Multipoint fishnet Feynman diagrams: sequential splitting

TL;DR

This paper develops a new solution method for multipoint fishnet Feynman diagrams using Separation of Variables, Mellin space, and combinatorics, providing explicit formulas and new insights into their structure and light-cone behavior.

Contribution

It introduces a recursive solution for n-point split-ladders, an elementary proof of the Basso-Dixon formula, and a combinatorial approach to analyze light-cone limits.

Findings

Explicit formulas for coefficient functions of leading logs

Elementary proof of the Basso-Dixon formula at 4 points

A vertex model approach for light-cone behavior analysis

Abstract

We study fishnet Feynman diagrams defined by a certain triangulation of a planar n-gon, with massless scalars propagating along and across the cuts. Our solution theory uses the technique of Separation of Variables, in combination with the theory of symmetric polynomials and Mellin space. The n-point split-ladders are solved by a recursion where all building blocks are made fully explicit. In particular, we find an elegant formula for the coefficient functions of the light-cone leading logs. When the diagram grows into a fishnet, we obtain new results exploiting a Cauchy identity decomposition of the measure over separated variables. This leads to an elementary proof of the Basso-Dixon formula at 4-points, while at n-points it provides a natural OPE-like stratification of the diagram. Finally, we propose an independent approach based on ``stampede" combinatorics to study the light-cone behaviour of the diagrams as the partition function of a certain vertex model.