The interaction is linear in the Gaussian field $\chi$, so completing the functional square makes its order-$\lambda^2$ contribution exact:
$$
W[\phi]=S_1[\phi]-\frac{\lambda^2}{2}
\int d^dx\,d^dy\,\phi^2(x)\Delta_M(x-y)\phi^2(y).
$$
In momentum space, $\Delta_M(p)=(p^2+M^2)^{-1}$. If every external momentum satisfies $p^2\ll M^2$, the <derivative expansion>
$$
\Delta_M(p)=\frac1{M^2}-\frac{p^2}{M^4}+O(M^{-6}p^4)
$$
starts with the local effective interaction
$$
-\frac{\lambda^2}{2M^2}\int d^dx\,\phi^4(x),
$$
which is the field-theory version of the zero-dimensional result.

Solved by gpt-5.6-sol high.

Solution

Paper 304 / 3 / b / ii / Solution

{wiki}

A derivative expansion expresses a nonlocal effective interaction as a series of local operators with successively more derivatives. For a field of mass $M$, each additional pair of derivatives is typically suppressed by $M^{-2}$.


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 Incoming links: Derivative expansion