diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzdvbd" "b/data_all_eng_slimpj/shuffled/split2/finalzzdvbd" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzdvbd" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction} \n\\label{sec_intro} \n\nAssociated absorbers are unique tools to probe the physical conditions of the gaseous environment in the immediate vicinity of the background quasar (QSOs). The abundance of heavy elements in these absorbers provides a direct measure of the star formation and chemical evolution in the center of galaxies hosting QSOs \\citep[]{Hamann97a}. Most importantly, good fraction of associated absorbers are believed to originate from the ejected material from the central engine of the QSOs \\citep[]{Gordon99}. These outflows are theoretically invoked to regulate the star formation of the host galaxies and growth of the supper massive black holes (SMBHs) at their centers \\citep[]{Silk98,King03,Bower06,Ostriker10}. \n\n\nThere is no firm definition in the literature for an associated absorber. The absorbers with velocity spread of few $\\times$ 100 km~s$^{-1}$, which appear within few $\\times$ 1000 km~s$^{-1}$\\ from the emission redshift of the QSO, are generally defined to be associated absorbers \\citep{Hamann97a}. In addition to the proximity to the QSOs, associated absorbers are also characterized by (a) time variable absorption line strength \\citep[]{Barlow89,Barlow92,Hamann95,Srianand01,Hall11,Vivek12} (b) very high metallicity \\citep[e.g. near solar abundances;][]{Petitjean94a,Hamann97a} and high ionization parameter \\citep[e.g. log~U $\\gtrsim 0.0$;][]{Hamann98,Hamann00,Muzahid12b} (c) partial coverage of the continuum source \\citep[e.g.][]{Barlow97,Srianand99,Ganguly99,Arav08} and (d) presence of excited fine structure lines \\citep[e.g.][]{Srianand00apm}. These above mentioned properties are unlikely to occur in intervening systems \\citep[however see][for a very special case]{Balashev11} and thus, they are believed to originate from gas very close to the QSO or a possible ejecta of the central engine. In the case of gas outflowing from the QSO, line driven radiative acceleration has often been suggested to be an important driving mechanisms however, only a handful of convincing evidences exists in the literature till date \\citep[see e.g.,][]{Arav94,Arav95,Srianand02}. \n \n\nBased on their line widths, the associated absorbers are broadly classified into two categories: (1) the broad absorption line (BAL) and (2) narrow absorption line (NAL) systems.\nBALs and NALs are predominantly detected through species (e.g., \\mbox{Mg\\,{\\sc ii}}, \\mbox{C\\,{\\sc iv}}, \\mbox{Si\\,{\\sc iv}}, \\mbox{N\\,{\\sc v}}\\ etc.) with low ionization potential (i.e. IP $\\lesssim 100$ eV) in the UV-optical regime. On the other hand, the soft X-ray spectra of $\\sim$ 40 -- 50\\% of the Seyfert galaxies and QSOs happen to show K-shell absorption edges of highly ionized oxygen \\citep[i.e., \\mbox{O\\,{\\sc vii}}, \\mbox{O\\,{\\sc viii}}\\ with IP $\\gtrsim 0.5$~keV;][]{Reynolds97a,George98,Crenshaw03}, known as X-ray ``warm absorbers\" (WAs). These X-ray WAs are often said to correlate with the presence of absorption in the UV regime \\citep[see e.g.,][]{Mathur94,Mathur95b,Mathur95a,Mathur98,Mathur99,Brandt00,Arav07}. \\citet{Telfer98} have argued that the BAL-like absorption seen towards SBS~1542$+$541 could be a potential X-ray WA candidate \\citep[see also][]{Hamann95}. However, QSOs known to have associated BAL absorption are generally found to be X-ray weak \\citep[]{Green95,Green96,Stalin11}. In few cases the physical conditions in the UV absorbers are shown to be incompatible with that of X-ray WAs \\citep[e.g.,][]{Srianand00apj,Hamann00}. Therefore, although a unified picture of X-ray and UV associated absorbers is desirable, it is not clear whether there is any obvious connection between them. Even in cases, where simultaneous occurrences of the X-ray and UV absorption are seen, the absorbing gas need not be co-spatial. For example, envisaging a disk-wind model, \\cite{Murray95b} have shown that the X-ray absorption originates very close to the accretion disk whereas UV absorption predominantly occurs in the accelerated gas farther away. \n\n\nThe study of the species with ionization potential intermediate between UV-optical and X-ray absorbers (i.e., few $\\times$ 100 eV) is important to understand the comprehensive nature of the ionization structure and thus the unified picture of QSO outflows detected in different wavebands. The resonant transitions of highly ionized species (e.g. \\mbox{Ne\\,{\\sc viii}}, \\nani, \\mgx, \\alel\\ and \\mbox{Si\\,{\\sc xii}}), that fall in the far-ultraviolet (FUV) regime, are ideally suited for studying the intermediate ionization conditions of the associated absorbers. However, only a handful of absorbers showing some of these species have been reported till date. For example, the first tentative detection of associated \\mbox{Ne\\,{\\sc viii}}\\ absorption was reported by \\citet{Korista92} towards Q~0226$-$1024. The three other tentative detections existing in literature are by \\citet{Petitjean96} towards HS~1700$+$6414, \\citet{Gupta05} towards 3C~48 and \\citet{Ganguly06} towards HE~0226$-$4110. We also note that a possible \\mbox{Ne\\,{\\sc viii}}\\ detection is reported in the composite {\\sl Far-Ultraviolet Spectroscopic Explorer} ($FUSE$) spectrum by \\citet{Scott04}. There are only six confirmed detections of \\mbox{Ne\\,{\\sc viii}}\\ absorption in associated absorbers reported till date [i.e., UM~675, \\citet{Hamann95}; SBS~1542$+$541, \\citet{Telfer98}; PG~0946$+$301, \\citet{Arav99a}; J2233$-$606, \\citet{Petitjean99}; 3C~288.1, \\citet{Hamann00}; HE~0238$-$1904, \\citet{Muzahid12b}]. Among these, both SBS~1542$+$541 and PG~0946$+$301 are BALQSOs. While multiphase photoionization models are generally used to explain most of these observations, \\citet{Muzahid12b} have shown that the models of collisional ionization equilibrium can also reproduce the observed column density ratios of high ions like \\mbox{O\\,{\\sc vi}}, \\mbox{Ne\\,{\\sc viii}}\\ and \\mgx. Therefore the collisional ionization could be an equally important ionizing mechanism in these absorbers. \n\n\nAlthough the highly ionized UV absorbers are of prime interest to probe the ``missing link\" between UV and X-ray continuum absorbers, the low rest frame wavelengths (i.e. $\\lambda_{\\rm rest}<$ 912 \\AA) of the diagnostic species (e.g. \\mbox{Ne\\,{\\sc viii}}, \\nani, \\mgx\\ etc.) make them difficult to detect. This is partly because of the Galactic Lyman-limit absorption in the spectra of low redshift sources and the Ly$\\alpha$\\ forest contamination in the spectra of high redshift sources. Hence the intermediate redshift (i.e. 0.5 $<$ $z_{\\rm em}$\\ $<$ 1.5) UV bright QSOs are ideal for this study. Note that such a study is only feasible with far-ultra-violet (FUV) sensitive space based telescopes like {\\sl Hubble Space Telescope} ($HST$). In this paper we present a sample of new class of associated absorbers detected through \\mbox{Ne\\,{\\sc viii}}\\ absorption in the FUV spectra of intermediate redshift QSOs obtained with the {\\sl Cosmic Origins Spectrograph} (COS) on board $HST$. \n\n \nThis paper is organized as follows. In Section~2 we describe the observations and data reduction techniques for the sample of QSOs studied here. In Section~3 we discuss the effects of partial coverage in column density measurements and how we correct for it. Data sample and analysis of individual absorption systems are presented in Section~4. In Section~5 we explore ionization models for some of these systems detected with number of different ions. In Section~6 we discuss the overall properties of these absorbers. We summarize our main results in Section~7. \nThroughout this paper we use flat $\\Lambda$CDM cosmology with ($\\Omega_{\\rm M}$, $\\Omega_{\\Lambda}$) = (0.27,0.73) and a Hubble parameter of $H_{0}$ = 71 km~s$^{-1}$ Mpc$^{-1}$. The solar relative abundances of heavy elements are taken from \\citet{Asplund09}. \n\n\n\n\n\\input{table1.tex} \n\n\n\n\\section{Observations and data reduction} \n\\label{sec_obs} \n\n\nThe sample in which we searched for \\mbox{Ne\\,{\\sc viii}}\\ absorbers had the following selection criteria: (1) archived $HST$\/COS FUV spectra (G130M+G160M) of quasars which were public as of February 2012, (2) QSOs with emission redshift $z_{\\rm em} \\ge 0.45$ so that the the \\mbox{Ne\\,{\\sc viii}}\\ doublet transitions (770\\AA\\ and 780\\AA) are redshifted into the wavelength coverage of the COS G130M and G160M gratings, and (3) spectra with signal-to-noise ratio ($S\/N)$ per resolution element $>$10. \nThe properties of COS and its in-flight operations are discussed by \\citet{Osterman11} and \\citet{Green12}. The data were retrieved from the $HST$ archive and reduced using the STScI {\\sc CalCOS} v.2.12 pipeline software. The reduced data were flux calibrated. The alignment and addition of the separate G130M and G160M exposures were done using the software developed by the COS team\\footnote{http:\/\/casa.colorado.edu\/$\\sim$danforth\/science\/cos\/costools.html}. The exposures were weighted by the integration time while coadding in flux units. The procedures followed for data reduction are described in greater detail in \\citet{Narayanan11}. \nThe unabsorbed QSO continuum is fitted using low-order polynomials interpolated between wavelength ranges devoid of strong absorption lines. We use the standard procedure that propagates the continuum placement uncertainty to the normalized flux. \n\nThe medium resolution ($R \\sim$ 20,000) with $S\/N \\ge 10$ COS data, covering 1150 -- 1800 \\AA\\ wavelength range, allow us to search for \\mbox{Ne\\,{\\sc viii}} $\\lambda\\lambda$770,780 doublets in the redshift range $\\sim$~0.45$-$1.31. Observational details of our final sample of 20 quasar sight lines are listed in Table~\\ref{tab:data}. Half of these sight lines were part of a blind survey to detect the warm-hot intergalactic gas (prop. ID 11741). Of the remaining, majority are from the COS-GTO program (prop. ID 11541) to probe the gas phases in the low redshift IGM and galaxy halos. For the QSO PKS~0405$-$123, we have combined spectra obtained under the GTO program of the COS science team from December 2009 and the $HST$ Early Release Observations (prop. ID 11508) of August 2009. While weak radio emission (i.e. flux density $\\le$ 1 mJy) is detected in most of the QSOs in our sample, only seven of them (called radio bright from now on) have radio flux density in excess of 50 mJy at 5 GHz. \n\n\n\\begin{figure} \n\\centerline{\n\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[height=4.4cm,width=8.8cm,angle=00]{fc_fun.ps} \n}}\n}}\n\\caption{Demonstration of the effect of partial coverage in case of a heavily saturated \n({\\sl left}) and an unsaturated ({\\sl right}) \\mbox{Ne\\,{\\sc viii}}\\ lines. Partially covered saturated \nline will appear as flat bottom profile with nonzero flux. For a partially covered \nunsaturated line, column density measurement can lead to much lower value compared to the \ntrue value if we do not correct for the covering fraction. \n} \n\\label{fc_fun} \n\\end{figure} \n\n\n\n\\section{Partial Coverage and Uncertainty in Column density measurement} \n\\label{sec_parcov} \n\n\nBecause of the close physical association between the background QSO and the associated absorber, in many cases it so happens that the latter does not cover the former entirely. In such cases, the observed residual intensity at any frequency can be written as, \n\\begin{equation} \nI(\\nu) = I_{0}(\\nu)(1-f_{c})+f_{c} I_{0}(\\nu) {\\rm exp}[-\\tau(\\nu)]~ . \n\\label{eqn:covf1}\n\\end{equation} \nHere $I_{0}(\\nu)$ is the incident intensity, $\\tau(\\nu)$ is the true optical depth, and $f_{c}$ is the covering fraction. In the case of doublets with rest frame wavelengths $\\lambda_{1}$ \\& $\\lambda_{2}$ and oscillator strengths $f_{1}$ \\& $f_{2}$, the residual intensities $R_{1}$ and $R_{2}$ in the normalized spectra, at any velocity $v$ with respect to the line centroid are related by \n\\begin{equation} \nR_{2}(v) = (1-f_{c})+f_{c}\\times \\left(\\frac{R_{1}(v)-1+f_{c}}{f_{c}}\\right)^{\\gamma}, \n\\label{eqn:covf2}\n\\end{equation} \nwhere $\\gamma = f_{2}\\lambda_{2}\/f_{1}\\lambda_{1}$. The value of $\\gamma$ is very close to 2 for doublets \\citep[see e.g.,][]{Srianand99,Petitjean99}. This equation in principle allows us to calculate the covering fraction of the absorbing gas. \n\n\n\n\\begin{table}\n\\caption{List of important Extreme-UV (EUV) lines used in this paper$^{1}$.} \n\\centering \n\\begin{tabular}{crrrcc} \n\\hline \n Ion & IP(1)$^{a}$ & IP(2)$^{b}$ & $\\lambda^{c}$ (\\AA) & $f_{\\rm osc}^{d}$ & log~$T_{\\rm max}^{e}$ \\\\ \n (1) & (2) & (3) & (4) & (5) & (6) \\\\ \n\\hline \n\\mbox{O\\,{\\sc iv}} & 54.93 & 77.41 & 787.7105 & 1.11$\\times10^{-1}$ & 5.20 \\\\ \n & & & 608.3968 & 6.70$\\times10^{-2}$ & \\\\ \n\\mbox{O\\,{\\sc v}}\\ & 77.41 & 113.90 & 629.7320 & 5.15$\\times10^{-1}$ & 5.40 \\\\ \n\\mbox{N\\,{\\sc iv}}\\ & 47.45 & 77.47 & 765.1467 & 6.16$\\times10^{-1}$ & 5.15 \\\\ \n\\mbox{Ne\\,{\\sc v}}\\ & 97.12 & 126.22 & 572.3380 & 7.74$\\times10^{-2}$ & 5.45 \\\\ \n\\mbox{Ne\\,{\\sc vi}}\\ & 126.22 & 157.93 & 558.5940 & 9.07$\\times10^{-2}$ & 5.65 \\\\ \n\\mbox{Ne\\,{\\sc viii}}\\ & 207.28 & 239.10 & 770.4089 & 1.03$\\times10^{-1}$ & 5.85 \\\\ \n & & & 780.3240 & 5.05$\\times10^{-2}$ & \\\\ \n\\arei\\ & 124.32 & 143.45 & 700.2450 & 3.85$\\times10^{-1}$ & 5.75 \\\\ \n & & & 713.8100 & 1.88$\\times10^{-1}$ & \\\\ \n\\nani\\ & 264.19 & 299.88 & 681.7190 & 9.24$\\times10^{-2}$ & 5.90 \\\\ \n & & & 694.1460 & 4.54$\\times10^{-2}$ & \\\\ \n\\mgx\\ & 328.24 & 367.54 & 609.7930 & 8.42$\\times10^{-2}$ & 6.05 \\\\ \n & & & 624.9410 & 4.10$\\times10^{-2}$ & \\\\ \n\\alel\\ & 399.37 & 442.08 & 550.0310 & 7.73$\\times10^{-2}$ & 6.15 \\\\ \n & & & 568.1200 & 3.75$\\times10^{-2}$ & \\\\ \n\\mbox{Si\\,{\\sc xii}}$^{\\dagger}$ & 476.08 & 523.52 & 499.4060 & 7.19$\\times10^{-2}$ & 6.35 \\\\ \n & & & 520.6650 & 3.45$\\times10^{-2}$ & \\\\ \n\\hline \n\\end{tabular}\n~\\\\ \n\\flushleft \n$^{1}$From \\citet{Verner94} \\\\ \n$^{a}$Creation ionization potential \\\\ \n$^{b}$Destruction ionization potential \\\\ \n$^{c}$Rest frame wavelength in \\AA \\\\ \n$^{d}$Oscillator strength \\\\ \n$^{e}$Temperature at which collisional ionization fraction \\citep[]{Sutherland93} peaks \\\\ \n$^{\\dagger}$Not covered for any of the systems reported here \n\\label{tab:atomic_data} \n\\end{table} \n\n \n\n\n\nThe effects of partial coverage in case of a heavily saturated and an unsaturated line are shown in Fig.~\\ref{fc_fun}. In the left panel of the figure we plot synthetic profiles of \\mbox{Ne\\,{\\sc viii}}$\\lambda$770 line (true line center optical depth $\\tau_0$= 21.0) with $N(\\mbox{Ne\\,{\\sc viii}})$ = 10$^{16}$ cm$^{-2}$ and $b$-parameter of 100 km~s$^{-1}$\\ for $f_c = 1.0$ (solid profile) and $f_c = 0.8$ (dashed profile). The heavy saturation in the profile with complete coverage suggests large optical depth (i.e. $e^{-\\tau(\\nu)}$ = 0). The dashed curve showing flat bottom profile but flux level not reaching to zero, clearly suggests a partial coverage scenario with $f_c = 1 - I(\\nu)\/I_{0}$. Evidently, presence of only one line is sufficient to compute $f_c$ in such a situation. \nIn the right hand panel of Fig.~\\ref{fc_fun}, we show synthetic profiles of \\mbox{Ne\\,{\\sc viii}}$\\lambda$770 line ($\\tau_0$= 2.1) with $N$ = 10$^{15}$ cm$^{-2}$ and $b$-parameter of 100 km~s$^{-1}$\\ for $f_c = 1.0$ (solid), $f_c = 0.8$ (short dashed) and $f_c = 0.5$ (long dashed). For the same column density, profiles with different covering fraction look different. The line center becomes shallower for lower $f_c$ values. The line with a column density of 10$^{15}$ cm$^{-2}$ will appear as $N(\\mbox{Ne\\,{\\sc viii}})$ = 10$^{14.85}$ cm$^{-2}$ ($\\tau_0$= 1.5) and 10$^{14.60}$ cm$^{-2}$ ($\\tau_0$= 0.8) for covering fractions of $f_c =$ 0.8 and 0.5 respectively. Evidently, the observed optical depth in this case is degenerate between the true optical depth and the covering fraction. Therefore, unlike the saturated case, we need at least two lines from the same ground state to estimate the true column density. After estimating the covering fraction (either by flat bottom approximation or from doublet transitions) for a given species we recover the true optical depth by inverting Eq.~\\ref{eqn:covf1}. We then use the partial coverage corrected flux for Voigt profile fitting. Here we make an explicit assumption that the individual Voigt profile components in a blend all have same $f_c$ for a given ion, \nbut note that $f_c$ can be strongly dependent on the velocity along the absorption trough \\citep[]{Srianand99,Arav99b,Gabel05,Arav08}. In addition inhomogeneous absorption models were shown to produce good fits for the absorption troughs as well \\citep[]{Arav08,Borguet12a}. However, given the survey nature of this work and the limited $S\/N$ of the data, we deem it adequate to treat the absorber with simple covering fraction models and to reserve the use of more elaborate models for future investigations and high $S\/N$ observations. \n\n\n\n\\section{Data Sample and Analysis} \n\\label{sec_data} \n \n\\input{tab1.tex} \n\n\\begin{figure} \n\\centerline{\n\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[height=8.4cm,width=9.2cm,angle=00]{outflow_HE0226_0.49253.ps} \n}}\n}}\n\\caption{Velocity plot of the associated \\mbox{Ne\\,{\\sc viii}}\\ absorption system at $z_{\\rm abs}$\\ = 0.49272 \ntowards HE~0226$-$4110. The zero velocity corresponds to the emission redshift \n($z_{\\rm em}$\\ = 0.493) of the QSO. In case of \\mbox{H\\,{\\sc i}}, \\mbox{O\\,{\\sc vi}}\\ and \\mbox{Ne\\,{\\sc viii}}, the smooth curves are the \nbest fitting Voigt profiles, overplotted on top of the data. The vertical ticks mark \nthe centroids of the individual Voigt profile components. Absorption features unrelated \nto this system are marked by the shaded regions.} \n\\label{vp_he0226} \n\\end{figure} \n\n\n\nFor the analysis presented in this paper, we concentrate on associated absorbers detected through \\mbox{Ne\\,{\\sc viii}}$\\lambda\\lambda$770,780 doublets. In Table~\\ref{tab:atomic_data} we have summarised some of the important EUV lines used in this paper including \\mbox{Ne\\,{\\sc viii}}. Here we define, an associated absorbers as (a) those with ejection velocities $|v_{\\rm ej}|$ $\\lesssim$ 8000 km~s$^{-1}$\\ with respect to the QSO emission redshift \\citep[see e.g.][]{Fox08}, or (b) show clear signatures of partial coverage (see e.g. Section \\ref{sec_parcov}) even when having higher ejection velocities (i.e. $|v_{\\rm ej}|$~$\\ge 8000$ km~s$^{-1}$). Here, the ejection velocity $v_{\\rm ej}$ is defined as the velocity separation between the emission redshift of the QSO and the \\mbox{Ne\\,{\\sc viii}}\\ optical depth weighted redshift of the absorber. The $-ve$ sign in the ejection velocity is used whenever absorber redshift is less than the emission redshift of the QSO (i.e. $z_{\\rm abs}$\\ $\\le$ $z_{\\rm em}$). However, in subsequent discussions we will use the term ``higher velocity\" assuming modulus of the ejection velocity. \n\n\nWe have searched for the \\mbox{Ne\\,{\\sc viii}}\\ doublets in the relevant spectral range by imposing the doublet matching criteria. For each identified coincidences we checked the consistency of the profile shape. However, we do not impose the condition of optical depth ratio consistency for the \\mbox{Ne\\,{\\sc viii}}\\ doublets, keeping in mind the effects of partial coverage as discussed in Section \\ref{sec_parcov}. We then checked for the presence of all other species at the redshift of the identified \\mbox{Ne\\,{\\sc viii}}\\ doublets. We find the signatures of associated \\mbox{Ne\\,{\\sc viii}}\\ absorption only in 8 out of 20 (40\\%) lines of sight. We have detected 12 associated \\mbox{Ne\\,{\\sc viii}}\\ absorption systems in total towards 8 lines of sight. Note that any continuous absorption comprised of single\/multiple component(s) are treated as system. Apart from the system detected towards PG~1206+459, all other systems are detected within $\\sim$8000 km~s$^{-1}$\\ with respect to the QSOs. Because of clear signature of partial coverage in the \\mbox{Ne\\,{\\sc viii}}\\ doublet we have included the system in our sample. Based on the number per unit redshift of \\mbox{Ne\\,{\\sc viii}}\\ absorbers \\citep{Narayanan09} we expect to detect only 2 \\mbox{Ne\\,{\\sc viii}}\\ absorbers from the intervening gas. Interestingly none of these \\mbox{Ne\\,{\\sc viii}}\\ absorption detected is towards the 7 radio bright QSOs. Although we search up to 8000 km~s$^{-1}$, 67 per cent of the absorbers are detected within 5000 km~s$^{-1}$\\ from the emission redshift of the QSO. \n\nDetails of the sight lines and the \\mbox{Ne\\,{\\sc viii}}\\ absorbers are summarized in Table~\\ref{tab_list}. Apart for $z_{\\rm abs}$\\ = 0.94262 towards HB89~0107$-$025 (marked as ``Tentative\" in column \\#11 of Table~\\ref{tab_list}), all other associated system in our sample show at least one other species which indeed makes our \\mbox{Ne\\,{\\sc viii}}\\ identification robust. Next, we provide details of each individual \\mbox{Ne\\,{\\sc viii}}\\ systems detected in our sample. \n\n\n\n\\subsection{$z_{\\rm abs}$ = 0.49272 towards HE~0226$-$4110} \n\\label{sec_discript_HE0226_0.49272} \n\n\n\n \nThe ejection velocity of the absorber is only $\\sim -$56 km~s$^{-1}$. The velocity plot of this system clearly shows that the \\mbox{Ne\\,{\\sc viii}}\\ absorption is spread over $\\sim$~226 km~s$^{-1}$\\ (see Fig.~\\ref{vp_he0226}). However, as \\mbox{Ne\\,{\\sc viii}}\\ doublets occur in the extreme blue end of the COS spectrum, the $S\/N$ is not high. A tentative detection of \\mbox{Ne\\,{\\sc viii}}\\ in this system in the $FUSE$ data was reported earlier by \\citet{Ganguly06}. Here we confirm their detection. Apart from the weak \\mbox{Ne\\,{\\sc viii}}, other ions detected in the COS spectrum are \\mbox{C\\,{\\sc iii}}, \\mbox{O\\,{\\sc iii}}, \\mbox{N\\,{\\sc iv}}, \\mbox{O\\,{\\sc iv}}, \\mbox{O\\,{\\sc vi}}\\ and possibly \\mbox{S\\,{\\sc v}}. The detection of \\mbox{O\\,{\\sc v}}\\ is also reported in the $FUSE$ spectrum by \\citet{Ganguly06} which is not covered by the COS spectrum. \nAs the \\mbox{O\\,{\\sc vi}}~$\\lambda 1037$ line is severely blended (see Fig.~\\ref{vp_he0226}), we could not use \\mbox{O\\,{\\sc vi}}\\ doublets to estimate the \\mbox{O\\,{\\sc vi}}\\ covering fraction. \\mbox{Ne\\,{\\sc viii}}, on the other hand, is very weak. \\nani\\ and \\mgx\\ doublets as well as Ly$\\alpha$\\ line are not covered by the COS spectrum. Nevertheless, unblended profiles of Ly$\\beta$ and Ly$\\delta$ transitions are found to be consistent with covering fraction ($f_c$) being 1.0, suggesting complete occultation of the background source by the absorber. The measured column density is log~$N(\\mbox{H\\,{\\sc i}})$ [cm$^{-2}$] = 14.49$\\pm$0.01. The unblended \\mbox{O\\,{\\sc vi}}\\ $\\lambda 1031$ profile is fitted with four Voigt profile components. The total column density (i.e. the summed up column densities measured in four components) is log~$N(\\mbox{O\\,{\\sc vi}})$ [cm$^{-2}$] = 14.76$\\pm$0.13. Because of the low $S\/N$ ratio we use the component structure of \\mbox{O\\,{\\sc vi}}\\ absorption to fit the \\mbox{Ne\\,{\\sc viii}}\\ doublets keeping the $b$-parameter tied with the corresponding \\mbox{O\\,{\\sc vi}}\\ component. The estimated total column densities of \\mbox{Ne\\,{\\sc viii}}\\ is log~$N(\\mbox{Ne\\,{\\sc viii}})$ [cm$^{-2}$] = 14.09$\\pm$0.19. The total column densities for \\mbox{Ne\\,{\\sc viii}}\\ and \\mbox{O\\,{\\sc vi}}\\ as reported by \\citet{Ganguly06}, using apparent optical depth technique, (i.e. log~$N(\\mbox{Ne\\,{\\sc viii}})$ [cm$^{-2}$] = 14.25$\\pm$0.15 and log~$N(\\mbox{O\\,{\\sc vi}})$ [cm$^{-2}$] = 14.84$\\pm$0.08) are very similar to our measurements. The difference in profile between high ions (e.g. \\mbox{O\\,{\\sc vi}}, \\mbox{Ne\\,{\\sc viii}}) and low ions (e.g. \\mbox{H\\,{\\sc i}}, \\mbox{C\\,{\\sc iii}}, \\mbox{O\\,{\\sc iii}}, etc.) is clearly evident from the system plot. Only the strongest \\mbox{O\\,{\\sc vi}}\\ component is accompanied by these low ions. Such a difference in profiles possibly suggests multiphase nature of the absorbing gas. A detailed discussion on the absorbing system and the QSO properties can be found in \\citet{Ganguly06}, therefore we do not discuss this system in detail. \n\n\\begin{figure} \n\\centerline{\n\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[height=8.4cm,width=9.0cm,angle=00]{outflow_HS1102_0.48432.ps} \n}}\n}}\n\\caption{Velocity plot of the associated \\mbox{Ne\\,{\\sc viii}}\\ absorption system at $z_{\\rm abs}$\\ = 0.48518 \ntowards HS~1102$+$3411. The zero velocity corresponds to the emission redshift \n($z_{\\rm em}$\\ = 0.509) of the QSO. The smooth curves overplotted on top of the data are the \nbest fitting Voigt profiles. The vertical ticks mark the centroids of the individual \nVoigt profile components. Absorption features unrelated to this absorber are marked by \nthe shaded regions. \n} \n\\label{vp_hs1102} \n\\end{figure} \n\n\n\\subsection{$z_{\\rm abs}$ = 0.48518 towards HS~1102$+$3441} \n\\label{sec_discript_HS1102_0.48518} \n\n\n\nThe ejection velocity of this system is $v_{\\rm ej} \\sim -4768$ km~s$^{-1}$\\ and is \ndetected through \\mbox{O\\,{\\sc vi}}\\ and \\mbox{Ne\\,{\\sc viii}}\\ absorption, kinematically spread over $\\sim$700 km~s$^{-1}$\\ (see Fig.~\\ref{vp_hs1102}). The \\mbox{Ne\\,{\\sc viii}}$\\lambda$770 is blended with unknown contaminants whereas \\mbox{O\\,{\\sc vi}}\\ $\\lambda 1031$ is found to be blended with Ly$\\alpha$\\ absorption of $z_{\\rm abs}$\\ = 0.26165 system. Unblended profiles of \\mbox{Ne\\,{\\sc viii}}\\ $\\lambda 780$ and \\mbox{O\\,{\\sc vi}}\\ $\\lambda 1037$ clearly show multicomponent structures with at least five components contributing to the absorption. Because of the blending we do not attempt to estimate the covering fraction for either of the detected ions. The Voigt profile fitting assuming complete coverage seems to give reasonably good fit to the unaffected pixels of the blended profiles. The estimated total column densities are log~$N(\\mbox{O\\,{\\sc vi}}) [{\\rm cm^{-2}}] = 15.00 \\pm 0.18$ and log~$N(\\mbox{Ne\\,{\\sc viii}})[{\\rm cm^{-2}}] = 15.22 \\pm 0.20$. Ly$\\alpha$\\ is not covered by the COS spectrum. Ly$\\beta$\\ and Ly$\\gamma$\\ lines are contaminated. Nevertheless, we use the contaminated Ly$\\beta$\\ profile to put an upper limit on $N(\\mbox{H\\,{\\sc i}})$. Assuming component structure and $b$-parameters similar to \\mbox{Ne\\,{\\sc viii}}, we find $N(\\mbox{H\\,{\\sc i}})< 10^{14.52}$~cm$^{-2}$. \n\n\n\n\n\n\\subsection{$z_{\\rm abs}$ = 0.59795, 0.60406 \\& 0.60989 towards HE~0238$-$1904} \n\\label{sec_discript_HE0238} \n\nThese systems are detected at $v_{\\rm ej} \\sim -4500$ km~s$^{-1}$\\ away from the emission redshift of the QSO, in seven absorption components kinematically spread over $\\sim$~1800 km~s$^{-1}$. We have presented a detailed analysis of this absorber in an earlier paper \\citep[see][]{Muzahid12b}. \\mgx\\ lines from this system are severely affected by the Galactic H$_{2}$ absorption and we were able to measure $N(\\mgx)$ only in some of the components showing \\mbox{Ne\\,{\\sc viii}}\\ detection. The \\nani\\ doublets are not covered by the COS spectrum for this system. We looked at $FUSE$ LiF2a data covering the \\nani\\ doublets but do not find any clear signature of \\nani\\ absorption. \n\n\n\n\\begin{figure} \n\\centerline{\n\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[height=8.4cm,width=8.4cm,angle=00]{outflow_FBQS0751_0.91980.ps} \n}}\n}}\n\\caption{Velocity plot of the associated \\mbox{Ne\\,{\\sc viii}}\\ absorption system at $z_{\\rm abs}$\\ = 0.91983 \ntowards FBQS~0751$+$2919. The zero velocity corresponds to the emission redshift \n($z_{\\rm em}$\\ = 0.915) of the QSO. The smooth curves overplotted on top of the data in the \n\\mbox{Ne\\,{\\sc viii}}\\ panel are the best fitting Voigt profiles. The vertical ticks mark the centroids \nof the individual Voigt profile components.} \n\\label{vp_fqbs0751} \n\\end{figure} \n\n\n\n\\subsection{$z_{\\rm abs}$ = 0.91983 towards FBQS~0751$+$2919} \n\\label{sec_discript_FBQS0751_0.91983} \n\n\nThe ejection velocity of this system is $v_{\\rm ej} \\sim +598$ km~s$^{-1}$, suggesting $z_{\\rm abs}$~$>$~$z_{\\rm em}$. \\mbox{Ne\\,{\\sc viii}}\\ doublets in this system clearly show multicomponent structure spreads over $\\sim$~170~km~s$^{-1}$\\ (see Fig.~\\ref{vp_fqbs0751}). Apart from \\mbox{Ne\\,{\\sc viii}}, other ions detected in this system are \\mbox{O\\,{\\sc iv}}, \\mbox{O\\,{\\sc v}}\\ and \\mbox{S\\,{\\sc v}}. \\mbox{Ne\\,{\\sc viii}}$\\lambda$770 seems to be mildly blended in both the wings. The optical depth ratios in the core pixels of \\mbox{Ne\\,{\\sc viii}}\\ absorption are consistent with $f_c$=1.0. The Voigt profile fitting of the \\mbox{Ne\\,{\\sc viii}}\\ doublets leads to a total column density of log~$N(\\mbox{Ne\\,{\\sc viii}}) [{\\rm cm^{-2}}] = 14.59 \\pm 0.09$. For this system \\mbox{O\\,{\\sc vi}}\\ lines are not covered by the COS spectrum. The clear non-detection of \\nani\\ $\\lambda681$ transition is consistent with log~$N(\\nani)[{\\rm cm^{-2}}]<$ 13.94 at 3$\\sigma$ confidence level. The expected positions of both the members of \\mgx\\ doublet are heavily blended and hence we do not have any estimate on $N(\\mgx)$. Very high order Lyman series lines (i.e. with $\\lambda_{\\rm rest}<$930 \\AA) are covered by the COS spectrum where we do not find any clear signature of \\mbox{H\\,{\\sc i}}\\ absorption. Non-detection of Ly$-9$ transition is consistent with $N(\\mbox{H\\,{\\sc i}})<10^{14.34}$~cm$^{-2}$. \n\n \n\\subsection{$z_{\\rm abs}$ = 0.93287 towards PG~1407$+$265} \n\\label{sec_discript_PG1407_0.93287} \n\n\\begin{figure} \n\\centerline{\n\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[height=8.4cm,width=8.4cm,angle=00]{outflow_PG1407_0.93295.ps} \n}}\n}}\n\\caption{Velocity plot of the associated \\mbox{Ne\\,{\\sc viii}}\\ absorption system at $z_{\\rm abs}$\\ = 0.93287 \ntowards PG~1407$+$265. The zero velocity corresponds to the emission redshift ($z_{\\rm em}$\\ = 0.940) \nof the QSO. The smooth curves overplotted on top of the data are the best fitting Voigt \nprofiles. The vertical tick marks the centroids of the individual Voigt profile components.} \n\\label{vplot_PG1407} \n\\end{figure} \nThe ejection velocity of this system is $v_{\\rm ej} \\sim -1103$ km~s$^{-1}$. \\mbox{Ne\\,{\\sc viii}}\\ absorption has two components spread over $\\sim$~100~km~s$^{-1}$\\ (see Fig.~\\ref{vplot_PG1407}). \\mbox{Ne\\,{\\sc viii}}$\\lambda$770 line is found to be contaminated in both the wings. Nevertheless, the unblended core pixels are consistent with covering fraction $f_c = 1.0$. We estimate log~$N(\\mbox{Ne\\,{\\sc viii}})[{\\rm cm^{-2}}] = 14.36\\pm0.22$. \\mgx\\ doublet is fitted with two components slightly off-centered with respect to the \\mbox{Ne\\,{\\sc viii}}\\ components. Estimated total column density of \\mgx\\ absorption is log~$N(\\mgx)[{\\rm cm^{-2}}] = 14.29\\pm0.18$. The non-detection of \\nani\\ $\\lambda 681$ transition is consistent with log~$N(\\nani)[{\\rm cm^{-2}}]<13.60$ at 3$\\sigma$ confidence level. \\mbox{O\\,{\\sc vi}}\\ doublets are not covered by the COS spectrum. \nVery high order Lyman series lines (i.e. with $\\lambda_{\\rm rest}<$ 930 \\AA) are covered by the COS spectrum where we do not find any clear signature of \\mbox{H\\,{\\sc i}}\\ absorption. In addition, no convincing Ly$\\beta$\\ (or Ly$\\gamma$) absorption is seen in archival $HST$\/FOS spectrum, obtained with the G190H grating. We note that the non-detection of Ly$\\beta$\\ is consistent with $N(\\mbox{H\\,{\\sc i}})<10^{13.71}$~cm$^{-2}$. \n\n \n\n\\subsection{$z_{\\rm abs}$ = 0.94262 towards HB89~0107--025} \n\\label{sec_discript_HB890107_0.94262} \n\nThe ejection velocity of this system is $v_{\\rm ej} \\sim -2057$~km~s$^{-1}$\\ and is detected only through \\mbox{Ne\\,{\\sc viii}}\\ absorption spread over $\\sim 120$~km~s$^{-1}$\\ (see Fig.~\\ref{aod_hb890107}). The covering fraction of \\mbox{Ne\\,{\\sc viii}}\\ is consistent with $f_c = 1$ within the continuum placement uncertainty. We measure log~$N(\\mbox{Ne\\,{\\sc viii}})$[cm$^{-2}$] = $14.27\\pm0.04$, whereas the non-detection of \\nani~$\\lambda 681$ transition in the COS spectrum is consistent with $N(\\nani) < 10^{13.75}$~cm$^{-2}$ at 3$\\sigma$ confidence level. The expected positions of \\mgx\\ doublets are contaminated and thus we cannot confirm its presence. In addition, we do not detect any other ion in the COS spectrum, corresponding to this system. \\mbox{O\\,{\\sc vi}}\\ is not covered by COS and we do not find any signature of \\mbox{O\\,{\\sc vi}}\\ lines in the FOS\/G190H spectra. Therefore, we treat this system as tentative one. The non-detection of Ly$\\beta$\\ in FOS\/190H spectra is consistent with $N(\\mbox{H\\,{\\sc i}})< 10^{14.75}$~cm$^{-2}$. \n\n\n\\begin{figure} \n\\centerline{\n\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[height=8.4cm,width=8.4cm,angle=00]{HB890107_aod.ps} \n}}\n}}\n\\caption{Velocity plot of the associated \\mbox{Ne\\,{\\sc viii}}\\ absorber at $z_{\\rm abs}$\\ = 0.94262 \ntowards HB89~0107--025. The zero velocity corresponds to the emission redshift \n($z_{\\rm em}$\\ = 0.956) of the QSO. The smooth curves overplotted on top of the data are \nthe best fitting Voigt profiles. The vertical tick marks the line centroid. The \napparent column density profiles of \\mbox{Ne\\,{\\sc viii}}\\ doublets \n[in units of 10$^{12}$ cm$^{-2}$(km~s$^{-1}$)$^{-1}$] are plotted in the top panel.} \n\\label{aod_hb890107} \n\\end{figure} \n\n\n\n\\subsection{$z_{\\rm abs}$ = 1.02854 towards PG~1206$+$459} \n\\label{sec_discript_PG1206_1.02854} \n\n\n\\begin{figure} \n\\centerline{\n\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[height=10.4cm,width=9.4cm,angle=00]{PG1206_1.02863.ps} \n}}\n}}\n\\caption{Velocity plot of the associated \\mbox{Ne\\,{\\sc viii}}\\ absorption system at $z_{\\rm abs}$\\ = 1.02854 \ntowards PG~1206$+$459. The zero velocity corresponds to the emission redshift \n($z_{\\rm em}$\\ = 1.163) of the QSO. The smooth curves overplotted on top of the data are the \nbest fitting Voigt profiles after correcting for the effect of partial coverage. \nThe vertical ticks mark the centroids of the individual Voigt profile components. \nLy$\\alpha$\\ and \\mbox{N\\,{\\sc v}}\\ are from STIS E230M spectrum. Absorption lines unrelated to \nthis system are marked by the shaded regions.} \n\\label{vp_pg1206} \n\\end{figure} \n\n\nThis is the highest ejection velocity associated system detected in our sample, with $v_{\\rm ej}$ $\\sim -19,228$ km~s$^{-1}$, and \\mbox{Ne\\,{\\sc viii}}\\ absorption is spread over $\\sim360$~km~s$^{-1}$. This system is part of our sample, despite having large ejection velocity, as it shows clear signature of partial coverage. In Fig.~\\ref{vp_pg1206} we show absorption profiles of different species as a function of their outflow velocity with respect to the QSO emission redshift ($z_{\\rm em}$\\ = 1.214). The highly ionized species like \\arei$\\lambda\\lambda$700,713; \\mbox{Ne\\,{\\sc viii}}$\\lambda\\lambda$770,780; \\nani$\\lambda\\lambda$681,694 and \\mgx$\\lambda\\lambda$609,624, originating from this absorber are detected in the COS spectrum. In addition, we also detect species like \\mbox{O\\,{\\sc iv}}, \\mbox{O\\,{\\sc v}}, \\mbox{N\\,{\\sc iv}}\\ in COS and \\mbox{H\\,{\\sc i}}, and \\mbox{N\\,{\\sc v}}\\ in the $HST$\/STIS E230M spectrum. The STIS spectrum does not cover Ly$\\beta$, \\mbox{C\\,{\\sc iii}}\\ or \\mbox{C\\,{\\sc iv}}\\ lines. However, expected wavelength range of \\mbox{Si\\,{\\sc iv}}, \\mbox{Si\\,{\\sc iii}}~$\\lambda$1206, \\mbox{Si\\,{\\sc ii}}~$\\lambda$1260 and \\mbox{C\\,{\\sc ii}}~$\\lambda$1334 lines are covered in the STIS data, but we do not detect any of these species. The profiles of \\mbox{O\\,{\\sc iv}}, \\mbox{N\\,{\\sc iv}}, \\mbox{N\\,{\\sc v}}\\ and \\mbox{Ne\\,{\\sc viii}}\\ doublets are flat over $\\sim$~300 km~s$^{-1}$, indicating partial coverage and heavy saturation of these lines. The flat bottom assumption (see Section~\\ref{sec_parcov}) gives the covering fractions for \\mbox{O\\,{\\sc iv}}, \\mbox{N\\,{\\sc iv}}, \\mbox{N\\,{\\sc v}}, and \\mbox{Ne\\,{\\sc viii}}\\ as $f_c$ = 0.21, 0.40, 0.32 and 0.59 respectively. Unlike these species, the doublets of \\mgx\\ absorption are unsaturated. The uncontaminated profile of \\mgx\\ $\\lambda 609$ clearly shows component structure with at least two components contributing to the absorption. For the subsequent discussions on this system (see section~\\ref{sec_phot_model_pg1206}) we will refer the higher and lower velocity components (i.e. blue and red) as component-1 and component-2 respectively. The blue wing of the \\mgx\\ $\\lambda 624$ is blended with S~{\\sc iv} $\\lambda 657$ line from $z_{\\rm abs}$\\ = 0.9275. The core pixels of \\mgx\\ $\\lambda 624$ which are not affected by this blending are consistent with $f_c = 0.68$. For the singlet transition of \\mbox{O\\,{\\sc v}}, we have taken $f_c$ = 0.59 which seems to be consistent with (nearly) flat bottom seen in the profile. We note that, \\mbox{O\\,{\\sc v}}\\ profile is unusually broad which could possibly due to unknown contamination. Therefore, the actual $f_c$ for \\mbox{O\\,{\\sc v}}\\ could be even less. Because of the weak line strength we do not attempt to estimate $f_c$ for \\nani, instead we use \\mbox{Ne\\,{\\sc viii}}\\ covering fraction for fitting. We note that unlike \\mbox{Ne\\,{\\sc viii}}, profiles of \\nani\\ are not saturated. \n\n\nIn Fig.~\\ref{IP_fc}, we have plotted the covering fractions ($f_c$) of different species detected in this system as a function of ionization potentials. It is clear from the figure that we have two sets of covering fractions for this system. The species with high ionization potentials (i.e. \\mbox{Ne\\,{\\sc viii}}, \\mgx) are showing covering fraction $f_c \\gtrsim 0.6$, whereas, the low ionization species (i.e. \\mbox{O\\,{\\sc iv}}, \\mbox{N\\,{\\sc iv}}, \\mbox{N\\,{\\sc v}}) show $f_c \\lesssim 0.4$. Ionization potential dependent covering fraction have already been reported by \\citet[]{Telfer98,Muzahid12b}, where, the idea of multiphase structure of the absorbing gas has been put forward, with different species having different projected area. In view of this, the covering fraction of \\nani\\ should be similar to that of \\mbox{Ne\\,{\\sc viii}}, as they have ionization potentials of the same order. This also justifies our use of \\mbox{Ne\\,{\\sc viii}}\\ covering fraction for the fitting of \\nani\\ doublets. Such an assumption indeed gives good fit to \\nani\\ doublets. \n\n\n\\begin{figure} \n\\centerline{\n\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[height=4.4cm,width=8.4cm,angle=00]{IP_fc.ps} \n}}\n}}\n\\caption{Covering fractions of different species detected in $z_{\\rm abs}$\\ = 1.02854 \ntowards PG~1206$+$459 as a function of ionization potential. The energy range \nbetween the creation and destruction ionization potentials of a given species \nare shown by the two stars connected by a solid line. \n} \n\\label{IP_fc} \n\\end{figure} \n\n\n\n\\begin{table}\n\\caption{Partial coverage corrected Voigt profile fit parameters for the absorber at $z_{\\rm abs}$\\ = 1.02854 towards PG~1206$+$459.} \n\\begin{tabular}{ccccc} \n\\hline \n\\hline \n $v_{\\rm ej}$(km~s$^{-1}$) & Ion & $b$(km~s$^{-1}$) & log~$N$(cm$^{-2}$) & $f_c$ \\\\ \n (1) & (2) & (3) & (4) & (5) \\\\ \n\\hline \n\\hline \n $-$19290 & \\mgx\\ & 65 $\\pm$ 4 & 15.31 $\\pm$ 0.06 & 0.68 ($db$) \\\\\n $-$19314 & \\nani\\ & 93 $\\pm$12 & 14.93 $\\pm$ 0.05 & 0.59 ($aa$) \\\\ \n $-$19285 & \\mbox{Ne\\,{\\sc viii}}\\ & 83 $\\pm$ 4 & $>$15.90 $\\pm$ 0.06 & 0.59 ($fb$) \\\\\n $-$19285 & \\arei\\ & 83 $\\pm$ 0 & $<$14.19 $\\pm$ 0.02 & 0.59 ($aa$) \\\\ \n $-$19285 & \\mbox{O\\,{\\sc v}}\\ & 141 $\\pm$ 5 & $>$14.92 $\\pm$ 0.03 & 0.59 ($fb$) \\\\\n $-$19339 & \\mbox{N\\,{\\sc v}}\\ & 72 $\\pm$18 & $>$15.38 $\\pm$ 0.40 & 0.32 ($fb$) \\\\\n $-$19285 & \\mbox{O\\,{\\sc iv}}\\ & 57 $\\pm$11 & $>$15.85 $\\pm$ 0.23 & 0.21 ($fb$) \\\\\n $-$19272 & \\mbox{N\\,{\\sc iv}}\\ & 94 $\\pm$17 & $>$14.77 $\\pm$ 0.18 & 0.40 ($fb$) \\\\ \n $-$19285 & \\mbox{H\\,{\\sc i}}\\ & 83 $\\pm$ 0 & $\\sim$13.78 & 0.59 ($aa$) \\\\ \n $-$19285 & \\mbox{H\\,{\\sc i}}\\ & 83 $\\pm$ 0 & $\\sim$14.42 & 0.30 ($aa$) \\\\ \n\\hline \n $-$19163 & \\mgx\\ & 88 $\\pm$ 9 & 15.28 $\\pm$ 0.06 & 0.68 ($db$) \\\\\n $-$19150 & \\nani\\ & 70 $\\pm$11 & 14.68 $\\pm$ 0.08 & 0.59 ($aa$) \\\\ \n $-$19140 & \\mbox{Ne\\,{\\sc viii}}\\ & 68 $\\pm$ 5 & $>$15.69 $\\pm$ 0.11 & 0.59 ($fb$) \\\\\n $-$19140 & \\arei\\ & 68 $\\pm$ 0 & $<$13.66 $\\pm$ 0.07 & 0.59 ($aa$) \\\\ \n $-$19140 & \\mbox{O\\,{\\sc v}}\\ & 113 $\\pm$11 & $>$14.97 $\\pm$ 0.05 & 0.59 ($fb$) \\\\\n $-$19202 & \\mbox{N\\,{\\sc v}}\\ & 70 $\\pm$26 & $>$15.21 $\\pm$ 0.38 & 0.32 ($fb$) \\\\\n $-$19140 & \\mbox{O\\,{\\sc iv}}\\ & 53 $\\pm$12 & $>$15.40 $\\pm$ 0.13 & 0.21 ($fb$) \\\\\n $-$19140 & \\mbox{N\\,{\\sc iv}}\\ & 46 $\\pm$15 & $>$14.28 $\\pm$ 0.36 & 0.40 ($fb$) \\\\ \n $-$19140 & \\mbox{H\\,{\\sc i}}\\ & 68 $\\pm$ 0 & $\\sim$13.61 & 0.59 ($aa$) \\\\\n $-$19140 & \\mbox{H\\,{\\sc i}}\\ & 68 $\\pm$ 0 & $\\sim$14.01 & 0.30 ($aa$) \\\\\n\\hline \n\\hline \n\\end{tabular}\n\\vfill ~\\\\ \nNote -- Listed errors on all the quantities in this paper only include the statistical \nerrors. For $v_{\\rm ej}$ the COS calibration uncertainty is $\\sim \\pm$~10 km~s$^{-1}$. In addition, \nthe uncertainty in the COS LSF introduces errors of at least 1 to 3 km~s$^{-1}$\\ in these profile \nfit line widths. Zero error implies that the parameter was tied\/fixed during fitting. \nCovering fraction, $f_c$ used to estimate the column density is given in column~5. Method used \nto compute $f_c$ is mentioned in parenthesis. ``$fb$\"-- from flat bottom profile, ``$db$\"-- from \ndoublets, ``$aa$\"-- physically motivated assumed value. Note that the column density estimated \nfrom the flat bottom profile (i.e. ``$fb$\") should be taken as lower limit. \n\\label{tab_pg1206} \n\\end{table} \n\n\nParameters estimated through Voigt profile fitting after correcting for partial coverage are given in Table~\\ref{tab_pg1206}. We treat column densities of all the species as lower limits in the case of ions showing flat bottom profiles. We note that, both the members of \\arei\\ doublets are partially blended by unknown contaminants which made the covering fraction estimation impossible. However, since the ionization potentials (creation$+$destruction) of \\arei\\ are comparable to those of \\mbox{O\\,{\\sc v}}, we take $f_c = 0.59$. We note that, because of blend $N(\\arei)$ should be taken as upper limit. \n\n\nAt the expected position of \\mbox{O\\,{\\sc vi}}, some absorption is seen in the low resolution $HST\/$FOS G190H spectrum. However, due to severe blending in both members of the doublets, we do not attempt to estimate the covering fraction. Estimated conservative upper limit on \\mbox{O\\,{\\sc vi}}\\ column density is log~$N(\\mbox{O\\,{\\sc vi}})$ [cm$^{-2}$] $<$ 14.80, assuming $f_c$ = 0.59. In addition, we do not detect any clear signature of Ly$\\beta$\\ absorption in G190H spectrum. Weak Ly$\\alpha$\\ absorption line, seen in STIS\/E230M spectrum, is fitted with two different values of covering fractions (i.e. $f_c$ = 0.59 and 0.30; see Table~\\ref{tab_pg1206}), in order to estimate the maximum \\mbox{H\\,{\\sc i}}\\ content associated with the high and low ionization phases. However, in both the phases $N(\\mbox{H\\,{\\sc i}})$ found to be $< 10^{14.5}$ cm$^{-2}$. All these suggest a very little neutral hydrogen content in this absorber. \n \n \n\n\\subsection{$z_{\\rm abs}$ = 1.21534 towards PG~1338$+$416} \n\\label{sec_discript_PG1338_1.21534} \n\n\nThe ejection velocity of this system is $v_{\\rm ej} \\sim +181$~km~s$^{-1}$\\ suggesting $z_{\\rm abs}$~$>$~$z_{\\rm em}$. This absorber (see the rightmost panel of Fig.~\\ref{vp_pg1338}) is primarily detected through the presence of \\mbox{O\\,{\\sc vi}}\\ doublets in FOS\/G270H spectrum and subsequently confirmed with various other low ionization species (e.g. \\mbox{O\\,{\\sc iii}}, \\mbox{N\\,{\\sc iv}}, \\mbox{O\\,{\\sc iv}}, \\mbox{O\\,{\\sc v}}\\ etc.) in COS spectrum. Weak absorption from high ionization species like \\mgx\\ and \\mbox{Ne\\,{\\sc viii}}\\ are also detected. However, \\mbox{Ne\\,{\\sc viii}}~$\\lambda780$ profile is blended with strong Ly$\\beta$\\ absorption from $z_{\\rm abs}$\\ = 0.6863 system. The \\mgx\\ $\\lambda 609$ line is heavily blended, possibly with low redshift Ly$\\alpha$\\ line and hence not shown in the figure. The non-detection of \\nani\\ $\\lambda 681$ is consistent with log~$N(\\nani)[{\\rm cm^{-2}}] < 14.18$ at 3$\\sigma$ confidence level. \\mbox{O\\,{\\sc iv}}~$\\lambda608$ line is severely blended with \\mgx~$\\lambda624$ line from $z_{\\rm abs}$\\ = 1.15456. \\mbox{O\\,{\\sc iii}}~$\\lambda702$ line is partially blended with unknown contaminants. \nThe uncontaminated low ionization species (i.e. \\mbox{N\\,{\\sc iv}}~$\\lambda765$, \\mbox{O\\,{\\sc iv}}~$\\lambda787$) clearly show multicomponent structure. In both cases, at least two Voigt profile components (shown by vertical dashed lines) are required to get best fitted $\\chi^{2}$ close to 1. Unlike low ionization species, \\mbox{Ne\\,{\\sc viii}}$\\lambda$770 absorption shows smooth and\/or broad profile which is well fitted by a single component. Due to poor spectral resolution, all the ions detected in FOS can be fitted with a single component. \n\n\nWe do not find a clear signature of partial coverage in any line. For example, \\mbox{N\\,{\\sc v}}\\ and \\mbox{O\\,{\\sc vi}}\\ doublets are well fitted with $f_c$ = 1.0 and do not show non-zero flat bottom profiles. The Ly$\\alpha$\\ and (weak) Ly$\\beta$\\ absorption are also consistent with complete coverage of the background source by the absorber. The Voigt profile fit parameters for this absorber are given in Table~\\ref{tab3_pg1338}. We would like to mention here that, because of blending in \\mbox{O\\,{\\sc iii}}\\ line and saturation in \\mbox{O\\,{\\sc v}}\\ line, $N(\\mbox{O\\,{\\sc iii}})$ and $N(\\mbox{O\\,{\\sc v}})$ should be taken as upper and lower limits respectively. We will use these bounds in section~\\ref{sec_phot_model2_pg1338}, where we discuss the photoionization modelling of this system. In passing, we note that \\mbox{H\\,{\\sc i}}\\ and \\mbox{O\\,{\\sc vi}}\\ line centroids are offset by $\\sim$40 km~s$^{-1}$. This could be a signature of multiphase gas. We also note that, \\mbox{H\\,{\\sc i}}\\ and \\mbox{C\\,{\\sc iii}}\\ line centroids are offset by $\\sim$60 km~s$^{-1}$. However, as they are detected in spectra taken with two different instruments (i.e. FOS G270H and G190H), such an offset could also be attributed to the systematic uncertainties. \n\n\n\n\\begin{table}\n\\begin{center}\n\\caption{Voigt profile fit parameters for the absorber at $z_{\\rm abs}$\\ = 1.21534 towards \nPG~1338$+$416 using $f_c = 1$ for all the species.} \n\\begin{tabular}{rccc} \n\\hline \n $v_{\\rm ej}$(km~s$^{-1}$) & Ion & $b$(km~s$^{-1}$) & log~$N$(cm$^{-2}$) \\\\ \n (1) & (2) & (3) & (4) \\\\ \n\\hline \n\\hline \n $+$81 & \\mbox{H\\,{\\sc i}}\\ & 137 $\\pm$14 & 14.04 $\\pm$ 0.04 \\\\\n $+$101 & \\mbox{N\\,{\\sc v}}\\ & 144 $\\pm$15 & 14.36 $\\pm$ 0.04 \\\\\n $+$121 & \\mbox{O\\,{\\sc vi}}\\ & 158 $\\pm$49 & 14.74 $\\pm$ 0.12 \\\\\n $+$126 & \\mbox{N\\,{\\sc iv}}\\ & 44 $\\pm$ 2 & 14.25 $\\pm$ 0.03 \\\\\n $+$126 & \\mbox{O\\,{\\sc iv}}\\ & 44 $\\pm$ 2 & 15.08 $\\pm$ 0.03 \\\\\n $+$126 & \\mbox{O\\,{\\sc iii}} & 44 $\\pm$ 0 & 14.82 $\\pm$ 0.01 \\\\\n $+$136 & \\mbox{O\\,{\\sc v}}\\ & 45 $\\pm$ 3 & 14.78 $\\pm$ 0.05 \\\\\n $+$139 & \\mbox{C\\,{\\sc iii}}\\ & 105 $\\pm$17 & 13.91 $\\pm$ 0.06 \\\\\n $+$181 & \\mbox{Ne\\,{\\sc viii}}\\ & 91 $\\pm$12 & 14.42 $\\pm$ 0.05 \\\\\n $+$181 & \\mgx\\ & 91 $\\pm$ 0 & 14.62 $\\pm$ 0.06 \\\\\n $+$214 & \\mbox{N\\,{\\sc iv}}\\ & 40 $\\pm$ 5 & 13.80 $\\pm$ 0.06 \\\\\n $+$214 & \\mbox{O\\,{\\sc iv}}\\ & 40 $\\pm$ 5 & 14.53 $\\pm$ 0.07 \\\\\n $+$214 & \\mbox{O\\,{\\sc iii}}\\ & 40 $\\pm$ 0 & 14.26 $\\pm$ 0.03 \\\\\n $+$223 & \\mbox{O\\,{\\sc v}}\\ & 32 $\\pm$ 3 & 14.55 $\\pm$ 0.06 \\\\\n\\hline \n\\hline \n\\end{tabular}\n\\label{tab3_pg1338} \n\\end{center}\n\\end{table} \n\n\n\n\n\\begin{figure*} \n\\begin{sideways}\n\\begin{minipage}{24cm} \n\\centerline{\\vbox{\n\\centerline{\\hbox{\n\\includegraphics[height=11.0cm,width=8.0cm,angle=00]{outflow2_PG1338.ps} \n\\includegraphics[height=11.0cm,width=8.0cm,angle=00]{outflow3_PG1338.ps} \n\\includegraphics[height=11.0cm,width=8.0cm,angle=00]{outflow1_PG1338.ps} \n}}\n}} \n\\caption{Velocity plot of the three associated \\mbox{Ne\\,{\\sc viii}}\\ systems detected towards PG~1338$+$416. \nThe zero velocity corresponds to the emission redshift ($z_{\\rm em}$\\ = 1.214) of the QSO. The \nsmooth curves overplotted on top of the data are the best fitting Voigt profiles after \ncorrecting for the partial coverage whenever needed. The shaded regions mark the contamination \ndue to unrelated absorption to the system of interest. The vertical dashed lines mark the \npositions of individual Voigt profile components. \n{\\sl Left :} The system at $z_{\\rm abs}$\\ = 1.15456. {\\sl Middle :} The system at $z_{\\rm abs}$\\ = 1.16420. \nThe smooth dashed curves are not fit to the data, but the synthetic profiles corresponding \nto the maximum allowed column density assuming complete coverage (see text). \n{\\sl Right :} The system at $z_{\\rm abs}$\\ = 1.21534. Apart from \\mbox{C\\,{\\sc iii}}\\ all other species plotted in \nthe left hand sub-panel are from FOS\/G270H spectrum whereas \\mbox{C\\,{\\sc iii}}\\ is from FOS\/G190H spectrum. \nAll the species plotted in the right hand sub-panel are from COS spectrum. Two components are \nclearly seen in \\mbox{O\\,{\\sc iii}}, \\mbox{O\\,{\\sc iv}}, \\mbox{N\\,{\\sc iv}}\\ and \\mbox{O\\,{\\sc v}}\\ absorption (shown by two vertical dashed lines). \nIn all other cases single component is needed as shown by (green) solid tick. \n} \n\\label{vp_pg1338} \n\\end{minipage}\n\\end{sideways} \n\\end{figure*}\n \n\n\\subsection{$z_{\\rm abs}$ = 1.16420 towards PG~1338$+$416} \n\\label{sec_discript_PG1338_1.16420} \n\n\nThe ejection velocity of this system is $v_{\\rm ej} \\sim -6818$~km~s$^{-1}$. The velocity plot of this systems is shown in the middle panel of Fig.~\\ref{vp_pg1338}. Apart from \\mbox{O\\,{\\sc v}}\\ and \\nani~$\\lambda 681$, all other detected ions in this system show complex blend in their profiles. The overall similarity in profiles of various ions clearly assures their presence. We do not find any other contamination in \\nani~$\\lambda 681$ absorption and it shows very similar profile like \\mbox{O\\,{\\sc v}}. Therefore, we believe \\nani\\ detection is robust, although the blue wing of \\nani~$\\lambda 694$ line is severely blended. \\mbox{Ne\\,{\\sc vi}}~$\\lambda 558$ absorption falls near the Galactic Ly$\\alpha$ absorption and hence the continuum around this absorption is not well constrained. Since \\mbox{O\\,{\\sc v}}\\ is singlet transition and \\nani~$\\lambda 694$ is blended, we did not estimate covering fraction for any of these ions. We assume $f_c = 1$ to get a lower limit on column densities. \nAt least four Voigt profile components are required to fit the unblended \\mbox{O\\,{\\sc v}}\\ and \\nani~$\\lambda 681$ profiles. The \\nani\\ to \\mbox{O\\,{\\sc v}}\\ column density ratio in all four components are consistent within factor $\\sim$~2 (e.g. log~$N(\\nani)\/N(\\mbox{O\\,{\\sc v}})$ = 0.27$\\pm$0.21). Because of contamination in the case of \\mgx\\ and \\mbox{Ne\\,{\\sc viii}}\\ lines and poorly constrained continuum in the case of \\mbox{Ne\\,{\\sc vi}}~$\\lambda 558$ line we do not perform Voigt profile fitting for these absorption. Instead, we check the consistency of synthetic profiles generated using the component structure and the $b$-parameters similar to \\mbox{O\\,{\\sc v}}\\ line, assuming $f_c = 1.0$. The synthetic profiles are shown in smooth dashed curves on top of data, in the middle panel of Fig.~\\ref{vp_pg1338}. The highest optical depth pixels in \\mbox{Ne\\,{\\sc viii}}\\ doublets are roughly consistent with $f_c \\gtrsim 0.8$. The line measurements for this system are presented in Table~\\ref{tab2_pg1338}. \nThe low resolution FOS\/G190H spectrum shows absorption in the expected position of \\mbox{O\\,{\\sc vi}}. However, contamination of \\mbox{O\\,{\\sc vi}}\\ lines from $z_{\\rm abs}$\\ = 1.15456 absorber do not allow any reliable column density estimation. Ly$\\alpha$\\ and Ly$\\beta$\\ absorption from this absorber are covered by the G270H and G190H spectra respectively. However, Ly$\\beta$\\ is found to be stronger than Ly$\\alpha$, suggesting a possible contamination in Ly$\\beta$. Ly$\\alpha$, on the other hand, is contaminated with Galactic Fe~{\\sc ii} lines. Therefore, we do not present any measurement for \\mbox{H\\,{\\sc i}}\\ in Table~\\ref{tab2_pg1338}. Due to poorly constrained $f_c$, the column density measurements are highly uncertain and hence we do not discuss the ionization modelling for this system, in spite of the presence of \\nani. \n\n\\subsection{$z_{\\rm abs}$ = 1.15456 towards PG~1338$+$416} \n\\label{sec_discript_PG1338_1.15456} \n\n\n\\begin{table}\n\\begin{center} \n\\caption{Voigt profile fit parameters for $z_{\\rm abs}$\\ = 1.16420 towards PG~1338$+$416 \nassuming $f_c = 1$ for all the species.} \n\\begin{tabular}{cccc} \n\\hline \n $v_{\\rm ej}$(km~s$^{-1}$) & Ion & $b$(km~s$^{-1}$) & log~$N$(cm$^{-2}$) \\\\ \n (1) & (2) & (3) & (4) \\\\ \n\\hline \n\\hline \n$-7022$ & \\nani\\ & 100$\\pm$ 13 & $>$ 14.50 $\\pm$ 0.05 \\\\\n & \\mbox{O\\,{\\sc v}}\\ & 40 $\\pm$ 3 & $>$ 14.03 $\\pm$ 0.02 \\\\\n & \\mgx\\ & 40 & $\\sim$ 14.67 \\\\\n & \\mbox{Ne\\,{\\sc viii}}\\ & 40 & $\\sim$ 14.94 \\\\\n & \\mbox{Ne\\,{\\sc vi}}\\ & 40 & $\\sim$ 14.68 \\\\ \n\\hline \n$-6932$\t & \\nani\\ & 34 $\\pm$ 11 & $>$ 13.91 $\\pm$ 0.15 \\\\ \t\n & \\mbox{O\\,{\\sc v}}\\ &\t 27 $\\pm$ 4 & $>$ 13.68 $\\pm$ 0.07 \\\\ \n & \\mgx\\ &\t 27 & $\\sim$ 14.67 \\\\ \n\t & \\mbox{Ne\\,{\\sc viii}}\\ &\t 27 & $\\sim$ 14.50 \\\\ \n & \\mbox{Ne\\,{\\sc vi}}\\ &\t 27 & $\\sim$ 14.14 \\\\ \n\\hline \n$-6832$\t & \\nani\\ & 74 $\\pm$ 7 & $>$ 14.65 $\\pm$ 0.04 \\\\ \t \n & \\mbox{O\\,{\\sc v}}\\ & 66 $\\pm$ 4 & $>$ 14.17 $\\pm$ 0.02 \\\\ \n & \\mgx\\ & 66 & $\\sim$ 15.20 \\\\ \n\t & \\mbox{Ne\\,{\\sc viii}}\\ & 66 & $\\sim$ 15.14 \\\\ \n & \\mbox{Ne\\,{\\sc vi}}\\ & 66 & $\\sim$ 14.87 \\\\ \n\\hline \n$-6613$\t & \\nani\\ & 86 $\\pm$ 14 & $>$ 14.40 $\\pm$ 0.06 \\\\ \n & \\mbox{O\\,{\\sc v}}\\ & 57 $\\pm$ 3 & $>$ 14.04 $\\pm$ 0.02 \\\\ \n & \\mgx\\ & 57 & $\\sim$ 15.09 \\\\ \n\t & \\mbox{Ne\\,{\\sc viii}}\\ & 57 & $\\sim$ 14.91 \\\\ \n & \\mbox{Ne\\,{\\sc vi}}\\ & 57 & $\\sim$ 14.75 \\\\ \n\\hline \n\\hline \n\\end{tabular}\n\\label{tab2_pg1338} \n\\end{center} \n\\end{table} \n\n \nThe ejection velocity of this system is $v_{\\rm ej} \\sim -8156$~km~s$^{-1}$, with \\mbox{Ne\\,{\\sc viii}}\\ absorption spread over $\\sim 340$~km~s$^{-1}$. The profiles of different species originating from this system are plotted as a function of outflow velocity in the leftmost panel of Fig.~\\ref{vp_pg1338}. This is the highest \\mgx\\ column density system in our sample. The core pixels of \\mgx\\ doublets are free from any blend and clearly show broad multicomponent structure with at least two components contributing to the absorption. We will refer to the highest velocity component as component-1 and the other as component-2\\ in subsequent discussions regarding this system (e.g. in section~\\ref{sec_phot_model1_pg1338}). There is only a mild contamination from Ly$\\gamma$ of $z_{\\rm abs}$\\ = 0.3488 absorber in the blue wing of the \\mgx~$\\lambda609$ line as shown by shaded region. The red wing of the \\mgx~$\\lambda624$, on the other hand, is blended with \\mbox{O\\,{\\sc iv}}~$\\lambda608$ transitions from another associated absorber (i.e. $z_{\\rm abs}$\\ = 1.21534) along this sight line. We use the uncontaminated core pixels (i.e., between $-8350 < v~(\\rm km s^{-1}) < -8150$) of \\mgx\\ doublets and estimate the covering fraction $f_c = 0.8\\pm0.1$ (see Fig.~\\ref{covf_pg1338}). \\nani~$\\lambda681$ line is completely free from any contamination and shows remarkable similarity with \\mgx\\ profiles. This possibly means \\mgx\\ and \\nani\\ are originating from the same phase of the absorbing gas. The red wing of \\nani~$\\lambda694$ line, however, is blended by Ly$-9$ transition from a previously known DLA at $z_{\\rm abs}$\\ = 0.6214 \\citep[]{Rao06}. Therefore we use covering fraction for \\nani\\ similar to that of \\mgx. We note that such an assumption gives remarkably good fit to \\nani\\ doublets. \n\\begin{figure} \n\\centerline{\n\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[height=8.4cm,width=8.8cm,angle=00]{covf_pg1338.ps} \n}}\n}}\n\\caption{Profiles of \\mgx\\ doublet for the system at $z_{\\rm abs}$\\ = 1.15456 towards PG~1338$+$416 \nare shown in the two bottom panels. Corresponding apparent column density distributions \n[in units of 10$^{13}$ cm$^{-2}$ (km s$^{-1}$)$^{-1}$] are shown in the second panel from \nthe top. The covering fraction distribution is shown in the topmost panel. The dashed line \nindicates the median value of covering fraction $f_c = 0.8 \\pm 0.1$, as measured in the core \npixels. The shaded regions show the velocity range affected by unrelated absorption.\n} \n\\label{covf_pg1338} \n\\end{figure} \n\n\nBoth transitions of \\mbox{Ne\\,{\\sc viii}}\\ doublet show very strong, albeit blended, absorption with flat bottom profiles consistent with $f_c = 0.8$. The strong uncontaminated \\mbox{O\\,{\\sc v}}\\ and \\mbox{Ne\\,{\\sc vi}}\\ lines also show flat bottom profiles. The covering fraction in these two cases, as calculated from the flat bottom, are very similar and lower (i.e. $f_c$ = 0.67) than that of very highly ionized species (i.e. \\mbox{Ne\\,{\\sc viii}}, \\mgx). Clearly, like the previous case (i.e. $z_{\\rm abs}$\\ = 1.02854 towards PG~1206$+$459), here also we find two sets of covering fraction for the detected species suggesting ionization potential dependent phase separation of the absorbing gas. \\mbox{Ne\\,{\\sc v}}\\ line seen in this absorber is unsaturated and shows two possible velocity components. However, due to severe blending in both the wings of \\mbox{Ne\\,{\\sc v}}\\ absorption, we only estimate the upper limit on $N(\\mbox{Ne\\,{\\sc v}})$ assuming $f_c$ and $b$-parameters similar to those of \\mbox{O\\,{\\sc v}}\\ line. In section~\\ref{sec_phot_model}, we will show that, under photoionization equilibrium \\mbox{O\\,{\\sc v}}\\ and \\mbox{Ne\\,{\\sc v}}\\ trace each other for the whole range of ionization parameters. Therefore, using the \\mbox{O\\,{\\sc v}}\\ covering fraction for \\mbox{Ne\\,{\\sc v}}\\ is legitimate. We also estimate upper limits on the weak absorption seen in the expected position of \\alel~$\\lambda550$ transition assuming $f_c$ and $b$-parameters similar to those of \\mgx, as they have ionization potentials of similar order. However, as both the wings of \\alel~$\\lambda550$ line is blended the measured column density is merely a upper limit. The other member of \\alel\\ doublet with $\\lambda_{\\rm rest} = 568$~\\AA, is severely affected by the Galactic Ly$\\alpha$\\ absorption and complex blend. In addition, we do not detect any clear signature of Ly$\\alpha$\\ absorption, in the FOS\/G270H spectrum. Some absorption is seen in the expected positions of \\mbox{O\\,{\\sc vi}}\\ doublets in the FOS\/G190H spectrum. However, the contamination of \\mbox{O\\,{\\sc vi}}\\ lines from $z_{\\rm abs}$\\ = 1.16420 absorber and the poor data quality prevent us from any reliable column density estimations. \nThe partial coverage corrected Voigt profile fit parameters are given in Table~\\ref{tab1_pg1338}. In the case of non-detections (i.e. \\mbox{H\\,{\\sc i}}, \\mbox{O\\,{\\sc iv}}\\ and \\arei), we present 3$\\sigma$ upper limits on column densities as estimated from the error in the continuum. \n\n\\section{Ionization models} \n\\label{sec_phot_model} \n\n\\begin{figure} \n\\centerline{\n\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[height=8.4cm,width=8.4cm,angle=00]{sed_mf.ps} \n}}\n}}\n\\caption{Typical shapes of spectral energy distributions of an AGN with arbitrary \nnormalization. The solid line gives the spectrum by \\citet{Mathews87} whereas the \ndotted curve is generated assuming a blackbody with temperature \n$T_{\\rm BB} \\sim 1.5\\times10^{5}$~K and power laws with typical slopes, \n$\\alpha_{\\rm uv} = -0.5$, $\\alpha_{\\rm x} = -0.7$ and $\\alpha_{\\rm ox} = -1.8$ \n(see text). The vertical lines with different line styles mark the frequency \ncorresponding to the ionization potential of the species mentioned in the plot. \n} \n\\label{sed_mf} \n\\end{figure} \n \n\\begin{table}\n\\caption{Partial coverage corrected Voigt profile fit parameters for $z_{\\rm abs}$\\ = \n1.15456 towards PG~1338$+$416.} \n\\begin{tabular}{ccccc} \n\\hline \n $v_{\\rm ej}$(km~s$^{-1}$) & Ion & $b$(km~s$^{-1}$) & log~$N$(cm$^{-2}$) & $f_c^{a}$ \\\\ \n (1) & (2) & (3) & (4) & (5) \\\\ \n\\hline \n\\hline \n$-$8195 & \\nani\\ & 165 $\\pm$ 12 & 15.05 $\\pm$ 0.03 & 0.80 ($aa$) \\\\ \n & \\mgx\\ & 165 $\\pm$ 7 & 15.62 $\\pm$ 0.02 & 0.80 ($db$) \\\\ \n\t & \\alel\\ & 165 & $\\le$14.77 $\\pm$ 0.07 & 0.80 ($aa$) \\\\ \n & \\mbox{Ne\\,{\\sc viii}}\\ & 89 $\\pm$ 35 & $>$15.94 $\\pm$ 0.41 & 0.80 ($fb$) \\\\ \n & \\mbox{O\\,{\\sc v}}\\ & 91 $\\pm$ 6 & $>$14.97 $\\pm$ 0.06 & 0.67 ($fb$) \\\\ \n & \\mbox{Ne\\,{\\sc vi}}\\ & 87 $\\pm$ 5 & $>$15.57 $\\pm$ 0.04 & 0.67 ($fb$) \\\\ \n\t & \\mbox{O\\,{\\sc iv}}\\ & 91 & $\\le$ 13.69 & 0.67 ($aa$) \\\\ \n\t & \\mbox{Ne\\,{\\sc v}}\\ & 91 & $\\le$ 15.26 & 0.67 ($aa$) \\\\ \n\t & \\mbox{Ne\\,{\\sc iv}}\\ & 91 & $\\le$ 13.66 & 0.67 ($aa$) \\\\ \n & \\arei\\ & 91 & $\\le$ 13.94 & 0.67 ($aa$) \\\\ \n\t & \\mbox{H\\,{\\sc i}}\\ & 165 & $\\le$ 13.64 & 0.67 ($aa$) \\\\ \n\t & \\mbox{H\\,{\\sc i}}\\ & 165 & $\\le$ 13.56 & 0.80 ($aa$) \\\\ \n\\hline \n$-$8055 & \\nani\\ & 90 $\\pm$ 8 & 14.82 $\\pm$ 0.04 & 0.80 ($aa$) \\\\ \n & \\mgx\\ & 86 $\\pm$ 5 & 15.33 $\\pm$ 0.03 & 0.80 ($db$) \\\\ \n\t & \\alel\\ & 86 & $\\le$14.66 $\\pm$ 0.06 & 0.80 ($aa$) \\\\ \n & \\mbox{Ne\\,{\\sc viii}}\\ & 62 $\\pm$ 6 & $>$15.45 $\\pm$ 0.12 & 0.80 ($fb$) \\\\ \n & \\mbox{O\\,{\\sc v}}\\ & 65 $\\pm$ 5 & $>$14.85 $\\pm$ 0.08 & 0.67 ($fb$) \\\\ \n & \\mbox{Ne\\,{\\sc vi}}\\ & 66 $\\pm$ 5 & $>$15.60 $\\pm$ 0.07 & 0.67 ($fb$) \\\\ \n\t & \\mbox{O\\,{\\sc iv}}\\ & 65 & $\\le$ 13.90 & 0.67 ($aa$) \\\\ \n\t & \\mbox{Ne\\,{\\sc v}}\\ & 65 & $\\le$ 14.87 & 0.67 ($aa$) \\\\ \n\t & \\mbox{Ne\\,{\\sc iv}}\\ & 65 & $\\le$ 13.10 & 0.67 ($aa$) \\\\ \n\t & \\arei\\ & 65 & $\\le$ 13.30 & 0.67 ($aa$) \\\\ \n\t & \\mbox{H\\,{\\sc i}}\\ & 91 & $\\le$ 13.50 & 0.67 ($aa$) \\\\ \n\t & \\mbox{H\\,{\\sc i}}\\ & 91 & $\\le$ 13.42 & 0.80 ($aa$) \\\\ \n\\hline \n\\hline \n\\end{tabular}\n\\vfill ~\\\\ \nTable Note -- $^{a}$Same as Table~\\ref{tab_pg1206}\n\\label{tab1_pg1338} \n\\end{table} \n\n\n\\begin{figure*} \n\\centerline{\n\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[height=6.4cm,width=8.4cm,angle=00]{fig1.ps} \n\\includegraphics[height=6.4cm,width=8.4cm,angle=00]{fig2.ps} \n}}\n\\centerline{\\hbox{ \n\\includegraphics[height=6.4cm,width=8.4cm,angle=00]{fig3.ps} \n\\includegraphics[height=6.4cm,width=8.4cm,angle=00]{fig4.ps} \n}}\n}} \n\\caption{Results of photoionization model calculation in optically thin condition with \n$N(\\mbox{H\\,{\\sc i}})$ = 10$^{14}$ cm$^{-2}$, incident ionizing continuum given by \\citet{Mathews87}, \nand with solar metallicity $(Z = Z_{\\odot})$. In each panel column densities of various \nspecies (bottom) and their ratios (top) are plotted as a function of ionization parameter. \nA species will be detectable for the ionization parameter range in which it has column density \n$N\\gtrsim10^{13}$~cm$^{-2}$. In the detectable range if two species show constant ratio (i.e. \ninsensitive to ionization parameter), they are likely to originate from the same phase of \nthe absorbing gas. Such a pair of ions is good to estimate relative abundance of the elements. \nIonization parameter should be estimated from the pair of ions whose ratio is sensitive to the \nlog~U. \nNote that, according to the ionization potentials, different species are grouped and plotted in \ndifferent panels for convenience. The notch seen in the $N(\\mgx)\/N(\\nani)$ ratio in panel-(D) is \nan artifact created by {\\sc cloudy} in the low column density limit of $N(\\mgx)$. \n} \n\\label{cloudy1} \n\\end{figure*} \n\n\nIn this section, we try to determine the ionization structure and the physical conditions in the outflowing gas with the help of photoionization equilibrium models using {\\sc cloudy v(07.02)} \\citep[first described in][]{Ferland98}. First, we describe results of a general photoionization model to understand the variation of column densities of different high and low ions and their ratios over a wide range in ionization parameter. We then describe more detailed models (both PI and CI) only for those individual absorbers showing absorption lines from several ions with adequate column density measurements. \n\n\nOur photoionization models assume the absorbing gas to be an optically thin (i.e. stopping H~{\\sc i} column density of 10$^{14}$ cm$^{-2}$ as measured in most cases) plane parallel slab with solar metallicity and relative solar abundances, illuminated by the AGN spectrum. To draw some general conclusions we use the mean spectrum of \\citet{Mathews87} (hereafter MF87, see solid curve in Fig.~\\ref{sed_mf}). However, it is well known that the results of photoionization modeling is very sensitive to the shape of the ionizing radiation. In order to minimize the uncertainties, while modelling individual absorbers, we use the QSO SED of the form: \n\\begin{equation} \nf_{\\nu} = \\nu^{\\alpha_{\\rm uv}} {\\rm exp} (-h\\nu\/kT_{\\rm BB}){\\rm exp}(-kT_{\\rm IR}\/h\\nu)+B\\nu^{\\alpha_{\\rm x}}, \n\\label{agn_cont}\n\\end{equation} \nwhile discussing individual systems. Here, $T_{\\rm BB}$, ${\\alpha_{\\rm uv}}$ and ${\\alpha_{\\rm x}}$ are disk black body temperature, UV spectral index and X-ray spectral index respectively. The normalization constant $B$ is fixed using the optical-to-X-ray powerlaw slope $\\alpha_{\\rm ox}$ and we use $kT_{\\rm IR}$ = 0.01 Rydberg. We use the SED defined by the Eq.~\\ref{agn_cont} with appropriate values for the parameters (based on available observations) when we discuss the photoionization models of individual absorbers. \n\nThe model predictions for MF87 incident continuum are plotted in Fig.~\\ref{cloudy1}. In the bottom of each panel, we plot the column densities of different species having similar ionization potential, as a function of ionization parameter. For the sensitivity of our COS spectra, we find that the column density of individual species has to be $\\ge10^{13}$ cm$^{-2}$ to produce detectable absorption lines which are as broad as $\\sim$~100 km~s$^{-1}$. \n\n\nIn panel {\\bf (A)} of Fig.~\\ref{cloudy1}, we plot the model predictions for the species \\mbox{N\\,{\\sc iv}}\\ (I.P = 47.5 eV), \\mbox{C\\,{\\sc iv}}\\ (I.P = 47.9 eV), \\mbox{O\\,{\\sc iv}}\\ (I.P = 54.9 eV) and \\mbox{Ne\\,{\\sc iv}}\\ (I.P = 63.5 eV). Among all these species, \\mbox{O\\,{\\sc iv}}\\ seems to be the dominant in the range $-2.0\\le$~log~U~$\\le 0.0$ and apart from \\mbox{C\\,{\\sc iv}}\\ all of them showing peak round log~U~$\\sim$~$-1.0$ (see bottom panel). \\mbox{C\\,{\\sc iv}}, however, shows relatively flat distribution over the above mentioned ionization parameter range. For log~U~$>0.0$, column densities of almost all these species become $<10^{13}$ cm$^{-2}$ and hence, they will not be detectable. \nFrom the top panel it is clear that the ionization parameter range where all the species are detectable (i.e., $-2.0\\le$~log~U~$\\le 0.0$), $N(\\mbox{O\\,{\\sc iv}})\/N(\\mbox{N\\,{\\sc iv}})$ and $N(\\mbox{O\\,{\\sc iv}})\/N(\\mbox{Ne\\,{\\sc iv}})$ ratios show remarkable constancy. The ratios where $N(\\mbox{C\\,{\\sc iv}})$ is involved [i.e. $N(\\mbox{Ne\\,{\\sc iv}})\/N(\\mbox{C\\,{\\sc iv}})$ and $N(\\mbox{O\\,{\\sc iv}})\/N(\\mbox{C\\,{\\sc iv}})$], on the other hand, show similar constancy for $-2.0 \\le$ log~U $\\le -1.0$ and fall by a factor of $\\ge$ 0.84 dex in the range $-1.0 \\le$ log~U $\\le 0.0$. \n\n\nIn panel {\\bf (B)} we plot the model predictions for the species \\mbox{S\\,{\\sc vi}}\\ (I.P = 72.7 eV), \\mbox{O\\,{\\sc v}}\\ (I.P = 77.4 eV), \\mbox{N\\,{\\sc v}}\\ (I.P = 77.5 eV) and \\mbox{Ne\\,{\\sc v}}\\ (I.P = 97.1 eV). From the bottom panel, it is apparent that \\mbox{O\\,{\\sc v}}\\ is the dominant species for the whole range in ionization parameters. In addition, all of them show roughly similar $N$ distribution with a peak around log~U $\\sim -0.5$. We also find that most of these species are detectable in the range $-1.5 \\le$ log~U $\\le 0.5$. From the top panel, it is interesting to note that, apart from $N(\\mbox{O\\,{\\sc v}})\/N(\\mbox{S\\,{\\sc vi}})$ ratio, all other ratios are exceptionally constant over the ionization parameter range where these species are detectable (i.e. $-1.5 \\le$ log~U $\\le 0.5$). \n\n\nIn panel {\\bf (C)} we plot the model predictions for the species \\mbox{O\\,{\\sc v}}\\ (I.P = 77.4 eV), \\mbox{O\\,{\\sc vi}}\\ (I.P = 113.9 eV), \\arei\\ (I.P = 124.3 eV) and \\mbox{Ne\\,{\\sc vi}}\\ (I.P = 126.2 eV). From the bottom panel, it is evident that, apart from \\arei\\ all other species are detectable roughly in the range $-1.0\\le$ log~U $\\le 1.0$. In addition, \\mbox{O\\,{\\sc vi}}\\ is found to be the dominant species in this ionization parameter range. \\arei, on the other hand, is detectable in a very narrow range in ionization parameter around log~U $\\sim 0.0$, where all these species show peak column densities. \nFrom the top panel, in is interesting to note that the $N(\\mbox{O\\,{\\sc vi}})\/N(\\mbox{O\\,{\\sc v}})$ ratio keeps on increasing with the increase of ionization parameter whereas $N(\\mbox{O\\,{\\sc vi}})\/N(\\mbox{Ne\\,{\\sc vi}})$ ratio remains constant for the entire range in log~U (i.e. $-2.0 \\le$ log~U $\\le 1.0$). The $N(\\mbox{Ne\\,{\\sc vi}})\/N(\\arei)$ ratio also remains constant in the range $-1.0 \\le$ log~U $\\le 1.0$. $N(\\mbox{O\\,{\\sc vi}})\/N(\\arei)$ ratio, on the contrary, varies by a factor of $\\gtrsim$ 6\\ in the same ionization parameter range. \n\n\nIn panel {\\bf (D)}, we plot the model predictions for the high ionization species e.g., \\mbox{O\\,{\\sc vi}}\\ (I.P = 113.9 eV), \\mbox{Ne\\,{\\sc viii}}\\ (I.P = 207.3 eV), \\nani\\ (I.P = 264.2 eV) and \\mgx\\ (I.P = 328.2 eV). From the bottom panel, we note that \\nani\\ and \\mgx\\ are detectable only for log~U~$\\gtrsim$~0.5 and their column densities show peak at log~U $\\sim$~1.4. \\mbox{Ne\\,{\\sc viii}}, on the other hand, shows peak at log~U $\\sim$~1.0 and $N(\\mbox{Ne\\,{\\sc viii}})>10^{13}$ cm$^{-2}$ for log~U~$\\gtrsim -0.5$. \\mbox{O\\,{\\sc vi}}, in contrast, shows relative flat distribution and is detectable for the entire range in ionization parameter (e.g. $-1.5 \\le$ log~U $\\le 1.8$). \nIt is interesting to note that the ratios plotted in the top panel show smooth variation over the whole range in ionization parameter. For example, $N(\\mgx)\/N(\\nani)$ ratio varies by a factor $\\sim$ 3\\ in the range $0.0 \\le$ log~U $\\le 1.0$ and by a factor $\\sim$ 5\\ in the range 1.0 $\\le$ log~U $\\le$ 2.0. Note that notch seen in $N(\\mgx)\/N(\\nani)$ ratio around log~U = $-1.2$ is not real but a numerical artifact where $N(\\mgx)$ become \nnegligibly small. \n\n\nThe above analysis clearly provides the rough range in the ionization parameter where species with similar ionization potentials are most likely to originate from the same phase of the absorber. In this U range the ratios of such ionic column densities are also useful in constraining the relative abundances of the heavy elements. On a different note, we wish to point out here that all these species originating from same phase (or density) will have similar projected area and hence they will show very similar covering fractions. \nThe ratios of very highly ionized species (i.e. \\mbox{O\\,{\\sc vi}}, \\mbox{Ne\\,{\\sc viii}}, \\nani\\ and \\mgx\\ that are the main focus of this work) show smooth variation over ionization parameter. These ratios are sensitive probes of the ionization parameter provided these\nspecies originate from the same phase of the absorbing gas. The nature of absorption profiles (e.g. velocity alignment, line spread, component structure etc.) can be used to decide whether these species originate from the same phase of the absorbing gas. \n\n\n\n\\begin{figure} \n\\centerline{\n\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[height=8.4cm,width=8.4cm,angle=00]{CIE_ratio.ps} \n}}\n}}\n\\caption{ {\\sl Bottom :} Column densities of various high ionization species \nas a function of gas temperature under collisional ionization equilibrium \n\\citep[]{Sutherland93}. Column densities are calculated for $N(\\mbox{H\\,{\\sc i}}) = 10^{14}$ \ncm$^{-2}$, assuming solar metallicity. \n{\\sl Top :} Column density ratios are plotted as a function of gas temperature.} \n\\label{CIE_ratio} \n\\end{figure} \n \n\\citet{Muzahid12b} have shown that the near constancy of $N$(O~{\\sc vi})\/$N$(Ne~{\\sc viii}) between different components in the associated absorber towards HE~0238--1904 can be explained if collisional excitation plays an important role. Therefore, we now consider the collisional ionization equilibrium (CIE) model \\citep[]{Sutherland93}. The model predicted column densities of high ionization species discussed in the panel-{\\bf (D)} of Fig.~\\ref{cloudy1} are plotted as a function of gas temperature, in the lower panel of Fig.~\\ref{CIE_ratio}. The column densities are calculated for $N(\\mbox{H\\,{\\sc i}})$ = 10$^{14}$ cm$^{-2}$ and $Z = Z_{\\odot}$, typically seen in most of the cases in our sample. It is clear from the figure that for log~$T >$ 6.0, all these high ionization species become fairly insensitive to the gas temperature. This fact is also manifested in the column density ratios, plotted in the top panel. \n\nIn what follows we provide detailed models for some individual systems (specially the ones that show \\nani) in the framework of photoionization and CIE models.\n\n\n\n\\subsection{Models for the system $z_{\\rm abs}$\\ = 1.02854 towards PG~1206$+$459}\n\\label{sec_phot_model_pg1206} \n\n\\begin{figure} \n\\centerline{\n\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[height=8.4cm,width=8.8cm,angle=00]{model1_pg1206.ps} \n}}\n}}\n\\caption{Photoionization model for the system $z_{\\rm abs}$\\ = 1.02854 towards PG~1206$+$459. \n{\\sc cloudy} predicted column density ratios of various high ionization species as \na function of ionization parameter are plotted in different panels. The horizontal \ndashed line in the bottom panel indicates the measured value of $N(\\mgx)\/N(\\nani)$ \nin component-1. In all other cases the horizontal dashed line marks the upper\/lower \nlimit on the ratio as shown by an arrow. \nThe dotted (green) curves are the model prediction in case of $\\rm Na$ is overabundant by \na factor of 7 relative to $\\rm Mg$ and\/or $\\rm Ne$. The dotted vertical line represents a \npossible solution for the ionization parameter. \n} \n\\label{model_pg1206} \n\\end{figure} \n\n\n\\begin{figure*} \n\\centerline{\n\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[height=9.2cm,width=9.2cm,angle=00]{PG1206_CIE.ps} \n\\includegraphics[height=9.2cm,width=9.2cm,angle=00]{PG1338_CIE.ps} \n}}\n}}\n\\caption{{\\sl Left:} CIE model for the system at $z_{\\rm abs}$\\ = 1.02854 towards PG~1206$+$459. \n{\\sl Right:} CIE model for the system at $z_{\\rm abs}$\\ = 1.15456 towards PG~1338$+$416. \nIn each panel model predicted column density ratios of various high ionization species \nare plotted as a function of gas temperature. In the bottommost panels the horizontal dotted \nline represents the measured $N(\\mgx)\/N(\\nani)$ ratios in component-1 and component-2. In all \nother panels horizontal dotted lines followed by an arrow mark the observed upper\/lower limits \non the plotted ratios.\n} \n\\label{CIE_model} \n\\end{figure*} \n\n\n\n\nHere we discuss the physical conditions in the $z_{\\rm abs}$\\ = 1.02854 system towards PG~1206$+$459. The ionizing background is characterized by the Eq.~\\ref{agn_cont} with $T_{\\rm BB} \\sim 1.5 \\times 10^{5}$~K, $\\alpha_{ox} = -1.7$, $\\alpha_{x} = -0.7$ and $\\alpha_{uv} = -0.5$. \nThe value of $\\alpha_{ox}$ has been calculated assuming power law shape of X-ray spectrum with photon index, $\\Gamma_{\\rm 0.3-12 keV} = -1.74\\pm0.09$ (or $\\alpha_{x} \\sim -0.7$) and normalization at 1keV is $A_{\\rm PL} = (2.4\\pm0.2)\\times10^{-4}$ $\\rm photons~cm^{-2} keV^{-1} s^{-1}$, as estimated for this source by \\citet{Piconcelli05}. Using the black hole mass, M$_{\\rm BH} = 1.0 \\times10^{9}$ M$_{\\odot}$, and $\\rm L_{Bol}\/L_{Edd} = 0.84$ from \\citet{Chand10}, the inner disk temperature ($T_{\\rm BB}$) is found to be very similar to the value used here. \n\nFrom Fig.~\\ref{vp_pg1206} (and Table~\\ref{tab_pg1206}) we notice that the covering fractions for \\nani, \\mbox{Ne\\,{\\sc viii}}, \\mbox{O\\,{\\sc v}}, \\mgx\\ and \\arei\\ are similar. It is clear that \\mbox{Ne\\,{\\sc viii}}\\ and \\mbox{O\\,{\\sc v}}\\ column density estimation are lower limits as they are affected by saturation effects. The photoionization model predictions for the above mentioned SED is given in Fig.~\\ref{model_pg1206}. The horizontal dashed line in each panel represents the observed values for the component 1 (i.e., $v_{\\rm ej} \\sim -19,250$ km~s$^{-1}$ component in Table~\\ref{tab_pg1206}). The upper limit on the observed $N(\\mgx)\/(\\mbox{Ne\\,{\\sc viii}})$ ratio, suggests log~U~$\\le$~1.0. The lower limit on the observed $N(\\mgx)\/(\\arei)$ ratio, on the other hand suggests log~U~$\\ge$~0.7. We notice that in this ionization parameter range (i.e. 0.7 $\\le$ log~U $\\le$ 1.0) the model over-predicts the observed $N(\\mgx)\/N(\\nani)$ ratio. The observed column density ratios involving \\nani\\ can be reproduced by the models if we assume ${\\rm Na}$ is enhanced by a factor of 0.85 dex with respect to ${\\rm Mg}$ and\/or ${\\rm Ne}$ [see the dotted (green) curves in Fig.~\\ref{model_pg1206}].\nWe estimate the upper limit for log~$N(\\mbox{H\\,{\\sc i}})$ (cm$^{-2}$) = 13.78 using $f_c = 0.59$ (see Table~\\ref{tab_pg1206}). For this model predicts log~$N$(\\mgx) (cm$^{-2}$) = 15.32, which is very close to the observed value implying the metallicity of the gas phase producing \\mbox{Ne\\,{\\sc viii}}\\ and \\mgx\\ is higher than solar.\nAmong the other species detected in this component only \\mbox{O\\,{\\sc iv}}\\ column density and covering fraction are well measured. We find $N$(\\mbox{O\\,{\\sc iv}}) predicted by our model for log~U$\\sim 1$ is a factor 25 times smaller than what is observed. This confirms that \\mbox{O\\,{\\sc iv}}\\ is originating from a distinctly different phase as suggested by the low covering fraction as well. The observed column density of \\mbox{O\\,{\\sc iv}}\\ for solar metallicity and log~$N(\\mbox{H\\,{\\sc i}})$ (cm$^{-2}$) = 14.42 (for the similar covering fraction measured for \\mbox{O\\,{\\sc iv}}) we find log~U $\\sim 0$. This ionization parameter also produces the correct value of observed $N(\\mbox{N\\,{\\sc iv}})$. If both these phases are at the same distance from the QSO then we can conclude that there is a factor ten change in the density along the transverse direction for the absorbing gas. \n\n \nIn the case of component-2 (i.e., $v_{\\rm ej}\\sim -19,150$ km~s$^{-1}$\\ component in Table~\\ref{tab_pg1206}), the upper limit on observed $N(\\mgx)\/(\\mbox{Ne\\,{\\sc viii}})$ ratio, suggests log~U~$\\le$~1.0. The lower limit on observed $N(\\mgx)\/(\\arei)$ ratio, on the other hand suggests log~U~$\\ge$~0.8. As in the case of component-1 for this ionization parameter range the photoionization model over predicts the observed $N(\\mgx)\/N(\\nani)$. We find that the observed column density ratios involving \\nani\\ can be reproduced by the model if we assume $\\rm Na$ is enhanced by a factor of 0.60 dex with respect to $\\rm Mg$ and\/or $\\rm Ne$. Like in the previous case the model that reproduces the high ions under-predicts the \\mbox{O\\,{\\sc iv}}\\ column density. We also find \\mbox{O\\,{\\sc iv}}\\ is originating from a phase that is up to a factor 10 lower density if both phases are at same distance from the QSO. \n\nIn both the components \\mbox{N\\,{\\sc v}}\\ absorption is detected. The measured column densities are consistent with an ionization parameter intermediate between the gas traced by \\mbox{N\\,{\\sc iv}}\/\\mbox{O\\,{\\sc iv}} and \\mgx. All this suggests that the outflow having smooth density gradients in the transverse direction.\n\nFor log~U=1, the inferred total column density of system is $N(\\rm H)$ = 4.7~$\\times$~10$^{20}$ cm$^{-2}$ (when log~$N(\\mbox{H\\,{\\sc i}})$ (cm$^{-2}$) = 14.0), whereas, \\mbox{O\\,{\\sc vii}}\\ and \\mbox{O\\,{\\sc viii}}\\ column densities are $N(\\mbox{O\\,{\\sc vii}})$~ = 1.0~$\\times$~10$^{17}$ cm$^{-2}$ and $N(\\mbox{O\\,{\\sc viii}})$~ = 9.5~$\\times$~10$^{16} \\rm cm^{-2}$, suggesting continuum optical depths of \\mbox{O\\,{\\sc vii}}\\ and \\mbox{O\\,{\\sc viii}}\\ are much less than 0.1. Therefore, this system may not be a potential X-ray WA candidate. \n \n\nIn the left hand panel of Fig.~\\ref{CIE_model}, the column density ratios of various high ionization species predicted by the CIE models, are plotted as a function of gas temperature. The horizontal dashed lines followed by arrows, in each sub-panel except for the bottom one, indicate the upper limit on the column density ratios measured in component-1 and component-2. The measured values of $N(\\mgx)\/N(\\nani)$ ratio, shown in the bottom panel, are found to be very similar for both the components which corresponds to a temperature of log~$T \\sim 5.9$. Note that the upper limits on $N(\\mgx)\/N(\\mbox{Ne\\,{\\sc viii}})$ ratios observed in both the components suggests log~$T \\lesssim 5.9$. The observed value of $N(\\mbox{O\\,{\\sc v}})$, on the other hand, suggests a temperature $T\\sim10^{5.8}$~K. On the other hand we notice that low ionization species like O~{\\sc iv} and N~{\\sc iv} require $T\\sim10^{5.2}$ K. In order for the two phases to be in pressure equilibrium the density of the low ionization phase needs to be a factor 4 higher. We next run {\\sc cloudy} model keeping the gas temperature to be constant at $T\\sim10^{5.8}$~K and found that the ionization parameter of the gas log~U$\\le-2$ so that the ratio of \\mgx\\ and \\mbox{Ne\\,{\\sc viii}}\\ are not affected by the QSO radiation. Given the luminosity of the QSO this corresponds to a radial separation of $\\gtrsim 2600\/\\sqrt{(n_{\\rm H}\/10^5)}$~pc between the absorbing gas and the QSO, so that the ionization state can be dominated by collisions. \n\n\n\n\n\\subsection{Models for the system $z_{\\rm abs}$\\ = 1.15456 towards PG~1338$+$416} \n\\label{sec_phot_model1_pg1338} \n\n\n\\begin{figure} \n\\centerline{\n\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[height=9.2cm,width=9.2cm,angle=00]{PG1338_mod.ps} \n}}\n}}\n\\caption{Photoionization model for the system $z_{\\rm abs}$\\ = 1.15456 towards PG~1338$+$416. \n{\\sc cloudy} predicted column density ratios of different ions are plotted as a \nfunction of ionization parameter in different panels. The horizontal (red) dashed line \nfollowed by an arrow in each panel represents the observed upper\/lower limit on the \nplotted ratio as measured in component-1. \n{\\sl Left :} Ratios of different ionization states of neon are plotted. \n{\\sl Right :} Same as {\\sl Left} but for different ionization states of different elements. \nThe dotted (green) curves are the model prediction in case of $\\rm Na$ is overabundant by a \nfactor of 5 relative to $\\rm Mg$ and\/or $\\rm Ne$. The shaded region represents the allowed \nrange in ionization parameter as suggested by various ionic ratios. \n} \n\\label{phot_pg1338} \n\\end{figure} \n \n\n\nHere we discuss the photoionization model for the system $z_{\\rm abs}$\\ = 1.15456 towards PG~1338$+$416. For SED we use $T_{\\rm BB}\\sim1.0\\times10^{5}$~K, $\\alpha_{\\rm x} = -1.5$, $\\alpha_{\\rm uv} = -0.5$ and $\\alpha_{\\rm ox} = -1.8$ \\citep[from][]{Anderson07}. From the \\mbox{Mg\\,{\\sc ii}}\\ emission line width \\citet{Chand10} have estimated the black hole mass for this source to be log~M$_{\\rm BH}$\/M$_{\\odot} \\sim$ 8.96 and 9.47 using the method by \\citet{McLure04} and \\citet{Dietrich09} respectively. They also find $L_{\\rm bol}\/L_{\\rm Edd} = 0.34$ for this source. Using these we calculate the inner disk temperature for this QSO to be $T_{\\rm BB} \\sim 1.2 \\times 10^{5}$~K and $9.1 \\times 10^{4}$~K for log~M$_{\\rm BH}$\/M$_{\\odot} \\sim$ 8.96 and 9.47 respectively. This is close to what we use to generate the SED. \n\n\nIn Fig.~\\ref{phot_pg1338} we show the results of our photoionization model. In the left hand panel of the figure we have shown the column density ratios of different ionization states of neon. All these ratios gives lower limits on ionization parameter. The best constraint comes from $N(\\mbox{Ne\\,{\\sc viii}})\/N(\\mbox{Ne\\,{\\sc v}})$ ratio, which suggests log~U~$\\ge$~0.2. All other ratios are consistent with this lower limit. In the right hand panel we have plotted ionic ratios of different species of different elements which can provide useful constraints on the ionization parameter (see section \\ref{sec_phot_model}). The observed upper limit on $N(\\mgx)\/N(\\mbox{Ne\\,{\\sc viii}})$ ratio suggests log~U~$\\le$~0.8. Hence the physically allowed range in ionization parameter becomes $0.2 \\le$~log~U~$\\le 0.8$, as marked by the shaded region. We note that the observed limits on $N(\\mgx)\/N(\\alel)$ and\/or $N(\\mgx)\/N(\\mbox{O\\,{\\sc v}})$ are also consistent with this range. However, it is apparent from the right-bottom panel, that our model cannot reproduce the observed $N(\\mgx)\/N(\\nani)$ ratio for the whole range in ionization parameter, where the individual species (i.e. \\nani\\ and \\mgx) are detectable. Similarly, $N(\\nani)\/N(\\mbox{Ne\\,{\\sc viii}})$ ratio also suggests a very high log~U, which is not in the allowed range of ionization parameter (i.e. shaded region). Like the previous case (see section~\\ref{sec_phot_model_pg1206}), such a discrepancy can be easily avoided if $\\rm Na$ is overabundant by factor of $\\sim$~5--6, as can be seen from the dotted (green) curves in the figure. \n\n\nAssuming log~U = 0.5 and using the estimated upper limit on $N(\\mbox{H\\,{\\sc i}})$ in the component-1 [i.e. log~$N(\\mbox{H\\,{\\sc i}})$ (cm$^{-2}$) $\\le$ 13.64; see Table~\\ref{tab1_pg1338}], we estimate the metallicity of the gas to be $\\gtrsim$ 10 $Z_{\\odot}$. The total column density of the system at log~U~$\\sim$~0.5 is log~$N(\\rm H)$ (cm$^{-2}$) = 20.09. Predicted column densities of \\mbox{O\\,{\\sc vii}}\\ and \\mbox{O\\,{\\sc viii}}\\ are log $N$ (cm$^{-2}$) = 17.47 and 17.02 respectively. Continuum optical depth of oxygen corresponding to these values is again much less than 0.1, suggesting that the system may not be a potential X-ray WA candidate. \n\n\nFrom the right hand panel of Fig.~\\ref{CIE_model}, we can conclude that the observed ratios and limits of high ions can be explained if the gas temperature is $T\\sim 10^{5.9}$ K without the enhancement of $\\rm Na$ as required by the photoionization models. However, in order for the QSO radiation field to not affect the ionization state of the absorbing gas the ionization parameter has to be log~U~$\\le-1.0$. For the inferred luminosity this corresponds to a separation of $\\gtrsim400\/\\sqrt{(n_{\\rm H}\/10^5)}$~pc of the absorbing cloud from the QSO. \n\n\n\n\\subsection{Model for the system $z_{\\rm abs}$\\ = 1.21534 towards PG~1338$+$416} \n\\label{sec_phot_model2_pg1338} \n\n \nIn this section we discuss the photoionization model for $z_{\\rm abs}$\\ = 1.21534 absorber towards PG~1338$+$416. This system has $z_{\\rm abs}$\\ very similar to $z_{\\rm em}$\\ = 2.2145$\\pm$0.0019. This is the only associated \\mbox{Ne\\,{\\sc viii}}\\ system along the line of sight without detectable \\nani\\ absorption. Unlike this system the other two have large outflow velocities and show signatures of partial coverage. The column density of \\mbox{Ne\\,{\\sc viii}}\\ is also high in the other two systems.\n\n\nIn section~\\ref{sec_discript_PG1338_1.21534}, we have seen that the low ionization species, detected in COS, originating from this system show 2 possible components. However, because of the poor spectral resolution, species detected in $HST\/$FOS spectra can be well fitted by a single Voigt profile component. Because of this disparity in the data quality, we use the total column densities (i.e. summed up component column densities), for the photoionization model. We run {\\sc cloudy} with same set of parameters as described in section~\\ref{sec_phot_model1_pg1338}. The results of our photoionization model are shown in Fig.~\\ref{fig:phot_1.21534_pg1338}. \n\n\nThe column density ratios of different ionization states of same element are very important diagnostics of ionization parameter. Therefore, we make use of simultaneous presence of \\mbox{O\\,{\\sc iii}}, \\mbox{O\\,{\\sc iv}}, \\mbox{O\\,{\\sc v}}\\ and \\mbox{O\\,{\\sc vi}}\\ lines of oxygen and \\mbox{N\\,{\\sc iv}}\\ and \\mbox{N\\,{\\sc v}}\\ lines of nitrogen to estimate the ionization parameter of the gas. Since we treat $N(\\mbox{O\\,{\\sc iii}})$ and $N(\\mbox{O\\,{\\sc v}})$ as upper and lower limits (see discussions in section~\\ref{sec_discript_PG1338_1.21534}), the ionization parameter is primarily decided by $N(\\mbox{O\\,{\\sc vi}})\/N(\\mbox{O\\,{\\sc iv}})$ and $N(\\mbox{N\\,{\\sc v}})\/N(\\mbox{N\\,{\\sc iv}})$ ratios. It is clear from Fig.~\\ref{fig:phot_1.21534_pg1338} that, both the ratios are remarkably consistent with log~U $\\sim -1.0$. We also note that, the upper limit on $N(\\mbox{O\\,{\\sc vi}})\/N(\\mbox{O\\,{\\sc v}})$ and lower limits on $N(\\mbox{O\\,{\\sc iv}})\/N(\\mbox{O\\,{\\sc iii}})$ ratios are also suggestive of such an ionization parameter. Using the ionization fractions at log~U $\\sim -1.0$, we find that the metallicity of the gas to be near solar, e.g. log~$Z\/Z_{\\odot}$ = 0.40$^{+0.90}_{-0.25}$. \n\n\n\\begin{figure} \n\\centerline{\n\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[height=7.2cm,width=8.8cm,angle=00]{phot_1.21534.ps} \n}}\n}}\n\\caption{Photoionization model for the system $z_{\\rm abs}$\\ = 1.21534 towards PG~1338$+$416. \n{\\sc cloudy} predicted column density ratios of different ions are plotted as a function \nof ionization parameter in different panels. The horizontal dashed line in the bottom two \npanels mark the measured ionic ratios. The dashed lines with arrows in the top two panels \nshow measured limits on the ionic ratios. The vertical dotted line at log~U = $-1$ marks \na possible solution for the ionization parameter. \n} \n\\label{fig:phot_1.21534_pg1338} \n\\end{figure} \n \n\n\n\n\nIn the section~\\ref{sec_phot_model}, we have seen that the species \\mbox{N\\,{\\sc iv}}\\ and \\mbox{O\\,{\\sc iv}}\\ traces each other for a wide range in ionization parameter. Therefore, $N(\\mbox{N\\,{\\sc iv}})\/N(\\mbox{O\\,{\\sc iv}})$ ratio is a sensitive probe of the relative abundances. From the observed $N(\\mbox{N\\,{\\sc iv}})\/N(\\mbox{O\\,{\\sc iv}})$ ratio we find that nitrogen is overabundant compared to oxygen by a factor of 0.93 dex (i.e., $\\rm [N\/O] = 0.07$). Furthermore, nitrogen is found to be overabundant compared to carbon by a factor of 0.89 dex (i.e. $\\rm [N\/C] = 0.29$), from the measured $N(\\mbox{N\\,{\\sc iv}})\/N(\\mbox{C\\,{\\sc iii}})$ ratio. Since \\mbox{O\\,{\\sc iii}}\\ line is blended, we use $N(\\mbox{C\\,{\\sc iii}})\/N(\\mbox{O\\,{\\sc iv}})$ ratio to estimate $\\rm [C\/O]$ and found that the carbon and oxygen roughly follow solar abundance pattern. For example, estimated $\\rm [C\/O] = -0.22$, whereas, in sun $\\rm (C\/O) = -0.26$ \\citep[]{Asplund09}. Such an enhanced nitrogen abundance is seen in high redshift ($z\\ge2.0$) QSOs \\citep[]{Hamann92,Korista96,Petitjean99}. These authors suggested a rapid star formation scenario which produces a super solar metallicity in order to boost the nitrogen abundance through enhanced secondary production in massive stars. We would like to mention that, with the estimated ionization parameter and metallicity, neither \\mbox{Ne\\,{\\sc viii}}\\ nor \\mgx\\ would be detectable [e.g., reproduced log~$N(\\mbox{Ne\\,{\\sc viii}})$ (cm$^{-2}$) $\\ll$ 14.0 at log~U = $-1.0$]. \n\n\nThe observed $N(\\mbox{Ne\\,{\\sc viii}})\/N(\\mgx)$ ratio require a different phase with fairly high ionization parameter (i.e., log~U~$\\sim$~1.3). If we assume most of the \\mbox{O\\,{\\sc v}}\\ originate from \\mbox{Ne\\,{\\sc viii}}\\ phase then we get log~U $\\ge$~0.8. In equality in this case is because some part of \\mbox{O\\,{\\sc v}}\\ will originate from \\mbox{O\\,{\\sc iv}}\\ phase. If we use $N$(\\mbox{O\\,{\\sc vi}})\/$N$(\\mbox{Ne\\,{\\sc viii}}) ratio then we get log~U $\\ge$~0.9. Thus one can conclude that the \\mbox{Ne\\,{\\sc viii}}\\ absorption is originating from a gas having log~U~$\\sim$ 1 (as we have seen in the other cases discussed above).\nIf we assume the \\mbox{Ne\\,{\\sc viii}}\\ phase has same metallicity as the low ionization phase discussed above then we can conclude that \\mbox{H\\,{\\sc i}}\\ associated with \\mbox{Ne\\,{\\sc viii}}\\ is $\\le 10^{12}$ cm$^{-2}$. We can conclude that the low hydrogen column density in this component is the reason for the lack of \\nani\\ absorption in this system. \n\n\nLike in the previous cases our model suggests that the absorbing gas will not have sufficient optical depth to be a X-ray warm absorber. \n\n\n\n\n\\begin{table*}\n\\caption{Summary of associated \\mbox{Ne\\,{\\sc viii}}\\ absorbers from literature (only secure detections are listed here)} \n\\centering \n\\begin{tabular}{crrcccccccc} \n\\hline \n & & & \\multicolumn {5}{c} {log~$N$~(cm$^{-2}$)} & \\\\ \\cline{4-8} \nQSO & $z_{\\rm em}$ & $v_{\\rm ej}$(km~s$^{-1}$) & \\mbox{O\\,{\\sc vi}} & \\mbox{Ne\\,{\\sc viii}} & \\mgx & \\mbox{H\\,{\\sc i}} & $\\rm H$ & Type & QSO Type & Reference \\\\ \\hline \n (1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) & (9) & (10) & (11) \\\\ \n\\hline \nUM~675 & 2.150 & $-$1500 & 15.5 & 15.4 & .... & 14.8 & 20.0 & NAL & RQ$^{b}$ & \\citet{Hamann95} \\\\ \nSBS~1542$+$541 & 2.631 & $-$11360 & 15.8 & 16.0 & 15.9 & 14.9 & 22.7 & BAL & RQ & \\citet{Telfer98} \\\\ \nJ~2233$-$606 & 2.240 & $-$3900 & 15.4 & 15.1 & .... & 14.0 & 22.0 & NAL & ... & \\citet{Petitjean99} \\\\ \nPG~0946$+$301 & 1.221 & $-$10000 & 16.6 & 16.7 & 16.6 & 15.3 & .... & BAL & RQ & \\citet{Arav99a} \\\\ \n3C~288.1 & 0.965 & $+$250 & 15.8 & 15.4 & 15.0 & 15.8 & 20.2 & NAL & RL$^{a}$ & \\citet{Hamann00} \\\\ \n\\hline \n\\hline \n\\end{tabular}\n~\\\\ \nTable Note -- $^{a}$ Radio Loud; ~~$^{b}$ Radio Quiet \n\\label{tab:summary} \n\\end{table*} \n\n\n\\begin{figure} \n\\centerline{\\hbox{ \n\\centerline{\\vbox{\n\\includegraphics[height=8.0cm,width=8.0cm,angle=00]{b_dist.ps} \n}}\n}}\n\\caption{ {\\sl Bottom:} The distribution of Doppler parameter as measured in individual \n\\mbox{Ne\\,{\\sc viii}}\\ components. {\\sl Top:} Number of \\mbox{Ne\\,{\\sc viii}}\\ components against the spread of \\mbox{Ne\\,{\\sc viii}}\\ \nabsorption in each system. \n} \n\\label{b_dist} \n\\end{figure} \n \n\\section{Discussions} \n\\label{sec_diss}\n\nIn this section, we try to draw a broad physical picture of the associated \\mbox{Ne\\,{\\sc viii}}\\ absorbers. \n\n\n\\subsection{Incidence of associated \\mbox{Ne\\,{\\sc viii}}\\ absorbers} \n\n \nThe fraction of AGNs that show associated absorption is important for understanding the global covering fraction and the overall geometry of the absorbing gas \\citep[]{Crenshaw03,Ganguly08}. In low redshift Seyfert galaxies, surveys in the UV \\citep[]{Crenshaw99}, FUV \\citep[]{Kriss02}, and X-rays \\citep[]{Reynolds97a} found $\\sim$ 50 -- 70\\% incidence of associated absorbers. For quasars, the fraction of occurrence has been found to be somewhat lower. For example \\citet[]{Ganguly01} found signature of associated \\mbox{C\\,{\\sc iv}}\\ absorption in $\\sim$~25\\% of QSOs. On the other hand, \\citep[]{Dai08} have found occurrence of BAL in $\\sim$~40\\% cases. However, we note that depending upon the selection criteria (e.g. cutoff velocity, rest frame equivalent width etc.) these numbers could be very different. An exposition on the incidence of different forms of the associated absorbers can be found in \\citet{Ganguly08}. We have found 12 associated \\mbox{Ne\\,{\\sc viii}}\\ systems in 8 out of 20 QSOs in our sample while only 2 is expected based on the statistics of intervening systems. Even if we restrict ourself to $|v_{\\rm ej}|$ up to 5000 km~s$^{-1}$\\ instead of 8000 km~s$^{-1}$, we have 8 associated systems which is factor 4 higher compared to what is expected from statistics of intervening systems. Such an enhanced occurrence of associated absorbers have also been noticed in the case of high-$z$ \\citep[]{Fox08} and low-$z$ \\citep[]{Tripp08} \\mbox{O\\,{\\sc vi}}\\ absorbers. \nThe incidence of associated \\mbox{Ne\\,{\\sc viii}}\\ absorbers in our sample is $\\sim$40\\% ($\\sim$35\\% if we do not include the tentative system towards HB89 0107$-$025 or restrict to systems with $|v_{\\rm ej}|<$ 5000 km~s$^{-1}$). It is also interesting to note only 5\/12 systems along 3\/20 sightlines show signature of partial coverage. Therefore the incidence of partially covered associated \\mbox{Ne\\,{\\sc viii}}\\ absorber is 15\\%. No associated \\mbox{Ne\\,{\\sc viii}}\\ system is detected towards 7 radio bright QSOs in our sample. There are 5 \\mbox{Ne\\,{\\sc viii}}\\ absorption reported in the literature (see Table~\\ref{tab:summary}) and only one of them ($z_{\\rm abs}$ = 0.965 towards 3C~288.1) is towards radio bright QSO. Confirming the high detection rate of associated \\mbox{Ne\\,{\\sc viii}}\\ systems and relatively less incidence rate towards radio bright QSOs is very important to understand the possible influences of radio jets. \n\n\n\n \n\\subsection{Line broadening} \nIn the bottom panel of Fig.~\\ref{b_dist} we show the distribution of Doppler parameter as measured in individual \\mbox{Ne\\,{\\sc viii}}\\ components. The median value of $b(\\mbox{Ne\\,{\\sc viii}})$ is $\\sim$~58.7 km~s$^{-1}$. The upper limit on temperature corresponding to this value is 10$^{6.6}$~K. Under CIE, even \\mbox{Ne\\,{\\sc viii}}\\ will not be a dominant species at such high temperatures. The collisional ionization fraction of \\mbox{Ne\\,{\\sc viii}}\\ becomes only $\\sim 3\\times10^{-3}$ at $T \\sim 10^{6.6}$~K. Therefore the width of individual Voigt profile components are most probably dominated by non-thermal motions. We note that $b(\\mbox{Ne\\,{\\sc viii}}) \\sim$~22~km~s$^{-1}$\\ corresponds to a temperature of 10$^{5.8}$~K, at which $N(\\mbox{Ne\\,{\\sc viii}})$ peaks under CIE (see bottom panel of Fig.~\\ref{CIE_ratio}). This indeed suggests that, based on the observed $b$-values of \\mbox{Ne\\,{\\sc viii}}, we cannot rule out the possibility of gas temperature being 6--7$\\times$~10$^{5}$~K at which collisional ionization becomes important. \nNote that we use minimum number of Voigt profile components needed to have a reduced $\\chi^2\\sim 1$. The discussions presented above are based on $b$-parameters derived this way. While we can not rule out each of our Voigt profile component being made of a blended large number of components, our analysis suggests that the observed line profiles allow for the gas temperature being higher than the typical photoionization equilibrium temperature. \nIn the top panel of Fig.~\\ref{b_dist} we have plotted the number of components required to fit \\mbox{Ne\\,{\\sc viii}}\\ absorption against the velocity spread of the line. Lack of any significant correlation between these two suggests that the line spread may not dominated by the presence of multiple number of narrow components \\citep[as seen in the case of high redshift \\mbox{O\\,{\\sc vi}}\\ absorbers, e.g.][]{Muzahid12a} but the line spread is related to the large scale velocity field. Further, $\\delta v (\\mbox{Ne\\,{\\sc viii}})$ lies roughly between 100 -- 800~km~s$^{-1}$\\ suggesting that these absorbers are intermediate of BAL and NAL. This type of associated absorbers are also known as mini-BAL. \n \n\n\n\n\\subsection{Ejection velocities and correlations} \nThe ejection velocity is defined as the velocity separation between the emission redshift of the QSO and the \\mbox{Ne\\,{\\sc viii}}\\ optical depth weighted redshift of the absorber. The distribution of ejection velocities in our sample are shown in panel {\\bf (A)} of Fig.~\\ref{beta_dist}. Clearly most of these associated \\mbox{Ne\\,{\\sc viii}}\\ absorbers are detected within $-5000$ km~s$^{-1}$\\ from the emission redshift of the QSO. The highest velocity absorber is detected at a ejection velocity of $\\sim-19,000$ km~s$^{-1}$. In panel {\\bf (B)} we have plotted covering fraction corrected total column densities of \\mbox{Ne\\,{\\sc viii}}\\ in our sample ({\\sl stars}) as a function of ejection velocity. The hexagons in this panel are from literature (see Table~\\ref{tab:summary}). The overall sample shows a possible correlation between $N(\\mbox{Ne\\,{\\sc viii}})$ and $v_{\\rm ej}$. If we consider all the limits as detections we find a 2.1$\\sigma$ correlation for the systems in our sample. When we consider the measurements from the literature the significance of the correlation increase to 2.7$\\sigma$. However, we note that the top two ejection velocity systems from the literature (filled hexagons) are BAL in nature. The (green) arrows in the bottom, identify the systems with \\nani\\ detection. It is apparent that these are the ones having top three ejection velocities with $|v_{\\rm ej}| > 5000$ km~s$^{-1}$, in our sample. Interestingly, we note that the only possible \\nani\\ detection was reported before, by \\citet{Arav99a} towards BALQSO PG~0946$+$301 where the system has an ejection velocity of $-10,000$ km~s$^{-1}$, which is consistent with the trend seen in our sample. \n \n \nIn panel {\\bf (C)} we have plotted the Lyman continuum luminosity (i.e. $L_{912\\rm \\AA}$ in ergs s$^{-1}$ Hz$^{-1}$) of the sources in our sample as a function of $v_{\\rm ej}$. We do not find any obvious correlation between them. However, the highest velocity system, which also show \\nani\\ absorption, originates from the highest UV luminosity source. Here we note that, the sources with higher UV luminosities are found to be the ones with higher outflow velocities in the sample of SDSS BALQSOs \\citep[see e.g.][]{Gibson09}. \nThe estimated \\mbox{Ne\\,{\\sc viii}}\\ covering fractions in different systems in our sample are plotted against the ejection velocity in panel {\\bf (D)}. It is to be noted that majority of the systems at smaller ejection velocities show nearly 100\\% coverage of the background source whereas the systems with higher ejection velocity tend to have lower covering fractions. In panel {\\bf (E)} the line spreads of \\mbox{Ne\\,{\\sc viii}}\\ absorption in each system are plotted as function of $v_{\\rm ej}$. A mild 2$\\sigma$ level correlation is seen between $\\delta v(\\mbox{Ne\\,{\\sc viii}})$ and $v_{\\rm ej}$, suggesting systems with higher outflow velocity are likely to show wider spread. \n\n\n\\begin{figure} \n\\centerline{\\hbox{ \n\\centerline{\\vbox{\n\\includegraphics[height=10.0cm,width=8.6cm,angle=00]{beta_dist.ps} \n}}\n}}\n\\caption{{\\bf (A):} The distribution of ejection velocity for the associated \\mbox{Ne\\,{\\sc viii}}\\ \nabsorbers presented in this paper. {\\bf (B):} The \\mbox{Ne\\,{\\sc viii}}\\ column density as a function \nof ejection velocity. The hexagons are from literature listed in Table~\\ref{tab:summary}. \nThe filled hexagons represent BALQSOs. The (green) upward arrows mark the systems where \nwe detect \\nani. {\\bf (C):} $L_{912\\AA}$ as a function of $v_{\\rm ej}$. {\\bf (D):} \\mbox{Ne\\,{\\sc viii}}\\ \ncovering fractions in individual systems against $v_{\\rm ej}$. {\\bf (E):} The line spread \nof \\mbox{Ne\\,{\\sc viii}}\\ absorbers against $v_{\\rm ej}$. \n} \n\\label{beta_dist} \n\\end{figure} \n\n\n\\begin{table}\n\\vfill \n\\caption{List of intervening \\mbox{Ne\\,{\\sc viii}}\\ absorbers that exist in literature} \n\\begin{tabular}{cccccc} \n\\hline \n & & & \\multicolumn{2}{c}{log~$N$ (cm$^{-2}$)} & \\\\ \\cline{4-5}\n QSO & $z_{\\rm em}$\\ & $z_{\\rm abs}$ & \\mbox{Ne\\,{\\sc viii}} & \\mbox{O\\,{\\sc vi}} & Ref.$^{a}$ \\\\ \\hline \n (1) & (2) & (3) & (4) & (5) & (6) \\\\ \n\\hline \n\\hline \nPG~1148$+$549 & 0.9754 & 0.6838 & 13.95 & 14.52 & 1 \\\\ \nPG~1148$+$549 & 0.9754 & 0.7015 & 13.86 & 14.37 & 1 \\\\ \nPG~1148$+$549 & 0.9754 & 0.7248 & 13.81 & 13.86 & 1 \\\\ \nPKS~0405$-$123 & 0.5726 & 0.4951 & 13.96 & 14.41 & 2 \\\\ \n3C~263\t & 0.646 & 0.3257 & 13.98 & 13.98 & 3 \\\\ \nHE~0226$-$4110 & 0.495 & 0.2070 & 13.89 & 14.37 & 4 \\\\ \n\\hline \n\\hline \n\\end{tabular} \n~\\\\ ~\\\\ \nNote-- $^{a}$Reference (1) \\citet{Meiring12}; (2) \\citet{Narayanan11}; (3) \n\\citet{Narayanan09,Narayanan12} (4) \\citet{Savage05a} \n\\label{ne8_int} \n\\end{table} \n\n\n\\subsection{Distribution of column densities} \n\n\nIn Fig.~\\ref{analysis1}, we show the column density distributions of the \\mbox{O\\,{\\sc vi}}, \\mbox{Ne\\,{\\sc viii}}, and \\mgx, as measured in intervening and associated \\mbox{Ne\\,{\\sc viii}}\\ absorbers in our sample and from the existing literature (i.e. using Table~\\ref{tab_list}, \\ref{tab:summary} and \\ref{ne8_int}). The (blue) 120$^\\circ$ and (red) 60$^\\circ$ hashed histograms show the distributions corresponding to the intervening \\mbox{Ne\\,{\\sc viii}}\\ systems (i.e. from Table~\\ref{ne8_int}) and the associated \\mbox{Ne\\,{\\sc viii}}\\ systems from this paper (i.e. from Table~\\ref{tab_list}) respectively. The histograms clearly show that the column densities of \\mbox{O\\,{\\sc vi}}\\ and \\mbox{Ne\\,{\\sc viii}}\\ are systematically higher in case of associated absorbers compared to those of intervening absorbers. For example, the median values of log~$N(\\mbox{Ne\\,{\\sc viii}})$ ($\\rm cm^{-2}$) are 13.95$\\pm$0.10 and 15.40$\\pm$0.78 for the intervening and the associated absorbers respectively. The median values of log~$N(\\mbox{O\\,{\\sc vi}})$ ($\\rm cm^{-2}$), on the other hand, are 14.37$\\pm$0.27 and 15.40$\\pm$0.58 for intervening and associated absorbers respectively. However we note that, both at high and low redshifts, \\mbox{O\\,{\\sc vi}}\\ absorbers do not show any compelling evidence of having different column density distribution for intervening and associated systems \\citep[see e.g.][]{Tripp08,Fox08}. The apparent discrepancy is mainly because of the fact that we consider \\mbox{O\\,{\\sc vi}}\\ column densities measured in the \\mbox{Ne\\,{\\sc viii}}\\ absorbers, both in the cases of intervening and associated systems. To our knowledge no \\mgx\\ absorption has ever been reported in intervening systems. The median value of log~$N(\\mgx)$ ($\\rm cm^{-2}$) in associated systems turns out to be 15.47$\\pm$0.66. The median values of \\mbox{O\\,{\\sc vi}}, \\mbox{Ne\\,{\\sc viii}}, and \\mgx\\ in our sample are very similar. But we caution here that we have assumed all the upper\/lower limits as measurements. \n\n\nIt is evident from the middle panel of Fig.~\\ref{analysis1} that all the associated absorbers show $N(\\mbox{Ne\\,{\\sc viii}}) > 10^{14} \\rm cm^{-2}$. This could primarily be due to the fact that we are not sensitive enough to detect a broad line with $N(\\mbox{Ne\\,{\\sc viii}}) < 10^{14} \\rm cm^{-2}$ in the COS spectra used here. For example, for a typical $S\/N$ ratio of $\\sim 10$, the 5$\\sigma$ upper limit for non-detection of \\mbox{Ne\\,{\\sc viii}}$\\lambda$770 line is log~$N(\\mbox{Ne\\,{\\sc viii}})~(\\rm cm^{-2}) < 13.66$ for $b$-parameter of 100 km~s$^{-1}$. Here the assumed $b$ value (i.e. 100 km~s$^{-1}$) is typical for mini-BAL system. The previously reported associated \\mbox{Ne\\,{\\sc viii}}\\ systems (see e.g. Table~\\ref{tab:summary}) are all showing $N(\\mbox{Ne\\,{\\sc viii}})>$ 10$^{15}$ cm$^{-2}$. \n\n\\begin{figure} \n\\centerline{\\hbox{ \n\\centerline{\\vbox{\n\\includegraphics[height=4.4cm,width=8.4cm,angle=00]{ana2.ps} \n}}\n}}\n\\caption{Distribution of total column densities of \\mbox{O\\,{\\sc vi}}\\ (left), \\mbox{Ne\\,{\\sc viii}}\\ (middle) and \n\\mgx\\ (right) in all the associated \\mbox{Ne\\,{\\sc viii}}\\ systems (i.e. using Table~\\ref{tab_list} \\& \n\\ref{tab:summary}). The 60$^\\circ$ hashed (red) histogram shows the data points from this \npaper (i.e. using Table~\\ref{tab_list} only) whereas as 120$^\\circ$ hashed (blue) \nhistograms are for measurements in intervening systems as given in Table~\\ref{ne8_int}. \n} \n\\label{analysis1} \n\\end{figure} \n\n\n\n\\subsection{Column density ratios and ionization state} \nIn different sub-panels of Fig.~\\ref{analysis2}, various column density ratios are plotted as a function of log~$N(\\mbox{Ne\\,{\\sc viii}})$. The {\\sl stars} and the {\\sl circles} in the bottom panel of Fig.~\\ref{analysis2} are representing the $N(\\mbox{Ne\\,{\\sc viii}})\/N(\\mbox{O\\,{\\sc vi}})$ ratios in associated and intervening \\mbox{Ne\\,{\\sc viii}}\\ absorbers respectively. A Spearman rank correlation analysis shows ($\\rho_s = 0.75$) a 2.3$\\sigma$ level correlation between \\mbox{Ne\\,{\\sc viii}}\\ and \\mbox{O\\,{\\sc vi}}\\ column densities in associated absorbers. When we include the intervening absorbers in the analysis, the correlation becomes even tighter (e.g. $\\rho_s = 0.91$ and $\\rho_s\/\\sigma$ = 3.5). \nThe median value of log~$N(\\mbox{Ne\\,{\\sc viii}})\/N(\\mbox{O\\,{\\sc vi}})$ ratio for the associated absorbers is 0.11$\\pm$0.50. Under photoionization equilibrium it corresponds to the ionization parameter log~U = 0.4$\\pm$0.2. Under collisional ionization equilibrium the above ratio is reproduced when T $\\sim$ $10^{5.8}$ K and log~U~$\\le-$2. Based on the present data we are not in a position to disentangle among different ionization mechanisms. However, detection of absorption line variability and its relationship to the continuum variation will enable us to distinguish between the two alternatives. For a flat SED, the ionization parameter, density ($n_{\\rm H}$) and distance between the absorber and the QSO are related by, \n\\begin{equation} \n{\\rm log} \\left(\\frac{n_{\\rm H}}{10^{5} \\rm \/cc}\\right) \n= {\\rm log}~L_{912\\rm\\AA}^{30} -{\\rm log} \\left( \\frac{r}{100 \\rm pc}\\right)^{2} - {\\rm log~U} -1.25 \n\\label{eqn:nH_rC}\n\\end{equation} \nwhere, log~$L_{912\\rm\\AA}^{30}$ is the monochromatic luminosity of the QSO at the Lyman continuum in units of 10$^{30}$ erg s$^{-1}$ Hz$^{-1}$. The density estimation using absorption line variability or fine-structure excitations will enable us to get the location of the absorbing gas with respect to the central engine. This will allow us to estimate the kinetic luminosity of the outflow which is very crucial for probing the AGN feedback \\citep[e.g.,][]{Moe09,Dunn10,Bautista10,Borguet12b}. \n\n\nWe have mentioned earlier that the intervening absorbers are showing systematically lower values of $N(\\mbox{Ne\\,{\\sc viii}})$. But, as far as $N(\\mbox{Ne\\,{\\sc viii}})\/N(\\mbox{O\\,{\\sc vi}})$ ratio is concerned there is very little difference between associated and intervening absorbers. For example, the median values of log~$N(\\mbox{Ne\\,{\\sc viii}})\/N(\\mbox{O\\,{\\sc vi}})$ ratios in intervening and associated absorbers are, $-$0.45$\\pm$0.50 and 0.11$\\pm$0.50 respectively, consistent within 1$\\sigma$ level. Naively this implies a large ionization parameter even for the intervening systems. In case of intervening \\mbox{Ne\\,{\\sc viii}}\\ absorbers models of collisional ionization are generally proposed, as photoionization by the extragalactic UV background \\citep[]{Haardt96} requires unusually large cloud sizes \\citep[]{Savage05a,Narayanan09,Narayanan11}. Thus similar ratios seen between associated and intervening systems and between different components in an associated system as in the case of \\citet{Muzahid12b} favour collisional ionization in the associated absorbers as well.\n\n\n\nFurther, we notice a strong correlation between $N(\\mbox{Ne\\,{\\sc viii}})$ and $N(\\mgx)$ (i.e. $\\rho_s$ = 0.90 and $\\rho_s\/\\sigma$ = 2.4). The median value of log~$N(\\mbox{Ne\\,{\\sc viii}})\/N(\\mgx)$ is found to be 0.11$\\pm$0.36 (see middle panel of Fig.~\\ref{analysis2}). These observed ratios correspond to a very narrow range in gas temperature (i.e.~$T \\sim 10^{5.95\\pm0.03}$~K) under CIE or very narrow range in ionization parameter (i.e. log~U $\\sim0.8\\pm0.2$) under photoionization [see panel~(D) of Fig.~\\ref{cloudy1}]. Given the high ionization parameter and low neutral hydrogen column density (i.e. $< 10^{14.5}$~cm$^{-2}$ in most of the cases), the predicted total hydrogen column density is too low (i.e. $N({\\rm H}) < 10^{20.5}$~cm$^{-2}$) to produce significant continuum optical depth in the soft X-ray regime. \n \nIn the top panel of Fig.~\\ref{analysis2} we show $N(\\mbox{Ne\\,{\\sc viii}})\/N(\\nani)$ ratio as a function of $N(\\mbox{Ne\\,{\\sc viii}})$ in logarithmic scale. It is interesting to note that the three systems, where we detect \\nani\\ absorption (i.e. solid hexagons in the plot), are all showing log~$N(\\mbox{Ne\\,{\\sc viii}}) \\gtrsim$ 15.60. The solid (green) triangles in this panel represents the tentative detection of \\nani\\ reported by \\citet{Arav99a}, which also shows log~$N(\\mbox{Ne\\,{\\sc viii}})>$15.60. We notice that log~$N(\\mbox{Ne\\,{\\sc viii}})\/N(\\mgx)$ and log~$N(\\mbox{Ne\\,{\\sc viii}})\/N(\\mbox{O\\,{\\sc vi}})$ are roughly similar between the systems with and without detectable \\nani\\ absorption. This clearly means that the lack of \\nani\\ detection can be attributed to low $N\\rm (H)$. \n\n\\begin{figure} \n\\centerline{\\hbox{ \n\\centerline{\\vbox{\n\\includegraphics[height=8.4cm,width=8.4cm,angle=00]{ana4.ps} \n}}\n}}\n\\caption{Column density ratios (\\mbox{Ne\\,{\\sc viii}}\/\\mbox{O\\,{\\sc vi}}, {\\sl bottom}; \\mbox{Ne\\,{\\sc viii}}\/\\mgx, {\\sl middle}; \n\t\\mbox{Ne\\,{\\sc viii}}\/\\nani, {\\sl top}) as a function of $N(\\mbox{Ne\\,{\\sc viii}})$. The open triangles \n\tand (blue) filled circles are from Table~\\ref{tab:summary} and Table~\\ref{ne8_int} \n\trespectively. In the top panel all the points apart from (green) triangle is from this \n\tpaper. The solid hexagon indicate \\nani\\ detections. The solid (green) triangle represents \n\tthe tentative \\nani\\ detection by \\citet{Arav99a}. The mean values and \n\tcorresponding scatters, only for data points from our sample, are shown in each panel by \n\thorizontal dashed and dotted lines respectively. \n\tHere we assumed all the limits as measurements. \n} \n\\label{analysis2} \n\\end{figure} \n\n\\subsection{Multiple phases in \\nani\\ absorbers} \n\nWe report secure detections of \\nani\\ absorption in three associated \\mbox{Ne\\,{\\sc viii}}\\ systems for the first time. These systems show signatures of multiple component structure. Photoionization models with log~U $\\sim$ 1 explain the \\nani\\ phase of the absorbers. However these models require $\\rm Na$ abundance being enhanced by a factor of 4--7 with respect to $\\rm Mg$. Standard chemical evolution models do not predict such large enhancement of $\\rm Na$ over $\\rm Mg$ \\citep[see Fig. 18 of][and Fig. 6 of \\citet{Venn04}]{Timmes95}. The photoionization models also suggest a typical density of the absorbing region varying by up to a factor 10 along the transverse direction. As photoionization predicts roughly same temperature for the range of ionization parameters probed by low and high ions, different phases cannot be in pressure equilibrium. On the contrary, if collisional excitations are important then one may not need an enhancement of $\\rm Na$, provided the gas temperature $T = 10^{5.9}$ K. In the case of CIE the low ionization phase requires $T\\sim 10^{5.2}$ K. Therefore, a factor $\\sim$~5 density difference between two phases is needed for the gas to be in pressure equilibrium. In the case of CIE, absorbing gas has to be far away from the QSO for the gas to be unaffected by the QSO radiation. Therefore it is important to identify the source of energy that maintains the high temperature of the gas. Probing the optical depth variability and presence of fine-structure transitions with new $HST\/$COS observations will allow us to make good progress in this direction. \n\nAt last, we note that the element $\\rm Na$ has not been incorporated in the non-equilibrium collisional ionization calculations so far. For the metallicity as measured in our sample, non-equilibrium effects would be important and can provide more realistic models of \\nani\\ absorbers. Therefore, inclusion of $\\rm Na$ in non-equilibrium calculations will be very useful. \n \n\n\n\n\\section{Summary \\& Conclusions} \n\\label{con} \n\nWe present a sample of new class of associated absorbers, detected through \\mbox{Ne\\,{\\sc viii}}$\\lambda\\lambda$770,780 absorption, in $HST\/$COS spectra of intermediate redshift (0.45~$\\le z \\le$~1.21) quasars. We searched for \\mbox{Ne\\,{\\sc viii}}\\ absorption in the public $HST\/$COS archive of QSOs with $S\/N \\ge 10$ and emission redshift $z_{\\rm em}$~$ > 0.45$. There were total 20 QSO sight lines in the $HST\/$COS archive before February 2012, satisfying these criteria. Seven of these QSOs are radio bright. The signatures of associated \\mbox{Ne\\,{\\sc viii}}\\ absorption are seen in 40\\% (i.e. 8 out of 20) of the lines of sight, with 10 secured and 2 tentative \\mbox{Ne\\,{\\sc viii}}\\ systems detected in total. None of them are towards radio bright QSOs. The associated absorbers detected towards QSO HE~0226$-$4110 and QSO HE~0238$-$1904 were previously reported by \\citet{Ganguly06} and \\citet{Muzahid12b} respectively. Here we summarize our main results. \n\n\n\\vskip 0.2cm \\noindent {\\bf (1)} \nMajority of the \\mbox{Ne\\,{\\sc viii}}\\ absorbers are detected with outflow velocities $\\lesssim$5000 km~s$^{-1}$. The highest velocity system shows $|v_{\\rm ej}| \\sim 19,000$~km~s$^{-1}$. Medium resolution COS spectra allow us to probe the component structure of \\mbox{Ne\\,{\\sc viii}}\\ absorption in most of the systems. The line spread of \\mbox{Ne\\,{\\sc viii}}\\ absorption is found to be in the range 100~$\\le \\delta v (\\rm km~s^{-1}) \\le$~1000, suggesting that these absorbers are most likely mini-BALs. The Doppler parameters measured in individual components (with median 58.7$\\pm$31.7 km~s$^{-1}$) indicates domination of non-thermal motions. \n\n\n\\vskip 0.2cm \\noindent {\\bf (2)} \nWe detect \\mgx\\ absorption in 7 of 8 \\mbox{Ne\\,{\\sc viii}}\\ systems when the lines are not blended and are covered by the observations. Moreover, we report first secure detections of \\nani\\ absorption in three highest velocity systems in our sample. All three \\nani\\ systems show high $N(\\mbox{Ne\\,{\\sc viii}})$ (i.e.$ > 10^{15.6}$ cm$^{-2}$). The measurements and\/or limits on the column densities of different ions, detected in these \\nani\\ absorbers, require very high ionization parameter (i.e. log~U $\\ge 0.5$) and high metallicity (i.e. $Z \\ge Z_{\\odot}$) when we consider single phase photoionization models. However, ionization potential dependent covering fraction seen in these absorbers suggests kinematic coincidence of multiphase gas with higher ionization species having higher projected area. Given the high value of ionization parameter (log~U) and observed low $N(\\mbox{H\\,{\\sc i}})$, the model predicted $N(\\rm H)$ is too low (i.e. $<10^{20.5}$~cm$^{-2}$) to produce any significant continuum optical depth in the soft X-ray regime. The observed $N(\\mgx)\/N(\\nani)$ ratios, under single phase photoionization scenario, require a factor $\\gtrsim 5$ enhancement of $\\rm Na$ abundance with respect to $\\rm Mg$. However, such enhancement is not required in CIE models provided gas temperature is $T \\ge 10^{5.9}$~K. In the case of CIE, the low ions require a different phase with temperature $T \\sim 10^{5.2}$ K suggesting a factor of $\\sim 5$ difference in density between two gas phases to be in pressure equilibrium. \n\n\n\\vskip 0.2cm \\noindent {\\bf (3)} \nWe notice a very narrow range in the column density ratios of high ions (i.e. \\mbox{O\\,{\\sc vi}}, \\mbox{Ne\\,{\\sc viii}}, \\mgx\\ etc.). This suggests a narrow range in ionization parameter (temperature) under photoionization (CIE). The median value of log~$N(\\mbox{Ne\\,{\\sc viii}})\/N(\\mbox{O\\,{\\sc vi}})$ $= 0.11\\pm0.50$ as measured in our sample is comparable to that measured in the intervening \\mbox{Ne\\,{\\sc viii}}\\ absorbers within the measurement uncertainties. In case of intervening \\mbox{Ne\\,{\\sc viii}}\\ absorbers collisional ionization is generally proposed, as photoionization by the extragalactic UV background requires unusually large cloud sizes. Indeed, CIE can play an important role in deciding the ionization structure of the absorbing gas in our sample as well. However, for CIE to be dominant, gas cloud has to be far away from the QSO. In that case it is crucial to understand sources of thermal and mechanical energy and the stability of the absorber. Variability study with repeated $HST\/$COS observation is needed to make further progress on these issues. \n\n\n\n\\section{acknowledgment} \n \nWe thank anonymous referee for useful comments. We appreciate the efforts of the people involved with the design and construction of COS and its deployment on the $HST$. Thanks are also extended to the people responsible for determining the orbital performance of COS and developing the {\\sc calcos} data processing pipeline. We thankfully acknowledge Dr. Jane Charlton for providing the STIS E230M spectrum of PG~1206$+$459. We thank Dr. Gulab C. Dewangan and Dr. Durgesh Tripathi for useful discussions. SM thanks Sibasish Laha for useful discussions on {\\sc cloudy} modelling. SM also thanks CSIR for providing support for this work. RS wish to thank Indo-French Centre for the Promotion of Advanced Research under the programme No. 4304--2. NA acknowledge support from NASA STScI grants AR-12653. \n\n\\def\\aj{AJ}%\n\\def\\actaa{Acta Astron.}%\n\\def\\araa{ARA\\&A}%\n\\def\\apj{ApJ}%\n\\def\\apjl{ApJ}%\n\\def\\apjs{ApJS}%\n\\def\\ao{Appl.~Opt.}%\n\\def\\apss{Ap\\&SS}%\n\\def\\aap{A\\&A}%\n\\def\\aapr{A\\&A~Rev.}%\n\\def\\aaps{A\\&AS}%\n\\def\\azh{AZh}%\n\\def\\baas{BAAS}%\n\\def\\bac{Bull. astr. Inst. Czechosl.}%\n\\def\\caa{Chinese Astron. Astrophys.}%\n\\def\\cjaa{Chinese J. Astron. Astrophys.}%\n\\def\\icarus{Icarus}%\n\\def\\jcap{J. Cosmology Astropart. Phys.}%\n\\def\\jrasc{JRASC}%\n\\def\\mnras{MNRAS}%\n\\def\\memras{MmRAS}%\n\\def\\na{New A}%\n\\def\\nar{New A Rev.}%\n\\def\\pasa{PASA}%\n\\def\\pra{Phys.~Rev.~A}%\n\\def\\prb{Phys.~Rev.~B}%\n\\def\\prc{Phys.~Rev.~C}%\n\\def\\prd{Phys.~Rev.~D}%\n\\def\\pre{Phys.~Rev.~E}%\n\\def\\prl{Phys.~Rev.~Lett.}%\n\\def\\pasp{PASP}%\n\\def\\pasj{PASJ}%\n\\def\\qjras{QJRAS}%\n\\def\\rmxaa{Rev. Mexicana Astron. Astrofis.}%\n\\def\\skytel{S\\&T}%\n\\def\\solphys{Sol.~Phys.}%\n\\def\\sovast{Soviet~Ast.}%\n\\def\\ssr{Space~Sci.~Rev.}%\n\\def\\zap{ZAp}%\n\\def\\nat{Nature}%\n\\def\\iaucirc{IAU~Circ.}%\n\\def\\aplett{Astrophys.~Lett.}%\n\\def\\apspr{Astrophys.~Space~Phys.~Res.}%\n\\def\\bain{Bull.~Astron.~Inst.~Netherlands}%\n\\def\\fcp{Fund.~Cosmic~Phys.}%\n\\def\\gca{Geochim.~Cosmochim.~Acta}%\n\\def\\grl{Geophys.~Res.~Lett.}%\n\\def\\jcp{J.~Chem.~Phys.}%\n\\def\\jgr{J.~Geophys.~Res.}%\n\\def\\jqsrt{J.~Quant.~Spec.~Radiat.~Transf.}%\n\\def\\memsai{Mem.~Soc.~Astron.~Italiana}%\n\\def\\nphysa{Nucl.~Phys.~A}%\n\\def\\physrep{Phys.~Rep.}%\n\\def\\physscr{Phys.~Scr}%\n\\def\\planss{Planet.~Space~Sci.}%\n\\def\\procspie{Proc.~SPIE}%\n\\let\\astap=\\aap\n\\let\\apjlett=\\apjl\n\\let\\apjsupp=\\apjs\n\\let\\applopt=\\ao\n\\bibliographystyle{mn}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{ INTRODUCTION }\n In the presently known energy range the spectrum of the\n elementary particles is {\\em ``chiral''} in the sense that no\n explicit fermion mass terms are allowed by the symmetry.\n Fermion masses, as well as all other masses, are entirely\n generated by spontaneous symmetry breaking, due to the nonzero\n vacuum expectation value $v_R \\simeq 250$ GeV of the Higgs-boson\n field.\n One of the consequences is the V-A structure of weak interactions,\n resulting in the breaking of space-reflection (parity) symmetry.\n\n A natural question is whether ``chirality'' and the accompanying\n parity breaking is perhaps only a low-energy phenomenon, and at\n high energy the space-reflection symmetry is restored by the\n existence of opposite chirality {\\em ``mirror fermions''}\n \\cite{LEEYAN}.\n If the presently known (almost complete) three fermion families\n were duplicated at the electroweak energy scale, in the range\n 100-1000 GeV, by three mirror fermion families with opposite\n chiralities and hence V+A couplings to the weak gauge vector\n bosons \\cite{MIRFAM}, then the whole fermion spectrum would be\n ``vectorlike''.\n This would very much simplify the nonperturbative lattice formulation\n of the Standard Model \\cite{CHFER,TSUKPR}.\n Of course, since no effects of the mirror fermions are experimentally\n observed up to now, first one has to ask whether the limits implied\n by the presently known experimental data allow their existence at all.\n\n\\section{ MIRROR FERMION PHENOMENOLOGY }\n The mirror partners of fermions have the same\n $\\rm SU(3) \\otimes SU(2)_L \\otimes U(1)_Y$ quantum numbers but\n opposite chiralities.\n For instance, the right-handed chiral components of mirror leptons\n form a doublet with $Y=-1$ with respect to\n $\\rm SU(2)_L \\otimes U(1)_Y$.\n Such particles appear in several extensions of the minimal Standard\n Model, for instance, in grand unified theories with large groups\n as O(16) or SU(15) \\cite{MAAROO}.\n In general mirror fermion models there might be some other quantum\n numbers which are different for fermions and mirror fermions, and the\n set of representations containing the mirror fermions may also be\n different.\n The lattice formulation of the Standard Model suggests a simple\n doubling of the fermion spectrum \\cite{NIENIN}, resulting in three\n mirror pairs of fermion families \\cite{MIRFAM}.\n\n\\subsection{ Present limits }\n The direct pair production of mirror fermions is not observed at LEP.\n This puts a lower limit on their masses of about 45 GeV.\n Heavier mirror fermions could be produced via their mixing to\n ordinary fermions.\n This implies some constraints on the mixing angles versus the masses.\n\n In order to discuss the mixing schemes, let us first consider the\n simplest case of a single fermion ($\\psi$) mirror fermion ($\\chi$)\n pair.\n The mass matrix on the $(\\overline{\\psi}_R,\\overline{\\psi}_L,\n \\overline{\\chi}_R,\\overline{\\chi}_L) \\otimes\n (\\psi_L,\\psi_R,\\chi_L,\\chi_R)$ basis is\n\\begin{equation} \\label{eq01}\nM = \\left( \\begin{array}{cccc}\n\\mu_\\psi & 0 & \\mu_R & 0 \\\\\n0 & \\mu_\\psi & 0 & \\mu_L \\\\\n\\mu_L & 0 & \\mu_\\chi & 0 \\\\\n0 & \\mu_R & 0 & \\mu_\\chi\n\\end{array} \\right) \\ .\n\\end{equation}\n Here $\\mu_{(L,R)}$ are the fermion mirror fermion mixing mass\n parameters, and the diagonal elements are produced by spontaneous\n symmetry breaking:\n\\begin{equation} \\label{eq02}\n \\mu_\\psi=G_{R\\psi}v_R \\ , \\hspace{2em} \\mu_\\chi=G_{R\\chi}v_R \\ ,\n\\end{equation}\n with the renormalized Yukawa-couplings $G_{R\\psi}$, $G_{R\\chi}$.\n\n For $\\mu_R \\ne \\mu_L$ the mass matrix $M$ in (\\ref{eq01}) is not\n symmetric, hence one has to diagonalize $M^T M$ by\n $O^T_{(LR)} M^T M O_{(LR)}$, and\n $M M^T$ by $O^T_{(RL)} M M^T O_{(RL)}$, where\n$$\nO_{(LR)} =\n$$\n$$\n \\left( \\begin{array}{cccc}\n\\cos\\alpha_L & 0 & \\sin\\alpha_L & 0 \\\\\n0 & \\cos\\alpha_R & 0 & \\sin\\alpha_R \\\\\n-\\sin\\alpha_L & 0 & \\cos\\alpha_L & 0 \\\\\n0 & -\\sin\\alpha_R & 0 & \\cos\\alpha_R\n\\end{array} \\right) \\ ,\n$$\n$$\nO_{(RL)} =\n$$\n\\begin{equation} \\label{eq03}\n \\left( \\begin{array}{cccc}\n\\cos\\alpha_R & 0 & \\sin\\alpha_R & 0 \\\\\n0 & \\cos\\alpha_L & 0 & \\sin\\alpha_L \\\\\n-\\sin\\alpha_R & 0 & \\cos\\alpha_R & 0 \\\\\n0 & -\\sin\\alpha_L & 0 & \\cos\\alpha_L\n\\end{array} \\right) \\ .\n\\end{equation}\n The rotation angles of the left-handed, respectively, right-handed\n components satisfy\n$$\n\\tan(2\\alpha_L) = \\frac{2(\\mu_\\chi \\mu_L + \\mu_\\psi \\mu_R)}\n{\\mu_\\chi^2 + \\mu_R^2 - \\mu_\\psi^2 - \\mu_L^2} \\ ,\n$$\n\\begin{equation} \\label{eq04}\n\\tan(2\\alpha_R) = \\frac{2(\\mu_\\chi \\mu_R + \\mu_\\psi \\mu_L)}\n{\\mu_\\chi^2 + \\mu_L^2 - \\mu_\\psi^2 - \\mu_R^2} \\ ,\n\\end{equation}\n and the two (positive) mass-squared eigenvalues are given by\n$$\n\\mu_{1,2}^2 = \\half \\left\\{\n\\mu_\\chi^2 + \\mu_\\psi^2 + \\mu_L^2 + \\mu_R^2\n\\right.\n$$\n$$\n\\mp \\left[ (\\mu_\\chi^2 - \\mu_\\psi^2)^2 + (\\mu_L^2 - \\mu_R^2)^2\n\\right.\n$$\n\\begin{equation} \\label{eq05}\n\\left.\\left.\n+ 2(\\mu_\\chi^2 + \\mu_\\psi^2) (\\mu_L^2 + \\mu_R^2)\n+ 8\\mu_\\chi \\mu_\\psi \\mu_L \\mu_R \\right]^\\half \\right\\} .\n\\end{equation}\n The mass matrix itself is diagonalized by\n$$\nO^T_{(RL)} M O_{(LR)} = O^T_{(LR)} M^T O_{(RL)}\n$$\n\\begin{equation} \\label{eq06}\n= \\left( \\begin{array}{cccc}\n\\mu_{1} & 0 & 0 & 0 \\\\\n0 & \\mu_{1} & 0 & 0 \\\\\n0 & 0 & \\mu_{2} & 0 \\\\\n0 & 0 & 0 & \\mu_{2}\n\\end{array} \\right) \\ .\n\\end{equation}\n This shows that for $\\mu_\\psi,\\mu_L,\\mu_R \\ll \\mu_\\chi$ there\n is a light state with mass $\\mu_{1}=O(\\mu_\\psi,\\mu_L,\\mu_R)$ and a\n heavy state with mass $\\mu_{2}=O(\\mu_\\chi)$.\n In general, both the light and heavy states are mixtures of\n the original fermion and mirror fermion.\n According to (\\ref{eq04}), for $\\mu_L \\ne \\mu_R$ the\n fermion-mirror-fermion mixing angle in the left-handed sector is\n different from the one in the right-handed sector.\n\n In case of three mirror pairs of fermion families the diagonalization\n of the mass matrix is in principle similar but, of course, more\n complicated.\n A particular class of mixing schemes will be discussed in the next\n subsection.\n The mirror fermions can be produced through their mixing to ordinary\n fermions.\n The upper limits on the absolute value of mixing angles depend on the\n mixing scheme.\n\n Indirect limits on the existence of heavy mirror fermions can also be\n deduced from the absence of observed effects in 1-loop radiative\n corrections \\cite{PESTAK,ALTBAR}, because of the non-decoupling of\n heavy fermions.\n The question of non-decoupling in higher loop orders is, however,\n open.\n In fact, one of the goals of nonperturbative lattice studies is to\n investigate this in the nonperturbative regime of couplings.\n\n\\subsection{ Mixing schemes }\n The strongest constraints on mixing angles between ordinary\n fermions and mirror fermions arise from the\n conservation of $e$-, $\\mu$- and $\\tau$- lepton numbers and from\n the absence of flavour changing neutral currents \\cite{MAAROO}.\n In a particular scheme these constraints can be avoided \\cite{MIRFAM}.\n In this {\\em ``monogamous mixing''} scheme the structure of the\n mass matrix is such that there is a one-to-one correspondence between\n fermions and mirror fermions, due to the fact that the family\n structure of the mass matrix for mirror fermions is closely related\n to the one for ordinary fermions.\n\n Let us denote doublet indices by $A=1,2$, colour indices by\n $c=1,2,3$ in such a way that the leptons belong to the fourth value of\n colour $c=4$, and family indices by $K=1,2,3$.\n In general the entries of the mass matrix for three mirror pairs\n of fermion families are diagonal in isospin and colour, hence they\n have the form\n$$\n\\mu_{(\\psi,\\chi);A_2c_2K_2,A_1c_1K_1} = \\delta_{A_2A_1}\n\\delta_{c_2c_1} \\mu^{(A_1c_1)}_{(\\psi,\\chi);K_2K_1} \\ ,\n$$\n$$\n\\mu_{L;A_2c_2K_2,A_1c_1K_1} = \\delta_{A_2A_1}\n\\delta_{c_2c_1} \\mu^{(c_1)}_{L;K_2K_1} \\ ,\n$$\n\\begin{equation} \\label{eq07}\n\\mu_{R;A_2c_2K_2,A_1c_1K_1} = \\delta_{A_2A_1}\n\\delta_{c_2c_1} \\mu^{(A_1c_1)}_{R;K_2K_1} \\ .\n\\end{equation}\n The diagonalization of the mass matrix can be achieved for given\n indices $A$ and $c$ by two $6 \\otimes 6$ unitary matrices $F_L^{(Ac)}$\n and $F_R^{(Ac)}$ acting, respectively, on the L-handed and R-handed\n subspaces:\n$$\nF_L^{(Ac)\\dagger}(M^\\dagger M)_L F_L^{(Ac)} \\ ,\n$$\n\\begin{equation} \\label{eq08}\nF_R^{(Ac)\\dagger}(M^\\dagger M)_R F_R^{(Ac)} \\ .\n\\end{equation}\n\n The main assumption of the ``monogamous'' mixing scheme is that\n in the family space $\\mu_\\psi,\\mu_\\chi,\\mu_L,\\mu_R$ are hermitian\n and simultaneously diagonalizable, that is\n\\begin{equation} \\label{eq09}\nF_L^{(Ac)} = F_R^{(Ac)} = \\left(\n\\begin{array}{cc}\nF^{(Ac)} & 0 \\\\ 0 & F^{(Ac)}\n\\end{array} \\right) \\ ,\n\\end{equation}\n where the block matrix is in $(\\psi,\\chi)$-space.\n The Kobayashi-Maskawa matrix of quarks is given by\n\\begin{equation} \\label{eq10}\nC \\equiv F^{(2c)\\dagger} F^{(1c)} \\ ,\n\\end{equation}\n independently for $c=1,2,3$.\n The corresponding matrix with $c=4$ and $A=1 \\leftrightarrow 2$\n describes the mixing of neutrinos, if the Dirac-mass of the neutrinos\n is nonzero.\n (Majorana masses of the neutrinos are not considered here, but\n in principle, they can also be introduced.)\n\n A simple example for the ``monogamous'' mixing is the following:\n$$\n\\mu^{(Ac)}_{\\chi;K_2K_1} = \\lambda^{(Ac)}_\\chi\n\\mu^{(Ac)}_{\\psi;K_2K_1} + \\delta_{K_2K_1}\\Delta^{(Ac)} \\ ,\n$$\n$$\n\\mu^{(c)}_{L;K_2K_1} = \\delta_{K_2K_1}\\delta^{(c)}_L \\ ,\n$$\n\\begin{equation} \\label{eq11}\n\\mu^{(Ac)}_{R;K_2K_1} = \\lambda^{(Ac)}_R\n\\mu^{(Ac)}_{\\psi;K_2K_1} + \\delta_{K_2K_1}\\delta^{(Ac)}_R \\ ,\n\\end{equation}\n where $\\lambda_\\chi^{(Ac)},\\; \\Delta^{(Ac)},\\; \\delta_L^{(c)},\\;\n \\lambda_R^{(Ac)},\\; \\delta_R^{(Ac)}$ do not depend on the family\n index.\n\n The general case can be parametrized by the eigenvalues\n $\\mu^{(AcK)}_\\psi,\\; \\mu^{(AcK)}_\\chi,\\; \\mu^{(AcK)}_R,\\;\n \\mu^{(cK)}_L$ and matrices $F^{(Ac)}$:\n$$\n\\mu^{(Ac)}_{(\\psi,\\chi,R);K_2K_1} = \\sum_K\nF^{(Ac)}_{K_2K} \\mu^{(AcK)}_{(\\psi,\\chi,R)} F^{(Ac)\\dagger}_{KK_1} \\ ,\n$$\n\\begin{equation} \\label{eq12}\n\\mu^{(c)}_{L;K_2K_1} = \\sum_K\nF^{(Ac)}_{K_2K} \\mu^{(cK)}_L F^{(Ac)\\dagger}_{KK_1} \\ .\n\\end{equation}\n Here in the second line the left hand side has to be independent of\n the value of $A=1,2$.\n\n The full diagonalization of the mass matrix on the\n $(\\psi_L,\\psi_R,\\chi_L,\\chi_R)$ basis of all three family pairs is\n achieved by the $96 \\otimes 96$ matrix\n$$\n{\\cal O}^{(LR)}_{A^\\prime c^\\prime K^\\prime,AcK}\n= \\delta_{A^\\prime A}\\delta_{c^\\prime c}\nF^{(Ac)}_{K^\\prime K}\n$$\n$$\n\\cdot \\left(\n\\begin{array}{cc}\n \\cos\\alpha^{(AcK)}_L & 0 \\\\\n 0 & \\cos\\alpha^{(AcK)}_R \\\\\n-\\sin\\alpha^{(AcK)}_L & 0 \\\\\n 0 & -\\sin\\alpha^{(AcK)}_R\n\\end{array} \\right.\n$$\n\\begin{equation} \\label{eq13}\n\\left.\n\\begin{array}{cc}\n \\sin\\alpha^{(AcK)}_L & 0 \\\\\n 0 & \\sin\\alpha^{(AcK)}_R \\\\\n \\cos\\alpha^{(AcK)}_L & 0 \\\\\n 0 & \\cos\\alpha^{(AcK)}_R\n\\end{array} \\right) \\ .\n\\end{equation}\n $M^\\dagger M$ is diagonalized by\n\\begin{equation} \\label{eq14}\n{\\cal O}^{(LR)\\dagger} M^\\dagger M {\\cal O}^{(LR)} \\ ,\n\\end{equation}\n and $M M^\\dagger$ by\n\\begin{equation} \\label{eq15}\n{\\cal O}^{(RL)\\dagger} M M^\\dagger {\\cal O}^{(RL)} \\ ,\n\\end{equation}\n where ${\\cal O}^{(RL)}$ is obtained from ${\\cal O}^{(LR)}$ by\n $\\alpha_L \\leftrightarrow \\alpha_R$.\n\n In case of $\\mu_R=\\mu_L$, which happens for instance in (\\ref{eq11})\n if $\\lambda_R=0$ and $\\delta_R=\\delta_L$, the left-handed and\n right-handed mixing angles are the same:\n\\begin{equation} \\label{eq16}\n\\alpha^{(AcK)} \\equiv \\alpha_L^{(AcK)} =\n\\alpha_R^{(AcK)} \\ .\n\\end{equation}\n In Ref.~\\cite{MIRFAM} only this special case was considered.\n The importance of the left-right-asymmetric mixing was pointed out in\n Ref.~\\cite{CSIFOD}, where the constraints arising from the measured\n values of anomalous magnetic moments were determined.\n It turned out that for the electron and muon the upper limit is\n\\begin{equation} \\label{eq17}\n|\\alpha_L \\alpha_R| \\le 0.0004 \\ ,\n\\end{equation}\n which is much stronger than the limits obtained from all other\n data \\cite{LANLON}:\n\\begin{equation} \\label{eq18}\n\\alpha_L^2,\\; \\alpha_R^2 \\le 0.02 \\ .\n\\end{equation}\n In case of the L-R asymmetric mixing the constraint (\\ref{eq17})\n can be satisfied, if either the left- or right-handed mixing exactly\n vanishes (or is very small): $\\alpha_L=0$ or $\\alpha_R=0$.\n\n\\subsection{ Future colliders }\n The hypothetical mirror fermions can be discovered at the next\n generation of high energy colliders.\n At HERA the first family mirror fermions can be produced via mixing\n to ordinary fermions up to masses of about 200 GeV, if the mixing\n angles are close to their present upper limits \\cite{CSIMON,HERA}.\n At SSC and LHC mirror lepton pair production can be observed up to\n masses of a few hundred GeV \\cite{CSIKOR}.\n This has the advantage of being essentially independent of the small\n mixing.\n At a high energy $e^+e^-$ collider, e.~g. LEP-200 or NLC,\n mirror fermions can be pair produced and easily identified up to\n roughly half of the total energy, and also produced via mixing\n almost up to the total energy \\cite{NLC}.\n\n\\section{ LATTICE SIMULATION OF YUKAWA MODELS }\n\n\\subsection{ Lattice actions }\n The lattice formulation of the electroweak Standard Model is\n difficult because of the doubler fermions \\cite{NIENIN}.\n In fact, at present no completely satisfactory formulation is known\n \\cite{TSUKPR}: if one insists on explicit chiral gauge invariance,\n then mirror fermion fields have to be introduced \\cite{CHFER},\n otherwise one has to fix the gauge as in the ``Rome-approach''\n \\cite{ROMA2}.\n\n The situation is different if the\n $\\rm SU(3)_{colour} \\otimes U(1)_{hypercharge}$\n gauge couplings are neglected.\n In this case, as a consequence of the pseudo-reality of SU(2)\n representations, mirror fermions can be transformed to normal fermions\n by charge conjugation.\n This allows the gauge invariant lattice formulation of $\\rm SU(2)_L$\n symmetric models with an even number of fermion doublets.\n For simplicity, let us also neglect here the $\\rm SU(2)_L$ gauge\n interaction, and consider a chiral Yukawa-model of two fermion\n doublets.\n A global $\\rm SU(2)_L \\otimes U(1)$ symmetric chiral Yukawa-model\n can be formulated by the lattice action\n\\begin{equation} \\label{eq19}\nS = S_{scalar}+ S_{fermion} \\ ,\n\\end{equation}\n where the pure scalar part in terms of the $2 \\otimes 2$ matrix\n Higgs-boson field $\\varphi$ is\n$$\nS_{scalar} = \\frac{1}{4}\\sum_x \\left\\{\nm_0^2 {\\rm Tr\\,}(\\varphi^\\dagger_x\\varphi_x)\n+ \\lambda \\left[ {\\rm Tr\\,}(\\varphi^\\dagger_x\\varphi_x) \\right]^2\n\\right.\n$$\n\\begin{equation} \\label{eq20}\n\\left.\n+ \\sum_\\mu\n[ {\\rm Tr\\,} (\\varphi^\\dagger_x \\varphi_x)\n- {\\rm Tr\\,} (\\varphi^\\dagger_{x+\\hat{\\mu}}\\varphi_x) ]\n\\right\\} \\ ,\n\\end{equation}\n and the fermionic part with two doublet fields $\\psi_{1,2}$ is\n$$\nS_{fermion} = \\sum_x \\left\\{ \\frac{\\mu_0}{2} \\left[\n (\\psi^T_{2x}\\epsilon C \\psi_{1x}) - (\\psi^T_{1x}\\epsilon C \\psi_{2x})\n\\right.\\right.\n$$\n$$\n\\left.\n+ (\\overline{\\psi}_{2x}\\epsilon C \\overline{\\psi}^T_{1x})\n- (\\overline{\\psi}_{1x}\\epsilon C \\overline{\\psi}^T_{2x}) \\right]\n$$\n$$\n- \\half \\sum_\\mu \\left[\n (\\overline{\\psi}_{1 x+\\hat{\\mu}} \\gamma_\\mu \\psi_{1x})\n+ (\\overline{\\psi}_{2 x+\\hat{\\mu}} \\gamma_\\mu \\psi_{2x})\n\\right.\n$$\n$$\n- \\frac{r}{2} \\left(\n (\\psi^T_{2x}\\epsilon C \\psi_{1x})\n- (\\psi^T_{2 x+\\hat{\\mu}}\\epsilon C \\psi_{1x})\n\\right.\n$$\n$$\n- (\\psi^T_{1x}\\epsilon C \\psi_{2x})\n+ (\\psi^T_{1 x+\\hat{\\mu}}\\epsilon C \\psi_{2x})\n$$\n$$\n+ (\\overline{\\psi}_{2x}\\epsilon C \\overline{\\psi}^T_{1x})\n- (\\overline{\\psi}_{2 x+\\hat{\\mu}}\\epsilon C \\overline{\\psi}^T_{1x})\n$$\n$$\n\\left.\\left.\n- (\\overline{\\psi}_{1x}\\epsilon C \\overline{\\psi}^T_{2x})\n+ (\\overline{\\psi}_{1 x+\\hat{\\mu}}\\epsilon C \\overline{\\psi}^T_{2x})\n\\right)\\right]\n$$\n$$\n+ (\\overline{\\psi}_{1Rx} G_1\\varphi^+_x \\psi_{1Lx})\n+ (\\overline{\\psi}_{1Lx} \\varphi_x G_1 \\psi_{1Rx})\n$$\n\\begin{equation} \\label{eq21}\n\\left.\n+ (\\overline{\\psi}_{2Rx} G_2\\varphi^+_x \\psi_{2Lx})\n+ (\\overline{\\psi}_{2Lx} \\varphi_x G_2 \\psi_{2Rx}) \\right\\} \\ .\n\\end{equation}\n Here the summations $\\sum_\\mu$ always go over eight directions of the\n neighbouring sites, $C$ is the Dirac matrix for charge conjugation and\n $\\epsilon = i\\tau_2$ is the antisymmetric unit matrix in isospin space.\n In the scalar part of the action $m_0^2$ is the bare mass squared and\n $\\lambda$ the bare quartic coupling.\n In the fermionic part $\\mu_0$ is an off-diagonal Majorana mass term,\n $r$ is the Wilson parameter for removing lattice fermion doublers,\n which is usually chosen to be $r=1$, and $G_{(1,2)}$ are diagonal $2\n \\otimes 2$ matrices in isospin space for the bare Yukawa-couplings:\n\\begin{equation} \\label{eq22}\nG_{(1,2)} \\equiv \\left(\n\\begin{array}{cc}\nG_{(1,2)u} & 0 \\\\ 0 & G_{(1,2)d}\n\\end{array} \\right) \\ .\n\\end{equation}\n\n The global $\\rm SU(2)_L$ symmetry is acting on the fields as\n$$\n\\varphi_x^\\prime = U_L^{-1}\\varphi_x \\ ,\n$$\n$$\n\\psi_{(1,2)Lx}^\\prime = U_L^{-1}\\psi_{(1,2)Lx} \\ ,\n$$\n\\begin{equation} \\label{eq23}\n\\overline{\\psi}_{(1,2)Lx}^\\prime =\n\\overline{\\psi}_{(1,2)Lx} U_L \\ .\n\\end{equation}\n The right-handed components of the fermion fields $\\psi_{(1,2)Rx}$ and\n $\\overline{\\psi}_{(1,2)Rx}$ are, of course, invariant.\n\n The global U(1) symmetry corresponds to the conservation of the\n difference of the fermion number of $\\psi_1$ minus the fermion\n number of $\\psi_2$.\n (This means that if it is identified by the hypercharge $\\rm U(1)_Y$,\n then $\\psi_1$ and $\\psi_2$ have opposite hypercharges.)\n\n An equivalent form of the fermionic part of the above action is\n obtained, if one introduces the mirror fermion fields by charge\n conjugation:\n\\begin{equation} \\label{eq24}\n\\chi_x \\equiv \\epsilon^{-1} C \\overline{\\psi}_{2x}^T \\ , \\hspace{1em}\n\\overline{\\chi}_x \\equiv \\psi_{2x}^T \\epsilon C \\ .\n\\end{equation}\n Since under charge conjugation the left- and right-handed components\n are interchanged, and $\\epsilon^{-1}U_L^T\\epsilon=U_L^{-1}$, under\n $\\rm SU(2)_L$ transformations we have\n\\begin{equation} \\label{eq25}\n\\chi_{Rx}^\\prime = U_L^{-1}\\chi_{Rx} \\ , \\hspace{1em}\n\\overline{\\chi}_{Rx}^\\prime = \\overline{\\chi}_{Rx} U_L \\ ,\n\\end{equation}\n and now the left-handed components $\\chi_{Lx}$ and\n $\\overline{\\chi}_{Lx}$ are invariant.\n Omitting the index on $\\psi_1$, one gets the action in terms of the\n mirror pair of fermion fields \\cite{CHFER}\n$$\nS_{fermion} = \\sum_x \\left\\{\n \\mu_0 \\left[ (\\overline{\\chi}_x\\psi_x)\n+ (\\overline{\\psi}_x\\chi_x) \\right]\n\\right.\n$$\n$$\n- \\half \\sum_\\mu \\left[\n (\\overline{\\psi}_{x+\\hat{\\mu}}\\gamma_\\mu\\psi_x)\n+ (\\overline{\\chi}_{x+\\hat{\\mu}}\\gamma_\\mu\\chi_x)\n\\right.\n$$\n$$\n- r \\left(\n (\\overline{\\chi}_x\\psi_x)\n- (\\overline{\\chi}_{x+\\hat{\\mu}}\\psi_x)\n\\right.\n$$\n$$\n\\left.\\left.\n+ (\\overline{\\psi}_x\\chi_x)\n- (\\overline{\\psi}_{x+\\hat{\\mu}}\\chi_x) \\right) \\right]\n$$\n$$\n+ (\\overline{\\psi}_{Rx} G_\\psi\\varphi^\\dagger_x \\psi_{Lx})\n+ (\\overline{\\psi}_{Lx} \\varphi_x G_\\psi \\psi_{Rx})\n$$\n\\begin{equation} \\label{eq26}\n\\left.\n+ (\\overline{\\chi}_{Lx} G_\\chi\\varphi^\\dagger_x \\chi_{Rx})\n+ (\\overline{\\chi}_{Rx} \\varphi_x G_\\chi \\chi_{Lx}) \\right\\} \\ .\n\\end{equation}\n The Yukawa-coupling of the fermion doublet is denoted here by\n $G_\\psi = G_1$, and the Yukawa-coupling of the mirror fermion\n doublet is $G_\\chi = \\epsilon^{-1}G_2 \\epsilon$.\n This means that in $G_\\chi$ the isospin components are interchanged.\n The mass term proportional to $\\mu_0$ and the off-diagonal\n Wilson term multiplied by $r$ look in the second form (\\ref{eq26}) not\n Majorana-like but Dirac-like.\n\n Note that if the doublets are degenerate, that is the Yukawa-couplings\n are proportional to the unit matrix in isospin space, then the\n $\\rm SU(2)_L \\otimes U(1)$ symmetry is enlarged\n to $\\rm SU(2)_L \\otimes SU(2)_R$ defined by\n$$\n\\varphi_x^\\prime = U_L^{-1}\\varphi_x U_R \\ ,\n$$\n$$\n\\psi_{(L,R)x}^\\prime = U_{(L,R)}^{-1}\\psi_{(L,R)x} \\ ,\n$$\n$$\n\\overline{\\psi}_{(L,R)x}^\\prime =\n\\overline{\\psi}_{(L,R)x} U_{(L,R)} \\ ,\n$$\n$$\n\\chi_{(R,L)x}^\\prime = U_{(L,R)}^{-1}\\chi_{(R,L)x} \\ ,\n$$\n\\begin{equation} \\label{eq27}\n\\overline{\\chi}_{(R,L)x}^\\prime =\n\\overline{\\chi}_{(R,L)x} U_{(L,R)} \\ .\n\\end{equation}\n\n\\subsection{ Chirality and decoupling }\n An important property of the lattice action in the previous\n subsection is the possibility of decoupling half of the fermions from\n the interacting sector \\cite{ROMA1}.\n Let us formulate this in the mirror fermion langauge corresponding\n to (\\ref{eq26}).\n For vanishing Yukawa-coupling of the mirror fermion doublet $G_\\chi=0$\n and fermion mirror fermion mixing mass $\\mu_0=0$ the action is\n invariant with respect to the Golterman-Petcher shift symmetry\n\\begin{equation} \\label{eq28}\n\\chi_x \\rightarrow \\chi_x + \\epsilon \\ , \\hspace{1em}\n\\overline{\\chi}_x \\rightarrow \\overline{\\chi}_x + \\overline{\\epsilon} \\ .\n\\end{equation}\n This implies \\cite{GOLPET,LINWIT} that all higher vertex functions\n containing the $\\chi$-field identically vanish, and the $\\chi$-$\\chi$\n and $\\chi$-$\\psi$ components of the inverse propagator are equal to\n the corresponding components of the free inverse propagator:\n\\begin{equation} \\label{eq29}\n\\tilde{\\Gamma}_{\\psi\\chi} = \\mu_0 + \\frac{r}{2} \\hat{p}^2 \\ ,\n\\hspace{1em}\n\\tilde{\\Gamma}_{\\chi\\chi} = i\\gamma \\cdot \\bar{p} \\ ,\n\\end{equation}\n where, as usual, $\\bar{p}_\\mu \\equiv \\sin p_\\mu$ and\n $\\hat{p}_\\mu \\equiv 2\\sin \\half p_\\mu$.\n\n The consequence of (\\ref{eq29}) is that the fermion mirror fermion\n mixing mass $\\mu_0$ is not renormalized by the Yukawa-interaction\n of the $\\psi$-field.\n This is very useful in numerical simulations, because the\n corresponding bare parameter (usually the fermionic hopping parameter\n $K \\equiv (2\\mu_0+8r)^{-1}$) is fixed, and the number of bare\n parameters to be tuned is less.\n\n There is also another possible interpretation of the fermion\n decoupling.\n Namely, interchanging the r\\^oles of $\\psi$ and $\\chi$, in the case\n of $G_\\psi=\\mu_0=0$ the ordinary fermions are decoupled.\n This decoupling scenario is in fact a rather good approximation to the\n situation in phenomenological models with mirror fermions discussed in\n the previous section.\n This is due to the fact that all known physical fermions have very\n small Yukawa-couplings.\n The only fermion states with strong Yukawa-coupling would be the\n members of the mirror families, if they would exist.\n In fact, the smallness of the known fermion masses on the electroweak\n scale could then be explained by the approximate validity of the\n Golterman-Petcher shift symmetry.\n {\\em Low energy chirality would be the consequence of the\n approximate decoupling of light fermions.}\n In this sense the mirror fermion model is natural, because the\n smallness of some of its parameters is connected to an approximate\n symmetry \\cite{THOOFT}.\n\n Let us shortly discuss the form of the broken Golterman-Petcher\n identities.\n They are broken in general by the small Yukawa-coupling $G_\\chi$, by\n small mixing mass $\\mu_0$, and by the small\n $\\rm SU(3) \\otimes SU(2) \\otimes U(1)$ gauge couplings.\n For definiteness, let us consider here the case of $G_\\chi \\simeq 0$.\n Consider the generating function of the connected Green's functions\n $W[\\eta,\\overline{\\eta},\\zeta,\\overline{\\zeta},j]$, where the\n external sources $\\eta,\\zeta,j$ belong, respectively, to\n $\\psi,\\chi,\\varphi$.\n The identities obtained by shifting the $\\chi$- and\n $\\overline{\\chi}$-fields are\n$$\n\\overline{\\zeta}_x\n+ \\half \\sum_{\\mu=1}^4 (\\Delta^f_\\mu + \\Delta^b_\\mu)\n\\frac{\\partial W}{\\partial\\zeta_x}\\gamma_\\mu\n$$\n$$\n+ \\frac{r}{2} \\sum_{\\mu=1}^4 \\Delta^f_\\mu\\Delta^b_\\mu\n\\frac{\\partial W}{\\partial\\eta_x}\n= \\mu_0 \\frac{\\partial W}{\\partial\\eta_x}\n$$\n$$\n+ G_\\chi \\left(\n\\frac{\\partial^2 W}{\\partial j_{Rx}\\partial\\zeta_x}\n+ \\frac{\\partial W}{\\partial j_{Rx}}\n\\frac{\\partial W}{\\partial\\zeta_x} \\right) \\Gamma^\\dagger_R \\ ,\n$$\n$$\n- \\zeta_x\n- \\half \\sum_{\\mu=1}^4 \\gamma_\\mu (\\Delta^f_\\mu + \\Delta^b_\\mu)\n\\frac{\\partial W}{\\partial\\overline{\\zeta}_x}\n$$\n$$\n+ \\frac{r}{2} \\sum_{\\mu=1}^4 \\Delta^f_\\mu\\Delta^b_\\mu\n\\frac{\\partial W}{\\partial\\overline{\\eta}_x}\n= \\mu_0 \\frac{\\partial W}{\\partial\\overline{\\eta}_x}\n$$\n\\begin{equation} \\label{eq30}\n+ G_\\chi \\left(\n\\frac{\\partial^2 W}{\\partial j_{Rx}\\partial\\overline{\\zeta}_x}\n+ \\frac{\\partial W}{\\partial j_{Rx}}\n\\frac{\\partial W}{\\partial\\overline{\\zeta}_x} \\right) \\Gamma^\\dagger_R\n\\ .\n\\end{equation}\n Here $\\Delta^f_\\mu$ and $\\Delta^b_\\mu$ denote, respectively, forward\n and backward lattice derivatives, and real scalar field components\n $\\phi_R$ $(R=0,1,2,3)$ are introduced by\n$$\n\\half \\left[ \\varphi_x (1+\\gamma_5)\n+ \\varphi^\\dagger_x (1-\\gamma_5) \\right]\n$$\n\\begin{equation} \\label{eq31}\n= \\phi_{0x} + i\\gamma_5 \\tau_r\\phi_{rx} \\equiv \\Gamma_R\\phi_{Rx} \\ .\n\\end{equation}\n\n Let us define the composite fermion field $\\Psi_x$ by\n\\begin{equation} \\label{eq32}\n\\Gamma_R\\phi_{Rx}\\chi_x\n= \\varphi_x\\chi_{Lx} + \\varphi^\\dagger_x\\chi_{Rx} \\equiv \\Psi_x \\ .\n\\end{equation}\n The mixed $\\psi$-$\\Psi$ two point function is\n\\begin{equation} \\label{eq33}\n\\langle \\psi_y \\Psi_x \\rangle \\equiv \\frac{1}{\\Omega}\n\\sum_k e^{ik \\cdot (y-x)} \\tilde{\\Delta}^{\\psi\\Psi}_k \\ .\n\\end{equation}\n Taking derivatives at zero sources, with the notation\n $\\langle \\phi_{0x} \\rangle \\equiv v$ for the vacuum expectation\n value one obtains, for instance,\n$$\n0 = (\\mu_0 + \\frac{r}{2} \\hat{k}^2) \\tilde{\\Delta}^{\\psi\\psi}_k\n+ \\tilde{\\Delta}^{\\psi\\chi}_k (i\\gamma \\cdot \\bar{k} + G_\\chi v)\n+ G_\\chi \\tilde{\\Delta}^{\\psi\\Psi}_k \\ ,\n$$\n$$\n1 = (\\mu_0 + \\frac{r}{2} \\hat{k}^2) \\tilde{\\Delta}^{\\chi\\psi}_k\n$$\n\\begin{equation} \\label{eq34}\n+ \\tilde{\\Delta}^{\\chi\\chi}_k (i\\gamma \\cdot \\bar{k} + G_\\chi v)\n+ G_\\chi \\tilde{\\Delta}^{\\chi\\Psi}_k \\ .\n\\end{equation}\n For $G_\\chi=0$ this is equivalent to (\\ref{eq29}).\n The case of small gauge couplings can be treated similarly.\n\n\\subsection{ A simple $SU(2)_L \\otimes SU(2)_R$ model }\n The lattice actions of Yukawa-models in the form (\\ref{eq21}) or\n (\\ref{eq26}) can be used for numerical simulation studies of chiral\n Yukawa-models.\n In order to apply Monte Carlo simulation methods one needs, however,\n a fermion determinant which is positive.\n For instance, in the Hybrid Monte Carlo algorithm \\cite{DKPR} one\n has to duplicate the number of fermionic degrees of freedom.\n Let us denote the fermion matrix corresponding to (\\ref{eq26}) by\n $Q$, then the replica fermions have $Q^\\dagger$, and the fermion\n determinant is $\\det(QQ^\\dagger)$, which is positive.\n Due to the adjoint, for the replica fermions $\\psi_x$ describes\n a mirror fermion doublet and $\\chi_x$ an ordinary fermion doublet.\n By charge conjugation as in (\\ref{eq24}) one can transform the\n $\\psi$-field of replica fermions to an ordinary doublet, and\n the $\\chi$-field of replica fermions to a mirror doublet.\n In this way one can consider two doublets described by the\n $\\psi$-fields and two mirror doublets described by the $\\chi$-fields.\n For simplicity, let us consider only degenerate doublets, that is,\n let the Yukawa-couplings $G_{(\\psi,\\chi)}$ be proportional to the\n unit matrix in isospin space.\n In this case the Hybrid Monte Carlo simulation describes two\n equal mass degenerate doublets plus two equal mass degenerate mirror\n doublets with exact global $\\rm SU(2)_L \\otimes SU(2)_R$ symmetry.\n In the phase with spontaneously broken symmetry the mass of the\n doublets is proportional to $G_\\psi$, and the mass of the mirror\n doublets to $G_\\chi$.\n\n As it was noted in the previous subsection, the limit $G_\\chi=\\mu_0=0$\n is particularly interesting, because it has less bare parameters to\n tune.\n In this case the $\\chi$-fields are exactly decoupled, and one is\n left in Hybrid Monte Carlo with two degenerate fermion doublets\n described by the $\\psi$-fields.\n This is the simplest model of heavy fermion doublets one can simulate\n by present day fermionic simulation techniques \\cite{SU2XSU2}.\n Besides the two bare parameters of the pure scalar sector\n $(m_0^2,\\lambda)$ there is only one additional bare parameter\n ($G_\\psi$) in the fermionic part of the action:\n$$\nS_{fermion} = \\sum_x \\left\\{ -\\half \\sum_\\mu \\left[\n (\\overline{\\psi}_{x+\\hat{\\mu}}\\gamma_\\mu\\psi_x)\n\\right.\\right.\n$$\n$$\n+ (\\overline{\\chi}_{x+\\hat{\\mu}}\\gamma_\\mu\\chi_x)\n$$\n$$\n\\left. - \\left(\n (\\overline{\\chi}_x\\psi_x)\n- (\\overline{\\chi}_{x+\\hat{\\mu}}\\psi_x)\n+ (\\overline{\\psi}_x\\chi_x)\n- (\\overline{\\psi}_{x+\\hat{\\mu}}\\chi_x) \\right) \\right]\n$$\n\\begin{equation} \\label{eq35}\n\\left. + G_\\psi \\left[\n (\\overline{\\psi}_{Rx} \\varphi^+_x \\psi_{Lx})\n+ (\\overline{\\psi}_{Lx} \\varphi_x \\psi_{Rx}) \\right] \\right\\} \\ .\n\\end{equation}\n To have the smallest possible number of parameters is very important,\n in order to keep parameter tuning as easy as possible.\n\n By a further duplication of the fermion fields one can also\n simulate four degenerate fermion doublets, which correspond to\n a heavy degenerate family.\n The $\\rm SU(4)_{Pati-Salam}$ symmetry \\cite{PATSAL} of the four\n $\\psi$-doublets, including $\\rm SU(3)_{colour}$ for the quarks,\n is exact in the continuum limit, but for finite lattice spacings\n it is broken by the off-diagonal Wilson-terms which mix the $\\psi$- and\n $\\chi$-fields.\n Note, however, that there is an exact SU(4) symmetry also at finite\n lattice spacings, if one transforms the $\\psi$- and $\\chi$-fields\n simultaneously.\n Of course, the $\\chi$'s are mirror fermion fields, which mix with the\n $\\psi$'s through the nonzero $\\tilde{\\Gamma}_{\\psi\\chi}$ in\n (\\ref{eq29}).\n This mixing goes to zero only in the continuum limit.\n The decoupling in the continuum limit is exact in the Yukawa-model,\n but for nonzero gauge couplings decoupling the $\\chi$'s in a gauge\n invariant way does not work.\n\n Another way of simulating a heavy degenerate fermion family with\n only two pairs of $(\\psi,\\chi)$-doublet fields is to choose\n $G_\\chi = \\pm G_\\psi$ (the opposite sign is preferred by the study\n of the $K=0$ limit \\cite{LIMAMO}).\n Since without $\\rm SU(3) \\otimes U(1)$ gauge fields the mirror\n fermion doublets are equivalent to ordinary fermion doublets, this\n describes the same model in the continuum limit as the one with\n twice as much fields and decoupling.\n In this case, however, the fermion hopping parameter $K$ has to be\n tuned, too, which can be worse than having more field components per\n lattice sites.\n\\begin{figure}[tb]\n\\vspace{9cm}\n\\caption{ \\label{fig1}\n Phase structure of the $\\rm SU(2)_L\\otimes SU(2)_R$ symmetric Yukawa\n model at $\\lambda=\\infty$ and $G_\\chi=\\mu_0=0$ in the\n ($G_\\psi,\\,\\kappa$)-plane.\n Open circles denote points in the PM~phase, crosses represent\n points in the FM~phase.\n The points in the AFM and FI~phases are denoted by full circles and\n open squares, respectively.\n The dashed lines labelled~R,S,T each show the range of~$\\kappa$ used\n for a systematic scan of renormalized parameters at fixed~$G_\\psi$.\n The crosses along those lines denote the kappa values where the\n minimum scalar mass in the broken phase is encountered.\n Solid lines connect the critical values for $\\kappa$ estimated from\n the behaviour of the magnetization on $4^3\\cdot8$.\n Dashed lines around the FI~phase show the expected continuation of the\n critical lines. }\n\\end{figure}\n\n\\subsection{ Phase structure }\n The first step in a recent numerical simulation of the\n $\\rm SU(2)_L \\otimes SU(2)_R$ symmetric Yukawa-model with $N_f=2$\n fermion doublets in the decoupling limit $G_\\chi=0$ \\cite{SU2XSU2}\n was to check the phase structure at infinite bare quartic coupling\n $\\lambda=\\infty$.\n On the basis of experience in several different lattice Yukawa models\n \\cite{GOLREV,SHIREV,LIMOWI}, this is expected to possess several phase\n transitions between the {\\it ``ferromagnetic'' (FM)}, {\\it\n ``antiferromagnetic'' (AFM)}, {\\it ``paramagnetic'' (PM)} and {\\it\n ``ferrimagnetic'' (FI)} phases.\n The resulting picture in the ($G_\\psi,\\kappa$)-plane is shown in\n fig.~\\ref{fig1}.\n ($\\kappa \\equiv (1-2\\lambda)\/(m_0^2+8)$ is the bare parameter which\n is usually taken in numerical simulations instead of $m_0^2$.)\n\\begin{figure}[tb]\n\\vspace{9cm}\n\\caption{ \\label{fig2}\n The cut-off dependent allowed region in the ($G_{R\\psi}^2,g_R$) plane\n for cut-off values equal to some multiples of the Higgs-boson mass\n $m_\\phi \\equiv m_{R\\sigma}$.\n In the Yukawa-model describing two degenerate heavy fermion doublets\n without gauge couplings the perturbative 1-loop $\\beta$-functions are\n assumed. }\n\\end{figure}\n\n\\subsection{ Allowed region in renormalized couplings }\n An important question for numerical simulations is the determination\n of the nonperturbative cut-off dependent {\\em allowed region} in the\n space of renormalized quartic and Yukawa-couplings.\n If the continuum limit of Yukawa-models is trivial, then there\n are cut-off dependent upper bounds on both the renormalized quartic\n and Yukawa-couplings, which tend to zero in the continuum limit.\n In pure $\\phi^4$ models the upper bound is qualitatively well\n described by the 1-loop perturbative $\\beta$-function, if the\n Landau-pole in the renormalization group equations is assumed to\n occur at the scale of the cut-off.\n The same might be true for scalar-fermion models with Yukawa-couplings.\n For instance, in the model with $\\rm SU(2)_L \\otimes SU(2)_R$ symmetry\n and $N_f$ degenerate fermion doublets the 1-loop $\\beta$-functions for\n the quartic ($g_R$) and Yukawa- ($G_{R\\psi}$) couplings are:\n$$\n\\beta_{g_R} = \\frac{1}{16\\pi^2} \\left(\n4g_R^2 + 16N_fg_RG_{R\\psi}^2 - 96N_fG_{R\\psi}^4 \\right) \\ ,\n$$\n\\begin{equation} \\label{eq36}\n\\beta_{G_{R\\psi}} = \\frac{1}{16\\pi^2} \\cdot\n4N_fG_{R\\psi}^3 \\ .\n\\end{equation}\n\n Since in the region where $G_{R\\psi}^2 \\gg g_R$ the 1-loop\n $\\beta$-function of the quartic coupling $\\beta_{g_R}$ is negative,\n besides the upper bounds there is also a lower bound on $g_R$ for fixed\n $G_{R\\psi}$, which is called in the literature {\\it vacuum stability\n bound} \\cite{SHER}.\n On the lattice, if one assumes the qualitative behaviour of the 1-loop\n $\\beta$-function to be valid also nonperturbatively, the vacuum\n stability lower bound occurs at zero bare quartic coupling $\\lambda=0$,\n whereas the upper bound at $\\lambda=\\infty$ \\cite{LMMW,LMMWTA,SHENTA}.\n Negative $\\lambda$-values are excluded, because there the path\n integral is divergent.\n For the 1-loop $\\beta$-functions in (\\ref{eq36}) with $N_f=2$\n the bounds in the plane of ($G_{R\\psi}^2,g_R$) are shown in\n fig.~\\ref{fig2}.\n\\begin{figure}[tb]\n\\vspace{9cm}\n\\caption{ \\label{fig3}\n The fermion mass in lattice units $\\mu_{R\\psi}$ plotted versus\n $\\kappa$ for $G_\\psi=0.3$ (triangles) and $G_\\psi=0,6$ (squares) on\n lattices of size $4^3\\cdot8$ (open symbols) and $6^3\\cdot12$\n (filled-in symbols).\n Errorbars are omitted when the variation is of the size of the symbols.\n It is seen that larger bare couplings $G_\\psi$ in general yield larger\n fermion masses. }\n\\end{figure}\n\n These curves have to be confronted with the results of the numerical\n simulations.\n It turned out (see fig.~\\ref{fig3}) that on $4^3 \\cdot 8$ and\n $6^3 \\cdot 12$ lattices the fermion mass in lattice units tends to\n zero, if one is approaching the FM-PM phase transition from above\n (i.~e. from the FM-side).\n This is as expected in the continuum limit on an infinite lattice.\n\\begin{figure}[tb]\n\\vspace{9cm}\n\\caption{ \\label{fig4}\n The scalar mass in lattice units $m_{R\\sigma}$ plotted versus $\\kappa$.\n The explanation of symbols is similar to the previous figure.\n When approaching the phase transition the scalar masses increase again\n after going through a minimum. }\n\\end{figure}\n\\begin{figure}[tb]\n\\vspace{9cm}\n\\caption{ \\label{fig5}\n The obtained numerical estimate of the upper bound on the renormalized\n quartic coupling $g_R$ (or Higgs-boson mass $m_{R\\sigma}$) as a\n function of the renormalized Yukawa-coupling squared $G_{R\\psi}^2$.\n Two points with the smaller errors are on $6^3 \\cdot 12$, the\n third one on $8^3 \\cdot 16$ lattice.\n The lines are the upper bounds according to 1-loop perturbation\n theory at a cut-off $\\Lambda \\simeq 3m_{R\\sigma}$ and $4m_{R\\sigma}$,\n respectively. }\n\\end{figure}\n\n The behaviour of the Higgs-boson mass $m_{R\\sigma}$ on finite lattices\n is more involved.\n This is because the decrease of $m_{R\\sigma}$ is stopped by a minimum,\n and instead of a further decrease there is an increase (see\n fig.~\\ref{fig4}).\n This can be understood as a finite size effect.\n The value at the minimum is smaller on larger lattices.\n Nevertheless, for increasing bare Yukawa-coupling $G_\\psi$ the\n required lattice size is growing.\n Reasonably small masses $m_{R\\sigma} \\simeq 0.5-0.7$ can be\n achieved at $G_\\psi=0.3$ on $6^3 \\cdot 12$, at $G_\\psi=0.6$ on\n $8^3 \\cdot 16$ lattices \\cite{SU2XSU2}.\n At $G_\\psi=1.0$ presumably lattices with spatial extension of\n at least $16^3$ are necessary.\n Taking the values of the Higgs-boson mass at $\\kappa$-values above the\n minimum, one obtains for the upper limit on the renormalized quartic\n coupling the estimates in fig.~\\ref{fig5}.\n These agree within errors with the 1-loop estimates, although the\n values of the renormalized couplings are close to the tree\n unitarity limits.\n The continuation of the upper bound on $g_R$ towards larger\n Yukawa-couplings can only be obtained on larger lattices.\n Particularly interesting is the behaviour in the vicinity of\n the FM-FI phase transition near $G_\\psi=1.0-1.5$.\n A further interesting question is the phase structure near\n $\\lambda \\simeq 0$, which has an influence on the vacuum stability\n lower bound.\n\n\\begin{acknowledge}\n It is a pleasure to thank the organizers for giving me the opportunity\n to participate in this workshop.\n I benefited a lot from the discussions with the participants in a\n very pleasant atmosphere.\n\\end{acknowledge}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nHarmonic analysis on a Riemannian symmetric space $G\/K$ of the noncompact type \nis by now well developed (cf. \\cite{Hel2}). \nA natural extension is \nto study harmonic analysis on homogeneous vector bundles over $G\/K$. \nOne of fundamental problems \nin harmonic analysis is to establish the Plancherel theorem. \nHarish-Chandra establishes a general theory of \n the Eisenstein integrals and \nthe Plancherel theorem for noncompact real semisimple Lie groups (cf. \\cite{HC76,Kn0,Wal,War}). \nThe Plancherel theorem on a homogeneous vector bundle over $G\/K$ \nassociated with an irreducible representation $\\pi$ of $K$ follows from \nHarish-Chandra's result by restricting the Plancherel measure to \n$K$-finite functions of type $\\pi$. But \nit is a highly nontrivial and important problem to determine the Plancherel measure \non the associated vector bundle as \nexplicitly as in the case of the trivial $K$-type. \nThere are several studies in this direction (cf. \\cite{Camporesi2, CP, vDP, \nFJ, Hec, OS, SPlancherel, Shyperbolic}). \n\nIn our previous paper \\cite{OS}, we study elementary \nspherical functions on $G$ with a \nsmall $K$-type $\\pi$ (in the sense of Wallach \\cite[\\S 11.3]{Wal}). \nNamely, we identify elementary spherical functions with the Heckman-Opdam \nhypergeometric function (cf. \\cite{Hec, Op:lecture}) and apply the \ninversion formula and the Plancherel formula \nfor the hypergeometric Fourier transform (\\cite{Op:Cherednik}) \nto obtain the inversion formula and the Plancherel formula for the \n$\\pi$-spherical transform. But there is an exception in \\cite{OS}. \nNamely, for a certain small $K$-type of a noncompact Lie group \nof type $G_2$, elementary spherical functions can not be expressed by \nthe Heckman-Opdam hypergeometric function. \n\nIn this paper we give a complete treatment of harmonic analysis of \n$\\pi$-spherical transform for each small $K$-type $\\pi$ of \n $G_2$. Namely, we give an explicit formula for the Harish-Chandra $c$-function \n$c^{\\pi}(\\lambda)$ \nand determine the Plancherel measure explicitly. The most continuous part \nof the Plancherel measure\n is $|c^\\pi(\\lambda)|^{-2}d\\lambda$ on $\\sqrt{-1}\\mathfrak{a}^*$ and the other \n spectra with supports of lower dimensions are given explicitly by \n using residue calculus. \nAs indicated by Oshima \\cite{Oshima81a} and as was done for one-dimensional $K$-types \nby the second author \\cite{SPlancherel}, we could prove the inversion formula for the \n$\\pi$-spherical tranform in the case of $G_2$ by \n extending Rosenberg's method of a proof of the inversion formula in the case of \n the trivial $K$-type (\\cite{R}). Instead of doing this, \n we utilize general results on the Plancherel theorem and residue calculus on \n $G$ due to Harish-Chandra and Arthur (cf. \\cite{HC76, A, Kn0,Wal}) \n and devote ourselves to the determination of the Plancherel measure. \n\nThis paper is organized as follows. In Section~2 we give general results \nfor elementary $\\pi$-spherical functions, the \nHarish-Chandra $c$-function, the inversion formula for $\\pi$-spherical transform \nwith respect to a small $K$-type $\\pi$ \non a noncompact real semisimple Lie group of finite center. \n \n In Section~3 we study the case of $G_2$. We give an explicit formula of \nthe $c$-function (Theorem~\\ref{thm:cfg2}), \nthe inversion formula, and the Plancherel formula (Theorem~\\ref{thm:main}, \n Corollary~\\ref{cor:main}) for each small $K$-type. \nIn particular, they cover the small $K$-type that is not treated in \\cite{OS}. \n\n\\section{Elementary spherical functions \nfor small $K$-types}\n\n\\subsection{Notation}\nLet $\\mathbb{N}$ denote the set of the nonnegative integers. \nLet $G$ be a non-compact connected real semisimple Lie group of \nfinite center and $K$ a maximal compact subgroup of $G$. \nLet $e$ denote the identity element of $G$. \nLie algebras of Lie groups $G,\\,K$, etc. are denoted by the corresponding German letter $\\mathfrak{g},\\,\\mathfrak{k}$, etc. \nLet $\\mathfrak{g}=\\mathfrak{k}+\\mathfrak{p}$ be \nthe Cartan decomposition and $\\mathfrak{a}$ a maximal abelian subspace \nof $\\mathfrak{p}$. Let $\\varSigma$ denote the root system for $(\\mathfrak{g},\\mathfrak{a})$. \nFor $\\alpha\\in\\varSigma$, let $\\mathfrak{g}_\\alpha$ denote the corresponding \nroot space and $\\boldsymbol{m}_\\alpha=\\dim \\mathfrak{g}_\\alpha$. \nFix a \npositive system $\\varSigma^+\\subset \\varSigma$ and let \n$\\varPi=\\{\\alpha_1,\\dots,\\alpha_r\\}$ denote the set of simple roots in $\\varSigma^+$. \nDefine $\\mathfrak{n}=\\sum_{\\alpha\\in\\varSigma^+}\\mathfrak{g}_\\alpha$ and \n$N=\\exp \\mathfrak{n}$. Then we have the Iwasawa decomposition $G=K\\exp\\mathfrak{a}\\,N$. \nDefine $\\rho=\\frac12\\sum_{\\alpha\\in\\varSigma^+}\\boldsymbol{m}_\\alpha \\alpha$. \n\nLet $W$ denote the Weyl group of $\\varSigma$ and \n $s_i$ the reflection across $\\alpha_i^\\perp$ ($1\\leq i\\leq r$). \nWe have $W\\simeq M'\/M$, where $M'$ (resp. $M$) is the normalizer (resp. centralizer) \nof $\\mathfrak{a}$ in $K$. \n\nDefine\n\\begin{align*}\n& \\mathfrak{a}_+\n=\\{H\\in\\mathfrak{a}\\,|\\,\\alpha(H)>0 \\text{ for all } \\alpha\\in\\varSigma^+\\},\n\\\\\n& \\mathfrak{a}_\\text{reg}\n=\\{H\\in\\mathfrak{a}\\,|\\,\\alpha(H)\\not=0 \\text{ for all } \\alpha\\in\\varSigma^+\\}.\n\\end{align*}\nWe have the Cartan decomposition $G=K\\exp \\overline{\\mathfrak{a}_+}\\,K$. \n\nLet $\\langle\\,\\,,\\,\\,\\rangle$ denote the inner product on $\\mathfrak{a}^*$ \ninduced by the Killing form on $\\mathfrak{g}$ and $||\\,\\,||$ the corresponding norm. \nDefine\n\\[\n\\mathfrak{a}_+^*\n=\\{\\lambda\\in\\mathfrak{a}^*\\,|\\,\\langle\\lambda,\\alpha\\rangle>0 \\text{ for all } \\alpha\\in \\varSigma^+\\}.\n\\]\n\n\\subsection{Elementary $\\pi$-spherical function}\n\nIn this subsection, we review elementary $\\pi$-spherical functions for small $K$-types \naccording to \\cite{OS}. \n\nLet $(\\pi,V)$ be a small $K$-type, that is, $\\pi|_M$ is irreducible. \nWe call an $\\text{End}_\\mathbb{C} V$-valued function $f$ on $G$ satisfying \n\\[\nf(k_1gk_2)=\\pi(k_2^{-1})f(g)\\pi(k_1^{-1})\\quad (k_1,\\,k_2\\in K,\\,g\\in G)\n\\]\na $\\pi$-spherical function. \n\nLet $\\boldsymbol{D}^\\pi$ denote the algebra of the invariant differential operators on the homogeneous \nvector bundle over $G\/K$ associated with $\\pi$. \nLet $U(\\mathfrak{g}_\\mathbb{C})$ denote the universal enveloping algebra of \n$\\mathfrak{g}_\\mathbb{C}=\\mathfrak{g}\\otimes_\\mathbb{R}\\mathbb{C}$ and \n$U(\\mathfrak{g}_\\mathbb{C})^K$ the set of the \n$\\text{Ad}(K)$-invariant elements in $U(\\mathfrak{g}_\\mathbb{C})$. \nLet $J_{\\pi^*}=\\text{ker}\\,\\pi^*$ in $U(\\mathfrak{k}_\\mathbb{C})$. \nWe have \n\\[\n\\boldsymbol{D}^\\pi\\simeq U(\\mathfrak{g}_\\mathbb{C})^K\/\nU(\\mathfrak{g}_\\mathbb{C})^K\\cap U(\\mathfrak{g}_\\mathbb{C})J_{\\pi^*}.\n\\] \n\nLet $S(\\mathfrak{a}_\\mathbb{C})$ denote the symmetric algebra \nof $\\mathfrak{a}_\\mathbb{C}=\\mathfrak{a}\\otimes_\\mathbb{R}\\mathbb{C}$ and \n$S(\\mathfrak{a}_\\mathbb{C})^W$ the set of the $W$-invariant elements in $S(\\mathfrak{a}_\\mathbb{C})$. \nThere exists an algebra homomorphism \n\\[\n\\gamma^\\pi:U(\\mathfrak{g}_\\mathbb{C})^K\\rightarrow S(\\mathfrak{a}_\\mathbb{C})^W\n\\]\nwith the kernel $U(\\mathfrak{g}_\\mathbb{C})^K\\cap U(\\mathfrak{g}_\\mathbb{C})\nJ_{\\pi^*}$ (cf. \\cite[Lemma~11.3.2, Lemma~11.3.3]{Wal}). \nNotice that the homomorphism $\\gamma^\\pi$ is independent of the choice of \n$\\varSigma^+$. \nThus we have the generalized Harish-Chandra isomorphism \n$\\gamma^\\pi:\\boldsymbol{D}^\\pi\\xrightarrow{\\smash[b]{\\lower 0.7ex\\hbox{$\\sim$}}} S(\\mathfrak{a}_\\mathbb{C})^W$. \nTherefore, any algebra homomorphism \nfrom $\\boldsymbol{D}^\\pi$ to $\\mathbb{C}$ is of the form $D\\mapsto \\gamma^\\pi(D)(\\lambda)\\,\\,(D\\in\\boldsymbol{D}^\\pi)$ \nfor some $\\lambda\\in \\lower0.8ex\\hbox{$W$}\\backslash \\mathfrak a_{\\mathbb C}^*$. \n\n\nFor $\\lambda\\in \\lower0.8ex\\hbox{$W$}\\backslash \\mathfrak a_{\\mathbb C}^*$ \nthere exists a unique \nsmooth $\\pi$-spherical \nfunction $f=\\phi_\\lambda^\\pi$ satisfying $f(e)=\\text{id}_V$ and \n$Df=\\gamma^\\pi(D)(\\lambda)f\\,\\,\\,(D\\in\\boldsymbol{D}^\\pi)$ (cf. \\cite[Theorem~1.4]{OS}). \nWe call $\\phi_\\lambda^\\pi$ \nthe \nelementary $\\pi$-spherical function. \nSince $\\boldsymbol{D}^\\pi$ contains an elliptic operator, $\\phi_\\lambda^\\pi$ is \nreal analytic. Moreover, \nit has an integral representation \n\\begin{equation}\n\\phi^\\pi_{\\lambda}(g) =\n\\int_Ke^{(\\lambda-\\rho)(H(gk))}\\pi(k\\kappa(gk)^{-1})dk. \\label{eq:eisenstein}\n\\end{equation}\nHere given $x\\in G$, define $\\kappa(x)\\in K$ and $H(x)\\in\\mathfrak{a}$ by $x\\in \\kappa(x)e^{H(x)}N$. \nNotice that $\\phi_\\lambda^\\pi$ is independent of the choice of \n$\\varSigma^+$, though the right hand side of \\eqref{eq:eisenstein} depends on \n$\\varSigma^+$ at first glance. \nMoreover, $\\phi_\\lambda^\\pi$ depends \nholomorphically on $\\lambda\\in\\mathfrak{a}_\\mathbb{C}^*$. \n\nFormula \\eqref{eq:eisenstein} is a special case of the integral representations of elementary spherical functions (or more generally the \n{Eisenstein integral}s) given by Harish-Chandra (cf. \\cite[\\S 6.2.2, \\S 9.1.5]{War}, \\cite[(42)]{Camporesi2}, \\cite[(14.20)]{Kn0}).\n\n\\subsection{Harish-Chandra series}\n\nIn this subsection, we review the Harish-Chandra expansion of the \nelementary spherical function according to \\cite[\\S~9.1]{War}. \nWe assume $(\\pi,V)$ is a small $K$-type. \n\nLet $C^\\infty(G,\\pi,\\pi)$ denote the space of the \nsmooth $\\pi$-spherical functions. \nIf $f\\in C^\\infty(G,\\pi,\\pi)$ then $f|_A$ takes values in $\\text{End}_M V\\simeq \\mathbb{C}$.\nHence we regard \n$\\varUpsilon^\\pi(f):=f|_A \\circ\\exp$ as a scalar valued function on $\\mathfrak{a}$. \nLet $C^\\infty(\\mathfrak{a})^W$ denote the space of the $W$-invariant smooth functions on $\\mathfrak{a}$.\nThe restriction map \n$\\varUpsilon^\\pi$ gives an isomorphism $C^\\infty(G,\\pi,\\pi)\\xrightarrow{\\smash[b]{\\lower 0.7ex\\hbox{$\\sim$}}}\nC^\\infty(\\mathfrak{a})^W$ (\\cite[Theorem~1.5]{OS}). \n\nLet $\\mathscr R$ be the unital algebra of functions on $\\mathfrak a_{\\mathrm{reg}}$\ngenerated by $(1\\pm e^{\\alpha})^{-1}$ ($\\alpha\\in\\varSigma^+$).\nFor any $D \\in U(\\mathfrak g_{\\mathbb C})^K$\nthere exists a unique $W$-invariant \ndifferential operator $\\varDelta^\\pi(D) \\in \\mathscr R \\otimes S(\\mathfrak a_{\\mathbb C})$\nsuch that for any $f \\in C^\\infty(G,\\pi,\\pi)$\n\\[\n\\varUpsilon^\\pi(Df)\n=\\varDelta^\\pi(D)\\varUpsilon^\\pi(f)\n\\]\non $\\mathfrak a_{\\mathrm{reg}}$ (\\cite[Proposition 3.10]{OS}). \nWe call $\\varDelta^\\pi(D)$ the $\\pi$-radial part of $D$. The function \n$\\Phi=\\varUpsilon^\\pi(\\phi^\\pi_\\lambda)$ satisfies differential equations\n\\begin{equation}\\label{eqn:derad}\n\\varDelta^\\pi(D)\\Phi=\\gamma^\\pi(D)(\\lambda)\\Phi\\quad \n(D\\in U(\\mathfrak{g}_\\mathbb{C})^K).\n\\end{equation}\n\nLet $\\mathbb{N}\\varSigma^+$ denote the set of $\\mu\\in\\mathfrak{a}^*$ of the form \n$\\mu=n_1\\alpha_1+\\cdots +n_r\\alpha_r\\,\\,(n_i\\in\\mathbb{N})$. \nFor $\\mu\\in\\mathbb{N}\\varSigma^+\\setminus\\{0\\}$, let \n$\\sigma_\\mu$ denote the hyperplane\n\\[\n\\sigma_\\mu=\\{\\lambda\\in\\mathfrak{a}_\\mathbb{C}^*\\,|\\,\n\\langle 2\\lambda-\\mu,\\mu\\rangle=0\\}.\n\\]\nIf $\\lambda\\not\\in\\sigma_\\mu$ for any $\\mu\\in\\mathbb{N}\\varSigma^+\\setminus\\{0\\}$, \nthen there exists a unique convergent series solution\n\\begin{equation}\\label{eqn:hcs1}\n\\Phi_\\lambda(H)=e^{(\\lambda-\\rho)(H)}\\sum_{\\mu\\in\\mathbb{N}\\varSigma^+}\n\\Gamma_\\mu(\\lambda)e^{-\\mu(H)}\n\\quad (H\\in \\mathfrak{a}_+)\n\\end{equation}\nof \\eqref{eqn:derad} \nwith $\\Gamma_\\mu(\\lambda)\\in\\mathbb{C}$ and $\\Gamma_0(\\lambda)=1$. \nThis is a special case of \\cite[Theorem~9.1.4.1]{War}. \n\nBy using differential equations \\eqref{eqn:derad}, apparent singularities of $\\Phi_\\lambda(H)$ \nas a function of $\\lambda$ is removable unless $\\mu=n\\alpha$ for some $n\\in \\mathbb{Z}_{>0}$ and \n$\\alpha\\in\\varSigma^+$ (cf. \\cite[Corollary~6.3]{A}, see also \n\\cite[Lemma~6.5]{Op:lecture} and \\cite[Proposition~7.5]{Hec}). \nFor $\\mu=n\\alpha$, $\\lambda\\not\\in \\sigma_{\\mu}$ if and only if $\\langle\\lambda,\\alpha^\\vee\\rangle\n\\not=n$. Here $\\alpha^\\vee =2\\alpha\/\\langle\\alpha,\\alpha\\rangle$. \nThus $\\Phi_\\lambda$ is defined if $\\langle\\lambda,\\alpha^\\vee\\rangle\\not\\in \\mathbb{Z}_{>0}$ for \nall $\\alpha\\in\\varSigma^+$. \n\nIf \n$\\langle\\lambda,\\alpha^\\vee\\rangle\\not\\in \\mathbb{Z}$ for all $\\alpha\\in\\varSigma^+$, then \n$\\{\\Phi_{w\\lambda}\\,|\\,w\\in W\\}$ forms a basis of the solution space \nof \\eqref{eqn:derad} on $\\mathfrak{a}_+$. Thus \n$\\varUpsilon^\\pi(\\phi_\\lambda^\\pi)$ is a linear combination of \n$\\Phi_{w\\lambda}\\,\\,(w\\in W)$. \nSince $\\phi_{w\\lambda}^\\pi=\\phi_\\lambda^\\pi$, there exists a constant \n$c^\\pi(\\lambda)$ such that\n\\begin{equation}\\label{eqn:hcs2}\n\\varUpsilon^\\pi(\\phi_\\lambda^\\pi)(H)\n=\\sum_{w\\in W}c^\\pi(w\\lambda)\\Phi_{w\\lambda}(H)\n\\quad (H\\in \\mathfrak{a}_+).\n\\end{equation}\n\n\\subsection{Harish-Chandra ${c}$-function}\n\nIn this subsection, \nwe review \nthe Harish-Chandra $c$-function. We refer to \\cite{Schiffman}, \n \\cite[\\S 9.1.6]{War}, \\cite[Chapter 8]{Wallach73}, \n and \\cite[\\S 5]{Sekiguchi} for details. \n\nLet $H\\in\\mathfrak{a}_+$ and \n$\\lambda\\in\\mathfrak{a}_\\mathbb{C}^*$ satisfying $\\text{Re}\\,\\lambda\\in \\mathfrak{a}_+^*$. \nThe leading \ncoefficient $c^\\pi(\\lambda)$ \nof $\\varUpsilon^\\pi(\\phi^\\pi_\\lambda)$ at infinity in $A_+=\\exp\\mathfrak{a}_+$ is \ngiven by the Harish-Chandra ${c}$-function \n(cf. \\cite[Theorem 9.1.6.1]{War}, \\cite[Theorem~14.7, (14.29)]{Kn0}):\n\\begin{align}\n& \\lim_{t\\to\\infty}e^{t(-\\lambda+\\rho)(H)}\\varUpsilon^\\pi(\\phi^\\pi_\\lambda)(e^{tH})={c}^\\pi(\\lambda), \n\\label{eq:limcf1} \\\\\n& {c}^\\pi(\\lambda)=\\int_{\\bar{N}}e^{-(\\lambda+\\rho)(H(\\bar{n}))}\\pi(\\kappa(\\bar{n}))d\\bar{n}, \n\\label{eq:cfgroup}\n\\end{align}\nwhere the Haar measure on $\\bar{N}$ is normalized so that \n\\[\n\\int_{\\bar{N}}e^{-2\\rho(H(\\bar{n}))}d\\bar{n}=1.\n\\]\nThe integral (\\ref{eq:cfgroup}) \nconverges absolutely for $\\text{Re}\\,\\lambda\\in \\mathfrak{a}_+^*$ and extends to a meromorphic function \non $\\mathfrak{a}_\\mathbb{C}^*$. \nNotice that ${c}^\\pi(\\lambda)\\in \\text{End}_M V\\simeq \\mathbb{C}$. \n\nDefine \n\\[\n\\varSigma_0=\\{\\alpha\\in\\varSigma\\,|\\,\\tfrac12\\alpha\\not\\in\\varSigma\\}\n\\]\nand $\\varSigma_0^+=\\varSigma_0\\cap \\varSigma^+$. \nFor $\\alpha\\in\\varSigma_0^+$ let $\\mathfrak{g}_{(\\alpha)}$ denote the Lie subalgebra of $\\mathfrak{g}$ \ngenerated by $\\mathfrak{g}_\\alpha$ and $\\mathfrak{g}_{-\\alpha}$. \nPut $\\mathfrak{k}_\\alpha=\\mathfrak{k}\\cap \\mathfrak{g}_{(\\alpha)}$, $\\mathfrak{p}_\\alpha=\\mathfrak{p}\\cap \\mathfrak{g}_{(\\alpha)}$, \n $\\mathfrak{a}_\\alpha=\\mathfrak{a}\\cap \\mathfrak{g}_{(\\alpha)}$, $\\mathfrak{n}_\\alpha=\\mathfrak{g}_\\alpha+\\mathfrak{g}_{2\\alpha}$, and \n$\\bar{\\mathfrak{n}}_{\\alpha}=\\theta\\mathfrak{n}_\\alpha$. \nFor $\\alpha\\in \\varSigma_0^+$, let $G_\\alpha,\\,K_\\alpha,\\,A_\\alpha,\\,N_\\alpha$, and $\\bar{N}_{\\alpha}$ denote the analytic subgroups of $G$ \ncorresponding to $\\mathfrak{g}_{(\\alpha)},\\,\\mathfrak{k}_\\alpha,\\,\n\\mathfrak{a}_\\alpha,\\,\\mathfrak{n}_\\alpha$, and $\\bar{\\mathfrak{n}}_{\\alpha}$, respectively. \nWe have the Iwasawa decomposition $G_\\alpha=K_\\alpha A_\\alpha N_\\alpha$. Put \n$\\rho_\\alpha=\\frac12(\\boldsymbol{m}_\\alpha+2\\boldsymbol{m}_{2\\alpha})\\alpha$. \nLet $d\\bar{n}_\\alpha$ denote the Haar measure on $\\bar{N}_\\alpha$ \nnormalized so that \n\\[\n\\int_{\\bar{N}_\\alpha}e^{-2\\rho_\\alpha(H(\\bar{n}_\\alpha))}d\\bar{n}_\\alpha=1. \n\\]\n\nLet $w^*\\in W$ be the element such that $w^*(\\varSigma^+)=-\\varSigma^+$. \nLet $w^*=s_{i_m}\\cdots s_{i_2}s_{i_1}$ be a reduced expression, where $m$ denotes \nthe length of $w^*$ and $1\\leq i_k\\leq r\\,\\,(1\\leq k\\leq m)$. Put \n$\\beta_k=s_{i_1}\\cdots s_{i_{k-1}}\\alpha_{i_k}$. Then we have \n$\\varSigma_0^+=\\{\\beta_1,\\beta_2,\\dots,\\beta_m\\}$ (cf. \\cite[Ch. IV Corollary 6.11]{Hel2}). \nWe have the decomposition $\\bar{N}=\\bar{N}_{\\beta_1}\\cdots \\bar{N}_{\\beta_m}$, \nthe product map being a diffeomorphism. \nMoreover, there exists a positive constant $c_0$ such that \n\\begin{equation}\\label{eqn:normalize}\nd\\bar{n}=c_0\\, d\\bar{n}_{\\beta_1}\\cdots d\\bar{n}_{\\beta_m}.\n\\end{equation}\n\nFor $\\alpha\\in\\varSigma_0^+$ define\n\\[\nc_\\alpha^\\pi(\\lambda)=\\int_{\\bar{N}_\\alpha}e^{-(\\lambda+\\rho_\\alpha)(H(\\bar{n}_\\alpha))}\n\\pi(\\kappa(\\bar{n}_\\alpha))d\\bar{n}_\\alpha.\n\\]\n\nWe have the following product formula (\\cite[Theorem 1.2]{Schiffman}, \n\\cite[\\S 8.11.6]{Wallach73}, \\cite[\\S 9.1.6]{War}, \\cite[Theorem 5.1]{Sekiguchi}).\n\n\\begin{thm}\\label{thm:prod}\n$c^\\pi(\\lambda)=c_0 \\,c_{\\beta_1}^\\pi\\!(\\lambda)\\cdots c_{\\beta_m}^\\pi\\!(\\lambda)$.\n\\end{thm}\n\nFor the case of the trivial $K$-type $\\pi=\\text{triv}$, \n$c_{\\beta_i}^\\text{triv}$ can be written\n explicitly by the classical Gamma function and we have \n the Gindikin and Karpelevi\\v{c} product formula for ${c}^\\text{triv}(\\lambda)$ \n (cf. \\cite{GK}, see also \\cite[Ch IV, \\S 6]{Hel2} and \\cite[\\S 9.1.7]{War}). \n Note that the constant $c_0$ in \n \\eqref{eqn:normalize} is determined explicitly by the \n Gindikin and Karpelevi\\v{c} formula for $c^{\\text{triv}}(\\lambda)$. \n \nThe ${c}$-function for a one-dimensional $K$-type $\\pi$ \nof a group $G$ of Hermitian type is also given \nexplicitly by the Gamma function (\\cite{Sc}, \\cite{S:eigen}). \n\nIn \\cite{OS} we give an explicit formula \nof $c^\\pi(\\lambda)$ for each \nsimple $G$ and each small $K$-type $\\pi$, with one exception for $G$ of type $G_2$ \nand a certain small $K$-type $\\pi$. \nThe method we use is to relate the $\\pi$-elementary spherical \nfunction $\\phi_\\lambda^\\pi$ with \nthe Heckman-Opdam hypergeometric function, \ninstead of computing the integral (\\ref{eq:cfgroup}) by using Theorem~\\ref{thm:prod}. \nHeckman~\\cite[Chapter 5]{Hec} gives in this way an explicit formula of ${c}^\\pi(\\lambda)$ \nfor a one-dimensional $K$-type $\\pi$ when the group $G$ is of Hermitian type. \n\nIn \\S~\\ref{subsec:g2} we give an explicit formula of $c^\\pi(\\lambda)$ \nfor $G$ of type $G_2$ and each small $K$-type $\\pi$ by using Theorem~\\ref{thm:prod}. \n\n\\subsection{$\\pi$-spherical transform}\n\nLet $dH$ denote the Euclidean measure on $\\mathfrak{a}$ with respect to the Killing form. \nDefine $\\delta_{G\/K}=\\prod_{\\alpha \\in \\varSigma^+}\n|2\\sinh \\alpha|^{\\boldsymbol m_\\alpha}$. \nWe normalize the Haar measure $dg$ on $G$ so that\n\\begin{equation*}\n\\int_G \\psi(g)dg=\\frac1{\\#W}\\int_{\\mathfrak a} \\psi(e^H)\\, \\delta_{G\/K}(H)dH\n\\end{equation*}\nfor any compactly supported continuous $K$-bi-invariant function $\\psi$ on $G$\n(cf.~\\cite[Ch.~I, Theorem 5.8]{Hel2}). \n\nLet $C^\\infty_c(G,\\pi,\\pi)$ be the subspace of $C^\\infty(G,\\pi,\\pi)$\nconsisting of the compactly supported smooth $\\pi$-spherical functions. \nThe $\\pi$-spherical transform of $f\\in C_c^\\infty(G,\\pi,\\pi)$ is the $\\text{End}_M V\\simeq \\mathbb{C}$-valued \nfunction ${f}^\\wedge$ on $\\mathfrak{a}_\\mathbb{C}^*$ defined by\n\\begin{equation}\n{f}^\\wedge(\\lambda)=\\int_G \\phi_\\lambda^\\pi(g^{-1})f(g)dg.\n\\end{equation}\nThe $\\pi$-spherical transform $f\\mapsto f^\\wedge(\\lambda)$ is a homomorphism from the commutative convolution \nalgebra $C_c^\\infty(G,\\pi,\\pi)$ to $\\mathbb{C}$ (cf. \\cite{Camporesi2}). \nIt \n is a special case of the Fourier transform given by \nArthur~\\cite{A} (see also \\cite[\\S 3]{BS3}). %\nBy the identification $C_c^\\infty(G,\\pi,\\pi)\\simeq C_c^\\infty(\\mathfrak{a})^W$, \nthe \n$\\pi$-spherical transform $f^\\wedge$ of $f\\in C_c^\\infty(\\mathfrak{a})^W$ is given by \n\\begin{equation}\n{f}^\\wedge(\\lambda)=\\frac{1}{\\# W}\\int_{\\mathfrak{a}}f(H)\n\\varUpsilon^\\pi(\\phi_{-\\lambda}^\\pi)(H)\\delta_{G\/K}(H)\\,dH. \n\\end{equation}\nWe normalize the Haar measure $d\\lambda$ on $\\sqrt{-1}\\mathfrak{a}^*$\n so that the Euclidean Fourier transform and its inversion are given by \n\\[\n\\tilde{f}(\\lambda)=\\int_{\\mathfrak{a}}f(H)e^{-\\lambda(H)}dH, \\qquad\nf(H)=\\int_{\\sqrt{-1}\\mathfrak{a}^*} \\tilde{f}(\\lambda)e^{\\lambda(H)}d\\lambda. \n\\]\n\nLet $\\eta_1$ be a point in $-\\overline{\\mathfrak{a}_+^*}$ such \nthat $c^\\pi(-\\lambda)^{-1}$ is a regular \nfunction of $\\lambda$ for $\\text{Re}\\,\\lambda\\in \\eta_1-\\overline{\\mathfrak{a}_+^*}$. \nThe existence of such $\\eta_1$ follows from an \nexplicit formula of the $c$-function for each small $K$-type, \nwhich is determined by \\cite{OS} and \\S\\ref{sec:g2} for $G_2$.\nIt also follows from a general result on the Harish-Chandra $c$-function \ndue to Cohn \\cite{Cohn}. \n\nLet $F=f^\\wedge$. \nFollowing \\cite[Chapter II, \\S 1]{A}, \ndefine the function $F^\\vee(H)$ on $\\mathfrak{a}_+$ by\n\\begin{equation}\\label{eqn:invft}\nF^\\vee(H)=\n\\int_{\\eta_1+\\sqrt{-1}\\mathfrak{a}^*} F(\\lambda)\\Phi\n_\\lambda(H)c^\\pi(-\\lambda)^{-1}d\\lambda.\n\\end{equation}\nThe integral \\eqref{eqn:invft} converges and is independent of $\\eta_1$ \n(cf. \\cite[Chapter II, \\S 1]{A}). \n$F^\\vee$ defined above coincides with that given by Arthur, \nbecause $\\phi_\\lambda^\\pi$ is $W$-invariant in $\\lambda$ and the \nHarish-Chandra $\\mu$-function associate with a minimal parabolic subgroup \nin our case is given by \n$c^\\pi(\\lambda)^{-1}c^\\pi(-\\lambda)^{-1}$ (cf. \\cite[\\S~10.5]{Wal}). \nThe following theorem is a special case of \\cite[Chapter III, Theorem 3.2]{A}.\nIt is also a special case of \\cite[Theorem 1.1]{BS2}, since $G$ is a \nsemisimple \nsymmetric space for $G\\times G$ (cf. \\cite{BS3}). \n\n\n\n\\begin{thm}\\label{thm:inv}\nFor $f\\in C_c^\\infty(\\mathfrak{a})^W$ we have\n\\[\nf(H)=({f}^\\wedge)^\\vee(H)\\quad (H\\in\\mathfrak{a}_+).\n\\]\n\\end{thm}\n\nIf $c^\\pi(-\\lambda)^{-1}$ is a regular function of $\\lambda$ for \n$\\text{Re}\\,\\lambda\\in -\\overline{\\mathfrak{a}_+^*}$, \nthen we can take $\\eta_1=0$ and by \\eqref{eqn:hcs2} and $W$-invariance of \n$c^\\pi(\\lambda)^{-1}c^\\pi(-\\lambda)^{-1}$ in \n$\\lambda\\in\\sqrt{-1}\\mathfrak{a}^*$, we have \n\\begin{equation}\\label{eqn:invcont}\nf(H)\n=\n\\frac{1}{\\# W}\\int_{\\sqrt{-1}\\mathfrak{a}^*}\nf^\\wedge (\\lambda)\\varUpsilon^\\pi(\\phi^\\pi_\\lambda)(H)\n|c^\\pi(\\lambda)|^{-2}d\\lambda\\quad (H\\in\\mathfrak{a}^*). \n\\end{equation}\n\nIn \\cite[Corollary 7.6]{OS} \nwe prove the formula \\eqref{eqn:invcont} by using the inversion formula \nof the hypergeometric Fourier transform due to Opdam~\\cite{Op:Cherednik},\nunder the assumption that $\\pi$ is a small $K$-type of\na real simple $G$ which is not in the following list:\n\n\\smallskip\n\\noindent\n(1) $\\mathfrak{g}=\\mathfrak{sp}(p,1),\\,\\pi=\\pi_n\\circ \\text{pr}_2$ \n($\\pi_n$ : $n$-dimensional irred. rep. of $\\text{Sp}(1))$, \n\\\\\n(2) $\\mathfrak{g}=\\mathfrak{so}(2r,1)$,\\\\ \n\\phantom{(3)} $\\pi=\\pi_s^{\\pm}$ : \nirred. rep. of $\\text{Spin}(2r)$ with h.w. $(s\/2,\\cdots s\/2,\\pm s\/2)\\,\\,\n(s\\in \\mathbb{N})$,\\\\\n(3) $\\mathfrak{g}=\\mathfrak{so}(p,q)\\quad (p>q\\geq 3,\\,\\,\\text{$p$ : even, \n$q$ : odd)}$,\\\\\n\\phantom{(3)} $\\pi=\\sigma\\circ\\text{pr}_1\\,\\,(\\sigma$ : one of half spin representation of $\\text{Spin}(p)$),\\\\\n(4) $\\mathfrak{g}$\\,:\\,Hermitian type, $\\pi$ : one dimensional $K$-type, \n\\\\\n(5) $\\mathfrak{g}=\\mathfrak{g}_2$, $\\pi=\\pi_2$ (see \\S \\ref{sec:g2} for the definition).\n\n\\medskip\nThough the case (3) is not covered by \\cite[Corollary 7.6]{OS}, \nthe formula \\eqref{eqn:invcont} holds in this case, since \n$c^\\pi(-\\lambda)^{-1}$ is a regular function of $\\lambda$ for \n$\\text{Re}\\,\\lambda\\in -\\overline{\\mathfrak{a}_+^*}$ as we mention in \nthe final part of \\cite{OS}. \n\nIf the parameter of the small $K$-type is \n``large enough'' in the cases (1), (2), and (4), \nthen $c^\\pi(-\\lambda)^{-1}$ has singularities in \n$\\text{Re}\\,\\lambda\\in -\\overline{\\mathfrak{a}_+^*}$ and \nwe must take account of residues during the contour \nshift $\\eta_1+\\sqrt{-1}\\mathfrak{a}^*\\to \\sqrt{-1}\\mathfrak{a}^*$. \nThe most continuous part of the spectrum is given by the right hand side \nof \\eqref{eqn:invcont}. In addition, there are spectra with low dimensional \nsupports. The residue calculus in the case (4) is done by \\cite{SPlancherel}. \nFor the cases (1) and (2), $\\dim\\mathfrak{a}=1$ and the residue calculus is \neasy to proceed. Also these cases are covered by the inversion formula \nof the Jacobi transform (cf. \\cite[Appendix 1]{FJ}, \\cite{Koornwinder}). \nSee also \\cite{vDP} and \\cite{Shyperbolic} for the case (1). \n\nWe will discuss the case (5) in the next section. \n\n\\section{The case of $G_2$}\n\\label{sec:g2}\n\n\\subsection{Notation and preliminary results}\\label{subsec:g2prem}\nLet $\\mathfrak{g}$ be the simple split real Lie algebra of type $G_2$ and $G$ the connected simply connected \nLie group with the Lie algebra $\\mathfrak{g}$. \n$G$ is the double cover of the adjoint group of $\\mathfrak{g}$. \nLet $K$ be a maximal compact subgroup of $G$ and \n$\\mathfrak{k}$ the Lie algebra of $K$. Then \n$K\\simeq \\SU(2)\\times \\SU(2)$ and $\\mathfrak{k}\\simeq \\mathfrak{su}(2)\\oplus \\mathfrak{su}(2)$. \n\nLet $\\mathfrak{t}$ be a maximal abelian subalgebra of $\\mathfrak{k}$. \nThen $\\mathfrak{t}$ is a Cartan subalgebra \nof $\\mathfrak{g}$. Let $\\Delta$ and $\\Delta_K$ denote the root system for $(\\mathfrak{g}_\\mathbb{C},\\mathfrak{t}_\\mathbb{C})$ \nand $(\\mathfrak{k}_\\mathbb{C},\\mathfrak{t}_\\mathbb{C})$, respectively. \nThen $\\Delta$ is a root system of type $G_2$. \nWe choose a positive system $\\Delta^+\\subset \\Delta$\nso that its base contains a short compact root $\\beta_1$.\nThe other simple root, say $\\beta_2$, is a long noncompact root.\nIf we put $\\Delta_K^+=\\Delta_K\\cap \\Delta^+$ then\n\\begin{align*}\n& \\Delta^+=\\{\\beta_1,\\beta_2,\\beta_1+\\beta_2,2\\beta_1+\\beta_2,3\\beta_1+\\beta_2,3\\beta_1+2\\beta_2\\},\\\\\n& \\Delta_K^+=\\{\\beta_1,3\\beta_1+2\\beta_2\\}.\n\\end{align*}\nWe fix an inner product on $\\sqrt{-1}\\mathfrak{t}^*$ such that \n$(\\beta_1,\\beta_1)=2$. Then $(\\beta_2,\\beta_2)=6$ and $(\\beta_1,\\beta_2)=-3$.\nWe let $K=K_1\\times K_2$ with $K_i\\simeq \\SU(2)\\,\\,(i=1,2)$\nassuming that\n$\\beta_1$ (resp. $3\\beta_1+2\\beta_2$) is a \nroot for $((\\mathfrak{k}_1)_\\mathbb{C}, \n(\\mathfrak{t}\\cap\\mathfrak{k}_1)_\\mathbb{C})$ (resp. \n$((\\mathfrak{k}_2)_\\mathbb{C}, \n(\\mathfrak{t}\\cap\\mathfrak{k}_2)_\\mathbb{C})$).\nLet \n$\\pr_i$ denote the projection of $K$ to $K_i\\,\\,(i=1,2)$. \n\nThe classification of the small $K$-types for $G$ is given as follows:\n\\begin{thm}[{\\cite[Theorem 1]{SWL}}]\\label{thm:G2}\nThe non-trivial small $K$-types are $\\pi_1:=\\sigma\\circ\\pr_1$ and $\\pi_2:=\\sigma\\circ\\pr_2$. \nHere $\\sigma$ is the two-dimensional irreducible representation of $\\SU(2)$. \n\\end{thm}\n\nA discrete series representation of $G$ is an irreducible representation of $G$ realized as a closed \n$G$-invariant subspace of the \nleft regular representation on $L^2(G)$. \n\n\\begin{lem}\nLet $G$ be as above.\nThen no small $K$-type appears in any discrete series representation of $G$.\n\\end{lem}\n\\begin{proof}\n\n\nIf $\\pi$ is the trivial $K$-type or $\\pi=\\pi_1$, then \nit follows from the Plancherel formula for the \n$\\pi$-spherical functions (cf. \\cite{Hel2}, \\cite{OS}) \nthat there are no discrete series representations having \nthe $K$-type $\\pi$. \n \nNext let us discuss the case of $\\pi_2$.\nThe positive open chamber $(\\sqrt{-1}\\mathfrak{t}^*)^+$ for $\\Delta_K^+$\ncontains the following three open chambers for $\\Delta$:\n\\begin{align*}\n(\\sqrt{-1}\\mathfrak{t}^*)^+_1&:=\\{\\lambda\\in (\\sqrt{-1}\\mathfrak{t}^*)^+\\,|\\, (\\lambda,\\beta_2)>0\\},\\\\\n(\\sqrt{-1}\\mathfrak{t}^*)^+_2&:=\\{\\lambda\\in (\\sqrt{-1}\\mathfrak{t}^*)^+\\,|\\, (\\lambda,\\beta_2)<0\\text{ and }(\\lambda,\\beta_1+\\beta_2)>0\\},\\\\\n(\\sqrt{-1}\\mathfrak{t}^*)^+_3&:=\\{\\lambda\\in (\\sqrt{-1}\\mathfrak{t}^*)^+\\,|\\, (\\lambda,\\beta_1+\\beta_2)<0\\}.\n\\end{align*}\nLet $\\Delta_i^+$ be the corresponding positive systems ($i=1,2,3$).\nNote that $\\Delta_1^+=\\Delta^+$. \nIf we put $\\delta_i=\\frac12\\sum_{\\beta\\in\\Delta^+_i}\\beta$ then\n\\[\n\\delta_1=5\\beta_1+3\\beta_2,\\quad\n\\delta_2=5\\beta_1+2\\beta_2,\\quad\n\\delta_3=4\\beta_1+\\beta_2.\n\\]\nOn the other hand,\n$\\delta_K:=\\frac12\\sum_{\\beta\\in\\Delta_K^+}\\beta=2\\beta_1+\\beta_2.$\nNow suppose $\\pi_2$ appears in a discrete series representation with Harish-Chandra parameter \n$\\lambda\\in \\sqrt{-1}\\mathfrak{t}^*$.\nWe may assume $\\lambda\\in (\\sqrt{-1}\\mathfrak{t}^*)^+_i$\nfor $i=1,2$, or $3$.\nSince the highest weight of $\\pi_2$ is $\\frac32\\beta_1+\\beta_2$,\nit follows from \\cite[Theorem~8.5]{AS} that \n\\[\n\\frac32\\beta_1+\\beta_2=\\lambda+\\delta_i-2\\delta_K+\\sum_{\\beta\\in\\Delta^+_i}\nn_\\beta \\beta\\quad \\text{for some }n_\\beta\\in\\mathbb{N}. \n\\]\nIf $i=1$ then this reduces to\n\\[\n\\lambda=\\left(\\frac12-c_1\\right)\\beta_1-c_2\\beta_2\\quad\\text{for some }c_1,\\,c_2\\in\\mathbb{N}.\n\\]\nSince $(\\lambda,\\beta_j)>0\\,\\,(j=1,2)$, we have $1-2c_1+3c_2>0$ and $-\\frac32+3c_1-6c_2>0$, \nwhich are impossible.\nIf $i=2$ or $3$ then we can also deduce a contradiction in a similar way.\n\\end{proof}\n\n\\subsection{Harish-Chandra ${c}$-function for $G_2$\n}\n\\label{subsec:g2}\n\nThe restricted root system $\\varSigma=\\varSigma(\\mathfrak{g},\\mathfrak{a})$ is \na root system of type $G_2$. \nFor all $\\alpha\\in\\varSigma$ we have \n$\\mathfrak{g}_{(\\alpha)}\\simeq \\mathfrak{sl}(2,\\mathbb{R})$, since \n$\\boldsymbol{m}_\\alpha=\\dim\\mathfrak{g}_\\alpha=1$ and $2\\alpha\\not\\in\\varSigma$. \n\nWe recall the ${c}$-function for $\\mathfrak{g}=\\mathfrak{sl}(2,\\mathbb{R})$. \nPut \n\\[\nh=\\begin{pmatrix} 1 & 0 \\\\ 0 & -1\\end{pmatrix},\\quad \ne=\\begin{pmatrix} 0 & 1 \\\\ 0 & 0\\end{pmatrix},\\quad \nf=\\begin{pmatrix} 0 & 0 \\\\ 1 & 0\\end{pmatrix}. \n\\]\nThen $\\{h,e,f\\}$ is a basis of $\\mathfrak{sl}(2,\\mathbb{R})$ and also \nforms an $\\mathfrak{sl}_2$-triple. \nWe put $\\mathfrak{a}=\\mathbb{R}h$ and $\\mu=\\lambda(h)$ for $\\lambda\\in\\mathfrak{a}_\\mathbb{C}^*$. \nBy \\cite[Remark 7.3]{Sc}, \nthe ${c}$-function for $\\mathfrak{sl}(2,\\mathbb{R})$ with a one-dimensional \n$\\mathfrak{k}$-type $\\pi$ of the weight $\\nu\\in\\mathbb{Q}$ for $\\sqrt{-1}(e-f)$ is given by \n\\begin{equation}\\label{eq:cfsl2}\n{c}^\\pi(\\lambda)=\\frac{2^{1-\\mu}\\varGamma(\\mu)}{\\varGamma(\\frac12(\\mu+1+\\nu))\n\\varGamma(\\frac12(\\mu+1-\\nu))}.\n\\end{equation}\n\nNow we come back to the case of $G_2$. For $\\alpha\\in\\varSigma^+$ \nchoose $e_\\alpha\\in \\mathfrak{g}_{\\alpha}$ so \nthat $\\{\\alpha^\\vee,\\,e_\\alpha,\\,-\\theta e_\\alpha\\}$ is an $\\mathfrak{sl}_2$-triple. \nPut $t_\\alpha:=e_\\alpha+\\theta e_\\alpha\\in \\sqrt{-1}\\mathfrak{k}_\\alpha$. \nIf $\\alpha\\in \\varSigma^+$ is a long root, then the \npossible weights of $t_\\alpha$ for \n$\\pi_i\\,(i=1,2)$ are $\\pm \\frac12$ by \\cite[Lemma 4.2]{SWL}. \nIf $\\alpha\\in \\varSigma^+$ is a short root, then the possible weights of $t_\\alpha$ \nfor $\\pi_1$ (resp. $\\pi_2$) are $\\pm\\frac12$ \n(resp. $\\pm\\frac32$) by \\cite[Lemma 4.3]{SWL}. \nSince \\eqref{eq:cfsl2} remains unchanged if we replace $\\nu$ by $-\\nu$, \n$c_\\alpha^{\\pi_i}(\\lambda)$ is a scalar operator for each $\\alpha\\in \\varSigma$ and $i=1,\\,2$. \n\nLet $\\varSigma_\\text{long}^+$ and $\\varSigma_\\text{short}^+$ denote \nthe sets of the long and short positive roots, respectively. \nDefine $\\lambda_\\alpha=\\langle\\lambda,\\alpha^\\vee\\rangle$ \nfor $\\lambda\\in\\mathfrak{a}_\\mathbb{C}^*$ and \n$\\alpha\\in\\varSigma^+$. \nIt follows from Theorem~\\ref{thm:prod}, \n\\eqref{eq:cfsl2}, and the proof of \\cite[Ch. IV, Theorem~6.13]{Hel2} \nthat\n\\begin{align*}\n{c}^{\\text{triv}}(\\lambda) & \n=\nc_0\\!\n\\prod_{\\alpha\\in\\varSigma^+}\\frac{2^{1-\\lambda_\\alpha}\n\\varGamma(\\lambda_\\alpha)}\n{\\varGamma(\\frac12\\lambda_\\alpha+\\frac12)^2}\n=\nc_0\\!\n\\prod_{\\alpha\\in\\varSigma^+}\\frac{\\pi^{-\\frac12}\n\\varGamma(\\frac12\\lambda_\\alpha)}\n{\\varGamma(\\frac12\\lambda_\\alpha+\\frac12)}, \n\\\\\n{c}^{\\pi_1}(\\lambda) & =c_0\\!\\prod_{\\alpha\\in\\varSigma^+} \n\\frac{2^{1-\\lambda_\\alpha}\\varGamma(\\lambda_\\alpha)}{\\varGamma(\\frac12(\\lambda_\\alpha+\\frac32))\n\\varGamma(\\frac12(\\lambda_\\alpha+\\frac12))} \n =c_0\\!\\prod_{\\alpha\\in\\varSigma^+} \n\\frac{2^{\\frac12}\\pi^{-\\frac12}\\varGamma(\\lambda_\\alpha)}\n{\\varGamma(\\lambda_\\alpha+\\frac12)}, \n\\\\\n{c}^{\\pi_2}(\\lambda) & \n=c_0\\!\\!\n\\prod_{\\alpha\\in\\varSigma_\\text{long}^+} \\!\\!\n\\frac{2^{1-\\lambda_\\alpha}\\varGamma(\\lambda_\\alpha)}{\\varGamma(\\frac12(\\lambda_\\alpha+\\frac32))\n\\varGamma(\\frac12(\\lambda_\\alpha+\\frac12))} \n\\prod_{\\beta\\in\\varSigma_\\text{short}^+} \\!\\!\n\\frac{2^{1-\\lambda_\\beta}\\varGamma(\\lambda_\\beta)}{\\varGamma(\\frac12(\\lambda_\\beta+\\frac52))\n\\varGamma(\\frac12(\\lambda_\\beta-\\frac12))} \\\\\n&=c_0\\!\\!\n\\prod_{\\alpha\\in\\varSigma_\\text{long}^+} \\!\\!\n\\frac{2^{\\frac12}\\pi^{-\\frac12}\\varGamma(\\lambda_\\alpha)}\n{\\varGamma(\\lambda_\\alpha+\\frac12)}\n\\prod_{\\beta\\in\\varSigma_\\text{short}^+} \\!\\!\n \\frac{2^{\\frac12}\\pi^{-\\frac12}\\left(\\lambda_\\beta-\\frac12\\right)\\varGamma(\\lambda_\\beta)}\n{\\varGamma(\\lambda_\\beta+\\frac32)} .\n\\end{align*}\n\nThe value of the constant $c_0$ is determined by $c^{\\text{triv}}(\\rho)=1$. \nWe have $c_0=2\\pi^2$ by direct computation. \nThus we have the following \ntheorem. \n\n\\begin{thm}\\label{thm:cfg2}\n\\begin{align*}\n{c}^{\\text{\\rm triv}}(\\lambda) & \n=\n\\frac{2}{\\pi}\n\\prod_{\\alpha\\in\\varSigma^+}\\frac{\n\\varGamma(\\frac12\\lambda_\\alpha)}\n{\\varGamma(\\frac12\\lambda_\\alpha+\\frac12)}, \n\\\\\n{c}^{\\pi_1}(\\lambda) & \n =\\frac{16}{\\pi}\\prod_{\\alpha\\in\\varSigma^+} \n\\frac{\\varGamma(\\lambda_\\alpha)}\n{\\varGamma(\\lambda_\\alpha+\\frac12)}, \n\\\\\n{c}^{\\pi_2}(\\lambda) \n&=\\frac{16}{\\pi}\n\\prod_{\\alpha\\in\\varSigma_\\text{\\rm long}^+} \n\\frac{\\varGamma(\\lambda_\\alpha)}\n{\\varGamma(\\lambda_\\alpha+\\frac12)}\n\\prod_{\\beta\\in\\varSigma_\\text{\\rm short}^+} \n \\frac{\\left(\\lambda_\\beta-\\frac12\\right)\\varGamma(\\lambda_\\beta)}\n{\\varGamma(\\lambda_\\beta+\\frac32)} .\n\\end{align*}\n\\end{thm}\n\nThe formula for $c^{\\text{triv}}(\\lambda)$ in Theorem~\\ref{thm:cfg2} is a special case of the \nGindikin-Karpelevi\\v{c} formula (cf. \\cite{GK}, \\cite[Ch.~IV, Theorem~6.13]{Hel2}). \nThe formula for $c^{\\pi_1}(\\lambda)$ \nis given in \\cite{OS} by use of a different method. \nThe formula for $c^{\\pi_2}(\\lambda)$ is new. \n\n\\subsection{$\\pi$-spherical transform}\n\nAn inversion formula for the $\\pi$-spherical transform is given by \nTheorem~\\ref{thm:inv}. We must shift the contour of integration from \n$\\eta_1+\\sqrt{-1}\\mathfrak{a}^*$ to $\\sqrt{-1}\\mathfrak{a}^*$ and get a \nglobally defined function on $\\mathfrak{a}$. \nWe refer to \\cite[Ch. II, Ch. III]{A} for the general residue scheme \n(see also \\cite{Oshima81a,BS,BS2,BS2000,BS3,BS4}). \n\nFor $\\pi=\\text{triv}$ and \n$\\pi_1$, $c^\\pi(-\\lambda)^{-1}$ is a regular function of $\\lambda$ for \n$\\text{Re}\\,\\lambda\\in -\\overline{\\mathfrak{a}_+^*}$, hence\n the inversion formula is given by \\eqref{eqn:invcont} for these \nsmall $K$-types (cf. \\cite[Ch~IV, Theorem~7.5]{Hel2}, \\cite[Corollary~7.6]{OS}). \n\nFor $\\pi=\\pi_2$, there appear singularities during the contour shift and \nwe must take account of residues. \nThe function $c^{\\pi_2}(-\\lambda)^{-1}$ for $\\text{Re}\\,\\lambda\\in\n\\mathfrak{a}_+^*$ \nhas singularities along \nlines $\\lambda_\\beta=-\\frac12\\,\\,(\\beta\\in\\varSigma^+_{\\text{short}})$. \nFigure~1 illustrates singular lines \n$\\lambda_{\\alpha_1}=-\\frac12,\\,\\lambda_{\\alpha_1+\\alpha_2}=-\\frac12$, and \n$\\lambda_{2\\alpha_1+\\alpha_2}=-\\frac12$ \n (dashed) for $c^{\\pi_2}(-\\lambda)^{-1}$ \nand $-\\overline{\\mathfrak{a}_+^*}$ \n(shaded region). \nHere $\\alpha_1$ and $\\alpha_2$ are the simple roots of $\\varSigma^+$ ($||\\alpha_1||<||\\alpha_2||$).\nThese singular lines divide $-\\overline{\\mathfrak{a}_+^*}$ into \nthe following four regions (indicated in Figure 1):\n\\begin{figure}\n\\begin{center}\n\\includegraphics[trim=0 105 0 105,width=9.2cm]{shift2b.pdf}\n\\caption{singular lines}\n\\end{center}\n\\end{figure}\n\\begin{align*}\n& \\text{\\emph{I}} \\,:\\, \\lambda_{\\alpha_1}<-\\frac12,\\quad \\lambda_{\\alpha_2}\\leq 0, \\\\\n& \\text{\\emph{II}} \\,:\\, -\\frac12<\\lambda_{\\alpha_1}\\leq 0,\\quad \\lambda_{\\alpha_1+\\alpha_2}<-\\frac12, \\\\\n& \\text{\\emph{III}} \\,:\\, -\\frac12<\\lambda_{\\alpha_1+\\alpha_2},\\quad \n\\lambda_{2\\alpha_1+\\alpha_2}<-\\frac12,\\quad \\lambda_{\\alpha_2}\\leq 0, \\\\\n& \\text{\\emph{IV}} \\,:\\,\\lambda_{2\\alpha_1+\\alpha_2}>-\\frac12,\\quad \\lambda_{\\alpha_1}\\leq 0,\\quad \n\\lambda_{\\alpha_2}\\leq 0. \n\\end{align*}\n\nFirst $\\eta_1\\in-\\mathfrak{a}_+^*$ in \\eqref{eqn:invft} \nlies in the region \\emph{I}. \nWe choose $\\eta_2,\\,\\eta_3$, and $\\eta_4$ \nin the regions \\emph{II,\\,III}, and \\emph{IV}, respectively. We may take $\\eta_4=0$. \nWe shift the contour of \nintegration from $\\eta_1+\\sqrt{-1}\\mathfrak{a}^*$ to $\\eta_2+\\sqrt{-1}\\mathfrak{a}^*$ \nand so on, and finally to $\\eta_4+\\sqrt{-1}\\mathfrak{a}^*=\n\\sqrt{-1}\\mathfrak{a}^*$, picking up residues. \nDefine\n\\[\nF^\\vee_i(H)\n=\\int_{\\eta_i+\\sqrt{-1}\\mathfrak{a}^*}f^\\wedge (\\lambda)\\Phi_{\\lambda}(H)\nc^{\\pi_2}(-\\lambda)^{-1}d\\lambda\\quad (1\\leq i\\leq 4).\n\\]\n\nWe regard \n$(\\lambda_{\\alpha_1},\\lambda_{3\\alpha_1+2\\alpha_2})\\in\\mathbb{C}^2$ as \na coordinate on $\\mathfrak{a}_\\mathbb{C}^*$. \nDefine \n\\begin{equation}\nc_1=\\frac{||\\alpha_1||\\,||3\\alpha_1+2\\alpha_2||}{4}.\n\\end{equation}\nFor $\\eta\\in\\mathfrak{a}^*$, the normalized measure $d\\lambda$ on $\\eta+\\sqrt{-1}\\mathfrak{a}^*$ is given by\n\\[\nd\\lambda=\\frac{c_1}{(2\\pi\\sqrt{-1})^2}d\\lambda_{\\alpha_1}d\\lambda_{3\\alpha_1+2\\alpha_2}.\n\\]\n\nFirst we change the contour of integration of $F^\\vee(H)=F^\\vee_1(H)$ from $\\eta_1+\\sqrt{-1}\\mathfrak{a}^*$ \nto $\\eta_2+\\sqrt{-1}\\mathfrak{a}^*$ with respect to the integration in the variable \n$\\lambda_{\\alpha_1}$. \nBy the explicit formula of $c^{\\pi_2}(\\lambda)$ in Theorem~\\ref{thm:cfg2}, \n$f^\\wedge (\\lambda)\\Phi_{\\lambda}(H)\nc^{\\pi_2}(-\\lambda)^{-1}$ has a possible simple pole during the change of \nthe contour coming from the factor \n$(-\\lambda_{\\alpha_1}-\\frac12)^{-1}$ of $c^{\\pi_2}_{\\alpha_1}(-\\lambda)^{-1}$. \nThus the difference $F^\\vee_1(H)-F^\\vee_2(H)$ is\n\\[\n-\\frac{c_1}{2\\pi\\sqrt{-1}}\\int_{\\mu+\\sqrt{-1}\\mathbb{R}}\n\\Res_{\n\\lambda_{\\alpha_1}=-\\frac12}\\!\\left(\nf^\\wedge (\\lambda)\\Phi_{\\lambda}(H)\nc^{\\pi_2}(-\\lambda)^{-1}\\right)\\!\nd\\lambda_{3\\alpha_1+2\\alpha_2}\n\\]\nfor some $\\mu$ with $\\mu_{\\alpha_1}=-\\frac12,\\,\\mu_{\\alpha_2}<0$. \nNext we move $\\mu_{3\\alpha_1+2\\alpha_2}$ to $0$ along the \nline $\\lambda_{\\alpha_1}=-\\frac12$. \nSingularities coming from $(-\\lambda_\\beta-\\frac12)^{-1}\n\\,\\,(\\beta=\\alpha_1+\\alpha_2,\\,2\\alpha_1+\\alpha_2)$ are on the walls and \nthey are canceled by $\\varGamma(-\\lambda_\\alpha)^{-1}\\,\\,\n(\\alpha=\\alpha_2,\\,\\alpha_1+\\alpha_2, \\text{respectively})$. Thus \nthe integrand \n$\\Res_{\n\\lambda_{\\alpha_1}=-\\frac12}\\!\\left(\nf^\\wedge (\\lambda)\\Phi_{\\lambda}(H)\nc^{\\pi_2}(-\\lambda)^{-1}\\right)$ is regular for $\\lambda_{3\\alpha_1+2\\alpha_2}\\leq 0$. \nHence we have \n\\[\nF^\\vee_1(H)-F^\\vee_2(H)=\n-\\frac{c_1}{2\\pi\\sqrt{-1}}\\int_{\\sqrt{-1}\\mathbb{R}}\n\\Res_{\n\\lambda_{\\alpha_1}=-\\frac12}\\!\\left(\nf^\\wedge (\\lambda)\\Phi_{\\lambda}(H)\nc^{\\pi_2}(-\\lambda)^{-1}\\right)\\!\nd\\lambda_{3\\alpha_1+2\\alpha_2}.\n\\]\nSimilarly, we have\n\\begin{align*}\n& F^\\vee_2(H)-F^\\vee_3(H)=\n-\\frac{c_1}{2\\pi\\sqrt{-1}}\\int_{\\sqrt{-1}\\mathbb{R}}\n\\Res_{\n\\lambda_{\\alpha_1+\\alpha_2}=-\\frac12}\\!\\left(\nf^\\wedge (\\lambda)\\Phi_{\\lambda}(H)\nc^{\\pi_2}(-\\lambda)^{-1}\\right)\\!\nd\\lambda_{3\\alpha_1+\\alpha_2}, \\\\\n& F^\\vee_3(H)-F^\\vee_4(H)=\n-\\frac{c_1}{2\\pi\\sqrt{-1}}\\int_{\\sqrt{-1}\\mathbb{R}}\n\\Res_{\n\\lambda_{2\\alpha_1+\\alpha_2}=-\\frac12}\\!\\left(\nf^\\wedge (\\lambda)\\Phi_{\\lambda}(H)\nc^{\\pi_2}(-\\lambda)^{-1}\\right)\\!\nd\\lambda_{\\alpha_2}.\n\\end{align*}\nBy summing up and changing variables, we have\n\\begin{align*}\nF_1^\\vee & (H)-F_4^\\vee (H) \\\\ & =-\\frac{c_1}{2\\pi\\sqrt{-1}}\n\\!\\!\n\\sum_{w\\in\\{e,s_1,s_2 s_1\\}}\\!\n\\int_{\\sqrt{-1}\\mathbb{R}}\n\\Res_{\n\\lambda_{2\\alpha_1+\\alpha_2}=-\\frac12}\\!\\left(\nf^\\wedge (\\lambda)\\Phi_{w\\lambda}(H)\nc^{\\pi_2}(-w\\lambda)^{-1}\\right)\\!\nd\\lambda_{\\alpha_2}. \n\\end{align*}\nLet $W^{2\\alpha_1+\\alpha_2}=\\{e,s_1,s_2,s_1s_2,s_2s_1,s_2 s_1 s_2\\}$. \nBy changing variables, we have\n\\begin{align}\nF_1^\\vee & (H) -F_4^\\vee (H) \\label{eqn:invtemp}\n \\\\ & =-\\frac{c_1}{4\\pi\\sqrt{-1}}\\!\\!\\!\n\\sum_{w\\in W^{2\\alpha_1+\\alpha_2}}\n\\int_{\\sqrt{-1}\\mathbb{R}}\n\\Res_{\n\\lambda_{2\\alpha_1+\\alpha_2}=-\\frac12}\\!\\left(\nf^\\wedge (\\lambda)\\Phi_{w\\lambda}(H)\nc^{\\pi_2}(-w\\lambda)^{-1}\\right)\\!\nd\\lambda_{\\alpha_2} . \\notag\n\\end{align}\n\nSince $W^{2\\alpha_1+\\alpha_2}=\\{w\\in W\\,|\\,w(2\\alpha_1+\\alpha_2)\\in\\varSigma^+\\}$, $c^{\\pi_2}(w\\lambda)|_\n{\\lambda_{2\\alpha_1+\\alpha_2}=-\\frac12}=0$ for \nany $w\\in W\\setminus W^{2\\alpha_1+\\alpha_2}$ by Theorem~\\ref{thm:cfg2}. \nNotice that the Harish-Chandra expansion \\eqref{eqn:hcs2} is valid for \n$\\lambda_{2\\alpha_1+\\alpha_2}=-\\frac12,\\,\n\\lambda_{\\alpha_2}\\in\\sqrt{-1}\\mathbb{R}\\setminus\\{0\\}$. \nHence \n\\begin{equation}\\label{eqn:hcstemp}\n\\varUpsilon^{\\pi_2}(\\phi_\\lambda^{\\pi_2})|_{\\lambda_{2\\alpha_1+\\alpha_2}=-\\frac12}\n=\\sum_{w\\in W^{2\\alpha_1+\\alpha_2}}c^{\\pi_2}(w\\lambda)\\Phi_{w\\lambda}|_{\\lambda_{2\\alpha_1+\\alpha_2}=-\\frac12}\n\\end{equation}\nfor $\\lambda_{\\alpha_2}\\in\\sqrt{-1}\\mathbb{R}\\setminus\\{0\\}$. \n\nWe write the $c$-function $c^{\\pi_2}(\\lambda)$ in Theorem~\\ref{thm:cfg2} as \n\\[\nc^{\\pi_2}(\\lambda)=\\frac{16}{\\pi}c_l(\\lambda)c_s(\\lambda)\n\\]\nwith\n\\[\nc_l(\\lambda)=\n\\prod_{\\alpha\\in\\varSigma_\\text{\\rm long}^+} \n\\frac{\\varGamma(\\lambda_\\alpha)}\n{\\varGamma(\\lambda_\\alpha+\\frac12)}, \n\\qquad \nc_s(\\lambda)=\n\\prod_{\\beta\\in\\varSigma_\\text{\\rm short}^+} \n \\frac{\\left(\\lambda_\\beta-\\frac12\\right)\\varGamma(\\lambda_\\beta)}\n{\\varGamma(\\lambda_\\beta+\\frac32)} .\n\\] \nNotice that the functions \n$(c_l(\\lambda)c_l(-\\lambda))^{-1}$ and $(c_s(\\lambda)c_s(-\\lambda))^{-1}$ \nare $W$-invariant.\n\\begin{lem}\nWe have\n\\begin{equation}\\label{eqn:restemp0}\n(c_l(\\lambda)c_l(-\\lambda))^{-1}|_\n{\\lambda_{2\\alpha_1+\\alpha_2}=-\\frac12}\n=\\frac{(4\\lambda_{\\alpha_2}^3-\\lambda_{\\alpha_2})\\sin\\pi\\lambda_{\\alpha_2}}\n{16\\cos\\pi\\lambda_{\\alpha_2}}\n\\end{equation}\nand \n\\begin{equation}\\label{eqn:restemp}\nc_s(w\\lambda)^{-1}|_\n{\\lambda_{2\\alpha_1+\\alpha_2}=-\\frac12}\n\\Res_{\n\\lambda_{2\\alpha_1+\\alpha_2}=-\\frac12}(c_s(-w\\lambda)^{-1})\n=\\frac{36\\lambda_{\\alpha_2}^2-1}{32\\pi}\n\\end{equation}\nfor any $w\\in W^{2\\alpha_1+\\alpha_2}$.\n\\end{lem}\n\\begin{proof}\nWe show only \\eqref{eqn:restemp} because \\eqref{eqn:restemp0} can be deduced in a similar way.\nSince the left hand side of \\eqref{eqn:restemp} is the residue of \nthe $(c_s(\\lambda)c_s(-\\lambda))^{-1}$ \nas a function of $\\lambda_{2\\alpha_1+\\alpha_2}$ \nat $\\lambda_{2\\alpha_1+\\alpha_2}=-\\frac12$,\nit suffices to show \\eqref{eqn:restemp} for $w=1$.\nBy elementary calculation we have\n\\[\n \\frac{\\left(z-\\frac12\\right)\\varGamma(z)}\n{\\varGamma(z+\\frac32)}\n\\cdot\n \\frac{\\left(-z-\\frac12\\right)\\varGamma(-z)}\n{\\varGamma(-z+\\frac32)}\n=-\\frac{\\cos\\pi z}{z\\sin\\pi z}\\quad(z\\in{\\mathbb C}).\n\\]\nUsing\n$\\lambda_{\\alpha_1}=\\frac12\\lambda_{2\\alpha_1+\\alpha_2}-\\frac32\\lambda_{\\alpha_2}$\nand\n$\\lambda_{\\alpha_1+\\alpha_2}=\\frac12\\lambda_{2\\alpha_1+\\alpha_2}+\\frac32\\lambda_{\\alpha_2}$\nwe calculate\n\\begin{align*}\n\\Res_{\n\\lambda_{2\\alpha_1+\\alpha_2}=-\\frac12}&((c_s(\\lambda)c_s(-\\lambda))^{-1})\\\\\n&=\\biggl.\\biggl(-\\frac{z\\sin\\pi z}{\\cos\\pi z}\\biggr)\\biggr|_{z=-\\frac14-\\frac32\\lambda_{\\alpha_2}}\n\\biggl.\\biggl(-\\frac{z\\sin\\pi z}{\\cos\\pi z}\\biggr)\\biggr|_{z=-\\frac14+\\frac32\\lambda_{\\alpha_2}}\n\\Res_{z=-\\frac12}\\biggl(-\\frac{z\\sin\\pi z}{\\cos\\pi z}\\biggr)\\\\\n&=\n\\frac{1-36\\lambda_{\\alpha_2}^2}{16}\n\\cdot\\frac{\\sin\\pi\\Bigl(-\\frac14-\\frac32\\lambda_{\\alpha_2}\\Bigr)\n\\sin\\pi\\Bigl(-\\frac14+\\frac32\\lambda_{\\alpha_2}\\Bigr)}%\n{\\cos\\pi\\Bigl(-\\frac14-\\frac32\\lambda_{\\alpha_2}\\Bigr)\n\\cos\\pi\\Bigl(-\\frac14+\\frac32\\lambda_{\\alpha_2}\\Bigr)}\n\\cdot\\biggl(-\\frac1{2\\pi}\\biggr).\n\\end{align*}\nIn the final expression the second factor reduces to $1$.\n\\end{proof}\n\nThus we have the following inversion formula for $\\pi_2$-spherical transform. \n\\begin{thm}\\label{thm:main}\nFor $f\\in C_c^\\infty(\\mathfrak{a})^W$, we have\n\\begin{align*}\nf(H) = \\frac{1}{12} & \\int_{\\sqrt{-1}\\mathfrak{a}^*}\n f^\\wedge(\\lambda)\\varUpsilon^{\\pi_2}\n (\\phi^{\\pi_2}_\\lambda)(H)|c^{\\pi_2}(\\lambda)|^{-2}d\\lambda \\\\\n &\n -\\frac{c_1}{4\\pi\\sqrt{-1}}\\int_{\\sqrt{-1}\\mathbb{R}}\n(f^\\wedge(\\lambda)\\varUpsilon^{\\pi_2}(\\phi^{\\pi_2}_\\lambda(H)))\n|_{\\lambda_{2\\alpha_1+\\alpha_2}=-\\frac12}\\,\np(\\lambda_{\\alpha_2})d\\lambda_{\\alpha_2}\n\\end{align*}\nfor $H\\in \\mathfrak{a}$, \nwhere \n\\begin{align*}\np(\\lambda_{\\alpha_2}) & =\n\\Res_{\n\\lambda_{2\\alpha_1+\\alpha_2}=-\\frac12}\n(c^{\\pi_2}(\\lambda)^{-1}c^{\\pi_2}(-\\lambda)^{-1})\n=\\frac{\\pi(36\\lambda_{\\alpha_2}^2-1)\n(4\\lambda_{\\alpha_2}^3-\\lambda_{\\alpha_2})\\sin\\pi\\lambda_{\\alpha_2}}\n {2^{17}\\cos\\pi\\lambda_{\\alpha_2}}\\\\\n& =-2^{-17}\\pi\\Bigl(36\\left(\\tfrac{\\lambda_{\\alpha_2}}{\\sqrt{-1}}\\right)^2+1\\Bigr)\n\\Bigl(4\\left(\\tfrac{\\lambda_{\\alpha_2}}{\\sqrt{-1}}\\right)^2+1\\Bigr)\n\\left(\\tfrac{\\lambda_{\\alpha_2}}{\\sqrt{-1}}\\right)\n\\tanh \\pi\\left(\\tfrac{\\lambda_{\\alpha_2}}{\\sqrt{-1}}\\right)\n.\n\\end{align*}\n\\end{thm}\n\nThe Plancherel formula follows from Theorem~\\ref{thm:main} by a \nstandard argument as in the case \nof $\\pi=\\text{triv}$ (cf. the proof of \\cite[Theorem~6.4.2]{GV} and \n\\cite[Ch~IV Theorem~7.5]{Hel2}). \n\n\\begin{cor}\\label{cor:main}\nFor $f\\in C_c^\\infty(\\mathfrak{a})^W$, we have\n\\begin{align*}\n\\frac{1}{12}\n\\int_\\mathfrak{a}|f(H)|^2\\delta_{G\/K} & (H)dH = \n\\frac{1}{12} \\int_{\\sqrt{-1}\\mathfrak{a}^*}\n |f^\\wedge(\\lambda)|^2|c^{\\pi_2}(\\lambda)|^{-2}d\\lambda \\\\\n - & \\frac{c_1}{4\\pi\\sqrt{-1}}\\int_{\\sqrt{-1}\\mathbb{R}}\n|f^\\wedge(\\lambda)|_{\\lambda_{2\\alpha_1+\\alpha_2}=-\\frac12}|^2\\,\np(\\lambda_{\\alpha_2})d\\lambda_{\\alpha_2}\n\\end{align*}\n\\end{cor}\n\nAs we see in \\S~\\ref{subsec:g2prem}, no discrete spectrum appears in \nthe inversion formula and the Plancherel formula. \nIn addition to the most continuous spectrum, there \nis a contribution of a principal series representation associated with a \nmaximal parabolic subgroup whose Levi part corresponds to a short restricted \nroot. \n\n\\section*{Acknowledgement}\nThe first author was supported by JSPS KAKENHI Grant Number 18K03346. \nThe authors \n thank anonymous reviewers for careful reading our manuscript and for giving useful comments.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\n \nOnline learning is a general model for sequential prediction. Within\nthat framework, the setting of prediction with expert advice has\nreceived widespread attention \\citep{LittlestoneWarmuth1994,\n CesaBianchiLugosi2006,CesaBianchiMansourStoltz2007}. In this\nsetting, the algorithm maintains a distribution over a set of experts,\nor selects an expert from an implicitly maintained distribution. At\neach round, the loss assigned to each expert is revealed. The\nalgorithm incurs the expected loss over the experts and then updates\nits distribution on the set of experts. Its objective is to minimize\nits expected regret, that is the difference between its cumulative\nloss and that of the best expert in hindsight.\n\nHowever, this benchmark is only significant when the best expert in\nhindsight is expected to perform well. When that is not the case, then\nthe learner may still play poorly. As an example, it may be that no\nsingle baseball team has performed well over all seasons in the past\nfew years. Instead, different teams may have dominated over different\ntime periods. This has led to a definition of regret against the best\nsequence of experts with $k$ shifts in the seminal work of\n\\cite{HerbsterWarmuth1998} on \\emph{tracking the best expert}. The\nauthors showed that there exists an efficient online learning\nalgorithm for this setting with favorable regret guarantees.\n\nThis work has subsequently been improved to account for broader expert\nclasses \\citep{GyorgyLinderLugosi2012}, to deal with unknown\nparameters \\citep{MonteleoniJaakkola2003}, and has been further\ngeneralized \\citep{CesaBianchiGaillardLugosiStoltz2012, Vovk1999}.\nAnother approach for handling dynamic environments has consisted of\ndesigning algorithms that guarantee small regret over any subinterval\nduring the course of play. This notion, coined as \\emph{adaptive\n regret} by \\cite{HazanSeshadhri2009}, has been subsequently\nstrengthened and generalized \\citep{DanielyGonenShalevShwartz2015,\n AdamskiyKoolenChernovVovk2012}. Remarkably, it was shown by\n\\cite{AdamskiyKoolenChernovVovk2012} that the algorithm designed by\n\\cite{HerbsterWarmuth1998} is also optimal for adaptive regret.\n\\cite{KoolenDeRooij2013} described a Bayesian framework for online\nlearning where the learner samples from a distribution of expert\nsequences and predicts according to the prediction of that expert\nsequence. They showed how the algorithms designed for $k$-shifting\nregret, e.g.\\ \\citep{HerbsterWarmuth1998, MonteleoniJaakkola2003}, can\nbe interpreted as specific priors in this formulation. There has also\nbeen work deriving guarantees in the bandit setting when the losses\nare stochastic \\citep{BesbesGurZeevi2014, WeiHongLu2016}.\n\n\\begin{figure*}[t]\n\\vskip -.15in\n\\centering\n\\begin{tabular}{ccc}\n\\includegraphics[scale=0.4]{kshift} &\n {\\includegraphics[scale=0.4]{wshift}} &\n \\includegraphics[scale=0.4]{hierarchy} \\\\ \n (i) &\n (ii) &\n (iii) \\\\ \n\\end{tabular}\n\\caption{WFAs representing sequences of experts in\n $\\Sigma = \\set{a, b, c}$. (i) $\\sC_\\text{$k$-shift}$ with\n $k = 2$ shifts, all weights are equal to one and not indicated; (ii)\n $\\sC_\\text{weighted-shift}$ with $\\alpha, \\beta, \\gamma \\in [0, 1]$;\n (iii) $\\sC_\\text{hierarchy}$ a hierarchical family of expert \n sequences: the learner must select expert $a$ from the start, can only shift \n onto $b$ once, and can only shift onto $c$ twice. }\n\\label{fig:kshift}\n\\vskip -.15in\n\\end{figure*}\n\nThe general problem of online convex optimization in the presence of\nnon-stationary environments has also been studied by many\nresearchers. One perspective has been the design of algorithms that\nmaintain a guarantee against sequences that do not vary too much\n\\citep{MokhtariShahrampourJadbabaieRibeiro2016,\n ShahrampourJadbabaie2016, JadbabaieRakhlinShahrampourSridharan2015,\n BesbesGurZeevi2015}. Another assumes that the learner has access to\na dynamical model that is able to capture the benchmark sequence\n\\citep{HallWillett2013}. \\cite{GyorgySzepesvari2016} reinterpreted the\nframework of \\cite{HallWillett2013} to recover and extend the results\nof \\cite{HerbsterWarmuth1998}.\n\nIn this paper, we generalize the framework just described to the case\nwhere the learner's cumulative loss is compared to that of sequences\naccepted by a weighted finite automaton (WFA). This strictly\ngeneralizes the notion of $k$-shifting regret, since $k$-shifting\nsequences can be represented by an automaton (see\nFigure~\\ref{fig:kshift}), and further extends it to a notion of\n\\emph{weighted regret} which takes into consideration the sequence\nweights. Our framework covers a very rich class of competitor classes,\nincluding WFAs learned from past observations. \n\nOur contributions are mainly algorithmic but also include\nseveral theoretical results and guarantees. We first describe an\nefficient online algorithm using automata operations that achieves\nboth favorable weighted regret and unweighted regret\n(Section~\\ref{sec:AWM}). Next, we present and analyze more efficient\nsolutions based on an approximation of the WFA representing the set of competitor\nsequences (Section~\\ref{sec:approx}), including a specific analysis of\napproximations via $n$-gram models both when minimizing the\n$\\infty$-R\\'enyi divergence and the relative entropy. Finally, we\nextend the results above to the sleeping expert setting\n\\citep{FreundSchapireSingerWarmuth1997}, where the learner may not\nhave access to advice from every expert at each round\n(Section~\\ref{sec:sleep}).\n\n\\ignore{\nIn this paper, we significantly generalize the framework just\ndescribed and consider prediction with expert advice in a setting\nwhere the learner's cumulative loss is compared against that of\nsequences represented by an \\emph{arbitrary weighted family of\n sequences}. We model this family using a weighted finite automaton\n(WFA). This strictly generalizes the notion of $k$-shifting regret and\nextends it to the notion of regret against a WFA.\n\nMeasuring regret against an automaton is both natural and flexible. In\nfact, it may often be sensible to \\emph{learn} the set of competitor\nsequences using data before competing against it. For instance, the\ncompetitor automaton could be a language model trained over best\nsequences of baseball teams in the past. Moreover, the competitor\nautomaton could be learned and reset incrementally. After each epoch,\nwe could choose to learn a new competitor model and seek to perform\nwell against that.\n\nWe show that not only it is possible to achieve favorable regret\nagainst a WFA but that there exist \\emph{computationally efficient}\nalgorithms to achieve that. We give a series of algorithms for this\nproblem. Our first algorithm (Section~\\ref{sec:autalg}) is an\nautomata-based algorithm extending weighted-majority and using\nautomata operations such as composition and shortest-distance; its\ncomputational cost is exponentially better than that of a na\\\"{i}ve\nmethod.\n\nWe further present efficient algorithms based on a compact\napproximation of the competitor automaton\n(Section~\\ref{sec:min-Renyi}), in particular efficient $n$-gram models\nobtained by minimizing the R\\'enyi divergence, and present an\nextensive study of the approximation properties of such models. We\nalso show how existing algorithms for minimizing $k$-shifting regret\ncan be recovered by learning a Maximum-Likelihood bigram language\nmodel over the $k$-shifting competitor automaton. To the best of our\nknowledge, this is the first instance of recovering the algorithms of\n\\cite{HerbsterWarmuth1998} by way of solely focusing on minimizing the\n$k$-shifting regret. Since approximating the competitor automaton is\nsubject to a trade-off between computational efficiency and\napproximation accuracy, we also design a model selection algorithm\nadapted to this problem.\n\nWe further improve that algorithm by using the notion of failure\ntransitions ($\\phi$-transitions) for a more compact and therefore more\nefficient automata representation. Here, we design a new algorithm\n(Appendix~\\ref{app:phiaut}) that can convert any weighted finite\nautomaton into a weighted finite automaton with $\\phi$-transitions\n($\\phi$-WFA). We then extend the classical composition and\nshortest-distance algorithms for WFAs to the setting of\n$\\phi$-WFAs. The shortest-distance algorithm is designed by extending\nthe probability semiring structure of the $\\phi$-WFA to that of a\nring. We show that if the number of consecutive $\\phi$-transitions is\nnot too large, then these algorithms have a computational complexity\nthat is comparable to those for standard WFAs. At the same time, our\nconversion algorithm can dramatically reduce the size of a WFA.\n\nFinally, we extend the results above to the sleeping expert setting\n\\citep{FreundSchapireSingerWarmuth1997}, where the learner may not\nhave access to advice from every expert at each round\n(Section~\\ref{sec:sleep}). \\ignore{Finally, we extend the ideas for\nprediction with expert advice to online convex optimization and a\ngeneral mirror descent setting (Section~\\ref{sec:oco}). Here, we\ndescribe a related framework that parallels the previous discussion\nand also recovers existing algorithms for $k$-shifting regret.}\n}\n\n\\section{Learning setup}\n\\label{sec:setup}\n\nWe consider the setting of prediction with expert advice over\n$T \\in \\Nset$ rounds. Let $\\Sigma = \\set{a_1, \\ldots, a_N}$ denote a set\nof $N$ experts. At each round $t \\in [T]$, an algorithm $\\cA$ specifies\na probability distribution $\\sfp_t$ over $\\Sigma$, samples an expert\n$i_t$ from $\\sfp_t$, receives the vector of losses of all experts\n$\\bfl_t \\in [0, 1]^N$, and incurs the specific loss $l_t[i_t]$. A\ncommonly adopted goal for the algorithm is to minimize its static\n(expected) regret $\\Reg_T(\\cA, \\Sigma)$, that is the difference\nbetween its cumulative expected loss and that of the best expert in\nhindsight:\n\\begin{align}\n\\label{eq:staticregret}\n\\Reg_T(\\cA, \\Sigma) = \n\\max_{x \\in \\Sigma} \\sum_{t = 1}^T \\sfp_t \\cdot \\bfl_t - \\sum_{t = 1}^T l_t[x].\n\\end{align}\nHere, we will consider an alternative benchmark, typically more\ndemanding, where the cumulative loss of the algorithm is compared\nagainst the loss of the best sequence of experts\n$\\bx \\in \\Sigma^T$ among those accepted by a weighted\nfinite automaton (WFA) $\\sC$ over the semiring\n$(\\Rset_+ \\cup \\set{+\\infty}, +, \\times, 0, 1)$.\\footnote{Thus, the\n weights in $\\sC$ are non-negative; the weight of a path is\n obtained by multiplying the transition weights along that path and\n the weight assigned to a sequence is obtained by summing the weights\n of all accepting paths labeled with that sequence.} The sequences\n$\\bx$ accepted by $\\sC$ are those which are assigned a positive\nvalue by $\\sC$, $\\sC(\\bx) > 0$, which we will assume to be\nnon-empty. We will denote by $K \\geq 1$ the cardinality of that set.\n\nWe will take into account the probability distribution $\\sfq$ defined\nby the weights assigned by $\\sC$ to sequences of length $T$:\n$\\sfq(\\bx) = \\frac{\\sC(\\bx)}{\\sum_{\\bx \\in \\Sigma^T}\n \\sC(\\bx)}$. This leads to the following definition of\n\\emph{weighted regret} at time $T$ given a WFA $\\sC$:\n\\begin{align}\n\\label{eq:weightedregret}\n& \\Reg_T(\\cA, \\sC) \\\\\\nonumber\n& = \\max_{\\substack{\\bx \\in \\Sigma^T\\\\ \\sC(\\bx) > 0}} \\set[\\Bigg]{ \\sum_{t = 1}^T \\sfp_t\n\\cdot \\bfl_t - \\sum_{t = 1}^T l_t[\\bx[t]] \n + \\log [\\sfq(\\bx) K ] },\n\\end{align}\nwhere $\\bx[t]$ denotes the $t$th symbol of $\\bx$. The presence of the\nfactor $K$ only affects the regret definition by a constant additive\nterm $\\log K$ and is only intended to make the last term\nvanish when the probability distribution $\\sfq$ is uniform, i.e.\n$\\sfq(\\bx) = \\frac{1}{K}$ for all $\\bx$. The last term in the\nweighted regret definition can be interpreted as follows: for a given\nvalue of an expert sequence loss $\\sum_{t = 1}^T l_t[\\bx[t]]$, the\nregret is larger for sequences $\\bx$ with a larger probability\n$\\sfq(\\bx)$. Thus, with this definition of regret, the learning\nalgorithm is pressed to achieve a small cumulative loss compared \nto expert sequences with small loss and high\nprobability. Notice that when $\\sC$ accepts only constant sequences,\nthat is sequences $\\bx$ with $\\bx[1] = \\ldots = \\bx[T]$ and assigns\nthe same weight to them, then the notion of weighted regret coincides\nwith that of static regret (Formula~\\ref{eq:staticregret}).\n\nWe also define the \\emph{unweighted regret} $\\Reg_T^0(\\cA, \\sC)$ of\nalgorithm $\\cA$ at time $T$ given the WFA $\\sC$ as:\n\\begin{align}\n\\label{eq:unweightedregret}\n \\Reg_T^0(\\cA, \\sC) \n= \\max_{\\substack{\\bx \\in \\Sigma^T\\\\ \\sC(\\bx) > 0}} \\set[\\Bigg]{\n \\sum_{t = 1}^T \\sfp_t \\cdot \\bfl_t - \\sum_{t = 1}^T l_t[\\bx[t]] }.\n\\end{align}\nThe weights of the WFA $\\sC$ play no role in this notion of regret.\nWhen $\\sC$ has uniform weights, then the unweighted regret and\nweighted regret coincide.\n\nAs an example, the sequences of experts with $k$ shifts studied by\n\\cite{HerbsterWarmuth1998} can be represented by the WFA\n$\\sC_\\text{$k$-shift}$ of\nFigure~\\ref{fig:kshift}(i). Figure~\\ref{fig:kshift}(ii) shows an\nalternative weighted model of shifting experts, and\nFigure~\\ref{fig:kshift}(iii) shows a hierarchical family of expert\nsequences.\n\n\\section{Automata Weighted-Majority algorithm}\n\\label{sec:AWM}\n\nIn this section, we describe a simple algorithm, \\emph{Automata\n Weighted-Majority} (\\AWM), that can be viewed as an enhancement of\nthe weighted-majority algorithm \\citep{LittlestoneWarmuth1994} to the\nsetting of experts paths represented by a WFA. \\footnote{This\n algorithm is in fact closer to the EXP4 algorithm\n \\citep{AuerCesaBianchiFreundSchapire2002}. However, EXP4 is\n designed for the bandit setting, so we use the weighted-majority\n naming convention.} We will show that it benefits from favorable\nweighted and unweighted regret guarantees.\n\nAs with standard weighted-majority, \\AWM\\ maintains a distribution\n$\\sfq_t$ over the set of expert sequences $\\bx \\in \\Sigma^T$ accepted\nby $\\sC$ at any time $t$ and admits a learning parameter $\\eta >\n0$. The initial distribution $\\sfq_1$ is defined in terms of the\ndistribution $\\sfq$ induced by $\\sC$ over $\\Sigma^T$, and\n$\\sfq_{t + 1}$ is defined from $\\sfq_t$ via an exponential update:\nfor all $\\bx \\in \\Sigma^T, t \\geq 1$,\n\\begin{align}\n & \\sfq_1[\\bx] = \\frac{\\sfq[\\bx]^\\eta}{\\sum_{\\bx \\in \\Sigma^T}\n \\sfq[\\bx]^\\eta}, \\nonumber \\\\\n & \\sfq_{t + 1}[\\bx] = \\frac{e^{-\\eta \\, l_t[\\bx[t]]}\\sfq_t[\\bx]}{\\sum_{\\bx\n \\in \\Sigma^T} e^{-\\eta \\, l_t[\\bx[t]]}\\sfq_t[\\bx]},\n\\end{align}\nwhere we denote by $\\bx[t] \\in \\Sigma$ the $t$th symbol in $\\bx$.\n$\\sfq_t$ induces a distribution $\\sfp_t$ over the expert set $\\Sigma$\ndefined for all $a \\in \\Sigma$ by\n\\begin{equation}\n\\sfp_t[a] = \\frac{\\sum_{\\bx \\in \\Sigma^T} \\sfq_t[\\bx] 1_{\\bx[t] =\n a}}{\\sum_{a \\in \\Sigma} \\sum_{\\bx \\in \\Sigma^T} \\sfq_t[\\bx] 1_{\\bx[t] = a}}.\n\\end{equation}\nThus, $\\sfp_t[a]$ is obtained by summing up the $\\sfq_t$-weights of\nall sequences with the $t$th symbol equal to $a$ and normalization.\nThe distributions $\\sfp_t$ define the \\AWM\\ algorithm. Note that the\nalgorithm cannot be viewed as weighted-majority with $\\sfq$-priors on\nexpert sequences as $\\sfq_1$ is defined in terms of $\\sfq^\\eta$.\n\nThe following regret guarantees hold for \\AWM.\n\n\\begin{theorem}\n\\label{th:awm}\nLet $\\sfq$ denote the probability distribution over expert sequences\nof length $T$ defined by $\\sC$ and let $K$ denote the cardinality of\nits support. Then, the following upper bound holds for the weighted\nregret of \\AWM:\n\\begin{align*}\n \\Reg_T(\\AWM, \\sC) \n & \\leq \\frac{\\eta T}{8} + \\frac{1}{\\eta} \\log \\bigg[ K^\\eta \\sum_{\\bx\n \\in \\Sigma^T} \\sfq[\\bx]^\\eta \\bigg] \\\\\n &\\leq \\frac{\\eta T}{8} + \\frac{1}{\\eta} \\log K. \n\\end{align*}\nFurthermore, when $K \\geq 2$, for any $T > 0$, there exists\n$\\eta^* > 0$, decreasing as a function of $T$, such that:\n\\begin{align*}\n \\Reg_T(\\AWM, \\sC) \n & \\leq \\sqrt{\\frac{T H_{\\eta^*}(\\sfq)}{2}} - H_{\\eta^*}(\\sfq) + \\log\n K,\n\\end{align*}\nwhere\n$H_\\eta(\\sfq) = \\frac{1}{1 - \\eta} \\log\\left(\\sum_{\\bx \\in \\Sigma^T}\n \\sfq[\\bx]^\\eta \\right)$ is the $\\eta$-R\\'{e}nyi entropy of $\\sfq$. The\nunweighted regret of \\AWM\\ can be upper-bounded as follows:\n\\begin{align*}\n \\Reg_T^0(\\AWM, \\sC) \n & \\leq \\frac{\\eta T}{8} + \\frac{1}{\\eta} \\log K.\n\\end{align*}\n\n\\end{theorem}\n\nThe proof is an extension of the standard proof for the\nweighted-majority algorithm and is given in Appendix~\\ref{app:awm}.\nThe bound in terms of the R\\'enyi entropy shows that the regret\nguarantee can be substantially more favorable than standard bounds of\nthe form $O(\\sqrt{T \\log K})$, depending on the properties of the\ndistribution $\\sfq$. First, since the $\\eta$-R\\'enyi entropy is\nnon-increasing in $\\eta$ \\citep{VanErvenHarremos2014}, we have\n$H_{\\eta^*}(\\sfq) \\leq H_0(\\sfq) = \\log(|\\supp(\\sfq)|) \\leq \\log\nK$. Second, if the distribution $\\sfq$ is concentrated on a subset\n$\\Delta$ with a small cardinality, $|\\Delta| \\ll K$, that is\n$\\sum_{\\bx \\not \\in \\Delta} \\sfq[\\bx]^{\\eta^*} < \\e (1-\\eta^*)\n\\sum_{\\bx \\in \\Delta} \\sfq[\\bx]^{\\eta^*}$ for some $\\e > 0$ and for\n$\\eta^* < 1$, then, by Jensen's inequality, the following upper bound\nholds:\n\\begin{align*}\n H_\\eta^*(\\sfq)\n\\ignore{\n & \\leq \\frac{1}{1 - \\eta^*} \\log \\bigg( \\sum_{\\bx \\in \\Delta}\n \\sfq[\\bx]^{\\eta^*\\!} + \\e (1 - \\eta^*) \\sum_{\\bx \\in \\Delta}\n \\sfq[\\bx] \\bigg) \\\\\n}\n & \\leq \\frac{1}{1 - \\eta^*} \\log \\bigg(\\sum_{\\bx \\in \\Delta}\n \\sfq[\\bx]^{\\eta^*\\!}\\bigg) + \\e \\\\ \n &\\leq \\frac{1}{1 - \\eta^*} \\log \\bigg(|\\Delta| \\bigg(\\frac{1}{|\\Delta|}\\sum_{\\bx \\in \\Delta}\n \\sfq[\\bx]\\bigg)^{\\eta^*\\!}\\bigg) + \\e \\\\ \n \n &\\leq \\log(|\\Delta|) + \\e. \n\\end{align*}\n\n{\\bf Efficient algorithm}. We now present an efficient computation\nof the distributions $\\sfp_t$. Algorithm~\\ref{alg:awm} gives the\npseudocode of our algorithm. We will assume throughout that $\\sC$ is\ndeterministic, that is it admits a single initial state and no two\ntransitions leaving the same state share the same label. We can\nefficiently compute a WFA accepting the set of sequences of length $T$\naccepted by $\\sC$ by using the standard intersection algorithm for\nWFAs (see Appendix~\\ref{app:intersection} for more detail on this\nalgorithm). Let $\\sS_T$ be a deterministic WFA accepting the\nset of sequences of length $T$ and assigning weight one to each (see\nFigure~\\ref{fig:S_T}). Then, the intersection of $\\sC$ and $\\sS_T$ is\na WFA denoted by $\\sC \\cap \\sS_T$ which, by definition, assigns to\neach sequence $\\bx \\in \\Sigma^T$ the weight\n\\begin{equation}\n (\\sC \\cap \\sS_T) (\\bx) = \\sC(\\bx) \\times \\sS_T(\\bx) = \\sC(\\bx),\n\\end{equation}\nand assigns weight zero to all other sequences. Furthermore, the WFA\n$\\sB = (\\sC \\cap \\sS_T)$ returned by the intersection algorithm is\ndeterministic since both $\\sC$ and $\\sS_T$ are deterministic. Next,\nwe replace each transition weight of $\\sB$ by its $\\eta$-power. Since\n$\\sB$ is deterministic, this results in a WFA that we denote by\n$\\sB^\\eta$ and that associates to each sequence $\\bx$ the weight\n$\\sC[\\bx]^\\eta$. Normalizing $\\sB^\\eta$ results in a WFA $\\sA$\nassigning weight\n$\\sA[\\bx] = \\frac{\\sB[\\bx]^\\eta}{\\sum_x \\sB[\\bx]^\\eta} = \\sfq_1[x]$ to\nany $\\bx \\in \\Sigma^T$. This normalization can be achieved in time\nthat is linear in the size of the WFA $\\sB^\\eta$ using the\n\\textsc{Weight-Pushing} algorithm \\citep{Mohri1997bis,Mohri2009}. For\ncompleteness, we describe this algorithm in\nAppendix~\\ref{app:weightpush}. Note that since $\\sB^\\eta$ is acyclic,\nits size is in $\\cO(|E_\\sA|)$.\\footnote{The \\textsc{Weight-Pushing}\n algorithm has been used in many other contexts to make a directed\n weighted graph stochastic. This includes network normalization in\n speech recognition \\citep{MohriRiley2001}, and online learning with\n large expert sets\n \\citep{TakimotoWarmuth2003,CortesKuznetsovMohriWarmuth2015}, where\n the resulting stochastic graph enables efficient sampling. The\n problem setting, algorithms and objectives in the last two\n references are completely distinct from ours. In particular, (a) in\n those, each path of the graph represents a single expert, while in\n our case each path is a sequence of experts; (b) in those,\n weight-pushing is applied at every round, while in our case it is\n used once at the start of the algorithm; (c) the regret is with\n respect to a static expert, while in our case it is with respect to\n a WFA of expert sequences.} We will denote by $\\sA$ the resulting\nWFA.\n\nFor any state $u$ of $\\sA$, we will denote by $\\bbeta[u]$ the sum of\nthe weights of all paths from $u$ to a final state. The vector\n$\\bbeta$ can be computed in time that is linear in the number of states and\ntransitions of $\\sA$ using a simple \\emph{single-source\n shortest-distance} algorithm in the semiring\n$(\\Rset_+ \\cup \\set{+\\infty}, +, \\times, 0, 1)$ \\citep{Mohri2009}, or\nthe forward-backward algorithm. We call this subroutine \\textsc{BwdDist} in \nthe pseudocode.\n\n\\begin{algorithm2e}[t]\n \\TitleOfAlgo{\\AWM($\\sC$, $\\eta$)}\n $\\sB \\gets \\sC \\cap \\sS_T$ \\\\\n $\\sA \\gets \\textsc{Weight-Pushing}(\\sB^\\eta)$ \\\\\n $\\bbeta \\gets \\textsc{BwdDist}(\\sA)$ \\\\\n $\\balpha \\gets 0$; $\\balpha[I_\\sA] \\gets 1$ \\\\ \n \\ForEach{$e \\in E_{\\sA}^{0 \\to 1}$}{\n $\\sfp_1[\\lab[e]] \\gets \\weight[e]$.\n }\n \\For{$t \\gets 1$ \\KwTo $T$}{\n $i_t \\gets $\\textsc{Sample}($\\sfp_t$); \\textsc{Play}($i_t$); \\textsc{Receive}($\\bfl_t$)\\\\\n $Z \\gets 0$; $\\bw \\gets 0$\\\\ \n \\ForEach{$e \\in E_{\\sA}^{t \\to t + 1}$}{\n $\\weight[e] \\gets \\weight[e] \\, e^{-\\eta l_t[\\lab[e]]}$ \\\\\n $\\bw[\\lab[e]] \\gets \\bw[\\lab[e]] + \\balpha[\\src[e]] \\, \\weight[e] \\, \\bbeta[\\dest[e]]$\\\\\n $Z \\gets Z + \\bw[\\lab[e]]$\\\\\n $\\balpha[\\dest[e]] \\gets \\balpha[\\dest[e]] + \\balpha[\\src[e]] \\,\n \\weight[e] $\n }\n $\\sfp_{t + 1} \\gets \\frac{\\bw}{Z}$ \\\\ \n }\n\\caption{\\textsc{AutomataWeightedMajority}(AWM).} \n\\label{alg:awm}\n\\end{algorithm2e}\n\n\\begin{figure}[t]\n\\vskip -.15in\n\\centering\n\\includegraphics[scale=0.55]{S_T}\n\\caption{WFA $\\sS_T$, for $\\Sigma = \\set{a, b, c}$ and $T = 3$.}\n\\label{fig:S_T}\n\\vskip .1in\n\\centering\n\\includegraphics[scale=0.65]{awm}\n\\caption{Illustration of algorithm \\AWM.}\n\\label{fig:awm}\n\\vskip -.15in\n\\end{figure}\n \nWe will denote by $Q_t$ the set of states in $\\sA$ that can be reached\nby sequences of length $t$ and by $E_{\\sA}^{t \\to t + 1}$ the set of\ntransitions from a state in $Q_t$ to a state in $Q_{t + 1}$. For each\ntransition $e$, let $\\src[e]$ denote its source state, $\\dest[e]$ its\ndestination state, $\\lab[e] \\in \\Sigma$ its label, and\n$\\weight[e] \\geq 0$ its weight. Since $\\sA$ is normalized, the expert\nprobabilities $\\sfp_1[a]$ for $a \\in \\Sigma$ can be read off the\ntransitions leaving the initial state: $\\sfp_1[a]$ is the weight of\nthe transition in $E_{\\sA}^{0 \\to 1}$ labeled with $a$.\n\nLet $\\balpha_t[u]$ denote the \\emph{forward weights}, that is the sum\nof the weights of all paths from the initial state to state $u$ just\nbefore the $t$th round. At round $t$, the weight of each transition\n$e$ in $E_{\\sA}^{t \\to t + 1}$ is multiplied by\n$e^{-\\eta l_t[\\lab[e]]}$. This results in new forward weights\n$\\balpha_{t + 1}[u]$ at the end of the $t$-th iteration. $\\balpha_{t + 1}$\ncan be straightforwardly derived from $\\balpha_t$ since for\n$u \\in Q_{t + 1}$, $\\balpha_{t + 1}[u]$ is given by\n$\\balpha_{t + 1}[u] = \\sum_{e\\colon \\dest[e] = u} \\balpha_t[\\src[e]]\n\\weight[e]$. \n\nObserve that for any $t \\in [T]$ and $\\bx$, $\\sfq_t[\\bx]$ can be\nwritten as follows by unwrapping its recursive update definition:\n$$\\sfq_t[\\bx] = \\frac{e^{-\\eta \\sum_{s = 1}^{t - 1} l_s[\\bx[s]]}\n \\sfq_1[\\bx]}{\\sum_{\\bx \\in \\Sigma^T} e^{-\\eta \\sum_{s = 1}^{t - 1}\n l_s[\\bx[s]]} \\sfq_1[\\bx]}.$$\n In view of that, for any\n$a \\in \\Sigma$, $\\sfp_{t + 1}[a]$ can be written as follows:\n\\begin{align*}\n\\sfp_{t + 1}[a] \n& = \\frac{\\sum_{\\bx \\in \\Sigma^T} e^{-\\eta \\sum_{s = 1}^t\n l_s[\\bx[s]]} \\sfq_1[\\bx] 1_{\\bx[t] =\n a}}{\\sum_{a \\in \\Sigma} \\sum_{\\bx \\in \\Sigma^T} e^{-\\eta \\sum_{s = 1}^t\n l_s[\\bx[s]]} \\sfq_1[\\bx] 1_{\\bx[t] =\n a}}.\n\\end{align*}\nSince the WFA $\\sA$ is deterministic, for any $\\bx$ accepted by $\\sA$\nthere is a unique accepting path $\\pi$ in $\\sA$ labeled with\n$\\bx$. The numerator of the expression of $\\sfp_{t + 1}[a]$ is then\nthe sum of the weights of all paths in $\\sA$ with the $t$th symbol $a$\nat the end of $t$th iteration. This can be expressed as the sum over\nall transitions $e$ in $E_{\\sA}^{t \\to t + 1}$ with label $a$ of the\ntotal \\emph{flow} through $e$, that is the sum of the weights of all\naccepting path going through $e$:\n$\\balpha_t[\\src[e]] \\, \\weight[e] \\, \\bbeta[\\dest[e]]$ (see\nFigure~\\ref{fig:awm}). This is precisely the formula \ndetermining $\\sfp_{t + 1}$ in the pseudocode, where $Z$ is the\nnormalization factor.\n\nThe \\AWM\\ algorithm is closely related to the Expert Hidden Markov\nModel of \\cite{KoolenDeRooij2013} given for the log loss. It can be\nviewed as a generalization of that algorithm to arbitrary loss\nfunctions. A key difference between our setup and the perspective\nadopted by \\cite{KoolenDeRooij2013} is that they assume a Bayesian\nsetting where a prior distribution over expert sequences is given and\nmust be used. We assume the existence of a competitor automaton\n$\\sC$, but do not necessarily need to sample from it for making\npredictions. This will be crucial in the next section, where we use a\ndifferent WFA than $\\sC$ to improve computational efficiency while\npreserving regret performance. Also, \nthe prior distribution in \\citep{KoolenDeRooij2013} would be over\n$\\sC_T$ (for a large $T$) and not $\\sC$.\n\nThe computational complexity of \\AWM\\ at each round $t$ is\n$\\cO\\big(|E_{\\sA}^{t \\to t + 1}|\\big)$, that is the time to update the\nweights of the transitions in $E_{\\sA}^{t \\to t + 1}$ and to\nincrementally compute $\\balpha$ for states reached by paths of length\n$t + 1$. The total computational cost of the algorithm is thus\n$\\cO\\big (\\sum_{t = 1}^T |E_{\\sA}^{t \\to t + 1}| \\big) =\n\\cO(|E_\\sA|)$, where $E_\\sA$ is the set of transitions of\n$\\sA$.\\footnote{By the discussion above and\n Appendix~\\ref{app:intersection}, the total complexity of the\n intersection and weight-pushing operations is also in\n $\\cO(|E_\\sA|)$, so that they do not add any additional\n cost. Moreover, these two operations need only be carried out once\n and can be performed offline.} Note that $\\sA$ and $\\sC \\cap \\sS_T$\nadmit the same topology, thus the total complexity of the algorithm\ndepends on the number of transitions of the intersection WFA\n$\\sC \\cap \\sS_T$, which is at most $|\\sC| NT$. This can be\nsubstantially more favorable than a na\\\"{i}ve algorithm, whose\nworst-case complexity is exponential in $T$.\n\nWhen the number of transitions of the intersection WFA\n$\\sC \\cap \\sS_T$ is not too large compared to the number of experts\n$N$, the \\AWM\\ algorithm is quite efficient. However, it is natural\nto ask whether one can design efficient algorithms even if the number\nof transitions $E_{\\sA}^{t \\to t + 1}$ to process per round is large\n(which may be the case even for a \\emph{minimized} WFA\n$\\sC \\cap \\sS_T$ \\citep{Mohri2009}).\n\n\n\nWe will give two sets\nof solutions to derive a more efficient algorithm, which can be\ncombined for further efficiency. In the next section, we\ndiscuss a solution that consists of using an\napproximate WFA with a smaller number of transitions.\nIn Appendix~\\ref{app:phiaut}, we show\nthat the notion of \\emph{failure transition}, originally used in the\ndesign of string-matching algorithms and recently employed for \nparameter estimation in backoff $n$-gram language models\n\\citep{RoarkAllauzenRiley2013}, can be used to derive a more compact representation\nof the WFA $\\sC \\cap \\sS_T$, thereby resulting in a significantly\nmore efficient online learning algorithm that still admits compelling regret\nguarantees.\n\n\\section{Approximation algorithms}\n\\label{sec:approx}\n\nIn this section, we present approximation algorithms for the problem\nof online learning against a weighted sequence of experts represented\nby a WFA $\\sC$. Rather than using the intersection WFA\n$\\sC_T = \\sC \\cap \\sS_T$, we will assume that \\AWM\\ is run with an\napproximate WFA $\\h \\sC_T$. The main motivation for doing so is that\nthe algorithm can be substantially more efficient if $\\h \\sC_T$ admits\nsignificantly fewer transitions than $\\sC_T$. Of course, this comes at\nthe price of a somewhat weaker regret guarantee that we now analyze in\ndetail.\n\\ignore{In Appendix~\\ref{app:timeindepapprox}, we will also discuss\nhow to perform approximations directly on the WFA $\\sC$.}\n\n\\subsection{Effect of WFA approximation}\n\\label{sec:WFAapprox}\n\nWe first analyze the effect of automata approximation on the regret of \\AWM.\nAs in the previous section, we denote by $\\sfq$ the distribution\ndefined by $\\sC_T$ over sequences of length $T$. We will similarly\ndenote by $\\h \\sfq$ the distribution defined by $\\h \\sC_T$ over the\nsame set. The effect of the WFA approximation on the regret can be\nnaturally expressed in terms of the $\\infty$-R\\'{e}nyi divergence\n$D_\\infty(\\sfq \\| \\h \\sfq)$ between the distributions $\\sfq$ and\n$\\h \\sfq$, which is defined by\n$D_\\infty(\\sfq \\| \\h \\sfq) = \\sup_{\\bx \\in \\Sigma^T} \\log [\n\\sfq(\\bx)\/\\h \\sfq(\\bx) ]$\\ignore{ \\citep{Renyi1961}}.\n\n\\begin{theorem}\n\\label{th:WFAapprox}\nThe weighted regret of the \\AWM\\ algorithm with respect to the WFA\n$\\sC$ when run with $\\h \\sC_T$ instead of $\\sC_T$ can be\nupper bounded as follows:\n\\begin{align*}\n \\Reg_T(\\cA, \\sC) \n & \\leq \\frac{\\eta T}{8} + \\frac{1}{\\eta} \\log \\Big[K^\\eta \\sum_{\\bx} \\h\n \\sfq[\\bx]^\\eta \\Big] + D_\\infty(\\sfq \\| \\h \\sfq) \\\\ \n & \\leq \\frac{\\eta T}{8} + \\frac{1}{\\eta} \\log K + D_\\infty(\\sfq \\| \\h \\sfq). \n\\end{align*}\nIts unweighted regret can be upper bounded as follows:\n\\begin{align*}\n \\Reg_T^0(\\cA, \\sC) &\\leq \\max_{\\sC(\\bx) > 0} \\frac{\\eta T}{8} + \\frac{1}{\\eta} \\log \\bigg[\n \\frac{1}{\\sfq[\\bx]}\\bigg] + \\frac{1}{\\eta}\nD_\\infty(\\sfq \\| \\h \\sfq).\n\\end{align*}\n\\end{theorem}\nThe proof is given in Appendix~\\ref{app:WFAapprox}.\nTheorem~\\ref{th:WFAapprox} shows that the extra cost of using an\napproximate WFA $\\h \\sC_T$ instead of $\\sC_T$ is\n$D_\\infty(\\sfq \\| \\h \\sfq)$ for the weighted regret and similarly\n$\\frac{1}{\\eta} D_\\infty(\\sfq \\| \\h \\sfq)$ for the unweighted regret.\nThe bound is tight since the best sequence in hindsight in the regret\ndefinition may also be the one maximizing the log-ratio.\n\nThe theorem suggests a general algorithm for selecting an approximate\nWFA $\\h \\sC$ out of a family $\\cC$ of WFAs with a relatively small\nnumber of transitions. This consists of choosing $\\h \\sC$ to\nminimize the R\\'enyi divergence as defined by the following program:\n\\begin{equation}\n\\label{eq:autapprox}\n\\min_{\\h \\sC \\in \\cC} D_\\infty( \\sfq \\| \\h \\sfq),\n\\end{equation}\nwhere $\\h q$ is the distribution induced by $\\h \\sC$ over $\\Sigma^T$\n(the one obtained by computing $\\h \\sC_T = \\h \\sC \\cap \\sS_T$ and\nnormalizing the weights). The theorem ensures that the solution\nbenefits from the most favorable regret guarantee among the WFAs in\n$\\cC$. When the set of distributions associated to $\\cC$ is convex,\nthen the set of distributions defined over $\\Sigma^T$ is also\nconvex. This is then a convex optimization problem, since\n$\\h \\sfq \\mapsto \\log(\\sfq\/\\h \\sfq)$ is a convex function and the\nsupremum of convex functions is convex.\n\nThe choice of the family $\\cC$ is subject to a trade-off: approximation\naccuracy versus computational efficiency of using WFAs in $\\cC$.\nThis raises a model selection question for which we discuss in\ndetail a solution in Section~\\ref{sec:min-Renyi}:\ngiven a sequence of families $(\\cC_n)_{n \\in \\bN}$ with growing\ncomplexity and computational cost, the problem consists of selecting\nthe best $n$.\n\nIn the following, we will consider the case where the family $\\cC$ of\nweighted automata is that of \\emph{$n$-gram models}, for which we can\nupper bound the computational complexity.\n\n\n\\subsection{Minimum R\\'{e}nyi divergence $n$-gram models}\n\\label{sec:min-Renyi}\n\n\n\nLet $\\Sigma^{\\leq n - 1}$ denote the set of sequences of length at\nmost $n - 1$. An $n$-gram language model is a Markovian model of\norder $(n - 1)$ defined over $\\Sigma^*$, which can be compactly\nrepresented by a WFA with each state identified with a sequence\n$\\bx \\in \\Sigma^{\\leq n - 1}$, thereby encoding the sequence\n\\emph{just read} to reach that state. The WFA admits a transition\nfrom state $(\\bx[1] \\cdots \\bx[n - 1])$ to state\n$(\\bx[2] \\cdots \\bx[n - 1] a)$ with weight\n$\\sfw \\big[a \\, | \\, \\bx[1] \\cdots \\bx[n - 1] \\big]$, for any\n$a \\in \\Sigma$, and, for any $k \\leq n - 1$, a transition from state\n$(\\bx[1] \\cdots \\bx[k - 1])$ to state $(\\bx[1] \\cdots \\bx[k - 1] a)$\nwith weight $\\sfw \\big[ a \\, | \\, \\bx[1] \\cdots \\bx[k - 1] \\big]$, for\nany $a \\in \\Sigma$. It admits a unique initial state which is the one\nlabeled with the empty string $\\e$ (sequence of length zero) and all\nits states are final. The WFA is stochastic, that is outgoing\ntransition weights sum to one at every state: thus,\n$\\sum_{a \\in \\Sigma} \\sfw[a | \\bx ] = 1$ for all\n$\\bx \\in \\Sigma^{\\leq n - 1}$. Notice that this WFA is also\ndeterministic since it admits a unique initial state and no two\ntransitions with the same label leaving any\nstate. Figure~\\ref{fig:bigram} illustrates this definition in the case\nof a simple bigram model.\\ignore{\\footnote{Notice that $n$-gram models of this\n form are commonly used in language and speech processing. In these\n applications, the models are typically \\emph{smoothed} since they\n are trained on a finite sample. In our case, smoothing will not be\n needed since we can train directly on $\\sC_T$, which we interpret as\n the full language.}}\n\nNote that the transition weights $\\sfw[a | \\bx]$, with $a \\in \\Sigma$\nand $\\bx \\in \\cup_{k \\leq n - 1} \\Sigma^k$ fully specify an $n$-gram\nmodel. Since for a fixed $\\bx \\in \\cup_{k \\leq n - 1} \\Sigma^k$,\n$\\sfw[\\cdot | \\bx]$ is an element of the simplex, an $n$-gram model\ncan be viewed as an element of the product of\n$\\sum_{k = 0}^{n - 1} \\Sigma^k$ simplices, a convex set. We will denote\nby $\\cW_n$ the family of all $n$-gram models.\n\n\\begin{figure}[t]\n\\vskip -.15in\n\\centering\n\\includegraphics[scale=0.5]{nbigram} \n \\caption{A bigram language \n model over the alphabet $\\Sigma =\\set{a, b, c}$.\n }\n\\label{fig:bigram}\n\\vskip -.15in\n\\end{figure}\n\nOne key advantage of $n$-gram models in this context is that the\nper-iteration complexity can be bounded in terms of the number of\nsymbols. Since an $n$-gram model has at most $|\\Sigma|^{n - 1}$\nstates, its per-iteration computational cost is in\n$\\cO\\big(|\\Sigma|^{n}\\big)$ as each state can take one of $|\\Sigma|$\npossible transitions. For $n$ small, this can be very advantageous\ncompared to the original $\\sC_T$, since in general the maximum\nout-degree of states reached by sequences of length $t$ in the latter\ncan be very large. For instance, the automaton $\\sC_\\text{weighted-shift}$ \nin Figure~\\ref{fig:kshift} (ii) can itself be viewed as a bigram model and \nadmits efficient computation. \n\nFor $n$-gram models, our approximation algorithm\n(Problem~\\ref{eq:autapprox}) can be written as follows:\n\\begin{equation}\n\\label{eq:minrenyingram}\n\\min_{\\sfw \\in \\cW_n} D_\\infty(\\sfq \\| \\sfq_{\\sfw})\n= \\min_{\\sfw \\in \\cW_n} \\sup_{\\bx \\in \\Sigma^T} \n\\log\\bigg[ \\frac{\\sfq[\\bx]}{\\sfq_{\\sfw}[\\bx]} \\bigg],\n\\end{equation}\nwhere $\\sfq_{\\sfw}$ is the distribution induced by the $n$-gram model\n$\\sfw$ on sequences in $\\Sigma^T$. By definition of the $n$-gram\nmodel, for any $\\bx \\in \\Sigma^T$, $\\sfq_{\\sfw}[\\bx]$ is given by the\nfollowing:\n\\begin{equation*}\n\\sfq_{\\sfw}[\\bx] = \\prod_{t = 1}^T \\sfw \\big[\\bx[t] \\big| \\bx_{\\max(t - n +\n 1, 1)}^{t - 1} \\big],\n\\end{equation*}\nsince the weights of sequences of any fixed\nlength sum to one in an $n$-gram model. Problem~\\ref{eq:minrenyingram} is a convex\noptimization problem over $\\cW_n$. The problem can be solved\nusing as an an extension of the Exponentiated Gradient (EG) algorithm\nof \\cite{KivinenWarmuth1997}, which we will refer to as\n\\textsc{Prod-EG}. The pseudocode of \\textsc{Prod-EG}, a general convergence\nguarantee, and its convergence guarantee in\nthe specific case of $n$-gram models are given in detail in\nAppendix~\\ref{app:Prod-EG} as Algorithm~\\ref{alg:prodeg}, Theorem~\\ref{th:prodeg},\nand Corollary~\\ref{cor:prodegngram} respectively.\n\n{\\bf Model selection}. In practice, we seek an $n$-gram model that\nbalances the tradeoff between approximation error and computational\ncost. Assume that we are given a maximum per-iteration computational\nbudget $B$. We therefore wish to determine an $n$-gram approximation\nmodel affordable within our budget and with the most favorable regret\nguarantee. Let $F(\\sfq, \\sfq_\\sfw)$ denote the objective function of\nProblem~\\eqref{eq:minrenyingram}:\n$F(\\sfq, \\sfq_\\sfw) = D_\\infty(\\sfq \\| \\sfq_{\\sfw})$. By the\nconvergence guarantee of Corollary~\\ref{cor:prodegngram}, if\n$\\sfq_\\sfw$ is the $n$-gram model returned by \\textsc{Prod-EG} after\n$\\tau$ iterations, we can write\n$F(\\sfq, \\sfq_\\sfw) - F(\\sfq, \\sfq_{\\sfw^*}) \\leq \\Delta(\\tau, n)$,\nwhere $\\sfw^*$ is the $n$-gram model minimizing\nProblem~\\eqref{eq:minrenyingram} over $\\cW_n$ and $\\Delta(\\tau, n)$\nthe upper bound given by Corollary~\\ref{cor:prodegngram}. Thus, if\n$F(\\sfq, \\sfq_\\sfw) - \\Delta(\\tau, n) > \\sqrt{T}$ for some $n$, then,\neven the optimal $n$-gram model for this $n$ will cause an increase in\nthe regret.\n\nLet $n^*$ be the smallest $n$ such that\n$F(\\sfq, \\sfq_\\sfw) - \\Delta(\\tau, n) \\leq \\sqrt{T}$ (or the smallest\nvalue that exceeds our budget). We can find this value in $\\log(n^*)$\ntime using a two-stage process. In the first stage, we double $n$\nafter every violation until we find an upper bound on $n^*$, which we\ndenote by $n_{\\text{max}}$. In the second stage, we perform a binary\nsearch within $[1, n_{\\text{max}}]$ to determine $n^*$. Each stage\ntakes $\\log(n^*)$ iterations, and each iteration is the cost of\nrunning \\textsc{Prod-EG} for that specific value of $n$. Thus, the\noverall complexity of the algorithm is\n$\\cO\\left(\\log(n^*) \\, \\text{Cost}(\\textsc{Prod-EG})\\right)$, where\n$\\text{Cost}(\\textsc{Prod-EG})$ is the cost of a call to\n\\textsc{Prod-EG}. The full pseudocode of this algorithm, \n\\textsc{$n$-GramModelSelect}, is presented as Algorithm~\\ref{alg:ngramselect},\nwhere $\\sfu_n$ denotes the\nuniform $n$-gram model and $\\textsc{Prod-EG-Update}(\\sfq_\\sfw, \\cW_n)$\ndenotes one update made by \\textsc{Prod-EG} when optimizing over\n$\\cW_n$.\n\n\\begin{algorithm2e}[t]\n \\TitleOfAlgo{\\textsc{$n$-GramModelSelect}($\\sfq$, $\\tau$, $B$)}\n $n \\gets 1$;\n $\\sfq_\\sfw \\gets \\sfq_{\\sfu_n}$;\n $s \\gets 0$ \\\\\n \\While{$s \\leq \\tau$}{\n $\\sfq_\\sfw \\gets \\textsc{Prod-EG-Update}(\\sfq_\\sfw, \\cW_n)$ \\\\\n $s \\gets s + 1$ \\\\\n \\If{$F(\\sfq, \\sfq_\\sfw) - \\Delta(s, n) > \\sqrt{T}$ and $|\\Sigma|^n \\leq B$}{\n $n \\gets 2n$;\n $s \\gets 0$;\n $\\sfq_\\sfw \\gets \\sfq_{\\sfu_n}$\n }\n }\n $n_{\\max} \\gets n$.\\\\\n $\\sfq_\\sfw \\gets \\textsc{BinarySearch}([1, n_\\text{max}], F(\\sfq, \\sfq_\\sfw) -\n \\Delta(\\tau, n) \\leq \\sqrt{T})$\\\\\n \\RETURN{$\\sfq_\\sfw$}\n\\caption{\\textsc{$n$-GramModelSelect}.}\n\\label{alg:ngramselect}\n\\end{algorithm2e}\n\n\n\nIn the simple case of a unigram automaton model over two symbols and when the\ndistribution $\\sfq$ defined by the intersection WFA $\\sC_T$ is uniform,\nwe can give an explicit form of the solution of Problem~\\ref{eq:minrenyingram}.\nThe solution is obtained from the paths with the\nsmallest number of occurrences of each symbol, which can be\nstraightforwardly found via a shortest-path algorithm in linear time.\n\n\\begin{theorem}\n\\label{th:inftyrd1gram}\nAssume that $\\sC_T$ admits uniform weights over all paths and\n$\\Sigma = \\set{a_1, a_2}$. For $j \\in \\set{1, 2}$, let $n(a_j)$ be the smallest\nnumber of occurrences of $a_j$ in a path of $\\sC_T$. For any $j \\in\n\\set{1, 2}$, define\n\\begin{equation*}\n\\sfq[a_j] = \\frac{\\max\\left\\{1, \\frac{n(a_j)}{T -\n n(a_j)}\\right\\}}{1+\\max\\left\\{1, \\frac{n(a_j)}{T -\n n(a_j)}\\right\\}}.\n\\end{equation*}\n\nThen, the unigram model $\\sfw \\in \\cW_1$ solution of\n$\\infty$-R\\'{e}nyi divergence optimization problem is defined by\n$\\sfw[a_{j^*}] = \\sfq[a_{j*}]$, $\\sfw[a_{j'}] = 1 - \\sfw[a_{j^*}]$,\nwith\n$j^* = \\argmax_{j \\in \\set{1, 2}} \\ n(a_j) \\log \\sfq[a_j] + \\left[T -\n n(a_j) \\right] \\log\\big[ 1 - \\sfq[a_j] \\big]$.\n\\end{theorem}\n\nThe proof of this result is provided in Appendix~\\ref{app:unigram}.\n\nTheorem~\\ref{th:inftyrd1gram} shows that the solutions of the\n$\\infty$-R\\'enyi divergence optimization are based on the $n$-gram\ncounts of sequences in $\\sC_T$ with ``high entropy''. This can be\nvery different from the maximum likelihood solutions, which are based\non the average $n$-gram counts. For instance, suppose we are under\nthe assumptions of Theorem~\\ref{th:inftyrd1gram}, and specifically,\nassume that there are $T$ sequences in $\\sC_T$. Assume that one of\nthe sequences has $\\left(\\frac{1}{2} + \\gamma\\right)T$ occurrences of\n$a_1$ for some small $\\gamma > 0$ and that the other $T-1$ sequences\nhave $T-1$ occurrences of $a_1$. Then,\n$n(a_1) = \\left(\\frac{1}{2} + \\gamma\\right)T$, and the solution\nof the $\\infty$-R\\'enyi divergence optimization problem is given by\n$\\sfq_\\infty(a_1) = \\frac{1 + 2\\gamma}{2}$ and\n$\\sfq_\\infty(a_2) = \\frac{1 - 2\\gamma}{2}$. On the other hand,\nthe maximum-likelihood solution would be\n$\\sfq_1(a_1) = 1 + \\frac{\\gamma}{T} - \\frac{3}{2T} + \\frac{1}{T^2} \\approx 1$\nand $\\sfq_1(a_2) = \\frac{3}{2T} - \\frac{\\gamma}{T} - \\frac{1}{T^2} \\approx 0$\nfor large $T$.\n\n\n\n\n\\subsection{Maximum-Likelihood $n$-gram models}\n\\label{subsec:mlapprox}\n\nA standard method for learning $n$-gram models is via\nMaximum-Likelihood, which is equivalent to minimizing the\nrelatively entropy between the target distribution $\\sfq$ and the\nlanguage model, that is via\n\\begin{align}\n \\label{eq:mlngram}\n\\min_{\\sfw \\in \\cW_n} D(\\sfq \\| \\sfq_{\\sfw}),\n\\end{align}\nwhere, $D(\\sfq \\| \\sfq_{\\sfw})$ denotes the relative entropy,\n$D(\\sfq \\| \\sfq_{\\sfw}) = \\sum_\\bx \\sfq[\\bx]\n\\log\\Big[\\frac{\\sfq[\\bx]}{\\sfq_\\sfw[\\bx]} \\Big]$. Maximum likelihood\n$n$-gram solutions are simple. For standard text data, the weight of each\ntransition is the frequency of appearance of the corresponding\n$n$-gram in the text. For a probabilistic $\\sC_T$, the weight can be\nsimilarly obtained from the expected count of the $n$-gram in the\npaths of $\\sC_T$, where the expectation is taken over the probability\ndistribution defined by $\\sC_T$ and can be computed efficiently\n\\citep{AllauzenMohriRoark2003}. In general, the solution of this\noptimization problem does not benefit from the guarantee of\nTheorem~\\ref{th:WFAapprox} since the $\\infty$-R\\'{e}nyi divergence is\nan upper bound on the relative entropy.\\ignore{\\footnote{The relative entropy\n also coincides with the R\\'{e}nyi divergence of order $1$ and the\n R\\'{e}nyi divergence is non-decreasing function of the order.}}\nHowever, in some cases, maximum likelihood solutions do benefit from\nfavorable regret guarantees. In particular, as shown by the following\ntheorem, remarkably, the maximum-likelihood bigram approximation to\nthe $k$-shifting automaton coincides with the \\textsc{Fixed-Share}\nalgorithm of \\cite{HerbsterWarmuth1998} and benefits from a constant\napproximation error. Thus, we can view and motivate the design of the\n\\textsc{Fixed-Share} algorithm as that of a bigram approximation of\nthe desired competitor automaton, which represents the family of\n$k$-shifting sequences.\n\n\n\\begin{theorem}\n\\label{th:bigramkshift}\nLet $\\sC_T$ be the $k$-shifting automaton for some $k$. Then, the\nbigram model $\\sfw_2$ obtained by minimizing relative entropy is\ndefined for all $a_1, a_2 \\in \\Sigma$ by\n\\begin{align*}\n\\sfp_{\\sfw_2}[a_1a_2]\n\\! = \\! \\frac{\\big[ 1 - \\frac{k}{(T - 1)}\\big] \\, 1_{a_1 = a_2} \n+ \\big[ \\frac{k}{(T - 1)(N - 1)} \\big] \\, 1_{a_1 \\neq a_2}}{N} .\n\\end{align*}\nMoreover, its approximation error can be bounded by a constant\n(independent of $T$):\n\\begin{equation*}\n D_\\infty(\\sfq \\| \\sfq_{\\sfw_2}) \\leq - \\log \\big[ 1 - 2e^{-\\frac{1}{12k}} \\big].\n\\end{equation*}\n\\end{theorem}\nThe proof of the theorem as well as other details about\nMaximum-Likelihood are given in Appendix~\\ref{app:mlapprox}.\nThe proof technique is illustrative\nbecause it reveals that the maximum likelihood $n$-gram model has low\napproximation error whenever (1) the model's distribution is\nproportional to the distribution of $\\sC_T$ on $\\sC_T$'s support and\n(2) most of the model's mass lies on the support of $\\sC_T$. When the\nautomaton $\\sC_T$ has uniform weights, then condition (1) is satisfied\nwhen the $n$-gram model is uniform on $\\sC_T$. This is true whenever\nall sequences in $\\sC_T$ have the same set of $n$-gram counts, and\nevery permutation of symbols over these counts is a sequence that lies\nin $\\sC_T$, which is the case for the $k$-shifting\nautomaton. Condition (2) is satisfied when $n$ is large enough, which\nnecessarily exists since the distribution is exact for $n = T$. On the\nother hand, note that a unigram approximation would have satisfied\ncondition (1) but not condition (2) for the $k$-shifting automaton.\n\nTo the best of our knowledge, this is the first framework that\nmotivates the design of \\textsc{Fixed-Share} with a focus on\nminimizing tracking regret. Other works that have recovered\n\\textsc{Fixed-Share} (e.g. \\citep{CesaBianchiGaillardLugosiStoltz2012,\n KoolenDeRooij2013, GyorgySzepesvari2016}) have generally viewed the\nalgorithm itself as the main focus.\n\nOur derivation of \\textsc{Fixed-Share} also allows us to naturally\ngeneralize the setting of standard $k$-shifting experts to\n$k$-shifting experts with non-uniform weights. Specifically, consider\nthe case where $\\sC_T$ is an automaton accepting up to $k$-shifts but\nwhere the shifts now occur with probability\n$\\sfq[a_2 | a_1, a_1 \\neq a_2] \\neq \\frac{1}{N-1} 1_{\\{a_2 \\neq\n a_1\\}}$. Since the bigram approximation will remain exact on\n$\\sC_T$, we recover the exact same guarantee as in\nTheorem~\\ref{th:bigramkshift}.\n\nMaximum likelihood $n$-gram models can further benefit from our use of\nfailure transitions and the $\\phi$-conversion algorithm presented in\nAppendix~\\ref{app:phiaut}. This can reduce the size of the automaton\nand often dramatically improve its computational efficiency without\naffecting its accuracy.\n\n\\section{Extension to sleeping experts}\n\\label{sec:sleep}\n\nIn many real-world applications, it may be natural for some experts to\nabstain from making predictions on some of the rounds. For instance,\nin a bag-of-words model for document classification, the presence of a\nfeature or subset of features in a document can be interpreted as an\nexpert that is awake. This extension of standard prediction with\nexpert advice is also known as the \\emph{sleeping experts framework}\n\\citep{FreundSchapireSingerWarmuth1997}. The experts are said to be\nasleep when they are inactive and awake when they are active and available\nto be selected. This framework is distinct from the\npermutation-based definitions adopted in the studies in \\citep{KleinbergNiculescuMizilSharma2010, KanadeMcMahanBryan2009, KanadeSteinke2014}. \n\nFormally, at each round $t$, the adversary chooses an awake set\n$A_t \\subseteq \\Sigma$ from which the learner is allowed to query an\nexpert. The algorithm then (randomly) chooses an expert $i_t$ from\n$A_t$, receives a loss vector $l_t \\in [0,1]^{|\\Sigma|}$ supported on\n$A_t$ and incurs loss $l_t[i_t]$. Since some experts may not be\navailable in some rounds, it is not reasonable to compare the loss\nagainst that of the best static expert or sequence of experts. In\n\\citep{FreundSchapireSingerWarmuth1997}, the comparison is made\nagainst the best fixed mixture of experts normalized at each round\nover the awake set:\n$\\min_{\\sfu \\in \\Delta_N}\\sum_{t = 1}^T \\frac{\\sum_{a \\in A_t} \\sfu[a] l_t[a]\n}{\\sum_{a' \\in A_t} \\sfu[a']}$, where $\\Delta_N$ is the $(N-1)$-dimensional \nsimplex.\n\nWe extend the notion of sleeping experts to the path setting, so that\ninstead of comparing against fixed mixtures over experts, we compare\nagainst fixed mixtures over the family of expert sequences. With some\nabuse of notation, let $A_t$ also represent the automaton accepting\nall paths of length $T$ whose $t$-th transition has label in $A_t$.\nThus, we want to design an algorithm that performs well with respect\nto the following quantity:\n$$\\min_{\\sfu \\in \\Delta_K}\\sum_{t = 1}^T \\frac{\\sum_{\\bx \\in \\sC_T \\cap A_t} \\sfu[\\bx] l_t[\\bx[t]] }{\\sum_{\\bx \\in \\sC_T \\cap A_t} \\sfu[\\bx]},$$\nwhere $K$ is the number of accepting paths of $\\sC_T$.\n\nThis motivates the design of \\textsc{AwakeAWM}, a \npath-based weighted majority algorithm that generalizes the algorithms\nin \\citep{FreundSchapireSingerWarmuth1997} to arbitrary families of\nexpert sequences. Like \\AWM, \\textsc{AwakeAWM} maintains a set of\nweights over all the paths in the input automaton. At each round $t$,\nthe algorithm performs a weighted majority-type update. However, it\nnormalizes the weights so that the total weight of the awake set\nremains unchanged. This prevents the algorithm from ``overfitting'' to\nexperts that have been asleep for many rounds. The pseudocode of this\nalgorithm and the proof of its accompanying guarantee,\nTheorem~\\ref{th:awakeawm}, \nare provided in Appendix~\\ref{app:sleep}.\n\n\\begin{theorem}[Regret Bound for \\textsc{AwakeAWM}]\n \\label{th:awakeawm}\n Let $K$ denote the number of accepting paths of $\\sC_T = \\sC \\cap \\sS_T$,\n and for each $t \\in [T]$, let $A_t\\subseteq \\Sigma$ denote the set of experts \n that are awake at time $t$.\n Then for any distribution $\\sfu\\in \\Delta_K$, \\textsc{AwakeAWM} admits\n the following unweighted regret guarantee:\n \\begin{align*}\n &\\sum_{t = 1}^T \\sum_{\\bx \\in \\sC_T \\cap A_t} \\sfu[\\bx] \\E_{a \\sim \\sfp_t^{A_t}} [l_t[a]] - \n \\sum_{t=1}^T \\sum_{\\bx \\in \\sC_T \\cap A_t} \\sfu[\\bx] l_t[\\bx[t]] \\\\\n &\\leq \\frac{\\eta}{8} \\sum_{t = 1}^T \\sfu(A_t) + \\frac{1}{\\eta} \\log(K). \n \\end{align*}\n\\end{theorem}\n\nAs with \\textsc{AWM}, \\textsc{AwakeAWM} is an efficient algorithm\nwith a total computational cost that is linear in the number of transitions \nof $\\sA$ (or equivalently, $\\sC_T$). \nMoreover, as in the non-sleeping expert setting, we can further improve the\ncomputational complexity by applying $\\phi$-conversion to arrive at a \nor $n$-gram approximation and then $\\phi$-conversion.\nAll other improvements in the sleeping expert setting will similarly mirror\nthose for the non-sleeping expert algorithms.\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nWe studied a general framework of online learning against a competitor\nclass represented by a WFA and presented a number of algorithmic\nsolutions for this problem achieving sublinear regret guarantees using\nautomata approximation and failure transitions. We also extended our\nalgorithms and results to the sleeping experts framework\n(Section~\\ref{sec:sleep}).\\ignore{and to the online convex optimization\nsetting (Appendix~\\ref{app:oco}).} Our results can be\nstraightforwardly extended to the adversarial bandit scenario using\nstandard surrogate losses based on importance weighting techniques and\nto the case where more complex formal language families such as\n(probabilistic) context-free languages over expert sequences are\nconsidered.\n\n\\ignore{\nWe gave a series of algorithms for this problem, including an\nautomata-based algorithm extending weighted-majority whose\ncomputational cost at round $t$ depends on the total number of\ntransitions leaving the states of the competitor automaton reachable\nat time $t$, which substantially improves upon a na\\\"ive algorithm\nbased on path updates. We used the notion of failure transitions to\nprovide a compact representation of the competitor automaton or its\nintersection with the set of strings of length $t$, thereby resulting\nin significant efficiency improvements. This required the\nintroduction of new failure-transition-based composition and\nshortest-distance algorithms that could be of independent interest.\n\nWe further gave an extensive study of algorithms based on a compact\napproximation of the competitor automata. We showed that the key\nquantity arising when using an approximate weighted automaton is the\nR\\'enyi divergence of the original and approximate automata. We\npresented a specific study of approximations based on $n$-gram models\nby minimizing the R\\'enyi divergence and studied the properties of\nmaximum likelihood $n$-gram models. We pointed out the efficiency\nbenefits of such approximations and provides guarantees on the\napproximations and the regret. We also extended our algorithms and\nresults to the framework of sleeping experts. We further described the\nextension of the approximation methods to online convex optimization\nand a general mirror descent setting.\n\nOur description of this general (weighted) regret minimization\nframework and the design of algorithms based on automata provides a\nunifying view of many similar problems and leads to general\nalgorithmic solutions applicable to a wide variety of problems with\ndifferent competitor class automata. In general, automata \nlead to a more general and cleaner analysis. \nAn alternative approximation method consists of directly\nminimizing the competitor class automaton before intersection\nwith the set of strings of length $t$. We have also studied\nthat method, presented guarantees for its success, and illustrated \nthe approach in a special case.\n\nNote that, instead of automata and regular languages, we could have\nconsidered more complex formal language families such as\n(probabilistic) context-free languages over expert sequences.\nHowever, more complex languages can be handled in a similar way since\nthe intersection with $\\sS_T$ would be a finite language. The method\nbased on a direct approximation would require approximating a\nprobabilistic context-free language using weighted automata, a problem\nthat has been extensively studied in the past\n\\citep{PereiraWright1991, Nederhof2000, MohriNederhof2001}.\n\n}\n\n\n\\newpage\n\\bibliographystyle{abbrvnat} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \n\\label{sec:intro}\n\\footnotetext{\\footnotemark[1]Work was done while Gunnar was an intern at AI2.}\n\nConsider the video shown in Figure~\\ref{fig:teaser1}: A man walks through a doorway, stands at a table, holds a cup, pours something into it, drinks it, puts the cup on the table, and finally walks away. Despite depicting a simple activity, the video involves a rich interplay of a sequence of actions with underlying goals and intentions. For example, the man stands at the table `to take a cup', he holds the cup `to drink from it', etc. Thorough understanding of videos requires us to model such interplay between activities as well as to reason over extensive time scales and multiple aspects of actions (objects, scenes, etc). \n\nMost contemporary deep learning based methods have treated the problem of video understanding as that of only appearance and motion (trajectory) modeling~\\cite{Simonyan13,Tran15,Georgia15,Snoek16}. While this has fostered interesting progress in this domain, these methods still struggle to outperform models based on hand-crafted features, such as Dense Trajectories~\\cite{WangIDT13}. \nWhy such a disconnect? We argue that video understanding requires going beyond appearance modeling, and necessitates reasoning about the activity sequence as well as higher-level constructs such as intentions. The recent emergence of large-scale datasets containing rich sequences of realistic activities~\\cite{charades,yeung2015every,weinzaepfel2016towards} comes at a perfect time facilitating us to explore such complex reasoning. \n\nBut what is the right way to model and reason about temporal relations and goal-driven behaviour? Over the last couple of decades, graphical models such as Conditional Random Fields (CRFs) have been the prime vehicles for structured reasoning. \nTherefore, one possible alternative is to use ConvNet-based approaches~\\cite{alexnet12} to provide features for a CRF training algorithm. \nAlternatively, it has been shown that integrating CRFs with ConvNet architectures and training them in an end-to-end manner provides substantial improvements in tasks such as segmentation and situation recognition~\\cite{raqueldense2016,ChenSchwingICML2015,yatskarsituation}. \n\n\\begin{figure}[t]\n\\includegraphics[width=\\linewidth]{teaser2f.pdf}\n\\caption{Understanding human activities in videos requires jointly reasoning about multiple aspects of activities, such as `what is happening', `how', and `why'. In this paper, we present an end-to-end deep structured model over time trained in a stochastic fashion. The model captures rich semantic aspects of activities, including \\emph{Intent} (why), \\emph{Category} (what), \\emph{Object} (how). The figure shows video frames and annotations used in training from the {\\em Charades}~\\cite{charades} dataset. }\n\\label{fig:teaser1}\n\\end{figure}\n\n\nInspired by these advances, we present a deep-structured model that can reason temporally about multiple aspects of activities. For each frame, our model infers the activity category, object, action, progress, and scene using a CRF, where the potentials are predicted by a jointly end-to-end trained ConvNet over all predictions in all frames. This CRF has a latent node for the intent of the actor in the video and pair-wise relationships between all individual frame predictions. \n\nWhile our model is intuitive, training it in an end-to-end manner is a non-trivial task. Particularly, end-to-end learning requires computing likelihoods for individual frames and doing joint inference about all connected frames with a CRF training algorithm. \nThis is in stark contrast with the standard stochastic gradient descent (SGD) training algorithm (backprop) for deep networks, where we require mini-batches with a large number of independent and uncorrelated samples, not just a few whole videos.\nIn order to handle this effectively: (1) we relax the Markov assumption and choose a fully-connected temporal model, such that each frame's prediction is influenced by all other frames, and (2) we propose an asynchronous method for training fully-connected structured models for videos. Specifically, this structure allows for an implementation where the influence (messages) from other frames are approximated by emphasizing influence from frames computed in recent iterations. They are more accurate, and show advantage over being limited to only neighboring frames. In addition to being more suitable for stochastic training, fully-connected models have shown increased performance on various tasks~\\cite{densecrfsegmentation,raqueldense2016}.\n\nIn summary, our key contributions are: (a) a deep CRF based model for structured understanding and comprehensive reasoning of videos in terms of multiple aspects, such as action sequences, objects, and even intentions; (b) an asynchronous training framework for expressive temporal CRFs that is suitable for end-to-end training of deep networks; and, (c) substantial improvements over state-of-the-art, increasing performance from 17.2\\% mAP to 22.4\\% mAP on the challenging Charades~\\cite{charades} benchmark.\n\n\\section{Related Work}\n\nUnderstanding activities and actions has an extensive history~\\cite{poppe2010survey,weinland2011survey,STIP05,HOG3D,HOF,MBH06,Matikainen09,WangIDT13,Peng14,Lan15}. Interestingly, \nanalyzing actions by their appearance has gone through multiple iterations. Early success was with hand-crafted representations such as Space Time Interest Points (STIP)~\\cite{STIP05}, 3D Histogram of Gradient (HOG3D)~\\cite{HOG3D}, Histogram of Optical Flow (HOF)~\\cite{HOF}, and Motion Boundary Histogram~\\cite{MBH06}. These methods capture and analyze local properties of the visual-temporal datastream. In the past years, the most prominent hand-crafted representations have been from the family of trajectory based approaches~\\cite{Matikainen09,WangIDT13,Peng14,Lan15}, where the Improved Dense Trajectories (IDT)~\\cite{WangIDT13} representation is in fact on par with state-of-the-art on multiple recent datasets~\\cite{THUMOS15,charades}. \n\nRecently there has been a push towards mid-level representations of video~\\cite{Corso12,YaleS13,ArpitJ13,lan2015iccv}, that capture beyond local properties. However, these approaches still used hand-crafted features. With the advent of deep learning, learning representations from data has been extensively studied~\\cite{3DCNN,Karpathy14,2stream14,TDD15,Taylor10,Tran15,Le11,Georgia15,Xu15,Vondrick16_repr,scnn_shou_wang_chang_cvpr16,Souza16}. Of these, one of the most popular frameworks has been the approach of Simonyan et al.~\\cite{2stream14}, who introduced the idea of training separate color and optical flow networks to capture local properties of the video. \n\nMany of those approaches were designed for short clips of individual activities and hence do not generalize well to realistic sequences of activities. \nCapturing the whole information of the video in terms of temporal evolution of the video stream has been the focus of some recent approaches~\\cite{KevinT12,Basura15,Izadinia12,Rohrbach12,Sun13,pirsiavash2014parsing}.\nMoving towards more expressive deep networks such as LSTM has become a popular method for encoding such temporal information~\\cite{Srivastava15,Donahue15,cnnlstm,Sun_2015_ICCV,Wang_Transformation,sigurdsson2016learning,yeung2015end}. Interestingly, while those models move towards more complete understanding of the full video stream, they have yet to significantly outperform local methods~\\cite{2stream14} on standard benchmarks.\n\nA different direction in understanding comes from reasoning about the complete video stream in a complementary direction --- Structure. Understanding activities in a human-centric fashion encodes our particular experiences with the visual world. Understanding activities with emphasis on objects has been a particularly fruitful direction~\\cite{li2007and,ryoo2007hierarchical,gupta2009observing,prest2012weakly,Vondrick16_actns}. In a similar vein, some works have also tried modeling activities as transformations~\\cite{Wang_Transformation} or state changes~\\cite{Fathi13}. Recently, there has been significant progress in modelling the complete human-centric aspect, where image recognition is phrased in terms of objects and their roles~\\cite{yatskarsituation,VSRL_gupta15}. Moving beyond appearance and reasoning about the state of {\\em agents} in the images requires understanding human intentions~\\cite{kitani2012activity,pirsiavash2014inferring}. This ability to understand people in terms of beliefs and intents has been traditionally studied in psychology as the Theory of mind~\\cite{premack1978does}.\n\nHow to exactly model structure of the visual and temporal world has been the pursuit of numerous fields. Of particular interest is work that combines the representative power of deep networks with structured modelling. Training such models is often cumbersome due to the differences in jointly training deep networks (stochastic sampling) and sequential models (consecutive samples)~\\cite{mnih2013playing,raqueldense2016}.\nIn this work, we focus on fully-connected random fields, that have been popular in image segmentation~\\cite{densecrfsegmentation}, where image filtering was used for efficient message passing, and later extended to use CNN potentials~\\cite{schwing2015fully}.\n\n\n\\section{Proposed Method}\n\\label{sect:app}\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=1.0\\linewidth]{model5.pdf}\n\\caption{An overview of our structured model. The semantic part captures \\emph{object}, \\emph{action}, etc. at each frame, and temporal aspects captures those over time. On the left side, we show how for each timepoint in the video, a Two-Stream Network predicts the potentials. Our model jointly reasons about multiple aspects of activities in all video frames. The \\emph{Intent} captures groups of activities of the person throughout the whole sequence of activities, and fine-grained temporal reasoning is through fully-connected temporal connections.}\n\\label{fig:model}\n\\end{figure*}\n\nGiven a video with multiple activities, our goal is to understand the video in terms of activities. Understanding activities requires reasoning about objects being interacted with, the place where the interaction is happening, what happened before and what happens after this current action and even the intent of the actor in the video. We incorporate all these by formulating a deep Conditional Random Field (CRF) over different aspects of the activity over time. That is, a video can be interpreted as a graphical model, where the components of the activity in each frame are nodes in the graph, and the model potentials are the edges in the graph. \n\nIn particular, we create a CRF which predicts activity, object, etc., for every frame in the video. For reasoning about time, we create a \\emph{fully-connected temporal CRF}, referred as Asynchronous Temporal Field in the text. That is, unlike a linear-chain CRF for temporal modelling (the discriminative counterpart to Hidden Markov Models), each node depends on the state of every other node in the graph. We incorporate intention as another latent variable which is connected to all the action nodes. \nThis is an unobserved variable that influences the sequence of activities. This variable is the common underlying factor that guides and better explains the sequence of actions an agent takes. Analysis of what structure this latent variable learns is presented in the experiments.\nOur model has three advantages: (1) it addresses the problem of long-term interactions; (2) it incorporates reasoning about multiple parts of the activity, such as objects and intent; and (3) more interestingly, as we will see, it allows for efficient end-to-end training in an asynchronous stochastic fashion.\n\n\\subsection{Architecture}\nIn this work we encode multiple components of an activity. Each video with $T$ frames is represented as $\\{X_1, \\dots, X_T, I\\}$ where $X_t$ is a set of frame-level random variables for time step $t$ and $I$ is an unobserved random variable that represent global intent in the entire video. We can further write $X_t = \\{C_t,O_t,A_t,P_t,S_t\\}$, where $C$ is the activity category (e.g., `drinking from cup'), $O$ corresponds to the object (e.g., `cup'), $A$ represents the action (e.g., `drink'), $P$ represents the progress of the activity \\{start, middle, end\\}, and $S$ represents the scene (e.g. `Dining Room'). \nFor clarity in the following derivation we will refer to all the associated variables of $X_t$ as a single random variable $X_t$. A more detailed description of the CRF is presented in the appendix.\n\nMathematically we consider a random field $\\{X,I\\}$ over all the random variables in our model ($\\{X_1, \\dots, X_T, I\\}$). Given an input video $V{=}\\{V_1,\\dots,V_T\\}$, where $V_t$ is a video frame, our goal is to estimate the maximum a posteriori labeling of the random field by marginalizing over the intent $I$. This can be written as:\n\\begin{equation}\n\\mathbf{x}^* = \\mathrm{arg}\\,\\max_x \\sum_I P(x,I|V).\n\\end{equation}\nFor clarity in notation, we will drop the conditioning on $V$ and write $P(X,I)$. We can define $P(X,I)$ using Gibbs distribution as: $P(X,I) {=} \\frac{1}{Z(\\mathbf{V})} \\exp \\left( -E(x,I) \\right)$ where $E(x,I)$ is the Gibbs energy over $x$. In our CRF, we model all unary and pairwise cliques between all frames $\\{X_1,\\dots,X_T\\}$ and the intent $I$. The Gibbs energy is:\n\\begingroup\\makeatletter\\def\\f@size{8}\\check@mathfonts\n\\def\\maketag@@@#1{\\hbox{\\m@th\\normalsize\\normalfont#1}}%\n\\begin{align}\nE(\\mathbf{x},I) = \\underbrace{\\sum_i \\phi_\\mathcal{X}(x_i)}_{\\mathrm{Semantic}} + \\underbrace{\\sum_{i} \\phi_{\\mathcal{X}\\!\\mathcal{I}}(x_i,I) + \\sum_{\\substack{i,j\\\\i \\neq j}} \\phi_{\\mathcal{X}\\!\\mathcal{X}}(x_i,x_j)}_{\\mathrm{Temporal}},\n\\end{align}\n\\endgroup\nwhere $\\phi_{\\mathcal{X}\\!\\mathcal{X}}(x_i,x_j)$ is the potential between frame $i$ and frame $j$, and $\\phi_{\\mathcal{X}\\!\\mathcal{I}}(x_i,I)$ is the potential between frame $i$ and the intent. For notational clarity $\\phi_\\mathcal{X}(x_i)$ incorporates all unary and pairwise potentials for ${C_t,O_t,A_t,P_t,S_t}$. \nThe model is best understood in terms of two aspects: Semantic aspect, which incorporates the local variables in each frame (${C_t,O_t,A_t,P_t,S_t}$); and Temporal aspect, which incorporates interactions among frames and the intent $I$. This is visualized in Figure~\\ref{fig:model}. \nWe will now explain the semantic, and temporal potentials.\n\n\\myparagraph{Semantic aspect}\nThe frame potential $\\phi_\\mathcal{X}(x_i)$ incorporates the interplay between activity category, object, action, progress and scene, and could be written explicitly as $\\phi_\\mathcal{X}(C_t,O_t,A_t,P_t,S_t)$. In practice this potential is composed of unary, pairwise, and tertiary potentials directly predicted by a CNN. We found predicting only the following terms to be sufficient without introducing too many additional parameters:\n$\\phi_\\mathcal{X}(C_t,O_t,A_t,P_t,S_t){=}\\phi(O_t,P_t){+}\\phi(A_t,P_t){+}\\phi(O_t,S_t)+ \\phi(C_t,O_t,A_t,P_t)$ where we only model the assignments seen in the training set, and assume others are not possible.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1.0\\linewidth]{messages2.pdf}\n\\caption{Illustration of the learning algorithm, and the message passing structure. Each timepoint that has been processed has a message (Blue highlights messages that have recently been computed). The loss receives a combination of those messages, uses those to construct new messages, and updates the network. }\n\\label{fig:learning}\n\\end{figure}\n\n\\myparagraph{Temporal aspect}\nThe temporal aspect of the model is both in terms of the frame-intent potentials $\\phi_{{\\mathcal{X}\\!\\mathcal{I}}}(x_i,I)$ and frame-frame potentials $\\phi_{{\\mathcal{X}\\!\\mathcal{X}}}(x_i,x_j)$. The frame-intent potentials are predicted with a CNN from video frames (pixels and motion). The pairwise potentials $\\phi_{{\\mathcal{X}\\!\\mathcal{X}}}(x_i,x_j)$ for two time points $i$ and $j$ in our model have the form:\n\\begin{align}\n\\phi_{{\\mathcal{X}\\!\\mathcal{X}}}(x_i,x_j) = \\mu(x_i,x_j)\\sum_{m} w^{(m)}k^{(m)}(v_i,v_j),\n\\end{align}\nwhere $\\mu$ models the asymmetric affinity between frames, $w$ are kernel weights, and each $k^{(m)}$ is a Gaussian kernel that depends on the videoframes $v_i$ and $v_j$. In this work we use a single kernel that prioritises short-term interactions:\n\\begin{align}\n\\label{eq:sigma}\nk(v_i,v_j) = \\exp \\left( - \\frac{(j-i)^2}{2\\sigma^2}\\right)\n\\end{align}\nThe parameters of the general asymmetric compatibility function $\\mu(x_i,x_j)$ are learned from the data, and $\\sigma$ is a hyper-parameter chosen by cross-validation.\n\n\\subsection{Inference}\n\nWhile it is possible to enumerate all variable configurations in a single frame, doing so for multiple frames and their interactions is intractable. Our algorithm uses a structured variational approximation to approximate the full probability distribution. In particular, we use a mean-field approximation to make inference and learning tractable. With this approximation, we can do inference by keeping track of message between frames, and asynchronously train one frame at a time (in a mini-batch fashion). \n\nMore formally, instead of computing the exact distribution $P(X,I)$ presented above, the structured variational approximation finds the distribution $Q(X,I)$ among a given family of distributions that best fits the exact distribution in terms of KL-divergence. By choosing a family of tractable distributions, it is possible to make inference involving the ideal distribution tractable. Here we use $Q(X,I)=Q_{\\mathcal{I}}(I) \\prod_i Q_i(x_i)$, the structured mean-field approximation. Minimizing the KL-divergence between those two distributions yields the following iterative update equation:\n\\begingroup\\makeatletter\\def\\f@size{8}\\check@mathfonts\n\\def\\maketag@@@#1{\\hbox{\\m@th\\normalsize\\normalfont#1}}%\n\\begin{align}\nQ_i(x_i) \\propto \\exp \\bigg\\{ & \\phi_\\mathcal{X}(x_i) + \\mathrm{E}_{U\\sim Q_{\\mathcal{I}}} \\left[ \\phi_{\\mathcal{X}\\!\\mathcal{I}}(x_i,U) \\right] \\nonumber \\\\ \n&+ \\sum_{j > i} \\mathrm{E}_{U_j\\sim Q_j} \\left[ \\phi_{{\\mathcal{X}\\!\\mathcal{X}}}(x_i,U_j) \\right] \\bigg\\} \\nonumber \\\\\n&+ \\sum_{j < i} \\mathrm{E}_{U_j\\sim Q_j} \\left[ \\phi_{{\\mathcal{X}\\!\\mathcal{X}}}(U_j,x_i) \\right] \\bigg\\} \\\\\nQ_{\\mathcal{I}}(I) \\propto \\exp \\bigg\\{ & \\sum_j \\mathrm{E}_{U_j\\sim Q_j} \\left[ \\phi_{\\mathcal{X}\\!\\mathcal{I}}(U_j,I) \\right] \\bigg\\}\n\\label{eq:Q}\n\\end{align}\n\\endgroup\nwhere $Q_i$ is marginal distribution with respect to each of the frames, and $Q_{\\mathcal{I}}$ is the marginal with respect to the intent. An algorithmic implementation of this equation is as presented in Algorithm~\\ref{alg:inference}.\n\n\\begin{algorithm}\n \\caption{Inference for Asynchronous Temporal Fields\n \\label{alg:inference}}\n {\\footnotesize\n \\begin{algorithmic}[1]\n \\State Initialize Q \\Comment{Uniform distribution}\n \\While{$\\textrm{not converged}$}\n \\State Visit frame $i$\n \\State Get $\\sum_{j > i} \\mathrm{E}_{U_j\\sim Q_j} \\left[ \\phi_{\\mathcal{X}\\!\\mathcal{X}} (x_i,U_j) \\right]$ \n \\State Get $\\sum_{j < i} \\mathrm{E}_{U_j\\sim Q_j} \\left[ \\phi_{\\mathcal{X}\\!\\mathcal{X}} (U_j,x_i) \\right]$ \n \\State Get $\\sum_{j} \\mathrm{E}_{U_j \\sim Q_j} \\left[ \\phi_{\\mathcal{X}\\!\\mathcal{I}} (U_j,I) \\right]$ \n \\While{$\\textrm{not converged}$}\n \\State Update $Q_i$ and $Q_{\\mathcal{I}}$ using Eq.~\\ref{eq:Q}\n \\EndWhile\n \\State Send $\\mathrm{E}_{U \\sim Q_i} \\left[ \\phi_{\\mathcal{X}\\!\\mathcal{X}} (x,U) \\right]$\n \\State Send $\\mathrm{E}_{U \\sim Q_i} \\left[ \\phi_{\\mathcal{X}\\!\\mathcal{X}} (U,x) \\right]$\n \\State Send $\\mathrm{E}_{U \\sim Q_i} \\left[ \\phi_{\\mathcal{X}\\!\\mathcal{I}} (U,I) \\right]$\n \\EndWhile\n \\end{algorithmic}\n }\n\\end{algorithm}\n\n\n\\noindent Here `Get' and `Send' refer to the message server, and $f(x)$ is a message used later by frames in the same video. The term message server is used for a central process that keeps track of what node in what video sent what message, and distributes them accordingly when requested. In practice, this could be implemented in a multi-machine setup.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1.0\\linewidth]{messages.pdf}\n\\caption{Evolution of prediction with increasing messages passes. The first row shows the initial prediction for the category {\\em tidying with a broom} without any message passing, where darker colors correspond to higher likelihood, blue is then an increase in likelihood, and brown decrease. In the first message pass, the confidence of high predictions gets spread around, and eventually increases the confidence of the whole prediction.}\n\\label{fig:messages}\n\\end{figure}\n\n\n\\subsection{Learning}\n\nTraining a deep CRF model requires calculating derivatives of the objective in terms of each of the potentials in the model, which in turn requires inference of $P(X,I|V)$. The network is trained to maximize the log-likelihood of the data $l(X) = \\log \\sum_I P(x,I|V)$. \nThe goal is to update the parameters of the model, for which we need gradients with respect to the parameters. Similar to SGD, we find the gradient with respect to one part of the parameters at a time, specifically with respect to one potential in one frame. That is, $\\phi_\\mathcal{X}^i(\\hat{x})$ instead of $\\phi_\\mathcal{X}(\\hat{x})$.\nThe partial derivatives of this loss with respect to each of the potentials are as follows:\n\\begingroup\\makeatletter\\def\\f@size{8}\\check@mathfonts\n\\def\\maketag@@@#1{\\hbox{\\m@th\\normalsize\\normalfont#1}}%\n\\begin{align}\n\\frac{\\partial l(X)}{\\partial \\phi_\\mathcal{X}^i(\\hat{x})} &=\n\\mathbf{1}_{x=\\hat{x}} - Q_i(\\hat{x}) \\label{eq:gradients1}\\\\ \n\\frac{\\partial l(X)}{\\partial \\phi_{\\mathcal{X}\\!\\mathcal{I}}^i(\\hat{x},\\hat{I})} &= \\frac{\\exp \\sum_j \\phi_{\\mathcal{X}\\!\\mathcal{I}}(x_j,\\hat{I})}{\\sum_I \\exp \\sum_j \\phi_{\\mathcal{X}\\!\\mathcal{I}}(x_j,I)}\\mathbf{1}_{x=\\hat{x}} - Q_i(\\hat{x}) Q_{\\mathcal{I}}(\\hat{I}) \\label{eq:gradients2}\\\\\n\\frac{\\partial l(X)}{\\partial \\mu^i(a,b)} &= \n\\sum_{j>i} \\mathbf{1}_{x=a} k(v_i,v_j) - Q_i(\\hat{x}) \\sum_{j>i} Q_{\\mathcal{I}}(b) k(v_i,v_j) \\nonumber \\\\\n&+ \\sum_{j