Lowest-order electron-electron and electron-muon scattering in a strong magnetic field

In this work we have investigated how the much studied scattering processes in vacuum at the lowest-order, {\em viz.} electron-muon ($e$-$\mu$) scattering in both $s$- and $t$-channel, Bhabha scattering, and M{\o}ller scattering, have been modified in the presence of a strong magnetic field ($|eB|>>m^2$, $m$ is the mass of electron or muon). For that purpose, we have first calculated the square of the matrix element by summing over the spin states, using the spinors in the presence of a strong magnetic field and then obtain the crosssection by integrating over the available phase space for the final states and averaging over the initial states. The first noticeable observation in the spin-summed of the matrix element squared is that the interference term between $s$- and $t$-channel and $t$- and $u$-channel in Bhabha and M{\o}ller scattering, respectively are missing in the presence of strong magnetic field. We have found that in the presence of strong magnetic field, the crosssection of $e^- e^+ \rightarrow \mu^- \mu^+$ annihilation process in the lowest order decreases inversely proportional to the fourth power of the center-of-mass energy ($\sqrt{s}$), compared to the inversely proportional to the square of center-of-mass energy in vacuum alone. Like in vacuum, the crosssection for $e$-$\mu$ scattering in ${\rm t}$ channel, {\em i.e.} for $e^- \mu^- \rightarrow e^- \mu^-$ process, even in a strong magnetic field too diverges but the finite part decreases with $\sqrt{s}$ much faster than in vacuum alone. Similarly the crosssections for both Bhabha and M{\o}ller scattering at the lowest-order diverges in the infrared limit. However, the finite term decreases with $\sqrt{s}$ much faster than the vacuum alone. In addition there is finite negative contribution, which is independent of $\sqrt{s}$ and decreases with the magnetic field.