Spinning Energy-Charge Correlators

• Spinning energy-charge correlators are quantum observables defined by energy and charge flux operators at asymptotic directions, capturing detailed angular momentum and spin transfers.
• They decompose into angular momentum channels using Wigner D-matrices and invariant angles, providing infrared-safe probes for hard scattering dynamics in gauge theories.
• Applications span collider phenomenology, deep inelastic scattering, and holographic dualities, linking experimental measurements to theoretical frameworks like SCET and conformal symmetry.

Spinning energy-charge correlators are quantum field–theoretic observables that simultaneously track the angular pattern of both energy flow and conserved charge transfer through idealized detectors, retaining the full spin structure from both the microscopic source and the geometry of the measurement. These correlators are defined by inserting a combination of energy flux operators $\mathcal{E}(n)$ and charge flux operators $\mathcal{Q}(n)$ at asymptotic directions $n$ on the celestial sphere, with the states created by local operators of definite spin, and are normalized by the total event rate. Their nontrivial angular structure resolves detailed angular momentum transfer between the initial state and the final measured configuration, in sharp contrast with inclusive or unpolarized observables. These objects generalize traditional event shapes and provide new, infrared-insensitive probes of QCD and gauge theory dynamics, applicable to collider phenomenology, conformal field theory, and holography (Riembau et al., 18 Dec 2025).

1. Operator Definition and Angular Decomposition

The general $N$-point spinning energy-charge correlator is defined as the normalized, polarization-resolved expectation value of a sequence of $N$ detectors placed at asymptotic directions $n_i$: $$\langle \mathcal{E}(n_1)\cdots\mathcal{E}(n_{N_E})\,\mathcal{Q}(n_{N_E+1})\cdots\mathcal{Q}(n_N)\rangle_{h'h} = \frac{1}{\mathcal{N}}\,\epsilon_{h'}^*\cdot\langle O(x)\,\mathcal{E}(n_1)\cdots\mathcal{Q}(n_N)\,O(0)\rangle \cdot\epsilon_h$$ where $O$ is a local operator of spin $J_S$ with polarizations $\epsilon_{h}, \epsilon_{h'}$, and $\mathcal{N}$ is the total event rate. The operators measure the summed energy and charge flow through infinitesimal solid angles in the asymptotic directions. In terms of field operators, energy and charge fluxes are

$$\mathcal{E}(n) = \lim_{r\to\infty}\int_0^\infty dt\, r^2\, n^i\,T_{0i}(t,rn),\quad \mathcal{Q}(n) = \lim_{r\to\infty}\int_0^\infty dt\, r^2\, n^i\,J^{0}(t, r n)$$

where $T_{\mu\nu}$ is the stress tensor and $J^\mu$ is a conserved current (Riembau et al., 18 Dec 2025, 0803.1467).

All nontrivial angular dependence can be encoded by splitting the $2N$ detector-sphere coordinates into $2N-3$ internal invariants $z_{ij}\equiv [1-n_i\cdot n_j]/2$ plus the three Euler angles $(\Phi, \Theta, \phi)$ describing rigid rotations. The correlators decompose into angular momentum channels labeled by $J$, the SO(3) spin transfer, with each Wigner D-matrix $D^J_{\Delta h, \Delta m}(\Phi, \Theta, \phi)$ multiplying a function $H^J_{\Delta h, \Delta m}(z_{ij})$ of the internal angles. For vector currents ($J_S=1$), the hadronic tensor $H^{ab}(z)$ decomposes into a singlet and two independent normalized spin-2 structures $c(z), b(z)$, each functionally distinct in (canonical) detector frames (Riembau et al., 18 Dec 2025).

2. Unitarity, Positivity, and Bounds

The tensorial correlators $$H^{ab} = \sum_{\alpha} \langle 0 | O^{\dagger a} | \alpha \rangle \langle \alpha | O^b | 0 \rangle$$ are positive-definite matrices for all angular configurations, a direct consequence of unitarity and positivity of physical energy and charge. Projected onto the basis of traceless symmetric tensors $S_m$ ($m=-2,\cdots,2$), the normalized coefficients $c_m(z_{ij})$ are confined to a convex region. For the $N=2$ vector-current case, the allowed region in the $(c,b)$ plane is a triangle defined by

$$1 + 2c - b \geq 0, \qquad 1 - c + 2b \geq 0, \qquad 1 - c - b \geq 0$$

with vertices realized by extremal spin states (e.g. free scalar, free fermion, WZW model) (Riembau et al., 18 Dec 2025, 0803.1467). These bounds constrain the possible angular patterns even before dynamical calculation.

3. Spinning Energy-Charge Correlators in Gauge Theory and Infrared Robustness

In QCD, the two-point spinning energy-charge correlator can be computed perturbatively. For a vector current creating a $q\bar{q}$ pair, the tree-level hadronic tensor components are: $$H^{\mathbf{1}}_{\mathcal{E}\mathcal{Q}}(z) = \tfrac{1}{2}(q_q + q_{\bar{q}})[\delta(z)+\delta(1-z)],\quad H^c_{\mathcal{E}\mathcal{Q}}(z) = -\tfrac{1}{4}(q_q + q_{\bar{q}})\delta(z),\quad H^b_{\mathcal{E}\mathcal{Q}}(z) = -\tfrac{1}{4}(q_q + q_{\bar{q}})\delta(1-z)$$ with additional components for $J=1$ and $m=\pm 1$ associated to charge-anticharge asymmetry.

Radiative corrections ($\mathcal{O}(\alpha_s)$) preserve the same tensorial structure, and their explicit forms involve distributions with endpoint and finite collinear/back-to-back contributions. The normalized spinning ratios such as $$a^{(2,0)}_{\mathcal{E}\mathcal{Q}}(z) = [H^c-1/2H^b]/H^{\mathbf{1}}$$, and similar forms, are finite for $0 < z < 1$: the cancellation of infrared (soft and collinear) singularities is exact, a direct reflection of the fact that charge flow is insensitive to soft emissions (which carry no net charge). These ratios thus probe the hard event structure and are more stable against hadronization corrections than standard, inclusive observables (Riembau et al., 18 Dec 2025).

4. Light-Ray Operator Product Expansion and Transverse Spin

Small-angle (collinear) limits of spinning energy and charge correlators are governed by light-ray operator product expansions (OPEs) (Chen et al., 2021). In this framework, products of $\mathcal{E}(n)$ operators at nearby angles are expanded into towers of nonlocal operators with definite spin and twist. For example: $$\mathcal{E}(n_1)\mathcal{Q}(n_2) \sim \sum_{k,n} C_{kn}\frac{1}{|n_1 - n_2|^{\tau_{kn} - 3}} U^{(k)}_{j=2,n}(n_2) + \ldots$$ where $U^{(k)}_{j=2,n}$ are leading-twist ($\tau\approx 2$) light-ray primaries with spin $j=2$. In perturbative gauge theory, these dominate the small-angle singularity. At strong coupling, the twist of these operators increases, softening singularities and yielding a distinct scaling regime.

Explicit construction of the light-ray OPE allows resummation of leading logarithmic and power corrections, with transverse-spin-2 terms (arising from gluon operators) generating characteristic $\cos 2\phi$ azimuthal modulations in multi-point correlators. The universality of the highest-spin collinear block ($j=2n$ at twist $2+n$) ensures that these angular patterns are analytic and controlled by perturbative RG (Chen et al., 2021).

5. Spin-Dependent Energy Correlators in Hadron Structure and Phenomenology

Spin-dependent (spinning) energy correlators have been applied to resolving the spin structure of the proton in deep inelastic scattering (DIS) (Gao et al., 22 Sep 2025). Here, two-point spinning energy correlators measured in lepton-nucleon collisions encode detailed correlations between the proton's spin and the angular flow of energy. Using soft-collinear effective theory (SCET), the correlators factorize:

• For back-to-back (jets) kinematics, they probe transverse-momentum dependent distributions (TMDs), including polarized ($g_1$) and unpolarized ($f_1$) functions, in the perturbative regime, with Sudakov-resummed predictions at $\text{N}^3$LL.
• In the forward (target fragmentation) region, the relevant objects are nucleon energy correlators (NECs), inclusive analogs of fracture functions with spin structure, evolving via modified DGLAP equations.

The comprehensive framework enables mapping experimental measurements of energy–charge correlation asymmetries (e.g., $$\Delta\Sigma(\theta) = [\Sigma(\theta; S=+) - \Sigma(\theta; S=-)]/2$$) directly onto polarized TMDs and NECs at the Electron-Ion Collider (EIC) (Gao et al., 22 Sep 2025).

6. Special Cases: T-odd Correlators and C-odd Tagging for Odderon Physics

By measuring track-based energy–charge correlators, specifically on charged hadrons, one can construct T-odd and C-odd observables sensitive to the spin-dependent odderon—a C-odd gluon operator in the small-$x$ regime (Mäntysaari et al., 26 Mar 2025). The single transverse spin asymmetry (SSA) for positive versus negative tracks directly reflects the odderon's dynamics, with the energy sum rule connecting the observable to partonic fracture functions weighted by fragmentation moments. Charge-weighted energy correlators

$$\mathcal{E}_\mathbb{Q} = \sum_h Q_h\,\frac{E_h}{E_N}\, \delta(\theta-\theta_h)\delta(\phi-\phi_h)$$

eliminate C-even contamination, providing clean access to C-odd QCD dynamics. Numerical studies indicate measurable asymmetries at the EIC, robust against soft and hadronization uncertainties (Mäntysaari et al., 26 Mar 2025).

7. Extensions: Conformal, Strong Coupling, and Holography

In conformal field theories (CFT), spinning energy–charge correlators are constrained by symmetry, conformal Ward identities, and anomaly coefficients (notably $a$ and $c$ in 4d ${\cal N}=1$ SCFTs), with energy positivity imposing bounds such as $1/2 \leq a/c \leq 3/2$ (0803.1467). At strong coupling, as realized in AdS/CFT dualities, the corresponding correlators reduce to trivial (angle-independent) patterns in the supergravity limit, but stringy corrections restore anisotropies and relate higher derivative bulk terms to angular moments in the event shape.

In lower-dimensional holography (AdS$_3$/CFT$_2$), spinning energy–charge correlators generalize to correlators of spinning line (shell) defects. The computation involves both a gravitational (first-order formalism) construction, tracking the backreaction and junction conditions of spinning shells, and Virasoro block computations in the dual CFT, with precise matching via the Eigenstate Thermalization Hypothesis (ETH). Higher point correlators and ordering effects capture the nonlocal character of these defects (Liu et al., 15 Jun 2025).

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In summary, spinning energy–charge correlators constitute a highly structured, angular-momentum-resolving extension of event-shape observables, providing a bridge between hard scattering dynamics, spin-dependent parton radiation, infrared/collinear safety, and deep theoretical structures in QCD, CFT, and holography. Their computation combines perturbative QCD, effective field theory factorization, conformal symmetry, and gravitational dual descriptions, enabling both precision phenomenology and fundamental insight into quantum field dynamics (Riembau et al., 18 Dec 2025, 0803.1467, Chen et al., 2021, Gao et al., 22 Sep 2025, Mäntysaari et al., 26 Mar 2025, Liu et al., 15 Jun 2025).