Equation of state in $2+1$ flavor QCD at high temperatures

We calculate the equation of state at high temperatures in $2+1$ flavor QCD using the highly improved staggered quark (HISQ) action. We study the lattice spacing dependence of the pressure at high temperatures using lattices with temporal extent $N_{\tau }=6$, 8, 10 and 12 and perform continuum extrapolations. We also give a continuum estimate for the equation of state up to temperatures $T=2\mathrm{GeV}$, which are then compared with results of the weak-coupling calculations. We find a reasonably good agreement with the weak-coupling calculations at the highest temperatures.

Figure 1

The fermionic part (left) and the gauge part (right) of the trace anomaly obtained with HISQ action. The open symbols correspond to the HotQCD results for $m_l=m_s/20$ [8].

Figure 2

The numerical results for the trace anomaly at two different quark masses (see text). The open symbols correspond to $m_l/m_s=1/20$, while the filled symbols correspond to $m_l/m_s=1/5$. The bursts correspond to calculations with p4 action and $m_l=m_s/10$ [3].

Figure 3

The trace anomaly calculated in the low temperature region using $N_{\tau }=10$ and $N_{\tau }=12$ lattices. The bands correspond to interpolations (see text). The dashed line corresponds to HRG-PDG, while the solid line corresponds to HRG-QM.

Figure 4

Left: The pressure in the low temperature region. The lines correspond to HRG with distorted hadron spectrum (see text). Right: The pressure in the entire temperature range. The horizontal lines correspond to the free theory result. Also shown are the results for the pressure obtained with p4 action and $N_{\tau }=6$ or 8 [3, 4].

Figure 5

The pressure calculated with HISQ action for different $N_{\tau }$ and corrected for cutoff effects. The filled squares are the continuum results for the pressure. For comparison we also plot the p4 results for the pressure corrected for cutoff effects at high temperatures.

Figure 6

The interpolation of the lattice results for $(\epsilon -3p)/T^4$ for $N_{\tau }=8$, 10 and 12 compared with the interpolations for $(\epsilon -3p)/T^4$ obtained with $N_{\tau }=4$ and 6. The latter have been multiplied by 1.2 and 1.4 to bring them into agreement with the former interpolation.

Figure 7

The comparison of the lattice results for the trace anomaly with three-loop HTL perturbation theory shown as the line and the band. The size of the band corresponds to the scale variation from $\mu =\pi T$ to $4\pi T$. Also shown as a dashed line is the EQCD result for the trace anomaly (see text). The lattice results for $N_{\tau }=4$ and 6 for $T>600\mathrm{MeV}$ have been scaled by 1.2 and 1.4, respectively.

Figure 8

The pressure (left) and the entropy density (right) in the high temperature region compared with the weak-coupling calculations. The filled symbols correspond to the continuum results obtained from lattice calculations on $N_{\tau }=6$, 8, 10 and 12 lattices. The open symbols correspond to continuum estimate (see text). The errors of the continuum estimate have been enlarged by factor two to indicate additional systematic errors that might be present. The red line and the band correspond to the three-loop HTL perturbation theory [22], the blue band corresponds to the resummed calculation in next-to-leading log approximation (NLA) [23]. The width of the bands correspond to the scale variation from $\mu =\pi T$ to $4\pi T$. Also shown is the weak-coupling result obtained in EQCD [10].

Figure 9

The potentials calculated for $m_l=m_s/5$ at $\beta =7.030$, 7.825, 8.00, 8.20 and 8.40.

Figure 10

The ratio of the potentials calculated for $m_l=m_s/20$ and $m_l=m_s/5$ for $\beta =7.03$.

Figure 11

The ratio of the ratio of scales $r_1$ and $r_2$ as function of $\beta$. Also shown is the fit (solid line) and its uncertainty (dashed lines).

Figure 12

The lattice spacing in units of $r_1$ as function of $\beta$. For better visibility we divided $a/r_1$ by the 2-loop beta function given by Eq. (a4). The horizontal line shows the asymptotic 2-loop result. Also shown as a dashed line is the old parametrization of $a/r_1$ from Ref. [8].

Figure 13

The logarithmic derivative of the nonperturbative beta function.

Figure 14

The lattice spacing dependence of vector meson (left) and baryon masses (right). The lines show fits using Eq. (b3).

Figure 15

The trace anomaly calculated in the HRG-QM and in the HRG-QM with modified hadron spectrum for $N_{\tau }=6$, 8, 10 and 12, and compared with the lattice results, see Sec. 3 for details. The solid thick line corresponds to the continuum HRG-QM, while the top, middle and bottom colored thin lines correspond to the lattice HRG-QM, where only the ps mesons are modified, all ground states hadron are modified, and all ground state and excited state hadrons are modified, respectively.

Figure 16

The pressure calculated in the HRG-QM and in the HRG-QM with modified hadron spectrum for $N_{\tau }=6$, 8, 10 and 12 from top left to bottom right. The solid thick line corresponds to the continuum HRG-QM. The top, middle and bottom colored thin lines correspond to the lattice HRG-QM, where only the ps mesons are modified, all ground states hadron are modified and all the ground state and excited state hadrons are modified, respectively.