Three-Point Energy Correlators

Updated 30 August 2025

Three-point energy correlators are observables that measure weighted triple energy flows to capture multiparticle correlations across distinct angular separations.
Their analytic computation employs phase space integration, conformal symmetry, and celestial block expansion to extract detailed quantum field dynamics.
They offer practical applications in jet substructure analysis, precision QCD studies, and probing both perturbative and nonperturbative phenomena in collider experiments.

Three-point energy correlators are a class of quantum field theoretical and collider observables that measure triple energy flow correlations across well-separated angular directions. Arising as three-point functions of the energy flux operator or as weighted cross sections over products of three detected energies, three-point correlators provide access to multi-particle correlations, operator product expansion data, and probe both perturbative and nonperturbative aspects of field theories with high angular and kinematic resolution. In conformal and gauge theories, their computation and analytic structure reveal deep connections to the geometry of correlation functions, the representation theory of the Lorentz/conformal group, jet substructure, and the holographic description of operator dynamics.

1. Formal Definitions and Physical Setting

Three-point energy correlators (often denoted EEEC or E3C) generalize the two-point energy correlator (EEC) by measuring weighted triple sums over all triplets of particles or energy flows in a final state. In collider QCD, for electron-positron annihilation or hadronic decays, the most general definition at leading order is

$\frac{1}{\sigma_\text{tot}}\frac{d^3\Sigma}{dx_1\,dx_2\,dx_3} = \sum_{i,j,k} \int \frac{E_i E_j E_k}{Q^3}\, d\sigma\, \delta\big(x_1 - \frac{\theta_{ij}^2}{4}\big)\, \delta\big(x_2 - \frac{\theta_{jk}^2}{4}\big)\, \delta\big(x_3 - \frac{\theta_{ki}^2}{4}\big),$

where $E_i$ are final-state energies, $\theta_{ij}$ are pairwise angles on the sphere of detectors, and $x_i$ parameterize the angular separations (Chen et al., 2019, Yang et al., 2022).

In conformal field theory, the corresponding object is the normalized correlation function of three energy flow operators: $\mathrm{EEEC}(\vec{n}_1, \vec{n}_2, \vec{n}_3) = \frac{\langle \mathcal{O}(p) | \mathcal{E}(\vec{n}_1) \mathcal{E}(\vec{n}_2) \mathcal{E}(\vec{n}_3) | \mathcal{O}(p) \rangle}{\langle \mathcal{O}(p)|\mathcal{O}(p)\rangle},$ with $\mathcal{E}(\vec{n})$ the averaged null energy condition (ANEC) operator. The geometrical variables are mapped onto conformally invariant cross ratios, often via “celestial sphere” coordinates (Chang et al., 2022).

2. Theoretical Construction and Symmetry Structure

The analytic calculation of three-point energy correlators employs several key ingredients:

Phase space decomposition: Integration over the final-state kinematics, often using energy fractions (denoted $z_i$ ), and explicit mapping to cross ratios on the celestial sphere (e.g., $|z_{ij}|^2 \sim \theta_{ij}^2/4$ ) (Chen et al., 2019, Yang et al., 2022).
Splitting functions and factorization: In the collinear (small-angle) limit, the dominant contribution factorizes into a hard process and universal splitting functions, such as the triple-collinear QCD splitting kernels. The leading power expansion is tied to the OPE of energy flow operators and operator “twist” (Chen et al., 2019, Barata et al., 17 Mar 2025).
Conformal symmetry and celestial block expansion: The Lorentz group (or conformal group in CFT) acts as SO( $d-1$ ,1) on the celestial sphere, enabling a harmonic analysis in terms of “celestial blocks,” i.e., special ( $d{-}2$ )-dimensional conformal blocks. The three-point correlator expands as

$\mathrm{EEEC} = \zeta_{12}^{(\Delta'_0-5)/2} \sum_{\delta,j} R_{\delta,j;\delta'_0} \, g_{\delta,j}^{(\mathcal{EEE}\mathcal{P}_{\delta'_0})}(z, \bar z),$

where $R_{\delta,j;\delta'_0}$ encode OPE data, and $g_{\delta,j}$ are conformal blocks (Chang et al., 2022).

Kinematic regimes: Several regimes are analyzed for phenomenological and theoretical significance:
- Collinear/triple-collinear limit: All detector directions are nearly aligned, yielding OPE-dictated scaling and providing sensitivity to splitting dynamics.
- Squeezed limit: Two detectors become coincident, corresponding to singular behavior driven by double light-ray OPEs.
- Coplanar limit: All detectors lie in a plane, enhancing sensitivity to multi-jet and soft emissions (Yang et al., 7 Feb 2024, Gao et al., 14 Nov 2024).
- Equilateral and general shape dependence: The dependence of the correlator on three independent angular variables exposes multi-particle correlations not accessible to lower-point observables.

3. Analytic Calculation in QCD and Conformal Theories

Analytic computation of the three-point energy correlator is achieved with advanced techniques (Chen et al., 2019, Yang et al., 2022, Yang et al., 7 Feb 2024):

Direct phase space integration: At leading order, the four-particle phase space is integrated using Mandelstam variables $s_{ij}$ , with delta functions encoding the measurement. Changes of variables rationalize complicated square roots in the phase space (e.g., via “celestial coordinates” or `` $z$ -parameterization'').
Function space: After integration, the results are expressed in a basis of classical polylogarithms (up to weight two for three-point correlators), with their symbols built from a finite “alphabet” of rational letters (e.g., a 16-letter alphabet in $\mathcal{N}=4$ SYM (Yan et al., 2022)).
Color structure and transcendentality: In QCD, different color structures yield more intricate rational coefficients than in conformal theories such as $\mathcal{N}=4$ SYM, where cancellations lead to “simpler” transcendental structures (Chen et al., 2019).
Differential equation and IBP methods: For generic angles or higher-point correlators, the canonical basis of master integrals is systematically obtained using integration-by-parts with syzygy (algebraic geometry) constraints, guaranteeing IR finiteness and block-triangular (nilpotent) canonical differential equations (Ma et al., 1 Jun 2025).

4. Physical Implications and Applications

Three-point energy correlators have multiple uses across high-energy and theoretical physics:

Jet substructure and collider physics: E3C provides the first analytic three-prong jet observable, carrying nontrivial information about multi-particle correlations and event shapes at LHC or lepton colliders (Chen et al., 2019, Yang et al., 2022). In hadronic Higgs decays, the EEEC can be used to probe gluon jet structure and to assist in extractions of $\alpha_s$ and studies of QCD radiation patterns (Yang et al., 7 Feb 2024).
Factorization and resummation: The presence of enhanced logarithms in collinear and coplanar limits (e.g., $\log R_L$ for the largest angle in the “projected” correlator) requires resummation. Recent work achieves resummation up to N $^3$ LL accuracy for EEEC in the coplanar/three-jet limit using TMD factorization (Gao et al., 14 Nov 2024).
Jet tomography in heavy-ion collisions: The full shape information of EEEC can disentangle medium-induced modifications—perturbative splitting modifications and collective (hydrodynamic) QGP response. The “equilateral” region of the correlator is found to be populated predominantly by particles from the jet-induced wake in the medium, providing a unique imaging tool for diagnosing QGP response (Bossi et al., 18 Jul 2024, Barata et al., 17 Mar 2025).
Conformal bootstrap and holography: The block expansion and its coefficients, when matched to operator data, encode OPE and anomalous dimension information, and at strong coupling, E3C calculations in AdS/CFT capture minimal surfaces associated to string solutions (Georgiou, 2010, Chang et al., 2022).

5. Mathematical Structures, Universality, and Extensions

The mathematical structure of three-point energy correlators reveals nontrivial features and universal behavior:

Transcendental function space: Across $\mathcal{N}=4$ SYM, QCD, and decay/inclusive processes, the same function space of logarithms and dilogarithms appears. At higher points (four-point and beyond), elliptic and hyperelliptic integrals naturally enter due to the geometric complexity of phase space (Ma et al., 1 Jun 2025).
Celestial block universality: The leading term in the celestial block expansion in the collinear regime maps to a traditional four-point block in ( $d{-}2$ ) dimensions, emphasizing the geometric and conformal symmetry constraints on the correlator (Chang et al., 2022).
Parity structures: In three dimensions, the three-point function tensor structures can include unique parity-odd components, providing diagnostics for parity-violating (e.g., Chern–Simons matter) theories (Giombi et al., 2011).

6. Experimental Realization and Challenges

Experimental paper of the E3C is emerging, with detailed analyses by ALICE and ongoing efforts at ATLAS and CMS (Liang-Gilman, 28 Jun 2025):

Measurement procedure: The E3C is extracted from reconstructed jets by summing over all track triplets, weighting by their energies, and binning by the largest pairwise angular separation ( $R_L$ ).
Strong coupling determination: The ratio E3C/EEC provides sensitivity to the strong coupling via characteristic scaling $\alpha_s \log R_L$ at wide angles, as both correlators are sensitive to different twist-two operators (spin-3 for EEC, spin-4 for E3C).
Systematic uncertainties: Extraction of $\alpha_s$ and deconvolution of perturbative and nonperturbative effects require high control over detector, background, and theoretical uncertainties, especially at the transition between perturbative and nonperturbative regimes.
Medium effects and multiplicity dependence: In heavy-ion and proton-nucleus systems, distinguishing genuine jet signal from medium background is a significant challenge.

7. Outlook and Future Directions

Three-point energy correlators serve as a bridge between theoretical field-theoretic constructs and experimental observables that probe multi-scale, multi-particle correlations. Their analytic calculability, rich symmetry structure, sensitivity to both perturbative and collective dynamics, and their role in mapping jet–medium interactions point toward several ongoing lines of research:

Full analytic higher-order and higher-point correlator calculations (e.g., four-point, see (Ma et al., 1 Jun 2025)).
Nonperturbative studies leveraging celestial block expansions and conformal bootstrap methods.
Experimental tomography of the QGP using high-statistics, shape-resolved E3C measurements.
Precision extractions of QCD parameters (e.g., $\alpha_s$ ) from multi-point correlator ratios.
Integration of three-point observables into the broader program of event shape and substructure analysis at future lepton and hadron colliders.

Three-point energy correlators, by leveraging the intersection of conformal symmetry, perturbative QCD, analytic methods, and experimental measurement, have become central to the next generation of fundamental and phenomenological studies of energy flow in quantum field theory.