Production of axion-like particles from photon conversions in large-scale solar magnetic fields

The Sun is a well-studied astrophysical source of axion-like particles (ALPs), produced mainly through the Primakoff process. Moreover, in the Sun there exist large-scale magnetic fields that catalyze an additional ALP production via a coherent conversion of thermal photons. We study this contribution to the solar ALP emissivity, typically neglected in previous investigations. Furthermore, we discuss additional bounds on the ALP-photon coupling from energy-loss arguments, and the detection perspectives of this new ALP flux at future helioscope and dark matter experiments.


I. INTRODUCTION
Axion-like particles (ALPs) are ultralight pseudoscalar bosons a with a two-photon vertex aγγ, predicted by several extensions of the Standard Model (see [1,2] for comprehensive reviews). The two-photon coupling allows the conversion of ALPs into photons, a ↔ γ, in external electric or magnetic fields. In stars, this leads to the Primakoff process that induces the production of low mass ALPs in the microscopic electric fields of nuclei and electrons. An ALP flux would then cause a novel source of energy-loss in stars, altering their evolution. In this context, the strongest bound comes from helium-burning stars in globular clusters, giving g aγ < 6.6×10 −11 GeV −1 for m a 1 keV [3]. Another interesting possibility is the conversion in a macroscopic field, usually a large scale magnetic field. In this case, the momentum transfer is small, the interaction is coherent over a large distance, and the conversion is best viewed as an axion-photon oscillation phenomenon in analogy to neutrino flavor oscillations. This effect is exploited to search for generic ALPs in light-shining-through-the-wall experiments (see e.g. the ALPS [4] and OSQAR [5] experiments), for solar ALPs (see e.g. the CAST experiment [6,7]) and for ALP dark matter [8] in micro-wave cavity experiments (e.g., the ADMX experiment [9]). In particular, the solar ALP search in a helioscope, like CAST, exploits the production of an ALP flux with E ∼ O(few) keV via Primakoff process in the Sun core and its back-conversion into X-rays in the large-scale magnetic field of the detector. The absence of an ALP signal allows one to get the best experimental bound on the photon-ALP coupling, g aγ 6.6 × 10 −11 GeV −1 for m a 0.02 eV [10], comparable with the one placed from helium-burning stars.
The Sun can be a source of intense magnetic fields that can be relevant for ALP conversions. Notably, in [11] it was studied the possibility to trigger ALP conversions in X-rays in the intense magnetic fields of the sunspots on the Sun surface. Observations of the Soft X-rays Telescope (SXT) on the Yohkoh satellites allows to obtain bounds on g aγ O(10 −10 ) GeV −1 . Presumably, this bound can be strengthened with a dedicated Sun observation by the current NuStar satellite experiment [12]. Furthermore, even though in the Standard Solar Models (SSMs) [13,14] the Sun is assumed as a quasi-static environment, seismic solar models have been developed including large-scale magnetic fields in different regions of the solar interior [15,16]. The presence of these Bfields may trigger conversions of the thermal photons into ALPs, creating an additional ALP flux besides the one produced by the Primakoff conversions. Some preliminary characterization of this flux has been presented in talks [17,18]. However, a detailed calculation is still lacking in the literature. Only recently there appear dedicated studies of the ALP flux produced in the solar interior via conversions in the B-fields of longitudinal plasmons [19,20] (see also [21][22][23]). Our work complements these results by taking into account the conversions into ALPs for transverse, as well as longitudinal photon modes in the solar plasma.
The plan of our paper is as follows. In Sec. II we describe the model of the solar B-fields we will use to characterize the photon conversions into ALPs. In Sec. III we revise the conversions of photon into ALPs for longitudinal and transverse modes. In Sec. IV we solve the ALP-photon kinetic equations in order to compute the ALP production rate, taking into account the photon absorption in the solar plasma. In Sec. V we calculate the solar ALP fluxes from magnetic conversions. In Sec. VI we present a new bound on the ALP-photon coupling g aγ from energy-loss argument associated with ALPs emitted from photon conversions in B-fields. In Sec. VII we discuss the detection perspectives for these fluxes. Finally, in Sec. VIII we summarize our results and we conclude. There follow Appendix A, where we give details of the calculation of conversion probabilities into ALPs for transverse and longitudinal photons; Appendix B, where we describe the solution of the ALP-photon kinetic equations; and Appendix C, where we present the thermal field theory approach. We show that the kinetic and the thermal field theory approach lead to the same ALP production rates.

II. SOLAR MAGNETIC FIELDS
The magnetic field of the Sun is most important in three different regions, the radiative zone (r 0.7 R ), the exterior (convective) zone (r 0.9 R ), and the intermediate region between these two, called the tachocline (r ∼ 0.7 R ). Here, we are using the standard notation R = 6.9598 × 10 10 m for the solar radius [14]. In our work for simplicity we assume spherical symmetry for the solar magnetic field, described by a radial profile B(r) (see the seismic model in [24] for a toroidal magnetic field).
The radiative zone (for r ≤ r 0 = 0.712R ) is characterized by the following profile [25] where K λ = (1+λ)(1+1/λ) λ , with λ = 1+10 r 0 /R , and 1×10 7 G B rad 3×10 7 G. This range was determined by Couvidat et al. [16]. They used the precision on solar sound speed and density to rule out fields with intensity B 0 ∼ 10 4 T and arguments on the solar oblateness to set the upper value. The field profile in the tachocline is simulated as where r 0 = 0.712R is the center of the zone and d is its half-width. As benchmark parameters in the tachocline we set d = 0.02 R , while 3 × 10 5 G B m ≡ B tach 5 × 10 5 G. These bounds were set by Antia et al. by the observation of the splittings of solar oscillation frequencies [26]. Similarly, the field profile in the upper layers is simulated as in Eq. (2), with r 0 = 0.96 R , d = 0.035 R and 2 × 10 4 G B m ≡ B conv 3 × 10 4 G. These bounds were also set in Ref. [26], from an analysis of the Global Oscillation Network Group (GONG). The radial profile of the solar magnetic field described above is shown in Fig. 1.

A. Photon dispersion in a plasma
The dispersion relation of a photon in a plasma has the form [27] where ω and k are the photon frequency and wavenumber and π T,L (ω, k) are the projection of the photon polarization tensor for the transverse (T) and longitudinal (L) modes, respectively. In particular, for the energies we are interested in, the dispersion relation for transverse photons (TP) becomes [27] where ω p = 4παn e m e = 1.31 × 10 18 n e 10 26 cm −3 R −1 , (5) is the plasma frequency, with n e the electron density, the numerical expression referring to typical solar conditions, in units of the solar radius R .
Instead, the longitudinal mode, the so-called longitudinal plasmon (LP), has a dispersion relation [27] ω ω p where p is the equilibrium pressure. In the typical conditions of the solar plasma, the second term on the right hand side is negligible. Therefore, in the Sun the dispersion relation for LP reduces to We now discuss how TP and LP mix with ALPs in a plasma. The derivation of the equations of motion for such a system is given in Appendix A.

B. Transverse photons
The ALP-photon interaction Lagrangian is given by [28] where a is the ALP field, g aγ is the ALP-photon coupling, F µν is the electromagnetic field tensor andF µν = 1/2 µνρσ F ρσ its dual. Equation (8) is responsible for the mixing among ALP and photons. Transverse photons mix with ALPs only through an external transverse magnetic field B ext ≡ B T . We denote with A ⊥ and A the components of the vector potential A perpendicular and parallel to B T respectively. Assuming a uniform magnetic field we can reduce the general 3 × 3 mixing problem into a 2 × 2 system involving only A and a, described by a Schrödinger like equation [28,29] where the Hamiltonian for the transverse modes reads (up to an overall phase diagonal term) The TP-ALP conversion probability after traveling a distance z in a uniform magnetic field B T is given by [28] P where we have introduced In solar units, The radial behavior of these quantities in the Sun is shown in Fig. 2. The plasma frequency has been characterized taking as reference the Solar Model AGSS09 [14], which we will use as benchmark for our estimations. For the solar B-fields we used the model of Sec. II.
Remarkably, the TP-ALP conversion probability exhibits a resonant behavior. Indeed, it is easy to see from Eq. (11)-(12) that the probability is maximal when ∆ a = ∆ p , i.e. when m 2 a = ω 2 p . In this situation the ALP dispersion relation (dot-dashed curve in Fig. 3) matches the one of the TP (continuous curve). From Fig. 2, one realizes that the resonance occurs in the radiative zone at r ∼ 0.3R for m a ∼ 100 eV. Instead, the resonance  in the tachocline occurs at r ∼ 0.7R for m a ∼ 10 eV. In the following we will take these values of ALP mass as benchmark for the calculation of resonant ALP production. In principle, we may have also a resonant conversion in the convective zone at r ∼ 0.9R for m a ∼ 1 eV. However, due to the lower local temperature and to the smaller magnetic field the resultant ALP flux would be smaller than the previous ones and we will neglect it hereafter. Finally, we need to take into account that thermal photons are continuously emitted and re-absorbed in the Sun. This process is characterized by the transverse photon absorption coefficient rate Γ abs , defined as the inverse of the photon mean free path λ mfp . An explicit calculation of this rate is presented in [30]. We take its numerical values, shown in Fig. 2, from [31]. One realizes that the photon absorption coefficient Γ abs in the solar plasma is not negligible with respect to the other oscillation parameters. Rather, it is always larger than ∆ T aγ . Therefore, it needs to be included in the treatment of the problem. We will account for this effect using a kinetic approach which includes simultaneously photon conversion and absorption in order to characterize the ALP emission rate, as we will show in Sec. IV.
Here we limit ourselves to a simple estimation of the conversion probability P (γ T → a). First, we consider the resonant case and we assume that the distance traveled by the photon in the B-field is equal to its mean free path λ mfp = Γ −1 abs , since for larger distances the photon scatterings break the coherence of the oscillations. For the photon transverse modes the resonance occurs when m 2 a = ω 2 p , thus ∆ T osc ≡ ∆ T aγ Γ abs as we see from Fig. 2. Therefore in Eq. (11) we have that Then the TP-ALP conversion probability reads Next, we consider the off-resonance conversion probability. For the photon transverse modes, when the resonance condition does not apply we can take m a ∼ 0, so that ∆ T osc ≈ ∆ p . From Fig. 2, we see that ∆ p Γ abs , thus in Eq. (11) we can approximate sin 2 (∆ T osc Γ −1 abs ) ≈ 1/2, since there are many photon oscillations within a mean free path Γ −1 abs and we can just consider the average of the oscillatory term. In this case the TP-ALP conversion probability becomes Thus, the off-resonance TP-ALP conversion probability is much smaller than the resonant one. However, the resonant conversions involve only a small fraction of photons in a given shell of the Sun. Therefore, a complete calculation of the ALP flux is necessary to determine which contribution would dominate.

C. Longitudinal photons
Longitudinal plasmons mix with ALPs in a plasma through an external longitudinal magnetic field. Therefore, we consider a uniform external magnetic field along the z-direction B ext = B Lẑ . If we take ω ω p ω a , we can linearize the Maxwell's equations for longitudinal modes, obtaining [22] i∂ where A L is the longitudinal plasmon field and a the ALP field and the Hamiltonian of the system is Then, the LP-ALP conversion probability is [23] with where the numerical value of ∆ L aγ can be calculated as for ∆ T aγ in Eq. (13). The LP-ALP conversion presents a resonance for ω = ω p , when the ALP dispersion relation (dot-dashed curve) crosses the LP one (dashed curve), as shown in Fig. 3.
Concerning the LP absorption, in [30] it has been shown that for these modes one finds the same expression as for the TP absorption rate. Therefore, also in this case Γ abs ∆ L aγ . Concerning the resonant conversion probability, one finds the same numerical expression of Eq. (14) for the TP.

IV. ALP PRODUCTION RATE
As we have seen in the previous Section, the photon absorption rate Γ abs in the Sun is not negligible with respect to the others oscillation parameters. Thus, TP-ALP and LP-ALP oscillations are interrupted by collisions and we have to face the problem of treating simultaneously oscillations and collisions. A suitable formalism is provided by the kinetic approach developed for relativistic mixed neutrinos in the presence of collisions [32]. This formalism has been applied to different mixing problems, such as the mixing of photons with hidden photons (HP) in the Sun [30], which we closely follow in our derivation. Details are given in Appendix B. In Appendix C, we show that this approach is equivalent to the thermal field theory formalism.
To begin, we present a completely general formalism, starting from two bosonic fields A and S, which evolve according to the linearized equation of motion where ω A is the energy associated with the field A, ω S the one associated with the field S, and µ is a mixing term which we assume to be small with respect to the diagonal terms. The Hamiltonian in Eq. (21) can be written as ; (22) where ∆ω = ω A − ω S . For such a system, we can define the oscillation frequency We consider the case in which collisions occur for the A quanta (i.e. the photons in our case). We assume that the field A interacts with the medium, namely with the solar plasma, which can absorb a quantum with rate Γ abs and produce one with rate Γ prod . The equations of motion for such a system, described by the density matrix ρ, are the Liouville equations [32] where G prod = Γ prod 0 0 0 ; We remind the reader that the diagonal components of the density matrix contain the occupation numbers of A and S quanta, while the off-diagonal components take into account the coherence between these two states. Note that in Eq. (24) the commutator on right-handside describes the dynamical evolution of the system while the anticommutators correspond to the collisional terms. In thermal equilibrium Γ prod = e −ω/T Γ abs and the S type particles are not excited, while we assume that the A type particles obey the Bose-Einstein statistics f BE = (e ω/T − 1) −1 . A non-equilibrium situation is described with a small deviation δρ from the thermal equilibrium state ρ eq . In this limit one finds a steady state solution for the S quanta production rate with Γ = (1 − e −ω/T )Γ abs , i.e. the total collisional rate. From Eq. (26) we obtain that the A − S mixing process is resonant, i.e. it is maximal for ω A = ω S . The result in Eq. (26) is completely general and it is valid both at the resonance and off-resonance, since it was obtained under the only assumption that the mixing term µ is small relatively to the diagonal terms ∼ ∆ω. This condition always applies in the solar plasma, for both TP and LP. Thus, we can adopt Eq. (26) to describe the TP-ALP and LP-ALP conversion rates.

A. Photon transverse modes
The Hamiltonian for TP-ALP system in Eq. (10) can be written (up to a term proportional to the identity matrix) as in Eq. (22) where here q = (ω 2 p − m 2 a )/2ω. Thus, with the substitutions µ → ∆ aγ and ∆ω → q, from Eq. (26) we obtain the TP-ALP conversion rate The expression in Eq. (28) is valid both on resonance and off-resonance and we will use it to estimate the ALP flux expected at Earth arising from these conversion processes.

B. Photon longitudinal modes
The Hamiltonian for LP-ALP system in Eq. (17) can be written as in Eq. (22) ; (29) where in this case ∆ω = ω p − ω a . If we insert this expression of ∆ω and we replace µ → ∆ L aγ in Eq. (26) we obtain the LP-ALP conversion rate This expression has been recently obtained in [19,20] from a thermal field theory calculation (Cfr. Appendix C). Contrarily to the photon transverse modes, the expression in Eq. (30) is valid only on resonance, since it is based on the evolution of on-shell LPs, i.e. it is obtained assuming that ω ∼ ω p ∼ ω a and it is not applicable for ω very far from ω p .

V. SOLAR ALP FLUXES
The solar ALP flux on Earth is given by [33] where D = 1.49 × 10 11 m is the Earth-Sun distance, Γ prod a is the ALP production rate expressed by Eq. (26), the factor g is the number of the photon polarization states (g = 1 for LP and g = 2 for TP), and the integral is performed over the photon momenta k and over the solar volume. From Eq. (31) we recover the differential ALP spectrum expected at Earth where we assumed relativistic states ω ≈ k and Ω k is the solid angle around the direction of photon momentum k.
In the following we will perform the radial integral over the SSM AGSS09 [14]. We now focus on the estimation of the ALP flux at Earth from different conversion processes in the solar magnetic fields for TP and LP modes.

A. Flux from TP-ALP conversions
The TP-ALP conversion process is dominated by the resonance, where the TP-ALP conversion probability [Eq. (11)] is maximal. Due to the extremely peaked nature of the resonant condition, we can approximate the ALP production rate [Eq. (28)] with a delta function If we insert the last expression in Eq. (32), we note that the integration over the solar volume gives To evaluate the above expression, we model the electron density in the region r 0.8 R with a simple exponential form n e = n 0 e e −r/Re .
We thus find Furthermore, we notice that in the case of resonance ∆ T abs Γ abs (see Fig. 3). Therefore, during the resonance the photons are continuously re-scattered such that information about their polarization is lost. The photon trajectories can form any angle with the magnetic field B. Since the photon trajectories are not straight, this angle is not correlated with the magnetic field direction and the photon polarization. Therefore, we have to perform a local angular average in the resonance shell before performing the integral in dΩ k of Eq. (32). For a generic photon polarization, the B T strength entering the conversion probability is where z is the position vector of the resonance region in a particular directionẑ,ˆ is the photon polarization vector (|ˆ | = 1,ˆ ×ẑ = 0), ϑ is the angle between the magnetic field B(z) and the photon propagation directionx and ϕ the angle between B T (the component of the magnetic field perpendicular to z) andˆ . We define where r res is the position in the Sun where the resonance condition occurs for the fixed value of m a and T res is the temperature at the same position.
Let us first consider the ALP spectrum for the resonance in the tachocline at r ∼ 0.7 R , where m a ∼ 10 eV and B ∼ 3 × 10 5 G, where the homogeneous B-field has its peak, as shown in Fig. 1. An analytic approximation to the solar ALP spectrum is provided by a fit with the three-parameter function [33] dΦ a,T dω = g 2 10 C where C is a normalization constant and the energy ω is expressed in keV. Numerical values of C, α and ω 0 are shown in Table I.
We can obtain the total ALP flux Φ a,T from resonant conversions in the solar magnetic fields expected at Earth by integrating Eq. (41) over the energies ω. The flux parameters for m a = 10 eV are found to be Φ a,T = 2.48 × 10 10 g 2 10 cm −2 s −1 ; (43) ω = 0.6 keV ; L a,T = 1.51 × 10 −5 g 2 10 L ; where L a,T is the ALP luminosity, L = 3.8418 × 10 33 erg is the Sun luminosity, ω is the average energy of the ALP flux spectrum and g 10 = g aγ /10 −10 GeV −1 . We now focus on the resonant production at r ∼ 0.25 R , i.e. the one with m a = 130 eV. Here we present results obtained assuming B = 3 × 10 7 G. We find the following flux parameters Φ a,T = 1.63 × 10 14 g 2 10 cm −2 s −1 keV −1 ; (46) ω = 2.76 keV ; L a,T = 0.2 g 2 10 L .
Finally, if we are far from resonance we can assume m a ≈ 0. In this case ∆ T osc ≈ ∆ p Γ abs , as shown in Fig. 2. Thus, the rate in Eq. (28) reduces to If we insert Eq. (49) in Eq. (32) and we integrate over the solar profile we obtain the ALP flux spectrum at Earth from off-resonant production. In this case the direction ϑ between the field B and the photon direction of propagation does not change during the conversions, since many oscillations occur into a single photon mean free path. However, the azimuthal angle ϕ between the transverse field B T and the photon polarizationˆ would change. Thus, we perform an average over ϕ before the integral over dΩ k in Eq. (32), i.e. The dominant contribution for the off-resonant flux comes from the radiative zone, since here the B-field amplitude reaches the highest value. Taking the peak value B = 3 × 10 7 G we find the following flux parameters for non-resonant ALP spectrum flux Φ a,T = 5.2 × 10 9 g 2 10 cm −2 s −1 ; (51) ω = 3.24 keV ; (52) L a,T = 1.92 × 10 −8 g 2 10 L .
We resport in Table I the fitting parameters of the energy spectrum of Eq. (42) for the three cases we considered. In Fig. 4, we compare the fluxes from TP-ALP conversions in solar magnetic fields. As expected, we see that the non-resonant contribution is always subdominant with respect to the resonant ones.

B. Flux from LP-ALP conversions
The ALP production rate for LP-ALP conversions in the solar magnetic fields is given in Eq. (30). Since the largest contribution arises from conversions in the radiative zone, we will present results just for the largest value of the B-field in this region, i.e. B = 3 × 10 7 G. Also in the case of LP-ALP conversions the process is extremely peaked around the resonance, thus we can approximate Γ prod a,L with a delta function In the case of LP there is just one projection of the magnetic field which is longitudinal to the photon propagation direction, i.e. B L = |B cos ϑ|. Then, in the resonant shell we should average If we insert this expression in Eq. (32) we obtain the ALP flux from LP-ALP conversion in the Sun dΦ a,L dω = 1 12πD 2 r 2 res ω 2 g 2 aγ |B(r res )| 2 e ω/T − 1 where r res is the position in the Sun where the resonance ω = ω p occurs, |ω (r res )| = |dω p /dr| computed at r = r res and we have denoted with ω ≈ ω a ≈ ω p the ALP and photon energies. The result in Eq. (56) agrees with the one recently obtained in Ref. [20]. Using Eq. (38), the derivative in Eq. (56) can be expressed as Adopting the Standard Solar Model AGSS09 [14] to compute the plasma frequencies we obtain the ALP flux spectrum shown in Fig. 5. The flux shows a peak at ω ∼ 0.12 keV. The flux parameters are found to be Φ a,L = 2.18 × 10 10 g 2 10 cm −2 s −1 ; (58) ω = 0.13 keV ; (59) L a,L = 3.34 × 10 −6 g 2 10 L .

VI. ENERGY-LOSS BOUNDS
We now discuss the phenomenological consequences of these solar ALP fluxes. We start considering the possibility to place a new bound on ALP-photon coupling g aγ based on the ALP emissivity from photon conversions in B-fields. On the basis of the energy-loss argument one can set a bound on the coupling g aγ imposing the condition [34] L a 0.03 L , which is obtained from the combination of helioseismology (sound speed, surface helium and convective radius) and solar neutrino observations. Assuming the usual ALP emission by Primakoff process, whose estimated luminosity is L a = 1.8 × 10 −3 g 2 10 L [33], the quoted bound is g aγ 4.1 × 10 −10 GeV −1 . Now we consider the case of the ALP flux from resonant conversions for masses m a ∼ O(100) eV. Using the luminosity associated with this flux we can set the upper limit on g aγ as shown in Fig. 6 where the blue region is the excluded one in the parameter space (m a , g aγ ) by resonant processes in the radiative zone of the Sun assuming a field with amplitude B = 3 × 10 7 G. The orange region is the excluded one by the same process, assuming a field with amplitude B = 1 × 10 7 G. In particular, This new bound is even more stringent than the constraint derived from Helium burning stars in GCs, g aγ < 6.6 × 10 −11 GeV −1 [3]. For ALP masses outside the previous range, the bound worsens, as shown in Fig. 6. For instance, for m a ∼ 30 eV we obtain a limit comparable with the one from the Primakoff process. For conversions of TP in the case of m a ∼ 10 eV and m a = 0 and for the LP conversions, the associated ALP luminosity is much smaller than the Primakoff one, as results from Eq. (45) and Eq. (60), respectively. Therefore, the contribution to the energy-loss is subleading with respect to the Primakoff process.

VII. DETECTION PERSPECTIVES
In order to assess the possibility to detect the solar ALP fluxes from conversions in B-fields we show in Fig. 7 the different ALP fluxes from conversions in B-fields and we compare them with the Primakoff flux (dotted curve, see, e.g. [35] for a recent calculation), which represents a benchmark for experimental searches on solar ALPs. Starting from the ALP flux from TP conversions, we see that for m a ∼ 10 eV the ALP flux (continuous curve) is peaked at energies below the CAST threshold [33] (ω < 2 keV), shown as vertical line in the Figure. There are plans to lower the threshold in the sub-keV region in the future helioscope IAXO [44]. However, masses larger than m a ∼ 1 eV are not accessible even to this experiment. For these large masses, the coherence of ALP-photon conversions in the magnetic field of the helioscopes is lost and consequently the sensitivity is rapidly reduced. In principle, there are ideas for a new class of helioscopes, like the proposed AMELIE (An Axion Modulation hELIoscope Experiment), which could be sensitive to ALPs with masses from a few meV to several eV, thanks to the use of a Time Projection Chamber [37]. Studies for low mass WIPMs are already being carried out by the TREX-DM experiment [38,39], which is taking data at the Canfranc Underground Laboratory (LSC) [40]. The project aims at demonstrating the feasibility to reach low backgrounds at low energy thresholds for dark matter searches, which require similar detection conditions as for ALPs. Concerning the ALP flux coming from resonant TP-ALP conversions in the radiative zone of the Sun, corresponding to an axion mass m a ∼ 130 eV (shortdashed curve), the flux is expected to be much larger than the Primakoff one above the CAST threshold. However, CAST cannot detect it due to the loss of coherence of ALP-photon conversions in the detector. In principle, ALPs with mass m a ∼ 100 eV could be detected with a dark matter detector like the Cryogenic Underground Observatory for Rare Events (CUORE), which exploits the inverse Bragg-Primakoff effect to detect solar axions [41]. CUORE is expected to cover a mass range m a 100 eV. Notice, that for m a ≥ 10 eV there are some cosmological constraints to be taken into account. Indeed, the ionization of primordial hydrogen (x ion ) from ALPs decaying into photons sets the bound g aγ 5 × 10 −13 GeV −1 for m a ∼ 10 2 eV [42]. However, in cosmological models with low-reheating temperature these bounds can be easily evaded (see, e.g. [43], for a discussion). On the contrary, our ALP signal from the Sun is not affected by the cosmological model. Therefore, its possible detection would also point towards a nonstandard cosmological scenario.
Moving now to the case of the non-resonant conversions of TP modes (m a = 0, long-dashed curve) we realize that this contribution can reach a few % of the Primakoff process in the same energy range. In case of a positive detection of an ALP flux in a future helioscope, precision spectral studies in principle might determine it as an excess with respect to the expected Primakoff flux.
Finally, the case of the ALP flux from LP-ALP con-versions (dot-dashed curve) has been recently discussed in Ref. [20]. There, the authors suggest the possibility of detecting ALPs from LP conversions in the energy range 10 −2 keV ω 10 −1 keV through an upgraded version of IAXO [44]. They forecast to have a sensitivity down to m a 10 −2 eV. We address the interested reader to this interesting and detailed work for further details.

VIII. CONCLUSIONS
We have characterized the ALP production in the large-scale solar magnetic fields and discussed the perspectives for their detection in helioscopes and dark matter detectors. In particular, we have characterized both resonant and non-resonant conversions of transverse photons, which have not been taken into account so far. At this regard, we have considered realistic models for the solar B-field in the radiative zone and in the tachcoline of the Sun. We first studied the problem from a theoretical point of view, using a kinetic approach based on the evolution of the density matrix for the photon-ALP ensemble. With this approach, we estimated the production rate of ALPs in the Sun and we used it to estimate the ALP flux expected at Earth. The expression of the ALP production rate obtained in this way is completely general and has been specialized to study ALPs production from both LP and TP conversions.
In the case of the ALP flux from LP-ALP conversions, we reproduce the result recently obtained in [19], using the thermal quantum field theory approach. This flux results peaked at E ∼ 100 eV, and might be detectable with an upgraded version of IAXO [20]. The ALP flux from TP conversions, for ALPs with mass m a ∼ 10 eV, associated to resonant conversions in the tachochline, is found to be peaked below the CAST threshold. A dedicate investigation is necessary to assess the experimental possibility to dectect such a low-energy flux. Conversely, the ALP flux arising from transverse photon-ALP conversions for ALPs with mass m a ∼ 100 eV in the radiative zone, is dominant above the CAST threshold and it is larger that the Primakoff one. In principle, this flux might be detected using the dark matter detector CUORE. Furthermore, this ALP flux allows to improve the bound on g aγ from energy-loss in the Sun, even exceeding the bound from Helium-burning stars in Globular Clusters. The ALP flux from non-resonant conversions of TP modes can reach a few % of the Primakoff process in the same energy range. Therefore, it might produce a distortion of this flux, possibly producing observable signatures in the case of a precise measurement of the solar ALP spectrum.
In conclusion, our work completes the recent studies of Ref. [19,20] about the production of ALPs in the solar magnetic fields via longitudinal plasmons, including also the analysis of the photon transverse mode. Despite challenges in measuring this flux, it is intriguing to realize that the Sun can be the source of additional ALP fluxes beyond the well-studied one from Primakoff conversions. A positive measurement of this flux would shed new light not only on ALPs, but also on the magnetic field in the Sun.

Acknowlegments
We warmly thank Andrea Caputo for valuable comments on our manuscript. The work of P.C. and A.M. is partially supported by the Italian Istituto Nazionale di Fisica Nucleare (INFN) through the "Theoretical Astroparticle Physics" project and by the research grant number 2017W4HA7S "NAT-NET: Neutrino and Astroparticle Theory Network" under the program PRIN 2017 funded by the Italian Ministero dell'Università e della Ricerca (MUR).

Appendix A: Photon-ALP mixing in a plasma
Let us now consider an ALP-photon system in a plasma. We start from the Lagrangian of a photon coupled with the pseudoscalar field a, i.e. the ALP field (63) where g aγ is the axion-photon coupling, J µ is the electromagnetic current, A µ is the vector potential, F µν is the electromagnetic field tensor andF µν = 1/2 µνρσ F ρσ its dual. From Eq. (63) one recovers Maxwell's equations Transverse modes Transverse modes are characterized by an electric field E transverse to the photon momentum and a magnetic field B transverse to both. We consider a strong external magnetic field B ext such that the total field is B ≈ B ext . According to the discussion of Section III A the photon dispersion relation for TP is where ω p is the plasma frequency. For the purpose of our discussion we rewrite just three Maxwell's equations [Eqs. (64)] in a non-explictly covariant form where ρ = −en e is the electron charge density, A is the time-varying part of the vector potential for the external magnetic field and = ∂ 2 t −∇ 2 . Note that we are considering only the electrons in plasma equations, since we are assuming that ions provide a uniform background which does not partecipate in plasma motion. For the transverse mode k · B ext = 0. Thus for clarity of notation we identify the magnetic field B ext ≡ B T , to denote that it is a transverse field. Moreover, we make the assumption that E B T . Then, Maxwell's equations [Eqs. (66)] become We specialize our calculation to a wave of frequency ω propagating in the z-direction and we denote with A ⊥ and A the components of the vector potential A perpendicular and parallel to B T respectively. Thus the equations of motion for the TP-ALP system become [28] (68) where n R corresponds to the so called Faraday effect, which denotes the possibility of rotation of the plane of polarization in optically active media with a consequent mixing of A ⊥ and A . Moreover The terms ∆ CM ⊥, describe the Cotton-Mouton effect, i.e. the birifrangence of fluids in the presence of a transverse magnetic field. The vacuum Cotton-Mouton effect arises from QED one-loop corrections to the photon polarization when an external magnetic field is present. In this case we define ∆ QED = |∆ CM ⊥ − ∆ CM | and it is defined as This QED correction is found to be negligible with respect to ∆ p in the case of solar plasma. For transverse modes only the component A of the vector potential couples to the ALP. Moreover, if we neglect the Faraday effect (n R = 0) and we assume that the magnetic field is uniform, we can reduce the general 3 × 3 problem of Eq. (68) to the 2 × 2 system involving only A and a.
In the ultrarelativistic limit, i.e. for energies ω m a and ω ω p , we can linearize Eq. (68). As a result of linearization we obtain a linear Schrödinger-like equation [28] i∂ z A a = −ωI + ω 2 p /2ω g aγ B T /2 g aγ B T /2 m 2 a /2ω A a .
(71) Thus the Hamiltonian for the transverse modes reads (up to an overall phase diagonal term) Finally, we obtain the TP-ALP conversion probability after traveling a distance z in a uniform magnetic field B T [28] where we have introduced (74)

Longitudinal modes
Longitudinal modes are allowed only in presence of a medium, that is in our discussion is the solar plasma. The photon dispersion relation for LP is In this case ∇ · B = 0, thus the plasma equations of motion and the relevant Maxwell's equations are [22,23,45] ∂n e ∂t + ∇ · (n e v) = 0 ; 77) ∇ · (E + g aγ Ba) = e(n e − n 0 e ) ; The goal is now compute the LP-ALP conversion probability, assuming that no LP absorption exists in the plasma. In the case of longitudinal modes we have to combine Eqs. (76) and (77) with Maxwell's equations. We consider a uniform external magnetic field along the z-direction B = B Lẑ and we consider the plane wave approximation, i.e. we assume that all the fields vary as e i(k·x−ωt) . Moreover, we take into account a small perturbation to the electron number density n e = n 0 e + δn ; where n 0 e is the equilibrium value and δn n 0 . If we finally take ω = ω p = ω a , i.e. the photon energy coincident with the plasma frequency and with the ALP energy, we can linearize Maxwell's equations Eqs. (78)-(79) for longitudinal modes, obtaining [22] i∂ where we have introduced the fields δn = in e A L and a = ω p m e a k /ek. Thus the Hamiltonian of the system is The LP-ALP conversion probability is where

Appendix B: Kinetic Approach
In order to determine the ALP emission rate in the Sun due to the conversions of photons in magnetic fields we closely follow the kinetic approach developed in [30] for the case of hidden photons. We consider the system introduced in Sec. IV constituted by two bosonic fields A and S. We assume that the field A interacts with the medium, namely with the solar plasma, which can absorb a quantum with rate Γ abs and produce one with rate Γ prod . The kinetic equation for the density matrix ρ of the A − S ensemble is given by [32] where and G prod = Γ prod 0 0 0 ; In thermal equilibrium Γ prod = e −ω/T Γ abs and the S type particles are not excited, while we assume that the A type particles obey the Bose-Einstein statistics f BE = (e ω/T − 1) −1 . A non equilibrium situation of Eq. (86) is described with a small deviation δρ from the thermal equilibrium state ρ eq , thus In Eq. (86) the collision terms, i.e. the anticommutators, vanish for ρ eq , thus the Liouville equation reduces tȯ where we have introduced G = diag(Γ, 0), with Γ = (1 − e −ω/T )Γ abs , i.e. the total collisional rate. We can write δρ as δρ = n A g g n S ; where n A is the occupation numbers of A quanta, n S the occupation numbers of S quanta and g represents the mixing between the two levels. If we insert Eqs. (89)-(91) into Eq. (90) we obtain the equations of motioṅ n A = −Γn A − 2µIm(g) ; (92) n S = 2µIm(g) ; (93) The mixing µ is always small so basically we are never far from the thermal equilibrium, i.e. f BE n A and f BE n S . In this limit Eq. (94) readṡ Assuming the initial condition g(0) = 0 the solution of Eq. (95) is then After an initial transient, Eq. (96) approaches the steadystate solution g ∞ = ∆ω + iΓ/2 If we insert this latter in Eq. (93) we finally obtain the S quanta production rate Γ prod S ≡ṅ S = Γµ 2 (ω A − ω S ) 2 + Γ 2 /4 1 e ω/T − 1 .
From Eq. (98) we obtain that the A − S mixing process is resonant, i.e. it is maximal for ω A = ω S . The result in Eq. (98) is completely general and it is valid both at the resonance and off-resonance, since it has been obtained on the only assumption that the mixing term µ is small relative to the diagonal terms ∼ ∆ω. This condition always applies in the solar plasma, both for photon TP and LP modes.
Hence, we can define the projection operator along the longitudinal direction as Q αβ = − 1 K 2 k 2 k 2 u α + ω K α k 2 u β + ω K β . (105) The orthogonal projector is defined as Notice that P and Q satisfy the properties P 2 = P , Q 2 = Q, and (P Q) µ α = (QP ) µ α = 0. With these definitions, the exact photon propagator in an isotropic medium has the form where π T,L are eigenvalues of the photon self-energy corresponding to the transverse and longitudinal direction, while α is the gauge parameter.
We can now proceed to extract the tensorial form of the axion self-energy from Eq. (100). First, let's notice that K = p + q t , where p is the axion momentum and q t = (0, q t ) is the momentum transferred from the magnetic field. The energies of the axion and of the photon are the same. Moreover, the axion and photon momenta have the same direction. Thus, we can calculate the momentum transfer as | q t | = | q γ | − | q a | = ω 2 − m 2 γ − ω 2 − m 2 a . Since | q a | ∼ ω, we can assume that p ∼ k, in our conditions. With this simplification, the term p µ F µα ext , which appears in Eq. (100), reduces to (− k · B, −ω B) in the plasma frame. Thus, we find p µ F µα ext q α = √ K 2 B T , and p µ F µα ext p µ F µβ ext P αβ = ω 2 B 2 − B 2 T = ω 2 B 2 L , where B L,T refer, respectively, to the direction parallel and perpendicular to the photon momentum. Thus, Π a,T = g 2 aγ ω 2 B 2 T K 2 − π T , Π a,L = g 2 We can now proceed to extract the imaginary parts. For the transverse mode we find: In a non-relativistic plasma, Re π T = ω 2 p . Moreover, we should interpret Im π T = −ωΓ T . So, finally, Im Π a,T = −g 2 aγ ω 2 B 2 T ωΓ T m 2 a − ω 2 p 2 + (ωΓ T ) 2 , (110) and so, For the longitudinal mode we find: Im Π a,L = g 2 aγ K 2 B 2 In a non-relativistic plasma, Re π L = K 2 ω 2 p /ω 2 . Thus, multiplying numerator and denominator by (ω 2 /K 2 ), we find Im Π a,L = g 2 We should therefore interpret (ω 2 /K 2 )Im π L = −ωΓ L . So, finally, Im Π a,L = −g 2 aγ ω 2 B 2 (115)