Heavy quarkonium dynamics at next-to-leading order in the binding energy over temperature

TL;DR

The paper develops a next-to-leading order Lindblad equation for heavy quarkonium dynamics in a quark-gluon plasma within the pNRQCD framework, valid for $Mv\gg T \gg E$ and expanded in $E/T$. By projecting onto a spherical basis and mapping the 3D problem to a 1D radial system, the authors implement a quantum trajectories algorithm to solve the NLO Lindblad equation, including six jump operators and a drag term, and compare ground and excited state dynamics to LO and to LHC data. They compute singlet-octet widths, survival probabilities, and $R_{AA}$ for $\Upsilon(1S)$, $2S$, and $3S$ states, finding improved agreement with data for the ground state and a need to include quantum jumps for excited states in future work. The work extends the applicability of the EFT+OQS approach to lower temperatures, clarifies the role of NLO corrections in state mixing and thermalization, and provides a framework for refined heavy-quarkonium phenomenology in heavy-ion collisions.

Abstract

Using the potential non-relativistic quantum chromodynamics (pNRQCD) effective field theory, we derive a Lindblad equation for the evolution of the heavy-quarkonium reduced density matrix that is accurate to next-to-leading order (NLO) in the ratio of the binding energy of the state to the temperature of the medium. The resulting NLO Lindblad equation can be used to more reliably describe heavy-quarkonium evolution in the quark-gluon plasma at low temperatures compared to the leading-order truncation. For phenomenological application, we numerically solve the resulting NLO Lindblad equation using the quantum trajectories algorithm. To achieve this, we map the solution of the three-dimensional Lindblad equation to the solution of an ensemble of one-dimensional Schrödinger evolutions with Monte-Carlo sampled quantum jumps. Averaging over the Monte-Carlo sampled quantum jumps, we obtain the solution to the NLO Lindblad equation without truncation in the angular momentum quantum number of the states considered. We also consider the evolution of the system using only the complex effective Hamiltonian without stochastic jumps and find that this provides a reliable approximation for the ground state survival probability at LO and NLO. Finally, we make comparisons with our prior leading-order pNRQCD results and experimental data available from the ATLAS, ALICE, and CMS collaborations.

Figures (8)

• Figure 1: The singlet to octet widths of the 1S, 2S, and 3S states including $E/T$ corrections. $\Gamma_{m}(nS)$ represents the singlet to octet width of the state $nS$ including terms up to order $(E/T)^{m}$. The black line at unity represents the limit of perfect convergence of the $E/T$ expansion, i.e., $E/T \to 0$. The plot spans from $T=158$ MeV to $T=600$ MeV; the gray shaded area on the left represents the temperature region $T<T_{f}=190$ MeV not included in the phenomenological results presented in this work.
• Figure 2: The survival probability as a function of $N_{\text{part}}$ of the $1S$ state calculated using $H_{\rm eff}$ evolution without jumps. In the top row, we compare results obtained using LO and NLO evolution both with $T_{f}=190$ MeV to highlight the effect of the inclusion of higher order terms in the $E/T$ expansion. In the bottom row, we compare our current state of the art NLO evolution with $T_{f}=190$ MeV to LO evolution with $T_{f}=250$ MeV as used in previous works. The bands indicate variation with respect to $\hat{\kappa}(T)$ (left) and $\hat{\gamma}$ (right). The central curves represent the central values of $\hat{\kappa}(T)$ and $\hat{\gamma}$, and the dashed and dot-dashed lines represent the lower and upper values, respectively, of $\hat{\kappa}(T)$ and $\hat{\gamma}$.
• Figure 3: Jumps vs. no jumps. Top row is $\hat{\kappa}$ variation; bottom row is $\hat{\gamma}$ variation. Solid, dashed, and dot-dashed curves represent the central, lower, and upper values, respectively, of $\hat{\kappa}$ and $\hat{\gamma}$.
• Figure 4: $R_{AA}$ for the $\Upsilon(1S)$, $\Upsilon(2S)$, and $\Upsilon(3S)$ as a function of $N_{\rm part}$. The left panel shows variation of $\hat{\kappa} \in \{ \kappa_L(T), \kappa_C(T), \kappa_U(T) \}$ and the right panel shows variation of $\hat{\gamma}$ in the range $-3.5 \leq \hat{\gamma} \leq 0$. In both panels, the solid line corresponds to $\hat{\kappa} = \hat{\kappa}_C(T)$ and the best fit value of $\hat{\gamma} = -2.6$. The experimental measurements shown are from the ALICE Acharya:2020kls, ATLAS ATLAS5TeV, and CMS Sirunyan:2018nszCMSupsilonQM2022 collaborations.
• Figure 5: $R_{AA}$ for the $\Upsilon(1S)$, $\Upsilon(2S)$, and $\Upsilon(3S)$ as a function of $p_T$. The bands and experimental data sources are the same as fig. \ref{['fig:raavsnpart1']}.