Covariant Chiral Kinetic Equation in Non-Abelian Gauge field from "covariant gradient expansion"

TL;DR

This paper derives a covariant chiral kinetic equation in non-Abelian gauge fields using covariant gradient expansion, revealing the structure of phase-space distributions and anomaly-induced currents in a unified framework.

Contribution

It introduces a novel derivation of the non-Abelian covariant chiral kinetic equation in 8D phase space using covariant gradient expansion, including the treatment of color space and Lorentz transformations.

Findings

Derived the non-Abelian chiral kinetic equation in 8D phase space.

Identified the Lorentz transformation properties of phase-space distributions.

Revealed the origin of chiral anomaly and derived anomalous currents in non-Abelian gauge fields.

Abstract

We derive the chiral kinetic equation in 8 dimensional phase space in non-Abelian $SU(N)$ gauge field within the Wigner function formalism. By using the "covariant gradient expansion", we disentangle the Wigner equations in four-vector space up to the first order and find that only the time-like component of the chiral Wigner function is independent while other components can be explicit derivative. After further decomposing the Wigner function or equations in color space, we present the non-Abelian covariant chiral kinetic equation for the color singlet and multiplet phase-space distribution functions. These phase-space distribution functions have non-trivial Lorentz transformation rules when we define them in different reference frames. The chiral anomaly from non-Abelian gauge field arises naturally from the Berry monopole in Euclidian momentum space in the vacuum or Dirac sea contribution. The anomalous currents as non-Abelian counterparts of chiral magnetic effect and chiral vortical effect have also been derived from the non-Abelian chiral kinetic equation.