2012"/> 2012-06-21"/> 2012"/> 2012-06-21"/> 236/jmp.2012.36065"/> 2153-120X"/> 20105"/> 20105"/> 2012-06-21"/> 2012-06-21"/> 236/jmp.2012.36065"/> 2153-1196"/> 2012-06-21"/> 2012"/>

Journal of Modern Physics
Vol.3 No.6(2012), Article ID:20105,9 pages 20105"> DOI:10.4236/jmp.2012.36065

χc and χb States in Hot Quark-Gluon Plasma

Binoy Krishna Patra1, Lata Devi1, Uttam Kakade1, Vineet Agotiya1, Vinod Chandra2

1Department of Physics, Indian Institute of Technology Roorkee, Roorkee, India

2Department of Theoretical Physics, Tata Institute of Fundamental Research, Mumbai, India

Email: binoyfph@iitr.ernet.in

Received January 23, 2012; revised February 15, 2012; accepted February 23, 2012

Keywords: Quarkonium; Debye mass; medium-modified; Heavy quark potential; Binding energy; Dissociation temperature


We have studied the dissociation phenomenon of 1p states (χc and χb) of the charmonium and bottomonium spectra in a hot QCD medium. This study employed a medium modified heavy quark potential encoding the medium effects in the dielectric function to the full Cornell potential. The medium modified potential has a quite different form in the sense that it has a long range Coulomb tail in addition to the usual Yukawa term even above the deconfinement temperature. We further study the flavor dependence of their binding energies and explore the nature of dissociation by employing the perturbative, non-perturbative, and the lattice parametrized form of the Debye masses in the medium-modified potential. Interestingly, perturbative result of the Debye mass predicts the dissociation temperatures closer to the results obtained in lattice correlator studies whereas the lattice parametrized form of the Debye masses gives the results closer to the current theoretical works based on potential studies.

1. Introduction

One of the amazing discoveries of experimental measurements at RHIC is the surprising amount of both radial [1-3] and elliptic flow [4-6] exhibited by the outgoing hadrons. Theoretical calculations cannot generate sufficient flow to explain the observations unless partonic cross sections are artificially enhanced by more than an order of magnitude over perturbative QCD predictions [7]. Thus the matter created in these collisions is strongly interacting, unlike the type of weakly interacting quarkgluon plasma expected to occur at very high temperatures on the basis of asymptotic freedom [8-10]. The behavior of the heavy quarkonium states in hot strongly interacting matter was proposed as test of its confinement status, since a sufficiently hot deconfined medium will dissolve any binding between the quark-antiquark pair [11]. Another possibility of dissociation of certain quarkonium states (sub-threshold states at212-9d24-9d546e8402f7.jpg width=49.019998550415 height=22.4200003623962  />) is the decay into open charm (beauty) mesons due to in-medium modification of quarkonia and heavy-light meson masses [12].

Many attempts have been made to understand the dissociation phenomenon of 26-8868-b43ad6b1d486.jpg width=34.675 height=30.5900007247925  /> states in the deconfined medium, using either lattice calculations of quarkonium spectral functions [13-19] or non-relativistic calculations based upon some effective potential [20-25]. These two approaches show poor matching between their predictions because of the uncertainties coming from a variety of sources. None of the approaches give a complete framework to study the properties of quarkonia states at finite temperature. However, some degree of qualitative agreement had been achieved for the S-wave correlators. The finding was somehow ambiguous for the P-wave correlators and the temperature dependence of the potential model was even qualitatively different from the lattice one. Refinement in the computations of the spectral functions have recently been done by incorporating the zero modes both in the Sand P-channels [26,27]. It was shown that, these contributions cure most of the previously observed discrepancies with lattice calculations. This supports the use of potential models at finite temperature as an important tool to complement lattice studies.

The production of 26f472.jpg width=39.805001449585 height=28.5950003623962  /> and 23-3db8-4200-9b7a-0a77d516b4de.jpg width=19.3799996376038 height=20.425  /> mesons in hadronic reactions occurs in part through production of higher excited 220b-429b-b89c-7322d2c12fa1.jpg width=26.5049996376038 height=22.4200003623962  /> (or28.5950003623962 height=28.5950003623962  />) states and their decay into quarkonia ground state. Since the lifetime of different subthreshold quarkonium states is much larger than the typical life-time of the medium which may be produced in nucleus-nucleus collisions; their decay occurs almost completely outside the produced medium. This means that the produced medium can be probed not only by the ground state quarkonium but also by different excited quarkonium states. Since, different quarkonium states have different sizes (binding energies), one expects that higher excited states will dissolve at smaller temperature as compared to the smaller and more tightly bound ground states. These facts may lead to a sequential suppression pattern in 29d-c6a1d789e401.jpg width=39.805001449585 height=28.5950003623962  /> and 24e9fc6-bf0d-4d1c-9c71-68c15915dcac.jpg width=19.3799996376038 height=20.425  /> yield in nucleus-nucleus collision as the function of the energy density. So, if one wants to interpret the 272-44e6-b1c3-d6bbb275e8ec.jpg width=39.805001449585 height=28.5950003623962  /> suppression pattern observed in nuclear collisions at CERN SPS and RHIC, as a signature of the formation of the QGP, one requires a right understanding of the dissociation of 29dbfbd-53cf-44eb-b5d7-c35641257d7a.jpg width=24.5099992752075 height=28.5950003623962  /> and 2976a-be90-474d-9428-13948ac53802.jpg width=24.5099992752075 height=28.5950003623962  /> in the QGP medium. This is due to the fact that a significant fraction (~30%) of the 28-4fa7-bd7a-7b8e87514165.jpg width=39.805001449585 height=28.5950003623962  /> yield observed in the collisions is produced by 2-25307c2135a2.jpg width=24.5099992752075 height=28.5950003623962  /> decays [28-30]. The 26-d971-4812-a414-89255e5c54f2.jpg width=39.805001449585 height=28.5950003623962  /> yield could show a significant suppression even if the energy density of the system is not enough to melt directly produced 2-64bd-4644-9cf4-71297eee7968.jpg width=39.805001449585 height=28.5950003623962  /> but is sufficient to melt the higher resonance states because they are loosely bound compared to the ground state2.jpg width=39.805001449585 height=28.5950003623962  />. This motivates the special attention to the excited states 21-cb51-44ed-aad7-152a0209db11.jpg width=24.5099992752075 height=28.5950003623962  /> and27-3914fff81522.jpg width=24.5099992752075 height=28.5950003623962  />.

In the studies of the bulk properties of the QCD plasma phase [31-34], deviations from perturbative calculations were found at temperatures much larger than the deconfinement temperature. This calls for quantitative non-perturbative calculations. The phase transition in full QCD appears as a crossover rather than a “true” phase transition with related singularities in thermodynamic observables (in the high-temperature and low density regime) [35-40]. Therefore, it is not reasonable to assume that the string-tension vanishes abruptly at or above 2da61504-fe8d-4955-9a4f-ae24ca46e466.jpg width=20.425 height=28.5950003623962  /> and one should study its effect on the behavior of quarkonia even above the deconfinement temperature. This issue, usually overlooked in the literature, was certainly worth to investigate. This is exactly what we have done in our recent work [41,42] where we have obtained the medium-modified form of the heavy quark potential by correcting the full Cornell potential (linear plus Coulomb), not only its Coulomb part alone as usually done in the literature, with a dielectric function encoding the effects of the deconfined medium. We found that this approach led to a long-range Coulomb potential with an (reduced) effective charge [41] in addition to the usual Debye-screened form employed in most of the literature. With this effective potential, we investigated the effects of perturbative and non-perturbative contributions to the Debye mass on the dissociation of quarkonium states. We subsequently used this study to determine the binding energies and the dissociation temperatures of the ground and the first excited states of charmonium and bottomonium spectra.

Our starting potential at 2db-4adf-abe0-bcc827d9ef62.jpg width=49.019998550415 height=22.4200003623962  /> was Cornell potential which has no terms to account the spin-dependence forces in QCD [43], so the medium-modified potential [41] also has no spin-dependent terms. As a consequence, Schrodinger equation with the above medium-modified potential gives the same energy eigenvalues for the first excited states 202da-6020-4af3-ae61-204c4cf5c850.jpg width=24.5099992752075 height=28.5950003623962  /> and 25417ff46b1.jpg width=24.5099992752075 height=28.5950003623962  /> in charmonium spectroscopy (29-69a7-4fd0-aa8c-aff45d162df8.jpg width=24.5099992752075 height=22.4200003623962  />and 201-4b10-4253-94f4-1679ac89a285.jpg width=24.5099992752075 height=28.5950003623962  /> states in bottomonium spectroscopy) making them degenerate. This is certainly not desirable since their masses are not the same (in fact, mass of 2-985c-4a2a-bf03-8151d80039dd.jpg width=24.5099992752075 height=28.5950003623962  /> is slightly higher than 244-b68e-432917717a98.jpg width=24.5099992752075 height=28.5950003623962  /> whereas the latter state is more tightly bound than the former). Therefore the determination of the binding energies of 1p states, viz., 2ba4e270-bb0c-4415-b1ba-5a8761edecb7.jpg width=24.5099992752075 height=28.5950003623962  />, 24.5099992752075 height=28.5950003623962  />and their dissociation temperatures like the ground and first excited states is not directly possible as had been done in our earlier work by employing the medium modified potential [41]. The principal quantum number (26347.jpg width=17.3849992752075 height=19.3799996376038  />) of 25db7-b967-4e9b-b5d0-bc4ea3ded502.jpg width=24.5099992752075 height=28.5950003623962  /> and 22a-b3cb4bb876ee.jpg width=24.5099992752075 height=28.5950003623962  /> are same but their spin quantum number and as well as their total angular momentum are not the same. So, their quantum states should be denoted by all four quantum numbers (25-c30b53532381.jpg width=35.719998550415 height=28.5950003623962  />) and the difference in their binding energies (or in their total masses) should be originated from a spin-dependent correction terms.

We have done this job in two fold way. First, we have determined the binding energy for 2.jpg width=24.5099992752075 height=28.5950003623962  /> by employing the medium-modified potential [41] into the Schrodinger equation. Then we obtain the binding energy for 24.5099992752075 height=28.5950003623962  /> by adding the correction terms to the binding energy of2ccded-bed4-4ae7-9f7c-a40f6667e0a9.jpg width=24.5099992752075 height=28.5950003623962  />. In our analysis, correction terms will be obtained by adopting a variational treatment of the relativistic twofermion bound-states in quantum electrodynamics (QED) [44-47] taking into account the spin-dependent terms for the corresponding quantum numbers of 2-4d66-4461-8517-996522c14b41.jpg width=24.5099992752075 height=28.5950003623962  /> and 29-1fd7-4776-9a08-e254d4a6df51.jpg width=24.5099992752075 height=28.5950003623962  /> states. In this endeavor, coupled integral equations for a relativistic two-fermion system are derived variationally within the Hamiltonian formalism of QED using an improved ansatz that is sensitive to all terms in the Hamiltonian [45,46].

The paper is organized as follows. In Section 2, we review the work on the medium modified Cornell potential and dissociation of 1s and 2s states of charmonium and bottomonium spectra. In Section 3, we discuss how to determine the binding energies of 2-a585-08eaed89edb8.jpg width=24.5099992752075 height=28.5950003623962  /> and2b64-e726-4017-95e6-3e738ac92c8f.jpg width=24.5099992752075 height=28.5950003623962  />. In Section 4, we study the melting of 282e2e-d05c-4e73-9654-79d86236ebea.jpg width=24.5099992752075 height=28.5950003623962  /> and 24.5099992752075 height=28.5950003623962  /> in a hot QCD medium and determine their dissociation temperatures. Finally, we conclude in Section 5.

2. In-Medium Modifications to Heavy-Quark Potential

The interaction potential between a heavy quark and antiquark gets modified in the presence of a medium and it plays a vital role in understanding the fate of quarkantiquark bound states in the QGP medium. This issue has well been studied and several excellent reviews exist [48,49] which dwell both on the phenomenology as well as on the lattice QCD. In these studies, they assumed the melting of the string motivated by the fact that there is a phase transition from a hadronic matter to a QGP phase. As a consequence they modified the Coulomb part of the potential only so they used a much simpler form (screened Coulomb) for the medium modified potential in the deconfined phase. But recent lattice results indicates that there is no genuine phase transition at vanishing baryon density, it is rather a cross-over, so there is no reason to assume the melting of string at the deconfinement temperature. We have addressed this issue in our recent work [41] where we developed an effective potential once one corrects the full Cornell potential with a dielectric function embodying medium effects. We recall the basic details which are relevant for the present demonstration.

Usually, in finite-temperature QFT, medium modification enters in the Fourier transform of heavy quark potential as

2fdf0df4a5.jpg width=115.330001449585 height=60.230001449585 /> (1)

where 2-5a3c-41f3-b86e-14080855d02a.jpg width=43.8900007247925 height=32.680001449585  /> is the dielectric permittivity given in terms of the static limit of the longitudinal part of gluon selfenergy [50]:

2-fbdbfd8ec3a4.jpg width=290.98498840332 height=62.319998550415 /> (2)

The quantity 2b5-c16e7a38914f.jpg width=46.930001449585 height=32.680001449585  /> in Equation (1) is the Fourier transform (FT) of the Cornell potential. The evaluation of the FT of the Cornell potential is not so straightforward and can be done by assuming2b1-bfef-7c0e5e09f950.jpg width=15.2950003623962 height=17.3849992752075  />—as distribution (2-a7e1-bc34035cbc42.jpg width=92.9100028991699 height=32.680001449585  />258a769-5f6f-4e98-a013-e7a184cf06e8.jpg width=37.8099992752075 height=19.3799996376038  />). After the evaluation of FT we let 202-594130ffd9ec.jpg width=17.3849992752075 height=22.4200003623962  /> tends to zero. Now the FT of the full Cornell potential can be written as

233.79500579834 height=55.1 /> (3)

Substituting Equations (2) and (3) into (1) and then evaluating its inverse FT one obtains the r-dependence of the medium modified potential [42]:

28.jpg width=252.12999420166 height=119.41499710083 /> (4)

This potential has a long range Coulombic tail in addition to the standard Yukawa term. Interestingly, high temperature behavior of quarkonia is rather governed by the former term in the potential with the (reduced) effective charge in analogous to the fine structure constant in QED. The constant terms are introduced to yield the correct limit of 23736e30-a2b2-4f1c-8653-11089e70fe51.jpg width=70.394998550415 height=32.680001449585  /> as 22.jpg width=55.1 height=22.4200003623962  /> (it should reduce to the Cornell form). Such terms could arise naturally from the basic computations of real time static potential in hot QCD [51] and from the real and imaginary time correlators in a thermal QCD medium [52].

It is worth to note that the potential in a hot QCD medium is not the same as the lattice parametrized heavy quark free-energy in the deconfined phase which is basically a screened Coulomb form [53-55] because onedimensional Fourier transform of the Cornell potential in the medium yields the similar form as used in the lattice QCD to study the quarkonium properties which assumes the one-dimensional color flux tube structure [56]. However, at finite temperature that may not be the case since the flux tube structure may expand in more dimensions [53]. Therefore, it is better to consider the three-dimensional form of the medium modified Cornell potential which have been done exactly in the present work. We have compared our in-medium potential with the colorsinglet free-energy [57] extracted from the lattice data and found that it agrees with the lattice results except for the non-perturbative result of the Debye masses [41].

We have thus employed this medium-modified effective potential (4) to study the binding energies and the dissociation temperatures for the ground and the first excited states of 2b2543d73224.jpg width=26.5049996376038 height=22.4200003623962  /> and 2191a65aa3.jpg width=28.5950003623962 height=28.5950003623962  /> spectroscopy in our earlier work [41]. Let us now proceed to the determination of the binding energies and the dissociation temperatures for 24.5099992752075 height=28.5950003623962  /> and 232608572c96.jpg width=24.5099992752075 height=28.5950003623962  /> states in Sections 3 and 4, respectively.

3. Binding Energy of χc and χb

The solution of the Schrödinger equation with the potential (4) numerically gives the energy eigenvalues for the ground states and the first excited states in charmonium (25e33ea7f.jpg width=39.805001449585 height=28.5950003623962  />,20ce86d-7879-4071-a4f8-65261ddce65e.jpg width=24.5099992752075 height=28.5950003623962  />etc.) and bottomonium (20.425  />,22b1868-fdd7-4d97-9dc2-cd5c91a45fe5.jpg width=24.5099992752075 height=22.4200003623962  />etc.) spectra. Theses energy-eigen values are known as ionization potentials/binding energies which become temperaturedependent through the Debye masses and decrease with the increase in temperature.

Apart from the ground and the first excited states, there are other important states (1p) in the charmonium and bottomonium spectra viz 24.5099992752075 height=28.5950003623962  /> and 23ad5a94.jpg width=24.5099992752075 height=28.5950003623962  /> which contribute significantly in the suppression of the ground state quarkonia (2167f4e3-35f7-4370-850a-7ef13d7891a1.jpg width=39.805001449585 height=28.5950003623962  />and24-9c1b-e0eb8a8f1728.jpg width=19.3799996376038 height=20.425  />) in RHIC experiments through their decays into2593-3c78-4106-b69f-cb5186206172.jpg width=39.805001449585 height=28.5950003623962  />’s and211.jpg width=19.3799996376038 height=20.425  />’s. Although both 27-4596-abec-2484120a6158.jpg width=24.5099992752075 height=28.5950003623962  /> and 2f-455b-8058-dc82c024837b.jpg width=24.5099992752075 height=28.5950003623962  /> are the first excited states of the charmonium spectra but they are not degenerate. In fact, 2-4289-873d-1c26c91e8f78.jpg width=24.5099992752075 height=28.5950003623962  />is more massive than 266b399-ef49-4dfc-8e46-2cba1f8064f6.jpg width=24.5099992752075 height=28.5950003623962  /> but 2be7d5.jpg width=24.5099992752075 height=28.5950003623962  /> is more tightly bound than2a4df7e609.jpg width=24.5099992752075 height=28.5950003623962  />. So the entire binding energy of 2a7-8580-dcc0d34e65bd.jpg width=24.5099992752075 height=28.5950003623962  /> will not come from the above calculation, the additional contribution will come from the spin-dependent quantum corrections.

Some authors have studied the relativistic two-particle Coulomb problem, based on approximations to the BetherSalpter equations. Others have started with effective Lagrangians based on perturbative expansions of the relativistic Lagrangians. However, we choose the variational methods [44-47] where coupled integral equations for a relativistic two-fermion system are derived variationally within the Hamiltonian formalism of quantum electrodynamics, using an improved ansatz that is sensitive to all terms in the Hamiltonian. The equations are solved approximately to determine the eigenvalues and eigenfunctions, at arbitrary coupling, for various states of the two-particle system. In the variational treatment of the relativistic two-fermion bound-state system in QED, the total energy in a quantum state (20d6b7cc-d8ae-4eef-b689-acc2523235f7.jpg width=35.719998550415 height=28.5950003623962  />) consists of Bohr like terms, relativistic correction in the kinetic energy and most importantly the spin-dependent terms which take into account the non-degeneracy between the sub-states. The total energy up to the fourth order in fine-structure constant 2f85d78154.jpg width=20.425 height=19.3799996376038  /> is written [45,46]:

2a13d28-a83e-4b79-82a8-b9ab5743e72f.jpg width=284.80998840332 height=55.1 /> (5)


204-bc48-1c1904e6713c.jpg width=267.52000579834 height=66.405001449585 /> (6)

is the 2786-f3a4-43da-92f7-38463b12582a.jpg width=26.5049996376038 height=28.5950003623962  /> correction to the kinetic energy and the correction to the spin-dependent potential energy is

23-44d8-b310-39bf50ba17e2.jpg width=161.31000289917 height=55.1 /> (7)

where the coefficients 2111d52c-90ff-46fe-a8c3-0948719d324f.jpg width=32.680001449585 height=30.5900007247925  /> for the different quantum states (2504dd2122b.jpg width=35.719998550415 height=28.5950003623962  />) are tabulated in Ref. [45,46]. The fine structure constant (2dd-2965374d66a2.jpg width=20.425 height=19.3799996376038  />) in QED will be replaced by the effective charge (2eef0c9-48c4-4f67-8cd0-69376b9d3296.jpg width=62.319998550415 height=32.680001449585  />) in our model. Using the appropriate values of the quantum numbers and the coefficients corresponding to 2-7920-408c-b2ff-a412c8d9c7fe.jpg width=24.5099992752075 height=28.5950003623962  /> and 23512a-23cc-430b-afb3-e52c712f7628.jpg width=24.5099992752075 height=28.5950003623962  /> states in charmonium spectra (2c-83b9-ffce4d95729d.jpg width=24.5099992752075 height=22.4200003623962  />and 246c21.jpg width=24.5099992752075 height=28.5950003623962  /> in bottomonium spectra), we obtain the correction term which is to be added to the binding energy of 2dc2bd51c71a.jpg width=24.5099992752075 height=28.5950003623962  /> is 

2.71499710083 height=60.230001449585 /> (8)

So the binding energy of 2c5-6970-43ae-83df-924ba4829a63.jpg width=64.3149971008301 height=32.680001449585  /> is

23d-36dd-444c-ae50-42a40780ef75.jpg width=292.97999420166 height=39.805001449585 /> (9)

where 2a7-a27e-33d8f5388d3c.jpg width=30.5900007247925 height=28.5950003623962  /> is the Debye mass for which we choose a gauge invariant, non-perturbative form by Kajantie et al. [58] which is obtained by computing the non-perturbative contributions of 2-8866-b971eaa24fc6.jpg width=68.4 height=35.719998550415  /> and from a 3-D effective field theory for QCD as

28ab.jpg width=320.52999420166 height=77.6149971008301 /> (10)

where the leading-order (LO) perturbative result,203.20499420166 height=59.1849992752075  />, has been known for a long time [59-61]. The logarithmic part of the 2f3.jpg width=59.1849992752075 height=35.719998550415  /> correction can be extracted perturbatively [62], but 210-4341-8c41-a043c4562c68.jpg width=30.5900007247925 height=34.675  /> and the higher order corrections are non-perturbative. We wish to explore the effects of the different terms in the Debye mass on the binding energy of 2-4a81-bc8c-0f9fa73b7f59.jpg width=24.5099992752075 height=28.5950003623962  /> and25-4584-42cb-8fd5-fca43589829f.jpg width=24.5099992752075 height=28.5950003623962  />. We have used the two-loop expression for the QCD coupling constant at finite temperature from Ref. [63] and the renormalization scale from Ref. [64].

The effects of each terms in the Debye mass (10) cannot always be explored separately due to the following reason: in the weak coupling regime, the soft scale (26.5049996376038  />) at the leading-order related to the screening of electrostatic fields is well separated from the ultra-soft scale (28ba0e-7d14-4a84-b03a-f0bf328af551.jpg width=53.105001449585 height=32.680001449585  />) related to the screening of magnetostatic fields. In such regime, it appears meaningful to see the contribution of each terms in the Debye mass separately. But when the coupling becomes large enough (which is indeed the case), the two scales are no longer well separated. So while looking for the next-to-leading corrections to the leading-order result from the ultra-soft scale, it is not a wise idea to stop at the logarithmic term, since it becomes crucial the number multiplying the factor 22a-80a6-10604dac1710.jpg width=32.680001449585 height=28.5950003623962  /> to establish the correction to the LO result. In fact the Debye mass in the NLO term is always smaller than the LO term because of the negative (logarithmic) contribution (28-93c8-4628-a227-7f4983034af9.jpg width=74.480001449585 height=32.680001449585  />) to the leadingorder term, while the full correction (all 228b.jpg width=37.8099992752075 height=32.680001449585  /> terms) to the Debye mass results positive. So, we consider only three forms of the Debye masses, viz. leading-order result (2585-4f96-b09c-aad02ad626f3.jpg width=37.8099992752075 height=32.680001449585  />), non-perturbative form (20cc0556e.jpg width=37.8099992752075 height=32.680001449585  />), and lattice parametrized form (22-65f6-419d-8e36-90c8c09293a0.jpg width=104.119998550415 height=32.680001449585  />) extracted from lattice free energy [25] to study the dissociation phenomena.

Thus we have finally computed the binding energies for 24.5099992752075 height=28.5950003623962  /> and 2b552acb.jpg width=24.5099992752075 height=28.5950003623962  /> and plotted them in Figures 1 and 2, respectively where different curves denote the choice of the Debye masses used to calculate the binding energy from Equation (9). We consider three cases for our analysis: pure gluonic, 2-flavor and 3-flavor QCD. There is a common observation in all figures that the binding energies show strong decrease with the increase in temperature. In particular, binding energies obtained from 2a1aa6fe1dc.jpg width=37.8099992752075 height=32.680001449585  /> and 2cd683c-6050-48c9-b4f6-b4fa2a00b535.jpg width=30.5900007247925 height=32.680001449585  /> give realistic variation with the temperature. The temperature dependence of the binding energies show a quantitative agreement with the results based on the spectral function technique calculated in a potential model for the non-relativistic Green’s function [64]. On the other hand, when we employ non-perturbative form of the Debye mass (25f-8324-5e23dc6c065c.jpg width=37.8099992752075 height=28.5950003623962  />) the binding energies become unrealistically small compared to its zero temperature value and also compared to the binding energies employing 2-a85d-430f-972c-c15ba48a3da2.jpg width=37.8099992752075 height=32.680001449585  /> and24f518ad-0312-4a1d-8377-cf5d130fde05.jpg width=30.5900007247925 height=32.680001449585  />. This anomaly can

2c60-85d7-4a6e-8b38-f7ee544ced07.jpg width=764.655023193359 height=258.30499420166  />

Figure 1. Dependence of χc binding energy (in GeV) on temperature T/Tc.

2f7-3091-4195-b0cc-8221038f0a45.jpg width=764.655023193359 height=258.30499420166  />

Figure 2. Dependence of χb binding energy (in GeV) on T/Tc.

be understood by the fact that the value of 2a822602-fdde-4d25-9dd8-9d9299179336.jpg width=37.8099992752075 height=32.680001449585  /> is significantly larger than both 2ea-bc23-b4f3e0cd1742.jpg width=37.8099992752075 height=32.680001449585  /> and 247925 height=32.680001449585  /> so that the binding energies become substantially smaller. This observation indicates that the present form of the nonperturbative corrections to the Debye mass may not be the complete one, the situation may change once the non-perturbative contributions to Debye mass are incorporated and then evaluate the binding energy. Thus, the study of temperature dependence of binding energy is poised to provide a wealth of information about the nature of dissociation of quarkonium states in a thermal medium which will be reflected in their dissociation temperatures discussed in the next section.

In addition, we take advantage of all the available lattice data, obtained not only in quenched QCD (2a-435a-95d0-7abe8296e7db.jpg width=62.319998550415 height=30.5900007247925  />), but also including two and, more recently, three light flavors. We are then in a position to study also the flavor dependence of the dissociation process.

4. Dissociation Temperatures

It has been customary to consider a state dissociated when its binding energy becomes zero. In principle, a state is dissociated when no peak structure is seen, but the widths shown in the spectral functions from the current potential model calculations are not physical. Broadening of the states as the temperature increases is not included in any of these models. In [64] authors have argued that no need to reach zero binding energy (2abb.jpg width=66.405001449585 height=28.5950003623962  />) to dissociate, but when 28.5950003623962  /> a state is weakly bound and thermal fluctuations can destroy it. However, others have set a more conservative condition for dissociation [65]:211-4a8e-840b-413025c07286.jpg width=134.80499420166 height=32.680001449585  />, where 2-603c-4330-9868-ff7fceb1e34d.jpg width=46.930001449585 height=32.680001449585  /> is the thermal width of state. However, we now calculate the upper bound of the dissociation temperature by the condition for dissociation:

2fdd-49e1-bc32-ec5f18200dc8.jpg width=136.8 height=43.8900007247925 /> (11)

where the string tension (2-ef14ffe3c3e7.jpg width=20.425 height=19.3799996376038  />) is 0.184 2-8a10-51ec7ce81738.jpg width=49.019998550415 height=28.5950003623962  /> and critical temperatures (2fccd8927f.jpg width=20.425 height=28.5950003623962  />) are taken as22-fbce-4167-8318-3d439fec8dec.jpg width=79.6100028991699 height=26.5049996376038  />, 297a-7e58-4477-8ec7-fc20819b77bb.jpg width=79.6100028991699 height=26.5049996376038  />and 203-4b4a-a01e-0c3194095294.jpg width=77.6149971008301 height=26.5049996376038  /> for pure gluonic, 2-flavor and 3-flavor QCD medium, respectively [66]. However, the choice of the mean thermal energy (246dc-fd29-491f-8681-0f3e628af5da.jpg width=19.3799996376038 height=20.425  />) is not rigid because even at low temperatures 268e-9d13-4680-8cac-90dd49285c1c.jpg width=57.1900007247925 height=28.5950003623962  /> (say) the Bose/Fermi distributions of partons will have a high energy tail with partons of mechanical energy250-2bf3-49c6-a530-5d1eea98541d.jpg width=125.58999710083 height=35.719998550415  />.

The dissociation temperatures for 24.5099992752075 height=28.5950003623962  /> and 24.5099992752075 height=28.5950003623962  /> are listed in Table 1 with the Debye mass in the leading-order. It is found from Table 1 that2858f31fd3.jpg width=24.5099992752075 height=28.5950003623962  />’s are dissociated at24a-c046-4a26-8eb6-bf63387e8803.jpg width=53.105001449585 height=28.5950003623962  />, 245261ab.jpg width=53.105001449585 height=28.5950003623962  />, and 2dc535fec.jpg width=53.105001449585 height=28.5950003623962  /> for the pure, 2-flavor, and 3-flavor QCD, respectively whereas247351a5.jpg width=24.5099992752075 height=28.5950003623962  />’s are dissociated comparatively much higher temperature which seems justifiable because of their higher binding energy. This

2c4851d.jpg width=438.99500579834 height=111.244998550415  />

Table 1. Upper(lower) bound on the dissociation temperature (TD) for χc and χb (in unit of TC) using the leading-order term in Debye mass,20323a6c-85e3-4c25-9a28-886bac83cc81.jpg width=35.719998550415 height=28.5950003623962  />.

finding agrees with the results obtained from the lattice correlator studies [15]. This is an interesting observation in the literature on the flavor (system) dependence of the dissociation temperature. This dependence is essential while calculating the screening energy density (energy density of the system at the dissociation temperature) in various descriptions of QGP (2604eba-a989-4e1c-bad3-5756c9650a09.jpg width=106.21000289917 height=30.5900007247925  />) for the study of 2-b44a15c40ac5.jpg width=39.805001449585 height=28.5950003623962  /> survival within the screening scenario in an expanding QGP. On the other hand, employing lattice parametrized form, 2-2842-4f15-8a25-8bdc888bf064.jpg width=30.5900007247925 height=32.680001449585  />for the Debye mass we obtain the values (Table 2) much smaller than the leading-order results where 24.5099992752075 height=28.5950003623962  /> is dissociated below 23-4285-bdd9-f4f3240732f3.jpg width=20.425 height=28.5950003623962  /> and 2556db46f.jpg width=24.5099992752075 height=28.5950003623962  /> is dissociated just above2.jpg width=20.425 height=28.5950003623962  />. However, this finding agrees with the results from the potential model studies [64] but does not agree with the lattice correlator studies. Finally, when we use non-perturbative form of the Debye mass, the dissociation temperatures come out to be unrealistically small which defy physical interpretation. Summarizing the results, we conclude that as we move from perturbative to non-perturbative domain, the binding energies become smaller and smaller. As a consequence the dissociation temperatures obtained also become smaller. This is due to the hierarchy in the Debye masses:221b6963-76ad-4aee-b0bb-a1a07b430146.jpg width=150.1 height=32.680001449585  />. In fact, 2-9530-48b6-a4e9-3d25a4fe6fbe.jpg width=30.5900007247925 height=32.680001449585  />is 1.4 times greater than 259056fc-d0d5-4239-9b2a-5d58d8d9a4cd.jpg width=41.894998550415 height=32.680001449585  /> while 264a-4ab6-8b70-cbd96e905fd3.jpg width=41.894998550415 height=32.680001449585  /> is much greater than both 22-a5af-5357323c8ccb.jpg width=37.8099992752075 height=32.680001449585  /> and2540fde.jpg width=30.5900007247925 height=32.680001449585  />.

However, if we treat the partons in high temperature to be relativistic, we could replace the mean thermal energy by 2b2c6-d655-456e-81c7-d96fd9ffe6fe.jpg width=28.5950003623962 height=22.4200003623962  /> (instead of2a1-920e-1957e4b6c8e5.jpg width=19.3799996376038 height=20.425  />) to obtain the lower bound for the dissociation temperatures. It is found that all entries in Tables 1 and 2 have been decreased by 20% - 25% approximately giving the lower bound of the dissociation temperatures (inside the first bracket). To compare our results quantitatively with the recent results [64] based on the spectral function technique calculated in a potential model with a similar description of the system (for 3-flavor QCD with 20.425 height=28.5950003623962  /> = 192 MeV), we tabulated the upper limit on the dissociation temperatures with the same form of Debye mass used in Ref. [64] in Table 3 giving a good agreement with their results.

Finally, it is learnt that the inclusion of non-perturbative corrections to the Debye mass (2752075 height=32.680001449585  />) lead to unusually smaller value of the dissociation temperatures for both 24.5099992752075 height=28.5950003623962  /> and2-a31f-6a2f45ae3887.jpg width=24.5099992752075 height=28.5950003623962  />. This does not immediately imply that the non-perturbative effects should be ignored. It is

2da6-4e27-8680-3ede78fed27f.jpg width=437.95 height=113.33500289917  />

Table 2. Same as Table 1 but with the lattice parametrized form of the Debye mass257ab05e-0cdf-467a-96c1-405b4ef071bb.jpg width=28.5950003623962 height=28.5950003623962  />.

27ab1da-b9f0-4720-9d9f-f140a901e88b.jpg width=438.99500579834 height=82.65  />

Table 3. Upper bound of the dissociation temperatures with TC = 192 MeV and the lattice parametrized form of the Debye mass [64].

rather interesting to investigate the disagreement between the non perturbative result obtained with a dimensionalreduction strategy and the Debye mass arising from the Polyakov-loop correlators. Only future investigations may throw more light on this issue.

5. Conclusions

In conclusion, we have studied the dissociation of 213d-44f0-8c4c-1fa2efa1600b.jpg width=26.5049996376038 height=26.5049996376038  /> states (2390dded06.jpg width=24.5099992752075 height=28.5950003623962  />and2ae-4f95-4c05-aa3a-dc13c54d19ef.jpg width=24.5099992752075 height=28.5950003623962  />) in the charmonium and bottomonium spectra in the hot QCD medium. We have employed the medium modified form of the heavy quarkpotential in which the medium modification causes the dynamical screening of both the (color) charge and the range of the potential which, in turn, lead to the temperature dependent binding energy of 2e5e3-cad0-4740-b780-1cc8b715300f.jpg width=24.5099992752075 height=28.5950003623962  /> and24.5099992752075 height=22.4200003623962  />. We have then studied the temperature dependence of the binding energy of 2b8-b4d5-472e-b08c-e9ac7f8afec5.jpg width=26.5049996376038 height=34.675  /> and 2-3493-4278-800c-8e6db0dedf0e.jpg width=26.5049996376038 height=34.675  /> states in the pure gauge and realistic QCD medium by incorporating the fourthorder corrections (in the screened charge) coming from the spin-dependent terms to the binding energies of 2b6-a30a-ec2e97e0194e.jpg width=24.5099992752075 height=28.5950003623962  /> and 2be4-4c96-a388-ebf197376486.jpg width=24.5099992752075 height=22.4200003623962  /> states, respectively. For this purpose, we have adopted a formulation [45,46], in which a variational treatment of the relativistic two-fermion boundstate system in QED [45,46] has been developed to compute the spin-dependent corrections.

Next we have determined the dissociation temperatures of 29-6e4e-4482-91f2-d2c06df7f81f.jpg width=24.5099992752075 height=28.5950003623962  /> and 213-433b-8cb7-bc25a2422a88.jpg width=24.5099992752075 height=28.5950003623962  /> employing the Debye mass in the leading-order and the lattice parametrized form. Our estimates with the Debye mass extracted from the lattice free energy are consistent with the finding of recent works based on potential models [64] whereas the Debye mass in the leading-order gives the identical results based on the lattice correlator studies. We have further shown that the inclusion of non-perturbative contributions to the Debye mass lower the dissociation temperatures substantially which looks unfeasible to compare to the spectral analysis of lattice temporal correlator of mesonic current [15]. This leaves an open problem of the agreement between these two kind of approaches. This could be partially due to the arbitrariness in the criteria/definition of the dissociation temperature. To examine this point we have estimated both the upper and lower bound on the dissociation temperatures by fixing the mean thermal energy 201ced-b980-486b-9bd9-1be708548c2f.jpg width=19.3799996376038 height=20.425  /> and252-f75e-42b2-bc40-d037684d2078.jpg width=28.5950003623962 height=22.4200003623962  />, respectively. Thus, this study provides us a handle to decipher the extent upto which non-perturbative effects should be incorporated into the Debye mass.

In brief, we obtained the binding energies and the dissociation temperatures of 24.5099992752075 height=28.5950003623962  /> and26192e-a5fa-41d1-a1bd-09bfe5f5f7ef.jpg width=24.5099992752075 height=28.5950003623962  />. This enables us to investigate their flavor dependence and temperature dependence. We have estimated the upper bound on the dissociation temperatures of 2-4bc0-afd4-7f1c6a1ee8e1.jpg width=24.5099992752075 height=28.5950003623962  /> and206194bd4c8.jpg width=24.5099992752075 height=28.5950003623962  />. We found that these estimates obtained by employing the lattice parametrized Debye mass show good agreement with the prediction in [64]. On the other hand, estimates with the perturbative result of the Debye mass are consistent with the predictions of lattice correlator studies [15,19,24,28].


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