Heavy scalar exchange in effective field theory ($\Delta \mathcal{L}=\frac{a^2}{8}{\phi }^2(M^2+\square {)}^{-1}{\phi }^2$)

A heavy real scalar $S$ with quadratic mass $M$ and interaction $-aS{\phi }^2/2$ can be removed at tree level using $(M^2+\square )S=-a{\phi }^2/2$. Substituting into both its source and quadratic terms gives the displayed interaction. Its derivative expansion starts with $a^2{\phi }^4/(8M^2)-a^2{\phi }^2\square {\phi }^2/(8M^4)$. For a light quartic written as $-\lambda {\phi }^4/4!$, the matched quartic is $\lambda -3a^2/M^2$. The expansion is valid for momentum invariants small compared with $M^2$.