Neutrino photoproduction on electron in dense magnetized medium

The effect of a strongly magnetized cold plasma on the Compton-like photoproduction of a neutrino-antineutrino pair on an electron, $\gamma e \to e \nu \bar \nu$, has been considered. The contribution of this process to the neutrino emissivity in both the non-resonance and the resonance cases has been calculated with taking account of the photon dispersion properties in medium. Our results show that the neutrino emissivity owing to the $\gamma e \to e \nu \bar \nu$ reaction is significantly modified as compared to the previously reported data.


Introduction
The problem of a correct description of effects of an active external medium (a strong magnetic field and/or dense plasma) on quantum processes is of current interest during a long time, for a review see e.g. [1]. First, these effects are caused by the sensitivity of charged fermions (primarily, electrons as particles with the largest specific charge) to the field. Second, a strongly magnetized plasma significantly changes the dispersion properties of photons and, thereby, the kinematics of the processes.
The conditions of the strongly magnetized plasma can exist in the interiors of magnetars, i.e., isolated neutron stars with magnetic fields much higher than the critical value B e = m 2 /e ≃ 4.41 × 10 13 G 1 . Recent observations (see [2] and the papers cited therein) make it possible, in particular, to identify some astrophysical objects (SGR and AXP) as magnetars.
At the same time, all known theoretical models of the internal structure of neutron stars give the parameters of the medium (density and temperature) at which the magnetized plasma is transparent to neutrinos. In this case, reactions with a neutrino-antineutrino pair in the final state are decisive for neutrino cooling. In this connection, the neutrino photoproduction (the so-called photoneutrino process) γe → eνν was studied by different authors [3][4][5][6]. The formulas were obtained for the neutrino emissivity, i.e., the energy carried by the neutrino from the unit volume of a star per unit time, for both the nonrelativistic and relativistic plasmas. However, in those papers the anisotropy in the dispersion of photons was disregarded, which can change the results essentially. Furthermore, the expressions for the neutrino emissivity owing to the Compton-like process γe → eνν in nonrelativistic and relativistic plasmas contain some inaccuracies [4][5][6].
In this work, we study in detail the Compton-like photoproduction of the neutrino-antineutrino pair γe → eνν and the neutrino emissivity owing to this process with an accurate inclusion of the dispersion properties of photons in dense magnetized medium in both resonance and nonresonance cases.

Photon dispersion in the magnetized medium
The propagation of electromagnetic radiation in any active medium is conveniently described in terms of normal modes (eigenmodes). In turn, the polarization and dispersion properties of normal modes are connected with the eigenvectors and the eigenvalues of the polarization operator P αβ , respectively. It is convenient to decompose the tensor P αβ in terms of the basis of 4-vectors [7] constructed of the electromagnetic field tensor reduced to a dimensionless form, and the 4-momentum of a photon q α = (ω, k): which are the eigenvectors of the polarization operator in a static uniform magnetic field. In Hereafter we use the following notations: the 4-vectors with the indices ⊥ and belong to the Euclidean {1, 2} subspace and the Minkowski {0, 3} subspace correspondingly. Then for arbitrary 4-vectors A µ , B µ one has where the matrices Λ µν = (ϕϕ) µν , Λ µν = (φφ) µν are constructed with the dimensionless tensor of the external magnetic field, ϕ µν = F µν /B, and the dual tensor,φ µν = 1 2 ε µνρσ ϕ ρσ . The matrices Λ µν and Λ µν are connected by the relation Λ µν − Λ µν = g µν = diag(1, −1, −1, −1), and play the role of the metric tensors in the perpendicular (⊥) and the parallel ( ) subspaces respectively.
We emphasize that, in contrast to the magnetic field, the value E e = B e for the electric field is limiting because the generation of the nearly critical electric field in a macroscopic region of space would result in the intense production of electron-positron pairs from vacuum. At the same time, if the electric field is perpendicular to the magnetic field, the electric field E e can exceed the critical value B e , remaining below B. However, even in this case, it is always possible to perform a Lorentz transformation to a reference frame where only the magnetic field exists. This statement can be generalized to the case where the plasma with a temperature T and a chemical potential µ moves as a whole along the magnetic field. To this end, it is sufficient to represent the electron distribution function in an explicit Lorentz-invariant form in terms of the 4-velocity of the medium u α : The condition of the absence of the electric field in this frame can be written in the relativistically covariant form (uΛ) µ = 0. Consequently, when the plasma moves as a whole along the magnetic field, it is possible to consider the situation where only the magnetic field exists.
In this case we can obtain the following expansion of P αβ over the eigenvectors r (λ) α with the eigenvalues κ (λ) [9][10][11][12]: where A (λ) i are some complex coefficients. As we can see from Eq. (3), it is rather difficult to determine the dispersion properties of photons for all three polarizations, and it is true even in the strongly magnetized plasma approximation, β ≫ m 2 , µ 2 , T 2 (hereafter β = eB). However, as the analysis shows (see, e.g., [4,13]), in the case β ≫ m 2 , where electrons occupy the ground Landau level, only photons with the polarization corresponding to the vector r α will provide the leading contributions to the amplitude of the γe → eνν process in the external field. In the cold plasma approximation, ω, T ≪ µ − m, the expression for the eigenvalue κ (2) can be obtained analytically and represented in the form where ω 2 p = (2αβ/π) v F is the so-called plasma frequency squared, p F is the Fermi momentum. In this case, the solution of the dispersion equation for a photon of the mode 2 propagating at a nonzero angle θ to the magnetic field direction can be found analytically as the function ω = ω(k) taking the form In the case of a nonrelativistic plasma, where v F ≪ 1, we obtain The corresponding expression for the relativistic plasma, where v F → 1, has the form One more question which appears to be important in some cases, is the renormalization of the wavefunction of a photon. We note that this renormalization becomes insignificant in the case of a cold plasma because the main contribution to the physically observed characteristics (e.g., emittance) comes from the photon energy range ω ≪ m.

Neutrino emissivity in the non-resonance case
Our main goal is to obtain the expression for the neutrino emissivity caused by the process γe → eνν. In turn, the neutrino emissivity can be defined as the zero component of the fourvector of the energy-momentum carried away by the neutrino pair due to this process from a unit volume of plasma per unit time. Here, we neglect the inverse effect of the energy and momentum loss on the state of plasma. In the conditions of a strongly magnetized plasma the neutrino emissivity can be represented in the form [8]:  (2)); q ′ 0 = ε 1 + ε 2 is the neutrino pair energy, ε 1,2 = |p 1,2 |; d 2 p = dp y dp z ; V = L x L y L z is the plasma volume, P µ = (p + q) µ .
In calculating the amplitude M γe→eνν of the process γe → eνν, we consider the case of relatively small momentum transfers compared with the W boson mass, |q ′2 | ≪ m 2 W . Then the corresponding interaction Lagrangian can be written as follows: where j ′ α =νγ α (1 − γ 5 )ν is the current of left-handed neutrinos, A α is the four-potential of the photon field, C V = ±1/2 + 2 sin 2 θ W , C A = ±1/2, θ W is the Weinberg angle. Here, the upper sign corresponds to electron neutrinos (ν = ν e ), when there is an exchange reaction both of W and Z bosons. The lower sign corresponds to the µ and τ neutrinos, when there is only Z boson exchange.
The amplitude of the process γe → eνν in the tree approximation is described by the Feynman diagrams shown in Fig. 1 and in the non-resonance case (i.e. when the virtual electron is on the ground Landau level) has the form [4] where q ′µ = (p 1 + p 2 ) µ is the total 4-momentum of the neutrino-antineutrino pair, κ = The resulting expression for the emissivity of the photoneutrino process can be significantly simplified in two limiting cases.
(i) In the case of the nonrelativistic plasma, µ ∼ m, and at an arbitrary relation between the plasma frequency and the temperature, the emissivity can be presented in the form where is the emissivity in the limit ω p ≪ T [3]. In Eq. (13) B 15 = B/(10 15 G), ρ 6 = ρ/(10 6 g/cm 3 ), T 8 = T /(10 8 K), and the constants C 2 V = 0.93 and C 2 A = 0.75 are the results of summation over all channels of the neutrino production of the types ν e , ν µ , ν τ .
The function F (ω p /T ) depending on the ratio of the plasma frequency to the temperature can be represented in the form of the single integral This integral can be approximated by the formula F (y) ≃ 3e −y 4π 8 2y 7 + 15y 6 + 95y 5 + 495y 4 + 2040y 3 + 6240y 2 + 12600y + 12600 .
The plot of the function F (y) is shown in Fig. 2. We note that the numerical analysis of the integral (14) as compared to the approximation (15) gives a discrepancy no more than 0.5%. Our results (12)-(15) for the case of the nonrelativistic plasma should be compared with the results of the previous calculations, see Eq. (56) of Ref. [3], Eq. (32) of Ref. [4] and Eq. (32) of Ref. [6]. There was a discrepancy between those formulas in the numerical factor of π/2. Now we should confess that the factor of π/2 was lost in Eq. (32) of Ref. [4]. However, it should be noted that all those formulas, with the indicated correction, were obtained in the limit ω p ≪ T , i.e. when the approximation for the function (14) was taken: F (y) → F (0) = 1. As one can see from Fig. 2, this approximation is valid for the case ω p T , but it would give an essentially overestimated result in the case ω p > T . The results (12)-(15) obtained can be used for an arbitrary relation between the plasma frequency and the temperature.
(ii) In the case of relativistic plasma, µ ≫ m, and at an arbitrary relation between the plasma frequency and the temperature, the emissivity can be presented in the form where is the emissivity in the limit ω p ≫ T [5,6].
The function R(z) can be represented in the form of the double integral  The plot of the function R(z) is shown in Fig. 3. We note that the numerical analysis of the integral (18) as compared to the approximation (19) gives a discrepancy no more than 0.8%. Therefore, the approximation (19) can be used for applications at an arbitrary ratio ω p /T . According to Fig. 3 and approximate formula (19), for the realistic parameters of the field and plasma of neutron stars, B ≃ 50 B e , T ≃ 10 8 K [8], the factor is R(14) ≃ 3 and the asymptotic expression R(∞) ≃ 1 is inapplicable. Furthermore, it follows from Eq. (16) that the numerical estimate presented in Ref. [4] is strongly overstated. At the same time, for the parameters, used in Ref. [5]: B ≃ 10 16 G, ρ ≃ 10 9 g/cm 3 , T ≃ 10 9 K, we obtain from Eq. (17): Q r ≃ 10 11 erg/(cm 3 s), but for the same parameters one has R(2.8) ≃ 260. Thus, the estimate obtained in Ref. [5] is understated by several orders of magnitude. However, for the realistic parameters of the inner crust of neutron stars (see above) we obtain from Eq. (16): Q γe→eνν ≃ 6 × 10 −7 erg/(cm 3 s). It is reasonable to neglect the neutrino photoproduction, γe → eνν, in the models of cooling of neutron stars at the matter density ρ ≃ 10 9 g/cm 3 . But at the same time, the resonance on the virtual electron in the process γe → eνν becomes possible at the same density. Because of this, the value Q γe→eνν would increase (see the next section).

Resonance in the process γe → eνν
On the boundary between the outer and the inner crust of a magnetar (where the electron density is estimated as ρ 10 9 g/cm 3 ), the higher Landau levels for a virtual electron begin to be excited. In this case the denominator of the electron propagator P 2 − m 2 − 2eBn can be equal to zero and the resonance on the virtual electron become possible. In addition, the virtual electron resonance occurs only in the s-channel diagram (the first diagram in Fig. 1). On the other hand, to accurately take into account the resonance behavior in the process γe → eνν, it is necessary to calculate radiative corrections to the electron mass, caused by the combined action of a magnetic field and plasma. This calculation is a separate challenge. However, because of the smallness of these corrections, we can approximately replace m 2 → m 2 − iP 0 Γ n in the denominator of the electron propagator, such that In this case, the main contribution to the amplitude arises from the resonance region, so that we can approximately replace the corresponding part of the amplitude (see [14]) by the δ function: where Γ n is the total width of the change of the electron state. This width can be represented in the form [15] Γ n = Γ abs + Γ cr ≃ Γ cr e 0 γ→en 1 + e (E ′′ n −µ)/T .
Here Γ cr e 0 γ→en = is the width of the electron creation in the nth Landau level.
On the other hand, in the case of resonance the expression for |R n | 2 being averaged over the photon polarizations can be factored in the strong field limit β ≫ m 2 as follows (see, for example, [16]): where is the amplitude of the absorption of a photon in the process e 0 γ → e n , when an electron passes from the ground Landau level to a higher Landau level n. The parameter σ = (pφq)/(pΛq) = ±1 determines the direction of the photon propagation with respect to the magnetic field direction. The values A (λ) i are introduced in Eq. (3). Finally, the amplitude squared of the electron transition from the nth Landau level to the ground level with the creation of the neutrinoantineutrino pair takes the form: With taking account of Eq. (22), the amplitude squared of the process γe → eνν takes the form: Here we have used the property of the δ function: where E ′′ n = p ′′2 z + m 2 + 2βn. Substituting Eq. (27) into the expression for the luminosity (9), and taking into account Eqs. (23) and (24), we obtain: where is the neutrino luminosity due to the process e n → e 0 νν [8].

Conclusion
In conclusion, let us summarize some of our results. We have considered the neutrino photoproduction on an electron, eγ → eνν, in dense magnetized medium in both resonance and non-resonance cases. The changes of the photon dispersion properties in a magnetized medium are investigated. It has been shown, that taking into account of the photon dispersion anisotropy in the limit of non-relativistic plasma leads to substantial modification of the neutrino emissivity due to the process γe → eνν, if compared with the previously obtained results. We have obtained the most general expression for the neutrino emissivity due to the process γe → eνν in relativistic and non-relativistic plasma at an arbitrary relation between the plasma frequency and the temperature. It has been shown that the result known in the literature for the contribution of the photoneutrino process to the neutrino emissivity in the limit of a relativistic plasma was understated by several orders of magnitude. It has been shown that in the case of resonance on the virtual electron, the neutrino emissivity due to the process γe 0 → e 0 νν can be expressed in terms of the neutrino emissivity due to the process e n → e 0 νν.