We calculate the evolution of quark-gluon-plasma droplets during the
hadronization in a thermodynamical model. It is speculated that cooling as well
as strangeness enrichment allow for the formation of strangelets even at very
high initial entropy per baryon $S/A^{\rm init}\approx 500$ and low initial
baryon numbers of $A_{\rm B}^{\rm init}\approx 30$. It is shown that the
droplet with vanishing initial chemical potential of strange quarks and a very
moderate chemical potential of up/down quarks immediately charges up with
strangeness. Baryon densities of $\approx 2\rho_0$ and strange chemical
potentials of $\mu_s>350$~MeV are reached if strangelets are stable. The
importance of net--baryon and net--strangeness fluctuations for the possible
strangelet formation at RHIC and LHC is emphasized

Coffin, J. P.

Greiner, C.

Spieles, C.

Stoecker, H.

English

arXiv

We calculate the evolution of quark-gluon-plasma droplets during the hadronization in a thermodynamical model. It is speculated that cooling as well as strangeness enrichment allow for the formation of strangelets even at very high initial entropy per baryon S/Ainit H 500 and low initial baryon numbers of Ainit B H 30. It is shown that the droplet with vanishing initial chemical potential of strange quarks and a very moderate chemical potential of up/down quarks immediately charges up with strangeness. Baryon densi- ties of H 2 0 and strange chemical potentials of µs > 350 MeV are reached if strangelets are stable. The importance of net baryon and net strangeness fluctuations for the possible strangelet formation at RHIC and LHC is em- phasized. Pacs-Classif.: 25.15.tr, 12.38.Mh, 24.85.t

Spieles, Christian

Greiner, Carsten

Stöcker, Horst

Coffin, Jean Pierre

Hochschulschriftenserver - Universität Frankfurt am Main

arXiv:nucl-th/9508016 v1   9 Aug 1995Distillation of Strangelets for low initial µ/TC. Spielesa, C. Greinerb, H. St¨ ockera, and J. P. Coﬃnca Institut f¨ ur Theoretische Physik, J. W. Goethe-Universit¨ at, 60054 Frankfurt am Main, Germanyb Department of Physics, Duke University, Durham, NC 27708, U.S.Ac Centre de Recherches Nucl´ eaires de Strasbourg et Universit´ e Louis Pasteur Strasbourg, France(January 4, 2006)AbstractWe calculate the evolution of quark-gluon-plasma droplets during thehadronization in a thermodynamical model. It is speculated that coolingas well as strangeness enrichment allow for the formation of strangelets evenat very high initial entropy per baryon S/Ainit ≈ 500 and low initial baryonnumbers of AinitB ≈ 30. It is shown that the droplet with vanishing initialchemical potential of strange quarks and a very moderate chemical potentialof up/down quarks immediately charges up with strangeness. Baryon densi-ties of ≈ 2ρ0 and strange chemical potentials of µs > 350 MeV are reachedif strangelets are stable. The importance of net–baryon and net–strangenessﬂuctuations for the possible strangelet formation at RHIC and LHC is em-phasized.25.15.tr, 12.38.Mh, 24.85.tpTypeset using REVTEX1Strangelets can be thought of as strange multiquark clusters which should be more com-pressed than ordinary nuclei and may exist as (meta-)stable exotic isomers of nuclear matter[1]. It was speculated [2] that strange matter might resolve the dark matter issue, if it wouldbe absolutely stable.The possible creation — in heavy ion collisions — of such long-lived remnants of thequark-gluon-plasma, cooled and charged up with strangeness by the emission of pions andkaons, was proposed by Liu and Shaw [3] and Greiner et al. [4,5]. Thus, strangelets canserve as unambiguous signatures for the creation of a quark gluon plasma. The detectionof strangelets would verify exciting theoretical ideas with consequenses for our knowledge ofthe evolution of the early universe, the dynamics of supernova explosions and the underlyingtheory of strong interactions [6].Here we want to point out that such exotic states of matter can be created in heavy ioncollisions even at collider energies, where such a process has received no attention so far,because common belief was that the (strange) baryon densities vanish at midrapidity, bothat RHIC and LHC. We argue, however, that this conclusion was premature. This is due tothe following eﬀects:• ﬂuctuations of the stopping power can provide ﬁnite baryochemical potential µB atmid–rapidity in a small fraction of all events;• ﬂuctuations of the net–baryon and –strangeness content between diﬀerent rapidity binswithin one event can be quite large;• strange (anti–)baryon enhancement due to collective eﬀects (e. g. a chiral phase tran-sition);• strangeness and baryon distillery, which are inherent for the two–phase system (hadrongas/quark gluon plasma) for a wide parameter range.The last point stresses the signiﬁcance of the ’chemistry‘ of the system in the evolution ofthe phase transition.2In the following we adopt a model [5] for the dynamical creation of strangelets via thestrangeness separation mechanism [4]. Consider a ﬁrst order phase transition of the QGPto hadron gas. Strange and antistrange quarks do not hadronize at the same time fora baryon–rich system [4]. The separation mechanism can be viewed as being due to theassociated production of kaons (containing ¯ s quarks) in the hadron phase, because of thesurplus of massless quarks compared to their antiquarks. The strange quarks can combineto Λ-particles, but it is energetically favourable that s-quarks remain in the plasma, whenhadronization proceeds. The ratio fs of the net strangess over the net baryon numberquantiﬁes the excess of net strangeness. Both the hadronic and the quark matter phasesenter the strange sector fs  = 0 of the phase diagram almost immediately, which has up tonow been neglected in almost all calculations of the time evolution of the system. Earlierstudies addressed the case of a baryon–rich QGP with rather moderate entropy per baryon[4,5,7]. Now we focus on low initial baryon densities and high speciﬁc entropies, to matchthe expected conditions of heavy ion collisions at RHIC and LHC, where the search forstrangeness enters in the objective of the ALICE experiment [8].The hadronization transition has been described by geometric and statistical models,where the matter is assumed to be in partial or complete equilibrium during the whole(quasi-)isentropic expansion. A more realistic scenario must take into account the particleradiation from the surface of the hadronic ﬁreball before ‘freeze out’. Our model [5] com-bines these two pictures. The expansion of the QGP droplet during the phase transitionis described as a two-phase equilibrium; in particular the strangeness degree of freedomstays in chemical equilibrium because the complete hadronic particle production is drivenby the plasma phase. The nonequilibrium radiation is incorporated by rapid freeze-out ofhadrons from the outer layer of the hadron phase surrounding the QGP droplet. During theexpansion, the volume increase of the system thus competes with the decrease due to thefreeze–out. The global properties like (decreasing) S/A and (increasing) fs of the remainingtwo-phase system then change in time according to the following diﬀerential equations forthe baryon number, the entropy, and the net strangeness number of the total system:3ddtAtot = −ΓAHGddtStot = −ΓSHG (1)ddt(Ns − Ns)tot = −Γ(Ns − Ns)HG ,where Γ = 1AHG￿∆AHG∆t￿ev is the eﬀective (‘universal’) rate of particles (of converted hadrongas volume) evaporated from the hadron phase. The equation of state consists of the bagmodel for the quark gluon plasma and a mixture of relativistic Bose–Einstein and Fermi–Dirac gases of well established strange and non–strange hadrons up to in Hagedorn’s eigen-volume correction for the hadron matter [4]. Thus, one solves simultaneously the equationsof motion (1) and the Gibbs phase equilibrium conditions for the intrinsic variables, i.e. thechemical potentials and the temperature, as functions of time.Fig. 1 illustrates the increase of baryon concentration in the plasma droplet as an in-herent feature of the dynamics of the phase transition (cf. [2]). The origin of this resultlies in the fact that the baryon number in the quark–gluon phase is carried by quarks withmq ≪ TC, while the baryon density in the hadron phase is suppressed by a Boltzmannfactor exp(−mbaryon/TC) with mbaryon ≫ TC. Mainly mesons (pions and kaons) are createdin the hadronic phase. More relative entropy S/A than baryon number is carried awayin the hadronization and evaporation process [5], i.e. (S/A)HG ≫ (S/A)QGP. Ultimately,whether (S/A)HG is larger or smaller than (S/A)QGP at ﬁnite, nonvanishing chemical po-tentials might theoretically only be proven rigorously by lattice gauge calculations in thefurure. However, model equations of state do suggest such a behaviour, which would opensuch intriguing possibilities as baryon inhomogenities in ultrarelativistic heavy ion collisions.In the early universe shrinking quark droplets may — in analogy — contain the accumu-lated baryon number with possibly very high baryon density [2]. This mechanism yields aprimeval inhomegeneous nucleosynthesis [6,9], which is signaled by the abundances of thelight elements.What ‘initial’ conditions do we expect at collider energies? At RHIC energies one mightsee baryon stopping, dNB/dy > 0, on the average, at midrapidity. This can be due to4multiple rescattering, leading to a nonvanishing, positive quarkchemical potential µq [10]. Onthe other hand, relativistic meson–ﬁeld models, which, at high temperature, qualitatitivelysimulate chiral behaviour of the nuclear matter, exhibit a transition into a phase of masslessbaryons [11]. Including hyperons and Y Y –interaction [12] it shows that at µ ≈ 0 thedensities for all baryon species are of the order of ρ0 near the critical temperature. Thus,the fraction of (anti–)strange quarks increases drastically. Several hundred (anti–)baryons,many of them (anti–)hyperons may then ﬁll the hot midrapidity region (with net baryonnumber ≈ 0). Districts of non–vanishing net baryon (respectively anti–baryon) density withﬁnite s (¯ s) content will then occur stochastically. Thus, the ﬁnite chemical potential is locallycaused by the ﬂuctuations of newly produced particles, not by the stopped matter. If sucha phenomenon persists also for the deconﬁned phase, the eﬀect of baryon concentration andstrangeness separation may then result in the production of strangelets and anti-strangeletsin roughly equal amounts.Let us try to give a rough estimate of the possible size of ﬂuctuations in the net baryonand net strangeness number to be expected at midrapidity (or ﬂuctuations along diﬀerentrapidity intervals) around their mean. The average number of initial quarks and antiquarks(before hadronization) in a rapidity interval is approximately 1/2dNπ/dy  ∆y, if half of thepions are made by the quarks and the other half by the gluons. For RHIC energies dNπ/dyhas been estimated to lie between 1100 and 1600 and for LHC energies to lie up to nearly4000 in central Au+Au collisions [13]. Hence the quark number is roughly 500 for RHICand up to 2000 for LHC in a rapidity interval ∆y ∼ 1. (In an equilibrated plasma thetotal number of quarks is N ∼ ρqπR2∆z within ∆z = 1 − 2 fm in the early stage of thehydrodynamical expansion. According to ρq = g 34π2T 3ζ(3) ≈ 1.1T 3 for a degeneracy of g=12these numbers correspond to temperatures of T ∼ 250−500 MeV.) We take the number tobe N = 500. A similar consideration holds for strange and antistrange quarks, and we takehere Ns = 200. We now assume independent ﬂuctuations according to Poissonians withinthis rapidity interval. In fact the actual width of the ﬂuctuation at collider energies could bemuch broader (KNO–scaling of particle multiplicity distributions in elementary pp–collisions5[14]). To justify the assumption of independent ﬂuctuations of B and ¯ B despite of the localcompensation of quantum numbers, one has to estimate the typical relative momenta withina quark–antiquark pair. If one follows the parton cascade concept embodying perturbativeQCD [15], the average√ˆ s of ﬁrst parton–parton interactions (gg → q¯ q being the mostimportant contribution) should be of the order of 5−10 GeV at LHC energies. The producedB and ¯ B, carrying about 0.4 of the (anti–)quark momenta, would thus be separated inrapidity by at least one unit (assuming a transverse momentum of about 500 MeV).The net baryon number in the box described above will be |B| > 30 with a probability of0.5 %. About 0.1 % of the events will reach |B| > 30 with a strangeness fraction |fs| > 0.7.Hence ﬂuctuations are not negligible. If each pion carries about 3.6 units of entropy (whichis true for massless bosons), the entropy per baryon content in the ﬁreball isSAB≈ 3.6dNπ/dydNB/dy, (2)and thus for dNB/dy = 30 a range of 60 to 250 is formed. If the plasma is equilibrated,the ratio of the quarkchemical potential and the temperature |µ|/T is directly related to theentropy per baryon number via S|AB|!QGP≈3715π2 |µ|T!−1. (3)Accordingly the ratio then varies between 0.1 to 0.4.We now consider various ﬁreballs with an initial net baryon number of AB = 30 anda net strangeness fraction fs of either 0 or 0.7. The initial entropy per baryon ratios arechosen between 50 and 500. Table 1 summarizes the initial conditions (adjusted by the(initial) chemical potentials µq, µs and temperature) used to start the hadronization. It alsoshows the ﬁnal parameters of the quark droplet like the saturated strangeness content andbaryon number. One further, yet crucial input, is the bag constant employed to describethe equation of state of the (strange) quark matter droplet. Only for the bag constantsB1/4 ≤ 180 MeV strange matter does exist as a metastable state at zero temperature [4],being absolutely stable only for B1/4 < 150 MeV [2].6Fig. 2 shows the time evolution of the baryon number for S/Ainit = 200 and finits = 0.7 forvaryous bag constants. For B1/4 < 180 MeV a cold strangelet emerges from the expansionand evaporation process, while the droplet completely hadronizes for bag constants B1/4 ≥180 MeV (for B1/4 = 210 MeV hadronization proceeds without any signiﬁcant cooling ofthe quark phase, although the speciﬁc entropy S/A decreases by a factor of 2 from 200 toonly 100). The strangeness separation works also in these cases, as can be read oﬀ the largeﬁnal values of the net strangeness content, fs> ∼ 1.5 − 2. However, then the volume of thedrop becomes small, it decays and the strange quarks hadronize into Λ-particles and otherstrange hadrons.Fig. 3 shows the evolution of the two-phase system for S/Ainit = 200, finits = 0 and for abag constant B1/4 = 160 MeV in the plane of the strangeness fraction vs. the baryon density.The baryon density increases by more than one order of magnitude! Correspondingly, thechemical potential rises as drastically during the evolution, namely from µi = 16 MeV toµf > 200 MeV. The strangeness separation mechanism drives the chemical potential of thestrange quarks from µis = 0 up to µfs ≈ 400 MeV. Thus, the thermodynamical and chemicalproperties during the time evolution are quite diﬀerent from the initial conditions of thesystem.Even for high initial entropies, S/A ≈ 100 − 500, in the quark blob the entropy in theremaining droplet approaches zero at the end of the evolution (assuming B1/4 = 160 MeV).High initial entropies per baryon require more time for kaon and pion evaporation in orderto end up ﬁnally with the same conﬁguration of (meta–)stable strange quark matter.In conclusion, we have shown in the present model that the evolution of quark-gluon-plasma droplets during their hadronization may result in the formation of strangelets evenat very high initial entropy per baryon S/Ainit ≈ 500 and low initial baryon numbers ofAinitB ≈ 30. The distillation of very small strangelets of a size AB ≤ 10 (see Table 1) ispossible. We note that ﬁnite size eﬀects of describing small strangelets neglected here mightbecome of crucial importance [16]. Special (meta-)stable candidates are the quark-alpha[17] with AB = 6 and the H-Dibaryon state with AB = 2 [18]. Local net–baryon and net–7strangeness ﬂuctuations can provide suitable initial conditions for the possible strangeletcreation at RHIC and LHC. Droplets with vanishing initial chemical potential of strangequarks and a small chemical potential of up/down quarks quickly charge up with strangenessand baryon–number: if strangelets are stable, the droplet reaches strange chemical potentialsof µs > 350 MeV and two times ground state nuclear matter density!ACKNOWLEDGMENTSC.G. thanks the Alexander von Humboldt Stiftung for his support by a Feodor Lynenscholarship. This work was supported in part by the U.S. Department of Energy (grant DE-FG05-90ER40592), by the Gesellschaft f¨ ur Schwerionenforschung, Darmstadt, Germany, theBundesministerium f¨ ur Forschung und Technologie (F. R. G.), Deutsche Forschungsgemein-schaft and the Graduiertenkolleg Schwerionenphysik.8REFERENCES[1] A. R. Bodmer, Phys. Rev. D4, 1601 (1971); S. A. Chin and A. K. Kerman, Phys. Rev.Lett. 43, 1292 (1979); J. D. Bjorken and L. D. McLerran, Phys. Rev. D20, 2353 (1979)[2] E. Witten, Phys. Rev. D30, 272 (1984); E. Farhi and R. L. Jaﬀe, Phys. Rev. D30, 2379(1984)[3] H.-C. Liu and G.L. Shaw, Phys. Rev. D30, 1137 (1984)[4] C. Greiner, P. Koch and H. St¨ ocker, Phys. Rev. Lett. 58, 1825 (1987); C. Greiner, D. H.Rischke, H. St¨ ocker and P. Koch, Phys. Rev. D38, 2797 (1988)[5] C. Greiner and H. St¨ ocker, Phys. Rev. D44, 3517 (1992)[6] Proc. Int. Workshop on Strange Quark Matter in Phys. and Astr., Nucl. Phys. (Proc.Suppl.) 24B, 1-299 (1991), see H. Kurki–Suonio 67, and S. Bonometto 54[7] K. S. Lee, U. Heinz, Phys. Rev. D47, 2068 (1993)[8] Letter of Intent, CERN/LHCC/93–16/LHCC/I 4; J. P. Coﬃn, project presentation,meeting of the ALICE technical board, April 23 (1995)[9] J. H. Applegate and C. J. Hogan, Phys. Rev. D31, 3037 (1985)[10] A. Dumitru, D. H. Rischke, T. Sch¨ onfeld, L. Winckelmann, H. St¨ ocker and W. Greiner,Phys. Rev. Lett. 70, 2860 (1993)[11] J. Theis, G. Graebner, G. Buchwald, J. Maruhn, W. Greiner, H. St¨ ocker, J. Polonyi,Phys. Rev. D28 (1983) 2286[12] J. Schaﬀner, C. Greiner, C. B. Dover, A. Gal, H. St¨ ocker, Phys. Rev. Lett. 71 (1993)1328[13] X.-N. Wang and M. Gyulassy, Phys. Rev. D.44, 3501 (1991); K. Geiger, Phys. Rev.D47, 133 (1992)9[14] UA5 collab., G. J. Alner et al., Phys. Rep. 154 (1987) 247[15] K. Geiger and B. M¨ uller, Nucl. Phys. B 369 (1992) 600[16] J. Madsen, Phys. Rev. Lett. 70, 391 (1993) and Conf. on Strangeness and Quark Matter,Crete, Greece (1994)[17] F. C. Michel, Phys. Rev. Lett. 60, 677 (1988)[18] R. L. Jaﬀe, Phys. Rev. Lett. 38, 195 (1977)10FIGURES0 20 40 60 80 100 120 140t (fm/c)0.00.51.01.52.02.53.0/0S/Ainit=50S/Ainit=100S/Ainit=200S/Ainit=500FIG. 1. Time evolution of the net baryon density of a QGP droplet. The initial conditions arefinits = 0 and AinitB = 30. The bag constant is B1/4 = 160 MeV.110 20 40 60 80 100 120 140 160 180t (fm/c)051015202530ABB1/4=210 MeVB1/4=180 MeVB1/4=160 MeVB1/4=145 MeVFIG. 2. Time evolution of the baryon number for a QGP droplet with AinitB = 30, S/Ainit = 200,finits = 0.7 and diﬀerent bag constants.1210-12 5 1002/ 00.00.51.01.52.0fs10 fm/c20 fm/c30 fm/c40 fm/c50 fm/c60 fm/c70 fm/c80 fm/cFIG. 3. Evolution of a QGP droplet with baryon number AinitB = 30 for S/Ainit = 200 andfinits = 0. The bag constant is B1/4 = 160 MeV. Shown is the baryon density and the correspondingstrangeness fraction.13TABLESTABLE I. Various situations of an hadronizing plasma droplet. In the ﬁrst column the bagconstant for describing the plasma phase is listed. Then the initial conditions follow. The ﬁnalvalues for the baryon number, the strangeness fraction and the two chemical potentials at the end(or after) hadronization are listed in the last four columns.B1/4 S/Ainit finits AinitB ρinitB µinitq µinits µ/Tinit fﬁnals AﬁnalB ρﬁnalB µﬁnalq µinits(MeV) (fm−3) (MeV) (MeV) (fm−3) (MeV) (MeV)160 500 0 30 0.0067 6.55 0 0.060 1.99 2.36 0.352 224.3 396.0160 500 0.7 30 0.0068 5.02 4.01 0.046 1.96 2.71 0.339 216.9 386.0160 200 0 30 0.017 16.36 0 0.150 2.0 2.79 0.349 224.7 396.9160 200 0.7 30 0.017 12.55 10.04 0.115 2.0 3.37 0.350 223.7 396.3160 100 0 30 0.034 32.60 0 0.300 1.99 2.93 0.350 225.2 396.7160 100 0.7 30 0.034 25.07 20.09 0.185 2.0 3.96 0.352 223.4 396.2160 50 0 30 0.066 64.23 0 0.599 1.94 2.75 0.344 219.7 385.8160 50 0.7 30 0.067 49.81 40.26 0.463 1.99 4.56 0.350 223.3 395.6145 200 0.7 30 0.012 11.23 9.52 0.114 1.60 9.33 0.270 234.6 347.6180 200 0.7 30 0.024 14.20 10.76 0.116 (1.83) 0 (0.349) (146.8) (315.7)210 200 0.7 30 0.039 16.74 11.99 0.117 (1.58) 0 (0.063) (19.5) (50.5)14

Distillation of strangelets for low initial mu/T

We calculate the evolution of quark-gluon-plasma droplets during the hadronization in a thermodynamical model. It is speculated that cooling as well as strangeness enrichment allow for the formation of strangelets even at very high initial entropy per baryon S/A^{\rm init}\approx 500 and low initial baryon numbers of A_{\rm B}^{\rm init}\approx 30. It is shown that the droplet with vanishing initial chemical potential of strange quarks and a very moderate chemical potential of up/down quarks immediately charges up with strangeness. Baryon densities of \approx 2\rho_0 and strange chemical potentials of \mu_s>350~MeV are reached if strangelets are stable. The importance of net--baryon and net--strangeness fluctuations for the possible strangelet formation at RHIC and LHC is emphasized

Spieles, C

Greiner, C

Stöcker, H

Coffin, J P

CERN Document Server

Distillation of Strangelets for low initial =T
C. Spieles
a
, C. Greiner
b
, H. Stocker
a
, and J. P. Con
c
a
Institut fur Theoretische Physik, J. W. Goethe-Universitat, 60054 Frankfurt am Main, Germany
b
Department of Physics, Duke University, Durham, NC 27708, U.S.A
c
Centre de Recherches Nucleaires de Strasbourg et Universite Louis Pasteur Strasbourg, France
(August 10, 1995)
Abstract
We calculate the evolution of quark-gluon-plasma droplets during the
hadronization in a thermodynamical model. It is speculated that cooling
as well as strangeness enrichment allow for the formation of strangelets even
at very high initial entropy per baryon S=A
init
 500 and low initial baryon
numbers of A
init
B
 30. It is shown that the droplet with vanishing initial
chemical potential of strange quarks and a very moderate chemical potential
of up/down quarks immediately charges up with strangeness. Baryon densi-
ties of  2
0
and strange chemical potentials of 
s
> 350 MeV are reached
if strangelets are stable. The importance of net{baryon and net{strangeness
uctuations for the possible strangelet formation at RHIC and LHC is em-
phasized.
25.15.tr, 12.38.Mh, 24.85.tp
Typeset using REVT
E
X
1
Strangelets can be thought of as strange multiquark clusters which should be more com-
pressed than ordinary nuclei and may exist as (meta-)stable exotic isomers of nuclear matter
[1]. It was speculated [2] that strange matter might resolve the dark matter issue, if it would
be absolutely stable.
The possible creation | in heavy ion collisions | of such long-lived remnants of the
quark-gluon-plasma, cooled and charged up with strangeness by the emission of pions and
kaons, was proposed by Liu and Shaw [3] and Greiner et al. [4,5]. Thus, strangelets can
serve as unambiguous signatures for the creation of a quark gluon plasma. The detection
of strangelets would verify exciting theoretical ideas with consequenses for our knowledge of
the evolution of the early universe, the dynamics of supernova explosions and the underlying
theory of strong interactions [6].
Here we want to point out that such exotic states of matter can be created in heavy ion
collisions even at collider energies, where such a process has received no attention so far,
because common belief was that the (strange) baryon densities vanish at midrapidity, both
at RHIC and LHC. We argue, however, that this conclusion was premature. This is due to
the following eects:
 uctuations of the stopping power can provide nite baryochemical potential 
B
at
mid{rapidity in a small fraction of all events;
 uctuations of the net{baryon and {strangeness content between dierent rapidity bins
within one event can be quite large;
 strange (anti{)baryon enhancement due to collective eects (e. g. a chiral phase tran-
sition);
 strangeness and baryon distillery, which are inherent for the two{phase system (hadron
gas/quark gluon plasma) for a wide parameter range.
The last point stresses the signicance of the 'chemistry` of the system in the evolution of
the phase transition.
2
In the following we adopt a model [5] for the dynamical creation of strangelets via the
strangeness separation mechanism [4]. Consider a rst order phase transition of the QGP
to hadron gas. Strange and antistrange quarks do not hadronize at the same time for
a baryon{rich system [4]. The separation mechanism can be viewed as being due to the
associated production of kaons (containing s quarks) in the hadron phase, because of the
surplus of massless quarks compared to their antiquarks. The strange quarks can combine
to -particles, but it is energetically favourable that s-quarks remain in the plasma, when
hadronization proceeds. The ratio f
s
of the net strangess over the net baryon number
quanties the excess of net strangeness. Both the hadronic and the quark matter phases
enter the strange sector f
s
6= 0 of the phase diagram almost immediately, which has up to
now been neglected in almost all calculations of the time evolution of the system. Earlier
studies addressed the case of a baryon{rich QGP with rather moderate entropy per baryon
[4,5,7]. Now we focus on low initial baryon densities and high specic entropies, to match
the expected conditions of heavy ion collisions at RHIC and LHC, where the search for
strangeness enters in the objective of the ALICE experiment [8].
The hadronization transition has been described by geometric and statistical models,
where the matter is assumed to be in partial or complete equilibrium during the whole
(quasi-)isentropic expansion. A more realistic scenario must take into account the particle
radiation from the surface of the hadronic reball before `freeze out'. Our model [5] com-
bines these two pictures. The expansion of the QGP droplet during the phase transition
is described as a two-phase equilibrium; in particular the strangeness degree of freedom
stays in chemical equilibrium because the complete hadronic particle production is driven
by the plasma phase. The nonequilibrium radiation is incorporated by rapid freeze-out of
hadrons from the outer layer of the hadron phase surrounding the QGP droplet. During the
expansion, the volume increase of the system thus competes with the decrease due to the
freeze{out. The global properties like (decreasing) S=A and (increasing) f
s
of the remaining
two-phase system then change in time according to the following dierential equations for
the baryon number, the entropy, and the net strangeness number of the total system:
3
ddt
A
tot
=   A
HG
d
dt
S
tot
=   S
HG
(1)
d
dt
(N
s
 N
s
)
tot
=    (N
s
 N
s
)
HG
;
where   =
1
A
HG

A
HG
t

ev
is the eective (`universal') rate of particles (of converted hadron
gas volume) evaporated from the hadron phase. The equation of state consists of the bag
model for the quark gluon plasma and a mixture of relativistic Bose{Einstein and Fermi{
Dirac gases of well established strange and non{strange hadrons up to in Hagedorn's eigen-
volume correction for the hadron matter [4]. Thus, one solves simultaneously the equations
of motion (1) and the Gibbs phase equilibrium conditions for the intrinsic variables, i.e. the
chemical potentials and the temperature, as functions of time.
Fig. 1 illustrates the increase of baryon concentration in the plasma droplet as an in-
herent feature of the dynamics of the phase transition (cf. [2]). The origin of this result
lies in the fact that the baryon number in the quark{gluon phase is carried by quarks with
m
q
 T
C
, while the baryon density in the hadron phase is suppressed by a Boltzmann
factor exp( m
baryon
=T
C
) with m
baryon
 T
C
. Mainly mesons (pions and kaons) are created
in the hadronic phase. More relative entropy S=A than baryon number is carried away
in the hadronization and evaporation process [5], i.e. (S=A)
HG
 (S=A)
QGP
. Ultimately,
whether (S=A)
HG
is larger or smaller than (S=A)
QGP
at nite, nonvanishing chemical po-
tentials might theoretically only be proven rigorously by lattice gauge calculations in the
furure. However, model equations of state do suggest such a behaviour, which would open
such intriguing possibilities as baryon inhomogenities in ultrarelativistic heavy ion collisions.
In the early universe shrinking quark droplets may | in analogy | contain the accumu-
lated baryon number with possibly very high baryon density [2]. This mechanism yields a
primeval inhomegeneous nucleosynthesis [6,9], which is signaled by the abundances of the
light elements.
What `initial' conditions do we expect at collider energies? At RHIC energies one might
see baryon stopping, dN
B
=dy > 0, on the average, at midrapidity. This can be due to
4
multiple rescattering, leading to a nonvanishing, positive quarkchemical potential 
q
[10]. On
the other hand, relativistic meson{eld models, which, at high temperature, qualitatitively
simulate chiral behaviour of the nuclear matter, exhibit a transition into a phase of massless
baryons [11]. Including hyperons and Y Y {interaction [12] it shows that at   0 the
densities for all baryon species are of the order of 
0
near the critical temperature. Thus,
the fraction of (anti{)strange quarks increases drastically. Several hundred (anti{)baryons,
many of them (anti{)hyperons may then ll the hot midrapidity region (with net baryon
number  0). Districts of non{vanishing net baryon (respectively anti{baryon) density with
nite s (s) content will then occur stochastically. Thus, the nite chemical potential is locally
caused by the uctuations of newly produced particles, not by the stopped matter. If such
a phenomenon persists also for the deconned phase, the eect of baryon concentration and
strangeness separation may then result in the production of strangelets and anti-strangelets
in roughly equal amounts.
Let us try to give a rough estimate of the possible size of uctuations in the net baryon
and net strangeness number to be expected at midrapidity (or uctuations along dierent
rapidity intervals) around their mean. The average number of initial quarks and antiquarks
(before hadronization) in a rapidity interval is approximately 1=2dN

=dy y, if half of the
pions are made by the quarks and the other half by the gluons. For RHIC energies dN

=dy
has been estimated to lie between 1100 and 1600 and for LHC energies to lie up to nearly
4000 in central Au+Au collisions [13]. Hence the quark number is roughly 500 for RHIC
and up to 2000 for LHC in a rapidity interval y  1. (In an equilibrated plasma the
total number of quarks is N  
q
R
2
z within z = 1   2 fm in the early stage of the
hydrodynamical expansion. According to 
q
= g
3
4
2
T
3
(3)  1:1T
3
for a degeneracy of g=12
these numbers correspond to temperatures of T  250  500 MeV.) We take the number to
be N = 500. A similar consideration holds for strange and antistrange quarks, and we take
here N
s
= 200. We now assume independent uctuations according to Poissonians within
this rapidity interval. In fact the actual width of the uctuation at collider energies could be
much broader (KNO{scaling of particle multiplicity distributions in elementary pp{collisions
5
[14]). To justify the assumption of independent uctuations of B and

B despite of the local
compensation of quantum numbers, one has to estimate the typical relative momenta within
a quark{antiquark pair. If one follows the parton cascade concept embodying perturbative
QCD [15], the average
p
s^ of rst parton{parton interactions (gg ! qq being the most
important contribution) should be of the order of 5 10 GeV at LHC energies. The produced
B and

B, carrying about 0:4 of the (anti{)quark momenta, would thus be separated in
rapidity by at least one unit (assuming a transverse momentum of about 500 MeV).
The net baryon number in the box described above will be jBj > 30 with a probability of
0.5 %. About 0:1 % of the events will reach jBj > 30 with a strangeness fraction jf
s
j > 0:7.
Hence uctuations are not negligible. If each pion carries about 3.6 units of entropy (which
is true for massless bosons), the entropy per baryon content in the reball is
S
A
B
 3:6
dN

=dy
dN
B
=dy
; (2)
and thus for dN
B
=dy = 30 a range of 60 to 250 is formed. If the plasma is equilibrated,
the ratio of the quarkchemical potential and the temperature jj=T is directly related to the
entropy per baryon number via
 
S
jA
B
j
!
QGP

37
15

2
 
jj
T
!
 1
: (3)
Accordingly the ratio then varies between 0.1 to 0.4.
We now consider various reballs with an initial net baryon number of A
B
= 30 and
a net strangeness fraction f
s
of either 0 or 0.7. The initial entropy per baryon ratios are
chosen between 50 and 500. Table 1 summarizes the initial conditions (adjusted by the
(initial) chemical potentials 
q
; 
s
and temperature) used to start the hadronization. It also
shows the nal parameters of the quark droplet like the saturated strangeness content and
baryon number. One further, yet crucial input, is the bag constant employed to describe
the equation of state of the (strange) quark matter droplet. Only for the bag constants
B
1=4
 180 MeV strange matter does exist as a metastable state at zero temperature [4],
being absolutely stable only for B
1=4
< 150 MeV [2].
6
Fig. 2 shows the time evolution of the baryon number for S=A
init
= 200 and f
init
s
= 0:7 for
varyous bag constants. For B
1=4
< 180 MeV a cold strangelet emerges from the expansion
and evaporation process, while the droplet completely hadronizes for bag constants B
1=4

180 MeV (for B
1=4
= 210 MeV hadronization proceeds without any signicant cooling of
the quark phase, although the specic entropy S=A decreases by a factor of 2 from 200 to
only 100). The strangeness separation works also in these cases, as can be read o the large
nal values of the net strangeness content, f
s
>
 1:5   2. However, then the volume of the
drop becomes small, it decays and the strange quarks hadronize into -particles and other
strange hadrons.
Fig. 3 shows the evolution of the two-phase system for S=A
init
= 200, f
init
s
= 0 and for a
bag constant B
1=4
= 160 MeV in the plane of the strangeness fraction vs. the baryon density.
The baryon density increases by more than one order of magnitude! Correspondingly, the
chemical potential rises as drastically during the evolution, namely from 
i
= 16 MeV to

f
> 200 MeV. The strangeness separation mechanism drives the chemical potential of the
strange quarks from 
i
s
= 0 up to 
f
s
 400 MeV. Thus, the thermodynamical and chemical
properties during the time evolution are quite dierent from the initial conditions of the
system.
Even for high initial entropies, S=A  100   500, in the quark blob the entropy in the
remaining droplet approaches zero at the end of the evolution (assuming B
1=4
= 160 MeV).
High initial entropies per baryon require more time for kaon and pion evaporation in order
to end up nally with the same conguration of (meta{)stable strange quark matter.
In conclusion, we have shown in the present model that the evolution of quark-gluon-
plasma droplets during their hadronization may result in the formation of strangelets even
at very high initial entropy per baryon S=A
init
 500 and low initial baryon numbers of
A
init
B
 30. The distillation of very small strangelets of a size A
B
 10 (see Table 1) is
possible. We note that nite size eects of describing small strangelets neglected here might
become of crucial importance [16]. Special (meta-)stable candidates are the quark-alpha
[17] with A
B
= 6 and the H-Dibaryon state with A
B
= 2 [18]. Local net{baryon and net{
7
strangeness uctuations can provide suitable initial conditions for the possible strangelet
creation at RHIC and LHC. Droplets with vanishing initial chemical potential of strange
quarks and a small chemical potential of up/down quarks quickly charge up with strangeness
and baryon{number: if strangelets are stable, the droplet reaches strange chemical potentials
of 
s
> 350 MeV and two times ground state nuclear matter density!
ACKNOWLEDGMENTS
C.G. thanks the Alexander von Humboldt Stiftung for his support by a Feodor Lynen
scholarship. This work was supported in part by the U.S. Department of Energy (grant DE-
FG05-90ER40592), by the Gesellschaft fur Schwerionenforschung, Darmstadt, Germany, the
Bundesministerium fur Forschung und Technologie (F. R. G.), Deutsche Forschungsgemein-
schaft and the Graduiertenkolleg Schwerionenphysik.
8
REFERENCES
[1] A. R. Bodmer, Phys. Rev. D4, 1601 (1971); S. A. Chin and A. K. Kerman, Phys. Rev.
Lett. 43, 1292 (1979); J. D. Bjorken and L. D. McLerran, Phys. Rev. D20, 2353 (1979)
[2] E. Witten, Phys. Rev. D30, 272 (1984); E. Farhi and R. L. Jae, Phys. Rev. D30, 2379
(1984)
[3] H.-C. Liu and G.L. Shaw, Phys. Rev. D30, 1137 (1984)
[4] C. Greiner, P. Koch and H. Stocker, Phys. Rev. Lett. 58, 1825 (1987); C. Greiner, D. H.
Rischke, H. Stocker and P. Koch, Phys. Rev. D38, 2797 (1988)
[5] C. Greiner and H. Stocker, Phys. Rev. D44, 3517 (1992)
[6] Proc. Int. Workshop on Strange Quark Matter in Phys. and Astr., Nucl. Phys. (Proc.
Suppl.) 24B, 1-299 (1991), see H. Kurki{Suonio 67, and S. Bonometto 54
[7] K. S. Lee, U. Heinz, Phys. Rev. D47, 2068 (1993)
[8] Letter of Intent, CERN/LHCC/93{16/LHCC/I 4; J. P. Con, project presentation,
meeting of the ALICE technical board, April 23 (1995)
[9] J. H. Applegate and C. J. Hogan, Phys. Rev. D31, 3037 (1985)
[10] A. Dumitru, D. H. Rischke, T. Schonfeld, L. Winckelmann, H. Stocker and W. Greiner,
Phys. Rev. Lett. 70, 2860 (1993)
[11] J. Theis, G. Graebner, G. Buchwald, J. Maruhn, W. Greiner, H. Stocker, J. Polonyi,
Phys. Rev. D28 (1983) 2286
[12] J. Schaner, C. Greiner, C. B. Dover, A. Gal, H. Stocker, Phys. Rev. Lett. 71 (1993)
1328
[13] X.-N. Wang and M. Gyulassy, Phys. Rev. D.44, 3501 (1991); K. Geiger, Phys. Rev.
D47, 133 (1992)
9
[14] UA5 collab., G. J. Alner et al., Phys. Rep. 154 (1987) 247
[15] K. Geiger and B. Muller, Nucl. Phys. B 369 (1992) 600
[16] J. Madsen, Phys. Rev. Lett. 70, 391 (1993) and Conf. on Strangeness and Quark Matter,
Crete, Greece (1994)
[17] F. C. Michel, Phys. Rev. Lett. 60, 677 (1988)
[18] R. L. Jae, Phys. Rev. Lett. 38, 195 (1977)
10
FIGURES
0 20 40 60 80 100 120 140
t (fm/c)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
/
0
S/Ainit=50
S/Ainit=100
S/Ainit=200
S/Ainit=500
FIG. 1. Time evolution of the net baryon density of a QGP droplet. The initial conditions are
f
init
s
= 0 and A
init
B
= 30. The bag constant is B
1=4
= 160 MeV.
11
0 20 40 60 80 100 120 140 160 180
t (fm/c)
0
5
10
15
20
25
30
A
B
B1/4=210 MeV
B1/4=180 MeV
B1/4=160 MeV
B1/4=145 MeV
FIG. 2. Time evolution of the baryon number for a QGP droplet with A
init
B
= 30, S=A
init
= 200,
f
init
s
= 0:7 and dierent bag constants.
12
10-1 2 5 100 2
/ 0
0.0
0.5
1.0
1.5
2.0
f s
10 fm/c
20 fm/c
30 fm/c
40 fm/c
50 fm/c
60 fm/c
70 fm/c
80 fm/c
FIG. 3. Evolution of a QGP droplet with baryon number A
init
B
= 30 for S=A
init
= 200 and
f
init
s
= 0. The bag constant is B
1=4
= 160 MeV. Shown is the baryon density and the corresponding
strangeness fraction.
13
TABLES
TABLE I. Various situations of an hadronizing plasma droplet. In the rst column the bag
constant for describing the plasma phase is listed. Then the initial conditions follow. The nal
values for the baryon number, the strangeness fraction and the two chemical potentials at the end
(or after) hadronization are listed in the last four columns.
B
1=4
S=A
init
f
init
s
A
init
B

init
B

init
q

init
s
=T
init
f
nal
s
A
nal
B

nal
B

nal
q

init
s
(MeV) (fm
 3
) (MeV) (MeV) (fm
 3
) (MeV) (MeV)
160 500 0 30 0.0067 6.55 0 0.060 1.99 2.36 0.352 224.3 396.0
160 500 0.7 30 0.0068 5.02 4.01 0.046 1.96 2.71 0.339 216.9 386.0
160 200 0 30 0.017 16.36 0 0.150 2.0 2.79 0.349 224.7 396.9
160 200 0.7 30 0.017 12.55 10.04 0.115 2.0 3.37 0.350 223.7 396.3
160 100 0 30 0.034 32.60 0 0.300 1.99 2.93 0.350 225.2 396.7
160 100 0.7 30 0.034 25.07 20.09 0.185 2.0 3.96 0.352 223.4 396.2
160 50 0 30 0.066 64.23 0 0.599 1.94 2.75 0.344 219.7 385.8
160 50 0.7 30 0.067 49.81 40.26 0.463 1.99 4.56 0.350 223.3 395.6
145 200 0.7 30 0.012 11.23 9.52 0.114 1.60 9.33 0.270 234.6 347.6
180 200 0.7 30 0.024 14.20 10.76 0.116 (1.83) 0 (0.349) (146.8) (315.7)
210 200 0.7 30 0.039 16.74 11.99 0.117 (1.58) 0 (0.063) (19.5) (50.5)
14


Distillation of Strangelets for low initial mu/T

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