$ρ$ mesons in finite magnetic field and finite temperature

Abstract

The mass spectra of $ρ$ mesons ($ρ_{Q=\pm 1}^{s_z=0,\pm 1}$ and $ρ_{Q=0}^{s_z=0,\pm 1}$) at finite magnetic field and temperature are studied in frame of the two-flavor Nambu-Jona-Lasinio model. Fully considering the breaking of translational invariance induced by external magnetic field, the analytical form of $ρ$ meson propagators have been derived in the Ritus scheme and Schwinger scheme, which gives the same algebraic formula. When solving the pole equation of $ρ$ meson propagators, multiple solutions of the meson mass appear due to the dimension reduction of their constituent quarks in magnetic fields. At vanishing temperature, we focus on the $ρ$ meson masses $M_ρ$ corresponding to the lowest value solution of the pole equation. $M_{ρ^{-}_+}$, $M_{ρ^{0}_+}$ and $M_{ρ^{\pm}_0}$ increase with magnetic field. $M_{ρ^{+}_+}$ firstly decreases and then becomes saturated with increasing magnetic field. $M_{ρ^0_0}$ is not sensitive to magnetic field. These results are consistent with the available LQCD simulations. At finite temperature, we discuss the lowest four/five solutions of $ρ$ meson masses $M^{i=0,1,2,3,4}_ρ$. With fixed magnetic field, they decrease with temperature, and approach the mass sum of their constituent quarks at high temperature. The mass solution $M^{i}_ρ$ for different mesons $ρ_+^{0,\pm}$ and $ρ_0^{0,\pm}$ may become degenerate at finite magnetic field and temperature.