Hot QCD equations of state and response functions for quark-gluon plasma
aa r X i v : . [ h e p - ph ] J u l Hot QCD equations of state and response functions for quark-gluon plasma
Vinod Chandra , ∗ Akhilesh Ranjan , † and V. Ravishankar , ‡ Department of Physics, Indian Institute of Technology Kanpur, UP, India, 208 016 and Raman Research Institute, C V Raman Avenue, Sadashivanagar, Bangalore, 560 080, India (Dated: November 17, 2018)We study the response functions (chromo-electric susceptibilities) of quark-gluon plasma as afunction of temperature in the presence of interactions. We consider two equations of state forhot QCD. The first one is fully perturbative, of O ( g ) EOS and, and the second one which is O [ g ln(1 /g ) + δ ], incorporates some non-perturbative effects. Following a recent work (PhysicalReview C 76 , 054909(2007)), the interaction effects contained in the EOS are encapsulated interms of effective chemical potentials(˜ µ ) in the equilibrium distribution functions for the partons.Byusing them in another recent formulation of the response functions( arXiv:0707.3697 ), we determineexplicitly the chromo-electric susceptibilities for QCD plasma. We find that it shows large deviationsfrom the ideal behavior. We further study the modification in the heavy quark potential due tothe medium effects. In particular, we determine the temperature dependence of the screeninglengths by fixing the effective coupling constant Q which appears in the transport equation bycomparing the screening in the present formalism with exact lattice QCD results. Finally, westudy the dissociation phenomena of heavy quarkonium states such as c ¯ c and b ¯ b , and determine thedissociation temperatures. Our results are in good agreement with recent lattice results. Keywords:
Response function; non-Abelian permittivity; Quark-Gluon plasma; hot QCDequation of state; equilibrium distribution function; chemical potential; RHIC.
PACS : 25.75.-q; 24.85.+p; 05.20.Dd; 12.38.Mh
I. INTRODUCTION
It is expected that at high temperatures ( T ∼ − M eV ) and high densities ( ρ ∼ Gev/f m ) nu-clear matter undergoes a deconfinement transition to thequark-gluonic phase. This phase is under intense inves-tigation in heavy ion collisions, and already, interest-ing results have been reported by Relativistic Heavy IonCollider(RHIC) experiments [1]. As an important de-velopment, flow measurements[2] suggest that close tothe transition temperature T c , the quark-gluon plasma(QGP) phase is strongly interacting — showing an al-most perfect liquid behavior, with very low viscosity toentropy ratio — rather than showing a behavior close tothat of an ideal gas. See Ref. [3] for a comprehensive re-view of experimental observations from RHIC, and Ref.[1, 4, 5, 6, 7] for other recent experimental results. On theother hand, lattice computations [8, 9] also suggest thatQGP is strongly interacting even at T = 2 T c . This find-ing has been reproduced by a number of other theoret-ical studies — by employing AdS/CFT correspondencein the strongly interacting regime of QCD[10], by molec-ular dynamical simulations for classical strongly coupledsystems[11], and by model calculations with Au-Au datafrom RHIC [12, 13]. [55]If this be the case, as it indeed appears to be, then ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected],[email protected] the plasma interactions would be largely in the non-perturbative regime; in this regime, few analytic tech-niques are available for a robust theoretical analysis. Ef-fective interaction approaches are needed. In this direc-tion, considerable work has already been done and werefer the reader to Ref. [14, 17, 18, 19, 20, 21, 22, 23] forsome of the theoretical results.The effective approaches emphasize the collective ori-gin of the plasma properties which can be best under-stood within a semi-classical framework. Indeed, in arecent work [24], the successes of hydrodynamics in inter-preting and understanding the experimental observationsfrom RHIC has been reviewed. Since more exciting anddiscerning data is expected from LHC experiments soon,and given the above context, it is worthwhile exploringsemi-classical techniques to understand the properties ofQGP in heavy ion collisions. In this context, it is knownby now [25, 26, 27, 28] that a classical behavior emergesnaturally when one considers hard thermal loop(HTL)contributions. A local formulation of HTL effective ac-tion has been obtained by Blaizot and Iancu who havesucceeded in rewriting the HTL effective theory as a ki-netic theory with a Vlasov term [29, 30, 31, 32]. A signif-icant development in this direction is the realization thatthe HTL effects are, in fact, essentially classical and thatthey are much easier to handle within the frame work ofclassical transport equations [33, 34, 35]. Thus, the semi-classical techniques appear hold the promise of providingtools to understand the bulk properties of QGP.The present paper continues the theme, and its centralaim is to combine the kinetic equation approach whichyields the transport properties, with the hot QCD equa-tions of state to make predictions which can be perhapstested in heavy ion collisions. Recently, Ranjan and Rav-ishankar have developed a systematic approach to deter-mine fully the response functions of QGP, with a specialemphasis on the color charge as a dynamical variable[14]. In parallel, Chandra, Kumar and Ravishankar havesucceeded in adapting two hot QCD EOS to make pre-dictions for heavy ion collisions [41]. They have shownthat the interaction effects which modify the equationsof state can be expressed by absorbing them into effec-tive fugacities ( z q,g ) of otherwise free or weakly interact-ing quasi quarks and gluons. Since the analysis in Ref.([14]) was illustrated only for (the academically interest-ing) case of ideal quarks and gluons, it is but natural tobring the two studies together and explore what the hotQCD EOS have to predict for heavy ion collisions. Wetake up this program in this paper.The main result of this paper is the determination ofthe modification that the heavy quark potential under-goes in a medium constituted by interacting QGP, aspredicted by the two EOS which we consider. After de-termining the screening length as a function of tempera-ture, we focus on the Cornell potential[36] and study thedissociation mechanism for c ¯ c and b ¯ b states. The resultsare rather surprising and may as well signal the inapplica-bility of these EOS to describe the deconfined phase. Onthe other hand, if the transition from the confined to thedeconfined state is not a phase transition as several stud-ies predict [37], it may still be possible to attribute somephysical significance to the predictions of these EOS. Weundertake the project here. We show that, by using oneof the phenomenological EOS is quite a good approxi-mation to the more rigorous lattice results, the value ofthe phenomenological coupling constant that occurs inthe Boltzmann equation can be fixed. Ultimately, thephysical viability or otherwise of the results need to beestablished by comparing them repeating the analysis of[41] with the lattice EOS. That will be taken up in aseparate paper.We consider two specific hot QCD equations of state:The first, which we call EOS1 is perturbative, with con-tributions up to O ( g )[38, 39]. The second EOS has a freeparameter δ , and is evaluated upto O [ g log(1 /g )][42].We denote it by EOS δ . δ may be fine tuned to get areasonably good agreement [42] with the lattice results[40], which we exploit here. Both the EOS are expectedto be valid for T > T c [42], and EOS δ is reliable beyond T ∼ T c .The paper is organized as follows: In section II, weintroduce the two hot QCD equations of state and out-line the recently developed method[41] to adapt them formaking definite predictions for QGP at RHIC and theforthcoming experiments at LHC. In section III, we ob-tain the expressions for the response functions of interact-ing QGP and in section IV, we study their temperaturedependence in detail. In section V, we study the mod-ifications in heavy quark potential due to the hot QCDmedium. We further study the temperature dependenceof the Debye screening lengths in hot QCD. We investi-gate the “melting phenomena” of heavy quarkonia suchas J/ Ψ and b ¯ b in the medium, and extract the dissoci- ation temperature. In doing so we also relate the phe-nomenological charge that occurs in the transport equa-tion to lattice and experimental observables. We con-clude the paper in section VI. II. HOT QCD EQUATIONS OF STATE ANDTHEIR QUASI-PARTICLE DESCRIPTION
There are various equations of state proposed for QGPat RHIC. These include non-perturbative lattice EOS[40], hard thermal loop(HTL) resumed EOS[43] and per-turbative hot QCD equations of state [38, 39, 42]. In thepresent paper, we seek to determine the chromo-electricresponse functions for QGP by employing two EOS: (i)the fully perturbative O ( g ) hot QCD EOS proposed byArnold and Zhai[38] and Zhai and Kastening [39], and(ii) The EOS of O [ g (ln(1 /g ) + δ )] determined by Ka-jantie et al [42], by incorporating contributions from non-perturbative scales, gT and g T . We employ the methodrecently formulated by Ranjan and Ravishankar[14] toextract the chromo-electric permittivities of the medium.EOS1 reads P g = 8 π β (cid:26) (1 + 21 N f
32 ) −
154 (1 + 5 N f
12 ) α s π + 30(1 + N f α s π ) + (cid:20) (237 . . N f − . N f + 1352 (1 + N f α s π × (1 + N f N f
12 )(1 − N f
33 ) ln[ µ MS β π ] (cid:21) ( α s π ) +(1 + N f (cid:20) − . − . N f − . N f + 4952 (1 + N f N f
33 ) ln[ µ MS β π ] (cid:21) ( α s π ) (cid:27) + O ( α s ) ln( α s )) . (1)while EOS δ is given by P g ln(1 /g ) = P g + 8 π T (cid:20) . . N f + 7 . N f − (cid:18) N f (cid:19) (cid:18) − N f (cid:19) ln( µ MS πT ) (cid:21) × (cid:16) α s π (cid:17) (ln 1 α s + δ ) . (2)As mentioned earlier, δ is an empirical parameter, in-troduced to incorporate phenomenologically the undeter-mined contributions at O ( g ). It also acts as a fittingparameter to get the best agreement with the lattice re-sults. .. δ = 1 . δ = 1 . δ = 1 . δ = 0 . δ = 0 . δ = 0 . TT c R FIG. 1: (Color online) Relative equation of state( wrt idealEOS) for pure gauge theory plasma as a function of
T /T c forvarious values of δ . N f = 3, δ = 1 . N f = 3, δ = 1 . N f = 3, δ = 1 . N f = 3, δ = 0 . N f = 3, δ = 0 . N f = 2, δ = 1 . N f = 2, δ = 1 . N f = 2, δ = 1 . N f = 2, δ = 0 . N f = 2, δ = 0 . TT c R FIG. 2: (Color online) Relative equation of state wrt ideal EOSfor full QCD plasma with N f = 2 , T /T c forvarious values of δ . A. The underlying distribution functions
The construction of the distribution functions that un-derlie the EOS, in terms of effective quarks and gluonswhich act as quasi-excitations, has been discussed byChandra et. al., [41] in the specific context of EOS1 andEOS δ . To review the method briefly, all the terms thatrepresent interactions are collected together by recastingthem as effective fugacities ( z q,g ≡ exp( µ q,g )) for the oth-erwise free quarks and gluons. Of course, the pure gaguetheory case is simply obtained by putting the numberof flavors, N F = 0 in the EOS. Thus, µ g represents theself interactions of the gluons, while µ f encapsulates thequark-quark and the quark gluon interaction terms. Im-portantly, the two EOS of interest to us are valid when T > T c , and in this range, the quantities ˜ µ q,g ≡ βµ q,g are perturbative parameters. Thus, it is possible to solvefor ˜ µ f,g self consistently through a systematic iterativeprocedure. In this procedure, all the temperature effectsare contained in the effective fugacities z ≡ z ( α s ( T /T c )),where we display the dependence on the temperature andcoupling constant explicitly. It has been shown in Ref.[41] (where the details can be found) that one can tradeoff the dependence of the effective fugacities on the renor-malization scale ( µ ¯ MS ) by their dependence on the crit- ical temperature T c . For that purpose, one utilizes theone loop expression of α s ( T ) at finite temperature givenby [15] α s ( T ) = 18 πb log( T /λ T ) = α s ( µ ) | µ = µ ¯ MS ( T ) µ ¯ MS ( T ) = 4 πT exp( − ( γ E + 1 / λ T = exp( γ E + 1 / π λ MS . (3)employing which the dependence on µ ¯ MS is eliminated,in favor of T c . Consequently, the effective chemical po-tentials get to depend only on T /T c . Note that effectivefugacities have merely been introduced to capture the in-teraction effects present in hot QCD equations of state.Once the distribution functions are in hand, the studyof transport properties is a straight forward exercise ifwe employ the analysis put forth by Ranjan et. al. [14].In Figs. 1 and 2, we display the behavior of EOS δ forvarious values of the parameter δ . The figures show thepure gauge theory contributions to the EOS and full QCDseparately. We remark parenthetically that the studiesin the earlier work [41] were confined to EOS1 and thespecial case δ = 0 in EOS δ . For the details on EOS1 andEOS δ for δ = 0, we refer the reader to Ref. [41] (see Fig.1-7 of Ref.[41]). First of all, we see that as δ increases inmagnitude, the EOS, for both pure gauge theory and fullQCD, become softer, with P/P I taking smaller values, wedenote the the ratio P/P I by R . Kajantie[42] obtainsthe best fit with the lattice results of Boyd et. al.[44] bychoosing a value δ = 0 .
7. We find that to get agreementwith the more recent results of Karsch [40], δ ≈ . T > T c . In short, we findthat the range of values 0 . ≤ δ ≤ . δ . This follows from thefreedom in choosing the QCD renormalization scale athigh temperature. This has been investigated in detailby Blaizot, Iancu and Rebhan [45]. The value of δ in thepresent paper has been obtained by employing the oneloop expression for the running coupling constant and theQCD renormalization scale determined in Ref.[15]. Weintend to study the quasi-particle content of HTL andHDL equations of state[45, 46] and lattice equation ofstate in future.The behavior of the corresponding fugacities, as a func-tion of temperature, is shown in Fig.3. It may be seenthat 0 < z g,q < . N f = 3 N f = 2Pure gauge theory T/T c E ff ec t i v e f u ga c i t y FIG. 3: (Color online) Effective parton fugacities ( z g,q ) quarksdetermined from EOS δ as a function of temperature. Notethat the behavior is shown for δ = 1 . III. RESPONSE FUNCTIONS FORINTERACTING QGP
Recently Ranjan and Ravishankar [14] have deter-mined the form of chromo-electric response functions forcollision less quark-gluon plasma within the frameworkof semi-classical transport theory. They have set up thetransport equation in the extended phase space includ-ing the SU(3) group space corresponding to dynamicalcolor degree of freedom. They have taken the distri-bution function in a coherent state basis defined overthe extended single particle phase space R ⊗ C G , where C G = G / H is the phase space corresponding to the colordegree of freedom, obtained as a coset space by factoringthe group space by the stabilizer group H of any refer-ence state in the Hilbert space. Having been employed tostudy the ideal case, the formalism has not been appliedto examine the behavior of the plasma with a realisticEOS. We employ the results of the previous section andrectify this drawback, by incorporating the interactioneffects as represented by EOS1 and EOS δ .A brief comment on the response functions. In contrastto electrodynamic plasma, the chromo-electric responsehas a richer structure. Apart from the standard permit-tivity which we shall call Abelian and denote by ǫ A , thereare additional response functions, their number depend-ing on the color carried by the partons. Thus, quarkshave an additional response function which affects thenon-Abelian coupling. The corresponding permittivitywill be called non-Abelian, and denoted by ǫ N . The twofunctions exhaust the response in the quark sector. Thegluonic sector, arising from the adjoint representation ofthe gauge group admits yet another kind of response,corresponding to tensor excitations. These excitationsare not allowed in the quark sector (which emerges fromthe fundamental representation of the gauge group). Weconsider each of these response functions for the inter-acting QGP. The response functions are obtained in thetemporal gauge.Consider first the familiar Abelian component of theresponse ǫ A . For an isotropic plasma(in the absence of chromo-magnetic fields), its expression is given by [14]˜ ǫ A ( ω, ~k ) = 1 + Q I ( ω, ~k ) (4)where Q = Q a Q a is the color charge magnitude squared,and I is determined by the equilibrium distribution func-tion thus: Z ω − ~k · ~pε ∂f eq ∂p i d ~p ≡ k i I ( ω, ~k ) , The non-Abelian response function, which has beenevaluated in the long wavelength limit, is given by˜ ǫ N ( ω, ω ′ ) = { Q I ( ω ′ , ~k ′ ) (cid:12)(cid:12)(cid:12) ~k ′ =0 ω } (5)where I is defined as I ( ω, ~k ) = 13 T r Z p j ε ( ω − ~k · ~pε ) ∂f eq ∂p i d ~p ! . We recall that the new constitutive Yang-Mills equations,in the presence of the medium, are given by˜ ρ a ( ω, ~k ) + iQ ˜ E ai ( ω, ~k ) k i I o ( ω, ~k ) − Q f alm ω Z I ( ω ′ , ~k ′ ) (cid:12)(cid:12)(cid:12) ~k ′ =0 ˜ A li ( ω − ω ′ , ~k − ~k ′ ) × ˜ E mi ( ω ′ , ~k ′ ) dω ′ d ~k ′ = 0 . (6)˜ j aj ( ω, ~k ) + iQ ˜ E ai ( ω, ~k ) δ ij I ( ω, ~k ) (cid:12)(cid:12)(cid:12) ~k =0 = 0 . (7)As pointed out in [14], the Abelian and non-Abelianresponses are not independent of each other. Gauge in-variance relates them, by virtue of which we can obtainboth from a common generating function as follows: I = 1 k ∂∂ω Z ln ( ω − ~k · ~pε ) k i ∂ p i f eq d pI = − T r (cid:18) ∂∂k j Z ln ( ω − ~k · ~pε ) ∂ p i f eq d p (cid:19) . (8)We further recall that these expansions are determinedwhen the system is displaced slightly from its equilib-rium, in the collisionless limit. A. Ideal response
It is convenient to first write the expressions for the re-sponses of ideal distributions for quarks and gluons. Theresponses due to EOS1 and EOS δ get a simple modifica-tion over their ideal forms since we have mapped success-fully the interaction effects into quasi free partons witheffective fugacities. Thus, in the ideal case we have, forthe quarks,˜ ǫ ( q ) A = [1 + 2 π Q T N f k {− ωk ln (cid:12)(cid:12)(cid:12)(cid:12) ω + kω − k (cid:12)(cid:12)(cid:12)(cid:12) + 2 } ] (9)and the non-Abelian response function is given by˜ ǫ ( q ) N = { − π Q T N f ωω ′ } . (10)The imaginary part of Abelian(˜ ǫ A ) and non-Abeliancomponent (˜ ǫ N ) of the chromo-electric permittivity canbe easily evaluated by the standard Landau iǫ prescrip-tion. These are needed to obtain landau damping whichwe do not study here.The contribution to the permittivity from the gluons isclosely related, and not independent of the contributionof the quarks written above. Indeed, if we define thesusceptibilities A ( q,g ) = ˜ ǫ ( q,g ) A − N ( q,g ) = ˜ ǫ ( q,g ) N − A ( q ) = N f A ( g ) , N ( q ) = N f N ( g ) . (11)where N F is the number of flavors. In short, for thetotal susceptibility, we have the simple relation χ A,Nq = N F χ A,Ng , B. Interaction effects
We now consider the modification that the above ex-pressions undergo permittivities arising because of thenew EOS. Recall that the corresponding equilibrium dis-tribution functions differ from each other only in theirform for the chemical potentials µ q,g . The responses thusdepend on the interactions implicitly through an explicitdependence on z q,g .Considering the gluonic case, i. e. , pure gauge theoryfirst, we get the expressions for the two permittivities as˜ ǫ A = [1 + 2 π Q T g ′ ( z g )3 k {− ωk ln (cid:12)(cid:12)(cid:12)(cid:12) ω + kω − k (cid:12)(cid:12)(cid:12)(cid:12) + 2 } ] , (12)and the non-Abelian response function is ˜ ǫ N = { − π Q T g ′ ( z g )9 1 ωω ′ } . (13)The function g ′ ( z g ) ≡ π g ( z g ) where g ( z g ) is definedvia the integral below. Z ∞ x ν − z − g exp ( x ) − dx = Γ( ν ) g ν ( z g ) g ν ( z g ) has the series expansion g ν ( z g ) = ∞ X l =1 z lg l ν for z g ≪ . Note that g ′ (1) = 1 gives the ideal limit.Similarly, the corresponding expressions for in thequark sector are obtained as˜ ǫ A = [1 + 2 π Q T N f f ′ ( z f )3 k {− ωk ln (cid:12)(cid:12)(cid:12)(cid:12) ω + kω − k (cid:12)(cid:12)(cid:12)(cid:12) + 2 } ] (14)and the non-Abelian response for effective quarksreads: ˜ ǫ N = { − π Q T N f f ′ ( z f )9 1 ωω ′ } . (15)The function f ′ ( z f ) ≡ π f ( z f ) where f ( z f ) is definedvia the integral below. Z ∞ x ν − z − f exp ( x ) + 1 dx = Γ( ν ) f ν ( z f ) f ν ( z f ) = ∞ X l =1 ( − l − z lf l ν for z f ≪ f ′ (1) = 1. IV. EFFECTIVE CHARGES AND RELATIVESUSCEPTIBILITIES
Eq.(12-15) admit a simple physical interpretation,when compared with their counterparts Eq.(9 -10). In-deed, the sole effect of the interactions on the transportproperties is to merely renormalize the the quark and thegluon charges Q g,q as shown below: Q g → ¯ Q g = Q g ′ ( z g ); Q q → ¯ Q q = Q q f ′ ( z f ) . The renormalization factors g ′ ( z g ) , f ′ ( z f ) further possessthe significance of chromo-electric susceptibilities, rela-tive to the ideal values. To see that, we note that theAbelian and the non-Abelian strengths for gluons as wellas quarks suffer the same renormalization reflecting theunderlying gauge invariance. Furthermore, the expres-sions for the relative susceptibilities are given by, R = χ ( z ) χ (1) ≡ A ( z ) A (1) = N ( z ) N (1) = (cid:26) f ′ ( z f ) for quarks, g ′ ( z g ) for gluons (16)and R q,g = χ ( q ) ( z f ) χ ( g ) ( z g ) ≡ A ( q ) ( z f ) A ( g ) ( z g ) = N ( q ) ( z f ) N ( g ) ( z g ) = f ′ ( z f ) N f g ′ ( z g ) . (17)Note that the relative susceptibilities are entirely func-tions of the single variable T /T c , and are independent of( ω, k ). The dependence of the susceptibilities on ( ω, k )has already been studied in detail in Ref.[14]. We merelyconcentrate on the temperature dependence below.Before we go on to discuss the susceptibilities and otherbulk properties, we point out an essential care to be takenin using the above susceptibilities for determining theresponse of the plasma. For pure gauge theory, only thegluonic part contributes, while for the full QCD, we haveto necessarily take the contribution from both the quarkand the gluonic sector. We discuss both the cases below.The response functions for the full QCD is obtained byaveraging up the above calculated response functions forquark as well as gluon plasma. The relative susceptibilityfor full QCD plasma is given by R ′ = χ (˜ z ) χ (1) ≡ A (˜ z ) A (1) = N (˜ z ) N (1)= N f f ′ ( z f ) + 2 g ′ ( z g ) N f + 2 , (18)where ˜ z is the effective fugacity of partons in full QCDplasma. A. Behavior of the susceptibilities
We now proceed to study the behavior of the relativesusceptibilities displayed in Eqs.(16), (17) and (18) asfunctions of temperature. As observed, relative suscepti-bilities for both quarks and gluons scale with
T /T c . Wehave plotted the relative susceptibilities R , R qg and R ′ as functions of T /T c (See Figs.4-7), for both EOS1 andEOS δ . Please note that we have chosen δ = 1 . δ .Fig.4 shows the relative susceptibility of a purely glu-onic plasma as a function of temperature for EOS1 andEOS δ .We see From Fig. 4 that the susceptibility of a purelygluonic plasma is weaker in the presence of interactions,approaching its ideal value asymptotically with increas-ing temperatures. Equivalently, there is a decrease in thevalue of the phenomenological coupling Q , relative to itsideal value.The behavior of quark gluon plasma is not qualitativelydifferent from that of a purely gluonic plasma, as may be EOS δ, δ = 1 . TT c R FIG. 4: (Color online) Relative susceptibility, g ′ ( z g ) (see Eq.(16), for pure gauge theory plasma as a function of T /T c forEOS1 and EOS δ ( δ = 1). . N f = 3, EOS δN f = 2, EOS δN f = 3, EOS1 N f = 2, EOS1 TT c R ′ FIG. 5: (Color online) Relative susceptibility, defined in Eq.(18), for the full QCD plasma as a function of
T /T c , for EOS1and EOS δ ( δ = 1). We have studied the cases N f = 2 , seen from Fig.5. In other words, the quark contribu-tion is of the same order as the purely gluonic contribu-tion. However, the relative contribution from the quarksand the gluons does depend on the EOS considered. In-deed, with EOS1 (where interactions up to O ( g ) areincluded), Fig.6 shows that the quark contribution dom-inates slightly over the gluonic contribution for N F = 2.The dominance is more pronounced for the more realisticcase N F = 3. In contrast, we see from Fig. 7, that EOS δ (with δ = 1) predicts that the gluonic contribution ismarginally larger for N F = 2 and becomes sub dominantwhen N F = 3. This distinction between the two EOS isof no practical consequence since, given T c ∼ M eV ,one has to necessarily work with N F = 3 at T = 2 T c . . N f = 3 N f = 2 TT c R q g FIG. 6: (Color online) Ratio of the quark to gluonic contribu-tions to the susceptibility (see Eq.(17) as a function of
T /T c ,as predicted by EOS1. N f = 3 N f = 2 TT c R q g FIG. 7: (Color online) Ratio of the quark to gluonic contribu-tions to the susceptibility (see Eq.(17) as a function of
T /T c ,for EOS δ , δ = 1.. V. THE HEAVY QUARK POTENTIAL
Now we shall apply the results of the previous sectionsto discuss the heavy quark potential in the presence ofinteracting medium. We consider the Cornell potential φ ( r ) = − αr + Λ r where α and Λ are phenomenological constants. Thefirst term shows the Coulombic behavior and dominatesat small distance while the second term causes linear con-finement, dominating at large distances.It had been expected earlier that the long range partof the Cornell potential does not survive in the quarkgluon phase. This expectation assumes a phase transi-tion from the hadronic to deconfined phase. More recentstudies[37] indicate that in all likelihood, deconfinementis not a phase transition, but a crossover. If such to bethe case, there is no reason to expect the linear part ofthe potential to disappear completely. With this in mind,we study the modifications of both the Coulomb and lin-ear terms, and examine how reasonable the EOS underconsideration are. Since the potential has no explicit color dependence, itis sufficient to employ the Abelian components of the per-mittivities. At ω = 0, the quark and gluon permittivitieshave the form˜ ǫ q ( k, T ) = 1 + 16 πQ T k f ( z q )˜ ǫ g ( k, T ) = 1 + 16 πQ T k g ( z g ) . (19)Therefore the full permittivity reads˜ ǫ ( k, T ) = (˜ ǫ g + ˜ ǫ q )2= 1 + 8 π Q T k (cid:20) N f f ( z q ) + g ( z g ) (cid:21) ≡ m D k , (20)in terms of the Debye mass m D = 8 πQ T (cid:2) N f f ( z q ) + g ( z g ) (cid:3) .The q ¯ q potential undergoes a modification due to themedium via ˜ ǫ ( k, T ), as given by ˜ φ ( k ) → ˜ φ ( k ) / ˜ ǫ ( k, T ) ≡ ˜ φ s ( k, T ). We note that in determining the Fourier trans-form of Cornell potential, we regulate the linear termexactly the same way we regulate the Coulomb term, bymultiplying with an exponential damping factor. Thedamping is switched off after the Fourier transform isevaluated. The Fourier transform is thus obtained as˜ φ ( k ) = − r π k − k √ π (21)The modified potential thus acquires the form˜ φ s ( k, T ) = − r π αk + m D − √ π Λ k ( k + m D ) . (22)We note that for a gluonic plasma, m D =16 πQ T g ( z g ).On comparing Eq.22 with Eq.21 we infer the renormal-ization of the couplingsΛ eff = Λ1 + m D k ; α eff = α m D k . A. Screening of the heavy quark potential
Of interest to us is the form of the potential in the realspace, as a function of spatial separation. The inverseFourier transform yields it to be φ s ( r, T ) = ( 2Λ m D − α ) exp ( − m D r ) r − m D r + 2Λ m D − αm D . (23)It follows from the above equation that the mediumtransforms the linear potential to the long range Coulombform, just as it modifies the bare Coulomb term to theshort ranged Yukawa.. The modified potential is notshort ranged; it is not confining either. To appreciatethis, note that at large T , the above expression reducesto φ s ( r, T ) ∼ − m D r − αm D (24)Thus, contrary to the Maxwellian plasmas which supportonly short range interactions, the two EOS predict thatthe heavy quark potential continues to be long ranged, al-though absolute confinement, which was a quintessentialfeature of the unscreened potential, is lost. It might aswell be that the above results signify that the EOS fail todescribe the hadronic matter in its deconfined state, Onthe other hand, since the transition from the hadronic toQGP phase could be a cross over, and not a phase tran-sition [37], it could be possible that the above result isnot entirely devoid of physical significance. If we adoptthe latter view, if only for the purposes of analysis, adiscussion of screening cannot, therefore rely entirely onthe interpretation of inverse Debye mass as a screeninglength in its usual sense. We address the issue below .Let us consider the high temperature limit of the po-tential, given by Eq.(24). Ignoring the additive contribu-tion, the energy of the q ¯ q in the ground state is simplygiven by E g = m q Λ m D , where m q is the mass of heavy quark. N f = 3 N f = 2Pure gauge TT c λ D FIG. 8: (Color online) Debye screening length for gluonicand quark-gluon plasmas as a function of
T /T c for EOS1. The binding energy is, of course, temperature depen-dent and approaches zero as T → ∞ . At any finite tem-perature though, the quarks possess a thermal energy E T ∼ T (by equipartition theorem), leading to an ion-ization of the quarkonium when E T matches the bindingenergy. The dissociation temperature T d is determined δ = 1 . δ = 1 . δ = 0 . TT c λ D FIG. 9: (Color online) Debye screening length for pure gaugetheory plasma in EOS δ as a function of T /T c for various valuesof δ . by the matching conditions. In the case of pure gaugetheory, m q Λ π Q T c = ( T D T c ) g ( z g ) . (25)And for full QCD: m q Λ π Q T c = 14 ( T D T c ) (cid:18) N f f ( z q ) + g ( z g ) (cid:19) (26) B. Estimation of Q and a determination of thescreening length The above equation is still not amenable to comparisonwith experiments since it has the undetermined parame-ter Q . To estimate Q , we need an additional input whichwe obtain by comparing the screening length obtained asa solution of Eqs. (25) and (26) with the lattice results,reported by Kazmarek and Zantow [47]. Note that thescreening lengths for gluonic and quark gluonic plasmasare respectively given by λ gD = 1 QT s(cid:18) πg ( z g ) (cid:19) , (27) λ D = 1 QT s(cid:18) π ( N f f ( z q ) + g ( z g )) (cid:19) . (28)Recalling that our results are valid for T > T c , wematch the pure gauge theory result with the lattice values λ ∼ .
15 fm, and T c = 0 .
27 GeV. We obtain λ gD = 0 . .
27 14 Q ( T /T c ) 1 p ( πg ( z g )) . (29)This leads to the estimate Q ∼ .
15. The temperaturedependence of the screening lengths can be thereafterdetermined for the two equations of state. We emphasizethat the choice δ = 0 . δ and the lattice EOS for gluonic plasma [40]. C. Dissociation temperatures for quarkonia
Since there are no free parameters left, it is a straightforward task to determine the dissociation temperaturesfor the heavy quark bound states. We are principally in-terested in J/ Ψ and b ¯ b states, for which we have gatheredthe results in Table. 1, after obtaining graphical solutionsfor Eq.(25) and Eq.(26). We have employed the values m c = 1 . GeV, m b = 4 . GeV and Λ = 0 . GeV for thequark masses and the strength of the Cornell potential.It is noteworthy that the dissociation temperatures areall roughly in the range T D ≈ (2 − T c , which is higherthan the temperatures achieved so far. Since the temper-atures expected at LHC is in the range T ∼ T c − T c ,one may expect to test these predictions there.We now turn our attention to compare hot QCD es-timates for dissociation temperatures with other theo-retical works. In a recent paper, Satz[48] has studiedthe dissociation of quarkonia states by studying their in-medium behavior. These estimates were based on theSchr¨odinger equation for Cornell potential. In a morerecent work, Alberico et al [49] reported the dissociationtemperatures for charmonium and bottomonium statesfor N f = 0 and N f = 2 QCD. In this work, they havesolved the Schr¨odinger equation for the charmonium andbottomonium states at finite temperature in the presenceof temperature dependent potential– computed from thelattice QCD. We have quoted these results in Table 2.The estimates for N f = 0 and N f = 2 cases for bothEOS1 and EOS δ (Table 1) are closer to Ref.[48]. On theother hand the estimates for J/ Ψ dissociation temper-atures for both EOS1 and EOS2 are larger than thatof Ref.[49] while bottomonium dissociation temperatureestimates are slightly smaller. We do not have latticeestimates at present to compare the dissociation temper-atures for N f = 3 QCD. However the hot QCD esti-mates are consistent with the lattice predictions[50] onthe survival of heavy quarkonia states near 2 T c and pre-dictions of dynamical N f = 2 QCD by Aarts et al [51].Along these results, we wish to mention the very recentestimates on dissociation temperature reported by M´ocsyand P´etreeczky[52]. Their estimates for J/ Ψ dissociationtemperature is 1 . T c and for Υ is 2 T c . The estimates forboth EOS1 and EOS δ are larger as compare to these re-sults.
1. Comparison of Debye screening length with lattice results
Finally, with a view to benchmark our estimates of thescreening lengths, by comparing them with the recent lat-
TABLE I: The dissociation temperature( T D ) for variousquarkonia states (in unit of T c ).Hot EOS Quarkonium Pure QCD N f = 2 N f = 3EOS1 J/ Ψ 2.2 2.62 2.46Υ 2.5 3.14 2.94EOS δ J/
Ψ 1.86 2.38 2.24 δ = 0 . δ J/ Ψ 1.95 2.45 2.32 δ = 1 . δ J/ Ψ 2.03 2.52 2.40 δ = 1 . T D ) for variousquarkonia states (in unit of T c ) from Ref.[48] and Ref.[49].The first and third rows are the estimated values for disso-ciation temperature from Ref.[48] and second and fourth arefrom Ref.[49] Quarkonium N f =0 N f = 2 J/ Ψ 2.1 > J/ Ψ 1.40 1.45Υ >
2Υ 2.96 3.9 tice results reported by Kazmarek and Zantow [47], weplot 2 λ D as a function of T /T c . The results are shown forEOS1 as well as EOS δ , for three values δ = 0 . , . , . δ ∼ a pos-teriori our choice for the parameter. N f = 3, δ = 1 . N f = 3, δ = 1 . N f = 3, δ = 0 . N f = 2, δ = 1 . N f = 2, δ = 1 . N f = 2, δ = 0 . TT c λ D FIG. 10: (Color online) Debye screening length for full QCDplasma as predicted by EOS δ as a function of T /T c for variousof δ . VI. CONCLUSIONS AND OUTLOOK
In conclusion, we have successfully extracted the quasi-free particle content of two hot QCD equations of statesand used them to determine the chromo-electric permit-tivities within the standard Boltzmann-Vlasov kineticapproach. The Abelian and the non-Abelian componentsof the permittivity are obtained, for pure gauge theoryand the full QCD. We have shown that the effect of theinteractions is to merely renormalize the magnitude ofthe effective color charge, Q . We have used the permit-tivities to study critically the modifications in a realisticheavy quark potential. The dissociation temperaturesare carefully estimated, by fixing the magnitude of Q byan explicit matching with a lattice result. The valuesobtained are quite close to the exact lattice results. Theviability of the two EOS, especially EOS δ is thus phe-nomenologically well supported. Our analysis suggests strongly, and in agreement with the lattice results, that J/ Ψ suppression can be seen in QGP only for T ≥ T c .A true test of the above predictions would be possible ifwe succeed in extracting a quasi particle description fromthe lattice EOS. Studies are under way in this direction.It should also be of interest to extend the analysis toother signatures like strangeness enhancement, and alsofor QGP with a finite baryonic chemical potential [53,54]. These will be taken up in a later work. Acknowledgments:
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