Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 301 / 1 / a / Solution

In four dimensions, $[\psi ]=3/2$, $[A_{\mu }]=1$, and $[F_{\mu \nu }]=2$. The minimal-coupling operator $\bar{\psi }{\gamma }^{\mu }{A}_{\mu }\psi$ has dimension four, so $[e]=0$ and $e$ is a marginal coupling. The Pauli term $\bar{\psi }[{\gamma }^{\mu },{\gamma }^{\nu }]F_{\mu \nu }\psi$ has dimension five, so $[\lambda ]=-1$ and $\lambda$ is an irrelevant coupling by power counting in quantum field theory.