Hints of the existence of Axion-Like-Particles from the gamma-ray spectra of cosmological sources
M. A. Sanchez-Conde, D. Paneque, E. Bloom, F. Prada, A. Dominguez
aa r X i v : . [ a s t r o - ph . C O ] A ug Hints of the existence of Axion-Like-Particles from the gamma-ray spectraof cosmological sources
M. A. S´anchez-Conde , , D. Paneque , E. Bloom , F. Prada , and A. Dom´ınguez , Instituto de Astrof´ısica de Andaluc´ıa (CSIC), E-18008, Granada, Spain ∗ Kavli Institute for Particle Astrophysics and Cosmology (KIPAC),SLAC National Accelerator Center, Sand Hill Road 2575, CA 94025, USA † Visiting student at SLAC National Accelerator Center Departamento de F´ısica At´omica, Molecular y Nuclear,Universidad de Sevilla, E-41012, Sevilla, Spain and Visiting research physicist at the Santa Cruz Institute for Particle Physics (SCIPP),University of California, Santa Cruz CA 95064, USA (Dated: October 24, 2018)Axion Like Particles (ALPs) are predicted to couple with photons in the presence of magneticfields. This effect may lead to a significant change in the observed spectra of gamma-ray sourcessuch as AGNs. Here we carry out a detailed study that for the first time simultaneously considers inthe same framework both the photon/axion mixing that takes place in the gamma-ray source andthat one expected to occur in the intergalactic magnetic fields. An efficient photon/axion mixing inthe source always means an attenuation in the photon flux, whereas the mixing in the intergalacticmedium may result in a decrement and/or enhancement of the photon flux, depending on thedistance of the source and the energy considered. Interestingly, we find that decreasing the value ofthe intergalactic magnetic field strength, which decreases the probability for photon/axion mixing,could result in an increase of the expected photon flux at Earth if the source is far enough. We alsofind a 30% attenuation in the intensity spectrum of distant sources, which occurs at an energy thatonly depends on the properties of the ALPs and the intensity of the intergalactic magnetic field,and thus independent of the AGN source being observed. Moreover, we show that this mechanismcan easily explain recent puzzles in the spectra of distant gamma-ray sources, like the possibledetection of TeV photons from 3C 66A (a source located at z=0.444) by MAGIC and VERITAS,which should not happen according to conventional models of photon propagation over cosmologicaldistances. Another puzzle is the recent published lower limit to the EBL intensity at 3.6 µ m (whichis almost twice larger as the previous one), which implies very hard spectra for some detected TeVgamma-ray sources located at z=0.1-0.2. The consequences that come from this work are testablewith the current generation of gamma-ray instruments, namely Fermi (formerly known as GLAST)and imaging atmospheric Cherenkov telescopes like CANGAROO, HESS, MAGIC and VERITAS. PACS numbers: 95.35.+d; 95.55.Ka; 95.85.Pw; 98.70.Vc; 98.70.Rz; 14.80.Mz
I. INTRODUCTION
The existence of axions is predicted by the Peccei-Quinn mechanism, which is currently the most com-pelling explanation to solve the CP problem in QCD[1]. Moreover, amongst all the valid candidates pro-posed to constitute a portion or the totality of the non-barionic cold dark matter content predicted to exist inthe Universe, hypothetical non-thermal axions, or in amore generic way, Axion-Like Particles (ALPs), wherethe mass and the coupling constant are not related toeach other, may represent a good option: they might ex-ist in sufficient quantities to account for the estimateddark matter density and they might interact very weaklywith the rest of the particles [2]. There is an additionalproperty of ALPs that makes them even more attractiveand that could have important implications for its de- ∗ Electronic address: [email protected] † Electronic address: [email protected] tectability, i.e. they can oscillate into photons and vice-versa in the presence of an electric or magnetic field [3, 4].This is analogous to that predicted to occur between neu-trinos of different flavors, and a similar behavior is ex-pected in the case of the recently proposed chameleonsas well [5]. This characteristic is the main vehicle usedat present to carry out an exhaustive search of ALPs byexperiments like CAST [6], PVLAS [7] and ADMX [8].The oscillation of photons to ALPs (and vice-versa)could have important implications for astronomical ob-servations. This argument was first investigated in theoptical band by Ref. [9], where authors proposed the ex-istence of axions to be the cause of the observed super-nova Ia dimming. In this context, the observed dimmingmight be explained as a result of an efficient photon toaxion conversion instead of a cosmic acceleration (albeitthis proposal was rejected some time later due to somechromatic problems, pointed out e.g. in Ref. [10]). Pho-ton/axion oscillations were also studied by the same au-thors in Ref. [11] as an alternative explanation for thosephotons arriving Earth from very distant sources at en-ergies above the GZK cutoff.Recently, it has been proposed that, if ALPs exist, theycould distort the spectra of gamma-ray sources, such asActive Galactic Nuclei (AGNs) [12, 13, 14, 15] or galacticsources in the TeV range [22], and that their effect may bedetected by current gamma-ray experiments. In [23], forexample, it is stated that also the scatter in AGN lumi-nosity relations could be used to search for ALPs. Otherastrophysical environments have been proposed in orderto detect ALPs, such as the magnetic field of the Sun [24],pulsars [25], the galactic halo [26] or GRBs and QSOs bycarefully studying their polarized gamma-ray emissions[27, 28]. In particular, these predictions are very relevantfor gamma-ray astronomy, where recent instrumentationdevelopments in the last few years have increased the ob-servational capabilities by more than one order of mag-nitude. On the ground, we have the new generation ofImaging Atmospheric Cherenkov Telescopes (IACTs) likeMAGIC [29], HESS [30], VERITAS [31] or CANGAROO-III [32], covering energies in the range 0.1-20 TeV. Inspace we have Fermi (previously called GLAST) [33], inoperation since Summer 2008 and covering energies inthe range 0.02-300 GeV[74].In this work we revisit the photon/axion mixing, forthe first time handling under the same consistent frame-work the mixing that takes place inside or near thegamma-ray sources together with that one expected tooccur in intergalactic magnetic field (IGMF). In the liter-ature, both effects have been considered separately. De-pending on the source dimension, magnetic field, ALPmass and coupling constant, both effects might producesignificant spectral distortions, or one effect could bemore important than the other. In any case, we believethat both effects could be relevant and hence need tobe considered simultaneously. We neglect, however, themixing that may happen inside the Milky Way due togalactic magnetic fields. At present, a concise modelingof this effect is still very dependent on the largely un-known morphology of the magnetic field in the galaxy.Furthermore, in the most idealistic/optimistic case, thiseffect would produce an enhancement of the photon fluxarriving at Earth of about 3% of the initial photon fluxemitted by the source [15]. This is in contrast with whatwe found for the IGMFs: although there is also little in-formation on the strength and morphology of the IGMFs,the derived photon/axion mixing in this case we show tobe crucial for a correct interpretation of the observed flux.It is worth mentioning that we will come to this conclu-sion using a conservative value of B=0.1 nG for the IGMFstrength, well below the current upper limits of ∼ e − e + pair production thatcomes from the interaction of the gamma-ray source pho-tons with infrared and optical-UV background photons[35]. Amongst all the EBL models that exist in the lit-erature, in this work we will make use of the Primack[36] and Kneiske best-fit [37] EBL models. They repre-sent respectively one of the most transparent and one ofthe most opaque models for gamma-rays, but still withinthe limits imposed by the observations. The EBL modelwill play a crucial role in our formalism and results: as wewill see, the more attenuating the EBL model considered,the more relevant the effect of photon/axion oscillationsin the IGMF.We also explore in this work the detection prospects forcurrent gamma-ray instruments (Fermi and IACTs). Wewill show that the signatures of photon/axion oscillationsmay be observationally detectable provided light ALPswith masses smaller than a given value for typical valuesof the IGMF. In order to study the detection prospects,we will propose an observational strategy. We can an-ticipate here that the main challenge for our proposedformalism to be testable comes from the lack of knowl-edge of the intrinsic source spectrum and EBL density.However, we note that there is the possibility that wecould be already detecting the first hints of axions withcurrent experiments. In this context, the potential detec-tion of TeV photons from very distant (z ∼ II. THE FORMALISM
At present, the Peccei-Quinn mechanism remains asthe most convincing solution to solve the CP violation ofQCD. As early as in 1978, Weinberg [43] and Wilczek[44] realized independently that a consequence of thismechanism is the existence of a pseudo-scalar boson, theaxion. One generic property of axions is a two-photoninteraction of the form:
FIG. 1: Sketch of the formalism used in this work, where both mixing inside the source and mixing in the IGMF are consideredunder the same consistent framework. Photon to axion oscillations (or vice-versa) are represented by a crooked line, while thesymbols γ and a mean gamma-ray photons and axions respectively. This diagram collects the main physical scenarios that wemight identify inside our formalism. Each of them are squematically represented by a line that goes from the source to theEarth. L aγ = − M F µν F µν a = 1 M E · B a (1)where a is the axion field, M is the inverse of the pho-ton/axion coupling strength, F is the electromagneticfield-strength tensor, F its dual, E the electric field,and B the magnetic field. The axion has the importantfeature that its mass m a and coupling constant are in-versely related to each other. There are, however, otherpredicted states where this relation does not hold; suchstates are known as Axion Like Particles (ALPs). Animportant and intriguing consequence of Eq. (1) is thatALPs oscillate into photons and vice-versa in the pres-ence of an electric or magnetic field. In fact this effectrepresents the keystone in ongoing ALP searches carriedout by current experiments.In this work, we will make use of the photon/axionmixing as well, but this time by means of astrophysicalmagnetic fields. As already mentioned, we will accountfor the mixing that takes place inside or near the gamma-ray sources together with that one expected to occur inthe IGMFs. We will do it under the same consistentframework. Furthermore, it is important to remark thatit will be necessary to include the EBL in our formal-ism, in particular in the equations that describe the in-tergalactic mixing. Its main effect we should rememberis an attenuation of the photon flux, especially at ener-gies above 100 GeV. We show in Fig. 1 a diagram thatoutlines our formalism. Very squematically, the diagramshows the travel of a photon from the source to the Earthin a scenario with axions. In the same Figure, we showthe main physical cases that one could identify inside ourformalism: mixing in both the source and the IGMF,mixing in only one of these environments, the effect ofthe EBL, axion to photon reconversions in the IGMF,etc. A quantitative description of the photon/axion mix-ing phenomenon in both the source and the IGMFs canbe found in the next two subsections. A. Mixing inside and near the source
It has been recently proposed that an efficient conver-sion from photons to ALPs (and vice-versa) could takeplace in or near some astrophysical objects that shouldhost a strong magnetic field [12].Given a domain of length s , where there is a roughlyconstant magnetic field and plasma frequency, the prob-ability of a photon of energy E γ to be converted into anALP after traveling through it can be written as [14, 22]: P = (∆ B s ) sin (∆ osc s/ osc s/ (2)Here ∆ osc is the oscillation wave number:∆ osc ≃ (∆ CM + ∆ pl − ∆ a ) + 4∆ B , (3)∆ B that gives us an idea of how effective is the mixing,i.e. ∆ B = B t M ≃ . × − M B mG cm − , (4)where B t the magnetic field component along the polar-ization vector of the photon and M the inverse of thecoupling constant.∆ CM is the vacuum Cotton-Mouton term, i.e.∆ CM = − α π (cid:18) B t B cr (cid:19) E γ ≃ − . × − B mG (cid:18) E γ T eV (cid:19) cm − , (5)where B cr = m e /e ≃ . × G the critical magneticfield strength ( e is the electron charge).∆ pl is the plasma term:∆ pl = w pl E ≃ . × − (cid:16) n e cm − (cid:17) (cid:18) T eVE γ (cid:19) cm − , (6)where w pl = p παn e /m e = 0 . × − µeV p n e /cm − the plasma frequency, m e the electron mass and n e theelectron density.Finally, ∆ a is the ALP mass term:∆ a = m a E γ ≃ . × − m a,µeV (cid:18) T eVE γ (cid:19) cm − . (7)Note that in Eqs.(4-7) we have introduced the dimen-sionless quantities B mG = B/ − G, M = M/ GeV and m µeV = m/ − eV.Since we expect to have not only one coherence do-main but several domains with magnetic fields differ-ent from zero and subsequently with a potential pho-ton/axion mixing in each of them, we can derive a totalconversion probability [22] as follows: P γ → a ≃
13 [1 − exp( − N P / P is given by Eq.(2) and N represents the numberof domains. Note that in the limit where N P → ∞ , thetotal probability saturates to 1/3, i.e. one third of thephotons will convert into ALPs.It is useful here to rewrite Eq. (2) following Ref. [12],i.e. P = 11 + ( E crit /E γ ) sin B s M s (cid:18) E crit E γ (cid:19) (9)so that we can define a characteristic energy, E crit , givenby: E crit ≡ m M B (10)or in more convenient units: E crit ( GeV ) ≡ m µeV M . B G (11)where the subindices refer again to dimensionless quan-tities: m µeV ≡ m/µeV , M ≡ M/ GeV and B G ≡ B/Gauss; m is the effective ALP mass m ≡ | m a − ω pl | .Recent results from the CAST experiment [6] give a valueof M ≥ .
114 for axion mass m a ≤ .
02 eV. Althoughthere are other limits derived with other methods or ex-periments, the CAST bound is the most general andstringent limit in the range 10 − eV ≪ m a ≪ − eV.At energies below E crit the conversion probability issmall, which means that the mixing will be small. There-fore we must focus our detection efforts at energies abovethis E crit , where the mixing is expected to be large( strong mixing regime ). As pointed out in Ref. [12], in thecase of using typical parameters for an AGN in Eq. (11), E crit will lie in the GeV range given an ALP mass of theorder of ∼ µ eV. To illustrate how the photon/axion mixing inside thesource works, we show in Figure 2 an example for anAGN modeled by the parameters listed in Table II (ourfiducial model, see Section III). The only difference is theuse of an ALP mass of 1 µ eV instead of the value thatappear in that Table, so that we obtain a critical energythat lie in the GeV energy range; we get E crit = 0 . >
10 GeV) gradually. The reason for this behavior isthe crucial role of the Cotton-Mouton term at those highenergies, which makes the efficiency of the source mixingto decrease as the energy increases (see Eq. (5) and howit affects to Eq. (3)). Indeed, the photon attenuation in-duced by the mixing in the source completely dissapearsat energies above around 200 GeV in this particular ex-ample. On the other hand, one can see in Figure 2 asinusoidal behavior just below the critical energy as wellas just below the energy at which the source mixing dis-sapears due to the Cotton-Mouton term. However, itmust be noted that a) the oscillation effects are small; b)these oscillations only occur when using photons polar-ized in one direction while, in reality, the photon fluxesare expected to be rather non-polarized; and c) the abovegiven expressions are approximations and actually onlytheir asymptotic behavior should be taken as exact andwell described by the formulae. Therefore, the chancesof observing sinusoidally-varying energy spectra in as-trophysical source, due to photon/axion oscillations, areessentially zero. I n t en s i t y FIG. 2: Example of photon/axion oscillations inside thesource or vicinity, and its effect on the source intensity (solidline), which was normalized to 1 in the Figure. We used theparameters given in Table II to model the AGN source, butwe adopted an ALP mass of 1 µ eV. This gives E crit = 0 . B. Mixing in the IGMFs
The strength of the Intergalactic Magnetic Fields(IGMFs) is expected to be many orders of magnitudeweaker ( ∼ nG) than that of the source and its surround-ings ( ∼ G). Consequently, as described by Eq. (11), theenergy at which photon/axion oscillation occurs in theIGM is many orders of magnitude larger than that atwhich oscillation can occur in the source and its vicinity.Despite the low magnetic field B , the photon/axion os-cillation can take place due to the large distances, sincethe important quantity defining the probability for thisconversion is the product B × s, as described by Eq (9).Assuming B ∼ M = 0 .
114 (co-incident with the upper limit reported by CAST [6]),then the effect can be observationally detectable ( E crit < a < × − eV. Ifthe axion mass m a was larger than this value, then theconsequences of this oscillation could not be probed withthe current generation of IACTs, that observe up to fewtens of TeV [75]. In our fiducial model (see Table II) weused m a = 10 − eV, which implies E crit = 28 . e − e + pair production that comes fromthe interaction of the gamma-ray source photons with in-frared and optical-UV background photons for the ener-gies under consideration [35]. Therefore, it will be neces-sary to modify the above equations to properly accountfor the EBL in our calculations. These equations can befound in Ref. [11], where the photon/axion mixing in theIGMF was also studied, although for other purposes anda different energy range. We note that the same equa-tions were also used in Ref. [13] to study for the firsttime the photon/axion mixing in the presence of IGMFsfor the same energy range that we are considering in thiswork.There is little information on the strength and mor-phology of the IGMFs. As for the morphology, several authors reported that space should be divided into sev-eral domains, each of them with a size for which the mag-netic field is coherent. Different domains will have ran-domly changing directions of B field of about the samestrength [45, 46]. The IGMF strength is constrained tobe smaller than 1 nG [47], which is somewhat supportedby the estimates of ∼ B fields innearby galaxy clusters [50, 51]. Given this controversy,we decided to use a mid-value of 0.1nG in our fiducialmodel (Table II).In our model, we assume that the photon beam prop-agates over N domains of a given length, the modulus ofthe magnetic field B roughly constant in each of them.We will take, however, randomly chosen orientations,which in practice will be also equivalent to a variation inthe strength of the component of the magnetic field in-volved in the photon/axion mixing. If the photon beamis propagating along the y axis, the oscillation will occurwith magnetic fields in the x and z directions since thepolarization of the photon can only be along those axis.Therefore, we can describe the beam state by the vector( γ x , γ z , a ). The transfer equation will be [11]: γ x γ z a = e iEy (cid:2) T e λ y + T e λ y + T e λ y (cid:3) γ x γ z a (12)where: λ ≡ − λ γ ,λ ≡ − λ γ h p − δ i λ ≡ − λ γ h − p − δ i (13) T ≡ sin θ − cos θ sin θ − cos θ sin θ cos θ
00 0 0 T ≡ √ − δ √ − δ cos θ √ − δ √ − δ cos θ sin θ − δ √ − δ cos θ √ − δ √ − δ cos θ sin θ √ − δ √ − δ sin θ − δ √ − δ sin θ δ √ − δ cos θ δ √ − δ sin θ − −√ − δ √ − δ T ≡ − −√ − δ √ − δ cos θ − −√ − δ √ − δ cos θ sin θ δ √ − δ cos θ − −√ − δ √ − δ cos θ sin θ − −√ − δ √ − δ sin θ δ √ − δ sin θ − δ √ − δ cos θ − δ √ − δ sin θ √ − δ √ − δ (14) θ being the angle between the x -axis and B in each single domain. δ a dimensionless parameter equal to: δ ≡ B λ γ M ≃ . (cid:18) B − G (cid:19) (cid:18) GeV M (cid:19) (cid:18) λ γ Mpc (cid:19) (15) I n t en s i t y I n t en s i t y I n t en s i t y I n t en s i t y I n t en s i t y I n t en s i t y FIG. 3: Effect of intergalactic photon/axion mixing on photon and ALP intensities versus distance to the source, computed forour fiducial model, i.e. for 3C 279 and those parameters given in Table II but taking B= 1 nG, and using the Primack EBLmodel. The black thick solid line represents the total photon intensity, while the blue dotted line is the ALP intensity. Thephoton intensity as given only by the EBL (i.e. without including photon/axion mixing) is shown as the red dashed line.
Toppanels : mixing computed for M = 4 GeV and an initial photon energy of 50 GeV (left), 500 GeV (middle) and 2 TeV (right); bottom panels: M = 0 . that represents the number of photon/axion oscillationswithin the mean free path of the photon λ γ . Notice thatif there was no EBL, the quanta beam would be equipar-titioned between the ALP component and the two photonpolarizations after crossing a large number of domains.However, the EBL introduces an energy dependent meanfree path λ γ for the photon.Amongst all the EBL models that exist in the liter-ature, we chose the Primack [36] and Kneiske best-fit[37] to fix the λ γ parameter. They represent respectivelyone of the most transparent and one of the most opaquemodels for gamma-rays, but still within the limits im-posed by the observations (galaxy counts for the lowerlimit and observations of distant blazars for the upperone). The model proposed by Kneiske et al. was initiallydisfavored by some TeV observations of distant AGNs,using the assumption that the intrinsic spectral indexneeds to be softer than 1.5 (see [52, 53]). On the otherhand, in the literature we also find work where this as-sumption is strongly criticized, as reported by [54, 55]and especially in [42]. Therefore, we will consider theKneiske best-fit EBL model as still valid. In the case ofthe Primack EBL model, we did not take into accountthe redshift evolution of the EBL, which effect may beparticularly important for sources located at z > λ γ as the distancegiven by the so-called gamma-ray horizon for the energyconsidered. Additionally, we have to take into accountthat the energy of each photon will change continuouslyfor a photon traveling towards us from cosmological dis-tances, due to the cosmological redshift. This effect mayhave a very important role in the calculations of the pho-ton/axion mixing, since e.g. for a source at a distance of1000 Mpc (i.e. z ∼ .
3) every photon arrives at Earthwith 30% less energy. We account here for this effectfor the first time by computing at each step (distance)the new energy of the photon due to cosmological red-shift, and then using this new energy as the input energyneeded for the calculation of λ γ . We did not includein the formalism, however, those secondary photons thatmay arise from the interaction of the primary source pho-tons with the EBL.To illustrate how the mixing in the IGM works, weshow in Figure 3 various examples of the evolution of thetotal photon and ALP intensities as a function of the dis-tance to the source when varying some of the critical pa-rameters, using the Primack EBL model in all cases. Weuse the parameters listed in Table II, that correspondsto our fiducial model, but using an IGMF strength of1 nG (instead of 0.1 nG), which is still consistent withupper limits. For a photon with an initial energy of 50GeV (left top panel) and coupling constant M = 4,which yields E crit ∼
100 GeV, there is not significantphoton/axion oscillations. Since the role of the EBL isalmost negligible at these energies, the total photon in-tensity remains almost constant traveling to the Earth.For 500 GeV photons (middle top panel), the total pho-ton intensity initially decreases as expected due to theEBL absorption, but also an early extra attenuation dueto a photon to ALP conversion is clearly observed. At thesame time, the ALP intensity, which was initially equalto zero (we neglect here the mixing inside the source forsimplicity), grows rapidly. At larger distances the ten-dency in the total photon intensity is just the opposite;the intensity increases slightly, since an efficient ALP tophoton reconversion (although operative since the verybeginning) is taking place and becomes relevant speciallyat these large distances, where the expected photon in-tensity is already very low due to the EBL absorption.In the case of photons with higher initial energy (e.g. 2TeV, right top panel in Fig. 3), the expected attenua-tion due to the EBL becomes very important even forsmall distances from the source, which makes more rele-vant the impact of ALP to photon reconversions on thephoton intensity. As a result, the photon/axion mixingimplies an enhancement in the photon intensity at almostall distances. The situation changes when using a slightlyhigher coupling constant, but still within the CAST con-straints (see bottom panels in Fig. 3). In this case, boththe attenuation and the enhancement in intensity becomemore pronounced, as expected. For relatively small dis-tances, the photon/axion mixing produces an attenuationin the photon flux, while for relatively large distances, themixing produces an enhancement in the photon flux. Thesame argument is essentially valid for any initial photonenergy, the results only changing depending on the rela-tive relevance of the EBL in each case, which will modifythe distance at which the photon-intensity enhancementstarts occurring.
III. RESULTS
Up to now, previous works have focused only in study-ing the photon/axion mixing either inside the source orin the IGMFs. Instead, for the first time we carried outa detailed study of the mixing in both regimes under thesame consistent framework. We neglect for the moment,however, the mixing that may happen inside the Milky Way due to its galactic magnetic fields. We believe thata concise modeling of this effect is still very dependenton the largely unknown morphology of the B field in ourGalaxy. In the most idealistic/optimistic case, in which30% of the photons convert to ALPs within the sourceand 10% of the ALPs convert to photons in the MilkyWay, as reported in Ref. [15], this effect would producean enhancement of the photon flux arriving at Earth ofabout 3% of the initial photon flux emitted by the source.As mentioned in the previous section, in order for thephoton/axion oscillation to be observationally noticeableby current instruments, that is E crit < · − eV for typical val-ues of the IGM. Larger ALP mass values translate intohigher E crit (e.g. m a = 10 − eV would yield E crit ∼ PeVin the IGM, when using B ∼ − eV, as mentioned in Ref. [15].We use an ALP mass of 10 − eV in our fiducial model(Table II), which implies E crit ∼
30 GeV in the IGM (forB ∼ E crit ∼ ∼ A. Photon/axion oscillation in our framework
In this section we show the results obtained when tak-ing into account the mixing inside the source and in theIGMF simultaneously. Since we expect the intergalacticmixing to be more important for larger distances, due tothe more prominent role of the EBL, we chose two dis-tant astrophysical sources (as our benchmark AGNs) thatare relatively well characterized at gamma-ray energies;namely the radio quasar 3C 279 (z=0.536), most distantdetected gamma-ray source at the VHE range, and theBL Lac PKS 2155-304 at z=0.117. In order to computethe photon/axion oscillation in the source we used theparameters reported in Ref. [56] for 3C 279 and Ref. [57]for PKS 2155-304. As for the size of the region with B field (where photons can convert to ALPs) we chose aregion 10 times larger than the radius of the gamma-rayemitting blob given in the above mentioned references,the reason being that the blob radius represents only alower limit for the region where B is confined. We notethat this parameter, as well as the number of domainswhere the B is coherent, play an important role in thephoton attenuations due to the photon/axion mixing inthe source. Table I shows the different attenuations thatare obtained when varying the size of the region where B is confined (what we called “B region”) and the lengthsof coherent B domains inside that region. One can seethat, once the number of domains is fixed, the photonattenuation increases when increasing the size of the “Bregion”. On the other hand, when fixing the size of the“B region” and scanning the size of the domains we findthat, as we increase the number of domains, the attenua-tion increases until the size of the domain is “too small”.At this point, the probability of photon/axion conversionis almost zero for the single domains, which reduces theoverall photon/axion conversion. TABLE I: Maximum attenuations due to photon/axion oscil-lations in the source obtained for different sizes of the regionwhere the magnetic field is confined (“B region”) and differ-ent lengths for the coherent domains. Only length domainssmaller than the size of the B region are possible. The B field strength used is 1.5 G (see Table II). The photon fluxintensity without ALPs was normalized to 1. In bold face, isthe attenuation given by our fiducial model.B region (pc) Length domains (pc)3 × − × − × − We summarize in Table II the parameters we have con-sidered in order to calculate the total photon/axion con-version in both the source (for the two benchmark AGNs)and in the IGM. As we already mentioned, these valuesrepresent our fiducial model.
TABLE II: Parameters used to calculate the total pho-ton/axion conversion in both the source (for the two AGNsconsidered, 3c279 and PKS 2155-304) and in the IGM. Thevalues related to 3C 279 were obtained from Ref. [56], whilethose ones for PKS 2155-304 were obtained from Ref. [57].As for the IGM, e d,int was obtained from [58], and B int waschosen to be well below the upper limit typically given in theliterature (see discussion in the text). This Table representsour fiducial model.Parameter 3C 279 PKS 2155-304B (G) 1.5 0.1Source e d (cm − ) 25 160parameters L domains (pc) 0.003 3 × − B region (pc) 0.03 0.003z 0.536 0.117Intergalactic e d,int (cm − ) 10 − − parameters B int (nG) 0.1 0.1L domains (Mpc) 1 1ALP M (GeV) 1.14 × × parameters ALP mass (eV) 10 − − The effect of existence of ALPs on the total photonflux coming from 3C 279 and from PKS 2155-304 (usingthe fiducial model presented in Table II) can be seen in -6 -5 -4 -3 -2 -1 I n t en s i t y I n t en s i t y FIG. 4: Effect of photon/axion conversions both inside thesource and in the IGM on the total photon flux coming from3C 279 (z=0.536) and PKS 2155-304 (z=0.117) for two EBLmodels: Kneiske best-fit (dashed line) and Primack (solidline). The expected photon flux without including ALPs isalso shown for comparison (dotted line for Kneiske best-fitand dot-dashed line for Primack).
Figure 4. We carried out the calculations for the twoEBL models cited above: Kneiske best-fit and Primack.The inferred critical energies for the mixing in the sourceare E crit = 4 . E crit = 69 eV forPKS 2155-304, while for the mixing in the IGM we ob-tain E crit = 28 . ∼
100 GeV),which means that the intensity curves for the two EBLmodels agree to this energy.The situation changes above ∼
100 GeV, where thephoton attenuation due to the EBL is noticeable. Atthis point, the results depend substantially on the sourcedistance and the EBL model used. A stronger photonattenuation is obtained for the Kneiske best-fit modelagainst the Primack EBL model, as expected. Becausethe strong photon attenuation due to the EBL, the ALPsthat later convert to photons imply a further enhance-ment of the expected photon flux. Therefore, as one cannotice from Figure 4, the existence of ALPs translatesinto a relatively small ( ∼ ax-ion boost factor ). Again, this was done for our fiducialmodel (Table II) and for the two EBL models describedabove. The plots show differences in the axion boost fac-tors obtained for 3C 279 and PKS 2155-304 due mostlyto the redshift difference.In the case of 3C 279, the axion boost is an atten-uation of about 16% below the critical energy (due tomixing inside the source). Above this critical energy andbelow 200-300 GeV, where the EBL attenuation is stillsmall, there is an extra attenuation of about 30% (mix-ing in the IGMF). Above 200-300 GeV the axion boostreaches very high values: at 1 TeV, a factor of ∼ ∼
340 for the Kneiske best-fitmodel. As already discussed, the more attenuating theEBL model considered, the more relevant the effect ofphoton/axion oscillations in the IGMF, since any ALPto photon reconversion might substantially enhance theintensity arriving at Earth. We note that the axion boostfactor may vary when changing the parameters we usedto model the source (as shown in Table I) and the IGM(see next section). The results we find in this work arein disagreement with those reported by De Angelis etal. [13]. We always find that the photon intensity below200-300 GeV decreases when including the oscillation toALPs regardless of the ALP and/or IGM parameters,while De Angelis et al. find that the photon intensityincreases for a large range of the phase space they tried(see their Fig. 1 in Ref. [13]). At those low energies thephoton attenuation due to pair conversion in the EBL isrelatively low (see Fig. 4) and thus the few ALPs thatconvert to gamma photons do not imply any substantialrelative increase in the photon intensity. On the otherhand, 1/3 of the photons oscillate to ALPs, which causesa substantial decrement in the amount of gamma pho-tons with respect to those we would have in the absenceof ALPs. Therefore, we think it is very difficult to get aphoton enhancement at energies ∼
100 GeV. On the otherhand, the axion boost factors we find at high energies( >
300 GeV) are substantially lower than those obtainedin [13]. As an example, in the case of a Kneiske best-fit EBL model with B=1 nG, we find a boost ∼ ∼
20 for the samephoton energy and the same redshift (note that, in orderto carry out a one-to-one comparison with that work, wealso used M = 4, as they do). One of the reasons for thediscrepancy in the axion boost factors is the used EBLmodel. We noted that the EBL model shown in Fig. 1 ofRef. [13] is substantially more attenuating than the onefrom Kneiske best-fit EBL model, which is the one we areusing. Consequently, the axion boost factors reported inRef. [13] are larger than the ones they would have ob-tained if they had used the Kneiske best-fit EBL model.Besides that, it is not clear to us whether the change inphoton energy due to cosmological redshift (see SectionII B) was taken into account in [13]; this is not explicitelymentioned in their work.In the case of PKS 2155-304, the situation is differ-ent from that of 3C 279 due to the very low photon-attenuation at the source and, mostly, due to the smallersource distance. The low redshift location decreases theimpact of the EBL absorption and thus the effect of therelative photon-flux enhancement due to photon/axionoscillation. When using B=0.1 nG, the axion boost fac-tor is larger than 1 only for the Kneiske best-fit modeland only above 1.3 TeV. In the case of Primack EBL, theaxion boost factor is always smaller than 1, thus implyingno photon-flux enhancement. Note however that the 30%drop in the photon intensity occurs at the same energy asthat of 3C 279. This drop in the photon intensity shouldoccur at the same energy for all sources located at rel-atively medium redshifts (0.1 < z < < >
100 GeV) of distant sources(z > A x i on boo s t f a c t o r A x i on boo s t f a c t o r FIG. 5: Boost in intensity due to ALPs for the Kneiske best-fit (dashed line) and Primack (solid line) EBL models, com-puted using the fiducial model presented in Table II for 3C 279(z=0.536) and PKS 2155-304 (z=0.117). parameters (essentially B strength and size of the “B re-gion”) for the modeling of the gamma-ray emission fromAGN sources, we find that the attenuation in the sourcedue to photon/axion conversion is relatively low; 16% forour model of 3C 279 and 1% for that of PKS 2155-304.These low photon-flux attenuation (equivalent to ALP-enhancement) would decrease significantly the effect ofthe mechanism proposed in [15].Finally, it is worth mentioning that we checked thatour results are robust against the randomness of the B field. We ran 100 different realizations of the same phys-ical scenario, randomly varying the orientation of B ineach coherent domain and each realization. We did so forthe four cases studied along this work, i.e. 3C 279 andPKS 2155-304, Primack and Kneiske best-fit EBL mod-els. Furthermore, we repeated the same exercise using0.1, 1 and 10 Mpc as the length of the coherent domainsin order to explore the dependence of our results on thisparameter. In all cases, we chose our fiducial value of 0.1 nG for the B field strength. We found that the maximumdifferences are typically well below 10%, implying thatthe results obtained are not sensitive to the randomnessof the B field. We increased the number of realizations to1000 for some cases and found no differences with respectto the results obtained with 100 realizations.A larger effect on the computed axion-boost factors oc-curs when changing the size of the domains being used.The computed axion-boost factors are sensitive to choiceof the size of the coherent domains to be used. Togetherwith the choice of the EBL model (which is also uncer-tain), the choice of the domain sizes modifies the resultsobtained by factors of a few. B. The impact of changing B
A very interesting result has been found when vary-ing the modulus of the intergalactic magnetic field. InRef. [15], the intergalactic photon/axion mixing was re-jected arguing that its effect on the final intensity atEarth would be negligible when using a more realisticvalue for B , which should be substantially lower thanthe value of 1 nG adopted in Ref. [13]. However, as wasshown in the previous section, when using B=0.1 nG wefind significant effects even for sources located at red-shifts as low as z ∼ B in our fiducial model by one order of mag-nitude (above and below). We do that for both 3C 279and PKS 2155-304.In the case of 3C 279, we see in the left top panel ofFig. 6 that higher intensities (or equivalently, higher ax-ion boost factors in the right top panel), are obtainedwhen using B=0.1 nG instead of taking B=1 nG. Thisseems to contradict the intuitive idea of getting higherintensities for larger magnetic fields, that make the pho-ton/axion mixing more efficient. The reason for this re-sult is the strong attenuation due to the EBL. If thephoton/axion mixing is efficient, then many ALPs con-vert to photons which soon disappear due to the EBLabsorption. Consequently, if the source distance is large,we end up having a very small number of photons arriv-ing to the Earth. On the other hand, if the photon/axionmixing is not that strong, then we can keep a higher num-ber of ALPs traveling towards the Earth, which act as apotential reservoir of photons. When decreasing B to0.01 nG, then the axion boost factors are lower than forthe other two cases. On the other hand, in the case ofPKS 2155-304, we see that the highest axion boost fac-tors are obtained with B=1 nG, because the source isnot as distant as 3C 279. If we had considered a sourcelocated at a much further distance than 3C 279, thenwe would have found the highest axion boost factors forB=0.01 nG.In summary, higher B values do not necessarily trans-late into higher photon flux enhancements. There is al-ways a B value that maximizes the axion boost factors;1 -6 -5 -4 -3 -2 -1 I n t en s i t y A x i on boo s t f a c t o r I n t en s i t y A x i on boo s t f a c t o r FIG. 6: Same as in Figures 4 and 5 but for different values of IGMF. Upper panels: 3C 279 using those parameters listed inTable II, only changing B . Lower panels: Same exercise for PKS 2155-304, using the corresponding parameters that can befound in the same Table II. this value is sensitive to the source distance, the consid-ered energy and the EBL adopted model. C. The impact of using the smallest photon/ALPcoupling constant
The most stringent limits on the ALP-photon couplingconstant were derived using the non-detection of gamma-rays (by the Solar Maximum Mission Gamma-Ray Spec-trometer) from the supernova (SN) 1987A during the ∼ M to values larger than 1 [16] and 3 [17]. Those limits areonly valid for ultralight ALPs. In both works the value m a < − eV is quoted, although this value holds onlyfor some specific situations. Indeed, a more robust valueis m a < − eV (see Refs. [12, 15]), i.e. the energy be-low which the exact value of the ALP mass is irrelevant because the “plasma frequency” dominates (see defini-tion of ALP effective mass in Section II A). Various au-thors (see Refs. [11, 13]) used M = 4 when dealing with m a < eV. Since the ALP mass in our fiducial modelis 10 − eV, and hence close to this limit, we decided torepeat the calculations using this value for M , which is35 times larger than the value we used in the previoussections (see Table II).Before we continue, it is worth pointing out that thelimits to the ALP-photon coupling constant given inRefs. [16, 17] are subject to large uncertainties that arenot fully discussed in those papers. Both the flux of ALPsproduced in the SN explosion and the back-conversionof ALPs to gamma photons can vary by large factors,and hence the upper limits computed with those num-bers have to be taken with caveats.The calculated flux of ALPs produced and releasedduring the SN explosion depends on the knowledge ofthe size, temperature and density of the proto-neutronstar. Those numbers are subject to large uncertainties2because we still do not know how stars explode. Eventhough there is general agreement that the ultimate en-ergy source is gravity, the relative roles of neutrinos,fluid instabilities, rotation and magnetic fields continueto be debated. In particular, back in the 90s it was be-lieved that neutrinos would be able to reheat the outgo-ing shock-wave and produce the explosion. Nowadays,with far more powerful computer simulations, we knowthat neutrino-driven explosions are only possible whenthe star has a small iron core and low density in the sur-rounding shells, as being found in stars near or below 10solar masses [18]. The progenitor of SN1987A was a bluesupergiant and hence it is expected to be somewhere be-tween 10-50 solar masses. A possibility to explain thoseexplosions might require the proper inclusion of rotationand magnetic fields (see Refs. [19, 20, 21] and referencestherein). Both B field and rotation are present in starsas well as in pulsars, which are the products of successfulSN explosions; thus it is very natural to consider themin SN explosion models. In particular, the rotation ofthe proto-neutron star can change substantially the tem-perature and, specially, the density of the inner core; in[21] it is shown that the density can vary by more thanone order of magnitude, which would change by a similarfactor the flux of ALPs being produced. Refs. [16, 17]did not consider such level of complexity (and uncertain-ties) in the parameters used to compute the flux of ALPs,mostly because 15 years ago we lacked that knowledge.On the other hand, the back-conversion of ALPs tophotons relies on the structure of the galactic magneticfield which is, again, not well known. Different modelspredict B fields that could differ substantially and hencethey would predict different values for the amount ofgamma photons we would obtain for a given flux of ALPs.This is clearly shown in Fig. 1 from Ref. [15], where theprobability of ALP-photon conversion is given for var-ious locations of the sky. Therefore, even if we couldaccurately predict the number of ALPs from SN1987A,the number of photons would be subject to large uncer-tainties.Therefore, we conclude that the limit in the inverse ofthe ALP-photon coupling constant given in Refs. [16, 17]is subject to large (orders of magnitude) uncertainties,and thus the limit given by the CAST collaboration re-mains as the most robust one up to date. However, forthe sake of comparison with other works, we computedthe axion-boost factors when using M = 4 eV. This isshown in Figure 7 for both 3C 279 and PKS 2155-304 fortwo values of the B field, 0.1 nG and 1 nG. For this lowcoupling constant, the effect due to the photon/ALP os-cillation in the source is negligible. The effect due to pho-ton/ALP oscillation in the IGMF in not negligible, butsubstantially lower than the one shown in the previoussection. Besides, such effect shows up at larger energiesnow (see Eq. 11); 100 GeV and 1000 GeV respectivelyfor 1 nG and 0.1 nG. A x i on boo s t f a c t o r A x i on boo s t f a c t o r FIG. 7: Boost in intensity due to ALPs for the Kneiske best-fitand Primack EBL models, computed using the fiducial modelpresented in Table II for 3C 279 (z=0.536) and PKS 2155-304(z=0.117), but with M = 4 GeV and B=0.1 nG (dashedand solid lines for Kneiske best-fit and Primack EBL modelsrespectively) and B=1 nG (dot-dashed and dotted lines). IV. DETECTION PROSPECTS FOR FERMIAND IACTS
As mentioned in Section II, for photon/axion couplingconstants close to the current published limits, and forrealistic ALP mass values, the energy at which the pho-ton/axion oscillation starts to become important is ex-pected to lie in the gamma-ray range. Consequently,the combination of the Fermi/LAT instrument and theIACTs, which cover 6 decades in energy (from 20 MeVto 20 TeV) is very well suited to study the photon/axionmixing effect. Because of the rapid change in the pre-dicted photon intensity attenuation close to E crit , theenergy resolution of the instrument is very relevant inorder to detect the ALP signatures. The Fermi/LAT in-strument has an energy resolution of about 10%, whereasIACTs, above 150 GeV, have an energy resolution of3about 20-25%. The photon intensity attenuation wefound in this work goes from 0 to 30% due to the mixingin the source, plus essentially 30% due to the mixing inthe IGM. This implies that one needs to be able to de-termine photon fluxes with a precision better than 10%.Such level of precision might not be achievable for ener-gies >
10 GeV with Fermi, or for energies > ∼ TeV energies and is located at large distances, the pho-ton/axion oscillation in the IGM could translate into anintensity enhancement of more than one order of magni-tude (see Figures 4, 5, 6), which should be certainly easyto detect with current IACTs.Therefore, if we accurately knew the intrinsic spectrumof the sources and/or the density of the EBL, we shouldbe able to observationally detect ALP signatures for awide range of the parameter space (photon/axion cou-pling constant and ALP mass). The main problem isthat we do neither know accurately the intrinsic spec-trum of the sources nor the EBL density. Thus the po-tential detection of those ALP signatures become quitechallenging, but not impossible. In order to study thisscenario, we propose the following strategy:1. Observe several AGNs located at different redshifts,as well as the same AGN undergoing different flar-ing states (low/high fluxes), at different energyranges, from radio to TeV. This is important be-cause the modeling of the gamma-ray emission de-pends critically on the emission at lower energies(specially infrared, optical, UV and X-rays), andalso because we do not know a priori the energyat which the photon/axion oscillation will start tooperate.2. Try to describe the observational data with“conventional” theoretical models for the broadband emission (Synchrotron Self-Compton, Exter-nal Compton, Proton synchrotron, etc) and the at-tenuation of the gamma-rays in the EBL (Primack,Kneiske best-fit or other EBL models). The currentmodels (sometimes very simplistic) will definitelyrequire some modifications to fit the observationaldata.3. Look for intensity drops in the residuals (“best-model”-data). We want to stress that the drop inthe photon flux due to the attenuation in the IGMonly depends on the IGMF and the properties ofthe ALPs (mass and coupling constant), i.e. it isindependent of the gamma-ray sources. Therefore,a detection of such photon flux drops at the sameenergy in a numerous of different sources would bea clear signature for the existence of photon/axionoscillation, because we do not expect that the in-trinsic spectrum from different sources (or samesource at different flux levels) have a rapid drop of ∼
30% in the emission at the same energy. Thedetection (or no-detection) of this photon intensitydrop implies a constraint for the product m a · M .In this specific search, the Fermi-LAT instrumentis expected to play a key role since it will detectthousands of AGN sources located at various red-shifts (up to z ∼ >
300 GeV) and thus only de-tectable with IACTs. The origin of the potentialphoton flux “excess” might be due to a wrong EBLmodel and/or wrong model for the source emis-sion, the last being very important because it intro-duces differences between the different sources (orthe same source under different activity levels). Inthis case what we need is to detect distant (z > > ? ], an intermediate-frequency-peaked BL Lac object located at redshift 0.444. Thisclaim coincides with the detection, at GeV energies, ofthis source in active state by the Fermi-LAT instru-ment during the same time window [59]. In addition,the MAGIC Collaboration reported gamma-rays above1 TeV coming from a location consistent with the posi-tion of 3C 66A [39]. Those observations would confirmthe earlier claims from the Crimean Astrophysical Ob-servatorys GT-48 IACT of several detections from this4source above 0.9 TeV [40, 41]. Those detections werenot confirmed by the HEGRA and Whipple telescopes,which are more sensitive instruments, but which observedthe source at different time windows [60, 61]. As men-tioned above, a detection of TeV photons from a sourcelocated at z=0.444 would pose serious problems to con-ventional models of photon propagation over cosmolog-ical distances, where the high energy gammas are ex-pected to disappear due to pair electron-positron produc-tion in the EBL. On the other hand, the recent publishedlower limits to the EBL at 3.6 microns [62], which is al-most twice larger as the previous ones, enhances even fur-ther the attenuation of gamma-rays at TeV energies andthus increases even more the magnitude of the mystery.Furthermore, as reported in [42], this fact extends theproblems to sources located at medium redshifts (z=0.1-0.2) whose intrinsic energy spectra appear to be harderthan previously anticipated. Those observations presentblazar emission models with the challenge of producingextremely hard intrinsic spectra (differential spectral in-dex in the spectrum smaller than 1.5) in the sub-TeV tomulti-TeV regime. As mentioned in the previous section,the photon/axion oscillation in the IGM would naturallyexplain these two puzzles; the detection of TeV photonsfrom very distant (z ∼ > V. CONCLUSIONS
If ALPs exist, then we should expect photon to ALPconversions (and vice-versa) in the presence of magnetic fields. This photon/axion mixing will occur in gamma-ray sources as well as in the IGM. We have explored indetail both mixing scenarios together in the same frame-work. The main conclusions on this work can be sum-marized as follows: • If photons oscillate into ALPs in the IGM, thenphoton/axion mixing in the source is also at workfor lower photon energies. In this picture, both ef-fects should be taken into account using the sameframework, since they will be governed by the sameset of physical parameters (ALP mass and couplingconstant). In the case of ALP masses m a >> − eV, the energies at which the photon/axion os-cillation occur in the IGMF are >> • The photon/axion oscillation in the source (andits vicinity) can produce photon-flux attenuationsup to 30%, as previously stated in the literature[12, 15]. However, when using available models forgamma-ray emitting blob regions to set values ofthe B field strength and the size of the region wherethe conversion can take place (we took a radius 10times the size of the blob), we obtain photon-fluxattenuations that are significantly lower. • The photon/axion oscillation in the IGM producesa photon-flux attenuation up to 30% below the en-ergies at which the EBL is important (but above E crit for the oscillation to be efficient). If the sourceredshift is larger than ∼ m a · M , under the assumption of a given IGMFstrength. If such an intensity drop is not seen inthe spectra, lower limits could be set. • Above energies at which the absorption of gamma-rays in the EBL become important, the pho-ton/axion oscillation in the IGMF could produceboth attenuation and enhancement in the photonflux, depending on the source distance and energyunder consideration. • We find that decreasing the intensity of the IGMFstrength does not necessarily decreases the photon-flux enhancements (axion boost factors). For asource located at z=0.5, B=0.1 nG produces higher5photon-flux enhancements that B=1 nG. This re-sult is somewhat unexpected since stronger B fieldsallow for a more efficient photon/axion mixing.The reason for this result is the strong attenuationdue to the EBL. If the photon/axion mixing is ef-ficient, then many ALPs convert to photons whichsoon disappear due to the EBL absorption. Conse-quently, if the source distance is large, it ends uphaving a very small number of photons arriving atthe Earth. On the other hand, if the photon/axionmixing is not that efficient (lower B field), thenthere is a higher number of ALPs traveling (to-wards the Earth), which act as a potential reservoirof photons. The net balance between the two pro-cesses is sensitive to the source distance, the energyconsidered and the EBL intensity. Given those pa-rameters, there is always a B value that maximizesthe photon flux enhancements.We have shown that the signatures of photon/axion os-cillations may be observationally detectable with currentgamma-ray instruments (Fermi/LAT and IACTs). Sincephoton/axion mixings in both the source and the IGMare expected to be at work over several decades in en-ergy, it is clear that a meticulous search for ALPs in the(sub)GeV-(multi)TeV regime will be greatly enhanced bymeans of a joint effort of Fermi and current IACTs.The main challenge in such detection comes from thelack of knowledge in conventional physics; namely theintrinsic source spectrum and EBL density and the in-tensity and configuration of the intergalactic magneticfield. In other words, the effect of the photon/axion os-cillations could be attributed to conventional physics inthe particular source and/or propagation of the gamma-rays towards the Earth. However, we believe that such photon/axion oscillations could be studied using severaldistant AGNs located at different redshifts, as well asthe same distant AGN detected at distinct activity levels.The signatures of such effect being attenuations (at rela-tively low energies) and/or enhancements (at the highestenergies) in the photon fluxes, that could be visible inthe residuals from the “Best-Model-Fit” and the obser-vational data.Recent work, like the potential detection of TeV pho-tons from very distant (z ∼ ∼ m a ≤ − eV) witha photon/axion coupling constant close to current upperlimits ( M ∼ Acknowledgments
We thank P. Serpico for useful discussions and com-ments. We also acknowledge the anonymous refereefor useful comments that allowed us to improve themanuscript. M.A.S.C. is very grateful for the hospitalityof the SLAC during his visit, where most of this workwas done. This work was supported by the Spanish I3P-CSIC and AYA2005-07789 grants, by the SLAC/DOEContract DE-AC02-76-SF00515, the Spanish Ministeriode Educaci´on y Ciencia and the European regional de-velopment fund (FEDER) under project FIS-2008-04189,and Junta de Andaluc´ıa under project P07-FQM-02894. [1] Peccei R. D. and Quinn H. R., 1977,
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