We present the first lattice quantum chromodynamics (QCD) calculation of the pion valence-quark transverse-momentum-dependent parton distribution function (TMDPDF) within the framework of large-momentum effective theory (LaMET). Using correlators fixed in the Coulomb gauge (CG), we computed the quasi-TMD beam function for a pion with a mass of 300 MeV, a fine lattice spacing of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>a</mi><mo>=</mo><mn>0.06</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math>, and multiple large momenta up to 3 GeV. The intrinsic soft functions in the CG approach are extracted from form factors with large momentum transfer, and as a byproduct, we also obtain the corresponding Collins-Soper (CS) kernel. Our determinations of both the soft function and the CS kernel agree with perturbation theory at small transverse separations (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>b</mi><mo>⊥</mo></msub></math>) between the quarks. At larger <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>b</mi><mo>⊥</mo></msub></math>, the CS kernel remains consistent with recent results obtained using both CG and gauge-invariant TMD correlators in the literature. By combining next-to-leading logarithmic factorization of the quasi-TMD beam function and the soft function, we obtain an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math>-dependent pion valence-quark TMDPDF for transverse separations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>b</mi><mo>⊥</mo></msub><mo>≳</mo><mn>1</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math>. Interestingly, we find that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>b</mi><mo>⊥</mo></msub></math> dependence of the phenomenological parametrizations of TMDPDF for moderate values of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> are in reasonable agreement with our QCD determinations. In addition, we present results for the transverse-momentum-dependent wave function for a heavier pion with 670 MeV mass.

United States

Dennis Bollweg

Xiang Gao

Jinchen He

Swagato Mukherjee

Yong Zhao

Transverse-momentum-dependent pion structures from lattice QCD: Collins-Soper kernel, soft factor, TMDWF, and TMDPDF

We calculate the perturbative corrections in the strong coupling to the unpolarized quark transverse-momentum dependent parton distribution function (TMDPDF) operator within a background-field framework, extending the approach of Mukherjee et al. [<a href="http://dx.doi.org/10.1103/PhysRevD.109.034035">Phys. Rev. D 109, 034035 (2024)</a>]. We focus on ensuring gauge invariance, identifying two key components needed: a gauge-invariant TMDPDF operator with a transverse gauge link at spatial infinity, and accounting of the equations of motion (EoM) of the background fields. We go beyond next-to-leading order in strong coupling expansion, considering multiple interactions with the background field at all orders of strong coupling. By examining the interplay between quark and gluon contributions, we show that spurious singularities, proportional to EoM, can be misinterpreted as genuine divergences in quantum chromodynamics factorization unless properly identified and removed.

Vladimir V. Skokov

Andrey Tarasov

Shaswat Tiwari

Perturbative corrections to quark TMDPDFs in the background-field method: Gauge invariance, equations of motion, and multiple interactions

We calculate the temperature dependence of bottomonium correlators in (2+1)-flavor lattice QCD with the aim to constrain in-medium properties of bottomonia at high temperature. The lattice calculations are performed using HISQ action with physical strange quark mass and light quark masses twenty times smaller than the strange quark mass at two lattice spacings $\textit{a}$ = 0$\textit{.}$0493 fm and 0$\textit{.}$0602 fm, and temporal extents $\textit{N}$ = 16 − 30, corresponding to the temperatures $\textit{T}$ = 133 − 250 MeV. We use a tadpole-improved NRQCD action including spin-dependent $\textit{v}$ corrections for the heavy quarks and extended meson operators in order to be sensitive to in-medium properties of the bottomonium states of interest. We find that within estimated errors the bottomonium masses do not change compared to their vacuum values for all temperatures under our consideration; however, we find different nonzero widths for the various bottomonium states.

China

Heng-Tong Ding

Wei-Ping Huang

Germany

Rasmus Larsen

Stefan Meinel

Peter Petreczky

Zhanduo Tang

In-medium bottomonium properties from lattice NRQCD calculations with extended meson operators

Hadronic vacuum polarization (HVP) and light-by-light scattering (HLBL) are crucial for evaluating the Standard Model predictions concerning the muon’s anomalous magnetic moment. However, direct first-principle lattice gauge theory-based calculations of these observables in the timelike region remain challenging. Discrepancies persist between lattice quantum chromodynamics (QCD) calculations in the spacelike region and dispersive approaches relying on experimental data parametrization from the timelike region. Here, we introduce a methodology employing 1+1-dimensional quantum electrodynamics (QED), i.e. the Schwinger Model, to investigate the HVP and HLBL. To that end, we use both tensor network techniques, specifically matrix product states, and classical emulators of digital quantum computers. Demonstrating feasibility in a simplified model, our approach sets the stage for future endeavors leveraging digital quantum computers.

João Barata

Kazuki Ikeda

Jonathan Raghoonanan

Towards a real-time computation of timelike hadronic vacuum polarization and light-by-light scattering: Schwinger Model tests

In this work, we report a lattice calculation of -dependent valence pion generalized parton distributions (GPDs) at zero skewness with multiple values of the momentum transfer − . The calculations are based on an = 2 + 1 gauge ensemble of highly improved staggered quarks with Wilson-Clover valence fermion. The lattice spacing is 0.04 fm, and the pion valence mass is tuned to be 300 MeV. We determine the Lorentz-invariant amplitudes of the quasi-GPD matrix elements for both symmetric and asymmetric momenta transfers with similar values and show the equivalence of both frames. Then, focusing on the asymmetric frame, we utilize a hybrid scheme to renormalize the quasi-GPD matrix elements obtained from the lattice calculations. After the Fourier transforms, the quasi-GPDs are then matched to the light-cone GPDs within the framework of large momentum effective theory with improved matching, including the next-to-next-to-leading order perturbative corrections, and leading renormalon and renormalization group resummations. We also present the 3-dimensional image of the pion in impact-parameter space through the Fourier transform of the momentum transfer − .

Qi Shi

Sergey Syritsyn

Three-dimensional imaging of pion using lattice QCD: generalized parton distributions

Motivated by recent developments in the application of light-ray operators (LROs) in high energy physics, we propose a new strategy to study correlation functions of LROs through real-time quantum simulations. We argue that quantum simulators provide an ideal laboratory to explore the properties LROs in lower-dimensional quantum field theories. This is exemplified in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>1</mn><mo>+</mo><mn>1</mn></mrow></math>-d Schwinger model, employing tensor network methods, focusing on the calculation of energy and charge correlators. Despite some challenges in extracting the necessary correlation functions from the lattice, the methodology used can be extended to real quantum devices.

Probing celestial energy and charge correlations through real-time quantum simulations: Insights from the Schwinger model

In this work, we present a lattice QCD calculation of the Mellin moments of the twist-2 axial-vector generalized parton distribution (GPD), $$ \overset{\sim }{H}\left(x,\xi, t\right) $$ , at zero skewness, , with multiple values of the momentum transfer, . Our analysis employs the short-distance factorization framework on ratio-scheme renormalized quasi-GPD matrix elements. The calculations are based on an = 2 + 1 + 1 twisted mass fermions ensemble with clover improvement, a lattice spacing of = 0 093 fm, and a pion mass of = 260 MeV. We consider both the iso-vector and iso-scalar cases, utilizing next-to-leading-order perturbative matching while omitting the disconnected contributions and gluon mixing in the iso-scalar case. For the first time, we determine the Mellin moments of $$ \overset{\sim }{H} $$ up to the fifth order. From these moments, we discuss the quark helicity and orbital angular momentum contributions to the nucleon spin, as well as the spin-orbit correlations of the quarks. Additionally, we perform a Fourier transform over the momentum transfer, which allows us to explore the spin structure in the impact-parameter space.

Shohini Bhattacharya

Poland

Krzysztof Cichy

Martha Constantinou

Andreas Metz

Joshua Miller

Fernanda Steffens

Moments of axial-vector GPD from lattice QCD: quark helicity, orbital angular momentum, and spin-orbit correlation

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

Nikhil Karthik

Rui Zhang

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

We report the first lattice QCD computation of pion and kaon electromagnetic form factors, <math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>(</mo><msup><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></math>, at large momentum transfer up to 10 and <math><mn>28</mn><mtext> </mtext><mtext> </mtext><msup><mi>GeV</mi><mn>2</mn></msup></math>, respectively. Utilizing physical masses and two fine lattices, we achieve good agreement with JLab experimental results at <math><msup><mi>Q</mi><mn>2</mn></msup><mo>≲</mo><mn>4</mn><mtext> </mtext><mtext> </mtext><msup><mi>GeV</mi><mn>2</mn></msup></math>. For <math><msup><mi>Q</mi><mn>2</mn></msup><mo>≳</mo><mn>4</mn><mtext> </mtext><mtext> </mtext><msup><mi>GeV</mi><mn>2</mn></msup></math>, our results provide ab initio QCD benchmarks for the forthcoming experiments at JLab 12 GeV and future electron-ion colliders. We also test the QCD collinear factorization framework utilizing our high-<math><msup><mi>Q</mi><mn>2</mn></msup></math> form factors at next-to-next-to-leading order in perturbation theory, which relates the form factors to the leading Fock-state meson distribution amplitudes. Comparisons with independent lattice QCD calculations using the same framework demonstrate, within estimated uncertainties, the universality of these nonperturbative quantities.

Andrew D. Hanlon

QCD Predictions for Meson Electromagnetic Form Factors at High Momenta: Testing Factorization in Exclusive Processes

We present some lattice QCD results on first (<math><mrow><msubsup><mrow><mi>χ</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>i</mi></mrow></msubsup></mrow></math>) and second (<math><msubsup><mi>χ</mi><mn>2</mn><mi>i</mi></msubsup></math>) cumulants of and correlations (<math><mrow><msubsup><mrow><mi>χ</mi></mrow><mrow><mn>11</mn></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msubsup></mrow></math>) among net baryon-number (<math><mi>B</mi></math>), strangeness (<math><mi>S</mi></math>) and electric charge (<math><mi>Q</mi></math>) along the pseudocritical line [<math><msub><mi>T</mi><mrow><mi>p</mi><mi>c</mi></mrow></msub><mo>(</mo><msub><mi>μ</mi><mi>B</mi></msub><mo>)</mo></math>] in the temperature (<math><mi>T</mi></math>)–baryon chemical potential (<math><msub><mi>μ</mi><mi>B</mi></msub></math>) phase diagram of (<math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math>)-flavor QCD. We point out that violations of sum rules among second order cumulants, which hold in the isospin symmetric limit of vanishing electric charge chemical potential, are small along the <math><msub><mi>T</mi><mrow><mi>p</mi><mi>c</mi></mrow></msub><mo>(</mo><msub><mi>μ</mi><mi>B</mi></msub><mo>)</mo></math> for the entire range of <math><msub><mi>μ</mi><mi>B</mi></msub></math> covered in the RHIC beam energy scan. For the strangeness neutral matter produced in heavy-ion collisions this leads to a close relation between <math><msubsup><mi>χ</mi><mn>11</mn><mrow><mi>B</mi><mi>S</mi></mrow></msubsup></math> and <math><msubsup><mi>χ</mi><mn>11</mn><mrow><mi>Q</mi><mi>S</mi></mrow></msubsup></math>. We compare lattice QCD results for <math><msubsup><mi>χ</mi><mn>11</mn><mrow><mi>B</mi><mi>S</mi></mrow></msubsup><mo>/</mo><msubsup><mi>χ</mi><mn>2</mn><mi>S</mi></msubsup></math> along the <math><msub><mi>T</mi><mrow><mi>p</mi><mi>c</mi></mrow></msub><mo>(</mo><msub><mi>μ</mi><mi>B</mi></msub><mo>)</mo></math> line with preliminary experimental measurements of <math><msubsup><mi>χ</mi><mn>11</mn><mrow><mi>B</mi><mi>S</mi></mrow></msubsup><mo>/</mo><msubsup><mi>χ</mi><mn>2</mn><mi>S</mi></msubsup></math> for collision energies <math><mrow><mn>7.7</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi><mo>≤</mo><msqrt><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>N</mi><mi>N</mi></mrow></msub></mrow></msqrt><mo>≤</mo><mn>62.4</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></mrow></math>. While we find good agreements for <math><msqrt><msub><mi>s</mi><mrow><mi>N</mi><mi>N</mi></mrow></msub></msqrt><mo>≥</mo><mn>39</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>, differences are sizeable at smaller values of <math><msqrt><msub><mi>s</mi><mrow><mi>N</mi><mi>N</mi></mrow></msub></msqrt></math>. Moreover, we compare lattice QCD results for the ratio of the strangeness (<math><msub><mi>μ</mi><mi>S</mi></msub></math>) to baryon (<math><msub><mi>μ</mi><mi>B</mi></msub></math>) chemical potentials, which define a strangeness neutral system with fixed electric charge to baryon number density, with experimental results obtained by the STAR collaboration for <math><msub><mi>μ</mi><mi>S</mi></msub><mo>/</mo><msub><mi>μ</mi><mi>B</mi></msub></math> using strange baryon yields on the freeze-out line. Finally, we determine the baryon chemical potential at the freeze-out (<math><msubsup><mi>μ</mi><mi>B</mi><mi>f</mi></msubsup></math>) by comparing <math><msubsup><mi>χ</mi><mn>1</mn><mi>B</mi></msubsup><mo>/</mo><msubsup><mi>χ</mi><mn>2</mn><mi>B</mi></msubsup></math> along the <math><msub><mi>T</mi><mrow><mi>p</mi><mi>c</mi></mrow></msub><mo>(</mo><msub><mi>μ</mi><mi>B</mi></msub><mo>)</mo></math> with the experimentally measured net-proton cumulants <math><msubsup><mi>χ</mi><mn>1</mn><mi>p</mi></msubsup><mo>/</mo><msubsup><mi>χ</mi><mn>2</mn><mi>p</mi></msubsup></math>. We find that <math><mrow><mo>{</mo><msubsup><mrow><mi>μ</mi></mrow><mrow><mi>B</mi></mrow><mrow><mi>f</mi></mrow></msubsup><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>c</mi></mrow></msub><mo>(</mo><msubsup><mrow><mi>μ</mi></mrow><mrow><mi>B</mi></mrow><mrow><mi>f</mi></mrow></msubsup><mo>)</mo><mo>}</mo></mrow></math> are consistent with the freeze-out parameters of the statistical-model fits to experimentally measured hadron yields for <math><msqrt><msub><mi>s</mi><mrow><mi>N</mi><mi>N</mi></mrow></msub></msqrt><mo>≥</mo><mn>11.5</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>.

D. Bollweg

H.T. Ding

Japan

J. Goswami

F. Karsch

P. Petreczky

C. Schmidt

Strangeness-correlations on the pseudocritical line in (<math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math>)-flavor QCD