Nuclear Physics B (Proc . Suppl.) 24B (1991) 239-245North-Holland

C. GREINERinGtibit fiir Theoretische Physik . Universität Frlanaen . 17-R520 Erlangen . Germany

and

J . SCHAFFNER and H. STÖCKERInstitut für Theoretische Physik, J.W . Goethe-Universität, D-6000 Frankfurt, Germany

1 . INTRODUCTION

One important goal in ultrarelativistic heavy-ionphysics is the observation of a temporarily created quark-gluon plasma (QGP). An enhanced production of strangeparticles from the (QGP. especially an enhanced K+/7r+-ratio, and the J/~, -;suppression were proposed as possiblesignatures for such a novel state of deconfined, stronglyinteracting matter . The observed J/0-suppression andthe observed K+/7r+-ratio had first raised premature eu-phoria that. such a state has indeed been snag-1-iguouslyobserved . However, it was subsequently shown that thesedata - and also most of the other proposed signals forQGP.-creation - can be understood as being due to theformation of a very (energy-) dense and hot region ofmatter consisting of confined hadronsi .

Therefore, it seems that perhaps the only unambigu-ous way to detect the transient existence of a QGP mightbe the experimental observation of remnants (`ashes' ofthe QGP), like the formation of strange quark matterdroplets . It has been proposed that strange quart. mat-ts d:oplets (`strangelets') at zero temperature and in,Q-equili :brum might be absolutely stable' . We have re-cently argued3,4.5 that such strangelets might play animportant role in ultrarelativistic heavy-ion collisions .Here one expects a total net strangeness of zero, a high

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NUCLEAR PHYSIC_ E

PROCEEDINGS

SUPPLEMENTS

STRANGE QUARK MATTER DROPLETS IN ULTRARELATIVISTIC HEAVY ION COLL151ûivS

We present a model describing the hadronisation of the quark gluon plasma through particle emission . Piousand K+s and I:'°s carry wvay entropy and antistrangeness from the system, thus facilitating the coolingprocess and the strangelet formation . The strangeness separation mechanism and the formation processworks well even for higher initial entropies per baryon, tantamount to higher bombarding energies . Thesurviving strangelets have a rather high strangeness content, f, ti 1 .2 - 2 (i .e. Z/A ti (-0.1)-(-0.5)) .Hence droplets of strange quark matter with baryon number ti 10 - 30 and with negative charge may beproduced .

temperature and no fl-chemical equilibrium . How canthese speculative objects then be produced in such ex-periments?

Let us briefly summarize how a stable or metastablestrangelet might look like: Think of bulk objects, con-taining a large number of quarks (u . . . it, d...d, s...s), socalled multiquark droplets . Multiquark states consistingonly of u- and d-quarks must have a mass larger thanordinary nuclei, otherwise normal nuclei would be un-stable . However, the situation is different for dropletsof strange quark matter, which would contain approxi-mately the same amount of u-, d- and s-quarks .

If the mass of a strangelet is smaller than the massof the corresponding ordinary nucleus with the samebaryon number, the strangelet would be absolutely sta-ble and thus be the true groundstate of nuclear matter' .Presently such a scenario cannot be ruled out .

It is also conceivable that the mass per baryon of astrange droplet is lower than the mass of the strange A-baryon, but larger than the nucleon mass. The dropletis then in a metastable state, it cannot decay into A's',s .For bag parameters B' /4 lower than 190 McV strangequark droplets can only decay via weak interactions (inthe calculation of Ref. 5 quark matter is simply treatedas a gas of non-interacting particles in a MIT-bag) . (Wit-ten's idea of absolute stable strange quark matter droplets

240 C. Greiner et al . /Strange quark matter droplets in ultrarelativistic heavy ion collisions

does work in this approach only for bag parameters around145 MeV.) For larger H-values strange?ets are instable .

There are now two related arguments why strangeletscould be produced in ultrarelativistic heavy ion colli-sions, although here tla initial conditions are such thatthere is no strangeness in the system at all, and thateven after the collision, there is no net strangeness:rqüai am.ouràts o f strange and antistrange quarks areabundantly produced in a hot QGP'. The strangenesssaturates the phase space in a baryon-rich environmentafter a very short equilibration time, t = 10'" sec, whichmay actually be shortcr than the duration of a nuclearcollision . It has been demonstrated"' that the phasetransition from such a QGP to hadron matter will yielda large antistrangeness content in the hadron phase whilethe quark-gluon plasma retains a large net strangenessexcess . This distillation happens even if the total netstrangeness of the combined QGP and hadron fluid isequal to zero. The separation process takes place, if theplasma phase (when the combined systern carries a netpo-:f "-e baryon number) is energetically favoured for thes-quark as compared to the confined phase where thes-quarks reside in antikaons (K- ) and heavy hyperons.The distillation mechanism can be viewed as being due tothe associated production of kaons (cuntaining, a-qu a.rl.s)

in the hadron phase with the s-quarks remaining in theQGP-droplets, which therefore develop a net strangenesssurplus .

Furthermore, rapid kaon emission leads to a finitenet strangeness of the expanding (combined hadron plusquark) system. This, in turn, results in an even strongerenhancement of the s-quark abundance in the quark phase .This prompt kwon (and, of course, pion) emission maycool the quark phase, which then condenses into metastableor stable droplets of strange quark matters .

2. A MODEL, FOR HADRONISATIONAND PARTICLE EMISSION

The hadronisation transition ofan0P droplet formedin heavy ion collisions has often been described by simple

geometric and statistical equilibrium models . A more re-alistic scenario must take into account the competition ofthe collective expansion with the particle radiation fromthe surface of the hadronic fireball before `freeze out'.Pions, nucleons and strange particles (e .g. kaons andhyperons) are particularly important . One importantoutcome of a cascade simulation9 was the observationthat the total entropy is approximately constant dur-ing the hadronisation . Chemical non-equilibrium modelcalculations 7 suggested furthermore that chemical equi-librium (in strangeness) is reached in the hadron phase,if the system evolves from the déconfined phase : gluonfusion yields fast equilibration of strangeness .

Hence, a two phase equilibrium description, which wewill employ, gives results which agree with non-equilibriummodels . In addition, the non-equilibrium evaporation isincorporated by rapid freeze-out of hadrons from the di-lute hadron phase surrounding the QGP droplet at theGibbs phase transition point . This scenario is visualizedin Fig . 1 .

FIGURE 1

iî,A0

Hadron fluid surrounds the QGP at the phase transi-tion . Particles evaporate from the hadronic region . Newhâlûrons :'-.merge out of the plasma by ai.dronisâtioîi .

The global properties of the two phase system thenchange in time, in accord with the following differentialequations for the baryon number, the entropy and thenet strangeness number of the total system:

_d Atot

=dtd stot

=

-r SHaduE

-1 AHad

T(N, - Ns)tot

=

-r (N, - N8)Had

where I' = A-~(°ôead\

is the effective (`unique') rate/1 evof particle (of converted hadron gas volume) evaporatedfrom the hadron phase . The three equations constitutea set of coupled differential equations for At", (S/A)tot

and the net strangeness fraction f ot = (N, - Ns)tot/At°t.

The latter two are given in terms of the hadron and quarkphase contents .

Although the baryon number A and the strangenessf, are conserved under strong interactions, their valuein the system changes with time according to the aboveequations due to the evaporation process . This calcula-tion requires solving simultaneously the equations of mo-tion and the Gibbs phase equilibrium conditions, whichspecify the intrinsic variables (e .g . the chemical poten-tials) of the functions (I)QGPIHad

(y, it., T) andQG.P/Had

f,

Still the effective rate I' and thebaryon fraction AHad are not specified, when,AHad = AHad/Atot is the ratio of baryon number con-tained in the hadron phase ever the total baryon numberin the system . Intuitively, the plasma is surrounded bythe hadror. phase, thus we take it as a shell of constantthickness of Ar - 1 fm (this is an ad hoc assumption,however, the results depend only weakly on this thick-ncsc) . Some hadron particle volume is then evaporatedin a small time interval At . For this unknown evapo-ration time we use an averaged rate of 'all particles atthe considered temperature, dictating the time-scale forthe evolution of the hadronic particles out of the hadronphase . The final (total) yield3 are not affected by theansatz .

3 . STRANGELET DISTILLATION

C. Greiner et al . /Strange quark matter droplets in ultrarelativistic heavy ion collisions

For the following arguments it is important to studythe expansion of an initially hot QGP fireball with spe-cial attention to the evolution of the strangeness dur-

241

ing the phase "transition . Early non-equilibrium parti-cle radiation off the hot, dense (baryon-rich) QGP fire-ball is expected to be important in the initial phase ofthe reaction"' . Pions and kaons (containing an ,§-quark)(and only a minor number of antikaons) are ernitted fromthe surfaceof the plasma . Thus the QGP is enriched with~Prcte net strangeness even before the phase transitionpoint is reached . More ltiaons (qs) than antikaons (qs)are emitted because the Ks containing an q-quark areexpected to be suppressed in baryon-rich matter (largepq) by a factor ti e-ZjA91T as compared to the Ifs . Thenet strangeness enrichment resulting from the early blackbody radiation off the pure QGP-droplet has been esti-mated to be in the range fa < 0.5 5.s. These valueshave been used in the pre -.Sent model as input for theinitial condition f,(t o).

Fig. 2 gives an impression how the hadronisationproceeds in the present model for a large bag constant(B*a = 235 MeV - no strangelet in the groundstateland a small bag constant (B 1/4 = 145 MeV). Hopefully,in the second case one may expect a distilled strangelet .The initial parameters are a net strangeness content off,(to) = 0.25 and an entropy per baryon ratio Ä(to) = 25in both cases : For the large bag constant the systemhadronizes completely in t - s L-, which is not toosurprising . A strong increase of the net strangeness ofthe system is found in both situations, which is an im-portant confirmation of our previous work3,5, and theplasma drop reaches a strangeness fraction of f, - 1 .4when the volume becomes small . Indeed, for the smallbag constant, however, a cold strangelet emerges fromthe expansion and evaporation process with an approxi-mate baryon number of A ti 22, a radius of R - 2.5fm,and a net strangeness fraction of f, > 1 .5, i .e . a chargeto baryon ratio Z/A - -0.25! This would be an inter-esting object also from its global properties: It wouldcomprise a nucleus of positive baryon number, but neg-ative charge.

In the first case, at high critical temperatures, thetemperature of the system increases slightly, althoughone might have expected same cooing by the ernirsion

242

170-,

165

160

!SF

150 r~

C. Greiner et al./Strange quark matter droplets in ultrareiativistic heavy ion collisions

S/A j,;t=25; fs=0.25

0

. ; 0

eo

so

`o

20

. ;0 [s e . 10 . 12 1`

0

t[fm/CI t[fm/CIFIGURE 2

50 100 150 200 . 250

a) Baryon number, strangeness content and temperatureof the quark glob during complete hadronisation as afunction of time for a very large bag constant B+ = 235AleV. The initial values a,re an initial baryon content ofA(to ) = 100, an entropy per baryon ratio of Â(to) = 25and an ',_nitial net strangeness fraction of f,(to) = 0.25 .Please note the strong increase of the strangeness con-tent with t;mc .b) The same situation as in a ., however, for d small bagconstant B14 = 145 MeV, when a strangelet is distilled .One observes a strong decrease in the evolving tempera-t L, re .

of particles . This reheating, however, is well-known"for baryon-rich matter at large bag constants ; it is dueto the large entropy content in the plasma, which leadsto a heating of the system in order to conserve entropy .More rigorously, whether reheating or cooling does oc-c~,ur, dprc-nds only on whether ÂQcp is larger or smaller

_10o F

than SHad at the phase transition point . Indeed, it waspointed out in Ref. 4 that the strangelet formation goeshand in hand with strong cooling rather than reheatingwhen the bag constant B+ is sinail. As one notices inFig . 2, temperature (and entropy) decreases drasticallythroughout the hadronisation process and condensation .Thus the pressure goes to zero and an absolutely sta-ble, cold strangelet is created . The energy per baryonof the remaining strange quark droplet converges to itszero temperature value .

Let us now understand one basic idea for the possibleformation of strange quark matter droplets in heavy-ioncollisions (which can be demonstrated in a conspicuousmanner already in Fig . 1 of Ref . 5) : Take a bag constantof 160 MeV and a strange quark mass of 150 MeV . Con-sider a totally equilibrated system of cold nuclear mat-ter in the ground-state with a finite total strangeness of0.4 . One would now naively argue that strange quarkmatter with f,=0.4 is the ground-state of the system,because the energy per baryon of this state is lower thanthat of the hyperonic state . However, the total energyper baryon can be lowered by additional -50 1AeV byassembling the non-strange quarks into pure nucleonicdegrees of freedom (i .e . `a - particle emission'), leav-

ing the strange quarks in a strange matter droplet, itsstrangeness fraction enriched to f, ;:e, l . The minimaltotal energy is obtained with the tangent constructionused in Ref. 5. In the above situation, the surplus inthermal energy provides additional nucleon evaporation,so that the tangent point will be `overshooted' and thestrangelet turns out to have a strangeness fraction largerthan 1 .

The particle yields of the most frequently emittedspecies are shown in Fig . 3 for the case, where a. strangeletcondenses after - 200 L. Their time dependence ismore or less as expected : the pions (- 300) are by farthe most quickly escaping and abundant particles fol-lowed by the nucleons (- 7?), kaon,-. ( .. . 32), hyperons(rr 6) acid antikaons ( .. . 3) . The total K+/7;+-ratio is

then approxinaaLCiy 0.16 . The surplus of 17 ir+ baryonnuzriber and of 23 in strangeness are contained in the

100

10

C. Greiner et al.IStrange quark matter droplets in ultrarelativistic heavy ion collisions

S/A,,t=25, fs.irot=0.00, 811=145 MeV

0 5o 100 150 200 250t [ fm/c lFIGURE 3

The number of emitted particles are shown versus timein a situation, when a strangelet is disti11e4 . The condi-tions are the same as in Fig . 2, however, no initial netstrangeness is assumed .

missing stre^âeiet, ;rhirl' has to hâ~" e a strangeness frac-tion of - 1.4!

In Fig . 4 the strangelet mass after a complete evolu-tion is plotted versus the initial, ad hoc unknown param-eters of the fireball, namely the initial entropy per baryonnumber and the net strangeness ratio of the plasma fire-ball due to the preequilibrium kaon evaporation out ofthe plasma before phase transition takes place . Observethat the higher the initial entropy, the larger the baryoncontent A of the surviving strangelet! This is also truefor the initial net strangeness fraction . Realistic parame-ters for the entropy may be larger than .., 10 and for thenet strangeness, as mentioned earlier, probably less orequal than 0.5 . Hence, either in the stable or metastabl<,-situation, one might expect a reasonable strangelet witha baryon number - 10 - 30 .

4 . CONCLUSIONS :WHERE TO LOOK FOR STRANGELETS?

We conclude tl: ntt there are two essential processeswhich favour the formation of strangelets from a baryon-

0)

100 [

e0

NmQ 4

B'/£=145 MeV

FIGURE 4

case ki .e . taking B+ = 160 MPV) .

SIA init c 10

0

0.0 0.2 0.4 0.6 0.6 1.0 1 .2 1.4fSWt

B1/`=160 MeV

5 10 15 20 25S/A"t

243

a) The surviving mass number of the strangelet are de-picted for the case of an absolute stahle strangelet (i .e.taking B+ = 145 MeV) for several initial configurations .b) The same situation as in a ., but for the metastable

rich QGP formed in ultrarelativistic heavy ion collisions .The first (as a prerequisite) is the s-,§-separation mech-anism". This process is not too sensitive to the bagr. r.7`ant used : It even works for B< = 210 MCV3,s .

This leaves us with a quark phase in the coexistenceregion of hadronic and quark matter, which is chargedup with strangeness . The s-quarks, created in pairs

244 C. Greiner et al./Strange quark matter droplets in ultrarelativistic heavy ion collisions

with s-quarks in the early quark-gluon fireball, remain

in the quark phase during the phase transition, the 9-

quarks materialize mainly into kaons . The second is the

evaporation of the hadronic gas with its antistrangenessexcess, which charges the remaining system with net

strangeness . We emphasize the importance of kaon, pionand nucleon evaporation 5.12 (which allow for the ncces-sary cooling) .

Recently Barz et al." presented nonequilibrium rateequation calculations, which seemed to suggest that nu-cleon evaporation heats up (non)strange quark matterdroplets, which in response evaporate completely. How-ever, only a very large bag constant is employed in theirdynamical calculation, for which in fact a strange quarkdroplet would be by far unstable 45.12. So, their con-clusicns are rather misleading. Instead, we have shownhere and before 45,12, that, when applying lower bag con-stants needed to make strangelets (meta-)stable in thegrounstai,e, nucleon evapurai,iun also cuois the ;eniairl-ing droplet at the !_ate stage cr the expansion . This isnow confirmed by Friman et al . 13 .

Two observations are remarkable: The distillery workseven for larger initial entropies S/A=50 or 100 (compareFig. 4a), contrary to our earlier remarks 3,a. A high ini-tial entropy does not necessarily prohibit strangelet for-mation . Abundant kwon production enriches the plasmarapidly with net strangeness at high entropies . This of-fers to look for strangelet production at the highest bom-barding energies available in the future for very heavysystems, e.g. at the CERN SPS (ELAB - 200~N) or atRHIC (FLAB ?- 20TH ) . The second feature is the factthat the charge of the strangelet (with positive baryonnumber) can be negative! It is found that the charge tomass ratio is Z/A - -0.1 for absolutely stable strangeletsand Z/A - -0.45 for metastable ones .

Such an object would be particularly easy to stparateexperimentally from the background : Due to its oppositecharge, it could easily be separated magnetically fromthe whole nuclear background with positive charge, andit would deposite much less energy (about 2 AGeV) in acalorimeter than an antinucleus of roughly the same A

and velocity.The question remains, on which time-scale weak de-

cay or flavour changing modes will appear. How woulda strangelet then look like when passing through the de-tector? A magnetic spectrometer with dimension - 10m sets a natural time scale - 10-'sec . If the producedstrangelet is absolutely stable (is - 1 .1 - 1.2, B+ = 145MeV) the only energet.caiiy possible decay rnecic is theweak leptonic decay (s --~ d, Q --~ Q' -I- e + v), whichwill turn the strangelet to its minimum value in energy:: .Î, j 0.8 . Chin e,ilu IiCiinnarP P_stt'niated fir_ > >?rf:ascale for this weak process to be - 10'`' sec. Hence,the strangelet will remain in its initial condition . Thesituation turns out to be a little bit more complicated,if the strangelet would be metastable (fe ti 1 .7 - 1.9,B+ = 160 MeV). The weak nucleon decay ( s --~ d,Q -~ Q' + n ) is energetically possible in the rangeof

0.5 < f, < 1 .7 . For fe > 1, it would enrichthe quark droplet with strangeness . The strangelet willlien also remain approximately at its initial cc..ditions,fe - 1 .7 -1 .9, since the weak leptonic decay mode is tooslow . Hence, metastable strangelets may be seen withZ/A r� -0.45 .We must be aware of the fact that these global es-

timates are valid only for large systems. However, forA r� 5 - 25, the finite strangelet might, of course, settlein a deeply bound state occuring in the strong shell struc-ture of stable quark matter with low mass number. Alldecay modes are then strongly suppressed.

REFERENCES

1 . Proceedings of Quark Matter 90, to appear inNucl. Phys. A (1991), and previous QuarkPlatter proceedings ;The Nuclear Equation o State, W.Greiner andH.St6cker, eds ., NASI Series B : Physics, Vol .216 A 1t (Plenum : New York) 1989 .

2 . E.Witten, Phys. Rev . D 30 (1984) 2723 . C.Greiner, P.Koch, H.St6cker, Phys- Rev. Lett .

53 (198s) 18254. R.Matiello, C.Hartnack, A.v.Keitz, J.Schaffner,H.Sorge, H.Stöcker and C .Greiner: this vol-unie

C. Greiner et al . /,Strange quark matter droplets in ultrarelativistic heavy ion collisions

245

5. C.Greiner, D.H.Rischke H.St6cker and P.Koch,

10. P.R.Subramanian, H.Stöcker and W.Greiner,Phys . Rev . D 38 (1988 2797

Phys. Lett . 137B (1986) 12026 . S.A.Chin and A.K.Kerman, Phys . Rev . Lett .

11. H.Barz, B.Friman J.Knoll and H.Schulz, Phys.43 (1979) 1292

Lett. 242B (1990 3287 . P.Koch, B.Miiller, and J.Rafelsk ,' , Phys. Rep.

12. C.Greiner and H.St6cker, University of Frank-142 (1986) 167

furt preprint, UFTP 258 (1991)8. H.Liu and G.L.Shaw, Phys . Rev . D 30,1137

(1984)

volume,9 . G.Beitsch,M:Gong, L.D.McLi,rran, V.Ruuskanen

and E.Sarkkinen, Phys. Rev . D 37 (1988)1202

13 . H.Barz, B.Friman, J.Knoll and H.Schulz : this