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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.

Cross the second incoming heavy particle and one outgoing heavy particle in i to obtain $\Phi \bar{\Phi }\to \Phi \bar{\Phi }$. Panel (e) of b's figure shows the resulting box diagram. Its four light-charge arrows again alternate source and sink corners; it is the only connected order-$g^4$ topology. There is no tree exchange because every cubic interaction vertex has only one heavy leg. With $|\mathbf{p}|\ll M\ll m$, crossing does not affect the zero-external-momentum loop integral, soAn independent check is to integrate out the complex scalar field $\varphi$ at constant $\Phi$. Its two real-component squared masses are $m^2\pm g|\Phi |$. The one-loop quantum effective potential is thereforeExpanding the logarithms to fourth order in $\Phi$ givesThus the effective Lagrangian has the positive local coefficient $g^4/(384{\pi }^2{m}^4)$. Its $2!2!$ external attachments give the common scattering amplitude $g^4/(96{\pi }^2{m}^4)$ in both channels. This also checks the box multiplicity and sign.