Gluon-Proton Spin Correlations in QCD

• Gluon–proton spin correlations are defined by the interplay between gluon helicity and orbital angular momentum, fundamental to the proton spin sum rule in QCD.
• Experimental studies using longitudinal double-spin and transverse single-spin asymmetries at RHIC, LHC, and EIC quantify these correlations across various kinematic regimes.
• Global QCD analyses and advanced nonperturbative models combine DIS, SIDIS, and top decay measurements to provide quantitative constraints on gluon spin contributions.

Gluon–Proton Spin Correlations

Gluon–proton spin correlations encode the fundamental interplay between gluon spin degrees of freedom and the total angular momentum of the proton in quantum chromodynamics (QCD). This subject is central to resolving the proton spin decomposition problem, quantifying how gluon helicity and orbital angular momentum contribute to the proton’s total spin in both longitudinally and transversely polarized regimes. The field combines experimental measurements—especially at facilities such as RHIC and the LHC—with sophisticated global QCD analyses, nonperturbative modeling, and operator-based field theory. This article provides a comprehensive technical overview of the operational definitions, measurement strategies, phenomenological status, and theoretical frameworks underpinning gluon–proton spin correlations.

1. Operator Definitions and Spin Decomposition

The polarized gluon parton distribution function (PDF), denoted $\Delta g(x, Q^2) = g_+(x, Q^2) - g_-(x, Q^2)$, quantifies the net helicity of gluons with momentum fraction $x$ in a proton of given longitudinal spin. The correlator is defined through light-cone field strength operators: $$\Delta g(x, Q^2) = \frac{1}{2P^+}\int\frac{dz^-}{2\pi}e^{ixP^+z^-}\langle P, S_L| G^{+ \mu}(0)\,\tilde{G}_{\mu}^{~+}(z^-) |P, S_L\rangle~,$$ with $$\tilde{G}^{\mu\nu} \equiv \frac{1}{2}\epsilon^{\mu\nu\alpha\beta}G_{\alpha\beta}$$ and $S_L=\pm1$ the proton helicity.

The proton spin sum rule is formulated as: $$\frac{1}{2} = \frac{1}{2}\Delta\Sigma(Q^2) + \Delta G(Q^2) + L_q(Q^2) + L_g(Q^2)~,$$ where $\Delta\Sigma$ is the total quark (and antiquark) helicity contribution, $\Delta G(Q^2) = \int_0^1 dx\, \Delta g(x, Q^2)$ the gluon helicity, and $L_q$, $L_g$ the quark and gluon orbital angular momentum (OAM). The decomposition can be formulated in both the Ji (gauge-invariant, kinetic) and Jaffe–Manohar (canonical) schemes (Lin, 23 Feb 2024, Tan et al., 2023, Bhattacharya et al., 2022).

For transverse spin–momentum correlations, three-gluon (“trigluon”) twist–3 correlators and transverse-momentum-dependent (TMD) functions such as the gluon Sivers function $f_{1T}^{\perp\,g}(x, k_T^2)$ enter in field-theory definitions, capturing PT–odd correlations between the proton's spin and intrinsic gluon kinematics (Lewis, 2020, Acharya et al., 2021, Abdulameer et al., 2022).

2. Experimental Probes of Gluon Spin Correlations

2.1 Longitudinal Double–Spin Asymmetries at RHIC

The principal observable constraining the gluon helicity in the proton is the longitudinal double–spin asymmetry: $$A_{LL} = \frac{\sigma^{++}-\sigma^{+-}}{\sigma^{++}+\sigma^{+-}}~,$$ where $\sigma^{++}$ ($\sigma^{+-}$) is the cross section for like (opposite) proton helicities. In leading-twist collinear factorization,

$$A_{LL} = \frac{\sum_{ab}\int dx_1dx_2\,\Delta f_a(x_1,Q^2)\Delta f_b(x_2,Q^2)\Delta\hat{\sigma}_{ab\to{\rm jet}+X}}{\sum_{ab}\int dx_1dx_2\,f_a(x_1,Q^2)f_b(x_2,Q^2)\hat{\sigma}_{ab\to{\rm jet}+X}}~.$$

RHIC measurements of inclusive jet, dijet, and pion-photon production at $\sqrt{s}=200$–$510\,\text{GeV}$ sample gluon $x$ in $0.01 \lesssim x \lesssim 0.3$, and have yielded nonzero $A_{LL}$, establishing a positive $\Delta g(x)$ for $x>0.05$ (Surrow, 2013, Li, 2014, Walker, 2011, Lin, 23 Feb 2024).

2.2 Transverse Single–Spin Asymmetries and Three-Gluon Correlators

Transverse spin–momentum correlations are probed via the single–spin asymmetry $A_N$ for inclusive and semi-inclusive channels sensitive to gluon subprocesses. Processes such as open–heavy–flavor production and midrapidity direct photon production in $p^\uparrow p$ collisions are predominantly gluon-initiated and isolate twist–3 three-gluon correlators and the gluon Sivers function. The null results for $A_N$ at RHIC, with sensitivities down to $|A_N|\sim 10^{-3}$, place strong constraints on gluon trigluon matrix elements and the first $k_T$ moment of $f_{1T}^{\perp\,g}$ (Abdulameer et al., 2022, Lewis, 2020, Acharya et al., 2021).

2.3 Heavy–Flavor and LHC Spin Correlations

At the LHC, top–antitop pair production is dominated by $gg\to t\bar t$, and spin correlation observables in top decays provide access to $\Delta G(x)$ at high scales. The extraction relies on measuring double–spin and single–spin asymmetries in dilepton, lepton+jets, and fully hadronic channels, mapping out gluon polarizations in the relevant $x$ and $Q^2$ domains (Goldstein et al., 2017).

2.4 Electron–Ion Collider and Exclusive/Dijet Channels

Exclusive dijet production in deep inelastic scattering (ep→e' jj p') at the planned EIC enables direct and differential measurement of the gluon orbital angular momentum via azimuthal double–spin or single–spin asymmetries, with the relevant modulations linked to CFFs of the gluon GTMDs and OAM distributions (Bhattacharya et al., 2022, Ji et al., 2016).

3. Global QCD Analysis and Gluon Helicity Extraction

Global fits incorporating DIS, SIDIS, and RHIC $A_{LL}$ data—exemplified by DSSV, NNPDFpol, and JAM frameworks—simultaneously constrain quark and gluon helicity distributions (Lin, 23 Feb 2024, Li, 2014, Binder et al., 2011). Typical results at $Q^2=10\,\text{GeV}^2$ find:

• Truncated integral: $$\Delta G(0.05 < x < 1) \approx 0.20^{+0.06}_{-0.07}$$,
• Extending to lower $x$ via dijet topologies: $\Delta G(x > 0.01) \approx 0.3 \pm 0.1$.

These results establish the gluon spin contribution as a major component of the proton's angular momentum, comparable in magnitude to the total quark plus antiquark spin $\Delta\Sigma \approx 0.3$ (Surrow, 2013, Lin, 23 Feb 2024, Li, 2014).

The uncertainties at $x<0.01$ remain large, underscoring the need for next-generation measurements at the EIC to fully map $\Delta g(x)$ to small $x$ and complete the spin puzzle.

4. Nonperturbative Structure and Spin–Orbit Correlations

4.1 GPDs, TMDs, and Wigner Distributions

Theoretical modeling via light-front quark–gluon spectator models, AdS/QCD, and BLFQ Hamiltonians has yielded explicit expressions for gluon GPDs ($H^g$, $\tilde H^g$, $E^g$), TMDs, and GTMDs. The GTMD $F_{1,4}^g(x, k_\perp, \Delta_\perp)$ encodes the gluon OAM density as the $k_\perp^2$ moment, while the GPD $E^g(x, \xi, t)$ is directly linked to OAM through Ji's sum rule: $$J^g = \frac{1}{2} \int_0^1 dx\, x [H^g(x, 0, 0) + E^g(x, 0, 0)],\qquad L^g = J^g - \Delta G.$$ Phase-space (Wigner) distributions and GTMDs additionally reveal spin–orbit correlations, quantified by canonical OAM and spin–OAM correlators such as $$C_{LS}^g = \int dx\, d^2k_\perp\, (k_\perp^2/M^2) G_{1,1}^g(x, 0, k_\perp, 0)$$ (Chakrabarti et al., 17 Sep 2025, Tan et al., 2023, Choudhary et al., 13 Aug 2024).

4.2 Quantum Entanglement Measures

Light-front Hamiltonian approaches (BLFQ) demonstrate that the inclusion of dynamical gluons in Fock sector expansions enhances spin–spin entanglement entropy among proton constituents, shifting the maximum entanglement to mid-$x$ and directly connecting entanglement entropy to measured helicity distributions via binary entropy relations. These quantum-informational diagnostics provide new avenues for probing multi-parton correlations in future experiments (Qian et al., 16 Dec 2024).

4.3 Spin–Orbit Dynamics and Model Uncertainties

Model calculations indicate that the gluon spin and OAM can be sizeable and often opposite in sign, with kinetic OAM $L^g$ typically negative for positive $\Delta G$ (Tan et al., 2023, Choudhary et al., 13 Aug 2024). These findings are robust across soft-wall AdS/QCD, holographic light-front models, and statistical parton frameworks (Bourrely et al., 2014, Gurjar et al., 2022), with extracted $\Delta G$ compatible with both lattice QCD and phenomenological fits.

5. Constraints, Implications, and Future Prospects

5.1 Summary of Present Knowledge

A cohesive picture emerges: gluon helicity $\Delta G$ is established as a positive, non-negligible component of proton spin for $x\gtrsim0.05$, with substantial theoretical and experimental evidence (Surrow, 2013, Li, 2014, Lin, 23 Feb 2024). The corresponding OAM $L_g$ is inferred to be negative and large enough to nearly balance $\Delta G$ in certain models, conforming to the proton spin sum rule across multiple theoretical frameworks (Zhu et al., 2015, Tan et al., 2023, Choudhary et al., 13 Aug 2024). Constraints on transverse spin–momentum correlations, including the gluon Sivers function and trigluon twist-3 correlators, are strong: current data indicate minimal transverse spin–momentum correlations for gluons at moderate $x$ (Lewis, 2020, Abdulameer et al., 2022, Acharya et al., 2021).

5.2 Outstanding Questions and the Role of the EIC

The central open issue is the behavior of $\Delta g(x)$ for $x < 0.01$ and the detailed mapping of $L_g(x)$ and spin–orbit correlations at small $x$. The upcoming Electron–Ion Collider is designed to address these gaps using exclusive and semi–inclusive channels sensitive to gluon GTMDs and OAM distributions, exploiting azimuthal spin asymmetries and reconstructing Wigner phase-space distributions (Bhattacharya et al., 2022, Ji et al., 2016). Such measurements will enable direct experimental determination of the elusive gluon OAM and provide a definitive test of the QCD spin sum rule.

6. Tables: Key Experimental Results and Theoretical Constructs

Table 1. RHIC Constraints on Integrated Gluon Helicity

$x$ Range	$\Delta G(x)$ Estimate	Source / Analysis
$0.05	$0.20^{+0.06}_{-0.07}$	DSSV14 Global Fits (Li, 2014, Lin, 23 Feb 2024)
$0.01	$0.3 \pm 0.1$	STAR (incl. forward dijets) (Lin, 23 Feb 2024)

Table 2. Theoretical Model Results for Gluon Angular Momentum

Model / Approach	$\Delta G$	$L_g$	Reference
Light-cone spectator	0.19	$-0.123$	(Tan et al., 2023)
Light-front AdS/QCD	$0.221^{+0.056}_{-0.044}$	—	(Gurjar et al., 2022)
LF spectator (DIS2024)	0.48	$-0.42$	(Choudhary et al., 13 Aug 2024)
BLFQ (spin entropy)	—	—	(Qian et al., 16 Dec 2024)

These results reinforce the central finding that gluon spin and OAM are dynamically entangled components of the proton's spin budget, with both experimental and theoretical efforts converging towards a detailed, quantitative understanding of gluon–proton spin correlations.