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 Nτ = 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 Tc = 154 ± 9 MeV.

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 Nτ = 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 MeV ≤ T ≤ 163 MeV, defined by the chiral transition, is c = (0.18 − 0.5) GeV/fm 3 , i.e., (1.2 − 3.1) 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 r0 from the static quark potential and w0 from the gradient flow.

Dramatic progress has been made over the last decade in the numerical study of quantum chromodynamics (QCD) through the use of improved formulations of QCD on the lattice (improved actions), the development of new algorithms and the rapid increase in computing power available to lattice gauge theorists. In this article we describe simulations of full QCD using the improved staggered quark formalism, "asqtad" fermions. These simulations were carried out with two degenerate flavors of light quarks (up and down) and with one heavier flavor, the strange quark. Several light quark masses, down to about 3 times the physical light quark mass, and six lattice spacings have been used. These enable controlled continuum and chiral extrapolations of many low energy QCD observables. We review the improved staggered formalism, emphasizing both advantages and drawbacks. In particular, we review the procedure for removing unwanted staggered species in the continuum limit. We then describe the asqtad lattice ensembles created by the MILC Collaboration.All MILC lattice ensembles are publicly available, and they have been used extensively by a number of lattice gauge theory groups. We review physics results obtained with them, and discuss the impact of these results on phenomenology. Topics include the heavy quark potential, spectrum of light hadrons, quark masses, decay constant of light and heavy-light pseudoscalar mesons, semileptonic form factors, nucleon structure, scattering lengths and more. We conclude with a brief look at highly promising future prospects. PACS numbers: 12.38.Gc, 11.15.Ha 3. Staggered fermions 16 4. Chirally invariant fermions 21 C. Numerical simulations 25 D. Asqtad improved staggered fermions 29 E. Highly improved staggered fermions 32 III. Staggered chiral perturbation theory and "rooting" 34 A. Chiral effective theory for staggered quarks 34 B. Extensions of staggered chiral perturbation theory 41 C. The issue of rooting 45 IV. Overview of the MILC lattice ensembles 56 A. Algorithms and algorithm tests 57 B. The static potential and determining the lattice spacing 62 C. Tuning the strange quark mass 68 D. The topological susceptibility 68 V. Spectroscopy of light hadrons 71 A. Hadron mass computations 72 B. Correlated fits 76 C. Results for some light hadrons 79 3 D. Flavor singlet spectroscopy 83 E. Scalar mesons f 0 and a 0 84 F. Summary 88 VI. Results for the light pseudoscalar mesons 88 A. Motivation 88 B. From correlators to lattice masses and decay constants 88 C. Other computations of f π and f K 95 VII. Heavy-light mesons: masses and decay constants 96 A. Heavy quarks on the lattice 97 1. Nonrelativistic QCD 98 2. Wilson fermions with the Fermilab interpretation 98 3. The HISQ action 99 B. Lattice calculations of masses and decay constants 100 C. Results for masses, decay constants, and CKM matrix elements 104 VIII. Semileptonic form factors 107 A. D → πℓν and D → Kℓν 107 B. B → πℓν and |V ub | 109 C. B → Dℓν and B → D * ℓν 113 IX. Other computations using MILC lattices 116 A. Determination of ...

We calculate the equation of state in 2+1 flavor QCD at finite temperature with physical strange quark mass and almost physical light quark masses using lattices with temporal extent Nτ = 8. Calculations have been performed with two different improved staggered fermion actions, the asqtad and p4 actions. Overall, we find good agreement between results obtained with these two O(a 2 ) improved staggered fermion discretization schemes. A comparison with earlier calculations on coarser lattices is performed to quantify systematic errors in current studies of the equation of state. We also present results for observables that are sensitive to deconfining and chiral aspects of the QCD transition on Nτ = 6 and 8 lattices. We find that deconfinement and chiral symmetry restoration happen in the same narrow temperature interval. In an Appendix we present a simple parametrization of the equation of state that can easily be used in hydrodynamic model calculations. In this parametrization we also incorporated an estimate of current uncertainties in the lattice calculations which arise from cutoff and quark mass effects. We estimate these systematic effects to be about 10 MeV.

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