} &:=& \hat e_{k,k+1}\underset{Λ}{\otimes} \leftidx{_Λ}{\bar{Λ}}{_{\bar{Λ}}}&∈\Hom_{\fp\Sp}(P_{k+1},P_k) &\text{for $k∈[1,p-2]$,} \\
ε &:=& \hat{ε}\underset{Λ}{\otimes} \leftidx{_Λ}{\bar{Λ}}{_{\bar{Λ}}}&∈\Hom_{\fp\Sp}(P_1,\fp) .
\end{array}
The complex
\begin{align}
\label{prfp}
\pres \fp := (\pres \zp)\underset{Λ}{\otimes} \leftidx{_Λ}{\bar{Λ}}{_{\bar{Λ}}}= (\cdots\xrightarrow{d_3} {\pr}_2\xrightarrow{ d_2}{\pr}_1\xrightarrow{ d_1}{\pr}_0\xrightarrow{0=d_0} 0\rightarrow\cdots),
\end{align}
\begin{align*}
{\pr}_i :\eqs& \begin{cases} P_{ω(i)} & i\geq 0 \\ 0 & i<0 \end{cases} &
d_i :\eqs& \begin{cases} e_{ω(i-1),ω(i)}: P_{ω(i)} →P_{ω(i-1)} & i\geq 1\\ 0 & i\leq 0, \end{cases}
\end{align*}
is a projective resolution of $\fp$ with augmentation $ε:P_1→\fp$.
More explicitly, $\pres \fp$ is
\begin{align*}
…\rightarrow & \underbrace{P_2}_{\mathclap{l+1}}\xrightarrow{ e_{1,2}} \underbrace{P_1}_{\mathclap{l=2(p-1)}} \xrightarrow{ e_{1,1}} \underbrace{P_1}_{\mathclap{(p-2)+p-1}} \xrightarrow{ e_{2,1}} \underbrace{P_2}_{\mathclap{(p-2)+p-2}} \rightarrow …
\rightarrow \underbrace{P_{p-2}}_{\mathclap{p=(p-2)+2}} \xrightarrow{ e_{p-1,p-2}} \underbrace{P_{p-1}}_{\mathclap{(p-2)+1}} \nonumber \\
&\xrightarrow{ e_{p-1,p-1}} \underbrace{P_{p-1}}_{\mathclap{p-2}} \xrightarrow{ e_{p-2,p-1}} \underbrace{P_{p-2}}_{p-3}\rightarrow … \rightarrow \underbrace{P_2}_{1} \xrightarrow{ e_{1,2}} \underbrace{P_1}_0 \rightarrow 0.
\end{align*}
Recall that $p\geq 3$ is a prime.
* We have the relations
\begin{equation*}
\begin{array}{lclr}
e_{1,1} + e_{1,2} \circ e_{2,1} &=& 0\\
e_{k,k-1}\circ e_{k-1,k} + e_{k,k+1}\circ e_{k+1,k} &=& 0 &\text{ for } k∈[2,p-2]\\
e_{p-1,p-2} \circ e_{p-2,p-1} + e_{p-1,p-1} &=& 0\\
ε \circ e_{1,1} &=& 0
\end{array}
\end{equation*}
and $e_k$ is the identity on $P_k$ for $k∈[1,p-1]$.
* Given $k∈[2,p-1]$, we have $\Hom_{\fp\!\Sp}(P_k,\fp) = \{0\}$.
* Given $k,k'∈[1,p-1]$ such that $|k-k'|>1$, we have $\Hom_{\fp\!\Sp}(P_k,P_{k'}) = \{0\}$.
* The set $\{ε\}$ is an $\fp$-basis of $\Hom_{\fp\!\Sp}(P_1,\fp)$.
Assertion (a) results from <ref>.
Assertions (b), (c) and (d) are derived from corresponding assertions over $\zp\!\Sp$ using $\Hom_{\fp\!\Sp}(P/pP, M/pM) \simeq \Hom_{\zp\!\Sp}(P,M)/p\Hom_{\zp\!\Sp}(P,M)$ for $\zp\!\Sp$-modules $P$ and $M$, where $P$ is projective.
§ A(OO)-ALGEBRAS
§.§ Definitions, General theory
In this subsection, we review results presented in [12] and we fix notation.
Let $R$ be a commutative ring.
We understand linear maps between $R$-modules to be $R$-linear. Tensor products are tensor products over $R$. By graded $R$-modules we understand $\Z$-graded $R$-modules.
A graded $R$-module $V$ is a $R$-module of the form $V = \oplus_{q∈\Z} V^q$.
An element $v_q∈V^q$, $q∈\Z$ is said to be of degree $q$. An element $v∈V$ is called homogeneous if there is an integer $q∈\Z$ such that $v∈V^q$. For homogeneous elements $v$ resp. graded maps $g$ (see below), we denote their degrees by $|v|$ resp. $|g|$.
Let $A = \oplus_{q∈\Z}A^q$, $B = \oplus_{q∈\Z}B^q$ be two graded $R$-modules. A graded map of degree $z∈\Z$ is a linear map $g:A→B$ such that for $q∈\Z$.
Let $A=\oplus_{q∈\Z}A^q$, $B=\oplus_{q∈\Z}B^q$ be two graded $R$-modules. We have
\begin{align*}
A\otimes B \eqs& \bigoplus_{z_1,z_2∈\Z} A^{z_1}\otimes B^{z_2}
= \bigoplus_{q∈\Z}\left(\bigoplus_{z_1+z_2=q} A^{z_1}\otimes B^{z_2}\right).
\end{align*}
As we understand the direct sums to be internal direct sums in $A\otimes B$ and understand $A^{z_1}\otimes B^{z_2}$ to be the linear span of the set $\{a\otimes b∈A\otimes B \mid a ∈ A^{z_1},b∈A^{z_2}\}$, we have equations in the above, not just isomorphisms.
We then set $A\otimes B$ to be graded by $A\otimes B = \bigoplus_{q∈\Z}(A\otimes B)^q$, where $(A\otimes B)^q:= \bigoplus_{z_1+z_2=q} A^{z_1}\otimes B^{z_2}$.
Moreover, we grade the direct sum
\begin{align*}
A\oplus B \eqs& \bigoplus_{q∈\Z} (A^q\oplus B^q)
%\bigoplus_{z_1,z_2∈\Z} A^{z_1}\oplus B^{z_2}
%= \bigoplus_{q∈\Z}\left(\bigoplus_{z_1+z_2=q} A^{z_1}\oplus B^{z_2}\right)
\end{align*}
by $(A\oplus B)^q := A^q\oplus B^q$.
In the definition of the tensor product of graded maps, we implement the Koszul sign rule: Let $A_1,A_2,B_1,B_2$ be graded $R$-modules and $g:A_1→B_1$, $h:A_2→B_2$ graded maps. Then we set for homogeneous elements $x∈A_1,y∈A_2$
\begin{align}
\label{koszulraw}
(g\otimes h)(x\otimes y) := (-1)^{|h|\cdot|x|} g(x)\otimes h(y).
\end{align}
It is known that for graded $R$-modules $A,B,C$, the map
\begin{align}
\label{theta}
\begin{array}{rccc}
Θ:&(A\otimes B) \otimes C& \longrightarrow & A\otimes (B\otimes C)\\
&(a\otimes b) \otimes c &\longmapsto & a\otimes (b\otimes c)
\end{array}
\end{align}
is an isomorphism of $R$-modules. Because of the following, $Θ$ is homogeneous of degree $0$.
\begin{align*}
((A\otimes B)\otimes C)^q \eqs& \bigoplus_{\mathclap{y+z_3=q}} (A\otimes B)^y \otimes C^{z_3} = \bigoplus_{y+z_3=q}\bigoplus_{z_1+z_2= y} (A^{z_1}\otimes B^{z_2})\otimes C^{z_3}\\
\eqs& \bigoplus_{\mathclap{z_1+z_2+z_3=q}} (A^{z_1}\otimes B^{z_2})\otimes C^{z_3}\\
(A\otimes (B\otimes C))^q \eqs& \bigoplus_{\mathclap{z_1+y=q}} A^{z_1}\otimes (B\otimes C)^y = \bigoplus_{z_1+y=q}\bigoplus_{z_2+z_3=y} A^{z_1}\otimes (B^{z_2}\otimes C^{z_3})\\
\eqs& \bigoplus_{\mathclap{z_1+z_2+z_3=q}} A^{z_1}\otimes (B^{z_2}\otimes C^{z_3})
\end{align*}
Let $A_1,A_2,B_1,B_2,C_1,C_2$ be graded $R$-modules, $f:A_1→A_2$, $g:B_1→B_2$, $h:C_1→C_2$ graded maps. For homogeneous elements $x∈A_1
$, $y∈B_1$, $z∈C_1$, we have
\begin{align*}
((f\otimes g)\otimes h)((x\otimes y)\otimes z) \eqs& (-1)^{|x\otimes y|\cdot|h|}((f\otimes g)(x\otimes y))\otimes h(z)\\
\eqs& (-1)^{(|x|+|y|)|h|+ |x|\cdot |g|} (f(x)\otimes g(y))\otimes h(z)\\
(f\otimes(g\otimes h))(x\otimes (y\otimes z)) \eqs& (-1)^{|x|\cdot |g\otimes h|} f(x)\otimes((g\otimes h)(y\otimes z))\\
\eqs& (-1)^{|x|(|g|+|h|) + |y|\cdot|h|} f(x)\otimes(g(y)\otimes h(z)) \\
\eqs& (-1)^{(|x|+|y|)|h|+ |x|\cdot |g|} f(x)\otimes(g(y)\otimes h(z)).
\end{align*}
Thus we have the following commutative diagram ($Θ_1$ and $Θ_2$ are derived from (<ref>))
\begin{align*}
\xymatrix@C=2pc{
(A_1\otimes B_1)\otimes C_1\ar[r]^{Θ_{1}}\ar[d]^{(f\otimes g)\otimes h} & A_1\otimes (B_1\otimes C_1)\ar[d]^{f\otimes (g\otimes h)}\\
(A_2\otimes B_2)\otimes C_2\ar[r]^{Θ_{2}} & A_2\otimes (B_2\otimes C_2)
\end{align*}
It is therefore valid to use $Θ$ as an identification and to omit the brackets for the tensorization of graded $R$-modules and the tensorization of graded maps.
Concerning the signs in the definition of $\A_∞$-algebras and $\A_∞$-morphisms, we follow the variant given e.g. in [16] and [8].
Let $n∈\Z_{\geq 0}\cup \{∞\}$.
* Let $A$ be a graded $R$-module. A pre-$\A_n$-structure on $A$ is a family of graded maps $(m_k:A^{\otimes k}→A)_{k∈[1,n]}$ with $|m_k|=2-k$ for $k∈[1,n]$. The tuple $(A,(m_k)_{k∈[1,n]})$ is called a pre-$\A_n$-algebra.
* Let $A$, $A'$ be graded $R$-modules. A pre-$\A_n$-morphism from $A'$ to $A$ is a family of graded maps $(f_k:A'^{\otimes k}→A)_{k∈[1,n]}$ with $|f_k|=1-k$ for $k∈[1,n]$.
Let $n∈\Z_{\geq 0}\cup\{∞\}$.
An $\A_n$-algebra is a pre-$\A_n$-algebra $(A,(m_k)_{k∈[1,n]})$ such that for
\begin{align}
\sum_{\natop{k=r+s+t,}{r,t\geq 0, s\geq 1}} (-1)^{rs+t}m_{r+1+t}\circ (1^{\otimes r}\otimes m_s\otimes 1^{\otimes t})=0 .
\tag*{(\theequation)[$k$]}
\end{align}
In abuse of notation, we sometimes abbreviate $A = (A,(m_k)_{k\geq 1})$ for $\A_∞$-algebras.
Let $(A',(m'_k)_{k∈[1,n]})$ and $(A,(m_k)_{k∈[1,n]})$ be $\A_n$-algebras. An $\A_n$-morphism or morphism of $\A_n$-algebras from $(A',(m'_k)_{k∈[1,n]})$ to $(A,(m_k)_{k∈[1,n]})$ is a pre-$\A_n$-morphism $(f_k)_{k∈[1,n]}$ such that for $k∈[1,n]$, we have
\begin{align}
%\sum_{\mathclap{n=r+s+t}} (-1)^{r+st} f_u(1^{\otimes r}\otimes m_s\otimes 1^{\otimes t}) = \sum_{\mathclap{\natop{1\leq r\leq n}{i_1+…+i_r=n}}} (-1)^s m_r(f_{i_1}\otimes f_{i_2}\otimes … \otimes f_{i_r})
\sum_{\mathclap{\substack{k=r+s+t\\r,t\geq 0,s\geq 1}}} (-1)^{rs+t} f_{r+1+t}\circ (1^{\otimes r}\otimes m'_s\otimes 1^{\otimes t}) = \sum_{\mathclap{\substack{1\leq r\leq k\\ i_1+…+i_r=k\\ i_s\geq 1}}} (-1)^v m_r\circ (f_{i_1}\otimes f_{i_2}\otimes … \otimes f_{i_r}),
\tag*{(\theequation)[$k$]}
\end{align}
$v := %\sum_{2\leq s\leq r}\left((1-i_s)\sum_{1\leq t\leq s} i_t\right) \equiv_2
\sum_{1\leq t < s\leq r}(1-i_s)i_t$.
bsp]ex:dg Let $(A,(m_k)_{k\geq 1})$ be an $\A_\infty$-algebra.
If $m_n=0$ for $n\geq 3$ then $A$ is called a differential graded algebra or dg-algebra. In this case the equations (<ref>)[$n$] for $n\geq 4$ become trivial: We have $(r+1+t)+s = n+1$ ⇒ $(r+1+t)+s\geq 5$ ⇒ $m_{r+1+t}=0$ or $m_s=0$. So all summands in (<ref>)[$n$] are zero for $n\geq 4$. Here are the equations for $n∈\{1,2,3\}$:
\begin{align*}
\eqref{ainfrel}[1]: && 0 \eqs& m_1\circ m_1\\
\eqref{ainfrel}[2]: && 0 \eqs& m_1\circ m_2 - m_2\circ (m_1\otimes 1+1\otimes m_1)\\
\eqref{ainfrel}[3]: && 0 \eqs& m_1\circ m_3 + m_2\circ (1\otimes m_2-m_2\otimes1)\\
&&& + m_3\circ (m_1\otimes 1^{\otimes 2}+1\otimes m_2\otimes 1+1^{\otimes 2}\otimes m_1)
\\&&
\ovs{m_3=0}& m_2\circ (1\otimes m_2-m_2\otimes1)
\end{align*}
So (<ref>)[$1$] ensures that $m_1$ is a differential. Moreover, (<ref>)[$3$] states that $m_2$ is an associative binary operation, since for homogeneous $x,y,z∈A$ we have $0=m_2\circ \mbox{$(1\otimes m_2-m_2\otimes1)(x\otimes y\otimes z)$} = m_2(x\otimes m_2(y\otimes z) - m_2(x\otimes y)\otimes z)$, where because of $|m_2|=0$ there are no additional signs caused by the Koszul sign rule. Equation (<ref>)[$2$] is the Leibniz rule.
[$\A_n$-morphisms induce complex morphisms]
bsp]ex:cc $ $
Let $n∈\Z_{\geq 1}\cup \{∞\}$. Let $(A', (m'_k)_{k∈[1,n]})$ and $(A, (m_k)_{k∈[1,n]})$ be two $\A_n$-algebras and let $(f_k)_{k∈[1,n]}: (A', (m'_k)_{k∈[1,n]}) → (A, (m_k)_{k∈[1,n]})$ be an $\A_n$-morphism.
By (<ref>)[$1$], $(A',m'_1)$ and $(A,m_1)$ are complexes. Equation (<ref>)[$1$] is
\begin{align*}
%\eqref{finfrel}[1]: &&
f_1\circ m'_1 \eqs& m_1\circ f_1.
\end{align*}
Thus $f_1:(A',m'_1)→(A,m_1)$ is a complex morphism.
For $n\geq 2$, we have also (<ref>)[$2$]:
\begin{align}
\label{finfrel2}
%\eqref{finfrel}[2]: &&
f_1\circ m'_2 - f_2\circ(m'_1\otimes 1 + 1\otimes m'_1) \eqs& m_1\circ f_2 + m_2\circ (f_1\otimes f_1)
\end{align}
Recall the conventions concerning $\Hom_B^k(C,C')$.
Let $B$ be an (ordinary) $R$-algebra and $M=((M_i)_{i∈\Z}, (d_i)_{i∈\Z})$ a complex of $B$-modules, that is a sequence $(M_i)_{i∈\Z}$ of $B$-modules and $B$-linear maps $d_i:M_i→M_{i-1}$ such that $d_{i-1}\circ d_i = 0$ for all $i∈\Z$.
\begin{align*}
\Hom^i_B(M,M):\eqs& \prod_{z∈\Z}\Hom_B(M_{z+i},M_z)\\
\eqs& \{g=(g_z)_{z∈\Z} \mid g_z∈\Hom_B(M_{z+i},M_z)\text{ for }z∈\Z\}.
\end{align*}
\begin{align*}
A=\Hom^*_B(M,M) := \bigoplus_{i∈\Z} \Hom^i_B(M,M)
\end{align*}
is a graded $R$-module.
We have $d := (d_{z+1})_{z∈\Z}=\sum_{i∈\Z} \ls d_{i+1}\rs_{i+1}^{i} ∈\Hom^1_B(M,M)$. We define $m_1:=d_{\Hom^*(M,M)}:A→A$, that is for homogeneous $g∈A$ we have
\begin{align*}
m_1(g) = d \circ g -(-1)^{|g|} g\circ d.
\end{align*}
We define $m_2:A^{\otimes 2}→A$ for homogeneous $g,h∈A$ to be composition, i.e.
\begin{align*}
m_2(g\otimes h) := g \circ h.
\end{align*}
For $n\geq 3$ we set $m_n:A^{\otimes n}→A$, $m_n=0$. Then $(m_n)_{n\geq 1}$ is an $\A_\infty$-algebra structure on $A=\Hom^*_B(M^*,M^*)$. More precisely, $(A, (m_n)_{n\geq 1})$ is a dg-algebra.
Because of $|d|=1$ we have $|m_1| = 1 = 2-1$. The graded map $m_2$ has degree $0 = 2-2$. The other maps $m_n$ are zero and have therefore automatically correct degree. As discussed in <ref> we only need to check (<ref>)[$n$] for $n=1,2,3$.
Equation (<ref>)[$1$] holds because for homogeneous $g∈A$ we have
\begin{align*}
m_1(m_1(g)) \eqs& m_1[d\circ g - (-1)^{|g|}g\circ d]\\
\eqs& d\circ[d\circ g - (-1)^{|g|}g\circ d] - (-1)^{|g|+1}[d\circ g - ({-}1)^{|g|}g\circ d]\circ d\\
\ovs{d^2=0}& -(-1)^{|g|}d\circ g\circ d - (-1)^{|g|+1}d\circ g\circ d = 0.
\end{align*}
Concerning (<ref>)[$2$], we have for homogeneous $g,h∈A$
\begin{align*}
(m_2\circ (m_1\otimes 1+&1\otimes m_1))(g\otimes h)= m_2(m_1(g)\otimes h + (-1)^{|g|}g\otimes m_1(h))\\
\eqs& (d\circ g - (-1)^{|g|}g\circ d)\circ h + (-1)^{|g|}g\circ(d\circ h - (-1)^{|h|}h\circ d)\\
\eqs& d\circ g \circ h - (-1)^{|g|+|h|}g\circ h\circ d\\
\eqs& (m_1\circ m_2 )(g\otimes h).
\end{align*}
The map $m_2$ is induced by the composition of morphisms which is associative. As discussed in <ref>, equation (<ref>)[$3$] holds.
In $\Hom^*(\pres\fp,\pres\fp)$ we have (cf. (<ref>))
\begin{align*}
d \eqs& \sum_{i\geq 0} \ls e_{ω(i),ω(i+1)}\rs_{i+1}^i\,.
\end{align*}
[Homology of $\A_∞$-algebras, quasi-isomorphisms, minimality, minimal models]
As $m_1^2=0$ (cf. (<ref>)[$1$]) and
$|m_1|=1$, we have the complex
\[\cdots → A^{i-1}\xrightarrow{m_1|_{A^{i-1}}}A^{i}\xrightarrow{m_1|_{A^i}} A^{i+1}→\cdots\,.\]
We define $\Hm^k A:= \ker(m_1|_{A^k})/\im(m_1|_{A^{k-1}})$ and $\Hm^* A := \bigoplus_{k∈\Z} \Hm^k A$, which gives the homology of $A$ the structure of a graded $R$-module.
A morphism of $\A_∞$-algebras $(f_k)_{k\geq 1}: (A', (m'_k)_{k\geq 1}) → (A, (m_k)_{k\geq 1})$ is called a quasi-isomorphism if the morphism of complexes $f_1:(A', m'_1)→(A,m_1)$ (cf. <ref>) is a quasi-isomorphism.
An $\A_∞$-algebra is called minimal, if $m_1=0$. If $A$ is an $\A_∞$-algebra and $A'$ is a minimal $\A_∞$-algebra quasi-isomorphic to $A$, then $A'$ is called a minimal model of $A$.
The existence of minimal models is assured by the following theorem.
(minimality theorem, cf. [13] (history), [9], [8], [21], [5], [7], [18], … )
Let $(A,(m_k)_{k\geq 1})$ be an $\A_\infty$-algebra such that the homology $\Hm^* A$ is a projective $R$-module. Then there exists an $\A_\infty$-algebra structure $(m'_k)_{k\geq 1}$ on $\Hm^*A$ and a quasi-isomorphism of $\A_∞$-algebras $(f_k)_{k\geq 1}:(\Hm^* A,(m'_k)_{k\geq 1})→(A,(m_k)_{k\geq 1})$, such that
* $m'_1 = 0$ and
* the complex morphism $f_1:(\Hm^*A,m'_1)→(A, m_1)$ induces the identity in homology. I.e. each element $x∈\Hm^* A$, which is a homology class of $(A,m_1)$, is mapped by $f_1$ to a representing cycle.
For constructing $\A_∞$-structures induced by another $\A_∞$-algebra, we have the following
Let $n∈\Z_{\geq 1}\cup \{∞\}$. Let $(A',(m'_k)_{k∈[1,n]})$ be a pre-$\A_n$-algebra. Let $(A,(m_k)_{k∈[1,n]})$ be an $\A_n$-algebra. Let $(f_k)_{k∈[1,n]}$ be a pre-$\A_n$-morphism from $A'$ to $A$ such that (<ref>)$[k]$ holds for $k∈[1,n]$. Suppose $f_1$ to be injective.
Then $(A',(m'_k)_{k∈[1,n]})$ is an $\A_n$-algebra and $(f_k)_{k∈[1,n]}$ is a morphism of $\A_n$-algebras from $(A',(m'_k)_{k∈[1,n]})$ to $(A,(m_k)_{k∈[1,n]})$.
This results from the bar construction and a straightforward induction on $n$.
We have the corresponding pre-$\A_n$-triple $((m'_k)_{k∈[1,n]}, (b'_k)_{k∈[1,n]}, b')$, the corresponding pre-$\A_n$-triple $((m_k)_{k∈[1,n]}, (b_k)_{k∈[1,n]}, b)$ and the corresponding pre-$\A_n$-morphism triple $((f_k)_{k∈[1,n]}, (F_k)_{k∈[1,n]}, F)$.
It suffices to prove by induction on $k∈[0,n]$ that $(b')^2\big|_{TSA'_{\leq k}} = 0$, cf. <ref>.
For $k=0$, there is nothing to prove.
For the induction step, suppose that $b'\,^2\big|_{TSA'_{\leq k}} = 0$. Then by <ref>(i), we have $\im(b'\,^2 \circ ι'_{k+1})\subseteq SA$. Thus $0 = b^2\circ F \circ ι'_{k+1} \overset{\text{L.}\ref{lem:stasheff2}}{=} F\circ b'\,^2 \circ ι'_{k+1} = F_1\circ b'\,^2 \circ ι'_{k+1}$. As the injectivity of $f_1$ implies the injectivity of $F_1$, we have $b'\,^2\circ ι'_{k+1} =0$ and thus $b'\,^2\big|_{TSA'_{\leq k+1}} = 0$.
Let $R$ be a commutative ring and $(A,(m_n)_{n\geq 1})$ be a dg-algebra (over $R$). Suppose given a graded $R$-module $B$ and graded maps $f_n:B^{\otimes n}→A$, $m'_n:B^{\otimes n}→B$ for $n\geq 1$. Suppose given $k\geq 1$ such that
we have $f_i=0$ for $i\geq k$, we have $m'_i=0$ for $i\geq k+1$,
and (<ref>)$[n]$ is satisfied for $n∈[1,2k-2]$. Then (<ref>)$[n]$ is satisfied for all $n\geq 1$.
§.§ The homology of Hom*(FpSp)(PResFp,PResFp)
We need a well-known result of homological algebra in a particular formulation:
Let $F$ be a field. Let $B$ be an $F$-algebra. Let $M$ be a $B$-module. Let $Q = (\cdots \rightarrow Q_2 \xrightarrow{d_2} Q_1 \xrightarrow{d_1} Q_0 → 0→
\cdots)$ be a projective resolution of $M$ with augmentation $ε:Q_0→M$. Then we have maps for $k∈\Z$
\begin{align*}
Ψ_k:\Hom^k_B(Q,Q) &\xrightarrow{}\Hom^k_B(Q, M) := \Hom_B(Q_k,M)\\
(g_i:Q_{i+k}→Q_i)_{i∈\Z} &\mapsto ε\circ g_0
\end{align*}
The right side is equipped with the differentials (dualization of $d_k$)
\begin{align*}
(d_k)^*:\Hom_B(Q_k,M) &\rightarrow \Hom_B(Q_{k+1},M)\\
g & \mapsto (-1)^k g\circ d_k
\end{align*}
and the left side is equipped with the differential $m_1$ of its dg-algebra structure, cf. <ref>.
Then $(Ψ_k)_{k∈\Z}$ becomes a complex morphism from the complex $\Hom^*_B(Q,Q)$ to the complex $\Hom^*_B(Q, M)$ that induces isomorphisms $\bar{Ψ}_k$ of $F$-vector spaces on the homology
\begin{align*}
\bar{Ψ}_k:\Hm^k\Hom^*_B(Q,Q) & \xrightarrow{\simeq}\Hm^k \Hom^*_B(Q, M)\\
\overline{(g_i:Q_{i+k}→Q_i)_{i∈\Z}}&\mapsto \overline{ε\circ g_0}
\end{align*}
<ref> is <cit.> applied to the quasi-isomorphism induced by the augmentation, cf. <cit.>.
Recall the notation $\ls x\rs_y^z$ for the description of elements of $\Hom_B^k(C,C')$.
Recall that $p\geq 3$ is a prime and $l=2(p-1)$.
Write .
\begin{align*}
ι:\eqs&\dum_{i\geq 0} \ls e_{ω(i)} \rs_{i+l}^i =\dum_{i\geq 0}\dum_{k=0}^{l-1}\ls e_{ω(k)} \rs_{(i+1)l+k}^{il+k} ∈ A^l
\\
% \end{align*}
% \begin{align*}
χ :\eqs& \dum_{i\geq 0}\left(\ls e_1 \rs_{il+l-1}^{il} + \left\lgroup\dum_{k=1}^{p-2}\ls e_{k+1,k}\rs_{il+l-1+k}^{il + k}\right\rgroup\right.\\
&\left. +
\ls e_{p-1}\rs_{il+l-1 +(p-1)}^{il+(p-1)}
+ \left\lgroup\dum_{k=1}^{p-2}\ls e_{p -k-1,p-k}\rs_{il+l-1+(p-1)+k}^{il +(p-1)+ k}\right\rgroup\right)∈A^{l-1}.
\end{align*}
* For $j\geq 0$, we have
ι^j = \dum_{i\geq 0} \ls e_{ω(i)} \rs_{i+jl}^i= \dum_{i\geq 0}\dum_{k=0}^{l-1}\ls e_{ω(k)} \rs_{(i+j)l+k}^{il+k}\,.
* Suppose given $y\geq 0$. Let $h∈A^y$ be $l$-periodic, that is
$h=\dum_{i\geq 0} \dum_{k=0}^{l-1} \ls h_k\rs_{il+k+y}^{il + k}$.
Then for $j\geq 0$, we have
\begin{align*}
h\circ ι^j = ι^j\circ h = \dum_{i\geq 0} \dum_{k=0}^{l-1} \ls h_k\rs_{(i+j)l + k +y}^{il+k}∈A^{y+jl}.
\end{align*}
* Suppose given $y∈\Z$. For $h∈A^y$ and $j\geq 0$, we have $m_1(h\circ ι^j) = m_1(h)\circ ι^j$.
* For $j\geq 0$, we have $m_1(ι^j)=0$. Thus $ι^j$ is a cycle.
* For $j\geq 0$, we have
\begin{align*}
χι^j :\eqs& χ\circ ι^j = ι^j\circ χ\\
\eqs& \dum_{i\geq 0}\left(\ls e_1 \rs_{(i+j+1)l-1}^{il} + \left\lgroup\dum_{k=1}^{p-2}\ls e_{k+1,k}\rs_{(i+j+1)l-1+k}^{il + k}\right\rgroup\right.\\
&\left. + \ls e_{p-1}\rs_{(i+j+1)l-1 +(p-1)}^{il+(p-1)}
+ \left\lgroup\dum_{k=1}^{p-2}\ls e_{p -k-1,p-k}\rs_{(i+j+1) l-1+(p-1)+k}^{il +(p-1)+ k}\right\rgroup\right)∈A^{jl+l-1}.
\end{align*}
For convenience, we also define $χ^0ι^j:= ι^j$ and $χ^1ι^j:=χι^j=χ\circ ι^j$ for $j\geq 0$.
* For $j\geq 0$, we have $m_1(χι^j) = 0$. Thus $χι^j$ is a cycle.
Suppose given $k∈\Z$. A $\fp$-basis of $\Hm^kA$ is given by
\begin{align*}
\{\overline{ι^j}\} & \text{ if $k=jl$ for some $j\geq 0$}\\
\{\overline{χι^j}\} & \text{ if $k=jl+l-1$ for some $j\geq 0$} \\
∅ &\text{ else}.
\end{align*}
Thus the set $\mathfrak{B}:=\{\overline{ι^j}\mid j\geq 0\}\sqcup \{\overline{χι^j}\mid j\geq 0\}$ is an $\fp$-basis of $\Hm^*A = \bigoplus_{z∈\Z} \Hm^zA$.
Before we proceed we display $ι$ and $χ$ for the case $p = 5$ as an example:
The period is of length $l = 2p - 2 = 2\cdot 5 - 2 = 8$. The terms inside circles denote the degrees.
For $χ$, the differentials on the right side are negated to obtain a commutative diagram (We have $0=m_1(χ) = d\circ χ + χ\circ d$ because $|χ|$ is odd)
\[
\xymatrix@C=0.5pc@!R=0.43pc{
& \pres(\ff)[8]\ar[r]^-\iota \ar@{}[d]^*--{\rotatebox{90}{\hspace{3mm}$=$\ru{2}}} & \pres(\ff) \ar@{}[d]^*--{\rotatebox{90}{\hspace{3mm}$=$\ru{2}}}\\
& \rotatebox{90}{$\cdots)$}\ar[d] & \rotatebox{90}{$\cdots)$}\ar[d] \\
*+[F-:<5pt>]{ 8} & P_1\ar[r]^{e_1} \ar[d]_{e_{1,1}} & P_1\ar[d]^{e_{1,1}} \\
*+[F-:<5pt>]{ 7} & P_1\ar[r]^{e_1} \ar[d]_{e_{2,1}} & P_1\ar[d]^{e_{2,1}} \\
*+[F-:<5pt>]{ 6} & P_2\ar[r]^{e_2} \ar[d]_{e_{3,2}} & P_2\ar[d]^{e_{3,2}} \\
*+[F-:<5pt>]{ 5} & P_3\ar[r]^{e_3} \ar[d]_{e_{4,3}} & P_3\ar[d]^{e_{4,3}} \\
*+[F-:<5pt>]{ 4} & P_4\ar[r]^{e_4} \ar[d]_{e_{4,4}} & P_4\ar[d]^{e_{4,4}} \\
*+[F-:<5pt>]{ 3} & P_4\ar[r]^{e_4} \ar[d]_{e_{3,4}} & P_4\ar[d]^{e_{3,4}} \\
*+[F-:<5pt>]{ 2} & P_3\ar[r]^{e_3} \ar[d]_{e_{2,3}} & P_3\ar[d]^{e_{2,3}} \\
*+[F-:<5pt>]{ 1} & P_2\ar[r]^{e_2} \ar[d]_{e_{1,2}} & P_2\ar[d]^{e_{1,2}} \\
*+[F-:<5pt>]{ 0} & P_1\ar[r]^{e_1} \ar[d]_{e_{1,1}} & P_1\ar[d] \\
*+[F-:<5pt>]{-1} & P_1\ar[r] \ar[d]_{e_{2,1}} & 0 \ar[d] \\
*+[F-:<5pt>]{-2} & P_2\ar[r] \ar[d]_{e_{3,2}} & 0 \ar[d] \\
*+[F-:<5pt>]{-3} & P_3\ar[r] \ar[d]_{e_{4,3}} & 0 \ar[d] \\
*+[F-:<5pt>]{-4} & P_4\ar[r] \ar[d]_{e_{4,4}} & 0 \ar[d] \\
*+[F-:<5pt>]{-5} & P_4\ar[r] \ar[d]_{e_{3,4}} & 0 \ar[d] \\
*+[F-:<5pt>]{-6} & P_3\ar[r] \ar[d]_{e_{2,3}} & 0 \ar[d] \\
*+[F-:<5pt>]{-7} & P_2\ar[r] \ar[d]_{e_{1,2}} & 0 \ar[d] \\
*+[F-:<5pt>]{-8} & P_1\ar[r] \ar[d] & 0 \ar[d] \\
*+[F-:<5pt>]{-9} & 0 \ar[r]\ar[d] & 0 \ar[d] \\
& \rotatebox{-90}{$\cdots)$} & \rotatebox{-90}{$\cdots)$} \\
\hspace{2cm}
\xymatrix@C=0.5pc@!R=0.43pc{
& \pres(\ff)[8-1]\ar[r]^-\chi \ar@{}[d]^*--{\rotatebox{90}{\hspace{3mm}$=$\ru{2}}} & \pres(\ff) \ar@{}[d]^*--{\rotatebox{90}{\hspace{3mm}$=$\ru{2}}}\\
& \rotatebox{90}{$\cdots)$}\ar[d] & \rotatebox{90}{$\cdots)$}\ar[d] \\
*+[F-:<5pt>]{ 9} & P_1\ar[r]^{e_{2,1}}\ar[d]_{e_{1,1}} & P_2\ar[d]^{-e_{1,2}} \\
*+[F-:<5pt>]{ 8} & P_1\ar[r]^{e_1} \ar[d]_{e_{2,1}} & P_1\ar[d]^{-e_{1,1}} \\
*+[F-:<5pt>]{ 7} & P_2\ar[r]^{e_{1,2}}\ar[d]_{e_{3,2}} & P_1\ar[d]^{-e_{2,1}} \\
*+[F-:<5pt>]{ 6} & P_3\ar[r]^{e_{2,3}}\ar[d]_{e_{4,3}} & P_2\ar[d]^{-e_{3,2}} \\
*+[F-:<5pt>]{ 5} & P_4\ar[r]^{e_{3,4}}\ar[d]_{e_{4,4}} & P_3\ar[d]^{-e_{4,3}} \\
*+[F-:<5pt>]{ 4} & P_4\ar[r]^{e_4} \ar[d]_{e_{3,4}} & P_4\ar[d]^{-e_{4,4}} \\
*+[F-:<5pt>]{ 3} & P_3\ar[r]^{e_{4,3}}\ar[d]_{e_{2,3}} & P_4\ar[d]^{-e_{3,4}} \\
*+[F-:<5pt>]{ 2} & P_2\ar[r]^{e_{3,2}}\ar[d]_{e_{1,2}} & P_3\ar[d]^{-e_{2,3}} \\
*+[F-:<5pt>]{ 1} & P_1\ar[r]^{e_{2,1}}\ar[d]_{e_{1,1}} & P_2\ar[d]^{-e_{1,2}} \\
*+[F-:<5pt>]{ 0} & P_1\ar[r]^{e_1} \ar[d]_{e_{2,1}} & P_1\ar[d] \\
*+[F-:<5pt>]{-1} & P_2\ar[r] \ar[d]_{e_{3,2}} & 0 \ar[d] \\
*+[F-:<5pt>]{-2} & P_3\ar[r] \ar[d]_{e_{4,3}} & 0 \ar[d] \\
*+[F-:<5pt>]{-3} & P_4\ar[r] \ar[d]_{e_{4,4}} & 0 \ar[d] \\
*+[F-:<5pt>]{-4} & P_4\ar[r] \ar[d]_{e_{3,4}} & 0 \ar[d] \\
*+[F-:<5pt>]{-5} & P_3\ar[r] \ar[d]_{e_{2,3}} & 0 \ar[d] \\
*+[F-:<5pt>]{-6} & P_2\ar[r] \ar[d]_{e_{1,2}} & 0 \ar[d] \\
*+[F-:<5pt>]{-7} & P_1\ar[r] \ar[d] & 0 \ar[d] \\
*+[F-:<5pt>]{-8} & 0 \ar[r]\ar[d] & 0 \ar[d] \\
& \rotatebox{-90}{$\cdots)$} & \rotatebox{-90}{$\cdots)$} \\
\]
The element $ι$ is well-defined since $ω(y) = ω(l+y)$ for $y\geq 0$.
In the definition of $χ$ we need to check that the "$\ls * \rs_*^*$" are well defined. This is easily proven by calculating the $ω(y)$ where $y$ is the lower respective upper index of "$\ls * \rs_*^*$".
(a): As $\pr_i = \{0\}$ for $i<0$, the identity element of $A$ is given by
$ι^0 = \dum_{i\geq 0}\ls e_{ω(i)}\rs_{i}^i$,
which agrees with the assertion in case $j=0$. So we have proven the induction basis for induction on $j$. So now assume that for some $j\geq 0$ the assertion holds. Then
\begin{align*}
ι^{j+1} \eqs& ι\circ ι^j = \left(\dum_{i\geq 0} \ls e_{ω(i)} \rs_{i+l}^i\right)\circ\left(\dum_{i'\geq 0} \ls e_{ω(i')} \rs_{i'+jl}^{i'}\right) \\
\eqs& \dum_{i\geq 0} \ls e_{ω(i)}\circ e_{ω(i+l)} \rs_{i+l+jl}^i = \dum_{i\geq 0} \ls e_{ω(i)}\rs_{i+(j+1)l}^i\,.
\end{align*}
Thus the proof by induction is complete.
We have
\begin{align*}
ι^j \circ h \eqs& \left(\sum_{i\geq 0} \sum_{k=0}^{l-1}\ls e_{ω(il+k)} \rs_{(i+j)l+k}^{il+k}\right) \circ \left(\sum_{i'\geq 0}\sum_{k'=0}^{l-1}\ls h_{k'}\rs_{i'l+k'+ y}^{i'l+k'}\right)
\overset{\natop{i'\leadsto i+j}{k'\leadsto k}}{=} \sum_{i\geq 0} \sum_{k=0}^{l-1} \ls h_k\rs_{(i+j)l+k+y}^{il+k}\\
h\circ ι^j \eqs& \left(\sum_{i\geq 0}\sum_{k=0}^{l-1}\ls h_k\rs_{il+k+y}^{il+k}\right) \circ
\left(\sum_{i'\geq 0} \ls e_{ω(i')} \rs_{i'+jl}^{i'}\right)
= \sum_{i\geq 0}\sum_{k=0}^{l-1} \ls h_k\rs_{(i+j)l + k+y}^{il+k}\, .
\end{align*}
So we have proven (b).
(c): The differential $d$ of $\pres\fp$ is $l$-periodic (cf. <ref>) and thus
\begin{align*}
m_1(h)\circ ι^j \eqs& (d\circ h - (-1)^y h\circ d)\circ ι^j\\
\ovs{(b), |ι^j|\equiv_2 0}& d\circ h\circ ι^j - (-1)^{y+|ι^j|} h\circ ι^j\circ d = m_1(h\circ ι^j).
\end{align*}
(d): We have
m_1(ι^j) \overset{(c)}{=} m_1(ι^0)\circ ι^j = (d\circ ι^0 - (-1)^0 ι^0 d)\circ ι^j = (d - d)\circ ι^j = 0.
(e) is implied by (b) using the fact that $χ$ is $l$-periodic.
(f): Because of (c) we have $m_1(χι^j) = m_1(χ)\circ ι^j$. Because $|χ|=l-1$ is odd we have
\begin{align*}
m_1&(χ) = d\circ χ - (-1) χ\circ d = χ\circ d + d\circ χ\\
\ovs{\text{R.}\ref{bem:di}}& \Big(\dum_{i\geq 0}\left(\ls e_1 \rs_{il+l-1}^{il} + \left\lgroup\dum_{k=1}^{p-2}\ls e_{k+1,k}\rs_{il+l-1+k}^{il + k}\right\rgroup+ \ls e_{p-1}\rs_{il+l-1 +(p-1)}^{il+(p-1)}\right.\\
+ \left\lgroup\dum_{k=1}^{p-2}\ls e_{p -k-1,p-k}\rs_{il+l-1+(p-1)+k}^{il +(p-1)+ k}\right\rgroup\right)\Big)
\circ \left(\dum_{y\geq 0} \ls e_{ω(y),ω(y+1)}\rs_{y+1}^y\right)\\
&+\left(\dum_{y\geq 0} \ls e_{ω(y),ω(y+1)}\rs_{y+1}^y\right) \circ \Big(
\dum_{i\geq 0}\left(\ls e_1 \rs_{il+l-1}^{il} + \left\lgroup\dum_{k=1}^{p-2}\ls e_{k+1,k}\rs_{il+l-1+k}^{il + k}\right\rgroup\right.\\
&\left. + \ls e_{p-1}\rs_{il+l-1 +(p-1)}^{il+(p-1)}
+ \left\lgroup\dum_{k=1}^{p-2}\ls e_{p -k-1,p-k}\rs_{il+l-1+(p-1)+k}^{il +(p-1)+ k}\right\rgroup\right)\Big)\\
\eqs&
\dum_{i\geq 0}\Big(
\ls e_1\circ e_{1,1}\rs_{il+l}^{il} + \left\lgroup\dum_{k=1}^{p-2}\ls e_{k+1,k}\circ e_{k,k+1}\rs_{il+l+k}^{il+k}\right\rgroup\\
&+\ls e_{p-1}\circ e_{p-1,p-1}\rs_{il+l+(p-1)}^{il+(p-1)}
+ \left\lgroup\dum_{k=1}^{p-2}\ls e_{p-k-1,p-k}\circ e_{p-k,p-k-1}\rs_{il+l+(p-1)+k}^{il+(p-1)+k}\right\rgroup\Big)\\
&+\dum_{i\geq 1} \ls e_{1,1}\circ e_1\rs_{il+l-1}^{il-1}
+ \dum_{i\geq 0}\Big( \left\lgroup\dum_{k=1}^{p-2}\ls e_{k,k+1}\circ e_{k+1,k}\rs_{il+l+k-1}^{il+k-1}\right\rgroup\\
&+ \ls e_{p-1,p-1}\circ e_{p-1}\rs_{il+l-1+(p-1)}^{il-1+(p-1)}
+ \left\lgroup\dum_{k=1}^{p-2}\ls e_{p-k,p-k-1}\circ e_{p-k-1,p-k}\rs_{il+l-1+(p-1)+k}^{il-1+(p-1) + k}\right\rgroup\Big)\\
\overset{*}{\eqs}& \dum_{i\geq 0}\Big(
\ls e_{1,1} + e_{1,2}\circ e_{2,1}\rs_{il+l}^{il} + \left\lgroup\dum_{k=1}^{p-3}\ls e_{k+1,k}\circ e_{k,k+1}+ e_{k+1,k+2}\circ e_{k+2,k+1}\rs_{il+l+k}^{il+k}\right\rgroup\\
&+ \ls e_{p-1,p-2}\circ e_{p-2,p-1} + e_{p-1,p-1}\rs_{il+l+p-2}^{il+p-2}
+ \ls e_{p-1,p-1}+e_{p-1,p-2}\circ e_{p-2,p-1}\rs_{il+l+p-1}^{il+p-1}\\
&+ \left\lgroup\dum_{k=1}^{p-3}\ls e_{p-k-1,p-k}\circ e_{p-k,p-k-1} + e_{p-k-1,p-k-2}\circ e_{p-k-2,p-k-1}\rs_{il+l+p-1+k}^{il+p-1+k}\right\rgroup\\
&+ \ls e_{1,2}\circ e_{2,1} + e_{1,1}\rs_{(i+1)l+l-1}^{(i+1)l-1}\Big)
\overset{\text{L.}\ref{lem:relfp}(a)}{=} 0
\end{align*}
In the step marked by "$*$" we sort the summands by their targets. Note that when splitting sums of the form $\dum_{k=1}^{p-2}(…)_k$ into $(…)_1+\dum_{k=2}^{p-2}(…)_k$ or into $(…)_{p-2}+\dum_{k=1}^{p-3}(…)_k$, the existence of the summand that is split off is ensured by $p\geq 3$.
(g): We first show that the differentials of the complex $\Hom^*(\pres \fp, \fp)$ (cf. <ref>) are all zero:
By <ref>, $\{ε\}$ is an $\fp$-basis of $\Hom_{\fp\!\Sp}(P_1,\fp)$, and for $k∈[2,p-1]$ we have $\Hom_{\fp\!\Sp}(P_k,\fp) = 0$. So the only non-trivial $(d_k)^*$ are those where $\pr_k = \pr_{k+1} = P_1$. This is the case only when $k=lj + l-1$ for some $j\geq 0$. Then $d_k= e_{1,1}$. For $ε∈\Hom(P_1,\fp)$, we have $(d_k)^* (ε) = (-1)^kε\circ e_{1,1} \overset{\text{L.}\ref{lem:relfp}(a)}{=} 0$. As $ \Hom(P_1,\fp)= \langle ε\rangle_\fp$, we have $(d_k)^* = 0$.
So $\Hm^k \Hom^*(\pres \fp, \fp) = \Hom^k(\pres \fp, \fp)$. We use <ref>.
For $k=jl$, $j\geq 0$, we have $\bar{Ψ}^k(\overline{ι^j}) \overset{(a)}{=} ε$, and $\{ε\}$ is a basis of $\Hm^k \Hom^*(\pres \fp, \fp)$.
For $k=jl+l-1$, $j\geq 0$, we have $\bar{Ψ}^k(\overline{χι^j})\overset{(e)}{=}ε$, and $\{ε\}$ is a basis of $\Hm^k \Hom^*(\pres \fp, \fp)$.
Finally, for $k=jl+r$ for some $j\geq 0$ and some $r∈[1,l-2]$ and for $k<0$, we have $\Hm^k \Hom^*(\pres \fp, \fp)=\{0\}$.
§.§ An A(oo)-structure on Ext*(FpSp)(Fp,Fp) as a minimal model of Hom*(FpSp)(PResFp,PresFp)
Recall that $p\geq 3$ is a prime. Write $A := \Hom^*_{\fp\!\Sp}(\pres\fp,\pres\fp)$, which becomes an $\A_∞$-algebra $(A, (m_n)_{n\geq 1})$ over $R=\fp$ via <ref>. We implement $\Ext^*_{\fp\!\Sp}(\fp,\fp)$ as $\Ext^*_{\fp\!\Sp}(\fp,\fp):=\Hm^*A$.
Our goal in this section is to construct an $\A_\infty$-structure $(m'_n)_{n\geq 1}$ on $\Hm^*A$ and a morphism of $\A_\infty$-algebras $f=(f_n)_{n\geq 1}:(\Hm^*A,(m'_n)_{n\geq 1})→(A, (m_n)_{n\geq 1})$ which satisfy the statements of <ref>. I.e. we will construct a minimal model of $A$.
In preparation of the definitions of the $f_n$ and $m'_n$, we name and examine certain elements of $A$:
Suppose given $k∈[2,p-1]$. We set
\begin{align*}
γ_k:= \dum_{i\geq 0}\left( \ls e_k \rs_{k(l-1)+li}^{k-1+li}
+\ls e_{p-k} \rs_{k(l-1) + (p-1) + li}^{k-1+(p-1)+li}\right)∈A^{k(l-2)+1}.
\end{align*}
For $j\geq 0$, we have
\begin{align*}
γ_kι^j := γ_k\circ ι^j = ι^j\circ γ_k = \dum_{i\geq 0}\left( \ls e_k \rs_{k(l-1)+l(i+j)}^{k-1+li}
+ \ls e_{p-k} \rs_{k(l-1) + (p-1) + l(i+j)}^{k-1+(p-1)+li}\right) ∈A^{k(l-2)+1+jl}
\end{align*}
\begin{align*}
m_1(γ_kι^j)\eqs&\dum_{i\geq 0}\left( \ls e_{k-1,k} \rs_{k(l-1)+l(i+j)}^{k-2+li}
+ %\dum_{i\geq 0}
\ls e_{p-k+1,p-k}\rs_{k(l-1) + (p-1) + l(i+j)}^{k-2+(p-1)+li}\right.\\
&\left.+%\dum_{i\geq 0}
\ls e_{k,k-1}\rs_{k(l-1)+1+l(i+j)}^{k-1+li}
+ %\dum_{i\geq 0}
\ls e_{p-k,p-(k-1)}\rs_{k(l-1) + p + l(i+j)}^{k-1+(p-1)+li}\right).
\end{align*}
We need to prove that $γ_k$ is well-defined. Let $i\geq 0$.
We consider the first term. The complex $\pres \fp$ (cf. (<ref>), (<ref>)) has entry $P_k$ at position $k(l-1)+li$ and at position $k-1+li$:
We have $k(l-1)+li = (k-1+i)l + l - k$. So $ω(k(l-1)+li) = l - (l - k) = k$ since $p-1\leq l-k \leq l-1$.
We have $ω(k-1+li) = (k-1)+1 = k$ since $0\leq k-1 \leq p-2$. As $k(l-1)+li, k-1+li\geq 0$, we have $\pr_{k(l-1)+li}=P_{ω(k(l-1)+li)}=P_k$ and $\pr_{k-1+li} = P_{ω(k-1+li)} = P_k$. So the first term is well-defined.
Now consider the second term. The complex $\pres \fp$ has entry $P_{p-k}$ at position $k(l-1) + (p-1) + li$ and at position $k-1+(p-1)+li$:
We have $k(l-1) + (p-1) + li = (i+k)l + (p-1)-k$, so $ω(k(l-1) + (p-1)+li) = (p-1)-k + 1 = p-k$ since $0\leq (p-1)-k \leq p-2$. We have $ω(k-1+(p-1)+li) = 2(p-1) - (k-1) -(p-1) = p-k$ since $p-1\leq k-1 + (p-1) \leq 2(p-1)-1$. As $k(l-1) + (p-1) + li, k-1+(p-1)+li\geq 0$, we have $\pr_{k(l-1)+(p-1)+li} = P_{ω(k(l-1)+(p-1)+li)} = P_{p-k}$ and $\pr_{k-1+(p-1)+li} = P_{ω(k-1+(p-1)+li)} = P_{p-k}$. So the second term is well-defined.
The degree of the tuple of maps is computed to be $(k(l-1) + li) - (k-1+li) = k(l-2) + 1 = (k(l-1)+(p-1)+li) - (k-1+(p-1)+li)$.
The explicit formula for $γ_kι^j$ is an application of <ref>(b).
The degree $|γ_kι^j| = k(l-2) + 1$ is odd, so
\begin{align*}
m_1(γ_kι^j)\,\, \ovs{\text{L.}\ref{lem:cai}}& d \circ γ_kι^j + γ_kι^j \circ d\\
\ovs{\text{R.}\ref{bem:di}}& \dum_{i\geq 0} \ls e_{ω(k-2),ω(k-1)}\rs_{k-1+li}^{k-2+li} \circ \dum_{i\geq 0} \ls e_k \rs_{k(l-1)+l(i+j)}^{k-1+li} \\
&+ \dum_{i\geq 0} \ls e_{ω(p-1+k-2),ω(p-1+k-1)}\rs_{k-1+(p-1)+li}^{k-2+(p-1)+li}\circ \dum_{i\geq 0} \ls e_{p-k} \rs_{k(l-1) + (p-1) + l(i+j)}^{k-1+(p-1)+li} \\
&+\dum_{i\geq 0} \ls e_k \rs_{k(l-1)+l(i+j)}^{k-1+li} \circ \dum_{i\geq 0} \ls e_{ω(l-k),ω(l-k+1)}\rs_{k(l-1)+1+l(i+j)}^{k(l-1)+l(i+j)}\\
&+ \dum_{i\geq 0} \ls e_{p-k} \rs_{k(l-1) + (p-1) + l(i+j)}^{k-1+(p-1)+li}\circ
\dum_{i\geq 0} \ls e_{ω(p-1-k),ω(p-k)}\rs_{k(l-1) + p + l(i+j)}^{k(l-1) + (p-1) + l(i+j)}\\
\eqs&\dum_{i\geq 0} \ls e_{k-1,k} \rs_{k(l-1)+l(i+j)}^{k-2+li}
+ \dum_{i\geq 0} \ls e_{p-k+1,p-k}\rs_{k(l-1) + (p-1) + l(i+j)}^{k-2+(p-1)+li}\\
&+\dum_{i\geq 0} \ls e_{k,k-1}\rs_{k(l-1)+1+l(i+j)}^{k-1+li}
+ \dum_{i\geq 0} \ls e_{p-k,p-(k-1)}\rs_{k(l-1) + p + l(i+j)}^{k-1+(p-1)+li}\\
\end{align*}
Note that in the second line $k-2 +li\geq 0$ as $i\geq 0$ and $k\geq 2$.
For $j,j'\geq 0$, we have
χι^j \circ χι^{j'} = m_1(γ_2ι^{j+j'}).
It suffices to prove that $χ\circ χ = m_1(γ_2)$ since then $χι^j \circ χι^{j'} \overset{\text{P.}\ref{pp:iota}(e)}{=} χ\circ χ\circ ι^{j+j'} = m_1(γ_2)\circ ι^{j+j'} \overset{\text{P.}\ref{pp:iota}(c)}{=} m_1(γ_2ι^{j+j'})$.
To determine when a composite is zero, we will need the following. For , we examine the condition
\begin{align}
\label{mulccg}
il +l - 1 + k = i'l + k'.
\end{align}
If $k=0$ then (<ref>) holds iff $i=i'$ and $k' = l-1$.
If $k\geq 1$ then (<ref>) holds iff $i+1=i'$ and $k'=k-1$.
\begin{align*}
χ\circ χ \ovs{p\geq 3\vphantom{I_{I_y}}}&%\overset{p\geq 3\vphantom{I_{I_y}}}{=}
\left(\dum_{i\geq 0}\left(\ls e_1 \rs_{il+l-1}^{il} + \ls e_{2,1}\rs_{il+l}^{il+1}+\left\lgroup\dum_{k=2}^{p-2}\ls e_{k+1,k}\rs_{il+l-1+k}^{il + k}\right\rgroup\right.\right.\\
& + \ls e_{p-1}\rs_{il+l-1 +(p-1)}^{il+(p-1)}
+ \ls e_{p-2,p-1}\rs_{il+l+p-1}^{il+p}
+\left.\left.\left\lgroup\dum_{k=2}^{p-2}\ls e_{p -k-1,p-k}\rs_{il+l-1+(p-1)+k}^{il +(p-1)+ k}\right\rgroup\right)\right)\\
&\circ\left(\dum_{i'\geq 0}\left(\ls e_1 \rs_{i'l+l-1}^{i'l} + \left\lgroup\dum_{k'=1}^{p-3}\ls e_{k'+1,k'}\rs_{i'l+l-1+k'}^{i'l + k'}\right\rgroup\right.\right.
+ \ls e_{p-1,p-2}\rs_{i'l+l+p-3}^{i'l+p-2}\\
& + \ls e_{p-1}\rs_{i'l+l-1 +(p-1)}^{i'l+(p-1)}
\left.\left.+ \left\lgroup\dum_{k'=1}^{p-3}\ls e_{p -k'-1,p-k'}\rs_{i'l+l-1+(p-1)+k'}^{i'l +(p-1)+ k'}\right\rgroup+ \ls e_{1,2}\rs_{i'l+l+2(p-2)}^{i'l+l-1}\right)\right)\\
\eqs&
\dum_{i\geq 0}\Big(\ls e_1\circ e_{1,2} \rs_{il+l+2(p-2)}^{il} + \ls e_{2,1}\circ e_1\rs_{il+2l-1}^{il+1}\\
& +\left\lgroup\dum_{k=2}^{p-2}\right.\!\underbrace{\ls e_{k{+}1,k}\circ e_{k,k{-}1}\rs_{il+2l-1+k-1}^{il + k}}_{=0\text{ by L.\ref{lem:relfp}(c)}}\left.\vphantom{\dum_k^p}\!\!\right\rgroup + \ls e_{p-1}\circ e_{p-1,p-2}\rs_{il+2l+p-3}^{il+(p-1)}\\
&+ \ls e_{p-2,p-1}\circ e_{p-1}\rs_{il+2l+p-2}^{il+p}
+\left\lgroup\dum_{k=2}^{p-2}\right.\underbrace{\ls e_{p -k-1,p-k}\circ e_{p-k,p-k+1}\rs_{il+2l-1+p-1+k-1}^{il +(p-1)+ k}}_{=0\text{ by L.\ref{lem:relfp}(c)}}\left.\vphantom{\dum_k^p}\!\right\rgroup\Big)\\
\eqs& \dum_{i\geq 0}\Big(\ls e_{1,2} \rs_{(i+2)l-2}^{il} + \ls e_{2,1}\rs_{(i+2)l-1}^{il+1}
+\ls e_{p-1,p-2}\rs_{(i+2)l+p-3}^{il+p-1} + \ls e_{p-2,p-1}\rs_{(i+2)l+p-2}^{il+p}
\Big)\\
\ovs{\text{L.}\ref{lem:gamma}}& m_1(γ_2)
\end{align*}
Below are the definitions which will give a minimal $\A_∞$-algebra structure on $\Hm^*A$ and a quasi-isomorphism of $\A_∞$-algebras $\Hm^*A → A$.
Recall from <ref> that $\mathfrak{B}= \mbox{$\{\overline{ι^j}\mid j\geq 0\}\sqcup \{\overline{χι^j}\mid j\geq 0\}$} = \{\overline{χ^aι^j} \mid j\geq 0, a∈\{0,1\}\}$ is a basis of $\Hm^*A$. For $n∈\Z_{\geq 1}$, we set
\begin{align*}
\mathfrak{B}^{\otimes n} :\eqs& \{\overline{χ^{a_1}ι^{j_1}}\otimes … \otimes \overline{χ^{a_n}ι^{j_n}} ∈ (\Hm^*A)^{\otimes n} \mid a_i∈\{0,1\} \text{ and } j_i∈\Z_{\geq 0} \text{ for all }i∈[1,n] \},%\\
% \eqs& \{b_1\otimes … \otimes b_n \mid b_1,…,b_n∈\mathfrak{B}\},
\end{align*}
which is a basis of $(\Hm^* A)^{\otimes n}$ consisting of homogeneous elements.
For $n\geq 1$, we define the $\fp$-linear map $f_n:(\Hm^*A)^{\otimes n}→ A$ as follows:
Case $n=1$: $f_1$ is given on $\mathfrak{B}$ by $f_1(\overline{ι^j}):=ι^j$ and $f_1(\overline{χι^j}):= χι^j$.
Case $n∈[2,p-1]$: $f_n$ is given on elements of $\mathfrak{B}^{\otimes n}$
\begin{align*}
f_n(\overline{χ^{a_1}ι^{j_1}}\otimes … \otimes \overline{χ^{a_n}ι^{j_n}}) := \begin{cases}
0 & \text{if }∃i∈[1,n]: a_i = 0 \\
(-1)^{n-1}γ_nι^{j_1+…+j_n} & \text{if } 1=a_1 = a_2 = … = a_n
\end{cases}
\end{align*}
Case $n\geq p$: We set $f_n:= 0$.
For $n\geq 1$, we define the $\fp$-linear map $m'_n:(\Hm^*A)^{\otimes n}→ \Hm^*A$ by defining it on elements $\overline{χ^{a_1}ι^{j_1}}\otimes … \otimes \overline{χ^{a_n}ι^{j_n}}∈\mathfrak{B}^{\otimes n}$:
Case $∃i∈[1,n]: a_i = 0$: $ $
$m'_n(\overline{χ^{a_1}ι^{j_1}}\otimes … \otimes \overline{χ^{a_n}ι^{j_n}}) := 0$ for $n\neq 2$ and
m'_2(\overline{χ^{a_1}ι^{j_1}} \otimes \overline{χ^{a_2}ι^{j_2}}) := \overline{χ^{a_1+a_2}ι^{j_1+j_2}}% \text{ note $a_1+a_2∈[0,1]$}
$ (Note that $a_1+a_2∈\{0,1\}$).
Case $a_1 = a_2 = … = a_n=1$: $ $
$m'_n(\overline{χι^{j_1}}\otimes … \otimes \overline{χι^{j_n}}):= 0$ for $n\neq p$ and
$m'_p(\overline{χι^{j_1}}\otimes … \otimes \overline{χι^{j_p}}) := (-1)^p\overline{ι^{p-1+j_1+…+j_p}}=-\overline{ι^{p-1+j_1+…+j_p}}$.
Note that since $p\geq 3$, we have $m'_2(\overline{χι^{j_1}} \otimes \overline{χι^{j_2}}) = 0$ for $j_1,j_2\geq 0$.
The pair $(\Hm^*A, (m'_n)_{n\geq 1})$ is a minimal $\A_∞$-algebra.
The tuple $(f_n)_{n\geq 1}$ is an quasi-isomorphism of $\A_∞$-algebras from $(\Hm^*A, (m'_n)_{n\geq 1})$ to $(A, (m_n)_{n\geq 1})$. More precisely, induces the identity in homology.
The proof of <ref> will take the remainder of <ref>. We will use <ref>.
The maps $f_n$ and $m'_n$ have degree $|f_n| = 1-n$ and $|m'_n| = 2-n$.
I.e. $(f_n)_{n\geq 1}$ is a pre-$A_∞$-morphism from $\Hm^* A$ to $A$, and $(\Hm^*A, (m'_n)_{n\geq 1})$ is a pre-$\A_∞$-algebra.
We have $|f_1|=0$ as $|\overline{ι^j}| = |ι^j|$ and $|\overline{χι^j}|=|χι^j|$. For $n\geq p$ the map $f_n$ is of degree $1-n$ as $f_n=0$. For $n∈[2,p-1]$ the statement $|f_n|=1-n$ is proven by checking the degrees for the elements of the basis $\mathfrak{B}^{\otimes n}$ whose image under $f_n$ is non-zero:
\begin{align*}
|f_n(\overline{χι^{j_1}}\otimes … \otimes \overline{χι^{j_n}})| \eqs&
\overset{\text{L.}\ref{lem:gamma}}{=} (j_1+…+j_n)l + n(l-1) + 1-n\\
\eqs& 1-n+\dum_{x=1}^n |\overline{χι^{j_x}}| = 1-n + |\overline{χι^{j_1}}\otimes … \otimes \overline{χι^{j_n}}|
\end{align*}
Thus $|f_n|=1-n$ for all $n$ and we have proven the first statement.
Now we show $|m'_n|=2-n$. As before, we only need check the degrees for basis elements whose image is non-zero: For $\overline{χ^{a_1}ι^{j_1}}\otimes\overline{χ^{a_2}ι^{j_2}}$, $j_1,j_2\geq 0$, $a_1,a_2∈\{0,1\}$, $0∈\{a_1,a_2\}$, we have
\begin{align*}
|m'_2(\overline{χ^{a_1}ι^{j_1}}\otimes\overline{χ^{a_2}ι^{j_2}})| \eqs& |\overline{χ^{a_1+a_2}ι^{j_1+j_2}}| = (a_1+a_2)(l-1)+l(j_1+j_2)\\
\eqs& a_1(l-1) + j_1l + a_2(l-1) + j_2l = |\overline{χ^{a_1}ι^{j_1}}\otimes\overline{χ^{a_2}ι^{j_2}}| + (2-2).
\end{align*}
For $\overline{χι^{j_1}}\otimes \cdots \otimes \overline{χι^{j_p}}$, $j_x\geq 0$ for $x∈[1,p]$, we have
\begin{align*}
|m'_p(\overline{χι^{j_1}}\otimes \cdots \otimes \overline{χι^{j_p}})| \eqs& |\overline{ι^{p-1+j_1+…+j_p}}| = l(p-1+j_1+…+j_p)\\
\eqs& lp - l + l(j_1+…+j_p) = lp - 2p+2 + l(j_1+…+j_p)\\
\eqs& p(l-1) + l(j_1+…+j_p) + 2-p
= |\overline{χι^{j_1}}\otimes \cdots \otimes \overline{χι^{j_p}}| + 2-p
\end{align*}
We have $m'_1=0$. The equation (<ref>)$[1]$ holds. The complex morphism $f_1:(A',m'_1)→(A,m_1)$ is a quasi-isomorphism inducing the identity in homology.
The equality $m'_1=0$ follows immediately from the definition. Thus $m_1\circ f_1 = 0 = f_1\circ m'_1$. Moreover $f_1$ is a quasi-isomorphism inducing the identity in homology by construction, cf. <ref>(g).
The map $f_1$ is injective.
The set $X:=\{ χ^aι^j \mid a∈\{0,1\}, j∈\Z_{\geq 1}\}\subseteq A$ is linearly independent, since it consists of nonzero elements of different summands of the direct sum $A=\bigoplus_{k∈\Z}\Hom^k(\pres\fp,\pres\fp)$.
The set $\mathfrak{B}$, which is a basis of $\Hm^* A$, is mapped bijectively to $X$ by $f_1$, so $f_1$ is injective.
The equation (<ref>)$[2]$ holds.
As $m'_1=0$, equation (<ref>)[$2$] is equivalent to (cf. (<ref>))
\begin{align*}
%&&f_1\circ m'_2 -f_2\circ (1\otimes m'_1) -f_2\circ (m'_1\otimes 1) \eqs& m_1\circ f_2 + %m_2\circ (f_1\otimes f_1)\\
f_1\circ m'_2 \eqs& m_1\circ f_2 + m_2\circ (f_1\otimes f_1).
\end{align*}
We check this equation on $\mathfrak{B}^{\otimes 2}$: Recall <ref>.
\begin{align*}
f_1m'_2(\overline{ι^j}\otimes \overline{ι^{j'}}) \eqs& ι^{j+j'}
= m_2(f_1\otimes f_1)(\overline{ι^j}\otimes \overline{ι^{j'}})
= (m_1\circ f_2 + m_2\circ (f_1\otimes f_1))(\overline{ι^j}\otimes \overline{ι^{j'}})\\
f_1m'_2(\overline{ι^j}\otimes \overline{χι^{j'}})\eqs& χι^{j+j'}
= m_2(f_1\otimes f_1)(\overline{ι^j}\otimes \overline{χι^{j'}})\\
\eqs& (m_1\circ f_2 + m_2\circ (f_1\otimes f_1)) (\overline{ι^j}\otimes \overline{χι^{j'}})\\
f_1m'_2(\overline{χι^j} \otimes \overline{ι^{j'}}) \eqs& χι^{j+j'}
= m_2(f_1\otimes f_1)(\overline{χι^j} \otimes \overline{ι^{j'}})\\
\eqs& (m_1\circ f_2 + m_2\circ (f_1\otimes f_1))(\overline{χι^j} \otimes \overline{ι^{j'}})\\
f_1m'_2(\overline{χι^j}\otimes \overline{χι^{j'}}) \eqs &0 \overset{\text{L.}\ref{lem:chichi}}{=}
-m_1(γ_2ι^{j+j'}) + m_2(f_1\otimes f_1)(\overline{χι^j}\otimes \overline{χι^{j'}})\\
\eqs& (m_1\circ f_2 + m_2\circ (f_1\otimes f_1))(\overline{χι^j}\otimes \overline{χι^{j'}})
\end{align*}
Note that there are no additional signs due to the Koszul sign rule since $|f_1|=0$.
The following results directly from <ref>.
For $n\geq 2$ and $a_1,…,a_n∈\{0,1\}$, $j_1,…,j_n\geq 0$, we have
\begin{align*}
f_n(\overline{χ^{a_1}ι^{j_1}}\otimes … \otimes \overline{χ^{a_n}ι^{j_n}})= f_n(\overline{χ^{a_1}}\otimes … \otimes \overline{χ^{a_n}}) \circ ι^{j_1+…+j_n}.
\end{align*}
If there is additionally an $x∈[1,n]$ with $a_x=0$ then
\[f_n(\overline{χ^{a_1}ι^{j_1}}\otimes … \otimes \overline{χ^{a_n}ι^{j_n}})=0.\]
Equation (<ref>)[$n$] can be reformulated as
\begin{align*}
&f_1\circ m'_n+\underbrace{\sum_{\substack{n=r+s+t \\ r,t\geq 0,s\geq 1\\ s\leq n-1}} (-1)^{rs+t} f_{r+1+t}\circ (1^{\otimes r}\otimes m'_s\otimes 1^{\otimes t})}_{=:Φ_n}\\
&= m_1\circ f_n +\underbrace{\sum_{\substack{2\leq r\leq n \\ i_1+…+i_r=n\\ i_s\geq 1}} (-1)^v m_r\circ (f_{i_1}\otimes f_{i_2}\otimes … \otimes f_{i_r})}_{=:Ξ_n}\,,
\end{align*}
where $v= \sum_{1\leq t < s\leq r}(1-i_s)i_t$.
A term of the form $f_{r+1+t}\circ (1^{\otimes r}\otimes m'_s\otimes 1^{\otimes t})$, $s\geq 3$, $r+t\geq 1$, is zero because of <ref> and the definition of $m'_p$. Also recall $m'_1=0$. Thus
\begin{align}
\label{eqphi}
Φ_n %\eqs& \sum_{\mathclap{n=r+2+t}} (-1)^{r+2t}f_{n-1}(1^{\otimes r}\otimes m'_2\otimes 1^{\otimes t}) \nonumber \\
% \eqs& \sum_{r=0}^{n-2} (-1)^r f_{n-1}(1^{\otimes r}\otimes m'_2\otimes 1^{\otimes n-r-2})
\eqs&\sum_{\mathclap{\substack{n=r+2+t\\ r,t\geq 0}}} (-1)^{2r+t}f_{n-1}\circ (1^{\otimes r}\otimes m'_2\otimes 1^{\otimes t}) = \sum_{r=0}^{n-2} (-1)^{n-r} f_{n-1}\circ (1^{\otimes r}\otimes m'_2\otimes 1^{\otimes n-r-2}).
\end{align}
Because of $m_k=0$ for $k\geq 3$, we have
\begin{align}
\label{eqxi}
Ξ_n %\eqs& \sum_{i_1+i_2 = n}(-1)^s m_2(f_{i_1}\otimes f_{i_2}) = \sum_{i=1}^{n-1}(-1)^{i-1}m_2(f_i\otimes f_{n-i})
\eqs& \sum_{\substack{i_1+i_2 = n \\ i_1,i_2\geq 1}}(-1)^{(1-i_2)i_1} m_2\circ (f_{i_1}\otimes f_{i_2}) = \sum_{i=1}^{n-1}(-1)^{ni}m_2\circ (f_i\otimes f_{n-i}).
\end{align}
We have proven:
For $n\geq 1$, condition (<ref>)$[n]$ is equivalent to $f_1\circ m'_n + Φ_n = m_1\circ f_n + Ξ_n$ where $Φ_n$ and $Ξ_n$ are as in (<ref>) and (<ref>).
Condition (<ref>)$[n]$ holds for $n\geq 3$ and arguments $\overline{χ^{a_1}ι^{j_1}}\otimes…\otimes\overline{χ^{a_n}ι^{j_n}}∈\mathfrak{B}^{\otimes n}=\{\overline{χ^{a_1}ι^{j_1}}\otimes … \otimes \overline{χ^{a_n}ι^{j_n}} ∈ (\Hm^*A)^{\otimes n} \mid a_i∈\{0,1\} \text{ and } j_i∈\Z_{\geq 0} \text{ for all }i∈[1,n] \}$ where $0∈\{a_1,…,a_n\}$.
Because of <ref> it is sufficient to show
\[Φ_n(\overline{χ^{a_1}ι^{j_1}}\otimes…\otimes\overline{χ^{a_n}ι^{j_n}}) = Ξ_n(\overline{χ^{a_1}ι^{j_1}}\otimes…\otimes\overline{χ^{a_n}ι^{j_n}})\]
if at least one $a_x$ equals $0$.
Case 1 At least two $a_x$ equal $0$:
To show $Φ_n(\overline{χ^{a_1}ι^{j_1}}\otimes…\otimes\overline{χ^{a_n}ι^{j_n}})=0$, we show
$f_{n-1}(1^{\otimes r}\otimes m'_2\otimes 1^{\otimes n-r-2})(\overline{χ^{a_1}ι^{j_1}}\otimes…\otimes\overline{χ^{a_n}ι^{j_n}})=0$ for $r∈[0,n-2]$: In case both components of the argument of $m'_2$ are of the form $\overline{χ^0ι^j}$, the result of $m'_2$ is of the form $\overline{ι^{j'}}$ (see <ref>). Since $2\leq n-1$, <ref> implies the result of $f_{n-1}$ is zero. Otherwise at least one of the components of the argument of $f_{n-1}$ must be of the form $\overline{ι^j}$ and the result of $f_{n-1}$ is zero as well. So $Φ_n(\overline{χ^{a_1}ι^{j_1}}\otimes…\otimes\overline{χ^{a_n}ι^{j_n}})=0$.
To show $Ξ_n(\overline{χ^{a_1}ι^{j_1}}\otimes…\otimes\overline{χ^{a_n}ι^{j_n}})=0$, we show $m_2(f_i\otimes f_{n-i})(\overline{χ^{a_1}ι^{j_1}}\otimes…\otimes\overline{χ^{a_n}ι^{j_n}})=0$ for $i∈[1,n-1]$:
* Suppose given $i∈[2,n-2]$: The statements $a_1=…=a_i=1$ and $a_{i+1}=…=a_n=1$ cannot be true at the same time, so $f_i(…)=0$ or $f_{n-i}(…)=0$ and we have $m_2(f_i\otimes f_{n-i})(\overline{χ^{a_1}ι^{j_1}}\otimes…\otimes\overline{χ^{a_n}ι^{j_n}})=0$.
* Suppose that $i=1$. Because at least two $a_x$ equal $0$ the statement $a_2=…=a_n=1$ cannot be true. Since $n-1\geq 2$, we have $f_{n-1}(…)=0$ and $m_2(f_1\otimes f_{n-1})(\overline{χ^{a_1}ι^{j_1}}\otimes…\otimes\overline{χ^{a_n}ι^{j_n}})=0$.
* The case $i=n-1$ is analogous to the case $i=1$.
So we have $Φ_n(\overline{χ^{a_1}ι^{j_1}}\otimes…\otimes\overline{χ^{a_n}ι^{j_n}}) = 0=Ξ_n(\overline{χ^{a_1}ι^{j_1}}\otimes…\otimes\overline{χ^{a_n}ι^{j_n}})$.
Case 2a Exactly one $a_x$ equals $0$, where $x∈[2,n-1]$.
We have $Φ_n(\overline{χ^{a_1}ι^{j_1}}\otimes…\otimes\overline{χ^{a_n}ι^{j_n}})=0$: In case $n\geq p+1$, it follows from $f_{n-1}=0$. Let us check the case $n∈[3,p]$: Because of <ref>, we have unless $r∈\{x-2,x-1\}$. So
\begin{align*}
\eqs& (-1)^{n-x+2}f_{n-1}(1^{\otimes x-2}\otimes m'_2\otimes 1^{\otimes n-x} - 1^{\otimes x-1}\otimes m'_2\otimes 1^{n-x-1})\\
\eqs& (-1)^{n-x}f_{n-1}(\overline{χι^{j_1}}\otimes …\otimes\overline{χι^{j_{x-2}}}\otimes m'_2(\overline{χι^{j_{x-1}}}\otimes \overline{ι^{j_x}})\otimes \overline{χι^{j_{x+1}}}\otimes…\otimes \overline{χι^{j_n}}\\
&- \overline{χι^{j_1}}\otimes …\otimes\overline{χι^{j_{x-1}}}\otimes m'_2(\overline{ι^{j_{x}}}\otimes \overline{χι^{j_{x+1}}})\otimes \overline{χι^{j_{x+2}}}\otimes…\otimes \overline{χι^{j_n}})\\
\eqs& (-1)^{n-x}f_{n-1}(\overline{χι^{j_1}}\otimes …\otimes\overline{χι^{j_{x-2}}}\otimes \overline{χι^{j_{x-1}+j_x}}\otimes \overline{χι^{j_{x+1}}}\otimes…\otimes \overline{χι^{j_n}}\\
&- \overline{χι^{j_1}}\otimes …\otimes\overline{χι^{j_{x-1}}}\otimes \overline{χι^{j_{x}+j_{x+1}}}\otimes \overline{χι^{j_{x+2}}}\otimes…\otimes \overline{χι^{j_n}})\\
\eqs& (-1)^{n-x}((-1)^{n-2}γ_{n-1}ι^{j_1+…+j_n} - (-1)^{n-2}γ_{n-1}ι^{j_1+…+j_n}) = 0
\end{align*}
To show $Ξ_n(\overline{χ^{a_1}ι^{j_1}}\otimes…\otimes\overline{χ^{a_n}ι^{j_n}})=0$, we show $m_2(f_i\otimes f_{n-i})(\overline{χ^{a_1}ι^{j_1}}\otimes…\otimes\overline{χ^{a_n}ι^{j_n}})=0$ for $i∈[1,n-1]$: The element $χ^{a_x}ι^{j_x}$ is a tensor factor of the argument of $f_i$ or of $f_{n-i}$. We write $y=i$ or $y=n-i$ accordingly. Then $y\geq 2$ since $x\notin\{1,n\}$, so $f_y(…) = 0$ and thus $m_2(f_i\otimes f_{n-i})(\overline{χ^{a_1}ι^{j_1}}\otimes…\otimes\overline{χ^{a_n}ι^{j_n}})=0$.
So $Φ_n(\overline{χ^{a_1}ι^{j_1}}\otimes…\otimes\overline{χ^{a_n}ι^{j_n}}) = 0=Ξ_n(\overline{χ^{a_1}ι^{j_1}}\otimes…\otimes\overline{χ^{a_n}ι^{j_n}})$.
Case 2b Only $a_1=0$, all other $a_x$ equal $1$.
We have $f_{n-1}(1^{\otimes r}\otimes m'_2\otimes 1^{\otimes n-r-2})(\overline{χ^{a_1}ι^{j_1}}\otimes…\otimes\overline{χ^{a_n}ι^{j_n}})=0$ unless $r=0$. So
\begin{align*}
\eqs&(-1)^nf_{n-1}(1^{\otimes 0}\otimes m'_2\otimes 1^{\otimes n-2})(\overline{χ^{a_1}ι^{j_1}}\otimes…\otimes\overline{χ^{a_n}ι^{j_n}})\\
\eqs&(-1)^n f_{n-1}(m'_2(\overline{ι^{j_1}}\otimes \overline{χι^{j_2}})\otimes \overline{χι^{j_3}}\otimes …\otimes \overline{χι^{j_n}})\\
\eqs&(-1)^n f_{n-1}(\overline{χι^{j_1+j_2}}\otimes \overline{χι^{j_3}}\otimes …\otimes \overline{χι^{j_n}})\\
%\eqs& s_{n-1}γ_{n-1}ι^{j_1+…+j_n}
\eqs& \begin{cases} γ_{n-1}ι^{j_1+…+j_n} & 3\leq n\leq p\\
0 & n\geq p+1 \end{cases}
\end{align*}
We have $(f_i\otimes f_{n-1})(\overline{χ^{a_1}ι^{j_1}}\otimes…\otimes\overline{χ^{a_n}ι^{j_n}})=0$ if $i\geq 2$. So
\begin{align*}
\eqs& (-1)^{1\cdot n}m_2(f_1\otimes f_{n-1})(\overline{χ^{a_1}ι^{j_1}}\otimes…\otimes\overline{χ^{a_n}ι^{j_n}})\\
\ovs{\eqref{koszulraw}}& (-1)^nm_2\left( (-1)^{n\cdot |ι^{j_1}|}f_1(\overline{ι^{j_1}})\otimes f_{n-1}(\overline{χι^{j_2}}\otimes…\otimes\overline{χι^{j_n}})\right)\\
\eqs& (-1)^n m_2\left(ι^{j_1}\otimes f_{n-1}(\overline{χι^{j_2}}\otimes…\otimes\overline{χι^{j_n}})\right)\\% = s_{n-1}γ_{n-1}ι^{j_1+…+j_n}
\eqs& \begin{cases} (-1)^nm_2(ι^{j_1}\otimes (-1)^{n-2}γ_{n-1}ι^{j_2+…+j_n}) & 3\leq n\leq p\\
0 & n\geq p+1 \end{cases}\\
\eqs& \begin{cases} γ_{n-1}ι^{j_1+…+j_n} & 3\leq n\leq p\\
0 & n\geq p+1 \end{cases}
% = ι^{j_1}\circ f_{n-1}(χι^{j_2}\otimes…\otimesχ^{j_n})
\end{align*}
So $Φ_n(\overline{χ^{a_1}ι^{j_1}}\otimes…\otimes\overline{χ^{a_n}ι^{j_n}}) =Ξ_n(\overline{χ^{a_1}ι^{j_1}}\otimes…\otimes\overline{χ^{a_n}ι^{j_n}})$.
Case 2c Only $a_n=0$, all other $a_x$ equal $1$.
Argumentation analogous to case 2b gives
\begin{align*}
\eqs&(-1)^{2}f_{n-1}(1^{\otimes n-2}\otimes m'_2\otimes 1^{\otimes 0})(\overline{χ^{a_1}ι^{j_1}}\otimes…\otimes\overline{χ^{a_n}ι^{j_n}})\\
\ovs{|m'_2|=0}&f_{n-1}(\overline{χι^{j_1}}\otimes …\otimes \overline{χι^{j_{n-2}}}\otimes m'_2(\overline{χι^{j_{n-1}}}\otimes \overline{ι^{j_n}}))\\
\eqs& \begin{cases} (-1)^{n-2}γ_{n-1}ι^{j_1+…+j_n} & 3\leq n\leq p\\
0 & n\geq p+1 \end{cases}
\end{align*}
\begin{align*}
\eqs& (-1)^{n(n-1)}m_2(f_{n-1}\otimes f_{1})(\overline{χ^{a_1}ι^{j_1}}\otimes…\otimes\overline{χ^{a_n}ι^{j_n}})\\
\ovs{|f_1|=0}& m_2\left(f_{n-1}(\overline{χι^{j_1}}\otimes…\otimes\overline{χι^{j_{n-1}}})\otimes f_{1}(\overline{ι^{j_n}})\right)\\
%\eqs& m_2( s_{n-1}γ_{n-1}ι^{j_1+…+j_{n-1}} \otimes ι^{j_n})\\
\eqs& \begin{cases} (-1)^{n-2}γ_{n-1}ι^{j_1+…+j_n} & 3\leq n\leq p \\
0& n\geq p+1 \end{cases}
\end{align*}
So $Φ_n(\overline{χ^{a_1}ι^{j_1}}\otimes…\otimes\overline{χ^{a_n}ι^{j_n}}) =Ξ_n(\overline{χ^{a_1}ι^{j_1}}\otimes…\otimes\overline{χ^{a_n}ι^{j_n}})$.
Now we examine the cases where $a_1=…=a_n=1$:
For $n\geq 3$, we have $Φ_n(\overline{χι^{j_1}}\otimes…\otimes\overline{χι^{j_n}})=0$ for $\overline{χι^{j_1}}\otimes…\otimes\overline{χι^{j_n}}∈\mathfrak{B}^{\otimes n}=\{\overline{χ^{a_1}ι^{j_1}}\otimes … \otimes \overline{χ^{a_n}ι^{j_n}} ∈ (\Hm^*A)^{\otimes n} \mid a_i∈\{0,1\} \text{ and } j_i∈\Z_{\geq 0} \text{ for all }i∈[1,n] \}$.
We have $Φ_n(\overline{χι^{j_1}}\otimes…\otimes\overline{χι^{j_n}})=0$ since $Φ_n=\sum_{r=0}^{n-2} (-1)^{n-r} f_{n-1}(1^{\otimes r}\otimes m'_2\otimes 1^{\otimes n-r-2})$ and the argument of $m'_2$ is always of the form $\overline{χι^x}\otimes \overline{χι^y}$, whence its result is zero.
Condition (<ref>)$[n]$ holds for $n∈[3,p-1]$ and arguments $\overline{χι^{j_1}}\otimes…\otimes\overline{χι^{j_n}}∈\mathfrak{B}^{\otimes n}=\{\overline{χ^{a_1}ι^{j_1}}\otimes … \otimes \overline{χ^{a_n}ι^{j_n}} ∈ (\Hm^*A)^{\otimes n} \mid a_i∈\{0,1\} \text{ and } j_i∈\Z_{\geq 0} \text{ for all }i∈[1,n] \}$.
For computing $Ξ_n$, we first show that $m_2(f_k\otimes f_{n-k})(\overline{χι^{j_1}}\otimes…\otimes\overline{χι^{j_n}})=0$ for $k∈[2,n-2]$. We will need the following congruence.
\begin{align}
\label{gradesmismatch}
\underbrace{(k(l-1)+l(i+x))}_{\equiv_{p-1} k(l-1)+(p-1)+l(i+x)} - \underbrace{(n-k-1+li')}_{\equiv_{p-1} n-k-1+(p-1)+li'} &\equiv_{p-1} -k+k-n+1 = -(n-1) \nonumber \\&
\not\equiv_{p-1} 0
\end{align}
The last statement results from $2\leq n\leq p-1$.
We set "$\pm$" as a symbol for the (a posteriori irrelevant) signs in the following calculation. For $k∈[2,n-2]$, we have
\begin{align*}
\hphantom{XXX}m_2&(f_k\otimes f_{n-k})(\overline{χι^{j_1}}\otimes…\otimes\overline{χι^{j_n}})\\
\eqs& \pm m_2((-1)^{k-1} γ_kι^{j_1+…+j_k}\otimes (-1)^{n-k-1}γ_{n-k}ι^{j_{k+1}+…+j_n})\\
\ovs{\substack {j_1+…+j_k=:x,\\j_{k+1}+…+j_n=:y}}&
\pm γ_kι^x\circ γ_{n-k}ι^{y}\\
\eqs& \pm \left(\dum_{i\geq 0} \ls e_k \rs_{k(l-1)+l(i+x)}^{k-1+li}
+ \dum_{i\geq 0} \ls e_{p-k} \rs_{k(l-1) + (p-1) + l(i+x)}^{k-1+(p-1)+li}\right)\\
&\circ \left(\dum_{i'\geq 0} \ls e_{n-k} \rs_{(n-k)(l-1)+l(i'+y)}^{n-k-1+li'}
+ \dum_{i'\geq 0} \ls e_{p-n+k} \rs_{(n-k)(l-1) + (p-1) + l(i'+y)}^{n-k-1+(p-1)+li'}\right)\\
\ovs{\eqref{gradesmismatch}} 0.
\end{align*}
\begin{align*}
\eqs& m_2((-1)^nf_1\otimes f_{n-1}+(-1)^{n(n-1)}f_{n-1}\otimes f_1)(\overline{χι^{j_1}}\otimes…\otimes\overline{χι^{j_n}})\\
\eqs& m_2((-1)^{n+n|\overline{χι^{j_1}}|}f_1(\overline{χι^{j_{1}}})\otimes f_{n-1}(\overline{χι^{j_2}}\otimes…\otimes\overline{χι^{j_n}})\\
&+ f_{n-1}(\overline{χι^{j_1}}\otimes…\otimes\overline{χι^{j_{n-1}}})\otimes f_1(\overline{χι^{j_n}})) \\
\eqs& m_2(χι^{j_1}\otimes (-1)^{n-2}γ_{n-1}ι^{j_2+…+j_n} + (-1)^{n-2}γ_{n-1}ι^{j_1+…+j_{n-1}}\otimes χι^{j_n})\\
\eqs& (-1)^n (χι^{j_1}\circ γ_{n-1}ι^{j_2+…+j_n} + γ_{n-1}ι^{j_1+…+j_{n-1}}\circ χι^{j_n})\\
\ovs{\text{P.}\ref{pp:iota}(e),\text{L.}\ref{lem:gamma}}& (-1)^n (χ\circ γ_{n-1} + γ_{n-1}\circ χ)\circ ι^{j_1+…+j_n}
\end{align*}
\begin{align*}
χ\circ γ_{n-1}
\eqs& \Big(\dum_{i\geq 0}\left(\ls e_1 \rs_{(i+1)l-1}^{il} + \left\lgroup\dum_{k=1}^{p-2}\ls e_{k+1,k}\rs_{(i+1)l-1+k}^{il + k}\right\rgroup\right.\\
& \left. + \ls e_{p-1}\rs_{(i+1)l-1 +(p-1)}^{il+(p-1)}
+ \left\lgroup\dum_{k=1}^{p-2}\ls e_{p -k-1,p-k}\rs_{(i+1)l-1+(p-1)+k}^{il +(p-1)+ k}\right\rgroup\right)\Big) \\
&\circ\Big(\dum_{i'\geq 0} \ls e_{n-1} \rs_{(n-1)(l-1)+li'}^{n-2+li'}
+ \dum_{i'\geq 0} \ls e_{p-n+1} \rs_{(n-1)(l-1) + (p-1) + li'}^{n-2+(p-1)+li'}\Big)\\
\overset{\hphantom{\displaystyle{=}}\mathllap{3\leq n\leq p-1}}{\underset{\hphantom{\displaystyle{=}}\mathllap{\natop{k\leadsto n-1}{i'\leadsto i+1}}}{=}}\eqsp&
\dum_{i\geq 0} \ls e_{n,n-1}\circ e_{n-1} \rs_{(n-1)(l-1)+l(i+1)}^{il+n-1}\\
&+ \dum_{i\geq 0} \ls e_{p-n,p-n+1}\circ e_{p-n+1} \rs_{(n-1)(l-1) + (p-1) + l(i+1)}^{il+p-1+n-1}
\\
\eqs&\dum_{i\geq 0}\left(\ls e_{n,n-1} \rs_{n(l-1) + 1 +li}^{il+n-1}+ \ls e_{p-n,p-n+1}\rs_{n(l-1)+p+li}^{il+p-1+n-1}\right)\\
γ_{n-1}\circ χ \eqs&\Big(\dum_{i'\geq 0} \ls e_{n-1} \rs_{(n-1+i'-1)l+2(p-1)-(n-1)}^{n-2+li'}
+ \dum_{i'\geq 0} \ls e_{p-n+1} \rs_{(n-1+i')l-n + p}^{n-2+(p-1)+li'}\Big) \\
&\circ\Big(\dum_{i\geq 0}\left(\ls e_1 \rs_{(i+1)l-1}^{il} + \left\lgroup\dum_{k=1}^{p-2}\ls e_{k+1,k}\rs_{(i+1)l-1+k}^{il + k}\right\rgroup\right.\\
& \left. + \ls e_{p-1}\rs_{(i+1)l-1 +(p-1)}^{il+(p-1)}
+ \left\lgroup\dum_{k=1}^{p-2}\ls e_{p -k-1,p-k}\rs_{(i+1)l-1+(p-1)+k}^{il +(p-1)+ k}\right\rgroup\right)\Big)\\
\ovs{\substack{k\leadsto p-n}}& \dum_{i'\geq 0} \ls e_{n-1} \circ e_{n-1,n} \rs_{(n-1+i')l-1+(p-1)+(p-n)}^{n-2+li'} \\
&+ \dum_{i'\geq 0} \ls e_{p-n+1}\circ e_{p-n+1,p-n} \rs_{(n+i')l-1+p-n}^{n-2+(p-1)+li'}\\
\eqs& \dum_{i'\geq 0} \ls e_{n-1,n} \rs_{n(l-1)+ i'l}^{n-2+li'} + \dum_{i'\geq 0} \ls e_{p-n+1,p-n} \rs_{n(l-1) +(p-1)+i'l }^{n-2+(p-1)+li'}
\end{align*}
So $χ\circ γ_{n-1} + γ_{n-1}\circ χ = m_1(γ_n)$ by <ref>. Therefore
\begin{align*}
\eqs& (-1)^n m_1(γ_n)\circ ι^{j_1+…+j_n}
\overset{\text{P.}\ref{pp:iota}(c)}{=} (-1)^n m_1(γ_n ι^{j_1+…+j_n})\\
\eqs& - m_1((-1)^{n-1}γ_n ι^{j_1+…+j_n})\\
\eqs& - m_1\circ f_n(\overline{χι^{j_1}}\otimes…\otimes\overline{χι^{j_n}}).
\end{align*}
We conclude using <ref> by
\begin{align*}
(f_1\circ m'_n + Φ_n)(\overline{χι^{j_1}}\otimes…\otimes\overline{χι^{j_n}}) \overset{\text{L.}\ref{lem:phichi},\text{D.}\ref{defall}}{=} 0 = (m_1\circ f_n + Ξ_n)(\overline{χι^{j_1}}\otimes…\otimes\overline{χι^{j_n}}).
\end{align*}
Condition (<ref>)$[p]$ holds for arguments $\overline{χι^{j_1}}\otimes…\otimes\overline{χι^{j_p}}∈\mathfrak{B}^{\otimes p}=\{\mbox{$\overline{χ^{a_1}ι^{j_1}}\otimes … \otimes \overline{χ^{a_p}ι^{j_p}}$} ∈ (\Hm^*A)^{\otimes p} \mid a_i∈\{0,1\} \text{ and } j_i∈\Z_{\geq 0} \text{ for all }i∈[1,p] \}$.
Recall that $|ι|=l=2(p-1)$ is even, $|χ|=l-1$ is odd and $|f_i| = 1-i$ by <ref>. We have
\begin{align*}
Ξ_p(\overline{χι^{j_1}}\otimes…&\otimes\overline{χι^{j_p}}) = \dum_{i=1}^{p-1}(-1)^{pi}m_2(f_i\otimes f_{p-i})
\eqs& \dum_{i=1}^{p-1}(-1)^{pi+i(1-(p-i))}m_2(f_i(\overline{χι^{j_1}}\otimes…\otimes\overline{χι^{j_i}})\otimes f_{p-i}(\overline{χι^{j_{i+1}}}\otimes…\otimes\overline{χι^{j_p}}))\\
\eqs& \dum_{i=1}^{p-1}f_i(\overline{χι^{j_1}}\otimes…\otimes\overline{χι^{j_i}})\circ f_{p-i}(\overline{χι^{j_{i+1}}}\otimes…\otimes\overline{χι^{j_p}})\\
% The last equality involves $(-1)^{(i-1)+i(1-(p-i))} = (-1)^{-1-i(p-i)}$; as $p$ is odd, $i(p-i)$ is even so there is left only $(-1)^{-1} = -1$.\\
% We will need the fact that
% \begin{align}
% \label{spme}
% s_is_{p-i} \eqs& s_{p-1} & 1\leq i\leq p-1
% \end{align}
% The induction base $i=1$ is trivial.\\
% Now let $s_ks_{p-k}=s_{p-1}$ for some $1\leq k<p-1$. As $s_n = (-1)^{n-1}s_{n-1}$, $n\geq 2$ or equivalently $s_{n+1} = (-1)^n s_n$, $n\geq 1$
% \begin{align*}
% s_{p-1}\eqs& s_ks_{p-k} = (-1)^ks_{k+1}(-1)^{p-k-1}s_{p-k-1}\\
% \eqs& (-1)^{p-1} s_{k+1} s_{p-(k+1)} \overset{p\text{ odd}}{=} s_{k+1}s_{p-(k+1)}
% \end{align*}
% and we have proven the above statement.
% Now
% \begin{align*}
% Ξ_p(\overline{χι^{j_1}}\otimes…&\otimes\overline{χι^{j_p}})= - \dum_{i=1}^{p-1}f_i(\overline{χι^{j_1}}\otimes…\otimes\overline{χι^{j_i}})\circ f_{p-i}(\overline{χι^{j_{i+1}}}\otimes…\otimes\overline{χι^{j_p}})\\
\ovs{p\geq 3\vphantom{I_{I_I}}}& χι^{j_1}\circ (-1)^{p-2}γ_{p-1}ι^{j_2+…+j_p} + (-1)^{p-2}γ_{p-1}ι^{j_1+…+j_{p-1}}\circ χι^{j_p}\\
& + \dum_{i=2}^{p-2} (-1)^{i-1}γ_iι^{j_1+…+j_i}\circ (-1)^{p-i-1}γ_{p-i}ι^{j_{i+1}+…+j_p}\\
\ovs{\text{P.}\ref{pp:iota}(b)}&
(-1)^p\left(χ\circ γ_{p-1} + γ_{p-1}\circ χ + \dum_{k=2}^{p-2} γ_k\circ γ_{p-k}\right)\circ ι^{j_1+…+j_p}
\end{align*}
\begin{align*}
χ\circγ_{p-1}\eqs& \Big(\dum_{i\geq 0}\left(\ls e_1 \rs_{(i+1)l-1}^{il} + \left\lgroup\dum_{k=1}^{p-2}\ls e_{k+1,k}\rs_{(i+1)l-1+k}^{il + k}\right\rgroup\right.\\
&\left. + \ls e_{p-1}\rs_{(i+1)l-1 +(p-1)}^{il+(p-1)}
+ \left\lgroup\dum_{k=1}^{p-2}\ls e_{p -k-1,p-k}\rs_{(i+1)l-1+(p-1)+k}^{il +(p-1)+ k}\right\rgroup\right)\Big)\\
&\circ\left(\dum_{i'\geq 0} \ls e_{p-1} \rs_{(p-1)(l-1)+li'}^{(p-1)-1+li'}+ \dum_{i'\geq 0} \ls e_{1} \rs_{(p-1)(l-1) + (p-1) + li'}^{-1+2(p-1)+li'}\right)\\
\eqs& \dum_{i\geq 0} \ls e_{p-1} \rs_{(p-1)(l-1)+l(i+1)}^{il+(p-1)}+ \dum_{i\geq 0} \ls e_{1} \rs_{(p-1)(l-1) + (p-1) + li}^{il}\\
\eqs& \dum_{i\geq 0} \ls e_{p-1} \rs_{(p+i-1)l+(p-1)}^{il+(p-1)}+ \dum_{i\geq 0} \ls e_{1} \rs_{(p+i-1)l}^{il}
\end{align*}
\begin{align*}
γ_{p-1}\circ χ\eqs& \left(\dum_{i'\geq 0} \ls e_{p-1} \rs_{(p+i'-2)l+(p-1)}^{(p-1)-1+li'}+ \dum_{i'\geq 0} \ls e_{1} \rs_{(p+i'-1)l}^{-1+2(p-1)+li'}\right) \\
&\circ\Big(\dum_{i\geq 0}\left(\ls e_1 \rs_{(i+1)l-1}^{il} + \left\lgroup\dum_{k=1}^{p-2}\ls e_{k+1,k}\rs_{(i+1)l-1+k}^{il + k}\right\rgroup\right.\\
&\left. + \ls e_{p-1}\rs_{(i+1)l-1 +(p-1)}^{il+(p-1)}
+ \left\lgroup\dum_{k=1}^{p-2}\ls e_{p -k-1,p-k}\rs_{(i+1)l-1+(p-1)+k}^{il +(p-1)+ k}\right\rgroup\right)\Big)\\
\eqs& \dum_{i'\geq 0} \ls e_{p-1} \rs_{(p+i'-1)l-1+(p-1)}^{(p-1)-1+li'}+ \dum_{i'\geq 0} \ls e_{1} \rs_{(p+i')l-1}^{-1+2(p-1)+li'}\\
\eqs& \dum_{i'\geq 0} \ls e_{p-1} \rs_{(p+i'-1)l+p-2}^{p-2+i'l}+ \dum_{i'\geq 0} \ls e_{1} \rs_{(p+i'-1)l+l-1}^{i'l+l-1}
\end{align*}
\begin{align*}
γ_k\circ γ_{p-k}\eqs& \left(\dum_{i\geq 0} \ls e_k \rs_{(i+k-1)l+l-k}^{k-1+li}
+ \dum_{i\geq 0} \ls e_{p-k} \rs_{(i+k)l+ (p-1)-k}^{k-1+(p-1)+li}\right)\\
&\circ\left(\dum_{i'\geq 0} \ls e_{p-k} \rs_{(p-k)(l-1)+li'}^{p-k-1+li'}
+ \dum_{i'\geq 0} \ls e_{k} \rs_{(p-k)(l-1) + (p-1) + li'}^{-k+2(p-1)+li'}\right)\\
\eqs&\dum_{i\geq 0} \ls e_k \rs_{(p-k)(l-1)+(p-1)+l(i+k-1)}^{k-1+li}
+ \dum_{i\geq 0} \ls e_{p-k} \rs_{(p-k)(l-1)+l(i+k)}^{k-1+(p-1)+li}\\
\eqs&\dum_{i\geq 0} \ls e_k \rs_{(p-k+i+k-1)l -(p-k)+(p-1)}^{k-1+li}
+ \dum_{i\geq 0} \ls e_{p-k} \rs_{(p-k+i+k)l-(p-k)}^{k-1+(p-1)+li}\\
\eqs&\dum_{i\geq 0} \ls e_k \rs_{(p+i-1)l + k-1}^{k-1+li}
+ \dum_{i\geq 0} \ls e_{p-k} \rs_{(p+i-1)l+ k-1 + (p-1)}^{k-1+(p-1)+li}\,.
\end{align*}
\begin{align*}
χ\circ γ_{p-1} +& γ_{p-1}\circ χ + \dum_{k=2}^{p-2} γ_k\circ γ_{p-k}\\
\eqs&\dum_{i\geq 0} \dum_{k=0}^{p-2}\left(\ls e_{k+1} \rs_{(p+i-1)l + k}^{k+li}
+ \ls e_{p-k-1} \rs_{(p+i-1)l+ k + (p-1)}^{k+(p-1)+li}\right)\\
\eqs& \dum_{i\geq 0} \dum_{k'=0}^{l-1} \ls e_{ω(k')}\rs_{(p-1+i)l+k'}^{k'+li}
\overset{\text{P.}\ref{pp:iota}(a)}{=} ι^{p-1}
\end{align*}
\begin{align*}
Ξ_p(\overline{χι^{j_1}}\otimes…\otimes\overline{χι^{j_p}})=(-1)^pι^{p-1+j_1+…+j_p}\,.% = -ι^{p-1+j_1+…+j_n}.
\end{align*}
So we conclude using <ref> by
\begin{align*}
\begin{array}{rcl}
(f_1\circ m'_p + Φ_p)(\overline{χι^{j_1}}\otimes…\otimes\overline{χι^{j_p}}) &\overset{\text{L.}\ref{lem:phichi},\text{D.}\ref{defall}}{=} &(-1)^p ι^{p-1+j_1+…+j_p} \\
&\overset{\text{D.}\ref{defall}}{=}& (m_1\circ f_p + Ξ_p)(\overline{χι^{j_1}}\otimes…\otimes\overline{χι^{j_p}}).
\end{array}
\end{align*}
Condition (<ref>)$[n]$ holds for $n∈[p+1,2(p-1)]$ and arguments
$\mbox{$\overline{χι^{j_1}}\otimes…\otimes\overline{χι^{j_n}}$} ∈\mathfrak{B}^{\otimes n}=\{\overline{χ^{a_1}ι^{j_1}}\otimes … \otimes \overline{χ^{a_n}ι^{j_n}} ∈ (\Hm^*A)^{\otimes n} \mid a_i∈\{0,1\} \text{ and } j_i∈\Z_{\geq 0} \text{ for all }i∈[1,n] \}$.
As $f_k=0$ for $k\geq p$, we have
\begin{align*}
Ξ_n&(\overline{χι^{j_1}}\otimes…\otimes\overline{χι^{j_n}}) = \dum_{k=n-p+1}^{p-1} (-1)^{nk}m_2(f_k\otimes f_{n-k})
\end{align*}
The right side is a linear combination of terms of the form $γ_k\circ γ_{n-k}$ for $k∈[n-p-1,p-1]$. We have
\begin{align*}
γ_k\circ γ_{n-k} \eqs& \left(\dum_{i\geq 0} \ls e_k \rs_{k(l-1)+li}^{k-1+li}
+ \dum_{i\geq 0} \ls e_{p-k} \rs_{k(l-1) + (p-1) + li}^{k-1+(p-1)+li}\right)\\
&\circ\left(\dum_{i'\geq 0} \ls e_{n-k} \rs_{(n-k)(l-1)+li'}^{n-k-1+li'}
+ \dum_{i'\geq 0} \ls e_{p-n+k} \rs_{(n-k)(l-1) + (p-1) + li'}^{n-k-1+(p-1)+li'}\right)
\end{align*}
A necessary condition for that term to be non-zero is $k(l-1)\equiv_{p-1}n-k-1$ as $l=2(p-1)$. We have
\begin{align*}
k(l-1)-(n-k-1) \equiv_{p-1}\eqsp& -k-n+k+1 = 1-n
\not\equiv_{p-1} 0,
\end{align*}
since $p\leq n-1\leq 2(p-1)-1$. So $γ_k\circ γ_{n-k}= 0$ and $Ξ_n(\overline{χι^{j_1}}\otimes…\otimes\overline{χι^{j_n}})=0$.
We conclude using <ref> by
\begin{align*}
(f_1\circ m'_n + Φ_n)(\overline{χι^{j_1}}\otimes…\otimes\overline{χι^{j_n}}) \overset{\text{L.}\ref{lem:phichi}, \text{D.}\ref{defall}}{=} 0 \overset{\text{D.}\ref{defall}}{=} (m_1\circ f_n + Ξ_n)(\overline{χι^{j_1}}\otimes…\otimes\overline{χι^{j_n}}).
\end{align*}
One could formulate a lemma similar to <ref> for the case $n> 2(p-1)$ as then the sum $ \sum_{k=n-p+1}^{p-1} (-1)^{nk} m_2(f_k\otimes f_{n-k})
(\overline{χι^{j_1}}\otimes…\otimes\overline{χι^{j_n}})$ is in fact empty.
Instead we use <ref> to prove (<ref>)[$n$] for $n>2p-2$:
<ref> ensure that (<ref>)[$n$] holds for . Then <ref> with $k=p$ proves that (<ref>)[$n$] holds for all $n∈[1,∞]$, cf. also <ref>. By <ref>, $f_1$ is injective. By <ref>, the degrees are as required in <ref>. <ref> proves that $(\Hm^* A, (m'_n)_{n\geq 1})$ is an $\A_∞$-algebra and $(f_n)_{n\geq 1}$ is an $\A_∞$-morphism from $(\Hm^* A, (m'_n)_{n\geq 1})$ to $(A, (m_n)_{n\geq 1})$.
By <ref>, we have $m'_1=0$. Thus $(\Hm^* A, (m'_n)_{n\geq 1})$ is a minimal $\A_∞$-algebra. By <ref>, the complex morphism $f_1:(\Hm^* A,m'_1)→(A,m_1)$ is a quasi-isomorphism which induces the identity in homology. So the $\A_∞$-morphism $(f_n)_{n\geq 1}: (\Hm^* A, (m'_n)_{n\geq 1}) → (A, (m_n)_{n\geq 1})$ is a quasi-isomorphism and the proof of <ref> is complete.
§.§ At the prime 2
We examine the case at the prime $2$.
We use a direct approach.
Note that $\Sy_2$ is a cyclic group so the theory of cyclic groups applies as well.
We have $\fs\Sy_2= \{0, (\text{id}),(1,2),(\text{id})+(1,2)\}$. We have maps given by
\begin{align*}
\begin{array}{rrcl}
ε:&\fs\Sy_2 & \longrightarrow & \fs\\
& a(\text{id})+b(1,2) &\longmapsto & a+b \\
% \end{array}\\
% \begin{array}{rrcl}
D:&\fs\Sy_2 & \longrightarrow & \fs\Sy_2\\
& a(\text{id})+b(1,2) &\longmapsto & (a+b)\left((\text{id})+(1,2)\right).
\end{array}
\end{align*}
We see that $ε$ is surjective and $\ker ε = \ker D = \im D = \{0, (\text{id})+(1,2)\}$. The maps $ε$ and $D$ are $\fs\Sy_2$-linear, where $\fs$ is the $\fs\Sy_2$-module that corresponds to the trivial representation of $\Sy_2$. So we have a projective resolution of $\fs$ with augmentation $ε$ by
\begin{align*}
\pres \fs := (\cdots \xrightarrow{D} \underbrace{\fs\Sy_2}_{1} \xrightarrow{D} \underbrace{\fs\Sy_2}_{0} \rightarrow \underbrace{0}_{-1} \rightarrow \cdots),
\end{align*}
where the degrees are written below.
We set $e_1$ to be the identity on $\fs \Sy_2$.
Let $A:=\Hom^*_{\fs\Sy_2}(\pres\fs,\pres\fs)$ and let the $\A_∞$-structure on $A$ be $(m_n)_{n\geq 1}$ (cf. <ref>). Recall the conventions concerning $\Hom_B^k(C,C')$ for complexes $C,C'$ and $k∈\Z$.
An $\fs$-basis of $\Hm^*A$ is given by $\{\overline{ξ^j} \mid j\geq 0\}$ where
\begin{align*}
ξ := \dum_{i\geq 0} \ls e_1\rs_{i+1}^i∈A.
\end{align*}
Straightforward induction yields, for $j\geq 0$,
\begin{align*}
ξ^j = \dum_{i\geq 0} \ls e_1 \rs_{i+j}^i\,.
\end{align*}
We have
\begin{align*}
m_1(ξ^j) \eqs& d \circ ξ^j - (-1)^j ξ^j \circ d \overset{} = d \circ ξ^j + ξ^j \circ d\\
\eqs& \left(\dum_{i\geq 0} \ls D\rs_{i+1}^i\right)\circ \left(\dum_{i\geq 0} \ls e_1 \rs_{i+j}^i\right) + \left(\dum_{i\geq 0} \ls e_1 \rs_{i+j}^i\right)\circ \left(\dum_{i\geq 0} \ls D\rs_{i+1}^i\right)\\
\eqs& \dum_{i\geq 0} \ls D\rs_{i+j+1}^i + \dum_{i\geq 0} \ls D\rs_{i+j+1}^i = 0,
\end{align*}
so $ξ^j$ is a cycle.
As $\Hom_{\fs \Sy_2}(\fs \Sy_2, \fs) = \{0,ε\}$ and $ε\circ D=0$, the differentials of $\Hom^*(\pres \fs, \fs)$ (cf. <ref>) are all zero. So $\{ε\}$ is an $\fs$-basis of $\Hm^k \Hom^*(\pres \fs, \fs)$ for $k\geq 0$. Since in the notion of <ref>, $\bar{Ψ}_k(\overline{ξ^k}) = ε$, the set $\{\overline{ξ^k}\}$ is an $\fs$-basis of $\Hm^k\Hom^*(\pres\fs,\pres\fs)$ for $k\geq 0$. For $k<0$ we have $\Hm^k\Hom^*(\pres\fs,\pres\fs) \cong \Hm^k \Hom^*(\pres \fs, \fs) = 0$. So $\{\overline{ξ^j} \mid j\geq 0\}$ is an $\fs$-basis of $\Hm^* A$.
We define families of maps $(f_n:(\Hm^*A)^{\otimes n}→A)_{n\geq 1}$ and $(m'_n:(\Hm^*A)^{\otimes n} → \Hm^* A)_{n\geq 1}$ as follows. $f_1$ and $m'_2$ are given on a basis by
\begin{align*}
f_1(\overline{ξ^j}) :=\eqsp& ξ^j &&\text{for $j\geq 0$}\\
m'_2(\overline{ξ^j}\otimes \overline{ξ^k}) :=\eqsp& \overline{ξ^{j+k}} &&\text{for $j,k\geq 0$}.
\end{align*}
All other maps are set to zero.
It is straightforward to check that $(\Hm^* A, (m'_n)_{n\geq 1})$ is a pre-$\A_∞$-algebra and $(f_n)_{n\geq 1}$ is a pre-$A_∞$-morphism from $\Hm^*A$ to $A$. As $m'_2$ is associative, $(\Hm^* A, (m'_n)_{n\geq 1})$ is a dg-algebra, so in particular an $\A_∞$-algebra.
As $f_k=0$ for $k\neq 1$, (<ref>)[$n$] simplifies to
\begin{align*}
f_1\circ m'_n \eqs& m_n\circ (\underbrace{f_1\otimes \cdots \otimes f_1}_{n\text{ factors}}).
\end{align*}
As $m'_n=0$ and $m_n=0$ for $n\geq 3$, (<ref>)[$n$] is satisfied for all $n\geq 3$. For $n∈\{1,2\}$, we have
\begin{align*}
f_1\circ m'_1 \eqs& m_1\circ f_1\\
f_1\circ m'_2 \eqs& m_2(f_1\otimes f_1).
\end{align*}
The second equation follows immediately from the definition of $m'_2$ and $f_1$. The first equation holds as $m'_1=0$ and the images of $f_1$ are all cycles. So (<ref>)[$n$] holds for all $n$ and $(f_n)_{n\geq 1}$ is an $\A_∞$-morphism from $(\Hm^* A, (m'_n)_{n\geq 1})$ to $(A, (m_n)_{n\geq 1})$. By the construction of $f_1$, it induces the identity on homology. Thus $(\Hm^* A, (m'_n)_{n\geq 1})$ is a minimal model of $(A, (m_n)_{n\geq 1})$.
[Comparison with primes $p\geq 3$]
At a prime $p \geq 3$, we have constructed a projective resolution with period length $l=2(p-1)$ in (<ref>).
If one constructs a projective resolution of $\Z_{(2)}$ analogous to the case $p\geq 3$, we have a sequence of the form
\[\cdots \rightarrow \Z_{(2)}\Sy_2 \xrightarrow{\hat e_{2,2}^*} \Z_{(2)}\Sy_2\xrightarrow{\hat e_{2,2}}\Z_{(2)}\Sy_2 \xrightarrow{\hat e_{2,2}^*} \Z_{(2)}\Sy_2\xrightarrow{\hat e_{2,2}}\Z_{(2)}\Sy_2 \rightarrow 0 \rightarrow \cdots\]
with a period length of $2$, where
\begin{align*}
\hat e_{2,2}\colon&\,(\text{id})\,\longmapsto\, (\text{id})-(1,2)\\
\hat e_{2,2}^*\colon&\,(\text{id})\,\longmapsto\, (\text{id})+(1,2).
\end{align*}
However, modulo $2$ the differentials $\hat e_{2,2}$ and $\hat e_{2,2}^*$ reduce to the same map $D: \fs \Sy_2\rightarrow \fs \Sy_2$, so we obtain a period length of $1$.
The maps $ι$ resp. $χ$ from <ref> may be identified with $ξ^2$ resp. $ξ$. This way, the definition of $m'_2$ at the prime $2$ is readily compatible with <ref>.
On the bar construction
§.§ Applications. Kadeishvili's algorithm and the minimality theorem.
In this subsection we will discuss the construction of minimal models of $\A_∞$-algebras. Firstly, <ref> states that certain pre-$\A_n$-structures and pre-$\A_n$-morphisms that arise in the construction of minimal models are actually $\A_n$-structures and $\A_n$-morphisms. Secondly, we give a proof of <ref>. We will review Kadeishvili's original proof of [9] as it gives a an algorithm for constructing minimal models which can be used for the direct calculation of examples.
Note that Lefèvre-Hasegawa has given a generalization of the minimality theorem, see <cit.>, which we will not cover.
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[26] Vejdemo-Johansson, M., Blackbox computation of $\A_∞$-algebras, Georgian Mathematical Journal Volume 17 Issue 2, p. 391-404, 2010
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arxiv-papers
| 2013-02-01T10:03:20 |
2024-09-04T02:49:41.161254
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Stephan Schmid",
"submitter": "Stephan Schmid",
"url": "https://arxiv.org/abs/1302.0121"
}
|
1302.0136
|
11institutetext: Centre for Digital Music, Queen Mary, University of London
11email: [email protected]
# Maximum a posteriori estimation of piecewise arcs in tempo time-series
Dan Stowell and Elaine Chew 1111
###### Abstract
In musical performances with expressive tempo modulation, the tempo variation
can be modelled as a sequence of tempo arcs. Previous authors have used this
idea to estimate series of piecewise arc segments from data. In this paper we
describe a probabilistic model for a time-series process of this nature, and
use this to perform inference of single- and multi-level arc processes from
data. We describe an efficient Viterbi-like process for MAP inference of arcs.
Our approach is score-agnostic, and together with efficient inference allows
for online analysis of performances including improvisations, and can predict
immediate future tempo trajectories.
###### Keywords:
tempo, expression, Viterbi, time series
## 1 Introduction
In various types of musical performance, one component of the musical
expression is conveyed in the short-term manipulation of tempo, with tempo
modulation reflecting musical phrase structure (McAngus Todd, 1992; Widmer and
Tobudic, 2003). This has motivated various authors to construct automatic
analyses of the arc-shaped tempo modulations in recorded musical performances,
with or without score-derived information to supplement the analysis (McAngus
Todd, 1992; Widmer and Tobudic, 2003; Chuan and Chew, 2007). (See also
Dannenberg and Mohan (2011) who fit piecewise linear arcs to rock and jazz
data, applying similar techniques but to genres in which the underlying tempo
is held more fixed.)
Machine understanding of tempo, including its variability, can be useful in
live human-machine interaction (Allen and Dannenberg, 1990; Robertson and
Plumbley, 2007). However most current online tempo-tracking systems converge
to an estimate of the current tempo, modelling expressive variations as
deviations rather than as components of an unfolding tempo expression. In this
paper we work towards the understanding of tempo arcs in a real-time system,
paving the way for automatic accompaniment systems which follow the expressive
tempo modulation of players in a more natural way.
We also consider tempo arcs within a probabilistic framework. Previous authors
have approached piecewise arc estimation using Dynamic Programming (DP) with
cost functions based on squared error (Chuan and Chew, 2007; Dannenberg and
Mohan, 2011). These are useful and can provide efficient estimation, but by
setting the problem in a probabilistic framework (and providing the
corresponding Viterbi-like DP estimator), we gain some advantages: prior
beliefs about the length and shape of arcs can be expressed coherently as
prior distributions; measurement noise is explicitly modelled; and the
goodness-of-fit of models is represented meaningfully as posterior
probabilities, which allows for model comparison as well as integration with
other workflow components which can make use of estimates annotated with
probability values. Note that while we describe a fully probabilistic model,
for efficient inference we will develop a Maximum A Posteriori (MAP)
estimator, which returns only the maximum probability parameter settings given
the priors and the data.
In the following we will describe our model of arcs in time-series data, and
develop an efficient MAP estimation technique based on least-squares
optimisation and Viterbi-like DP. The approach requires some kind of
unsmoothed instantaneous tempo estimate as its input, which may come from a
tempo tracker or from a simple measurement such as inter-onset interval (IOI).
We will then discuss how the estimator can be used for immediate-future tempo
prediction, and how it can be applied to multiple levels simultaneously.
Finally we will apply the technique to tempo data from three professional
piano performances, and discuss what the analysis reflects in the
performances.
## 2 Modelling and Estimation
For our basic model, we consider tempo to evolve as a function of metrical
position (beat number) $x$ in a musical piece as a series of connected arcs,
where each arc’s duration, curvature and slope are independently drawn from
prior distributions (to be described shortly). Our model is deliberately
simple, and agnostic of any score information that might be available. To
sample from this model, we pick an initial tempo at the starting time, then
define a single upwards tempo arc which starts from that point, and the tempo
trajectory (speeding up and then slowing down) over a number of measures. Any
tempo data which may be measured during this interval is modelled as being
drawn from the arc plus some amount of gaussian noise. Once the ending
breakpoint of this arc is reached, the next arc is sampled from the same
priors, using the ending tempo as the new starting tempo. Hence each tempo arc
is conditionally independent of all previous observations once the starting
tempo is determined, i.e. once the previous arc’s parameters are fixed. This
assumption of conditional independence is slightly unrealistic, since it
ignores long-range relationships between tempo arcs, but it accounts for the
most important interactions and makes inference tractable.
Our basic model is also only single-level, assuming that a single arc
contributes to the current tempo at any moment, rather than considering for
example contributions from multiple timescales such as piece-level, movement-
level, phrase-level and bar-level combined. In Section 2.4 we will consider a
simple multi-scale extension of our technique, which we will apply in our
analysis of piano performance data. (For an alternative approach in which
various components can be simultaneously active see McAngus Todd (1992).)
### 2.1 Fitting a Single Arc
To fit a single arc shape to data, one can use standard quadratic regression,
fitting a function of the form
$f(x)=a+bx+cx^{2},$ (1)
and minimising the $L_{2}$ prediction error over the supplied data for
$y\approx f(x)$. In the Bayesian context, we wish to incorporate our prior
beliefs about the regression parameters (here $a$, $b$ and $c$), which is
related to the optimisation concept of regularisation, the class of techniques
which aims to prevent overfitting by favouring certain parameter settings. In
fact, a gaussian prior on a regression parameter can be shown to be equivalent
to the conventional $L_{2}$-norm regularisation of the parameters (Bishop,
2006, p. 153), summarised as:
$\textrm{regularisation coefficient}=\frac{\textrm{variance of gaussian
noise}}{\textrm{variance of gaussian prior}}.$ (2)
This equivalence is useful because it allows us to use common convex
optimisation algorithms to perform the equivalent regularised least squares
optimisation, and they will yield the MAP estimate for the probabilistic
model.
However, in this context a standard gaussian prior is not exactly what we
require, since we are expecting upwards arcs and not troughs – we are
expecting $c$ in Equation 1 to be negative. A more appropriate choice of prior
might be a negative log-gaussian distribution, which allows us to specify a
“centre of mass” for the arc shapes (expressed through the log-mean and log-
standard-deviation parameters), yet better represents our expectation that
tempo arcs will always have negative curvature, (almost) flat and extremely
strongly curved arcs being equally rare.
The unconventional choice of prior might seem to remove the equivalence of the
MAP regression technique with standard regularised least squres. Yet if we
rewrite our function to be
$f(x)=a+bx-e^{c}x^{2},$ (3)
then our prior belief about this modified parameter $c$ becomes a gaussian,
yielding a negative-log-gaussian in combination with our function. In
addition, we will use a standard gaussian prior on $b$. We could do the same
for $a$ but instead we will use an improper uniform prior, for reasons which
will be described in Section 2.2. Therefore, our priors for Equation 3 will be
gaussian priors on $b$ and $c$, which can easily be converted to the
equivalent $L_{2}$-regularisation terms for optimisation.
The strength of the regularisation (the value of the regularisation
coefficient) reflects the specificity of our priors versus our data –
specifically, the regularisation parameter is given by the noise variance
divided by the prior variance (Bishop, 2006, p. 153). Again, we see how the
probabilistic setting helps to ground our problem, connecting the strength of
the regularisation directly to our prior beliefs about the model and the data
rather than manually-tuned parameters.
### 2.2 Fitting Multiple Arcs
Figure 1: A selection of piecewise arc fits performed on a synthetic dataset.
For illustration purposes, we have manually specified a sequence of possible
breakpoint locations, and then performed a single-arc fit within each
subsection. The “logMAP” (log of MAP probability) values quoted with each plot
indicate the relative likelihood assigned to the depicted fit, given the prior
parameters chosen. Prior parameters are the same for each of these plots. The
best-fitting plots have correspondingly higher (less negative) logMAP values.
If a time-series is composed of multiple arcs and the breakpoints are known,
then fitting multiple arcs is as simple as performing the above single-arc fit
for each subsection of the time series (as in Figure 1). Additionally, one
should take care of the arc’s dependence upon its predecessor (to enforce that
they meet up), which is not shown in these plots. In our case, we want to
estimate the breakpoint locations as well as the arc shapes between those
breakpoints. This can be performed by iterating over all possible combinations
of one breakpoint, two breakpoints, three (…) for the dataset, and choosing
the result with the lowest cost (the highest posterior likelihood).
The Bayesian setting makes it possible to compare these different alternatives
(e.g. one single arc vs. one arc for every datapoint) without having to add
arbitrary terms to counter overfitting; instead, we specify a prior
distribution over the arc durations, which in combination with the other
priors and data likelihoods yields a MAP probability for any proposed set of
arcs. In this paper we choose a log-normal prior distribution over arc
durations. See Figure 1 for some examples of different sets of arcs fitting to
a synthetic dataset, and the posterior (log-)probabilities associated.
In order for only a single tempo value to exist at each breakpoint (and not a
discontinuous leap from one tempo to another), we fit each arc under the
constraint that its starting value equals the ending value of the previous
arc. This removes one degree of freedom from the function to be fit (Equation
3) which otherwise has three free parameters. We implement this by
constraining the value of $a$ in the optimisation so that the function
evaluates to the predetermined value at the appropriate time-point. The least-
squares optimisation therefore only operates on $b$ and $c$.
### 2.3 Viterbi-like Algorithm
The number of possible combinations of arcs for even a small time-series (such
as Figure 1) grows quickly very large, and so it is impractical to iterate all
combinations. This is where Dynamic Programming (DP) can help. Here we
describe our DP algorithm, which, like the well-known Viterbi algorithm,
maintains a record of the most likely route that leads to each of a set of
possible states. Rather than applying it to the states of a Hidden Markov
Model, we apply it to the possibility that each incoming datum represents a
breakpoint.
Assume that the first incoming datum is a breakpoint. (This assumption can be
relaxed, in a similar way to the treatment of the final datum which we
consider later.) Then, for each incoming datum ($x_{n}$, $y_{n}$), we find
what would be the most likely path if it were certainly a breakpoint. We do
this by finding the most appropriate past datum ($x_{n-k}$, $y_{n-k}$) which
could begin an arc to the current datum – where the appropriateness is judged
from the MAP probability of said arc, combined with the MAP probability of the
whole multiple-arc history that leads up to that past datum (recursively
defined).
With our lognormal prior on the arc lengths (and with many common choices of
prior), the probability mass is concentrated at an expected time-scale, and
very long arcs are highly improbable a priori. Hence in practice we truncate
the search over potential previous arc points to some maximum limit $K$ (i.e.
$k\leq K$).
Thus, for every incoming data point we perform no more than $K$ single-arc
fits, then store the details of the chosen arc, the MAP probability so far,
and a pointer back to the datapoint at the start of the chosen single arc. The
simplest way to choose the overall MAP estimate is then to pick another
definite breakpoint (for example, the last datum if the performance has
finished) and backtrack from there to recover the MAP arc path.
#### 2.3.1 Complexity
The time complexity of the algorithm depends strongly on that of the convex
optimisation used to perform a single-arc fit. Assume that the complexity of a
single-arc fit is proportional to the number of data points $k$ included in
the fit, where $k\leq K$. Then for each incoming data point a search is
performed for one subset each of 2, 3, …K data points, which essentially
yields an order $\mathcal{O}$($K^{2}$) process. For online processing this is
manageable if $K$ is not too large. Analysing a whole dataset of $M$ points
then has time complexity $\mathcal{O}$($K^{2}M$). (Compare this to the broadly
similar complexity analysis of Dannenberg and Mohan (2011).) The space
complexity is simply $\mathcal{O}$($M$), or $\mathcal{O}$($K$) if the full arc
history since the very beginning does not need to be stored. This is because a
small fixed amount of data is stored per datapoint.
#### 2.3.2 Predicting Immediate Future Arcs
As discussed, if we know the performance has finished then we can find the
Viterbi path leading to a breakpoint at the final data point received.
However, we would also like to determine the most likely set of arcs in cases
where the performance might not have finished (e.g. for real-time interactive
systems), and thus where we do not wish to assert that the latest datum is a
breakpoint. We wish to be able to estimate an arc which may still be in
progress. If we can, this has a specific benefit of predicting the immediate
future evolution of the tempo modulations (until the end of the present arc),
which may be particularly useful for real-time interaction.
We can carry this out in our current approach as follows. Since an arc’s
duration (as well as the curve-fit) affects its MAP probability, in the case
where the latest arc may or may not be terminating we must iterate over the
arc’s possible durations and pick the most likely. To do this we choose a set
of future time-points as candidate breakpoints, $x_{n+1}$ …$x_{n+J}$ (e.g. an
evenly-spaced tatum grid of $J=K$ future points). Then we supply these data to
the Viterbi update process exactly as is done with actual data, but with no
associated $y$ values. These “hypothetical” Viterbi updates will use these
time-points to determine the arc-lengths being estimated, and in normalising
the data subset, but will not include them in the arc-fitting process. It will
therefore yield a MAP probability estimate for each of the time-points as if
an arc extended from the real data as far as this hypothetical breakpoint. Out
of these possibilities, the one with the highest MAP probability is the MAP
estimate for an arc which includes the latest real datum and some portion of
the hypothetical future points. (The hypothetical Viterbi updates are not
preserved: if more data comes in, it is appended to the Viterbi storage
corresponding only to the actual data.)
### 2.4 Multi-scale Estimation
The model we describe operates at one level, with expected arc durations given
by the corresponding prior. Our model is adaptable to any time-scale by simply
adapting the prior. It does not however automatically lend itself to
simultaneous consideration of multiple active timescales.
Multi-scale analysis can be carried out by analysing a dataset with one
timescale, then analysing the residual at a second timescale. This residual-
based decomposition has been used previously in the literature (e.g. Widmer
and Tobudic (2003)); it requires a strong hierarchical assumption that the
arcs at the first timescale do not depend at all on those at the second
timescale, while the second is subordinate to the first. We consider this to
be unrealistic, since there may well be interactions between the different
timescales on which a performer’s expression evolves. However this assumption
leads to a tractable analysis.
Note also that this approach to multi-scale estimation requires the first
analysis to be completed (so that the residual is known) before the second
scale can be analysed. Some DP approach may be possible to enable both to be
calculated online, but we have not developed that here. For the present work,
the single-scale Viterbi tracking is applicable and useful for online
tracking, while multi-scale analysis is an offline process, which we will next
apply to modelling of pre-recorded tempo data.
## 3 Analysis of Expressive Piano Performance
We applied our analysis to an existing set of annotations of three
performances of Beethoven’s Moonlight Sonata. The annotations by Elaine Chew
have previously been analysed by Chew with reference to observations noted by
Jeanne Bamberger (Chew, 2012). For each of three well-known performances of
the piece—by Daniel Barenboim (1987), Maurizio Pollini (1992) and Artur
Schnabel (2009)—the first 15 bars have been annotated with note onset times,
which correspond to regular triplet eighth-note timings.
We implemented the algorithm in Python, using the scipy.optimize.fmin
optimiser to solve individual regressions. Source code is
available.111https://code.soundsoftware.ac.uk/projects/arcsml (Note that this
development implementation is not generally fast enough for real-time use.)
Instantaneous tempo was derived from these inter-onset intervals, then
analysed using a two-pass version of our algorithm: first the data was
analysed using an arc-duration prior centred on four bars; then the residual
was analysed using an arc-duration prior centred on one bar. This choice of
timescales is a relatively generic choice which might reasonably be considered
to reflect a performer’s short-term and medium-term state; however it might
also be said to be a form of basic contextual information about the relevant
timescales in the current piece. For the current study, we confine ourselves
to priors with log-normal shapes, though an explicitly score-derived or
corpus-derived prior could have a more tailored and perhaps multimodal shape.
Figure 2: Manual analyses of performances by each of three pianists
(Barenboim, Pollini, Schnabel). Two to three levels of arcs are drawn by
visual inspection for each tempo time series. No higher level arcs are drawn
when it is uncertain that one exists.
Figure 3: Two-level analysis of performances by each of three pianists
(Barenboim, Pollini, Schnabel). In each plot, the first long-scale fit
(centred on the four-bar timescale) is depicted in red, and the second
shorter-scale fit (centred on the one-bar timescale) is given in blue. The
second fit is pre-offset by the first, meaning the blue arcs display the
combined model produced by both timescales combined. Annotated data finish at
tatum 180; where the MAP choice extends beyond that, we show the predicted
immediate future arc.
Figure 4: As Figure 3 but with the standard deviation of the noise prior set
at 4.0 rather than 3.0.
Figure 2 shows a set of manual annotations of hierarchically embedded phrases,
and Figure 3 shows the automatically computed results. The automatic analyses
show some notable similarities with the manual one at the shorter time scale,
and significant differences at the longer time scale. The difficulty of the
longer time scale analysis may be explained by Figure 5, which shows one
plausible set of phrase groupings for this excerpt; the overlapping {5,5,7}
bar phrases do not fit easily into a four-bar duration framework.
Figure 5: Possible set of overlapping phrases in the first 15 bars of the
Moonlight Sonata.
Nevertheless, the longer time scale analysis (centred on four bars) highlights
differences between the performances: Pollini’s performance appears to contain
relatively little variation on this level, as the fit yields long and shallow
arcs, with breakpoints near positions 48, 96 and 168 (structurally important
positions; 96 is where the key-change occurs). On the other hand, both
Barenboim and Schnabel’s tempo curves exhibit fairly deep and varied arcs.
Schnabel’s performance exhibits the most dramatic variation in the first four
bars until around measure 48: this first four-bar section corresponds to the
opening statement of the basic progression, before the melody enters in the
fifth bar (and the underlying progression repeats). Bamberger described
Schnabel as performing them “as if in one long breath” (quoted in Chew
(2012)), not quite reflected in our automatic analysis.
On the shorter time scale, the analysis tends to group phrases into one-bar or
two-bar arcs. Aspects of the musical structure are reflected in the arcs
observed. Sections of the melody which lend themselves to two-bar phrasing
(e.g. 72–96) are generally reflected in longer arcs crossing bar lines.
Conversely, in the region 96–132 the change to the new key unfolds as each new
chord enters at the start of a bar, and the tempo curves for all three
performers reflect an expressive focus on this feature, with one-bar arcs
which are more closely locked to the bar-lines than elsewhere. Note that in
this section Schnabel matches Pollini in exhibiting a long and shallow arc on
the slow timescale, with all the expressive variation concentrated on the one-
bar arcs.
Over the excerpt generally, the breakpoints for Schnabel are further away from
the barline than the others, as was observed in Chew’s manual analysis. We can
quantify this by measuring the mean deviances of arc endpoints from the
barlines in each performance. The resulting mean deviances confirm our
observations (Table 1).
Performer | Mean deviance (bars)
---|---
Barenboim | 13.9%
Pollini | 8.3%
Schnabel | 15.5%
Table 1: Mean deviance from the barlines of the arc endpoints inferred for
each performance, averaged over the short-timescale arcs in each case.
We have extended the plots slightly beyond the 180 annotated data points, to
illustrate the immediate-future predictions made by the model. (This is done
for both timescales, though only the longer timescale (in red) shows
noticeable extended arcs.) All the performers, and especially Schnabel,
exhibit an acceleration towards the end of the annotated data, reflected in
the predictions of an upward arc followed by a gradual slowing over the next
bar. This type of prediction is plausible for such expressively-timed music.
To illustrate the effect that the prior parameters have upon the regression,
Figure 4 shows the same analysis as Figure 3 but with the standard deviation
of the noise prior set at 4.0 rather than 3.0. The increase in the assumed
noise variance leads the algorithm to “trust” the data less and the prior
slightly more (cf. Equation 2). In our example, some of the breakpoints for
the long-term arcs (in red) have changed, losing some detail, though most of
the detail of the second-level analysis (in blue) is consistent.
## 4 Conclusions
We have described a model with similarities to some previous piecewise-arc
models of musical expression, but with a Bayesian formulation which
facilitates model comparison and the principled incorporation of prior
beliefs. We have also described an efficient Viterbi-like Dynamic Programming
approach to estimation of the model from data. The approach provides scope to
apply the model to real-time score-free performance tracking, including
prediction of immediate future tempo modulation. Source code for the algorithm
(in Python) is available.
We have applied the model in a two-level analysis to data from expressive
piano performance, illustrating the algorithm’s capacity to operate at
different time-scales, and to recover expressive arc information that
corresponds with some musicological observations regarding phrasing and
timing.
Further research would be needed to develop a model of multiple
simultaneously-active levels of expression which can be applied online as with
our single-level Viterbi-like algorithm. Similar arcs have been observed and
analysed in loudness information extracted from performances (Cheng and Chew,
2008). It would also be useful to combine loudness information with tempo
information in this model.
## References
* Allen and Dannenberg (1990) P. E. Allen and R. B. Dannenberg. Tracking musical beats in real time. In _Proc. International Computer Music Conference (ICMC)_ , pages 140–143, Hong Kong, 1990.
* Bishop (2006) C. M. Bishop. _Pattern Recognition and Machine Learning_ , volume 4. Springer, New York, 2006.
* Cheng and Chew (2008) E. Cheng and E. Chew. Quantitative analysis of phrasing strategies in expressive performance: computational methods and analysis of performances of unaccompanied bach for solo violin. _Journal of New Music Research_ , 37(4):325–338, 2008. doi: 10.1080/09298210802711660.
* Chew (2012) E. Chew. About time: Strategies of performance revealed in graphs. _Visions of Research in Music Education_ , 20(1), Jan 2012\. URL http://www-usr.rider.edu/~vrme/v20n1/.
* Chuan and Chew (2007) C. H. Chuan and E. Chew. A dynamic programming approach to the extraction of phrase boundaries from tempo variations in expressive performances. In _Proceedings of the International Conference on Music Information Retrieval (ISMIR)_ , pages 305–308, Vienna, Austria, 2007. URL http://ismir2007.ismir.net/proceedings/ISMIR2007_p305_chuan.pdf.
* Dannenberg and Mohan (2011) R. B. Dannenberg and S. Mohan. Characterizing tempo change in musical performances. In _Proceedings of the International Computer Music Conference (ICMC)_ , pages 650–656, 2011.
* McAngus Todd (1992) N. P. McAngus Todd. The dynamics of dynamics: A model of musical expression. _Journal of the Acoustical Society of America_ , 91(6), 1992\. doi: 10.1121/1.402843. URL http://www.iro.umontreal.ca/~pift6080/H09/documents/papers/todd_dyn.pdf%.
* Robertson and Plumbley (2007) A. Robertson and M. D. Plumbley. B-Keeper: A beat-tracker for live performance. In _Proc. International Conference on New Interfaces for Musical Expression (NIME), New York, USA_ , pages 234–237, 2007. doi: 10.1145/1279740.1279787.
* Widmer and Tobudic (2003) G. Widmer and A. Tobudic. Playing Mozart by analogy: Learning multi-level timing and dynamics strategies. _Journal of New Music Research_ , 32(3):259–268, 2003. doi: 10.1076/jnmr.32.3.259.16860.
|
arxiv-papers
| 2013-02-01T10:51:20 |
2024-09-04T02:49:41.177272
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dan Stowell and Elaine Chew",
"submitter": "Dan Stowell",
"url": "https://arxiv.org/abs/1302.0136"
}
|
1302.0175
|
# Criteria of positivity for linear maps constructed from permutation pairs
Haili Zhao, Jinchuan Hou Faculty of Mathematics, Taiyuan University of
Technology, Taiyuan 030024, P. R. of China [email protected];
[email protected]
###### Abstract.
In this paper, we show that a $D$-type map $\Phi_{D}:M_{n}\rightarrow M_{n}$
with $D=(n-2)I_{n}+P_{\pi_{1}}+P_{\pi_{2}}$ induced by a pair
$\\{\pi_{1},\pi_{2}\\}$ of permutations of $(1,2,\ldots,n)$ is positive if
$\\{\pi_{1},\pi_{2}\\}$ has property (C). The property (C) is characterized
for $\\{\pi_{1},\pi_{2}\\}$, and an easy criterion is given for the case that
$\pi_{1}=\pi^{p}$ and $\pi_{2}=\pi^{q}$, where $\pi$ is the permutation
defined by $\pi(i)=i+1$ mod $n$ and $1\leq p<q\leq n$.
2010 Mathematics Subject Classification. 15A86; 47B49; 47N50.
Key words and phrases. matrix algebras, positive linear maps, permutations,
inequalities
This work is partially supported by Natural Science Foundation of China
(11171249, 11271217)
## 1\. Introduction
Denote by $M_{m,n}$ the set of all $m\times n$ complex matrices, and write
$M_{n,n}=M_{n}$ and $M_{n}^{+}$ the set of all positive semi-definite matrices
in $M_{n}$. A linear map $L:M_{n}\rightarrow M_{n}$ is positive if
$L(M_{n}^{+})\subseteq M_{n}^{+}$. The study of positive maps have been the
central theme for many pure and applied topics. For example, see [3, 6, 9, 10,
11]. In particular, the study is pertinent in quantum information science
research; see [1, 2, 4, 7, 8, 12, 13]. So, the study of positive maps is
significant.
Suppose $\Phi_{D}:{M_{n}}\rightarrow{M_{n}}$ is a linear map of the form
$None$ $(a_{ij})\longmapsto{{\rm diag}(f_{1},f_{2},...,f_{n})-(a_{ij})}$
with $(f_{1},f_{2},...,f_{n})=(a_{11},a_{22},...,a_{nn})D$ for an $n\times n$
nonnegative matrix $D=(d_{ij})$ (i.e., $d_{ij}\geq 0$ for all $i,j$). The map
$\Phi_{D}$ of the form Eq.(1.1) defined by a nonnegative matrix $D$ is called
a $D$-type map. The question of when a $D$-type map is positive was studied
intensively by many authors and applied in quantum information theory to
detect entangled states and construct entanglement witnesses (ref., for
instance, [9, 14] and the references therein).
A very interesting class of $D$-type maps is the class of maps constructed
from permutations.
Assume that $\pi$ is a permutation of $(1,2,\ldots,n)$. Recall that the
permutation matrix $P_{\pi}=(p_{ij})$ of $\pi$ is a $n\times n$ matrix
determined by
$p_{ij}=\begin{cases}1\ \ \ \ \ \ {\rm if}\ \ i=\pi(j)\ ({\rm mod}\ n),\\\ 0\
\ \ \ \ \ {\rm if}\ \ i\neq{\pi(j)}\ ({\rm mod}\ n).\end{cases}$
Recall also that a subset $\\{i_{1},\dots,i_{l}\\}\subseteq\\{1,2,\ldots,n\\}$
is an $l$-cycle of the permutation $\pi$ if $\pi(i_{j})=i_{j+1}$ for
$j=1,\dots,l-1$ and $\pi(i_{l})=i_{1}$. Note that every permutation $\pi$ of
$(1,\dots,n)$ has a disjoint cycle decomposition
$\pi=(\pi_{1})(\pi_{2})\cdots(\pi_{r})$, that is, there exists a set
$\\{F_{s}\\}_{s=1}^{r}$ of disjoint cycles of $\pi$ with
$\cup_{s=1}^{r}F_{s}=\\{1,2,\ldots,n\\}$ such that $\pi_{s}=\pi|_{F_{s}}$ and
$\pi(i)=\pi_{s}(i)$ whenever $i\in F_{s}$.
The positivity of $D$-type maps generated by one permutation were discussed in
[9] recently. Let $\pi$ be a permutation of $(1,2,\ldots,n)$ with disjoint
cycle decomposition $(\pi_{1})\cdots(\pi_{r})$ such that the maximum length of
$\pi_{h}$s is equal to $l>1$ and $P_{\pi}=(\delta_{i\pi(j)})$ is the
permutation matrix associated with $\pi$. For $t\geq 0$, let
$\Phi_{t,\pi}:M_{n}\rightarrow M_{n}$ be the $D$-type map of the form in
Eq.(1.1) with $D=(n-t)I_{n}+tP_{\pi}$. It is shown in [9] that $\Phi_{t,\pi}$
is positive if and only if $0\leq t\leq\frac{n}{l}$. Particularly,
$\Phi_{\pi}=\Phi_{1,\pi}$ is always positive.
Motivated by the above result, let us consider the $D$-type maps constructed
from several permutations. Generally speaking,
$\Phi_{D_{\pi_{1},\ldots,\pi_{k}}}$ is not positive if
$None$
$D_{\pi_{1},\ldots,\pi_{k}}=(n-k)I_{n}+P_{\pi_{1}}+P_{\pi_{2}}+\cdots+P_{\pi_{k}}$
with $1<k\leq n-1$. For example, take $\pi$ defined by $\pi(i)=i+1$ mod $n$
and let $D=(n-2)I_{n}+2P_{\pi}$,; then $\Phi_{D}$ is not positive by the
result in [9] mentioned above. However there do exist $D$-type positive linear
maps of the form $\Phi_{D_{\pi_{1},\ldots,\pi_{k}}}$ with $k>1$. For the
permutation $\pi$ defined by $\pi(i)=i+1$ mod $n$, and for any $1\leq k\leq
n-1$, it is known that $\Phi_{D}:M_{n}\rightarrow M_{n}$ is positive, where
$D=(n-k)I_{n}+P_{\pi}+P_{\pi^{2}}+\cdots+P_{\pi^{k}}$ (ref. [15]).
So it is an natural and interesting topic to study the $D$-type linear maps
$\Phi_{D}$ with $D={D_{\pi_{1},\ldots,\pi_{k}}}$ of the form Eq.(1.2) and
establish positivity criteria for them.
The purpose of this paper is to start the discussion of this topic. We
introduce a concept of property (C) for a pair of permutations (Definition
3.2), and show that, if $\\{\pi_{1},\pi_{2}\\}$ has the property (C), then the
$D$-type map $\Phi_{D_{\pi_{1},\pi_{2}}}$ is positive (Theorem 3.3 and Theorem
3.4). To do this, we develop some inequalities (Lemma 2.1, Lemma 2.2), which
are interesting themselves. Furthermore, we give a necessary and sufficient
condition for two permutations to have property (C) (Proposition 3.5). This
results then applied to give some criteria of positivity of $D$-type maps
induced by pair of permutations and to construct some new positive $D$-type
maps, especially from pair of permutations $\\{\pi^{p},\pi^{q}\\}$, where
$\pi$ is the cyclic permutation defined by $\pi(i)=i+1$ mod $n$. Some examples
are also presented to illustrate how to use the criteria.
## 2\. Preliminary inequalities
In this section, we first establish two cyclic like inequalities, which are
needed to establish the criteria for a $D$-type linear maps induced from a
pair of permutations to be positive.
Lemma 2.1. Let $s,M$ be positive numbers and $f(u_{1},u_{2},...,u_{m})$ be a
function in $m$-variable defined by
$f(u_{1},u_{2},...,u_{m})=\frac{1}{s+u_{1}}+\frac{1}{s+u_{2}}+...+\frac{1}{s+u_{m}}$
on the region $u_{i}>0$ with $u_{1}u_{2}...u_{m}=M^{m}$, $i=1,2,...,m$. Then
we have
(1) $f$ has extremum values
$\frac{rs^{\frac{m}{2r-m}}+(m-r)M^{\frac{m}{2r-m}}}{s(s^{\frac{m}{2r-m}}+M^{\frac{m}{2r-m}})}$
with $\frac{m}{2}<r\leq m$ at the points that $r$ of $u_{i}$s are
$(\frac{M^{m}}{s^{2m-2r}})^{\frac{1}{2r-m}}$ and others are
$(\frac{s^{2r}}{M^{m}})^{\frac{1}{2r-m}}$;
(2) $f$ may also achieve the extremum $\frac{m}{s+M}$ when $m$ is even, at
points $\frac{m}{2}$ of $u_{i}$s are $u$ and others are $\frac{s^{2}}{u}$, in
this case we must have $s=M$;
(3) $\sup f(u_{1},u_{2},...,u_{m})=\max\\{\frac{m-1}{s},\frac{m}{s+M}\\}.$
Proof. The case $m=1$ is obvious. For the case that $m=2$, we have
$u_{2}=\frac{M^{2}}{u_{1}}$ and
$f(u_{1},u_{2})=\frac{1}{s+u_{1}}+\frac{1}{s+\frac{M^{2}}{u_{1}}}=\frac{1}{s+u_{1}}+\frac{u_{1}}{su_{1}+M^{2}}=g(u_{1})$.
Letting
$g^{\prime}(u_{1})=\frac{M^{2}}{(su_{1}+M^{2})^{2}}-\frac{1}{(s+u_{1})^{2}}=0$,
we get $(M^{2}-s^{2})(u_{1}^{2}-M^{2})=0$. If $M\not=s$, we must have
$u_{1}=M$ and $g$ has an extremum at $u_{1}=M$ with $g(M)=\frac{2}{s+M}$; if
$s=M$, then $g(u_{1})=\frac{1}{s+u_{1}}+\frac{u_{1}}{s(s+u_{1})}=\frac{1}{s}$.
Also note that $\lim_{u_{1}\rightarrow 0}g(u_{1})=\frac{1}{s}$. So, $\max
f(u_{1},u_{2})\leq\max\\{\frac{1}{s},\frac{2}{s+M}\\}$.
Assume that $m\geq 3$. $f$ approximates its supremum at the singular boundary
of its domain or at some of its extremum values. It is obvious that if
$(u_{1j},u_{2j},\ldots,u_{mj})$ converges to the boundary of the domain
$\\{\Pi_{i=1}^{m}u_{i}=M^{m}\\}$, then some $u_{i}$ tend to 0 and some tend to
$\infty$. Say $k$ of them tend to 0 and $m-k-r$ of them tend to $\infty$. Thus
there are $r$ nonnegative numbers $t_{h}$ so that
$\overline{\lim}_{j\rightarrow\infty}f(u_{1j},u_{2j},\ldots,u_{mj})=\frac{k}{s}+\sum_{h=1}^{r}\frac{1}{s+t_{h}}\leq\frac{k+r}{s}\leq\frac{m-1}{s}$.
Next let us consider the extremum values of $f$. Let
$\varphi(u_{1},u_{2},...,u_{m})=u_{1}u_{2}...u_{m}-M^{m},$ and
$\begin{array}[]{rl}L(u_{1},u_{2},...,u_{m},\lambda)=&f(u_{1},u_{2},...,u_{m})+\lambda\varphi(u_{1},u_{2},...,u_{m})\\\
=&\sum_{i=1}^{m}\frac{1}{s+u_{i}}+\lambda(u_{1}u_{2}...u_{m}-M^{m}).\end{array}$
By the method of Lagrange multipliers, we have the system
$None$
$L_{u_{i}}^{\prime}=\frac{-1}{(s+u_{i})^{2}}+\lambda{u_{1}u_{2}...u_{i-1}u_{i+1}...u_{m}}=\frac{-1}{(s+u_{i})^{2}}+\frac{\lambda
M^{m}}{u_{i}}=0,$
$i=1,2\ldots,m$. Solving this system gives
$None$ ${\lambda M^{m}}=\frac{u_{i}}{(s+u_{i})^{2}}$
for $i=1,2,...,m.$ Thus, for any $i,j$ with $i\not=j$,
$\frac{u_{i}}{(s+u_{i})^{2}}={\lambda M^{m}}=\frac{u_{j}}{(s+u_{j})^{2}}$
which yields
$u_{i}[s+u_{j}]^{2}=u_{j}[s+u_{i}]^{2}.$
It follows that
${s^{2}}{u_{i}}+2{u_{i}}{u_{j}}{s}+{u_{i}}{u_{j}}^{2}={s^{2}}{u_{j}}+{2u_{i}u_{j}s}+{u_{j}}{u_{i}^{2}},$
${s^{2}}(u_{i}-u_{j})+u_{i}u_{j}(u_{j}-u_{i})=0,$
$(u_{i}-u_{j})[s^{2}-u_{i}u_{j}]=0.$
Hence we get
$None$ ${u_{i}=u_{j}}\ \ \ {\rm or}\ \ \ {u_{i}u_{j}=s^{2}}.$
Eq.(2.3) implies that, there exists a positive integer $\frac{m}{2}\leq r\leq
m$ such that $r$ of $u_{1},u_{2},\ldots,u_{m}$ are the same, denoted by $u$,
and other $m-r$ of them are the same, denoted by $v$, with $uv=s^{2}$. Note
that
$None$ $u^{(2r-m)}s^{2(m-r)}=u^{r}v^{m-r}=u_{1}u_{2}\cdots u_{m}=M^{m}.$
If $2r=m$ (in this case $m$ is even), then we get $s^{m}=M^{m}$ and hence
$s=M$. Consequently, $f$ has a possible extremum value
$\frac{m}{2(s+u)}+\frac{m}{2(s+\frac{s^{2}}{u})}=\frac{m}{2(s+u)}+\frac{mu}{2s(s+u)}=\frac{m}{2s}=\frac{m}{s+M}$
at each point $(u_{1},u_{2},\ldots,u_{m})$ satisfying that $\frac{m}{2}$ of
$u_{i}$s are $u$ for some $u>0$ and other $\frac{m}{2}$ of them are
$\frac{s^{2}}{u}$.
If $m<2r\leq 2m$, we get by Eq.(2.4) that
$None$ $u=\left(\frac{M^{m}}{s^{2(m-r)}}\right)^{\frac{1}{2r-m}}\quad{\rm
and}\quad v=\left(\frac{s^{2r}}{M^{m}}\right)^{\frac{1}{2r-m}},$
and hence $f$ has an extremum value
$None$
$\begin{array}[]{rl}\frac{r}{s+u}+\frac{m-r}{s+v}=&\frac{r}{s+\left(\frac{M^{m}}{s^{2(m-r)}}\right)^{\frac{1}{2r-m}}}+\frac{m-r}{s+\left(\frac{s^{2r}}{M^{m}}\right)^{\frac{1}{2r-m}}}\\\
=&\frac{rs^{\frac{2m-2r}{2r-m}}+\frac{m-r}{s}M^{\frac{m}{2r-m}}}{s^{\frac{m}{2r-m}}+M^{\frac{m}{2r-m}}}=\frac{rs^{\frac{m}{2r-m}}+(m-r)M^{\frac{m}{2r-m}}}{s(s^{\frac{m}{2r-m}}+M^{\frac{m}{2r-m}})}\end{array}$
at those points $(u_{1},u_{2},\ldots,u_{m})$ that $r$ of $u_{i}$s equal
$\left(\frac{M^{m}}{s^{2(m-r)}}\right)^{\frac{1}{2r-m}}$ and other $m-r$ of
$u_{i}$s equal $\left(\frac{s^{2r}}{M^{m}}\right)^{\frac{1}{2r-m}}$.
Note that $1<\frac{m}{2}\leq r\leq m$ and
$\frac{m-1}{s}-\frac{rs^{\frac{m}{2r-m}}+(m-r)M^{\frac{m}{2r-m}}}{s(s^{\frac{m}{2r-m}}+M^{\frac{m}{2r-m}})}=\frac{(m-1-r)s^{\frac{m}{2r-m}}+(r-1)M^{\frac{m}{2r-m}}}{s(s^{\frac{m}{2r-m}}+M^{\frac{m}{2r-m}})}.$
So, if $r\leq m-1$, then we always have
$\frac{m-1}{s}\geq\frac{rs^{\frac{m}{2r-m}}+(m-r)M^{\frac{m}{2r-m}}}{s(s^{\frac{m}{2r-m}}+M^{\frac{m}{2r-m}})}$.
If $r=m$, then
$\frac{rs^{\frac{m}{2r-m}}+(m-r)M^{\frac{m}{2r-m}}}{s(s^{\frac{m}{2r-m}}+M^{\frac{m}{2r-m}})}=\frac{m}{s+M}$.
These, together with the cases that $m$ is even and $r=\frac{m}{2}$ entails
that $\sup
f(u_{1},u_{2},\ldots,u_{m})\leq\max\\{\frac{m-1}{s},\frac{m}{s+M}\\}$,
completeing the proof. $\Box$
The following lemma is crucial for our purpose. Though we only need the
special case of $k=2$ in this paper, we present the inequality for any $k\geq
1$ because it may be useful to discussing $D$-type maps induced by $k$
permutations.
Lemma 2.2. Let $s$ be a positive number, $n,k$ be positive integers with
$s>k$. Then for any $nk$ positive real numbers
$\\{x_{hi},h=1,2,...,k;i=1,2,...,n,\\}$ satisfying $x_{h1}x_{h2}...x_{hn}=1$
for each $h$ with $1\leq h\leq k$, we have
$None$
$\begin{array}[]{rl}f(x_{11},\ldots,x_{1n},x_{21},\ldots,x_{k1},\ldots,x_{kn})=&\sum_{i=1}^{n}\frac{1}{s-k+x_{1i}+x_{2i}+...+x_{ki}}\\\
\leq&\max\\{\frac{n-1}{s-k},\frac{n}{s}\\}.\end{array}$
Moreover, the extremum values of $f$ are
$None$
$\begin{array}[]{ll}\delta_{r}=\frac{r(s-k)^{\frac{n}{2r-n}}+(n-r)k^{\frac{n}{2r-n}}}{(s-k)((s-k)^{\frac{n}{2r-n}}+k^{\frac{n}{2r-n}})},&[\frac{n}{2}]+1\leq
r<n;\\\ \delta_{\frac{n}{2}}=\frac{n}{s}&\mbox{\rm if }n\ \mbox{\rm is
even},\end{array}$
where $[t]$ stands for the integer part of real number $t$.
Proof. The question is reduced to find the supremum of the $kn$-variable
function
$f(x_{11},...,x_{1n},x_{21},...,x_{2n},...,x_{k1},...,x_{kn})=\sum_{i=1}^{n}\frac{1}{s-k+x_{1i}+x_{2i}+...+x_{ki}}$
on the region $x_{hi}>0$ with $x_{h1}x_{h2}...x_{hn}=1$, $h=1,2,\ldots,k$.
Considering the behavior of $f$ near the boundary of its domain, it is obvious
that the upper bound of $f$ near the boundary is $\frac{n-1}{s-k}$.
To find the extremum values of $f$, let
$\begin{array}[]{rl}&L(x_{11},...x_{1n},x_{21},...,x_{2n},...,x_{k1},...,x_{kn},\lambda_{1},...\lambda_{k})\\\
=&f(x_{11},...x_{1n},x_{21},...,x_{2n},...,x_{k1},...,x_{kn})+\sum_{h=1}^{k}\lambda_{h}\varphi_{h},\end{array}$
where $\varphi_{h}=x_{h1}x_{h2}...x_{hn}-1$, $h=1,2,...k.$ By the method of
Lagrange multipliers, we have the system
$None$
$\begin{cases}L_{x_{11}}^{\prime}=\frac{-1}{[(s-k)+x_{11}+x_{21}+...+x_{k1}]^{2}}+\frac{\lambda_{1}}{x_{11}}=0,\\\
\ \vdots\\\
L_{x_{k1}}^{\prime}=\frac{-1}{[(s-k)+x_{11}+x_{21}+...+x_{k1}]^{2}}+\frac{\lambda_{k}}{x_{k1}}=0,\\\
\\\
L_{x_{12}}^{\prime}=\frac{-1}{[(s-k)+x_{12}+x_{22}+...+x_{k2}]^{2}}+\frac{\lambda_{1}}{x_{12}}=0,\\\
\ \vdots\\\
L_{x_{k2}}^{\prime}=\frac{-1}{[(s-k)+x_{12}+x_{22}+...+x_{k2}]^{2}}+\frac{\lambda_{k}}{x_{k2}}=0,\\\
\ \vdots\\\
L_{x_{1n}}^{\prime}=\frac{-1}{[(s-k)+x_{1n}+x_{2n}+...+x_{kn}]^{2}}+\frac{\lambda_{1}}{x_{1n}}=0,\\\
\ \vdots\\\
L_{x_{kn}}^{\prime}=\frac{-1}{[(s-k)+x_{1n}+x_{2n}+...+x_{kn}]^{2}}+\frac{\lambda_{k}}{x_{kn}}=0.\end{cases}$
Solving this system, one obtains that
$None$
$\begin{cases}\frac{\lambda_{1}}{x_{11}}={\frac{\lambda_{2}}{x_{21}}}=\ldots={\frac{\lambda_{k}}{x_{k1}}},\\\
{\frac{\lambda_{1}}{x_{12}}}={\frac{\lambda_{2}}{x_{22}}}=\ldots={\frac{\lambda_{k}}{x_{k2}}},\\\
\vdots\\\
{\frac{\lambda_{1}}{x_{1n}}}={\frac{\lambda_{2}}{x_{2n}}}=\ldots={\frac{\lambda_{k}}{x_{kn}}},\end{cases}$
which implies that
$None$
$\frac{{\lambda_{1}}^{n}}{{x_{11}}{x_{12}}\ldots{x_{1n}}}=\frac{{\lambda_{2}}^{n}}{{x_{21}}{x_{22}}\ldots{x_{2n}}}=\ldots=\frac{{\lambda_{k}}^{n}}{{x_{k1}}{x_{k2}}\ldots{x_{kn}}}.$
Now since ${x_{h1}}{x_{h2}}\ldots{x_{hn}}=1$, for each $h=1,2,...,k$, we get
$None$ ${{\lambda_{1}}^{n}}={{\lambda_{2}}^{n}}=\ldots={{\lambda_{k}}^{n}}.$
Furthermore, by Eq.(2.8) we see that
${\lambda_{h}}={\frac{x_{h1}}{{[(s-k)+({x_{11}}+{x_{21}}+\ldots+{x_{k1}})]}^{2}}}=\ldots={\frac{x_{hn}}{{[(s-k)+({x_{1n}}+{x_{2n}}+\ldots+{x_{kn}})]}^{2}}}>0$
for each $h=1,2,...,k.$ Therefore, we must have
${\lambda_{1}}={\lambda_{2}}=\ldots={\lambda_{k}}>0,$
which forces, by the Eq.(2.9), that
$None$ $\begin{cases}{x_{11}}=x_{21}=\ldots=x_{k1},\\\
{x_{12}}=x_{22}=\ldots=x_{k2},\\\ \ \vdots\\\
{x_{1n}}=x_{2n}=\ldots=x_{kn}.\\\ \end{cases}$
Let $v_{i}=x_{1i}$, $i=1,2,...,n.$ Then, the question reduces to find the
supremum of the function in $n$-variable
$g(v_{1},v_{2},...,v_{n})=\frac{1}{s-k+kv_{1}}+\frac{1}{s-k+kv_{2}}+...+\frac{1}{s-k+kv_{n}}$
on the region $v_{i}>0$ and $v_{1}v_{2}\cdots v_{n}=1$. Applying Lemma 2.1
with $m=n$, $s$ replaced by $s-k$, $M=k$ and $u_{i}=kv_{i}$, $i=1,2,\ldots n$,
we obtain that
$\sup
g(v_{1},v_{2},...,v_{n})=\max\\{\frac{n-1}{s-k},\frac{n}{s-k+k}\\}=\max\\{\frac{n-1}{s-k},\frac{n}{s}\\}.$
Hence $\sup f=\max\\{\frac{n-1}{s-k},\frac{n}{s}\\}$. Moreover, the extremum
values of $f$ are $\\{\delta_{r}:n\leq 2r\leq 2n\\}$, where
$\delta_{n}=\frac{n}{s},\
\delta_{r}=\frac{r(s-k)^{\frac{n}{2r-n}}+(n-r)k^{\frac{n}{2r-n}}}{(s-k)((s-k)^{\frac{n}{2r-n}}+k^{\frac{n}{2r-n}})}\quad{\rm
for}\ [\frac{n}{2}]+1\leq r\leq n-1,$
and
$\delta_{\frac{n}{2}}=\frac{n}{s}\quad{\rm if}\ n\ {\rm is\ even}.$
It is easily checked by Lemma 2.1 that $f$ achieves the extremum $\delta_{n}$
at $x_{11}=x_{21}=\ldots=x_{k,n}=1$; $f$ achieves $\delta_{r}$ with
$\frac{n}{2}<r\leq n-1$ at those points that $r$ of $v_{i}$s equal
$(\frac{k}{s-k})^{\frac{2n-2r}{2r-n}}$ and other $n-r$ of $v_{i}$s equal
$(\frac{s-k}{k})^{\frac{2r}{2r-n}}$; and, if $n$ is even, $f$ achieves the
extremum $\delta_{\frac{n}{2}}$ at those points that $\frac{n}{2}$ of $v_{i}$s
are the same, denoted by $v$, and other $\frac{n}{2}$ $v_{i}$s are equal to
$\frac{(s-k)^{2}}{v}$, in this case we must have $k=\frac{n}{2}=r$. $\Box$
## 3\. Criteria of positivity of $D$-type maps induced by two permutations
In [9], the authors established a criterion of positivity for $D$-type maps
constructed from one permutation, and proved that $\Phi_{D}:M_{n}\rightarrow
M_{n}$ is positive if $D=(n-1)I_{n}+P_{\pi}$, where $\pi$ is any permutation
of $\\{1,2,...,n\\}$. For those constructed from more than one permutations,
it is known that, the $D$-type map $\Phi_{D}$ with
$D=(n-k)I_{n}+P_{\pi_{0}}+P_{{\pi_{0}}^{2}}+...+P_{{\pi_{0}}^{k}}$ is
positive, where ${\pi_{0}}(i)=i-1$ mod $n$ [2] or $\pi_{0}(i)=i+1$ mod $n$
[15]. In this section we give a criterion of positivity of $D$-type linear
maps constructed from pairs of general permutations.
The following lemma comes from [9].
Lemma 3.1. Suppose $\Phi_{D}:{M_{n}}\rightarrow{M_{n}}$ is a $D$-type linear
map of the form
$None$ $(a_{ij})\longmapsto{{\rm diag}(f_{1},f_{2},...,f_{n})-(a_{ij})}$
with $(f_{1},f_{2},...,f_{n})=(a_{11},a_{22},...,a_{nn})D$ for an $n\times n$
nonnegative matrix $D=(d_{ij})$ (i.e., $d_{ij}\geq 0$ for all $i,j$). Then,
$\Phi_{D}$ is positive if and only if, for any unit vector
$u=(u_{1},u_{2},...,u_{n})^{t}\in{\mathbb{C}}^{n}$, we have
$f_{j}(u)=\sum_{i=1}^{n}d_{ij}|u_{i}|^{2}\neq 0$ whenever $u_{j}\neq 0$, and
that $\sum_{u_{j}\neq 0}{\frac{|u_{j}|^{2}}{f_{j}(u)}}\leq 1.$
Before stating our main result, we introduce a conception of property (C) for
a pair of permutations.
Definition 3.2. A pair $\\{\pi_{1},\pi_{2}\\}$ of permutations of
$(1,2,\ldots,n)$ is said to have property (C) if, for any given
$i\in\\{1,2,\ldots,n\\}$ and for any $j\not=i$, there exists
$\pi_{h_{j}}(j)\in\\{\pi_{1}(j),\pi_{2}(j)\\}$ with $h_{j}\in\\{1,2\\}$ such
that
$\\{\pi_{h_{j}}(j):j=1,2,\ldots,i-1,i+1,\ldots,n\\}=\\{1,2,\ldots,i-1,i+1,\ldots,n\\}$.
Let $\pi$ be the permutation of $(1,2,\ldots,n)$ defined by $\pi(i)=i+1$ mod
$n$. Then it is easily checked that, for any $1\leq p\leq n-1$,
$\\{\pi^{p},\pi^{p+1}\\}$ has Property (C). There are other kinds of examples.
For instance, $\\{\pi,\pi^{4}\\}$ has the property (C) if $n=5$ and $7$, but
does not have the property (C) if $n=4$ and $6$.
The following is our main general result which gives a criterion of positivity
for $D$-type maps induced by pair of general permutations.
Theorem 3.3. For any two permutations ${\pi_{1}}$ and $\pi_{2}$ of
$(1,2,...,n)$ with $n\geq 3$, let $\Phi_{D}:M_{n}\rightarrow M_{n}$ be the
$D$-type map of the form
$(a_{ij})\longmapsto{{\rm diag}(f_{1},f_{2},...,f_{n})-(a_{ij})},$
where $(f_{1},f_{2},...,f_{n})=(a_{11},a_{22},...,a_{nn})D$ and
$D=(n-2)I_{n}+{P_{\pi_{1}}}+{P_{\pi_{2}}}$ with $P_{\pi_{h}}$ the permutation
matrix of $\pi_{h}$, $h=1,2$. If $\\{\pi_{1},\pi_{2}\\}$ has the property (C),
then $\Phi_{D}$ is positive.
Proof. For any unit vector $x=(x_{1},x_{2},...,x_{n})^{t}\in{\mathbb{C}}^{n}$,
we have
$X=xx^{*}={(x_{1},x_{2},...,x_{n})^{t}}(\bar{x_{1}},\bar{x_{2}},...\bar{x_{n}})=\left(\begin{array}[]{cccc}|x_{1}|^{2}&{x_{1}\bar{x_{2}}}&\cdots&{x_{1}\bar{x_{n}}}\\\
{x_{2}\bar{x_{1}}}&|x_{1}|^{2}&\cdots&{x_{2}\bar{x_{n}}}\\\
\vdots&\vdots&\ddots&\vdots\\\
{x_{n}\bar{x_{1}}}&{x_{n}\bar{x_{2}}}&\ldots&|x_{n}|^{2}\end{array}\right).$
As $D=(d_{ij})=(n-k)I_{n}+{P_{\pi_{1}}}+{P_{\pi_{2}}}$, we get
$f_{j}(x)=\sum_{i=1}^{n}d_{ij}|x_{i}|^{2}\\\
=(n-2)|x_{j}|^{2}+|x_{\pi_{1}(j)}|^{2}+|x_{\pi_{2}(j)}|^{2}.$
Then by Lemma 3.1, $\Phi_{D}$ is positive if the following inequality
$None$
$\sum_{i=1}^{n}\frac{|x_{i}|^{2}}{f_{i}(x)}=\sum_{i=1}^{n}{\frac{|x_{i}|^{2}}{(n-2)|x_{i}|^{2}+|x_{\pi_{1}(i)}|^{2}+|x_{\pi_{2}(i)}|^{2}}}\leq
1$
holds for any unit vector
$x=(x_{1},x_{2},\ldots,x_{n})^{t}\in{\mathbb{C}}^{(n)}$.
Case 1. $n\geq 4$.
Let us consider first the case that $n\geq 4$. Assume that all $x_{i}\neq 0$
and let $u_{hi}={\frac{|x_{\pi_{h}(i)}|^{2}}{|x_{i}|^{2}}}$, $i=1,2,...,n$;
$h=1,2.$ Then, $u_{h1}u_{h2}\cdots u_{hn}=1$ for each $h=1,2$, and thus, by
Lemma 2.2, we have
$\begin{array}[]{rl}&f(x_{1},x_{2},\ldots,x_{n})=\sum_{i=1}^{n}{\frac{|x_{i}|^{2}}{(n-2)|x_{i}|^{2}+|x_{\pi_{1}(i)}|^{2}+|x_{\pi_{2}(i)}|^{2}}}\\\
=&g(u_{11},u_{12},\ldots,u_{1n},u_{21},u_{22},\ldots,u_{2n})=\sum_{i=1}^{n}{\frac{1}{n-2+u_{1i}+u_{2i}}}\\\
\leq&\max\\{\frac{n-1}{n-2},1\\}.\end{array}$
and, as a restriction of $g$, the maximum of all extremum values of $f$ is
bounded by the maximum of all extremum values of $g$. Thus, by Lemma 2.2, we
have
$\max\\{\mbox{\rm extreme values of
}f\\}\leq\max\\{1,\delta_{r}:r=[\frac{n}{2}]+1,[\frac{n}{2}]+1,\ldots,n-1\\},$
where
$\delta_{r}=\frac{r(n-2)^{\frac{n}{2r-n}}+(n-r)2^{\frac{n}{2r-n}}}{(n-2)((n-2)^{\frac{n}{2r-n}}+2^{\frac{n}{2r-n}})}$
and $[t]$ is the integer part of $t$. If $[\frac{n}{2}]+1\leq r\leq n-2$, then
$\begin{array}[]{rl}\delta_{r}=&\frac{r(n-2)^{\frac{n}{2r-n}}+(n-r)2^{\frac{n}{2r-n}}}{(n-2)((n-2)^{\frac{n}{2r-n}}+2^{\frac{n}{2r-n}})}\\\
=&\frac{(2r-n)(n-2)^{\frac{n}{2r-n}}}{(n-2)((n-2)^{\frac{n}{2r-n}}+2^{\frac{n}{2r-n}})}+\frac{n-r}{n-2}\leq\frac{r}{n-2}\leq
1.\end{array}$
We claim that we also have $\delta_{n-1}\leq 1$. In fact,
$\delta_{n-1}=\frac{(n-1)(n-2)^{\frac{n}{2r-n}}+2^{\frac{n}{2r-n}}}{(n-2)((n-2)^{\frac{n}{2r-n}}+2^{\frac{n}{2r-n}})}=\frac{(n-2)^{\frac{n}{2r-n}}}{(n-2)^{\frac{n}{2r-n}}+2^{\frac{n}{2r-n}}}+\frac{1}{n-2}\leq
1$
if and only if
$None$ $\psi(n)=\frac{n-2}{(n-3)^{\frac{n-2}{n}}}\leq 2.$
Let
$\varphi(t)=\ln(\frac{t-2}{(t-3)^{\frac{t-2}{t}}})=\ln(t-2)-\frac{t-2}{t}\ln(t-3)$
for $t\geq 4$. Then
$\varphi^{\prime}(t)=\frac{t^{2}-4t-(2t^{2}-10t+12)\ln(t-3)}{t^{2}(t-2)(t-3)}.$
If $t\geq 9$, then
$t^{2}-4t-(2t^{2}-10t+12)\ln(t-3)<-(t^{2}-10t+12)\ln(t-3)<0$
and hence $\varphi(t)$ is decreasing on $[9,\infty)$. It follows that
$\psi(n)=\frac{n-2}{(n-3)^{\frac{n-2}{n}}}$ is decreasing for $n\geq 9$. Since
$\psi(4)=2,\ \psi(5)\approx 1.9793,\ \psi(6)\approx 1.9230,\ \psi(7)\approx
1.8575,\ \psi(8)\approx 1.7944,\ \psi(9)\approx 1.7373,$
we see that the inequality in Eq.(3.3) is true for all $n\geq 4$ and hence
$\delta_{n-1}\leq 1$ holds for all $n\geq 4$, as desired.
Therefore, for any $n\geq 4$, 1 is the maximum extremum value of
$f(x_{1},x_{2},\ldots,x_{n})$ achieving at
$f(\frac{1}{\sqrt{n}},\frac{1}{\sqrt{n}},\ldots,\frac{1}{\sqrt{n}})$.
In the following we show that the supremum of $f$ on the boundary of its
domain $|x_{1}|^{2}+|x_{2}|^{2}+\cdots+|x_{n}|^{2}=1$ is not greater than 1,
either. Assume that $(x_{1},x_{2},\ldots,x_{n})^{t}$ lies in the boundary of
the region $|x_{1}|^{2}+|x_{2}|^{2}+\cdots+|x_{n}|^{2}=1$; then at least one
of $x_{i}$ is zero. With no loss of generality, say $x_{i}\not=0$ for
$i=1,2,\ldots r$, but $x_{r+1}=\ldots=x_{n}=0$, where $1<r<n$. Then we have
$\begin{array}[]{rl}&\sum_{i=1}^{n}{\frac{|x_{i}|^{2}}{(n-2)|x_{i}|^{2}+|x_{\pi_{1}(i)}|^{2}+|x_{\pi_{2}(i)}|^{2}}}=\sum_{i=1}^{r}{\frac{1}{(n-2)+\frac{|x_{\pi_{1}(i)}|^{2}}{|x_{i}|^{2}}+\frac{|x_{\pi_{2}(i)}|^{2}}{|x_{i}|^{2}}}}.\end{array}$
If $r=n-1$, by the assumption that $\\{\pi_{1},\pi_{2}\\}$ has the property
(C), one can choose $\pi_{h_{i}}(i)\in\\{\pi_{1}(i),\pi_{2}(i)\\}$ so that
$\\{\pi_{h_{i}}(i)\\}_{i=1}^{n-1}=\\{1,2,\ldots,n-1\\}$. So
$\Pi_{i=1}^{n-1}\frac{|x_{\pi_{h_{i}}(i)}|^{2}}{|x_{i}|^{2}}=1,$
and thus, by Lemma 2.2 with $k=1$ and $s=n-1$, we get
$None$
$\begin{array}[]{rl}&\sum_{i=1}^{n}{\frac{|x_{i}|^{2}}{(n-2)|x_{i}|^{2}+|x_{\pi_{1}(i)}|^{2}+|x_{\pi_{2}(i)}|^{2}}}=\sum_{i=1}^{n-1}{\frac{1}{(n-2)+\frac{|x_{\pi_{1}(i)}|^{2}}{|x_{i}|^{2}}+\frac{|x_{\pi_{2}(i)}|^{2}}{|x_{i}|^{2}}}}\\\
\leq&\sum_{i=1}^{n-1}{\frac{1}{((n-1)-1)+\frac{|x_{\pi_{h_{i}}(i)}|^{2}}{|x_{i}|^{2}}}}\leq\frac{n-1}{n-1}=1.\end{array}$
If $r\leq n-2$, then we always have
$None$
$\begin{array}[]{rl}&\sum_{i=1}^{n}{\frac{|x_{i}|^{2}}{(n-2)|x_{i}|^{2}+|x_{\pi_{1}(i)}|^{2}+|x_{\pi_{2}(i)}|^{2}}}=\sum_{i=1}^{r}{\frac{1}{(n-2)+\frac{|x_{\pi_{1}(i)}|^{2}}{|x_{i}|^{2}}+\frac{|x_{\pi_{2}(i)}|^{2}}{|x_{i}|^{2}}}}\leq\frac{r}{n-2}\leq
1.\end{array}$
The inequalities in Eqs.(3.4) and (3.5) ensure us that $f$ is upper bounded by
1 on the boundary of its domain, and hence Eq.(3.2) is true for any unit
vectors. Applying Lemma 3.1, $\Phi_{D}$ is a positive linear map for the case
$n\geq 4$.
Case 2. $n=3$.
Finally, assume that $n=3$. If one of $\pi_{1}$ and $\pi_{2}$ is the identity,
then the question reduce to the case of $D$-type maps generalized by one
permutation, and hence the corresponding $D$-type map is positive by [9]. So
we may assume that both $\pi_{1}$ and $\pi_{2}$ are not the identity. It is
easily checked that we have four possible cases so that
$\\{\pi_{1},\pi_{2}\\}$ has the property (C):
$\begin{array}[]{llll}\\{(2,3,1),(3,1,2)\\};&\\{(3,2,1),(2,1,3)\\};&\\{(3,2,1),(1,3,2)\\};&\\{(1,3,2),(2,1,3)\\}.\end{array}$
The case $\\{(2,3,1),(3,1,2)\\}$ is implied by the result in [15]. By Lemma
3.1, the last three cases induce to prove the following inequality
$\frac{1}{1+u+v}+\frac{u}{2u+1}+\frac{v}{2v+1}\leq 1$
for any $u,v>0$. This is true because
$\begin{array}[]{rl}&(2u+1)(2v+1)+u(1+u+v)(2v+1)+v(1+u+v)(2u+1)\\\
\leq&(1+u+v)(2u+1)(2v+1)\end{array}$
by $2uv\leq u^{2}+v^{2}$. So the theorem is also true for the case $n=3$,
finishing the proof. $\Box$
It is clear that the less $n$ is the easier to check the property (C) of
$\\{\pi_{1},\pi_{2}\\}$. This motivates us to decompose the permutations into
small ones.
For a nonempty subset $F$ of $\\{1,2,\ldots,n\\}$, if $F$ is a common
invariant subset of $\pi_{s}$s, i.e., if $\pi_{h}(F)=F$ holds for all
$h=1,2,\ldots k$, we say that $F$ is an invariant subsets of
$\\{\pi_{1},\pi_{2},\ldots,\pi_{k}\\}$. Obvious, there exist disjoint minimal
invariant subsets $F_{1},F_{2},\ldots,F_{l}$ of
$\\{\pi_{1},\pi_{2},\ldots,\pi_{k}\\}$ such that $\sum_{s=1}^{l}\\#F_{s}=n$
(i.e., $\cup_{s=1}^{l}F_{s}=\\{1,2,\ldots,n\\}$). We say
$\\{F_{1},F_{2},\ldots,F_{l}\\}$ is the complete set of minimal invariant
subsets of $\\{\pi_{1},\ldots,\pi_{k}\\}$. Thus one can reduce the set
$\\{\pi_{1},\pi_{2},\ldots,\pi_{k}\\}$ of permutations into $l$ sets
$\\{\pi_{1s},\pi_{2s},\ldots,\pi_{ks}\\}_{s=1}^{l}$ of small ones, where
$\pi_{hs}=\pi_{h}|_{F_{s}}$. It is easily checked that $\\{\pi_{1},\pi_{2}\\}$
has the property (C) if and only if each pair $\\{\pi_{1s},\pi_{2s}\\}$ of the
sub-permutations has the property (C).
The following is a version of Theorem 3.3.
Theorem 3.4. Let $\\{F_{s}\\}_{s=1}^{l}$ be the complete set of minimal
invariant subsets of a pair $\\{\pi_{1},\pi_{2}\\}$ of permutations of
$\\{1,2,\ldots,n\\}$. Let $\pi_{is}=\pi_{i}|_{F_{s}}$. If
$\\{\pi_{1s},\pi_{2s}\\}$ has Property (C) for every $s=1,2,\ldots,l$, then
the $D$-type map $\Phi_{D}$ defined as in Theorem 3.3 is positive.
Theorem 3.3 and Theorem 3.4 give a possible way to construct new positive maps
but it is not very clear how to check whether or not a given pair of
permutations has the property (C). To get an insight of this, in the
following, we give a characterization of a pair of permutations to have
property (C), which will be used to construct new positive maps in the next
section. Note that $\\{\pi_{1},\pi_{2}\\}$ has property (C) if and only if
each pair $\\{\pi_{1s},\pi_{2s}\\}$ on the common invariant subset $F_{s}$ has
property (C). So, to check whether or not $\\{\pi_{1},\pi_{2}\\}$ has property
(C), we may assume that $\pi_{1}$ and $\pi_{2}$ has no common proper invariant
subsets. It is clear that, if $n=1$, $\\{\pi_{1},\pi_{2}\\}$ always has
property (C); if $n=2$, $\\{\pi_{1},\pi_{2}\\}$ has property (C) if and only
if one of $\pi_{1},\pi_{2}$ is the identity and other is
$(1,2)\rightarrow(2,1)$.
Proposition 3.5. Let $\pi_{1},\pi_{2}$ be two permutations of $(1,2,\ldots,n)$
with $n\geq 2$ having no proper common invariant subsets. Then
$\\{\pi_{1},\pi_{2}\\}$ has the property (C) if and only if the following
conditions are satisfied:
(1) For any distinguished $i,j$, $\pi_{1}(i)\not=\pi_{2}(i)$ and
$\\{\pi_{1}(i),\pi_{2}(i)\\}\not=\\{\pi_{1}(j),\pi_{2}(j)\\}$;
(2) For any $i$ and $j_{1},j_{2}$ with $\pi_{1}(j_{2})=\pi_{2}(j_{1})=i$, if
distinct $j_{3},\ldots,j_{m}\not\in\\{i,j_{1},j_{2}\\}$ satisfy that
$\pi_{2}(j_{3})=\pi_{1}(j_{1})$,
$\pi_{2}(j_{4})=\pi_{1}(j_{3}),\ldots,\pi_{2}(j_{m})=\pi_{1}(j_{m-1})$, then
$\pi_{1}(j_{m})\not=\pi_{2}(j_{2})$.
Proof. Assume that $\\{\pi_{1},\pi_{2}\\}$ satisfy the conditions (1)-(2). For
any $i$, we have to show that we can choose one element in
$\pi_{h_{j}}(j)\in\\{\pi_{1}(j),\pi_{2}(j)\\}$ for each $j\not=i$ so that
$\\{\pi_{h_{j}}(j),j\not=i\\}=\\{1,2,\ldots,i-1,i+1,\ldots,n\\}$.
Case (i). $i\in\\{\pi_{1}(i),\pi_{2}(i)\\}$, say $\pi_{1}(i)=i$, then
obviously the choice
$\\{\pi_{1}(j):j\not=i\\}=\\{1,2,\ldots,i-1,i+1,\ldots,n\\}$.
Case (ii). $i\not\in\\{\pi_{1}(i),\pi_{2}(i)\\}$.
Let $j_{1},j_{2}$ such that $\pi_{1}(j_{2})=i=\pi_{2}(j_{1})$. By the
condition (1), we must gave $\pi_{1}(j_{1})\not=\pi_{2}(j_{2})$. In fact, if
$\pi_{1}(j_{1})=\pi_{2}(j_{2})$, then
$\\{\pi_{1}(j_{1}),\pi_{2}(j_{1})\\}=\\{\pi_{1}(j_{2}),\pi_{2}(j_{2})\\}$,
which contradicts to the condition (1). If
$\\{\pi_{1}(j_{1}),\pi_{2}(j_{2})\\}=\\{\pi_{1}(i),\pi_{2}(i)\\}$, then
$\\{\pi_{1}(j_{1}),\pi_{2}(j_{2})\\}\cup\\{\pi_{1}(j):j\not\in\\{i,j_{1},j_{2}\\}\\}=\\{1,\ldots,i-1,i+1,\ldots,n\\}$
and we finished the proof.
Assume that
$\\{\pi_{1}(j_{1}),\pi_{2}(j_{2})\\}\not=\\{\pi_{1}(i),\pi_{2}(i)\\}$.
If $\pi_{1}(j_{1})=\pi_{2}(i)$ or $\pi_{2}(j_{2})=\pi_{1}(i)$, saying
$\pi_{2}(j_{2})=\pi_{1}(i)$, then we have
$\\{\pi_{2}(j_{2})\\}\cup\\{\pi_{1}(j):j\not\in\\{i,j_{2}\\}\\}=\\{\pi_{1}(j):j\not=j_{2}\\}=\\{1,2,\ldots,i-1,i+1,\ldots,n\\}$,
and then the proof is finished.
Thus we may in the sequel assume that
$\\{\pi_{1}(j_{1}),\pi_{2}(j_{2})\\}\cap\\{\pi_{1}(i),\pi_{2}(i)\\}=\emptyset$.
Take $j_{3}$ so that $\pi_{2}(j_{3})=\pi_{1}(j_{1})$. As
$\pi_{1}(j_{1})\not=\pi_{2}(j_{2})$, we have $j_{3}\not=j_{2}$. Also
$\pi_{2}(j_{3})=\pi_{1}(j_{1})\not=\pi_{2}(i)$ and
$\pi_{2}(j_{3})=\pi_{1}(j_{1})\not=\pi_{2}(j_{1})=i$ ensures that
$j_{3}\not\in\\{i,j_{1}\\}$. So have $j_{3}\not\in\\{i,j_{1},j_{2}\\}$ and by
the condition (3), we have $\pi_{1}(j_{3})\not=\pi_{2}(j_{2})$. Thus we get a
set $\\{\pi_{1}(j_{1}),\pi_{2}(j_{2}),\pi_{1}(j_{3})\\}$ of distinct elements.
If $\pi_{1}(j_{3})=\pi_{2}(i),$ then
$\\{\pi_{1}(j_{1}),\pi_{1}(j_{3})\\}\cup\\{\pi_{2}(j):j\not\in\\{i,j_{1},j_{3}\\}=\\{\pi_{2}(j):j\not=j_{1}\\}=\\{1,2,\ldots,i-1,i+1,\ldots,n\\}$
and we finish the proof. Suppose that $\pi_{1}(j_{3})\not=\pi_{2}(i)$ and take
$j_{4}$ so that $\pi_{2}(j_{4})=\pi_{1}(j_{3})$. Then $j_{4}\not=i$ and
$j_{4}\not=j_{3}$ by the previous proof. In fact, because
$\pi_{2}(j_{3})=\pi_{1}(j_{1})\not=\pi_{1}(j_{3})$, so
$\pi_{2}(j_{3})\not=\pi_{1}(j_{3})$, and $j_{4}\not=j_{3}$. As
$\pi_{2}(j_{4})=\pi_{1}(j_{3})\not=\pi_{2}(j_{2})$, thus we have
$j_{4}\not=j_{2}$. Also
$\pi_{2}(j_{4})=\pi_{1}(j_{3})\not=\pi_{1}(j_{2})=\pi_{2}(j_{1})=i$ implies
that $j_{4}\not=j_{1}$. So $j_{4}\not\in\\{i,j_{1},j_{2},j_{3}\\}$ and by the
condition (3), $\pi_{1}(j_{4})\not=\pi_{2}(j_{2})$. Therefore, we have
$\pi_{1}(j_{4})\not\in\\{\pi_{1}(j_{1}),\pi_{2}(j_{2}),\pi_{1}(j_{3})\\}$.
Again, if $\pi_{1}(j_{4})=\pi_{2}(i),$ then
$\\{\pi_{1}(j_{1}),\pi_{1}(j_{3}),\pi_{1}(j_{4})\\}\cup\\{\pi_{2}(j):j\not\in\\{i,j_{1},j_{3},j_{4}\\}=\\{\pi_{2}(j):j\not=j_{1}\\}=\\{1,2,\ldots,i-1,i+1,\ldots,n\\}$,
and the proof is finished. If $\pi_{1}(j_{4})\not=\pi_{2}(i)$, then, by the
condition (3), $\pi_{1}(j_{4})\not=\pi_{2}(j_{2})$ and
$\pi_{1}(j_{4})\not=\pi_{2}(j_{1})=i$. It follows that we can take
$j_{5}\not\in\\{i,j_{1},j_{2},j_{3},j_{4}\\}$ such that
$\pi_{2}(j_{5})=\pi_{1}(j_{4})$, and
$\pi_{1}(j_{5})\not\in\\{\pi_{1}(j_{1}),\pi_{2}(j_{2}),\pi_{1}(j_{3}),\pi_{1}(j_{4})\\}$.
We continue the above process until getting $j_{m}$ such that
$\pi_{2}(j_{m})=\pi_{1}(j_{m-1})$ and $\pi_{1}(j_{m})=\pi_{2}(i)$. Then we get
$\\{\pi_{1}(j_{1}),\pi_{1}(j_{3}),\ldots,\pi_{1}(j_{m})\\}\cup\\{\pi_{2}(j):j\not\in\\{i,j_{1},j_{3},\ldots,j_{m}\\}\\}=\\{\pi_{2}(j):j\not=j_{1}\\}=\\{1,2,\ldots,i-1,i+1,\ldots,n\\}$.
Hence, the conditions (1) and (2) imply $\\{\pi_{1},\pi_{2}\\}$ has Property
(C).
By checking the arguments above, one sees that, if any one of the conditions
(1) and (3) is broken, then $\\{\pi_{1},\pi_{2}\\}$ can not have Property (C).
Therefore, the conditions are also necessary. $\Box$
By the symmetry of $\pi_{1}$ and $\pi_{2}$, the condition (2) in Proposition
3.5 may be replaced by the following
(2′) For any $i$ and $j_{1},j_{2}$ with $\pi_{1}(j_{2})=\pi_{2}(j_{1})=i$, if
distinct $j_{3},\ldots,j_{m}\not\in\\{i,j_{1},j_{2}\\}$ satisfy that
$\pi_{1}(j_{3})=\pi_{2}(j_{2})$,
$\pi_{1}(j_{4})=\pi_{2}(j_{3}),\ldots,\pi_{1}(j_{m})=\pi_{2}(j_{m-1})$, then
$\pi_{2}(j_{m})\not=\pi_{1}(j_{1})$.
Corollary 3.6. Let $\pi_{1},\pi_{2}$ be permutations of $(1,2,\ldots,n)$ and
let $D=(n-2)I_{n}+P_{\pi_{1}}+P_{\pi_{2}}$. Then the $D$-type map $\Phi_{D}$
is positive if for any minimal invariant subset $F$ of $\\{\pi_{1},\pi_{2}\\}$
with $\\#F\geq 2$, $\\{\pi_{1}|_{F},\pi_{2}|_{F}\\}$ satisfies the conditions
(1) and (2) in Proposition 3.5.
Proof. The corollary is an immediate consequence of Proposition 3.5 and
Theorem 3.4. $\Box$
Before going to next section, we give a simple example of how to using the
results in this section to construct new positive linear maps.
Example 3.7. The map $\Phi:M_{5}\rightarrow M_{5}$ defined by
$(a_{ij})\mapsto\left(\begin{array}[]{ccccc}3a_{11}+a_{55}&-a_{12}&-a_{13}&-a_{14}&-a_{15}\\\
-a_{21}&2a_{22}+a_{11}+a_{44}&-a_{23}&-a_{24}&-a_{25}\\\
-a_{31}&-a_{32}&3a_{33}+a_{22}&-a_{34}&-a_{35}\\\
-a_{41}&-a_{42}&-a_{43}&2a_{44}+a_{33}+a_{55}&-a_{45}\\\
-a_{51}&-a_{52}&-a_{53}&-a_{54}&2a_{55}+a_{44}\end{array}\right)$
is positive. In fact, $\Phi$ is a $D$-type map with
$D=3I_{5}+P_{\pi_{1}}+P_{\pi_{2}}$, where $\pi_{1}$ and $\pi_{2}$ are
permutations of $(1,2,3,4,5)$ determined respectively by $(2,3,4,5,1)$ and
$(1,5,3,2,4)$. It is easily checked by Definition 3.2 or Proposition 3.5 that
$\\{\pi_{1},\pi_{2}\\}$ has the property (C). Hence the positivity of $\Phi$
follows from Theorem 3.3. We remark here that, this example is different from
those treated in the next section.
## 4\. Constructing positive $D$-type maps from cyclic permutations
The class of positive linear maps may detected by Theorem 3.3 is quite large,
which contains many known positive maps of $D$-type such as that induced by
just one permutation in [9] and that in [15] with $k=2$. Applying the results,
especially Theorem 3.3, Proposition 3.5 in the previous section, we construct
in this section some new classes of $D$-type positive maps induced by two
powers of the cyclic permutation.
In the sequel, for any integer $n\geq 3$, we denote $\pi$ the cyclic
permutation of $(1,2\ldots,n)$ defined by $\pi(i)=i+1$ mod $n$. Let $1\leq
p<q\leq n$ be two integers. Consider the $D$-type map induced by
$\pi^{p},\pi^{q}$ with $D=(n-2)I_{n}+P_{\pi^{p}}+P_{\pi^{q}}$. We shall give
an easily handled criterion for $\Phi_{n,p,q}=\Phi_{D}$ to be positive. Note
that $q=n$ implies $\pi^{q}={\rm id}$ and
$D=(n-2)I_{n}+P_{\pi^{p}}+P_{\pi^{n}}=(n-1)I_{n}+P_{\pi^{p}}$. So, $\Phi_{D}$
is positive by [9, 14] whenever $q=n$ and it is also clear that
$\\{\pi^{p},{\rm id}\\}$ has the property (C). Thus we may assume $q<n$ in the
following lemma.
Lemma 4.1. Let $\pi$ be the permutation of $(1,2,\ldots,n)$ defined by
$\pi(i)=i+1$ (mod $n$). For any $1\leq p<q<n$, $\\{\pi^{p},\pi^{q}\\}$ has the
property (C) if and only if one of the following conditions holds.
(1) $q-p=1$.
(2) $1<q-p<n-1$ and if there are relatively prime positive integers $k,m$ with
$m<n$ and $k<q-p$ such that $m(q-p)={kn}$, then $p=n-d(q-p)$ for some integer
$1\leq d\leq m-1$.
Proof. If $q-p=1$, then it is clear that $\\{\pi^{p},\pi^{q}\\}$ has property
(C).
So, to prove the lemma, it suffices to show that, for any $p,q$ with
$1<q-p<n-1$ and $1\leq p<q\leq n-1$, $\\{\pi^{p},\pi^{q}\\}$ does not have
property (C) if and only if there exist relatively prime integers $k,m$ with
$0<k<q-p$ and $0<m<n$ such that $m(q-p)=kn$ and $p\not=n-d(q-p)$ for any
integer $d$ with $2\leq d\leq m-1$. By Proposition 3.5, it suffices to check
the following two claims.
Claim 1. $\\{\pi_{1}=\pi^{p},\pi_{2}=\pi^{q}\\}$ does not satisfy the
condition (1) of Proposition 3.5 if and only if $2(q-p)=n$ and
$p\not=n-(q-p)$.
In fact, $\\{\pi_{1},\pi_{2}\\}$ does not meet the condition (1) of
Proposition 3.5 if and only if
$i+p\ {\rm mod}\ n=\pi_{1}(i)=\pi_{2}(j)=j+q\ {\rm mod}\ n$
and
$i+q\ {\rm mod}\ n=\pi_{2}(i)=\pi_{1}(j)=j+p\ {\rm mod}\ n$
for some distinct $i,j$. This happens if and only if $2(q-p)=0$ mod $n$, and
in turn, if and only if $n=2(q-p)$ as $q\leq n$. It is clear that $p$ can not
be $n-(q-p)$, because this would imply that $q=n$.
Claim 2. $\\{\pi_{1}=\pi^{p},\pi_{2}=\pi^{q}\\}$ does not satisfy the
condition (2) of Proposition 3.5 if and only if $m(q-p)=kn$ for some
relatively prime positive integers $k,m$ with $1\leq k<q-p$, $2<m<n$, and
$p\not=n-d(q-p)$ for any $1<d<m-1$.
$\\{\pi_{1}=\pi^{p},\pi_{2}=\pi^{q}\\}$ does not satisfy the condition (2) of
Proposition 3.5 if and only if for some $i$, there exist distinct
$j_{1},\ldots,j_{m}\not\in\\{i\\}$ such that
$\pi_{1}(j_{2})=\pi_{2}(j_{1})=i$, $\pi_{2}(j_{3})=\pi_{1}(j_{1})$,
$\pi_{2}(j_{4})=\pi_{1}(j_{3}),\ldots,\pi_{2}(j_{m})=\pi_{1}(j_{m-1})$ and
$\pi_{1}(j_{m})=\pi_{2}(j_{2})$, and in turn, if and only if for some $i$
there exist distinct $j_{1},\ldots,j_{m}\not\in\\{i\\}$ such that
$None$ $\begin{array}[]{rl}j_{1}+q=&j_{2}+p\ {\rm mod}\ n,\\\
j_{3}+q=&j_{1}+p\ {\rm mod}\ n,\\\ j_{4}+q=&j_{3}+p\ {\rm mod}\ n,\\\
\vdots&\\\ j_{m}+q=&j_{m-1}+p\ {\rm mod}\ n,\\\ j_{2}+q=&j_{m}+p\ {\rm mod}\
n.\end{array}$
Summing up all equations in Eq.(4.1) we obtain that
$m(q-p)=0\ \ {\rm mod}\ n.$
Note that $2\leq m<n$; thus there exists positive integer $k<q-p$ such that
$None$ $m(q-p)=kn.$
Moreover, $j_{h}=j_{1}-(q-p)(h-2)=j_{2}-(q-p)(h-1)$ mod $n$ for
$h=3,4,\ldots,m$. Obviously, $j_{1},j_{2}\not\in\\{i\\}$. While
$j_{3},\ldots,j_{m}\not\in\\{i\\}$ implies that $j_{h}\not=j_{1}+q=j_{2}+p$
mod $n$ for any $h=3,4,\ldots,m$. It is clear that $j_{h}=i=j_{1}+q=j_{2}+p$
mod $n$ for some $j_{h}=j_{1}-(q-p)(h-2)=j_{2}-(q-p)(h-1)$ mod $n$ if and only
if $j_{1}+q=j_{1}-(q-p)(h-2)$ and $j_{2}+p=j_{2}-(q-p)(h-1)$, and in turn, if
and only if $p=n-(q-p)(h-1)=n-d(q-p)$ for some $2\leq d\leq m-1$. So,
$j_{1},\ldots,j_{m}\not\in\\{i\\}$ implies that $p\not=n-d(q-p)$ for any
$2\leq d<m$. Thus we proved that $\\{\pi^{p},\pi^{q}\\}$ does not satisfies
Condition (2) of Proposition 3.6 implies that $m(q-p)=kn$ for some relatively
prime positive integers $m,k$ with $1\leq k<q-p$, $1\leq m<n$, and
$None$ $p\not=n-d(q-p)\quad{\rm for\ every}\ 2\leq d\leq m-1.$
Conversely, assume that Eq.(4.2) and Eq.(4.3) hold. Take any $j_{1},j_{2}$ so
that $j_{2}-j_{1}=q-p$ mod $n$. Then $j_{2}=j_{1}+(q-p)$ mod $n$. If $m=2$, we
must have $2(q-p)=n$ and hence $j_{2}+(q-p)=j_{1}+2(q-p)=j_{1}$ mod $n$, which
gives $j_{2}+q=j_{1}+p$ mod $n$. If $m\geq 3$, for $3\leq h\leq m$, let
$None$ $j_{h}=j_{1}-(q-p)(h-2)=j_{2}-(q-p)(h-1)\ \ {\rm mod}\ n.$
Then $j_{2}=j_{m}+(q-p)(m-1)$ mod $n$ and hence $j_{2}+q-p=j_{m}$ mod $n$ as
$(q-p)m=kn$. This ensures that $j_{2}+q=j_{m}+p$ mod $n$. We claim that
$j_{h}\not=j_{s}$ whenever $t\not=s$. In fact, it is clear that
$j_{1}\not=j_{2}$; for $3\leq h,s\leq m$,
$j_{h}=j_{s}\Leftrightarrow j_{1}-(q-p)(h-2)=j_{1}-(q-p)(s-2)\Leftrightarrow
h=s.$
Moreover, the Eqs.(4.3) and (4.4) imply that
$j_{1},\ldots,j_{m}\not\in\\{i\\}$. Hence Eq.(4.2) and Eq.(4.3) imply that
$\\{\pi^{p},\pi^{q}\\}$ does not satisfies the condition (2) of Proposition
3.5.
Now, by Proposition 3.5, Claims 1-2 ensure that the lemma holds, completing
the proof. $\Box$
Using Lemma 4.1, the following criterion of positivity of $\Phi_{n,p,q}$ is
immediate, which is very easily applied.
Theorem 4.2. Let $n\geq 3$ and $\pi$ be the permutation defined by
$\pi(i)=i+1$ mod $n$. For any integers $1\leq p<q\leq n$, let
$D=(n-2)I_{n}+P_{\pi^{p}}+P_{\pi^{q}}$. Then the $D$-type map
$\Phi_{n,p,q}=\Phi_{D}:M_{n}\rightarrow M_{n}$ of the form Eq.(1.1) is
positive if one of the following conditions holds.
(1) $q-p=1$ or $q=n$.
(2) $q<n$, $1<q-p<n-1$ and if there are relatively prime positive integers
$k,m$ with $m<n$ and $k<q-p$ such that $m(q-p)={kn}$, then $p=n-d(q-p)$ for
some integer $1\leq d\leq m-1$.
We remark that in general the condition $q-p=\frac{kn}{m}$ does not imply that
$q-p$ is a factor of $n$. For example, let $n=8$, $q-p=6$; then $m=3$ and
$k=4$. If $p=8-6=2,q=8$, then $\\{\pi^{2},\pi^{8}\\}$ has property (C). But,
$\\{\pi^{1},\pi^{7}\\}$ does not posses property (C).
The following corollary is immediate.
Corollary 4.3. The $D$-type map $\Phi_{n,p,q}:M_{n}\rightarrow M_{n}$ in
Theorem 4.2 is positive if any one of the following conditions holds.
(1) $q-p=1$ or $q=n$.
(2) $n$ is prime.
(3) There are no relatively prime $m,k$ with $1\leq m<n$ and $1\leq k<q-p$ so
that $m(q-p)=kn$.
(4) $q-p$ is prime and is not a factor of $n$. Particularly, if $q-p=2$ and
$n$ is odd.
(5) $q-p$ is a prime factor of $n$ and $p=d(q-p)$ for some $1\leq
d\leq\frac{n}{q-p}-2$.
Proof. The implication (1) $\Rightarrow\Phi_{D}$ is positive is clear by
Theorem 4.2.
If (2) holds, that is, if $n$ is prime, then there are no relative prime
positive integers $m<n,k<q-p$ such that $m(q-p)=kn$. Hence, for any $1\leq
p<q\leq n$, $\\{\pi^{p},\pi^{q}\\}$ has property (C).
(3) entails that $\\{\pi^{p},\pi^{q}\\}$ meets the condition (2) in Theorem
4.2 and hence $\Phi_{n,p,q}$ is positive.
If (4) holds, that is, if $q-p$ is prime and is not a factor of $n$, then
$n=\frac{m(q-p)}{k}$ implies that $m=kr$ for some integer $r$ and $n=r(q-p)$,
which is impossible. So $\\{\pi^{p},\pi^{q}\\}$ meets the condition (2) of
Theorem 4.2 and thus $\Phi_{n,p,q}$ is positive.
The condition (5) implies that $m_{1}(q-p)=kn$ and the greatest common divisor
of $m_{1}$ and $k$ is 1 if and only if $m_{1}=m$ and $k=1$ as $q-p$ is prime.
As $p=d(q-p)=n-(m-d)(q-p)$, $\\{\pi^{p},\pi^{q}\\}$ has property (C) by Lemma
4.1. $\Box$
Since, in quantum information theory, an $N$-qbit quantum system corresponds
to a complex Hilbert space of dimension $2^{N}$, the case $n=2^{N}$ is of
special importance.
Corollary 4.4. For $n=2^{N}$ with $N\geq 2$ and $1\leq p<q<n$, the $D$-type
map $\Phi_{{2^{N}},p,q}:M_{n}\rightarrow M_{n}$ in Theorem 4.2 is positive if
one of the following holds:
(1) $q-p$ is odd.
(2) $q-p=2^{b}$ with $1\leq b\leq N-1$ and $p=d2^{b}$ for some $1\leq d\leq
2^{N-b}-1$.
(3) $q-p=2^{b}r$ with $r$ odd and $p=2^{b}(2^{N-b}-dr)$ for some $1\leq
d\leq\frac{2^{N-b}-1}{r}$.
In the following we present some simple examples to illustrate how to use the
results in this section.
Example 4.5. For $n=5$ or $7$ and any $1\leq p<q\leq n$, the $D$-type maps
$\Phi_{5,p,q}:M_{5}\rightarrow M_{5}$ and $\Phi_{7,p,q}:M_{5}\rightarrow
M_{5}$ are always positive.
Observe that $\Phi_{5,1,3}$ is of the form
$\left(\begin{array}[]{ccccc}a_{11}&a_{12}&a_{13}&a_{14}&a_{15}\\\
a_{21}&a_{22}&a_{23}&a_{24}&a_{25}\\\ a_{31}&a_{32}&a_{33}&a_{34}&a_{35}\\\
a_{41}&a_{42}&a_{43}&a_{44}&a_{45}\\\
a_{51}&a_{52}&a_{53}&a_{54}&a_{55}\end{array}\right)\mapsto$
$\small\left(\begin{array}[]{ccccc}2a_{11}+a_{22}+a_{44}&-a_{12}&-a_{13}&-a_{14}&-a_{15}\\\
-a_{21}&2a_{22}+a_{33}+a_{55}&-a_{23}&-a_{24}&-a_{25}\\\
-a_{31}&-a_{32}&2a_{33}+a_{44}+a_{11}&\\-a_{34}&-a_{35}\\\
-a_{41}&-a_{42}&-a_{43}&2a_{44}+a_{55}+a_{22}&-a_{45}\\\
-a_{51}&-a_{52}&-a_{53}&a_{54}&2a_{55}+a_{11}+a_{33}\end{array}\right),$
which is positive by Corollary 4.3.
It is also obvious that when $n=4$, $\Phi_{4,p,q}$ is positive if $q-p\not=2$;
when $n=6$, $\Phi_{6,p,q}$ is positive if $q-p\not\in\\{2,3,4\\}$.
Example 4.6. For $n=8=2^{3}$, and any $1\leq p<q\leq 8$, the $D$-type map
$\Phi_{8,p,q}:M_{8}\rightarrow M_{8}$ is positive if one of the following is
true:
(i) $q-p\in\\{1,3,5,7\\}$.
(ii) $q-p=2$, $p\in\\{2,4,6\\}$.
(iii) $q-p=4$, $p=4$.
(iv) $q-p=6$, $p=2$.
Example 4.7. For $n=16=2^{4}$, and any $1\leq p<q\leq 16$, the $D$-type map
$\Phi_{16,p,q}:M_{16}\rightarrow M_{16}$ is positive if one of the following
is true:
(i) $q-p\in\\{1,3,5,7,9,11,13,15\\}$.
(ii) $q-p=2$ and $p\in\\{2,4,6,8,10,12,14\\}$.
(iii) $q-p=4$ and $p\in\\{4,8,12\\}$.
(iv) $q-p=6$ and $p\in\\{4,10\\}$.
(v) $q-p=8$ and $p=8$.
(vi) $q-p=10$ and $p=6$.
(vii) $q-p=12$ and $p=4$.
(viii) $q-p=14$ and $p=2$.
Remark 4.8. The condition that $\\{\pi^{p},\pi^{q}\\}$ has property (C) in
Theorem 4.2 is sufficient for the associated $D$-type map $\Phi_{n,p,q}$ to be
positive. The condition is not necessary, and thus, the condition in Theorem
3.3 is not necessary for $\Phi_{D}$ to be positive. For example, let us
consider the case of $n=4$. Then, $\\{\pi^{p},\pi^{q}\\}$ does not have
property (C) if and only if $p=1,q=3$. By Lemma 3.1, the $D$-type map
$\Phi_{4,1,3}=\Phi_{D}$ with $D=2I_{4}+P_{\pi}+P_{\pi^{3}}$ is positive if and
only if
$f(x_{1},x_{2},x_{3},x_{4})=\frac{x_{1}}{2x_{1}+x_{2}+x_{4}}+\frac{x_{2}}{2x_{2}+x_{3}+x_{1}}+\frac{x_{3}}{2x_{3}+x_{4}+x_{2}}+\frac{x_{4}}{2x_{4}+x_{1}+x_{3}}\leq
1$
holds for all non-negative $x_{1},\ldots,x_{4}\in{\mathbb{R}}$ with
$x_{1}+\cdots+x_{4}=1$. By Lemma 2.2, all extremum values of
$f(x_{1},x_{2},x_{3},x_{4})$ is $1$ because $s=M=2$. Consider the supremum of
$f$ on the boundary. Assume that only one of $x_{i}$ is zero, say $x_{4}=0$;
then
$f(x_{1},x_{2},x_{3},0)=\frac{x_{1}}{2x_{1}+x_{2}}+\frac{x_{2}}{2x_{2}+x_{3}+x_{1}}+\frac{x_{3}}{2x_{3}+x_{2}}$
with $x_{1},x_{2},x_{3}$ nonzero. Consider the function
$g(s,t)=\frac{1}{2+s+t}+\frac{1}{2+\frac{1}{s}}+\frac{1}{2+\frac{1}{t}}=\frac{1}{2+s+t}+\frac{s}{2s+1}+\frac{t}{2t+1},$
where $s>0$ and $t>0$. As
$\begin{array}[]{rl}&(2s+1)(2t+1)+s(s+t+2)(2t+1)+t(s+t+2)(2s+1)\\\
=&4s^{2}t+4st^{2}+14st+s^{2}+t^{2}+4s+4t+1,\end{array}$
$(2s+1)(2t+1)(s+t+2)=4s^{2}t+4st^{2}+12st+2s^{2}+2t^{2}+5s+5t+2$
and $2st<s^{2}+t^{2}+s+t+1$, it is easily checked that
$g(s,t)=1-\frac{(s-t)^{2}+s+t+1}{(2s+1)(2t+1)(s+t+2)}<1$. So we still have
$f(x_{1},x_{2},x_{3},0)<1$. If there are more than one $x_{i}=0$, it is clear
that $f(x_{1},x_{2},x_{3},x_{4})<1$. Therefore, $\sup
f(x_{1},x_{2},x_{3},x_{4})=1$ on the region $x_{1}+\cdots+x_{4}=1$ and
$\Phi_{D}$ is positive.
It is then interesting to ask
Question 4.9. What is the necessary and sufficient condition for
$\Phi_{n,p,q}=\Phi_{D}:M_{n}\rightarrow M_{n}$ with
$D=(n-1)I_{n}+P_{\pi^{p}}+P_{\pi^{q}}$ to be positive? Where $\pi$ is the
permutation defined by $\pi(i)=i+1$ mod $n$ and $1\leq p<q<n$.
## References
* [1] P. Albert and A. Uhlmanm, A problem relating to positive linear maps on matrix algebras, Rep. Math. Phys. 18 (1980), 163.
* [2] R. Augusiak, J. Bae, L. Czekaj, M. Lewenstein, On structural physical approximations and entanglement breaking maps, J. Phys. A: Math. Theor. 44 (2011) 185308.
* [3] A. Chefles, R. Jozsa, and A. Winter, On the existence of physical transformations between sets of quantum states, International J. Quantum Information, 2 (2004), 11-21.
* [4] M.-D. Choi, Completely Positive Linear Maps on Complex Matrix, Lin. Alg. Appl. 10 (1975), 285-290.
* [5] D. Chru$\acute{s}$ci$\acute{n}$ski and A. Kossakowski, Spectral conditions for positive maps, Comm. Math. Phys. 290 (2009) 1051.
* [6] Roger A. Horn, Charles R. Johnson, Matrix Analysis, Cambridge Univ. Press, 1985, New York.
* [7] J.-C. Hou, A characterization of positive linear maps and criteria for entangled quantum states, J. Phys. A: Math. Theor. 43 (2010) 385201..
* [8] J.-C. Hou, Acharacterization of positive elementary operators, J. Operator Theory, 39 (1998), 43-58.
* [9] J.-C. Hou, C.-K.Li, Y.-T.Poon, X.F. Qi, and N.-S. Sze, Criteria of $k$-positivity of linear maps, ArXiv: 1211.0386v1[quant-ph].
* [10] Z. Huang, C.-K. Li, E. Poon, N.-K. Sze, Physical transformation between quantum states, arXiv:1203.5547
* [11] K. Kraus, States, Effects, and Operations: Fundamental Notions of Quantun Theory, Lecture Notes in Physics, Vol.190.Spring-Verlag,Berlin, 1983.
* [12] C.-K. Li and Y.-T. Poon, Interpolation by Completely Positive Maps, Linear and Multilinear Algebra 59 (2011), 1159-1170.
* [13] M. A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information ,Cambridge University Press, Cambridge, 2000
* [14] X.-F. Qi and J.-C. Hou, Positive finite rank elementary operators and characterizing entanglement of states, J.Phys. A: Math.Theor. 44 (2011) 215305
* [15] S. Yamagami, Cyclic inequalities, Proc. Amer. Math. Sco., 118 (1993), 521-527.
|
arxiv-papers
| 2013-02-01T13:15:18 |
2024-09-04T02:49:41.184816
|
{
"license": "Public Domain",
"authors": "Haili Zhao, Jinchuan Hou",
"submitter": "Jinchuan Hou",
"url": "https://arxiv.org/abs/1302.0175"
}
|
1302.0240
|
# Henry Norris Russell and the Expanding Universe
David DeVorkin Senior Curator, National Air and Space Museum, Smithsonian
Institution, Washington D.C., USA
###### Abstract
Henry Norris Russell, one of the most influential American astronomers of the
first half of the 20th Century, had a special place in his heart for the
Lowell Observatory. Although privately critical of the founder for his
pronouncements about life on Mars and the superiority of the Mars Hill
observing site, he always supported the Observatory in public and professional
circles. He staunchly supported Tombaugh’s detection of a planet as leading
from Lowell’s prediction, and always promoted V. M. Slipher’s spectroscopic
investigations of planetary and stellar phenomena. But how did he react to
Slipher’s puzzling detection of the extreme radial velocities of spiral
nebulae starting in 1912, and how did he regard the extension and
interpretation of those observations by Hubble and others in following
decades? Here we describe the arc of Russell’s reactions, dating from
Slipher’s first detection, as an indicator of how mainstream stellar
astronomers reacted to the concept of an expanding universe.
## 1 Introduction
Henry Norris Russell was one of the most influential American astronomers of
the first half of the 20th Century (DeVorkin 2000). During his most active
years, dating roughly from the eve of the first World War and lasting till the
eve of the second, his influence was deeply felt not only as a frequent
referee and arbiter of standards for American professional journals relating
to astronomy, but as a spokesman for the profession, reaching popular and
technically informed audiences in a monthly astronomy column for _Scientific
American_. His column provided news of upcoming celestial events surrounding a
simple star map, but it also became a bully pulpit for promoting the deeds and
discoveries of astronomers worldwide, as did his two-volume textbook that was
an introduction to the profession for over a generation of astronomers
(Russell et al. 1938).
Russell’s career spanned what Peter van de Kamp, Richard Berendzen and others
have called, first, the “Galactocentric Revolution” and then the “Acentric
Revolution” (van de Kamp 1965; Berendzen 1975). History will remember the 20th
Century as the period when these revolutions in the human comprehension of the
universe, in conjunction with the establishment of relativistic cosmology by
Einstein and others, were discovered, elaborated and reinforced.111An
excellent summation of present historical knowledge, with an extensive
bibliography, can be found in Smith (2009).
Today, these concepts are in place, but how did those who were born and raised
in a static stellar universe react at the time? Observational astronomers rose
to the challenge, pragmatically conducting observational tests suggested by
Einstein or inferred from his theories.222There is an extensive literature,
from, for instance, Wright (1966) to Crelinsten (2006). Those without the
means to make the delicate observations, or without the inclination, however,
had fewer options to participate: among them, either ignore the implications
and continue on studying the stellar universe, or incorporate the implications
when they might be useful.
Russell was in the last category. The Astrophysics Data System suggests that
Russell invoked the term “expanding universe” only twice in refereed
publications from 1931 - 1940, less frequently than de Sitter, C. A. Chant, W.
H. McCrea and G. C. McVittie (9 times each), Eddington (4 times) and Hubble (3
times).333There were 224 hits total using the Astrophysics Data System’s full-
text search (Beta version), searching on refereed papers with synonyms
disabled. And distinct from the others (save Chant, who was acting in his
capacity as editor of the _Journal of The Royal Astronomical Society of
Canada_), Russell did not try to contribute to the new dynamics other than to
speculate on its implications for the formation of planetary systems.
For these reasons, recounting how Russell reacted first to V. M. Slipher’s
observations of the extreme velocities of the spiral nebulae and then to their
elaboration and correlation with distance by Hubble and others, will offer
insight into how “normal” astronomers engaged the new framework. Of course,
following Russell alone will not provide a full historical picture, but
considering his influence, his example is a good place to start, on the
occasion of this celebration of the centennial of Slipher’s initial detection
of what ultimately was interpreted as the expansion of the Universe.
## 2 First Reactions
In April 1920, the Dearborn Observatory astronomer Philip Fox wrote his friend
V. M. Slipher, congratulating him on receiving the 1919 Lalande Prize of the
Paris Academy for his nebular work. Fox also took the occasion to recall for
Slipher how astronomers reacted when, at the 16th meeting of the American
Astronomical Society (AAS) in Atlanta in December 1913, he read one of
Slipher’s first announcements of the great radial velocities of a few bright
spiral nebulae. “It has been a matter of very great gratification to me to see
the recognition which your splendid work has been receiving,” Fox (1920)
began, but then pointedly recalled:
> I believe you remember that at the time of the Atlanta meeting of the
> Astronomical Society I presented your paper on the high velocity of the
> nebulae, and it was greeted with some expression of incredulity, especially
> on the part of Professor Russell, to which I made a heated rejoinder… I note
> that he makes a confident reference to your work, however, in “Some Problems
> of Sidereal Astronomy,” published by the National Research Council.
In the some four years between Russell’s initial reaction to Slipher’s work
and when he prepared his influential essay “Some Problems of Sidereal
Astronomy,” he came to accept the velocities as measured quantities. As his
essay was intended to be a referendum on the state of astronomy, pointing to
work that had to be done, he clearly identified the velocities of the spirals
as a problem that needed to be addressed in the context of the “Sidereal
Problem.” He, along with J. C. Kapteyn, A. Eddington and most mainstream
astronomers, regarded the defining problem of sidereal astronomy as the
spatial and kinematic arrangement of stars in the visible universe. The
Sidereal Problem was then the search for order, trying to understand Kapteyn’s
two star streams as the motions of particles in a spiraling vortex (Paul
1993).
What of the spirals then? Were they part of the stellar system, or external to
it? Russell was aware of the problem, but to appreciate where they (or any
unexpected observation) fit in his larger scheme of things, we need to
appreciate the state of the profession of astronomy at the time. This was a
time of war, and for some, including Russell, there were questions of what was
critical for the future of the profession after the war.
The most contentious issue for the National Research Council (NRC)444Of the
National Academy of Sciences, was what would be the nature of science after
the Great War. In George Ellery Hale’s view, would international relations be
re-established? Who was in, and who was out?555See Kevles (1995, 1971, 1968).
Hale organized the deliberations around disciplines, and asked Russell to take
the lead for astronomers, concentrating on what the nature of the discipline
should be like. As Russell (1917b) explained his position to Harvard’s E. C.
Pickering in November 1917,
> [T]here are two sides of astronomical research, one of which has to do with
> the collections of facts, and the other with their interpretation.
The former was routine; the latter was not, and this was why, Russell (1917a)
claimed, none of Pickering’s Harvard staff had been asked to write reviews for
the NRC. Pickering’s own earlier reports, he pointed out to his old patron,
had already covered the routine aspects. Now the other side needed airing:
> [I]t is upon studies of this sort that the future advances of any science
> must very largely depend.
Considering that it had been Pickering who made the vast holdings of Harvard
spectra and magnitudes available to Russell in past years, which led to his
landmark study of the relations between the characteristics of the stars,
Russell’s views could be thought of as insensitive.
Figure 1.: Starting in the late 1920s, Roger Lowell Putnam, the Observatory
Trustee (r), was receptive to Russell’s (l) advice and intellectual patronage.
Photograph by Clyde Fisher, circa late 1920s, Neg. #280362, courtesy
Department Library Services, American Museum of Natural History.
Russell indeed gave highest marks to interpretation, but he was always very
deferential to those who collected the most interesting and useful
observations, like Pickering, and indeed like Slipher, for it was upon their
work that his career was based. His patronage of Slipher and Lowell
Observatory needs to be understood in this way (as does his patronage of other
observatories like Mount Wilson). Russell’s essay (Russell 1920c) had to wait
till war’s end, and there he expressed his opinion very forcibly:
> The main object of astronomy, as of all science, is not the collection of
> facts, but the development, on the basis of collected facts, of satisfactory
> theories regarding the nature, mutual relations, and probable history and
> evolution of the objects of study.
When those facts were collected properly, and presented clinically, they had
to be dealt with. Indeed, for Russell, observation guided theory as much as
theory informed plans for additional observations. This was a very new idea
for American astronomers, one that few could assimilate. In fact, “Some
Problems” was an appeal to observational astronomers to appreciate the power
of actually designing observational programs that answer specific questions.
Very much due to Percival Lowell’s problem-oriented style, the Lowell
Observatory staff were more receptive than most.
Russell’s feeling was that one of the most important results of a wider
sampling of spiral radial velocities was to better determine solar motion:
best determined once radial velocities of many nebulae are secured in all
directions. This was a classical application he shared with most others who
commented at the time.666See Smith (1982). So through 1916 to early 1917, at
least, he expressed little interest in the nature of that larger realm, though
he was rather open to the possibility that the nebulae were at great
distances, even beyond the Milky Way. In fact, as early as 1903, in a
_Scientific American_ column, prior to the rise of the Chamberlin–Moulton
encounter theory and still feeling safe in a planetary system rendered common
by the Nebular Hypothesis, Russell (1903) went so far as to speculate that,
viewed from a distance of a million light years, “our own Galaxy would appear
as a spiral or a ring nebula, something like the Great Nebula…But it must be
clearly borne in mind that the evidence available at present is too scanty to
justify us in making any definite statement…” Thus for Russell in 1903, at
least, the scale of the universe was not an issue.
He maintained this view, more or less. In a series of lectures in the
Princeton Chapel in 1916 and 1917, Russell pondered the “Scientific Approach
to Christianity” mainly to argue that there need be no antagonism between
“Genesis and Geology” and that those at play were due to mutual
misunderstandings that he hoped his lectures would put to rest. In one
lecture, “What Nature itself looks like to the man of science,” Russell
speculated that some of the fainter spirals could be upwards of 200,000 light
years distant and that there was “No evidence that we have met with the
limit.” Echoing the emerging findings of his former graduate student, Harlow
Shapley, who was now at Mount Wilson, Russell (1916b) wholly accepted that “We
are some 60,000 ly from the center of the galactic system is now known….” He
saw no inconsistency between the growing evidence of the nebulae being beyond
the Milky Way and of the vastness of the Milky Way itself.777Shapley was
coming to this distance estimate in late 1917. See Shapley (1918, p. 48). Nor
did he express any concern for divergent time scales in Genesis and Geology at
this time, evidently of a generation for which the “gap theory” was no longer
necessary.888On the gap theory, see Moore (1986). But what of the motions of
the spirals?
## 3 Adjusting to Shifting Evidence
Indeed, one can easily trace Russell’s initial growing acceptance of the
extragalactic status of the spirals through his _Scientific American_
articles. Despite his initial skepticism, Russell was sufficiently impressed
by June 1914 to encourage his _Scientific American_ editors to invite Slipher
to state his case for nebular motion and especially rotation, discussing its
importance for “stellar and nebular evolution.” And by April 1915, Russell
(1915) felt that Slipher had collected enough evidence for the phenomenon to
devote most of his column to his work, opening his discussion saying “Among
the most difficult of the yet unsolved problems of astronomy is that of the
distances, dimensions and nature of the nebulae.”
Photography, Russell emphasized, had recently shown that the spectra of the
spiral forms of these objects resembled stellar spectra, wholly unlike the
spectra of the brighter “green” nebulae.999At the time, astronomers still
called diffuse nebulae, like Orion, green nebulae because of the visual
dominance of a greenish tinge caused by the oxygen emission spectrum. White
nebulae, like spirals, displayed continuous spectra. An explanation is given
in: Thompson (2011). See also de Sitter (1932, p. 91). This meant they were
either single stars cloaked in gas and dust, or assemblages of stars at vast
distances. Russell then introduced Slipher’s accumulating observations of
spiral radial velocities, now confirmed by “others” that showed them to be in
the range of many hundreds of miles per second.
What was important for Russell was that the average of the radial velocities
for the spirals was some 25 times the average for stars. If there was such
great motion in the line of sight, one would expect there to be some
observable translational motion as well. But Russell (1915) then noted that
none of these nebulae showed any sensible proper motions, none exceeding 0.1
arc second per year, “though one out of every 4 stars exceeds this limit…”
Taking the planetary nebulae observed by the Lick astronomers, and the spirals
observed by Slipher, as two classes, their measured radial velocities
indicated statistically even higher space velocities, which Russell translated
into AUs/year. The planetaries travelled 15 AUs/year and the “white spirals”
were travelling an astounding 120 AUs per year or more. Given this estimate,
and the vanishingly small proper motions of these diffuse objects, or their
stellar cores, they had to be at least 8 to 10 thousand light years away, but
more likely in the ten to twenty thousands of light years. “The dimensions
which the nebulae must have, to appear as big as they do at such distances,
are astounding.”
Comparing Ejnar Hertzsprung’s recent estimate of the distance to the Small
Magellanic Cloud, some 30,000 light years, Russell (1915) further speculated
that the distances to those unresolved nebulae must be even greater:
> …it may well be that we are on the brink of an expansion of our conception
> of the extent of the universe almost comparable to that which resulted from
> the first measurements of the distances of the nearer
> stars.”101010Hertzsprung (1913, p. 204) converts a parallax of 0.0001
> seconds of arc into a distance of 3000 light years, where it would be closer
> to 32,600 light years. Russell (1915) related it as 30,000 light years.
In September Russell (1916a) hailed V. M. Slipher’s observations of the great
velocities of the spiral nebulae, which, along with the work of the other
“Pacific Coast Observatories…has so greatly widened, at a single bound, the
limits of distance at which our investigations may operate.”
But by late 1917, influenced somewhat by Shapley, who was excitedly reporting
Adriaan van Maanen’s provocative conclusions about the rotation of spirals,
Russell (1917a) started reversing his views. Responding to Shapley in
November, Russell accepted van Maanen’s observations, which immediately threw
doubt on the Island Universe theory: “But if they are not star clouds,”
Russell mused, “what the Dickens are they?” 111111Also quoted by Smith (1982,
pp. 36-7).
Russell never overtly criticized van Maanen’s results. Indeed, since 1916 he
had gained high respect for his astrometric abilities, convinced that van
Maanen’s deft re-analysis and refinement of stellar parallaxes, especially his
analyses of internal errors, was a very strong step forward in that delicate
art. He looked upon van Maanen as a new breed of astrometrist, more adept at
mathematical analysis than his predecessors, like Benjamin Boss or even J. C.
Kapteyn.
As much as Russell evidently respected van Maanen’s abilities, his early
results for rotation in spirals were so extreme that Russell remained
cautious, despite Shapley’s promotional efforts. However, in October 1920,
after van Maanen reported directly to Russell that he had found much the same
rotational properties for another spiral, M 33, Russell expressed both
excitement and great relief:
> I have been waiting with great interest to see what you would get on
> additional spiral nebulae and this confirmation of your early discovery is
> most gratifying.
Russell (1920a) looked forward to more confirmations from Mount Wilson; in
conjunction with Pease’s spectroscopic efforts, he encouraged van Maanen to
obtain distances to these objects, so that they all could proclaim:
> Good-bye to Island Universidus!
Privately to van Maanen, he was even willing to play down the nagging question
of the intrinsic brightnesses of various novae events seen in the spirals by
Curtis at Lick, admitting that they were dim compared to galactic novae. They
might well be intrinsically dim, Russell speculated, which he felt was well
within bounds for discounting this odd class of object.
Thus in the two years between the time Russell first drafted the essay for the
NRC, and when it was finally published in October 1919, and reprinted in
_Popular Astronomy_ in 1920, van Maanen’s rotational proper motions had
already raised serious questions about the existence of a hierarchy beyond the
Milky Way, questions Russell was evidently happy to raise. Russell clearly
wanted van Maanen’s results to be confirmed, and so in his essay he went so
far as to suggest, without mentioning de Sitter, that the distances the light
had to travel from the spirals might also be causing the effects Slipher was
seeing.
Shapley, of course, was wholly convinced by his own work on the distances to
globulars and their association with the Milky Way, but it was not the
deciding factor for Russell. Even without the confirmation that would come in
October 1920 from van Maanen, Russell exhibited his enthusiasm in his National
Academy essay, concluding for the moment that the distances to the spirals,
and their intrinsic sizes, could not be “extragalactic” largely owing to van
Maanen’s remarkable proper motion measurements.
Russell’s rejection of the extragalactic framework came after, as Robert Smith
and other historians have related so well, Slipher’s continuing efforts, now
bolstered by Pease’s spectra with the 60 inch, had won many astronomers over.
There was strong evidence that spirals were not solar systems in formation:
they were not stars. To Hertzsprung and others these speeds meant that the
spirals could not be bound to the Milky Way. International acclaim soon
followed: In his Presidential address at the Annual General Meeting of the
Societé Astronomique de France in September 1917, even in the depths of war,
the Count de La Baume Pluvinel (1917) rejoiced that much work had been done
recently, especially noting Slipher’s spiral radial velocities which “give a
further proof of the independence of the spiral nebulae of our Universe.”
In the early 1920s, therefore, Russell remained an agnostic over the meaning
of Slipher’s velocities. He never doubted the observations themselves any more
than he doubted van Maanen’s measurements; at every turn, Russell indicated
that Slipher’s careful work had to be factored into any discussion, for just
as van Maanen had proven to be a pioneer in refining astrometric technique,
Slipher had certainly done so for spectroscopy. In 1920 Russell became
especially excited by Slipher’s recent work on the Orion stars (Slipher 1919),
writing to Adams that it “looks to me as if the luminescence of the nebula was
due to some influence emanating from the hot stars within it…” This was
another piece of evidence that “the nebular lines are produced by known atoms,
under unknown conditions of excitation” (Russell 1920b). Russell never doubted
Slipher’s observations, and supported them through speculating about what they
revealed about the stars, something Slipher rarely if ever tried to do.
Several months before Hubble’s revelatory announcement of the distance to M31
based upon Cepheids of some 285,000 parsecs, Russell gave a series of popular
lectures in February 1924 at the University of Toronto. During the 14th and
final lecture, on the “other classes of great nebulae” - those of spiral and
globular forms, he very much followed the evidence van Maanen offered for 7
spirals showing they were in rapid rotation, throwing off material, very much
in line, Russell believed, with James Jeans’ theoretical models. Given all the
evidence now at hand, Russell concluded, the spirals themselves could not be
more than some 10,000 light years distant, which was not large compared to the
size of Shapley’s galaxy: “… if [van Maanen’s] measurements are right, these
things cannot be anything at all like as big as the Milky Way.” The Andromeda
nebula, however, was for Russell (1924c, p. 259) still one of the most
“remarkable objects known to astronomical science.” Given its enormous size
and luminosity, and its apparent violent motions throwing off matter, it was a
clue to formative processes in his universe.
Russell’s highly convoluted presentation appears in hindsight to be due to his
attempt to reconcile the wild observations at hand. But in fact he was most
influenced by how, in his mind, Jeans’ evolutionary model provides “a
beautiful picture of the spiral nebulae” as evidence once again of his
cherished nebular hypothesis. “It looks as if we have stages of evolution
shown” Russell believed; Jeans could even account for the lumpiness of spiral
arms. But Russell admitted that a very recent paper by Jeans which “came out
only a week or so ago shows that when we want to be quantitative it is much
more difficult.” Jeans’ latest model suggested, in fact, that spirals become
more tightly wound with time.121212Jeans (1923) discusses the tightening of
the arms. Russell (1924c, p. 265) characteristically reminded his audience
that science was tasked with how spirals “got into their present shape” but
was not equipped to say: “…how they got there, because of that we do not know
anything at all. It is no business of science to explain how matter got into
space; only supposing there was matter in space, what would be the history of
it.” As Russell (1924c) concluded, “Only a portion of Genesis is accessible to
the realm of science.”
Everything changed, of course, by the end of the year and Hubble’s work
announcing the distance to M31. Russell heard the news first from James Jeans
when they happened to meet in New Haven sometime in the fall, and then later
in November. Finally, Hubble reported directly in December and Russell (1924a)
responded heartily: “They are certainly quite convincing…It is a beautiful
piece of work…”; Russell urged Hubble to send his paper to the Washington
meetings of the American Association for the Advancement of Science in
February 1925, if for no other reason than his paper would no doubt “bang that
$1000 prize.” Russell read Hubble’s paper at that meeting131313Smith (1982,
pp. 111-120) and Berendzen et al. (1976, Chapter 5) have explored in some
detail the impact of Hubble’s first paper on Cepheids on the acceptance of van
Maanen’s rotational measurement work. saying nothing of the implications this
new result had on van Maanen’s measurements.
Jeans and Russell did meet in New Haven in August 1924, presumably because
there were two major meetings starting with the American Astronomical Society,
held at Dartmouth and timed specifically to coincide with the special annual
meeting of the British Association for the Advancement of Science in Toronto.
Russell reported on both in his _Scientific American_ column, highlighting the
recent studies of Yale’s E. W. Brown on the osculating orbits of “particles
composing the arms” in spiral nebulae. Both Jeans and Russell later
acknowledged that they spoke of Hubble’s observations and their implications
for Jeans’ theory of spirals in New Haven. And after Jeans returned to England
in late October he wrote to Hubble based upon his conversations with Russell,
sending a copy to Russell, setting out points that Jeans believed supported
Hubble’s new estimate of the distance to M31. Jeans made his points rather
dramatically. Using Hubble’s new distance to M31, and F. Pease’s spectroscopic
observations of its rotational motion, “we found a period of rotation of
17,000,000 years.” Then Jeans (1924) speculated to Hubble: “Suppose that Van
Maanen had measured motions in M.31 and had announced this period. If I had
calculated the parallax by the method I used for M.101 on p. 217 of my
“Cosmogony” (assuming, as I did there, that the condensations are 3′′ of arc
apart) the result I should have obtained would be (17,000,000/85,000) $\times$
5/3 $\times$ 1,000 parsecs = 333,000 parsecs. Which is near enough (rather too
near!) to your estimate.”
At first glance, it seems as if Jeans employed Hubble’s distance post hoc but
in fact, he found it by a very independent method: using Eddington’s just
announced mass/luminosity law for stars. Using the observed brightnesses of
the average condensations in the nebula, their angular distances apart, and
Pease’s radial velocities, he found the distance to M31 to be 337,000 parsecs,
employing Hubble’s estimate of the absolute magnitude for the Cepheids of M =
-4. To which Jeans (1924) again noted: “which again is near (too near) to your
estimate” 141414Also noted in (Smith 1982, p. 120). and bluntly concluded: “In
view of these calculations, I feel no further suspicions of the period-
luminosity law, or of my physical interpretation of the condensations – on the
other hand, I fear van Maanen’s measures have to go.”
Jeans was now able to show that Eddington’s mass-luminosity law, as well as
the period- luminosity relation, combined with Hubble’s distances, supported
his own concept of the nature of the spirals.151515Jeans described his
thinking in greater detail in Jeans (1925). Here Jeans (1925) increased his
earlier estimate of the distances to spirals like M31 (originally 5000 light
years), in light of Hubble’s and Shapley’s work, to some 950,000 light years,
and concluded: “If the same correlation [Eddington’s M/L relation] is assumed
to hold for the stellar condensations in a nebula, we can dispense with v.
Maanen’s measurements altogether and (in theory at least) determine the
distance of the nebulae from the observed stellar magnitudes of its
condensations.” Russell (1924b) agreed, writing Jeans hastily in early
November 1924 that “the agreement of the new data with your theory is
beautiful…” Russell did not elaborate, however, saying there was nothing he
could add off the top of his head.
Figure 2.: Russell and Shapley remained life long colleagues and friends. Here
they are captured aboard a steamer tour at the 1938 International Astronomical
Union General Assembly in Stockholm. Photography by Dorothy Davis Locanthi,
Courtesy E. Segrè Visual Archives, American Institute of Physics.
Throughout 1925, Jeans explored the ramifications of Hubble’s distances,
clearly delighted with the new framework. Russell attended to his solar and
stellar spectroscopic studies, experimenting with the emerging explanatory
framework from quantum physics to find useful links between the laboratory and
the stars. He could not have relished Jeans’ enthusiasm for Eddington’s mass-
luminosity relation, of course, because it meant that, in Russell’s own words
(Russell 1926), it put his theory of stellar evolution “in a state of chaos.”
Later Russell (1928b) admitted openly that “[t]he path of this idea is strewn
with the wreckage of abandoned theories,” and that the field was at an
impasse. In like manner, even though Russell could not have been comfortable
with the implications of Hubble’s vast distances, when he reported on them for
_Scientific American_ in March 1925, he also credited Shapley’s distances to
globulars and hence his estimate of the size of the Milky Way as having
“expanded our existing ideas of the universe ten-fold.” He did this,
evidently, to assert that Shapley’s universe was still viable (see Figure 2).
He now fully accepted the vast distance to M31, admitting that it “leads to
conclusions which are enough to make even a case-hardened astronomer gasp.”
M31 was now, at Hubble’s distance of a million light years, at least 35,000
light years in diameter, which would have made it larger than the Milky Way on
Kapteyn’s old model. But, Russell (1925) hastened to point out, it was still
only 1/10th the size of the Milky Way given Shapley’s work. This lingering
attachment to the idea of the Milky Way as being somehow special provides
insight into the nature of Russell’s personal universe.
## 4 Russell’s Universe
In addition to his scientific writings, as noted above, Russell actively wrote
and lectured on science and religion for a widening audience, starting on
campus after he attained the status of a Princeton professorship in 1908,
promoted to full professor in 1911 and Director of the Halsted Observatory in
June 1912. He espoused liberal theological views, reflecting his upbringing as
the eldest son of a New School Presbyterian pastor, and by the 1920s took on
the role of a modernist, intent on arguing that there was much theology could
learn from science in, what he was coming to accept evermore, was an age of
relativism. Russell’s efforts to explore the interface between science and
religion thus promises further insight into his reaction to the concept of an
expanding universe.
In 1925 he was invited to Yale to deliver the annual set of Terry Lectures,
which he titled “Fate and Freedom.” Keenly aware that Presbyterian theological
orthodoxy was still struggling to kill the liberalism of New School reformers,
Russell asserted his modernism, proclaiming that “The old hope of finding
final and perfect statements of the truth about a simple universe has fled.”
In its place was a striving for closer and closer approximations to the truth,
“a steadily increasing accuracy of approximation in the description and
interpretation of an incredibly and magnificently complex universe.” In its
constant striving, science had freed itself from absolutes, whereas theology
had not (Russell 1927; Longfield 1991).
During these years, as he gained prominence, he maintained a lengthy
correspondence with inquirers seeking illumination about God from the world of
science. In late November 1928, he responded to the inquiries of a reader of
“Fate and Freedom,” the Beacon Hill doyenne Margaret Deland, an American
novelist and poet and particularly articulate inquirer. Deland, inspired by
her reading, hoped her views as a “philosophically uneducated” admirer, in the
spirit of Moliere’s “average mind,” might be of some use to the Princeton
savant. The revelation she wanted to share, without attribution, was “that in
this universe of evident order and purpose, God is still ‘becoming’.” Only in
this way, she felt, could she understand that “our imperfections may be steps
in His (I wish I didn’t have to use the personal pronoun!) process of
‘becoming’”(Deland 1928). Although she makes no mention of Alfred North
Whitehead’s influence here, especially his widely read 1925 “Science and the
Modern World” there are similarities with what he later espoused as “process
philosophy.” If there is a connection and insights to be had, I leave them for
others. Thanks though are due to Matt Stanley for raising this question during
the conference.
Russell (1928a) responded at length, clarifying his position, and his
universe:
> I suppose it is because I am a physicist that I find it hard to think of God
> as “becoming.” Of course, we can none of us hope for a full understanding of
> the Power behind the universe. But those of us who study the universe
> itself, at least from the material side, do not find evidence of progress at
> the heart of things. There is plenty of development and evolution of
> individual parts of the universe, from stars to souls; but these are all
> particular cases of the operation of invariable laws of nature.
Russell took the laws of physics as invariant in a universe that, on the
whole, was unchanging, though the things in it do change. Russell (1928a)
thought of his “God as manifesting Himself rather than evolving.”
Just over a month later, in early January 1929, Russell’s unchanging universe
was again challenged. Soon after he reached Pasadena in what had become a
yearly trek cross country to advise and counsel, he heard the local news about
what Hubble was now up to. In a long letter to Lowell astronomer Carl
Lampland, who was then in residence at Princeton in a scheme Russell created
that hopefully would better acquaint Lampland with mainstream astrophysics,
Russell (1929b) reported that Hubble:
> … has definite evidence, at last, that the radial velocities of non-galactic
> nebulae increase with distance … the explanation “may” involve a “de Sitter
> Universe” or some new mathematical invention - it is too early to say…
Russell’s expression “at last” highlights the fact, as Robert Smith has well
shown (Smith 1982), that the velocity/distance relation was very much in the
air in the 1920s. But he mentioned it only in passing, devoting most of his
long letter reporting on stellar astronomy and new instrumentation at Mount
Wilson.
Within a few months, as Russell gathered his thoughts for his _Scientific
American_ column, the idea of an expanding universe did not sit too well,
although, as usual, the observed facts were not in question. It was the
implication. In a column titled “The Highest Known Velocity” Russell made much
of Milton Humason’s tedious work, taking exposures over several nights, of
between 33 and 45 hours duration, and the resulting “very pronounced distance-
effect.” Regarding it all as very “strange” Russell asked “what does all this
mean?” Could this be a “peculiar form of the theory of relativity suggested by
…. de Sitter” or was it something wholly different. Conversations with Richard
Tolman helped Russell appreciate how a de Sitter universe could look like a
“now familiar” Einstein universe on small scales, but the two differed, as he
reported to his readers, “in a larger view, [where] space and time have
unexpected properties.” On small scale, two particles at rest would remain at
rest save for a mutual Newtonian attraction. But at vast distances, such as
those between the nebulae, they would recede from one another and scatter
“ever more and more widely and recede from one another faster and faster.”
Even though de Sitter’s model literally predicted what Hubble had found,
Russell (1929a) concluded that it was “premature” to adopt de Sitter’s ideas:
“The notion that all nebulae were originally close together is philosophically
rather unsatisfactory…”
Russell’s reaction was distinct only insofar as he did not question the data.
Others certainly did. At about the same time that Russell wrote his column, V.
M. Slipher, along with his friends at Mount Wilson, notably John C. Duncan and
Walter Sydney Adams, questioned the reality of Hubble’s and Humason’s
observations. The fact that two nebulae in the Coma group had velocities
differing by some 3000 km/sec made Duncan (1929) skeptical; he happily
reported to Slipher that Adams felt this discrepancy was throwing “doubt on
the de Sitter theory and making the whole situation more interesting.”
Hubble’s continued confirmation and strengthening of the velocity-distance
relation had to be faced by astronomers, one way or another. I’ll end by
discussing briefly how Russell dealt with it through the rest of his career.
The first edition of Russell, Dugan and Stewart’s 2-volume Astronomy textbook
was completed in 1926 and published in 1926 and 1927. A second edition
appeared in 1938 (Russell et al. 1938). In the first edition he outlined
Slipher’s observations in some detail, and, not questioning the observations
themselves, still asked what it all meant: “Whether this represents a real
scattering of the nebulae away from this region where the sun happens to be is
very doubtful,” Russell asserted in a smaller pitched paragraph (a tactic he
used to highlight speculative sections), adding that some form of “the
generalized theory of relativity” that space is bounded and “returns into
itself” might explain the spectral shifts.161616Russell (1927, p. 850) In the
second edition a decade later, Russell voiced this same concern in his chapter
on “Nebulae,” but in a later chapter on the “Evolution of the Stars” that had
been extensively rewritten, he included a long passage on the problem of
reconciling various time scales: the earth, the stars, and the universe. Here
Russell did mention Hubble’s 1929 work in a supplementary section, but it was
highly terse, clinical and with little elaboration. He made no mention of
Lemaître.
Russell also wrote popular essays on Einstein’s theories of relativity and
toyed with the visible consequences of intense gravitational fields for his
_Scientific American_ audience. Even though Russell never tried to contribute
to general relativity or theories of the expanding universe, he was fully
aware of the potential impact both might have on modern astrophysics, writing
about them with clarity and style, making them accessible to the general
reader and rank-and-file astronomer alike (Kaiser 1998, p. 321).171717As
Kaiser and others discuss, in the 1930s, Princeton harbored more interest in
General Relativity than most American campuses, yet it was not a central topic
in physics until the 1960s.
Russell’s most revealing response to the expanding universe was how he used it
to restore the possibility that we are not alone in the universe. Even though
he initially was repulsed by the idea of the nebulae being closer together in
the distant past, a rather surprisingly rash statement to make in a
_Scientific American_ column, once he had the benefit of years and time for
rationalization, he realized that in fact he could see something useful in the
framework.
Deprived of the Nebular Hypothesis by the popularity of the Chamberlin–Moulton
encounter theory, Russell always looked for ways to reinstate the theory (see
Brush (1978b, a)). On its influence on the ubiquity of planetary systems, see
Dick (2001). The expanding universe gave him an opportunity. The solar system
was formed, he argued in his 1935 University of Virginia lectures “The Solar
System and its Origin,” at a time when the universe would have been much
smaller than it is now, at a time when collisions were more probable. He
thought more (Russell 1935), in fact, of the idea that the solar system was
formed at a time when, some two billion years ago:
> …a very great event occurred. This would have been the time par excellence
> for encounters of all sorts. Or it may be that a cosmic New Deal occurred,
> and that, just afterwards, matter was distributed more widely, but more
> thinly, through space, to settle down into the stars.181818pp. 137-8
This was the approximate epoch when the presently observed expansion of the
universe had its start. It also agreed, he then thought, with the radioactive
age dating of rocks, the tidal evolution of the earth-moon system, and was
well within the possible lifetimes of the stars. In this distant epoch, the
probability of encounters may have been far greater than today. Not only were
stars like the sun larger than at present in his lingering view of the course
of stellar evolution, but the universe was far more compact. Russell
speculated that at this distant epoch, encounters and collisions could have
been a common occurrence. Following a suggestion he attributed to Willem de
Sitter, he was willing to speculate (Russell 1935, p. 25) that the stars
themselves pre-existed this moment of crisis in the universe, many coming
through it unscathed. Collisions would have been more frequent in the past,
relieving all of the encounter theories of the burden of making mankind feel
alone in the universe. Envisioning a time when “all the matter of the Universe
was tighly packed together, perhaps in the one great Atom, which forms the
starting point of Lemaître’s hypothesis,–which deserves no less respect
because of its picturesqueness,” Russell (1935, p. 138) thus ended his
lectures on a hopeful note for the ubiquity of life in the universe. In
commentary in _Scientific American_ , popular speeches, and radio talks,
Russell harbored a deep desire to believe that human beings were not an
accident in the universe. For a talk at Bowdoin College in Maine in April
1941, Russell relied on the humor of Gilbert Chesterton, who once defended his
belief that he had been born, even though he had no recollection of it and
could not test it.
Russell always remained optimistic that somehow, in time, with effort,
discrepancies would be reduced, and consensus would come, and though consensus
would come a deeper appreciation of the universe we live in. He delighted in
describing the process of science as building on what was, for him, “a tissue
of approximations,” as he described it in his 1937 AAS Presidential
address.191919Quoted in DeVorkin (2000, p. 246). There was great heuristic
value in the process, as long as physical insight was not lost, and the clues
that observational data provided were faithfully followed up and tested for
veracity, but never swept under the rug. Thus, throughout this period when
others looked askance at the disparity in the time scale for the age of the
universe computed from Hubble’s recessional velocities, compared to the ages
of all the things in it as derived by radioactive determinations of the age of
the earth, the ages of stars allowed by nuclear energy sources recently
uncovered by Bethe and Marshak, the time scales from dynamical studies of
binary stars and star clusters, Russell searched as usual for a silver lining.
In his May 1940 James Arthur Foundation lecture at New York University,
Russell could at least hope that the inference of Hubble’s evidence, best
described, he thought, “a little speculatively” by the Abbe Lemaître, could
make it possible to have a universe filled with planetary systems:
> The principal attraction of this scheme is that it pictures a short but
> tumultuous time in the early days of our present universe during which all
> sorts of things which can never happen now might have occurred, such as the
> origin of the planetary system and … the formations of the heavy radioactive
> atoms.202020Russell (1949, p. 16). Russell did wonder privately if the heavy
> elements were older than the stars, in letters to van Maanen and Shapley in
> 1939. See DeVorkin (2000, p. 254, ref. 83). For a contemporary review of
> time scales, see Bok (1946). Good secondary sources that address one or more
> aspects of the time scale problem include Brush (1978b); Dick (2001);
> Hufbauer (1991); Burchfield (1975).
Russell acknowledged that if this were so, then the ages of everything had to
be limited to less than 2 billion years time. Even so, in concluding his
lecture, Russell rejoiced that science had at least succeeded in showing that
the time scale of the universe can be measured in billions of years, not
millions, “and in very few of them” as concluded from “four independent lines
of evidence.” 212121Russell (1949, p. 29). Russell identified the “outstanding
difficulties” remaining as the persistence of giant stars and the premature
appearance of white dwarfs. This was at a time before Baade’s populations were
articulated, and when giants were still considered, by Russell at least, to be
young stars, in spite of the initial evidence provided by Strömgren, Kuiper
and Gamow. See DeVorkin (2006).
But this was as far as he was willing to go. He ended his essay acknowledging
that there was a “widespread desire” that this expanding universe might
someday reverse itself and contract, leading to “some cyclical restoration of
activity.” But he was not in sympathy with this view, expressing surprise that
it had such a “strong aesthetic hold” on so many people. Here Russell enjoyed
a bit of Eddingtonian wordplay to make his point: “I am an evolutionist, not a
multiplicationist. It seems so tiresome to me doing the same things over and
over again.” 222222Russell (1949, p. 30)
Indeed, this is the same Russell who in 1914 chided a critic’s rejection of
his penchant for making inferences based upon statistical behavior and not
direct measurement: “A hundred years hence all this work of mine will be
utterly superseded: but I am getting the fun of it now” (Russell
1914).232323Doolittle had rejected Russell’s technique of hypothetical
parallaxes. So even while he could marvel at Slipher’s high radial velocities,
van Maanen’s proper motions or Hubble’s vast distances, following the
observations to see where they lead, he knew only too well, and was
comfortable with the fact, that the implications one might draw from them were
fleeting.
Nevertheless, in his last known commentary on the implications of the
expanding universe, in a foreword to the English edition of Lemaître’s “The
primeval atom, an essay on cosmogony,” Russell kept his distance from the
question. Describing it as an exercise in “pure intellect;” as an hypothesis
and not a theory, he felt it “goes far beyond the limits of present proof.”
Indeed, not apparently aware of, nor concerned with, the recent prediction of
a redshifted remnant by Ralph Alpher and Robert Herman in 1948, nor making any
effort in his foreword to explicitly link the theory to the observed redshifts
themselves and that the universe must be expanding, Russell slyly regarded
Lemaître’s retrospective essay as “…a fascinating view of the birth and growth
of a noteworthy hypothesis regarding the origin of the material universe.” The
only reference Russell made explicitly to observational evidence dealt with
the recent detection of high-energy cosmic rays which emulated Lemaître’s
“atom.” So as he did throughout much of his career, Russell held highest
evidence from physical observation, whether or not it tested his
preconceptions. Just why he was asked to provide the foreword, and not someone
closer to the subject matter, indicates his prominence in the field of
professional astronomy in that day.242424Russell (1950, pp. v-vii)
### Acknowledgments
This research has made use of NASA’s Astrophysics Data System Bibliographic
Services, and the kindnesses of the U S Naval Observatory Library. The author
has also benefited from comments by Robert Smith, Michael Neufeld and Joseph
Tenn, as well as by the editors.
## References
* Berendzen (1975) Berendzen, R. 1975, Geocentric to Heliocentric to Galactocentric to Acentric: The Continuing Assault to the Egocentric, Vistas in Astronomy, 17, 65
* Berendzen et al. (1976) Berendzen, R., Hart, R., & Seeley, D. 1976, Man Discovers the Galaxies (Science History Publications Inc)
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|
arxiv-papers
| 2013-02-01T18:20:49 |
2024-09-04T02:49:41.195554
|
{
"license": "Public Domain",
"authors": "David H. DeVorkin",
"submitter": "David DeVorkin",
"url": "https://arxiv.org/abs/1302.0240"
}
|
1302.0251
|
# First detection of $c$-C3H2 in a circumstellar disk
Chunhua Qi Harvard-Smithsonian Center for Astrophysics, 60 Garden Street,
Cambridge, MA 02138, USA Karin I. Öberg Departments of Chemistry and
Astronomy, University of Virginia, Charlottesville, VA 22904, USA David J.
Wilner, Katherine A. Rosenfeld Harvard-Smithsonian Center for Astrophysics,
60 Garden Street, Cambridge, MA 02138, USA
###### Abstract
We report the first detection of $c$-C3H2 in a circumstellar disk. The
$c$-C3H2 J=$6-5$ line (217.882 GHz) is detected and imaged through Atacama
Large Millimeter Array (ALMA) Science Verification observations toward the
disk around the Herbig Ae star HD 163296 at 0.8′′ resolution. The emission is
consistent with that arising from a Keplerian rotating disk. Two additional
$c$-C3H2 transitions are also tentatively detected, bolstering the
identification of this species, but with insufficient signal-to-noise ratio to
constrain the spatial distribution. Using a previously developed model for the
physical structure of this disk, we fit a radial power-law distribution model
to the $c$-C3H2 $6-5$ emission and find that $c$-C3H2 is present in a ring
structure from an inner radius of about 30 AU to an outer radius of about 165
AU. The column density is estimated to be 1012–1013 cm-2. The clear detection
and intriguing ring structure suggest that $c$-C3H2 has the potential to
become a useful probe of radiation penetration in disks.
protoplanetary disks; astrochemistry; stars: formation; ISM: molecules;
techniques: high angular resolution; radio lines: ISM
## 1 Introduction
Pre-main sequence stars are commonly surrounded by disks, which serve the dual
purpose of funneling accretion onto the central star and transporting away
angular momentum. These disks are also the formation sites of planets and
connect protostars and planets physically and chemically. The main component
of gas-rich circumstellar disks, cold H2, cannot be observed directly because
of a lack of transitions at suitable energy levels. Instead, constraints on
disk mass, density and temperature structures depend on observations of trace
species, most commonly dust and CO gas (e.g. Qi et al., 2006; Piétu et al.,
2007; Andrews et al., 2009). To access other disk properties, such as
ionization levels, X-ray and UV radiation fields, turbulent mixing, chemical
age and bulk composition requires the development of additional molecular
probes whose formation and destruction depend critically on one or several of
these properties.
The past decade has witnessed significant progress on disk chemistry theory,
resulting in large numbers of predictions on how molecular abundance
structures should depend on different disk properties (e.g. Aikawa & Nomura,
2006; Willacy, 2007; Semenov & Wiebe, 2011; Fogel et al., 2011; Walsh et al.,
2012). A lack of observational sensitivity has however limited detections to
the simplest molecules in the millimeter/submillimeter range which
characterizes the bulk of the gas disk, including CO, HCO+, DCO+, CN, HCN,
DCN, C2H, H2CO, CS, SO, CH2, N2H+ and HC3N (Dutrey et al., 1997; Aikawa et
al., 2003; Thi et al., 2004; Semenov et al., 2005; Dutrey et al., 2007; Qi et
al., 2008; Fuente et al., 2010; Henning et al., 2010; Öberg et al., 2010,
2011). This list is expected to expand with the arrival of ALMA and its
unprecedented sensitivity and spatial resolution, which will enable the
development of a more diverse set of molecular probes. In this Letter, we use
ALMA Science Verification observations of HD 163296 to report the first
detection of cyclopropenylidene, $c$-C3H2, in a circumstellar disk. At a
distance of $\sim$122 pc (Perryman et al., 1997), this Herbig Ae star (stellar
mass 2.3 M⊙; spectral type A1; age $\sim$4 Myr) harbors a large disk (a
scattered light pattern extending out to a radius of $\sim$500 AU, Grady et
al., 2000) with strong millimeter continuum and molecular line emission
(Mannings & Sargent, 1997; Thi et al., 2004; Natta et al., 2004; Isella et
al., 2007; Qi et al., 2011). The large disk size and strong molecular emission
make this system an ideal target to search for new molecules.
The $c$-C3H2 molecule is a small cyclic hydrocarbon, which was first detected
in space by Thaddeus et al. (1985) towards Sgr B2. Since then, $c$-C3H2 has
been detected in a wide range of astrophysical environments, including diffuse
and dense clouds, protostars, planetary nebulae and extragalactic sources
(e.g., Matthews & Irvine, 1985; Cox et al., 1988, 1989; Madden et al., 1989;
Benson et al., 1998; Menten et al., 1999; Fossé et al., 2001; Sakai et al.,
2008; Tenenbaum et al., 2009; Liszt et al., 2012), and is one of the most
abundant molecules with 3 carbon atoms in the interstellar medium (Teyssier et
al., 2004). In the context of disks, $c$-C3H2 has some favorable
characteristics that may make it a powerful molecular probe, especially when
observed in combination with isotopologues and chemically related molecules:
1. 1.
$c$-C3H2 has two equivalent H nuclei, which couple to generate ortho (nuclear
spin of 1) and para (nuclear spin of 0) states. Deviations from the
statistical weights of 3 to 1 may be used to constrain the formation
conditions of the molecule (Park et al., 2006).
2. 2.
$c$-C3H2 is also one of two isomers; the other is the linear molecule
$l$-C3H2. The isomer ratio can be used to assess the fractional ionization in
clouds (Fossé et al., 2001) and may provide similar constraints on disks.
3. 3.
The formation and destruction chemistry of $c$-C3H2 (and carbon chains) are
sensitive to high energy radiation. The vertical and radial abundance
structure should provide strong constraints on the penetration depth of X-rays
and UV photons (Semenov & Wiebe, 2011).
4. 4.
Within the family of hydrocarbons, the sensitivity to turbulent mixing
increases dramatically from C2 to C3H2, and from C3H2 to larger hydrocarbons.
Observed ratios and limits have the potential to constrain transport in disks
(Semenov & Wiebe, 2011).
Finally, the chemistry that produces $c$-C3H2 is important to constrain in its
own right, since $c$-C3H2 shares precursors with other, more complex carbon
chains, which may be an important source of carbon during planet formation. In
light of these possibilities, we aim to provide first constraints on the
abundance, distribution and $o/p$ ratio of $c$-C3H2 in the HD 163296 disk.
## 2 Observations
HD 163296 (RA: 17h56m21$\fs$287, DEC: $-$21°57′22$\farcs$39; J2000.0) was
observed on 2012 June 9, June 23 and July 7 as part of the ALMA Science
Verification (SV) program in band 6. At the time of observations, 25 12-m
antennae were used (2 7-m ACA antennae were present in the dataset but
flagged) in the array with a resulting synthesized beam size of
0.9$\times$0.7$\arcsec$ (PA=84°). The source was observed with a total
integration time of 84 minutes. The ALMA correlator was configured to
simultaneously observe four spectral windows (two in each sideband). One of
those windows (SpwID #1) covered both 13CO and C18O $2-1$ lines (220.399 and
219.560 GHz, respectively). The data were calibrated in the CASA software
package (v3.4), following the detailed calibration and imaging scripts
provided by the ALMA science verification team. Since those scripts, along
with fully calibrated measurement sets, are publicly available
online111https://almascience.nrao.edu/almadata/sciver/HD163296Band6, we do not
repeat the details here. The 218 GHz (1.37mm) continuum was generated from the
line-free channels in the two spectral windows (SpWID #0 and #1) of the lower
sideband, and the flux is determined to be 608.5$\pm$2.5 mJy, which agrees
with the SMA observations (Qi et al., 2011). Excellent agreement is also seen
between the SMA and ALMA spectra for the 13CO and C18O $2-1$ lines with fluxes
within 10% (Qi et al., 2011).
All of the $c$-C3H2 lines are continuum-subtracted and retrieved from SpwID #0
with a channel width of 0.488 MHz (0.67 km s-1) covering the total 1.875 GHz
bandwidth from 216.167 to 218.042 GHz. We have binned the lines in channels of
1 km s-1 to achieve higher signal-to-noise ratios and still keep the key
kinematic features of the disk. The resulting line rms noise level is
estimated to be 2.2 mJy beam-1 per 1 km s-1 channel in line free channels.
## 3 Results and Analysis
### 3.1 $c$-C3H2 detection
Figure 1 shows that the $c$-C3H2 $6-5$ line is clearly detected towards HD
163296 and that the position angle of its resolved velocity field is
consistent with that of 13CO from the same data set, which shows a clear
pattern of Keplerian rotation. Interestingly, the $c$-C3H2 $6-5$ moment map
displays a double-peaked morphology, indicative of an inner hole. The $c$-C3H2
$3-2$ and $5-4$ lines are only tentatively detected, insufficient to resolve
the velocity field or any spatial structure. To improve the sensitivity in
imaging these lines, we applied a Gaussian taper to the visibility data that
corresponds to 1″ FWHM in the image domain, which resulted in a beam of
1.3$\times$1.2$\arcsec$ (PA=78°).
Because the line emission changes position with channels, using a single mask
for all channels to extract spectra will lead to a degradation of the signal.
Instead, we create separate masks to cover the $c$-C3H2 $6-5$ line emission at
each channel and use these masks to extract the line spectra. The same masks
are also used for other $c$-C3H2 lines. Figure 2 shows the resulting spectra
of all three $c$-C3H2 $3-2$ lines together with 13CO $2-1$ line (with its own
masks covering the 13CO $2-1$ emission at each channel). The $6-5$ line is
clearly detected with the expected line shape and Vlsr consistent with 13CO.
The $5-4$ line presents a consistent spectral profile, at lower signal-to-
noise ratio. In contrast, the $3-2$ line is both narrower and clearly blue-
shifted, suggestive of a different physical origin, perhaps in a disk wind.
However, both the $3-2$ and $5-4$ lines are observed at the edge of the
spectral window, so this apparent asymmetry should be confirmed by independent
observations before any strong conclusions are drawn.
The integrated fluxes are reported in Table 1. These are just below the upper
limits reported in two previous failed searches of $c$-C3H2 in disks (Fuente
et al., 2010; Öberg et al., 2010).
### 3.2 Ring structure model
We explore the distributions of $c$-C3H2 in HD 163296 based on a previously
developed accretion disk model with well-defined temperature and density
structures (Qi et al., 2011). To simulate and compare with the data, we assume
the disk material orbits the central star in Keplerian motion, and fix the
disk geometric and kinematic parameters that affect the observed spatio-
kinematic behavior of the disk. The details of the stellar and accretion
properties and the disk geometric and kinematic parameters are summarized in
Table 3 of Qi et al. (2011). We have developed a physically self-consistent
accretion disk model with an exponentially tapered edge that matches the
spectral energy distribution and spatially resolved millimeter dust continuum
emission. The disk temperature and density structures are further constrained
by multiple CO and CO isotopologue line observations. Such analysis provides
the essential framework for constraining the distribution of molecular
abundances in protoplanetary disks.
Within this model framework, we assume that $c$-C3H2 is present with a
constant abundance in a layer with boundaries toward the midplane and toward
the surface of the disk (Qi et al., 2008; Öberg et al., 2012). This assumption
is motivated by chemical models (e.g. Aikawa & Nomura, 2006) that predict a
three-layered structure where most molecules are photodissociated in the
surface layer, frozen out in the midplane, and have an abundance that peaks at
intermediate disk heights. The surface ($\sigma_{s}$) and midplane
($\sigma_{m}$) boundaries are presented in terms of
$\Sigma_{21}=\Sigma_{H}/(1.59\times 10^{21}cm^{-2})$, where $\Sigma_{H}$ is
the hydrogen column density measured from the disk surface in the adopted
physical model. This simple model approach approximates the vertical location
where $c$-C3H2 is most abundant. Due to the limited signal-to-noise of the
multiple $c$-C3H2 line detections, we cannot constrain the location of this
vertical layer based on the $c$-C3H2 data. We therefore fix the vertical
surface boundary $\sigma_{s}$ as 0.32 ($\log_{10}(\sigma_{s})=-0.5$) and the
midplane boundary $\sigma_{m}$ as 3.2 ($\log_{10}(\sigma_{m})=0.5$), between
which lies the expected location of most molecules with peak abundances in the
warm molecular layer (Aikawa & Nomura, 2006).
We model the radial column density distribution of $c$-C3H2 as a power law
N${}_{100}\times(r/100)^{p}$, where N100 is the column density at 100 AU in
cm-2, $r$ is the distance from the star in AU, and $p$ is the power-law index.
Since the emission shows a clear indication of a central hole, we fit for both
an inner radius Rin and outer radius Rout together with the power-law
parameters (N100 and $p$). Using the structure model and the fixed vertical
distributions, we compute a grid of synthetic $c$-C3H2 J=$6-5$ visibility
datasets over a range of Rout, Rin, $p$ and N100 values and compare with the
observations. The best-fit model is obtained by minimizing $\chi^{2}$, the
weighted difference between the real and imaginary part of the complex
visibility measured in the ($u,v$)-plane sampled by the ALMA observations. We
use the two-dimensional Monte Carlo model RATRAN (Hogerheijde & van der Tak,
2000) to calculate the radiative transfer and molecular excitation. The
collisional rates are taken from Chandra & Kegel (2000) and the molecular data
file is retrieved from the Leiden Atomic and Molecular Database (Schöier et
al., 2005).
Figure 3 shows the $\chi^{2}$ surfaces for the Rin and $R_{out}$ versus the
power law index $p$. We find the $p$ is constrained between -2.5 and -1.5, Rin
15–40 AU and Rout 150–200 AU (within 1$\sigma$) with best-fit values at -2, 30
and 165 AU, respectively (reduced $\chi^{2}=1.07$). By fixing the above
parameters at their best-fit values, N($c$-C3H2) is determined to be
2.2$\pm$0.2 $\times$1012 cm-2 at 100 AU. Figure 4 presents comparisons between
the observed channel maps and the best-fit model. The residual image doesn’t
show any significant emission, which indicates that the $c$-C3H2 emission is
consistent with that arising from a Keplerian rotating disk. The $6-5$ line is
actually blended with the $6_{1,6}-5_{0,5}$ (ortho) and $6_{0,6}-5_{1,5}$
(para) lines at the same frequency. Since the Einstein $A$-coefficients for
both ortho and para lines are the same, 5.393$\times$10-4 s-1 (Schöier et al.,
2005), we fit the data with the ortho line only and the resulting column
density can be accounted as the sum of both ortho and para forms of the
molecule.
For the weaker $5-4$ ortho line, the signal-to-noise ratio is too low to
provide any constraints on the $c$-C3H2 the distribution. By assuming the
$c$-C3H2 distribution derived from the $6-5$ line, however, we can use the
unresolved line flux to constrain the $o/p$ ratio. The resulting best-fit
N(C3H2-($o$))=1.6$\pm$0.3 $\times$1012 and N(C3H2-($o+p$)=2.2$\pm$0.2
$\times$1012 cm-2 at 100 AU (as shown above). Hence $o/p$=2.7$\pm$1.7, which
is close to the statistical value of 3, but the large uncertainty implies that
better data are required to use this ratio as a probe of formation conditions.
## 4 Discussion
We have clearly detected $c$-C3H2 in the disk around HD 163296. This is one of
the two largest molecules detected in disks so far, the other being HC3N, also
a carbon-chain molecule, detected recently using deep single-dish observations
(Chapillon et al., 2012).
Using the spatially and spectrally resolved line emission, we have constrained
the radial distribution of $c$-C3H2 in the HD 163296 disk as a ring structure
ranging from $\sim$30 to $\sim$165 AU. The outer edge of $<$250 AU (2$\sigma$
limit) is much smaller than the gas disk size which extends to 500 AU (Qi et
al., 2011), which suggests that $c$-C3H2 formation is slow or that $c$-C3H2
destruction is fast in the outer disk. In terms of formation, the best-fit
outer radius coincides with the previously determined location of the CO
midplane snow line, and it is possible that CO freeze-out limits the carbon
available in the gas phase to form hydrocarbons in general and $c$-C3H2 in
particular. A second characteristic of the outer disk is that it is rather
tenuous because of a rapidly decreasing column density outside of 200 AU (Qi
et al., 2011). This may result in efficient penetration of radiation and thus
destruction of the easily dissociated $c$-C3H2. Explicit model predictions are
required to resolve which effect drives the disappearance of $c$-C3H2 in the
outer disk.
No existing disk chemistry model in the literature contains predictions for
the radial distribution of $c$-C3H2. Based on strong correlations found
between $c$-C3H2 and CCH in several astrophysical environments (Lucas & Liszt,
2000; Pety et al., 2005; Gerin et al., 2011), predictions on CCH in disks can
be used to shed further light on the origins of $c$-C3H2. Aikawa & Herbst
(2001) predicts a CCH central cavity in the absence of X-ray radiation; when
X-rays are present, and HD 163296 is a known X-ray emitter (Swartz et al.,
2005; Günther & Schmitt, 2009), the CCH column is instead centrally peaked.
Despite the presence of X-rays, the central cavity could be due to a UV
dominated radiation field. This may be tested by observing $c$-C3H2 towards T
Tauri stars with weak UV fields and strong X-rays, and towards additional
Herbig Ae stars with weak X-rays.
The inner cavity could also be a product of the main formation pathway of
$c$-C3H2. Carbon chains can form through at least three different pathways:
(1) ion-neutral reactions at low temperatures, (2) photo-erosion of larger
carbonaceous compounds, and (3) neutral-neutral reactions at luke-warm
temperatures following CH4 ice evaporation at $\sim$30 K (Herbst et al., 1984;
Teyssier et al., 2004; Sakai et al., 2008; Gerin et al., 2011). Detailed
modeling of $c$-C3H2 radial distributions in each of these formation scenarios
are clearly needed to use $c$-C3H2 rings as tracers of the radiation field and
other disk characteristics.
In addition to more detailed modeling, additional observations are needed to
constrain whether different hydrocarbons, especially CCH and $c$-C3H2, are
correlated in disk environments, and to constrain the vertical distribution of
these molecules. The latter requires multiple lines with a range of excitation
energies. With the combined advance in theory and observations, $c$-C3H2 may
become one of the more useful probes of penetration of the radiation fields in
disks.
Facility: ALMA
This paper makes use of the following ALMA data: ADS/JAO.ALMA#2011.0.00010.SV.
ALMA is a partnership of ESO (representing its member states), NSF (USA) and
NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan), in
cooperation with the Republic of Chile. The Joint ALMA Observatory is operated
by ESO, AUI/NRAO and NAOJ.
We would like to thank an anonymous referee for thoughtful suggestions for the
paper. We also acknowledge NASA Origins of Solar Systems grant No. NNX11AK63.
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Table 1: $c$-C3H2 line results. Transition | Frequency (GHz) | Eu (K) | Beam | $\int Fdv$ (mJy km s-1)
---|---|---|---|---
$3_{3,0}-2_{2,1}$ | 216.279 | 19 | $1\farcs 3\times 1\farcs 2$(76°) | 53[9]
$6_{1,6}-5_{0,5}/6_{0,6}-5_{1,5}$ | 217.822 | 39 | $0\farcs 9\times 0\farcs 7$(83°) | 185[10]
$5_{1,4}-4_{2,3}$ | 217.940 | 35 | $1\farcs 3\times 1\farcs 2$(78°) | 74[9]
Figure 1: The integrated intensity maps summed between 0 and 11 km s-1) and
intensity-weighted mean velocity fields of 13CO $2-1$ and $c$-C3H2 $6-5$ lines
(left panel), $c$-C3H2 $5-4$ and $3-2$ lines (right panel) toward HD 163296.
The resolved velocity field of the $c$-C3H2 $6-5$ line agrees with the CO
kinematics. In the $c$-C3H2 maps, the first contour marks 3$\sigma$ followed
by 1$\sigma$ contour increases. The rms varies between 6 and 9 mJy km s-1 per
beam. Synthesized beams are presented in the lower left corners. The star
symbol indicates the continuum (stellar) position. The axes are offsets from
the pointing center in arcseconds. Figure 2: Extracted spectra of the three
$c$-C3H2 transitions toward HD 163296, plotted together with 13CO $2-1$. The
dashed line marks VLSR toward HD 163296. Figure 3: Iso-$\chi^{2}$ surfaces of
$R_{\rm out}$ and $R_{\rm in}$ versus $p$. Contours correspond to the 1–5
$\sigma$ errors. Figure 4: The top panel shows the velocity channel map of
the $c$-C3H2 $6-5$ emission toward HD 163296 (velocities binned in 1 km s-1).
Contours are 0.0022 Jy Beam-1 (1$\sigma$) $\times[3,4,5,6,7,8,9]$. The middle
and bottom panels show the best-fit model and the difference between the best-
fit model and data on the same contour scale. The star symbol indicates the
continuum (stellar) position. The axes are offsets from the pointing center in
arcseconds.
|
arxiv-papers
| 2013-02-01T19:10:35 |
2024-09-04T02:49:41.206235
|
{
"license": "Public Domain",
"authors": "Chunhua Qi, Karin Oberg, David Wilner and Katherine Rosenfeld",
"submitter": "Chunhua Qi",
"url": "https://arxiv.org/abs/1302.0251"
}
|
1302.0325
|
# A Dual-polarized Broadband Planar Antenna and Channelizing Filter Bank for
Millimeter Wavelengths
Roger O’Brient1, Peter Ade2, Kam Arnold1, Jennifer Edwards3, Greg Engargiola4,
William L. Holzapfel1, Adrian T Lee1, Michael J. Myers1,Erin Quealy1,Gabriel
Rebeiz3, Paul Richards1, Aritoki Suzuki1 1Dept. of Physics, University of
California, Berkeley, CA, USA.
2School of Physics and Astronomy, Cardiff University, Cardiff, Wales, UK.
3Dept. of Electrical Engineering, UCSD, La Jolla, CA USA
4 Radio Astronomy Laboratory, University of California, Berkeley, CA USA.
###### Abstract
We describe the design, fabrication, and testing of a broadband log-periodic
antenna coupled to multiple cryogenic bolometers. This detector architecture,
optimized here for astrophysical observations, simultaneously receives two
linear polarizations with two octaves of bandwidth at millimeter wavelengths.
The broad bandwidth signal received by the antenna is divided into sub-bands
with integrated in-line frequency-selective filters. We demonstrate two such
filter banks: a diplexer with two sub-bands and a log-periodic channelizer
with seven contiguous sub-bands. These detectors have receiver efficiencies of
20-40% and percent level polarization isolation. Superconducting transition-
edge sensor bolometers detect the power in each sub-band and polarization. We
demonstrate circularly symmetric beam patterns, high polarization isolation,
accurately positioned bands, and high optical efficiency. The pixel design is
applicable to astronomical observations of intensity and polarization at
millimeter through sub-millimeter wavelengths. As compared with an imaging
array of pixels measuring only one band, simultaneous measurements of multiple
bands in each pixel has the potential to result in a higher signal-to-noise
measurement while also providing spectral information. This development
facilitates compact systems with high mapping speeds for observations that
require information in multiple frequency bands.
The steady improvement in sensitivity of millimeter and sub-millimeter
wavelength bolometer arrays has enabled significant advances in the
measurement of primary Cosmic Microwave Background (CMB) anisotropies,
secondary CMB anisotropies such at the Sunyaev-Zel’dovich (SZ) effect, and
dust emission from star-forming galaxies. Bolometers cooled to sub-Kelvin
temperatures can achieve sensitivity limited by fluctuations in the arrival
rate of photons Richards . At this point, increasing sensitivity requires
receiving more electromagnetic modes. Over the past decade, this has been
achieved using telescopes with large optical throughput (etendue) coupled to
kilopixel bolometer arrays where each pixel received an independent sky signal
or spatial mode. Each detector in these arrays typically has a frequency
bandwidth defined by a single band-defining filter which limits the number of
spectral modes received by a telescope of fixed etendue.
However, spectral information is required by many of these imaging
applications. CMB polarization measurements require spectral information to
identify and remove polarized galactic foregrounds based on spectral
signatures foreground_removal . The thermal SZ (tSZ) effect has a null at 220
GHz and measurements in this band are particularly useful for separating the
tSZ signal from primary CMB anisotropy and sources of foreground emission.
Surveys of star forming regions in high-redshift galaxies Blain require
spectral information to constrain redshift. In principle, CII emissions
redshifted into the sub-millimeter range can trace matter during reionization
Gong_reionization . With spectral information, maps of this emission line at
different redshifts could provide a history of this epoch.
In many current bolometric receivers, each pixel is sensitive to only one
frequency Kuo Niemak myers . In these systems, multifrequency response can
be achieved by dedicating parts of a single focal plane to different
frequencies, repeating observations with different band pass filters, or even
by using multiple telescopes for simultaneous observations. Because optical
design constraints limit the etendue of a given telescope, in these designs
there is a necessary tradeoff between the number of spectral bands and the
mapping speed of each band. In principle, dichroic beam splitters and multiple
focal planes can provide multichannel response, but such complex optical
arrangements are challenging to implement while maintaining low loss and low
polarization systematics. They also tend to be large, making them poor options
for upgrading existing telescopes or for proposed satellites.
In this letter, we present a fully integrated prototype pixel lithographed in
thin films. The planar antenna receives two linear polarizations over a
bandwidth of two octaves. This bandwidth is partitioned with a superconducting
Niobium microstrip channelizer into sub-bands which are measured with separate
Transition Edge Sensor (TES) bolometric detectors. The bolometers operate at
$0.25\,$K and simultaneously measure the power in each sub-band and
polarization. We first present the design of each subcomponent in the pixel,
then fabrication details of the prototype pixels, and finally the results of
laboratory tests.
The antenna shown in Figures 1 and 2, is a four-armed dual-polarized sinuous
antennaDuHamel with a self similar, or log-periodic, spiral geometry that
repeats every 30% in radius. This geometry results in continuous bandwidth
from 60-240 GHz, limited only by the outer and inner radii. Our first
generation devices had an outer and inner radii of 600 and
$60\,\upmu\mbox{m}$. To obtain an antenna pattern with high circularity, we
find extra cells at the outside and inside of the antenna are required for the
low and high frequency range respectively. In three-dimensional
electromagnetic simulations (HFSS), the extra cells reduce spurious radiating
current density at the ends of the antenna, suggesting that these cells change
the boundary conditions for the active cells. For CMB polarimetry, we require
percent-level ellipticities and find that we need a larger aspect ratio of
inner to outer radius than was chosen for our first generation devices . For
the antenna in the diplexer device described below, we used conservative outer
and inner radii of 1500 and $15\,\upmu\mbox{m}$.
Figure 1: Microscope photograph of the channelizer pixel. Bolometers are
beyond the bottom of the photo. Upper-left: Close-up microstrip feed-points in
the antenna center. Upper-right: Close-up of one channelizer filter. Figure
2: Microscope photograph of the Diplexer pixel.
We fabricated this antenna by etching slots through an otherwise continuous
ground plane of 0.3 $\upmu\mbox{m}$ thick superconducting niobium on a silicon
substrate. We simulated the input impedance in Agilent’s planar
electromagnetics method of moments software (ADS momentum). For a semi-
infinite substrate, it lies on a complex circle of 10 $\Omega$ radius around
104 $\Omega$, which matches the impedance of the lithographed microstrip
lines, leading to a return loss at the antenna feed point of less than -15 dB
across the band. This relatively constant impedance is the result of the self-
complimentary geometry of the sinuous antenna in free space (no dielectrics)
which is only slightly changed with the introduction to the silicon substrate
DuHamel .
We mounted a 14 mm diameter extended hemispherical silicon lens over the
antenna in each device to produce beams that would match typical telescope
optics and also pass through the window of our test cryostat with minimal
vignetting. We offset the hemisphere’s center from the antenna center along
boresight with a 2.7 mm silicon extension to approximate an ellipsoidal
lensFilipovic Edwards . This extension was formed by the detector substrate
and an additional silicon wafer. Both the high index of the lens and the wide
beamwidth of the unlensed antenna suppress the backlobe response; ADS momentum
simulations suggest it is only 9% of the total beam. Lens-coupled detector
arrays may require lenses smaller than 14mm to optimize for mapping speed, but
such lens designs still provide ample room to fit the filters and bolometers
under the lens in addition to the antenna. For example, the Polarbear
Experiment’s f/1.8 150GHz telescope uses 6.5mm diameter contacting lenses over
its detectors, which greatly exceeds the 3mm diameter of this sinuous antenna.
Kam
Each pair of opposite antenna arms couples one of two linear polarizations to
microstrip transmission lines with 0.6 $\upmu\mbox{m}$ superconducting niobium
(Nb) upper conductors separated from the ground plane by an insulating 0.5
$\upmu\mbox{m}$ thick silicon dioxide ($\mbox{SiO}_{2}$) film. The antenna
arms are defined by slots in the ground plane. The microstrip transmission
lines carry millimeter waves from the central feed points shown in Figure 1,
using the back of the Nb antenna arms as their ground plane. While traditional
feeds for the sinuous antenna use transmission lines perpendicular to the
antenna plane, this feed architecture makes the entire microwave design
planar. Opposite arms are driven differentially to receive only the odd mode
and reflect away the even mode.
At the exterior of the antenna, each polarization’s transmission lines enter a
filter bank that partitions the bandwidth into narrow channels. For
terrestrial experiments which must observe through the Earth’s atmosphere,
each band must be located in a spectral window of high transmittance to avoid
undesirable atmospheric emission and noiseOBrientLTD12 . In one such design,
we extract bands at 90 and 150 GHz using a diplexer circuit that splits the
line in a microstrip T-junction leading to two three-pole filters. We have
also fabricated and tested a three-band triplexer version (90, 150, and 220
GHz), with similar band positions and efficiencies. The inductors in these
filters are short stretches of high impedance transmission line separated by
pi-networks of capacitors. This architecture allowed us to form the capacitors
from the same Nb and $\mbox{SiO}_{2}$ films used for the microstrip lines in a
way that does not require vias between the Nb films. We terminate the filtered
signal from a differential pair of transmission lines by differentially
driving a 20 $\Omega$ lumped resistive load in close thermal contact with a
TES on a thermally isolated bolometer island on the silicon wafer. Both the
resistive load and TES are etched from the same aluminum-titanium bilayer with
a superconducting transition between 0.5 and 0.6 K.
In an alternative design for satellite and balloon applications where
atmospheric loading is less of a concern, each polarization’s transmission
line enters a channelizer with three-pole band-pass filters of geometrically
decreasing resonant frequencies Galbraith_3pol , where the individual resonant
filter design is nearly identical to that used in the diplexers and triplexers
described above. We connect these filters by a trunkline of series inductors
formed from short stretches of 15 $\Omega$ microstrip. For frequencies that
resonate in filters in the interior of the manifold, the filters near the
entrance port will appear capacitive. This shunt capacitance acts with the
series trunk-line inductors to create an effective transmission line circuit
that passes the mode along until it reaches its resonant filter. The log-
periodic nature of the channelizer guaranties a low return loss over the seven
bands, provided that the input port is tuned with a shunt capacitor. We
designed this channelizer to function as a low-resolution spectrometer with
low cross-talk between channels; we could alternatively allow increased
spectral cross-talk between tighter packed channels in a design more suitable
for spectroscopyGalbraith_1pol . We terminated each filter with a lossy
aluminum-titanium transmission line on the released bolometer, also formed
from the same bilayer as the TES. In all of our designs, we voltage-bias the
TES bolometers into the superconducting transition and measure current changes
with a SQUID ammeterIrwin . Strong electrothermal feedback holds the TES in
the transition, elevated above the thermal bath temperature of 0.3 K. While we
chose to use TES bolometers because of their proven low noise performance, in
principle we could replace the bolometers with other detectors, such as
kinetic inductance detectorsZmuidzinas_KIDS_review .
Our test cryostat has a foam window and a stack of metal-mesh filters to
reduce IR loading. We scanned a thermal source modulated between 77 and 300 K
on a pair of translation stages in front of the cryostat to map the far-field
beam patterns from each channel in our test pixels. The first device we
characterized was designed to detect power in each polarization through a
diplexer with bands centered at 90 and 150 GHz. Two-dimensional maps from one
polarization in each band are shown in Figure 3. The other devices that we
mapped contained only a single band-pass filters between the antenna and
bolometer so we could demonstrate acceptable antenna performance up to the 250
GHz upper band edge. Figure 3 shows measured and simulated single-plane beam
cuts through those single filter devices. All beams were highly circular with
an ellipticity (difference divided by sum of major and minor axis widths of
the best fit Gaussian) of only $\sim 1$ percent.
We find good agreement between beam measurements and simulations, which we co-
plotted in Figure 3. We simulated the antenna on a semi-infinite silicon
substrate, and then accounted for the lens surface with an optics software
that modeled refraction and diffraction at the vacuum-silicon surface
Filipovic . The software ignores reflections which, for a synthesized ellipse,
only slightly alter the beam and are greatly suppressed by an anti-reflection
coating. The beam opening angle, or antenna gain, varies with wavelength since
the lens acts as a diffracting aperture with a constant physical size. In the
shorter-wavelength channels, the lens is an electrically larger diffracting
surface, which narrows the beam relative to the longer-wavelength
channelsEdwards . When used in a focal plane array where the highest frequency
beam matches the telescope optics, the lower frequencies will have a
relatively large beam and lower spillover efficiency. But provided that the
aperture stop is cold, this loss is offset by the higher pixel density of
those channels relative to the pixel density of monochromatic arrays.
Figure 3: Far-field antenna patterns averaged over the 30% bandwidth
integrated filters. (a) Measured 2-D Diplexer maps through a diplexer, stamped
with ellipticity. (b) Measured beam profiles compared to simulations at four
frequency channels show acceptable beams over the entire antenna operating
band. The simulations average the beams over the filter bandwidth.
We positioned the chopped source on boresight and rotated a wire-grid
polarizer between the window and source aperture, searching for the minimum
power position in each channel. This test showed polarization isolation of
2.8% or lower, and that highest value occurred in the higher frequency
channels. Based on our experience with adding exterior cells to improve the
lower band edge performance, we expect that this leakage will decrease with
the addition of more high-frequency cells. The diplexer device showed 0.3% and
1.6% polarization isolation in the 90 and 150 GHz channels.
Finally, we measured the spectral response of all the channels in our
channelizer with a Fourier transform spectrometer that filled the antenna beam
for all but the lowest frequency channel. Figure 4 shows the results of this
spectroscopy for both the diplexer and log-periodic channelizer, where we have
divided away the spectral response of the interferometer’s 0.01 inch thick
mylar beam-splitter.
Figure 4: Measured spectra of each channel. (a) shows two channels from one polarization of a diplexer device. (b) shows one polarizations of a log-periodic channelizer. We show the designed -3 dB bandwidths in the horizontal bars above the plots and have printed the band-averaged efficiency above each. We measured the Receiver Efficiency using the power received from a temperature modulated beam-filling thermal load and used these to normalize the spectra Table 1: Sources of loss in Diplexer Measurement $Component$ | $Efficiency$
---|---
| 90GHz | 150GHz
Cryostat Thermal Filters | $\approx$70% | $\approx$70%
Lens-vacuum interface | 90% | 90%
Antenna Front-lobe | 91% | 91%
Dielectric Loss ($\tan(\delta)$=0.005) | 81% | 70%
Product | 45% | 39%
We normalize the peak of each spectrum to the receiver efficiency measured
with a chopped 77–300 K beam-filling thermal source and label each curve in
figure 4 with the band averaged efficiency. Table 1 summarizes the losses that
explain the observed efficiencies of 45% in the diplexer. The AR-coating was
optimized for 120 GHz, which reduced the efficiency by about 5% in both 90 GHz
and 150 GHz. Internal reflections within the coating further reduce the
efficiency by another 5% vandervost . The microstrip dielectric losses are a
function of frequency with an average loss of 20% between the diplexed
channels. The log-periodic channelizer had lower efficiency because we did not
anti-reflection coat the contacting lens and there was an impedance mismatch
between the microstrip and antenna. We will correct these in future versions
as we did with the diplexer device.
We envision using focal planes of these multi-colored pixels in CMB and sub-
millimeter telescopes in lieu of focal planes with single color pixels. Focal
planes of these multichroic pixels can provide the same experimental
sensitivity as several monochromatic focal planes, enabling a significant
increase in sensitivity of both ground- and space-based instruments.
We fabricated these devices in the Berkeley Microlab. NASA grant NNG06GJ08G
supported this work.
## References
* (1) P.L. Richards, Journal of Applied Physics, 76, 1, 1-24, 1994.
* (2) J. Dunkley, A. Amblard, C. Baccigalupi, M. Betoule, D. Chuss, A. Cooray, J. Delabrouille, C. Dickinson, G. Dobler, J. Dotson, _et al._ , American Institute of Physics Conference Series, 1141, 222, 2009.
* (3) A. Blain, I. Smail, R. Ivison, J. Kneib, D. Frayer, Physics Reports, 369 (2), 111-176, 2002.
* (4) Y. Gong, A. Cooray, M. Silva, M. Santos, J. Bock, M. Bradford, and M. Zemcov, The Astrophysical Journal, 745 (1), 49-64. 2012.
* (5) C. Kuo, J. Bock, J. Bonetti, J. Brevik, G, Chattopadhyay, P. Day, S Golwala, M Kenyon, A Lange, H LeDuc, _et all_ , Proc SPIE, 7020, 1L:1-14 2008
* (6) M. D. Niemack, J. Beall, D. Becker, H.-M. Cho, A. Fox, G. Hilton, J. Hubmayr, K. Irwin, D. Li, J. McMahon, _et all_ , Journal of Low Temperature Physics, 167:917-22, 2012
* (7) R. DuHamel, United States Patent 4658262, 1987.
* (8) D. Filipovic, S. Gearhart, and G. Rebeiz, Microwave Theory and Techniques, IEEE Transactions on, 4 (10), 1738–1749, 1993.
* (9) J. Edwards, R. O’Brient, A. Lee, G. Rebeiz, Antennas and Propagation, IEEE Transactions on , vol. 60, no. 9, pp. 4082-4091. 2012
* (10) K. Arnold, P. Ade, A. Anthony, D. Barron, D. Boettger, J. Borrill, S. Chapman, Y. Chinone, M. Dobbs, J. Errard, _et al_. Proc SPIE, 8452 (49), 1D:1-12. 2012.
* (11) R. O Brient, P. Ade, K, Arnold, G. Engargiola,W. Holzapfel, A. Lee, M. Myers, X. Meng, E. Quealy, P. Richards, _et al_ , Journal of Low Temperature Physics, 151 (1). 450–463 2008.
* (12) C. Galbraith and G. Rebeiz, Microwave Theory and Techniques, IEEE Transactions on, 56 (7), 1675–1683, 2008.
* (13) Zmuidzinas, J., Annual Review of Condensed Matter Physics, 3 (1), 169–214, 2012.
* (14) M. Van der Vorst,P. de Maagt, M. Herben, Microwave Theory and Techniques, IEEE transactions on, 47 (9), 1696 - 1704, 1999.
* (15) C. Galbraith, G. Rebeiz, Microwave Theory and Techniques, IEEE Transactions on, 56 (7), 1675–1683, 2008.
* (16) K. Irwin and G. Hilton,Topics in Applied Physics. Vol 99, 63-150, 2005
* (17) M. Myers, W. Holzapfel, A. Lee, R. O’Brient, P. Richards, H. Tran, P. Ade, G. Engargiola, A. Smith, H. Spieler, Applied Physics Letters, 86 (11), p. 114103 2005.
|
arxiv-papers
| 2013-02-02T00:50:32 |
2024-09-04T02:49:41.216357
|
{
"license": "Public Domain",
"authors": "Roger OBrient, Peter Ade, Kam Arnold, Jennifer Edwards, Greg\n Engargiola, William L Holzapfel, Adrian T Lee, Michael J Myers, Erin Quealy,\n Gabriel Rebeiz, Paul Richards, Aritoki Suzuki",
"submitter": "Roger O'Brient",
"url": "https://arxiv.org/abs/1302.0325"
}
|
1302.0394
|
∎
11institutetext: Xiaogang Liu 22institutetext: Department of Computer Science
and Engineering, Shanghai Jiao Tong University, China
22email: [email protected] 33institutetext: Yuan Luo (Corresponding author)
44institutetext: Department of Computer Science and Engineering, Shanghai Jiao
Tong University, China
Addr.: 800 Dongchuan Road, Min Hang District, Shanghai 200240, China.
Tel.:+86 21 34205477 44email: [email protected]
# The weight distributions of some cyclic codes with three or four nonzeros
over $\mathbb{F}_{3}$ ††thanks: This work is supported in part by the National
Key Basic Research and Development Plan of China under Grant 2012CB316100, and
the National Natural Science Foundation of China under Grants 61271222,
60972033.
Xiaogang Liu Yuan Luo
(Received: date / Accepted: date)
###### Abstract
Because of efficient encoding and decoding algorithms, cyclic codes are an
important family of linear block codes, and have applications in communication
and storage systems. However, their weight distributions are known only for a
few cases mainly on the codes with one or two nonzeros. In this paper, the
weight distributions of two classes of cyclic codes with three or four
nonzeros are determined.
###### Keywords:
Association scheme Cyclic code Exponential sum Quadratic form Weight
distribution
###### MSC:
94B05 94B65
††journal: Designs, Codes and Cryptography
## 1 Introduction
An $[n,k,d;p]$ code is a $k$-dimensional subspace of $\mathbb{F}_{p}^{n}$ with
minimum(Hamming) distance $d$. Let $A_{i}$ denote the number of codewords with
Hamming weight $i$ in a code $\mathcal{C}$ of length $n$. The weight
enumerator of $\mathcal{C}$ is defined by
$1+A_{1}x+A_{2}x^{2}+\cdots+A_{n}x^{n}.$
The sequence $(1,A_{1},\cdots,A_{n})$ is called the weight distribution of the
code, which is an important parameter of a linear block code. In fact, the
minimum distance $d$ determines the error correcting capability of the code
$\mathcal{C}$. Furthermore, under some algorithms, we can compute the error
probability of error detection and correction.
An $[n,k,d;p]$ linear code $\mathcal{C}$ is called cyclic if
$(c_{0},c_{1},\cdots,c_{n-1})\in\mathcal{C}$ implies that
$(c_{n-1},c_{0},c_{1},\cdots,c_{n-2})\in\mathcal{C}$ $(\mbox{gcd}(n,p)=1)$.
Using the vector space isomorphism from $\mathbb{F}_{p}^{m}$ to the principal
ideal ring $R_{n}:=\mathbb{F}_{p}[x]/(x^{n}-1)$
$(a_{0},a_{1},\ldots,a_{n-1})\longrightarrow a_{0}+a_{1}x+\cdots
a_{n-1}x^{n-1},$
$\mathcal{C}$ is an ideal. The generator $g(x)$ of this ideal is called the
generating polynomial of $\mathcal{C}$, which satisfies that $g(x)|(x^{n}-1)$.
When the ideal is minimal, the code $\mathcal{C}$ is called an irreducible
cyclic code. For any $v=(c_{0},c_{1},\cdots,c_{n-1})\in\mathcal{C}$, the
weight of $v$ is $wt(v)=\\#\\{c_{i}\not=0,i=0,1,\ldots,n-1\\}$.
Many authors have studied how to determine the weight distributions of cyclic
codes. MacWilliams and Seery MS01 gave a procedure for binary cyclic codes,
but it can be implemented only on a powerful computer. The problem of
computing weight distributions is connected to the evaluation of certain
exponential sums McEliece, Rumsey MR001 and Van Der Vlugt M01 , which are
generally hard to determine explicitly. In R001 , Schoof studied its relation
with the rational points of certain curves. For the weight distributions of
the cyclic codes with three nonzeros, please refer to Feng, Luo JH001 and
Zeng, Hu, etc. JHK001 . The related problems in the binary cases with two
nonzeros, were analyzed in Johansen, Helleseth, Kholosha JH001 ; JHK001 .
Assume that $p$ is an odd prime, $q=p^{m}$ for a positive odd integer $m$. Let
$\pi$ be a primitive element of $\mathbb{F}_{q}$. This paper determines the
weight distributions of cyclic codes $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$
over $\mathbb{F}_{3}$ with nonzeros $\pi^{-2},\pi^{-4},\pi^{-10}$ and
$\pi^{-1},\pi^{-2},\pi^{-4},\pi^{-10}$ respectively, the weight distributions
of which are verified by two examples using matlab. Note that the length of
the cyclic codes is $l=q-1=3^{m}-1$ and $3\nmid m$.
In the following, Section 2 presents the basic notations and results about
cyclic codes. Section 3 focus on the class of cyclic code $\mathcal{C}_{1}$.
Section 4 is on the class of cyclic code $\mathcal{C}_{2}$. Final conclusion
is in Section 5.
## 2 Preliminaries
In this section, relevant knowledge from finite fields is presented first for
our study of cyclic codes in Section 2.1. Then some results about the
calculations of exponential sums are presented in Section 2.2. Section 2.3
concerns the sizes of cyclotomic cosets and the ranks of certain quadratic
forms.
### 2.1 Finite fields and cyclic codes
Here, some known properties about the codeword weight are listed, and the
mathematical tools exponential sums and quadratic forms are introduced. For
more researches about cyclic codes, refer to D001 ; DY001 ; FY001 ; M001 for
the irreducible case, and FL001 ; LF001 ; V001 ; YC01 for the reducible case.
For an odd prime $p$ and a positive integer $m$, let the cyclic code
$\mathcal{C}$ over $\mathbb{F}_{p}$ be of length $l=q-1=p^{m}-1$ with non-
conjugate nonzeros $\pi^{-s_{\lambda}}$, where $1\leq s_{\lambda}\leq
q-2(1\leq\lambda\leq\iota)$ and $\pi$ is a primitive element of
$\mathbb{F}_{q}$. Then the codewords in $\mathcal{C}$ can be expressed by
$c(\alpha_{1},\ldots,\alpha_{\iota})=(c_{0},c_{1},\ldots,c_{l-1})\ \
(\alpha_{1},\ldots,\alpha_{\iota}\in\mathbb{F}_{q}),$ (1)
where
$c_{i}=\sum\limits_{\lambda=1}^{\iota}\mbox{Tr}(\alpha_{\lambda}\pi^{is_{\lambda}})(0\leq
i\leq l-1)$ and $\mbox{Tr}:\mathbb{F}_{q}\rightarrow\mathbb{F}_{p}$ is the
trace mapping from $\mathbb{F}_{q}$ to $\mathbb{F}_{p}$. Therefore the Hamming
weight of the codeword $c=c(\alpha_{1},\ldots,\alpha_{\iota})$ is:
$\begin{array}[]{ll}w_{H}(c)&=\\#\\{i|0\leq i\leq l-1,c_{i}\not=0\\}\\\
&=l-{l\over p}-{1\over
p}\sum\limits_{a=1}^{p-1}\sum\limits_{x\in\mathbb{F}_{q}^{*}}{\zeta}_{p}^{\mbox{Tr}(af(x))}\\\
&=p^{m-1}(p-1)-{1\over
p}\sum\limits_{a=1}^{p-1}S(a\alpha_{1},\ldots,a\alpha_{\iota})\\\
&=p^{m-1}(p-1)-{1\over p}R(\alpha_{1},\ldots,\alpha_{\iota})\end{array}$ (2)
where ${\zeta_{p}}=e^{{2\pi i}\over p}$ ($i$ is imaginary unit),
$f(x)=\alpha_{1}x^{s_{1}}+\alpha_{2}x^{s_{2}}+\cdots+\alpha_{\iota}x^{s_{\iota}}\in\mathbb{F}_{q}[x],\mathbb{F}_{q}^{*}=\mathbb{F}_{q}\backslash\\{0\\}$,
$S(\alpha_{1},\ldots,\alpha_{\iota})=\sum\limits_{x\in\mathbb{F}_{q}}{\zeta}_{p}^{\mbox{Tr}\left(\alpha_{1}x^{s_{1}}+\cdots+\alpha_{\iota}x^{s_{\iota}}\right)},$
(3)
and
$R(\alpha_{1},\ldots,\alpha_{\iota})=\sum\limits_{a=1}^{p-1}S(a\alpha_{1},\ldots,a\alpha_{\iota})$.
For general functions of the form $f_{\alpha,\ldots,\gamma}(x)=\alpha
x^{p^{i}+1}+\cdots+\gamma x^{p^{j}+1}$ where $0\leq
i,\ldots,j\leq\lfloor{{m}\over 2}\rfloor$, there are quadratic forms
$F_{\alpha,\ldots,\gamma}(X)$ (4)
and corresponding symmetric matrices
$H_{\alpha,\ldots,\gamma}$ (5)
satisfying that
$F_{\alpha,\ldots,\gamma}(X)=XH_{\alpha,\ldots,\gamma}X^{T}=\mbox{Tr}(f_{\alpha,\ldots,\gamma}(x)).$
It is known that there exists
$M_{\alpha,\ldots,\gamma}\in\mbox{GL}_{m}(\mathbb{F}_{p})$ such that
$H_{\alpha,\ldots,\gamma}^{\prime}=M_{\alpha,\ldots,\gamma}H_{\alpha,\ldots,\gamma}M_{\alpha,\ldots,\gamma}^{T}=\mbox{diag}(a_{1},\ldots,a_{r_{\alpha,\ldots,\gamma}},0,\ldots,0),$
where $a_{i}\in\mathbb{F}_{p}^{*}(1\leq i\leq r_{\alpha,\ldots,\gamma})$ and
$r_{\alpha,\ldots,\gamma}=\mbox{rank}H_{\alpha,\ldots,\gamma}$. Let
$\Delta=a_{1}\cdots a_{r_{\alpha,\ldots,\gamma}}$ (set $\Delta=1$ for
$r_{\alpha,\ldots,\gamma}=0$), and
$\left({\Delta\over p}\right)$ (6)
denotes the Legendre symbol. The following result is about the exponential sum
corresponding to the symmetric matrix $H_{\alpha,\ldots,\gamma}$ FL001 , see
also LH001 .
###### Lemma 1
(Lemma 1, FL001 )
1. $($i$)$
For the quadratic form $F(X)=XHX^{T}$,
$\sum\limits_{X\in\mathbb{F}_{p}^{m}}\zeta_{p}^{F(X)}=\begin{cases}\left({\Delta\over
p}\right)p^{m-r/2}&\mbox{if}\ \ p\equiv 1\ (\mbox{mod}\ 4),\\\
i^{r}\left({\Delta\over p}\right)p^{m-r/2}&\mbox{if}\ \ p\equiv 3\
(\mbox{mod}\ 4).\end{cases}$
2. $($ii$)$
For $A=(a_{1},\ldots,a_{m})\in\mathbb{F}_{p}^{m}$, if $2YH+A=0$ has solution
$Y=B\in\mathbb{F}_{p}^{m}$, then
$\sum_{X\in\mathbb{F}_{p}^{m}}{\zeta}_{p}^{F(X)+AX^{T}}={\zeta}_{p}^{c}\sum_{X\in\mathbb{F}_{p}^{m}}{\zeta}_{p}^{F(X)}\
\mbox{where}\ \ c={1\over 2}AB^{T}\in\mathbb{F}_{p}.$ (7)
Otherwise $\sum\limits_{X\in\mathbb{F}_{p}^{m}}{\zeta}_{p}^{F(X)+AX^{T}}=0$.
### 2.2 Results about exponential sums
In this subsection Lemma 2 and Remark 1 are from LL001 , also refer to FL001
for the calculations of exponential sums that will be needed in the sequel.
###### Lemma 2
For the quadratic form
$F_{\alpha,\ldots,\gamma}(X)=XH_{\alpha,\ldots,\gamma}X^{T}$ corresponding to
$f_{\alpha,\ldots,\gamma}(x)$, see (4)
1. $($i$)$
if the rank $r_{\alpha,\ldots,\gamma}$ of the symmetric matrix
$H_{\alpha,\ldots,\gamma}$ is even, which means that
$S(\alpha,\ldots,\gamma)=\varepsilon p^{m-{r_{\alpha,\ldots,\gamma}\over 2}}$,
then
$R(\alpha,\ldots,\gamma)=\varepsilon(p-1)p^{m-{r_{\alpha,\ldots,\gamma}\over
2}};$
2. $($ii$)$
if the rank $r_{\alpha,\ldots,\gamma}$ of the symmetric matrix
$H_{\alpha,\ldots,\gamma}$ is odd, which means that
$S(\alpha,\ldots,\gamma)=\varepsilon\sqrt{p^{*}}p^{m-{{r_{\alpha,\ldots,\gamma}+1}\over
2}}$, then
$R(\alpha,\ldots,\gamma)=0$
where $\varepsilon=\pm 1$ and $p^{*}=\left({{-1}\over p}\right)p$.
###### Lemma 3
Let $F_{\alpha,\ldots,\gamma}(X)=XH_{\alpha,\ldots,\gamma}X^{T}$ be the
quadratic form corresponding to $f_{\alpha,\ldots,\gamma}(x)$, see (4). If the
rank $r_{\alpha,\ldots,\gamma}$ of the symmetric matrix
$H_{\alpha,\ldots,\gamma}$ is odd, then the number of quadratic forms with
exponential sum $\sqrt{p^{*}}p^{m-{{r_{\alpha,\ldots,\gamma}+1}\over 2}}$
equals the number of quadratic forms with exponential sum
$-\sqrt{p^{*}}p^{m-{{r_{\alpha,\ldots,\gamma}+1}\over 2}}$ where
$p^{*}=\left({{-1}\over p}\right)p$.
###### Proof
Choose a quadratic nonresidue $a\in\mathbb{F}_{p}^{*}$, then the symmetric
matrix corresponding to $af_{\alpha,\ldots,\gamma}(x)$ is
$aH_{\alpha,\ldots,\gamma}$ which also has rank $r_{\alpha,\ldots,\gamma}$,
and
$M_{\alpha,\ldots,\gamma}\left(aH_{\alpha,\ldots,\gamma}\right)M_{\alpha,\ldots,\gamma}^{T}=\mbox{diag}(aa_{1},\ldots,aa_{r_{\alpha,\ldots,\gamma}},0,\ldots,0).$
Since $r_{\alpha,\ldots,\gamma}$ is odd,
$\left({\Delta\cdot a^{r_{\alpha,\ldots,\gamma}}\over
p}\right)=\left({\Delta\cdot a\over p}\right)=\left({\Delta\over
p}\right)\cdot\left({a\over p}\right)=-\left({\Delta\over p}\right)$
where $\Delta=a_{1}\cdots a_{r_{\alpha,\ldots,\gamma}}$. The result follows
from Lemma 1 and the statement above it. ∎
###### Remark 1
For the exponential sum $S(\alpha,\ldots,\gamma)$ corresponding to
$f_{\alpha,\ldots,\gamma}(x)=\alpha x^{p^{i}+1}+\cdots+\gamma x^{p^{j}+1}$
with quadratic form $F_{\alpha,\ldots,\gamma}(X)$ and symmetric matrix
$H_{\alpha,\ldots,\gamma}$ (equation (4)), consider
$S^{\prime}(\alpha,\ldots,\gamma,\delta)$ with respect to
$f^{\prime}_{\alpha,\ldots,\gamma,\delta}(x)=f_{\alpha,\ldots,\gamma}(x)+\delta
x,$ (8)
and
$R^{\prime}(\alpha,\ldots,\gamma,\delta)=\sum\limits_{a=1}^{p-1}S^{\prime}(a\alpha,\ldots,a\gamma,a\delta)$
(equation (2)). From Lemma 1, there are four cases to be considered where the
first two equations are for the case with symmetric matrices $H$ of even rank
and the last two equations for the case of odd rank.
* *
If $S^{\prime}(\alpha,\ldots,\gamma,\delta)=\varepsilon p^{r^{\prime}}$, then
$R^{\prime}(\alpha,\ldots,\gamma,\delta)=\varepsilon(p-1)p^{r^{\prime}};$
* *
If
$S^{\prime}(\alpha,\ldots,\gamma,\delta)=\varepsilon\zeta_{p}^{c}p^{r^{\prime}}$,
then $R^{\prime}(\alpha,\ldots,\gamma,\delta)=-\varepsilon p^{r^{\prime}};$
* *
If
$S^{\prime}(\alpha,\ldots,\gamma,\delta)=\varepsilon\sqrt{p^{*}}p^{r^{\prime}}$,
then $R^{\prime}(\alpha,\ldots,\gamma,\delta)=0;$
* *
If
$S^{\prime}(\alpha,\ldots,\gamma,\delta)=\varepsilon\zeta_{p}^{c}\sqrt{p^{*}}p^{r^{\prime}}$,
then $R^{\prime}(\alpha,\ldots,\gamma,\delta)=\varepsilon\left({-c}\over
p\right)p^{{r^{\prime}}+{1}}$.
In the above, $r^{\prime}$ is a positive integers, $c\in\mathbb{F}_{p}^{*}$,
$p^{*}=\left({-1}\over p\right)p$ and $\varepsilon=\pm 1$.
### 2.3 Cyclotomic cosets and the ranks of certain quadratic forms
The cyclotomic coset containing $s$ is defined to be
$\mathcal{D}_{s}=\\{s,sp,sp^{2},\ldots,sp^{m_{s}-1}\\}$ (9)
where $m_{s}$ is the smallest positive integer such that $p^{m_{s}}\cdot
s\equiv s\ (\mbox{mod}\ p^{m}-1)$.
In the following, Lemma 4 and Lemma 5 are from LL001 , also refer to CH001
for the binary case of Lemma 4.
###### Lemma 4
If $m=2t+1$ is odd, then for $l_{i}=1+p^{i}$, the cyclotomic coset
$\mathcal{D}_{l_{i}}$ has size
$|\mathcal{D}_{l_{i}}|=m,\ \ \ 0\leq i\leq t.$
If $m=2t+2$ is even, then for $l_{i}=1+p^{i}$, the cyclotomic coset
$\mathcal{D}_{l_{i}}$ has size
$|\mathcal{D}_{l_{i}}|=\begin{cases}m,&0\leq i\leq t\\\
m/2,&i=t+1.\end{cases}$
For
$f_{d}^{\prime}(x)=\alpha_{0}x^{2}+\alpha_{1}x^{p+1}+\cdots+\alpha_{d}x^{p^{d}+1}$
with corresponding quadratic form
$F_{d}^{\prime}(X)=\mbox{Tr}(f_{d}^{\prime}(x))=XH_{d}^{\prime}X^{T}$ where
$(\alpha_{0},\alpha_{1},\ldots,\alpha_{d})\in\mathbb{F}_{q}^{d+1}\backslash\\{(0,0,\ldots,0)\\}$,
the following result is about its rank.
###### Lemma 5
Let $m$ be a positive integer, $0\leq d\leq\lfloor{m\over 2}\rfloor$. The rank
$r_{d}^{\prime}$ of the symmetric matrix $H_{d}^{\prime}$ satisfies
$r_{d}^{\prime}\geq m-2d$.
The following corollary is a special case of Lemma 5.
###### Corollary 1
The rank $r_{2}^{\prime}$ of the symmetric matrix $H_{2}^{\prime}$
corresponding to
$f_{2}^{\prime}(x)=\alpha_{0}x^{2}+\alpha_{1}x^{p+1}+\alpha_{2}x^{p^{2}+1}$
has five possible values:
$m,m-1,m-2,m-3,m-4.$
## 3 The cyclic code $\mathcal{C}_{1}$
This section investigates the weight distribution of the cyclic code
$\mathcal{C}_{1}$ over $\mathbb{F}_{3}$ with length $l=3^{m}-1$ and nonzeros
$\pi^{-2},\pi^{-4}$ and $\pi^{-10}$, where $\pi$ is a primitive element of the
finite field $\mathbb{F}_{3^{m}}$ for an odd integer $m$ satisfying $3\nmid
m$.
First, Lemma 6 and Lemma 7 are stated about the number of solutions of
quadratic equations over finite field. Secondly moments of exponential sum
$S(\alpha,\beta,\gamma)$ are calculated in Section 3.1 and Section 3.2, which
provide four equations and one equation for the weight distributions
respectively. Finally, some relevant results about quadratic forms are
presented in Section 3.3 which provides another two equations using
association schemes, and main results are provided by using the seven
equations in Theorem 3.1 of Section 3.4.
###### Definition 1
For any finite field $\mathbb{F}_{q}$ the integer-valued function $\upsilon$
on $\mathbb{F}_{q}$ is defined by $\upsilon(b)=-1$ for
$b\in\mathbb{F}_{q}^{*}$ and $\upsilon(0)=q-1$.
###### Lemma 6
(Theorem 6.26., LH001 ) Let $f$ be a nondegenerate quadratic form over
$\mathbb{F}_{q}$, $q$ odd, in an even number $n$ of indeterminates. Then for
$b\in\mathbb{F}_{q}$ the number of solutions of the equation
$f(x_{1},\ldots,x_{n})=b$ in $\mathbb{F}_{q}^{n}$ is
$q^{n-1}+\upsilon(b)q^{(n-2)/2}\eta\left((-1)^{n/2}\Delta\right)$
where $\eta$ is the quadratic character of $\mathbb{F}_{q}$ and
$\Delta=\mbox{det}(f)$.
###### Lemma 7
(Theorem 6.27., LH001 ) Let $f$ be a nondegenerate quadratic form over
$\mathbb{F}_{q}$, $q$ odd, in an odd number $n$ of indeterminates. Then for
$b\in\mathbb{F}_{q}$ the number of solutions of the equation
$f(x_{1},\ldots,x_{n})=b$ in $\mathbb{F}_{q}^{n}$ is
$q^{n-1}+q^{(n-1)/2}\eta\left((-1)^{(n-1)/2}b\Delta\right)$
where $\eta$ is the quadratic character of $\mathbb{F}_{q}$ and
$\Delta=\mbox{det}(f)$.
### 3.1 Moments of the exponential sum $S(\alpha,\beta,\gamma)$
For an odd prime $p$, this subsection calculates the first three moments of
the exponential sum $S(\alpha,\beta,\gamma)$ (equation 3).
###### Lemma 8
Let $p$ be an odd prime satisfying $p\equiv 3\ \mbox{mod}\ 4$, and $q=p^{m}$
where $m$ is an odd integer with property $3\nmid m$. Then there are the
following results about the exponential sum $S(\alpha,\beta,\gamma)$ (equation
3) corresponding to $f_{2}^{\prime}(x)=\alpha x^{2}+\beta x^{p+1}+\gamma
x^{p^{2}+1}$
1. $($i$)$
$\sum\limits_{\alpha,\beta,\gamma\in\mathbb{F}_{q}}S(\alpha,\beta,\gamma)=p^{3m}$
2. $($ii$)$
$\sum\limits_{\alpha,\beta,\gamma\in\mathbb{F}_{q}}S(\alpha,\beta,\gamma)^{2}=p^{3m}$
3. $($iii$)$
$\sum\limits_{\alpha,\beta,\gamma\in\mathbb{F}_{q}}S(\alpha,\beta,\gamma)^{3}=\left((p+1)(p^{m}-1)+1\right)p^{3m}$.
###### Proof
From definition, changing the order of summations, (i) can be calculated as
follows
$\begin{array}[]{ll}\sum\limits_{\alpha,\beta,\gamma\in\mathbb{F}_{q}}S(\alpha,\beta,\gamma)&=\sum\limits_{\alpha,\beta,\gamma\in\mathbb{F}_{q}}\sum\limits_{x\in\mathbb{F}_{q}}\zeta_{p}^{\mbox{Tr}\left(\alpha
x^{2}+\beta x^{p+1}+\gamma x^{p^{2}+1}\right)}\\\
&=\sum\limits_{x\in\mathbb{F}_{q}}\sum\limits_{\alpha\in\mathbb{F}_{q}}\zeta_{p}^{\mbox{Tr}\left(\alpha
x^{2}\right)}\sum\limits_{\beta\in\mathbb{F}_{q}}\zeta_{p}^{\mbox{Tr}\left(\beta
x^{p+1}\right)}\sum\limits_{\gamma\in\mathbb{F}_{q}}\zeta_{p}^{\mbox{Tr}\left(\gamma
x^{p^{2}+1}\right)}\\\
&=\sum\limits_{\stackrel{{\scriptstyle\alpha\in\mathbb{F}_{q}}}{{x=0}}}\zeta_{p}^{\mbox{Tr}\left(\alpha
x^{2}\right)}\sum\limits_{\stackrel{{\scriptstyle\beta\in\mathbb{F}_{q}}}{{x=0}}}\zeta_{p}^{\mbox{Tr}\left(\beta
x^{p+1}\right)}\sum\limits_{\stackrel{{\scriptstyle\gamma\in\mathbb{F}_{q}}}{{x=0}}}\zeta_{p}^{\mbox{Tr}\left(\gamma
x^{p^{2}+1}\right)}\\\ &=q^{3}=p^{3m}.\end{array}$
Equation (ii) can also be calculated in this way
$\begin{array}[]{lll}&&\sum\limits_{\alpha,\beta,\gamma\in\mathbb{F}_{q}}S(\alpha,\beta,\gamma)^{2}\\\
&=&\sum\limits_{x,y\in\mathbb{F}_{q}}\sum\limits_{\alpha\in\mathbb{F}_{q}}\zeta_{p}^{\mbox{Tr}\left(\alpha\left(x^{2}+y^{2}\right)\right)}\sum\limits_{\beta\in\mathbb{F}_{q}}\zeta_{p}^{\mbox{Tr}\left(\beta\left(x^{p+1}+y^{p+1}\right)\right)}\sum\limits_{\gamma\in\mathbb{F}_{q}}\zeta_{p}^{\mbox{Tr}\left(\gamma\left(x^{p^{2}+1}+y^{p^{2}+1}\right)\right)}\\\
&=&M_{2}\cdot p^{3m}\end{array}$
where $M_{2}$ is the number of solutions to the equation system
$\left\\{\begin{array}[]{ll}x^{2}+y^{2}&=0\\\ x^{p+1}+y^{p+1}&=0\\\
x^{p^{2}+1}+y^{p^{2}+1}&=0.\end{array}\right.$
Since it is assumed that $p\equiv 3\ \mbox{mod}\ 4$ and $m$ is an odd integer,
there is not an element $x_{0}\in\mathbb{F}_{q}$ satisfying $x_{0}^{2}=-1$.
The only solution to above system is $x=y=0$, that is $M_{2}=1$.
As to (iii), we have
$\sum\limits_{\alpha,\beta,\gamma\in\mathbb{F}_{q}}S(\alpha,\beta,\gamma)^{3}=M_{3}\cdot
p^{3m}$ (10)
where
$\begin{array}[]{lll}M_{3}&=\\#\\{(x,y,z)\in\mathbb{F}_{q}^{3}|&x^{2}+y^{2}+z^{2}=0,\\\
&&x^{p+1}+y^{p+1}+z^{p+1}=0,x^{p^{2}+1}+y^{p^{2}+1}+z^{p^{2}+1}=0\\}\\\
&=M_{2}+T_{3}\cdot(q-1),\end{array}$
and $T_{3}$ is the number of solutions of
$\left\\{\begin{array}[]{ll}x^{2}+y^{2}+1&=0\\\ x^{p+1}+y^{p+1}+1&=0\\\
x^{p^{2}+1}+y^{p^{2}+1}+1&=0.\end{array}\right.$ (11)
To study equation system (11), consider the last two equations. Canceling $y$
there is
$(x^{p+1}+1)^{p^{2}+1}=(x^{p^{2}+1}+1)^{p+1},$
after simplification, it becomes
$(x^{p^{2}}-x^{p})(x^{p^{3}}-x)=(x^{p}-x)^{p}(x^{p^{3}}-x)=0.$ (12)
Since $3$ is not a divisor of $m$, from (12) it can be checked that
$x\in\mathbb{F}_{p}$. In the same way, it implies that $y\in\mathbb{F}_{p}$.
Since $a^{p}=a$ for any $a\in\mathbb{F}_{p}$, we only need to consider the
first one of system (11). In case of Lemma 6, $\Delta=1,b=-1,n=2$ and $-1$ is
a quadratic nonresidue of $\mathbb{F}_{p}$. So $|T_{3}|=p+1$, and then
$M_{3}=(p+1)(q-1)+1.$ (13)
Substituting to equation (10), the third statement of the lemma is obtained. ∎
Corresponding to Lemma 1 and Corollary 1, we introduce the following notations
for convenience. Let
$N_{\varepsilon,j}=\left\\{(\alpha,\beta,\gamma)\in\mathbb{F}_{q}^{3}\backslash\\{(0,0,0)\\}|S(\alpha,\beta,\gamma)=\varepsilon
p^{{m+j}\over 2}\right\\}$ (14)
where $\varepsilon=\pm 1$ and $j=1,3$. Also, denote
$n_{\varepsilon,j}=|N_{\varepsilon,j}|$ for $j=1,3$. And
$N_{\varepsilon,j}=\left\\{(\alpha,\beta,\gamma)\in\mathbb{F}_{q}^{3}\backslash\\{(0,0,0)\\}|S(\alpha,\beta,\gamma)=\varepsilon
ip^{{m+j}\over 2}\right\\}$ (15)
for $j=0,2,4$, where $i$ is the imaginary unit. By Lemma 3, set
$n_{j}=n_{\varepsilon,j}=|N_{\varepsilon,j}|$
for $j=0,2,4$, since $m-j$ is odd. Using the above notations, Lemma 8 can be
restated as follows.
###### Lemma 9
Let $p$ be an odd prime satisfying $p\equiv 3\ \mbox{mod}\ 4$, and $q=p^{m}$
where $m$ is an odd integer with property $3\nmid m$.
$\begin{array}[]{ll}2(n_{0}+n_{2}+n_{4})+n_{-1,1}+n_{1,1}+n_{-1,3}+n_{1,3}&=p^{3m}-1\\\
n_{1,1}-n_{-1,1}+p(n_{1,3}-n_{-1,3})&=p^{{m-1}\over 2}\left(p^{2m}-1\right)\\\
-2\left(n_{0}+p^{2}n_{2}+p^{4}n_{4}\right)+p(n_{1,1}+n_{-1,1})+p^{3}(n_{1,3}+n_{-1,3})&=p^{m}\left(p^{m}-1\right)\\\
n_{1,1}-n_{-1,1}+p^{3}(n_{1,3}-n_{-1,3})&=(p+1)p^{{3(m-1)}\over
2}\left(p^{m}-1\right)\end{array}$
###### Proof
Substituting the symbols of (14) and (15) to Lemma 8, we have the following
four equations
$\begin{array}[]{lll}&&2(n_{0}+n_{2}+n_{4})+n_{-1,1}+n_{1,1}+n_{-1,3}+n_{1,3}\\\
&=&p^{3m}-1\\\ \\\
&&\sum\limits_{\alpha,\beta,\gamma\in\mathbb{F}_{q}}S(\alpha,\beta,\gamma)\\\
&=&i{p^{m\over 2}}(n_{1,0}-n_{-1,0})+p^{{m+1}\over 2}(n_{1,1}-n_{-1,1})\\\
&&+i{p^{{m+2}\over 2}}(n_{1,2}-n_{-1,2})+p^{{m+3}\over
2}(n_{1,3}-n_{-1,3})+i{p^{{m+4}\over 2}}(n_{1,4}-n_{-1,4})+p^{m}\\\
&=&p^{{m+1}\over 2}(n_{1,1}-n_{-1,1})+p^{{m+3}\over
2}(n_{1,3}-n_{-1,3})+p^{m}\\\ &=&p^{3m}\end{array}$
$\begin{array}[]{lll}&&\sum\limits_{\alpha,\beta,\gamma\in\mathbb{F}_{q}}S(\alpha,\beta,\gamma)^{2}\\\
&=&-{p^{m}}(n_{1,0}+n_{-1,0})+p^{{m+1}}(n_{1,1}+n_{-1,1})\\\
&&-{p^{{m+2}}}(n_{1,2}+n_{-1,2})+p^{{m+3}}(n_{1,3}+n_{-1,3})-{p^{{m+4}}}(n_{1,4}+n_{-1,4})+p^{2m}\\\
&=&p^{3m}\\\ \\\
&&\sum\limits_{\alpha,\beta,\gamma\in\mathbb{F}_{q}}S(\alpha,\beta,\gamma)^{3}\\\
&=&-i{p^{{3m}\over 2}}(n_{1,0}-n_{-1,0})+p^{{3(m+1)}\over
2}(n_{1,1}-n_{-1,1})\\\ &&-i{p^{{3(m+2)}\over
2}}(n_{1,2}-n_{-1,2})+p^{{3(m+3)}\over 2}(n_{1,3}-n_{-1,3})-i{p^{{3(m+4)}\over
2}}(n_{1,4}-n_{-1,4})+p^{3m}\\\ &=&p^{{3(m+1)}\over
2}(n_{1,1}-n_{-1,1})+p^{{3(m+3)}\over 2}(n_{1,3}-n_{-1,3})++p^{3m}\\\
&=&\left((p+1)\left(p^{m}-1\right)+1\right)p^{3m}\end{array}$
where the first one comes from the fact that there are $p^{3m}-1$ elements in
the set $\mathbb{F}_{q}^{3}\backslash\\{(0,0,0)\\}$. Also, note that
$S(\alpha,\beta,\gamma)=p^{m}$ when $\alpha=\beta=\gamma=0$.
Using $n_{j}=n_{\varepsilon,j}=|N_{\varepsilon,j}|$ for $j=0,2,4$, the result
is obtained by simplification. ∎
### 3.2 The fourth moment of $S(\alpha,\beta,\gamma)$
For the fourth moment of $S(\alpha,\beta,\gamma)$ in the particular case of
$p=3$, we calculate the number of solutions of the following equation system
$\left\\{\begin{array}[]{ll}x^{2}+y^{2}+z^{2}+1&=0\\\
x^{p+1}+y^{p+1}+z^{p+1}+1&=0\\\
x^{p^{2}+1}+y^{p^{2}+1}+z^{p^{2}+1}+1&=0\end{array}\right.$ (16)
in Lemma 10, which is denoted by $T_{4}$.
###### Lemma 10
Let $p=3$ and $q=p^{m}$, then
$T_{4}=4\left(2p^{m}-3\right).$
###### Proof
The following process is composed of three parts: Part I is to find the values
of the possible solutions $(x_{0},y_{0},z_{0})$; Part II is to verify that
they are actually a solution of equation (16); Part III is to find the number
of the solutions.
Part I: To study equation system (16), this part tries to get the formula (20)
by using (17,18,19), and then obtain the solution cases (21,22). Consider the
first two equations
$\left\\{\begin{array}[]{ll}x^{2}+y^{2}+z^{2}+1&=0\\\
x^{p+1}+y^{p+1}+z^{p+1}+1&=0,\end{array}\right.$ (17)
we find that
$z^{2}=-(x^{2}+y^{2}+1).$ (18)
Substituting (18) to the second one of (17)
$\begin{array}[]{ll}x_{0}^{p+1}+y_{0}^{p+1}+z_{0}^{p+1}+1&=x^{4}+y^{4}+\left(-(x^{2}+y^{2}+1)\right)^{2}+1\\\
&=x^{4}+y^{4}+x^{4}+y^{4}+1+2x^{2}+2y^{2}+2x^{2}y^{2}+1\\\
&=2x^{4}+2y^{4}+2x^{2}y^{2}+2x^{2}+2y^{2}+2\\\ &=0,\end{array}$
that is
$x^{4}+y^{4}+x^{2}y^{2}+x^{2}+y^{2}+1=0.$ (19)
For equation (19), set $x=x_{0}$ and consider $y$ as the variable to be
determined, then
$x_{0}^{4}+y^{4}+x_{0}^{2}y^{2}+x_{0}^{2}+y^{2}+1=y^{4}+\left(x_{0}^{2}+1\right)y^{2}+\left(x_{0}^{4}+x_{0}^{2}+1\right).$
Set $y^{\prime}=y^{2}$, the above equation becomes
$y^{\prime 2}+(x_{0}^{2}+1)y^{\prime}+(x_{0}^{4}+x_{0}^{2}+1)$ (20)
which is a quadratic polynomial about variable $y^{\prime}$ over the finite
field $\mathbb{F}_{q}$. Set $b=x_{0}^{2}+1$ and $c=x_{0}^{4}+x_{0}^{2}+1$,
then
$\begin{array}[]{ll}\Delta&=b^{2}-4c\\\
&=(x_{0}^{2}+1)^{2}-4(x_{0}^{4}+x_{0}^{2}+1)\\\
&=x_{0}^{4}+2x_{0}^{2}+1-(x_{0}^{4}+x_{0}^{2}+1)\\\ &=x_{0}^{2}.\end{array}$
Note that the characteristic of the finite field $\mathbb{F}_{q}$ is $3$ and
the elements in $\mathbb{F}_{3}$ is $0,1,2=-1$. Corresponding to the solutions
of equation (19)
$\begin{array}[]{ll}y^{\prime}&={{-b\pm\sqrt{\Delta}}\over 2}\\\
&=b\pm\sqrt{\Delta}\\\ &=x_{0}^{2}+1\pm x_{0}\\\ &=(x_{0}\pm
1)^{2}\end{array}$
where we have used the fact that
$x_{0}^{2}+x_{0}+1=x_{0}^{2}-2x_{0}+1=(x_{0}-1)^{2}$ and
$x_{0}^{2}-x_{0}+1=x_{0}^{2}+2x_{0}+1=(x_{0}+1)^{2}$. That is to say
$y^{\prime}=y^{2}=(x_{0}\pm 1)^{2},$ so
$y=x_{0}\pm 1\ \ \ \mbox{or}\ \ \ y=-x_{0}\pm 1.$ (21)
Let’s consider the first case of (21) where $y=y_{0}=x_{0}\pm 1$. By the first
equation of (17)
$\begin{array}[]{ll}x_{0}^{2}+y_{0}^{2}+z^{2}+1&=x_{0}^{2}+(x_{0}\pm
1)^{2}+z^{2}+1\\\ &=x_{0}^{2}+x_{0}^{2}\pm 2x_{0}+1+z^{2}+1\\\
&=2\left(x_{0}^{2}\pm x_{0}+1\right)+z^{2}\\\ &=-(x_{0}\mp 1)^{2}+z^{2}\\\
&=0,\end{array}$
i.e., $z^{2}=(x_{0}\mp 1)^{2}$ and
$z=z_{0}=x_{0}\mp 1\ \ \ \mbox{or}\ \ \ z=z_{0}=-x_{0}\pm 1$ (22)
note that the symbols $+,-$ in (22) are taken with respect to the symbols
$\pm$ of (21) when we set $y=y_{0}$.
Part II: For (21) and (22), this part considers two first cases. Now, for the
possible solutions $x=x_{0},y=y_{0}=x_{0}\pm 1$ and $z=z_{0}=x_{0}\mp 1$,
substituting to the second equation of (16) we verify that
$\begin{array}[]{ll}x_{0}^{p+1}+y_{0}^{p+1}+z_{0}^{p+1}+1&=x_{0}^{4}+(x_{0}\pm
1)^{4}+(x_{0}\mp 1)^{4}+1\\\ &=x_{0}^{4}+\left(x_{0}^{3}\pm 1\right)(x_{0}\pm
1)+\left(x_{0}^{3}\mp 1\right)(x_{0}\mp 1)+1\\\ &=x_{0}^{4}+x_{0}^{4}\pm
x_{0}^{3}\pm x_{0}+1+x_{0}^{4}\mp x_{0}^{3}\mp x_{0}+1+1\\\
&=3\left(x_{0}^{4}+1\right)\\\ &=0\end{array}$
which satisfies the second equation of (16).
Substituting the above values of $x_{0},y_{0}$ and $z_{0}$ to the third
equation of (16),
$\begin{array}[]{ll}x_{0}^{p^{2}+1}+y_{0}^{p^{2}+1}+z_{0}^{p^{2}+1}+1&=x_{0}^{10}+(x_{0}\pm
1)^{10}+(x_{0}\mp 1)^{10}+1\\\ &=x_{0}^{10}+\left(x_{0}^{9}\pm
1\right)(x_{0}\pm 1)+\left(x_{0}^{9}\mp 1\right)(x_{0}\mp 1)+1\\\
&=x_{0}^{10}+x_{0}^{10}\pm x_{0}^{9}\pm x_{0}+1+x_{0}^{10}\mp x_{0}^{9}\mp
x_{0}+1+1\\\ &=3\left(x_{0}^{10}+1\right)\\\ &=0\end{array}$
which implies that the third equation of (16) is also satisfied. Therefore the
possible solutions $(x_{0},y_{0},z_{0})$ satisfy system (16).
As to the first case of (21) and the second case of (22), since
$a^{2}=(-a)^{2}$ for any $a\in\mathbb{F}_{q}$, it can be checked that
$x_{0},y_{0}=x_{0}\pm 1,z_{0}=-x_{0}\pm 1$ also satisfy (16). For other cases,
similar results can be obtained.
Part III: For the values in (21), easy to see that $x_{0}+1\not=x_{0}-1$ and
$-x_{0}+1\not=-x_{0}-1$.
1. $($i$)$
If $x_{0}+1=-x_{0}+1$ then $x_{0}=0$;
2. $($ii$)$
If $x_{0}+1=-x_{0}-1$ then $x_{0}=-1$;
3. $($iii$)$
If $x_{0}-1=-x_{0}+1$ then $x_{0}=1$;
4. $($iv$)$
If $x_{0}-1=-x_{0}-1$ then $x_{0}=0$.
Therefore the possible values of $x_{0}$ which can produce same $y_{0}$ in
(21), are $0,1,-1$. And all other values of $x_{0}\in\mathbb{F}_{q}$ will
reduce to different values of $y_{0}$. The same situation occurs for the cases
of $z_{0}$. Let’s consider these particular values of $x_{0}$.
1. $($i$)$
Suppose $x_{0}=0$.
* *
In equation (21), if $y_{0}=x_{0}+1=1$, then $z_{0}=x_{0}-1=-1$ or
$-x_{0}+1=1$ in (22).
* *
If $y_{0}=x_{0}-1=-1$, then $z_{0}=x_{0}+1=1$ or $-x_{0}-1=-1$.
* *
If $y_{0}=-x_{0}+1=1$, then $z_{0}=-x_{0}-1=-1$ or $z_{0}=x_{0}+1=1$.
* *
If $y_{0}=-x_{0}-1=-1$, then $z_{0}=-x_{0}+1=1$ or $z_{0}=x_{0}-1=-1$.
So there are $4$ solutions of $(x_{0},y_{0},z_{0})$ when $x_{0}=0$.
2. $($ii$)$
For $x_{0}=1$ and $x_{0}=-1$, there are also $4$ solutions respectively.
Altogether there are $12$ solutions for $x_{0}=0,1$ and $-1$.
For each $x_{0}\not\in\\{0,1,-1\\}$, there are $4$ cases of $y_{0}$ in (21),
and for each selected $y_{0}$, there are two choices for $z_{0}$ in (22),
which leads to $4\times 2=8$ solutions of $(x_{0},y_{0},z_{0})$. So the number
of solutions of (16) is $T_{4}=12+\left(p^{m}-3\right)\cdot
8=4\left(2p^{m}-3\right)$. ∎
###### Lemma 11
Let $p=3$ and $q=p^{m}$ where $m$ is an odd integer satisfying $3\nmid m$. The
number of solutions of the following equation system
$\left\\{\begin{array}[]{ll}x^{2}+y^{2}+z^{2}+w^{2}&=0\\\
x^{p+1}+y^{p+1}+z^{p+1}+w^{p+1}&=0\\\
x^{p^{2}+1}+y^{p^{2}+1}+z^{p^{2}+1}+w^{p^{2}+1}&=0\\\ \end{array}\right.$ (23)
is $M_{4}=8\left(p^{m}-1\right)^{2}+1$.
###### Proof
For $w\not=0$, divide the three equations in (23) by $w^{2},w^{p+1}$ and
$w^{p^{2}+1}$ respectively, then equation system (16) is obtained, and the
number of solutions of which is $T_{4}$ by Lemma 10. For $w=0$, the number of
solutions of (23) is $M_{3}$ (equation 13) which assume that $m$ is an odd
integer satisfying $3\nmid m$. Altogether, the number of solutions of equation
system (23) is
$M_{4}=\left(p^{m}-1\right)T_{4}+M_{3}=8\left(p^{m}-1\right)^{2}+1$. ∎
Applying Lemma 11, the following result about the fourth moment of the
exponential sum $S(\alpha,\beta,\gamma)$ can be obtained.
###### Lemma 12
Let $p=3$ and $q=p^{m}$ where $m$ is an odd integer satisfying $3\nmid m$.
Then
$\sum\limits_{\alpha,\beta,\gamma\in\mathbb{F}_{q}}S(\alpha,\beta,\gamma)^{4}=M_{4}\cdot
p^{3m}=\left(8(p^{m}-1)^{2}+1\right)p^{3m}.$
Using the symbols of (14) and (15), Lemma 12 can be rewritten as the following
corollary.
###### Corollary 2
Let $p=3$ and $q=p^{m}$ where $m$ is an odd integer satisfying $3\nmid m$.
Then
$2n_{0}+p^{2}(n_{-1,1}+n_{1,1})+p^{4}\cdot
2n_{2}+p^{6}(n_{-1,3}+n_{1,3})+p^{8}\cdot
2n_{4}=\left(8(p^{m}-1)^{2}-p^{m}+1\right)p^{m}.$
### 3.3 Association schemes
The following introduction is about skew-symmetric matrix (Lemma 13) and
symmetric matrix (Proposition 1), and a relevant discussion LL001 . In fact,
they correspond to two association schemes, the fundamental properties of
which are referred to CV ; D02 ; MS001 ; S01 . Note that there exists a one-
to-one correspondence between the set of alternating bilinear forms and the
set of skew-symmetric matrices, and a one-to-one correspondence from the set
of quadratic forms to the set of symmetric matrices.
A skew-symmetric matrix $B=[b_{i,j}]$ of order $m$ is a matrix which satisfies
$b_{i,i}=0,\ \ \ b_{i,j}+b_{j,i}=0,$
and it has even rank. Let $Y_{m}=Y(m,p)$ denote the set of skew-symmetric
matrices of order $m$ over $\mathbb{F}_{p}$. It can be checked that $Y_{m}$ is
an ${m(m-1)\over 2}$-dimensional vector space over $\mathbb{F}_{p}$.
Set $n=\lfloor m/2\rfloor$. For $k=0,1,\ldots,n$, the partition
$R^{\prime}=\left\\{R_{0}^{\prime},R_{1}^{\prime},\ldots,R_{n}^{\prime}\right\\}$
of $Y_{m}^{2}=Y_{m}\times Y_{m}$ is defined by
$R_{k}^{\prime}=\\{(A,B)\in Y_{M}^{2}|\mbox{rank}(A-B)=2k\\}.$
Then $(Y_{m},R^{\prime})$ is an association scheme with $n$ classes DG01 . The
distance distribution of a nonempty subset $Y$ of $Y_{m}$ in the scheme
$(Y_{m},R^{\prime})$ is the $(n+1)$-tuple
$\mathbf{a}=(a_{0},a_{1},\ldots,a_{n})$ of rational numbers $a_{i}$, where
$|Y|a_{i}=|Y^{2}\bigcap R_{i}^{\prime}|$. Easy to see that
$a_{0}=1,\ \ \ \mbox{and}\ \ \ a_{0}+a_{1}+\cdots+a_{n}=|Y|.$
An $(m,d)$-set $Y$ is a subset of $Y_{m}$ satisfying that
$\mbox{rank}(A-B)\geq 2d,\ \ \ \forall A,B\in Y,\ \ \ A\not=B,$
where $1\leq d\leq n$. In other words,
$a_{1}=a_{2}=\cdots=a_{d-1}=0.$
For a real number $b\not=1$ and all nonnegative integers $k$, denote the
Gaussian binomial coefficients with basis $b$ by $\left[\begin{array}[]{c}x\\\
k\end{array}\right]_{b}$:
$\left[\begin{array}[]{c}x\\\ 0\end{array}\right]_{b}=1,\ \ \
\left[\begin{array}[]{c}x\\\
k\end{array}\right]_{b}=\prod\limits_{i=0}^{k-1}{{(b^{x}-b^{i})}/{(b^{k}-b^{i})}},\
\ \quad k=1,2,\ldots.$
Set $b=p^{2}$ and $c=p^{m(m-1)/{2n}}$. The following result is about
$(m,d)$-set DG01 .
###### Lemma 13
(Theorem 4., DG01 )
1. $($i$)$
For any $(m,d)$-set $Y$, we have (the Singleton bound)
$|Y|\leq c^{n-d+1}.$
2. $($ii$)$
In case of equality, the distance distribution of $Y$ is uniquely determined
by
$a_{n-i}=\sum\limits_{j=i}^{n-d}(-1)^{j-i}{b^{\left(\begin{array}[]{c}j-i\\\
2\end{array}\right)}{\left[\begin{array}[]{c}j\\\
i\end{array}\right]_{b}}{\left[\begin{array}[]{c}n\\\
j\end{array}\right]_{b}}(c^{n-d+1-j}-1),}$
for $i=0,1,\ldots,n-d$. Here, $\left(\begin{array}[]{c}j-i\\\
2\end{array}\right)$ represents the general binomial coefficient.
The set of symmetric matrices $X_{m}=X(m,p)$ forms a vector space of dimension
$m(m+1)/2$ over $\mathbb{F}_{p}$. Let
$R=\\{R_{i}\\},\ \ \ i=0,1,2,\ldots,\lfloor{{m+1}\over 2}\rfloor,$
be the set of symmetric relations $R_{i}$ on $X_{m}$ defined by
$R_{i}=\\{(A,B)|A,B\in X_{m},\mbox{rank}(A-B)=2i-1\ \mbox{or}\ 2i\\}.$
Comparing to the association scheme $(Y_{m},R^{\prime})$ of skew-symmetric
matrices, the following two lemmas are about the scheme $(X_{m},R)$ E001 .
###### Lemma 14
(Theorem 1., E001 ) $(X_{m},R)$ forms an association scheme of class
$\lfloor(m+1)/2\rfloor$.
###### Lemma 15
(Theorem 2., E001 ) All the parameters (and consequently all the eigenvalues)
of the two association schemes $(X_{m},R)$ and $(Y_{m+1},R^{\prime})$ of class
$\lfloor(m+1)/2\rfloor$ are exactly the same.
Distance distribution of a nonempty subset $X$ of $X_{m}$ can be defined as
that of $(Y_{m},R^{\prime})$. A set $X\subset X_{m}$ is called an $(m,d)$-set
if it satisfies
$\mbox{rank}(A-B)\geq 2d-1,\ \ \ \forall A,B\in X,\ \ \ A\not=B,$
where $1\leq d\leq\lfloor(m+1)/2\rfloor$.
Similar to Lemma 13, Proposition 1 is about the association scheme
$(X_{m},R)$.
###### Proposition 1
Set $b=p^{2},c=p^{m(m+1)/{2n}},n=\lfloor(m+1)/2\rfloor$.
1. $($i$)$
For any $(m,d)$-set $X\subset X_{m}$, we have (the Singleton bound)
$|X|\leq c^{n-d+1}.$
2. $($ii$)$
In case of equality, the distance distribution of $X$ is uniquely determined
by
$a_{n-i}=\sum\limits_{j=i}^{n-d}(-1)^{j-i}{b^{\left(\begin{array}[]{c}j-i\\\
2\end{array}\right)}{\left[\begin{array}[]{c}j\\\
i\end{array}\right]_{b}}{\left[\begin{array}[]{c}n\\\
j\end{array}\right]_{b}}(c^{n-d+1-j}-1),}$
for $i=0,1,\ldots,n-d.$ Here, $\left(\begin{array}[]{c}j-i\\\
2\end{array}\right)$ represents the general binomial coefficient.
###### Proof
Refer to the proof of Lemma 13 DG01 . ∎
### 3.4 Main results
This subsection contains the main results about the cyclic code
$\mathcal{C}_{1}$. Before going on, Lemma 16 is for the distance distribution
of relevant subset $X$ of $X_{m}$. With notations as in Proposition 1 for
$m=2t+1$, we have $n=t+1$ and $c=p^{m}$. In fact, Lemma 9, Corollary 2 and
Corollary 3 result in Corollary 4, and then lead to Theorem 3.1.
###### Lemma 16
Let $m=2t+1,t\geq 2$ and $q=p^{m}-1$ where $p$ is an odd prime. Let $X$ denote
the set of quadratic forms corresponding to
$f_{2}^{\prime}(x)=\alpha_{0}x^{2}+\alpha_{1}x^{p+1}+\alpha_{2}x^{p^{2}+1},\alpha_{0},\alpha_{1},\alpha_{2}\in\mathbb{F}_{q}$
(equation 4). Then $X$ is an $(m,t-1)$-set, and $a_{n-i}$ is the number of
quadratic forms in $X$ with rank $2(n-i)$ or $2(n-i)-1$ for $i=0,1,2$.
###### Proof
By Corollary 1, the ranks of the quadratic forms contained in $X$ satisfy
$r\geq m-4=2t+1-4=2(t-1)-1$. So $X$ is an $(m,t-1)$-set. For $i=0,1,2$, by
definition
$|X|a_{n-i}=|X^{2}\bigcap R_{n-i}|.$ (24)
Let $X_{i}^{\prime}$ denote the set of quadratic forms of rank $2(n-i)$ or
$2(n-i)-1$ in $X$. Easy to find that the sum of two quadratic forms from $X$
also lies in $X$. For given $A\in X$, we find that $(A,C)=(A,A+B)\in R_{n-i}$
for any $B\in X_{i}^{\prime}$, here $C=A+B$. And no element else in $X$
satisfies this property. So
$|X^{2}\bigcap R_{n-i}|=|X||X_{i}^{\prime}|,$ (25)
and $a_{n-i}=|X_{i}^{\prime}|$ by comparing (24) with (25). ∎
###### Corollary 3
Let $m=2t+1,t\geq 2$ and $q=p^{m}-1$ where $p$ is an odd prime. The notations
in (14) and (15) satisfy
$2n_{0}=a_{n},n_{-1,1}+n_{1,1}+2n_{2}=a_{n-1},\ \mbox{and}\ \
n_{-1,3}+n_{1,3}+2n_{4}=a_{n-2}$
where $a_{n-i}$ is as defined in Proposition 1 for $i=0,1,2$.
###### Proof
With notations as stated in Lemma 16, $X$ is an $(m,d)$-set with $d=t-1$ and
$n-d=(t+1)-(t-1)=2$. By the first statement of Proposition 1, $|X|\leq
c^{n-d+1}=p^{3m}$. Since the size of $X$ is $p^{3m}$, applying the second
statement of Proposition 1, the result is obtained from Lemma 16. ∎
###### Corollary 4
Let $p=3$ and $q=p^{m}$ where $m>1$ is an odd integer satisfying $3\nmid m$.
The notations in (14) and (15) satisfy
$\begin{array}[]{ll}2n_{0}&=a_{n}\\\ 2n_{4}&={{B-A}\over{p^{2}(p^{2}-1)}}\\\
2n_{2}&={{p^{2}A-B}\over{p^{2}-1}}\\\ 2n_{1,3}&=p^{{m-3}\over
2}{{(p^{m}-1)(p^{m-1}-1)}\over{p^{2}-1}}+p^{3m}-1-a_{n}-a_{n-1}-2n_{4}\\\
2n_{-1,3}&=p^{3m}-1-a_{n}-a_{n-1}-2n_{4}-p^{{m-3}\over
2}{{(p^{m}-1)(p^{m-1}-1)}\over{p^{2}-1}}\\\
2n_{1,1}&={{p^{m+2}-4p^{m-1}+p^{2}}\over{p^{2}-1}}\cdot p^{{m-1}\over
2}(p^{m}-1)+a_{n-1}-2n_{2}\\\
2n_{-1,1}&=a_{n-1}-2n_{2}-{{p^{m+2}-4p^{m-1}+p^{2}}\over{p^{2}-1}}\cdot
p^{{m-1}\over 2}(p^{m}-1)\end{array}$
where
$\begin{array}[]{ll}A&={{p^{3m+2}-p^{m-1}(p^{m}-1)-p^{2}}\over{p+1}}-{{a_{n}(p^{2}-p+1)}\over
p}-a_{n-1}(p-1),\\\
B&={{(p^{m}-1)(8p^{m}-9)p^{m-2}+p^{4}-p^{3m+4}}\over{p^{2}-1}}+a_{n}\cdot{{p^{4}+p^{2}+1}\over
p^{2}}+a_{n-1}\cdot(p^{2}+1)\\\
&={{(p^{m}-1)(8p^{m}-9)p^{m-2}-p^{3m}+1}\over{p^{2}-1}}-(p^{2}+1)(p^{3m}-1)+a_{n}\cdot{{p^{4}+p^{2}+1}\over
p^{2}}+a_{n-1}\cdot(p^{2}+1),\end{array}$
and $a_{n-i}$ is as defined in Proposition 1 for $i=0,1,2$.
###### Proof
From Lemma 9, Corollary 2 and Corollary 3, there are the following equations
$\begin{array}[]{ll}2n_{0}&=a_{n}\\\ n_{-1,1}+n_{1,1}+2n_{2}&=a_{n-1}\\\
2(n_{0}+n_{2}+n_{4})+n_{-1,1}+n_{1,1}+n_{-1,3}+n_{1,3}&=p^{3m}-1\\\
n_{1,1}-n_{-1,1}+p(n_{1,3}-n_{-1,3})&=p^{{m-1}\over 2}\left(p^{2m}-1\right)\\\
-2(n_{0}+p^{2}n_{2}+p^{4}n_{4})+p(n_{1,1}+n_{-1,1})+p^{3}(n_{1,3}+n_{-1,3})&=p^{m}\left(p^{m}-1\right)\\\
n_{1,1}-n_{-1,1}+p^{3}(n_{1,3}-n_{-1,3})&=4p^{{3(m-1)}\over
2}\left(p^{m}-1\right)\\\ 2n_{0}+p^{2}(n_{-1,1}+n_{1,1})+p^{4}\cdot
2n_{2}+p^{6}(n_{-1,3}+n_{1,3})+p^{8}\cdot
2n_{4}&=\left(8(p^{m}-1)^{2}-p^{m}+1\right)p^{m}.\end{array}$
After simplification, we find that
$\begin{array}[]{ll}2n_{2}+p^{2}\cdot 2n_{4}&=A\\\ 2n_{2}+p^{4}\cdot
2n_{4}&=B.\\\ \end{array}$
From which $2n_{2}$ and $2n_{4}$ can be calculated, then all the other
notations can be obtained. ∎
###### Remark 2
It can be calculated in above paragraph that
$\begin{array}[]{ll}a_{n}&=p^{3m}-1-{{p^{m+1}-1}\over{p^{2}-1}}(p^{2m}-1)+p^{2}\cdot{{p^{m+1}-1}\over{p^{4}-1}}\cdot{{p^{m-1}-1}\over{p^{2}-1}}\cdot(p^{m}-1);\\\
a_{n-1}&={{p^{m+1}-1}\over{p^{2}-1}}(p^{2m}-1)-(p^{2}+1)\cdot{{p^{m+1}-1}\over{p^{4}-1}}\cdot{{p^{m-1}-1}\over{p^{2}-1}}\cdot(p^{m}-1);\\\
a_{n-2}&={{p^{m+1}-1}\over{p^{4}-1}}\cdot{{p^{m-1}-1}\over{p^{2}-1}}\cdot(p^{m}-1).\end{array}$
###### Lemma 17
Let $p=3$ and $q=p^{m}$ where $m>1$ is an odd integer satisfying $3\nmid m$.
The number of solutions of the equation system
$\left\\{\begin{array}[]{ll}x^{2}+y^{2}+z^{2}+w^{2}+u^{2}&=0\\\
x^{p+1}+y^{p+1}+z^{p+1}+w^{p+1}+u^{p+1}&=0\\\
x^{p^{2}+1}+y^{p^{2}+1}+z^{p^{2}+1}+w^{p^{2}+1}+u^{p^{2}+1}&=0,\end{array}\right.$
(26)
is
$p^{2m}+p^{{5-m}\over 2}\left(n_{1,1}-n_{-1,1}+p^{5}(n_{1,3}-n_{-1,3})\right)$
where $n_{1,1},n_{-1,1},n_{1,3}$ and $n_{-1,3}$ are those of Corollary 4.
###### Proof
The following moment of exponential sum $S(\alpha,\beta,\gamma)$ satisfies
$\begin{array}[]{ll}\sum\limits_{\alpha,\beta,\gamma\in\mathbb{F}_{q}}S(\alpha,\beta,\gamma)^{5}&=p^{{5(m+1)}\over
2}(n_{1,1}-n_{-1,1})+p^{{5(m+3)}\over 2}(n_{1,3}-n_{-1,3})+{(p^{m})}^{5}\\\
&=M_{5}\cdot p^{3m}\end{array}$
where $M_{5}$ is the number of solutions of (26). Solving the above equation
for $M_{5}$, the result is obtained. ∎
Equation (26) considers the case for $5$ variables. Using the sixth moment of
$S(\alpha,\beta,\gamma)$, the number of solutions can be calculated when there
are $6$ variables, etc.
###### Theorem 3.1
Let $p=3$ and $q=p^{m}$ where $m>1$ is an odd integer satisfying $3\nmid m$.
The cyclic code $\mathcal{C}_{1}$ with nonzeros $\pi^{-2},\pi^{-(p+1)}$ and
$\pi^{-(p^{2}+1)}$ has five nonzero weights with distribution
$\begin{array}[]{ll}A_{p^{m-1}(p-1)}&=2(n_{0}+n_{2}+n_{4}),\\\
A_{p^{m-1}(p-1)-{{p-1}\over p}p^{{m+1}\over 2}}&=n_{1,1},\\\
A_{p^{m-1}(p-1)+{{p-1}\over p}p^{{m+1}\over 2}}&=n_{-1,1},\\\
A_{p^{m-1}(p-1)-{{p-1}\over p}p^{{m+3}\over 2}}&=n_{1,3},\\\
A_{p^{m-1}(p-1)+{{p-1}\over p}p^{{m+3}\over 2}}&=n_{-1,3},\end{array}$
where the notations on the right hand side are defined in Corollary 4.
###### Proof
There is a relation between the weight of a codeword in $\mathcal{C}_{1}$ and
the corresponding exponential sum (equation 2)
$w_{H}(c)=p^{m-1}(p-1)-{1\over p}R(\alpha,\beta,\gamma).$
The possible values of corresponding exponential sums $S(\alpha,\beta,\gamma)$
are indicated in (14) and (15). By Lemma 2, the values of the sums
$R(\alpha,\beta,\gamma)$ can be calculated, and then the corresponding
codeword weights and distribution. ∎
###### Example 1
Let $m=5,p=3$ and $q=p^{m}$. Let $\mathcal{C}_{1}$ be the cyclic code with
nonzeros $\pi^{-(p^{0}+1)}=\pi^{-2}$, $\pi^{-(p^{1}+1)}=\pi^{-4}$ and
$\pi^{-(p^{2}+1)}=\pi^{-10}$ where $\pi$ is a primitive element of the finite
field $\mathbb{F}_{q}$. Using Matlab, $\mathcal{C}_{1}$ has five nonzero
weights
$\begin{array}[]{l}A_{162}=9740258,A_{144}=2548260,A_{180}=2038608,\ \
\mbox{and}\\\ A_{108}=14520,A_{216}=7260,\end{array}$
which verifies the result of Theorem 3.1.
## 4 The cyclic code $\mathcal{C}_{2}$
Let $p=3$ and $q=p^{m}$ where $m\equiv 1\ \mbox{mod}\ 4$ satisfying $3\nmid
m$. In this section, we study the cyclic code $\mathcal{C}_{2}$ with nonzeros
$\pi^{-1},\pi^{-2},\pi^{-(p+1)}$ and $\pi^{-(p^{2}+1)}$ where $\pi$ is a
primitive element of the finite field $\mathbb{F}_{p^{m}}$. Lemma 18 is for
the calculation of the multiplicities of exponential sum
$S^{\prime}(\alpha,\beta,\gamma,\delta)$ which leads to the result of weight
distribution in Theorem 4.1.
For this, let’s consider the possible values of the exponential sums
$S^{\prime}(\alpha,\beta,\gamma,\delta)$,
$R^{\prime}(\alpha,\beta,\gamma,\delta)$ (Remark 1) and their multiplicities.
By Lemma 1
$S(\alpha,\beta,\gamma)=i^{r}\left({\Delta\over p}\right)p^{m-r/2}$
where $r$ is the rank of the corresponding quadratic form, and $\Delta$ is
defined in equation (6). For clearness, we list the notations of (14) and (15)
in Table I.
Table I: rank | m | m-1 | m-2 | m-3 | m-4
---|---|---|---|---|---
$S(\alpha,\beta,\gamma)$ | $ip^{m\over 2}$ | $-ip^{m\over 2}$ | $p^{{m+1}\over 2}$ | $-p^{{m+1}\over 2}$ | $ip^{{m+2}\over 2}$ | $-ip^{{m+2}\over 2}$ | $p^{{m+3}\over 2}$ | $-p^{{m+3}\over 2}$ | $ip^{{m+4}\over 2}$ | $-ip^{{m+4}\over 2}$
multiplicity | $n_{0}$ | $n_{0}$ | $n_{1,1}$ | $n_{-1,1}$ | $n_{2}$ | $n_{2}$ | $n_{1,3}$ | $n_{-1,3}$ | $n_{4}$ | $n_{4}$
For quadratic form $F(X)$ with corresponding symmetric matrix $H$ of rank $r$,
depending on Table I, Lemma 18 considers the exponential sum
$S^{\prime}(\alpha,\beta,\gamma,\delta)=\sum\limits_{X\in\mathbb{F}_{p}^{m}}\zeta_{p}^{\mbox{Tr}\left(F(X)+AX^{T}\right)}$
(27)
where $A$ varies over $\mathbb{F}_{p}^{m}$.
###### Lemma 18
Let $p=3$ and $q=p^{m}$ where $m\equiv 1\ \mbox{mod}\ 4$ satisfying $3\nmid
m$. The exponential sums $S^{\prime}(\alpha,\beta,\gamma,\delta)$ and their
multiplicities are listed in Table II where
$(\alpha,\beta,\gamma)\in\mathbb{F}_{q}^{3}\backslash\\{(0,0,0)\\}$.
Table II: $\Delta=1$ | $\Delta=2$
---|---
$S^{\prime}(\alpha,\beta,\gamma,\delta)$ | multiplicity | $S^{\prime}(\alpha,\beta,\gamma,\delta)$ | multiplicity
0 | 0 | 0 | 0
$ip^{m\over 2}$ | $n_{0}\cdot p^{m-1}$ | $-ip^{m\over 2}$ | $n_{0}\cdot p^{m-1}$
$\zeta_{p}ip^{m\over 2}$ | $n_{0}\cdot(p^{m-1}-p^{{m-1}\over 2})$ | $-\zeta_{p}ip^{m\over 2}$ | $n_{0}\cdot(p^{m-1}+p^{{m-1}\over 2})$
$\zeta_{p}^{2}ip^{m\over 2}$ | $n_{0}\cdot(p^{m-1}+p^{{m-1}\over 2})$ | $-\zeta_{p}^{2}ip^{m\over 2}$ | $n_{0}\cdot(p^{m-1}-p^{{m-1}\over 2})$
r=m-1
$\Delta=1$ | $\Delta=2$
$S^{\prime}(\alpha,\beta,\gamma,\delta)$ | multiplicity | $S^{\prime}(\alpha,\beta,\gamma,\delta)$ | multiplicity
0 | $n_{1,1}(p^{m}-p^{m-1})$ | 0 | $n_{-1,1}(p^{m}-p^{m-1})$
$p^{{m+1}\over 2}$ | $n_{1,1}\cdot(p^{m-2}+(p-1)p^{{m-3}\over 2})$ | $-p^{{m+1}\over 2}$ | $n_{-1,1}\cdot(p^{m-2}-(p-1)p^{{m-3}\over 2})$
$\zeta_{p}p^{{m+1}\over 2}$ | $n_{1,1}\cdot(p^{m-2}-p^{{m-3}\over 2})$ | $-\zeta_{p}p^{{m+1}\over 2}$ | $n_{-1,1}\cdot(p^{m-2}+p^{{m-3}\over 2})$
$\zeta_{p}^{2}p^{{m+1}\over 2}$ | $n_{1,1}\cdot(p^{m-2}-p^{{m-3}\over 2})$ | $-\zeta_{p}^{2}p^{{m+1}\over 2}$ | $n_{-1,1}\cdot(p^{m-2}+p^{{m-3}\over 2})$
r=m-2
$\Delta=1$ | $\Delta=2$
$S^{\prime}(\alpha,\beta,\gamma,\delta)$ | multiplicity | $S^{\prime}(\alpha,\beta,\gamma,\delta)$ | multiplicity
0 | $n_{2}(p^{m}-p^{m-2})$ | 0 | $n_{2}(p^{m}-p^{m-2})$
$-ip^{{m+2}\over 2}$ | $n_{2}\cdot p^{m-3}$ | $ip^{{m+2}\over 2}$ | $n_{2}\cdot p^{m-3}$
$-i\zeta_{p}p^{{m+2}\over 2}$ | $n_{2}\cdot(p^{m-3}+p^{{m-3}\over 2})$ | $i\zeta_{p}p^{{m+2}\over 2}$ | $n_{2}\cdot(p^{m-3}-p^{{m-3}\over 2})$
$-i\zeta_{p}^{2}p^{{m+2}\over 2}$ | $n_{2}\cdot(p^{m-3}-p^{{m-3}\over 2})$ | $i\zeta_{p}^{2}p^{{m+2}\over 2}$ | $n_{2}\cdot(p^{m-3}+p^{{m-3}\over 2})$
r=m-3
$\Delta=1$ | $\Delta=2$
$S^{\prime}(\alpha,\beta,\gamma,\delta)$ | multiplicity | $S^{\prime}(\alpha,\beta,\gamma,\delta)$ | multiplicity
0 | $n_{-1,3}(p^{m}-p^{m-3})$ | 0 | $n_{1,3}(p^{m}-p^{m-3})$
$-p^{{m+3}\over 2}$ | $n_{-1,3}\cdot(p^{m-4}-(p-1)p^{{m-5}\over 2})$ | $p^{{m+3}\over 2}$ | $n_{1,3}\cdot(p^{m-4}+(p-1)p^{{m-5}\over 2})$
$-\zeta_{p}p^{{m+3}\over 2}$ | $n_{-1,3}\cdot(p^{m-4}+p^{{m-5}\over 2})$ | $\zeta_{p}p^{{m+3}\over 2}$ | $n_{1,3}\cdot(p^{m-4}-p^{{m-5}\over 2})$
$-\zeta_{p}^{2}p^{{m+3}\over 2}$ | $n_{-1,3}\cdot(p^{m-4}+p^{{m-5}\over 2})$ | $\zeta_{p}^{2}p^{{m+3}\over 2}$ | $n_{1,3}\cdot(p^{m-4}-p^{{m-5}\over 2})$
r=m-4
$\Delta=1$ | $\Delta=2$
$S^{\prime}(\alpha,\beta,\gamma,\delta)$ | multiplicity | $S^{\prime}(\alpha,\beta,\gamma,\delta)$ | multiplicity
0 | $n_{4}(p^{m}-p^{m-4})$ | 0 | $n_{4}(p^{m}-p^{m-4})$
$ip^{{m+4}\over 2}$ | $n_{4}\cdot p^{m-5}$ | $-ip^{{m+4}\over 2}$ | $n_{4}\cdot p^{m-5}$
$i\zeta_{p}p^{{m+4}\over 2}$ | $n_{4}\cdot(p^{m-5}-p^{{m-5}\over 2})$ | $-i\zeta_{p}p^{{m+4}\over 2}$ | $n_{4}\cdot(p^{m-5}+p^{{m-5}\over 2})$
$i\zeta_{p}^{2}p^{{m+4}\over 2}$ | $n_{4}\cdot(p^{m-5}+p^{{m-5}\over 2})$ | $-i\zeta_{p}^{2}p^{{m+4}\over 2}$ | $n_{4}\cdot(p^{m-5}-p^{{m-5}\over 2})$
###### Proof
In the following, there are three parts for the proof: Part I is for general
analysis and the case where (28) is not solvable; Part II gives the counting
formula for the times that $S^{\prime}(\alpha,\beta,\gamma,\delta)$ takes each
possible value according to the rank of corresponding quadratic form; Part III
presents the calculation of the counting formula.
Part I: Lemma 1 implies that the value of exponential sum
$S^{\prime}(\alpha,\beta,\gamma,\delta)$ (27) is
$\zeta_{p}^{c}S(\alpha,\beta,\gamma)$
where $c={1\over 2}AB^{T}\in\mathbb{F}_{p}$. And the value is $0$ if
$2YH+A=0$ (28)
does not have a solution $Y=B\in\mathbb{F}_{p}^{m}$. For more details of (28),
see follows.
Denote by $M\in\mbox{GL}_{m}(\mathbb{F}_{p})$ such that
$MHM^{T}=H^{\prime}=\mbox{diag}(h_{1},h_{2},\ldots,h_{r},0,0,\ldots,0)$
where $h_{i}\in\mathbb{F}_{p}^{*}$. Let $Y^{\prime}=2YM^{-1}$ and
$A^{\prime}=-AM^{T}$, then (28) is equivalent to
$Y^{\prime}H^{\prime}=A^{\prime}.$ (29)
Since $A$ varies over the elements of $\mathbb{F}_{p}^{m}$ and $M$ is
nonsingular, $A^{\prime}$ also varies over the elements of
$\mathbb{F}_{p}^{m}$. And if solvable, it can be checked that
$\begin{array}[]{ll}Y^{\prime}A^{\prime
T}&=2YM^{-1}\cdot\left(-AM^{T}\right)^{T}\\\
&=2YM^{-1}\cdot\left(-MA^{T}\right)=YA^{T}\\\ &=-\left({1\over
2}AB^{T}\right)\\\ &=-c\end{array}$ (30)
where $B=-Y^{\prime}M$ is a solution of equation (28). Note that equation (30)
can also be written as follows
$\begin{array}[]{ll}Y^{\prime}A^{\prime T}&=Y^{\prime}H^{\prime}Y^{\prime
T}\\\ &=h_{1}y_{1}^{\prime 2}+h_{2}y_{2}^{\prime 2}+\cdots+h_{r}y_{r}^{\prime
2}\\\ &=-c\end{array}$ (31)
where $y_{1}^{\prime},y_{2}^{\prime},\ldots,y_{r}^{\prime}$ are the elements
of the first $r$ coordinates of the vector $Y^{\prime}\in\mathbb{F}_{p}^{m}$.
It can be checked that equation (29) is solvable only when the elements of the
last $m-r$ coordinates of $A^{\prime}$ are $0$, and in this case for any such
$A^{\prime}$ the number of solutions of equation (29) is $p^{m-r}$. The number
of such vectors $A^{\prime}$ is $p^{r}$. So
$S^{\prime}(\alpha,\beta,\gamma,\delta)$ is zero for
$p^{m}-p^{r}$ (32)
vectors $A^{\prime}\in\mathbb{F}_{p}^{m}$ when equation (29) is not solvable.
Part II: In addition to (32), let’s consider the case where
$S^{\prime}(\alpha,\beta,\gamma,\delta)$ is not zero. From Lemma 1, we find
that
$S^{\prime}(\alpha,\beta,\gamma,\delta)=\zeta_{p}^{c}S(\alpha,\beta,\gamma)$
with $c$ as explained above by equations (30) and (31). There are two cases to
be considered.
1. $($i$)$
The rank $r$ of the corresponding quadratic form is even.
* *
For the case of $c=0$, equation (31) becomes
$h_{1}y_{1}^{\prime 2}+h_{2}y_{2}^{\prime 2}+\cdots+h_{r}y_{r}^{\prime 2}=0.$
By Lemma 6 the number of solutions of the above equation is
$p^{r-1}+(p-1)p^{{r-2}\over 2}\eta\left((-1)^{r\over 2}\Delta\right)$ (33)
where $\Delta=h_{1}^{\prime}h_{2}^{\prime}\cdots h_{r}^{\prime}$ and $\eta$ is
the quadratic character of $\mathbb{F}_{p}$. Note that, when
$y_{1}^{\prime},y_{2}^{\prime},\ldots,y_{r}^{\prime}$ are set, $A^{\prime}$ is
determined by equation (29) with
$Y^{\prime}=(y_{1}^{\prime},y_{2}^{\prime},\ldots,y_{r}^{\prime},0,0,\ldots,0)$,
and then $A$ can be calculated by $A^{\prime}=-AM^{T}$.
* *
For the cases of $c=1$ and $c=2$, the number of solutions is
$p^{r-1}-p^{{r-2}\over 2}\eta\left((-1)^{r\over 2}\Delta\right).$
2. $($ii$)$
The rank $r$ of the corresponding quadratic form is odd.
* *
For the case of $c=0$, equation (31) becomes
$h_{1}y_{1}^{\prime 2}+h_{2}y_{2}^{\prime 2}+\cdots+h_{r}y_{r}^{\prime 2}=0.$
By Lemma 7 the number of solutions of the above equation is
$p^{r-1}.$
* *
For the case of $c=1$, the number of solutions is
$p^{r-1}+p^{{r-1}\over 2}\eta\left((-1)^{{r-1}\over 2}(-1)\Delta\right).$ (34)
* *
For the case of $c=2$, the number of solutions is
$p^{r-1}+p^{{r-1}\over 2}\eta\left((-1)^{{r-1}\over 2}\Delta\right).$
Part III: This part considers the calculation of above counting formulas in
the two cases when $r=m$ and $r=m-1$ for odd and even ranks respectively.
1. $($i$)$
For the case of $r=m$, assume that $\Delta=1$, then
$S(\alpha,\beta,\gamma)=ip^{m\over 2}$ and the number of such quadratic forms
is $n_{0}$ (Table I).
* *
By equation (32), the number of times that
$S^{\prime}(\alpha,\beta,\gamma,\delta)=0$ is equal to
$n_{0}\left(p^{m}-p^{m}\right)=0.$
* *
The number of times that $S^{\prime}(\alpha,\beta,\gamma,\delta)=ip^{m\over
2}(c=0)$ is equal to
$n_{0}\cdot p^{m-1}.$
* *
The number of times that
$S^{\prime}(\alpha,\beta,\gamma,\delta)=\zeta_{p}\cdot ip^{m\over 2}(c=1)$ is
equal to
$n_{0}\cdot\left(p^{m-1}-p^{{m-1}\over 2}\right)$
where $\eta\left((-1)^{{m-1}\over 2}(-1)\Delta\right)=\eta(-1)=-1$ using 34.
* *
The number of times that
$S^{\prime}(\alpha,\beta,\gamma,\delta)=\zeta_{p}^{2}\cdot ip^{m\over 2}(c=2)$
is equal to
$n_{0}\cdot\left(p^{m-1}+p^{{m-1}\over 2}\right)$
where $\eta\left((-1)^{{m-1}\over 2}\Delta\right)=1$.
Similarly, the case of $\Delta=2$ can be analyzed.
2. $($ii$)$
Now, let’s consider the case of $r=m-1$, and assume that $\Delta=1$. In this
case $S(\alpha,\beta,\gamma)=p^{{m+1}\over 2}$, and the number of such
quadratic forms is $n_{1,1}$.
* *
By equation (32), the number of times that
$S^{\prime}(\alpha,\beta,\gamma,\delta)=0$ is equal to
$n_{1,1}\cdot\left(p^{m}-p^{m-1}\right).$
* *
The number of times that $S^{\prime}(\alpha,\beta,\gamma,\delta)=p^{{m+1}\over
2}(c=0)$ is equal to
$n_{1,1}\cdot\left(p^{m-2}+(p-1)p^{{m-3}\over 2}\right),$
noting that $\eta((-1)^{{m-1}\over 2}\Delta)=1$ using 33.
* *
The number of times that
$S^{\prime}(\alpha,\beta,\gamma,\delta)=\zeta_{p}p^{{m+1}\over 2}(c=1)$ or
$\zeta_{p}^{2}p^{{m+1}\over 2}(c=2)$ is equal to
$n_{1,1}\cdot\left(p^{m-2}-p^{{m-3}\over 2}\right).$
The case of $\Delta=2$ can also be investigated in this way.
Using similar ideas on the other cases of $r$, the lemma is obtained. ∎
The weight distribution of the cyclic code $\mathcal{C}_{2}$ is obtained in
Theorem 4.1 by analyzing exponential sum
$R^{\prime}(\alpha,\beta,\gamma,\delta)$ from Lemma 18 and Remark 1.
###### Theorem 4.1
Let $p=3$ and $q=p^{m}$ where $m\equiv 1\ \mbox{mod}\ 4$ satisfying $3\nmid
m$. The multiplicities of the exponential sums
$R^{\prime}(\alpha,\beta,\gamma,\delta)$ and the weight distribution of the
cyclic code $\mathcal{C}_{2}$ are listed in Table III.
Table III: $R^{\prime}(\alpha,\beta,\gamma,\delta)$ | weight | multiplicity
---|---|---
| | $2n_{0}p^{m-1}+(n_{-1,1}+n_{1,1})(p^{m}-p^{m-1})+2n_{2}(p^{m}-2p^{m-3})$
0 | $p^{m-1}(p-1)$ | $+(n_{-1,3}+n_{1,3})(p^{m}-p^{m-3})+2n_{4}(p^{m}-2p^{m-5})+p^{m}-1$
$-p^{{m+1}\over 2}$ | $p^{m-1}(p-1)+p^{{m-1}\over 2}$ | $2n_{0}(p^{m-1}-p^{{m-1}\over 2})+2n_{1,1}(p^{m-2}-p^{{m-3}\over 2})$
$p^{{m+1}\over 2}$ | $p^{m-1}(p-1)-p^{{m-1}\over 2}$ | $2n_{0}(p^{m-1}+p^{{m-1}\over 2})+2n_{-1,1}(p^{m-2}+p^{{m-3}\over 2})$
$p^{{m+3}\over 2}$ | $p^{m-1}(p-1)-p^{{m+1}\over 2}$ | $2n_{2}(p^{m-3}+p^{{m-3}\over 2})+2n_{-1,3}(p^{m-4}+p^{{m-5}\over 2})$
$-p^{{m+3}\over 2}$ | $p^{m-1}(p-1)+p^{{m+1}\over 2}$ | $2n_{2}(p^{m-3}-p^{{m-3}\over 2})+2n_{1,3}(p^{m-4}-p^{{m-5}\over 2})$
$(p-1)p^{{m+1}\over 2}$ | $p^{m-1}(p-1)-(p-1)p^{{m-1}\over 2}$ | $n_{1,1}(p^{m-2}+(p-1)p^{{m-3}\over 2})$
$-(p-1)p^{{m+1}\over 2}$ | $p^{m-1}(p-1)+(p-1)p^{{m-1}\over 2}$ | $n_{-1,1}(p^{m-2}-(p-1)p^{{m-3}\over 2})$
$(p-1)p^{{m+3}\over 2}$ | $p^{m-1}(p-1)-(p-1)p^{{m+1}\over 2}$ | $n_{1,3}(p^{m-4}+(p-1)p^{{m-5}\over 2})$
$-(p-1)p^{{m+3}\over 2}$ | $p^{m-1}(p-1)+(p-1)p^{{m+1}\over 2}$ | $n_{-1,3}(p^{m-4}-(p-1)p^{{m-5}\over 2})$
$-p^{{m+5}\over 2}$ | $p^{m-1}(p-1)+p^{{m+3}\over 2}$ | $2n_{4}(p^{m-5}-p^{{m-5}\over 2})$
$p^{{m+5}\over 2}$ | $p^{m-1}(p-1)-p^{{m+3}\over 2}$ | $2n_{4}(p^{m-5}+p^{{m-5}\over 2})$
###### Example 2
Let $m=5,p=3$ and $q=p^{m}$. Let $\mathcal{C}_{2}$ be the cyclic code with
nonzeros $\pi^{-1},\pi^{-(p^{0}+1)}=\pi^{-2}$, $\pi^{-(p^{1}+1)}=\pi^{-4}$ and
$\pi^{-(p^{2}+1)}=\pi^{-10}$ where $\pi$ is a primitive element of the finite
field $\mathbb{F}_{q}$. Using Matlab, $\mathcal{C}_{2}$ has weight
distribution
$\begin{array}[]{l}A_{162}=1618713316,A_{171}=782825472,A_{153}=947952720,A_{135}=6853440,\\\
A_{189}=3455760,A_{144}=84092580,A_{180}=42810768,A_{108}=72600,A_{216}=7260,\\\
A_{81}=484,\end{array}$
which verifies the result of Theorem 4.1.
## 5 Conclusions
In this paper, we describe the weight distributions of some cyclic codes: the
codes $\mathcal{C}_{1}$ with nonzeros $\pi^{-2},\pi^{-(p+1)}$,
$\pi^{-(p^{2}+1)}$ has five nonzero weights, and the code $\mathcal{C}_{2}$
with nonzeros $\pi^{-1},\pi^{-2},\pi^{-4},\pi^{-10}$ has ten nonzero weights
respectively, where $p=3$, $q=p^{m}$ and $m$ is an odd integer satisfying
$3\nmid m$. As can be expected, in general the weight formulas of cyclic codes
are rather complicated.
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|
arxiv-papers
| 2013-02-02T15:57:49 |
2024-09-04T02:49:41.224170
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiaogang Liu and Yuan Luo",
"submitter": "Xiaogang Liu",
"url": "https://arxiv.org/abs/1302.0394"
}
|
1302.0438
|
1Navy Air Warfare Center Aircraft Division (NAWCAD), Joint Base McGuire-Dix-
Lakehurst, Lakehurst NJ 08733, USA 2Department of Mechanical Engineering,
Columbia University in the City of New York, New York NY 10027, USA 3Tech-
Physical Research Center Science College, National University of Defense
Technology, Changsha, Hunan 410073, PRC ∗Corresponding author:
[email protected]
# Soliton Propagation with Cross Phase Modulation in Silicon Photonic Crystal
Waveguides
Matthew Marko,1,2,∗ Xiujian Li,2,3 and Jiangjun Zheng2
###### Abstract
An effort was conducted to numerically determine, using the Nonlinear
Schrödinger Split-Step Fourier method, if using cross phase modulation could
cause temporal soliton pulse propagation in a silicon slow-light photonic
crystal waveguide shorter than a millimeter. The simulations demonstrated that
due to the higher powers and shorter scales of photonic crystals, two-photon
absorption would cause an optical soliton pulse to be extremely dissipative.
The model demonstrated, however, that by utilizing cross-phase modulation, it
is possible to sustain a compressed soliton pulse within a silicon photonic
crystal waveguide subjected to two-photon absorption over longer relative
distances.
350.4238, 190.4720.
## 1 Introduction
Optical solitons are undistorted standing waves that result as part of the
interplay between the nonlinear Kerr effect and anomalous dispersion [1, 2].
One method of studying solitons numerically is to use the Split-Step Fourier
Nonlinear Schrödinger Equation (NLSE) [1, 2, 3, 4]. The NLSE is one of the
most effective methods for numerically analyzing soliton pulse propagation in
a nonlinear dispersive waveguide.
Photonic crystal waveguides (PhCWG) have extraordinary nonlinear effects [5],
where measurements of the dispersion have shown an increase of five orders of
magnitude in the Group Velocity Dispersion (GVD) compared to the material
dispersion [7, 6]. PhCWG, with their vastly increased nonlinear effects, are
an ideal medium for reducing the nonlinear length scales from meters and
kilometers typical of fibers, to sub-millimeter PhCWG. A soliton is formed by
the interplay of anomalous GVD, as well as the 3rd-order Kerr nonlinear
optical effect. When the pulse is focused by its own intensity, it is known as
Self Phase Modulation (SPM); whereas when it is focused by a second pulse, it
is referred to as Cross Phase Modulation (XPM) [1, 2]. In addition to large
GVD, PhCWG’s can be fabricated out of highly 3rd-order-nonlinear dielectrics,
dramatically reducing nonlinear length scales.
The first such experimental measurement of soliton pulse compression in a
PhCWG was conducted using Gallium Indium Phosphate (GaInP) PhCWG [7], which
have the advantage of not having intensity-dependent two-photon absorption
(TPA) and TPA-induced free-carrier absorption (FCA) [8]. The applications of
this technology are numerous, as two-dimensional PhCWG may be fabricated into
optical waveguides on a device the size of a computer chip, to be utilized for
all-optical computing [9]. Optical computing has the advantage of
significantly greater bandwidth compared to current solid-state devices, with
reduced power and cooling, which can allow vastly improved computational
performance [10].
Another highly nonlinear dielectric material is silicon [11, 12], which may
also potentially be used to fabricate chip-scale optical waveguides. Silicon
is advantageous, as it is cheap, durable, readily available, and there already
exists an infrastructure to mass-produce silicon devices. Pulse compression
was previously observed experimentally in novel silicon nanowires [13, 14],
but previous analytical and numerical studies have demonstrated that with SPM
alone, TPA and FCA will prevent a pulse from self-focusing itself sufficiently
to generate a soliton-like wave [8]. Even if the soliton pulse were to
compress, it would simply dissipate after propagating relatively short
nonlinear lengths. One of the advantages of XPM is that a pulse can be focused
without needing to be highly energetic and intense [1]; therefore there was
interest in seeing if a second pump pulse could be used to focus the primary
signal pulse, to induce the soliton. If XPM can be used to overcome the
limitations of TPA and FCA and enable chip-scale soliton propagation in
silicon, it may very well develop into practical all-optical computing.
## 2 Nonlinear Schrödinger Equation
In order to accurately model the optical soliton, two crucial nonlinear
characteristics are necessary, the GVD and the nonlinear Kerr effect [1]. The
GVD is determined from the derivatives of the refractive index over the
angular frequency. The other component of soliton pulse propagation is the
nonlinear Kerr effect, which results in a change of refractive index
proportional to the square of the electric field, or proportional to the
optical intensity. When there is strong SPM and/or XPM interplaying with
strong anomalous GVD, the SPM/XPM will cause the leading edge frequencies to
be lowered, and the trailing edge frequencies to be raised. At the same time,
the anomalous GVD will cause the lowered leading edge frequencies to slow
down, and the raised trailing edge frequencies to speed up. This results in
the pulse compressing temporally into a soliton pulse [1, 2].
The GVD is determined by knowing the derivatives of the refractive index over
the angular frequency. These terms are found using the following equations [1,
2, 4]:
$\displaystyle\beta(\omega)$ $\displaystyle=$ $\displaystyle
n(\omega)\frac{\omega}{c_{0}},$ (1) $\displaystyle{\beta_{M}(\omega)}$
$\displaystyle=$ $\displaystyle\frac{d^{M}\beta(\omega)}{d\omega^{M}},$ (2)
$\displaystyle\beta_{1}(\omega)$ $\displaystyle=$
$\displaystyle\frac{1}{v_{G}}=\frac{n_{G}}{c_{0}}=\frac{1}{c_{0}}(n(\omega)+\omega\frac{dn(\omega)}{d\omega}),$
(3) $\displaystyle\beta_{3}(\omega)$ $\displaystyle=$
$\displaystyle\frac{d{\beta_{2}(\omega)}}{d\omega}=\frac{d^{2}{\beta_{1}(\omega)}}{d\omega^{2}}=\frac{d^{3}{\beta(\omega)}}{d\omega^{3}},$
(4)
where $n(\omega)$ is the refractive index as a function of angular velocity,
$\omega$ ($\mathrm{{rad}/{s}}$) is the angular frequency, $c_{0}$ (m/s) is the
speed of light, $v_{G}$ (m/s) is the group velocity, and $n_{G}$ is the group
index. These values can be determined either experimentally [11, 15],
approximated with the Sellmeier equation [1, 2], or engineered to specific
values with geometric dispersion, as is the case of photonic crystals [5, 6].
The other component of generating soliton pulse propagation and compression is
the nonlinear Kerr effect. The change in refractive index is determined by:
$\displaystyle\Delta n$ $\displaystyle=$
$\displaystyle{n_{2}}{\cdot}({I_{1}}+{2{\cdot}{I_{2}}}),$ (5) $\displaystyle
I$ $\displaystyle=$ $\displaystyle\frac{{|}{A}{|}^{2}}{2{(\eta_{0}/n)}},$ (6)
where $n_{2}$ ($\mathrm{{m^{2}}/{W}}$) is the nonlinear Kerr coefficient,
$\eta_{0}$ is the vacuum impedance ($\approx{120{\pi}}$ $\Omega$), n is the
refractive index, _A_ (V/m) is the electric field, and $I_{1}$ and $I_{2}$
($\mathrm{{W}/{m^{2}}}$) are, respectively, the intensities of the pulse and
another coupled pulse. This nonlinear term can be determined from the 3rd-
order nonlinear susceptibility, which is also responsible for four-wave mixing
[2, 16]:
$\displaystyle{n_{2}}$ $\displaystyle=$
$\displaystyle{3{\eta_{0}}{\cdot}}{\frac{{\chi}^{(3)}}{{{\varepsilon}_{0}}{\cdot}n}},$
(7)
where $\epsilon_{0}$ is the electric permittivity in a vacuum (8.854 $\cdot$
10-12 F/m), and $\chi^{(3)}$ $\mathrm{({{m^{2}}/{V^{2}}})}$ is the
$\mathrm{3^{rd}}$-order nonlinear susceptibility coefficient [2]. Throughout
this study, a Kerr nonlinearity of $\mathrm{4{\cdot}{10^{-18}}{\hskip
5.69054pt}{m^{2}}/{W}}$ [11, 6, 17] is used. The nonlinear phase shift is thus
[1]:
$\displaystyle{\Delta}{{\phi}_{NL}}$ $\displaystyle=$
$\displaystyle{n_{2}}{\cdot}{k_{0}}{\cdot}L{\cdot}({I_{1}}+{2{\cdot}{I_{2}}})={n_{2}}{\cdot}{\frac{2{\pi}}{\lambda}}{\cdot}L{\cdot}({I_{1}}+{2{\cdot}{I_{2}}}),$
(8)
where $k_{0}$ (m-1) is the optical wave-number, ${\lambda}$ (m) is the
wavelength, and _L_ (m) is the optical propagation length. These two nonlinear
optical effects can be combined to form the NLSE, which in its complete form
is represented as:
$\displaystyle\frac{\partial A}{{\partial
z}}={\frac{i}{2}{\beta_{2}}{\frac{\partial A^{2}}{{\partial
t^{2}}}}}+{\frac{i}{6}{\beta_{3}}{\frac{\partial A^{3}}{{\partial
t^{3}}}}}+{{i}{\gamma{|A|^{2}}A{\cdot}{exp(-\alpha z)}}},$ (9)
$\displaystyle\gamma={\frac{2{\pi}{n_{2}}}{A_{e}{\cdot}{\lambda}}}{\cdot}{(\frac{n_{g}}{n})^{2}},$
(10)
where $\beta_{2}$ ($\mathrm{s^{2}/m}$) is the $\mathrm{2^{nd}}$-order GVD
coefficient, $\beta_{3}$ ($\mathrm{s^{3}/m}$) is the $\mathrm{3^{rd}}$-order
GVD coefficient, $A_{e}$ $\mathrm{(m^{2})}$ is the effective cross-sectional
area of the waveguide, $\gamma$ $\mathrm{(m{\cdot}W)^{-1}}$ is the nonlinear
parameter, $\alpha$ ($\mathrm{m^{-1}}$) is the linear loss coefficient, _i_ =
$\sqrt{-1}$, and _A_ (V/m) is the pulse electric field envelope function.
The NLSE is also known as the Split-Step Fourier method because each
incremental distance step involves two fundamental steps. The first step is to
account for the GVD. A code will convert the pulse information into the
spectral domain by using the Fast Fourier Transforms (FFT) method [1, 4, 18,
19], apply the GVD, and then convert the pulse back from the spectral domain
to the temporal domain to calculate the Kerr nonlinearity. The equation for
the GVD is as follows [1, 4]:
$\displaystyle{\bar{A}}(z,\omega)$ $\displaystyle=$
$\displaystyle{{\bar{A}}(0,\omega)}{\cdot}exp[({\frac{i}{2}}{\cdot}{{\beta}_{2}}{\cdot}{{\omega}^{2}}{\cdot}z)+({\frac{i}{6}}{\cdot}{{\beta}_{3}}{\cdot}{{\omega}^{3}}{\cdot}z)+{...}],$
(11) $\displaystyle{\bar{A}}(z,\omega)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}{{A}(z,T)}{\cdot}{{exp}[-2{\pi}{i}{\omega}{T}]}\,\mathrm{d}T,$
(12) $\displaystyle T$ $\displaystyle=$ $\displaystyle t-{\frac{z}{v_{g}}},$
(13)
where $\emph{A}(z,T)$ and $\bar{A}(z,\omega)$ are the pulse envelope functions
in the temporal and spectral domains, T is the normalized time, $v_{g}$ is the
group velocity, z is the propagation distance, and $\omega$ is the normalized
angular frequency. The second step is to take the pulse, now in the temporal
domain, and apply the effects of SPM/XPM. The equation for the nonlinear phase
shift is [1, 4]:
$\displaystyle{{A_{1}}(z,T)}$ $\displaystyle=$
$\displaystyle{{{A_{1}}}(0,T)}{\cdot}exp[i{\cdot}{n_{2}}{\cdot}{\frac{2{\pi}}{\lambda}}{\cdot}L{\cdot}{\frac{({{|}{A_{1}(0,T)}{|}^{2}}+{2{\cdot}{{|}{A_{2}(0,T)}{|}^{2}}})}{2({\eta_{0}}/{n})}}],$
(14)
After completely these two steps, one can apply the effects of linear and
nonlinear loss for the specified distance increment. By following this
algorithm, soliton pulse propagation can be simulated and studied.
## 3 Two Photon Absorption
One big challenge to soliton pulse propgation in silicon is the nonlinear
effect of multi-photon absorption [8]. Dielectric materials have an electronic
band gap, where photons with energy greater than the electronic band gap get
absorbed by the material [20]. The electronic band gap for silicon is 1.1
electron-volts (eV) [7, 8]. As it is desired to use light pulses at a
wavelength of 1550 nm, this corresponds to a photon energy of 0.8 eV, and thus
silicon is transparent at this wavelength. Multi-photon absorption, however,
is the result of multiple photons interacting, and their combined energy
exceeding the electronic band gap. For example, at 1550 nm, two photons will
result in a total energy of 0.8$\times$2=1.6 eV, which exceeds the 1.1 eV
electronic band gap of silicon. Therefore, silicon is subject to TPA at 1550
nm.
In order to calculate the impacts of TPA on silicon, one must find the
coefficient of TPA at the given wavelength by the following equations [21]:
$\displaystyle{{\beta}_{TPA}}(\omega)$ $\displaystyle=$ $\displaystyle
K{\cdot}{\sqrt{\frac{EP}{{{n_{0}}^{2}}{\cdot}{{E_{g}}^{3}}}}}{\cdot}{F_{2}}(\frac{2{\pi}{\hbar}{\cdot}{\nu}}{E_{g}}),$
(15) $\displaystyle{F_{2}}(x)$ $\displaystyle=$
$\displaystyle\frac{{({{2x}-1})}^{\frac{3}{2}}}{{(2x)}^{5}},$ (16)
$\displaystyle{2{\pi}{\hbar}{\cdot}{\nu}}$ $\displaystyle=$
$\displaystyle{\frac{1240}{\lambda(nm)}}{\indent}(eV),$ (17)
where EP is equal to 21, $\hbar$ is Plank’s constant over $2{\pi}$, $\nu$ is
the frequency, $n_{0}$ is the refractive index, and $E_{g}$ is equal to the
1.1 eV electronic band-gap of silicon [7, 8, 12]. The constant K was
calculated based on the experimentally measured value of
$4.4{\cdot}{10^{-12}}$ (m/W) at 1550 nm [22]. With the value of
${\beta}_{TPA}$, the TPA can be simulated with [12, 23]:
$\displaystyle{I(z,t)}$ $\displaystyle=$
$\displaystyle{I(0,t)}{\cdot}{exp[{-}{{\beta}_{TPA}}{\cdot}{I(0,t)}{\cdot}{z}]},$
(18)
Once the two-photon absorption coefficient is determined, one can determine
the impact from the free carriers generated by the TPA. For ultrashort pulses,
where the pulse is substantially shorter than the free-carrier lifetime, the
FCA tends to be substantially less than the effects of TPA [12]. The model
calculates the FCA at each distance step separately, assuming that there were
no free-carriers there at the beginning. This is an approximation, as carriers
will in fact drift over to the different distance steps, but this is difficult
to accurately predict. For this reason, the assumption that carriers stay
localized will be used. After the SPM/XPM and dispersion was applied to the
pulse, the model predicts the rate of free-carrier generation from the two-
photon absorption [22, 12, 24, 23],
$\displaystyle\frac{dN(t)}{dt}$ $\displaystyle=$ $\displaystyle
G-{\frac{N(t)}{{\tau}_{FCA}}},$ (19) $\displaystyle G$ $\displaystyle=$
$\displaystyle{\frac{{{\beta}_{TPA}}{\cdot}{I^{2}}}{4{\pi}{\hbar}{\nu}}},$
(20)
where ${\beta}_{TPA}$ (m/W) is the coefficient of two photon absorption, I
($\mathrm{W/m^{2}}$) is the optical intensity (either from the signal or
coupled pulse), $\nu$ is the frequency, ${\hbar}$ is Plank’s Constant over
2$\pi$, G is the rate of free-carrier generation, ${\tau}_{TPA}$ (s) is the
free carrier lifetime, and N(t) is the free-carrier density. Once the rate of
free-carrier generation over time has been determined at the specific distance
step, the next step is to calculate the density of free-carriers. While the
simulation goes through the process of generating carriers over time, it also
reduces carriers based on the specified free-carrier lifetime. After the
carrier density is determined as a function of time (at the given distance
increment), the appropriate free-carrier absorption is applied to the pulse in
the temporal domain. The equation is as follows [12, 23]:
$\displaystyle
I(z)={I(0)}{\cdot}exp[{-{\frac{1}{2}}}{{\cdot}(1.45{\cdot}{10^{-21}}){{\cdot}N(t){\cdot}z}}],$
(21)
By implementing these equations within the NLSE simulation, an accurate
analysis of soliton pulse propagation in a silicon waveguide subjected to TPA
and FCA can be achieved.
## 4 Simulations of Cross Phase Modulation
The NLSE model that will be used throughout this study was first set to a
single pulse with SPM. First, the model propagated a 10 pJ hyperbolic secant
squared pulse down a 10 cm silicon nanowire; the results of this simulation
are shown in Figure 1-a. A compression ratio factor of 7 was clearly observed.
The model then scaled the pulse energy and waveguide length to match the
soliton number [7] and nonlinear length [1] as the nanowire study, using the
following relations:
$\displaystyle
L^{\prime}={L}{\cdot}{|{\frac{{\beta}_{2}}{{\beta}^{\prime}_{2}}}|},$ (22)
$\displaystyle
E^{\prime}={E}{\cdot}{|{\frac{{\beta}^{\prime}_{2}}{{\beta}_{2}}}|}{\cdot}{[{\frac{A^{\prime}_{e}}{A_{e}}}]}{\cdot}{[{\frac{{n_{G}}{\cdot}{n^{\prime}}}{{n^{\prime}_{G}}{\cdot}{n}}}]},$
(23)
where L (m) is the waveguide length, E (J) is the input pulse energy,
${\beta_{2}}$ $\mathrm{(s^{2}/m)}$ is the 2nd-order GVD coefficient, ${A_{e}}$
$\mathrm{(m^{2})}$ is the waveguide effective cross-sectional area, and
$n_{G}$ and n are the group and refractive indices, respectively. Based on
these relations, it is clear that by using a PhCWG, the soliton length scales
can become reduced dramatically. The caveat is that it is necessary to use
higher pulse-energies, which in turn have greater TPA and FCA, which attenuate
the soliton, and disturb the pulse, as observed in Figure 1b. While
compression has been observed at the early onset of soliton propagation within
the PhCWG, these compressed pulses simply cannot sustain themselves in the
presence of intense TPA and FCA.
[width=]Figure1a.jpg
[width=]Figure1b.jpg
Figure 1: SPM-induced soliton propagation, in a (a) silicon nanowire and a (b)
silicon PhCWG.
The NLSE model was modified to study both XPM as well as the effects of cross-
TPA (XTPA) and the resulting FCA [12]. In this XPM simulation the process is
completed twice, first for the primary pulse, and second for the secondary
pulse. It uses a modified split-step, since there are two coupled pulses, to
take into account differences in the group velocity which result from the
different wavelengths between the pump and the signal.
Part of the modified aspect is a loop for the nonlinear effects that require
considering both pulses, the two include XPM, XTPA, and FCA induced from XTPA.
The split-step calculations are used in a normalized time domain [1], however,
with different wavelengths and thus different group velocities, this
normalized time is not always applicable. In fact, one concern with XPM is the
case of walk-off where a faster traveling pulse fully clears away from the
slower pulse within the waveguide [25]. A time shift factor is determined,
based on the difference in group velocities, defined as ${\delta}$T, where
$\displaystyle{\delta}T$ $\displaystyle=$
$\displaystyle{({n^{\prime}_{G}}-{n_{G}})}{\cdot}{(\frac{{\delta}z}{{\delta}t{\cdot}{c_{0}}})},$
(24)
where ${\delta}z$ is the distance increment, ${\delta}t$ is the time step used
in the simulation, and $c_{0}$ is the speed of light in a vacuum. The value of
${\delta}T$ is rounded down, as it is an integer number of time steps to
shift, and added to calculate the effects of pulse 2 onto pulse 1 (and
subtracted for pulse 1 onto pulse 2).
## 5 Soliton Propagation with Cross Phase Modulation
This model was run using the parameters experimentally observed in silicon
PhCWG [6], with a negligibly low pulse energy (to avoid SPM) of 10 fJ pulse
for the signal pulse near the highly-dispersive photonic band-gap wavelength
at 1544 nm, coupled temporally with an intense 105 pJ pump pulse at 1536 nm.
In this particular example, the GVD at the signal wavelength was a highly
dispersive anomalous -4 ps2/mm, whereas the pump wavelength had a normal GVD
coefficient of 100 ps2/m. The signal was also set to a third-order GVD
coefficient of 0.1 ps3/mm, a value typical of these engineered photonic
crystal waveguides [7]. While higher-order dispersion is known to affect
soliton propagation, especially at wavelengths very close to the highly
dispersive photonic crystal band-gap [26], there was negligible difference in
outcomes for the practical lengths modeled in this study. Finally, a linear
loss of 2.4 dB/mm was added to the simulation.
By the phenomena of XPM, temporal compression was observed; the signal pulse
underwent compression by a factor of 5 after propagating through 120
$\mathrm{\mu}$m of the PhCWG. This result is displayed both as a function of
normalized time and propagation distance in Figure 2-a; as well as a
comparison of the normalized temporal intensity function in Figure 2-b.
Finally, the pump pulse did not experience any compression, as it was
subjected to much less GVD; this is demonstrated in Figure 2-c. As a result of
this, the simulation clearly demonstrates that the compression is the result
of XPM, as no compression was observed when the same 10 fJ signal pulse
propagates through the waveguide without the pump pulse.
[width=]Figure2a.jpg
[width=]Figure2b.jpg
[width=]Figure2c.jpg
Figure 2: XPM induced compression: (a) is the signal pulse compression as
function of waveguide propagation distance and time; (b) is the waveguide
output of weak signal pulse under XPM; and (c) is the waveguide output of weak
signal pulse without XPM, thus proving the compression is the result of XPM.
When running the simulation, a numerical scale was developed to represent XPM-
induced solitons. Much like a SPM-soliton [7, 8], the XPM-soliton number N is
a ratio of the fundamental soliton energy:
$\displaystyle{N}$ $\displaystyle=$
$\displaystyle{\sqrt{\frac{E_{P}}{E_{f}}}},$ (25) $\displaystyle{E_{f}}$
$\displaystyle=$
$\displaystyle{{\frac{2|{\beta_{2}}|}{|{\gamma}|{\cdot}{\tau}}}},$ (26)
where ${\beta}_{2}$ $\mathrm{(s^{2}/m)}$ is the 2nd-order GVD coefficient,
${\gamma}$ $\mathrm{(W{\cdot}m)^{-1}}$ is the nonlinear Kerr coefficient,
${\tau}$ (s) is the temporal duration of the signal pulse in seconds, and
$E_{p}$ (J) is the pump pulse energy. The pulse compression ratio requires
higher XPM soliton numbers than with SPM; this is expected due to the effect
of pulse walk-off, which becomes more profound at longer waveguide lengths. At
the practical length scale of 0.1 nonlinear lengths (similar length studied
with SPM), which equated to 0.4 mm (a typical photonic crystal waveguide [11,
7, 6, 8]), a pulse compression of nearly 3 was observed from an XPM-soliton
number of 15; the results of these studies can be seen in Figure 3-a. The Time
Bandwidth Product (TBP) was tabulated and is represented by Figure 3-b. Under
the practical conditions mentioned earlier, the TBP remained relatively
unchanged; evidence of XPM-induced soliton pulse propagation. At longer length
scales, the pulse eventually underwent broadening and the TBP increased; this
can be attributed to TPB attenuating the pump pulse, as well as pulse walk-
off. The nonlinear loss (in dB) was also observed in figure 3-c. In all cases,
the longer the nonlinear propagation length, the more loss was observed; this
is to be expected. It was interesting to note, that in shorter propagation
lengths, higher XPM-Soliton numbers had an ability to actually _increase_ the
peak energy as the signal pulse was compressed by XPM. In all cases studied,
the optical pulses were all by-definition dissipative solitons; it was
demonstrated that XPM can sustain the pulse over much longer propagation
lengths scales than SPM alone.
[width=]Figure3a.jpg
[width=]Figure3b.jpg
[width=]Figure3c.jpg
Figure 3: Results of numerical XPM studies: (a) the signal pulse compression
factor, (b) the output signal pulse TBP, and (c) the nonlinear peak intensity
loss (dB).
## 6 Practical Applications of Cross Phase Modulation
The previous study was on the assumption that both the pump and signal
wavelengths were subjected to TPA and FCA; this may be practically realized by
utilizing a spectrally broad pulse and narrow bandpass filters [25]. While
this may be simple to implement, it has the limitation of significant power
attenuation from the filters. Another potential option to generate optical
pulses of different wavelengths coupled temporally and spatially is to use an
Optical Parametric Oscillator (OPO), where a broad range of wavelengths may be
generated by adjusting the length of the resonating cavity [27, 28]. In an
OPO, signal and idler pulses are generated with a combined wavelength energy
equal to the pump wavelength energy:
$\displaystyle\frac{1}{{\lambda}_{pump}}={\frac{1}{{\lambda}_{signal}}}+{\frac{1}{{\lambda}_{idler}}}.$
(27)
By utilizing this technology, one may realistically generate pulses coupled
temporally and spectrally, where the pump pulse (not to be confused with the
pump pulse for the OPO) has a wavelength not subjected to TPA within silicon.
In this next study, the authors investigate pulse compression resulting from a
low-powered signal pulse at 1550 nm; and a pump pulse at a wavelength of 2300
nm, which is not subjected to TPA due to silicon’s electronic band gap. This
can be achieved with a properly aligned OPO pumped 926 nm, a realistic
wavelength that can be achieved in a titanium sapphire laser [29].
[width=]Figure4a.jpg
[width=]Figure4b.jpg
[width=]Figure4c.jpg
Figure 4: Results of numerical XPM studies, with a low-energy C-band pulse
coupled with a high-energy 2300 nm _pump_ pulse: (a) signal pulse compression
as function of waveguide propagation distance and time; (b) numerical
predicted free-carrier density as a function of waveguide propagation distance
and time; and (c) waveguide output of weak signal pulse under XPM.
The study was conducted with the same dispersion and energy parameters as
before, but without the pump pulse subjected to TPA, and a compression ratio
over 5 was observed at 90 $\mathrm{\mu}$m of propagation. This is shorter than
the 120 $\mathrm{\mu}$m of propagation needed for the study in Figure 2; and
the results of this study can be observed both in Figure 4-a and Figure 4-b.
It was interesting to observe that the generation of free carriers, as shown
in Figure 4-c, actually increased as the signal pulse, which is subjected to
TPA, compressed and increased in temporal intensity due to the XPM from the
pump pulse. In addition, it was observed that the compression remained
consistent for even the longer propagation studies. This study was repeated
for a host of different XPM Soliton Numbers (Figure 5-a) and nonlinear length
ratios (Figure 5-b), and the nonlinear losses were observed (Figure 5-c).
These nonlinear losses were very close to the previous non-OPO study (Figure
3-c). The data clearly shows that by utilizing a substantially longer
wavelength pulse, practical soliton pulse propagation in silicon can be both
achieved and sustained for low-energy pulses at wavelengths subjected to
intensity-dependent TPA.
[width=]Figure5a.jpg
[width=]Figure5b.jpg
[width=]Figure5c.jpg
Figure 5: Results of numerical XPM studies, with a low-energy C-band pulse
coupled with a high-energy 2300 nm _pump_ pulse: (a) the signal pulse
compression factor, (b) the output signal pulse TBP, and (c) the nonlinear
peak intensity loss (dB).
## 7 Conclusion
The simulations conducted have demonstrated the physical ability of soliton
pulse compression and propagation via XPM in a silicon PhCWG with TPA and FCA
present at both the pump and signal wavelengths. Use of a pump pulse and XPM
to focus the pulse enables the soliton phenomena to occur on pulses with much
lower intensities and thus much lower TPA. One challenge that was successfully
overcome was to develop a NLSE Split-Step Fourier algorithm that would take
into account the differences in group velocity and pulse walk-off that will
occur between two pulses of different wavelengths. The simulation demonstrated
that through the use of XPM, temporal soliton pulse propagation can be
realized over practical length scales with low-intensity ultrashort pulses in
a silicon PhCWG with TPA. This phenomenon might be practically used to sustain
a data pulse within a silicon waveguide, a capability with numerous potential
technological applications.
Sources of funding for this effort include Navy Air Systems Command
(NAVAIR)-4.0T Chief Technology Officer Organization as an Independent
Laboratory In-House Research (ILIR) Basic Research Project (Nonlinear Analysis
of Ultrafast Pulses with Modeling and Simulation and Experimentation); the
National Science Foundation (NSF), the National Science Foundation of China
(NSFC) award (61070040 and 60907003), and the Science Mathematics And Research
for Transformation (SMART) fellowship. The author’s thank Chee Wei Wong, Jiali
Liao, James McMillan, Tingyi Gu, Kishore Padmaraju, Noam Ofir, and the
laboratory of Karen Bergman for fruitful discussions.
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|
arxiv-papers
| 2013-02-03T00:10:40 |
2024-09-04T02:49:41.235748
|
{
"license": "Public Domain",
"authors": "Matthew Marko, Xiujian Li, and Jiangjun Zheng",
"submitter": "Matthew Marko PE",
"url": "https://arxiv.org/abs/1302.0438"
}
|
1302.0589
|
# Subgroups of a Subnormal Subgroup in Division Rings
Nguyen Anh Tu and Bui Xuan Hai111Corresponding author; e-mail address:
[email protected]
Faculty of Mathematics and Computer Science, University of Science, Vietnam
National University - Ho Chi Minh City (VNU-HCMC), 227 Nguyen Van Cu Str.,
Dist. 5, HCM-City, Vietnam
Abstract
Let $D$ be a division ring with center $F$ and the multiplicative group
$D^{*}$. Our main purpose in this paper is to investigate the subgroup
structure of an arbitrary subnormal subgroup $G$ of $D^{*}$. In particular, we
prove that if $D$ is locally finite, then $G$ contains a noncyclic free
subgroup. The structure of maximal subgroups of $G$ is also investigated. The
new obtained results carry over some previous results about maximal subgroups
of $D^{*}$ to maximal subgroups of an arbitrary subnormal subgroup $G$ of
$D^{*}$.
Key words: division ring, maximal subgroup, subnormal subgroup, noncyclic free
subgroup, $FC$-element.
Mathematics Subject Classification 2010: 16K20, 16K40.
## 1 Introduction
Let $D$ be a division ring with center $F$ and $D^{*}$ be the multiplicative
group of $D$. The subgroup structure of $D^{*}$ is one of subjects which
attract the attention of many authors for last time. A particular interest is
that to study maximal subgroups of $D^{*}$, see for example, [1], [2], [4],
[8], [14], [21]. In this paper we replace $D^{*}$ by its arbitrary subnormal
subgroup $G$ and we study the subgroup structure of $G$. Recall that in [1],
[4], [21], Akbari et al. and Mahdavi-Hezavehi study maximal subgroups of
$D^{*}$ and many nice properties of such subgroups were obtained. In the
present paper, studying maximal subgroups of $G$, we get in many cases the
similar results for these subgroups as the results obtained in [1], [4], [21]
… for maximal subgroups of $D^{*}$. Other problem we study is the problem of
the existence of noncyclic free groups in a subnormal subgroups of a division
ring. This problem plays an important role for the understanding of the
structure of division rings. For more information we refer to the works
[5]-[7], [16]-[18], and [25]. Section 4 is devoted to this problem and the new
result we get in Theorem 4.4 gives the affirmative answer to Conjecture 2 from
[7] for the case of locally finite division rings.
In Section 2, we give some generalization of the well-known Wedderburn’s
Factorization theorem about polynomials over division rings. This
generalization and its corollaries will be used as important tools for our
study in the next sections of the paper. In the last part of this section, as
some illustration, we use them to generalize some previous results by Mahdavi-
Hezavehi et al. in [19].
Section 3 is devoted to the study of maximal subgroups of a subnormal subgroup
$G$ of $D^{*}$. Firstly, we consider the case when a maximal subgroup $M$ of
$G$ contains some non-central $FC$-element. The properties obtained in Theorem
3.5 are principal tools for our study in the next parts of the paper. Also, in
Theorem 3.7, we give a perfect description of the structure of $D$ when $M$ is
metabelian. In Section 5, we concentrate our study to the particular case when
$M$ is a maximal subgroup of $D^{*}$. The results we get are generalizations
of the result by Akbari et al. in [1, Theorem 6] and the result by B. X. Hai
and N. V. Thin in [8, Theorem 3.2] . More exactly, we replace the condition of
algebraicity of $M$ in [1, Theorem 6] and the condition of algebraicity of $D$
in [8, Theorem 3.2] by the weaker condition of algebraicity of derived
subgroup $M^{\prime}$ and the obtained results are the same. We refer to [2],
[10], and [14] for more information about the existence of maximal subgroups
in division rings.
Throughout this paper, for a ring $R$, the symbol $R^{*}$ denotes the group of
all units in $R$. For a non-empty subset $S$ of $D$, we denote by $F[S]$ and
$F(S)$ the subring and the division subring of $D$ respectively, generated by
the set $F\cup S$, where $F=Z(D)$ is the center of $D$. We say that a subgroup
$G$ of $D^{*}$ is absolutely irreducible if $F[G]=D$. Given a group $G$, a
subset $S$ and a subgroup $H$ of $G$, we denote by $Z(G)$, $G^{\prime}$,
$C_{G}(S)$ and $H_{G}=\bigcap_{x\in G}xHx^{-1}$ the center of $G$, the derived
group of $G$, the centralizer of $S$ in $G$ and the core of $H$ in $G$
respectively. If $x\in G$ and $H$ is a subgroup of $G$, then we denote by
$x^{H}$ the set of all elements $x^{h}=hxh^{-1},h\in H$. If $x^{G}$ is finite,
then we say that $x$ is an $FC$-element of $G$. The set of all $FC$-elements
of $G$ is called the $FC$-center of $G$. If $x,y\in G$ then
$[x,y]=xyx^{-1}y^{-1}$ and $[H,K]$ is the subgroup of $G$ generated by all
elements $[h,k],h\in H,\,k\in K$. We say that an element $x\in D$ is radical
over $F$ if there exists a positive integer $n(x)$ depending on $x$ such that
$x^{n(x)}\in F$. A non-empty subset $S$ of $D$ is radical over $F$ if every
element of $S$ is radical over $F$. We write $H\leq G$ and $H<G$ if $H$ is a
subgroup and proper subgroup of $G$ respectively. All other notation and
symbols in this paper are standard and one can find, for example, in [15],
[22], [23], [26], [27].
## 2 Algebraicity over a division subring and the generalization of
Wedderburn’s Factorization theorem
Let $D$ be a division ring with center $F$, and let $A$ be a conjugacy class
of $D$ which is algebraic over $F$ with minimal polynomial $f(t)\in F[t]$ of
degree $n$. Then, there exist $a_{1},\ldots,a_{n}\in A$ such that
$f(t)=(t-a_{1})\ldots(t-a_{n})\in D[t].$
This factorization theorem due to Wedderburn (see [15, (16.9), p. 265]) plays
an important role in the theory of polynomials over a division ring and its
applications in different problems are well-known. In this section, firstly we
consider the similar question in more general circumstance in order to get the
analogous theorem which could be used to generalize some previous results of
other authors about subgroups in division rings.
Let $K\subseteq D$ be division rings and $\alpha\in D$. We say that $\alpha$
is (right) algebraic over $K$ if there exists some nonzero polynomial $f(t)\in
K[t]$ having $\alpha$ as a right root. A monic polynomial from $K[t]$ with
smallest degree having $\alpha$ as a right root is called a minimal polynomial
of $\alpha$ over $K$. Throughout this paper we consider only right roots and
right algebraicity, so we shall always omit the prefix “right”. The minimal
polynomial of $\alpha$ over $K$ is unique, but it may not be irreducible as
the following example shows:
Let $\mathbb{H}$ be the division ring of real quaternions. Then,
$f(t)=t^{2}+1\in\mathbb{C}[t]$ is the minimal polynomial of $j$ and $k$ over
$\mathbb{C}$. Here, $\\{1,i,j,k\\}$ is standard basis of $\mathbb{H}$ over
$\mathbb{R}$.
The proof of the following lemma is a simple modification of the proof of
Lemma (16.5) in [15], so we now omit it.
###### Lemma 2.1.
Let $R$ be a ring, $D$ a division subring of $R$ and suppose $M$ is a subgroup
of $R^{*}$ normalizing $D^{*}$. If $K=C_{D}(M)$ and $x\in D^{*}$ is algebraic
over $K$ with the minimal polynomial $f(t)\in K[t]$, then a polynomial
$h(t)\in D[t]$ vanishes on $x^{M}$ if and only if $h(t)\in D[t]f(t)$.
Using Lemma 2.1, we get the following theorem which can be considered as some
generalization of Wedderburn’s Factorization theorem.
###### Theorem 2.2.
Let $R$ be a ring, $D$ a division subring of $R$ and suppose $M$ is a subgroup
of $R^{*}$ normalizing $D^{*}$. If $K=C_{D}(M)$ and $x\in D^{*}$ is algebraic
over $K$ with the minimal polynomial $f(t)$ of degree n, then there exist
$x_{1},\ldots,x_{n-1}\in x^{MD^{*}}$ such that
$f(t)=(t-x_{n-1})\cdots(t-x_{1})(t-x)\in D[t].$
###### Proof.
Take a factorization
$f(t)=g(t)(t-x_{r})\cdots(t-x_{1})(t-x)$
with $g(t)\in D[t]$, $x_{1},\ldots,x_{r}\in x^{MD^{*}}$, where $r$ is chosen
as large as possible. We claim that $h(t):=(t-x_{r})\cdots(t-x_{1})(t-x)$
vanishes on $x^{M}$. In fact, consider an arbitrary element $y\in x^{M}$. If
$h(y)\neq 0$, then, by [15, (16.3), p. 263], $g(x_{r+1})=0$, where
$x_{r+1}=aya^{-1}\in x^{MD^{*}},a=h(y)$. It follows that
$g(t)=g_{1}(t)(t-x_{r+1})$ for some $g_{1}(t)\in D[t]$, and so
$f(t)=g_{1}(t)(t-x_{r+1})(t-x_{r})\cdots(t-x_{1})(t-x).$
Since this contradicts to the choice of $r$, we have $h(x^{M})=0$; so, in view
of Lemma 2.1, $r=n-1$. Hence, $f(t)=(t-x_{n-1})\cdots(t-x_{1})(t-x)$, as it
was required to prove. ∎
We note that, the Wedderburn’s Factorization theorem is the special case of
Theorem 2.2, when we take $R=D$ and $M=D^{*}$.
From Theorem 2.2 and [15, (16.3), p. 263], we get the following corollaries.
###### Corollary 2.3.
Let $R$ be a ring, $D$ a division subring of $R$ and suppose $M$ is a subgroup
of $R^{*}$ normalizing $D^{*}$. Assume that $K=C_{D}(M)$ and $x\in D^{*}$ is
algebraic over $K$ with the minimal polynomial $f(t)$. If $y$ is a root of
$f(t)$ in $D$, then $y\in x^{MD^{*}}$.
###### Corollary 2.4.
Let $R$ be a ring, $D$ a division subring of $R$ and suppose $M$ is a subgroup
of $R^{*}$ such that $D^{*}\trianglelefteq M$. Assume that $K=C_{D}(M)$ and
$x\in D^{*}$ is algebraic over $K$ with the minimal polynomial $f(t)$ of
degree n. Then, $K$ is contained in the center of $D$ and there exists an
element $c_{x}\in[M,x]\cap K(x)$ such that $x^{n}=N_{K(x)/K}(x)c_{x}$ with
$N_{K(x)/K}(c_{x})=1$, where $N_{K(x)/K}$ is the norm of $K(x)$ to $K$.
###### Proof.
Since $D^{*}\leq M$, $K$ is contained in the center of $D$ and $K(x)$ is a
field. Taking $b=N_{K(x)/K}(x)$, by Theorem 2.2 we have $b=x^{r_{1}}\cdots
x^{r_{n}}$, with $r_{1},\ldots,r_{n}\in M$. We can write $b$ in the following
form:
$b=[r_{1},x][r_{2},x]^{x}[r_{3},x]^{x^{2}}\cdots[r_{n},x]^{x^{n-1}}x^{n}.$
Putting
$c^{-1}_{x}=[r_{1},x][r_{2}^{x},x][r_{3}^{x^{2}},x]\cdots[r_{n}^{x^{n-1}},x],$
we have $c_{x}=b^{-1}x^{n}\in[M,x]\cap K(x)$. So,
$N_{K(x)/K}(c_{x})=N_{K(x)/K}(b^{-1})N_{K(x)/K}(x)^{n}=b^{-n}b^{n}=1,$
as it was required to prove. ∎
This corollary can be reformulated in the following form which should be
convenient in some cases of its application in the next sections.
###### Corollary 2.5.
Let $R$ be a ring, $D$ a division subring of $R$ and suppose $M$ is a subgroup
of $R^{*}$ normalizing $D^{*}$. Assume that $K=C_{D}(M)$ is a field and $x\in
Z(D)^{*}\cap M$ is algebraic over $K$ with the minimal polynomial $f(t)$ of
degree n. Then, there exists an element $c_{x}\in[M,x]\cap K(x)$ such that
$x^{n}=N_{K(x)/K}(x)c_{x}$ with $N_{K(x)/K}(c_{x})=1$.
###### Proof.
Let $b=N_{K(x)/K}(x)\in K$, by Theorem 2.2, we have $b=x^{r_{1}d_{1}}\cdots
x^{r_{n}d_{n}}$, with $r_{1},\ldots,r_{n}\in M$ and $d_{1},\ldots,d_{n}\in
D^{*}$. We can write $b$ in the following form:
$b=[r_{1}d_{1},x][r_{2}d_{2},x]^{x}[r_{3}d_{3},x]^{x^{2}}\cdots[r_{n}d_{n},x]^{x^{n-1}}x^{n}.$
Since $x\in Z(D)^{*}$, we get
$b=[r_{1},x][r_{2},x]^{x}[r_{3},x]^{x^{2}}\cdots[r_{n},x]^{x^{n-1}}x^{n}.$
Putting
$c^{-1}_{x}=[r_{1},x][r_{2}^{x},x][r_{3}^{x^{2}},x]\cdots[r_{n}^{x^{n-1}},x],$
we have $c_{x}=b^{-1}x^{n}\in[M,x]\cap K(x)$. So,
$N_{K(x)/K}(c_{x})=N_{K(x)/K}(b^{-1})N_{K(x)/K}(x)^{n}=b^{-n}b^{n}=1,$
as it was required to prove. ∎
In view of Corollary 2.4, the following fact is evident.
###### Corollary 2.6.
Let $R$ be a ring, $D$ a division subring of $R$ and suppose $M$ is a subgroup
of $R^{*}$ such that $D^{*}\trianglelefteq M$. If $K=C_{D}(M)$, then
$K\subseteq Z(D)$ and the following assertions hold:
(i) If D is algebraic over K, then $D^{*}/K^{*}[M,D^{*}]$ is a torsion group.
(ii) If $[D:K]<\infty$, then $D^{*}/K^{*}[M,D^{*}]$ is a torsion group of a
bounded exponent dividing $[D:K]$.
For the illustration of the use of the generalization of Wedderburn’s
Factorization theorem, we are now ready to get the following result, which can
be considered as a generalization of Theorem 1 in [19].
###### Theorem 2.7.
Let $R$ be a ring, $D$ a division subring of $R$ and suppose $M$ is a subgroup
of $R^{*}$ such that $D^{*}\trianglelefteq M$. If $K=C_{D}(M)$ and $H/K$ is a
finite separable field extension in $D$, then the following statements hold:
(i) There exists an element $c\in[M,H^{*}]$ such that $N_{H/K}(c)=1$ and
$H=K(c)$. (ii) Either $H$ is algebraic over the prime subfield
$\mathbb{Z}_{p}$ or there exists an element $c\in[M,H^{*}]$ such that
$N_{H/K}(c)=1$ and $H=K(c)=K(c^{2})=K(c^{3})=\cdots$. (iii) If $H$ has a
nontrivial $K$-automorphism, then there exists an element $z=[m,h]$ with $m\in
M,\ h\in H^{*}$ such that $N_{H/K}(z)=1$ and $H=K(z)$.
###### Proof.
The proof of this theorem is similar to the proof of Theorem 1 in [19]. To get
the proof of our theorem, it suffices to replace the use of Skolem-Noether
theorem and Lemma A in the proof of Theorem 1 in [19] by the use of Corollary
2.3 and Corollary 2.4 respectively. ∎
## 3 Maximal subgroups of subnormal subgroups
The main purpose in this section is to describe the structure of maximal
subgroups in an arbitrary subnormal subgroup of $D^{*}$ and show their
influence to the whole structure of $D$. In the first, we prove the following
useful lemmas we need for further study.
###### Lemma 3.1.
If $D$ is a division ring with center $F$ and $G$ is a solvable-by-locally
finite subnormal subgroup of $D^{*}$, then $G\subseteq F$.
###### Proof.
By assumption there is a solvable normal subgroup $S$ of $G$ such that $G/S$
is locally finite. So, $S$ is a solvable subnormal subgroup of $D^{*}$ and in
view of [23, 14.4.4, p. 440], we have $S\subseteq F$. Hence, $G/Z(G)$ is
locally finite, so by [1, Lemma 3], $G^{\prime}$ is locally finite. Thus,
$G^{\prime}$ is a torsion subnormal subgroup of $D^{*}$ and by [12, Theorem
8], $G^{\prime}\subseteq F$. Hence, $G$ is solvable and by [23, 14.4.4, p.
440], $G\subseteq F$. ∎
###### Lemma 3.2.
Let $D$ be a division ring with center $F$ and $G$ a subnormal subgroup of
$D^{*}$. If $G$ is locally polycyclic-by-finite (e.g. if $G$ is locally
nilpotent) then $G\subseteq F$.
###### Proof.
If $G\not\subseteq F$, then by Stuth’s theorem (see, for example, [23, 14.3.8,
p. 439]) we have $D=F(G)$. By a result of Lichtman [24, 4.5.2, p. 155]
together with an exercise in [24, p. 162] or with the fact that $G$ is
contained in a unique maximal locally polycyclic-by-finite normal subgroup of
$D^{*}$, it follows that $G$ contains a noncyclic free subgroup, but this
contradicts to the fact that $G$ is locally polycyclic-by-finite. ∎
###### Lemma 3.3.
Let $D$ be a division ring with center $F$, $G$ a subnormal subgroup of
$D^{*}$ and suppose that $M$ is a maximal subgroup of $G$. If
$[M:Z(M)]<\infty$, then $M$ is abelian.
###### Proof.
Suppose that $M$ is non-abelian. Let $\\{x_{1},\ldots,x_{t}\\}$ be a
transversal of $Z(M)$ in $M$, $E$ a subfield of $D$ generated by $Z(M)$ and
$K=E[M]=\\{\sum_{i=1}^{t}a_{i}x_{i}|a_{i}\in E\\}$. Then, $K$ is a finite
dimensional vector space over $E$ and this implies that $K$ is a division
ring. Clearly, $E\subseteq Z(K)$, so we have $[K:Z(K)]<\infty$ and
$C_{K}(M)=Z(K)$. Since $M$ is maximal in $G$ and $M$ normalizes $K^{*}$, it
follows that either $G$ normalizes $K^{*}$ or $K^{*}\cap G\subseteq M$.
Case 1: $G$ normalizes $K^{*}$.
By Stuth’s theorem, either $K\subseteq F$ or $K=D$. If $K\subseteq F$ then
$M\subseteq F$, a contradiction. Thus, $D=K$; hence $Z(M)\subseteq
C_{D}(M)=Z(D)=F$ and, consequently, $Z(M)=F^{*}\cap M$. Take an $x\in
G\setminus M$ and put $H=\langle x,x_{1},\ldots,x_{t}\rangle$. By maximality
of $M$ in $G$, we get $G=HZ(M)=H(F^{*}\cap M)$. It follows that
$G^{\prime}\leq H$, so $H$ is normal in $G$. By [20, Theorem 1], we have
$H\subseteq F$, hence $G\subseteq F$. In particular, $M$ is abelian, a
contradiction.
Case 2: $K^{*}\cap G\subseteq M$.
In this case $K^{*}\cap G=M$, so $M$ is a subnormal subgroup of $K^{*}$. For
$N=\langle x_{1},\ldots,x_{t}\rangle$, we have $M=NZ(M)$, hence
$M^{\prime}\leq N$. Thus, $N$ is a finitely generated subnormal subgroup of
$K^{*}$. Recall that $[K:Z(K)]<\infty$, so in view of [20, Theorem 1], $N$ is
abelian and, consequently $M$ is abelian, a contradiction. This completes our
proof. ∎
It is well-known that the conjugacy class of a non-central element in $D$ has
the same cardinality with $D$ (see, for example, [23, 14.2.2, p. 429]). This
fact can be generalized by the following theorem which should be useful for
the next study.
###### Theorem 3.4.
If $D$ is a division ring with center $F$ and $G$ is a non-central subnormal
subgroup of $D^{*}$, then $|x^{G}|=|D|$ for any $x\in D\setminus F$.
###### Proof.
We claim that $|x^{G}|$ is infinite for any $x\in D\setminus F$. Thus, assume
that $|x^{G}|<\infty$, or, equivalently that $[G:C_{G}(x)]<\infty$. Since $G$
normalizes $F(x^{G})$, by Stuth’s theorem it follows that $F(x^{G})=D$.
Putting $H=(C_{G}(x))_{G}$, we have $H\subseteq C_{D}(x^{G})=C_{D}(D)=F$. On
the other hand, by $[G:C_{G}(x)]<\infty$, $[G:H]<\infty$, so $G\subseteq F$ by
Lemma 3.1, a contradiction. Therefore $|x^{G}|$ is infinite. Let $R$ be a
division ring generated by $x^{G}$, by Stuth’s theorem, $R=D$, so
$|x^{G}|=|D|$. ∎
Using Theorem 3.4, in the following we show the role of an $FC$-element in
maximal subgroups of a subnormal subgroup of $D^{*}$. The results obtained in
this theorem will be used as principal tools in the proofs of next theorems.
###### Theorem 3.5.
Let $D$ be a division ring with center $F$, $G$ a subnormal subgroup of
$D^{*}$ and suppose that $M$ is a maximal subgroup of $G$, containing a non-
central $FC$-element $\alpha$. By setting $K=F((\alpha)^{M})$ and
$H=C_{D}(K)$, the following conditions hold:
(i) $K$ is a field, $[K:F]=[D:H]<\infty$ and $F(M)=D$. (ii) $K^{*}\cap G$ is
the $FC$-center and also the Fitting subgroup of $M$. (iii) $K/F$ is a Galois
extension, $M/H^{*}\cap G\cong\mathrm{Gal}(K/F)$ is a finite simple group.
(iv) If $H^{*}\cap G\subseteq K$, then $H=K$ and $[D:F]<\infty$.
###### Proof.
First, we claim that $F(M)=D$. In fact, if $F(M)^{*}\cap G=M$, then $M$ is a
subnormal subgroup of $F(M)^{*}$ containing the non-central $FC$-element
$\alpha$, so we have a contradiction in view of Theorem 3.4. Thus,
$F(M)^{*}\cap G=G$ by maximality of $M$ in $G$; hence, by Stuth’s theorem, we
have $F(M)=D$.
Since $\alpha$ is an $FC$-element of $M$, $[M:C_{M}(\alpha)]<\infty$. Taking
$N=(C_{M}(\alpha))_{M}$, $K=F(\alpha^{M})$ and $H=C_{D}(K)$, we have
$N\vartriangleleft M$, $N\leq H^{*}$ and $[M:N]$ is finite. Since $M$
normalizes $K^{*}$, by maximality of $M$ in $G$, it follows that, either $G$
normalizes $K^{*}$ or $K^{*}\cap G\leq M$. If $G$ normalizes $K^{*}$, then by
Stuth’s theorem we have $K=D$, so $H=F$. Therefore, $N\leq F^{*}\cap M\leq
Z(M)$. The inequality $[M:C_{M}(\alpha)]<\infty$ implies $[M:N]<\infty$ and
consequently, $[M:Z(M)]<\infty$. Now, in view of Lemma 3.3, $M$ is abelian, a
contradiction. Thus $K^{*}\cap G=K^{*}\cap M\trianglelefteq M$, hence
$K^{*}\cap G$ is a subnormal subgroup of $K^{*}$ containing the set
$\alpha^{M}$ of $FC$-elements in $M$. By Theorem 3.4, $\alpha^{M}\subseteq
Z(K)$, hence $K$ is a field. Since $H=C_{D}(K)$ and $M$ normalizes $K^{*}$,
$M$ also normalizes $H^{*}$. By maximality of $M$ in $G$, either $G$
normalizes $H^{*}$ or $H^{*}\cap G\leq M$. If $G$ normalizes $H^{*}$, then by
Stuth’s theorem, either $H=D$ or $H\subseteq F$. However, both cases are
impossible since $K\subseteq C_{D}(K)=H$ and $K$ contains an element
$\alpha\not\in F=Z(D)$. Therefore, $H^{*}\cap G=H^{*}\cap M\trianglelefteq M$.
Since $N\leq H^{*}\cap G$ and $[M:N]<\infty$, we have $[M:H^{*}\cap G]<\infty$
and $M=\bigcup_{i=1}^{t}x_{i}(H^{*}\cap G)$ for some transversal
$\\{x_{1},\ldots,x_{t}\\}$ of $H^{*}\cap G$ in $M$. Put
$R=\sum_{i=1}^{t}x_{i}H$. Since $M$ normalizes $H^{*}$, it is easy to see that
$R$ is a ring and also, it is a finite-dimensional right vector space over its
division subring $H$. This fact implies that $R$ is a division subring of $D$
containing $F(M)$. Therefore $R=D$ and $[D:H]<\infty$. By using Double
Centralizer theorem [15, 15.4, p. 253] we get $[K:F]=[D:H]<\infty$ and
$Z(H)=C_{D}(H)=K$. Thus $(i)$ is established.
Since $M$ normalizes $K^{*}$, for all $a\in M$ the mapping $\theta_{a}:K\to K$
given by $\theta_{a}(x)=axa^{-1}$ is an automorphism of $K/F$. Now, consider
the mapping $\psi:M\to\mathrm{Gal}(K/F)$ given by $\psi(a)=\theta_{a}$.
Clearly, $\psi$ is a group homomorphism with
$\mathrm{ker}\psi=C_{M}(K^{*})=C_{D}(K)^{*}\cap M=H^{*}\cap M=H^{*}\cap G$.
Since $F(M)=D$, we obtain $C_{D}(M)=F$ and it follows that the fixed field of
$\psi(M)$ is $F$. By Galois correspondence, we conclude that $\psi$ is a
surjective homomorphism and $K/F$ is a Galois extension. Hence, $M/H^{*}\cap
G\cong\mathrm{Gal}(K/F)$ is a finite group. Assume that $\mathrm{Gal}(K/F)$ is
not simple. Then, there exists some intermediate subfield $L,F\subset L\subset
K$, such that $\theta(L)=L$ for all $\theta\in\mathrm{Gal}(K/F)$. Thus, $L$
and $E=C_{D}(L)$ are normalized by $M$ and clearly, $E\neq D$ and
$E\not\subseteq F$. Therefore, by Stuth’s theorem, $E^{*}$ is not normalized
by $G$. If $E^{*}\cap G\not\subseteq M$, then $G=M(E^{*}\cap G)$ normalizes
$E^{*}$, a contradiction. Hence, $E^{*}\cap G\leq M$. Since $K^{*}\cap
G\trianglelefteq M$ and $K^{*}\cap G\leq E^{*}\cap G$, we have $K^{*}\cap
G\trianglelefteq E^{*}\cap G$ and so $K^{*}\cap G$ is an abelian subnormal
subgroup of $E^{*}$. By Lemma 3.1, we have $K^{*}\cap G\subseteq Z(E)=L$.
Hence, $K=F(\alpha^{M})\subseteq F(K^{*}\cap G)\subseteq L$, a contradiction.
Thus $\mathrm{Gal}(K/F)$ is simple, and the proof of $(iii)$ is complete.
If $H^{*}\cap G\subseteq K$, then $H^{*}\cap G=K^{*}\cap G$. Recall that
$[M:H^{*}\cap G]<\infty$, hence $[M:K^{*}\cap G]<\infty$. Suppose that
$\\{y_{1},\ldots,y_{l}\\}$ is a transversal of $K^{*}\cap G$ in $M$. Since $M$
normalizes $K^{*}$, $Q=\sum_{i=1}^{l}y_{i}K$ is a division ring. Clearly, $Q$
contains both $F$ and $M$, so in view of $(i)$, we have $Q=D$. It is easy to
see that $[D:K]=[Q:K]\leq|M/H^{*}\cap G|=|\mathrm{Gal}(K/F)|=[K:F]$. Hence, by
Double Centralizer theorem, $K$ is a maximal subfield of $D$, $K=H$ and
$[D:F]<\infty$. Thus $(iv)$ is established.
To prove $(ii)$, first we claim that $K^{*}\cap G$ is a maximal abelian normal
subgroup of $M$. Thus, suppose $C$ is a maximal abelian normal subgroup of $M$
containing $K^{*}\cap G$. Then, $\alpha^{M}\subseteq C$ and it follows $C\leq
H^{*}$. Recall that $H^{*}\cap G\leq M$, so we have $C\trianglelefteq
H^{*}\cap G$. Hence, $C$ is an abelian subnormal subgroup of $H^{*}$, and by
Lemma 3.1, $C\subseteq Z(H)=K$. So $K^{*}\cap G=C$ is a maximal abelian normal
subgroup of $M$.
To prove $K^{*}\cap G$ is the Fitting subgroup of $M$, it suffices to show
that $K^{*}\cap G$ is a maximal nilpotent normal subgroup of $M$. Now, assume
that $A$ is a nilpotent normal subgroup of $M$ which strictly contains
$K^{*}\cap G$. Then, $B=H^{*}\cap G\cap A$ is a nilpotent subnormal subgroup
of $H^{*}$. Hence, by Lemma 3.1, we conclude that $B\subseteq Z(H)=K$ and
consequently, $B=K^{*}\cap G\cap A$. If $A\subseteq H^{*}\cap G$, then
$A=B\subseteq K^{*}\cap G$, a contradiction. Therefore, $A\not\subseteq
H^{*}\cap G$. Thus, $A(H^{*}\cap G)/H^{*}\cap G$ is a nontrivial normal
subgroup of $M/H^{*}\cap G$. Since $M/H^{*}\cap G$ is simple, $M/H^{*}\cap
G=A(H^{*}\cap G)/H^{*}\cap G\cong A/B$. Suppose $S=\sum_{i=1}^{m}z_{i}K$,
where $\\{z_{1},\ldots,z_{m}\\}$ is a transversal of $B$ in $A$. Since $A$
normalizes $K^{*}$ and $B\subseteq K$, $S$ is a division ring and
$[S:K]_{r}\leq m$. Recall that $M$ normalizes $A$ and $K$; so, $M$ also
normalizes $S$. By maximality of $M$ in $G$, it follows either $G$ normalizes
$S$ or $S^{*}\cap G\leq M$. If the second case occurs, then $A$ is a nilpotent
subnormal subgroup of $S^{*}$. By Lemma 3.1, $A$ is abelian and this
contradicts to the fact that $K^{*}\cap G$ is a maximal abelian normal
subgroup of $M$. Thus, $G$ normalizes $S$, and by Stuth’s theorem we have
$S=D$. Therefore, $[D:K]_{r}\leq m=|M/H^{*}\cap G|=|\mathrm{Gal}(K/F)|=[K:F]$.
This implies that $[D:F]=m^{2}$ and $K=H$ is a maximal subfield of $D$. From
$M/H^{*}\cap G=A(H^{*}\cap G)/H^{*}\cap G$ and $K^{*}\cap G<A$, it follows
$M=A$. Since $[D:F]<\infty$, $M$ can be considered as a nilpotent linear group
no containing unipotent elements ($\neq 1$). By a result in [3, p. 114],
$[M:Z(M)]$ is finite, which contradicts to Lemma 3.3. Thus, $K^{*}\cap G$ is
the Fitting subgroup of $M$.
For any $x\in K^{*}\cap G$, the elements of $x^{M}\subseteq K$ have the same
minimal polynomial over $F$, so $|x^{M}|<\infty$. For any $x\in M\setminus
K^{*}\cap G$, if $|x^{M}|<\infty$ then, by what we proved, $F(x^{M})\cap G$ is
the Fitting subgroup of $M$ which is different from $K^{*}\cap G$, a
contradiction. Hence, $|x^{M}|=\infty$ and $(ii)$ follows. Thus, the proof of
the theorem is now complete. ∎
From Theorem 3.5 we get the following corollary which is more convenient for
further applications.
###### Corollary 3.6.
Let $D$ be a division ring with center $F$, $G$ a subnormal subgroup of
$D^{*}$ and suppose $M$ is a maximal subgroup of G. If $M$ contains an abelian
normal subgroup $A$ and an element $\alpha\in A\setminus Z(M)$ algebraic over
$F(Z(M))$, then $K=F(A)$ and $H=C_{D}(K)$ satisfy (i) - (iv) in Theorem 3.5.
Moreover, $F(A)=F[A]$.
###### Proof.
Since $A\trianglelefteq M$, the elements of $\alpha^{M}$ in the field
$F(Z(M)A)$ have the same minimal polynomial over $F(Z(M))$. Hence,
$|\alpha^{M}|<\infty$. Now, by using Theorem 3.5, $K=F(\alpha^{M})$ and
$H=C_{D}(K)$ satisfy (i) - (iv) in Theorem 3.5. By (ii), $A\subseteq K$ and
$K=F(A)=F[A]$ since $[K:F]<\infty$. ∎
###### Theorem 3.7.
Let $D$ be a division ring with center $F,G$ a subnormal subgroup of $D^{*}$
and suppose $M$ is a non-abelian metabelian maximal subgroup of $G$. Then
$[D:F]=p^{2}$, $M/M^{\prime}Z(M)\cong\bigoplus_{i\in I}\mathbb{Z}_{p}$, where
$p$ is a prime number, and there exists a maximal subfield $K$ of $D$ such
that $K/F$ is a Galois extension, $\mathrm{Gal}(K/F)\cong M/K^{*}\cap
G\cong\mathbb{Z}_{p}$ and $K^{*}\cap G$ is the $FC$-center and also the
Fitting subgroup of $M$. Furthermore, for any $x\in M\setminus K$ we have
$x^{p}\in F$ and $D=F[M]=\bigoplus_{i=1}^{p}Kx^{i}$.
###### Proof.
Denote by $A$ a maximal abelian normal subgroup of $M$ containing $M^{\prime}$
(we note that such a subgroup exists since $M^{\prime}$ is abelian). Consider
an arbitrary subgroup $N$ of $M$ which properly contains $A$. Since
$M^{\prime}\leq N$, $N\trianglelefteq M$; so the maximality of $M$ in $G$
implies that either $G$ normalizes $F(N)^{*}$ or $F(N)^{*}\cap G\leq M$. If
the second case occurs, then $N\trianglelefteq F(N)^{*}\cap G$, so $N$ is a
metabelian subnormal subgroup of $F(N)^{*}$. By Lemma 3.1, $N$ is abelian,
which contradicts to the maximality of $A$. Hence, $G$ normalizes $F(N)^{*}$
and by Stuth’s theorem we conclude that $F(N)=D$.
Now let $a$ be an element from $M\setminus A$ and assume that $a$ is
transcendental over $F(A)$. Put $T=F(A)^{*}\langle a^{2}\rangle$; since $a$
normalizes $F(A)^{*}$, it is not hard to see that
$F[T]=\bigoplus_{i\in\mathbb{Z}}F(A)a^{2i}$ and $(F[T],F(A),T,T/F(A)^{*})$ is
a crossed product. But $T/F(A)^{*}\cong\langle a^{2}\rangle$ is an infinite
cyclic group and by [[wehfritz], 1.4.3, p. 26], $F[T]$ is an Ore domain. On
the other hand, by what we proved before, we have $F(T)=D$. Therefore, there
exist two elements $s_{1},s_{2}\in F[T]$ such that $a=s_{1}s_{2}^{-1}$.
Writing $s_{1}=\sum_{i=l}^{m}k_{i}a^{2i}$ and
$s_{2}=\sum_{i=l}^{m}k^{\prime}_{i}a^{2i}$, with $k_{i},\ k^{\prime}_{i}\in
F(A)$, for any $l\leq i\leq m$, we have
$\sum_{i=l}^{m}ak^{\prime}_{i}a^{2i}=\sum_{i=l}^{m}k_{i}a^{2i}$. If we set
$l_{i}=ak^{\prime}_{i}a^{-1}$, for any $l\leq i\leq m$, then $l_{i}$’s are
elements of $F(A)$ and we have
$\sum_{i=l}^{m}l_{i}a^{2i+1}=\sum_{i=l}^{m}k^{\prime}_{i}a^{2i}$. This shows
that $a$ is algebraic over $F(A)$, say of degree $n$. Using the fact that $a$
normalizes $F(A)^{*}$, we see that $R=\bigoplus_{i=0}^{n-1}F(A)a^{i}$ is a
domain, which is a finite-dimensional left vector space over $F(A)$. Therefore
$R$ is a division ring and $R=F(A\langle a\rangle)$. By what we proved before
we conclude that $R=D$, hence $[D:F(A)]_{l}<\infty$. So, by Double Centralizer
theorem, $[D:F]<\infty$.
If $A\leq Z(M)$, then, since $M^{\prime}\leq A$, $\langle A,x\rangle$ is an
abelian normal subgroup of $M$ properly containing $A$, for any $x\in
M\setminus A$. But, this contradicts to the maximality of $A$; hence
$A\not\subseteq Z(M)$. Since $[D:F]<\infty$, all elements of $A\setminus Z(M)$
is algebraic over $F$. In view of Corollary 3.6, there exists a subfield $K$
of $D$ such that $K$ and $H=C_{D}(K)$ satisfy $(i)-(iv)$ in Theorem 3.5. So,
$H^{*}\cap G$ is a metabelian subnormal subgroup of $H^{*}$. By Lemma 3.1,
$H^{*}\cap G\subseteq Z(H)=K$. Thus, $(iv)$ implies that $K=H$ is a maximal
subfield of $D$. Since $M$ is metabelian, $M/K^{*}\cap G$ is simple and
metabelian. We conclude that $\mathrm{Gal}(K/F)\cong M/K^{*}\cap
G\cong\mathbb{Z}_{p}$, where $p$ is a prime number and so $[D:F]=p^{2}$. Since
$[M:K^{*}\cap G]=p$ and $D$ is algebraic over $F$, for any $x\in M\setminus
K$, we have $D=F(M)=F[M]=\sum_{i=1}^{p}Kx^{i}$ and, since $[D:K]=p$,
$D=\bigoplus_{i=1}^{p}Kx^{i}$. Suppose $x^{p}\not\in F$. By $F(M)=D$, we have
$Z(M)=M\cap F^{*}$, and it follows $C_{M}(x^{p})\neq M$. On the other hand, we
note that $\langle x,K^{*}\cap G\rangle\leq C_{M}(x^{p})$ and $[M:K^{*}\cap
G]$ is prime, so we get $C_{M}(x^{p})=M$, a contradiction. Thus, $x^{p}\in F$.
Now, by Corollary 2.5, for any $y\in K^{*}\cap M=K^{*}\cap G$, we have
$y^{p}\in M^{\prime}F^{*}$. Therefore $y^{p}\in M^{\prime}(M\cap
F^{*})=M^{\prime}Z(M)$, so $M/M^{\prime}Z(M)$ is an abelian group of exponent
$p$ and by Bear-Prufer’s theorem [22, p. 105],
$M/M^{\prime}Z(M)\cong\bigoplus_{i\in I}\mathbb{Z}_{p}$. This completes the
proof. ∎
In [4], Ebrahimian proved that if $M$ is a nilpotent maximal subgroup of
$D^{*}$, then $M$ is abelian. The following theorem shows that this remains
also true if we replace $D^{*}$ by any its subnormal subgroup.
###### Theorem 3.8.
Let $D$ be a division ring and $G$ a subnormal subgroup of $D^{*}$. If $M$ is
a nilpotent maximal subgroup of $G$, then $M$ is abelian.
###### Proof.
Assume that $M$ is non-abelian. Then, there exists an element $x\in
Z_{2}(M)\setminus Z(M)$. By considering the homomorphism $f:M\to Z(M)$ given
by $f(y)=xyx^{-1}y^{-1}$ we see that $M/C_{M}(x)$ is an abelian group, so
$M^{\prime}\leq C_{M}(x)$. Put $F=Z(D)$; since $x\not\in Z(M)$, we have
$F(M^{\prime})\neq D$. Now we claim that $M^{\prime}$ is abelian. If $G$
normalizes $F(M^{\prime})$, then, by Stuth’s theorem, we have
$F(M^{\prime})\subseteq F$, so $M^{\prime}$ is abelian, as claimed. Now,
suppose that $G$ does not normalize $F(M^{\prime})$. Since $M$ normalizes
$F(M^{\prime})$ and $M$ is maximal in $G$, it follows that $F(M^{\prime})\cap
G\subseteq M$. Thus, $M^{\prime}\trianglelefteq F(M^{\prime})\cap G$ is a
nilpotent subnormal subgroup of $F(M^{\prime})^{*}$ and so, by Lemma 3.1,
$M^{\prime}$ is abelian, as claimed. Therefore $M$ is metabelian. By Theorem
3.7, we conclude that the Fitting subgroup of $M$ is a proper subgroup of $M$,
a contradiction. Thus, $M$ is abelian as it was required to prove. ∎
###### Lemma 3.9.
Let $D$ be a centrally finite division ring with center $F$. Then,
$D^{\prime}\cap F^{*}$ is finite.
###### Proof.
Suppose $[D:F]=n^{2}$. By taking the reduced norm, we obtain $x^{n}=1$ for all
$x\in D^{\prime}\cap F^{*}$. Since $F$ is a field, $D^{\prime}\cap F^{*}$ is
finite. ∎
###### Lemma 3.10.
Let $D$ be a division ring with center $F,G$ a subnormal subgroup of $D^{*}$
and suppose $M$ is a non-abelian maximal subgroup of $G$. If $M^{\prime}$ is
locally finite, then $M/Z(M)$ is locally finite, $M^{\prime}$ is locally
cyclic and the conclusions of Theorem 3.7 follow.
###### Proof.
Suppose $M^{\prime}$ is non-abelian, so we need only to consider the case
${\rm Char}D$=0.
First, we claim that there exists a torsion abelian normal subgroup of $M$
which is not contained in $Z(M)$. By maximality of $M$ in $G$, either
$F(M^{\prime})\cap G\subseteq M$ or $G$ normalizes $F(M^{\prime})$. If
$F(M^{\prime})\cap G\subseteq M$, then $M^{\prime}$ is a locally finite
subnormal subgroup of $F(M^{\prime})^{*}$, so $M^{\prime}$ is abelian by Lemma
3.1, a contradiction. So, $G$ normalizes $F(M^{\prime})$ and by Stuth’s
theorem we have $F[M^{\prime}]=F(M^{\prime})=D$. On the other hand, by [24,
2.5.5, p. 74], there exists a metabelian normal subgroup of $M^{\prime}$ of
finite index $n$. Let $Q=\langle\\{x^{n}|x\in M^{\prime}\\}\rangle$. Then, $Q$
is a metabelian normal subgroup of $M$ and $Q^{\prime}$ is a torsion abelian
normal subgroup of $M$. If $Q^{\prime}$ is not contained in $Z(M)$ then we are
done. Hence, we may assume that $Q^{\prime}\leq Z(M)$, so $Q$ is locally
finite and nilpotent. Thus, by [24, 2.5.2, p. 73], $Q$ contains an abelian
subgroup $B$ such that $[Q:B]<\infty$. Clearly, $C=\cap_{x\in M}xBx^{-1}$ is a
torsion abelian normal subgroup of $M$ and we may assume that $C\leq Z(M)$. On
the other hand, $[Q:B]<\infty$ and $Q\trianglelefteq M$, so it follows that
$Q/C$ has bounded exponent. From definition of $Q$ and $C\leq Z(M)$,
$M^{\prime}/Z(M^{\prime})$ has also bounded exponent. Therefore, by the
classification theorem of locally finite groups in division ring [24, 2.5.9,
p. 75], we conclude that $M^{\prime}$ is abelian-by-finite. Now, let $A$ be an
abelian normal subgroup of $M^{\prime}$ of finite index. Since
$F[M^{\prime}]=D$, $D$ is finite-dimensional over a field $F[A]$. The Double
Centralizer theorem implies that $[D:F]<\infty$ and by Lemma 3.9, $|A\cap
F^{*}|<\infty$. Also by $F[M^{\prime}]=D$, we have $Z(M^{\prime})=F\cap
M^{\prime}$ and hence, $A/A\cap F^{*}$ has bounded exponent, so is $A$. Since
$A$ can be considered as the multiplicative subgroup of a field, we conclude
that $A$ is finite, so is $M^{\prime}$. Hence, $M$ is an $FC$-group and so it
is abelian by Theorem 3.5, a contradiction. Therefore, there exists a torsion
abelian normal subgroup of $M$ which is not contained in $Z(M)$.
Now, by Corollary 3.6, there exists a field $K$ such that $K$ and $H=C_{D}(K)$
satisfy $(i)-(iv)$ in Theorem 3.5. Taking $N=M^{\prime}\cap H^{*}$, we have
$N\trianglelefteq M\cap H^{*}=G\cap H^{*}$, so $N$ is a locally finite
subnormal subgroup of $H^{*}$. By Lemma 3.1, $N=M^{\prime}\cap H^{*}$ is
abelian and so $M^{\prime}\not\subseteq M\cap H^{*}$. If $M/M\cap H^{*}$ is
abelian, then $M^{\prime}\subseteq M\cap H^{*}$, a contradiction. Thus, from
$(iii)$ in Theorem 3.5, $M^{\prime}/N\cong M^{\prime}(M\cap H)/M\cap H=M/M\cap
H$ is a non-abelian finite simple group. Let $\\{x_{1},\ldots,x_{m}\\}$ be a
transversal of $N$ in $M^{\prime}$ and $S=\langle x_{1},\ldots,x_{m}\rangle$ a
finite subgroup of $M^{\prime}$. Since $S$ has a quotient group which is a
finite simple non-abelian group, $S$ is unsolvable. The classification theorem
of finite groups in division ring [24, 2.1.4, p. 46] states that the only
unsolvable finite subgroup of a division ring is $SL(2,5)$, so
$M^{\prime}=S\cong SL(2,5)$ is a finite group. Thus, $M$ is an $FC$-group and
so it is abelian by Theorem 3.5, a contradiction. Therefore, $M^{\prime}$ is
abelian and it can be considered as a torsion multiplicative subgroup of a
field. Hence $M^{\prime}$ is locally cyclic and Theorem 3.7 completes the
proof. ∎
## 4 The existence of noncyclic free subgroups
The question on the existence of noncyclic free subgroups in the
multiplicative group of a division ring was posed by Lichtman in [16]. In [5],
Goncalves proved that if $D$ is centrally finite division ring (i.e. $D$ is
finite dimensional vector space over its center $F$), then $D^{*}$ contains a
noncyclic free group. Later, in [7], Goncalves and Mandel posed the more
general question: is it true that every subnormal subgroup of the
multiplicative group of a division ring contains a noncyclic free subgroup? In
this work, the authors show that this is true in some particular cases. More
exactly, they proved that a subnormal subgroup $G$ of $D^{*}$ contains a
noncyclic free subgroup if $G$ contains some element $x\in D\setminus F$,
which is algebraic over the center $F$ of $D$ such that either (a) ${\rm
Gal}(F(x)/F)\neq 1$ or (b) $x^{p}\in F$, where $p=2$ or $p={\rm char}(F)>0$.
The affirmative answer was also obtained for centrally finite division rings
by Goncalves in his precedent work [6]. Note that the affirmative answer to
the question above would imply some known theorems like the commutativity
theorems of Kaplansky, Jacobson, Hua, Herstein, Stuth,… For more information
we refer to [11]. In this section we shall prove that the question above has
the affirmative answer for locally finite division rings. Recall that a
division ring $D$ with the center $F$ is locally finite if for every finite
subset $S$ of $D$, the division subring $F(S)$ is finite dimensional vector
space over $F$. It can be prove that in a locally finite division ring every
finite subset generates the centrally finite division subring. From this
observation, the following definition seems to be quite natural: a division
ring $D$ is called weakly locally finite if for every finite subset $S$ of
$D$, the division subring of $D$ generated by $S$ is centrally finite. This
notion is introduced firstly in [9]. Clearly, every locally finite is weakly
locally finite. The converse is not true. In [9], using the well-known method
due to Mal’cev and von Neumann, the authors constructed the example of weakly
locally finite division ring which is not even algebraic. In the next two
lemmas we prove some facts for weakly locally finite division rings.
###### Lemma 4.1.
Let $D$ be a weakly locally finite division ring of characteristic $0$ and $G$
be a subgroup of $D^{*}$. If $G$ contains no noncyclic free subgroup, then $G$
is locally solvable -by-finite.
###### Proof.
Take an arbitrary finite subset $S$ of $G$, and let $K$ be a division subring
of $D$ generated by $S$. Then, $K$ is centrally finite , so we have
$m=[K:Z(K)]<\infty$. Therefore, the subgroup $<S>$ generated by $S$ in $G$ can
be considered as a subgroup of ${\rm GL}_{m}(Z(K))$. By [25, Theorem 1], there
exists a solvable subgroup $H$ of $<S>$ of finite index. If $N=H_{G}$, then
$N$ is a solvable normal subgroup of finite index in $<S>$. Hence, $G$ is
locally solvable-by-finite. ∎
###### Lemma 4.2.
Let $D$ be a weakly locally finite division ring and $G$ be a subgroup of
$D^{*}$. If $G$ contains no noncyclic free subgroup, then $G$ is (locally
solvable) -by-(locally finite).
###### Proof.
Firstly, we claim that $G$ is locally solvable-by-locally finite. If ${\rm
char}(D)=0$, then this claim follows from Lemma 4.1. Assume that ${\rm
char}(D)=p>0$. Then, by the same argument as in the proof of Lemma 4.1, for
any finite subset $S$ of $G$, the subgroup $<S>$ generated by $S$ can be
viewed as a linear group over the field of characteristic $p>0$. By [25,
Theorem 2], $<S>$ contains a solvable normal subgroup $H$ such that $<S>/H$ is
locally finite. Hence, $G$ is locally solvable-by-locally finite. Now, by [24,
3.3.9, p. 103], $G$ is (locally solvable) -by-(locally finite). ∎
###### Lemma 4.3.
Let $D$ be a locally finite division ring and G an arbitrary subgroup of
$D^{*}$. Then either $G$ contains a noncyclic free subgroup or $G$ is abelian-
by-locally finite.
###### Proof.
Assume that $G$ does not contain any noncyclic free subgroups. By Lemma 4.2,
$G$ is (locally solvable)-by-(locally finite). The Wehrfritz’s theorem [24,
5.5.1, p. 199] implies that there exists an abelian normal subgroup $A$ of $G$
such that $G/A$ is locally finite. Therefore, $G$ is abelian-by-locally
finite. ∎
From Lemma 3.1 and Lemma 4.3 we get the following result, which shows that
Conjecture 2 in [7] is true for locally finite division rings.
###### Theorem 4.4.
Let $D$ be a locally finite division ring with center $F$ and $G$ a subnormal
subgroup of $D^{*}$. If $G$ is noncentral, then $G$ contains a noncyclic free
subgroup.
The next theorem can be considered as a broad generalization of the main
theorem in [21].
###### Theorem 4.5.
Let $D$ be a locally finite division ring with center $F,G$ a subnormal
subgroup of $D^{*}$ and suppose $M$ is a non-abelian maximal subgroup of $G$.
If $M$ contains no noncyclic free subgroups, then $[D:F]<\infty$, $F(M)=D$ and
there exists a maximal subfield $K$ of $D$ such that $K/F$ is a Galois
extension, $\mathrm{Gal}(K/F)\cong M/K^{*}\cap G$ is a finite simple group and
$K^{*}\cap G$ is the $FC$-center and also the Fitting subgroup of $M$.
###### Proof.
First, we show that $F(M)=D$. If $F(M)\neq D$, then, by Stuth’s theorem,
$F(M)$ is not normalized by $G$. So, by maximality of $M$ in $G$, we have
$F(M)\cap G=M$. Thus, $M$ is a non-abelian subnormal subgroup of $F(M)^{*}$ no
containing a noncyclic free subgroup, which contradicts to Theorem 4.4. So, we
have $F(M)=D$; hence $Z(M)=M\cap F^{*}$. Now, by Lemma 4.3, there exists an
abelian normal subgroup $A$ of $M$ such that $M/A$ is locally finite. If
$A\subseteq Z(M)$, then $M/Z(M)$ is locally finite, so is $M^{\prime}$ and the
result follows from Lemma 3.10. Assume that $A\not\subseteq Z(M)$. By applying
Corollary 3.6, there exists a subfield $K$ of $D$ such that $H=C_{D}(K)$ and
$K$ satisfy $(i)-(iv)$ in Theorem 3.5. Since $H^{*}\cap G$ is a subgroup of
$M$ and also a subnormal subgroup of $H^{*}$, by Theorem 4.4 we have
$H^{*}\cap G\subseteq K$. Now, the condition $(iv)$ in Theorem 3.5 completes
our proof. ∎
## 5 Maximal subgroups of $GL_{1}(D)$
In this section we investigate the similar problem as in Section 3 for the
particular case, when $G=D^{*}$.
###### Theorem 5.1.
Let $D$ be a division ring with center $F$ and $M$ a maximal subgroup of
$D^{*}$, containing a non-central $FC$-element $\alpha$. By setting
$K=F(\alpha^{M})$, the following conditions hold:
(i) $K$ is a field, $[K:F]<\infty$ and $F[M]=D$. (ii) $K^{*}$ is the Hirsch-
Plotkin radical of $M$. (iii) $F^{*}<K^{*}\leq C_{D}(K)^{*}<M<D^{*}$. (iv)
$K/F$ is a Galois extension, $M/C_{D}(K)^{*}\cong\mathrm{Gal}(K/F)$ is a
finite simple group. (v) For any $x\in K,\ |x^{M}|\leq[K:F]$ and for any $y\in
D\setminus K,\ |y^{M}|=|D|$.
###### Proof.
We note that $K$ and $H=C_{D}(K)$ satisfy $(i)-(iii)$ in Theorem 3.5 (for
$G=D^{*}$). As in first paragraph of the proof of Theorem 3.5, we can write
$D=\sum_{i=1}^{t}x_{i}H$, where $x_{1},\ldots,x_{t}\in M$. In this case, we
have $H^{*}\leq M$, and this implies $F[M]=D$. It remains to prove $(ii)$ and
$(v)$. Let $x$ be an element of $K^{*}$. Since $K$ is algebraic over $F$ and
$K^{*}\vartriangleleft M$, $x^{M}\subseteq K$ and the elements of $x^{M}$ have
the same minimal polynomial $f$ over $F$. So
$|x^{M}|\leq\textrm{deg}(f)\leq[K:F]$. For any $y\in D\setminus K$, $C_{H}(y)$
is a proper division subring of $H$ since $y\not\in K=C_{D}(H)$. By [23,
14.2.1, p. 429] and $[D:H]<\infty$ we have
$|y^{M}|=[M:C_{M}(y)]\geq[H^{*}:C_{H}(y)^{*}]=|H|=|D|$. Thus, $(i),(iii),(iv)$
and $(v)$ hold.
Now we prove $(ii)$. Let $A$ be the Hirsch-Plotkin radical of $M$ and suppose
$K^{*}<A$. Then, $B=H^{*}\cap A$ is a locally nilpotent normal subgroup of
$H^{*}$. Hence, by Lemma 3.2, we conclude that $B\subseteq Z(H)=K$ and
consequently, $B=K^{*}$. If $A\subseteq H^{*}$, then $A=B=K^{*}$, a
contradiction. Therefore, $A\not\subseteq H^{*}$. Thus, $AH^{*}/H^{*}$ is a
nontrivial normal subgroup of $M/H^{*}$. Since $M/H^{*}$ is simple,
$M/H^{*}=AH^{*}/H^{*}\cong A/B$. Suppose $R=\sum_{i=1}^{m}y_{i}K$, where
$\\{y_{1},\ldots,y_{m}\\}$ is a transversal of $B$ in $A$. Since $A$
normalizes $K^{*}$ and $B\subseteq K$, $R$ is a division ring and
$[R:K]_{r}\leq m$. Since $R$ is a division ring generated by $A$ and $K$, $M$
normalizes $R$; so by maximality of $M$ in $D^{*}$, it follows that either
$D^{*}$ normalizes $R$ or $R^{*}\leq M$. If the second case occurs, then $A$
is a locally nilpotent normal subgroups of $R^{*}$. By Lemma 3.2, $A$ is
abelian and this contradicts to the fact that $K^{*}$ is the Fitting subgroup
of $M$ (Theorem 3.5 $(ii)$). Thus, $D^{*}$ normalizes $R$, and by Stuth’s
theorem we have $R=D$. Therefore, $[D:K]_{r}\leq
m=|M/H^{*}|=|\mathrm{Gal}(K/F)|=[K:F]$. This implies that $[D:F]=m^{2}$ and
$K=H$ is a maximal subfield of $D$. From $M/H^{*}=AH^{*}/H^{*}$ and $K^{*}<A$,
it follows $M=A$. Since $F[M]=D$, $M$ is a locally nilpotent absolutely
irreducible subgroup of $D^{*}$ and by [24, 5.7.11 p. 215], $M/Z(M)$ is
locally finite. Thus, $K/F$ is a nontrivial radical Galois extension and by
[15, 15.13, p. 258], $F$ is algebraic over a finite field, so is $D$.
Therefore, by Jacobson’s theorem [15, p. 219], $D$ is a field, a
contradiction. Thus, $K^{*}=A$ is the Hirsch-Plotkin radical of $M$. ∎
The proof of the following corollary is similar as the proof of Corollary 3.6.
###### Corollary 5.2.
Let $D$ be a division ring with center $F$ and suppose $M$ is a maximal
subgroup of $D^{*}$. Assume that $M$ contains an abelian normal subgroup $A$
and there exists some element $\alpha\in A\setminus Z(M)$ such that $\alpha$
is algebraic over $F(Z(M))$. Then, $K=F[A]$ satisfies (i) - (v) in Theorem
5.1.
###### Lemma 5.3.
Let $D$ be a division ring with center $F$ and suppose $M$ is a maximal
subgroup of $D^{*}$ such that $C_{D}(M)=F$ and $F[M]^{*}=M$. If $M^{\prime}$
is algebraic over $F$, then $F[M^{\prime}]$ is an algebraic division
$F$-algebra and $F[M^{\prime}]^{*}\trianglelefteq M$.
###### Proof.
First, we claim that if $x\in M$ and $g(t)=(t-r_{n})\cdots(t-r_{1})\in D[t]$
with $r_{n},\ldots,r_{1}\in x^{M}$, then $g(x)\in M\cup\\{0\\}$. Let
$h(t)=(t-r_{n-1})\cdots(t-r_{1})$, by induction we have $h(x)\in
M\cup\\{0\\}$. If $h(x)=0$, then $g(x)=0$ as claimed. If $h(x)\neq 0$, then,
by [15, (16.3), p. 263], we have $g(x)=(x^{h(x)}-r_{n})h(x)$. Take
$r_{n}=x^{m}$ with $m\in M$, we have
$x^{h(x)}-r_{n}=(x^{m^{-1}h(x)}-x)^{m}=([m^{-1}h(x),x]-1)^{m}x^{m}$. On the
other hand, $[m^{-1}h(x),x]-1\in M$ since $M^{\prime}$ is algebraic over $F$
and $F[M]^{*}=M$. Thus, $x^{h(x)}-r_{n}\in M$ and so $g(x)\in M$.
Next, by the same argument as in the proof of Theorem 2.2 and what we proved,
if $x\in M$ is algebraic over $F$ with minimal polynomial $f(t)$ of degree
$n$, then there exists $x_{1},\ldots,x_{n-1}\in x^{M}$ such that
$f(t)=(t-x_{n-1})\cdots(t-x_{1})(t-x)\in D[t].$
Also by the same argument as in the proof of Corollary 2.4, we have $x^{n}\in
M^{\prime}F^{*}$. Thus, if $x,y\in M$ are algebraic over $F$, then
$xM^{\prime}F^{*}$ and $yM^{\prime}F^{*}$ are two elements of
$M/M^{\prime}F^{*}$ of finite order and so $xyM^{\prime}F^{*}$ is of finite
order. Since $M^{\prime}$ is algebraic over $F$, so is $xy$. If $x,y\in M$ are
algebraic over $F$, then, by $F[M]^{*}=M$, $x+y=x(1+x^{-1}y)$ is algebraic
over $F$ and belongs to $M$. Thus, $F[M^{\prime}]$ is an algebraic division
$F$-algebra and $F[M^{\prime}]^{*}\subseteq F[M]^{*}=M$. This completes our
proof. ∎
In the next theorem we get some result as in [1, Theorem 6], but with a weaker
condition. In fact, we replace the condition of algebraicity of $M$ by the
condition of algebraicity of derived subgroup $M^{\prime}$.
###### Theorem 5.4.
Let $D$ be a division ring with center $F$ and $M$ a non-abelian locally
solvable maximal subgroup of $D^{*}$ with $M^{\prime}$ is algebraic over $F$.
Then, $[D:F]=p^{2}$, $M/M^{\prime}F^{*}\cong\bigoplus_{i\in I}\mathbb{Z}_{p}$,
where $p$ is a prime number, and there exists a maximal subfield $K$ of $D$
such that $K/F$ is a Galois extension, $\mathrm{Gal}(K/F)\cong
M/K^{*}\cong\mathbb{Z}_{p}$ and $K^{*}$ is the $FC$-center and also the
Hirsch-Plotkin radical of $M$. Furthermore, for any $x\in M\setminus K^{*}$ we
have $x^{p}\in F,\ M=\bigcup_{i=1}^{p}K^{*}x^{i}$ and
$D=F[M]=\bigoplus_{i=1}^{p}Kx^{i}$.
###### Proof.
If $F(M)\neq D$, then by maximality of $M$, $M\cup\\{0\\}=F(M)$ is a division
ring. By Remark 2 in [4], $M$ is abelian, a contradiction. Hence $F(M)=D$ and
$C_{D}(M)=F$. Suppose that $F[M]\neq D$, so $F[M]^{*}=M$ by maximality of $M$.
By Lemma 5.3, $F[M^{\prime}]$ is a division ring whose multiplicative group is
locally solvable. The Remark 2 in [4] implies that $M^{\prime}$ is abelian, so
by Theorem 3.7, we have $F[M]=D$, a contradiction. Therefore, $M$ is
absolutely irreducible and by Corollary 4 of [27], $M$ is abelian-by-locally
finite. Thus, there exists an abelian normal subgroup $A$ of $M$ such that
$M/A$ is locally finite. We need to show that $D$ is locally finite over $F$.
Let us examine two possible cases.
Case 1: $A\cap M^{\prime}\subseteq F$
We note that $F(M^{\prime})^{*}$ is normalized by $M$. By maximality of $M$,
either $F(M^{\prime})\cap G\subseteq M$ or $G$ normalizes $F(M^{\prime})^{*}$.
In the first case, $M^{\prime}$ is a subnormal subgroup of
$F(M^{\prime})^{*}$. On the other hand, $M$ is abelian-by-locally finite, so
is $M^{\prime}$. By Lemma 3.1, $M^{\prime}$ is abelian; hence, by Theorem 3.7,
$[D:F]<\infty$, as claimed. In the second case, $G$ normalizes
$F(M^{\prime})^{*}$ and by Stuth’s theorem, either $F(M^{\prime})\subseteq F$
or $F(M^{\prime})=D$. If $F(M^{\prime})\subseteq F$, then, by Theorem 3.7,
$[D:F]<\infty$ and we are done. Suppose that $F(M^{\prime})=D$. We know that
$M^{\prime}/M^{\prime}\cap A\cong AM^{\prime}/A\leq M/A$ is locally finite, so
$M^{\prime}/M^{\prime}\cap F$ is locally finite since $M^{\prime}\cap
A\subseteq F$. Therefore, $D=F(M^{\prime})$ is locally finite over $F$.
Case 2: $A\cap M^{\prime}\not\subseteq F$
Since $M^{\prime}$ is algebraic over $F$, there exists an element $x\in A\cap
M^{\prime}\setminus F$ algebraic over $F$. By Corollary 5.2, there exists a
subfield $K_{1}$ of $D$ such that $K_{1}$ satisfies $(i)-(v)$ in Theorem 5.1.
So, $C_{D}(K_{1})^{*}\leq M$ is abelian-by-locally finite and by Lemma 3.1,
$C_{D}(K_{1})^{*}$ is abelian. Now, by Double Centralizer theorem,
$[D:F]<\infty$. Therefore, $D$ is locally finite. Since $M$ is locally
solvable, $M$ contains no noncyclic free subgroups. By Theorem 4.5, there
exists a maximal subfield $K$ of $D$ as in Theorem 4.5. Since $M$ is locally
solvable and $M/K^{*}$ is a finite simple group, we have
$M/K^{*}\cong\mathbb{Z}_{p}$, where $p$ is a prime number. Thus,
$M^{\prime}\leq K^{*}$ and $M$ is metabelian. Now, the proof is complete by
applying of Theorems 3.7 and 5.1. ∎
In [8, Theorem 3.2], it was proved that, if $D$ is a non-commutative division
ring, algebraic over its center, then any locally nilpotent maximal subgroup
of $D^{*}$ is the multiplicative group of some maximal subfield of $D$. Here,
we are now ready to generalize this result by the following theorem.
###### Theorem 5.5.
Let $D$ be a non-commutative division ring with center $F$ and $M$ a locally
nilpotent maximal subgroup of $D^{*}$. If $M^{\prime}$ is algebraic over $F$,
then $M$ is abelian. Consequently, $M$ is the multiplicative group of some
maximal subfield of $D$.
###### Proof.
If $M$ is non-abelian. Using Theorem 5.4, we conclude that the Hirsch-Plotkin
radical of $M$ is a properly subgroup of $M$, this contradicts to the fact
that $M$ is locally nilpotent. Thus $M$ is abelian. The last conclusion is
evident. ∎
## References
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* [7] Jairo Z. Goncalves and Arnando Mandel, Are there free groups in division rings? Israel J. of Math., Vol. 55, No. 1, 1986, 69-80.
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* [19] M. Mahdavi-Hezavehi, S. Akbari-Feyzaabaadi, M. Mehraabaadi, H. Hajie-Abolhassan, On derived groups of division rings II, Comm. Algebra, Volume 23 (1995), Issue 8, 2881 - 2887.
* [20] M. Mahdavi-Hezavehi, M. G. Mahmudi and S. Yasamin, Finitely generated subnormal subgroups of $GL_{n}(D)$ are central, J. Algebra 225 (2000), 517-521.
* [21] M. Mahdavi-Hezavehi, Free subgroups in maximal subgroups of $GL_{1}(D)$, J. Algebra 241 (2001), 720-730.
* [22] Derek J. S. Robinson, A Course in the Theory of Groups, Second Edition, Springer, 1995.
* [23] W. R. Scott, Group Theory, Dover Publication, INC, 1987.
* [24] M. Shirvani and B. A. F. Wehrfritz, Skew Linear Groups, Cambridge Univ. Press, Cambridge, 1986.
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|
arxiv-papers
| 2013-02-04T06:04:10 |
2024-09-04T02:49:41.247196
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Bui Xuan Hai and Nguyen Anh Tu",
"submitter": "Hai Bui Xuan",
"url": "https://arxiv.org/abs/1302.0589"
}
|
1302.0649
|
# Correlation effects in the electronic structure of the Ni-based
superconducting $KNi_{2}S_{2}$
Feng Lu1, Wei-Hua Wang2, Xinjian Xie3, and Fu-Chun Zhang1,4 1 Department of
Physics, and Center of Theoretical and Computational Physics, The University
of Hong Kong, Hong Kong, China
2 Department of Electronics and Tianjin Key laboratory of Photo-Electronic
Thin film Device and Technology, Nankai University, Tianjin, China
3 School of materials and engineering, Hebei University of Technology,Tianjin
300130, People s Republic of China 4 Department of Physics, Zhejiang
University, Hangzhou, China
(today)
###### Abstract
The LDA plus Gutzwiller variational method is used to investigate the ground-
state physical properties of the newly discovered superconducting
$KNi_{2}S_{2}$. Five Ni-3d Wannier-orbital basis are constructed by the
density-functional theory, to combine with local Coulomb interaction to
describe the normal state electronic structure of Ni-based superconductor. The
band structure and the mass enhanced are studied based on a multiorbital
Hubbard model by using Gutzwiller approximation method. Our results indicate
that the correlation effects lead to the mass enhancement of $KNi_{2}S_{2}$.
Different from the band structure calculated from the LDA results, there are
three energy bands across the Fermi level along the X-Z line due to the
existence of the correlation effects, which induces a very complicated Fermi
surface along the X-Z line. We have also investigated the variation of the
quasi-particle weight factor with the hole or electron doping and found that
the mass enhancement character has been maintained with the doping.
###### pacs:
74.20.-z, 74.20.Pq, 71.15.Mb
## I Introduction
The discovery of the second class of high-transition temperature
superconductors in LaFeAsO1-xFxKamihara-1 has stimulated great interest in
the study of iron pnictides. Over the past five years, many iron-based
compounds were reported to show superconductivity after doping or under high
pressuresKamihara-1 ; Chen-G-F-1 ; Lei-H-1 ; C-1 ; and the superconducting
(SC) transition temperature exceeds 50K in the $RFeAsO_{1-x}F_{x}$($R$: Rare
earth atoms)Wug-1 . Recently, one family $AFe_{2}Se_{2}(A=K,Rb$, or $Tl)$
compounds with the $ThCr_{2}Si_{2}$-type crystal structure, such as
$K_{y}Fe_{2}Se_{2}$Guo-Jin-1 and $(Tl,K)_{y}Fe_{2-x}Se_{2}$ Fang-Wang-1 ;
Wang-Dong-1 , have been found to become superconductors at about 30 K.
Different from other crystal structures of the iron-based superconductor,
these compounds $AFe_{2}Se_{2}(A=K,Rb$, or $Tl)$ have many unique physical
properties, such as absence of hole-like Fermi surface (FS)in
$K_{0.8}Fe_{1.7}Se_{2}$Qian-Wang-1 and the intrinsic $\sqrt{5}\times\sqrt{5}$
Fe vacancies superstructure in $K_{0.8}Fe_{1.6}Se_{2}$ Wei-Bao-1 ; Wei-Bao-2 ;
Wei-Bao-3 . Thus, searching for superconductors with the $ThCr_{2}Si_{2}$-type
crystal structure has become one of the interesting topics in condensed matter
physicsLee-P-A-1 ; C-2 .
Figure 1: (Color online) $KNi_{2}S_{2}$ with the $ThCr_{2}Si_{2}$-type
structure.
Very recently, a novel group of superconductors with $ThCr_{2}Si_{2}$-type
crystal structure, $KNi_{2}A_{2}(A=S,Se)$, have been discoveredNeilson-1 ;
Neilson-2 . The experiments show that the Ni-based superconductor exhibits a
rich and unusual electronic and structural phase diagramNeilson-1 ; Neilson-2
. For example, based on high-resolution synchrotron X-ray diffraction and
time-of-flight neutron scattering experiments, James et al. have found that
the mixed valence compound $KNi_{2}S_{2}$ displays a number of highly unusual
structural transitions and presents the charge density wave
fluctuationsNeilson-2 . The specific heat measurement reveals that
$KNi_{2}S_{2}$ exhibits the correlated behavior at low-temperature with an
enhanced effective electron mass, $m^{*}=11m_{b}$ to $24m_{b}$. Below
$T_{c}=0.46K$, $KNi_{2}S_{2}$ becomes superconducting without the partial
substitution of K or Ni deficientNeilson-2 . As discussed above, the Ni-based
superconductor $KNi_{2}S_{2}$ has a similar crystal structure as the iron-
based superconductor $AFe_{2}S_{2}$ compound. However, there are also many
differences between these two kinds of compounds. For example, the localized
magnetism is absent in the Ni-based superconductor $KNi_{2}S_{2}$Neilson-2 .
Thus, it is critical to understand the electronic structures of the parent
$ANi_{2}S_{2}$ compounds at first in order to investigate the mechanism of
superconductivity in this material.
In this paper, we have performed the first-principle electronic structure
calculations on $KNi_{2}S_{2}$. To overcome the problem of dealing with
Coulomb interaction in the local density approximation (LDA) calculation, we
use LDA plus the Gutzwiller variational approach to investigate this typical
correlated material. This method has proved to be effective in dealing with
Coulomb interaction and has been widely applied to strongly correlated
systemsDeng_Wang_1 ; Wang_Qian-2 . The calculation shows that $KNi_{2}S_{2}$
is a kind of multi-orbital system with five Ni-3d orbital near the Fermi
level. Our results indicate that the correlation effects of the 3d local
electrons lead to mass enhancement of $KNi_{2}S_{2}$. This behavior is always
present with the hole or electron doping, which is well agreement with the
experimental resultsNeilson-2 . The correlation effect is not only presented
by the narrowing of the bandwidth, but also by the change of the FS near the
high symmetry line. In the LDA calculation, there are two bands across the X-Z
line. Actually, three electron bands across the Fermi level along the X-Z
line, which induce a very complicated FS structure along the X-Z line. The
relationship among the quasi-particle weight factors of the different orbital
is also discussed in this paper.
## II Model and method
The first-principles calculations were performed within LDA using the plane
wave basis methodP-Giannozzi-1 . The generalized gradient approximation (GGA)
was adopted for the exchange-correlation potential of the density function
theory (DFT)Perdew-1 . We used the ultrasoft pseudopotentials to model the
electron-ion interactionsD-Vanderbilt-1 . The kinetic energy cutoff and the
charge density cutoff were taken as 4800 and 6400 eV, respectively. A mesh of
$18\times 18\times 9$ k points were used to perform the Brillouin zone
integration. $KNi_{2}S_{2}$ has $ThCr_{2}Si_{2}$-type structure (space group
I4$/$mmm, ($\\#139$, Z=1)) and each unit cell contains two formula units, as
shown in Fig.1. In our calculation, the structure parameters optimized by the
energy minimization, a=3.7792Å, c=12.7139Å, and $z(S)$=0.35, were adopted,
which are in good agreement with the experimental values a= 3.80899Å,
c=12.67609Å, and $z(S)$=0.34642 determined by synchrotron X-ray powder
diffractionNeilson-2 . Since the LDA can not describe the Coulomb interaction
adequately, in order to take into account more explicitly the correlated
effects of the d electrons, we have performed Gutzwiller variational approach
to deal with the Coulomb interaction.
Figure 2: (Color online) (a) Energy band structure of $KNi_{2}S_{2}$ by
LDA(black line) and Wannier function(dotted line) ; (b) The Fermi surface of
$KNi_{2}S_{2}$ by LDA calculation.
## III Results and discussions
We first present the electronic structure of $KNi_{2}S_{2}$ by LDA
calculation. The corresponding band structures and FS are given in Fig. 2a and
Fig. 2b, respectively. Similar to the iron-based superconductorMazin-I-I-1 ;
Singh-D-1 ; Boeri-L-1 ; C-Cao-1 ; F-Ma-1 ; lufeng-1 , there are several bands
across the Fermi level, which indicates that the $KNi_{2}S_{2}$ is a kind of
multi-orbital superconducting system. The multi-orbital character can also be
seen from the complicated FS sheets. As shown in Fig.2 b, there is a large
hole-like FS around the $\Gamma$ point, two 2D-cylinder-like ones around the
corners of the Brillouin zone and a very complicated pocket at the middle of
the four boundaries of the Brillouin zone. As mentioned above, the structure
of the FS is very complicated due to the multi-orbital characters, which
implies that the $KNi_{2}S_{2}$ should be a kind of multiple-gap
superconductors similar to the iron based superconductor, in contrast to the
single-orbital copper-based superconductors.
Figure 3: (Color online) (a) DOS and PDOS of $KNi_{2}S_{2}$ . (b) PDOS of
$KNi_{2}S_{2}$ for the five Fe-3d orbitals.
To investigate the orbital characters near the FS, the corresponding density
of states (DOS) and the projecting the density of states (PDOS) of
$KNi_{2}S_{2}$ are plotted in Fig.3. The basic shape of the DOS and PDOS is
very similar to that of $AFe_{2}Se_{2}$Yan_Miao_3 . However, the position of
the Fermi energy is higher due to the more electron count. As shown by the DOS
in Fig.3, the total DOS near the $E_{f}$ is manly contributed by the Ni-3d
orbitals, which mix with the S-3p orbitals near the $E_{f}$. In detail, the
PDOS of the Ni-3d orbitals consist of the DOS from -3 eV to 1 eV around the
$E_{f}$, and at the same time there is some contribution from S-3p orbitals.
Evidently, the contribution of S-3p orbitals is smaller than that from the
Ni-3d orbitals. The value of DOS at the Fermi energy N($E_{f}$) is about 6.9
states per eV per formula unit both spins.
Figure 4: (Color online) The quasi-particle weight factor Z as a function of
Coulomb interaction U at J=0.8eV. (a) The quasi-particle weight factor Z for
$d_{xy}$ orbital; (b) The quasi-particle weight factor Z for $d_{yz}$ orbital;
(c) The quasi-particle weight factor Z for $d_{xz}$ orbital; (d) The quasi-
particle weight factor Z for $d_{x^{2}-y^{2}}$ orbital; (e) The quasi-particle
weight factor Z for $d_{z^{2}}$ orbital; (f) The quasi-particle weight factor
Z as a function of doping $x$ for $K_{x}Ni_{2}S_{2}$ at U=8 eV and J=0.8eV.
As we know, both LDA and GGA fail to depict the strongly correlated electron
systems due to the insufficient treatment of electron correlation. In the last
20 years, a lot of new methods and new computational tools have been developed
to improve the calculation of strongly correlated electron materials, such as
the LDA+UAnisimov_Zaanen-1 , LDA+DMFT(dynamical mean-field theory)
Kotliar_Savrasov-1 and LDA+G (Gutzwiller method)Deng_Wang_1 . Among all these
methods, LDA+G is thought as one that is practically efficient and can capture
the key feature of the correlation effect as well in the ground-state studies.
The Gutzwiller variational approachGutzwiller-1 was firstly introduced by
Gutzwiller and has been proved to be quite efficient and accurate for the
ground-state physical properties studies, such as the the Mott transition,
ferromagnetism, and superconductivityDeng_Wang_1 ; Gutzwiller-1 ; Vollhardt-1
; Brinkman-1 ; Zhan_Gros_1 . In our calculation, we also used the LDA+G
framework to treat the correlation effects between the localized d electrons.
In other words, we firstly construct the multi-orbital Hubbard model and then
solve this model by the Gutzwiller variational approach (GW). Firstly, the
Wannier function is constructed by the maximally localized Wannier functions
(MLWFs) approachNicola_Arash-1 , as shown by the dotted line in Fig.2 a.
Secondly, combining with the Coulomb interaction, the multi-orbital model can
effectively describe the multi-orbital characters of $KNi_{2}S_{2}$. Then the
GW method is used to deal with this many-body model and the physical
properties in this system Deng_Wang_1 ; Gutzwiller-1 ; Vollhardt-1 ;
Brinkman-1 ; Zhan_Gros_1 .
We start from this five-orbital Hubbard model,
$\displaystyle H_{Tot}$ $\displaystyle=$ $\displaystyle H_{wannier}+H_{int},$
(1)
where $H_{wannier}$ describes the on-site energies and hopping integrals of
electrons between the Ni-3d Wannier orbitals; and $H_{int}$ is the local
interaction term for the Coulomb interaction between the d electron described
by the parameters $U$, $U^{\prime}$, $J$. The parameters $U$, $U^{\prime}$,
and $J$ denote the intra-orbital Coulomb interaction, inter-orbital Coulomb
interaction and Hund’s coupling. In what follows, considering the realistic
wavefunctions of 3d-orbitals Castellani and the spin rotational symmetry, we
adopt the relationships $U=U^{\prime}+2J$.
The true ground state of $H_{Tot}$ is described by the Gutzwiller wave
function $|\Psi_{G}>$, which is constructed by a many-particle projection
operator $\hat{P}_{G}$ on the uncorrelated wave function $|\Psi_{0}>$
$\displaystyle|\Psi>$ $\displaystyle=$
$\displaystyle\hat{P}_{G}|\Psi_{0}>=\prod_{i}\hat{P}_{i}|\Psi_{0}>$ (2)
with Gutzwiller projector operator
$\displaystyle\hat{P}_{i}$ $\displaystyle=$
$\displaystyle\sum_{\Gamma}\lambda_{\Gamma}|\Gamma><\Gamma|$ (3)
where $\lambda_{\Gamma}$ is the variational parameter for the i-th site with
the atomic eigenstates $|\Gamma>$, which is determined by minimizing the
ground state total energy. And the renormalization of kinetic energy can be
described by the bandwidth renormalization or quasi-particle weight factor Z,
as defined in Ref. 17.
Generally speaking, the essence of the correlation effects under Gutzwiller
approach is to suppress the double occupancy and to renormalize the kinetic
energy. Specifically, the Coulomb interaction U will enhance the orbital
polarization; however, the interorbital Hund s coupling J favors even
distribution of electrons among five orbitals. To understand the role of
Coumlomb interaction and the change of the band structure in the presence of
interactions, the value of the quasi-particle weight factor Z of the kinetic
energy as a function of U at fixed J=0.8 eV is shown in Fig. 4. As shown by
the solid line in Figs. 4(a), the Z factor decreases with the increase of the
Coulomb interaction U, due to the suppression of the double occupancy by U.
Because of the valence state of $Ni^{1.5+}$, each five Ni-3d orbitals have 8.5
electrons and are non-integer filling. Thus, the metal insulator transition
does not occur in this system and $KNi_{2}S_{2}$ is a multi-orbital metal
system, which is agreement with the experimentNeilson-2 . In the iron-based or
the copper-based superconductors, different doping with holes ($p$ type) or
electrons ($n$ type) has important influence on the superconducting transition
temperature, which lead to a rich phase diagram of these high temperature
superconductors. For example, superconductivity only occurs in a doping range,
which indicates that electron doping or hole doping may affect the electronic
properties in these materials. To study the doping effect on the physical
properties of $KNi_{2}S_{2}$ , we investigate the quasi-particle weight factor
Z at different doping situations, as shown in Fig. 4(a-e). The dotted line
shows Z at the $20\%$ hole doping, the black solid line shows Z without doping
and the dashed line shows Z at the $20\%$ electron doping, respectively. When
the Coulomb interaction U is not too large, the electron or hole doping almost
do not affect the quasi-particle weight factor Z. But, the difference of the
quasi-particle weight factor Z with different doping become more and more
obvious with the increase of the U in the large range; and the value of Z is
also dependent on the orbital degree of freedom due to the orbital anisotropy.
For example, the doping effect is very weak for the $d_{xy}$ orbital but
important for the $d_{z^{2}}$ orbital in the large U range, as shown in Fig.
4(a-e). At fixed Coulomb interaction value U=7 eV,J=0.8 eV, the variation of
the quasi-particle weight factor Z with the different doping is also
investigated and plotted in Fig. 4f. The results show that the correlated
electron character of the $KNi_{2}S_{2}$ has been maintained with the
electron-like or hole-like doping for all the Ni 3-$d$ orbitals, which is well
agreement with the experimental measurementslei-1 .
Figure 5: (Color online) Electronic band structure of $KNi_{2}S_{2}$
calculated by the Gutzwiller method.
Figure 6: (Color online) DOS and PDOS of $KNi_{2}S_{2}$ calculated by the
Gutzwiller method.
The corresponding band structures and PDOS of the Ni-based superconducting
compound $KNi_{2}S_{2}$ calculated by the GW method are presented in Fig. 5
and Fig. 6. Since the electron occupation number is 8.5 for each Ni-3d
orbitals and the five Ni-3d orbitals are almost occupied by the electrons, the
added Coulomb interactions can not make $KNi_{2}S_{2}$ become a Mott insulator
state, which is agreement with the experimentNeilson-2 . At a typical value of
Coulomb interaction U=8 eV and J=0.8 eV Lu_Zhao-1 ; Stewart_Liu-1 ; Han_Wang-2
, the quasi-particle weight factor is about 0.5. In other words, the overall
bandwidth calculated by the GW method is smaller than that by the LDA
calculation due to the strong correlation effect among the Ni-3d orbitals
electrons. In the LDA results, the band structure near Fermi level along the
X-Z line is very simple, which causes a two FS structure along the X-Z line.
Actually, our GW calculation shows that there are three Bloch bands across the
Fermi level along the X-Z line due to the existence of the correlation effect,
which would induce a very complicated FS along the X-Z line. The corresponding
electron FS will dominate the low energy properties along the X-Z line, which
is easier to be observed by the Angle-resolved photoemission spectroscopy
experiments (ARPES). As shown above, comparing with the LDA calculation, the
GW calculation can improve the correct band narrowing and mass
renormalization, which can also be seen from the DOS calculated by the LDA+G
method, as shown in the Fig.6. The corresponding orbital character is also
plotted in Fig.6. Comparing the orbital character of Ni-based superconductor
with that of the iron based superconductorWang_Qian-2 , we can clearly find
that all the Ni-3d orbitals are renormalized by the correlation effect. We
obtain the value of DOS at the Fermi energy N($E_{f}$) is about 11.5 states
per eV per formula unit both spins, which corresponds to a bare specific heat
coefficient, $\gamma=27$ mJ mol f.u.-1 K-2 based on the formula
$\gamma=\pi^{2}k^{2}N(E_{f})/3$Singh_2 ; Beck_Claus_1 . This $\gamma$ value is
2.5 times smaller than the experimental value $\gamma=68$ mJ mol f.u.-1
K-2Neilson-2 , but it is better than that of the LDA calculation,
$\gamma=16.2$ mJ mol f.u.-1.
## IV Conclusions
In conclusion, we have presented the results of the electronic structure of
$KNi_{2}S_{2}$ based on LDA plus Gutzwiller variational calculations. The
results show that the ground state of $KNi_{2}S_{2}$ is a strongly correlated
metal and the energy bands near Fermi level are dominated by the Ni-3d
electron states. The multi-orbital correlation effect between the Ni-3d
electrons determines the main physical properties of newly discovered Ni-based
superconductor $KNi_{2}S_{2}$, such as the mass enhancement behavior and the
multi-orbital characters. Different from the LDA understanding, the band
structure has a lot of changes under correlation effect, especially for the
band structure along the X-Z high symmetry line. In particular, the bands are
narrowed by a factor of about 2, and the five Ni-3d orbital are all almost
filled, which determine the low energy physical properties of $KNi_{2}S_{2}$.
The calculation about the electronic structure provides a reasonable starting
point for the investigations of the Ni-based superconductor. The variation of
the quasi-particle weight factor with the doping is also investigated and
compared to the experiments. Our results confirm that $KNi_{2}S_{2}$ is a
multi-orbital strongly correlated system with a reduced renormalized bandwidth
by the correlation effect.
###### Acknowledgements.
The authors thank X. Dai for helpful discussions. FCZ thanks the support of
RGC HKU707211 and AOE/P-04/08.
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|
arxiv-papers
| 2013-02-04T11:15:14 |
2024-09-04T02:49:41.256670
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Feng Lu, Wei-Hua Wang, Xinjian Xie, and Fu-Chun Zhang",
"submitter": "Feng Lu",
"url": "https://arxiv.org/abs/1302.0649"
}
|
1302.0702
|
# Segregation of impurities in GaAs and InAs nanowires
M. Galicka Institute of Physics, Polish Academy of Sciences, Al. Lotników
32/46, 02-668 Warsaw, Poland R. Buczko Institute of Physics, Polish Academy
of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland P. Kacman Institute
of Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw,
Poland [email protected]
###### Abstract
Using ab initio methods based on the density functional theory, we investigate
the segregation and formation energies for various dopants (Si, Be, Zn, Sn),
commonly used to obtain p- or n-type conductivity in GaAs and InAs nanowires.
The distribution of Au and O atoms, which may be unintentionally incorporated
during the wire growth, is also studied. The calculations performed for
nanowires of zinc blende and wurtzite structure show that the distribution of
most of the impurities depends on the crystal structure of the wires. For
example, it is shown that the same growth conditions can lead to lower energy
for Si substituting Ga (donor) in the wire of wurtzite structure and
substituting As (acceptor) in the wire of zinc blende structure. In contrast,
we obtain that gold and oxygen atoms always tend to stay at the lateral
surfaces of GaAs and InAs nanowires, in agreement with experimental findings,
while for beryllium the lowest energies are found when the impurities are
located in sites in the center of the wurtzite wire or along the
$[1,0,\overline{1}]$ axis from surface to the center of the zinc blende wire,
what can explain the recently observed diffusion of this impurity into the
volume of GaAs wires.
###### pacs:
63.20dk, 61.72uj, 62.23Hj, 61.72sh
## I Introduction
Successful realization of most of electronic nanodevices requires a high
carrier concentration so as to reach sufficient conductivity. The excellent
properties of III-V semiconductors, in particular the high electron mobility
which can be obtained in GaAs and InAs, turns nanowires (NWs) from these
materials into natural candidates for building blocks of such high speed
applications. The properties of semiconductor NWs differ from those of their
parent bulk materials due to quantum confinement and surface related effects –
pure semiconductor NWs usually have low concentration of carriers. Thus, the
application of semiconductor NWs in novel electronic devices requires
controllable p-type and n-type doping. This task still remains a big challenge
to the growers, despite substantial efforts, both experimental Piccin _et
al._ (2007); Tambe _et al._ (2010); Thelander _et al._ (2010); Ford _et
al._ (2010) and theoretical. Leao _et al._ (2008); Rurali and Cartoixà
(2009); Peelaers _et al._ (2007); Ghaderi _et al._ (2010); dos Santos _et
al._ (2011); Shu _et al._ (2011, 2012) Most of the publications describing
doping of III-V NWs focus on InAs NWs.Thelander _et al._ (2010) In these
structures the surface Fermi level is pinned in the conduction band. This
makes n-type conductivity in InAs NWs easy to obtain at the expense of
difficulties in p-type doping.Hasegawa and Akazawa (2008) In contrast, in GaAs
NWs the Fermi level at the surface is pinned approximately in the center of
the band gap and both, controlled p- and n-type doping, can be produces.
Doping of GaAs NWs grown by molecular beam epitaxy (MBE) has been demonstrated
in different means. Czaban et al. used Be and Te as p- and n-type dopant
precursors respectively,Czaban _et al._ (2009) while Fontcuberta i Morral et
al. pointed out that Si may act as both, acceptor and donor, by just changing
the operating temperature during growth as shown before for 2D epitaxial
growth of GaAs. Colombo _et al._ (2009); Dufouleur _et al._ (2010) The
incorporation of Si and Be into GaAs NWs was investigated in a further study.
Hilse _et al._ (2010) Recently, the use of tetraethyl tin as a precursor
material for successful n-type doping of GaAs NWs, grown by metal organic
vapor phase epitaxy, was reported.Gutsche _et al._ (2011) Moreover, it
occurred that not only specific growth conditions but also the crystal
structure of the wires may lead to different incorporation behavior than known
for planar layers.Dufouleur _et al._ (2010); Gutsche _et al._ (2011); Erwin
_et al._ (2005); Hilse _et al._ (2010) This was shown in particular for the
group IV element, silicon, which is known to be amphoteric in III-V compounds.
Namely, it was found that although under typical MBE growth conditions Si
acted as n-type dopant for GaAs, p-type conductivity was obtained in Si-doped
GaAs NWs which adapted wurtzite (wz) structure in similar growth
conditions.Piccin _et al._ (2007)
As mentioned above, doping of NWs was studied theoretically by several groups.
They focused, however, mainly on silicon NWs (see Ref. Chen _et al._ , 2012
and the references therein). There are very few publications which describe
theoretically, incorporation of non-magnetic dopants in GaAs and InAs NWs. To
our best knowledge, doping of GaAs NWs was considered only in Ref. Ghaderi
_et al._ , 2010, where only Si doping of wz structure GaAs NWs was studied. In
contrast, Clàudia dos Santos and co-authors considered Cd and Zn doping of
InAs NWs having the zinc-blende (zb) structure.dos Santos _et al._ (2011) In
Ref. dos Santos _et al._ , 2011 the studied NWs were extremely thin (with
diameters of 1 nm) and their lateral surfaces were passivated with hydrogen
atoms. Recently, the role of the surface dangling bonds and the effect of
molecular passivation on doping of InAs NWs was also studied by Haibo Shu et
al.Shu _et al._ (2011, 2012)
In this paper we report a systematic theoretical study of several dopants,
which are usually used to obtain p- or n-type conductivity in GaAs and InAs
NWs, i.e., Si, Be, Sn and Zn. In contrast to the previous papers, we assume
that the surface atoms are not saturated by foreign atoms. This assumption is
justified by the fact, that we consider incorporation of dopants during the
growth process in high vacuum conditions, e.g., MBE. To check what is the
preferential location of a given dopant in the wire, i.e., to answer the
question whether it is possible to explain the problems of doping of III-V NWs
by segregation phenomena, we study the energy of the wire with the impurity
situated in different sites. Such analysis proved already to be very useful,
e.g., for predicting conditions to obtain ferromagnetism in GaAs NWs with
magnetic Mn ions.Galicka _et al._ (2011) In Ref. Galicka _et al._ , 2011 it
was shown that the distribution of Mn ions and thus the electric and magnetic
properties of (Ga,Mn)As NWs depend strongly on the crystal structure. Since
the III-V semiconductor NWs can grow in both, zb and wz structures, we
consider here both types of wires to check whether the crystal structure has
an impact on the distribution of the studied impurities in GaAs and InAs NWs.
Finally, we consider the distribution of Au and O atoms in the studied NWs.
Oxygen can be used for n-type doping of GaAs Huang and Kuech (1994); Salem
_et al._ (2006) but also unintentional incorporation of O atoms in GaAs wires
has been reported in the literature.Zhi _et al._ (2005) On the other hand,
gold can defuse into the wire from the catalyst particle during Au-assisted
vapor - liquid - solid (VLS) growth. This is known to be a serious problem in
Si NWs.Allen _et al._ (2008) Although it seems that the unintentional
incorporation of Au in GaAs and InAs wires is much lower Perea _et al._
(2006) the measured Au doping values can make ballistic transport through the
nanowires difficult to obtain.Bar-Sadan _et al._ (2012) This makes the study
of the distribution of Au atoms in the wires very important. It should be
mentioned here that to avoid Au incorporation self-catalyzed growth of GaAs
and InAs NWs has been pursued by several groups (see, e.g., Mandl _et al._ ,
2006; Jabeen _et al._ , 2008; Heiß _et al._ , 2009; Krogstrup _et al._ ,
2010).
## II Methodology
In our calculations the doping of the wires is performed by substituting one
cation or anion in the elementary cell of the NW with an impurity atom. One
foreign atom in the elementary cell of the studied NWs corresponds to a
concentration of impurities in the range from $\sim\,$0.5% to $\sim\,$1.6%,
depending on the crystal structure and diameter of the wire. We study the
distribution of impurities by comparing the total energies of the NWs with an
impurity at various non-equivalent substitutional sites. In Fig. 1 we present
the dopant positions considered here. To a given cation site and the anion
above this cation we ascribe the same number. It should be emphasized that in
the cross section corners, i.e., in the positions 1 and 6, the geometry of
bonds is different for cations and anions. Namely, in site 1 the cation has
two dangling bonds while the anion has one dangling bond, and vice versa in
site 6.
a) b)
Figure 1: (Color online) Cross section of an a) wz NW with diameter d=2.8 nm
and b) zb NW (d=2.4 nm). The green balls show the 10 nonequivalent sites,
which we consider as substituted by a dopant. The blue balls denote Ga(In)
atoms, gray - As atoms. The darker colors denote atoms in the deeper double
layer.
The energies of the NWs are calculated by _ab initio_ methods, based on the
density functional theory (DFT). The energy of the wire with the dopant
occupying the site exactly at the center of the wire cross section is taken as
a reference point, i.e., as zero on the energy scale. The segregation energy
is defined by the difference between the energy of the NW with an impurity at
a given site and the reference energy:
${}E_{S}=E_{doped\ NW}-E_{center\ doped\ NW}.$ (1)
Thus, the negative value of the segregation energy for a dopant atom at a
given site means that this impurity prefers to substitute the host atom at
this particular site rather than be at the center of the wire. In contrast,
positive segregation energy values denote the situation when substitution of
an atom in the center of the NW will be energetically more favorable. Let us
mention that in a study of doped Ge wires,Peelaers _et al._ (2007) performed
with use of a similar method, the values of formation energy, i.e., energy
needed to insert the impurity atom (taken from a reservoir) into a given NW s
site, from which the original atom has been removed (to a reservoir), were
shown to depend strongly on the size of the simulated supercell and that
segregation energy, being a relative quantity, is free of these systematic
inaccuracies. However, by the study of segregation energies we can only answer
the question where an impurity prefers to stay within the wire. To answer the
question for which impurity the energy cost of substituting a host atom is
lower, one has to compare their formation energies, which depend on atomic and
electronic chemical potentials. The formation energy $\Omega$ of a neutral
impurity $X$ can be calculated according to the formalism proposed by Northrup
and Zhang, Northrup and Zhang (1993)
${}\Omega=E_{tot}[X]-E_{tot}[NW]+\mu_{Ga/As}-\mu_{X},$ (2)
where $E_{tot}[X]$ is the total energy derived from the calculation of the
nanowire with an impurity $X$, $E_{tot}[NW]$ is the total energy of undoped
nanowire. The term $\mu_{Ga/As}$ denotes the chemical potential of the
corresponding substituted Ga or As atom, while $\mu_{X}$ denotes the chemical
potential of the impurity atom. Our analysis is performed with an assumption
of thermal equilibrium conditions of the impurity region with the surrounding
nanowire, i.e.,
$\mu^{bulk}_{GaAs}=\mu_{Ga}+\mu_{As}.$ (3)
Using $\mu_{i}\leq\mu_{i}^{bulk}$ with $i\in{Ga,As,X}$ and the definition of
the heat of formation of GaAs
$\Delta H=\mu^{bulk}_{Ga}+\mu^{bulk}_{As}-\mu^{bulk}_{GaAs},$ (4)
we can limit the range of possible values of $\Delta\mu$ to $-\Delta
H\leq\Delta\mu\leq+\Delta H$. The quantity $\Delta\mu$ is the difference
between the chemical potentials of Ga and As and is given by
$\Delta\mu=(\mu_{Ga}-\mu_{As})-(\mu^{bulk}_{Ga}-\mu^{bulk}_{As}).$ (5)
Thereby, the situation $\Delta\mu=+\Delta H$ corresponds to Ga-rich growth
conditions, while $\Delta\mu=-\Delta H$ corresponds to As-rich conditions.
As mentioned above, we incorporate the dopants into GaAs and InAs wires of
both wz and zb crystal structures. The only considered NWs are those grown
along [0001] direction (wz) and along [111] axis (zb), because a vast majority
of III-V NWs grow only along these directions. It has been also shown
theoretically that these growth directions and the types of facets presented
in Fig. 1 constitute the most stable configurations of GaAs and InAs
NWs.Galicka _et al._ (2008); Glas _et al._ (2007); Dubrovskii _et al._
(2008); Cahangirov and Ciraci (2009); Shtrikman _et al._ (2009) It is well
known that with _ab initio_ methods only very thin NWs, much thinner than the
real ones, can be simulated. This can lead to overestimation of the side
surfaces’ effects in the results, deceptive when such calculations are used to
determine, e.g., the crystal structure of thicker wires. This problem is,
however, not so important in our study, in which we try to answer only the
question: is there any energy difference between the wires with an impurity
situated at the side facets and with the impurity inside the core of the wire.
The diameter test, in which we considered zb wires of several different
diameters, 0.9 nm, 1.4 nm, 1.8 nm and 2.4 nm, has shown that for the two
highest diameters the obtained segregation energies are practically the same.
Similarly, the results do not depend considerably on the diameter in wz NWs
with diameters from 2.0 nm to 2.8 nm. Congruous studies reported in Ref. Leao
_et al._ , 2008 also indicate that for diameters larger than approximately 3
nm the P impurity levels inside the Si NWs recover bulk characteristics. In
the following, all the presented results are for the thickest wz NWs with
diameters of 2.8 nm and the thickest (2.4 nm) zb wires. As already mentioned,
we assume that the surface atoms are not saturated by any foreign, for
example, hydrogen atoms, because we study the process of NWs growth and
incorporation of the impurities. Still, for the calculations of the density of
states of already doped wires and discussion about the positions of the Fermi
level, the side facets of the wires have been passivated by partial atomic
charges of hydrogen atoms.
The calculations are performed using the Vienna _ab initio_ simulation package
(VASP).Kresse and Hafner (1993); Kresse and Furthmüller (1996) Each of the
initial structures is cut out from an appropriate bulk material, in either zb
or wz structure. The wz elementary cell is doubled along the $c$ direction, to
guarantee that there is enough separation between the impurity ion and its
periodic images. Thus, the smallest supercell used in our study contains 122
atoms (zb) whereas the largest (wz) 384 atoms. For different dopants a plane
wave basis set is adopted with different cutoff energies: for Si the cutoff
energy was equal to 307 eV, for Be – 375 eV, Sn – 261 eV, Zn – 346 eV, Au –
287 eV and O – 500 eV. In each case all the energies and chemical potentials
are calculated with these cutoff values. These cutoff energies are sufficient
to ensure that the results do not depend on the size of the plane wave basis.
The k-points are generated with a ($1\times 1\times n$) mesh, where $n\geq
50~{}$Å$/c$ and $c$ is the unit cell dimension in the growth direction. The
interaction between the valence electrons and the ionic cores is included
using the projected augmented wave method (PAW).Blöchl (1994) The exchange
correlation energy is calculated within the generalized gradient approximation
(GGA). The atomic positions leading to minimum energy are determined with
relaxation of all atomic positions as well as of the unit cell dimension in
the growth direction, and with full reconstruction of the NW’s surfaces. In
each simulation process the atomic coordinates are relaxed with a conjugate
gradient technique. The criterion that the maximum force is smaller than 0.01
eV/Å is used to determine the equilibrium configurations. Neither pressure nor
temperature are included in our calculations.
## III Results
The distribution of gold in the cross-section of InAs wires was studied
experimentally using a technique called pulsed-laser atom probe
tomography.Perea _et al._ (2006) In Ref. Perea _et al._ , 2006 it was shown
that Au atoms accumulate in a shell at the lateral surfaces of the wire. We
start our study by considering Au impurities in InAs NWs to check if our
calculations can describe this experimental finding. We analyze InAs NWs with
an Au atom in various cation sites and focus on the search of the impurity
positions for which the segregation energy is the most negative, i.e., the
sites in which the energy cost of substituting the cation by an Au atom is
lower than in the other nonequivalent sites. In Fig. 2 the results for both wz
and zb InAs as well as GaAs NWs are presented. In this and all following
Figures the diameters of the wz NWs equal to 2.8 nm and of zb NWs to 2.4 nm.
In all Figures the calculated values of segregation energy are connected by a
line to guide the eye.
a) b)
Figure 2: (Color online) The segregation energy in a) wz (diameter 2.8 nm); b)
zb (diameter 2.4 nm) InAs (green) and GaAs (blue) NWs as a function of
different Au atom positions.
As one can see in Fig. 2, in both wz and zb NWs the Au atoms tend to
accumulate near the lateral surfaces. In wz structure the energy is the lowest
for the dopant occupying the threefold coordinated sites 1 and 4, the corner
sites of the sidewall facets. In NWs of zb structure the most energetically
favorable for Au atom is to substitute the cation in position 1, i.e., the
cation with extra dangling bond in the corner of the cross-section of the
wire. Analogous results are obtained for oxygen, another element, which can be
unintentionally incorporated during the growth, as shown for GaAs wires in
Fig. 3. The difference between the energy of the wire with an Au or O atom at
the lateral surface and at the center of the wire is significant, from ca 0.6
eV for gold in wz InAs NWs to nearly 3 eV for oxygen in GaAs zb NWs. Thus,
both these impurities should indeed be trapped at the lateral surfaces of the
wires. In zb NWs we note that the segregation energy decreases with the
distance from the center of the wire monotonically along the different
crystallographic axes, as shown for example in the inset to Fig. 3 for two
nonequivalent $[1,0,\overline{1}]$ directions.
a) b)
Figure 3: (Color online) The segregation energy in a) wz; b) zb GaAs NW as a
function of different O atom positions. In the inset to b) the segregation
energies for the O atom in subsequent sites along the two nonequivalent
$[1,0,\overline{1}]$ lines are shown
Next, we consider Si doping in GaAs NWs. As already mentioned, since Si is a
group IV element in III-V compounds it can serve either as a donor, by
substituting the cation, or as an acceptor, by occupying the anion site. We
therefore analyze the NWs with the Si atom in various cation and anion sites.
In Fig. 4 the results for both wz and zb NWs are presented.
a) b)
Figure 4: (Color online) The segregation energy in a) wz; b) zb GaAs NW as a
function of different Si atoms positions.
We recall that Si-doped GaAs NWs of wz structure were already studied by
Ghaderi et al. in Ref. Ghaderi _et al._ , 2010. These authors considered only
four dopant positions in a wz NW, i.e., one center-like, two surface threefold
(distinction is made for corner or middle of facets positions), and one
subsurface fourfold coordinated position. Our results presented in Fig. 4a,
which are obtained for many more Si positions in a thicker wz GaAs NW, fully
confirm the conclusion of Ref. Ghaderi _et al._ , 2010. Namely, the four-
coordinated subsurface positions are the most energetically favorable
locations for Si acting as a donor as well as an acceptor. The segregation in
these sites is, however, not as strong as for Au or O atoms (ca -0.2 eV). The
calculated segregation energies for Si-doped GaAs NWs of zb structure, which,
we recall, were not considered by the authors of Ref. Ghaderi _et al._ ,
2010, are presented in Fig. 4b. As one can notice by comparing Fig. 4a and b,
the spread of the segregation energy values is considerably larger for zb
wires ($\pm 0.7$ eV). It means that one can expect much more homogeneous
distribution of Si atoms in wz wires than in those of zb structure. For both
Si donors and acceptors the most preferable position in zb GaAs wires is site
7, i.e., the fourfold coordinated site next to site 1. However, for Si
substituting Ga (donor) low energy is also obtained for site 11, in the
vicinity of the corner anion with an extra dangling bond. We can thus conclude
that our calculations suggest that while in wz GaAs NWs the energy barrier at
the lateral surface is not very high and Si can be incorporated more or less
homogeneously, in zb GaAs NWs Si doping should result in accumulation of Si at
the shell.
The impact of the crystal structure on the incorporation of Si into the III-V
NWs is presented in Fig. 5. In this figure the formation energies for Si
substituting Ga and As in the center (site 16) of the wz GaAs NW (Fig. 5 a)
and center (site 21) of zb GaAs NW (Fig. 5 b) are compared. We observe that
there exists a region of $\Delta\mu$ in which the same growth conditions lead
to lower energy for Si substituting Ga in the wire of wz structure and
substituting As in the wire of zb structure.
a) b)
Figure 5: (Color online) The formation energy $\Omega$ of GaAs NW with Si
impurity in the center of a) wz NW, b) zb NW as a function of the value
$\Delta\mu$.
Next, we have calculated the density of states (DOS) for Si-doped GaAs
nanowires. In wz structure wires Si acts as expected, i.e., as a donor when
substituting Ga and as acceptor when substituting As. Impurity bands are
located close to valence band maximum (VBM) and conduction band minimum (CBM)
for Si acceptors and donors, respectively. The passivation of the lateral
surface have almost no influence on the position of these states, similarly to
what was shown for CdTe NWs of wz structure in the paper of T.
Sadowski.Sadowski and Ramprasad (2010) In contrast, in not passivated zb NWs
we observed surface states in the band gap and the impurity states (for both
Si at As and Ga sites) located at these surface states.
a) b) c)
Figure 6: (Color online) The density of states calculated for zb GaAs NW with
Si substituting Ga atom in the site 21 a) and site 1, c) and b), (site numbers
as denoted in Fig.1). The DOS presented in part b) is for the NW with
unpassivated lateral surfaces; the results in parts a) and c) are for NWs with
lateral surfaces passivated by H atoms. Zero on energy scale denotes always
the Fermi level.
After saturation the surface states disappear and the impurity states move
close to VBM and CBM, for Si atom substituting an anion or cation atom,
respectively, inside the wire core. In Fig. 6 a) we show the density of states
calculated for H-passivated zb GaAs nanowire with Si substituting the Ga atom
in the wire’s center position 21. The situation is quite different when Si
substitutes the Ga atom in position 1, in the corner of the wire cross-
section. For unpassivated nanowires (part b of Fig. 6) Si states and the Fermi
level are again pinned at the surface states in the band gap, as for the other
impurity positions in zb NWs. However, after passivation the Fermi level moves
to the VBM (compare part c of Fig. 6), what indicates that in this position Si
will not behave as an n-type dopant. As the segregation energies suggest that
most of the Si atoms are trapped around these corner positions, this behavior
can explain the observed dominance of p-type in Si-doped GaAs NWs. Piccin _et
al._ (2007)
In the next step we compare the segregation energy obtained for Si-doped GaAs
NWs with the segregation energy in wires doped with beryllium, one of the most
widely used acceptors in GaAs. Results obtained for Be-doped GaAs NW of wz and
zb structure are presented in Fig. 7.
a) b)
Figure 7: (Color online) The segregation energy in a) wz; b) zb GaAs NW as a
function of different Be atom positions.
As shown in the Figure, in wz NWs the distribution of Be atoms should be again
quite homogenous, but in this case the lowest energy corresponds to the
situation where the impurity substitutes an atom inside the wire. Again, in zb
GaAs NWs the lowest value of the segregation energy (-0.4 eV) was obtained
when Be substitutes the Ga atom with an extra dangling bond in site 1. It
should be noted, however, that this is much weaker segregation than obtained
in zb GaAs NWs for Si, and of course Au and O. Therefore, these results seem
to be in agreement with the experimental observation that even though Be atoms
accumulate in the shell of zb GaAs NWs, it should be possible to dope the NWs
almost homogeneously during axial growth.Casadei _et al._ (2013) It is also
interesting to note that for some reason the lowest energies of the zb GaAs
wires with Be atoms are found when the impurities are located in sites along
the $[1,0,\overline{1}]$ axis (from the center of the wire to the cation with
extra dangling bond at the corner of the lateral surface). This is in contrast
to the situation presented in Fig. 3 for oxygen and gold impurities. The low
energy cost of incorporation of Be along the $[1,0,\overline{1}]$ axis can
explain the recently observed diffusion of this impurity into the volume of
the zb GaAs NWs, interpreted in Ref. Casadei _et al._ , 2013 as an
alternative incorporation path.
Finally, the comparison of formation energies for Be incorporation into the
GaAs NWs, presented in Fig. 8, shows that it should be much easier to dope
with beryllium the NWs with zb structure than those with wz crystal structure.
It shows also that the formation energies for Be are much lower than for Si.
Figure 8: (Color online) The formation energy $\Omega$ of zb and wz GaAs NW
with Be impurity as a function of the value $\Delta\mu$.
As mentioned before, the as grown InAs NWs are usually n-type. Doping with
several acceptors, like Be or Zn, have been tried to obtain p-type
conductivity in these NWs.Sørensen _et al._ (2008); Ford _et al._ (2010) In
Fig. 9 we present the comparison of segregation energies obtained for InAs NWs
doped with Si, Be and Zn atoms as a function of their different locations. The
study of segregation energy for different acceptors in InAs NWs shows that in
these wires, similarly to GaAs NWs, wz structure should lead to a more
homogeneous distribution of impurities. Inside the wire, in the core, all
impurity positions are energetically equivalent in both structures. In the
shell, Be and Zn prefer to be in the middle of the sidewall of the wire,
substituting threefold coordinated cations, while Si, substitutes fourfold
coordinated anion at the NW’s lateral surface (site 6), as presented in Fig.
10. One can understand this difference by recalling that Zn and Be occupy the
cation positions while Si, should substitute an anion to serve as an acceptor.
The relaxation of the atom positions at the wire lateral surfaces leads to
different reconstruction of bonds for cations and anions, as shown in Ref.
Galicka _et al._ , 2008 for pure III-V NWs. The most important result of this
study is, however, the fact that in InAs NWs of wz structure the segregation
energy for Be atoms is
a) b)
Figure 9: (Color online) The segregation energy for Si, Be and Zn acceptors in
various positions in wz (a) and zb (b) InAs NW. Figure 10: (Color online) The
most energetically favorable positions of Zn, Be and Si acceptors at the
lateral surfaces of wz InAs NW.
not negative for any of the considered positions. Also for Si in wz InAs NWs
the segregation energy for the impurity in site 6, although negative, is close
to zero (-0.06 eV). Thus, we can conclude that the best precursor for p-type
doping of InAs NWs should be beryllium but using Si can also be effective,
provided the wz crystal structure of the wires is assured.
a) b)
Figure 11: (Color online) The segregation energy of a) wz; b) zb InAs NW as a
function of different Sn positions.
In Ref. Astromskas _et al._ , 2010 it was shown, by capacitance measurements,
that the concentration of carriers in InAs NWs can be increased by n-type
doping with Sn. In Fig. 11 the segregation energy obtained for InAs NWs doped
with Sn is presented. The results for Sn in wz InAs wires are very similar to
that for Si-doped InAs NWs, presented above. Tin ions distribute fairly
homogeneously across the wz wire. Although for Sn the most favorable is to
substitute the fourfold coordinated cations at the wz NW’s lateral surface,
the segregation energy is again very small. This result seems to agree with
the observed increase of the carrier concentration and the surface charge
density in Sn-doped wz InAs NWs.Astromskas _et al._ (2010) It should be noted
that in Ref. Thelander _et al._ , 2010 it was shown that the Sn precursor
increases the stacking fault density in wz NWs, ultimately at high flows
leading to a zb crystal structure with strong overgrowth and very low
resistivity. In agreement with this observation we obtain that in zb NWs,
analogously to other dopants in this material, Sn atoms do not incorporate
into the wire but stay at the lateral surface, with particular reference to
atoms with extra dangling bond (Fig. 11b).
In contrast, our calculations performed for GaAs NWs show that in this
material Sn atoms tend to segregate to the lateral surfaces in both wz and zb
structures (see Fig. 12). Thus, although our calculations performed for Sn
impurities in III-V wires agree with the observation that doping with tin can
be used to increase the concentration of electrons in InAs wires,Astromskas
_et al._ (2010) they suggest that using Sn for n-type doping of GaAs NWs can
be much less efficient.
a) b)
Figure 12: (Color online) The segregation energy of a) wz; b) zb GaAs NW as a
function of different Sn atoms positions.
## IV Conclusions
Nanowire growth processes usually rely on the vapor - liquid - solid (VLS)
mechanism normally assisted by gold droplets or self assisted on a SiO/SiO2
surface. Doping these structures during such complex growth process is
difficult to understand and control. It is, however, already widely accepted
that there are two distinct pathways for dopant incorporation into
semiconductor NWs: the VLS mechanism, in which the dopant atoms first dissolve
in the catalyst and are then transported across the liquid - solid interface
to the core of the NW, and the vapor - solid (VS) mechanism, when dopant atoms
are directly deposited on the lateral NW surfaces. In Ref. Perea _et al._ ,
2009 it has been suggested that these two mechanisms proceed at different
rates – namely, the latter predominates and leads to a much higher
concentration of dopants in the shell of the NW than in the core. This
conclusion was based on the experimentally observed accumulation of gold atoms
close to the lateral surfaces of InAs NWs Perea _et al._ (2006) as well as
phosphorus in Ge NWs.Perea _et al._ (2009) Radovanovic and co-workers, when
studying GaN wires doped with Co and Cr, have also concluded that adsorption
on the lateral surface is the main doping mechanism of the semiconductor
NWs.Stamplecoskie _et al._ (2008) Finally, the conclusion that Be atoms are
mostly incorporated into the zb GaAs NWs from the side facets and that the
incorporation through the Ga droplet is negligible are made in Ref. Casadei
_et al._ , 2013. It should be also mentioned that the very recently reported
beautiful model,Chen _et al._ (2012) in which B-doped Si NWs were considered,
has shown that 2D diffusivity along the solid-liquid NW interface plays also a
considerable role in impurity incorporation. In this paper we show, by means
of first-principles calculations of the total energy of doped GaAs and InAs
NWs, that indeed most of the commonly used impurities for p- and n-type doping
should accumulate near the surfaces of the NW. It should be, however,
emphasized that this result does not depend on the growth mechanism, because
in our calculations only the differences in formation energies for various
dopant positions have been taken into account. Thus, it shows that even for
incorporation of the dopants into the core, via VLS mechanism, the segregation
of impurities to the surfaces should take place. The only exceptions are wz
GaAs as well as wz InAs NWs doped with Be – in these the segregation energy is
positive for all studied cation sites substituted by Be atoms.
We note that the degree of segregation depends considerably on the NW’s host
material, crystal structure of the wire and of course the particular dopant.
Our calculations show that in NWs of zb structure the segregation for the most
common dopants is much higher than in NWs of wz structure. The studied
impurities in zb NWs should remain either at the lateral surfaces, where they
prefer to substitute the atom with extra dangling bond, or should be trapped
at the subsurface, in the vicinity of the atom with additional dangling bond.
At these positions the segregation energies in zb GaAs NWs are lower than in
the core by ca 0.4 eV for beryllium, but by even 1.5-3 eV for gold and oxygen.
It is also shown that at these positions Si atoms do not act as donors even
when substituting Ga cations - this result is in agreement with the observed
predominance of p-type in Si-doped GaAs NWs. For gold and oxygen in wz wires
the results are similar to those obtained for zb structure but with lower,
though still high, segregation. With our calculations we have thus confirmed
the experimentally observed segregation of Au atoms to lateral surfaces.Perea
_et al._ (2006) According to our results the same should be also observed for
oxygen. In contrast, all other studied impurities distribute much more
homogeneously across the wz wire and Be even prefers to substitute the Ga(In)
ion at the center of the wire, as mentioned above. We can thus conclude that
our calculations suggest that growth conditions leading to wz structure can
help to avoid accumulation of impurity atoms at the surface during the growth
of doped III-V wires. It seems that the best choice for effective p-type
doping of both GaAs and InAs wz NWs is to use beryllium as a dopant, due to
it’s low segregation. However, the cost of substituting the host cation by Be
atom, i.e., the formation energy, is lower in the zb than in the wz NWs.
Finally, we observe that the different from other impurities behavior of Au
and O can be attributed to the large electronegativity of these elements (3.5
for O and 2.4 for Au) as compared to electronegativities of Ga, In and all
other studied dopants, which are in the range of 1.5 – 1.8.
As shown above, our results suggest that the energy of a doped NW can be very
different when the impurity position inside the wire changes, especially in zb
structure. Only in the case of oxygen and beryllium in GaAs NWs some
regularities in these differences can be observed. In other cases the
fluctuations of the segregation energy values seem to be random. These results
were fully confirmed by several tests, i.e., by repeating the calculations for
the impurities in equivalent sites, by changing the NW diameter and finally by
moving slightly the initial atomic positions. Recently, thanks to laser
assisted atom probe tomography the studies of the distribution of impurities
in semiconductor wires have become possible. Our results may suggest that the
observed inhomogeneities (compare, e.g., Ref. Chen _et al._ , 2012) may have
a more fundamental origin than just disorder. Neither the noticed trends nor
the other segregation fluctuations, which we are not able to explain in our
approach, were reported before. It should be noted, however, that in all
previous studies the segregation and/or formation energies in doped NWs have
been calculated for at most 3-4 different impurity positions. In our study the
results for wz NWs with the impurities in 16 different sites and for zb NWs in
21 different sites have been compared - to our best knowledge, most of the
nonequivalent sites considered here have not been studied before for any
impurity.
###### Acknowledgements.
We thank Hadas Shtrikman, Peter Krogstrup, Hanka Przybylinska and Ewa
Bialkowska-Jaworska for fruitful discussions and valuable comments. The
research leading to these results has received funding from the European
Community’s 7th Framework Programme [FP7/2007-2013] under grant agreement
n°215368, from Foundation for Polish Science and Ministry of Higher Education
(Poland) Grant IP 2011013671. All computations were carried out in the
Informatics Center Tricity Academic Computer Net (CI TASK) in Gdansk.
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|
arxiv-papers
| 2013-02-04T14:28:05 |
2024-09-04T02:49:41.264604
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Galicka and R. Buczko and P. Kacman",
"submitter": "Marta Galicka",
"url": "https://arxiv.org/abs/1302.0702"
}
|
1302.0716
|
# Langmuir dark solitons in dense ultrarelativistic electron-positron gravito-
plasma in pulsar magnetosphere
U. A. Mofiz1111Corresponding author; email address: [email protected]
telephone:880-2-8824051-4 ext.4078, fax: 880-2-8810383. and M. R. Amin2
1Department of Mathematics and Natural Sciences, BRAC University, 66
Mohakhali, Dhaka-1212, Bangladesh
2Department of Electronics and Communication Engineering, East West
University, Jahurul Islam City. Aftabnagar, Dhaka-1212, Bangladesh
###### Abstract
Abstract
Nonlinear propagation of electrostatic modes in ultrarelativistic dense
elelectron-positron gravito-plasma at the polar cap region of pulsar
magnetosphere is considered. A nonlinear Schrödinger equation is obtained from
the reductive perturbation method which predicts the existence of Langmuir
dark solitons. Relevance of the propagating dark solitons to the pulsar radio
emission is discussed.
Keywords: Electron-positron plasma; Langmuir solitons; pulsar magnetosphere.
## I. Introduction
Pulsars are celestial sources that believe to be rotating neutron stars
producing light-house like beams of radio emissions from the magnetic poles.
As shown by Goldreich and Julian (1969) the rotating magnetic dipole produces
a quadrupole electric field whose component parallel to the open magnetic
field lines at the poles extracts particles very effectively from neutron star
surface and accelerates them to highly relativistic energies. Thus, the
magnetosphere is filled with plasma which shields the electric field. Complete
shielding is established when the net charge reaches $n_{GJ}$ \- the Goldreich
-Julian charge density. The Lorentz factors of the accelerated particles reach
about $10^{6}$ and they emit hard curvature radiations that propagate at a
sufficient angle to the magnetic field, so that significant pair production of
electron-positron can occur (Erber 1966). It is commonly accepted that the
newly created particles produce more pairs by emitting energetic synchrotron
or curvature radiation. As a result an avalanche of secondary particles
populates the magnetosphere with densities $10^{4}n_{GJ}$ (Ruderman and
Sutherland, 1975) . Here, we extend our earlier research on pulsar
microstructure, soliton formation, wakefield accelerations, gravitational
waves , and growing modes (Mofiz, et al. 1985-2011) to account the pair
ultrarelativistic pressure and the gravity
The paper is organized as follows. Section II describes the fluid model of the
dense ultrarelativistic electron-positron plasma under gravity. Considering a
Lorentz invariant frame moving with group velocity of the wave, a Nonlinear
Schrödinger Equation (NLSE) is derived using the reductive perturbation method
(Gardner and Morikawa, 1960). A linear dispersion relation is obtained showing
the existence of Langmuir waves under gravity and with ultrarelativistic
temperature for wave propagation. The solution of NLSE shows the generation of
Langmuir dark solitons. Results are discussed in Sec. III. Finally, Sec. IV
concludes the paper.
## II. The Mathematical Model
We consider two-fluid magnetohydrodynamic (MHD) equations to describe the
electron-positron plasma in the pulsar magnetosphere of the neutron star. The
equations are the usual continuity and momentum balance equations for the
plasma species, electrons and positrons, supplemented by the Poisson’s
equation for electrostatic wave propagation . Thus, the required set of
equations are as follows (Mofiz and Ahmedov, 2000):
$\displaystyle\frac{\partial}{\partial
t}\left(\gamma_{s}n_{s}\right)+\nabla\cdot\left(\sqrt{1-\frac{r_{g}}{r}}\gamma_{s}n_{s}\bf{v}_{s}\right)=0,$
(1) $\displaystyle\frac{\partial}{\partial
t}\left(\gamma_{s}\bf{v}_{s}\right)$ $\displaystyle+$
$\displaystyle\sqrt{1-\frac{r_{g}}{r}}\>\bf{v}_{s}\cdot\nabla\left(\gamma_{s}\bf{v}_{s}\right)$
(2) $\displaystyle=$
$\displaystyle-\frac{q_{s}}{m}\nabla\phi-\frac{1}{m\gamma_{s}n_{s}}\nabla
p_{s},\>\>\>\>$
$\displaystyle\nabla\cdot\left(\frac{1}{\sqrt{1-\frac{r_{g}}{r}}}\nabla\phi\right)=-4\pi\sum_{s}\gamma_{s}q_{s}n_{s},$
(3)
where, $\gamma_{s}=1/\sqrt{1-v_{s}^{2}/c^{2}}$, $n_{s}$, $\bf{v}_{s}$, and
$q_{s}$ are respectively the particle number density, particle velocity, and
particle charge of plasma species $s$; $q_{s}=-e$ for $s=e$ (electron) and
$q_{s}=+e$ for $s=e^{+}$ (positron); $\phi$ is the electrostatic potential,
$r_{g}$ is the Schwarzschild radius of the neutron star; $m$ is the
electron/positron mass; $e$ is the absolute value of the electronic charge. In
Eq. (2), the pressure $p_{s}$ is given by the expression for the
ultrarelativistic pressure (Chandrasekhar, 1938):
$p_{s}=n_{0}k_{B}T\left(n_{s}/n_{0}\right)^{4/3}$, where $T_{e}=T_{p}=T$ has
been assumed; $k_{B}$ is the Boltzmann constant, $n_{0}$ is the equilibrium
particle number density.
In the polar cap region of the pulsar, we consider $\theta=0$,
$\nabla=\widehat{z}\partial/\partial z$, ${\bf{v}}_{s}=v_{sz}\widehat{z}$ and
adopt the following normalization of different quantities: $z\rightarrow
z\omega_{pe}/c$, $t\rightarrow t\omega_{pe}$, $n_{s}\rightarrow n_{s}/n_{0}$,
$u_{s}\rightarrow v_{sz}/c$, $\phi\rightarrow e\phi/mc^{2}$,
$\sigma_{T}\rightarrow k_{B}T/mc^{2}$, $r_{g}\rightarrow r_{g}\omega_{pe}/c$,
and $v_{g}\rightarrow v_{g}/c$, where $\omega_{pe}=\left(4\pi
n_{0}e^{2}/m\right)^{1/2}$ is the electron plasma frequency, $c$ is the speed
of light. To study the nonlinear dynamics, we consider the following stretched
coordinates in the moving frame (Melikidze et al. 2000):
$\displaystyle\xi=\epsilon\gamma_{0}\left(z-v_{g}t\right),$ (4)
and
$\displaystyle\tau=\epsilon^{2}\gamma_{0}\left(t-v_{g}z\right),$ (5)
where, $v_{g}$ is the group velocity, $\gamma_{0}=\gamma_{e}=\gamma_{p}$ is
the average relativistic Lorentz factor and is given by the following
expression: $\gamma_{0}=\left(1-v_{g}^{2}\right)^{-1/2}$, $\epsilon$ is a
small quantity, the perturbation parameter with $\epsilon<1$. With the
transformations, given by Eqs. (4) and (5), the derivatives $\partial/\partial
t$ and $\partial/\partial z$ transform to $\partial/\partial
t\rightarrow\partial/\partial
t-\epsilon\gamma_{0}v_{g}\partial/\partial\xi+\epsilon^{2}\gamma_{0}\partial/\partial\tau$
and $\partial/\partial z\rightarrow\partial/\partial
z+\epsilon\gamma_{0}\partial/\partial\xi-\epsilon^{2}\gamma_{0}v_{g}\partial/\partial\tau$
respectively. With the above considerations, the set of equations, Eqs.
(1)-(3) take the following forms:
$\displaystyle\frac{\partial n_{s}}{\partial t}$ $\displaystyle+$
$\displaystyle\frac{\partial}{\partial z}\left(g(z)n_{s}u_{s}\right)$ (6)
$\displaystyle+$ $\displaystyle\epsilon\left[-\gamma_{0}v_{g}\frac{\partial
n_{s}}{\partial\xi}+\gamma_{0}\frac{\partial}{\partial\xi}\left(g(z)n_{s}u_{s}\right)\right]$
$\displaystyle+$ $\displaystyle\epsilon^{2}\left[\gamma_{0}\frac{\partial
n_{s}}{\partial\tau}-\gamma_{0}v_{g}\frac{\partial}{\partial\tau}\left(g(z)n_{s}u_{s}\right)\right]=0,$
$\displaystyle\frac{\partial u_{s}}{\partial t}$ $\displaystyle+$
$\displaystyle g(z)u_{s}\frac{\partial u_{s}}{\partial
z}+\frac{q_{s}}{e\gamma_{0}}\frac{\partial\phi}{\partial
z}+\frac{4\sigma_{T}}{3\gamma_{0}^{2}}n_{s}^{-2/3}\frac{\partial
n_{s}}{\partial z}$ (7) $\displaystyle+$
$\displaystyle\epsilon\left[-\gamma_{0}v_{g}\frac{\partial
u_{s}}{\partial\xi}+g(z)u_{s}\gamma_{0}\frac{\partial
u_{s}}{\partial\xi}\right.$ $\displaystyle+$
$\displaystyle\left.\frac{q_{s}}{e}\frac{\partial\phi}{\partial\xi}+\frac{4\sigma_{T}}{3\gamma_{0}}n_{s}^{-2/3}\frac{\partial
n_{s}}{\partial\xi}\right]$ $\displaystyle+$
$\displaystyle\epsilon^{2}\left[\gamma_{0}\frac{\partial
u_{s}}{\partial\tau}-g(z)\gamma_{0}v_{g}u_{s}\frac{\partial
u_{s}}{\partial\tau}\right.$ $\displaystyle-$
$\displaystyle\left.\frac{q_{s}v_{g}}{e}\frac{\partial\phi}{\partial\tau}-\frac{4\sigma_{T}v_{g}}{3\gamma_{0}}n_{s}^{-2/3}\frac{\partial
n_{s}}{\partial\tau}\right]=0,$
$\displaystyle\frac{\partial}{\partial
z}\left(\frac{1}{g(z)}\frac{\partial\phi}{\partial z}\right)$ $\displaystyle-$
$\displaystyle\gamma_{0}\left(n_{e}-n_{p}\right)$ (8) $\displaystyle+$
$\displaystyle\epsilon\left[\gamma_{0}\frac{\partial}{\partial\xi}\left(\frac{1}{g(z)}\frac{\partial\phi}{\partial
z}\right)\right.$ $\displaystyle+$
$\displaystyle\left.\frac{\partial}{\partial
z}\left(\frac{\gamma_{0}}{g(z)}\frac{\partial\phi}{\partial\xi}\right)\right]$
$\displaystyle+$
$\displaystyle\epsilon^{2}\left[-\gamma_{0}v_{g}\frac{\partial}{\partial\tau}\left(\frac{1}{g(z)}\frac{\partial\phi}{\partial
z}\right)\right.$ $\displaystyle-$
$\displaystyle\left.\gamma_{0}v_{g}\frac{\partial}{\partial
z}\left(\frac{1}{g(z)}\frac{\partial\phi}{\partial\tau}\right)\right.$
$\displaystyle+$
$\displaystyle\left.\gamma_{0}\frac{\partial}{\partial\xi}\left(\frac{\gamma_{0}}{g(z)}\frac{\partial\phi}{\partial\xi}\right)\right]$
$\displaystyle+$
$\displaystyle\epsilon^{3}\left[-2\gamma_{0}^{2}\frac{\partial}{\partial\xi}\left(\frac{v_{g}}{g(z)}\frac{\partial\phi}{\partial\tau}\right)\right.$
$\displaystyle-$
$\displaystyle\left.\gamma_{0}^{2}v_{g}\frac{\partial}{\partial\tau}\left(\frac{1}{g(z)}\frac{\partial\phi}{\partial\xi}\right)\right]$
$\displaystyle+$
$\displaystyle\epsilon^{4}\left[\gamma_{0}^{2}v_{g}^{2}\frac{\partial}{\partial\tau}\left(\frac{1}{g(z)}\frac{\partial\phi}{\partial\tau}\right)\right]=0,$
where, the factor $g(z)=\left(1-r_{g}/z\right)^{1/2}$ accounts for the
gravitational effect. Now we expand the quantities $n_{s}$, $u_{s}$, $\phi$ as
$\displaystyle n_{s}$ $\displaystyle=$ $\displaystyle 1+\epsilon^{2}n_{s0}$
(9) $\displaystyle+$
$\displaystyle\sum_{l=1}^{\infty}\epsilon^{l}\left(n_{sl}e^{il(kz-\omega
t)}+n_{sl}^{*}e^{-il(kz-\omega t)}\right),$ $\displaystyle
u_{s}=\epsilon^{2}u_{s0}+\sum_{l=1}^{\infty}\epsilon^{l}\left(u_{sl}e^{il(kz-\omega
t)}+u_{sl}^{*}e^{-il(kz-\omega t)}\right),$ (10)
$\displaystyle\phi=\epsilon^{2}\phi_{0}+\sum_{l=1}^{\infty}\epsilon^{l}\left(\phi_{l}e^{il(kz-\omega
t)}+\phi_{l}^{*}e^{-il(kz-\omega t)}\right).$ (11)
Here,
$(n_{s0}$, $u_{s0}$, $\phi_{0}$ , $n_{sl}$, $u_{sl}$, $\phi_{l})$ $\equiv$
$A^{(1)}+\epsilon A^{(2)}+\epsilon^{2}A^{(3)}+.......$ are functions of
stretched coordinates $(\xi,\tau)$.
II. A. Linear Dispersion Relation for the Langmuir Wave
Now considering $\left|1/g(z)\cdot dg(z)/dz\right|<<k$ for the first harmonic
($l=1$)in the first order ($\epsilon=1$ ), we have the following equations for
the first-order quantities:
$\displaystyle-i\omega n_{s1}^{(1)}+ikg(z)u_{s1}^{(1)}=0,$ (12)
$\displaystyle-i\omega
u_{s1}^{(1)}+\frac{4ik\sigma_{T}}{3\gamma_{0}^{2}}n_{s1}^{(1)}+\frac{ikq_{s}}{e\gamma_{0}}\phi_{1}^{(1)}=0,$
(13)
$\displaystyle-\frac{k^{2}}{g(z)}\phi_{1}^{(1)}-\gamma_{0}\left(n_{e1}^{(1)}-n_{p1}^{(1)}\right)=0.$
(14)
Eliminating $u_{s1}^{(1)}$ from Eqs. (12) and (13), we obtain the following
equation relating $n_{s1}^{(1)}$ and $\phi_{1}^{(1)}$:
$\displaystyle
n_{s1}^{(1)}=-\frac{k^{2}g(z)}{-\omega^{2}+4\sigma_{T}g(z)k^{2}/3\gamma_{0}^{2}}\frac{q_{s}}{e\gamma_{0}}\phi_{1}^{(1)},$
(15)
from which we obtain
$\displaystyle
n_{e1}^{(1)}-n_{p1}^{(1)}=\frac{2k^{2}g(z)}{-\omega^{2}+4\sigma_{T}g(z)k^{2}/3\gamma_{0}^{2}}\frac{1}{\gamma_{0}}\phi_{1}^{(1)}.$
(16)
Using Eq. (16) into Eq. (14), we obtain the following linear dispersion
relation:
$\displaystyle\omega^{2}=2g^{2}(z)+\frac{4\sigma_{T}g(z)}{3\gamma_{0}^{2}}k^{2},$
(17)
which in the dimensional form is
$\displaystyle\omega^{2}=\omega_{pe}^{2}g^{2}(z)+\frac{2}{3}k^{2}v_{th}^{2}g(z),$
(18)
with $v_{th}^{2}=\frac{2k_{B}T}{m\gamma_{0}^{2}}$. The group velocity
$v_{g}=\partial\omega/\partial k$ is obtained from the linear dispersion
relation Eq. (17) as:
$\displaystyle v_{g}=\frac{4\sigma_{T}g(z)}{3\gamma_{0}^{2}}\frac{k}{\omega}.$
(19)
The same expression for $v_{g}$ is also obtained from the compatibility
condition and shown in the Appendix A.
The group dispersion is found to be
$\displaystyle
v_{g}^{{}^{\prime}}=\frac{dv_{g}}{dk}=\frac{\frac{4}{3}\frac{\sigma_{T}g(z)}{\gamma_{0}^{2}}-v_{g}^{2}}{\omega}.$
(20)
Eq.(17) represents the dispersion relation for Langmuir waves in
ultrarelativistic $e,e^{+}$ plasma under gravity. Pair production in the polar
cap region of pulsar magnetosphere occurs through curvature radiation which
happens for $\mathcal{E}_{\parallel}>>m_{e}c^{2}$, where
$\mathcal{E}_{\parallel}$ is the energy of electron along the magnetic field.
It is estimated that for cascade generation of pair plasma $\gamma_{0}\sim
10^{6}-10^{7}$,$\mathcal{E}_{\parallel}\sim 10^{12}-10^{13}eV$ (Beskin et al.
1993). Here, $\gamma_{0}=\mathcal{E}_{\parallel}/m_{e}c^{2}$,then considering
$\mathcal{E}_{\parallel}=k_{B}T$, we find
$\gamma_{0}=k_{B}T/m_{e}c^{2}\equiv\sigma_{T}$. Using Eqs.(17),(19) and
Eq.(20), we perform an analysis of the dispersion relation, group velocity and
group dispersion of Langmuir waves at the ultrarelativistic temperature of the
$e,e^{+}$ plasma under gravity. The analysis is shown graphically in Fig.1-4,
respectively.
Figure 1: The variation of the normalized frequency $\omega$ of the linear
Langmuir wave with respect to the normalized pump wavenumber $k$ for different
values of plasma parameters : $r_{g}=1$, $z=2$, $\gamma_{0}=\sigma_{T}=10^{6}$
with the corresponding temperature $T=5\times 10^{15}\>K$ . Figure 2: The
variation of the normalized group velocity $v_{g}$ of the linear Langmuir wave
with respect to the normalized pump wavenumber $k$ for different values of
plasma parameters : $r_{g}=1$, $z=2$, $\gamma_{0}=\sigma_{T}=10^{6}$ with the
corresponding temperature $T=5\times 10^{15}\>K$ . Figure 3: The variation of
the normalized group dispersion $v_{g}^{\prime}$ of the linear Langmuir wave
with respect to the normalized pump wavenumber $k$ for different values of
plasma parameters : $r_{g}=1$, $z=2$, $\gamma_{0}=\sigma_{T}=10^{6}$ with the
corresponding temperature $T=5\times 10^{15}\>K$ . Figure 4: The variation of
the normalized frequency $\omega$ of the linear Langmuir wave with respect to
the normalized distance $z$ for parameters : $r_{g}=1$,
$k=1000$,$\gamma_{0}=\sigma_{T}=10^{6}$ with the corresponding temperature
$T=5\times 10^{15}\>K$ .
II.B Nonlinear Evolution Equation for the Langmuir Wave
Finding the zeroth harmonic and second harmonic of the second order quantities
in terms of the first harmonic of the first order quantities and using these
into the first harmonic of the third order quantities , we easily obtain the
following NLSE for the evolution of the potential
$\displaystyle i\frac{\partial
a}{\partial\tau}+P\frac{\partial^{2}a}{\partial\xi^{2}}+Q\left|a\right|^{2}a=0,$
(21)
where $a\equiv\phi_{1}^{(1)}$, and the coefficients $P$ and $Q$ are given by
the following expressions:
$\displaystyle P$ $\displaystyle=$
$\displaystyle\frac{\frac{4\sigma_{T}g(z)}{3\gamma_{0}^{2}}-v_{g}^{2}}{\frac{2}{\gamma_{0}}\left[\omega-\frac{v_{g}}{kg^{3}(z)}\left\\{2\omega^{2}+v_{g}k\omega+2g^{2}(z)\right\\}\right]},$
(22)
$\displaystyle Q$ $\displaystyle=$
$\displaystyle\frac{\frac{g^{2}(z)}{\gamma_{0}}\left[\frac{\omega}{k}f_{11}+h_{11}\right]}{\frac{2}{\gamma_{0}}\left[\omega-\frac{v_{g}}{kg^{3}(z)}\left\\{2\omega^{2}+v_{g}k\omega+2g^{2}(z)\right\\}\right]},$
(23)
where
$\displaystyle f_{11}$ $\displaystyle=$ $\displaystyle-\frac{\omega
k}{g^{2}(z)\gamma_{0}}\left[b_{1}\left(1+\frac{kv_{g}}{\omega}\right)+2b_{2}\right]$
(24) $\displaystyle+$ $\displaystyle\frac{3\omega
k^{5}}{4g^{4}(z)\gamma_{0}^{3}},$ $\displaystyle h_{11}$ $\displaystyle=$
$\displaystyle-\frac{\omega^{2}}{g^{2}(z)\gamma_{0}}\left[\frac{kv_{g}}{\omega}\frac{b_{1}}{g(z)}+b_{2}-\frac{3k^{4}}{4g^{3}(z)\gamma_{0}^{2}}\right]$
(25) $\displaystyle+$
$\displaystyle\frac{8\sigma_{T}k^{2}}{9g^{2}(z)\gamma_{0}^{3}}\left(b_{1}+b_{2}\right),$
with
$\displaystyle b_{1}$ $\displaystyle=$
$\displaystyle-\frac{1}{\frac{4\sigma_{T}g(z)}{3\gamma_{0}^{2}}-v_{g}^{2}}\left[g^{2}(z)\left(\frac{\omega
k}{2g^{2}(z)\gamma_{0}}\right)^{2}\right.$ (26) $\displaystyle-$
$\displaystyle\left.\frac{2\sigma_{T}k^{4}}{9g(z)\gamma_{0}}+\frac{v_{g}\omega
k^{3}}{2g^{2}(z)\gamma_{0}^{2}}\right],$
$\displaystyle b_{2}$ $\displaystyle=$
$\displaystyle\frac{3\omega^{2}k^{4}}{16g^{4}(z)\gamma_{0}^{2}}-\frac{\sigma_{T}k^{4}}{9g^{2}(z)\gamma_{0}^{3}}.$
(27)
Here, the coefficient $P$ can be written as
$\displaystyle P=\frac{1}{2\alpha}v_{g}^{{}^{\prime}},$ (28)
where,
$\displaystyle\alpha=\frac{1}{\gamma_{0}}[1-\frac{4\sigma_{T}}{\gamma_{0}^{2}g^{2}(z)}],$
(29)
represents the effects of ultrarelativistic temperature and gravity ,
neglecting of which we recover the usual results. The coefficients $P$ and $Q$
appearing in the NLSE, given by Eqs. (22), (23)are known as the dispersion and
nonlinear coefficients, respectively. The signs of $P$ and $Q$ determine
whether the slowly varying wave pulse is stable or not (Lighthill condition;
Lighthill, 1967). If the signs of $P$ and $Q$ are such that $PQ<0$, the wave
pulse is modulationally stable and the corresponding solution of the NLSE is
called the dark soliton. On the other hand, if $PQ>0$, then the pulse may be
modulationally unstable and the solution of the NLSE in this case is called
the bright soliton. Graphically, we study the nature of $P$ and $Q$ for
continuous values of the wave number $k$ with particular values of plasma
parameters, which are shown graphically in Fig.5-6, respectively.
Figure 5: The variation of the dispersion coefficient $P$ of the linear
Langmuir wave with respect to the normalized pump wavenumber $k$ for different
values of plasma parameters : $r_{g}=1$, $z=2$, $\gamma_{0}=\sigma_{T}=10^{6}$
with the corresponding temperature $T=5\times 10^{15}\>K$ . Figure 6: The
variation of the nonlinear coefficient $Q$ of the linear Langmuir wave with
respect to the normalized pump wavenumber $k$ for different values of plasma
parameters : $r_{g}=1$, $z=2$, $\gamma_{0}=\sigma_{T}=10^{6}$ with the
corresponding temperature $T=5\times 10^{15}\>K$ .
From the graphical analysis, we find that $PQ<0$. Then applying the standard
technique, and by taking $a(\xi,\tau)=a(\xi)\>\exp{[i(K\xi-\Omega\tau)]}$, the
following solution of the NLSE, Eq. (21) (Mofiz,2007)is easily obtained:
$\displaystyle a(\xi,\tau)$ $\displaystyle=$ $\displaystyle
a_{0}\;\mbox{tanh}\left[\left|\frac{Q}{4P}\right|^{1/2}a_{0}\xi\right]\exp{[i(K\xi-\Omega\tau)]},$
(30)
Here,$K\xi-\Omega\tau$ is the modulation phase with $K(<<k)$ and
$\Omega(<<\omega)$, respectively. Eq.(30) represents a dark soliton with
amplitude
$a_{0}=\left|\frac{\Omega+PK^{2}}{Q}\right|^{1/2}$ and width
$\delta=\left|\frac{4P}{Qa_{0}^{2}}\right|^{1/2}$, respectively. The dark
soliton (Eq.(30)) in the ultrarelativistic $e,e+$ plasma is shown graphically
in Fig.7.
Figure 7: Dark soliton in ultrarelativistic $e,e^{+}$ plasma at the polar cap
region of pulsar magnetosphere. The parameters are : $r_{g}=1$, $z=2$,
$k=1000$,$\Omega=0$,$K=1$,, $\gamma_{0}=\sigma_{T}=10^{6}$ with the
corresponding temperature $T=5\times 10^{15}\>K$ .
## III. Results and Discussion
In this section, we analyze the linear dispersion as well as the nonlinear
Langmuir dark soliton in the pulsar magnetosphere. Eq.(17) shows that the
Langmuir frequency depends on ultrarelativistic temperature and it is
redshifted due to gravity near the Schwarzchild radius. Similarly, the group
velocity and group dispersion, shown by Eq.(19) and Eq. (20), are also depend
on temperature and gravity.
For numerical appreciation of the dark soliton ,we consider the two cases of
ultrarelativistic temperatures: $6\times 10^{10}K-1.8\times 10^{11}K$ (Crab
pulsar) with the corresponding energies $5-15MeV$ (Nanobashvilli,2004) and
$5\times 10^{15}K-5\times 10^{16}K$ (x-ray pulsar) with the corresponding
energies $10^{12}-10^{13}eV$ (Beskin et al., 1993).
The solution of the NLSE (Eq.(21)) is a stable dark soliton (Eq.(30))whose
amplitude and width depend on temperature. The amplitude is increased and the
width is decreased with the increase of ultrarelativistic temperature. Thus,
stable spiky Langmuir solitons are possible in the ultrarelativistic electron-
positron plasma.
## IV. Conclusion
To summarize, we have investigated the nonlinear propagation of electrostatic
modes in a dense ultrarelativistic electron-positron gravito-plasma at the
polar cap region of pulsar magnetosphere. A multiscale perturbation analysis
of the fluid equations shows that stable dark Langmuir solitons are produced
due to the balnce of dispersion and nonlinearity in the wave propagation. As
the amplitude of the soliton increaes and width of the soliton decreases with
the increase of ultrarelativistic temperature, so spiky stable dark Langmuir
solitons may propagate along the open field lines of the pulsar magnetosphere,
which may have some relation with pulsar radio emission and its
microstructure.
## Acknowledgement
This work has been supported by the Ministry of Education of the Government of
Bangladesh under Grants for Advanced Research in Science: MOE.ARS.PS.2011.
No.-86.
## Appendix A: The compatibility condition
It can be shown that the 1st harmonic of the 2nd-order electron and positron
densities can be found to be
$n_{s1}^{(2)}=\frac{q_{s}k^{2}}{2eg^{2}(z)\gamma_{0}}\phi_{1}^{(2)}+\frac{i\gamma_{0}}{2g^{2}(z)}$
$\times\left[\left(\omega
v_{g}-\frac{4\sigma_{T}g(z)k}{3\gamma_{0}^{2}}\right)\frac{\partial
n_{s1}^{(1)}}{\partial\xi}+g(z)\left(kv_{g}-\omega\right)\frac{\partial
u_{s1}^{(1)}}{\partial\xi}\right]$
$-\frac{iq_{s}k}{2eg(z)}\frac{\partial\phi_{1}^{(1)}}{\partial\xi}.$
After finding $n_{e1}^{(2)}-n_{p1}^{(2)}$ and substituting it in the following
1st-harmonic of the 2nd-order part of the Poisson’s equation:
$-\frac{k^{2}}{g(z)}\phi_{1}^{(2)}-\gamma_{0}\left(n_{e1}^{(2)}-n_{p1}^{(2)}\right)+\frac{2ik\gamma_{0}}{g(z)}\frac{\partial\phi_{1}^{(1)}}{\partial\xi}=0,$
we obtain the following compatibility condition:
$v_{g}=\frac{4\sigma_{T}g(z)}{3\gamma_{0}^{2}}\frac{k}{\omega},$
which is exactly the same as the expression of the group velocity, Eq. (19),
obtained by differentiating $\omega$ with respect to $k$ from the linear
dispersion relation, Eq. (17).
## References
* (1) Beskin, V. S., Gurevich, A. V. and Istomin, Ya. N. 1993, Physics of the Pulsar Magnetosphere, Cambridge University Press.
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arxiv-papers
| 2013-02-04T15:18:14 |
2024-09-04T02:49:41.273114
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "U. A. Mofiz and M. R. Amin",
"submitter": "Mofiz Ahmed",
"url": "https://arxiv.org/abs/1302.0716"
}
|
1302.0763
|
Hadron Collider Physics symposium 2012
11institutetext: Centro Studi e Ricerche e Museo Storico della Fisica âEnrico
Fermiâ, Rome, Italy 22institutetext: Sezione INFN, Bologna, Italy
# ALICE Physics Summary
Roberto Preghenella for the ALICE Collaboration 1122 [email protected]
###### Abstract
The ALICE experiment at the LHC has collected data in proton-proton (pp)
collisions at $\sqrt{s}$ = 0.9, 2.76, 7 and 8 TeV, in lead-lead (Pb–Pb)
collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV and in proton-lead (p-Pb)
collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV. A summary overview of recent
experimental physics results obtained by ALICE is presented in this paper with
a selection of few representative measurements.
## 1 Introduction
ALICE (A Large Ion Collider Experiment) is a general-purpose heavy-ion
detector at the CERN LHC (Large Hadron Collider) whose design has been chosen
in order to fulfill the requirements to track and identify particles from very
low ($\sim$100 MeV/$c$) up to quite high ($\sim$100 GeV/$c$) transverse
momenta in an environment with large charged-particle multiplicities as in the
case of central lead-lead (Pb–Pb) collisions at extremely high energies.
;
Figure 1: Schematic layout of the ALICE detector with its main subsystems.
The ALICE experiment, shown in Figure 1, consists of a central-barrel detector
system and several forward detectors. The central system covers the mid-
rapidity region ($\left|\eta\right|\leq$ 0.9) over the full azimuthal angle
and it is installed inside a large solenoidal magnet with a moderate magnetic
of 0.5 T. It includes a six-layer high-resolution inner-tracking system (ITS),
a large-volume time-projection chamber TPC) and electron and charged-hadron
identification detectors which exploit transition-radiation (TRD) and time-of-
flight (TOF) techniques, respectively. Small-area detectors for high-$p_{\rm
T}$ particle-identification (HMPID), photon and neutral-meson measurements
(PHOS) and jet reconstruction (EMCal) complement the central barrel. The
large-rapidity systems include a single-arm muon spectrometer covering the
pseudorapidity range -4.0 $\leq\eta\leq$ -2.4 and several smaller detectors
(VZERO, TZERO, FMD, ZDC, and PMD) for triggering, multiplicity measurements
and centrality determination. A detailed description of the ALICE detector
layout and of its subsystems can be found in Aamodt:2008zz .
Since November 2009 when the first LHC collisions occured, the ALICE detector
collected proton-proton data at several centre-of-mass energies ($\sqrt{s}$ =
0.9, 2.76, 7 and 8 TeV). During the first two LHC heavy-ion runs, in the falls
of 2010 and 2011, the ALICE experiment recorded Pb–Pb collisions at a centre-
of-mass energy per nucleon pair of $\sqrt{s_{\rm NN}}$ = 2.76 TeV and could
profit of an integrated luminosity of about 10 $\mu\rm b^{-1}$ and 100 $\mu\rm
b^{-1}$, respectively. The instant luminosity exceeded $10^{26}\rm
cm^{-2}s^{-1}$ in the second run, higher than the design value. Proton-lead
(p–Pb) collision data were also recorded with the ALICE detector. This occured
during a short run performed in September 2012 in preparation for the main
p–Pb run scheduled at the beginning of 2013. Each beam contained 13 bunches; 8
pairs of bunches were colliding in the ALICE interaction region, providing a
luminosity of about $8\times 10^{25}\rm cm^{-2}s^{-1}$. Beam 1 consisted of
protons at 4 TeV energy circulating in the negative z-direction in the ALICE
laboratory system, while beam 2 consisted of fully stripped 208Pb ions at
$82\times 4$ TeV energy circulating in the positive z-direction. This
configuration resulted in collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV in the
nucleon-nucleon centre-of-mass system, which moves with a rapidity of $\Delta
y_{\rm NN}$ = 0.465 in the direction of the proton beam.
## 2 First results in proton-lead collisions
Particle production in proton-lead (p–Pb) collisions allows to study and
understand QCD at low parton fractional momentum $x$ and high gluon density.
Moreover it is expected to be sensitive to nuclear effects in the initial
state. For this reason p–Pb measurements provide an essential reference tool
to discriminate between initial and final state effects and they are crucial
for the studies and the understanding of deconfined matter created in nucleus-
nucleus collisions.
### 2.1 Charged-particle pseudorapidity density
Figure 2: Pseudorapidity density of charged particles measured in NSD p–Pb
collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV compared to theoretical
predictions.
The measurement of primary charged-particle pseudorapidity density performed
in non-single diffractive (NSD) p–Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02
TeV is reported in ALICE:2012xs . The resulting pseudorapidity density is
presented in Figure 2 as measured in the laboratory system for
$\left|\eta_{\rm lab}\right|<2$. A forward-backward asymmetry between the
proton and the lead hemispheres is clearly visible. The data is also compared
to several model predictions of particle production which have been shifted in
the laboratory system when needed. The comparison shows that the
pseudorapidity dependence is best described by the models DPMJET and HIJING
2.1 (where the gluon shadowing parameter $s_{g}$ was tuned on experimental
$\sqrt{s_{\rm NN}}$ = 200 GeV d–Au data at RHIC), whereas the saturation
models (KNL, rcBK, IP-Sat) exibit a steeper $\eta_{\rm lab}$ dependence. The
pseudorapidity density in the centre-of-mass system at mid-rapidity
$\left|\eta_{\rm cms}\right|<0.5$ is ${\rm d}N_{\rm ch}/{\rm d}\eta_{\rm
cms}=16.81\pm 0.71$, corresponding to $2.14\pm 0.17$ charged particles per
unit pseudorapidity per participant when scaled by the number of participating
nucleons, determined using the Glauber model Alver:2008aq .
### 2.2 Transverse momentum distribution
Figure 3: The nuclear modification factor of charged particles as a function
of transverse momentum in NSD p–Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02
TeV compared to measurements in central (0-5%) and peripheral (70-80%) Pb–Pb
collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV.
The measurement of the transverse momentum $p_{\rm T}$ distributions of
charged particle in p–Pb collisions were also reported ALICE:2012mj . It was
previously shown that the production of charged hadrons in central Pb–Pb
collisions at the LHC is strongly suppressed Aamodt:2010jd ; :2012eq . The
suppression remains substantial up to 100 GeV/$c$ and is also seen in
reconstructed jets :2012is . Proton-lead collisions provide a control
experiment to establish whether the initial state of the colliding nuclei
plays a role in the observed high-$p_{\rm T}$ hadron production in Pb–Pb
collisions. In order to quantify nuclear effects, the $p_{\rm T}$-differential
yield relative to the proton-proton reference, the so-called nuclear
modification factor, is calculated. The nuclear modification factor is unity
for hard processes which are expected to exhibit binary collision scaling.
This has been recently confirmed in Pb–Pb collisions at the LHC by the
measurements of direct photon, Z0 and W± production, observables which are not
affected by hot QCD matter. In Figure 3 the measurement of the nuclear
modification factor in p–Pb collisions RpPb is compared to that in central
(0-5% centrality) and peripheral (70–80%) Pb–Pb collisions RPbPb. RpPb is
observed to be consistent with unity for transverse momenta higher that about
2 GeV/$c$. This demonstrates that the strong suppression observed in central
Pb–Pb collisions at the LHC is not due to an initial-state effect, but it is
rather a final state effect related to the hot matter created in high-energy
heavy-ion collisions.
## 3 Recent physics results
A limited selection of recent physics results obtained by the ALICE experiment
at the LHC is reported in the following section. They include both
measurements performed in proton-proton (pp) and in lead-lead (Pb–Pb)
collisions and are meant to give a feeling of the physics capabilities of the
experiment, though this cannot be done in a complete way in this report.
### 3.1 Exclusive $\rm J/\psi$ photoproduction
Figure 4: Measured differential cross section of coherent $\rm J/\psi$
photoproduction compared with models. The point at positive rapidity is the
reflected result from the measurement at negative rapidities.
Exclusive vector meson production in heavy-ion interactions is expected to
probe the nuclear gluon distribution for which there is considerable
uncertainty in the low-$x$ region and it has been studied so far in gold-gold
(Au–Au) collisions at RHIC. The first LHC results on exclusive photoproduction
of $\rm J/\psi$ vector mesons measured at forward rapidities in ultra-
peripheral Pb–Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV has been
reported here Abelev:2012ba . The forward and mid-rapidity measured
differential cross section for coherent $\rm J/\psi$ production is compared
with calculations from various models in Figure 4. Best agreement is found
with models which include nuclear gluon shadowing consistent with the EPS09
parametrizations of the nuclear gluon distribution functions (AB-EPS09).
### 3.2 Direct photons
Figure 5: Direct-photon invariant yield in 0-40% central Pb–Pb collisions at
$\sqrt{s_{\rm NN}}$ = 2.76 TeV with NLO pQCD prediction and exponential fit in
low-pT region.
Direct-photon production has been measured by ALICE in proton-proton and in
Pb–Pb collisions Wilde:2012wc . The measurements when compared to NLD pQCD
predictions are found to be in good agreement for pp collisions and peripheral
Pb–Pb collisions (40-80% centrality). On the other hand, in the case of
central (0-40% centrality) Pb–Pb collisions the direct photon signal is well
reproduced by NLO pQCD only for photon momenta above 4 GeV/$c$, as shown in
Figure 5. The low-$p_{\rm T}$ excess, of about 20% at around 2 GeV/$c$, is
attributed to thermal photons, that is photons produced in the QGP phase by
the scattering of thermalized partons. The thermalized nature of the
production medium should be reflected in the $p_{\rm T}$ distribution of
thermal photons. The excess over NLO pQCD is fit in Figure 5 with an
exponential in the $p_{\rm T}$ range 0.8–2.2 GeV/$c$. The inverse slope of
this exponential is found to be T = (304 $\pm$ 51) MeV. In a similar analysis
performed in central (0-20%) Au–Au collisions at $\sqrt{s_{\rm NN}}$ = 200
GeV, the PHENIX experiment at RHIC measures an inverse slope parameter of T =
(221 $\pm$ 19 $\pm$ 19) MeV. The LHC value is about 40% higher than the one
measured at RHIC.
### 3.3 Light-flavour hadron production
Figure 6: Transverse momentum distributions of the sum of positive and
negative pions, kaons and protons for central Pb–Pb collisions. The results
are compared to RHIC data and hydrodynamic models. Figure 7: Mid-rapidity
particle ratios compared to RHIC results and predictions from thermal models
for central Pb–Pb collisions at the LHC.
Figure 8: $\rm K/\pi$ (top) and $\rm p/\pi$ production ratio in pp collisions
compared with PYTHIA Monte Carlo predictions and NLO calculations.
ALICE has measured the production yields of primary charged pions, kaon and
(anti)protons in a wide momentum range. Primary particles are defined as
prompt particles produced in the collision and all decay products, except
products from weak decay of strange particles. The measurements have been
performed both in proton-proton collisions at several centre-of-mass enegies
($\sqrt{s}$ = 0.9, 2.76 and 7 TeV) and in Pb–Pb collisions at $\sqrt{s_{\rm
NN}}$ = 2.76 TeV as a function of collision centrality.
In Pb–Pb collisions the $p_{\rm T}$ distributions and yields are compared to
previous results at RHIC and expectations from hydrodynamic and thermal
models. The results obtained for central Pb–Pb collisions are shown in Figure
6 and 7 and are also reported in :2012iu . The spectral shapes are harder than
those measured at RHIC, indicating a strong increase of the radial flow
velocity with the centre-of-mass energy. The radial flow at the LHC is found
to be about 10% higher than at RHIC energy. While the K/$\pi$ integrated
production ratio is measured to be in line with lower energy results and
predictions from the thermal model, both the p/$\pi$ and the $\Lambda/\pi$
ratios are lower than at RHIC and significantly lower (a factor $\sim$ 1.5)
than predictions. A possible explanation of these deviations from the thermal-
model predictions may be re-interactions in the hadronic phase due to large
cross sections for antibaryon-baryon annihilation Steinheimer:2012rd ;
Becattini:2012sq ; Pan:2012ne .
The $p_{\rm T}$-dependent production of charged kaons and protons normalized
to charged pions, respectively K/$\pi$ and p/$\pi$, are shown in Figure 8 for
pp collisions at $\sqrt{s}$ = 2.76 and 7 TeV where they are compared with
theoretical model predictions. No energy dependence is observed in the data
within the systematic uncertainties. The observed production ratios are not
reproduced by NLO calculations. PYTHIA Monte Carlo generator underpredicts the
proton-to-pion ratio at intermediate $p_{\rm T}$. Pion, kaon and (anti)proton
production in Pb–Pb collisions were compared to that of proton-proton
interactions and all show a suppression pattern which is similar to that of
inclusive charged hadrons at high momenta ($p_{\rm T}$ above $\simeq$ 10
GeV/$c$) OrtizVelasquez:2012te . This suggests that the dense medium formed in
Pb–Pb collisions does not affect the fragmentation. A similar conclusion can
be drawn by observing the proton-to-pion ratio measured in Pb–Pb collisions:
for intermediate momenta (3–7 GeV/$c$) it exibits a relatively strong
enhancement, a factor 3 higher than proton-proton collisions at $p_{\rm
T}\approx$ 3 GeV/$c$ and gets back to the proton-proton value at higher
momenta ($p_{\rm T}$ above $\simeq$ 10 GeV/$c$) OrtizVelasquez:2012te . A
similar observation is reported also for the $\rm\Lambda/K^{0}_{s}$ ratio and
possible explanations include among those proposed so far particle production
via quark recombination Fries:2003vb .
### 3.4 Heavy-flavour production
Figure 9: $p_{\rm T}$-differential invariant cross sections of electrons from
beauty hadron decays. The ratio of the data and the FONLL calculations are
shown in in the bottom panel and the dashed lines indicate the calculation
uncertainty. Figure 10: Average D meson suppression versus number of
participants compared with CMS non prompt $\rm J/\psi$ suppression.
ALICE has studied heavy-flavour production in pp and Pb–Pb collisions at the
LHC. Their prodution in proton-proton collisions is a tool to test pQCD
calculations in a new energy domain. The $p_{\rm T}$-differential production
cross section of prompt charmed mesons (D0, D+, D∗+, D${}^{+}_{s}$), heavy-
flavour decay electron and muons in pp collisions at $\sqrt{s}$ = 7 and 2.76
TeV are reported in Abelev:2012xe ; Abelev:2012pi ; Abelev:2012qh ;
ALICE:2011aa ; :2012sx ; :2012ik and the measurements are well described by
pQCD prediction. The production cross section of electrons from semileptonic
decays of beauty hadrons was also measured at mid-rapidity
($\left|y\right|<0.8$) in pp collisions at $\sqrt{s}$ = 7 TeV :2012ic . The
results are shown in Figure 9 together with the pQCD FONLL prediction.
Heavy-flavuor production studies in Pb–Pb collisions at $\sqrt{s_{\rm NN}}$ =
2.76 TeV were also performed ALICE:2012ab ; delValle:2012qw . In particular,
the relative production of heavy-flavour particles with respect to that in
nucleon-nucleon interaction show a suppression of up to a factor of 5 at
$p_{\rm T}\sim$ 10 GeV/$c$ in central Pb–Pb collisions (0-7.5% centrality).
The suppression is observed to be similar for the three studied decay
channels, that is heavy-flavour electrons, heavy-flavour muons and prompt D
mesons. The average D meson production suppression is shown in Figure 10 as a
function of the number of participating nucleons and it is compared with the
measurement of non-prompt $\rm J/\psi$ suppression performed by CMS
CMS:2012vxa . The suppression of non-prompt $\rm J/\psi$ from B meson decays
reflects the in-medium energy loss of $b$ quarks. The two measurements provide
a first indication of different suppression for charm and beauty in central
collisions, that is of different in-medium energy loss.
### 3.5 Quarkonia
Figure 11: $\rm J/\psi$ production yield ${\rm d}N_{\rm J/\psi}/{\rm d}y$ as a
function of the charged particle multiplicity densities at mid-rapidity ${\rm
d}N_{\rm ch}/{\rm d}\eta$. Both values are normalized by the corresponding pp
minimum bias value. Figure 12: Inclusive $\rm J/\psi$ suppression $R_{\rm AA}$
versus the number of participant nucleons $N_{\rm part}$ in Pb–Pb collisions,
compared with results from PHENIX at RHIC.
ALICE has measured the inclusive $\rm J/\psi$ production in pp and Pb–Pb
collisions at the LHC, down to zero $p_{\rm T}$ Aamodt:2011gj ; Abelev:2012kr
. A unique study performed by ALICE in pp collisions is the determination of
the inclusive $\rm J/\psi$ production yield as a function of the charged
multiplicity at central rapidity. These results Abelev:2012rz show a linear
increase of the yield with charged multiplcity which is not explained by
theory and model predictions yet. A similar increase is observed both a
forward and mid-rapidity, as shown in Figure 11.
Results on inclusive $\rm J/\psi$ production in Pb–Pb collisions clearly
indicate a saturation of the suppression at both central and forward rapidity
moving towards central collisions, a phenomenon not observed at lower energy
where the suppression increases as shown in Figure 12. There is in fact a
clear evidence for a smaller suppression at LHC with respect to RHIC energy
Abelev:2012rv . Differential studies of the suppression versus collision
centrality for various momentum bins seem to favour a scenario where
(re)combination processes play a sizeble role BraunMunzinger:2000px ;
Thews:2000rj . The observed hint for a non-zero anisotropic production signal
(elliptic flow) are in agreement with such a picture.
## 4 Summary
ALICE has obtained so far a wealth of physics results both from the analysis
of proton-proton collision data and from the first two LHC heavy-ion runs.
First results from a short pilot run with proton-lead have been already
obtained and the coming dedicated p-Pb run at the beginning of 2013 will set
the beginning of precision carachterization of the matter formed in heavy-ion
collisions at the LHC. A clear detector upgrade strategy plan for the LHC
luminosity upgrade has also been presented at this conference ken .
## References
* (1) K. Aamodt et al. [ALICE Collaboration], JINST 3 (2008) S08002.
* (2) B. Abelev et al. [ALICE Collaboration], arXiv:1210.3615 [nucl-ex].
* (3) B. Alver, M. Baker, C. Loizides and P. Steinberg, arXiv:0805.4411 [nucl-ex].
* (4) B. Abelev et al. [ALICE Collaboration], arXiv:1210.4520 [nucl-ex].
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* (25) [CMS Collaboration], CMS-PAS-HIN-12-014.
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|
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| 2013-02-04T17:24:22 |
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|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Roberto Preghenella (for the ALICE Collaboration)",
"submitter": "Roberto Preghenella",
"url": "https://arxiv.org/abs/1302.0763"
}
|
1302.0869
|
# The Neutron Star Zoo
Alice K. Harding [email protected] Astrophysics Science Division
NASA Goddard Space Flight Center
Greenbelt, MD 20771
###### Abstract
Neutron stars are a very diverse population, both in their observational and
their physical properties. They prefer to radiate most of their energy at
X-ray and gamma-ray wavelengths. But whether their emission is powered by
rotation, accretion, heat, magnetic fields or nuclear reactions, they are all
different species of the same animal whose magnetic field evolution and
interior composition remain a mystery. This article will broadly review the
properties of inhabitants of the neutron star zoo, with emphasis on their
high-energy emission.
Stars: Neutron stars, Pulsars, Binary stars
###### pacs:
97.60.Jd,97.60.Gb
###### Contents
1. I Introduction
2. II Rotation-Powered Pulsars
3. III Magnetars
4. IV Compact Central Objects
5. V Isolated Neutron Stars
6. VI Accreting Neutron Stars
1. VI.1 Low-Mass X-Ray Binaries
2. VI.2 Intermediate- and High-Mass X-Ray Binaries
3. VI.3 Microquasars and Gamma-Ray Binaries
7. VII Summary
## I Introduction
Neutron stars are the remnants of massive stars whose cores collapse during
the supernova explosions at the end of their nuclear fusion lifetimes.
Conservation of both the angular momentum and the magnetic flux of the
progenitor star during the collapse gives the neutron star an extremely high
spin rate and magnetic field. The collapse ends when the degeneracy pressure
of neutrons balances the gravitational forces of the matter. At that point the
core radius is about 10 km and with a mass $\sim 1.4M_{\odot}$ the center of
the star has already reached nuclear densities. Neutron stars thus possess the
highest spin frequencies, magnetic fields and densities of any known objects
in the Universe. Since their theoretical conception by Baade1934 they have
been fascinating celestial objects, both for study of their exotic interiors
and environments and for their important place in stellar evolution.
Figure 1: Schematic diagram of neutron star populations with respect to their
periods and derived surface magnetic field strengths.
Neutron stars (NSs) all typically have detectable pulsations, since they are
rapidly spinning and their emission patterns are highly anisotropic. This may
be one of the few observational properties that they have in common, because
they otherwise show an amazing variety of pulsating and bursting behaviors.
Neutron star types are classified according to the primary power source for
their emission and spin evolution. Rotation-powered pulsars (RPP) derive their
energy primarily from the rotation of the NS, magnetars from magnetic field
energy, isolated NSs (INS) from the latent heat of the NS matter, and
accretion-powered NSs from the energy released by matter accreting onto the NS
from a binary companion. A subclass of accreting NSs are X-ray bursters whose
bursts are powered by thermo-nuclear explosions. An additional class, Central
Compact Objects (CCO), are seen as soft X-ray point sources inside supernova
remnants and seem to be quiet at all other wavelengths. Figure 1 shows where
these different NS types roughly fall in period and surface magnetic field
space. Although period is a measured quantity, surface magnetic field is
derived in different ways for the various NS types. Magnetic fields in RPPs,
magnetars, INS and CCO are derived from measured period derivatives,
$\dot{P}$, assuming magnetic dipole radiation spin down, although in magnetars
this can only give an approximate value since their spin behavior is
complicated by magnetically-driven NS heating and bursting. The surface dipole
field at the pole determined assuming the observed $\dot{P}$ is from magnetic
dipole radiation is
$B_{d}=\left({\frac{{3Ic^{3}P\dot{P}}}{{2\pi^{2}R^{6}}}}\right)^{1/2}\simeq
2\times 10^{12}{\rm G}\,(P\dot{P}_{15})^{1/2},$ (1)
where $\dot{P}_{15}\equiv\dot{P}/(10^{-15}\,\rm s\,s^{-1})$, $P$ is in units
of seconds, and $I$ ($\simeq 10^{45}$ g cm2) and $R$ ($\simeq 10^{6}$ cm) are
the neutron star moment of inertia and radius. Magnetic fields of accretion-
powered NSs cannot be measured from their $\dot{P}$ since their spin evolution
is governed by accretion torques. For accreting X-ray pulsars, the cyclotron
lines seen in their spectra give good estimates of surface field strength:
$B_{\rm cyc}\simeq\left({E_{c}\over 11.6{\rm keV}}\right)\,10^{12}\,{\rm G}$
(2)
where $E_{c}$ is the cyclotron line energy. For low-mass X-ray binaries (LMXB)
and bursting sources, the Alfven radius where the NS magnetic pressure
balances that of the accretion flow gives an estimate of the surface magnetic
field strength:
$B_{\rm A}\sim 10^{12}\,{\rm G}\,P_{\rm eq}\,\left({\dot{M}\over
10^{-9}M_{\odot}\,{\rm yr^{-1}}}\right)^{1/2}$ (3)
Ghosh1979 where $P_{\rm eq}$ (in seconds) is the equilibrium spin period of
the NS and $\dot{M}$ is the mass accretion rate. The NS types generally occupy
different areas of $P-B$ space, although there is a good deal of overlap. The
LMXB population, being progenitors of rotation-powered MSPs, overlap the MSP
population. CCOs lie at the lower end of the RPPs, coinciding with some of the
MSP population. The lower end of the magnetar population overlaps the upper
end of the RPPs, and the INS overlaps both. The accreting X-ray pulsars have
magnetic fields and periods similar to some of the older RPPs. The magnetars
have the highest surface magnetic fields, while the rotation-powered
millisecond pulsars and LMXBs have very low surface fields but the shortest
periods.
Figure 2: Plot of period vs. period derivative for the presently known
rotation-powered pulsars, Isolated Neutron Stars (INS), Compact Central
Objects (CCO), Rotating Radio Transients (RRATs) and magnetars (from
http://www.atnf.csiro.au/people/pulsar/psrcat/). Lines of constant
characteristic age, $P/2\dot{P}$, and dipole spin-down luminosity,
$\dot{E}_{d}$, are superposed.
Figure 2 shows the distribution of non-accreting NS in measured period and
period derivative. The radio pulsars occupy the largest region of this phase
space with their population extending from the very short period, low
$\dot{P}$ MSPs up to the high $\dot{P}$, high magnetic field pulsars that
border the magnetar range. The magnetars have the highest $\dot{P}$ and some
of the longest periods in the NS zoo. with some having $P\sim 11$ s (the
longest period of a radio pulsar is about 8 s). The INS have periods very
similar to those of magnetars, also reaching up to 11 s, but with $\dot{P}$
and magnetic fields about a factor of ten lower than those of the lower field
magnetars. The CCOs have very low $\dot{P}$, almost as low as the MSPs, but
their spin periods are more like those of young RPPs. High energy pulsars
(RPPs with X-ray or gamma-ray pulsations) typically have the highest spin-down
power (see Eq [4]) but are not necessarily the youngest, as many of the very
old MSPs are efficient X-ray and gamma-ray pulsars.
## II Rotation-Powered Pulsars
Figure 3: Skymap in Galactic coordinates showing the gamma-ray pulsars of
different types detected by Fermi: Blue squares: radio quiet pulsars. Red
triangles: millisecond gamma-ray pulsars. Green circles: radio loud gamma-ray
pulsars. Black dots: Gamma-rays were phase-folded using a rotation ephemeris.
Gray dots: Pulsars for which no rotation ephemeris was available. From
Abdo2013 .
Neutron stars that are spinning down as a result of torques from magnetic
dipole radiation and particle emission are known as rotation-powered pulsars
(RPP). The energy from their spin down appears as broad-band pulsations from
radio to gamma-ray wavelengths and as a wind of energetic particles flowing
into their surrounding pulsar wind nebulae. Since the discovery of RPP through
their radio pulsations in 1967 Hewish1968 , more than 2000 radio pulsars are
now known with periods ranging from a few ms to several seconds Manchester2005
, http://www.atnf.csiro.au/people/pulsar/psrcat/). X-ray, gamma-ray and
optical pulsations were soon discovered in a few of these pulsars by folding
the time series obtained at these wavelengths at the radio periods. At
present, there are over 100 RPP detected at X-ray energies and over 130 gamma-
ray pulsars Abdo2013 ; most were discovered using known radio ephemerides, but
many were also discovered through their X-ray or gamma-ray pulsations and are
radio quiet. The spin down of RPPs is typically smooth and predictable, but
they have occasionally been observed to undergo sudden changes in spin called
‘glitches’, where the period decreases and then recovers back to its normal
spin-down rate on a timescale of days to weeks.
There are two main populations of RPPs: normal pulsars having characteristic
ages $\tau=P/2\dot{P}$ $<$ 100 Myr, and millisecond pulsars (MSP) with
$\tau\mathrel{\mathchoice{\lower
2.0pt\vbox{\halign{$\m@th\displaystyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
2.0pt\vbox{\halign{$\m@th\textstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
2.0pt\vbox{\halign{$\m@th\scriptstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
2.0pt\vbox{\halign{$\m@th\scriptscriptstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}}100$
Myr. The periods of the normal pulsars are thought to have increased at a
steady rate from their birth periods at the time of supernova core collapse.
The birth periods of RPPs are not well known, and inferences of the period
range from tens of ms to hundreds of seconds Kaspi2005 Gotthelf2010 . MSPs are
thought to have originally been members of the normal RPP population, spun
down for tens of Myr and then spun up by accretion from a binary companion
Alpar1982 . MSPs make up about 10% of the RPP population and about 80% of them
are in binary systems. A large number of new MSPs have recently been found
through radio searches at positions of Fermi unidentified gamma-ray sources
(e.g. Ransom2011 ), doubling the known numbers of radio MSPs in the Galactic
disk. Their radio ephemerides can then be used to find the gamma-ray
pulsations. They are extremely good clocks since they rarely glitch, they have
very little of the timing noise seen in young RPPs, and their spin down is
very stable. These characteristics make them potentially valuable sources for
use in celestial navigation Coll2009 and gravitational wave detection
Hobbs2010 .
Figure 4: Gamma-ray (black) and radio (red) pulse profiles for a selection of
pulsars detected by Fermi, from Abdo2013 .
The rotating dipole model that seems to well describe RPPs gives estimates of
their main electrodynamic properties. The Poynting flux of a rotating dipole
in vacuum gives an estimate of the spin-down power:
$\dot{E}_{d}={4\pi^{2}I\dot{P}\over P^{3}}={2B_{0}^{2}\Omega^{4}\over
3R^{6}c^{5}}=10^{31}\,{\rm erg\,s^{-1}}\,B_{12}^{2}\,P^{-4}$ (4)
where $\Omega=2\pi/P$ is the spin angular velocity, $R$ the NS radius, $B_{0}$
is the surface magnetic field and $B_{12}\equiv B_{0}/10^{12}$ G. The rotating
dipole induces strong electric fields, ${\bf E=\Omega\times r\times B}$ that
in vacuum are orders of magnitude larger than the gravitational force, pulling
charges out of the NS surface to fill the magnetosphere GJ1969 . The maximum
charge density, which screens the electric field along magnetic field lines,
is
$\rho_{\rm GJ}\simeq-{{\bf\Omega\cdot B}\over 2\pi c}$ (5)
which gives a scale for the current density flowing along the open field
lines, $J_{\rm GJ}=\rho_{\rm GJ}c$. The particles and magnetic field corotate
with the NS out to the light cylinder radius, $R_{\rm LC}=c/\Omega$, where the
corotation velocity reaches the speed of light. The magnetosphere is divided
into regions of open field lines that cross $R_{\rm LC}$ and regions of field
lines that close within $R_{\rm LC}$. The footpoints on the NS surface of the
open/closed boundary define the polar caps. The induced potential drop in
vacuum across the open field lines is
$V_{\rm pc}\simeq{1\over 2}\left({2\pi\over cP}\right)^{2}\,B_{0}R^{3}=6\times
10^{12}\,{\rm Volts}\,B_{12}\,P^{-2}$ (6)
The open field lines and the particles that flow along them become the pulsar
wind, which carries the bulk of the Poynting flux beyond $R_{\rm LC}$.
The pulsed radiation of RPPs carries less than about 10% of the total spin-
down power. Most of the power in pulsed emission is in gamma rays around a GeV
for all but a few RPPs, notably the Crab pulsar whose pulsed emission power
peaks in hard X rays. The radio pulsations make up typically only $10^{-4}$ or
less of the spin-down power, but are easier to detect since the photon flux is
much higher. Since the gamma-ray pulsations typically carry the highest
percentage of $\dot{E}_{d}$, they can reveal the most information about the
particle acceleration in pulsar magnetospheres. The discovery of gamma-ray
pulsations from over 130 RPPs by the Fermi Gamma-Ray Space Telescope Abdo2010
; Abdo2013 , where there were only seven known previously Thompson2001 , has
thus revolutionized the study of this type of NS (see Figure 3). Two new
populations of gamma-ray pulsars have been established with Fermi: radio quiet
pulsars discovered through blind searches for gamma-ray pulsations Abdo2009b ;
Pletsch2012 , and gamma-ray MSPs whose pulsations are found using radio
ephemerides Abdo2009c . These are presently equal in number to the radio-loud
gamma-ray pulsars. The gamma-ray pulse profiles show recurring patterns of two
narrow peaks separated by phase intervals of 0.1 - 0.5, with occasional single
peaks. The most striking pattern is that the gamma-ray peaks are not in phase
with the radio pulses, but typically arrive later in phase, with a lag that is
correlated with the gamma-ray peak separation (smaller lags for larger peak
separation, see Figure 4). The pulsed gamma-ray spectra are power laws with
high-energy exponential cutoffs in the range 1 - 10 GeV. The pulsed gamma-ray
emission of RPPs results from radiation of particles accelerated to energies
of order 10 TeV by electric fields parallel to the magnetic fields in the open
magnetosphere. Acceleration could take place near the polar caps Arons1979 ,
in the outer magnetosphere near the light cylinder Cheng1986 in slot gaps
along the last open field lines Arons1983 ; MH2004 or from reconnection in
the striped wind outside the light cylinder Petri2005 . The Fermi measurement
of an exponential shape for the spectral cutoff of the Vela pulsar Abdo2009a
ruled out emission in the intense magnetic field near the polar caps, whose
spectra would have much sharper, super-exponential cutoffs DH1996 , as being
the primary source of the observed gamma-rays. Instead, the softer cutoffs as
well as the observed profile shapes and peak separation vs. radio phase-lag
correlation Abdo2010 predicted by outer magnetosphere models Romani1995 ,
strongly suggest that the gamma-ray emission comes from high altitudes up to
and possibly beyond the light cylinder. In these models, the narrow gamma-ray
peaks are caustics formed by phase bunching of the emission along the open
field boundary through aberration and light travel time effects Dyks2004 ;
Harding2005 . The main radiation mechanism at GeV energies is thought to be
curvature radiation by the primary particles that have been accelerated to 10
TeV energies, but inverse-Compton emission may also contribute radiation at
the higher energies (up to 400 GeV) seen from the Crab pulsar Aliu2011 .
Many RPPs (over 100 at present) also show X-ray emission, with pulsations
detected in many of these. This emission is usually made up of two components:
non-thermal emission that is probably magnetospheric, and 0.05 - 0.1 keV
thermal emission from surface cooling or heated polar caps (see Kaspi2005 for
a detailed review). Some, like the Crab pulsar, show only non-thermal X-ray
emission, but in these cases the thermal components are probably buried under
the strong non-thermal emission. Others, like MSP J2124-3358, show only
thermal components with statistically insignificant non-thermal emission. The
thermal and non-thermal peaks in the pulse profiles are usually not in phase,
but the non-thermal peaks are sometimes in phase with one or two gamma-ray
peaks. In the youngest RPPs, emission from cooling of the NS surface is
thought to dominate the thermal radiation. In contrast, middle-aged RPPs like
Vela, Geminga and PSR J0659+1414 show a two-component thermal spectrum from
both heating and cooling, and a non-thermal power-law component. MSPs show all
of this behavior but, since these sources are too old to have detectable
emission from cooling, the thermal components, some of which are multiple
blackbodies, must be due to polar cap heating. The non-thermal emission is in
most models due to synchrotron radiation from electron-positron pairs, either
from the polar cap or the outer gap, emitting at high altitude Hirotani2006 ;
Harding2008 . The non-cooling thermal emission could come from polar caps
heated by high-energy particles flowing toward the NS from the polar cap
HM2001 or outer gap Halpern1993 accelerators.
There is an unusual and growing subpopulation of RPPs known as Rotating Radio
Transients (RRATS) (see Keane2011 , for review). They were discovered only
very recently McLaughlin2006 , through detections of their single, isolated
radio pulses. They show a variety of transient radio behavior, ranging from
nulling (or turning off for up to $\sim 10^{4}$ s) for long time periods to
steady pulsations that are highly modulated. While nulling behavior in normal
radio pulsars has been known for many years, the nulling of RRATS is extreme.
Over 70 RRATS have been identified and $\dot{P}$s have been measured for about
20 of these sources. Their $P$ and $\dot{P}$ are scattered throughout the
normal RPP population (see Figure 2), but they tend to be older
($\tau\mathrel{\mathchoice{\lower
2.0pt\vbox{\halign{$\m@th\displaystyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
2.0pt\vbox{\halign{$\m@th\textstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
2.0pt\vbox{\halign{$\m@th\scriptstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
2.0pt\vbox{\halign{$\m@th\scriptscriptstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}}10^{5}$
yr) and a number of them have higher than average surface magnetic fields.
X-ray pulsations have been detected from one RRAT, PSR J1819-1458
McLaughlin2007 , revealing a thermal spectrum and an absorption line at 0.5
keV, possibly due to proton cyclotron resonant scattering in a field of
$2\times 10^{14}$ G. The causes of such radio transient behavior is unknown,
but global changes in the currents or charge density in the pulsar
magnetosphere has been suggested Kala2012 ; Li2012 . It is not yet clear how
RRATS fit into the normal RPP population. They share some similar properties
with INS (see Section V), but if they are a separate evolutionary group they
would significantly raise the birthrate of NSs in the Galaxy Keane2008 .
In addition to pulsed emission, broadband, un-pulsed emission from radio to
high-energy (TeV) gamma-rays is detected from the pulsar wind nebulae (PWNe)
associated with RPPs. The PWN is a repository of high-energy particles and
fields from the RPP, accumulated over the pulsar’s lifetime and trapped by the
surrounding supernova shell that is moving more slowly than the relativistic
pulsar wind, or by the bow shock driven by the supersonic motion of the pulsar
through the interstellar medium. The particles are mostly electron-positron
pairs from the pair cascades DH1982 that occur in the magnetosphere, but
there could be some protons or positively-charged ions as well. Most of the
Poynting flux of the RPP must at some point, either in the wind or at the wind
termination shock, be transferred to the particles. In the Crab nebula, the
highest energy particles receive nearly the entire open-field voltage, as
deduced from the $\sim 100$ MeV cutoff in the synchrotron spectrum DJH1992 ;
Arons2011 . Many PWNe are detected at energies up to several or even tens of
TeV deJager2005 . This VHE emission is thought to be inverse-Compton emission
from the nebular particles, scattering either their own synchrotron radiation
as in the Crab PWN or, in most cases, the microwave and infrared background
radiation. Powerful gamma-ray flares from the Crab nebula that appear only in
the synchrotron emission component have recently been observed by Fermi
Abdo2011a and AGILE Tavani2011 . Their short timescales of hours to days
indicate that they are coming from a very small region ($<1$ arcsec) and their
energies of up to several GeV challenge traditional models of acceleration in
pulsar wind nebulae Buehler2012 .
## III Magnetars
NSs whose primary power source is the tapping of energy stored in their
magnetic fields are known as magnetars (see Woods2006 , for review). There are
two sub-classes of magnetars, Anomalous X-Ray Pulsars (AXPs) and Soft Gamma-
Ray Repeaters (SGRs), that were thought for many years to be separate and
unrelated objects. Today, we know that SGRs and AXPs are both NSs possessing
magnetic fields of unprecedented strength of $10^{14}-10^{16}$ G, and that
show both steady X-ray pulsations as well as soft $\gamma$-ray bursts. Their
inferred steady X-ray luminosities are about one hundred times higher than
their spin-down luminosities, requiring a source of power way beyond the
magnetic dipole spin-down that powers RPPs. New high-energy components
discovered in the spectra of a number of AXPs and SGRs require non-thermal
particle acceleration and look very similar to high-energy spectral components
of young rotation-powered pulsars denHartog2008 (see Figure 6).
Figure 5: Light curve of the 2005 superburst from SGR 1806-20, seen by the
Burst Alert Telescope on Swift. Strong pulsations are seen in the decaying
tail of the burst. (from Palmer2005 )
SGRs were first detected around 1979 as $\gamma$-ray transients and were
thought to be a type of classical $\gamma$-ray burst. They undergo repeated
bursts with several tenths of second duration and average energy
$10^{40}-10^{41}\,\rm erg$, and their bursting often occurs in episodes spaced
years apart. They more rarely undergo giant superflares of total energy
$10^{45}-10^{47}\,\rm erg$, consisting of an initial spike of duration several
hundred ms followed by a longer decay of duration several hundred seconds
showing pulsations (Figure 5). Such superflares have been observed in three
SGR sources, SGR0526-66 (the famous 5th March 1979 event) Cline1980 ,
SGR1900+14 Hurley1999 and SGR1806-20 Palmer2005 . In 1998, SGR1806-20 was
discovered to have 7.47 s pulsations in its quiescent X-ray emission
Kouveliotou1998 and a large $\dot{P}$ that implies a huge surface magnetic
field of $\simeq 10^{15}$ G if due to dipole spin-down. Quiescent
periodicities of 8 s and 5.16 s and large $\dot{P}$ were subsequently detected
in SGR0526-66 and SGR1900+14, implying similarly high surface magnetic fields.
In all three sources, the quiescent periods are the same periods seen in the
decay phases of their superflares. The quiescent pulse profiles are very broad
and undergo dramatic changes just before and after superflares. The profiles
are often more complex, with multiple peaks before flares, changing to more
simple single peaked profiles following the flares. Since the modulation is
thought to result from beaming along magnetic field lines, the profile changes
signal a re-arrangement of the magnetic field structure during the flares. All
of the SGRs lie near the Galactic plane and are thought to have distances
around 10-15 kpc (except for SGR0526-66, which is in the LMC). Recently, a
transient SGR was discovered with an apparent surface magnetic field strength
$<7.5\times 10^{12}$ G Rea2010 . This could be an aging magnetar that has
experienced significant field decay over its lifetime.
The first AXPs were discovered as pulsating X-ray sources in the early 1980s
by EXOSAT and were thought to be a strange (anomalous) type of accreting X-ray
pulsar. They are bright X-ray sources possessing luminosities (in their
highest states) of $L_{X}\sim 10^{35}\,\rm erg\;s^{-1}$, but show no sign of
any companion or accretion disks that would be required to support the
accretion hypothesis. The AXPs have pulsation periods in a relatively narrow
range of 5 - 11 s and are observed to be spinning down with large period
derivatives Vasisht1997 . Their pulse profiles are broad and very similar to
those of SGR sources. The very high surface magnetic fields of
$10^{14}-10^{15}$ G implied by dipole spin-down were originally controversial,
but have come to be accepted after the quiescent periods were found in SGRs
and especially following the recent discovery of SGR-like bursts from several
AXPs Kaspi2003 . It is now believed that SGRs and AXPs are two varieties of
the same type of object, very strongly magnetized, isolated NSs possibly
powered by magnetic field decay. In both sources, the high-state quiescent
luminosities of $L_{X}\sim 10^{35}\,\rm erg\;s^{-1}$ are much higher than
their spin-down luminosities of $\dot{E}_{d}\sim 2-6\times 10^{33}\,\rm
erg\;s^{-1}$, demanding an alternative power source.
Figure 6: Magnetar spectral components. Left: quiescent phase-averaged
spectrum of AXP 4U 0142+61 showing lower energy thermal and power law
components below 10 keV and high-energy component above 10 keV (from
denHartog2008 ). Right: typical burst spectra from AXP/SGR, compared with
burst spectra from X-ray and gamma-ray bursts (from Woods2006 ).
The quiescent spectra of AXPs and SGRs (see Figure 6) consist of a thermal
component fit by $\sim$ 0.5-1 keV blackbodies and one or more non-thermal
components. Until recently, non-thermal spectra were seen only below 10 keV
and were fit with steep power laws having photon indices $\sim 3-4$. When
INTEGRAL and RXTE recently measured the spectra above 10 keV for the first
time hard, non-thermal components were discovered in three AXPs, and also SGR
1806-20. In two of the AXPs, the differential spectra between 10 keV and 50
keV are extremely flat: 1E 1841-045 Kuiper2004 has a power-law photon index
of $s=0.94$ and 4U 0142+61 denHartog2004 displays a photon index of $s=0.45$,
both much flatter than the steep non-thermal components in the classic X-ray
band. RXS J1708-40 possesses a slightly steeper continuum with $s=1.18$. The
non-thermal tail of quiescent emission in SGR 1806-20 is similarly pronounced
Mereghetti2005 ; Molkov2005 , but somewhat steeper, with a photon index of
$s=1.6-1.9$ extending to 100 keV. Such hard non-thermal components require
continuous particle acceleration during the quiescent state. Transient, highly
variable radio emission has been detected from several magnetars Camilo2006 ;
Camilo2007 and may be correlated with their X-ray luminosity and
outburstsRea2012 .
It was proposed early on Ramaty1980 ; Katz1982 that SGRs were NSs with strong
magnetic fields in the range $10^{11}-10^{13}$ G to confine the super-
Eddington burst radiation. That SGRs and AXPs had much higher magnetic fields
above $10^{14}$ G was proposed by Duncan & Thompson [DT] Duncan1992 before
their existence was observationally verified. In this model, some NSs generate
huge magnetic fields by dynamo action soon after their birth in supernova
explosions. Such high fields have several properties that can cause the NSs to
behave differently from NS with lower fields. These fields can decay on much
shorter timescales, due to ambipolar diffusion Heyl1998 .
$t_{amb}\cong 10^{5}yr\left({\frac{{B_{core}}}{{10^{15}G}}}\right)^{-2}$ (7)
Diffusion of magnetic flux out of the NS core on these timescales provides the
power to magnetars in the DT model. Magnetar-strength fields also apply higher
stresses to the stellar crust, so that the yield strain can exceed the crustal
strength. This property is responsible for the small SGR and AXP bursts in the
DT model Thompson1996 . If a toroidal component of the field
$B_{core}>10^{15}$ G develops in the interior of the star, it can twist the
external field Duncan2001 . Such action can cause the superflares if the
twisted field lines reconnect. Finally, due to the much faster heat transport
in very strong magnetic fields, there is a greater heat flux through the crust
Heyl1998 . Such a property may explain the much hotter surface temperatures of
magnetars and the high quiescent X-ray emission.
Magnetar fields also produce a variety of different behavior of radiative
processes (see Duncan2000 and Harding&Lai2006 for reviews). In general, in
magnetic fields approaching and exceeding the quantum critical field $B_{\rm
cr}=4.4\times 10^{13}$ G, for radiative processes such as Compton scattering,
cyclotron and synchrotron emission and absorption, and pair production and
annihilation, classical descriptions are largely inaccurate and QED
descriptions must be used. In addition, new processes become possible in
strong fields, such as one-photon pair production and annihilation, vacuum
polarization and photon splitting, that cannot take place in field-free
environments. These processes, in particular vacuum polarization Lai2005 ,
strongly influence the propagation of photons in the NS atmosphere and the
spectrum of the emergent radiation.
According to the DT model Thompson1995 , the magnetar superflares result from
reconnection of sheared or twisted external field lines, leading to particle
acceleration and radiation of hard emission. The estimated luminosity of such
events,
$\frac{{B_{core}^{2}}}{{8\pi}}R^{3}\approx 4\times 10^{46}{\rm
erg}\left({\frac{{B_{core}}}{{10^{15}G}}}\right)^{2},$ (8)
is similar to observed luminosities of superflares. The smaller bursts result
from cracking of the crust, which is continually overstressed by diffusion of
magnetic flux from the NS interior. The shaking of magnetic footpoints then
excites Alfven waves that accelerate particles. The energy radiated in such
events would be
$E_{SGR}\cong 10^{41}{\rm
erg}\left({\frac{{B_{0}}}{{10^{15}G}}}\right)^{-2}\left({\frac{l}{{\rm
1km}}}\right)^{2}\left({\frac{{\theta_{\max}}}{{10^{-3}}}}\right)^{2},$ (9)
where $l$ is the length scale of the displacements, $B_{0}$ is the crustal
field and $\theta_{\max}$ is the yield strain of the crust. The quiescent
emission in the DT model is powered by magnetic field decay through conduction
of heat from the core. The NS crust is heated to a temperature of
$T_{crust}\cong 1.3\times
10^{6}K\left({\frac{{T_{core}}}{{10^{8}K}}}\right)^{5/9}$ (10)
where $T_{core}$ is the core temperature, and luminosity
$L_{\rm{x}}\cong 6\times 10^{35}{\rm
ergs^{-1}}\left({\frac{{B_{core}}}{{10^{16}G}}}\right)^{4.4}.$ (11)
Thompson2005 have proposed that the hard, non-thermal quiescent component is
due to the creation of a strong $E_{\parallel}$ induced by twisting of field
in the closed magnetosphere (see also Belo2009 ), producing synchrotron
radiation from electron acceleration at high altitude. An alternative model
for magnetar activity and emission has been discussed by Heyl2005 . The burst
and quiescent radiation are a result of shocks from fast-mode plasma waves and
the hard quiescent component is due to a pair-synchrotron cascade. On the
other hand, Baring Baring2004 and Baring & Harding Baring2007 propose that
resonant Compton upscattering of thermal X-rays by accelerated particles
produces the quiescent hard emission.
## IV Compact Central Objects
Compact Central Objects (CCOs) are X-ray sources detected close to the centers
of young supernova remnants (SNRs) that have no apparent emission in other
wavebands and no binary companions. Although these sources have been known and
studied for several decades without much understanding of their nature,
exciting results over the past few years have brought them into the forefront
of NS studies. They have soft, exclusively thermal spectra in the few hundred
eV range and X-ray luminosities around $10^{33}-10^{34}\,{\rm erg\,s^{-1}}$.
About ten CCOs are presently known, including the central sources of CasA
(Figure 7), Puppis A and Kes 79 supernova remnants. Several, J1852+0040 in
Kes79 Gotthelf2010 , J0822.0-4300 in Puppis A Gotthelf2009 and 1E 1207.4
-5209 in PKS 1209-51/52 Zavlin2000 have detected pulsations in the hundreds
of ms range. J1852+0040 has a detected $\dot{P}$ Gotthelf2010 , indicating
that it is spinning down like a RPP. The measured $P$ and either measurements
or constraints on $\dot{P}$ indicate that these sources have very low magnetic
fields in the range $10^{10}-10^{11}$ G assuming magnetic dipole braking, so
that their birth periods were close to their present values. Since their SNRs
are all young, $\sim 10^{3}-10^{4}$ yr, they were probably born with unusually
low magnetic fields, which makes them ‘anti-magnetars’ Gotthelf2010 .
Figure 7: X-ray image of the supernova remnant Cassiopeia A taken with the
Chandra X-Ray Observatory, color coded for energy (Red: 0.5-1.5 keV; Green:
1.5-2.5 keV; Blue 4.0-6.0 keV). The CCO is the central white point source.
Credit: NASA/CXC/MIT/UMass Amherst/M.D.Stage et al.
The observed spectra are purely thermal, but some have multiple blackbody
components. The measured radii of the emitting areas, assuming blackbody
spectra or even H atmospheres, are less that 1 km, much smaller than a NS
radius. This is surprising if the emission is from cooling of the NS,
especially since the very low magnetic fields do not preferentially conduct
the cooling to small hotspots at the poles Zavlin1995 . Recently though, fits
of the spectrum of the CCO in CasA with C atmosphere models, that are harder
than those of H or He, give emitting radii that are consistent with a NS
radius of 10 - 12 km Ho2009 . This could be an indication of recent accretion
activity that would deposit some heavy elements on the NS surface. Even more
recently, the temperature of CasA is observed to be decreasing, by 4% in just
10 years Heinke2010 . If this is in fact the first measured cooling of a NS,
the rate of cooling is much more rapid than standard NS cooling models
predict. The measurements require proton or neutron superfluid models
Shternin2011 that predict sudden cooling at around the age of CasA (300 yr).
These measurements could thus also be the strongest evidence for superfluidity
in NS cores. Some CCOs show absorption lines in their spectra. 1E 1207.4 -5209
has 2-4 harmonically spaced lines at 0.7, 0.14, 0.21 and 0.28 keV Bignami2003
and the spectrum of J0822.0-4300 may have an absorption feature at 0.8 keV
Gotthelf2009 . These could be electron cyclotron lines in a field around
$8\times 10^{10}$ G.
Several mysteries surround these objects. Were they really born with such low
magnetic fields or were higher interior fields submerged by mass fallback
accretion to emerge later, enabling them to turn on as normal RPPs Ho2011 ?
This hypothesis would fit with the need for C atmospheres that make their
spectra consistent with full-surface NS cooling.
## V Isolated Neutron Stars
Another class of NSs that appear to be thermally cooling with no emission
outside the soft X-ray band, except for faint optical/UV counterparts, are
known as Isolated Neutron Stars (INS) (see review by Kaplan2008 ). Although
these properties are similar to those of CCOs, they are a distinct class
because they lack any observable associated supernova remnant or nebula. There
are presently seven confirmed INS (sometimes referred to as The Magnificent
Seven), six of which have measured weakly modulated X-ray pulsations with
periods between 3 s and 11 s, much longer than those of CCOs. Two have high
measured $\dot{P}$ near $10^{-13}$, giving surface magnetic fields around
$10^{13}$ G, characteristic ages $\tau\sim 2$ Myr and spin-down luminosity
around $4\times 10^{30}\,{\rm erg\,s^{-1}}$ assuming dipole spindown. They
have significant proper motions, indicating distances of less than 500 pc. The
INS thus appear to be very nearby, cooling middle-aged NSs.
The X-ray spectra of INS are soft blackbodies with kT in the range 40 - 100
eV, and X-ray luminosities in the range $10^{30}-10^{32}\,{\rm erg\,s^{-1}}$,
with no measurable power law or non-thermal components. Four have observed
optical and/or UV counterparts that appear at first to be Rayleigh-Jeans
extensions of the X-ray blackbodies. However the optical spectra are about a
factor of 10 above the extrapolation of the X-ray blackbody. Alternatives to
simple blackbody fits have been explored, such as a thin magnetic, ionized H
atmosphere for the spectrum of J1856.5–3754, the original INS Ho2007 , or high
B-field atmospheres Potekhin2012 . Most INS show multiple or complex
absorption lines in their spectra at energy between 0.3 and 1 keV (e.g.
Kaplan2009 ). It is not clear what causes these features, but some
possibilities are proton cyclotron, and neutral or molecular H absorption. The
X-ray luminosities of several INS are larger than their spin down
luminosities, by a factor of 60 in the case of RX J0720, so that rotation
cannot power the observed X-ray emission. Furthermore, the temperatures and
thermal luminosities of INS are too high (by about a factor of 10) for
conventional cooling at their characteristic ages. This could imply that their
magnetic fields have not been constant since birth but have decayed Heyl1998 ,
providing another power source in addition to cooling and making them younger
than their spin-down ages.
It is not clear how INS are connected to the rest of the RPP population, other
than being NSs with thermal emission. Do they lack detectable radio emission
because our viewing angle does not cross the radio beam? A fit of the X-ray
profile of J1856.5–3754 Ho2007 constrains magnetic inclination and viewing
angles to be far apart. Or are they dead radio pulsars or extreme RRATS? They
do lie very near or beyond the empirical radio death line. But why do they not
have detectable non-thermal high energy emission? Other RPPs of similar ages
or older have detectable X-ray or gamma-ray pulsations. If their magnetic
fields are decaying or have decayed from magnetar-strength fields, then they
could be old magnetars Heyl1998 that wound up in the normal pulsar part of
$P$-$\dot{P}$ space Pons2009 . At the moment the population of INS is too
small to draw any firm conclusions about their nature and origin, but searches
for new INS are underway.
## VI Accreting Neutron Stars
NSs in binary systems can accrete matter from the companion stars, either from
the stellar winds or from an accretion disk that forms if the companion
overflows its Roche Lobe. The gravitational energy from the infalling matter
provides at least part of the energy for the observed radiation and the
accretion torques dominate the spin evolution. Despite these common
properties, accreting NSs display a wide variety of behaviors, depending on
the NS magnetic field strength, mass of the companion and properties of the
accretion.
### VI.1 Low-Mass X-Ray Binaries
LMXBs are binary systems in which one member is a NS or black hole and the
other star is a low-mass main sequence star, white dwarf or red giant that
fills its Roche Lobe, transferring matter onto the compact object through an
accretion disk. Almost all of the radiation is emitted as X rays with a very
small amount (about 1%) in optical light. The NS in LMXB are thought to be
old, have weak magnetic fields ($\sim 10^{8}$ G) and are being spun up by
torques from the accretion disk. One explanation for their very weak fields is
that the accretion has reduced or submerged the magnetic flux Romani1993 ;
Cumming2001 ; Ruderman2004 . Depending on the accretion rate $\dot{M}$, they
will reach an equilibrium where the pressure of the NS magnetic field balances
the pressure of the accretion flow after about 1 Myr, at which point the NS
period will have reached an equilibrium period $P_{\rm eq}$. The value of
$P_{\rm eq}$ depends on the NS field and $\dot{M}$ given by Eq (3). For
magnetic fields around $10^{8}$ G and $\dot{M}$ near the Eddington limit,
$P_{\rm eq}$ will reach millisecond periods. X-ray pulsations at millisecond
periods were in fact found in a number of LMXBs with the Rossi X-Ray Timing
Explorer (RXTE), the first being SAX J1808.4-3658 Wijnands1998 . Around 24 of
the more than 100 LMXB sources have shown ms X-ray pulsations, with
frequencies in the range 100 - 700 Hz (see review by Chak2005 ). Discovery of
these long-sought ms X-ray pulsations in LMXB finally established them as the
progenitors of the rotation-powered MSPs. The transition between an LMXB and a
MSP was actually observed when ms radio pulsations were discovered from the
source J102347.6+003841 which less than ten years earlier had shown optical
emission indicating an accretion disk Arch2009 .
Some LMXBs have steady X-ray emission, with occasional bursting behavior while
others are detectable only during their bursts. A good fraction are transient,
with outbursts on month to year timescales. The outbursts typically last weeks
to months. The X-ray bursts are thought to be thermonuclear explosions on the
NS surface, (see Strohmayer2004 for review) when the density and temperature
of the accreted material reach the critical point for igniting nuclear
reactions. The bursts have typical timescales of around 10 - 100 s and total
energy of $10^{39}-10^{40}\,{\rm erg\,s^{-1}}$. LMXBs also show kHz
quasiperiodic oscillations (QPO) in their X-ray emission (see vanderKlis2000
for review), thought to originate in the inner accretion disk flow. In most
models of QPO, the oscillation frequencies are the orbital frequencies of
accreting matter. Since these are stable orbits around a NS, they must be
outside the innermost stable circular orbit (ISCO) determined by General
Relativity, so their frequencies can be used to place limits on the NS mass
Miller1998 . Since the maximum measured QPO frequencies are about 1200 Hz,
using the ISCO of $6GM/c^{2}$ in the Schwarzschild metric gives the NS mass
$M<2M_{\odot}$ Zhang1997 .
Figure 8: X-ray burst oscillations in 4U 1702-43 (from Strohmayer1999 ). The
histogram shows the X-ray intensity of the X-ray burst. The contours show the
Fourier power level as a function of frequency and time, indicating a drifting
oscillation starting at 328 Hz (t = 7 s) and ending at 330.5 Hz.
A subset ($\sim 13$) of LMXB show ms pulsations in their persistent emission.
The rest (11) show the pulsations only during X-ray bursts Strohmayer1996 ;
Watts2012 or in QPO behavior. During the X-ray bursts, oscillations have been
observed, usually during the decay phase (see Fig 8), whose frequency rises to
an asymptotic limit as the intensity drops Strohmayer1999 . The interpretation
of this behavior is that the rotation of the NS is modulating the burst hot
spot which expands and rises in altitude, then drops back onto the surface.
The true rotation frequency of the NS is then the observed asymptotic
frequency. The NS spin frequency can also appear as the frequency difference
between pairs of QPO that move up and down in frequency together. Since the
difference in the frequencies of the QPO pair stays the same and is in the
range 200 - 400 Hz, it is thought to be the NS rotation frequency. In some
cases though, the frequency difference appears to be closer to 1/2 of the spin
frequency, and the relation between QPO and spin frequency has been questioned
Mendez2007 .
The distribution of NS spin frequencies from these various measurements cuts
off suddenly at a maximum frequency around 620 Hz Chak2005 , which is well
below the $\sim 1000$ Hz break-up frequency of a NS. This indicates that there
are some mechanisms that halt or counteract the spin up process. Several
possibilities have been suggested, including gravitational wave damping
Harding1983 ; Wagoner1984 , making these good sources of kHz gravitational
radiation for possible detection with Advanced LIGO.
### VI.2 Intermediate- and High-Mass X-Ray Binaries
Binary systems consisting of a neutron star or black hole and high-mass O or B
star ($M\mathrel{\mathchoice{\lower
2.0pt\vbox{\halign{$\m@th\displaystyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
2.0pt\vbox{\halign{$\m@th\textstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
2.0pt\vbox{\halign{$\m@th\scriptstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
2.0pt\vbox{\halign{$\m@th\scriptscriptstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}}5M_{\odot}$
) are known as High-Mass X-Ray Binaries (HMXB), an example being Vela X-1. In
a large subclass of HMXBs, the NS orbits a Be star in a very eccentric orbit,
accreting material only occasionally in outbursts when the NS crosses a disk
of material surrounding the Be star. An example of a Be X-ray binary is
A0535+26. If the donor star is an intermediate-mass star
($1.0M_{\odot}\mathrel{\mathchoice{\lower
2.0pt\vbox{\halign{$\m@th\displaystyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
2.0pt\vbox{\halign{$\m@th\textstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
2.0pt\vbox{\halign{$\m@th\scriptstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
2.0pt\vbox{\halign{$\m@th\scriptscriptstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}}M\mathrel{\mathchoice{\lower
2.0pt\vbox{\halign{$\m@th\displaystyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
2.0pt\vbox{\halign{$\m@th\textstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
2.0pt\vbox{\halign{$\m@th\scriptstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
2.0pt\vbox{\halign{$\m@th\scriptscriptstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}}5M_{\odot}$),
the system is an intermediate-mass X-ray binary (IMXB) Pod2002 , a well-known
example being Her X-1. In HMXBs, the compact object accretes material from the
wind of the companion star or from a Be star disk, while in IMXBs the NS
accretes from a disk as in LMXBs. Both of these classes, numbering about
several hundred, are bright X-ray sources, and a large number show either
persistent or transient X-ray pulsations (in the case of Be star binaries). In
contrast to the ms pulsations of LMXBs, pulsation periods of HMXBs are much
longer, in the range 1 - 1000 s. The relation in Eq (3) then requires that the
NS magnetic field in these sources are near $10^{12}$ G. In fact, many X-ray
pulsars in IMXBs and HMXBs (about 20) show absorption features in their
spectra Heindl2004 that are almost certainly due to electron cyclotron
resonance scattering. The energies of these cyclotron resonant scattering
features (CRSF) are in the range 10 - 50 keV, also suggesting (Eq. [2]) that
the surface magnetic of the NSs are around $10^{12}-10^{13}$ G.
Because the NS magnetic fields are high in these systems, the Alfven radius
$r_{\rm A}=3\times 10^{8}\,{\rm cm}\,B_{12}^{4/7}\,\left({\dot{M}\over
10^{17}\,{\rm g\,s^{-1}}}\right)^{-2/7}\,\left({M/M_{\odot}}\right)^{-1/7},$
(12)
lies at a much larger distance from the NS than in LMXBs, and the accreting
material is strongly channeled via the magnetic field lines onto a small polar
cap. The infalling material then heats the NS atmosphere above the polar cap
to temperatures around $10^{8}$ K . The proposed mechanisms by which the
kinetic energy of the infalling material is decelerated and transferred to
heat depends on the mass accretion rate, which in turn determines its pressure
and the source luminosity $L_{\rm X}=GM\dot{M}$. If the accretion pressure
exceeds the radiation pressure, which is much higher than the spherical
Eddington limit due to the channeling of the flow, a radiative shock will form
at some distance above the NS surface Basko1976 . If the pressure of the
accretion flow does not exceed the radiation pressure, the atmosphere is
heated either by Coulomb collisions Pavlov1976 or a collisionless shock. In
the case where a stand-off shock forms, a slowly sinking accretion column
radiates mostly perpendicular to the magnetic axis in a fan beam Becker2005 ,
while in the Coulomb-heated case, the atmosphere is a thin slab at the NS
surface and radiates primarily along the magnetic axis Harding1984 . The
continuum spectrum of the X-ray pulsar radiation is thermal bremsstrahling and
Compton scattering emission (e.g. Becker2007 ). The CRSFs appear as absorption
lines and as many as four harmonics also appear in some X-ray pulsars such as
4U 0115+63 Heindl1999 . They are formed through resonant scattering of
continuum photons by electrons that have a thermal distribution of momenta
along the strong magnetic field lines (see Harding&Lai2006 , for a detailed
description of the physics of CRSFs). Since the resonant scattering cross
section has a strong dependence on angle, CRSFs can provide a good diagnostic
of the geometry and physical conditions of the emitting atmosphere Araya1999 ;
Schonherr2007 .
### VI.3 Microquasars and Gamma-Ray Binaries
A subclass of accreting X-ray binaries are called microquasars since they
display properties similar to those of quasars, including rapid variability of
their X-ray emission and radio jets. Strong, broadband emission as well as
broad emission lines are also observed from their accretion disks. They
consist of a compact object, either neutron star or black hole, with a normal
companion. These sources undergo repeated and sometimes periodic radio,
optical and X-ray flaring with associated formation of relativistic jets.
Cygnus X-3 is a well-known microquasar that could be a black hole or neutron
star in a 4.8 hr orbit around a massive hot star. One of the strongest X-ray
sources in the sky, it is also detected from radio to GeV gamma-ray
wavelengths Abdo2009d .
The term gamma-ray binary has come to include binary systems that contain a
compact object and emit high energy gamma rays. Since several microquasars
have been detected in gamma rays, they are one of the gamma-ray binary source
types. The other gamma-ray binary source type is a system containing a RPP
orbiting a massive star. The prototype of this source class is PSR B1259-63,
which is a 48 ms radio pulsar in a 3.4 yr eccentric orbit around a Be star. It
appears as a normal radio pulsar at apastron, but at periastron the pulsar is
thought to make several crossings through the Be star disk, producing X-rays
and gamma-rays at GeV Abdo2011b and TeV Ahar2005 energies. The X-ray
binaries LSI+61 303 and LS5039 have been detected by Fermi in high-energy
gamma rays Abdo2009e ; Abdo2009f , have characteristics similar to PSR
B1259-63 and are thought to contain undetected pulsars. The Fermi source 1FGL
J1018.6 5856 is the first Galactic binary system to have been discovered at
gamma-ray energies Abdo2012 but appears similar to LS5039, as both X-ray and
radio modulation at the orbital period were subsequently detected.
## VII Summary
Neutron stars are found in a wide variety of sources, displaying an amazing
array of behavior. They can be isolated or in binary systems, accreting,
heating, cooling, spinning down, spinning up, pulsing, flaring and bursting.
The one property that seems to determine their behavior most strongly is their
magnetic field strength, structure and evolution. The hot polar caps, bursts
and flares of magnetars are likely due to the rapid decay and twisting of
their superstrong magnetic fields, whose very existence requires some kind of
early dynamo activity. The intermediate-strength magnetic fields of RPPs
determines their spin-down behavior and radiation properties. However, the
overlap of the magnetar and RPP populations is not understood at present. Why
don’t high-field RPPs burst or flare? Why don’t lower-field magnetars
sometimes behave more like RPPs? INS may be old magnetars whose high fields
have decayed, but they do not account for the existence of younger RPPs with
magnetar-strength fields. Not only the strength of the magnetic field but also
its configuration may be important in making a NS a magnetar or a RPP.
Magnetic field decay is a critical link between other NS populations as well.
“Decay” of the magnetic field is necessary for normal RPPs to evolve into MSPs
through accretion and spin up in LMXBs. Some kind of accretion-driven field
reduction is the most likely mechanism, but it is controversial since it is
not clear how effective it is or on what timescale a buried field might re-
emerge. One piece of evidence in favor of accretion-driven field reduction is
the fact that NSs in LMXBs, which are older systems ($>10^{8}$ yr), have
mostly low fields and NSs in HMXBs, which are younger systems ($10^{7}-10^{8}$
yr), have higher fields. This may be an indication that accretion-driven field
reduction or decay has not had enough time to operate in HMXBs but has in
LMXBs. However, there does not seem to be any evidence of decaying fields in
either the LMXB or HMXB populations; e.g. smaller magnetic fields in older
systems. On the other hand, CCOs are very young so if they acquired their low
fields through mass fallback accretion, the field submergence would have had
to operate on much faster timescales than it apparently does in LMXBs. But as
we continue to find new species in the NS zoo, one of these may someday be the
“Rosetta Stone” that will give us the clues for solving these puzzles.
I would like to thank Megan Decesar, Isabelle Grenier, Peter Meszaros, Tod
Strohmayer and Bing Zhang for providing helpful comments on the manuscript.
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|
arxiv-papers
| 2013-02-04T21:25:30 |
2024-09-04T02:49:41.287782
|
{
"license": "Public Domain",
"authors": "Alice K. Harding",
"submitter": "Alice K. Harding",
"url": "https://arxiv.org/abs/1302.0869"
}
|
1302.0923
|
# Regular circle actions on 2-connected 7-manifolds
Yi Jiang
Institute of Mathematics, Chinese Academy of Sciences,
Beijing 100190, China The author’s research is partially supported by 973
Program 2011CB302400 and NSFC 11131008.
###### Abstract
We determine the homeomorphism (resp. diffeomorphism) types of those
2-connected 7-manifolds (resp. smooth 2-connected 7-manifolds) that admit
regular circle actions (resp. smooth regular circle actions).
2010 Mathematics subject classification:
55R15 (55R40)
Key Words and phrases:
group actions, Kirby–Siebenmann invariant, $\mu-$invariant, characteristic
classes
Email address:
[email protected]
## 1 Introduction
In this paper all manifolds under consideration are closed, oriented and
topological, unless otherwise stated. Moreover, all homeomorphisms and
diffeomorphisms are to be orientation preserving. Given a positive integer $n$
let $S^{n}$ (resp. $D^{n+1}$) be the diffeomorphism type of the unit
$n$–sphere (resp. the unit $(n+1)$–disk) in $(n+1)$–dimensional Euclidean
space $\mathbb{\mathbb{R}}^{n+1}$.
Definition 1.1. A circle action $S^{1}\times M\rightarrow M$ on a manifold $M$
is called regular if this action is free and the orbit space $N:=M/S^{1}$
(with quotient topology) is a manifold.
Similarly, a smooth circle action $S^{1}\times M\rightarrow M$ on a smooth
manifold $M$ is called regular if this action is free (see [26, p.38,
Proposition 5.2]).
For a given manifold $M$ one can ask
Problem 1.2. Does $M$ admit a regular circle action?
Solutions to Problem 1.2 can have direct implications in contact topology. For
example, the Boothby-Wang theorem implies that the existence of a smooth
regular circle action on a smooth manifold $M$ is a necessary condition to the
existence of a regular contact form on $M$ (see [8, p.341]).
Problem 1.2 has been solved for all $1$–connected $5$–manifolds by Duan and
Liang[6]. In particular, it was shown that all $1$–connected $4$–manifolds
with second Betti number $r$ can be realized as the orbit spaces of some
regular circle actions on the single $5$–manifold $\\#_{r-1}S^{2}\times
S^{3}$, the connected sums of $r$–$1$ copies of the product $S^{2}\times
S^{3}$. In this paper we study Problem 1.2 for the $2$–connected
$7$–manifolds.
Our main result is stated in terms of a family
$M_{l,k}^{c},c\in\left\\{0,1\right\\},l,k\in\mathbb{Z}$ of $2$–connected
$7$–manifolds. The manifolds $M_{l,k}^{0}$ are the total spaces of the
$S^{3}$–bundles $\pi_{M}:M_{l,k}^{0}\rightarrow S^{4}$ with characteristic map
$[f_{l,k}]\in\pi_{3}(SO_{4})$ defined by
> $f_{l,k}(u)v=u^{l+k}vu^{-l}$, $v\in\mathbb{\mathbb{R}}^{4}$, $u\in S^{3}$
where the space $\mathbb{\mathbb{R}}^{4}$ and the sphere $S^{3}$ are
identified with the algebra of quaternions and the space of unit quaternions
respectively, and where quaternion multiplication is understood on the right
hand side of the formula. Complete classification on the manifolds
$M_{l,k}^{0}$ has been obtained by Milnor[19], Crowley and Escher[3].
The manifold $M_{l,k}^{1}$ is an analogue of the manifold $M_{l,k}^{0}$ in the
non–smooth category when $k\equiv 0\mathop{\mathrm{m}od}2$. Write
$\mathcal{S}^{TOP}(M_{l,k}^{0})$ for the set of all equivalence classes
$[M,h]$ of the pairs $(M,h)$ with $h:M\rightarrow M_{l,k}^{0}$ a homotopy
equivalence of $7$–manifolds. Two such pairs $(M,h)$ and
$(M^{{}^{\prime}},h^{{}^{\prime}})$ are called equivalent if there is a
homeomorphism $f:M\rightarrow M^{{}^{\prime}}$ such that $h$ is homotopic to
$h^{{}^{\prime}}\circ f$ (see [18, Chapter 2]). Adapting the arguments of [3,
Section 5] from the PL case to the TOP case we have the composition of
isomorphisms
$\mathcal{S}^{TOP}(M_{l,k}^{0})\overset{\eta}{\rightarrow}[M_{l,k}^{0},G/TOP]\overset{d}{\rightarrow}H^{4}(M_{l,k}^{0};\pi_{4}(G/TOP))\cong
H^{4}(M_{l,k}^{0})$,
where the space $G$ (resp. $TOP$) is the direct limit of the set of self
homotopy equivalences of $S^{n-1}$ (resp. the topological monoid of origin-
preserving homeomorphisms of $\mathbb{R}^{n}$), $\eta$ is the one to one
correspondence in the surgery exact sequence (see [18, p.40-44]) and where the
isomorphism $d$ is induced by the primary obstruction to null-homotopy. Write
$\left[M_{l,k}^{1},h_{M}\right]$ for the element
$(d\circ\eta)^{-1}(\pi_{M}^{\ast}(\iota))\in\mathcal{S}^{TOP}(M_{l,k}^{0})$
where $\iota\in H^{4}(S^{4})$ is the generator as in [3] and
$H^{4}(M_{l,k}^{0})\cong\mathbb{Z}_{k}$ is generated by
$\pi_{M}^{\ast}(\iota)$. Clearly, the 7-manifold $M_{l,k}^{1}$ is
$2$–connected and unique up to homeomorphism.
The group $\Gamma_{7}$ of exotic $7$–spheres is cyclic of order $28$ with
generator $M_{1,1}^{0}$ [7, Section 6]. Let
$\Sigma_{r}:=rM_{1,1}^{0}\in\Gamma_{7}$, $r\in\mathbb{Z}$. Our main result is
stated below, where $\mathbb{\mathbb{N}}$ is the set of all nonnegative
integers.
Theorem 1.3. All homeomorphism classes of the $2$–connected $7$–manifolds that
admit regular circle actions are represented by
$\\#_{2r}S^{3}\times S^{4}\\#M_{6m,(1+c)k}^{c}$,
$c\in\left\\{0,1\right\\},r\in\mathbb{\mathbb{N}}$ and
$m,k\in\mathbb{\mathbb{Z}}$.
All diffeomorphism classes of the smooth $2$–connected $7$–manifolds that
admit smooth regular circle actions are represented by
$\\#_{2r}S^{3}\times S^{4}\\#M_{6(a+1)m,(a+1)k}^{0}\\#\Sigma_{(1-a)m}$,
$a\in\left\\{0,1\right\\}$, $r\in\mathbb{\mathbb{N}}$,
$m,k\in\mathbb{\mathbb{Z}}$.
In the course to establish Theorem 1.3 we obtain also a classification of the
$6$–manifolds that can appear as the orbit spaces of some regular circle
actions on $2$–connected $7$–manifolds, see Lemmas 2.1 and 2.2 in Section 2.
In addition, Theorem 1.3 has some direct consequences which are discussed in
Section 4.
## 2 The homeomorphism types of the orbit spaces
In this section we determine the homeomorphism types of those $6$–manifolds
which can appear as the orbit spaces of some regular circle actions on
$2$–connected $7$–manifolds.
A $2$–connected $7$–manifold $M$ with a regular circle action defines a
principal $S^{1}$–bundle $M\rightarrow N$ with base space $N=M/S^{1}$ [9].
Fixing an orientation on $S^{1}$ once and for all, and let $N$ be furnished
with the orientation compactible with that on $M$. From the homotopy exact
sequence
$0\rightarrow\pi_{2}(M)\rightarrow$
$\pi_{2}(N)\rightarrow\pi_{1}(S^{1})\rightarrow\pi_{1}(M)\rightarrow\pi_{1}(N)\rightarrow
0$
of the fibration one finds that
$\pi_{1}(N)=0$; $\pi_{2}(N)\cong\pi_{1}(S^{1})\cong\mathbb{\mathbb{Z}}.$
Consequently $N$ is a $1$–connected $6$–manifold with
$H_{2}(N)\cong\mathbb{\mathbb{Z}}$.
Conversely, for a $1$–connected $6$–manifold $N$ with
$H_{2}(N)\cong\mathbb{\mathbb{Z}}$ let $t\in H^{2}(N)\cong\mathbb{\mathbb{Z}}$
be a generator and let
$S^{1}\hookrightarrow N_{t}\rightarrow N$
be the oriented circle bundle over $N$ with Euler class $t$. From the homotopy
exact sequence of this fibration we find that $N_{t}$ is $2$–connected with
the canonical orientation as the total space of the circle bundle. Summarizing
we get
Lemma 2.1. Let $S^{1}\times M\rightarrow M$ be a regular circle action on a
$2$–connected $7$–manifold $M$ with orbit space $N$. Then $N$ is a
$1$–connected $6$–manifold with $H_{2}(N)\cong\mathbb{\mathbb{Z}}$.
Conversely, every $1$–connected $6$–manifold $N$ with
$H_{2}(N)\cong\mathbb{\mathbb{Z}}$ can be realized as the orbit space of some
regular circle action on a $2$–connected $7$–manifold.$\square$
In view of Lemma 2.1 the classification of those $1$–connected $6$–manifolds
$N$ with $H_{2}(N)\cong\mathbb{\mathbb{Z}}$ amounts to a crucial step toward a
solution to Problem 1.2. In terms of the known invariants for $1$–connected
$6$–manifolds due to Jupp[14] and Wall[27] we can enumerate all these
manifolds in the next result.
Denote by $\Theta$ the set of equivalence classes $[N,t]$ of the pairs
$\left(N,t\right)$ with $N$ a $1$–connected $6$–manifold whose integral
cohomology satisfies
$H^{r}(N)=\left\\{\begin{tabular}[]{l}$\mathbb{\mathbb{Z}}$ if $r=0,2,4,6$;\\\
$0$ otherwise,\end{tabular}\right.$
and with $t\in H^{2}(N)$ a fixed generator. Two elements
$(N_{1},t_{1}),(N_{2},t_{2})$ are called equivalent if there is a
homeomorphism $f:N_{1}\rightarrow N_{2}$ such that $f^{\ast}t_{2}=t_{1}$. For
each $(N,t)$ fix a generator $x\in H^{4}(N)$ such that the value $\left\langle
t\cup x,[N]\right\rangle$ of the product $t\cup x$ on the fundamental class
$[N]$ is equal to $1$. Consider the functions
$k,p:\Theta\rightarrow\mathbb{\mathbb{Z}}$;
$\varepsilon:\Theta\rightarrow\\{0,1\\}$; $\delta:\Theta\rightarrow\\{0,1\\}$
determined by the following properties
> i) $t^{2}=k([N,t])x$;
>
> ii) the second Stiefel-Whitney class $w_{2}(N)$ and the first Pontrjagin
> class $p_{1}(N)$ of $N$ are given by $\varepsilon([N,t])t$
> $\mathop{\mathrm{m}od}2$ and $p([N,t])x$, respectively;
>
> iii) the class $\Delta(N)\equiv\delta([N,t])x\mathop{\mathrm{m}od}2\in
> H^{4}(N;\mathbb{\mathbb{Z}}_{2})$ is the Kirby-Siebenmann invariant of $N$,
where the Kirby-Siebenmann invariant $\Delta(V)$ of a manifold $V$ is the
obstruction to lift the classifying map $V\rightarrow BTOP$ for the stable
tangent bundle of $V$ to $BPL$, and where $BTOP$ and $BPL$ are the classifying
spaces for the stable $TOP$ bundles and $PL$ bundles, respectively[15].
Lemma 2.2. For each $1$–connected $6$–manifold $M$ with
$H_{2}(M)\cong\mathbb{\mathbb{Z}}$ there exists an $r\in\mathbb{\mathbb{N}}$
and an element $[N,t]\in\Theta$ such that $M\cong\\#_{r}S^{3}\times
S^{3}\\#N$.
Moreover, the system $\\{k,p,\varepsilon,\delta\\}$ is a set of complete
invariants for elements $\left[N,t\right]\in\Theta$ that is subject to the
following constraints:
> i)If $k(\left[N,t\right])\equiv 1\mathop{\mathrm{m}od}2$, then
> $\varepsilon(\left[N,t\right])=0$ and
>
> $p(\left[N,t\right])=24m+4k(\left[N,t\right])+24\delta(\left[N,t\right])$
> for some $m\in\mathbb{Z}$;
>
> ii)If $k(\left[N,t\right])\equiv 0\mathop{\mathrm{m}od}2$, then for some
> $m\in\mathbb{Z}$
>
>
> $p(\left[N,t\right])=\left\\{\begin{array}[]{cc}24m+4k(\left[N,t\right])+24\delta(\left[N,t\right])&\text{if
> }\varepsilon(\left[N,t\right])=0\\\
> 48m+k(\left[N,t\right])+24\delta(\left[N,t\right])&\text{if
> }\varepsilon(\left[N,t\right])=1\end{array}\right.$.
In addition, the manifold $N$ is smoothable if and only if
$\delta(\left[N,t\right])=0.$
Proof. This is a direct consequence of [14, Theorem 0; Theorem 1]. In
particular, the expressions of the function $p$ are deduced from the following
relation on $H^{6}(N)$ which holds for all $d\in\mathbb{\mathbb{Z}}$:
$(2dt+\varepsilon(\left[N,t\right])t)^{3}\equiv(p(\left[N,t\right])x+24\delta(\left[N,t\right])x)(2dt+\varepsilon(\left[N,t\right])t)\mathop{\mathrm{m}od}48$.$\square$
## 3 Circle bundles over $\left[N,t\right]\in\Theta$
Lemma 2.2 singles out the family $\Theta$ of $1$–connected $6$–manifolds which
plays a key role in presenting the orbit spaces of regular circle actions on
$2$–connected $7$–manifolds. In this section we determine the homeomorphism
and diffeomorphism type of the total space $N_{t}$ of the circle bundle over
$\left[N,t\right]\in\Theta$, i.e. the oriented circle bundle over $N$ with
Euler class $t$. For this purpose we shall recall in Section 3.1 that the
definition of the known invariant system for $2$–connected $7$–manifolds. The
main results in this section are Lemmas 3.3 and 3.5, which identify the
homeomorphism and diffeomorphism types of the manifolds $N_{t}$ with certain
$M_{l,k}^{c}$.
### 3.1 Invariants for 2–connected 7–manifolds
Recall from Eells, Kuiper[7], Kreck, Stolz[16] and Wilkens[29] that associated
to each $2$–connected $7$–manifold $M$ there is a system
$\\{H,\frac{p_{1}}{2},b,\Delta,\mu,s_{1}\\}$ of invariants characterized by
the following properties:
> i) $H$ is the forth integral cohomology group $H^{4}(M)$[29];
>
> ii) $\frac{p_{1}}{2}(M)\in H$ is the first spin characteristic class [25]
> introduced by Wilkens [29] in smooth category and extended to topological
> category by Kreck–Stolz[16, Lemma 6.5];
>
> iii)
> $b:\tau(H)\otimes\tau(H)\rightarrow\mathbb{\mathbb{Q}}/\mathbb{\mathbb{Z}}$
> is the linking form on the torsion part $\tau(H)$ of the group $H$[29];
>
> iv) $\Delta(M)\in H^{4}(M;\mathbb{\mathbb{Z}}_{2})$ is the Kirby–Siebenmann
> invariant of $M$ [16].
Furthermore, if the manifold $M$ is smooth and bounds a smooth $8$–manifold
$W$ with the induced map $j^{\ast}:H^{4}(W,M;\mathbb{\mathbb{Q}})\rightarrow
H^{4}(W;\mathbb{\mathbb{Q}})$ an isomorphism, then
> v) the invariant $\mu\in\mathbb{\mathbb{Q}}/\mathbb{\mathbb{Z}}$ is firstly
> defined in [7] for a spin $W$ and extended in [16] for a general $W$, whose
> value is given by the formula
>
> $\mu(M)\equiv$–$\frac{1}{2^{5}\cdot 7}\sigma(W)+\frac{1}{2^{7}\cdot
> 7}p_{1}^{2}(W)$–$\frac{1}{2^{6}\cdot 3}z^{2}\cdot
> p_{1}(W)+\frac{1}{2^{7}\cdot 3}z^{4}\mathop{\mathrm{m}od}\mathbb{Z}$,
where $z\in H^{2}(W)$ satisfies $w_{2}(W)=z\mathop{\mathrm{m}od}2$,
$\sigma(W)$ is the signature of the intersection form on
$H^{4}(W,M;\mathbb{Q})$, and where $p_{1}^{2}(W)$, $z^{2}\cdot p_{1}(W)$ and
$z^{4}$ are the characteristic numbers
> $\langle p_{1}(W)\cup j^{\ast-1}p_{1}(W),[W,M]\rangle$, $\langle z^{2}\cup
> j^{\ast-1}p_{1}(W),[W,M]\rangle$,
>
> $\langle z^{2}\cup j^{\ast-1}z^{2},[W,M]\rangle$,
respectively. Finally, if $M$ is topological and bounds a topological
8–manifold $W$ with the induced map
$j^{\ast}:H^{4}(W,M;\mathbb{\mathbb{Q}})\rightarrow
H^{4}(W;\mathbb{\mathbb{Q}})$ an isomorphism, then
> vi) the topological invariant
> $s_{1}\in\mathbb{\mathbb{Q}}/\mathbb{\mathbb{Z}}$ is defined in [16] whose
> value is given by
>
>
> $s_{1}(M)\equiv$–$\frac{1}{2^{3}}\sigma(W)+\frac{1}{2^{5}}p_{1}^{2}(W)$–$\frac{7}{2^{4}\cdot
> 3}z^{2}\cdot p_{1}(W)+\frac{7}{2^{5}\cdot
> 3}z^{4}\mathop{\mathrm{m}od}\mathbb{Z}$.
Example 3.1. Let $N_{t}$ be the total space of the circle bundle over
$\left[N,t\right]\in\Theta$. Then the system
$\\{H,\frac{p_{1}}{2},b,\Delta,\mu,s_{1}\\}$ of invariants for the manifold
$N_{t}$ can be expressed in terms of the invariants for $\Theta$ introduced in
Lemma 2.2 as follows. For simplicity we write $p,k,\varepsilon$ and $\delta$
in place of $p(\left[N,t\right])$, $k(\left[N,t\right])$,
$\varepsilon(\left[N,t\right])$ and $\delta(\left[N,t\right])$, respectively.
> i) $H^{4}(N_{t})\cong\mathbb{\mathbb{Z}}_{k}$ with generator
> $\pi^{\ast}(x)$, where $\pi:N_{t}\rightarrow N$ is the bundle projection and
>
> $\mathbb{\mathbb{Z}}_{k}=\left\\{\begin{array}[]{cc}\mathbb{Z}&\text{if
> }k=0\\\ \mathbb{Z}/k\mathbb{Z}&\text{if }k\neq 0\end{array}\right.$;
>
> ii)
> $\Delta(N_{t})\equiv\frac{1+(-1)^{k}}{2}\cdot\delta\pi^{\ast}(x)\mathop{\mathrm{m}od}2$;
>
> iii) $\frac{p_{1}}{2}(N_{t})\equiv\frac{p+\varepsilon
> k}{2}\pi^{\ast}(x)\mathop{\mathrm{m}od}k$;
>
> iv)
> $b(\pi^{\ast}(x),\pi^{\ast}(x))\equiv\frac{1}{k}\mathop{\mathrm{m}od}\mathbb{\mathbb{Z}}$;
>
> v) $\mu(N_{t})\equiv$–$\frac{\left|k\right|}{2^{5}\cdot
> 7k}+\frac{(p+k)^{2}}{2^{7}\cdot 7k}+\frac{(\varepsilon-1)(2p+k)}{2^{7}\cdot
> 3}\mathop{\mathrm{m}od}\mathbb{Z}$;
>
> vi)
> $s_{1}(N_{t})=$–$\frac{\left|k\right|}{2^{3}k}+\frac{(p+k)^{2}}{2^{5}k}+\frac{7(\varepsilon-1)(2p+k)}{2^{5}\cdot
> 3}\mathop{\mathrm{m}od}\mathbb{Z}$.
Firstly, from the section $H^{2}(N)\overset{\cup
t}{\rightarrow}H^{4}(N)\overset{\pi^{\ast}}{\rightarrow}H^{4}(N_{t})\rightarrow
0$ in the Gysin sequence of the fibration $N_{t}\overset{\pi}{\rightarrow}N$
and from the relation $t^{2}=kx$ on $H^{4}(N)$ we find that
$H^{4}(N_{t})\cong\mathbb{\mathbb{Z}}_{k}$ with generator $\pi^{\ast}(x)$.
This shows i).
Next, let $f:N\rightarrow BTOP$ be the classifying map for the stable tangent
bundle of $N$. In view of the decomposition
$TN_{t}\cong\pi^{\ast}TN\oplus\varepsilon^{1}$($\varepsilon^{1}$ denotes the
trivial line bundle) for the tangent bundle of $N_{t}$ the classifying map for
the stable tangent bundle of $N_{t}$ is given by the composition
$f\circ\pi:N_{t}\rightarrow N\rightarrow BTOP$. It follows that the Kirby-
Siebenmann invariant $\Delta(N_{t})$ of the manifold $N_{t}$ is
$\pi^{\ast}\Delta(N)\equiv\delta\pi^{\ast}(x)\mathop{\mathrm{m}od}2$. This
shows ii).
To calculate the remaining invariants $\frac{p_{1}}{2},b,\mu,s_{1}$ of the
manifold $N_{t}$ we make use of the associated disk bundle
$W_{t}\overset{\pi_{0}}{\rightarrow}N$ of the oriented $2$–plane bundle
$\xi_{t}$ over $N$ with Euler class $t$. If $\varepsilon=1$, It follows from
the decomposition $TW_{t}\cong\pi_{0}^{\ast}TN\oplus\pi_{0}^{\ast}\xi_{t}$
that $w_{2}(W_{t})=0$ and
$\frac{p_{1}}{2}(W_{t})=\frac{p+k}{2}\pi_{0}^{\ast}x.$ From the relation
$\partial W_{t}=N_{t}$ we get
> $\frac{p_{1}}{2}(N_{t})=\frac{p+k}{2}\pi^{\ast}x$.
If $\varepsilon=0$, from the decomposition
$TN_{t}\cong\pi^{\ast}TN\oplus\varepsilon^{1}$ we get
> $\frac{p_{1}}{2}(N_{t})=\frac{p}{2}\pi^{\ast}(x)$.
This shows iii).
To compute the linking form $b$ of $N_{t}$ we can assume that $k\neq 0$.
Consider the commutative ladder of exact sequences
$\begin{array}[]{ccccccccc}0&\rightarrow&H^{4}(W_{t},N_{t})&\overset{j^{\ast}}{\rightarrow}&H^{4}(W_{t})&\overset{i^{\ast}}{\rightarrow}&H^{4}(N_{t})&\rightarrow&0\\\
&&\cong\uparrow\phi&&\uparrow\pi_{0}^{\ast}&&\uparrow id&&\\\
0&\rightarrow&H^{2}(N)&\overset{\cup
t}{\rightarrow}&H^{4}(N)&\overset{\pi^{\ast}}{\rightarrow}&H^{4}(N_{t})&\rightarrow&0\end{array}$
with $\phi$ the Thom isomorphism. Since
$\pi^{\ast}(x)=i^{\ast}\pi_{0}^{\ast}(x)$ and $y:=\phi(t)$ is a generator of
$H^{4}(W_{t},N_{t})$ with
$j^{\ast}(y)=\pi_{0}^{\ast}(t^{2})=k\pi_{0}^{\ast}(x)$ we get
>
> $b(\pi^{\ast}(x),\pi^{\ast}(x))\equiv\frac{1}{k}<y\cup\pi_{0}^{\ast}x,[W_{t},N_{t}]>\equiv\frac{1}{k}\mathop{\mathrm{m}od}\mathbb{\mathbb{Z}}$.
This shows iv).
Since the induced map
$j^{\ast}:H^{4}(W_{t},N_{t};\mathbb{\mathbb{Q}})\rightarrow
H^{4}(W_{t};\mathbb{\mathbb{Q}})$ is clearly an isomorphism when $k\neq 0$,
the invariants $\mu$ and $s_{1}$ are defined for $N_{t}$. Moreover, from the
Lefschetz duality and the relation $j^{\ast}(y)=k\pi_{0}^{\ast}(x)$ we get
$\sigma(W_{t})=\frac{\left|k\right|}{k}.$ From the decomposition
$TW_{t}\cong\pi_{0}^{\ast}(TN\oplus\xi_{t})$ we get, in addition to
> $p_{1}(W_{t})=\pi_{0}^{\ast}(p_{1}(N)+t^{2})=(p+k)\pi_{0}^{\ast}(x)$,
that
> $w_{2}(W_{t})\equiv(\varepsilon+1)\pi_{0}^{\ast}(t)\mathop{\mathrm{m}od}2$.
Therefore we can take $z=(1$–$\varepsilon)\pi_{0}^{\ast}(t)$ in the formulae
for $\mu$ and $s_{1}$, and as a result
>
> $z^{2}=(1$–$\varepsilon)^{2}\pi_{0}^{\ast}(t^{2})=k(1$–$\varepsilon)^{2}\pi_{0}^{\ast}(x)$.
As the group $H^{4}(W_{t},N_{t})\cong\mathbb{\mathbb{Z}}$ is generated by
$y=\phi(t)$ with the relation $j^{\ast}(y)=k\pi_{0}^{\ast}(x),$ the
isomorphism
> $H^{4}(W_{t},N_{t})\otimes
> H^{4}(W_{t})\overset{\cup}{\rightarrow}H^{8}(W_{t},N_{t})$
by the Lefschetz duality, together with the formulae for $p_{1}(W_{t})$ and
$z^{2}$ above, implies the relations below
> $z^{2}p_{1}(W_{t})=(1$–$\varepsilon)^{2}(p+k)$;
> $p_{1}^{2}(W_{t})=\frac{1}{k}(p+k)^{2}$; $z^{4}=k(1$–$\varepsilon)^{4}$.
Substituting these values in the formulae for $\mu$ and $s_{1}$ yields v) and
vi) respectively. This completes the computation of the invariant system for
the manifolds $N_{t}$.
Example 3.2. The invariant system $\\{H,\Delta,\frac{p_{1}}{2},b,s_{1},\mu\\}$
of the manifolds $M_{l,k}^{c}$ has been computed by Crowley and Escher [3] for
the case of $c=0$. We extend their calculation as to include the exceptional
case of $c=1$.
> i) $H^{4}(M_{l,k}^{c})\cong\mathbb{Z}_{k}$ with generator
> $\kappa=\left\\{\begin{array}[]{cc}\pi_{M}^{\ast}(\iota)&\text{if }c=0\\\
> (\pi_{M}\circ h_{M})^{\ast}(\iota)&\text{if }c=1\end{array}\right.$;
>
> ii) $b(\kappa,\kappa)\equiv\frac{1}{k}\mathop{\mathrm{m}od}\mathbb{Z}$;
>
> iii) $\Delta(M_{l,k}^{c})\equiv\frac{1+(-1)^{k}}{2}\cdot
> c\kappa\mathop{\mathrm{m}od}2$;
>
> iv)
> $\frac{p_{1}}{2}(M_{l,k}^{c})\equiv(2l+12c)\kappa\mathop{\mathrm{m}od}k$;
>
> v)
> $s_{1}(M_{l,k}^{c})\equiv\frac{(2l+k+12c)^{2}-\left|k\right|}{8k}\mathop{\mathrm{m}od}\mathbb{Z}$;
>
> vi) $\mu(M_{l,k}^{0})\equiv\frac{(k+2l)^{2}-\left|k\right|}{28\cdot
> 8k}\mathop{\mathrm{m}od}\mathbb{Z}$.
Firstly, since $h_{M}:M_{l,k}^{1}\rightarrow$ $M_{l,k}^{0}$ is a homotopy
equivalence we get i) and ii) from the relations
$H^{4}(M_{l,k}^{0})\cong\mathbb{Z}_{k}$ (with generator
$\pi_{M}^{\ast}(\iota)$) and
$b(\pi_{M}^{\ast}(\iota),\pi_{M}^{\ast}(\iota))\equiv\frac{1}{k}\mathop{\mathrm{m}od}\mathbb{Z}$
when $c=0$.
Next, since the map
$\mathcal{S}^{TOP}(M_{l,k}^{0})\overset{\Delta}{\rightarrow}H^{4}(M_{l,k}^{0};\mathbb{Z}_{2})$
of taking Kirby– Siebenmann class is a surjective homomorphism [24, Theorem
15.1], and since $\left[M_{l,k}^{1},h_{M}\right]$ is a generator of the cyclic
group $\mathcal{S}^{TOP}(M_{l,k}^{0})\cong\mathbb{Z}_{k}$ we have
>
> $\Delta(M_{l,k}^{1})=\Delta(\left[M_{l,k}^{1},h_{M}\right])=\frac{1+(-1)^{k}}{2}\kappa\mathop{\mathrm{m}od}2$.
This shows iii).
To calculate the remaining invariants $\frac{p_{1}}{2},\mu,s_{1}$ of the
manifold $M_{l,k}^{c}$ we construct an 8–manifold $W_{l,k}^{c}$ with boundary
$\partial W_{l,k}^{c}\cong M_{l,k}^{c}$ as follows. Let
$\pi_{W}:W_{l,k}^{0}\rightarrow S^{4}$ be the associated disk bundle of the
sphere bundle $\pi_{M}:M_{l,k}^{0}\rightarrow S^{4}$. Then
$M_{l,k}^{0}=\partial W_{l,k}^{0}$. Write $\mathcal{S}^{TOP}(W_{l,k}^{0})$ for
the set of equivalence classes $[W,h]$ of the pairs $(W,h)$ with
$h:(W,\partial W)\rightarrow(W_{l,k}^{0},M_{l,k}^{0})$ a homotopy equivalence
between $8$-manifolds with boundary. Consider the following commutative
diagram analoguing to the one [3, (7)] due to Crowley and Escher in the
PL–category(see also Section 1)
$\begin{array}[]{ccccc}\mathcal{S}^{TOP}(W_{l,k}^{0})&\overset{\eta}{\underset{\cong}{\rightarrow}}&[W_{l,k}^{0},G/TOP]&\underset{\cong}{\overset{d}{\rightarrow}}&H^{4}(W_{l,k}^{0})\cong\mathbb{Z}\\\
i^{\ast}\downarrow&&i^{\ast}\downarrow&&i^{\ast}\downarrow\\\
\mathcal{S}^{TOP}(M_{l,k}^{0})&\overset{\eta}{\underset{\cong}{\rightarrow}}&[M_{l,k}^{0},G/TOP]&\underset{\cong}{\overset{d}{\rightarrow}}&H^{4}(M_{l,k}^{0})\cong\mathbb{Z}_{k}\end{array}$
where
$i^{\ast}:\mathcal{S}^{TOP}(W_{l,k}^{0})\rightarrow\mathcal{S}^{TOP}(M_{l,k}^{0})$
sends each $\left[W,h\right]$ to the restriction $\left[\partial
W,h|_{\partial W}\right]$. Writing $\left[W_{l,k}^{1},h_{W}\right]$ for the
element
$(d\circ\eta)^{-1}(\pi_{W}^{\ast}(\iota))\in\mathcal{S}^{TOP}(W_{l,k}^{0})$ we
get $M_{l,k}^{1}\cong\partial W_{l,k}^{1}$ from the diagram above.
To find the formula of $\frac{p_{1}}{2}(M_{l,k}^{c})$ we compute the first
Pontrjagin class $p_{1}(W_{l,k}^{c})$ of $W_{l,k}^{c}$. Let $\alpha$ denote
the generator of $H^{4}(W_{l,k}^{c})\cong\mathbb{Z}$ satisfies
$\alpha=\left\\{\begin{array}[]{cc}\pi_{W}^{\ast}(\iota)&\text{if
}c=0\text{;}\\\ (\pi_{W}\circ h_{W})^{\ast}(\iota)&\text{if
}c=1\text{;}\end{array}\right.$
and associate an integer $p(W_{l,k}^{c})$ to $W_{l,k}^{c}$ such that
$p_{1}(W_{l,k}^{c})=p(W_{l,k}^{c})\alpha$. Let $\overline{i}:G/TOP\rightarrow
BTOP$ be the natural inclusion and let $f_{c}:W_{l,k}^{c}\rightarrow BTOP$ be
the classifying map for the stable tangent bundle of $W_{l,k}^{c}$. It follows
from the isomorphism
$\mathcal{S}^{TOP}(W_{l,k}^{0})\overset{\eta}{\rightarrow}[W_{l,k}^{0},G/TOP]$
and the proof of [18, Theorem 2.23] that
>
> $\overline{i}_{\ast}\eta([W_{l,k}^{1},h_{W}])=h_{W}^{\ast-1}[f_{1}]$–$[f_{0}]$
and hence
>
> $p(W_{l,k}^{1})\pi_{W}^{\ast}(\iota)=h_{W}^{\ast-1}p_{1}(W_{l,k}^{1})=p_{1}(W_{l,k}^{0})+f^{\ast}\overline{i}^{\ast}p_{1}$,
where $f$ $=d^{-1}(\pi_{W}^{\ast}(\iota))=\eta([W_{l,k}^{1},h_{W}])$ is the
generator of $[W_{l,k}^{0},G/TOP]$ and $p_{1}\in H^{4}(BTOP)$ is the first
Pontrjagin class [14]. It is shown in [24, Lemma 13.3,Proposition 13.4] that a
generator $g$ of $[S^{4},G/TOP]$ corresponds to a topological bundle $\xi$
with classifying map $\overline{i}\circ g$ and Pontrjagin class
$p_{1}(\xi)=g^{\ast}\overline{i}^{\ast}p_{1}=\pm 24\iota$. With an appropriate
choice of $\pm d:[W_{l,k}^{0},G/TOP]\rightarrow H^{4}(W_{l,k}^{0})$ applying
$\pi_{W}^{\ast}$ to this equation we get
> $f^{\ast}\overline{i}^{\ast}p_{1}=24\pi_{W}^{\ast}(\iota).$
This, together with the fact $p(W_{l,k}^{0})=2(k+2l)$ [19] and the formula for
$p(W_{l,k}^{1})$ above, implies that $p(W_{l,k}^{c})=2k+4l+24c$. Consequently
from $M_{l,k}^{c}\cong\partial W_{l,k}^{c}$ we get iv).
Finally, we compute $s_{1}(M_{l,k}^{c})$. The exact sequence
>
> $H^{4}(W_{l,k}^{c},M_{l,k}^{c})\overset{j^{\ast}}{\rightarrow}H^{4}(W_{l,k}^{c})\rightarrow
> H^{4}(M_{l,k}^{c})\rightarrow 0$,
together with the isomorphisms $H^{4}(M_{l,k}^{c})\cong\mathbb{Z}_{k}$ and
$H^{4}(W_{l,k}^{c},M_{l,k}^{c})\cong\mathbb{Z}$ by the Lefschetz duality,
implies that we can take a generator $\beta$ of
$H^{4}(W_{l,k}^{c},M_{l,k}^{c})$ such that $j^{\ast}(\beta)=k\alpha$. Since
$j^{\ast}:H^{4}(W_{l,k}^{c},M_{l,k}^{c};\mathbb{Q})\rightarrow
H^{4}(W_{l,k}^{c};\mathbb{Q})$ is an isomorphism for $k\neq 0$ the invariant
$s_{1}$ is defined for $M_{l,k}^{c}$. It follows from the Lefschetz duality
and the relation $j^{\ast}(\beta)=k\alpha$ that
$\sigma(W_{l,k}^{c})=\frac{\left|k\right|}{k}$. On the other hand, the formula
for $p(W_{l,k}^{c})$, together with the relation $j^{\ast}(\beta)=k\alpha$ and
the isomorphism
> $H^{4}(W_{l,k}^{c},M_{l,k}^{c})\otimes
> H^{4}(W_{l,k}^{c})\overset{\cup}{\rightarrow}H^{8}(W_{l,k}^{c},M_{l,k}^{c})$
by the Lefschetz duality, implies that
> $p_{1}^{2}(W_{l,k}^{c})=\frac{4}{k}(k+2l+12c)^{2}$.
In addition, the relation $w_{2}(W_{l,k}^{1})=w_{2}(W_{l,k}^{0})=0$ indicates
that we can take $z=0$ in the formula of $s_{1}$. Substituting the values of
$\sigma(W_{l,k}^{c}),z,p_{1}^{2}(W_{l,k}^{c})$ in the formula for $s_{1}$
shows v).
Similarly, we refer vi) to Crowley and Escher [3]. This completes the
computation of the invariant system for the manifolds $M_{l,k}^{c}$.
### 3.2 Circle bundles over $[N,t]\in\Theta$
In this section we will prove Lemmas 3.3 and 3.5 which identify the
homeomorphism and diffeomorphism types of the manifolds $N_{t}$ with certain
$M_{l,k}^{c}$.
Lemma 3.3. Let $N_{t}$ be the total space of the circle bundle over
$[N,t]\in\Theta$. Then there is a homeomorphism $N_{t}\cong M_{l,k}^{c}$ where
>
> $(k,c)=(k(\left[N,t\right]),\frac{1+(-1)^{k(\left[N,t\right])}}{2}\cdot\delta(\left[N,t\right]));$
>
> $l=\frac{p(\left[N,t\right])+(3\varepsilon(\left[N,t\right])-4)\cdot
> k(\left[N,t\right])-(1+(-1)^{k(\left[N,t\right])})\cdot
> 12\delta(\left[N,t\right])}{4}.$
Proof. We divide the proof into two cases depending on $\Delta(N_{t})\equiv 0$
or $1$ $\mathop{\mathrm{m}od}2$.
Case 1. $\Delta(N_{t})\equiv 0\mathop{\mathrm{m}od}2$ (i.e. the manifold
$N_{t}$ is smoothable, see [18, p.33], [12] and [24, Theorem 5.4]): Module by
a $\mathbb{Z}_{2}$ ambiguity Wilkens[29] showed that the system
$\\{H,\frac{p_{1}}{2},b\\}$ of invariants classifies $N_{t}$ and $M_{l,k}^{0}$
up to homeomorphism. Moreover, Crowley and Escher [3] proved that this
ambiguity can be realized by some $M_{l,k}^{0}$ whose homeomorphism types can
be distinguished by the invariant $s_{1}$ and hence the system
$\\{H,\frac{p_{1}}{2},b,s_{1}\\}$ classifies the manifolds $N_{t}$ and
$M_{l,k}^{0}$. Therefore the proof is completed by comparing these invariants
for $N_{t}$ and $M_{l,k}^{0}$ obtained in Example 3.1 and 3.2.
Case 2. $\Delta(N_{t})\equiv 1\mathop{\mathrm{m}od}2$: We only need to show
that $N_{t}$ is homeomorphic to $M_{l,k}^{1}$ with $\left[N,t\right]\in\Theta$
and
>
> $(k,l)=(k(\left[N,t\right]),\frac{p(\left[N,t\right])+(3\varepsilon(\left[N,t\right])-4)\cdot
> k(\left[N,t\right])-24}{4})$.
It suffices to construct a homotopy equivalence $q:N_{t}\rightarrow
M_{l,k}^{0}$ with
>
> $\left[N_{t},q\right]=\left[M_{l,k}^{1},h_{M}\right]\in\mathcal{S}^{TOP}(M_{l,k}^{0})$.
According to Lemma 2.2 there exists a manifold $N^{\prime}$ with
$[N^{\prime},t^{\prime}]\in\Theta$ whose invariant system
$(k([N^{\prime},t^{\prime}]),p([N^{\prime},t^{\prime}]),\varepsilon([N^{\prime},t^{\prime}]),\delta([N^{\prime},t^{\prime}])$
is
> $(k([N,t]),p(\left[N,t\right])$–$24,\varepsilon([N,t]),0)$.
Consider the map
$\eta:\mathcal{S}^{TOP}(N^{{}^{\prime}})\rightarrow[N^{{}^{\prime}},G/TOP]$ in
the surgery exact sequence of $N^{{}^{\prime}}$. By the argument at the end of
the proof of [14, Theorem 1] we find a homotopy equivalence
$h_{N}:N\rightarrow N^{{}^{\prime}}$ such that
> i) the class $\eta(\left[N,h_{N}\right])$ is trivial on the $2$ skeleton of
> $N^{{}^{\prime}}$;
>
> ii) the primary obstruction to finding a null-homotopy of
> $\eta(\left[N,h_{N}\right])$ is the generator $x^{\prime}\in
> H^{4}(N^{{}^{\prime}};\pi_{4}(G/TOP))$ with $\left\langle x^{\prime}\cup
> t^{\prime},[N^{\prime}]\right\rangle$=$1$.
Pulling $h_{N}$ back by the bundle projection
$\pi^{\prime}:N_{t}^{{}^{\prime}}\rightarrow N^{{}^{\prime}}$ induces a
homotopy equivalence $h_{t}:N_{t}\rightarrow N_{t}^{{}^{\prime}}$. On the
other hand, by the result of Case 1 we get a homeomorphism
$u:M_{l,k}^{0}\rightarrow N_{t}^{{}^{\prime}}$ such that
$u^{\ast}(\pi^{\prime\ast}(x^{\prime}))=\pi_{M}^{\ast}(\iota)$. So it remains
to show that $\left[N_{t},u^{-1}\circ
h_{t}\right]=\left[M_{l,k}^{1},h_{M}\right]$.
Let $[N^{{}^{\prime}},G/TOP]_{2}$ denote the subset of
$[N^{{}^{\prime}},G/TOP]$ whose elements are trivial on the $2$ skeleton of
$N^{{}^{\prime}}$ and consider the following two commutative diagrams:
$\begin{array}[]{ccc}\mathcal{S}^{TOP}(N^{{}^{\prime}})&\overset{\pi^{\prime\ast}}{\rightarrow}&\mathcal{S}^{TOP}(N_{t}^{{}^{\prime}})\\\
\downarrow\eta&&\cong\downarrow\eta\\\
\left[N^{{}^{\prime}},G/TOP\right]&\overset{\pi^{\prime\ast}}{\rightarrow}&\left[N_{t}^{{}^{\prime}},G/TOP\right]\end{array}\quad$
$\begin{array}[]{ccccc}&&\mathcal{S}^{TOP}(N_{t}^{{}^{\prime}})&\underset{\cong}{\overset{u^{\ast}}{\rightarrow}}&\mathcal{S}^{TOP}(M_{l,k}^{0})\\\
&&\cong\downarrow\eta&&\cong\downarrow\eta\\\
\left[N^{{}^{\prime}},G/TOP\right]_{2}&\overset{\pi^{\prime\ast}}{\rightarrow}&\left[N_{t}^{{}^{\prime}},G/TOP\right]&\underset{\cong}{\overset{u^{\ast}}{\rightarrow}}&\left[M_{l,k}^{0},G/TOP\right]\\\
\downarrow d&&\cong\downarrow d&&\cong\downarrow d\\\
H^{4}(N^{{}^{\prime}})&\overset{\pi^{\prime\ast}}{\rightarrow}&H^{4}(N_{t}^{{}^{\prime}})&\underset{\cong}{\overset{u^{\ast}}{\rightarrow}}&H^{4}(M_{l,k}^{0})\end{array}$
where
> i)
> $\mathcal{S}^{TOP}(N^{{}^{\prime}})\overset{\pi^{\prime\ast}}{\rightarrow}\mathcal{S}^{TOP}(N_{t}^{{}^{\prime}})$
> maps $\left[N^{{}^{\prime\prime}},h^{{}^{\prime\prime}}\right]$ to
> $\left[N_{t}^{{}^{\prime\prime}},h_{t}^{{}^{\prime\prime}}\right]$ with
> $h_{t}^{{}^{\prime\prime}}$ a pull-back of $h^{{}^{\prime\prime}}$ by the
> bundle projection $\pi^{\prime}:N_{t}^{{}^{\prime}}\rightarrow
> N^{{}^{\prime}};$
>
> ii) $\mathcal{S}^{TOP}(N_{t}^{{}^{\prime}})\overset{u^{\ast}}{\rightarrow}$
> $\mathcal{S}^{TOP}(M_{l,k}^{0})$ maps
> $\left[M^{{}^{\prime}},g^{{}^{\prime}}\right]$ to
> $\left[M^{{}^{\prime}},u^{-1}\circ g^{{}^{\prime}}\right];$
>
> iii) the maps $d$ send a homotopy class to its primary obstruction to null-
> homotopy.
The diagrams above, together with the relations
> $\pi^{\prime\ast}\left[N,h_{N}\right]=\left[N_{t},h_{t}\right]$,
> $u^{\ast}(\pi^{\prime\ast}(x^{\prime}))=\pi_{M}^{\ast}(\iota)$ and
> $d(\eta\left[N,h_{N}\right])=x^{\prime}$,
imply that $u^{\ast}\left[N_{t},h_{t}\right]=\left[M_{l,k}^{1},h_{M}\right]$,
i.e. $\left[N_{t},u^{-1}\circ h_{t}\right]=\left[M_{l,k}^{1},h_{M}\right]$.
This completes the proof of Case 2.$\square$
The following lemma plays a key role in the proof of Lemma 3.5 and its proof
will be postponed to the end of this section.
Lemma 3.4. Let $N_{t}$ be the total space of the circle bundle over
$[N,t]\in\Theta$ with $\delta([N,t])=0$ and $k([N,t])=0$. Then there exists an
$8$–manifold $W$ homotopy equivalent to $S^{4}$ whose boundary satisfies
> $\partial W\cong\left\\{\begin{array}[]{cc}N_{t}&\text{if
> }\varepsilon([N,t])=1\\\ N_{t}\\#\Sigma_{\frac{p([N,t])}{24}}&\text{if
> }\varepsilon([N,t])=0\end{array}\right..$
Lemma 3.5. Let $N_{t}$ be the total space of the circle bundle over
$[N,t]\in\Theta$ with $\delta([N,t])=0$. Then one has a diffeomorphism
$N_{t}\cong M_{l,k}^{0}\\#\Sigma_{r}$ where $N_{t}$ has the smooth structure
as the total space of the circle bundle and where
$(k,l,r)=(k(\left[N,t\right]),\frac{p(\left[N,t\right])+(3\varepsilon(\left[N,t\right])-4)\cdot
k(\left[N,t\right])}{4},\frac{(1-\varepsilon(\left[N,t\right]))\cdot(p(\left[N,t\right])-4k(\left[N,t\right]))}{24})$.
Proof. In the case of $k([N,t])\neq 0$ it is shown in [3] that the system
$\\{H,\frac{p_{1}}{2},b,\mu\\}$ classifies $N_{t}$ and $M_{l,k}^{0}$ up to
diffeomorphism. Hence the proof is done by comparing those invariants for
$N_{t}$ and $M_{l,k}^{0}$ obtained in Examples 3.1 and 3.2.
Assume next that $k([N,t])=0$ and let $W$ be the $8$–manifold given by Lemma
3.4. We can take a closed tubular neighborhood $E$ of an embedding
$h:S^{4}\hookrightarrow$ Interior $W$ which is also a homotopy equivalence
[20, Lemma 6]. As $H_{i}(W\backslash$Interior $E,\partial E)\cong
H_{i}(W,E)=0$ for all $i$ by the excision theorem, $W\backslash$Interior $E$
is an h–cobordism between $\partial W$ and $\partial E$. Hence we get the
diffeomorphisms $\partial W\cong\partial E\cong M_{l,k}^{0}$ for some
$l,k\in\mathbb{Z}$ by the h-cobordism theorem[21, Theorem 9.1] and the fact
that $E$ is the total space of the normal disk bundle of the embedding $h$.
Comparing the invariants $\\{H,\frac{p_{1}}{2}\\}$ of $\partial W$ and
$M_{l,k}^{0}$ given in Examples 3.1 and 3.2 we find that $\partial W$ is
diffeomorphic to $M_{\frac{p([N,t])}{4},0}^{0}$[3]. This shows
>
> $N_{t}\cong\left\\{\begin{array}[]{cc}M_{\frac{p([N,t])}{4},0}^{0}\\#\Sigma_{\frac{p([N,t])}{24}}&\text{if
> }\varepsilon([N,t])=0\\\ M_{\frac{p([N,t])}{4},0}^{0}&\text{if
> }\varepsilon([N,t])=1\end{array}\right.$
which completes the proof.$\square$
Proof of Lemma 3.4. The construction of $W$ and the corresponding calculations
will be divided into two cases depending on $\varepsilon([N,t])=0$ or $1$. Let
$\pi_{0}:W_{t}\rightarrow N$ be the associated disk bundle of the circle
bundle $\pi:N_{t}\rightarrow N$.
Case 1. $\varepsilon([N,t])=1$: Take an embedding $f:S^{2}\hookrightarrow$
Interior $W_{t}$ that represents a generator of the group
$H_{2}(W_{t})\cong\mathbb{Z}$. Since $w_{2}(W_{t})=0$(see Example 3.1), the
$f$ extends to an embedding $\overline{f}:S^{2}\times D^{6}\hookrightarrow$
Interior $W_{t}$. The $8$–manifold $W$ is obtained from $W_{t}$ by surgery
along $\overline{f}$. On one hand, it is clear that $\partial W\cong\partial
W_{t}\cong N_{t}$. On the other hand, from the homotopy equivalences
> $X\simeq W_{t}\cup_{f}D^{3}\simeq W\cup D^{6}$
with $X$ the trace of the surgery [1, P.83-84] we find that $W$ is
$3$–connected with $H_{4}(W)\cong H_{4}(X)\cong\mathbb{Z}$. Moreover, since
$N_{t}$ is $2$–connected and $W$ is $3$–connected we can conclude that
$H_{i}(W)\cong H^{8-i}(W,N_{t})=0$ for $i\geq 5$ by the Lefschetz duality and
the cohomology exact sequence. Hence from the Whitehead theorem we get
$W\simeq S^{4}$.
Case 2. $\varepsilon([N,t])=0$: The desired manifold $W$ is constructed as
follows. Represent the generator $x\cap[N]\in H_{2}(N)$ by an embedding
$\overline{g}:S^{2}\times D^{4}\hookrightarrow N$ ($w_{2}(N)=0$). Let
$\overline{h}$ be the pull-back of $\overline{g}$ by the projection $\pi$ as
shown in the diagram
1. (3.1)
$\begin{array}[]{ccc}S^{3}\times
D^{4}&\overset{\overline{h}}{\hookrightarrow}&N_{t}\\\
\downarrow&&\pi\downarrow\\\ S^{2}\times
D^{4}&\overset{\overline{g}}{\hookrightarrow}&N\end{array}$,
and let $\widetilde{W}:=N_{t}\times[0,1]\cup_{(\overline{h},1)}D^{4}\times
D^{4}$. Then
1. (3.2)
$\partial\widetilde{W}\cong N_{t}\sqcup($–$\Sigma_{r})$ for some
$r\in\mathbb{Z}$
since in (3.1) the map $\overline{h}$ induces an isomorphism
$\pi_{3}(S^{3}\times D^{4})\rightarrow\pi_{3}(N_{t})$ [28, Lemma 1]. The
manifold $W$ is obtained from $\widetilde{W}$ by removing the tubular
neighborhood of a smooth arc $\alpha:$ $[0,1]\rightarrow\widetilde{W}$ with
$\alpha(0)\in N_{t}$, $\alpha(1)\in\partial W^{{}^{\prime}}$ and
$\alpha(0,1)\subset$Interior $\widetilde{W}$.
It remains to show that
> i) $W\simeq S^{4}$; ii) $r=\frac{p([N,t])}{24}$(in (3.2)).
The property i) follows from the facts that the trace $\widetilde{W}$ of the
surgery along $\overline{h}$ has the homotpy type $\Sigma_{r}\cup D^{4}$ and
the homeomorphism type of $W$ is obtained from $\widetilde{W}$ to collapse the
component $\Sigma_{r}$ of $\partial\widetilde{W}$ to a point.
For the property ii) we only need to show
$\mu(\Sigma_{r})\equiv\frac{p(N)}{24\cdot 28}\mathop{\mathrm{m}od}\mathbb{Z}$.
By the collar neighborhood theorem, the homotopy sphere $\Sigma_{r}$ in (3.2)
bounds an 8–manifold $W^{{}^{\prime}}:=W_{t}\cup_{\overline{h}}D^{4}\times
D^{4}$. For the convenience of calculation, we make use of an alternative
decomposition
$W^{{}^{\prime}}=W_{t}\cup_{\overline{i_{0}}}\mathbb{C}P^{2}\times D^{4}$
where $\overline{i_{0}}$ is the pull-back of $\overline{g}$ by the projection
$\pi_{0}$ as in the diagram
> $\begin{array}[]{ccc}V\times
> D^{4}&\overset{\overline{i_{0}}}{\hookrightarrow}&W_{t}\\\
> \downarrow&&\pi_{0}\downarrow\\\ S^{2}\times
> D^{4}&\overset{\overline{g}}{\hookrightarrow}&N\end{array}$.
From the isomorphism
>
> $\begin{array}[]{ccccc}H_{4}(\mathbb{C}P^{2})&\oplus&H_{4}(W_{t})&\underset{\cong}{\overset{i_{1\ast}\oplus
> i_{2\ast}}{\rightarrow}}&H_{4}(W^{{}^{\prime}})\end{array}$
by the Mayer–Vietoris sequence with $\alpha\in H_{4}(W_{t})\cong\mathbb{Z}$
the generator satisfies $\left\langle\pi_{0}^{\ast}x,\alpha\right\rangle=1$
(see Section 2) and $i_{1}:\mathbb{C}P^{2}\rightarrow
W^{{}^{\prime}},i_{2}:W_{t}\rightarrow W^{{}^{\prime}}$ the inclusions, we can
see below that the intersection matrix of $W^{{}^{\prime}}$ with respect to
the basis $x_{1},x_{2}\in H^{4}(W^{{}^{\prime}},\partial W^{{}^{\prime}})$ is
$\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right)$
where $i_{1\ast}[\mathbb{C}P^{2}]=x_{1}\cap[W^{{}^{\prime}},\partial
W^{{}^{\prime}}],i_{2\ast}\alpha=x_{2}\cap[W^{{}^{\prime}},\partial
W^{{}^{\prime}}]$. First observe that
> $\left\langle x_{1}\cup x_{1},[W^{{}^{\prime}},\partial
> W^{{}^{\prime}}]\right\rangle=0$
as the normal bundle of $i_{1}$ is trivial. Next since the self-intersection
number of $i_{2\ast}\alpha$ is the same as that of $\alpha$ and the
homomorphism $j^{\ast}:H^{4}(W_{t},N_{t})\rightarrow H^{4}(W_{t})$ is trivial
(see Example 3.1), this implies
> $\left\langle x_{2}\cup x_{2},[W^{{}^{\prime}},\partial
> W^{{}^{\prime}}]\right\rangle=\left\langle j^{\ast}D_{W_{t}}\alpha\cup
> D_{W_{t}}\alpha,[W_{t},N_{t}]\right\rangle=0$[1, p.115]
where $D_{W_{t}}\alpha$ denotes the Lefschetz duality of $\alpha$. Finally we
have
$\left\langle x_{1}\cup x_{2},[W^{{}^{\prime}},\partial
W^{{}^{\prime}}]\right\rangle=\left\langle
j^{{}^{\prime}\ast}x_{1},i_{2\ast}\alpha\right\rangle=\left\langle
i_{2}^{\ast}j^{{}^{\prime}\ast}x_{1},\alpha\right\rangle=\left\langle\pi_{0}^{\ast}x,\alpha\right\rangle=1$
with $j^{{}^{\prime}}:W^{{}^{\prime}}\rightarrow(W^{{}^{\prime}},\partial
W^{{}^{\prime}})$ the inclusion and where
$\pi_{0}^{\ast}x=i_{2}^{\ast}j^{{}^{\prime}\ast}x_{1}$ follows from the
relations $\pi_{0}^{-1}f[S^{2}]=i_{2}^{-1}i_{1}[\mathbb{C}P^{2}]$,
$x=D_{N}f_{\ast}[S^{2}]$ and
$x_{1}=D_{W^{{}^{\prime}}}i_{1\ast}[\mathbb{C}P^{2}]$. Thus the signature
$\sigma(W^{{}^{\prime}})$ is $0$.
We can take $z$ to be a generator of $H^{2}(W^{{}^{\prime}})\cong\mathbb{Z}$
since $i_{2}^{\ast}w_{2}(W^{{}^{\prime}})=w_{2}(W_{t})\neq 0$ by Example 3.1
and the isomorphism $i_{2}^{{}^{\ast}}TW^{{}^{\prime}}\cong TW_{t}$ . To get
the values of $z^{2},p_{1}(W^{{}^{\prime}})$, it is necessary to compute the
images of $z^{2},p_{1}(W^{{}^{\prime}})$ under the isomorphism
> $i_{1}^{\ast}\oplus i_{2}^{\ast}:H^{4}(W^{{}^{\prime}})\rightarrow
> H^{4}(\mathbb{C}P^{2})\oplus H^{4}(W_{t})$,
whose matrix with respect to the basis
$\\{j^{{}^{\prime\ast}}x_{1},j^{{}^{\prime\ast}}x_{2}\\}$ and
$\\{[\mathbb{C}P^{2}]^{\ast},\pi_{0}^{\ast}x\\}$ is the same as the
intersection matrix of $W^{{}^{\prime}}$ with respect to the basis
$x_{1},x_{2}$, where $[\mathbb{C}P^{2}]^{\ast}\in H^{4}(\mathbb{C}P^{2})$
satisfies
$\left\langle[\mathbb{C}P^{2}]^{\ast},[\mathbb{C}P^{2}]\right\rangle=1$. Since
$i_{1}^{\ast}z\in H^{2}(\mathbb{C}P^{2}),i_{2}^{\ast}z\in H^{2}(W_{t})$ are
generators, it follows that
> $i_{1}^{\ast}\oplus
> i_{2}^{\ast}(z^{2})=(i_{1}^{\ast}z^{2},\pi_{0}^{\ast}t^{2})=([\mathbb{C}P^{2}]^{\ast},0)$.
Moreover, according to the isomorphisms
$i_{2}^{{}^{\ast}}TW^{{}^{\prime}}\cong TW_{t}$ and
$i_{1}^{{}^{\ast}}TW^{{}^{\prime}}\cong
T\mathbb{C}P^{2}\oplus\varepsilon^{4}$, the relations $\left\langle
p_{1}(\mathbb{C}P^{2}),[\mathbb{C}P^{2}]\right\rangle=3$ and $p_{1}(W_{t})$
$=p([N,t])\pi_{0}^{\ast}x$ imply that
> $i_{1}^{\ast}\oplus
> i_{2}^{{}^{\ast}}p_{1}(W^{{}^{\prime}})=(3[\mathbb{C}P^{2}]^{\ast},p([N,t])\pi_{0}^{\ast}x)$.
Therefore we can see that
>
> $z^{2}=j^{{}^{\prime\ast}}x_{2};p_{1}(W^{{}^{\prime}})=p([N,t])j^{{}^{\prime\ast}}x_{1}+3j^{{}^{\prime\ast}}x_{2}$.
Again from these relations and the intersection form of $W^{{}^{\prime}}$ we
get
>
> $p_{1}^{2}(W^{{}^{\prime}})=6p([N,t]);z^{2}p_{1}(W^{{}^{\prime}})=p([N,t]);z^{4}=0.$
Consequently, substituting these values in the formula of $\mu,$ this implies
> $\mu(M^{{}^{\prime}})=\frac{p(N)}{24\cdot
> 28}\mathop{\mathrm{m}od}\mathbb{Z}$.$\square$
Remark 3.6. In a communication concerning this work Diarmuid Crowley pointed
out that according to a result of Wilkens [30, Theorem 1 (ii)] the
decomposition $N_{t}\cong M_{l,k}^{0}\\#\Sigma_{r}$ in Lemma 3.4 can be
simplified as $N_{t}\cong M_{l,k}^{0}$ when $k([N,t])=0$, which will play a
role in the proof of Corollary 4.5 in the coming section.
In the recent paper [4] (see also [2][5]) Crowley and Nordstrom generalised
the classical Eells-Kuiper invariant $\mu$. Their new invariant can be applied
to give a simple proof of the diffeomorphism $N_{t}\cong M_{l,k}^{0}$ when
$k([N,t])=0$.
## 4 Proof of Theorem 1.3 and applications
We establish Theorem 1.3 and present some applications.
Proof of Theorem 1.3. Let $M$ be a $2$–connected $7$–manifold with a regular
circle action. By Lemmas 2.1 and 2.2 $M$ is the total space of the oriented
circle bundle over $N\\#_{r}S^{3}\times S^{3}$ with Euler class
$\overline{t}\in H^{2}(N\\#_{r}S^{3}\times S^{3})\cong\mathbb{Z}$ a generator,
where $[N,t]\in\Theta$, $r\in\mathbb{N}$. Identify $\overline{t}$ with the
generator $t\in H^{2}(N)\cong\mathbb{Z}$ under the isomorphism
$H^{2}(N)\rightarrow H^{2}(N\\#_{r}S^{3}\times S^{3})$ induced by the map
$N\\#_{r}S^{3}\times S^{3}\rightarrow N$ collapsing $\\#_{r}S^{3}\times S^{3}$
to a point. By Lemmas 3.3 and 3.4 it suffices to show that $M\cong
N_{t}\\#_{2r}S^{3}\times S^{4}$.
Consider the decomposition
> $N\\#_{r}S^{3}\times
> S^{3}=(N\backslash\overset{\circ}{D}_{1})\cup_{f}(\\#_{r}S^{3}\times
> S^{3}\backslash\overset{\circ}{D}_{2})$
with $D_{i}\cong D^{6}$ and $f:\partial D_{2}\rightarrow$ $\partial D_{1}$ a
diffeomorphism. Since the restriction of the bundle $N_{t}\rightarrow N$ on
$D_{1}$ is trivial and $N_{t}\cong N_{t}\\#S^{7}$ one has the corresponding
decomposition
> $M\cong(N_{t}\backslash\overset{\circ}{D}_{1}\times S^{1})\cup_{f\times
> id}((\\#_{r}S^{3}\times S^{3}\backslash\overset{\circ}{D}_{2})\times
> S^{1})\cong N_{t}\\#M_{0}$
where $id$ is the identity on $S^{1}$, and where
> $M_{0}=(S^{7}\backslash\overset{\circ}{D}_{1}\times S^{1})\cup_{f\times
> id}((\\#_{r}S^{3}\times S^{3}\backslash\overset{\circ}{D}_{2})\times
> S^{1}).$
Since $M_{0}$ can be easily identified with the total space of the oriented
circle bundle over $CP^{3}\\#_{r}S^{3}\times S^{3}$ with Euler class a proper
generator of $H^{2}(CP^{3}\\#_{r}S^{3}\times S^{3})\cong\mathbb{Z}$, a
calculation similar to that in Example 3.1 shows that the invariant system
$\\{H,\frac{p_{1}}{2},b,\mu\\}$ for $M_{0}$ and $\\#_{2r}S^{3}\times S^{4}$
coincides. Consequently $M_{0}$ is diffeomorphic to $\\#_{2r}S^{3}\times
S^{4}.$ This shows that $M\cong N_{t}\\#_{2r}S^{3}\times S^{4}$ which
completes the proof.$\square$
A classical topic is to decide which homotopy spheres admit smooth regular
circle actions ([13] [17] [22] [23]). Combining Theorem 1.3 with Example 3.2
we recover the classical computation of Montgomery and Yang [22] .
Corollary 4.1. Among the $28$ homotopy $7$–spheres $\Sigma_{r},0\leq r\leq 27$
the following ones admit smooth regular circle actions
> $\Sigma_{r},r=0,4,6,8,10,14,18,20,22,24$.$\square$
In term of our notation the unit tangent bundle of the sphere $S^{4}$ is
$M_{-1,2}^{0}$. The additive property of the Eells-Kuiper invariant $\mu$
shows that $M_{-1,2}^{0}\\#\Sigma_{r}$ with $0\leq r\leq 27$ represent all the
diffeomorphism types of the smooth manifolds homeomorphic to $M_{-1,2}^{0}$.
One can deduce from Theorem 1.3 and Example 3.2 that
Corollary 4.2. All the smooth manifolds homeomorphic to the unit tangent
bundle of the sphere $S^{4}$ and admitting smooth regular circle actions are
> $M_{-1,2}^{0}\\#\Sigma_{r},r=0,2,6,7,8,12,14,15,16,19,20,23,26.\square$
In [11] Grove, Verdiani and Ziller constructed on the manifold
$M_{-1,2}^{0}\\#\Sigma_{27}$ a metric with positive sectional curvature (see
Goette[10, p.34-35]). According to Corollary 4.2 this manifold does not admit
any smooth regular circle action.
Definition 4.3 Two regular (resp. smooth regular) circle actions
$S^{1}\times M_{i}\rightarrow M_{i},i=1,2,$
on two manifolds (resp. smooth manifolds) $M_{i}$ are called equivalent if
there is a equivariant homeomorphism (diffeomorphism) $f:M_{1}\rightarrow
M_{2}$. Let $\rho_{T}(M)$ (resp.$\rho_{S}(M)$) be the number of all
equivalence classes of regular (resp. smooth regular) circle actions on a
given manifold (resp. smooth manifold) $M$.
Since the number $\rho_{T}(M)$ can be seen as the number of those elements
$[N,t]\in\Theta$ satisfying $N_{t}\cong M$, we get from Lemmas 2.2 and 3.3
that
Corollary 4.4. For the family
> $M=M_{6m,(1+c)k}^{c}\\#_{2r}S^{3}\times
> S^{4},c\in\\{0,1\\},r\in\mathbb{N},m,k\in\mathbb{Z}$
of manifolds that represent all homeomorphism classes of the $2$–connected
$7$–manifolds with regular circle actions (see Theorem 1.3) we have
> $\rho_{T}(M)=\left\\{\begin{array}[]{cc}1&\text{if }k=0\text{ and }m\equiv
> 1\mathop{\mathrm{m}od}2,\\\ 2&\text{if }k=0\text{ and }m\equiv
> 0\mathop{\mathrm{m}od}2,\\\ \infty&\text{if }k\neq
> 0\end{array}\right..\square$
Similarly, in the smooth category we get from Lemmas 2.2 and 3.5, together
with Remark 3.6, that
Corollary 4.5. For the family
$M=M_{6(1+a)m,(1+a)k}^{0}\\#\Sigma_{(1-a)m}\\#_{2r}S^{3}\times
S^{4},a\in\\{0,1\\},r\in\mathbb{N},m,k\in\mathbb{Z}$
of manifolds that represent all diffeomorphism classes of the smooth
$2$–connected $7$–manifolds with smooth regular circle actions (see Theorem
1.3) we have
> $\sigma(M)=\left\\{\begin{array}[]{cc}1&\text{if }k=0,a=0\text{ and }m\equiv
> 1\mathop{\mathrm{m}od}2,\\\ 2&\text{if }k=0\text{ and }(1+a)m\equiv
> 0\mathop{\mathrm{m}od}2,\\\ \infty&\text{if }k\neq
> 0\end{array}\right..\square$
>
Acknowledgement The author is grateful to the referee for many improvements
over the first version of this paper. In particular, the results in
Corollaries 4.2, 4.4 and 4.5 are suggested by him.
The author would also like to thank to Haibao Duan for bring her attention to
the topic, and to Diarmuid Crowley for communication concerning this work (see
Remark 3.6). Thanks are also due to Yang Su and Yueshan Xiong for valuable
discussions.
## References
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|
arxiv-papers
| 2013-02-05T02:37:21 |
2024-09-04T02:49:41.299308
|
{
"license": "Public Domain",
"authors": "Yi Jiang",
"submitter": "Yi Jiang",
"url": "https://arxiv.org/abs/1302.0923"
}
|
1302.0973
|
arxiv-papers
| 2013-02-05T09:39:06 |
2024-09-04T02:49:41.307491
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Martin Avanzini and Georg Moser",
"submitter": "Martin Avanzini",
"url": "https://arxiv.org/abs/1302.0973"
}
|
|
1302.1072
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-013 LHCb-PAPER-2012-048 February 5, 2013
Measurement of the $\mathchar 28931\relax^{0}_{b}$, $\mathchar
28932\relax^{-}_{b}$ and $\mathchar 28938\relax^{-}_{b}$ baryon masses
The LHCb collaboration†††Authors are listed on the following pages.
Bottom baryons decaying to a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
meson and a hyperon are reconstructed using $\rm 1.0~{}fb^{-1}$ of data
collected in 2011 with the LHCb detector. Significant $\mathchar
28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$, $\mathchar
28932\relax^{-}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28932\relax^{-}$ and $\mathchar
28938\relax^{-}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28938\relax^{-}$ signals are observed and the corresponding
masses are measured to be
$M(\mathchar 28931\relax^{0}_{b})$ $~{}=~{}$ | 5619.53 $\,\,\pm\,$ | 0.13 (stat) $\pm\,$ | 0.45 (syst) MeV/$c^{2}$ | ,
---|---|---|---|---
$M(\mathchar 28932\relax^{-}_{b})$ $~{}=~{}$ | 5795.8 $\,\,\pm\,$ | 0.9 (stat) $\pm\,$ | 0.4 (syst) MeV/$c^{2}$ | ,
$M(\mathchar 28938\relax^{-}_{b})$ $~{}=~{}$ | 6046.0 $\,\,\pm\,$ | 2.2 (stat) $\pm\,$ | 0.5 (syst) MeV/$c^{2}$ | ,
while the differences with respect to the $\mathchar 28931\relax^{0}_{b}$ mass
are
$M(\mathchar 28932\relax^{-}_{b})-M(\mathchar 28931\relax^{0}_{b})$ $~{}=~{}$ | 176.2 $\,\,\pm\,$ | 0.9 (stat) $\pm\,$ | 0.1 (syst) MeV/$c^{2}$ | ,
---|---|---|---|---
$M(\mathchar 28938\relax^{-}_{b})-M(\mathchar 28931\relax^{0}_{b})$ $~{}=~{}$ | 426.4 $\,\,\pm\,$ | 2.2 (stat) $\pm\,$ | 0.4 (syst) MeV/$c^{2}$ | .
These are the most precise mass measurements of the $\mathchar
28931\relax^{0}_{b}$, $\mathchar 28932\relax^{-}_{b}$ and $\mathchar
28938\relax^{-}_{b}$ baryons to date. Averaging the above $\mathchar
28931\relax^{0}_{b}$ mass measurement with that published by LHCb using $\rm
35~{}pb^{-1}$ of data collected in 2010 yields $M(\mathchar
28931\relax^{0}_{b})=\rm 5619.44\pm 0.13\,(stat)\pm 0.38\,(syst)$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
Submitted to Physical Review Letters
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht9, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves
Jr22,35, S. Amato2, Y. Amhis7, L. Anderlini17,f, J. Anderson37, R.
Andreassen57, R.B. Appleby51, O. Aquines Gutierrez10, F. Archilli18, A.
Artamonov 32, M. Artuso53, E. Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J.
Back45, C. Baesso54, V. Balagura28, W. Baldini16, R.J. Barlow51, C.
Barschel35, S. Barsuk7, W. Barter44, Th. Bauer38, A. Bay36, J. Beddow48, I.
Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28, E. Ben-Haim8, M.
Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, A. Berezhnoy29, R.
Bernet37, M.-O. Bettler44, M. van Beuzekom38, A. Bien11, S. Bifani12, T.
Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36, C. Blanks50,
J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N. Bondar27, W.
Bonivento15, S. Borghi51, A. Borgia53, T.J.V. Bowcock49, E. Bowen37, C.
Bozzi16, T. Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M.
Britsch10, T. Britton53, N.H. Brook43, H. Brown49, I. Burducea26, A.
Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo
Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14,c, G. Carboni21,k, R.
Cardinale19,i, A. Cardini15, H. Carranza-Mejia47, L. Carson50, K. Carvalho
Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, P. Chen3,36, N. Chiapolini37, M. Chrzaszcz 23, K. Ciba35, X.
Cid Vidal34, G. Ciezarek50, P.E.L. Clarke47, M. Clemencic35, H.V. Cliff44, J.
Closier35, C. Coca26, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins35, A.
Comerma-Montells33, A. Contu15,52, A. Cook43, M. Coombes43, S. Coquereau8, G.
Corti35, B. Couturier35, G.A. Cowan36, D. Craik45, S. Cunliffe50, R. Currie47,
C. D’Ambrosio35, P. David8, P.N.Y. David38, I. De Bonis4, K. De Bruyn38, S. De
Capua51, M. De Cian37, J.M. De Miranda1, L. De Paula2, W. De Silva57, P. De
Simone18, D. Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C.
Deplano15, D. Derkach14, O. Deschamps5, F. Dettori39, A. Di Canto11, J.
Dickens44, H. Dijkstra35, M. Dogaru26, F. Domingo Bonal33,n, S. Donleavy49, F.
Dordei11, A. Dosil Suárez34, D. Dossett45, A. Dovbnya40, F. Dupertuis36, R.
Dzhelyadin32, A. Dziurda23, A. Dzyuba27, S. Easo46,35, U. Egede50, V.
Egorychev28, S. Eidelman31, D. van Eijk38, S. Eisenhardt47, U. Eitschberger9,
R. Ekelhof9, L. Eklund48, I. El Rifai5, Ch. Elsasser37, D. Elsby42, A.
Falabella14,e, C. Färber11, G. Fardell47, C. Farinelli38, S. Farry12, V.
Fave36, D. Ferguson47, V. Fernandez Albor34, F. Ferreira Rodrigues1, M. Ferro-
Luzzi35, S. Filippov30, C. Fitzpatrick35, M. Fontana10, F. Fontanelli19,i, R.
Forty35, O. Francisco2, M. Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, E.
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Gao3, J. Garofoli53, P. Garosi51, J. Garra Tico44, L. Garrido33, C. Gaspar35,
R. Gauld52, E. Gersabeck11, M. Gersabeck51, T. Gershon45,35, Ph. Ghez4, V.
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P.F. Harrison45, T. Hartmann55, J. He7, V. Heijne38, K. Hennessy49, P.
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Huse49, N. Hussain52, D. Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12,
R. Jacobsson35, A. Jaeger11, E. Jans38, F. Jansen38, P. Jaton36, F. Jing3, M.
John52, D. Johnson52, C.R. Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M.
Karacson35, T.M. Karbach35, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A.
Keune36, B. Khanji20, O. Kochebina7, I. Komarov36,29, R.F. Koopman39, P.
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Kreps45, G. Krocker11, P. Krokovny31, F. Kruse9, M. Kucharczyk20,23,j, V.
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Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E. Lanciotti35, G.
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van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O.
Leroy6, Y. Li3, L. Li Gioi5, M. Liles49, R. Lindner35, C. Linn11, B. Liu3, G.
Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33, N. Lopez-March36, H.
Lu3, J. Luisier36, H. Luo47, F. Machefert7, I.V. Machikhiliyan4,28, F.
Maciuc26, O. Maev27,35, S. Malde52, G. Manca15,d, G. Mancinelli6, N.
Mangiafave44, U. Marconi14, R. Märki36, J. Marks11, G. Martellotti22, A.
Martens8, L. Martin52, A. Martín Sánchez7, M. Martinelli38, D. Martinez
Santos39, D. Martins Tostes2, A. Massafferri1, R. Matev35, Z. Mathe35, C.
Matteuzzi20, M. Matveev27, E. Maurice6, A. Mazurov16,30,35,e, J. McCarthy42,
R. McNulty12, B. Meadows57,52, F. Meier9, M. Meissner11, M. Merk38, D.A.
Milanes8, M.-N. Minard4, J. Molina Rodriguez54, S. Monteil5, D. Moran51, P.
Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K. Müller37, R. Muresan26,
B. Muryn24, B. Muster36, P. Naik43, T. Nakada36, R. Nandakumar46, I. Nasteva1,
M. Needham47, N. Neufeld35, A.D. Nguyen36, T.D. Nguyen36, C. Nguyen-Mau36,o,
M. Nicol7, V. Niess5, R. Niet9, N. Nikitin29, T. Nikodem11, S. Nisar56, A.
Nomerotski52, A. Novoselov32, A. Oblakowska-Mucha24, V. Obraztsov32, S.
Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M. Orlandea26, J.M.
Otalora Goicochea2, P. Owen50, B.K. Pal53, A. Palano13,b, M. Palutan18, J.
Panman35, A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J. Parkinson50, G.
Passaleva17, G.D. Patel49, M. Patel50, G.N. Patrick46, C. Patrignani19,i, C.
Pavel-Nicorescu26, A. Pazos Alvarez34, A. Pellegrino38, G. Penso22,l, M. Pepe
Altarelli35, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo34, A. Pérez-
Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G. Pessina20, K.
Petridis50, A. Petrolini19,i, A. Phan53, E. Picatoste Olloqui33, B. Pietrzyk4,
T. Pilař45, D. Pinci22, S. Playfer47, M. Plo Casasus34, F. Polci8, G. Polok23,
A. Poluektov45,31, E. Polycarpo2, D. Popov10, B. Popovici26, C. Potterat33, A.
Powell52, J. Prisciandaro36, V. Pugatch41, A. Puig Navarro36, W. Qian4, J.H.
Rademacker43, B. Rakotomiaramanana36, M.S. Rangel2, I. Raniuk40, N.
Rauschmayr35, G. Raven39, S. Redford52, M.M. Reid45, A.C. dos Reis1, S.
Ricciardi46, A. Richards50, K. Rinnert49, V. Rives Molina33, D.A. Roa Romero5,
P. Robbe7, E. Rodrigues51, P. Rodriguez Perez34, G.J. Rogers44, S. Roiser35,
V. Romanovsky32, A. Romero Vidal34, J. Rouvinet36, T. Ruf35, H. Ruiz33, G.
Sabatino22,k, J.J. Saborido Silva34, N. Sagidova27, P. Sail48, B. Saitta15,d,
C. Salzmann37, B. Sanmartin Sedes34, M. Sannino19,i, R. Santacesaria22, C.
Santamarina Rios34, E. Santovetti21,k, M. Sapunov6, A. Sarti18,l, C.
Satriano22,m, A. Satta21, M. Savrie16,e, D. Savrina28,29, P. Schaack50, M.
Schiller39, H. Schindler35, S. Schleich9, M. Schlupp9, M. Schmelling10, B.
Schmidt35, O. Schneider36, A. Schopper35, M.-H. Schune7, R. Schwemmer35, B.
Sciascia18, A. Sciubba18,l, M. Seco34, A. Semennikov28, K. Senderowska24, I.
Sepp50, N. Serra37, J. Serrano6, P. Seyfert11, M. Shapkin32, I. Shapoval40,35,
P. Shatalov28, Y. Shcheglov27, T. Shears49,35, L. Shekhtman31, O.
Shevchenko40, V. Shevchenko28, A. Shires50, R. Silva Coutinho45, T.
Skwarnicki53, N.A. Smith49, E. Smith52,46, M. Smith51, K. Sobczak5, M.D.
Sokoloff57, F.J.P. Soler48, F. Soomro18,35, D. Souza43, B. Souza De Paula2, B.
Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S. Stahl11, O. Steinkamp37,
S. Stoica26, S. Stone53, B. Storaci37, M. Straticiuc26, U. Straumann37, V.K.
Subbiah35, S. Swientek9, V. Syropoulos39, M. Szczekowski25, P. Szczypka36,35,
T. Szumlak24, S. T’Jampens4, M. Teklishyn7, E. Teodorescu26, F. Teubert35, C.
Thomas52, E. Thomas35, J. van Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39,
D. Tonelli35, S. Topp-Joergensen52, N. Torr52, E. Tournefier4,50, S.
Tourneur36, M.T. Tran36, M. Tresch37, A. Tsaregorodtsev6, P. Tsopelas38, N.
Tuning38, M. Ubeda Garcia35, A. Ukleja25, D. Urner51, U. Uwer11, V. Vagnoni14,
G. Valenti14, R. Vazquez Gomez33, P. Vazquez Regueiro34, S. Vecchi16, J.J.
Velthuis43, M. Veltri17,g, G. Veneziano36, M. Vesterinen35, B. Viaud7, D.
Vieira2, X. Vilasis-Cardona33,n, A. Vollhardt37, D. Volyanskyy10, D. Voong43,
A. Vorobyev27, V. Vorobyev31, C. Voß55, H. Voss10, R. Waldi55, R. Wallace12,
S. Wandernoth11, J. Wang53, D.R. Ward44, N.K. Watson42, A.D. Webber51, D.
Websdale50, M. Whitehead45, J. Wicht35, J. Wiechczynski23, D. Wiedner11, L.
Wiggers38, G. Wilkinson52, M.P. Williams45,46, M. Williams50,p, F.F. Wilson46,
J. Wishahi9, M. Witek23, S.A. Wotton44, S. Wright44, S. Wu3, K. Wyllie35, Y.
Xie47,35, F. Xing52, Z. Xing53, Z. Yang3, R. Young47, X. Yuan3, O.
Yushchenko32, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang53, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, A. Zvyagin35.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25National Center for Nuclear Research (NCBJ), Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
56Institute of Information Technology, COMSATS, Lahore, Pakistan, associated
to 53
57University of Cincinnati, Cincinnati, OH, United States, associated to 53
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pMassachusetts Institute of Technology, Cambridge, MA, United States
Hadrons are systems bound by the strong interaction, described at the
fundamental level by quantum chromodynamics (QCD). While QCD is well
understood at high energy in the perturbative regime, low-energy phenomena
such as the binding of quarks and gluons within hadrons are more difficult to
predict. Several models and techniques, such as constituent quark models or
lattice QCD calculations, attempt to reproduce the spectrum of the measured
hadron masses (for a review, see Ref. [1]). While the masses of all expected
ground-state mesons are now well measured, baryon data are still sparse. In
particular, only six out of the sixteen $b$-baryon ground states predicted by
the quark model have been observed so far [2]. A complete and reliable
experimental mass spectrum would allow for precision tests of a variety of QCD
models [3, *Day:2012yh, *Liu:2007fg, *Jenkins:2007dm, *Roncaglia:1995az,
*Mathur:2002ce, *Ebert:2005xj].
The mass measurement of the heaviest observed $b$ baryon, the $\mathchar
28938\relax^{-}_{b}$ state with $bss$ valence quark content, is of particular
interest. While both the D0 and CDF collaborations have claimed the
observation of the $\mathchar
28938\relax^{-}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28938\relax^{-}$ decay, the reported mass values, ${\rm
6165\pm 10\,(stat)\pm 13\,(syst)}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
from D0 [10] and ${\rm 6054.4\pm 6.8\,(stat)\pm
0.9\,(syst)}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ from CDF [11], differ by
more than 6 standard deviations. On the other hand, there is good agreement
between the mass measurements of the $\mathchar 28932\relax^{-}_{b}$ ($bsd$)
baryon, which has also been observed by D0 [12] and CDF [13] in the $\mathchar
28932\relax^{-}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28932\relax^{-}$ mode and, more recently, by CDF [14] in the
$\mathchar 28932\relax^{-}_{b}\rightarrow\mathchar 28932\relax^{0}_{c}\pi^{-}$
mode. These measurements average to
$5791.1\,\pm\,2.2{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [2].
This letter presents mass measurements of the weakly decaying $\mathchar
28931\relax^{0}_{b}$ ($bud$), $\mathchar 28932\relax^{-}_{b}$ and $\mathchar
28938\relax^{-}_{b}$ baryons using the decay modes $\mathchar
28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$, $\mathchar
28932\relax^{-}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28932\relax^{-}$ and $\mathchar
28938\relax^{-}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28938\relax^{-}$ (charge-conjugated modes are implied
throughout). The mass differences with respect to the $\mathchar
28931\relax^{0}_{b}$ mass are also reported. This analysis uses data
corresponding to an integrated luminosity of $\rm 1.0~{}fb^{-1}$ and collected
in $pp$ collisions at a centre-of-mass energy of $\sqrt{s}=7$ TeV with the
LHCb detector in 2011.
The LHCb detector [15] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift tubes placed downstream. The
combined tracking system has a momentum resolution $\Delta p/p$ that varies
from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an impact parameter resolution
of 20$\,\upmu\rm m$ for tracks with high transverse momentum. Charged hadrons
are identified using two ring-imaging Cherenkov detectors. Photon, electron
and hadron candidates are identified by a calorimeter system consisting of
scintillating-pad and preshower detectors, an electromagnetic calorimeter and
a hadronic calorimeter. Muons are identified by a system composed of
alternating layers of iron and multiwire proportional chambers. The trigger
[16] consists of a hardware stage, based on information from the calorimeter
and muon systems, followed by a software stage which applies a full event
reconstruction.
Precision mass measurements require the momenta of the final state particles
to be determined accurately. Therefore, an important feature of this analysis
is the calibration of the tracker response. This accounts for imperfect
knowledge of the magnetic field and tracker alignment [17]. In order to reduce
these dominant contributions to the systematic uncertainty, a two-step
momentum calibration procedure is applied. Firstly, inclusive
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$ decays
are used to account for the changes in the relative momentum scale between
different data taking periods. Secondly, the absolute scale is derived from
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decays,
taking the known masses [2] as references. In this procedure, the use of a
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass constraint allows the
momentum scale to be determined as a function of the $K^{+}$ track kinematics.
The resulting calibration is checked with a variety of fully reconstructed
decays listed in Fig. 1. For each mode the mass distribution is modelled
taking into account the effect of QED radiative corrections, resolution and
background, and the mean mass value is determined. Following the procedure
described in Ref. [17], the deviation of the measured mass from the expected
value is converted into an estimate of an average momentum scale bias
independent of time and track kinematics. The bias is referred to as $\alpha$,
which is defined such that the measured mass becomes equal to the expected
value if all particle momenta are multiplied by $1-\alpha$. By definition, one
expects $\alpha=0$ to a good precision for the
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
calibration mode. Figure 1 shows the resulting values of $\alpha$. The nominal
mass measurements are performed with $\alpha=0$ and the largest value of
$|\alpha|$ amongst the considered modes, $0.3\times 10^{-3}$, is
conservatively taken as the systematic uncertainty on the calibrated momentum
scale. This uncertainty is somewhat larger than the $0.2\times 10^{-3}$
achieved with the 2010 data [17], due to changes in the alignment of the
tracking devices.
Figure 1: Average momentum scale bias $\alpha$ determined from the
reconstructed mass of various decay modes after the momentum calibration
procedure. The $K^{0}_{\rm\scriptscriptstyle S}$ decays are divided into two
categories according to whether both daughter tracks (a) have hits or (b) do
not have hits in the vertex detector. The black error bars represent the
statistical uncertainty whilst the (yellow) filled areas also include
contributions to the systematic uncertainty from the fitting procedure, the
effect of QED radiative corrections, and the uncertainty on the mass of the
decaying meson [2]. The (red) dashed lines show the assigned uncertainty of
$\pm 0.3\times 10^{-3}$ on the momentum scale.
In this analysis, the $b$-baryon decays $\mathchar
28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$, $\mathchar
28932\relax^{-}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28932\relax^{-}$ and $\mathchar
28938\relax^{-}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28938\relax^{-}$, followed by
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$,
$\mathchar 28931\relax\rightarrow p\pi^{-}$, $\mathchar
28932\relax^{-}\rightarrow\mathchar 28931\relax\pi^{-}$ and $\mathchar
28938\relax^{-}\rightarrow\mathchar 28931\relax K^{-}$, are reconstructed. The
topology of these decays is characterised by the long-lived particles in the
decay chain. The lifetime of weakly-decaying $b$ baryons is $\sim$1.5 ps and
their decay vertex is expected to be separated from the primary $pp$
interaction vertex by $\sim$6 mm on average. The
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates are reconstructed
from pairs of oppositely-charged tracks, originating from the secondary
vertex, that have hits in the muon detector. In the $\mathchar
28932\relax^{-}_{b}$ ($\mathchar 28938\relax^{-}_{b}$) decay chain the long-
lived $\mathchar 28932\relax^{-}$ ($\mathchar 28938\relax^{-}$) decays into a
$\mathchar 28931\relax$ and a charged pion (kaon) at a tertiary vertex and the
$\mathchar 28931\relax$ decays at a quaternary vertex. Since $\sim$90$\%$ of
the decays are not fully contained in the vertex detector, tracks that have no
hits in the vertex detector are also considered in the reconstruction of the
tertiary and quaternary vertices.
The selections of $\mathchar 28932\relax^{-}_{b}$ and $\mathchar
28938\relax^{-}_{b}$ candidates are identical apart from the choice of the
$\mathchar 28932\relax^{-}$, $\mathchar 28932\relax^{-}_{b}$, $\mathchar
28938\relax^{-}$, $\mathchar 28938\relax^{-}_{b}$ invariant mass ranges and
particle identification requirements on the pion (kaon) from the $\mathchar
28932\relax^{-}$ ($\mathchar 28938\relax^{-}$) vertex. The $\mathchar
28931\relax^{0}_{b}$ selection is slightly different, owing to the different
topology.
The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates are required to
satisfy $|M_{\mu\mu}-M_{J/\psi}|<4.2\sigma$ where $M_{\mu\mu}$ is the
reconstructed di-muon mass, $M_{J/\psi}$ the $J/\psi$ mass [2] and $\sigma$
the estimated event-by-event uncertainty on $M_{\mu\mu}$ (typically
$10{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$). The invariant mass windows for
the $\mathchar 28931\relax$, $\mathchar 28932\relax^{-}$ and $\mathchar
28938\relax^{-}$ candidates are $\pm
6{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, $\pm
11{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and $\pm
11{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the expected masses [2],
respectively. Particle identification requirements are applied to the kaon
from the $\mathchar 28938\relax^{-}$ candidate and the proton from the
$\mathchar 28931\relax$ decay to improve the purity of the selected daughter
particles, but none is placed on the pion from the $\mathchar 28932\relax^{-}$
candidate. In addition, the hyperon decay vertices are required to be
downstream of the $b$-hadron decay vertex.
The $\mathchar 28931\relax^{0}_{b}$, $\mathchar 28932\relax^{-}_{b}$ and
$\mathchar 28938\relax^{-}_{b}$ mass resolutions are improved by performing a
fit of the decay topology and vertices [18] while constraining the masses of
the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$, $\mathchar 28931\relax$,
$\mathchar 28932\relax^{-}$ and $\mathchar 28938\relax^{-}$ hadrons to have
their known values [2], the final-state and intermediate long-lived particles
to originate from common vertices according to the decay chain, and the $b$
baryon to originate from the primary vertex. Three additional variables are
considered for the selection. These are the $\chi^{2}$ per degree of freedom
($\chi^{2}/{\rm ndf}$) from the fit, the reconstructed decay time and the
$\chi^{2}_{\rm IP}$ of the $b$ baryon from the primary vertex. The
$\chi^{2}_{\rm IP}$ is defined as the difference in the $\chi^{2}$ of the
primary vertex fit with and without the $b$-baryon candidate. In the case of
the $\mathchar 28932\relax^{-}_{b}$ candidates, the selection requirements for
these variables are chosen to maximise the expected significance of the
$\mathchar 28932\relax^{-}_{b}$ signal; the same selection is used for the
$\mathchar 28938\relax^{-}_{b}$ candidates. To determine the significance for
a set of selection criteria the background yield is estimated from the yield
of $\mathchar 28932\relax^{-}_{b}$ candidates found in mass side-bands in the
ranges 5600–5700${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
5900–6100${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The expected signal yield
is estimated using the product of the world average hadronisation fraction for
$b\rightarrow\mathchar 28932\relax^{-}_{b}$ and branching fractions for
$\mathchar 28932\relax^{-}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28932\relax^{-}$ and subsequent daughter particle decays [2],
the $b\bar{b}$ production cross-section in the LHCb acceptance [19] and the
selection efficiencies obtained from simulation. The selection criteria giving
the highest expected signal significance correspond to a decay time greater
than 0.25 ps, a $\chi^{2}/{\rm ndf}$ smaller than 4 and a $\chi^{2}_{\rm IP}$
smaller than 16. Amongst these, the decay time requirement is the most
powerful given the high level of background close to the interaction point. In
the case of the $\mathchar 28931\relax^{0}_{b}$ candidates, the decay time is
required to be greater than 0.3 ps and the $\chi^{2}/{\rm ndf}$ smaller than 5
(no requirement on the $\chi^{2}_{\rm IP}$ is made). The possibility of a
cross-feed background between $\mathchar 28932\relax^{-}_{b}$ and $\mathchar
28938\relax^{-}_{b}$ is investigated using simulation and found to be
negligible in comparison with the combinatorial background.
Figure 2: Invariant mass distribution for (a) $\mathchar
28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$, (b) $\mathchar
28932\relax^{-}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28932\relax^{-}$ and (c) $\mathchar
28938\relax^{-}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28938\relax^{-}$ candidates. The results of the unbinned
maximum likelihood fits are shown with solid lines.
The invariant mass distributions of the selected $\mathchar
28931\relax^{0}_{b}$, $\mathchar 28932\relax^{-}_{b}$ and $\mathchar
28938\relax^{-}_{b}$ candidates are shown in Fig. 2. In each case, the mass is
measured by performing an unbinned extended maximum likelihood fit. The
$\mathchar 28931\relax^{0}_{b}$, $\mathchar 28932\relax^{-}_{b}$, $\mathchar
28938\relax^{-}_{b}$ candidates are retained for the mass fit if their
invariant mass lies in the range 5500–5750, 5600–6000, 5800–6300
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, respectively. The signal
component is described with a single Gaussian function (or the sum of two
Gaussian functions with common mean in the case of the $\mathchar
28931\relax^{0}_{b}$ baryon) and the background is modelled with an
exponential function. The widths of the $\mathchar 28931\relax^{0}_{b}$ and
$\mathchar 28932\relax^{-}_{b}$ signals are left unconstrained in the fit. Due
to the low expected yield for the $\mathchar 28938\relax^{-}_{b}$ signal, the
width of the Gaussian function describing the $\mathchar 28938\relax^{-}_{b}$
signal is fixed to the measured $\mathchar 28932\relax^{-}_{b}$ signal width
multiplied by the ratio of $\mathchar 28938\relax^{-}_{b}$ and $\mathchar
28932\relax^{-}_{b}$ widths from the simulation (8.2
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for $\mathchar
28938\relax^{-}_{b}$ and 8.9 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for
$\mathchar 28932\relax^{-}_{b}$). The fit results are given in Table 1.
Table 1: Results of the fits to the invariant mass distributions. The quoted uncertainties are statistical. The $\mathchar 28931\relax^{0}_{b}$ signal is described by a double Gaussian function with widths $\sigma_{1}$ and $\sigma_{2}$; the fraction of the yield described by the first component is $0.58\pm 0.11$. | Signal yield | Mass [${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ ] | Width(s) [${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ ]
---|---|---|---
$\mathchar 28931\relax^{0}_{b}$ | $6870\pm 110$ | $5619.53\pm 0.13$ | $\sigma_{1}=~{}\,6.4\pm 0.5$
$\sigma_{2}=12.5\pm 1.3$
$\mathchar 28932\relax^{-}_{b}$ | $111\pm 12$ | $5795.8\pm 0.9$ | $7.8\pm 0.7$
$\mathchar 28938\relax^{-}_{b}$ | $19\pm 5$ | $6046.0\pm 2.2$ | $7.2$ (fixed)
The statistical significance of the $\mathchar 28938\relax^{-}_{b}$ signal is
determined using simulated pseudo-experiments with background only. We
determine the probability that, anywhere in the mass range between 5800 and
6300 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, a peak appears with the
expected width and a yield at least as large as that observed in the data.
This probability corresponds to 6 standard deviations, which we interpret as
the statistical significance of the $\mathchar 28938\relax^{-}_{b}$ signal.
The systematic uncertainties are evaluated by repeating the complete analysis
(including the track fit and the momentum scale calibration when needed),
varying in turn within its uncertainty each parameter to which the mass
determination is sensitive. The observed changes in the central values of the
fitted masses relative to the nominal results are then assigned as systematic
uncertainties and summed in quadrature. The systematic uncertainties are
summarised in Table 2.
The dominant systematic uncertainty is due to the momentum scale calibration
described previously, which is assigned an uncertainty of $\pm 0.3\times
10^{-3}$. A significant contribution to this uncertainty comes from the
overall detector length scale along the beam axis, which is known to a
relative precision of $10^{-3}$ [20]. This translates into a $\pm 0.13\times
10^{-3}$ uncertainty on the momentum scale, and is included in the overall
$\pm 0.3\times 10^{-3}$ uncertainty. Most of the uncertainty related to the
momentum scale is removed in the measurements of the mass differences.
The uncertainty on the amount of material assumed in the track reconstruction
for the energy loss ($dE/dx$) correction has been found to be small [17]. It
translates into an uncertainty on the $\mathchar 28931\relax^{0}_{b}$ mass of
0.09${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, which we apply to all masses.
The invariant mass of $\mathchar 28938\relax^{-}_{b}$ candidates is computed
assuming the central value of the $\mathchar 28938\relax^{-}$ world-average
mass [2]. The uncertainty of $\pm
0.29{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ on this value is propagated as a
systematic uncertainty. A similar, but smaller, uncertainty is estimated for
the $\mathchar 28932\relax^{-}_{b}$ and $\mathchar 28931\relax^{0}_{b}$ masses
from the imperfect knowledge of the $\mathchar 28932\relax^{-}$ and $\mathchar
28931\relax$ masses, respectively.
Two alternative fits for the $\mathchar 28931\relax^{0}_{b}$ signal are
performed: a first fit where the candidates are split into two categories
depending on whether the daughter tracks have vertex detector information or
not, each category being described with a single Gaussian function where the
two Gaussian functions have a common mean, and a second fit using the sum of
two Crystal Ball functions [21] with common peak value and otherwise
unconstrained parameters. The second fit allows to take into account possible
QED radiative corrections.
The $\mathchar 28932\relax^{-}_{b}$ mass fit is repeated using as an
alternative model either the sum of two Gaussian functions with a common mean,
or a single Crystal Ball function. In the $\mathchar 28938\relax^{-}_{b}$ mass
fit, the fixed Gaussian width is varied within both the uncertainty of the
fitted $\mathchar 28932\relax^{-}_{b}$ width and the statistical uncertainty
of the width ratio from simulation.
An alternative background model assuming a linear shape leads to negligible
changes. We also repeat the $\mathchar 28932\relax^{-}_{b}$ and $\mathchar
28938\relax^{-}_{b}$ mass fits in a restricted mass range of
5650–5950${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
5900–6200${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, respectively, and assign
the resulting change as a systematic uncertainty.
Table 2: Systematic uncertainties (in ${\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$) on the mass measurements and their differences. The total systematic uncertainty is obtained from adding all uncertainties in quadrature. Source | $\mathchar 28931\relax^{0}_{b}$ | $\mathchar 28932\relax^{-}_{b}$ | $\mathchar 28938\relax^{-}_{b}$ | $\mathchar 28932\relax^{-}_{b}$–$\mathchar 28931\relax^{0}_{b}$ | $\mathchar 28938\relax^{-}_{b}$–$\mathchar 28931\relax^{0}_{b}$
---|---|---|---|---|---
Momentum scale | 0.43 | 0.43 | 0.31 | 0.01 | 0.12
$dE/dx$ correction | 0.09 | 0.09 | 0.09 | 0.01 | 0.01
Hyperon mass | 0.01 | 0.07 | 0.25 | 0.07 | 0.25
Signal model | 0.07 | 0.01 | 0.24 | 0.07 | 0.25
Background model | 0.01 | 0.01 | 0.02 | 0.01 | 0.02
Total | 0.45 | 0.45 | 0.47 | 0.10 | 0.37
In summary, the $\mathchar 28931\relax^{0}_{b}$, $\mathchar
28932\relax^{-}_{b}$ and $\mathchar 28938\relax^{-}_{b}$ baryons are observed
in the $\mathchar
28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$, $\mathchar
28932\relax^{-}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28932\relax^{-}$ and $\mathchar
28938\relax^{-}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28938\relax^{-}$ decay modes using 1.0 fb-1 of $pp$ collisions
collected in 2011 at a centre-of-mass energy of $\sqrt{s}=7$ TeV. The
statistical significance of the observed $\mathchar
28938\relax^{-}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28938\relax^{-}$ signal is 6 standard deviations. The masses
of the $b$ baryons are measured to be
$\displaystyle M(\mathchar 28931\relax^{0}_{b})\,$ $\displaystyle=$
$\displaystyle 5619.53\pm 0.13\pm
0.45{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}},$ $\displaystyle M(\mathchar
28932\relax^{-}_{b})$ $\displaystyle=$ $\displaystyle 5795.8~{}\,\pm
0.9~{}\,\pm 0.4~{}\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}},$ $\displaystyle
M(\mathchar 28938\relax^{-}_{b})$ $\displaystyle=$ $\displaystyle
6046.0~{}\,\pm 2.2~{}\,\pm 0.5~{}\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}},$
where the first (second) quoted uncertainty is statistical (systematic). The
dominant systematic uncertainty, due to the knowledge of the momentum scale,
partially cancels in mass differences. We obtain
$\displaystyle M(\mathchar 28932\relax^{-}_{b})-M(\mathchar
28931\relax^{0}_{b})$ $\displaystyle=$ $\displaystyle 176.2\pm 0.9\pm
0.1{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}},$ $\displaystyle M(\mathchar
28938\relax^{-}_{b})-M(\mathchar 28931\relax^{0}_{b})$ $\displaystyle=$
$\displaystyle 426.4\pm 2.2\pm 0.4{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}.$
A measurement of the $\mathchar 28931\relax^{0}_{b}$ mass based on the 2010
data sample, $M(\mathchar 28931\relax^{0}_{b})=\rm 5619.19\pm 0.70\pm
0.30$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, has been previously reported
by LHCb [17]. Since the new alignment and momentum calibration procedures
differ from those applied in the previous study, a possible correlation
between the systematic uncertainties related to the momentum scale can be
neglected. Considering that the only correlated systematic uncertainties are
those due to energy loss correction and mass fitting, the weighted average of
the two $\mathchar 28931\relax^{0}_{b}$ mass measurements that minimizes the
total uncertainty is
$M(\mathchar 28931\relax^{0}_{b})=5619.44\pm 0.13\pm
0.38{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}\,.$
These $\mathchar 28931\relax^{0}_{b}$, $\mathchar 28932\relax^{-}_{b}$ and
$\mathchar 28938\relax^{-}_{b}$ mass measurements are the most precise to
date. They are compared in Table 3 with the single most precise measurements
from ATLAS, CDF and D0, and with the current world averages [2]. The
$\mathchar 28931\relax^{0}_{b}$ and $\mathchar 28932\relax^{-}_{b}$ results
are in agreement with previous measurements. The $\mathchar
28938\relax^{-}_{b}$ result is in agreement with the CDF measurement [11], but
in disagreement with the D0 measurement [10].
Table 3: Comparison of the $b$-baryon mass measurements using the full 2011 data sample with the single most precise results from the ATLAS [22], CDF [11, 23] and D0 [10, 12] collaborations, and with the PDG averages [2]. The PDG averages contain the results from CDF and D0 as well as the $\mathchar 28931\relax^{0}_{b}$ measurement from LHCb performed with the 2010 data sample. The quoted errors include statistical and systematic uncertainties. All values are in ${\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$. | $M(\mathchar 28931\relax^{0}_{b})$ | $M(\mathchar 28932\relax^{-}_{b})$ | $M(\mathchar 28938\relax^{-}_{b})$
---|---|---|---
ATLAS | $5619.7\pm 1.3~{}$ | – | –
CDF | $5619.7\pm 1.7~{}$ | $5790.9\pm 2.7$ | $6054.4\pm 6.9$
D0 | – | $5774~{}~{}\,\pm 19$ | $6165~{}~{}\,\pm 16$
PDG | $5619.4\pm 0.7~{}$ | $5791.1\pm 2.2$ | $6071~{}~{}\,\pm 40$
LHCb | $5619.5\pm 0.5~{}$ | $5795.8\pm 1.0$ | $6046.0\pm 2.3$
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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|
arxiv-papers
| 2013-02-05T15:55:16 |
2024-09-04T02:49:41.315539
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, A. Adametz, B. Adeva,\n M. Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio,\n M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S.\n Amato, Y. Amhis, L. Anderlini, J. Anderson, R. Andreassen, R.B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W. Baldini, R.J.\n Barlow, C. Barschel, S. Barsuk, W. Barter, Th. Bauer, A. Bay, J. Beddow, I.\n Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G.\n Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M.\n van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T.\n Blake, F. Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov, V. Bocci, A.\n Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, E.\n Bowen, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M.\n Britsch, T. Britton, N.H. Brook, H. Brown, I. Burducea, A. Bursche, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, H. Carranza-Mejia,\n L. Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo, Ch. Cauet, M. Charles,\n Ph. Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal,\n G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V.\n Coco, J. Cogan, E. Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A.\n Cook, M. Coombes, S. Coquereau, G. Corti, B. Couturier, G.A. Cowan, D. Craik,\n S. Cunliffe, R. Currie, C. D'Ambrosio, P. David, P.N.Y. David, I. De Bonis,\n K. De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, W. De\n Silva, P. De Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi, L. Del Buono,\n C. Deplano, D. Derkach, O. Deschamps, F. Dettori, A. Di Canto, J. Dickens, H.\n Dijkstra, M. Dogaru, F. Domingo Bonal, S. Donleavy, F. Dordei, A. Dosil\n Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, A.\n Dzyuba, S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, S.\n Eisenhardt, U. Eitschberger, R. Ekelhof, L. Eklund, I. El Rifai, Ch.\n Elsasser, D. Elsby, A. Falabella, C. F\\\"arber, G. Fardell, C. Farinelli, S.\n Farry, V. Fave, D. Ferguson, V. Fernandez Albor, F. Ferreira Rodrigues, M.\n Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M. Fontana, F. Fontanelli, R.\n Forty, O. Francisco, M. Frank, C. Frei, M. Frosini, S. Furcas, E. Furfaro, A.\n Gallas Torreira, D. Galli, M. Gandelman, P. Gandini, Y. Gao, J. Garofoli, P.\n Garosi, J. Garra Tico, L. Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M.\n Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V.V. Gligorov, C. G\\\"obel, D.\n Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa G\\'andara, R.\n Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E.\n Greening, S. Gregson, O. Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C.\n Hadjivasiliou, G. Haefeli, C. Haen, S.C. Haines, S. Hall, T. Hampson, S.\n Hansmann-Menzemer, N. Harnew, S.T. Harnew, J. Harrison, P.F. Harrison, T.\n Hartmann, J. He, V. Heijne, K. Hennessy, P. Henrard, J.A. Hernando Morata, E.\n van Herwijnen, E. Hicks, D. Hill, M. Hoballah, C. Hombach, P. Hopchev, W.\n Hulsbergen, P. Hunt, T. Huse, N. Hussain, D. Hutchcroft, D. Hynds, V.\n Iakovenko, P. Ilten, R. Jacobsson, A. Jaeger, E. Jans, F. Jansen, P. Jaton,\n F. Jing, M. John, D. Johnson, C.R. Jones, B. Jost, M. Kaballo, S. Kandybei,\n M. Karacson, T.M. Karbach, I.R. Kenyon, U. Kerzel, T. Ketel, A. Keune, B.\n Khanji, O. Kochebina, I. Komarov, R.F. Koopman, P. Koppenburg, M. Korolev, A.\n Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F.\n Kruse, M. Kucharczyk, V. Kudryavtsev, T. Kvaratskheliya, V.N. La Thi, D.\n Lacarrere, G. Lafferty, A. Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G.\n Lanfranchi, C. Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van\n Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, O. Leroy, Y.\n Li, L. Li Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben,\n J.H. Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, J. Luisier, H. Luo, F.\n Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, S. Malde, G. Manca, G.\n Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G. Martellotti,\n A. Martens, L. Martin, A. Mart\\'in S\\'anchez, M. Martinelli, D. Martinez\n Santos, D. Martins Tostes, A. Massafferri, R. Matev, Z. Mathe, C. Matteuzzi,\n M. Matveev, E. Maurice, A. Mazurov, J. McCarthy, R. McNulty, B. Meadows, F.\n Meier, M. Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez,\n S. Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F. Muheim, K.\n M\\\"uller, R. Muresan, B. Muryn, B. Muster, P. Naik, T. Nakada, R. Nandakumar,\n I. Nasteva, M. Needham, N. Neufeld, A.D. Nguyen, T.D. Nguyen, C. Nguyen-Mau,\n M. Nicol, V. Niess, R. Niet, N. Nikitin, T. Nikodem, S. Nisar, A. Nomerotski,\n A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O.\n Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora Goicochea, P. Owen, B.K.\n Pal, A. Palano, M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, C.\n Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel, G.N. Patrick, C.\n Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A. Pellegrino, G. Penso, M.\n Pepe Altarelli, S. Perazzini, D.L. Perego, E. Perez Trigo, A. P\\'erez-Calero\n Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina, K. Petridis, A.\n Petrolini, A. Phan, E. Picatoste Olloqui, B. Pietrzyk, T. Pila\\v{r}, D.\n Pinci, S. Playfer, M. Plo Casasus, F. Polci, G. Polok, A. Poluektov, E.\n Polycarpo, D. Popov, B. Popovici, C. Potterat, A. Powell, J. Prisciandaro, V.\n Pugatch, A. Puig Navarro, W. Qian, J.H. Rademacker, B. Rakotomiaramanana,\n M.S. Rangel, I. Raniuk, N. Rauschmayr, G. Raven, S. Redford, M.M. Reid, A.C.\n dos Reis, S. Ricciardi, A. Richards, K. Rinnert, V. Rives Molina, D.A. Roa\n Romero, P. Robbe, E. Rodrigues, P. Rodriguez Perez, G.J. Rogers, S. Roiser,\n V. Romanovsky, A. Romero Vidal, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino,\n J.J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann, B.\n Sanmartin Sedes, M. Sannino, R. Santacesaria, C. Santamarina Rios, E.\n Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D.\n Savrina, P. Schaack, M. Schiller, H. Schindler, S. Schleich, M. Schlupp, M.\n Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R.\n Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska,\n I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P.\n Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V.\n Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, N.A. Smith, E.\n Smith, M. Smith, K. Sobczak, M.D. Sokoloff, F.J.P. Soler, F. Soomro, D.\n Souza, B. Souza De Paula, B. Spaan, A. 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Wilson, J. Wishahi,\n M. Witek, S.A. Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, F. Xing, Z. Xing,\n Z. Yang, R. Young, X. Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev, F.\n Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong, A. Zvyagin",
"submitter": "Raphael M\\\"arki",
"url": "https://arxiv.org/abs/1302.1072"
}
|
1302.1192
|
11institutetext: School of Computer Science and Statistics,
Trinity College Dublin
# Homomorphic Encryption with Access Policies: Characterization and New
Constructions***A preliminary version of this work appeared in Africacrypt
2013 [39]. This is the full version.
Michael Clear111The author’s work is funded by the Irish Research Council
EMBARK Initiative. Arthur Hughes and Hitesh Tewari
###### Abstract
A characterization of predicate encryption (PE) with support for homomorphic
operations is presented and we describe the homomorphic properties of some
existing PE constructions. Even for the special case of IBE, there are few
known group-homomorphic cryptosystems. Our main construction is an XOR-
homomorphic IBE scheme based on the quadratic residuosity problem (variant of
the Cocks’ scheme), which we show to be strongly homomorphic. We were unable
to construct an anonymous variant that preserves this homomorphic property,
but we achieved anonymity for a weaker notion of homomorphic encryption, which
we call _non-universal_. A related security notion for this weaker primitive
is formalized. Finally, some potential applications and open problems are
considered.
## 1 Introduction
There has been much interest recently in encryption schemes with homomorphic
capabilities. Traditionally, malleability was avoided to satisfy strong
security definitions, but many applications have been identified for
cryptosystems supporting homomorphic operations. More recently, Gentry [1]
presented the first fully-homomorphic encryption (FHE) scheme, and several
improvements and variants have since appeared in the literature [2, 3, 4, 5].
There are however many applications that only require a scheme to support a
single homomorphic operation. Such schemes are referred to as _partial
homomorphic_. Notable examples of unbounded homomorphic cryptosystems include
Goldwasser-Micali [6] (XOR), Paillier [7] and ElGamal [8].
Predicate Encryption (PE) [9] enables a sender to embed a hidden descriptor
within a ciphertext that consists of attributes describing the message
content. A Trusted Authority (TA) who manages the system issues secret keys to
users corresponding to predicates. A user can decrypt a ciphertext containing
a descriptor $\mathbf{a}$ if and only if he/she has a secret key for a
predicate that evaluates to true for $\mathbf{a}$. This construct turns out to
be quite powerful, and generalizes many encryption primitives. It facilitates
expressive fine-grained access control i.e. complex policies can be defined
restricting the recipients who can decrypt a message. It also facilitates the
evaluation of complex queries on data such as range, subset and search
queries. Extending the class of supported predicates for known schemes is a
topic of active research at present.
PE can be viewed in two ways. It can be viewed as a means to delegate
computation to a third party i.e. allow the third party to perform a precise
fixed function on the encrypted data, and thus limit what the third party
learns about the data. In the spirit of this viewpoint, a generalization known
as Functional Encryption has been proposed [10], which allows general
functions to be evaluated.
PE can also be viewed as a means to achieve more fine-grained access control.
It enables a stronger separation between sender and recipient since the former
must only describe the content of the message or more general conditions on
its access while decryption then depends on whether a recipient’s access
policy matches these conditions.
Why consider homomorphic encryption in the PE setting? It is conceivable that
in a multi-user environment such as a large organization, certain computations
may be delegated to the cloud whose inputs depend on the work of multiple
users distributed within that organization. Depending on the application, the
circuit to be computed may be chosen or adapted by the cloud provider, and
thus is not fixed by the delegator as in primitives such as non-interactive
_verifiable computing_ [11]. Furthermore, the computation may depend on data
sets provided by multiple independent users. Since the data is potentially
sensitive, the organization’s security policy may dictate that all data must
be encrypted. Accordingly, each user encrypts her data with a PE scheme using
relevant attributes to describe it. She then sends the ciphertext(s) to the
cloud. It is desirable that the results of the computation returned from the
cloud be decryptable only by an entity whose access policy (predicate)
satisfies the attributes of _all_ data sets used in the computation. Of course
a public-key homomorphic scheme together with a PE scheme would be sufficient
if the senders were able to interact before contacting the cloud, but we would
like to remove this requirement since the senders may not be aware of each
other. This brings to mind the recent notion of multikey homomorphic
encryption presented by López-Alt, Tromer and Vaikuntanathan [12].
Using a multikey homomorphic scheme, the senders need not interact with each
other before evaluation takes place on the cloud. Instead, they must run an
MPC decryption protocol to jointly decrypt the result produced by the cloud.
The evaluated ciphertexts in the scheme described in [12] do not depend on the
circuit size, and depend only polynomially on the security parameter and the
number of parties who contribute inputs to the circuit. Therefore, the problem
outlined above may be solved with a multikey fully homomorphic scheme used in
conjunction with a PE scheme if we accept the evaluated ciphertext size to be
polynomial in the number of parties. In this work, we are concerned with a
ciphertext size that is independent of the number of parties. Naturally, this
limits the composition of access policies, but if this is acceptable in an
application, there may be efficiency gains over the combination of multikey
FHE and PE.
In summary, homomorphic encryption in the PE setting is desirable if there is
the possibility of multiple parties in a large organization (say) sending
encrypted data to a semi-trusted †††We assume all parties are semi-honest.
evaluator and access policies are required to appropriately limit access to
the results, where the “composition” of access policies is “lossy”. We assume
the semi-honest model in this paper; in particular we do not consider
verifiability of the computation.
The state of affairs for homomorphic encryption even for the simplest special
case of PE, namely identity-based encryption (IBE), leaves open many
challenges. At his talk at Crypto 2010, Naccache [13] mentioned “identity-
based fully homomorphic encryption” as one of a list of theory questions.
Towards this goal, it has been pointed out in [14] that some LWE-based FHE
constructions can be modified to obtain a weak form of an identity-based FHE
scheme using the trapdoor functions from [15]; that is, additional information
is needed (beyond what can be non-interactively derived from a user’s
identity) in order to evaluate certain circuits and to perform bootstrapping.
Therefore, the valued non-interactivity property of IBE is lost whereby no
communication between encryptors and the TA is needed. To the best of our
knowledge, fully-homomorphic or even “somewhat-homomorphic” IBE remains open,
and a variant of the BGN-type scheme of Gentry, Halevi and Vaikuntanathan [16]
is the only IBE scheme that can compactly evaluate quadratic formulae
(supports 2-DNF).
As far as the authors are aware, there are no $(\mathbb{Z}_{N},+)$ (like
Paillier) or $(\mathbb{Z}_{p}^{\ast},\ast)$ (like ElGamal) homomorphic IBE
schemes. Many pairings-based IBE constructions admit multiplicative
homomorphisms which give us a limited additive homomorphism for small ranges;
that is, a discrete logarithm problem must be solved to recover the plaintext,
and the complexity thereof is $O(\sqrt{M})$, where $M$ is the size of the
message space. Of a similar variety are public-key schemes such as BGN [17]
and Benaloh [18]. It remains open to construct an unbounded additively
homomorphic IBE scheme for a “large” range such as Paillier [7]. Possibly a
fruitful step in this direction would be to look at Galbraith’s variant of
Paillier’s cryptosystem based on elliptic curves over rings [19].
One of the contributions of this paper is to construct an additively
homomorphic IBE scheme for $\mathbb{Z}_{2}$, which is usually referred to as
XOR-homomorphic. XOR-homomorphic schemes such as Goldwasser-Micali [6] have
been used in many practical applications including sealed-bid auctions,
biometric authentication and as the building blocks of protocols such as
private information retrieval, and it seems that an IBE XOR-homomorphic scheme
may be useful in some of these scenarios.
We faced barriers however trying to make our XOR-homomorphic scheme anonymous.
The main obstacle is that the homomorphism depends on the public key. We pose
as an open problem the task of constructing a variant that achieves anonymity
and retains the homomorphic property. Inheriting the terminology of Golle et
al. [20] (who refer to re-encryption without the public key as _universal_ re-
encryption), we designate homomorphic evaluation in a scheme that does not
require knowledge of the public key as _universal_. We introduce a weaker
primitive that explicitly requires additional information to be passed to the
homomorphic evaluation algorithm. Our construction can be made anonymous and
retain its homomorphic property in this context; that is, if the attribute
(identity in the case of IBE) is known to an evaluator. While this certainly
is not ideal, it may be plausible in some scenarios that an evaluator is
allowed to be privy to the attribute(s) encrypted by the ciphertexts, and it
is other parties in the system to whom the attribute(s) must remain concealed.
An adversary sees incoming and outgoing ciphertexts, and can potentially
request evaluations on arbitrary ciphertexts. We call such a variant _non-
universal_. We propose a syntax for a non-universal homomorphic primitive and
formulate a security notion to capture attribute-privacy in this context.
### 1.1 Related Work
There have been several endeavors to characterize homomorphic encryption
schemes. Gjøsteen [21] succeeded in characterizing many well-known group
homomorphic cryptosystems by means of an abstract construction whose security
rests on the hardness of a subgroup membership problem. More recently,
Armknecht, Katzenbeisser and Peter [22] gave a more complete characterization
and generalized Gjøsteen’s results to the IND-CCA1 setting. However, in this
work, our focus is at a higher level and not concerned with the underlying
algebraic structures. In particular, we do not require the homomorphisms to be
unbounded since our aim to provide a more general characterization for
homomorphic encryption in the PE setting. Compactness, however, is required;
that is, informally, the length of an evaluated ciphertext should be
independent of the _size_ of the computation.
The notion of receiver-anonymity or key-privacy was formally established by
Bellare et al. [23], and the concept of universal anonymity (any user can
anonymize a ciphertext) was proposed in [24]. The first universally anonymous
IBE scheme appeared in [25]. Prabhakaran and Rosulek [26] consider receiver-
anonymity for their definitions of homomorphic encryption.
Finally, since Cocks’ IBE scheme [27] appeared, variants have been proposed
([28] and [25]) that achieve anonymity and improve space efficiency. However,
the possibility of constructing a homomorphic variant has not received
attention to date.
### 1.2 Organization
Notation and background definitions are set out in Section 2. Our
characterization of homomorphic predicate encryption is specified in Section
3; the syntax, correctness conditions and security notions are established,
and the properties of such schemes are analyzed. In Section 4, some
instantiations are given based on inner-product PE constructions. Our main
construction, XOR-homomorphic IBE, is presented in Section 5. Non-universal
homomorphic encryption and the abstraction of universal anonymizers is
presented in Section 6 towards realizing anonymity for our construction in a
weaker setting. Conclusions and future work are presented in Section 7.
## 2 Preliminaries
A quantity is said to be negligible with respect to some parameter $\lambda$,
written $\mathsf{negl}(\lambda)$, if it is asymptotically bounded from above
by the reciprocal of all polynomials in $\lambda$.
For a probability distribution $D$, we denote by $x\xleftarrow{\$}D$ that $x$
is sampled according to $D$. If $S$ is a set, $y\xleftarrow{\$}S$ denotes that
$y$ is sampled from $x$ according to the uniform distribution on $S$.
The support of a predicate $f:A\to\\{0,1\\}$ for some domain $A$ is denoted by
$\mathsf{supp}(f)$, and is defined by the set $\\{a\in A:f(a)=1\\}$.
###### Definition 1 (Homomorphic Encryption)
A homomorphic encryption scheme with message space $M$ supporting a class of
$\ell$-input circuits $\mathbb{C}\subseteq M^{\ell}\to M$ is a tuple of PPT
algorithms
$(\mathbf{\mathsf{Gen}},\mathbf{\mathsf{Enc}},\mathbf{\mathsf{Dec}},\mathbf{\mathsf{Eval}})$
satisfying the property:
$\forall(\mathsf{pk},\mathsf{sk})\leftarrow\mathbf{\mathsf{Gen}}(1^{\lambda}),\;\forall
C\in\mathbb{C},\forall m_{1},\ldots,m_{\ell}\in M$
$\forall
c_{1},\ldots,c_{\ell}\leftarrow\mathbf{\mathsf{Enc}}(\mathsf{pk},m_{1}),\ldots,\mathbf{\mathsf{Enc}}(\mathsf{pk},m_{\ell})$
$C(m_{1},\ldots,m_{\ell})=\mathbf{\mathsf{Dec}}(\mathsf{sk},\mathbf{\mathsf{Eval}}(\mathsf{pk},C,c_{i},\ldots,c_{\ell}))$
The following definition is based on [29],
###### Definition 2 (Strongly Homomorphic)
Let $\mathcal{E}$ be a homomorphic encryption scheme with message space $M$
and class of supported circuits $\mathbb{C}\subseteq\\{M^{\ell}\to M\\}$.
$\mathcal{E}$ is said to be _strongly homomorphic_ iff $\;\forall
C\in\mathbb{C},\;\forall(\mathsf{pk},\mathsf{sk})\leftarrow\mathbf{\mathsf{Gen}},\;\forall
m_{1},\ldots,m_{\ell},\;\forall
c_{1},\ldots,c_{\ell}\leftarrow\mathbf{\mathsf{Enc}}(\mathsf{pk},m_{1}),\ldots,\mathbf{\mathsf{Enc}}(\mathsf{pk},m_{\ell})$,
the following distributions are statistically indistinguishable
$\displaystyle\mathbf{\mathsf{Enc}}(\mathsf{pk},C(m_{1},\ldots,m_{\ell}))$
$\displaystyle\approx$
$\displaystyle(\mathbf{\mathsf{Eval}}(\mathsf{pk},C,c_{1},\ldots,c_{\ell}).$
###### Definition 3 (Predicate Encryption (Adapted from [9] Definition 1))
A predicate encryption (PE) scheme for the class of predicates $\mathcal{F}$
over the set of attributes $A$ and with message space $M$ consists of four
algorithms $\mathbf{\mathsf{Setup}}$, $\mathbf{\mathsf{GenKey}}$,
$\mathbf{\mathsf{Encrypt}}$, $\mathbf{\mathsf{Decrypt}}$ such that:
* $-$
$\mathbf{\mathbf{\mathsf{PE.Setup}}}$ takes as input the security parameter
$1^{\lambda}$ and outputs public parameters $\mathsf{PP}$ and master secret
key $\mathsf{MSK}$.
* $-$
$\mathbf{\mathbf{\mathsf{PE.GenKey}}}$ takes as input the master secret key
$\mathsf{MSK}$ and a description of a predicate $f\in\mathcal{F}$. It outputs
a key $\mathsf{SK}_{f}$.
* $-$
$\mathbf{\mathbf{\mathsf{PE.Encrypt}}}$ takes as input the public parameters
$\mathsf{PP}$, a message $m\in M$ and an attribute $a\in A$. It returns a
ciphertext $c$. We write this as
$c\leftarrow\mathbf{\mathsf{Encrypt}}(\mathsf{PP},a,m)$.
* $-$
$\mathbf{\mathbf{\mathsf{PE.Decrypt}}}$ takes as input a secret key
$\mathsf{SK}_{f}$ for a predicate $f$ and a ciphertext $c$. It outputs $m$ iff
$f(a)=1$. Otherwise it outputs a distinguished symbol $\bot$ with all but
negligible probability.
###### Remark 1
Predicate Encryption (PE) is known by various terms in the literature. PE
stems from Attribute-Based Encryption (ABE) with Key Policy, or simply KP-ABE,
and differs from it in its support for attribute privacy. As a result,
“ordinary” KP-ABE is sometimes known as PE with public index. Another variant
of ABE is CP-ABE (ciphertext policy) where the encryptor embeds her access
policy in the ciphertext and a recipient must possess sufficient attributes in
order to decrypt. This is the reverse of KP-ABE. In this paper, the emphasis
is placed on PE with its more standard interpretation, namely KP-ABE with
attribute privacy.
## 3 Homomorphic Predicate Encryption
### 3.1 Syntax
Let $M$ be as message space and let $A$ be a set of attributes. Consider a set
of operations $\Gamma_{M}\subseteq\\{M^{2}\to M\\}$ on the message space, and
a set of operations $\Gamma_{A}\subseteq\\{A^{2}\to A\\}$ on the attribute
space. We denote by $\gamma=\gamma_{A}\times\gamma_{M}$ for some
$\gamma_{A}\in\Gamma_{A}$ and $\gamma_{M}\in\Gamma_{M}$ the operation
$(A\times M)^{2}\to(A\times M)$ given by
$\gamma((a_{1},m_{1}),(a_{2},m_{2}))=(\gamma_{A}(a_{1},a_{2}),\gamma_{M}(m_{1},m_{2}))$.
Accordingly, we define the set of permissible “gates”
$\Gamma\subseteq\\{\gamma_{A}\times\gamma_{M}:\gamma_{A}\in\Gamma_{A},\gamma_{M}\in\Gamma_{M}\\}\subseteq\\{(A\times
M)^{2}\to(A\times M)\\}$‡‡‡It is assumed that $\Gamma_{A}$ and $\Gamma_{M}$
are minimal insofar as
$\forall\gamma_{A}\in\Gamma_{A}\exists\gamma_{M}\in\Gamma_{M}\text{ s.t.
}\gamma_{A}\times\gamma_{M}\in\Gamma$ and the converse also holds. In
particular, we later assume this of $\Gamma_{A}$.. Thus, each operation on the
plaintexts is associated with a single (potentially distinct) operation on the
attributes. Finally, we can specify a class of permissible circuits
$\mathbb{C}$ built from $\Gamma$.
###### Definition 4
A homomorphic predicate encryption (HPE) scheme for the non-empty class of
predicates $\mathcal{F}$, message space $M$, attribute space $A$, and class of
$\ell$-input circuits $\mathbb{C}$ consists of a tuple of five PPT algorithms
$\mathbf{\mathsf{Setup}}$, $\mathbf{\mathsf{GenKey}}$,
$\mathbf{\mathsf{Encrypt}},\mathbf{\mathsf{Decrypt}}$ and
$\mathbf{\mathsf{Eval}}$. such that:
* $-$
$\mathbf{\mathbf{\mathsf{HPE.Setup}}}$,
$\mathbf{\mathbf{\mathsf{HPE.GenKey}}}$,
$\mathbf{\mathbf{\mathsf{HPE.Encrypt}}}$ and
$\mathbf{\mathbf{\mathsf{HPE.Decrypt}}}$ are as specified in Definition 3.
* $-$
$\mathbf{\mathbf{\mathsf{HPE.Eval}}}(\mathsf{PP},C,c_{1},\ldots,c_{\ell})$
takes as input the public parameters $\mathsf{PP}$, an $\ell$-input circuit
$C\in\mathbb{C}$, and ciphertexts
$c_{1}\leftarrow\mathbf{\mathsf{HPE.Encrypt}}(\mathsf{PP},a_{1},m_{1}),\ldots,c_{\ell}\leftarrow\mathbf{\mathsf{HPE.Encrypt}}(\mathsf{PP},a_{\ell},m_{\ell})$.
It outputs a ciphertext that encrypts the attribute-message pair
$C((a_{1},m_{1}),\ldots,(a_{\ell},m_{\ell}))$.
Accordingly, the correctness criteria are defined as follows:
Correctness conditions:
For any
$(\mathsf{PP},\mathsf{MSK})\leftarrow\mathbf{\mathsf{HPE.Setup(1^{\lambda}}})$,
$f\in\mathcal{F}$,
$\mathsf{SK}_{f}\leftarrow\mathbf{\mathsf{HPE.GenKey}}(\mathsf{PP},\mathsf{MSK},f)$,
$C\in\mathbb{C}$:
1. 1.
For any $a\in A,m\in
M,c\leftarrow\mathbf{\mathsf{HPE.Encrypt}}(\mathsf{PP},m,a)$:
$\mathbf{\mathsf{HPE.Decrypt}}(\mathsf{SK}_{f},c)=m\iff f(a)=1$
2. 2.
$\forall m_{1},\ldots,m_{\ell}\in M,\;\forall a_{1},\ldots,a_{\ell}\in A,\;$
$\forall
c_{1},\ldots,c_{\ell}\leftarrow\mathbf{\mathsf{HPE.Encrypt}}(\mathsf{PP},a_{1},m_{1}),\ldots,\mathbf{\mathsf{HPE.Encrypt}}(\mathsf{PP},a_{\ell},m_{\ell}):$
$\forall
c^{\prime}\leftarrow\mathbf{\mathsf{HPE.Eval}}(\mathsf{PP},C,c_{1},\ldots,c_{\ell})$
1. (a)
$\mathbf{\mathsf{HPE.Decrypt}}(\mathsf{SK}_{f},c^{\prime})=m^{\prime}\iff
f(a^{\prime})=1$
where $(m^{\prime},a^{\prime})=C((a_{1},m_{1}),\ldots,(a_{\ell},m_{\ell}))$
2. (b)
$|c^{\prime}|<L(\lambda)$
where $L(\lambda)$ is a fixed polynomial derivable from $\mathsf{PP}$.
The special case of “predicate only” encryption [9] that excludes plaintexts
(“payloads”) is modelled by setting $M\triangleq\\{\mathbf{0}\\}$ for a
distinguished symbol $\mathbf{0}$, and setting
$\Gamma\triangleq\\{\gamma_{A}\times\mathbf{id_{M}}:\gamma_{A}\in\Gamma_{A}\\}$
where $\mathbf{id_{M}}$ is the identity operation on $M$.
### 3.2 Security Notions
The security notions we consider carry over from the standard notions for PE.
The basic requirement is IND-CPA security, which is referred to as “payload-
hiding”. A stronger notion is “attribute-hiding” that additionally entails
indistinguishability of attributes. The definitions are game-based with non-
adaptive and adaptive variants. The former prescribes that the adversary
choose its target attributes at the beginning of the game before seeing the
public parameters, whereas the latter allows the adversary’s choice to be
informed by the public parameters and secret key queries.
###### Definition 5
A (H)PE scheme $\mathcal{E}$ is said to be (fully) attribute-hiding (based on
Definition 2 in [9]) if an adversary $\mathcal{A}$ has negligible advantage in
the following game:
1. 1.
In the non-adaptive variant, $\mathcal{A}$ outputs two attributes $a_{0}$ and
$a_{1}$ at the beginning of the game.
2. 2.
The challenger $\mathcal{C}$ runs Setup($1^{\lambda}$) and outputs
$(\mathsf{PP},\mathsf{MSK})$
3. 3.
Phase 1
$\mathcal{A}$ makes adaptive queries for the secret keys for predicates
$f_{1},\ldots,f_{k}\in\mathcal{F}$ subject to the constraint that
$f_{i}(a_{0})=f_{i}(a_{1})$ for $1\leq i\leq k$.
4. 4.
###### Remark 2
In the stronger _adaptive_ variant, $\mathcal{A}$ only chooses attributes
$a_{0}$ and $a_{1}$ at this stage.
5. 5.
$\mathcal{A}$ outputs two messages $m_{0}$ and $m_{1}$ of equal length. It
must hold that $m_{0}=m_{1}$ if there is an $i$ such that
$f_{i}(a_{0})=f_{i}(a_{1})=1$.
6. 6.
$\mathcal{C}$ chooses a random bit $b$, and outputs
$c\leftarrow\mathbf{\mathsf{Encrypt}}(\mathsf{PP},a_{b},m_{b})$
7. 7.
Phase 2
A second phase is run where $\mathcal{A}$ requests secret keys for other
predicates subject to the same constraint as above.
8. 8.
Finally, $\mathcal{A}$ outputs a guess $b^{\prime}$ and is said to win if
$b^{\prime}=b$.
A weaker property referred to as _weakly_ attribute-hiding [9] requires that
the adversary only request keys for predicates $f$ obeying
$f(a_{0})=f(a_{1})=0$.
We propose another model of security for non-universal homomorphic encryption
in Section 6.
### 3.3 Attribute Operations
We now characterize HPE schemes based on the properties of their attribute
operations (elements of $\Gamma_{A}$).
###### Definition 6 (Properties of attribute operations)
$\forall f\in\mathcal{F},\quad\forall a_{1},a_{2}\in
A,\quad\forall\gamma_{A}\in\Gamma_{A}$:
1. 1.
$f(\gamma_{A}(a_{1},a_{2}))\Rightarrow f(a_{1})\land f(a_{2})$ (3.1)
_(Necessary condition for IND-CPA security)_
2. 2.
__
$f(\gamma_{A}(a_{1},a_{1}))=f(a_{1})$ (3.2)
3. 3.
$\forall d\in A$:
$\displaystyle f(a_{1})=f(a_{2})$ $\displaystyle\Rightarrow$ $\displaystyle
f(\gamma_{A}(d,a_{1}))=f(\gamma_{A}(d,a_{2}))$ $\displaystyle\land$
$\displaystyle f(\gamma_{A}(a_{1},d))=f(\gamma_{A}(a_{2},d))$ (3.3)
_(Non-monotone Indistinguishability)_
4. 4.
$f(\gamma_{A}(a_{1},a_{2}))=f(a_{1})\land f(a_{2})$ (3.4)
_(Monotone Access)_
Property 3.1 is a minimal precondition for payload-hiding i.e. IND-CPA
security under both adaptive and non-adaptive security definitions.
Property 3.2 preserves access under a homomorphic operation on ciphertexts
with the same attribute.
Property 3.3 is a necessary condition for full attribute-hiding.
Property 3.4 enables monotone access; a user only learns a function of a
plaintext if and only if that user has permission to learn the value of that
plaintext. This implies that $(A,\gamma_{A})$ cannot be a group unless
$\mathcal{F}$ is a class of constant predicates. In general, 3.4 implies that
$\mathcal{F}$ is monotonic. Monotone access is equivalent to the preceding
three properties collectively; that is
$\ref{eq:minimal}\land\ref{eq:idempotence}\land\ref{eq:nonmonotone}\iff\ref{eq:monotone}$
#### 3.3.1 Non-Monotone Access
Non-monotone access is trickier to define and to suitably accommodate in a
security definition. It can arise from policies that involve negation. As an
example, suppose that it is permissible for a party to decrypt data sets
designated as either “geology” or “aviation”, but is not authorized to decrypt
results with both designations that arise from homomorphic computations on
both data sets. Of course it is then necessary to strengthen the restrictions
on the adversary’s choice of $a_{0}$ and $a_{1}$ in the security game. Let
$a_{0}$ and $a_{1}$ be the attributes chosen by the adversary. Intuitively,
the goal is to show that any sequence of transitions that leads $a_{0}$ to a
an element outside the support of $f$, also leads $a_{1}$ to an element
outside the support of $f$, and vice versa. Instead of explicitly imposing
this non-triviality constraint on the adversary’s choice of attributes, one
may seek to show that there is no pair of attributes distinguishable under any
$\gamma_{A}$ and $f\in\mathcal{F}$. This is captured by the property of non-
monotone indistinguishability (3.3). Trivially, the constant operations
satisfy 3.3. Of more interest is an operation that limits homomorphic
operations to ciphertexts with the same attribute. This captures our usual
requirements for the (anonymous) IBE functionality, but it is also
satisfactory for many applications of general PE where computation need only
be performed on ciphertexts with matching attributes. To accomplish this, the
attribute space is augmented with a (logical) absorbing element $z$ such that
$f(z)=0\;\forall f\in\mathcal{F}$. The attribute operation is defined as
follows:
$\delta(a_{1},a_{2})=\begin{cases}a_{1}&\text{ if }a_{1}=a_{2}\\\ z&\text{ if
}a_{1}\neq a_{2}\end{cases}$ (3.5)
$\delta$ models the inability to perform homomorphic evaluations on
ciphertexts associated with unequal attributes (identities in the case of
IBE). A scheme with this operation can only be fully attribute-hiding in a
vacuous sense (it may be such that no restrictions are placed upon the
adversary’s choice of $f$ but it is unable to find attributes $a_{0}$ and
$a_{1}$ satisfying $f(a_{0})=f(a_{1})=1$ for any $f$.) This is the case for
anonymous IBE where the predicates are equality relations, and for the
constant map $(a_{1},a_{2})\mapsto z$ that models the absence of a homomorphic
property, although this is preferably modeled by appropriately constraining
the class of permissible circuits. More generally, such schemes can only be
weakly attribute-hiding because their operations $\gamma_{A}$ only satisfy a
relaxation of 3.3 given as follows:
_Necessary condition for weakly attribute-hiding_ $\forall a_{1},a_{2},d\in
A$:
$\displaystyle f(a_{1})=f(a_{2})=0$ $\displaystyle\Rightarrow$ $\displaystyle
f(\gamma_{A}(d,a_{1}))=f(\gamma_{A}(d,a_{2}))$ $\displaystyle\land$
$\displaystyle f(\gamma_{A}(a_{1},d))=f(\gamma_{A}(a_{2},d))$ (3.6)
###### Remark 3
In the case of general schemes not satisfying 3.3, placing constraints on the
adversary’s choice of attributes weakens the security definition. Furthermore,
it must be possible for the challenger to efficiently check whether a pair of
attributes satisfies such a condition. Given the added complications, it is
tempting to move to a simulation-based definition of security. However, this
is precluded by the recent impossibility results of [30] in the case of both
weakly and fully attribute-hiding in the NA/AD-SIM models of security.
However, for predicate encryption with public index (the attribute is not
hidden), this has not been ruled out for 1-AD-SIM and many-NA-SIM where “1”
and “many” refer to the number of ciphertexts seen by the adversary. See [30,
31] for more details. In the context of non-monotone access, it thus seems
more reasonable to focus on predicate encryption with public index. Our main
focus in this work is on schemes that facilitate attribute privacy, and
therefore we restrict our attention to schemes that at least satisfy 3.6.
#### 3.3.2 Delegate Predicate Encryption
A primitive presented in [32] called “Delegate Predicate Encryption” (DPE)
§§§Not to be confused with the different notion of Delegatable Predicate
Encryption. enables a user to generate an encryption key associated with a
chosen attribute $a\in A$, which does not reveal anything about $a$. The user
can distribute this to certain parties who can then encrypt messages with
attribute $a$ obliviously. The realization in [32] is similar to the widely-
used technique of publishing encryptions of “zero” in a homomorphic
cryptosystem, which can then be treated as a key. In fact, this technique is
adopted in [33] to transform a strongly homomorphic private-key scheme into a
public-key one. Generalizing from the results of [32], this corollary follows
from the property of attribute-hiding
###### Corollary 1
An attribute-hiding HPE scheme is a DPE as defined in [32] if there exists a
$\gamma\in\Gamma$ such that $(A\times M,\gamma)$ is unital.
## 4 Constructions with Attribute Aggregation
In this section, we give some meaningful examples of attribute homomorphisms
(all which satisfy monotone access) for some known primitives. We begin with a
special case of PE introduced by Boneh and Waters [34], which they call
_Hidden Vector Encryption_. In this primitive, a ciphertext embeds a vector
$\mathbf{w}\in\\{0,1\\}^{n}$ where $n$ is fixed in the public parameters. On
the other hand, a secret key corresponds to a vector $\mathbf{v}\in
V\triangleq\\{\ast,0,1\\}^{n}$ where $\ast$ is interpreted as a “wildcard”
symbol or a “don’t care” (it matches any symbol). A decryptor who has a secret
key for some $\mathbf{v}$ can check whether it matches the attribute in a
ciphertext.
To formulate in terms of PE, let $A=\\{0,1\\}^{n}$ and define
$\mathcal{F}\subseteq\\{(w_{1},\ldots,w_{n})\mapsto\bigwedge_{i=1}^{n}(v_{i}=w_{i}\lor
v_{i}=\ast):\mathbf{v}\in V\\}$
Unfortunately, we cannot achieve a non-trivial homomorphic variant of HVE that
satisfies 3.4. To see this, consider the HVE class of predicates $\mathcal{F}$
and an operation $\gamma_{A}$ satisfying 3.4. For any
$\mathbf{x},\mathbf{y}\in A$, let
$\mathbf{z}=\gamma_{A}(\mathbf{x},\mathbf{y})$. Now for 3.4 to hold, we must
have that $f(\mathbf{z})=f(\mathbf{x})\land f(\mathbf{y})$ for all
$f\in\mathcal{F}$. Suppose $\mathbf{x}_{i}\neq\mathbf{y}_{i}$ and
$\mathbf{z}_{i}=\mathbf{x}_{i}$. Then there exists an $f\in\mathcal{F}$ with
$f(\mathbf{z})=f(\mathbf{x})$ and $f(\mathbf{z})\neq f(\mathbf{y})$. It is
necessary to restrict $V$. Accordingly, let $V=\\{\ast,1\\}^{n}$ Setting the
non-equal elements to 0 yields associativity and commutativity. Such an
operation is equivalent to component-wise logical AND on the attribute
vectors, and we will denote it by $\land^{n}$. $(A,\land^{n})$ is a
semilattice.
Recall that a predicate-only scheme does not incorporate a payload into
ciphertexts. Even such a scheme $\mathcal{E}$ with the $\land^{n}$ attribute
homomorphism might find some purpose in real-world scenarios. One particular
application of $\mathcal{E}$ is secure data aggregation in Wireless Sensor
Networks (WSNs), an area which has been the target of considerable research (a
good survey is [35]). It is conceivable that some aggregator nodes may be
authorized by the sink (base station to which packets are forwarded) to read
packets matching certain criteria. An origin sensor node produces an outgoing
ciphertext as follows: (1). It encrypts the attributes describing its data
using $\mathcal{E}$. (2) It encrypts its sensor reading with the public key of
the sink using a _separate_ additively (say) homomorphic public-key
cryptosystem. (3) Both ciphertexts are forwarded to the next hop.
Since an aggregator node receives packets from multiple sources, it needs to
have some knowledge about how to aggregate them. To this end, the sink can
authorize it to apply a particular predicate to incoming ciphertexts to check
for matching candidates for aggregation. One sample policy may be [“REGION1”
$\land$ “TEMPERATURE”’]. It can then aggregate ciphertexts matching this
policy. Additional aggregation can be performed by a node further along the
route that has been perhaps issued a secret key for a predicate corresponding
to the more permissive policy of [“TEMPERATURE”]. In the scenario above, it
would be more ideal if $\mathcal{E}$ were also additively homomorphic since
besides obviating the need to use another PKE cryptosystem, more control is
afforded to aggregators; they receive the ability to decrypt partial sums, and
therefore, to perform (more involved) statistical computations on the data.
It is possible to achieve the former case from some recent inner-product PE
schemes that admit homomorphisms on both attributes and payload. We focus on
two prominent constructions with different mathematical structures. Firstly, a
construction is examined by Katz, Sahai and Waters (KSW) [9], which relies on
non-standard assumptions on bilinear groups, assumptions that are justified by
the authors in the generic group model. Secondly, we focus on a construction
presented by Agrawal, Freeman and Vaikuntanathan (AFV) [36] whose security is
based on the learning with errors (LWE) problem.
In both schemes, an attribute is an element of $\mathbb{Z}_{m}^{n}$¶¶¶In [9],
$m$ is a product of three large primes and $n$ is the security parameter. In
[36], $n$ is independent of the security parameter and $m$ may be polynomial
or superpolynomial in the security parameter; in the latter case $m$ is the
product of many “small” primes. We require that $m$ be superpolynomial here.
and a predicate also corresponds to an element of $\mathbb{Z}_{m}^{n}$. For
$\mathbf{v}\in\mathbb{Z}_{m}^{n}$, a predicate
$f_{\mathbf{v}}:\mathbb{Z}_{m}^{n}\to\\{0,1\\}$ is defined by
$f_{\mathbf{v}}(\mathbf{w})=\begin{cases}1&\text{ iff
}\langle\mathbf{v},\mathbf{w}\rangle\\\ 0&\text{ otherwise }\end{cases}$
Roughly speaking, in a ciphertext, all sub-attributes (in $\mathbb{Z}_{m}$)
are blinded by the same uniformly random “blinding” element $b$ ∥∥∥a scalar in
KSW and a matrix in AFV. The decryption algorithm multiplies each component by
the corresponding component in the predicate vector, and the blinding element
$b$ is eliminated when the inner product evaluates to zero with all but
negligible probability, which allows decryption to proceed.
Let $\mathbf{c_{1}}$ and $\mathbf{c_{2}}$ be ciphertexts that encrypt
attributes $a_{1}$ and $a_{2}$ respectively. It can be easily shown that the
sum $\mathbf{c^{\prime}}=\mathbf{c_{1}}\boxplus$******$\boxplus$ denotes a
pairwise sum of the ciphertext components in both schemes $\mathbf{c_{2}}$
encrypts both $a_{1}$ and $a_{2}$ in a somewhat “isolated” way. The lossiness
is “hidden” by the negligible probability of two non-zero inner-products
summing to 0. For linear aggregation, this can be repeated a polynomial number
of times (or effectively unbounded in practice) while ensuring correctness
with overwhelming probability. While linear aggregation is sufficient for the
WSN scenario, it is interesting to explore other circuit forms. For the KSW
scheme, we observe that all circuits of polynomial depth can be evaluated with
overwhelming probability. For AFV, the picture is somewhat similar to the
fully homomorphic schemes based on LWE such as [4, 5] but without requiring
multiplicative gates.
While there are motivating scenarios for aggregation on the attributes, in
many cases it is adequate or preferable to restrict evaluation to ciphertexts
with matching attributes; that is, by means of the $\delta$ operation defined
in Section 3.3.1. Among these cases is anonymous IBE. In the next section, we
introduce an IBE construction that supports an unbounded XOR homomorphism,
prove that it is strongly homomorphic and then investigate anonymous variants.
## 5 Main Construction: XOR-Homomorphic IBE
In this section, an XOR-homomorphic IBE scheme is presented whose security is
based on the quadratic residuosity assumption. Therefore, it is similar in
many respects to the Goldwasser-Micali (GM) cryptosystem [6], which is well-
known to be XOR-homomorphic. Indeed, the GM scheme has found many practical
applications due to its homomorphic property. In Section 6.3, we show how many
of these applications benefit from an XOR-homomorphic scheme in the identity-
based setting.
Our construction derives from the IBE scheme due to Cocks [27] which has a
security reduction to the quadratic residuosity problem. To the best of our
knowledge, a homomorphic variant has not been explored to date.
### 5.1 Background
Let $m$ be an integer. A quadratic residue in the residue ring
$\mathbb{Z}_{m}$ is an integer $x$ such that $x\equiv y^{2}\mod{m}$ for some
$y\in\mathbb{Z}_{m}$. The set of quadratic residues in $\mathbb{Z}_{m}$ is
denoted $\mathbb{QR}(m)$. If $m$ is prime, it easy to determine whether any
$x\in\mathbb{Z}_{m}$ is a quadratic residue.
Let $N=pq$ be a composite modulus where $p$ and $q$ are prime. Let
$x\in\mathbb{Z}$. We write $\genfrac{(}{)}{}{0}{x}{N}$ to denote the Jacobi
symbol of $x\mod{N}$. The subset of integers with Jacobi symbol +1 (resp. -1)
is denoted $\mathbb{Z}_{N}[+1]$ (resp. $\mathbb{Z}_{N}[-1])$. The quadratic
residuosity problem is to determine, given input $(N,x\in\mathbb{Z}_{N}[+1])$,
whether $x\in\mathbb{QR}(N)$, and it is believed to be intractable.
Define the encoding $\nu:\\{0,1\\}\to\\{-1,1\\}$ with $\nu(0)=1$ and
$\nu(1)=-1$. Formally, $\nu$ is a group isomorphism between
$(\mathbb{Z}_{2},+)$ and $(\\{-1,1\\},\ast)$.
In this section, we build on the results of [25] and therefore attempt to
maintain consistency with their notation where possible. As in [25], we let
$H:\\{0,1\\}^{\ast}\to\mathbb{Z}^{\ast}_{N}[+1]$ be a full-domain hash. A
message bit is mapped to an element of $\\{-1,1\\}$ via $\nu$ as defined
earlier (0 (1 resp.) is encoded as 1 (-1 resp.)).
### 5.2 Original Cocks IBE Scheme
* $-$
$\mathbf{\mathbf{\mathsf{CocksIBE}}.Setup}(1^{\lambda})$:
1. 1.
Repeat: $p,q\xleftarrow{\$}\mathbf{\mathsf{RandPrime}}(1^{\lambda})$ Until:
$p\equiv q\equiv 3\pmod{4}$
2. 2.
$N\leftarrow pq$
3. 3.
Output $(\mathsf{PP}:=N,\mathsf{MSK}:=(p,q))$
* $-$
$\mathbf{\mathbf{\mathsf{CocksIBE}}.KeyGen}(\mathsf{PP},\mathsf{MSK},\mathsf{id})$:
1. 1.
Parse $\mathsf{MSK}$ as $(p,q)$.
2. 2.
$a\leftarrow H(\mathsf{id})$
3. 3.
$r\leftarrow a^{\frac{N+5-p-q}{8}}\pmod{N}$
($\therefore r^{2}\equiv a\pmod{N}$ or $r^{2}\equiv-a\pmod{N}$)
4. 4.
Output $\mathsf{sk}_{\mathsf{id}}:=(\mathsf{id},r)$
* $-$
$\mathbf{\mathbf{\mathsf{CocksIBE}}.Encrypt}(\mathsf{PP},\mathsf{id},b)$:
1. 1.
$a\leftarrow H(\mathsf{id})$
2. 2.
$t_{1},t_{2}\xleftarrow{\$}\mathbb{Z}^{\ast}_{N}[\nu(b)]$
3. 3.
Output $\mathbf{\psi}:=(t_{1}+at_{1}^{-1},t_{2}-at_{2}^{-1})$
* $-$
$\mathbf{\mathbf{\mathsf{CocksIBE}}.Decrypt}(\mathsf{PP},\mathsf{sk}_{\mathsf{id}},\mathbf{\psi})$:
1. 1.
Parse $\mathbf{\psi}$ as $(\mathbf{\psi}_{1},\mathbf{\psi}_{2})$
2. 2.
Parse $\mathsf{sk}_{\mathsf{id}}$ as $(\mathsf{id},r)$
3. 3.
$a\leftarrow H(\mathsf{id})$
4. 4.
If $r^{2}\equiv a\pmod{N}$, set $d\leftarrow\mathbf{\psi}_{1}$. Else if
$r^{2}\equiv-a\pmod{N}$, set $d\leftarrow\mathbf{\psi}_{2}$. Else output
$\bot$ and abort.
5. 5.
Output $\nu^{-1}(\genfrac{(}{)}{}{0}{d+2r}{N})$
The above scheme can be shown to be adaptively secure in the random oracle
model assuming the hardness of the quadratic residuosity problem.
#### 5.2.1 Anonymity
Cocks’ scheme is not anonymous. Boneh et al. [37] report a test due to
Galbraith that enables an attacker to distinguish the identity of a
ciphertext. This is achieved with overwhelming probability given multiple
ciphertexts. It is shown by Ateniese and Gasti [25] that there is no “better”
test for attacking anonymity. Briefly, let $a=H(\mathsf{id})$ be the public
key derived from the identity $\mathsf{ID_{a}}$. Let $c$ be a ciphertext in
the Cocks’ scheme. Galbraith’s test is defined as
$\mathsf{GT}(a,c,N)=\genfrac{(}{)}{}{0}{c^{2}-4a}{N}$
Now if $c$ is a ciphertext encrypted with $a$, then $\mathsf{GT}(a,c,N)=+1$
with all but negligible probability. For $b\in\mathbb{Z}_{N}^{\ast}$ such that
$b\neq a$, the value $\mathsf{GT}(b,c,N)$ is statistically close to the
uniform distribution on $\\{-1,1\\}$. Therefore, given multiple ciphertexts,
it can be determined with overwhelming probability whether they correspond to
a particular identity.
### 5.3 XOR-homomorphic Construction
Recall that a ciphertext in the Cocks scheme consists of two elements in
$\mathbb{Z}_{N}$. Thus, we have
$(c,d)\leftarrow\mathbf{\mathsf{CocksIBE.Encrypt}}(\mathsf{PP},\mathsf{id},b)\in\mathbb{Z}_{N}^{2}$
for some identity $\mathsf{id}$ and bit $b\in\\{0,1\\}$. Also recall that only
one element is actually used for decryption depending on whether
$a:=H(\mathsf{id})\in\mathbb{QR}(N)$ or $-a\in\mathbb{QR}(N)$. If the former
holds, it follows that a decryptor has a secret key $r$ satisfying
$r^{2}\equiv a\pmod{N}$. Otherwise, a secret key $r$ satisfies
$r^{2}\equiv-a\pmod{N}$. To simplify the description of the homomorphic
property, we will assume that $a\in\mathbb{QR}(N)$ and therefore omit the
second “component” $d$ from the ciphertext. In fact, the properties hold
analogously for the second “component” by simply replacing $a$ with $-a$.
In the homomorphic scheme, each “component” of the ciphertext is represented
by a pair of elements in $\mathbb{Z}_{N}^{2}$ instead of a single element as
in the original Cocks scheme. As mentioned, we will omit the second such pair
for the moment. Consider the following encryption algorithm $E_{a}$ defined by
$\mathbf{\mathsf{E_{a}}}(b:\\{0,1\\}):$
$t\xleftarrow{\$}\mathbb{Z}^{\ast}_{N}[\nu(b)]$
return $(t+at^{-1},2)\in\mathbb{Z}^{2}_{N}$.
Furthermore, define the decryption function
$D_{a}(\mathbf{c})=\nu^{-1}(c_{0}+rc_{1})$. The homomorphic operation
$\boxplus:\mathbb{Z}^{2}_{N}\times\mathbb{Z}^{2}_{N}\to\mathbb{Z}^{2}_{N}$ is
defined as follows:
$\mathbf{c}\boxplus\mathbf{d}=(c_{0}d_{0}+ac_{1}d_{1},c_{0}d_{1}+c_{1}d_{0})$
(5.1)
It is easy to see that
$D_{a}(\mathbf{c}\boxplus\mathbf{d})=D_{a}(\mathbf{c})\oplus
D_{a}(\mathbf{d})$:
$\displaystyle D_{a}(\mathbf{c}\boxplus\mathbf{d})$ $\displaystyle=$
$\displaystyle D_{a}((c_{0}d_{0}+ac_{1}d_{1},c_{0}d_{1}+c_{1}d_{0}))$ (5.2)
$\displaystyle=$
$\displaystyle\nu^{-1}((c_{0}d_{0}+ac_{1}d_{1})+r(c_{0}d_{1}+c_{1}d_{0}))$
$\displaystyle=$
$\displaystyle\nu^{-1}(c_{0}d_{0}+rc_{0}d_{1}+rc_{1}d_{0}+r^{2}c_{1}d_{1})$
$\displaystyle=$ $\displaystyle\nu^{-1}((c_{0}+rc_{1})(d_{0}+rd_{1}))$
$\displaystyle=$
$\displaystyle\nu^{-1}(c_{0}+rc_{1})\oplus\nu^{-1}(d_{0}+rd_{1})$
$\displaystyle=$ $\displaystyle D_{a}(\mathbf{c})\oplus D_{a}(\mathbf{d})$
Let $R_{a}=\mathbb{Z}_{N}[x]/(x^{2}-a)$ be a quotient of the polynomial ring
$R=\mathbb{Z}_{N}[x]$. It is more natural and convenient to view ciphertexts
as elements of $R_{a}$ and the homomorphic operation as multiplication in
$R_{a}$. Furthermore, decryption equates to evaluation at the point $r$. Thus
the homomorphic evaluation of two ciphertext polynomials $c(x)$ and $d(x)$ is
simply $e(x)=c(x)\ast d(x)$ where $\ast$ denotes multiplication in $R_{a}$.
Decryption becomes $\nu^{-1}(e(r))$. Moreover, Galbraith’s test is generalized
straightforwardly to the ring $R_{a}$:
$\mathsf{GT}(a,c(x))=\genfrac{(}{)}{}{0}{c_{0}^{2}-c_{1}^{2}a}{N}.$
We now formally describe our variant of the Cocks scheme that supports an XOR
homomorphism.
###### Remark 4
We have presented the scheme in accordance with Definition 4 for consistency
with the rest of the paper. Therefore, it uses the circuit formulation, which
we would typically consider superfluous for a group homomorphic scheme.
Let
$\mathbb{C}\triangleq\\{\mathbf{x}\mapsto\langle\mathbf{t},\mathbf{x}\rangle:\mathbf{t}\in\mathbb{Z}^{\ell}_{2}\\}\subset\mathbb{Z}^{\ell}_{2}\to\mathbb{Z}_{2}$
be the class of arithmetic circuits characterized by linear functions over
$\mathbb{Z}_{2}$ in $\ell$ variables. As such, we associate a representative
vector $V(C)\in\mathbb{Z}^{\ell}_{2}$ to every circuit $C\in\mathbb{C}$. In
order to obtain a strongly homomorphic scheme, we use the standard technique
of re-randomizing the evaluated ciphertext by homomorphically adding an
encryption of zero.
* $-$
$\mathbf{\mathbf{\mathsf{xhIBE}}.Encrypt}(\mathsf{PP},\mathsf{id},b)$:
1. 1.
$a\leftarrow H(\mathsf{id})$
2. 2.
As a subroutine (used later), define
$E(\mathsf{PP},a,b)$:
1. (a)
$t_{1},t_{2}\xleftarrow{\$}\mathbb{Z}^{\ast}_{N}[\nu(b)]$
2. (b)
$g_{1},g_{2}\xleftarrow{\$}\mathbb{Z}^{\ast}_{N}$
3. (c)
$c(x)\leftarrow(t_{1}+ag_{1}^{2}t_{1}^{-1})+2g_{1}x\in\mathbb{Z}_{N}[x]$
4. (d)
$d(x)\leftarrow(t_{2}+ag_{2}^{2}t_{2}^{-1})+2g_{2}x\in\mathbb{Z}_{N}[x]$
5. (e)
Repeat steps (a) - (d) until $\mathsf{GT}(a,c(x))=1$ and
$\mathsf{GT}(-a,d(x))=1$.
6. (f)
Output $(c(x),d(x))$
3. 3.
Output $\mathbf{\psi}:=(E(\mathsf{PP},a,b),a)$
* $-$
$\mathbf{\mathbf{\mathsf{xhIBE}}.Decrypt}(\mathsf{PP},\mathsf{sk}_{\mathsf{id}},\mathbf{\psi})$:
1. 1.
Parse $\mathbf{\psi}$ as $(c(x),d(x),a)$
2. 2.
Parse $\mathsf{sk}_{\mathsf{id}}$ as $(\mathsf{id},r)$
3. 3.
If $r^{2}\equiv a\pmod{N}$ and $\mathsf{GT}(a,c(x))=1$, set $e(x)\leftarrow
c(x)$. Else if $r^{2}\equiv-a\pmod{N}$ and $\mathsf{GT}(-a,c(x))=1$, set
$e(x)\leftarrow d(x)$. Else output $\bot$ and abort.
4. 4.
Output $\nu^{-1}(\genfrac{(}{)}{}{0}{e(r)}{N})$
$\mathbf{\mathbf{\mathsf{xhIBE}}.Eval}(\mathsf{PP},C,\mathbf{\psi_{1}},\ldots,\mathbf{\psi}_{\ell})$:
1. 1.
Parse $\mathbf{\psi_{i}}$ as $(c_{i}(x),d_{i}(x),a_{i})$ for $1\leq i\leq\ell$
2. 2.
If $a_{i}\neq a_{j}$ for $1\leq i,j\leq\ell$, abort with $\bot$.
3. 3.
Let $a=a_{1}$ and let $R_{a}=\mathbb{Z}_{N}[x]/(x^{2}-a)$
4. 4.
$v\leftarrow V(C)$
5. 5.
$J\leftarrow\\{1\leq i\leq\ell:v_{i}=1\\}$
6. 6.
$(c^{\prime}(x),d^{\prime}(x))\leftarrow(\prod_{i\in
J}c_{i}(x)\mod{(x^{2}-a)},\prod_{i\in I}d_{i}(x))\mod{(x^{2}+a)}$
7. 7.
$(c_{z}(x),d_{z}(x))\leftarrow E(\mathsf{PP},a,0)$ ($E$ is defined as a
subroutine in the specification of $\mathbf{\mathsf{xhIBE.Encrypt}}$)
8. 8.
Output $(c^{\prime}(x)\ast c_{z}(x)\mod{(x^{2}-a)},d^{\prime}(x)\ast
d_{z}(x)\mod{(x^{2}+a)},a)$.
We now prove that our scheme is group homomorphic and strongly homomorphic. A
formalization of group homomorphic public-key schemes is given in [38]. Our
adapted definition for the PE setting raises some subtle points. The third
requirement in [38] is more difficult to formalize for general PE; we omit it
from the definition here and leave a complete formalization to Appendix 0.A.
We remark that this property which relates to distinguishing “illegitimate
ciphertexts” during decryption is not necessary to achieve IND-ID-CPA
security.
###### Definition 7 (Adapted from Definition 1 in [38])
Let $\mathcal{E}=(G,K,E,D)$ be a PE scheme with message space $M$, attribute
space $A$, ciphertext space $\hat{\mathcal{C}}$ and class of predicates
$\mathcal{F}$. The scheme $\mathcal{E}$ is group homomorphic with respect to a
non-empty set of attributes $A^{\prime}\subseteq A$ if for every
$(\mathsf{PP},\mathsf{MSK})\leftarrow G(1^{\lambda})$, every
$f\in\mathcal{F}:A^{\prime}\subseteq\mathsf{supp}(f)$, and every
$\mathsf{sk}_{f}\leftarrow K(\mathsf{MSK},f)$, the message space $(M,\cdot)$
is a non-trivial group, and there is a binary operation
$\boxdot:\hat{\mathcal{C}}^{2}\to\hat{\mathcal{C}}$ such that the following
properties are satisfied for the restricted ciphertext space
$\hat{\mathcal{C}_{f}}=\\{c\in\hat{\mathcal{C}}:D_{\mathsf{sk}_{f}}(c)\neq\bot\\}$:
1. 1.
The set of all encryptions $\mathcal{C}:=\\{c\in\hat{\mathcal{C}_{f}}\mid
c\leftarrow E(\mathsf{PP},a,m),a\in A^{\prime},m\in M\\}$ under attributes in
$A^{\prime}$ is a non-trivial group under the operation $\boxdot$.
2. 2.
The restricted decryption
$D_{\mathsf{sk}_{f}}^{\ast}:=D_{\mathsf{sk}_{f}|\mathcal{C}}$ is surjective
and $\forall c,c^{\prime}\in\mathcal{C}\quad D_{\mathsf{sk}_{f}}(c\boxdot
c^{\prime})=D_{\mathsf{sk}_{f}}(c)\cdot D_{\mathsf{sk}_{f}}(c^{\prime})$.
3. 3.
IBE only (generalized in Appendix 0.A) If $\mathcal{E}$ is an IBE scheme, then
$\hat{\mathcal{C}_{f}}$ is also required to be a group, and it is required to
be computationally indistinguishable from $\mathcal{C}$; that is:
$\\{(\mathsf{PP},f,\mathsf{sk}_{f},S,c)\mid
c\xleftarrow{\$}\mathcal{C},S\subset\\{\mathsf{sk}_{g}\leftarrow
K(g):g\in\mathcal{F}\\}\\}\underset{C}{\approx}\\{(\mathsf{PP},f,\mathsf{sk}_{f},S,\hat{c})\mid\hat{c}\xleftarrow{\$}\hat{\mathcal{C}_{f}},S\subset\\{\mathsf{sk}_{g}\leftarrow
K(g):g\in\mathcal{F}\\}\\}.$
Informally, the above definition is telling us that for a given subset of
attributes $A^{\prime}$ satisfying a predicate $f$, the set of honestly
generated encryptions under these attributes forms a group that is epimorphic
to the plaintext group. It does not say anything about ciphertexts that are
not honestly generated except in the case of IBE, where we require that all
ciphertexts that do not decrypt to $\bot$ under a secret key are
indistinguishable.
For the remainder of this section, we show that $\mathbf{\mathsf{xhIBE}}$
fulfills the definition of a group homomorphic scheme, and that it is IND-ID-
CPA secure under the quadratic residuosity assumption in the random oracle
model. To simplify the presentation of the proofs, additional notation is
needed. In particular, we inherit the notation from [25], and generalize it to
the ring $R_{a}$.
Define the subset $G_{a}\subset R_{a}$ as follows:
$G_{a}=\\{c(x)\in R_{a}:\mathsf{GT}(a,c(x))=1\\}$
Define the subset $S_{a}\subset G_{a}$††††††This definition is stricter than
its analog in [25] in that all elements are in $G_{a}$. This definition here
corrects an error in [39] where $h\in\mathbb{Z}^{\ast}_{N}$ instead of
$h\in\mathbb{Z}_{N}$.:
$S_{a}=\\{2hx+(t+ah^{2}t^{-1})\in G_{a}\mid
h\in\mathbb{Z}_{N},t,(t+ah^{2}t^{-1})\in\mathbb{Z}_{N}^{\ast}\\}$
We have the following simple lemma:
###### Lemma 1
1. 1.
$(G_{a},\ast)$ is a multiplicative group in $R_{a}$.
2. 2.
$(S_{a},\ast)$ is a subgroup of $G_{a}$
###### Proof
We must show that $G_{a}$ is closed under $\ast$. Let $c(x),d(x)\in G_{a}$,
and let $e(x)=c(x)\ast d(x)$.
$\displaystyle\mathsf{GT}(a,e(x))$ $\displaystyle=$
$\displaystyle\genfrac{(}{)}{}{0}{e_{0}^{2}-ae_{1}^{2}}{N}$ $\displaystyle=$
$\displaystyle\genfrac{(}{)}{}{0}{(c_{0}d_{0}+ac_{1}d_{1})^{2}-a(c_{0}d_{1}+c_{1}d_{0})^{2}}{N}$
$\displaystyle=$
$\displaystyle\genfrac{(}{)}{}{0}{(c_{0}^{2}-ac_{1}^{2})(d_{0}^{2}-ad_{1}^{2})}{N}$
$\displaystyle=$
$\displaystyle\genfrac{(}{)}{}{0}{(c_{0}^{2}-ac_{1}^{2})}{N}\genfrac{(}{)}{}{0}{(d_{0}^{2}-ad_{1}^{2})}{N}$
$\displaystyle=$ $\displaystyle\mathsf{GT}(a,c(x))\cdot\mathsf{GT}(a,d(x))$
$\displaystyle=$ $\displaystyle 1$
Therefore, $e(x)\in G_{a}$.
It remains to show that every element of $G_{a}$ is a unit. Let
$z=c_{0}^{2}-ac_{1}^{2}\in\mathbb{Z}_{N}$. An inverse $d_{1}x+d_{0}$ of $c(x)$
can be computed by setting $d_{0}=\frac{c_{0}}{z}$ and
$d_{1}=\frac{-c_{1}}{z}$ if it holds that $z$ is invertible in
$\mathbb{Z}_{N}$. Indeed such a $d_{1}x+d_{0}$ is in $G_{a}$. Now if $z$ is
not invertible in $\mathbb{Z}_{N}$ then $p|z$ or $q|z$, which implies that
$\genfrac{(}{)}{}{0}{z}{p}=0$ or $\genfrac{(}{)}{}{0}{z}{q}=0$. But
$\mathsf{GT}(a,c(x))=\genfrac{(}{)}{}{0}{z}{N}=\genfrac{(}{)}{}{0}{z}{p}\genfrac{(}{)}{}{0}{z}{q}=1$
since $c(x)\in G_{a}$. Therefore, $z$ is a unit in $\mathbb{Z}_{N}$, and
$c(x)$ is a unit in $G_{a}$.
Finally, to prove (2), note that the members of $S_{a}$ are exactly the
elements $c(x)$ such that $c_{0}^{2}-c_{1}^{2}a$ is a square, and it is easy
to see that this is preserved under $\ast$ in $R_{a}$. ∎
We will also need the following corollary
###### Corollary 2 (Extension of Lemma 2.2 in [25])
The distributions
$\\{(N,a,t+ah^{2}t^{-1},2h):N\leftarrow\mathbf{\mathsf{Setup}}(1^{\lambda}),a\xleftarrow{\$}\mathbb{Z}^{\ast}_{N}[+1],t,h\xleftarrow{\$}\mathbb{Z}^{\ast}_{N})\\}$
and
$\\{(N,a,z_{0},z_{1}):N\leftarrow\mathbf{\mathsf{Setup}}(1^{\lambda}),a\xleftarrow{\$}\mathbb{Z}^{\ast}_{N}[+1],z_{0}+z_{1}x\xleftarrow{\$}G_{a}\setminus
S_{a}\\}$ are indistinguishable assuming the hardness of the quadratic
residuosity problem.
###### Proof
The corollary follows immediately from Lemma 2.2 in [25] Let $\mathcal{A}$ be
an efficient adversary that distinguishes both distributions. Lemma 2.2 in
[25] shows that the distributions
$d_{0}:=(\\{(N,a,t+at^{-1}):N\leftarrow\mathbf{\mathsf{Setup}}(1^{\lambda}),a\xleftarrow{\$}\mathbb{Z}^{\ast}_{N}[+1],t\\}$
and
$d_{1}:=\\{(N,a,z_{0}):N\leftarrow\mathbf{\mathsf{Setup}}(1^{\lambda}),a\xleftarrow{\$}\mathbb{Z}^{\ast}_{N}[+1],z_{1}x+z_{0}\xleftarrow{\$}G_{a}\setminus
S_{a}\mid z_{2}=2\\}$ are indistinguishable. Given a sample $(N,a,c)$, the
simulator generates $h\xleftarrow{\$}\mathbb{Z}^{\ast}_{N}$ and computes
$b:=h^{-2}a$. It passes the element $(N,b,c,2h)$ to $\mathcal{A}$. The
simulator aborts with the output of $\mathcal{A}$. ∎
###### Theorem 5.1
$\mathbf{\mathsf{xhIBE}}$ is a group homomorphic scheme with respect to the
group operation of $(\mathbb{Z}_{2},+)$.
###### Proof
Let $a=H(\mathsf{id})$ for any valid identity string $\mathsf{id}$. Assume
that the secret key $r$ satisfies $r^{2}\equiv a\mod{N}$. The analysis holds
analogously if $r^{2}\equiv-a\mod{N}$; therefore, we omit the second component
of the ciphertexts for simplicity.
By definition, $S_{a}=\\{c(x)\in
R_{a}\mid\mathbf{\psi}:=(c(x),d(x),a)\leftarrow\mathbf{\mathsf{xhIBE.Encrypt}}(\mathsf{PP},\mathsf{id},m),m\in
M\\}$. By corollary 2, it holds that $S_{a}\underset{C}{\approx}G_{a}$ without
the master secret key. The decryption algorithm only outputs $\bot$ on input
$\mathbf{\psi}:=(c(x),d(x),a)$ if $c(x)\notin G_{a}$ or $d(x)\notin G_{-a}$.
Thus, omitting the second component, we have that $S_{a}$ corresponds to
$\mathcal{C}$ and $G_{a}$ corresponds to $\hat{\mathcal{C}}_{f}$ in Definition
7 (in this case $f$ is defined as $f(\mathsf{id}^{\prime})=1$ iff
$\mathsf{id^{\prime}}=\mathsf{id}$). It follows that the third requirement of
Definition 7 is satisfied.
By Lemma 1, $G_{a}$ is a group and $S_{a}$ is a non-trivial subgroup of
$G_{a}$. The surjective homomorphism between $\mathcal{C}:=S_{a}$ and
$M:=\mathbb{Z}_{2}^{\ast}$ has already been shown in the correctness
derivation in equation 5.2. This completes the proof. ∎
###### Remark 5
It is straightforward to show that $\mathbf{\mathsf{xhIBE}}$ also meets the
criteria for a shift-type homomorphism as defined in [38].
###### Corollary 3
$\mathbf{\mathsf{xhIBE}}$ is _strongly homomorphic_.
###### Proof
Any group homomorphic scheme can be turned into a strongly homomorphic scheme
by rerandomizing an evaluated ciphertext. Indeed this follows from Lemma 1 in
[38]. Rerandomization is achieved by multiplying the evaluated ciphertext by
an encryption of the identity, as in $\mathbf{\mathsf{xhIBE.Eval}}$. Details
follow for completeness.
Let $\mathsf{id}$ be an identity and let $a=H(\mathsf{id})$. For any circuit
$C\in\mathbb{C}$, any messages $b_{1},\ldots,b_{\ell}$ and ciphertexts
$\mathbf{\psi_{1}},\ldots,\mathbf{\psi_{\ell}}\leftarrow\mathbf{\mathsf{xhIBE.Encrypt}}(\mathsf{PP},b_{1},\mathsf{id}),\ldots,\linebreak[1]\mathbf{\mathsf{xhIBE.Encrypt}}(\mathsf{PP},b_{\ell},\mathsf{id})$,
we have
$(c^{\prime}(x),d^{\prime}(x),a)\leftarrow\mathbf{\mathbf{\mathsf{xhIBE}}.Eval}(\mathsf{PP},C,\mathbf{\psi_{1}},\ldots,\mathbf{\psi_{\ell}}).$
From the last step of $\mathbf{\mathsf{xhIBE.Eval}}$, we see that
$c^{\prime}(x)\leftarrow c^{\prime\prime}(x)\ast r(x)$ where
$r(x)\xleftarrow{\$}{S_{a}}^{(0)}$ and $c^{\prime\prime}(x)$ is the result of
the homomorphic evaluation. Suppose that $c^{\prime\prime}(x)$ encrypts a bit
$b$. Since $S_{a}$ is a group, it follows that $c^{\prime}(x)$ is uniformly
distributed in the coset $S_{a}^{(b)}$ (of the subgroup $S_{a}^{(0)}$) and is
thus distributed according to a “fresh” encryption of $b$. ∎
###### Theorem 5.2
$\mathbf{\mathsf{xhIBE}}$ is IND-ID-CPA secure in the random oracle model
under the quadratic residuosity assumption.
###### Proof
Let $\mathcal{A}$ be an adversary that breaks the IND-ID-CPA security of
$\mathbf{\mathsf{xhIBE}}$. We use $\mathcal{A}$ to construct an algorithm
$\mathcal{S}$ to break the IND-ID-CPA security of the Cocks scheme with the
same advantage. $\mathcal{S}$ proceeds as follows:
1. 1.
Uniformly sample an element $h\xleftarrow{\$}\mathbb{Z}_{N}^{\ast}$. Receive
the public parameters $\mathsf{PP}$ from the challenger $\mathcal{C}$ and pass
them to $\mathcal{A}$.
2. 2.
$\mathcal{S}$ answers a query to $H$ for identity $\mathsf{id}$ with
$H^{\prime}(\mathsf{id})\cdot h^{-2}$ where $H^{\prime}$ is $\mathcal{S}$’s
random oracle. The responses are uniformly distributed in
$\mathbb{Z}_{N}[+1]$.
3. 3.
$\mathcal{S}$ answers a key generation query for $\mathsf{id}$ with the
response $K(\mathsf{id})\cdot h^{-1}$ where $K$ is its key generation oracle.
4. 4.
When $\mathcal{A}$ chooses target identity $\mathsf{id}^{\ast}$, $\mathcal{S}$
relays $\mathsf{id}^{\ast}$ to $\mathcal{C}$. Assume w.l.o.g that $H$ has been
queried for $\mathsf{id}$, and that $\mathcal{A}$ has not made a secret key
query for $\mathsf{id}^{\ast}$. Further key generation requests are handled
subject to the condition that $\mathsf{id}\neq\mathsf{id}^{\ast}$ for a
requested identity $\mathsf{id}$.
5. 5.
Let $a=H(\mathsf{id}^{\ast})$. On receiving a challenge ciphertext $(c,d)$
from $\mathcal{C}$, compute $c(x)\leftarrow 2hx+c\in R$ and
$d(x)\leftarrow(2hx+d)\ast r(x)\in R$ where $r(x)\xleftarrow{\$}S_{-a}^{(0)}$
and $S_{-a}^{(0)}$ is the second component of the set of legal encryptions of
0. From corollary 3, $d(x)$ is uniformly distributed in $S_{-a}^{(b)}$ where
the ciphertext $(c,d)$ in the Cocks scheme encrypts the bit $b$. It follows
that $(c(x),d(x))$ is a perfectly simulated encryption of $b$ under identity
$\mathsf{id}^{\ast}$ in $\mathbf{\mathsf{xhIBE}}$. Give $(c(x),d(x))$ to
$\mathcal{A}$.
6. 6.
Output $\mathcal{A}$’s guess $b^{\prime}$.
Since the view of $\mathcal{A}$ in an interaction with $\mathcal{S}$ is
indistinguishable from its view in the real game, we conclude that the
advantage of $\mathcal{S}$ is equal to the advantage of $\mathcal{A}$.
∎
In the next section, attention is drawn to obtaining an anonymous variant of
our construction.
## 6 Anonymity
Cocks’ scheme is notable as one of the few IBE schemes that do not rely on
pairings. Since it appeared, there have been efforts to reduce its ciphertext
size and make it anonymous. Boneh, Gentry and Hamburg [28] proposed a scheme
with some elegant ideas that achieves both anonymity and a much reduced
ciphertext size for multi-bit messages at the expense of performance, which is
$O(n^{4})$ for encryption and $O(n^{3})$ for decryption (where $n$ is the
security parameter). Unfortunately the homomorphic property is lost in this
construction.
As mentioned earlier (cf. Section 5.2.1), another approach due to Ateniese and
Gasti [25] achieves anonymity and preserves performance, but its per-bit
ciphertext expansion is much higher than in [28]. However, an advantage of
this scheme is that it is universally anonymous (anyone can anonymize the
message, not merely the encryptor [24]).
On the downside, anonymizing according to this scheme breaks the homomorphic
property of our construction, which depends crucially on the public key $a$.
More precisely, what is forfeited is the _universal homomorphic_ property
mentioned in the introduction (i.e. anyone can evaluate on the ciphertexts
without additional information). There are applications where an evaluator is
aware of the attribute(s) associated with ciphertexts, but anonymity is
desirable to prevent any other parties in the system learning about such
attributes. This motivates a variant of HPE, which we call non-universal HPE,
denoted by $\mathsf{HPE}_{\bar{U}}$.
### 6.1 Non-Universal HPE
Motivation “Non-universal” homomorphic encryption is proposed for schemes that
support attribute privacy but require some information that is derivable from
the public key (or attribute in the case of PE) in order to perform
homomorphic evaluation. Therefore, attribute privacy must be surrendered to an
evaluator. If this is acceptable for an application, while at the same time
there is a requirement to hide the target recipient(s) from other entities in
the system, then “non-universal” homomorphic encryption may be useful.
Consider the following informal scenario. Suppose a collection of parties
$P_{1},\ldots,P_{\ell}$ outsource a computation on their encrypted data sets
to an untrusted remote server $S$. Suppose $S$ sends the result (encrypted) to
an independent database $\mathsf{DB}$ from which users can retrieve the
encrypted records. For privacy reasons, it may be desirable to limit the
information that $\mathsf{DB}$ can learn about the attributes associated with
the ciphertexts retrieved by certain users. Therefore, it may desirable for
the encryption scheme to provide attribute privacy. However, given the
asymmetric relationship between the delegators $P_{1},\ldots,P_{\ell}$ and the
target recipient(s), it might be acceptable for $S$ to learn the target
attribute(s) provided there is no collusion between $S$ and $\mathsf{DB}$. In
fact, the delegators may belong to a different organization than the
recipient(s).
In this paper, we introduce a syntax and security model for non-universal
homomorphic IBE. The main change in syntax entails an additional input
$\alpha$ that is supplied to the $\mathbf{\mathsf{Eval}}$ algorithm. The input
$\alpha\in\\{0,1\\}^{d}$ (where $d=poly(\lambda)$) models the additional
information needed to compute the homomorphism(s). A description of an
efficient map $Q_{A}:A\to\\{0,1\\}^{d}$ is included in the public parameters.
We say that two attributes(i.e. identities in IBE) $a_{1},a_{2}\in A$
satisfying $Q_{A}(a_{1})=Q_{A}(a_{2})$ belong to the same attribute class.
One reason that the proposed syntax is not general enough for arbitrary PE
functionalities is that it only facilitates evaluation on ciphertexts whose
attributes are in the same attribute class, which suffices for (relatively)
simple functionalities such as IBE.
We now formulate the security notion of attribute-hiding for non-universal
homomorphic IBE. Our security model provides the adversary with an evaluation
oracle whose identity-dependent input $\alpha$ is fixed when the challenge is
produced. Accordingly, for a challenge identity $\mathsf{id}\in A$, and binary
string $\alpha=Q_{A}(\mathsf{id})\in\\{0,1\\}^{d}$, the adversary can query
$\mathbf{\mathsf{{IBE}_{\bar{U}}.Eval}}(\mathsf{PP},\alpha,\cdot,\cdot)$ for
any circuit in $\mathbb{C}$ and any $\ell$-length sequence of ciphertexts.
Formally, consider the experiment
Experiment
$\boldsymbol{\mathbf{\mathsf{{\bar{U}Priv}}}(\mathcal{A}_{1},\mathcal{A}_{2})}$‡‡‡‡‡‡In
the random oracle model, the adversary is additionally given access to a
random oracle. This is what the results in this paper will use.
$(\mathsf{PP},\mathsf{MSK})\leftarrow\mathbf{\mathsf{IBE.Setup(1^{\lambda}}})$
$(\mathsf{id}_{0},m_{0}),(\mathsf{id}_{1},m_{1}),\sigma\leftarrow\mathcal{A}_{1}^{\mathbf{\mathsf{{IBE}_{\bar{U}}.KeyGen}}(\mathsf{MSK},\cdot)}(\mathsf{PP})$
$\triangleright$ $\sigma$ denotes the adversary’s state
$b\xleftarrow{\$}\\{0,1\\}$
$\alpha\leftarrow Q_{A}(\mathsf{id}_{b})$
$c\leftarrow\mathbf{\mathsf{IBE.Encrypt}}(\mathsf{PP},\mathsf{id}_{b},m_{b})$
$b^{\prime}\leftarrow\mathcal{A}_{2}^{\mathbf{\mathsf{{IBE}_{\bar{U}}.KeyGen}}^{\ast}(\mathsf{MSK},\cdot),\mathbf{\mathsf{{IBE}_{\bar{U}}.Eval}}(\mathsf{PP},\alpha,\cdot,\cdot)}(\mathsf{PP},c,\sigma)$
return 1 iff $b^{\prime}=b$ and 0 otherwise.
Define the advantage of an adversary
$\mathcal{A}:=(\mathcal{A}_{1},\mathcal{A}_{2})$ in the above experiment for a
$\mathsf{IBE}_{\bar{U}}$ scheme $\mathcal{E}$ as follows:
$\mathbf{Adv}_{\mathcal{E}}^{\mathbf{\mathsf{{\bar{U}Priv}}}}(\mathcal{A})=\mathsf{Pr}\big{[}\mathbf{\mathsf{\bar{U}Priv}}(\mathcal{A})\Rightarrow
1\big{]}-\frac{1}{2}.$
A $\mathsf{IBE}_{\bar{U}}$ scheme $\mathcal{E}$ is said to be attribute-hiding
if for all pairs of PPT algorithms
$\mathcal{A}:=(\mathcal{A}_{1},\mathcal{A}_{2})$, it holds that
$\mathbf{Adv}_{\mathcal{E}}^{\mathbf{\mathsf{{\bar{U}Priv}}}}(\mathcal{A})\leq\mathsf{negl}(\lambda)$.
Note that the above definition assumes adaptive adversaries, but can be easily
modified to accommodate the non-adaptive case.
### 6.2 Universal Anonymizers
We now present an abstraction called a _universal anonymizer_. With its help,
we can transform a universally-homomorphic, non-attribute-hiding IBE scheme
$\mathcal{E}$ into a non-universally homomorphic, attribute-hiding scheme
$\mathcal{E}^{\prime}$. In accordance with the property of universal anonymity
proposed in [24], any party can anonymize a given ciphertext.
Let
$\mathcal{E}:=(\mathbf{\mathsf{Setup}},\mathbf{\mathsf{KeyGen}},\mathbf{\mathsf{Encrypt}},\mathbf{\mathsf{Decrypt}},\mathbf{\mathsf{Eval}})$
be a PE scheme parameterized with message space $M$, attribute space $A$,
class of predicates $\mathcal{F}$, and class of circuits $\mathbb{C}$. Denote
its ciphertext space by $\mathcal{C}$. Note that this definition of a
universal anonymizer only suffices for simple functionalities such as IBE.
###### Definition 8
A _universal anonymizer_ $U_{\mathcal{E}}$ for a PE scheme $\mathcal{E}$ is a
tuple $(\mathcal{G},\mathcal{B},\mathcal{B}^{-1},Q_{A},Q_{\mathcal{F}})$ where
$\mathcal{G}$ is a deterministic algorithm, $\mathcal{B}$ and
$\mathcal{B}^{-1}$ are randomized algorithms, and $Q_{A}$ and
$Q_{\mathcal{F}}$ are efficient maps, defined as follows:
* $-$
$\mathbf{\mathbf{\mathsf{\mathcal{G}}}(\mathsf{PP})}$:
On input the public parameter of an instance of $\mathcal{E}$, output a
parameters structure $\mathsf{params}$. This contains a description of a
modified ciphertext space $\hat{\mathcal{C}}$ as well as an integer
$d=\mathsf{poly}(\lambda)$ indicating the length of binary strings
representing an attribute class.
* $-$
$\mathbf{\mathbf{\mathsf{\mathcal{B}}}(\mathsf{params},\mathbf{c})}$:
On input parameters $\mathsf{params}$ and a ciphertext
$\mathbf{c}\in\mathcal{C}$, output an element of $\hat{\mathcal{C}}$.
* $-$
$\mathbf{\mathbf{\mathsf{\mathcal{B}}}^{-1}(\mathsf{params},\alpha,\hat{\mathbf{c})}}$:
On input parameters $\mathsf{params}$, a binary string
$\alpha\in\\{0,1\\}^{d}$ and an element of $\hat{\mathcal{C}}$, output an
element of $\mathcal{C}$.
* $-$
Both maps $Q_{A}$ and $Q_{\mathcal{F}}$ are indexed by $\mathsf{params}$:
${Q_{A}}_{\mathsf{params}}:A\to\\{0,1\\}^{d}$ and
${Q_{\mathcal{F}}}_{\mathsf{params}}:\mathcal{F}\to\\{0,1\\}^{d}$.
Note: $\mathsf{params}$ can be assumed to be an implicit input; it will not be
explicitly specified to simplify notation.
The binary string $\alpha$ is computed by means of a map
$Q_{A}:A\to\\{0,1\\}^{d}$. In order for a decryptor to invert $\mathcal{B}$,
$\alpha$ must also be computable from any predicate that is satisfied by an
attribute that maps onto $\alpha$. Therefore, the map
$Q_{\mathcal{F}}:\mathcal{F}\to\\{0,1\\}^{d}$ has the property that for all
$a\in A$ and $f\in\mathcal{F}$:
$f(a)=1\Rightarrow Q_{A}(a)=Q_{\mathcal{F}}(f).$
We define an equivalence relation $\sim$ on $\mathcal{F}$ given by
$f_{1}\sim f_{2}\triangleq Q_{\mathcal{F}}(f_{1})=Q_{\mathcal{F}}(f_{2}).$
We have that
$f\sim g\iff\exists
h_{1},\ldots,h_{k}\in\mathcal{F}\quad\mathsf{supp}(f)\cap\mathsf{supp}(h_{1})\neq\emptyset\land\ldots\land\mathsf{supp}(h_{k})\cap\mathsf{supp}(g)\neq\emptyset.$
It follows that each $\alpha$ is a representative of an equivalence class in
$\mathcal{F}/\sim$. As a result, as mentioned earlier, our definition of a
universal anonymizer above is only meaningful for “simple” functionalities
such as IBE. For example, $|\mathcal{F}/\sim|=|\mathcal{F}|$ for an IBE scheme
whose ciphertexts leak the recipient’s identity.
Let $c$ be a ciphertext associated with an attribute $a$. Let
$\alpha=Q_{A}(a)$. Informally,
$c^{\prime}:=\mathcal{B}^{-1}(\alpha,\mathcal{B}(c))$ should “behave” like
$c$; that is, (1) it should have the same homomorphic “capacity” and (2)
decryption with a secret key for any $f$ should have the same output as that
for $c$. A stronger requirement captured in our formal correctness criterion
defined Appendix 0.B is that $c$ and $c^{\prime}$ should be indistinguishable
even when a distinguisher is given access to $\mathsf{MSK}$.
A universal anonymizer is employed in the following generic transformation
from a universally-homomorphic, non-attribute-hiding IBE scheme $\mathcal{E}$
to a non-universally homomorphic, attribute-hiding scheme
$\mathcal{E}^{\prime}$.
The transformation is achieved by setting:
* $-$
$\mathbf{\mathsf{\mathcal{E}^{\prime}.Encrypt}}(\mathsf{PP},a,m):=$
$\mathcal{B}(\mathbf{\mathsf{\mathcal{E}.Encrypt}}(\mathsf{PP},a,m))$
* $-$
$\mathbf{\mathsf{\mathcal{E}^{\prime}.Decrypt}}(\mathsf{SK}_{f},c):=$
$\mathbf{\mathsf{\mathcal{E}.Decrypt}}(\mathsf{SK}_{f},\mathcal{B}^{-1}(Q_{\mathcal{F}}(f),c))$
* $-$
$\mathbf{\mathsf{\mathcal{E}^{\prime}.Eval}}(\mathsf{PP},\alpha,C,c_{1},\ldots,c_{\ell}):=$
return
$\mathcal{B}(\mathbf{\mathsf{\mathcal{E}.Eval}}(\mathsf{PP},C,\mathcal{B}^{-1}(\alpha,c_{1}),\ldots,\mathcal{B}^{-1}(\alpha,c_{\ell})))$
Denote the above transformation by $T_{U_{\mathcal{E}}}(\mathcal{E})$. We
leave to future work the task of establishing (generic) sufficient conditions
that $\mathcal{E}$ must satisfy to ensure that
$\mathcal{E}^{\prime}:=T_{U_{\mathcal{E}}}(\mathcal{E})$ is an attribute-
hiding $\mathsf{HPE}_{\bar{U}}$ scheme.
An instantiation of a universal anonymizer for our XOR homomorphic scheme is
given in Appendix 0.C.
### 6.3 Applications (Brief Overview)
It turns out that XOR-homomorphic cryptosystems have been considered to play
an important part in several applications. The most well-known and widely-used
_unbounded_ XOR-homomorphic public-key cryptosystem is Goldwasser-Micali (GM)
[6], which is based on the quadratic residuosity problem. Besides being used
in protocols such as private information retrieval (PIR), GM has been employed
in some specific applications such as:
* $-$
Peng, Boyd and Dawson (PBD) [40] propose a sealed-bid auction system that
makes extensive use of the GM cryptosystem.
* $-$
Bringer et al. [41] apply GM to biometric authentication. It is used in two
primary ways; (1) to achieve PIR and (2) to assist in computing the hamming
distance between a recorded biometric template and a reference one.
Perhaps in some of these applications, a group-homomorphic identity-based
scheme may be of import, although the authors concede that no specific usage
scenario has been identified so far.
With regard to performance, our construction requires $8$ multiplications in
$\mathbb{Z}_{N}$ for a single homomorphic operation in comparison to a single
multiplication in GM. Furthermore, the construction has higher ciphertext
expansion than GM by a factor of $4$. Encryption involves 2 modular inverses
and 6 multiplications (only 4 if the strongly homomorphic property is
forfeited). In comparison, GM only requires $1.5$ multiplications on average.
## 7 Conclusions and Future Work
We have presented a characterization of homomorphic encryption in the PE
setting and classified schemes based on the properties of their attribute
homomorphisms. Instantiations of certain homomorphic properties were presented
for inner-product PE. However, it is clear that meaningful attribute
homomorphisms are limited. We leave to future work the exploration of
homomorphic encryption with access policies in a more general setting .
In this paper, we introduced a new XOR-homomorphic variant of the Cocks’ IBE
scheme and showed that it is strongly homomorphic. However, we failed to fully
preserve the homomorphic property in anonymous variants; that is, we could not
construct an anonymous universally-homomorphic variant. We leave this as an
open problem. As a compromise, however, a weaker primitive (non-universal IBE)
was introduced along with a related security notion. Furthermore, a
transformation strategy adapted from the work of Ateniese and Gasti [25] was
exploited to obtain anonymity for our XOR-homomorphic construction in this
weaker primitive.
In future work, it is hoped to construct other group homomorphic IBE schemes,
and possibly for more general classes of predicates than the IBE
functionality.
Noteworthy problems, which we believe are still open:
1. 1.
Somewhat-homomorphic IBE scheme (even non-adaptive security in the ROM)
2. 2.
(Unbounded) Group homomorphic IBE schemes for $(\mathbb{Z}_{m},+)$ where
$m=O(2^{\lambda})$ and $(\mathbb{Z}^{\ast}_{p},\ast)$ for prime $p$.
Extensions include anonymity and support for a wider class of predicates
beyond the IBE functionality.
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## Appendix 0.A Group-Homomorphic Encryption Generalized for PE
###### Definition 9 (Extension of Definition 7)
Let $\mathcal{E}=(G,K,E,D)$ be a PE scheme with message space $M$, attribute
space $A$, ciphertext space $\hat{\mathcal{C}}$ and class of predicates
$\mathcal{F}$. The scheme $\mathcal{E}$ is group homomorphic if for every
$(\mathsf{PP},\mathsf{MSK})\leftarrow G(1^{\lambda})$, every
$f\in\mathcal{F}:\mathsf{supp}(f)\neq\emptyset$, and every
$\mathsf{sk}_{f}\leftarrow K(\mathsf{MSK},f)$, the message space $(M,\cdot)$
is a non-trivial group, and there is a binary operation
$\ast:\hat{\mathcal{C}}^{2}\to\hat{\mathcal{C}}$ such that the following
properties are satisfied for the restricted ciphertext space
$\hat{\mathcal{C}_{f}}=\\{c\in\hat{\mathcal{C}}:D_{\mathsf{sk}_{f}}(c)\neq\bot\\}$:
1. 1.
$(\hat{\mathcal{C}_{f}},\ast)$ is a non-trivial group.
2. 2.
The set of all encryptions $\mathcal{C}_{f}=\\{c\in\hat{\mathcal{C}_{f}}\mid
c\leftarrow E(\mathsf{PP},a,m),a\in\mathsf{supp}(f),m\in M\\}$ is a subgroup
of $\hat{\mathcal{C}_{f}}$ with respect to the operation $\ast$.
3. 3.
The restricted decryption
$D_{\mathsf{sk}_{f}}^{\ast}:=D_{\mathsf{sk}_{f}|\hat{\mathcal{C}_{f}}}$ is
surjective and $\forall c,c^{\prime}\in\hat{\mathcal{C}_{f}}\quad
D_{\mathsf{sk}_{f}}(c\ast c^{\prime})=D_{\mathsf{sk}_{f}}(c)\cdot
D_{\mathsf{sk}_{f}}(c^{\prime})$.
4. 4.
The following distributions are computationally indistinguishable:
$\\{(\mathsf{PP},f,\mathsf{sk}_{f},S,c)\mid
c\xleftarrow{\$}\mathcal{C}_{f},S\subset\\{\mathsf{sk}_{g}\leftarrow
K(g):g\in\mathcal{F}\\}\\}\underset{C}{\approx}\\{(\mathsf{PP},f,\mathsf{sk}_{f},S,\hat{c})\mid\hat{c}\xleftarrow{\$}\hat{\mathcal{C}_{f}},S\subset\\{\mathsf{sk}_{g}\leftarrow
K(g):g\in\mathcal{F}\\}\\}.$
5. 5.
There is an efficient function $\tau:A\to\mathcal{F}$ such that for any $a\in
A$, $f=\tau(a)$ satisfies
$g(a^{\prime})=g(a)\text{ for all
}a^{\prime}\in\mathsf{supp}(f),g\in\mathcal{F}.$
Armknecht, Katzenbeisser and Peter give a characterization of group-
homomorphic public key cryptosystems [22]. Their characterization includes the
condition that the secret key contain an efficient predicate, or _decision
function_ , $\delta:\hat{\mathcal{C}}\to\\{0,1\\}$ satisfying
$\delta(c)=1\iff c\in\mathcal{C}$
where $\hat{\mathcal{C}}$ denotes the ciphertext space and
$\mathcal{C}\subseteq\hat{\mathcal{C}}$ denotes the set of legally-generated
ciphertexts under the public key (i.e. the image of the encryption algorithm
over all messages and random coins). Now generalizing this to PE in the above
definition yields a decision function
$\delta_{f}:\hat{\mathcal{C}}\to\\{0,1\\}$ with
$\delta_{f}(c)=1\iff c\in\mathcal{C}_{f}.$
We can show that such a decision function does not always exist. A
counterexample is our XOR-homomorphic IBE system from Section 5. Let
$a=H(\mathsf{id})$ and for some identity $\mathsf{id}$. Let $f:A\to\\{0,1\\}$
be the point function that is nonzero at exactly $\mathsf{id}\in A$. Then
$\mathcal{C}_{f}$ corresponds to $\\{(c(x),d(x),a):c(x)\in S_{a},d(x)\in
S_{-a}\\}$ and $\hat{\mathcal{C}}_{f}$ corresponds to
$\\{(c(x),d(x),a):c(x)\in G_{a},d(x)\in G_{-a}\\}$. However, there is no
efficient distinguisher that can distinguish between $S_{a}$ and $G_{a}$ (or
$S_{-a}$ and $G_{-a}$) without access to the factorization of $N$ (i.e. the
master secret key). It follows that there is no efficient decision function.
This necessitates property 3 in the above definition in order to extend the
abstract characterizations of IND-CCA1 security in [22] to the PE setting.
Because of property 3, it suffices to define $\delta_{f}$ as
$\delta_{f}(c)=1\iff c\in\hat{\mathcal{C}}_{f}.$ (0.A.1)
We also extend the notion of GIFT (Generic shIFt-Type) from Definition 3 in
[22]. We defer the reader to this paper for a formal definition of GIFT.
Informally, a GIFT PE scheme satisfies the following:
* $-$
The public parameters $\mathsf{PP}$ contains information to determine a non-
trivial, proper normal subgroup $\mathcal{N}_{f}$ for every group
$\mathcal{C}_{f}$.
* $-$
It holds that for every $f,g\in\mathcal{F}$, the systems of representatives
$\mathcal{R}_{f}=\mathcal{C}_{f}/\mathcal{N}_{f}$ and
$\mathcal{R}_{f}=\mathcal{C}_{g}/\mathcal{N}_{g}$ have the same cardinality;
that is, $|\mathcal{R}_{f}|=|\mathcal{R}_{g}|$.
* $-$
$\mathsf{PP}$ contains an efficient function $\psi:\mathcal{F}\times
M\to\hat{\mathcal{C}}$ with the property that $\psi_{f}=\psi(f,\cdot)$ for any
$f\in\mathcal{F}$ is an isomorphism between $M$ and $\mathcal{R}_{f}$.
* $-$
To encrypt a message $m\in M$ under attribute $a\in A$, an encryptor:
1. 1.
computes $f^{\prime}\leftarrow\tau(a)$,
2. 2.
chooses a random $n\xleftarrow{\$}\mathcal{N}_{f^{\prime}}$,
3. 3.
and outputs the ciphertext $\psi_{f^{\prime}}(m)\ast
n\in\mathcal{C}_{f^{\prime}}$.
* $-$
A secret key $\mathsf{sk}_{f}$ for predicate $f\in\mathcal{F}$ contains an
efficient description of $\psi_{f}^{-1}\circ\mu_{f}$ where
$\mu_{f}:\hat{\mathcal{C}_{f}}\to\mathcal{R}_{f}$ such that $r=\mu(c)$ is the
unique representative with $c=r\ast n$ where $n\in\mathcal{N}_{f}$.
### 0.A.1 Interactive Splitting Oracle-Assisted Subgroup Membership Problem
(ISOAP)
Let $G$ be a PPT algorithm that takes as input a security parameter $\lambda$
and outputs a tuple $(\hat{\mathcal{G}},\mathcal{I},\mathbb{G},k)$ where
$\hat{\mathcal{G}}$ is a finite semigroup, $\mathcal{I}$ is a set of indices
and $\mathbb{G}$ and $k$ are defined momentarily. Firstly, $\mathbb{G}$ is a
family
$\\{(\mathcal{G}_{i},\mathcal{N}_{i},\mathcal{R}_{i})\\}_{i\in\mathcal{I}}$
where $\mathcal{G}_{i}\subseteq\hat{\mathcal{G}}$ is a non-trivial group,
$\mathcal{N}_{i}$ is a proper, non-trivial subgroup of $\mathcal{G}_{i}$ and
$\mathcal{R}_{i}\subset\mathcal{G}_{i}$ is a finite set of representatives of
$\mathcal{G}_{i}/\mathcal{N}_{i}$. It is required that $|R_{i}|=|R_{j}|$ for
all $i,j\in\mathcal{I}$. Finally, $k$ is efficient trapdoor information that
allows us to efficiently solve the splitting problem (SP) in any group
$\mathcal{G}_{i}$; that is, given some $c\in\mathcal{G}_{i}$, the goal of SP
is to find the unique $r\in\mathcal{R}_{i}$ and $n\in\mathcal{N}_{i}$ such
that $c=r\ast n$. We let $K$ be a PPT algorithm that uses $k$ and takes an
index $i\in\mathcal{I}$ as input, and outputs a description of an efficient
function $\sigma_{i}:\mathcal{G}_{i}\to\mathcal{R}_{i}\times\mathcal{N}_{i}$.
Such a function solves SP in $\mathcal{G}_{i}$.For brevity, we set
$K^{\prime}:=K_{k}$.
We define an interactive version of the problem $\mathbf{\mathsf{SOAP}}$ from
[22], which we refer to as $\mathbf{\mathsf{ISOAP}}$. This is a subgroup
membership problem relating to a group chosen by the adversary who in addition
is granted access to a “splitting oracle” for that group.
The game that defines $\mathbf{\mathsf{ISOAP}}$ proceeds as follows. Prior to
the challenge phase, the adversary is granted access to a “splitting oracle”
$\mathcal{O}_{\mathsf{SP}}^{\hat{\mathcal{G}},\mathcal{I},\mathbb{G},K}$ that
takes an index $i\in\mathcal{I}$ and an element $c$ of $\hat{\mathcal{G}}$,
and answers with $\bot$ if $c\notin\mathcal{G}_{i}$; otherwise, it answers
with $\sigma_{i}(c)$. In addition, the adversary is given access to another
oracle $\mathcal{O}_{K_{1}}^{\hat{\mathcal{G}},\mathcal{I},\mathbb{G},K,Q}$ in
the first phase which responds to a query for an index $i\in\mathcal{I}$ by
storing $i$ in a cache $\mathcal{Q}$ and returning $K(i)$.
Then the adversary chooses a “challenge” group by specifying an index
$\mathsf{ind}\in\mathcal{I}$ subject to the condition that
$\mathcal{G}_{\mathsf{ind}}\cap\mathcal{G}_{j}=\emptyset$ for every $j\in Q$.
It receives a challenge element $c^{\ast}\in\mathcal{G}_{\mathsf{ind}}$.
In the second phase, the adversary is given access to a more restricted oracle
$\mathcal{O}_{K_{2}}^{\hat{\mathcal{G}},\mathcal{I},\mathbb{G},K,c^{\ast}}$
that when queried on index $i\in\mathcal{I}$, returns $K(i)$ if
$c^{\ast}\notin\mathcal{G}_{i}$, and returns $\bot$ otherwise.
#####
Experiment
$\mathbf{Exp}_{(\mathcal{A}_{1},\mathcal{A}_{2}),G,K}^{\mathbf{\mathsf{ISOAP}}}(\lambda)$:
1. 1.
$(\hat{\mathcal{G}},\mathcal{I},\mathbb{G},k)\leftarrow G(\lambda)$.
$K^{\prime}:=K_{k}$.
2. 2.
$s,\mathsf{ind}\leftarrow\mathcal{A}_{1}^{\mathcal{O}_{\mathsf{SP}}^{\hat{\mathcal{G}},\mathcal{I},\mathbb{G},K^{\prime}}(\cdot,\cdot),\mathcal{O}_{K_{1}}^{\hat{\mathcal{G}},\mathcal{I},\mathbb{G},K^{\prime},Q}}(\hat{\mathcal{G}},\mathcal{I},\mathbb{G})$.
3. 3.
Choose $b\xleftarrow{\$}\\{0,1\\}$. If $b=1$:
$c^{\ast}\xleftarrow{\$}\mathcal{G}_{\mathsf{ind}}$. Otherwise,
$c^{\ast}\xleftarrow{\$}\mathcal{N}_{\mathsf{ind}}$.
4. 4.
$b^{\prime}\leftarrow\mathcal{A}_{2}^{\mathcal{O}_{K_{2}}^{\hat{\mathcal{G}},\mathcal{I},\mathbb{G},K^{\prime},c^{\ast}}}(\hat{\mathcal{G}},\mathcal{I},\mathbb{G},s,c^{\ast})$
5. 5.
Output $1$ if $b^{\prime}=b$. Output $0$ otherwise.
###### Theorem 0.A.1
Let $\mathcal{E}=(G,K,E,D)$ be a GIFT PE scheme. Then $\mathcal{E}$ is IND-AD-
CCA1 secure if and only if $\mathbf{\mathsf{ISOAP}}$ is hard relative to an
algorithm $G^{\prime}$ that derives the tuple it outputs, namely
$(\hat{\mathcal{C}},\mathcal{F},\mathbb{G}:=\\{(\hat{\mathcal{C}_{f}}.\mathcal{R}_{f},\hat{\mathcal{N}_{f})}\\}_{f\in\mathcal{F}},\mathsf{MSK})$,
from $(\mathsf{PP},\mathsf{MSK})\leftarrow G(\lambda)$ where
$\hat{\mathcal{C}_{f}}=\mathcal{R}_{f}\ast\hat{\mathcal{N}_{f}}$ for every
$f\in\mathcal{F}$.
###### Proof (sketch)
The proof is similar to the proof of Theorem 3 in [22].
Firstly, we show that the hardness of $\mathbf{\mathsf{ISOAP}}$ implies the
IND-AD-CCA1 security of $\mathcal{E}$. Suppose $\mathbf{\mathsf{ISOAP}}$ is
hard. Assume that $\mathcal{E}$ is not IND-AD-CCA1 secure. Then there is an
algorithm $\mathcal{B}$ that has a non-negligible advantage $\epsilon$
attacking the IND-AD-CCA1 security of $\mathcal{E}$. This algorithm can be
used to construct an adversary
$\mathcal{A}^{\mathsf{ISOAP}}=(\mathcal{A}^{\mathsf{ISOAP}}_{1},\mathcal{A}^{\mathsf{ISOAP}}_{2})$
that obtains an advantage of $\frac{1}{2}\epsilon$ against
$\mathbf{\mathsf{ISOAP}}$. Now $\mathcal{A}^{\mathsf{ISOAP}}_{1}$ can simulate
$\mathsf{PP}$ and forward it to $\mathcal{B}$. It handles a secret key query
for $f\in\mathcal{F}$ by querying its oracle $\mathcal{O}_{K_{1}}$.
Furthermore, it handles a decryption query for $(f,c)$ where $f\in\mathcal{F}$
and $c\in\hat{\mathcal{C}}$ by returning $\bot$ if $\delta_{f}(c)=0$ (see the
definition of $\delta_{f}$ in Equation 0.A.1) and responding with
$\psi_{f}^{-1}(r)$ otherwise, where
$(r,n)\leftarrow\mathcal{O}_{\mathsf{SP}}(f,c)$. When $\mathcal{B}$ chooses a
target attribute $a^{\ast}$ and two messages $(m_{0},m_{1})$,
$\mathcal{A}^{\mathsf{ISOAP}}_{1}$ computes its target index
$f^{\ast}=\tau(a^{\ast})$ and forwards it to the $\mathbf{\mathsf{ISOAP}}$
challenger. $\mathcal{A}^{\mathsf{ISOAP}}_{2}$ derives an IND-AD-CCA1
challenge ciphertext from its $\mathbf{\mathsf{ISOAP}}$ challenge $c^{\ast}$
by choosing a random bit $t\xleftarrow{\$}\\{0,1\\}$ and computing
$c^{\prime}=\psi_{f^{\ast}}(m_{t})\ast c^{\ast}$. It hands $c^{\prime}$ to
$\mathcal{B}$. $\mathcal{A}^{\mathsf{ISOAP}}_{2}$ responds to secret key
queries by using its oracle $\mathcal{O}_{K_{2}}$ in a similar manner to
$\mathcal{A}^{\mathsf{ISOAP}}$. Finally, it outputs $t\oplus b^{\prime}$ where
$b^{\prime}$ is $\mathcal{B}$’s guess. Now let $b$ be the bit chosen by the
$\mathbf{\mathsf{ISOAP}}$ challenger. If $b=0$, then $c^{\prime}$ is
indistinguishable from a correctly distributed encryption of $m_{t}$. It is
indistinguishable due to property 3 in Definition 0.A since
$c^{\prime}\in\hat{\mathcal{C}}_{f^{\ast}}$ whereas a legally-generated
ciphertext lies within $\mathcal{C}_{f^{\ast}}$. Denote $\mathcal{B}$’s
advantage distinguishing both cases by
$\mathbf{Adv}_{\mathcal{B},\mathcal{E}}^{\mathbf{\mathsf{IND-CT}}}$. If $b=1$,
then $c^{\prime}$ is an encryption of a random element of $M$, which contains
no information about $t$, forcing $\mathcal{B}$’s advantage to zero.
Therefore, the overall advantage of $\mathcal{A}^{\mathsf{ISOAP}}$ is
$\mathbf{Adv}_{\mathcal{B},\mathcal{E}}^{\mathbf{\mathsf{IND-
CT}}}+\frac{1}{2}\epsilon$.
Now we prove the reverse direction. Suppose that $\mathcal{E}$ is IND-AD-CCA1
secure. Assume for the purpose of contradiction that there is an adversary
$\mathcal{A}^{\mathsf{ISOAP}}=(\mathcal{A}^{\mathsf{ISOAP}}_{1},\mathcal{A}^{\mathsf{ISOAP}}_{2})$
whose advantage is $\alpha$ against $\mathbf{\mathsf{ISOAP}}$. We can use
$\mathcal{A}^{\mathsf{ISOAP}}$ to construct an adversary $\mathcal{B}$ to
attack the IND-AD-CCA1 security of $\mathcal{E}$. Firstly, $\mathcal{B}$
derives $\hat{\mathcal{C}}$, $\mathcal{F}$ and $\mathbb{G}$ from $\mathsf{PP}$
and passes them to $\mathcal{A}^{\mathsf{ISOAP}}_{1}$. It simulates
$\mathcal{O}_{K_{1}}$ by forwarding a query for $f$ to its secret key oracle
and responding with a description of $\sigma_{f}$ derived from the secret key
$\mathsf{sk}_{f}$ it receives. It simulates a query to
$\mathcal{O}_{\mathsf{SP}}$ for $(f,c)$ by (1). querying its decryption oracle
for $c$ to obtain $m^{\prime}$; (2). computing
$r\leftarrow\psi_{f}(m^{\prime})$ and $n\leftarrow r^{-1}\ast c$; and (3).
responding with $(r,n)$.
Let $f^{\ast}$ be the target index outputted by
$\mathcal{A}^{\mathsf{ISOAP}}_{1}$. Subsequently, $\mathcal{B}$ chooses an
attribute $a^{\ast}\in\mathsf{supp}(f^{\ast})$ and forwards $a^{\ast}$ as its
challenge attribute. Furthermore, it chooses messages
$m_{0},m_{1}\xleftarrow{\$}M$ and forwards them to its challenger who responds
with a challenge ciphertext $c^{\ast}$. Next $\mathcal{B}$ computes
$c^{\prime}\leftarrow c^{\ast}\ast E(a^{\ast},m_{0})^{-1}\ast n^{\prime}$
where $n^{\prime}\xleftarrow{\$}\mathcal{N}_{f^{\ast}}$ and hands $c^{\prime}$
to $\mathcal{A}^{\mathsf{ISOAP}}_{2}$. Let $b^{\prime}$ be the bit guessed by
$\mathcal{A}^{\mathsf{ISOAP}}$. Then $\mathcal{B}$ outputs $b^{\prime}$ as its
guess. Recall how a ciphertext is generated by the IND-AD-CCA1 challenger.
Firstly, the challenger samples bit $t\xleftarrow{\$}\\{0,1\\}$. Then the
challenger computes $f\leftarrow\tau(a)$ and sets
$c^{\ast}:=\psi_{f}(m_{t})\ast n$ where $n\xleftarrow{\$}\mathcal{N}_{f}$.It
follows by definition of $\tau$ that
$\mathcal{N}_{f}\subseteq\mathcal{N}_{f^{\ast}}$. This immediately implies
that $\mathcal{R}_{f}=\mathcal{R}_{f^{\ast}}$. If $t=0$, then $c^{\prime}$ is
distributed according to a uniformly random element from
$\mathcal{N}_{f^{\ast}}$, which results in an advantage of $\frac{1}{2}\alpha$
provided that $\mathcal{A}^{\mathsf{ISOAP}}_{2}$ cannot distinguish between
$\mathcal{C}_{f}$ and $\hat{\mathcal{C}_{f}}$ (property 3 of Definition 0.A).
If $t=1$, then $c^{\prime}$ is a uniform in $\hat{\mathcal{C}}_{f}$, and the
advantage of $\mathcal{B}$ in this case is also $\frac{1}{2}\alpha$ provided
that $\mathcal{A}^{\mathsf{ISOAP}}_{2}$ cannot distinguish between
$\mathcal{C}_{f}$ and $\hat{\mathcal{C}_{f}}$. The overall advantage is
therefore
$\mathbf{Adv}_{\mathcal{A}^{\mathsf{ISOAP}}_{2},\mathcal{E}}^{\mathbf{\mathsf{IND-
CT}}}+\alpha$. ∎
## Appendix 0.B Correctness Condition for a Universal Anonymizer
Let $\mathcal{E}$ be a H(PE) scheme with public index, and let
$U_{\mathcal{E}}:=(\mathcal{G},\mathcal{B},\mathcal{B}^{-1},Q_{A},Q_{\mathcal{F}})$
be a universal anonymizer for $\mathcal{E}$. Define the distributions
$D_{1}:=\\{(\mathsf{PP},\mathsf{MSK},\mathsf{params},c)\mid(\mathsf{PP},\mathsf{MSK})\leftarrow\mathbf{\mathsf{\mathcal{E}.Setup}}(1^{\lambda}),\mathsf{params}\leftarrow\mathcal{G}(\mathsf{PP}),c\xleftarrow{\$}\mathcal{C}\\}$
and
$D_{2}:=\\{(\mathsf{PP},\mathsf{MSK},\mathsf{params},c^{\prime})\mid(\mathsf{PP},\mathsf{MSK})\leftarrow\mathbf{\mathsf{\mathcal{E}.Setup}}(1^{\lambda}),\mathsf{params}\leftarrow\mathcal{G}(\mathsf{PP}),c\xleftarrow{\$}\mathcal{C},c^{\prime}\leftarrow\mathcal{B}^{-1}(Q_{A}(\mathsf{attr}(c)),\mathcal{B}(c))\\}$
where $\mathsf{attr}(c)$ returns the attribute associated with $c$. The
correctness condition for a universal anonymizer $U_{\mathcal{E}}$ is that
$D_{1}\underset{C}{\approx}D_{2}$ (computationally indistinguishability).
## Appendix 0.C Instantiation of a Universal Anonymizer for Main Construction
The techniques from [25] can be employed to construct a universal anonymizer
for $\mathbf{\mathsf{xhIBE}}$. In this paper, the basic version of their
construction is adapted.
Let $L(\lambda)$ be the maximum bit-length of identities in
$\mathbf{\mathsf{xhIBE}}$. A universal anonymizer
$\mathsf{AG}_{\mathbf{\mathsf{xhIBE}}}:=$
$(\mathbf{\mathsf{\mathsf{AG}_{\mathbf{\mathsf{xhIBE}}}.\mathcal{G}}},\mathbf{\mathsf{\mathsf{AG}_{\mathbf{\mathsf{xhIBE}}}.\mathcal{B}}},\mathbf{\mathsf{\mathsf{AG}_{\mathbf{\mathsf{xhIBE}}}.{\mathcal{B}^{-1}}}},Q_{A}:=H,Q_{f}:=f_{\mathbf{id}}\mapsto
H(\mathbf{id}))$
for $\mathbf{\mathsf{xhIBE}}$ based on the techniques of Ateniese and Gasti is
given as follows:
Let $\mathsf{Geom}(p)$ be a geometric distribution with parameter $p$.
$m\leftarrow\lambda$ $\triangleright$ $\lambda$ can be derived from
$\mathsf{PP}$
$\mathsf{params}:=(m,\lg{N})$ $\triangleright$ (length of members of
$\hat{\mathcal{C}}$ is $2(m+1)\cdot\lg{N}$ bits, length of $\alpha$)
return $\mathsf{params}$
Algorithm 1
$\mathbf{\mathsf{\mathsf{AG}_{\mathbf{\mathsf{xhIBE}}}.\mathcal{G}}}(\mathsf{PP})$
Parse $\mathsf{params}$ as $(m,L)$
Parse $\mathbf{\psi}$ as $(c(x),d(x),a)$
$k_{1},k_{2}\xleftarrow{\$}\mathsf{Geom}(\frac{1}{2})$
$k_{1}\leftarrow\mathsf{min}(k_{1},m)$.
$k_{2}\leftarrow\mathsf{min}(k_{2},m)$.
$t(x),v(x)\xleftarrow{\$}\mathbb{Z}_{N}[x]$
$z_{1}(x)\leftarrow c(x)+t(x)$
$z_{2}(x)\leftarrow d(x)+v(x)$
for $1\leq i<k_{1}$ do
repeat
$t_{i}(x)\xleftarrow{\$}\mathbb{Z}_{N}[x]$
until $\mathsf{GT}(a,z_{1}(x)-t_{i}(x),N)=-1$
end for
$t_{k_{1}}\leftarrow t(x)$
for $1\leq i<k_{2}$ do
repeat
$v_{i}(x)\xleftarrow{\$}\mathbb{Z}_{N}[x]$
until $\mathsf{GT}(-a,z_{2}(x)-v_{i}(x),N)=-1$
end for
$v_{k_{2}}\leftarrow v(x)$
for $k_{1}<i\leq m$ do
$t_{i}(x)\xleftarrow{\$}\mathbb{Z}_{N}[x]$
end for
for $k_{2}<i\leq m$ do
$v_{i}(x)\xleftarrow{\$}\mathbb{Z}_{N}[x]$
end for
return
$\mathbf{\hat{\psi}}:=((z_{1}(x),t_{1}(x),\ldots,t_{m}(x)),(z_{2}(x),v_{1}(x),\ldots,v_{m}(x)))\in\mathbb{Z}_{N}[x]^{2m+2}$
Algorithm 2
$\mathbf{\mathsf{\mathsf{AG}_{\mathbf{\mathsf{xhIBE}}}.\mathcal{B}}}(\mathsf{params},\mathbf{\psi})$
Parse $\mathsf{params}$ as $(m,L)$
Parse $\mathbf{\hat{\psi}}$ as
$((z_{1}(x),t_{1}(x),\ldots,t_{m}(x)),(z_{2}(x),v_{1}(x),\ldots,v_{m}(x)))$
$i\leftarrow 1$
while $\mathsf{GT}(a,t_{i}(x)-z_{1}(x),N)\neq 1$ do
$i\leftarrow i+1$
end while
$c(x)\leftarrow t_{i}(x)-z_{1}(x)$
$i\leftarrow 1$
while $\mathsf{GT}(-a,v_{i}(x)-z_{2}(x),N)\neq 1$ do
$i\leftarrow i+1$
end while
$d(x)\leftarrow v_{i}(x)-z_{2}(x)$
return $(c(x),d(x),a)$
Algorithm 3
$\mathbf{\mathsf{\mathsf{AG}_{\mathbf{\mathsf{xhIBE}}}.\mathcal{B}}}^{-1}(\mathsf{params},a,\mathbf{\hat{\psi}})$
Let the set of valid ciphertexts $\mathcal{C}$ be defined as
$\\{(c(x),d(x),a)\in\mathbb{Z}_{N}[x]^{2}\times\mathbb{Z}_{N}:c(x)\in
G_{a},d(x)\in G_{-a}\\}$. Then for any
$(\mathsf{PP},\mathsf{MSK})\leftarrow\mathbf{\mathsf{xhIBE.Setup}}(1^{\lambda})$
and
$\mathsf{params}\leftarrow\mathbf{\mathsf{\mathsf{AG}_{\mathbf{\mathsf{xhIBE}}}.\mathcal{G}}}(\mathsf{PP})$:
the correctness condition in Appendix 0.B is trivially satisfied since
$\forall\mathbf{\psi}:=(c(x),d(x),a)\in\mathcal{C}$
$\mathbf{\psi}=\mathbf{\mathsf{\mathsf{AG}_{\mathbf{\mathsf{xhIBE}}}.\mathcal{B}}}^{-1}(a,\mathbf{\mathsf{\mathsf{AG}_{\mathbf{\mathsf{xhIBE}}}.\mathcal{B}}}(\mathbf{\psi}))$
We can apply the transformation
$\mathbf{\mathsf{xhIBE}}^{\prime}\leftarrow
T_{\mathsf{AG}_{\mathbf{\mathsf{xhIBE}}}}(\mathbf{\mathsf{xhIBE}})$
described in the last section to obtain a scheme
$\mathbf{\mathsf{xhIBE}}^{\prime}$. The scheme in [25] is shown to satisfy a
security definition (ANON-IND-ID-CPA) in the random oracle model that is
stronger than the attribute-hiding definition for IBE in the random oracle
model. It can be easily shown with the help of Corollary 2 that
$\mathbf{\mathsf{xhIBE}}^{\prime}$ is an attribute-hiding
$\mathsf{HPE}_{\bar{U}}$ scheme for the IBE functionality supporting the group
homomorphism $(\mathbb{Z}_{2},+)$.
|
arxiv-papers
| 2013-02-05T20:58:46 |
2024-09-04T02:49:41.327357
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Michael Clear, Arthur Hughes, Hitesh Tewari",
"submitter": "Michael Clear",
"url": "https://arxiv.org/abs/1302.1192"
}
|
1302.1206
|
# Thermality and Heat Content of horizons from infinitesimal coordinate
transformations
Bibhas Ranjan Majhi and T. Padmanabhan
IUCAA, Pune University Campus,
Ganeshkhind, Pune- 411 007.
email: [email protected]; email: [email protected]
###### Abstract
Thermal properties of a static horizon, (like the entropy $S$, heat content
$TS$ etc.) can be obtained either from the surface term of the Einstein-
Hilbert action or by evaluating the Noether charge, corresponding to the
diffeomorphisms generated by the timelike Killing vector field. We show that,
for a wide class of geometries, the same results can be obtained using the
vector field which produces an infinitesimal coordinate transformation between
two physically relevant reference frames, viz. the freely falling frame near
the horizon and the static, accelerated, frame. In particular, the
infinitesimal coordinate transformation from inertial coordinates to uniformly
accelerated frame can be used to obtain the heat content and entropy of the
Rindler horizon. This result offers insight into understanding the observer
dependent degrees of freedom which contribute to the entropy of null surfaces.
## 1 Introduction
Classical general relativity is built on the principle of general covariance
which denies special status to any particular class of coordinate systems or
associated observers. Phenomena which are specifically coordinate dependent
are usually treated with suspicion by classical relativists. The situation,
however, is different when one brings in quantum effects through the study of
quantum field theory in non-trivial background spacetimes. The pioneering work
by Davies and Unruh [1] showed that an accelerating observer will attribute a
non-zero temperature to the quantum state, interpreted as a zero-temperature
vacuum state by the inertial observers. Similarly the thermal phenomenon
associated with a black hole will also depend on the observer; while a
stationary observer outside the event horizon will attribute an entropy and
temperature to the event horizon, another observer freely falling through the
event horizon will perceive the physics quite differently. It appears
inevitable that these results require one to attribute an observer dependence
to all thermal phenomena in nature. (For more details on this point of view,
see e.g.,[2, 3]).
The two physical contexts mentioned above — the black hole horizon and the
Rindler horizon — are, in fact, related to each other in a fairly simple
manner. The metric near the event horizon of a massive ($M\gg M_{Planck}$)
black hole can be approximated by the metric of the accelerated (Rindler)
frame in flat spacetime. The thermal phenomena perceived by observers close to
the event horizon can then be mapped to those perceived by Rindler observers
in flat spacetime and one obtains the correct expression for the temperature
by this mapping. In this correspondence, the freely falling observers through
the event horizon will be analogous to inertial observers in flat spacetime
while the stationary observers outside the event horizon will be similar to
uniformly accelerated observers in flat spacetime. Given the fact that
stationary observers in black hole spacetime attribute an entropy density
(entropy per unit area) of 1/4 in Planck units to the event horizon, it seems
inevitable that accelerated observers in flat spacetime will also attribute an
entropy density of 1/4 to the Rindler horizon. The quasi-local nature of
physics implies that any null surface which acts as a local Rindler horizon to
a class of observers should have an observer dependent entropy density of 1/4
vis-a-vis this class of observers.
While the above results do not seem to lead to any clear-cut paradoxes (like
violation of conservation laws or perpetual motion) they do appear somewhat
perplexing and counter-intuitive. In non-gravitational physics, we are
accustomed to thinking of entropy as an observer-independent quantity being
related to certain unobserved degrees of freedom. It is, therefore tempting to
think of the entropy of a black hole as being absolute and is observer
independent and related to some specific degrees of freedom. This, however,
cannot be true in view of the argument given above, if the quasi-local nature
of physics has to be maintained. The fact that an observer freely falling
through the black hole event horizon has access to more spacetime region than
the observer eternally confined to the outside of the black hole shows how
such an observer dependence can indeed arise in this context. While this is
intuitively understandable, it would be nice to see how this effect arises in
the standard procedures we adopt for the computation of the entropy of the
horizons. We will illustrate one class of such examples in this paper.
A possible mechanism behind the origin of observer dependent gravitational
degrees of freedom was suggested in Ref.[4] based on the approach pioneered by
Carlip [[5]; also see [6]]. The standard description of gravity is invariant
under the set $\mathcal{C}$ of all possible diffeomorphisms. These
diffeomorphisms allow us, in principle, to remove all the gauge degrees of
freedom retaining only the diffeomorphism invariant physical degrees of
freedom. When we limit our attention to a set of observers who perceive a null
surface as a horizon, the physical context gets restricted to a situation in
which only those diffeomorphisms in the set $\mathcal{C}^{\prime}$, which
retain the horizon structure, are allowed. While it is a difficult (and
unsolved) problem to precisely quantify what this restricted class
$\mathcal{C}^{\prime}$ is, it seems reasonable to assume that
$\mathcal{C}^{\prime}$ is a proper subset of $\mathcal{C}$. This, in turn,
implies that using the diffeomorphisms in $\mathcal{C}^{\prime}$, we cannot
remove all the redundant gravitational degrees of freedom which we could have
originally removed using the full set $\mathcal{C}$. In other words, certain
degrees of freedom which would have been treated as pure gauge (when the
theory was invariant under $\mathcal{C}$) now gets ‘upgraded’ to physical
degrees of freedom (when the theory is invariant under
$\mathcal{C}^{\prime}$). The entropy attributed to the null surface, by the
observers who perceive it as a horizon, could arise from these degrees of
freedom. (See [7] for related ideas in a different context.)
The above description is purely intuitive and mathematically ill-defined. To
make progress towards a somewhat concrete realization of the above ideas and
test their conceptual veracity, we will investigate the following issue: In a
flat spacetime, the inertial observers do not attribute any thermal properties
to any null surface. Consider now an infinitesimal coordinate transformation
$x^{a}\to x^{a}+q^{a}(x)$ from the inertial coordinate system to the Rindler
coordinate system. We then ask: Is it possible to obtain the thermal
properties, associated with a null surface in flat spacetime, in terms of the
vector field $q^{a}$? We will show that this is indeed possible and — in this
sense — one can consider the infinitesimal transformations described by the
vector field $q^{a}$ as having upgraded some of the gauge degrees of freedom
to physical degrees of freedom. We stress that $q^{a}(x)$ is not the usual
Killing vector field associated with the Rindler time translation; in fact, we
are not aware of any previous discussion in the literature of this particular
vector field. The same ideas work in a much wider class of spacetimes with
horizons, when we consider the infinitesimal coordinate transformations
between the freely falling frame near the horizon and the frame of the static
observers (like, for e.g., in terms of the infinitesimal coordinate
transformation from Kruskal to Schwarzschild coordinates) and even in a more
general context.
The paper is organized as follows: In Section 2 we review and rephrase some of
the standard known results to motivate and show how the surface Hamiltonian,
$H_{\textrm{sur}}\equiv TS$ (which represents the heat containt of the
horizon), is related to the surface action and the Noether charge. Next, we
introduce the infinitesimal coordinate transformation from inertial to Rindler
frame and the vector field corresponding to this transformation. In Section 4,
both the surface term of the action and the Noether charge are evaluated in
terms of this vector field and shown to lead to identical expression. We
explain how this arises and provide a discussion of related issues. Section 5
generalizes our findings for a general static, spherically symmetric metric
and describes how a more general spacetime can be handled by this approach. In
the final section we discuss our results.
## 2 Heat content and thermality of horizons
We begin by briefly reviewing some of the standard results rephrasing them in
a useful manner. A convenient physical quantity which captures the thermality
of a any system is the difference between the energy $E$ and the free energy
$F=E-TS$, viz. $TS$. In the case of a spacetime horizon, as we will see, this
heat content can be related [3, 8] to the surface Hamiltonian
$H_{\rm sur}\equiv TS=\frac{\kappa}{2\pi}\ S=\frac{\kappa A_{\perp}}{8\pi}$
(1)
where $S=(A_{\perp}/4)$ is the entropy of the horizon with $A_{\perp}$ being
its area, $T=(\kappa/2\pi)$ is the temperature with $\kappa$ being the
relevant acceleration (Rindler acceleration or the surface gravity). In the
case of non-compact horizons we will interpret $(H_{\rm sur}/A_{\perp})$ as
$Ts$ where $s$ is the entropy density, viz. entropy per unit area. The
acceleration $\kappa$ can be defined through the relation
$\xi^{b}\nabla_{b}\xi^{a}=\kappa\xi^{a}$ in the case of Killing horizons
associated with a time-like Killing vector field $\xi^{a}$, which becomes null
on the horizon surface. Defined in this manner, the numerical value of
$\kappa$ changes if we rescale $\xi^{a}$. This ambiguity is avoided by either
defining the normalization of $\xi^{a}$ through some physical consideration or
by dealing with quantities which are invariant under the rescaling.
The heat content $H_{\rm sur}$ can be computed in two equivalent ways, both of
which capture its physical meaning [3, 8]. The first method is to define
$H_{\rm sur}$ as the Hamiltonian associated with the surface term of the
Einstein-Hilbert action through $H_{\rm sur}=-(\partial\mathcal{A}_{\rm
sur}/\partial t)$. As is well-known, the Einstein-Hilbert Lagrangian can be
written as a sum of a bulk term ($L_{bulk}$, which is quadratic in $\Gamma$)
and a surface term $L_{sur}$ with
$L_{\rm sur}=\frac{1}{16\pi}\partial_{c}(\sqrt{-g}V^{c});\qquad
V^{c}\equiv-\frac{1}{g}\partial_{b}(gg^{bc})$ (2)
(see e.g., eq (6.15) of [9]). The surface action $\mathcal{A}_{\rm sur}$ is
defined as the integral over $L_{sur}$. If the near horizon metric is
approximated as a Rindler metric (with $-g_{00}=1/g_{xx}=N^{2}=2\kappa x$ to
evaluate these expressions on $N=$const surface, we will get:
$\mathcal{A}_{\rm sur}=\frac{1}{16\pi}\int_{x}dtd^{2}x_{\perp}V^{x}=\pm
t\left(\frac{\kappa A_{\perp}}{8\pi}\right)$ (3)
where $A_{\perp}$ is the transverse area. (The sign depends on the convention
chosen for the outward normal or whether the contribution of the integral is
taken at the inner or outer boundaries; see e.g., the discussion in [15]. We
will choose the negative sign in Eq. (3).) More generally, a static, near-
horizon, geometry can be described by the metric [16, 17]
$ds^{2}=-N^{2}dt^{2}+dl^{2}+\sigma_{AB}dx^{A}dx^{B};$ (4)
where $N$ and $\sigma_{AB}$ have the near-horizon behaviour of the form
$N=\kappa l+O(l^{3});\quad\sigma_{AB}=\mu_{AB}(x^{A})+O(l^{2})$ (5)
with $l=0$ surface being the horizon. The integrals in Eq. (3) again leads to
the same result. Near the static horizon, the integrand is independent of $t$
and transverse coordinates and hence $\mathcal{A}_{\rm sur}$ must be
proportional to $tA_{\perp}$. The proportionality constant turns out to be
$\kappa/8\pi$.
With future applications in mind, we stress that the $A_{sur}$ used in Eq. (3)
is not the full surface action but only the contribution of it on the horizon.
This is relevant when $L_{sur}\propto\partial_{c}(\sqrt{-g}V^{c})$ vanishes
because $\sqrt{h}n_{c}V^{c}=$ constant. If we define the action as the
integral of $L_{sur}$ over a spacetime domain $\mathcal{V}$, the action can be
expressed as the surface integral over the boundary $\partial\mathcal{V}$ of
this domain and this will also vanish. But the contribution of the surface
integral from part of the boundary surface (like the horizon, say) will be
given by the integral of $\sqrt{h}n_{c}V^{c}$ over that part of the surface
and need not vanish. It is this contribution, computed over the horizon, which
we have defined as $A_{sur}$ and this can be nonzero even when
$L_{sur}\propto\partial_{c}(\sqrt{-g}V^{c})=0$ provided $V^{c}$ itself is not
zero. We will have occasion to consider such a case later on.
Given $A_{sur}$, the surface Hamiltonian is defined in a standard manner as:
$H_{sur}\equiv-\frac{\partial\mathcal{A}_{sur}}{\partial
t}=\frac{1}{16\pi}\int_{x}d^{2}x_{\perp}V^{x}=\left(\frac{\kappa
A_{\perp}}{8\pi}\right)=TS$ (6)
with suitable choice of sign. The $H_{sur}$ is the difference between the
energy $E$ and the free energy $F=E-TS$ of a finite temperature system and
measures the heat content of the horizon.
The second method [3, 8] to compute the heat content is by using the Noether
potential. In any gravitational theory which is invariant under the
diffeomorphism $x^{a}\to x^{a}+q^{a}(x)$, one can define a conserved Noether
current $J^{a}$ (which depends on $q^{a}$) related to an anti-symmetric
Noether potential $J^{ab}$ through $J^{a}=\nabla_{b}J^{ab}$. In Einstein’s
theory one can choose the Noether potential to be:
$J_{ab}=\frac{1}{16\pi}\left(\nabla_{a}q_{b}-\nabla_{b}q_{a}\right)=\frac{1}{16\pi}\left(\partial_{a}q_{b}-\partial_{b}q_{a}\right)$
(7)
The surface Hamiltonian can then be defined as the surface integral over a co-
dimension-2 cross-section of the null surface:
$H_{\rm sur}=TS\equiv\int d^{D-2}\Sigma_{ab}\;J^{ab}$ (8)
where the Noether potential is computed for $q^{a}=\xi^{a}$ with $\xi^{a}$
being the local Killing vector corresponding to time translation symmetry of
the local Rindler frame near the horizon. (The connection between Noether
charge and energy has been previously noted in, e.g., Refs. [10]) Note that
both $J_{ab}$ and $T$ rescale in the same manner, if $\xi^{a}$ is rescaled,
leaving the entropy invariant.
While the two ways of computing $H_{\rm sur}$ described above — from the
surface term of the Einstein-Hilbert action in Eq. (6) or from the Noether
potential related to the time-like Killing vector in Eq. (8) — lead to the
same expression for $H_{\rm sur}$, there is a subtle difference between the
two results. The surface term of Einstein-Hilbert action defined through
$L_{sur}$ is not generally covariant. In flat spacetime, $A_{sur}$ will vanish
in inertial coordinates while it can be non-zero in the non-inertial
coordinates. To obtain a non-zero value for $H_{\rm sur}$ from
$\mathcal{A}_{\rm sur}$ using Eq. (6) we need to explicitly work with the
noninertial, Rindler, coordinates in flat spacetime. More generally, in a
curved spacetime, the expression for $\mathcal{A}_{\rm sur}$ can depend on the
coordinate system which is used. The expression for $H_{\rm sur}$ in terms of
the Noether potential in Eq. (8), on the other hand, is manifestly covariant
and will lead to the same result in all coordinate systems. While this may be
surprising at first sight, it should be noted that — while the value of the
integral in Eq. (8) will be the same in all coordinate systems — the physical
interpretation of $\xi^{a}$ used to define the Noether potential is different
in different coordinate systems. In flat spacetime, $\xi^{a}$ will be thought
of as generator of time translation in Rindler frame but it is (proportional
to) the Lorentz boost generator in the inertial frame.
It is worth mentioning at this stage that there is another (York-Gibbons-
Hawking, [11]) surface term which is often discussed in the gravitational
context and is given by
$\mathcal{A}_{\rm
YGH}\equiv\pm\frac{1}{16\pi}\int_{\mathcal{S}}d^{3}x\,\sqrt{h}\,(2K)$ (9)
where $K$ is the extrinsic curvature of the surface $\mathcal{S}$ and $h_{ab}$
is the metric induced on this surface. (The overall sign depends on whether
the surface is spacelike or timelike.) This term is usually added to the
Einstein-Hilbert action to cancel the variations of the metric normal to the
boundary. Contrary to the popular myths: (i) The action $\mathcal{A}_{\rm
YGH}$ does not cancel the surface term $\mathcal{A}_{\rm sur}$ in general.
(ii) Even their variations do not cancel each other for arbitrary variations
of the metric but they do cancel each other if the metric is held fixed on the
boundary [see e.g., pages 249-250 of Ref. [9]]. This property is shared by an
infinite set of surface functionals [12] and the choice of $\mathcal{A}_{\rm
YGH}$ is merely a convenient one to make the variational principle in gravity
well defined. In this sense, $\mathcal{A}_{\rm sur}$ has a better standing as
a well defined surface term arising naturally from the Einstein-Hilbert action
itself than $\mathcal{A}_{\rm YGH}$. Nevertheless, given the popularity
enjoyed by $\mathcal{A}_{\rm YGH}$ in the literature, we will briefly mention
the corresponding results for this action.
It is easy to show that (see, for e.g., Ex. (6.3) of Ref. [9]), in general,
the integrands of the surface integral in $\mathcal{A}_{\rm sur}$ and
$\mathcal{A}_{\rm YGH}$ are related as follows:
$V^{a}n_{a}=2K+2h^{ab}\partial_{b}n_{a}-n^{m}h^{ns}\partial_{n}g_{sm}$ (10)
The second and third terms on the right hand side will not vanish for an
arbitrary surface and coordinates because of which $n_{c}V^{c}$ is obviously
not a scalar even though $2K$ is. However, if the coordinate system is chosen
such that the surface corresponds to, say, $x^{1}=$ constant with the metric
being block diagonal with respect to the $x^{1}$ coordinate (i.e., $g_{1k}=0$
for $k\neq 1$) then it is easy to show that $V^{a}n_{a}=2K$ (see, for e.g.,
the Appendix of Ref. [13]). Because most of the metrics we deal with (and
certainly the Rindler metric which is valid close to the horizon) belongs to
this class, we can use $2K$ and $V^{c}n_{c}$ interchangeably for our purpose.
This implies that, in the Rindler frame, we should get the same answer for
$\mathcal{A}_{\rm YGH}$ and $\mathcal{A}_{\rm sur}$ when evaluated on a
stretched horizon corresponding to $x=\epsilon$ with the limit $\epsilon\to 0$
taken in the end. This is indeed true since in the Rindler frame
$K\sqrt{h}=\kappa$=constant, independent of $\epsilon$, leading to the result:
$\mathcal{A}_{\rm YGH}=-t\left(\frac{\kappa A_{\perp}}{8\pi}\right)$ (11)
So, in the Rindler frame, we can also write the result in Eq. (6) as
$H_{\rm sur}=-\xi^{a}\nabla_{a}\mathcal{A}_{\rm
YGH}=\xi^{a}\nabla_{a}\left[\frac{1}{8\pi}\int_{\mathcal{S}}d^{3}x\,\sqrt{h}\,K\right]$
(12)
This provides a generally covariant expression for the heat content as long as
the expression on the right hand side is well defined. If we repeat the
calculation in the inertial coordinates, it turns out that we obtain the same
result provided: (a) We perform the differentiation in Eq. (12) before taking
the $\epsilon\to 0$ limit. (b) We choose $\xi^{a}$ with components
$\xi^{a}=\kappa(X,T,\mathbf{0})$ in the inertial frame. This is the standard
Killing vector for generation of the Lorentz boosts scaled up by $\kappa$. The
inertial frame has no intrinsic $\kappa$ in it and we can obtain the same
result for $H_{\rm sur}$ in both frames only through this scaling. We will
come back to this issue a little later.
Given the fact that $\mathcal{A}_{\rm sur}=0$ in the inertial coordinates
(because $\Gamma$s vanish), while $\mathcal{A}_{\rm sur}\neq 0$ and is given
by Eq. (3) in the Rindler coordinates, one might wonder what we get for
$\mathcal{A}_{\rm YGH}$ if evaluated in inertial and Rindler frames. As we
said before, we get $\mathcal{A}_{\rm YGH}=\mathcal{A}_{\rm sur}$ if we
evaluate both in the Rindler frame. If we compute $\mathcal{A}_{\rm YGH}$ in
the inertial coordinates taking the surface $\mathcal{S}$ to be given by
$X^{2}-T^{2}=\epsilon^{2}$ then we get the result
$\mathcal{A}_{\rm YGH}=-\frac{A_{\perp}}{8\pi}\,{\rm
tanh}^{-1}\left(\frac{T}{X}\right)\Bigg{|}_{\mathcal{S}}$ (13)
which, of course, matches with the result in Eq. (11) for non-zero $\epsilon$
because ${\rm tanh}^{-1}(T/X)=\kappa t$ ensuring general covariance. But a
naive limit of $\epsilon\to 0$ corresponding to $(T/X)\to\pm 1$ leads to a
divergence in Eq. (13) so that this expression is not well defined when we
take the null surface limit. (One can do slightly better by working in
Euclidean coordinates when one would have got $\tan^{-1}(T_{E}/X)$ instead of
the tanh${}^{-1}(T/X)$. The null surface limit now corresponds to $T_{E}\to
0,X\to 0$ and the results depend on the order in which these limits are taken.
A choice of taking $T_{E}\to 0$ first with $X$ finite will give
$\mathcal{A}_{\rm YGH}=0$ which will match with $\mathcal{A}_{\rm sur}=0$ in
inertial coordinates but not with Eq. (11).) Thus we find that
$\mathcal{A}_{\rm YGH}$ is not well defined in the null surface limit when
evaluated in the inertial coordinates even though the heat content, obtained
after the differentiation is carried out in Eq. (12), remains well defined in
both coordinates. Because of these issues and the comments made earlier about
the ad-hoc nature of $\mathcal{A}_{\rm YGH}$, we will not use it further in
our discussion.
We will next consider the corresponding aspects in the case of $H_{\rm sur}$
evaluated from Eq. (8) by explicitly computing it from Eq. (8) in flat
spacetime in both Rindler and inertial coordinates. In the local Rindler frame
with $-g_{00}=1/g_{xx}=N^{2}=2\kappa x$ if we take
$\xi^{a}=\delta^{a}_{0},\quad
d^{2}\Sigma^{ab}=\frac{1}{2}(n^{a}m^{b}-n^{b}m^{a})\sqrt{\sigma}\,d^{2}x_{\perp},$
(14)
where $n^{a}=(2\kappa x)^{1/2}\,\delta^{a}_{1}$, $m^{a}=-(2\kappa
x)^{-1/2}\delta^{a}_{0}$, then a straightforward calculation gives
$H_{\rm sur}=\int_{\mathcal{H}}d^{2}\Sigma_{ab}\,J^{ab}=\frac{1}{16\pi}\int
d^{2}x_{\perp}\,\sqrt{\sigma}\,(2\kappa)=\frac{\kappa A_{\perp}}{8\pi}$ (15)
The above calculation was done in the Rindler coordinates which has the
parameter $\kappa$ in the metric. But since the integral defining $H_{\rm
sur}$ is generally covariant we could have evaluated the same in the inertial
coordinates as well. In the inertial coordinates ($T,X,\mathbf{X}_{\perp}$),
the components of $\xi^{a}$ will be $\kappa(X,T,\mathbf{0})$ which represents
the Lorentz boost vector with an arbitrary scaling factor $\kappa$. This
vector becomes null on $X=T$ and hence $\xi^{b}\nabla_{b}\,\xi^{a}$ must be
proportional to $\xi^{a}$ on this surface; that is,
$\xi^{b}\nabla_{b}\,\xi^{a}=\lambda\xi^{a}$ on $X=T$. An elementary
calculation shows that $\lambda=\kappa$ which can be thought of as the non-
affinity parameter associated with the generators of the null surface $X=T$.
(Obviously there is no thermal interpretation for $\kappa$ in the inertial
coordinates.) The relevant non-zero component of the Noether potential in the
inertial frame is $J_{TX}=2\kappa$ while the measure for transverse
integration is just $d\Sigma^{TX}=\frac{1}{2}\sqrt{\sigma}d^{2}x_{\perp}$. The
integral in Eq. (8) now becomes
$H_{\rm
sur}=\frac{1}{16\pi}\,\int_{\mathcal{H}}d^{2}x_{\perp}\,\sqrt{\sigma}\,(2\kappa)=\frac{\kappa
A_{\perp}}{8\pi}$ (16)
Thus we get the same result, as we should, when we use the same vector
$\xi^{a}$ in either coordinate system. If we had known that non-affinity
parameter $\kappa$ has the interpretation as horizon temperature in the
Rindler frame, we could have done the entire computation in the inertial
coordinate itself. (In Section 4 we will describe a similar procedure but with
a different vector field.)
There is a related issue, regarding the dimensions of $\xi^{a}$, which is
worth noting. If we think of the coordinates $x^{a}$ having the dimensions of
length, then it is natural to assume that the deformation vector $q^{a}$
appearing in the diffeomorphism $x^{a}\to x^{a}+q^{a}$ also has the dimensions
of length. But it is conventional to take the time-like Killing vector
$\xi^{a}$ in static spacetimes to be dimensionless with
$\xi^{a}=\delta^{a}_{0}$ as we did in the above calculation. In that case, one
cannot take $q^{a}=\xi^{a}$ unless we change the dimensions of either one of
them. In the inertial coordinates it is more natural to work with
$\chi^{a}=\xi^{a}/\kappa=(X,T,\mathbf{0})$ (which is the usual Killing vector
corresponding to Lorentz boosts with the dimension of length). This is
equivalent to setting $\kappa=1$ and the non-affinity parameter for the null
congruence generating the $X=T$ surface becomes unity; that is,
$\xi^{b}\nabla_{b}\,\xi^{a}=\xi^{a}$ on $X=T$. This rescales $H_{\rm sur}$ but
note that the corresponding Rindler temperature is now just $(1/2\pi)$. The
simultaneous rescaling of $H_{\rm sur}$ and $\kappa$ keeps the entropy $S=2\pi
H_{\rm sur}/\kappa=A_{\perp}/4$ invariant.
Based on these results, we can give two prescriptions for computing the
entropy. The first one is from the surface term in the action:
$S=A_{\rm sur}\Big{|}_{t=2\pi/\kappa}$ (17)
which of course has a natural interpretation in the Euclidean Rindler sector
since the periodicity of Euclidean time is $2\pi/\kappa$; but as a
prescription it can be used even with Lorentzian Rindler time. The second
prescription is essentially that of Wald entropy [14]:
$S=\frac{2\pi}{\kappa}\,\int d^{D-2}\Sigma_{ab}\;J^{ab}$ (18)
In terms of $H_{\rm sur}$ the two definitions given above are equivalent to
$S=\int_{0}^{2\pi/\kappa}dt\,H_{\rm sur}=\frac{2\pi}{\kappa}\,H_{\rm sur}$
(19)
This shows that, in either way, the central quantity which captures the
thermality of the null surface is its heat content, $H_{\rm
sur}=(\kappa/2\pi)S$.
From the heat content $H_{sur}$ of the horizon we can obtain the heat energy
per unit area of the horizon, $H_{sur}/A_{\perp}=\kappa/8\pi=P$, which also
appears as the pressure term in the Navier-Stokes equation obtained by
projecting Einstein’s equation on to the null surface [18, 19] and leads to
the equation of state $PA_{\perp}=TS$. This heat energy density on the
stretched horizon at $x^{1}=const$ with $n_{c}=\delta_{c}^{1}$ is
$\mathcal{H}=\frac{NK}{8\pi}=\frac{1}{16\pi}\sqrt{-g}V^{c}n_{c}=-\frac{1}{16\pi}\,n_{c}(f^{ab}N^{c}_{ab})$
(20)
where $f^{ab}=\sqrt{-g}\,g^{ab}$ and $N^{c}_{ab}$ is the corresponding
conjugate momentum defined by
$N^{i}_{jk}\equiv\frac{\partial(\sqrt{-g}L_{bulk})}{\partial(\partial_{i}f^{jk})}=-[\Gamma^{i}_{jk}-\frac{1}{2}(\delta^{i}_{j}\Gamma^{a}_{ka}+\delta^{i}_{k}\Gamma^{a}_{ja})].$
(21)
Closing the loop, one can express the surface term in the Hilbert action in
terms of the heat energy density of the horizon as:
$A_{sur}=\frac{1}{16\pi}\int dt\int d^{2}x\ n_{c}(f^{ab}N^{c}_{ab})=-\int
dt\int d^{2}x\mathcal{H}$ (22)
These results show that the heat content is directly related to gravitational
degrees of freedom $f^{ab}$ and their conjugate momenta $N^{c}_{ab}$. When
expressed in terms of $N^{c}_{ab}$ the expression is not generally covariant;
the $N^{a}_{bc}$ vanishes in the inertial coordinates but not in the Rindler
coordinates.
We conclude this section with a brief mention of the physical importance of
$H_{sur}$, which also arises in the study of black hole horizons [15] and is
closely related to the phase of the semiclassical wave function of the black
hole. The phase of the semiclassical wave function describing a geometry with
a horizon will pick up a term from the surface term in the action given by:
$\Psi\propto\exp(iA_{\rm sur})=\exp\left(-i\int dtH_{\rm sur}\right)$ (23)
When a semiclassical black hole interacts with matter fields, the probability
for its area to change by $\Delta A_{\perp}$ is determined by the integral of
the form
$\mathcal{P}(\Delta A_{\perp})=\int_{-\infty}^{\infty}dtF_{m}(t)\exp[-it\Delta
H_{sur}]=\int_{-\infty}^{\infty}dtF_{m}(t)\exp[-it\frac{\kappa}{8\pi}\Delta
A_{\perp}]$ (24)
where $F_{m}(t)$ is a suitable matter variable. The exponential redshift near
the horizon will result in the time evolution of $F_{m}(t)$ having the
asymptotic form $\exp[-iC\exp(-\kappa t)]$ where $C$ is a constant. An
elementary calculation now shows that the relative probability for the horizon
to change its area by $\Delta A_{\perp}$ is given by $\exp[\Delta
A_{\perp}/4]$.
## 3 Infinitesimal coordinate transformation from the inertial to the Rindler
frame
The calculation of $H_{\rm sur}=TS$ in the Rindler frame, performed in the
last section, used the full non-linear transformation between the inertial and
Rindler coordinates in flat spacetime (or between the freely falling and
static coordinate systems near a horizon in a curved spacetime). The
calculation based on the Noether potential also used the Killing vector field
associated with the (local) Rindler frame. These results bring to focus the
issue we raised in Sec. 1 viz., how certain degrees of freedom contributing to
the heat content of a null surface arises when we switch from one coordinate
system to another. Because degrees of freedom are closely associated with the
symmetries of the theories — which in this case are infinitesimal coordinate
transformations — it would be nice to see whether the results in Sec. 2 can be
obtained using only infinitesimal coordinate transformations. As a preliminary
to this investigation in Sec.4, we will now set up the necessary
transformation.
Let us begin with a flat spacetime described in the inertial coordinates
($T,X,\mathbf{X}_{\perp}$) with the metric
$ds^{2}=-dT^{2}+dX^{2}+dL_{\perp}^{2}~{},$ (25)
where $dL_{\perp}^{2}$ denotes the flat transverse metric which will not play
any significant role in the discussion. The usual coordinate transformation
from the inertial coordinates ($T,X,\mathbf{X}_{\perp}$) to the Rindler
coordinates $(t,x,\mathbf{X}_{\perp})$ can be taken to be
$\displaystyle T=\kappa^{-1}(1+2\kappa x)^{1/2}\sinh(\kappa t);\quad
X=\kappa^{-1}(1+2\kappa x)^{1/2}\cosh(\kappa t)-\kappa^{-1}.$ (26)
leading to the metric:
$\displaystyle ds^{2}=-(1+2\kappa x)dt^{2}+\frac{dx^{2}}{1+2\kappa
x}+dL_{\perp}^{2}~{}.$ (27)
These transformations are structured such that they reduce to identity when
the acceleration $\kappa=0$ making the metric in Eq. (27) reduce to the metric
in Eq. (25).
We are interested in the infinitesimal form of these coordinate
transformations which we will define as the ones obtained by retaining only
terms up to linear order in $\kappa$ in Eq. (26). It is easy to verify that,
to this order, the transformations in Eq. (26) become:
$T=t+\kappa xt+{\cal{O}}(\kappa^{2});\,\,\,\ X=x+\frac{1}{2}\kappa
t^{2}-\frac{1}{2}\kappa x^{2}+{\cal{O}}(\kappa^{2})~{}.$ (28)
The inverse transformations are:
$t=T-\kappa XT+{\cal{O}}(\kappa^{2});\,\,\,\ x=X-\frac{1}{2}\kappa
T^{2}+\frac{1}{2}\kappa X^{2}+{\cal{O}}(\kappa^{2})~{}.$ (29)
(Since $\kappa$ is a dimensionfull parameter, the smallness or otherwise of
$\kappa$ is somewhat ill-defined. To handle this, we can replace $\kappa$ by
$\epsilon\kappa$ in all expressions where $\epsilon$ is a dimensionless
infinitesimal parameter and do the Taylor series expansion in $\epsilon$. We
will not bother to do this since it clutters up the notation and ultimately
leads to the same results.) In general, an infinitesimal transformation
between two coordinate systems $x^{a}\to\bar{x}^{a}=x^{a}+q^{a}(x)$ is
implemented by a vector field $q^{a}(x)$ in the spacetime. (Once again,
$q^{a}$ should be thought of as $\epsilon q^{a}$ with all expansions treated
as Taylor series in $\epsilon$.) In the case of infinitesimal transformation
from the inertial to Rindler coordinate system the vector field $q^{a}\equiv
x^{a}-X^{a}$ has the components in the inertial frame given by:
$\displaystyle q^{T}=t-T=-\kappa XT;$ $\displaystyle
q^{X}=x-X=-\frac{1}{2}\kappa T^{2}+\frac{1}{2}\kappa X^{2}~{}.$ (30)
It can be easily verified that $q^{a}$ is not a Killing vector; its integral
curves are given by
$\displaystyle X^{2}T-\frac{T^{3}}{3}=C~{},$ (31)
where $C$ is a constant (In fact, as far as we know, this vector field has not
been studied in the literature).
To avoid possible misunderstanding, we emphasize again how the vector field
$q^{a}$ is related to the transformation from inertial to Rindler coordinate
system. This is important because, this transformation in Eq. (26) does not
reduce to identity at infinity; similarly, the Rindler metric in Eq. (27) is
not asymptotically flat. So one cannot obtain an infinitesimal version of the
transformations at asymptotic infinity. Instead, we imagine a family of
Rindler transformations, parametrized by a dimensionless parameter $\epsilon$
with an acceleration $\epsilon\kappa$. When we vary the parameter $\epsilon$
from zero to unity, we move from identity transformation (at $\epsilon=0$) to
the final Rindler transformation (at $\epsilon=1$). Now we can consider
transformations in this family, close to the identity by doing a Taylor series
expansion of Eq. (26) in the parameter $\epsilon$. To the linear order in
$\epsilon$ we will get $x^{a}=X^{a}+\epsilon q^{a}$ which defines the vector
field $q^{a}$ we work with. Once this exercise is carried out we can set
$\epsilon$ to unity. As mentioned earlier, it is obvious that — since
$\epsilon$ occurs only in the combination $\epsilon\kappa$ — the same $q^{a}$
is obtained by simply expanding the transformation in Eq. (26) in a Taylor
series in $\kappa$. This procedure, in fact, is quite general. If we consider
any finite coordinate transformation $\bar{x}^{a}=f^{a}(x^{i},\lambda)$ where
$\lambda$ is a (in general, dimensionfull) parameter defined such that we get
identity transformation when $\lambda=0$, then by considering the family of
transformations $\bar{x}^{a}=f^{a}(x^{i},\epsilon\lambda)$ we can smoothly
interpolate between identity transformation and finite transformation when
$\epsilon$ varies between 0 and 1. Expanding
$\bar{x}^{a}=f^{a}(x^{i},\epsilon\lambda)$ in a Taylor series in $\epsilon$ we
will get $\bar{x}^{a}=x^{a}+\epsilon q^{a}$ which identifies the vector field
$q^{a}$.
## 4 Thermality from the infinitesimal coordinate transformation
We will now relate the infinitesimal coordinate transformations between
inertial and Rindler coordinates to the thermality of the null surface $X=T$.
This is probably the most instructive example because it shows such an effect
can arise even in the flat spacetime. The idea generalizes in a straight
forward manner to all static horizons and we will comment on these
generalizations later on.
How does the surface $X=T$ acquire an interpretation in terms of a heat
content $H_{\rm sur}$ when we made a coordinate transformation from inertial
to Rindler frame? Of the two methods described in Sec.2 for computing $H_{\rm
sur}$, the one based on surface term in the action is conceptually simpler to
understand, which we will discuss first.
We know that neither the bulk nor the surface term of the Einstein-Hilbert
action is generally covariant individually though, of course, $R=0$ in all
coordinate systems. In the inertial coordinates $\Gamma$s vanish making
$A_{\rm sur}=0=A_{\rm bulk}$ and hence $H_{\rm sur}=0$; so we cannot attribute
any heat content to the null surface $T=X$. But since $A_{\rm sur}$ is not
generally covariant, it acquires non-zero value under infinitesimal coordinate
transformations $x^{a}\to x^{a}+q^{a}(x)$. The connection generated to the
lowest order by this transformation is given by
$\Gamma^{a}_{bc}=-\partial_{b}\partial_{c}q^{a}.$ (32)
For the infinitesimal transformation between inertial and Rindler coordinates
in Eq. (30), this gives the non-zero components (in the Rindler frame) to be
$\Gamma^{t}_{tx}=\Gamma^{x}_{tt}=\Gamma^{x}_{xx}=\kappa$. Using this we find
that $V^{c}$ now has the component $V^{x}=-2\kappa$. Therefore, the non-zero
surface term in the action which is generated is now given by
$\mathcal{A}_{\rm sur}=-t\left(\frac{\kappa A_{\perp}}{8\pi}\right)$ (33)
leading to the standard results
$H_{\rm sur}=-\frac{\partial A_{\rm sur}}{\partial t}=\left(\frac{\kappa
A_{\perp}}{8\pi}\right)$ (34)
and, using Eq. (17) we get:
$S=A_{\rm sur}\Big{|}_{t=2\pi/\kappa}=\frac{A_{\perp}}{4}$ (35)
In this approach, it is not surprising that we get $H_{\rm sur}\neq 0$ because
$A_{\rm sur}$ is not generally covariant. The infinitesimal coordinate
transformation from inertial to Rindler coordinates generates a non-zero
$V^{c}$ and thus a nonzero contribution to $A_{\rm sur}$ on the horizon
leading to a non-zero $H_{\rm sur}$. (Aside: Note that this corresponds to the
situation mentioned earlier, in the para after Eq. (5). When we make an
infinitesimal coordinate transformation $x^{a}\to x^{a}+\epsilon q^{a}$, the
$\Gamma$ that is produced are of $\mathcal{O}(\epsilon)$. But from the
structure of the scalar curvature
$R\sim\Gamma^{2}+\partial\Gamma=\mathcal{O}(\epsilon^{2})+\mathcal{O}(\epsilon)$
we see that the two terms are of different order. When $R=0$, this will
require the two terms to vanish separately, which — in turn — implies that
$L_{sur}\propto\partial_{c}(\sqrt{-g}V^{c})=0$. The integral of $L_{sur}$,
which is a surface term, will also vanish when the contribution from the
entire surface is taken into account. However, the contribution of this
surface term from just on one surface need not vanish as long as
$\sqrt{-g}V^{c}$ itself is not zero — but a constant, which is the current
situation.)
What is surprising is that the infinitesimal coordinate transformations
captures the exact result! The transformations in Eq. (28) are obtained by
linearising the full transformations in Eq. (26) in $\kappa$. The $A_{\rm
sur}$ was computed correct only to linear order in $\kappa$ because we used
the linearized expressions in Eq. (32) in its calculation. In spite of this,
the result in Eq. (33) matches with the exact result in Eq. (3).
A partial, mathematical reason for this result is the following. Since the
exact form of $A_{\rm sur}$ has to be proportional to
$t(A_{\perp}/L_{P}^{2})$, dimensional considerations show that the
proportionality constant must have the dimensions of inverse length. The only
such constant available to us is $\kappa$ and hence it follows that the
expression should have the form $C(tA_{\perp}\kappa/L_{P}^{2})$ where $C$ is a
dimensionless numerical factor. Therefore, $A_{\rm sur}$ must be linear in
$\kappa$ and an infinitesimal transformation accurate to linear order in
$\kappa$ can reproduce the result except for a numerical factor.
This argument, however, does not explain why we obtain the correct numerical
factor which is not guaranteed by the dimensional analysis. A more physical
reasoning will be to attribute this result to the fact that infinitesimal
coordinate transformations are all that is required to study the nature of
gauge degrees of freedom. As we conjectured in Sec.1, the observer dependent
thermality of null surfaces has to do with upgrading of some gauge degrees of
freedom to true degrees of freedom. Infinitesimal coordinate transformations
should be capable of capturing this effect, which is what has happened.
We will now turn into the second procedure described in Sec.2 for computing
$H_{\rm sur}$ using Noether potential. A natural vector field to use in this
context is, of course, $q^{a}\equiv x^{a}-X^{a}$, which defines the
infinitesimal coordinate transformation between the inertial and Rindler
coordinates. We will now compute the right hand side of Eq. (8) using $J^{ab}$
defined for $q^{a}=x^{a}-X^{a}$ and working entirely in inertial coordinates.
The only relevant non-trivial component of $J^{ab}$ is
$J^{TX}=(\kappa/8\pi)T\approx(\kappa/8\pi)t$ to order $\mathcal{O}(\kappa)$.
Using $d\Sigma_{TX}=-\frac{1}{2}\sqrt{\sigma}\,d^{2}X_{\perp}$ we find that
the Noether integral now gives 111Incidentally, the null horizon surface for
the exact transformations in Eq (26) is at $T=X+1/\kappa$. If we evaluate the
charge on this surface, instead of on the surface $X=T$, and express the
result in terms of Rindler time $t$, we will get the same result in Eq (36).,
$Q=\int d^{2}\Sigma_{ab}\;J^{ab}=-T\,\frac{\kappa
A_{\perp}}{8\pi}\approx-t\,\frac{\kappa A_{\perp}}{8\pi}$ (36)
This is not the $H_{\rm sur}$ computed earlier; and indeed it is not expected
to be since the vector field we are using now is $q^{a}=x^{a}-X^{a}$ and not
$\xi^{a}$ and the result of the integration, of course, will depend on the
vector field. What is curious is that the expression matches with the
expression for $A_{\rm sur}$. This could be, at first sight, surprising for
two reasons: (i) It is not obvious why the Noether integral for the
infinitesimal coordinate transformation, calculated in the inertial frame,
should give $A_{\rm sur}$, computed in the Rindler frame. (ii) We have now
provided a generally covariant expression for $A_{\rm sur}$ which — when
treated as the surface term of the action — is obviously not generally
covariant.
It is possible to understand both these features by probing a little deeper
into the structure of the Noether charge and Noether potential. In general
relativity, the Noether current for a diffeomorphism generated by a vector
field $q^{a}$ is given by
$J^{c}=\nabla_{l}(\nabla^{c}q^{l}-\nabla^{l}q^{c})=2R^{c}_{m}q^{m}+\mathcal{V}^{c};\quad\mathcal{V}^{c}\equiv
g^{lm}\pounds_{q}N^{c}_{lm}$ (37)
where $N^{c}_{lm}$ is the canonical momentum defined in Eq. (21). In flat
spacetime, $R^{a}_{b}=0$ and we have $J^{c}=\mathcal{V}^{c}$. Note that even
though $N^{c}_{lm}=0$ in inertial coordinates, its Lie derivative along a
vector field:
$\pounds_{q}N^{c}_{lm}=-\partial_{l}\partial_{m}q^{c}+\frac{1}{2}\Big{(}\delta^{c}_{l}\partial_{m}\partial_{i}q^{i}+\delta^{c}_{m}\partial_{l}\partial_{i}q^{i}\Big{)}$
(38)
is nonzero! To the lowest order we can interpret the Lie derivative in
$\pounds_{q}N^{c}_{lm}$ as giving the variation $\delta N^{c}_{lm}$ under the
diffeomorphism generated by $q^{a}$. But since $N^{c}_{lm}=0$ in inertial
coordinates, we find that $\delta N^{c}_{lm}=\pounds_{q}N^{c}_{lm}$ is equal
to $N^{c}_{lm}$ in the the non-inertial coordinates. Thus, in this case,
$\mathcal{V}^{c}=g^{lm}\pounds_{q}N^{c}_{lm}=g^{lm}N^{c}_{lm}$ where
$N^{c}_{lm}$ now denotes the canonical momenta in the noninertial frame
generated by the transformation. It then follows that, in our case,
$J^{c}=\mathcal{V}^{c}=g^{lm}N^{c}_{lm}$ (39)
The integral in Eq. (36) over Noether potential $J^{ab}$ can be expressed as
an integral over Noether current $J^{c}$ using Stokes theorem:
$Q=\int d^{2}\Sigma_{ab}\;J^{ab}=\int d^{3}\Sigma_{c}\;J^{c}\to\int
d^{3}\Sigma_{c}\,\mathcal{V}^{c}=A_{\rm sur}$ (40)
In arriving at the third equality we have restricted ourselves to the
contribution from the horizon surface, as mentioned earlier; in obtaining the
last equality, we have used Eq. (22) giving $A_{\rm sur}$ in terms of
$N^{c}_{lm}$. This explains why in this case the Noether integral leads to
$A_{\rm sur}$.
As an aside, we note the following: The dimensions of $A_{\rm sur}$ and
$H_{\rm sur}$ are different and it may be surprising at first sight that the
integral over $d\Sigma_{ab}J^{ab}$ leads to $H_{\rm sur}$ in Eq. (8) and
$A_{\rm sur}$ in Eq. (36). This arises because $\xi^{a}$ used to define
$J^{ab}$ appearing in Eq. (8) is dimensionless while $q^{a}$ used to define
$J^{ab}$ appearing in Eq. (36) has the dimensions of length. Any deformation
vector field, appearing in $x^{a}\to x^{a}+q^{a}$ with length dimensions,
cannot lead to a result with dimensions of $H_{\rm sur}$ but will lead to a
result with dimensions of $A_{\rm sur}$ in the Noether integral. So the
dimensions are consistent.
We can now define the heat content of the null surface through the relation:
$H_{\rm sur}=-\frac{\partial}{\partial t}\,\int
d^{2}\Sigma_{ab}\;J^{ab}=-\xi^{a}\nabla_{a}\int d^{2}\Sigma_{ab}\;J^{ab}$ (41)
which shows that the heat content can be given a generally covariant
definition even with our $q^{a}$. In the Rindler coordinates we can also write
$\int dt\,H_{\rm sur}=-\int d^{2}\Sigma_{ab}\;J^{ab};$ (42)
Since the right hand side of the first relation is generally covariant, we
expect the integral of $H_{\rm sur}dt$ also to be generally covariant. In the
static situation we are considering, we again get the standard result that
$H_{\rm sur}=(\kappa A_{\perp}/8\pi)$.
It should be stressed that the entire calculation given above was carried out
in inertial coordinates. The only hint of the Rindler frame was through the
vector field $q^{a}$ which was defined by the infinitesimal coordinate
transformation connecting the inertial and Rindler frames. After defining
$q^{a}$ by this procedure, we computed the Noether potential corresponding to
this infinitesimal diffeomorphism. This Noether integral, evaluated over the
null surface $X=T$ in the inertial frame, then reproduces the surface term of
the action evaluated using the exact metric in the Rindler frame. From this
expression, we can obtain the heat content by Eq. (41) working entirely in the
inertial frame. The fact that the Noether potential for the infinitesimal
coordinate transformation to Rindler frame is directly related to the degrees
of freedom — which ‘come alive’ through non-zero $N^{a}_{bc}$ — is the
physical reason why the procedure works. We believe this is a first step in
demonstrating in a concrete setting the relationship between coordinate
transformations and observer dependent notion of thermality.
Before we conclude the section, we clarify certain aspects of the above
analysis in order to avoid possible misunderstanding. We begin by noting that
the Noether charge
$Q=\int_{\mathcal{S}}d^{2}\Sigma_{ab}J^{ab}[q]$ (43)
is well-defined for any 2D surface $\mathcal{S}$ and any diffeomorphism
$x^{a}\to x^{a}+\epsilon q^{a}$ which introduces the vector field $q^{a}$.
Normally, in the case static horizons, one takes $\mathcal{S}$ to be the
horizon surface (usually treated as a limit of a timelike stretched horizon)
and takes $q^{a}$ to be the timelike Killing vector field; i.e
$q^{a}=\xi^{a}=(1,0,0,0)$. One then finds that $Q$ can be related to the
entropy of the horizon.
But one can certainly explore the integral in Eq. (43) over the horizon
surface $\mathcal{S}$ for any diffeomorphism $x^{a}\to x^{a}+\epsilon q^{a}$.
The diffeomorphism can be chosen independently of the surface $\mathcal{S}$.
In our approach, we make these choices as follows:
1. 1.
We choose $\mathcal{S}$ to be the null surface $X=T$ which acts as the horizon
for a class of Rindler observers with acceleration $\kappa$. This allows one
to define the temperature of the horizon using the standard, finite,
coordinate transformations in Eq. (26), using, say, periodicity in Euclidean
continuation.
2. 2.
We obtain $q^{a}$ from the finite Rindler transformation by a well-defined
Taylor series expansion described at the end of Sec.3. This essentially
involves introducing a book-keeping parameter $\epsilon$, defining a family of
transformations and picking $q^{a}$ from the structure of transformations near
the identity transformation.
3. 3.
The choice of $\mathcal{S}$ and choice of $q^{a}$ in the above two items are
independent of each other. The thermodynamics of $\mathcal{S}$ (like e.g.,
definition of temperature) arises from the full Rindler transformation and
could not be generated just from $q^{a}$. This is obvious from the fact that
several finite transformations might have the same infinitesimal limit.
4. 4.
So, a priori, if we use $\mathcal{S}$ and $q^{a}$ defined by the above
procedure and evaluate the integral in Eq. (43) we do not expect anything
physically meaningful to emerge. One would have naively thought that the
infinitesimal limit of this transformation does not contain adequate structure
to reproduce the temperature etc. originally defined by the full
transformation.
5. 5.
The key point of this paper is that, contrary to this naive expectation, the
$Q$ in Eq. (43) evaluated by this procedure captures the thermodynamic
features of $\mathcal{S}$, determined originally via the full Rindler
transformations. That is, one can indeed reconstruct the thermality of the
horizon by a prescription which uses only the infinitesimal form of the
transformation. If the $\kappa$ of the infinitesimal coordinate transformation
can be directly related to the temperature associated with the Rindler horizon
at $X=T$, then it is not surprising if we get the horizon thermality using
only the infinitesimal transformation. But this is not the case; one does not
expect any relation between the parameter $\kappa$ in the infinitesimal
coordinate transformation and the temperature associated with the Rindler
horizon at $X=T$ using full Rindler transformation, which makes our result
nontrivial and interesting.
We have tried to give detailed reasoning as to why this happens and how it
connects up with the broader picture of degrees of freedom etc. We will now
study some generalizations of this approach to curved spacetimes.
## 5 Generalization to curved spacetime
The above ideas can be generalized to a wide class of curved, static
spacetimes with horizon. One such class is described by metrics of the form
$\displaystyle ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2}~{}.$
(44)
with $f(r=r_{H})=0$ defining the location of the horizon and the surface
gravity given by $\kappa=f^{\prime}(r_{H})/2\neq 0$. It is well known that
(see e.g., page 343 of Ref. [9]) all such metrics allow a transformation to
Kruskal-like coordinates with
$\displaystyle\kappa T=e^{\kappa r_{*}}\sinh(\kappa t);\quad\kappa X=e^{\kappa
r_{*}}\cosh(\kappa t)~{},$ (45)
where the tortoise coordinate $r_{*}$ is defined by the relation
$dr_{*}=[dr/f(r)]$. Comparison with the transformation between inertial and
Rindler coordinates shows that the $T,X$ coordinate system is similar to
inertial coordinates and $t,r$ coordinate system is similar to Rindler
coordinates. In fact, the $T,X$ coordinate system is a freely falling
coordinate system near the horizon and allows us to define locally inertial
frames in that region.
Using the Taylor series expansion in $\kappa$ of the coordinate
transformations in Eq. (45) we can define the infinitesimal version of the
transformations and obtain $q^{a}=x^{a}-X^{a}$. We then define the Noether
potential for this $q^{a}$ and evaluate the Noether integral as before. The
straight forward but somewhat long computation (see Appendix A) now gives the
result that
$\int d^{2}\Sigma_{ab}\;J^{ab}=-t\,\frac{\kappa A_{\perp}}{8\pi}$ (46)
to the lowest order in $\kappa$.
The rest of the interpretation is similar. The $T,X$ coordinate system and the
associated freely falling observers near the horizon do not attribute any
special properties to the horizon just as the inertial observers do not
attribute any thermal properties to the null surface $X=T$. We however know
that the static observers outside the horizon attribute a heat content and
entropy to the horizon. The above result shows that infinitesimal coordinate
transformation from the Kruskal-like coordinates to Schwarzschild-like
coordinates captures the essence of this phenomena.
In the case of flat spacetime $N^{a}_{bc}$ was zero in the inertial frame and
non-zero in the Rindler frame. In this case Kruskal-like coordinates we have,
in general, non-zero values for connection at an arbitrary event. However,
near the horizon since Kruskal coordinates reduce to a freely falling frame,
we have $N^{a}_{bc}\approx 0$ near the horizon in the $T-X$ plane in Kruskal
coordinates. The coordinate transformation to Schwarzschild-like coordinates
now generates $N^{a}_{bc}$ near the horizon, the integral over which leads to
the result in Eq. (46). We believe this provides a fairly local description of
the heat content of the horizons and an alternate way of interpreting the
results even in the well known cases like the Schwarzschild metric, de Sitter
metric etc..
A natural question to ask would be how general these results are. We describe
below one possible direction of analysis and some preliminary results leaving
further details for a future investigation. Consider a spacetime described by
a metric of the form
$ds^{2}=-\Omega(u,v,x_{\perp})dudv+dx_{\perp}^{2}~{},$ (47)
where $u=T-X$, $v=T+X$ are the standard null coordinates. We will consider the
spacetime around the null surface $u=0$. Consider an infinitesimal coordinate
transformations: $\bar{u}=u+\xi^{u}(u,v,x_{\perp})$ and
$\bar{v}=v+\xi^{u}(u,v,x_{\perp})$, such that
$\displaystyle\xi^{u}(u,v,x_{\perp})=A_{1}+uA_{2}+u^{2}A_{3}+\dots;$
$\displaystyle\xi^{v}(u,v,x_{\perp})=B_{1}+uB_{2}+u^{2}B_{3}+\dots~{},$ (48)
where $A_{i}$ and $B_{i}$ are functions of $(v,x_{\perp})$ and we have done a
Taylor series expansion around $u=0$. If the null congruence associated with
the surfaces $u=$ constant is $l^{a}$, then the non-affinity parameter is
defined by the relation $l^{a}\nabla_{a}l_{b}=\lambda l_{b}$. This leads to
$\lambda(u,v,x_{\perp})=\partial_{v}\ln\Omega$ (see Appendix B for details).
We can also do a Taylor series expansion of $\lambda$ as
$\displaystyle\lambda(u,v,x_{\perp})$ $\displaystyle=$
$\displaystyle\lambda_{0}(v,x_{\perp})+uC_{1}(v,x_{\perp})+u^{2}C_{2}(v,x_{\perp})+\dots~{},$
(49)
Repeating the analysis described in the previous sections, one can show that
we again get the result:
$\int d^{2}\Sigma_{ab}\;J^{ab}=-t\,\frac{\kappa A_{\perp}}{8\pi}$ (50)
provided the coefficients of Taylor series satisfies a particular condition
(see Eq. (B.9) of Appendix B). Here $\kappa$ is an average over the horizon of
the surface gravity given by
$\kappa=\int_{u=0}\frac{\sqrt{\sigma}d^{2}x}{A_{\perp}}\lambda_{0}(v,x_{\perp})$
(51)
This shows that our results will hold for a wide class of infinitesimal
transformations near a null surface provided the condition in Eq. (B.9) of
Appendix B holds. The physical meaning of this condition is unclear and
deserves further analysis.
## 6 Discussion
We provided an explicit example which throws more light on the issues raised
in Sec. 1 related to observer dependence of thermal phenomenon. The central
idea is: (i) to introduce a freely falling coordinate system ($X^{a}$) near
the horizon and a more standard coordinate system ($x^{a}$) appropriate to
observers who attribute thermality to the horizon; (ii) determine the
infinitesimal coordinate transformations connecting these two coordinate
systems and thereby determine the vector field $q^{a}=x^{a}-X^{a}$ and (iii)
study the integral over a codimension-2 cross-section of the horizon of the
Noether potential $J^{ab}$ appropriate for $q^{a}$. We found that one can
determine the heat content $H_{\rm sur}=TS=E-F$ of the horizon in terms of
these quantities and also relate it to the surface term of the gravitational
action. The Christoffel symbols vanish in the freely falling frame but are
non-zero in the static coordinates natural to the horizon. This difference is
closely related to the degrees of freedom contributing to the heat content and
entropy of the horizon and remarkably enough, can be captured just from
studying the infinitesimal version of the coordinate transformation.
These results are easy to interpret and understand in the case of flat
spacetime where the freely falling coordinates are globally defined static
coordinates. We showed that the results also generalize, in a straight forward
manner, to the standard spherically symmetric static horizons including
Schwarzschild, de Sitter spacetimes etc. One possible direction of further
work will be to study the infinitesimal coordinate transformation near a
general null surface as indicated at the end of last section.
## Acknowledgments
The research of one of the authors (TP) is partially supported by a J.C. Bose
research grant of DST, India.
## Appendix A Static, spherically symmetric spacetimes
We express the Kruskal-like metric using the coordinates ($T,X$) which have
dimensions of length as:
$\displaystyle ds^{2}=f(r)e^{-2\kappa
r_{*}}(-dT^{2}+dX^{2})+r^{2}d\Omega^{2}~{},$ (A.1)
with the transformations:
$T=\frac{e^{\kappa r_{*}}}{\kappa}\sinh(\kappa t);\,\,\
X+\frac{1}{\kappa}=\frac{e^{\kappa r_{*}}}{\kappa}\cosh(\kappa t)~{}.$ (A.2)
Expanding Eq. (A.2) in a Taylor series in $\kappa$ and keeping the terms upto
${\cal{O}}(\kappa)$, we find
$T=t+\kappa r_{*}t;\,\,\,\ X=r_{*}+\frac{1}{2}\kappa t^{2}+\frac{1}{2}\kappa
r_{*}^{2}~{}.$ (A.3)
The inverse transformations are
$t=T-\kappa XT;\,\,\,\ r_{*}=X-\frac{1}{2}\kappa T^{2}-\frac{1}{2}\kappa
X^{2}~{}.$ (A.4)
So the components of the generator $q^{a}\equiv x^{a}-X^{a}$ for this
transformation are
$q^{T}=-\kappa XT;\,\,\,\ q^{X}=-\frac{1}{2}\kappa T^{2}-\frac{1}{2}\kappa
X^{2}~{}.$ (A.5)
For the metric in (A.1), the essential component of the Noether potential,
using (A.5), is found to be:
$\displaystyle J^{TX}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi}\Big{[}\frac{2e^{2\kappa r_{*}}}{f(r)}\kappa
T+\frac{e^{4\kappa r_{*}}}{f^{2}(r)}\Big{\\{}\kappa
XT\partial_{X}\Big{(}f(r)e^{-2\kappa r_{*}}\Big{)}$ (A.6) $\displaystyle+$
$\displaystyle\Big{(}\frac{1}{2}\kappa T^{2}+\frac{1}{2}\kappa
X^{2}\Big{)}\partial_{T}\Big{(}f(r)e^{-2\kappa
r_{*}}\Big{)}\Big{\\}}\Big{]}~{},$
and the unit normals are
$n_{a}=\Big{(}0,\sqrt{f(r)}e^{-\kappa r_{*}},0,0\Big{)};\,\,\,\
m_{a}=\Big{(}\sqrt{f(r)}e^{-\kappa r_{*}},0,0,0\Big{)}~{}.$ (A.7)
Therefore,
$\displaystyle d^{2}\Sigma_{TX}=-d^{2}x\sqrt{\sigma}\frac{1}{2}f(r)e^{-2\kappa
r_{*}}~{},$ (A.8)
and so
$\displaystyle d^{2}\Sigma_{TX}J^{TX}$ $\displaystyle=$
$\displaystyle-d^{2}x\sqrt{\sigma}\frac{\kappa
T}{16\pi}-\frac{d^{2}x\sqrt{\sigma}}{32\pi}\frac{e^{2\kappa
r_{*}}}{f(r)}\Big{[}\kappa XT\partial_{X}\Big{(}f(r)e^{-2\kappa r_{*}}\Big{)}$
(A.9) $\displaystyle+$ $\displaystyle\Big{(}\frac{1}{2}\kappa
T^{2}+\frac{1}{2}\kappa X^{2}\Big{)}\partial_{T}\Big{(}f(r)e^{-2\kappa
r_{*}}\Big{)}\Big{]}~{}.$
Now in the following we shall show that near the horizon ${\cal{H}}$, the
second and the last term in Eq. (A.9) are of the order $\kappa^{2}$ and
$\kappa^{3}$, respectively. The second term can be expressed as
$\displaystyle\frac{e^{2\kappa r_{*}}}{f(r)}\Big{(}\kappa
XT\Big{)}\partial_{X}\Big{(}f(r)e^{-2\kappa
r_{*}}\Big{)}=\Big{(}\frac{\partial_{X}f(r)}{f(r)}-2\kappa\partial_{X}r_{*}\Big{)}\kappa
TX~{}.$ (A.10)
Since $X=X(r_{*},t)$,
$\displaystyle\frac{\partial_{X}f(r)}{f(r)}$ $\displaystyle=$
$\displaystyle\frac{1}{f(r)}\Big{[}\frac{\partial r^{*}}{\partial
X}\frac{\partial}{\partial r_{*}}+\frac{\partial t}{\partial
X}\frac{\partial}{\partial t}\Big{]}f(r)$ (A.11) $\displaystyle=$
$\displaystyle\frac{1}{f(r)}\frac{\partial r^{*}}{\partial X}\frac{\partial
f(r)}{\partial r}\frac{\partial r}{\partial r_{*}}$ $\displaystyle=$
$\displaystyle f^{\prime}(r)\Big{(}1-\frac{1}{2}\kappa
X+{\cal{O}}(\kappa^{2})\Big{)}~{},$
where in the last step (A.4) has been used. So, near the horizon
$\displaystyle\frac{\partial_{X}f(r)}{f(r)}\kappa
TX|_{\cal{H}}={\cal{O}}(\kappa^{2})~{}.$ (A.12)
Similarly, $2\kappa(\partial_{X}r_{*})(\kappa TX)={\cal{O}}(\kappa^{2})$.
Therefore the second term in (A.9), near the horizon, is of the order
$\kappa^{2}$. A similar analysis shows that the third term in Eq. (A.9) is of
the order $\kappa^{3}$. So both the terms in (A.9) can be ignored giving:
$\displaystyle
d^{2}\Sigma_{TX}J^{TX}|_{\cal{H}}=-d^{2}x\sqrt{\sigma}\frac{\kappa
T}{16\pi}=-d^{2}x\sqrt{\sigma}\frac{\kappa t}{16\pi}~{},$ (A.13)
upto linear order in $\kappa$. This has been used in computing the integration
in Eq. (46).
## Appendix B A more general spacetime
For the metric Eq. (47) we find that:
$d^{2}\Sigma^{uv}=\frac{\sqrt{\sigma}}{\Omega}d^{2}x$ (B.1)
and
$\displaystyle
J_{uv}=-\frac{\Omega}{2}\Big{[}A_{2}-\partial_{v}B_{1}+2uA_{3}-u\partial_{v}B_{2}+{\cal{O}}(u^{2})\Big{]}$
$\displaystyle-\frac{1}{2}\Big{[}(A_{1}+uA_{2}+{\cal{O}}(u^{2}))\partial_{u}\Omega-(B_{1}+uB_{2}+{\cal{O}}(u^{2}))\partial_{v}\Omega\Big{]}~{}.$
(B.2)
Therefore,
$\displaystyle
d^{2}\Sigma^{uv}J_{uv}=-\frac{\sqrt{\sigma}}{2}d^{2}x\Big{[}A_{2}-\partial_{v}B_{1}+2uA_{3}-u\partial_{v}B_{2}+{\cal{O}}(u^{2})\Big{]}$
$\displaystyle-\frac{\sqrt{\sigma}}{2}d^{2}x\Big{[}(A_{1}+uA_{2}+{\cal{O}}(u^{2}))\partial_{u}\ln\Omega-(B_{1}+uB_{2}+{\cal{O}}(u^{2}))\partial_{v}\ln\Omega\Big{]}~{}.$
(B.3)
In the following we will show that $\partial_{v}\ln\Omega$ is related to the
non-affinity parameter corresponding to the null surface $u=$ constant. Since
$u=0$ is the null surface, the normal to it is a null vector. The explicit
expression for the null vector $l^{a}$ in a ($u,v,x_{\perp}$) coordinate
system is:
$\displaystyle l^{a}=(0,1,0,0);\,\,\,\ l_{a}=(-\frac{\Omega}{2},0,0,0)~{}.$
(B.4)
The non-affinity parameter $\lambda$ is defined by the relation
$l^{a}\nabla_{a}l_{b}=\lambda l_{b}$. This leads to
$\lambda(u,v,x_{\perp})=\partial_{v}\ln\Omega$ so that:
$\displaystyle\ln\Omega=\int\lambda(u,v,x_{\perp})dv~{}.$ (B.5)
Expressing since $\lambda(u,v,x_{\perp})$ in terms of the expansion in Eq.
(49) we get:
$\displaystyle\partial_{u}\ln\Omega=\int\partial_{u}\lambda(u,v,x_{\perp})dv=D_{1}(v,x_{\perp})+uD_{2}(v,x_{\perp})+\dots~{},$
(B.6)
where $D_{1}(v,x_{\perp})=\int C_{1}(v,x_{\perp})dv$, etc. Therefore (B.3)
leads to
$\displaystyle
d\Sigma^{uv}J_{uv}=-\frac{1}{2}d^{2}x\Big{[}A_{2}-\partial_{v}B_{1}+2uA_{3}-u\partial_{v}B_{2}+{\cal{O}}(u^{2})\Big{]}$
(B.7)
$\displaystyle-\frac{1}{2}d^{2}x\Big{[}(A_{1}+uA_{2}+{\cal{O}}(u^{2}))(D_{1}+uD_{2}+\dots)$
$\displaystyle-$
$\displaystyle(B_{1}+uB_{2}+{\cal{O}}(u^{2}))\lambda\Big{]}~{}.$
In the near null surface limit, the above reduces to
$\displaystyle d^{2}\Sigma^{uv}J_{uv}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{\sigma}}{2}d^{2}x\Big{[}A_{2}-\partial_{v}B_{1}+A_{1}D_{1}-B_{1}\lambda_{0}\Big{]}~{}.$
(B.8)
Now to obtain the required value of the Noether charge given in Eq. (50), we
need to impose the following condition on the coefficients:
$\displaystyle\Big{[}A_{2}-\partial_{v}B_{1}+A_{1}D_{1}-B_{1}\lambda_{0}\Big{]}_{(u=0)}=v|_{(u=0)}\lambda_{0}=2T\lambda_{0}\simeq
2t\lambda_{0}~{}.$ (B.9)
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|
arxiv-papers
| 2013-02-05T21:06:17 |
2024-09-04T02:49:41.341946
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bibhas Ranjan Majhi, T. Padmanabhan",
"submitter": "Bibhas Majhi Ranjan",
"url": "https://arxiv.org/abs/1302.1206"
}
|
1302.1213
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-008 LHCb-PAPER-2012-040 February 5, 2013
Amplitude analysis and branching fraction measurement of $\kern
2.59189pt\overline{\kern-2.59189ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}K^{-}$
The LHCb collaboration†††Authors are listed on the following pages.
An amplitude analysis of the final state structure in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi K^{+}K^{-}$
decay mode is performed using $1.0~{}\rm fb^{-1}$ of data collected by the
LHCb experiment in 7 TeV center-of-mass energy $pp$ collisions produced by the
LHC. A modified Dalitz plot analysis of the final state is performed using
both the invariant mass spectra and the decay angular distributions. Resonant
structures are observed in the $K^{+}K^{-}$ mass spectrum as well as a
significant non-resonant S-wave contribution over the entire $K^{+}K^{-}$ mass
range. The largest resonant component is the $\phi(1020)$, accompanied by
$f_{0}(980)$, $f_{2}^{\prime}(1525)$, and four additional resonances. The
overall branching fraction is measured to be $\mathcal{B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi
K^{+}K^{-})=(7.70\pm 0.08\pm 0.39\pm 0.60)\times 10^{-4}$, where the first
uncertainty is statistical, the second systematic, and the third due to the
ratio of the number of $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
to $B^{-}$ mesons produced. The mass and width of the $f_{2}^{\prime}(1525)$
are measured to be $1522.2\pm 2.8^{+5.3}_{-2.0}~{}\rm MeV$ and $84\pm
6^{+10}_{-~{}5}~{}\rm MeV$, respectively. The final state fractions of the
other resonant states are also reported.
Submitted to Physical Review D
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22,
S. Amato2, Y. Amhis36, L. Anderlini17,f, J. Anderson37, R.B. Appleby51, O.
Aquines Gutierrez10, F. Archilli18,35, A. Artamonov 32, M. Artuso53, E.
Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J. Back45, C. Baesso54, W.
Baldini16, R.J. Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48,
Th. Bauer38, A. Bay36, J. Beddow48, I. Bediaga1, S. Belogurov28, K. Belous32,
I. Belyaev28, E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J.
Benton43, A. Berezhnoy29, R. Bernet37, M.-O. Bettler44, M. van Beuzekom38, A.
Bien11, S. Bifani12, T. Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35,
F. Blanc36, C. Blanks50, J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A.
Bondar31, N. Bondar27, W. Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V.
Bowcock49, C. Bozzi16, T. Brambach9, J. van den Brand39, J. Bressieux36, D.
Brett51, M. Britsch10, T. Britton53, N.H. Brook43, H. Brown49, A. Büchler-
Germann37, I. Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O.
Callot7, M. Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A.
Carbone14,c, G. Carboni21,k, R. Cardinale19,i, A. Cardini15, L. Carson50, K.
Carvalho Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, P. Chen3,36, N. Chiapolini37, M. Chrzaszcz 23, K. Ciba35, X.
Cid Vidal34, G. Ciezarek50, P.E.L. Clarke47, M. Clemencic35, H.V. Cliff44, J.
Closier35, C. Coca26, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins35, A.
Comerma-Montells33, A. Contu52,15, A. Cook43, M. Coombes43, G. Corti35, B.
Couturier35, G.A. Cowan36, D. Craik45, S. Cunliffe50, R. Currie47, C.
D’Ambrosio35, P. David8, P.N.Y. David38, I. De Bonis4, K. De Bruyn38, S. De
Capua21,k, M. De Cian37, J.M. De Miranda1, L. De Paula2, P. De Simone18, D.
Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C. Deplano15, D.
Derkach14, O. Deschamps5, F. Dettori39, A. Di Canto11, J. Dickens44, H.
Dijkstra35, P. Diniz Batista1, F. Domingo Bonal33,n, S. Donleavy49, F.
Dordei11, A. Dosil Suárez34, D. Dossett45, A. Dovbnya40, F. Dupertuis36, R.
Dzhelyadin32, A. Dziurda23, A. Dzyuba27, S. Easo46, U. Egede50, V.
Egorychev28, S. Eidelman31, D. van Eijk38, S. Eisenhardt47, R. Ekelhof9, L.
Eklund48, I. El Rifai5, Ch. Elsasser37, D. Elsby42, D. Esperante Pereira34, A.
Falabella14,e, C. Färber11, G. Fardell47, C. Farinelli38, S. Farry12, V.
Fave36, V. Fernandez Albor34, F. Ferreira Rodrigues1, M. Ferro-Luzzi35, S.
Filippov30, C. Fitzpatrick35, M. Fontana10, F. Fontanelli19,i, R. Forty35, O.
Francisco2, M. Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, A. Gallas
Torreira34, D. Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J-C.
Garnier35, J. Garofoli53, P. Garosi51, J. Garra Tico44, L. Garrido33, C.
Gaspar35, R. Gauld52, E. Gersabeck11, M. Gersabeck35, T. Gershon45,35, Ph.
Ghez4, V. Gibson44, V.V. Gligorov35, C. Göbel54, D. Golubkov28, A.
Golutvin50,28,35, A. Gomes2, H. Gordon52, M. Grabalosa Gándara33, R. Graciani
Diaz33, L.A. Granado Cardoso35, E. Graugés33, G. Graziani17, A. Grecu26, E.
Greening52, S. Gregson44, O. Grünberg55, B. Gui53, E. Gushchin30, Yu. Guz32,
T. Gys35, C. Hadjivasiliou53, G. Haefeli36, C. Haen35, S.C. Haines44, S.
Hall50, T. Hampson43, S. Hansmann-Menzemer11, N. Harnew52, S.T. Harnew43, J.
Harrison51, P.F. Harrison45, T. Hartmann55, J. He7, V. Heijne38, K.
Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van Herwijnen35, E.
Hicks49, D. Hill52, M. Hoballah5, P. Hopchev4, W. Hulsbergen38, P. Hunt52, T.
Huse49, N. Hussain52, D. Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12,
J. Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E. Jans38, F.
Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D. Johnson52, C.R.
Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35, T.M. Karbach35,
J. Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A. Keune36, B.
Khanji20, Y.M. Kim47, O. Kochebina7, V. Komarov36,29, R.F. Koopman39, P.
Koppenburg38, M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M.
Kreps45, G. Krocker11, P. Krokovny31, F. Kruse9, M. Kucharczyk20,23,j, V.
Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G.
Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E. Lanciotti35, G.
Lanfranchi18,35, C. Langenbruch35, T. Latham45, C. Lazzeroni42, R. Le Gac6, J.
van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O.
Leroy6, T. Lesiak23, Y. Li3, L. Li Gioi5, M. Liles49, R. Lindner35, C. Linn11,
B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33, N. Lopez-
March36, H. Lu3, J. Luisier36, A. Mac Raighne48, F. Machefert7, I.V.
Machikhiliyan4,28, F. Maciuc26, O. Maev27,35, J. Magnin1, M. Maino20, S.
Malde52, G. Manca15,d, G. Mancinelli6, N. Mangiafave44, U. Marconi14, R.
Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52, A. Martín
Sánchez7, M. Martinelli38, D. Martinez Santos35, A. Massafferri1, Z. Mathe35,
C. Matteuzzi20, M. Matveev27, E. Maurice6, A. Mazurov16,30,35,e, J.
McCarthy42, G. McGregor51, R. McNulty12, M. Meissner11, M. Merk38, J. Merkel9,
D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S. Monteil5, D. Moran51,
P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K. Müller37, R.
Muresan26, B. Muryn24, B. Muster36, J. Mylroie-Smith49, P. Naik43, T.
Nakada36, R. Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35, A.D.
Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, N. Nikitin29, T. Nikodem11,
A. Nomerotski52,35, A. Novoselov32, A. Oblakowska-Mucha24, V. Obraztsov32, S.
Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M. Orlandea26, J.M.
Otalora Goicochea2, P. Owen50, B.K. Pal53, A. Palano13,b, M. Palutan18, J.
Panman35, A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J. Parkinson50, G.
Passaleva17, G.D. Patel49, M. Patel50, G.N. Patrick46, C. Patrignani19,i, C.
Pavel-Nicorescu26, A. Pazos Alvarez34, A. Pellegrino38, G. Penso22,l, M. Pepe
Altarelli35, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo34, A. Pérez-
Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G. Pessina20, K.
Petridis50, A. Petrolini19,i, A. Phan53, E. Picatoste Olloqui33, B. Pie
Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, S. Playfer47, M. Plo Casasus34,
F. Polci8, G. Polok23, A. Poluektov45,31, E. Polycarpo2, D. Popov10, B.
Popovici26, C. Potterat33, A. Powell52, J. Prisciandaro36, V. Pugatch41, A.
Puig Navarro36, W. Qian3, J.H. Rademacker43, B. Rakotomiaramanana36, M.S.
Rangel2, I. Raniuk40, N. Rauschmayr35, G. Raven39, S. Redford52, M.M. Reid45,
A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, V. Rives
Molina33, D.A. Roa Romero5, P. Robbe7, E. Rodrigues48,51, P. Rodriguez
Perez34, G.J. Rogers44, S. Roiser35, V. Romanovsky32, A. Romero Vidal34, J.
Rouvinet36, T. Ruf35, H. Ruiz33, G. Sabatino21,k, J.J. Saborido Silva34, N.
Sagidova27, P. Sail48, B. Saitta15,d, C. Salzmann37, B. Sanmartin Sedes34, M.
Sannino19,i, R. Santacesaria22, C. Santamarina Rios34, R. Santinelli35, E.
Santovetti21,k, M. Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M.
Savrie16,e, P. Schaack50, M. Schiller39, H. Schindler35, S. Schleich9, M.
Schlupp9, M. Schmelling10, B. Schmidt35, O. Schneider36, A. Schopper35, M.-H.
Schune7, R. Schwemmer35, B. Sciascia18, A. Sciubba18,l, M. Seco34, A.
Semennikov28, K. Senderowska24, I. Sepp50, N. Serra37, J. Serrano6, P.
Seyfert11, M. Shapkin32, I. Shapoval40,35, P. Shatalov28, Y. Shcheglov27, T.
Shears49,35, L. Shekhtman31, O. Shevchenko40, V. Shevchenko28, A. Shires50, R.
Silva Coutinho45, T. Skwarnicki53, N.A. Smith49, E. Smith52,46, M. Smith51, K.
Sobczak5, F.J.P. Soler48, F. Soomro18,35, D. Souza43, B. Souza De Paula2, B.
Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S. Stahl11, O. Steinkamp37,
S. Stoica26, S. Stone53, B. Storaci38, M. Straticiuc26, U. Straumann37, V.K.
Subbiah35, S. Swientek9, M. Szczekowski25, P. Szczypka36,35, T. Szumlak24, S.
T’Jampens4, M. Teklishyn7, E. Teodorescu26, F. Teubert35, C. Thomas52, E.
Thomas35, J. van Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39, D.
Tonelli35, S. Topp-Joergensen52, N. Torr52, E. Tournefier4,50, S. Tourneur36,
M.T. Tran36, A. Tsaregorodtsev6, P. Tsopelas38, N. Tuning38, M. Ubeda
Garcia35, A. Ukleja25, D. Urner51, U. Uwer11, V. Vagnoni14, G. Valenti14, R.
Vazquez Gomez33, P. Vazquez Regueiro34, S. Vecchi16, J.J. Velthuis43, M.
Veltri17,g, G. Veneziano36, M. Vesterinen35, B. Viaud7, I. Videau7, D.
Vieira2, X. Vilasis-Cardona33,n, J. Visniakov34, A. Vollhardt37, D.
Volyanskyy10, D. Voong43, A. Vorobyev27, V. Vorobyev31, H. Voss10, C. Voß55,
R. Waldi55, R. Wallace12, S. Wandernoth11, J. Wang53, D.R. Ward44, N.K.
Watson42, A.D. Webber51, D. Websdale50, M. Whitehead45, J. Wicht35, D.
Wiedner11, L. Wiggers38, G. Wilkinson52, M.P. Williams45,46, M. Williams50,p,
F.F. Wilson46, J. Wishahi9, M. Witek23,35, W. Witzeling35, S.A. Wotton44, S.
Wright44, S. Wu3, K. Wyllie35, Y. Xie47, F. Xing52, Z. Xing53, Z. Yang3, R.
Young47, X. Yuan3, O. Yushchenko32, M. Zangoli14, M. Zavertyaev10,a, F.
Zhang3, L. Zhang53, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, A.
Zvyagin35.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25National Center for Nuclear Research (NCBJ), Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pMassachusetts Institute of Technology, Cambridge, MA, United States
## 1 Introduction
The study of $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays to
$J/\psi h^{+}h^{-}$, where $h$ is either a pion or kaon, has been used to
measure mixing-induced $C\\!P$ violation in $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays [1, 2, 3, 4, 5, 6,
7].‡‡‡Mention of a particular mode implies use of its charge conjugate
throughout this paper. In order to best exploit these decays a better
understanding of the final state composition is necessary. This study has been
reported for the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ channel [8]. Here we perform a similar analysis for
$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}K^{-}$. While a large $\phi(1020)$ contribution is well known [9]
and the $f_{2}^{\prime}(1525)$ component has been recently observed [10] and
confirmed [11], other components have not heretofore been identified including
the source of S-wave contributions [12]. The tree-level Feynman diagram for
the process is shown in Fig. 1.
Figure 1: Leading order diagram for $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}_{s}\rightarrow J/\psi K^{+}K^{-}$.
In this paper the $J/\psi K^{+}$ and $K^{+}K^{-}$ mass spectra and decay
angular distributions are used to study resonant and non-resonant structures.
This differs from a classical “Dalitz plot” analysis [13] since the $J/\psi$
meson has spin-1, and its three helicity amplitudes must be considered.
## 2 Data sample and detector
The event sample is obtained using $1.0~{}\rm fb^{-1}$ of integrated
luminosity collected with the LHCb detector [14] using $pp$ collisions at a
center-of-mass energy of 7 TeV. The detector is a single-arm forward
spectrometer covering the pseudorapidity range $2<\eta<5$, designed for the
study of particles containing $b$ or $c$ quarks. Components include a high
precision tracking system consisting of a silicon-strip vertex detector
surrounding the $pp$ interaction region, a large-area silicon-strip detector
located upstream of a dipole magnet with a bending power of about
$4{\rm\,Tm}$, and three stations of silicon-strip detectors and straw drift-
tubes placed downstream. The combined tracking system has momentum§§§We work
in units where $c=1$. resolution $\Delta p/p$ that varies from 0.4% at
5$\mathrm{\,Ge\kern-1.00006ptV}$ to 0.6% at
100$\mathrm{\,Ge\kern-1.00006ptV}$. The impact parameter (IP) is defined as
the minimum distance of approach of the track with respect to the primary
vertex. For tracks with large transverse momentum with respect to the proton
beam direction, the IP resolution is approximately 20$\,\upmu\rm m$. Charged
hadrons are identified using two ring-imaging Cherenkov detectors. Photon,
electron and hadron candidates are identified by a calorimeter system
consisting of scintillating-pad and preshower detectors, an electromagnetic
calorimeter and a hadronic calorimeter. Muons are identified by a system
composed of alternating layers of iron and multiwire proportional chambers.
The trigger [15] consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage that applies a full
event reconstruction. Events selected for this analysis are triggered by a
$J/\psi\rightarrow\mu^{+}\mu^{-}$ decay, where the $J/\psi$ is required at the
software level to be consistent with coming from the decay of a $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ meson by use either of IP
requirements or detachment of the $J/\psi$ from the primary vertex. Monte
Carlo simulations are performed using Pythia [16] with the specific tuning
given in Ref. [17], and the LHCb detector description based on Geant4 [18,
*Allison:2006ve] described in Ref. [20]. Decays of $B$ mesons are based on
EvtGen [21].
## 3 Signal selection and backgrounds
We select $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi K^{+}K^{-}$ candidates trying to simultaneously maximize the signal
yield and reduce the background. Candidate $J/\psi\rightarrow\mu^{+}\mu^{-}$
decays are combined with a pair of kaon candidates of opposite charge, and
then requiring that all four tracks are consistent with coming from a common
decay point. To be considered a $J/\psi\rightarrow\mu^{+}\mu^{-}$ candidate,
particles identified as muons of opposite charge are required to have
transverse momentum, $p_{\rm T}$, greater than 500 MeV, and form a vertex with
fit $\chi^{2}$ per number of degrees of freedom (ndf) less than 11. These
requirements give rise to a large ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ signal over a small background [22]. Only candidates with a dimuon
invariant mass between $-$48 MeV to +43 MeV relative to the observed $J/\psi$
mass peak are selected. The asymmetric requirement is due to final-state
electromagnetic radiation. The two muons are subsequently kinematically
constrained to the known $J/\psi$ mass [9].
Our ring-imaging Cherenkov system allows for the possibility of positively
identifying kaon candidates. Charged tracks produce Cherenkov photons whose
emission angles are compared with those expected for electrons, pions, kaons
or protons, and a likelihood for each species is then computed. To identify a
particular species, the difference between the logarithm of the likelihoods
for two particle hypotheses (DLL) is computed. There are two criteria used:
loose corresponds to DLL$(K-\pi)>0$, while tight has DLL$(K-\pi)>10$ and
DLL$(K-p)>-3$. Unless stated otherwise, we require the tight criterion for
kaon selection.
We select candidate $K^{+}\kern-1.60004ptK^{-}$ combinations if each particle
is inconsistent with having been produced at the primary vertex. For this test
we require that the $\chi^{2}$ formed by using the hypothesis that the IP is
zero be greater than 9 for each track. Furthermore, each kaon must have
$p_{\rm T}>250$ MeV and the scalar sum of the $p_{\rm T}$ of the kaon
candidates must be greater than 900 MeV. To select $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ candidates we further require
that the two kaon candidates form a vertex with $\chi^{2}<10$, and that they
form a candidate $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ vertex
with the $J/\psi$ where the vertex fit $\chi^{2}$/ndf $<5$. We require that
this $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ vertex be more
than $1.5$ mm from the primary vertex, and the angle between the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ momentum vector and the vector
from the primary vertex to the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ vertex must be less than 11.8
mrad.
The $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ candidate invariant
mass distribution is shown in Fig. 2. The vertical lines indicate the signal
and sideband regions, where the signal region extends to $\pm 20$ MeV around
the nominal $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass [9]
and the sidebands extend from 35 MeV to 60 MeV on either side of the peak. The
small peak near 5280 MeV results from $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays, and will be subject to
future investigation.
Figure 2: Invariant mass spectrum of $J/\psi K^{+}K^{-}$ combinations. The
vertical lines indicate the signal (black-dotted) and sideband (red-dashed)
regions.
The background consist of combinations of tracks, which have a smooth mass
shape through the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}\kern-1.60004ptK^{-}$ region, and peaking contributions caused by
the reflection of specific decay modes where a pion is misidentified as a
kaon. The reflection background that arises from the decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow J/\psi K^{-}\pi^{+}$,
where the $\pi^{+}$ is misidentified as a $K^{+}$, is determined from the
number of $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ candidates in the
control region $25-200$ MeV above the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass peak.
For each of the candidates in the $J/\psi K^{+}K^{-}$ control region, we
reassign each of the two kaons in turn to the pion mass hypothesis. The
resulting $J/\psi K\pi$ invariant mass distribution is shown in Fig. 3. The
peak at the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mass has
$906\pm 51$ candidates, determined by fitting the data to a Gaussian function
for the signal, and a polynomial function for the background. From these
events we estimate the number in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ signal region, based on a
simulation of the shape of the reflected distribution as a function of $J/\psi
K^{-}K^{+}$ mass. Using simulated $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow J/\psi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}(892)$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow J/\psi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*}_{2}(1430)$ samples, we calculate
$309\pm 17$ reflection candidates within $\pm 20$ MeV of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ peak. This number is used as a
constraint in the mass fit described below.
Figure 3: Invariant mass distribution for $J/\psi K^{+}K^{-}$ candidates
$25-200$ MeV above the $\kern 1.61993pt\overline{\kern-1.61993ptB}{}^{0}_{s}$
mass, reinterpreted as $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}\rightarrow J/\psi K^{\mp}\pi^{\pm}$
events. The fit is to a signal Gaussian whose mass and width are allowed to
vary as well as the polynomial background.
To determine the number of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ signal candidates we perform a
fit to the candidate $J/\psi K^{+}K^{-}$ invariant mass spectrum shown in Fig.
4.
Figure 4: Fit to the invariant mass spectrum of $J/\psi K^{+}K^{-}$
combinations. The dotted (black) line is the combinatorial background, the
dashed (red) shape shows the misidentified $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}\rightarrow J/\psi
K^{-}\pi^{+}$decays, and the solid (blue) curve shows the total. The vertical
dashed lines indicate the signal region.
The fit function is the sum of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ signal component,
combinatorial background, and the contribution from the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow J/\psi K^{-}\pi^{+}$
reflections. The signal is modeled by a double-Gaussian function with a common
mean. The combinatorial background is described by a linear function. The
reflection background is constrained as described above. The mass fit gives
19,195$\pm$150 signal together with $894\pm 24$ combinatorial background
candidates within $\pm 20$ MeV of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass peak.
We use the decay $B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-}$ as the normalization channel for branching fraction
determinations. The selection criteria are similar to those used for
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}\kern-1.60004ptK^{-}$,
except for particle identification as here a loose kaon identification
criterion is used. Figure 5 shows the $J/\psi K^{-}$ mass distribution. The
signal is fit with a double-Gaussian function and a linear function is used to
fit the combinatorial background. There are 342,786$\pm$661 signal and
10,195$\pm$134 background candidates within $\pm 20$ MeV of the $B^{-}$ peak.
Figure 5: Fit to the invariant mass spectrum of $J/\psi K^{-}$ candidates. The
dotted line shows the combinatorial background and the solid (blue) curve is
the total.
## 4 Analysis formalism
One of the goals of this analysis is to determine the intermediate states in
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi
K^{+}K^{-}$ decay within the context of an isobar model [23, 24], where we sum
the resonant and non-resonant components testing if they explain the invariant
mass squared and angular distributions. We also determine the absolute
branching fractions of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi(1020)$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{2}^{\prime}(1525)$ final states and the mass and width of the
$f_{2}^{\prime}(1525)$ resonance. Another important goal is to understand the
S-wave content in the $\phi(1020)$ mass region.
Four variables completely describe the decay of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi K^{+}K^{-}$
with $J/\psi\rightarrow\mu^{+}\mu^{-}$. Two are the invariant mass squared of
$J/\psi K^{+}$, $s_{12}\equiv m^{2}(J/\psi K^{+})$, and the invariant mass
squared of $K^{+}K^{-}$, $s_{23}\equiv m^{2}(K^{+}K^{-})$. The other two are
the $J/\psi$ helicity angle, $\theta_{J/\psi}$, which is the angle of the
$\mu^{+}$ in the $J/\psi$ rest frame with respect to the $J/\psi$ direction in
the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ rest frame, and the
angle between the $J/\psi$ and $K^{+}K^{-}$ decay planes, $\chi$, in the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ rest frame. To simplify
the probability density function (PDF), we analyze the decay process after
integrating over the angular variable $\chi$, which eliminates several
interference terms.
### 4.1 The model for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi K^{+}K^{-}$
In order to perform an amplitude analysis a PDF must be constructed that
models correctly the dynamical and kinematic properties of the decay. The PDF
is separated into two components, one describing signal, $S$, and the other
background, $B$. The overall PDF given by the sum is
$\displaystyle F(s_{12},s_{23},\theta_{J/\psi})$ $\displaystyle=$
$\displaystyle\frac{1-f_{\rm com}-f_{\rm refl}}{{\cal{N}}_{\rm
sig}}\varepsilon(s_{12},s_{23},\theta_{J/\psi})S(s_{12},s_{23},\theta_{J/\psi})$
$\displaystyle+$ $\displaystyle B(s_{12},s_{23},\theta_{J/\psi}),$
where $\varepsilon$ is the detection efficiency. The background is described
by the sum of combinatorial background, $C$, and reflection, $R$, functions
$B(s_{12},s_{23},\theta_{J/\psi})=\frac{f_{\rm com}}{{\cal{N}}_{\rm
com}}C(s_{12},s_{23},\theta_{J/\psi})+\frac{f_{\rm refl}}{{\cal{N}}_{\rm
refl}}R(s_{12},s_{23},\theta_{J/\psi}),$ (2)
where $f_{\rm com}$ and $f_{\rm refl}$ are the fractions of the combinatorial
background and reflection, respectively, in the fitted region. The fractions
$f_{\rm com}$ and $f_{\rm refl}$ obtained from the mass fit are fixed for the
subsequent analysis.
The normalization factors are given by
$\displaystyle{\cal{N}}_{\rm sig}$ $\displaystyle=$
$\displaystyle\int\\!\varepsilon(s_{12},s_{23},\theta_{J/\psi})S(s_{12},s_{23},\theta_{J/\psi})\,ds_{12}\,ds_{23\,}d\cos\theta_{J/\psi},$
$\displaystyle{\cal{N}}_{\rm com}$ $\displaystyle=$
$\displaystyle\int\\!C(s_{12},s_{23},\theta_{J/\psi})\,ds_{12\,}ds_{23\,}d\cos\theta_{J/\psi},$
(3) $\displaystyle{\cal{N}}_{\rm refl}$ $\displaystyle=$
$\displaystyle\int\\!R(s_{12},s_{23},\theta_{J/\psi})\,ds_{12}\,ds_{23\,}d\cos\theta_{J/\psi}.$
This formalism similar to that used by Belle in their analysis of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow K^{-}\pi^{+}\chi_{c1}$
[25], and later used by LHCb for the analysis of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\pi^{+}\pi^{-}$ [8].
The invariant mass squared of $J/\psi K^{+}$ versus $K^{+}K^{-}$ is shown in
Fig. 6 for $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi K^{+}K^{-}$ candidates. No structure is seen in
$m^{2}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+})$. There are however
visible horizontal bands in the $K^{+}K^{-}$ mass squared spectrum, the most
prominent of which correspond to the $\phi(1020)$ and $f_{2}^{\prime}(1525)$
resonances. These and other structures in $m^{2}(K^{+}K^{-})$ are now
examined.
Figure 6: Distribution of $m^{2}(K^{+}K^{-})$ versus $m^{2}(J/\psi K^{+})$ for
$\kern 1.61993pt\overline{\kern-1.61993ptB}{}^{0}_{s}$ candidate decays within
$\pm 20$ MeV of the $\kern 1.61993pt\overline{\kern-1.61993ptB}{}^{0}_{s}$
mass. The horizontal bands result from the $\phi(1020)$ and
$f_{2}^{\prime}(1525)$ resonances.
The signal function is given by the coherent sum over resonant states that
decay into $K^{+}K^{-}$, plus a possible non-resonant S-wave
contribution¶¶¶The interference terms between different helicities are zero
because we integrate over the angular variable $\chi$.
$S(s_{12},s_{23},\theta_{J/\psi})=\sum_{\lambda=0,\pm
1}\left|\sum_{i}a^{R_{i}}_{\lambda}e^{i\phi^{R_{i}}_{\lambda}}\mathcal{A}_{\lambda}^{R_{i}}(s_{12},s_{23},\theta_{J/\psi})\right|^{2},$
(4)
where $\mathcal{A}_{\lambda}^{R_{i}}(s_{12},s_{23},\theta_{J/\psi})$ describes
the decay amplitude via an intermediate resonance state $R_{i}$ with helicity
$\lambda$. Note that the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ has
the same helicity as the intermediate $K^{+}K^{-}$ resonance. Each $R_{i}$ has
an associated amplitude strength $a_{\lambda}^{R_{i}}$ and a phase
$\phi_{\lambda}^{R_{i}}$ for each helicity state $\lambda$. The amplitude for
resonance $R$, for each $i$, is given by
$\mathcal{A}_{\lambda}^{R}(s_{12},s_{23},\theta_{J/\psi})=F_{B}^{(L_{B})}\;A_{R}(s_{23})\;F_{R}^{(L_{R})}\;T_{\lambda}(\theta_{KK})\Big{(}\frac{P_{B}}{m_{B}}\Big{)}^{L_{B}}\;\Big{(}\frac{P_{R}}{\sqrt{s_{23}}}\Big{)}^{L_{R}}\;\Theta_{\lambda}(\theta_{J/\psi}),$
(5)
where $P_{R}$ is the momentum of either of the two kaons in the di-kaon rest
frame, $m_{B}$ is the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
mass, $P_{B}$ is the magnitude of the $J/\psi$ three-momentum in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ rest frame, and
$F_{B}^{(L_{B})}$ and $F_{R}^{(L_{R})}$ are the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ meson and $R_{i}$ resonance
decay form factors. The orbital angular momenta between the $J/\psi$ and
$K^{+}K^{-}$ system is given by $L_{B}$, and the orbital angular momentum in
the $K^{+}K^{-}$ decay is given by $L_{R}$; the latter is the same as the spin
of the $K^{+}K^{-}$ system. Since the parent $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ has spin-0 and the $J/\psi$ is
a vector, when the $K^{+}K^{-}$ system forms a spin-0 resonance, $L_{B}=1$ and
$L_{R}=0$. For $K^{+}K^{-}$ resonances with non-zero spin, $L_{B}$ can be 0, 1
or 2 (1, 2 or 3) for $L_{R}=1(2)$ and so on. We take the lowest $L_{B}$ as the
default value and consider the other possibilities in the systematic
uncertainty.
The Blatt-Weisskopf barrier factors $F_{B}^{(L_{B})}$ and $F_{R}^{(L_{R})}$
[26] are
$\displaystyle F^{(0)}$ $\displaystyle=$ $\displaystyle 1,$ $\displaystyle
F^{(1)}$ $\displaystyle=$ $\displaystyle\frac{\sqrt{1+z_{0}}}{\sqrt{1+z}},$
(6) $\displaystyle F^{(2)}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{z_{0}^{2}+3z_{0}+9}}{\sqrt{z^{2}+3z+9}}.$
For the $B$ meson $z=r^{2}P_{B}^{2}$, where $r$, the hadron scale, is taken as
5.0 GeV-1; for the $R$ resonance $z=r^{2}P_{R}^{2}$, and $r$ is taken as 1.5
GeV-1 [27]. In both cases $z_{0}=r^{2}P_{0}^{2}$ where $P_{0}$ is the decay
daughter momentum at the pole mass; for the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decay the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ momentum is used, while for the
$R$ resonances the kaon momentum is used.
In the helicity formalism, the angular term, $T_{\lambda}(\theta_{KK})$ is
defined as
$T_{\lambda}(\theta_{KK})=d^{J}_{\lambda 0}(\theta_{KK}),$ (7)
where $d$ is the Wigner $d$-function, $J$ is the resonance spin, $\theta_{KK}$
is the helicity angle of the $K^{+}$ in the $K^{+}K^{-}$ rest frame with
respect to the $K^{+}K^{-}$ direction in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ rest frame, and may be
calculated directly from the other variables as
$\cos\theta_{KK}=\frac{\left[m^{2}(J/\psi K^{+})-m^{2}(J/\psi
K^{-})\right]m(K^{+}K^{-})}{4P_{R}P_{B}m_{B}}.$ (8)
The $J/\psi$ helicity dependent term $\Theta_{\lambda}(\theta_{J/\psi})$ is
defined as
$\displaystyle\Theta_{\lambda}(\theta_{J/\psi})$ $\displaystyle=$
$\displaystyle\sqrt{\sin^{2}\theta_{J/\psi}}\;\;\;\;\;\;\;\;\;\text{for}\;\;\lambda=0$
(9) $\displaystyle=$
$\displaystyle\sqrt{\frac{1+\cos^{2}\theta_{J/\psi}}{2}}\;\;\text{for}\;\;|\lambda|=1.$
The mass squared shape of each resonance, $R$ is described by the function
$A_{R}(s_{23})$. In most cases this is a Breit-Wigner (BW) amplitude. When a
decay channel opens close to the resonant mass, complications arise, since the
proximity of the second threshold distorts the line shape of the amplitude.
The $f_{0}(980)$ can decay to either $\pi\pi$ or $KK$. While the $\pi\pi$
channel opens at much lower masses, the $K^{+}K^{-}$ decay channel opens near
the resonance mass. Thus, for the $f_{0}(980)$ we use a Flatté model [28] that
takes into account these coupled channels.
We describe the BW amplitude for a resonance decaying into two spin-0
particles, labeled as 2 and 3, as
$A_{R}(s_{23})=\frac{1}{m^{2}_{R}-s_{23}-im_{R}\Gamma(s_{23})}~{},$ (10)
where $m_{R}$ is the resonance mass, $\Gamma(s_{23})$ is its energy-dependent
width that is parametrized as
$\Gamma(s_{23})=\Gamma_{0}\left(\frac{P_{R}}{P_{R_{0}}}\right)^{2L_{R}+1}\left(\frac{m_{R}}{\sqrt{s_{23}}}\right)F^{2}_{R}~{}.$
(11)
Here $\Gamma_{0}$ is the decay width when the invariant mass of the daughter
combinations is equal to $m_{R}$.
The Flatté mass shape is parametrized as
$A_{R}(s_{23})=\frac{1}{m_{R}^{2}-s_{23}-im_{R}(g_{\pi\pi}\rho_{\pi\pi}+g_{KK}\rho_{KK})},$
(12)
where the constants $g_{\pi\pi}$ and $g_{KK}$ are the $f_{0}(980)$ couplings
to $\pi^{+}\pi^{-}$ and $K^{+}K^{-}$ final states, respectively. The $\rho$
factors are given by Lorentz-invariant phase space
$\displaystyle\rho_{\pi\pi}$ $\displaystyle=$
$\displaystyle\frac{2}{3}\sqrt{1-\frac{4m^{2}_{\pi^{\pm}}}{s_{23}}}+\frac{1}{3}\sqrt{1-\frac{4m^{2}_{\pi^{0}}}{s_{23}}},$
(13) $\displaystyle\rho_{KK}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{1-\frac{4m^{2}_{K^{\pm}}}{s_{23}}}+\frac{1}{2}\sqrt{1-\frac{4m^{2}_{K^{0}}}{s_{23}}}.$
(14)
For non-resonant processes, the amplitude
$\mathcal{A}(s_{12},s_{23},\theta_{J/\psi})$ is constant over the variables
$s_{12}$ and $s_{23}$, but has an angular dependence due to the $J/\psi$
decay. The amplitude is derived from Eq. (5), assuming that the non-resonant
$K^{+}K^{-}$ contribution is an S-wave (i.e. $L_{R}=0$, $L_{B}=1$) and is
uniform in phase space (i.e. $A_{R}=1$),
$\mathcal{A}(s_{12},s_{23},\theta_{J/\psi})=\frac{P_{B}}{m_{B}}\sqrt{\sin^{2}\theta_{J/\psi}}.$
(15)
### 4.2 Detection efficiency
The detection efficiency is determined from a phase space simulation sample
containing $3.4\times 10^{6}$ $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi K^{+}K^{-}$
events with $J/\psi\rightarrow\mu^{+}\mu^{-}$. We also use a separate sample
of $1.3\times 10^{6}$ $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi\phi$ events.
The $p$ and $p_{\rm T}$ distributions of the generated $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mesons are weighted to match
the distributions found using $J/\psi\phi$ data. The simulation is also
corrected by weighting for difference between the simulated kaon detection
efficiencies and the measured ones determined by using a sample of
$D^{*+}\rightarrow\pi^{+}(D^{0}\rightarrow K^{-}\pi^{+})$ events.
Next we describe the efficiency in terms of the analysis variables. Both
$s_{12}$ and $s_{13}$ range from 12.5 $\rm GeV^{2}$ to 24.0 $\rm GeV^{2}$,
where $s_{13}$ is defined below, and thus are centered at $s_{0}=18.25~{}{\rm
GeV}^{2}$. We model the detection efficiency using the dimensionless symmetric
Dalitz plot observables
$x=\left(s_{12}-s_{0}\right)/\left(1~{}{\rm
GeV}^{2}\right),~{}~{}~{}~{}~{}~{}y=\left(s_{13}-s_{0}\right)/\left(1~{}{\rm
GeV}^{2}\right),$ (16)
and the angular variable $\theta_{J/\psi}$. The observables $s_{12}$ and
$s_{13}$ are related to $s_{23}$ as
$s_{12}+s_{13}+s_{23}=m^{2}_{B}+m^{2}_{J/\psi}+m^{2}_{K^{+}}+m^{2}_{K^{-}}.$
(17)
To parametrize this efficiency, we fit the $\cos\theta_{J/\psi}$ distributions
of the $J/\psi K^{+}K^{-}$ and $J/\psi\phi$ simulation samples in bins of
$m^{2}(K^{+}K^{-})$ with the function
$\varepsilon_{2}(s_{23},\theta_{J/\psi})=\frac{1+a\cos^{2}\theta_{J/\psi}}{2+2a/3},$
(18)
giving values of $a$ as a function of $m^{2}(K^{+}K^{-})$. The resulting
distribution, shown in Fig. 7, is described by an exponential function
$a(s_{23})=\exp(a_{1}+a_{2}s_{23}),$ (19)
with $a_{1}=-0.76\pm 0.18$ and $a_{2}=(-1.02\pm 0.15)~{}\rm GeV^{-2}$.
Equation (18) is normalized with respect to $\cos\theta_{J/\psi}$.
Figure 7: Exponential fit to the efficiency parameter $a(s_{23})$. The point
near the $\phi(1020)$ meson mass is determined more precisely due to the use
of a large simulation sample.
The efficiency in $\cos\theta_{J/\psi}$ depends on $s_{23}$, and is observed
to be independent of $s_{12}$. Thus the detection efficiency can be expressed
as
$\varepsilon(s_{12},s_{23},\theta_{J/\psi})=\varepsilon_{1}(x,y)\times\varepsilon_{2}(s_{23},\theta_{J/\psi}).$
(20)
After integrating over $\cos\theta_{J/\psi}$, Eq. (20) becomes
$\int_{-1}^{+1}\varepsilon(s_{12},s_{23},\theta_{J/\psi})d\cos\theta_{J/\psi}=\varepsilon_{1}(x,y),$
(21)
and is modeled by a symmetric fifth order polynomial function given by
$\displaystyle\varepsilon_{1}(x,y)$ $\displaystyle=$ $\displaystyle
1+\epsilon^{\prime}_{1}(x+y)+\epsilon^{\prime}_{2}(x+y)^{2}+\epsilon^{\prime}_{3}xy+\epsilon^{\prime}_{4}(x+y)^{3}+\epsilon^{\prime}_{5}xy(x+y)$
(22)
$\displaystyle+\epsilon^{\prime}_{6}(x+y)^{4}+\epsilon^{\prime}_{7}xy(x+y)^{2}+\epsilon^{\prime}_{8}x^{2}y^{2}+\epsilon^{\prime}_{9}(x+y)^{5}$
$\displaystyle+\epsilon^{\prime}_{10}xy(x+y)^{3}+\epsilon^{\prime}_{11}x^{2}y^{2}(x+y),$
where $\epsilon^{\prime}_{i}$ are the fit parameters.
Figure 8: Parametrized detection efficiency as a function of
$m^{2}(K^{+}K^{-})$ versus $m^{2}(J/\psi K^{+})$. The z-axis scale is
arbitrary.
The $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi
K^{+}K^{-}$ phase space simulation sample is modeled with the polynomial
function. The fitted function is shown in Fig. 8, and the projections of the
fit are shown in Fig. 9. The efficiency is well described by the
parametrization.
Figure 9: Projections of the invariant mass squared (a) $K^{+}K^{-}$ and (b)
$J/\psi K^{+}$ from the simulation used to measure the efficiency parameters.
The points represent the generated event distributions and the curves the
polynomial fit.
For the region within $\pm 20$ MeV of the $\phi(1020)$ mass, the
$\cos\theta_{KK}$ acceptance is used separately, due to the large number of
signal events. Here the $\cos\theta_{KK}$ distribution shows a variation in
efficiency, which can be parametrized using the efficiency function
$A(\theta_{KK})=\frac{1+\epsilon^{\prime}_{12}\cos^{2}\theta_{KK}}{1+\epsilon^{\prime}_{12}/3},$
(23)
where the parameter $\epsilon^{\prime}_{12}$ is measured from a fit to the
simulated $J/\psi\phi$ sample with $\varepsilon_{1}(x,y)\times
A(\theta_{KK})$, giving $\epsilon^{\prime}_{12}=-0.099\pm 0.010$, as shown in
Fig. 10.
Figure 10: Distribution of $\cos\theta_{KK}$ for the $J/\psi\phi$ simulated
sample fitted with $\varepsilon_{1}(x,y)\times A(\theta_{KK})$, within $\pm
20$ MeV of the $\phi(1020)$ mass.
The mass resolution is $\sim 0.7$ MeV at the $\phi(1020)$ mass peak, which is
added to the fit model by increasing the Breit-Wigner width of the
$\phi(1020)$ to 4.59 MeV.
### 4.3 Background composition
The shape of the combinatorial background is modeled as
$C(s_{12},s_{23},\theta_{J/\psi})=\left[C_{1}(s_{12},s_{23})\frac{P_{B}}{m_{B}}+\frac{c_{0}}{(m^{2}_{0}-s_{23})^{2}+m^{2}_{0}\Gamma_{0}^{2}}\right]\times\left(1+\alpha\cos^{2}\theta_{J/\psi}\right),$
(24)
where $C_{1}(s_{12},s_{23})$ is parametrized as
$C_{1}(s_{12},s_{23})=1+c_{1}(x+y)+c_{2}(x+y)^{2}+c_{3}xy+c_{4}(x+y)^{3}+c_{5}xy(x+y),$
(25)
with $c_{i}$, $m_{0}$, $\Gamma_{0}$ and $\alpha$ as the fit parameters. The
variables $x$ and $y$ are defined in Eq. (16).
Figure 11 shows the mass squared projections from the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass sidebands with the fit
projections overlaid. The $\chi^{2}/\rm ndf$ of the fit is 291/305. The value
of $\alpha$ is determined by fitting the $\cos\theta_{J/\psi}$ distribution of
background, as shown in Fig. 12, with a function of the form
$1+\alpha\cos^{2}\theta_{J/\psi}$, yielding $\alpha=-0.14\pm 0.08$.
Figure 11: Invariant mass squared projections of (a) $K^{+}K^{-}$ and (b)
$J/\psi K^{+}$ from the background Dalitz plot of candidates in the $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}_{s}$ mass sidebands. Figure 12:
Distribution of $\cos\theta_{J/\psi}$ from the background sample fit with the
function $1+\alpha\cos^{2}\theta_{J/\psi}$.
The reflection background is parametrized as
$R(s_{12},s_{23},\theta_{J/\psi})=R_{1}(s_{12},s_{23})\times\left(1+\beta\cos^{2}\theta_{J/\psi}\right),$
(26)
where $R_{1}(s_{12},s_{23})$ is modeled using the simulation; the projections
of $s_{12}$ and $s_{23}$ are shown in Fig. 13.
Figure 13: Projections of the reflection background in the variables (a)
$m^{2}(K^{+}K^{-})$ and (b) $m^{2}(J/\psi K^{+})$, obtained from $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}\rightarrow J/\psi\kern
1.79997pt\overline{\kern-1.79997ptK}{}^{*0}(892)$ and $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}\rightarrow J/\psi\kern
1.79997pt\overline{\kern-1.79997ptK}{}^{*}_{2}(1430)$ simulations. Figure 14:
Distribution of $\cos\theta_{J/\psi}$ for the reflection fit with the function
$1+\beta\cos^{2}\theta_{J/\psi}$.
The $J/\psi$ helicity angle dependent part of the reflections is modeled as
$1+\beta\cos^{2}\theta_{J/\psi}$, where the parameter $\beta$ is obtained from
a fit to the simulated $\cos\theta_{J/\psi}$ distribution, shown in Fig. 14,
giving $\beta=-0.19\pm 0.01$.
## 5 Final state composition
### 5.1 Resonance models
The resonances that are likely to contribute are produced from the $s\bar{s}$
system in Fig. 1, and thus are isoscalar ($I=0$). The $K^{+}K^{-}$ system in
the decay $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi K^{+}K^{-}$ can, in principle, have zero or any positive integer
angular momentum. Both the $P$-parity and $C$-parity of $K^{+}K^{-}$ pair in a
state of relative angular momentum $L$ are given by $(-1)^{L}$. Therefore the
allowed resonances decaying to $K^{+}K^{-}$ are limited to
$J^{PC}=0^{++},~{}1^{--},~{}2^{++},~{}...,$ with isospin $I=0$. In the
kinematically accessible mass range up to 2 GeV, resonances with
$J^{PC}=3^{--}$ or higher are not expected and thus the subsequent analysis
only uses spins up to $J=2$. Possible resonance candidates are listed in Table
1. There could also be a contribution from non-resonant events which we assume
to be S-wave and evenly distributed over the available phase space.
Table 1: Possible resonance candidates in the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi K^{+}K^{-}$ decay mode. Spin | Helicity | Resonance | Amplitude
---|---|---|---
0 | 0 | $f_{0}(980)$ | Flatté
0 | 0 | $f_{0}(1370)$, $f_{0}(1500)$, $f_{0}(1710)$ | BW
1 | $0,\pm 1$ | $\phi(1020)$, $\phi(1680)$ | BW
2 | $0,\pm 1$ | $f_{2}(1270)$, $f_{2}^{\prime}(1525)$, $f_{2}(1640)$, | BW
| | $f_{2}(1750)$, $f_{2}(1950)$ |
To study the resonant structures of the decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi K^{+}K^{-}$
we use 20, 425 candidates with an invariant mass within $\pm 20$ MeV of the
observed $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass peak.
This includes both signal and background, with $94\%$ signal purity. We begin
our analysis considering only the resonance components
$\phi(1020),~{}f_{2}^{\prime}(1525)$ and a non-resonant component, established
in our earlier measurement [10], and add resonances until no others are found
with more than two standard deviation statistical significance (2$\sigma$).
The significance is estimated from the fit fraction divided by its statistical
uncertainty. Our best fit model includes a non-resonant component and 8
resonance states: $\phi(1020)$, $f_{0}(980)$, $f_{0}(1370)$,
$f_{2}^{\prime}(1525)$, $f_{2}(1640)~{}(\rm|\lambda|=1)$,
$\phi(1680)~{}(\rm|\lambda|=1),$∥∥∥The $f_{2}(1640)~{}(\lambda=0)$ and
$\phi(1680)~{}(\lambda=0)$ components have less than two standard deviation
significance when added separately to the fit, and therefore are not included
in the best fit model. $f_{2}(1750)$, and $f_{2}(1950)$. Most of the
resonances considered here are well established except for the modes
$f_{2}(1640)$, $f_{2}(1750)$, and $f_{2}(1950)$. Although the existence of
$f_{2}(1640)$ is not confirmed yet [9], the right shoulder of
$f_{2}^{\prime}(1525)$ fits better when we add this state. The presence of
multiple broad overlapping resonances in this region may indicate a failure of
the isobar model used in this analysis, but with the present data sample
alternative descriptions are not feasible. Indeed, the situation is not clear
for the resonance states in the vicinity of 1750 MeV. The PDG lists a spin-0
resonance, $f_{0}(1710)$, around 1.72 GeV of $K^{+}K^{-}$ invariant mass [9].
The Belle collaboration observed a resonance in the vicinity of 1.75 GeV with
$J^{PC}=(\rm even)^{++}$ in their study of $\gamma\gamma\rightarrow
K^{+}K^{-}$ [29], but could not establish its spin. A state of mass
1767$\pm$14 MeV was seen by the L3 collaboration decaying into
$K^{0}_{S}K^{0}_{S}$ with $J=2$ [30]. We find that our data are better fit
including the $f_{2}(1750)$ mode. If we substitute either the $f_{0}(1710)$ or
$f_{0}(1750)$ resonance the fit is worsened, as the $\rm-ln\mathcal{L}$
increase by 59 and 7 units, respectively.
In the same analysis of $\gamma\gamma\rightarrow K^{+}K^{-}$, Belle also
observed the $f_{2}(1950)$ [29] resonance. We include this state in our best
fit model. Furthermore, we do not expect significant contributions from the
$f_{2}(1270)$ and $f_{0}(1500)$ resonances, since the PDG branching fractions
are much larger in the $\pi^{+}\pi^{-}$ final state than in $K^{+}K^{-}$ [9]
and we did not see significant contributions from these two resonances in the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\pi^{+}\pi^{-}$ final state [8]. Therefore, these two resonances are not
considered in the best fit model. However, we add these states, in turn, to
the best fit model in order to test for their possible presence.
The masses and widths of the BW resonances are listed in Table 2. When used in
the fit they are fixed to the central values, except for the
$f_{2}^{\prime}(1525)$, whose mass and width are allowed to vary.
Table 2: Breit-Wigner resonance parameters. Resonance | Mass (MeV) | Width (MeV) | Source
---|---|---|---
$\phi(1020)$ | $1019.46$ | $\pm$ | $0.02$ | $4.26$ | $\pm$ | $0.04$ | PDG [9]
$f_{2}(1270)$ | $1275$ | $\pm$ | $1$ | $185$ | $\pm$ | $3$ | PDG [9]
$f_{0}(1370)$ | $1475$ | $\pm$ | $6$ | $113$ | $\pm$ | $11$ | LHCb [8]
$f_{0}(1500)$ | $1505$ | $\pm$ | $6$ | $109$ | $\pm$ | $7$ | PDG [9]
$f_{2}^{\prime}(1525)$ | $1525$ | $\pm$ | $5$ | $73$ | $\pm$ | $6$ | PDG [9]
$f_{2}(1640)$ | $1639$ | $\pm$ | $6$ | $99$ | $\pm$ | $60$ | PDG [9]
$\phi(1680)$ | $1680$ | $\pm$ | $20$ | $150$ | $\pm$ | $50$ | PDG [9]
$f_{0}(1710)$ | $1720$ | $\pm$ | $6$ | $135$ | $\pm$ | $8$ | PDG [9]
$f_{2}(1750)$ | $1737$ | $\pm$ | $9$ | $151$ | $\pm$ | $33$ | Belle [29]
$f_{2}(1950)$ | $1980$ | $\pm$ | $14$ | $297$ | $\pm$ | $13$ | Belle [29]
The $f_{0}(980)$ is described by a Flatté resonance shape, see Eq. (12). The
parameters describing the function are the mass, and the couplings
$g_{\pi\pi}$ and $g_{KK}$, which are fixed in the fit from the previous
analysis of $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\pi^{+}\pi^{-}$ [8]. The parameters are $m_{0}=939.9\pm 6.3$ MeV,
$g_{\pi\pi}=199\pm 30$ MeV and $g_{KK}/g_{\pi\pi}=3.0\pm 0.3$. All background
and efficiency parameters are fixed in the fit.
To determine the complex amplitudes in a specific model, the data are fitted
maximizing the unbinned likelihood given as
$\mathcal{L}=\prod_{i=1}^{N}F(s^{i}_{12},s^{i}_{23},\theta^{i}_{J/\psi}),$
(27)
where $N$ is the total number of candidates, and $F$ is the total PDF defined
in Eq. (4.1). The PDF normalization is accomplished by first normalizing the
$J/\psi$ helicity dependent part by analytical integration, and then for the
mass dependent part using numerical integration over 400$\times$800 bins.
The fit determines the relative values of the amplitude strengths,
$a_{\lambda}^{R_{i}}$, and phases, $\phi_{\lambda}^{R_{i}}$, defined in Eq.
(4). We choose to fix $a_{0}^{\phi(1020)}=1$. As only relative phases are
physically meaningful, one phase in each helicity grouping must be fixed. In
addition, because $J/\psi K^{+}\kern-1.60004ptK^{-}$ is a self-charge-
conjugate mode and does not determine the initial $B$ flavor, the signal
function is an average of $B^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$. If we consider no
$K^{+}\kern-1.60004ptK^{-}$ partial-waves of a higher order than D-wave, then
we can express the differential decay rate
($d\Gamma/dm_{KK}\,d\cos\theta_{KK\,}d\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$) derived from Eq. (4) in terms of S-, P-, and D-waves including
helicity 0 and $\pm 1$ components. The differential decay rates for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ and $B^{0}_{s}$, respectively
are
$\displaystyle\frac{d\overline{\Gamma}}{dm_{KK}\,d\cos\theta_{KK\,}d\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}=$ $\displaystyle\hskip 11.38109pt\left|{\cal
A}^{s}_{S_{0}}e^{i\phi^{s}_{S_{0}}}+{\cal
A}^{s}_{P_{0}}e^{i\phi^{s}_{P_{0}}}\cos\theta_{KK}+{\cal
A}^{s}_{D_{0}}e^{i\phi^{s}_{D_{0}}}\left(\frac{3}{2}\cos^{2}\theta_{KK}-\frac{1}{2}\right)\right|^{2}\sin^{2}\theta_{J/\psi}$
$\displaystyle+\left|{\cal A}^{s}_{P_{\pm 1}}e^{i\phi^{s}_{P_{\pm
1}}}\sqrt{\frac{1}{2}}\sin\theta_{KK}+{\cal A}^{s}_{D_{\pm
1}}e^{i\phi^{s}_{D_{\pm
1}}}\sqrt{\frac{3}{2}}\sin\theta_{KK}\cos\theta_{KK}\right|^{2}\frac{1+\cos^{2}\theta_{J/\psi}}{2},$
(28)
and
$\displaystyle\frac{d\Gamma}{dm_{KK}\,d\cos\theta_{KK\,}d\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}=$ $\displaystyle\hskip 11.38109pt\left|{\cal
A}^{s}_{S_{0}}e^{i\phi^{s}_{S_{0}}}-{\cal
A}^{s}_{P_{0}}e^{i\phi^{s}_{P_{0}}}\cos\theta_{KK}+{\cal
A}^{s}_{D_{0}}e^{i\phi^{s}_{D_{0}}}\left(\frac{3}{2}\cos^{2}\theta_{KK}-\frac{1}{2}\right)\right|^{2}\sin^{2}\theta_{J/\psi}$
$\displaystyle+\left|{\cal A}^{s}_{P_{\pm 1}}e^{i\phi^{s}_{P_{\pm
1}}}\sqrt{\frac{1}{2}}\sin\theta_{KK}-{\cal A}^{s}_{D_{\pm
1}}e^{i\phi^{s}_{D_{\pm
1}}}\sqrt{\frac{3}{2}}\sin\theta_{KK}\cos\theta_{KK}\right|^{2}\frac{1+\cos^{2}\theta_{J/\psi}}{2},$
(29)
where ${\cal A}^{s}_{k_{\lambda}}$ and $\phi^{s}_{k_{\lambda}}$ are the sum of
amplitudes and reference phases, for the spin-$k$ resonance group,
respectively. The decay rate for $B^{0}_{s}$ is similar to that of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$, except
$\theta_{K^{+}\kern-1.12003ptK^{-}}$ and $\theta_{J/\psi}$ are now changed to
$\pi-\theta_{K^{+}\kern-1.12003ptK^{-}}$ and $\pi-\theta_{J/\psi}$
respectively, as a result of using $K^{-}$ and $\mu^{-}$ to define the
helicity angles and hence the signs change in front of the ${\cal
A}^{s}_{P_{0}}$ and ${\cal A}^{s}_{D_{\pm 1}}$ terms.
Summing Eqs. (28) and (29) results in cancellation of the interference
involving $\lambda=0$ terms for spin-1, and the $\lambda=\pm 1$ terms for
spin-2, as they appear with opposite signs for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ and $B^{0}_{s}$ decays.
Therefore we have to fix one phase in the spin-1 ($\lambda=0$) group
($\phi^{s}_{P_{0}}$) and one in the spin-2 ($\lambda=\pm 1$) group
($\phi^{s}_{D_{\pm 1}}$). The other phases in each corresponding group are
determined relative to that of the fixed resonance.
### 5.2 Fit results
The goodness of fit is calculated from 3D partitions of $s_{12}$, $s_{23}$ and
$\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$. We use the
Poisson likelihood $\chi^{2}$ [31] defined as
$\chi^{2}=2\sum_{i=1}^{N}\left[x_{i}-n_{i}+n_{i}\text{ln}\left(\frac{n_{i}}{x_{i}}\right)\right],$
(30)
where $n_{i}$ is the number of candidates in the three dimensional bin $i$ and
$x_{i}$ is the expected number of candidates in that bin according to the
fitted likelihood function. An adaptive binning algorithm is used, requiring a
minimum of 25 entries in each bin. The associated number of degrees of freedom
(ndf) is $N-k-1$, where $k$ is the number of free parameters in the likelihood
function. The $\chi^{2}/\text{ndf}$ and the negative of the logarithm of the
likelihood, $\rm-ln\mathcal{L}$, of the fits are given in Table 3. Starting
values of parameters are varied in order to ensure that global likelihood
minimums are found rather than local minimums.
Table 3: $\chi^{2}/\text{ndf}$ and $\rm-ln\mathcal{L}$ of different resonance models. Resonance model | $\rm-ln\mathcal{L}$ | $\chi^{2}/\text{ndf}$
---|---|---
Best fit | 29,275 | 649/545 = 1.1908
Best fit + $f_{2}(1270)$ | 29,273 | 644/541 = 1.1911
Best fit + $f_{0}(1500)$ | 29,274 | 647/543 = 1.1915
Attempts to add one more resonance such as $f_{2}(1270)$ and $f_{0}(1500)$
improve the $-\rm ln\mathcal{L}$ marginally, but the $\chi^{2}/\rm ndf$ are
worse than the best fit model. We retain only those resonances that are more
than $2\sigma$ significant, except for the $f_{2}(1750)$ where we allow the
$|\lambda|=1$ component, since the $\lambda=0$ component is significant. For
models with one more resonance, the additional components never have more than
2$\sigma$ significance. Figure 15 shows the projection of $m^{2}(K^{+}K^{-})$
for the best fit model, the $m^{2}(J/\psi K^{+})$ and $\cos\theta_{J/\psi}$
projections are displayed in Fig. 16. The projection of the $K^{+}K^{-}$
invariant mass spectrum is shown in Fig. 17.
Figure 15: Dalitz plot fit projection of $m^{2}(K^{+}K^{-})$ using a
logarithmic scale. The points with error bars are data, the (black) dotted
curve shows the combinatorial background, the (red) dashed curve indicates the
reflection from the misidentified $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}\rightarrow J/\psi K^{-}\pi^{+}$
decays, and the (blue) solid line represents the total. Figure 16: Dalitz plot
fit projections of (a) $m^{2}(J/\psi K^{+})$ and (b) $\cos\theta_{J/\psi}$.
The points with error bars are data, the (black) dotted curve shows the
combinatorial background, the (red) dashed curve indicates the reflection from
the misidentified $\kern 1.61993pt\overline{\kern-1.61993ptB}{}^{0}\rightarrow
J/\psi K^{-}\pi^{+}$ decays, and the (blue) solid line represents the total
fit results. Figure 17: Dalitz fit projection of $m(K^{+}K^{-})$. The points
represent the data, the dotted (black) curve shows the combinatorial
background, and the dashed (red) curve indicates the reflection from
misidentified $\kern 1.61993pt\overline{\kern-1.61993ptB}{}^{0}\rightarrow
J/\psi K^{-}\pi^{+}$ decays. The largest three resonances $\phi(1020)$,
$f_{2}^{\prime}(1525)$ and $f_{0}(980)$ are shown by magenta, brown and green
long-dashed curves, respectively; all other resonances are shown by thin black
curves. The dashed (cyan) curve is the non-resonant contribution. The dot-
dashed (black) curve is the contribution from the interferences, and the solid
(blue) curve represents the total fit result.
While a complete description of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}K^{-}$ decay is given in terms of the fitted amplitudes and phases,
knowledge of the contribution of each component can be summarized by defining
a fit fraction, ${\cal{F}}^{R}_{\lambda}$. To determine
${\cal{F}}^{R}_{\lambda}$ we integrate the squared amplitude of $R$ over the
Dalitz plot. The yield is then normalized by integrating the entire signal
function over the same area. Specifically,
${\cal{F}}^{R}_{\lambda}=\frac{\int\left|a^{R}_{\lambda}e^{i\phi^{R}_{\lambda}}\mathcal{A}_{\lambda}^{R}(s_{12},s_{23},\theta_{J/\psi})\right|^{2}ds_{12}\;ds_{23}\;d\cos\theta_{J/\psi}}{\int
S(s_{12},s_{23},\theta_{J/\psi})~{}ds_{12}\;ds_{23}\;d\cos\theta_{J/\psi}}.$
(31)
Note that the sum of the fit fractions is not necessarily unity due to the
potential presence of interference between two resonances. Interference term
fractions are given by
${\cal{F}}^{RR^{\prime}}_{\lambda}=\mathcal{R}e\left(\frac{\int
a^{R}_{\lambda}\;a^{R^{\prime}}_{\lambda}e^{i(\phi^{R}_{\lambda}-\phi_{\lambda}^{R^{\prime}})}\mathcal{A}_{\lambda}^{R}(s_{12},s_{23},\theta_{J/\psi}){\mathcal{A}_{\lambda}^{R^{\prime}}}^{*}(s_{12},s_{23},\theta_{J/\psi})ds_{12}\;ds_{23}\;d\cos\theta_{J/\psi}}{\int
S(s_{12},s_{23},\theta_{J/\psi})~{}ds_{12}\;ds_{23}\;d\cos\theta_{J/\psi}}\right),$
(32)
and
$\sum_{\lambda}\left(\sum_{R}{\cal{F}}^{R}_{\lambda}+\sum^{R\neq
R^{\prime}}_{RR^{\prime}}{\cal{F}}^{RR^{\prime}}_{\lambda}\right)=1.$ (33)
If the Dalitz plot has more destructive interference than constructive
interference, the sum of the fit fractions will be greater than unity.
Conversely, the sum will be less than one if the Dalitz plot exhibits
constructive interference. Note that interference between different spin-$J$
states vanishes because the $d^{J}_{\lambda 0}$ angular functions in
$\mathcal{A}^{R}_{\lambda}$ are orthogonal.
The determination of the statistical uncertainties of the fit fractions is
difficult because they depend on the statistical uncertainty of every fitted
magnitude and phase. Therefore we determine the uncertainties from simulated
experiments. We perform 500 experiments: each sample is generated according to
the model PDF, input parameters are taken from the fit to the data. The
correlations of fitted parameters are also taken into account. For each
experiment the fit fractions are calculated. The distributions of the obtained
fit fractions are described by Gaussian functions. The r.m.s. widths of the
Gaussian functions are taken as the statistical uncertainties on the
corresponding parameters. The fit fractions and phases of the contributing
components are given in Table 4, while the fit fractions of the interference
terms are quoted in Table 5.
Table 4: Fit fractions (%) and phases of contributing components. For P- and D-waves $\lambda$ represents the helicity. Component | Fit fraction (%) | Phase (degree)
---|---|---
$\phi(1020)$, $\lambda=0$ | $32.1\pm 0.5\pm 0.8$ | 0(fixed)
$\phi(1020)$, $|\lambda|=1$ | $34.6\pm 0.5\pm 1.3$ | 0(fixed)
$f_{0}(980)$ | $12.0\pm 1.8^{+2.8}_{-2.5}$ | $-294\pm 8\pm 25$
$f_{0}(1370)$ | $1.2\pm 0.3^{+0.3}_{-1.2}$ | $-81\pm 8\pm 8$
$f_{2}^{\prime}(1525)$, $\lambda=0$ | $9.9\pm 0.7^{+2.4}_{-1.6}$ | 0(fixed)
$f_{2}^{\prime}(1525)$, $|\lambda|=1$ | $5.1\pm 0.9^{+1.8}_{-1.4}$ | 0(fixed)
$f_{2}(1640)$, $|\lambda|=1$ | $1.5\pm 0.7^{+0.7}_{-0.9}$ | $165\pm 27^{+13}_{-44}$
$\phi(1680)$, $|\lambda|=1$ | $3.4\pm 0.3^{+4.4}_{-0.3}$ | $106\pm 14^{+260}_{-210}$
$f_{2}(1750)$, $\lambda=0$ | $2.6\pm 0.5^{+1.0}_{-0.6}$ | $238\pm 8\pm 9$
$f_{2}(1750)$, $|\lambda|=1$ | $1.8\pm 1.0^{+2.2}_{-1.8}$ | $45\pm 30^{+16}_{-70}$
$f_{2}(1950)$, $\lambda=0$ | $0.4\pm 0.2^{+0.2}_{-0.4}$ | $46\pm 17^{+110}_{-~{}20}$
$f_{2}(1950)$, $|\lambda|=1$ | $1.7\pm 0.5^{+2.5}_{-1.7}$ | $-53\pm 26^{+150}_{-~{}80}$
Non-resonant | $6.0\pm 1.6^{+2.0}_{-2.2}$ | $-39\pm 6\pm 23$
Total | $112.3$ |
Table 5: Fit fractions matrix for the best fit in units of %. The diagonal elements correspond to the decay fractions in Table 4. The off-diagonal elements give the fit fractions of the interference. The null values originate from the fact that any interference contribution between different spin-$J$ state integrates to zero. Here the resonances are labeled by their masses in MeV and the subscripts denote the helicities. | $1020_{0}$ | $1020_{1}$ | 980 | 1370 | $1525_{0}$ | $1525_{1}$ | $1640_{1}$ | $1680_{1}$ | $1750_{0}$ | $1750_{1}$ | $1950_{0}$ | $1950_{1}$ | NR
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$1020_{0}$ | 32.1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$1020_{1}$ | | 34.6 | 0 | 0 | 0 | 0 | 0 | 2.0 | 0 | 0 | 0 | 0 | 0
980 | | | 12.0 | -2.3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4.7
1370 | | | | 1.2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1.5
$1525_{0}$ | | | | | 9.9 | 0 | 0 | 0 | -4.5 | 0 | 0.9 | 0 | 0
$1525_{1}$ | | | | | | 5.1 | -0.9 | 0 | 0 | 2.5 | 0 | -1.8 | 0
$1640_{1}$ | | | | | | | 1.5 | 0 | 0 | -2.4 | 0 | 0.7 | 0
$1680_{1}$ | | | | | | | | 3.4 | 0 | 0 | 0 | 0 | 0
$1750_{0}$ | | | | | | | | | 2.6 | 0 | -0.4 | 0 | 0
$1750_{1}$ | | | | | | | | | | 1.8 | 0 | -2.2 | 0
$1950_{0}$ | | | | | | | | | | | 0.4 | 0 | 0
$1950_{1}$ | | | | | | | | | | | | 1.7 | 0
NR | | | | | | | | | | | | | 6.0
Table 6 shows a comparison of the fit fractions when a different
parametrization is used for the $f_{0}(980)$ resonance shape. The BES
$f_{0}(980)$ functional form is the same as ours with the parameters
$m_{0}=965\pm 10$ MeV, $g_{\pi\pi}=165\pm 18$ MeV and
$g_{KK}/g_{\pi\pi}=4.21\pm 0.33$ [32]. The BaBar collaboration assumes that
the non-resonant S-wave is small and consistent with zero [33]. The BaBar
functional form is different and parametrized as
$A_{R}(s_{23})=\frac{1}{m^{2}_{R}-s_{23}-im_{R}\Gamma_{R}\rho_{KK}},$ (34)
with $\rho_{KK}=2P_{R}/\sqrt{s_{23}}$. The parameters are $m_{R}=922\pm 3$ MeV
and $\Gamma_{R}=240\pm 80$ MeV, taken from BaBar’s Dalitz plot analysis of
$D_{s}^{+}\rightarrow K^{+}K^{-}\pi^{+}$ [34]. The $f_{0}(980)$ fraction is
smaller in the BaBar parametrization, while the total S-wave fraction is
consistent in the three different parameterizations. In all cases, the
dominant component is the $\phi(1020)$, the second largest contribution is the
$f_{2}^{\prime}(1525)$, and the third the $f_{0}(980)$ resonance. There are
also significant contributions from the $f_{0}(1370)$, $f_{2}(1640)$,
$\phi(1680)$, $f_{2}(1750)$, $f_{2}(1950)$ resonances, and non-resonant final
states. The amount of $f_{0}(980)$ is strongly parametrization dependent, so
we treat these three models separately and do not assign any systematic
uncertainty based on the use of these different $f_{0}(980)$ shapes. Therefore
we refrain from quoting a branching fraction measurement for the decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi f_{0}(980)$.
The determination of the parameters of the $f_{2}^{\prime}(1525)$ resonance
are not dependent on the $f_{0}(980)$ parametrization. The parameters of the
$f_{2}^{\prime}(1525)$ are determined to be:
$\displaystyle m_{f_{2}^{\prime}(1525)}$ $\displaystyle=$ $\displaystyle
1522.2\pm 2.8^{+5.3}_{-2.0}~{}\rm MeV,$
$\displaystyle\Gamma_{f_{2}^{\prime}(1525)}$ $\displaystyle=$ $\displaystyle
84\pm 6^{+10}_{-~{}5}~{}\rm MeV.$
Whenever two or more uncertainties are quoted, the first is the statistical
and the second systematic. The latter will be discussed in Section 5.6. These
values are the most accurate determinations of the $f_{2}^{\prime}(1525)$
resonant parameters [9]. Note that our determination of the mass has the same
uncertainty as the current PDG average.
Table 6: Comparison of the fit fractions (%) with the LHCb, BES and BaBar $f_{0}(980)$ parameterizations described in the text. For P- and D-waves, $\lambda$ represents the helicity. Component | LHCb | BES | BaBar
---|---|---|---
$\phi(1020)$, $\lambda=0$ | $32.1\pm 0.5~{}$ | $32.1\pm$0.5 | $32.0\pm 0.5$
$\phi(1020)$, $|\lambda|=1$ | $34.6\pm 0.5~{}$ | $34.6\pm 0.5~{}$ | $34.5\pm 0.5$
$f_{0}(980)$ | $12.0\pm 1.8~{}$ | $9.2\pm 1.4$ | $4.8\pm 1.0$
$f_{0}(1370)$ | $1.2\pm 0.3$ | $1.2\pm 0.3$ | $1.3\pm 0.3$
$f_{2}^{\prime}(1525)$, $\lambda=0$ | $9.9\pm 0.7$ | $9.8\pm 0.7$ | $9.5\pm 0.7$
$f_{2}^{\prime}(1525)$, $|\lambda|=1$ | $5.1\pm 0.9$ | $5.1\pm 0.9$ | $4.9\pm 0.9$
$f_{2}(1640)$, $|\lambda|=1$ | $1.5\pm 0.7$ | $1.5\pm 0.7$ | $1.5\pm 0.7$
$\phi(1680)$, $|\lambda|=1$ | $3.4\pm 0.3$ | $3.4\pm 0.3$ | $3.4\pm 0.3$
$f_{2}(1750)$, $\lambda=0$ | $2.6\pm 0.5$ | $2.5\pm 0.5$ | $2.2\pm 0.5$
$f_{2}(1750)$, $|\lambda|=1$ | $1.8\pm 1.0$ | $1.8\pm 1.0$ | $1.9\pm 1.0$
$f_{2}(1950)$, $\lambda=0$ | $0.4\pm 0.2$ | $0.4\pm 0.2$ | $0.4\pm 0.2$
$f_{2}(1950)$, $|\lambda|=1$ | $1.7\pm 0.5$ | $1.8\pm 0.5$ | $1.8\pm 0.5$
Non-resonant S-wave | $6.0\pm 1.4$ | $4.7\pm 1.2$ | $8.6\pm 1.7$
Interference between S-waves | $-$5.5 | $-$1.7 | $-$1.1
Total S-wave | 13.7 | 13.4 | 13.6
-ln$\mathcal{L}$ | 29,275 | 29,275 | 29,281
$\chi^{2}/\rm ndf$ | 649/545 | 653/545 | 646/545
### 5.3 $K^{+}K^{-}$ S-wave in the $\phi(1020)$ mass region
It was claimed by Stone and Zhang [12] that in the decay of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi\phi$, the
$K^{+}K^{-}$ system can have S-wave contributions under the $\phi(1020)$ peak
of order 7% of the total yield. In order to investigate this possibility we
calculate the S-wave fractions as given by the fit in 4 MeV mass intervals
between $990<m(K^{+}\kern-1.60004ptK^{-})<1050$ MeV. The resulting behavior is
shown in Fig. 18. Here we show the result from our preferred model and also
from the alternative $f_{0}(980)$ parameterizations discussed above. The
observation of significant S-wave fractions in this region means that this
contribution must be taken into account when measuring $C\\!P$ violation in
the $\phi$ mass region. The total S-wave fraction as a function of the mass
interval around the $\phi$ mass is also shown in Fig. 19.
Figure 18: S-wave fraction as a function of $m(K^{+}K^{-})$ starting from 990
MeV up to 1050 MeV in 4 MeV mass intervals. The squares (blue), triangles
(red), and circles (green) represent the LHCb, BES and BaBar parameterizations
of $f_{0}(980)$, respectively. The experimental statistical uncertainties are
only shown for the LHCb model; they are almost identical for the other cases.
The experimental mass resolution is not unfolded. Figure 19: S-wave fractions
in different $m(K^{+}K^{-})$ intervals centered on the $\phi$ meson mass. The
squares (blue), triangles (red), and circles (green) represent the LHCb, BES
and BaBar parameterizations of $f_{0}(980)$, respectively. The experimental
statistical uncertainties are only shown for the LHCb model; they are almost
identical for the other cases. The experimental mass resolution is not
unfolded.
Using a time dependent analysis of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi\phi(1020)$,
LHCb reported $(2.2\pm 1.2\pm 0.07)\%$ [3] of S-wave within $\pm 12$ MeV of
the $\phi(1020)$ mass peak. We measure the S-wave fraction within the same
mass window as a consistent, and more precise $(1.1\pm 0.1^{+0.2}_{-0.1})\%$.
CDF measured the S-wave fraction as $(0.8\pm 0.2)$% for $m(K^{+}K^{-})$ within
about $\pm$9.5 MeV of the $\phi$ mass [6], while ATLAS quotes (2$\pm$2)% for
an 11 MeV interval [7]. These results are consistent with ours. The D0
collaboration, however, claimed a (14.7$\pm$3.5)% S-wave fraction within
approximately $\pm$10 MeV of the $\phi$ meson mass [5], in disagreement with
all of the other results.
### 5.4 Helicity angle distributions
The decay angular distributions or the helicity angle distributions are
already included in the signal model via Eqs. (8) and (9). In order to test
the fit model we examine the $\cos\theta_{J/\psi}$ and $\cos\theta_{KK}$
distributions in two different $K^{+}K^{-}$ mass regions: one is the
$\phi(1020)$ region defined within $\pm 12$ MeV of the $\phi(1020)$ mass peak
and the other is defined within one full width of the $f_{2}^{\prime}(1525)$
mass. The background-subtracted efficiency-corrected distributions are shown
in Figs. 20 and 21. The distributions are in good agreement with the fit
model.
Figure 20: Background-subtracted efficiency-corrected $\cos\theta_{J/\psi}$
helicity distributions: (a) in $\phi(1020)$ mass region ($\chi^{2}/\rm
ndf=54.4/40$), (b) in $f_{2}^{\prime}(1525)$ mass region ($\chi^{2}/\rm
ndf=34.4/40$).
Figure 21: Background-subtracted efficiency-corrected $\cos\theta_{KK}$
helicity distributions: (a) in $\phi(1020)$ mass region ($\chi^{2}/\rm
ndf=57.4/40$), (b) in $f_{2}^{\prime}(1525)$ mass region ($\chi^{2}/\rm
ndf=43.4/40$). The distributions are compatible with expectations for spin-1
and spin-2, respectively.
### 5.5 Angular moments
The angular moment distributions provide an additional way of visualizing the
effects of different resonances and their interferences, similar to a partial
wave analysis. This technique has been used in previous studies [8, 35].
We define the angular moments $\langle Y_{l}^{0}\rangle$ as the efficiency-
corrected and background-subtracted $K^{+}K^{-}$ invariant mass distributions,
weighted by orthogonal and normalized spherical harmonic functions
$Y_{l}^{0}(\cos\theta_{KK})$,
$\langle
Y_{l}^{0}\rangle=\int_{-1}^{1}d\Gamma(m_{KK},\cos\theta_{KK})Y_{l}^{0}(\cos\theta_{KK})d\cos\theta_{KK}.$
(35)
If we assume that no $K^{+}K^{-}$ partial-waves of a higher order than D-wave
contribute, then we can express the differential decay rate, derived from Eq.
(4) in terms of S-, P-, and D-waves including helicity 0 and $\pm 1$
components as
$\displaystyle\frac{d\overline{\Gamma}}{dm_{KK}\,d\cos\theta_{KK}}=$
$\displaystyle\hskip 11.38109pt2\pi\left|{\cal
S}_{0}Y_{0}^{0}(\cos\theta_{KK})+{\cal
P}_{0}e^{i\Phi_{P_{0}}}Y_{1}^{0}(\cos\theta_{KK})+{\cal
D}_{0}e^{i\Phi_{D_{0}}}Y_{2}^{0}(\cos\theta_{KK})\right|^{2}$
$\displaystyle+2\pi\left|{\cal P}_{\pm 1}e^{i\Phi_{P_{\pm
1}}}\sqrt{\frac{3}{8\pi}}\sin\theta_{KK}+{\cal D}_{\pm 1}e^{i\Phi_{D_{\pm
1}}}\sqrt{\frac{15}{8\pi}}\sin\theta_{KK}\cos\theta_{KK}\right|^{2},$ (36)
where ${\cal S}_{\lambda}$, ${\cal P}_{\lambda}$, ${\cal D}_{\lambda}$ and
$\Phi_{k_{\lambda}}$ are real-valued functions of $m_{KK}$, and we have
factored out the S-wave phase. We can then calculate the angular moments
$\displaystyle\sqrt{4\pi}\langle Y_{0}^{0}\rangle$ $\displaystyle=$
$\displaystyle{\cal S}_{0}^{2}+{\cal P}_{0}^{2}+{\cal D}_{0}^{2}+{\cal P}_{\pm
1}^{2}+{\cal D}_{\pm 1}^{2}$ $\displaystyle\sqrt{4\pi}\langle
Y_{1}^{0}\rangle$ $\displaystyle=$ $\displaystyle 2{\cal S}_{0}{\cal
P}_{0}\cos\Phi_{P_{0}}+\frac{4}{\sqrt{5}}{\cal P}_{0}{\cal
D}_{0}\cos(\Phi_{P_{0}}-\Phi_{D_{0}})$ (37)
$\displaystyle+8\sqrt{\frac{3}{5}}{\cal P}_{\pm 1}{\cal D}_{\pm
1}\cos(\Phi_{P_{\pm 1}}-\Phi_{D_{\pm 1}})$ $\displaystyle\sqrt{4\pi}\langle
Y_{2}^{0}\rangle$ $\displaystyle=$ $\displaystyle\frac{2}{\sqrt{5}}{\cal
P}_{0}^{2}+2{\cal S}_{0}{\cal D}_{0}\cos\Phi_{D_{0}}+\frac{2\sqrt{5}}{7}{\cal
D}_{0}^{2}-\frac{1}{\sqrt{5}}{\cal P}_{\pm 1}^{2}+\frac{\sqrt{5}}{7}{\cal
D}_{\pm 1}^{2}$ $\displaystyle\sqrt{4\pi}\langle Y_{3}^{0}\rangle$
$\displaystyle=$ $\displaystyle 6\sqrt{\frac{3}{35}}{\cal P}_{0}{\cal
D}_{0}\cos(\Phi_{P_{0}}-\Phi_{D_{0}})+\frac{6}{\sqrt{35}}{\cal P}_{\pm 1}{\cal
D}_{\pm 1}\cos(\Phi_{P_{\pm 1}}-\Phi_{D_{\pm 1}})$
$\displaystyle\sqrt{4\pi}\langle Y_{4}^{0}\rangle$ $\displaystyle=$
$\displaystyle\frac{6}{7}{\cal D}_{0}^{2}-\frac{4}{7}{\cal D}_{\pm 1}^{2}.$
The angular moments for $l>4$ vanish. Figures 22 and 23 show the distributions
of the angular moments for the fit model around $\pm 30$ MeV of the
$\phi(1020)$ mass peak and above the $\phi(1020)$, respectively. In general
the interpretation of these moments is that $\langle Y^{0}_{0}\rangle$ is the
efficiency-corrected and background-subtracted event distribution, $\langle
Y^{0}_{1}\rangle$ the sum of the interference between S-wave and P-wave, and
P-wave and D-wave amplitudes, $\langle Y^{0}_{2}\rangle$ the sum of P-wave,
D-wave and the interference of S-wave and D-wave amplitudes, $\langle
Y^{0}_{3}\rangle$ the interference between P-wave and D-wave amplitudes, and
$\langle Y^{0}_{4}\rangle$ the D-wave. As discussed in Section 5.1, the
average of $B^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ cancels the interference terms
that involve ${\cal P}_{0}$ and ${\cal D}_{\pm}$. This causes the angular
moments $\langle Y^{0}_{1}\rangle$ and $\langle Y^{0}_{3}\rangle$ to be zero
when averaging over $B^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays. We observe that the
fit results well describe the moment distributions, except for the $\langle
Y^{0}_{1}\rangle$ and $\langle Y^{0}_{4}\rangle$ values below 1.2 GeV. This
may be the result of statistical fluctuations or imperfect modeling.
Figure 22: Dependence of the spherical harmonic moments of $\cos\theta_{KK}$
as a function of the $K^{+}K^{-}$ mass around the $\phi(1020)$ mass peak after
efficiency corrections and background subtraction. The points with error bars
are the data and the solid curves are derived from the fit model. Figure 23:
Dependence of the spherical harmonic moments of $\cos\theta_{KK}$ as a
function of the $K^{+}K^{-}$ mass above 1050 MeV, after efficiency corrections
and background subtraction. The points with error bars are the data and the
solid curves are derived from the fit model.
### 5.6 Systematic uncertainties
The sources of the systematic uncertainties on the results of the Dalitz plot
analysis are summarized in Table 7. The uncertainties due to the background
parametrization are estimated by comparing the results from the best fit model
with those when the background shape parameters are obtained from a fit to the
lower sideband region only. The uncertainties in the efficiency are estimated
by comparing the fit results when the efficiency parameters are changed by
their statistical uncertainties and are added in quadrature. The effect on the
fit fractions of changing the efficiency function is evaluated using a similar
method to that used previously [8]. Briefly, we change the efficiency model by
increasing the minimum IP $\chi^{2}$ requirement from 9 to 12.5 on both of the
kaon candidates. This has the effect of increasing the $\chi^{2}$ of the fit
to the angular distributions of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ data by 1 unit. The new efficiency function is then applied to the
data with the original minimum IP $\chi^{2}$ selection of 9, the likelihood is
re-evaluated and the uncertainties are estimated by comparing the results with
the best fit model. The largest variations among these two efficiency
categories are included in the uncertainty.
Table 7: Absolute systematic uncertainties on the fit results. Item | Efficiency | Background | Fit model | Total |
---|---|---|---|---|---
$m_{f_{2}^{\prime}(1525)}$ (MeV) | 1.2 | 0.4 | ${}^{+5.2}_{-1.5}$ | ${}^{+5.3}_{-2.0}$ |
$\Gamma_{f_{2}^{\prime}(1525)}$ (MeV) | 4.7 | 0.5 | ${}^{+8.6}_{-1.8}$ | ${}^{+9.8}_{-5.0}$ |
Fit fractions (%) |
$\phi(1020)~{}\lambda=0$ | 0.8 | 0 | ${}^{+0.06}_{-0.04}$ | $\pm 0.8$ |
$\phi(1020)~{}|\lambda|=1$ | 1.3 | 0 | ${}^{+0.4}_{-0.1}$ | $\pm 1.3$ |
$f_{0}(980)$ | 1.7 | 0.4 | ${}^{+2.1}_{-1.7}$ | ${}^{+2.8}_{-2.5}$ |
$f_{0}(1370)$ | 0.3 | 0.02 | ${}^{+0.2}_{-1.2}$ | ${}^{+0.3}_{-1.3}$ |
$f_{2}^{\prime}(1525)~{}\lambda=0$ | 1.5 | 0.2 | ${}^{+1.8}_{-0.5}$ | ${}^{+2.4}_{-1.6}$ |
$f_{2}^{\prime}(1525)~{}|\lambda|=1$ | 1.1 | 0.4 | ${}^{+1.3}_{-0.8}$ | ${}^{+1.8}_{-1.4}$ |
$f_{2}(1640)~{}|\lambda|=1$ | 0.1 | 0.1 | ${}^{+0.7}_{-0.9}$ | ${}^{+0.7}_{-0.9}$ |
$\phi(1680)~{}|\lambda|=1$ | 0.3 | 0.1 | ${}^{+4.4}_{-0.1}$ | ${}^{+4.4}_{-0.3}$ |
$f_{2}(1750)~{}\lambda=0$ | 0.3 | 0.1 | ${}^{+1.0}_{-0.5}$ | ${}^{+1.0}_{-0.6}$ |
$f_{2}(1750)~{}|\lambda|=1$ | 1.6 | 0.3 | ${}^{+1.5}_{-2.9}$ | ${}^{+2.2}_{-3.3}$ |
$f_{2}(1950)~{}\lambda=0$ | 0.1 | 0.04 | ${}^{+0.2}_{-0.4}$ | ${}^{+0.2}_{-0.4}$ |
$f_{2}(1950)~{}|\lambda|=1$ | 2.1 | 0.6 | ${}^{+1.1}_{-3.1}$ | ${}^{+2.5}_{-3.8}$ |
Non-resonant | 1.7 | 0.4 | ${}^{+0.9}_{-1.5}$ | ${}^{+2.0}_{-2.2}$ |
S-wave within $\pm$12 MeV | 0.02 | 0.02 | ${}^{+0.2}_{-0.1}$ | ${}^{+0.2}_{-0.1}$ |
of $\phi(1020)$ peak (%) | | | | |
Phases (degrees) |
$f_{0}(980)$ | 19 | 8 | ${}^{+14}_{-~{}8}$ | ${}^{+25}_{-22}$ |
$f_{0}(1370)$ | 6 | 1 | ${}^{+6}_{-5}$ | $\pm 8$ |
$f_{2}(1640)~{}|\lambda|=1$ | 7 | 11 | ${}^{+~{}0}_{-42}$ | ${}^{+13}_{-44}$ |
$\phi(1680)~{}|\lambda|=1$ | 1 | 1 | ${}^{+260}_{-210}$ | ${}^{+260}_{-210}$ |
$f_{2}(1750)~{}\lambda=0$ | 2 | 2 | ${}^{+9}_{-9}$ | $\pm 9$ |
$f_{2}(1750)~{}|\lambda|=1$ | 12 | 10 | ${}^{+~{}2}_{-70}$ | ${}^{+16}_{-70}$ |
$f_{2}(1950)~{}\lambda=0$ | 13 | 1 | ${}^{+108}_{-~{}14}$ | ${}^{+110}_{-~{}20}$ |
$f_{2}(1950)~{}|\lambda|=1$ | 77 | 26 | ${}^{+130}_{-~{}~{}1}$ | ${}^{+150}_{-~{}80}$ |
Non-resonant | 18 | 7 | ${}^{+12}_{-~{}5}$ | ${}^{+23}_{-20}$ |
We estimate additional uncertainties by comparing the results when one more
resonance is added to the best fit model. The uncertainties due to the line
shape of the contributing resonances with fixed mass and width parameters are
estimated by varying them individually in the fit according to their combined
statistical and systematic uncertainties added in quadrature. We compare the
results with the best fit and add them in quadrature to estimate the
uncertainties due to the line shape.
Another source of systematic uncertainty is the value we choose for $L_{B}$,
the orbital angular momentum in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decay. If $L_{R}$ equals zero
then $L_{B}$ equals zero. If, however, $L_{R}$ is 1 then $L_{B}$ can either be
0 or 1, and if $L_{R}$ is 2, $L_{B}$ can be 1, 2 or 3. For our best fit we
don’t allow multiple values for $L_{B}$, but choose the lowest allowed value.
To estimate the systematic uncertainties due to the choice of $L_{B}$, we
repeat the fit changing the default value of $L_{B}$, in turn, to each higher
allowed value and compare the fit results with the best fit. The differences
are grouped into the fit model category, and we assign the largest variations
as the systematic uncertainties. These later two categories often give in
asymmetric uncertainties.
## 6 Absolute branching fractions
Branching fractions are measured from ratios of the decay rates of interest
normalized to the well established decay mode $B^{-}\rightarrow J/\psi K^{-}$.
This decay mode, in addition to having a well measured branching fraction, has
the advantage of having two muons in the final state and hence the same
triggers as the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decay.
However, we require knowledge of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}/B^{-}$ production ratio. For
this we assume isospin invariance and use the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}/\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ production ratio
$f_{s}/f_{d}=0.256\pm 0.020$, given in Ref. [36, *Aaij:2011jp]. The branching
fractions are calculated using
$\mathcal{B}(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi X)=\frac{N_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}}/\epsilon_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}}}{N_{B^{-}}/\epsilon_{B^{-}}}\times\mathcal{B}(B^{-}\rightarrow
J/\psi K^{-})\times\frac{1}{f_{s}/f_{d}},$ (38)
where $X$ indicates a specific $K^{+}K^{-}$ state, $N$ represents the yield of
the decay of interest, and $\epsilon$ corresponds to the overall efficiency.
We form an average of
$\mathcal{B}(B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-})=(10.18\pm 0.42)\times 10^{-4}$ using the recent Belle [38] and
BaBar [39] measurements, corrected to take into account different rates of
$B^{+}B^{-}$ and $B^{0}\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ pair
production from $\Upsilon(4S)$ using
$\frac{\Gamma(B^{+}B^{-})}{\Gamma(B^{0}\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0})}=1.055\pm 0.025$ [9].
The detection efficiency is obtained from simulation and is a product of the
geometrical acceptance of the detector, the combined reconstruction and
selection efficiency and the trigger efficiency. The efficiency also includes
the efficiency of the Dalitz plot model for the case of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi K^{+}K^{-}$,
where the best fit model is used. The detection efficiencies and their various
correction factors are given in Table 8. To ensure that the $p$ and $p_{\rm
T}$ distributions of the generated $B$ meson are correct we weight the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ simulations using $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi\phi(1020)$
data and the $B^{-}$ simulations using $B^{-}\rightarrow J/\psi K^{-}$ data.
Since the control channel has a different number of charged tracks than the
decay channel, we weight the simulations with the tracking efficiency ratio by
comparing the data and simulations in bins of the track’s $p$ and $p_{\rm T}$.
we further weight the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}\kern-1.60004ptK^{-}$ simulation, using the PDG value of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ lifetime, $(1.497\pm
0.015)\times 10^{-12}~{}\rm s$ [9], as input.
Table 8: Detector efficiencies determined from simulation and the correction factors. Item | $J/\psi K^{+}K^{-}$ | $J/\psi K^{-}$
---|---|---
Detection efficiency (%) | $1.061\pm 0.004$ | $2.978\pm 0.011$
Correction factors
Tracking efficiency | $0.999\pm 0.010$ | $1.003\pm 0.010$
PID | $0.819\pm 0.008$ | $0.974\pm 0.005$
$p$ and $p_{\rm T}$ | $1.077\pm 0.005$ | $1.053\pm 0.005$
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ lifetime | $0.993\pm 0.015$ | -
Total efficiency (%) | $0.887\pm 0.004\pm 0.018$ | $3.065\pm 0.012\pm 0.038$
The resulting branching fractions are
$\displaystyle\mathcal{B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi
K^{+}K^{-})~{}~{}$ $\displaystyle=$ $\displaystyle~{}~{}\\!(7.70\pm 0.08\pm
0.39\pm 0.60)\times 10^{-4},$ $\displaystyle\mathcal{B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\phi(1020))~{}$ $\displaystyle=$ $\displaystyle(10.50\pm 0.13\pm 0.64\pm
0.82)\times 10^{-4},$ $\displaystyle\mathcal{B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi
f_{2}^{\prime}(1525))$ $\displaystyle=$ $\displaystyle~{}~{}\\!(2.61\pm
0.20^{+0.52}_{-0.46}\pm 0.20)\times 10^{-4},$
where the branching fractions ${\cal{B}}(\phi(1020)\rightarrow
K^{+}K^{-})=(48.9\pm 0.5)\%$ and ${\cal{B}}(f_{2}^{\prime}(1525)\rightarrow
K^{+}K^{-})=(44.4\pm 1.1)\%$ are used [9]. Here the first uncertainty in each
case is statistical, the second is systematic and the third reflects the
uncertainty due to $f_{s}/f_{d}$. Note that these are the time-integrated
branching fractions. Results on the polarization fractions of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi(1020)$ from a time-dependent analysis will be forthcoming in a
separate publication [40]. The ratio of $\mathcal{B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi
f_{2}^{\prime}(1525))/\mathcal{B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi\phi(1020))$
is consistent with our previous result [10], D0 [11], and the Belle result
[41]. The current PDG value of $\mathcal{B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\phi(1020))=(1.4\pm 0.5)\times 10^{-3}$ is dominated by the CDF
measurement [42]. Our measured value is in good agreement with this
measurement and also the most recent yet unpublished values measured by CDF
[43] and Belle [41]. The Belle collaboration has also recently reported the
branching fraction of $\mathcal{B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi K^{+}K^{-})$
[41], where $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\phi(1020)$ is excluded.
Table 9: Relative systematic uncertainties on branching fractions (%). Item | $J/\psi K^{+}K^{-}$ | $J/\psi\phi(1020)$ | $J/\psi f_{2}^{\prime}(1525)$
---|---|---|---
Tracking efficiency | 1.0 | 1.0 | 1.0
Material and physical effects | 2.0 | 2.0 | 2.0
PID efficiency | 1.0 | 1.0 | 1.0
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ $p$ and $p_{\rm T}$ distributions | 0.5 | 0.5 | 0.5
$B^{-}$ $p$ and $p_{\rm T}$ distributions | 0.5 | 0.5 | 0.5
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ lifetime | 1.5 | 1.5 | 1.5
Efficiency function | 0.02 | 0.02 | 0.02
$\mathcal{B}(\phi(1020)\rightarrow K^{+}K^{-})$ | - | 1.0 | -
$\mathcal{B}(f_{2}^{\prime}(1525)\rightarrow K^{+}K^{-})$ | - | - | 2.5
$\mathcal{B}(B^{-}\rightarrow J/\psi K^{-})$ | 4.1 | 4.1 | 4.1
Contributions from Dalitz analysis
Efficiency | - | 3.1 | 12.3
Background | - | 0 | 1.3
Fit model | - | ${}^{+0.7}_{-0.2}$ | ${}^{+14.6}_{-11.2}$
Sum in quadrature of items above | 5.0 | ${}^{+6.1}_{-6.0}$ | ${}^{+20.0}_{-17.7}$
$f_{s}/f_{d}$ | $7.8$ | $7.8$ | $7.8$
The systematic uncertainty on the branching fraction has several contributions
listed in Table 9. Since the branching fractions are measured with respect to
$B^{-}\rightarrow J/\psi K^{-}$ which has a different number of charged tracks
than the decays of interest, a 1% systematic uncertainty is assigned due to
differences in the tracking performance between data and simulation. Another
2% uncertainty is assigned for the additional kaon which is due to decay in
flight, large multiple scatterings and hadronic interactions along the track.
Using the PDG value for the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ lifetime [9] as input gives
rise to an additional 1.5% systematic uncertainty. Small uncertainties are
introduced if the simulation does not have the correct $B$ meson kinematic
distributions. We are relatively insensitive to any of these differences in
the $B$ meson $p$ and $p_{\rm T}$ distributions since we are measuring the
relative rates. By varying the $p$ and $p_{\rm T}$ distributions we see at
most a change of 0.5%. There is a 1% systematic uncertainty assigned for the
relative particle identification efficiencies. An uncertainty of 0.02% is
included due to the change of the efficiency function Eq. (20). Three
additional uncertainties are considered in the branching fractions of
$\mathcal{B}(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\phi(1020))$ and $\mathcal{B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi
f_{2}^{\prime}(1525))$ as these are measured from the fit fractions of the
Dalitz plot analysis. The total systematic uncertainty is obtained by adding
each source of systematic uncertainty in quadrature as they are uncorrelated.
## 7 Conclusions
We have determined the final state composition of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi K^{+}K^{-}$
decay channel using a modified Dalitz plot analysis where we include the decay
angle of the $J/\psi$. The largest contribution is the $\phi(1020)$ resonance,
along with other S-, P- and D-wave $K^{+}\kern-1.60004ptK^{-}$ states, and a
non-resonant $K^{+}\kern-1.60004ptK^{-}$ contribution. All of the components
are listed in Table 4. The mass and width of the $f_{2}^{\prime}(1525)$
resonance are measured as
$\displaystyle m_{f_{2}^{\prime}(1525)}$ $\displaystyle=$ $\displaystyle
1522.2\pm 2.8^{+5.3}_{-2.0}~{}\rm MeV,$
$\displaystyle\Gamma_{f_{2}^{\prime}(1525)}$ $\displaystyle=$ $\displaystyle
84\pm 6^{+10}_{-~{}5}~{}\rm MeV.$
We also observe a significant S-wave component that is present over the entire
$K^{+}K^{-}$ mass region. Within $\pm$12 MeV of the $\phi(1020)$ mass it is
$(1.1\pm 0.1^{+0.2}_{-0.1})\%$ of the yield, and can affect precision $C\\!P$
violation measurements [12]. Finally we determine the absolute branching
fractions
$\displaystyle\mathcal{B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi
K^{+}K^{-})~{}~{}$ $\displaystyle=$ $\displaystyle~{}~{}\\!(7.70\pm 0.08\pm
0.39\pm 0.60)\times 10^{-4},$ $\displaystyle\mathcal{B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\phi(1020))~{}$ $\displaystyle=$ $\displaystyle(10.50\pm 0.13\pm 0.64\pm
0.82)\times 10^{-4},$ $\displaystyle\mathcal{B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi
f_{2}^{\prime}(1525))$ $\displaystyle=$ $\displaystyle~{}~{}\\!(2.61\pm
0.20^{+0.52}_{-0.46}\pm 0.20)\times 10^{-4},$
where the first uncertainty in each case is statistical, the second is
systematic and the third due to $f_{s}/f_{d}$. These results provide a good
understanding of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}K^{-}$
final state in $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays
over the entire kinematically allowed region. The $J/\psi
f_{2}^{\prime}(1525)$ results supersede those of Ref. [10]. This decay mode
offers the opportunity for additional measurements of $C\\!P$ violation [44,
*Zhang:2012zk].
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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* [43] CDF collaboration, Measurement of ${\cal{B}}\left(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi\right)$ and extraction of the fragmentation functions, Public Note 10795
* [44] C. Sharma and R. Sinha, Angular analysis of $B$ decaying into $J/\psi$ tensor, $J/\psi$ vector and $J/\psi$ scalar modes, Phys. Rev. D73 (2006) 014016, arXiv:hep-ph/0504178
* [45] L. Zhang and S. Stone, Time-dependent Dalitz-plot formalism for $B_{q}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}h^{+}h^{-}$, Phys. Lett. B719 (2013) 383, arXiv:1212.6434
|
arxiv-papers
| 2013-02-05T21:42:47 |
2024-09-04T02:49:41.353741
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, A. Adametz, B. Adeva,\n M. Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio,\n M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S.\n Amato, Y. Amhis, L. Anderlini, J. Anderson, R. Andreassen, R.B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W. Baldini, R.J.\n Barlow, C. Barschel, S. Barsuk, W. Barter, A. Bates, Th. Bauer, A. Bay, J.\n Beddow, I. Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M.\n Benayoun, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O.\n Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M.\n Bj{\\o}rnstad, T. Blake, F. Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov,\n V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V.\n Bowcock, E. Bowen, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D.\n Brett, M. Britsch, T. Britton, N.H. Brook, H. Brown, A. B\\\"uchler-Germann, I.\n Burducea, A. Bursche, J. Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo\n Gomez, A. Camboni, P. Campana, A. Carbone, G. Carboni, R. Cardinale, A.\n Cardini, H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, M.\n Cattaneo, Ch. Cauet, M. Charles, Ph. Charpentier, P. Chen, N. Chiapolini, M.\n Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic,\n H.V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, E. Cogneras, P. Collins,\n A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, G. Corti, B. Couturier,\n G.A. Cowan, D. Craik, S. Cunliffe, R. Currie, C. D'Ambrosio, P. David, P.N.Y.\n David, I. De Bonis, K. De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L.\n De Paula, W. De Silva, P. De Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi,\n L. Del Buono, C. Deplano, D. Derkach, O. Deschamps, F. Dettori, A. Di Canto,\n J. Dickens, H. Dijkstra, P. Diniz Batista, M. Dogaru, F. Domingo Bonal, S.\n Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis,\n R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V. Egorychev, S.\n Eidelman, D. van Eijk, S. Eisenhardt, U. Eitschberger, R. Ekelhof, L. Eklund,\n I. El Rifai, Ch. Elsasser, D. Elsby, A. Falabella, C. F\\\"arber, G. Fardell,\n C. Farinelli, S. Farry, V. Fave, D. Ferguson, V. Fernandez Albor, F. Ferreira\n Rodrigues, M. Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M. Fontana, F.\n Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M. Frosini, S. Furcas,\n E. Furfaro, A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini, Y. Gao,\n J. Garofoli, P. Garosi, J. Garra Tico, L. Garrido, C. Gaspar, R. Gauld, E.\n Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V.V. Gligorov, C.\n G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa\n G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G. Graziani,\n A. Grecu, E. Greening, S. Gregson, O. Gr\\\"unberg, B. Gui, E. Gushchin, Yu.\n Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S.C. Haines, S. Hall, T.\n Hampson, S. Hansmann-Menzemer, N. Harnew, S.T. Harnew, J. Harrison, P.F.\n Harrison, T. Hartmann, J. He, V. Heijne, K. Hennessy, P. Henrard, J.A.\n Hernando Morata, E. van Herwijnen, E. Hicks, D. Hill, M. Hoballah, C.\n Hombach, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, N. Hussain, D.\n Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J. Imong, R. Jacobsson, A.\n Jaeger, E. Jans, F. Jansen, P. Jaton, F. Jing, M. John, D. Johnson, C.R.\n Jones, B. Jost, M. Kaballo, S. Kandybei, M. Karacson, T.M. Karbach, I.R.\n Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji, O. Kochebina, I. Komarov,\n R.F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K.\n Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V.\n Kudryavtsev, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G. Lafferty, A.\n Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch,\n T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre,\n A. Leflat, J. Lefran\\c{c}ois, O. Leroy, Y. Li, L. Li Gioi, M. Liles, R.\n Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben, J.H. Lopes, E. Lopez Asamar,\n N. Lopez-March, H. Lu, J. Luisier, H. Luo, A. Mac Raighne, F. Machefert, I.V.\n Machikhiliyan, F. Maciuc, O. Maev, S. Malde, G. Manca, G. Mancinelli, N.\n Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, L.\n Martin, A. Mart\\'in S\\'anchez, M. Martinelli, D. Martinez Santos, D. Martins\n Tostes, A. Massafferri, R. Matev, Z. Mathe, C. Matteuzzi, M. Matveev, E.\n Maurice, A. Mazurov, J. McCarthy, R. McNulty, B. Meadows, F. Meier, M.\n Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez, S.\n Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller,\n R. Muresan, B. Muryn, B. Muster, P. Naik, T. Nakada, R. Nandakumar, I.\n Nasteva, M. Needham, N. Neufeld, A.D. Nguyen, T.D. Nguyen, C. Nguyen-Mau, M.\n Nicol, V. Niess, R. Niet, N. Nikitin, T. Nikodem, S. Nisar, A. Nomerotski, A.\n Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O.\n Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora Goicochea, P. Owen, B.K.\n Pal, A. Palano, M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, C.\n Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel, G.N. Patrick, C.\n Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A. Pellegrino, G. Penso, M.\n Pepe Altarelli, S. Perazzini, D.L. Perego, E. Perez Trigo, A. P\\'erez-Calero\n Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina, K. Petridis, A.\n Petrolini, A. Phan, E. Picatoste Olloqui, B. Pietrzyk, T. Pila\\v{r}, D.\n Pinci, S. Playfer, M. Plo Casasus, F. Polci, G. Polok, A. Poluektov, E.\n Polycarpo, D. Popov, B. Popovici, C. Potterat, A. Powell, J. Prisciandaro, V.\n Pugatch, A. Puig Navarro, W. Qian, J.H. Rademacker, B. Rakotomiaramanana,\n M.S. Rangel, I. Raniuk, N. Rauschmayr, G. Raven, S. Redford, M.M. Reid, A.C.\n dos Reis, S. Ricciardi, A. Richards, K. Rinnert, V. Rives Molina, D.A. Roa\n Romero, P. Robbe, E. Rodrigues, P. Rodriguez Perez, G.J. Rogers, S. Roiser,\n V. Romanovsky, A. Romero Vidal, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino,\n J.J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann, B.\n Sanmartin Sedes, M. Sannino, R. Santacesaria, C. Santamarina Rios, E.\n Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D.\n Savrina, P. Schaack, M. Schiller, H. Schindler, S. Schleich, M. Schlupp, M.\n Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R.\n Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska,\n I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P.\n Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V.\n Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, N.A. Smith, E.\n Smith, M. Smith, K. Sobczak, M.D. Sokoloff, F.J.P. Soler, F. Soomro, D.\n Souza, B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S.\n Stahl, O. Steinkamp, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U.\n Straumann, V.K. Subbiah, S. Swientek, V. Syropoulos, M. Szczekowski, P.\n Szczypka, T. Szumlak, S. T'Jampens, M. Teklishyn, E. Teodorescu, F. Teubert,\n C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S. Tolk, D.\n Tonelli, S. Topp-Joergensen, N. Torr, E. Tournefier, S. Tourneur, M.T. Tran,\n M. Tresch, A. Tsaregorodtsev, P. Tsopelas, N. Tuning, M. Ubeda Garcia, A.\n Ukleja, D. Urner, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P.\n Vazquez Regueiro, S. Vecchi, J.J. Velthuis, M. Veltri, G. Veneziano, M.\n Vesterinen, B. Viaud, D. Vieira, X. Vilasis-Cardona, A. Vollhardt, D.\n Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo\\ss, H. Voss, R. Waldi,\n R. Wallace, S. Wandernoth, J. Wang, D.R. Ward, N.K. Watson, A.D. Webber, D.\n Websdale, M. Whitehead, J. Wicht, J. Wiechczynski, D. Wiedner, L. Wiggers, G.\n Wilkinson, M.P. Williams, M. Williams, F.F. Wilson, J. Wishahi, M. Witek, W.\n Witzeling, S.A. Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, F. Xing, Z.\n Xing, Z. Yang, R. Young, X. Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev,\n F. Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, A. Zhokhov, L. Zhong,\n A. Zvyagin",
"submitter": "Sheldon Stone",
"url": "https://arxiv.org/abs/1302.1213"
}
|
1302.1224
|
# Key Choices in the Design of Simple Knowledge Organization System (SKOS)
Thomas Baker Sean Bechhofer [email protected] Antoine Isaac
Alistair Miles Guus Schreiber Ed Summers Dublin Core Metadata Initiative,
c/o National Library Board Singapore, 100 Victoria Street 14-01, Singapore
188064 University of Manchester, Oxford Road, Manchester M13 9PL, UK Vrije
Universiteit Amsterdam, De Boelelaan 1081a 1081 HV Amsterdam, NL Europeana,
c/o Koninklijke Bibliotheek, PO Box 90407, 2509 LK The Hague, NL Wellcome
Trust Centre for Human Genetics, Roosevelt Drive, Oxford OX3 7BN, UK Library
of Congress, 101 Independence Ave, SE Washington, DC 20540, US
###### Abstract
Simple Knowledge Organization System (SKOS) provides a data model and
vocabulary for expressing Knowledge Organization Systems (KOSs) such as
thesauri and classification schemes in Semantic Web applications. This paper
presents the main components of SKOS and their formal expression in Web
Ontology Language (OWL), providing an extensive account of the design
decisions taken by the Semantic Web Deployment (SWD) Working Group of the
World Wide Web Consortium (W3C), which between 2006 and 2009 brought SKOS to
the status of W3C Recommendation. The paper explains key design principles
such as ”minimal ontological commitment” and systematically cites the
requirements and issues that influenced the design of SKOS components.
By reconstructing the discussion around alternative features and design
options and presenting the rationale for design decisions, the paper aims at
providing insight into how SKOS turned out as it did, and why. Assuming that
SKOS, like any other successful technology, may eventually be subject to
revision and improvement, the critical account offered here may help future
editors approach such a task with deeper understanding.
## Introduction
Simple Knowledge Organization System (SKOS) — a vocabulary and data model for
expressing Knowledge Organization Systems (KOSs) such as thesauri and
classification schemes for referencing and re-use in Semantic Web applications
— was developed by successive projects and working groups from the late 1990s
through its publication in August 2009 as a World Wide Web Consortium (W3C)
Recommendation.111http://www.w3.org/2009/07/skos-pr This paper describes the
work of the W3C Semantic Web Deployment Working Group, which was chartered in
2006 to carry SKOS Core, a W3C Working Draft, through the rigorous review
required by the W3C Recommendation Track
process.222http://www.w3.org/2006/07/swdwg-charter
The final results of that process are recorded in the formal specification for
SKOS [14]. This paper, in contrast, focuses on the process itself. By
reconstructing the discussion around alternative features and design options
and presenting the rationale for key decisions, the paper aims at providing
insight into how SKOS turned out as it did, and why. Assuming that SKOS, like
any other successful technology, may eventually be subject to revision and
improvement, the critical account offered here may help future editors
approach such a task with deeper understanding.
After presenting a brief history of SKOS from 1997 through 2009, the paper
outlines the rationale for a language, other than existing formal ontology
languages, for expressing Knowledge Organization Systems. Drawing a contrast
between logically precise conceptual structures and more intuitive, pragmatic
knowledge representations, the section describes the principle of “minimal
ontological commitment” that guided the design of SKOS.
The middle section of the paper walks through the components of the SKOS model
— SKOS _Concepts_ (and how they differ from formal-ontological Classes),
_Concept Schemes_ , _Semantic Relations_ between concepts, _Lexical Labels_ ,
_Documentation Properties_ , and _Collections_ of concepts. The section
considers several proposed features of SKOS deemed by the working group to be
out of scope.
While SKOS was developed for expressing KOSs as “concept schemes” — sets of
interrelated concepts — without modeling those concepts as formal “classes,”
the data model for SKOS itself is defined as an ontology, i.e., as a set of
formal properties and classes expressed using the W3C Web Ontology Language
(OWL). The final section of the paper reviews the semantics of SKOS properties
and classes as defined by axioms supporting inference and “integrity
conditions” for when data can be considered “not consistent” with the SKOS
data model. The section also considers the compatibility of SKOS with
different variants of OWL and with previous versions of SKOS itself. The paper
concludes by highlighting issues that could provide starting points for a
future revision of the specification.
## 1 History of SKOS from the late 1990s through 2009
Today’s SKOS can be traced back to work on improving search interfaces in the
European project Desire (1997–2000). The original W3C Resource Description
Framework (RDF) Model and Syntax Working Group (1997–1999), aware of Desire,
raised the question of expressing thesauri in RDF as an
issue.333https://www.w3.org/RDF/Group/Schema/openissues.html Phil Cross, Dan
Brickley, and Traugott Koch turned the Desire results into a proposal,
published jointly by the Institute for Learning and Research Technology (ILRT)
in the UK and the Lund University Library Netlab in Sweden, “for encoding a
core set of thesaurus relationships using an RDF
schema.”444http://www.ilrt.bristol.ac.uk/publications/researchreport/rr1011/report_html?ilrtyear=00
This draft schema was picked up by the European project LIMBER (Language
Independent Metadata Browsing of European Resources, 1999–2001), which defined
a vocabulary based more explicitly on “concepts” labeled by terms in multiple
languages.555http://journals.tdl.org/jodi/article/view/32/33
The results of the LIMBER Project fed into the SWAD Europe project (Semantic
Web Advanced Development, 2001–2004).666http://www.w3.org/2001/sw/Europe/ In
SWAD Europe, Alistair Miles of Rutherford Labs solicited input from experts on
thesaurus and classification standards, creating a community of interested
users, for whom W3C set up a community mailing list, public-esw-
thes,777http://lists.w3.org/Archives/Public/public-esw-thes/ and the revised
vocabulary was published under the name “Simple Knowledge Organization
System.” This draft was picked up in 2004 by the W3C Semantic Web Best
Practice and Deployment Working Group (2004–2006), whose Porting Thesauri Task
Force888http://www.w3.org/2004/03/thes-tf/mission created a home page for what
was now called “SKOS Core.” In 2005, the working group published “SKOS Core
Vocabulary Specification” as a W3C Working
Draft.999http://www.w3.org/TR/2005/WD-swbp-skos-core-spec-20051102/ SKOS Core
was taken as a starting point for the review process described in this paper.
The Semantic Web Deployment Working Group began by distilling requirements for
SKOS out of use cases solicited from early adopters about present and future
applications [7]. Successive revisions of the 2005 SKOS Core specification
were posted for public comment as Working Drafts, then as Candidate and
Proposed Recommendations, prior to finalization as a W3C Recommendation. The
editors of the specification were supported by two working-group chairs with
active input from a dozen or two working-group members and a wider circle of
external reviewers and mailing-list followers. Discussion took place on the
working group’s mailing list101010http://lists.w3.org/Archives/Public/public-
swd-wg/ and on public-esw-thes for the wider
community.111111http://lists.w3.org/Archives/Public/public-esw-thes/ The group
met over a period of 35 months in three face-to-face meetings and 110 near-
weekly teleconferences. Teleconferences used W3C’s bot-supported telephone
bridge, which assigned URIs to actions “scribed” into a shared chat channel
and automatically generated draft minutes, complete with pointers to the
agenda, previous minutes, actions past and current, mailing-list postings, and
document drafts. As technical or design issues were formally raised they were
assigned URIs and added to an Issue Tracker
121212http://www.w3.org/2006/07/SWD/track/issues/ that automatically collected
links to any minutes or postings in which the issues were mentioned. Each such
URL cited in this paper leads the interested reader into a web of richly
interlinked working-group resources.
The following discussion will make reference to online resources produced
during the WG process. Rather than peppering the narrative text with URIs,
references to issues, requirements and axioms will be handled as follows:
* 1.
Issues. Details of all issues are documented in the Working Group’s issue
tracker at http://www.w3.org/2006/07/SWD/track/issues. Issues will be cited in
the text by number, e.g., Issue 27.
* 2.
Requirements. Requirements are documented in the SKOS Use Cases and
Requirement document at http://www.w3.org/TR/skos-ucr. Requirements will be
referred to in the text by their handles, e.g.,
R-GroupingInConceptHierarchies.
* 3.
Axioms. SKOS axioms are listed in Tables 2 and 3. Details of these axioms are
given in the SKOS Reference document at http://www.w3.org/TR/skos-reference.
Axioms will be referred to in the text by their “S” handle, e.g., S1.
For all of the above, full URI references will be available in digital
versions of the paper.
## 2 Rationale for SKOS
Many institutions develop and maintain Knowledge Organization Systems (KOSs) —
thesauri, classification systems, subject heading lists, folksonomies, and the
like, holding concepts and terminologies for a wide range of domains — as
backbone structures for their information systems. The potential of such KOSs
to serve as components in knowledge-rich applications has been recognized
since the rise of the Web in the 1990s.
Porting an existing KOS for use in Semantic Web applications, however, is not
a trivial problem. The Semantic Web languages for expressing domain knowledge
are mathematically formal in nature. The vocabulary description language of
the Resource Description Framework (RDFS) and the Web Ontology Language (OWL),
in particular, provide ways to define classes and properties and to associate
those classes and properties with formal reasoning rules that enforce
constraints or produce new knowledge by inference. KOSs, on the other hand,
have typically been designed not as formally precise representations of domain
knowledge, but as informal structures reflecting the intuitive knowledge of
human users in a form useful for resource discovery (e.g., through supporting
query expansion). KOSs have variously been classified as “term-based” or
“concept-based” depending on how explicitly they are intended to represent
conceptual structures.131313The evolution of standards such as ISO2788 [9]
(into ISO25964 [10]) illustrates the shift, but also the continuity from one
representation approach to the other. Traditional KOS standards have never
included the sort of formal axioms expressed by Semantic Web ontology
languages.
Informally defined KOSs cannot typically be translated into the language of
RDFS and OWL properties and classes, with their formal-logical implications,
without introducing potentially false or misleading logical precision.
Informal KOSs may be converted into formal ontologies (see [6]), but the
process of assigning appropriate formal semantics to the elements of a KOS may
require a long, hard modeling effort. Hierarchical relationships, for example,
must be disambiguated into relationships of class instantiation, class
subsumption, part-whole, or other types — a process that cannot usually be
automated. An analysis of the thesaurus of the National Cancer Institute [4]
(as reported in [3]), for example, found conceptual structures that are
incompatible with formalized frameworks that assume stricter modeling
principles. The AGROVOC thesaurus of multilingual agricultural terminology,
the product of many people over many years working from multiple perspectives,
was straightforwardly converted into a hierarchy of OWL classes many years
before the finalization of SKOS. While the maintainers of AGROVOC-in-OWL
intended to increase its ontological precision over time, through editorial
correction and refinement, it eventually proved to be more practical simply to
convert AGROVOC back into the formally less “committed” form of a SKOS concept
scheme, leaving it to designers of specific implementations to upgrade parts
of the thesauri into class-based ontologies when required to support reasoning
[2].
The traditional use cases for which KOSs were typically designed are still
relevant in the Web context. One key role of a controlled vocabulary, for
example, is to improve precision when retrieving objects from an indexed
collection. The hierarchical and associative relationships of thesauri enable
users to browse for search terms, and information retrieval applications can
use this structure to automatically expand queries, which improves recall.
Applications such as simple search or browsing of documents or “conceptual
spaces” can all benefit from a shared basis for data exchange and linking. For
such purposes, Semantic Web technology is indeed a game changer, as it allows
users and developers to seamlessly re-use data from different contexts, or to
link together multiple KOSs, in order to achieve broader or deeper search,
even across languages. Expressing KOSs as Linked Data allows the library
community to create pools of trusted URIs citable by catalogers in resource
descriptions in support of such applications [1, 19].
SKOS aims at providing a path for migrating KOSs to a Semantic Web context at
low cost by expressing features common to a wide range of KOS types. The SKOS
properties for “broader,” “narrower,” and “related,” for example, are intended
to capture the native, sometimes ambiguous semantics of existing thesauri and
similar structured vocabularies. Using SKOS, no additional intellectual work
is required to represent these relationships in RDF, allowing the maintainers
of controlled structured vocabularies to leverage their existing investments.
The design of SKOS followed the principle of making a _minimal ontological
commitment_ to the nature of concepts and of relationships between concepts.
As explained by Thomas Gruber [5]:
> An ontology should require the minimal ontological commitment sufficient to
> support the intended knowledge sharing activities. An ontology should make
> as few claims as possible about the world being modeled, allowing the
> parties committed to the ontology freedom to specialize and instantiate the
> ontology as needed.
The principle of avoiding over-commitment guided many of the discussions about
possible extensions to SKOS. Where the use cases collected by the working
group demonstrated no clear requirement for a candidate feature, or in the
absence of clear usage experience, the group tended to opt for a “safe” course
of action. As a result, SKOS captures the basic, informal semantics most
commonly required by the use cases. Where there was doubt that a particular
feature would be easy to understand or use, the working group generally chose
to omit the feature from the specification.
The working group was particularly focused on keeping SKOS compatible with the
thesaurus standards ISO 2788 and ISO 5964,141414The SKOS Primer includes a
table of correspondences with ISO 2788 and ISO 5964 http://www.w3.org/TR/skos-
primer/#seccorrespondencesISO. with the result that the SKOS data model
reflects standard thesaurus construction principles. SKOS does not, however,
express all of the best practices described in the ISO standards, nor does it
include the elements needed to capture all of the features of any given,
existing KOS standard, such as specializations of broader and narrower
hierarchical relations (see Section 3.3). Experience indeed shows that best
practices are not always followed—a problem revealed, for example, when
generic “See also” references in the Library of Congress Subject Headings were
converted into standardized thesaurus relations [18]—and that some KOSs use
idiosyncratic constructs for meeting very specific requirements. The working
group felt that fully committing SKOS to supporting the creation and
validation of any particular type of concept scheme, such as a standard
thesaurus, would create an obstacle to the wide-spread adoption of SKOS by
users of other types.
Lightly specified by design, SKOS is intended to prevent data publishers from
introducing false precision into their data and to prevent inference engines
from drawing unwarranted conclusions. In some cases, however, the
specification recommends usage conventions, such as best practices for KOS
design. The SKOS model thus presents two layers of specification: formal,
enforceable axioms, along with weaker “guidelines.” Guidelines are not
represented formally, nor they are considered to be inviolable integrity
constraints; rather, they are considered to be advisory.
Opting for such a minimal approach is made dramatically easier by the
vocabulary extension mechanisms offered natively by Semantic Web technology.
Applications that require more constrained behaviour may define compatible
extensions to SKOS [8]. For example, modelers may coin sub-classes and sub-
properties of SKOS properties or associate those properties with specific
formal axioms. The RDF data model allows properties from such extension
vocabularies to be used alongside properties from SKOS in expressing data.
Where properties seen as required were already provided elsewhere, such as the
Dublin Core property dc:subject, the working group deferred to existing
vocabularies.
## 3 Components of SKOS
Using SKOS, concepts can be identified using URIs, labeled with lexical
strings in one or more natural languages, assigned notations (lexical codes),
documented with various types of note, linked to other concepts and organized
into informal hierarchies and association networks, aggregated into concept
schemes, grouped into labeled and/or ordered collections, and mapped to
concepts in other schemes.
The SKOS data model enables features listed above—identifying, labeling,
documenting, linking, and mapping concepts, and aggregating concepts into
concept schemes or collections—by defining the elements depicted in Figure 1.
This section looks at the design choices made in modeling those components.
Figure 1: Main elements of the SKOS data model
### 3.1 SKOS Concepts (and how they differ from OWL Classes)
#### A wide diversity of concepts.
SKOS is designed to express, in an interoperable way, different types of
Knowledge Organization System—sets of terms or concepts, whether listed with
definitions (glossaries), in hierarchical structures (basic classifications or
taxonomies), or characterized by more complex semantic relations (thesauri,
subject heading lists, or other advanced structures). Each type of KOS has its
own specific characteristics. Yet they all organize knowledge by gathering a
coherent set of lexical entities (terms, words, headings, captions…) around
more abstract notions that the SKOS model represents as _Concepts_. In a
thesaurus, for example, a concept is the construct that clusters a _preferred
term_ (the one used for describing resources in a document retrieval system)
with near-synonymous _alternative terms_ (or variants). A KOS may link such
concepts among themselves with various types of semantic relations, such as
class-subclass, part-whole, or looser associative links.
SKOS leaves ample room for interpreting the notion of concept, and many
artifacts from information science and other fields fall in scope. As the SKOS
Reference puts it [14], “a SKOS concept can be viewed as an idea or notion; a
unit of thought. However, what constitutes a unit of thought is subjective,
and this definition is meant to be suggestive, rather than restrictive.”
If the objective is information retrieval via the use of a knowledge
organisation system as a subject indexing language, then one can take an
operational view and define concepts as units of indexing and retrieval [21].
The subject indexing process can then be viewed as the action of linking
documents (such as a textbook about butterflies) to concepts (such as a
concept labeled “butterflies”), and the retrieval process involves selecting
one or more concepts to use as a subject query and retrieving the sets of
documents linked to those concepts. One possible formulation of this view is
to describe the set of documents linked to the same concept as a kind of
“document extension” of that concept. Some approaches to mapping are based on
this notion, as when conceptual equivalences between concepts are derived by
measuring the overlap between the document extensions with which they are
associated [25].
Information retrieval use cases are an important motivation for SKOS, and the
SKOS data model is perfectly compatible with this view. However, subject
indexing and retrieval are not the only uses for knowledge organisation
systems, so SKOS does not attempt to normatively define or formalize any
relationship between documents and concepts. This flexibility enables SKOS to
represent knowledge organisation systems used in a variety of applications, as
well as enabling implementers of information retrieval systems to explore
alternative retrieval strategies and algorithms such as query expansion.
Moreover, conceptual vocabularies need not be intended primarily for
describing documents for information retrieval. At the most basic level,
applications merely require that a concept have identity and that it have
features which distinguish it from other concepts within a KOS, such as
natural-language labels, definitions, and semantic relations to other
concepts. The group felt that this simple, flexible model would cover most of
the available use cases and requirements while enabling a broad range of
applications, whereas formally reconciling traditional KOS models at a higher
degree of granularity would have been both more difficult and of less obvious
utility.
#### SKOS concepts vs. OWL classes.
The concepts from Knowledge Organization Systems (and hence, SKOS concepts)
are often wrongly interpreted as classes from formal ontology languages like
OWL. Some SKOS concepts indeed reflect universal categories that also appear
in in OWL ontologies, such as “animals” and “cats” in a vocabulary animals.
Yet, as seen previously, SKOS concepts are not by default provided with
precise extensional semantics, while an OWL class explicitly describes a
collection of individuals. Following the example above, the interpretation of
an OWL Class “Butterfly” would be the set of butterflies.
From a technical perspective, SKOS _Concepts_ are simply individuals in an OWL
ontology (see Section 4). These individuals are interpreted as arbitrary
elements in the domain that _might_ (or might not) correspond to collections
of indexed documents.
Hierarchical relationships between SKOS _Concepts_ , moreover, _might_ (or
might not) correspond to sub-class relationships between OWL classes
(owl:Class). Where a KOS might assert _broader_ links between “Dog,” “Collie”
(a type of dog), and “Lassie” (a particular dog), an OWL ontology might
specify that “Collie” is a sub-class of “Dog” (rdfs:subClassOf), then “Lassie”
could be described as an instance of “Collie” (rdf:type). Some KOS standards
distinguish a class-instance variant of “broader,” such as the “broader
instantive” relationship of ISO 2788 [9]. (Indeed, a preliminary draft SKOS
Extensions Vocabulary Specification with “broader instantive” and other such
variants of the semantic relation properties was posted for discussion in
2004.151515http://www.w3.org/2004/02/skos/extensions/spec/2004-10-18.html)
Many KOSs, however, use the same general relation for all hierarchical links.
As the SKOS Reference says, concept data could be “facts about the thesaurus
or classification scheme itself, such as ‘concept X has preferred label “Y” ’;
not facts about the way the world is arranged within a particular subject
domain, as might be expressed in a formal ontology.” As hinted in Section 2,
converting a traditional KOS into an OWL ontology may thus require some hard
“cleaning.” Conversion into a straightforward, lightweight representation is
often the most appropriate course of action [2, 24]. One may see formal (OWL)
ontologies as KOSs, as they organize knowledge in a domain; but clearly not
all KOSs can be directly interpreted as formal ontologies.
#### Disjointness of the SKOS Concept Class.
The previous paragraph highlights that KOS concepts function in a quite
specific way: one can view them as proxies that establish an “indirection
layer” between lexical entities and “real-world” ones, either represented as
individuals (say, a person) or classes (say, all persons with a specific role)
in the OWL sense. A test for identifying such resources could be for example
the “date of creation” associated with them. The resource that stands for a
person in a name authority file (thus, represented as an instance of SKOS
concept) will probably have a different date associated to it than the one
associated to the resource that stands for that person as a “real person”
(represented using the foaf:Person class).161616For example, the data
available for an authority name for Michelle Obama indicate a creation date of
2008, quite some time after Michelle Obama was born. See
http://id.loc.gov/authorities/names/n2008054754 and a discussion on the SKOS
community mailing list, http://lists.w3.org/Archives/Public/public-esw-
thes/2009Nov/0000.html
This observation leads to a first kind of representation pattern, which
distinguishes separate KOS concepts from entities in the real world (or
classes thereof) and represent the referential link between them using
properties such as foaf:focus (defined as “The underlying or ‘focal’ entity
associated with some SKOS-described
concept.”171717http://xmlns.com/foaf/spec/#term_focus). This is for example
what was retained for the Virtual International Authority File
(VIAF),181818http://viaf.org/ which creates for each cluster of authority
records an instance of foaf:Person and (at least) one instance of
skos:Concept, linking the latter (e.g.,
http://viaf.org/viaf/sourceID/SELIBR%7C317488#skos:Concept) to the former
(http://viaf.org/viaf/85312226) using foaf:focus.
Such approaches would fit well a modeling choice making skos:Concept disjoint
with other classes of entities, such as foaf:Person or meta-modeling classes
like owl:Class. This would rule out that different “modelling streams,” each
coming with different kind of possibly incompatible data, are “crossed” within
one same graph. (The cultural reference in working-group discussions on this
topic was that of the dire warning, from the 1984 film _Ghostbusters_
,191919imdb:ghostbusters never to “cross the streams” of proton beams from
multiple particle throwers because, vaguely but ominously, “it would be bad.”)
However, the working group opted for not asserting explicit disjointness
between SKOS concepts and non-SKOS classes. The first reason is quite
pragmatic: the world of ontologies beyond SKOS is wide, and choosing specific
classes for disjointness statements would have been an incomplete, biased
effort. skos:Concept is only formally disjoint with other classes in the SKOS
namespace: skos:Collection, skos:ConceptScheme, skosxl:Label, which will be
introduced below. Advocates of the disjointness option may argue that SKOS
could have featured a new class of “Non-Concepts” to handle the case, next to
a property to relate the two disjoint categories.202020The SKOS Primer
suggests to use a dedicated (annotation) property like
ex:correspondingConcept. A skos:it was also proposed. It was only after SKOS
was published as a recommendation that foaf:focus emerged as possible standard
candidate. At that time, however, the group felt that the proposed patterns
were still not mature enough and quite out of the scope defined in our
charter, because such information is usually not present in the data defining
concepts in existing KOSs.
The second reason for not declaring SKOS concepts to be disjoing with “non-
concepts” stems, again, from the requirement for minimal commitment, as well
as from a concern not to rule out valid patterns. As pointed out above, a
skos:Concept is intended to provide a neutral target for migrating a wide
diversity of KOS concepts to the Web of Data. This includes cases where
concepts are also elements of formalized ontologies. Some OWL properties and
classes can be seen as members of a concept scheme as, for example, in
applications that would not handle the full complexity of OWL reasoning but
would require lexical annotations richer than those supported by OWL. The
Library of Congress, for example, represents MARC
relators212121http://id.loc.gov/vocabulary/relators both as SKOS concepts and
OWL properties. The SKOS Primer discusses cases in which it might make sense
to treat an instance of SKOS _Concept_ also as a
class.222222http://www.w3.org/TR/skos-primer/#secskosowl
### 3.2 Concept Schemes
Using SKOS, concepts can be identified using URIs, labeled with lexical
strings in one or more natural languages, assigned notations (lexical codes),
documented with various types of note, linked to other concepts and organized
into informal hierarchies and association networks, aggregated into concept
schemes, grouped into labeled and/or ordered collections, and mapped to
concepts in other schemes.
Sets of concepts were referred to as “schemes” as early as the Limber Project
(1999–2001).232323http://www.w3.org/2001/sw/Europe/reports/thes/8.2/ It is
worth noting that the Dublin Core community began talking in 1997 about
“schemes,”242424http://www.dlib.org/dlib/june97/metadata/06weibel.html one
type of which came to be called a “vocabulary encoding
scheme”252525http://dublincore.org/usage/documents/2003/02/07/principles/—a
notion much less specific than, but not incompatible with, the SWAD Europe
project’s notion of a SKOS _Concept Scheme_.
The “ability to explicitly represent the containment of any SKOS individual or
statement within a concept scheme” was accepted by the working group as a
candidate requirement for SKOS (R-ConceptSchemeContainment). The ability to
express the containment not only of particular concepts, but also of
particular statements using SKOS predicates, such as skos:broader, was seen as
necessary for tracking the provenance of a concept scheme’s informational
content, for example to establish trust. Two properties from the 2005 SKOS
Core specification—skos:inScheme and skos:hasTopConcept—already provided a way
to relate SKOS concepts (along with instances of other classes, such as
skos:Collection) to a given concept scheme.
Expressing the containment of statements, on the other hand, implied a
mechanism for denoting an entire set of statements as a named entity—a
challenge faced by any RDF-based application and thus not specific to SKOS.
The use cases for containing relations between concepts also seemed more
marginal than for containing concepts. The text used to close this issue
(Issue 36)and Section 5.3 of the SKOS Primer point to ongoing work on named
graphs and RDF Datasets in
SPARQL262626http://http://lists.w3.org/Archives/Public/public-swd-
wg/2008Jan/0093.html—work which at the time of writing in 2012 remains on the
agenda of the W3C working group developing RDF
1.1.272727http://www.w3.org/2011/rdf-wg/
### 3.3 Semantic Relations
Using SKOS, concepts can be identified using URIs, labeled with lexical
strings in one or more natural languages, assigned notations (lexical codes),
documented with various types of note, linked to other concepts and organized
into informal hierarchies and association networks, aggregated into concept
schemes, grouped into labeled and/or ordered collections, and mapped to
concepts in other schemes.
The properties skos:broader, skos:narrower and skos:related are referred to
collectively as the SKOS semantic relation properties. They have their origins
in thesauri—controlled structured vocabularies used primarily for keyword
indexing of collections of documents or other objects [9]. Thesaurus standards
provide guidance on the use of hierarchical and associative relationships when
constructing a thesaurus. However, because these relationships exist primarily
as aids to information retrieval, some ambiguity is permitted. There has been
no need for them to support precise formal entailments (such as those
supported, for example, by a class subsumption hierarchy in an ontology).
Given this context, a number of design decisions needed to be made during the
standardisation of SKOS concerning the formal definition of the properties
skos:broader, skos:narrower and skos:related.
Some constraints were deemed uncontroversial. For example, skos:broader and
skos:narrower form an inverse property pair (S25), describing the two
directions of a hierarchical relationship. If the concept “mammals” is linked
to the broader concept “animals,” then the concept “animals” is linked to the
narrower concept “mammals.” The property skos:related is symmetric (S23)
because the fact that two concepts are associated with each other is
independent of direction (although sub-properties of skos:related may be
defined as directional, i.e., non-symmetric). If the concept “birds” is
related to the concept “ornithology,” then the concept “ornithology” is
related to the concept “birds.” In SKOS, hierarchical and associative
relationships are declared to be disjoint (S27).
Other constraints, less obvious, concerned transitivity, sub-properties of
semantic relations, reflexivity and cycles.
#### Hierarchies and transitivity.
It was decided that the properties skos:broader and skos:narrower would not be
transitive, and that by convention these properties should only be used to
assert direct (i.e., immediate) links between concepts. This decision was made
to simplify implementation. For example, many applications will render
hierarchical relationships as a tree, and so need some convenient way to
differentiate immediate links (parent/child) from indirect links
(ancestor/descendant). To support the fairly common use case where
applications do want to use the transitive closure of skos:broader or
skos:narrower (e.g., to expand a search query), transitive super-properties
skos:broaderTransitive and skos:narrowerTransitive were defined. Note that
exactly how retrieval applications make use of skos:broader, skos:narrower and
skos:related to improve recall is not defined by the SKOS specifications and
is left to the application. Some applications will take the view that if
document D is indexed with concept A, then document D will always be relevant
to a query for any concept that is an ancestor of concept A in the hierarchy.
This is equivalent to the view that if a document is about growing vegetables,
then the document is necessarily also about gardening. If the property
dc:subject is taken to represent the “aboutness” relationship between document
and concept, then this behaviour may easily be implemented, for example, by
computing the transitive closure of skos:broaderTransitive. However, other
applications may take the less categorical view that relevance is likely to
degrade as a query is expanded away from some focal concept, and that the
different properties skos:narrower, skos:broader, and skos:related might
correspond to different shapes or rates of degradation [23].
#### Sub-properties of skos:broader and skos:narrower.
Some thesauri disambiguate the hierarchical relationship into one of class
subsumption, instantiation, or part-whole relationship. The working group
discussed whether the SKOS standard should define sub-properties of
skos:broader, such as the putative properties broaderGeneric (for class
subsumption), broaderInstantive (for class instantiation) and broaderPartitive
(for part-whole relations), corresponding to distinctions made in thesaurus
standards [9, 10] (see also Issue 56, Issue 150, Issue 178). There would be
obvious value in having a standard set of properties, rather than leaving it
to third parties to define their own extensions to the detriment of
interoperability. However, the working group was also conscious that there is
overlap here with RDFS and OWL and was reluctant to define new properties that
might be redundant with existing standards. For instance, one might have seen
broaderGeneric as equivalent to rdfs:subClassOf and broaderInstantive to
rdf:type.282828http://lists.w3.org/Archives/Public/public-swd-
wg/2008Mar/0037.html Whether or not it would be appropriate to use such RDFS
or OWL properties in these cases would require a deeper understanding of use
cases in which SKOS and OWL are used in combination. The working group decided
to postpone this decision, leaving it open for a future revision of the
standard.
#### Reflexivity and cycles.
For a conventional thesaurus or similar vocabulary, it is an error for a
concept to be in a hierarchical relationship with itself, or to be associated
with itself (reflexivity). It could be argued that these constraints should
become part of the SKOS data model by stating formally that skos:broader and
skos:related are irreflexive properties, and that this would promote
consistent implementation. Similarly, in a thesaurus it is an error for there
to exist any cycles within the concept hierarchy, which could be enforced by
declaring skos:broaderTransitive to be irreflexive. However, the working group
was also conscious that there are possible advanced usage patterns (or
extensions to SKOS) where both SKOS and OWL would be used together within the
description of the same knowledge organisation system, and that more work was
required to understand these patterns. Although this may be an uncommon edge
case, under some of these patterns, it is conceivable that inferences such as
“<A>skos:broader <A>” could arise (for example, if someone were to assert that
rdfs:subClassOf is a sub-property of skos:broader). It was therefore decided
that no formal statements on the reflexivity of the SKOS semantic relation
properties would be made, although some informal guidance would be given to
application developers on how to detect structural features that are likely to
represent errors in the majority of use cases.
### 3.4 Mapping Relations
Using SKOS, concepts can be identified using URIs, labeled with lexical
strings in one or more natural languages, assigned notations (lexical codes),
documented with various types of note, linked to other concepts and organized
into informal hierarchies and association networks, aggregated into concept
schemes, grouped into labeled and/or ordered collections, and mapped to
concepts in other schemes.
The use cases for SKOS confirmed a strong requirement for mapping between
related concepts in different concept schemes. Indeed, the prospect of
enabling machine-readable mappings between concept schemes developed in a
diversity of contexts, and possibly on the basis of different modeling
principles, was expected to be a key advantage of expressing those systems in
the common language of SKOS. Taking as its starting point an unfinished SKOS
Mapping Vocabulary Specification from
2004,292929http://www.w3.org/2004/02/skos/mapping/spec/2004-11-11.html the
working group settled on five mapping properties: skos:broadMatch,
skos:narrowMatch, skos:relatedMatch, skos:closeMatch, and skos:exactMatch, all
of which were declared, either directly or by inference, to be sub-properties
of skos:mappingRelation, itself a sub-property of skos:semanticRelation.
Much of the discussion about mapping properties revolved around clarifying how
they differed from analogous semantic relation properties. The mapping
properties skos:broadMatch, skos:narrowMatch, and skos:relatedMatch were
declared to be sub-properties, respectively, of skos:broader, skos:narrower,
and skos:related. However, these “parallel” properties were not otherwise
distinguished in a formal sense. The question indeed arose whether, given this
lack of formal-semantic distinction, separate properties for broader,
narrower, and related matches were needed at all.
The dilemma, as the working group saw it, was that large parts of the KOS
community saw inter-KOS mapping relations and intra-KOS semantic relations as
fundamentally different things, perhaps even disjoint from each other. From
the standpoint of the working group, the intended distinction between mapping
relations and semantic relations depended, conceptually, on the ability to
“contain” a concept scheme, along with its intra-KOS relations, as an entity
distinct from other concept schemes — an issue, as discussed in Section 3.2
above, which the working group considered to be out of scope for SKOS per se.
Even if a distinction between mapping relations and semantic relations might,
in principle, be anchored in a formally solid notion of concept scheme
containment, the group recognized that the evolution of concept schemes over
time could mean that related concepts in two separate schemes could become
aggregated into the same scheme, or vice versa — situations in which the use
of formally disjoint mapping and semantic properties would prove to be most
inconvenient.
The solution adopted by the group was to make the formal-semantic distinction
between mapping and semantic properties very weak while emphasizing the
“conventional” difference between the two types. As explained in the SKOS
Primer, “By convention, mapping properties are used to represent links that
have the same intended meaning as the ‘standard’ semantic properties, but with
a different application scope. One might say that mapping relationships are
less inherent to the meaning of the concepts they involve. …By convention,
mapping relationships are expected to be asserted between concepts that belong
to different concept schemes.” The authors of this paper are not aware that
the lack of a strong formal distinction between mapping and semantic
properties has been flagged as a problem in the three years since the
publication of SKOS as a W3C Recommendation.
The two other mapping properties, skos:closeMatch and skos:exactMatch, were
positioned in part as alternatives to owl:sameAs, at the time much overused as
a mapping predicate for Linked Data. skos:closeMatch was intended for use with
concepts sufficiently similar to be used interchangeably in a given context.
The property was not defined as transitive in order to avoid the uncontrolled
propagation of the similarity relation to further contexts. skos:exactMatch,
defined as a transitive sub-property of skos:closeMatch, was intended to
express a degree of similarity close enough to justify such propagation.
### 3.5 Lexical labels
Using SKOS, concepts can be identified using URIs, labeled with lexical
strings in one or more natural languages, assigned notations (lexical codes),
documented with various types of note, linked to other concepts and organized
into informal hierarchies and association networks, aggregated into concept
schemes, grouped into labeled and/or ordered collections, and mapped to
concepts in other schemes.
The ability to annotate a concept for purposes of display or search is met by
properties for preferred, alternative, and hidden labels (skos:prefLabel,
skos:altLabel, and skos:hiddenLabel), all sub-properties of rdfs:label. These
properties are typically used to link an instance of skos:Concept to an RDF
plain literal, which the working group took to mean a character string (such
as the word “love”) combined with an optional language tag (e.g., “en-US”).
Note that because RDF lacked a class for RDF Plain Literal, the property
definition axiom S12 could not be expressed as a formal range assertion in the
normative RDF/XML or informative OWL 1 DL expressions of SKOS, as shown in
Table 2 (see also Section 4.3). The 2005 SKOS Core specification included
properties for symbolic labels—skos:symbol and skos:prefSymbol—which allowed
for resource types other than RDF plain literals as labels; however, these
were dropped for the 2009 SKOS Recommendation due to a lack of clear
requirements (see Issue 76 and Issue 180).
The notion of “preferred label” derived from what the thesaurus community
calls a “preferred term” or “descriptor”—i.e., a “term specified by a
controlled vocabulary for use to represent a concept when
indexing.”303030http://www.willpowerinfo.co.uk/glossary.htm#preferred_term
Preferred terms are in principle unique within the representation of a concept
scheme in a given natural language. In order to give formal expression to this
convention, the working group operationalized the notion of language as
meaning language tag, noting that language tags can be extended to distinguish
arbitrarily specific regional variants of, say, English, Portuguese, French,
or Chinese. The integrity condition axiom S14, therefore, specifies that a
resource “has no more than one value of skos:prefLabel per language tag” (see
Section 4.2 and Table 2). The properties for alternative and hidden labels
were intended for non-preferred indexing terms, whether displayed to users or
not. The three labeling properties are considered pairwise disjoint (S13),
such that assigning the same literal as both a preferred and alternative label
is formally considered an error.
In keeping with the principle of minimal ontological commitment, the SKOS
labeling properties have no explicit domain contraints. This follows the
example of Dublin Core, which does not specify domains for many of its
properties. The lack of specific domains allows the SKOS labeling properties
to be used in contexts other than concept schemes, providing Semantic Web
applications with a generic vocabulary for labels—a usage already seen in
various OWL ontologies [13] and supported by non-SKOS-centric tools such as
Protégé.
In order to address the need for associating concepts with alphanumeric codes
such as “M1495-2199” (meaning “Vocal music” in Library of Congress
Classification), the working group introduced a property skos:notation (see
Issue 79). A SKOS notation is intended to uniquely identify a concept within a
given concept scheme. It differs from a lexical label “in that a notation is
not normally recognizable as a word or sequence of words in any natural
language.” As explained in SKOS Reference, Section 6.5.1., “By convention, the
property skos:notation is only used with a typed literal in the object
position of the triple, where the datatype URI denotes a user-defined datatype
corresponding to a particular system of notations or classification
codes.”313131http://www.w3.org/TR/skos-reference/#L2613
#### Relations between labels.
The ability to model binary relations between lexical labels was identified as
a candidate requirement for SKOS (R-RelationshipsBetweenLabels). It should be
possible, for example, to assert that the label “FAO” is related to the label
“Food and Agriculture Organization” via a relation “acronym for.” The
proposals initially considered for enabling such assertions offered
combinations of three basic ideas: creating a class for instantiating a “term”
to which a plain-literal label could be associated; dropping range
restrictions on the SKOS labeling properties so that they could be associated
with either RDF plain literals or with instances of such a class; and viewing
relations, such as the “acronym for” relation, as classes. Instances of such
classes would be linked from a concept, via (for example) a seeLabelRelation
property, and would link, via an n-ary relation pattern [15], both to a full
form (ex:fullForm “Food and Agriculture Organization”) and to an acronym form
(ex:acronymForm “FAO”@en)—a pattern which, it was recognized, would involve
replicating the label
literals.323232http://www.w3.org/2006/07/SWD/wiki/SkosDesign/RelationshipsBetweenLabels.html
The solution that emerged—split off into an optional appendix, “SKOS eXtension
for Labels (SKOS-XL),” with its own SKOS-XL namespace URI, in order to keep
the main SKOS specification as simple as
possible333333http://www.w3.org/TR/skos-reference/#xl—defined a class,
skosxl:Label, instances of which are associated with exactly one literal form
(see SKOS-XL axiom S52 in Table 3). The properties skosxl:prefLabel,
skosxl:altLabel, and skosxl:hiddenLabel were coined, with the class
skosxl:Label as their range. The property skosxl:labelRelation was coined as a
common super-property for applications defining their own specific label
relations. The working group felt that defining properties for specific types
of label relation was out of scope due to insufficient consensus on what would
comprise a reasonably complete set.
In order to ensure the interoperability of data created using the SKOS and
SKOS-XL labeling properties, three axioms were formulated to declare a
property chain composed of a SKOS-XL labeling property with a literal form.
For example, the chain “(skosxl:prefLabel, skosxl:literalForm)” is a sub-
property of the corresponding SKOS labeling property (in this case,
skos:prefLabel) (see axioms S55, S56, and S57 in Table 3). In other words,
SKOS-XL labels can be “dumbed down” to corresponding SKOS labels. It is worth
noting that the skosxl:literalForm property chain is analogous to a pattern
described in the 1999 W3C Recommendation for RDF, whereby one of the
properties of a “structured value” is marked, using the property rdf:value, as
“the principal value of the main relation” of a subject to a value
resource.343434http://www.w3.org/TR/1999/REC-rdf-syntax-19990222/#ex-NonBinary
Defining labels, optionally, as individuals that could be annotated or related
among themselves in arbitrary ways allowed the working group to resolve an
issue raised with regard to the assertion of mapping relations between the
labels of different concept schemes (Issue 49) and an issue requiring the
capability of applying annotations to the lexical items used as labels (Issue
27). Two concerns that arose during discussions of modeling alternatives for
label relations were: identity conditions (When are two instances of the class
skosxl:Label the same individual?), and the formal relationship between the
class skosxl:Label and the set of RDF plain literals (Can instances of the
class skosxl:Label have more than one literal form?). The working group
decided to assert that instances of skosxl:Label have exactly one literal form
in order to avoid ambiguity, but that sharing a common literal form should not
be sufficient to infer that two instances of the class skosxl:Label were the
same individual. In other words, two distinct instances of skosxl:Label might
have the same literal form; there is no one-to-one mapping between the class
extension of skosxl:Label and the set of RDF plain literals.
### 3.6 Documentation Properties
Using SKOS, concepts can be identified using URIs, labeled with lexical
strings in one or more natural languages, assigned notations (lexical codes),
documented with various types of note, linked to other concepts and organized
into informal hierarchies and association networks, aggregated into concept
schemes, grouped into labeled and/or ordered collections, and mapped to
concepts in other schemes.
SKOS provides a number of documentation (or note) properties. These allow for
a variety of annotations including general notes, change notes, definitions,
editorial notes, examples, historical notes, and scope notes. These seven note
types provided are not intended to be exhaustive, and it is expected that
specific application domains may extend the documentation properties
(potentially via sub-properties of the given properties, thus allowing generic
SKOS machinery access to information asserted using bespoke properties). As
with labeling properties, no domains are given for these properties, allowing
their usage outside of SKOS concept schemes.
In addition, the documentation properties have no ranges asserted (in contrast
to labels). As discussed in the SKOS Primer,353535http://www.w3.org/TR/skos-
primer/#secadvanceddocumentation this allows for a number of different
documentation patterns, including the use of literals, the use of blank nodes
for structured annotations, and the use of document references.
### 3.7 Concept Collections
Using SKOS, concepts can be identified using URIs, labeled with lexical
strings in one or more natural languages, assigned notations (lexical codes),
documented with various types of note, linked to other concepts and organized
into informal hierarchies and association networks, aggregated into concept
schemes, grouped into labeled and/or ordered collections, and mapped to
concepts in other schemes.
In thesauri and other structured KOSs, concepts can be grouped into
semantically meaningful bundles. For example, arrays are used to group
specializations of a concept that share a common feature: the concept “cups”
might be specialized into a first group of “cups by form” (“stemware”,
“tumbler”…) and a second group of “cups by function” (“coffee cups”, “ice
cream cups”…) [10]. This is especially useful for displaying KOSs: these
groups are indeed most often meant as a navigation aid in a conceptual
network, not to be used for describing resources. SKOS supports the
requirement (R-GroupingInConceptHierarchies), discussed in Issue 33 for
representing such constructs using the skos:Collection class and its subclass
skos:OrderedCollection for groups where the ordering of concepts matters.
Note that SKOS defines skos:Collection as disjoint with skos:ConceptScheme and
skos:Concept. This has important consequences. First, it can raise issues when
representing “subsets of vocabularies” such as micro-thesauri in the Eurovoc
thesaurus363636http://eurovoc.europa.eu/drupal/?q=node/555 or subdivision
lists in the Library of Congress Subject
Headings373737http://id.loc.gov/authorities/subjects. The disjointness
constraint forces data modelers to opt for using (a sub-class of) either
skos:Collection or skos:ConceptScheme, a choice that can be hard to make in
the absence of clear guidance in the SKOS documentation. Eurovoc thus now
represents microthesauri as concept schemes, while LCSH represents subdivision
lists as collections. Fortunately, the KOS community has realized this and
started to address the problem, as witnessed by recent advocacy on how to
relate ISO 25964 thesaurus standard’s “concept groups” to skos:Collection and
skos:ConceptScheme [11].
Note, too, that collections cannot be used in combination with semantic
relations to assign them a position in the semantic structure of a KOS. It is
not consistent with the SKOS data model to declare a collection to be a
semantic generalization or refinement of a “normal” SKOS concept with
statements using skos:broader. In SKOS, concepts are merely grouped into
collections using the properties skos:member and skos:memberList. It may be
seen as an obstacle to represent simply semantic hierarchies with collections,
and a deviation from the minimal commitment approach. But it is in fact the
consequence of a conscious choice to keep data on semantic relations between
concepts clearly separate from the display-related considerations that usually
motivate the creation of collections. SKOS takes the stance that fitting
collections into KOS hierarchies must be handled by specific display
algorithms that reflect the need of users in a given navigation environment
(see Issue 84).
### 3.8 Issues deemed out of scope
Originally chartered for just 20 months,383838http://www.w3.org/2006/07/swdwg-
charter the Semantic Web Deployment Working Group needed 35 months to complete
its work. In order to focus its efforts and keep the specification as short
and simple as possible, the group declared several topics to be out of scope.
* 1.
Concept coordination. Many KOSs are intended to be used as building blocks for
constructing “coordinated” concepts, for example to aggregate the “simple”
concepts “aspirin” and “side effect” into a “compound” concept “aspirin –
side-effects.” Compound concepts can be created on a one-off basis by
catalogers, as they are needed in resource description, or they can be added
as concepts to the KOS itself by its maintainers (which is known as “pre-
coordination,” as with the Library of Congress subject heading “China –
history”). The working group recognized this well-known pattern — “the ability
to create new concepts from existing ones, e.g. by using special qualifiers
that add a shade of meaning to a normal concept”—as a candidate requirement
(R-ConceptCoordination). The group also considered a common practice in the
thesaurus world [9] whereby two simple concepts (such as “Road transport” and
“Safety”) are designated to be used in combination instead of minting a new
compound concept such as “Road safety” (see Issue 45).
After much discussion, the group decided to postpone these issues (Issue 40,
Issue 131). While the requirements for coordination were not questioned, the
group considered them to be relevant more for particular thesaurus and subject
heading applications than to the interchange of KOSs generally. The group also
noted that the patterns proposed to represent concept combinations were rather
complex and largely untested. Finally, it was felt that allowing the core SKOS
model to handle such constructs could be seen as a potentially confusing move
towards supporting some functions of formal ontology languages such as
OWL—languages which support the definition of complex classes or properties
from more primitive vocabulary elements.
In retrospect, the authors feel that the decision to postpone was sound. It
not only kept untested patterns out of SKOS, avoiding delays in finalizing the
standard; it also motivated the community to tackle the issue itself. By the
end of 2010, for example, the Library of Congress had developed a first
version of MADS/RDF [16], an extension to SKOS which, among other things,
supports concept coordination within library subject heading lists.
* 2.
Subject indexing. As defined by Leonard Will, subject indexing involves
“intellectual analysis of the subject matter of a document to identify the
concepts represented in it, and allocation of the corresponding preferred
terms to allow the information to be
retrieved.”393939http://www.willpowerinfo.co.uk/glossary.htm The working group
recognized as a candidate requirement the “ability to represent the indexing
relationship between a resource and a concept that indexes it,” whereby the
SKOS model would include “mechanisms to attach a given resource (e.g.
corresponding to a document) to a concept the resource is about, e.g. to query
for the resources described by a given concept” (see R-IndexingRelationship,
Issue 77). Noting the existence of indexing relation properties in other
vocabularies, such as Dublin Core’s dc:subject, the working group declared
such properties to be out of scope and decided not to carry forward the
property skos:subject from the 2005 SKOS Core
specification.404040http://www.w3.org/TR/2005/WD-swbp-skos-core-
spec-20051102/#subject (It should be noted that recent Web crawls show
skos:subject to be one of the most-used properties from SKOS, due primarily to
its use in
DBPedia.414141http://sindice.com/search?q=&nq=%28*%20%3Chttp%3A%2F%2Fpurl.org%2Fdc%2Fterms%2Ftitle%3E%20*%29&fq=&interface=advanced)
For lack of a SKOS indexing vocabulary, a candidate requirement for
distinguishing between indexing and non-indexing concepts was also declared
out of scope (see R-IndexingAndNonIndexingConcepts, Issue 46).
* 3.
Provenance information about mappings. The ability “to record provenance
information on mappings between concepts in different concept schemes” was
recognized as a candidate requirement for SKOS
(R-MappingProvenanceInformation). The issue was resolved with a decision not
to introduce specific SKOS vocabulary about the provenance of mappings (Issue
47). Rather, the group felt that this issue depended on the use of standard
containment mechanisms for encompassing mapping assertions within a context
that could be denoted with a URI—an issue relevant for RDF in general,
specifically for the future development of standards regarding “named graphs”
(see also the discussion of containment in Section 3.2).
* 4.
Describing concept schemes. Concept schemes have authors, titles, publishers,
dates issued, subject coverage, and the like. The working group felt that the
question of what properties to use in describing a concept scheme was an issue
best left to communities of practice. Shortly after the publication of SKOS in
2009, for example, a joint DCMI–NKOS task group was formed between the Dublin
Core Metadata Initiative and Networked Knowledge Organization Systems
community to develop an application profile and a KOS Type Vocabulary for
describing KOSs.424242http://dublincore.org/groups/nkos/
* 5.
Concept evolution. The working group acknowledged the importance of mechanisms
for representing the temporal evolution of concept schemes—an issue that
raises questions of granularity (whether to version individual statements,
concept descriptions, or entire concept schemes) and of how to represent such
versioning information in interoperably machine-readable ways. The group
considered this topic best left to the community for research and testing (see
[22]).
## 4 Formal semantics
This section discusses aspects of SKOS relating to its formal semantics, in
particular highlighting the use of OWL. The working group434343In this
section, references to “the working group” refer to the Semantic Web
Deployment Working Group. Other working groups will be referred to by their
full name. was tasked to specify SKOS in accordance with OWL, so as to allow
for applications to validate SKOS datasets or to infer new facts from the ones
explicitly encoded by publishers of SKOS data. The SKOS model is thus
specified by defining OWL classes and properties, which can be interpreted
using OWL’s formal semantics. A particular SKOS concept scheme is an
instantiation of the OWL ontology that defines SKOS in which SKOS concepts are
instances of the class skos:Concept with characteristics expressed using the
SKOS properties.
### 4.1 Axioms supporting inference
As described above, the SKOS data model contains a number of axioms (stated as
S1 to S46 in the Recommendation444444SKOS-XL includes additional axioms S47 to
S62) relating to the classes and properties of the SKOS vocabulary.
All but six of these axioms, as listed in Table 1, describe how the classes
and properties of SKOS are defined, primarily by stating subclass or sub-
property relationships or domain and range assertions. These axioms allow the
use of inference engines (“reasoners”) to derive additional information about
the nature of, and relationships among, components of a concept scheme. Note
that such inference concerns the concept scheme as an information artefact in
itself and says nothing about the nature of the resources or “real-world”
entities to which the concepts of a concept scheme may refer. For example, the
axiom S4 allows the inference that an object of a triple using skos:inScheme
is an instance of the class skos:ConceptScheme.454545Note the semantics of
rdf:range here. A common misconception is that a concept scheme that does not
explicitly type an object of an skos:inScheme as a skos:ConceptScheme would be
in error. This is not the case though—rdfs:range assertions are not
_constraints_ , but are conditions on interpretations providing inferences.
Axiom S25 allows an application that is OWL-semantics aware to infer the
presence of skos:broader relationships in a concept scheme that asserts only
skos:narrower relationships.
The SKOS-XL extension (see Section3.5) includes axioms relating to property
chains, for example S55 would allow an application given the triples:
>
> ex:concept-1234 skosxl:prefLabel ex:label-5678.
> ex:label-5678 skosxl:literalForm "love".
>
to infer the triple
>
> ex:concept-1234 skos:prefLabel "love".
>
### 4.2 Integrity conditions
In addition to the axioms described above, a number of _integrity conditions_
(labeled as S9, S13, S14, S27, S37, and S46) are also given. The integrity
conditions serve a different purpose to the other axioms stated, in that they
are intended to facilitate and promote interoperability by defining
circumstances under which data are not consistent with respect to the SKOS
data model. Details of integrity conditions are given in Table 2.
The working group was chartered to create a machine-readable specification of
the SKOS axioms using the OWL language, which forms the base for exchanging
and exploiting formal specifications of ontologies on the Web of Data, as
envisioned in the W3C Semantic Web technology stack. The SKOS Recommendation
makes no assumptions, however, as to _how_ implementation of the checking of
integrity conditions for a particular concept scheme are performed. They could
be checked through inference, but other mechanisms could be used, for example
querying for particular graph patterns or the use of rule driven approaches
such as SPIN464646http://spinrdf.org/ or Pellet’s Integrity
Constraints.474747http://clarkparsia.com/pellet/icv
### 4.3 SKOS as an OWL Ontology
#### Historical context.
The Web Ontology Language (OWL) was first published as a collection of W3C
Recommendations in 2004484848http://www.w3.org/2004/OWL/ developed by the Web
Ontology working group,494949http://www.w3.org/2001/sw/WebOnt/ first convened
in 2001. One key aspect of OWL was the definition of three sublanguages known
as OWL Lite, OWL DL and OWL Full. OWL DL supported those users who wanted
maximum expressiveness while still retaining computational completeness. OWL
Full provided greater expressiveness and syntactic freedom, but with a lack of
computational guarantees. OWL Lite was a subset of OWL DL intended to support
users needing a classification hiearchy and simple constraints. The working
group was tasked to specify SKOS in accordance with OWL Full.
In 2007, the OWL Working Group505050http://www.w3.org/2007/OWL was convened,
with a charter to produce an update to OWL, resulting (in 2009) in a
collection of recommendations defining OWL
2515151http://www.w3.org/TR/owl2-overview/ (also earlier known as OWL 1.1
during the process). The work of the OWL Working Group overlapped with the
work of the Semantic Web Deployment Group, with the consequence that the SKOS
recommendation did not have the opportunity of using OWL 2 features in the
SKOS recommendation (this point and the related issue of defining SKOS within
the limits of OWL DL in is covered in more detail in Section 4.4).
To avoid confusion, this section refers explicitly to the original (2004)
recommendation as OWL 1 and the revision (2009) as OWL 2.
#### SKOS as an OWL ontology.
The SKOS data model is represented as an OWL 1 ontology, i.e. a collection of
classes and properties with associated axioms.
The SKOS Namespace Document RDF/XML Variant525252{http://www.w3.org/TR/skos-
reference/skos.rdf} provides definitions of the classes and properties of this
model using OWL 1, along with axioms that represent integrity conditions on
the data represented using SKOS. As there are limits to the expressivity of
OWL 1 (and of its subspecies or fragments), not all of the desired constraints
can be fully expressed using OWL 1. This is further discussed below. Where
this is the case, the constraint is expressed as a comment in the schema.
### 4.4 Compatibility with OWL 1 DL and OWL 2
SKOS is defined as an OWL 1 ontology, and a requirement
R-CompatibilityWithOWL-DL made to the working group was that SKOS should
provide a legal OWL 1 DL ontology, primarily to ensure compatibility with
editing tools and to facilitate the use of reasoners, many of which operate in
the OWL 1 DL space.
This was problematic as OWL 1 DL lacked the expressivity needed to capture
some of the assertions. For example, OWL 1 DL has no facility to express
hierarchies of annotation properties. Nor does OWL 1 provide a mechanism for
stating axioms concerning property chains as used in axiom S55. A further
complication was that the work of the Semantic Web Deployment Group overlapped
with that of the OWL Working Group, which was defining the OWL 2
Recommendation (also referred to as OWL 1.1 during the process). OWL 2 was
likely to introduce features that would support some of these assertions, but
as the OWL Working Group was scheduled to finish _after_ SKOS delivery, the
normative SKOS reference could not make reference to OWL 2\. For example, the
particular feature supporting hierarchies of annotation properties was
ultimately introduced into OWL 2.
In order to provide some support for reasoning engines and those applications
working in the OWL 1 DL space, a “pruned” RDF schema was produced, providing a
non-normative resource. This is made available (in a non-normative fashion) as
the SKOS RDF Schema - OWL 1 DL Sub-set.535353http://www.w3.org/TR/skos-
reference/skos-owl1-dl.rdf In particular, the pruning removed axioms stating
that SKOS labeling properties are sub-properties of rdfs:label as sub-property
axioms are not applicable to annotation properties in OWL 1 DL.
This particular pruning of the schema in order to provide a valid OWL 1 DL
ontology is only one of a number of possible ways in which the OWL 1 Full RDF
Schema for SKOS can be adjusted in order to sit in the OWL 1 DL space—each of
which would have differing semantic consequences. As a result the OWL 1 DL
prune was considered non-normative.
Other constraints were also problematic in terms of OWL 1 representation. S14
states that “A resource has no more than one value of skos:prefLabel per
language tag.” This was not expressible in OWL 1. Nor were property
disjointness constraints as expressed in S13, S27 and S46. Issues relating to
compatibility with OWL 1 DL—Issue 38, Issue 137, Issue 138—were thus formally
postponed by the working group, indicating that, should work resume on an
updated recommendation, this should be the focus of attention.
Comments from members of the OWL Working Group (raised as Issue 155 and Issue
157) highlighted areas where an adjustment to the model would potentially
provide better alignment with the emerging OWL 2 recommendation. Issue 157 was
formally postponed. Following the resolution of Issue 135, labeling properties
were defined as owl:AnnotationProperty.
### 4.5 Machine-readable Formalizations, Formal Semantics and Data Quality
As discussed earlier, the SKOS data model is represented as a collection of
axioms, some providing definitions of classes and properties, which then
support inference, others asserting integrity conditions. When representing
these axioms in a machine-readable way (which was the main mission of the SWD
group), the implementation creates two “layers” orthogonal to this question of
definition versus integrity:
1. 1.
Axioms formally represented in the ontology, for example, sub-property
relations to skos:semanticRelation;
2. 2.
Axioms that are not explicitly represented in the ontology, primarily due to a
lack of expressivity in the representations, for example assertions about
disjoint properties.
The SKOS RDF/OWL representation thus proposes a core layer for inference and
validation of SKOS data. However, as mentioned in Section 4.2, the working
group did not assume a specific technique for checking the integrity
conditions of the SKOS data model.
This flexibility can be explained by the difficulty of representing all
integrity conditions in the OWL language (as discussed above). But it is also
in line with a more fundamental stance of the SWD group, which allows for a
flexible approach to data quality in SKOS, generally. In addition to the two
layers described (formal versus informal axioms), the SKOS reference includes
what one might call _guidelines_ which are weaker recommendations, for example
that skos:closeMatch should be used to relate concepts from different schemes.
There is no attempt at formal representation of the latter, nor is it
considered an integrity constraint that should not be violated. These
assertions are more “advisory,” but are still somehow part of the SKOS model.
It is left to SKOS implementations to adapt these guidelines—or others from
specific domains, such as thesaurus design guidelines [9], which can provide
useful “checks” for SKOS data.
Example approaches to validation of SKOS data include the Poolparty Thesaurus
Consistency Checker,545454http://poolparty.punkt.at/ which runs custom
validation rules derived from the SKOS axioms. The qSKOS tool by Mader et al.
[12] is used to identify a number of quality issues in SKOS vocabularies, in
particular fifteen “guideline” violations. The Skosify tool [20] identifies an
overlapping (but slightly different) set of criteria, some of which correspond
to SKOS integrity conditions (e.g., S13 concerning disjointness of alternate
and preferred labels).
Tables 1, 2 and 3 provide a summary of the axioms in the SKOS and SKOS-XL data
models. It also highlights those axioms that lack a formal machine
representation in either the normative RDF Schema or the non-normative OWL 1
DL prune (note there is no corresponding OWL 1 DL prune of SKOS-XL).
Axiom | Content | RDF Schema | OWL Prune
---|---|---|---
S1 | skos:Concept is an instance of owl:Class. | |
S2 | skos:ConceptScheme is an instance of owl:Class. | |
S3 | skos:inScheme, skos:hasTopConcept and skos:topConceptOf are each instances of owl:ObjectProperty. | |
S4 | The rdfs:range of skos:inScheme is the class skos:ConceptScheme. | |
S5 | The rdfs:domain of skos:hasTopConcept is the class skos:ConceptScheme. | |
S6 | The rdfs:range of skos:hasTopConcept is the class skos:Concept. | |
S7 | skos:topConceptOf is a sub-property of skos:inScheme. | |
S8 | skos:topConceptOf is owl:inverseOf the property skos:hasTopConcept. | |
S10 | skos:prefLabel, skos:altLabel and skos:hiddenLabel are each instances of owl:AnnotationProperty. | |
S11 | skos:prefLabel, skos:altLabel and skos:hiddenLabel are each sub-properties of rdfs:label. | |
S12 | The rdfs:range of each of skos:prefLabel, skos:altLabel and skos:hiddenLabel is the class of RDF plain literals. | |
S15 | skos:notation is an instance of owl:DatatypeProperty. | |
S16 | skos:note, skos:changeNote, skos:definition, skos:editorialNote, skos:example, skos:historyNote and skos:scopeNote are each instances of owl:AnnotationProperty. | |
S17 | skos:changeNote, skos:definition, skos:editorialNote, skos:example, skos:historyNote and skos:scopeNote are each sub-properties of skos:note. | |
S18 | skos:semanticRelation, skos:broader, skos:narrower, skos:related, skos:broaderTransitive and skos:narrowerTransitive are each instances of owl:ObjectProperty. | |
S19 | The rdfs:domain of skos:semanticRelation is the class skos:Concept. | |
S20 | The rdfs:range of skos:semanticRelation is the class skos:Concept. | |
S21 | skos:broaderTransitive, skos:narrowerTransitive and skos:related are each sub-properties of skos:semanticRelation. | |
S22 | skos:broader is a sub-property of skos:broaderTransitive, and skos:narrower is a sub-property of skos:narrowerTransitive. | |
S23 | skos:related is an instance of owl:SymmetricProperty. | |
S24 | skos:broaderTransitive and skos:narrowerTransitive are each instances of owl:TransitiveProperty. | |
S25 | skos:narrower is owl:inverseOf the property skos:broader. | |
S26 | skos:narrowerTransitive is owl:inverseOf the property skos:broaderTransitive. | |
S28 | skos:Collection and skos:OrderedCollection are each instances of owl:Class. | |
S29 | skos:OrderedCollection is a sub-class of skos:Collection. | |
S30 | skos:member and skos:memberList are each instances of owl:ObjectProperty. | |
S31 | The rdfs:domain of skos:member is the class skos:Collection. | |
S32 | The rdfs:range of skos:member is the union of classes skos:Concept and skos:Collection. | |
S33 | The rdfs:domain of skos:memberList is the class skos:OrderedCollection. | |
S34 | The rdfs:range of skos:memberList is the class rdf:List. | |
S35 | skos:memberList is an instance of owl:FunctionalProperty. | |
S36 | For any resource, every item in the list given as the value of the skos:memberList property is also a value of the skos:member property. | |
S38 | skos:mappingRelation, skos:closeMatch, skos:exactMatch, skos:broadMatch, skos:narrowMatch and skos:relatedMatch are each instances of owl:ObjectProperty. | |
S39 | skos:mappingRelation is a sub-property of skos:semanticRelation. | |
S40 | skos:closeMatch, skos:broadMatch, skos:narrowMatch and skos:relatedMatch are each sub-properties of skos:mappingRelation. | |
S41 | skos:broadMatch is a sub-property of skos:broader, skos:narrowMatch is a sub-property of skos:narrower, and skos:relatedMatch is a sub-property of skos:related. | |
S42 | skos:exactMatch is a sub-property of skos:closeMatch. | |
S43 | skos:narrowMatch is owl:inverseOf the property skos:broadMatch. | |
S44 | skos:relatedMatch, skos:closeMatch and skos:exactMatch are each instances of owl:SymmetricProperty. | |
S45 | skos:exactMatch is an instance of owl:TransitiveProperty. | |
Table 1: SKOS Class and Property Definition Axioms. A green cell denotes that an axiom is present in the corresponding formalisation, while a red cell denotes absence. Axiom | Content | RDF Schema | OWL Prune
---|---|---|---
S9 | skos:ConceptScheme is disjoint with skos:Concept. | |
S13 | skos:prefLabel, skos:altLabel and skos:hiddenLabel are pairwise disjoint properties. | |
S14 | A resource has no more than one value of skos:prefLabel per language tag. | |
S27 | skos:related is disjoint with the property skos:broaderTransitive. | |
S37 | skos:Collection is disjoint with each of skos:Concept and skos:ConceptScheme. | |
S46 | skos:exactMatch is disjoint with each of the properties skos:broadMatch and skos:relatedMatch. | |
Table 2: SKOS Integrity Condition Axioms. A green cell denotes that an axiom is present in the corresponding formalisation, while a red cell denotes absence. Axiom | Content | RDF Schema
---|---|---
S47 | skosxl:Label is an instance of owl:Class. |
S48 | skosxl:Label is disjoint with each of skos:Concept, skos:ConceptScheme and skos:Collection. |
S49 | skosxl:literalForm is an instance of owl:DatatypeProperty. |
S50 | The rdfs:domain of skosxl:literalForm is the class skosxl:Label. |
S51 | The rdfs:range of skosxl:literalForm is the class of RDF plain literals. |
S52 | skosxl:Label is a sub-class of a restriction on skosxl:literalForm cardinality exactly 1. |
S53 | skosxl:prefLabel, skosxl:altLabel and skosxl:hiddenLabel are each instances of owl:ObjectProperty. |
S54 | The rdfs:range of each of skosxl:prefLabel, skosxl:altLabel and skosxl:hiddenLabel is the class skosxl:Label. |
S55 | The property chain (skosxl:prefLabel, skosxl:literalForm) is a sub-property of skos:prefLabel. |
S56 | The property chain (skosxl:altLabel, skosxl:literalForm) is a sub-property of skos:altLabel. |
S57 | The property chain (skosxl:hiddenLabel, skosxl:literalForm) is a sub-property of skos:hiddenLabel. |
S58 | skosxl:prefLabel, skosxl:altLabel and skosxl:hiddenLabel are pairwise disjoint properties. |
S59 | skosxl:labelRelation is an instance of owl:ObjectProperty. |
S60 | The rdfs:domain of skosxl:labelRelation is the class skosxl:Label. |
S61 | The rdfs:range of skosxl:labelRelation is the class skosxl:Label. |
S62 | skosxl:labelRelation is an instance of owl:SymmetricProperty. |
Table 3: SKOS XL Axioms. A green cell denotes that an axiom is present in the
corresponding formalisation, while a red cell denotes absence.
### 4.6 SKOS Namespace URI
A question that was the focus of much attention during the Recommendation
process was that of the URI to be used for SKOS, formally raised as Issue 153
and Issue 175. Earlier work from the Semantic Web Best Practices and
Deployment Group resulted in a SKOS Core Working
Draft.555555http://www.w3.org/TR/swbp-skos-core-spec This Working Draft was a
key input to the work of the working group, and much of the content of the
original Core was preserved in the final Recommendation. The original core
defined vocabulary using the namespace URI
http://www.w3.org/2004/02/skos/core.
Various possibilities were open to the working group:
1. 1.
Provide a new namespace URI for the SKOS Recommendation;
2. 2.
Use the existing SKOS Core namespace URI for the SKOS Recommendation,
potentially redefining or changing the semantics of URIs defined in that
namespace.
3. 3.
Use the existing SKOS Core namespace URI for the SKOS Recommendation, minting
new URIs for those vocabulary elements where semantics had been changed.
As an example of a situation where the semantics of a vocabulary element had
changed, consider the hierarchical semantic relations skos:broader and
skos:narrower.
In the original core, these properties were declared to be transitive, while
in the final SKOS recommendation, as discussed in Section 3.3, they were not
(instead a transitive reduction [17] design pattern was used, introducing
transitive superproperties skos:broaderTransitive and
skos:narrowerTransitive).
Each option had pros and cons. The introduction of a new namespace URI would
reduce the problems of inconsistent interpretations of existing vocabularies
that may have been producing using the original semantics. However, a new
namespace URI would then potentially require changes to existing tools,
infrastructure and concept schemes.
The final decision made was for option 2. It was felt that disruption to the
existing body of data that had been published using SKOS Core would have been
significant if the namespace URI or property names of key elements have been
changed. Although this resulted in a change of semantics to some properties,
applications should, in principle, be able to make use of the machine-readable
published schema to access those semantics.
On a similar note, elements were removed from the SKOS Core vocabulary (see
discussion in the SKOS Reference565656http://www.w3.org/TR/skos-
reference/#namespace) although historical versions of the schemas remain
available.575757http://www.w3.org/2004/02/skos/history
Although SKOS Core had at that point been deployed by early adopters for
several years, changing the semantics associated with the URIs was
unproblematic strictly from the standpoint of process because the 2005
specification had only attained the status of Working Draft—a type of
specification by definition subject to change.
It is worth noting that following publication of SKOS as a Recommendation, it
was observed that the the SKOS OWL 1 DL prune ontology had no version IRI,
thus breaking a rule specified in the 2009 OWL 2 recommendation, that “If an
ontology has an ontology IRI but no version IRI, then a different ontology
with the same ontology IRI but no version IRI SHOULD NOT exist.” In order to
address this, an additional owl:versionIRI triple was added to the
ontology585858http://www.w3.org/2006/07/SWD/SKOS/reference/20090811-errata.
## 5 Conclusion
The intellectual roots of Knowledge Organization Systems go back decades, even
centuries. The goal of expressing Knowledge Organization Systems in a
generically interoperable way was raised already as a goal when W3C working
groups began developing the Semantic Web language, Resource Description
Framework (RDF), in the late 1990s. In the twelve years from the beginnings of
RDF in 1997 through the finalization of SKOS as a W3C Recommendation in 2009,
the torch for this work was passed among a succession of UK and European
research projects and of W3C working groups, each of which added features,
dropped others, and progressively clarified its underlying concepts. This
process illustrated the challenge of developing specifications that depend on
related specifications which, in today’s continually evolving environment, are
inevitably subject to change. As discussed above, SKOS would have looked
slightly different if OWL 2, published as a W3C Recommendation just two months
after SKOS,595959http://www.w3.org/TR/owl2-overview/ had been finalized just
half a year earlier.
This paper highlights a number of issues that were “postponed” – a status
which marks them as being of potential interest to future working groups:
* 1.
SKOS and OWL. In the three years since the publication of the W3C
Recommendation for SKOS, one of the most Frequently Asked Questions has been
that of the relationship between information KOSs, expressed using SKOS, and
OWL ontologies. As discussed in Section 3.1, almost anything can be considered
a SKOS Concept (as long as it is not a SKOS Concept Scheme, Collection, or
Label), and foaf:focus provides a way to link SKOS concepts to things in the
world to which those concepts refer. The working group defined SKOS this way
so as not to preclude experimentation with usage patterns as yet unforeseen.
* 2.
The formal expression of SKOS. The formal expression of SKOS axioms could be
enhanced in light of OWL 2 (see Issue 38, Issue 136, Issue 137, Issue 138,
Issue 155, and Section 4.4). Such enhancements could help consolidate a more
consistent approach to validating SKOS concept schemes. The choice of axioms
pruned to create the non-normative OWL DL 1 Prune should also be revisited in
light of implementation experience (see Section 4.4). A future working group
might also want to formulate recommendations on the use of non-OWL semantics
based, perhaps, on constraints with a closed-world interpretation.
* 3.
Inference, validation, and quality control. A future working group might want
to incorporate work being done in the implementation community on extending
the definition of “conformance,” for example to enhance support for specific
types of KOS (see Issue 35).
* 4.
Extending SKOS with additional properties. Extending SKOS with richer
semantics relations (Issue 56, Issue 149, Issue 150, and Issue 178) remains a
very popular topic and has resulted already in proposals, e.g., in the ISO
25964 standard [10]. The issue of extending SKOS with symbolic labels remains
unaddressed, though the authors see the potential for experimentation related,
for example, to Web accessibility. The potential refinement of SKOS mapping
properties perhaps awaits a stronger push on formulating best practice for
mapping in the Semantic Web context generally (see Issue 176).
* 5.
Concept coordination. Patterns defined by MADS and ISO 25964 for concept
coordination could be evaluated in light of implementation experience,
especially as pre- and post-coordination patterns are tested in the context of
different types of KOS and in information retrieval applications (see Issue
40, Issue 45, Issue 131, and Section 3.8).
* 6.
Concept scheme containment and provenance. As of late 2012, the RDF Working
Group is working towards standardizing an approach to naming graphs and
datasets. As pointed out in Section 3.2, the identification of graphs is
relevant to all issues which require that concept schemes be delimited, or
“contained,” for the purpose of tracking provenance or expressing precise
alignments (see R-ConceptSchemeContainment and
R-MappingProvenanceInformation). The Linked Data community is developing
relevant practices and vocabularies, such as
VoID606060http://vocab.deri.ie/void, which addresses the provenance both of
generic datasets and of more specific “linksets.” The SKOS community has made
some progress in the past years on modeling concept
evolution616161http://www.w3.org/2001/sw/wiki/SKOS/Issues/ConceptEvolution—an
issue of particular interest to builders of SKOS “registries” and APIs.
* 7.
Best practices for modeling SKOS Concept Schemes. As discussed in Section 3.7
there are some concept groupings, such as micro-thesauri, to which the
architecture of SKOS Collections versus SKOS Concept Schemes does not neatly
fit, suggesting a need to clarify best practices.
The W3C Semantic Web Deployment Working Group, which carried SKOS forward
during the final three years of this process, began its work with a draft
specification, at the time called SKOS Core, which had already been widely
deployed and tested by early adopters. In the three years since its
publication in 2009, SKOS has become one of the most widely used vocabularies
in the Linked Data cloud626262http://thedatahub.org/dataset?tags=format-skos—a
context to which its flexible, generic design, based on the principle of
minimal ontological commitment, is uniquely well-suited. As its designers
intended, SKOS continues to be adapted and extended to meet more specialized
requirements. Some of this work is discussed on the public-esw-thes mailing
list636363http://lists.w3.org/Archives/Public/public-esw-thes and tracked on
the W3C SKOS community wiki646464http://www.w3.org/2001/sw/wiki/SKOS. The
development of SKOS has been the collective result of several dozen
contributors working, typically, in the context of working groups or projects
of two or three years’ duration. The accumulated impact of such incremental
contributions becomes clear only in retrospect, looking back with the
perspective of a decade or two.
### Acknowledgements
The authors would like to acknowledge the contributions of all of the members
of the W3C Semantic Web Deployment Working Group in the production of SKOS.
## References
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* [9] ISO TC46/SC9. ISO 2788:1986 Documentation - Guidelines for the establishment and development of monolingual thesauri. Technical report, International Organization for Standardization, 1986\.
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|
arxiv-papers
| 2013-02-05T22:21:52 |
2024-09-04T02:49:41.368537
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Thomas Baker, Sean Bechhofer, Antoine Isaac, Alistair Miles, Guus\n Schreiber and Ed Summers",
"submitter": "Ed Summers",
"url": "https://arxiv.org/abs/1302.1224"
}
|
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