We present the first lattice QCD calculation of the quark transverse spin-momentum correlation, i.e., the T-odd Boer-Mulders function of the pion, using large-momentum effective theory. The calculation is done at three lattice spacings <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>a</mi><mo>=</mo><mo stretchy="false">{</mo><mn>0.098</mn><mo>,</mo><mn>0.085</mn><mo>,</mo><mn>0.064</mn><mo stretchy="false">}</mo><mtext> </mtext><mtext> </mtext><mi>fm</mi></mrow></math> and pion masses <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo>∼</mo><mn>350</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>, with pion momenta up to 1.8 GeV. The matrix elements are renormalized in a state-of-the-art scheme and extrapolated to the continuum and infinite momentum limit. We have implemented the perturbative matching up to the next-to-next-to-leading order and carried out a renormalization-group resummation. Our results provide valuable input for phenomenological analyses of the Boer-Mulders single-spin asymmetry.

Germany

Lisa Walter

China

Jun Hua

Sebastian Lahrtz

Lingquan Ma

Andreas Schäfer

Hai-Tao Shu

United States

Yushan Su

Peng Sun

Wei Wang

Xiaonu Xiong

Yi-Bo Yang

Jian-Hui Zhang

Qi-An Zhang

Quark transverse spin-momentum correlation of the pion from lattice QCD: The Boer-Mulders function

We report a lattice QCD calculation of the parton distribution function (PDF) of a deuteronlike dibaryon system using large-momentum effective theory. The calculation is done on three Wilson-clover ensembles with a fixed lattice spacing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>a</mi><mo>=</mo><mn>0.105</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></mrow></math> and two pion masses. The lattice matrix elements are computed at proton momenta up to 2.46 GeV, with the signal of high-momentum modes being improved by applying the momentum smearing technique. The state-of-the-art renormalization, matching, and extrapolation are then applied to obtain the final result of the light-cone PDF. A comparison between the result of the dibaryon system and the sum of the proton and neutron PDFs is also given.

Chen Chen

Yiqi Geng

Liuming Liu

Fei Yao

Kuan Zhang

Parton distribution function of a deuteronlike dibaryon system from lattice QCD

We calculate the leading-twist light cone distribution amplitudes of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Λ</mi><mo stretchy="false">(</mo><mi>u</mi><mi>d</mi><mi>s</mi><mo stretchy="false">)</mo></math> baryon using lattice QCD methods within the framework of large-momentum effective theory. Our numerical computations are carried out with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></math> stout smeared clover fermions and a Symanzik gauge action on a lattice with spacing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>a</mi><mo>=</mo><mn>0.077</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math> and a pion mass of 303 MeV. To approach the large-momentum region, we simulate the equal-time correlations with the hadron momentum <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>P</mi></mrow><mrow><mi>z</mi></mrow></msup><mo>=</mo><mo stretchy="false">{</mo><mn>2.52</mn><mo>,</mo><mn>3.02</mn><mo>,</mo><mn>3.52</mn><mo stretchy="false">}</mo><mtext> </mtext><mtext> </mtext><mi>GeV</mi></mrow></math>. We investigate the analytic properties of the baryon quasidistribution amplitude in coordinate space and validate them through our lattice calculations. After renormalization and extrapolation, we present results for the three-dimensional distribution in momentum fractions of the two light quarks. Based on these findings, we briefly discuss the phenomenological impact on weak decays of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Λ</mi><mi>b</mi></msub></math> and outline potential systematic uncertainties that can be improved in the future. This work lays the theoretical foundation for accessing baryon light cone distribution amplitudes from lattice QCD.

Poland

Min-Huan Chu

Haoyang Bai

Jian Liang

Xiangdong Ji

Jun Zeng

Light cone distribution amplitude for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Λ</mi></math> baryon from lattice QCD

We develop an approach for calculating heavy quark effective theory (HQET) light-cone distribution amplitudes (LCDAs) by employing a sequential effective theory methodology. The theoretical foundation of the framework is established, elucidating how the quasidistribution amplitudes (quasi-DAs) with three scales can be utilized to compute HQET LCDAs. We provide theoretical support for this approach by demonstrating the rationale behind devising a hierarchical ordering for the three involved scales, discussing the factorization at each step, clarifying the underlying reason for obtaining HQET LCDAs in the final phase, and addressing potential theoretical challenges. The lattice QCD simulation aspect is explored in detail, and the computations of quasi-DAs are presented. We employ three fitting strategies to handle contributions from excited states and extract the bare matrix elements. For renormalization purposes, we apply hybrid renormalization schemes at short and long distance separations. To mitigate long-distance perturbations, we perform an extrapolation in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>λ</mi><mo>=</mo><mi>z</mi><mo>·</mo><msup><mi>P</mi><mi>z</mi></msup></math> and assess the stability against various parameters. After two-step matching, our results for HQET LCDAs are found in agreement with existing model parametrizations. The potential phenomenological implications of the results are discussed, shedding light on how these findings could impact our understanding of the strong interaction dynamics and physics beyond the standard model. It should be noted, however, that systematic uncertainties have not been accounted for yet.

Xue-Ying Han

Cai-Dian Lü

Ji Xu

Yibo Yang

Shuai Zhao

Calculation of heavy meson light-cone distribution amplitudes from lattice QCD

It has been recently demonstrated that light cone distribution amplitudes (LCDAs) for a heavy meson, defined in the heavy quark limit, can be extracted by simulating quasidistribution amplitudes on the lattice with a large meson momentum <math><msup><mi>P</mi><mi>z</mi></msup></math>. This extraction involves a two-step procedure. In the first step, the quasidistribution amplitudes are matched onto the QCD LCDAs in the large <math><msup><mi>P</mi><mi>z</mi></msup></math> limit. In the second step, the QCD LCDAs are matched onto the desired LCDAs defined in heavy quark effective theory. In this work, we present the <math><msubsup><mi>m</mi><mi>H</mi><mrow><mn>2</mn><mi>n</mi></mrow></msubsup><mo>/</mo><mo>(</mo><msup><mi>P</mi><mi>z</mi></msup><msup><mo>)</mo><mrow><mn>2</mn><mi>n</mi></mrow></msup></math> and <math><msubsup><mi>Λ</mi><mi>QCD</mi><mn>2</mn></msubsup><mo>/</mo><mo>(</mo><mi>x</mi><msup><mi>P</mi><mi>z</mi></msup><msup><mo>)</mo><mn>2</mn></msup></math> and <math><msubsup><mi>Λ</mi><mi>QCD</mi><mn>2</mn></msubsup><mo>/</mo><mo>(</mo><mo>(</mo><mn>1</mn><mo>−</mo><mi>x</mi><mo>)</mo><msup><mi>P</mi><mi>z</mi></msup><msup><mo>)</mo><mn>2</mn></msup></math> corrections in the first step with <math><msub><mi>m</mi><mi>H</mi></msub></math> being the heavy meson mass and <math><mi>x</mi></math> being the momentum fraction of the light quark. To account for <math><msubsup><mi>m</mi><mi>H</mi><mrow><mn>2</mn><mi>n</mi></mrow></msubsup><mo>/</mo><mo>(</mo><msup><mi>P</mi><mi>z</mi></msup><msup><mo>)</mo><mrow><mn>2</mn><mi>n</mi></mrow></msup></math> corrections, we employ two methods: the moment relation that can provide mass corrections to all orders, and a leading-twist projector that gives the leading-order mass corrections. For the <math><msubsup><mi>Λ</mi><mi>QCD</mi><mn>2</mn></msubsup><mo>/</mo><mo>(</mo><msup><mi>P</mi><mi>z</mi></msup><msup><mo>)</mo><mn>2</mn></msup></math> corrections, we employ a renormalon model that addresses the renormalon ambiguity stemming from the divergence of the perturbative series. We adopt two parametrizations for QCD LCDAs to conduct a numerical analysis. The results suggest that these corrections are typically smaller than 20% in the region <math><mn>0.2</mn><mo>&lt;</mo><mi>x</mi><mo>&lt;</mo><mn>0.8</mn></math>, and should be included in a precision analysis. These findings will serve as a valuable guide for future studies on heavy meson LCDAs, particularly in the context of lattice QCD, where accounting for such corrections is crucial for improving the accuracy and reliability of the results.

Chao Han

Jia-Lu Zhang

Hong Kong

Power corrections to quasidistribution amplitudes of a heavy meson

We analyze the gauge fixing precision dependence of some nonlocal quark-bilinear lattice operators interesting in computing parton physics for several measurements, using five lattice spacings ranging from 0.032 to 0.121 fm. Our results show that gauge-dependent nonlocal measurements are significantly more sensitive to the precision of gauge fixing than anticipated. The impact of imprecise gauge fixing is significant for fine lattices and long distances. For instance, even with the typically defined precision of Landau gauge fixing of <math><msup><mn>10</mn><mrow><mo>−</mo><mn>8</mn></mrow></msup></math>, the deviation caused by imprecise gauge fixing can reach 12%, when calculating the trace of Wilson lines at 1.2 fm with a lattice spacing of approximately 0.03 fm. Similar behavior has been observed in <math><mi>ξ</mi></math> gauge and Coulomb gauge as well. For both quasi-parton-distribution functions (quasi-PDFs) and quasi-transverse-momentum-dependent PDFs operators renormalized using the regularization independent momentum subtraction (RI/MOM) scheme, convergence for different lattice spacings at long distance is only observed when the precision of Landau gauge fixing is sufficiently high. To describe these findings quantitatively, we propose an empirical formula to estimate the required precision.

Yi-Kai Huo

Chun-Jiang Shi

Impact of gauge fixing precision on the continuum limit of nonlocal quark-bilinear lattice operators

We propose two approaches for extracting the total gluon helicity contribution to proton spin from lattice QCD, one from local operator matrix elements in a fixed gauge accessible on lattice with feasible renormalization, and the other from gauge-invariant nonlocal gluon correlators. Neither of these approaches requires a matching procedure when converted to the <math> <mover> <mi>MS</mi> <mo>¯</mo> </mover> </math> $$ \overline{\textrm{MS}} $$ scheme. Our proposal resolves a long-standing inconsistency in the literature regarding lattice calculations of the total gluon helicity, and has the potential to greatly facilitate these calculations.

Zhuoyi Pang

Total gluon helicity from lattice without effective theory matching

We present a first lattice QCD calculation of the unpolarized nucleon’s isovector transverse-momentum-dependent parton distribution functions (TMDPDFs), which are essential to predict observables of multiscale, semi-inclusive processes in the standard model. We use a <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> MILC ensemble with valence clover fermions on a highly improved staggered quark (HISQ) sea to compute the quark momentum distributions in a large-momentum nucleon on the lattice. The state-of-the-art techniques in renormalization and extrapolation in the correlation distance on the lattice are adopted. The perturbative kernel up to next-to-next-to-leading order is taken into account, and the dependence on the pion mass and the hadron momentum is explored. Our results are qualitatively comparable with phenomenological TMDPDFs, which provide an opportunity to predict high energy scatterings from first principles.

Jin-Chen He

Unpolarized transverse momentum dependent parton distributions of the nucleon from lattice QCD

We present a lattice QCD calculation of the transverse-momentum-dependent wave function (TMDWF) of the pion using large-momentum effective theory. Numerical simulations are based on one ensemble with <math><mrow><mn>2</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>1</mn></mrow></math> flavors of highly improved staggered quarks with lattice spacing <math><mi>a</mi><mo>=</mo><mn>0.121</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math> from the MILC collaboration, and one with <math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math> flavor clover fermions and tree-level Symanzik gauge action generated by the CLS collaboration with <math><mi>a</mi><mo>=</mo><mn>0.098</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math>. As a key ingredient, the soft function is first obtained by incorporating the one-loop perturbative contributions and a proper normalization. Based on this and the equal-time quasi-TMDWF simulated on the lattice, we extract the TMDWF. The results for both lattice ensembles are compatible and a comparison with a phenomenological parametrization is made. Our studies provide a first attempt of ab initio calculation of TMDWFs which will eventually lead to crucial theory inputs for making predictions for exclusive processes under QCD factorization.

Ji-Hao Wang

Transverse-momentum-dependent wave functions of the pion from lattice QCD

We report a state-of-the-art lattice QCD calculation of the isovector quark transversity distribution of the proton in the continuum and physical mass limit using large-momentum effective theory. The calculation is done at four lattice spacings <math><mrow><mi>a</mi><mo>=</mo><mo>{</mo><mn>0.098</mn><mo>,</mo><mn>0.085</mn><mo>,</mo><mn>0.064</mn><mo>,</mo><mn>0.049</mn><mo>}</mo><mtext> </mtext><mtext> </mtext><mi>fm</mi></mrow></math> and various pion masses ranging between 220 and 350 MeV, with proton momenta up to 2.8 GeV. The result is nonperturbatively renormalized in the hybrid scheme with self-renormalization, which treats the infrared physics at large correlation distance properly, and extrapolated to the continuum, physical mass, and infinite momentum limit. We also compare with recent global analyses for the nucleon isovector quark transversity distribution.

Taiwan

Jiunn-Wei Chen

Luchang Jin

India

Protick Mohanta

Nucleon Transversity Distribution in the Continuum and Physical Mass Limit from Lattice QCD