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DOE OSTI2025

TMD Handbook

This handbook provides a comprehensive review of transverse-momentum-dependent parton distribution functions and fragmentation functions, commonly referred to as transverse momentum distributions (TMDs). TMDs describe the distribution of partons inside the proton and other hadrons with respect to both their longitudinal and transverse momenta. They provide unique insight into the internal momentum and spin structure of hadrons, and are a key ingredient in the description of many collider physics cross sections. Understanding TMDs requires a combination of theoretical techniques from quantum field theory, nonperturbative calculations using lattice QCD, and phenomenological analysis of experimental data. The handbook covers a wide range of topics, from theoretical foundations to experimental analyses, as well as recent developments and future directions. It is intended to provide an essential reference for researchers and graduate students interested in understanding the structure of hadrons and the dynamics of partons in high energy collisions.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2023

Detailed study of quark-hadron duality in spin structure functions of the proton and neutron

Background: The response of hadrons, the bound states of the strong force (QCD), to external probes can be described in two different, complementary frameworks: as direct interactions with their fundamental constituents, quarks and gluons, or alternatively as elastic or inelastic coherent scattering that leaves the hadrons in their ground state or in one of their excited (resonance) states. The former picture emerges most clearly in hard processes with high momentum transfer, where the hadron response can be described by the perturbative expansion of QCD, while at lower energy and momentum transfers, the resonant excitations of the hadrons dominate the cross section. The overlap region between these two pictures, where both yield similar predictions, is referred to as quark-hadron duality and has been extensively studied in reactions involving unpolarized hadrons. Some limited information on this phenomenon also exists for polarized protons, deuterons, and 3 He nuclei, but not yet for neutrons. Purpose: In this paper, we present comprehensive and detailed results on the correspondence between the extrapolated deep inelastic structure function g 1 of both the proton and the neutron with the same quantity measured in the nucleon resonance region. Thanks to the fine binning and high precision of our data, and using a well-controlled perturbative QCD (pQCD) fit for the partonic prediction, we can make quantitative statements about the kinematic range of applicability of both local duality and global duality. Method: We use the most updated QCD global analysis results at high x from the Jefferson Lab Angular Momentum Collaboration to extrapolate the spin structure function g 1 into the nucleon resonance region and then integrate over various intervals in the scaling variable x. We compare the results with the large data set collected in the quark-hadron transition region by the CLAS Collaboration, including, for the first time, deconvoluted neutron data, integrated over the same intervals. We present this comparison as a function of the momentum transfer Q 2 . Results: We find that, depending on the integration interval and the minimum momentum transfer chosen, a clear transition to quark-hadron duality can be observed in both nucleon species. Furthermore, we show, for the first time, the approach to scaling behavior for g 1 measured in the resonance region at sufficiently high momentum transfer. Conclusions: Here, our results can be used to quantify the deviations from the applicability of pQCD for data taken at moderate energies and can help with extraction of quark distribution functions from such data.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2024

Tagged DVCS on neutron with the CLAS12 Experiment and the detector BONuS12 at Jefferson Lab (USA)

Hadronic physics studies the properties of the proton and neutron. The theory of Quantum Chromodynamics (QCD) governs the structure of hadrons. Non-perturbative structure functions are used to describe the hadrons? structure. Generalized Parton Distributions (GPDs) are a set of structure functions providing access to a 3D view of the nucleon. GPDs contain information about the longitudinal momentum and transverse position of quarks and their correlation. They are related to the spin structure and play a role in solving the proton spin puzzle. They provide access to the quarks? orbital angular momentum, a crucial puzzle piece. For this, measurements on the proton, already present, and neutron are necessary. This thesis aims to explore Compton scattering on the neutron with a deuterium target by measuring the spectator proton with the CLAS12 detector and the BONuS12 RTPC at Jefferson Lab. This work first describes the implementation, which I carried out, of a Kalman filter to reconstruct spectator protons in the BONuS12 detector. The second part of the manuscript is dedicated to the work on the analysis of DVCS on the neutron with the subtraction of background noises. Two different methods for the subtraction of ?0 will be detailed. Finally, the obtained results are presented and discussed

Ouillon, Mathieu↗
DOE OSTI2020

Strong QCD Insights from Excited Nucleon Structure Studies with CLAS and CLAS12

Studies of the spectrum of hadrons and their structure in experiments with electromagnetic probes offer unique insight into many facets of the strong interaction in the regime of large quark-gluon running coupling, i.e. the regime of strong QCD. The experimental program within Hall B at Jefferson Laboratory based on data acquired with the CLAS spectrometer using electron and photon beams with energies up to 6 GeV has already considerably extended the scope of research in hadron physics in joint efforts between experiment and phenomenological data analysis. Impressive progress in relating the hadron structure observables inferred from the data to the strong QCD mechanisms underlying hadron mass generation has been achieved in the past decade. Furthermore, these results will be considerably extended with data from the experimental program with the new CLAS12 spectrometer that has begun data taking using electron beams with energies up to 11 GeV. With this extended kinematic reach the structure of nucleon resonances will be probed at the highest photon virtualities ever achieved in the studies of exclusive electroproduction, which will allow for the exploration of the distance scale where>98% of light hadron mass emerges from QCD in the transition of the strong interaction from the regime of quark-gluon confinement to perturbative QCD.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2022

Physics with CEBAF at 12 GeV and future opportunities

Here, we summarize the ongoing scientific program of the 12 GeV Continuous Electron Beam Accelerator Facility (CEBAF) and give an outlook into future opportunities. The program addresses important topics in nuclear, hadronic, and electroweak physics, including nuclear femtography, meson and baryon spectroscopy, quarks and gluons in nuclei, precision tests of the standard model and dark sector searches. Potential upgrades of CEBAF and their impact on scientific reach are discussed, such as higher luminosity, the addition of polarized and unpolarized positron beams, and doubling the beam energy.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2021

Forward and Off-Forward Parton Distributions from Lattice QCD

The interpretation of (semi-)inclusive and certain exclusive scattering processes relies on the factorization of hard parton level cross sections from long-range and non-perturbative parton correlations. The familiar Parton Distribution Functions (PDFs) and Generalized Parton Distributions quantify the non-perturbative dynamics in these situations and address a number of key questions surrounding the structure of hadrons. A certain class of matrix elements accessible in lattice QCD, so called Lattice Cross Sections, have been shown to factorize into these collinear distributions in a manner akin to the factorization of hadronic cross sections. In the short-distance regime, matrix elements of space-like separated two-current operators and parton bilinears can be expressed as the convolution of perturbative coefficient functions and the PDFs. Matrix elements of this type are isolated in the pion and nucleon, each offering a glimpse of the unpolarized valence quark content of these phenomenologically important hadronic states. The calculations within the nucleon represent the first application of the distillation spatial smearing paradigm to the collinear structure of hadrons, and is found to offer higher precision data compared to similar calculations in the literature. A novel method to obtain PDFs from these lattice data, while simultaneously controlling systematic effects, is developed and applied to the nucleon dataset. The coordinate space factorization of space-like separated parton bilinears has also recently been extended to include Generalized Parton Distributions. Preliminary results in off-forward nucleon matrix elements using distillation are explored.

Egerer, Colin↗
DOE OSTI2026

2025 EIC-France Workshop: Physics Highlights and Perspectives

This document presents a synthesis of the theory contributions and discussions from the 2nd EIC-France Workshop, held at IJCLab (Orsay) on 1-3 December 2025. The workshop brought together members of the French hadron-physics community to review recent theoretical developments relevant to the future Electron-Ion Collider (EIC) and to coordinate national efforts in preparation for its early physics program. The report first summarizes the collider's initial running conditions and luminosity performance, as outlined in the EIC Early Science Matrix. It then provides concise overviews of the theoretical presentations on inclusive, semi-inclusive, exclusive, heavy-flavor, and small-x physics. Based on these discussions, two measurements emerged as especially well suited for early EIC operation and strongly aligned with areas of established French expertise: inclusive diffraction and inclusive quarkonium production. These channels offer clean signatures, robust theoretical interpretability, and direct sensitivity to fundamental QCD phenomena such as gluon saturation, heavy-quark dynamics, and the small-x structure of hadrons and nuclei. In addition, the workshop identified longer-term physics opportunities that will benefit from the full capabilities of the EIC after its ramp-up phase. These include accessing the three-dimensional structure of the pion through the Sullivan process and a broader program of exclusive three-body final states, both of which represent high-impact avenues for exploring hadronic structure and non-perturbative QCD. Together, the elements summarized in this report provide a coherent overview of the strategic priorities and scientific ambitions shaping the French community's contribution to the EIC physics program.

FOS: Physical sciences↗
DOE OSTI2026

Roper Resonance Structure and Exploration of Emergent Hadron Mass from CLAS Electroproduction Data

The N(1440)1/2 + nucleon resonance, first identified in 1964 by Roper and collaborators in analyses of πN hadroproduction data, has continued to provide pivotal insights that serve to advance our understanding of nucleon excited states. In this contribution, we present results from studies of the structure of the Roper resonance based on exclusive πN and π + π − p electro production data measured with the CLAS detector at Jefferson Lab. These analyses have revealed the Roper resonance as a complex interplay between an inner core of three dressed quarks and an external meson–baryon cloud. Analyses of the CLAS results on the evolution of the Roper resonance electroexcitation amplitudes with photon virtuality Q 2 , within the framework of the Continuum Schwinger Method, have conclusively demonstrated the capability to gain insight into the strong interaction dynamics responsible for generating more than 98% of hadron mass. Further extension of such studies to higher Q 2 , through experiments currently underway with the CLAS12 detector and in the future with a potential CEBAF energy upgrade to 22 GeV, offers the only foreseeable opportunity to explore the full range of distances where the dominant portion of hadron mass and resonance structure emerge.

Mokeev, Victor I. [Thomas Jefferson National Accel↗
DOE OSTI2022

Electron-Hadron Colliders: EIC, LHeC and FCC-eh

Electron-hadron colliders are the ultimate tool for high-precision quantum chromodynamics studies and provide the ultimate microscope for probing the internal structure of hadrons. The electron is an ideal probe of the proton structure because it provides the unmatched precision of the electromagnetic interaction, as the virtual photon or vector bosons probe the proton structure in a clean environment, the kinematics of which is uniquely determined by the electron beam and the scattered lepton, or the hadronic final state accounting appropriately for radiation. The Hadron Electron Ring Accelerator HERA (DESY, Hamburg, Germany) was the only electron-hadron collider ever operated (1991–2007) and advanced the knowledge of quantum chromodynamics and the proton structure, with implications for the physics studied in RHIC (BNL, Upton, NY) and the LHC (CERN, Geneva, Switzerland). Recent technological advances in the field of particle accelerators pave the way to realize next-generation electron-hadron colliders that deliver higher luminosity and enable collisions in a much broader range of energies and beam types than HERA. Electron-hadron colliders combine challenges from both electron and hadron machines besides facing their own distinct challenges derived from their intrinsic asymmetry. This review paper will discuss the major features and milestones of HERA and will examine the electron-hadron collider designs of the Electron-Ion Collider (EIC) currently under construction at BNL, the CERN’s Large Hadron electron Collider (LHeC), at an advanced stage of design and awaiting approval, and the Future Circular lepton-hadron Collider (FCC-eh).

43 PARTICLE ACCELERATORS↗
DOE OSTI2024

Extracting the femtometer structure of strange baryons using the vacuum polarization effect

Abstract One of the fundamental goals of particle physics is to gain a microscopic understanding of the strong interaction. Electromagnetic form factors quantify the structure of hadrons in terms of charge and magnetization distributions. While the nucleon structure has been investigated extensively, data on hyperons are still scarce. It has recently been demonstrated that electron-positron annihilations into hyperon-antihyperon pairs provide a powerful tool to investigate their inner structure. We present a method useful for hyperon-antihyperon pairs of different types which exploits the cross section enhancement due to the effect of vacuum polarization at theJ/ψresonance. Using the 10 billionJ/ψevents collected with the BESIII detector, this allows a precise determination of the hyperon structure function. The result is essentially a precise snapshot of the$$\bar{\Lambda }{\Sigma }^{0}\,(\Lambda {\bar{\Sigma }}^{0})$$ Λ ¯ Σ 0 ( Λ Σ ¯ 0 ) transition process, encoded in the transition form factor ratio and phase. Their values are measured to beR = 0.860 ± 0.029(stat.) ± 0.015(syst.),$$\Delta {\Phi }_{\bar{\Lambda }{\Sigma }^{0}}=(1.011\pm 0.094({{\rm{stat.}}})\pm 0.010({{\rm{syst.}}}))\,{{\rm{r}}}ad$$ Δ Φ Λ ¯ Σ 0 = ( 1.011 ± 0.094 ( stat. ) ± 0.010 ( syst. ) ) r a d and$$\Delta {\Phi }_{\Lambda {\bar{\Sigma }}^{0}}=(2.128\pm 0.094({{\rm{stat.}}})\pm 0.010({{\rm{syst.}}}))\,{{\rm{r}}}ad$$ Δ Φ Λ Σ ¯ 0 = ( 2.128 ± 0.094 ( stat. ) ± 0.010 ( syst. ) ) r a d . Furthermore, charge-parity (CP) breaking is investigated in this reaction and found to be consistent with CP symmetry.

Science & Technology - Other Topics↗
DOE OSTI2024

Towards unpolarized GPDs from pseudo-distributions

We present an exploration of the unpolarized isovector proton generalized parton distributions (GPDs) H u−d (x, ξ, t) and E u−d (x, ξ, t) in the pseudo-distribution formalism using distillation. Taking advantage of the large kinematic coverage made possible by this approach, we present results on the moments of GPDs up to the order x 3 — including their skewness dependence — at a pion mass m π = 358 MeV and a lattice spacing a = 0.094 fm.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2026

A unified neural-network framework for nucleon imaging from numerical simulations of QCD

Parton distributions encode the momentum-space structure and, in their generalizations, the spatial tomography of quarks and gluons inside hadrons, the building blocks of visible matter. We present a unified neural-network approach that learns these distributions directly from matrix elements calculated via numerical simulations of quantum chromodynamics (QCD) on the lattice by fitting two complementary inputs simultaneously: data matched to physical quantities via known momentum-space and coordinate-space formalisms. Utilizing data from both methods stabilizes the extraction and mitigates biases that can arise when either is used alone. We validate the method on controlled mock data and apply it to lattice-QCD matrix elements to extract parton distribution functions (PDFs). We show benefits of such an approach for determining the physical quantities. We further extend the framework to zero-skewness generalized parton distributions and demonstrate nucleon tomography within the same neural-network parameterization. Our results provide an adaptable and systematically improvable approach for extracting partonic distributions from Euclidean correlators. It can incorporate polarization, additional channels, and future experimental constraints from current and future facilities, such as the Electron-Ion Collider.

Hadronic Spectroscopy↗
DOE OSTI2021

The Drell-Yan process with pions and polarized nucleons

The Drell-Yan process provides important information on the internal structure of hadrons including transverse momentum dependent parton distribution functions (TMDs). In this work we present calculations for all leading twist structure functions de- scribing the pion induced Drell-Yan process. The non-perturbative input for the TMDs is taken from the light-front constituent quark model, the spectator model, and available parametrizations of TMDs extracted from the experimental data. TMD evolution is implemented at Next-to-Leading Logarithmic precision for the first time for all asymmetries. Our results are compatible with the first experimental information, help to interpret the data from ongoing experiments, and will allow one to quantitatively assess the models in future when more precise data will become available.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2020

Efficient Fourier transforms for transverse momentum dependent distributions

Hadron production at low transverse momenta in semi-inclusive deep inelastic scattering can be described by transverse momentum dependent (TMD) factorization. This formalism has also been widely used to study the Drell-Yan process and back-to-back hadron pair production in $e^+e^-$ collisions. These processes are the main ones for extractions of TMD parton distribution functions and TMD fragmentation functions, which encode important information about nucleon structure and hadronization. One of the most widely used TMD factorization formalism in phenomenology formulates TMD observables in coordinate $b_\perp$-space, the conjugate space of the transverse momentum. The Fourier transform from $b_\perp$-space back into transverse momentum space is sufficiently complicated due to oscillatory integrands that it requires a careful and computationally intensive numerical treatment in order to avoid potentially large numerical errors. Within the TMD formalism, the azimuthal angular dependence is analytically integrated and the two-dimensional $b_\perp$ integration reduces to a one-dimensional integration over the magnitude $b_\perp$. In this paper we develop a fast numerical Hankel transform algorithm for such a $b_\perp$-integration that improves the numerical accuracy of TMD calculations in all standard processes. Libraries for this algorithm are implemented in Python 2.7 and 3, C++, as well as FORTRAN77. All packages are made available open source.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Elastic and resonance structures of the nucleon from the hadronic tensor in lattice QCD: Implications for neutrino-nucleon scattering and hadron physics

We compute the Euclidean hadronic tensor from charge density operators and extract elastic and resonance structures by employing exponential fits to the four-point function correlator, as well as a Bayesian reconstruction inverse algorithm to obtain the corresponding spectral density for qualitative comparison. We present the determination of the nucleon’s Sachs electric form factor using the hadronic tensor formalism and verify that it is consistent with that from the conventional three-point function calculation. Beyond the elastic peak, we observe a structure located approximately 0.5–0.7 GeV above the nucleon mass in the Bayesian reconstruction. This structure is interpreted as a mixture of the Roper resonance [𝑁⁡(1440)], and states with both positive and negative parities in this mass region, as well as multihadron states. Assuming the observed structure is dominated by 𝐽 𝑃 = 1/2 ± states, we extract the transition electric form factor 𝐺$^*_𝐸$⁡(𝑄 2 ) and the corresponding longitudinal helicity amplitude 𝑆 1/2 ⁡(𝑄 2 ), and compare them with those determined from the CLAS experimental data of nucleon-to-Roper transition. Although fitting to the four-point correlation function or using the inverse algorithm does not resolve individual resonances, it nevertheless enables the determination of total inclusive lepton–nucleon scattering cross sections in appropriate energy bins. This lattice QCD calculation presents the first major step toward studying the inclusive 𝑁 → 𝑋 contributions within the hadronic tensor formalism.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2025

Measurement of the Lund jet plane in hadronic decays of top quarks and W bosons with the ATLAS detector

The Lund jet plane (LJP) is measured for the first time in $t\bar{t}$ events, using 140 $\textrm{fb}^{-1}$ of $\sqrt{s} = 13$ TeV pp collision data collected with the ATLAS detector at the LHC. The LJP is a two-dimensional observable of the sub-structure of hadronic jets that acts as a proxy for the kinematics of parton showers and hadron formation. The observable is constructed from charged particles and is measured for R = 1.0 anti-$k_t$ jets with transverse momentum above 350 GeV containing the full decay products of either a top quark or a daughter W boson. The other top quark in the event is identified from its decay into a b-quark, an electron or a muon and a neutrino. The measurement is corrected for detector effects and compared with a range of Monte Carlo predictions sensitive to different aspects of the hadronic decays of the heavy particles. In the W-boson-initiated jets, all the predictions are incompatible with the measurement. In the top quark initiated jets, disagreement with all predictions is observed in smaller subregions of the plane, and with a subset of the predictions across the fiducial plane. The measurement could be used to improve the tuning of Monte Carlo generators, for better modelling of hadronic decays of heavy quarks and bosons, or to improve the performance of jet taggers.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2023

The Present and Future of QCD: QCD Town Meeting White Paper – An Input to the 2023 NSAC Long Range Plan

It is currently understood that there are four fundamental forces in nature: gravitational, electromagnetic, weak and strong forces. The strong force governs the interactions between quarks and gluons, elementary particles whose interactions give rise to the vast majority of visible mass in the universe. The mathematical description of the strong force is provided by the non-Abelian gauge theory Quantum Chromodynamics (QCD). While QCD is an exquisite theory, constructing the nucleons and nuclei from quarks, and furthermore explaining the behavior of quarks and gluons at all energies, remain to be complex and challenging problems. Such challenges, along with the desire to understand all visible matter at the most fundamental level, position the study of QCD as a central thrust of research in nuclear science. Experimental insight into the strong force can be gained using large particle accelerator facilities, which are necessary to probe the very short distance scales over which quarks and gluons interact. The Long Range Plans (LRPs) exercise of 1989 and 1996 led directly to the construction of two world-class facilities: the Continuous Electron Beam Accelerator Facility (CEBAF) at Jefferson Lab (JLab) that is focused on studying how the structure of hadrons emerges from QCD (cold QCD research), and the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Lab (BNL) that aims at the discovery and study of a new state of matter, the quark-gluon plasma (QGP), at extremely high temperatures (hot QCD research). These past investments have produced major advances. Nucleons and nuclei are being studied with increasing precision with a unified description of the partonic structure utilizing multi-dimensional imaging. Significant progress has been made, paving the way towards a complete picture of how quarks and gluons give rise to the mass, spin, and momentum of the nucleon. In hot QCD, the QGP is created in the collisions of nuclei at RHIC and the Large Hadron Collider (LHC) and is observed to behave like a fluid with very low specific shear viscosity; the current goals are to understand how the fluid behavior emerges from QCD and to characterize the temperature (and chemical potential) dependence of the properties of the QGP. As this White Paper is written, current experimental programs at CEBAF, RHIC and the LHC continue to provide exciting near term opportunities to capitalize on the investments in experimental equipment and accelerator operations. Most importantly, the QCD community looks forward to the construction of the Electron Ion Collider (EIC) as a major new facility to push forward QCD research in the next decades, with significant focus on exploring the properties of gluons, the mediators of the strong force.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2020

Deeply Virtual Compton Scattering on the Proton in Hall A at Jefferson Lab

The structure of nucleons as they are composed of quarks and gluons is not well understood. Non-perturbative methods (such as lattice-QCD) describing quark/gluon interactions are ongoing in development, but still cannot fully reproduce the structure of hadrons. Among the observables which describe longitudinal momentum and transverse position of partons, generalized parton distributions (GPDs) describe the correlations between the two, and are accessible through p(right-arrow e;e'y)p scattering experiments in the Bjorken limit when factorization holds. A complete understanding of these functions will help us to better understand the structure of nucleons. Experiment E12-06-114 was performed by JLab's DVCS Hall A Collaboration in 2014 and 2016. Using a polarized electron beam and a liquid hydrogen target, the collaboration measured cross sections for the exclusive reaction p(right-arrow e;e'y)p at xB=0.36, 0.48, and 0.60,3 .2 lte Q2 lte 9.00, and -1.28 lte t lte -0.69. This dissertation details the analysis and results of measuring the p(right-arrow e;e'y)p cross section at xB=0.48. The measured cross section results allow us to demonstrate the validity of factorization in our kinematic regime, and extract Compton Form Factor combinations important for understanding GPDs. Additionally, the cross section results are used to test existing GPD models, KM10a and KM15.

Johnson, Alexa N.↗