Differential Equations for Energy Correlators in Any Angle

TL;DR

This paper develops a systematic algorithm to compute differential equations for energy correlators at any angle in $ abla$=4 super Yang-Mills theory, revealing complex function spaces including elliptic and hyperelliptic curves.

Contribution

It introduces a novel algorithm for calculating differential equations of energy correlators, including their master integrals and function space analysis, with an accompanying computational package.

Findings

Derived canonical basis for three-point energy correlators.

Computed full set of master integrals for four-point energy correlators.

Identified elliptic and hyperelliptic curves in the integral functions.

Abstract

Energy Correlators (EC) are the simplest IR finite observables, which connect theories and experiments. In this paper, we provide a systematic algorithm to calculate the canonical differential equations for energy correlators at generic angle in $\mathcal{N}=4$ super Yang-Mills theory. The integrand is obtained from the 5-point form factor square for scalar half-BPS operators. Applying the algorithm, we obtain the canonical basis for three-point EC and the full set of master integrals for four-point EC. We analyze the function space for four-point case. For multiple polylogrithmic (MPLs) integrals, we calculate their symbols, and for integrals beyond MPLs, we make further investigation by Picard-Fuchs operators. We find two elliptic curves and one genus 2 hyperelliptic curve. The results are achieved by means of integration by part (IBP) reduction and differential equations powered by computational algebraic geometry methods. We provide a package that implements the algorithm. The data is a valuable reference for exploring the structure of physical observables in perturbation theories.