The electromagnetic field effects in in-out and in-in formalisms

TL;DR

This paper compares the effects of electromagnetic fields on strong coupling systems using in-out and in-in formalisms, revealing divergence at high electric fields and differences in chiral phase transitions and collective modes.

Contribution

It introduces a detailed comparison of in-out and in-in formalisms in electromagnetic field effects within the Nambu--Jona-Lasinio model, highlighting their differences in chiral restoration and collective mode behavior.

Findings

Formalisms diverge when electric field exceeds chiral mass scale.

Chiral restoration is more abrupt in the in-in formalism.

Nonmonotonic behavior of neutral collective modes due to parity mixing.

Abstract

In this work, we compare the effects of electromagnetic (EM) fields on strong coupling systems in two formalisms: in-out and in-in, which are different from each other when a finite electric field is present. The chiral effective Nambu--Jona-Lasinio model is adopted for this study, and two kinds of EM field distributions are considered: pure electric field and parallel EM (PEM) field with equal electric and magnetic components. For both distributions, we find that the results of in-out and in-in formalisms start to diverge when the electric field $(qE)^{1/2}$ is larger than the chiral effective mass $M=[m^2+(\pi^0)^2]^{1/2}$, that is, when the Schwinger pair production mechanism becomes important. Besides, the chiral restorations are stiffer in the in-in formalism, especially the transition shifts to first-order instead of the second-order for the PEM field. The neutral collective modes are also explored accordingly: For the PEM field, more precise calculations show nonmonotonic features of their pole masses due to parity mixing, and then the Goldstone-like mode is found to be noneffective at the end of chiral rotation because of chiral anomaly.