Instability of magnetic fields in electroweak plasma driven by neutrino asymmetries

The magnetohydrodynamics (MHD) is modified to incorporate the parity violation in the Standard Model leading to a new instability of magnetic fields in the electroweak plasma in the presence of nonzero neutrino asymmetries. The main ingredient for such a modified MHD is the antisymmetric part of the photon polarization tensor in plasma, where the parity violating neutrino interaction with charged leptons is present. We calculate this contribution to the polarization tensor connected with the Chern-Simons term in effective Lagrangian of the electromagnetic field. The general expression for such a contribution which depends on the temperature and the chemical potential of plasma as well as on the photon's momentum is derived. The instability of a magnetic field driven by the electron neutrino asymmetry for the $\nu$-burst during the first second of a supernova explosion can amplify a seed magnetic field of a protostar, and, perhaps, can explain the generation of strongest magnetic fields in magnetars. The growth of a cosmological magnetic field driven by the neutrino asymmetry density $\Delta n_{\nu}=n_{\nu} - n_{\bar{\nu}}\neq 0$ is provided by a lower bound on $|\xi_{\nu_e}|=|\mu_{\nu_e}|/T$ which is consistent with the well-known Big Bang nucleosynthesis (upper) bound on neutrino asymmetries in a hot universe plasma.


Introduction
The generation of the cosmological magnetic field (CMF) as a seed of observable galactic magnetic fields is still an open problem [1]. The two facts enhanced a new interest to such a problem. The first observational indications of the presence of CMF in intergalactic medium which may survive even till the present epoch [2,3] were as a new incitement to the conception of CMF and its helicity. Secondly, there appeared some new models of the magnetic field instability leading to the generation of CMF. In particular, in a hot universe plasma (T > 10 MeV) the generation of CMF having a maximum magnetic helicity was based on the quantum chiral (Adler) anomaly in relativistic QED plasma for which the difference of rightand left-chiral electron chemical potentials ∆µ = µ R − µ L is not equal to zero, ∆µ = 0 [4]. In the Standard Model (SM) plasma accounting for weak interactions one suggests the magnetic field generation based on the parity violation Chern-Simons (CS) term in the photon selfenergy (PSE) [5,6].
Another problem concerns strong magnetic fields existing in neutron stars as remnants of supernovae (SN). In particular, we are interested here how the strongest magnetic fields observed in magnetars [7] can be generated. To solve this problem, it was recently suggested to use the chiral plasma instability [8,9] caused by an imbalance between right-and lefthanded electrons µ R − µ L = µ 5 = 0 arising via the left-handed electron capture by protons inside the SN core (urca process). Obviously this mechanism is similar to the generation of helical magnetic fields in a hot plasma [4]. The chirality flip in both dense media (cases of a hot plasma T ≫ [m e , ∆µ] and a degenerate ultrarelativistic electron gas µ 5 ≫ [T, m e ]) leads to the damping ∆µ → 0, µ 5 → 0 due to collisions that should be taken into account for estimates of the magnetic field generation efficiency.
In the present work we study magnetic field generation problems (both in the early universe and in a supernova) based on the use of the photon PSE in electroweak plasma where a parity violating neutrino interaction with charged leptons is present. Such a contribution to PSE is equivalent to the appearance of the CS term Π 2 (A · B) in the effective Lagrangian of the electromagnetic field, where Π 2 is a master parameter we are looking for to solve a problem of the magnetic field generation, B = ∇ × A is the magnetic field, and A is the vector potential.
We shall describe the interaction between neutrinos and charged leptons in frames of the Fermi theory which is a good approximation at low energies. Since we study a νν gas embedded into lepton plasma we can treat neutrinos (antineutrinos) as proper combinations of the external neutrino hydrodynamic currents coming from the effective SM Lagrangian for the νl interaction that is linear in the Fermi constant (∼ G F ) being averaged over the neutrino ensemble. Thus, our approach is analogous to the generalized Furry representation in quantum electrodynamics.
Our work is organized as follows. First, in section 2 we derive the contribution of virtual charged leptons to the one loop PSE using the exact propagator of a charged lepton calculated in Appendix A via the effective νl interaction in the presence of the neutrino-antineutrino gas. Then, in section 3, using the imaginary time perturbation theory, we calculate the most general plasma contribution to PSE. We analyze our results for the cases of a classical plasma with low temperature and density as well as for hot and degenerate relativistic plasmas.
In the main section 4 we consider some applications of PSE for the evolution of magnetic fields in relativistic plasmas of a supernova and a hot plasma of the early universe (sections 4.2 and 4.3). For such media filled by a plenty of neutrinos we reveal the instability of B-field driven by the neutrino asymmetry ∆n ν = n ν − nν = 0. The analysis of the instabilities of CMF and magnetic fields in a supernova relies on a particular solution of the Faraday equation modified in SM that governs the B-field evolution. Such a solution for a 3D configuration of magnetic field with the maximum magnetic helicity is derived in Appendix D. In section 4.4 we compare our results with some issues in papers based on chiral properties of ultrarelativistic plasmas.
Finally, in section 5 we summarize our results and compare our calculations of the master parameter Π 2 valid for any plasma with the similar results obtained by other authors. Some useful formulas of the dimensional regularization are provided in Appendix B and the example of the calculation of an integral involving plasma effects is given in Appendix C.

Photon polarization tensor in a νν gas
In this section we calculate the parity violating term in the polarization tensor in the presence of a νν gas. It should be noted that photons do not interact directly with neutrinos since latter particles are neutral. Thus the νγ interaction should be mediated by charged leptons, denoted as l, which are taken to be virtual particles in this section. We shall take into account the νl interaction in propagators of l's as the external mean fields f µ We shall be mainly interested in the case of an isotropic νν gas when f L = f R = 0 and the nonzero f 0 L,R are given in eq. (A.5). In this situation the most general expression for the polarization tensor reads [12] where k µ = (k 0 , k) is the photon momentum, g µν = diag(+1, −1, −1, −1) is the Minkowski metric tensor, ε µναβ is the absolute antisymmetric tensor having ε 0123 = +1, and Π T,L,P are the form factors of a photon. Since we study parity violating effects, we should analyze the form factor Π 2 = (f 0 L − f 0 R )Π P . Since only real particles, considered in this section, are neutrinos, we add the superscript "ν" to photon form factors, e.g., Π 2 → Π (ν) 2 etc. The one loop contribution to PSE is schematically shown in figure 1. The lepton propagators are represented as broad lines since we take into account f µ L,R in our calculations. Note that we shall consider only the contribution to Π where e is the electric charge of l and the propagators S 0,1 (p) are given in eq. (A.8).
The traces of Dirac matrices in eq. (2.2) can be evaluated using the following expressions: To derive eq. (2.3) we use the fact that σ αβ γ 5 = i 2 ε αβλρ σ λρ . Using the dimensional regularization and eq. (B.2) we can express Π where M 2 = m 2 − k 2 x(1− x) and Γ(z) is the Euler Gamma function. The parameters ε and λ are defined in Appendix B. Considering the limit ε → 0 and using the fact that Γ(ε) ≈ 1 ε + γ, where γ ≈ 0.577, we can represent Π (ν) 2 in the form, .
It should be noted that eq. (2.5) does not contain ultraviolet divergencies. As shown in ref. [10], the contribution to PSE, calculated in frames of an effective theory which contains a parity violating interaction, is finite but it can depend on the regularization scheme used. Basing on eq. (2.5) we find that Π (ν) 2 = 0 at k 2 = 0, which agrees with the general analysis made in ref. [11] for a CPT-odd gauge invariant effective theory. We also note that Π (ν) 2 in eq. (2.5) coincides with the result of ref. [12], where the more fundamental Weinberg-Salam theory was used. Moreover, the fact that Π (ν) 2 vanishes at k 2 = 0 also agrees with the finding of ref. [13], where it was shown that the neutrino-photon interaction is absent in the lowest order in the Fermi constant. Nevertheless, as demonstrated in ref. [14], the amplitude for νγ → νγ has the nonzero value in two loops.

Plasma contribution to polarization tensor
In this section we study the direct contribution of charged leptons to the photon form factor Π 2 corresponding to their parity violating interaction with background neutrinos. We take into account a lepton mass that is absolutely necessary, e.g., for the classical nonrelativistic plasma. Thus leptons are not chirally polarized. For relativistic plasmas we again substitute an effective lepton mass m eff (T, µ) [15] in photon dispersion characteristics, see below in eq. (3.8), that also differs our approach from the use of the lepton chirality.
Thus, in this section we obtain the general expression for Π 2 taking into account both the temperature and the chemical potential of the charged leptons. It means that these leptons now are not virtual particles. We also exactly account for the photon's dispersion relation k 0 = k 0 (k) in this plasma. On the basis of the general results we discuss the cases of low temperature and low density classical plasma, as well as hot relativistic and degenerate relativistic plasmas.
If we study the photon propagation in a plasma of charged leptons with nonzero temperature and density, the photon's dispersion relation differs from the vacuum one, k 2 = (k 2 0 − k 2 ) = 0. As seen in eq. (2.5), in this case Π (ν) 2 = 0. However, we should also evaluate the direct contribution of plasma particles to the parity violating form factor of a photon. We can define it as Π (νl) 2 analogously to section 2. Therefore we shall study the system consisting of a real l's plasma and a real νν gas. The presence of ν's andν's is essential since it is these particles which provide the nonzero contribution to the parity violating form factor based on the νl interaction.
The expression for the contribution to PSE from the plasma of not virtual leptons can be obtained if we make the following replacement in eq. (2.2) (see ref. [16]): i dp 0 2π → T n , p 0 = (2n + 1)πT i + µ, n = 0, ±1, ±2, . . . , (3.1) where T and µ are the temperature and the chemical potential of the l's plasma. In principle, we can discuss a general situation when T and µ are different from T να and µ να defined in eq. (A.4). However, in section 4, where we study the application of our calculations, the system in the thermodynamic equilibrium is considered. Thus, in the following we shall suppose that T = T ν , where T ν is the νν gas temperature equal for all neutrino flavors. However, we shall keep different µ and µ να . Using eqs. (2.2), (2.3), (3.1), and (A.8), as well as defining the effective chemical poten- in the following form: Here E p = p 2 + M 2 , β = 1/T , and M 2 is defined in section 2. To obtain eqs. (3.2) and (3.3) we assume that k 2 < 4m 2 , i.e. no creation of ll-pairs occurs.
It should be noted that in deriving eqs. (3.2) and (3.3) we exactly account for the l's mass m. Thus charged leptons are not taken to be chirally polarized. It means that the magnetic field instability discussed later in section 4, which results from the nonzero Π 2 = Π is generated rather by the neutrino asymmetry (n ν − nν) = 0 than by the chiral asymmetry of charged leptons ∼ (µ R − µ L ) = 0 studied, e.g., in refs. [4,8,9].

Low density classical plasma
Let us first discuss the case of a low density plasma of l's, that corresponds to k 2 ≪ m 2 . Using the general eqs. (3.2) and (3.3) in the limit max(k 2 0 , k 2 ) ≪ m 2 we obtain that where E p = p 2 + m 2 since we neglect the photon's dispersion in plasma. Note that Π 2 , we shall consider the low temperature limit: T ≪ m. We will identify l with an electron and assume that the electron gas has a classical Maxwell distribution. For this medium we get that k 2 = 4πα em n e /m, where α em = e 2 /4π = 1/137 is the fine structure constant and n e is the background electron density. Moreover for a classical electron gas one has that µ = m + T ln ne gs 2π mT 3/2 , where g s = 2 is the number of spin degrees of freedom of an electron. Using eqs. (2.5) and (3.4), we get that where we add a tilde over Π (ν,νl) 2 to stress that these quantities correspond to real photons in plasma (plasmons) rather to the virtual photons. In the following we shall omit the tilde in order not to encumber notations.
One can see that Π (νl) 2 in eq. (3.5) is 7π 2αem ∼ 10 3 times greater than Π (ν) 2 . Note that for a classical nonrelativistic plasma, corresponding to m ≫ max(|p|, T ), the integrals in last two lines in eq. (3.4) cancel each other while the integral in the first line leads to the term Π (νl) 2 in eq. (3.5). Let us study the derived Π 2 in the static limit k 0 = 0. If we discuss the situation when only virtual charged leptons contribute to PSE, we should set n e → 0 in eq. (3.5). This limit is equivalent to k 2 → 0. Using eq. (3.5), we obtain that Π 2 → 0. This our result is in agreement the findings of ref. [13], where it was found that the one loop contribution to νγ-interaction should be vanishing. The leading nonzero contribution to Π 2 in the case when charged leptons are virtual particles, i.e. when we neglect the plasma contribution given by eqs. (3.2) and (3.3), was obtained in ref. [14]. Using the results of ref. [14], one gets that in

Hot relativistic plasma
The dispersion relation for transverse waves in relativistic plasma reads [15], The plasma frequency ω p can be found from the following expression: where v = |p|/E p . Using eq. (3.7) in the relativistic limit T ≫ max(m, µ), we get that ω 2 p = 4α em πT 2 /9. The transcendant eq. (3.6) can be explicitly solved if long waves with k 2 0 ≫ k 2 are considered. In this situation the dispersion relation is k 2 = ω 2 p . It should be noted that the electron's mass in plasma can significantly differ from its vacuum value. The radiative corrections to the electron's mass were studied in ref. [17]. Thus, if we consider a dense and hot plasma, we should replace 1 in eqs. (2.5), (3.2), and (3.3). Note that eq. (3.8) is valid for both T ≫ µ and µ ≫ T . Accounting for the dispersion relation and the expression for ω p , we get that k 2 < 4m 2 eff in a hot relativistic plasma.
Let us represent Π 2 as where F is the dimensionless function which depends on k 0 /T . Note that Π 2 in eq. (3.9) includes the contributions from eqs. (2.5) and (3.2). Accounting for eq. (3.8), we present the behaviour of F versus we k 0 /T in figure 2(a). We study long waves limit when k 0 ≈ ω p ≈ 0.1T . Thus we should be interested in the values of F corresponding to k 0 ≪ T . One can see in figure 2(a) that for a hot relativistic plasma Π 2 is nonvanishing in the static limit: F (k 0 → 0) ≈ −0.18. However this nonzero value strongly depends on the photon's dispersion law in such a plasma.

Degenerate relativistic plasma
In case of a degenerate plasma the dispersion relation for transverse waves is [15], where v F is the Fermi velocity. The plasma frequency ω p can be found from eq. (3.7) if we make the following replacement: is the Heaviside step function. Let us discuss the degenerate plasma in the relativistic limit. In this situation v F = 1 and ω 2 p = 4α em µ 2 /3π. The general dispersion relation in eq. (3.10) transforms into k 2 = ω 2 p if we study long waves. Using eq. (3.8), we also get the effective electron mass in a degenerate plasma. One can check that the inequality k 2 < 4m 2 eff is valid. Using eqs. (2.5), (3.2), and (3.3) in the limit µ ≫ T , as well as the following representations of the Dirac delta function and its derivative: we can derive the expression for the function F = F (k 0 /µ), see eq. (3.9), in case of a relativistic degenerate plasma. In this situation the integration over momenta can be performed explicitly. However, here we do not give the expression for F since it is very cumbersome. The function F versus k 0 /µ is shown in figure 2(b). We discuss the long waves limit. Thus k 0 ≈ ω p ≈ 0.06µ. It means that for our purposes we should consider F at k 0 ≪ µ. One can see in figure 2 05. Therefore, as in case of a hot relativistic plasma, for a degenerate relativistic plasma Π 2 is nonvanishing at k 0 → 0, but its actual value Π 2 (0) is different from that found in section 3.2.

Instability of magnetic fields in relativistic plasmas driven by neutrino asymmetries
We consider below two cases for which the CS term Π 2 in the photon polarization operator Π µν plays a crucial role. A nonzero Π 2 leads to the α-dynamo amplification (instability) of a seed magnetic field even without fluid vortices or any rotation Ω in plasma which are usually exploited in the standard MHD approach for αΩ-dynamo [18]. The first case considered here concerns the magnetic field growth in a degenerate ultrarelativistic electron plasma, µ ≫ max(T, m e ), during the collapse and deleptonization phases of a supernova burst. In the second case we consider below a hot plasma of the early universe with the temperatures T ≫ max(m e , µ) before the neutrino decoupling at T > T dec ≃ 2 ÷ 3 MeV. In both cases neutrinos are in equilibrium with a plasma environment. For these applications we use our result in eq. (3.9). First, we derive in subsection 4.1 the Faraday equation generalized in SM to find the key parameters leading to the B-field instability. The corresponding evolution equations for the spectra of the magnetic helicity density h(k, t) and magnetic energy density ρ B (k, t) presented in Appendix D allow us to interpret the simplest solution of Faraday equation for the case of the maximum helicity density obeying the inequality h(k, t) ≤ 2ρ B (k, t)/k [19]. Here is the magnetic helicity density and ρ B (t) = dkρ B (k, t) = B 2 (t)/2 is the magnetic energy density for an uniform isotropic medium.
An excess of electron neutrinos during a first second of a supernova explosion 2 allow us to put n νe − nν e = 0 in the problem of the magnetic field amplification considered in subsection 4.2. In subsection 4.3 we find the lower bound on the neutrino asymmetry providing the growth of CMF field in our causal scenario. It would be interesting to compare such a limit with the upper bound on the electron neutrino-antineutrino asymmetry |ξ νe | ≤ 0.07 given by the Big Bang nucleosynthesis (BBN) constraint [21]. Thus, we shall consider magnetic fields in media with a plenty of neutrinos (antineutrinos) where a nonzero neutrino asymmetry exists. Finally, in section 4.4 we compare our findings with what other authors found in similar problems.

Generalized Faraday equation in the Standard Model
The existence of a neutrino asymmetry accounting for the difference in eq. (A.5), leads to a non-zero parity violation term in the photon polarization operator Π ij (ω, k) = iε ijn k n Π 2 (ω, k), where Π 2 is given by eq. (3.9) and ω ≡ k 0 . The CS polarization term in eq. (3.9) corresponds to the induced pseudovector current in the Fourier representation,  Expressing the ohmic current as j(ω, k) = σ cond E(ω, k), then neglecting the displacement current in the l.h.s. of eq. (4.3), that is a standard assumption in the MHD approach for which ω ≪ σ cond 3 , and finally using the Bianchi identity k×E = ωB, one gets the generalized Faraday equation in SM in the coordinate representation, where α is the magnetic helicity parameter, and η = (σ cond ) −1 is the magnetic diffusion coefficent.
Here we use the long-wave approximation for large-scale magnetic fields where the operator Π 2 (k 0 , k = 0) is at least uniform, k → 0, and almost stationary since the function F (x) 2 Neutrino emission prevails over the antineutrino one during first milliseconds of a supernova burst due to the reaction e − + p → n + νe (urca-process) before its equilibrium with beta-decays n → p + e − +νe is settled in (see figure 11.3 in ref. [20]). 3 The conductivity σ cond = ω 2 p /ν coll = 4παemT 2 /9ν coll ∼ T /αem ∼ 100T depends on the Coulomb collision frequency ν coll = σ Coul ne = [4πLα 2 em /9T 2 ]ne ∼ α 2 em T . Here we use the values for the electron density ne = 0.183T 3 in a hot plasma and L ∼ 10 for the Coulomb logarithm. Obviously the MHD condition ω = ωt ≪ σ cond is fulfilled to obtain eq. (4.4).
depends on a small ratio x = k 0 /T ≪ 1 or x = k 0 /µ ≪ 1. For instance, in the long-wave limit k ≪ ω t the transversal plasmons (photons) have the spectrum k 2 0 ≡ ω 2 t = ω 2 p + k 2 ≈ ω 2 p = 4πα em T 2 /9 in a hot plasma (T ≫ max[µ, m e ]) and k 2 0 ≡ ω 2 t = ω 2 p + k 2 ≈ ω 2 p = 4α em µ 2 /3π in the ultrarelativistic degenerate electron gas (µ ≫ max[m e , T ]) [15] (see spectra in eqs. (3.6) and (3.10) above). In a relativistic plasma this approximation corresponds to the negligible spatial dispersion, k 0 ≫ k v ∼ k, where we put v = 1 both in hot and degenerate relativistic plasmas. Here k = |k| is the wave number. Thus, the ratio k 0 /T ∼ 0.1 or k 0 /µ ∼ 0.06 allows us to consider Π 2 ≈ const without temporal and spatial dispersion as a function of the temperature T (a hot plasma in the early universe) or the chemical potential µ (a degenerate electron gas in a supernova) only.

Amplification of a seed magnetic field in a supernova
During the collapse (time t < 0.1 s after onset of collapse, see figure 11.1 in ref. [20]) one can neglect ν µ,τ emission and Π 2 reads where the function F (x) is shown in figure 2(b) for a degenerate ultrarelativistic electron gas with µ ≫ max(T, m e ). Let us give some estimates for Π 2 in a collapsing SN with the progenitor stellar mass M ∼ 8M ⊙ considered in ref. [20] (see there the plots for evolution stages in figures 11.1-11.3). In order to obtain Π 2 we should find the appropriate neutrino asymmetry density ∆n νe .
At the stage just after collapse neutrinos are captured, their free path does not exceed the core radius λ ν ≪ R ∼ 10 km. For instance, for the nuclear core density ρ 0 = M N (n n + n p ) = 3 × 10 14 g · cm −3 one gets λ ν ∼ 300 m if E ν = 30 MeV, or λ ν ∼ 2.7 km for E ν = 10 MeV. Here, using the nucleon mass M N = 940 MeV, one gets the baryon density n B = n p + n p = 1.8 × 10 38 cm −3 .
The lepton abundance Y L = 0.3 is typical for the material in a SN core just after the collapse, so that the equilibrium condition Y L n B = n e + n νe = [p 3 Fe + p 3 Fν e ]/3π 2 = 5.4 × 10 37 cm −3 allows us to look for the Fermi momenta for degenerate electrons and neutrinos, p Fe and p Fν e . The second equation for these quantities comes from the consideration of figure D7(a) in ref. [20], where for the same matter density ρ 0 = 3 × 10 14 g · cm −3 one finds the difference µ n − µ p = µ e − µ νe = 40 MeV corresponding to the temperature T ≃ 10 MeV. Note that leptons are ultrarelativistic, µ e − µ νe = p Fe − p Fν e = 40 MeV, while nucleons are degenerate and nonrelativistic, MeV. Eventually we get all Fermi momenta in such dense core: p Fν e = 163 MeV, p Fn = 341 MeV, p Fe = p Fp = 203 MeV. Here the last equality comes from the electroneutrality condition n e = n p . Thus, we get the electron neutrino density at this stage of the SN evolution, n νe = 1.46×10 5 MeV 3 = 1.9×10 37 cm −3 , which should be substituted into eq. (4.6) neglecting antineutrino contribution.
The magnetic diffusion time t diff = Λ 2 /η seen from the Faraday eq. (4.4), is given by the electrical conductivity for degenerate ultrarelativistic electrons and degenerate nonrelativistic protons, σ cond = ω 2 p /ν coll [22]. Note that the combined effects of the degeneracy and the shielding reduce the collision frequency ν coll ∼ T 2 . Thus collisions of charged particles are blocked due to the Pauli principle since states p < p F are busy and ν coll → 0 at T → 0. The electrical conductivity was found in ref. [22], σ cond = 1.6 × 10 28 (T /10 8 K) 2 n e 10 36 cm −3 3/2 s −1 . (4.8) For p Fe = 203 MeV and the corresponding electron density n e = p 3 Fe /3π 2 = 3.7 × 10 37 cm −3 , as well as the temperature T = 10 MeV ≃ 10 11 K in SN core we have just estimated, eq. (4.8) gives σ cond = 2250 MeV. This result leads to the estimate t diff = 0.023 s. It means that any seed magnetic field B 0 existing in plasma does not dissipate ohmically during first milliseconds after onset of collapse, t ≪ t diff , and evolves for a given wave number k through the α-dynamo driven by neutrino asymmetries (see in Appendix D) as (4.9) If k < |α|/η = |Π 2 |, the seed magnetic field in eq. (4.9) will grow exponentially. The fastest growth corresponds to the α 2 -dynamo with k = |α|/2η for which Unfortunately, under the same conditions (for large n νe = 1.9 × 10 37 cm −3 ) the scale of the magnetic field occurs to be rather small, Λ = k −1 ≃ η/|α| = |Π 2 | −1 ∼ 1.25 × 10 −3 cm. Here we use the fact that |F | = 2, see figure 2(b). However, such a scale grows when the neutrino asymmetry diminishes due to a significant involvement of antineutrinos somewhere later at t ≤ 0.03 − 0.1 s, ∆n νe = n νe − nν e → 0 (see figure 11.3 in ref. [20]). It reaches the core radius Λ → R 0 = 10 km, k = |Π 2 | = R −1 0 , for the neutrino asymmetry density The magnetic diffusion time should be recalculated for this stage of SN burst separately (the release of prompt ν e -burst due to the shock propagation and the following matter accretion, see in figure 11.1 in ref. [20]). However this task is beyond the scope of the present work. The suggested mechanism of the B-field growth in a supernova driven by the electron neutrino asymmetry could lead to an additional amplification of a strong seed magnetic field (B 0 = 10 10 ÷ 10 12 G) during the first second of a SN explosion when the asymmetry n νe − nν e = 0 remains appreciable. Here a strong seed magnetic field can arise from a small magnetic field of a protostar, e.g., B proto ∼ 1 ÷ 10 2 G, due to the conservation of the magnetic field flux, B 0 = B proto (R proto /R 0 ) 2 , during the protostar collapse. The question whether this new mechanism can explain the strongest magnetic field of observed magnetars (B = 10 14 ÷ 10 15 G) deserves a separate study (see also in section 4.4).

Growth of primordial magnetic fields provided by the lower bound on neutrino asymmetries
In a hot plasma of the early universe the magnetic helicity parameter α in Faraday eq. (4.4) reads as where we substituted the dimensionless neutrino asymmetries ξ να = µ να /T for the asymmetry densities ∆n να = ξ να T 3 /6 and used the hot plasma conductivity σ cond = σ c T , with σ c ≃ 100. The magnetic field evolution with the parameter α in eq. (4.11) obeys the causal scenario, where the magnetic field scale is less than the horizon, Λ B ≃ η/|α| < l H = H −1 , if the sum of neutrino asymmetries −2 α c Here we take into account that c (A) = ∓0.5 (upper sign stays for electron neutrinos) is the SM axial coupling constant for νe interaction corresponding to the difference f 0 L − f 0 R in eq. (4.1). In eq. (4.11) we use that |F | ≃ 0.2, which results from figure 2(a). Moreover we account for that GeV is the Plank mass, g * = 106.75 is the number of relativistic degrees of freedom above the QCD phase transition, T > T QCD ≃ 150 MeV. Let us remind that to get eq. (4.12) we applied the photon polarization term in eq. (3.9) for ultrarelativistic leptons with T ≫ max(m e , µ).
One can see that the inequality in eq. (4.12) does not contradict to the well-known BBN bounds on the neutrino asymmetries at the lepton stage of the universe expansion corresponding to g * = 10.75, |ξ να | < 0.07, (see ref. [21]) and gives an additive (lower) bound on the neutrino asymmetry which supports the growth of CMF in our causal scenario. Here different flavors equilibrate due to neutrino oscillations before BBN, ξ νe ∼ ξ νµ ∼ ξ ντ , somewhere at the neutrino decoupling time T = 2−3 MeV 4 , accounting for all active neutrino flavors with the non-zero mixing angles (including sin 2 θ 13 = 0.04), see in ref. [23].
We also obtain that the magnetic field diffusion time t diff is bigger than the expansion time ∼ H −1 , t diff = σ cond /Π 2 2 > M 0 /T 2 , or ohmic losses are not danger, if the opposite inequality for neutrino asymmetries is valid, Here just after the electroweak phase transition T ≤ T EW = 100 GeV the combined asymmetry in eq. (4.13) seems to be resonable, |ξ νe − ξ νµ − ξ ντ | < 0.013, while at lower temperatures m e ≪ T ≤ O(GeV) the condition in eq. (4.13) is obviously fulfilled and consistent with the BBN bound obtained in ref. [21].

4.4
Comparison with the chiral magnetic mechanism in refs. [4,8,9] In ref. [4], the magnetic helicity coefficient analogous to that in eq. (4.11) in our work, is proportional to the magnetic chiral parameter ∆µ = µ e L − µ e R where µ e L (µ e R ) are the left (right) electron chemical potentials. In QED plasma such a parameter arises due to the Adler anomaly in external electromagnetic fields, ∂(j µ L − j µ R )/∂x µ = (2α/π)E · B, evolving in a self-consistent way with the magnetic field B. However, it tends to a small value ∆µ/T ∼ 10 −6 − 10 −7 for a small wave number 10 −10 ≤ k/T ≤ 3 × 10 −9 at temperatures T ≥ 10 MeV (see figure F.1 in ref. [4]) and vanishes later at all due to the chirality flip with the increasing rate Γ f ∼ (m 2 e /T 2 ) in cooling universe, n e L − n e R → 0. This is not the case for the helicity parameter given in eq. (4.11) based on neutrino asymmetries ξ να for which there are no triangle anomalies in Maxwellian fields contrary to charged leptons 5 . Moreover, after the neutrino decoupling and relic neutrino oscillations before BBN, there are no ways to change the equivalent asymmetries ξ να = const = 0, α = e, µ, τ .
In ref. [9] one suggests a new mechanism for the production of strong magnetic fields in magnetars [7] based on the chiral instability for electrons with the difference of chemical potentials for right-and left-handed electrons, µ 5 = µ R − µ L = 0. The chirality imbalance of electrons is produced via the same electron capture inside a core we considered above (urca-process), p + e − L → n + ν e L , where the subscript L stands for left-handedness. In ref. [9] the typical scales of wave number k and vector potential A relevant to such instability were obtained (see eq. (27) there): where an estimate µ 5 = 200 MeV gives huge B max ∼ µ 2 5 ∼ 10 18 G. The authors also show that instability proceeds faster than the danger chirality flip, Γ inst = α 2 em µ 5 ≫ Γ flip ∼ α 2 em (m e /µ 5 ) 2 µ 5 , or the process µ 5 → 0 due to collisions is negligible because µ 5 ≫ m e . In the proposed mechanism of the magnetic field amplification it remains unclear how a largescale magnetic field is produced in this scenario since the magnetic field generated seems to be microscopic. Indeed, the scale k −1 ∼ MeV −1 found in ref. [9] is much smaller than we obtain in section 4.2. More realistic calculations using MHD for chiral plasma were suggested in ref. [9] to reach in future a definite conclusion.

Conclusion
In this work, we have shown a way to bridge between the photon self-energy (PSE) calculated by FTFT methods and its macroscopic consequences in magnetohydrodynamics (MHD). The latter is modified due to the appearance in SM of the parity violation CS term Π 2 as a part of PSE. This leads to the new α-helicity parameter in eq. (4.5) in Faraday equation that is scalar in modified dynamo theory instead of the well-known pseudoscalar α MHD ∼ v · (∇ × v) [18] which corresponds to the parity conservation in standard MHD.
While our astrophysical applications in the modified MHD are very preliminary, and at the present stage, for instance, does not involve Navier-Stokes equation for the fluid velocity v, our calculations of PSE in electroweak plasma are exact and completed, at least, with the inclusion of νl interaction in Fermi approximation. Therefore we discuss below main steps in our calculations of PSE in quantum FTFT and compare the master CS term Π 2 we calculated here with results obtained by other authors.
We have started with the derivation of the exact propagator of a charged lepton in the presence of the neutrino background. This propagator was used for the calculation of the loop contribution to PSE. We have obtained the contributions of both virtual l's, cf. eq. (2.5), and plasma of particles having nonzero temperature and density, cf. eqs. (3.2) and (3.3).
To obtain the plasma contribution to PSE the imaginary time perturbation theory has been used. The expression for Π (νl) 2 has been derived under the assumption that k 2 < 4m 2 , which means that no creation of ll-pairs occurs in plasma. Indeed, as shown in ref. [15], the plasmon decay is forbidden.
In eq. (3.5) we have derived the expression for Π (νl) 2 in case of a low temperature classical plasma of electrons. We have obtained that Π em . Accounting for the additional constant factor, we get that Π (νl) 2 is three orders of magnitude greater than Π (ν) 2 . Therefore one cannot neglect Π (νl) 2 compared to Π (ν) 2 . Note that the plasma contribution to the polarization tensor was overlooked in ref. [12], where the optical activity of the relic neutrino gas was studied and only Π (ν) 2 was accounted for. The computation of Π 2 for eē plasma and νν gas was made in ref. [5], where, like in our work, the approximation of the Fermi interaction was used. In frames of the real time perturbation theory, the expression for the antisymmetric contribution to PSE was derived in ref. [5] on the basis of the modified electron's propagator. The cases of hot and degenerate relativistic plasmas were studied in ref. [5]. It was found that the contribution of virtual l's to Π 2 is much smaller than Π (νl) 2 . However, using eq. (2.5) and the photon's dispersion laws found in sections 3.2 and 3.3, we get that Π in these media. The absolute values for the function F in the static limit, obtained in ref. [5], are different from these calculated in our work (see figures 2(a) and 2(b)). This discrepancy is because the radiative corrections to the electron mass in a hot and dense plasma, cf. eq. (3.8), were not taken into account in ref. [5].
Recently the two loop contribution to Π 2 was computed in ref. [6]. It was found there that It should be noted that the value of Π 2 in ref. [6] depends on the type of plasma where a photon propagates with conservation of global charges B/3 − L a = const appropriate for a given medium, a = e, µ, τ .
We have explicitly demonstrated that the static value of Π 2 can have different values depending on what kind of plasma of charged leptons is studied. In the lowest order over G F and in the case of a classical electron plasma with low temperature and density the total → 0 at k 2 → 0 because in vacuum filled by neutrinos (antineutrinos) there are no real electrons, n e = 0. This result is in agreement with ref. [13]. However, in the same approximation, ∼ G F , in the case of hot relativistic and degenerate relativistic plasmas coexisting with neutrino background Π 2 = 0 at k 0 → 0. Nevertheless, the values of Π 2 (0) are different for these plasmas (see eq. (3.9) and figure 2). Therefore our study of the parity violating effects in PSE generalizes the results of ref.
[6] since our expression for Π 2 exactly accounts for charged lepton plasma characteristics, like T and µ, the lepton mass, which should not be omitted in hot and dense matter (see, e.g., ref. [17]), as well as the photon dispersion law in this matter. Moreover, unlike ref. [6], our method of calculations allows to reproduce the value of Π 2 = 0 corresponding to the case of νν gas with the nonzero asymmetry, n ν − nν = 0, and purely virtual charged leptons, T = 0 and µ = 0.
In section 4 we have applied our main result in eq. (3.9) to the magnetic field evolution in both (i) ultrarelativistic degenerate electron gas of a supernova during its neutrino burst and (ii) hot plasma of early universe at T ≫ m e . In both cases one finds a possibility of the magnetic field growth driven by the neutrino asymmetries and avoiding the magnetic diffusion. Of course, a seed field B 0 should be assumed at the initial time instance t 0 which then evolves through the α-dynamo mechanism as B(k, t) = B 0 exp[ given wave number k and the magnetic field scale Λ B = k −1 . The question whether the new B-field amplification mechanism in a supernova driven by a nonzero (n νe − nν e )-asymmetry could lead to an explanation of strongest magnetic fields observed in magnetars deserves a separate study.
We have found an interesting lower bound in eq. (4.12) on the combined neutrino asymmetry providing the CMF growth in a hot plasma of the early universe which is consistent with the well-known BBN (upper) bound on the electron neutrino asymmetry [21]. We suppose that the new mechanism suggested in section 4.3 for a B-field growth driven by neutrino asymmetries in a hot plasma can be more productive in comparison with that involving the chiral electron asymmetry ∼ (µ R − µ L ) [4]. For the latter the chirality flip due to collisions with the rate Γ f ∼ m 2 e /T 2 , washes out the corresponding α-magnetic helicity parameter stronger and stronger in cooling universe. In contrast to that the neutrino asymmetries ∼ ξ α equilibrate before BBN due to neutrino oscillations being conserved even after neutrino decoupling if the total lepton number is conserved.

A Propagator of a charged lepton interacting with a neutrino gas
In this Appendix we briefly describe the interaction between charged leptons and neutrinos in frames of the Fermi theory. Then we derive the exact propagator of a charged lepton in the presence of the neutrino-antineutrino gas.
The evolution of a charged lepton l, represented as a bispinor ψ, interacting with the νν gas, is described by the following Dirac equation [26]: where m is the mass of l, γ µ = (γ 0 , γ) are the Dirac matrices, P L,R = (1 ∓ γ 5 )/2 are the chiral projectors, γ 5 = iγ 0 γ 1 γ 2 γ 3 . The lν interaction in eq. (A.1) can be described in the mean field approximation via the external neutrino macroscopic currents f µ L,R = (f 0 L,R , f L,R ). To find the explicit form of f µ L,R we shall consider a background matter consisting of the ν ανα gas, α = e, µ, τ , and identify l with an electron. The effective Lagrangian for the νe-interaction has the form [27], and θ W is the Weinberg angle. The symbol δ e,α in eq. (A. 3) equals to one if α = e and to zero otherwise. To derive eq. (A.2) we use the Fierz transformation and take into account that ψ and ν α are anticommuting operator valued spinors. For applications above we calculate Π 2 in the case of the isotropic νν gas. It means that, in averaging over the neutrino ensemble, the only nonzero quantity is ν α γ 0 (1 − γ 5 )ν α . Nowadays it is generally believed that neutrinos possess nonzero masses leading to observed neutrino oscillations in numerous underground experiments. In section 4, where we discuss applications of our results, we consider for simplicity the case of the νν gas consisting of massless neutrinos hence neglecting neutrino oscillations influence the magnetic field generation. In this situation ν α γ 0 (1 − γ 5 )ν α = 2∆n να , where ∆n να = n να − nν α and Using the results of ref. [26] one can derive the expression for S, is the canonical momentum and σ µν = i 2 [γ µ , γ ν ] − . Note that the denominator D of S in eq. (A.7) is the fourth order polynomial with respect to p 0 . In general the equation D = 0 has four different roots p 0 1,...,4 , which determine the poles of S. The Green function in eq. (A.7) corresponds to a propagator if we bypass its poles in the complex plane in a standard manner: positive roots should be added a small negative imaginary contribution −i0 and negative ones acquire +i0.
It is convenient to represent S as a series S = S 0 + S 1 + · · · , keeping only the terms linear in f µ L,R which contain γ 5 since they are responsible for the parity violation. The explicit forms of S 0 and S 1 are where we should take into account that m 2 → m 2 − i0 in the denominators.

B Dimensional regularization
The dimensional regularization is introduced in the following way [28]: where N = 4 − 2ε and ε → 0. The convolution of the metric tensor reads g µν g µν = N . Some useful momentum integrals have the form [29], where Γ(z) is the Euler Gamma function and λ is the parameter having the dimension of mass.

C Technique for the integrals calculation
In this appendix we give an example of the calculation of one of the integrals which contribute to eqs. (3.2) and (3.3). To account for the plasma effects we use the technique for the summation over Matsubara frequencies.
Let us consider the Feynman integral in Minkowski space where M 2 is a positive real parameter. To get the analog of I in the presence of a fermionic plasma, we should transform eq. (C.1) to (see eq. (3.1) and ref. [16]) where p 2 = [(2n+1)πT i+µ] 2 −p 2 , T is the plasma temperature, and µ is the plasma chemical potential. The summation over the Matsubara frequencies in eq. (C.2) for fermions can be replaced by the integration in a complex plane [16], 3) D General system of evolution equations for the spectra of the helicity density and the magnetic energy density We derive this system for the case of helical magnetic fields in the early universe starting from the Faraday eq. (4.4) to show how its solution in eq. (4.9) can be obtained for maximum helical magnetic field . The analogous system can be presented for the case of magnetic fields in a supernova. The method is similar to that in Appendix E in ref. [4] for CMF and in ref. [30] for the helical hypermagnetic field. The magnetic helicity H = d 3 x (A · B) is an inviscid invariant of motion in the contemporary universe. The corresponding conservation law dH/dt = 0 in an ideal plasma (σ cond → ∞) severely constraints the magnetic field generation by the dynamo process and its further evolution. Multiplying eq. (4.4) by the corresponding vector potential and adding the analogous construction produced by the evolution equation governing the vector potential (multiplied by a magnetic field), after the integration over space we get the evolution equation for the magnetic helicity H = d 3 x (A · B), where α is given by eq. (4.5), β = (σ cond ) −1 is the magnetic diffusion coefficient given by the plasma conductivity σ cond . We changed here the notation η = (σ cond ) −1 used above in section 4 to avoid its confusion with the standard notation for the conformal time (see below). The surface integral (. . . ) was omitted in the last line in eq. (D.1) since electromagnetic fields vanish at infinity. Let us change physical variables to the conformal ones using the conformal time η = M 0 /T , M 0 = M Pl /1.66 √ g * , where M Pl = 1.2 × 10 19 GeV is the Plank mass, g * = 106.75 is the effective number of relativistic degrees of freedom.
In Friedmann-Robertson-Walker metric ds 2 = a 2 (η)(dη 2 − dx 2 ), using the definitions a = T −1 , where a 0 = 1 at the present temperature T now , and dη = dt/a(t) , we input the following notations:k = ka = const is the conformal momentum (giving a red shift for the physical one, k ∼ T = T now [1 + z]),Π 2 (η) = aΠ 2 = Π 2 /T is the dimensional CS term in PSE proportional to the neutrino asymmetry ∆n ν and changing over time,B = a 2 B, and A = aA are the conformal dimensionless counterparts of the magnetic field and the vector potential correspondingly.