A consistent description of kinetic equation with triangle anomaly

TL;DR

The paper develops a consistent relativistic kinetic description of systems with a triangle anomaly by incorporating first-order corrections to the phase-space distribution that couple to vorticity and magnetic fields, while enforcing the second-law entropy principle and charge/energy-momentum conservation. It introduces a momentum-dependent correction $\chi(x,p)=\lambda(p)p\cdot\omega+\lambda_B(p)p\cdot B$, derives entropy-constraint relations that fix the anomaly-induced transport coefficients ($D$, $D_B$, $\xi$, $\xi_B$), and demonstrates that a nonzero anomalous source term in the collision integral is required to preserve conservation laws. The authors solve for the correction coefficients in one-charge and two-charge scenarios, obtaining explicit results for massless fermions at small chemical potentials, and revealing simple, symmetric structures in the two-charge case that resemble CME-like transport. The work provides a kinetic-theory foundation for anomaly-induced transport, with potential implications for chiral magnetic/vortical effects in heavy-ion collisions and connections to holographic hydrodynamics.

Abstract

We provide a consistent description of the kinetic equation with triangle anomaly which is compatible with the entropy principle of the second law of thermodynamics and the charge/energy-momentum conservation equations. In general an anomalous source term is necessary to ensure that the equations for the charge and energy-momentum conservation are satisfied and that the correction terms of distribution functions are compatible to these equations. The constraining equations from the entropy principle are derived for the anomaly-induced leading order corrections to the particle distribution functions. The correction terms can be determined for minimum number of unknown coefficients in one charge and two charge cases by solving the constraining equations.