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

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 $\varphi$ 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 givesThe Euclidean massive loop integral is convergent. For a constant field, ${\mathcal{L}}_{\mathrm{eff}}=-V_1$, 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 $i{g}^4/(96{\pi }^2{m}^4)$ 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 $\varphi$ 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.