In recent work Amorosso et al. (2024), we computed a novel flux tube entanglement entropy (FTE) of the color flux tube stretched between a heavy quark-antiquark pair on a Euclidean lattice in (2+1)D Yang–Mills theory. Our numerical results suggested that FTE can be partitioned into an internal color entanglement entropy and a vibrational entropy corresponding to the transverse excitations of a QCD string, with the latter described by a thin string model. Since the color flux tube does not have transverse excitations in (1+1)D Yang–Mills theory, we use this simpler framework to perform an exact analytical computation of the contribution of the internal color degrees of freedom to FTE. For the multipartite partitioning of the color flux tube, we find the remarkable result that FTE depends only on the dimension of the color group representation and the number of times the flux tube crosses the boundary between the traced and untraced spatial regions but not on the string length. Our proof is independent of whether the replica and region boundaries on the lattice are placed on vertices or in plaquette centers.

United States

Rocco Amorosso

Sergey Syritsyn

Raju Venugopalan

Entanglement entropy of a color flux tube in (1+1)D Yang–Mills theory

In this work, we report a lattice calculation of $\textit{x}$-dependent valence pion generalized parton distributions (GPDs) at zero skewness with multiple values of the momentum transfer −$\textit{t}$. The calculations are based on an $\textit{N}$ = 2 + 1 gauge ensemble of highly improved staggered quarks with Wilson-Clover valence fermion. The lattice spacing is 0.04 fm, and the pion valence mass is tuned to be 300 MeV. We determine the Lorentz-invariant amplitudes of the quasi-GPD matrix elements for both symmetric and asymmetric momenta transfers with similar values and show the equivalence of both frames. Then, focusing on the asymmetric frame, we utilize a hybrid scheme to renormalize the quasi-GPD matrix elements obtained from the lattice calculations. After the Fourier transforms, the quasi-GPDs are then matched to the light-cone GPDs within the framework of large momentum effective theory with improved matching, including the next-to-next-to-leading order perturbative corrections, and leading renormalon and renormalization group resummations. We also present the 3-dimensional image of the pion in impact-parameter space through the Fourier transform of the momentum transfer −$\textit{t}$.

China

Heng-Tong Ding

Xiang Gao

Swagato Mukherjee

Peter Petreczky

Qi Shi

Yong Zhao

Three-dimensional imaging of pion using lattice QCD: generalized parton distributions

We construct a novel flux tube entanglement entropy (FTE2), defined as the excess entanglement entropy relative to the vacuum of a region of color flux stretching between a heavy quark-anti-quark pair in pure gauge Yang-Mills theory. We show that FTE2 can be expressed in terms of correlators of Polyakov loops, is manifestly gauge-invariant, and therefore free of the ambiguities in computations of the entanglement entropy in gauge theories related to the choice of the center algebra. Employing the replica trick, we compute FTE2 for SU(2) Yang-Mills theory in (2+1)D and demonstrate that it is finite in the continuum limit. We explore the properties of FTE2 for a half-slab geometry, which allows us to vary the width and location of the slab, and the extent to which the slab cross-cuts the color flux tube. Following the intuition provided by computations of FTE2 in (1+1)D, and in a thin string model, we examine the extent to which our FTE2 results can be interpreted as the sum of an internal color entropy and a vibrational entropy corresponding to the transverse excitations of the string.

Entanglement entropy of a color flux tube in (2+1)D Yang-Mills theory

We present a lattice quantum chromodynamics (QCD) calculation of the <math><mi>x</mi></math>-dependent pion and kaon distribution amplitudes (DA) in the framework of large momentum effective theory. This calculation is performed on a fine lattice of <math><mi>a</mi><mo>=</mo><mn>0.076</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math> at physical pion mass, with the pion boosted to 1.8 GeV and kaon boosted to 2.3 GeV. We renormalize the matrix elements in the hybrid scheme and match to <math><mover><mi>MS</mi><mo>¯</mo></mover></math> with a subtraction of the leading renormalon in the Wilson-line mass. The perturbative matching is improved by resumming the large logarithms related to the small quark and gluon momenta in the soft-gluon limit. After resummation, we demonstrate that we are able to calculate a range of <math><mi>x</mi><mo>∈</mo><mo>[</mo><msub><mi>x</mi><mn>0</mn></msub><mo>,</mo><mn>1</mn><mo>−</mo><msub><mi>x</mi><mn>0</mn></msub><mo>]</mo></math> with <math><msub><mi>x</mi><mn>0</mn></msub><mo>=</mo><mn>0.25</mn></math> for pion and <math><msub><mi>x</mi><mn>0</mn></msub><mo>=</mo><mn>0.2</mn></math> for kaon with theoretical systematic errors under control. The kaon DA is shown to be slighted skewed, and narrower than pion DA. Although the <math><mi>x</mi></math>-dependence cannot be direct calculated beyond these ranges, we estimate higher moments of the pion and kaon DAs by complementing our calculation with short-distance factorization.

Ian Cloët

Nikhil Karthik

Rui Zhang

Lattice QCD calculation of <math><mi>x</mi></math>-dependent meson distribution amplitudes at physical pion mass with threshold logarithm resummation

We report the first lattice QCD computation of pion and kaon electromagnetic form factors, <math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>(</mo><msup><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></math>, at large momentum transfer up to 10 and <math><mn>28</mn><mtext> </mtext><mtext> </mtext><msup><mi>GeV</mi><mn>2</mn></msup></math>, respectively. Utilizing physical masses and two fine lattices, we achieve good agreement with JLab experimental results at <math><msup><mi>Q</mi><mn>2</mn></msup><mo>≲</mo><mn>4</mn><mtext> </mtext><mtext> </mtext><msup><mi>GeV</mi><mn>2</mn></msup></math>. For <math><msup><mi>Q</mi><mn>2</mn></msup><mo>≳</mo><mn>4</mn><mtext> </mtext><mtext> </mtext><msup><mi>GeV</mi><mn>2</mn></msup></math>, our results provide ab initio QCD benchmarks for the forthcoming experiments at JLab 12 GeV and future electron-ion colliders. We also test the QCD collinear factorization framework utilizing our high-<math><msup><mi>Q</mi><mn>2</mn></msup></math> form factors at next-to-next-to-leading order in perturbation theory, which relates the form factors to the leading Fock-state meson distribution amplitudes. Comparisons with independent lattice QCD calculations using the same framework demonstrate, within estimated uncertainties, the universality of these nonperturbative quantities.

Andrew D. Hanlon

QCD Predictions for Meson Electromagnetic Form Factors at High Momenta: Testing Factorization in Exclusive Processes

The second Mellin moments <math><mo>⟨</mo><mi>x</mi><mo>⟩</mo></math> of the nucleon’s unpolarized, polarized, and transversity parton distribution functions are computed. Two lattice QCD ensembles at the physical pion mass are used: these were generated using a tree-level Symanzik-improved gauge action and <math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math> flavor tree-level improved Wilson Clover fermions coupling via 2-level HEX-smearing. The moments are extracted from forward matrix elements of local leading twist operators. We determine renomalization factors in RI-(S)MOM and match to <math><mover><mi>MS</mi><mo>¯</mo></mover></math> at scale 2 GeV. Our findings show that operators that exhibit vanishing kinematics at zero momentum can have significantly reduced excited-state contamination. The resulting polarized moment is used to quantify the longitudinal contribution to the quark spin-orbit correlation. All our results agree within two sigma with previous lattice results.

Germany

Marcel Rodekamp

Michael Engelhardt

Jeremy R. Green

Stefan Krieg

Simonetta Liuti

Stefan Meinel

John W. Negele

Andrew Pochinsky

Moments of nucleon unpolarized, polarized, and transversity parton distribution functions from lattice QCD at the physical point

We present a lattice QCD calculation of the transversity isovector- and isoscalar-quark parton distribution functions (PDFs) of the proton utilizing a perturbative matching at next-to-leading-order (NLO) accuracy. Additionally, we determine the isovector and isoscalar tensor charges for the proton. In both calculations, the disconnected contributions to the isoscalar matrix elements have been ignored. The calculations are performed using a single 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></mrow></math> highly improved staggered quarks simulated with physical-mass quarks and a lattice spacing of <math><mi>a</mi><mo>=</mo><mn>0.076</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math>. The Wilson-clover action, with physical quark masses and smeared gauge links obtained from one iteration of hypercubic smearing, is used in the valence sector. Using the NLO operator product expansion, we extract the lowest four to six Mellin moments and the PDFs via a neural network from the matrix elements in the pseudo-PDF approach. In addition, we calculate the PDFs in the quasi-PDF approach with hybrid-scheme renormalization and the recently developed leading-renormalon resummation technique, at NLO with the resummation of leading small-<math><mi>x</mi></math> logarithms.

Transversity PDFs of the proton from lattice QCD with physical quark masses

We present a lattice QCD calculation of the unpolarized isovector quark parton distribution function (PDF) of the proton utilizing a perturbative matching at next-to-next-to-leading-order (NNLO). The calculations are carried out using a single ensemble of gauge configurations generated with <math><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></math> highly-improved staggered quarks with physical masses and a lattice spacing of <math><mi>a</mi><mo>=</mo><mn>0.076</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math>. We use one iteration of hypercubic smearing on these gauge configurations, and the resulting smeared configurations are then used for all aspects of the subsequent calculation. For the valence quarks, we use the Wilson-clover action with physical quark masses. We consider several methods for extracting information on the PDF. We first extract the lowest four Mellin moments using the leading-twist operator product expansion approximation. Then, we determine the <math><mi>x</mi></math> dependence of the PDF through a deep neural network within the pseudo-PDF approach and additionally through the framework of large-momentum effective theory utilizing a hybrid renormalization scheme. This is the first application of the NNLO matching coefficients for the nucleon directly at the physical point.

Jack Holligan

Unpolarized proton PDF at NNLO from lattice QCD with physical quark masses

We present lattice QCD calculations of the valence parton distribution function (PDF) of pion employing next-to-next-leading-order (NNLO) perturbative QCD matching. Our calculations are based on three gauge ensembles of <math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math> flavor highly improved staggered quarks and Wilson-Clover valance quarks, corresponding to pion mass <math><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>π</mi></mrow></msub><mo>=</mo><mn>140</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math> at a lattice spacing <math><mi>a</mi><mo>=</mo><mn>0.076</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math> and <math><msub><mi>m</mi><mi>π</mi></msub><mo>=</mo><mn>300</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> at <math><mi>a</mi><mo>=</mo><mn>0.04</mn></math>, 0.06 fm. This enables us to present, for the first time, continuum-extrapolated lattice QCD results for the NNLO valence PDF of the pion at the physical point. Applying leading-twist expansion for renormalization group invariant (RGI) ratios of bilocal pion matrix elements with NNLO Wilson coefficients, we extract second, fourth, and sixth Mellin moments of the PDF. We reconstruct the Bjorken-<math><mi>x</mi></math> dependence of the NNLO PDF from real-space RGI ratios using a deep neural network as well as from momentum-space matrix elements renormalized using a hybrid scheme. All our results are in broad agreement with the results of global fits to the experimental data carried out by the xFitter and JAM collaborations.

Philipp Scior

Shuzhe Shi

Kai Zhou

Continuum-extrapolated NNLO valence PDF of the pion at the physical point

We present a lattice QCD determination of the distribution amplitude (DA) of the pion and the first few Mellin moments from an analysis of the quasi-DA matrix element within the leading-twist framework. We perform our study on a HISQ ensemble with <math><mi>a</mi><mo>=</mo><mn>0.076</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math> lattice spacing with the Wilson-clover valence quark mass tuned to the physical point. We analyze the ratios of pion quasi-DA matrix elements at short distances using the leading-twist Mellin operator product expansion (OPE) at the next-to-leading order and the conformal OPE at the leading-logarithmic order. We find a robust result for the first nonvanishing Mellin moment <math><mo>⟨</mo><msup><mi>x</mi><mn>2</mn></msup><mo>⟩</mo><mo>=</mo><mn>0.287</mn><mo>(</mo><mn>6</mn><mo>)</mo><mo>(</mo><mn>6</mn><mo>)</mo></math> at a factorization scale <math><mi>μ</mi><mo>=</mo><mn>2</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>. We also present different Ansätze-based reconstructions of the <math><mi>x</mi></math>-dependent DA, from which we determine the perturbative leading-twist expectations for the pion electromagnetic and gravitational form-factors at large momentum transfers.