3 December 1998

Ž .Physics Letters B 442 1998 449–452

Quark model and strange baryon productionin heavy ion collisions

A. Bialas 1

M. Smoluchowski Institute of Physics, Jagellonian UniÕersity, Reymonta 4, 30-059 Krakow, Poland

Received 16 August 1998; revised 4 September 1998Editor: P.V. Landshoff

Abstract

It is pointed out that the recent data on strange baryon and antibaryon production in Pb–Pb collisions at 159 GeVrcagree well with the hypothesis of an intermediate state of quasi-free and randomly distributed constituent quarks andantiquarks. Also the S–S data are consistent with this hypothesis. The p–Pb data follow a different pattern. q 1998Published by Elsevier Science B.V. All rights reserved.

Recently, rather precise data on strange baryonand antibaryon production in the central rapidityregion of Pb–Pb and p–Pb collisions were pre-

w xsented by the WA97 collaboration 1 . In this note Iwould like to point out that these Pb–Pb data agreerather well with a simple quark-counting rule whereasthe p–Pb data follow a different pattern. This obser-vation implies that the quark degrees of freedom aremuch more relevant in collisions of two heavy nucleithan in ‘‘elementary’’ hadronic interactions. It thussupports the interpretation of the data on strangenessproduction in Pb–Pb collisions as an evidence for

w xcreation of the quark–gluon plasma 2,3 .Our argument is an application of the old idea

w xproposed first by Rafelski 4 . Considering a systemŽof partons in thermodynamic equilibrium i.e.

. Žquark–gluon plasma , he observed that strange and.antistrange particle abundances must satisfy a host

of simple relations. Below we consider some of these

1 E-mail: bialas@thp1.if.uj.edu.pl

relations which have a virtue of being rather general,independent of the assumption of thermal equilib-rium but – on the other hand – sensitive to the quarkdegrees of freedom.

To explain the argument, let us formulate thequark counting rule we are talking about. We simply

Žassume that probability of creation of a baryon or an.antibaryon with a given quark content is propor-

Žtional to the probability that three quarks or anti-.quarks with appropriate quantum numbers happen

to meet at a certain region of phase-space – neces-sary for the binding to take place. Assuming further-more that the quarks are uncorrelated 2, we obtainthe following relative probabilities:

psv q3 ; LrS 0 sv q2s ; Jsv qs2 ;p L J

Vsv s3 1Ž .V

where q and s are relative probabilities to find a

2 Both these assumptions are valid in thermal equilibrium. Theinverse is not true, however.

0370-2693r98r$ - see front matter q 1998 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 98 01250-7

( )A. BialasrPhysics Letters B 442 1998 449–452450

light quark and a strange quark in the suitablephase-space region. Analogous formulae are validfor antibaryons. v are the proportionality factorsi

taking into account the effects of resonance structureand of the binding energy in formation of variousbaryons. These factors, generally different for differ-ent baryons, are rather difficult to calculate and

Ž .therefore the comparison of Eq. 1 with experimentis rather involved and depends on further assump-

w xtions 2,5 .One may observe, however, that these v-factors

are identical for a baryon and the correspondingantibaryon. Consequently, if one considers only theratios of the antibaryon to baryon rates, the v-fac-

w xtors cancel 4,6 and the discussion becomes muchsimpler. This is what we are going to do. We thushave

3p qs 2Ž .3p q

and

LrS p J p V p2 3s D ; s D ; s D 3Ž .

LrS p J p V p

where

qsDs 4Ž .

qs

From these equations we see that the four ratios inŽ . Ž .2 and 3 are expressed in terms of two parameters.Therefore we have two constraints which must besatisfied by the data.

Let us first discuss the data for Pb–Pb collisionsw xat CERN SPS. The data of 1 give the following

values for strange antibaryonrbaryon ratios in thecentral rapidity region

LrS Js .133" .007; s .249" .019;

LrS J

Vs .383" .081 5Ž .

V

w xThe data of NA44 7 give

ps .07" .01 6Ž .

p

Ž . Ž . Ž .Dividing the ratios 5 by the ratio 6 and using 3we have

D s1.9" .3; D s1.89" .15;L J

D s1.76" .15 7Ž .V

and thus we see that the three values of the parame-ter D obtained from the data are in good agreementwith each other up the experimental accuracy ofabout 10 percent.

Ž .From 6 we also deduce that

qs .41" .02 8Ž .

q

Ž .and thus employing 7

ss .75" .06 9Ž .

s

where we have used the average of the three valuesŽ .for D given in 7 , i.e. Ds1.83" .10. The ratios

Ž . Ž .8 and 9 are in good agreement with those ob-w x 3tained in 8 from a thermal fit to the data

This completes the analysis of the Pb–Pb data.Let us now turn to the p–Pb collisions.

w xThe data of WA97 coll. 1 give

LrS Js .20" .03; s .33" .03; 10Ž .

LrS J

w xThe VrV is not given in 1 . The prp ratio wasw xmeasured by NA44 collaboration 9 , with the result

ps .31" .03 11Ž .

p

Ž . Ž .Using these values and the formulae 2 , 3 we thusobtain

D s .65" .11; D s1.03" .07 12Ž .L J

in clear disagreement. We must conclude that thequark counting rule is apparently in contradictionwith p–Pb data, indicating that in this case thequark degrees of freedom do not represent a decisivefactor in the production mechanism.

Ž . Ž .Taken together, Eqs. 7 and 12 show that theresult obtained for Pb–Pb collisions is likely notaccidental but indeed indicates existence of an inter-

3 These ratios can be calculated as the inverse square of thew xcorresponding fugacities given in 8 .

( )A. BialasrPhysics Letters B 442 1998 449–452 451

Ž .mediate step in baryon antibaryon production pro-cess: a system of quasi-free constituent quarks andantiquarks 4 distributed randomly in phase-space. Anatural interpretation seems to be that this intermedi-ate q–q system is the first step of the chiral symme-try breaking transition from the earlier quark–gluonplasma phase.

Another interesting issue is: which pattern is fol-lowed in collisions of lighter nuclei. Answering thisquestion could bring new arguments to the contro-versy as to where the transition to the quark–gluonplasma phase takes place.

w x w xThe data of 9 and 12 on S–S collisions give:

p LrSs .12" .01; s .22" .01;

p LrS

Js .55" .07. 13Ž .

J

So that we obtain

D s1.83" .17; D s2.14" .16 14Ž .L J

Ž . Ž .Thus the agreement with Eqs. 2 and 3 is not bad,although not as good as in the case of Pb–Pbcollisions.

Using the average value Ds1.99" .12 and qrqs .49" .02 we obtain srss .98" .07 in good

w xagreement with the analysis of 13 where the dataon central rapidity region in S–S collisions werediscussed using the thermal model. It is also interest-ing to note that the obtained value of the parameterD is not inconsistent with that found from the Pb–Pbdata. This certainly supports the idea that already inS–S collisions the baryon and antibaryon productionprocess proceeds through an intermediate randomq–q system. This observation, in turn, strengthensthe evidence for quark–gluon plasma phase presentalready in collisions of light nuclei.

We would like to close this paper with the follow-ing comments.

Ž .i Although we consider only baryon and an-tibaryon production, it is tempting to extend the

4 Dynamical models which explicitly introduce the q – q inter-mediate system are being developed since some time by the

w x w xgroups in Budapest 10 and in Bratislava 11 .

argument also to K and K production. Taking intoŽ .account 4 , we obtain

K qss 'D. 15Ž .

sqKw x w xThe data 12 and 14 give

Ks1.91" .37;ž /K S–S

Kf1.8 no error given 16Ž . Ž .ž /K P p–P b

We see that these results are not in disagreementwith the values of D found from baryon-antibaryondata in S–S and Pb–Pb collisions 5.

Ž .ii The advantage of our argument is that theŽ . Ž .Eqs. 2 – 4 can be applied to any phase spaceŽregion of course the specific parameters may de-

.pend on the region . When the appropriate data areavailable, it shall be thus possible to test the rele-vance of the quark degrees of freedom also outsidethe central rapidity region considered here. One mayhope in this way to determine the kinematic domainwhere the particles are dominantly produced by theintermediate step of quark–gluon plasma. Further-more, this should allow to determine the rapiditydependence of the basic ratio srs. This last point isparticularly interesting in view of the recent sugges-

w xtion by Letessier and Rafelski 15 that the observeddeviation of srs from unity is a reflection of theCoulomb interactions.

Ž .iii Our argument assumes that the production ofall baryons and antibaryons in the central rapidityregion of heavy ion collisions proceeds by a com-mon mechanism, i.e. ‘‘coalescence’’ of the indepen-dently distributed quarks and antiquarks. Recentanalysis of the thermal model by Letessier and

w xRafelski 15 , based on the Pb–Pb data extrapolatedŽto full phase-space, and including Coulomb correc-

Ž . Ž ..tions which modify somewhat the relations 2 – 4 ,indicates that the conditions for production of V andV may differ from those of other baryons. This iscertainly a serious possibility. The present experi-mental accuracy does not yet allow, however, todraw definite conclusions about this problem.

5 I could not find the data for p – Pb collisions. The p – S dataw x q y12 give K rK s2.02".14 is strong disagreement both values

Ž .of D in 12 .

( )A. BialasrPhysics Letters B 442 1998 449–452452

Ž .iv We would like to repeat that the relationsŽ . Ž .2 – 4 are independent of the assumption of thermalequilibrium. When thermal equilibrium is addition-ally assumed, one may produce many more specific

w xpredictions, as discussed in detail in 6,13 . In partic-ular, it is possible to calculate the ratios of the ratesof particles with different strangeness content, a taskwhich is clearly beyond the scope of the presentinvestigation. We feel, however, that our simpleargument can still serve a useful purpose of convinc-ing a layman that the quark degrees of freedom areessential for a correct description of particle produc-tion in heavy ion collisions.

Acknowledgements

I would like to thank B.Muller for raising myinterest in the subject. The discussions with F. Bec-catini, R. Caliandro, R. Lietava, E. Quercigh, J.Rafelski and K. Zalewski are highly appreciated.Thanks are also due to M. Morando and E. Quercighfor a very kind hospitality at the Padova meeting onStrangeness in Quark Matter, where part of this workwas done. This investigation was supported in partby the KBN Grant No. 2 P03B 086 14.

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