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Nuclear Physics A NuclearPhysicsA00(2016)1–4 www.elsevier.com/locate/procedia Upsilon suppression in the Schro¨dinger-Langevin approach P.B. Gossiaux1 and R. Katz1 1SUBATECH,UMR6457,EcoledesMinesdeNantes,Universite´deNantes,IN2P3/CNRS. 6 4rueAlfredKastler,44307Nantescedex3,France 1 0 2 n a J Abstract 7 Wetreatthequestionofbottomoniasuppressioninultrarelativisticheavyioncollisions(URHIC)asadynamicalopen ] quantum problem, tackled for the first time using the Schro¨dinger-Langevin equation. Coupling this equation to the h EPOS2eventgenerator,predictionsaremadeforthenuclearmodificationfactorofΥ(1S)andΥ(2S). p - p Keywords: Quarkgluonplasma,bottomoniasuppression,Schro¨dinger-Langevinequation,openquantumsystem e h [ 1. MotivationandmodelforQQ¯ dynamics 1 QuarkoniaproductioninURHIC–see[1]forarecentreview–isoneofthebestprobesofthecolor- v deconfined QCD medium of high temperature – the quark-gluon plasma (QGP) – achieved in these col- 3 lisions. The production of bottomonia is of particular interest as it is much less impacted by exogenous 4 4 recombinationthancharmoniaproductionandthenlessdifficulttomodel. Adynamicalcalculationofthe 1 bb¯ evolutionsatisfyingthebasicprinciplesofquantummechanicsisagenuineopenquantumsystem. As 0 apossiblemethodtotacklethisproblem,weusetheso-calledSchro¨dinger-Langevinequation(SLE)intro- . 1 ducedbyKostin[2]forthedynamicsoftheinternaldegreesoffreedom(relativecoordinatesofthepair) 0 16 i(cid:126)∂∂ψt =Hˆ0+(cid:126)A(cid:18)S(x,t)−(cid:90) ψ∗S(x,t)ψ dx(cid:19)−x·FR(t)ψ. (1) : v i Inthisequation,thenonlinearterm∝ (cid:126)AisadissipativefrictiontermbasedonthewavefunctionphaseS X whiletheterm−x·F (t)representsastochasticdipolarforce. Thehamiltonian Hˆ encodesthedynamics R 0 r a in the absence of such interactions with the medium. The SLE admits some important properties: norm conservation–despiteanonlineartermandcontrarilytosimilartreatmentsrelyingonimaginarypotentials –,respectoftheHeisenbergprinciple,violationofthesuperpositionprinciple,...Thestochasticforcesdrive thesystemfromapurestatetoamixedstate. Observablesarebuiltusinganensembleaverageoverthese forces,whichnecessitatestoevaluatereplicationsofthetimeevolution,afeaturewellsuitedtoMonteCarlo simulations.TheSLEnaturallyallowstodealwiththetransitionsfromtheboundstatestothecontinuum,at thecoreofquarkoniadeconfinement. Theasymptoticthermalizationofasubsystemcoupledtoaheatbath isaparticularissueoftheSLE,whosemasteringisessentialtoobtainreliablepredictions. In[3],wehave investigatedthisproblemforthe1Dcase,bothforaharmonicandalinearpotentialandforwhite–where 2 P.B.GossiauxandR.Katz/NuclearPhysicsA00(2016)1–4 (cid:104)F (t)F (t+τ)(cid:105) = Bδ(τ)–andcoloredstochasticforces. EventhoughthestateweightsW (definedas R R stat n W (t) = (cid:104)|(cid:104)n|ψ(t)|2(cid:105) ) do not systematically converge towards some Boltzmann distribution exp(−E /T) n stat (cid:104) (cid:16) (cid:17) (cid:105) n forthe”canonical”expressionofB–B=2mAE coth E0 −1 –,itisindeedthecaseforasubsetoflow 0 T lyingeigenstatesprovidedoneadaptstheautocorrelationBas B=2mAT˜(T), (2) wheretherelationT˜(T)dependsontheparticularpotential. ForafirstapplicationoftheSLEtotheproblemofbottomoniasuppressioninURHIC,weadopta1D modeling of the bb¯ pairs, with even (resp. odd) 1D-states mocking S (resp. P) 3D-states. In vacuum, the potentialinHˆ istakenasV (x)= K|x|,truncatedtoV =1.2GeV(accordingtolQCDcalculations[4]). 0 vac max m is chosen as 4.575 GeV, while the string parameter K = 1.375 GeV/fm is chosen to obtain an energy b difference between the first two even states E − E = E(Υ(cid:48))− E(Υ) = 563 MeV. Even and odd bound 2 0 statesarefoundforE =9.41,9.97,10.33GeVandE =9.74,10.18GeVwhichareingoodagreementwith correspondingexperimentalvaluesforΥ(nS)andχ (nP). Onenoticeshoweverthatthebindingenergyof b theΥ(3S)isjust20MeVinthismodel,muchsmallerthanthereal3D-case. Thus,littleconfidenceisput into predictions concerning this precise state, a default that will be corrected in a future parametrization of the potential. Investigating the time evolution asymptotics with V in Hˆ as well as with the friction vac 0 termandthestochasticforcesturnedon,adetailedstudyshowsthatacorrectthermalizationisachievedby using T˜(T) = 0.83T −0.08 GeV in the expression (2) of B. At finite temperature, however, the potential should be Debye-screened. This feature is implemented in our model by letting the maximal value V max be T-dependent in Hˆ – which then acquires the status of a mean field (MF) Hamiltonian – and taken as 0 the asymptotic (r → +∞) value of the finite-T lQCD potential. As some ambiguity subsists on the most appropriatechoiceatfinite-T,twoprescriptionshavebeeninvestigatedforV : theinternalenergyU [5] max (whichshowssomeextrabindingaroundT )aswellasapotentialsuggestedbyMocsyandPetreczky[4] c (ourprivilegedchoiceintheseproceedings)characterizedbysomeweakerbindingascomparedtoU. The corresponding1D-potentialsarereferredtoasU andV . IntheSLEapproach,thefriction/dragcoefficient W Afortherelativemotionisconsideredtobeidenticaltotheoneforsingleheavyflavor(afeaturethatcan beintuitedstartingfromthe2-bodySLEandintroducingthecenterofmassandtherelativecoordinates). It istakenfrom[6]–A (p = 0,T) ≈ 0.46T +0.32T2 (c/fm)–applyingonthetopafactorK = 1.5tomatch b theopenheavyflavorexperimentaldataatLHC. 2. Bottomoniasuppressioninastationarymedium Contrarilytosomecommonanalysis, wedonotpaymuchattentiontheprecisetemperaturesatwhich the various bound states ”melt” under the influence of the MF in Hˆ . We believe that such an analysis 0 is not very relevant for the problem of bottomonia suppression in URHIC, not only because the medium is not stationary (see section 3) but also because a) the evolution time of the bb¯ is finite so that b and b¯ quarkswhicharefoundclosetoeachotherattheendoftheevolutionhaveapossibilitytorecombineinto some bound state and b) if one waits long enough, the stochastic forces may destroy bound states at any temperatureanyhow. Instead,weinvestigatethecapabilityoftheseforcesaswellasofthe”screening”to modifythe(vacuum)statecontentofsomeinitialwavefunctionψ(t =0)throughdynamicalevolution. We providesuchananalysisinfig.1, fortypicalQGPtemperatureshappeninginURHICandstartingfroma pure (vacuum) eigenstate considered at rest. For each temperature and each initial state, three models are investigatedtounderstandtherespectivecontributionsofthescreeningandofthestochasticforces. Inall cases,thetimeevolutionpopulatesalargevarietyofbb¯ eigenstates(attheexceptionoftheevolutionwith theMFonlywhichconservestheparity). Whenstochasticforcesareturnedon, oneobserves, aftersome transient phase lasting ∝ A−1, an asymptotic regime where all considered weights decay uniformly. This is due to the global probability flow driven out of the potential well by these forces (while for the pure MFcase,afiniteprobabilityistrappedinthiswell). Duringthisasymptoticregime,theweightratiostend towards values which solely depend on T (as can be seen by comparing both evolutions at T = 0.3 GeV, starting either from a 1S or a 2S state). We therefore interpret the transient phase as a re-equilibration of P.B.GossiauxandR.Katz/NuclearPhysicsA00(2016)1–4 3 thebb¯ d.o.f. leadingtoamixedquantumstateinthermalequilibriumwiththesurroundingQGP.Fromthe comparisonbetweenthesolidandthedashedcurvesonfig.1,werealizethatthestochasticforcesaremore efficientindestroyingtheΥ-likestateatsmallT andthescreeningtendtomemoreefficientathighT. The largestsuppressionisobtainedbycombiningbotheffects. Wi t Wi t 2.00 T(cid:61)0.2GeV;Ψ0(cid:61)(cid:85)1S(cid:45)like 2.00 T(cid:61)0.3GeV;Ψ0(cid:61)(cid:85)1S(cid:45)like (cid:85)1S(cid:45)like 1.00 (cid:72)(cid:76) 1.00 (cid:72)(cid:76) 0.50 (cid:85)1S(cid:45)like (cid:72) (cid:76) (cid:72) (cid:76) 0.50 (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) 0.20 0.20 Χb1P(cid:45)like (cid:72) (cid:76) 0.10 0.10 Χb1P(cid:45)like 0.05 (cid:85)2S(cid:45)lik(cid:72)e (cid:76) 0.05 (cid:85)2S(cid:45)like 0.02 (cid:72) (cid:76) 5 10 15 20 (cid:72)5 (cid:76) 10 15 20 0.02 t fmc t fmc (cid:72) (cid:76) Wi t (cid:64) (cid:144) (cid:68) Wi t (cid:64) (cid:144) (cid:68) 2.00 T(cid:61)0.3GeV;Ψ0(cid:61)(cid:85)2S(cid:45)like 2.00 T(cid:61)0.6GeV;Ψ0(cid:61)(cid:85)1S(cid:45)like 1.00 (cid:72)(cid:76) 1.00 (cid:72)(cid:76)(cid:85)1S(cid:45)like 0.50 (cid:72) (cid:76) (cid:72) (cid:76) 0.50 (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) Χb1P(cid:45)like 0.20 0.20 0.10 Χb(cid:45)like (cid:85)1S(cid:45)like 0.10 (cid:72) (cid:76) 0.05 0.05 (cid:85)2S(cid:45)like (cid:85)2S(cid:45)like (cid:72) (cid:76) 5 10 15 20 5 10 15 20 0.02 t fmc 0.02 (cid:72) (cid:76) t fmc (cid:72) (cid:76) Fig.1.(Coloronline)Evolutionof(cid:64)the(cid:144)w(cid:68)eightsofthe3lowestlyingvaccumeigenstates(blue:1S-(cid:64)like(cid:144),(cid:68)brown:1P-like,purple:2S-like) inastationarymedium.SolidlinescorrespondtoVW(T)withoutfrictionnorstochasticforces,whiledashedandthicklinescorrespond respectivelytoVW(0)andVW(T)withfrictionandstochasticforces. 3. BottomoniasuppressioninURHIC We now consider dynamical bb¯ in genuine URHIC, modeled by ideal fluid dynamics ensuing EPOS initialconditions[? 7](hereafternamedasEPOS2). Forthispurpose, wegeneratetheinitialpositionsof the bb¯ pairs according to the Glauber model. Essential bb¯ contributing to the production of final bound states are assumed to be global color-neutral states throughout the full evolution. The original center of massmomentump isthuspreservedandbb¯ pairsfollowstraighttrajectoriesalongwhichthetemperature CM entering the SLE ingredients is obtained from EPOS2 profiles, while the friction coefficient is evaluated from [6] at p = p /2. Typical time evolution are shown in fig. 2. In the left panel, we show the b/b¯ CM same information as in fig. 1, albeit with a more realistic initial bb¯ state – a Gaussian of width 0.045 fm, adjustedtotheΥ(cid:48)/Υratioinpp. Contrarilytothestationarymediumcase,oneobservesasaturationofthe weightsatlargetime,correspondingtothedisappearanceofthescreeningandtothefreezingofthethermal forces. FocusingontheΥ(1S),theevolutionseemstobedrivenbytheMFonly,afeaturethatisobviously not generic (see fig. 1) and deserves further investigation. For excited states, the role of stochastic forces is crucial, as one can judge from the evolution of the Υ(2S)-like content: Whereas these forces have the tendency to reduce it up to t ≈ 5 fm/c, they lead to a significant content afterwards (as compared to the pure MF evolution). This can be interpreted as a continuous repopulation of the Υ(2S)-like content from thelowestlyingstates,aneffectthatisusuallynotcorrectlyimplementedinotherapproaches. Ontheright paneloffig.2,weexploretheroleoftheinitialstateontheweightsofΥ(nS)andontheassociated”survival” S , defined as the ratio W (t)/W (0). We notice, in particular, that S is much smaller if one initiates the n n n 2 bb¯ stateasapure2Sstate. Thisisdue,onceagain,totherepopulationofhigherstatesfromthe1Sifone startsfromaGaussianwavepacket. Infig.3,wepresentthesesurvivalsSΥ andSΥ(cid:48) atthelaststageofthe 4 P.B.GossiauxandR.Katz/NuclearPhysicsA00(2016)1–4 evolution–whichcorrespond,inourmodel,tothenuclearmodificationfactorR measuredinexperiments AA –bothasafunctionofbottomoniatransversemomentum p andasafunctionofthenumberofparticipant T intheAAcollision. Globaltrendsarefoundtobeingoodagreementwithexperimentalmeasurements[8], inparticulartheincompletesuppressionoftheΥgroundstateaswellastheremarkablyflat p dependence. T Fig.2. (Coloronline)timeevolutionoftheaverageΥcontentofbb¯stateatsmallpCM,TfordifferentvariationsoftheSLEapproach embeddedintheEPOS2profile(seetextfordetails). 1.4 b(cid:61)12fm Tx fromEPOS2 1.4 Y2S11..02 (cid:76) bb(cid:61)(cid:61)48ffmm (cid:72) (cid:76) VW 2S11..02 (cid:225)(cid:76)T(cid:72)xp(cid:76)T(cid:225)f(cid:60)ro6mGEeVPOcS2 No(cid:224)(cid:225)Fe&&ed(cid:230)(cid:231)d::oVVwWUn(cid:33)(cid:33)(cid:33) &0.8 &Y0.8 (cid:224) (cid:144)(cid:225) (cid:225) (cid:225) (cid:225) RY1SAA00..46 (cid:72) RY1SAA00..46 (cid:231)(cid:72) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) 0.2 0.2 (cid:231) (cid:231) (cid:231) (cid:231) (cid:231) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) 0.0 0.0 0 5 10 15 20 0 100 200 300 400 p GeVc N T part Fig.3.(Coloronline)(cid:72)Nuclea(cid:144)r(cid:76)modificationfactorofΥandΥ(cid:48)asafunctionofpT (left)andofNparr(right). 4. Conclusion Toconclude,thisfirststudyshowsthepromisingfeaturesoftheSchro¨dinger-Langevinequationapplied tothetopicofbottomoniasuppressioninURHIC.Noticethatnofeeddownhasbeenincludedinthepresent work,alackwhichwillbecorrectedinanupcomingreviewpapercontainingmoredetailedanalysis. Acknowledgement WegratefullyacknowledgethesupportfromtheTOGETHERproject,Re´gionPaysdelaLoire(France). References [1] A.Andronic,etal.,arXiv:1506.03981[nucl-ex]. [2] M.D.Kostin,J.Chem.Phys.57(1972)3589. [3] R.KatzandP.B.Gossiaux,arXiv:1504.08087[quant-ph]. [4] A.MocsyandP.Petreczky,Phys.Rev.D77(2008)014501. [5] O.KaczmarekandF.Zantow,Eur.Phys.J.C43(2005)63. [6] P.B.GossiauxandJ.Aichelin,Phys.Rev.C78(2008)014904. [7] K.Werner,Iu.Karpenko,T.Pierog,M.Bleicher,K.Mikhailov,Phys.Rev.C82(2010)044904. [8] CMSCollaboration,CMSPASHIN-15-001(2015).