Feasibility of measuring the speed of sound of the quark-gluon plasma from the multiplicity and mean $p_T$ of ultracentral heavy-ion collisions

TL;DR

This study questions the use of the ultracentral-scale exponent $b_{\rm UC} = \frac{d\ln\langle p_{T,\text{ch}}\rangle}{d\ln N_{\rm ch}}$ as a direct measure of the QCD speed of sound. By performing boost-invariant hydrodynamic simulations with the lattice QCD equation of state and analyzing how hypersurface energy $E$, entropy $S$, effective temperature $T_{\rm eff}$, and effective volume $V_{\rm eff}$ map to observables, they show that under the key assumption of a constant $V_{\rm eff}$ one would obtain $b_{\rm UC} = P(T_{\rm eff})/\varepsilon(T_{\rm eff})$, not $c_s^2$. However, realistic factors—nonconserved longitudinal energy/entropy windows, $N_{\rm ch} \propto S$, imperfect proxies $\langle p_T\rangle \npropto T_{\rm eff}$, and the nontrivial dependence of $V_{\rm eff}$ on the EOS—mean that the observed $b_{\rm UC}$ can deviate from both $c_s^2$ and $P/\varepsilon$, sometimes aligning with $c_s^2$ only in narrow temperature ranges or under contrived conditions. Varying the EOS further amplifies these deviations, highlighting that $b_{\rm UC}$ should not be treated as a universal thermodynamic thermometer, but rather as a correlated observable that requires careful, EOS-aware interpretation via numerical simulations constrained by data.

Abstract

The mean transverse momentum $\langle p_T \rangle$ of hadrons has been observed experimentally and in numerical simulations to have a power-law dependence on the hadronic multiplicity $N$ in ultracentral relativistic heavy-ion collisions: $\langle p_{T} \rangle \propto N^{b_{\rm UC}}$. It has been put forward that this exponent $b_{\rm UC}$ is the speed of sound of quark-gluon plasma measured at a temperature determined from $\langle p_T \rangle$. We study step by step the connection between (i) the energy and entropy of hydrodynamic simulations and (ii) experimentally measurable observables. We show that an argument based on energy and entropy should yield an exponent equal to the pressure over energy density $P/\varepsilon$, rather than the speed of sound $c_s^2$; however, we also observe that $\langle p_T \rangle$ and $N$ are not sufficiently accurate proxies for the energy and entropy to make this possible in practice. From simulations, we find that the exponent $b_{\rm UC}$ is significantly different whether the ''effective volume'' is strictly constant or not, a condition that cannot be enforced experimentally. Additional tests using a modified equation of state find that the exponent $b_{\rm UC}$ exhibits a variable degree of correlations with the speed of sound and with $P/\varepsilon$, but is not an accurate measurement of either quantity in general.

Figures (4)

• Figure 1: The energy over entropy ratio $E/S$ and the effective temperature $T_{\rm eff}$ computed on a 130 MeV constant temperature hypersurface, as a function of mean transverse momentum of charged hadrons for a QCD equation of state. The dashed black line is a straight line added as a reference.
• Figure 2: Accumulation of effects that appears to coincidentally lead from $P/\varepsilon$ to $c_s^2$ for temperatures between 200 and 250 MeV. Panel a) Assuming $V_\text{eff}$ constant, the derivative $d\ln(E/S)/d\ln S$ computed from the energy $E$ and entropy $S$ on constant-temperature hypersurfaces leads to $P/\varepsilon$, and not $c_s^2$. Panel b) When we replace the hydrodynamic energy on the hypersurface with the Cooper-Frye particle energy with midrapidity cuts, a first error is generated, which scatters the point above $P/\varepsilon$. Panel c) When we replace the mean energy $E/S$ with $\langle p_T \rangle$, the points are scattered even more randomly above $P/\varepsilon$. Panel d) When we finally relax the assumption that $V_\text{eff}=\text{const}$ (which was a central ingredient in the original derivation by Gardim:2019xjsGardim:2019brr), and fix instead the width $\sigma$ of the initial profile, the points rearrange on a curve that happens to fall on top of the $c_s^2$-curve between temperatures 200 and 250 MeV.
• Figure 3: Same as Fig \ref{['fig']} but with a modified equation of state for temperatures $T>180$ MeV, see Eq. \ref{['eq:cs2-modif']}. The systematics for panels a), b) and c) are comparable to those seen earlier, while observable $b_{\rm UC}$ in panel d) shows much larger deviations from the speed of sound $c_s^2$.
• Figure 4: The $\langle p_T \rangle /T_{\rm eff}$ ratio as a function of charged hadron multiplicity for QCD equation of state (left-hand panel) and the modified equation of state (right-hand panel).