$\ddot{I}$ mesons in finite magnetic field and finite temperature

Abstract

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