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

TL;DR

This work derives an all-orders SCET factorization for the energy-energy correlation in the back-to-back region, revealing that the soft function is universal with the color-singlet qT soft function up to Wilson-line directions. By relating EEC jet and soft functions to known TMDFFs and the qT soft function, the authors establish RG evolution that enables N^3LL resummation and NNLO matching, with explicit three-loop soft-function boundary conditions. The analysis highlights the universality and crossing symmetry of soft functions for recoil-sensitive observables and lays the groundwork for precision α_s extractions and non-perturbative studies. Overall, the paper provides a concrete operator-based framework that unifies EEC and qT physics and pushes perturbative control to the highest currently achieved orders for a recoil-sensitive e+e− observable.

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.

Figures (3)

• Figure 1: An illustration of the EEC observable in $e^+e^-$ annihilation, which is defined as the energy-energy correlation of two calorimeter cells with opening angle $\chi$.
• Figure 2: A schematic of the kinematics relevant for the description of the EEC observable in the back-to-back limit, $z\to 1$. The total transverse momentum of final-state hadrons perpendicular to the thrust axis (the black dashed line) is denoted as $\vec{k}_{\perp,s}^h$. The three-momentum of a pair of collinear hadrons which enter weighted sum in Eq. (\ref{['eq:eec_defz']}) are denoted as $\vec{k}_{i,j}^h$. The corresponding transverse components perpendicular to the jet axis $\vec{n}_{a,b}$ are $\vec{k}_{\perp,i}^h$ and $\vec{k}_{\perp,j}^h$. Soft radiation acts only to recoil the two collinear sectors, denoted by $\vec{n}_{a,b}$, with respect to each other, but does not contribute directly to the observable.
• Figure 3: The soft functions for $q_T$ for color singlet production in (a) and for the EEC in (b), as defined in the exponential rapidity regulator of Li:2016axz. Solid black lines denotes Wilson lines. Both the rapidity regulator and the measurement are defined in terms of space-time shifts of the Wilson lines, allowing an all orders proof of the equivalence of these two soft functions.