Covariant chiral kinetic equation in the Wigner function approach

Abstract

The covariant chiral kinetic equation (CCKE) is derived from the 4-dimensional Wigner function by an improved perturbative method under the static equilibrium conditions. The chiral kinetic equation in 3-dimensions can be obtained by integration over the time component of the 4-momentum. There is freedom to add more terms to the CCKE allowed by conservation laws. In the derivation of the 3-dimensional equation, there is also freedom to choose coefficients of some terms in dx0/dτ and dx/dτ (τ is a parameter along the worldline, and (x0, x) denotes the timespace position of a particle) whose 3-momentum integrals are vanishing. So the 3-dimensional chiral kinetic equation derived from the CCKE is not uniquely determined in the current approach. To go beyond the current approach, one needs a new way of building up the 3-dimensional chiral kinetic equation from the CCKE or directly from covariant Wigner equations.

Anomalous current from the covariant Wigner function

“…This is one of the main results of the present paper. We have reproduced in a new way the well-known formula for axial current, corresponding to CVE, which is linear in vorticity, with the necessary coefficient, depending on μ 2 and T 2 [1][2][3][4][5][9][10][11]24]. Thus, we confirmed the similar calculation recently made in [24].…”

“…The most important effects (see e.g. [1] for reference), chiral magnetic effect (CME) and chiral vortical effect (CVE) [2][3][4][5][6][7][8][9][10][11] effectively modify the equations of hydrodynamics [2]. Chiral effects can have different experimental consequences, in particular, resulting in baryon polarisation [7,8,12], approached also in the different context [13,14].…”

“…The existence of CVE was proven in many different ways, e.g. in hydrodynamic approximation as the result of joint consideration of triangle axial anomaly and second law of thermodynamics [2,16], exploring Kubo formulas and thermal field theory loop calculation [5,17,18], kinetics [9,10], calculations of triangle anomaly in effective quantum field theory and generalization of the form of axial charge in hydrodynamics [3,7,19].…”

Quantum kinetic theory of spin polarization of massive quarks in perturbative QCD: Leading log

“…The 2) for the free streaming part was recently studied in Ref. [21,22,23,24] for massive quarks, extending the recent development of chiral kinetic theory for massless chiral quarks [36,37,38,39,40,41,42], for which a Berry's curvature in momentum space due to spin projection plays a critical role. The 1) is also expected to be intimately related to the "side-jump" phenomenon in chiral kinetic theory [43,44,45].…”

Local spin polarization in high energy heavy ion collisions

“…So the average spin polarization is proportional to the so-called thermal vorticity (see definition in next section) if the thermal vorticity is small. Another theoretical approach is the Wigner function (WF) formalism [24][25][26][27][28][29][30], which has been revived [31][32][33][34][35][36][37][38][39] to study the chiral magnetic effect (CME) [40][41][42] (for reviews, see, e.g., Ref. [43][44][45][46]) and chiral vortical effect (CVE) [31,[47][48][49][50][51] for massless fermions.…”