Let $K_M=-\partial^2+M^2$ and let $\Delta_M=K_M^{-1}$ be its Euclidean <quantum field theory propagator>. With the source-sign convention suited to the expression in the question, define
$$
Z_0[J]=N_0\int\mathcal D\chi\,
\exp\left[-S_2[\chi]-\int d^dx\,J(x)\chi(x)\right]
=\exp\left[\frac12\int d^dx\,d^dy\,
J(x)\Delta_M(x-y)J(y)\right],
$$
where $N_0$ makes $Z_0[0]=1$. Since inserting $\chi(x)$ is equivalent to acting with $-\delta/\delta J(x)$, the <integrating out a field>[integral over $\chi$] gives
$$
W[\phi]=S_1[\phi]-\log\left.
\left\{\exp\left[-S_3\left(\phi,-\frac{\delta}{\delta J}\right)\right]Z_0[J]\right\}\right|_{J=0}.
$$

Solved by gpt-5.6-sol high.

Solution

Paper 304 / 3 / b / i / Solution

Integrating out a field means performing its path integral while retaining the other fields as backgrounds. The result is generally a nonlocal <quantum effective action>; for a field much heavier than all external momenta, a derivative expansion turns it into local effective interactions.

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