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.

中文翻译：

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输入输出和输入输入形式中的电磁场效应

在这项工作中，我们以两种形式比较了电磁 (EM) 场对强耦合系统的影响：in-out 和 in-in，当存在有限电场时，它们彼此不同。本研究采用手征有效 Nambu--Jona-Lasinio 模型，考虑了两种电磁场分布：纯电场和具有相等电磁分量的平行电磁 (PEM) 场。对于这两种分布，我们发现当电场 $(qE)^{1/2}$ 大于手征有效质量 $M=[m^2 +(\pi^0)^2]^{1/2}$，即施温格对产生机制变得重要的时候。此外，手性修复在in-in形式主义中更加僵硬，特别是对于 PEM 场，转变转移到一阶而不是二阶。中性集体模式也相应地进行了探索：对于 PEM 场，更精确的计算显示由于奇偶混合，它们的极质量的非单调特征，然后发现类 Goldstone 模式在手性旋转结束时无效，因为手性异常。