The QCD Equation of State with almost Physical Quark Masses

TL;DR

This work computes the QCD equation of state for 2+1 flavors with almost physical quark masses using improved lattice actions on Nt=4 and 6 lattices, plus Nt=8 at high temperature, along a line of constant physics defined by $m_{\bar{s}s} r_0=1.59$ and $\hat m_l/\hat m_s=0.1$. By evaluating the trace anomaly and integrating it, the authors obtain the pressure, energy density, and entropy density across a wide temperature range, and analyze the deconfinement and chiral-restoration aspects via the renormalized Polyakov loop and chiral condensates. They show moderate cut-off effects at high temperature for Nt≥6, locate a peak in the trace anomaly near the crossover region around Tc ≈ 196 MeV, and provide isentropic fits for the high-temperature behavior, offering a solid lattice-based baseline for heavy-ion phenomenology and future improvements toward physical light masses and charm contributions.

Abstract

We present results on the equation of state in QCD with two light quark flavors and a heavier strange quark. Calculations with improved staggered fermions have been performed on lattices with temporal extent Nt =4 and 6 on a line of constant physics with almost physical quark mass values; the pion mass is about 220 MeV, and the strange quark mass is adjusted to its physical value. High statistics results on large lattices are obtained for bulk thermodynamic observables, i.e. pressure, energy and entropy density, at vanishing quark chemical potential for a wide range of temperatures, 140 MeV < T < 800 MeV. We present a detailed discussion of finite cut-off effects which become particularly significant for temperatures larger than about twice the transition temperature. At these high temperatures we also performed calculations of the trace anomaly on lattices with temporal extent Nt=8. Furthermore, we have performed an extensive analysis of zero temperature observables including the light and strange quark condensates and the static quark potential at zero temperature. These are used to set the temperature scale for thermodynamic observables and to calculate renormalized observables that are sensitive to deconfinement and chiral symmetry restoration and become order parameters in the infinite and zero quark mass limits, respectively.

Figures (11)

• Figure 1: The static quark potential in units of the scale $r_0$ versus distance $r/r_0$ (left) and dimensionless combinations of the potential shape parameters $r_0/r_1$ and $r_0\sqrt{\sigma}$ extracted from fits to these potentials (right). The left hand figure shows potentials for several values of $\beta$ taken from our entire simulation interval, $\beta \in [3.15:4.08]$. The lowest curve in this figure combines all potentials by matching them to the string potential (solid line) as explained in the text. Curves in the right hand figure show quadratic fits and a fit to a constant with a 1% error band. The lattice spacing has been converted to physical units using $r_0=0.469$ fm.
• Figure 2: The scale parameter $\hat{r}_0 \equiv r_0/a$ versus $\beta=6/g^2$ (left) and its product with the bare light quark mass on the LCP (right). The two curves shown in the left hand part of this figure correspond to two different fit ansätze. As explained in the text in addition to the renormalization group motivated ansatz given in Eq. \ref{['fit']} the result from a 3-interval fit is shown. The curve in the right hand part of the figure shows a fit based on the ansatz given in Eqs. \ref{['mr0']} and \ref{['poly']}.
• Figure 3: The $\beta$-function on the LCP (Eq. \ref{['r0beta']}) (left) and the product $R_\beta R_m$ (right). The horizontal lines show the weak coupling behavior given in Eqs. \ref{['betaasym']} and \ref{['Rm2loop']}. The two curves result from two different fits of $\hat{r}_0$ as discussed in the text.
• Figure 4: The trace anomaly $\Theta^{\mu\mu}(T) \equiv \epsilon -3p$ in units of $T^4$ versus temperature obtained from calculations on lattices with temporal extent $N_\tau =4$, $6$, and $8$. The temperature scale, $Tr_0$ (upper x-axis) has been obtained using the parametrization given in Eq. \ref{['fit']}, and $T$ [MeV] (lower x-axis), has been extracted from this using $r_0=0.469$ fm.
• Figure 5: The fermionic contribution to the trace anomaly (left) and the ratio of the light and strange quark contributions to $\Theta_F^{\mu\mu}/T^4$ (right).