$\rho$ mesons in finite magnetic field and finite temperature

The mass spectra of $\rho$ mesons ($\rho_{Q=\pm 1}^{s_z=0,\pm 1}$ and $\rho_{Q=0}^{s_z=0,\pm 1}$) 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 $\rho$ meson propagators have been derived in the Ritus scheme and Schwinger scheme, which gives the same algebraic formula. When solving the pole equation of $\rho$ 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 $\rho$ meson masses $M_{\rho}$ corresponding to the lowest value solution of the pole equation. $M_{\rho^{-}_+}$, $M_{\rho^{0}_+}$ and $M_{\rho^{\pm}_0}$ increase with magnetic field. $M_{\rho^{+}_+}$ firstly decreases and then becomes saturated with increasing magnetic field. $M_{\rho^0_0}$ 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 $\rho$ meson masses $M^{i=0,1,2,3,4}_{\rho}$. With fixed magnetic field, they decrease with temperature, and approach the mass sum of their constituent quarks at high temperature. The mass solution $M^{i}_{\rho}$ for different mesons $\rho_+^{0,\pm}$ and $\rho_0^{0,\pm}$ may become degenerate at finite magnetic field and temperature.