QCD equation of state with 2+1 flavors of improved staggered quarks

TL;DR

The paper addresses deriving the QCD equation of state for 2+1 flavors from first-principles lattice QCD calculations. It employs the integral method along trajectories of constant physics with a Symanzik improved gauge action and Asqtad staggered quarks on lattices with $N_t=4$ and $6$, extracting $I$, $p$, and $\varepsilon$ by integrating the interaction measure and applying zero-temperature subtractions. The results show the EOS remains non-perturbative at high temperatures, with $\varepsilon$ and $p$ approaching but staying below the Stefan–Boltzmann limit, and only mild dependence on light-quark mass within the studied range. These findings provide more realistic EOS inputs for hydrodynamic models of heavy-ion collisions and early-universe cosmology, highlighting the impact of realistic quark masses and lattice actions on QGP thermodynamics.

Abstract

We report results for the interaction measure, pressure and energy density for nonzero temperature QCD with 2+1 flavors of improved staggered quarks. In our simulations we use a Symanzik improved gauge action and the Asqtad $O(a^2)$ improved staggered quark action for lattices with temporal extent $N_t=4$ and 6. The heavy quark mass $m_s$ is fixed at approximately the physical strange quark mass and the two degenerate light quarks have masses $m_{ud}\approx0.1 m_s$ or $0.2 m_s$. The calculation of the thermodynamic observables employs the integral method where energy density and pressure are obtained by integration over the interaction measure.

Figures (11)

• Figure 1: Pion taste splitting relative to the Goldstone pion mass in units of $r_1 =0.318\, (7)\, (4)$ fm vs. the lattice scaling variable $(a/r_1)^2 \alpha_V(a)^2$ in a log-log plot. Here we take $\alpha_V(a)= 12 \pi/ [54 \ln[(3.33/a \Lambda)]$ with $\Lambda = 319$ MeV. The rising line has slope 1. The fancy diamonds locate the kaon splittings $(m_K^2 - m_G^2)r_1^2$ for $m_{ud}\approx 0.2\, m_s$. The fancy crosses do the same for $m_ {ud}\approx 0.1\, m_s$. The vertical lines indicate the approximate lattice spacing at the crossover temperature for various $N_t$. Data are from Aubin:2004wf and unpublished simulation results. The pion taste assignments are given in the gamma matrix basis. The taste singlet is denoted $\pi_s$.
• Figure 2: Plot of the two trajectories of constant physics in the $(am_{ud}, \beta)$ plane.
• Figure 3: Inverse lattice spacing in units of $f(g^2)r_1$vs. gauge coupling $\beta = 10/g^2$, based on the best fit parameterization Eq. (\ref{['eq:a']}). The fitting function is evaluated along the two lines of constant physics, namely $m_{ud} \approx 0.1 m_s$ and $0.2 m_s$. It is derived from forty measured values of $r_1/a$. Eight of them lie on the trajectories of constant physics and are plotted here.
• Figure 4: The interaction measure is shown for both of the trajectories of constant physics and the different $N_t$.
• Figure 5: The temperature dependence of the pressure for both of the trajectories of constant physics and the different $N_t$. The continuum Stefan-Boltzmann limit for 3 massless flavors is also shown.