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

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 $N+8$.