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.

 Ancestors (11)

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