Light-cone limits of large rectangular fishnets

TL;DR

This paper studies the behavior of rectangular fishnet Feynman graphs in various double scaling limits, including the null limit, revealing how coordinate dependence emerges and can be explicitly evaluated.

Contribution

It introduces the most general double scaling limit for rectangular fishnets, capturing all singular behaviors including the null limit, with explicit evaluations.

Findings

Double scaling limit captures all singular behaviors

Coordinate dependence reappears in specific limits

Explicit evaluation in null limit for any grid size

Abstract

Basso-Dixon integrals evaluate rectangular fishnets -- Feynman graphs with massless scalar propagators which form a $m\times n$ rectangular grid -- which arise in certain one-trace four-point correlators in the `fishnet' limit of $\mathcal{N}=4$ SYM. Recently, Basso {\it et al} explored the thermodynamical limit $m\to\infty$ with fixed aspect ratio $n/m$ of a rectangular fishnet and showed that in general the dependence on the coordinates of the four operators is erased, but it reappears in a scaling limit with two of the operators getting close in a controlled way. In this note I investigate the most general double scaling limit which describes the thermodynamics when one of two pairs of operators become nearly light-like. In this double scaling limit, the rectangular fishnet depends on both coordinate cross ratios. I show that all singular limits of the fishnet can be attained within the double scaling limit, including the null limit with the four points approaching the cusps of a null square. A direct evaluation of the fishnet in the null limit is presented any $m$ and $n$.