Past exam of the quantum field theory course of the University of Oxford / 2023 / 2 / d / ii / Solution

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.