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

For the Euclidean quartic scalar field theory, expansion of $e^{-S}$ gives the momentum-space rules

At one loop, the four-point one-particle-irreducible correlation function receives the three bubble diagrams in the $s$, $t$, and $u$ channels. If $P$ is the momentum through one channel, its contribution isUsing a Feynman parameter, shifting the loop momentum, and writing ${\Delta }_x=m^2+x(1-x)P^2$ gives, up to terms caused by shifting the boundary of a hard cutoff,Thus every channel has the logarithmic ultraviolet divergenceThe complete one-loop vertex isLet $I_{m,\mathrm{os}}$ denote the bracket evaluated at the chosen on-shell kinematic point. The on-shell renormalization scheme requires $V_a^{(4)}{|}_{\mathrm{os}}=-{\lambda }_{\mathrm{phys}}$, hence the perturbative solution isThe cutoff dependence of ${\lambda }_a$ is the coupling counterterm needed to hold the measured coupling fixed.

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.

After ${\lambda }_a$ and ${\lambda }_b$ have been matched to the same measured coupling, the nonanalytic low-energy dependence from the light scalar loops is identical. For $|P|\ll M$, a heavy loop has the local expansionIts constant term is already absorbed into the matched quartic coupling, while the remaining terms are higher-dimensional local interactions suppressed by powers of $M$. Dependence on the finite ultraviolet cutoff is similarly suppressed by powers of $\Lambda$. This is heavy-field decoupling: low-energy scattering agrees up to $O(p^2/M^2,p^2/{\Lambda }^2)$ corrections after all relevant parameters are matched. At energies comparable to $M$, the two theories differ sharply because the second theory contains wrong-statistics heavy states.