Effects of equation of state on nuclear suppression and the initial entropy density of quark gluon plasma
aa r X i v : . [ nu c l - t h ] O c t Effects of equation of state on nuclear suppression and the initial entropy density ofquark gluon plasma
Surasree Mazumder and Jan-e Alam
Theoretical Physics Division, Variable Energy Cyclotron Centre, 1/AF, Bidhan Nagar , Kolkata - 700064 (Dated: June 16, 2018)We study the effects of the equation of state on the nuclear suppression of heavy flavours in quarkgluon plasma and estimate the initial entropy density of the system produced at the highest RHICenergy. For this purpose we have used the experimental data on the charged particle multiplicityand the nuclear suppression of single electron spectra originating from the semi-leptonic decays ofopen charm and beauty mesons. We have used inputs from lattice QCD to minimize the modeldependence of the results. We obtain the value of the initial entropy density which varies from20 to 59 /fm depending on the value of the velocity of sound that one uses for the analysis. Ourinvestigation leads to a conservative value of the initial entropy density ∼ / fm with correspondinginitial temperature ∼
210 MeV well above the value of the transition temperature predicted by latticeQCD.
PACS numbers: 12.38.Mh,25.75.-q,24.85.+p,25.75.Nq
A thermalized system of quarks and gluons, calledquark gluon plasma (QGP) is expected to be formedin the collisions of two nuclei at ultra-relativistic ener-gies [1]. Rigorous experimental and theoretical efforts areon to create and characterize this novel, deconfined phaseof quarks and gluons. Lattice QCD (LQCD) calculationsindicate that at a temperature ∼
175 MeV the entropydensity ( s ) of the hadronic matter rises significantly dueto the release of colour degrees of freedom which are con-fined within the hadrons at zero temperature. Therefore,it is of foremost importance to determine the value of theinitial entropy density ( s i ) / initial temperature ( T i ) forthe system formed in nuclear collisions at RelativisticHeavy Ion Collider (RHIC) and Large Hadron Collider(LHC) and assess whether the system is formed in colourdeconfined phase or not. The focus of the present studyis to estimate s i or T i of the system formed at Au+Aucollisions at √ s N N = 200 GeV. For this purpose we strictto take inputs from experimental data and LQCD calcu-lations to minimize the model dependence of the outcomeof the present analysis.One of the possible way to estimate the value of the ini-tial entropy density is the extrapolation of the measured(final) observables backward in time through a suitabledynamical model. In absence of viscous loss the timereversal symmetry of the system is valid, therefore, themeasured multiplicity at the freeze-out of the system canbe used to estimate s i . The s i and the thermalizationtime ( τ i ) are constrained by the measured (final) hadronmultiplicity ( dN/dy ) by the following relation [2]: s i τ i = κ A ⊥ dNdy (1)where A ⊥ is the transverse area of the system can be de-termined from the collision geometry and κ is a knownconstant (=3.7 for massless bosons like pions). The valueof dN/dy which is connected to s i through Eq. 1 is read-ily available for different collision centralities [1]. In Eq. 1there are two unknown quantities, τ i and s i both of which can not be determined from a single equation involvinga single measured the dN/dy . Therefore, we choose an-other experimentally measured quantity the nuclear sup-pression of heavy quarks (HQ), R A A [3, 4], which is sen-sitive to the initial condition and hence is very useful toestimate s i .The advantage of choosing the heavy flavour are (a)they are produced in early hard collisions and hence canwitness the entire evolution of the QGP and (b) HQare Boltzmann suppressed at the temperature range ex-pected to be achieved in heavy ion collisions at RHIC,therefore, the HQ do not determine the bulk features ofthe QGP. The magnitude of R A A depends on the amountof drag the heavy quark faces during its propagationthrough QGP. The heavy quarks being the witness of theearly condition and the drag (and diffusion) coefficient isa temperature dependent quantity that makes the R A A agood probe for the measurement of initial temperature.Earlier various attempts have been made to explainthe experimental results on R AA - some of these are ad-dition of non-perturbative effects from the quasi-hadronicbound state [5], three-body scattering process [6], dissoci-ation of open heavy flavoured mesons by the thermal par-tons [7] and inclusion of temperature dependent strongcoupling [8].We briefly outline the procedure of evaluating the R AA for single electrons originating from the semi-leptonic de-cays of heavy mesons produced from the fragmentationof the HQ. The HQ while propagating through the QGPdissipates energy in the medium and hence its momen-tum gets attenuated. The magnitude of the momentumdegradation gets reflected in the experimentally mea-sured quantity, R AA mentioned above. Theoretically themomentum evolution of the HQ in the expanding QGPbackground can be described by using Fokker PlanckEquation (FPE) [9–19]. The evolution of the probe i.e. the HQ is described by the FPE: ∂f∂t = ∂∂p i (cid:20) A i ( p ) f + ∂∂p j [ B ij ( p ) f ] (cid:21) , (2)where A i = Z d k w ( p , k ) k i = γp i , (3)and B ij = 12 Z d k w ( p , k ) k i k j = Dδ ij . (4) γ and D are called the drag and diffusion coefficients,contain the interaction of the probe with the medium.To solve the FPE we need to supply the drag and diffu-sion coefficients and the initial HQ (charm and beauty)momentum distributions.There are two main processes through which the HQdissipates energy in the QGP: (i) energy dissipation cantake place due to the elastic collisions of the HQ withthe quarks, antiquarks and gluons in the thermal bath,(ii) the radiative process due to which the HQ emits softgluons (which subsequently get absorbed in the QGP)due to its interaction with the QGP. The details solu-tion of FPE with temperature and momentum depen-dent drag and diffusion coefficients including both theprocesses (i) and (ii) have been discussed in our earlierworks [18]. The initial momentum distributions of charmand bottom quarks at RHIC energy ( √ s N N = 200 GeV)have been taken from MNR code [20]. The method ofsolving the FPE numerically with temperature and mo-mentum dependent transport coefficients including otherissues are discussed in [18], therefore we do not repeatthose details here.Classically the induced radiation takes place due tothe jiggling motion of the propagating particle in themedium. Since the heavier particle jiggle less conse-quently induced energy loss is expected to be smaller[dead cone effect [21] (see also [22, 23])] for HQ com-pared to light particles. However, the experimental datafrom RHIC indicates similar amount of energy loss byheavy quarks and light partons in the measured kine-matic range. Various reasons like the anomalous massdependence of the radiative process due to the finite sizeof the QGP [24], development of dead cone due to highvirtuality of the partons resulting from the dismantlingof colour fields during the initial hard collisions [25] havebeen proposed as reasons for this observation. The au-thors in [26] concluded that the reduction in the energyloss of HQ due to radiative process is due to the deadcone effect but it is fair to mention that the issue is yetto be settled.The other effect which influences the radiative lossis the Landau-Pomeranchuk-Migdal(LPM) effect. Thiseffect originate due to the interplay between two timescales of the system [27]: the formation time ( τ F ) andthe mean scattering time ( τ sc ) of the gluon radiated from the HQ. LPM effect imposes certain constrain on thephase space of the emitted gluon [28, 29]. Both the deadcone and the LPM effects have been taken in to accountin evaluating the drag and diffusion coefficients in thepresent work (see [18, 19] for details).Now we discuss the QGP background with which theHQ interacts. The equation of state (EoS) plays a cru-cial role in describing the space time evolution of the ex-panding QGP from the initial state to the quark-hadrontransition point. We use boost invariant hydrodynamicmodel [30] with the LQCD calculation EoS [31] for thespace time description of the matter. The velocity ofsound ( c s ) as obtained in [31] from LQCD calculationsshows a significant variation with temperature (Fig. 1).It starts with a very low value of c s at T ∼ T c and thenincreases with T to reach the maximum value ( c s = 1 / P = c s ǫ . The EoS sets the expansion timescale for the system as τ e xp ∼ [(1 /ǫ ) dǫ/dτ ] − ∼ τ / (1+ c s )indicating the fact that lower value of c s makes the expan-sion time scale longer i.e. the rate of expansion slower.Therefore, for given values of T i and T c the life time ofthe QGP will be longer for smaller c s . The value of T c isfixed at 175 MeV.We solve the FPE numerically [18] for momentum de-pendent drag and diffusion coefficients to get the charmand beauty quarks momentum distribution. The solutionthen convoluted with the fragmentation functions [32] toobtain the transverse momentum distribution of the D and B mesons which subsequently decay to create lep-tons [33, 34].In the same way the lepton spectra from the heavyflavours produced in p-p collisions can be calculated fromthe charm and beauty quark distributions which enteras initial conditions to the FPE. The solution of FPEcontains the effects of drag (quenching) on the HQ whereas the the initial distributions of HQ does not contain anysuch effects, therefore the ratio of these two quantities,the nuclear suppression factor, R AA act as a marker forthe medium. This is observed experimentally through thedepletion of R AA at high transverse momentum ( p T ).As discussed above for larger c s the expansion timescale is shorter i.e. the QGP life time is smaller. con-sequently the HQ spends less time in the QGP whichultimately leads to less suppression of the single electronspectra originating from the decays of HQ. Therefore, wetake the following strategy to obtain the allowed range ofinitial temperature. We take the highest possible valueof c s (= 1 /
3) for the space-time description of the flow-ing QGP background, in the present approach this willlead to the maximum value of T i . In this case the HQ willspend the lesser amount of time in the QGP. Therefore, toachieve the experimentally measured R A A one will needlarger drag or in other word larger initial temperature.The results for c s = 1 / T i obtained from the analysis for this case is 300 MeV,the corresponding value of s i = 2 π g e ff T i / ∼ / fm .The value of g e ff ∼
38 is extracted from the variation of s/T with T provided by the LQCD calculations [31].In Fig. 3 results for c s = 1 / c s = 1 / c s = 1 / T i = 240MeV and s i ∼ . / fm , the data can be reproduced. c s ( T ) FIG. 1: Velocity of sound squared as a function tempera-ture [31]
For c s = 1 / c s = 1 / T i = 210MeV and s i ∼ . / fm .Further lowering of c s will make the value of τ i large(for given dN/dy ) enough to contradict other results likethe observation of large hadronic elliptic flow which re-quires small τ i (see [35] for review). That will also resultsin lower T i with which it will be difficult explain otherexperimental results.For all the theoretical results displayed in Figs. 2, 3and 4 we have kept the quantity dN/dy constant conse-quently the value of τ i changes to 0.6, 1.17 and 1.7 fm/cfor T i = 300 ,
240 and 210 MeV respectively. The changesin T i is forced by the change in the EoS. In Fig. 5 weshow the variation of T i with c s obtained by constraintsimposed by the experimental data on R A A and dN/dy .The value of T i varies from 210 to 300 MeV. In this con-text we compare the value of T i obtained in the work withsome of those reported earlier. In Refs. [5] the value of T i is obtained as ∼
375 MeV from the study of heavyquark suppression. From the simultaneous analysis oflight and heavy quarks suppressions in Ref. [36] a valueof T i = 400 MeV is obtained. The authors in Ref. [37–39]mentioned the values of the initial gluon rapidity distri-bution, dN g /dy , which may be converted to T i = 290 , T i obtained from the present analysis is well above thequark-hadron phase transition temperature, indicating T (GeV)00.511.52 R AA ( p T ) STAR (0−5%)C−−>eB−−>ePHENIX (0−10%)total
FIG. 2: (colour online) Variation of R AA with p T for c s = 1 / T i = 300 MeV. T (GeV)00.511.52 R AA ( p T ) STAR (0−5%)D−−>eB−−>ePHENIX (0−10%)total
FIG. 3: (colour online) Variation of R AA with p T for for c s =1 / T i = 240 MeV. the fact that the system formed in Au+Au collisions at √ s N N = 200 GeV might be formed in the partonic phase.In summary, we have studied the effects of the EoSon the suppression of single electrons originating fromthe decays of heavy flavours produced in Au+Au colli-sions at √ s N N = 200 GeV. We found that the initialtemperature may vary from 210 to 300 MeV dependingon the velocity of sound, which sets the scale for the ex-pansion that one uses. We have used experimental data(charged particle multiplicity and R A A of heavy flavours)and LQCD results ( c s , g e ff etc.) to keep the model de-pendence minimum. The effects of transverse expansionis neglected here. With the transverse expansion the HQwill (1) travel longer path (2) with diluted density. How-ever, the two competing effects (1) and (2) will have somesort of cancellation due to which our final conclusion may T (GeV)00.511.52 R AA ( p T ) STAR (0−5%)D−−>eB−−>ePHENIX (0−10%)total
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