= Solution
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.
Back to article page
