Leading singularities and chambers of Correlahedron

TL;DR

This paper investigates the geometric structure of loop integrands in four-point correlators of planar N=4 super Yang-Mills, revealing a chamber decomposition pattern that simplifies understanding of leading singularities, including elliptic functions.

Contribution

It demonstrates that the chamber decomposition pattern of the Correlahedron persists at four loops, suggesting a universal structure for all loop orders and simplifying the analysis of complex elliptic functions.

Findings

Loop integrand at four loops expressed as sum over chamber-forms and local integrands.

Chamber structures are identical at three and four loops, indicating potential all-loop completeness.

Elliptic functions appear in specific chambers, and a diagonalized form yields pure functions.

Abstract

In this paper, we explore the chamber dissection of the loop-geometry of Correlahedron, which encodes the loop integrand of four-point stress-energy correlators in planar $\mathcal{N}=4$ super Yang-Mills. We demonstrate that at four loops, continuing the pattern of lower loops, the integrand of the four-point correlation function can be written as a sum over products of chamber-forms and local loop integrands. The chambers and their associated forms are identical to those of three loops, indicating that the dissection may be complete to all loop orders. Furthermore, this suggests that the leading singularities at all loops are simply linear combinations of these chamber forms. This is especially intriguing at four loops since it contains elliptic functions. Interestingly, each elliptic function appears in a subset of chambers. Our geometric approach motivates us to ``diagonalize" the representation, where the local integrals only possess a single leading singularity or elliptic cut. In such a representation, all integrands must evaluate to pure functions, including a single pure elliptic integrand. Inspired by this picture, we also present a simplified form of the three-loop correlator in terms of two independent pure functions (weight-$6$ single-valued multiple polylogarithms), which are directly computed from local integrands with unit leading singularities, multiplied by the leading singularities from chamber forms.