Centrality and transverse momentum dependent suppression of \(\Upsilon (1S)\) and \(\Upsilon (2S)\) in p–Pb and Pb–Pb collisions at the CERN Large Hadron Collider
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Abstract
Deconfined QCD matter in heavyion collisions has been a topic of paramount interest for many years. Quarkonia suppression in heavyion collisions at the relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) experiments indicate the quarkgluon plasma (QGP) formation in such collisions. Recent experiments at LHC have given indications of hot matter effect in asymmetric p–Pb nuclear collisions. Here, we employ a theoretical model to investigate the bottomonium suppression in Pb–Pb at \(\sqrt{s_{NN}}=2.76\), 5.02 TeV, and in p–Pb at \(\sqrt{s_{NN}}=5.02\) TeV centerofmass energies under a QGP formation scenario. Our present formulation is based on an unified model consisting of suppression due to color screening, gluonic dissociation along with the collisional damping. Regeneration due to correlated \(Q\bar{Q}\) pairs has also been taken into account in the current work. We obtain here the net bottomonium suppression in terms of survival probability under the combined effect of suppression plus regeneration in the deconfined QGP medium. We mainly concentrate here on the centrality, \(N_\text {part}\) and transverse momentum, \(p_{T}\) dependence of \(\Upsilon (1S)\) and \( \Upsilon (2S)\) states suppression in Pb–Pb and p–Pb collisions at midrapidity. We compare our model predictions for \(\Upsilon (1S)\) and \(\Upsilon (2S)\) suppression with the corresponding experimental data obtained at the LHC energies. We find that the experimental observations on \(p_t\) and \(N_\text {part}\) dependent suppression agree reasonably well with our model predictions.
1 Introduction
The medium formed in heavyion collision experiments at the Large Hadron Collider (LHC) at CERN and the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory (BNL) shows collectivity and probably indicates the existence of deconfined QCD matter commonly known as QuarkGluon Plasma (QGP). Such a partonic state is considered as a phase of QCD matter at extremely high temperature and/or baryon density [1, 2, 3]. More precisely QGP is considered as a thermalized state of quarks and gluons which are asymptotically free inside a range which is of the order of the strong interaction (2–3 fm). It is believed that QGP existed in nature until a few micro seconds after the Bigbang when hadrons began to form and that it can be recreated for a much shorter timespan of about \(10^{23}\) s in relativistic heavyion collisions at sufficiently high energy. Due to very short spatial and temporal extension of the QGP in heavyion collisions, its direct observation becomes impossible. There are, however, many suggested observables to validate the QGP formation in the heavyion collision at RHIC and LHC experiments [4, 5, 6, 7, 8, 9]. Quarkonium suppression is one such observable of QGP formation in heavyion collisions experiments. The mass scale of quarkonia (\(m=3.1\) GeV for \(J/\psi \) and \({m = 9.46}\) GeV for \(\Upsilon \)) is of the order of, but larger than the QCD scale (\(\Lambda _{QCD} \le 1\) GeV). In particular the measurement of the suppression of the heavy \(\Upsilon \)mesons in the quarkgluon plasma is therefore a clean probe to investigate QGP properties. Based on the scales involved, the production of quarkonia is assumed to be factorized into two parts: first, quark and antiquark (\(q\bar{q}\)) production through nucleon–nucleon collision as a perturbative process [10]. Second, the formation and evolution of bound state meson from \(q\bar{q}\) governed by nonperturbative QCD. Hence, heavy quarkonia provide a unique laboratory which enables us to explore the interplay of perturbative and nonperturbative QCD effects. A variety of theoretical approaches have been proposed in the literature to calculate the heavy quarkonium production in nucleon–nucleon collisions [11, 12, 13, 14, 15, 16, 17]. Potential nonrelativistic QCD (pNRQCD) [11, 12] and fragmentation approaches [13, 14] are the theoretical frameworks based on the QCD which are being frequently employed in many of the quarkonium production and suppression model calculations. Quarkonia (\(J/\psi ,~\Upsilon \), etc.) formed in the initial collision interact with the partonic QGP medium. This interaction leads to the dissociation of quarkonia through various mechanisms [18, 19]. The theoretical study of quarkonia suppression in the QGP medium has gone through many refinements over the past few decades and it is still under intense investigation.
Charmonium or bottomonium suppression in heavyion collision consists of two distinct processes: The first one is the cold nuclear matter (CNM) effect and second is the hot nuclear matter effect, commonly named as QGP effect. The quarkonia suppression due to CNM processes gets strongly affected by the nuclear environment [20]. There are three kinds of CNM effects generally utilized in the calculations. The first and dominant CNM effect in the case of quarkonium production is shadowing. It corresponds to the change in parton distribution function (PDF) in the nucleus as compared to its value in the nucleon which controls the initial parton behaviour. The shadowing effect strongly depends on the collisional kinematics, as parton distribution functions are different in A–A collision compared to p–p and/or p–Pb collision. Quarkonia production in A–A collision may be suppressed due to change in nuclear parton distribution function in the small x region to that of nucleon [21]. Shadowing causes the quarkonia production crosssection to become less in A–A case to that of pure p–p collision. The Cronin effect is another CNM contribution [22, 23]. It signifies the initial gluon multiscattering with the neighbouring nucleons presented in the nucleus prior to the hard scattering and the quarkonia formation. This results in the broadening of transverse momentum distribution of produced quarkonia. In the current model calculation, we have not incorporated the Cronin effect. Nuclear absorption [24] is another CNM contribution to the quarkonia production. The interaction between quarkonia and the primary nucleons leads to the absorption of quarkonia in nuclear environment which causes suppression of quarkonia in A–A collisions. It is the dominant CNM effect at lower energies. The crosssection for nuclear absorption decreases with the increase in energy and hence it is negligible at LHC energies [25].
Hot matter effects on quarkonia production, include “color screening” which was first proposed by Matsui and Satz in a seminal work [18]. Color screening suggests more suppression of quarkonia at mid rapidity in comparison to that at forward rapidity in heavyion collisions and more suppression at RHIC than at SPS, but experimental data is on contrary. Gluonic dissiciation [19, 26, 27] corresponds to the absorption of a E1 gluons (soft gluons) (where E1 is the lowest electric mode for the spinorbital wavefunction of gluons) by a quarkonium. This absorption induces transition of quarkonia from color singlet state to color octet state (an unbound state of quark antiquark; correlated quarks pairs) [28, 29, 30]. Collisional damping arises due to the inherent property of the complex potential between (\(q\bar{q}\)) located inside the QCD medium. The imaginary part of the potential in the limit of \(t\rightarrow \infty \), represents the thermal decay width induced due to the low frequency gauge fields that mediate interaction between two heavy quarks [31].
Apart from the dissociation of quarkonia in the QGP, recombination is also possible at LHC energies. There are two ways by which quarkonia can be reproduced within the QGP medium. The first possibility is through uncorrelated \(q\bar{q}\) pairs present in the medium. They can recombine within the QGP medium at a later stage [32, 33, 34, 35, 36, 37, 38, 39, 40, 41]. This regeneration process is thought to be significant for charmonium states \((J/\psi , \chi _{c}, \psi ^{'}, etc.)\) at LHC energies because \(c\bar{c}\) are produced just after the collisions in abundant numbers in QGP medium. While the regeneration of bottomonium states (\(\Upsilon (1S), \Upsilon (2S)\), etc.) due to uncorrelated \(b\bar{b}\) pairs is almost negligible because \(b\bar{b}\) pairs produced in the QGP medium are scarce even at the LHC energies.
The calculation of regeneration of quarkonia through uncorrelated \(q\bar{q}\) pair is usually based either on the statistical hadronization model [32, 33, 34, 35], or on kinetic models in which the production is described via dynamical melting and regeneration over the whole temporal evolution of the QGP [36, 37, 38, 41]. Some transport calculations are also performed to calculate the number of regenerated \(J/\psi \)s [42, 43]. The second regeneration mechanism i.e., recombination due to correlated \(q\bar{q}\) pairs is just the reverse of gluonic dissociation, in which correlated \(q\bar{q}\) pairs may undergo transition from color octet state to color singlet state in the due course of time in QGP medium. Bottomonium as a color singlet bound state of \(b\bar{b}\) pair, with b and \(\bar{b}\) separated by distances \(\sim 1/m_{b}v\), is smaller than \(1/\Lambda _{QCD}\). Here, \(v\sim \alpha _{s}(m_{b}v)\) is the relative velocity between \(q\bar{q}\). The size of bottomonium states (\(\Upsilon (1S), \Upsilon (2S)\)) is thus smaller than the corresponding charmonium states (\(J/\psi (1S), \psi ^{'}(2S)\)). Due to this its melting temperature or dissociation temperature, \(T_{D}\) (\(T_{D}\sim 670\) MeV for \(\Upsilon (1S)\)) is large compared to the charmonia (\(T_{D} \sim 350\) MeV for \(J/\psi \)). Thus, one may think that other suppression mechanisms such as sequential melting of bottomonia is merely possible in QGP. Although one may observe that the melting of higher states of bottomonia in QGP as their dissociation temperature is not much as \(\Upsilon (1S)\) [19]. High \(T_{D}\) of \(\Upsilon (1S)\) favors recombination due to correlated \(b\bar{b}\) pairs. In this scenario, regeneration of bottomonium is also possible because of the deexcitation of correlated \(b\bar{b}\) or octet state to the singlet states. All these dissociation and regeneration mechanisms indicate that the quarkonia production in heavyion collisions is a consequence of the complex interplay of various physical processes.
An interesting/puzzling category of collisional system is p–A collision (asymmetric nuclear collision system). The p–A collisions has been thought to serve as an important baseline for the understanding and the interpretation of the nucleusnucleus data. These measurements allow us to separate out the hot nuclear matter effect from the CNM effects. The p–A collision was used to quantify the CNM effect, when the QGP was not expected to be formed in such a small asymmetric collision systems. Till the last few years, the p–A experimental data corresponding to quarkonia suppression have been effectively explained by considering CNM effects only at various rapidity, \(p_T\) and centrality [44]. For instance, the suppression pattern obtained for charmonium (\(J/\psi \)) in d–Au collisions at RHIC is well explained by CNM effects. Recent experimental data for p–Pb collision at \(\sqrt{s_{NN}}\, =\, 5.02\) TeV at LHC open up the possibility of the hot matter i.e., QGP formation in such a small asymmetric systems [45, 46]. It may be possible since the number of participants (\(N_{part}\)) in p–Pb collision at centrality class 0–5% is approximately equal to the \(N_{part}\) in Pb–Pb collision at centrality class 80–100%. At this centrality class, there is a finite chance of QGP formation even in p–Pb collisions at the available LHC energies [47]. If QGP exists in such a small system, its lifetime would obviously be comparatively less (\(\sim 23\) fm) than the QGP lifetime (\(\sim 69\) fm) formed in Pb–Pb collisions.
It is quite a nontrivial task to explain the quarkonia suppression data available from various heavyion collision experiments obtained at different energies and collision systems. Various models [18, 19, 48, 49] have been employed to explain the centrality and transverse momentum (\(p_T\)) dependent suppression at mid rapidity. Moreover, only few models are available that can explain simultaneously \(p_T\), rapidity y and centrality dependent quarkonia suppression data in A–A collisions [50].
Here our current formulation of gluonic dissociation and collisional damping is based on the model that has originally been developed (mainly for the centrality dependence suppression [19]) by the Heidelberg group [19, 26, 72, 73], but implement refinements such as dilated formation time and simplifications such as the neglect of the running of the strong coupling. We have incorporated the transverse momentum dependence in the currently used gluonic dissociation in a different way (see Eq. 18). Regeneration of bottomonium due to correlated \(b\bar{b}\) pairs has been incorporated in the present work. Its net effect is to reduce the effective gluonic dissociation. We then used the formulation to analyze centrality and transverse momentum (\(p_T\)) dependence data from Pb–Pb collision at \(\sqrt{s_{NN}}=2.76\) and 5.02 TeV LHC energies and p–Pb collision data at \(\sqrt{s_{NN}}=5.02\) TeV have also been analyzed in the present article.
The current work is an attempt to explain \(p_T\) and centrality dependent suppression data obtained at LHC energies in A–A and p–A collisions systems utilizing a modified version of a ‘Unified Model of quarkonia suppression (UMQS)’ [51] that has been used to mainly explain the centrality dependence. The modifications in the UMQS have been carried out in order to account for the \(p_T\) dependence in the formalism. The current model includes the suppression mechanisms such as shadowing (as a CNM effect), color screening, gluonic dissociation and collisional damping (as a hot matter effect) along with the regeneration of bottomonium within QGP medium due to the correlated \(b\bar{b}\) pairs.
We determine the centrality and \(p_{T}\) dependent bottomonium suppression in Pb–Pb as well as in p–Pb collisions at mid rapidity at energies \(\sqrt{s_{NN}} = 2.76\;\) and 5.02 TeV at CERN LHC [46, 52, 53, 54, 55, 56]. We then compare our model predictions for \(\Upsilon (1S)\;\) and \(\; \Upsilon (2S)\) suppression with the corresponding experimental data. We find that the experimental observations agree reasonably well with our model predictions over a wide range of LHC energies and at different collision systems.
The organization of the paper is as follows. In Sect. 2, the time evolution of QGP medium and corresponding bottomonium kinematics are discussed. In Sect. 3, the details of key ingredients of UMQS model such as color screening, gluonic dissociation, collisional damping, regeneration and shadowing mechanisms are described. Their effects on \(\Upsilon (1S)\) and \(\Upsilon (2S)\) production is also discussed in this section. In Sect. 4, we describe our results and discussions on \(\Upsilon (1S)\) and \(\Upsilon (2S)\) yield at mid rapidity. Finally, in Sect. 5, we summarize and conclude our research work.
2 Time evolution of QGP and bottomonium kinematics
The formulation of the current work is based on our recent work [51]. Here we describe the model in brief for the sake of completeness emphasizing the modifications wherever incorporated.
2.1 Bottomonium transport in evolving QGP
\(\sqrt{s_{NN}}\) TeV  \(\sigma _{\Upsilon (1S)}^{NN}\)  \(\sigma _{\chi _{b}(1P)}^{NN}\)  \(\sigma _{\Upsilon (2S)}^{NN}\)  \(\sigma _{\Upsilon (2P)}^{NN}\)  \(\sigma _{\Upsilon (3S)}^{NN}\)  \(\sigma _{b\bar{b}}^{NN}\) 

pp@2.76  72nb  20nb  24nb  3.67nb  0.72nb  \(23.28\mu b\) 
pp@5.02  78nb  25nb  26nb  3.97nb  0.78nb  \(47.5\mu b\) 
Due to lack of the experimental data of \(\sigma _{\Upsilon (nl)}^{NN}\) at 5.02 TeV in p–p collision at mid rapidity, we extracted the same at 5.02 TeV by doing the linear interpolation between 2.76 and 7.00 TeV. We obtain \(\sigma _{\chi _{b}(1P)}^{NN} \), \(\sigma _{\Upsilon (2S)}^{NN}\), \(\sigma _{\Upsilon (2P)}^{NN}\) and \(\sigma _{\Upsilon (3S)}^{NN}\) by considering the feeddown fraction \(\sim 28\%\) (\(\sigma _{\chi _{b}(1P)}^{NN} \simeq \frac{1}{4} \sigma _{\Upsilon (1S)}^{NN}\)), \(\sim 35\%\) (\(\sigma _{\Upsilon (2S)}^{NN} \simeq \frac{1}{3} \sigma _{\Upsilon (1S)}^{NN}\)), \(\sim 5\%\) and \(\sim 1\%\) of \(\sigma _{\Upsilon (1S)}^{NN}\), respectively.
We obtained the initial temperature, \(T_{0}\), using initial (thermalization) time (\(\tau _{0}\)) and QGP lifetime (\(\tau _{QGP}\)) for \(T_{c} = 170\) MeV, corresponding to collision system and their respective center of mass collision energy \(\sqrt{s_{NN}}\) at most central collisions, i.e. \(\frac{N_{part}(b)}{N_{part}(b_{0})} = 1\)
\(\sqrt{s_{NN}}\) (TeV)  \(\tau _{0}\) (fm)  \(T_{0}\) (MeV)  \(\tau _{QGP}\) (fm)  \(T_{tr}(\tau _{0})\) (MeV)  \(\tau _{qgp}^{tr}\) (fm) 

PbPb@2.76  0.3  485  7.0  455  3.7 
PbPb@5.02  0.13  723  10.0  620  4.3 
pPb@5.02  0.3  366  3.0  342  1.63 
2.2 Temperature gradient
In Fig. 1, we have compared the temperature cooling law for QGP medium corresponding to the bag model (BM) and quasiparticle model (QPM) equation of states. Initially at \(\tau \sim 0.11.0 \) fm, QGP medium cools down with the same rate for both, BM as well as QPM EoS based expansion. In the due course of time, R decreases (i.e., \(R^{1}\gg 0\)), which leads to the faster cooling of QGP medium corresponding to QPM EoS based expansion, as shown in Fig. 1. In the case of symmetric ultrarelativistic nucleusnucleus collisions, \((1+1)\)dimensional Bjorken’s scaling solution seems to give a satisfactory results. In order to get a tentative estimate of the impact of the transverse expansion on our results, transverse expansion can be incorporated as a correction in \((1+1)\)dimensional hydrodynamics using QPM EoS, by assuming that transverse expansion starts at time \(\tau _{tr} > \tau _{0}\). The \(\tau _{tr}\) is estimated by considering that thermodynamical densities are homogeneous in the transverse direction, so \(\tau _{tr}\) can be written as: \(\tau _{tr} \cong \tau + \frac{r}{c_{s}} (\frac{\sqrt{2}  1}{\sqrt{2}})\) [66, 67]. Here r is the transverse distance and \(c_{s}\) is speed of sound in the QGP medium. Using \(\tau _{tr}\), we calculated the cooling rate of temperature corresponding to transverse expansion (\(T_{tr}(\tau )\)) correction in \((1+1)\)dimensional expansion based on QPM EoS. As expected, Fig. 1 depicts that the transverse expansion makes the cooling of QGP medium faster as compared to that in \((1+1)\)dimensional scaling solution case. As a result of this, QGP lifetime (\(\tau _{qgp}^{tr}\) corresponding to transverse expansion correction) would be reduced as given in Table 2.
The values of \(T_{0}\) at LHC energies mentioned in the Table 2 are comparable with \(T_{0}\) values used to explain the bulk observables (hadron spectra, flow coefficients, etc.) and dynamical evolution of the QGP medium [65, 68, 69, 70, 71]. \(T_{tr}(\tau _{0})\) mentioned in the Table 2, is the temperature at \(\tau = \tau _{0}\) but at some finite initial transverse position, \(r=0.45\) fm (say) and that is why we obtained \(T_{tr}(\tau _{0}) < T_{0}\). Also the time taken by the QGP to reach its temperature to \(T_c\) from \(T_0\) i.e., QGP lifetime would be reduced if transverse expansion is included in the calculation. For p–Pb collisions, \(T_{0}\) reached in the most central bin are considerably higher than the temperature reached in peripheral ones in Pb–Pb collisions. It supports the idea of QGP like medium formation even in asymmetric p–Pb collisions at \(\sqrt{s_{NN}} = 5.02\) TeV.
2.3 Volume expansion
3 Inmedium \(\Upsilon (1S)\) and \(\Upsilon (2S)\) production
\(\Upsilon (1S)\)  \(\chi _{b}(1P)\)  \(\Upsilon (2S)\)  \(\chi _{b}(2P)\)  \(\Upsilon (3S)\)  

\(M_{nl}\) (GeV)  9.46  9.99  10.02  10.26  10.36 
\(T_{D}\) (MeV)  668  206  217  185  199 
\(\tau _{f}\) (fm)  0.76  2.6  1.9  3.1  2.0 
3.1 Color screening
The centrality dependent dissociation of bottomonia in QGP due to collisional damping and gluonic dissociation mechanisms was originally formulated by Wolschin et al. [19, 26, 72, 78, 79]. In the present work, we modified their gluonic dissociation and collisional damping model and incorporated the transverse momentum dependence.
3.2 Collisional damping

\(\sigma \) is the string tension constant between \(b\bar{b}\) bound state, given as \(\sigma = 0.192\) GeV\(^2\).

\(m_{D}\) is Debye mass, \(m_D = T_{eff} \sqrt{4\pi \alpha _s^T \left( \frac{N_c}{3} + \frac{N_f}{6} \right) }\), and \(\alpha _s^T\) is coupling constant at hard scale, as it should be \(\alpha _{s}^{T} = \alpha _{s}(2\pi T)\le 0.50\). We have taken \(\alpha _s^T \simeq 0.4430\). \(N_{c} = 3\), \(N_{f} = 3\).

\(\alpha _{eff}\) is effective coupling constant, depending on the strong coupling constant at soft scale \(\alpha _{s}^{s} = \alpha _{s} (m_{b} \alpha _{s}/2) \simeq 0.48\), given as \(\alpha _{eff} = \frac{4}{3}\alpha _{s}^{s}\).
3.3 Gluonic dissociation

\(m_{b} = 4.89 \) GeV, is the mass of bottom quark.

\(\alpha _s^u \simeq 0.59\) [19], is coupling constant, scaled as \(\alpha _{s}^{u} = \alpha _{s}(\alpha _{s}m_{b}^{2}/2)\).

\(E_{nl}\) is energy eigen values corresponding to the bottomonia wave function, \(g_{nl}(r)\).

the octet wave function \(h_{ql'}(r)\) has been obtained by solving the Schrödinger equation with the octet potential \(V_{8} = \alpha _{eff}/8r\). The value of q is determined using conservation of energy, \(q = \sqrt{m_{b}(E_{g}+E_{nl})}\).
To obtain the gluonic dissociation decay rate, \(\Gamma _{gd,nl}\) of a bottomonium moving with speed \(v_{\Upsilon }\), we have calculated the mean of gluonic dissociation crosssection by taking its thermal average over the modified Bose–Einstein distribution function for gluons in the rest frame of bottomonium, as suggested in [26]. The modified gluon distribution function is given as, \(f_{g} = 1/(\exp [\frac{\gamma E_{g}}{T_{eff}}(1 + v_{\Upsilon }\cos \theta )]  1)\), where \(\gamma \) is a Lorentz factor and \(\theta \) is the angle between \(v_{\Upsilon }\) and incoming gluon with energy \(E_{g}\).
3.4 Regeneration factor
Here, s is the Mandelstam variable, related with the centerofmass energy of \(b\bar{b}\) pair, given as; \(s = ({\mathbf{p}_{b} + \mathbf{p}_{\bar{b}}})^{2}\), where \(\mathbf{p}_{b}\) and \(\mathbf{p}_{\bar{b}}\) are four momentum of b and \(\bar{b}\), respectively.
3.5 Cold nuclear matter effect
We have already discussed shadowing, absorption and Cronin effect as the three main nuclear effects on the charmonium production. Only shadowing has been incorporated in the current work since it is the dominant CNM effect.

\(f_{g}^{i}(A,\;x_{1},\;\mu ,\;r,\; z_{1}) = \rho _{A}(s) S^{i} (A, x_{1}, \mu , r, z)\;f_{g}(p,\; x_{1},\;\mu )\).

\(f_{g}^{j}(A,\;x_{2}, \; \mu , \;br,\; z_{2}) = \rho _{A}(s) S^{j} (A, x_{2}, \mu , br, z)\;f_{g}(p,\; x_{2},\;\mu )\).
In Fig. 7, initial suppression of \(\Upsilon (1S)\) due to shadowing effect is plotted as the function of transverse momentum \(p_{T}\), it shows effective shadowing effect at low \(p_{T}\) which decreases with increasing \(p_{T}\). The suppression of \(\Upsilon (1S)\) due to shadowing is more in same collision system at \(\sqrt{s_{NN}} = 5.02\) TeV as compared with \(\sqrt{s_{NN}} = 2.76\) TeV, indicates that the medium formed in Pb–Pb collision at \(\sqrt{s_{NN}} = 5.02\) TeV is much hot and dense. The same explains the shadowing pattern of \(\Upsilon (1S)\) in p–Pb collision at \(\sqrt{s_{NN}} = 5.02\) TeV.
3.6 Final yield
4 Results and discussions
In the present work, we have compared our model predictions on bottomonium suppression with the corresponding experimental results obtained at LHC energies. Our UMQS model determines the \(p_{T}\) and centrality dependent survival probability of bottomonium states at mid rapidity in Pb–Pb collisions at \(\sqrt{s_{NN}}\;=\;2.76\) and 5.02 TeV [52, 89, 90, 91] and in p–Pb collisions at \({\sqrt{s_{NN}}\;=\;5.02}\) TeV [92]. We have also calculated the \(S^{\Upsilon (2S)/\Upsilon (1S)}_{P} = S^{\Upsilon (2S)}_{P}/S^{\Upsilon (1S)}_{P}\) yield ratio and compared with the available double ratio of nuclear modification factor, \(R_{AA}^{\Upsilon (2S)}/R_{AA}^{\Upsilon (1S)}\). The abbreviation “FD” used in all the figures stands for feeddown correction. The results are compared to the respective experimental data with and without feeddown correction, as mentioned in the figures.
4.1 \(p_{T}\) Dependent suppression
Figure 9 depicts the suppression for Pb–Pb collision at \(\sqrt{s_{NN}}\;=\;5.02\) TeV, else it is very similar to what is shown in Fig. 8. Above plot shows that 2S suppression and its variation with \(p_T\) is very much similar to what was observed at \(\sqrt{s_{NN}}=2.76\) TeV energy. But 1S is more suppressed in the whole \(p_T\) range as compared to the corresponding suppression at \(\sqrt{s_{NN}}=2.76\) TeV energy. This enhancement in the suppression of 1S is due to the combined effects of color screening and gluonic dissociation along with the collisional damping. Energy deposited in Pb–Pb collisions at \(\sqrt{s_{NN}}\;=\;5.02\) TeV generates the initial temperature, \(T_{0}\sim 700\) MeV, which enables dissociation of \(\Upsilon (1S)\) due to color screening.
In Fig. 11, we have plotted our model predictions in terms of survival probability of \(\Upsilon (1S)\) and \(\Upsilon (2S)\) versus \(p_T\) along with a small suppression in \(\Upsilon (1S)\) at low \(p_{T}\) and a bit enhancement or almost no suppression at high \(p_{T}\) observed in central rapidity region in \(pPb\) collision at 5.02 TeV energy. Our model calculation showing small suppression of \(\Upsilon (1S)\) at low \(p_{T}\) which decreases at high \(p_{T}\) is consistent with the observed suppression data. The less suppression in bottomonia in p–Pb as compared to Pb–Pb collisions is due to the short life span of QGP in such a small collision system. Dissociation mechanisms depend on the bottomonium velocity \(v_{\Upsilon }\) in the QGP medium, so the low \(p_{T}\) mesons take more time to traverse through medium as compared to high \(p_{T}\) at the same QGP medium velocity. Thus, high \(p_T\) bottomonium would be less suppressed as observed in \(pPb\) collision at LHC energy. Feed down of higher states into 1S boost the suppression at \(p_{T}\) range 1–3 GeV which suggest that higher resonances are much more suppressed than \(\Upsilon (1S)\) at very low \(p_{T}\) while at mid and high \(p_{T}\) they are only bit more suppress than 1S. Our model predictions for \(\Upsilon (2S)\) depicts more suppression at very low \(p_T\) while a bit more suppression in the high \(p_T\) regions as compared to the \(\Upsilon (1S)\) predicted suppression. After taking feed down of \(\Upsilon (3S)\) and \(\chi _{b}(2P)\) into \(\Upsilon (2S)\), suppression of \(\Upsilon (2S)\) increases but follow the suppression pattern of 2S plotted without feed down. It shows that all the higher resonances are highly suppressed at very low \(p_{T}\) and at high \(p_{T}\) their suppression remains invariant with \(p_{T}\). Direct \(\Upsilon (2S)\) suppression versus \(p_T\) data in \(pPb\) collisions are needed in order to do a better comparison with our model prediction for \(\Upsilon (2S)\) correction.
In Figs. 8, 9 feed down correction to \(\Upsilon (1S)\) rises the suppression at low \(p_{T}\) regime which suggest, higher resonances are more suppressed at low \(p_{T}\) and their suppression decreases with increasing \(p_{T}\). For \(\Upsilon (2S)\) suppression, feed down correction is less significant at very low \(p_{T}\) because \(\Upsilon (2S)\), \(\chi _{b}(2P)\) and \(\Upsilon (3S)\) are almost equally suppressed at very low \(p_{T}\). The differences in suppression of higher resonances can be observed at high \(p_{T}\) regime through the feeddown correction to \(\Upsilon (2S)\). Feed down correction for double ratio plotted in Fig. 10 shows much suppression at very low \(p_{T}\) which is decreasing with increasing \(p_{T}\) but still it predicts over suppression for double ratio. The above plot shows that our model predictions for \(\Upsilon (1S)\) and \(\Upsilon (2S)\) matches reasonably well with the experimentally observed \(p_T\) dependent suppression data at mid rapidity in Pb–Pb and p–Pb collisions at LHC energies.
4.2 Centrality dependent suppression
5 Conclusions
We have employed our Unified Model of Quarkonia Suppression (UMQS) in order to analyze the \(\Upsilon \) suppression data obtained from Pb–Pb and p–Pb collisions at \(\sqrt{s_{NN}} = 2.76\) and 5.02 TeV LHC energies. Outcomes of UMQS model show that the bottomonium suppression is the combined effect of hot and cold nuclear matters. We have observed that color screening effect is almost insignificant to suppress the \(\Upsilon (1S)\) production since it only gives suppression in Pb–Pb central collision at \(\sqrt{s_{NN}} = 5.02\) TeV. While \(\Upsilon (2S)\) production is suppressed in Pb–Pb and p–Pb collisions at all the LHC energies. The gluonic dissociation along with the collisional damping mechanisms play an important role in \(\Upsilon (1S)\) dissociation as they suppress the \(\Upsilon (1S)\) production at less number of participants in Pb–Pb and p–Pb collisions. Our model suggests an effective regeneration of \(\Upsilon (1S)\) in sufficiently hot and dense medium formed at much higher collision energies e.g., Pb–Pb at \(\sqrt{s_{NN}} = 5.02\) TeV. This regeneration reduces the \(\Upsilon (1S)\) suppression in Pb–Pb collisions at \(\sqrt{s_{NN}} = 2.76\) and 5.02 TeV energies, while the regeneration for \(\Upsilon (2S)\) is found almost negligible for all the collision systems. We found that the UMQS results for \(\Upsilon (1S)\) and \(\Upsilon (2S)\) yields of bottomonium states agree well with the centrality and \(p_{T}\) dependent \(\Upsilon (1S)\) and \(\Upsilon (2S)\) experimental results in Pb–Pb collisions at \(\sqrt{s_{NN}} = 2.76\) and 5.02 TeV. Based on the above suppression results, the UMQS model strongly supports the QGP formation in Pb–Pb collisions. QGP formation in p–Pb collision may not be clearly explained by bottomonium suppression, because experimental results for \(\Upsilon (1S)\) suppression are around unity with large uncertainty and no direct experimental results are available for \(\Upsilon (2S)\) suppression. However, an indirect experimental information of \(\Upsilon (2S)\) suppression is available in the form of double ratio. The UMQS model predicted the \(\Upsilon (2S)\) suppression in p–Pb collisions. The experimental results for \(\Upsilon (2S)\) to \(\Upsilon (1S)\) double ratio support our prediction since observed yield ratio of \(\Upsilon (2S)\) to \(\Upsilon (1S)\) agrees quite well with our model predictions. Based on the above facts, it can be concluded that UMQS model advocates the formation of QGP like medium in p–Pb collisions at \(\sqrt{s_{NN}} = 5.02\) TeV. Here, it is worthwhile to note that in our UMQS model, not even a single parameter is varied freely in order to explain the suppression data. Although there are few parameters in the model, yet their values have been taken from the works done by the earlier researchers. It is also to be noted here that more precise calculation should use the (\(3+1\))dimensional hydrodynamical expansion contrary to the (\(1+1\))dimensional expansion employed in the current work. Although transverse expansion in the (\(3+1\))dimensional expansion would slightly enhance the cooling rate and therefore finally affect the dissociation as well as regeneration rate yet not very significantly.
Furthermore, work on additional observables is required to better constrain theoretical models and study the interplay between suppression and regeneration mechanisms. The elliptic flow pattern of charmonium observed in ultrarelativistic heavyion collisions at LHC energies is one such observable. It is important to test the degree of thermalization of heavy quarks. It is also of paramount interest in discriminating between quarkonium production from initial hard collisions and from recombination in the QGP medium. In our future work, we will attempt to concentrate on the above mentioned issue.
Notes
Acknowledgements
M. Mishra is grateful to the Department of Science and Technology (DST), New Delhi for financial assistance. M. Mishra thanks Prof. G. Wolschin for useful discussions/suggestions/comments on the present research work and providing hospitality at the Institute of Theoretical Physics, University of Heidelberg, Germany during his visit in summer 2016. Captain R. Singh is grateful to the BITS–Pilani, Pilani for the financial assistance.
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