Symmetric equal-mass relativistic elastic scattering

A particle of total incident energy $E$ and rest energy $E_0$ strikes an identical stationary particle. If the two outgoing paths make equal and opposite nonzero angles $\theta$ to the incoming direction in the laboratory frame, transverse conservation of momentum makes their outgoing energies equal to $(E+E_0)/2$. Longitudinal conservation of momentum and the mass shell relation imply ${\tan }^2\theta =2E_0/(E+E_0)$. The nonrelativistic limit is $\theta =45^{\circ }$; at large incident energy both paths become forward directed.