Data-Driven Analysis for the Bottomonium Potential in the Quark-Gluon Plasma

TL;DR

This work tackles the problem of extracting the in-medium bottomonium potential in the quark-gluon plasma by marrying a nonrelativistic, complex $V(r,T)$ with a data-driven Bayesian framework. The authors solve the time-dependent Schrödinger equation for $b\bar b$ dipoles in a hydrodynamic QGP background, relate the potential parameters to observables $R_{AA}^{n\ell}$ and $v_2^{n\ell}$ through a detailed production and feed-down model, and use Latin Hypercube sampling, PCA, Gaussian Process emulation, and Hamiltonian Monte Carlo to infer the five-parameter potential $\{a_m,f_T,f_1,f_2,f_3\}$. A closure test confirms the framework’s ability to recover input potentials, while application to LHC Pb-Pb data reveals a bimodal posterior for $a_m$ at $T_d=160~\mathrm{MeV}$ (two physically distinct scenarios) that becomes monomodal at higher switching temperatures, suggesting that excited-state measurements are key to breaking degeneracies. The results underscore the potential’s sensitivity to the balance between real-screening and color-octet transitions and highlight the role of $T_d$ in shaping the inferred in-medium dynamics, with implications for future experiments and lattice comparisons.

Abstract

We present a data-driven analysis within a quantum evolutionary microscopic framework to constrain the in-medium bottomonium potential. In relativistic heavy-ion collisions, bottomonium bound states serve as invaluable probes of the quark-gluon plasma (QGP) owing to their negligible production in the QGP phase. Meanwhile, their non-relativistic nature allows a straightforward theoretical description via effective field theories such as potential models. Recent lattice QCD calculations of the bottomonium interaction potential have yielded qualitatively distinct results. These discrepancies motivate a data-driven extraction of the potential based on heavy-ion experiments. In this work, we perform a Bayesian analysis to constrain the bottomonium interaction potential. The relationship between potential parameters and observables is established by numerically solving the non-relativistic time-dependent Schr"odinger equation. By comparing these simulations with experimental measurements, our Bayesian framework provides the effective potential that is readily testable in future experiments.

Figures (7)

• Figure 1: The comparisons between the results of closure test and the given analytical potential where $T_d = 160$ MeV, including the real (upper) and the imaginary part (lower) as functions of the radius $r$, where the temperature is chosen as $T=200$ MeV. The $68\%$ and $95\%$ creditable intervals are represented as the lightblue and blue bands, respectively. The black-solid lines represent the mock, analytical potential corresponding to the target median parameter. The parameter for mock test come from the MCMC sampling, where the selection of the emulators is the same as that of the real estimation.
• Figure 2: The posterior distributions of five parameters of Bayesian analysis for the switching temperature $T_d = 160~\mathrm{MeV}$. The histograms on the diagonal are the marginal distributions for each parameter, and the off-diagonal subgraphs correspond to the 2D joint probability distributions of parameter-pairs. Noting the clear bimodal distribution, we show median parameters separately for two peaks, indicated by red and black dash-dot lines, respectively.
• Figure 3: The $68\%$ (darker blue) and $95\%$ (lighter blue) creditable intervals and median values (black lines) of the effective potential as functions of radius $r$ for temperature $T = 200~\mathrm{MeV}$. The two peaks in the bimodal posterior distribution are show separately with filled bands ($a_m < 1.5$) and hatch shading ($a_m \geq 1.5$), respectively.
• Figure 4: Nuclear modification factors of $1S$ (purple), $2\mathrm{S}$ (orange) and $3\mathrm{S}$ (green) states of $\Upsilon$ as functions of centrality(left) and the transverse momentum(right). Conventions for peak separation and posterior distribution is the same as in Fig. \ref{['fig:EffectvePotential_160']}. Experimental data from ALICE ALICE:2018wzm, ATLAS ATLAS:2022exb and CMS CMS:2018zzaCMS:2023lfu Collaborations are also shown from comparison.
• Figure 5: Same as Figure. \ref{['fig:Posterior_160']} but for switching temperatures set to be $T_d = 180~\mathrm{MeV}$ (left) and $T_d = 200~\mathrm{MeV}$ (right), respectively.