Non-Markovian dynamics of bottomonia in the QGP

TL;DR

This work develops and solves a general non-Markovian master equation for bottomonium evolution in the QGP within the open quantum systems framework, emphasizing the competition between the bottomonium intrinsic time scale $τ_S\sim 1/E_b$ and the environmental time scale $τ_E\sim 1/T$. Using pNRQCD, singlet–octet dynamics and dipole transitions are encoded via transition operators $T_n(l\to l',t)$, with memory kernels derived from chromoelectric correlators $Γ(t,s)$. A stochastic unraveling approach (neglecting jumps for Υ(1S)) is employed to numerically propagate the system on a discretized radial grid across static, Bjorken, and realistic hydrodynamic backgrounds, exploring $ξ_E=τ_E T$ values in {1/1.5, 1, 1.5} and κ̂=4.0, γ̂=0. The results show that non-Markovian memory significantly mitigates suppression compared to LO Lindblad predictions, with NLO Lindblad capturing much of this reduction at higher temperatures; in realistic heavy-ion collisions the model can describe LHC Υ(1S) data for reasonable $ξ_E$, but underpredicts RHIC suppression, indicating missing physics such as excited-state feed-down and cold nuclear matter effects. The work highlights κ as a key input and motivates future extensions to include jumps and excited bottomonium states for a complete phenomenology.

Abstract

The evolution of quarkonia in the QGP medium can be described through the formalism of Open Quantum Systems (OQS). In previous works with OQS, the quarkonium evolution was studied by working in either quantum Brownian or optical regime. In this paper, we set up a general non-Markovian master equation to describe the evolution of the quarkonia in the medium. Due to the non-Markovian nature, it cannot be cast in Lindblad form which makes it challenging to solve. We numerically solve the master equation for $Υ(1S)$ without considering stochastic jumps for Bjorken and viscous hydrodynamic background at energies relevant to LHC and RHIC. We quantify the effect of the hierarchy between the system time scale, $τ_{\textrm{S}}\sim 1/E_b$, and the environment time scale, $τ_{\textrm{E}}$, on quarkonium evolution and show that it significantly affects the nuclear modification factor. A comparison of our results with the existing experimental data from LHC and RHIC is presented.

Figures (7)

• Figure 1: (Colour online) The time dependence of the survival probabilities of $\Upsilon(1S)$ in a static medium at a temperature of $350$ MeV. The three solid curves correspond to the calculations using the master equation Eq. \ref{['eq:ang']}. The dotted curve (black online) corresponds to LO Brambilla:2020qwo and the dot-dashed curve (black online) to NLO Brambilla:2022ynh.
• Figure 2: $\Upsilon(1S)$ survival probabilities as a function of time in Bjorken expanding media with $(t_0,~T(t_0))$ taken to be $(0.6~\textrm{fm},~480~\textrm{MeV})$ (left panel) and $(1.0~\textrm{fm},~333~\textrm{MeV})$ (right panel). The vertical line corresponds to the time at which the temperature 190 MeV, which corresponds to the $T_\textrm{min}$ used for NLO simulations in Brambilla:2022ynh. The choice of the style of curves for the curves is the same as Fig. \ref{['fig:constantT']}.
• Figure 3: Final survival probability as a function of $\langle N_\textrm{part}\rangle$ for $\Upsilon(1S)$ with a viscous hydrodynamic background for Pb-Pb collisions at $5.02$ TeV (left) and Au-Au collisions at $200$ GeV (right). In each plot, three sets of results corresponding to $\xi_{{\rm{E}}}=\{1.5,\;1,\;1/1.5\}$ are shown which were obtained from the master equation Eq. \ref{['eq:ang']}. The error bars correspond to the statistical errors from the sampling of quarkonium trajectories.
• Figure 4: $R_\textrm{AA}$ as a function of $\langle N_\textrm{part}\rangle$ for $\Upsilon(1S)$ in Pb-Pb collisions at 5.02 TeV (left) and Au-Au collisions at 200 GeV energy (right) after estimating the feed-down contributions as discussed in the text. Experimental data from CMS CMS:2019270 and STAR STAR:2022rpk is plotted for comparison. Data from ATLAS ATLAS:2022exb at $5.02$ TeV is consistent with the CMS data and are not shown here for cleaner plots.
• Figure 5: Static medium at $T=250$ MeV (left) and $T=450$ MeV (right).