Scattering amplitudes in planar   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 super Yang-Mills theory exhibit singularities which reflect various aspects of the cluster algebras associated to the Grassmannians Gr(4, n) and their tropical counterparts. Here we investigate the potential origins of such structures and examine the extent to which they can be recovered from the Gröbner structure of the underlying Plücker ideals, focussing on the Grassmannians corresponding to finite cluster algebras.

Mexico

L. Bossinger

United Kingdom

J. Drummond

R. Glew

Adjacency for scattering amplitudes from the Gröbner fan

We present the first physical-continuum limit <math><mi>x</mi></math>-dependent nucleon gluon distribution from lattice QCD using the pseudo-PDF approach, on lattice ensembles with <math><mn>2</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>1</mn></math> flavors of highly improved staggered quarks (HISQ), generated by MILC Collaboration. We use clover fermions for the valence action on three lattice spacings <math><mi>a</mi><mo>≈</mo><mn>0.9</mn></math>, 0.12 and 0.15 fm and three pion masses <math><msub><mi>M</mi><mi>π</mi></msub><mo>≈</mo><mn>220</mn></math>, 310 and 690 MeV, with nucleon two-point measurements numbering up to <math><mi>O</mi><mo>(</mo><msup><mn>10</mn><mn>6</mn></msup><mo>)</mo></math> and nucleon boost momenta up to 3 GeV. We study the lattice-spacing and pion-mass dependence of the reduced pseudo–Ioffe-time distribution matrix elements obtained from the lattice calculation, then extrapolate them to the continuum-physical limit before extracting <math><mi>x</mi><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>/</mo><mo>⟨</mo><mi>x</mi><msub><mo>⟩</mo><mi>g</mi></msub></math>. We use the gluon momentum fraction <math><mo>⟨</mo><mi>x</mi><msub><mo>⟩</mo><mi>g</mi></msub></math> calculated from the same ensembles to determine the nucleon gluon unpolarized PDF <math><mi>x</mi><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> for the first time entirely through lattice-QCD simulation. We compare our results with previous single-ensemble lattice calculations, as well as selected global fits.

United States

Zhouyou Fan

William Good

Huey-Wen Lin

Gluon parton distribution of the nucleon from (<math><mrow><mn>2</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>1</mn></mrow></math>)-flavor lattice QCD in the physical-continuum limit

We present the nonperturbatively renormalized nucleon gluon momentum fraction using ensembles with <math><mn>2</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>1</mn></math> flavors of highly improved staggered quarks (HISQ), generated by the MILC Collaboration. The calculation is done using clover fermions for the valence action with three pion masses, 220, 310, and 690 MeV, and three lattice spacings, 0.09, 0.12, and 0.15 fm. The renormalization is done using RI/MOM nonperturbative renormalization and using cluster-decomposition error reduction (CDER) to enhance the signal-to-noise ratio of the renormalization constant. We find the CDER technique is particularly important to improve the signal at the finer lattice ensembles where the lattice volume is larger. We extrapolate the gluon momentum fraction to the continuum-physical limit and obtain <math><mo>⟨</mo><mi>x</mi><msub><mo>⟩</mo><mi>g</mi></msub><mo>=</mo><mn>0.502</mn><mo>(</mo><mn>53</mn><msub><mo>)</mo><mrow><mi>stat</mi><mo>+</mo><mi>NPR</mi></mrow></msub><mo>(</mo><mn>50</mn><msub><mo>)</mo><mtext>mixing</mtext></msub></math> in the <math><mover><mi>MS</mi><mo>¯</mo></mover></math> scheme at 2 GeV, where first error includes the statistical error and uncertainties in nonperturbative renormalization, while the latter systematic error accounts for ignoring quark mixing. Our gluon momentum fraction is consistent with other recent lattice-QCD results at physical pion mass.

Matthew Zeilbeck

Nonperturbatively renormalized nucleon gluon momentum fraction in the continuum limit 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> lattice QCD

In this work, we present the first results on the gluon parton distribution of the kaon from lattice quantum chromodynamics. We carry out the lattice calculation at pion mass around 310 MeV and two lattice spacings, 0.15 and 0.12 fm, using <math><mrow><mn>2</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>1</mn></mrow></math>-flavor HISQ ensembles generated by MILC Collaboration. The kaon correlators are calculated using clover fermions and momentum-smearing sources with maximum boost momentum around 2 GeV and high statistics (up to 324,000 measurements). We study the dependence of the resulting reduced Ioffe-time pseudodistributions at multiple boost momenta and lattice spacings. We then extract the kaon gluon distribution function in the <math><mover><mi>MS</mi><mo>¯</mo></mover></math> scheme at <math><mrow><mi>μ</mi><mo>=</mo><mn>2</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></mrow></math>, neglecting the mixing between the gluon and singlet-quark sectors. Our results at the smaller lattice spacing are consistent with phenomenological determinations.

Alejandro Salas-Chavira

First glimpse into the kaon gluon parton distribution using lattice QCD

We present the first determination of the x-dependent pion gluon distribution from lattice QCD using the pseudo-PDF approach. We use lattice ensembles with 2+1+1 flavors of highly improved staggered quarks (HISQ), generated by MILC Collaboration, at two lattice spacings <math><mi>a</mi><mo>≈</mo><mn>0.12</mn></math> and 0.15 fm and three pion masses <math><msub><mrow><mi>M</mi></mrow><mrow><mi>π</mi></mrow></msub><mo>≈</mo><mn>220</mn></math>, 310 and 690 MeV. We use clover fermions for the valence action and momentum smearing to achieve pion boost momentum up to 2.29 GeV. We find that the dependence of the pion gluon parton distribution on lattice spacing and pion mass is mild. We compare our results from the lightest pion mass ensemble with the determination by JAM and xFitter global fits.

Gluon parton distribution of the pion from lattice QCD

We present the first lattice-QCD calculation of the kaon valence-quark distribution functions using the large-momentum effective theory (LaMET) approach. The calculation is performed with multiple pion masses with the lightest one around 220 MeV, 2 lattice spacings <math><mi>a</mi><mo>=</mo><mn>0.06</mn></math> and 0.12 fm, <math><mo>(</mo><msub><mi>M</mi><mi>π</mi></msub><msub><mo>)</mo><mi>min</mi></msub><mi>L</mi><mo>≈</mo><mn>5.5</mn></math>, and high statistics ranging from 11,600 to 61,312 measurements. We also calculate the valence-quark distribution of pion and find it to be consistent with the FNAL E615 experimental results, and our ratio of the <math><mi>u</mi></math> quark PDF in the kaon to that in the pion agrees with the CERN NA3 experiment. We also make predictions of the strange-quark distribution of the kaon.

Taiwan

Jiunn-Wei Chen

China

Jian-Hui Zhang

Rui Zhang

Valence-quark distribution of the kaon and pion from lattice QCD

We study isovector unpolarized and helicity parton distribution functions (PDF) of the proton within the framework of large-momentum effective theory. We use a gauge ensemble, generated by the MILC Collaboration, with a superfine lattice spacing of 0.042 fm and a pion mass of 310 MeV, enabling us to simultaneously reach sub-Fermi spatial separations and larger nucleon momenta. We compare the spatial dependence of quasi-PDF matrix elements in different renormalization schemes with the corresponding results of the global fits, obtained using one-loop perturbative matching. We present determinations of the first four moments of the unpolarized and helicity PDFs of proton from the Ioffe-time dependence of the isovector matrix elements, obtained by employing a ratio-based renormalization scheme.

Xiang Gao

Ruizi Li

Nikhil Karthik

Swagato Mukherjee

Peter Petreczky

Sergey Syritsyn

Yi-Bo Yang

Isovector parton distribution functions of the proton on a superfine lattice

There have been rapid developments in the direct calculation in lattice QCD (LQCD) of the Bjorken-<math><mi>x</mi></math> dependence of hadron structure through large-momentum effective theory (LaMET). LaMET overcomes the previous limitation of LQCD to moments (that is, integrals over Bjorken <math><mi>x</mi></math>) of hadron structure, allowing LQCD to directly provide the kinematic regions where the experimental values are least known. LaMET requires large-momentum hadron states to minimize its systematics and allow us to reach small-<math><mi>x</mi></math> reliably. This means that very fine lattice spacing to minimize lattice artifacts at order <math><mo>(</mo><msub><mi>P</mi><mi>z</mi></msub><mi>a</mi><msup><mo>)</mo><mi>n</mi></msup></math> will become crucial for next-generation LaMET-like structure calculations. Furthermore, such calculations require operators with long Wilson-link displacements, especially in finer lattice units, increasing the communication costs relative to that of the propagator inversion. In this work, we explore whether machine-learning algorithms can make predictions of correlators to reduce the computational cost of these LQCD calculations. We consider two algorithms, gradient-boosting decision tree and linear models, applied to LaMET data, the matrix elements needed to determine the kaon and <math><msub><mi>η</mi><mi>s</mi></msub></math> unpolarized parton distribution functions (PDFs), meson distribution amplitude (DA), and the nucleon gluon PDF. We find that both algorithms can reliably predict the target observables with different prediction accuracy and systematic errors. The predictions from smaller displacement <math><mi>z</mi></math> to larger ones work better than those for momentum <math><mi>p</mi></math> due to the higher correlation among the data.

Boram Yoon

Machine-learning prediction for quasiparton distribution function matrix elements

Within QCD reggeon field theory we study the formation of two subsequent triple pomeron vertices in the process P→PP→PPP. We make use of an earlier investigation [1] of the six-reggeon amplitude in deep inelastic scattering and show that in the large-N  c  limit pomeron fan diagrams emerge with the same triple pomeron vertex in all places. We thus confirm the BK-equation, but we also find additional terms related to the reggeization of the gluon, and we discuss their potential significance. Our analysis also includes the general pomeron → two odderon vertex: a particular version of this vertex has been included into earlier generalizations the BK equation.