Effects of a parallel electromagnetic field in three-flavor Nambu--Jona-Lasinio model

In this work, we explore the ground state of QCD system and the relevant collective modes in a parallel electromagnetic (EM) field within the effective three-flavor Nambu--Jona-Lasinio model. From the features of neutral chiral condensates, three critical EM fields are identified in our study: $e\bar{I}_2^{c_i}~(i=1,2,3)$. Moreover, competition between QCD and QED anomalies is found: the negativeness of $\pi_{\rm d}^*$ and $\pi_{\rm s}^*$ is a signal of QCD anomaly dominance for all the flavors, and the positiveness of $\pi_{\rm s}^*$ beyond $e\bar{I}_2^{c_1}$ indicates QED anomaly dominance for strange quark. For the lowest-lying neutral collective modes, the masses of $\Pi$ and $H$ modes reduce to zero around the critical EM fields and the scalar-pseudoscalar components are exchanged between $\Sigma$ and $H$ modes, as can be seen in the crossing structure.