Generation of cosmic magnetic fields in electroweak plasma

We study the generation of strong magnetic fields in magnetars and in the early universe. For this purpose we calculate the antisymmetric contribution to the photon polarization tensor in a medium consisting of an electron-positron plasma and a gas of neutrinos and antineutrinos, interacting within the Standard Model. Such a contribution exactly takes into account the temperature and the chemical potential of plasma as well as the photon dispersion law in this background matter. It is shown that a nonvanishing Chern-Simons parameter, which appears if there is a nonzero asymmetry between neutrinos and antineutrinos, leads to the instability of a magnetic field resulting to its growth. We apply our result to the description of the magnetic field amplification in the first second of a supernova explosion. It is suggested that this mechanism can explain strong magnetic fields of magnetars. Then we use our approach to study the cosmological magnetic field evolution. We find a lower bound on the neutrino asymmetries consistent with the well-known Big Bang nucleosynthesis bound in a hot universe plasma. Finally we examine the issue of whether a magnetic field can be amplified in a background matter consisting of self-interacting electrons and positrons.

The origin of magnetic fields (B fields) in some astrophysical and cosmological media is still a puzzle for the modern physics and astrophysics. There are multiple models for the generation of strong B fields in magnetars [1]. The observable galactic B field can be a remnant of a strong primordial B field existed in the early universe [2]. Recently the indication on the existence of the inflationary B field was claimed basing on the analysis of BICEP2 data [3]. In the present work we analyze the possibility for the strong B field generation in an electroweak plasma. First, we study the B field generation driven by neutrino asymmetries. Then, we apply our results for the description of strong B fields in magnetars and in the early universe. Finally, we analyze the evolution of a B field in a self-interacting electronpositron plasma.
To study the B field evolution we start with the anal-Email address: maxim.dvornikov@usp.br (Maxim Dvornikov) ysis of the electromagnetic properties of an electroweak plasma consisting of electrons e − , positrons e + , neutrinos ν, and antineutrinosν of all types. These particles are supposed to interact in frames of the Fermi theory. This interaction is parity violating. Thus the photon polarization tensor Π µν acquires a contribution [4], where Π 2 = Π 2 (k) is the new form factor, or the Chern-Simons (CS) parameter, we will be looking for and k µ = (k 0 , k) is the photon momentum. Here we adopt the notation of [5] First, we will be interested in the contribution to Π 2 arising from the interaction of a e − e + plasma with a νν gas. In this case the most general analytical expression for Π 2 can be obtained on the basis of the Feynman diagram shown in Fig. 1. We shall represent Π 2 as Π 2 = Π (ν) 2 + Π (νe) 2 , where Π (ν) 2 is the contribution of only the neutrino gas and Π (νe) 2 is the contribution of the e − e + plasma with the nonzero temperature T and the chemi-  cal potential µ.
The expression for Π (ν) 2 can be obtained using the standard quantum field theory technique [6], where e is the electron charge, m is the electron mass, can be obtained using the technique for the summation over the Matsubara frequencies [6], where Here . Note that in Eqs. (3) and (4) we assume that k 2 < 4m 2 , i.e. no creation of e − e + pairs occurs [7].
It is convenient to represent Π 2 as Π 2 = 2 α em π V 5 F, where F is the dimensionless function and α em = e 2 4π is the fine structure constant. Using Eqs. (2)-(4), in Fig. 2 we show the behavior of F versus k 0 in relativistic plasmas. It should be noted that in the static limit F(k 0 = 0) 0. To plot Fig. 2 we take into account the dispersion law of long electromagnetic waves in plasma k 2 = k 2 (T, µ) [6] and the fact that an electron acquires the effective mass m 2 eff = e 2 8π 2 (µ 2 + π 2 T 2 ) in a hot and dense matter [7]. As shown in [6], the nonzero Π 2 (0) = Π 2 (k 0 = 0) results in the instability of a B field leading to the exponential growth of a seed field.
We can apply our results for the description of the B field evolution in a dense relativistic electron gas in a supernova explosion. It is known that, just after the core collapse, a supernova is a powerful source of ν e whereas the fluxes of ν µ,τ andν e,µ,τ are negligible [8]. Thus V 5 0 and we get that Π 2 (0) = √ 2 π α em G F n ν e F(0) 0, where G F is the Fermi constant and |F(0)| ≈ 2, see Fig. 2(a), since electrons are degenerate. The magnetic diffusion time t diff = σΠ −2 2 (0) ≈ 2.3 × 10 −2 s for n e = 3.7 × 10 37 cm −3 and n ν e = 1.9 × 10 37 cm −3 in the supernova core [6]. Here σ is the electron gas conductivity. Thus at t ∼ 10 −3 s t diff , when the flux of ν e is maximal, no seed magnetic field dissipates. Therefore the neutrino driven instability can result in the growth of the B field. It should be noted that the scale of the B field turns out to be small Λ ∼ 10 −3 cm. However, at later stages of the star evolution V 5 diminishes and Λ can be comparable with the magnetar radius. Thus our mechanism can be used to explain strong B fields of magnetars. Figure 3: The Feynman diagrams contributing to the photon polarization tensor in case of a e − e + self-interacting plasma. Here A µ is the potential of the electromagnetic field. Now let us apply out results to study the B field evolution in the primordial plasma. At the stages of the early universe evolution before the neutrino decoupling at T > (2 − 3) MeV, the e − e + plasma is hot and relativistic. Assuming the causal scenario, in which Λ < H −1 , where H is the Hubble constant, we get that |ξ ν e − ξ ν µ − ξ ν τ | > 1.1 × 10 −6 g * /106.75 × (T/MeV) −1 , see [6], where ξ α = µ α /T , g * is the number of relativistic degrees of freedom, and µ α is the chemical potential of neutrinos of the type α = ν e , ν µ , ν τ . Here we use that |F(0)| ≈ 0.2, see Fig. 2(b). Assuming that before the Big Bang nucleosynthesis at T ∼ (2 − 3) MeV all neutrino flavors equilibrate owing to neutrino oscillations ξ ν e ∼ ξ ν µ ∼ ξ ν τ , we get the lower bound on the neutrino asymmetries, which is consistent with the well-known Big Bang nucleosynthesis upper bound on |ξ α |, see [9].
Finally, let us examine the issue of whether a B field can be amplified in a e − e + plasma self-interacting within the Fermi model, i.e. when a νν gas is not present. In this case the contributions to Π 2 are schematically depicted in Fig. 3. The analytical expression for Π (ee) 2 can be obtained analogously to the previous case [10], where n e,ē are the electron and positron densities, θ W is the Weinberg angle, and J 0,2 = J + 0,2 in Eq. (4), with µ = µ + . The expressions for J 0,1 can be obtained from J 0,2 if we make the replacement µ → µ = µ + k 0 (1 − x) there. As in deriving of Eqs. (3) and (4), here we also assume that k 2 < 4m 2 .
Let us express Π 2 in Eq. (5) as Π 2 = α em √ 2π 1 − 4 sin 2 θ W G F (n e − nē) F, where F is the dimensionless function. We shall analyze this function in the static limit k 0 → 0. We mention that, if we neglect k 0 in Eq. (5), then J 0,2 = J 0,2 and Π 2 → 0. The behavior of F for relativistic plasmas is shown in Fig. 4, where one can see that Π 2 (0) = 0. In Fig. 4 we also account for the thermal corrections to the photon dispersion and to the electron mass. It means that a e − e + plasma does not reveal the instability of a B field leading to its growth. Therefore, contrary to the claim of [5], one can use this mechanism for neither the explanation of strong B fields of magnetars nor the B field amplification in the early universe.
In conclusion we mention that we have derived the CS term Π 2 in an electroweak plasma consisting of e − and e + as well as ν andν of all flavors. These particles are involved in the parity violating interaction. It makes possible the existence of a nonzero CS term. In case of a e − e + plasma interacting with a νν background, the CS term is nonvanishing in the static limit when k 0 = 0. Therefore, a B field becomes unstable in this system. We have shown that a seed field can be exponentially amplified. This feature of an electroweak plasma in question can be used to explain strong B fields of magnetars and to study the evolution of a primordial B field. We have also demonstrated that there is no B field instability in a self-interacting e − e + plasma.
I am thankful to the organizers of 37 th ICHEP for the invitation and a financial support, to V.B. Semikoz for helpful discussions, and to FAPESP (Brazil) for a grant.