The Hadrons — Question 2

Question 2

Pion production on a stationary proton pays a recoil-energy penalty

Consider $p+p\to p+p+\pi^0$ with one initial proton at rest. Use $m_pc^2=1.00$ GeV and $m_\pi c^2=.140$ GeV for this model.

1. Derive the threshold kinetic energy of the incident proton.

2. Evaluate it and compare it with the pion rest energy.

Question 2 - Solution

Pion production on a stationary proton pays a recoil-energy penalty

(a) Method and calculation

$$W^2=4m_p^2c^4+2m_pc^2K,\quad W_{\mathrm{th}}=(2m_p+m_\pi)c^2.$$$$\boxed{K_{\mathrm{th}}=2m_\pi c^2+\frac{m_\pi^2}{2m_p}c^2}.$$

(b) Interpretation and further calculation

$$\boxed{K_{\mathrm{th}}=.280+.0196/2=.2898\,\mathrm{GeV}}.$$$$\boxed{K_{\mathrm{th}}/(m_\pi c^2)=2.07>1}.$$

Physical check

At threshold the three final particles are at rest relative to one another, not relative to the lab. Initial momentum consumes part of the available energy.