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

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.