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.

United States

Ian Cloët

Xiang Gao

Swagato Mukherjee

Sergey Syritsyn

Nikhil Karthik

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

We present a direct lattice QCD calculation of the x-dependence of the pion distribution amplitude (DA), which is performed using the quasi-DA in large momentum effective theory on a domain-wall fermion ensemble at physical quark masses and spacing a ≈ 0.084 fm. The bare quais-DA matrix elements are renormalized in the hybrid scheme and matched to   <math> <mover> <mi>MS</mi> <mo>¯</mo> </mover> </math>  $$ \overline{\textrm{MS}} $$  with a subtraction of the leading renormalon in the Wilson-line mass. For the first time, we include threshold resummation in the perturbative matching onto the light-cone DA, which resums the large logarithms in the soft gluon limit at next-to-next-to-leading log. The resummed results show controlled scale-variation uncertainty within the range of momentum fraction x ∈ [0.25, 0.75] at the largest pion momentum P  z  ≈ 1.85 GeV. In addition, we apply the same analysis to quasi-DAs from a highly-improved-staggered-quark ensemble at physical pion mass and a = 0.076 fm. By comparison we find with 2σ confidence level that the DA obtained from chiral fermions is flatter and lower near x = 0.5.

Ethan Baker

Dennis Bollweg

Peter Boyle

Lattice QCD calculation of the pion distribution amplitude with domain wall fermions at physical pion mass

The feasibility of extracting generalized parton distributions (GPDs) from deeply virtual Compton scattering (DVCS) data has recently been questioned because of the existence of an infinite set of so-called “shadow GPDs” (SGPDs). These SGPDs depend on the process and manifest as multiple solutions (at a fixed scale <math><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math>) to the inverse problem that needs to be solved to infer GPDs from DVCS data. SGPDs therefore pose a significant challenge for extracting GPDs from DVCS data. With this motivation we study the extent to which QCD evolution can provide constraints on SGPDs. This is possible because the known classes of SGPDs begin to contribute to observables after evolution, and can then be constrained (at the input scale <math><msubsup><mi>Q</mi><mn>0</mn><mn>2</mn></msubsup></math>) by data that has a finite <math><msup><mi>Q</mi><mn>2</mn></msup></math> range. The impact that SGPDs could have on determining the total angular momentum, pressure and sheer force distributions, and tomography is also discussed. We find that <math><msup><mi>Q</mi><mn>2</mn></msup></math> evolution, given empirical data over a wide range of skewness (<math><mi>ξ</mi></math>) and <math><msup><mi>Q</mi><mn>2</mn></msup></math> scales, constrains the SGPDs studied at low <math><mi>x</mi></math> and low <math><mi>ξ</mi></math>. If this finding generalizes to all SGPDs—and we have not been able to find any SGPDs with qualitatively different behavior—then this potentially makes extraction of GPDs from DVCS possible over a limited range of <math><mi>x</mi></math> and <math><mi>ξ</mi></math>.

Eric Moffat

Adam Freese

Thomas Donohoe

Leonard Gamberg

Australia

W. Melnitchouk

Andreas Metz

Alexei Prokudin

Nobuo Sato

Shedding light on shadow generalized parton distributions

We present a calculation of the scalar, vector, and tensor form factors for the pion and kaon in lattice QCD. We use an ensemble of two degenerate light, a strange and a charm quark (<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>) of maximally twisted mass fermions with clover improvement. The corresponding pion and kaon masses are about 265 MeV and 530 MeV, respectively. The calculation is done in both rest and boosted frames obtaining data for four-vector momentum transfer squared up to <math><mo>−</mo><msup><mi>q</mi><mn>2</mn></msup><mo>=</mo><mn>2.5</mn><mtext> </mtext><mtext> </mtext><msup><mrow><mi>GeV</mi></mrow><mrow><mn>2</mn></mrow></msup></math> for the pion and <math><mn>3</mn><mtext> </mtext><mtext> </mtext><msup><mi>GeV</mi><mn>2</mn></msup></math> for the kaon. The excited-states effects are studied by analyzing six values of the source-sink time separation for the rest frame (1.12–2.23 fm) and for four values for the boosted frame (1.12–1.67 fm). The lattice data are renormalized nonperturbatively and the results for the scheme- and scale-dependent scalar and tensor form factors are presented in the <math><mover><mi>MS</mi><mo>¯</mo></mover></math> scheme at a scale of 2 GeV. We apply different parametrizations to describe <math><msup><mi>q</mi><mn>2</mn></msup></math>-dependence of the form factors to extract the scalar, vector, and tensor radii, as well as the tensor anomalous magnetic moment. We compare the pion and kaon form factors to study SU(3) flavor symmetry breaking effects. By combining the data for the vector and tensor form factors we also obtain the lowest moment of the densities of transversely polarized quarks in the impact parameter space. Finally, we give an estimate for the average transverse shift in the <math><mi>y</mi></math> direction for polarized quarks in the <math><mi>x</mi></math> direction.

Cyprus

Constantia Alexandrou

Simone Bacchio

Martha Constantinou

Joseph Delmar

Kyriakos Hadjiyiannakou

Giannis Koutsou

Colin Lauer

Alejandro Vaquero

Scalar, vector, and tensor form factors for the pion and kaon from lattice QCD

We present a calculation of the pion and kaon Mellin moment <math><mo>⟨</mo><msup><mi>x</mi><mn>3</mn></msup><mo>⟩</mo></math> extracted directly in lattice QCD using a three-derivative local operator. We use one ensemble of gauge configurations with two degenerate light, a strange and a charm quark (<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>) of maximally twisted mass fermions with clover improvement. The ensemble reproduces a pion mass <math><mo>∼</mo><mn>260</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>, and a kaon mass <math><mo>∼</mo><mn>530</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>. Excited-states contamination is evaluated using four values of the source-sink time separation within the range of 1.12–1.67 fm. We use an operator that is free of mixing, and apply a multiplicative renormalization function calculated nonperturbatively. Our results are converted to the <math><mover><mi>MS</mi><mo>¯</mo></mover></math> scheme and evolved at a scale of 2 GeV, using three-loop expressions in perturbation theory. The final values are <math><mo>⟨</mo><msup><mi>x</mi><mn>3</mn></msup><msubsup><mo>⟩</mo><mi>π</mi><msup><mi>u</mi><mo>+</mo></msup></msubsup><mo>=</mo><mn>0.024</mn><mo>(</mo><mn>18</mn><msub><mo>)</mo><mrow><mi>stat</mi></mrow></msub><mo>(</mo><mn>2</mn><msub><mo>)</mo><mrow><mi>syst</mi></mrow></msub></math>, <math><mo>⟨</mo><msup><mi>x</mi><mn>3</mn></msup><msubsup><mo>⟩</mo><mi>K</mi><msup><mi>u</mi><mo>+</mo></msup></msubsup><mo>=</mo><mn>0.035</mn><mo>(</mo><mn>6</mn><msub><mo>)</mo><mrow><mi>stat</mi></mrow></msub><mo>(</mo><mn>3</mn><msub><mo>)</mo><mrow><mi>syst</mi></mrow></msub></math>, and <math><mo>⟨</mo><msup><mi>x</mi><mn>3</mn></msup><msubsup><mo>⟩</mo><mi>K</mi><msup><mi>s</mi><mo>+</mo></msup></msubsup><mo>=</mo><mn>0.075</mn><mo>(</mo><mn>5</mn><msub><mo>)</mo><mrow><mi>stat</mi></mrow></msub><mo>(</mo><mn>1</mn><msub><mo>)</mo><mrow><mi>syst</mi></mrow></msub></math>, where the systematic error is the uncertainty due to excited state contamination. We combine <math><mo>⟨</mo><msup><mi>x</mi><mn>3</mn></msup><mo>⟩</mo></math> with the two lower moments, namely <math><mo>⟨</mo><mi>x</mi><mo>⟩</mo></math> and <math><mo>⟨</mo><msup><mi>x</mi><mn>2</mn></msup><mo>⟩</mo></math>, to obtain the ratios <math><mo>⟨</mo><msup><mi>x</mi><mn>3</mn></msup><mo>⟩</mo><mo>/</mo><mo>⟨</mo><mi>x</mi><mo>⟩</mo></math> and <math><mo>⟨</mo><msup><mi>x</mi><mn>3</mn></msup><mo>⟩</mo><mo>/</mo><mo>⟨</mo><msup><mi>x</mi><mn>2</mn></msup><mo>⟩</mo></math>, as well as <math><mo>⟨</mo><msup><mi>x</mi><mn>3</mn></msup><msubsup><mo>⟩</mo><mi>K</mi><msup><mi>u</mi><mo>+</mo></msup></msubsup><mo>/</mo><mo>⟨</mo><msup><mi>x</mi><mn>3</mn></msup><msubsup><mo>⟩</mo><mi>π</mi><msup><mi>u</mi><mo>+</mo></msup></msubsup></math> and <math><mo>⟨</mo><msup><mi>x</mi><mn>3</mn></msup><msubsup><mo>⟩</mo><mi>K</mi><msup><mi>u</mi><mo>+</mo></msup></msubsup><mo>/</mo><mo>⟨</mo><msup><mi>x</mi><mn>3</mn></msup><msubsup><mo>⟩</mo><mi>π</mi><msup><mi>u</mi><mo>+</mo></msup></msubsup></math>. In addition, we reconstruct the <math><mi>x</mi></math>-dependence of the pion and kaon PDFs via 2- and 3-parameter fits to our results. We find that the reconstruction is feasible and that our lattice data favor a large <math><mi>x</mi></math>-dependence that falls as <math><mo>(</mo><mn>1</mn><mo>−</mo><mi>x</mi><msup><mo>)</mo><mn>2</mn></msup></math> for both the pion and kaon PDFs. We integrate the reconstructed PDFs to extract the higher moments with <math><mn>4</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>6</mn></math>. Finally, we compare the pion and kaon PDFs, as well as the ratios of their moments, to address the effect of SU(3) flavor symmetry breaking.

Pion and kaon <math><mo>⟨</mo><msup><mi>x</mi><mn>3</mn></msup><mo>⟩</mo></math> from lattice QCD and PDF reconstruction from Mellin moments

We present a calculation of the pion quark momentum fraction, <math><mo>⟨</mo><mi>x</mi><mo>⟩</mo></math>, and its third Mellin moment <math><mo>⟨</mo><msup><mi>x</mi><mn>2</mn></msup><mo>⟩</mo></math>. We also obtain directly, for the first time, <math><mo>⟨</mo><mi>x</mi><mo>⟩</mo></math> and <math><mo>⟨</mo><msup><mi>x</mi><mn>2</mn></msup><mo>⟩</mo></math> for the kaon using local operators. We use an ensemble of two degenerate light, a strange, and a charm quark (<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>) of maximally twisted mass fermions with clover improvement. The quark masses are chosen so that they reproduce a pion mass of about 260 MeV and a kaon mass of 530 MeV. The lattice spacing of the ensemble is 0.093 fm and the lattice has a spatial extent of 3 fm. We analyze several values of the source-sink time separation within the range of 1.12–2.23 fm to study and eliminate excited-state contributions. The necessary renormalization functions are calculated nonperturbatively in the <math><mrow><msup><mrow><mrow><mi>RI</mi></mrow></mrow><mrow><mo>′</mo></mrow></msup></mrow></math> scheme and are converted to the modified minimal subtraction scheme at a scale of 2 GeV. The final values for the momentum fraction are <math><mrow><mo>⟨</mo><mi>x</mi><msubsup><mrow><mo>⟩</mo></mrow><mrow><msup><mrow><mi>u</mi></mrow><mrow><mo>+</mo></mrow></msup></mrow><mrow><mi>π</mi></mrow></msubsup><mo>=</mo><mspace></mspace><mn>0.261</mn><mo>(</mo><mn>3</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>stat</mi></mrow></msub><mo>(</mo><mn>6</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>syst</mi></mrow></msub></mrow></math>, <math><mo>⟨</mo><mi>x</mi><msubsup><mo>⟩</mo><msup><mi>u</mi><mo>+</mo></msup><mi>K</mi></msubsup><mo>=</mo><mn>0.246</mn><mo>(</mo><mn>2</mn><msub><mo>)</mo><mrow><mi>stat</mi></mrow></msub><mo>(</mo><mn>2</mn><msub><mo>)</mo><mrow><mi>syst</mi></mrow></msub></math>, and <math><mo>⟨</mo><mi>x</mi><msubsup><mo>⟩</mo><msup><mi>s</mi><mo>+</mo></msup><mi>K</mi></msubsup><mo>=</mo><mn>0.317</mn><mo>(</mo><mn>2</mn><msub><mo>)</mo><mrow><mi>stat</mi></mrow></msub><mo>(</mo><mn>1</mn><msub><mo>)</mo><mrow><mi>syst</mi></mrow></msub></math>. For the third Mellin moments, we find <math><mo>⟨</mo><msup><mi>x</mi><mn>2</mn></msup><msubsup><mo>⟩</mo><msup><mi>u</mi><mo>+</mo></msup><mi>π</mi></msubsup><mo>=</mo><mn>0.082</mn><mo>(</mo><mn>21</mn><msub><mo>)</mo><mrow><mi>stat</mi></mrow></msub><mo>(</mo><mn>17</mn><msub><mo>)</mo><mrow><mi>syst</mi></mrow></msub></math>, <math><mo>⟨</mo><msup><mi>x</mi><mn>2</mn></msup><msubsup><mo>⟩</mo><msup><mi>u</mi><mo>+</mo></msup><mi>K</mi></msubsup><mo>=</mo><mn>0.093</mn><mo>(</mo><mn>5</mn><msub><mo>)</mo><mrow><mi>stat</mi></mrow></msub><mo>(</mo><mn>3</mn><msub><mo>)</mo><mrow><mi>syst</mi></mrow></msub></math>, and <math><mo>⟨</mo><msup><mi>x</mi><mn>2</mn></msup><msubsup><mo>⟩</mo><msup><mi>s</mi><mo>+</mo></msup><mi>K</mi></msubsup><mo>=</mo><mn>0.134</mn><mo>(</mo><mn>5</mn><msub><mo>)</mo><mrow><mi>stat</mi></mrow></msub><mo>(</mo><mn>2</mn><msub><mo>)</mo><mrow><mi>syst</mi></mrow></msub></math>. The reported systematic uncertainties are due to excited-state contamination. We also give the ratio <math><mo>⟨</mo><msup><mi>x</mi><mn>2</mn></msup><mo>⟩</mo><mo>/</mo><mo>⟨</mo><mi>x</mi><mo>⟩</mo></math> which is an indication of how quickly the parton distribution functions lose support at large <math><mi>x</mi></math>.

Mellin moments <math><mrow><mo>⟨</mo><mi>x</mi><mo>⟩</mo></mrow></math> and <math><mrow><mo>⟨</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>⟩</mo></mrow></math> for the pion and kaon from lattice QCD