Towards unpolarized GPDs from pseudo-distributions

The study investigates unpolarized isovector proton GPDs H and E, which encode 3D information on the nucleon’s internal structure as functions of the parton momentum fraction x, skewness ξ, and momentum transfer t. Experimental access via DVCS/DVMP provides only indirect, moment-like constraints and suffers from ill-posed deconvolution, especially in the ERBL region. The research aims to compute GPD information directly and systematically from lattice QCD by employing the pseudo-distribution (space-like nonlocal operator) formalism. The central objective is to extract Mellin moments up to O(x^3), including their ξ dependence, using wide kinematic coverage enabled by distillation, thereby advancing nucleon tomography beyond what is achievable with current experiments.

The paper reviews GPDs and their role in nucleon structure (angular momentum, spatial distributions), and experimental probes (DVCS, DVMP, and emerging channels like double DVCS and diphoton production). It highlights the limitations of phenomenological extractions due to convolution kernels, limited sensitivity near x≈ξ, and the inverse problem (e.g., shadow GPDs), with particular challenges in the ERBL region and for gravitational form factors. It situates the work within recent advances enabling x-dependent parton distributions from Euclidean correlators (quasi-/pseudo-distributions) and prior lattice GPD efforts (pion and nucleon). Theoretical ingredients summarized include light-cone to space-like matching, Ioffe-time formalism, polynomiality and D-term, twist-3 effects, and perturbative matching kernels in coordinate space for Ioffe-time GPDs. Recent studies on evolution/matching in Ioffe-time versus x-space and on higher-twist/renormalon effects inform the adopted strategies.

• Theory: Define off-forward Euclidean matrix elements of a nonlocal quark bilinear with a straight Wilson line and spacelike separation z. Decompose the vector current matrix element into eight Lorentz-invariant amplitudes A_1…A_8 depending on Ioffe time ν=z·P, ξ, t, and z^2. Identify leading-twist combinations that approach light-cone GPDs in z^2→0 (H from A_1−ξA_5; E from A_4+νA_6−2ξνA_7+ξA_5). Use ratio renormalization to form an RGI reduced matrix element.
• Matching: Employ one-loop matching of Ioffe-time pseudo-GPDs to MS at scale μ with kernels written in Ioffe-time variables, and provide a leading-log resummation in Mellin-moment space using matrices that diagonalize LO GPD evolution. Derive moment-level matching/mixing coefficients c_{n+1,k}(z^2 μ^2) (skewness-independent and ξ^2-dependent) and a LL-resummed form involving evolution factors A(μ^2,z^2), diagonalization matrix G(ξ), and nonlogarithmic matching B(ξ,z^2). Show that the D-term (from A_5) matches independently of the rest due to polynomiality.
• Extraction of moments: Use derivatives at ν=0 of Ioffe-time distributions to obtain Mellin moments F_{n}(ξ,t,z^2). Fit small-ν behavior of Re/Im parts with forms dictated by polynomiality: Re A_1(ν)=F_1−(ν^2/2)(A_{2,0}+ξ^2 A_{3,2})+…, Im A_1(ν)=−ν A_{2,0}+(ν^3/6)(A_{4,0}+ξ^2 A_{4,2})+…, and similarly for the E-combination to extract B_{n,k}. For the D-term, fit Im A_5(ν,ξ)≈ξ ν C_2(t,z^2) at small ν ξ.
• Lattice setup: 2+1 flavor Wilson-clover ensemble a094m358: a=0.094(1) fm, m_π=358(3) MeV, volume 32^3×64; 348 configs, 4 time sources/config; distillation rank 64. Momenta chosen to cover 186 (p_i,p_f) pairs yielding 116 distinct (ξ,t) with t up to ≈−2.5 GeV^2, separations z∈{0,…,6a} (up to 0.6 fm), hadron momenta up to ~1.4 GeV, ν up to ~3.5.
• Distillation and correlators: Compute two- and three-point functions with distillation, enabling efficient reuse of perambulators and elementals and broad momentum coverage. Use symmetric operator insertion about the origin. Source-sink separations T∈{4,6,8,10,12,14} and all 1≤τ≤T−1.
• Amplitude reconstruction: Build kinematic matrices using subduced helicity operators to relate 16 measured matrix elements (4 spin combinations × 4 Γ) to 8 Lorentz amplitudes at fixed kinematics. Solve the overconstrained linear system with SVD pseudoinverse (validated with LAD). Exclude problematic kinematics (e.g., q parallel to z) that yield singular/ill-conditioned systems.
• Excited-state control: Extract bare matrix elements with two methods: (i) summation method over τ with linear-in-T fit; (ii) multi-state (n=0,1) exponential fits constrained by two-point function analyses. Use cuts excluding short Euclidean times (τ, T−τ<3) and quote combined estimates using cuts t_s=3 and 4 to assess excited-state systematics.
• Binning and t-dependence: Bin data into 15 t-bins. Within each bin, fit amplitudes versus z and ν (and ξ) to extract moments; then fit t-dependence with dipole form F(t)=F(0)(1−t/Λ^2)^{-2} and, for cross-checks, a z-expansion up to quadratic order. For elastic form factors F_1, F_2, also extract from nonlocal data by evaluating special kinematics with ν=ν ξ=0 and fitting mild z^2 dependence.
• Matching choice: Apply LL-resummed matching to μ=2 GeV as default (scale variations tested). For moments, after matching the z-dependence is taken as constant over z∈[a,6a] (robust to using [a,4a]).
• Radial distributions: Using zero-skewness dipole fits of A_{n,0}(t) and B_{n,0}(t), perform 2D Fourier transforms to impact-parameter space to obtain first three moments of unpolarized and transversely polarized transverse densities (model dependent).

• Broad kinematic coverage achieved: 186 (p_i,p_f) combinations giving 116 distinct (ξ,t); separations up to z=0.6 fm; hadron momenta up to ~1.4 GeV; maximal Ioffe time ν≈3.5.
• Elastic form factors (isovector, from local operator; dipole fits with t-binning): F_1(0)=0.97±0.004(stat)±0.006(excited-state), Λ≈1.246±0.003±0.006 GeV; F_2(0)=3.53±0.01±0.06, Λ≈0.966±0.002±0.007 GeV. Extraction from nonlocal data (after accounting for correlated z^2 drift) is consistent with local results within small O(z^2) effects.
• Mellin moments of GPDs up to n=4 obtained at μ=2 GeV, including ξ-dependent coefficients (first lattice extraction of A_{3,2}, B_{3,2}, A_{4,2}, B_{4,2} in this framework). Representative values at t=0 (dipole fits; uncertainties include statistical and excited-state systematics): • H (u−d): A_{1,0}≈0.974; A_{2,0}≈0.206; A_{3,0}≈0.064; A_{3,2}≈0.39; A_{4,0}≈0.065; A_{4,2}≈0.5. • E (u−d): B_{1,0}≈3.40; B_{2,0}≈0.370; B_{3,0}≈0.063; B_{3,2}≈1.1; B_{4,0}≈0.06; B_{4,2} poorly constrained. Dipole masses are O(1) GeV and vary with moment; skewness-dependent moments generally exhibit larger magnitudes than skewness-independent ones at the same n.
• D-term (isovector): Extracted C_2(t) from Im A_5(ν,ξ)≈ξ ν C_2; small magnitude consistent with prior local-operator lattice results. At t=0, C_2≈0.025 with sizeable uncertainty; dipole mass poorly constrained (LL matching applied; scale variation subdominant at current precision).
• Matching behavior: In Ioffe-time space, evolution and scheme-matching kernels often partially cancel over accessible ν; LL-resummed matching shows the most stable behavior across z, with scale-variation uncertainty generally smaller than statistical/excited-state systematics for moments beyond A_{2,0}, B_{2,0}.
• Radial (impact-parameter) distributions at ξ=0 (model-dependent via dipole fits): higher Mellin moments (larger n) yield more localized transverse profiles, consistent with dominance of large-x partons and the expectation that GPDs become t-independent as x→1.
• Systematics: Clear signals of discretization and potential higher-twist (z^2) effects (e.g., residual A_5 at z=0; z^2 drift in nonlocal EFF). Excited-state contamination can materially shift results; using both summation and 2-state fits with t_s=3,4 provides a realistic systematic envelope.

The results demonstrate that pseudo-distribution methods combined with distillation can access skewness-dependent Mellin moments of nucleon GPDs with broad kinematic coverage. By fitting the small-ν behavior of Ioffe-time amplitudes and applying perturbative matching, the study extracts A_{n,k}(t) and B_{n,k}(t) up to n=4, including ξ^2 coefficients, directly addressing the difficulty of constraining skewness dependence from experiment alone. The consistency between elastic form factors extracted from local and nonlocal data validates the methodology and supports the handling of z^2 and matching effects within quoted uncertainties. The small isovector D-term moment C_2 aligns with prior lattice findings and theoretical expectations (1/N_c suppression). The observed trends—larger skewness-dependent coefficients relative to skewness-independent ones and shrinking impact-parameter distributions for higher moments—are physically reasonable and useful for constraining GPD models and phenomenology. Matching uncertainties at leading-log are generally subleading compared to statistical and excited-state systematics for most moments, suggesting that improved control of excited states and discretization will directly sharpen physics conclusions. However, evident discretization artifacts and potential higher-twist contamination at z up to 0.6 fm limit quantitative precision and complicate clean separation of leading-twist behavior, underlining the need for continuum-limit studies and finer lattices.

This work establishes a practical and scalable strategy to compute unpolarized isovector proton GPD moments on the lattice using the pseudo-distribution formalism with distillation. It achieves extensive kinematic coverage, extracts Mellin moments up to n=4—including, for the first time in this framework, ξ-dependent coefficients A_{3,2}, B_{3,2}, A_{4,2}, B_{4,2}—and provides consistent determinations of elastic form factors from both local and nonlocal operators. It further presents initial constraints on the isovector D-term and model-dependent impact-parameter distributions at zero skewness. Future work should prioritize: (i) continuum-limit studies and simulations at/near the physical pion mass; (ii) improved excited-state control via variational/GEVP analyses with expanded operator bases; (iii) higher hadron momenta to access larger ν and ultimately x-dependent GPDs; (iv) refined perturbative matching (higher orders, resummation) and possibly lattice-determined evolution kernels in z^2; and (v) enlarged kinematic coverage, including more ξ and t points, to reduce model dependence in t-extrapolations and improve D-term constraints.

• Unphysical pion mass m_π=358 MeV; no chiral extrapolation to the physical point.
• No continuum limit; clear signs of discretization artifacts (e.g., nonvanishing A_5 at z=0) and potential z^2 (higher-twist) contamination, especially at larger separations (z up to 0.6 fm).
• Limited ν-range (ν≲3.5) restricts sensitivity to higher Mellin moments and to higher D-term moments (only C_2 constrained).
• Excited-state contamination is significant; although mitigated via two complementary extraction methods and time-separation cuts, residual systematics remain.
• Matching performed at one-loop with LL resummation; higher-order perturbative effects and scheme/systematic ambiguities may impact precision.
• Some kinematics removed due to ill-conditioning; limited coverage at large |t| and large |ξ|.