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I would be very grateful if anyone would be willing to check my understanding of this stuff as it has been several years since I used it in undergrad calculations.

If a proton and a photon collide head-on with known energies, then the energy in the center of mass frame will be given by the invariant mass [itex]E_\mathrm{com} = W[/itex]. Thus [itex]E_\mathrm{com} = \sqrt{(E_\gamma+ E_p)^2-(p_\gamma + p_p)^2} = \sqrt{(E_\gamma+ E_p)^2-\left(E_\gamma + \sqrt{E_p^2 - m_p^2}\right)^2} [/itex].

Does this sound reasonable?