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

The interaction is linear in the Gaussian field

χ

, so completing the functional square makes its order-

λ^{2}

contribution exact:

W [ϕ] = S_{1} [ϕ] - \frac{λ ^{2}}{2} \int d^{d} x d^{d} y ϕ^{2} (x) Δ_{M} (x - y) ϕ^{2} (y) .

In momentum space,

Δ_{M} (p) = (p^{2} + M^{2})^{- 1}

. If every external momentum satisfies

p^{2} ≪ M^{2}

, the derivative expansion

Δ_{M} (p) = \frac{1}{M ^{2}} - \frac{p ^{2}}{M ^{4}} + O (M^{- 6} p^{4})

starts with the local effective interaction

- \frac{λ ^{2}}{2 M ^{2}} \int d^{d} x ϕ^{4} (x),

which is the field-theory version of the zero-dimensional result.

Solved by gpt-5.6-sol high.

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