Analysis of anisotropic flow with Lee-Yang zeroes

arXiv:nucl-th/0310016v13Oct2003AnalysisofanisotropicflowwithLee-YangzeroesR.S.Bhalerao,1,∗N.Borghini,2,†andJ.-Y.Ollitrault2,‡1DepartmentofTheoreticalPhysics,TataInstituteofFundamentalResearch,HomiBhabhaRoad,Colaba,Mumbai400005,India2ServicedePhysiqueTh´eorique,CEA-Saclay,F-91191Gif-sur-Yvettecedex,France(Dated:February6,2008)Wepresentanewmethodtoextractanisotropicflowinheavyioncollisionsfromthegenuinecorrelationamongalargenumberofparticles.Anisotropicflowisobtainedfromthezeroesinthecomplexplaneofageneratingfunctionofazimuthalcorrelations,incloseanalogywiththetheoryofphasetransitionsbyLeeandYang.Flowisfirstestimatedglobally,i.e.,averagedoverthephasespacecoveredbythedetector,andthendifferentially,asafunctionoftransversemomentumandrapidityforidentifiedparticles.Thecorrespondingestimatesarelessbiasedbynonflowcorrelationsthanwithanyothermethod.Thepracticalimplementationofthemethodisratherstraightforward.Furthermore,itautomaticallytakesintoaccountmostcorrectionsduetoazimuthalanisotropiesinthedetectoracceptance.Themainlimitationofthemethodisstatisticalerrors,whichcanbesignificantlylargerthanwiththe“standard”methodofflowanalysisiftheflowand/ortheeventmultiplicitiesaretoosmall.Inpractice,weexpectthistobethemostaccuratemethodtoanalyzedirectedandellipticflowinfixed-targetheavy-ioncollisionsbetween100MeVand10GeVpernucleon(attheDarmstadtSISsynchrotronandtheBrookhavenAlternatingGradientSynchrotron),andellipticflowatultrarelativisticenergies(attheBrookhavenRelativisticHeavyIonCollider,andtheforthcomingLargeHadronCollideratCERN).PACSnumbers:25.75.Ld,25.75.Gz,05.70.FhI.INTRODUCTIONStudyofanisotropicflowofparticles[1]producedinrelativisticheavy-ioncollisionshasemergedasanimpor-tanttooltoprobetheearlyhistory,especiallythether-malization,ofthedensefireballproducedinthesecol-lisions.Anisotropicflowmeansthattheazimuthaldis-tributionofparticlesproducedinnon-centralcollisions,measuredwithrespecttothedirectionofimpactparam-eter,isnotflat.ThisischaracterizedbytheFourierharmonics[2]:vn≡hcosn(φ−ΦR)i,(1)whereφdenotestheazimuthalangleofanoutgoingpar-ticle,ΦRistheazimuthalangleoftheimpactparameter(ΦRisalsocalledtheorientationofthereactionplane),nisapositiveinteger,andangularbracketsdenoteanav-erageovermanyparticlesbelongingtosomephase-spaceregion,andovermanyevents.Throughoutthispaper,weassumethatcollidingnucleiaresphericalandparityisconserved:then,thesystemissymmetricwithrespecttothereactionplaneandhsinn(φ−ΦR)ivanishesforalln.Thetwolowestharmonicsv1andv2arenameddirectedandellipticflows.Theyhavebeenstudiedextensivelyoverthelastseveralyears,withreferencetothewealthofdataproducedatGANIL[3],attheDarmstadtSIS∗Electronicaddress:[e-mail]†Electronicaddress:[e-mail]‡Electronicaddress:[e-mail];alsoatL.P.N.H.E.,Uni-versit´ePierreetMarieCurie,4placeJussieu,F-75252Pariscedex05,Francesynchrotron[4–7],theBrookhavenAlternatingGradientSynchrotron(AGS)[8–14],andtheSuperProtonSyn-chrotron(SPS)atCERN[15–19],aswellasthenewandupcomingdatafromtheRelativisticHeavyIonCollider(RHIC)atBrookhaven[20–25].SincethereferencedirectionΦRisnotknownexperi-mentally,measuringvnisadifficulttask.Analternateideaistoextractitfromtheinterparticlecorrelationswhichariseindirectlyduetothecorrelationofeachpar-ticlewiththereactionplane.However,standardmethodsofcorrelatingaparticlewithanestimateofthereactionplane[26–28],orofcorrelatingtwoparticleswitheachother[29]wereshowntobeinadequateatultrarelativis-ticSPSenergies[30]duetothesmallnessoftheflow,andthecomparativelylargemagnitudeof“nonflow”correla-tions[28,31]duetotransversemomentumconservation,resonancedecays,etc.,whichareusuallyneglectedinthesemethods.AtthehigherRHICenergies,itwasar-guedthatnonflowcorrelationsduetominijetscouldevendominatethemeasuredcorrelations[32].Theseshortcomingsofconventionalmethodsmoti-vatedthedevelopmentofnewmethodsbasedonacu-mulantexpansionofmultiparticlecorrelations[33–35].Thecumulantofthek-particlecorrelation,wherekisapositiveinteger,isolatesthegenuinek-particlecorrela-tionbysubtractingthecontributionoflower-ordercor-relations.Nonflowcorrelations,whichgenerallyinvolveonlyasmallnumberofparticles(typically,twoorthreeinthecaseofresonancedecays,possiblymoreinthecaseofminijets)contributelittletothecumulantifkislargeenough.Notethatthisisalsotrueforcorrelationsfromglobalmomentumconservation,althoughthelatterin-volvesallparticles[36].Ontheotherhand,anisotropic2flowisagenuinecollectiveeffect,inthefollowingspecificsense:inagivenevent,allazimuthalanglesarecorre-latedwiththereactionplaneazimuthΦR,whichvariesrandomlyfromoneeventtotheother.Inthelaboratoryframe,whereΦRisunknown,thisresultsinazimuthalcorrelationsbetweenallparticles,oratleastasignificantfractionoftheparticles.Therefore,unlikenonflowcor-relations,anisotropicflowcontributestocumulantsofallordersk.Neglectingthecontributionofnonflowcorrela-tionstothecumulant,onethusobtainsanestimateofvn,whichwasdenotedbyvn{k}inRef.[34],andbecomesmoreandmorereliableaskincreases.Cumulantsoffour-particlecorrelationswerefirstusedtomeasureellipticflowatRHIC[24].However