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Recent lattice quantum chromodynamics (lQCD) computations of bottomonium correlation functions with extended sources provide new insights into heavy-quark dynamics at distance scales which are of the order of the inverse temperature. We analyze these results employing the thermodynamic T-matrix approach, in a continued effort to interpret lQCD data for quarkonium correlation functions in a non-perturbative framework suitable for strongly coupled systems. Its key inputs are the in-medium driving kernel (potential) of the scattering equation and an interference function which implements 3-body effects in the quarkonium coupling to the thermal medium. A simultaneous description of lQCD results for the bottomonium correlators with extended operators and the previously analyzed Wilson line correlators only requires minor refinements of the potential but calls for stronger interference effects at larger separation of the bottom quark and antiquark. We then analyze the poles of the self-consistent T-matrices on the real axis to assess the survival of the various bound states. We estimate the pertinent temperatures where the poles disappear for the various bottomonium states and discuss the relation to the corresponding peaks in the bottomonium spectral functions. We also recalculate the spatial diffusion coefficient of the QGP and find it to be similar to that in our previous study., Comment: 30 pages, 17 figures

In this work, we present a lattice QCD calculation of the Mellin moments of the twist-2 axial-vector generalized parton distribution (GPD), $\widetilde{H}(x,\xi,t)$, at zero skewness, $\xi$, with multiple values of the momentum transfer, $t$. Our analysis employs the short-distance factorization framework on ratio-scheme renormalized quasi-GPD matrix elements. The calculations are based on an $N_f=2+1+1$ twisted mass fermions ensemble with clover improvement, a lattice spacing of $a = 0.093$ fm, and a pion mass of $m_\pi = 260$ MeV. We consider both the iso-vector and iso-scalar cases, utilizing next-to-leading-order perturbative matching while ignoring the disconnected contributions and gluon mixing in the iso-scalar case. For the first time, we determine the Mellin moments of $\widetilde{H}$ 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., Comment: 17 pages, 13 figures

In this work, we report a lattice calculation of $x$-dependent valence pion generalized parton distributions (GPDs) at zero skewness with multiple values of the momentum transfer $-t$. The calculations are based on an $N_f=2+1$ 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 $-t$., Comment: 33 pages, 14 figures

We present a lattice QCD calculation of the $x$-dependent pion and kaon distribution amplitudes (DA) in the framework of large momentum effective theory. This calculation is performed on a fine lattice of $a=0.076$~fm 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 $\overline{\rm MS}$ 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 $x\in[x_0,1-x_0]$ with $x_0=0.25$ for pion and $x_0=0.2$ for kaon with systematics under control. The kaon DA is shown to be slighted skewed, and narrower than pion DA. Although the $x$-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.

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 $a\approx 0.084$ fm. The bare quais-DA matrix elements are renormalized in the hybrid scheme and matched to $\overline{\rm MS}$ 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 $x\in[0.25,0.75]$ at the largest pion momentum $P_z\approx 1.85$~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 $2\sigma$ confidence level that the DA obtained from chiral fermions is flatter and lower near $x=0.5$., Comment: update to match the version published in journal

We report the first lattice QCD computation of pion and kaon electromagnetic form factors, $F_M(Q^2)$, at large momentum transfer up to 10 and 28 $\mathrm{GeV}^2$, respectively. Utilizing physical masses and two fine lattices, we achieve good agreement with JLab experimental results at $Q^2 \lesssim 4~\mathrm{GeV}^2$. For $Q^2 \gtrsim 4~\mathrm{GeV}^2$, our results provide $\textit{ab-initio}$ 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-$Q^2$ 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., Comment: Phys. Rev. Lett. 133, 181902; 15 pages, 9 figures

First lattice QCD calculations of $x$-dependent GPD have been performed in the (symmetric) Breit frame, where the momentum transfer is evenly divided between the initial and final hadron states. However, employing the asymmetric frame, we are able to obtain proton GPDs for multiple momentum transfers in a computationally efficient setup. In these proceedings, we focus on the helicity twist-2 GPD at zero skewness that gives access to the $\widetilde{H}$ GPD. We will cover the implementation of the asymmetric frame, its comparison to the Breit frame, and the dependence of the GPD on the squared four-momentum transfer, $-t$. The calculation is performed on an $N_f = 2+1+1$ ensemble of twisted mass fermions with a clover improvement. The mass of the pion for this ensemble is roughly 260 MeV., Comment: 9 pages, 4 figures, Contribution to The 40th International Symposium on Lattice Field Theory (Lattice 2023)

We reveal that the criticality of the chiral phase transition in QCD at the macroscale arises from the microscopic energy levels of its fundamental constituents, the quarks. We establish a novel relation between cumulants of the chiral order parameter (i.e., chiral condensate) and correlations among the energy levels of quarks (i.e., eigenspectra of the massless Dirac operator), which naturally leads to a generalization of the Banks-Casher relation. Based on this novel relation and through (2+1)-flavor lattice QCD calculations using the HISQ action with varying light quark masses in the vicinity of the chiral phase transition, we demonstrate that the correlations among the infrared part of the Dirac eigenspectra exhibit same universal scaling behaviors as expected of the cumulants of the chiral condensate. We find that these universal scaling behaviors extend up to the physical values of the up and down quark masses., Comment: 4 pages, 2 figures, talk presented at the 30th International Conference on Ultra-relativistic Nucleus-Nucleus Collisions (Quark Matter 2023), September 3-9, 2023, Houston, Texas, USA. arXiv admin note: text overlap with arXiv:2401.10263

We employ a thermodynamic $T$-matrix approach to study Wilson line correlators (WLCs) for a static quark-antiquark pair within the quark-gluon plasma (QGP). By refining earlier determined input parameters, self-consistent $T$-matrix calculations can approximately reproduce the WLCs from 2+1-flavor lattice-QCD (lQCD) computations, while also satisfying constraints on the bulk medium from the QGP equation of state. The main difference from earlier results is a reduced screening in the input potential toward higher temperatures. When elaborating the consequences of this scenario on transport properties we find a more pronounced temperature dependence of the low-momentum relaxation rate of heavy quarks in the QGP. This, in particular, leads to a rather weak increase of the (temperature-scaled) spatial diffusion coefficient, which turns out to be in fair agreement with the most recent pertinent lQCD data., Comment: 4 pages, 2 figures, Quark Matter 2023 proceeding

We study the QCD equation of state and the chiral condensate using the hadron resonance gas model with repulsive mean-field interactions. We find that the repulsive interactions improve the agreement with the lattice results on the derivatives of the pressure with respect to the baryon chemical potential up to eighth order. From the temperature dependence of the chiral condensate we estimate the crossover temperature as a function of baryon chemical potential, $T_{pc}(\mu_B)$. We find that the chiral crossover line starts to deviate significantly from the chemical freeze-out line already for $\mu_B>400$ MeV. Furthermore, we find that the chiral pseudocritical line can be parametrized as $T_{pc}(\mu_B)/T_{pc}(0)=1-\kappa_2 (\mu_B/T_{pc} (0))^2-\kappa_4 (\mu_B/T_{pc} (0))^4$ with $\kappa_2=0.0150(2)$ and $\kappa_4=3.1(6) \times 10^{-5}$, which are in agreement with lattice QCD results for small values of $\mu_B$. For the first time we find a tiny but non-zero value of $\kappa_4$ in our study., Comment: Matched with published version

We reveal that the universal scaling properties of the chiral phase transition in quantum chromodynamics (QCD) at the macroscale are, in fact, encoded within the microscopic energy levels of its fundamental constituents, the quarks. We introduce a novel relation between the cumulants of the chiral order parameter, i.e., the chiral condensate, and the correlations among the energy levels of quarks, i.e., the eigenspectra of the massless QCD Dirac operator. This relation elucidates how the fluctuations of the chiral condensate arise from the correlations within the infrared part of the energy spectra of quarks, and naturally leads to a generalization of the Banks-Casher relation for the cumulants of the chiral condensate. Then, through (2+1)-flavor lattice QCD calculations using HISQ action with varying light quark masses around the chiral phase transition temperature, we demonstrate that the correlations among the infrared part of the Dirac eigenvalue spectra exhibit same universal scaling behaviors as expected of the cumulants of the chiral condensate. We find that these universal scaling behaviors extend up to the physical values of the up and down quark masses., Comment: 9 pages, 3 figures, talk presented at the 40th International Symposium on Lattice Field Theory (Lattice2023), July 31st - August 4th, 2023, Fermi National Accelerator Laboratory

We present the first study of the quark mass dependence of the heavy quark momentum and spatial diffusion coefficients using lattice QCD with light dynamical quarks corresponding to a pion mass of 320 MeV. We find that, for the temperature range 195 MeV $

We present a lattice QCD calculation of the transversity isovector- and isoscalar-quark parton distribution functions (PDFs) of the proton utilizing a perturbative matching at next-to-leading-order (NLO) accuracy. Additionally, we determine the isovector and isoscalar tensor charges for the proton. In both calculations, the disconnected contributions to the isoscalar matrix elements have been ignored. The calculations are performed using a single ensemble of $N_f = 2 +1$ highly-improved staggered quarks simulated with physical-mass quarks and a lattice spacing of $a = 0.076$ fm. The Wilson-clover action, with physical quark masses and smeared gauge links obtained from one iteration of hypercubic smearing, is used in the valence sector. Using the NLO operator product expansion, we extract the lowest four to six Mellin moments and the PDFs via a neural network from the matrix elements in the pseudo-PDF approach. In addition, we calculate the PDFs in the quasi-PDF approach with hybrid-scheme renormalization and the recently developed leading-renormalon resummation technique, at NLO with the resummation of leading small-$x$ logarithms., Comment: 23 pages, 22 figures, and 2 tables; version accepted for publication in PRD

We utilize a previously constructed thermodynamic $T$-matrix approach to the quark-gluon plasma (QGP) to calculate Wilson line correlators (WLCs) of a static quark-antiquark pair and apply them to the results from 2+1-flavor lattice-QCD (lQCD) computations with realistic pion mass. The self-consistent $T$-matrix results, which include constraints from the lQCD equation of state in the light-parton sector, can describe the lQCD data for WLCs fairly well once refinements of the input parameters are implemented. In particular, the input potential requires less screening than used in previous $T$-matrix analyses. Pertinent predictions for the spectral and transport properties of the QGP are discussed, including the spatial diffusion coefficient for heavy quarks which turns out to have a rather weak temperature dependence, in approximate agreement with recent lQCD results., Comment: 11 pages, 10 figures

Recently, we made significant advancements in improving the computational efficiency of lattice QCD calculations for Generalized Parton Distributions (GPDs). This progress was achieved by adopting calculations of matrix elements in asymmetric frames, deviating from the computationally-expensive symmetric frame typically used, and allowing freedom in the choice for the distribution of the momentum transfer between the initial and final states. A crucial aspect of this approach involves the adoption of a Lorentz covariant parameterization for the matrix elements, introducing Lorentz-invariant amplitudes. This approach also allows us to propose an alternative definition of quasi-GPDs, ensuring frame independence and potentially reduce power corrections in matching to light-cone GPDs. In our previous work, we presented lattice QCD results for twist-2 unpolarized GPDs ($H$ and $E$) of quarks obtained from calculations performed in asymmetric frames at zero skewness. Building upon this work, we now introduce a novel Lorentz covariant parameterization for the axial-vector matrix elements. We employ this parameterization to compute the axial-vector GPD $\widetilde{H}$ at zero skewness, using an $N_f=2+1+1$ ensemble of twisted mass fermions with clover improvement. The light-quark masses employed in our calculations correspond to a pion mass of approximately 260 MeV., Comment: 32 pages, 20 figures. Version accepted for publication in Physical Review D

We study the correlator of temporal Wilson lines at non-zero temperature in 2+1 flavor lattice QCD with the aim to define the heavy quark-antiquark potential at non-zero temperature. For temperatures $153~{\rm MeV} \leq T \leq 352~{\rm MeV}$ the spectral representation of this correlator is consistent with a broadened peak in the spectral function, position or width of which then defines the real or imaginary parts of the heavy quark-antiquark potential at non-zero temperature, respectively. We find that the potential's real part is not screened contrary to the widely-held expectations. We comment on how this fact may modify the picture of quarkonium melting in the quark-gluon plasma.

The rise of exascale supercomputers has fueled competition among GPU vendors, driving lattice QCD developers to write code that supports multiple APIs. Moreover, new developments in algorithms and physics research require frequent updates to existing software. These challenges have to be balanced against constantly changing personnel. At the same time, there is a wide range of applications for HISQ fermions in QCD studies. This situation encourages the development of software featuring a HISQ action that is flexible, high-performing, open source, easy to use, and easy to adapt. In this technical paper, we explain the design strategy, provide implementation details, list available algorithms and modules, and show key performance indicators for SIMULATeQCD, a simple multi-GPU lattice code for large-scale QCD calculations, mainly developed and used by the HotQCD collaboration. The code is publicly available on GitHub., Comment: 17 pages, 7 figures

We outline the physics opportunities provided by the Electron Ion Collider (EIC). These include the study of the parton structure of the nucleon and nuclei, the onset of gluon saturation, the production of jets and heavy flavor, hadron spectroscopy and tests of fundamental symmetries. We review the present status and future challenges in EIC theory that have to be addressed in order to realize this ambitious and impactful physics program, including how to engage a diverse and inclusive workforce. In order to address these many-fold challenges, we propose a coordinated effort involving theory groups with differing expertise is needed. We discuss the scientific goals and scope of such an EIC Theory Alliance., Comment: 44 pages, ReVTeX, White Paper on EIC Theory Alliance

For the first time, we present a lattice QCD determination of Mellin moments of unpolarized generalized parton distributions (GPDs) of the proton from an analysis of the quasi-GPD matrix elements within the short-distance factorization framework. We perform our calculation on an $N_f$=2+1+1 twisted mass fermions ensemble with a clover improvement at lattice spacing $a=0.093$ fm and a pion mass of $m_\pi=260$ MeV. Focusing on the zero-skewness case, the iso-vector and iso-scalar quasi-GPDs are calculated from the $\gamma_0$ definition, as well as a recently proposed Lorentz-invariant definition. We utilize data on both symmetric and asymmetric kinematic frames, which allows us to obtain the Mellin moments for several values of the momentum transfer, $-t$, in the range 0.17 to $2.77~\rm{GeV}^2$. We use the ratio scheme for GPDs, i.e. renormalization group invariant ratios with leading-twist factorization formula and perturbatively calculated matching coefficients up to the next-next-to-leading order (NNLO) to extract Mellin moments of GPDs, which are consistent with renormalization-group improved results. We compare our determination from quasi-GPDs with the results extracted using standard calculations of Mellin moments of local operators, specifically those related to the electromagnetic and gravitational form factors. We estimated the moments of GPDs up to the fifth ones for the first time. By extrapolating the Mellin moments to $-t=0$, we obtained the quark charges, momentum fraction, as well as the angular momentum contributions to the proton spin. The impact parameter space interpretation of the GPD moments is discussed, which provides insights into the spatial distribution of unpolarized quarks and their correlations in the transverse plane of an unpolarized or transversely polarized proton., Comment: 28 pages, 22 figures

Macroscopic properties of the strong interaction near its chiral phase transition exhibit scaling behaviors, which are the same as those observed close to the magnetic transition in a 3-dimensional classical spin system with $O(4)$ symmetry. We show that the universal scaling properties of the chiral phase transition in Quantum Chromodynamics (QCD) at the macroscale are, in fact, encoded within the microscopic energy levels of its fundamental constituents, the quarks. We establish a connection between the cumulants of the chiral order parameter, i.e., the chiral condensate, and the correlations among the energy levels of quarks, i.e., the eigenspectra of the massless QCD Dirac operator. This relation elucidates how the fluctuations of the chiral condensate arise from the correlations within the infrared part of the energy spectra of quarks, and naturally leads to a generalization of the Banks-Casher relation for the cumulants of the chiral condensate. Then, through (2+1)-flavor lattice QCD calculations with varying light quark masses near the QCD chiral transition, we demonstrate that the correlations among the infrared part of the Dirac eigenvalue spectra exhibit same universal scaling behaviors as expected of the cumulants of the chiral condensate. We find that these universal scaling behaviors extend up to the physical values of the up and down quark masses. Our study reveals how the hidden scaling features at the microscale give rise to the macroscopic universal properties of QCD., Comment: 5 pages main text + 3 pages supplemental material, version to appear in PRL

We present the first calculations of the heavy flavor diffusion coefficient using lattice QCD with light dynamical quarks. For temperatures $195\,\mathrm{MeV}

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 $N_f = 2 + 1$ highly-improved staggered quarks with physical masses and a lattice spacing of $a = 0.076$ fm. 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 $x$ 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., Comment: 20 pages, 17 figures, and 3 tables; version accepted for publication in PRD

The heavy quark diffusion coefficient is encoded in the spectral functions of the chromo-electric and the chromo-magnetic correlators, of which the latter describes the T/M contribution. We study these correlators at two different temperatures $T=1.5T_c$ and $T=10^4T_c$ in the deconfined phase of SU(3) gauge theory. We use gradient flow for noise reduction. We perform both continuum and zero flow time limits to extract the heavy quark diffusion coefficient. Our results imply that the mass suppressed effects in the heavy quark diffusion coefficient are 20% for bottom quarks and 34% for charm quark at $T=1.5T_c$., Comment: 9 pages, 6 figures, Contribution to the 39th International Symposium on Lattice Field theory (LATTICE2022),8th-13th August, 2022, Bonn, Germany

We report our analysis for the static energy in (2+1+1)-flavor QCD over a wide range of lattice spacings and several quark masses. We obtain results for the static energy out to distances of nearly 1 fm, allowing us to perform a simultaneous determination of the lattice scales $r_2$, $r_1$ and $r_0$ as well as the string tension, $\sigma$. While our results for ${r_0}/{r_1}$ and $r_0$ $\sqrt{\sigma}$ agree with published (2+1)-flavor results, our result for ${r_1}/{r_2}$ differs significantly from the value obtained in the (2+1)-flavor case, likely due to the effect of the charm quark. We study in detail the effect of the charm quark on the static energy by comparing our results on the finest lattices with the previously published (2+1)-flavor QCD results at similar lattice spacing. The lattice results agree well with the two-loop perturbative expression of the static energy incorporating finite charm mass effects., Comment: 9 pages, 4 figures, The 39th International Symposium on Lattice Field Theory (Lattice2022),8-13 August, 2022,Bonn, Germany

We present results on the in-medium interactions of static quark anti-quark pairs using realistic 2+1 HISQ flavor lattice QCD. Focus is put on the extraction of spectral information from Wilson line correlators in Coulomb gauge using four complementary methods. Our results indicate that on HISQ lattices, the position of the dominant spectral peak associated with the real-part of the interquark potential remains unaffected by temperature. This is in contrast to prior work in quenched QCD and we present follow up comparisons to newly generated quenched ensembles., Comment: 7 pages, 4 figures, talk given at the XVth Quark confinement and the Hadron spectrum conference, Aug. 1st - 6th, 2022, Stavanger, Norway. Date of presentation Aug. 5th 2022

We present lattice QCD calculations of valence parton distribution function (PDF) of pion employing next-to-next-leading-order (NNLO) perturbative QCD matching. Our calculations are based on three gauge ensembles of 2+1 flavor highly improved staggered quarks and Wilson--Clover valance quarks, corresponding to pion mass $m_\pi=140$~MeV at a lattice spacing $a=0.076$~fm and $m_\pi=300$~MeV at $a=0.04, 0.06$~fm. This enables us to present, for the first time, continuum-extrapolated lattice QCD results for NNLO valence PDF of the pion at the physical point. Applying leading-twist expansion for renormalization group invariant (RGI) ratios of bi-local pion matrix elements with NNLO Wilson coefficients we extract $2^{\mathrm{nd}}$, $4^{\mathrm{th}}$ and $6^{\mathrm{th}}$ Mellin moments of the PDF. We reconstruct the Bjorken-$x$ dependence of the NNLO PDF from real-space RGI ratios using a deep neural network (DNN) as well as from momentum-space matrix elements renormalized using a hybrid-scheme. All our results are in broad agreement with the results of global fits to the experimental data carried out by the xFitter and JAM collaborations., Comment: 25 pages, 29 figures, version published in PRD

Contribution from the USQCD Collaboration to the Proceedings of the US Community Study on the Future of Particle Physics (Snowmass 2021)., Comment: 27 pp. main text, 4 pp. appendices, 29 pp. references, 1 p. index

In this work we address the question of how well the chiral crossover transition can be understood in terms of a noninteracting hadron resonance gas model. Using the latest results on the variation of hadron masses as a function of the pion mass from lattice quantum chromodynamics, we study the temperature dependence of the renormalized chiral condensate in 2+1 flavor QCD. Furthermore, we suggest a better criterion to estimate of the pseudocritical temperature, which gives $T_c = 161.2 \pm 1.7$ MeV, which is much improved compared to all the earlier results within the hadron resonance gas model or chiral perturbation theory. For the curvature of the pseudocritical line we find $\kappa_2 = 0.0203(7)$, which is in very good agreement with continuum extrapolated lattice results., Comment: v2: published version

We present a lattice QCD determination of the distribution amplitude (DA) of the pion and the first few Mellin moments from an analysis of the quasi-DA matrix element within the leading-twist framework. We perform our study on a HISQ ensemble with $a=0.076$ fm lattice spacing with the Wilson-Clover valence quark mass tuned to the physical point. We analyze the ratios of pion quasi-DA matrix elements at short distances using the leading-twist Mellin operator product expansion (OPE) at the next-to-leading order and the conformal OPE at the leading-logarithmic order. We find a robust result for the first non-vanishing Mellin moment $\langle x^2 \rangle = 0.287(6)(6)$ at a factorization scale $\mu=2$ GeV. We also present different Ans\"atze-based reconstructions of the $x$-dependent DA, from which we determine the perturbative leading-twist expectations for the pion electromagnetic and gravitational form-factors at large momentum transfers., Comment: 26 pages, 13 figures; published version

We present results for the static energy in ($2+1+1$)-flavor QCD over a wide range of lattice spacings and several quark masses, including the physical quark mass, with ensembles of lattice-gauge-field configurations made available by the MILC Collaboration. We obtain results for the static energy out to distances of nearly $1$~fm, allowing us to perform a simultaneous determination of the scales $r_{1}$ and $r_{0}$, as well as the string tension $\sigma$. For the smallest three lattice spacings we also determine the scale $r_{2}$. Our results for $r_{0}/r_{1}$ and $r_{0}\sqrt{\sigma}$ agree with published ($2+1$)-flavor results. However, our result for $r_{1}/r_{2}$ differs significantly from the value obtained in the ($2+1$)-flavor case, which is most likely due to the effect of the charm quark. We also report results for $r_{0}$, $r_{1}$, and $r_{2}$ in~fm, with the former two being slightly lower than published ($2+1$)-flavor results. We study in detail the effect of the charm quark on the static energy by comparing our results on the finest two lattices with the previously published ($2+1$)-flavor QCD results at similar lattice spacing. We find that for $r > 0.2$~fm our results on the static energy agree with the ($2+1$)-flavor result, implying the decoupling of the charm quark for these distances. For smaller distances, on the other hand, we find that the effect of the dynamical charm quark is noticeable. The lattice results agree well with the two-loop perturbative expression of the static energy incorporating finite charm mass effects. This is the first time that the decoupling of the charm quark is observed and quantitatively analyzed on lattice data of the static energy., Comment: 51 pages, 37 figures, journal version

We calculate chromo-electric and chromo-magnetic correlators in quenched QCD at $1.5T_c$ and $10^4 T_c$ with the aim to estimate the heavy quark diffusion coefficient at leading order in the inverse heavy quark mass expansion, $\kappa_E$, as well as the coefficient of first mass suppressed correction, $\kappa_B$. We use gradient flow for noise reduction. At $1.5T_c$ we obtain: $1.70 \le \kappa_E/T^3 \le 3.12$ and $1.03< \kappa_B/T^3 < 2.61$. The latter implies that the mass suppressed effects in the heavy quark diffusion coefficient are 20% for bottom quarks and 34% for charm quark at this temperature., Comment: 18 pages, 20 figures, updated to match the published version

In this document, we summarize the status and challenges of calculating parton physics in lattice QCD for the US Particle Physics Community Planning Exercise (a.k.a. "Snowmass"). While PDF-moments calculations have been very successful and been continuously improved, new methods have been developed to calculate distributions directly in $x$-space. Many recent lattice studies have been focused on calculating isovector PDFs of the pion and nucleon, learning to control systematics associated with excited-state contamination, renormalization and continuum extrapolations, pion-mass and finite-volume effects, etc. Although in some cases, the lattice results are already competitive with experimental data, to reach the level of precision in a wide range of $x$ for unpolarized nucleon PDFs impactful for future collider physics remains a challenge, and may require exascale supercomputing power. The new theoretical methods open the door for calculating other partonic observables which will be the focus of the experimental program in nuclear physics, including generalized parton distributions and transverse-momentum dependent PDFs. A fruitful interplay between experimental data and lattice-QCD calculations will usher in a new era for parton physics and hadron structure., Comment: 14 pages

The sequential melting of the bottomonium states is one of the important signals for the existence of a quark gluon plasma. The study of bottomonia spectral functions on the lattice is a difficult task for many reasons. Calculations based on NRQCD, that are commonly used for such purpose, are not applicable at high temperatures. In this work we propose a new method to study this problem by calculating the spatial screening masses of bottomonium states. We calculate the spatial meson correlators and extract the screening masses for mesons in different quantum channels using highly improved staggered quark (HISQ) action for bottom quarks and dynamical $2+1$ flavor QCD HISQ gauge configurations. The typical lattices we choose are of size $N_s^3 \times N_\tau$ where $N_s=4 N_\tau$ and $N_\tau=8, 10, 12$. We consider the temperature range $T = 300$-$1000$ MeV. We show that for $T > 500$ MeV the temperature dependence of the screening masses of the ground state bottomonia are compatible with the expectations based on uncorrelated quark anti-quark pairs., Comment: 9 pages, 3 figures, contribution to the 38th International Symposium on Lattice Field Theory, Lattice 2021, 26-30th July, 2021, Zoom/Gather@Massachusetts Institute of Technology

We report the first lattice QCD calculation of pion valence quark distribution with next-to-next-to-leading order perturbative matching correction, which is done using two fine lattices with spacings $a=0.04$ fm and $0.06$ fm and valence pion mass $m_\pi=300$ MeV, at boost momentum as large as $2.42$ GeV. As a crucial step to control the systematics, we renormalize the pion valence quasi distribution in the recently proposed hybrid scheme, which features a Wilson-line mass subtraction at large distances in coordinate space, and develop a procedure to match it to the $\overline{\rm MS}$ scheme. We demonstrate that the renormalization and the perturbative matching in Bjorken-$x$ space yield a reliable determination of the valence quark distribution for $0.03\lesssim x \lesssim 0.80$ with 5-20\% uncertainties., Comment: 4 pages, 4 figures and appendix

We study Polyakov loop as well as correlators of real and imaginary parts of the Polyakov loop in 2+1 flavor QCD at finite temperature. We use hypercubic (HYP) smearing to improve the signal in the lattice calculations and to obtain reliable results for the correlators at large distances. From the large distance behavior of the correlators we estimate the chromo-electric screening length to be (0.38-44)/T. Furthermore, we show that the short distance distortions due to HYP smearing do not affect the physics of interest, Comment: Contribution to the Proceedings of The 38th International Symposium on Lattice Field Theory, LATTICE2021 26th-30th July, 2021 Zoom/Gather@Massachusetts Institute of Technology , 8 pages, 3 figures

We have studied finite temperature complex static quark-antiquark potentials for 2+1 flavor QCD using highly improved staggered action with physical strange quark masses and light quark masses corresponding to a pion mass of 161 MeV. We calculated the potential using Wilson line correlators fixed in Coulomb gauge. For the extraction, we have used HTL motivated parametrization of the correlators. We found that the real part of the potential is screened above the crossover temperature and it's close to singlet free energies, whereas the imaginary part is increasing with both distance and temperature., Comment: 9 pages, 4 figures, Contribution to the 38th International Symposium on Lattice Field Theory, 26th-30th July 2021, Zoom/Gather@Massachusetts Institute of Technology

We investigate the criticality of chiral phase transition manifested in the first and second order derivatives of Dirac eigenvalue spectrum with respect to light quark mass in (2+1)-flavor lattice QCD. Simulations are performed at temperatures from about 137 MeV to 176 MeV on $N_{\tau}=8$ lattices using the highly improved staggered quarks and the tree-level improved Symanzik gauge action. The strange quark mass is fixed to its physical value $m_s^{\text{phy}}$ and the light quark mass is set to $m_s^{\text{phy}}/40$ which corresponds to a Goldstone pion mass $m_{\pi}=110$ MeV. We find that in contrast to the case at $T\simeq 205$ MeV $m_l^{-1} \partial \rho(\lambda, m_l)/\partial m_l$ is no longer equal to $\partial ^2\rho(\lambda, m_l)/\partial m_l^2$ and $\partial ^2\rho(\lambda, m_l)/\partial m_l^2$ even becomes negative at certain low temperatures. This means that as temperature getting closer to $T_c$ $\rho(\lambda, m_l)$ is no longer proportional to $m_l^2$ and thus dilute instanton gas approximation is not valid for these temperatures. We demonstrate the temperature dependence can be factored out in $\partial \rho(\lambda, m_l)/ \partial m_l$ and $\partial^2 \rho(\lambda, m_l)/ \partial m_l^2$ at $T \in [137, 153]$ MeV, and then we propose a feasible method to estimate the power $c$ given $\rho \propto m_l^{c}$., Comment: Talk presented at the 38th International Symposium on Lattice Field Theory - LATTICE2021, 26th-30th, July, 2021, Zoom/Gather@Massachusetts Institute of Technology

We explore the interactions between a quark anti-quark pair in a thermal medium based on lattice QCD ensembles with $N_f = 2+1$ dynamical HISQ flavors. Our dataset spans the phenomenologically relevant temperature range between T=140MeV-2GeV based on lattice sizes $N_\tau=10,12$ and $16$, with an aspect ratio of $N_\sigma/N_\tau=4$. The peak position $\Omega$ and the width $\Gamma$ of the spectral function of Wilson-line correlators in Coulomb gauge is computed. We assess the information content in the correlation functions and deploy three complementary strategies to reconstruct spectral information: model fits, Pad\'e approximation and the Bayesian BR method. Limitations of each approach are carefully assessed., Comment: Contribution to The 38th International Symposium on Lattice Field Theory, LATTICE2021 26th-30th July, 2021 Zoom/Gather@Massachusetts Institute of Technology, 9 pages, 7 figure, LaTeX, uses PoS class

The heavy quark diffusion coefficient is encoded in the spectral functions of the chromoelectric and the chromomagnetic correlators that are calculable on the lattice. We study the chromoelectric and the chromomagnetic correlator in the deconfined phase of SU(3) gauge theory using Symanzik flow at two temperatures $1.5T_c$ and $10000 T_c$, with $T_c$ being the phase transition temperature. To control the lattice discretization errors and perform the continuum limit we use several temporal lattice extents $N_t=16,20,24$ and 28. We observe that the flow time dependence of the chromomagnetic correlator is quite different from chromoelectric correlator most likely due to the anomalous dimension of the former as has been pointed out recently in the literature., Comment: 9 pages, 6 figures, The 38th International Symposium on Lattice Field Theory, LATTICE2021, 26th-30th July, 2021, Zoom/Gather@Massachusetts Institute of Technology

We report on the status of the analysis of the static energy in $2+1+1$-flavor QCD. The static energy is obtained by measuring Wilson line correlators in Coulomb gauge using the HISQ action, yielding the scales $r_{0}/a$, $r_{1}/a$, $r_{2}/a$, their ratios, and the string tension $\sigma r_{i}^{2}$. We put emphasis on the possible effects due to the dynamical charm-quark by comparing the lattice results to continuum results of the static energy with and without a massive flavor at two-loop accuracy. We employ gauge-field ensembles from the HotQCD and MILC Collaborations., Comment: 9 pages, 2 figures, The 38th International Symposium on Lattice Field Theory, LATTICE2021, 26th-30th July, 2021, Zoom/Gather@Massachusetts Institute of Technology

We study the interactions of a static quark antiquark pair at non-zero temperature using realistic 2+1 flavor lattice QCD calculations. The study consists of two parts: the first investigates the properties of Wilson line correlators in Coulomb gauge and compares to predictions of hard-thermal loop perturbation theory. As a second step we extract the spectral functions underlying the correlators using four conceptually different methods: spectral function fits, a HTL inspired fit for the correlation function, Pad\'e rational approximation and the Bayesian BR spectral reconstruction. We find that our high statistics Euclidean lattice data are amenable to different hypotheses for the shapes of the spectral function and we compare the implications of each analysis method for the existence and properties of a well defined ground state spectral peak.

We report on preliminary results from the calculations of the QCD equation of state for 2+1+1 flavors using HISQ action. The calculations are performed on lattices with temporal extents $N_{\tau}=6,~8,~10$, and $12$ and aspect ratio $N_{\sigma}/N_{\tau}=4$. We find that there is a significant contribution to the pressure from charm quarks at temperatures $T> 300$ MeV., Comment: 8 pages, 3 figures; poster presented at "The 38th International Symposium on Lattice Field Theory", LATTICE2021, 26th-30th July, 2021 Zoom/Gather@Massachusetts Institute of Technology

We present here continuum extrapolated results for all 2nd order cumulants using the most recent results obtained by the HotQCD collaboration in (2+1)-flavor QCD. We constrain the applicability of various HRG models by presenting a detailed comparison of our results based on different sets of hadron spectra as well as with virial expansion based model calculations. A comparison with our lattice QCD results for 2nd order cumulants with models that parametrize repulsive interactions among baryons and anti-baryons in a hadron resonance gas through a single excluded volume parameter (EVHRG) is also shown., Comment: prepared for the proceedings of "SQM2021"

We study the bottomonium screening masses in a 2+1 flavor QCD medium on the lattice using the Highly Improved Staggered Quark(HISQ) discretization for the quarks. We focus on a wide temperature range in the region $350~{\rm MeV}\le T < 1000 ~{\rm MeV}$, and perform our calculations on three different lattice spacings to control the lattice cut-off effects, corresponding to temporal lattice extent of $N_\tau=8,10,12$. From a detailed study of the temperature dependence of screening masses we conclude that while the $\eta_b(1S)$ and $\Upsilon(1S)$ states melt at $T>500$ MeV, the scalar and axial-vector states $\chi_{b0}(1P)$ and $h_b(1P)$ melt already at $T>350$ MeV., Comment: 6 pages, RevTEX, 3 figures, matches the published version

We present our results on the electromagnetic form factor of pion over a wide range of $Q^2$ using lattice QCD simulations with Wilson-clover valence quarks and HISQ sea quarks. We study the form factor at the physical point with a lattice spacing $a=0.076$ fm. To study the lattice spacing and quark mass effects, we also present results for 300 MeV pion at two different lattice spacings $a=0.04$ and 0.06 fm. The lattice calculations at the physical quark mass appear to agree with the experimental results. Through fits to the form factor, we estimate the charge radius of pion for physical pion mass to be $\langle r_{\pi}^2 \rangle=0.42(2)~{\rm fm}^2$., Comment: 14 pages, 14 figures, revised version accepted by PRD

We present an exploratory lattice QCD investigation of the differences between the valence quark structure of pion and its radial excitation $\pi(1300)$ in a fixed finite volume using the leading-twist factorization approach. We present evidences that the first pion excitation in our lattice computation is a single particle state that is likely to be the finite volume realization of $\pi(1300)$. An analysis with reasonable priors result in better estimates of the excited state PDF and the moments, wherein we find evidence that the radial excitation of pion correlates with an almost two-fold increase in the momentum fraction of valence quarks. This proof-of-principle work establishes the viability of future lattice computations incorporating larger operator basis that can resolve the structural changes accompanying hadronic excitation., Comment: 11 pages, 7 figures; References added

We revisit previous determination of the strong coupling constant from moments of quarkonium correlators in (2+1)-flavor QCD. We use previously calculated moments obtained with Highly Improved Staggered Quark (HISQ) action for five different quark masses and several lattice spacings. We perform a careful continuum extrapolations of the moments and from the comparison of these to the perturbative result we determine the QCD Lambda parameter, $\Lambda_{\overline{MS}}^{n_f=3}=332 \pm 17 \pm 2(scale)$ MeV. This corresponds to $\alpha_s^{n_f=5}(\mu=M_Z)=0.1177(12)$., Comment: 24 pages, 6 figures; data of Figs. 2-4 among the source files; primary data (figs. 1 and 6) available through arXiv:1901.06424)

We present results from lattice QCD calculations for $2^{nd}$ and $4^{th}$ order cumulants of conserved charge fluctuations and correlations, and compare these with various HRG model calculations. We show that differences between HRG and QCD calculations already show up in the second order cumulants close to the pseudo-critical temperature for the chiral transition in (2+1)-flavor QCD and quickly grow large at higher temperatures. We also show that QCD results for strangeness fluctuations are enhanced over HRG model calculations which are based only on particles listed in the Particle Data Group tables as 3-star resonances. This suggests the importance of contributions from additional strange hadron resonances. We furthermore argue that additional (repulsive) interactions, introduced either through excluded volume (mean field) HRG models or the S-matrix approach, do not improve the quantitative agreement with $2^{nd}$ and $4^{th}$ order cumulants calculated in lattice QCD. HRG based approaches fail to describe the thermodynamics of strongly interacting matter at or shortly above the pseudo-critical temperature of QCD., Comment: 11 pages, 11 figures, Prepared for the proceedings of workshop on Criticality in QCD and the Hadron Resonance Gas; 29-31 July 2020, Wroclaw, Poland