Correlations and fluctuations in high energy heavy ion collision experiments

Correlationsandfluctuationsinhighenergy

heavyioncollisionexperiments*

ZHOUDai-Mei(󰀁󰀁󰀁)

1)

WANGYa-Ping(󰀂󰀁󰀁)

2)

WEILi-Hua(󰀁󰀁󰀁)CAIXu(󰀁󰀁)

3)

(InstituteofParticlePhysics,CentralChinaNormalUniversity,Wuhan430079,China)

AbstractAnoverviewofresearchstatusofsoftphysicsinhighenergyheavy-ioncollisionexperimentsandrecentexperimentalresultsarepresented.TheexperimentalstatusonfluctuationsandcorrelationshasbeenreviewedandtheoutlookforresearchstatusofsoftphysicsinLHC/ALICEhasbeenintroducedinthispaper.Keywordsquark-gluonplasma,softphysics,correlation,fluctuationPACS25.75.-q

1Introduction

ThispaperpresentsanoverviewofsoftphysicsfromSPStoRHICanddiscusseswhatnewknowledgecanbelearnedfromtheresultsonthebulkproper-tiesofhighenergyheavyioncollisions.Thisoverviewfocusesonfluctuationsandcorrelations.Sevenareaswillbediscussed:

1)Fluctuationsofparticlemultiplicity.Theen-ergydependenceofmultiplicityfluctuationsinhighenergyheavyioncollisionscanbeusedtolookforex-perimentalsignatureofincreasedfluctuationsduetoaphasetransitionorthecriticalpointandforthepre-dictedreductionoffluctuationsinrelativistichadrongasduetoconservationlow.

2)Fluctuationsofparticleratios,whichhavebeenconsideredasapossiblesignatureofquarkgluonplasmaformation.

3)Transversemomentumptfluctuationsandcor-relations,whichstudiedveryextensively.Thestudyofnuclearmatteratlargeenergydensityandthepossibilityofatransformationtocolor-deconfinedorQCDmatterhavebeenthecentralgoals.Byanalogywiththethermodynamicsofordinarymattercriticalfluctuationshavebeenviewedasameanstodemon-stratetransitionsacrosstheQCDphaseboundary.Inparticular,criticalfluctuationof󰀌pt󰀍orevent-wisemeanptasananalogtotemperaturehavebeen

sought.

4)HanburyBrown-Twiss(HBT)correlations,whichcanbeusedtoexplorethespace-timeevolu-tionandfreeze-outofthesystem.

5)Forwardbackwardmultiplicitycorrelation.Thestudyofcorrelationsamongparticlesproducedindifferentrapidityregionsmayprovideunderstand-ingofthemechanismofparticleproduction.

6)Fluctuationofellipticflow,whichisverysen-sitivetotheinitialeccentricityfluctuations.

7)Jet-mediuminteraction,whichprobestheearly-stageofthemedium.

Theoverviewofglobalpropertiesofsoftphysics,includingcollisiongeometry,particleproduction,hadronizationandflowwillbepresentedbyWANGYa-ping,etal.[1,2].Wealsosuggestreadersrefertoanoverviewthattitled“ExperimentalStatusofUltra-highEnergyInducedNuclearReactions”presentedbyCAIXuandZHOUDai-mei[3].

2Fluctuationsandcorrelations

Thestudyoffluctuationandcorrelationhasspe-cialimportancewhileinvestigatingtheexistenceofpossiblephasetransitioninrelativisticheavyioncol-lisions.Eventbyeventfluctuationsinthethermody-namicquantitiesprovideimportantinsighttowardsthephysicalpropertiesofthedensenuclearmatter

No.5ZHOUDai-Meietal󰀁Correlationsandfluctuationsinhighenergyheavyioncollisionexperiments401

whereVar(n)and󰀌n󰀍arethevarianceandmeanofmultiplicitydistributions,respectively.

Thescaledvarianceforpositively(ω(h+)),nega-tively(ω(h−))andallchargedhadrons(ω(h±))arepresentedhere.ThescaledvarianceofaPoissondis-tributionis1,independentofitsmeanmultiplicity.Alargerωmightindicateadditionalnon-statisticalfluctuations,asmallerωmightbeahintforasup-pressionoffluctuatione.g.duetoconservationlaws.Fig.1showsthecentralitydependenceofmultipli-cityfluctuationinPb+Pbcollisionsatdifferenten-ergiesforcentralcollisions.Ingeneralωdecreaseswithincreasingthecentrality,thistrendisstrongerforhigherenergies.

Figure2showstheenergydependenceofmulti-plicityfluctuationsinPb+Pbcollisions.Itcanbeseenthatatallenergiesthescaledvarianceforposi-tivelyandnegativelychargedhadronsissmallerthan1,thevalueforaPoissoniandistribution.

󰀌n󰀍

Fig.1.Centralitydependenceofω(h+)(top),ω(h−)(middle)andω(h±)(bottom)forPb+Pbcollisionsatdifferentenergies.C<1%correspondstothemostcentralcollisions.TheshowncentralityrangeofC<10%

proj

correspondsapproximatelytothenumberofprojectileparticipantsNp>160.Onlystatisticalerrorsareshown.

402ChinesePhysicsC(HEP&NP)Vol.32

sNN=200GeVwithinthepseudo-rapidity

rangeof−3<η<3[7,8].Intheiranalysisanevent-by-eventobservableCisusedtostudythemultiplicityfluctuations.

N1−N2

,C=

N1+N2whereN1andN2arethemultiplicitiesinapairofηbinswiththesamebinsizeandsymmetricwithre-specttoη=0.ThewidthoftheCdistribution(σ(C))isusedastheirfluctuationsobservables.Becausethedifferenceofthetwomultiplicitiesisusedinthedef-initionofCandthetwomultiplicitieschangeinthesamedirectionwhenNpartvariesfromeventtoevent,theNpartfluctuationsaresuppressedinthemeasuredσ(C).Also√becauseoftheevent-by-eventnormaliza-tionfactor

No.5ZHOUDai-Meietal󰀁Correlationsandfluctuationsinhighenergyheavyioncollisionexperiments403

preliminarydatafromtheSTARCollaborationintheRHICenergyrange.Asignificantincreaseofthefluc-tuationsignalofthekaontopionratioat20and30AGeVisobservedwhileitstaysconstantfromthehighestSPSenergiesouttotheRHICenergyrange[10](SeeFig.5andFig.6).

σ2data−σ2mixed.

Ingeneral,processesleadingtoacorrelatedpro-ductionofoneortheotherparticlespeciesorto

acorrelationintheirmultiplicitieswouldresultinσnon-stat>0,andtheyobtain

σnon-stat=2.8.

TheresultsoftheNA49energyscanprogramshowasharpmaximumoftheratioofK+toπ+yieldsinthecentralPb+Pbcollisionsatbeamener-giesof20—30AGeV.(seeFig.4bottom)

ThisobservationwasinterpretedasanindicationofaphasetransitionatlowSPSenergies.TheNA49Collaborationpresentedtheresultsonenergydepen-denceofevent-by-eventfluctuationsofthekaontopionandprotontopionratiosatbeamenergiesclosetothismaximumandcomplementedthisstudywith

Fig.5.Energydependenceoftheevent-by-event−fluctuationsignalofthe[K++K−]/[π++π]ratio(top)andthe[p+p¯]/[π++π−]ra-tio(bottom).Thesystematicerrorsofthemeasurementsareshownasgraybands.

Fig.6.PreliminarydataobtainedbytheSTARCollaboration.Top:DistributionofK/πra-tio√

fromdataandmixedeventsforAu+Auat404ChinesePhysicsC(HEP&NP)Vol.32

󰀁

Σi=j(pti−pˆt)(ptj−pˆt)

2

σptestsinvarianceofˆt

󰀌pt󰀍fluctuationsundersuperpositionofindependent

systems[15],e.g.,p-plinearsuperpositioncomparedwithA-Acollisions[16—18].Aclosely-relatedmeasureisbasedonPearson’snormalizedcovariance[19]

rab≡

2σab

→2σ2σab

2

σpˆt

√

2

(pt−npˆt)2/n¯−σpˆt

acomparisonbetweenanormalizedvarianceandits

2

central-limit(CLT)Refs.[16,20].∆σpisavari-t:n

ancedifferencewhereasΦptisadifferencebetweenr.m.sterms.Ingeneral,variancesandcovariances

2

obeyalinearalgebra,and∆σpissimplyrelatedtot:ntwo-particlecorrelations.

Σptismotivatedbyaspecificmodelofglobaltem-peraturefluctuationsinthermalizedevents.Therearetwoversions:

󰀅Σpt≡

󰀌δpti.δptj󰀍/pˆt

2[14]

.

Ifthehypothesisofglobalthermalizationunderlying

thesedefinitionsisnotvalidthemeaningofeitherΣptisnotclear.TheSTARCollaborationfindsplentifulevidencethattheglobalthermalizationisnotsatis-fiedinRHICcollisions.

2.3.2󰀌pt󰀍fluctuationmeasurements

Thefirst󰀌pt󰀍fluctuationmeasurement,madebyNA49attheSPSforthecentralPb+Pbcollisionsat17.3GeV[21],isshowninFig.8.AfrequencycurveonM(pt)=󰀌pt󰀍(points)iscomparedwithamixed-pairreference(histogram).Aquantitativecomparisonbe-tweendataandreferencewasmadewithΦpt[15].TheygotΦpt=0.6±1.0MeV/c,compatiblewithzero.Thismeansthatnosignificantnon-statisticalfluctuationswereobservedintherapidityacceptance.

No.5ZHOUDai-Meietal󰀁Correlationsandfluctuationsinhighenergyheavyioncollisionexperiments405

406ChinesePhysicsC(HEP&NP)Vol.32

sNN=130GeVattheRHIC.By

comparingthedistributionswithwhatisexpectedforN.Thelargestatisticalsignificanceofthevarianceexcessisindicatedbydeviationsofupto20stan-darddeviationsineachhistogrambin.ThatSTAR

statisticallyindependenceparticleemission,themag-nitudeofnonstatisticalfluctuationinmeantrans-versemomentumisdeterminedtobeconsistentwithzero(seeFig.10).

Afteraninitialnullresult[12],measurementsbyPHENIXprovidedthefirstindicationofnonzero󰀌ptfluctuationsatRHIC[13](seeFig.11andFig.12).The󰀍increaseofFptwithincreasingptimpliesthatthema-jorityofthefluctuationsareduetocorrelatedhighptparticles.AMonteCarlosimulationthatincludesel-lipticflowandaPYTHIA-basedhard-scatteringde-scriptioncanconsistentlydescribecontributionstothesignalasafunctionofcentralityandptwithasimpleimplementationofjetsuppression.

Fig.11.Fpt(inpercent,0.2GeV/cExtensivemeasurementsofseveralaspectsof󰀌STARpt󰀍fluctuationsCollaborationhave[14,16,17,23]been.Fig.carried13(upperoutbypanel)thepresentsaSTARmeasurementofthefrequencydis-tributionon

√

Fig.12.Fpt(inpercent)ofnonrandomfluctu-ationasafunctionoftheptrangeoverwhichMptiscalculated,0.2GeV/c,forthe20%—25%centralityclass(Npart=181.6).ThecurveistheresultofaMonteCarlosim-ulationwithhard-scatteringprocessmodeledusingPYTHIAwithSprob(Npart)=0.075andRAA=0.41.Theerrorbarsincludestatisticalandsystematicerrorsandaredominatedbythelatter.Thecontributionofellipticflowisestimatedtobenegligibleatthiscentrality.

Fig.13.Upperpanel:Eventfrequencydistri-butionon

√Nevtineach

bin.

No.5ZHOUDai-Meietal󰀁Correlationsandfluctuationsinhighenergyheavyioncollisionexperiments407

panel),thecorrelationsmultipliedbythemultiplicitydensity(middlepanel)andthesquarerootofthecor-relationsdividedbytheevent-wiseaveragetransversemomentumperevent(bottompanel),asafunctionofeventcentralityforAu+Aucollision[14].

Giventhecloseconnectionbetweenpartonscat-teringand󰀌pt󰀍fluctuationatRHICthecollisionen-ergydependenceof󰀌pt󰀍fluctuationcouldrevealpre-viouslyinaccessiblepartondynamicsatlower(e.g.,SPS)collisionenergies.TheSTARCollaborationgivesthefirststudyoftheenergydependenceofptangularcorrelationsinferredfromevent-wisemeantransversemomentum󰀌pt󰀍fluctuationinheavyioncollisions[23].Theycomparetheirlarge-acceptance

√

measurementsatCMenergies

sNN=130GeVvsrelativemultipli-cityN/N0,whichisapproximatelyNpart/Npart,max,therelativefractionofparticipantnucleons.Charge-independent(CI)(solidtri-angularpoints)andcharge-dependent(CD)(opentriangularpoints,multipliedby3forclarity)differencefactorsincludesta-tisticalerrorsonly(smallerthansymbols).Parametrization(dashedcurves),extrapola-tionofparametrizationtotrueprimarypar-ticlenumber(solidcurves),andsystematicuncertainties(bands)arediscussed.Differ-encefactorsforthe15%most-centralcollisioneventsareshownbythesolidcircleandopencirclesymbols.

sNN[23].The󰀌pt󰀍fluc-tuationincentalcollisionsvariesalmostlinearlyas

√log{

Figure15showstheenergydependenceofthetransversemomentumcorrelations,󰀌∆pt,i∆pt,j󰀍(top

󰀅∆pt,i∆pt,j󰀆/󰀅󰀅pt󰀆󰀆(right)asafunctionof

centralityandincidentenergyforAu+AucollisionscomparedwithHIJINGresults.

408ChinesePhysicsC(HEP&NP)Vol.32

󰀄

ρref(n)isa

normalizedcovariancedensity.Thatequationcanbesolvedtoobtaintheptangularautocorrelation.

Figure18(toprightpanel)showstheangularau-tocorrelationondifferentaxes(η∆,φ∆)(e.g.,η∆=η1−η2)obtainedbyinvertingthefluctuationscalede-pendenceinthefirstpanel.Therearetwomajorfea-tures:asinusoidcorrespondingto“ellipticflow”andnon-sinusoidalstructurecalled“non-flow”inconven-tionalflowterminology.Thisisthefirstobservationofflowasaptcorrelationorvelocitystructure[17].Thesinusoidcanberemovedprecisely,leavingthestructureinthebottomleftpanelwhichisdominatedbymini-jetcorrelations,especiallyasame-sideposi-tivepeak[17].Inthebottomrightpaneltheyplotthesameangularautocorrelationon(η∆,φ∆)inacylin-derformat.

FromtheexampleinFig.18wecanseethatin-versionofptfluctuationscaledependencetoanauto-correlationprovidesdirectphysicalinterpretationofptfluctuationmechanisms.Partonfragmentdistribu-tions(mini-jets)arevisualizedasevent-wisetemper-ature/velocitystructureson(η,φ).Acomprehensivepictureofpartonscattering,dissipationandfragmen-tationinheavyioncollisionsistherebyestablished.

No.5ZHOUDai-Meietal󰀁Correlationsandfluctuationsinhighenergyheavyioncollisionexperiments409

sNN=200GeVmid-centralAu-Aucollisionsmeasured

bySTAR(topleft);correspondingptangularautocorrelationobtainedbyinversion(topright);thesameautocorrelationaftersubtractingtheellipticflowcontribution(bottomleft);thesamedataplottedincylinderformat(bottomright).

2.4HBT

Theexperimentaltechniqueofusingtwo-particleinterferometrytorelatethemomentumspacesepa-rationofparticlestotheirseparationinspace-timeiswellestablished[25].Inthecaseofidenticalbosons,e.g.π+mesons,quantuminterferenceamongthepar-ticlesleadstoanenhancementofpairswithsmallmo-mentumdifferenceq(Bose-Einsteinenhancement).Toisolatethesmallsetofcorrelatedpairsthatun-dergothisquantuminterferencefromtheenormousamountofuncorrelatedpairsinanevent,acorre-lationfunctionC(q)isformedinwhichpairsfromrealeventsaredividedbypairsfromdifferentevents.Inheavy-ioncollisions,C(q)isoftenconstructedinthreedimensionsandfittoathree-dimensionalGaus-sian:

real−pairs

C(q)=

Diff.quantitywhatitinvestigates

Wewillhighlightthesefivedifferentialstudieshere.

2.4.1Beamenergy

CERESCollaborationhadpresentedasystem-aticstudyoftwo-pioninterferometrydataatSPSenergies[27].AdetailedstudyoftheBertsch-PrattBHTradiusparametershasbeenperformedasfunc-tionofthemeanpairtransversemomentumktand

410ChinesePhysicsC(HEP&NP)Vol.32

sNN[30—32].NosuddenjumpsinHBTradiiwereobserved(seeFig.21),butlowerenergyRHICmea-surementswereneededtocompletethesearchforapredictedincreaseinemissiontimescalerelatedtothepossibleonsetofQGPformation.2.4.2Transversemomentum

Thetransversemomentum(kt)dependenceoftheHBTradiiforidenticalpionsprobablyisstudiedmostoften,underthemodel-dependenceviewthatspace-momentumcorrelationsinthesourceareduemostlytocollectiveexpansion[25].Asthesourceexpands,radialflowpusheshigherptparticlemoreatsurface.Withinthispicture,analyticalexpressionshavebeen

No.5ZHOUDai-Meietal󰀁Correlationsandfluctuationsinhighenergyheavyioncollisionexperiments411

isstartlinglysimilarforallenergies[44].

2p2t+m)depen-

denceoftheHBTradii.

ThehydrodynamicalapproachtounderstandingHBTismotivatedatRHICbythemodel’sdemon-stratedabilitytodescribesoftptspectraandellipticflowconsistentlyforseveralparticlespecies[33].“Hy-dro”calculationsfortheseobservablespointtofastthermalizationinapartonicphase,followedbyhy-drodynamicexpansionfor∼15fm/cwithaninter-mediatephasetransition.However,thesecalculationsyieldstrongdisagreementwithHBTradii[34]:RoutandRlongareoverpredictedbyasmuchasafactorof2,andRsideissomewhatunderpredicted.Inparticular,themeasuredktdependenceofRsideisincontrasttohydroandothermodelsthatpredictlittle(ifany)ktdependence.Thisdisagreement,andthelackofenergydependenceoftheBHTradiiforafixedktbin[29],isknownasthe“HBTPuzzle”.

Collectiveflowgeneratesacharacteristicfall-offofthepionsourceradiiwithkt,whichisubiquitouslyobservedindata.Finalresultsforthekt-dependenceofGaussianradiifromcentralAu+Au(Pb+Pb)col-lisionsexistattheAGS[28,35],SPS[30,36—39],andRHIC[29,40—43].AsisclearfromFig.22,asidefromasmallvariationinoverallscale,thektdependence

Fig.21.Theenergydependenceofπ−HBTparametersforcentralAu+Au(Pb+Pb)collisionsatmidrapidityandpt≈

[28,30—32]

0.17GeV/c.TheSPSdataareoffset

√

slightlyin

412ChinesePhysicsC(HEP&NP)Vol.32

mtscaling.

Figure23collectsthemtdependenceofho-mogeneitylengthsforseveralenergies.TheleftpanelsshowtheresultsforSi+Aucollisionsat√

No.5ZHOUDai-Meietal󰀁Correlationsandfluctuationsinhighenergyheavyioncollisionexperiments413

sNN=200GeV,

forcomparisonwiththecentrality-binnedAu+Auanalysis[].Liketheheavy-ioncase,thethreeHBTradiiinp+pexhibitacharacteristicdecreasewithin-creasingkt(seeFig.24),thoughtforp+pithasbeenattributedtostringandmultistringfragmentationinearlierstudies.

AsomewhatsurprisingresultcomesaboutwhendividingtheAu+Auandd+Auradiibythep+pradii.ThedividedtrendsareroughlyflatwithktforallradiithatindicatinganapparentscalinginthektdependenceoftheHBTradiiforthesethreesystems.Giventhatthektdependencepresumablyarisesinverydifferentways,theseresultsalsoareabitpuzzling[26].2.4.5Azimuthalangle

HBTstudiesrelativetothereactionplaneinnon-centralcollisionallowthepossibilitytocomparetheexpandedsystem’stransverseeccentricityatfreeze-outwithitsinitialeccentricityfromanuclearoverlapmodelcalculation.

STARhadcompletedananalysisoftheazimuthaldependenceofHBTradiirelativetothereaction

planeat

√

D2bf

󰀌Nf2󰀍−󰀌Nf󰀍2

=

sNN=200GeV[76]

(0.8<|η|<1.0).

TheSTARCollaborationgavethefirstworkonthemeasurementofthelong-rangecorrelationstrength(b),inultrarelativisticnucleus-nucleuscollisions[77].

Thecentralityofthecollisionplaysanimpor-tantroleinthegrowthoflongrangecomponentofthetotalcorrelationstrength.Datafrom10%—20%,20%—30%,and30%—40%mostcentralAu+Aucol-lisionshavebeenanalyzed,followingthesamepro-

414ChinesePhysicsC(HEP&NP)Vol.32

sNN=200GeV.Dataarecomparedwith

PSMcalculationswithandwithoutstringfusion.

Fig.27.Correlationstrengthbasafunctionof∆η.(a)forAu+Auatfourcentralitybins;(b)forp+p;(c)for40%—50%Au+Au.

Fig.28.(a)Shortrangecorrelationstrengthobtainedfromthescaledbfromp+pasafunctionofpseudora-piditygap;(b)GrowthoflongrangecorrelationformidcentralAu+Auevents.

No.5ZHOUDai-Meietal󰀁Correlationsandfluctuationsinhighenergyheavyioncollisionexperiments415

sNN=200GeVobtainedwiththePHOBOSdetec-tor.

Fig.29.Relativeflowfluctuations,σv2/󰀅v2󰀆,

√

asafunctionofcentrality,for

416ChinesePhysicsC(HEP&NP)Vol.32

No.5ZHOUDai-Meietal󰀁Correlationsandfluctuationsinhighenergyheavyioncollisionexperiments417

sNN=200GeV.

Animportantaspectoftheanalysisisthesubtractionofcombinatorialbackgrounds,seeFig.33.

Fig.33.Backgroundsubtractedjet-like3-particlecorrelationsforp+p(topleft),d+Au(topmiddle),andAu+Au50%—80%(topright),30%—50%(bottomleft),10%—30%(bottomcenter),andZDCtriggered

√

0%—12%(bottomright)collisionsat

References

1WANGYa-Pingetal.Chin.Phys.C(HEP&NP),2008,32(4):308

2WANGYa-Ping,ZHOUDai-Mei,CAIXu.CollectiveEx-pansionandHadronizationinHighEnergyHeavyIonCol-liaionExperiments.AcceptedbyChin.Phys.C(HEP&NP)

3CAIXu,ZHOUDai-Mei.HEP&NP,2002,26(9):971—990(inChinese)

4HeiselbergH.PhysicsReport,2001,351:161—194

5StephanovM,MajagopalK,ShuryakE.Phys.Rev.D,1999,60:114028

6LungwitzBetal.(NA49Collaboration).nucl-ex/06100467CHAIZWetal.(PHOBOSCollaboration).J.Phys.Conf.

Ser.,2005,27:128

8WozniakK(PHOBOSCollaboration).J.Phys.G:Nucl.Part.Phys.,2004,30:S1377—S1380

9AfanasievSVetal.(NA49Collaboration).Phys.Rev.Lett.,2001,86:1965—1969

10DasSetal.(STARCollabration).nucl-ex/0503023

11VoloshinSA,KochV,RitterHG.Phys.Rev.C,1999,60:

024901

12AdcoxKetal.(PHENIXCollabration).Phys.Rev.C,

2002,66:024901

13AdlerSSetal.(PHENIXCollabration).Phys.Rev.Lett.,

2004,93:092301

14AdamsJetal.(STARCollaboration).Phys.Rev.C,2005,

72:044902

15GazdzickiM,MrowczynskiS.Z.Phys.C,1992,:127

418ChinesePhysicsC(HEP&NP)Vol.32