PDF《BNL-NT-07/38, BI-TP 2007/20, CU-TP-》

4 operators. These di?erences are particularly di?cult to evaluate because both terms being subtracted contain the pressure or energy density of the vacuum, an unphysical quantity that is approximately 1/(aT )4 larger than the sought-after di?erence. Numerical signals thus rapidly decrease with the fourth power of the lattice spacing, a, when one tries to approach the continuum limit at ?xed temperature (T ). For this reason improved actions, which allow one to perform calculations on rather coarse lattices with relatively small lattice discretization errors, are quite useful in thermodynamic calculations. Indeed, the early calculations of bulk thermodynamics with standard staggered [7] and Wilson [8] fermion discretization schemes have shown that at high temperature bulk thermodynamic observables are particularly sensitive to lattice

2 discretization errors. This closely follows observations made in studies of the thermodynamics of SU(3) gauge theories [9]. In order to minimize discretization errors at high temperature, improved staggered fermion actions - the p4-action [10] and the asqtad action [11] - have been used to study the QCD equation of state. Recent studies, performed with the asqtad action with almost physical quark mass values on lattices with two di?erent values of the lattice cut-o? [11], indeed show much smaller discretization errors than similar studies performed with the 1-link, stout smeared staggered fermion action [12]. Another source for cut-o? errors arises, however, from the explicit breaking of ?avor symmetry in the staggered fermion formulation. While this is not of much concern in the chirally symmetric high temperature phase of QCD, it leads to cut-o? dependent modi?cations of the hadron spectrum and thus may in?uence the calculation of thermodynamic observables in the low temperature hadronic phase of QCD. Techniques to reduce ?avor symmetry breaking through the introduction of so-called '

fat links'

are thus generally exploited in numerical calculations with staggered fermions [14]. In this paper we report on a calculation of bulk thermodynamics in QCD with almost physical light quark masses and a physical value of the strange quark mass. Our calculations have been performed with a tree level Symanzik- improved gauge action and an improved staggered fermion action, the p4-action with 3-link smearing (p4fat3), which removes O(a2 ) cut-o? e?ects at tree-level and also leads to small cut-o? e?ects in O(g2 ) perturbation theory [13]. At each temperature, we perform simulations with two degenerate light quark masses and a heavier strange quark mass for two di?erent values of the lattice cut-o?, corresponding to lattices with temporal extent Nτ =

4 and 6. In these calculations we explore a wide range of temperatures varying from about

140 MeV to about

800 MeV. This corresponds to the temperature interval relevant for current experimental studies of dense matter in heavy ion collisions at RHIC as well as the forthcoming experiments at the LHC. Bare quark masses have been adjusted to keep physical masses approximately constant when the lattice-cut o? is varied. At high temperatures, T >

?350 MeV, we also performed calculations on lattices with temporal extent Nτ =

8 to get control over cut-o? e?ects in the high temperature limit. We will start in the next section by reviewing basic thermodynamic relations in the continuum valid for thermody- namic calculations on such lines of constant physics (LCP). In section III we outline details of our calculational set-up with improved staggered fermions. In section IV we present our zero temperature calculations needed to de?ne the line of constant physics and the temperature scale deduced from properties of the static quark-antiquark potential. Section V is devoted to the presentation of our basic result, the di?erence between energy density and three times the pressure from which we obtain all other thermodynamic observables, e.g. the pressure, energy and entropy densities as well as the velocity of sound. Section VI is devoted to a discussion of the temperature dependence of Polyakov loop expectation values and chiral condensates which provides a comparison between the decon?ning and chiral symmetry restoring features of the QCD transition. We ?nally present a discussion of our results and a comparison with other improved staggered fermion calculations of bulk thermodynamics in Section VII. II. THERMODYNAMICS ON LINES OF CONSTANT PHYSICS To start our discussion of QCD thermodynamics on the lattice we recall some basic thermodynamic relations in the continuum. For large, homogeneous media the basic bulk thermodynamic observables we will consider here can be obtained from the grand canonical partition function with vanishing quark chemical potentials, Z(T, V ). We introduce the grand canonical potential, ?(T, V ), normalized such that it vanishes at vanishing temperature, ?(T, V ) = T lnZ(T, V ) ? ?0 , (1) with ?0 = lim T →0 T lnZ(T, V ). With this we obtain the thermal part of the pressure (p) and energy density (?) p =