Meson mass spectrum in isospin QCD medium from the $N_f=2+1$ quark-meson model

Abstract: We study phase structures, meson mass spectra, and sound velocities at finite temperature and density in QCD with isospin chemical potential ($μ<em>I$). We employ the quark-meson model with $N_f=2+1$, incorporating the Kobayashi-Maskawa-'t Hooft type coupling to capture dynamical effects of the $U(1)_A$ axial anomaly. Within the mean-field approximation at quark one-loop, we analyze the onset of pion condensation, which manifests as a second-order phase transition at low temperatures and may exhibit a first-order behavior at higher temperatures within this approximation. We examine the corresponding mass spectra of scalar and pseudoscalar singlet-octet mesons, in which the $π</em>+$ mass is exactly massless in the superfluid phase due to its Nambu-Goldstone boson nature. The neutral pion mass is, meanwhile, found to exhibit a strictly linear growth with $μ_I$ in the pion condensed phase. We also investigate the isothermal squared sound velocity and identify characteristic structures associated with the phase transitions. Furthermore, we highlight how enhanced $U(1)_A$ anomaly effects facilitate the pion condensate to generate a less-pronounced sound velocity peak in cold medium. Our findings are expected to provide future lattice simulations with useful information on meson mass spectra from symmetry aspects.