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We study the mass spectra and decay process of $\sigma$ and $\pi_0$ mesonsunder strong external magnetic field. For this purpose, we deduce thethermodynamic potential in a two-flavor, hot and magnetized Nambu-Jona-Lasiniomodel. We calculate the energy gap equation through the random phaseapproximation(RPA). Then we use Ritus method to calculate the decay trianglediagram and self-energy in the presence of a constant magnetic field B. Ourresults indicate that the magnetic field has little influence on the mass of$\pi_0$ at low temperatures. While for quarks and $\sigma$ mesons, their masschanges obviously, which reflects the influence of magnetic catalysis(MC). Thepresence of magnetic field accelerates the decay of the meson while thepresence of chemical potential will decrease the decay process.

Mass Spectra and Decay of Mesons under Strong External Magnetic Field