= Light-scalar box matching for a charge-two field
{title2=$\mathcal L_{\rm eff}\supset g^4(\Phi^\dagger\Phi)^2/(384\pi^2m^4)$}
The zero-external-momentum expansion of the <charge-two cubic complex-scalar interaction> determines a local four-$\Phi$ coefficient. For a constant background $\Phi$, the integrated <complex scalar field> $\phi$ has two real <squared masses> $m^2+g|\Phi|$ and $m^2-g|\Phi|$. Expanding the sum of their one-loop <functional determinants> to fourth order in the background gives
$$
V_1\supset-\frac{g^4|\Phi|^4}{4}I_4,\qquad
I_4=\int\frac{d^4k_E}{(2\pi)^4}\frac1{(k_E^2+m^2)^4}
=\frac1{96\pi^2m^4}.
$$
The <Euclidean massive loop integral> is convergent. For a constant field, $\mathcal L_{\rm eff}=-V_1$, which gives the displayed positive <effective Lagrangian> coefficient.
The local vertex has $2!2!$ external attachments. Thus its contribution to the convention $i\mathcal M$ is $i g^4/(96\pi^2m^4)$ for either the charge-two particle-particle or particle-antiparticle ordering. This matches the zero-momentum expansion of the charge-allowed <box diagram>. The background expansion assumes $|g\Phi|\ll m^2$; the <derivative expansion> additionally requires the relevant external <four-momenta> to be small compared with $m$.
Calling $\phi$ the light field and $\Phi$ the heavy field does not make this a low-energy approximation to arbitrary on-shell heavy-particle scattering. If the retained particle has <mass> $M\gg m$, its external <energies> remain at least $M$ even when its spatial <momenta> are small. The zero-four-momentum coefficient is then an off-shell background coefficient, rather than a controlled approximation to that on-shell amplitude. The full momentum-dependent <box diagram> is needed. For $M>2m$, the cubic interaction also permits the charge-two particle to decay into two charge-one particles, so a stable external-particle interpretation needs further qualification.
This example separates computing a local <effective action> term from establishing the scale hierarchy needed to use it for a physical <scattering amplitude>.
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