Complete the square in the heavy variable:
$$
S(\phi,\chi)=\frac12m^2\phi^2
+\frac12M^2\left(\chi+\frac{\lambda\phi^2}{M^2}\right)^2
-\frac{\lambda^2}{2M^2}\phi^4.
$$
The shifted <Gaussian integral> is independent of $\phi$. For $M>0$, choosing $N=M/\sqrt{2\pi}$ removes it and gives
$$
W(\phi)=\frac12m^2\phi^2-\frac{\lambda^2}{2M^2}\phi^4,
\qquad
\widetilde\lambda=-\frac{\lambda^2}{2M^2}.
$$

Solved by gpt-5.6-sol high.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-304/3/a/i/solution