Speed of Sound and Baryon Cumulants in Heavy-Ion Collisions

We present a method that may allow an estimate of the value of the speed of sound as well as its logarithmic derivative with respect to the baryon number density in matter created in heavy-ion collisions. To this end, we use well-known observables: cumulants of the baryon number distribution. In analyses aimed at uncovering the phase diagram of strongly interacting matter, cumulants gather considerable attention as their qualitative behavior along the explored range of collision energies is expected to aid in detecting the QCD critical point. We show that the cumulants may also reveal the behavior of the speed of sound in the temperature and baryon chemical potential plane. We demonstrate the applicability of such estimates within two models of nuclear matter and explore what might be understood from known experimental data.

Figure 1

Model study of regions of applicability of Eqs. (6) and (9). The left (right) panels show results obtained in the VDF (Walecka) model. The upper and lower panels show quantities entering Eq. (6) and Eq. (9), respectively. Results at $T=50$, 100, 150, 200 MeV are given by blue and green solid lines, dark and light purple long-dashed lines, red and pink short-dashed lines, and orange and brown dash-dotted lines, respectively. For each $T$, the thickest lines correspond to the exact results and the medium-thick lines correspond to the approximations given by the right-hand sides of Eqs. (6) and (9). Additionally, in the upper panels the thinnest lines correspond to $c_{\sigma }^2$. Upper panels: for both models, Eq. (6) is valid for $T\lesssim 100\mathrm{MeV}$ and ${\mu }_B\gtrsim 600\mathrm{MeV}$. Lower panels: for both models, Eq. (9) is valid for ${\mu }_B\gtrsim 200\mathrm{MeV}$; the exception is the Walecka model at $T=200\mathrm{MeV}$, where a phase transition to an almost massless gas of nucleons dramatically decreases the applicability of both Eqs. (6) and (9).

Figure 2

Comparison of the right-hand sides of Eq. (6) (upper panel) and Eq. (9) (lower panel) for experimental data (red triangles), ideal gas at the freeze-out (small gray circles), the VDF model at the freeze-out (light green stars), and the VDF model at a set of points chosen to reproduce the data (dark purple stars); exact results, that is, the left-hand sides of Eqs. (6) and (9), are shown for the two cases considered in the VDF model (green and purple circles). The data points for the matched VDF results (shown only for collisions at low energies, where using the model is justified) are chosen to reproduce experimental values of $1-{\kappa }_3{\kappa }_1/{\kappa }_2^2$ (see Fig. 3). We note that at $\sqrt{s}=2.4\mathrm{GeV}$, matching the value of $1-{\kappa }_3{\kappa }_1/{\kappa }_2^2$ exactly is possible but would place the matched point close to the nuclear liquid-gas CP, which we find unlikely.

Figure 3

Contour plot of $1-{\kappa }_3{\kappa }_1/{\kappa }_2^2$ in the VDF model. Yellow and black lines correspond to the spinodal and coexistence lines, respectively; white contours signify regions where $1-{\kappa }_3{\kappa }_1/{\kappa }_2^2=1\pm 0.03$. Light green stars denote experimentally measured freeze-out parameters $(T_{\mathrm{fo}},{\mu }_{\mathrm{fo}})$, while dark purple stars denote points where $1-{\kappa }_3{\kappa }_1/{\kappa }_2^2$, taken along lines informed by average phase diagram trajectories for STAR collision energies [22], matches the experimentally measured values for a given collision energy. The softening of the EOS, leading to negative values of $(d\ln c_T^2/d\ln n_B{)}_T$, occurs in two regions of the phase diagram, corresponding to the ordinary nuclear matter phase transition and to the conjectured QGP-like phase transition.