Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 304 / 1 / a

The Quantum effective action contains the following diagrams. For ${\Gamma }_2$, there are the tree inverse propagator, the one-loop tadpole at order $\lambda$, and at order ${\lambda }^2$ the two-loop sunset and the two-loop tadpole with a tadpole insertion. For ${\Gamma }_4$, there are the tree vertex $-i\lambda$ and, at order ${\lambda }^2$, the three one-loop bubble diagrams in the $s$, $t$, and $u$ channels. Disconnected and one-particle-reducible graphs do not occur in the quantum effective action.