 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-303/2/e/v/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Classify the connected contractions by the numbers

(r, 4 - r)

of external slow legs on the two sextic vertices and by the number

c

of fast propagators joining them. Besides the displayed

(2, 2; c = 4)

graph, the distinct topologies are:

$(2, 2; c = 2)$ : two joining lines and one tadpole on each vertex.
$(1, 3; c = 3)$ : three joining lines and one tadpole on the one-external-leg vertex.
$(0, 4; c = 2)$ : two joining lines and two tadpoles on the vertex with no external legs.
$(1, 3; c = 1)$ : one joining line, two tadpoles on the one-external-leg vertex, and one tadpole on the three-external-leg vertex.

Exchanging the two vertices gives no new topology. All four are connected Feynman diagrams selected by the logarithm in the cumulant expansion. With an ideal sharp momentum shell and a projection at exactly zero external momentum, the single joining line in the last topology cannot carry shell momentum, so that topology gives zero to the local quartic coupling; it is still the remaining formal connected contraction.

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