Charge-two cubic complex-scalar interaction (${\mathcal{L}}_I=-g({\varphi }^2{\Phi }^{†}+{\varphi }^{†2}\Phi )/2$)

This interaction preserves the global U(1) symmetry with relative charges $q_{\varphi }=1$, $q_{\Phi }=2$. The two conjugate interaction vertices have Feynman rule $-ig$, with either one charge-two arrow entering and two charge-one arrows leaving, or all arrows reversed. For $\varphi \varphi \to \varphi \varphi$ only the heavy $s$-channel exchange occurs at tree level. For $\varphi \bar{\varphi }\to \varphi \bar{\varphi }$, with outgoing particle labelled 3 and outgoing antiparticle labelled 4, only the heavy $u$-channel exchange occurs. For small momenta and heavy mediator mass, the amplitudes coincide but the identical-particle total scattering cross-section is smaller by the identical final-state symmetry factor $1/2$.

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.