Pole And Screening Masses Of Neutral Pions In A 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 model. And two equivalent formalisms in the presence of a magnetic field, i.e., the Landau level representation 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, i.e., the RPA in the LME. It is confirmed that the pole masses of pi(0) in the FRPA suffer a sudden mass jump at the Mott transition temperature when in the presence of external magnetic field, 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 noncovariant regularization scheme, and 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.