Relativistic scattering - determining bound on initial momentum

[Afonso Campos]

Homework Statement

A high-energy photon collides with a proton at rest. A neutral pi meson is produced according to the reaction $\gamma + p \to p + \pi^{0}$. What is the minimum energy the photon must have for this reaction to occur? (The rest mass of a proton is $938\ \text{MeV/c}^{2}$ and the rest mass of a $\pi^{0}$ is $135\ \text{MeV/c}^{2}$.)

Homework Equations

The Attempt at a Solution

I understand that the momentum of the photon will turn into the energy of the final-state proton and/or the muon.

I need to figure out if, for a minimum energy of the decayed photon, only the proton gets the energy of the photon or if the muon also gets a piece of the momentum of the photon.

My hunch is that only the final state proton (and not the muon) should move, because the final state proton has more rest energy than the muon, so that would translate into a smaller momentum for the final state proton and a consequently smaller energy for the photon.

 

[Orodruin]

Muon? There is no muon in this problem. Assuming that you mean pion.

You need to take conservation of total 4-momentum into account. The total 4-momentum should be equal before and after the interaction. Then there are certain manipulations you need to do to find the threshold energy. This should be described in your textbook (which do you use?).