A single quintic vertex has five half-edges, so <Wick theorem> cannot contract all of them in pairs. The leading connected <Vacuum Feynman diagrams> therefore have two quintic vertices and are of order $g^2$. If $c$ propagators join the two vertices, each vertex has $5-c$ half-edges left for tadpoles, so $c$ must be odd. This gives exactly three topologies:

* $c=1$: one line joins the vertices and each vertex carries two tadpole loops.
* $c=3$: three lines join the vertices and each vertex carries one tadpole loop.
* $c=5$: all five lines join the vertices.

The three diagrams are connected; every other pairing is either isomorphic to one of them or disconnected.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 304 / 1 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-304/1/a/solution