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 Incoming links: Standard perturbation theory velocity kernel

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 312 / 3 / iii / b / Solution Created 2026-09-24 Updated 2026-10-14

Define the standard perturbation theory velocity kernel by

θ^{(n)} (k) = \int_{q_{1} \dots q_{n}} (2 π)^{3} δ_{D} (k - \sum_{a} q_{a}) G_{n} (q_{1}, \dots, q_{n}) \prod_{a} δ_{1} (q_{a}),

where

\int_{q} = \int d^{3} q / (2 π)^{3}

and

P (q) = ⟨ ∣ δ_{1} (q) ∣^{2} ⟩^{'}

. For

k_{1} + k_{2} + k_{3} = 0

, one representative external labelling of the tree contribution is

B_{211} = 2 G_{2} (k_{1}, k_{2}) P (k_{1}) P (k_{2}) .

The four one-loop representatives are

B_{222} = 8 \int_{q} G_{2} (- q, q + k_{1}) G_{2} (- q - k_{1}, q - k_{2}) G_{2} (k_{2} - q, q) P (q) P (∣ q + k_{1} ∣) P (∣ q - k_{2} ∣),

B_{321}^{I} = 6 P (k_{3}) \int_{q} G_{3} (- k_{3}, - q, q - k_{2}) G_{2} (q, k_{2} - q) P (q) P (∣ k_{2} - q ∣),

B_{321}^{II} = 6 P (k_{1}) P (k_{3}) G_{2} (k_{1}, k_{3}) \int_{q} G_{3} (k_{1}, q, - q) P (q),

B_{411} = 12 P (k_{2}) P (k_{3}) \int_{q} G_{4} (- k_{2}, - k_{3}, q, - q) P (q) .

The full answer adds the distinct permutations of the external labels; the question asks for only one labelling of each topology.

Figure 1.
Tree and one-loop matter-bispectrum topologies
. The five panels show representative external labellings of B211, B222, the two B321 topologies, and B411. Filled dots denote perturbation-theory vertices and colored internal lines make the loop structures visible. Details.

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