Past exam of the particle physics course of the University of Oxford / 2022 / 1 / c / Solution

Define the Mandelstam variables for the scattering labels by $s=(p_1+p_2{)}^2$, $t=(p_1-p_3{)}^2=q^2$ and $u=(p_1-p_4{)}^2$. Dropping the electron and muon masses givesThese identities follow by expanding the corresponding squared four-momenta, using $p_i^2=0$ and four-momentum conservation. Substitution into the supplied spin-averaged matrix element yieldsThe $1/4$ initial spin average is already included in the given expression; no further averaging factor belongs here.