= Transverse radiative capture with equal opposite speeds
{title2=$Q^2=(R-r)^2+4rR\gamma^2-2\varepsilon\gamma(R+r)$}
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 $\chi=\varepsilon(r+R)/(4rR)$. An exact threshold requires the composite rest mass, not just the incoming constituent masses.
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