QCD equation of state with almost physical quark masses

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 $N_{\tau }=4$ 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, $140\mathrm{MeV}\le T\le 800\mathrm{MeV}$. We present a detailed discussion of finite cutoff 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 $N_{\tau }=8$. 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.

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 $r_0=0.469\mathrm{fm}$.

Figure 2

The scale parameter ${\widehat{r}}_0\equiv r_0/a$ 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. (22) 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. (25, 26).

Figure 3

The $\beta$-function on the LCP [Eq. (24)] (left) and the product $R_{\beta }{R}_m$ (right). The horizontal lines show the weak coupling behavior given in Eqs. (9, 19). The two curves result from two different fits of ${\widehat{r}}_0$ as discussed in the text.

Figure 4

The trace anomaly ${\Theta }^{\mu \mu }(T)\equiv ϵ-3p$ in units of $T^4$ versus temperature obtained from calculations on lattices with temporal extent $N_{\tau }=4$, 6 and 8. The temperature scale, $T{r}_0$ (upper x-axis) has been obtained using the parametrization given in Eq. (22), and $T$ [MeV] (lower x-axis), has been extracted from this using $r_0=0.469\mathrm{fm}$.

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).

Figure 6

Comparison of the low temperature part of $(ϵ-3p)/T^4$ calculated on lattices with temporal extent $N_{\tau }=4$ and 6 with a resonance gas model that includes all resonances up to mass 2.5 GeV (dashed curve). The solid curve shows a polynomial fit to the $N_{\tau }=6$ data obtained with the p4fat3 action. Data for calculations with the asqtad action are taken from 11.

Figure 7

The trace anomaly in the vicinity of the transition temperature calculated on lattices with temporal extent $N_{\tau }=4$ and 6 on lattices with aspect ratio $N_{\sigma }/N_{\tau }=4$. Data for calculations with the asqtad action are taken from 11 which have been performed on finite temperature lattices with a smaller physical volume corresponding to an aspect ratio $N_{\sigma }/N_{\tau }=2$.

Figure 8

The high temperature part of $(ϵ-3p)/T^4$ calculated on lattices with temporal extent $N_{\tau }=4$, 6 and 8. The curves show fits to the $N_{\tau }=4$ and 6 data with the Ansatz given in Eq. (29).

Figure 9

Energy density and 3 times the pressure as function of the temperature (left) and the ratio $p/ϵ$ as function of the fourth root of the energy density (right) obtained from calculations on lattices with temporal extent $N_{\tau }=4$ and 6. Temperature and energy density scales have been obtained using the parametrization of $r_0/a$ given in Eq. (22) and $r_0=0.469\mathrm{fm}$. The small vertical bar in the left-hand figure at high temperatures shows the estimate of the systematic uncertainty on these numbers that arises from the normalization of the pressure at $T_0=100\mathrm{MeV}$. The dashed curve (HRG) in the right-hand figure shows the result for $p/ϵ$ in a hadron resonance gas for temperatures $T<190\mathrm{MeV}$.

Figure 10

Entropy density as function of the temperature obtained from calculations on lattices with temporal extent $N_{\tau }=4$ and 6. Temperature and energy density scales have been obtained using the parametrization of $r_0/a$ given in Eq. (22) and $r_0=0.469\mathrm{fm}$. The small vertical bar at high temperatures shows the estimate of the systematic uncertainty on these numbers that arises from the normalization of the pressure at $T_0=100\mathrm{MeV}$.

Figure 11

Renormalized Polyakov loop on lattices with temporal extent $N_{\tau }=4$, 6 and 8 (left) and the normalized difference of light and strange quark chiral condensates defined in Eq. (36). The vertical lines show the location of the transition temperature determined in 18 on lattices with temporal extent $N_{\tau }=4$ (right line) and in this analysis for $N_{\tau }=6$ (left line).