---
title: Celestial Energy–Energy Correlator (cEEC)
url: https://www.emergentmind.com/topics/celestial-energy-energy-correlator-ceec
type: topic
---

# Celestial Energy–Energy Correlator (cEEC)

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 [2202.04085], [2601.21852].

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 $z_i$, $\bar z_i$ [2601.21852].

## 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”:
\[
\langle \mathcal{E}(n_1)\,\mathcal{E}(n_2)\,\mathcal{E}(n_3) \rangle \sim \sum_{j=0}^{\infty} \int_{-\infty}^{\infty} d\nu\, C_j(\nu)\, G_{j,\nu}(z, \bar z)\, \mathbb{O}_{j,\nu}(n_2)
\]
where $G_{j,\nu}(z, \bar z)$ solves the two-dimensional Casimir equation:
\[
\left[2 z^2(1-z)\partial_z^2 + 2 \bar z^2(1-\bar z)\partial_{\bar z}^2\right] G_{j,\nu}(z, \bar z) = [\delta(\delta-2) + j^2] G_{j,\nu}(z, \bar z),\quad\delta=1+i\nu
\]
The closed-form expression is:
\[
G_{j,\nu}(z, \bar z) = \frac{1}{1+\delta_{j,0}} \Big[k_{(δ-j)/2}(z)\,k_{(δ+j)/2}(\bar z) + k_{(δ+j)/2}(z)\,k_{(δ-j)/2}(\bar z)\Big]
\]
with $k_h(x)=x^h\,{}_2F_1(h, h-1, 2h; x)$ [2202.04085], [2202.04090], [2505.16753].

In collider setups with a preferred axis, the celestial block incorporates a Mellin label $\gamma$:
\[
F_{δ,j,\gamma}(z,\bar z,w) = w^\gamma\,G_{δ,j}^{(\gamma)}(z,\bar z)
\]
where $w$ diagonalizes the boost action and encodes interference among collinear spin states [2505.16753].

## 3. Analyticity, Lorentzian Inversion, and OPE Data

The OPE data $C_{j}(\nu)$—the expansion coefficients—are extracted using the Lorentzian inversion formula, involving double discontinuities of the collinear three-point function:
\[
C_j(\nu) = \kappa_{δ,j} \int_0^1 dz\,d\bar z\,\mu(z, \bar z)\, G_{2-\delta, j}(z,\bar z)\, \mathrm{dDisc}\,F(z,\bar z)
\]
This inversion, convergent for real $j > j^*$, yields analytic dependence on the transverse spin $j$—a feature fundamental to the rapid convergence of block expansions in data modeling and the summation of singularities in the crossed channels [2202.04085], [2202.04090], [2505.16753].

For weakly-coupled QCD and $\mathcal{N}=4$ SYM, explicit formulas for the block coefficients $R_{\Delta,\ell}$ and anomalous dimensions encode both leading and higher-twist behaviors in the collinear and double lightcone limits. At strong coupling (large ’t Hooft coupling $\lambda$), the cEEC admits analytic celestial block expansions matching the Hofman–Maldacena structure [2202.04090].

## 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 ($n_i\cdot n_j\to1$), producing $\sim 1/\theta^2$ singularities associated with on-shell parton exchange. The cEEC reduces to universal forms, e.g.,
  \[
  \langle E(n_1) E(n_2) E(n_3) \rangle \approx (Q/4\pi)^3 x_L^{-2} F(z, \bar z)
  \]
  with $F(z,\bar z)$ capturing the angular dependence [2202.04085], [2505.16753].
- **Opposite-coplanar/back-to-back limit**: Separation approaches $\pi$; soft and collinear radiation dominate, leading to Sudakov double-logarithmic behavior $\sim\ln\delta r/\delta r^2$ [2505.16753], [2601.21852].
- **Regge/forward limit**: Large rapidity separation, governed by multi-Regge kinematics and BFKL dynamics, resulting in exponential growth $\sim \Delta Y e^{3\Delta Y}$ [2505.16753].
- **Celestial frame unification**: In the celestial context, all angular limits are encoded in the analytic structure of the single function cEEC$(z, \bar z)$, connecting OPE, Sudakov, and Regge regimes [2601.21852].

## 5. Relation to Splitting Kernels and Jet Substructure

At leading power in perturbative QCD, the cEEC is computed via integration of $1\to3$ splitting kernels $P_{a\to bcd}(\xi_1, \xi_2, \xi_3)$ against phase space and kinematic weights. Standard splitting-function procedures resum nested $1\to2$ splittings but do not diagonalize the celestial Lorentz symmetry. The celestial block approach decomposes all kinematic power corrections explicitly:
\[
\frac{1}{s_{123}^2} = \frac{1}{[(\xi_1 \xi_2) u + ...]^2}
\]
allowing power corrections to be organized systematically in the block expansion. For example, in $q \to q' \bar{q}' q$, higher-spin series at leading twist resum into
\[
\sum_{j \text{ even}} a_j G_{4, j}(z, \bar z)
\]
with coefficients $a_j$ extracted from polarized splitting functions [2202.04085], [2505.16753].

## 6. Phenomenological and Experimental Implications

The block decomposition separates kinematics ($G_{\delta, j}$) from dynamics ($C_{\delta, j}$), allowing efficient parameterization of higher-order and nonperturbative corrections. Numerical studies show rapid convergence: only a few low-twist blocks (e.g., $\delta \leq 8$) suffice to model $F(z, \bar z)$ with high accuracy (10–20%) away from extreme squeezed limits [2202.04085], [2505.16753]. This framework enables precision fits and extraction of QCD parameters (e.g., $\alpha_s$, 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 $\alpha_s$ via angular ratios in the collinear regime
- Nonperturbative phenomena at small angular separation
- BFKL physics in the Regge region
- Transverse-spin effects through nonzero $j$-blocks [2505.16753], [2601.21852]

## 7. Celestial Bootstrap and Closed-Form Solutions

In highly symmetric theories, notably $\mathcal{N}=8$ supergravity, the cEEC is uniquely determined by imposing $SL(2,\mathbb{C})$ 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:
\[
cEEC_{\rm SUGRA}^{(0,0)}(z, \bar z) = c_0 \frac{(1+z\bar z)^3}{z\,\bar z\,(z-\bar z)} \left[ \frac{\bar z}{1-\bar z} \ln^2 z + \frac{z-\bar z}{2} \frac{\ln z}{1-z} \frac{\ln \bar z}{1-\bar z} - 2 \operatorname{Li}_2(1-z) \right] + (z \leftrightarrow \bar z)
\]
This result is robust against infrared and collinear divergences and matches all known kinematic asymptotics [2601.21852].

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: [2202.04085], [2202.04090], [2505.16753], [2601.21852]

Source: https://www.emergentmind.com/topics/celestial-energy-energy-correlator-ceec