Towards unpolarized GPDs from pseudo-distributions

Generalized Parton Distributions (GPDs) are crucial for characterizing the three-dimensional internal structure of hadrons. Unlike one-dimensional Parton Distribution Functions (PDFs), GPDs provide insights into the parton's longitudinal momentum fraction (x), the fraction of longitudinal momentum transfer (skewness ξ), and the invariant momentum transfer (t). This three-dimensional perspective is key to understanding crucial features like the orbital angular momentum carried by quarks and gluons. GPDs also generalize elastic form factors, allowing the definition of radial profiles of energy or pressure distribution within the partonic matter. Experimentally, GPDs are primarily accessed through exclusive processes like Deeply Virtual Compton Scattering (DVCS) and Deeply Virtual Meson Production (DVMP). While a significant experimental program is underway (Jefferson Lab, future Electron-Ion Collider), these experiments provide only indirect access to GPDs. The amplitudes of these processes depend on GPDs through a convolution with a perturbative kernel, limiting access to specific features or integrals of the GPDs, obscuring the full three-dimensional behavior. This inverse problem is further complicated by the existence of "shadow GPDs", where different GPDs can produce very similar experimental cross-sections. The deconvolution problem is particularly challenging in the ERBL region (|x| < |ξ|), where the lack of theoretical constraints like positivity makes extraction difficult. Lattice QCD offers a systematically controlled approach to calculating GPDs. While direct calculation of GPDs is not possible due to their light-cone definition, recent advancements have enabled the computation of the x-dependence using non-local operators with space-like separation. This approach, employing the pseudo-distribution formalism, forms the basis of this research. This study builds upon previous work by the HadStruc collaboration which successfully used this method for PDFs and gluon PDFs. The present paper extends this to GPDs, presenting a formalism for the computation of pseudo-GPDs using distillation and demonstrating the significant kinematic reach provided by this approach. The ultimate aim is to provide model-independent three-dimensional information of the nucleon, complementing experimental efforts like DVCS and DVMP.

The theoretical framework for GPDs has been extensively developed, starting with the work of Müller et al. [1] on light-ray operators and the seminal papers by Ji [2] and Radyushkin [3, 4] on DVCS and the mathematical formulation of GPDs. Ji's work [5] on the gauge-invariant decomposition of nucleon spin highlights the importance of GPDs for understanding nucleon structure. The connection between GPDs and the strong forces inside nucleons and nuclei, including concepts like pressure and mechanical radius, has been explored by Polyakov [6] and Polyakov and Schweitzer [7]. The factorization theorems for hard exclusive processes like DVCS and DVMP (Collins, Frankfurt, and Strikman [9]) are essential for interpreting experimental data. The difficulties in extracting GPDs from experimental data due to the convolution with perturbative kernels and the presence of shadow GPDs have been discussed extensively in the literature [14, 15, 23]. The potential of using small Bjorken-x to improve the extraction has also been investigated [16–18]. Positivity constraints on GPDs [19–22] provide valuable theoretical guidelines. The challenges in extracting gravitational form factors from DVCS data [24–26] underscore the need for alternative approaches, with lattice QCD providing a strong complement. Existing lattice QCD studies of GPD moments using local operators have provided valuable data [27–31], but only for low-order moments. The use of non-local operators and the pseudo-distribution formalism offers an avenue for calculating GPD x-dependence [43–51, 52–55, 56–67], leading to this current study.

This research utilizes lattice QCD calculations within the pseudo-distribution formalism to extract unpolarized isovector proton GPDs. The methodology builds upon previous work by the HadStruc collaboration, leveraging the distillation technique for computational efficiency. Distillation allows for the efficient calculation of many combinations of source and sink momenta, reusing expensive computational components. The isovector matrix elements of the space-like quark bilinear are isolated using two-point and connected three-point correlation functions, eliminating disconnected diagrams. The spectral content of the two- and three-point correlation functions is analyzed using optimized ratios of three-point and two-point functions. Two strategies are employed to address excited-state contamination: the summation method and exponential fitting. The summation method sums over insertion time slices, while the exponential fitting method incorporates both ground state and first excited state contributions to extract the bare matrix elements. The ratio normalization ensures the final quantity is renormalization group invariant. The Lorentz decomposition of the matrix elements involves eight invariant amplitudes (A1-A8). The authors developed an approach using singular value decomposition (SVD) to solve for these amplitudes from the over-constrained system of equations derived from the various spin and Dirac structures. The methodology also includes a procedure to mitigate lattice systematic uncertainties by grouping data into bins of t, correcting for intra-bin t-dependence. The extraction of elastic form factors is performed using both local (z=0) and non-local data. Perturbative matching is applied to the generalized form factors using several leading-order/leading-logarithmic matching procedures to connect the lattice data (at a z-dependent scale) to a standard MS scale (2 GeV). This involves careful consideration of the evolution kernels, anomalous dimensions, and scale dependence of the strong coupling constant. Finally, radial distributions of unpolarized quarks are extracted from the t-dependence of the GPDs.

The study presents a detailed analysis of unpolarized isovector proton GPDs using a lattice ensemble with a pion mass of 358 MeV and a lattice spacing of 0.094 fm. A large kinematic range in terms of skewness (ξ) and momentum transfer (t) was achieved. The research successfully extracted Mellin moments of GPDs up to x³, including skewness dependence. The analysis of the local matrix elements (z=0) yielded elastic form factors F₁(t) and F₂(t), with high precision and relatively small excited state contamination. These results were compared to a previous study [116] and show reasonable agreement. The amplitude A₅(t), which should vanish in the continuum limit, displayed a non-zero signal, likely indicating lattice artifacts or insufficient control over excited states. The authors investigated the impact of the treatment of excited states on various cuts of the data. Both dipole and z-expansion fits were performed on the t-dependence of the elastic form factors. Using non-local data, the elastic form factors were also extracted, demonstrating the feasibility of obtaining these from the non-local matrix elements. A systematic z² dependence was observed in these extractions, indicating the need for careful perturbative matching. The matching of the generalized form factors (A₂,₀, B₂,₀, A₃,₀, B₃,₀, A₄,₀, B₄,₀, A₃,₂, B₃,₂, A₄,₂, B₄,₂) to a common MS scale (2 GeV) was performed using different leading-order/leading-logarithmic matching procedures. The results show the importance of including the large z data to constrain the higher-order moments. The t-dependence of the generalized form factors was then analyzed via dipole and z-expansion fits. Radial distributions, or impact parameter distributions, were computed using the Fourier transforms of the fitted t-dependence. The distributions revealed a shrinking radial extent with increasing order of the moments. The transverse distributions exhibited subtle features related to the even/odd nature of the moments, reflecting the even/odd properties of the Ioffe-time distributions. Finally, the D-term form factor C₂(t) was extracted from the amplitude A₅. The results indicate that the D-term is small, consistent with other lattice calculations, although further work is needed to reduce lattice artifacts. The authors noted the challenge of separating lattice discretization effects from higher twist contributions.

The findings of this study significantly advance our understanding of nucleon structure through the use of lattice QCD and the pseudo-distribution formalism to extract three-dimensional information. The extraction of Mellin moments, including their skewness dependence, provides valuable input for theoretical models of GPDs and comparisons with experimental data. The large kinematic coverage enabled by the distillation technique is a substantial improvement over previous lattice calculations. The detailed analysis of excited state contamination is crucial for obtaining reliable results and serves as an important methodological contribution. The extraction of the D-term, while currently limited by statistical uncertainties and potential lattice artifacts, provides valuable insight into this fundamental aspect of nucleon structure. The comparison between the results from local and non-local data, along with the perturbative matching procedures, highlights the complexities of extracting three-dimensional information, which is important for future theoretical and experimental endeavors. The observed z-dependence in the elastic form factors and generalized form factors emphasizes the need for improved control of lattice systematic errors and a more robust treatment of perturbative matching at higher orders.

This research presents a significant step towards a systematically controlled calculation of nucleon tomography through GPDs. The methodology combines the pseudo-distribution formalism, distillation, and careful treatment of excited states to achieve an unprecedented kinematic coverage. The extraction of higher-order Mellin moments, including their skewness dependence, and the investigation of radial distributions, provides valuable new information. Future research should focus on improving control of excited states using more sophisticated operator constructions (e.g., GEVP analysis), achieving a continuum limit, and extending the analysis to higher orders in perturbation theory and more sophisticated resummation techniques. Further work on improving the perturbative matching, particularly at large z, is also crucial.

The current study is performed at an unphysical pion mass (358 MeV) and has not attempted an extrapolation to the physical pion mass. The observed non-zero values of A₅(t) at z=0, and the challenges in the perturbative matching at large z values, indicate that lattice discretization errors and higher-twist contributions may still impact the results. The limited range of Ioffe time achievable with the current lattice data restricts the sensitivity to higher moments of the D-term. While the methodology addresses excited state contamination, further improvements are needed, especially for higher moments. The radial distributions are extracted using model-dependent extrapolations of the t-dependence, and thus suffer from model dependence.