The shape of transverse momentum spectra in hybrid hydrodynamic models

Abstract

We study the scaled transverse momentum spectra over a wide parameter space of state-of-the-art hydrodynamic simulation models in order to learn what information can be obtained from the shape of identified-particle spectra -- previously observed to be surprisingly universal across centrality and collision systems in both experimental data and hydrodynamic simulations. We study its sensitivity to each of 17 model parameters in the context of 4 different models for particlization when switching from the hydro description to the kinetic theory afterburner. We find that the strongest sensitivity is to parameters relating to bulk viscosity, free-streaming time, and the $\texttt{T$_\mathrm{R}$ENTo}$ nucleon width parameter $w$. However, we find that the model generally has surprisingly little flexibility in describing the scaled spectrum observable, despite the large number of parameters. Within this small range of parameter dependence, we further find significant tension in a simultaneous description of momentum-integrated observables. In particular, while the mean transverse momentum prefers a large value of the nucleon width parameter $w$, a small value is required to obtain scaled spectra that are consistent with experimental measurements. We speculate on the origin of these model tensions and possible missing physics in the commonly-used $\texttt{T$_\mathrm{R}$ENTo}$+free streaming+hydro+afterburner simulation model.

Figures (9)

• Figure 1: Prior (gray hatched band) and posterior (orange and blue bands) model predictions for the scaled spectra $U(x_T)$ obtained with the Grad viscous correction, compared with ALICE Pb--Pb data at $\sqrt{s_{NN}} = 2.76~\mathrm{TeV}$ for the most central events. The narrow prior bands reflect the intrinsic universality of $U(x_T)$, while the posterior calibration sharpens these predictions, capturing the overall shape but revealing systematic deviations at low and high $x_T$. Experimental data points include combined uncorrelated statistical and systematic uncertainties. The original JETSCAPE posterior JETSCAPE:2020mzn calibrated to $p_T$-integrated observables are displayed by the orange bands. The scaled-spectra posterior, represented by the blue band, is consistent with data but differs systematically from the integrated-observable posterior, indicating a tension between the two calibration strategies and highlighting the sensitivity of $U(x_T)$ to distinct regions of parameter space.
• Figure 2: Posterior predictions (median and 90% credible intervals) from the scaled-spectra calibration using the Grad viscous correction, shown for $p_T$-integrated observables in Pb--Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV. The posterior reproduces the general centrality trends of particle yields, mean transverse momenta, $p_T$ fluctuations, and anisotropic flow coefficients $v_n\{2\}$, but systematic deviations appear in the mean $\langle p_T\rangle$ and its fluctuations. These discrepancies indicate the tension between the calibration to the scaled spectra $U(x_T)$ and the traditional $p_T$-integrated observables. Black triangles represent ALICE experimental data ALICE:2010mlfALICE:2016igkALICE:2013mezALICE:2011abALICE:2014gvd.
• Figure 3: First-order Sobol sensitivity indices for the scaled spectra $U(x_T)$, computed using Gaussian process emulators for the Grad viscous correction. The analysis identifies the free-streaming time scale $\tau_R$, the maximum bulk viscosity $(\zeta/s)_{\rm max}$, the position of the peak of the bulk viscosity $T_{\zeta}$, and the Gaussian nucleon width $w$ as the dominant parameters controlling the overall shape and scaling behavior of the spectra. Their combined influence underscores the role of pre-equilibrium dynamics and initial-state granularity in establishing the observed universality.
• Figure 4: Posterior distributions of the Gaussian nucleon width $w$ from Bayesian calibrations using scaled-spectra data (blue) and $p_T$-integrated observables (orange). The scaled-spectra posterior favors smaller values of $w \sim 0.6~\mathrm{fm}$, corresponding to more granular initial conditions, whereas the $p_T$-integrated calibration prefers smoother initial profiles with $w \sim 1.0~\mathrm{fm}$. This opposing behavior highlights the origin of the tension between the two calibration approaches.
• Figure 5: Model-to-data comparison of the scaled spectra $U(x_T)$ for 30-40% centrality using four different viscous correction prescriptions at particlization: Grad (a), Chapman--Enskog (CE) (b), Pratt--Torrieri--McNelis (PTM) (c), and Pratt--Torrieri--Bernhard (PTB) (d). While all models preserve the overall universality of $U(x_T)$, the PTB implementation exhibits noticeably larger posterior uncertainty and stronger sensitivity to bulk viscous parameters (see Fig. \ref{['fig:GSA_PTB']}), reflecting the nonlinear structure of its exponentiated viscous correction. In contrast, the linear Grad, CE, and PTM corrections display comparable behavior and a shared sensitivity to both the bulk viscosity peak $(\zeta/s)_{\rm max}$ and the free-streaming relaxation time $\tau_R$ (see Appendix section \ref{['appendix:GSA_viscous_corrections']}). These results demonstrate that the universal scaling of $p_T$ spectra is primarily governed by the collective expansion dynamics and remains robust under variations of the particlization scheme.