From squared amplitudes to energy correlators

TL;DR

This paper connects energy correlators in supersymmetric Yang-Mills theory to positive geometry, computes integrands up to N=11, and explores their complex mathematical structures including elliptic functions for higher N.

Contribution

It reveals the geometric structure of energy correlators via the squared amplituhedron and computes their integrands up to N=11, uncovering new mathematical features.

Findings

Integrand expressed as an integral of the 1→N splitting function.

Computed integrands up to N=11 using the f-graph construction.

Identified elliptic polylogarithmic functions for N≥5.

Abstract

The leading order $N$-point energy correlators of maximally supersymmetric Yang-Mills theory in the limit where the $N$ detectors are collinear can be expressed as an integral of the $1\to N$ splitting function, which is given by the $(N{+}3)$-point squared super-amplitudes at tree level. This provides yet another example that the integrand of certain physical observable -- $N$-point energy correlator -- is computed by the canonical form of a positive geometry -- the (tree-level) "squared amplituhedron". By extracting such squared amplitudes from the $f$-graph construction, we compute the integrand of energy correlators up to $N=11$ and reveal new structures to all $N$; we also show important properties of the integrand such as soft and multi-collinear limits. Finally, we take a first look at integrations by studying possible residues of the integrand: our analysis shows that while this gives prefactors in front of multiple polylogarithm functions of $N=3,4$, the first unknown case of $N=5$ already involves elliptic polylogarithmic functions with many distinct elliptic curves, and more complicated curves and higher-dimensional varieties appear for $N>5$.