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In this work, we investigate not only the pole masses but also the screeningmasses of neutral pions at finite temperature and magnetic field by utilizingthe random phase approximation (RPA) approach in the framework of thetwo-flavor Nambu--Jona-Lasinio (NJL) model. And two equivalent formalisms inthe presence of a magnetic field, i.e. the Landau level representation (LLR)and the proper-time representation (PTR), are applied to obtain thecorresponding analytical expressions of the polarization functions (except theexpressions for the pole masses in the PTR). In order to evaluate theapplicable region of the low-momentum expansion (LME), we compare the numericalresults 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 theFRPA suffer a sudden mass jump at the Mott transition temperature when in thepresence of external magnetic field, and the Mott transition temperature iscatalyzed by the magnetic field. And by analyzing the behaviors of thedirectional sound velocities of {\pi}0, which are associated with the breakingof the Lorentz invariance by the heat bath and the magnetic field, we clarifythe two problems existing in previous literatures: one is that the transversesound velocities in the medium are always larger than unity and thus violatethe law of causality on account of the non-covariant regularization scheme, theother is that the longitudinal sound velocities are identically equal unity atfinite temperature on account of the limitation of the derivative expansionmethod used.

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