Spinning Energy Correlators

• Spinning energy correlators are quantum observables that measure energy flow while resolving spin and angular momentum through energy‐weighted, angle-resolved detectors.
• They are constructed via energy-flow operators and factorize into contributions from polarization components, playing a pivotal role in perturbative QCD and ultracold gas analyses.
• Their application enables precise discrimination of jet substructure and phase transitions by linking energy distribution with spin dynamics in both collider and quantum many-body experiments.

Spinning energy correlators are a class of quantum field theory observables that provide energy-weighted, angle-resolved, and spin-sensitive measurements of the final state produced in processes ranging from collider scattering to many-body quantum gases. They generalize the standard energy-energy correlator (EEC) framework to track not only the distribution of energy but also the angular momentum (spin) structure, thereby probing dynamics inaccessible to inclusive observables. Spinning energy correlators enable direct access to multi-dimensional event shapes, polarization effects, quantum coherence, and the flow of both energy and spin across disparate physical systems, from high-energy QCD to ultracold atomic ensembles.

1. General Definition and Theoretical Framework

A spinning energy correlator measures the flux of energy (and potentially other conserved charges, such as electric or baryon number) through multiple detectors, while resolving the full quantum numbers of both source and detected states. For a local operator $\mathcal O_J$ of spin $J$ inserted at the origin to create an initial state from the vacuum, and $N$ energy-flow detectors at directions $\{\hat n_i\}$ with spin indices $\{\lambda_i\}$, the $N$-point spinning correlator is

$$\mathcal E_{N}^{(J)}\bigl(\{\hat n_i, \lambda_i\}; h, h'\bigr) = \langle 0 | \epsilon_{h'}^{* \mu_1 \cdots \mu_J} \mathcal O_{\mu_1 \cdots \mu_J}(0) \prod_{i=1}^N \mathcal E(\hat n_i, \lambda_i) \epsilon_{h}^{\nu_1 \cdots \nu_J} \mathcal O_{\nu_1 \cdots \nu_J}(0) | 0 \rangle$$

where $h, h'$ label source helicities and $\epsilon_h$ is the associated polarization tensor. The angles $\{\hat n_i\}$ decompose into internal cross-ratios and global (Euler) orientations. The correlator factorizes into a sum over polarization components, with Wigner matrices $D^J$ encoding spin rotations and structure functions $H^J_{\Delta h,\Delta m}$ (the spinning correlators) capturing the intrinsic event shape and polarization content (Riembau et al., 18 Dec 2025).

This construction admits extensions to charge-flow (using operators $\mathcal Q_n$), mixed energy-charge correlators, and conserved tensor currents.

2. Positivity, Spin Decomposition, and Extremal Boundaries

Unitarity and energy positivity require that the energy correlator matrix $H^{ab}$ is positive-definite. Upon decomposing into spin-$J$ representations, each correlator is bounded within a convex region defined by the allowed spin projections:

• For $N=1$, the spinning correlator is described by a single parameter $a$ as $$H^{ab}=\frac{1}{3}H^\mathbb{1}(\delta^{ab}+a(3n^a n^b-\delta^{ab}))$$, constrained by the Hofman–Maldacena bound $-1/2\le a \le 1$.
• For $N=2$, the correlator projects onto spin-2 components $H^c(z), H^b(z)$, with normalized ratios $c(z), b(z)$ restricted by triangle inequalities: $1+2c(z)-b(z)\geq0$, $1-c(z)+2b(z)\geq0$, $1-c(z)-b(z)\geq0$.

The boundaries of these allowed regions are saturated by elementary states with definite spin projections: scalars, equal-helicity fermion/antifermion pairs, or gluon states with definite helicity (Riembau et al., 18 Dec 2025).

3. Quantum Field Theory Realizations and IR Safety

In perturbative QCD, spinning energy correlators are calculated as weighted sums over final-state partons, projecting onto desired angular and spin quantum numbers. At leading order in processes such as $e^+e^-\to q\bar q g$ or deep-inelastic scattering, contributions are expressible in terms of partonic splitting functions and are IR safe (infrared divergences cancel in ratios of spinning to inclusive correlators). Operators inserted in the final state act as calorimeters, registering both energy and polarization features.

The angular dependence can be expressed using cross-ratios of detector directions. The ratios $a^{(2,0)}(z)$ and $a^{(2,2)}(z)$, constructed from appropriately normalized spinning correlators, isolate hard QCD splitting dynamics and are insensitive to soft and collinear radiation. This structure endows spinning energy correlators with enhanced robustness to IR effects compared to traditional observables (Riembau et al., 18 Dec 2025).

4. Methodologies for Measurement and Analysis

Energy-Resolved Spin Correlators in Quantum Gases

In weakly interacting Fermi gases, an effective synthetic energy-space lattice structure is realized, where each single-particle energy eigenstate serves as a "site" and the two hyperfine levels provide a spin-1/2 degree of freedom (Huang et al., 2023, Pegahan et al., 2020). The Hamiltonian,

$$H = \hbar \sum_{i<j} g_{ij}(a) \vec{s}_i \cdot \vec{s}_j + \hbar \sum_i \Omega' E_i s_{z,i}$$

models an infinite-range anisotropic Heisenberg system with inhomogeneous Zeeman terms.

Transverse spin correlators are defined as

$$C^\perp_{ij}(t) = \left\langle \frac{1}{2}[\tilde s_{x,i}(t)\tilde s_{x,j}(t) + \tilde s_{y,i}(t)\tilde s_{y,j}(t)] \right\rangle$$

measured experimentally via real-space spin-density imaging and inverse Abel transforms to recover energy-resolved profiles. Statistical control over the measurement phase ensures unbiased access to quantum correlations. Data analysis normalizes correlators by population to yield $c_{ij}^\perp(t)$, quantifying localization (diagonal peaking) versus global (off-diagonal) correlation spreading as a function of the collective interaction parameter $\zeta=\bar{g}(a)/(\sqrt{2}\sigma_{\Omega_z})$ (Huang et al., 2023).

Spin-Dependent EECs and Jet Physics

In hadronic and leptonic collisions, spinning EECs are formulated by inserting angle- and azimuthally-resolved energy flow operators, enabling event-level resolution of spin effects. Factorization theorems in the TMD regime relate these correlators to convolutions of TMD PDFs and fragmentation functions, including Collins-type components sensitive to azimuthal modulations such as $\cos2\phi$. Operator definitions and Sudakov resummation procedures (via Collins–Soper evolution) give precision-predicted distributions (Kang et al., 2023).

5. Applications in QCD, Spin Structure, and Quantum Phase Transitions

Collider Phenomenology

• Jet Substructure and Flavor Discrimination: Ratios of two-point spinning correlators discriminate between quark and gluon jets, and provide high-precision extraction of $\alpha_s$.
• Electroweak Boson Polarization: One-point off-diagonal spinning correlators in the orientation angle $\Phi$ separate longitudinal from transverse $W/Z$ production.
• Proton Spin Decomposition: In DIS with polarized proton beams, single-spin energy correlators resolve the angular flow of energy correlated with proton helicity; full factorization across perturbative and nonperturbative scales is achievable via TMDs and nucleon energy correlators (NECs) (Gao et al., 22 Sep 2025).

Quantum Gases and Emergent Order

• Decoding Phase Transitions: Spinning energy correlators in WIFGs serve as microscopic order parameters, revealing the transition from localized, dephasing-dominated (paramagnetic) to globally synchronized (ferromagnetic) phases. Correlation localization, spreading, and temporal rearrangements track microscopic dynamics not visible in bulk magnetization.
• Information Scrambling: Energy-resolved out-of-time-order correlators (OTOCs) enable direct imaging of information spreading and quantum coherence across energy bands, illuminating sector-specific scrambling dynamics in many-body quantum systems (Pegahan et al., 2020).

6. Evolution Equations, Operator Matching, and Resummation

Spinning energy correlators admit systematic operator product expansions (OPE) and renormalization group resummation:

• TMD Region: They factorize into convolutions of hard coefficients, soft functions, TMD beam functions (encoding polarization structure, e.g., $g_1$ for helicity), and jet functions—each with associated evolution equations. Cusp anomalous dimensions control double-logarithmic scaling via Sudakov factors; RG analysis is performed at N$^3$LL/NNLL precision (Gao et al., 22 Sep 2025).
• Forward Region/NECs: Nucleon Energy Correlators evolve according to modified DGLAP equations with explicit “hadronic” logarithms and are matched onto standard PDFs via twist-2 OPE.
• Scheme Issues: Spin-dependent cases require careful treatment of $\gamma_5$ via prescriptions like Larin+ or HVBM for consistency with axial anomaly constraints.

The joint theoretical machinery ensures reliable phenomenology, interpolation between perturbative and nonperturbative dynamics, and systematic uncertainties.

7. Extensions, Physical Interpretation, and Experimental Outlook

• Angular Momentum Structure: Spinning correlators restore event-level angular-momentum information lost in inclusive observables, enabling the study of interference between helicity amplitudes and the extraction of event-by-event polarization signatures.
• Sum Rules: Generalized sum rules, expressing integrated multi-point correlators in terms of lower-point quantities, enforce consistency between bulk observables and their angular-momentum decompositions.
• Experiment: Precise calorimetric and tracking detectors (colliders, EIC) or energy- and spin-resolved imaging (ultracold atoms) provide the necessary data.
• Impact: Spinning energy correlators unlock novel probes of quantum information dynamics, many-body criticality, QCD color and spin structure, and new classes of IR-safe, precision event-shape observables.

Spinning energy correlators form a foundational set of observables at the interface of quantum field theory, many-body physics, and experiment—enabling direct characterization of both energy and spin flow with unprecedented granularity and theoretical control (Riembau et al., 18 Dec 2025, Huang et al., 2023, Kang et al., 2023, Gao et al., 22 Sep 2025, Pegahan et al., 2020).