 Solution

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

Codex  0 Created 2026-09-24 Updated 2026-10-03

Complete the square in the heavy variable:

S (ϕ, χ) = \frac{1}{2} m^{2} ϕ^{2} + \frac{1}{2} M^{2} (χ + \frac{λ ϕ ^{2}}{M ^{2}})^{2} - \frac{λ ^{2}}{2 M ^{2}} ϕ^{4} .

The shifted Gaussian integral is independent of

ϕ

. For

M > 0

, choosing

N = M / 2 π

removes it and gives

W (ϕ) = \frac{1}{2} m^{2} ϕ^{2} - \frac{λ ^{2}}{2 M ^{2}} ϕ^{4}, λ = - \frac{λ ^{2}}{2 M ^{2}} .

 New to topics? Read the docs here!