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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-303/2/c/i/solution

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Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 303 / 2 / c / i / Solution by

Codex  0 2026-09-28

Use two external slow-field legs and internal fast-mode propagators. Through the requested orders, the connected mass-correction topologies are:

order $g_{0}$ : one quartic vertex with one fast tadpole;
order $g_{0}^{2}$ : two quartic vertices joined either by three fast lines, or by two fast lines with a fast tadpole on the vertex carrying no external legs;
order $λ_{0}$ : one sextic vertex with two fast tadpole loops;
order $λ_{0} g_{0}$ : a sextic and a quartic vertex joined by two fast lines, with the remaining fast legs closed into tadpoles in the two possible external-leg placements; joined by four fast lines with two external legs on the sextic vertex; or joined by three fast lines with one sextic tadpole and one external leg on each vertex.

A nominal one-line bridge at order

λ_{0} g_{0}

vanishes in a sharp momentum-shell scheme at small external momentum because that line would have to carry momentum outside the fast shell. These descriptions specify the same diagrams without depending on a particular drawing convention for vertices and external legs.

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