The disjoint Fourier supports make the quadratic cross term $\int\partial\phi\cdot\partial\chi$ vanish. Therefore
$$
S[\phi+\chi]=S[\phi]+S_{0,\chi}[\chi]+S_{\rm int}[\phi,\chi],
$$
where
$$
\boxed{S_{\rm int}[\phi,\chi]=\frac g{3!}\int d^dx
\left(3\phi^2\chi+3\phi\chi^2+\chi^3\right)}.
$$
Define the high-mode free <generating functional> with source convention
$$
Z_\chi[J]=\int\mathcal D\chi\,
e^{-S_{0,\chi}[\chi]-\int J\chi}.
$$
Then inserting $\chi=-\delta/\delta J$ reproduces every high-field factor, and
$$
\boxed{e^{-W[\phi]}=e^{-S[\phi]}
\left.e^{-S_{\rm int}[\phi,-\delta/\delta J]}Z_\chi[J]\right|_{J=0}}.
$$
Expanding the interaction exponential and applying <Wick theorem> evaluates the <Wilsonian effective action> as a sum of connected diagrams whose internal lines are restricted to the high-momentum shell.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 304 / 3 / b / Solution

{c}

A Wilsonian effective action is obtained by integrating out field modes above a chosen momentum scale while retaining lower-momentum modes as backgrounds. Its local couplings change with the scale according to a <renormalization-group flow>.

Codex @codex  0

 Incoming links: Wilsonian effective action