Mass spectra of neutral mesons $K_0,\ \pi_0,\ \eta,\ \eta'$ at finite magnetic field, temperature and baryon chemical potential

The mass spectra of neutral mesons $K_0, \pi_0, \eta, \eta'$ on temperature-quark chemical potential $(T-\mu)$ plane in the presence of a constant magnetic field is investigated in the $SU(3)$ NJL model. As a Goldstone boson of chiral symmetry breaking, the mass of $K_0$ meson increases with temperature and/or quark chemical potential, and we observe two kinds of mass jumps of $K_0$ meson in media, which is induced by the mass jump of constituent quarks and the magnetic field, respectively. Due to the breaking of isospin symmetry between $u$ and $d$ quarks in magnetic fields, the mixing of $\pi_0-\eta- \eta'$ mesons occurs and this leads to rich structures of their mass spectra. For instance, $\pi_0$ mass is influenced by the strange quark. There appear the change of increase ratio of $\pi_0$ mass at high $\mu$ and vanishing $T$ and the $\pi_0$ mass jump crossing over the threshold of two times of strange quark mass at finite $T$ and $\mu$. The mass ordering of $\pi_0, \ \eta,\ \eta'$ mesons varies in media, due to their mass jumps, which are induced by the mass jump of constituent quarks or the magnetic field.