Charge-dependent correlations from event-by-event anomalous hydrodynamics

0.15 MB·

by  Yuji Hirono

Nuclear
Physics A
NuclearPhysicsA00(2016)1–4
www.elsevier.com/locate/procedia
Charge-dependent correlations
from event-by-event anomalous hydrodynamics
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1
0 Yuji Hironoa, Tetsufumi Hiranob, Dmitri E. Kharzeeva,c
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n aDepartmentofPhysicsandAstronomy,StonyBrookUniversity,StonyBrook,NewYork11794-3800,USA
a bDepartmentofPhysics,SophiaUniversity,Tokyo102-8554,Japan
J cDepartmentofPhysicsandRIKEN-BNLResearchCenter,BrookhavenNationalLaboratory,Upton,NewYork11973-5000,USA

5

]
h
p
- Abstract
p
e WereportonourrecentattemptofquantitativemodelingoftheChiralMagneticEffect(CME)inheavy-ioncollisions.
h Weperform3+1dimensionalanomaloushydrodynamicsimulationsonanevent-by-eventbasis,withconstitutiveequa-
[ tionsthat containtheanomaly-induced effects. Wealsodevelop amodel of theinitialconditionfor theaxialcharge

densitythatcapturesthestatisticalnatureofrandomchiralityimbalancescreatedbythecolorfluxtubes. Basingonthe
1
event-by-eventhydrodynamicsimulationsforhundredsofthousandsofcollisions,wecalculatethecorrelationfunctions
v
thataremeasuredinexperiments,anddiscusshowtheanomaloustransportaffectstheseobservables.
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Keywords: Chiralmagneticeffect,Chiralanomaly,Heavy-ioncollisions,Hydrodynamics
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0 1. Introduction
.
1
TheChiralMagneticEffect(CME)[1,2,3,4]hasreceivedconsiderableattentioninrecentyears,par-
0
ticularly in the context of heavy-ion collisions. The anomaly-inducedtransport effects like the CME are
6
1 macroscopicandareincorporatedintohydrodynamicequationsgivingriseto“anomaloushydrodynamics”
: [5]. Theoretically, the CME is expected to occur in heavy-ion collision experiments. The data reported
v
by STAR [6, 7] and PHENIX [8] collaborationsat RHIC and ALICE collaboration [9] at the LHC show
i
X
a behavior consistent with the CME, but the quantitative understandingis still lacking. In order to reach
r a definitive conclusion, a reliable theoretical tool that can describe the charge-dependent observables is
a
indispensable.
Inthiswork[10],we quantitativelyevaluatetheobservablesto detecttheanomaloustransport,basing
on event-by-eventsimulations of anomalous hydrodynamics. The observable of interest in this talk is a
charge-dependenttwo-particlecorrelation[11],
γ = cos φ1 +φ2 2Ψ , (1)
αβ α β− RP
D (cid:16) (cid:17)E
whereφi isthe azimuthalangleofi thparticle(i = 1,2)withchargeα +, , andΨ isthe reaction
α − ∈ { −} RP
planeangleforv . Physicalmeaningofthisobservableisevidentifwedecomposeγ as
2 αβ
γ = cos φ1 Ψ cos φ2 Ψ sin φ1 Ψ sin φ2 Ψ vαvβ aαaβ , (2)
αβ α− RP β− RP − α− RP β− RP ≡ 1 1 − 1 1
D (cid:16) (cid:17) (cid:16) (cid:17)E D (cid:16) (cid:17) (cid:16) (cid:17)E D E D E
2 /NuclearPhysicsA00(2016)1–4
wherevα(aα)isthedirectedflowwhichisparallel(perpendicular)toΨ ,respectively.
1 1 RP
Letusseehowa ’sbehaveinthepresenceofanomalouseffects. Inoff-centralcollisions,themagnetic
1
fieldsperpendiculartoΨ (onaverage)arecreated.IftheCMEoccurs,acurrentshouldbegeneratedalong
RP
themagneticfield,whichwouldresultinfinitea+ anda . Thedirectionofthecurrentdependsonthesign
1 −1
of the initial axial charge, which is basically random, so the signs of a s are also random. However, the
1
signsofa+anda tendtobeopposite.Thus,theCMEexpectationsarethefollowing:(1) a+ = a =0,
1 −1 1 −1
2 D E D E
becausethe signofinitialaxialchargeis random;(2) aα becomeslargerinthepresenceoftheCME
1
(cid:28)(cid:16) (cid:17) (cid:29)
currents;(3) a+a <0,whichindicatestheanti-correlationbetweena+anda .
1 −1 1 −1
D E
2. Event-by-eventanomaloushydrodynamicmodelforheavy-ioncollisions
The model consists of three parts: (i) anomalous-hydroevolution, (ii) hadronizationvia Cooper-Frye
formula,and(iii)calculationoftheobservables. Forthehydropart, wesolvetheequationsofmotionfor
a dissipationless anomalousfluid, ∂ Tµν = eFνλj , ∂ jµ = 0, ∂ jµ = CE Bµ, whereC Nc q2 is
µ λ µ µ 5 − µ ≡ 2π2 f f
theanomalyconstant, Eµ Fµνu , Bµ F˜µνu with F˜µν = 1ǫµναβF . Theenergy-momentumtenPsorand
currentsarewrittenasTµν≡=(ε+νp)uµu≡ν pηµνν, jµ =nuµ+κ2 Bµ, jµαβ=n uµ+ξ Bµ,whereεistheenergy
− B 5 5 B
density, p is the hydrodynamicpressure, n and n are electric and axial charge densities, eκ Cµ [1
5 B 5
≡ −
µn/(ε+ p)]andeξ Cµ[1 µ n /(ε+ p)]aretransportcoefficientsforchiralmagnetic/separationeffects
B 5 5
≡ −
(CME/CSE),andηµν diag 1, 1, 1, 1 istheMinkowskimetric.Inthiswork,theelectromagneticfields
≡ { − − − }
arenotdynamicalandtreatedasbackgroundfields. Asfortheequationofstate (EOS),we use thatof an
idealgasofquarksandgluons.
Letusspecifytheelectromagneticfieldconfigurationsusedtogettheresultsshownlater. WetakeB to
y
be(x-axisischosentobethereactionplaneangleΨ )
RP
b x2 y2 η2 τ
eB (τ,η ,x )=eB exp s , (3)
y s ⊥ 02R −σ2x − σ2y − σ2ηs − τB
where σ , σ , and σ are the widths of the field in x, y, and η (space-time rapidity) directions, τ is
x y ηs s B
the duration time of the magnetic field, R = 6.38 fm is the radius of a gold nucleus, and b is the impact
parameter. Other elementsof B and E are setto zero. The widthsare taken so thatthe fields are applied
only in the region where matter exists as σ = 0.8 R b , σ = 0.8 R2 (b/2)2, and σ = √2. We
x − 2 y − η
set other parameters as τ = 3 fm and eB = 0.5Ge(cid:16)V2 in(cid:17)following caplculations, which is equivalentto
B 0
eB (τ ,0,0) (3m )2.
y in π
∼
Bysolvingthehydrodynamicequations,weobtainaparticledistributionviatheCooper-Fryeformula
withfreezeouttemperatureT = 160MeV. We producethehadronsbytheMonte-Carlosamplingbased
fo
onthatdistribution. Thus, onerandominitialconditionresultsintheparticlesinanevent. We repeatthis
proceduremanytimesandstorethedataofmanyevents,thatarelaterusedtocalculatethecharge-dependent
correlationfunctions.Wecalculatefluctuationsofv anda separately,withthefollowingexpressions,
1 1
vα 2 1 cos(φα Ψ )cos(φα Ψ ) , aα 2 1 sin(φα Ψ )sin(φα Ψ ) .
(cid:28)(cid:16) 1(cid:17) (cid:29)≡*MP2 <Xi,j> i − RP j − RP + (cid:28)(cid:16) 1(cid:17) (cid:29)≡*MP2 <Xi,j> i − RP j − RP +
(4)
for the same-chargecorrelation, where M is the number of producedparticles, P = M(M 1),
M 2 − <i,j>
indicatesthesum overallthepairs, andouterbracketmeansaveragingoverevents. Similar expressPionis
usedfortheopposite-chargecorrelation.
Itisanimportantissuetoestimatetheamountofaxialchargesatthebeginningofhydroevolutions.The
majorsourcesoftheinitialchiralitiesarecolorfluxtubesinheavy-ioncollisions. Whentwonucleicollide,
numerouscolorfluxtubesarespannedbetweenthem. Theanomalyequation,∂ jµ =CEa Ba,determines
µ 5 ·
therateoftheaxialchargegeneration,sotherateisdeterminedbythevalueofEa Ba.Thereisnopreferred
·
signofEa Ba anditcanbepositiveornegativefordifferentcolorfluxtubes.
·
/NuclearPhysicsA00(2016)1–4 3
Inordertoincorporatethisfeature,wehavemadeanextensiontotheso-calledMC-Glaubermodel.For
eachbinarycollision,we assign 1randomly. Eachsignindicatesto thesignof color Ea Ba ofthe flux
± ·
tube.Then,weinitializetheaxialchemicalpotentialas
Ncoll(xT)
µ (x ,η )=C f(η ) X , (5)
5 T s µ5 s j
Xj=1
where X are the signs of Ea Ba randomly assigned to binary collisions, and C is a constant which
j · µ5
expressesthetypicalstrengthoftheµ5,and f(ηs)=exp(cid:20)−θ(|ηs|−∆ηs)(|ηs|−σ∆2ηηs)2(cid:21). Thesumistakenoverthe
binarycollisionshappeningonthatpointinthetransverseplane. Thestrengthofthecolorfieldsareofthe
orderofthesaturationscale. Takingthisintoaccount,wechoseC =0.1GeV[10].
µ5
1.5 × 10-4
Non-anomalous Anomalous
(v1-)2
1.0 × 10-4 (a1-)2
5.0 × 10-5
0
5.0 × 10-5
0
-5.0 × 10-5 Non-anomalous Anomalous
v1+ v1-
a1+ a1-
-1.0 × 10-4
Fig.1. Thecorrelations v 2 , a 2 (upperfigure), v+v ,and a+a (lowerfigure)foranomalousandnon-anomalouscasesat
h −1 i h −1 i h 1 −1i h 1 −1i
b=7.2fm.Thosequantiti(cid:16)esa(cid:17)reca(cid:16)lcu(cid:17)latedfromthedataof10,000eventsforbothoftheanomalousandnon-anomalouscases.
3. Calculatedobservables
The valuesofthe observablesareshownin Fig. 1. The datafrom10,000eventsare usedto calculate
thoseobservablesforeachofanomalousandnon-anomalouscase. Impactparameterissetto7.2fm. The
2 2
upper figure of Fig. 1 shows the values of v and a . In the left figure, anomalous transport
h −1 i h −1 i
effectsareswitchedoff(noCMEandCSE).Th(cid:16)ep(cid:17)lotsinthe(cid:16) rig(cid:17)htfigurearefromanomaloushydrodynamic
simulations. In the non-anomalouscase, the valuesofthe fluctuationsof v and a are similar. When we
1 1
2 2
switchontheanomaly(rightfigure), v goesup,and a increasesfurther.Thelargefluctuationof
h −1 i h −1 i
(cid:16) (cid:17) (cid:16) (cid:17) 2 2
a isinlinewiththequalitativeexpectationfromtheCME.Theorderofmagnitudeofγ = vα aα
is1comparabletoexperimentallymeasuredvalues. αα (cid:28)(cid:16) 1(cid:17) (cid:29)−(cid:28)(cid:16) 1(cid:17) (cid:29)
In the lower figure of Fig. 1, we show the values of v+v , and a+a . In the absence of anomaly,
h 1 −1i h 1 −1i
they take similar positive values, but once we turn on the anomaly, a+a becomesnegative. This is the
indicationoftheanti-correlationbetweena+anda andisconsistenthw1ith−1tiheCMEexpectations.
1 −1
Ithasbeendiscussedthattheobservedvaluesofγ mightbereproducedbyothereffectsunrelatedtothe
αβ
CME, includingtransversemomentumconservation[12, 13], chargeconservation[14], orcluster particle
correlations[15]. Sucheffectsareabsentinthecalculationshere,becausetheparticlesaresampledbased
ontheCooper-Fryeformula,whichisone-particledistribution,whereasallofthebackgroundeffectsarise
4 /NuclearPhysicsA00(2016)1–4
frommulti-particlecorrelations. Thus,thedifferencebetweenanomalousandnon-anomalouscalculations
purely originatesfrom the CME and CSE. The contributionfrom the transverse momentum conservation
intheCMEsignalisrecentlyestimatedinRef.[16],inwhichthechargedeformationsaretreatedaslinear
perturbationsonthebulkevolutionsin2+1D.
4. Conclusionsandoutlook
Wereportedtheresultsofevent-by-eventsimulationsofananomaloushydrodynamicmodelforheavy-
ion collisions. We solved the hydrodynamic equations including anomalous transport effects (CME and
CSE) in 3+1D, and calculated the values of observables. We also developed a model of the initial axial
chargescreatedfrom the color fux tubes. The caluculatedvaluesof the observablesindicate thatthis ob-
servableworksasexpected,andtheorderofmagnitudeiscomparabletoexperimentallymeasuredvalues.
The largest uncertainty arises from the choice of the life-time of the magnetic fields. The existence
ofconductingmatter affectsthe durationofthe magneticfields. We thushaveto solvethe hydrodynamic
equationstogetherwiththeMaxwellequations–thisworkisdeferredtothefuture.
Acknowledgements
This work was supported in part by the U.S. Department of Energy under Contracts No. DE-FG-
88ER40388andDE-AC02-98CH10886. Y.H.issupportedbyJSPSResearchFellowshipsforYoungSci-
entists. TheworkofT.H.wassupportedbyJSPSKAKENHIGrantsNo. 25400269.
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