Spin-Dependent Energy Correlator

• Spin-dependent energy correlator is an observable that retains spin information in energy-flow measurements, revealing angular momentum details in collision processes.
• It is applied in collider QCD and polarized DIS to investigate azimuthal asymmetries and factorization regimes, linking detector geometry with intrinsic spin dynamics.
• The methodology extends to weakly interacting Fermi gases, using energy-resolved spin correlations to probe microscopic spin distributions and dephasing phenomena.

A spin-dependent energy correlator is an energy-flow observable in which the measured energy distribution retains explicit spin information rather than averaging it away. In collider QCD, the term covers several closely related constructions: a deep-inelastic-scattering energy-correlation cross section with a longitudinally polarized proton beam, nucleon energy-energy correlators in proton-proton collisions whose azimuthal structure is controlled by a spinning gluon distribution, and fully differential spinning energy correlators that retain the angular momentum of both the source and the detector configuration (Gao et al., 22 Sep 2025, Guo et al., 2024, Riembau et al., 18 Dec 2025). In a distinct but conceptually related usage, energy-resolved spin correlators are measured between energy bins of a weakly interacting trapped Fermi gas, where energy becomes the indexing variable for spin correlations rather than an angular detector weight (Huang et al., 2023).

1. Definitions and observable content

The common operator-theoretic starting point is the light-ray energy flow operator

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

or, equivalently for asymptotic states,

$$\mathcal E_n |\alpha\rangle = \sum_{i\in\alpha} E_i\,\delta^{(2)}(\Omega_i-\Omega_n)\,|\alpha\rangle.$$

Ordinary $N$-point energy correlators insert $N$ such detectors and sum inclusively over final states; spin-dependent variants preserve source polarization, detector-orientation dependence, or both, thereby restoring angular momentum information that inclusive correlators erase (Chen et al., 2022, Riembau et al., 18 Dec 2025).

In polarized DIS, the observable is defined by inserting an energy-weight operator into the hadronic tensor,

$$W_{\text{EC}}^{\mu\nu}(q,P_N,\theta) = \int d^4 x\, e^{iq\cdot x} \langle P_N,S_\parallel | J^{\dagger \mu}(x)\, \hat{\mathcal E}(\theta)\, J^\nu(0) |P_N,S_\parallel\rangle ,$$

with

$$\hat{\mathcal E}(\theta)\,|X\rangle = \sum_{h\in X}\frac{E_h}{E_N}\, \Theta(\theta-\theta_h)\,|X\rangle .$$

This measures the cumulative energy deposited inside an angular cone bounded by $\theta$, normalized to the target energy $E_N$, and makes the helicity dependence visible through the polarized part of the hadronic tensor (Gao et al., 22 Sep 2025).

In proton-proton collisions, the relevant object is a nucleon energy-energy correlator framework in which the gluon correlator has two independent structures because the gluon has spin $1$: an unpolarized piece $f_{g,\mathrm{EEC}}$ and an off-diagonal spinning-gluon correlator $$\mathcal E_n |\alpha\rangle = \sum_{i\in\alpha} E_i\,\delta^{(2)}(\Omega_i-\Omega_n)\,|\alpha\rangle.$$0. The latter is the component responsible for a nontrivial $$\mathcal E_n |\alpha\rangle = \sum_{i\in\alpha} E_i\,\delta^{(2)}(\Omega_i-\Omega_n)\,|\alpha\rangle.$$1 azimuthal asymmetry in the forward-backward energy-flow correlation (Guo et al., 2024).

2. Angular-momentum decomposition and transverse spin

A central development is the reorganization of energy correlators into irreducible angular-momentum channels. For a spinful source, the fully differential $$\mathcal E_n |\alpha\rangle = \sum_{i\in\alpha} E_i\,\delta^{(2)}(\Omega_i-\Omega_n)\,|\alpha\rangle.$$2-point correlator depends on detector directions $$\mathcal E_n |\alpha\rangle = \sum_{i\in\alpha} E_i\,\delta^{(2)}(\Omega_i-\Omega_n)\,|\alpha\rangle.$$3, but these coordinates can be split into $$\mathcal E_n |\alpha\rangle = \sum_{i\in\alpha} E_i\,\delta^{(2)}(\Omega_i-\Omega_n)\,|\alpha\rangle.$$4 internal angles

$$\mathcal E_n |\alpha\rangle = \sum_{i\in\alpha} E_i\,\delta^{(2)}(\Omega_i-\Omega_n)\,|\alpha\rangle.$$5

and three Euler angles $$\mathcal E_n |\alpha\rangle = \sum_{i\in\alpha} E_i\,\delta^{(2)}(\Omega_i-\Omega_n)\,|\alpha\rangle.$$6 describing the rigid-body orientation of the detector configuration. The correlator then admits a universal decomposition of the schematic form

$$\mathcal E_n |\alpha\rangle = \sum_{i\in\alpha} E_i\,\delta^{(2)}(\Omega_i-\Omega_n)\,|\alpha\rangle.$$7

where the Euler-angle dependence is fixed by Wigner $$\mathcal E_n |\alpha\rangle = \sum_{i\in\alpha} E_i\,\delta^{(2)}(\Omega_i-\Omega_n)\,|\alpha\rangle.$$8-matrices and the reduced functions $$\mathcal E_n |\alpha\rangle = \sum_{i\in\alpha} E_i\,\delta^{(2)}(\Omega_i-\Omega_n)\,|\alpha\rangle.$$9 are the spinning energy correlators proper (Riembau et al., 18 Dec 2025).

In the collinear limit of the three-point energy correlator, the same structure appears through a light-ray operator product expansion organized not only by twist or celestial dimension but also by transverse spin $N$0, the quantum number under rotations around the detector axis. The shape function is expanded as

$N$1

with celestial blocks $N$2 fixed by Lorentz symmetry and OPE coefficients $N$3 extracted by a Lorentzian inversion formula. A major result is that the OPE data is analytic in transverse spin $N$4, so the transverse-spin tower is not arbitrary discrete data but admits smooth analytic continuation (Chen et al., 2022).

These two formulations are complementary. The spinning-correlator language keeps the full detector orientation and source polarization manifest, while the celestial-block language isolates the transverse-spin content of the collinear limit. A plausible implication is that “spin dependence” in energy correlators is not restricted to polarized beams: it also includes the angular-momentum channels carried by detector geometry itself (Chen et al., 2022, Riembau et al., 18 Dec 2025).

3. Factorization regimes in polarized deep-inelastic scattering

For longitudinally polarized proton beams, the energy-correlation hadronic tensor decomposes as

$N$5

and the corresponding cross section is

$N$6

with

$N$7

The longitudinal-spin information is therefore isolated in the $N$8 term (Gao et al., 22 Sep 2025).

The same observable interpolates between two distinct angular limits. In the nearly back-to-back limit $N$9, corresponding to the current fragmentation region, $N$0 and the cross section is described by SCET/TMD factorization with a hard coefficient $N$1, soft function $N$2, TMD beam function $N$3, helicity-dependent beam function $N$4, and TMD fragmentation function $N$5. In the forward limit $N$6, corresponding to the target fragmentation region, the relevant nonperturbative objects are fracture functions or nucleon energy correlators (NECs), which encode the joint distribution of the struck parton and the forward spectator-fragmentation energy flow (Gao et al., 22 Sep 2025).

The formalism treats these limits with different but matched evolution structures. In the back-to-back region, the TMD functions obey Collins–Soper and $N$7-RG equations, and the paper gives an RG-improved prediction. In the forward region, NECs obey a modified DGLAP-type evolution and can be matched onto PDFs in the perturbative regime. The resulting quantitative description is given at joint $N$8LL precision in the current fragmentation region and NNLL precision in the target fragmentation region. A common misconception is that the observable has a single universal factorization form; the literature instead emphasizes a regime-dependent description, with TMD factorization near $N$9 and NEC/fracture-function-type factorization near $$W_{\text{EC}}^{\mu\nu}(q,P_N,\theta) = \int d^4 x\, e^{iq\cdot x} \langle P_N,S_\parallel | J^{\dagger \mu}(x)\, \hat{\mathcal E}(\theta)\, J^\nu(0) |P_N,S_\parallel\rangle ,$$0 (Gao et al., 22 Sep 2025).

4. Long-range azimuthal asymmetries in proton-proton collisions

A different spin-dependent energy correlator appears in proton-proton collisions as a forward-backward azimuthal correlation of energy flow associated with multi-jet production in the central rapidity region. The observable is explicitly long-range because the forward and backward energy measurements are widely separated in rapidity from the central hard process. In schematic form,

$$W_{\text{EC}}^{\mu\nu}(q,P_N,\theta) = \int d^4 x\, e^{iq\cdot x} \langle P_N,S_\parallel | J^{\dagger \mu}(x)\, \hat{\mathcal E}(\theta)\, J^\nu(0) |P_N,S_\parallel\rangle ,$$1

with

$$W_{\text{EC}}^{\mu\nu}(q,P_N,\theta) = \int d^4 x\, e^{iq\cdot x} \langle P_N,S_\parallel | J^{\dagger \mu}(x)\, \hat{\mathcal E}(\theta)\, J^\nu(0) |P_N,S_\parallel\rangle ,$$2

The $$W_{\text{EC}}^{\mu\nu}(q,P_N,\theta) = \int d^4 x\, e^{iq\cdot x} \langle P_N,S_\parallel | J^{\dagger \mu}(x)\, \hat{\mathcal E}(\theta)\, J^\nu(0) |P_N,S_\parallel\rangle ,$$3 term therefore requires both protons to supply the spinning-gluon correlator $$W_{\text{EC}}^{\mu\nu}(q,P_N,\theta) = \int d^4 x\, e^{iq\cdot x} \langle P_N,S_\parallel | J^{\dagger \mu}(x)\, \hat{\mathcal E}(\theta)\, J^\nu(0) |P_N,S_\parallel\rangle ,$$4 (Guo et al., 2024).

The asymmetry is traced to double helicity-flip interference. For dijet production, the relevant schematic structure is

$$W_{\text{EC}}^{\mu\nu}(q,P_N,\theta) = \int d^4 x\, e^{iq\cdot x} \langle P_N,S_\parallel | J^{\dagger \mu}(x)\, \hat{\mathcal E}(\theta)\, J^\nu(0) |P_N,S_\parallel\rangle ,$$5

so the hard scattering acts as a polarizer that determines whether the off-diagonal gluon correlator can survive. The paper emphasizes a distinctive power-counting rule: $$W_{\text{EC}}^{\mu\nu}(q,P_N,\theta) = \int d^4 x\, e^{iq\cdot x} \langle P_N,S_\parallel | J^{\dagger \mu}(x)\, \hat{\mathcal E}(\theta)\, J^\nu(0) |P_N,S_\parallel\rangle ,$$6 For dijets, tree-level Parke–Taylor helicity rules eliminate the required interference, and one-loop color structures still do not generate $$W_{\text{EC}}^{\mu\nu}(q,P_N,\theta) = \int d^4 x\, e^{iq\cdot x} \langle P_N,S_\parallel | J^{\dagger \mu}(x)\, \hat{\mathcal E}(\theta)\, J^\nu(0) |P_N,S_\parallel\rangle ,$$7 at $$W_{\text{EC}}^{\mu\nu}(q,P_N,\theta) = \int d^4 x\, e^{iq\cdot x} \langle P_N,S_\parallel | J^{\dagger \mu}(x)\, \hat{\mathcal E}(\theta)\, J^\nu(0) |P_N,S_\parallel\rangle ,$$8; the asymmetry first appears only at NNLO. For three jets it starts at $$W_{\text{EC}}^{\mu\nu}(q,P_N,\theta) = \int d^4 x\, e^{iq\cdot x} \langle P_N,S_\parallel | J^{\dagger \mu}(x)\, \hat{\mathcal E}(\theta)\, J^\nu(0) |P_N,S_\parallel\rangle ,$$9, and for four or more jets it is already present at leading order (Guo et al., 2024).

The dijet case also shows that the asymmetry is only meaningful after the full infrared-safe combination is assembled. Virtual corrections, soft radiation, and collinear radiation contribute with infrared poles that cancel in the final result, leaving a finite $$\hat{\mathcal E}(\theta)\,|X\rangle = \sum_{h\in X}\frac{E_h}{E_N}\, \Theta(\theta-\theta_h)\,|X\rangle .$$0. The same work also identifies a different harmonic,

$$\hat{\mathcal E}(\theta)\,|X\rangle = \sum_{h\in X}\frac{E_h}{E_N}\, \Theta(\theta-\theta_h)\,|X\rangle .$$1

which can be nonzero already in dijet production but vanishes after averaging over the dijet-plane angle $$\hat{\mathcal E}(\theta)\,|X\rangle = \sum_{h\in X}\frac{E_h}{E_N}\, \Theta(\theta-\theta_h)\,|X\rangle .$$2. With respect to the near-side ridge in high-multiplicity $$\hat{\mathcal E}(\theta)\,|X\rangle = \sum_{h\in X}\frac{E_h}{E_N}\, \Theta(\theta-\theta_h)\,|X\rangle .$$3 events, the stated claim is limited: the framework offers a new QCD mechanism to understand the ridge, not a complete solution of the puzzle (Guo et al., 2024).

5. Positivity, bounds, and perturbative structure

Spinning energy correlators obey nontrivial positivity constraints because the underlying hadronic tensor is positive semidefinite. For a spinful source,

$$\hat{\mathcal E}(\theta)\,|X\rangle = \sum_{h\in X}\frac{E_h}{E_N}\, \Theta(\theta-\theta_h)\,|X\rangle .$$4

and the detector weights $$\hat{\mathcal E}(\theta)\,|X\rangle = \sum_{h\in X}\frac{E_h}{E_N}\, \Theta(\theta-\theta_h)\,|X\rangle .$$5 are nonnegative. Unitarity and energy positivity therefore confine the reduced spinning correlators to bounded regions, with boundary points realized by rank-deficient hadronic tensors or, equivalently, pure spin states (Riembau et al., 18 Dec 2025).

For the one-point correlator of a spin-1 current, this reproduces the Hofman–Maldacena bound

$$\hat{\mathcal E}(\theta)\,|X\rangle = \sum_{h\in X}\frac{E_h}{E_N}\, \Theta(\theta-\theta_h)\,|X\rangle .$$6

For the two-point spinning energy correlator of a vector current, the independent reduced functions $$\hat{\mathcal E}(\theta)\,|X\rangle = \sum_{h\in X}\frac{E_h}{E_N}\, \Theta(\theta-\theta_h)\,|X\rangle .$$7 and $$\hat{\mathcal E}(\theta)\,|X\rangle = \sum_{h\in X}\frac{E_h}{E_N}\, \Theta(\theta-\theta_h)\,|X\rangle .$$8 are naturally interpreted as “collinear” and “back-to-back” structures. Defining

$$\hat{\mathcal E}(\theta)\,|X\rangle = \sum_{h\in X}\frac{E_h}{E_N}\, \Theta(\theta-\theta_h)\,|X\rangle .$$9

positivity implies

$\theta$0

which define a triangle in the $\theta$1 plane with vertices $\theta$2, $\theta$3, and $\theta$4 (Riembau et al., 18 Dec 2025).

The same work gives the first explicit perturbative QCD calculation of the spinning two-point energy correlator. At Born level,

$\theta$5

so the collinear endpoint probes $\theta$6 and the back-to-back endpoint probes $\theta$7. At NLO, both the inclusive correlator and the spinning correlators acquire explicit analytic corrections, and the normalized ratios

$\theta$8

are singled out because universal soft and collinear enhancement largely cancels. The paper describes these ratios as much less sensitive to infrared dynamics than the raw correlators and therefore more direct probes of the hard subprocess (Riembau et al., 18 Dec 2025).

6. Related energy-space usage in weakly interacting Fermi gases

Outside collider energy flow, a conceptually related observable appears in weakly interacting trapped Fermi gases as an energy-resolved spin correlator between transverse spin components associated with different energy bins. After mapping the collisionless or weakly interacting gas to a synthetic spin lattice in energy space, the fundamental measured correlator is

$\theta$9

Because the measured transverse signal mixes Bloch-frame $E_N$0 and $E_N$1 components through an uncontrolled RF detuning phase $E_N$2, the experiment selects shot pairs for which $E_N$3, yielding

$E_N$4

The normalized version,

$E_N$5

lies in the interval $E_N$6 (Huang et al., 2023).

This correlation matrix is directly related to the macroscopic transverse magnetization,

$E_N$7

but contains more microscopic information than $E_N$8 alone. The measured “flow of correlations” in energy space distinguishes localized correlations, associated with demagnetization and dephasing, from extended correlations, associated with spin-locking and persistent magnetization. The scalar measure $E_N$9, defined from the average magnitude of the discrete gradient of a maximally correlated row, quantifies this spread and drops sharply near the interaction-driven transition (Huang et al., 2023).

This suggests a broader methodological meaning for spin-dependent energy correlators: they are observables designed to resolve how spin information is distributed across an energy variable, whether that variable is an angle on the celestial sphere, a forward detector cone in DIS, a rapidity-separated energy flow in hadronic collisions, or an energy bin in a trapped quantum gas. Across these settings, the common theme is that spin-resolved correlations reveal structures that are hidden in fully inclusive or purely macroscopic measurements (Gao et al., 22 Sep 2025, Guo et al., 2024, Huang et al., 2023).