Fishnet four-point integrals: integrable representations and thermodynamic limits

TL;DR

This paper proves the equivalence of different integral representations of four-point fishnet integrals, analyzes their thermodynamic limit, and uncovers connections to matrix models and string theory, revealing boundary condition sensitivities.

Contribution

It establishes the equivalence of integral representations, derives a concise free-energy expression, and links the saddle-point equations to string theory in a novel scaling regime.

Findings

Proved equivalence of integral representations of fishnet integrals.

Derived a parametric free-energy expression involving elliptic integrals.

Identified a connection between saddle-point equations and Frolov-Tseytlin strings.

Abstract

We consider four-point integrals arising in the planar limit of the conformal "fishnet" theory in four dimensions. They define a two-parameter family of higher-loop Feynman integrals, which extend the series of ladder integrals and were argued, based on integrability and analyticity, to admit matrix-model-like integral and determinantal representations. In this paper, we prove the equivalence of all these representations using exact summation and integration techniques. We then analyze the large-order behaviour, corresponding to the thermodynamic limit of a large fishnet graph. The saddle-point equations are found to match known two-cut singular equations arising in matrix models, enabling us to obtain a concise parametric expression for the free-energy density in terms of complete elliptic integrals. Interestingly, the latter depends non-trivially on the fishnet aspect ratio and differs from a scaling formula due to Zamolodchikov for large periodic fishnets, suggesting a strong sensitivity to the boundary conditions. We also find an intriguing connection between the saddle-point equation and the equation describing the Frolov-Tseytlin spinning string in $AdS_{3}\times S^{1}$, in a generalized scaling combining the thermodynamic and short-distance limits.