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

Four external $\Phi$ lines require four cubic interaction vertices, since each such interaction vertex contains only one heavy field. Their other eight endpoints must form four light internal lines. A connected Feynman diagram therefore has $V=I=4$ and $L=1$: the leading contribution is a one-loop box diagram, of order $g^4$, with four $\varphi$ scalar propagators. Charge-two incoming and outgoing heavy lines attach alternately around the box, as shown in panel (d) of b's figure. Each incoming heavy line supplies the two outgoing light-charge arrows at its corner, and each outgoing heavy line absorbs them. With external labels fixed there is one such box and its symmetry factor is one.

For $|\mathbf{p}|\ll M\ll m$, the external energies are also small compared with $m$, so expand the loop integrand in all external four-momenta. Its leading term isA Wick rotation and the Euclidean massive loop integral giveHence, more precisely than just its requested scaling,The mass dimension $[g]=1$ makes $g^4/m^4$ dimensionless, as a four-dimensional scattering amplitude should be. The leading amplitude has no extra identical-particle multiplicity.