Right-angle Compton scattering in the centre-of-momentum frame

For a photon and a massive particle with zero total momentum, their momenta are opposite and equal. Conservation of total energy fixes the same momentum magnitude before and after elastic scattering, independently of direction. A right-angle photon turn therefore makes the recoiling particle move perpendicular to the original photon direction in that frame. Transform both velocity components, or the complete four-momentum, to obtain the laboratory recoil; a transverse component is reduced by the boost Lorentz factor when the particle has zero longitudinal velocity in the zero-momentum frame.