Search

We present some lattice QCD results on first ($\chi_1^i$) and second ($\chi_2^i$) cumulants of and correlations ($\chi_{11}^{ij}$) among net baryon-number ($B$), strangeness ($S$) and electric charge ($Q$) along the pseudo-critical line ($T_{pc}(\mu_B)$) in the temperature ($T$)--baryon chemical potential ($\mu_B$) phase diagram of (2+1)-flavor QCD. We point out that violations of the isospin symmetric limit of vanishing electric charge chemical potential are small along the $T_{pc}(\mu_B)$ for the entire range of $\mu_B$ covered in the RHIC beam energy scan. For the strangeness neutral matter produced in heavy-ion collisions this leads to a close relation between $\chi_{11}^{BS}$ and $\chi_{11}^{QS}$. We compare lattice QCD results for $\chi_{11}^{BS}/\chi_2^S$ along the $T_{pc}(\mu_B)$ line with preliminary experimental measurements of $\chi_{11}^{BS}/\chi_2^S$ for collision energies $7.7~{\rm GeV}\le \sqrt{s_{_{NN}}}\le 62.4~{\rm GeV}$. While we find good agreements for $\sqrt{s_{_{NN}}}\ge 39$~GeV, differences are sizeable at smaller values of $\sqrt{s_{_{NN}}}$. Moreover, we compare lattice QCD results for the ratio of the strangeness ($\mu_S$) to baryon ($\mu_B$) 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 $\mu_S/\mu_B$ using strange baryon yields on the freeze-out line. Finally, we determine the baryon chemical potential at the freeze-out ($\mu_B^f$) by comparing $\chi_1^B/\chi_2^B$ along the $T_{pc}(\mu_B)$ with the experimentally measured net-proton cumulants $\chi_1^p/\chi_2^p$. We find that $\{\mu_B^f, T_{pc}(\mu_B^f) \}$ are consistent with the freeze-out parameters of the statistical-model fits to experimentally measured hadron yields for $\sqrt{s_{_{NN}}} \geq 11.5$ GeV., Comment: 9 pages, 6 figures

We analyze the dependence of the chiral phase transition temperature on baryon number and strangeness chemical potentials by calculating the leading order curvature coefficients in the light and strange quark flavor basis as well as in the conserved charge ($B, S$) basis. Making use of scaling properties of the magnetic equation of state (MEoS) and including diagonal as well as off-diagonal contributions in the expansion of the energy-like scaling variable that enters the parametrization of the MEoS, allows to explore the variation of $T_c(\mu_B,\mu_S) = T_c ( 1 - (\kappa_2^B \hat{\mu}_B^2 + \kappa_2^S \hat{\mu}_S^2 + 2\kappa_{11}^{BS} \hat{\mu}_B \hat{\mu}_S))$ along different lines in the $(\mu_B,\mu_S)$ plane. On lattices with fixed cut-off in units of temperature, $aT=1/8$, we find $\kappa_2^B=0.015(1)$, $\kappa_2^S=0.0124(5)$ and $\kappa_{11}^{BS}=-0.0050(7)$. We show that the chemical potential dependence along the line of vanishing strangeness chemical potential is about 10\% larger than along the strangeness neutral line. The latter differs only by about $3\%$ from the curvature on a line of vanishing strange quark chemical potential, $\mu_s=0$. We also show that close to the chiral limit the strange quark mass contributes like an energy-like variable in scaling relations for pseudo-critical temperatures. The chiral phase transition temperature decreases with decreasing strange quark mass, $T_c(m_s)= T_c(m_s^{\rm phy}) (1 - 0.097(2) (m_s-m_s^{\rm phys})/m_s^{\rm phy}+{\cal O}((\Delta m_s)^2)$., Comment: 19 pages, 11 figures, updated to published version

We point out that chiral condensates at nonzero temperature and magnetic fields are in strict connection to the space-time integral of corresponding two-point neutral meson correlation functions in the pseudoscalar channel via the Ward-Takahashi identity. Screening masses of neutral pseudoscalar mesons, which are defined as the exponential decay of the corresponding spatial correlation functions in the long distance, thus are intrinsically connected to (inverse) magnetic catalysis of chiral condensates. To study this we performed lattice simulations of $(2+1)$-flavor QCD on $32^3\times N_t$ lattices with pion mass $M_\pi\simeq 220$ MeV in a fixed scale approach having temperature $T\in[17, 281]$ MeV and magnetic field strength $eB\in[0, 2.5]$ GeV$^2$. We find that screening lengths, i.e. inverses of screening masses of $\pi^0$, $K^0$ and $\eta^0_{s\bar{s}}$, turn out to have the similar complex $eB$ and $T$ dependences of the corresponding chiral condensates. Although the transition temperature is found to always decrease as $eB$ grows, we show that the suppression due to magnetic fields becomes less significant for hadron screening length and chiral condensates with heavier quarks involved, and ceases to occur for $\eta^0_{s\bar{s}}$ and strange quark chiral condensate. The complex $eB$ and $T$ dependences of both screening masses and chiral condensates, reflecting the crossover nature of the QCD transition, are attributed to the competition between sea and valence quark effects. These findings could be useful to guide low-energy models and effective theories of QCD., Comment: 12 pages, 10 figures, typos corrected, version to appear in PRD

We study the inverse problem of reconstructing spectral functions from Euclidean correlation functions via machine learning. We propose a novel neural network, SVAE, which is based on the variational autoencoder (VAE) and can be naturally applied to the inverse problem. The prominent feature of the SVAE is that a Shannon-Jaynes entropy term having the ground truth values of spectral functions as prior information is included in the loss function to be minimized. We train the network with general spectral functions produced from a Gaussian mixture model. As a test, we use correlators generated from four different types of physically motivated spectral functions made of one resonance peak, a continuum term and perturbative spectral function obtained using non-relativistic QCD. From the mock data test we find that the SVAE in most cases is comparable to the maximum entropy method (MEM) in the quality of reconstructing spectral functions and even outperforms the MEM in the case where the spectral function has sharp peaks with insufficient number of data points in the correlator. By applying to temporal correlation functions of charmonium in the pseudoscalar channel obtained in the quenched lattice QCD at 0.75 $T_c$ on $128^3\times96$ lattices and $1.5$ $T_c$ on $128^3\times48$ lattices, we find that the resonance peak of $\eta_c$ extracted from both the SVAE and MEM has a substantial dependence on the number of points in the temporal direction ($N_\tau$) adopted in the lattice simulation and $N_\tau$ larger than 48 is needed to resolve the fate of $\eta_c$ at 1.5 $T_c$., Comment: 25 pages, 14 figures. Investigations on the dependences of output spectral functions on the noise model of mock correlators, and detailed derivation of formulae for the output spectral function are added

We present results on the second-order fluctuations of and correlations among net baryon number, electric charge and strangeness in (2+1)-flavor lattice QCD in the presence of a background magnetic field. Simulations are performed using the tree-level improved gauge action and the highly improved staggered quark (HISQ) action with a fixed scale approach ($a\simeq$ 0.117 fm). The light quark mass is set to be 1/10 of the physical strange quark mass and the corresponding pion mass is about 220 MeV at vanishing magnetic field. Simulations are performed on $32^3\times N_\tau$ lattices with 9 values of $N_\tau$ varying from 96 to 6 corresponding to temperatures ranging from zero up to 281 MeV. The magnetic field strength $eB$ is simulated with 15 different values up to $\sim$2.5 GeV$^2$ at each nonzero temperature. We find that quadratic fluctuations and correlations do not show any singular behavior at zero temperature in the current window of $eB$ while they develop peaked structures at nonzero temperatures as $eB$ grows. By comparing the electric charge-related fluctuations and correlations with hadron resonance gas model calculations and ideal gas limits we find that the changes in degrees of freedom start at lower temperatures in stronger magnetic fields. Significant effects induced by magnetic fields on the isospin symmetry and ratios of net baryon number and baryon-strangeness correlation to strangeness fluctuation are observed, which could be useful for probing the existence of a magnetic field in heavy-ion collision experiments., Comment: 12 pages, 7 figures, invited contribution to The European Physical Journal A-Hadrons and Nuclei: Topical Issue on "QCD Phase Diagram in Strong Magnetic Fields", minor revision

In this proceedings we discuss the natural connection between the reduction of neutral pion mass in the vacuum, and the magnetic catalysis as well as the reduction of transition temperature in the external magnetic field. We also present the first results on fluctuations of and correlations among conserved charges in strong magnetic fields from lattice QCD computations., Comment: 12 pages, 7 figures, Invited talk in the online Workshop on Criticality in QCD and Hadron Resonance Gas, July 29-31, 2020, Wroclaw, Poland, to be published in the proceedings

We introduce novel relations between the derivatives [$\partial^{n}\rho(\lambda, m_{l})/\partial m_{l}^n$] of the Dirac eigenvalue spectrum [$\rho(\lambda, m_{l})$] with respect to the light sea quark mass ($m_{l}$) and the $(n+1)$-point correlations among the eigenvalues ($\lambda$) of the massless Dirac operator. Using these relations we present lattice QCD results for $\partial^{n}\rho(\lambda, m_{l})/\partial m_{l}^n$ ($n=1, 2, 3$) for $m_{l}$ corresponding to pion masses $m_\pi=160-55$ MeV, and at a temperature of about 1.6 times the chiral phase transition temperature. Calculations were carried out using (2+1) flavors of highly improved staggered quarks with the physical value of strange quark mass, three lattice spacings $a=0.12, 0.08, 0.06$ fm, and lattices having aspect ratios $4-9$. We find that $\rho(\lambda\to0, m_{l})$ develops a peaked structure. This peaked structure arises due to non-Poisson correlations within the infrared part of the Dirac eigenvalue spectrum, becomes sharper as $a\to0$, and its amplitude is proportional to $m_{l}^2$. We demonstrate that this $\rho(\lambda\to0,m_l)$ is responsible for the manifestations of axial anomaly in two-point correlation functions of light scalar and pseudoscalar mesons. After continuum and chiral extrapolations we find that axial anomaly remains manifested in two-point correlation functions of scalar and pseudoscalar mesons in the chiral limit., Comment: 17 pages, 18 figures, minor updates to match the published version in PRL

We present lattice QCD results for masses and magnetic polarizabilities of light and strange pseudoscalar mesons, chiral condensates, decay constants of neutral pion, and neutral kaon in the presence of background magnetic fields with $eB$ ranging up to around 3.35 GeV$^2$ ($\sim70~M_\pi^2$) in the vacuum. The computations were carried out in (2+1)-flavor QCD mostly on $32^3 \times 96$ lattices using the highly improved staggered quark action with $M_{\pi} \approx $ 220 MeV at zero temperature. We find that the masses of neutral pseudoscalar mesons monotonously decrease as the magnetic field strength grows and then saturate at a nonzero value, while there exists a nonmonotonous behavior of charged pion and kaon masses in the magnetic field. We observe a $qB$ scaling of the up and down quark flavor components of neutral pion mass, neutral pion decay constant as well as the quark chiral condensates at 0.05 $\lesssim eB\lesssim$ 3.35 GeV$^2$. We show that the correction to the Gell-Mann-Oakes-Renner relation involving the neutral pion is less than 6% and the correction for the relation involving neutral kaon is less than 30% at $eB\lesssim$ 3.35 GeV$^2$. We also derive the Ward-Takahashi identities for QCD in the magnetic field in the continuum formulation including the relation between integrated neutral pseudoscalar meson correlators and chiral condensates., Comment: 30 pages, 18 figures, updated to the published version in PRD

We present new results on up to $6^{th}$ order cumulants of net baryon-number fluctuations at small values of the baryon chemical potential, $\mu_B$, obtained in lattice QCD calculations with physical values of light and strange quark masses. Representation of the Taylor expansions of higher order cumulants in terms of the ratio of the two lowest order cumulants, $M_B/\sigma_B^2=\chi_1^B(T,\mu_B)/\chi_2^B(T,\mu_B)$, allows for a parameter free comparison with data on net proton-number cumulants obtained by the STAR Collaboration in the Beam Energy Scan at RHIC. We show that recent high statistics data on skewness and kurtosis ratios of net proton-number distributions, obtained at beam energy $\sqrt{s_{_{NN}}}=54.4$ GeV, agree well with lattice QCD results on cumulants of net baryon-number fluctuations close to the pseudo-critical temperature, $T_{pc}(\mu_B)$, for the chiral transition in QCD. We also present first results from a next-to-leading order expansion of $5^{th}$ and $6^{th}$ order cumulants on the line of pseudo-critical temperatures., Comment: 13 pages, 11 figures, minor corrections and updates of references

We present a lattice QCD based determination of the chiral phase transition temperature in QCD with two degenerate, massless quarks and a physical strange quark mass. We propose and calculate two novel estimators for the chiral transition temperature for several values of the light quark masses, corresponding to Goldstone pion masses in the range of $58~{\rm MeV}\lesssim m_\pi\lesssim 163~{\rm MeV}$. The chiral phase transition temperature is determined by extrapolating to vanishing pion mass using universal scaling analysis. Finite volume effects are controlled by extrapolating to the thermodynamic limit using spatial lattice extents in the range of $2.8$-$4.5$ times the inverse of the pion mass. Continuum extrapolations are carried out by using three different values of the lattice cut-off, corresponding to lattices with temporal extent $N_\tau=6,\ 8$ and $12$. After thermodynamic, continuum and chiral extrapolations we find the chiral phase transition temperature $T_c^0=132^{+3}_{-6}$ MeV., Comment: published version in journal

In this work, we strive to gain insight into thermal modifications of charmonium and bottomonium bound states as well as the heavy quark diffusion coefficient. The desired information is contained in the spectral function which can not be calculated on the lattice directly. Instead, the correlator given by an integration over the spectral function times an integration kernel is obtained. Extracting the spectral function is an ill-posed inversion problem and various different solutions have been proposed. We focus on a comparison to a spectral function obtained from combining perturbative and pNRQCD calculations. In order to get precise results, continuum extrapolated correlators originating from large and fine lattices are used. We first analyze the pseudoscalar channel since the absence of a transport peak simplifies the analysis. The knowledge gained from this is then used to extend the analysis to the vector channel, where information on heavy quark transport is encoded in the low frequency regime of the spectral function. The comparison shows a qualitatively good agreement between perturbative and lattice correlators. Quantitative differences can be explained by systematic uncertainties., Comment: Presented at Hot Quarks 2018 - Workshop for young scientists on the physics of ultrarelativistic nucleus-nucleus collisions, Texel, The Netherlands, September 7-14 2018, Submitted to MDPI Proceedings

We present results for pseudo-critical temperatures of QCD chiral crossovers at zero and non-zero values of baryon ($B$), strangeness ($S$), electric charge ($Q$), and isospin ($I$) chemical potentials $\mu_{X=B,Q,S,I}$. The results were obtained using lattice QCD calculations carried out with two degenerate up and down dynamical quarks and a dynamical strange quark, with quark masses corresponding to physical values of pion and kaon masses in the continuum limit. By parameterizing pseudo-critical temperatures as $ T_c(\mu_X) = T_c(0) \left[ 1 -\kappa_2^{X}(\mu_{X}/T_c(0))^2 -\kappa_4^{X}(\mu_{X}/T_c(0))^4 \right] $, we determined $\kappa_2^X$ and $\kappa_4^X$ from Taylor expansions of chiral observables in $\mu_X$. We obtained a precise result for $T_c(0)=(156.5\pm1.5)\;\mathrm{MeV}$. For analogous thermal conditions at the chemical freeze-out of relativistic heavy-ion collisions, i.e., $\mu_{S}(T,\mu_{B})$ and $\mu_{Q}(T,\mu_{B})$ fixed from strangeness-neutrality and isospin-imbalance, we found $\kappa_2^B=0.012(4)$ and $\kappa_4^B=0.000(4)$. For $\mu_{B}\lesssim300\;\mathrm{MeV}$, the chemical freeze-out takes place in the vicinity of the QCD phase boundary, which coincides with the lines of constant energy density of $0.42(6)\;\mathrm{GeV/fm}^3$ and constant entropy density of $3.7(5)\;\mathrm{fm}^{-3}$.

We present here results on the determination of the critical temperature in the chiral limit for (2+1)-flavor QCD. We propose two novel estimators of the chiral critical temperature where quark mass dependence is strongly suppressed compared to the conventional estimator using pseudo-critical temperatures. We have used the HISQ/tree action for the numerical simulation with lattices with three different temporal extent $N_{\tau}=$6, 8, 12 and varied the aspect ratio over the range $4 \leq N_{\sigma}/N_{\tau} \leq 8$. To approach the chiral limit, the light quark mass has been decreased keeping the strange quark mass fixed at its physical value. Our simulations correspond to the range of pion masses, 55 MeV $\leq m_{\pi} \leq$ 160 MeV., Comment: Prepared for the proceedings of Quark Matter 2018

The pseudoscalar correlator is an ideal lattice probe for thermal modifications to quarkonium spectra, given that it is not compromised by a contribution from a large transport peak. We construct a perturbative spectral function incorporating resummed thermal effects around the threshold and vacuum asymptotics above the threshold, and compare the corresponding imaginary-time correlators with continuum-extrapolated lattice data for quenched SU(3) at several temperatures. Modest differences are observed, which may originate from non-perturbative mass shifts or renormalization factors, however no resonance peaks are needed for describing the quenched lattice data for charmonium at and above T ~ 1.1Tc ~ 350 MeV. For comparison, in the bottomonium case a good description of the lattice data is obtained with a spectral function containing a single thermally broadened resonance peak., Comment: 28 pages. v2: published version

We present results for the ratios of mean ($M_B$), variance ($\sigma_B^2$), skewness ($S_B)$ and kurtosis ($\kappa_B$) of net baryon-number fluctuations obtained in lattice QCD calculations with a physical light to strange quark mass ratio. Using next-to-leading order Taylor expansions in baryon chemical potential we find that qualitative features of these ratios closely resemble the corresponding experimentally measured cumulants ratios of net proton-number fluctuations for beam energies down to $\sqrt{s_{_{NN}}} \ge 19.6$ GeV. We show that the difference in cumulant ratios for the mean net baryon-number, $M_B/\sigma_B^2=\chi_1^B(T,\mu_B)/\chi_2^B(T,\mu_B)$ and the normalized skewness, $S_B\sigma_B=\chi_3^B(T,\mu_B)/\chi_2^B(T,\mu_B)$, naturally arises in QCD thermodynamics. Moreover, we establish a close relation between skewness and kurtosis ratios, $S_B\sigma_B^3/M_B=\chi_3^B(T,\mu_B)/\chi_1^B(T,\mu_B)$ and $\kappa_B\sigma_B^2=\chi_4^B(T,\mu_B)/\chi_2^B(T,\mu_B)$, valid at small values of the baryon chemical potential., Comment: 14 pages, 13 figures, minor updates, final version

We calculated the QCD equation of state using Taylor expansions that include contributions from up to sixth order in the baryon, strangeness and electric charge chemical potentials. Calculations have been performed with the Highly Improved Staggered Quark action in the temperature range $T\in [135~{\rm MeV}, 330~{\rm MeV}]$ using up to four different sets of lattice cut-offs corresponding to lattices of size $N_\sigma^3\times N_\tau$ with aspect ratio $N_\sigma/N_\tau=4$ and $N_\tau =6-16$. The strange quark mass is tuned to its physical value and we use two strange to light quark mass ratios $m_s/m_l=20$ and $27$, which in the continuum limit correspond to a pion mass of about $160$ MeV and $140$ MeV espectively. Sixth-order results for Taylor expansion coefficients are used to estimate truncation errors of the fourth-order expansion. We show that truncation errors are small for baryon chemical potentials less then twice the temperature ($\mu_B\le 2T$). The fourth-order equation of state thus is suitable for the modeling of dense matter created in heavy ion collisions with center-of-mass energies down to $\sqrt{s_{NN}}\sim 12$ GeV. We provide a parametrization of basic thermodynamic quantities that can be readily used in hydrodynamic simulation codes. The results on up to sixth order expansion coefficients of bulk thermodynamics are used for the calculation of lines of constant pressure, energy and entropy densities in the $T$-$\mu_B$ plane and are compared with the crossover line for the QCD chiral transition as well as with experimental results on freeze-out parameters in heavy ion collisions. These coefficients also provide estimates for the location of a possible critical point. We argue that results on sixth order expansion coefficients disfavor the existence of a critical point in the QCD phase diagram for $\mu_B/T\le 2$ and $T/T_c(\mu_B=0) > 0.9$., Comment: Updated to correspond to journal version. The ancillary files, which contain tables of the basic thermodynamic observables calculated for baryon chemical potentials \mu_B/T < 2.2 in the temperature range 135 MeV < T < 280 MeV and for the cases \mu_Q = \mu_S = 0 and n_S=0, n_Q/n_B=0.4, can be found in version v1

We investigate the phase structure of QCD with 3 degenerate quark flavors as function of the degenerate quark masses at vanishing baryon number density. We use the Highly Improved Staggered Quarks on lattices with temporal extent $N_{t}=6$ and perform calculations for six values of quark masses, which in the continuum limit correspond to pion masses in the range $80~{\rm MeV} \lesssim m_{\pi} \lesssim 230~$MeV. By analyzing the volume and temperature dependence of the chiral condensate and chiral susceptibility we find no direct evidence for a first order phase transition in this range of pion mass values. Relying on the universal scaling behaviors of the chiral observables near an anticipated chiral critical point, we estimate an upper bound for the critical pion mass, $m_\pi^c \lesssim$ 50 MeV, below which a region of first order chiral phase transition is favored.

We study the free energy of a static quark in QCD with 2+1 flavors in a wide temperature region, 116 MeV $< T < $ 5814 MeV, using the highly improved staggered quark (HISQ) action. We analyze the transition region in detail, obtain the entropy of a static quark, show that it peaks at temperatures close to the chiral crossover temperature and also revisit the temperature dependence of the Polyakov loop susceptibilities using gradient flow. We discuss the implications of our findings for the deconfinement and chiral crossover phenomena at physical values of the quark masses. Finally a comparison of the lattice results at high temperatures with the weak-coupling calculations is presented., Comment: RevTeX, 21 pages, 20 figures

We discuss the next-to-leading order Taylor expansion of ratios of cumulants of net-baryon number fluctuations. We focus on the relation between the skewness ratio, $S_B\sigma_B = \chi_3^B/\chi_1^B$, and the kurtosis ratio, $\kappa_B\sigma_B^2 =\chi_4^B/\chi_2^B$. We show that differences in these two cumulant ratios are small for small values of the baryon chemical potential. The next-to-leading order correction to $\kappa_B\sigma_B^2$ however is approximately three times larger than that for $S_B\sigma_B$. The former thus drops much more rapidly with increasing beam energy, $\sqrt{s_{NN}}$. We argue that these generic patterns are consistent with current data on cumulants of net-proton number fluctuations measured by the STAR Collaboration at $\sqrt{s_{NN}}\ge 19.6$~GeV., Comment: 4 pages, 4 figures, contribution to the Quark Matter 2015 proceedings

We calculate the mean and variance of net-baryon number and net-electric charge distributions from Quantum Chromodynamics (QCD) using a next-to-leading order Taylor expansion in terms of temperature and chemical potentials. We compare these expansions with experimental data from STAR and PHENIX, determine the freeze-out temperature in the limit of vanishing baryon chemical potential, and, for the first time, constrain the curvature of the freeze-out line through a direct comparison between experimental data on net-charge fluctuations and a QCD calculation. We obtain a bound on the curvature coefficient, kappa_2^f < 0.011, that is compatible with lattice QCD results on the curvature of the QCD transition line., Comment: 5 pages, 2 figures

We present continuum extrapolated lattice QCD results for up to fourth-order diagonal and off-diagonal quark number susceptibilities in the high temperature region of 300-700 MeV. Lattice QCD calculations are performed using 2+1 flavors of highly improved staggered quarks with nearly physical quark masses and at four different lattice spacings. Comparisons of our results with recent weak coupling perturbative calculations yield reasonably good agreements for the entire temperature range., Comment: 7 pages, 5 figures, RevTeX, published version, typos corrected, minor modifications

Quarkonium properties at finite temperature have been studied with quark masses of the charm and bottom quarks. Our simulations have been performed in quenched QCD with the $O(a)$-improved Wilson quarks on large and fine isotropic lattices with the spatial lattice extents $N_\sigma =$ 96, 192 and the corresponding lattice spacings $a =$ 0.0190, 0.00967 fm, respectively, at temperatures in a range between about 0.7$T_c$ and 1.4$T_c$. We show temperature and quark mass dependence of quarkonium correlation functions and related physical quantities: the quark number susceptibility and the heavy quark diffusion constant., Comment: 7 pages, 2 figures, talk presented at The 32nd International Symposium on Lattice Field Theory, June 23-28, 2014, Columbia University, New York

We present results for the equation of state in (2+1)-flavor QCD using the highly improved staggered quark action and lattices with temporal extent $N_{\tau}=6,~8,~10$, and $12$. We show that these data can be reliably extrapolated to the continuum limit and obtain a number of thermodynamic quantities and the speed of sound in the temperature range $(130-400)$ MeV. We compare our results with previous calculations, and provide an analytic parameterization of the pressure, from which other thermodynamic quantities can be calculated, for use in phenomenology. We show that the energy density in the crossover region, $145~ {\rm MeV} \leq T \leq 163$ MeV, defined by the chiral transition, is $\epsilon_c=(0.18-0.5)~{\rm GeV}/{\rm fm}^3$, $i.e.$, $(1.2-3.1)\ \epsilon_{\rm nuclear}$. At high temperatures, we compare our results with resummed and dimensionally reduced perturbation theory calculations. As a byproduct of our analyses, we obtain the values of the scale parameters $r_0$ from the static quark potential and $w_0$ from the gradient flow., Comment: Minor modifications, published version

We compare lattice QCD results for appropriate combinations of net strangeness fluctuations and their correlations with net baryon number fluctuations with predictions from two hadron resonance gas (HRG) models having different strange hadron content. The conventionally used HRG model based on experimentally established strange hadrons fails to describe the lattice QCD results in the hadronic phase close to the QCD crossover. Supplementing the conventional HRG with additional, experimentally uncharted strange hadrons predicted by quark model calculations and observed in lattice QCD spectrum calculations leads to good descriptions of strange hadron thermodynamics below the QCD crossover. We show that the thermodynamic presence of these additional states gets imprinted in the yields of the ground-state strange hadrons leading to a systematic 5--8 MeV decrease of the chemical freeze-out temperatures of ground-state strange baryons., Comment: version published in Physical Review Letters

Ratios of cumulants of conserved net charge fluctuations are sensitive to the degrees of freedom that are carriers of the corresponding quantum numbers in different phases of strong interaction matter. Using lattice QCD with 2+1 dynamical flavors and quenched charm quarks we calculate second and fourth order cumulants of net charm fluctuations and their correlations with other conserved charges such as net baryon number, electric charge and strangeness. Analyzing appropriate ratios of these cumulants we probe the nature of charmed degrees of freedom in the vicinity of the QCD chiral crossover region. We show that for temperatures above the chiral crossover transition temperature, charmed degrees of freedom can no longer be described by an uncorrelated gas of hadrons. This suggests that the dissociation of open charm hadrons and the emergence of deconfined charm states sets in just near the chiral crossover transition. Till the crossover region we compare these lattice QCD results with two hadron resonance gas models --including only the experimentally established charmed resonances and also including additional states predicted by quark model and lattice QCD calculations. This comparison provides evidence for so far unobserved charmed hadrons that contribute to the thermodynamics in the crossover region., Comment: The published version

We report on the first lattice calculation of the QCD phase transition using chiral fermions at physical values of the quark masses. This calculation uses 2+1 quark flavors, spatial volumes between (4 fm$)^3$ and (11 fm$)^3$ and temperatures between 139 and 196 MeV . Each temperature was calculated using a single lattice spacing corresponding to a temporal Euclidean extent of $N_t=8$. The disconnected chiral susceptibility, $\chi_{\rm disc}$ shows a pronounced peak whose position and height depend sensitively on the quark mass. We find no metastability in the region of the peak and a peak height which does not change when a 5 fm spatial extent is increased to 10 fm. Each result is strong evidence that the QCD ``phase transition'' is not first order but a continuous cross-over for $m_\pi=135$ MeV. The peak location determines a pseudo-critical temperature $T_c = 155(1)(8)$ MeV. Chiral $SU(2)_L\times SU(2)_R$ symmetry is fully restored above 164 MeV, but anomalous $U(1)_A$ symmetry breaking is non-zero above $T_c$ and vanishes as $T$ is increased to 196 MeV., Comment: 6 pages and 4 figures

We present studies of universal properties of the chiral phase transition in $N_f$=2+1 QCD based on the simulations using Highly Improved Staggered fermions on lattices with temporal extent $N_\tau$=6. We analyze the quark mass and volume dependence of the chiral condensates and chiral susceptibilities in QCD with two degenerate light quarks and a strange quark. The strange quark mass is chosen to be fixed to its physical value ($m^{phy}_s$) and five values of light quark masses ($m_l$) that are varied in the interval 1/20$\gtrsim m_l/m^{phy}_s \gtrsim$1/80. Here various quark masses correspond to pseudo Goldstone pion masses ranging from about 160 MeV to about 80 MeV. The O(N) scaling of chiral observables and the influence of universal scaling on physical observables in the region of physical quark mass values are also discussed., Comment: 7 pages, talk presented at the 31st International Symposium on Lattice Field Theory (Lattice 2013), 29 July - 3 August 2013, Mainz, Germany

We report on a study of the finite-temperature QCD transition region for temperatures between 139 and 196 MeV, with a pion mass of 200 MeV and two space-time volumes: $24^3\times8$ and $32^3\times8$, where the larger volume varies in linear size between 5.6 fm (at T=139 MeV) and 4.0 fm (at T=195 MeV). These results are compared with the results of an earlier calculation using the same action and quark masses but a smaller, $16^3\times8$ volume. The chiral domain wall fermion formulation with a combined Iwasaki and dislocation suppressing determinant ratio gauge action are used. This lattice action accurately reproduces the $\sua$ and $\ua$ symmetries of the continuum. Results are reported for the chiral condensates, connected and disconnected susceptibilities and the Dirac eigenvalue spectrum. We find a pseudo-critical temperature, $T_c$, of approximately 165 MeV consistent with previous results and strong finite volume dependence below $T_c$. Clear evidence is seen for $\ua$ symmetry breaking above $T_c$ which is quantitatively explained by the measured density of near-zero modes in accordance with the dilute instanton gas approximation., Comment: 58 pages, 9 figures and 8 tables

We calculate second and fourth order quark number susceptibilities for 2+1 flavor QCD in the high temperature region. In our study we use two improved staggered fermion formulations, namely the highly improved staggered quark formulation, and the so-called p4 formulation, as well as several lattice spacings. Second order quark number susceptibilities are calculated using both improved staggered fermion formulations, and we show that in the continuum limit the two formulations give consistent results. The fourth order quark number susceptibilities are studied only using the p4 formulation and at non-zero lattice spacings. We compare our results on quark number susceptibilities with recent weak coupling calculations, and find that these agree reasonably well with the lattice calculations within the estimated uncertainties., Comment: RevTeX, 9 pages, 5 figures, the discussion of the comparison with the perturbative results is revised and the conclusion about the validity of the perturbative approaches is modified, published version

Appropriate combinations of up to fourth order cumulants of net strangeness fluctuations and their correlations with net baryon number and electric charge fluctuations, obtained from lattice QCD calculations, have been used to probe the strangeness carrying degrees of freedom at high temperatures. For temperatures up to the chiral crossover separate contributions of strange mesons and baryons can be well described by an uncorrelated gas of hadrons. Such a description breaks down in the chiral crossover region, suggesting that the deconfinement of strangeness takes place at the chiral crossover. On the other hand, the strangeness carrying degrees of freedom inside the quark gluon plasma can be described by a weakly interacting gas of quarks only for temperatures larger than twice the chiral crossover temperature. In the intermediate temperature window these observables show considerably richer structures, indicative of the strongly interacting nature of the quark gluon plasma., Comment: Published version

We present new lattice results on the continuum extrapolation of the vector current correlation function. Lattice calculations have been carried out in the deconfined phase at a temperature of 1.1 Tc, extending our previous results at 1.45 Tc, utilizing quenched non-perturbatively clover-improved Wilson fermions and light quark masses. A systematic analysis on multiple lattice spacings allows to perform the continuum limit of the correlation function and to extract spectral properties in the continuum limit. Our current analysis suggests the results for the electrical conductivity are proportional to the temperature and the thermal dilepton rates in the quark gluon plasma are comparable for both temperatures. Preliminary results of the continuum extrapolated correlation function at finite momenta, which relates to thermal photon rates, are also presented., Comment: 8 pages, 8 figures, Proceeding "Xth Quark Confinement and the Hadron Spectrum" (submitted)

We study the properties of charmonium states at finite temperature in quenched lattice QCD on large and fine isotropic lattices. We perform a detailed analysis of charmonium correlation and spectral functions both below and above Tc. Our analysis suggests that the S wave states disappear at about 1.5 Tc. The charm diffusion coefficient is estimated and found to be approximately 1/{\pi}T at 1.5Tc {\leq} T {\leq} 3Tc., Comment: 8 pages, 12 figures, talk presented at "International Conference on New Frontiers in Physics 2012" - Kolymbari, Crete

We present a determination of chemical freeze-out conditions in heavy ion collisions based on ratios of cumulants of net electric charge fluctuations. These ratios can reliably be calculated in lattice QCD for a wide range of chemical potential values by using a next-to-leading order Taylor series expansion around the limit of vanishing baryon, electric charge and strangeness chemical potentials. From a computation of up to fourth order cumulants and charge correlations we first determine the strangeness and electric charge chemical potentials that characterize freeze-out conditions in a heavy ion collision and confirm that in the temperature range 150 MeV < T < 170 MeV the hadron resonance gas model provides good approximations for these parameters that agree with QCD calculations on the (5-15)% level. We then show that a comparison of lattice QCD results for ratios of up to third order cumulants of electric charge fluctuations with experimental results allows to extract the freeze-out baryon chemical potential and the freeze-out temperature., Comment: 5 pages, 7 EPS files

We present results on both the restoration of the spontaneously broken chiral symmetry and the effective restoration of the anomalously broken U(1)_A symmetry in finite temperature QCD at zero chemical potential using lattice QCD. We employ domain wall fermions on lattices with fixed temporal extent N_\tau = 8 and spatial extent N_\sigma = 16 in a temperature range of T = 139 - 195 MeV, corresponding to lattice spacings of a \approx 0.12 - 0.18 fm. In these calculations, we include two degenerate light quarks and a strange quark at fixed pion mass m_\pi = 200 MeV. The strange quark mass is set near its physical value. We also present results from a second set of finite temperature gauge configurations at the same volume and temporal extent with slightly heavier pion mass. To study chiral symmetry restoration, we calculate the chiral condensate, the disconnected chiral susceptibility, and susceptibilities in several meson channels of different quantum numbers. To study U(1)_A restoration, we calculate spatial correlators in the scalar and pseudo-scalar channels, as well as the corresponding susceptibilities. Furthermore, we also show results for the eigenvalue spectrum of the Dirac operator as a function of temperature, which can be connected to both U(1)_A and chiral symmetry restoration via Banks-Casher relations., Comment: 80 pages, 14 figures, 4 appendices

We study the properties of charmonium states at finite temperature in quenched QCD on large and fine isotropic lattices. We perform a detailed analysis of charmonium correlation and spectral functions both below and above $T_c$. Our analysis suggests that both S wave states ($J/\psi$ and $\eta_c$) and P wave states ($\chi_{c0}$ and $\chi_{c1}$) disappear already at about $1.5 T_c$. The charm diffusion coefficient is estimated through the Kubo formula and found to be compatible with zero below $T_c$ and approximately $1/\pi T$ at $1.5 T_c\lesssim T\lesssim 3 T_c$., Comment: 32 pages, 19 figures, typo corrected, discussions on isotropic vs anisotropic lattices expanded, published version

We calculate the quadratic fluctuations of net baryon number, electric charge and strangeness as well as correlations among these conserved charges in (2+1)-flavor lattice QCD at zero chemical potential. Results are obtained using calculations with tree level improved gauge and the highly improved staggered quark (HISQ) actions with almost physical light and strange quark masses at three different values of the lattice cut-off. Our choice of parameters corresponds to a value of 160 MeV for the lightest pseudo scalar Goldstone mass and a physical value of the kaon mass. The three diagonal charge susceptibilities and the correlations among conserved charges have been extrapolated to the continuum limit in the temperature interval 150 MeV

We present results on the chiral and deconfinement properties of the QCD transition at finite temperature. Calculations are performed with 2+1 flavors of quarks using the p4, asqtad and HISQ/tree actions. Lattices with temporal extent N_tau=6, 8 and 12 are used to understand and control discretization errors and to reliably extrapolate estimates obtained at finite lattice spacings to the continuum limit. The chiral transition temperature is defined in terms of the phase transition in a theory with two massless flavors and analyzed using O(N) scaling fits to the chiral condensate and susceptibility. We find consistent estimates from the HISQ/tree and asqtad actions and our main result is T_c=154 +/- 9 MeV., Comment: Published version, typos corrected, minor revisions in section I and VII, conclusions unchanged

We studied the QCD phase transition as a function of quark mass in the $N_f=3$ QCD at vanishing baryon density. Lattice simulations have been performed using Highly Improved Staggered Quarks on $N_{\tau}=6$ lattices with quark masses that correspond to pion masses in the region $80 \lesssim m_{\pi} \lesssim 230 $MeV. We found no evidence of the first order phase transition in the current pion mass window. The pion mass at the critical point where the first order phase transition starts is estimated to be $m^c_{\pi}\lesssim 45$ MeV., Comment: 7 pages, presented at The XXIX International Symposium on Lattice Field Theory - Lattice 2011, July 10-16, 2011, Squaw Valley, Lake Tahoe, California

We analyze the low frequency part of charmonium spectral functions on large lattices close to the continuum limit in the temperature region $1.5\lesssim T/T_c\lesssim 3$ as well as for $T \simeq 0.75T_c$. We present evidence for the existence of a transport peak above $T_c$ and its absence below $T_c$. The heavy quark diffusion constant is then estimated using the Kubo formula. As part of the calculation we also determine the temperature dependence of the signature for the charmonium bound state in the spectral function and discuss the fate of charmonium states in the hot medium., Comment: 4 pages, Proceedings for Quark Matter 2011 Conference, May 23-28, 2011, Annecy, France

We calculate the vector current correlation function for light valence quarks in the deconfined phase of QCD. The calculations have been performed in quenched lattice QCD at T=1.45 Tc for four values of the lattice cut-off on lattices up to size 128^3x48. This allows to perform a continuum extrapolation of the correlation function in the Euclidean time interval tau*T -in [0.2, 0.5], which extends to the largest temporal separations possible at finite temperature, to better than 1% accuracy. In this interval, at the value of the temperature investigated, we find that the vector correlation function never deviates from the free correlator for massless quarks by more than 9%. We also determine the first two non-vanishing thermal moments of the vector meson spectral function. The second thermal moment deviates by less than 7% from the free value. With these constraints, we then proceed to extract information on the spectral representation of the vector correlator and discuss resulting consequences for the electrical conductivity and the thermal dilepton rate in the plasma phase., Comment: 25 pages, 20 EPS-figures, v2: final version accepted for publication (Phys. Rev. D)

We study the properties of charmonium states at finite temperature in quenched QCD on isotropic lattices. We measured charmonium correlators using non-perturbatively $\cO(a)$ improved clover fermions on fine ($a=0.01$ fm) lattices with a relatively large size of $128^{3}\times 96$, $128^3\times48$, $128^3\times32$ and $128^3\times24$ at $0.73~T_c$, $1.46~T_c$, $2.20~T_c$ and $2.93~T_c$, respectively. Our analysis suggests that $\Jpsi$ is melted already at $1.46~T_c$ and $\eta_c$ starts to dissolve at $1.46~T_c$ and does not exist at higher temperatures. We also identify the heavy quark transport contribution at the spectral function level for the first time., Comment: Talk presented at The XXVIII International Symposium on Lattice Field Theory, Lattice2010, June 14-19, 2010, Villasimius, Italy

We study charmonium correlators and spectral functions in quenched QCD, using Clover improved Wilson fermions on very fine (0.015 fm) isotropic lattices at 0.75 Tc and 1.5 Tc. We use a new approach to distinguish the zero mode contribution from the other contributions. Once this is removed, we find that the ratios of correlators to reconstructed correlators remain almost unity at all distances. The ground state peaks of spectral functions obtained at 0.75 Tc are reliable and robust. The present accuracy and limited number of points in the temporal direction at 1.5 Tc do not allow for a reliable conclusion about a possible melting of charmonium states in the QGP., Comment: 7 pages, 4 figures, talk given at The XXVII International Symposium on Lattice Field Theory, July 26-31, 2009, Peking University, Beijing, China

We study the nuclear modification factors RAA and RCP of the high transverse momentum 5

The effects of heavy quarks energy loss on the transverse momentum spectra of single muons are studied. The energy loss of heavy quarks when traversing the medium formed in PbPb collisions at $\sqrt{s_\mathrm{NN}}$=5.5 TeV is calculated by implementing the collision geometry and the quenching weights. The medium density is assumed to decrease at forward pseudo-rapidity and to be proportional with pseudo-rapidity multiplicity $dN/d\eta$. Muons from W decays can be used as a medium-blind reference to quantify the effect of heavy quarks energy loss on the single muon production., Comment: to appear in a special issue of Int. J. of Mod. Phys. E (the proceedings of the 19th International Conference on Ultrarelativistic Nucleus-Nucleus Collisions: Quark Matter 2006 (QM 2006), Shanghai, China, 14-20 Nov 2006)

The energy loss of heavy quarks in a quark-gluon plasma of finite size is studied within the light-cone integral approach. A simple analytical formulation of the radiative energy loss of heavy quarks is derived. This provides a convenient way to quantitatively estimate the quark energy loss. Our results show that if the energy of a heavy quark is much larger than its mass, the radiative energy loss approaches the radiative energy loss of light quarks., Comment: 4 pages, 2 figures. Accepted for publication in European Physical Journal A