Equation of state of quark-gluon matter from lattice QCD with two flavors of twisted mass Wilson fermions

We report on lattice QCD results for the thermodynamic equation of state of quark-gluon matter obtained with $N_f=2$ degenerate quark flavors. For the fermion field discretization we are using the Wilson twisted mass prescription. Simulations have been carried out at three values of the bare quark masses corresponding to pion masses of $\sim 360$, $\sim 430$ and $\sim 640\mathrm{MeV}$. We highlight the importance of a good control of the lattice cutoff dependence of the trace anomaly which we have studied at several values of the inverse temperature $T^{-1}=a{N}_{\tau }$ with a timelike lattice extent up to $N_{\tau }=12$. We contrast our results with those of other groups obtained for $N_f=0$ and $N_f=2+1$. At low temperature we also confront them with hadron resonance gas model predictions for the trace anomaly.

Figure 1

Left: Chirally extrapolated Sommer scale $r_{\chi }/a$ and a fit using Eq. (7). Right: Charged pion mass in physical units for the three ensembles together with a constant fit over all data points. Open symbols denote data that have been interpolated using ETMC results.

Figure 2

The disconnected part of the chiral susceptibility in the crossover temperature range together with Gaussian fit curves versus temperature. Left: For the B mass; Middle: C mass; and Right: D mass.

Figure 3

The renormalized real part of the Polyakov loop versus temperature. Left: For the B mass; Middle: C mass; and Right: D mass. Also shown are fits with $P(T)$ according to Eq. (12) (where evaluated).

Figure 4

Left: The dependence of the critical hopping parameter ${\kappa }_c$ on the coupling $\beta$. The curve represents a Padé interpolation. In the inlaid figure we show the interpolation for asymptotically large $\beta =6/g^2$, i.e. small $g^2$, where the fit has been constrained to ${\kappa }_c(g\equiv 0)=1/8$ representing the asymptotic free limit value. Right: The PCAC mass as a function of the hopping parameter $\kappa$ around the critical value at $\beta =3.85,a\mu =0.006$. This corresponds to a check of the validity of the spline interpolation of ${\kappa }_c$ (see text for details).

Figure 5

Left: Check of the continuum limit of the trace anomaly contribution originating from the derivative with respect to the untwisted quark mass $m$. We show data for the B-mass ensemble at two values of the temperature. Middle: The same for the C mass. Right: The same for the D mass.

Figure 6

The $\beta$-function obtained according to Eq. (18) from fitting expression (7) to the chirally extrapolated data of the Sommer scale $r_{\chi }/a$. We also show the perturbative two-loop expectation at large couplings as obtained from Eq. (8).

Figure 7

Left: Fit of $r_{\chi }\mu$ with Eq. (21) for all three masses. Right: Combination of $B$-functions $B_{\beta }B\mu$ for all three masses. The perturbative asymptotic behavior is indicated by the lines at high values of the coupling. The curves for the B and D mass have been shifted for better visibility.

Figure 8

Upper panels: Interpolation of ($T=0$) values in $\beta$ for the gauge action contributions $⟨S_g⟩$ to the trace anomaly for (from left to right) the B, C and D mass. We show the outcome of the average of fits of type A ($d_p=5$), B ($d_p=4$ or 5) and C. Middle panels: Residuals for the fit results in order to illustrate the quality of the interpolation. Lower panels: Magnitudes of the absolute errors of the fit results shown in the upper panel. In the figures of the middle and lower panels only those points are shown for which $T>0$ computations have been carried out.

Figure 9

Left: The trace anomaly for the B mass obtained for different values of the temporal extent $N_{\tau }$. Middle: The same quantity for the C mass. Right: The same quantity for the D mass. For the B mass the results obtained on the smaller spatial volume are superimposed slightly shifted for better visibility.

Figure 10

Continuum limit of the trace anomaly for three masses and for, in each case, two values of the temperature once with tree-level correction (blue circles and lines) and once without (red squares and lines). We compare the continuum limit results for the two extrapolations with the continuum estimate provided by the global fit Eq. (28) at the same temperature (green triangles).

Figure 11

Left: The tree-level corrected trace anomaly for the B mass obtained for different values of the temporal extent $N_{\tau }$. $T_{\chi }$ and $T_{\text{deconf}}$ are located at 193 and 219 MeV, respectively. Middle: The same quantity for the C mass with $T_{\chi }$ and $T_{\text{deconf}}$ located at 208 and 225 MeV, respectively. Right: The same quantity for the D mass with $T_{\chi }$ and $T_{\text{deconf}}$ located at 229 and 244 MeV, respectively. Also shown is the result of a combined fit of the interpolation formula Eq. (28) to the $N_{\tau }=8,10$ and 12 data ($N_{\tau }=6,8$ and 10 in the case of the D mass). For the B mass the results obtained on the smaller spatial volume are superimposed slightly shifted for better visibility. This data however have not been included in the fit.

Figure 12

Left: Final result for the pressure $p$ and the energy density $\epsilon$ in units of $T^4$ for the B-mass ensemble. We also show once more the interpolation of the trace anomaly used for integrating the pressure. The arrow in the upper right corner indicates the expected Stefan-Boltzmann limit for the pressure. On top of the panels we provide the temperature in units of $T_c\equiv T_{\chi }$. Middle: The same for the C mass. Right: The same for the D mass.

Figure 13

Left: The ratio $p/\epsilon$ for the B ensemble as a function of the energy density in units of $\mathrm{GeV}/{\mathrm{fm}}^3$. We also show the speed of sound squared $c_s^2$ obtained from $p/\epsilon$. Arrows indicate the expected large $T$ Stefan-Boltzmann limit given by $1/3$. Middle: The same for the C mass. Right: The same for the D mass.

Figure 14

Comparison of the interaction measure in the low temperature region to the predictions of several HRG model adaptations. Left: The data and analysis corresponding to the B mass is shown. Right: The same for the C mass. See text for details.

Figure 15

Left: The pressure as obtained from several HRG models is shown. See text for details. Right: The pressure and energy density for the B mass ($m_{\pi }\sim 360\mathrm{MeV}$) and for the C mass ($m_{\pi }\sim 430\mathrm{MeV}$) as obtained when using the lattice HRG model pressure for estimating the integration constant $p_0$.

Figure 16

A comparison of $I/T^4$ versus $T/T_c$ between $N_f=0$ obtained in Ref. [50], our data at $N_f=2$ for the B mass and $N_f=2+1$ obtained in Ref. [3]. We also show our curves for the C and D ($m_{\pi }\sim 640\mathrm{MeV}$) masses, however, suppressing the errors for better visibility. For $T_c$ we use our $T_{\chi }$ estimates obtained at the largest $N_{\tau }$ available.