Large-momentum effective theory (LaMET) provides an approach to directly calculate the $\textit{x}$-dependence of generalized parton distributions (GPDs) on a Euclidean lattice through power expansion and a perturbative matching. When a parton’s momentum becomes soft, the corresponding logarithms in the matching kernel become non-negligible at higher orders of perturbation theory, which requires a resummation. But the resummation for the off-forward matrix elements at nonzero skewness $\textit{ξ}$ is difficult due to their multi-scale nature. In this work, we demonstrate that these logarithms are important only in the threshold limit, and derive the threshold factorization formula for the quasi-GPDs in LaMET. We then propose an approach to resum all the large logarithms based on the threshold factorization, which is implemented on a GPD model. We demonstrate that the LaMET prediction is reliable for [−1 + $\textit{x}$ $\textit{,}$ −$\textit{ξ}$ − $\textit{x}$ ] ∪ [−$\textit{ξ}$ + $\textit{x}$ $\textit{, ξ}$ − $\textit{x}$ ] ∪ [$\textit{ξ}$ + $\textit{x}$ $\textit{,}$ 1 − $\textit{x}$ ], where $\textit{x}$ is a cutoff depending on hard parton momenta. Through our numerical tests with the GPD model, we demonstrate that our method is self-consistent and that the inverse matching does not spread the nonperturbative effects or power corrections to the perturbatively calculable regions.

United States

Jack Holligan

Huey-Wen Lin

Rui Zhang

Yong Zhao

Resummation for lattice QCD calculation of generalized parton distributions at nonzero skewness

We present a first study of the effects of renormalization-group resummation (RGR) and leading-renormalon resummation (LRR) on the systematic errors of the unpolarized isovector nucleon generalized parton distribution in the framework of large-momentum effective theory. This work is done using lattice gauge ensembles generated by the MILC Collaboration, consisting of <math><mrow><mn>2</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>1</mn></mrow></math> flavors of highly improved staggered quarks with a physical pion mass at lattice spacing <math><mi>a</mi><mo>≈</mo><mn>0.09</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math> and a box width <math><mi>L</mi><mo>≈</mo><mn>5.76</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math>. We present results for the nucleon <math><mi>H</mi></math> and <math><mi>E</mi></math> generalized parton distributions (GPDs) with average boost momentum <math><msub><mi>P</mi><mi>z</mi></msub><mo>≈</mo><mn>2</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math> at momentum transfers <math><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>0.97</mn><mo>]</mo><mtext> </mtext><mtext> </mtext><msup><mrow><mi>GeV</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math> at skewness <math><mi>ξ</mi><mo>=</mo><mn>0</mn></math> as well as <math><msup><mi>Q</mi><mn>2</mn></msup><mo>∈</mo><mn>0.23</mn><mtext> </mtext><mtext> </mtext><msup><mrow><mi>GeV</mi></mrow><mrow><mn>2</mn></mrow></msup></math> at <math><mi>ξ</mi><mo>=</mo><mn>0.1</mn></math>, renormalized in the modified minimal subtraction (<math><mover><mi>MS</mi><mo>¯</mo></mover></math>) scheme at scale <math><mi>μ</mi><mo>=</mo><mn>2.0</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>, with two- and one-loop matching, respectively. We demonstrate that the simultaneous application of RGR and LRR significantly reduces the systematic errors in renormalized matrix elements and distributions for both the zero and nonzero skewness GPDs, and that it is necessary to include both RGR and LRR at higher orders in the matching and renormalization processes.

Systematic improvement of <math><mi>x</mi></math>-dependent unpolarized nucleon generalized parton distributions in lattice-QCD calculation

We report the findings of our extensive study of the spectra of flavored mesons in lattice gauge theories with symplectic gauge group and fermion matter content treated in the quenched approximation. For the <math><mi>S</mi><mi>p</mi><mo>(</mo><mn>4</mn><mo>)</mo></math>, <math><mi>S</mi><mi>p</mi><mo>(</mo><mn>6</mn><mo>)</mo></math>, and <math><mi>S</mi><mi>p</mi><mo>(</mo><mn>8</mn><mo>)</mo></math> gauge groups, the (Dirac) fermions transform in either the fundamental, or the 2-index, antisymmetric or symmetric, representations. This study sets the stage for future precision calculations with dynamical fermions in the low-mass region of lattice parameter space. Our results have potential phenomenological applications ranging from composite Higgs models, to top (partial) compositeness, to dark matter models with composite, strong-coupling dynamical origin. Having adopted the Wilson flow as a scale-setting procedure, we apply Wilson chiral perturbation theory to extract the continuum and massless limits for the observables of interest. The resulting measurements are used to perform a simplified extrapolation to the large-<math><mi>N</mi></math> limit, hence drawing a preliminary connection with gauge theories with unitary groups. We conclude with a brief discussion of the Weinberg sum rules.

United Kingdom

Ed Bennett

South Korea

Deog Ki Hong

Jong-Wan Lee

Taiwan

C.J. David Lin

Biagio Lucini

Maurizio Piai

Davide Vadacchino

Spectrum of mesons in quenched <math><mrow><mi>S</mi><mi>p</mi><mo>(</mo><mn>2</mn><mi>N</mi><mo>)</mo></mrow></math> gauge theories

We present the first lattice calculation of the nucleon isovector helicity parton distribution function (PDF) in the framework of large-momentum effective theory (LaMET) that uses the hybrid scheme with self-renormalization. We use ensembles generated by the MILC collaboration at lattice spacings <math><mi>a</mi><mo>=</mo><mo>{</mo><mn>0.1207</mn><mo>,</mo><mn>0.0888</mn><mo>,</mo><mn>0.0582</mn><mo>}</mo></math> fm, with <math><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></math> flavors of highly improved staggered quarks at sea pion mass of <math><msub><mrow><mi>M</mi></mrow><mrow><mi>π</mi></mrow></msub><mo>≈</mo><mn>315</mn></math> MeV. We use clover-improved action for our valence quarks with nucleon boost momentum <math><msub><mrow><mi>P</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>≈</mo><mn>1.75</mn></math> GeV and high-statistics measurements for the LaMET matrix elements. We perform an extrapolation to the continuum limit and improve the handling of systematic errors using renormalization-group resummation (RGR) and leading-renormalon resummation (LRR). Our final nucleon helicity PDF is renormalized in the <math><mover><mrow><mtext>MS</mtext></mrow><mo>‾</mo></mover></math> scheme at energy scale <math><mi>μ</mi><mo>=</mo><mn>2.0</mn></math> GeV. We compare our results with and without the two systematic improvements of RGR and LRR at each lattice spacing as well as the continuum limit, and we see that the application of RGR and LRR greatly reduces the systematic errors across the whole x range. Our continuum results with both RGR and LRR show a small positive antiquark region for the nucleon helicity PDF as well as a change of as much as a factor of two in the central values compared to results with neither RGR or LRR. By contrast, the application of RGR and LRR only changes the central values by about 5% in the quark region. We compare our lattice results with the global fits by the JAM, NNPDF and DSSV collaborations, and we observe some tension between our results.

Nucleon helicity parton distribution function in the continuum limit with self-renormalization

We present a new Bjorken x-dependence analysis of a previous lattice quantum chromodynamics data for the pion distribution amplitude from MILC configurations with three lattice spacing <math><mi>a</mi><mo>=</mo><mn>0.06</mn><mo>,</mo><mn>0.09</mn><mo>,</mo><mn>0.12</mn></math> fm. A leading renormalon resummation in renormalization as well as the perturbative matching kernel in the framework of large momentum expansion generates the power accuracy of the matching to the light-cone amplitude. Meanwhile, a small momentum log resummation is implemented for both the quark momentum <math><mi>x</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>z</mi></mrow></msub></math> and the antiquark momentum <math><mo>(</mo><mn>1</mn><mo>−</mo><mi>x</mi><mo>)</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>z</mi></mrow></msub></math> inside a meson of boost momentum <math><msub><mrow><mi>P</mi></mrow><mrow><mi>z</mi></mrow></msub></math> up to 1.72 GeV along the z direction, allowing us to have more accurate determination of the x-dependence in the middle range. Finally, we use the complementarity between the short-distance factorization and the large momentum expansion to constrain the endpoint regions <math><mi>x</mi><mo>∼</mo><mn>0</mn><mo>,</mo><mn>1</mn></math>, thus obtaining the full-range x-dependence of the amplitude.

Xiangdong Ji

Yushan Su

Precision control in lattice calculation of -dependent Pion Distribution Amplitude

In lattice-QCD calculations of parton distribution functions (PDFs) via large-momentum effective theory, the leading power (twist-three) correction appears as <math><mi>O</mi><mo>(</mo><msub><mrow><mi>Λ</mi></mrow><mrow><mi>QCD</mi></mrow></msub><mo>/</mo><msup><mrow><mi>P</mi></mrow><mrow><mi>z</mi></mrow></msup><mo>)</mo></math> due to the linear-divergent self-energy of Wilson line in quasi-PDF operators. For lattice data with hadron momentum <math><msup><mrow><mi>P</mi></mrow><mrow><mi>z</mi></mrow></msup></math> of a few GeV, this correction is dominant in matching, as large as 30% or more. We show how to eliminate this uncertainty through choosing the mass renormalization parameter consistently with the resummation scheme of the infrared-renormalon series in perturbative matching coefficients. An example on the lattice pion PDF data at <math><msup><mrow><mi>P</mi></mrow><mrow><mi>z</mi></mrow></msup><mo>=</mo><mn>1.9</mn></math> GeV shows an improvement of matching accuracy by a factor of more than <math><mn>3</mn><mo>∼</mo><mn>5</mn></math> in the expansion region <math><mi>x</mi><mo>=</mo><mn>0.2</mn><mo>∼</mo><mn>0.5</mn></math>.

Leading power accuracy in lattice calculations of parton distributions

In the large-momentum expansion for parton distribution functions (PDFs), the natural physics scale is the longitudinal momentum (<math><msub><mrow><mi>p</mi></mrow><mrow><mi>z</mi></mrow></msub></math>) of the quarks (or gluons) in a large-momentum hadron. We show how to expose this scale dependence through resumming logarithms of the type <math><msup><mrow><mi>ln</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>⁡</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>/</mo><mi>μ</mi></math> in the matching coefficient, where μ is a fixed renormalization scale. The result enhances the accuracy of the expansion at moderate <math><msub><mrow><mi>p</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>&gt;</mo><mn>1</mn></math> GeV, and at the same time, clearly shows that the partons cannot be approximated from quarks with <math><msub><mrow><mi>p</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>∼</mo><msub><mrow><mi>Λ</mi></mrow><mrow><mi>QCD</mi></mrow></msub></math> which are not predominantly collinear with the parent hadron momentum, consistent with power counting of the large-momentum effective theory. The same physics mechanism constrains the coordinate space expansion at large distances z, the conjugate of <math><msub><mrow><mi>p</mi></mrow><mrow><mi>z</mi></mrow></msub></math>, as illustrated in the example of fitting the moments of the PDFs.

China

Fei Yao

Hong Kong

Jian-Hui Zhang

Resumming quark's longitudinal momentum logarithms in LaMET expansion of lattice PDFs

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.

Xiang Gao

Andrew D. Hanlon

Nikhil Karthik

Swagato Mukherjee

Peter Petreczky

Sergey Syritsyn

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

3D (3 dimensional) Young diagrams are a generalization of 2D Young diagrams. In this paper, We consider 3D Bosons and 3-Jack polynomials. We associate three parameters h 1, h 2, h 3 to y, x, z-axis respectively. 3-Jack polynomials are polynomials of P  n,j , n ≥ j with coefficients in ℂ(h 1, h 2, h 3), which are the generalization of Schur functions and Jack polynomials to 3D case. Similar to Schur functions, 3-Jack polynomials can also be determined by the vertex operators and the Pieri formulas.

Na Wang

Ke Wu

3D bosons, 3-Jack polynomials and affine Yangian of   <math> <mi>gl</mi> <mfenced> <mn>1</mn> </mfenced> </math>  $$ \mathfrak{gl}(1) $$

In this paper, we define one kind of new Jack polynomials denoted by <math><mi>J</mi><msub><mrow><mi>k</mi></mrow><mrow><mi>λ</mi></mrow></msub></math>, which equal the standard Jack polynomials <math><msubsup><mrow><mi>P</mi></mrow><mrow><mi>λ</mi></mrow><mrow><mi>α</mi></mrow></msubsup></math> multiplied by a coefficient. We show that the structure constant <math><msubsup><mrow><mi>C</mi></mrow><mrow><mi>λ</mi><mi>μ</mi></mrow><mrow><mi>ν</mi></mrow></msubsup></math> in <math><mi>J</mi><msub><mrow><mi>k</mi></mrow><mrow><mi>λ</mi></mrow></msub><mo>⋅</mo><mi>J</mi><msub><mrow><mi>k</mi></mrow><mrow><mi>μ</mi></mrow></msub><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>ν</mi></mrow></msub><msubsup><mrow><mi>C</mi></mrow><mrow><mi>λ</mi><mi>μ</mi></mrow><mrow><mi>ν</mi></mrow></msubsup><mi>J</mi><msub><mrow><mi>k</mi></mrow><mrow><mi>ν</mi></mrow></msub></math> can be given from affine Yangian of <math><mrow><mi>gl</mi></mrow><mo>(</mo><mn>1</mn><mo>)</mo></math>. Then we give the Boson-Fermion correspondence for Jack polynomials <math><mi>J</mi><msub><mrow><mi>k</mi></mrow><mrow><mi>λ</mi></mrow></msub></math> and show that 3-Schur functions <math><msub><mrow><mi>S</mi></mrow><mrow><mi>π</mi></mrow></msub></math> equal Jack polynomials <math><mi>J</mi><msub><mrow><mi>k</mi></mrow><mrow><mi>λ</mi></mrow></msub></math> in the special case of <math><msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mi>h</mi></math>, <math><msub><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mo>−</mo><msup><mrow><mi>h</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math>, <math><msub><mrow><mi>h</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>=</mo><mo>−</mo><mi>h</mi><mo>+</mo><msup><mrow><mi>h</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math>.

Zhennan Cui

Yang Bai

Jack polynomials and affine Yangian