Solution
= Solution
The induced four-$\phi$ interaction is the tree-level diagram with two $\lambda\phi^2\chi$ 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 $\widetilde\lambda$ found by the <Gaussian integral>.
Solved by gpt-5.6-sol high.