Magnetic field induced neutrino chiral transport near equilibrium

Based on the recently formulated chiral radiation transport theory for left-handed neutrinos, we study the chiral transport of neutrinos near thermal equilibrium in core-collapse supernovae. We first compute the near-equilibrium solution of the chiral radiation transport equation under the relaxation time approximation, where the relaxation time is directly derived from the effective field theory of the weak interaction. By using such a solution, we systematically derive analytic expressions for the nonequilibrium corrections of the neutrino energy-momentum tensor and neutrino number current induced by magnetic fields via the neutrino absorption on nucleons. In particular, we find the nonequilibrium neutrino energy current proportional to the magnetic field. We also discuss its phenomenological consequences such as the possible relation to pulsar kicks.


I. INTRODUCTION
One of the most important properties of neutrinos in the Standard Model of particle physics is the left-handedness. Although neutrinos are expected to play important roles in the explosion dynamics of core-collapse supernovae, this property has been neglected in the conventional neutrino radiation transport theory [1][2][3][4][5] applied so far; for recent reviews on the theoretical aspects of core-collapse supernovae, see, e.g., Refs. [6][7][8][9][10][11]. It is thus important to study the effects of chirality of neutrinos on the dynamics of the core-collapse supernova as pointed out in Ref. [12].
Recently, starting from the underlying quantum field theory, the authors of this paper have systematically constructed the neutrino radiation transport theory incorporating the effects of chirality. It is dubbed as the chiral radiation transport theory [13]. Unlike the conventional neutrino radiation hydrodynamics, this theory explicitly breaks the spherical symmetry and axisymmetry of the system by the quantum effects related to the chirality. Moreover, novel transport phenomena that have been missed in the conventional theory emerge, which may qualitatively change the time evolution of the system. The construction of such a theory was made possible thanks to the recent developments of the kinetic theory for chiral fermions, called chiral kinetic theory, in high-energy physics [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29].
In this paper, based on this chiral radiation transport theory for neutrinos, we study the chiral transport of neutrinos near thermal equilibrium in core-collapse supernovae. We first compute the near-equilibrium solution of the chiral radiation transport equation under the relaxation time approximation, where the relaxation time is directly derived from the effective field theory of the weak interaction. By using this solution, we then analytically derive the nonequilibrium corrections of the neutrino energy-momentum tensor and current induced by magnetic fields through the neutrino absorption on nucleons. In particular, we find the nonequilibrium neutrino energy current and neutrino number current proportional to the magnetic field; see Eqs. (73) and (74). Although the asymmetric neutrino emission induced by the strong magnetic field was also discussed in previous works in relation to the possible origin of the pulsar kicks [30][31][32][33][34][35][36][37][38][39], this work is the first, to the best of our knowledge, to derive the explicit form of the magnetic field induced energy-momentum tensor of neutrinos by systematically taking into account the effects of chirality of leptons. This work, together with our previous work [13], also explicitly bridges the gap between the microscopic theory of the weak interaction for neutrinos and the neutrino radiation hydrodynamics.
The paper is organized as follows: In Sec. II, we review the chiral radiation transport theory for neutrinos. In Sec. III, using the relaxation time derived from the effective theory of the weak interaction, we compute the nearequilibrium solution of the chiral radiation transport theory. In Sec. IV, we derive generic expressions for the neutrino energy-momentum tensor and current near equilibrium. In Sec. V, we compute the nonequilibrium corrections on the neutrino energy-momentum tensor and current induced by the magnetic field. Section VII is devoted to discussions and outlook.
Throughout this work, we assume massless neutrinos. We use the Minkowski metric η µν = diag{+, −, −, −}. We define the Levi-Civita tensor ǫ µναβ =ǫ µναβ / √ −g, whereǫ µναβ denotes the permutation symbol and g represents the determinant of the spacetime metric with the conventionǫ 0123 = −ǫ 0123 = 1. For a given vector V µ , the unit vector is denoted byV µ = V µ /|V | with V being the spatial component of V µ . We absorb the electric charge e into the definition of the gauge field A µ . We also introduce the notations After Sec. II, we take = c = k B = 1 except where the expansion is shown.

II. CHIRAL RADIATION TRANSPORT THEORY FOR NEUTRINOS
In this section, we review the chiral radiation transport theory for neutrinos developed in Ref. [13] that will be applied in the following sections. The general relativistic form of the chiral transfer equation with collisions for left-handed neutrinos is given by 1 Here, ∂ µ and ∂ qµ denote the spacetime and four-momentum derivatives, respectively, f is the distribution function of the left-handed neutrino which generically depends on the frame vector n µ (see below), is the spin tensor for spin 1/2 fermions with n µ the frame vector satisfying ρ is the Riemann tensor, and Σ ≶ µ are the lesser and greater self-energies. The emission and absorption rates are given by R emis = Γ < /q 0 and R abs = Γ > /q 0 , respectively. The terms related to the spin tensor S µν q(n) in Eqs. (1) and (2) that have been missed in the conventional neutrino transport theory explicitly break the spherical symmetry and axisymmetry of the system.
Note that the dependence of the spin tensor S µν q(n) on the frame vector n µ emerges as a choice of the spin basis, and consequently, f (ν) q(n) and Γ ≶ q(n) also depend on n µ [19,20]. However, the physical quantities do not depend on the choice of n µ at the end. Below we will always choose the frame vector n µ = ξ µ ≡ (1, 0) in the inertial frame, then we have ∇ µ n ν = 0, D µ S µν (n) = 0, and R λ ρµν = 0, and all the corrections due to the chirality of neutrinos appear in the collision term as Γ Accordingly, we will not hereafter highlight the frame dependence of the quantities, such as f (ν) q(n) . In this case, the chiral radiation transport equation reads where ✷ i is given by [5] Here, we adopt the spherical coordinate system (r, θ, φ) for the position and (E i ,θ i ,φ i ) for the momentum of the neutrino and the subscripts "i" stand for the quantities in the inertial frame. We also defined µ i ≡ cosθ i . Note that ✷ i may also be written in a more generic form via the horizontal lift, ✷ i = q · D/E i . For the collision term, we will focus on the neutrino absorption on nucleons ν e L (q) + n(k) ⇋ e L (q ′ ) + p(k ′ ). We are interested in the length scale much larger than the mean free path in the matter sector composed of electrons and nucleons. In this case, ignoring the viscous corrections and the gradients of the temperature and chemical potentials, we may decomposeΓ whereŌ stands for a quantity O in local thermal equilibrium, ω µ = ǫ µναβ u ν ∂ α u β /2 is the vorticity, and B µ = ǫ µναβ u ν F αβ /2 is the magnetic field defined in the fluid rest frame with u µ being the fluid four velocity and F αβ the field strength of the U(1) electromagnetic gauge field. The expression for the classical termΓ (0)≶ q was derived in Ref. [40], while the expressions for the quantum correctionsΓ where G F is the Fermi constant and g V = 1 and g A ≈ 1.27 are the nucleon vector and axial charges, respectively. We also introduced the Fermi-Dirac distributions where β = 1/(k B T ) with T being temperature and µ i chemical potentials for i = n, p, e, and n n/p = are neutron/proton densities. Although q · u ≈ E i ≡ q · ξ for the on-shell fermions, we rigorously distinguish between q · u and E i in the expressions ofΓ ≶ q above. This difference will become important in computing the neutrino energy-momentum tensor T µν (ν) and neutrino current J µ (ν) below since ∇ µ (q · u) = ∇ µ E i = 0. For a given f (ν) q , the energy-momentum tensor and current of neutrinos are given by [13] T µν where and we introduced the notation (with setting √ −g = 1 in flat spacetime) The energy-momentum transfer from neutrino radiation to matter is dictated by the energy-momentum conservation law where T µν mat is the energy-momentum tensor of the matter sector composed of electrons, neutrons, and protons. In the presence of the electromagnetic fields, the energy-momentum conservation law is modified to where J (p)µ is the electric current of protons and J µ (e) = J µ R(e) + J µ L(e) is the electric current of electrons including the contributions from both right-and left-handed electrons.
In addition, we also have the lepton current conservation, anomaly relation for the axial current, electric current conservation, and baryon current conservation, which are given by is the axial current of electrons, J µ (n) is the current of neutrons, and E µ = F µν u ν is the electric field defined in the fluid rest frame. When the matter sector is in equilibrium, its state is characterized by u µ , T , µ p , µ n , the electron (vector) chemical potential µ e = (µ eR + µ eL )/2 and electron chiral chemical potential µ e5 = (µ eR − µ eL )/2. 2 So far, the governing equations are generic and are applicable even when the neutrino sector is far away from equilibrium. In the following, we will consider the case where the neutrino sector is near equilibrium (which is the case near the core of the supernova), and then its evolution is further characterized by the neutrino chemical potential µ ν . Here, for simplicity, we assume that the matter sector and neutrino sector have the same temperature and fluid velocity. In this case, the time evolution of the system, when ignoring the evolution of the dynamical electromagnetic fields, is governed by Eqs. (13) and (15)- (18). In total, one has nine variables and eight conservative equations. To form a closure for the equations and variables, we have to incorporate the β equilibrium condition, µ e + µ p = µ ν + µ n . In the presence of dynamical electromagnetic fields, we need to solve Eqs. (14) and (15)-(18) coupled to Maxwell's equation simultaneously.

III. NEAR-EQUILIBRIUM SOLUTION FOR THE CHIRAL TRANSPORT EQUATION
Based on the chiral radiation transport equation above, let us solve for the near-equilibrium distribution function of neutrinos. In the following, we take = c = k B = 1 except where the expansion is shown.
We first consider the case of equilibrium state for neutrinos where the collision term vanishes, We decompose the neutrino distribution function asf 1,q denotes the quantum correction on the classical distribution function in equilibrium, f (ν) 0,q . It then follows that From Eqs. (5)-(8) on the other hand, we havē up to O( ). Comparing the right-hand sides of Eqs. (19) and (21) order by order in , we obtain We accordingly obtain the equilibrium distribution function for neutrinos, where with µ ν the neutrino chemical potential that satisfies the β equilibrium condition µ e + µ p = µ ν + µ n . For consistency, we here drop the q · u/M correction since the O(1/M ) corrections onΓ are already neglected based on the nonrelativistic approximation above. After dropping this term,f (ν) q above agrees with the equilibrium distribution function in Refs. [19,21].
When neutrinos are not in complete equilibrium but are close to equilibrium, we may rewrite the collision term in the relaxation time approximation, where δf is the fluctuation of the distribution function and τ = E i / Γ > q +Γ < q denotes a momentumdependent relaxation time which describes how long the system returns to the equilibrium state. From Eqs. (19) and (5), we find Solving Eq. (3), the perturbative solution of δf (ν) q is given by By decomposing the relaxation time as τ = τ (0) + τ (1) via the expansion, we have more explicit expressions with Here, we used Eq. (23) with dropping the O(1/M ) terms. Note that the relaxation time is directly derived from the Fermi theory, which is the low-energy effective field theory of the weak interaction. Some remarks are in order here. First, one may attempt to include the magnetic moments of nucleons neglected in Ref. [13]. Naively, we may take into account the effects of the nucleon magnetic moment by consistently replacing µ i by µ i − s i λ i |B|/(2M ) for i = n, p, where λ i /(2M ) is the magnetic moment and s i = ±1 denotes the spin up or down. This amounts to the replacement of f for M − µ i ≫ T and |B| ≪ M T . In such a case, one obtains an extra contribution from the magnetic field to the relaxation time, In this approximation, however, the nucleon wave functions do not include the magnetic field corrections. Hence, a more systematic inclusion of the magnetic field corrections in the nucleon Wigner functions (in addition to the distribution functions) would be necessary. 3 For this reason, we do not consider the magnetic moment contributions from nucleons in the present paper. Second, one may also consider the elastic neutrino-nucleon scattering ν ℓ L (q) + N(k) ⇋ ν ℓ L (q ′ ) + N(k ′ ). Nevertheless, an analytic form for the collision term in the relaxation time approximation linear to δf (ν) q cannot be derived by simply adopting the isoenergetic approximation. In light of Ref. [13], the collision term reads where the O( ) terms are dropped here. (The detailed structure of Π (NN) p,µλ obtained from the isoenergetic approximation can be found there.) When neutrinos are near equilibrium, one finds f where p is the momentum transfer. To obtain a nonvanishing collision term analytically, a further assumption for the hierarchy between the neutrino momentum |q| and the momentum transfer |p| has to be imposed. Moreover, it is necessary to consistently incorporate O(|p|/M ) corrections and the recoil momenta on nucleons, which are already neglected in the isoenergetic approximation. Therefore, we also do not include the elastic neutrino-nucleon scattering in the present work for consistency.
On the other hand, inserting Eq. (28) into Eqs. (37) and (39), the nonequilibrium corrections for the neutrino energy-momentum tensor and current become As D µ is defined such that D µ q ν = 0, it follows that D µ F (q · u) = ∇ µ F (q · u) for an arbitrary function F (q · u). Accordingly, we may replace D µ by ∇ µ when it acts on τ (0) or f (ν) 0,q in Eqs. (46) and (47). We now make a further decomposition, δT µν (ν) = δT correspond to the explicit classical and quantum fluctuations, respectively. However, as will be discussed later, from the corrections encoded in hydrodynamic equations of motion, δT  . Similarly, we decompose as δJ µ (ν) = δJ

V. NONEQUILIBRIUM CORRECTIONS FROM MAGNETIC FIELDS
In this section, we derive the explicit forms of nonequilibrium corrections on the neutrino energy-momentum tensor and current. Using whereμ i ≡ βµ i for i = ν, e, p, n, we can evaluate δT µν (ν) and δJ µ (ν) explicitly. [Note again that the difference between q · u and E i is essential here since ∇ µ (q · u) = ∇ µ E i = 0.] Nonetheless, the full δT µν (ν) and δJ µ (ν) are rather complicated, and here we will focus on the contributions due to magnetic fields in which τ (1) is involved.
In principle, the leading-order corrections δT and δJ (0)µ (ν) may also incorporate magnetic field corrections through the hydrodynamic equations of motion that determine the temporal derivatives on thermodynamic parameters up to O( ). Nevertheless, as will be shown in Sec. VI, the possible contributions are proportional to B · ∇ ⊥ T and B · ∇ ⊥ µ, which are different from the forms of the viscous corrections originating from τ (1) that we are interested in here. For the magnetic field induced corrections involving τ (1) , we find where Θ ρλ ≡ ∇ {ρ βu λ} . We can decompose Θ ρλ and q · ∇μ ν as where with π µν ≡ β∇ {µ ⊥ u ν} − ∆ µν θ, θ ≡ β∇ ⊥ν u ν /3, D ≡ u · ∇ the temporal derivative in the fluid rest frame, and v µ ⊥ ≡ ∆ µν v ν for an arbitrary vector v µ . By symmetry, we expect the following constitutive relations: where ∆ µν B = ∆ µν +B µBν , h ⊥ · B = 0, and σ µν B u µ = 0. The explicit forms of these transport coefficients read The details of the derivation are shown in Appendix B. Here, all the temporal derivatives D on the thermodynamic parameters should be replaced by spatial gradients via hydrodynamic equations shown in Sec. VI. Note that δN B in Eq. (65) logarithmically diverges, but this may be regularized by the screening mass of the neutrino in medium. By utilizing hydrodynamic equations shown in Eq. (102), one may replace Du µ by ∇ µ ⊥ T and ∇ µ ⊥μ forμ = (μ e ,μ p ,μ n ,μ ν ) and drop the terms coupled to Dμ ν = O( ) as higher order corrections in the expansion. For simplicity, we assume ∇ µ ⊥ T and ∇ µ ⊥μ are suppressed and omit δǫ B , δp B , h ν ⊥ , and δN B . The remaining terms are then given by where Note that the results in Eqs. (67) and (68) are independent of the nuclear equation of state.
Although the isoenergetic approximation may break down, it would be useful to extrapolate these results to the regimeμ e ≫ 1 andμ ν ≫ 1 to obtain more compact forms, which will be used for an order of estimate in Sec. VII. In this regime, we find When we further assume thatμ n −μ p =μ e −μ ν ≫ 1 and u µ ≈ (1, v) with |v| ≪ 1, the explicit expressions for δT Note that T 0i ∝ B i and J i ∝ B i are prohibited in usual parity-invariant matter by parity symmetry. However, these chiral transport become possible in the present case due to the parity-violating nature of the weak interaction.

VI. HYDRODYNAMIC EQUATIONS OF MOTION
In this section, we present an explicit derivation of the hydrodynamic equations of motion for the system composed of nucleons, electrons, and neutrinos. For simplicity, here we consider the hydrodynamic equations in the Lorentz covariant form, which can reduce to a nonrelativistic expression with appropriate change of variables. It is also sufficient to focus on the dissipationless terms for our purpose and we will ignore the dissipative terms, such as the viscosity and conductivity.
As briefly mentioned in Sec. V, however, the magnetic field can also be involved through the temporal derivatives D on the thermodynamic parameters when incorporating corrections. Therefore, we need to work out the leading-order corrections in expansion as well, which are shown in Appendix A. For simplicity, we here set µ 5e = 0. In this case, the magnetic field is only involved in Eq. (94) for the hydrodynamic equations when taking E µ = ω µ = 0. For such a correction, one finds Here, we used the relation which follows from the Bianchi identity ∇ µF µν = 0 and the decompositionF µν = ǫ µναβ u α E β − u µ B ν + u ν B µ . By further substituting the expression of Du µ from Eq. (102), we conclude that Eq. (103) only contains B · ∇ ⊥ T and B · ∇ ⊥ µ terms, and thus, δT µν (ν)B and δJ µ (ν)B are not affected when assuming ∇ ⊥µ T = ∇ ⊥µ µ = E µ = ω µ = 0.
(The reason why this should be regarded as the upper bound will be described shortly.) In order to account for the observed pulsar velocity v kick ∼ 10 2 km/s (see, e.g., Refs. [56][57][58][59]) solely from this contribution, the required magnetic field at the core is of order 10 15-16 G. 6 However, this estimate should be taken with care because it depends sensitively on the choice of the parameters. From Eq. (73), one might think that for a given magnetic field, v kick becomes arbitrarily large if (µ n − µ p )/T becomes sufficiently large. In fact, this is not the case because for a sufficiently large (µ n − µ p )/T , the mean free path ℓ mfp would become larger than the typical length scale of the system, as can been seen from Eq. (31), where κ increases when (µ n − µ p )/T increases. Then the assumption that neutrinos are near equilibrium would break down. This means that the kick velocity is bounded from above for a given magnetic field because of the hydrodynamic approximation. 7 On the other hand, the chiral radiation transport theory itself is applicable to neutrinos even far away from equilibrium, in which case such a limitation is not present. It is thus necessary to investigate the fully nonequilibrium contribution of this mechanism to provide a more realistic estimate.
Although we have highlighted the neutrino chiral transport induced by the magnetic field near equilibrium in this paper, there are also other neutrino chiral transport induced by the vorticity and gradients of temperature and chemical potential. One expects that these chiral effects would further modify the nonlinear hydrodynamic evolution of the supernova, such as the turbulent behavior. For example, chiral/helical transport phenomena lead to the tendency toward the inverse energy cascade even in three dimensions, as analytically and numerically shown in Refs. [60,61] (see also Refs. [62,63] in the context of the early Universe).
We also note that neutrino chiral transport far away from equilibrium is not captured by the relaxation time approximation adopted in the present paper. In fact, even though the net momentum flux is generated for nearequilibrium neutrinos by magnetic fields, it is not guaranteed that these neutrinos can escape from the protoneutron star. This neutrino momentum flux could be canceled by the back reaction of the matter sector, and then there could be no significant emission asymmetry. The emission asymmetry might rather be caused by neutrinos outside the neutrino sphere, where the near-equilibrium approximation is not applicable. In order to see the consequences of fully nonequilibrium chiral effects, it would be eventually important to perform numerical simulations of the chiral radiation transport theory for neutrinos in the future.  6 For the previous works that attempt to explain the pulsar kick by an asymmetric neutrino emission induced by strong magnetic fields, see Refs. [30][31][32][33][34][35][36][37][38][39]. Note that our work is the first to derive T 0i (ν)B explicitly and systematically. The parametric dependence of v kick here are also different from the previous results although the final order of estimate itself is comparable to Ref. [30] among others. 7 Parametrically, T i0 (ν)B may be expressed as Then the near-equilibrium condition of neutrinos (L ℓ mfp ) leads to the upper bound of v kick as v kick µ 3 ν B M ρcore ∼ B 10 13 G km/s .