  We study the mass spectra and decay process of $\sigma$ and $\pi_0$ mesons
under strong external magnetic field. For this purpose, we deduce the
thermodynamic potential in a two-flavor, hot and magnetized Nambu-Jona-Lasinio
model. We calculate the energy gap equation through the random phase
approximation(RPA). Then we use Ritus method to calculate the decay triangle
diagram and self-energy in the presence of a constant magnetic field B. Our
results indicate that the magnetic field has little influence on the mass of
$\pi_0$ at low temperatures. While for quarks and $\sigma$ mesons, their mass
changes obviously, which reflects the influence of magnetic catalysis(MC). The
presence of magnetic field accelerates the decay of the meson while the
presence of chemical potential will decrease the decay process.
