The Three-Point Energy Correlator in the Coplanar Limit

TL;DR

This paper develops a new theoretical framework for analyzing the three-point energy correlator in the coplanar limit at lepton colliders, enabling high-precision resummation of logarithms and advancing understanding of trijet configurations.

Contribution

It introduces a novel projection method and a TMD-based factorization theorem for the EEEC, achieving the first N$^3$LL resummation for a trijet event shape.

Findings

Derived a factorization theorem capturing soft and collinear logs.

Achieved N$^3$LL resummation accuracy for the EEEC.

Provided a method to study coplanar trijet shapes.

Abstract

Energy correlators are a type of observables that measure how energy is distributed across multiple detectors as a function of the angles between pairs of detectors. In this paper, we study the three-point energy correlator (EEEC) at lepton colliders in the three-particle near-to-plane (coplanar) limit. The leading-power contribution in this limit is governed by the three-jet (trijet) configuration. We introduce a new approach by projecting the EEEC onto the volume of the parallelepiped formed by the unit vectors aligned with three detected final-state particles. Analogous to the back-to-back limit of the two-point energy correlator probing the dijet configuration, the small-volume limit of the EEEC probes the trijet configuration. We derive a transverse momentum dependent (TMD) based factorization theorem that captures the soft and collinear logarithms in the coplanar limit, which enables us to achieve the next-to-next-to-next-to-leading logarithm (N$^3$LL) resummation. To our knowledge, this is the first N$^3$LL result for a trijet event shape. Additionally, we demonstrate that a similar factorization theorem can be applied to the fully differential EEEC in the three-particle coplanar limit, which provides a clean environment for studying different coplanar trijet shapes.