Bilinear Euclidean Quark & Gluon Correlators

• The paper establishes that bilinear Euclidean correlators, built with nonlocal operators and Wilson lines, robustly encode spatial and momentum correlations of quark and gluon fields.
• It demonstrates that these correlators exhibit exponential decay governed by the lightest mass gap, ensuring controlled Fourier transforms for extracting PDFs, GPDs, and DAs.
• The work employs systematic factorization and NLO perturbative matching to connect lattice QCD data to physical parton distributions with quantifiable uncertainties.

Bilinear Euclidean quark and gluon correlators are foundational objects in the nonperturbative study of partonic structures of hadrons. By encoding information on the spatial and momentum correlations of quark and gluon fields, these correlators serve as the lattice-accessible counterparts to physical, light-cone-defined parton distribution functions (PDFs), generalized parton distributions (GPDs), and distribution amplitudes (DAs). Their rigorous construction ensures gauge invariance via Wilson lines and enables systematic factorization and matching procedures, which are critical for extracting physically meaningful information from Euclidean lattice QCD and connecting to high-energy experimental observables.

1. Formal Definitions and Operator Structure

Bilinear Euclidean correlators are defined using nonlocal, gauge-invariant operators constructed from quark or gluon fields at spatially separated points, connected by Wilson lines to maintain color gauge invariance. For quark bilinears with equal-time, spatial separation $z=(0,0,0,z)$, the general form is

$$O_q(z_1,z_2) = \bar\psi(z_1)\,\Gamma\,[z_1,z_2]\,\psi(z_2),$$

where $\Gamma$ determines the Dirac structure (unpolarized, helicity, or transversity channels) and $[z_1,z_2]$ is a straight Wilson line. Similar constructions are made for gluon bilinears, involving field strength tensors, e.g.,

$$O_{g,u}(z_1,z_2) = F^{\rho\mu}(z_1)[z_1,z_2]F_{\rho}{}^{\nu}(z_2)\,(z_1-z_2)_\mu(z_1-z_2)_\nu.$$

These bilinear operators can be projected onto flavor singlet and nonsinglet combinations, allowing both quark and gluon dynamics (and their mixing) to be systematically incorporated (Yao et al., 2022).

2. Large-Distance Asymptotics and Exponential Decay

Color confinement dictates that bilinear Euclidean correlators with spatial separation $|z|$ exhibit exponential decay at large distances. The asymptotic expansion, derived from heavy-quark effective theory (HQET) reduction, dispersive analysis, and analyticity arguments, takes the form: $$\tilde C_{\Gamma}(z,P^z) \sim \sum_{i} e^{-\Lambda_i |z|}\,|z|^{-3N_i/2+...}\,A_i\,e^{i\phi_i}e^{\pm iP^z z},$$ where $\Lambda_i$ are binding energies determined by the spectrum of heavy-flavor hadrons, and $N_i$ counts the number of physical intermediate states. This behavior is crucial for ensuring that Fourier transforms to momentum space—needed to access PDFs and related quantities—are well-controlled and allow for systematic error estimation in lattice QCD calculations (Ji et al., 17 Jan 2026).

The leading asymptotics are dominated by the lightest mass gap (typically set by the physical lowest-mass hadron in the relevant channel), while next-to-leading terms introduce additional polynomial suppression and oscillatory phases. Explicit calculations in $\phi^3$ theory at two loops confirm the precise exponents and phase structure predicted by general field-theoretic analysis.

3. Factorization and Perturbative Matching

The extraction of physical PDFs, GPDs, and DAs from Euclidean correlators relies on factorization theorems: a short-distance operator product expansion relates nonlocal bilinears to their light-cone counterparts via matching kernels $C_{ij}$. For leading-twist operators,

$$(O_q, O_g)_B(z_1,z_2) = \int_0^1 d\alpha \int_0^1 d\beta \;C_{ij}(\alpha, \beta; \mu^2 z^2)\,(O_j)^{lt}_R(z'_1, z'_2) + \text{h.t.}$$

where $h.t.$ denotes higher-twist corrections. In the nonsinglet sector, only $C_{qq}$ is relevant; in the singlet sector, the full $2\times2$ kernel matrix—$C_{qq}$, $C_{qg}$, $C_{gq}$, $C_{gg}$—encodes quark-gluon mixing. At next-to-leading order (NLO) in $\alpha_s$, these kernels are known in explicit form in both coordinate and momentum space, including all logarithmic and plus-prescription structures. The results apply to both forward (PDF) and nonforward (GPD) kinematics, as well as the DA limit (Yao et al., 2022).

4. Renormalization Schemes and Anomalous Dimensions

The construction and matching of bilinear correlators require careful renormalization. Ultraviolet divergences are handled via dimensional regularization and the $\overline{\text{MS}}$ scheme. Wilson line self-energies induce additional ultraviolet structure, necessitating multiplicative renormalization factors $Z_q(\mu^2 z^2)$ or $Z_g(\mu^2 z^2)$, whose determination is complete up to three loops.

Various practical schemes exist:

• Ratio scheme (RI/MOM-inspired): Divides the correlator by its zero-momentum value $H(0,z^2)$, removing UV divergences and preserving IR structure.
• Hybrid scheme: Applies the ratio renormalization at short distances $|z|<z_s$ and a Wilson-line mass subtraction $e^{-\delta m |z|}$ at longer $|z|$ to avoid IR contamination (Yao et al., 2022).

The anomalous dimensions for local leading-twist operators (Mellin moments) coincide with the standard DGLAP splitting functions $P_{ij}(x)$, ensuring consistency with perturbative QCD evolution.

5. Reconstruction of Physical Distributions

By selecting kinematics, a unified operator framework suffices to access GPDs, PDFs, and DAs:

• GPDs: Nonforward matrix elements with skewness $\xi$ correspond to generalized distributions.
• PDFs: Forward limit $\xi\to0$ reduces the correlator to the standard parton distribution.
• DAs: The crossed kinematics $p_2 = -p_1$, $\xi\to1$ extract distribution amplitudes.

For practical lattice QCD calculations, two approaches are prominent:

• Pseudo-distributions: Fixed-$z^2$ correlators are Fourier-transformed in the $z\cdot P$ variable to yield pseudo-GPDs, which factorize onto light-cone distributions via known matching kernels.
• Quasi-distributions and LaMET: Large-momentum effective theory (LaMET) prescribes taking $P^z \gg \Lambda_\text{QCD}$ and Fourier transforming in $z$ to reconstruct quasi-distributions. These are then matched to their light-cone counterparts using perturbative kernels $C_{ij}$.

Systematic uncertainties in the large-distance extrapolation and Fourier transforms are quantifiable through explicit examination of asymptotic decay (via binding energy fits) and truncation errors, yielding percent-level control for moderate momentum fractions (Ji et al., 17 Jan 2026).

6. Practical and Theoretical Considerations in Lattice QCD

Gauge-invariant bilinear correlators are directly accessible in Euclidean lattice QCD, and their proper asymptotic treatment is crucial. Controlled extrapolation of the correlator's behavior outside the measured range is achieved by fitting the leading exponential and polynomial terms to available lattice data, with binding energies constrained by known heavy-flavor hadron masses.

Error propagation from coordinate to momentum space is governed by the oscillatory nature of the Fourier transform and the exponential suppression dictated by the mass gap. At moderate momentum fractions, the resulting uncertainties in extracted PDFs are typically within a few percent. End-point regions exhibit error enhancement proportional to $P^z/(\pi\Lambda |y|)$ but remain controllable. These prescriptions are validated through explicit numerical applications, such as the extraction of the pion valence PDF using state-of-the-art lattice ensembles (Ji et al., 17 Jan 2026).

7. Clarifications, Subtleties, and Comparison to Landau-Gauge Correlators

The coordinate-space factorization framework naturally respects all required symmetries (including Bose and charge conjugation) and yields correct analytic continuation between distinct kinematic regions (DGLAP and ERBL for GPDs). Unphysical support regions ($|x|>1$) are fixed by anomalous dimensions and do not affect physical convolutions.

Scheme dependence arising from differing quasi-operator definitions must be tracked when comparing different lattice implementations. The hybrid renormalization approach mitigates possible large-$|z|$ infrared distortions present in pure ratio-type schemes, which is essential for reliable LaMET factorization (Yao et al., 2022).

Comparatively, two-point Landau-gauge correlators (gluon and quark propagators) provide insight into the infrared and ultraviolet properties of QCD, and are accurately calculable at the one-loop level in the Curci–Ferrari model, matching lattice data at the percent level for propagator dressing functions. The analytic structure of these two-point correlators illustrates the general principles that underlie the exponential decay of spatially separated bilinear correlators in confining gauge theories (Peláez et al., 2014).

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These developments establish bilinear Euclidean quark and gluon correlators as the essential lattice-accessible link between nonperturbative QCD and partonic observables. The unified framework described ensures systematic perturbative control, rigorous renormalization, and reliable quantification of errors in extracting light-cone distributions from Euclidean data (Yao et al., 2022, Ji et al., 17 Jan 2026, Peláez et al., 2014).