Rho meson decay in presence of magnetic field

We find a general expression for the one-loop self-energy function of neutral $\rho$-meson due to $\pi^+\pi^-$ intermediate state in a background magnetic field, valid for arbitrary magnitudes of the field. The pion propagator used in this expression is given by Schwinger, which depends on a proper-time parameter. Restricting to weak fields, we calculate the decay rate $\Gamma(\rho^0 \rightarrow \pi^+ +\pi^-)$, which changes negligibly from the vacuum value.