The chiral vortical effect in Wigner function approach

TL;DR

This paper presents two derivations of the chiral vortical effect (CVE) using the Wigner function approach, clarifying its Lorentz covariance and resolving the 'one-third' puzzle in three-dimensional chiral kinetic theory.

Contribution

It provides a detailed, transparent derivation of the CVE in the Wigner function framework, including a generalization to higher orders and clarification of frame dependence.

Findings

The CVE current can be decomposed into normal and magnetization parts.

In the comoving frame, these parts contribute one-third and two-thirds of the total CVE current.

The total CVE current is frame-independent and consistent with previous results.

Abstract

It is more subtle to obtain the chiral vortical effect (CVE) than chiral magnetic effect (CME) in quantum transport approach. To investigate the subtlty of the CVE we present two different derivation in the Wigner function approach. The first one is based on the method in our previous work \cite{Gao:2012ix} in which the CVE was derived under static-equilibrium conditions without details. We provide a detailed derivation using a more transparent and powerful method, which can be easily generalized to higher order calculation. In this derivation of the CVE current, there is an explicit Lorentz covariance. The second derivation is based on a more general chiral kinetic theory in a semi-classical expansion of the Wigner function without assuming static-equilibrium conditions \cite{Gao:2018wmr}. In this derivation, there is a freedom to choose a reference frame for the CVE current, so the explicit Lorentz covariance seems to be lost. Howerver, under static-equilibrium conditions, we show that the CVE current in this derivation can be decomposed into two parts, identified as the normal and magnetization current. Each part depends on the reference frame, but the sum of two parts does give the total CVE current which is independent of the reference frame. In the comoving frame of the fluid, it can be shown that the normal and magnetization current give one-third and two-thirds of the total CVE current respectively. This gives a natural solution to the 'one-third' puzzle in the CVE current in three-dimensional version of the chiral kinetic theory in the literature.