Family of equations of state based on lattice QCD: Impact on flow in ultrarelativistic heavy-ion collisions

M. Bluhm, B. Kämpfer, R. Schulze, D. Seipt, and U. Heinz

Phys. Rev. C 76, 034901 – Published 17 September 2007

Abstract

We construct a family of equations of state within a quasiparticle model by relating pressure, energy density, baryon density, and susceptibilities adjusted to first-principles lattice QCD calculations. The relation between pressure and energy density from lattice QCD is surprisingly insensitive to details of the simulations. Effects from different lattice actions, quark masses, and lattice spacings used in the simulations show up mostly in the quark-hadron phase transition region, which we bridge over by a set of interpolations to a hadron resonance gas equation of state. Within our optimized quasiparticle model we then examine the equation of state along isentropic expansion trajectories at small net baryon densities, as relevant for experiments and hydrodynamic simulations at RHIC and LHC energies. We illustrate its impact on azimuthal flow anisotropies and on the transverse momentum spectra of various hadron species.

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Received 6 May 2007

DOI:https://doi.org/10.1103/PhysRevC.76.034901

Authors & Affiliations

M. Bluhm¹, B. Kämpfer^1,2, R. Schulze², D. Seipt², and U. Heinz³

¹Forschungszentrum Dresden-Rossendorf, PF 510119, D-01314 Dresden, Germany
²Institut für Theoretische Physik, TU Dresden, D-01062 Dresden, Germany
³Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA

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Vol. 76, Iss. 3 — September 2007

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Figure 1
Comparison of the QPM with lattice QCD results (symbols) for the scaled pressure $p / T^{4}$ as a function of $T / T_{c}$ for $N_{f} = 2$ and $μ_{q} = 0$ . The raw lattice QCD data from Ref. [17] have been continuum extrapolated as described in the text. The QPM parameters are $λ = 4.4, T_{s} = 0.67 T_{c}, b = 344.4,$ and $B (T_{c}) = 0.31 T_{c}^{4}$ , with $T_{c} = 175$ MeV as suggested in Ref. [39]. The horizontal line indicates the Stefan-Boltzmann value $p_{SB} / T^{4} = {\bar{c}}_{0} = (32 + 21 N_{f}) π^{2} / 180$ for $N_{f} = 2$ .Reuse & Permissions
Figure 2
Scaled baryon density $n_{B} / T^{3}$ (upper panel) and baryon number susceptibility $χ_{B} / T^{2}$ (lower panel) as a function of $T / T_{c}$ , for $μ_{B} / T_{c} =$ 2.4, 1.8, 1.2, and 0.6 (from top to bottom). QPM results from the truncated expansions (13) and (14) (solid lines) are compared with lattice QCD data (symbols) from ref. [41] for $N_{f} = 2$ . Dashed lines in the upper panel represent the full QPM result (7) for $n_{B} = n_{q} / 3$ . The QPM parameters are $λ = 12.0, T_{s} = 0.87 T_{c}$ , and $b = 426.05$ , for $T_{c} = 175$ MeV.Reuse & Permissions
Figure 3
Comparison of the Taylor series expansion coefficients for $e_{n} (T)$ (squares/dashed black lines) and $s_{n} (T)$ (circles/solid red lines) for $N_{f} = 2$ from Ref. [39] with the QPM (same parameters as in Fig. 2). Left panel: $n = 2$ . Right panel: $n = 4$ . For details see text.Reuse & Permissions
Figure 4
Comparison of the QPM result for $(χ_{uu} + χ_{ud}) / T^{2}$ (solid line) with lattice QCD data (circles) from Ref. [48] for $N_{f} = 2$ , extrapolated to the continuum as suggested in Ref. [49]. The QPM parameters are $λ = 7.0, T_{s} = 0.76 T_{c}$ , and $b = 431$ , with $T_{c} = 175$ MeV. For comparison, we also show lattice QCD data for $c_{2} (T)$ for $N_{f} = 2$ from Ref. [41] (squares) together with the corresponding QPM fit (dashed line), using the same parameters as in Fig. 2.Reuse & Permissions
Figure 5
Top panel: $c_{0} (T) = p (T, μ_{B} = 0) / T^{4}$ as a function of $T / T_{c}$ for $N_{f} = 2$ . Raw lattice QCD data from Ref. [17] (squares) and guesses for the continuum extrapolated data obtained by multiplying (for $T ⩾ T_{c} = 175$ MeV) by a factor $d = 1.1$ (circles) and $d = 1.25$ (triangles) [17, 40] are shown together with the corresponding QPM fits (dashed-dotted, dashed, and solid curves, respectively). The QPM parameters read $B (T_{c}) = 0.31 T_{c}^{4}, b = 344.4, λ = 2.7$ , and $T_{s} = 0.46 T_{c}$ for the dashed-dotted line (“fit 1”); they are the same as in Fig. 1 for the dashed line (“fit 2”) and the same as in Fig. 2 [with $B (T_{c}) = 0.61 T_{c}^{4}$ ] for the solid line (“fit 3”). Bottom panel: Corresponding QPM results compared with lattice results for $c_{2} (T)$ (squares) and $c_{4} (T)$ (circles) as a function of $T / T_{c}$ with the same line code as in the top panel. The horizontal lines indicate the Stefan-Boltzmann values.Reuse & Permissions
Figure 6
Isentropic evolutionary paths. Triangles and circles indicate $N_{f} = 2$ lattice QCD data from Ref. [39] for $s / n_{B} =$ 300 and 45, respectively. Corresponding QPM results are depicted in the left panel for a mixed fit where $c_{0} (T)$ and $c_{2} (T)$ were fitted independently (cf. Figs. 1 and 2). In the right panel we show results from “fit 3” from Fig. 5, with open squares indicating the corresponding continuum-extrapolated lattice results where the raw $c_{0} (T)$ lattice data were multiplied by a constant factor $d = 1.25$ at $T ⩾ T_{c}$ [17]. Full red squares show chemical freeze-out points deduced in Refs. [52, 53] from hadron multiplicity data, as summarized in Ref. [54].Reuse & Permissions
Figure 7
Lattice QCD data [39] of $p$ as a function of $e$ for $N_{f} = 2$ along isentropes with $s / n_{B} = 300$ (triangles) and 45 (circles), compared with the corresponding QPM results (solid blue and dashed black lines, respectively). These two thick lines employ the mixed fit shown in the left panel of Fig. 6 and are indistinguishable for $s / n_{B} = 300$ and 45. The thin solid lines show corresponding results for the self-consistent “fit 3” from Fig. 5. Again the curves for different $s / n_{B}$ are indistinguishable, and also the deviations from the mixed fit are minor.Reuse & Permissions
Figure 8
Comparison of the QPM with lattice QCD results (symbols) for the scaled pressure $p / T^{4}$ (top panel) and the scaled entropy density $s / T^{3}$ (bottom panel) as a function of $T / T_{c}$ for $N_{f} = 2 + 1$ and $μ_{B} = 0$ . The lattice QCD data [63] are already continuum extrapolated. The QPM parameters read $λ = 7.6, T_{s} = 0.8 T_{c}, b = 348.72,$ and $B (T_{c}) = 0.52 T_{c}^{4},$ where $T_{c} = 170$ MeV. In the top panel, the horizontal line indicates the Stefan-Boltzmann value $p_{SB} / T^{4} = {\bar{c}}_{0} = (32 + 21 N_{f}) π^{2} / 180$ , using $N_{f} = 2.5$ to account for the nonzero strange quark mass.Reuse & Permissions
Figure 9
Top panel: $N_{f} = 2 + 1$ QPM equation of state of strongly interacting matter for vanishing net baryon density (solid line) compared with $N_{f} = 2 + 1$ continuum-extrapolated lattice QCD data [63] (squares) at $n_{B} = 0$ . The dotted line represents $p (e)$ for a gas of massless noninteracting quarks and gluons with a bag constant $B^{1 / 4} = 230$ MeV. Bottom panel: QPM EoS for $N_{f} = 2$ (dashed line) employing “fit 2” in Figs. 1 and 5, compared with lattice data [63] (squares) and QPM results (solid line) for $N_{f} = 2 + 1$ , in logarithmic representation.Reuse & Permissions
Figure 10
Baryon number density dependence of the EoS $p (e, n_{B})$ at constant energy density $e$ as indicated. The curves end where the solution of the flow equation (8) is no longer unique.Reuse & Permissions
Figure 11
Upper left panel: QPM EoS for $N_{f} = 2 + 1$ (solid red) and its chiral extrapolation to physical quark masses (dashed blue). Squares show LQCD data for $N_{f} = 2 + 1$ quark flavors with unphysical masses [63]. Upper right panel: Comparison of the squared speed of sound, $c_{s}^{2} = \partial p / \partial e,$ as a function of $T / T_{c}$ from the QPM with lattice QCD data [69] (diamonds and triangles) deduced from the $N_{f} = 2$ data for $p (e)$ in Ref. [39]. Differences between the QPM fit to the LQCD data (solid red) and its extrapolation to physical quark masses (dashed blue) for $N_{f} = 2 + 1$ are almost invisible. Bottom panel: Same as upper right panel, but plotted as a function of energy density $e$ . In all three panels the solid green line shows the hadron resonance gas model EoS “aa1” from Ref. [68].Reuse & Permissions
Figure 12
Dependence of the EoS for $N_{f} = 2 + 1$ on the chosen value of the physical scale $T_{c}$ . Dashed, solid, and dash-dotted curves correspond to $T_{c} = 160, 170,$ and 180 MeV, respectively. Lattice data (squares) are from Ref. [63].Reuse & Permissions
Figure 13
Dependence of the EoS for $N_{f} = 2$ on the employed continuum extrapolation as performed in Fig. 5. Dash-dotted, dashed, and solid curves correspond to the QPM parametrizations of the raw lattice QCD data [17] and continuum extrapolations of these data by a factor $d = 1.1$ and $d = 1.25$ , respectively.Reuse & Permissions
Figure 14
Stability of the QPM EoS fitted to lattice QCD results for $N_{f} = 2 + 1$ . Top panel: The scaled pressure $p (T) / T^{4}$ at $μ_{B} = 0$ from different lattice QCD calculations (Ref. [63] (squares), Ref. [70] (diamonds and triangles), and Ref. [71] (circles)), together with corresponding QPM fits (solid, long-dashed and dash-dotted, and short-dashed lines, respectively). The fit parameters are optimized separately in each case, keeping, however, $B (T_{c}) = 0.52 T_{c}^{4}$ with $T_{c} = 170$ MeV in all four parametrizations fixed. Bottom panel: The EoS $p (e, n_{B} = 0)$ corresponding to the data and fits shown in the top panel.Reuse & Permissions
Figure 15
A family of equations of state for $N_{f} = 2 + 1$ , combining our QPM at high energy densities with a hadron resonance gas model (“res. gas”) in the low-energy-density regime through linear interpolation. We show the range of energy densities relevant for collisions at RHIC. The solid lines show $p (e)$ for QPM(4.0), QPM(2.0), QPM(1.25), and QPM(1.0) (from top to bottom), where the numerical label indicates the matching point $e_{m}$ in GeV/fm $^{3}$ . On the given scale, effects of varying $n_{B}$ between 0 and 0.5 fm $^{- 3}$ are not visible. Lattice QCD data (squares) are from Ref. [63]. For comparison a bag model EoS (“bag”) with a sharp first-order phase transition is also shown (dashed line). The bottom panel zooms in on the transition region, using a linear energy density scale.Reuse & Permissions
Figure 16
Squared speed of sound $c_{s}^{2} = \partial p / \partial e$ as a function of energy density $e$ along an isentropic expansion trajectory with $s / n_{B} = 100$ , for the EoS family QPM( $e_{m}$ ) depicted in Fig. 15. Baryon density effects are not visible on the given scale as long as $n_{B} < 0.5$ fm $^{- 3}$ .Reuse & Permissions
Figure 17
Transverse momentum spectra (top) and elliptic flow coefficient (bottom) for directly emitted strange baryons. The symbols represent STAR data [78] (see text for details). Solid and dashed curves are for EoS QPM(4.0) and the bag model EoS, respectively.Reuse & Permissions
Figure 18
Transverse momentum spectra (top and third panels) and azimuthal anisotropy (second and bottom panels) for pions, kaons, and protons (top two panels) and strange baryons (bottom two panels). Initial conditions are according to Eq. (22). The spectra show only directly emitted hadrons. Solid and dashed curves are for EoS QPM(4.0) and the bag model EoS being similar to QPM(1.0), respectively.Reuse & Permissions

Physical Review C

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