= Photon excitation threshold with a soft final photon
{title2=$E_{\rm th}=(M^2-m^2)c^2/(2m)$}
A photon hitting a resting mass $m$ has squared total energy invariant $m^2c^4+2mc^2E$. Producing a larger mass $M$ and an outgoing photon requires this invariant to exceed $M^2c^4$ if the photon has positive energy. The displayed threshold is therefore the limiting infimum as the final photon becomes soft, not an attained physical state with a zero-energy photon. Above threshold the outgoing photon energy in the centre-of-momentum frame is positive.
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