Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 304 / 4 / b / Solution

Up to Euclidean sign conventions, the parity-even renormalizable Lagrangian isThe Yang-Mills term is invariant because $F_{\mu \nu }$ transforms by conjugation and the matrix trace is cyclic. Covariance of $D_{\mu }\psi$ makes $\bar{\psi }{\gamma }_{\mu }{D}_{\mu }\psi$ invariant, and the fermion mass is invariant because the factors $U^{-1}U$ cancel. Likewise $D_{\mu }\varphi$ transforms by conjugation, so its trace norm, ${\varphi }^a{\varphi }^a=2\mathrm{tr}{\varphi }^2$, and every power of that norm are invariant. Finally,which proves gauge invariance of the Yukawa interaction.

Gauge invariance and power counting also permit the parity-odd Yukawa interaction $i{y}_5\bar{\psi }{\gamma }_5\varphi \psi$, a pseudoscalar fermion mass $i{m}_5\bar{\psi }{\gamma }_5\psi$, and the Yang-Mills theta term. They are absent if parity and CP are imposed. There is no nonzero cubic scalar invariant: $\mathrm{tr}{\varphi }^3$ vanishes for $SU(2)$, equivalently ${ϵ}^{abc}{\varphi }^a{\varphi }^b{\varphi }^c=0$ for commuting scalar components.