Dynamic solar Primakoff process

The Primakoff mechanism is one of the primary channels for the production of solar axion. In canonical estimation of the Primakoff photon-axion conversion rate, the recoil effect is neglected and a static structure factor is adopted. By use of the linear response theory, we provide a dynamic description of the solar Primakoff process. It is found that the collective electrons overtake ions as the dominant factor, in contrast to the static screening picture where ions contribute more to the photon-axion conversion. Nonetheless, the resulting axion flux is only 1-2% lower than the standard estimate based on the static structure factor.

The Sun is the primary natural source for terrestrial axion detection.With a coupling to standard model (SM) particles, axions can be produced in the solar interior through a number of channels, such as the Primakoff process [15,16] and the axio-recombination, bremsstrahlung and Compton scattering (ABC) process [17,18].For Kim-Shifman-Vainshtein-Zhakharov (KSVZ) axions [19,20], the former reaction dominates, while for the Dine-Fischler-Srednicki-Zhinitsky (DFSZ) axions [21,22], the latter mechanism dominates.In this paper, we focus on the Primakoff axion production mechanism, where photons convert to axions through the Coulomb field sourced by the charged particles (i.e., electrons and ions).
In conventional wisdom [8], the charged particles in the Sun are so heavy compared to the energies of ambient photons that they can be regarded as fixed, in which case the photon energy in a scattering event is considered equal to that of the emitted axion, and the differential cross section of the Primakoff process dσ γ→a (p γ ) /dΩ is proportional to |p γ × p a | 2 /Q 4 , with p γ and p a being the momenta of the incident photon and emitted axion, respectively, and Q = |Q| = |p γ − p a |.In the massless limit of an axion, the cross section is divergent due to the long-range Coulomb interaction.This Coulomb potential can be regulated if the solar in-medium screening effect is taken into account.Raffelt [8] argued that the implication of screening effect on the differential cross section is described with the substitution, where the Debye-Hückel scale κ can be as large as ∼ 9 keV at the solar center, and effectively provides a cutoff of the Coulomb interaction.Note that this description is based on the assumption of a negligible recoil effect in the solar medium, and thus a static structure factor S (Q) = 1 + κ 2 /Q 2 −1 is introduced to measure the correlation between the charged particle density [8,10].In Ref. [23], Raffelt further considered finite energy shifts in the axion production process in the presence of classical electric-field fluctuations in the solar plasma, so that the collective electron motion and the its implication for the axion conversion are unified under a general framework.Moreover, by use of the Kramers-Kronig relations that relate the spectral densities in the electromagnetic fluctuation description to the static structure factor S (Q), Raffelt reasoned that the Primakoff production rate in Ref. [8] agrees with the total rates of the decay process γ t (transverse plasmon) → γ l (longitudinal plasmon) + a (axion), the plasma coalescence process γ t + γ l → a, and the individual Primakoff process γ t + e/N (electron/ion) → a + e/N .Thus, as far as the calculation of the axion production rate is concerned, the static structure factor S (Q) has already included the collective behavior of the solar medium.These two seemingly different descriptions reflect the same electromagnetic properties of the solar medium.
Refs. [24][25][26] reproduced the same Primakoff production rate using the thermal field theory, by the same Kramers-Kronig relations argument in the last step.It should be noted that in order for the Kramers-Kronig sum rules to work, the energy shift between the axion and the photon ω is assumed to be remarkably smaller than the solar temperature T ⊙ such that 1 − e −ω/T⊙ ≃ ω/T ⊙ .Thus a detailed numerical examination of this assumption is one of the inspirations for this work.
Recently, in Ref. [27] we applied the nonrelativistic linear response theory to the dynamic screening effect in the nondegenerate gas of the solar plasma associated with the dark matter scattering.Under this framework, one no longer needs to add the dielectric function by hand, since both the finite temperature effect and the many-body effect are inherently encapsulated in the dynamic structure factor S (Q, ω)."Dynamic" means that a finite energy ω transfer and thus a temporal variation is taken into account in a scattering event, in contrast to the static case where the charged particles are regarded as fixed targets.This is important considering the thermal velocities of electrons can reach ∼ 0.1 c in the core of the Sun, which may bring a non-negligible Doppler energy shift in the Primakoff process.And more importantly, the screening and the collective effect (plasmon) are naturally incorporated into this dynamic structure factor S (Q, ω).This method could be an alternative to the electromagnetic fluctuation description [23] and the thermal field theory approach [24][25][26] mentioned above.
Therefore, the purpose of this work is to apply the linear response theory approach [27] to the photon-axion conversion process inside the Sun, in order to investigate the implication of the recoil effect and the collective effect, and especially to numerically explore in detail to what extent the Kramers-Kronig sum rule argument is reliable to validate the calculation of the Primakoff conversion rate based on the static structure factor.
Discussion will proceed in the natural units, where ℏ = c = k B = 1.

II. PRIMAKOFF EVENT RATE
We first introduce how we describe the Primakoff process in the context of the linear response theory that naturally encodes the relevant finite temperature physics and the many-body inmedium effect.
At the effective field theory (EFT) level, the interaction rel- evant for the Primakoff process is given as where a is the axion field, g aγ represents the axion-photon coupling, and F µν and Fµν = 1 2 ϵ µνρσ F ρσ are the electromagnetic field strength and its dual, respectively.
Since the electrons and ions move nonrelativistically in the Sun, we express the relevant interactions in the nonrelativistic EFT.For instance, the electromagnetic field-electron interaction is written as where ψ e is the nonrelativistic electron wavefunction.Considering that the second term on the right-hand side is subject to an electron velocity suppression ∇ me ∼ v e ∼ O 10 −2 ∼ 10 −1 in the solar medium (with m e being the electron mass), and the longitudinal and transverse photon propagators do not mix under the random phase approximation (RPA), only the longitudinal component in the nonrelativistic effective electron-photon interaction A 0 (or more specifically, the Coulomb interaction) is retained for the description of electron-electron (and electron-ion) interaction in this work.Thus, we only consider the components −g aγ a ϵ ijk0 ∂ i A j ∂ k A 0 of the Lagrangian in Eq. (2.1) in the estimate of the Primakoff process in the Sun.While the A 0 component is responsible for the Coulomb interaction, {A j } are relevant for the transverse photon external legs.
The calculation of the Primakoff process shown in Fig. 1 depends on an accurate description of the electronic and ionic in-medium effect inside the Sun.In this work, we invoke the linear response approach proposed in Ref. [27] to describe the screening effect in the Sun.Within this framework, the axion production rate for an incident photon with energy E γ and momentum p γ can be summarized by the following expression (see the Appendix A for further details): where α = e 2 /4π is the electromagnetic fine structure constant, ω and Q = |Q| denote the energy and the magnitude of the momentum transfer to the solar medium, respectively, V e (Q) = 4πα/Q 2 is the electron Coulomb interaction in momentum space, and E a (m a ) is the energy (mass) of the axion.The delta function represents the energy conservation in the scattering.
In this work, we only consider the case where the axion masses are so small (typically ≪ keV) compared to their energies that they can be effectively considered as massless.Besides, Eq. (2.3) does not take into account the fact that photons inside a plasma have a nontrivial dispersion relation , which means photons propagating in the solar medium have an effective mass ω p = 4παn e /m e .Consequently, the conversion process is only possible for E γ > ω p , and the plasmon mass effect becomes remarkable for energies E γ ≳ ω p .However, since ω p ≈ 0.3 keV is much smaller than the typical plasmon energy 3T ⊙ ≈ 4 keV in the solar core [28], where the majority of axions are produced, the photons are also treated as massless in Eq. (2.3), in line with the treatment in Ref. [8].Ref. [29] provided an analytical conversion rate that adds the plasma frequency to the original expression in Ref. [8], based on which we verify that the plasmon mass effect brings a correction at the order of 10 −4 between 1 and 12 keV.
The first and second terms in the square brackets in Eq. (2.3) correspond to the finite temperature many-body effect from the electrons and ions, respectively; while Im (Π e ) (Im (Π Ni )) is responsible for the thermal movement of the electrons (ions), the denominator describes the screening [27].Π e denotes the electron one-particle-irreducible diagram, in the RPA which is approximated as the bubble diagram.For the nondegenerate electron gas in the Sun, Π e can be expressed as [27] where n e is the number density of the electron gas, and the function Φ is defined as the Cauchy principal value of the integration [30] Φ medium, we introduce a nondimensional function F (Q, ω), which represents the second line in Eq. (2.3).Interestingly, from Fig. 2 a strong resonance structure is observed in the parameter area where the real part approaches zero in the denominator in Eq. (2.3), which corresponds to the absorption of a longitudinal plasmon in the process γ t + γ l → a.At the symmetric position in the upper half plane there is another pole corresponding to the emission of a plasmon in the process γ t → γ l + a.As long as kinematically allowed, such collective behavior may significantly alter the fixed-electron picture of the axion production process.F (Q, ω) provides a complete description of the longitudinal plasmon behavior far beyond the approximated dispersion relation Based on the axion production rate of Eq. (2.3), the differential axion flux reaching Earth can then be written as the convolution of the differential transition rate with the photon blackbody distribution in the Sun, with the Sun-Earth distance d ⊙ and the solar radius R ⊙ .
In the static screening prescription, since the energy of the incident photon equals that of the axion, the differential axion flux is given as [10] with the relevant static photon-axion conversion rate where κ 2 = (4πα/T ⊙ ) n e + i Z 2 i n Ni is the square of the Debye-Hückel scale.

III. AXION FLUX ON EARTH
Equipped with the above formulation that describes the solar in-medium effect with the linear response theory, now we are in the position to calculate the axion flux at terrestrial detectors.
In the left panel of Fig. 3 we compare the solar Primakoff axion fluxes on Earth computed with the linear response theory in Eq. (2.6) and with the static screening description of the Coulomb interaction in Eq. (2.7).
These spectra are obtained by integrating the contributions from the charged particles in every thin shell in the Sun, based on the Standard Sun Model AGSS09 [31].In practice, the solar radius is discretized into 100 slices, and 29 most common solar elements are included in our computation.Besides, we assume that these solar elements are fully ionized.
While in the left panel of Fig. 3 it is observed that the average (4.2 keV) and the maximum (3.0 keV) of the Primakoff axion energy distribution remain unchanged, the differential rate is only 1~2% lower than the calculation based on the  static structure factor in the energy range from 2 to 12 keV.
It is quite a surprising result, given that the denominator term 3) asymptotes to the Debye screening form 1 + κ 2 /Q 2 −2 in the limit ω → 0, where it brings a stronger screening than the static structure factor 1 + κ 2 /Q 2 −1 in Eq. (1.1); the contributions of the recoil effect and the collective effect must coincidentally make up this loss to keep the total rate unchanged.
Such a coincidence would be difficult if there were no intrinsic relation protecting the total rate, especially considering that in contrast to the static screening picture, where the contributions from the electrons and ions are in scale to the electric charge densities n e and i Z 2 i n Ni and hence a larger part of the conversion comes from the scattering with ions, it turns out that the collective electrons contribute dominantly to the total Primakoff axion flux in the dynamic screening picture.Thus, our results actually confirm the validity of the Kramers-Kronig sum rule argument in Ref. [23], up to a percent-level correction.
In the right panel of Fig. 3 we also present the differential solar axion flux (using the dynamic structure factor) as an apparent surface luminosity ϕ a (E a , ρ) of the solar disc [29,32,33], where the dimensionless quantities r = r/R ⊙ and ρ represent the radial position of the conversion process, and the distance from the center of the solar disc, respectively.

IV. DISCUSSIONS AND CONCLUSIONS
In order to further explore the many-body effect in the solar medium in detail, in Fig. 4 we present the differential axion production rates for photon energies E γ = 2 keV and 8 keV at the solar radius r = 0.1 R ⊙ , respectively, with the benchmark coupling g aγ = 10 −10 GeV −1 .While the spectra of heavy ions are found to narrowly center at the photon energies, behaving like static targets, it is intriguingly observed that the non-negligible electron movement in the inner part of the Sun can bring an energy shift up to O (0.1) keV from the initial photon energies.For one thing, the two peaks in Fig. 4 correspond to the absorption and emission of a longitudinal plasmon at ω ≃ ± 4παn e /m e in the Sun, respectively.That is, a considerable part of Primakoff processes proceed in company with absorbing and emitting a plasmon.For another, a broadening width of around 0.4 keV is also clearly seen due to the thermal movement of the electrons.While such a finite spread of the photon energy may not bring a noticeable change to the total spectrum of the solar axion, the strength of the resonance, i.e., the implication of the collective effect, can only be determined by concrete calculation.
To summarize, in this paper we have applied the linear response theory formalism for a delicate estimate of the Primakoff photon-axion conversion rate in the Sun.Based on this method, progress is gained in two aspects: (1) we provide an up-to-date panoramic description of the dynamic Primakoff process, which is explicitly shown as a combination of the decay process γ t → γ l + a, the plasma coalescence process γ t + γ l → a, and the individual Primakoff process γ t + e/N → a + e/N ; (2) beyond the approximate Kramers-Kronig sum rules, we numerically calculate the relevant terrestrial axion flux due to the Primakoff process, and the flux is found to be around 1~2% lower than the previous estimation based on the static structure factor.
Lastly, this dynamic response-oriented approach can be further applied to other axion production mechanisms such as electron-and ion-bremsstrahlung processes in the Sun, where a systematic treatment of the screening and collective effects are also highly useful.
Appendix A: Formulation for the Primakoff scattering event rate in the Sun In this appendix we give a detailed derivation of the formulas in the main text that describe the Primakoff photon-axion conversion process in the Sun.In the nonrelativistic regime, it is convenient to discuss in the Coulomb gauge.
We start with the T -matrix for the Primakoff process where a photon (with momentum p γ and polarization λ) scatters with a nonrelativistically moving electron (illustrated in Fig. 1), emitting an axion with momentum p a , i.e., where ελ (p γ ) is the polarization vector for the incident photon, which satisfies the complete and orthonormal relation, i.e., λ=±1 ελi (p γ ) ελj * (p γ ) = δ ij − p i γ p j γ / |p γ | 2 , and p γ • ε±1 (p γ ) = 0; E γ , E a , ε i , and ε j represent the energies of the photon, the emitted axion, the initial and the final state of the electron, respectively.
Then we take into account the many-body effect of the solar medium with the approach adopted in Refs.[27,34].To this end, we resort to the linear response theory, whereby the Primakoff event rate for a photon with momentum p γ (by averaging over the initial states and summing over the final states) is written as the following (for simplicity, here we assume only one type of ions with charge Ze and mass m N are present): where we introduce the density operators for electrons ρe (x) ≡ ψ † e (x) ψe (x) and ions ρN (x) ≡ ψ † N (x) ψN (x), p j represents the thermal distribution of the initial state |j⟩, the symbol ⟨• • • ⟩ represents the thermal average, ρeI (x ′ , t) ≡ e i Ĥ0t ρe (x ′ ) e −i Ĥ0t (ρ N I (x, t) ≡ e i Ĥ0t ρN (x) e −i Ĥ0t ), with Ĥ0 being the unperturbed Hamiltonian of the medium system, and V is the volume of the solar medium under consideration, which is only an intermediate quantity and is canceled in the final expression of the event rate.
Besides, in the above derivation we invoke the fluctuationdissipation theorem where T represents temperature, ρ generally stands for ρe and ρN , so S ρ ρ (Q, ω) represents the dynamic structure factor associated with the density-density correlation.In practice [35], one first evaluates the master function χ ρ ρ (Q, z) using the Matsubara Green's function within the framework of finite temperature field theory, and then obtains the retarded polarizability function χ r ρ ρ (Q, ω) by performing the analytic continuation χ r ρ ρ (Q, ω) = χ ρ ρ (Q, z → ω + i0 + ).Here we take the retarded correlation function χ r ρe ρN as an example to illustrate how the calculation is carried out, which is presented as the sum of all possible diagrams that connect the two density operators as follows, where and represent the electron and ion pairbubble diagrams, respectively, i.e., Π e and Π N at the RPA level (see Eq. (2.4)), and the double wavy line represents the electron Coulomb interaction screened by the ions (the single wavy line corresponds to the Coulomb interaction V e (Q)).Then Eq. (A4) can be explicitly written as The above discussion can be extended to obtain the following retarded correlation functions χ r ρN ρe and χ r ρN ρN such that, and In addition, in Ref. [27], we have already derived in Eq. (A2).As has been noted in Ref. [27], this expression encodes both the thermal movement, and the in-medium effect of the electrons and ions.
In practical computation of Eq. (A2), we first integrate out the polar angle of Q with respective to the direction of the photon momentum p γ , which is fixed as the z-axis in the spherical coordinate system.We then take a variable transformation from cos θ Qpγ to the variable E a = |p γ − Q| 2 + m 2 a , along with the corresponding Jacobian With this change of variable, the term proportional to sin 2 θ Qpγ in Eq. (A2), i.e., |p γ × Q| 2 can be rewritten as δ (E a − E γ + ω) where E ± = (p γ ± Q) 2 + m 2 a .In the last line, we take m a → 0, and Θ is the Heaviside step function.The integral area on the Q-ω plane corresponding to these step functions is shown in Fig. 5 for illustration.In the last step, we generalize the above expression to the multiple atom species in the Sun (as the sum over isotopes in the square brackets in Eq. (2.3)).

Figure 1 .
Figure 1.Diagram for the Primakoff scattering process where a photon is converted into an axion in the Coulomb potential of charged particles (electron and ions).See the text for details.

Figure 2 .
Figure 2. The factor F (Q, ω) that demonstrates the longitudinal plasmon resonance of the solar medium at radius r = 0.1 R⊙ (with R⊙ being the solar radius).The highly resonant "line" (below Q < 1 keV) is too narrow to be shown in the plot.It is evident that such plasmon suffers strong damping for Q 2 keV.See the text for details.

Figure 3 .
Figure3.Left: Solar Primakoff axion spectra on Earth calculated with the static structure factor (blue dashed), and with the dynamic structure factor (red) that consists of the contributions from electrons (green) and ions (orange), for a benchmark coupling strength gaγ = 10 −10 GeV −1 .∆ (in %) represents the relative difference between the two approaches, i.e., ∆ ≡ (static − dynamic) /static.Right: The contour plot of the solar axion surface luminosity, normalized to its maximum value, depending on the radius ρ on the solar disk and energy Ea.See the text for details.

Figure 4 .
Figure 4.The Primakoff differential photon-axion conversion rates for photon energies Eγ = 2 keV (left) and 8 keV (right) at the solar radius r = 0.1 R⊙, with contributions from electrons (green) and ions (orange), respectively.The plasmon absorption and emission peaks are clearly seen.

Figure 5 .
Figure 5.The effective integral area relevant for the Primakoff event rate of Eq. (A12).See the text for details.