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

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 $N+8$.