Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 305 / 1 / d / Solution

A gauge boson couples to the conserved current of a charged fermion throughwith the appropriate chiral projector when the gauge representation is chiral. At momentum transfer $|q^2|\ll M_X^2$, its propagator can be expanded aswhere the longitudinal term drops for a conserved current. The operation of integrating out a field, applied to $X_{\mu }$, gives the local four-fermion interactionup to the normalization used for $J^{\mu }$. Hence the effective Fermi coupling scales as $G\sim g_X^2/M_X^2$. For Standard Model charged currents, the conventional normalization is