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We analyze the color-magnetic (or "$B$") field two-point function that encodes the finite-mass correction to the heavy quark momentum diffusion coefficient. The simulations are done on fine isotropic lattices in the quenched approximation at $1.5\,T_c$, using a range of gradient flow times for noise suppression and operator renormalization. The continuum extrapolation is performed at fixed flow time followed by a second extrapolation to zero flow time. Perturbative calculations to next-to-leading order of this correlation function, matching gradient-flowed correlators to MS-bar, are used to resolve nontrivial renormalization issues. We perform a spectral reconstruction based on perturbative model fits to estimate the coefficient $\kappa_B$ of the finite-mass correction to the heavy quark momentum diffusion coefficient. The approach we present here yields high-precision data for the correlator with all renormalization issues incorporated at next-to-leading order, and is also applicable for actions with dynamical fermions., Comment: 12 pages, 9 figures

The heavy ion experiments in Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) are going through upgrade in the next five years, shifting their focus more on the hard processes in the new runs. One of the main goals is to draw a finer image for the quark gluon plasma (QGP). The heavy flavor probes , which witness the whole history of heavy ion collision are particularly sensitive to test the properties of QGP formed in such collisions. The lattice results for heavy flavor probes provide transport and phenomenological models crucial inputs to describe the experimental observations like the strong suppression of the nuclear modification factor $R_{AA}$ and the non-zero azimuthal anisotropy at low $p_T$. In the last two years we have seen significant advances in the lattice QCD studies of heavy flavor probes, including the in-medium quarkonium properties, the complex static quark-antiquark potential and the heavy quark diffusion from lattice simulations at nonzero temperature. These achievements substantially deepen our understanding of the fate of quarkonium, the screening/unscreening of the complex potential and the temperature and quark mass dependence of the heavy quark diffusion in thermal medium. In these proceedings, we review recent results and briefly discuss possible directions in these studies., Comment: 18 pages, 8 figures, plenary talk delivered at the 40th International Symposium on Lattice Field Theory (Lattice 2023), July 31st - August 4th, 2023, Fermilab

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 calculate the soft function using lattice QCD in the framework of large momentum effective theory incorporating the one-loop perturbative contributions. The soft function is a crucial ingredient in the lattice determination of light cone objects using transverse-momentum-dependent (TMD) factorization. It consists of a rapidity-independent part called intrinsic soft function and a rapidity-dependent part called Collins-Soper kernel. We have adopted appropriate normalization when constructing the pseudo-scalar meson form factor that is needed in the determination of the intrinsic part and applied Fierz rearrangement to suppress the higher-twist effects. In the calculation of CS kernel we consider a CLS ensemble other than the MILC ensemble used in a previous study. We have also compared the applicability of determining the CS kernel using quasi TMDWFs and quasi TMDPDFs. As an example, the determined soft function is used to obtain the physical TMD wave functions (WFs) of pion and unpolarized iso-vector TMD parton distribution functions (PDFs) of proton., Comment: 24 pages, 19 figures, published version

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

This paper discusses the charmonium and bottomonium correlators in the pseudoscalar channel and the corresponding spectral reconstruction on the lattice. The absence of a transport peak in the pseudoscalar channel spectral function allows for an easier study of the in-medium modification of bound states. However, extracting spectral information from Euclidean correlators is still a numerically ill-posed problem. To constrain the spectral reconstruction, we use an ansatz motivated from perturbation theory. The perturbative model spectral function has two main contributions: a thermal part around the threshold obtained from pNRQCD and the vacuum part well above the threshold. These two regions are matched continuously, and the model spectral function is obtained by introducing parameters that control the overall thermal shift of the peak and the overall amplitude. The lattice correlator data is computed using clover-improved Wilson valence fermions on large and fine gauge field configurations generated using $N_f=2+1$ flavors Highly Improved Staggered Quark (HISQ) action with physical strange quark mass $m_s$, and slightly heavy degenerate up and down quark masses $m_l=m_s/5$ that corresponds to $m_\pi\simeq 320$ MeV. Our results obtained at $T=220$ MeV and $T=251$ MeV suggest that no resonance peaks are needed to describe the charmonium lattice data at these temperatures, while for bottomonium thermally broadened resonance peaks persist., Comment: 13 pages, 4 figures, Contribution to a special issue of Few Body Systems: Emergence and Structure of Baryons - Selected Contributions from the International Conference Baryons 2022

We present a first lattice QCD calculation of the transverse-momentum-dependent wave functions (TMDWFs) of the pion using large-momentum effective theory. Numerical simulations are based on one ensemble with 2+1+1 flavors of highly improved staggered quarks action with lattice spacing $a=0.121$~fm from the MILC Collaboration, and one with 2 +1 flavor clover fermions and tree-level Symanzik gauge action generated by the CLS Collaboration with $a=0.098$~fm. 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-TMDWFs simulated on the lattice, we extract the light-cone TMDWFs. The results are comparable between the two lattice ensembles and a comparison with 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.

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

The Collins-Soper (CS) kernel is a nonperturbative function that characterizes the rapidity evolution of transverse-momentum-dependent parton distribution functions (TMDPDFs) and wave functions. In this Letter, we calculate the CS kernel for pion and proton targets and for quasi-TMDPDFs of leading and next-to-leading power. The calculations are carried out on the CLS ensemble H101 with dynamical $N_f=2+1$ clover-improved Wilson fermions. Our analyses demonstrate the consistency of different lattice extractions of the CS kernel for mesons and baryons, as well as for twist-two and twist-three operators, even though lattice artifacts could be significant. This consistency corroborates the universality of the lattice-determined CS kernel and suggests that a high-precision determination of it is in reach., Comment: 10 pages, 7 figures, published version

We present charmonium and bottomonium correlators and corresponding reconstructed spectral functions from full QCD calculations in the pseudoscalar channel. Correlators are obtained using a mixed-action approach, clover-improved Wilson valence quarks on gauge field configurations generated with $N_f=2+1$ HISQ sea quarks, with physical strange quark masses and light quark masses corresponding to $m_\pi=315$ MeV. The charm and bottom quark masses are tuned to reproduce the experimental mass spectrum of the spin averaged quarkonium vector mesons from the particle data group. For the spectral reconstruction, we use models based on perturbative spectral functions from different frequency regions like resummed thermal contributions around the threshold from pNRQCD and vacuum contributions well above the threshold. We show preliminary results of the reconstructed spectral function obtained for the first time in our study for full QCD.

We calculate shear viscosity and bulk viscosity in SU(3) gauge theory on the lattice at $1.5 \,T_c$. The viscosities are extracted via a Kubo formula from the reconstructed spectral function which we determine from the Euclidean-time dependence of the corresponding channel of the energy-momentum tensor correlators. We obtain unprecedented precision for the correlators by applying gradient flow and blocking methods. The correlators are extrapolated to the continuum and then to zero flow time. To extract the viscosities we fit theoretically inspired models to the lattice data and crosscheck the fit results using the Backus Gilbert method. The final estimates for shear and bulk viscosity are $\eta/s = 0.15-0.48$ and $\zeta/s = 0.017-0.059$., Comment: 16 pages, 12 figures, published version

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 $a=\{0.098,0.085,0.064,0.049\}$~fm and various pion masses ranging between $220$ and $350$ MeV, with proton momenta up to $2.8$ GeV. The result is non-perturbatively 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., Comment: 16 pages, 18 figures, 2 tables

In these lecture notes we will discuss recent progress in extracting spectral and transport properties from lattice QCD. We will focus on results of probes of the thermal QCD medium as well as transport coefficients which are important ingredients for hydrodynamic and transport models that describe the evolution of the produced medium. These include electromagnetic probes, like the rates of emitted photons and dileptons, quarkonium spectral functions as well as transport coefficients like the electrical conductivity or heavy flavor diffusion coefficients of the quark gluon plasma (QGP). All these real time quantities are encoded in the vector meson spectral function. A direct determination of the spectral functions is not possible in Euclidean lattice QCD calculations but they can be analytically continued from imaginary to real time. Therefore it is possible to relate the spectral function to the corresponding Euclidean correlation functions. In the following sections we will discuss the procedure to determine the required correlation functions and the extraction of the spectral functions from lattice QCD correlators. We will illustrate the concepts and methods to obtain spectral functions and related physical observables from continuum extrapolated correlation functions. We will focus here on results obtained from continuum extrapolated lattice correlation functions, which requires large and fine lattices, which so far was only possible in quenched approximation. We will only give a brief introduction to lattice QCD and refer to the textbooks [1,2,3,4] and lecture notes [5] for more detailed introductions to lattice field theory. For the topics addressed in this lecture note we also like to refer to the overview articles on QCD thermodynamics and the QCD phase transition [5,6,7] and quarkonium in extreme conditions [8]., Comment: Lectures delivered at the 53rd Karpacz Winter School of Theoretical Physics, February 26th - March 4th, 2017, Karpacz, Poland; submitted to Lecture Notes in Physics (LNP, volume 999), ISBN: 978-3-030-95490-1

In a recently published work we employ gradient flow on the lattice to extract the leading contribution of the heavy quark momentum diffusion coefficient in the heavy quark limit from calculations of a well-known two-point function of color-electric field operators. In this article we want to report the progress of calculating the recently derived color-magnetic correlator that encodes a finite mass correction to this transport coefficient. The calculations we present here are based on the same ensemble of quenched gauge configurations at $1.5\, T_c$ that we previously used for the color-electric correlator., Comment: 7 pages, 4 figures, presented at the 38th International Symposium on Lattice Field Theory

We present $\texttt{SIMULATeQCD}$, HotQCD's software for performing lattice QCD calculations on GPUs. Started in late 2017 and intended as a full replacement of the previous single GPU lattice QCD code used by the HotQCD collaboration, our software has been developed into an extensive framework for lattice QCD calculations distributed on multiple GPUs over many compute nodes. The code is built on C++, CUDA, and MPI and leverages modern C++ language features to provide high-level data structures, objects, and algorithms that allow users to express lattice QCD calculations in an intuitive way without sacrificing performance. Implemented algorithms range from gradient flow, correlator measurements, and mixed precision conjugate gradient solvers all the way to full HISQ gauge field configuration generation using RHMC. After successful deployment in large-scale computing projects, we want to share the result of our efforts with the lattice QCD community by making it publicly available. In these proceedings, we will present some of the key features of our code, demonstrate its ease of use, and show benchmarks of performance critical kernels on state-of-the-art supercomputers., Comment: 7 pages, 3 figures, presented at the 38th International Symposium on Lattice Field Theory

In a recently published work we provide a proof-of-concept of a novel method to extract the heavy quark momentum diffusion coefficient from color-electric correlators on the lattice using gradient flow. The transport coefficient can be found in the infrared limit of the corresponding spectral function which is reconstructed through perturbative model fits of the correlator data. In this proceedings report we want to give more detailed insights into the systematic uncertainties of this procedure and compare our results with other studies., Comment: Contribution to the proceedings for the 19th International Conference on Strangeness in Quark Matter (SQM 2021), online 17-22 May 2021 (Submission to EPJ). Published version

We present continuum extrapolated results of charmonium and bottomonium correlators in the vector channel at several temperatures below and above $T_c$. The continuum extrapolation jointly performed with the interpolations to have physical values of $J/\psi$ and $\Upsilon$ masses in the confined phase is based on calculations on several large quenched isotropic lattices using clover-improved Wilson valence fermions carrying different quark masses. The extrapolated lattice correlators are confronted with perturbation theory results incorporating resummed thermal effects around the threshold from pNRQCD and vacuum asymptotics above the threshold. An additional transport peak is modelled below the threshold allowing for an estimate of the diffusion coefficients for charm and bottom quarks. We find that charmonium correlators in the vector channel can be well reproduced by perturbative spectral functions above $T_c$ where no resonance peaks for $J/\psi$ are needed at and above 1.1 $T_c$, while for bottomonium correlators a resonance peak for $\Upsilon$ is still needed up to 1.5 $T_c$. By analyzing the transport contribution to the correlators we find that the drag coefficient of a charm quark is larger than that of a bottom quark., Comment: published version

We show how the sphaleron rate (the Minkowski rate for topological charge diffusion) can be determined by analytical continuation of the Euclidean topological-charge-density two-point function, which we investigate on the lattice, using gradient flow to reduce noise and provide improved operators which more accurately measure topology. We measure the correlators on large, fine lattices in the quenched approximation at $1.5\,T_c$ with high precision. Based on these data we first perform a continuum extrapolation at fixed physical flow time and then extrapolate the continuum estimates to zero flow time. The extrapolated correlators are then used to study the sphaleron rate by spectral reconstruction based on perturbatively motivated models.

We apply the gradient flow on a color-electric two-point function that encodes the heavy quark momentum diffusion coefficient. The simulations are done on fine isotropic lattices in the quenched approximation at $1.5\,T_c$. The continuum extrapolation is performed at fixed flow time followed by a second extrapolation to zero flow time. Perturbative calculations of this correlation function under Wilson flow are used to enhance the extrapolations of the non-perturbative lattice correlator. The final estimate for the continuum correlator at zero flow time largely agrees with one obtained from a previous study using the multi-level algorithm. We perform a spectral reconstruction based on perturbative model fits to estimate the heavy quark momentum diffusion coefficient. The approach we present here yields high-precision data for the correlator and is also applicable for actions with dynamical fermions., Comment: Published version, 14 pages, 15 figures

We investigate the in-medium modifications of heavy quarkonia in the vector channel and the heavy quark diffusion coefficient by comparing Euclidean correlators from the lattice to a perturbative spectral function. On the lattice side, we work with continuum extrapolated data from four different large and fine lattices with Clover-improved Wilson fermions in the quenched approximation at five temperatures (0.75, 1.1, 1.3, 1.5 and 2.25$T_c$). On the perturbative side, we use a combination of pNRQCD and vacuum asymptotics to describe the spectral function. After accounting for systematic errors, we obtain a spectral function that is suited to describe the bound state region. This spectral function describes charmonium well without a resonance peak at any of our analyzed temperatures above $T_c$, while we observe a thermally broadened resonance peak for bottomonium that is only melted at our largest temperature, $2.25T_c$. For the transport contribution we assume a Breit-Wigner shaped peak and find that the drag coefficient of charm quarks is larger than that of bottom quarks.

We present first results of our study on the Euclidean topological charge density correlation function. In order to get a well defined topological charge density and to improve the signal of the correlation function at large separations we make use of the gradient flow. We investigate the flow-time dependence on fine quenched lattices. The final goal of this study is to perform a continuum extrapolation for the pure SU(3) plasma and to extract the related transport coefficient, the sphaleron rate., Comment: 7 pages, 4 figures. Contribution to the 37th International Symposium on Lattice Field Theory - Lattice2019, 16-22 June 2019, Wuhan, China

We report on the progress of our study on the color-electric correlation functions under gradient flow on the lattice. This calculation is the first step of our long-term project to estimate a series of important transport coefficients, of which the heavy quark momentum diffusion coefficient is our first attempt, as it can be extracted from a color-electric correlation function that has been calculated non-perturbatively using noise reduction technique in the quenched approximation [1]. By comparing the flowed correlation function with those obtained by other signal-improving techniques, for instance the multi-level algorithm used in [1], one can gain insight into the applicability of the gradient flow and the renormalization of the correlation functions. We start with quenched, isotropic lattices. Currently we have finished measuring the color-electric correlation functions on 4 different lattices with $\beta$-values corresponding to a temperature of $T\approx 1.5T_c$. We perform a continuum extrapolation on the flowed correlators at fixed physical flow times followed by an extrapolation of the continuum estimate back to zero flow time. The next step is to extend the study to different temperatures and to extend the study to dynamical QCD., Comment: 7 pages, 4 figures. Contribution to the 37th International Symposium on Lattice Field Theory - Lattice2019, 16-22 June 2019, Wuhan, China

In this paper we report our results on quarkonium spectral functions in the vector channel obtained from quenched lattice QCD simulations at $T\in[0.75, 2.25]~T_c$. The calculations have been performed on very large and fine isotropic lattices where both charm and bottom quarks can be treated relativistically. The spectral functions are reconstructed using the Maximum Entropy Method. We study the dissociation of quarkonium states from the temperature dependence of the spectral functions and estimate heavy quark diffusion coefficients using the low-frequency behavior of the vector spectral functions., Comment: 4 pages, 5 figures, oral contribution to the XXVII International Conference on Ultrarelativistic Nucleus-Nucleus Collisions (Quark Matter 2018), Venezia, Italy, May 13-19, 2018

We present a detailed study of the applications of two stochastic approaches, stochastic optimization method (SOM) and stochastic analytical inference (SAI), to extract spectral functions from Euclidean correlation functions. SOM has the advantage that it does not require prior information. On the other hand, SAI is a more generalized method based on Bayesian inference. Under mean field approximation SAI reduces to the often-used maximum entropy method (MEM), and for a specific choice of the prior SAI becomes equivalent to SOM. To test the applicability of these two stochastic methods to lattice QCD, firstly, we apply these methods to various reasonably chosen model correlation functions, and present detailed comparisons of the reconstructed spectral functions obtained from SOM, SAI and MEM. Next, we present similar studies for charmonia correlation functions obtained from lattice QCD computations using clover-improved Wilson fermions on large, fine, isotropic lattices at $0.75$ and $1.5T_c$, $T_c$ being the deconfinement transition temperature of a pure gluon plasma. We find that SAI and SOM give consistent results to MEM at these two temperatures., Comment: 24 pages, 18 figures

We present our study on the thermal modifications of charmonia and bottomonia from spatial correlation functions at zero and nonzero momenta in quenched QCD. To accommodate the heavy quarks on the lattice we performed simulations on very fine lattices at a fixed beta value corresponding to a lattice spacing $a^{-1}=22.8$ GeV on $192^3\times32$, $192^3\times48$, $192^3\times56$, $192^3\times64$ and $192^3\times96$ lattices using clover-improved Wilson fermions. These lattices correspond to temperatures of $2.25 T_c$, $1.50 T_c$, $1.25 T_c$, $1.10 T_c$ and $0.75 T_c$. To increase the signal to noise ratio in the axial-vector and scalar channels we used multi-sources for the measurement of spatial correlation functions. By investigating on the differences between spatial and temporal correlators as well as the temperature dependence of screening masses we will discuss the thermal effects in different channels of quarkonium states. Besides this the dispersion relation of the screening mass at different momenta is also discussed., Comment: 8pages, 5 figures, Proceedings of the 35rd International Symposium on Lattice Field Theory, Granada, Spain

We present a Stochastic Optimization Method (SOM) for the reconstruction of the spectral functions (SPFs) from Euclidean correlation functions. In this approach the SPF is parameterized as a sum of randomly distributed boxes. By varying the width, location and height of the boxes stochastically an optimal SPF can be obtained. Using this approach we reproduce mock SPFs fairly well, which contain sharp resonance peaks, transport peaks and continuum spectra. We also analyzed the charmonium correlators obtained from $N_{\tau}$=96, 48, 32 lattices using SOM and found similar conclusion on the dissociation temperatures of charmonium ground states as that obtained using the Maximum Entropy Method., Comment: Proceedings of the 33rd International Symposium on Lattice Field Theory, Kobe, Japan