Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 304 / 2 / b / Solution

After integration by parts, the quadratic ghost operator is $c_1(-{\partial }^2)+c_2{M}^2$. Matching its propagator to the scalar propagator after $M\mapsto m$ requiresDifferentiating $c_3\lambda \bar{\eta }{\varphi }^2\eta$ with respect to its two identical scalar fields gives the ghost-scalar vertex $-2c_3\lambda$. Matching it to the four-scalar vertex therefore requiresThe additional rules are an oriented ghost propagator $(p^2+M^2{)}^{-1}$, a two-scalar two-ghost vertex $-\lambda$, and a minus sign for every closed loop of Grassmann-valued fields.

Besides the scalar bubbles from part a, each channel now has a closed heavy-ghost bubble. Its two directed internal lines cannot be interchanged, so it has no scalar bubble's factor $1/2$. ConsequentlyIf $I_{M,\mathrm{os}}$ denotes the corresponding sum of heavy integrals, on-shell matching givesThe root continuously connected to ${\lambda }_b={\lambda }_{\mathrm{phys}}$ is the perturbative, small positive root.