Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 303 / 3

The leading mass correction is the one-vertex tadpole diagram: two legs of the quartic $g_0({\mathbf{\varphi }}^2{)}^2$ vertex are external and the remaining two form one fast loop. One contraction has a freely summed component index and contributes $N$, while two exchange contractions have their index fixed by the external component. The result is proportional to $N+2$.

The leading quartic correction is the one-loop bubble diagram with two quartic vertices joined by two fast propagators. The four external legs can be distributed in the three exchange channels. One family of index contractions contains a freely summed closed component loop and contributes $N$; the other contractions contribute eight, giving the factor $N+8$.

Define the shell integralsThe index multiplicities described in part a and the canonical rescaling giveFor $N=1$, these reduce to the stated Ising coefficients $12$ and $36$.

PutFor $\zeta =e^s$, differentiating a thin radial shell at $s=0$ givesReplacing bare parameters by running parameters after each infinitesimal step yields the beta functions

For $d=4-ϵ$, define dimensionless variablesTo leading order, with $K_4=1/(8{\pi }^2)$,The Gaussian fixed point is $(r_{\ast },u_{\ast })=(0,0)$. Its thermal eigenvalue is $y_t=2$, soThe interacting Wilson-Fisher fixed point isLinearizing the flow givesSince the correlation-length critical exponent is $\nu =1/y_t$,

Corrections quadratic in the cubic-anisotropy coupling use two ${\lambda }_0{\sum }_a{\varphi }_a^4$ vertices joined by two fast propagators. Because all four legs at each anisotropic vertex carry the same component index, the internal propagators force the two vertices to have that same index. The resulting one-loop bubble renormalizes the anisotropic tensor ${\sum }_a{\varphi }_a^4$, hence $\lambda$, but does not generate the distinct-index structure in $({\mathbf{\varphi }}^2{)}^2$ needed for a correction to $g$. There is no freely summed closed component index, so the diagram is not proportional to $N$.

At order ${\lambda }_0{g}_0$, use one anisotropic quartic vertex and one $O(N)$-invariant quartic vertex joined by two fast propagators. With four equal external component labels, the bubble renormalizes $\lambda$; with external labels in two equal pairs, it also renormalizes $g$. The anisotropic vertex fixes the component index carried by the internal lines, so none of these mixed diagrams contains an independent component loop. Consequently no correction proportional to $N$ occurs at order ${\lambda }_0^2$ or ${\lambda }_0{g}_0$.