= Light-scalar box matching for a charge-two field
{title2=$\mathcal L_{\rm eff}\supset g^4(\Phi^\dagger\Phi)^2/(384\pi^2m^4)$}
For the <charge-two cubic complex-scalar interaction>, integrating out a <complex scalar field> of mass $m$ generates the displayed local <effective Lagrangian> term when external energies and momenta are small compared with $m$. A constant $\Phi$ gives two real squared masses $m^2\pm g|\Phi|$. Expanding half the sum of their one-loop logarithms gives $V_1\supset-g^4|\Phi|^4I_4/4$, with the <Euclidean massive loop integral> $I_4=1/(96\pi^2m^4)$. Since $\mathcal L_{\rm eff}=-V_1$ for a constant field, $2!2!$ external attachments give the <scattering amplitude> $g^4/(96\pi^2m^4)$ for either heavy-heavy or heavy-antiparticle elastic scattering. This agrees with the unique charge-allowed light <box diagram>.
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