Verifying Conservation of Energy and Momentum for γ + p → Δ+ → πo + p

[Liquidxlax]

Homework Statement

γ + p → Δ⁺ → π^o + p

incident photon with energy 0.34 GeV excites a photon into Δ⁺ which decays into π^o + p

Write down and apply the laws of conservation of energy and momentum to show taht the peak photon energy of 0.34 GeV is consistent with Δ⁺ mass of 1.23 GeV

Homework Equations

four momentum [E,p]

The Attempt at a Solution

just wondering what to start with exactly after writing down the equations.

M = 938.727MeV/c^2
so the rest energy is just Mc^2 with 0 momentum for the intial proton

do I assume that all the momentum from the photon is transferred to the Δ⁺ because the photon does not show up in the final state?

In the rest frame the pion and proton are ejected at 180 degrees from each other and have the same momentum, but not energy or velocity?

so

p' = 0.34/2c = γMv where gamma here is 1/sqrt(1-β²) and v is the velocity of the proton

 

[Nate810]

in the lab frame?then E' = sqrt(M2c4 + p'2c2) = 1.23GeVbut then how do I show that the peak photon energy of 0.34 GeV is consistent with this?