Past exam of the particle physics course of the University of Oxford / 2015 / 2 / b / Solution

Assume unpolarized beams, $s\gg m_{\mu }^2$, a single exchanged photon, full azimuthal coverage and no radiative corrections. Take $c_{\ast }=\cos {\theta }^{\ast }$ between incoming $e^{-}$ and outgoing ${\mu }^{-}$. The four nonzero helicity amplitudes have magnitudes $e^2(1+c_{\ast })$ for equal incoming/outgoing negative-particle helicities, and $e^2(1-c_{\ast })$ for opposite ones. Averaging four initial spin states and summing the final ones givesThe massless two-body phase space givesThe even distribution reflects the equal left/right photon couplings and is symmetric under interchange of forward and backward directions.