In this work, we present a lattice QCD calculation of the Mellin moments of the twist-2 axial-vector generalized parton distribution (GPD), $$ \overset{\sim }{H}\left(x,\xi, t\right) $$ , at zero skewness, , with multiple values of the momentum transfer, . Our analysis employs the short-distance factorization framework on ratio-scheme renormalized quasi-GPD matrix elements. The calculations are based on an = 2 + 1 + 1 twisted mass fermions ensemble with clover improvement, a lattice spacing of = 0 093 fm, and a pion mass of = 260 MeV. We consider both the iso-vector and iso-scalar cases, utilizing next-to-leading-order perturbative matching while omitting the disconnected contributions and gluon mixing in the iso-scalar case. For the first time, we determine the Mellin moments of $$ \overset{\sim }{H} $$ up to the fifth order. From these moments, we discuss the quark helicity and orbital angular momentum contributions to the nucleon spin, as well as the spin-orbit correlations of the quarks. Additionally, we perform a Fourier transform over the momentum transfer, which allows us to explore the spin structure in the impact-parameter space.

United States

Shohini Bhattacharya

Poland

Krzysztof Cichy

Martha Constantinou

Xiang Gao

Andreas Metz

Joshua Miller

Swagato Mukherjee

Peter Petreczky

Germany

Fernanda Steffens

Yong Zhao

Moments of axial-vector GPD from lattice QCD: quark helicity, orbital angular momentum, and spin-orbit correlation

We present an exploratory work on integrating lattice-QCD results with experimental data for elastic scattering. Within the framework of generalized parton distributions, this approach allows for the extraction of detailed information about nucleon tomography and the total angular momentum carried by valence quarks. To accomplish this reliably, we introduce a new quantity, the “double ratio,” which significantly reduces the systematic uncertainties inherent in lattice QCD computations. Moreover, we introduce a “shadow” term in the extraction procedure, which is sensitive only to lattice-QCD results. This term allows us to investigate the model dependence of the extraction, providing a more flexible description of the nucleon that goes beyond the previously considered bell-shaped distribution.

Paweł Sznajder

Jakub Wagner

Nucleon tomography and total angular momentum of valence quarks from synergy between lattice QCD and elastic scattering data

Generalized parton distributions (GPDs) are key quantities for the description of a hadron’s three-dimensional structure. They are the current focus of all areas of hadronic physics—phenomenological, experimental and theoretical, including lattice QCD. Synergies between these areas are desirable and essential to achieve precise quantification and understanding of the structure of, particularly, nucleons, as the basic ingredients of matter. In this paper, we investigate, for the first time, the numerical implementation of the pseudodistribution approach for the extraction of zero-skewness GPDs for unpolarized quarks. Pseudodistributions are Euclidean parton correlators computable in lattice QCD that can be perturbatively matched to the light-cone parton distributions of interest. Although they are closely related to the quasidistributions and come from the same lattice-extracted matrix elements, they are, however, subject to different systematic effects. We use the data previously utilized for quasi-GPDs and extend it with other momentum transfers and nucleon boosts, in particular a higher one (<math><mrow><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>=</mo><mn>1.67</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></mrow></math>) with eightfold larger statistics than the largest one used for quasidistributions (<math><msub><mi>P</mi><mn>3</mn></msub><mo>=</mo><mn>1.25</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>). We renormalize the matrix elements with a ratio scheme and match the resulting Ioffe time distributions to the light cone in coordinate space. The matched distributions are then used to reconstruct the <math><mi>x</mi></math> dependence with a fitting Ansatz. We investigate some systematic effects related to this procedure, and we also compare the results with the ones obtained in the framework of quasi-GPDs. Our final results involve the invariant four-momentum transfer squared (<math><mo>−</mo><mi>t</mi></math>) dependence of the flavor nonsinglet (<math><mi>u</mi><mo>−</mo><mi>d</mi></math>) <math><mi>H</mi></math> and <math><mi>E</mi></math> GPDs.

Niilo Nurminen

Generalized parton distributions from the pseudodistribution approach on the lattice

<p>Recently, we made significant advancements in improving the computational efficiency of lattice QCD calculations for generalized parton distributions (GPDs). This progress was achieved by adopting calculations of matrix elements in asymmetric frames, deviating from the computationally-expensive symmetric frame typically used, and allowing freedom in the choice for the distribution of the momentum transfer between the initial and final states. A crucial aspect of this approach involves the adoption of a Lorentz covariant parametrization for the matrix elements, introducing Lorentz-invariant amplitudes. This approach also allows us to propose an alternative definition of quasi-GPDs, ensuring frame independence and potentially reduce power corrections in matching to light cone GPDs. In our previous work, we presented lattice QCD results for twist-2 unpolarized GPDs (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>H</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>E</mi></math>) of quarks obtained from calculations performed in asymmetric frames at zero skewness. Building upon this work, we now introduce a novel Lorentz covariant parametrization for the axial-vector matrix elements. We employ this parametrization to compute the axial-vector GPD <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mi>H</mi><mo stretchy="false">˜</mo></mover></math> at zero skewness, using an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>1</mn></math> ensemble of twisted mass fermions with clover improvement. The light-quark masses employed in our calculations correspond to a pion mass of approximately 260 MeV.</p>

Jack Dodson

Generalized parton distributions from lattice QCD with asymmetric momentum transfer: Axial-vector case

Staple-shaped Wilson line operators are necessary for the study of transverse momentum-dependent parton distribution functions (TMDPDFs) in lattice QCD and beyond. In this work, we study the renormalization of such operators in the general case of an asymmetric staple. We analyze the mixing pattern of these operators using their symmetry properties, where we find that the possible mixing is restricted within groups of four operators. We then present numerical results using the regularization independent momentum subtraction (RI/MOM) scheme to study the importance of mixing using one operator in particular, the <math><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></math> operator. Based on these results, we consider the short distance ratio (SDR) scheme, which is desirable in the absence of mixing. Finally, we investigate a variant of the RI/MOM scheme, where the renormalization factors are computed at short distances.

Cyprus

Constantia Alexandrou

Simone Bacchio

China

Xu Feng

Karl Jansen

Chuan Liu

Aniket Sen

Gregoris Spanoudes

Jacopo Tarello

Nonperturbative renormalization of asymmetric staple-shaped operators in twisted mass lattice QCD

In this paper, we present lattice QCD results for the <math><mi>x</mi></math> dependence of the unpolarized gluon parton distribution functions (PDFs) for the proton. We use one ensemble of <math><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>1</mn></math> maximally twisted mass fermions with a clover improvement, and the Iwasaki improved gluon action. The quark masses are tuned to produce a pion with a mass of 260 MeV. The ensemble has a lattice spacing of <math><mi>a</mi><mo>=</mo><mn>0.093</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math> and a spatial extent of 3 fm. We employ the pseudodistribution approach, which relies on matrix elements of nonlocal operators that couple to momentum-boosted hadrons. In this work, we use five values of the momentum boost between 0 and 1.67 GeV. The gluon field-strength tensors of the nonlocal operator are connected with straight Wilson lines of varying length <math><mi>z</mi></math>. The light-cone Ioffe time distribution (ITD) is extracted utilizing data with <math><mi>z</mi></math> up to 0.56 fm and a quadratic parametrization in terms of the Ioffe time at fixed values of <math><mi>z</mi></math>. We explore systematic effects, such as the effect of the stout smearing for the gluon operator, excited states effects, and the dependence on the maximum value of <math><mi>z</mi></math> entering the fits to obtain the gluon PDF. Also, for the first time, the mixing with the quark singlet PDFs is eliminated using matrix elements with nonlocal quark operators that were previously analyzed within the quasi-PDF framework on the same ensemble. Here, we expand the dataset for the quark singlet and reanalyze within the pseudo-PDFs method eliminating the corresponding mixing in the gluon PDF.

Joseph Delmar

Kyriakos Hadjiyiannakou

Gluon PDF of the proton using twisted mass fermions

This work presents the first lattice-QCD calculation of the twist-3 axialquark generalized parton distributions (GPDs) for the proton using the large-momentum effective theory approach. We calculate matrix elements with momentum-boosted proton states and a nonlocal axial operator. The calculation is performed using one ensemble of two degenerate light, a strange and a charm quark (<math><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>1</mn></mrow></math>) of maximally twisted mass fermions with a clover term. The ensemble has a volume <math><msup><mn>32</mn><mn>3</mn></msup><mo>×</mo><mn>64</mn></math> and lattice spacing 0.0934 fm and corresponds to a pion mass of 260 MeV. The matrix elements are calculated for three values of the proton momentum, namely 0.83, 1.25, and 1.67 GeV. The light-cone GPDs are defined in the symmetric frame, which we implement here with a (negative) 4-momentum transfer squared of 0.69, 1.38, and <math><mn>2.76</mn><mtext> </mtext><mtext> </mtext><msup><mi>GeV</mi><mn>2</mn></msup></math>, all at zero skewness. We also conduct several consistency checks, including assessing the local limit of the twist-3 GPDs and examining the Burkhardt-Cottingham-type as well as Efremov-Teryaev-Leader-type sum rules.

Aurora Scapellato

Chiral-even axial twist-3 GPDs of the proton from lattice QCD

For the first time, we present a lattice QCD determination of Mellin moments of unpolarized generalized parton distributions (GPDs) of the proton from an analysis of the quasi-GPD matrix elements within the short-distance factorization framework. We perform our calculation on an <math><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>1</mn></mrow></math> twisted mass fermions ensemble with a clover improvement at lattice spacing <math><mi>a</mi><mo>=</mo><mn>0.093</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math> and a pion mass of <math><msub><mi>m</mi><mi>π</mi></msub><mo>=</mo><mn>260</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>. Focusing on the zero-skewness case, the isovector and isoscalar quasi-GPDs are calculated from the <math><msub><mi>γ</mi><mn>0</mn></msub></math> definition, as well as a recently proposed Lorentz-invariant definition. We utilize data on both symmetric and asymmetric kinematic frames, which allows us to obtain the Mellin moments for several values of the momentum transfer, <math><mo>−</mo><mi>t</mi></math>, in the range 0.17 to <math><mn>2.77</mn><mtext> </mtext><mtext> </mtext><msup><mi>GeV</mi><mn>2</mn></msup></math>. We use the ratio scheme for GPDs, i.e. renormalization group invariant ratios with leading-twist factorization formula and perturbatively calculated matching coefficients up to the next-next-to-leading order (NNLO) to extract Mellin moments of GPDs, which are consistent with renormalization-group improved results. We compare our determination from quasi-GPDs with the results extracted using standard calculations of Mellin moments of local operators, specifically those related to the electromagnetic and gravitational form factors. We estimated the moments of GPDs up to the fifth ones for the first time. By extrapolating the Mellin moments to <math><mo>−</mo><mi>t</mi><mo>=</mo><mn>0</mn></math>, we obtained the quark charges, momentum fraction, as well as the angular momentum contributions to the proton spin. The impact parameter space interpretation of the GPD moments is discussed, which provides insights into the spatial distribution of unpolarized quarks and their correlations in the transverse plane of an unpolarized or transversely polarized proton.

Moments of proton GPDs from the OPE of nonlocal quark bilinears up to NNLO

Traditionally, lattice QCD computations of generalized parton distributions (GPDs) have been carried out in a symmetric frame, where the transferred momentum is symmetrically distributed between the incoming and outgoing hadrons. However, such frames are inconvenient since they require a separate calculation for each value of the momentum transfer, increasing significantly the computational cost. In this work, by focusing on the quasidistribution approach, we lay the foundation for faster and more effective lattice QCD calculations of GPDs exploiting asymmetric frames, with freedom in the transferred momentum distribution. An important ingredient of our approach is the Lorentz covariant parametrization of the matrix elements in terms of Lorentz-invariant amplitudes, which allows one to relate matrix elements in different frames. We also use this amplitude approach to propose a new definition of quasi-GPDs that is frame independent and, more importantly, may lead to smaller power corrections in the matching relations to the light-cone GPDs. We demonstrate the efficacy of the formalism through numerical calculations using one ensemble of <math><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>1</mn></mrow></math> twisted-mass fermions with a clover improvement. The value of the light-quark masses lead to a pion mass of about 260 MeV. Concentrating on the proton, and limiting ourselves to a vanishing longitudinal momentum transfer to the target, we extract the invariant amplitudes from matrix element calculations in both the symmetric and asymmetric frame and obtain results for the twist-2 light-cone GPDs for unpolarized quarks, that is, <math><mi>H</mi></math> and <math><mi>E</mi></math>.

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

Precise quantification of the structure of nucleons is one of the crucial aims of hadronic physics for the coming years. The expected progress related to ongoing and planned experiments should be accompanied by calculations of partonic distributions from lattice QCD. While key insights from the lattice are expected to come for distributions that are difficult to access experimentally, it is important that lattice QCD can reproduce the well-known unpolarized parton distribution functions (PDFs) with full control over systematic uncertainties. One of the novel methods for accessing the partonic <math><mi>x</mi></math>-dependence is the pseudodistribution approach, which employs matrix elements of a spatially extended nonlocal Wilson line operator of length <math><mi>z</mi></math>. In this paper, we address the issue of discretization effects, related to the necessarily nonzero value of the lattice spacing <math><mi>a</mi></math>, which start at first order in <math><mi>a</mi></math> as a result of the nonlocal operator. We use twisted mass fermions simulated at three values of the lattice spacing, at a pion mass of 370 MeV, and extract the continuum limit of isovector unpolarized PDFs. We also test, for the first time in the pseudodistribution approach, the effects of the recently derived two-loop matching. Finally, we address the issue of the reliability of the extraction with respect to the maximal value of <math><mi>z</mi></math>.

Manjunath Bhat

Wojciech Chomicki

Ireland

Jeremy R. Green

Continuum limit of parton distribution functions from the pseudodistribution approach on the lattice