Equation of state in (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math>)-flavor QCD

Abstract

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 ${N}_{\ensuremath{\tau}}=6$, 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, $145\text{ }\text{ }\mathrm{MeV}\ensuremath{\le}T\ensuremath{\le}163\text{ }\text{ }\mathrm{MeV}$, defined by the chiral transition, is ${\ensuremath{\epsilon}}_{c}=(0.18--0.5)\text{ }\text{ }\mathrm{GeV}/{\mathrm{fm}}^{3}$, i.e., $(1.2--3.1)\text{ }{\ensuremath{\epsilon}}_{\text{nuclear}}$. 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.