Exclusivity of high‑order three‑particle ladder terms to a single particle–particle–hole channel

Determine whether, in the three‑particle ladder approximation constructed from two‑particle irreducible vertices, diagrammatic contributions of sufficiently high order (i.e., involving many two‑particle vertices) arise exclusively in a single particle–particle–hole (pph) channel rather than being shared across multiple pph channels under crossing or leg permutations. Establish a proof or provide a counterexample to assess the impact of averaging over the nine pph channels on higher‑order terms in the three‑particle vertex and the resulting second‑order response functions.

Background

The paper develops Bethe–Salpeter–like ladder equations for three‑particle vertices and proposes an approximate solution that builds the ladder solely from two‑particle irreducible vertices. Because the approximate ladder lacks several crossing symmetries, the authors average ladder and one‑particle‑reducible (1PR) contributions over the nine particle–particle–hole (pph) channels to restore symmetry and to avoid double counting of lower‑order diagrams.

They show that averaging ensures correctness up to second order in the two‑particle vertex and prevents overcounting that would occur if all channels were summed. However, their numerical results for second‑order response functions are systematically too small, leading them to conjecture that higher‑order ladder contributions might be exclusive to a single pph channel. If true, averaging across all channels would undercount these higher‑order diagrams by a factor of nine, and a different channel‑combination strategy would be needed.