Simplicity from Recoil: The Three-Loop Soft Function and Factorization for the Energy-Energy Correlation

TL;DR

This paper derives a three-loop factorization theorem for the energy-energy correlation (EEC) in electron-positron collisions, enabling high-precision QCD calculations and insights into non-perturbative effects.

Contribution

It provides the first three-loop analytic results for the soft function and anomalous dimensions of the EEC, advancing precision in QCD predictions.

Findings

Soft functions for EEC and transverse-momentum resummation are equivalent at all orders.

Analytic three-loop results for anomalous dimensions are obtained.

EEC can be computed at N3LL accuracy matched with NNLO.

Abstract

We derive an operator based factorization theorem for the energy-energy correlation (EEC) observable in the back-to-back region, allowing the cross section to be written as a convolution of hard, jet and soft functions. We prove the equivalence of the soft functions for the EEC and color singlet transverse-momentum resummation to all-loop order, and give their analytic result to three-loops. Large logarithms appearing in the perturbative expansion of the EEC can be resummed to all orders using renormalization group evolution. We give analytic results for all required anomalous dimensions to three-loop order, providing the first example of a transverse-momentum (recoil) sensitive $e^+e^-$ event shape whose anomalous dimensions are known at this order. The EEC can now be computed to next-to-next-to-next-to-leading logarithm matched to next-to-next-to-leading order, making it a prime candidate for precision QCD studies and extractions of the strong coupling constant. We anticipate that our factorization theorem will also be crucial for understanding non-perturbative power corrections for the EEC, and their relationship to those appearing in other observables.