How a spinning quark moves in the NJL type mean field

Abstract

The transport equation of the spinning quarks, moving in the Nambu-Jona-Lasinio (NJL) type mean field, is derived by solving the Schwinger-Dyson equations, up to the order of $\hbar$, under the mean field approximation. It shows that the scalar and the vector mean field potentials have different impacts on the quantum correction to the transport equation. Both the forces give rise to an anomalous velocity and an anomalous force where the latter would vanish if the vector force is turned off. The particle dispersion relation is modified by the vector force as well. Besides, the spin precession in the particle rest frame is purely governed by the vector force.