The equation of state in (2+1)-flavor QCD

TL;DR

This study delivers a continuum-extrapolated equation of state for (2+1)-flavor QCD using the HISQ/tree action, spanning 130–400 MeV in temperature. By combining high-statistics lattice simulations on multiple $N_\tau$ values with HRG inputs and perturbative insights, the authors obtain precise thermodynamic quantities, including the trace anomaly, pressure, energy, and entropy density, and provide a practical analytic parametrization for phenomenology. They locate the softest point near 145–150 MeV and identify a pseudo-critical temperature around 154 MeV, underscoring the crossover nature of the transition. The work also determines scale-setting parameters ($r_0$, $r_1$, $w_0$) and offers cross-checks against stout-action results and high-temperature perturbative theories, informing hydrodynamic modeling of heavy-ion collisions.

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 $N_τ=6,~8,~10$, 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, $145~ {\rm MeV} \leq T \leq 163$ MeV, defined by the chiral transition, is $ε_c=(0.18-0.5)~{\rm GeV}/{\rm fm}^3$, $i.e.$, $(1.2-3.1)\ ε_{\rm nuclear}$. 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.

Figures (19)

• Figure 1: The trace anomaly calculated with the HISQ/tree action at different $N_{\tau}$ and compared with results from previous calculations with the p4 and asqtad actions on $N_{\tau}=8$ lattices Bazavov:2009zn, except for the two highest temperatures, where we show the $N_{\tau}=6$ p4 data from Ref. Petreczky:2009at and Ref. Cheng:2007jq, respectively.
• Figure 2: The gluonic (left) and fermionic (right) parts of the trace anomaly for different $N_{\tau}$. See text for details.
• Figure 3: The trace anomaly in the low temperature region compared with the hadron resonance gas model (solid line).
• Figure 4: The data for the trace anomaly and the result (thick lines showing the $1 \sigma$ bootstrap error bands) of applying Eq. (\ref{['eq:SplineFit']}) with $N_\tau = 8$, $10$, and $12$. The parameters in Eq. (\ref{['eq:SplineFit']}) and their errors, defining this final fit, were determined from these data as discussed in the text. The error bands shown are generated by the same bootstrap process used to estimate the fit parameters and their errors. The additional $2\%$ error that is added to the final continuum result to account for the uncertainty in the determination of the temperature scale as discussed in the text is not included in these plots.
• Figure 5: Spline fits to the trace anomaly for several values of the lattice spacing $aT=1/N_\tau$ and the result of our continuum extrapolation (left). Note that the error bands shown here do not include the 2% scale error. The right hand panel shows suitably normalized pressure, energy density, and entropy density as a function of the temperature. In this case the 2% scale error is included in the error bands. The dark lines show the prediction of the HRG model. The horizontal line at $95 \pi^2/60$ in the right panel corresponds to the ideal gas limit for the energy density and the vertical band marks the crossover region, $T_c=(154\pm 9)$ MeV.