Revisiting longitudinal plasmon-axion conversion in external magnetic fields

In the presence of an external magnetic field the axion and the photon mix. In particular, the dispersion relation of a longitudinal plasmon always crosses the dispersion relation of the axion (for small axion masses), thus leading to a resonant conversion. Using thermal field theory we concisely derive the axion emission rate, applying it to astrophysical and laboratory scenarios. For the Sun, depending on the magnetic field profile plasmon-axion conversion can dominate over Primakoff production at low energies ($\lesssim 200\,$eV). This both provides a new axion source for future helioscopes and, in the event of discovery, would probe the magnetic field structure of the Sun. In the case of white dwarfs (WDs), plasmon-axion conversion provides a pure photon coupling probe of the axion, which may contribute significantly for low-mass WDs. Finally we rederive and confirm the axion absorption rate of the recently proposed plasma haloscopes.


I. INTRODUCTION
The absence of CP violation in the quantum chromodynamics (QCD) sector is still a pressing mystery in Particle Physics. The solution of the Strong CP problem based on the Peccei-Quinn mechanism makes the QCD axion a very well motivated extension of the Standard Model [1 -3]. While the QCD axion is a pseudo-Goldstone boson defined by the interaction with gluons through the QCD coupling aGG, it also has model independent couplings to electromagnetism and to matter [4,5]. The QCD axion is also a viable candidate for dark matter (DM) [6][7][8]. Inspired by the "leave no stone unturned" principle and supported by string theory predictions [9][10][11], axion-like-particles ("axions" in the rest of this paper) generalize the QCD axion, as their mass is not fixed by the QCD coupling.
In the last three decades an increasingly intense theoretical and experimental effort has been dedicated to the search of such particles [12][13][14][15]. Most of these efforts take advantage of the axion coupling to transverse photons via the electromagnetic tensor, in the context of both astrophysical and laboratory probes. We focus on a less explored production (and detection) channel: the coupling between axions and longitudinal plasmons, electromagnetic excitations allowed by the presence of a medium.
The conversion of longitudinal plasmons in the presence of strong magnetic fields has first been pioneered in Ref. [16] in the context of supernovae. Inspired by similar works which focused on scalar and vector resonant conversion [17][18][19], we recast the calculation involving a pseudoscalar and an external magnetic field using thermal field theory, applying it to both astrophysical and laboratory systems. While it has already been used once, the approach based on thermal field theory is not widely spread in the literature and has been applied only on the production of axions in the magnetosphere of a magnetar [20].
We aim to revitalize plasmon-axion conversion in astrophysical environments and expand the work of Ref. [20] to new systems. In particular we consider the experimentally relevant systems of the Sun, white dwarfs (WDs), and the recently proposed plasma haloscopes [21]. While for the Sun the total axion luminosity due to the plasmon-axion conversion process is subdominant, this is not true in the low energy regime for the differential flux. As axion production from plasmon conversion in strong magnetic fields dominates in some energy ranges over more studied processes (such as the Primakoff effect, the conversion of photons in the electric field generated by nuclei [22]), these results have important implications for building axion observatories, for example motivating 1 − 100 eV-scale helioscopes. In WDs, we identify low mass, highly magnetized WDs as an ideal target. For higher mass WDs, the large core density and correspondingly high plasma frequency prohibits plasmon-axion conversion except in an outer shell.

II. AXION PRODUCTION FROM A THERMAL BATH OF PHOTONS
The effective Lagrangian which describes the coupling between axions and photons reads where the effective coupling g aγ accounts for both the mixing of axions and pions as well as a field-induced part given by a loop of fermions which couples to both the axion and the photon. Using thermal field theory we can concisely and elegantly derive the axion emission rate, first calculated with other methods in Ref. [16]. To begin, we recall that the emission rate of a boson by a thermal medium is related to the self-energy of the particle in the medium [23,24] Im Π = −ωΓ , where Γ = Γ abs − Γ prod is the rate by which the considered particle distributions approach thermal equilibrium. Using the principle of detailed balance the desired thermal production is found to be Therefore, we need to calculate the self-energy of the axion in the medium and then take the imaginary part. The axion self-energy due to an external magnetic field at lowest order is depicted in Fig. 1. Each vertex brings a factor m a g aγ B and the self-energy is easily written as where K = (ω, k) is the four-momentum of the external axion, Π γ is the self-energy of the photon and where we used the basis vector for the longitudinal degree of freedom (in Lorentz gauge) [13] l ≡ (k 2 , ωk) The vertex factors B || and B ⊥ are the parallel and perpendicular projections of the B-field ontok. Thus depending on the projection of the B-field both transverse and longitudinal photon modes contribute to the selfenergy.
Thus the photon self-energy is the quantity we need to complete our computation; the real part in the nonrelativistic approximation is easily found in the literature and, to lowest order in electron velocity, is given by [13,25] Re Π γ,T = ω 2 p , (6a) where ω p is the plasma frequency. We see that the dispersion relation for the transverse plasmon gives ω 2 − k 2 = ω 2 p , with the usual interpretation of transverse excitations as particles with mass ω p . The latter is given in the nonrelativistic limit in terms of the electron density n e by The longitudinal plasmon, on the other hand, has a peculiar dispersion relation, so that in the nonrelativistic limit ω is independent from k (see Fig. 2). The imaginary part of the photon self-energy is related, as stated above, to the rate Γ γ associated to electron-nucleus bremsstrahlung, Compton scattering or other processes keeping photons in thermal equilibrium.
For the longitudinal channel we are interested in, we can define the vertex renormalization constant Z L [26] relevant for the coupling of external photons or plasmons to electrons in the medium. Working in the static limit one can deduce [13] that magnetic fields associated with stationary currents are the same at distance, whether or not the plasma is present. The same is of course not true for the electric field, which gets affected by screening effects.
Neglecting the transverse part one can write where the factor Z L can be interpreted as renormalizing the coupling to the axion. Then, using Eq.
(2) we can interpret −Z L Im Π γ,L /ω as the damping rate for longi- Given that Γ L ω p , we notice a resonance for ω ω p , which gives a δ-function peaked around the plasma frequency (we will always need to integrate over phasespace) Dispersion relation for an axion with mass ma (dotted-dashed line), which we make large enough to be distinguishable from that of an ordinary photon (solid line), and the two plasmon modes, longitudinal (dashed) and transverse (dotted). For ma ≤ ωp the dispersion relation of a longitudinal plasmon always crosses the dispersion relation of the axion, thus leading to a resonant conversion.
where we used the definition of the Dirac δ-function Interestingly, for the resonant production of axions we do not need to calculate a production rate for either axions or photons [18]. With this result and B || =k · B the energy loss due to axion emission reads where we assumed m a ω p . Our Eq. (13) agrees with expressions previously derived with a different formalism in Ref. [16,20]. Note that there is a misprint in Eq. (1) of Ref. [16] that gives factor of two relative to our Lagrangian in Eq. (1). This is an energy loss per unit volume, meaning we just need to integrate it over the volume of the astrophysical object we consider to obtain the total luminosity. Different objects will have different temperature, plasma frequency and magnetic field, each of those being in principle function of the position inside the star.

III. ENERGY LOSS IN STARS
In this Section we apply the results obtained in Section II to two of the most experimentally relevant astro-physical systems, namely the Sun and WDs.

A. Plasmon conversion in the Sun
While the energy lost to axions does not have a measurable effect on the Sun, the solar axion flux can be detected by helioscope searches [22,25,[29][30][31]. In order to calculate the axion luminosity associated with the axion production from plasmon conversion, we need to know the temperature, the plasma frequency and the magnetic field profiles of the star. All these quantities can depend strongly on the radius, and their values are crucial to determine the importance of the process. They are obtained from a solar model, evolving several initial conditions (mass, helium and metal abundances) through a stellar evolution code. The latter depends in turn on radiative opacities, convection, and so forth, to fit the present-day radius, luminosity, and photospheric composition. While photospheric composition estimations can vary, the temperature and the plasma frequency profiles of the Sun predicted by different solar models are consistent to a degree sufficient for our purposes. Anticipating that the most relevant effect will be at low energies, we follow Ref. [32] and choose the reference Saclay model [27,28], constructed when the surface chemical composition GS98 [33] was suitable to reproduce heliseismology measurements, as it is to our knowledge the most complete concerning external layers. A comparison with the model of Ref. [34] with AGSS09 abundances [35] shows that the plasma frequency and the temperature uncertainty stemming from the solar model is less than a few percent, so we the flux produced by processes which depend only on these quantities have theoretical uncertainties of less than 10%. For the magnetic field the situation is less clear. In fact, there is no wellestablished picture of the magnetic field in the interior of the Sun [36,37]; hence we will consider three scenarios: 1. The simplest scenario in which the magnetic field is constant over the entire star. We assumed different values for B. The range we considered spans from the most pessimistic to the most optimistic case: B = [10 5 , 7 × 10 6 ] G [36]. In the following we will show the results for the caseB = 10 5 G; 2. For a more nuanced model, we parametrize the magnetic field with a step function, taking B = 7 × 10 6 G [36] in the interior region of the Sun up to the beginning of the convective zone ( 0.75 R ), where we assume B = 10 3 G [37]; 3. Finally we considered the seismic Solar model of Ref. [38], where the authors studied in detail the solar neutrino fluxes and divided the magnetic structure of the Sun in three zones: the radiative interior, the tachocline and the upper layers. For these three regions different possibilities were considered; as an example we considered here the model of Ref. [38] named seismic-B 21 .
10 -3 10 -2 10 -1 1 10 -3 FIG. 3. Internal solar properties from the reference Saclay model [27,28]. We plot the temperature, the plasma frequency and the Debye screening scale. For the latter we plot the total contribution (solid blue line), as well as the individual ones from electrons (dotted blue) and ions (dashed blue).
In Fig. 4 we show the luminosity as a function of the effective coupling between axions and photons. The solid red curve corresponds to the seismic-B 21 model of Ref. [38]. The dashed red line corresponds to the luminosity from plasmon conversion with the magnetic field configuration where θ is the Heaviside step function. Finally the dotted red lines was drawn assuming a constant magnetic field ofB = 10 5 G.
We notice that in all the following computations we assume the plasma frequency to be given by the free electron contribution only. Species which are not completely ionized could contribute significantly to the plasma frequency [39], but we expect the error in neglecting the contribution of the bound-bound transition to be comparable or smaller than the uncertainty on the magnetic field in the interior of the Sun. This effect would be most sigificant in the external layers of the Sun.
To compare with the Primakoff effect, we estimate the total energy loss [13] where is the Debye screening scale in a nonrelativistic, nondegenerate plasma and Luminosity associated with the Primakoff effect (solid blue line) and plasmon conversion in the interior of the Sun (red curves). For the latter, the solid line corresponds to the seismic-B21 model of Ref. [38], the dotted red line corresponds to B(r) =B = 10 5 G and the dashed red line corresponds to B(r) = 7 × 10 6 θ(0.75R − r) + 10 3 θ(R − r) G.
The same expression can be found in the framework of thermal field theory [40,41]. We stress that while ions do not contribute to the plasma frequency, as forward scattering is suppressed by their large mass, they should be included when estimating the Debye screening scale. The Primakoff contribution is shown as a dashed blue line; unless the magnetic field takes unrealistic large values, the plasmon conversion is a negligible correction to the luminosity generated by the Primakoff effect. However, this does not mean that the Primakoff effect is the dominant source of axions all over the spectrum.
As the energy dependence is different between plasmon conversion and the Primakoff effect, it is worth to investigate the differential axion flux to the Earth. For the Primakoff effect the differential flux at Earth is usually expressed by [13,42] dΦ Pr dω = g 2 10 6.02 × 10 10 cm −2 s −1 keV −1 (ω/keV) 2.481 e (ω/keV)/1.205 , However, while this is a good approximation in the range [1 − 11] keV, it is not accurate for the low energy tail we are interested in. We therefore computed the differential flux using our solar model of reference [27,28] and the axion emission rate [13,31,43] where x = (k 2 a + k 2 γ )/2k a k γ and y = x + k 2 s /2k a k γ . The dispersion relation of the photon requires the frequency of the photon to be always larger than the plasma frequency at a given radius. This requirement further suppresses Primakoff contribution at low frequencies, as production only occurs in the outer layers of the sun. We show the Primakoff differential flux in Fig. 6 (solid blue curve), while in the Appendix. C we give some details of the calculation.
The flux produced by the longitudinal plasmon conversion reads instead where δ-function will be used to integrate over the radius. For a given axion frequency ω the equality ω = ω p (r 0 ) fixes the value of the radius r 0 at which the integrand needs to be evaluated. Therefore which we notice to have a different functional dependence on the energy with respect to Eq. (19). Compared to the Primakoff process, which produces a peak in the spectrum around 3 − 4 keV [13,44,45], axion-plasmon conversion has peaks at very low frequency, 1 − 100 eV depending on the assumed magnetic field. This shift is due to the fact that the axion frequency matches the plasma frequency, which is limited to relatively small values, ω p 0.3 keV. In Fig. 5 we show the differential axion flux from longitudinal plasmon conversion, which overcomes Primakoff conversion at low frequencies ( 200 eV). As for the luminosity in Fig. 4, we show the results for three different configurations of the the internal magnetic field. Interestingly enough the spectral features of axion emission map the magnetic structure of the sun. Consider the seismic model (solid red curves): three regions are evident from Fig. 5, which correspond to different shells of the sun and consequently to different plasma frequencies (hence axion energies) and different magnetic fields. For example, the radiative interior corresponds in Fig. 5 to the solid red curve at higher energies. In this region the density and the plasma frequency are high, therefore the produced axion will have large energies ω 2 × 10 −2 eV; furthermore the magnetic field is 5 × 10 7 G, which significantly enhances the conversion rate.
The lower energy peaks of plasmon-axion conversion are potentially very interesting for axion helioscope designs. Helioscope such as CAST or IAXO [46,47] are designed for X-ray energies, where cavities and optics are very difficult to build. While the flux in the high UV is lower, it may prove to be more easily instrumented, or be enhanced by a mildly resonant cavity. We plot the axion spectrum for g aγ = 10 −10 GeV −1 from 0.1 eV to 10 keV in Fig. 6. Here we assume that the axion can always be considered ultrarelativisitc. At the lowest energies, 1 eV, axions generated by Bremsstralung in the Earth dominate even if the axion electron coupling is suppressed [48]. In this case the differential flux is found to be thermal dΦ/dω ∝ ω 3 /(exp(ω/T c ) − 1), with T c being the temperature of the Earth's core [49]. We stress that this flux, often overlooked in previous plots of the "grand unified axion spectrum" (in analogy to photon [50] and neutrino [51] spectra), should fill the gap between axions from the Sun and a population of thermally produced axions constituting dark radiation [15]. For the comparison we fixed the axion-electron coupling to be g ae = 2 × 10 −4 |g aγ | GeV; this value can vary and may be larger depending on the considered model (see for example the recent review [52]). Here we considered a benchmark value for models where the axion-electron coupling is suppressed, and only occurs at one loop due to the presence of an axion-photon coupling. While relevant for eV scale experiments, the flux produced from the Earth has very different directionalty, meaning that it is experimentally distinct from longitudinal plasmon conversion in the Sun. At high frequencies 200 eV Primakoff production takes over, giving the traditional window for helioscopes. However, while the exact spectrum depends heavily on the magnetic field structure of the sun, longitudinal conversion is very important for intermediate energies. Here we have plotted the seismic-B 21 model, though this statement holds more generally.
Intriguingly, emission at this intermediate energy range would allow one to corroborate a signal by showing structure (a double peak in the simplest model) in the axion flux, and give further information on the internal structure of the Sun, as also shown in the context of neutrinos and of axions with electron coupling [25,32]. It is evident from Fig. 5 that at least an upper bound of the Sun's internal magnetic field can be obtained. In fact, as the spectrum produced by plasmons essentially maps the magnetic field as a function of plasma frequency, one should be able to largely reconstruct the internal magnetic field as a function of radius inside the Sun, further showing the potential of "axion astronomy" to investigate the interior of the Sun [53].

B. White Dwarfs
Now we turn our attention to another interesting astrophysical candidate, WDs.
These stars are in some ways a simpler system, being nearly isothermal and degenerate, with typical densities of order ρ WD 1.8 × 10 6 g cm −3 and temperatures T WD 3 × 10 6 − 2 × 10 7 K [13]. Moreover they often exhibit very strong magnetic fields, Before proceeding we have to adapt the previous formalism to the degenerate case. We will use expressions depending on the temperature T and the electron chemical potential µ valid in any plasma condition [13,54]. First, we introduce the sum of the phase space distributions for e ± , so that the plasma frequency ω p and the characteristic frequency ω 1 are given by where α em = e 2 /4π and v = p/E. Finally, the real part of the photon self-energy can be written as where v * is the "typical" electron velocity in the medium defined as the ratio v * ≡ ω 1 /ω p and G is an auxilary function, defined as The imaginary part of the axion self-energy now reads Interpreting again Im Π L γ = −ωΓ L , we are left with As before we can take the limit of small damping rate to find where we have defined the renormalization factor which generalizes Eq. (7) to a degenerate plasma [13] moreover, we defined The emission rate will thus be δ(ω−ω 0 (k)) .
(34) Degenerate stellar systems have long been used as probes of axions by studying the possibility of energy loss from axion emission [55,56]. Interestingly, several hints have been measured of a preference for an additional, unaccounted for, cooling channel of these stars [57,58]. The observations include the rates of period changes of several systems [59][60][61][62][63] and the luminosity function, the number of WDs per unit bolometric magnitude and unit volume, which tracks the cooling of these stars [64]. Other degenerate systems showing excessive cooling are the red giants [65,66]. More recently the authors of Ref. [67] considered X-ray signatures of axion conversion in magnetic WD stars: the axions are produced in the interior of the star via the coupling to electrons We report the axion flux from Earth (dark green) given in Ref. [48], and both the Primakoff (blue) and longitudinal plasmon (red) spectra from the Sun as considered in this work. For the longitudinal plasmon conversion we only report the results for the seismic-B21 model. As the focus of this work is on the photon coupling, we plot the results of Ref. [48] assuming a suppressed gae = 2 × 10 −4 |gaγ|GeV. and then converted to photons in the external magnetic field. The main process in this case is electron bremsstrahlung in electron-nuclei scattering. The luminosity associated to this process can be written as as [31,49,64] L a L 1.6 × 10 −4 g ae 10 −13 where F is a factor that depends on the density and composition of the star, but is usually ∼ O(1). Similarly to the axion geoflux, the differential axion flux is thermal dΦ/dω ∝ ω 3 /(exp(ω/T c ) − 1) with T c now being the temperature of the WD's core.
Here we want to consider the possibility of breaking the degeneracy between the couplings g ae and g aγ , producing the axion via the longitudinal plasmon conversion which relies only on the axion-photon coupling. Then, once the axions are produced, they travel from the WD center outwards, where they can be converted into photons in the magnetic field surrounding the star [67][68][69][70] with a probability p a→γ , which depends mainly on the magnetic field and the coupling g aγ (see Appendix B).
The electromagnetic flux at the Earth reads finally where d WD is the distance of the WD. While the axion luminosity produced by bremsstrahlung can be expressed with Eq. (36), to compute the contribution of plasmon conversion we will use a detailed WD model. We do this because the axion production depends very strongly on the plasma frequency, which varies strongly as a function of radius, as shown in the middle panel of Fig. 7. This model is built from asteroseismological observations [56,[71][72][73][74].
The model we will employ, 1 hereafter referred to as Model-1, describes a WD with mass M = 1.2919 M at effective surface temperature T eff = 40,960 K. While WDs are isothermal for large part of their profile due to the large electron degeneracy, which implies a long mean free path the electrons and consequently a large thermal conductivity, the Fermi energy (and correspondingly the electron density) will depend on the radius. We thus anticipate that the flux will be larger around ∼ 5 keV, as the axions will be produced in the outer shell of the star where the ratio between the plasma frequency and the temperature is smaller, softening the suppression due to the plasmon population (see Fig. 7). The vast majority of axion-plasmon conversion occurs in the outer ∼ 10% of the star.
We can now compute the total flux for the longitudinal plasmon conversion and compare it with that from bremsstrahlung using Eq. 36. In Fig. 8 we show the comparison between the two effects as a function of the magnetic field. We consider both Model-1 (solid line) and a more optimistic scenario Model-2 (dashed line) in which we re-scaled the central density to lower values ρ core ∼ 10 6 g/cm 3 . Again, for comparative purposes we fixed the axion-electron coupling to be g ae = 2 × 10 −4 |g aγ | GeV.
From Fig. 8 it is evident that there is a strong dependence over both the magnetic field and the density profile. In fact, keeping fixed the temperature profile, the smaller the density, the smaller the ratio between the plasma frequency and the temperature. This then implies a thicker shell in which plasmon production is effective. Further, lower density is typically reached in less massive WDs, which are actually larger in radius [75]; for Model-2 we thus considered a re-scaled mass of 0.4 M .
Furthermore, the closer you go to the core of the star the higher is the magnetic field, which is driving the resonant conversion. Indeed some theoretical studies consider the possibility that internal magnetic fields may even be as large as 10 12 − 10 13 G [76,77]. Theoretical models of WDs with very high core magnetic fields, but low surface magnetic fields, were made in Ref. [78]. We thus identify as an ideal target a strongly magnetic WD with small mass and central density. While rare, it may be possible for plasmon conversion to dominate in such a WD. In such a situation the observational strategy would be the same as the one discussed in Ref. [67], as the signal is an X-ray spectrum with the peak around 5 keV (see also Ref. [79,80] for other possible X-ray analyses).

IV. PLASMA HALOSCOPES
Lastly, we turn our attention away from the stars and into the lab. A recent use of plasmas in the literature is the proposal of a cryogenic plasma inside a strong external magnetic field to DM axions on Earth [21]. Such a device would be capable of exploring well motivated the high mass parameter space, inaccessible to other experiments. With the formalism used here it is easy to check the power produced by such an experiment, which so far has only been calculated classically. For simplicity we will consider the medium to be isotropic and relatively large, such that boundary effects are unimportant.
As DM axion is nonrelativistic the velocity is negligible. Thus to first approximation it becomes impossible to distinguish the couplings to longitudinal or transverse photons. Because of this both contributions must be calculated in order to get a correct rate.
The DM axion absorption rate Γ abs is simply given by where the latter holds in the limit that T → 0 (a good approximation considering cryogenic temperatures). The transverse part can be handled similarly to the longitudinal part studied above. Using the on-shell condition K 2 = m 2 a and treating the axion nonrelativistically, ω = m a , we see that where we have used equation (2) to define Γ T = −Im Π T /ω. As longitudinal and transverse photons are indistinguishable in the zero momentum limit, Γ T = Γ L ≡ Γ. As both contributions are now equal up to the projection of the magnetic field, we find on resonance (m a = ω p ) that Thus the power absorbed from a number of DM axions N is simply given by where Q = ω/Γ is the "quality factor" and ρ a is the local DM density. Equation (41) is in exact agreement with Ref. [21] in the limit where boundary conditions are negligible (in their notation, the "geometry factor" tends to unity). Thus we confirm the classical calculation of Ref. [21], and see that for a sufficiently large medium so that boundary effects are unimportant both transverse and longitudinal polarizations play a significant role in the generated signal.

V. CONCLUSION
In this paper we have reconsidered the calculation of axion emission from a thermal bath of photons using thermal field theory. As longitudinal plasmon-axion conversion is resonant, but largely neglected in the literature, many interesting physical environments are yet to be properly explored. In the interest of revitalising plasmonaxion conversion, we have applied our results to the most relevant astrophysical and laboratory targets, comparing the energy loss due to plasmon-axion conversion to other processes and to the present experimental bounds. In different energy regimes this new process dominates over Primakoff or bremsstrahlung processes.
We first considered the closest source of astrophysical axions, the Sun. While the luminosity of solar axions is largely set by the Primakoff effect, at low energies plasmon-axion conversion provides a new and dominant source of axions. This new flux motivates 1 − 100 eVscale helioscope experiments. Such an experiment would benefit from the improvements to optics at low energies. In the event of a discovery a low energy helioscope would provide an additional probe of stellar structure.
For WDs this resonant conversion mechanism provides purely photonic contribution to the axion flux. For high mass WDs, the high inner density suppresses axion plasmon conversion, so only the outer shell contributes and the flux is subdominant relative to bremsstrahlung. It is possible that in low mass, high temperature and high magnetic field WDs plasmon production may dominate for electrophobic axions, leading to stronger bounds on the axion-photon coupling.
Lastly we used our thermal field theoretic calculations to confirm the behaviour of plasma haloscopes in the large medium limit, demonstrating that both transverse and longitudinal modes contribute to the absorption of dark matter axions.
As plasmon-axion conversion is relatively unexplored compared to more traditional production mechanisms, there are still many astrophysical and laboratory environments to explore. A prime example is the magnetosphere of neutrons stars, where the density and the magnetic field strength are very promising. However the magnetic fields involved are too strong for the simple treatment outlined here and thus left for future work.
Here we derive the factors entering the vertices of the axion self energy, as used in Eq. (4). We restrict ourselves to the computation of the longitudinal component, which is the focus of the present work. An analogous calculation can be performed also for the transverse mode.
The Lagrangian is given by While it is tempting to use A ν as the electric field, and F µν to provide the magnetic field, this would actually undercount how many ways one can assign these fields, leading to a reduction by a factor of two. To calculate the vertex factor, the easiest starting point is to rewrite Eq. (A1) in terms of the electric and magnetic fields, where we have used that E i = F 0i and B i = 1 2 ijk F jk . This writing allows us to unambiguously assign B to be the external magnetic field. We can now calculate the contribution to the vertex factors, where in the second line we used that the longitudinal plasmon is a plane wave with polarization vector given by Eq. (5) and, as we are considering the axion self energy, √ ω 2 − k 2 = m a . Note that there are two vertices entering Eq. (4), leading to a mod squaring.