Specific heat of matter formed in relativistic nuclear collisions

We report the excitation energy dependence of specific heat ($c_v$) of hadronic matter at freeze-out in Au+Au and Cu+Cu collisions at the BNL Relativistic Heavy Ion Collider energies by analyzing the published data on event-by-event mean transverse momentum ($\langle p_T\rangle$) distributions. The $\langle p_T\rangle$ distributions in finite $p_T$ ranges are converted to distributions of effective temperatures, and dynamical fluctuations in temperature are extracted by subtracting widths of the corresponding mixed event distributions. The heat capacity per particle at the kinetic freeze-out surface is presented as a function of collision energy, which shows a sharp rise in $c_v$ below $\sqrt{s_{NN}}=62.4$ GeV. We employ the hadron resonance gas (HRG) model to estimate $c_v$ at the chemical and kinetic freeze-out surfaces. The experimental results are compared to the HRG and other theoretical model calculations. HRG results show good agreement with data. Model predictions for $c_v$ at the CERN Large Hadron Collider energy are presented.

Figure 1

Event-by-event $T_{\mathrm{eff}}$ distributions of pions for central Pb+Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV from the AMPT model within rapidity range of $-1.0$ to 1.0 and $0.15<p_T<2.0$ GeV. $T_{\mathrm{eff}}$ distributions, obtained by fitting the $p_T$ distribution of each event and from the $\langle p_T\rangle$ are presented.

Figure 2

Chemical and kinetic freeze-out temperatures for central Au+Au [44] and Cu+Cu [29] collisions at RHIC energies, and Pb+Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV [45]. Thermal model calculation [46] to $T_{\mathrm{ch}}$ is also shown.

Figure 3

Specific heat, $c_v$ as a function of collision energy from the HRG model for two temperature settings, $T_{\mathrm{ch}}$ with finite ${\mu }_{\mathrm{B}}$, and $T_{\mathrm{kin}}$, and two phase space volumes, $\mathrm{\Delta }=V{T}^3$ and $\langle N\rangle$. Lattice calculation for $c_v$ at $T=154\pm 9$ MeV [15] and Stefan-Boltzmann limit are indicated in the figure.

Figure 4

Left panels show event-by-event mean transverse momentum distributions for 5% most central Au+Au collisions at $\sqrt{s_{NN}}=20$, 62.4, 130, and 200 GeV within $\left|\eta \right|<1$ and $0.15<p_T<2.0$ GeV [28]. Distributions for mixed events are superimposed on the data. The solid and dashed lines show the fits with $\mathrm{\Gamma }$ functions. The right panels show the extracted $T_{\mathrm{eff}}$ distributions for each incident energy.

Figure 5

Similar distributions as in Fig. 4 for 10% most central Cu+Cu collisions at $\sqrt{s_{NN}}=62.4$ and 200 GeV [29].

Figure 6

Specific heat, $c_v$ as a function of collision energy for central Au+Au and Cu+Cu collisions at RHIC energies. Result from AMPT model is given for the LHC energy at $\sqrt{s_{NN}}=2.76$ TeV. HRG calculations at $T_{\mathrm{kin}}$ are shown in the figure. Model calculations for three different scenarios from Ref. [11] are superimposed on the experimental results.