A.Bialas
M.SmoluchowskiInstituteofPhysicsJagellonianUniversity,Cracow∗
February7,2008
Abstract
Itispointedoutthattherecentdataonstrangebaryonandan-tibaryonproductioninPb-Pbcollisionsat159GeV/cagreewellwiththehypothesisofanintermediatestateofquasi-freeandrandomlydistributedconstitutentquarksandantiquarks.AlsotheS-Sdataareconsistentwiththishypothesis.Thep-Pbdatafollowadifferentpattern.
Recently,ratherprecisedataonstrangebaryonandantibaryonproduc-tioninthecentralrapidityregionofPb-Pbandp-PbcollisionswerepresentedbytheWA97collaboration[1].InthisnoteIwouldliketopointoutthatthesePb-Pbdataagreeratherwellwithasimplequark-countingrulewhereasthep-Pbdatafollowadifferentpattern.Thisobservationimpliesthatthequarkdegreesoffreedomaremuchmorerelevantincollisionsoftwoheavynucleithanin”elementary”hadronicinteractions.Itthussupportsthein-terpretationofthedataonstrangenessproductioninPb-Pbcollisionsasanevidenceforcreationofthequark-gluonplasma[2,3].
OurargumentisanapplicationoftheoldideaproposedfirstbyRafelski[4].Consideringasystemofpartonsinthermodynamicequilibrium(i.e.quark-gluonplasma),heobservedthatstrange(andantistrange)particle
abundancesmustsatisfyahostofsimplerelations.Belowweconsidersomeoftheserelationswhichhaveavirtueofbeingrathergeneral,independentoftheassumptionofthermalequilibriumbut-ontheotherhand-sensitivetothequarkdegreesoffreedom.
Toexplaintheargument,letusformulatethequarkcountingrulewearetalkingabout.Wesimplyassumethatprobabilityofcreationofabaryon(oranantibaryon)withagivenquarkcontentisproportionaltotheprobabilitythatthreequarks(orantiquarks)withappropriatequantumnumbershappentomeetatacertainregionofphase-space-necessaryforthebindingtotakeplace.Assumingfurthermorethatthequarksareuncorrelated1,weobtainthefollowingrelativeprobabilities:
p=ωpq3;
Λ/Σ0=ωΛq2s;
Ξ=ωΞqs2;
Ω=ωΩs3
(1)
whereqandsarerelativeprobabilitiestofindalightquarkandastrangequarkinthesuitablephase-spaceregion.Analogousformulaearevalidforantibaryons.ωiaretheproportionalityfactorstakingintoaccounttheeffectsofresonancestructureandofthebindingenergyinformationofvariousbaryons.Thesefactors,generallydifferentfordifferentbaryons,areratherdifficulttocalculateandthereforethecomparisonofEqs(1)withexperimentisratherinvolvedanddependsonfurtherassumptions[2,5].
Onemayobserve,however,thattheseω-factorsareidenticalforabaryonandthecorrespondingantibaryon.Consequently,ifoneconsidersonlytheratiosoftheantibaryontobaryonrates,theω-factorscancel[4,6]andthediscussionbecomesmuchsimpler.Thisiswhatwearegoingtodo.Wethushave
p¯
(2)
q3and
¯/Σ¯Λ
p
where
D=
1
D;
¯Ξ
p
D;
2
¯Ω
p
D3
(3)
qs¯
Boththeseassumptionsarevalidinthermalequilibrium.Theinverseisnottrue,
however.
2
Fromtheseequationsweseethatthefourratiosin(2)and(3)areexpressedintermsoftwoparameters.Thereforewehavetwoconstraintswhichmustbesatisfiedbythedata.
LetusfirstdiscussthedataforPb-PbcollisionsatCERNSPS.Thedataof[1]givethefollowingvaluesforstrangeantibaryon/baryonratiosinthecentralrapidityregion
Λ
¯/Σ¯Ξ
=.249±.019;
Ω
¯p
=.07±.01(6)
Dividingtheratios(5)bytheratio(6)andusing(3)wehave
DΛ=1.9±.3;
DΞ=1.89±.15;
DΩ=1.76±.15
(7)
andthusweseethatthethreevaluesoftheparameterDobtainedfromthedataareingoodagreementwitheachotheruptheexperimentalaccuracyofabout10percent.
From(6)wealsodeducethat
q¯
s
=.75±.06(9)wherewehaveusedtheaverageofthethreevaluesforDgivenin(7),i.e.D=1.83±.10.Theratios(8)and(9)areingoodagreementwiththoseobtainedin[8]fromathermalfittothedata2
ThiscompletestheanalysisofthePb-Pbdata.Letusnowturntothep-Pbcollisions.
ThedataofWA97coll.[1]give
Λ
¯/Σ¯Ξ
=.33±.03;
(10)
¯/Ωisnotgivenin[1].ThepTheΩ¯/pratiowasmeasuredbyNA44collabo-ration[9],withtheresult
p¯
p
=.12±.01;
¯/Σ¯Λ
Ξ
=.55±.07.(13)
Sothatweobtain
DΛ=1.83±.17;
DΞ=2.14±.16
(14)
ThustheagreementwithEqs(2)and(3)isnotbad,althoughnotasgood
asinthecaseofPb-Pbcollisions.
UsingtheaveragevalueD=1.99±.12andq¯/q=.49±.02weobtains¯/s=.98±.07ingoodagreementwiththeanalysisof[13]wherethedataon
centralrapidityregioninS−Scollisionswerediscussedusingthethermalmodel.ItisalsointerestingtonotethattheobtainedvalueoftheparameterDisnotinconsistentwiththatfoundfromthePb-Pbdata.ThiscertainlysupportstheideathatalreadyinS−Scollisionsthebaryonandantibaryonproductionprocessproceedsthroughanintermediaterandomq−q¯system.Thisobservation,inturn,strenghtenstheevidenceforquark-gluonplasmaphasepresentalreadyincollisionsoflightnuclei.
Wewouldliketoclosethispaperwiththefollowingcomments.
(i)Althoughweconsideronlybaryonandantibaryonproduction,itis
¯production.TakingintotemptingtoextendtheargumentalsotoKandK
account(4),weobtain
K
≡D.(15)sq¯Thedata[12]and[14]give
K
¯K
Pp−Pb
≈1.8(noerrorgiven)(16)
WeseethattheseresultsarenotindisagreementwiththevaluesofD
foundfrombaryon-antibaryondatainS−SandPb−Pbcollisions4.
(ii)TheadvantageofourargumentisthattheEqs(2)-(4)canbeappliedtoanyphasespaceregion(ofcoursethespecificparametersmaydependontheregion).Whentheappropriatedataareavailable,itshallbethuspossibletotesttherelevanceofthequarkdegreesoffreedomalsooutsidethecentralrapidityregionconsideredhere.Onemayhopeinthiswaytodeterminethekinematicdomainwheretheparticlesaredominantlyproducedbytheintermediatestepofquark-gluonplasma.Furthermore,thisshouldallowtodeterminetherapiditydependenceofthebasicratios¯/s.ThislastpointisparticularlyinterestinginviewoftherecentsuggestionbyLetessierandRafelski[15]thattheobserveddeviationofs¯/sfromunityisareflectionoftheCoulombinteractions.
(iii)Ourargumentassumesthattheproductionofallbaryonsandan-tibaryonsinthecentralrapidityregionofheavyioncollisionsproceedsbyacommonmechanism,i.e.”coalescence”oftheindependentlydistributedquarksandantiquarks.RecentanalysisofthethermalmodelbyLetessier
andRafelski[15],basedonthePb−Pbdataextrapolatedtofullphase-space,(andincludingCoulombcorrectionswhichmodifysomewhattherelations
¯maydif-(2)-(4)),indicatesthattheconditionsforproductionofΩandΩ
ferfromthoseofotherbaryons.Thisiscertainlyaseriouspossibility.Thepresentexperimentalaccuracydoesnotyetallow,however,todrawdefiniteconclusionsaboutthisproblem.
(iv)Wewouldliketorepeatthattherelations(2)-(4)areindependentoftheassumptionofthermalequilibrium.Whenthermalequilibriumisaddi-tionallyassumed,onemayproducemanymorespecificpredictions,asdis-cussedindetailin[6,13].Inparticular,itispossibletocalculatetheratiosoftheratesofparticleswithdifferentstrangenescontent,ataskwhichisclearlybeyondthescopeofthepresentinvestigation.Wefeel,however,thatoursimpleargumentcanstillserveausefulpurposeofconvincingalaymanthatthequarkdegreesoffreedomareessentialforacorrectdescriptionofparticleproductioninheavyioncollisions.Acknowledgements
IwouldliketothankB.Mullerforraisingmyinterestinthesubject.ThediscussionswithF.Beccatini,R.Caliandro,R.Lietava,E.Quercigh,J.RafelskiandK.Zalewskiarehighlyappreciated.ThanksarealsoduetoM.MorandoandE.QuercighforaverykindhospitalityatthePadovameetingonStrangenessinQuarkMatter,wherepartofthisworkwasdone.ThisinvestigationwassupportedinpartbytheKBNGrantNo2P03B08614.
References
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