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All sources NASA NTRS DOE OSTI

DOE OSTI2026

Proton isovector helicity PDF at NNLO and the twist-3 moment $\tilde{d}$ 2 from lattice QCD at physical quark masses

We present a lattice quantum chromodynamics calculation of the 𝑥-dependent isovector quark helicity parton distribution function (PDF) of the proton in the large momentum effective theory (LaMET) framework. Through operator product expansion (OPE) we also extract the $\tilde{d}$ 2 moment of the twist-3 PDF 𝑔 𝑇 ⁡(𝑥) for the first time in the $\overline{MS}$ scheme, which is proportional to the average color Lorentz force experienced by the quark in the proton. This calculation is performed on a lattice of spacing 𝑎 =0.076 fm at physical quark masses. The quasi-PDF matrix elements are measured in proton states boosted to momenta 𝑃 𝑧 ={0,0.25,1.02,1.53} GeV. We first extract the lowest few helicity PDF moments from the renormalization-group (RG) invariant ratios of the matrix elements with OPE. Combined with the matrix elements relevant for 𝑔 𝑇 ⁡(𝑥), we obtain $\tilde{d}$$^{u-d}_2$⁡(2 GeV) =0.0024⁢(46) at next-to-leading order in $\overline{MS}$. Then, the helicity quasi-PDF matrix elements are renormalized in the hybrid scheme with linear renormalon resummation and Fourier transformed to the 𝑥-space after an asymptotic extrapolation. The quasi-PDF is perturbatively matched to the $\overline{MS}$ PDF with RG and threshold resummations at next-to-leading power and next-to-next-to-leading logarithmic accuracies. After resummations, we determine the PDF in the region 𝑥 ∈[0.25,0.75]. The end-point regions are then parametrized, combined with the LaMET prediction at moderate 𝑥, and fitted to the short-distance matrix elements in coordinate space.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2026

QCD+QED PDF implications for the Higgs sector

In this work, we examine the implications of electroweak corrections beyond leading order for processes of special interest in the Higgs sector. We especially explore the role of these corrections given the introduction of an explicit parton distribution function (PDF) for the photon in the proton, an object which emerges necessarily in global PDF fits which include QED effects (i.e., ‘QCD+QED PDFs’). We concentrate on several representative cases, including total Higgs-production cross sections through gluon fusion, gg → H, vector-boson fusion (VBFH), and associated production, pp → V H; we also examine differential distributions, taking a representative Higgs-strahlung process, pp → W + H. We find that the recently developed LUX formalism for the photon PDF significantly stabilizes the PDF dependence of both QED-PDF and electroweak corrections in the Higgs sector, while leaving overall ~ 3–4% cross-section-level variations, depending on the chosen QCD+QED PDF. We illustrate this QCD+QED PDF dependence by exploring predictions based upon recent analyses of the CTEQ-TEA, MSHT, and NNPDF analysis groups, fitted either at NNLO or approximate N3LO in QCD.

Higgs production↗

DOE OSTI2025

PDF Entity Annotation Tool (PEAT)

While different text mining approaches – including the use of Artificial Intelligence (AI) and other machine based methods - continue to expand at a rapid pace, the tools used by researchers to create the labeled datasets required for training, modeling, and evaluation remain rudimentary. Labeled datasets contain the target attributes the machine is going to learn; for example, training an algorithm to delineate between images of a car or truck would generally require a set of images with a quantitative description of the underlying features of each vehicle type. Development of labeled textual data that can be used to build natural language machine learning models for scientific literature is not currently integrated into existing manual workflows used by domain experts. Published literature is rich with important information, such as different types of embedded text, plots, and tables that can all be used as inputs to train ML/natural language processing (NLP) models, when extracted and prepared in machine readable formats. Currently, both normalized data extraction of use to domain experts and extraction to support development of ML/NLP models are labor intensive and cumbersome manual processes. Automatic extraction of data and information from formats such as PDFs that are optimized for layout and human readability, not machine readability. The PDF (Portable Document Format) Entity Annotation Tool (PEAT) was developed with the goal of allowing users to annotate publications within their current print format, while also allowing those annotations to be captured in a machine-readable format. One of the main issues with traditional annotation tools is that they require transforming the PDF into plain text to facilitate the annotation process. While doing so lessens the technical challenges of annotating data, the user loses all structure and provenance that was inherent in the underlying PDF. Also, textual data extraction from PDFs can be an error prone process. Challenges include identifying sequential blocks of text and a multitude of document formats (multiple columns, font encodings, etc.). As a result of these challenges, using existing tools for development of NLP/ML models directly from PDFs is difficult because the generated outputs are not interoperable. We created a system that allows annotations to be completed on the original PDF document structure, with no plain text extraction. The result is an application that allows for easier and more accurate annotations. In addition, by including a feature that grants the user the ability to easily create a schema, we have developed a system that can be used to annotate text for different domain-centric schemas of relevance to subject matter experts. Different knowledge domains require distinct schemas and annotation tags to support machine learning.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2026

First nucleon gluon PDF from large momentum effective theory

We report the first nucleon gluon parton distribution function (PDF) using Large-Momentum Effective Theory (LaMET). We focus on the gluon operator which was demonstrated to have the best signal-to-noise in the previous attempt [1] in computing gluon PDFs using LaMET. We compute the corresponding Wilson coefficients needed for the hybrid-renormalized matrix elements and the matching kernel to convert the quasi-PDF to the lightcone one at the one-loop level. We demonstrate that with the proper Wilson coefficients in place, the counterterms for the renormalization are independent of the hadron and mass within statistical error. Using the resulting renormalization, we then compute the nucleon PDF using a HISQ ensemble generated by the MILC collaboration with N f = 2 + 1 +1, a ≈ 0.12 fm, with valence pion masses of 310 and 690 MeV and two gauge link smearing techniques. Despite the physics effects of the heavier than physical pion masses and gauge link smearing, this calculation provides excellent proof of principle and compares reasonably with selected global fit results.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2026

Determination of proton PDF uncertainties with Markov chain Monte Carlo

We present an analysis of parton distribution functions (PDFs) of the proton using Markov chain Monte Carlo (MCMC) methods. The MCMC approach naturally implements Bayes’ theorem and, thus, provides a means to directly sample the underlying probability distribution—in this case, the probability distribution of the PDF parameters. This allows for a straightforward propagation of the resulting uncertainties into any PDF-dependent observable, preserving their simple probabilistic interpretation. In our analysis we include a broad set of deep inelastic scattering data from HERA, BCDMS and NMC experiments along with the Drell-Yan, 𝑊 and 𝑍 boson data from LHC and Tevatron experiments, which combined with theoretical calculations at next-to-next-to-leading order in QCD allow for realistic determination of PDFs. The main focus of this analysis is to explore alternative methods for PDF uncertainty estimation that are more firmly grounded in statistical principles. We show that the flexibility of the Bayes framework, allowing one, e.g., to account for non-Gaussianity or inconsistencies of datasets, is crucial to extract realistic uncertainties when such assumptions are not fulfilled. We also demonstrate that MCMC allows one to determine the Δ⁢𝜒 2 value corresponding to a given confidence level in the sample, which can, in turn, be used as a statistically well-founded tolerance criterion used in the Hessian method, thus addressing one of its main long-standing drawbacks.

Risse, Peter Clemens [Universität Münster (Germany↗

DOE OSTI2025

Systematic Study of the Self-Renormalized Nucleon Gluon PDF in Large-Momentum Effective Theory

We present a systematic study of the nucleon gluon parton distribution function (PDF) using the self-renormalized large-momentum effective theory (LaMET) approach in lattice QCD. This work extends previous gluon-PDF extractions by performing a detailed analysis of key systematic effects, including gauge-link smearing, lattice spacing, pion mass, and nucleon boost momentum. The self-renormalization framework mitigates ultraviolet divergences associated with Wilson-line self-energy and renormalon contributions by combining lattice matrix elements with perturbative short-distance information, thereby preserving the correct infrared structure. Calculations are performed on $N_f=2+1+1$ HISQ ensembles generated by the MILC Collaboration at three lattice spacings and two pion masses, with boosted nucleon states reaching momenta up to 2.2~GeV. We determine renormalization factors from zero-momentum matrix elements and apply hybrid renormalization to suppress discretization artifacts. After extrapolating large-separation behavior and performing Fourier transforms, we reconstruct quasi-PDFs and match them to lightcone PDFs using next-to-leading order Wilson coefficients. Our results demonstrate that smearing and lattice-spacing effects are under control, and pion-mass and lattice-spacing dependence is mild relative to the current $O(10^6)$ statistics; however, momentum dependence remains a significant source of uncertainty. Future work including even larger boost momenta will be essential to reduce systematics in lattice determinations of the gluon PDF and to advance toward precision QCD phenomenology at the LHC and the future Electron-Ion Collider.

FOS: Physical sciences↗

DOE OSTI2025

Ion Clusters in Multicomponent Solutions Determined from X-ray PDF and SAXS Analysis: The NaNO 2 –NaOH–H 2 O System

Here, this study explores ion cluster formation in the NaNO 2 –NaOH–H 2 O system to understand how ion cluster formation is influenced by the composition in multicomponent mixtures. X-ray pair distribution function (PDF) and small-angle X-ray scattering (SAXS) identified complex ion clusters in concentrated NaNO 2 and NaOH solutions as well as their mixtures. PDF analysis showed that the Na–O distance depended primarily on total Na + concentrations rather than anion composition, whereas the nitrite-water oxygen distance stayed the same regardless of concentration or composition. This result indicates that the nitrite ion hydration was relatively independent of composition. SAXS confirms local fluctuations and coherent clusters with notable size differences between NaOH and NaNO 2 solutions. SAXS analysis of mixed solutions shows that their clusters are an average of the individual solutions, indicating mixed clusters.

Reynolds, Jacob G. [Central Plateau Cleanup Compan↗

DOE OSTI2024

PDF DECODER ANALYSIS CODE

SF-24-038"PDFdecoder", as a new application to explore parametrizations of parton distribution functions (PDFs) of the proton or other hadrons. The PDFs are fundamental quantities in particle physics which are necessary inputs to precise theoretical predictions for experiments at the Large Hadron Collider (LHC) and other facilities. As such, understanding how the PDFs are parametrized and associated uncertainties is a pressing need. The specific problem PDFdecoder confronts is the need of having a tractable and interpretably machine-learning (ML) framework to parametrize the PDFs and their uncertainties so as to understand how a given preferred parametrization is obtained. This problem has not been significantly addressed in the current literature. While other groups have used ML-based approaches to parametrize PDFs in the form of feed-forward neural networks, the question of tractability has not been explored in a PDF context. Our solution makes significant progress in this problem by using an array of encoder-decoder (essentially, autoencoder) architectures with varying constraints to the intermediate latent spaces based on interpretable physics. As a consequence, the trained models can be used as generative networks to produce interpretable predictions for the PDFs in a way that can be refined and studied further.

Hobbs, Timothy↗

DOE OSTI2025

Effects of threshold resummation for large-x PDF in large momentum effective theory

Parton distribution functions (PDFs) at large x are challenging to extract from experimental data, yet they are essential for understanding hadron structure and searching for new physics beyond the Standard Model. Within the framework of the large momentum P z expansion of lattice quasi-PDFs, we investigate large x PDFs, where the matching coefficient is factorized into the hard kernel, related to the active quark momentum xP z , and the threshold soft function, associated with the spectator momentum (1 − x)P z . The renormalization group equation of the soft function enables the resummation of the threshold double logarithms α k ln 2k (1 − x), which is crucial for a reliable and controllable calculation of large x PDFs. Our analysis with pion valence PDFs indicates that perturbative matching breaks down when the spectator momentum (1 − x)P z approaches Λ QCD , but remains valid when both xP z and (1 − x)P z are much larger than Λ QCD . Additionally, we incorporate leading renormalon resummation within the threshold framework, demonstrating good perturbative convergence in the region where both spectator and active quark momenta are perturbative scales.

factorization↗

DOE OSTI2025

Investigating Quantum Materials with Half-Polarized Diffraction and magnetic PDF analysis at the HB-2A Neutron Powder Diffractometer

Local magnetic ordering and anisotropy is often central to the emergent behavior and subsequent functional properties in quantum materials and beyond. Neutron powder diffraction provides a straightforward yet extremely powerful technique for quantitative measurements of microscopic magnetic properties. The HB-2A powder diffractometer located at the High Flux Isotope Reactor in ORNL is traditionally utilized for long-range magnetic structure determination. Recently these capabilities have been extended to include methods aimed at accessing local magnetism: Half- polarized neutron powder diffraction (pNPD) and magnetic pair distribution function (mPDF) analysis. These two distinct techniques are possible on HB-2A due to the versatility of the instrument’s reciprocal space coverage, resolution and novel ultra-low temperature multi-sample changers that operate down to dilution refrigerator temperatures. This provides unique capabilities not found on any powder diffraction instrument and is particularly well suited to investigations of magnetic quantum materials. The development and implementation of these techniques will be discussed with a series of science case examples ranging from geometric frustrated magnets to magnetic metal-organic frameworks. Data reduction and analysis tools will be presented that enable the extraction of the local site susceptibility tensor and local spin-spin correlations in real space. Finally, potential combinations of these techniques in the form of half-polarized magnetic pair distribution function (pmPDF) analysis will be considered. Looking forward, HB-2A is undergoing a detector upgrade that will be in the user program by 2026. This will offer an order of magnitude increase in count rates to further aid the development of these often low signal measurements and provide new scientific capabilities.

Neutron Scattering↗

DOE OSTI2025

Small-𝑥 gluon GPD constrained from deeply virtual 𝐽/𝜓 production and gluon PDF through universal-moment parametrization

We phenomenologically constrain the small-𝑥 and small-𝜉 gluon generalized parton distributions (GPDs) with the deeply virtual 𝐽/𝜓 production (DV⁢𝐽/𝜓⁢P) in the framework of GPDs through universal moment parameterization (GUMP). We use a hybrid cross-section formula combining collinear factorization to the next-to-leading order (NLO) accuracy of the strong coupling 𝛼 𝑠 , with corrections from nonrelativistic QCD to account for the power corrections due to the heavy 𝐽/𝜓 mass. We reach reasonable fit to the measured differential cross sections of DV⁢𝐽/𝜓⁢P by H1 at the Hadron-Electron Ring Accelerator as well as forward gluon parton distribution functions (PDFs) from JAM22 global analysis. We find that both NLO and nonrelativistic corrections are significant for heavy vector meson productions. Of course, the gluon GPD we obtain still contains considerable freedom in need of inputs from other constraints, particularly in the distribution-amplitude-like region.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2025

Impact of LHC precision measurements of inclusive jet and dijet production on the CTEQ-TEA global PDF fit

In this study, we investigate the impact of new Large Hadron Collider (LHC) inclusive jet and dijet measurements on parton distribution functions (PDFs) that describe the proton structure, with a particular focus on the gluon distribution at large momentum fraction, x , and the corresponding partonic luminosities. We assess constraints from these datasets using next-to-next-to-leading-order (NNLO) theoretical predictions, accounting for a range of uncertainties from scale dependence and numerical integration. From the scale choices available for the calculations, our analysis shows that the central predictions for inclusive jet production show a smaller scale dependence than dijet production. We examine the relative constraints on the gluon distribution provided by the inclusive jet and dijet distributions, and also explore the phenomenological implications for inclusive H , t t ¯ , and t t ¯ H production at the LHC at 14 TeV. Published by the American Physical Society 2025

Ablat, Alim (ORCID:0009000519877755)↗

DOE OSTI2025

FY24 LDRD Annual Report PDF

The Laboratory Directed Research and Development (LDRD) Program at Lawrence Livermore National Laboratory (LLNL) is the Lab's most significant resource for supporting internally directed research and development. It provides investments in cutting-edge science, technology, and engineering. This program expands the frontiers of knowledge, creates capabilities required by our evolving mission needs, and attracts and retains the world's most talented scientists and engineers. In this annual report, we describe the LDRD investment portfolio, provide information to demonstrate the program's value and impact to LLNL's science, technology and engineering capabilities, and showcase LDRD accomplishments across the Lab's mission space.

42 ENGINEERING↗

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 OSTI2025

Voltage Probability Density Function Shaping Control Strategy Considering Grid Operational Uncertainties

It is well-known that power systems operation always affected by various uncertainties which make the bus voltage a random process that can be characterized by its probability density function (PDF) at any time instant. In this context, this paper presents a novel PDF-based voltage control framework for power systems. By modeling voltage as a stochastic process, we formulate a stochastic differential equationthat captures grid uncertainties. The associated Fokker-Planck-Kolmogorov equation is derived to describe the evolution of the voltage PDF, which enables the formulation of a PDF-shaping control strategy. To simplify the PDF control formulation, a B-spline neural network is introduced for real-time estimation and regulation of the voltage distribution. The proposed PDF control law updates voltage references for energy storage systems and synchronous generators using real-time PDF measurements and feedback signals. The proposed method is validated on a modified Kundur’s two-area system. Simulation results demonstrate that the controller can significantly improve the voltage stability under stochastic conditions, highlighting its effectiveness in modern inverter-rich grids.

Gui, Yonghao [ORNL] (ORCID:0000000250435534)↗

DOE OSTI2024

Operando pair distribution function analysis of nanocrystalline functional materials: the case of TiO 2 -bronze nanocrystals in Li-ion battery electrodes

Structural modelling of operando pair distribution function (PDF) data of complex functional materials can be highly challenging. To aid the understanding of complex operando PDF data, this article demonstrates a toolbox for PDF analysis. The tools include denoising using principal component analysis together with the structureMining , similarityMapping and nmfMapping apps available through the online service `PDF in the cloud' ( PDFitc , https://pdfitc.org/). The toolbox is used for both ex situ and operando PDF data for 3 nm TiO 2 -bronze nanocrystals, which function as the active electrode material in a Li-ion battery. The tools enable structural modelling of the ex situ and operando PDF data, revealing two pristine TiO 2 phases (bronze and anatase) and two lithiated Li x TiO 2 phases (lithiated versions of bronze and anatase), and the phase evolution during galvanostatic cycling is characterized.

Chemistry↗

DOE OSTI2025

Distinguishing isotropic and anisotropic signals for X-ray total scattering using machine learning

Understanding structure–property relationships is essential for advancing technologies based on thin films. X-ray pair distribution function (PDF) analysis can access relevant atomic structure details spanning local-, mid- and long-range structure. While X-ray PDF has been adapted for thin films on amorphous substrates, measurements on single-crystal substrates are necessary to accurately determine structure origins for some thin film materials, especially those for which the substrate changes the accessible structure and properties. However, when measuring films on single-crystal substrates, high-intensity anisotropic Bragg spots saturate 2D detector images, overshadowing the thin films' isotropic scattering signal. This renders previous data processing methods for films on amorphous substrates unsuitable for films on single-crystal substrates. To address this measurement need, we developed IsoDAT2D, an innovative data processing approach using unsupervised machine learning algorithms. The program combines dimensionality reduction and clustering algorithms to separate thin film and single-crystal substrate X-ray scattering signals. We use SimDAT2D , a program we developed to generate simulated thin film data, to validate IsoDAT2D . Here we also use IsoDAT2D to isolate X-ray total scattering signal from a thin film on a single-crystal substrate. The resulting PDF data are compared with similar data processed using previous methods, especially substrate subtraction for single-crystal and amorphous substrates. PDF data from IsoDAT2D -identified X-ray total scattering data are significantly better than from single-crystal substrate subtraction, but not as reliable as PDF data from amorphous substrate subtraction. With IsoDAT2D , there are new opportunities to expand PDF to a wider variety of thin films, including those on single-crystal substrates, with which new structure–property relationships can be elucidated to enable fundamental understanding and technological advances.

36 MATERIALS SCIENCE↗

DOE OSTI2025

Probability Density Function Control of Frequency Fluctuations in Renewable-Rich Power Systems

The stochastic nature of renewable energy sources (RESs) necessitates treating power system frequency response as a random process with a nonstationary probability density function (PDF). Here, based upon the stochastic distribution control theory originated by the second author, this paper proposes a novel stochastic controller to improve the frequency PDF in power grids when integrating a large amount of RESs, thereby minimizing the effects of uncertainties and enhancing overall system stability. The key idea is to manipulate the controllable power generation resources so that the frequency PDF is make to follow a target PDF by using the stochastic distribution control theory originated by the second author. The proposed method can easily be plugged into existing automatic generation controls for multi-area transmission grids. The proposed method is validated via a modified Kundar's two area system and 240-bus Western Electricity Coordinating Council systems. The simulation results show that the proposed control shapes the frequency PDF narrower and sharper, leading to a notable improvement toward minimizing the effects of randomness and uncertainty during grid operation.

frequency↗