Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 304 / 2 / b / Solution

For one channel with momentum $p$, introduce a Feynman parameter and shift the loop momentum:In $d=4-ϵ$, with dimensional-regularization scale $\bar{\mu }$ before the usual modified-minimal-subtraction redefinition, the bubble at $p^2=-M^2$ isSince ${\int }_0^1\log \{x(1-x)\}dx=-2$, summing the three equal channels gives the form displayed in the question, beginning with $6/ϵ-3\gamma$.

ConsequentlyThe momentum-subtraction scheme condition ${\lambda }_{\mathrm{eff}}(M)=\lambda$ is enforced byChoosing $\bar{\mu }=M$ removes the logarithm. A minimal-subtraction scheme keeps only the pole and therefore defines a different finite renormalized coupling.