The pole and screening masses of neutral pion in hot and magnetized medium: a comprehensive study in the Nambu--Jona-Lasinio model

In this work, we investigate not only the pole masses but also the screening masses of neutral pions at finite temperature and magnetic field by utilizing the random phase approximation (RPA) approach in the framework of the two-flavor Nambu--Jona-Lasinio (NJL) model. And two equivalent formalisms in the presence of a magnetic field, i.e. the Landau level representation (LLR) and the proper-time representation (PTR), are applied to obtain the corresponding analytical expressions of the polarization functions (except the expressions for the pole masses in the PTR). In order to evaluate the applicable region of the low-momentum expansion (LME), we compare the numerical results within the full RPA (FRPA) with those within the reduced RPA (RRPA), i.e. the RPA in the LME. It is confirmed that the pole masses of {\pi}0in the FRPA suffer a sudden mass jump at the Mott transition temperature when in the presence of external magnetic field, and the Mott transition temperature is catalyzed by the magnetic field. And by analyzing the behaviors of the directional sound velocities of {\pi}0, which are associated with the breaking of the Lorentz invariance by the heat bath and the magnetic field, we clarify the two problems existing in previous literatures: one is that the transverse sound velocities in the medium are always larger than unity and thus violate the law of causality on account of the non-covariant regularization scheme, the other is that the longitudinal sound velocities are identically equal unity at finite temperature on account of the limitation of the derivative expansion method used.