Light-scalar box matching for a charge-two field (${\mathcal{L}}_{\mathrm{eff}}\supset g^4({\Phi }^{†}\Phi {)}^2/(384{\pi }^2{m}^4)$)

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 $V_1\supset -g^4|\Phi {|}^4{I}_4/4$, with the Euclidean massive loop integral $I_4=1/(96{\pi }^2{m}^4)$. Since ${\mathcal{L}}_{\mathrm{eff}}=-V_1$ for a constant field, $2!2!$ external attachments give the scattering amplitude $g^4/(96{\pi }^2{m}^4)$ for either heavy-heavy or heavy-antiparticle elastic scattering. This agrees with the unique charge-allowed light box diagram.