Upsilon suppression in the Schr\"odinger-Langevin approach

0.79 MB·

by  Pol Bernard Gossiaux

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.
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4rueAlfredKastler,44307Nantescedex3,France
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Abstract
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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
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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).
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