Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 301 / 2 / a / Solution

In four spacetime dimensions the action is dimensionless, so the mass dimension of the Lagrangian density is $[\mathcal{L}]=4$. The kinetic term gives $[\varphi ]=1$, and $g{\varphi }^3$ then givesThus $g$ has mass dimension one and is a relevant coupling under power counting in quantum field theory.