Constraining the equation of state with identified particle spectra

We show that in a central nucleus-nucleus collision, the variation of the mean transverse mass with the multiplicity is determined, up to a rescaling, by the variation of the energy over entropy ratio as a function of the entropy density, thus providing a direct link between experimental data and the equation of state. Each colliding energy thus probes the equation of state at an effective entropy density, whose approximate value is 19 ${\mathrm{fm}}^{-3}$ for Au+Au collisions at 200 GeV and 41 ${\mathrm{fm}}^{-3}$ for Pb+Pb collisions at 2.76 TeV, corresponding to temperatures of 227 and 279 MeV if the equation of state is taken from lattice calculations. The relative change of the mean transverse mass as a function of the colliding energy gives a direct measure of the pressure over energy density ratio $P/\epsilon$, at the corresponding effective density. Using Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) data, we obtain $P/\epsilon =0.21\pm 0.10$, in agreement with the lattice value $P/\epsilon =0.23$ in the corresponding temperature range. Measurements over a wide range of colliding energies using a single detector with good particle identification would help reduce the error.

Figure 1

The pressure $P$ normalized by $T^4$ vs the temperature $T$. The curves correspond to various parametrizations obtained by varying the number of degrees of freedom (a), or the transition temperature (b). The solid line in both panels, labeled “L,” corresponds to the lattice result [9].

Figure 2

Energy over entropy vs entropy density for the equations of state shown in Fig. 1. Symbols correspond to Eq. (4), where $\langle m_T\rangle$ and $dN/dy$ are evaluated at $T_f=140$ MeV in ideal hydrodynamics, before resonance decays.

Figure 3

Comparison of results from various hydrodynamic calculations. From top to bottom: ideal hydrodynamics before and after decays, and viscous hydrodynamics without and with viscous correction at freeze-out.

Figure 4

Same as Fig. 2 after resonance decays.

Figure 5

Same as Fig. 4 with shear and bulk viscosity included.

Figure 6

Experimental data from Table 1, compared to value given by various equations of state and Eqs. (4), where $a$ and $b$ are taken from our viscous hydrodynamic calculation, and $R_0$ from a Glauber model (see text).

Figure 7

The trace anomaly normalized by $T^4$ vs the temperature $T$. The curves correspond to various parameterizations obtained by varying the number of degrees of freedom (a) or the transition temperature (b).

Figure 8

Transverse momentum distributions of identified particles in Pb+Pb collisions at the LHC and Au+Au collisions at RHIC. The centrality range is 0–5% for all sets of data except 130-GeV data, which are 0–6%. Symbols are data from ALICE [75], PHENIX [74], and STAR [73, 76]. Solid lines are blast-wave fits (see text). Each panel corresponds to a different particle species: positive (a) and negative (b) pions, positive (c) and negative (d) kaons, protons (e), and antiprotons (f). STAR data for protons and antiprotons also include secondary products of $\mathrm{\Lambda }$ and $\overline{\mathrm{\Lambda }}$ decays, which explain the larger values. Experimental errors are not shown for sake of readability, but they are taken into account in the fits.