  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.
