Energy-Flow Moments for Elliptic Gluon Wigner Tomography

Abstract: The elliptic small-$x$ gluon Wigner distribution correlates transverse momentum with impact parameter, but it is usually accessed through exclusive diffractive dijets whose recoil is sensitive to soft radiation. We propose instead an azimuthal energy-flow moment in DIS dijet production. Within leading-power small-$x$ dijet factorization, its normalized $\cos2φ$ moment is a linear projection of the elliptic Wigner harmonic after a calculable kinematic subtraction, while the final-state energy weighting is infrared and collinear safe. In conjugate recoil space a rotationally scalar Sudakov factor evolves the isotropic and elliptic channels through a fixed $J_0/J_2$ Hankel pair without $W_0\to W_1$ leakage. A proof-of-principle calculation gives a per-mille Sudakov-level moment in an unoptimized conservative window, while auxiliary perturbative-window scans reach several per mille. The observable therefore formulates elliptic Wigner tomography as a moment-level energy-flow measurement whose statistical reach can be assessed by simple angular-moment counting.

• The paper introduces a moment-based formulation using azimuthal energy-energy correlations to isolate the elliptic component of gluon Wigner distributions.
• It demonstrates an IRC-safe method that subtracts kinematic cos2ϕ modulations, ensuring robust QCD factorization and controlled Sudakov evolution.
• Numerical studies reveal that the normalized cos2ϕ moment sustains soft-gluon resummation at the per-mille level, offering realistic prospects for EIC measurements.

Energy-Flow Moments as a Probe of Elliptic Gluon Wigner Distributions

Motivation and Theoretical Background

The Wigner distribution framework provides a rigorous phase-space description of partonic correlations, retaining both transverse position ($b$) and momentum ($k$) information. In the small-$x$ regime of QCD, gluon Generalized Transverse-Momentum Dependent Distributions (GTMDs) encode nontrivial angular correlations, notably the elliptic (quadrupole) structure corresponding to the $n=2$ harmonic ($W_1$), which is sensitive to position–momentum correlations of gluons in the impact-parameter-dependent Color Glass Condensate (CGC). Accessing this elliptic component is essential for a detailed tomographic understanding of high-density gluon matter, a primary aim of the scientific program at the Electron-Ion Collider (EIC).

Traditionally, the elliptic gluon Wigner distribution has been probed through exclusive diffractive dijet production, where the angle between the jet pair and target recoil resolves this harmonic. However, exclusive measurements are highly susceptible to uncertainties from recoil reconstruction, sensitivity to soft QCD radiation, and potentially large Sudakov logarithms in the back-to-back dijet kinematics.

Azimuthal Energy-Energy Correlation Proposal

This work proposes a moment-based formulation utilizing azimuthal energy-energy correlations (EECs) in inclusive (non-diffractive) deep inelastic scattering (DIS) dijet production, specifically targeting the normalized $\cos2\phi$ moment of the EEC as a direct projection of the elliptic gluon Wigner harmonic. The observable is defined as the energy-weighted cross section for events sorted by the azimuthal angle $\phi$ between the dijet axis and the transverse recoiling momentum, employing an infrared and collinear (IRC) safe prescription.

A critical advance is the exact subtraction of a calculable, kinematic $\cos2\phi$ modulation generated by the phase-space weight, isolating the genuine Wigner harmonic. The analysis operates in the correlation limit $q_T \ll P_T$ (dijet imbalance much smaller than dijet hardness), where the QCD factorization is controlled and allows for robust calculation of leading-power contributions.

Factorization and Sudakov Evolution

In the factorized approach, the measured EEC moment is given as:

$$C_2(x,Q^2) = \frac{\int d\Phi_0\, \omega^{(0)} \sum_f {\cal N}_f\, 2 A_0 A_1}{\int d\Phi_0\, \omega^{(0)} \sum_f {\cal N}_f\, (A_0^2 + 2A_1^2)}$$

where $k$0 are transverse radial amplitudes corresponding to the isotropic and elliptic projections of the dipole operator in impact-parameter space, and $k$1 is the zeroth harmonic (kinematically subtracted) of the energy-weight.

Final-state soft radiation, central for EIC observables, is included via a global Sudakov factor implemented in recoil-conjugate space, with the notable property that it is a rotational scalar and thus affects only the radial but not the angular content of the harmonics. The $k$2 Hankel transform preserves the purity of the projected $k$3 channel, eliminating Sudakov-induced $k$4 leakage at leading power. Non-global logarithms and subleading corrections are acknowledged sources of further systematics but are outside the scope of the baseline calculation.

Numerical Results and Statistical Reach

A proof-of-principle numerical study is performed using an impact-parameter-dependent, HHU-inspired model for the Wigner input. The observable is computed using the full transverse energy weight with a perturbative-scale cut ($k$5) and the global Sudakov factor included. The dominant result is that the normalized $k$6 moment survives soft-gluon resummation at the $k$7 (full window) to several-per-mille (window-optimized) level, with the sign and size sensitive to the details of the small-$k$8 evolution kernel and fiducial integration window.

Theoretical analysis shows the extracted $k$9 is affected by the photon wavefunction, energy weight, fiducial cuts, and phase-space integrations, distinguishing it from a pointwise $x$0 ratio. Auxiliary scans indicate practical EIC measurements may achieve statistical sensitivity with $x$1--$x$2 effective events for a $x$3 observation in the relevant moment range.

A further model diagnostic demonstrates qualitative sensitivity to the details of the underlying QCD kernel (e.g., running coupling vs. collinearly improved) reflected in the sign and magnitude of the $x$4 moment, though caution is required due to the dependence on infrared regularization and fiducial region choice.

Implications and Future Directions

The formalism established in this work recasts elliptic gluon Wigner tomography into the language of moment-level, IRC-safe energy-flow measurements. This approach sidesteps several experimental and theoretical difficulties present in fully exclusive measurements, potentially broadening robust access to gluonic phase-space correlations at small $x$5. The statistical requirements for observing expected signals are within realistic reach for future high-luminosity data at the EIC, pending detailed studies of detector acceptance, reconstruction, and QCD corrections beyond the global Sudakov approximation.

Theoretically, this observable provides a complementary handle on angular features of the CGC and could be extended to study non-trivial correlations, non-global logarithms, and finite jet-radius effects impacting the measurement. Further global fits with realistic impact-parameter evolution and full detector-level simulations are required for precision predictions and a comprehensive tomography program.

Conclusion

This study presents a principled, leading-power projection of the elliptic gluon Wigner distribution via normalized $x$6 moments in DIS dijet energy-energy correlations, with IRC safety, calculable Sudakov suppression, and model sensitivity preserved. The formulation enables robust moment-level tomography of gluon angular correlations in the small-$x$7 regime, positioning the observable as a valuable addition to the experimental and theoretical toolbox for EIC-era QCD science, and motivating further development toward global phenomenology and experimental realization.

Citation: "Energy-Flow Moments for Elliptic Gluon Wigner Tomography" (2606.31708)