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From DGLAP to Sudakov: Precision Predictions for Energy-Energy Correlators

Max Jaarsma, Yibei Li, Ian Moult, Wouter J. Waalewijn, Hua Xing Zhu

TL;DR

The paper delivers the first full, high-precision calculation of the energy-energy correlator measured on tracks in $e^+e^-$ collisions across the entire spectrum, achieving NNLO fixed order plus NNLL collinear and NNNNLL back-to-back resummation, augmented by leading nonperturbative corrections including a lattice-determined Collins-Soper kernel. It introduces and leverages track functions to systematically separate perturbative and nonperturbative physics, and provides detailed factorization theorems for both the collinear and back-to-back regions, with smooth matching across regimes. The work couples perturbative QCD, SCET factorization, and lattice inputs to enable precise extractions of fundamental parameters like $ ext{alpha}_s$ and the nonperturbative Omega parameters, and it explores the physics of confinement through plateau and contact-term behavior in the EEC. Together with LEP track measurements, this represents a major advancement in precision QCD phenomenology, offering new handles on nonperturbative dynamics and potential cross-checks with lattice and other theoretical approaches. The results underscore the EEC’s potential as a versatile, high-precision probe of QCD across UV and IR scales, with implications for future collider phenomenology and nonperturbative theory.

Abstract

Correlations in the distribution of energy produced in collider experiments provide a snapshot of the microscopic dynamics of QCD, and its evolution from asymptotically free quarks and gluons, to confined hadrons. There has recently been considerable progress in the interpretation and precision calculation of these correlations, using a specific class of observables called energy correlators (EECs). These observables are most cleanly studied in $e^+e^-$ collisions, where they can be measured over their full angular range. Of particular interest are kinematic limits of the correlator, both collinear, and back-to-back, where the correlator exhibits scaling behaviors governed by specific operators in QCD. Resolving these scalings requires measurements with exceptional angular resolution, which can be achieved by performing measurements on tracks (charged particles). In this paper we perform the first calculation of the track-based EEC over its entire kinematic range, achieving a record precision of of NNLL (collinear) + NNLO (fixed order) + NNNNLL (back-to-back) for the track-based EEC, and additionally incorporate the leading non-perturbative corrections and their resummation, including the Collins-Soper kernel computed using lattice QCD. We describe the breadth of physics probed by this observable, and highlight the impact of different components of our factorization theorem on the final distribution. Combined with recent measurements of the track-based EEC with archival LEP data, our calculation initiates the precision study of track-based observables at LEP, which will lead to new insights into the dynamics of QCD, and the precision extraction of its underlying parameters.

From DGLAP to Sudakov: Precision Predictions for Energy-Energy Correlators

TL;DR

The paper delivers the first full, high-precision calculation of the energy-energy correlator measured on tracks in collisions across the entire spectrum, achieving NNLO fixed order plus NNLL collinear and NNNNLL back-to-back resummation, augmented by leading nonperturbative corrections including a lattice-determined Collins-Soper kernel. It introduces and leverages track functions to systematically separate perturbative and nonperturbative physics, and provides detailed factorization theorems for both the collinear and back-to-back regions, with smooth matching across regimes. The work couples perturbative QCD, SCET factorization, and lattice inputs to enable precise extractions of fundamental parameters like and the nonperturbative Omega parameters, and it explores the physics of confinement through plateau and contact-term behavior in the EEC. Together with LEP track measurements, this represents a major advancement in precision QCD phenomenology, offering new handles on nonperturbative dynamics and potential cross-checks with lattice and other theoretical approaches. The results underscore the EEC’s potential as a versatile, high-precision probe of QCD across UV and IR scales, with implications for future collider phenomenology and nonperturbative theory.

Abstract

Correlations in the distribution of energy produced in collider experiments provide a snapshot of the microscopic dynamics of QCD, and its evolution from asymptotically free quarks and gluons, to confined hadrons. There has recently been considerable progress in the interpretation and precision calculation of these correlations, using a specific class of observables called energy correlators (EECs). These observables are most cleanly studied in collisions, where they can be measured over their full angular range. Of particular interest are kinematic limits of the correlator, both collinear, and back-to-back, where the correlator exhibits scaling behaviors governed by specific operators in QCD. Resolving these scalings requires measurements with exceptional angular resolution, which can be achieved by performing measurements on tracks (charged particles). In this paper we perform the first calculation of the track-based EEC over its entire kinematic range, achieving a record precision of of NNLL (collinear) + NNLO (fixed order) + NNNNLL (back-to-back) for the track-based EEC, and additionally incorporate the leading non-perturbative corrections and their resummation, including the Collins-Soper kernel computed using lattice QCD. We describe the breadth of physics probed by this observable, and highlight the impact of different components of our factorization theorem on the final distribution. Combined with recent measurements of the track-based EEC with archival LEP data, our calculation initiates the precision study of track-based observables at LEP, which will lead to new insights into the dynamics of QCD, and the precision extraction of its underlying parameters.

Paper Structure

This paper contains 51 sections, 220 equations, 21 figures.

Table of Contents

  1. Introduction
  2. The Physics of the Energy-Energy Correlator
  3. Sum Rules
  4. The Energy-Energy Correlator in Different Theories
  5. A Brief History of Calculations of the Energy-Energy Correlator
  6. Overview of Theory Framework
  7. Track Functions
  8. Non-Perturbative Power Corrections
  9. Factorization Theorems and Resummation
  10. Matching
  11. Bulk Region: $z\sim 1/2$
  12. Perturbative Calculations on Tracks
  13. Fixed-Order Results
  14. LO
  15. NLO
  16. ...and 36 more sections

Figures (21)

  • Figure 1: The EEC as "measured" in toy theories. On the left, we show the distribution arising from a large charge operator in a free scalar theory. On the right, we show the distribution for a conformal gauge theory in $d=4$.
  • Figure 2: The EEC in $e^+e^-$ collisions: In purple we show data from the ALEPH experiment, and in light-blue we show the calculation developed in this paper. The inset figures show the underlying physics dominating each kinematic region of the energy correlator, and are discussed in the text. This figure originally appeared in Electron-PositronAlliance:2025fhk.
  • Figure 3: The perturbative convergence of the track-EEC at $Q=m_Z$ in the bulk region of the distribution, which is dominated by fixed order perturbation theory.
  • Figure 4: The NNLO fixed order result, along with the singular and non-singular contributions in the collinear limit. The shaded region indicates where we choose our profiles.
  • Figure 5: Convergence in the perturbative collinear limit of the energy correlator.
  • ...and 16 more figures