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DOE OSTI2024

Parametrization of Generalized Parton Distributions from 𝑡-Channel String Exchange in AdS Spaces

We introduce a string-based parametrization for nucleon quark and gluon generalized parton distributions (GPDs) that is valid for all skewness. Our approach leverages conformal moments, representing them as the sum of spin-𝑗 nucleon 𝐴-form factor and skewness-dependent spin-𝑗 nucleon 𝐷-form factor, derived from 𝑡-channel string exchange in AdS spaces consistent with Lorentz invariance and unitarity. This model-independent framework, satisfying the polynomiality condition due to Lorentz invariance, uses Mellin moments from empirical data to estimate these form factors. With just five Regge slope parameters, our method accurately produces various nucleon quark GPD types and symmetric nucleon gluon GPDs through pertinent Mellin-Barnes integrals. Our isovector nucleon quark GPD is in agreement with existing lattice data, promising to improve the empirical extraction and global analysis of nucleon GPDs in exclusive processes, by avoiding the deconvolution problem at any skewness, for the first time.

QCD phenomenology↗

DOE OSTI2026

Moments of parton distribution functions of the pion from lattice QCD using gradient flow

We present a nonperturbative determination of the pion valence parton distribution function (PDF) moment ratios ⟨𝑥 𝑛−1 ⟩/⟨𝑥⟩ up to 𝑛 = 6, using the gradient flow in lattice quantum chromodynamics (QCD). As a testing ground, we employ SU(3) isosymmetric gauge configurations generated by the OpenLat initiative with a pseudoscalar mass of 𝑚 𝜋 ≃ 411 MeV. Our analysis uses four lattice spacings and a nonperturbatively improved action, enabling full control over the continuum extrapolation, and the limit of vanishing flow time, 𝑡 →0. The flowed ratios exhibit O(𝑎 2 ) scaling across the ensembles, and the continuum-extrapolated results, matched to the $\overline{MS}$ scheme at 𝜇 = 2 GeV using next-to-next-to-leading order matching coefficients, show only mild residual flow-time dependence. The resulting ratios, computed with a relatively small number of configurations, are consistent with phenomenological expectations for the pion’s valence distribution, with statistical uncertainties that are competitive with modern global fits. These findings demonstrate that the gradient flow provides an efficient and systematically improvable method to access partonic quantities from first principles. Future extensions of this work will target lighter pion masses toward the physical point, and applications to nucleon structure such as the proton PDFs and the gluon and sea-quark distributions.

lattice QCD↗

DOE OSTI2024

Lattice QCD calculation of the pion generalized parton distribution

We present the results of a Lattice QCD computation of pion generalized parton distribution (GPD), employing perturbative matching up to next-to-next-to-leading order (NNLO). The computations are based on an ensemble of Nf=2+1 highly improved staggered quarks (HISQ) with a pion mass of 300 MeV and a lattice spacing of 0.04 fm. Centered on the zero-skewness limit, we utilize a recently proposed Lorentz-invariant definition of GPD, which is derived from Lorentz-invariant amplitudes. We analyze and compare these amplitudes in both Breit and non-Breit kinematic frames at comparable momentum transfers, validating their frame-independent nature. To obtain light-cone GPD, we integrate hybrid scheme renormalization with the large momentum effective theory (LaMET). Moreover, we determine the first three iso-vector generalized form factors (GFFs) of the pion using the ratio scheme renormalization and leading-twist factorization, achieving NNLO accuracy.

Ding, Heng-Tong↗

DOE OSTI2024

Overview on the theory and phenomenology of generalized parton distributions

We give a brief overview on the theory and phenomenology of generalized parton distributions (GPDs), including the recently developed framework of single-diffractive hard exclusive process for matching GPDs to experimental observables. We concentrate on the extraction of GPDs from experimental processes, especially on the challenges and potential solutions regarding the separation of different GPDs and the extraction of $x$-dependence of GPDs, which is critically important for constructing the tomographic images and matching the $x$-moments of GPDs to various emergent hadron properties.

Qiu, Jianwei↗

DOE OSTI2025

Chiral-odd generalized parton distributions in the large-𝑁 𝑐 limit of QCD: Spin-flavor structure, polynomiality, and sum rules

We study the nonperturbative properties of the nucleon’s chiral-odd generalized parton distributions (transversity GPDs) in the large-𝑁 𝑐 limit of QCD. This includes the parametric ordering of the spin-flavor components, the polynomiality property of the moments, and the sum rules connecting the GPDs with the tensor form factors. A multipole expansion in the transverse momentum transfer is used to enumerate and interpret the structures in the nucleon matrix element of the chiral-odd partonic operator, including monopole, dipole and quadrupole terms. The 1/𝑁 𝑐 expansion of the GPDs is performed using the abstract mean-field picture of baryons in the large-𝑁 𝑐 limit and its symmetries. We derive a large-𝑁 𝑐 relation between the flavor-nonsinglet GPDs 𝐸$^{𝑢−𝑑}_𝑇$ and $\tilde{𝐻}^{𝑢−𝑑}_𝑇$ and test it with recent lattice QCD results. We show that the polynomiality property and sum rules of the GPDs are fulfilled with the restricted realization of translational and rotational invariance in the mean-field picture. The results provide a basis for the phenomenological analysis of chiral-odd GPDs and hard exclusive processes in the large-𝑁 𝑐 limit, and for calculations in specific dynamical models.

generalized parton distributions↗

DOE OSTI2025

Generalized parton distributions from lattice QCD with asymmetric momentum transfer: Tensor case

The calculation of generalized parton distributions (GPDs) in lattice QCD was traditionally done by calculating matrix elements in the symmetric frame. Recent advancements have significantly reduced computational costs by calculating these matrix elements in the asymmetric frame, allowing us to choose the momentum transfer to be in either the initial or final states only. The theoretical methodology requires a new parametrization of the matrix element to obtain Lorentz-invariant amplitudes, which are then related to the GPDs. The formulation and implementation of this approachaveh already been established for the unpolarized and helicity GPDs. Building upon this idea, we extend this formulation to the four leading-twist quark transversity GPDs ($𝐻_𝑇$, $𝐸_𝑇$, $\tilde{𝐻}_𝑇$, $\tilde{𝐸}_𝑇$). We also present numerical results for zero skewness using an 𝑁 𝑓 = 2 +1 +1 ensemble of twisted mass fermions with a clover improvement. The light quark masses employed in these calculations correspond to a pion mass of about 260 MeV. Furthermore, we include a comparison between the symmetric and asymmetric frame calculations to demonstrate frame independence of the Lorentz-invariant amplitudes. Analysis of the matrix elements in the asymmetric frame is performed at several values of the momentum transfer squared, −𝑡, ranging from 0.17 to 2.29 GeV 2 .

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2025

Toward the first gluon parton distribution from the LaMET

Abstract We present progress towards the first unpolarized gluon quasi-parton distribution function (PDF) from lattice quantum chromodynamics using high-statistics measurements for hadrons at two valence pion masses M π ≈ 310 and 690 MeV computed on an a ≈ 0.12 fm ensemble with 2 + 1 + 1-flavors of highly improved staggered quark generated by the MILC collaboration. In this study, we consider two gluon operators for which the hybrid-ratio renormalization matching kernels have been recently derived and a third operator that has been used in prior pseudo-PDF studies of the gluon PDFs. We compare the matrix elements for each operator for both the nucleon and pion, at both pion masses, and using two gauge-smearing techniques. Focusing on the more phenomenologically studied nucleon gluon PDF, we compare the ratio and hybrid-ratio renormalized matrix elements at both pion masses and both smearings to those reconstructed from the nucleon gluon PDF from the CT18 global analysis. We identify the best choice of operator to study the gluon PDF and present the first gluon quasi-PDF under some caveats. Additionally, we explore the recent idea of Coulomb gauge fixing to improve signal at large Wilson-line displacement and find it could be a major help in improving the signal in the gluon matrix elements. This work helps identify the best operator for studying the gluon quasi-PDF, shows higher hadron boost momentum is needed to implement hybrid-ratio renormalization reliably, and suggests the need to study more diverse set of operators with their corresponding perturbative calculations for hybrid-ratio renormalization to further gluon quasi-PDF study.

Good, William (ORCID:0000000184281003)↗

DOE OSTI2026

Probing quantum entanglement with generalized parton distributions at the Electron-Ion Collider

Within the collinear factorization framework based on generalized parton distributions, we calculate the spin density matrix of exclusively produced quark and antiquark pairs $𝑢\bar{𝑢}$, $𝑑⁢\bar{d}$, $𝑠⁢\bar{𝑠}$, $𝑐\bar{⁢𝑐}$, $𝑏\bar{𝑏}$ in electron-proton scattering. The presence of both real and imaginary parts in the scattering amplitudes leads to a rich pattern of entanglement between the quark and the antiquark. We map out kinematical regions where the pairs exhibit entanglement, Bell nonlocality, and nonstabilizerness. We also predict that massive quarks and antiquarks are transversely polarized, similar to the well-known transverse hyperon polarization in unpolarized collisions. In strangeness, charm, and bottom productions, the polarization can reach 50%–80% in certain kinematic regions in the low-energy runs of the Electron-Ion Collider.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2025

Three-dimensional imaging of pion using lattice QCD: generalized parton distributions

In this work, we report a lattice calculation of x-dependent valence pion generalized parton distributions (GPDs) at zero skewness with multiple values of the momentum transfer −t. The calculations are based on an N f = 2 + 1 gauge ensemble of highly improved staggered quarks with Wilson-Clover valence fermion. The lattice spacing is 0.04 fm, and the pion valence mass is tuned to be 300 MeV. We determine the Lorentz-invariant amplitudes of the quasi-GPD matrix elements for both symmetric and asymmetric momenta transfers with similar values and show the equivalence of both frames. Then, focusing on the asymmetric frame, we utilize a hybrid scheme to renormalize the quasi-GPD matrix elements obtained from the lattice calculations. After the Fourier transforms, the quasi-GPDs are then matched to the light-cone GPDs within the framework of large momentum effective theory with improved matching, including the next-to-next-to-leading order perturbative corrections, and leading renormalon and renormalization group resummations. We also present the 3-dimensional image of the pion in impact-parameter space through the Fourier transform of the momentum transfer −t.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2024

Moments of parton distribution functions of any order from lattice QCD

We describe a procedure to determine moments of parton distribution functions of any order in lattice quantum chromodynamics (QCD). The procedure is based on the gradient flow for fermion and gauge fields. The flowed matrix elements of twist-2 operators renormalize multiplicatively, and the matching with the physical matrix elements can be obtained using continuum symmetries and the irreducible representations of Euclidean 4-dimensional rotations. We calculate the matching coefficients at one-loop in perturbation theory for moments of any order in the flavor nonsinglet case. We also give specific examples of operators that could be used in lattice QCD computations. It turns out that it is possible to choose operators with identical Lorentz indices and still have a multiplicative matching. One can thus use twist-2 operators exclusively with temporal indices, thus substantially improving the signal-to-noise ratio in the computation of the hadronic matrix elements.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2026

First simultaneous global QCD analysis of kaon and pion parton distributions with lattice QCD constraints

We perform the first simultaneous global QCD analysis of pion and kaon parton distribution functions (PDFs) constrained by pion- and kaon-induced Drell-Yan (DY) and leading neutron electroproduction data together with lattice QCD data on pion and kaon PDF moments. The analysis indicates a softer valence u ¯ distribution in the K − than in the π − and a significantly more peaked valence s -quark density in K − compared with the u ¯ . The effective exponent governing the high- x behavior of the PDF is found to be larger for u ¯ in the kaon, β u ¯ K − = 1.6 ( 2 ) , than in the pion, β u ¯ π − = 1.16 ( 4 ) , in the range 0.7 ≤ x ≤ 0.95 . From the gluon momentum fractions, we find the pion’s gluon content accounts for ≈ 1 / 3 of the mass budget of the pion at μ = 2 GeV , but only ≈ 1 / 4 for the kaon.

Barry, P. C. [Argonne National Laboratory] (ORCID:↗

DOE OSTI2024

New results in the CTEQ-TEA global analysis of parton distributions in the nucleon

This report summarizes the latest developments in the CTEQ-TEA global analysis of parton distribution functions (PDFs) in the nucleon. The focus is on recent NNLO fits to high-precision LHC data at 8 and 13 TeV, including Drell–Yan, jet, and top-quark pair production, pursued on the way toward the release of the new generation of CTEQ-TEA general-purpose PDFs. The report also discusses advancements in statistical and numerical methods for PDF determination and uncertainty quantification, highlighting the importance of robust and replicable uncertainties for high-stakes observables. Additionally, it covers phenomenological studies related to PDF determination, such as the interplay of experimental constraints, exploration of correlations between high-x nucleon sea and low-energy parity-violating measurements, fitted charm in the nucleon, the photon PDF in the neutron, and simultaneous SMEFT-PDF analyses.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2025

Window Observables for Benchmarking Parton Distribution Functions

Global analysis of collider and fixed-target experimental data and calculations from lattice quantum chromodynamics (QCD) are used to gain complementary information on the structure of hadrons. We propose novel “window observables” that allow for higher precision cross-validation between the different approaches, a critical step for studies that wish to combine the datasets. Global analyses are limited by the kinematic regions accessible to experiment, particularly in a range of Bjorken-𝑥, and lattice QCD calculations also have limitations requiring extrapolations to obtain the parton distributions. We provide two different window observables that can be defined within a region of 𝑥 where extrapolations and interpolations in global analyses remain reliable and where lattice QCD results retain sensitivity and precision.

lattice QCD↗

DOE OSTI2025

Description of di-hadron saturation signals within a universal nuclear parton distribution function approach

Di-hadron and di-jet correlation measurements in proton-nucleus (𝑝+𝐴) and electron-nucleus collisions are widely motivated as sensitive probes of novel, nonlinear quantum chromodynamices saturation dynamics in hadrons, which are particularly accessible in the dense nuclear environment at low values of Bjorken-𝑥 (𝑥 A ). Current measurements at the BNL Relativistic Heavy Ion Collider (RHIC) and the CERN Large Hadron Collider (LHC) observe a significant suppression in the per-trigger yield at forward rapidities compared to that in proton-proton collisions, nominally consistent with the “mono-jet” production expected in a saturation scenario. However, the width of the azimuthal correlation remains unmodified, in contradiction to the qualitative expectations from this physics picture. I investigate whether the construction of these observables leaves them sensitive to effects from simple nuclear shadowing as captured by, for example, universal nuclear parton distribution function (nPDF) analyses. I find that modern nPDF sets, informed by recent precision measurements sensitive to the shadowing of low-𝑥 A gluon densities in LHC and other data, can describe all or the majority of the di-hadron/jet suppression effects in 𝑝+𝐴 data at both RHIC and the LHC, while giving a natural explanation for why the azimuthal correlation width is unmodified. Notably, this is achieved via a (𝑥 A ,𝑄 2 )-differential suppression of overall cross sections only, without requiring additional physics dynamics which alter the interevent correlations.

Jets & heavy flavor physics↗

DOE OSTI2025

Generalized parton distributions from lattice QCD with asymmetric momentum transfer: Unpolarized quarks at nonzero skewness

We extend the formalism of asymmetric frames of reference for generalized parton distributions (GPDs) to the case of nonzero skewness, i.e., including longitudinal momentum transfer. The framework, based on Lorentz-invariant amplitudes and previously developed and numerically implemented for unpolarized, helicity and transversity GPDs at zero skewness, gives efficient access to a broad range of kinematics, making full mapping of GPDs from the lattice realistic. The general-skewness formalism is tested using lattice data with both transverse and longitudinal or only longitudinal momentum transfer, the latter being a special case with a reduced number of independent amplitudes. We extract the amplitudes in coordinate space and express the GPDs 𝐻 and 𝐸 in terms of these amplitudes. This is followed by reconstruction of quasidistributions and their matching to the light cone. We further identify and discuss the principal challenges for nonzero skewness GPDs.

Lattice QCD↗

DOE OSTI2025

Nonlocal chiral anomaly and generalized parton distributions

We discuss the nonlocal generalization of the QCD chiral anomaly along the light cone and derive relations between twist-two, twist-three, and twist-four generalized parton distributions (GPDs) mediated by the anomaly. We further establish the connection to the “anomaly pole” in the GPD E ˜ recently identified in the perturbative calculation of the Compton scattering amplitudes, and demonstrate its cancellation at the GPD level. Our work helps elucidate the previously unexplored connection between GPDs, the chiral anomaly, and the mass generation of the η ′ meson. Published by the American Physical Society 2025

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2026

Third moments of nucleon unpolarized, polarized, and transversity parton distribution functions from physical-point lattice QCD

Using forward matrix elements of local leading-twist operators, we present a determination of the isovector third Mellin moments ⟨𝑥 2 ⟩ of nucleon unpolarized, polarized, and transversity parton distribution functions. Two lattice QCD ensembles at the physical pion mass are used, which were generated using a tree-level Symanzik-improved gauge action and 2+1 flavor tree-level improved Wilson Clover fermions coupling via 2-level HEX-smearing. Leveraging a wide set of operators, two extraction methods for the matrix elements, and the automatic inclusion of model uncertainties via bootstrapped model averages, we extract values of the third Mellin moments. Furthermore, this is the first direct calculation of these observables performed at the physical pion mass.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2025

Exploring baryon resonances with transition generalized parton distributions: status and perspectives

QCD gives rise to a rich spectrum of excited baryon states. Understanding their internal structure is important for many areas of nuclear physics, such as nuclear forces, dense matter, and neutrino-nucleus interactions. Generalized parton distributions (GPDs) are an established tool for characterizing the QCD structure of the ground-state nucleon. They are used to create 3D tomographic images of the quark/gluon structure and quantify the mechanical properties such as the distribution of mass, angular momentum, and forces in the system. Transition GPDs extend these concepts to N → N* transitions and can be used to characterize the 3D structure and mechanical properties of baryon resonances. They can be probed in high-momentum-transfer exclusive electroproduction processes with resonance transitions e + N → e' + M + N*, such as deeply-virtual Compton scattering (M = γ) or meson production (M = π, K, etc.), and in related photon/hadron-induced processes. This White Paper describes a research program aiming to explore baryon resonance structure with transition GPDs. This includes the properties and interpretation of the transition GPDs, theoretical methods for structures and processes, first experimental results from JLab 12 GeV, future measurements with existing and planned facilities (JLab detector and energy upgrades, COMPASS/AMBER, EIC, EicC, J-PARC, LHC ultraperipheral collisions), and the theoretical and experimental developments needed to realize this program.

Experimental Nuclear Physics↗