Transverse radiative capture with equal opposite speeds ($Q^2=(R-r{)}^2+4rR{\gamma }^2-2\epsilon \gamma (R+r)$)

Two particles with unequal rest energies $r,R$ and equal opposite velocities have total energy $\gamma (r+R)$ but nonzero total momentum. If they form a composite while emitting a photon transverse to their initial axis, the final composite rest energy $Q$ satisfies the displayed identity. It follows by squaring the residual four-momentum. Neglecting binding, put $Q=r+R$; the minimum incoming energy is $(r+R)[\chi +\sqrt{1+{\chi }^2}]$, where $\chi =\epsilon (r+R)/(4rR)$. An exact threshold requires the composite rest mass, not just the incoming constituent masses.