  In this work the neutral meson properties have been investigated in the
presence of thermo-magnetic background using two-flavor Nambu--Jona-Lasinio
model. Mass, spectral function and dispersion relations are obtained in the
scalar ($\sigma$) and pseudo-scalar ($\pi^0$) channels as well as in the vector
($\rho^0$) and axial vector ($a^0_1$) channels. The general Lorentz structures
for the vector and axial-vector meson polarization functions have been
considered in detail. The ultra-violet divergences appearing in this work have
been regularized using a mixed regularization technique where the gamma
functions arising in dimensional regularization are replaced with incomplete
gamma functions as usually done in the proper time regularization procedure.
The meson spectral functions obtained in the presence of a magnetic field
possess nontrivial oscillatory structure. Similar to the scalar and
pseudo-scalar channel, the spectral functions for each of the modes of $\rho^0$
are observed to overlap with the corresponding modes of its chiral partner
$a_1^0$ mesons in the chiral symmetry restored phase. We observe
discontinuities in the masses of all the mesonic excitations for a non-zero
external magnetic field.
