Celestial Energy–Energy Correlator (cEEC)

• Celestial Energy–Energy Correlator (cEEC) is a novel observable that correlates energy flux on the celestial sphere using energy-flow operators and boost eigenstates.
• The framework decomposes energy deposition into celestial blocks, enabling systematic analysis of collinear, soft, and Regge dynamics in collider physics.
• Its formulation integrates conformal field theory techniques and operator product expansions to connect insights in QCD, gravity, and supergravity.

The Celestial Energy–Energy Correlator (cEEC) is a fundamental observable in collider physics, defined as the correlation function of energy-flow (Average Null Energy, ANE) operators measured on the celestial sphere via boost eigenstates. The cEEC provides an infrared- and collinear-safe partial-wave decomposition of energy flux, manifestly organizing energy deposition patterns in terms of celestial conformal symmetry. This framework enables systematic analysis of jet substructure, soft-collinear and Regge dynamics, and spin effects, with applications from QCD to gravity and supergravity. The mathematical structure of the cEEC connects higher-point event shapes with complex analytic techniques rooted in conformal field theory (CFT), celestial blocks, and operator product expansions.

1. Definition and Operator Formalism

The cEEC is constructed from ensemble averages of multiple energy-flow operators: $$\langle \mathcal{E}(\vec n_1)\,\mathcal{E}(\vec n_2)\,\cdots\mathcal{E}(\vec n_k) \rangle$$ where each $\mathcal{E}(\vec n)$ is given by the null limit of the stress tensor: $$\mathcal{E}(\vec n) = \lim_{r\to\infty} \int_0^{\infty} dt\,r^2 n^i T_{0i}(t, r \vec n)$$ for $n^i$ a unit vector on the celestial sphere. In practical terms, the cEEC describes the (weighted) probability of energy depositions in specified calorimeter cells corresponding to directions $\vec n_i$ for states produced at the interaction point (Chen et al., 2022, Ruan et al., 29 Jan 2026).

In the context of hadron colliders, beam eigenstates are prepared using “boost-eigenstate” or “beam operator” projections: $$\mathbb{P}^{(J)}(n) = \int_0^\infty \frac{dP}{P} P^{-J} \frac{|P n\rangle\langle P n|}{\langle P n| P n\rangle}$$ rendering the cEEC as a four-point function in a fictitious $\mathrm{CFT}_2$ on the celestial sphere: $$\mathrm{cEEC}^{(J_1,J_2)}(z_i,\bar z_i) = \left\langle \mathcal{E}(z_1,\bar z_1) \mathcal{E}(z_2,\bar z_2) \mathbb{P}^{(J_1)}(z_3,\bar z_3) \mathbb{P}^{(J_2)}(z_4,\bar z_4) \right\rangle$$ with celestial cross-ratios $u$ and $v$ constructed from the stereographic coordinates $\mathcal{E}(\vec n)$0, $\mathcal{E}(\vec n)$1 (Ruan et al., 29 Jan 2026).

2. Symmetry Structure and Celestial Block Decomposition

Lorentz invariance on null directions induces a conformal group action on the celestial sphere, enabling the partial wave decomposition of cEEC observables into “celestial blocks”: $\mathcal{E}(\vec n)$2 where $\mathcal{E}(\vec n)$3 solves the two-dimensional Casimir equation: $\mathcal{E}(\vec n)$4 The closed-form expression is: $\mathcal{E}(\vec n)$5 with $\mathcal{E}(\vec n)$6 (Chen et al., 2022, Chang et al., 2022, Chen et al., 22 May 2025).

In collider setups with a preferred axis, the celestial block incorporates a Mellin label $\mathcal{E}(\vec n)$7: $\mathcal{E}(\vec n)$8 where $\mathcal{E}(\vec n)$9 diagonalizes the boost action and encodes interference among collinear spin states (Chen et al., 22 May 2025).

3. Analyticity, Lorentzian Inversion, and OPE Data

The OPE data $$\mathcal{E}(\vec n) = \lim_{r\to\infty} \int_0^{\infty} dt\,r^2 n^i T_{0i}(t, r \vec n)$$0—the expansion coefficients—are extracted using the Lorentzian inversion formula, involving double discontinuities of the collinear three-point function: $$\mathcal{E}(\vec n) = \lim_{r\to\infty} \int_0^{\infty} dt\,r^2 n^i T_{0i}(t, r \vec n)$$1 This inversion, convergent for real $$\mathcal{E}(\vec n) = \lim_{r\to\infty} \int_0^{\infty} dt\,r^2 n^i T_{0i}(t, r \vec n)$$2, yields analytic dependence on the transverse spin $$\mathcal{E}(\vec n) = \lim_{r\to\infty} \int_0^{\infty} dt\,r^2 n^i T_{0i}(t, r \vec n)$$3—a feature fundamental to the rapid convergence of block expansions in data modeling and the summation of singularities in the crossed channels (Chen et al., 2022, Chang et al., 2022, Chen et al., 22 May 2025).

For weakly-coupled QCD and $$\mathcal{E}(\vec n) = \lim_{r\to\infty} \int_0^{\infty} dt\,r^2 n^i T_{0i}(t, r \vec n)$$4 SYM, explicit formulas for the block coefficients $$\mathcal{E}(\vec n) = \lim_{r\to\infty} \int_0^{\infty} dt\,r^2 n^i T_{0i}(t, r \vec n)$$5 and anomalous dimensions encode both leading and higher-twist behaviors in the collinear and double lightcone limits. At strong coupling (large ’t Hooft coupling $$\mathcal{E}(\vec n) = \lim_{r\to\infty} \int_0^{\infty} dt\,r^2 n^i T_{0i}(t, r \vec n)$$6), the cEEC admits analytic celestial block expansions matching the Hofman–Maldacena structure (Chang et al., 2022).

4. Kinematic Regimes: Collinear, Coplanar, Back-to-Back, and Regge

The cEEC interpolates smoothly between key kinematic regimes:

• Collinear limit: All detectors confined to a small angular patch ($$\mathcal{E}(\vec n) = \lim_{r\to\infty} \int_0^{\infty} dt\,r^2 n^i T_{0i}(t, r \vec n)$$7), producing $$\mathcal{E}(\vec n) = \lim_{r\to\infty} \int_0^{\infty} dt\,r^2 n^i T_{0i}(t, r \vec n)$$8 singularities associated with on-shell parton exchange. The cEEC reduces to universal forms, e.g.,

$$\mathcal{E}(\vec n) = \lim_{r\to\infty} \int_0^{\infty} dt\,r^2 n^i T_{0i}(t, r \vec n)$$9

with $n^i$0 capturing the angular dependence (Chen et al., 2022, Chen et al., 22 May 2025).

• Opposite-coplanar/back-to-back limit: Separation approaches $n^i$1; soft and collinear radiation dominate, leading to Sudakov double-logarithmic behavior $n^i$2 (Chen et al., 22 May 2025, Ruan et al., 29 Jan 2026).
• Regge/forward limit: Large rapidity separation, governed by multi-Regge kinematics and BFKL dynamics, resulting in exponential growth $n^i$3 (Chen et al., 22 May 2025).
• Celestial frame unification: In the celestial context, all angular limits are encoded in the analytic structure of the single function cEEC$n^i$4, connecting OPE, Sudakov, and Regge regimes (Ruan et al., 29 Jan 2026).

5. Relation to Splitting Kernels and Jet Substructure

At leading power in perturbative QCD, the cEEC is computed via integration of $n^i$5 splitting kernels $n^i$6 against phase space and kinematic weights. Standard splitting-function procedures resum nested $n^i$7 splittings but do not diagonalize the celestial Lorentz symmetry. The celestial block approach decomposes all kinematic power corrections explicitly: $n^i$8 allowing power corrections to be organized systematically in the block expansion. For example, in $n^i$9, higher-spin series at leading twist resum into

$\vec n_i$0

with coefficients $\vec n_i$1 extracted from polarized splitting functions (Chen et al., 2022, Chen et al., 22 May 2025).

6. Phenomenological and Experimental Implications

The block decomposition separates kinematics ($\vec n_i$2) from dynamics ($\vec n_i$3), allowing efficient parameterization of higher-order and nonperturbative corrections. Numerical studies show rapid convergence: only a few low-twist blocks (e.g., $\vec n_i$4) suffice to model $\vec n_i$5 with high accuracy (10–20%) away from extreme squeezed limits (Chen et al., 2022, Chen et al., 22 May 2025). This framework enables precision fits and extraction of QCD parameters (e.g., $\vec n_i$6, fragmentation functions, color-flow observables), the modeling of hadronization effects, and the investigation of spin interference.

At hadron colliders, convolution with parton distribution functions (PDFs) accommodates detector binning and initial-state complexities, “smearing out” rapidity divergences while preserving physical singularity patterns. The cEEC provides unique sensitivity to:

• Running of $\vec n_i$7 via angular ratios in the collinear regime
• Nonperturbative phenomena at small angular separation
• BFKL physics in the Regge region
• Transverse-spin effects through nonzero $\vec n_i$8-blocks (Chen et al., 22 May 2025, Ruan et al., 29 Jan 2026)

7. Celestial Bootstrap and Closed-Form Solutions

In highly symmetric theories, notably $\vec n_i$9 supergravity, the cEEC is uniquely determined by imposing $$\mathbb{P}^{(J)}(n) = \int_0^\infty \frac{dP}{P} P^{-J} \frac{|P n\rangle\langle P n|}{\langle P n| P n\rangle}$$0 invariance, crossing symmetry, boundary conditions, and the transcendental function alphabet. The bootstrap procedure reconstructs compact closed-form expressions for the cEEC without explicit phase-space integrals: $$\mathbb{P}^{(J)}(n) = \int_0^\infty \frac{dP}{P} P^{-J} \frac{|P n\rangle\langle P n|}{\langle P n| P n\rangle}$$1 This result is robust against infrared and collinear divergences and matches all known kinematic asymptotics (Ruan et al., 29 Jan 2026).

In future, the celestial conformal and block expansion methodology suggests broad utility for higher-point energy correlators, global event shapes, and gravitational observables, contingent on correct identification of celestial coordinates and constraint imposition.

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References: (Chen et al., 2022, Chang et al., 2022, Chen et al., 22 May 2025, Ruan et al., 29 Jan 2026)