diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzntlt" "b/data_all_eng_slimpj/shuffled/split2/finalzzntlt"
new file mode 100644--- /dev/null
+++ "b/data_all_eng_slimpj/shuffled/split2/finalzzntlt"
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n\nHost-parasite systems are widespread in nature. Understanding the selection mechanisms underlying virulence and pathogen transmissibility is essential to control outbreaks and disease progression in the long term~\\cite{messinger_consequences_2009}. Humans, livestocks, and crops are the most studied hosts due to their relevance for human activities. Despite some commonalities, different hosts differ profoundly in strategies that have coevolved with pathogens, such as immunity or avoiding behaviour, but also in intrinsic features, such as their degree of mobility. As evidence supports, pathogens specific of different host categories display diverse adaptive strategies~\\cite{dodds:2010,brunke:2014,lucia-sanz:2017}. \n\nEarly models in epidemiology were developed two centuries ago, at a time when empirical data was scarce~\\cite{hethcote:2000}. A modelling standard was much later set by compartmental epidemiological models~\\cite{anderson_coevolution_1982}, which classify the individuals in a population in a few states (susceptible, infected, recovered) and are usually formulated as a set of differential equations at the mean-field level --i.e. assuming a well-mixed population. In the last forty years, many efforts have been devoted to devise and analyse mathematical and computational models able to explain and predict the evolution of viral traits~\\cite{ewald_host-parasite_1983,anderson_coevolution_1982,alizon_transmissionrecovery_2008,kamo_role_2007}. Recently, the ability to obtain and process big data, along with the development of detailed metapopulation models, has allowed for the achievement of reasonably accurate estimations of epidemic spreading in the short term~\\cite{tizzoni_real-time_2012,zhang_spread_2017}.\n\nPathogens are in need of suitable strategies in order to persist in a host population, in a continuous arms race with the defenses of the host. Understanding which strategies are selected along evolution may help developing improved epidemiological models that include the long-term behaviour of the strains. The classical theory~\\cite{alizon_transmissionrecovery_2008,lion_evolution_2015} assumed that new pathogens, not adapted to the host, are more virulent (where virulence is understood as the number of deaths caused by the disease). Indeed, mean-field epidemiological models predict that the base reproductive number $R_{0}$, defined as the average number of infections produced by an infected individual, tends to increase along evolution~\\cite{anderson_coevolution_1982}. Maximization of $R_{0}$ in a finite population, however, exhausts the pool of susceptible individuals, leading to pathogen extinction. A possible way out of this runaway process would be for the pathogen to become less virulent as it co-evolves with its host; however, a revision of experimental data shows that this hypothesis does not hold in general~\\cite{ewald_host-parasite_1983,anderson_coevolution_1982}. As an alternative solution, it was proposed that pathogens should be subjected to a trade-off between virulence and transmission, such that virulent strains have a lower transmission rate~\\cite{alizon_transmissionrecovery_2008}. The equilibrium between an aggresive, virulent strategy and a low-virulence, highly transmissive strategy results in a limit to the reproduction speed, i.e. a maximum value of the base reproductive number~\\cite{alizon_transmissionrecovery_2008,kamo_role_2007}, yielding a possible explanation for the existence of strains with intermediate values of $R_0$. On the other hand, the trade-off hypothesis does not explain avirulent diseases. For this case, alternative trade-offs have been proposed, such as the recovery-transmission trade-off~\\cite{alizon_transmissionrecovery_2008,shrestha_evolution_2014}. The trade-off hypothesis has been questioned in the last years for several reasons. One of the most important issues regards a definition of virulence able to link observed data with theoretical models. As a consequence, as of yet, there is little available empirical evidence supporting the existence of such physiological trade-offs~\\cite{alizon_virulence_2009}. \n\nThe transmission of pathogenic diseases rarely occurs in a host population that is well mixed, such that an important issue in infection propagation is the role played by structured host populations~\\cite{lion:2008} and the evolutionary parameters characterizing pathogen strategies~\\cite{sun_pattern_2016}. Many models have studied the effect of space and network structure in disease spreading~\\cite{watts_multiscale_2005, keeling_implications_2005}. There is ample theoretical evidence that the evolutionarily stable traits of pathogens significantly differ in well-mixed or spatially structured scenarios~\\cite{messinger_consequences_2009} and that space has important effects on evolutionary dynamics~\\cite{boots:2004,webb:2007,boerlijst:2010} as well as on the effects of different functional forms of the {\\it a priori} trade-off between transmission and virulence~\\cite{webb:2013}.\n\nSeveral studies suggest that trade-offs do not necessarily come from the physiology of the pathogens, but might arise as an emergent property of a spatially structured host-pathogen system~\\cite{haraguchi:2000,rauch:2002,ballegooijen_emergent_2004,kamo_role_2007,lion_evolution_2015}. The phenomenology of spatially (self-)structured host-pathogen systems has important features that sets them apart from mean-field approaches. The exhaustion of hosts by highly pathogenic variants becomes localized both in space and in time; through evolution, a hierarchy of time scales related to mutants of decreasing virulence defines a collective and time-dependent invasion fitness~\\cite{rauch:2002}. Also, it has been found that the spatial structure is enough to select for intermediate virulence values~\\cite{kamo_role_2007}, and that the recovery-transmission trade-off can emerge due to spatial structuring of the hosts in a lattice, even when the infection characteristic time and the transmission rate evolve independently~\\cite{ballegooijen_emergent_2004}.\n\nThe effects of spatial structure on the evolutionary parameters of pathogens and the possible existence of emergent trade-offs has recently received empirical support.  Experiments with two variants of the phage $\\lambda$ infecting {\\it E. coli} have demonstrated that spatial structuring impedes the spread of the virulent variant, and selects for a prudent infection strategy~\\cite{berngruber:2015}. Also, it has been shown that high host availability favours parasites with lower virulence and higher transmissibility, while low host availability selects for the contrary relationship, experimentally demonstrating the existence of a trade-off promoted by spatial structure~\\cite{leggett:2017}. \n\nThough a few studies have emphasized the important role played by host-mobility fluxes under a metapopulation structure~\\cite{poletto_characterising_2015,poletto_host_2013}, host mobility is not typically included in evolutionary models. Interestingly, the effects of host mobility on pathogen infectivity have been empirically evaluated in a series of experiments where insect larvae living in environments that limited their mobility to different extents were infected with a species-specific virus~\\cite{boots:2007}: in agreement with expectations, infectivity was reduced concomitantly with host mobility. \n\nHere, we address the question of the limits to the emergence and the structural stability of the trade-off between transmissibility and virulence under a variety of scenarios that represent realistic features of different host or pathogen types: local diffusion and long-range jumps of hosts (as for cattle or foraging animals, e.g.), high mutation rates (as for virus or viroids), and metapopulation structure. To this end we implement the model by Ballegooijen and Boerlijst~\\cite{ballegooijen_emergent_2004} and first characterize, in terms of invasibility criteria, the feedback evolutionary mechanism that selects for a curve of constant $R_{0}$. We show that this curve is a critical self-evolved boundary~\\cite{rand_invasion_1995}, whose functional form we calculate in a one-dimensional approximation, that separates two regions with qualitatively different spatial structure. While that evolutionary outcome is mostly robust under variations in the parameters, it is very fragile under host mobility. Finally, we extend our main results to a metapopulation model, showing that the main mechanism of evolution in the metapopulation is infection propagation within each subpopulation. Self-structured and disordered (mean-field) states are characterized by different average lifetimes that entail an asymmetric invasion likelihood such that, in metapopulations with mobile hosts, convergence to the critical boundary is favoured by the metapopulation structure. \n\n\\section{Model}\n\nFollowing~\\cite{ballegooijen_emergent_2004}, we define a host-pathogen dynamical model on a two-dimensional lattice with Moore neighbourhood (each node has $8$ neighbours) and periodic boundary conditions. Each individual occupies a lattice site that can be either in a susceptible ($S$), infected ($I$) or recovered ($R$) state; transitions between these states are controlled by the stochastic reactions\n\n\\begin{subequations}\n\\begin{align}\nS+I\\overset{\\beta}{\\rightarrow}2I \\label{eq:model1} \\\\\nI\\overset{\\tau_{I}}{\\rightarrow}R \\label{eq:model2} \\\\\nR\\overset{\\tau_{R}}{\\rightarrow}S \\label{eq:model3} \\, ,\n\\end{align}\n\\end{subequations}\nwhere $\\beta$ is the rate of infection, and $\\tau_{I}$ and $\\tau_{R}$ are, respectively, the infection and recovery times (see Fig.~\\ref{fig:model}). Each node has an internal time counter in order to trigger reactions (\\ref{eq:model2}) and (\\ref{eq:model3}). In the well-mixed (mean-field) scenario reactions (\\ref{eq:model1})-(\\ref{eq:model3}) represent an SIRS model with delay. As in~\\cite{ballegooijen_emergent_2004}, we employ a fixed time-step algorithm with synchronous update for numerical simulations. At each time step, we loop over the nodes. If a node is in the susceptible state, we identify its infected neighbours, compute the total infection rate, and infect it with the corresponding probability. If a node is in the infected or recovered states, we check if the internal counter is greater than the infection or recovery times, respectively, in order to change its state. The detailed procedure is explained in Methods.\n\nEach infected node $j$ has its own transmission rate $\\beta_{j}$, as well as its own infection period $\\tau_{Ij}$. A strain is defined by a pair $s=\\left(\\beta,\\tau_{I}\\right)$ with no {\\it a priori} imposed trade-off. Both parameters are independently mutated at each time step with probability $\\mu\\Delta t$, where $\\mu$ is the mutation rate and $\\Delta t$ is the timestep. We assume superinfection exclusion, so a host individual in state $I$ cannot be infected by a second strain.\n\nTo introduce host local diffusion we employ the Toffoli-Margolus algorithm~\\cite{toffoli_cellular_1987}. This algorithm divides the lattice in $2\\times2$ squares that rotate either clockwise or counterclockwise. Briefly, squares are taken starting alternatively from $(0,0)$ and $(1,1)$ so the system becomes mixed, as illustrated in Figure~\\ref{fig:toffoli}, regardless the state of the site.\n\nIn a lattice, diffusion is defined as $D=\\Gamma(\\Delta x)^{2}$, where $\\Gamma$ is the rate at which particles hop and $\\Delta x=1$ the distance between sites. Since every time the mixing algorithm is applied every particle hops to a neighbouring site, $\\Gamma$ can be fixed in order to have the desired diffusion coefficient $D$ as $\\Gamma=1\/(D\\Delta t)$.\n\nTo recreate a metapopulation scenario, we use a scheme similar to that implemented in previous studies~\\cite{poletto_characterising_2015,poletto_host_2013}. We create a network of lattices of small sizes and connect them in a metapopulation network. After updating all lattices following the previous algorithm, we check the state of each population. Each host (for which the state of a site acts as a proxy) can jump with probability $\\lambda\\Delta t\/N$, where $\\lambda$ is the jump rate. If the jump occurs, we randomly select a site in a neighbouring lattice, as shown in Figure~\\ref{fig:metapop}. Then, both sites (their states) are exchanged. We fix $\\lambda=0.01$. \n\n\\section{Results}\n\n\\subsection{Spatial self-structuring results in an emergent trade-off between transmissibility and infectivity}\n\nWe start by simulating infection propagation and evolution without host mobility, i.e. with $D=0$, to recall the core results of Ballegooijen and Boerlijst's model~\\cite{ballegooijen_emergent_2004}. This delayed SIRS model has base reproductive number $R_{0}=8\\beta\\tau_{I}$, independent of $\\tau_R$, in the mean-field approximation. For fixed parameter values, the system exhibits different kinds of patterns, as depicted in Figure~\\ref{fig:patterns}a-d. Low values of $R_{0}$ produce short-lived local infection bursts, while high values of $R_{0}$ eventually lead to the formation of spiral waves, a pattern characteristic of two-dimensional excitable media~\\cite{cross_pattern_2009}. Thus, there are two different ``phases'', with wave patterns and without wave patterns, that we call ordered or self-structured and disordered or mean-field, respectively. A ``phase transition'' or ``critical boundary'' separates these two regimes.\n\nWhen the system is free to evolve through changes in virulence and transmissibility parameters (keeping the recovering time $\\tau_R$ fixed), it converges to trajectories of constant $R_{0}^{ev}=\\left(6.623\\pm0.003\\right)$ that, after a transient period of variable duration, become independent of the initial conditions.\n\nThe evolutionary trajectory of the system is illustrated in Figure~\\ref{fig:patterns}e. This result is in agreement with~\\cite{ballegooijen_emergent_2004}, where it was shown that the trade-off is a by-product of the spatial self-structuring of the system, and where the quantity that seems to be under positive selection is the frequency of emission of infection waves, and not $R_0$. Though not explicitly mentioned in~\\cite{ballegooijen_emergent_2004}, where the maximum allowed value of the infectivity was $\\beta=4$, the evolutionary trajectory continues indefinitely to higher values of $\\beta$ and lower values of $\\tau_I$ at an increasingly slower but non-arresting pace.\n\n\\subsection{Selection for higher frequency of emission of infection waves only succeeds at the local scale}\n\nIn order to delve into the feedback mechanism that drives the evolutionary process towards a fixed base reproductive number, we undertake two simulation experiments to evaluate the ability to invade of different strains once the system is spatially organized. No parameter evolution is considered here, since our aim is to quantify to which extent spatial pattern formation enhances or hinders the invasion of analogous populations. To this end we select a focal population with parameters corresponding to each of the four situations represented in Figure~\\ref{fig:patterns}a-d. Let us call this strain $s_{0}\\equiv\\left(\\beta,\\tau_{I}\\right)$. Then, we also select four strains that are nearby in an evolutionary sense. That is, if the evolutionary process would be on, these strains would be one mutational step away from fhe focal population; their parameters are $s_{++}\\equiv\\left(\\beta^{+},\\tau_{I}^{+}\\right)$, $s_{+-}\\equiv\\left(\\beta^{+},\\tau_{I}^{-}\\right)$, $s_{-+}\\equiv\\left(\\beta^{-},\\tau_{I}^{+}\\right)$, and $s_{--}\\equiv\\left(\\beta^{-},\\tau_{I}^{-}\\right)$, where we have defined $\\beta^\\pm = \\beta \\pm \\Delta \\beta$ and $\\tau _I ^\\pm = \\tau _I \\pm \\Delta \\tau _I$ to simplify the notation. All possible competitions between the focal strain $s_0$ and its mutants are assayed, and each competition is performed 10 times.\n\nNote that the five populations (one focal and four nearby mutants) can be ordered with respect to their base reproductive number. In the simulations, where initial conditions ensure that $\\beta > \\tau_I$ for all times, the ordering is given by\n\n\\begin{equation}\n  \\label{eq:R0}\nR_{0}\\left(s_{++}\\right)>R_{0}\\left(s_{-+}\\right)>R_{0}\\left(s_{0}\\right)>R_{0}\\left(s_{+-}\\right)>R_{0}\\left(s_{--}\\right) \\, ,\n\\end{equation}\nso $s_{++}$ and $s_{-+}$ increase the base reproductive number of the focal population $s_0$, while $s_{+-}$ and $s_{--}$ decrease it. On the other hand,  if $\\beta < \\tau_I$, the ordering is $R_{0}\\left(s_{++}\\right)>R_{0}\\left(s_{+-}\\right)>R_{0}\\left(s_{0}\\right)>R_{0}\\left(s_{-+}\\right)>R_{0}\\left(s_{--}\\right)$.\nIn a mean-field scenario, this ordering coincides in either case with the relative advantage of one population over the others and, thus, it determines the mutual ability to invade and predicts the outcome of competition experiments in a well-mixed scenario.\n\nHowever, results in~\\cite{ballegooijen_emergent_2004} suggest that, in cases where the population is spatially structured, the relative advantage corresponds not to the population with the larger $R_0$, but to the one with the higher frequency $w$ of emission of infective waves. The emission frequency is obviously larger for those strains with a faster infection rate which simultaneously produce waves with narrower fronts, but these two quantities do not bear a straight relationship with parameters $\\tau_I$ and $\\beta$.\n\nIn order to better understand how the invasibility criteria might change in spatially structured systems, let us briefly explore the one-dimensional version of model~\\cite{ballegooijen_emergent_2004}. The frequency $w_{1D}$ is the inverse of the average time elapsed between two consecutive infection events. A node with an infected neighbour becomes infected after a typical time $t_{\\beta}=\\beta^{-1}$, and stays itself infected for a time $\\tau_I$. Then it recovers to become susceptible again after a time $\\tau_R$. Therefore,\n\n\\begin{equation}\n  \\label{eq:w}\n  w_{1D}=\\frac{1}{1\/\\beta+ \\tau_I+\\tau_R} \\, ,\n\\end{equation}\nand the mutants and the focal population display the following ordering\n\n\\begin{equation}\n  \\label{eq:worder}\n  w_{1D}\\left(s_{+-}\\right)>w_{1D}\\left(s_{--}\\right)>w_{1D}\\left(s_{0}\\right)>w_{1D}\\left(s_{++}\\right)>w_{1D}\\left(s_{-+}\\right) \\, ,\n\\end{equation}\nwhich yields invadability criteria different from Eq.~(\\ref{eq:R0}). Again, as for the basic reproductive number, the precise ordering of the mutants $s_{--}$ and $s_{++}$ depends on the relative values of the parameters. The order in eq.~(\\ref{eq:worder}) corresponds to the case studied in simulations, $\\beta>\\tau_I$. On the other hand, if $\\beta<\\tau_I$,  then $w_{1D}\\left(s_{+-}\\right)>w_{1D}\\left(s_{++}\\right)>w_{1D}\\left(s_{0}\\right)>w_{1D}\\left(s_{--}\\right)>w_{1D}\\left(s_{-+}\\right)$. This calculation, however, cannot be straightforwardly extended to the two-dimensional case.\n\n\\subsubsection{Invadability experiment 1}\nTo go beyond the one-dimensional case, we considered a two-dimensional lattice of size $2L\\times L$; the two parts of the space are not connected initially. $s_0$ and each of its mutant strains are picked up pair-wise and placed either at the left or right half of the lattice. After a fixed time, such that spatial patterns have developed according to the parameters chosen, both parts of the space are allowed to interact.\n\nIn all simulations performed under the previous conditions, the system was eventually invaded by the strain with higher $R_{0}$, independently of any other condition. Strains $s_{++}$ and $s_{-+}$ were systematically selected in competition with $s_0$, while strains $s_{+-}$ and $s_{--}$ were always removed from the system, as would be predicted by a mean-field approximation. This is an {\\it a priori} unexpected result that apparently contradicts the dynamics of evolutionary trajectories in the spatially structured model. \n\n\\subsubsection{Invadability experiment 2}\nTake strain $s_{0}$ and let the system run enough time to develop patterns in a large $L \\times L$ lattice. Then, substitute any infected individual inside a randomly chosen area of size $10\\times10$ sites with strains of one of the nearby mutants. In this case, we observed two different behaviours:\n\n\\begin{enumerate}\n\\item If the focal strain $s_0$ was not able to develop patterns (that is, it is located in the disordered region of the parameter space, with a base reproductive number below $R_0^{ev}$), the mutant strain invades the whole system if its $R_{0}$ is higher: $s_{++}$ and $s_{-+}$ are at an advantage and therefore are selected in front of $s_0$;\n\\item If $s_0$ has clearly developed patterns (with a base reproductive number above $R_{0}^{ev}$), the only mutant strain able to invade the system is $s_{+-}$, following the direction observed in the evolutionary curve. \n\\end{enumerate}\n\n\\subsubsection{The critical boundary emerges as an equilibrium between two different selection mechanisms}\n\nThe results above highlight that the likelihood to invade a spatially organized population depends on the size and structure of the invading population. Very often, this size is small because it is a single mutant individual or a small sample of individuals from disconnected populations that attemp the invasion. If this is so, the scenario in our invadability experiment 2, whose dynamical properties coincide with those leading to the emergence of the trade-off, is the applicable one.\n\nLet us return to the evolutionary trajectory in Figure~\\ref{fig:patterns}e with the previous results in mind. We see that initial conditions with low $R_{0}$ first increase the base reproductive number by selecting mutants mainly in the $s_{++}$ direction, and later converge to the curve evolving along the $s_{+-}$ quadrant. All regions in the evolutionary trajectory where $\\tau_I$ increases (and also $R_0$, since $\\beta$ always grows) correspond to spatially disordered situations, either because the parameters correspond to the phase with $R_0 < R_0^{ev}$ or because the system is in the initial transient before spatial self-structuring sets in. The increase of $R_0$ progressively drives the system to a new regime where spiral waves start to develop. This qualitative change modifies the criteria for invadability, and selection for waves of higher frequenty $w$ sets in.\n\nAt this point, it seems reasonable to assume that the critical boundary results from the equilibration of two mechanisms: selection for larger $R_0$ in the disordered phase and selection for higher $w$ in the ordered phase (see Figure~\\ref{fig:patterns}f). Even though the frequency of emission is a complex function of the parameters in two dimensions, and results from 1D cannot be straightforwardly extrapolated, in general, to higher spatial dimensions, let us use the functional forms of $w_{1D} (\\beta,\\tau_I)$ and $R_0(\\beta,\\tau_I)$ previously derived to give an estimation of the curve where the two surfaces cross. Note that $R_0$ grows in the $s_{++}$ direction. If our understanding of the evolutionary feedback is correct, this estimation should resemble the numerical boundary $R_0^{ev}$. The relationship between $\\beta$ and $\\tau_I$ along the critical boundary is defined through $w_{1D}=R_0$,\n\n\\begin{equation}\n  \\label{eq:1dExact}\n  n \\tau_I \\beta = \\frac{c}{1\/\\beta+ \\tau_I+\\tau_R}  \\, \n\\end{equation}\nwhere $c$ is a constant required for correct dimensionalization and $n$ is the number of neighbours, which yields\n\n\\begin{equation}\n\\label{eq:expansion}\n\\beta = \\frac{c\/n - \\tau_I}{\\tau_I (\\tau_I+\\tau_R)} \\simeq \\frac{c}{n \\tau_I \\tau_R} + \\mathcal{O}\\left( \\frac{1}{\\tau_I ^2} \\right) \\, .   \n\\end{equation}\n\nFor $\\tau_R=1$, to first order in $\\tau_I$ we get $n \\beta \\tau_I \\sim c$, where we can identify $n=8$ and $c=R_0^{ev}$. The selection mechanism described puts the system at the edge of wave formation: higher $R_0$ strains tend to be selected in the disordered phase, while higher values of $w$ are selected in the spatially structured phase, such that the curve of constant $R_0$ is where these two mechanisms are at equilibrium (see Figure~\\ref{fig:patterns}f). This boundary also represents the limit of validity of mean-field calculations, providing a self-consistent, {\\it ad hoc} explanation of why its functional form verifies $R_0 =8 \\beta \\tau_I$. \n\n\\subsection{Spatial self-structuring is fragile}\n\nThe boundary $w_0^{ev}=R_0^{ev}$ separates two phases with qualitatively different spatial structure. Selection mechanisms, as revealed by the invasion criteria in either phase, are different and of opposing sense regarding mutations in $\\tau_I$, causing an evolutionary feedback loop that eventually drives the system to a self-evolved phase boundary. The emergence of the trade-off is critically dependent on the development and persistence of spatial self-structuring. Are there conditions under which the latter is not possible, even if evolutionary parameters are in the ordered region? In this section we explore the stability of the emergent trade-off under changes in the mobility of hosts as well as in two model parameters that have been kept constant so far: the mutation rate $\\mu$ and the recovery time $\\tau_R$.\n\n\\subsubsection{Host mobility prevents spatial self-structuring}\n\nBroadly speaking, infection propagation depends on the degree of mobility of hosts: propagation speed, endemicity or optimal evolutionary parameters vary whether hosts are sessile, diffuse locally or perform high-distance jumps. In this section, we explore how locally diffusing hosts and hosts able to jump to arbitrary sites in the lattice affect the formation of spatial structures.  \n\nDiffusing hosts are modeled by means of the Toffoli-Margolus algorithm (see Methods). In the high-diffusion limit, the system becomes well-mixed and the expectation is that it behaves as in mean-field, thus increasing its average $R_{0}$~\\cite{anderson_coevolution_1982}. Indeed, simulations with high diffusion follow an evolutionary trajectory along which $\\beta$ and $\\tau_{I}$ steadily increase. The process continues until all individuals become infected, eventually causing the extinction of the pathogen.\n\nOn the other hand, sufficiently low diffusion should recover the $D=0$ trajectory we analysed before. Therefore, there should be an intermediate value of the diffusion where the behaviour crosses over from $R_0=R_0^{ev}$ to an ever increasing $R_0$. Our simulations show that, for intermediate $D$ values, the convergence of the system to either phase depends on the initial conditions, as depicted in Figure~\\ref{fig:dif}. For an initial fixed value of $\\beta_{0}$, there exist a critical $\\tau_{I0}^{c}$ such that any $\\tau_{I0}<\\tau_{I0}^{c}$ will exhibit an evolutionary trajectory identical to $D=0$, while for $\\tau_{I0}>\\tau_{I0}^{c}$ the parameters will diverge as predicted by the mean-field theory. The position of the critical point $\\tau_{I0}^{c}=\\tau_{I0}^{c}\\left(\\beta_{0}\\right)$ increases as $\\beta_{0}$ increases. The persistent perturbation caused by diffusion distorts the spatial structure, which is developed only if diffusion is slower than wave formation speed. For a fixed value of $\\beta$, wave formation and propagation speed decrease as $\\tau_{I}$ increases. As a consequence, the system approaches the mean-field behaviour when $\\tau_{I}$ increases, as patterns are suppressed. For higher values of $\\beta_{0}$, it is more difficult to sufficiently distort the patterns, such that the value of the critical point $\\tau_{I0}^{c}$ increases. Higher values of $D$ lower the value of the critical point: for hosts with sufficiently high local mobility, pattern formation is not possible and all the phase space is in the mean-field regime.\n\nThe situation is qualitatively analogous if, instead of local diffusion, we assume that nodes can have arbitrary neighbours in the lattice --rather than just nearest neighbours-- with some given probability. Actually, the effect of long-distance transmission was already discussed in~\\cite{ballegooijen_emergent_2004}, where it was pointed out that mixing up to $2\\%$ of contacts yielded similar results, i.e. spatial structuring and an emergent trade-off with a slightly different value for $R_0^{ev}$. Our simulations indicate that, as in the case with diffusion, relatively low values of the fraction of long-distance contacts (below $10\\%$ in all considered cases) prevent the formation of waves; the precise value causing the transition depends on the initial parameters. Figure \\ref{fig:long_range} shows, for a fixed initial condition, the transition between the $D=0$ and the mean field behaviour, which is in fact analogous to the diffusive case.\n\n\\subsubsection{Waves develop in a finite range of $\\tau_R$ values}\n\nThe intrinsic self-excitability of Ballegoijeen and Boerlijst's model~\\cite{ballegooijen_emergent_2004} lies at the origin of the emerging spatial waves. This property is lost in simpler susceptible-infected (SI) models, which do not consider a recovery period after infection. In other words, spatial waves do not develop in the limit $\\tau_R \\to 0$. From a geometrical perspective, $\\tau_R>0$ has the effect of separating subsequent wave fronts, thus allowing the formation of well-defined infection waves.\n\nInterestingly, in the simulations, changes in $\\tau_R$ do not seem to affect the critical value $R_0^{ev}$. Nevertheless, the analytic reasoning made before suggests that the critical value should change, given the dependence of equations~(\\ref{eq:w}) and~(\\ref{eq:expansion}) with $\\tau_R$. We believe that the one dimensional argument we have used is not able to capture the full phenomenology observed in two dimensions. \n\nEven when $\\tau_R$ does not affect the critical point of the transition, it is relevant for the study of the stability of the patterns. Our computational results show that if $\\tau_R$ decreases, the transient time needed to converge to the curve $R_0=R_0^{ev}$ increases. This is due to the fact that the formation of wave fronts requires larger values of $\\tau_I$ the smaller $\\tau_R$ is. Therefore, for small $\\tau_R$, increasingly larger values of $R_0$ are required in order to develop patterns and start the selection for wave frequency. Conversely, larger values of $\\tau_R$ permit the formation of waves with lower values of $\\tau_I$, thus accelerating convergence to the critical boundary. \n\nWe have numerically studied the limit $\\tau_R \\rightarrow 0$, which corresponds to the conversion of our model to an SI-like model. For low values of $\\tau_R$, we have checked that the system is not able to develop patterns, exhibing the mean-field behaviour as described in the diffusion case. Moreover, we have checked that too large values of $\\tau_R$ also prevent the formation of waves. Intuitively, this is due to the blocking effect caused by individuals remaining for too long in the recovered state, and thus avoiding wave propagation. We cannot discard, however, that this is a finite size effect showing up when $\\tau_R \\simeq L$. \n\nIn summary, considering that self-structuring patterns are possible only for intermediate values of $\\tau_R$, too large or too small values of the recovery rate also jeopardize the stability of the emergent trade-off.\n\n\\subsubsection{Evolutionary trajectories are weakly affected by changes in the mutation rate}\n\nPathogenic organisms display a range of mutation rates that spans at least seven orders of magnitude, from the barely $10^{-10}$ substitutions per nucleotide and replication cycle of pathogenic fungi to the $10^{-4}$ of most RNA viruses~\\cite{sniegowski:2000}. Viroids, circular non-coding RNA molecules that are pathogenic to plants, affecting especially crops, might present mutation rates up to over $10^{-3}$ substitutions per genome copied, thanks to the small size of their genomes (a few hundred nucleotides)~\\cite{gago:2009}.\n\nLow mutation rates are not an issue in the context of the model here studied, since they cannot disrupt the emergence of the trade-off. In a more realistic scenario, though, low $\\mu$ entails long transients that would delay the convergence to the critical curve. The pertinent question, therefore, is if large values of $\\mu$ prevent in any way the development of spatial self-structuring. Actually, our simulations where performed for $\\mu=0.01$, which is not a small value for a phenotypic mutation rate, as implemented in the model. We have verified that mutation rates up to $\\mu=0.5$ do not prevent convergence to $R_0^{ev}$, though they increase the size of fluctuations away from the critical curve. Additionally, one could consider changes in the parameters that, in agreement with empirical observations~\\cite{eyre-walker:2007}, would be randomly drawn from a fat-tailed probability distribution, that is where mutations causing large changes in phenotype are not exceedingly rare. Though this possibility has not been tested, we do not expect it to modify the evolutionary trajectories in the light of our previous results. \n\n\\subsection{Role of a metapopulation structure in the evolutionary fate of populations}\n\nOur metapopulation model consists in a network of lattices that can exchange the states of a randomly chosen pair of individuals (see Methods and Figure~\\ref{fig:metapop}). Alternatively, we have also simulated a situation where infection could be propagated through vectors, and where an infected individual tries to transmit the disease to a second, randomly chosen individual in a different lattice. Since results are indistinguishable in these two formulations, here we present results for the first case (swapping of individual states). Simulations are performed with a fully-connected metapopulation (as in Figure~\\ref{fig:metapop}a).\n\n\\subsubsection{The metapopulation structure is irrelevant for sessile hosts}\n\nWe performed simulations of a metapopulation where all the patches were in the no-diffusion phase or in the mean-field phase, respectively. The evolutionary trajectories are identical to the ones displayed by a single population in the no-diffusion regime or in the mean-field. This is because all the subpopulations are subject to the same selection mechanism. In fact, evolution happens locally when the patches are in any of these two limits. This result is easy to understand if we think of two connected populations, A and B. Let us say A is in the high diffusion regime and B in the $D=0$ regime. Since the base reproductive number in B has a low value, a strain that jumps from B to A will not invade the well-mixed population, where the mechanism of selection relies on $R_0$. Moreover, a strain from A to B would not invade either, since it has a low wave emitting frequency, which is the main selection mechanism in B. At the end, the evolutionary trajectory in each population is the same as the isolated system's trajectory, since the dynamics only depends on the base reproductive number of the perturbation and the spatial structure, and not on where this strain comes from. Both populations will evolve with no apparent interaction between them. Therefore, evolutionary dynamics depends only on the local spatial structure of the populations, and not on the large scale metapopulation. Figure~\\ref{fig:metapop_results} illustrates this point.\n\n\\subsubsection{Well-mixed states are shortly lived in small lattices}\n\nWe have seen that, when hosts are mobile, the evolutionary dynamics does not always converge to the critical boundary but, depending on the initial conditions and the strength of diffusion, it might enter the runaway regime where $R_0$ steadily increases. The eventual fate of such populations is a pathogen-free system with all individuals in the susceptible state. While in the absence of other populations that may act as reservoirs of the pathogen this is the final state, pathogen-free populations can be rescued if a metapopulation structure is present. At this point, therefore, the fact that populations in the mean-field regime have a finite lifetime becomes an important evolutionary feature. This is in contrast with self-structured populations, which are in a state of endemic infection that is sustained in time. \n\nWith this motivation in mind, we have studied the average lifetime of populations for intermediate values of the diffusion as a function of their lattice size. Here, lifetime is defined as the number of timesteps required for all the individuals to reach the susceptible state. As expected, populations in the ordered phase did not decay in any of the performed simulations. In contrast, the lifetime of populations as a function of their lattice size can be fitted in the mean-field phase to a power-law with exponent $(2.6\\pm0.6)$ (Figure~\\ref{fig:scaling}). In the limit of infinitely large systems, the mean-field phase is stable and infection can survive for very large times whose duration diverges as $L \\to \\infty$.\n\n\\subsubsection{The metapopulation structure favours the emergence of the evolutionary trade-off if hosts diffuse}\n\nThe results in the previous section quantify differences in the lifetime of subpopulations that have reached the critical boundary, which are self-structured and thus stable in time, or entered the mean-field behaviour, therefore with a finite lifetime that depends on their size. At some point, the latter will become fully susceptible and can be re-infected by a neighbouring population. When hosts are mobile, the reinfected lattice can either be attracted again to the mean-field behaviour or develop waves and converge to the critical boundary. The likelihood of either outcome is dependent on the initial conditions, as we have shown. However, the initial conditions are not arbitrary now, but correspond to the state of a neighbouring lattice, which has already evolved to one of the two possible states. If the lattice from which the infecting individual is drawn is the mean-field regime, the newly infected lattice will also fall into that disordered phase, and again collapse in finite time to the fully susceptible situation. Instead, if the infecting individual is drawn from a subpopulation that has converged to the critical value $R_0^{ev}$, the initial conditions are such that the newly infected lattice will develop spatial patterns and the emergent trade-off. The ability of a neighbouring individual to infect follows the criteria derived for mutants in former sections (Figure~\\ref{fig:metapop_results}). \n\nIn a metapopulation structure, however, the global process functions as a ratchet. Once a lattice has settled in a self-structured state with fixed base reproductive number $R_0^{ev}$, it will not be kicked out of it by any attempt of invasion from neighbouring subpopulations. At the same time, populations in the disordered state regularly collapse until they are infected in such conditions that they fall into the ordered phase, at which point their dynamics are stable and long-lived. Therefore, a metapopulation structure selects for a globally ordered phase due to the finite lifetime of any subpopulation in the disordered phase, where $R_0$ diverges. \n\n\\section{Discussion}\n\nIn this work, we have revisited a spatial self-structuring model of host-pathogen evolution where an emergent trade-off between infectivity and transmissibility had been described~\\cite{ballegooijen_emergent_2004}. We have shown that an evolutionary feedback mechanism here characterized poises the system to a critical boundary where selection on the base reproductive number and selection on the frequency of emission of infective waves equilibrate. This critical, self-evolved state only emerges if the conditions of the system are such that spatial ordering in the form of epidemic waves can set in. The critical boundary separates two regions characterized by spatial order\/disorder through a phase transition where incipient spatial waves wax as disorder wanes. The precise, quantitative nature of the phase transition, however, needs to be formally characterized in future research. \n\nClassical definitions of fitness often fail in spatially extended evolutionary competition, as the inability of $R_0$ or $w$ to predict by themselves the winner in an invasion here demonstrates. Indeed, there have been other studies clearly pointing out that, in spatially structured evolving populations, the strategy of maximizing $R_0$ is not adaptive in the long run, either because fitness depends on the time-scale (different strategies are successful at different times)~\\cite{rauch:2002} or because, at odds with mean-field scenarios, a finite fraction of immune hosts might induce pathogen extinction even if $R_0$ is arbitrarily large~\\cite{cuesta:2011}. \n\nConvergence to the critical boundary is a far from trivial issue. First, the emergence of the trade-off can be severely delayed depending on initial conditions and specific evolutionary parameters. Second, the trade-off is structurally unstable under host mobility, such that moderate values of host difusion or of a fraction of long-distance host jumps cause a cross-over to a mean-field behaviour, where the base reproductive number grows unboundedly. Third, the evolutionary stable, finite $R_0^{ev}$ value can be achieved through successive invasions of the resident pathogen only if invasion is attempted at a local scale. This result is highly reminiscent of invasion experiments by mutant viral strains where it was observed that the substitution of the wild type by an in principle fitter mutant did not succeed unless the mutant was seeded above a minimum relative population size threshold~\\cite{delaTorre:1990}. In the latter case, the impossibility to displace the wild type if the invading population was too small was ascribed to its limited genotypic heterogeneity~\\cite{aguirre:2007}. Actually, space might play an important role in the emergence of heterogeneous populations and its properties~\\cite{aguirre:2008}, therefore conditioning also in this respect evolution~\\cite{rauch:2003} and invasion~\\cite{champagnat:2007}. Fourth, convergence to the critical boundary is dependent on the initial evolutionary parameters, a fact that might have important ecological implications. It indicates that the onset of spatial self-structuring, and therefore of the emergent trade-off, is contingent on the life-history of the system, and afects its evolutionary fate. Indeed, the jump of a pathogenic species to a new host is affected by multiple variables, among which ecological factors, viral genetic plasticity and host specificities~\\cite{elena:2011}. While sufficiently long coevolution with the original host species has probably selected for evolutionary parameters permitting coexistence, the aetiology of the disease might be completely different in the new host. If the pathogen turns out to be too virulent (i.e., it starts with a too high $\\tau_I$), persistent infection of this new host is prevented, and it can only infect in bursts that terminate with the death of the local host population and the erradication of the pathogen in a short time. Bursts of infection can however have different origins, and in particular result from prudent infective strategies~\\cite{boerlijst_spatial_2010}.\n\nIn a metapopulation organization, we have shown that the evolutionary dynamics happen at the scale of the local populations, and not at the scale of metapopulations, in contrast with other studies where the dynamics of the metapopulation cannot be extrapolated from the dynamics of a patch (see e.g.~\\cite{poletto_characterising_2015,poletto_host_2013}). Moreover, in the current case evolutionary trajectories in patches can be unrelated if the subpopulation properties are very different, leading to a sort of ecological speciation. Instead, the metapopulation structure is here responsible for driving the system to a globally stable phase in the presence of host mobility. In this study, we have kept the jump rate between subpopulations fixed for all strains, though, together with a complex network structure, it may affect the spreading of disease in the metapopulation at long time scales.\n\nThough attention is typically focused on the evolution of pathogenic traits and on the immune strategies of the host (be they intrinsic, through an immune coevolving system or extrinsic, as in avoiding behaviour), it cannot be discarded that the spatial pattern itself be a feature under selection~\\cite{jackson:2014}. In a different class of systems, it has been shown that disordered states are conductive to extinction, as in the case of spatially extended catalytic hypercycles, where the formation of spiral waves avoids the otherwise lethal effect of parasitic mutants~\\cite{boerlijst:1991}. The question therefore remains, whether selection for long-lived coexistence with the host mediated by the selection of specific spatial patterns may act as an additional force to promote emergent trade-offs.\n\n\\section*{Methods}\nHere we describe in detail the numerical algorithm used to simulate the dynamics of the model, including the diffusion scheme. The steps of the algorithm are:\n\n\\begin{enumerate}\n\\item Initialize the lattice with periodic boundary conditions, and a $5\\%$ of infected individuals, all of them with the same initial strain. Take $t=0$. Initialize internal counters $t_{j}=0$ for all nodes. \n\\item At each timestep, we iterate over the nodes. For node $j$, \n\\begin{enumerate}\n\\item If the node is susceptible, consider the set of infected neighbours $\\Omega_{j}$. Node $j$ is infected with probability $p=1-\\exp\\left(-\\Delta t\\sum_{n\\in\\Omega_{j}}\\beta_{n}\\right)$. Each neighbour has different transmission rate, so we have to select the origin of the infection computing all the probabilities $p_{m}=1-\\exp\\left(-\\Delta t\\beta_{m}\\right)$ and then selecting the source node with probability $q_{m}=p_{m}\/\\sum_{n\\in\\Omega_{j}}p_{n}$ to ensure normalization. If neighbour $m$ was selected, then set $\\beta_{j}=\\beta_{m}$ and $\\tau_{Ij}=\\tau_{Im}$. Using this method, we infect the node at the correct rate, and select the neighbour with a probability according to its transmission rate. \n\\item If the node is infected, check if $t_{j}\\geq\\tau_{Ij}$. In this case, the infection has finished, the node becomes recovered, and internal time is reset to $t_{j}=0$. In other case, with probability $\\mu\\Delta t$ make a mutation $(\\beta_{j},\\tau_{Ij})\\rightarrow(\\beta_{j}\\pm\\Delta\\beta,\\tau_{Ij}\\pm\\Delta\\tau)$, changing $\\beta_{j}$ and $\\tau_{Ij}$ independently. \n\\item If the node is recovered, check if $t_{j}\\geq\\tau_{R}$. If this happens, then make the node susceptible again. \n\\end{enumerate}\n\\item We have $t=k\\Delta t$. If $k=\\Gamma$ then apply the mixing algorithm. \n\\item Make $t=t+\\Delta t\\equiv(k+1)\\Delta t$ and return to step 2 until\nthe desired time. \n\\end{enumerate}\nThe next step is to select the parameters. We choose to fix $\\tau_{R}=1$ as  the basic timescale (except when this parameter is externally varied). All times are relative to this scale. We have fixed $\\Delta t=0.01$, $\\mu=0.01$, $\\Delta\\beta=0.01$ and $\\Delta\\tau=0.01$, as in~\\cite{ballegooijen_emergent_2004}. Lattice size varies through the study and is indicated in each case.  \n\n\n\n\n\n\n\n\n\\bibliographystyle{naturemag} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\nKnown extrasolar planetary systems orbiting main sequence stars consist of a few large \nplanets such as Jupiter \\citep{cum08}, and\/or, as demonstrated by studies of debris disks, \nnumerous minor planets analogous to solar system asteroids and Kuiper belt objects \n\\citep{zuc01}.  Apparently, the assembly of planets from planetesimals is inefficient, and \nstars possess complex populations of orbiting material (see \\citealt{ida08} and references\ntherein).\n\nRelative to main sequence stars, white dwarfs offer two advantages for the study of extrasolar \nplanetary systems. First, white dwarfs are earth-sized and their low luminosities permit the \ndirect detection of infrared emission from cool self-luminous companions such as brown \ndwarfs and massive jovian planets \\citep{far08a,bur06,far05a,zuc87a,pro83}.  Second, cool \nwhite dwarfs should be atmospherically free of heavy elements \\citep{zuc03,alc86,paq86}, \nand those stars with planetary system remnants can become spectroscopically contaminated \nby small, but detectable, amounts of accreted material.  Analysis of metal-polluted white dwarfs \nenables an indirect, yet detailed and powerful compositional analysis of extrasolar planetary \nmatter; \\citet{zuc07} found that the abundances in the spectacularly metal-rich white dwarf \nGD 362 are consistent with the accretion of a large asteroid with composition similar to the \nEarth-Moon system.\n \nThe most metal-contaminated white dwarfs often display evidence of circumstellar disks; \neither by infrared excess \\citep{far08b,kil07,von07,jur07a,kil06,bec05,kil05,gra90,zuc87b}, \nor by broad, double-peaked optical emission lines \\citep{gan08,gan07,gan06}, or both \n\\citep{mel08,bri08}.  Evidence is strong that these disks evolve from the tidal disruption \nof minor planets \\citep{jur08,jur03}.  To be tidally destroyed within the Roche limit of a \nwhite dwarf, an asteroid needs to be perturbed from its orbit, and hence unseen planets \nof conventional size are expected at white dwarfs with dusty disks.\n\nIncluding the disks around GD 16 and PG 1457$-$086, which are reported in this paper, \nthe number of white dwarfs with circumstellar disks is 14 (see Table \\ref{tbl1}).  Although \nat least half a dozen publications present {\\em Spitzer} observations of white dwarfs, there \nhas not yet been a thorough search of metal-rich white dwarfs for cool dust at longer, MIPS \nwavelengths, nor a focus on those contaminated degenerates with helium-rich atmospheres; \nthis study bridges that gap and analyzes all 53 metal-rich degenerates observed by {\\em \nSpitzer}.  The numbers are now large enough that one can investigate the presence of a \ndisk as a function of cooling age and metal accretion rate.\n\nThis paper presents the results of an IRAC $3-8$ $\\mu$m and MIPS 24 $\\mu$m photometric \nsearch for mid-infrared excess due to circumstellar dust at cool, metal-contaminated white \ndwarfs.  The goals of the study are: 1) to constrain the frequency of dust disks around white \ndwarfs as a function of cooling age; and 2) to combine all available {\\em Spitzer} data on \nmetal-rich white dwarfs to better understand their heavy element pollutions.\n\n\\section{OBSERVATIONS AND DATA}\n\nMetal-rich white dwarf targets were imaged over $3-8$ $\\mu$m with the InfraRed Array Camera \n(IRAC, $1.20''$ pixel$^{-1}$; \\citealt{faz04}) using all four bandpasses, and at 24 $\\mu$m with \nthe Multiband Imaging Photometer for {\\em Spitzer} (MIPS, $2.45''$ pixel$^{-1}$ ; \\citealt{rie04}). \nObservations analyzed here are primarily taken from {\\em Spitzer} Cycle 3 Program 30387 and \nthose previously published in \\citet{jur07a}, with several archival datasets also included.  Twenty \nnewly observed or analyzed white dwarf targets are listed in Table \\ref{tbl2}, representing near \nequal numbers of DAZ and DBZ stars, and including G180-57 and HS 2253$+$803, two examples \nof metal-rich yet carbon-deficient stars.  Several Table \\ref{tbl2} stars have been previously imaged \nwith IRAC in other programs, hence for those stars only the MIPS observations are unique to this \nwork; their IRAC data were extracted from the {\\em Spitzer} archive and analyzed independently \nhere.  For two metal-rich white dwarf targets of particular interest, vMa 2 and LTT 8452, both IRAC \nand MIPS data were extracted from the {\\em Spitzer} archive.  The primary datasets utilized IRAC \nindividual frame times of 30 s in a 20-point, medium-scale dither pattern, for a total exposure time \nof 600 s in each of the four filters at each star.  MIPS 24 $\\mu$m observations were executed using \n10 s individual frame times with the default 14-point dither pattern, repeated for 10 cycles, yielding \na 1400 s total exposure time for each science target.  Table \\ref{tbl3} lists the IRAC and MIPS fluxes \nwith errors and upper limits for all Table \\ref{tbl2} white dwarfs, and also the non-metal-rich white \ndwarfs NLTT 3915 and LHS 46; these two stars are discussed in \\S3.6.\n\nData reduction and photometry, including $3\\sigma$ upper limits for non-detections, were \nperformed as described in \\citet{far08a,far08b}, and \\citet{jur07a,jur07b}, but photometric errors \nwere not treated as conservatively.  To account for the faintest MIPS 24 $\\mu$m detections, and \nfor reasons of consistency, the photometric measurement errors for both IRAC and MIPS were \ntaken to be the Gaussian noise in an $r=2$ pixel aperture at all wavelengths; i.e. the sky noise \nper pixel multiplied by the square root of the aperture area.  IRAC data at 3.6 and 4.5 $\\mu$m \nare typically high signal-to-noise (S\/N $\\ga50$) and therefore dominated by a 5\\% calibration \nuncertainty \\citep{far08b,jur07a}, whereas data at 5.7 and 7.9 $\\mu$m have total errors which \nare a combination of calibration and photometric measurement errors.  Most of the MIPS 24 \n$\\mu$m detections are dominated by the photometric measurement errors, with a 10\\% \ncalibration uncertainty employed for these data \\citep{eng07}.  In actuality, the photometric \nerrors may be somewhat larger for reasons described in detail elsewhere; namely crowded \nfields for the two short wavelength IRAC channels, and non-uniform sky backgrounds for \nthe two longer wavelength IRAC channels and for MIPS \\citep{far08b}.  Among the 20 science \ntargets imaged at 24 $\\mu$m and analyzed here, the background noise varied significantly, \nas evidenced by the variance among the $1\\sigma$ errors and $3\\sigma$ upper limits in \nTable \\ref{tbl3} (and similar variance among the 24 $\\mu$m data in Table 1 of \\citealt{jur07a}).\n\n\\section{ANALYSIS AND RESULTS}\n\nFigures \\ref{fig1}$-$\\ref{fig8} plot the spectral energy distributions (SEDs) of the 20 metal-rich \nstars listed in Table \\ref{tbl2}, grouped by right ascension, including {\\em Spitzer} photometry \nand upper limits.  Optical and near-infrared photometric data were taken from: Table \\ref{tbl2} \nreferences; \\citet{mcc06} and references therein, \\citet{sal03}, \\citet{mon03}, and the 2MASS \npoint source catalog \\citep{skr06}.  The data were fitted with blackbodies of the appropriate \ntemperature (i.e. able to reproduce the photospheric flux), which are sufficient to recognize \nexcess emission at the $3\\sigma$ level.  Of all the targets, only GD 16 and PG 1457$-$086 \ndisplay photometric excess above this threshold in their IRAC or MIPS observations.\n\n\\subsection{GD 16}\n\nFigure \\ref{fig2} displays the {\\em Spitzer} photometry for GD 16, which reveals a prominent \nmid-infrared excess indicative of $T\\sim1000$ K circumstellar dust.  The 2MASS data on this \nstar appear to indicate an infrared excess at $H$ and $K$ bands, but it is likely these data are \nunreliable; independent near-infrared photometry at high S\/N (Farihi et al. 2009, in preparation) \nindicate an excess only at $K$-band, as seen in Figure \\ref{fig2}.  No published optical \nphotoelectric photometry exists, but photographic data indicate $V\\approx15.5$ mag from both \nUSBO-B1 and the original discovery paper \\citep{mon03,gic65}; this $V$-band flux is plotted \nand agrees well with the measured near-infrared fluxes for a white dwarf of the appropriate \ntemperature.\n\nA detailed optical spectral analysis of GD 16 by \\citet{koe05b} yielded atmospheric parameters \n$T_{\\rm eff}=11,500$ K, [H\/He] $=-2.9$, and $M_V=12.05$ mag, with the assumption of log \n$g=8.0$.  Using the measured $J=15.55$ mag together with the model-predicted $V-J=0.09$ \ncolor for the relevant effective temperature, surface gravity, and helium-rich composition, the \nwhite dwarf is expected to have $M_J=12.14$ mag and hence a nominal photometric distance \nof $d=48$ pc \\citep{ber95a,ber95b}.  Figure \\ref{fig9} shows a fit to the thermal dust emission \nusing the flat ring model of \\citet{jur03}.  This model does an excellent job of reproducing all \ninfrared data from 2.2 $\\mu$m onwards.  For an 11,500 K star, the Figure \\ref{fig9} inner and \nouter dust temperatures correspond to 12 and 30 stellar radii, or 0.15 and 0.39 $R_{\\odot}$ \nrespectively \\citep{chi97}.  The fractional disk luminosity, $\\tau=L_{\\rm IR}\/L=0.02$, is about \n$2\/3$ that of the disks at G29-38, GD 362, and GD 56 \\citep{far08b,jur07a}.  \n\n\\subsection{PG 1457$-$086}\n\n{\\em Spitzer} photometry for PG 1457$-$086 is shown in Figure \\ref{fig5}, and reveals flux \nexcess just over $3\\sigma$ at both 3.6 and 4.5 $\\mu$m, relative to the expected photospheric \nflux.  The 2MASS photometry for this white dwarf at $H$ and $K$ bands has large errors, and \nindependent near-infrared photometry at high S\/N (Farihi et al. 2009, in preparation) indicate \na probable, slight excess at $K$ band.  Overall, the infrared excess is mild but indicative of \nvery warm, $T\\sim1500$ K emission.  \\citet{koe05a} and \\citet{lie05} derive $T_{\\rm eff}=\n20,400$ K and 21,500 K, respectively for PG 1457$-$086, while both find a surface gravity \nlog $g\\approx8.0$.  The 20,400 K value was employed for the figures because it is more \nconservative, predicting slightly less infrared excess than 21,500 K and because the model \nof \\citet{koe05a} accounts for the measured high calcium abundance, while the \\citet{lie05} \nmodel does not.\n\nFigure \\ref{fig10} shows a narrow ring model fitted to the infrared excess.  To match the \nrelatively low fractional luminosity of the disk, $\\tau=0.0006$, the model ring is narrow in \nradial extent and highly inclined. For a 20,400 K white dwarf, the inner and outer dust \ntemperatures, in an optically thick disk, correspond to 19 and 21 stellar radii, or 0.25 \nand 0.28 $R_{\\odot}$ respectively \\citep{chi97}.\n\nDue to the shape of the mild infrared excess at PG 1457$-$086, it is the only metal-rich \nwhite dwarf {\\em Spitzer} target where a substellar companion might be considered as \nviable.  If the measured $J=16.07$ mag is solely due to the white dwarf photosphere, \nthen the difference between the measured and model-predicted $K$-band flux for such \na 20,000 K hydrogen white dwarf is $\\Delta K=0.24$ mag or an excess of $K=17.8$ mag, \nwhich corresponds to $M_K=12.6$ mag at the photometric distance of 110 pc \\citep{lie05}.  \nIf a self-luminous, substellar companion of this $K$-band brightness were orbiting PG \n1457$-$086, it would have an effective temperature near 1500 K and a spectral type \naround L7 \\citep{vrb04,dah02}.  It is worth noting that the white dwarf GD 1400 has an \nunresolved L7 brown dwarf companion, and the ratio of the 3.6 to 2.2 $\\mu$m excess \nthere is 1.3 \\citep{far05b}, whereas for PG 1457$-$086 this value is 0.7, although these \nratios are consistent within the uncertainties.  \n\nAs mentioned in the Appendix, winds from nearby M dwarf companions can, and occasionally \ndo, pollute the atmospheres of white dwarfs.  But for PG 1457$-$086 an M dwarf companion is \nruled out by the absence of significant near-infrared excess \\citep{far05a}.  As described in the \nAppendix, there is no known connection between brown dwarf companions and the presence \nof metal pollution in white dwarf photospheres.  Therefore, for PG 1457$-$086 to be metal-rich \nand to possess an L dwarf companion would require, simultaneously, two low probability events.  \nFor this reason, it is it highly unlikely that the excess infrared emission at PG 1457$-$086 is due \nto a brown dwarf companion.\n\n\\subsection{LTT 8452}\n\n\\citet{von07} reported the discovery of {\\em Spitzer} IRAC 4.5 and 7.9 $\\mu$m excess due \nto warm dust at LTT 8452.  Figure \\ref{fig7} shows the SED of this metal-rich white dwarf, \nnow plotted with its IRAC 3.6, 5.7, and MIPS 24 $\\mu$m data, which better constrain the \ninner and outer temperature of the disk.  Figure \\ref{fig11} shows a flat ring model fitted to \nthe mid-infrared emission of LTT 8452, with inner edge temperature $T_{\\rm in} =1000$ K, \nouter temperature $T_{\\rm out}=600$ K, and a modest inclination angle of $i=53 \\arcdeg$,\nwhereas \\citet{von07} derived a temperature range of $900-550$ K and $i=80 \\arcdeg$.  \nFrom the flat disk model, the fractional infrared luminosity of LTT 8452 is $\\tau= 0.008$.\n\n\\subsection{G238-44 and G180-57}\n\nFigure \\ref{fig4} includes all available {\\em Spitzer} data for G238-44, one of the most \nhighly contaminated, nearest, and brightest DAZ white dwarfs \\citep{hol97}.  The aperture \nlaid down for MIPS 24 $\\mu$m photometry was extrapolated to the expected location of the \nstar at the epoch 2007.3 observation, using the {\\em Hipparcos} measured J2000 position \nand proper motion \\citep{per97}.  The expected position on the MIPS 24 $\\mu$m array is in \nexcellent agreement with the measured position of G238-44 in the epoch 2004.9 IRAC \nimages, after accounting for proper motion, and coincides with a faint MIPS 24 $\\mu$m \nsource.  However, the potential for source confusion is high.\n\nIf associated with the white dwarf, the Figure \\ref{fig4} apparent 24 $\\mu$m excess at the \nlocation of G238-44 would be just slightly greater than 0.04 mJy ($2.0\\sigma$).  Based on \n{\\em Spitzer} MIPS 24 $\\mu$m source counts from deep imaging, the expected number of \nbackground galaxies of brightness $0.04\\pm0.02$ mJy is around 8000 per square degree \n\\citep{mar04}.  Hence within an area of diameter equal to one full width at half maximum \nintensity at this wavelength (about 2.3 pixels or approximately 25 square arcseconds), there \nshould be 0.015 galaxies of the right brightness to contaminate the MIPS aperture.  Using a\nbinomial probability, the chance of finding at least one in 20 metal-rich white dwarf targets \nconfused with a background source in this manner is then 26\\%; therefore the 24 $\\mu$m \nflux may not originate from G238-44.\n\nFurther, if the potential contamination area is widened to encompass two full widths at half \nmaximum, the probability of source confusion is then 71\\%, although such a large an area \nis perhaps overly conservative.  The expected position of G238-44 is offset from the centroid \nposition of the MIPS source by 0.35 pixels or $0.9''$, the error of which is hard to estimate \ndue to the relatively low S\/N.\n\nTurning to the white dwarf G180-57 (Figure \\ref{fig4}), it has a 0.06 mJy (or $1.5\\sigma$ excess) \n24 $\\mu$m source at its expected position on the array.  All the previous arguments and caveats \napply, and the chance that at least two in 20 MIPS targets are contaminated in this way is still \nrelatively high, somewhere between 4\\% and 34\\%.\n\nIn any event, cold dust alone at this 24 $\\mu$m luminosity level cannot explain the source \nof external metals in these white dwarfs.  Blackbody dust which reveals itself at 24 $\\mu$m but \nnot at 8 $\\mu$m would be around 200 K or cooler, and located too far away (at or beyond 10 \nand 70 $R_{\\odot}$ for G180-57 and G238-44 respectively) to be the source of accreted metals \nin the photospheres of either white dwarf.  Average-sized grains of 10 $\\mu$m with density 2.5 g \n${\\rm cm}^{-3}$ at these distances would imply (minimum) dust masses around $10^{15}-10\n^{17}$ g for $\\tau=10^{-5}$, appropriate for these detections, if real \\citep{far08b}.  This mass is \ninsufficient to sustain an accretion rate of $3\\times10^8$ g ${\\rm s}^{-1}$ for more than 10 years \nat G238-44 \\citep{koe06}.\n\n\\subsection{vMa 2}\n\nThe MIPS observations of vMa 2, plotted in Figure \\ref{fig1}, suggest a 24 $\\mu$m flux \nsignificantly lower than expected for a simple Rayleigh-Jeans extrapolation from its IRAC \nfluxes.  Rather than an excess, the SED of vMa 2 appears deficient at 24 $\\mu$m, at the \n$4\\sigma$ level; the measured 24 $\\mu$m flux is $0.11\\pm0.03$ mJy, while the predicted \nflux is 0.23 mJy (Figure \\ref{fig1}).  Blackbody models are essentially no different than pure \nhelium atmosphere white dwarf models at these long wavelengths (\\citealt{wol02}; see their \nFigure 1), hence the apparent deficit rests on the validity of the relatively low S\/N photometry \nrather than model predictions.  Still, this intriguing possibility requires confirmation with \nsuperior data, and if confirmed, vMa 2 would become by far the highest temperature white \ndwarf to display significant infrared flux suppression due to collision-induced absorption \n(B. Hansen 2007, private communication; \\citealt{far05c}).\n\n\\subsection{Notes on Individual Objects}\n\n{\\em 0108$+$277}.  NLTT 3915 belongs to an optical pair of stars separated by roughly $3''$ \non the sky in a 1995.7 POSS II plate scan.  The northeast star has proper motion $\\mu=0.22''$ \nyr$^{-1}$ at $\\theta=222\\arcdeg$ \\citep{lep05} which can be readily seen between the POSS I \nand POSS II epochs.  The IRAC images reveal two overlapping sources with a separation of \n$2.4''$ as determined by {\\sf daophot} at all four wavelengths.  Using this task to deconvolve \nthe pair of stars photometrically reveals the object to the southwest is likely a background red \ndwarf, based on its 2MASS and IRAC photometry, while the northeast star has colors consistent \nwith a very cool white dwarf.  \\citet{kaw06} identified this star as a 5200 K DAZ white dwarf \nwhose spectrum appears to exhibit sodium but not calcium; an anomalous combination for \na metal-rich degenerate.  Keck \/ HIRES observations confirm this object is a white dwarf, but \nwith a cool DA spectrum and no metal features (C. Melis 2008, private communication).  The \ndata tables include fluxes, and the appendix gives limits on substellar companions for this \ntarget, but it is excluded it from analyses for metal-rich white dwarfs.\n\n{\\em 0208$+$396}. G74-7 is the prototype DAZ star.  IRAC observations of this white dwarf, \npreviously reported in \\citet{deb07} were taken during a solar proton event, and individual \nframes are plagued by legion cosmic rays, making the reduction and photometric analysis \ndifficult, especially at the longer wavelengths.  Only the 4.5 and 7.9 $\\mu$m fluxes are plotted \nand tabled; some data were irretrievably problematic.\n\n{\\em 1202$-$232}.  This bright and nearby white dwarf is located about $8''$ away from \na luminous infrared galaxy in the IRAC 8 $\\mu$m images, and the white dwarf location is \nswamped by light from the galaxy at MIPS 24 $\\mu$m, where the diffraction limit is $6''$.  \nHence the upper limit on the white dwarf 24 $\\mu$m flux is about a factor of five worse than \nfor a typical star.\n\n{\\em 1334$+$039}.  LHS 46 has been (mistakenly) identified as spectral type DZ in several \npapers over the years (\\citealt{gre84}; see discussion in \\citealt{lie77}).  It was first (correctly) \nre-classified as a DC star by \\citet{sio90}, and modern, high resolution spectroscopy confirms \nthe white dwarf is featureless in the region of interest \\citep{zuc98}.  Unfortunately, some \nrelatively current literature \\citep{hol08,hol02,mcc99} still contains the incorrect spectral \ntype for this star and it was included in the {\\em Spitzer} observations.  The data tables \ninclude fluxes, and the appendix gives limits on substellar companions for this target, \nbut it is excluded it from analyses for metal-rich white dwarfs.\n\n\\section{DISCUSSION}\n\nWith the detection of mid-infrared excess at GD 16 and PG 1457$-$086, the number of \nexternally polluted, cool white dwarfs with warm circumstellar dust becomes 14, including \nEC 1150$-$153 \\citep{jur09}, SDSS 1228 \\citep{bri08} (see Table \\ref{tbl1}), SDSS 1043 \nand Ton 345 (C. Brinkworth 2008, private communication; \\citealt{mel08}).  The latter three \nstars, whose photospheric metals and circumstellar gas disks were discovered simultaneously, \nare not analyzed here.  Over 200 white dwarfs have been observed with {\\em Spitzer} IRAC \n\\citep{far08a,far08b,mul07,deb07,han06,jur07b,fri07}.  Of these, only stars with detected \nphotospheric metals display an infrared excess, presumably because metal pollution from \na circumstellar disk is inevitable and optical spectroscopy is a powerful tool for detecting \nthe unusual presence of calcium in a white dwarf atmosphere.  While the sample of white\ndwarfs observed with {\\em Spitzer} is quite heterogeneous, it is likely that at least half of \nthese stars have been observed in the Supernova Progenitor Survey (SPY; \\citealt{nap03}),\nand hence with high sensitivity to photospheric calcium.\n\n\\subsection{The Fraction of Single White Dwarfs with an Infrared Excess}\n\nThe fraction of white dwarfs cooler than about 20,000 K that display photospheric metals \ndepends on the stellar effective temperature and on whether its photospheric opacity is \ndominated by hydrogen or by helium.  For DA white dwarfs, it is easier to detect metals in \ncooler stars and, in a survey that focused primarily on DA white dwarfs cooler than 10,000 \nK, \\citet{zuc03} found that of order 25\\% displayed a calcium K line.  In the extensive SPY \nsurvey, more focused on warmer white dwarfs with $T_{\\rm eff}\\ga10,000$ K, \\citet{koe05a} \nfound that only 5\\% show a calcium K line.  To produce a detectable optical wavelength line \nat such high temperatures (and high opacities) typically requires a greater fractional metal \nabundance than at low temperatures.  Because extensive ground and, especially, {\\em \nSpitzer} surveys have failed to reveal infrared excess emission at any single white dwarf \nthat lacks photospheric metals \\citep{far08b,mul07,hoa07,han06,far05a}, one can regard \nthe above percentages as upper limits to the fraction of DA white dwarfs that possess dusty \ndisks; at least until more sensitive metal abundance measurements can be achieved.  \nPerhaps more instructive is the number DA stars without metals observed with {\\em \nSpitzer}; 121 such targets are reported between \\citet{mul07} and \\citet{far08a}, none \nof which show evidence for circumstellar dust, an upper limit of 0.8\\%.\n\nLower limits to the fraction of DA white dwarfs with dust disks can be estimated in the following \nway.  Consider four large surveys of white dwarfs: the Palomar-Green Survey (347 DA stars;\n\\citealt{lie05}), the Supernova Progenitor Survey (478 DA stars; \\citealt{koe05a}); the 371\nwhite dwarfs from \\citet{far05a}, and the 1321 non-binary stars present in both the 2MASS\n\\citep{hoa07} and \\citet{mcc99} catalogs.  The presently known frequency of dust disks \nfor white dwarfs in these four surveys, all of which tend to find warmer white dwarfs (due\nto luminosity), are 1.4, 1.5, 1.1 and 0.8\\%, respectively.  Thus, because not all stars in these \nsurveys have been searched for dust disks, one can say that, at a minimum, 1\\% of all warm \nwhite dwarfs have dust disks. \n\n{\\em Spitzer} surveys of cool metal-rich white dwarfs have now targeted 53 stars at $3-8$ \n$\\mu$m using IRAC (a few at 4.5 and 7.9 $\\mu$m only, but most at all wavelengths), while \n31 of these targets have also been observed at 24 $\\mu$m with MIPS (see Table \\ref{tbl4}).  \nIncluding the unlikely exceptions discussed above, there does not exist a clear-cut case of \nMIPS 24 $\\mu$m detection at any white dwarf without a simultaneous and higher IRAC flux \n-- a telling result by itself.  Hence, targets observed with IRAC only should accurately reflect \nthe frequency of dusty degenerates.  \n\nOf the 53 contaminated white dwarfs surveyed specifically for circumstellar disks, 21\\% \nhave mid-infrared data consistent with warm dust, regardless of atmospheric composition.\nWith more data, this estimate is somewhat larger than the result of \\citet{kil08} that at least \n14\\% of polluted white dwarfs have an infrared excess.  As shown in Figure \\ref{fig12}, the \nlikelihood of a white dwarf displaying an infrared excess is strongly correlated with effective \ntemperature and\/or its measured calcium pollution.  Only two of 34 stars with $T_{\\rm eff}\\leq\n10,000$ K have an excess, while that fraction is nine of 19 stars with $T_{\\rm eff}>10,000$ \nK.  Alternatively, nine (or ten) of 17 stars with [Ca\/H(e)] $\\geq-8.0$ possess an excess, \nwhereas that fraction is only one (or two) of 38 stars with [Ca\/H(e)]  $<-8.0$.\n\n\\subsubsection{The Role of Cooling Age}\n\nCooling ages for the white dwarfs in Table \\ref{tbl4} were calculated with the models of P. \nBergeron (2002, private communication; \\citealt{ber95a,ber95b}). The white dwarf masses \nused as input for the cooling ages were taken from the literature (see Table \\ref{tbl4} references), \nor log $g=8.0$ was assumed for those stars with no estimates available.  Because cooling age is \nsensitive to mass, and because within the literature discrepancies exist among white dwarf mass \nestimates, the values here should not be considered authoritative, but preference was given to \nvalues based on trigonometric parallaxes \\citep{hol08} and those with multiple, independent,\ncorroborating determinations.  Still, it should be the case that a typical uncertainty in cooling \nage is between 10\\% and 20\\% due to an error in white dwarf mass.\n\nFigure \\ref{fig12} indicates that the most metal-polluted white dwarfs are the warmest, and \nFigure \\ref{fig13} illustrates the same phenomenon, but now cast in terms of white dwarf \ncooling age.  In the context of a model of tidal destruction of minor planets, as described \nbelow, it is plausible that younger (i.e., shorter cooling time) white dwarfs would be the most \npolluted.  As indicated in the figure and tables, circumstellar dust disks are found at:  eight of \n17 stars (47\\%) with $t_{\\rm cool}<0.5$ Gyr, two of 12 stars (17\\%) with 0.5 Gyr $<t_{\\rm cool}\n<1.0$ Gyr, and only one of 24 (4\\%) stars with $t_{\\rm cool}>1.0$ Gyr.  The three dusty white \ndwarfs with cooling ages beyond 0.5 Gyr are: GD 362 and LTT 8452, both with cooling ages \nnear 0.8 Gyr; and G166-58 with cooling age 1.29 Gyr.\n\nTo estimate the percentage of white dwarfs with cooling ages less than 0.5 Gyr that might have \ndusty disks, one can consider each of the four large surveys mentioned in the previous section.\nFor example, the SPY survey contains 14 metal-contaminated DA white dwarfs with likely cooling \nages less than 0.5 Gyr ($T_{\\rm eff}\\ga11,000$ K).  Of these 14, eight have been observed with \nIRAC and six have dust disks, while the remaining six, unobserved stars are prime candidates \nfor dust disks (see \\S4.2).  The SPY sample contains approximately 400 DA stars in this range\nof cooling ages and therefore, based on these data, it is likely that between 2\\% and 3\\% of white \ndwarfs with $t_{\\rm cool}<0.5$ Gyr have detectable dust disks.  This estimate is also consistent \nwith the 72 DB white dwarfs (all with $T_{\\rm eff}\\ga11,000$ K) observed with SPY \\citep{vos07,\nkoe05b}, two of which are metal-contaminated and possess dust disks.\n\nIn a similar manner, the SPY survey contains 10 metal-contaminated DA white dwarfs with \nlikely cooling ages between 0.5 and 1.5 Gyr (7000 K $\\la$ $T_{\\rm eff}\\la11,000$ K).  Of these\n10, eight have been observed with IRAC and one has a dust disk, while the remaining two,\nunobserved stars both have [Ca\/H] $>-8.0$, and hence one of them may have a disk.  The \nSPY sample contains around 70 DA stars in this range of cooling ages, and therefore these \ndata suggest between 1\\% and 2\\% white dwarfs with 0.5 Gyr $<$ $t_{\\rm cool}<1.5$ Gyr \nmay have detectable dust disks.  However, the smaller number statistics for these cooler\nwhite dwarfs make this estimate somewhat uncertain.\n\nAssuming only metal-bearing white dwarfs can possess warm dust disks, the {\\em Spitzer}\nobservations suggest similar percentages.  The fraction of younger, $t_{\\rm cool}<0.5$ Gyr \nwhite dwarfs with dust disks can be estimated by observing the fraction of these with IRAC \nexcess is 0.47, and the fraction of SPY DA white dwarfs with metals in this cooling age range \nis 0.05, leading to a frequency between 2\\% and 3\\%, consistent with the above estimate.  For \nsomewhat older white dwarfs where 0.5 Gyr $<t_{\\rm cool}<1.5$ Gyr, the same fractions are \n0.10 with IRAC excess, and 0.13 with metal lines among SPY DA targets, yielding a frequency \nof 1\\%.  For white dwarfs with $t_{\\rm cool}>1.5$ Gyr, there there are not enough stars in \nappropriate age bins to make meaningful estimates.\n\n\\citet{hol08} have compiled a nearly (estimated at 80\\%) complete catalog of white dwarfs \nwithin 20 pc of the Sun.  In this local volume, there are 24 white dwarfs with 10,000 K $<\nT_ {\\rm eff}\\la 20,000$ K, and at least one white dwarf (G29-38), i.e. 4\\%, has an infrared \nexcess.  If the true percentage of warm dusty white dwarfs is 2.5\\%, then out to 50 pc there \nshould be nine such stars with infrared excess.  Currently, only GD 16 and GD 133 meet \nthese criteria; either this expectation is incorrect or several white dwarfs within 50 pc of \nthe Sun with an infrared excess remain to be discovered.\n\n\\subsubsection{Implications for Disk Lifetimes and Pollution Events}\n\nThe relative dearth of dust disks at the cooler metal-rich white dwarfs, as established by \n{\\em Spitzer} observations, suggests that dust disk lifetimes are likely to be shorter than \na typical $10^4-10^6$ yr heavy element diffusion timescale in either a hydrogen or helium \natmosphere white dwarf at these temperatures.  Whether a typical disk mass is fully consumed \nwithin that period or is rendered gaseous via collisions on much shorter timescales is unclear, \nand both mechanisms are likely to play a role to remove dust at white dwarfs \\citep{jur08,\nfar08b,jur07a}.  However, as noted above, the lack of dust at the cooler polluted stars is also \nrelated to their lower overall metal abundances, and partially an observational bias.\n\nMore important, this empirical result may have implications for the timescales of the \nevents which give rise to and\/or sustain circumstellar dust.  If stochastic perturbations within \na reservoir of planetesimals are the ultimate source of pollution events in metal-enriched \nwhite dwarfs, gradual depletion could give rise to a exponential decay in the number of \nevents per unit time \\citep{jur08}, and hence the likelihood of a pollution event decreases \nas a white dwarf ages.  Such a scenario is consistent with the {\\em Spitzer} observations \nof the cooler, metal-lined white dwarfs, and would predict that the frequency of disruption, \nand subsequent disk creation events increases with increasing stellar effective temperature, \ncorresponding to younger post-main sequence ages.\n\nFor typical white dwarfs of log $g=8.0$, it takes around 0.07 Gyr to cool to an effective \ntemperature of 20,000 K, about 0.2 Gyr to 15,000 K, and just over 0.6 Myr to achieve 10,000 \nK \\citep{ber95a,ber95b}.  The only known disk-bearing white dwarf which is likely to have a \ncooling age significantly older than several hundred Myr is G166-58; at 7400 K and assuming \nlog $g=8.0$, its cooling age should be near 1.3 Gyr.  Notably, the properties of G166-58 are \nrather anomalous compared to the other known disk-bearing white dwarfs: its infrared excess \ncomes up at 5 $\\mu$m \\citep{far08a}, while it has a modest calcium abundance and accretion \nrate.  One possibility is that planetary system remnants at white dwarfs tend to stabilize by \nroughly 1 Gyr.  Minor planet belts may become significantly depleted on these timescales or \ngravitational perturbations may subside on shorter timescales via dynamical rearrangement \n\\citep{jur08,bot05,deb02}.\n\nOn the other hand, the lack of dust at the relatively warm, but highly polluted white dwarf\nHS 2253$+$803 points to a disk lifetime shorter than the timescale for removal of accreted\nmetals.  This very metal-enriched and carbon-poor degenerate sits in the outlying region of \nFigure \\ref{fig12} where it is virtually the only star in this abundance range without a disk.  \nTherefore, it is likely this star had a dust disk which has been fully consumed within the \n$10^6$ yr diffusion timescale for this warm DBAZ.\n\n\\subsection{Metal Accretion Rates}\n\nIt has been previously argued that DAZ white dwarfs with the highest calcium abundances \nalso have an infrared excess and, thus, a circumstellar disk \\citep{kil06}.  This argument has \nbeen slightly recast \\citep{jur07b,jur08} to suggest that those DAZ white dwarfs with the highest\nmetal accretion rates are the stars with infrared excess.  However, put into this proper context, \nboth DAZ and DBZ white dwarfs should exhibit this correlation, if correct.  This connection \nbetween disk frequency and metal accretion rate is now re-evaluated for all 53 metal-polluted \nwhite dwarfs observed with the IRAC camera.\n\nAssuming accretion-diffusion equilibrium (i.e. a steady state), mass accretion rates were \ncalculated using Equation 2 of \\citet{koe06} with the best available photospheric calcium \nabundance determinations, together with the solar calcium abundance relative to either \nhydrogen or helium, as appropriate.  Settling times for various metals are usually within a \nfactor of 2 \\citep{koe06,dup93}, and calcium is employed here because it is the best studied \nelement.  Where available for DAZ stars, convective envelope masses and calcium diffusion \ntimescales were taken from Table 3 of \\citet{koe06}.  Otherwise log $g=8$ was assumed and \ndiffusion timescales were taken directly from their Table 2, while convective envelope masses \nwere interpolated using their Table 3 values for stars of similar effective temperature and surface \ngravity (log $g=8$ was also assumed for G166-58 for reasons discussed in \\citealt{far08b}).  For \nDBZ stars, convective envelope masses were read from Figure 1 of \\citet{paq86}, and calcium \ndiffusion times were interpolated using their Table 2 values; all assuming $M=0.6$ $M_{\\odot}$.  \n\nIn actuality, steady state accretion is likely for the warmer DAZ stars, but much less so for\nthe cooler DAZ and DBZ stars, which have relatively long metal dwell times.  Based on this\nwork and the results of \\citet{kil08}, it is likely that disks at white dwarfs typically dissipate \nwithin $10^4-10^5$ yr, a timescale at which photospheric metals persist in the cooler DAZ \nand DBZ stars.  Hence, the calculated accretion rates listed in Table \\ref{tbl4} should be \nconsidered {\\em time-averaged} over a single diffusion timescale, and may not accurately\ndescribe stars where detectable metals may dwell beyond $10^3$ yr.  An additional factor \nof $1\/100$ was introduced to ``correct'' the \\citet{koe06} Equation 2 rates to reflect that only \naccreted heavy elements are of interest, this factor being the typical dust-to-gas ratio in the \ninterstellar medium.  Figure \\ref{fig14} plots these accretion rates for all {\\em Spitzer} observed \nstars versus effective temperature.  Figure \\ref{fig15} plots the same metal accretion rates versus \nwhite dwarf cooling age.\n\nThe results show that the implied metal accretion rates are quite similar between the \ntwo atmospheric varieties of metal-rich white dwarf, strengthening the argument that \ntheir externally-polluted photospheres are caused by a common phenomenon; namely \ncircumstellar material.  In fact, at accretion rates $dM\/dt\\ga$ $3\\times10^8$ g s$^{-1}$, \nthe fraction of metal-rich white dwarfs with circumstellar dust -- regardless of atmospheric \ncomposition -- is over 50\\%.  If one associates accretion rate with circumstellar disk \nmass (a modest assumption), then this picture is consistent with white dwarf dust disks \nbeing strongly linked to tidal disruption events involving fairly large minor planets; while \nless massive disrupted asteroids give rise to more tenuous (possibly shorter-lived, possibly \ngaseous) disks, and modest accretion rates.\n\nWhile the white dwarfs with an infrared excess are likely accreting from tidally-disrupted \nminor planets, the origin of the pollution in high accretion rate white dwarfs without obvious \nevidence for a disk is uncertain.  \\citet{jur08} has proposed that if multiple, small asteroids \nare tidally-disrupted, their debris self-collides so the dust is efficiently destroyed, and matter \nthen accretes onto the white dwarf from an undetected gaseous disk.  Given that the typical\ntimescale for collisions within a white dwarf dust disk is hours, with disk velocities at hundreds\nof km s$^{-1}$, it is also conceivable that a single, modest-sized planetesimal could grind itself\ndown to gas-sized material via self-collisions \\citep{far08b}.  Another possibility is that the \ndwell time of the photospheric metals in the white dwarf is longer than the characteristic \ndisk dissipation time \\citep{kil08}. \n\n\\subsection{Circumstellar Versus Interstellar Accretion}\n\nTwo competing models to explain the metal-contaminated photospheres of white dwarfs \ninvolve interstellar accretion and circumstellar accretion \\citep{sio90,alc86}.  The SEDs of \nthe mid-infrared excess at GD 16 and PG 1457$-$086 are well explained by a circumstellar \ndust disk arising from a tidally disrupted asteroid or similar planetesimal, but are not consistent \nwith interstellar accretion.  The same argument applies to the other metal-rich white dwarfs with \nmid-infrared excess; simply stated, the compact size and olivine composition of the dust are at \nodds with expectations for disks formed through interstellar accretion \\citep{jur07a,jur07b,rea05}.\n\nWith Equation 3 of \\citet{jur07a} and following their method, predicted 24 $\\mu$m \nfluxes were calculated for all 32 metal-rich white dwarfs observed with MIPS, assuming \ninterstellar Bondi-Hoyle type grain accretion followed by Poynting-Robertson drag onto \nthe star.  Figure \\ref{fig16} displays these predicted infrared fluxes from interstellar accretion \nversus the MIPS 24 $\\mu$m upper limits for 25 white dwarfs; stars with fluxes consistent with \ncircumstellar disks were excluded, as were targets with contaminated photometry. The plot \nshows that the predicted fluxes are typically more than an order of magnitude greater than \nthe observed $3\\sigma$ upper limits.  With few possible exceptions, this simple interstellar \naccretion model cannot account for the observed infrared data.\n\nFurthermore, there is now a growing number of DB white dwarfs with evidence for external \npollution via carbon deficiency, relative to metals such as iron \\citep{des08,zuc07,jur06,wol02}.  \nThey have accreted rocky material, independent of any evidence for or against circumstellar \ndust.  Table \\ref{tbl5} lists the seven known helium- and metal-rich white dwarfs with low carbon \nabundances or upper limits.  Of these stars, only two of six observed by {\\em Spitzer} have strong \nevidence in favor of circumstellar dust, GD 40 and GD 362.  This raises the possibility that the \nremaining dust-poor and metal-rich stars are accreting gaseous heavy elements.\n\nFigure \\ref{fig14} reveals that the DBZ HS 2253$+$803 has one of the highest implied \n(time-averaged) accretion rates among all polluted white dwarfs, yet has no dust disk as \nevidenced by the {\\em Spitzer} photometry presented here.  This white dwarf is spectacularly \ncarbon-poor relative to its metal content, and is highly likely to have accreted rocky material \n\\citep{jur06}. In this particular case, its near pure helium nature gives it a likely diffusion \ntimescale of $10^6$ yr, and hence a dissipated disk is quite possible; yet its atmospheric \nmetal composition should nonetheless provide a measure of the extrasolar planetary \nmaterial it has accreted.\n\nThe preponderance of the evidence firmly favors the interpretation that heavily \nmetal-contaminated white dwarfs are currently accreting, or have in their recent history \naccreted, rocky planetesimal material in either dusty or gaseous form.  The origin of the\nmetals in white dwarfs at the low end of [Ca\/H(e)] ratios and metal accretion rates, such \nas a number of DAZs in the survey of \\citet{zuc03} remains unclear.\n\n\\subsection{Comparison with Main-Sequence Stars with Planetary Systems}\n\nThe circumstellar disks around white dwarfs likely arise from the tidal disruption \nof minor planets, with inferred metal accretion rates $dM\/dt\\ga$ $3\\times10^{8}$ g \ns$^{-1}$.  In the solar system, the dust production rate in the zodiacal cloud is near $10^{6}$ \ng s$^{-1}$ \\citep{fix02}.  Collisions among parent bodies result in dust production rates around \nmain-sequence A-type stars in excess of $10^{10}$g s$^{-1}$ \\citep{che06}.  Thus, the rate\nestimated for the erosion of minor planets around white dwarfs is within the range inferred for \nmain-sequence stars.  Because A-type stars evolve into white dwarfs, it appears there is an \nample population of parent bodies among the main-sequence progenitors of white dwarfs to \naccount for the observed pollutions.\n\nThe model in which an infrared excess around a white dwarf results from a tidally-disrupted \nminor planet requires that the orbit of an asteroid is perturbed substantially \\citep{deb02}.  It \nis plausible that a Jupiter-mass planet is such a perturber.  \\citet{cum08} find that 10.5\\% of \nsolar-type stars have gas giant planetary companions with periods between two and 2000 dy.\nClearly, many of these planets will be destroyed when the main-sequence star becomes a first \nascent and then asymptotic giant \\citep{far08a}, but half of these planets have periods longer \nthan one year and may survive.  At present, the known fraction of main-sequence stars with \nmassive planets that could persist beyond the post-main sequence evolution of their host is \ncomparable to the estimated fraction of white dwarfs with an infrared excess.  The upper mass \nlimits for self-luminous companions to white dwarfs, as found by this and previous {\\em Spitzer} \nstudies, are consistent with the scenario that asteroid orbits are perturbed by a typical gas giant \nplanet \\citep{far08a,far08b}.\n \n\\section{CONCLUSIONS}\n\n{\\em Spitzer} mid-infrared observations of metal-contaminated white dwarfs are extended to \na larger sample, and to longer wavelengths, including numerous DBZ targets.  This study \nbuilds on previous work which all together indicate that warm dust orbits 21\\% of all externally \npolluted white dwarfs observed by {\\em Spitzer}.  Several patterns are revealed among the \ndegenerates with and without warm circumstellar dust, which together lend support to the \nidea that tidally disrupted planetesimals are responsible for the heavy element abundances \nin many, if not most externally polluted white dwarfs. \n\n1.  Sufficient metal-rich targets now have been studied to estimate that between 1\\% and 3\\% of \nsingle white dwarfs with cooling ages less than around 0.5 Gyr possess an infrared excess that \nis likely the result of a tidally-disrupted asteroid.  Evidence is strong that white dwarfs can be used \nto study disrupted minor planets.\n\n2.  As yet no evidence for cool, $T<400$ K dust exists from MIPS 24 $\\mu$m observations \nof more than 30 metal-rich white dwarfs, including many with heavily polluted photospheres.\nAll stars with circumstellar disks detected at 24 $\\mu$m have coexisting, strong $3-8$ $\\mu$m \nIRAC excess fluxes.  The disks appear outwardly truncated; their mid-infrared spectral energy \ndistributions are clearly decreasing at wavelengths beyond 8 $\\mu$m.  These observed and \nmodeled infrared excesses indicate rings of dust where the innermost grains typically exceed \n$T=1000$ K, and outer edges which lie within the Roche limit of the white dwarf, consistent \nwith disks created via tidal disruption of rocky planetesimals. \n\n3.  Circumstellar disks at white dwarfs are vertically optically thick at wavelengths as long as \n20 $\\mu$m.  Particles in an optically thin disk would not survive Poynting-Robertson drag for \nmore than a few days to years.  Additionally, the warmest dust has been successfully modeled \nto lay within the radius at which blackbody grains in an optically thin cloud should sublimate \nrapidly.  With a single, notable exception (G166-58), the dusty circumstellar disks have inner \nedges which approach the sublimation region for silicate dust in an optically thick disk; precisely \nthe behavior expected for a dust disk which is feeding heavy elements to the photosphere of its \nwhite dwarf host.\n\n4. The majority of metal-contaminated white dwarfs do not have dust disks.  Circumstellar \ngas disks are a distinct possibility at dust-free, metal-rich stars with $dM\/dt$ $\\geq3\\times\n10^8$ g s$^{-1}$.  Fully accreted disks are a possibility for white dwarfs with metal diffusion \ntimescales approaching $10^6$ yr.  It is possible that a critical mass and density must be \nreached to prevent the dust disk from rapid, collisional self-annihilation, and when this \nmilestone is not reached, a gas disk results.  If correct, optical and ultraviolet spectroscopy \nof metal-rich white dwarfs are powerful tools with which to measure the bulk composition \nof extrasolar planetary material.\n\n5.  Cooling age is correlated with the frequency of dusty disks at white dwarfs; for $T_{\\rm eff}\\la\n20,000$ K, white dwarfs with younger cooling ages are more likely to be orbited by a dusty disk.  \nG166-58 is by far the coolest white dwarf with a (rather anomalous) infrared excess, and the \ntimescale for diffusion of metals out of the photosphere is relatively long for a DAZ white dwarf.  \n\n{\\em Spitzer} IRAC may soon bring a few more dust discoveries at polluted white dwarfs, \nbut the statistics are unlikely to change significantly without a commensurate increase in the \nnumber of surveyed stars.  Such a program remains feasible with IRAC in the post-cryogenic \nphase, provided that $T\\sim1000$ K dust is present in most white dwarf circumstellar disks.  \nUnfortunately, objects with somewhat cooler dust emission such as G166-58 will evade detection \nuntil {\\em JWST}, which should have photometric sensitivity similar to, or better than, {\\em Spitzer} \nat all relevant mid-infrared wavelengths.  Observations of a sizable number of metal-contaminated \nwhite dwarfs at longer wavelengths, where cold planetesimal belt debris might be seen {\\em in situ}, \nwill have to await future facilities.  If belts of rocky planetary remnants persist around metal-polluted \nwhite dwarfs at tens of AU, where Poynting-Robertson drag cannot remove large particles within \n$10^9$ yr, then sensitive submillimeter observations may have a chance to directly detect them.\n\n\\acknowledgments\n\nThe authors thank the referee for constructive comments which improved the manuscript.  \nJ. Farihi thanks D. Koester and R. Napiwotzki for sharing aspects of their spectroscopic\ndataset.  This work is based on observations made with the {\\em Spitzer Space Telescope}, \nwhich is operated by the Jet Propulsion Laboratory, California Institute of Technology \nunder a contract with NASA.  Support for this work was provided by NASA through an \naward issued by JPL\/Caltech to UCLA.  This work has also been partly supported by \nthe NSF.\n\n{\\em Facility:} \\facility{Spitzer (IRAC,MIPS)}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe nuclear reactions producing energy in the Sun also produce the well-known\nsolar neutrino flux of about $6.6\\times10^{10}~{\\rm cm}^{-2}~{\\rm s}^{-1}$ with\nMeV energies. At Earth this is the\nlargest neutrino flux, except perhaps in the immediate vicinity of a nuclear\npower reactor. The role of solar neutrinos for the discovery of leptonic\nflavor conversion and for pioneering the field of astroparticle physics\ncannot be overstated. It is a remarkable shift of paradigm that solar\nneutrinos today, fifty years after their first detection, are part of the\n``neutrino floor,'' the dominant background for direct searches of dark\nmatter in the form of weakly interacting massive particles (WIMPs).\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.90\\textwidth]{pics\/fig01.pdf}\n\\caption{Processes for thermal neutrino pair production in the Sun.}\n\\label{fig:processes}\n\\end{figure}\n\nAnother well-motivated dark matter candidate is a sterile neutrino in the keV\nmass range~\\cite{Adhikari:2016bei}. One idea for a direct search is the\nsterile-neutrino capture on a stable isotope of dysprosium if $m_s>2.83~{\\rm\nkeV}$ \\cite{Lasserre:2016eot}. Other searches for slightly heavier sterile neutrinos include unstable isotopes \\cite{Li:2010vy,Li:2011mw}, coherent inelastic scattering on atoms \\cite{Ando:2010ye} and electron scattering \\cite{Campos:2016gjh}. Once again, solar neutrinos could be a\nlimiting background, now those with keV energies that emerge from various\nthermal processes in the solar plasma which has a typical temperature of\n1~keV. While this idea is futuristic with present-day technology, it\nmotivates us to consider keV-range solar neutrinos. This is a standard\nneutrino flux, yet it is conspicuously absent from a popular plot of the\n``grand unified neutrino spectrum'' at Earth that ranges from cosmic\nbackground neutrinos to those from cosmic-ray sources at EeV energies\n\\cite{Spiering:2012xe}.\\footnote{See also the IceCube MasterClass at\n  \\url{http:\/\/masterclass.icecube.wisc.edu\/en\/learn\/detecting-neutrinos}.} The only detailed previous study of the keV range\nsolar flux \\cite{Haxton:2000xb} ignores bremsstrahlung production and\noverestimates photo production by a spurious plasmon resonance. This\nsituation motivates us to take a completely fresh look, taking advantage\nof recent progress in calculating the keV-range solar flux of other low-mass\nparticles such as axions and hidden photons\n\\cite{Pospelov:2008jk,Derevianko:2010kz,An:2013yfc,Redondo:2013lna,Redondo:2013wwa,Hardy:2016kme}.\n\nThermal neutrino emission from stars is an old topic, central to the physics\nof stellar evolution, and detailed studies exist as well as Computer routines\nto be coupled with stellar evolution codes \\cite{Itoh:1996}. However, in this\ncontext neutrinos play the role of a local energy sink for the stellar plasma\nand so the emission spectrum is not provided. Moreover, for a low-mass\nmain-sequence star like our Sun, energy loss by thermal neutrinos is\nnegligible. Therefore, standard energy-loss rates, which cover a large range\nof temperatures, densities and chemical compositions, may not be optimized\nfor solar conditions.\n\n\\begin{figure}[b!]\n\\centering\n\\hbox to\\textwidth{\\includegraphics[height=5.8cm]{pics\/fig02a.pdf}\n\\hfil\\includegraphics[height=5.8cm]{pics\/fig02b.pdf}}\n\\vskip12pt\n\\hbox to\\textwidth{\\includegraphics[height=5.8cm]{pics\/fig02c.pdf}\n\\hfil\\includegraphics[height=5.8cm]{pics\/fig02d.pdf}}\n\\caption{Solar neutrino flux at Earth in the keV range. The flavor\n  dependence is given in the mass basis for the 1, 2 and 3 mass\n  eigenstates (blue, orange and green). Thick lines are for $\\bar\\nu$,\n  thin lines for $\\nu$ which includes a contribution from the nuclear\n  pp reaction which produces only $\\nu_e$ at the source. The other\n  source channels are thermal reactions which produce $\\nu$ and\n  $\\bar\\nu$ in equal measure.  The bottom panels show the fractions of\n  the total flux provided by the individual mass eigenstates.}\n\\label{fig:summaryflux}\n\\end{figure}\n\nLow-energy neutrinos are produced in the solar plasma by the pair-production\nprocesses shown in figure~\\ref{fig:processes}, where nonrelativistic\nelectrons are the sources. Electron velocities and spins are ``kicked'' by the\nambient electromagnetic fields, leading to the emission of neutrino pairs. At\nlow energies, the weak interaction is sufficiently well described by an\neffective four-fermion local interaction\nproportional to the Fermi constant $G_{\\rm F}$. The effective coupling constants\nfor the vector and axial-vector interaction, $C_{\\rm V}$ and $C_{\\rm A}$, are different\nfor $\\nu_e$ and the other flavors, leading to a nontrivial flavor dependence\nof the emitted fluxes. The vector-current interaction leads essentially to\nelectric dipole radiation caused by the time variation of the electron\nvelocity, whereas the axial-vector current leads to magnetic dipole radiation\ncaused by fluctuations of the electron spin. Yet in the nonrelativistic\nlimit, the rates for both mechanisms are related by simple numerical factors\nand there is no interference between them, so all processes provide rates\nproportional to $(a\\, C_{\\rm V}^2+b\\,C_{\\rm A}^2)G_{\\rm F}^2$ with coefficients $a$ and $b$\nthat depend on the specific emission process. One consequence of this simple\nstructure is that the emission rates are closely related to those for axions\n(axial current interaction) or hidden photons (vector current interaction)\nand also closely related to photon absorption rates. We will take full\nadvantage of these similarities, i.e., the relation between these different\nprocesses by simple phase-space factors.\n\nIn figure~\\ref{fig:summaryflux} we show the overall low-energy solar\nneutrino and antineutrino flux at Earth from our calculation.  All\nthermal processes shown in figure~\\ref{fig:processes} produce $\\nu\\bar\\nu$ pairs\nand thus equal fluxes of neutrinos and antineutrinos.\nThis equipartition is another consequence\nof the nonrelativistic approximation, where weak magnetism effects\ndisappear along with $C_{\\rm V}C_{\\rm A}$ cross terms in the emission\nrate~\\cite{Horowitz:2001xf}. In addition, the low-energy tail of the\nneutrino spectrum produced in the nuclear pp reaction contributes\nsignificantly to the keV flux. At the source, this reaction\nproduces $\\nu_e$ which, like the other channels, have decohered into\ntheir mass components long before they reach Earth. The pp flux causes\nan overall asymmetry between the keV-range $\\nu$ and $\\bar\\nu$\nspectra. The fractional contribution of the 1, 2 and 3 mass\neigenstates arriving at Earth are shown in the\nlower panels of figure~\\ref{fig:summaryflux}.\nDifferent emission processes have different energy dependences and there are different coefficients $(a\\,C_{\\rm V}^2+b\\,C_{\\rm A}^2)$ for $\\nu_e$ and the other flavors,\nthus explaining the fractional flux variation.\n\nThe rest of the paper is devoted to deriving the results shown in\nfigure~\\ref{fig:summaryflux}. In\nsections~\\ref{sec:plasmon-decay}--\\ref{sec:freebound} we study\nindividual processes. In section~\\ref{sec:solarflux} we derive the\noverall flux at Earth after integration over a standard solar model\nwhich is detailed in appendix~\\ref{app:smm}.\nSection~\\ref{sec:discussion} is finally devoted to a summary and\ndiscussion.\n\n\n\\section{Plasmon decay}\n\\label{sec:plasmon-decay}\n\n\\subsection{Matrix element}\n\nWe begin our calculation of neutrino pair emission from the solar\ninterior with plasmon decay, $\\gamma\\to\\nu\\bar\\nu$, the process of\nfigure~\\ref{fig:processes} involving the smallest number of\nparticipating particles. This process is also special in that it has\nno counterpart for axion emission.\nIn any medium, electromagnetic excitations with\nwave vector $k=(\\omega,{\\bf k})$ acquire a nontrivial dispersion relation\nthat can be written in the form $\\omega^2-{\\bf k}^2=\\Pi({\\bf k})$, where\n$\\Pi_{\\bf k}=\\Pi({\\bf k})$ is the on-shell polarization function. We will always\nconsider an unmagnetized and isotropic plasma. It supports both\ntransverse (T) modes, corresponding to the usual photons, and\nlongitudinal (L) modes, corresponding to collective oscillations of\nelectrons against ions. Whenever $\\Pi_{\\bf k}>0$ (time like dispersion),\nthe decay into a neutrino pair, taken to be massless, is kinematically\nallowed. For both T and L modes, neutrino pairs are actually emitted\nby electrons which oscillate coherently as a manifestation of the\nplasma wave.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=6cm]{pics\/fig03.pdf}\n\\caption{Plasmon decay mediated by electrons of the medium.}\n\\label{fig:plasmon-graph}\n\\end{center}\n\\end{figure}\n\nTherefore, plasmon decay and actually all other processes relevant for thermal\npair emission in the Sun depend on the neutrino-electron interaction.\nAt low energies, it is given by the effective\nneutral-current interaction\n\\begin{equation}\\label{eq:NC-interaction}\n  {\\mathcal L}_{\\rm int}=\\frac{G_{\\rm F}}{\\sqrt{2}}\\,\n  \\bar\\psi_e\\gamma^\\mu(C_{\\rm V}-C_{\\rm A}\\gamma_5 )\\psi_e\\,\n  \\bar\\psi_\\nu\\gamma_\\mu(1-\\gamma_5)\\psi_\\nu\\,,\n\\end{equation}\nwhere $G_{\\rm F}$ is Fermi's constant. The effective vector (V) and axial-vector (A)\ncoupling constants include a neutral-current contribution and for\n$\\nu_e$ also a charged-current piece from $W^\\pm$ exchange. Altogether\none finds\n\\begin{subequations}\n\\begin{eqnarray}\n  \\hbox to 7.cm{$C_{\\rm V}=\\frac{1}{2}(4\\sin\\Theta_{\\rm W}+1)$\n    \\quad and\\quad$C_{\\rm A}=+\\frac{1}{2}$\\hfil}&&\n  \\hbox to 3.cm{for\\quad$\\nu_e$,\\hfil}\\\\\n  \\hbox to 7.cm{$C_{\\rm V}=\\frac{1}{2}(4\\sin\\Theta_{\\rm W}-1)$\n    \\quad and\\quad$C_{\\rm A}=-\\frac{1}{2}$\\hfil}&&\n  \\hbox to 3.cm{for\\quad$\\nu_\\mu$ and $\\nu_\\tau$,\\hfil}\n\\end{eqnarray}\n\\end{subequations}\nwhere $4\\sin^2\\Theta_{\\rm W}=0.92488$ in terms of the weak mixing\nangle.\nIn particular, this implies that the rates of A processes,\nproportional to $C_{\\rm A}^2=1\/4$, are the same for all flavors. On the\nother hand, the rates for V processes are proportional to\n\\begin{equation}\n   C_{\\rm V}^2=0.9263~~\\hbox{for}~\\nu_e\\bar\\nu_e\n   \\quad\\hbox{and}\\quad\n   C_{\\rm V}^2=0.0014~~\\hbox{for}~\\nu_{\\mu,\\tau}\\bar\\nu_{\\mu,\\tau}\\,.\n\\end{equation}\nThus for heavy-lepton neutrinos we may\nsafely ignore the vector-current interaction, i.e., such processes\nproduce an almost pure $\\nu_e\\bar\\nu_e$ flux.\n\nPlasmon decay has been extensively\nstudied in the literature \\cite{Adams:1963zzb,Zaidi:1963zzb,Braaten:1993jw,Haft:1993jt,Ratkovic:2003td}.\nThe squared matrix element for the\ntransition $\\gamma\\to\\nu\\bar\\nu$ with photon four-momentum\n$k=(\\omega,{\\bf k})$ and $\\nu$ and $\\bar\\nu$ four momenta\n$k_1=(\\omega_1,{\\bf k}_1)$ and $k_2=(\\omega_2,{\\bf k}_2)$ is found to be\n(see figure~\\ref{fig:plasmon-graph}),\n\\begin{equation}\n  |\\mathcal{M}_{\\gamma\\to\\nu\\bar\\nu}|^2=\n  \\frac{C_{\\rm V}^2G_{\\rm F}^2}{8\\pi\\alpha}\\,Z_{{\\bf k}}\\Pi^2_{{\\bf k}}\\,\n  \\epsilon_\\mu\\epsilon_\\nu^* N^{\\mu\\nu},\n\\end{equation}\nwhere $\\alpha=e^2\/4\\pi$ is the fine-structure constant.\n$Z_{\\bf k}$ is the on-shell\nwave-function renormalization factor and\n$\\Pi_{\\bf k}$ the polarization factor appropriate for the T or L\nexcitation. The photon polarization vector is $\\epsilon^\\mu$\nwith $\\epsilon^\\mu\\epsilon_\\mu^*=-1$. The neutrino tensor, appearing in all\npair emission processes, is\n\\begin{equation}\\label{eq:neutrino-tensor}\n  N^{\\mu\\nu}=8\\bigl(k_1^\\mu k_2^\\nu+k_1^\\nu k_2^\\mu\n  -k_1{\\cdot}k_2\\,g^{\\mu\\nu}+i\\varepsilon^{\\alpha\\mu\\beta\\nu}k_{1\\alpha}k_{2\\beta}\\bigr).\n\\end{equation}\nInserting this expression in the squared matrix element yields\n\\begin{equation}\n  |\\mathcal{M}_{\\gamma\\to\\nu\\bar\\nu}|^2=\n  \\frac{C_{\\rm V}^2G_{\\rm F}^2}{\\pi\\alpha}\\,Z_{{\\bf k}}\\Pi^2_{{\\bf k}}\\,\n  (\\epsilon^*{\\cdot}k_1\\,\\epsilon{\\cdot}k_2+\\epsilon{\\cdot}k_1\\,\\epsilon^*{\\cdot}k_2+k_1{\\cdot}k_2\\bigr)\\,.\n\\end{equation}\nNotice that the axial-vector interaction does not induce plasmon decay under the approximations described. This is\nparticularly obvious in the nonrelativistic limit where we can think\nof the emission process as dipole radiation from coherently\noscillating electrons, whereas the electron spins, responsible for\nnon-relativistic axial-current processes, do not oscillate coherently. The\nabsence of a sizeable axial-current rate implies that\nplasmon decay produces with high accuracy only $\\nu_e\\bar\\nu_e$ pairs.\n\n\\subsection{Nonrelativistic limit}\n\nIn a classical plasma (nonrelativistic and nondegenerate), the\nelectromagnetic dispersion relations for transverse (T) and\nlongitudinal (L) plasmons are found to be\n\\begin{equation}\n\\omega^2\\big|_{\\rm T}=\n    \\omega_{\\rm p}^2\\left(1+\\frac{{\\bf k}^2}{\\omega_{\\rm p}^2+{\\bf k}^2}\\,\\frac{T}{m_e}\\right)+{\\bf k}^2\n\\qquad\\hbox{and}\\qquad\n  \\omega^2\\big|_{\\rm L}=\\omega_{\\rm p}^2\\left(1+3\\,\\frac{{\\bf k}^2}{\\omega_{\\rm p}^2}\\,\\frac{T}{m_e}\\right).\n\\end{equation}\nThe plasma frequency is given in terms of the electron density\n$n_e$ by\n\\begin{equation}\n    \\omega_{\\rm p}^2=\\frac{4\\pi\\alpha\\,n_e}{m_e}\\,.\n\\end{equation}\nIn the Sun, $T\\lesssim 1.3~{\\rm keV}$ so that $T\/m_e\\lesssim 0.0025$ and\nwith excellent approximation we may limit our discussion to the\nlowest-order term. Moreover, the lowest-order expression pertains\nto any level of degeneracy as long as the electrons remain nonrelativistic.\nIn this case, T modes propagate in the same way\nas particles with mass $\\omega_{\\rm p}$, i.e., $\\omega^2={\\bf k}^2+\\omega_{\\rm p}^2$, whereas\nL modes oscillate with a fixed frequency $\\omega=\\omega_{\\rm p}$, independently\nof ${\\bf k}$.\nTherefore, the L-plasmon dispersion relation is\ntime-like only for $|{\\bf k}|<\\omega_{\\rm p}$, so only these soft quanta can decay\ninto neutrino pairs.\n\nIn the nonrelativistic limit and using Lorentz gauge one finds\n$Z_{\\rm T}=1$, $\\Pi_{\\rm T}=\\omega_{\\rm p}^2$,\n$Z_{\\rm L}=\\omega_{\\rm p}^2\/(\\omega_{\\rm p}^2-{\\bf k}^2)$ and\n$\\Pi_{\\rm L}=\\omega_{\\rm p}^2-{\\bf k}^2$. Without loss of generality,\nwe may assume the photon to move in the $z$ direction. The T\npolarization vectors are in this case $\\epsilon^\\mu=(0,1,0,0)$ and\n$\\epsilon^\\mu=(0,0,1,0)$, respectively, whereas the L case with\n$|{\\bf k}|<\\omega_{\\rm p}$ has $\\epsilon^\\mu=(|{\\bf k}|,0,0,\\omega_{\\rm p})\/(\\omega_{\\rm p}^2-{\\bf k}^2)^{1\/2}$.\nIn Coulomb gauge one finds different expressions for the L quantities.\n\n\n\\subsection{Decay rate and spectrum}\n\nNext we consider the decay rate of a transverse or longitudinal\non-shell plasmon with wave vector ${\\bf k}$ and ask for its decay rate\n\\begin{equation}\\label{eq:Gamma-plasmon}\n  \\Gamma_{\\gamma\\to\\nu\\bar\\nu}=\n  \\int\\frac{d^3{\\bf k}_1}{(2\\pi)^3}\\,\\frac{d^3{\\bf k}_2}{(2\\pi)^3}\\,\n  \\frac{|{\\cal M}_{\\gamma\\to\\nu\\bar\\nu}|^2}{2\\omega\\,2\\omega_1\\,2\\omega_2}\\,\n  (2\\pi)^4\\,\\delta^4(k-k_1-k_2)\\,.\n\\end{equation}\nIn a nonrelativistic plasma one easily finds the usual result\n\\begin{equation}\n  \\Gamma_{\\rm T}=\\Gamma_{\\rm p}\\,\\frac{\\omega_{\\rm p}}{\\omega_{\\bf k}}\n    \\qquad\\hbox{and}\\qquad\n  \\Gamma_{\\rm L}=\\Gamma_{\\rm p}\\,\\frac{(\\omega_{\\rm p}^2-{\\bf k}^2)^2}{\\omega_{\\rm p}^4}\n   \\qquad\\hbox{with}\\qquad\n \\Gamma_{\\rm p}=\\frac{C_{\\rm V}^2G_{\\rm F}^2\\omega_{\\rm p}^5}{48\\,\\pi^2\\alpha}\\,.\n\\end{equation}\nFor T plasmons $\\omega_{\\bf k}=(\\omega_{\\rm p}^2+{\\bf k}^2)^{1\/2}$\nwith $0\\leq|{\\bf k}|<\\infty$, whereas for L\nplasmons the decay is allowed for $0\\leq|{\\bf k}|<\\omega_{\\rm p}$. The T case\nis reminiscent of a decaying particle with mass $\\omega_{\\rm p}$ where the laboratory decay rate\nis time-dilated by the factor $\\omega_{\\rm p}\/\\omega_{\\bf k}$. In the limit ${\\bf k}\\to 0$ both rates are the same.\nIndeed, in the limit of vanishing wave number one cannot distinguish a transverse from\na longitudinal excitation.\n\nWe are primarily interested in the neutrino energy spectrum. The symmetry\nof the squared matrix element under the exchange $k_1\\leftrightarrow k_2$ implies\nthat it is enough to find the $\\nu$ spectrum which is identical to the one for $\\bar\\nu$.\nTherefore, in equation~(\\ref{eq:Gamma-plasmon}) we integrate over $d^3{\\bf k}_2$ to remove\nthe momentum delta function, and over $d\\Omega_1$ to remove the one for energy conservation.\nOverall, with $\\omega_\\nu=\\omega_1$ we write the result in the form\n\\begin{equation}\n\\frac{d\\Gamma}{d\\omega_\\nu}=\\Gamma\\,g(\\omega_\\nu)\\,,\n\\end{equation}\nwhere $g(\\omega_\\nu)$ is a normalized function. For T\nplasmons, averaged over the two polarization states, we find\n\\begin{equation}\\label{eq:T-spectrum}\ng_{\\rm T}(\\omega_\\nu)= \\frac{3}{4}\\,\\frac{{\\bf k}^2+\\(\\omega_{\\bf k}-2\\,\\omega_\\nu\\)^2}{|{\\bf k}|^3}\n\\quad\\hbox{for}\\quad \\frac{\\omega_{\\bf k}-|{\\bf k}|}{2} <\\omega_\\nu< \\frac{\\omega_{\\bf k}+|{\\bf k}|}{2}\n\\end{equation}\nand zero otherwise. If the T plasmon were an unpolarized massive\nspin-1 particle, this would be a top-hat spectrum on the shown\ninterval, corresponding to isotropic emission boosted to the\nlaboratory frame. However, the T plasmon misses the third polarization\nstate so that even unpolarized T plasmons do not show this\nbehavior. For L plasmons we find\n\\begin{equation}\ng_{\\rm L}(\\omega_\\nu)= \\frac{3}{2}\\,\\frac{{\\bf k}^2-\\(\\omega_{\\rm p}-2\\,\\omega_\\nu\\)^2}{|{\\bf k}|^3}\n\\quad\\hbox{for}\\quad \\frac{\\omega_{\\rm p}-|{\\bf k}|}{2} <\\omega_\\nu< \\frac{\\omega_{\\rm p}+|{\\bf k}|}{2}\n\\end{equation}\nand zero otherwise, with the additional constraint $0\\leq |{\\bf k}|<\\omega_{\\rm p}$.\n\nWe show these distributions in figure~\\ref{fig:plasmon-spectra}.\nAssuming equal $\\omega$ for both types of excitations and also\nequal ${\\bf k}$, the distributions add to a top-hat spectrum of the form\n$\\frac{2}{3}\\,g_{\\rm T}(\\omega_\\nu)+\\frac{1}{3}\\,g_{\\rm L}(\\omega_\\nu)=1\/|{\\bf k}|$\non the interval\n$(\\omega-|{\\bf k}|)\/2<\\omega_\\nu<(\\omega+|{\\bf k}|)\/2$, i.e.,\nthis average resembles the decay spectrum of an unpolarized spin-1\nparticle.\nHowever, the dispersion relations are different for T and L plasmons\nso that, for equal ${\\bf k}$, they have different $\\omega$ and these\ndistributions are not on the same $\\omega_\\nu$ interval. The only\nexception is the ${\\bf k}\\to0$ limit when $\\omega\\to\\omega_{\\rm p}$ for both\ntypes.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=7.5cm]{pics\/fig04.pdf}\n\\caption{Normalized $\\nu$ spectrum from transverse and longitudinal\nplasmon decay $\\gamma\\to\\nu\\bar\\nu$. For T plasmons $\\omega=(\\omega_{\\rm p}^2+{\\bf k}^2)^{1\/2}$,\nwhereas for L plasmons $\\omega=\\omega_{\\rm p}$.}\n\\label{fig:plasmon-spectra}\n\\end{center}\n\\end{figure}\n\n\\subsection{Thermal emission spectrum}\n\n\\begin{figure}[b!]\n\\centering\n\\hbox to \\textwidth{\\includegraphics[height=5.7cm]{pics\/fig05a.pdf}\\hfil\\includegraphics[height=5.7cm]{pics\/fig05b.pdf}}\n\\caption{Neutrino spectrum from thermal plasmon decay.\n  {\\em Left panel:\\\/} Transverse plasmons. The curves represent the dimensionless integral in equation~(\\ref{eq:T-spectrum-final}) and correspond to $\\omega_{\\rm p}\/T=0.25$ and 1 as indicated.\n  {\\em Right panel:\\\/} Longitudinal plasmons. The curve is the dimensionless integral in equation~(\\ref{eq:L-spectrum}). To make the vertical scale comparable to T plasmons, a factor\n  $(\\omega_{\\rm p}\/T)\/(e^{\\omega_{\\rm p}\/T}-1)=1+\\mathcal{O}(\\omega_{\\rm p}\/T)$ must be included.}\n\\label{fig:TL-spectrum}\n\\end{figure}\n\nAs our final result we determine the spectral emission density from a\nnonrelativistic plasma with temperature $T$. The number of neutrinos emitted\nper unit volume per unit time per unit energy interval from T plasmon decay is\n\\begin{equation}\\label{eq:T-spectrum-final}\n\\frac{d \\dot n_\\nu}{d\\omega_\\nu}\\Big|_{\\rm T}=\n\\int\\limits_{V_{\\bf k}}\\frac{d^3{\\bf k}}{(2\\pi)^3}\\,\\frac{2 \\Gamma_{\\rm T} g_{\\rm T}(\\omega_\\nu)}{e^{\\omega_{\\bf k}\/T}-1}\n=\\frac{3\\,\\Gamma_{\\rm p}\\omega_{\\rm p} T}{4\\pi^2}\\!\\!\\int\\limits_{\\omega_\\nu+\\frac{\\omega_{\\rm p}^2}{4\\omega_\\nu}}^{~\\infty}\n\\!\\!\\frac{d\\omega}{T}\\,\\frac{1}{e^{\\omega\/T}-1}\\,\\[1+\\frac{(\\omega-2\\omega_\\nu)^2}{\\omega^2-\\omega_{\\rm p}^2}\\]\\,.\n\\end{equation}\nThe integration is over the volume in ${\\bf k}$-space allowed by the decay kinematics\ngiven in equation~(\\ref{eq:T-spectrum}) and the factor of 2\naccounts for two transverse degrees of freedom.\nFor $\\omega_\\nu\\ll\\omega_{\\rm p}$ the required plasmon energy\nis large so that we may approximate $e^{\\omega\/T}-1\\to e^{\\omega\/T}$ and\n$(\\omega-2\\omega_\\nu)^2\/(\\omega^2-\\omega_{\\rm p}^2)\\to 1$. In this case the\ndimensionless integral is \\smash{$2 e^{-\\omega_{\\rm p}^2\/4\\omega_\\nu T}$},\ni.e., this neutrino flux is exponentially suppressed at low energies due to the exponential suppression of\nthe density of T-plasmons with sufficient energy.\nIn figure~\\ref{fig:TL-spectrum} we show\nthe dimensionless integral as a function of $\\omega_\\nu\/T$\nfor $\\omega_{\\rm p}\/T=0.25$ and 1. Notice that in the central solar region\n$T=1.3~{\\rm keV}$ and $\\omega_{\\rm p}=0.3~{\\rm keV}$ so that\n$\\omega_{\\rm p}\/T=0.25$ corresponds approximately to conditions of the central\nSun. The external shells of the Sun, where $\\omega_{\\rm p}^2\/T$ is smaller, turn out to be\nrelevant for the lowest energy neutrinos from T-plasmon decay. However, we show later that this contribution is subdominant.\n\nFor L plasmons, the integral over the initial photon distribution\nyields the spectrum of the number emission rate\n\\begin{eqnarray}\\label{eq:L-spectrum}\n\\frac{d \\dot n_\\nu}{d\\omega_\\nu}\\Big|_{\\rm L}&=&\n\\int\\limits_{V_{\\bf k}}\\frac{d^3{\\bf k}}{(2\\pi)^3}\\,\\frac{\\Gamma_{\\rm L}\n  g_{\\rm L}(\\omega_\\nu)}{e^{\\omega_{\\rm p}\/T}-1}\n\\nonumber\\\\\n&=&\\frac{3\\Gamma_{\\rm p}\\,\\omega_{\\rm p}^2}{4\\pi^2(e^{\\omega_{\\rm p}\/T}-1)}\\,\n\\int\\limits_{|\\omega_{\\rm p}-2\\omega_\\nu|}^{~\\omega_{\\rm p}}\n\\!\\!d|{\\bf k}|\\,\\frac{|{\\bf k}|\\,\\(\\omega_{\\rm p}^2-{\\bf k}^2\\)^2}{\\omega_{\\rm p}^6}\n\\,\\[1-\\frac{(\\omega_{\\rm p}-2\\omega_\\nu)^2}{{\\bf k}^2}\\]\n\\nonumber\\\\[1ex]\n&=&\\frac{3\\Gamma_{\\rm p}\\,\\omega_{\\rm p} T}{4\\pi^2}\\,\\frac{\\omega_{\\rm p}\/T}{e^{\\omega_{\\rm p}\/T}-1}\\,\\,\n\\(\\frac{2+3y^2-6y^4+y^6}{12}+y^2\\log|y|\\)\\,,\n\\end{eqnarray}\nwhere $y=(2\\omega_\\nu-\\omega_{\\rm p})\/\\omega_{\\rm p}$ equivalent to\n$\\omega_\\nu=(y+1)\\,\\omega_{\\rm p}\/2$. L plasmons have the fixed\nenergy $\\omega_{\\rm p}$, yet neutrinos from decay occupy the full\ninterval $0<\\omega_\\nu<\\omega_{\\rm p}$ owing to the peculiar L dispersion\nrelation. The neutrino distribution is symmetric relative to\n$\\omega_\\nu=\\omega_{\\rm p}\/2$. In figure~\\ref{fig:TL-spectrum}\nwe show the dimensionless $\\omega_\\nu$ distribution which\nis universal for any value of $\\omega_{\\rm p}$.\n\nFor $\\omega_\\nu\\ll\\omega_{\\rm p}$ the dimensionless integral can be expanded\nand yields $(32\/3)\\,(\\omega_\\nu\/\\omega_{\\rm p})^4$, i.e., the spectrum decreases as a power law.\nBecause the T-plasmon spectrum decreases exponentially,\nthe L-plasmon decay provides the dominant\nneutrino flux at very low $\\omega_\\nu$.\nWe illustrate this point in figure~\\ref{fig:plasmon-log-spectrum}\nwhere we show both spectra in common units of $3\\Gamma_{\\rm p}\\omega_{\\rm p} T\/4\\pi^2$\nfor $\\omega_{\\rm p}\/T=0.25$ on a log-log-plot. The central solar temperature is 1.3~keV,\nso L-plasmon decay takes over for sub-eV neutrinos where the overall rate is\nextremely small.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=7.5cm]{pics\/fig06.pdf}\n\\caption{Neutrino spectrum from thermal T and L plasmon decay for\n$\\omega_{\\rm p}\/T=0.25$ in units of $3\\Gamma_{\\rm p}\\omega_{\\rm p} T\/4\\pi^2$. At\nvery low energies, L-plasmon decay dominates.}\n\\label{fig:plasmon-log-spectrum}\n\\end{figure}\n\n\n\\subsection{Compton pole process?}\n\\label{sec:pole}\n\nThus far we have used kinetic theory in that we treat the excitations\nof the medium as free particles which propagate until they decay or\ncollide. Plasmons were treated as quasi-stable excitations,\ndistributed as an ideal Bose gas, which occasionally decay into a\nneutrino pair. In a previous study of solar thermal neutrino emission\n\\cite{Haxton:2000xb} another channel was considered in the form\n$\\gamma+e^-\\to e^-+\\gamma$ followed by $\\gamma\\to\\nu\\bar\\nu$, i.e.,\nthe decaying plasmon was treated as an intermediate virtual particle\nin a Compton-like process. In its propagator, an imaginary part (a\nwidth) was included, but it was stressed that this width is small and\nthat one needs to integrate over a narrow range of virtual\nenergy-momenta near the on-shell condition. This ``Compton plasmon\npole'' process was found to dominate thermal pair emission, a finding\nthat would change everything about neutrino energy losses from stars.\n\nHowever, we think this result is spurious. In a plasma, of course any\nparticle is an intermediate state between collisions and as such a\npole in a more complicated process. It is the very basis of the\nkinetic approach to treat particles as on-shell states coming from far\naway without memory of their previous history. This assumption need\nnot always be justified, but there is no particular reason why in the\npresent context it should not apply to the plasmon. Its width is very\nsmall as stressed in reference~\\cite{Haxton:2000xb}.\n\nOn the other hand, it is not wrong to trace the plasmon one step back\nin its collision history. In this case one must be consistent,\nhowever, as to which processes produce and absorb plasmons and are\nthus responsible for its width. In reference~\\cite{Haxton:2000xb} the\nplasmon width was taken as a complicated expression from the\nliterature based essentially on inverse bremsstrahlung, whereas the\nlast thing the plasmon did before decaying was taken to be Compton\nscattering. In this way, their equation~(11) includes in the numerator\nessentially the Compton production rate, in the denominator the\nimaginary part of the propagator based on inverse bremsstrahlung. The\nemission rate gets spuriously enhanced by a large ratio of plasmon\ninteraction rates based on different processes.\n\nIn summary, as long as the plasmon width is small, as everybody agrees\nit is, the ``pole process'' is identical with the plasmon decay\nprocess, not a new contribution. The only difference is that for a\ngiven momentum, the plasmon energy distribution is taken to follow a\ndelta function (plasmon decay) or a narrow resonance distribution\n(pole process). The overall normalization is the same in both cases.\n\n\n\\subsection{Solar neutrino flux}\n\nWe finally integrate the plasmon decay rate over a standard solar\nmodel and show the expected neutrino flux at Earth in\nfigure~\\ref{fig:plasma-flux}. We specifically use a solar model\nfrom the Saclay group which is described in more detail in\nappendix~\\ref{app:smm}. At this stage we do not worry about\nflavor oscillations and simply give the all-flavor flux at Earth,\nrecalling that plasmon decay produces pure $\\nu_e\\bar\\nu_e$ pairs\nat the source. We find\n\\begin{equation}\n\\Phi_{\\rm T}=4.12\\times10^5~{\\rm cm}^{-2}~{\\rm s}^{-1}\n\\qquad\\hbox{and}\\qquad\n\\Phi_{\\rm L}=4.67\\times10^3~{\\rm cm}^{-2}~{\\rm s}^{-1}\n\\end{equation}\nfor the integrated fluxes at Earth.\n\nThe T plasmon flux now reaches much smaller energies than\nin figure~\\ref{fig:plasmon-log-spectrum} when we considered\nconditions near the solar center. Very low-energy\nneutrinos from plasmon decay require the plasmon to be very\nrelativistic because the accessible energy range is\n$\\omega-|{\\bf k}|<\\omega_\\nu<\\omega+|{\\bf k}|$, so the low energy flux is\nexponentially suppressed due to the exponential suppression of the density of\nhigh-energy plasmons. At larger solar radii $T$ is\nsmaller, but $\\omega_{\\rm p}$ drops even faster\nand T plasmons are more relativistic. Therefore,\nlower-energy neutrinos become kinematically allowed, i.e.,\nlower-energy neutrinos derive from larger solar radii. From\nfigure~\\ref{fig:plasma-flux} we conclude that the L plasmon flux\nbegins to dominate at energies so low that the assumption of\nmassless neutrinos is not necessarily justified.\n\n\\begin{figure}[htbp]\n\\centering\n\\hbox to\\textwidth{\\includegraphics[height=5.8cm]{pics\/fig07a.pdf}\\hfil\n\\includegraphics[height=5.8cm]{pics\/fig07a.pdf}}\n\\caption{Solar neutrino flux at Earth from transverse and longitudinal plasmon decay. This is the total $\\nu$ flux produced as nearly pure $\\nu_e$ in the Sun. There is an equal $\\bar\\nu$ flux. We also show the\n  low-energy tail of the usual flux from the nuclear pp reaction which are\n  born as $\\nu_e$.}\n\\label{fig:plasma-flux}\n\\end{figure}\n\nFor comparison we also show the $\\nu$ flux from the pp reaction\nthat produces the lowest-energy flux from nuclear reactions.\nAll standard solar models agree on this flux within around 1\\%,\nso we may use as a generic number $6.0\\times10^{10}~{\\rm cm}^{-2}~{\\rm s}^{-1}$.\nThe reaction $p+p\\to d+e^++\\nu_e$ has\na neutrino endpoint energy of $Q=420.22~{\\rm keV}$. Including the thermal\nkinetic energy of the protons in the solar plasma, the overall endpoint of\nthe solar spectrum is $Q=423.41~{\\rm keV}$ \\cite{Bahcall:1997eg}. This reference\nalso gives a numerical tabulation of the solar $\\nu_e$ spectrum from the pp reaction.\nIgnoring a small\n$e^+$ final-state correction, the spectrum follows that of an allowed\nweak transition of the normalized form\n\\begin{eqnarray}\n\\frac{dN}{d\\omega_\\nu}&=&\n\\frac{\\omega_\\nu^2\\,(Q+m_e-\\omega_\\nu)\\sqrt{(Q+m_e-\\omega_\\nu)^2-m_e^2}}{A^5}\n\\nonumber\\\\[1ex]\n&=&\\frac{(Q+m_e)\\sqrt{Q(Q+2m_e)}}{A^5}\\,\\omega_\\nu^2+{\\cal O}(\\omega_\\nu^3)\n\\end{eqnarray}\nwhere $A=350.8~{\\rm keV}$ if we use the solar endpoint energy.\nThe analytic form of the normalization factor $A$ is too complicated to be shown here.\nThe low-energy pp flux spectrum at Earth is the power law\n\\begin{equation}\\label{eq:ppflux}\n\\frac{d\\Phi_{pp}}{d\\omega_\\nu}=8150~{\\rm cm}^{-2}~{\\rm s}^{-1}~{\\rm keV}^{-1}\\,\\(\\frac{\\omega_\\nu}{\\rm keV}\\)^2\\,,\n\\end{equation}\nan approximation that is good to about $\\pm1.5$\\% for energies below 10~keV.\nThis shallow power law is simply determined by the low-energy neutrino phase space.\nIt dominates over plasmon decay at very low energies, although, of course, it does\nnot produce antineutrinos. We will show later that both are subdominant compared to the neutrino flux produced in bremsstrahlung transitions.\n\n\n\\section{Photo production}\n\\label{sec:compton}\n\n\\subsection{Matrix element and decay rate}\n\nThe Compton process (figure~\\ref{fig:compton-graph}), also known as\nphoto production or photoneutrino production, was one of the first\nprocesses to be considered as an energy-loss mechanism for stars\n\\cite{Chiu:1961zza, Petrosian:1967alk, Ritus:1961alk}. In these older\npapers, only the energy-loss rate was calculated, whereas the neutrino\nspectrum was calculated in the nonrelativistic limit in\nreference~\\cite{Haxton:2000xb} and for general kinematics in\nreference~\\cite{Dutta:2003ny}. We restrict ourselves to the\nnonrelativistic limit where electron recoils are neglected. The\nprocess then amounts to the conversion $\\gamma\\to\\nu\\bar\\nu$,\ncatalyzed through bystander electrons which take up three-momentum,\nand as such is somewhat similar to plasmon decay.  However, no plasma\nmass is needed because momentum is taken up by the electrons. Even\nthough we neglect recoils, the process is not ``forward'' for the\nelectrons. We can interpret plasmon decay as the coherent version of\nphoto production.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=6cm]{pics\/fig08.pdf}\n\\caption{Photo production of neutrino pairs (Compton process).\n  A second diagram with vertices interchanged is not shown.\n}\n\\label{fig:compton-graph}\n\\end{figure}\n\nIn the nonrelativistic limit we find for the squared matrix element, averaged over initial spins\nand polarizations and summed over final ones,\n\\begin{equation}\n\\overline{|\\mathcal{M}|^2}=\\frac{1}{4}\\sum_{\\epsilon,s_1,s_2}|\\mathcal{M}|^2\n=\\frac{e^2 G_{\\rm F}^2}{\\omega^2}\\, M^{\\mu\\nu}N_{\\mu\\nu}\n\\end{equation}\nwhere the neutrino tensor was given in equation~(\\ref{eq:neutrino-tensor}). The nonrelativistic\nelectron tensor for the Compton process is~\\cite{Haxton:2000xb}\n\\begin{equation}\nM^{\\mu\\nu}=\\sum_\\epsilon\\Bigl\\lbrace\\(C_{\\rm V}^2+C_{\\rm A}^2\\)\\(-\\omega\\epsilon^\\mu+\\delta^{\\mu0}\\epsilon{\\cdot}q\\)\n\\(-\\omega\\epsilon^\\nu+\\delta^{\\nu0}\\epsilon{\\cdot}q\\)\n+C_{\\rm A}^2\\[k^\\mu k^\\nu-(\\epsilon{\\cdot}q)^2 g^{\\mu\\nu}\\]\\Bigr\\rbrace\\,,\n\\end{equation}\nwhere $\\epsilon$ is the photon polarization vector and $q=k_1+k_2$ the four momentum carried away by\nthe neutrino pair.\n\nIt is noteworthy that both the vector and axial-vector currents contribute\non comparable levels, in contrast to plasmon decay. This is heuristically understood if we observe that the vector part amounts\nto electric dipole emission by the electron being ``shaken'' by the incoming EM wave. The rate is\nproportional to $(G_{\\rm F}\/m_e)^2$ because the outgoing radiation couples with strength $G_{\\rm F}$ and the mass appears due to its inertia against the acceleration. Axial-current\nemission amounts to magnetic dipole emission caused by the electron spin. The coupling is through the electron dipole moment $\\propto 1\/m_e$, so the rate is also\nproportional to $(G_{\\rm F}\/m_e)^2$. In the case of axion emission,\n$\\gamma+e\\to e+a$, enabled by a derivative axial-vector coupling, the\nrate is suppressed by a factor $(\\omega\/m_e)^2$ relative to Compton\nscattering. For neutrinos the coupling structure is the same for both\naxial and vector coupling. Note however that these considerations require some handwaving and one should always check which terms in the nonrelativistic expansion of the Hamiltonian contribute to a certain order.\n\nWe use the symmetry under the exchange $1\\leftrightarrow 2$ and\nintegrate over the phase space of $\\bar\\nu$ and over the angles of\n$\\nu$. With $\\omega_{\\rm p}=0$ we find for the differential ``decay rate'' of T\nplasmons with energy $\\omega$ of either polarization\n\\begin{equation}\\label{comptonrate}\n\\frac{d\\Gamma_\\omega}{d\\omega_\\nu}= n_e\\, \\frac{2}{3}\\,\\frac{G_{\\rm F}^2 \\alpha}{\\pi^2 m_e^2}\\, \\(C_{\\rm V}^2+5C_{\\rm A}^2\\) \\,\\frac{(\\omega-\\omega_{\\nu})^2 \\omega_{\\nu}^2}{\\omega}\\[1-\\frac{2}{3}\\frac{(\\omega-\\omega_{\\nu})\\omega_{\\nu}}{\\omega^2}\\]\n\\quad\\hbox{for}\\quad \\omega_\\nu< \\omega \\,.\n\\end{equation}\nIntegrated over the photon distribution it gives the familiar result~\\cite{Haxton:2000xb}\n\\begin{eqnarray}\\label{eq:comptonrate-2}\n  \\frac{d\\dot n_{\\nu}}{d\\omega_\\nu}&=& n_e\\frac{2}{3}\\,\\frac{G_{\\rm F}^2 \\alpha}{\\pi^4 m_e^2} \\,\n  \\int_{\\omega_\\nu}^\\infty d\\omega\\,\n  \\omega_\\nu^2(\\omega-\\omega_\\nu)^2\\,\\frac{1}{e^{\\omega\/T}-1}\\nonumber\\\\[2ex]\n&&\\kern5em{}\\times\n  \\(C_{\\rm V}^2+5C_{\\rm A}^2\\) \\,\\omega\\[1-\\frac{2}{3}\\frac{(\\omega-\\omega_{\\nu})\\omega_{\\nu}}{\\omega^2}\\]\n\\,.\n\\end{eqnarray}\nIncluding the modified dispersion relation in the plasma with a\n  nonvanishing $\\omega_{\\rm p}$ leads to a more complicated expression that modifies the result for $\\omega$ near $\\omega_{\\rm p}$ and by up to\n  a few percent elsewhere, for us a negligible correction.  On the\n  other hand, at energies near $\\omega_{\\rm p}$, Compton emission is subdominant\n  relative to plasmon decay. Moreover, one should then also worry\nabout longitudinal plasmons which can be understood as collective\nelectron oscillations. One therefore would need to avoid double\ncounting between $\\gamma_{\\rm L}+e\\to e+\\nu\\bar\\nu$ and bremsstrahlung\n$e+e\\to e+e+\\nu\\bar\\nu$, see the related discussion in\nreference~\\cite{Raffelt:1987np}. Therefore, we use the emission\n  rate based on the $\\omega_{\\rm p}=0$ expression of\n  equation~\\eqref{eq:comptonrate-2}, but we will include $\\omega_{\\rm p}$\n  in the phase space of initial T plasmons, cutting off $\\omega<\\omega_{\\rm p}$\n  intial-state photons.\n\n\\subsection{Correlation effects}\n\\label{sec:correlatoins}\n\nSo far we have assumed that electrons are completely uncorrelated and\nthe overall neutrino emission rate is the incoherent sum from\nindividual scattering events. However, electrons are anticorrelated\nby the Pauli exclusion principle and by Coulomb repulsion, both\neffects meaning that at the location of a given electron it is less\nlikely than average to find another one. These anticorrelations lead\nto a reduction of the rate, i.e., we need to include a structure\nfactor $S({\\bf q}^2)$ where ${\\bf q}={\\bf k}-{\\bf k}_1-{\\bf k}_2$ is the\nthree-momentum transfer. For photon transport, exchange effects\nproduce a 7\\% correction in the solar center and less elsewhere,\nwhereas Coulomb correlations provide a 20--30\\% correction~\\cite{Boercker:1987jn}.\n\nBeginning with the exchange correlation, a simple approach is to\ninclude a Pauli blocking factor $(1-f_{\\bf p})$ for the final state\nelectron in the phase-space integration. For nonrelativistic electron\ntargets that barely recoil, the final-state ${\\bf p}$ can be taken the\nsame as the initial one, so the overall reduction is the average Pauli\nblocking factor\n\\begin{equation}\\label{eq:Reta-definition}\n  R_\\eta=\\frac{2}{n_e}\\,\\int\\frac{d^3{\\bf p}}{(2\\pi)^3}\\,f_{\\bf p}(1-f_{\\bf p})\n=\\int_0^\\infty \\frac{dx\\,x^2}{e^{x^2-\\eta}+1}\\(1-\\frac{1}{e^{x^2-\\eta}+1}\\)\n  \\Bigg\/\\int_0^\\infty \\frac{dx\\,x^2}{e^{x^2-\\eta}+1}\\,,\n\\end{equation}\nwhere the nonrelativistic degeneracy parameter $\\eta=(\\mu-m_e)\/T$ is given by\n\\begin{equation}\\label{eq:electron-density}\nn_e=2\\int\\frac{d^3{\\bf p}}{(2\\pi)^3}\\,\\frac{1}{e^{\\frac{{\\bf p}^2}{2m_eT}-\\eta}+1}\\,.\n\\end{equation}\nWe have checked that this expression is indeed the $|{\\bf q}|=\\kappa\\to 0$ limit of\n$1+h_x(\\kappa)$ given in equation~(6) of reference\n\\cite{Boercker:1987jn}. In this paper and the literature on solar\nopacities, the Pauli blocking effect is interpreted as an\nanticorrelation of the electrons in analogy to what is caused\nby a repulsive force. So we\ncan picture Pauli effects either as a blocking of final electron states in collisions\nor as an anticorrelation of initial-state electron targets.\n\nNext we turn to Coulomb repulsion where for solar conditions the\nstructure function can be reasonably approximated essentially\nby a Debye-H\\\"uckel screening prescription \\cite{Boercker:1987jn}\n\\begin{equation}\\label{eq:S-Coulomb}\nS_e({\\bf q}^2)=1-\\frac{k_e^2}{{\\bf q}^2+k_e^2+k_i^2}\\,.\n\\end{equation}\nThe screening scales are\n\\begin{equation}\\label{eq:screening-scales}\nk_e^2=R_\\eta\\,\\frac{4\\pi\\alpha n_e}{T}\n\\qquad\\hbox{and}\\qquad k_i^2=\\frac{4\\pi\\alpha}{T}\\sum_Z Z^2 n_Z\\,,\n\\end{equation}\nwhere $n_Z$ is the number density of ions with charge $Z e$. For\nelectrons, we have included the correction factor $R_\\eta$ for partial\ndegeneracy.\\footnote{Our $R_\\eta$ defined in\n  equation~(\\ref{eq:Reta-definition}) is the same as $R_\\alpha$ defined\n  in reference~\\cite{Boercker:1987jn} by a ratio of Fermi integrals. The\n  overall structure factor was written in the form\n  $1+h_x(\\kappa)+h_r(\\kappa)$ with $h_r(\\kappa)=-R_\\eta\n  k_e^2\/(\\kappa^2+k_e^2+k_i^2)$. However, for the exchange\n  correlations, the $\\kappa\\to0$ limit is justified and\n  $R_\\eta=1+h_x(0)$ so that $1+h_x+h_r=R_\\eta-R_\\eta\n  k_e^2\/(\\kappa^2+k_e^2+k_i^2)=R_\\eta S_e(\\kappa)$. In other words,\n  the exchange correlations indeed amount to a global factor $R_\\eta$ for\n  final-state Pauli blocking besides a reduction of the electron\n  screening scale $k_e^2$.}\nFor conditions of the central Sun we have an electron density of about\n$n_e=6.3\\times10^{25}~{\\rm cm}^{-3}$ and a temperature $T=1.3~{\\rm keV}$,\nproviding $\\eta=-1.425$, leading to $R_\\eta=0.927$. The Debye-H\\\"uckel\nscales are $k_e=5.4~{\\rm keV}$ and $k_i=7.0~{\\rm keV}$.\nAs these scales are comparable to a typical momentum\ntransfer there is no simple limit for Coulomb corrections.\nIn our numerical estimate of the solar emission we use a simple prescription\nto account for this effect: the largest possible momentum transfer for\nan initial photon of energy $\\omega$ is $|{\\bf q}_{\\rm max}|=2\\omega$. Using\n$S_e(4\\omega^2)$ to multiply  equation~(\\ref{comptonrate})\nprovides an upper limit to the suppression caused by\nCoulomb correlations, i.e., the neutrino flux will be slightly overestimated.\n\nCoulomb correlations apply to processes where the electron density\nis the crucial quantity, i.e., to the vector-current\npart proportional to $C_{\\rm V}^2$. The axial-current contributions, proportional to\n$C_{\\rm A}^2$, depend on the electron spins which are not correlated by\nCoulomb interactions. If an electron at a given location\nhas a certain spin, the chance of finding one with the same or opposite spin\nat some distance is the same, i.e., the spins are not correlated. Therefore,\nthe interference of spin-dependent scattering amplitudes from different\nelectrons average to zero and we do not need any Coulomb correlation correction.\nOnly the emission of $\\nu_e\\bar\\nu_e$ has any V contribution and\nin all cases, the A term strongly dominates. Therefore, overall\nCoulomb corrections are small for photo production.\n\nTreating exchange corrections as an average final-state Pauli blocking\nfactor reveals that it applies for both V and A processes. We can also see\nthis point in terms of initial-state correlations. Electrons of opposite spin\nare not correlated because they can occupy the same location, whereas those\nwith equal spin ``repel'' each other. Therefore, the interference\neffects between initial electrons of equal and opposite spins do not\naverage to zero.\n\nIn summary, the photo production rate is reduced by the overall Pauli\nblocking factor $R_\\eta$ given in equation~(\\ref{eq:Reta-definition})\nwhich in the Sun is a few percent.\nThe terms propoportional to $C_{\\rm V}^2$, on the other hand, require the\nadditional Coulomb structure factor given in\nequation~(\\ref{eq:S-Coulomb}) which can be a 30\\% correction.\nThe V channel essentially applies only to $\\nu_e\\bar\\nu_e$ emission,\nwhere Coulomb correlations provide an overall reduction of perhaps\n10\\%.\n\n\n\\subsection{Solar neutrino flux}\n\nWe now integrate the source reaction rate over our solar model and show the\nneutrino flux in figure~\\ref{fig:comptonflux1} on a linear scale. The blue\ncurve derives from the V channel and includes Coulomb correlations, whereas\nthe orange curve applies to the A channel. To obtain the proper flux\nthe curves need to be multiplied with $C_{\\rm V}^2$ and $5C_{\\rm A}^2$, respectively.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[height=5.8cm]{pics\/fig09.pdf}\n\\caption{Neutrino flux from Compton production for the vector (blue) and\naxial-vector (orange) interaction. For the proper flux, the V curve is to be\nmultiplied with $C_{\\rm V}^2$, the A curve with $5C_{\\rm A}^2$. The difference between\nthe blue and orange curves derives from Coulomb correlations which apply\nonly to the V channel. The Coulomb correlations were treated in an approximate\nway as described in the text and the suppression could be slightly larger.\n}\n\\label{fig:comptonflux1}\n\\end{figure}\n\n\\begin{figure}[b]\n\\centering\n\\hbox to\\textwidth{\\includegraphics[height=5.8cm]{pics\/fig10a.pdf}\\hfil\n\\includegraphics[height=5.8cm]{pics\/fig10b.pdf}}\n\\caption{Solar neutrino flux at Earth from the Compton process\n(axial-vector channel and only one flavor) compared with the pp flux\n(only $\\nu_e$) and transverse plasmon decay ($\\nu_e$ and equal flux $\\bar\\nu_e$).\nFlavor oscillations are not considered here.}\n\\label{fig:comptonflux2}\n\\end{figure}\n\nIn figure~\\ref{fig:comptonflux2} we compare the axial-vector Compton flux for\na single flavor with the fluxes from T-plasmon decay and with pp neutrinos. While\nwe have not included the plasma frequency in the squared matrix element for the\nCompton process, we do include it in the phase-space integration. In this way,\nthe lowest-energy Compton flux is suppressed and explains the kink in the\nlow-energy flux. As a consequence, the lowest-energy neutrino flux is dominated\nby plasmon decay. Notice that T-plasmon decay produces almost\nexclusively $\\nu_e\\bar\\nu_e$ pairs, whereas the axial-vector Compton process produces\nequal fluxes of all flavors. For $\\nu_e\\bar\\nu_e$, there is an additional\ncontribution from the V channel. Apart from Coulomb-correlation corrections\nand overall coefficients, the spectrum is the same as shown in\nfigure~\\ref{fig:comptonflux1}. The flavor dependence of fluxes at Earth will\nbe studied later.\n\n\n\\section{Bremsstrahlung}\n\\label{sec:bremsstrahlung}\n\n\\subsection{Matrix element}\n\nNext we consider bremsstrahlung production of neutrino pairs\n(figure~\\ref{fig:brems-graph}), also known as the free-free process,\nwhere we consider nuclei or ions with charge $Ze$ to provide a Coulomb\npotential without recoil. This process was not included in a previous\nstudy of low-energy solar neutrino emission \\cite{Haxton:2000xb}.  In\ngeneral, it is the dominant energy loss mechanism in stars with low\ntemperature and high electron density \\cite{Cazzola:1971ru,\n  Dicus:1976rj}. The first differential flux evaluation was carried\nout for our nonrelativistic and nondegenerate conditions a long time\nago in reference~\\cite{Gandelman:1960ex}, which has some flaws as described below,\nand recently also for general conditions~\\cite{Guo:2016vls}.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=6cm]{pics\/fig11.pdf}\n\\caption{Bremsstrahlung production of neutrino pairs. The Coulomb potential\nis provided by a heavy nucleus or ion with charge $Ze$ taking up the momentum transfer\n$q=(0,{\\bf q})$. The outgoing neutrino radiation carries the four-momentum\n$k=k_1+k_2=(\\omega,{\\bf k})$. There is a second diagram with the vertices exchanged.}\n\\label{fig:brems-graph}\n\\end{figure}\n\nThe scattering targets are taken to be very heavy (no recoil) with charge $Ze$ and\nnumber density $n_Z$ and the electrons are taken be nonrelativistic. The emission rate\nof neutrino pairs per unit volume and unit time is then\n\\begin{equation}\\label{eq:brems-emission-1}\n\\dot n_{\\nu}=n_Z \\int\\frac{d^3{\\bf p}_1}{(2\\pi)^3}\\frac{d^3{\\bf p}_2}{(2\\pi)^3}\n\\frac{d^3{\\bf k}_1}{(2\\pi)^3}\\frac{d^3{\\bf k}_2}{(2\\pi)^3}\\,f_1(1-f_2)\\,\n\\frac{\\sum_{s_1,s_2}|\\mathcal{M}|^2}{(2m_e)^2 2\\omega_1 2\\omega_2}\\,\n2\\pi\\delta(E_1-E_2-\\omega)\\,,\n\\end{equation}\nwhere the sum is taken over the electron spins and $f_1$ and $f_2$ are the initial and\nfinal-state electron occupation numbers. The final-state neutrino radiation is described by\n$k=(\\omega,{\\bf k})=k_1+k_2=(\\omega_1+\\omega_2,{\\bf k}_1+{\\bf k}_2)$.\nFor the squared matrix element we find\n\\begin{equation}\n\\sum_{s_1,s_2}|\\mathcal{M}|^2\n=\\frac{8 Z^2 e^4 }{|{\\bf q}|^4\\,\\omega^2}\\, \\(\\frac{G_{\\rm F}^2}{2}\\)\n\\Bigl(C_{\\rm V}^2 M_{\\rm V}^{\\mu\\nu}+C_{\\rm A}^2 M_{\\rm A}^{\\mu\\nu}\\Bigr)N_{\\mu\\nu}\\,,\n\\end{equation}\nwhere the neutrino tensor was given in equation~(\\ref{eq:neutrino-tensor})\nand ${\\bf q}$ is the momentum transfer to the nucleus.\n\nFor nonrelativistic electrons, as usual we can ignore the\ntransfer of three-momentum to the external radiation so that\n${\\bf q}={\\bf p}_1-{\\bf p}_2$. In this approximation we find\n\\begin{equation}\nM_{\\rm V}^{\\mu\\nu}=\n\\begin{pmatrix}\n\\(\\frac{{\\bf q}\\cdot{\\bf k}}{\\omega}\\)^2&&\\frac{{\\bf q}\\cdot{\\bf k}}{\\omega}\\,{\\bf q}\\\\[3ex]\n\\frac{{\\bf q}\\cdot{\\bf k}}{\\omega}\\,{\\bf q}&&{\\bf q}^i{\\bf q}^j\n\\end{pmatrix}\n\\qquad\\hbox{and}\\qquad\nM_{\\rm A}^{\\mu\\nu}=\n\\begin{pmatrix}\n{\\bf q}^2&~~&\\frac{{\\bf q}\\cdot{\\bf k}}{\\omega}\\,{\\bf q}\\\\[2ex]\n\\frac{{\\bf q}\\cdot{\\bf k}}{\\omega}\\,{\\bf q}&~~&\\(\\frac{{\\bf q}\\cdot{\\bf k}}{\\omega}\\)^2\\delta^{ij}\n\\end{pmatrix}\\,.\n\\end{equation}\nWith $C_{\\rm V}=C_{\\rm A}=1$ one should find the bremsstrahlung rates in the old literature before the discovery\nof neutral currents. However, the terms proportional to ${\\bf q}{\\cdot}{\\bf k}\/\\omega$\nare missing (see the steps from equation 5 to~6 in reference \\cite{Gandelman:1960ex}).\nFor the V-case,\nbremsstrahlung arises from the electron velocity abruptly changing in a collision.\nIn the nonrelativistic limit, the 0-component of the electron current remains unchanged.\nHowever, the squared matrix element is quadratic in the velocity change.\nTherefore,\na consistent nonrelativistic expansion requires to go to second order in the small\nvelocity everywhere. If one expands the electron current only to linear order\nbefore taking the matrix element one misses some of the terms.\nA similar issue explains the factor $2\/3$ difference in the axion\nbremsstrahlung calculation between reference~\\cite{Raffelt:1985nk} (equations 38 to 42)\nand \\cite{Krauss:1984gm} (equations 1 to 4).\n\nThe electron gas is assumed to be isotropic, so in equation~(\\ref{eq:brems-emission-1})\nwe may first perform an angle average over the electron direction,\nkeeping their relative angle fixed. So we average over the\nrelative angle between ${\\bf q}$ and ${\\bf k}$, leading to\n${\\bf q}^i{\\bf q}^j\\to\\frac{1}{3}{\\bf q}^2\\delta^{ij}$, $({\\bf q}{\\cdot}{\\bf k}){\\bf q}\\to\\frac{1}{3}\\,{\\bf q}^2\\,{\\bf k}$,\nand $({\\bf q}\\cdot{\\bf k})^2\\to\\frac{1}{3}\\,{\\bf q}^2{\\bf k}^2$.\nTherefore, in an isotropic medium we may write\n\\begin{equation}\n\\Bigl\\langle\\sum_{s_1,s_2}|\\mathcal{M}|^2\\Bigr\\rangle_{\\cos ({\\bf q},{\\bf k})}\n=\\frac{8 Z^2 e^4 }{3 {\\bf q}^2}\\,\\(\\frac{G_{\\rm F}^2}{2}\\)\\,\n\\frac{\\(C_{\\rm V}^2 \\bar M_{\\rm V}^{\\mu\\nu}+C_{\\rm A}^2 \\bar M_{\\rm A}^{\\mu\\nu}\\)N_{\\mu\\nu}}{\\omega^4}\\,,\n\\end{equation}\nwhere\n\\begin{equation}\n\\bar M_{\\rm V}^{\\mu\\nu}=\n\\begin{pmatrix}\n{\\bf k}^2&~&\\omega {\\bf k}\\\\[2ex]\n\\omega {\\bf k}&~&\\omega^2\\delta^{ij}\n\\end{pmatrix}\n\\qquad\\hbox{and}\\qquad\n\\bar M_{\\rm A}^{\\mu\\nu}=\n\\begin{pmatrix}\n3\\omega^2&~&\\omega {\\bf k}\\\\[2ex]\n\\omega {\\bf k}&~&{\\bf k}^2\\delta^{ij}\n\\end{pmatrix}\\,.\n\\end{equation}\nNotice that lowering the indices in this matrix changes the sign of\nthe time-space part (the 0j and j0 components), i.e., lowering the indices\namounts to $\\omega{\\bf k}\\to-\\omega{\\bf k}$. For the contractions we find\nexplicitly\n\\begin{subequations}\n  \\begin{eqnarray}\n   \\bar M_{\\rm V}^{\\mu\\nu}N_{\\mu\\nu}&=&\n   16\\[\\omega_1\\omega_2(\\omega_1^2+\\omega_2^2+\\omega_1\\omega_2)\n   +({\\bf k}_1{\\cdot}{\\bf k}_2)^2-\\omega^2\\,{\\bf k}_1{\\cdot}{\\bf k}_2\\]\n   \\nonumber\\\\[1ex]\n   &\\to&\n   \\frac{16}{3}\\,\\omega_1\\omega_2\\(3\\omega_1^2+3\\omega_2^2+4\\omega_1\\omega_2\\),\n   \\\\[2ex]\n      \\bar M_{\\rm A}^{\\mu\\nu}N_{\\mu\\nu}&=&\n   16\\[\\omega_1\\omega_2(2\\omega_1^2+2\\omega_2^2+\\omega_1\\omega_2)\n   -({\\bf k}_1{\\cdot}{\\bf k}_2)^2+4\\omega_1\\omega_2\\,{\\bf k}_1{\\cdot}{\\bf k}_2\\]\n   \\nonumber\\\\[1ex]\n   &\\to&\n    \\frac{32}{3}\\,\\omega_1\\omega_2\\(3\\omega_1^2+3\\omega_2^2+\\omega_1\\omega_2\\).\n  \\end{eqnarray}\n\\end{subequations}\nThe second expressions apply after an angle average over the\nrelative directions of ${\\bf k}_1$ and ${\\bf k}_2$ where\n${\\bf k}_1{\\cdot}{\\bf k}_2\\to 0$ and\n$({\\bf k}_1{\\cdot}{\\bf k}_2)^2\\to \\frac{1}{3}\\,\\omega_1^2\\omega_2^2$.\n\n\\subsection{Emission rate}\n\nWe may write the neutrino pair emission rate of equation~(\\ref{eq:brems-emission-1})\nin a way that separates the properties of the\nemitted radiation (the neutrino pairs) from the properties of the medium (thermal electrons\ninteracting with nuclei) and find\n\\begin{equation}\\label{eq:brems-emission-2}\n\\dot n_{\\nu}=n_Z n_e \\frac{8\\,Z^2 \\alpha^2}{3}\n\\int\\frac{d^3{\\bf k}_1}{2\\omega_1(2\\pi)^3}\\frac{d^3{\\bf k}_2}{2\\omega_2(2\\pi)^3}\n\\(\\frac{G_{\\rm F}}{\\sqrt{2}}\\)^2\n\\frac{\\(C_{\\rm V}^2 \\bar M_{\\rm V}^{\\mu\\nu}+C_{\\rm A}^2 \\bar M_{\\rm A}^{\\mu\\nu}\\)N_{\\mu\\nu}}{\\omega^4}\n\\,{\\cal S}(\\omega)\\,,\n\\end{equation}\nwhere $\\omega=\\omega_1+\\omega_2$ is the energy carried away by a neutrino pair. Collecting\ncoefficients in a slightly arbitrary way, the relevant\nresponse function of the medium is\n\\begin{equation}\\label{eq:brems-response-1}\n{\\cal S}(\\omega)=\\frac{(4\\pi)^2}{(2m_e)^2}\\frac{1}{n_e}\n\\int\\frac{d^3{\\bf p}_1}{(2\\pi)^3}\\frac{d^3{\\bf p}_2}{(2\\pi)^3}\\,f_1(1-f_2)\\,\\frac{1}{{\\bf q}^2}\\,\n2\\pi\\delta(E_1-E_2-\\omega)\\,,\n\\end{equation}\nwhere ${\\bf q}={\\bf p}_1-{\\bf p}_2$ is the momentum transfer in the electron-nucleus collision\nmediated by the Coulomb field. We see that for both vector and axial vector emission\nit is the same property of the medium causing the emission. We will\nreturn to this point later for the free-bound and bound-bound emission\nprocesses because we can relate both vector and axial-vector\nprocesses to the monochromatic photon opacities, the latter providing us\nessentially with ${\\cal S}(\\omega)$.\n\nWe now integrate over neutrino emission angles and find the\nneutrino emission spectrum by using $\\omega_1=\\omega_\\nu$ and\n$\\omega_2=\\omega-\\omega_\\nu$. Integrating over the anti-neutrino\nenergy we find\n\\begin{eqnarray}\\label{eq:brems-emission-3}\n  \\frac{d\\dot n_{\\nu}}{d\\omega_\\nu}&=&n_Z n_e\\,\\frac{8\\,Z^2 \\alpha^2}{3}\\,\n  \\(\\frac{G_{\\rm F}}{\\sqrt{2}}\\)^2\\frac{1}{3\\pi^4}\n  \\int_{\\omega_\\nu}^\\infty d\\omega\\,{\\cal S}(\\omega)\\,\n  \\frac{\\omega_\\nu^2(\\omega-\\omega_\\nu)^2}{\\omega^4}\\nonumber\\\\[2ex]\n&&\\kern5em{}\\times\n  \\Bigl[C_{\\rm V}^2\\(3\\omega^2-2\\omega\\omega_\\nu+2\\omega_\\nu^2\\)\n  +2C_{\\rm A}^2\\(3\\omega^2-5\\omega\\omega_\\nu+5\\omega_\\nu^2\\)\\Bigr]\\,,\n\\end{eqnarray}\nwhich is the rate of neutrino emission per unit volume, unit time,\nand unit energy interval. The same spectrum applies to antineutrinos\nbecause all expressions were symmetric under the exchange $1\\leftrightarrow2$.\n\n\n\\subsection{Photon and axion emission}\n\\label{sec:axion-emission}\n\n\nIt is useful to compare the bremsstrahlung\nemission rate of neutrino pairs with that of photons and axions to connect to the\nprevious literature and, more importantly, to relate the bremsstrahlung absorption\nrate of photons to the neutrino pair emission rate.\nFor photon emission, the neutral-current interaction of equation~(\\ref{eq:NC-interaction})\ngets replaced by $i e\\bar\\psi_e\\gamma^\\mu\\bar\\psi_e A_\\mu$. On the level of the squared\nmatrix element, or rather, on the level of the emission rate, this substitution\ntranslates to\n\\begin{equation}\\label{eq:brems-photon-emission-1}\n\\dot n_{\\gamma}=n_Z n_e\\,\\frac{8\\,Z^2 \\alpha^2}{3}\n\\int\\frac{d^3{\\bf k}}{2\\omega(2\\pi)^3}\\,\ne^2\\,\\frac{\\bar M_{\\rm V}^{\\mu\\nu}\\epsilon_\\mu\\epsilon_\\nu}{\\omega^4}\\,{\\cal S}(\\omega)\\,.\n\\end{equation}\nFor the contraction we find $\\bar M_{\\rm V}^{\\mu\\nu}\\epsilon_\\mu\\epsilon_\\nu=\\omega^2$.\nWe have used the polarization vector for a transverse photon (not a\nlongitudinal plasmon) so that ${\\bf k}{\\cdot}{\\bm \\epsilon}=0$,\n$\\epsilon^0\\epsilon^0=0$,\nand ${\\bm\\epsilon}{\\cdot}{\\bm\\epsilon}=1$. So the spectral photon\nproduction rate per transverse polarization degree of freedom is\n\\begin{equation}\\label{eq:brems-photon-emission-2}\n  \\frac{d\\dot n_{\\gamma}}{d\\omega}=n_Z n_e\\, \\frac{8\\,Z^2 \\alpha^2}{3}\\,\n  \\frac{\\alpha}{\\pi}\\,\\frac{{\\cal S}(\\omega)}{\\omega}\\,.\n\\end{equation}\nThis quantity is directly related to the medium response function\n${\\cal S}(\\omega)$. Therefore, we can express the neutrino\nemissivity of equation~(\\ref{eq:brems-emission-3})\nin terms of the photon emissivity as\n\\begin{eqnarray}\\label{eq:brems-emission-4}\n  \\frac{d\\dot n_{\\nu}}{d\\omega_\\nu}&=&\n  \\frac{G_{\\rm F}^2}{6\\pi^3\\alpha}\n  \\int_{\\omega_\\nu}^\\infty d\\omega\\,\\(\\frac{d\\dot n_{\\gamma}}{d\\omega}\\)\\,\n  \\frac{\\omega_\\nu^2(\\omega-\\omega_\\nu)^2}{\\omega^3}\\nonumber\\\\[2ex]\n&&\\kern4em{}\\times\n  \\Bigl[C_{\\rm V}^2\\(3\\omega^2-2\\omega\\omega_\\nu+2\\omega_\\nu^2\\)\n  +2C_{\\rm A}^2\\(3\\omega^2-5\\omega\\omega_\\nu+5\\omega_\\nu^2\\)\\Bigr]\\,.\n\\end{eqnarray}\nTherefore, given the spectral photon emissivity, e.g.\\ taken from the\nphoton opacity calculation, we can directly extract the neutrino\nemission spectrum. We will return to this point in section~\\ref{sec:freebound}.\n\n\nAxions couple to the electron axial current with an interaction of the\nderivative form\n$(C_e\/2f_a)\\,\\bar\\psi_e\\gamma^\\mu\\gamma_5\\psi_e\\,\\partial_\\mu a$,\nwhere $a$ is the axion field, $f_a$ the axion decay constant,\nand $C_e$ a model-dependent numerical coefficient. One often\nuses a dimensionless axion-electron Yukawa coupling\n$g_{ae}=C_e m_e\/f_a$ so that the interaction is\n$(g_{ae}\/2m_e)\\,\\bar\\psi_e\\gamma^\\mu\\gamma_5\\psi_e\\,\\partial_\\mu a$.\nThe bremsstrahlung emission rate is found to be\n\\begin{equation}\\label{eq:brems-axion-emission-1}\n\\dot n_{a}=n_Z n_e\\, \\frac{8\\,Z^2 \\alpha^2}{3}\n\\int\\frac{d^3{\\bf k}}{2\\omega(2\\pi)^3}\\,\n\\(\\frac{g_{ae}}{2m_e}\\)^2\n\\frac{\\bar M_{\\rm A}^{\\mu\\nu}k_\\mu k_\\nu}{\\omega^4}\\,{\\cal S}(\\omega)\\,.\n\\end{equation}\nFor massless axions with $\\omega=|{\\bf k}|$ we find for the contraction\n$\\bar M_{\\rm A}^{\\mu\\nu}k_\\mu k_\\nu=2\\omega^4$. Therefore, the\nspectral emissivity is\n\\begin{equation}\\label{eq:brems-axion-emission-2}\n  \\frac{d\\dot n_{a}}{d\\omega}=n_Z n_e\\,\\frac{8\\,Z^2 \\alpha^2}{3}\\,\n  \\(\\frac{g_{ae}}{2m_e}\\)^2\n    \\frac{\\omega}{2\\pi^2}\\,{\\cal S}(\\omega)\\,.\n\\end{equation}\nNotice that this spectrum is harder than the photon spectrum by a\nfactor $\\omega^2$ caused by the derivative structure of the axion\ninteraction. Still, fundamentally it depends on the same\nmedium response function ${\\cal S}(\\omega)$.\nWe have checked that in the nondegenerate limit this axion\nemission rate agrees with reference~\\cite{Raffelt:1985nk}.\n\n\\subsection{Medium response function and screening effects}\n\nThe medium response function defined in\nequation~(\\ref{eq:brems-response-1}) could be easily evaluated if the\nnuclei used as scattering centers were uncorrelated. However, their\nCoulomb interaction leads to anticorrelations encoded in an ion\nstructure factor $S_i({\\bf q}^2)$ similar to the case of electron-electron\ncorrelations discussed in section~\\ref{sec:correlatoins}.  Therefore,\nunder the integral in equation~(\\ref{eq:brems-response-1}) we need to\ninclude $S_i({\\bf q}^2)$, which we will discuss later. We mention in\npassing that ${\\cal S(\\omega)}$ given in\nequation~(\\ref{eq:brems-response-1}), with or without including\n$S_i({\\bf q}^2)$, fulfills the detailed-balancing condition\n${\\cal S}(-\\omega)={\\cal S}(\\omega)\\,e^{\\omega\/T}$. Here a negative $\\omega$ means\nenergy absorbed by the medium, whereas a positive $\\omega$ for us\nalways means energy emitted, although in the literature one usually uses\nthe opposite convention.\n\nTo write ${\\cal S}(\\omega)$ in a more compact form we first note that the\nelectron number density is given by\nequation~(\\ref{eq:electron-density}) in terms of the nonrelativistic\ndegeneracy parameter $\\eta$. We further write the kinetic electron\nenergy in dimensionless form as $u={\\bf p}^2\/(2m_e T)$ so that the\noccupation number is $f_u=1\/(e^{u-\\eta}+1)$. Then the structure\nfunction is\n\\begin{equation}\n  {\\cal S}(\\omega)=\\frac{\\pi}{m_e\\sqrt{2m_eT}}\\,s(\\omega\/T)\\,\n\\end{equation}\nwhere\n\\begin{equation}\n  s(w)=\n  \\int_0^\\infty\\!du_2\\,\\frac{\\sqrt{u_1 u_2}}{e^{u_1-\\eta}+1}\\,\n  \\frac{1}{e^{-(u_2-\\eta)}+1}\n\\int_{-1}^{+1}\\!d{\\rm c}_\\theta\\,\\frac{2m_eT}{{\\bf q}^2}\\,S_i({\\bf q}^2)\n\\bigg\/\\int_0^\\infty du\\,\\frac{\\sqrt{u}}{e^{u-\\eta}+1}\\,.\n\\end{equation}\nHere $u_1=u_2+w$ and $w=\\omega\/T$. Moreover,\n${\\bf q}^2=|{\\bf p}_1-{\\bf p}_2|^2={\\bf p}_1^2+{\\bf p}_2^2-2|{\\bf p}_1||{\\bf p}_2|{\\rm c}_\\theta$\nwith ${\\rm c}_\\theta=\\cos\\theta$. Therefore,\n${\\bf q}^2\/(2m_eT)=u_1+u_2-2\\sqrt{u_1u_2}\\,{\\rm c}_\\theta$.\n\nBesides the original squared matrix element, this function depends on\nelectron degeneracy effects and\nCoulomb correlation effects encoded in $S_i({\\bf q}^2)$. Anticipating that electron degeneracy is\nnot a large correction we first reduce this expression to Maxwell-Boltzmann rather than\nFermi-Dirac statistics. Formally this is the $\\eta\\to-\\infty$ limit, leading to the\nnon-degenerate (ND) structure function\n\\begin{equation}\n  s_{\\rm ND}(w)=\n  \\int_0^\\infty\\!du\\,e^{-u-w}\\sqrt{(u+w)\\,u}\\,\n  F_i(u,w)\n\\bigg\/\\int_0^\\infty du\\,e^{-u}\\sqrt{u}\\,,\n\\end{equation}\nwhere the integral in the denominator is simply $\\sqrt{\\pi}\/2$.\nThe integral kernel is\n\\begin{equation}\nF_i(u,w)=\\int_{-1}^{+1}\\!d{\\rm c}_\\theta\\,\\frac{2m_eT\\,S_i({\\bf q}^2)}{{\\bf q}^2}\\,.\n\\end{equation}\nThe ion structure function from Coulomb correlation effects will be an expression of the type given in equation~(\\ref{eq:S-Coulomb}), but with the role of electrons and ions interchanged. However, in a multi-component plasma, an exact treatment is difficult; because screening will be a relatively small correction, we use\n\\begin{equation}\\label{eq:screeningbrem}\nS_i({\\bf q}^2)=\\frac{{\\bf q}^2}{{\\bf q}^2+k_{\\rm s}^2}\\,,\n\\end{equation}\nwhere $k_{\\rm s}$ is a phenomenological screening scale. We use $k_i$ given in equation~(\\ref{eq:screening-scales}) for the ion correlations.\nWith $\\mu=k_{\\rm s}^2\/(2m_eT)$ we therefore use\n\\begin{eqnarray}\nF_i(u,w)&=&\\int_{-1}^{+1}\\!d{\\rm c}_\\theta\\,\\frac{1}{\\mu+2u+w-2\\sqrt{(u+w)\\,u}\\,{\\rm c}_\\theta}\n\\nonumber\\\\[2ex]\n&=&\\frac{1}{2\\sqrt{(u+w)\\,u}}\\log\\(\\frac{\\mu+2u+w+2\\sqrt{(u+w)\\,u}}{\\mu+2u+w-2\\sqrt{(u+w)\\,u}}\\)\\,.\n\\end{eqnarray}\nWithout screening ($\\mu=0$) this expression diverges logarithmically for small $w$. However, for axion emission and neutrino pair emission, this divergence is moderated by at least one power of $\\omega$, so even without correlation effects, the emission of soft radiation does not diverge. Near the solar center one finds $k_i=7~{\\rm keV}$ and with $T=1.3~{\\rm keV}$ one finds $\\mu=0.037\\ll 1$, so screening is not a strong effect. Overall then the ND structure function is\n\\begin{equation}\\label{eq:sND}\n  s_{\\rm ND}(w)=\\frac{e^{-w}}{\\sqrt{\\pi}}\n  \\int_0^\\infty\\!du\\,e^{-u}\\,\\log\\(\\frac{\\mu+2u+w+2\\sqrt{(u+w)\\,u}}{\\mu+2u+w-2\\sqrt{(u+w)\\,u}}\\)\n  \\to\\frac{2e^{-w}}{\\sqrt{w}}\\quad(\\hbox{large $w$})\n\\end{equation}\nIn figure~\\ref{fig:sND} we show $e^w s_{\\rm ND}(w)$ with and without Coulomb correlation effects\nand the asymptotic form for large $w$.\nWe also show as a red line $e^w s(w)$ including Fermi-Dirac statistics for the electrons with\n$\\eta=-1.4$, appropriate for the solar center.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=7cm]{pics\/fig12.pdf}\n\\caption{Structure function $e^w s(w)$ for bremsstrahlung.\n{\\em Blue line:} Non-degenerate electrons\ngiven in equation~(\\ref{eq:sND}) without Coulomb correlations ($\\mu=0$).\n{\\em Orange line:} Coulomb correlations included with $\\mu=0.04$\nappropriate for the solar center.\n{\\em Red line:} Electron degeneracy included with $\\eta=-1.4$ appropriate\nfor the solar center.\n{\\em Green line:} Asymptotic form for large $w=\\omega\/T$.}\n\\label{fig:sND}\n\\end{figure}\n\n\nWhile correlation effects strongly modify the structure function at low energy transfer, this effect is much smaller after folding with the neutrino phase space according to equation~\\eqref{eq:brems-emission-3}. Even without correlations, the neutrino spectrum does not diverge at low energies. Including Coulomb correlations reduces it at low energies by some 20\\% and only by very little in the main keV-range of the spectrum. Pauli blocking of final states provides a further 5\\% suppression effect at very low energies, so overall these are fairly minor effects.\n\n\\subsection{Electron-electron bremsstrahlung}\n\\label{sec:ebremsstrahlung}\n\nThe electron-electron bremsstrahlung process is similar to\nelectron-proton bremsstrahlung with a number of important\nmodifications \\cite{Raffelt:1985nk}. The vector-current emission rate\nis of higher order in velocity---the simple dipole term vanishes in\nthe scattering of equal-mass particles. In the non-degenerate and\nnon-relativistic limit and ignoring Coulomb correlations, the\naxial-vector rate is $1\/\\sqrt2$ that of the electron-proton rate.  In\nother words, we obtain the $ee$ bremsstrahlung rate from the\naxial-current part of equation~\\eqref{eq:brems-emission-3} with the\nsubstitution $Z^2 n_Z n_e\\to n_e^2\/\\sqrt{2}$.\n\nTaking degeneracy effects and Coulomb correlations exactly into\naccount would be very hard.  Instead we simply add the\nelectron-electron term to the electron-ion one and therefore use the\nsame treatment as for the latter. These corrections are rather small\nand $ee$ bremsstrahlung is subdominant, so the overall error\nintroduced by this approach is on the level of a few percent.\n\n\n\n\\subsection{Solar neutrino flux}\n\n\\begin{figure}[b!]\n\\centering\n\\includegraphics[width=7.2cm]{pics\/fig13.pdf}\n\\caption{Neutrino flux from bremsstrahlung production for the vector\n  (blue) and axial-vector (orange) electron-ion interaction, and from the $ee$ interaction (green) which contributes only in the\n  axial-vector channel. For the proper flux, the blue\n  curve is to be multiplied with $C_{\\rm V}^2$, the orange and green\n  curves with $2C_{\\rm A}^2$.}\n\\label{fig:brems1}\n\\end{figure}\n\n\n\nAs a final step we integrate the bremsstrahlung emission rate over our\nstandard solar model to obtain the neutrino flux at Earth. In\nfigure~\\ref{fig:brems1} we show separately the vector and axial-vector\ncontributions from electron-ion scattering as well as the one from\nelectron-electron scattering which only contributes in the axial\nchannel. These curves need to be multiplied with the flavor-dependent\nvalues of $C_{\\rm V}^2$ and $2C_{\\rm A}^2$ to obtain the proper fluxes. For the\nelectron-ion contributions, we include only hydrogen and helium as\ntargets. The contribution from metals is only a few percent and\nwill be included in the opacity-derived flux in\nsection~\\ref{sec:freebound}.\n\nFinally we show in figure~\\ref{fig:brems2} the axial-channel\nbremsstrahlung flux for one flavor in comparison with the $\\nu_e$ flux\nfrom the nuclear pp reaction and from T plasmon decay. Similar to the\nCompton process, the bremsstrahlung flux becomes important in the\ncross-over region between the T-plasmon and pp fluxes.\n\n\\begin{figure}[htbp]\n\\centering\n\\hbox to\\textwidth{\\includegraphics[height=5.8cm]{pics\/fig14a.pdf}\\hfil\n\\includegraphics[height=5.8cm]{pics\/fig14a.pdf}}\n\\caption{Solar neutrino flux at Earth from the axial-channel\n  bremsstrahlung process ($eI$ and $ee$ contributions) for one\n  flavor, compared with the pp flux (only $\\nu_e$) and transverse\n  plasmon decay ($\\nu_e$ and equal flux $\\bar\\nu_e$).  Flavor\n  oscillations are not considered here.}\n\\label{fig:brems2}\n\\end{figure}\n\nBremsstrahlung is the dominant contribution at very low energies.\nFrom equation~\\eqref{eq:brems-emission-3} we see that for very low neutrino energies, the spectrum varies as $\\omega_\\nu^2$, independently of details of the structure function ${\\cal S}(\\omega)$. This scaling remains true after integrating over the Sun, so the very-low energy thermal emission spectrum scales as $\\omega_\\nu^2$ and thus in the same way as the pp flux given in equation~\\eqref{eq:ppflux}.\n\n\n\\subsection{Beyond the Born approximation}\n\nTraditionally the bremsstrahlung emission rate of neutrinos and other particles is\ncalculated in Born approximation starting with the usual Feynman rules. However, the\nbremsstrahlung emission by a non-relativistic electron scattering on an ion receives a significant modification if one uses appropriately modified electron wave functions instead of plane waves,\na point first discussed in the context of x-ray production in free-free transitions a long time ago by Sommerfeld \\cite{Sommerfeld:1931}. Such an enhancement in bremsstrahlung emission arises because of the long-range Coulomb potential. It is the counterpart of what is known as Fermi-Coulomb function in the context of beta decay and of the so-called Sommerfeld enhancement, to be taken into account in dark matter annihilation processes \\cite{ArkaniHamed:2008qn}. In these cases, the correction is simply given by $f(v)=|\\psi(0)|^2$, i.e., by the normalization of the outgoing (or ingoing) Coulomb distorted wave function\n\\begin{equation}\n\\sigma=\\sigma_0 f(v)=\\sigma_0 \\frac{2\\pi Z \\alpha}{v}\\frac{1}{1-e^{\\frac{2\\pi Z\\alpha}{v}}}\\,,\n\\end{equation}\nwhere $\\sigma_0$ is the cross section evaluated with plane waves and $v$ is the velocity of the outgoing (or ingoing) particle.\n\nSuch corrections have been extensively studied also for bremsstrahlung (see e.g.\\ reference~\\cite{Karzas:1961}). Elwert found that a good approximation to correct the Born scattering formula is obtained by simply multiplying equation~(\\ref{eq:brems-response-1}) by the factor \\cite{Elwert:1939}\n\\begin{equation}\nf_E=\\frac{v_{\\rm i}}{v_{\\rm f}}\\frac{1-e^{\\frac{2\\pi Z \\alpha}{v_{\\rm i}}}}{1-e^{\\frac{2\\pi Z \\alpha}{v_{\\rm f}}}} ,\n\\end{equation}\nwhich is the ratio $|\\psi_{\\rm f}(0)|^2\/|\\psi_{\\rm i}(0)|^2$ of the final and initial state electron wave functions. In figure~\\ref{fig:bremssgaunt} (green line) we show the effect on the overall solar neutrino flux of including this factor in the bremsstrahlung rate, leading to a typical 20--30\\% enhancement. We also show as an orange line the effect of including Coulomb correlations which reduce the flux typically by some 5\\%. At very small energies, the Elwert factor becomes less important and Coulomb correlations more important.\n\n\n\\begin{figure}[b!]\n\\centering\n\\hbox to\\textwidth{\\includegraphics[scale=0.72]{pics\/fig15a.pdf}\\hfill\n\\includegraphics[scale=0.72]{pics\/fig15b.pdf}}\n\\vskip12pt\n\\hbox to\\textwidth{\\includegraphics[scale=0.72]{pics\/fig15c.pdf}\\hfill\n\\includegraphics[scale=0.72]{pics\/fig15d.pdf}}\n\\caption{Solar neutrino flux from bremsstrahlung production for axial-vector electron-ion interaction, without corrections (blue), and including respectively correlations (orange) and the Elwert factor (green). For the proper flux, the\n  curves in the upper panels are to be multiplied with $2C_{\\rm A}^2$.}\n\\label{fig:bremssgaunt}\n\\end{figure}\n\n\nAs noted by one of us in the context of solar axion emission~\\cite{Redondo:2013wwa}, the Sommerfeld correction is included in the photon opacity calculation. On the other hand, using unscreened Coulomb wave functions in a stellar plasma is not fully consistent---the true correction should be considerably smaller, especially for bremsstrahlung on hydrogen and helium. Therefore, as in reference~\\cite{Redondo:2013wwa} we calculate these rates directly, not from the opacities, and leave out the Elwert factor, possibly underestimating the true flux by some 10\\%. On the other hand, we will include the Coulomb correlation factor. At keV-range energies this makes no big difference, but should be a reasonable correction in the far sub-keV range where bremsstrahlung is the dominant flux\nand the Elwert factor is small.\n\n\n\\section{Free-bound and bound-bound transitions}\n\\label{sec:freebound}\n\n\\begin{figure}[b!]\n\\centering\n\\includegraphics[height=5.8cm]{pics\/fig16.pdf}\n\\caption{Solar neutrino flux at Earth from free-free (ff), free-bound (fb) and\n  bound-bound (bb) electron-ion transitions for the vector (V) and axial-vector (A)\n  contributions. The proper fluxes are found by multiplying the curves\n  with $C_{\\rm V}^2$ and $2C_{\\rm A}^2$, respectively. The ff curves\n  include bremsstrahlung on hydrogen and helium and are the same as in\n  figure~\\ref{fig:brems1}; they exclude\n  electron-electron bremsstrahlung. The fb+bb curves include\n  bremsstrahlung on metals.}\n\\label{fig:fb-bb}\n\\end{figure}\n\n\nThe nuclei of the solar plasma are imperfectly ionized, notably the\n``metals'' (elements heavier than helium). Therefore, in addition to\nbremsstrahlung (free-free electron transitions), free-bound (fb) and\nbound-bound (bb) transitions are also important for particle\nemission. In the context of axion emission by electrons, these\nprocesses imprint a distinctive line pattern on the expected solar\naxion flux \\cite{Redondo:2013wwa}. Likewise, the photon opacities, as\ninput to solar models, depend strongly on these processes. In\nreference~\\cite{Haxton:2000xb} free-bound processes were included by\nexplicit atomic transition calculations for a number of elements.\n\nHowever, following the approach taken by one of us in an earlier\npaper for calculating the solar axion flux \\cite{Redondo:2013wwa}, the\nemission rate can be related to the photon opacity\nby equation~\\eqref{eq:brems-emission-4}, i.e., the neutrino emissivity\nis the same as the phase-space weighted photon\nemissivity. Therefore, one can use the solar photon opacity from the\nliterature to derive the neutrino emissivity. In\nsection~\\ref{sec:axion-emission} we have derived the relation between photon and neutrino emissivity explicitly for bremsstrahlung, but it\napplies in this form to all processes where a nonrelativistic\nelectron makes a transition in the potential of an external scattering center\nwhich takes up momentum. In other words, it applies in the long-wavelength\napproximation with regard to the electron.  On the other hand, this relation does not apply to electron-electron bremsstrahlung, the Compton process, or plasmon decay.\n\nWhile we could have used this relation to extract the bremsstrahlung\nemissivity from the opacities, we have preferred to treat\nbremsstrahlung on hydrogen and helium as well as electron-electron\nbremsstrahlung explicitly in the interest of completeness. For the fb and bb transitions as well as\nbremsstrahlung on metals we proceed as in\nreference~\\cite{Redondo:2013wwa} to extract the neutrino\nemissivity. In figure~\\ref{fig:fb-bb} we show the resulting neutrino\nflux spectrum at Earth in comparison with the bremsstrahlung result on\nhydrogen and helium derived earlier. We show the vector and\naxial-vector contributions, each to be multiplied with the\nflavor-dependent $C_{\\rm V}^2$ or $2C_{\\rm A}^2$ to arrive at the proper flux. While the\nfb and bb contributions are subdominant relative to\nbremsstrahlung, they dominate at higher energies. This behavior\nwas to be expected because\nthey are more relevant than ff processes in the Rosseland opacities at\nany radius (see e.g.\\ figure~15 of reference~\\cite{Krief:2016znd}).\n\n\n\\section{Solar neutrino flux at Earth}\n\\label{sec:solarflux}\n\n\\subsection{Flavor-dependent fluxes}\n\nIn order to consolidate the solar flux results from different\nthermal processes we show the spectra at Earth in\nfigure~\\ref{fig:V-A-flux}. In the left panel we show all contributions\nrelevant for the vector coupling, where the true flux is found by\nmultiplication with the flavor-dependent value of $C_{\\rm V}^2$. At low\nenergies, plasmon decay dominates, at intermediate ones\nbremsstrahlung, and at the highest energies Compton process.\nIn the right panel we show the analogous axial-vector result which does not\nhave any significant plasmon-decay contribution.\n\n\\begin{figure}[htbp]\n\\centering\n\\hbox to\\textwidth{\\includegraphics[height=6.9cm]{pics\/fig17a.pdf}\n\\hfil\\includegraphics[height=6.9cm]{pics\/fig17b.pdf}}\n\\caption{Solar neutrino flux at Earth from the indicated\n  processes, where ``brems'' includes ff, fb and bb.\n  {\\em Left panel:} Vector coupling, for proper\n  flux multiply with $C_{\\rm V}^2$. {\\em Right panel:} Axial-vector coupling, for proper\n  flux multiply with $C_{\\rm A}^2$.}\n\\label{fig:V-A-flux}\n\\end{figure}\n\nThe intrinsic uncertainties of the various emissivity calculations\nshould not be larger than a few tens of percent concerning various\nissues of in-medium effects such as correlations or Coulomb wave\nfunctions of charged particles. In addition, every process has a\ndifferent dependence on temperature, density and chemical composition\nso that different standard solar models will produce somewhat\ndifferent relative weights of the different processes. We have not\nstudied the variation of the different flux contributions depending on\ndifferent standard solar models, but the overall uncertainty again\nshould be in the general ten percent range.\n\nWe may show the same results in a somewhat different form if we observe that the\nvector-current processes produce almost exclusively $\\nu_e\\bar\\nu_e$ pairs, whereas\nthe axial-vector processes produce all flavors in equal measure. In the upper panels of\nfigure~\\ref{fig:total-flavor-flux} we show these total fluxes, where now the coupling constants $C_{\\rm V}^2=0.9263$ and $C_{\\rm A}^2=1\/4$ are included. In addition we show the $\\nu_e$ flux from the nuclear pp reaction. In the bottom panels\nwe add the different source channels for every flavor and show the keV-range fluxes for $\\nu_e$, $\\bar\\nu_e$, and each of the other species, still ignoring flavor oscillations.\n\n\\begin{figure}[htbp]\n\\centering\n\\hbox to\\textwidth{\\includegraphics[height=5.8cm]{pics\/fig18a.pdf}\n\\hfil\\includegraphics[height=5.8cm]{pics\/fig18b.pdf}}\n\\vskip12pt\n\\hbox to\\textwidth{\\includegraphics[height=5.8cm]{pics\/fig18c.pdf}\n\\hfil\\includegraphics[height=5.8cm]{pics\/fig18d.pdf}}\n\\caption{Flavor-dependent solar neutrino fluxes at Earth, ignoring flavor mixing.\n{\\em Top panels:\\\/} Thermal flux from vector-current interactions producing only\n$\\nu_e\\bar\\nu_e$ pairs, axial-vector emission, producing equal fluxes of every flavor, where\n$x$ stands for $e$, $\\mu$ or $\\tau$, and $\\nu_e$ from the nuclear pp reaction.\n{\\em Bottom panels:\\\/} $\\nu_e$, $\\bar\\nu_e$, and other single-species fluxes after adding the source\nchannels.}\n\\label{fig:total-flavor-flux}\n\\end{figure}\n\n\n\\subsection{Including flavor mixing}\n\nFlavor-eigenstate neutrinos are mixtures of three different mass eigenstates. After propagating over a long distance, these mass eigenstates effectively decohere, so the neutrino flux arriving at Earth is best described as an incoherent mixture of mass eigenstates. It depends on the nature of a possible detector if these should be re-projected on interaction eigenstates or if these fluxes can be used directly if the detection channel is flavor blind. As we do not know the nature of such a future detector, the neutrino flux in terms of mass eigenstates is the most natural form of presentation.\n\nThe axial-current production channel has amplitude $C_{\\rm A}=+1\/2$ for $\\nu_e$ and $C_{\\rm A}=-1\/2$ for the other flavors, yet on the level of the rate (proportional to $C_{\\rm A}^2$) all flavors are produced in equal measure. Therefore, the density matrix in flavor space is proportional to the $3{\\times}3$ unit matrix and thus the same in any basis. Without further ado we can think of the axial-current processes as producing mass eigenstates, so in the upper panels of figure~\\ref{fig:total-flavor-flux}, the fluxes marked $\\nu_x$ and $\\bar\\nu_x$ can be interpreted as $x$ standing for the mass indices 1, 2 and~3.\n\nThe vector-current channels, on the other hand, have the peculiar property of producing almost exclusively $e$-flavored states as discussed earlier, which is also true of the charged-current pp nuclear reaction. These states oscillate in flavor space after production. The relevant oscillation scale is $\\omega_{\\rm osc}=\\Delta m^2\/2E=3.8\\times10^{-8}~{\\rm eV}\/E_{\\rm keV}$ for the solar mass difference of $\\Delta m^2=7.5\\times10^{-5}~{\\rm eV}^2$ and using $E_{\\rm keV}=E\/{\\rm keV}$. For the atmospheric mass difference $\\Delta m^2=2.5\\times10^{-3}~{\\rm eV}^2$ the oscillation scale is\n$\\omega_{\\rm osc}=1.25\\times10^{-3}~{\\rm eV}\/E_{\\rm keV}$. These numbers should be compared with the\nmatter-induced energy splitting between $\\nu_e$ and the other flavors of\n$\\Delta V=\\sqrt{2}G_{\\rm F} n_e= 7.6\\times10^{-12}~{\\rm eV}$ for the electron density $n_e=6\\times10^{25}~{\\rm cm}^{-3}$ of the solar center. Therefore, for keV-range neutrinos, the matter effect is very small and neutrino oscillations proceed essentially as in vacuum. The source region in the Sun is much larger than the oscillation length and is far away from Earth, so flavor oscillations effectively decohere long before reaching here. Therefore, we may think of the $e$-flavored channels as producing an incoherent mixture of mass eigenstates. On the probability level, the best-fit mass components of $\\nu_e$ are~\\cite{Esteban:2016qun}\n\\begin{equation}\np_1=67.9\\%,\\qquad p_2=29.9\\%,\\qquad p_3=2.2\\%.\n\\end{equation}\nOn this level of precision, these probabilities do not depend on the mass ordering.\nThe final fluxes in terms of mass eigenstates were shown in figure~\\ref{fig:summaryflux}\nin the introduction as our main result.\n\n\n\\section{Discussion and summary}\n\\label{sec:discussion}\n\nWe have calculated the solar neutrino flux produced by various thermal processes that produce\n$\\nu\\bar\\nu$ pairs with keV-range energies. A proposed dark matter detector for keV-mass sterile neutrinos might find this flux to be a limiting background and conversely, conceivably it could measure this solar flux, although these are somewhat futuristic ideas. Whatever these experimental developments, it is well motivated to provide a benchmark calculation of the thermal solar neutrino flux.\n\nOne complication is that there is not a single dominant production channel, but all the processes shown in figure~\\ref{fig:processes} are relevant in different ranges of energy. Each of them has its own idiosyncratic issues concerning in-medium many-body effects, yet we think that our flux calculations should be correct on the general 10\\% level of precision. Similar uncertainties arise from the variation between different standard solar models.\nThe only previous study of the keV-range solar neutrino flux in reference~\\cite{Haxton:2000xb} stressed the importance of a plasmon enhancement in the photo-production channel, but we think that this result is spurious as argued in section~\\ref{sec:pole} and we think that this particular collective effect is not relevant.\n\nIt is interesting that the free-bound and bound-bound emission processes on elements heavier than hydrogen and helium produce the dominant flux in the few-keV range. We have calculated this flux taking advantage of the solar opacity calculations available in the literature. The neutrino emissivity is related by a simple phase-space integration to the monochromatic photon emissivity provided by the opacity calculations. Our solar flux based on these processes roughly agrees with that of reference~\\cite{Haxton:2000xb} who estimated it by direct calculation on several characteristic elements. Therefore, this flux carries information about the solar metal abundances, quantities of crucial interest in the context of the ``solar opacity problem,'' a point stressed in  reference~\\cite{Haxton:2000xb}. As this flux is no longer overshadowed by the spurious plasmon resonance in the photo-production channel, this argument is resurrected by our study, i.e., a measurement of the keV-range flux could provide nontrivial information on the solar metal content.\n\n\n\\section*{Acknowledgments}\n\nWe thank Thierry Lasserre for bringing up the question of the\nlow-energy solar neutrino flux, and Alexander Millar for helpful\ndiscussions.  In Munich, we acknowledge partial support by the\nDeutsche Forschungsgemeinschaft through Grant No.\\ EXC 153 (Excellence\nCluster ``Universe'') and Grant No.\\ SFB 1258 (Collaborative Research\nCenter ``Neutrinos, Dark Matter, Messengers'') as well as by the\nEuropean Union through Grant No.\\ H2020-MSCA-ITN-2015\/674896\n(Innovative Training Network ``Elusives'').  J.R.\\ is supported by the\nRamon y Cajal Fellowship 2012-10597 and FPA2015-65745-P\n(MINECO\/FEDER).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn the last decade, the great improvement has been made in\ncosmology, which leads to the so-called ``concordant cosmological\nmodel\". The concordant cosmological model, together with the\n\\emph{Standard Model} of particle physics, represents our current\nunderstanding of the nature. The later can be extended slightly by\nneutrinos with nonzero masses, and the former is extended\nexcellently by inflation. However, there still exist lots of\nquestions and challenges to the concordant cosmological model and\nthe \\emph{Standard Model}, in spite of their impressive successes.\nFor example, why does this particular pattern of the fundmental\ninteractions (the electromagnetic, weak, strong and gravitational\ninteractions) exist in the nature? Why is the total rank of the\nelectroweak-strong gauge group just four? Why are there exactly\nthree generations of particles?\nWhat's the nature of the dark matter and the dark energy? What is\nthe mechanism that generates the baryon asymmetry? How to\nincorporate the inflation with the \\emph{Standard Model} properly?\nThen we can conclude that the concordant cosmological model and the\n\\emph{Standard Model} are not the end of the story, but the tip of\nthe iceberg. A more fundmental theory is expected.\n\nSeveral promising ideas have been put forward: Supersymmetry,\nSupergravity, Superstring\/M theory and Loop quantum gravity. Among\nthese ideas, the superstring\/M theory is supposed to be the most\ncompelling one. This theory has some excellent properties: no UV\ndivergence, including gravity and gauge interaction automatically,\nno free parameters before compactification, uniqueness (different\nstring theories are dual to each other, as a whole), etc.. However\nthe superstring\/M theory requires a definite number of space-time,\n10\/11. To reconcile with our empirically 4-dimensional universe, the\nextra 6\/7 dimensions have to be compacted. Unfortunately, it is\nargued that there may exist lots of approaches of compactification.\nThis even leads some authors to suggest the scenario, Landscape\n\\cite{landscape}. In this scenario, it is argued that there exist\nplenty of vacua, at the order of $10^{500}$, as the result of\ndifferent models of compactification \\cite{counting}, and the\n\\emph{Standard Model} can be constructed in some vacua.\n\nBriefly, there are three classes of models on building the\n\\emph{Standard Model} within the framework of the superstring\/M\ntheory: intersecting D-brane models \\cite{0502005}, Calabi-Yau\nmodels \\cite{heteroticCalabiY} and orbifold models\n\\cite{heteroticOrbifold1,heteroticOrbifold2}. Calabi-Yau (orbifold)\nmodels are based on the $E_8\\times E_8$ heterotic superstring theory\nwith Calabi-Yau (orbifold) compactification. In some sense, orbifold\nmodels can be taken as the limitation of Calabi-Yau models. Many\nmodels on building the \\emph{Standard Model} in string theory have\nbeen suggested. Although none of the models is accepted by the\nmajority, we think some models are worth studying further. For\nexample, in Ref.\\cite{0511035,kim}, some nice orbifold models have\nbeen suggested.\n\nIn orbifold models, due to the six extra-dimensions and the lattice\ngroup, $E_8\\times E_8$, the maximum rank of gauge group is limited\nto be 22, while the rank of gauge group in intersecting D-brane\nmodels is unlimited. Then This property reduces the arbitrary of\norbifold models greatly. Usually, in the orbifold models, the\norbifold is constructed from the torus $T^6$ by identifying the\npoints under a discrete point group. Once the point group is given,\nthere are only a few possible sets of the shift vectors and the\nWilson lines from which the unequivalent breaking patterns of $E_8$\nare deduced \\cite{Kang-Sin Choi}. When the shift vector and the\nWilson line are given, the unbroken gauge group derived from $E_8$\nis definite. Then by exhausting all the possibilities, we can select\nthe appropriate breaking patterns which lead the (supersymmetric)\nStandard Model gauge group and matter content.\n\nIn the models of Ref.\\cite{0511035,kim}, the authors obtain the 3\nchiral matter generations including the right-handed neutrinos, plus\nthe singlet and vector-like exotic matter. Particularly, in\n\\cite{kim}, the authors construct a three-family flipped SU(5) model\nfrom $Z_{12-I}$. By the Yukawa couplings, the model provides\nnaturally the $R$-parity, the doublet-triplet splitting, and one\npair of Higgs doublets. The model also contains some superheavy\nsinglets, which can be the excellent candidates for inflation\nfields. So far, in the model of \\cite{kim}, no serious\nphenomenological problem is encountered.\n\nBut a remarkable property of this model is that it predicts\nhalf-integer charged particles! From now on, We call half-integer\ncharged particle as \\emph{halfon}. In fact, this is a common\nproperty of orbifold models. However, it is well known that no\nsignal of the existence of fractional charged particles has been\nobserved. Fortunately, in the model of \\cite{kim}, due to the broken\nU(1) symmetries at the GUT scale, the halfons are superheavy\nnaturally with mass at the order of $10^{16}$Gev. Because of the\nhalf-integer charge, the lightest of halfons, labeled by\n$S^{\\pm1\/2}$, must be stable. Then it is possible that $S^{\\pm1\/2}$,\nas the stable particles predicted by orbifold models, may exist in\nour universe, and imprints of halfons in the universe would be\nobserved in the future.\n\nHowever, we know the existence of fractional electric charged\nparticles are limited severely by observations, particularly by the\nMillikan oil drop experiments. In \\cite{Millikan}, it is shown that\nThe concentration of particles with fractional charge more than\n0.16e (e being the magnitude of the electron charge) from the\nnearest integer charge is less than $4.17\\times 10^{-22}$ particles\nper nucleon with $95\\%$ confidence.\n\nThen, in order to alleviate the limitation, generally, it is argued\nthat most of halfons are diluted away by inflation if the mass of\nLHIC is much larger than the reheating temperature. The case is\nsimilar to monopoles. Because density of halfons becomes so low\nafter inflation, we can not observe halfons on the earth. So the\ncontradiction between the observation and the existence of halfons\nin physics is eliminated.\n\nBut, there exists another possibility that the density of halfons\nmay be not low. We can not observe halfons on the earth just because\nhalfons do not locate on the earth. Then, if halfons do not locate\non the earth, where can they locate? If halfons do exist in our\nuniverse, are there any observable imprints? Will halfons be helpful\nin solving some present puzzles in cosmology? After extensive\nconsideration of these questions, we find that, if we assume\nappropriate halfons generated by the reheating, halfons should\ncondense in the center of each galaxy. Our solar system is away from\nthe center of the Milk Way Galaxy, so halfons can not be observed on\nthe earth. Further we find that, at the centers of galaxies, the\nannihilation of $S^{1\/2}$ and $S^{-1\/2}$ can happen, and this may\nexplain the origin of the ultra-high energy cosmic rays (UHECRs)\nabove the GZK cutoff \\cite{uhecr}. Even, because of the electric\ncharge, $S^{\\pm1\/2}$ may form bound states by attracting protons or\nelectrons. Particularly, $S^{1\/2}$ and $e^-$ can form the bound\nstate $S^{1\/2}e^-$. The state has one spectral line with the\nwavelength of 4680\\AA, which is slightly different from the spectral\nline of Hydrogen atom with the wavelength of 4682\\AA. If this\nspectra line is observed, the interesting would be very great.\n\nBelow, let's show our consideration in detail. The paper is\norganized as follows. In Section \\ref{condensation}, by analyzing\nthe procession of the formation of the large scale structure, we\nshow that most of halfons should locate at the center of galaxies.\nIn Section \\ref{implication}, we consider some implications of\nhalfons in cosmology.\n\n\n\\section{Condensation of Halfons}\n\\label{condensation}\n\nFirstly, we assume the density of halfons generated by reheating is\nappropriate. In the model of \\cite{kim}, the particle content is\ndefinite. If the inflation fields are taken as the singlets in the\nmodel, the coupling between halfons and inflation fields can be\ndeduced naturally. Then halfons should be generated during reheating\nby the coupling. The number density of halfons is determined by the\nmass of halfon and the temperature of reheating. Then, by choosing\nparameters, appropriate halfons can be generated. So we think our\nassumption is reasonable. Denoting the present ratio of density of\nhalfons by $\\Omega_S$, we expect $10^{-6}<\\Omega_S<10^{-2}$. If\n$\\Omega_S>10^{-2}$, the big bang nucleosynthesis will be effected.\nIf $\\Omega_S<10^{-6}$, the imprints of halfons in the universe may\nbe too weak to be observed.\n\nWe know the formation of the large scale structure is dominated by\nthe cold dark matter. Presently, a galactic halo is mainly composed\nof baryonic matter and a huge cold dark matter halo. Generally, it\nis supposed that the configuration of the dark matter is virialized.\nThe virial velocity is about at the order of $10^2km\/s$. On the\nother hand, the baryonic matter, due to the ability of dissipation,\nforms objects of atrophysical size as individual and distinct\nentities in the core of the galactic halo. Then what about halfons?\n\n\nIn order to answer this question, let's recall the process of the\nformation of a galaxy in an ideal model. We know, the formation of\nthe large scale structure begins when the universe becomes the\nmatter-dominated. After the last scattering, the baryonic matter\nfalls into the well of the gravitational potential formed by the\ncold dark matter. When ($\\delta\\rho\/\\rho$) becomes of order unity,\nthe cluster which forms the galaxy eventually, separates from the\nexpansion of the universe. Then the cluster begins to contract and\ncollapse. At the beginning of the contraction, due to the\ncosmological expansion, the velocities of the cluster matter can be\ntaken as zero roughly. For simplicity, we assume the cluster is\nroughly spherically symmetric. Then the cluster matter, including\ncold dark matter, baryonic matter and halfons, begins to fall\ntowards the center of the cluster roughly along the radiuses under\nthe attraction of gravitation. If we neglect the interaction and\ncollision, the cluster particles would oscillate between the\nantipodal points for long time. In fact, the existence of the\ncollision and interaction will disturb the oscillating and virialize\nthe cluster inevitably.\nHowever, the process of virialization for cold dark matter, halfons\nor baryonic matter is different.\n\nFor baryonic matter, due to the strong and electromagnetic\ninteraction, the particles of the baryonic matter collide each other\nviolently and frequently as falling towards the center. Then the\nvirialization of the baryonic matter is completed long before the\ndark matter. The phase space distribution of the virialized baryonic\nmatter becomes roughly Maxwellian and their density varies roughly\nas $r^{-2}$. Such a configuration is often referred to as an\nisothermal sphere. The random distribution of the velocities of the\nvirialized particles would prevent the baryonic matter from\ncontracting further. But, due to the dissipative process--e.g.,\ncollisional excitation of atoms and molecules, and Compton\nscattering off the CMBR, the baryonic matter can lose energy and\ncondense further into the core of the cluster. Finally, objects of\nastrophysical size are formed as individual and distinct entities.\nIf the galactic halo has the angular momentum, after dissipation,\nthe baryonic matter will wind up in a disk-like structure\n\\cite{Kolb}.\n\nFor dark matter, the virialization is later than the baryonic\nmatter. Generally, the cold dark matter particles are supposed to be\nweak-interaction massive particles (WIMPs). Due to the WIMP model,\nthe cold dark matter cannot be virialized by colliding with each\nother or the baryonic matter. However, the galactic halo is not\nspherically symmetric definitely. And the time- and pace-varying\ngravitational field provides the means for WIMPs to change their\nmomentums and to become well mixed in phase space. Particularly, as\nthe condensation of the baryonic matter, Many local gravitational\ncenters are formed. Then the gravitational scattering of the cold\ndark matter particles by the local centers accelerates the\nvirialization greatly. After a few dynamical times\n($\\tau\\sim(G\\rho)^{1\/2}$), the virialization of the cold dark matter\nis finished \\cite{Kolb}. However, as the result of the WIMP model,\nthe virialized cold dark matter cannot lose energy to condense and\ncollapse further. So, now the configuration of the cold dark matter\nis still a virialized halo.\n\nFor halfons, the case is different from both baryonic matter and\ncold dark matter. The coupling between halfons and dark matter can\nhappen only by the weak neutral current, which is very weak. So\nhalfons almost do not collide with the cold dark matter. Since\nhalfon has the half-integer electric charge, it follows that halfons\ncan be involved in the electromagnetic interaction. So, as falling\ntowards the center of the cluster, halfons must collide with the\nbaryonic matter by the electromagnetic interaction. Due to the\nsuperheavy mass, the effect of one collision on the motion of halfon\nis unobvious. However, near the center of the cluster, the density\nof the baryonic matter becomes very high. Then, near the center, the\nfrequent collisions between halfons and baryonic matter can\nevidently reduce the kinetic energy of halfons. At the same time,\nthe configuration of the baryonic matter can be taken as a\nisothermal sphere. So, roughly, the motion of halfon may be taken as\na damped oscillator along a radius. Then only after several\noscillations, halfons may lose most part of their kinetic energy.\nThe velocities of halfons become so small that they cannot move away\nfrom the cluster center. In some sense, we can say that halfons are\nvirialized and become one part of the configuration of the\nvirialized baryonic matter at the cluster center. After then,\nhalfons, together with baryonic matter, condense and contract\nfurther. Finally, halfons become one part of the individual and\ndistinct objects with astrophysical size near the cluster center. We\nthink, due to the superheavy mass, the locations of halfons should\nbe at the centers of these objects.\n\nAdditionally, due to electromagnetic interaction, halfons and\nelectrons (protons) can form some bound states, $S^{-1\/2}p^+$,\n$S^{1\/2}e^-$, etc. These states make halfons have the ability to\nundergo dissipation, as the baryonic matter, to lose energy. Even,\ndue to the electron\/proton cloud around $S^{1\/2}$\/$S^{-1\/2}$, the\nscattering cross section of these states is much bigger than that of\nhalfons. Then it implies that, after having been combined with\nelectrons or protons, halfons collide with the baryonic matter much\nmore frequently. So these bound states can accelerate the\ncondensation of halfons.\n\nIn fact, we find that these bound states may have other remarkable\nimplications. For example, the ground state energy level of\n$S^{1\/2}e^-$ is about at the order of $10eV$, while the ground state\nenergy level of $S^{-1\/2}p^+$ is about at the order of $10keV$. Then\nthe collisionally excited state of $S^{-1\/2}p^+$ can emit photons\nwith much higher energy than that of $S^{1\/2}e^-$. It implies that\n$S^{-1\/2}p^+$ can lose energy quicker than $S^{1\/2}e^-$. Then,\nfinally, there may exist the segregation between the two states,\nfrom which some new physical phenomenons may arise.\n\nParticularly, the spectral line, $1s\\rightarrow2p$, of $S^{+1\/2}e^-$\nis very close to the spectral line, $2s\\rightarrow4p$, of $p^+e^-$.\nHowever, due to the superheavy mass of $S^{+1\/2}$, the reduced\nmasses of $S^{+1\/2}e^-$ and $p^+e^-$ are different. So the two\nspectral lines are not degenerate. The wavelength of the spectral\nline of Hydrogen atom is about 4862{\\AA}, while the wavelength of\nthe spectral line of $S^{+1\/2}e^-$ is about 4860{\\AA}. Then it\nbecomes very interesting to find whether, in our universe, there\nexists the new spectral line with the wavelength 4860{\\AA} or not.\n\nAdditionally, there may exist neutral states, e.g. $(S^{-1\/2})^2p^+$\nand $(S^{+1\/2})^2e^-$. We think that it should also be interesting\nto calculate the spectra of these states and then to try to find\nthese spectral lines in our universe.\n\nAbove, in an ideal model, we show that the location of halfons in a\ngalaxy should be at the center of the galaxy. For our Milky Way\ngalaxy, the case is more complicated and special. Our galaxy is very\nhuge, but the black hole at the center is small. This implies that\nthe formation of the Milky Way is special. The early stage of our\ngalaxy is the merging epoch of many dwarf galaxies, but the merging\nepoch ended early. The late stage of the Milky Way is astonishingly\npeaceable. In these dwarf protogalaxies, we think, halfons should\nlocate at the centers of the protogalaxies. Then, in the Milky Way,\nmost of halfons should locate in the spheroidal core of our galaxy.\nAlthough the core of the galaxy may eject the baryonic matter during\nthe active epoch, few halfons may exist in the spiral arm away from\nthe core of the galaxy. The solar system, which may be formed by\nsome ejected matter, is far away from the core. So very few halfons\nmay exist in the solar system. Then, we think, this is the reason\nthat, on the earth, no signal of fractional electric charged\nparticles has been observed.\n\n\n\\section{Implications of Halfons}\n\\label{implication}\n\n\n\nIn the last subsection, we have given one of the implications of\nhalfons, the spectrum line with the wavelength 4860{\\AA}. If the\nspectrum line is observed, the interesting is obvious.\n\nAdditionally, We find that halfons may be helpful in solving the\nUHECR puzzle. Due to the GZK cutoff, the UHECR spectrum should\ndramatically steepen above $E_{GZK}\\approx5\\times10^{19}eV$.\nHowever, a significant excess of events above $10^{20}eV$ has been\ndetected. Many proposals, including top-down models, have been\nsuggested to solve this puzzle (for a review, see \\cite{uhecr}).\nTop-down model is a generic name for all proposals in which that the\nobserved UHECR primaries are produced as decay products of some\nsuperheavy particles $X$ with mass\n $m_X\\gtrsim 10^{12}GeV$.\nThe superheavy particles may be produced by topological defects such\nas cosmic strings, monopoles and hybrid defects, or be superheavy\nmetastable relic particles. But the both approaches have the\nfine-tuning problem. For the latter, the lifetime of the superheavy\nparticle should be fine tuned to be in the range $10^{17}s\\lesssim\n\\tau_X \\lesssim 10^{28}s$. For topological defects, the case is even\nworse, because topological defects is constrained by observation\nseverely.\n\nHowever, if the superheavy particl is halfon, the fine-tuning\nproblem can be solved naturally.  Halfons are stable, and\n$S^{\\pm1\/2}$ can annihilate into photons or $Z$ bosons,\n$S^{1\/2}+S^{-1\/2}\\rightarrow\\gamma$ or $Z$. We know that it is very\ndifficult for the annihilation of $S^{\\pm1\/2}$ to happen because of\nthe small number density and the small annihilation cross section.\nBut we have shown in last subsection that most of halfons should\ncondense into the center of a galaxy. Then, at the centers of\ngalaxies, the number density of halfons may be large enough and make\nit possible for $S^{\\pm1\/2}$ to collide with each other and then to\nannihilate. The mass of $S^{\\pm1\/2}$ is at the order $10^{16}GeV$.\nThen the annihilation may produce photons or $Z$ bosons with the\nenergy above $10^{24}eV$. By colliding with particles around, these\nbosons can produce particles with ultra high energy (UHE) above\n$10^{23}eV$ as secondaries, part of which may be UHE neutrinos or\nneutralinos. These UHE neutrinos or netralinos can traverse the\nextragalactic space without attenuation, thus avoiding the GZK\ncutoff. Then the UHE neutrinos can collide with particles in the\ngalaxy, e.g. background neutrinos or neutralinos, and produce\nprotons with energy above $10^{20}eV$. The protons can be the UHE\nprimaries initiating the observed air showers and cause the UHECRs\nabove the GZK cutoff. Additionally, the annihilation of $S^{\\pm1\/2}$\nhappens at centers of galaxies. Then the isotropic distribution of\ngalaxies in the universe can explain the isotropy of UHECRs\nnaturally. So qualitatively, the idea works well. Of course, the\nfurther work is needed to make sure whether halfons can solve the\nUHECR puzzle quantitatively or not.\n\nAnd, due to the condensation and the large mass, halfons may explain\nthe formation of the supermassive black hole(SMBH) at the very high\nredshift. We know, the structure formation in the cold dark matter\nmodel proceeds hierarchically, ``from the bottom up\". This means\nbigger structures form through tidal interaction and mergers of\nsmaller objects. Then the formation of SMBH at the high redshift\nrequires that the efficiency of the mergers should be high enough or\norigin small objects should be heavy enough. Yet no one has proposed\na concrete mechanism for converting stellar mass objects into\nobjects 6 to 10 orders of magnitude larger or for generating origin\nsmall objects big enough \\cite{a0204486}. Halfons, because of large\nmass and quick condensation, may be the natural candidate to\ngenerate the massive seeds to form SMBH. So it is interesting for\nfurther work to make sure whether the idea works or not.\n\n\n\\section{Discussion and Summary}\n\nAbove, we have shown our study on halfons. We find that halfons can\nexist in out universe by condensing into the centers of galaxies.\nAnd, we have analysed several potential implications of halfons: the\nspectral line with wavelength 4860{\\AA}, solving the UHECR puzzle\nand explaining the formation of SMBH at the high redshift.\nSuperheavy halfons are the special prediction of the orbifold models\nbuilt within the framework of the superstring\/M theory. If the\nspectral line with wavelength 4860{\\AA} is observed, it will support\nstrongly the orbifold models and the superstring\/M theory. We know,\nup to date, no observational or experimental clue in supporting the\nsuperstring\/M theory is found. This may be the first observable\nsignal of the superstring\/M theory.\n\nAdditionally, besides the half integer charged particles, the\norbifold models also predict the superheavy singlet exotics and the\nparticles in the hidden sector. The singlets may be candidates as\ninflatoion fields. We think that it is interesting to construct\ninflation models using these singlets. The particles in the hidden\nsector can deduce the broken supersymmetry and are also worth\nresearching.\n\nSuperheavy half integer charged particles seem to be very strange.\nBut we should keep our mind open. Halfon is no more strange than\naxion or the dark matter. We have accepted the dark matter and\nsuppose the existence of axion. Why can not we assume the existence\nof $S^{\\pm 1\/2}$?\n\nFinally, we emphasize that, even if no signals of halfons in our\nuniverse are observed, this does not mean that the orbifold models\nshould be excluded. It is possible that the number density of LHIC\nis too small because of the very low reheating temperature, and then\nthe imprints of LHIC is too weak to be observed.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nStudy of diffuse background emission produced by faint sources with flux levels below the sensitivity of a telescope is commonly used to constrain the nature of source populations in the Universe and their cosmological evolution. In the high-energy $\\gamma$-ray\\ band (HE, 0.1-100 GeV) the diffuse extragalactic $\\gamma$-ray\\ background (EGB) was detected for the first time by {\\it SAS-2} satellite \\citep{fichtel78}, was further studied by EGRET telescope on board of {\\it CGRO} mission \\citep{sreekumar98,strong04} and, most recently, by the Large Area Telescope (LAT) on board of {\\it Fermi} satellite \\citep{fermi_background}. \n\nIt is often assumed that the dominant contribution to the EGB is given by distant Active Galactic Nuclei (AGN), in particular, blazars \\citep{padovani93,stecker93,chiang95,stecker96,mukherjee99,mucke00,inoue09}. However, a recent study by {\\it Fermi} collaboration reveals that blazars might contribute only a relatively small fraction of the HE EGB level \\citep{fermi_agn}, while a significant part of the EGB should be either explained by a yet unknown source population or have a truly diffuse nature (see, however, \\citet{stecker10}). \n\n\nThe EGB in the Very-High-Energy (VHE, $\\gamma$-ray s in the $E\\gtrsim 100$~GeV) range has never been measured. On one hand, the effective collection area of previous space-based $\\gamma$-ray\\ telescopes was not sufficient to achieve significant photon statistics in this energy band. On the other hand, the efficiency of cosmic ray background rejection in the ground-based Cherenkov $\\gamma$-ray\\ telescopes, like HESS, MAGIC and VERITAS is not sufficient for detection of the isotropic diffuse EGB on top of the cosmic ray background. Thus, the properties and the origin of VHE EGB remain largely unconstrained up to now.\n\nIt is clear that the VHE EGB should contain a contribution from the unresolved point sources. The main candidate source class is, as in the case of HE EGB, that of blazars. At the same time,  the VHE EGB could contain, apart from the contribution from unresolved extragalactic point sources, genuine diffuse components which could be produced via several mechanisms.  For example, if the spectra of a large number of  $\\gamma$-ray -loud AGNs extend to the energies above 300~GeV, all the power emitted initially in \n$\\gamma$-ray s with energies higher than $\\sim 300$~GeV  is absorbed in the pair\nproduction of $\\gamma$-ray s on the cosmological infrared and\/or  microwave backgrounds \\citep{gould67}.\nSecondary inverse Compton emission of electron-positron pairs deposited in the\nintergalactic space in result of the pair production leads to generation of\ndiffuse extragalactic emission in the VHE energy band \\citep{coppi97}.  Another mechanism\nwhich can lead to the generation of diffuse component of VHE EGB  is electromagnetic cascade initiated in the intergalactic space by ultra-high energy cosmic rays (UHECR) interacting with cosmic microwave background\nphotons \\citep{berezinsky75,semikoz09,berezinsky11}. The cascade channels the power from the highest energies of about\n$10^{20}$~eV down to the $\\sim 0.1$~GeV  band in which the mean free path of the\n$\\gamma$-ray s becomes comparable to the size of the visible part of the Universe. \n\nApart from the \"guaranteed\" (but, possibly, very weak) diffuse contributions, isotropic VHE $\\gamma$-ray\\ background might contain contributions from \"exotic\" diffuse sources, like diffuse emission from annihilation of Dark Matter particles in the outer halo of the Milky Way galaxy and the annihilation signal accumulated from the dark matter halos of all galaxies in the course of cosmological evolution \\citep{fermi_DM}.\n\nWhatever are the sources of VHE EGB, they are scattered across the Universe, so that a significant contribution to the flux is produced at redshifts $z\\sim 1$. A known effect of absorption of VHE $\\gamma$-ray s due to the interactions with infrared\/optical Extragalactic Background Light (EBL) should lead to attenuation of the $E>50$~GeV signal produced by the sources at large redshifts $z\\sim 1$ \\citep{gould67,kneiske04,franceschini08,stecker_ebl,gilmore09}. This should leave an \"imprint\" on the VHE EBL spectrum, which should have the form of a gradual suppression with the increasing photon energy. Detecting a EBL suppression feature in the EGB spectrum would provide an important constraint on the (largely uncertain) evolution of the EBL density and spectrum up to redshifts $z\\sim 1$. Such a constraint is otherwise difficult to obtain from the studies of individual extragalactic VHE $\\gamma$-ray\\ sources because of the limited signal statistics at the highest  energies, especially for the sources at significant redshifts.\n\nIn what follows we discuss the measurement of  the EGB  derived from the data of  {\\it Fermi}\/LAT telescope \\citep{fermi_description}. The measurement is obtained from the counting of  photons at high Galactic latitudes, after subtraction of the Galactic diffuse emission and the residual cosmic-ray background not rejected by the LAT data analysis software. We compare the measurement of EGB obtained in this way with the measurement previously derived from the likelihood analysis of all-sky data by \\citet{fermi_background}.\n\nThe EGB flux above 30~GeV turns out to be comparable to the flux in extragalactic VHE $\\gamma$-ray\\ sources resolved by {\\it Fermi}. We find that the spectrum of EGB in this energy range follows the cumulative spectrum of the resolved sources. \nDominant population of extragalactic VHE $\\gamma$-ray\\ sources is BL Lacs. Noticing the similarity of the spectrum of VHE EGB and of the cumulative BL Lac VHE $\\gamma$-ray\\ spectrum, we put forward a hypothesis that the VHE EBL is produced by unresolved BL Lacs with fluxes below the sensitivity of LAT. We explore this hypothesis and show that it could be valid if BL Lacs follow a positive cosmological evolution pattern, characteristic for other types of AGN, in particular for the parent population of BL Lacs objects, Fanaroff-Riley type I (FR I) radio galaxies.\n\n\\section{Data selection and data analysis} \n\nFor our analysis we consider all publicly available LAT data from August 4, 2008 to January 23, 2011. We process the data using {\\it Fermi} Science Tools\\footnote{\\tt http:\/\/fermi.gsfc.nasa.gov\/ssc\/data\/analysis\/scitools\/}. We filter the entire data set with {\\it gtselect} and {\\it gtmktime} tools following the recommendations of {\\it Fermi} team\\footnote{\\tt http:\/\/fermi.gsfc.nasa.gov\/ssc\/data\/analysis\/scitools\/} and  retain only events belonging to \"ultraclean\" ({\\tt P7ULTRACLEAN\\_V6}) event class, which has minimal residual cosmic ray contamination.  \n\nEstimate of the contribution of point sources to the total flux requires separation of the photons coming from the point sources from those produced by the diffuse emission. Such separation is most straightforward for the photons with narrow point-spread-function (PSF). Taking this into account, select two sub-classes (which provide dominant contribution to the ultraclean events) with the most compact PSF, the sub-classes selected by imposing the selection criterium {\\tt EVENT\\_CLASS=65311} or {\\tt 32543}. Other sub-classes of the ultraclean events have worse PSF. Point source contribution in these photons  suffers from  an additional uncertainty. Taking this into account, we restrict our attention to the subset of the ultraclean events with the best PSF.   \n\nWe retain events with Earth zenith angle $\\theta_z\\le 100^\\circ$. To estimate the flux from the photon counts we use {\\it gtexposure} tool.  We consider only events at high Galactic latitudes, in the regions $|b|\\ge 60^\\circ$. \n\nOur analysis is based on the so-called \"Pass 7\" selection of the LAT data (see {\\tt http:\/\/fermi.gsfc.nasa.gov\/ssc\/data\/access\/}). However, use use a comparison of the Pass 7 data with the previous Pass 6 data selection in the estimate residual cosmic ray contamination of the set of events chosen for the analysis. The residual cosmic ray fraction in the Pass 6 data was studied in details by \\citet{fermi_background}. Re-calculation of the residual cosmic ray fraction for any new selection of events, including the one considered in our analysis, could be done in a straightforward way as explained in Section \\ref{sec:cr}. \n\n\n\n\n\\section{Diffuse $\\gamma$-ray background}\n\nSignal detected by LAT at high Galactic latitudes contains four types of contributions: emission from point sources, diffuse $\\gamma$-ray\\ emission from the Galaxy, EGB and residual cosmic ray background not rejected by the analysis software. To measure the EGB flux, one needs to separate the contributions from the four components in the overall signal in a given energy band. \n\n\\subsection{Point source contribution}\n\\label{sec:ps}\n\nPoint source component could be singled out in a straightforward way if the set of sources detectable in a given energy band is known. To define the set of sources we find the sources correlating with the arrival directions of photons in each energy band, using the method described by \\citet{100GeV_sky}\\footnote{See {\\tt http:\/www.isdc.unige.ch\/vhe\/index.html} for an updated version of the VHE source list.}. To calculate the total number of photons associated to the sources, we construct a cumulative distribution of photons as a function of the distance $\\theta$ from the source and split it on the background and source contributions. The background contribution grows asymptotically as $\\theta^2$, while the source contribution asymptotically reaches constant. An example of the cumulative photon distribution around the source positions in the 12.5-25 GeV energy band is shown in Fig. \\ref{fig:PSF_cumulative}.\n\n\\begin{figure}\n\\includegraphics[height=\\linewidth,angle=-90]{fig1}\n\\caption{Cumulative front photon distributions around point sources in 12.5-25 GeV energy band. Red points show the source photons, blue points show the background. Horizontal lines (from top to bottom) show the 100\\%, 95\\% and 68\\% levels.}\n\\label{fig:PSF_cumulative}\n\\end{figure}\n\n\\subsection{Galactic diffuse emission contribution}\n\\label{sec:gal}\n\nContribution of the diffuse emission from the Galaxy should be found via a detailed fitting the all-sky photon distribution to an all-sky spatial and spectral template. This contribution is best constrained by the all-sky photon distribution in the 0.1-10~GeV energy band, where event statistics is very high. Detailed fitting of the Galactic diffuse emission to the data in the 0.1-10~GeV band was done by \\citet{fermi_background}. In our analysis we rely on the best-fit model of Galactic diffuse emission derived by \\citet{fermi_background}. This model is available in the sky region of interest, $|b|\\ge 60^\\circ$, see Fig. 6 of the Supplemental Material in the Ref. \\citet{fermi_background}. The uncertainties of the Galactic diffuse emission model are also discussed by \\citet{fermi_background}. We take these uncertainties into account.\n\nThe model consists of two main contributions: the \"atomic hydrogen\" component produced by interactions of cosmic rays with interstellar matter and \"inverse Compton\"  (IC) component produced by inverse Compton emission from cosmic ray electrons. Extrapolation of the atomic hydrogen component to the highest LAT energies is straightforward: the pion decay spectrum follows the cosmic ray spectrum and extrapolation has the form of a simple powerlaw with photon index $\\sim 2.7$, the same as the slope of the cosmic ray spectrum.  This component gives a sub-dominant contribution above 100~GeV. Extrapolation of the IC component depends on the unknown shape of the (average over interstellar medium) cosmic ray electron spectrum at the energies above TeV. We have checked that in the model of \\citet{fermi_background} the spectrum of IC component is consistent with the spectrum of IC scattering of the local interstellar radiation field \\citep{moskalenko06} by electrons with the spectrum $dN_e\/dE\\sim E^{-3}\\exp\\left(-E\/1\\mbox{ TeV}\\right)$. This electron spectrum consistent with the cosmic ray electron spectrum observed on the Earth \\citep{fermi_electrons,HESS_electrons}.  The IC spectrum produced by such electron population is shown by the cyan dotted line in Fig. \\ref{fig:Galactic}. The overall Galactic diffuse emission spectrum at high Galactic latitudes is then the sum of the atomic hydrogen and IC contributions, shown by the dashed black line in Fig. \\ref{fig:Galactic}.\n\nThe high-energy cut-off of the local cosmic ray electron spectrum is most likely determined by the distance to the closest cosmic ray electron sources, e.g. to the closest pulsar wind nebulae \\citep{aharonian04}, rather than by the intrinsic cut-off in the injection spectrum of electrons from the sources. This means that the local measurement of the high-energy cut-off of the cosmic electron spectrum does not provide a measurement of the high-energy cut-off in the injection spectrum of electrons. It is possible that Galactic cosmic electron sources inject electrons with energies much higher than $\\sim 1$~TeV. This possibility is shown by the solid grey line in Fig. \\ref{fig:Galactic} which shows the sum of the atomic hydrogen contribution with the IC emission from electrons without high-energy cut-off in the spectrum. The IC component still exhibits suppression at the energies $\\sim 1$~TeV because of the Klein-Nishina effect.\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth]{fig2}\n\\caption{Extrapolation of the spectrum of the Galactic diffuse emission at high Galactic latitudes $|b|\\ge 60^\\circ$ to the  100~GeV energy range. Blue and cyan dotted lines below 100~GeV show the contributions from cosmic ray interactions with interstellar medium and inverse Compton scattering from cosmic electrons calculated by \\citet{fermi_background}. Continuation of the cyan dotted line above 100~GeV is calculated assuming that inverse Compton emission is produced by electrons with a cut-off powerlaw spectrum with cut-off at 1~TeV. Grey dashed line is the sum of the cosmic ray and inverse Compton contributions. Solid grey line shows the overall diffuse emission spectrum in which the inverse Compton emission is produced by electron distribution without high-energy cut-off at 1~TeV. Red thick solid line shows the spectrum used for subtraction of Galactic component from the overall high Galactic latitude diffuse emission flux. }\n\\label{fig:Galactic}\n\\end{figure}\n\nTo take into account the above mentioned uncertainty of the IC component we adopt an approximation $dN_\\gamma\/dE\\sim E^{-2.5}\\exp\\left(-E\/2\\mbox{ TeV}\\right)$ for the high Galactic latitude diffuse emission spectrum at $E>100$~GeV. This approximation is shown by the red thick solid line in Fig. \\ref{fig:Galactic}. This spectrum lies exactly in the middle between the two extreme possibilities: TeV-scale high-energy cut-off in the cosmic electron spectrum and no cut-off in the cosmic electron spectrum. One should take into account that the uncertainty of this approximation reaches $\\simeq 50$\\% at the highest energies. We take this uncertainty into account in the calculation of the EGB spectrum, by adding it as a systematic error. The two extreme possibilities for the behavior of electron spectrum above 1~TeV (exponential cut-off exactly at 1~TeV and no cut-off at all) provide a good estimate of the overall uncertainty of electron spectrum in the interstellar medium in this range. The uncertainty of the shape of electron spectrum dominates the uncertainty of the inverse Compton component of Galactic diffuse emission at high Galactic latitudes.\n\n\\subsection{Residual cosmic ray background contribution}\n\\label{sec:cr}\n\nTo estimate the residual cosmic ray background in the set of events selected for the analysis, we rely on the knowledge of residual \nThe residual cosmic ray background in the {\\tt dataclean} event class of Pass 6 data is extensively discussed in \\citet{fermi_background}. The residual cosmic ray background in the subset of Pass 7 {\\tt superclean} events used in our analysis could be calculated from the known residual cosmic ray background in the Pass 6 {\\tt dataclean} events via a straightforard comparison of statistics of events on- and off-point-sources in the two classes.\n\nFirst, the residual cosmic ray fraction in the Pass 6 {\\tt dataclean} events should be calculated from the known suppression factor of cosmic ray event at transition from the {\\tt diffuse} event class to the {\\tt dataclean} event class in Pass 6 (see \\citet{fermi_background} for the detailed discussion of the suppression factor). \nIn each of the two event classes, the entire event set consists of a certain number of $\\gamma$-ray\\ events $N_{\\gamma, i}$ and a certain number of residual cosmic ray events, $N_{CR, i}$ where $i$ stands for ${3+4}$ or $4$. Cleaning of the event set done to produce the {\\tt dataclean} event set from {\\tt diffuse} set results in rejection of a large fraction of the cosmic ray events, $N_{CR,4}=\\alpha_{CR} N_{CR,3+4}$ with $\\alpha_{CR}\\ll 1$. However, it results also in rejection of a number of true $\\gamma$-ray\\ events, so that $N_{\\gamma,4}=\\alpha_\\gamma N_{\\gamma,3+4}$ with $\\alpha_\\gamma<1$. \n\nThe suppression factor $\\alpha_{CR}$ is known as a function of energy from the Monte-Carlo simulations of cosmic ray and $\\gamma$-ray\\ induced events in the LAT detector by \\citet{fermi_background}. The suppression factor $\\alpha_\\gamma$ could be found directly from the data set, by comparing statistics of events coming from the point sources in the {\\tt diffuse} and {\\tt dataclean} event classes (see section  \\ref{sec:ps} above). In the calculation of $\\alpha_\\gamma$ all the photons associated to $\\sim 10^3$ point sources listed in the Fermi 2-year catalog \\citep{fermi_catalog} could be used. This provides very large event statistics so that uncertainty of $\\alpha_\\gamma$ is negligible. Knowing the total numbers of events in the two event classes $N_{tot,i}$ one can resolve the system of equations \n\\begin{equation}\n\\left\\{\n\\begin{array}{l}\nN_{CR,4}\/\\alpha_{CR}+N_{\\gamma,4}\/\\alpha_{\\gamma}=N_{tot,3+4}\\\\\nN_{CR,4}+N_{\\gamma,4}=N_{tot,4}\n\\end{array}\n\\right.\n\\end{equation}\nwith respect to $N_{\\gamma,4}, N_{CR,4}$ to find the residual cosmic ray background in each energy bin  for the {\\tt dataclean} event class. \n\nThe residual cosmic ray fraction in the sub-class of the Pass 7 {\\tt superclean} events used in our analysis is then estimated in a similar way, once the residual cosmic ray fraction $\\kappa_4$ in the point-source-subtracted set of the Pass 6 events, $N_{CR,\\rm off,4}=\\kappa_{4}N_{\\rm off,4}$, is known.\n\nIndeed, the transition from the Pass 6 {\\tt dataclean} events to the Pass 7 events belonging to the event classes  65311 and 32543 leaves a fraction $\\alpha_{\\gamma,6\\rightarrow 7}$ of $\\gamma$-ray\\ events (actually, $\\alpha_{\\gamma, 6\\rightarrow 7}>1$ in a broad energy range around 10 GeV). It also suppresses or increases the residual cosmic ray background, so that the residual cosmic ray fraction in the off-source events changes from $\\kappa_4$ to $\\kappa_{7}$. The off-source events in the two classes are then the sum  of the diffuse $\\gamma$-ray\\ emission photons and of the residual cosmic rays:\n\\begin{equation}\n\\left\\{\n\\begin{array}{l}\n\\kappa_4 N_{\\rm off,4}+N_{\\gamma,\\rm off, 4}=N_{\\rm off,4}\\\\\n\\kappa_{7} N_{\\rm off, 7}+\\alpha_{\\gamma, 6\\rightarrow 7}N_{\\gamma,\\rm off, 4}=N_{\\rm off,7}\\\\\n\\end{array}\n\\right.\n\\end{equation}\nKnowing the statistics of the off-source events in the Pass 6 and Pass 7 events, $N_{\\rm off,4}$ and $N_{\\rm off,7}$, one could find the residual cosmic ray fraction in the Pass 7 data \n\\begin{equation}\n\\kappa_7=1-\\alpha_{\\gamma,6\\rightarrow 7}(1-\\kappa_6)\\frac{N_{\\rm off,4}}{N_{\\rm off,7}}\n\\end{equation}\nThe resulting estimates of the level of residual cosmic ray background for  the events selected in the Pass 7 data in energy bins between 3 and 100~GeV are shown by the grey data points in Fig. \\ref{fig:spectrum}. \n\nFrom Fig. \\ref{fig:spectrum} one could see that the contribution of the residual cosmic rays to the signal at 100~GeV is likely to be small. However, extrapolation of the estimate of efficiency of rejection of the residual cosmic ray background much above 100~GeV  is highly uncertain. It is possible that the efficiency of rejection of both the nuclear and electron\/positron component of the cosmic ray flux drops because of the similarity of the cosmic ray and $e^+e^-$ pair tracks with large Lorentz factors. Inefficient rejection of the residual cosmic rays might lead to the contaminate the diffuse background signal and lead to a large over-estimation of the diffuse background flux. Because of this problem, we are able to only derive an upper limit on the EGB at the energies much above 100~GeV (for the energy band at 100~GeV we show a comparison between the 95\\% confidence level upper limit and the measurement). A proper measurement of the EGB flux at the highest energies accessible to LAT would require extensive Monte-Carlo simulations taking into account detector response \\citep{ackerman_texas}.\n\n\n\\subsection{Extragalactic $\\gamma$-ray\\ background spectrum}\n\\label{sec:spectrum}\n\nEGB flux could be found by subtracting the point source, Galactic diffuse and residual cosmic ray contributions to the overall number of events at high Galactic latitudes in each energy bin. The  spectrum of EGB obtained in this way is shown by the red thick data points in Fig. \\ref{fig:spectrum}. The error in the measurement of the EGB flux at the energies below 100~GeV has contributions from the uncertainty of the level of the residual cosmic ray background as well as from the systematic uncertainties of the Instrument Response Functions (IRF) and of the Galactic diffuse background in the relevant sky region. An additional contribution to the error in the VHE band is given by  the statistical error arising from the low signal statistics. Finally, one more uncertainty stems from the uncertainty of the shape of the cosmic ray electron spectrum in the TeV energy range, which propagates to the uncertainty of extrapolation of the inverse Compton component of the Galactic diffuse emission above 100 GeV.\n\n\n\\begin{figure}\n\\includegraphics[height=\\columnwidth]{fig3}\n\\caption{Estimate of the flux of isotropic component of diffuse emission obtained by the direct photon counting method (red thick line, data points and upper limits). For comparison, the spectrum of isotropic component of diffuse sky emission, obtained using likelihood analysis at lower energies by \\citet{fermi_background}, is shown as a red shaded region. \nPink upper limits above 100~GeV are from  \\citet{ackerman_texas}. Black data points show the total point-source-subtracted flux from the North and South Galactic pole regions at $|b|\\ge 60^\\circ$. Solid line errorbars show  statistical error. Dashed line errorbar show the systematic error at the level of $\\simeq 20$\\% stemming from the uncertainty of the the Instrument Response Functions (IRF) of LAT (see {\\tt http:\/\/fermi.gsfc.nasa.gov\/ssc\/data\/analysis\/LAT\\_caveats.html}). Black horizontally shaded region shows the point source subtracted flux in the Galactic Pole regions found by \\citet{fermi_background}. Grey data points and grey curve show the estimate of the residual cosmic ray background in the event set used in this analysis. The residual cosmic ray background level in the data set considered by \\citet{fermi_background} is shown by the grey shaded region. Blue shading shows the Galactic diffuse emission in the North\/South Galactic \nPole regions $|b|>60^\\circ$ derived by \\citet{fermi_background}. Blue line shows the  Galactic diffuse background spectrum.}\n\\label{fig:spectrum}\n\\end{figure}\n\nIn the same figure we compare the measurement of EGB obtained from the direct photon counting at high Galactic latitudes with the results of the likelihood analysis of the all-sky data by \\citet{fermi_background}. The two measurements agree well. \n\nPink arrows at the energies above 100~GeV show the upper limit on the VHE EGB derived by \\citep{ackerman_texas} using the likelihood analysis of the all-sky data. These upper limits agree with the upper limits derived from the direct photon counting (shown by the red arrows in Fig. \\ref{fig:spectrum}. \n\nAs a matter of fact, the level of Galactic diffuse emission at high Galactic latitudes turns out to be comparable to the level of EGB in the entire energy range $E>10$~GeV.\\footnote{There is no a-priori reason why the two fluxes should be nearly equal. Thus, the equality of the two contributions poses a \"fine-tuning\" problem which requires further investigation. }\nThe Galactic diffuse emission contribution to the total flux at high Galactic latitude could not be negligibly small in the 100~GeV band. Taking this into account, it is not surprising that our estimate of the VHE EGB flux is  somewhat lower than the total diffuse emission flux at high Galactic latitudes and is, respectively, lower than the upper limit derived by \\citet{ackerman_texas}. \n\nIt is useful to note that extrapolating the EGB spectrum as a powerlaw spectrum to $E\\ge 100$~GeV band would give the spectrum consistent with the data above 100~GeV.  With the current LAT exposure, there is still no evidence for suppression of VHE EGB flux due to absorption on EBL. A larger exposure time is needed to verify the presence of the feature. As it is mentioned in the Introduction, suppression of the flux above 100~GeV due to the absorption of VHE $\\gamma$-ray s on the EBL is expected if the EGB is accumulated over the  cosmological distance scale. Detection of such suppression would be an important test of the origin of EGB.  \n\n\\section{$E>30$~GeV Extragalactic $\\gamma$-ray\\ background from point sources}\n\n As it is mentioned in the Introduction, different types of point and diffuse sources could contribute to the EGB in the VHE band. The main class of extragalactic point sources detected by {\\it Fermi} is blazars, which are divided onto two sub-classed: BL Lac type objects and Flat Spectrum Radio Quasars (FSRQ). Over the first year of operation LAT has detected some $\\sim 700$ such objects above 100~MeV energy \\citep{fermi_agn}. BL Lacs and FSRQ have somewhat different spectral characteristics in the $\\gamma$-ray\\ band, with the spectra of BL Lacs being systematically harder than the spectra of FSRQ \\citep{fermi_agn}. Hardness of the spectra of BL Lacs implies that they might produce significant contribution to the overall $\\gamma$-ray\\ flux in the VHE band. In fact, most of the extragalactic VHE $\\gamma$-ray\\ sources detected up to now by the ground based $\\gamma$-ray\\ telescopes sensitive above 100~GeV are BL Lacs\\footnote{For the catalogs of extragalactic VHE $\\gamma$-ray\\ sources see e.g. {\\tt http:\/\/tevcat.uchicago.edu} and {\\tt http:\/\/www.isdc.unige.ch\/vhe\/index.html}}. \n\nFig. \\ref{fig:sources} shows the breakdown of the point source contributions to the high Galactic latitude flux by the source type. One could clearly see that the dominant contribution is given by BL Lac objects which provide $\\ge 90\\%$ of the total point source flux above $30$~GeV. The cumulative spectrum of the other major blazar class, FSRQ has a high-energy cut-off at $\\sim 10$~GeV so that FSRQ contribution to the point source flux is negligible in the VHE band. From Fig. \\ref{fig:sources} one could see that the total point source flux calculated from the cumulative photon distribution around stacked point sources in the high Galactic latitude regions (see Section \\ref{sec:ps}) is in a good agreement with the total point source flux calculated using the likelihood analysis by \\citet{fermi_background}, shown by the green shaded region.\nAbove 50 GeV  $ 90\\%$  of source photons  come from BL Lacs and $10\\% $ from \"Other\" Fermi sources, which are dominated by not-identified sources with some contribution from nearby AGN's. \n\n\\begin{figure}\n\\includegraphics[height=\\columnwidth]{fig4}\n\\caption{Cumulative $\\gamma$-ray\\ flux from different classes of resolved  point sources. Source class is marked to the left from each curve. Green shaded area shows the point source flux at high Galactic latitudes found by \\citet{fermi_background}. Red curve shows the EGB spectrum from Fig. \\ref{fig:spectrum}, which includes only unresolved sources. }\n\\label{fig:sources}\n\\end{figure}\n\nThe EGB spectral shape above 30~GeV  follows the cumulative point source spectrum. This observation leads to a conjecture that the VHE EGB is produced by already known type of VHE $\\gamma$-ray\\ point sources with fluxes below the sensitivity of LAT. Since the dominant source class in the VHE band is that of BL Lacs, a more precise conjecture is that the VHE EGB is produced by the unresolved BL Lacs. \n\n\\section{BL Lac contribution to VHE EGB}\n\n\nIn the unification schemes of AGN BL Lac objects are identified with the Fanaroff-Riley type I  (FR I) radio galaxies with jets aligned with the line of sight \\citep{urry95}. This implies that the cosmological evolution of the BL Lacs should follow that of the FR I radio galaxies. Recent studies of the cosmological evolution of  FR I galaxies show that they experience \"positive\" cosmological evolution, which is usually described in terms of  luminosity or comoving source density evolution as the increase of either average source luminosity or the average comoving source density with the redshift $z$, as $(1+z)^k,\\ \\ k>0$. Different recent studies find somewhat different values of $k$, depending on the analyzed radio galaxy samples and different assumptions about the evolution type (luminosity or density), with $k$ ranging in $1\\lesssim k\\lesssim 3$ \\citep{sadler07,hodge09,smolcic09}. Since BL Lacs are just the FR I galaxies specially oriented with respect to the line of sight, their cosmological evolution follows the evolution of FR I galaxies, with the increasing source luminosity or spatial density with the redshift. This implies that significant flux should be produced by the sources at large redshifts, $z\\sim 1$. It is possible that most of  individual sources at large redshifts are too weak to be significantly detected by LAT, but collective emission from all the set of BL Lacs at high redshifts  gives a significant contribution to the EGB.\n\n\\begin{figure}\n\\includegraphics[angle=-90,width=\\columnwidth]{fig5}\n\\caption{Number of detected VHE photons as a function of redshift in the 6.25-12.5~GeV (blue dotted histogram) 25-50~GeV (green dashed histogram) and 100-200~GeV (red solid histogram) bands. Also shown are the distributions of photons with the redshift expected for different laws of BL Lac cosmological evolution. }\n\\label{fig:dNz}\n\\end{figure}\n\nDependence of the total flux of the BL Lac population on the redshift could be found from the following straightforward calculation. \nLet us consider the total flux produced by sources at redshift $z$ in a redshift interval $\\Delta z$. This redshift interval corresponds to the comoving distance interval $\\Delta r=\\Delta z\/H(z)$ where $H(z)\\sim\\sqrt{\\Omega_\\Lambda+\\Omega_m(1+z)^3}$ is the expansion rate of the Universe filled with matter and cosmological constant with today's densities $\\Omega_m$ and $\\Omega_\\Lambda$. \n\nAs an example, we take the case of \"pure luminosity\" evolution with the average source luminosity increasing as $(1+z)^k$ and conserved comoving source density $n(z)=n_0=const$. The number of sources in a spherical layer of thickness $\\Delta z$ is $\\Delta N_s=4\\pi n_0 r^2\\Delta r$. Each source produces the flux in a given energy band  $F\\sim (1+z)^{k-\\Gamma}\/(4\\pi r^2)$, where $\\Gamma$ is the photon index, the factor $(1+z)^{1-\\Gamma}$ describes the change in the number of photons in a given energy band due to the cosmological redshift of the photon energies. One power of $(1+z)$ is compensated by the time delay between subsequent photons. \n\nThe flux from the sources at large redshifts is affected by absorption of VHE photons on EBL. For example, at $z\\simeq 1.5$ the absorption modifies source spectrum above the energy $E\\simeq 50$ GeV, if one assumes the EBL evolution calculated by \\citet{franceschini08}. Absorption on EBL leads to suppression of the flux by a factor $\\exp\\left(-\\tau(E,z)\\right)$ where \n $\\tau(E,z)$ is the optical depth with respect to the pair production.\n \nThe overall flux from the sources in the redshift interval $\\Delta z$ is \n\\begin{equation}\n\\frac{\\Delta F(E,z)}{\\Delta z}=F\\Delta N_s\\sim \\frac{(1+z)^{k-\\Gamma}e^{-\\tau(E,z)}}{\\sqrt{\\Omega_\\Lambda+\\Omega_m(1+z)^3}}\n\\label{dfdz}\n\\end{equation}\n\nFig. \\ref{fig:dNz} shows the number of $\\gamma$-ray s as a function of source redshift. For this we used BL Lacs with known redshifts  from Veron\\&Veron \\citep{veron13} catalog complemented by BL Lacs detected by LAT, but not listed in the Veron\\&Veron catalog. Only sources with $|b|>10^\\circ$ were considered. \nHere we plot photon distributions from Fermi BL Lacs  in the  three energy bands: $6.25-12.5$ GeV, $25-50$ GeV and $100-200$ GeV.   One can see that at lower energies $E<50$ GeV a  significant flux is produced by BL Lacs at large redshifts  up to  $z=1.5$. At the highest energies only contribution from nearby sources at $z<0.7$ is present. Two effects might explain the deficit of high-redshift sources at high energies. First, the flux at the highest energies is suppressed by absorption on EBL. Next, the photon statistics in the highest energy bin is low so that  sources contributing to the flux in the 6.25-12.5~GeV bin produce less than one photon in the 100-200~GeV bin. \n\nIn the same figure we also show the expected dependence of the number of photons on the redshift expected in different  evolution models,  Eq.~(\\ref{dfdz}). The models for cases $k=1,2,3$ are shown with magenta lines for $6.25-12.5$ GeV energy band. We normalize the models to the number of photons in the first redshift bin, in which we have the most complete knowledge of the BL Lac population. \n\nFrom the comparison of the evolution models with the data one might get an impression that $(1+z)$ model is more consistent with the data than the models assuming faster evolution. However, the histogram on Fig. \\ref{fig:dNz} does not take into account photons from BL Lacs with unknown redshifts. These BL Lacs produce  about 30 \\% of all cumulative BL Lac flux. This means that at least 30\\% contribution to the overall flux (integrated over all redshifts) is missing in  Fig. \\ref{fig:dNz}.  \nThe model with evolution $k=1$ predicts the total number of photons which is $\\sim 3\\sigma$  below the total number of photons in BL Lacs with known and unknown redshift together in the $6.25-12.5$ GeV energy band. In the energy band 3.125-6.25~GeV the under-prediction of the total number of photons from BL Lacs in the  $k=1$ model  is at  $\\ge 5\\sigma$  level, which means that the model is efficiently ruled out. Thus, Fermi LAT observations of BL Lac objects indicate that BL Lacs have positive cosmological evolution with $k>1$. \n   \n\nIn all other cases $k>1$, the discrepancy between the observed and expected number of photons from BL Lacs starts already at small redshifts, $z\\ge 0.2$. The \"missing BL Lac\" $\\gamma$-ray s could come either form BL Lacs with unknown redshifts or from {\\it Fermi} sources which are not yet identified as BL Lacs or, finally, from BL Lacs with fluxes below the sensitivity of LAT. \n\n\n\\begin{figure}\n\\includegraphics[height=\\columnwidth]{fig6}\n\\caption{VHE EGB produced by unresolved BL Lacs under different assumptions about the cosmological evolution of BL Lac population (the evolution law is marked to the left of each curve). Red data points show the EGB spectrum from Fig. \\ref{fig:spectrum}.}\n\\label{fig:evolution}\n\\end{figure}\n\nAlthough individual high redshift BL Lacs would not be detectable by LAT, cumulative flux of these BL Lacs could give significant contribution to the EGB. Fig. \\ref{fig:evolution} shows the contributions from \"missing BL Lacs\" at high redshifts up to $z=1$ expected in four different models of cosmological evolution of BL Lac \/ FR I population. To calculate this contribution, we have normalized $\\Delta F\/\\Delta z$ distribution on the measured flux of BL Lacs in the redshift bin $0<z<0.1$. For the two last bins,100-200~GeV and  200-400~GeV, the statistics of the BL Lac signal is too low to normalize $\\Delta F\/\\Delta z$ by the flux in the first redshift bin. To estimate the normalization of $\\Delta F\/\\Delta z$ in these energy bins we have assumed that the average BL Lac spectrum extends as a powerlaw with photon index $-2$ up to the 400~GeV energy range. The resulting statistics of the signal in the last 200-400 bin in Fig. \\ref{fig:evolution} is low and is subject to large fluctuations.\n\nFrom Fig. \\ref{fig:evolution} one could see that the conjecture that EGB is produced by collective emission from distant BL Lacs is valid if BL Lacs experience positive luminosity or density evolution of the form  $(1+z)^k$ with $k\\simeq 3$. Slower or faster cosmological evolution with $k=2$ or $k=4$, respectively, would under- or over-produce the EGB.  Since evolution $k = 4$ predicts number of photons more then\n diffuse gamma-ray background (see Fig. \\ref{fig:evolution})  it is also excluded. Thus, the LAT data impose a constraint on the cosmological evolution of the BL Lac\/ FR I population \n \\begin{equation}\n 1< k < 4\n \\end{equation} \nFrom Fig. \\ref{fig:evolution} one could also see that if $k=3$, all the flux of EGB above 30~GeV could be explained by a cumulative emission from the BL Lac population. At the same time, if $k$ is significantly smaller than 3, as it is suggested by some recent studies of the evolution of the parent population of FR I galaxies \\citep{smolcic09}, significant part of the VHE EGB flux should come from a yet unknown source population or have a truly diffuse nature.  \n\n$\\gamma$-ray\\ flux from high-redshift BL Lacs is modified in the VHE band by the effect of absorption on EBL. The model spectra shown in Fig. \\ref{fig:evolution} take this effect into account. We use the model of \\citet{franceschini08} to estimate the attenuation of the VHE $\\gamma$-ray\\ flux from redshifts up to $z=1.5$. Model curves shown in Fig. \\ref{fig:evolution} assume that the average intrinsic spectrum of BL Lacs does not have a high-energy cut-off up to $\\sim 400$~GeV. The observed suppression of the flux above 100~GeV is explained only by the effect of absorption on EBL.\n\n\n\n\\section{Discussion and Conclusions}\n\nIn this paper we have derived a measurement of  EGB in the 10-400~GeV energy range from the analysis of {\\it Fermi}\/LAT data in the North and South Galactic Poles regions, $|b|>60^\\circ$. Our approach was to count all the protons  detected by LAT in this region and estimate the number of counts from the point sources, from the Galactic diffuse emission, form the residual cosmic ray background and from the EGB. Subtracting the source, Galactic diffuse and residual cosmic ray background counts from the total number of counts in the North and South Galactic Pole regions we derived the EGB spectrum shown in Fig. \\ref{fig:spectrum}.\n\nComparing the spectrum of EGB in the $>30$~GeV energy band with the  spectrum of extragalactic point sources in the same energy band (Fig. \\ref{fig:sources}), we have noticed that the two spectra closely follow each other. Based on this observation, we have put forward a conjecture that the EGB above 30~GeV is explained by the unresolved BL Lacs, which give dominant contribution to the extragalactic point source flux in this energy band. We have demonstrated that this conjecture is consistent with the EGB measurement provided that BL Lacs follow positive cosmological evolution with the overall power of emission from the source population increasing as  $(1+z)^3$ up to $z\\sim 1$ (Fig. \\ref{fig:evolution}). \nSuch cosmological evolution is roughly consistent with the measurements of cosmological evolution of FR I radio galaxies which are believed to be the parent population of BL Lac type objects and are also observed to have positive cosmological evolution of the form $(1+z)^k$ with an uncertain value of $k$ between 1 and 3  \\citep{rigby08,smolcic09,sadler07}. At the same time, it is opposite to the negative cosmological evolution of the high-energy-peaked BL Lacs \\citep{giommi99,giommi01}, which constitue a sub-class of the GeV-TeV $\\gamma$-ray\\ emitting BL Lacs considered in our analysis. \n\nIf the VHE EGB is indeed produced by distant BL Lacs at redshifts up to $z\\sim 1$, LAT will not be able to resolve it into point sources. Indeed, the brightest BL Lac on the sky, Mrk 421 produced $\\simeq 30$ photons above 100~GeV. If the positive cosmological evolution of BL Lac population is mostly due to the increase of the comoving source density rather than increase of the typical source luminosity, the brightest BL Lacs at redshift $z\\sim 1$ produce $\\sim 10^{-2}$ $\\gamma$-ray s in LAT over some 2.5 years of exposure. This means that LAT would not collect sufficient photon statistics to detect distant BL Lacs individually. If the $(1+z)^3$ evolution is mostly due to the increase of the average source luminosity, BL Lacs at redshift 1 have an order-of-magnitude higher luminosity than local BL Lacs. However, even with higher luminosity, they produce  on average 0.1 photon in LAT, so that they are still not individually detectable.\n\nIf the real value of $k$ is much below $k=3$, as indicated by a recent study by \\citet{smolcic09}, emission from the unresolved BL Lacs would not explain the VHE EGB flux and there should be another source class or a mechanism of production of diffuse emission which would account for the EGB. Such a mechanism might, in fact, be indirectly related to the BL Lac population. Most of the power output from BL Lacs at the energies above 100~GeV is converted into the electromagnetic emission from $\\gamma$-ray\\ induced cascade in intergalactic medium.  The intrinsic spectra of BL Lacs (and of FR I radio galaxies, such as M87 and Cen A \\citep{m87,cena}) are known extend up to  $\\sim 10$~TeV. Typical energy of the cascade photons which are produced via inverse Compton scattering of CMB photons by the $e^+e^-$ pairs deposited in the intergalactic medium is $E_\\gamma\\simeq 100\\left[E_{\\gamma_0}\/10\\mbox{ TeV}\\right]^2\\mbox{ GeV}$ \\citep{neronov09}. If the intrinsic source luminosities in the 1-10~TeV range are comparable to the luminosities in the 10-100~GeV, total flux of the cascade emission in the 10-100~GeV band is expected to be comparable to the point source flux, so that the cascade emission could give significant contribution to the EGB \\citep{coppi97}. This is consistent with the observation that the VHE EGB level is comparable to the cumulative extragalactic  point source flux in the 10-100~GeV band observed by LAT. \n\n\nThe only possibility to test the hypothesis of BL Lac origin of EGB would be to use deep observations with ground-based $\\gamma$-ray\\ telescopes. Ground based $\\gamma$-ray\\ telescopes, which are sensitive in the VHE energy band, have much larger collection area above several hundreds of GeV and, as a consequence, could detect much weaker sources, than LAT. The flux from Mrk 421-like BL Lacs at the redshift $z\\simeq 1$ is $\\sim 10^{-13}$~erg\/cm$^2$s. If the cosmological evolution of BL Lacs is due to increase of the average source luminosity with redshift, brightest BL Lacs at redshift $z\\sim 1$ might produce fluxes up to $10^{-12}$~erg\/cm$^2$s at 100~GeV. At the energies around $E\\lesssim 100$~GeV this flux is not strongly attenuated by the absorption on EBL. \nThe energy $E\\simeq 100$~GeV is around or below the low energy threshold of the  current generation Cherenkov telescopes, like HESS, MAGIC and VERITAS. However, next generation facilities, Cherenkov Telescope Array (CTA) \\citep{cta} or 5@5  \\citep{5at5} are expected to have an energy threshold significantly below 100~GeV. Their sensitivity could be sufficient to  resolve the VHE EGB into point sources, at least in the case when the cosmological evolution is mostly luminosity, rather than source density evolution.\n\n \\section*{Acknowledgements}\n \n We would like to thank I.Moskalenko, for the discussion of the issues related to the Galactic $\\gamma$-ray\\ background, and M.Ackermann for the clarification of the uncertainties of the residual cosmic ray background. The work of AN is supported by the Swiss National Science Foundation grant PP00P2\\_123426.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}