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Probing the Scale of ALP Interactions with Fermi Blazars

Gamma-rays from cosmological sources contain information about gamma-ray interactions. Standard model and non-standard model photon interactions along the path between the source and the observer can lead to changes in the energy or state of the photons, which in turn alters the observed energy spectrum of the source. In general, these interactions are a function of photon energy as well as source distance. Here we show how existing high energy (50 GeV -- 10TeV) gamma-ray observations of blazars can be used to constrain the coupling of axion-like-particles (ALPs) to the photon. The same ALP-photon coupling that has been invoked to explain the observations of TeV blazars beyond their pair production horizon is shown to have an effect of the data set of \textit{Fermi} blazars.


Introduction
Cosmological gamma-rays are an ideal probe of relatively rare photon interactions. In addition to expected standard model interactions, like those with the extragalactic background light (EBL) through pair production, any additional interaction which alters the energy or state of the photon can in principle be probed. The process of constraining gamma-ray interactions beyond the standard model depends on knowledge of the source gamma-ray spectra, the quality of rapidly growing data sets of high energy gamma-ray observations, and a sufficiently well understood modeling of the EBL. As a test case, we consider a group of gamma-ray sources at approximately the same distance, using the high redhift (z ∼ 1) portion of the F ermi blazar catalogue [1] as an example. In addition, we discuss a similar set of TeV blazars clustered at z ∼ 0.1 [2,3] . We expect blazars at the same distance to have their spectra be attenuated in a similar way by the EBL (for a recent review see [4]). If we assume a well-determined model for the EBL, then a comparison of the observed spectra to the spectra at the source can probe new photon couplings. This requires some assumptions about the intrinsic spectra, namely that they are a power law. In practice, we then use the residuals between the observed and expected EBL attenuation as an example of the range of constraints that can be placed on rare photon interactions. Such rare photon couplings have recently been proposed to explain the unexpected observation of TeV gammarays from sources beyond their EBL attenuation horizon [5][6][7][8][9][10][11][12], leading to general questions about novel couplings of cosmological photons, which we show could be probed by the Fermi blazars.
Specifically, we examine the mixing of the photon with a pseudoscalar axion-like particle (ALP). ALPs couple to photons via a two-photon vertex, just as the axion [13] does. Unlike the axion, which has a definite relationship between this coupling and its mass, the ALP's mass is independent of coupling [14,15]. In this paper we explore the effect of ALPS on the propagation of cosmological gamma-rays. In Sec.3 we will show how ALP-photon oscillations manifest in the observed energy spectra, in Sec.4 we discuss the data sets and proposed analysis technique, and finally we conclude with our results in Sec.5. We find that, under our assumptions, Fermi sources can exclude couplings above 1.7 × 10 −11 GeV −1 for both oscillation models considered, and above 0.6× 10 −11 GeV −1 for the extra attenuation scenario, only. These limits hold true for ALP masses below 1.4× 10 −4 µeV and 8.4×10 −5 µeV, respectively.

ALPs
ALPs, unlike axions, have a mass and coupling constant that are independent. A general property of pseudoscalar particles is their connection to the electromagnetic field via a twophoton vertex [16][17][18][19]. Because of this coupling, ALPs and photons can interconvert in the presence of a magnetic field, for example in earth based experiments or, as considered here, in the intergalactic magnetic field (IGMF). Earth-based exmperiments set constraints on the ALP/axion mass and coupling constant. These include CAST [20,21] which searches for solar axions, ADMX [22] which searches for axion dark matter, and laser regeneration experiments [23][24][25].
We are concerned with the important feature that to lowest order axions (or more generally, ALPs) can oscillate into photons through the interaction where g is the photon-axion coupling strength, a is the axion field, F is the electromagnetic field-strength tensor,F its dual, and B and E are the magnetic and electric field, respectively. This is a three state system: ALP, coupled photon, and decoupled photon. In order for the mixing to occur, there needs to be a coherent magnetic field; the coupled photon parallel to this magnetic field can mix with the ALP state via propogation eigenstates [26]. Driven by this interaction, photons and ALPs oscillate in and around astrophysical sources with strong magnetic fields as well as in the IGMF and the magnetic field of our own galaxy [27]. The IGMF strength is in the range 0.1-1.0 nG [28,29]. For a constant magnetic field, the oscillation probabilitiy is given by: where d is the distance traveled and Here, the subindices indicate dimensionless quantities: g 11 ≡ g/10 −11 GeV −1 , m µeV ≡ m/µeV , and B G ≡ B/Gauss. The effective ALP mass is defined as: m 2 ≡| m 2 a − w 2 pl | and the plasma frequency is: w pl = 0.37 × 10 −4 µeV n e /cm −3 and n e ∼ 10 −7 cm −3 . The critical energy is the energy above which oscillation mixing occurs. For our analysis, we consider parameters so that E crit is at or below the typical energy of the data set (e.g., 30 GeV for Fermi and 100 GeV for a very high energy blazar data set). 1 The probability for oscillations is driven by the product B × d. For oscillations around a source there needs to be a relatively high magnetic field given the relatively short distance scales; this is in contrast to oscillations occurring in the IGMF, which is a much weaker field, but is coherent over much longer distances. In general, ALP-photon couplings can be probed in a variety of astrophysical and/or cosmological environments provided B × d is of sufficient magnitude and this quantity is related to the energy of accelerated particles associated with various environments: E max ≃ 9.3 × 10 20 eV B G d pc , for more see [30].
For intergalactic propagation we consider the more physical case, where the IGMF is not constant but consists of a large number of coherent domains. In this case the ALP-photon oscillations reduce to semi-classical decay [26]. After traversing many magnetic domains, the system is balanced between the ALP and the two photon states and the conversion probability is given by where the decay length is defined as in the limit µL dom << 1 for µ = B × g [26]. The magnetic field domain length, L dom , is taken to be 1 Mpc. For B ∼ 1nG, the decay limit approximation corresponds to g 11 < 33. If we consider critical energies at or below E crit = 30 GeV, as we will in this paper, then we are also limited to m µeV < 6.3 × 10 −4 . We can also write the probability for the double conversion process γ → a → γ The probability for a → γ (the second part of Eq.2.6) is twice that for γ → a since two photon states are available [27]. Integrating over conversion locations accounts for all relative distance possibilities and the integrand is maximized for d 1 ∼ d 2 .
All possible oscillation scenarios are shown schematically in Fig.1. Gamma-ray sources at or above E crit will experience alterations to their energy spectra, an attenuation or enhancement, due to ALP-photon interconversion. The probability for a single (double) conversion plateaus at 1/3 (2/9), as evident by Eq.2.4 (Eq.2.6). This is shown in Fig.2 where we have plotted these probabilities as a function of source distance using a set of canonical parameters, chosen so as to distort the spectrum within typical allowed limits considered later in this paper. For scale, we focus on sources at z ∼ 1 (d ∼ 5000 Mpc), a distance at which both scenarios are possible, though a single conversion is more probable. This choice of coupling would have a nearly maximal effect in the Fermi data set (z ∼ 1) and a negligible effect on TeV blazars (z ∼ 0.1). These two redshifts also correspond to the apporximate pair production horizons of the two data sets (5000 Mpc for Fermi observed blazars and 500 Mpc for TeV blazars) and therfore the largest L decay 's (smallest couplings) possible.
To determine the approximate value of ALP coupling that will impact a given set of blazars, we set L source ∼ L decay . Using the canonical values for the domain length and B field (L dom = 1 Mpc, B = 1 nG), we get a relationship between the decay length and the coupling strength: or g 2 11 ∼ 1(10) for ALP effects in the Fermi (TeV) data. Beyond this rough constraint, ALP parameters are chosen so that the range of interconversion probabilities give rise to acceptable signatures (from expectation) observed in blazar energy spectra.

Blazar Spectra
We know that gamma-rays coming from distant blazars interact with the EBL, creating electron-positron pairs, and attenuating the energy spectrum [31,32]. This attenuation corresponds to an optical depth τ = τ (E, z) such that F obs = F s e −τ , where F s is the energy spectrum at the source. If the EBL and source spectra were known precisely, then the residual between the predicted and observed spectra could be used to constrain other photon interactions beyond the standard model. In practice, neither the EBL nor the source spectra are known to a high enough level of accuracy to meaningfully constrain new physics. Neverthe-less, we can pick a well-fit EBL model using all existing data (see [31,32]) and assign the errors in that fit to potential new physics in order to illustrate the possible constraints achievable. Here we will assume the theoretical EBL model given by [33] to be correct, and allow the error in the best-fit τ to be due to ALP physics. This allows us to place conservative constraints on ALP-photon coupling and mass.
We rewrite the spectrum as: where φ ALP s = e ±△τ . The plus sign indicates there was less attenuation than predicted by the model and thus gamma-rays are enhanced. Conversely, the negative sign indicates there was more attenuation than predicted by the model and thus extra extra attenuation.
Considering the enhanced gamma-rays we can write: It follows that the enhanced spectrum can be constrained by: Similarly, if we consider the extra attenuation gamma-rays we can write: And collectively the extra attenuation spectrum can be constrained by : Maximal constraints can be put on ALPs when L decay ∼ d source . Hence, we are only considering a minimal number of interconversions as the data we examine is not of high enough quality to probe less probable, smaller effect processes. The orange lines in Fig.3 show lines of constant decay length, chosen to bracket the Fermi blazars and TeV blazar data sets. For example, any gamma-ray above 30 GeV that originates in a source at z ∼ 0.1 would have significant conversion to an ALP for g 11 ≥ 2 GeV −1 . 2

Calculations
At present, attenuation due to the EBL is not precisely known at any energy. What is known is that a suite of reasonable models provide a good fit to the EBL opacity for photons in the energy range 50 GeV -10 TeV [32]. The error in the EBL model fit is driven primarily by the uncertainty in the intrinsic spectrum produced by blazars. This can be seen by comparing the 1 and 2 σ regions of the optical depth given in [31] to [32]. Never-the-less we can illustrate the expected range of applicable ALP physics by setting the absorption/amplification due to ALP-photon interconversion to be no larger than the error in the best fit optical depth. That is to say, EBL models provide a fairly accurate description of the absorption observed in the current sample of gamma-ray blazars. What ALP physics there is should not be so large as to spoil this agreement.  In order to constrain ALP physics from a given source, the critical energy needs to be at or below the range of gamma-ray energies produced by the source. Otherwise, no oscillations occur for some or all of the photons emitted by the source. In addition, the characteristic decay length must be chosen to accurately describe the data when the best fit EBL model is taken into account using the observational range in τ (Eq. 3.3 and Eq. 3.4 ) . This leads to a constraint on the coupling strength which in turn constrains the mass, for a given E crit . In Fig.3, the red (blue) slanted region corresponds to the allowed paramter space for E crit = 30 GeV (E crit = 10 TeV). The width corresponds to the allowed range in B field. Lines of constant L decay correspond to horizontal lines in this parameter space (Eq. 2.7) and serve to identify the region of maximal ALP conversion for a given source distance.

Results and Discussion
We have shown how to constrain the coupling of ALPS to high energy photons if we assume that the residual between the observed optical depth and a best fit model (Fermi collaboration [31]), can't be exceeded by the ALP-photon interconversion associated with this coupling. In Fig. 4 we illustrate the results of this method. As in Fig. 3, the slanted region corresponds to the paramter space such that E crit = 30 GeV and the spread is for a range of IGMF B field values. The actual data constraint calculation was done for B = 1 nG, causing the maximal allowed ALPS coupling to lie on the right side of the 30 GeV band. The two constraint points shown correspond to the exact limit from the Fermi data assuming a critical energy of 30 GeV. Constant L decay contours (Eq. 2.7) correspond to horizontal lines in this parameter space. Moving to lower m µeV corresponds to lower values of E crit , which would still allow all the photons from the source to oscillate. Therefore points to the left of the shown constraints result in the same ALP distortion. Similarly, points below the data  Figure 4: ALP photon coupling and ALP mass as constrained by HE blazar observations made by the Fermi collaboration. CAST line from [21].
constraints and along the lines of constant E crit correspond to longer decay lengths (lower probabilities for conversion) and are therefore allowed. The green shaded region is excluded by this data, under our assumptions. The red shaded region corresponds to points that are acceptable for enhancement, but unacceptable for extra attenuation. Our constraint regions continue up to g 11 = 33, into the already excluded region from CAST. 3 Historically, attempts at constraining ALPs from blazars have focused on examining the spectrum of an individual source and attributing a break in the assumed source power-law to ALPs. These analyses run Monte Carlo simulations over ALP parameter space, including a parameterization of magnetic field domains, among other quantities [5,7]. Our method instead illustrates the range of constraints possible if we integrate over the spectral variations of many sources, at approximately the same distance, at the same time. This is completely analogous to the EBL measurement made by Fermi and hence why we can use their result. In a perfect world, improving the data set for TeV blazars at z ∼ 0.1 could be used in concert with the Fermi blazars to better probe slightly larger ALP masses at roughly the same coupling, in particular in the region where both experiments are measuring the same EBL [32].
In [5], ALP mixing in both the source and the IGMF was considered for 3C 279 (z=0.536) and PKS 2155-304 (z=0.117). Their fiducial model used the values g 11 = 8.77 and m µeV = 10 −4 . ALP-photon mixing can also be used to reduce the photon opacity within the source itself -in a recent paper [7] the possibility of ALP-photon conversion in the source PKS 1222+216 (z=0.432) was explored as a possible explanation of reducing the source opacity. They considered the presence of a strong B field in/around the source that allowed for γ → a conversion such that the gamma-ray could escape the broad line region of the blazar. Once escaped, the ALP could convert back a → γ in either the host galaxy, or in the IGMF. In order to explain both the HE and VHE observations (from Fermi and MAGIC) using a standard blazar model, the authors needed to invoke ALPs with an inverse coupling g 11 =1.4 in a (source) magnetic field B G = 0.2.
That is to say, if ALPs are required to extend the TeV horizon and/or reduce the source opacity of several 100 GeV photons in blazars, we would expect correlated ALP effects, at a scale demonstrated in this paper, in the set of Fermi blazars clustered near redshift 1.