The mass spectra of $\rho$ mesons (${\rho }_{Q_m=\pm 1}^{s_z=0,\pm 1}$ and ${\rho }_{Q_m=0}^{s_z=0,\pm 1}$) 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 has 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 $M_{{\rho }_0^{\pm }}$ increase with the magnetic field. $M_{{\rho }_{+}^{+}}$ firstly decreases and then becomes saturated with increasing the magnetic field. $M_{{\rho }_0^0}$ is not sensitive to the magnetic field. These results are consistent with the available Lattice QCD simulations. At finite temperature, we discuss the lowest four/five solutions of $\rho$ meson masses $M_{\rho }^{i=0,1,2,3,4}$. With fixed magnetic field, they decrease with temperature, and approach the mass sum of their constituent quarks at high temperature. The mass solution $M_{\rho }^i$ for different mesons ${\rho }_{+}^{0,\pm }$ and ${\rho }_0^{0,\pm }$ may become degenerate at finite magnetic field and temperature.

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• High magnetic fields
• QCD phenomenology
• Quark model
• Vector mesons