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

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 $N+2$.