Chiral phase transition and Schwinger mechanism in a pure electric field

We systematically study the chiral symmetry breaking and restoration in the presence of a pure electric field in the Nambu--Jona-Lasinio (NJL) model at finite temperature and baryon chemical potential. In addition, we also study the effect of the chiral phase transition on the charged pair production due to the Schwinger mechanism. For these purposes, a general formalism for parallel electric and magnetic fields is developed at finite temperature and chemical potential for the first time. In the pure electric field limit $B\rightarrow0$, we compute the order parameter, the transverse-to-longitudinal ratio of the Goldstone mode velocities, and the Schwinger pair production rate as functions of the electric field. The inverse catalysis effect of the electric field to chiral symmetry breaking is recovered. And the Goldstone mode is find to disperse anisotropically such that the transverse velocity is always smaller than the longitudinal one, especially at nonzero temperature and baryon chemical potential. As expected, the quark-pair production rate is greatly enhanced by the chiral symmetry restoration.