Past exam of the particle and nuclear physics course of the University of Cambridge / 2016 / Paper 1 / 1 / c

For neutrino-electron elastic scattering, work in the laboratory frame with a stationary electron and a massless incoming neutrino of energy $E$. If the electron receives kinetic energy $T$ and momentum $\mathbf{p}$, its relativistic energy-momentum relation givesconservation of energy fixes the outgoing neutrino energy to $E-T$, and conservation of momentum fixes its momentum to $\mathbf{k}-\mathbf{p}$. Its zero mass therefore requiresSubstitution and cancellation giveFor $T>0$, this requires $\cos \theta >0$; squaring must not introduce a backward-electron branch. Squaring, cancelling $T$ and solving givesThe expression increases with ${\cos }^2\theta$ on the physical interval $0\le \theta \le \pi /2$. Maximum transfer occurs for a forward electron and a backward outgoing neutrino:The remaining $0.197\mathrm{MeV}$ is carried by the neutrino. An elastic collision cannot transfer the entire incident energy to a free massive electron.