Bottomonium production in heavy-ion collisions using quantum trajectories: Differential observables and momentum anisotropy

We report predictions for the suppression and elliptic flow of the $\mathrm{\Upsilon }(1S)$, $\mathrm{\Upsilon }(2S)$, and $\mathrm{\Upsilon }(3S)$ as a function of centrality and transverse momentum in ultrarelativistic heavy-ion collisions. We obtain our predictions by numerically solving a Lindblad equation for the evolution of the heavy-quarkonium reduced density matrix derived using potential nonrelativistic QCD and the formalism of open quantum systems. To numerically solve the Lindblad equation, we make use of a stochastic unraveling called the quantum trajectories algorithm. This unraveling allows us to solve the Lindblad evolution equation efficiently on large lattices with no angular momentum cutoff. The resulting evolution describes the full 3D quantum and non-Abelian evolution of the reduced density matrix for bottomonium states. We expand upon our previous work by treating differential observables and elliptic flow; this is made possible by a newly implemented Monte Carlo sampling of physical trajectories. Our final results are compared to experimental data collected in $\sqrt{s_{NN}}=5.02\mathrm{TeV}$ Pb-Pb collisions by the ALICE, ATLAS, and CMS collaborations.

Figure 1

Nuclear modification factor $R_{AA}$ of the $\mathrm{\Upsilon }(1S)$, $\mathrm{\Upsilon }(2S)$, and $\mathrm{\Upsilon }(3S)$ as a function of $N_{\mathrm{part}}$ compared to experimental measurements from the ALICE [69], ATLAS [71], and CMS [67] collaborations. The bands in the theoretical curves indicate variation with respect to $\widehat{\kappa }(T)$ (left) and $\widehat{\gamma }$ (right). The central curves represent the central values of $\widehat{\kappa }(T)$ and $\widehat{\gamma }$, and the dashed and dot-dashed lines represent the lower and upper values, respectively, of $\widehat{\kappa }(T)$ and $\widehat{\gamma }$.

Figure 2

Nuclear modification factor $R_{AA}$ of the $\mathrm{\Upsilon }(1S)$, $\mathrm{\Upsilon }(2S)$, and $\mathrm{\Upsilon }(3S)$ as a function of $p_T$ compared to experimental measurements. The experimental data are taken from the ALICE [69], ATLAS [71], and CMS [67] collaborations. The bands represent theoretical uncertainties as in Fig. 1.

Figure 3

Double ratio of the nuclear modification factor $R_{AA}[\mathrm{\Upsilon }(2S)]$ to $R_{AA}[\mathrm{\Upsilon }(1S)]$ as a function of $N_{\mathrm{part}}$ compared to experimental measurements of the ALICE [69], ATLAS [71], and CMS [72] collaborations. The bands in the theoretical curves indicate variation of $\widehat{\kappa }(T)$ and $\widehat{\gamma }$ as in Fig. 1. The black and red bars in the experimental data represent statistical and systematic uncertainties, respectively.

Figure 4

Double ratio of the nuclear modification factor $R_{AA}[\mathrm{\Upsilon }(3S)]$ to $R_{AA}[\mathrm{\Upsilon }(1S)]$ as a function of $N_{\mathrm{part}}$ compared to experimental measurements of the ATLAS [71] and CMS [72] collaborations. The bands and bars represent uncertainties as in Fig. 3; we note that the CMS measurements give only an upper bound at 95% confidence level.

Figure 5

Double ratio of the nuclear modification factor $R_{AA}[\mathrm{\Upsilon }(2S)]$ to $R_{AA}[\mathrm{\Upsilon }(1S)]$ as a function of $p_T$ compared to experimental measurements of the ATLAS [71] and CMS [72] collaborations. The bands and bars represent uncertainties as in Fig. 3.

Figure 6

Elliptic flow $v_2$ of the $\mathrm{\Upsilon }(1S)$ as a function of centrality compared to experimental measurements of the CMS [73] collaboration. The bands represent uncertainties as in Fig. 1.

Figure 7

Elliptic flow $v_2$ of the $\mathrm{\Upsilon }(1S)$ as a function of $p_T$ compared to experimental measurements of the ALICE [70] and CMS [73] collaborations. The bands represent uncertainties as in Fig. 3. Note the much larger range of the ordinate compared to Fig. 6.

Figure 8

Elliptic flow $v_2$ of the $\mathrm{\Upsilon }(2S)$ and $\mathrm{\Upsilon }(3S)$ as a function of centrality compared to experimental measurements of the CMS [73] collaboration. The bands represent uncertainties as in Fig. 1.

Figure 9

Survival probability of the $\mathrm{\Upsilon }(1S)$, $\mathrm{\Upsilon }(2S)$, and $\mathrm{\Upsilon }(3S)$ as a function of $N_{\mathrm{part}}$ obtained from evolution with only $H_{\mathrm{eff}}$, i.e., without applying the jump operators. Solid lines indicate results obtained by sampling approximately ${10}^6$ distinct physical trajectories; dashed lines indicate results obtained using an averaged physical trajectory as in Ref. [24]. The simulations were carried out using $\widehat{\kappa }(T)={\widehat{\kappa }}_C(T)$ and $\widehat{\gamma }(T)=-1.75$. We note that the statistical errors associated with the sampling of distinct physical trajectories are indeed shown in the figure but are of the order of the line width.

Figure 10

Nuclear modification factor $R_{AA}$ of the $\mathrm{\Upsilon }(2S)$ as a function of $p_T$ together with experimental measurements of $R_{AA}[\mathrm{\Upsilon }(2S)]$ from the ATLAS [71] and CMS [67] collaborations. We compare the results obtained using the full qtraj algorithm (red, solid) with results obtained using evolution with $H_{\mathrm{eff}}$ with no jumps (purple, dashed). The top row varies $\widehat{\kappa }(T)$ at $\widehat{\gamma }=-1.75$, and the bottom row varies $\widehat{\gamma }$ at ${\widehat{\kappa }}_C(T)$.

Figure 11

Double ratio of the nuclear modification factor $R_{AA}[\mathrm{\Upsilon }(2S)]$ to $R_{AA}[\mathrm{\Upsilon }(1S)]$ as a function of $p_T$ computed using qtraj plotted against experimental measurements of $R_{AA}[\mathrm{\Upsilon }(1S)]$ and $R_{AA}[\mathrm{\Upsilon }(2S)]$ from the ATLAS [71] and CMS [72] collaborations. We compare the results obtained using the full qtraj algorithm (blue) with results obtained using evolution with $H_{\mathrm{eff}}$ with no jumps (orange). The bands represent uncertainties as in Fig. 1.

Figure 12

Elliptic flow $v_2$ of the $\mathrm{\Upsilon }(1S)$ as a function of centrality computed using qtraj plotted against experimental measurements of the CMS collaboration [73]. We compare the results obtained using the full qtraj algorithm (blue) with results obtained using evolution with $H_{\mathrm{eff}}$ with no jumps (orange). The parameter variation is as in Fig. 10.