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

To implement crossing symmetry with the annihilation labels, replace the old scattering four-momenta by $(p_1,-p_3,-p_2,p_4)$. The scattering Mandelstam variables then become $(s_{\mathrm{old}},t_{\mathrm{old}},u_{\mathrm{old}})=(t,s,u)$, where now $s=(p_1+p_2{)}^2$, $t=(p_1-p_3{)}^2$ and $u=(p_1-p_4{)}^2$. ConsequentlyIn the centre-of-momentum frame, take $\theta$ between the incoming electron and outgoing positive muon, $p_3$. For negligible masses, $t=-s(1-\cos \theta )/2$ and $u=-s(1+\cos \theta )/2$. ThereforeDefining $\theta$ using the outgoing negative muon interchanges $t$ and $u$ and leaves the result unchanged.