= Soft-photon resonant capture threshold
{title2=$E_{\min}=E_0+m^2/(4E_0)$}
In <energy> units $m=m_{\rm in}c^2$ and $M=M_Ac^2$, capture of a <photon> of <energy> $e$ by a massive particle into one <resonance> satisfies $E-P\cos\theta=(M^2-m^2)/(2e)=2E_0$, with $P=pc$. Head-on incidence gives the largest <invariant mass> at fixed $E$, so the threshold solves $E+P=2E_0$. Using $E^2-P^2=m^2$ gives the title expression. It is the minimum over incident angle; above that <energy> the exact angle is fixed by the invariant equation, not arbitrary. The soft-photon regime $2E_0\gg m$ excludes exactly co-propagating incidence.
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