All-loop geometry for four-point correlation functions

TL;DR

This paper introduces a geometric framework in twistor space to encode four-point correlation functions in planar N=4 super Yang-Mills theory, providing a new way to understand loop integrands through positive geometry.

Contribution

It constructs a positive geometric model for loop integrands of four-point correlators, extending the understanding of their structure in twistor space and verifying it up to three loops.

Findings

Explicit construction of the loop geometry in twistor space.

Verification of the geometric model up to three loops.

Identification of chamber organization by point orderings.

Abstract

In this letter, we consider a positive geometry conjectured to encode the loop integrand of four-point stress-energy correlators in planar $\mathcal{N}=4$ super Yang-Mills. Beginning with four lines in twistor space, we characterize a positive subspace to which an $\ell$-loop geometry is attached. The loop geometry then consists of $\ell$ lines in twistor space satisfying positivity conditions among themselves and with respect to the base. Consequently, the $\textit{loop geometry}$ can be viewed as fibration over a $\textit{tree geometry}$. The fibration naturally dissects the base into chambers, in which the degree-$4 \ell$ loop form is unique and distinct for each chamber. Interestingly, up to three loops, the chambers are simply organized by the six ordering of $x^2_{1,2}x^2_{3,4}$, $x^2_{1,4}x^2_{2,3}$ and $x^2_{1,3}x^2_{2,4}$. We explicitly verify our conjecture by computing the loop-forms in terms of a basis of planar conformal integrals up to $\ell=3$, which indeed yield correct loop integrands for the four-point correlator.