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.
Highlights
In this paper, we will be dedicated to deriving the chiral kinetic equation in SU(N ) gauge field from the quantum transport theory [51–54, 58] based on the Wigner functions from quantum gauge field theory
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
It should be noted that in the definition of the Wigner function given by eq (2.8) and the Wigner equations (2.9) and (2.10) there is no normal ordering in the Wigner matrix because we did not make any manipulation on the order of the quark field
Summary
The gauge invariant density matrix for spin-1/2 quarks is defined as [51–53]. It should be noted that in the definition of the Wigner function given by eq (2.8) and the Wigner equations (2.9) and (2.10) there is no normal ordering in the Wigner matrix because we did not make any manipulation on the order of the quark field It has been demonstrated in [56, 57] that this plays a central role to give rise to the chiral anomaly in the quantum kinetic theory. Where we have recovered the dependence before the generalized derivative operators in the last equation in order to make perturbative expansion These Wigner equations will be the starting point of our present work in the following. We will suppress the subscript s of the left-hand or right-hand Wigner function Jsμ in the subsequent sections and reinstate it when it is necessary
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