We reexamine the scenario of homogeneous nucleation of the quark-gluon plasma produced in ultra-relativistic heavy ion collisions. A generalization of the standard nucleation theory to rapidly expanding system is proposed. The nucleation rate is derived via the new scaling parameter Z. It is shown that the size distribution of hadronic clusters plays an important role in the dynamics of the phase transition. The longitudinally expanding system is supercooled to about 3 6%, then it is reheated, and the hadronization is completed within 6 10 fm/c, i.e. 5 10 times faster than it was estimated earlier, in a strongly nonequilibrium way. PACS: 12.38.Mh; 12.39.Ba; 25.75.-q; 64.60.Q

Bravina, Larissa V.

Csernai, László Pal

Greiner, Walter

Stöcker, Horst

Zabrodin, Eugene E.

English

Bravina, Larissa

Hochschulschriftenserver - Universität Frankfurt am Main

arXiv:hep-ph/9806207 v1   31 May 1998Supercooling of rapidly expandingquark-gluon plasmaE.E. Zabrodina,b, L.V. Bravinaa,b,1, L.P. Csernaic, H. St¨ ockeraand W. Greineraa Institute for Theoretical Physics, University of Frankfurt,Robert-Mayer-Str. 8-10, D-60054 Frankfurt, Germanyb Institute for Nuclear Physics, Moscow State University, 119899 Moscow, Russiac Department of Physics, University of Bergen, All´ egaten 55, 5007 Bergen, Norway1 Alexander von Humboldt FellowAbstractWe reexamine the scenario of homogeneous nucleation of the quark-gluon plasmaproduced in ultra-relativistic heavy ion collisions. A generalization of the standardnucleation theory to rapidly expanding system is proposed. The nucleation rate isderived via the new scaling parameter λZ. It is shown that the size distribution ofhadronic clusters plays an important role in the dynamics of the phase transition.The longitudinally expanding system is supercooled to about 3 − 6%, then it isreheated, and the hadronization is completed within 6 − 10 fm/c, i.e. 5 − 10 timesfaster than it was estimated earlier, in a strongly nonequilibrium way.Key words: QGP−hadrons phase transition; Homogeneous nucleation; Clusterdistribution; Bjorken model.PACS: 12.38.Mh; 12.39.Ba; 25.75.-q; 64.60.QbPreprint submitted to Elsevier Preprint 12 December 2005Homogeneous nucleation of quark-gluon plasma (QGP) produced in ultra-relativistic heavy ion collisions has been subject of interest during the last fewyears. Obviously, the plasma must hadronize, but the mechanism of hadroniza-tion still remains an open question. In case of a second order phase transition,percolation model calculations are useful, also for strong supercooling of theplasma during a ﬁrst order phase transition, provided the expansion is rapidand the amount of the plasma converted into hadronic matter is not suﬃcientlylarge to reheat the growing volume.Results of simulations [1,2] show that the supercooling of the system, if quenchedrapidly into the metastable or even unstable region, may be as large as 20%.Then it is questionable whether the homogeneous nucleation scheme is appro-priate for the description of the hadronization process. Other processes whichmay cause a rapid hadronization of QGP have been studied recently [3,4].In the present paper we propose a generalization of the standard nucleationtheory to rapidly expanding systems. If the size distribution of hadronic bub-bles is taken into account, the total rate of the plasma conversion turns outto be high enough to prevent the system from strong supercooling. The lastcircumstance makes relevant the application of the homogeneous nucleationscenario.The paper is organized as follows: after a sketch of the nucleation theory thedynamics of the plasma−hadrons phase transition in the Bjorken hydrody-namic model [5] is discussed. Finally, conclusions will be drawn.The starting point of the homogeneous nucleation theory is the assumption offormation of nucleating clusters within the initially homogeneous metastablestate. The rate of conversion of the metastable state into a thermodynami-cally stable phase is called nucleation rate. According to Langer [6], who hasextended the classical nucleation theory (see, e.g. [7] and references herein) toﬁeld theories, the rate of the relaxation of a metastable state is given byI =κ2πΩ0 exp(−β∆F∗) , (1)where κ and Ω0 is a dynamical prefactor and a statistical prefactor, respec-tively, ∆F∗ is the excess free energy of the cluster of critical size, β ≡ (kBT)−1,kB is the Boltzmann constant, and T is the temperature of the system.Let us consider a classical system with N degrees of freedom described by aset of N collective coordinates ξi, i = 1,...,N. The energy function E({ξ})of the system should have two locally stable states separated by the energybarrier. The point with minimal energy on the barrier {ξS} is the so-calledsaddle point.2The statistical prefactor Ω0 is a measure of the available phase space volumeof the saddle point region. If the ﬂuctuation corrections to the (mean ﬁeld)excess free energy of the cluster are absorbed into ∆F, the statistical prefactor[6] can be written asΩ0 = V 2πβ|λ1|!1/2, (2)where V = V 8πσ3|λ1|!3/2is the available volume of the saddle point subspace,V is the volume of the system, σ is the surface tension, and λ1 is the onlynegative eigenvalue of the matrix∂2E∂ξi∂ξjdeﬁned at the saddle point {ξS}.The dynamical prefactor κ determines the growth rate of the critical clusterat the saddle point. For a relativistic system of particles, where the thermalconductivity is absent because of the absence of the rest frame deﬁned by thebaryon net charge, the dynamical prefactor has been calculated [8] to beκ =4σ￿ζ + 43η￿(∆ω)2R3∗. (3)Here ζ and η is the shear and bulk viscosity, ∆ω is the diﬀerence in theenthalpy densities of the quark-gluon phase and hadronic phase, and R∗ is theradius of the critical cluster.We will explore a model QGP, produced in the collision of ultrarelativisticheavy ions, and subsequently rapidly quenched into the metastable regionbelow the critical temperature. Then the ﬁrst order phase transition shouldoccur. It will be triggered by the creation of hadronic clusters as a result of thethermodynamic ﬂuctuations. The Fisher droplet model [9] yields the changein the free energy of the system due to the formation of a spherical cluster ofradius R∆F(R) = −4π3R3∆p + 4πR2σ + 3τFβ−1lnRr0. (4)Here ∆p = ph − pqgp is the diﬀerence of the pressures inside and outside thecluster, τF = 2.2 is the critical exponent, and r0 is the radius of the smallestcluster formed in the system. The maximum of the ∆F(R) will be reached atthe critical radiusR∗ =￿a1 +qa32 + a21￿1/3+￿a1 −qa32 + a21￿1/3+ a3 , (5)3containing a1 =9τFa34α2 + a33 , a2 = −a23 , a3 =2σ3∆p, α =q4πσβ .Clusters whose radii are smaller than R∗ are shrinking, clusters whose radiiare larger than R∗ are growing, and clusters of critical size are in metastableequilibrium. It is convenient to parametrize the excess free energy of a clusterby the similarity number λZ = R∗q4πσβ [10] and reduced radius r = R/R∗β∆F = −￿τF +23λ2Z￿r3 + λ2Zr2 + 3τF lnrR∗r0. (6)Then for the only negative eigenvalue λ1, associated with the instability of thecritical cluster against growth or shrinking, we haveλ1 = −(9τF + 2λ2Z)β−1 . (7)Substitution of this result in Eq. (2) gives ﬁnally for the statistical prefactorΩ0 =4√π3√3 λZR∗!3 V(9τF + 2λ2Z)2 . (8)The result for the total nucleation rate I readsI = V83√3π λZR2∗!3 σ￿ζ +43η￿(∆ω)2 (9τF + 2λ2Z)2 exp(−β∆F∗) , (9)where ∆F∗ = −τFβ￿1 − 3lnR∗r0￿+λ2Z3βis the free energy of the critical cluster.The formula (9) is similar to, but also diﬀerent from the earliar expression [8].Near the critical temperature this rate drops since the critical radius R∗ in-creases to inﬁnity. Then the nucleation rate rises quickly when the temperaturefalls, and the ﬁrst order phase transition begins.From the deﬁnition of the nucleation rate, this is just a number of viableclusters in volume V passed through the critical region per unit of time. Ifthe total volume of the hadronic phase is not large enough to maintain thetemperature of the expanding system at a constant level due to the release ofthe latent heat, the temperature will continue to decrease.On the other hand, the critical radius drops very quickly with decreasingtemperature. Therefore, the subcritical clusters which started to dissolve (Fig.1, upper panel) suddenly become of supercritical size with respect to the newcritical radius R′c of the system (Fig. 1, lower panel). The general picture of theconversion of QGP into hadronic clusters will be rather complex. Clusters large4enough will grow, the smallest ones will continue to shrink, while the clustersof medium size will grow or shrink depending on temperature variations andcurrent critical radii in the system.It is worth noting that the model presented above is valid for the hadronizationof thermalized QGP produced in a large volume (of about 500fm3 or more).For smaller systems, the ﬁnite size eﬀects [11] lead to the shift of the criticaltemperature and rounding of the phase transition, but the discussion of theseproblems lies out of scope of the present paper.At given temperature T, the size distribution of clusters in the droplet modelwith the curvature energy is found to be [12]f(R)=Iq2π(9τF + 2λ2Z)exp"9τF + 2λ2Z2(r − 1)2#(10)×∞ Zrah3τF(a2 + a + 1) + 2λ2Za2iexp"−9τF + 2λ2Z2(a − 1)2#da .Hadronic matter in our calculations is represented by a discrete spectrum ofspherical bubbles starting from r0 = 1fm.Within the framework of the bag model the equations of state for two-ﬂavourQGP and for hadronic gas arepqgp =3790π2T4 − B , (11)ph =130π2T4 . (12)With the bag constant B1/4 = 235MeV this gives us the critical temperatureTc = 169 MeV.The Bjorken model of longitudinal expansion [5] yields for the evolution of theenergy density ededτ= −e + pτ. (13)The energy density of the whole system, as well as the enthalpy, is a linearcombination of the energy density of the plasma eqgp in the part of volumeoccupied by plasma, νqgp = Vqgp/Vtot, and the energy density of hadronic phaseeh in the rest of the volume, νh = 1 − νqgp,e = eqgpνqgp + (1 − νqgp)eh . (14)5To determine the viscous term in Eq. (3), note that the bulk viscosity is muchsmaller as compared with the shear viscosity, which has been derived in leadinglogarithmic order of QCD [13] for a QGP with two ﬂavours asη =1.29T 3α2S ln(1/αS). (15)Here αS = 0.23 is the strong coupling constant.The last important parameter for our analysis is the value of the surfacetension σ. Recent lattice QCD calculations [14,15] predict 0.01 < σ/T 3c < 0.1,i.e. 1.25 < σ < 12.5 MeV/fm2 for the given Tc. We will use σ = 2 and 5MeV/fm2 in our simulations.Following [1,2], our scenario of longitudinal expansion presumes that the sys-tem reaches the critical temperature at the time τcr = 8τinit = 3fm/c. FromFig. 2 it is apparent that the large amount of latent heat, released duringthe conversion process, is suﬃcient to prevent the expanding system from thesupercooling below 3 − 6%.To investigate the eﬀect of the dissipative processes on the course of the phasetransition, Fig. 2 shows also results of calculations, in which Eq. (13) has beenreplaced [16] bydedτ= −e + pτ+ζ + 43ητ2 . (16)One can see that the supercooling of a viscous plasma is weaker than that ofan ideal QGP. This circumstance delays the homogeneous nucleation by onlyabout 2−3 fm/c. Creation of hadronic bubbles is the main mechanism of theQGP conversion at the earlier stage of the transition. Apart from the ﬁrst fm/cof cooling the mixed system is reheated continuously, and the temperatureapproaches the critical one.This occurs up to ≈ 6 fm/c, then the behaviour of the system changes drasti-cally: diﬀusion growth of the ”old” bubbles (Fig. 3, upper panels) dominatesthe creation of new bubbles, which is practically turned oﬀ. When T is closeto Tc, the growth process stops, but it starts again immediately after a smallextra-cooling of the system. This is a highly unstable region. Even an insigniﬁ-cant rise of the temperature is suﬃcient for the system to hit the critical point,then the homogeneous nucleation scheme is no longer relevant. Therefore, thespinodal decomposition or the percolation scenario may be appropriate todescribe the strongly non-equilibrium hadronization of the rest of the QGP,which constitutes about 30 − 35% of the total volume.6The size distribution of the hadronic bubbles at diﬀerent stages of the phasetransition is shown in lower panels of Fig. 3. We see that the initially broadplateau of the distribution function becomes narrower with the rise of theradius due to the increase of the number of bubbles per unit of radial interval.The distribution functions reach their maximum values with a radius of about3 ≤ r ≤ 4.5fm. At the very beginning of the nucleation almost all bubblesenlarge their volumes. Then the bubbles of r ≤ 1.5fm dissolve because theirradii are smaller than the critical one. In contrast, the bubbles of r ≥ 4fmwill stop to grow only if the temperature of the system will be very close tothe critical temperature. Bubbles of size 1.5 ≤ r ≤ 4fm are either growingor shrinking due to the variations of the critical radius in the system. Thisleads to the appearance of the irregularities in the spectrum of small andintermediate-size bubbles (Fig. 3, right lower panel).As a consequence, there will be no unique source of pion emission at the freeze-out. The individual pairs of pions are coming from sources of r ≈ 1.5 − 2fm,while the major part of pions should come from relatively large sources of3 ≤ r ≤ 4.5fm. This can be checked by the analysis of the particle correlationdata and by the rapidity distributions, if the hadronic clusters will be wellseparated along the expansion axis; this problem has to be investigated.In conclusion, we study two-ﬂavour QGP undergoing a ﬁrst order phase tran-sition during the longitudinal expansion. The nucleation rate is derived viathe new scaling parameter λZ. It is shown that the distribution of hadronicclusters in size plays an essential role in the dynamics of plasma hadronizationnear the critical temperature: there is a signiﬁcant variation of the value ofcritical cluster in the system. The supercooling of QGP is found to be rela-tively moderate, 3 − 6% only, therefore the application of the homogeneousnucleation scenario seems to be quite reasonable.The completion time of the transition varies from 6 to 10 fm/c. It is stronglydependent on the absolute value of the interfacial energy. The weaker thesurface tension σ of the interface is, the faster the hadronization will be, andvice versa. Since recent lattice QCD results [15] favour lower values of σ, itis likely that the QGP hadronizes within ﬁrst eight fm/c’s, i.e. very close tothe currant estimations of the freeze-out time scale based on HBT data andmicroscopic models. Thus the completion time of the homogeneous nucleationstage is about 5 − 10 times shorter than those of the standard nucleationscheme obtained before [1,2].The size distribution of the hadronic clusters indicates that most of the pionsare emitted from sources of about 3 ≤ r ≤ 4.5fm. This signal may be inves-tigated experimentally by means of the particle interferometry and, probably,by the rapidity spectra of secondaries on an event-by-event basis.7Acknowledgements. We thank M.I. Gorenstein for useful discussions. L.B.and E.Z. are grateful to the Institute for Theoretical Physics, University ofFrankfurt for the warm and kind hospitality. L.B. acknowledges support ofthe Alexander-von-Humboldt Foundation.8References[1] L.P. Csernai, J.I. Kapusta, Phys. Rev. Lett. 69 (1992) 737.[2] L.P. Csernai, J.I. Kapusta, Gy. Kluge, E.E. Zabrodin, Z. Phys. C 58 (1993) 453.[3] T. Cs¨ org¨ o, L.P. Csernai, Phys. Lett. B 333 (1994) 494.[4] L.P. Csernai, I. N. Mishustin, Phys. Rev. Lett. 74 (1995) 5005.[5] J.D. Bjorken, Phys. Rev. D 27 (1983) 140.[6] J.S. Langer, Ann. Phys. (N.Y.) 41 (1967) 108; 54 (1969) 258.[7] J. Frenkel, Kinetic Theory of Liquids (Clarendon, Oxford, 1946), Ch. 7.[8] L.P. Csernai, J.I. Kapusta, Phys. Rev. D 46 (1992) 1379.[9] M.E. Fisher, Physics (NY) 3 (1967) 255.[10] L.V. Bravina, E.E. Zabrodin, Phys. Lett. A 202 (1995) 61.[11] C. Spieles, H. St¨ ocker, C. Greiner, hep-ph/9708280.[12] L.V. Bravina, E.E. Zabrodin, Phys. Rev. C 54 (1996) 2493.[13] H. Heiselberg, Phys. Rev. D 49 (1994) 4739.[14] Y. Iwasaki et al., Phys. Rev. D 49 (1994) 3540.[15] F. Karsch et al., Nucl. Phys. B (Proc. Suppl.) 53 (1997) 413.[16] P. Danielewicz, M. Gyulassy, Phys. Rev. D 31 (1985) 53.9 t = t1 T = T1  Rc                    R Üf(R), a.u. clusters shrink clusters grow t = t1 + Dt T = T2 < T1  Rc                                      R  ¢f(R), a.u. clusters shrink clusters growFig. 1. Distribution of clusters as a function of their radius R. Due to the rapid falloﬀof the critical radius with temperature, the clusters of subcritical sizes (upper panel)formed at the moment τ1 become supercritically sized (lower panel) at τ1 + ∆τ.Consequently, the amount of matter converted into hadrons increases.100.91s = 2 MeV/fm2without viscous termwith viscous termT/Tc0.910 5 10 15s = 5 MeV/fm2without viscous termwith viscous termt   (fm/c)T/TcFig. 2. The temperature as a function of the proper time for the phase transitionQGP → hadrons during the longitudinal expansion. Upper and lower panel rep-resents the results of calculations with σ = 2 and 5 MeV/fm2, respectively. Thesolid/dashed curves correspond to calculations with Eq. (13)/Eq. (16).110.510 1 2 3 4 5s = 2total volumediffusion partcreation partVh / Vtott - tc  (fm/c)0 1 2 3 4 5s = 5total volumediffusion partcreation partt - tc  (fm/c)10-810-710-610-510-40 1 2 3 4 5s = 2s = 5Dt = 1 fm/cf (R)R   (fm)0 1 2 3 4 5s = 2s = 5final stageR   (fm)Fig. 3. Upper : The part of the QGP volume converted into hadrons for systemwith σ = 2 (left panel) and 5 MeV/fm2 (right panel). Solid lines correspond to thetotal volume fractions of hadrons, dashed lines denote the increase of the hadronicvolume due to the creation of new bubbles, and dash-dotted lines indicate theenlargement of the hadronic bubbles due to diﬀusion. Lower : Size distributionof the hadronic bubbles at ∆τ = 1fm/c after the beginning of the nucleation (leftpanel), and at the freeze-out (right panel) with σ = 2 (solid curves) and 5 MeV/fm2(dashed curves).12

Supercooling of rapidly expanding quark-gluon plasma

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