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

For massless phi-fourth theory in four dimensions, the momentum-space rules areWritingthe two-point counterterm insertion is $i({\delta }_Z{p}^2-\delta m^2)$ and the four-point counterterm is $-i\delta \lambda$.

The renormalized one-particle-irreducible correlation function ${\Gamma }_4$ through order ${\lambda }^2$ contains the tree quartic vertex, the quartic counterterm, and three one-loop bubble diagrams. The bubbles are the $s$, $t$, and $u$ channels and each has symmetry factor $1/2$.

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.

For a renormalized $n$-point one-particle-irreducible function, the Callan-Symanzik equation iswith a convention-dependent sign on ${\gamma }_{\varphi }$. At this order ${\gamma }_{\varphi }=0$.

At a general Euclidean momentum scale $Q$, the one-loop four-point function containsRequiring a physical amplitude to be independent of the arbitrary subtraction scale givesand therefore