# Neutrino induced vorticity, Alfvén waves and the normal modes

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## Abstract

We consider a plasma consisting of electrons and ions in the presence of a background neutrino gas and develop the magnetohydrodynamic equations for the system. We show that the electron neutrino interaction can induce vorticity in the plasma even in the absence of any electromagnetic perturbations if the background neutrino density is left–right asymmetric. This induced vorticity supports a new kind of Alfvén wave whose velocity depends on both the external magnetic field and on the neutrino asymmetry. The normal mode analysis show that in the presence of neutrino background the Alfvén waves can have different velocities. We also discuss our results in the context of dense astrophysical plasma such as magnetars and show that the difference in the Alfvén velocities can be used to explain the observed pulsar kick. We discuss also the relativistic generalisation of the electron fluid in presence of an asymmetric neutrino background.

## 1 Introduction

It is important to study the characteristics of the plasma in the presence of neutrinos, since such systems are important in understanding various physical phenomena during the evolution of the early Universe as well as the systems like core-collapsing supernovae and magnetars (see e.g. [1] for a brief overview). The presence of the cosmic neutrino background can influence cosmic microwave anisotropy and matter clustering [2, 3] and it can also influence dynamics of the primordial magnetic field [4, 5, 6]. There exist several studies in the literature where the neutrino plasma interaction has been analysed in a variety of physical situations. Non-linear coupling of intense neutrino flux with collective plasma oscillations is studied in Ref. [7]. The authors have shown that a neutrino flux as intense as that in supernovae core can cause parametric instabilities in the surrounding plasma. The effect of a neutrino medium in the evolution of the lepton plasma had been studied invoking a ponderomotive description [8, 9]. In these cases it was shown that the ponderomotive force is proportional to the gradient of neutrino density and the electrons are repelled from the regions where neutrino density is large. Interaction of very large number of neutrinos with collective plasma and oscillation and the excitation of the plasma turbulence is considered in Ref. [10]. Different kinds of the plasma–neutrino interactions using the ponderomotive force description and the effect on collective plasma properties can be found in Refs. [11, 12, 13, 14, 15, 16, 17]. In the above-mentioned ponderomotive force description, it was assumed that the neutrino field satisfies the naive Klein–Gordon equation with appropriate interaction terms. Thus in this formalism the information about the chiral structure of the weak interaction is absent. Here we note that, by Silva et al. in Refs. [18, 19], the problem of neutrino driven streaming instability, which in turn can cause significant energy transfer from neutrino to the plasma, was considered in the kinetic theory formalism. A formulation to study the plasma interaction with intense neutrino beam using the field theory techniques is developed in [20]. The photon polarisation tensor in a medium consistent with gauge and Lorentz invariance can be found in [21]. In this work it is shown that, in the presence of a medium, the photon polarisation tensor can have an anti-symmetric part, indicating *P* and *CP* violations. Further studies of such effect in the presence of neutrinos for different physical scenarios are found in [22, 23].

In the context of the early Universe, it has been shown by Shukla et al. [8] that the ponderomotive force of a non-uniform intense neutrino beam can be responsible for a large-scale quasi-stationary magnetic field. In fact, this was first to suggest the magnetic field generation in the plasma due to plasma–neutrino interactions. Further, large-scale magnetic field generation at the time of neutrino decoupling due to the evolution of the plasma in the presence of an asymmetric neutrino background is studied in [24, 25]. This field can act as a seed for generation of the galactic magnetic field via the galactic dynamo mechanism (see e.g [26] for the galactic dynamo mechanism). It is to be noted that at finite lepton/baryon density the loop corrections to the photon polarisation tensor are non-vanishing. With these corrections the photon polarisation tensor acquires a non-zero parity-odd contribution \(\Pi _2(\varvec{k})\) where \(\varvec{k}\) is the wave vector. A finite and non-zero value of \(\Pi _2(\varvec{k})\) in the photon polarisation tensor means that there can be single field derivative terms in the effective Lagrangian and free energy, which dominates the kinetic energy part of the free energy which is having a double derivative term. For e.g. the free energy for a static gauge field can be written as \(\mathcal {F}[A]=\int \mathrm{d}^3pA_i(k)\Pi _{ij}(k)A_j(-k)\) and with parity violating interactions \(\Pi _{ij}(k)\) can have a contribution \(i\Pi _2(p^2)\epsilon _{ijl}k_l\). Thus a non-zero value of \(\Pi _2(0)\) means a term \(\Pi _2(0)\varvec{A}\cdot \nabla \times \varvec{A}\) in the expression for the free energy. This in turn means that there can be a generation of a large-scale (\(k\rightarrow 0\)) magnetic field by an instability arising due to non-zero values of parity-odd contributions \(\Pi _2(0)\) to the polarisation tensor [24]. In Ref. [27], thermal field theory calculations were carried out to study the corrections to the photon polarisation in the presence of a background neutrino which is asymmetric in left–right number densities. The authors have shown that the axial part \(\Pi _{2}\) is proportional to the neutrino asymmetry parameter and argued that the contribution to \(\Pi _{2}\) due to the plasma which is interacting with the neutrino gas is \(\sim \) \(10^1\) times larger than the contribution to \(\Pi _2\) through the correction due to the virtual process. In Ref. [28] using a kinetic theory approach it was shown that the photon polarisation tensor can have a parity-odd contribution \(\Pi _2(\varvec{k})\) due to the asymmetric neutrino background in both the collision-less and the collision dominated regime. In the collision dominated regime the result for \(\Pi _2(\varvec{k})\) using the kinetic approach agrees with that in Ref. [27]. In a recent work [29] the authors have calculated the effective potential or refractive index for the cosmic neutrino background (CNB) and future experimental implications have been discussed.

Further, recent theoretical calculations showed that the asymmetry in the neutrino density can be transmuted to the fluid helicity for sufficiently large electron neutrino interaction [30]. This neutrino induced vorticity can act as axial chemical potential for the chiral electrons. This phenomenon can induce the *helical plasma instability*, generating a strong magnetic field [30]. In this work the plasma particles are considered to be massless and chirally polarised. Moreover, it was assumed, in this work, that the neutrino mean free path \(l_\nu \) is much smaller than the system dynamics at the length scale *L* i.e. \(L\gg l_\nu \). This allows one to write the equations for the neutrino hydrodynamics [30]. Though this assumption is justifiable for a core collapsing supernova, it is hard to satisfy in other scenarios like the early Universe. Electroweak plasma in a rotating matter is studied in [31]. In this work it is shown that the electric current can be induced in the direction of the rotation axis due to the parity violating nature of the interaction. This phenomenon is called the *galvano-rotational effect* (GRE). In a recent work [32], a spin paramagnetic deformation of neutron star has been studied and the authors have calculated the ellipticity of a strongly magnetised neutron star using the spin magnetohydrodynamic equations developed in [33].

In the present work we are interested in developing a magnetohydrodynamic description of the plasma in the presence of the left–right asymmetric neutrino background. The expression of the interaction Lagrangian of a charged lepton field and the asymmetric neutrinos suggests that the neutrino can couple with the spin of the electron [27, 34]. It is interesting to note here that there exists a lot of literature in on the usual electron–ion plasmas where the dynamics of spin degree can play a significant role. For example it was suggested that a spin polarised plasma in a fusion reactor can yield a higher nuclear reaction cross section [35] and the spin depolarisation process in the plasma can remain small [36]. The effect of spin dynamics using a single particle description, valid for a dilute gas, is studied in the context of laser plasma interaction in Ref. [37]. The collective effects within the framework of spin-magnetohydrodynamics has been studied in Refs. [33, 38] (for a general discussion see [39]). This work can have applications in studying environments with a strong external magnetic field like pulsars and magnetars. In the present work we generalise the spin-magnetohydrodynamics considered in [33] to incorporate the effect of the asymmetric neutrino background.

The report is organised in the following way. In Sect. 2 we consider the low energy Lagrangian for our system and the equations of motion and spin evolution equations are derived invoking the non-relativistic approximations. MHD equations are considered in Sect. 3. Velocity perturbations and electromagnetic perturbations in a magnetised plasma interacting with neutrino background are considered in Sect. 4. In Sect. 5 we apply our theory to a neutron star to calculate the kick and Sect. 6 is for a summary and for our conclusions. We provide a brief summary of a relativistic generalisation of the theory in Appendix A.

## 2 The Lagrangian and non-relativistic approximation

*m*is the mass of the lepton, \(\gamma ^\mu =(\gamma ^0, \varvec{\gamma })\) are the Dirac matrices and \(P_{L,R}=\frac{1\mp \gamma ^5}{2}\) are the chiral projection operators with \(\gamma ^5=i\gamma ^0\gamma ^1\gamma ^2\gamma ^3 \). \(f^\mu _{L,R}=(f^0_{L,R}, \varvec{f}_{L,R})\) are the neutrino currents and they are regarded as external macroscopic quantities.

*g*is the Landè g-factor. The first three terms on the right hand side are well known and very well studied in the literature. The fourth and fifth terms are due to the neutrino background. The last term might contribute to the energy of the system, but it will not enter into the equations of motion as the neutrino background is considered to be constant. If the neutrino background varies with space and time, this term would modify the force equation as \(\varvec{F}\propto \varvec{\nabla } f=\varvec{\nabla }\psi _\nu ^*\psi _\nu \). This force is called the ponderomotive force. Such a scenario was studied in Ref. [9]; however, in the formalism in this reference the fourth term was not considered.

## 3 The hydrodynamic equations

In this section we follow the methods developed in Ref. [33] to derive the hydrodynamic equations from the quantum Lagrangian for spin half particles. We consider a system of electrons and ions in the presence of a homogeneous neutrino background. The neutrino background is assumed to have a left–right asymmetry. Furthermore we treat electrons as quantum particles and ions as classical particles so that we can neglect the spin dynamics and other quantum effects for ions.

*q*and mass

*m*, so that the wave function for the entire system can be factorised as \(\psi =\psi _{(1)}\psi {(2)}\cdots \psi _{(N)}\). Then we can define the total particle density for charge

*q*as \(n_q=\sum _{\alpha }n_\alpha \) and the expectation value of any quantity

*f*as \(\langle f\rangle =\sum _\alpha \frac{n_\alpha }{n_q}f\).

*P*is the homogeneous part. \(\mathcal {C}_{qi}\) represents the collision between particle with charge

*q*and ion denoted using the letter

*i*and the quantum force density \(\varvec{F}_\mathrm{Q}\) and the force due to the interaction with the neutrino back ground \(\varvec{F}_{\nu e}\) are defined by

*q*with

*e*and

*i*for electrons and ions, respectively. Since we are considering ions as classical particles, we can neglect the contributions from spin and other quantum effects for ions. Thus, the fluid equations for ions read

## 4 Neutrino induced vorticity, Alfvén wave and normal modes

*induced*by the asymmetry in the neutrino background. The wave velocity not just depends on the strength of magnetic field but also on the neutrino asymmetry. This new mode is similar to the one found in the very high energy plasma with the chiral-anomaly [30]. In contrast to Ref. [30], in our work the electrons are not considered to be chirally polarised. However, the parity violating interaction in our work arises due to a neutrino–electron interaction. Furthermore, the effect of dissipation can easily be introduced by incorporating the contribution of the finite shear viscosity \(-ik^2\eta _\mathrm{vis}\) and the resistivity \(-i\sigma _1 B^2_0\) into the dispersion relation (42), where \(\eta _\mathrm{vis}\) is the kinematic viscosity and \(\sigma _1=\sigma /\rho _0\) with \(\sigma \) being the resistivity.

## 5 Neutrino asymmetry and the pulsar kick

We use our formalism for a qualitative calculation of the observed pulsar kick [43, 44, 45]. There have been several attempts to explain the reason for the kick, see e.g. Refs. [46, 47, 48, 49]. Recently there have been attempts to explain the pulsar kick using anomalous hydrodynamic theories (see e.g. Ref. [50]), but the exact reason for the pulsar kick is not yet resolved.

*A*is the magnitude of the vector potential \(\varvec{A}_{\omega ,\varvec{k}}\). Using Eq. (53) we write

*excess*photons leaving the pulsar per unit area per unit time. Therefore the change in velocity experienced by the pulsar can be expressed as

## 6 Discussion and conclusion

In conclusion we have developed spin magnetohydrodynamic equations in the presence of asymmetric background neutrinos and analysed the normal modes of the plasma in the presence of a constant magnetic field. We have shown that a new kind of wave (Alfvén) is generated as described by Eq. (42) and speed of this wave depends on the neutrino asymmetry. Such a wave can be generated in a dense astrophysical plasma such as a magnetar. For example for \(B_0^{15}\) Gauss, \(T\sim 10 MeV\) and \(\Delta n_{\nu e}\sim 1.6\times 10^{8}\) (MeV)\(^3\), one can estimate the velocity of the wave (in units of speed of light) around \(10^{-5}\). We have shown that the background neutrino asymmetry can modify the wave velocity in directions parallel and anti-parallel to the external magnetic field (as shown in Eq. (54)). We have used our formalism to calculate the kick received by a pulsar during its birth. An order of magnitude calculation matches with the observations \(\Delta V_\mathrm{NS}\simeq (10^2{-}10^3)~\mathrm{km}/\mathrm{s}\). In the appendix we have derived the relativistic hydrodynamical equation for the electrons using the Dirac equation. For the case when the electrons are relativistic the estimate given here for the Alfvén velocity can be suppressed by a factor \(1/\sqrt{2}\).

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