Past exam of the mathematics course of the University of Cambridge / 2007 / ib / Paper 4 / 17B (Special Relativity)

Let the photon momenta be ${\mathbf{p}}_1,{\mathbf{p}}_2$, with $|{\mathbf{p}}_i|=E_i/c$. Four-momentum conservation gives total energy $E_1+E_2$ and total momentum ${\mathbf{p}}_1+{\mathbf{p}}_2$ for the pion. Its mass shell relation therefore givesSince $0\le \theta \le \pi$, the sine is nonnegative, and taking the positive square root yields

In the pion rest frame, conservation of three-momentum makes the electron and massless neutrino momenta equal and opposite, of common magnitude $p$. Their energies obeySquaring $E_e=m_{\pi }{c}^2-pc$ and cancelling $p^2{c}^2$ givesThe electron four-momentum satisfies $p=E_{e}v/c^2$. Thus, for the physically allowed decay $m_{\pi }>m_e$,