Axion-photon multimessenger astronomy with giant flares

We treat prospects for multimessenger astronomy with giant flares (GFs), a rare transient event featured by magnetars that can be as luminous as a hundred of the brightest supernovae ever observed. The beamed photons could correlate with an axion counterpart via resonant conversion in the magnetosphere. In a realistic parameter space, we find that the sensitivity limit to galactic GFs for currently viable experiments is $\mathrm{g}_{\phi \gamma}\!\gtrsim\!\mathrm{several}\!\times\!10^{-13}$ GeV$^{-1}$ \&$\mathrm{g}_{\phi e}\!\gtrsim\!\mathrm{few}\!\times\!10^{-12}$. We rule out the compatibility of axion flares with the recent XENON1T excess only due to the time persistence of the signal.

where the first term on the right-hand side represents the effective axion-photon Lagrangian density and the second term the axion-electron coupling; being φ the axion field, F µν the photon field, ψ the electron field and γ 5 the usual matrix.

A. General dispersion relations
Aided by axion-electrodynamics [8] for the linearization of the equations of motion from Eq. 1, a set of Schrödinger-like dispersion relations emerges that can be solved analytically in idealized astrophysical boundaries [9]   ω+ (2) * javier.miguelhernandez@riken.jp The interaction term L φγ = g φγ E · B is obtained from Eq. 1 being E the photon and B the external magnetic field, establishing a preferential polarization. Now, obviating the Faraday rotation term -∆ R -the perpendicular component E ⊥ decouples and the mixing is reduced to a 2D system given by the low-right sector in the second term of the left-side hand of Eq. 2. The term ∆ = 1 2 ω(n − 1) vanishes in the weak dispersion limit, since the refraction index is n ∼1 for relativistic axions. The quantum-electrodynamics (QED) vacuum refraction parameter -∆ Q -can be treated independently. We have defined the refraction parameter ∆ p =− ω 2 p /2ω, with ω p = (e 2 n e /m e ε 0 ) 1 /2 the characteristic frequency of the plasma, and the axion mass parameter ∆ φ = − m 2 φ/2ω.
B. The role of the plasma density in the relative weight of QED corrections It has been pointed out that QED vacuum polarization effect acts as a suppressor of the axion-(keV)photon cross-section in a minimal model of neutron star adopting the Goldreich-Julian (GJ) corotation densityn GJ = 4πε 0 B(r)(eP ) −1 - [9,10]. However, a pair multiplicity factor, defined κ = n e /n GJ , emerges in modern approaches beyond the GJ paradigm [11]. The ratio of plasma effects to QED vacuum effects, manifested through the Euler-Heisenberg term L QED = , scales with the density profile across the gas [9,12] ∆ p ∆ Q ⊥ = 5×10 −10 keV ω From Eq. 3, it follows that the tension with QED results progressively relaxed above κ 10 5 at relatively large distances from the star for keV photons, while ∆ Q gradually vanishes near its surface for κ 10 8 . In the case of very energetic transitory events featured by magnetars, the charge density near the star surface is n e ∼10 30 cm −3 [13], many orders over the occupation number predicted by the GJ model. Although the density profile through the atmosphere of the 'fireball' is barely known, simulations suggest that pair multiplicity factors κ 10 10 are tenable at an altitude of few stellar radii [14]. That renders resonant conversion efficient in the parameter space dominated by plasma effects; including highly magnetic neutron stars, where 10 4−5 κ 10 7 [11,15], against previous expectations.

C. Magnetar axion
Magnetars are compact stellar remnants endowed with extreme magnetic fields, typically about 10 10−11 T, a 'canonical' mass around 10-20 solar masses and a rotation period of the order of seconds [16]. In the standard picture, their persistent emission mechanisms involve pair production and magnetic acceleration, catalyzing the migration of charges to higher altitudes from the star to give raise to curvature radiation, synchrotron emission and inverse Compton scattering [17,18]. The beamed photons could be converted to axion through non-adiabatic resonant mechanism at their path across the non-relativistic, 'cold', plasma of the magnetosphere [11,12,[19][20][21][22]. In the aligned-rotator approximation, where the magnetic axis and the rotation axis -ẑare superimposed, the axion-photon oscillation amplitude is derived from introducing the resonance condition ∆ p − ∆ φ = 0 in Eq. 2, yielding a conversion probability P γφ = ∆ 2 k/(2 ∂ ∂z ∆ p ). Since outgoing photons present a gradient ∂ ∂z ∆ p ∼ m 2 φ ω −1 z −1 c , it is straightforward to obtain [11] In Eq. 4, B(r) = B 0 (R/r) 3 expresses the magnetic field in the dipole approximation, stationary as a result from an aligned rotator, as a function of the distance from the surface of the star, z c is the conversion altitude, ω the pulse of the photon.

A. Magnetar flares
The so-called anomalous X-ray pulsars (AXPs) and soft gamma-ray repeaters (SGRs) today are thought to be transient events from magnetars. Their strong outbursts are frequently referred short-bursts (SBs), intermediate flares (IFs) and giant flares (GFs), whose more general characteristics are summarized in Table I. If the quiescent emission mechanisms of magnetars are poorly understood, the scenario for transient events is even more complex. Rapid magnetic field reconfiguration is thought to be an important part of the outbursts. The released magnetic energy is converted then to thermal energy in the magnetosphere -cf.
[24] for solved non-thermal spectral features -. However, the triggering mechanisms, and the exact role of the magnetic field remain unresolved. The time-scale of the magnetic reconnection t rec ∼ L/V , where L is the height of the reconnection point and V the inflow velocity -should be much larger than the typical time-scale of the system, τ s ∼|ω − m φ | −1 , to render plausible the conversion mechanism described by Eq. 4. Within this scenario, reasonable for a relatively high frequency or a sufficiently high distance from the star, we have τ s << t rec and the oscillation takes place in a pseudo-stationary regime. We include the spectra of an IF with energy-scale about 10 41 erg s −1 in Fig. 1 left. The correlated axion flare is quantified in units of the typical occupation of virialized dark matter in the Galactic halo, around 10 13 keVcm −2 s −1 .
We will confine our attention throughout the following sections to the study of axion giant flares (AGFs) triggered by GFs. Three 'galactic' GFs have been observed: from SGR 0526-66, actually hosted by Large Magellanic Cloud (LMC), in 1979 [25]; from SGR 1900+14 in 1998 [26] and, the most luminous to date, from SGR 1806-20 in 2004 [27,28]. Although there is consensus that GFs are rare events, the exact rate is source of controversy and is susceptible to observational bias [29]. Statistics in [30] suggests that the rate for events above 4 × 10 44 erg is ∼3.8 × 10 5 Gpc −3 yr −1 , resulting in a event rate per magnetar of the order of 0.02 yr −1 . These estimates rely on the strong evidence that 1-20% of the gamma-ray burst (GRB) sample are indeed extragalactic GFs, whose tails are undetectable due to the large distance [13].
In that manner, the catalogue of confirmed GFs is poor and, in addition, their spectral features are not adequately characterized due to saturation of the detectors and other systematics. Therefore, we fit and extend to low frequency a black-body (BB) curve at a temperature k B T BB ∼175 keV, with a fluence of 1.36 erg cm −2 of >30 keV photons on Earth consistent with the SGR 1806-20 event [27], to be employed in simulations throughout the following sections. Notwithstanding that empirical data are not available when referring to the polarization of  the GF, it is predicted that the strong magnetic field in magnetars linearly polarizes the photons in two normal modes, i.e., ordinary (O) -with the E field lying on the k-B plane -and extraordinary (X). The weight of the O-mode, which efficiently mix with axion, was established in a conservative 0.5 fraction for our estimates after consulting the literature [31][32][33][34].

B. The flight of axion to Earth
Obviously, both the photon and axion flux suffer from the inverse-square law with distance. Moreover, axions with a rest mass m φ 10 −5 eV crossing through several cosmic magnetic domains, with a typical length scales -of the order of Mpc, are reconverted efficiently into photons; first across their outgoing path through the intergalactic medium (IGM), and then upon reaching the Milky Way (MW) [35][36][37]. The photon-toaxion conversion probability, derived from Eq. 2, is read P γφ = 4∆ 2 M sin 2 ( ∆k/2)/∆ 2 k. In the strong-coupling limit, the inequality (∆ p +∆ Q −∆ φ ) 2 << 4∆ 2 M is obtained, resulting in the well-known expression [38] P φγ = g 2 φγ B 2 2 4 .
From a panoramic perspective, our galaxy takes the form of a thin disk of ∼30 kpc length -i.e., <<swith a magnetic field of the order of a fraction of nT, and a low electron density, about 10 −3 cm −3 . Crucially, from Eq. 5 it follows that only a marginal fraction of the magnetar axions released in the vicinity of the MW are reconverted into photons during their approximation to Earth. In other words, the astroparticle flux density at origin, determined by Eq. 4, approximately persists.
Photons are massless through the IGM. Axion is massive. In contrast to the plasma that envelopes the magnetosphere, where the resonant conversion takes place at ω p ∼ m φ , the relation ω ∼ γm φ is maintained through the vacuum, being γ the Lorentz factor. The axion velocity is β φ = 1 − 1/γ 2 . In that form, the delay time, at keV energies, for values of the Lorentz factor between 10 1−7 and pair multiplicity factor 10 2 κ 10 9 varies between about a year per traveled kpc and a few milliseconds. That would render photon-axion multimessenger astronomy constrained to a human-scale time frame and would motivate the tracking of confirmed outbursts for relatively massive axions.

A. Detection of axion flares through inverse Primakoff-effect
The solar axion spectra is formed by interactions of axion with SM particles in the internal plasma of the star. A spectra dominated by Primakoff-effect presents a maximal flux about 10 9 cm −2 s −1 at 3-4 keV for an axion-photon coupling strength g φγ ∼ 10 −11 GeV −1 . A solar spectra shaped by atomic recombination and deexcitation, bremsstrahlung, and Compton (ABC) processes peaks a similar fluence around 1 keV for an axion-electron FIG. 2. Limit on the sensitivity to axion flares by realistic helioscope-like detectors, based on the inverse Primakoffeffect. The CAST helioscope limit is established at 95% confidence level (CL) [39]. The horizontal branch (HB) exclusion region is established by indirect methods at 95% CL [40]. Astro-bounds established by indirect methods are also included in green [41,42]. The light brown solid line represents the KSVZ axion. The dot-dash yellow line projects newgeneration helioscope sensitivity at 95% CL from a campaign lasting several years [43], while the duration of the giant flare is set t = 0.125 s. Dark lines correspond to direct detection of GFs with 15 -dotted -, 3 -dashed -and 0.1 -solid -kpc distance, at 95% CL. The comparison is performed by integration of the spectral functions between 1-10 keV, where the detectors are more sensitive coinciding with the maximal flux of solar axion.
coupling constant g φe ∼10 −13 . Differently from magnetar axion, where one finds from Eq. 4 that the QCD axion flux is enhanced for X-ray photons, the emitted solar flux scales with g 2 mitigating the emission of axion-like particles (ALPs) with a low coupling strength, including both KSVZ [44,45] and DFSZ [46,47] axion.
Helioscopes are directional detectors tracking the Sun [48][49][50][51]. Their working-principle relies on inverse Primakoff axion-to-photon conversionφ + γ virt → γin a magnetized cryostat equipped with photomultiplier tubes (PMTs). The cross-section of helioscopes in the limit q << 1 is given by Eq. 5, being q the transfer of momenta between the axion and the photon and the length of flight across the vessel. The product of the inductance of the magnetic field and the length/area is restricted in a physical implementation. On the other hand, coherence is gradually lost for m φ 10 −2 eV. The sensitivity of helioscopes in terms of the axion-photon coupling strength is approximately [52] g φγ [GeV −1 ] 1.4×10 −9 b 1 /8 where b is the integrated background noise in the energy range 1-10 keV, t is exposure time, A is the transverse area of the detector and Φ φ is the received flux in units of Φ , the integrated flux of solar axions on Earth.
Helioscope-type apparatus can be employed to search for axion flares. In Fig. 2, we qualitatively compare solar and magnetar scanning performed by an IAXO-like instrument [43,53]. There, we find a volume of special interest in a 3 kpc radius from Earth, with around three dozen cataloged magnetars, being about 1 /3 of them within a 1 /2 kpc radius; while the closest object is about 0.1 kpc distant [54].

B. Observation of axion flares using liquid-xenon detectors
Experiments storing liquid-xenon are sensitive to axion through the axio-electric effectφ + e + Z → e + Zhold by Eq. 1. In the relativistic limit, the cross-section reads where the proton-electron cross-section σ pe (E) can be, e.g., interpolated from [63]. By the introduction of an expectation signal-to-noise ratio (SNR) in Eq. 7, it is possible to write the sensitivity of liquid-xenon detectors in terms of the axion-electron coupling strength [64] FIG. 3. Limit on the sensitivity to axion giant flares (AGFs) based on the axio-electric effect for realistic liquid-xenon detectors. The regions shaded in warm colors are established by direct detection of solar axion at 90% confidence level (CL) [55][56][57]. Dashed light-green lines project stellar hints [58][59][60]. The light reddish zone corresponds with methods whose hypothesize that all the dark matter in the Halo is formed of axion and a high homogeneity, overlapping other modeldependent, or indirect, results [55-57, 61, 62]. Dark lines correspond with direct detection of giant flares with 15dotted -, 3 -dashed -and 0.1 -solid -kpc distance, at 95% CL. The red solid line corresponds with DFSZ, nonrelativistic, axion. Parameters are t = 0.125 s, W=10 4 kg, = 0.3137 √ keV, b = 1.4×10 −5 s −1 kg −1 keV −1 .
is the spectral resolution of the instrument, b is the background event rate, t is integration time, W is weight of the stored xenon. In Fig. 3 we represent the sensitivity to AGFs for XAX-like experiments [65].

V. CONCLUSIONS
The conversion of the beam of light emitted from highly magnetic neutron stars into relativistic axions, first at their magnetospheres and then during their flight to Earth, opens a new window for the direct detection of axion, or for re-visiting and extending exclusion limits whose arbitrarily assume that axion-like particles form all the dark matter (DM) in the nearby universe, and that DM is distributed homogeneously. In this manuscript we pioneered the concept of photon-axion multimessenger astronomy with giant flares (GFs), a type of rare event featured by soft gamma-ray repeaters (SGRs) that can be, for a fraction of a second, more luminous than hundred times the brightest supernovae [66,67], or almost 10 14 suns radiating coherently; with an uncertain upper bound, as more magnetic magnetars could release GFs 1-2 orders brighter than the strongest event observed to date if we only take into account stored magnetic energy [27].
Giant flares could correlate with an axion counterpart via resonant mixing through the magnetosphere. Anticipating a short-term at which the SGR catalogue is more extensive and their characteristics better understood, we analyze sensitivities to axion bursts in the soft X-ray energy range for realistic experiments. Under a number of idealizations of recurring use in related works, we find that GFs originating in the vicinity of Earth would provide the detectors with sensitivity to axion-like particles with an axion-photon coupling g φγ several×10 −13 GeV −1 and axion-electron coupling g φe few×10 −12 at 95% confidence level (CL), over a broad range and for reasonable benchmark parameters. A confirmed detection would result in an absolute 'spectra', while for the projection of new exclusion bounds the sensitivity would be calibrated against data from X-ray telescopes and grounded by theory. The expected axion flux density on Earth results from the convolution of three terms: the GF photon spectral flux density, the photon-to-axion conversion probability and the geometric dilution with distance. As a reference, for an upper-limit photon-axion conver-sion probability O( 1 /2), the flare from SGR 1806-20 -10 46 erg, 15 kpc distance -would result in a transient fluence on Earth similar to the flux of solar axions for an axion-photon coupling strength g φγ ∼10 −11 GeV −1 .
Assuming axion, this work predicts the existence of axion giant flares (AGFs) across our universe, and perhaps their detectability by ground-based observatories. Transcendentally, the echo of a galactic flare might be received today in the form of AGF, due to the delay-time that massive particles suffer compared to ordinary photons during their flight through the interstellar medium. Thus, monitoring the confirmed sources of giant flares -SGR 0526-66, 48 kpc distant, in 1979; SGR 1900+14, 6 kpc distant, in 1998; and SGR 1806-20 in 2004 -with dark matter detectors, in addition to future events, can be motivated -e.g., during hours when helioscopes cannot track the Sun -. A state-of-the-art estimate of the event rate suggests that each magnetar releases up to one potent giant flare every fifty years [30]. There are currently more than four dozen cataloged magnetars within a radius of 15 kpc from Earth. Therefore, the probability of an observable AGF event would be in the order of a dozen in the next decade, with approximately 2/3 of them originating within a distance of 3 kpc. In addition, it is striking that, although the bursts themselves are not periodic, the activity might only occur during predictable periodic intervals [68]. As a consequence, the prediction and tracking of magnetars entering an active period using dedicated axio-telescopes, or a network, is not discardable in order to enhance the probability of detection through the observation of stars with active, non-overlapping windows.
Finally, the XENON1T Collaboration recently reported an electronic recoil excess below 7 keV compatible with solar axion at 3.4σ CL [69]. However, the precise parameter space is in tension with stellar evolution at 8σ [70]. Interestingly, magnetar axions could mimic solar axion at keV energies without conflicting with stellar physics. However, the signal persisted for a large time interval incompatible with the known nature of magnetar flares, while quiescent isolated sources would be too distant to provide the deposited energy. Magnetar axion count rate could be a factor to consider in future experiments.