The weak-field expansion for processes in a homogeneous background magnetic field

The weak-field expansion of the charged fermion propagator under a uniform magnetic field is studied. Starting from Schwingers proper-time representation, we express the charged fermion propagator as an infinite series corresponding to different Landau levels. This infinite series is then reorganized according to the powers of the external field strength $B$. For illustration, we apply this expansion to $gammato ubar{ u}$ and $ uto ugamma$ decays, which involve charged fermions in the internal loop. The leading and subleading magnetic-field effects to the above processes are computed.