Heavy Quarkonia in a Potential Model: Binding Energy, Decay Width, and Survival Probability

Recently a lot of progress has been made in deriving the heavy quark potential within a QCD medium. In this article we have considered heavy quarkonium in a hot quark gluon plasma phase. The heavy-quark potential has been modeled properly for short as well as long distances. The potential at long distances is modeled as a QCD string which is screened at the same scale as the Coloumb field. We have numerically solved the 1+1-dimensional Schrodinger equation for this potential and obtained the eigen wavefunction and binding energy for the $1S$ and $2S$ states of charmonium and bottomonium. Further, we have calculated the decay width and dissociation temperature of quarkonium states in the QCD plasma. Finally, we have used our recently proposed unified model with these new values of decay widths to calculate the survival probability of the various quarkonium states with respect to centrality at relativistic heavy ion collider (RHIC) and large hadron collider (LHC) energies. This study provides a unified, consistent and comprehensive description of spectroscopic properties of various quarkonium states at finite temperatures along with their nuclear modification factor at different collision energies.


I. INTRODUCTION
Heavy quarkonium production and suppression was one of the earliest proposed tool to study the properties of the medium created in heavy ion collisions. In mid 80's, Matsui and Satz [1]has proposed theoretically that quarkoinum suppression is the signal of the possible creation of quark gluon plasma (QGP) in collision experiments. From there onward, the physical picture of quarkonium dissociation in a thermal medium has undergone various theoretical and experimental refinements [2]. Recent experimental observations suggest that the charmonium suppression in QCD plasma is not the result of a single mechanism, but is a complex interplay of various physical processes. Heavy quarkonia (QQ) has a special edge over many other proposed tools since the heavy mass scale (m J/ψ = 3.1 GeV for J/ψ and m Υ = 9.2 GeV for Υ) makes this system possible for analytical treatment theoretically.
On the other side, decay of heavy quarkonia via dileptonic channel lead to relatively clean signal which can be precisely measured experimentally.
We can get the physical insight of the medium dependence by analyzing the behavior of spectral function of heavy quarkonium. The two useful approaches to study the production and suppression via spectral function of heavy quarkonium are potential method and lattice approach [3][4][5][6][7][8][9][10]. As we all know that lattice QCD method is first principle tool to study the properties and behavior of heavy quarkonium thus none of the potential method can be alternative to this approach. However, the lattice observations are suffering from discretization effects and statistical errors. In this scenario, potential models can be utilized to surve the purpose. As we now know that the problem of heavy quark bound state at zero temperature involves different energy scales, i.e., hard scale, which is the mass m Q of heavy quark, soft scale which is inverse size m Q v ∼ 1/r of bound state and ultrasoft scale, which is the binding energy mv 2 ∼ α s /r. After integrating out the hard scale modes, one obtains an effective field theory non-relativistic QCD (NRQCD) [11,12]. Subsequently, integrating out the modes related with the inverse size scale, potential NRQCD (pNRQCD) appears [13,14].
In this pNRQCD, the heavy quark-antiquark pair in singlet and octet state are included via dynamical singlet and octet fields (or potentials).
The generalization of this approach at finite temperature involves three different thermal scales : T , gT and g 2 T . In the static limit and if the binding energy is larger than the temperature T , the derivation of pNRQCD proceed in the same way as in zero temperature theory and heavy quark potential is not affected by the medium. However, the bound state properties can be affected through interaction of bound states with the ultrasoft gluons of the medium. The main effect of this interaction is the reduction of binding energy of the heavy quark bound state and emergence of a finite thermal width. In second case when one of the thermal scales is higher than binding energy, the singlet and octet potential become temperature dependent and will acquire an imaginary part [15]. It is important to state here that the real part of the potential leads to colour screening while imaginary part of the potential introduces the Landau damping to the heavy quark bound states [16][17][18][19].
Another observation from numerical lattice calculations show the crossover type of deconfinement transition from hadron gas to QGP [20]. Thus, we can expect some nonperturbative effects such as non-vanishing string tension in heavy quark-antiqurk potential above the critical temperature T C as well. So it is reasonable to assume the string term above T C [21][22][23]. Further this potential should also incorporate the effect of Landau damping induced thermal width by calculating the imaginary part of the potential. In the recent years, the real [24,25] and imaginary parts [17,26] of the heavy quark potential have been calculated by modifying both the perturbative and non-perturbative terms of the Cornell potential in the static as well as in a moving medium. The complex static interquark potential at finite temperature has also been derived in Ref. [27] by considering both the Coulombic and linear string terms. One can calculate the dissociation coefficient at a given temperature by solving the Schrödinger equation using the modified heavy quark complex potential [17,24,25,28,29]. Recently we have constructed a unified model for charmonium suppression in Ref. [30]. Here we want to incorporate this modified heavy quark potential from Ref. [17,24] in unified model to calculate the survival probability of heavy quarkonium states. The survival probability of heavy quarkonium states has been studied recently in Refs. [31,32]. Specially we will focus here on the double ratio of two states of charmonium since most of the suppression models are failed to reproduce the suppression pattern of this double ratio.
In this article we have modified our unified model to properly include perturbative as well as nonperturbative effects on quarkonium suppression. We have constructed this model based on the kinetic approach whose original ingredients was given by Thews et al. [33].
In this approach, there are two terms written on the basis of Boltzmann kinetic equation as shown in subsection II D. First term, which we call as dissociation term, includes the dissociation process like gluo-dissociation and collisional damping. The second term (formation term) provides the (re)generation of J/ψ due to the recombination of charm-anticharm quark. These two terms compete over the entire temporal evolution of the QGP and we get the multiplicity of finally survived quarkonia at freezeout temperature. To define the dynamics of the system created in the heavy ion collisions, we have used the 1+1 dimensional viscous hydrodynamics. Here we have included only the shear viscosity and neglected the bulk viscosity. We have also suitably incorporated the overall feed-down correction from the higher states to the low-lying states. Rest of the paper is organised as follows : In section II, model formulation, we have provided four subsections which discuss briefly about modified heavy quark potential at finite temperature, binding energy, decay width and calculation of survival probability, respectively. Further in section III, we have presented our results along with their discussions. In the end of this section, we have also summarized our present work.

A. Heavy Quark Complex Potential
In this section we discuss about the heavy quark-antiquark potential which have both the columbic and string-like parts. Authors in Ref. [16] have derived the static potential between heavy quark-antiquark pair at finite temperature by defining a suitable gaugeinvariant Green's function and computing it to first order in hard thermal loop (HTL) resummed perturbtion theory. In medium both the columbic and string-like part of potential receive modification. Further complications arise from the fact an imaginary part of the potential arises due to the presence of scattering of light medium degrees of freedom with the color string spanning in between the heavy quarks and antiquarks. It has been pointed out that the physics of the finite width originates from the Landau damping of low-frequency gauge fields. Further it has been studied non-perturbatively by making use of the classical approximation. In the view of above observations, a meaningful description of the relavant physics of quarkonium must therefore consist both the effects of screening of the real part of potential and imaginary part of the potential. There are several efforts to derive and/or phenomenologically construct HQ potential which can be used as an input in the quarkonium suppression models [16,27,[34][35][36][37]. The standard Polyakov loop correlator is fail to reproduce the expected Debye-screening potential at asymptotically large distances. Many modified descriptions of Polyakov loop correlator are affected by gauge ambiguities. There are studies based on generalized Gauss law and further its combination with the characterization of in-medium effects through the perturbative HTL permittivity. The use of Gauss law, non-local concept leads to a self-consistent descriptions of both screening and damping effects. Recently, direct lattice determination of the quarkonium spectral function have been attempted [10]. However, these calculations are again plagued by the model assumptions as there are very finite number of points in time direction and data is of statisctical nature in these lattice studies. On the other side, similar observables have been calculated for strongly-coupled N = 4 super Yang-Mills theory through AdS/CFT correspondence [38][39][40][41][42].
In these derivations, static potentials in real time can be calculated by computing the standard Wilson loop in Euclidean spacetime, and then carry out the analytic continuation. In other words, the expectation value of a particular timelike Wilson loop defines the potential between a static quark and antiquark at finite temperature. The medium modified heavy quark potential can be obtained by correcting both the Coulombic (short-distance) and string (long-distance) terms, not its Coulomb term alone, with a dielectric function encoding the effects of the deconfined medium as discussed in Refs. [17,24,25]. In the literature only a screened Coulomb potential was assumed above critical temperature (T c ) and the non-perturbative (string) term was usually overlooked (assumed zero), was certainly worth investigation. Recent lattice results indicate the phase transition in full QCD appears to be a crossover rather than a phase transition with the related singularities in thermodynamic observables as discussed in introduction section. The effects of string tension between the QQ pairs should not be ignored beyond T c . Therefore, it is important to incorporate the string term while setting up the criterion for the dissociation. In our approach, we make the assumption that medium potential can be derived from the vacuum potential by multiplying it with a field-theoretically determined complex permittivity in momentum space. It is possible to reproduce the real and imaginary part of the corresponding in-medium potential in by using the hard thermal loop permittivity. The real part of the medium modified heavy quark potential can be written as [24,25] wherer = rm D and α = 4/3 α s with α s as the one loop running coupling constant given as Here we take Λ MS = 0.1 GeV and the string tension, σ = 0.184 GeV 2 . m D is the Debye screening mass which is defined as with N f and N c as the number of flavours and colours, respectively.
In the small r limit, the real part of potential reduces to the Cornell potential.
On the other hand, in the large distance limit (where the screening occurs), potential is reduced to a long-range Coulomb potential with a dynamically screened-color charge. However, if we compare our QQ potential (Eq. 1) with the classical concept of Debye-Hückel theory by Digal et.al. [43], we found that in the asymptotic limit (r → ∞), Eq. (1) reduces to whereas in Ref. [43] free energy reduces to here the difference can be seen only in the string term only and may be due to the treatment of the problem classically or quantum mechanically. Also in the framework of Debye-Hückel theory, Digal et al. employed different screening functions, f c and f s for the Coulomb and string terms, respectively, to obtain the free energy. Here we have used the same screening scale, m D for both the Coulombic and linear terms.
The imaginary part of the medium modified heavy quark potential can be calculated in the similar way as in Ref. [17] and is given by where the functions φ(r) and ψ(r) are defined as In the small r limit, we can expand the potential and at leading logarithmic order inr we get solved the Schrodinger equation with the real part of heavy quark potential as described above and a angular momentum dependent part. The motivation behind using only the real part of the potential in solving Schrodinger equation is the large magnitude of real part over imaginary part (one can easily veriy it from Fig. 1 and Fig. 2). We have used the method of Ganesh and Mishra [29] to numerically solve the one dimensional Schrodinger equation on a logarithamic equally spaced one dimensional lattice. We have obtained the energy eigen values and eigen functions of different quarkonium states. Then we calculate the energy eigen value at infinity (U ∞ ). The binding energy of a given quarkonia state with principal quantum number n and orbital quantum number l is calculated by using the following relation [44] B.E.(n, l) = g nl − U ∞ .

C. Decay Width (Γ)
We have calculated the decay width (Γ) of 1S and 2S of quarkonia state numerically. As we know that the thermal width can be calculated from the imaginary part of the potential by using the following expression [17,26,39]  It is important to mention here that analytically one can calculate the decay width by folding the imaginary part of the potential with 1S and 2S hydrogen atom wavefunc- which are assumed to represent most of the properties of heavy quarkonia states. However as we will show in our results that the various quarkonia wavefunctions actually depends on temperature very strongly and thus assuming a temperature independent coloumbic wave function to calculate the decay width is not realistic. Thus we have used the wavefunctions as obtained by us, solving Schrodinger equation at different temperatures.
For the sake of comparison, we are providing here the expression of decay widths obtained by folding the imaginary part of the potential with the coloumbic wavefunctions of different quarkonia state as follows : and The dissociation temperature for the quarkonium states can be determined by using the conservative quantitative condition Γ n, l (T C ) ≈ 2 × B.E. n, l (T C ) [7], where T C is the disso- ciation temperature of that particular quarkonia state having principal quantum number n and azimuthal quantum number l.

D. Survival Probability including Regeneration
The survival probability or the nuclear modification factor of various quarkonia states with respect to centrality, rapidity and transverse momentum is key signatures to quantify the properties of medium created in heavy ion collision experiments. As we have the decay width of various quarkonia states from our calculation, we can calculate their survival probability by using our recently proposed unified model [30]. Here we show the variation of survival probability with respect to participant number (N part ) which actually measures the centrality of collision. To understand the evolution dynamics of the system created in heavy ion collisions, we have used 1+1 dimensional viscous hydrodynamics in which we have included the shear viscous effect. We have done our calculation for mid-rapidity region only where Bjorken scaling are applicable. We have derived the cooling law for temperature [45,46] which depends only on proper time τ and then extend this cooling law to make it τ and N part dependent using Ref. [29,47]. To calculate the survival probability we have used the following expression : where N i HM and N f HM is the initial and final multiplicity of heavy meson (quarkonia). The final multiplicity of quarkonia can be calculated as [30]: Here Γ f is the reactivity for the recomination of uncorrelated Q andQ quark to form a quarkonia and it can be calculated by using decay width [30]. N QQ is the number density of quark-antiquark pair. ǫ(τ f ) is the dissociation factor which can be calculated using the following expression : where τ 0 and τ f are initial and final proper time which actually spans over the QGP lifetime, i.e., τ 0 = 0.5 fm and τ f = 6.0 fm. We have used τ and N part dependent cooling law for temperature as follows [45,47]: where N part (bin 0 ) is the number of participant corresponding to the most central bin as used in our calculation and N part (bin) is the number of participant corresponding to the bin at which we want to calculate the temperature. τ QGP is the lifetime of QGP. The cooling law for volume is derived using the condition of isentropic evolution of the medium and can be expressed as follows [30]: where V (τ 0 , b) = π (r t − b/2) 2 τ 0 is volume at the initial time τ 0 and an impact of b fm.
N cc and N bb are calculated in our model using the help of Glauber model. The extrapolation to the nucleus-nucleus collisions is done via standard overlap integral scaling as follows : 20) where σ N N cc is the cross section for cc pair production in p+p collision. The σ N N cc has been calculated using pQCD approach for GRV HO hadronic structure function [33], we have

III. RESULTS AND DISCUSSIONS
The main ingredient of this paper is the heavy quark potential in QCD plasma [17,24].
We first show few characteristics of this potential. In Fig. 1, we demonstrate the variation of the real part of the heavy-quark potential with respect to the separation distance (r) between the QQ pair. We have plotted the real part of potential at three different values of temperature, i.e., T = 160, 320 and 480 MeV by solid, dashed and dash-dotted curve, respectively. The real potential starts from negative value and increases very sharply to zero as we increase the distance from zero to 0.5 fermi. Further the real part of potential increases from zero to 0.4 GeV as we increase the distance at T = 160 MeV. We choose this specific value of temperature since we take 160 MeV as the critical crossover temperature (T C ) in our calculation. We also show the variation in the saturation value of real potential with increase in temperature. As we increase the temperature the potential saturates at lower values, i.e., 0.4 GeV at T = 160 MeV to 0.05 GeV at T = 480 MeV. In Fig. 2, we have plotted the imaginary part of heavy-quark potential with respect to r. The imaginary potential starts from zero value at r = 0 fm and then decreases and became negative with increase in r.
As we increase the temperature the magnitude of imaginary potential also increases in the negative direction. At higher temperatures, we observed a flucuating behaviour for r > 0.4 fm. This is due to the sine term in imaginary potential.   we increase the temperature the peak-height of eigen function decreases and the peak-width increases which causes the binding between the heavy quark and anti-quark in the the bound state of charmonium to decrease. Similarly we have shown the change in eigen function of Υ(1S) with respect to temperature in Fig. 7.
In Fig. 8 and 9, we demonstrate the change in shape of eigen functions of Ψ ′ and Υ(2S) state with respect to temperature going from T C to 2.5 T C , respectively. is near T C , the binding energy of J/Ψ is large and thus charmonium can still survive after critical temperature. As we start to increase the temperature from T C to higher values, the binding energy of charmonium starts decreasing and acquires a low value which is near to zero. However, our calculation shows that even at 2.5 T C , there is a finite value of binding energy for J/Ψ. For Ψ ′ , the binding energy starts from a lower value in comparison to J/Ψ which is quite obvious and it decreases with increase in temperature and acquire almost zero We now present the decay width coming only due to the imaginary part of the heavy quark potential. In Fig. 13, we demonstrate the variation of decay width of 1S and 2S charmonia states with respect to T /T C . As we have earlier shown in Fig. 10 that the binding energy of J/Ψ is large at T C and thus the decay width should be small at T C . As the binding energy of Ψ ′ is less in comparison to J/Ψ over the entire temperature range therefore Ψ ′ has a larger decay width than J/Ψ as shown in this figure at each temperature. Further, the decay width increases with the increase in temperature. Furthermore, the difference between the decay width of J/Ψ and Ψ ′ increases with the temperature.
In Fig. 14, we have plotted the variation of decay width of Υ(1S) and Υ(2S) with respect to T /T C . The trend is quite similar with the charmonia states but the decay width of Υ(2S) increases rapidly after T = 3.5 T C . Fig. 15 presents a comparison between the decay width of J/Ψ and Υ(1S). This figure clearly shows that decay width of J/Ψ and Υ(1S) starts from almost similar value at T C . However, the width of J/Ψ increases more rapidly in comparison to Υ(1S). This means that the J/Ψ dissociates at lower temperatures in comparison to bottomonium which is a tightly bound state and thus survive to higher temperatures. In Fig. 16 we have plotted the decay width and two times of binding energy of J/Ψ with respect to T /T C . The two curves intersect each other at 2.4 T C . Thus the dissociation temperature of J/Ψ comes out as 2.4 T C in our calculation. We have plotted the width and the two times of binding energy of Ψ ′ with respect to T /T C in Fig. 17 In Fig. 20, we present the variation of survival probability of J/Ψ and Ψ ′ with respect to N part at highest RHIC energy, i.e., √ s N N = 200 GeV. As it is clear from the binding energy and decay width curve of J/ψ and ψ ′ , the dissociation probability of ψ ′ is large in comparison to J/ψ. Thus the survival probability of J/ψ is larger than the ψ ′ at each centrality. In other words, it means a less suppression of J/ψ in comparison to ψ ′ . We have compared our J/ψ results with the corresponding results obtained by STAR experiment [49]. Our model results underestimate the data in most peripheral collisions. However, it suitably describes the data for central and semi-peripheral collisions. Fig. 21 shows the variation of survival probability of J/ψ and ψ ′ with respect to centrality at LHC energy, i.e., √ s N N = 2.76 TeV.
As the energy increases, the corresponding temperature and energy density in each centrality class also increases and thus the survival probability of J/ψ and ψ ′ decreases in comparison We found in our calculation that the regeneration effect is negligibly small at RHIC energy. shown the STAR data [51] for Υ(1S) and combined suppression of Υ(1S) + Υ(2S) + Υ(3S).
Further in Fig. 23, we have plotted the CMS data [50] for Υ(1S) and Υ(2S). We observed that our model at RHIC energy is able to reproduce the experimental data of Υ(1S). Further at LHC energy, model results regarding Υ(1S) and Υ(2S) satisfy the experimental data satisfactorily. However our results underestimate the data in semi-peripheral collisions. In central collisions, one can again observe the clear effect of regeneration on the survival probability of Υ(1S) and Υ(2S) which is negligibly small at RHIC energy.
Finally we have calculated the double ratio between charmonia states (ψ ′ and J/ψ) and bottomonia states (Υ(2S) and Υ(1S)) at 2.76 TeV with respect to N part in Figs. 24 and 25, respectively. In Fig. 24, we have compared model result for the double ratio of charmonium states with the recent experimental data from CMS Collaboration at mid-rapidity as well as at forward rapidity. Since we have done our calculation for survival probability at midrapidity thus one can observe that our model with modified heavy quark potential satisfies the data well in most peripheral collisions while it underestimates the data at semicentral and central collisions. Similarly, in Fig. 25, we have compared our model results for the double ratio of bottomonium states with the experimental data. We observed that the model results satisfy the data satisfactorily well. Only in the semi-peripheral region, the model underestimates.
In summary, we have solved the 1+1 dimension Schrodinger equation using a modified heavy quark potential and obtained the eigen function and eigen values of the different charmonium and bottomonium states. We have also calculated the binding energy by solving the 1+1 dimensional Schrodinger equation for an infinite heavy quark potential. Further we have obtained the decay width of different quarkonium states using temperature dependent wavefunction obtained by us and demonstrate their variation with respect to temperature.
Furthermore we have obtained the dissociation temperature of diffferent quarkonium states by using the dissociation criteria, i.e., decay width = 2×binding energy. After that we feed these decay widths in our recently proposed unified model to calculate the survival probability of various quarkonium states. We have also obtained the nuclear modification factor for double ratios and observed that our model suitably describes the experimental data regarding nuclear modification factor (or survival probability).

IV. ACKNOWLEDGMENTS
PKS acknowledges IIT Ropar, India for providing an institute postdoctoral research grant.