QCD Quark-Gluon-Quark Correlations

• Quark-gluon-quark correlations are three-point functions in QCD that capture quantum interference and multipartonic dynamics beyond the independent-parton approximation.
• They manifest in twist-three PDFs, jet fragmentation observables, and color charge fluctuations, offering insight into effects such as the Sivers asymmetry and odderon dynamics.
• Recent theoretical and experimental advances have enabled precise extraction of qqg functions, refining quantum tomography of the proton and tests of QCD factorization.

Quark-gluon-quark (qqg) correlations are quantum mechanical phenomena in quantum chromodynamics (QCD) that encode the nontrivial dynamical interplay between quarks and gluons inside hadrons or jets. These correlations appear in a variety of observables, from twist-three parton distributions in the nucleon, to multiparticle cumulants in jet fragmentation and collective effects in heavy-ion collisions. They capture physics beyond the independent-parton approximation, providing access to genuine multipartonic and quantum interference effects that are suppressed at leading twist but contribute at next-to-leading twist or higher in QCD factorization theorems.

1. Formal Definitions and Operator Structures

qqg correlations are typically formulated as three-point functions involving quark fields and a gluon field-strength insertion. In the context of twist-three parton distribution functions (PDFs), the chiral-even quark–gluon–quark correlator is defined on the light cone as: $$\langle p,s|g\,\bar{q}(z_1 n)\,F^{\mu+}(z_2 n)\,\gamma^+\,q(z_3 n)|p,s\rangle = 2\,\epsilon_T^{\mu\nu}\,s_\nu\,(p^+)^2 M \int [dx]\,e^{-i p^+\sum_i z_i x_i} T(x_1, x_2, x_3)$$ with the momentum fractions satisfying $x_1 + x_2 + x_3 = 0$, where $[dx] = \delta(x_1 + x_2 + x_3)\,dx_1dx_2dx_3$ (Vladimirov et al., 6 Nov 2025). The function $T(x_1, x_2, x_3)$ encodes the quantum interference between a quark, a gluon, and a second quark absorption or emission.

In jet physics, qqg correlations correspond to three-particle cumulants, such as the energy–energy–energy correlations between hadrons that can be traced back to the emission kinematics of two quarks and one gluon fragmenting from a parent parton (Ramos et al., 2011, Ramos et al., 2011).

In color charge algebra, the cubic correlator

$$C_{abc}(\vec{x}_1, \vec{x}_2, \vec{x}_3) = \langle P | \rho^a(\vec{x}_1) \rho^b(\vec{x}_2) \rho^c(\vec{x}_3) | P \rangle$$

(where $\rho^a$ is the QCD color charge density) quantifies the spatial or momentum-domain correlation of color charge density in a proton, impacted by quark-gluon quantum fluctuations (Dumitru et al., 2021).

2. Phenomenological Manifestations

qqg correlations appear across a range of physical situations:

• Twist-three PDFs and Quantum Interference: The functions $T(x_1,x_2,x_3)$ and $\Delta T(x_1, x_2, x_3)$ represent nonperturbative distributions that describe the quantum coherence and interference in the proton wave function. The Qiu–Sterman function, $T_F(x,x)$, is a special case contributing to single-spin asymmetries (SSAs) via the Sivers effect (Vladimirov et al., 6 Nov 2025, Kanazawa et al., 2014).
• Jet Fragmentation and In-Jet Correlations: In the context of parton showers, normalized three-particle correlators such as

$$C_{qgq}(z_1, z_2, z_3; Y) = \frac{D_Q^{(3)}(z_1, z_2, z_3; Y)}{D_Q(z_1; Y) D_G(z_2; Y) D_Q(z_3; Y)}$$

measure the degree of correlation among two quark-like and one gluon-like hadron within the same jet at a given evolution “time” $Y$ (Ramos et al., 2011, Ramos et al., 2011). These correlations provide stringent tests of parton-shower dynamics and the hypothesis of Local Parton–Hadron Duality.

• Color Charge Fluctuations and Odderon Physics: The cubic color charge correlator decomposes into C-even and C-odd pieces. The C-odd part, $G_3^-$, acts as the “odderon” initial condition for small-x evolution of the dipole amplitude, entering high-energy scattering phenomena (Dumitru et al., 2021).
• Correlations as CME Backgrounds: In the Color Glass Condensate (CGC) framework, initial-state qqg correlations can account for observed three-particle cumulants (e.g., $\langle \cos(\phi_p + \phi_q - 2\phi_m) \rangle$) that can partially mimic signals attributed to the Chiral Magnetic Effect (CME) in proton–nucleus (p+A) collisions (Kovner et al., 2017).

3. Theoretical Approaches and Evolution

Several theoretical tools are used for calculating and evolving qqg correlations:

• Collinear and TMD Factorization: qqg correlators enter QCD factorization theorems at twist three, both in collinear and transverse-momentum-dependent (TMD) frameworks. The evolution equations are nontrivial, involving closed systems for all chiral-even twist-three PDFs. For instance, the evolution of $\vec{T}(x_1,x_2,x_3;\mu)$ is governed by a kernel matrix $\mathbf{H}$ through convolution on constrained momentum fractions (Vladimirov et al., 6 Nov 2025).
• CGC and Color Source Averaging: In the high parton density regime, the calculation employs CGC techniques, treating color sources as random classical fields and averaging over their distributions, e.g., through the McLerran–Venugopalan Gaussian weight (Kovner et al., 2017).
• Perturbative Calculations in Parton Showers: Inclusive three-particle correlators are calculated via generating-function methods and evolution equations in the double logarithmic (DLA) and modified leading logarithmic (MLLA) approximations. Analytic expressions are obtained, factoring in color algebra and kinematical dependencies (Ramos et al., 2011, Ramos et al., 2011).
• Light-Front Wave Function Techniques: In modeling the T-odd component of qqg TMD correlators, light-front wave functions (LFWFs) in the quark–diquark picture are employed to calculate relevant distributions such as $\tilde{e}_L$ and $\tilde{e}_T$ (Sharma et al., 2021).

4. Experimental Signatures and Global Analysis

Recent experimental developments and global QCD analyses have extended access to qqg correlations:

• Global Fits of Twist-Three Correlators: Direct determination from experimental data has been achieved. A recent global analysis incorporated collinear and TMD observables (including $g_2(x)$, the $d_2$ moment, and spin asymmetries in SIDIS) to extract the full flavor and momentum dependence of $T_f(x_1,x_2,x_3)$ and $\Delta T_f(x_1,x_2,x_3)$ for $f = u,d,s$. The best-fit parameters reveal that these distributions are nonzero at $2-3\sigma$ significance (Vladimirov et al., 6 Nov 2025).
• Connection to Single-Spin Asymmetries: In direct photon production ($p^\uparrow p \to \gamma X$), transverse SSAs ($A_N^\gamma$) are controlled by qqg correlators. Calculations show $A_N^\gamma \sim -(2\text{–}5)\%$ in the forward region, with dominance by the soft-gluon pole component of the Qiu–Sterman function, offering a clean extraction and resolving universality puzzles of the Sivers function (Kanazawa et al., 2014).
• Jet Correlation Observables at Colliders: At LHC scales, three-particle correlators inside jets have peak values $C_{qgq} \sim 2-3$ at DLA, reduced to $1.3-1.6$ including MLLA corrections, and are maximal for nearly equal soft hadrons. These are accessible via three-hadron clusters and can differentiate soft-gluon coherence from independent fragmentation (Ramos et al., 2011, Ramos et al., 2011).
• Angular Correlations and Collectivity in p+A Collisions: Three-particle qqg correlators in the CGC formalism reproduce the observable $$\gamma = \langle \cos(\phi_p + \phi_q - 2\phi_m) \rangle$$, which changes sign as a function of rapidity separation. The calculated initial-state background shares the same Δη-width and sign pattern as signals often attributed to the CME (Kovner et al., 2017).

5. Analytic and Numerical Properties

Key general features and theoretical subtleties include:

• Operator Symmetries and Sum Rules: The qqg correlators obey symmetry constraints, such as the interchange symmetry $T(x_1, x_2, x_3) = T(-x_3, -x_2, -x_1)$, as dictated by the field-theoretic structure (Vladimirov et al., 6 Nov 2025). Sum rules exist for their integrals, such as $d_2$ in DIS.
• Ultraviolet and Infrared Structure: In the calculation of color charge correlators, sums of real and virtual diagrams ensure cancellation of UV divergences (Ward identities), but soft and collinear divergences persist and require regularization or are absorbed into PDFs (Dumitru et al., 2021).
• Sign Changes and Rapidity Dependence: The rapidity-dependent Pauli-blocking contribution in the CGC calculation, and similar momentum-space dependencies in jet correlators, lead to sign-changing features that are crucial for phenomenological interpretation, such as the sign change in $\gamma(\Delta \eta)$ and in SSA observables (Kovner et al., 2017, Kanazawa et al., 2014).
• Parametric and Numerical Estimates: Correlated terms in p+A collisions, for typical CGC parameters ($$\mu \sim 1\,\mathrm{GeV}, m \sim 1\,\mathrm{GeV}, \alpha_s \sim 0.3, S_\perp \sim 10\,\mathrm{fm}^2$$), are $\mathcal{O}(10^{-5}–10^{-4})$ in normalized units (Kovner et al., 2017). In proton impact parameter models, NLO qqg corrections become significant at small $x$ (Dumitru et al., 2021).

6. Outlook and Broader Implications

The direct extraction of qqg correlations marks a shift from merely modeling such effects to empirical exploration of multipartonic quantum dynamics (Vladimirov et al., 6 Nov 2025). qqg functions now play a critical role in:

• Refining the "quantum tomography" of the proton, exposing the nonclassical, flavor-dependent, and multidimensional structure of QCD bound states.
• Providing necessary inputs for the understanding and estimation of backgrounds to new-physics searches (such as the CME in small-system collisions).
• Enabling precision tests of QCD factorization, evolution, and hadronization through multiparticle correlations in jets and hadronic final states.
• Constraining and cross-validating QCD's description of process-dependent TMDs, such as the Sivers and Boer–Mulders functions, particularly with regard to sign-changing behavior under different Wilson line structures.
• Supplying initial conditions and nontrivial spatial profiles ("hot spots") for small-x evolution and odderon dynamics, relevant to high-energy scattering.

These developments indicate that qqg correlations are an integral component in bridging perturbative and nonperturbative QCD, providing both a window into and a diagnostic of genuinely quantum multiparton dynamics across nuclear, hadronic, and collider physics.