Workshop Opening

• Yoshitaka Hatta (BNL, RBRC)

Toward a global analysis of GPDs through GUMP

• Xiangdong Ji (University of Maryland)

Global analysis of GPDs is challenging because they contain multiple kinematic variables and usable physics constraints are often dwarfed by the number of degrees of freedom in fitting. In this talk, I will describe the rationale behind an approach named as "GPDs through Universal Moment Parameterization" or GUMP. It adopts the standard PDF fitting strategies used by, e.g., CTEQ, but extends it smoothly into non-forward directions, allowing almost all of the constraints on GPDs be included in the fitting. Some level of success has been achieved in analysis of the existing data, including gluon GPDs.

What can we learn from amplitudes for DVCS?

• Paweł Sznajder (National Centre for Nuclear Research, Poland)

We discuss the problem of model dependency in the extraction of generalized parton distributions (GPDs). In processes like deeply virtual Compton scattering (DVCS), this problem is highlighted by the presence of 'shadow' GPDs, which do not contribute to observables for this process. The problem can be explicitly solved by considering double deeply virtual Compton scattering (DDVCS), or thanks to the recent breakthrough in evaluating GPDs from lattice QCD. In this talk we will argue, however, that analyses of amplitudes (not requiring the deconvolution) still allow us to learn a lot about the structure of hadrons. In particular, one can directly study the nuclear tomography at low $\xi$, mechanical properties via the subtraction constant appearing in the dispersion relation, and response of hadrons on probes carrying different angular momenta thanks to the recent development of techniques based on the Froissart-Gribov projections.

The D-term form factor from dispersion relations in deeply virtual Compton scattering

• Barbara Pasquini (University of Pavia and INFN, Pavia)

We review the dispersion analysis of deeply virtual Compton scattering and present a dispersive representation of the D-term form factor for hard exclusive reactions. We use unsubtracted $t$-channel dispersion relations, where the $t$-channel unitarity relation is saturated with the contribution of two-pion intermediate states, using the two-pion distributions amplitude for the $\gamma^*\gamma\rightarrow \pi\pi$ subprocess and reconstructing the $\pi\pi\rightarrow N\bar N$ subprocess from available information on pion-nucleon partial-wave helicity amplitudes. Results for the D-term form factor as function of $t$ as well as at $t=0$ are discussed in comparison with available model predictions and phenomenological parametrizations.

Exclusive and Diffractive Physics Opportunities at the Electron-Ion Collider

• Alexander Jentsch (Brookhaven National Laboratory)

The Electron-Ion Collider (EIC) will afford the opportunity to drastically advance our understanding of QCD and the multidimensional structure of both protons and nuclei. An essential component of the EIC physics program is the measurement and study of exclusive and diffractive final states, which yield insight into topics including partonic imaging, structure functions, proton spin, and saturation. The EIC Yellow Report [1] provides a comprehensive look at the exclusive physics program, and several studies have been published since the Yellow Report detailing the experimental needs for these measurements [2-5]. The primary challenge for these final states is measuring the particles they produce, which are generally near-collinear with the outgoing hadron beam and often cannot be seen by the central detectors of the EIC project detector, ePIC. It is therefore important to use subsystems integrated with the outgoing hadron beam-line, the so-called “far-forward” detectors. The ePIC experiment includes a suite of far-forward detectors designed to deliver the necessary geometric coverage and resolution required to achieve the full exclusive physics program envisioned at the EIC. In this presentation, the ePIC far-forward detectors will be briefly introduced, progress on technology selection/evaluation and accelerator integration will be discussed, and a few relevant physics topics will be addressed, in detail. References [1] R. Abdul Khalek et al., Nuclear Physics A Volume 1026, October 2022, 122447 [2] A. Jentsch, Z. Tu, and C. Weiss, Phys. Rev. C 104, 065205, (2021) (Editor’s Suggestion) [3] W. Chang, E.C. Aschenauer, M. D. Baker, A. Jentsch, J.H. Lee, Z. Tu, Z. Yin, and L.Zheng, Phys. Rev. D 104, 114030 (2021) [4] Z. Tu, A. Jentsch, et al., Physics Letters B, (2020) [5] I. Friscic, D. Nguyen, J. R. Pybus, A. Jentsch, et al., Phys. Lett. B, Volume 823, 136726 (2021)

Prospects for Double DVCS measurements

• Jakub Wagner (National Centre for Nuclear Research)

We revisit the phenomenology of the deep exclusive electroproduction of a lepton pair, i.e. double deeply virtual Compton scattering (DDVCS), in view of new experiments planned in the near future. The importance of DDVCS in the reconstruction of generalized parton distributions (GPDs) in their full kinematic domain is emphasized. We provide the formulation convenient for practical implementation in the PARTONS framework and EpIC Monte Carlo generator that we use in simulation studies.

Machine Learning for Inverse Problems in Femtography

• Brandon Kriesten (Argonne National Lab)

Deeply virtual exclusive reactions are theorized to be sensitive to the dynamics of bound partons in hadrons through 3D quantum mechanical phase space distributions - the generalized parton distributions; however, there are many layers of abstraction in the analysis from experimental data to information on hadron structure. The FemtoNet framework of the EXCLAIM collaboration was developed to analyze deeply virtual exclusive experimental data using deep learning models to quantify information loss and reconstruction through the many inverse problems encountered. Simultaneously, this framework leverages a suite of uncertainty quantification techniques to separate epistemic (reducible) and aleatoric (irreducible) errors from the analysis and properly propagate experimental uncertainty. I will demonstrate one such deep neural network, called the variational autoencoder inverse mapper (VAIM) and show what it is capable of in the context of reconstructing lost information from inverse problems in exclusive scattering experiments.

Generalized Parton Distributions through Universal Moment Parameterization: gluons at non-zero skewness

• Gabriel Santiago (SURA/Jefferson Lab)

We present a further step toward a global extraction of gluon generalized parton distributions (GPDs). In our previous work we performed the first global analysis of quark GPDs by including lattice quantum chromodynamics (QCD) calculations, global fitted forward parton distribution functions (PDFs), form factors (FFs), and Deeply Virtual Compton Scattering (DVCS) measurements from JLab and Hadron-Electron Ring Accelerator (HERA) to constrain two quark flavors with leading order QCD evolution. There, the inclusion of DVCS did not probe gluon structure at leading order, as the gluon GPDs only enter through evolution. Here, we include HERA measurements of Deeply Virtual Meson Production (DVMP) in order to study gluon GPDs at non-zero skewness using the same moment parameterization ansatz and obtaining results consistent with the previous global fit including lattice QCD calculations and experimental measurements. We concentrate our study on the production of J/Ψ mesons in order to limit quark contributions and thus allow for greater constraints on the gluon GPDs at non-zero skewness.

Gravitational form factors on the lattice

• Daniel Hackett (Fermilab)

The gravitational form factors (GFFs) of hadrons are the subject of ongoing and quickly developing theoretical and experimental investigation. These quantities, defined from hadronic matrix elements of the energy-momentum tensor, encode fundamental aspects of a hadron's structure, including how mass, spin, and internal forces are distributed both spatially and among the hadron's constituents. In this talk, I present and contextualize the results of a recent first lattice determination of the flavor decomposition of the GFFs of the proton and pion at close-to-physical parameters.

Probing the gravitational form factors of the nucleon from near threshold heavy quarkonium photo-production

• Yuxun Guo

The gravitational form factors (GFFs) are one of most fundamental quantities that carry information about the mechanic properties such as the mass and spin of the nucleon. Probing them, on the other hand, has been experimentally challenging due to the weak gravitational coupling of the nucleon. The generalized parton distribution (GPD) not only offers an alternative way to access these GFFs, but can also provides further information of the 3D structure of the nucleons, and therefore has captivated rising interest. In this talk, I will discuss our recent work on the near threshold heavy quarkonium photo-production in the GPD framework. I will talk about how the GFFs can be constrained/extracted from such processes in the large skewness limit and discuss the recent measurements of by the experimental groups at JLab as an example.

Lattice QCD calculation of the pion generalized parton distributions (GPDs)

• Qi Shi (BNL)

We present a Lattice QCD calculation of the generalized parton distributions (GPDs) for the pion. Focusing on the zero skewness, we obtain the matrix elements from both symmetric and asymmetric kinematic frames with the recently proposed Lorentz-invariant definition. The calculations are performed using a single ensemble of $N_f=2+1$ highly-improved staggered quarks with $m_\pi = 300$ MeV and a lattice spacing $a=0.04$ fm. We use the hybrid-scheme renormalization with a perturbative matching up to NNLO to obtain the valence light-cone GPDs.

Probing quark OAM in excluisve $\pi^0$ production in ep collisions

• Jian Zhou (Shandong University, QingDao, China)

We propose to extract quark orbital angular momentum (OAM) through exclusive $\pi^0$ production in electron-(longitudinally-polarized) proton collisions. Our analysis demonstrates that the $sin(2\phi)$ azimuthal angular correlation between the transverse momentum of the scattered electron and the recoil proton serves as a sensitive probe of quark OAM. Additionally, we present a numerical estimate of the asymmetry associated with this correlation for the kinematics accessible at EIC and EicC. This study aims to pave the way for the first measurement of quark OAM in relation to the JaffeManohar spin sum rule.

From nonsinglet to singlet GPDs in lattice QCD

• Jianhui Zhang (The Chinese University of Hong Kong, Shenzhen, China)

Exploring hadron structure with generalized parton distributions (GPDs) is an important goal of the EIC program. In recent years, there has been rapid progress on calculating the GPDs from lattice QCD, with a particular focus on the nonsinglet case. In this talk, I will discuss the computational strategy that is required when extending the calculation from nonsinglet to singlet combinations.

Recent advances on GPDs from lattice QCD

• Martha Constantinou (Temple University)

Generalized parton distributions (GPDs) play a crucial role in characterizing the 3-D structure of hadrons, offering valuable insights into the momentum and position space distributions of partons. These distributions, essential for understanding high-energy processes, are expanded in terms of the process's large energy scale, giving rise to a tower of distribution functions labeled by their twist. While leading twist (twist-2) contributions have been extensively studied, the recognition of sizable twist-3 contributions emphasizes the need to explore these aspects for a comprehensive understanding of the proton's structure. However, experimentally disentangling twist-3 contributions from their leading counterparts poses a significant challenge. Existing lattice QCD data primarily focus on the Mellin moments of GPDs, such as form factors and generalized form factors. Recent advancements in lattice QCD techniques now enable the calculation of the x-dependence of GPDs. This work utilizes the quasi-distribution approach, involving matrix elements of fast-moving hadrons coupled to non-local operators. These quasi-distributions are subsequently matched to light-cone distributions using Large Momentum Effective Theory (LaMET). This approach has been extensively studied for twist-2 PDFs. In this presentation, we demonstrate the extension of such methods to twist-2 GPDs and, more recently, to twist-3 PDFs and GPDs. This showcases the potential of lattice QCD calculations to complement theoretical and experimental efforts, offering a nuanced exploration of the 3-D structure of hadrons.

Twist-four quark/gluon gravitational form factor $\bar{C}_{q,g}$ at NNLO QCD from trace anomaly constraints

• Kazuhiro Tanaka (Juntendo University, Inzai, Chiba, Japan)

he gravitational form factors are related to a particular moment of the GPDs, and are relevant to EIC physics. The gravitational form factors are subject to the QCD constraints as the hadron matrix elements of the energy-momentum tensor $T^{\mu \nu}$, and we discuss the results for, in particular, the twist-four gravitational form factor for the quark as well as for the gluon, $\bar{C}_{q,g}$. It is known that the trace anomaly in the QCD energy-momentum tensor $T^{\mu \nu}$ can be attributed to the anomalies for each of the gauge-invariant quark part and gluon part of $T^{\mu \nu}$, and their explicit three-loop formulas have been derived in the $\overline{\rm MS}$ scheme in the dimensional regularization. The matrix elements of this quark/gluon decomposition of the QCD trace anomaly allow us to derive the QCD constraints on the twist-four quark/gluon gravitational form factor, $\bar{C}_{q,g}$. Using the three-loop quark/gluon trace anomaly formulas, we calculate the forward (zero momentum transfer) value of $\bar{C}_{q,g}$ at the next-to-next-to-leading-order (NNLO) accuracy. We present quantitative results for nucleon as well as for pion, leading to a model-independent determination of the forward value of $\bar{C}_{q,g}$. We find quite different pattern in the obtained results between the nucleon and the pion. In particular, for the nucleon, the present information from experiment and lattice QCD on the nonperturbative matrix elements arising in our NNLO formula allows us to obtain a prediction of the forward value of $\bar{C}_{q,g}$ at the accuracy of a few percent level. We also mention that the same framework based on the trace anomaly constraints also indicates quite different pattern in the mass structures between the nucleon and the pion. This talk is based on JHEP 03 (2023) 013.

Single diffractive hard exclusive processes for the study of generalized parton distributions

• Zhite Yu (Jefferson Lab)

Generalized parton distributions (GPDs) are important nonperturbative functions that provide tomographic images of the partonic structures of hadrons. We introduce a type of exclusive processes for a better study of GPDs, which we refer to as single diffractive hard exclusive processes (SDHEPs), and give a general proof for their factorization into GPDs. We demonstrate that the SDHEP is not only sufficiently generic to cover all known processes but also well motivated for searching for new processes for extracting GPDs. Importantly, we also examine the sensitivity of the SDHEP to the parton momentum fraction $x$ dependence of GPDs, and demonstrate quantitatively with two processes that can be readily measured at J-PARC/AMBER and JLab, respectively.

The electron-ion collider -- A world wide unique collider to unravel the 3d structure of visible matter

• E. C. Aschenauer (BNL)

Understanding the properties of nuclear matter and its emergence through the underlying partonic structure and dynamics of quarks and gluons requires a new experimental facility in hadronic physics known as the Electron-Ion Collider (EIC). The EIC will address some of the most profound questions concerning the emergence of nuclear properties by precisely imaging gluons and quarks inside protons and nuclei such as their distributions in space and momentum. In January 2020 the EIC received CD-0 and Brookhaven National Laboratory was selected as site, followed by CD-1 in June 2021 and CD-3A is expected by early spring 2024. This presentation will give highlights on the EIC 3d hadron structure science program, introduce the experimental equipment and accelerator capabilities needed to make the science program a success. If time permits the status of the EIC project, as well what are the next major steps will be presented.

Flavor decomposition of the proton gravitational form factors and mechanical structure of the proton

• Hyun-Chul Kim (nha University, Republic of Korea)

The gravitational form factors of the proton provide essential information on its mechanical structure such as its mass, spin, mechanical pressure, and shear force. In the current talk, we present recent results for the flavor decomposition of the gravitational form factors (GFFs) of the proton in a pion mean-field approach or the chiral quark-soliton model, which illuminate the distinctive role played by the separate quarks. We assess the quantitative fraction of light-front momentum carried by individual quark flavors within the nucleon. We analyze variations arising from the flavor decomposition of the mass distribution, as evidenced through the non-conserved $\bar{c}(q^2)$ form factor of the nucleon, which is also known as the proton cosmological term. We study not only the decomposition of the total angular momentum into the orbital angular momentum and intrinsic spin, but also its flavor decomposition. We then examine the intricate interplay between the D-term and $\bar{c}$ form factors and their collaborative impact on the stabilization of the nucleon system. Questioning the assumption of “large Nc blindness” concerning $D^{u−d} \sim 0$ in a recent experimental analysis, we compute it within both the flavor SU(2) and SU(3) framework. We conclude that such an assumption finds justification predominantly within the framework of flavor SU(3) symmetry.

Vector mesons in exclusive reaction $e+p \to {\rm VM} + p$

• Mark Strikman

We review the HERA data on production of vector mesons in exclusive reaction $e+p \to {\rm VM} + p$ and argue that corrections to the leading twist leading twist remain strong up to rather large $Q^2$. Also, we suggest the procedure for extracting information on the small $x$ GPDs of heavy nuclei within the leading twist shadowing framework using information on neutron production from a Zero Degree Calorimeter.

Nucleon GPDs from Basis Light-front Quantization

• Xingbo Zhao (hinese Academy of Sciences, Lanzhou, China)

Basis Light-front Quantization (BLFQ) is a recently constructed nonperturbative approach to quantum field theory based on light-front quantization. In BLFQ, the wave functions of bound states are found by solving the eigenvalue problem of the light-front Hamiltonian of quantum field theory. In this talk I will introduce our recent work of applying BLFQ to QCD and calculating the light-front wave function of the nucleon in a truncated Fock space containing one dynamical gluon and one pair of quark and antiquark. Based on the obtained light-front wave function, we calculated the GPDs of the nucleon at the leading twist. I will present our numerical results of the quark and gluon GPDs at both zero and nonzero skewness. In addition, I will show the corresponding Compton form factors by integrating out the parton’s longtudinal momenum fraction in the GPDs.

The role of the chiral anomaly in GPDs

• Andrey Tarasov (North Carolina State University)

It has long been understood that spin dependent observables in high-energy scattering are sensitive to the physics of the chiral anomaly in QCD. One example is the generalized parton distribution (GPD) $\tilde{E}$, another is the polarized proton structure function $g_1$. The anomaly manifests itself as an infrared pole $1/l^2$ of the one-loop quark box diagram for the Deeply Virtual Compton Scattering (DVCS) and polarized Deep Inelastic Scattering (DIS), where $l^\mu$ is the off-forward momentum transfer. The cancellation of the pole involves a subtle interplay of perturbative and non-perturbative physics that is deeply related to the $U_A(1)$ problem in QCD. This cancellation allows to relate the aforementioned functions to the topological properties of the QCD vacuum. However, the extraction of the anomaly effects is challenging because it requires calculation in full off-forward kinematics. In particular, it concerns the tensorial structure of the box diagram and derivation of the twist-four pseudoscalar operator $F\tilde{F}$ which is associated with the anomaly. In this talk I’ll present a novel approach based on the worldline formalism, which allows to unambiguously define the tensorial structure of the box diagram and corresponding kinematic factors. The application of the method goes beyond the anomaly effects and allows to fully determine the contribution of the box diagram to GPDs.

Deuteron structure in a covariant separable framework

• Adam Freese

I propose a separable but non-local nucleon-nucleon interaction Lagrangian, for which the Bethe-Salpeter equation can be solved covariantly. The deuteron wave function in the non-relativistic limit can be matched to precision wave functions (such as AV18, Paris, Nijmegen and Bonn) to fix the parameters in the Lagrangian. The separable interaction Lagrangian thus provides a means of covariantly calculating relativistic deuteron structure using realistic, well-established non-relativistic potentials as input. I present results for form factors and DIS structure functions to demonstrate feasibility of the framework, which will ultimately be used to calculate deuteron GPDs that observe polynomiality.

Gluon distributions in exclusive Compton Scattering from low to moderate x

• Renaud Boussarie (CPHT)

With similar techniques to the semi-classical high energy descriptions such as the Color Glass Condensate effective framework, we build an interpolating formula between the Bjorken regime and the saturated Regge limit of exclusive Compton Scattering processes (DDVCS and its DVCS and TCS limits). We find a new operator definition for an unintegrated gluon Generalized Parton Distribution, factorized in such a way that the Feynman x dependence is explicit in both the hard and the hadronic matrix elements. We obtain a compact factorized master formula that can be used as a top-down approach to collinear logarithms in the CGC.

GPDs and gluonic structure from the instanton vacuum

• Christian Weiss (Jefferson Lab)

GPDs represent hadronic matrix elements of QCD operators composed of quark and gluon fields. Predicting/explaining these structure from nonperturbative dynamics at the hadronic scale is a major challenge. It requires a picture of the relevant nonperturbative gluon fields in hadrons and their interactions with quarks. Topological fluctuations of the QCD gauge fields are known to cause the dynamical breaking of chiral symmetry ("mass generation") and account for most of the hadronic properties of light hadrons; they are also expected to play an essential role in partonic structure. We present a method that allows one to compute the quark/gluon structure caused by topological gauge fields in a systematic fashion (instanton vacuum, diluteness approximation). QCD operators involving gluon fields are converted to effective operators in the low-energy effective theory emerging after chiral symmetry breaking. As an application, we compute the twist-3 GPDs describing the spin-orbit correlations of quarks in the nucleon [1]. It is shown that the topological gauge fields have a large effect on the twist-3 QCD operators and induce spin-flavor dependent couplings in the effective chiral operators. The spin-orbit correlations in the nucleon are qualitatively different from naive expectations. We discuss other applications of the method (gluonic operators, trace anomaly) and its potential contribution to the future GPD program. [1] J.-Y. Kim, C. Weiss, Instanton effects in twist-3 generalized parton distributions Phys.Lett.B 848 (2024) 138387

Observables for scattering on targets with arbitrary spin

• Wim Cosyn (FIU)

Starting from the Weinberg formalism for fields for arbitrary spin, we discuss a construction for the decomposition of matrix elements of QCD operators (local currents, quark/gluon bilinears) for targets with arbitrary spin. This procedure is advantageous for the systematic study of the structure of hadrons and nuclei, particularly in the case of spin-dependent observables. As higher spin targets exhibit new features in their hadronic structure, the investigation of these properties can enhance our understanding of the strong force. The construction allows for a unified framework to discuss spin > 1/2 very similar to the spin 1/2 case, without subsidiary conditions for the wave functions. Different types of spinors (canonical, helicity, light-front helicity) can be easily accommodated. Its numerical implementation is simple and can be entirely reduced to objects familiar from the rotation group. An sl(2,C) multipole decomposition naturally appears and allows for a physical interpretation of non-perturbative objects multiplying spinor bilinears. To demonstrate the efficacy of this method, we apply it to the description of a spin 1 target, such as the deuteron. We discuss extensions of the formalism to hard exclusive processes on the deuteron and beyond.

Meson production at NLO and higher-twist revisited

• Kornelija Passek-Kumericki (Theoretical Physics Division, Rudjer Boˇskovi´c Institute, Croatia)

The phenomenological impact of NLO corrections to deeply virtual vector meson production and the role of higher-twist contributions in the decription of deeply virtual pseudoscalar mesons will be discussed. The deeply virtual production of $\rho_0$ mesons has been investigated using the conformal partial wave (CPaW) formalism, which allows for the straightforward inclusion of higher-order contributions and evolutionary effects. Our findings indicate that a description of the longitudinal component of the vector meson DVMP cross-section at high energies is achievable only at NLO within the standard collinear approach at twist-2. A simultaneous description of DIS, DVCS, and DVMP processes is demonstrated, providing insights into the proton structure by unique universal generalized parton distributions (GPDs). On the other hand, the standard collinear approach at lowest twist does not provide the good decription of the deeply virtual $\pi_0$ production at accessible energies. The twist-3 contribution, consisting of twist-2 transversity generalized parton distributions (GPDs) and a twist-3 meson wave function, to deeply virtual pion electroproduction has been determined by including both the $q\bar{q}$ and the $q\bar{q}g$ meson Fock components. Thus obtained cross sections for $\pi^0$ production were successfully confronted with the experimental data.

Renormalons and power corrections in pseudo- and quasi-GPDs

• Jakob Schoenleber (BNL, RBRC)

High-order behavior of the perturbative expansion for short-distance observables in QCD is inti- mately related to the contributions of small momenta in the corresponding Feynman diagrams and this correspondence provides one with a useful tool to investigate power-suppressed nonperturbative corrections. We use this technique to study the structure of power corrections to parton quasi- and pseudo-GPDs which are used in lattice calculations of generalized parton distributions. As the main result, we predict the functional dependence of the leading power corrections to quasi(pseudo)-GPDs on x variable for nonzero skewedness parameter ξ. The kinematic point x = ±ξ turns out to be spe- cial. We find that the nonperturbative corrections to quasiGPDs at this point are suppressed by the first power of the hard scale only. These contributions come from soft momenta and have nothing to do with the known UV renormalon in the Wilson line. We also show that power corrections can be affected strongly by the normalization procedure.