The QCD equation of state with dynamical quarks

TL;DR

The paper advances the lattice QCD determination of the QCD equation of state for 2+1 dynamical quark flavors by employing finer Nt lattices, physical quark masses, and improved actions to perform a continuum-extrapolated analysis. It introduces a two-pronged methodological innovation: a Lines of Constant Physics framework extended to high temperatures and a spline-based pressure reconstruction derived from lattice derivatives, complemented by tree-level improvement to reduce discretization errors. The results provide detailed thermodynamic quantities (p, I, ε, s, c_s^2) over 100–1000 MeV, quantify the charm contribution in a partially quenched setup, and offer a continuum-parametrized trace anomaly; comparisons with hotQCD highlight systematic differences likely due to discretization artifacts. The work establishes robust benchmarks for QCD thermodynamics, with implications for heavy-ion phenomenology and early-universe physics, and points to future work on dynamical charm and finer lattices.

Abstract

The present paper concludes our investigation on the QCD equation of state with 2+1 staggered flavors and one-link stout improvement. We extend our previous study [JHEP 0601:089 (2006)] by choosing even finer lattices. Lattices with $N_t=6,8$ and 10 are used, and the continuum limit is approached by checking the results at $N_t=12$. A Symanzik improved gauge and a stout-link improved staggered fermion action is utilized. We use physical quark masses, that is, for the lightest staggered pions and kaons we fix the $m_π/f_K$ and $m_K/f_K$ ratios to their experimental values. The pressure, the interaction measure, the energy and entropy density and the speed of sound are presented as functions of the temperature in the range $100 ...1000 \textmd{MeV}$. We give estimates for the pion mass dependence and for the contribution of the charm quark. We compare our data to the equation of state obtained by the "hotQCD" collaboration.

Figures (10)

• Figure 1: The trace anomaly $I=\epsilon-3p$ normalized by $T^4$ as a function of the temperature on $N_t=6,8,10$ and $12$ lattices.
• Figure 2: The pressure normalized by $T^4$ as a function of the temperature on $N_t=6,8$ and $10$ lattices. The Stefan-Boltzmann limit $p_{SB}(T) \approx 5.209 \cdot T^4$ is indicated by an arrow. For our highest temperature $T=1000$ MeV the pressure is almost 20% below this limit.
• Figure 3: The energy density normalized by $T^4$ as a function of the temperature on $N_t=6,8$ and $10$ lattices. The Stefan-Boltzmann limit $\epsilon_{SB}= 3p_{SB}$ is indicated by an arrow.
• Figure 4: The entropy density normalized by $T^3$ as a function of the temperature on $N_t=6,8$ and $10$ lattices. The Stefan-Boltzmann limit $s_{SB}= 4p_{SB}/T$ is indicated by an arrow.
• Figure 5: The squared of the speed of sound as a function of the temperature on $N_t=6,8$ and $10$ lattices. The Stefan-Boltzmann limit is $c_{s,SB}^2=1/3$ indicated by an arrow.