Past exam of the physics 1 paper of the University of Oxford / 2014 resit / 3 / b

Four-momentum conservation fixes the outgoing system's invariant mass to the same $2E/c^2$. In the rest frame of particle 3, let particle 4 have energy $E_4^{\prime }$ and momentum ${\mathbf{p}}_4^{\prime }$. Each outgoing particle has rest mass $M_2$, soConsequentlyHere $E_4^{\prime }$ is the total energy, including rest energy. At the production threshold $E=M_2{c}^2$, it reduces to $M_2{c}^2$, as it should. Production is kinematically possible only for $E\ge M_2{c}^2$. The initial rest mass $M_1$ drops out because the conserved total centre-of-mass energy, rather than the identity of the incoming particles, determines this outgoing relative motion.