In the lab frame where the proton is at rest, the total momentum is not zero due to the finite photon energy. This means that the kinetic energy of the particles after the reaction can't be all zero, so the photon must have more energy than the pion's mass. If we consider the center of mass frame then because the total momentum is zero, we do have conservation of momentum if both the pion and the proton are at rest. We can thus find the minimum photon energy by considering the total energy in the center of mass frame and equating that to the sum of the rest masses of the proton and pion. More energy would lead to kinetic energy for these particles while for zero kinetic energy we can satisfy both momentum conservation and energy conservation.
Working in c = 1 units, the energy E and momentum p of a ...
We explain in detail how one can compute the threshold energy for the process.