Equation of state and QCD transition at finite temperature

We calculate the equation of state in $2+1$ flavor QCD at finite temperature with physical strange quark mass and almost physical light quark masses using lattices with temporal extent $N_{\tau }=8$. Calculations have been performed with two different improved staggered fermion actions, the asqtad and p4 actions. Overall, we find good agreement between results obtained with these two $O(a^2)$ improved staggered fermion discretization schemes. A comparison with earlier calculations on coarser lattices is performed to quantify systematic errors in current studies of the equation of state. We also present results for observables that are sensitive to deconfining and chiral aspects of the QCD transition on $N_{\tau }=6$ and 8 lattices. We find that deconfinement and chiral symmetry restoration happen in the same narrow temperature interval. In an appendix we present a simple parametrization of the equation of state that can easily be used in hydrodynamic model calculations. In this parametrization we include an estimate of current uncertainties in the lattice calculations which arise from cutoff and quark mass effects.

Figure 1

The trace anomaly $(ϵ-3p)/T^4$ calculated on lattices with temporal extent $N_{\tau }=6$, 8. The upper $x$-axis shows the temperature scale in units of the scale parameter $r_0$ which has been determined in studies of the static quark potential. The lower $x$-axis gives the temperature in units of MeV, which has been obtained using for $r_0$ the value determined from the level splitting of bottomonium states, $r_0=0.469\mathrm{fm}$ [16]. The right-hand figure shows the region around the maximum of $(ϵ-3p)/T^4$, which also is the temperature region where results obtained with the two different discretization schemes show the largest differences. Open symbols for the $N_{\tau }=6$ denote previous asqtad data obtained with the $R$ algorithm [3]. All other data have been obtained with an RHMC algorithm.

Figure 2

The trace anomaly calculated with the p4 (left) and asqtad (right) actions. Shown is a comparison of results obtained on lattices with temporal extent $N_{\tau }=6$ and 8. The curves show interpolations discussed in the text. Open symbols for the $N_{\tau }=6$, asqtad data set denote data obtained with the $R$ algorithm. All other data have been obtained with an RHMC algorithm.

Figure 3

The trace anomaly at low temperatures calculated with the asqtad and p4 actions on lattices with temporal extent $N_{\tau }=6$ and 8. Open symbols for the $N_{\tau }=6$, asqtad data set denote data obtained with the $R$ algorithm. All other data have been obtained with an RHMC algorithm. Solid lines show interpolation curves for the p4 action discussed in the text. The dashed and dashed-dotted curves give the trace anomaly calculated in a hadron resonance gas model with two different cuts for the maximal mass, $m_{\max }=1.5\mathrm{GeV}$ (dashed-dotted) and 2.5 GeV (dashed).

Figure 4

The trace anomaly at high temperatures calculated with the asqtad and p4 actions on lattices with temporal extent $N_{\tau }=6$ and 8. For the p4 action we also show results obtained on lattices with temporal extent $N_{\tau }=4$ [4]. Open symbols for the $N_{\tau }=6$, asqtad data set denote data obtained with the $R$ algorithm. All other data have been obtained with an RHMC algorithm. Solid curves show fits to the data based on Eq. (5). Fit parameters are given in Table .

Figure 5

Gluon condensate and quark condensate contributions to the trace anomaly. Shown are results for the asqtad and p4 actions obtained on lattices of size $N_{\tau }=4$, 6 and 8. Some results shown for the asqtad and p4 actions on lattices of size $N_{\tau }=4$, 6 have been taken from earlier calculations [3, 4]. Open symbols for the $N_{\tau }=4$ and 6 asqtad data sets denote data obtained with the $R$ algorithm. All other data have been obtained with an RHMC algorithm.

Figure 6

The ratio of light and strange quark contributions to the fermionic part of the trace anomaly on lattices with temporal extent $N_{\tau }=4$, 6, and 8. Some results shown for the asqtad and p4 actions on lattices of size $N_{\tau }=4$, 6 have been taken from earlier calculations [3, 4]. Open symbols for the $N_{\tau }=4$ and 6 asqtad data sets denote data obtained with the $R$ algorithm. All other data have been obtained with an RHMC algorithm.

Figure 7

Energy density and 3 times the pressure calculated on lattices with temporal extent $N_{\tau }=4$, 6 [4], and 8 using the p4 action (left). The right-hand figure compares results obtained with the asqtad and p4 actions on the $N_{\tau }=8$ lattices. Crosses with error bars indicate the systematic error on the pressure that arises from different integration schemes as discussed in the text. The black bars at high temperatures indicate the systematic shift of data that would arise from matching to a hadron resonance gas at $T=100\mathrm{MeV}$. The band indicates the transition region $185\mathrm{MeV}<T<195\mathrm{MeV}$. It should be emphasized that these data have not been extrapolated to physical pion masses.

Figure 8

The entropy density obtained on lattices with temporal extent $N_{\tau }=6$ [3, 4] and 8. The band indicates the transition region $185\mathrm{MeV}<T<195\mathrm{MeV}$.

Figure 9

Pressure divided by energy density ($p/ϵ$) and the square of the velocity of sound ($c_s^2$) calculated on lattices with temporal extent $N_{\tau }=6$ (p4, [4]) and $N_{\tau }=8$ using the p4 as well as the asqtad action. Lines without data points give the square of the velocity of sound calculated analytically from Eq. (9) using the interpolating curves for $ϵ/T^4$ and $p/T^4$. The dashed-dotted line at low temperatures gives the result for $p/ϵ$ from a hadron resonance gas (HRG) calculation using $m_{\max }=2.5\mathrm{GeV}$.

Figure 10

The light (left) and strange (right) quark number susceptibilities calculated on lattices with temporal extent $N_{\tau }=6$ and 8. The $N_{\tau }=6$ results for the p4 action are taken from [4]. The band corresponds to a temperature interval $185\mathrm{MeV}\le T\le 195\mathrm{MeV}$.

Figure 11

The ratio of energy density and quark number susceptibilities (left) and the ratio of strange and light quark number susceptibilities (right) calculated on lattices with temporal extent $N_{\tau }=8$. For the energy densities the interpolating curves shown in Fig. 7 have been used. Curves in the right-hand figure show results for a hadron resonance gas including resonance up to $m_{\max }=1.5\mathrm{GeV}$ (upper branch) and 2.5 GeV (lower branch), respectively.

Figure 12

The subtracted chiral condensate normalized to the corresponding zero temperature value (right). In the right-hand figure data for $N_{\tau }=6$ calculations have been shifted by $-7\mathrm{MeV}$ (asqtad) and $-5\mathrm{MeV}$ (p4).

Figure 13

The renormalized Polyakov loop obtained with the asqtad and p4 actions from simulations on lattices with temporal extent $N_{\tau }=6$ and 8. Open symbols for the $N_{\tau }=6$, asqtad data set denote data obtained with the $R$ algorithm. All other data have been obtained with an RHMC algorithm.

Figure 14

The renormalization constant $Z(\beta )$. The error band is determined with bootstrap analysis.

Figure 15

Fits to the trace anomaly using Eq. (c1) for p4 and asqtad $N_{\tau }=8$ lattice data (solid lines), to lattice data merged with the hadron resonance gas calculation ($m_{\mathrm{res}}<2.5\mathrm{GeV}$) over the region $100<T<130\mathrm{MeV}$ (double-dot dashed), and to the lattice data shifted to lower temperature by 10 MeV (dashed).

Figure 16

Energy density and 3 times the pressure calculated from fitting Eq. (c1) to the trace anomaly. Fits to p4 and asqtad lattice data are solid lines, fits to lattice data shifted by negative 10 MeV are dashed lines, and combined fits to HRG and lattice data are double-dot dashed. The systematic error band associated with beginning the pressure integration at $T=100\mathrm{MeV}$ and plotted as a black shaded box in Fig. 7 is plotted as a grey band, with lower bound defined by the pressure integration derived from the p4 interpolation. The systematic errors associated the interpolation are plotted here as grey boxes. These were shown as narrow error bars in Fig. 7.

Figure 17

Square of the velocity of sound, $c_s^2$, for the Eq. (c1) parametrization of the trace anomaly for fits to the lattice data (solid), lattice data shifted by 10 MeV (dashed), and combined HRG and lattice calculations (double-dot dashed). The lattice curves are compared to two typically EoS inputs currently used in hydrodynamic codes: the qcdEOS used in vh2 (dotted), and an HRG calculation with a first-order transition at 180 MeV (double-dotted).