Particle Creation

Consider a photon of energy ε_γ incident on a stationary proton. For sufficiently large ε_γ, aπ meson can be produced in a reaction

What is ε_min the threshold photon energy for this reaction to occur?

SOLUTION

In this problem and in most of the problems involving particle reactions, we will use units where c = 1. Compose the invariant quantity for the system p^μ p_μ. Before the reaction in the rest frame of the proton

(1)

(2)

where we used p_γ = ε_γ. After the reaction, in the center of mass frame, the lowest energy products will take away no momentum, so

(3)

Setting (2) equal to (3), we find ε_γ = ε_min.

(4)

So the threshold for this reaction

(5)