Extracting the speed of sound of QCD from transverse momentum fluctuations

Abstract

We extract the speed of sound ($c_s$) in the quark-gluon plasma from ATLAS data on the probability distribution of the transverse momentum per particle, $[p_T]$, in ultra-central Pb+Pb collisions. With an ideal detector, $c_s$ can be inferred from the rise of the mean $[p_T]$ with the collision multiplicity. In practice, however, low-$p_T$ particles escape detection, which biases the analysis. We show how to correct for this bias by using data on the variance of $[p_T]$, as well as information from the recently-measured $v_0(p_T)$. We also introduce a systematic method for deblurring the noise from the hadronization process. Assuming that the size of the quark-gluon plasma is independent of the hadron multiplicity in collisions at zero impact parameter, which is the scenario preferred both by high-energy QCD and heavy-ion data, we obtain $c_s/c=0.496\pm 0.008$ at temperature $T=221\pm 13$ MeV, in perfect agreement with first-principles calculations from lattice QCD.

Figures (4)

• Figure 1: Variation of the acceptance factors $C_A$ (full lines) and $D_A$ (dashed lines), defined by Eq. (\ref{['defDA']}), as a function of the lower $p_T$ cut. We indicate by vertical lines the cuts implemented by ALICE ALICE:2025rtg, ATLAS ATLAS:2024jvf and CMS CMS:2024sgx. Our results are obtained from a hydrodynamic simulation with smooth initial conditions Parida:2024ckk. We set the bulk viscosity to zero and compare two values of the shear viscosity over entropy density ratio $\eta/s$.
• Figure 2: Symbols are ATLAS data ATLAS:2024jvf, lines are fits. (a) Probability of the charged-particle multiplicity $N_{ch}$. We only display results for $N_{ch}\gtrsim 3500$, corresponding to 10% of the total number of events. The full line is a fit assuming Gaussian fluctuations at fixed $b$. The dash-dotted line is the distribution of $N_A$ after unfolding Poisson fluctuations (Sec. \ref{['s:geometry']}). (b) Event-averaged transverse momentum per particle $\langle [p_{TA}]\rangle$. (c) Relative variance $k_2\equiv {\rm Var}([p_{TA}])/\langle [p_{TA}]\rangle^2$.
• Figure 3: Comparison between the value of $c_s$ obtained by fitting ATLAS data ATLAS:2024jvf and lattice QCD results by the BMW Borsanyi:2013bia and HotQCD HotQCD:2014kol Collaborations.
• Figure 4: (a) Line: Variation of $\langle[p_{TA}]\rangle$, reconstructed from ATLAS data, in GeV/$c$, as a function of the centrality fraction $c=\pi b^2\sigma_{\rm PbPb}$. For the sake of illustration, we also display values in our initial-state simulation, assuming Eq. (\ref{['effhydro']}), and taking into account the effect of the acceptance cuts. (b) Pearson correlation coefficient between $S$ and $R^2$ at fixed $b$ in our initial-state simulation.