= Heavy scalar exchange in effective field theory
{title2=$\Delta\mathcal L=\frac{a^2}{8}\varphi^2(M^2+\Box)^{-1}\varphi^2$}
A heavy real scalar $S$ with quadratic mass $M$ and interaction $-aS\varphi^2/2$ can be removed at tree level using $(M^2+\Box)S=-a\varphi^2/2$. Substituting into both its source and quadratic terms gives the displayed interaction. Its <derivative expansion> starts with $a^2\varphi^4/(8M^2)-a^2\varphi^2\Box\varphi^2/(8M^4)$. For a light quartic written as $-\lambda\varphi^4/4!$, the matched quartic is $\lambda-3a^2/M^2$. The expansion is valid for momentum invariants small compared with $M^2$.
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