Temperature-dependent formation-time approach for ${\rm Y}$ suppression at energies available at the CERN Large Hadron Collider

We present here a comprehensive model to describe the bottomonium suppression data obtained from the CERN Large Hadron Collider (LHC) at a center-of-mass energy of $\sqrt{s_{NN}}=2.76$ TeV. We employ a quasiparticle model (QPM) equation of state for the quark-gluon plasma (QGP) expanding under Bjorken's scaling law. The current model includes the modification of the formation time based on the temperature of the QGP, color screening during bottomonium production, gluon-induced dissociation, and collisional damping due to the imaginary part of the potential between the $b\overline{b}$ pair. We propose a method for determining the temperature-dependent formation time of bottomonia using the solution of the time-independent Schrödinger equation and compare it with another approach based on time-dependent Schrödinger wave equation simulation. We find that these two independent methods based on different axioms give similar results for the formation time. Cold nuclear matter effects and feed-down from higher resonance states of ${\rm Y}$ have also been included in the present work. The suppression of the bottomonium states at midrapidity is determined as a function of centrality. The results compare closely with the recent centrality-dependent suppression data at the energies available at the CERN LHC in the midrapidity region.

Figure 1

Formation time given by the intersection of the two curves $t_{\mathrm{qgp}}$ and ${\tau }_f\left[T\left(t_{\mathrm{qgp}}\right)\right]$ for ${\rm Y}\left(1S\right)$ for the most central bin.

Figure 2

Variation of mean separation between the $b\overline{b}$ pair with time for ${\rm Y}\left(1S\right)$ for the most central bin.

Figure 3

Comparison of the ${\rm Y}\left(1S\right)$ formation time obtained using the time-dependent Schrödinger equation (TDS), our method based on the time-independent Schrödinger equation (TIS), and the vacuum formation time.

Figure 4

Comparison of the ${\rm Y}\left(2S\right)$ formation time obtained using the time-dependent Schrödinger equation (TDS), our method based on the time-independent Schrödinger equation (TIS), and the vacuum formation time.

Figure 5

Comparison of the ${\rm Y}\left(1P\right)$ formation time obtained using the time-dependent Schrödinger equation (TDS), our method based on the time-independent Schrödinger equation (TIS), and the vacuum formation time.

Figure 6

Monotonic and smooth nature of shadowing versus atomic mass. The shadowing values shown here are for momentum fraction $=0.000312$.

Figure 7

CMS data [19] compared with simulation results with only color screening for ${\rm Y}\left(1S\right)$.

Figure 8

CMS data [19] compared with simulation results with only color screening for ${\rm Y}\left(2S\right)$.

Figure 9

Survival probability of ${\rm Y}\left(1S\right)$ and ${\rm Y}\left(2S\right)$ versus $N_{\mathrm{part}}$ by incorporating color screening and gluonic dissociation with collisional damping. CNM effects are not incorporated. Temperature-dependent formation time using our proposed method is incorporated while calculating color screening.

Figure 10

Survival probability of ${\rm Y}\left(1S\right)$ and ${\rm Y}\left(2S\right)$ versus $N_{\mathrm{part}}$ by incorporating color screening, gluonic dissociation, and collisional damping along with CNM effects. The CNM calculation used here is based on the Vogt approach [24]. The bands given by the shaded regions on the top and bottom indicate the maximum and minimum range of CNM for ${\rm Y}\left(1S\right)$ and ${\rm Y}\left(2S\right)$, respectively. The temperature-dependent formation time using our proposed method is incorporated while calculating color screening.

Figure 11

Survival probability of ${\rm Y}\left(1S\right)$ and ${\rm Y}\left(2S\right)$ versus $N_{\mathrm{part}}$ by incorporating color screening, gluonic dissociation, and collisional damping along with CNM effects. The formation time has been determined by simulation of the time-dependent Schrödinger equation.

Figure 12

Comparison between the two CNM approaches for ${\rm Y}\left(1S\right)$ and ${\rm Y}\left(2S\right)$. The subscripts 1 and 2 in ${\mathrm{CNM}}_1$ and ${\mathrm{CNM}}_2$ refer to the two CNM approaches. ${\mathrm{CNM}}_1$ is the “Vogt approach” [24], while ${\mathrm{CNM}}_2$ is “our approach.”

Figure 13

Our prediction for the ${\rm Y}\left(1P\right)$ bottomonium state using eps09 and cteq6 after including color screening, gluon dissociation, collisional damping, and the CNM effect.

Figure 14

Comparison between the two CNM approaches for ${\rm Y}\left(1P\right)$. ${\mathrm{CNM}}_1$ is the approach used by Vogt [24]. ${\mathrm{CNM}}_2$ is our approach.