Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 304 / 3 / a / ii / Solution

The induced four-$\varphi$ interaction is the tree-level diagram with two $\lambda {\varphi }^2\chi$ vertices joined by one internal $\chi$ propagator. The zero-dimensional propagator is $M^{-2}$, and the second-order expansion of the exponential supplies the factor $1/2$. The resulting effective-action coefficient is therefore $-{\lambda }^2/(2M^2)$, exactly the value of $\tilde{\lambda }$ found by the Gaussian integral.