# Instability of magnetic fields in electroweak plasma driven by neutrino asymmetries

###### Abstract

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 -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 is provided by a lower bound on which is consistent with the well-known Big Bang nucleosynthesis (upper) bound on neutrino asymmetries in a hot universe plasma.

a,b,c]Maxim Dvornikov c]Victor B. Semikoz \affiliation[a]Research School of Physics and Engineering, Australian National University, 2601 Canberra, ACT, Australia \affiliation[b]Institute of Physics, University of São Paulo, CP 66318, CEP 05315-970 São Paulo, SP, Brazil \affiliation[c]Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radiowave Propagation of the Russian Academy of Sciences (IZMIRAN), 142190 Troitsk, Moscow, Russia \emailAdd \emailAdd \keywordsneutrinos, Chern-Simons term, thermal field theory, magnetic fields, supernova, early universe

## 1 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 () 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 right- and left-chiral electron chemical potentials is not equal to zero, [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 self-energy (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 left-handed electrons 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 and a degenerate ultrarelativistic electron gas ) leads to the damping , 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 in the effective Lagrangian of the electromagnetic field, where is a master parameter we are looking for to solve a problem of the magnetic field generation, is the magnetic field, and 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 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 interaction that is linear in the Fermi constant () 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 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 -field driven by the neutrino asymmetry . 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 -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 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.

## 2 Photon polarization tensor in a gas

In this section we calculate the parity violating term in the polarization tensor in the presence of a gas. It should be noted that photons do not interact directly with neutrinos since latter particles are neutral. Thus the interaction should be mediated by charged leptons, denoted as , which are taken to be virtual particles in this section. We shall take into account the interaction in propagators of ’s as the external mean fields (see Appendix A).

We shall be mainly interested in the case of an isotropic gas when and the nonzero are given in eq. (36). In this situation the most general expression for the polarization tensor reads [12]

(1) |

where is the photon momentum, is the Minkowski metric tensor, is the absolute antisymmetric tensor having , and are the form factors of a photon. Since we study parity violating effects, we should analyze the form factor . Since only real particles, considered in this section, are neutrinos, we add the superscript “” to photon form factors, e.g., 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 in our calculations. Note that we shall consider only the contribution to linear in the external fields . The expression for , which leads to the nonzero in eq. (1), reads

(2) |

where is the electric charge of and the propagators are given in eq. (A).

The traces of Dirac matrices in eq. (2) can be evaluated using the following expressions:

(3) |

To derive eq. (2) we use the fact that .

Using the dimensional regularization and eq. (B) we can express as

(4) |

where and is the Euler Gamma function. The parameters and are defined in Appendix B. Considering the limit and using the fact that , where , we can represent in the form,

(5) |

It should be noted that eq. (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. (5) we find that at , which agrees with the general analysis made in ref. [11] for a CPT-odd gauge invariant effective theory. We also note that in eq. (5) coincides with the result of ref. [12], where the more fundamental Weinberg-Salam theory was used. Moreover, the fact that vanishes at 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 has the nonzero value in two loops.

## 3 Plasma contribution to polarization tensor

In this section we study the direct contribution of charged leptons to the photon form factor 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 [15] in photon dispersion characteristics, see below in eq. (13), that also differs our approach from the use of the lepton chirality.

Thus, in this section we obtain the general expression for 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 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, . As seen in eq. (5), in this case . 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 analogously to section 2. Therefore we shall study the system consisting of a real ’s plasma and a real gas. The presence of ’s and ’s is essential since it is these particles which provide the nonzero contribution to the parity violating form factor based on the 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) (see ref. [16]):

(6) |

where and are the temperature and the chemical potential of the ’s plasma. In principle, we can discuss a general situation when and are different from and defined in eq. (35). 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 , where is the gas temperature equal for all neutrino flavors. However, we shall keep different and .

Using eqs. (2), (2), (6), and (A), as well as defining the effective chemical potentials , we can express in the following form:

(7) |

where

(8) |

Here , , and is defined in section 2. To obtain eqs. (3) and (3) we assume that , i.e. no creation of -pairs occurs.

It should be noted that in deriving eqs. (3) and (3) we exactly account for the ’s mass . 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 , is generated rather by the neutrino asymmetry than by the chiral asymmetry of charged leptons studied, e.g., in refs. [4, 8, 9].

### 3.1 Low density classical plasma

Let us first discuss the case of a low density plasma of ’s, that corresponds to . Using the general eqs. (3) and (3) in the limit we obtain that

(9) |

where since we neglect the photon’s dispersion in plasma. Note that in eq. (3.1) exactly accounts for and .

To estimate the values of and , we shall consider the low temperature limit: . We will identify with an electron and assume that the electron gas has a classical Maxwell distribution. For this medium we get that , where is the fine structure constant and is the background electron density. Moreover for a classical electron gas one has that

(10) |

where we add a tilde over 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 in eq. (10) is times greater than . Note that for a classical nonrelativistic plasma, corresponding to , the integrals in last two lines in eq. (3.1) cancel each other while the integral in the first line leads to the term in eq. (10).

Let us study the derived in the static limit . If we discuss the situation when only virtual charged leptons contribute to PSE, we should set in eq. (10). This limit is equivalent to . Using eq. (10), we obtain that . This our result is in agreement the findings of ref. [13], where it was found that the one loop contribution to -interaction should be vanishing. The leading nonzero contribution to in the case when charged leptons are virtual particles, i.e. when we neglect the plasma contribution given by eqs. (3) and (3), was obtained in ref. [14]. Using the results of ref. [14], one gets that in this situation , where is the -boson mass.

### 3.2 Hot relativistic plasma

The dispersion relation for transverse waves in relativistic plasma reads [15],

(11) |

The plasma frequency can be found from the following expression:

(12) |

where .

Using eq. (12) in the relativistic limit , we get that . The transcendant eq. (11) can be explicitly solved if long waves with are considered. In this situation the dispersion relation is .

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}^{1}1Under intermediate conditions in plasma with or (or both) the effective mass of an electron should be , see in ref. [15].

(13) |

in eqs. (5), (3), and (3). Note that eq. (13) is valid for both and . Accounting for the dispersion relation and the expression for , we get that in a hot relativistic plasma.

Let us represent as

(14) |

where is the dimensionless function which depends on . Note that in eq. (14) includes the contributions from eqs. (5) and (3). Accounting for eq. (13), we present the behaviour of versus we in figure 2(a). We study long waves limit when . Thus we should be interested in the values of corresponding to .

One can see in figure 2(a) that for a hot relativistic plasma is nonvanishing in the static limit: . However this nonzero value strongly depends on the photon’s dispersion law in such a plasma.

### 3.3 Degenerate relativistic plasma

In case of a degenerate plasma the dispersion relation for transverse waves is [15],

(15) |

where is the Fermi velocity. The plasma frequency can be found from eq. (12) if we make the following replacement: and , where is the Heaviside step function.

Let us discuss the degenerate plasma in the relativistic limit. In this situation and . The general dispersion relation in eq. (15) transforms into if we study long waves. Using eq. (13), we also get the effective electron mass in a degenerate plasma. One can check that the inequality is valid.

Using eqs. (5), (3), and (3) in the limit , as well as the following representations of the Dirac delta function and its derivative:

(16) |

we can derive the expression for the function , see eq. (14), 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 since it is very cumbersome.

The function versus is shown in figure 2(b). We discuss the long waves limit. Thus . It means that for our purposes we should consider at . One can see in figure 2(b) that . Therefore, as in case of a hot relativistic plasma, for a degenerate relativistic plasma is nonvanishing at , but its actual value is different from that found in section 3.2.

## 4 Instability of magnetic fields in relativistic plasmas driven by neutrino asymmetries

We consider below two cases for which the CS term in the photon polarization operator plays a crucial role. A nonzero leads to the -dynamo amplification (instability) of a seed magnetic field even without fluid vortices or any rotation in plasma which are usually exploited in the standard MHD approach for -dynamo [18]. The first case considered here concerns the magnetic field growth in a degenerate ultrarelativistic electron plasma, , 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 before the neutrino decoupling at . In both cases neutrinos are in equilibrium with a plasma environment. For these applications we use our result in eq. (14).

First, we derive in subsection 4.1 the Faraday equation generalized in SM to find the key parameters leading to the -field instability. The corresponding evolution equations for the spectra of the magnetic helicity density and magnetic energy density 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 [19]. Here is the magnetic helicity density and is the magnetic energy density for an uniform isotropic medium.

An excess of electron neutrinos during
a first second of a supernova explosion^{2}^{2}2Neutrino emission prevails over the antineutrino one during first milliseconds of a supernova burst due to the reaction (urca-process) before its equilibrium with beta-decays is settled in (see figure 11.3 in ref. [20]). allow us to put 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 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.

### 4.1 Generalized Faraday equation in the Standard Model

The existence of a neutrino asymmetry accounting for the difference in eq. (36),

(17) |

leads to a non-zero parity violation term in the photon polarization operator , where is given by eq. (14) and .

The CS polarization term in eq. (14) corresponds to the induced pseudovector current in the Fourier representation,

(18) |

entering the generalized Maxwell equation in the standard model (SM)

(19) |

Expressing the ohmic current as ,
then neglecting the displacement current in the l.h.s. of eq. (19), that is a standard assumption in the MHD approach for which ^{3}^{3}3The conductivity depends on the Coulomb collision frequency . Here we use the values for the electron density in a hot plasma and for the Coulomb logarithm. Obviously the MHD condition is fulfilled to obtain eq. (20)., and finally using the Bianchi identity , one gets the generalized Faraday equation in SM in the coordinate representation,

(20) |

where is the magnetic helicity parameter,

(21) |

and is the magnetic diffusion coefficent.

Here we use the long-wave approximation for large-scale magnetic fields where the operator is at least uniform, , and almost stationary since the function depends on a small ratio or . For instance, in the long-wave limit the transversal plasmons (photons) have the spectrum in a hot plasma (]) and in the ultrarelativistic degenerate electron gas () [15] (see spectra in eqs. (11) and (15) above). In a relativistic plasma this approximation corresponds to the negligible spatial dispersion, , where we put both in hot and degenerate relativistic plasmas. Here is the wave number. Thus, the ratio or allows us to consider without temporal and spatial dispersion as a function of the temperature (a hot plasma in the early universe) or the chemical potential (a degenerate electron gas in a supernova) only.

### 4.2 Amplification of a seed magnetic field in a supernova

During the collapse (time after onset of collapse, see figure 11.1 in ref. [20]) one can neglect emission and reads

(22) |

where the function is shown in figure 2(b) for a degenerate ultrarelativistic electron gas with .

Let us give some estimates for in a collapsing SN with the progenitor stellar mass considered in ref. [20] (see there the plots for evolution stages in figures 11.1-11.3). In order to obtain we should find the appropriate neutrino asymmetry density .

At the stage just after collapse neutrinos are captured, their free path does not exceed the core radius . For instance, for the nuclear core density one gets if , or for . Here, using the nucleon mass , one gets the baryon density .

The lepton abundance is typical for the material in a SN core just after the collapse, so that the equilibrium condition allows us to look for the Fermi momenta for degenerate electrons and neutrinos, and . The second equation for these quantities comes from the consideration of figure D7(a) in ref. [20], where for the same matter density one finds the difference corresponding to the temperature . Note that leptons are ultrarelativistic, , while nucleons are degenerate and nonrelativistic, . Eventually we get all Fermi momenta in such dense core: , , . Here the last equality comes from the electroneutrality condition . Thus, we get the electron neutrino density at this stage of the SN evolution, , which should be substituted into eq. (22) neglecting antineutrino contribution.

The magnetic diffusion time seen from the Faraday eq. (20),

(23) |

is given by the electrical conductivity for degenerate ultrarelativistic electrons and degenerate nonrelativistic protons, [22]. Note that the combined effects of the degeneracy and the shielding reduce the collision frequency . Thus collisions of charged particles are blocked due to the Pauli principle since states are busy and at .

The electrical conductivity was found in ref. [22],

(24) |

For and the corresponding electron density , as well as the temperature in SN core we have just estimated, eq. (24) gives . This result leads to the estimate . It means that any seed magnetic field existing in plasma does not dissipate ohmically during first milliseconds after onset of collapse, , and evolves for a given wave number through the -dynamo driven by neutrino asymmetries (see in Appendix D) as

(25) |

If , the seed magnetic field in eq. (25) will grow exponentially. The fastest growth corresponds to the -dynamo with for which .

Unfortunately, under the same conditions (for large ) the scale of the magnetic field occurs to be rather small, . Here we use the fact that , see figure 2(b). However, such a scale grows when the neutrino asymmetry diminishes due to a significant involvement of antineutrinos somewhere later at , (see figure 11.3 in ref. [20]). It reaches the core radius , , for the neutrino asymmetry density

(26) |

The magnetic diffusion time should be recalculated for this stage of SN burst separately (the release of prompt -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 -field growth in a supernova driven by the electron neutrino asymmetry could lead to an additional amplification of a strong seed magnetic field () during the first second of a SN explosion when the asymmetry remains appreciable. Here a strong seed magnetic field can arise from a small magnetic field of a protostar, e.g., , due to the conservation of the magnetic field flux, , during the protostar collapse. The question whether this new mechanism can explain the strongest magnetic field of observed magnetars ( G) deserves a separate study (see also in section 4.4).

### 4.3 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 in Faraday eq. (20) reads as

(27) |

where we substituted the dimensionless neutrino asymmetries for the asymmetry densities and used the hot plasma conductivity , with . The magnetic field evolution with the parameter in eq. (27) obeys the causal scenario, where the magnetic field scale is less than the horizon, , if the sum of neutrino asymmetries satisfies the inequality

(28) |

Here we take into account that (upper sign stays for electron neutrinos) is the SM axial coupling constant for interaction corresponding to the difference in eq. (17). In eq. (27) we use that , which results from figure 2(a). Moreover we account for that , with , where is the Plank mass, is the number of relativistic degrees of freedom above the QCD phase transition, . Let us remind that to get eq. (28) we applied the photon polarization term in eq. (14) for ultrarelativistic leptons with .

One can see that the inequality in eq. (28) does not contradict to the well-known BBN bounds on the neutrino asymmetries at the lepton stage of the universe expansion corresponding to