<p>Lattice quantum chromodynamics (QCD) calculations share a defining challenge by requiring a small finite range of spatial separation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>z</mi></math> between quark/gluon bilinears for controllable power corrections in the perturbative QCD factorization, and a large hadron boost <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>p</mi><mi>z</mi></msub></math> for a successful determination of collinear parton distribution functions (PDFs). However, these two requirements make the determination of PDFs from lattice data very challenging. We present the application of generative machine learning algorithms to estimate the polarized and unpolarized gluon correlation functions utilizing short-distance data and extending the correlation up to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>z</mi><msub><mi>p</mi><mi>z</mi></msub><mo>≲</mo><mn>14</mn></math>, surpassing the current capabilities of lattice QCD calculations. We train physics-informed machine learning algorithms to learn from the short-distance correlation at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>z</mi><mo>≲</mo><mn>0.36</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math> and take the limit, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>p</mi><mi>z</mi></msub><mo stretchy="false">→</mo><mi>∞</mi></math>, thereby minimizing possible contamination from the higher-twist effects for a successful reconstruction of the polarized gluon PDF. We also expose the bias and problems with underestimating uncertainties associated with the use of model-dependent and overly constrained functional forms, such as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>x</mi><mi>α</mi></msup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mi>β</mi></msup></math> and its variants to extract PDFs from the lattice data. We propose the use of generative machine learning algorithms to mitigate these issues and present our determination of the polarized and unpolarized gluon PDFs in the nucleon.</p>

United States

Bangladesh

Talal Ahmed Chowdhury

Taku Izubuchi

Methun Kamruzzaman

Nikhil Karthik

Tanjib Khan

China

Tianbo Liu

Arpon Paul

Jakob Schoenleber

Raza Sabbir Sufian

Polarized and unpolarized gluon PDFs: Generative machine learning applications for lattice QCD matrix elements at short distance and large momentum

The quantum numbers of monopoles in <math><mrow><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></math> in the presence of massless fermions have been analyzed using a uniform flux background in <math><msup><mi>S</mi><mn>2</mn></msup><mo>×</mo><mi>R</mi></math> coupled to fermions. An analogous study in <math><msup><mi>T</mi><mn>2</mn></msup><mo>×</mo><mi>R</mi></math> is performed by studying the discrete symmetries of the Dirac Hamiltonian in the presence of a static uniform field on <math><msup><mi>T</mi><mn>2</mn></msup></math> with a total flux of <math><mi>Q</mi></math> in the continuum. The degenerate ground states are classified based on their transformation properties under <math><mfrac><mi>π</mi><mn>2</mn></mfrac></math> rotations of <math><msup><mi>T</mi><mn>2</mn></msup></math> that leave the background field invariant. We find that the lattice analysis with overlap fermions exactly reproduces the transformation properties of the single-particle zero modes in the continuum. Whereas the transformation properties of the single-particle negative energy states can be studied in the continuum and the lattice, we are also able to study the transformation properties and the particle number (charge) of the many-body ground state on a finite lattice, and we show that the contributions from the fully filled single-particle states cannot be ignored.

Rajamani Narayanan

Ray Romero

Massless fermions in uniform flux background on <math><mrow><msup><mrow><mi>T</mi></mrow><mn>2</mn></msup><mo>×</mo><mi>R</mi></mrow></math>: Vacuum quantum numbers from single-particle filled modes using lattice regulator

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

Xiang Gao

Swagato Mukherjee

Sergey Syritsyn

Peter Petreczky

Rui Zhang

Yong Zhao

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

Two-dimensional QCD with <math><msub><mi>N</mi><mi>c</mi></msub></math> colors and <math><msub><mi>N</mi><mi>f</mi></msub></math> flavors of massless fermions in the fundamental representation is expected to exhibit conformal behavior in the infrared governed by a <math><mi>u</mi><mo>(</mo><msub><mi>N</mi><mi>f</mi></msub><mo>)</mo></math> Wess–Zumino–Witten model with level <math><msub><mi>N</mi><mi>c</mi></msub></math>. Using numerical analysis within the lattice formalism with exactly massless overlap fermions, we show the emergence of such behavior in the infrared limit. Both the continuum extrapolated low-lying eigenvalues of the massless Dirac operator and the propagator of scalar mesons exhibit a flow from the ultraviolet to the infrared. We find that the amplitude of the conserved current correlator remains invariant under the flow, while the amplitude of the scalar correlator approaches <math><msub><mi>N</mi><mi>f</mi></msub></math>-independent values in the infrared.

Canada

Sruthi A. Narayanan

Approaching the conformal WZW behavior in the infrared limit of two-dimensional massless QCD: A lattice study

We numerically address the issue of which monopole operators are relevant under renormalization group flow in three-dimensional parity-invariant noncompact QED with four flavors of massless two-component Dirac fermion. Using lattice simulation and finite-size scaling analysis of the free energy to introduce monopole-antimonopole pairs in <math><mi>N</mi><mo>=</mo><mn>4</mn></math> and <math><mi>N</mi><mo>=</mo><mn>12</mn></math> flavor noncompact <math><mrow><msub><mrow><mi>QED</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math>, we estimate the infrared scaling dimensions of monopole operators that introduce <math><mn>2</mn><mi>π</mi></math> and <math><mn>4</mn><mi>π</mi></math> fluxes around them. We first show that the estimates for the monopole scaling dimensions are consistent with the large-<math><mi>N</mi></math> expectations for <math><mi>N</mi><mo>=</mo><mn>12</mn></math> <math><mrow><msub><mrow><mi>QED</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math>. Applying the same procedure in <math><mi>N</mi><mo>=</mo><mn>4</mn></math> <math><mrow><msub><mrow><mi>QED</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math>, we estimate the scaling dimension of <math><mn>4</mn><mi>π</mi></math> flux monopole operator to be 3.7(3), which allows the possibility of the operator being irrelevant. This finding offers support to the scenario in which higher-flux monopoles are irrelevant deformations to the Dirac spin liquid phase that could be realized on certain non-bipartite lattices by forbidding <math><mn>2</mn><mi>π</mi></math>-flux monopoles.

Scaling dimension of the <math><mrow><mn>4</mn><mi>π</mi></mrow></math>-flux monopole operator in four-flavor three-dimensional QED using lattice simulation

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.

Andrew D. Hanlon

Jack Holligan

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

The non-singlet helicity quark parton distribution functions (PDFs) of the nucleon are determined from lattice QCD, by jointly leveraging pseudo-distributions and the distillation spatial smearing paradigm. A Lorentz decomposition of appropriately isolated space-like matrix elements reveals pseudo-distributions that contain information on the leading-twist helicity PDFs, as well as an invariant amplitude that induces an additional z 2 contamination of the leading-twist signal. An analysis of the short-distance behavior of the space-like matrix elements using matching coefficients computed to next-to-leading order (NLO) exposes the desired PDF up to this additional z 2 contamination. Due to the non-conservation of the axial current, we elect to isolate the helicity PDFs normalized by the nucleon axial charge at the same scale μ 2. The leading-twist helicity PDFs as well as several sources of systematic error, including higher-twist effects, discretization errors, and the aforementioned z 2 contaminating amplitude are jointly determined by characterizing the computed pseudo-distribution in a basis of Jacobi polynomials. The Akaike Information Criterion is exploited to effectively average over distinct model parameterizations and cuts on the pseudo-distribution. Encouraging agreement is observed with recent global analyses of each non-singlet quark helicity PDF, notably a rather small non-singlet anti-quark helicity PDF for all quark momentum fractions.

Robert Edwards

Colin Egerer

Joseph Karpie

Christopher Monahan

Wayne Morris

Kostas Orginos

Anatoly Radyushkin

David Richards

Eloy Romero

Raza Sufian

France

Savvas Zafeiropoulos

Non-singlet quark helicity PDFs of the nucleon from pseudo-distributions

We perform perturbative computations in a lattice gauge theory with a conformal measure that is quadratic in a noncompact Abelian gauge field and is nonlocal, as inspired by the induced gauge action in massless <math><mrow><msub><mrow><mi>QED</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math>. In a previous work, we showed that coupling fermion sources to the gauge model led to nontrivial conformal data in the correlation functions of fermion bilinears that are functions of charge <math><mi>q</mi></math> of the fermion. In this paper, we compute such gauge invariant fermionic observables to order <math><msup><mi>q</mi><mn>2</mn></msup></math> in lattice perturbation theory with the same conformal measure. We reproduce the expectations for scalar anomalous dimension from previous estimates in dimensional regularization. We address the issue of the lattice regulator dependence of the amplitudes of correlation functions.

Matthew Klein

Perturbative computation in a <math><mrow><msub><mrow><mi>QED</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math>-inspired conformal Abelian gauge model on the lattice

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

Germany

Kai Zhou

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

We present the first exploratory lattice quantum chromodynamics (QCD) calculation of the polarized gluon Ioffe-time pseudodistribution in the nucleon. The Ioffe-time pseudodistribution provides a frame-independent and gauge-invariant framework to determine the gluon helicity in the nucleon from first principles. We employ a high-statistics computation using a <math><msup><mn>32</mn><mn>3</mn></msup><mo>×</mo><mn>64</mn></math> lattice ensemble characterized by a 358 MeV pion mass and a 0.094 fm lattice spacing. We establish the pseudodistribution approach as a feasible method to address the proton spin puzzle with successive improvements in statistical and systematic uncertainties anticipated in the future. Within the statistical precision of our data, we find a good comparison between the lattice determined polarized gluon Ioffe-time distribution and the corresponding expectations from the state-of-the-art global analyses. We find a hint for a nonzero gluon spin contribution to the proton spin from the model-independent extraction of the gluon helicity pseudodistribution over a range of Ioffe-time, <math><mi>ν</mi><mo>≲</mo><mn>9</mn></math>.

Bálint Joó

Christopher J. Monahan

David G. Richards

Toward the determination of the gluon helicity distribution in the nucleon from lattice quantum chromodynamics