S-Matrix approach to Compton scattering at the tree level in a strong magnetic field

We have studied the Compton scattering ($\gamma e^- \longrightarrow \gamma e^-$) at the tree level in a homogeneous background of strong magnetic field ($|eB| \gg { m }^{ 2 },~\text{m is the mass of electron}$) through the S-matrix approach. For that purpose, using the Schwinger propagator for the electron, we have first calculated the square of the S-matrix element in the Landau gauge by summing over the final states of electron and photon and averaging over the initial states of the same. In the strong magnetic field, only the lowest Landau level for electron is considered. Finally we have computed the crosssection for Compton scattering as a function of initial photon energy for the different strengths of strong magnetic fields, where we have found that the crosssection in vacuum gets decreased due to the presence of strong magnetic field. However, for a fixed initial photon energy, the crosssection increases linearly with the magnetic field.