Enhancement of jet quenching around phase transition: result from the dynamical holographic model

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by  Danning Li

Enhancement ofjet quenching around phasetransition:
result from the dynamical holographicmodel
Danning Li1, Jinfeng Liao2,3, Mei Huang 1,4
1 Institute of High Energy Physics, Chinese Academy of Sciences, Beijing, China
2 Physics Department and Center for Exploration of Energy and Matter,
Indiana University, 2401 N Milo B. Sampson Lane, Bloomington, IN 47408, USA
3 RIKENBNLResearchCenter, Bldg. 510A,BrookhavenNationalLaboratory, Upton, NY11973, USA
4 Theoretical PhysicsCenterforScience Facilities, ChineseAcademyof Sciences, Beijing, China
(Dated:January15,2014)
Thephasetransitionandjetquenching parameter qˆ havebeen investigatedintheframework ofdynamical
holographicQCDmodel. Itisfoundthatboththetraceanomalyandtheratioofthejetquenchingparameter
4 over cubic temperature qˆ/T3 show a peak around the critical temperature Tc, and the ratio of jet quenching
1 parameter over entropy density qˆ/s sharply rises at Tc. This indicates that the jet quenching parameter can
characterizethephasetransition.Theeffectofjetquenchingparameterenhancementaroundphasetransitionon
0
nuclearmodificationfactorR andellipticflowv havealsobeenanalyzed,anditisfoundthatthetemperature
2 AA 2
dependent jet quenching parameter from dynamical holographic QCD model can considerably improve the

n descriptionofjetquenchingazimuthalanisotropyascomparedwiththeconformalcase.
a
J
PACSnumbers:

4
1
I. INTRODUCTION substances [10]. It has been shown h /s is suppressed near

] thecriticaltemperatureinthesemiquarkgluonplasma[11],
h
andh /scancharacterizefirst-,second-orderphasetransitions
p StudyingQuantumChromodynamics(QCD) phasetransi-
- tionandpropertiesofhot/densequarkmatterathightemper- andcrossover[12],i.e. h /sshowsacusp,ajumpatTc anda
p shallowvalleyaroundT ,respectively.
ature has been the main targetof heavy ion collision experi- c
e
h ments at the Relativistic Heavy Ion collider (RHIC) and the In[13]ithasbeensuggestedthatthejetquenchingparame-
[ LargeHadronCollider(LHC).Itisnowbelievedthatthesys- ter can also be used to measure the coupling strength of the

tem created at RHIC and LHC is a strongly coupled quark- medium and a general relation between the shear viscosity
2
gluonplasma(sQGP)andbehaveslikeanearly”perfect”fluid h /sandthejetquenchingparameterqˆ/T3foraquasi-particle
v
5 [1,2]. dominatedquark-gluonplasma has been derived,i.e. h /s
3 The collective flow v of the highly excited and strongly T3/qˆ. Therelationassociatesasmallratioofshearviscosi∼ty
2
0 interacting matter formed at RHIC can be well describedby to entropy density to a large value of the jet quenching pa-
2 relativistichydrodynamicswithanegligibleratioofshearvis- rameter. Ifwenaivelyextendthisrelationtothecriticaltem-
1. cosity over entropy density h /s [3]. Lattice QCD calcula- peratureregion,wewouldexpectthatqˆ/T3 willshowapeak
0 tionconfirmedthath /sforthepurelygluonicplasmaisrather around the critical temperature Tc. Phenomenologically, the
4 smallandintherangeof0.1 0.2[4].Shearviscosityh char- strongnear-Tc-enhancement(NTcE)scenario of jet-medium
1 acterizeshowstronglypartic−lesinteractandmovecollectively interaction [14] was proposed in the efforts to explain the
:
v inamany-bodysystem. Ingeneral,thestrongertheinterpar- large jet quenching anisotropy at high pt at RHIC [15–18].
i ticle interaction, the smaller the ratio of shear viscosity over More recently it was shown in Refs. [19, 20] that the NTcE
X
entropydensity.Anotherunusualfeatureofthestronglyinter- modelnaturally inducesa reduction ( 30%) of jet-medium
ar actingmatterformedatRHICisthattheemissionofhadrons interactionstrengthfromRHICtoLHC∼.
with large transverse momentum is strongly suppressed in Itisworthyofmentioningthatanothertransportcoefficient,
central collisions [5]. The suppression of hadrons at large the bulk viscosity z /s, also exhibits a sharp rising behavior
transverse momentum is normally referred to as jet quench- around the critical temperature T as shown in Lattice QCD
c
ing, which characterizes the squared average transverse mo- [21–23], thelinearsigmamodel[24], thePolyakov-looplin-
mentum exchange between the medium and the fast parton earsigmamodel[25]andtherealscalarmodel[26]. Theris-
per unit path length [6]. Current knowledge on jet quench- ingofbulkviscositynearphasetransitioncorrespondstopeak
ingisthatitiscausedbygluonradiationinducedbymultiple oftraceanomalyaroundT,whichshowstheequationofstate
c
collisionsoftheleadingpartonwithcolorchargesinthenear- ishighlynon-conformal[27,28]aroundphasetransition.
thermal medium [6–8]. Therefore, jet quenchingcan tell us DuetothecomplexityofQCDintheregimeofstrongcou-
thepropertiesofthecreatedhotdensematterbytheenergetic pling,inrecentyears,theanti-deSitter/conformalfieldtheory
partonpassingthroughthemedium. (AdS/CFT)correspondence[29–31]hasgeneratedenormous
Itisveryinterestingtoaskwhethertransportquantitiescan interest in using thermal N = 4 super-Yang-Mills theory
characterizephasetransitions. (SYM) to understand sQGP. However, a conspicuous short-
It has been observed that the shear viscosity over entropy coming of this approach is the conformality of SYM: the
density ratio h /s has a minimum in the phase transition re- squareofthespeedofsoundc2alwaysequalsto1/3,thebulk
s
gioninsystemsofwater,helium,nitrogen[9]andmanyother viscosityisalwayszeroatalltemperaturesinthistheory,and
2
h /s= 1 [32]andqˆ/T3 7.53√l (l =g2 N the’tHooft (FRGs). In recent decades, an entirely new method based
4p ≃ YM c
coupling)[33]keepsaconstantforalltemperatures. Inorder on the anti-de Sitter/conformal field theory (AdS/CFT) cor-
todescribethenonconformalpropertiesnearphasetransition, respondence and the conjecture of the gravity/gauge duality
and mimic the QCD equation of state, much efforthas been [29–31] provides a revolutionarymethod to tackle the prob-
puttofindthegravitydualofgaugetheorieswhichbreakthe lemofstronglycoupledgaugetheories.
conformal symmetry, e.g, [34–37], where a real scalar dila- Ingeneral,holographyrelatesquantumfieldtheory(QFT)
ton field backgroundhas been introducedto couple with the in d-dimensions to quantum gravity in (d + 1)-dimensions,
graviton. Refs.[34] and [35] have used different dilaton po- withthegravitationaldescriptionbecomingclassicalwhenthe
tentialsasinput,Ref.[36]hasusedQCDb -functionasinput, QFT is strongly-coupled. The extra dimension can be inter-
andRef.[37]hasintroducedadeformedmetricbackground. pretedasanenergyscaleorrenormalizationgroup(RG)flow
On the other hand, much efforts have also been paid to intheQFT[46].
establish a more realistic holographic QCD model for glue-
ball spectra and meson spectra [38–42]. Recently, a dynam-
icalholographicQCD model[43–45]hasbeendevelopedby
resembling the renormalization group from ultraviolet (UV)
to infrared (IR). The dynamical holographic QCD model is
constructed in the graviton-dilaton-scalar framework, where
the dilaton background field F and scalar field X are re-
sponsibleforthegluodynamicsandchiraldynamics, respec-
tively. At the UV boundary, the dilaton field is dual to the
dimension-4 gluon operator, and the scalar field is dual to
the dimension-3quark-antiquarkoperator. The metric struc-
ture at IR is automatically deformed by the nonperturbative
gluon condensation and chiral condensation in the vacuum.
The produced scalar glueball spectra in the graviton-dilaton
framework agree well with lattice data, and the light-flavor
FIG. 1: Duality between d-dimension QFT and d+1-dimension
mesonspectrageneratedinthegraviton-dilaton-scalarframe-
gravityasshownin[46](Left-handedside).Dynamicalholographic
workareinwellagreementwithexperimentaldata. Boththe
QCDmodel resemblesRG fromUVtoIR (Right-handed side): at
chiralsymmetrybreakingandlinearconfinementarerealized UVboundarythedilatonbulkfieldF (z)andscalarfieldX(z)aredual
inthisdynamicalholographicQCDmodel. tothedimension-4gluonoperatoranddimension-3quark-antiquark
Inthiswork,wewillinvestigatethephasetransition,equa- operator,whichdevelopcondensatesatIR.
tionofstate andcalculatethejetquenchingparameterin the
dynamicalholographicQCD model. The paper is organized
asfollows.InSec.II,webrieflyintroducethedynamicalholo-
TherecentlydevelopeddynamicalholographicQCDmodel
graphic QCD model for pure gluon system and light-flavor
[43,44]canresembletherenormalizationgroupfromultravi-
system.InSec.III,wewillinvestigatethephasetransitionand
olet(UV)toinfrared(IR)asshowninFig.1[45]. Thedilaton
equationof state, includingthe entropydensity, the pressure backgroundF (z)isintroducedtodescribethegluodynamics,
density, the energydensity for the pure gluon system. Then
and the scalar field X(z) is responsible for chiral dynamics,
in Sec.IV, Sec.V andSec.VI, we calculate jet quenchingpa-
respectively.
rameter and investigate the nuclear modification factor R
AA For the pure gluon system, we construct the quenched
andellipticflowv . Thesummaryanddiscussionisgivenin
2 dynamical holographic QCD model in the graviton-dilaton
Sec.VII. frameworkbyintroducingonescalardilatonfieldF (z)inthe
bulk. The 5D graviton-dilaton coupled action in the string
frameisgivenbelow:
II. DYNAMICALHOLOGRAPHICQCDMODEL
1
Quantum chromodynamics (QCD) in terms of quark and SG= 16p G5Z d5x√gse−2F (cid:0)Rs+4¶ MF ¶ MF −VGs(F )(cid:1).(1)
gluondegreesoffreedomisacceptedasthefundamentalthe-
oryofthestronginteraction. Intheultraviolet(UV)orweak Where G5 is the 5D Newton constant, gs, F andVGs are the
coupling regime of QCD, the perturbative calculations for 5Dmetric,thedilatonfieldanddilatonpotentialinthestring
deep inelastic scattering (DIS) agree well with experimental frame,respectively.Themetricansatzisoftenchosentobe
data. However, in the infrared (IR) regime, the description
ofQCD vacuumaswellashadronpropertiesandnonpertur- ds2=e2As(z)(dz2+h mn dxm dxn ). (2)
bativeprocessesstillremainsasoutstandingchallengeinthe
formulation of QCD as a local quantum field theory. In the Inthispaper,thecapitalletterslike”M,N”wouldstandforall
pasthalfcentury,variousnon-perturbativemethodshavebeen the coordinates(0,1,..,4), and the greek indexes would stand
developed,inparticularlatticeQCD,Dyson-Schwingerequa- forthe4Dcoordinates(0,...,3). Wewouldusetheconvention
tions(DSEs),andfunctionalrenormalizationgroupequations h 00=h = 1,h ij=h =d .)
00 ij ij
−
3
To avoidthe gaugenon-invariantproblemandto meetthe III. PHASETRANSITIONANDEQUATIONOFSTATES
requirementofgauge/gravityduality,wetakethedilatonfield
intheformof The chiral and deconfinement phase transitions for the
F (z)=m 2z2tanh(m 4 z2/m 2). (3) graviton-dilaton-scalarsystemwillbeinvestigatedinthenear
G G2 G future. In this section, we will focus on the deconfinement
In this way, the dilaton field at UV behavesF (z)z→0m 4 z4, phasetransitionforthepuregluonsystemdescribedbyEq.(1).
→ G2 The finite temperature dynamics of gauge theories, has
andisdualtothedimension-4gaugeinvariantgluonoperator
TrG2, whileatIRittakesthequadraticformF (z)z→¥ m 2z2. a natural holographic counterpart in the thermodynamics of
→ G black-holesonthegravityside. Addingtheblack-holeback-
Theequationsofmotioncanbederivedas
groundtotheholographicQCDmodelconstructedfromvac-
4 2 uumproperties,inthestringframewehave
A′′ F ′A′ +A2+ F ′′ =0, (4)
− s−3 s s 3
F ′′+(3A′s−2F ′)F ′−38e2As−34F ¶ F (e43F VGs(F ))=0. (5) ds2S=e2As(cid:18)−f(z)dt2+ fd(zz2)+dxidxi(cid:19). (11)
Byself-consistentlysolvingtheEinsteinequations,themetric
The metric in the string frame is useful to calculate the jet
structureA willbeautomaticallydeformedatIRbythedila-
s quenchingparameter.
tonbackgroundfieldorthenonperturbativegluodynamics. It
Thethermodynamicalpropertiesofequationofstateiscon-
isfoundin[44]thatthescalarglueballspectrainthequenched
venienttobederivedintheEinsteinframe,
dynamicalmodelis inverywellagreementwithlattice data.
FoTrodedtaeislcsr,ipbleeathseertewfeor-fltoavRoerfss.y[4st4e]m. , we then add light fla- ds2E =e2As−43F (cid:18)−f(z)dt2+ fd(zz2)+dxidxi(cid:19). (12)
vors in terms of meson fields on the gluodynamical back-
ground. The total 5D action for the graviton-dilaton-scalar Undertheframetransformation
systemtakesthefollowingform:
gE =gs e 2F /3, VE =e4F /3Vs, (13)
N mn mn − G G
f
S=S + S . (6)
G KKSS
Nc Eq.(1)becomes
HereS isthe5DactionforgluonsintermsofdilatonfieldF
G 1 4
faonrdmtaekseosntsheprsoapmaegaftoirnmgoansEthqe.(d1)il,aatnodnSbKacKkSSgriosuthned5aDndatcatkioens SGE = 16p G5Z d5x√gE(cid:18)RE−3¶ mF ¶ mF −VGE(F )(cid:19).(14)
thesameformasintheKKSSmodel[39]
We can derive the following equations from the Einstein
SKKSS = d5x√gse−F Tr(DX 2+VX(X+X,F ) equationsof(t,t),(z,z)and(x1,x1)components:
−Z | |
2 4
+41g2(FL2+FR2)). (7) −A′s′+A′s2+3F ′′−3A′sF ′ = 0, (15)
5 f (z)+ 3A (z) 2F (z) f (z) = 0 (16)
The difference here is that the metric structure A is solved ′′ ′s − ′ ′
s (cid:0) (cid:1)
fromthefollowingcoupledequationsofmotion: TheEOMofthedilatonfieldisgivenasfollowing
−A′s′+As′2+23F ′′−34A′sF ′−l60eF c ′2 = 0, (8) 83¶ z(cid:16)e3As(z)−2F f(z)¶ zF (cid:17)−e5As(z)−130F ¶ F VGE =0. (17)
3l
F ′′+(3A′ 2F ′)F ′ 0eF c ′2 TogetthesolutionsweimposetheasymptoticAdS condi-
s− − 16 tion f(0)=1 nearthe UV boundaryz 0, and requ5ireF ,f
3e2As−34F ¶ F VG(F )+l 0e73F VC(c ,F ) = 0, (9) tobefiniteatz=0,zh withzh theblack∼-holehorizon. Fortu-
−8 (cid:16) (cid:17) nately,wefindthatthesolutionoftheblack-holebackground
c ′′+(3A′s−F ′)c ′−e2AsVC,c (c ,F ) = 0. (10) takestheformof,
H16eprGe5Nwf e hlav.e defined VC = Tr(VX) and VC,c = ¶¶cVC. f(z)=1−fchZ0ze−3As(z′)+2F (z′)dz′, (18)
L3Nc → 0
Fortwoflavorsysteminthegraviton-dilaton-scalarframe- with
work,thedeformedmetricisself-consistentlysolvedbycon-
1
sideringboththechiralcondensateandnonperturbativegluo- fh= , (19)
dynamicsinthevacuum,whichareresponsibleforthechiral c 0zhe−3As(z′)+2F (z′)dz′
symmetrybreakingandlinearconfinement,respectively. The . R
mixing between the chiral condensate and gluon condensate A black-hole solution with a regular horizon is character-
isimportanttoproducethecorrectlightflavormesonspectra ized by the existence of a surface z=z , where f(z )=0.
h h
[44]. The Euclidean version of the solution is defined only for
4
0<z<z , in order to avoid the conical singularity, the pe- T(cid:144)GeV
trThioheFdistrieocdmimetypteEeorhrqmfa.ttiuh(n1reee8sE)at,unhocdenlitpedeotemcsaa→iTpnntieote=ritnaamts+|oueiflfry|c′e4(tfahprzo′n4ehe(fapz)bbtd|hhe.l)aoe|ficu.sxktoetlhdhuoetblieoryenhlaoatrsiioznonb.etw((e22e10n)) 000000......123456++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ +++ ++++++++++ ++++++++++ΜΜ+++ΜP+++GGG22+++u2r+++===e+++ A3¥0+++..d07S+++5GG+++eVe+++V+++ +++ +++ +++ +++ +++ +++ +++
T = 4p 0ezh−e3−As3(Azhs()z+′)2+F 2(Fzh()z′)dz′ (22) 1 2 3 4 zh(cid:144)GeV-1
R
For numerical calculations, we take m = 0.75GeV in
G
Eq.(3) so that the transition temperature is around 255MeV,
a0n.7d5GweeVt,amkGe2t=hr3eeGedViffaenrdenmtGv2a=lu¥es.WfohremnGm2G:2=m G¥2,=themdGil=a- FsmpIGeGc=.ti3v0:e.7lTy5h.GeTetheVmeapbnelduraemtulGirn2ee=asss0at.a7fn5udGncfeotViro,CnmaoGsf2eh=Io,r3IizIGoaennVdfoatrnhGde5mreG=d21l=i.n2e¥5s,aarnerde-
ton field Eq.(3) takes the form of quadratic term F =m 2z2, resultsofAdS-Schwardzblackhole.
G
and the model can be regarded as the self-consistent KKSS
model. The only difference is that in this model the met-
ricstructureisselfconsistentlydeformedbythedilatonback- From Eq.(16) and Eq.(22), we can get the f(z) solution
ground,whileintheKKSSmodel,themetricstructuretakes and the temperature behavior. The temperature v.s. hori-
thesameasAdS5. zon for m G = 0.75GeV and m G2 = 0.75GeV, m G2 = 3GeV
and m =¥ are shown in Fig.3. As long as m is large,
G2 G2
theIRphysicsisnotsensitivetolargem , thebehaviorsfor
As+Log@z(cid:144)LD m =3GeVandm =¥ arealmosttheGs2ame. FromFig. 3,
4 G2 G2
itisnoticedthatforpureAdS Schwarzblack-hole,thetem-
5
perature monotonically decreases with the increasing of the
3 horizon. If one solves the dual black-hole background self-
Μ =¥
G2 consistently,onecanfindthatthereisaminimaltemperature
2 ΜG2=3.0GeV Tmin=255MeVatcertainblack-holehorizonz0h. Thisissim-
ilartothecasefortheconfiningtheory(atzerotemperature)
ΜG2=0.75GeV discussedinRef. [35]. ForT <Tmin,therearenoblack-hole
solutions. ForT >T ,therearetwobranchesofblack-hole
1 min
+ + + Pure AdS solutions. When z <z0, the temperature increases with the
h h
decreasingofz ,whichmeansthatthetemperatureincreases
h
0+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ when the horizon moves close to UV, this phase is thermo-
dynamicallystable. Whenz >z0, thetemperatureincreases
h h
withtheincreaseofz ,whichmeansthatthetemperaturebe-
h
z(cid:144)GeV-1
comeshigherandhigherwhenthehorizonmovestoIR.This
0 5 10 15 20
indicates that the solution for the branch z > z0 is unsta-
h h
ble and thus not physical. In order to determine the critical
temperature, we have to compare the free energy difference
FIG.2:AsconfigurationscomparedwithAdS5metricforG5=1.25
and m G =0.75GeV and m G2 =m G =0.75GeV, m G2 =3GeV and between the stable black hole solution and the thermal gas.
m =¥ ,respectively.Toshowtheconfigurationsmoothly,wehave Following the discussion in [37], the transition temperature
G2
subtractedthelog(z)divergenceinAs. would be near this minimal temperature and we would just
takeitasthetransitiontemperatureT =255MeV,whichisin
c
agreementwithlatticeresultforpuregluonsystem.
WecansolveA fromEq.(15),andtheresultsofA config-
s s
urationsform =0.75GeVandm =0.75GeV,m =3GeV FromtheBekenstein-Hawkingformula,onecaneasilyread
G G2 G2
and m = ¥ are shown in Fig.2. To show the configura- theblack-holeentropydensitys,whichisdefinedbythearea
G2
A ofthehorizon:
tion smoothly, we have subtracted the log(z) divergence in area
Amse.trCicosmtrpuacrtuinrgeiwsiltahrgAedlySd5emfoertmriecd, iattiIsRebaysythteodfiilnadtotnhabtacthke- s= 4AGreVa |zh = 4G1 e3As(zh)−2F (zh). (23)
ground field or gluodynamics. The two cases m =3GeV 5 3 5
G2
andm =¥ arealmostthesame. Where G is the Newton constant in 5D curved space and
G2 5
5
s Ε-3p
T3 T4
56 ++++++++++++++++++++++++++++++++++++++++++ΜG+++2=+++¥++++++++++++++++++++++++++++++ 33..05 ΜΜGG22==¥3.0GeV
++++++ ΜG2=3.0GeV +++++++++
4 ++++++ 2.5 +++ +++ ΜG2=0.75GeV
+++ ΜG2=0.75GeV
+++ 2.0 +++ + + + SU(3) Lattice
3 +++ +++
+++ + + + SU(3) Lattice
1.5
+++ +++
2 +++ +++
++++++ 1.0 +++
1 +++++++++++++++ 0.5 +++ ++++++ +++
++++++++++++ T 0.0 +++++++++ +++ +++ T
1 2 3 4 Tc 1 2 3 4 Tc
FIG. 4: The entropy density as a function of T/Tc for G5 =1.25 FIG. 5: Trace anomaly as a function of T/Tc for G5 =1.25 and
andm G=0.75GeVandm G2=0.75GeV,m G2=3GeVandm G2=¥ , m G=0.75GeVandm G2 =0.75GeV,m G2 =3GeVandm G2 =¥ ,re-
respectively.Theredcrossesarelatticeresultsfrom[27]. spectively.Theredcrossesarelatticeresultsfrom[27].
V is the volume of the spatial directions. It is noticed that
3
the entropy density is closely related to the metric in the c2 will deviate from1/3. FromEq.(26), we can see that the
s
Einstein frame. The results of scaling entropy density s/T3 speedofthesoundisindependentofthenormalizationofthe
for m = 0.75GeV and m = 0.75GeV, m = 3GeV and 5DNewtonconstantG andthespacevolumeV .
G G2 G2 5 3
m =¥ areshowninFig.4comparedwithlatticeresultsfor
G2 Thenumericalresultofthesquareofthesoundvelocityis
the pure gluon system [27]. It can be seen that when m is
G2 showninFig.6. AtT, the soundvelocitysquareisaround0
largeenough,theresultisnotsensitivetothevalueofm ,and c
G2 whichisinagreementwithlatticedata[27]. Athightempera-
ittakesalmostthesameasthatintheselfconsistKKSSmodel.
ture,thesoundvelocitysquaregoesto1/3,whichmeansthat
Theentropydensityform =3GeVto m =¥ agreeswell
G2 G2 thesystemisasymptoticallyconformal.
withthelatticeresultforpureSU(3)gaugetheory.
The pressure density p(T) can be calculated from the en-
tropydensitys(T)bysolvingtheequation
C2
s
dp(T) 0.35
andtheenergydensityeis=dreT−lapte+=dtssoT(T.th)e,entropydensityby((2245)) 00..2350 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++Μ+++ΜG+++G2+++2==+++¥3+++.0+++G+++eV+++ ++++++ +++ ++++++++++++
Fm iGgT2.5h=eco0trm.a7cp5eaGraeendVow,mimtahGly2la=(tetic3−eG3reepVs)u/latTsn4dfofmroGrth2me=Gpu=¥re0ag.rl7eu5osGnheosVwysntaenmind 00..1250 +++++++++++++++++++++++++++ + + + SΜUG(23=) L0a.7tt5icGeeV
+++
[27]. The trace anomaly shows a peak around T/Tc =1.1. 0.10 ++++++
When m =0.75GeV, the heightof the peak is around3.7, ++++++
G2 +++
andwhenm G2 =3GeV∼¥ ,theheightreducesto2.7,which 0.05 ++++++
agreeswithlatticedataforpureSU(3)gaugetheoryasshown
T
in[27]. Atveryhightemperature,thetraceanomalygoesto 0.00
zero,whichindicatesthesystemisasymptoticallyconformal 1 2 3 4 Tc
athightemperature.
Thesoundvelocityc2canbeobtainedfromthetemperature
s
andentropy:
FIG.6: Thesquareofthesoundvelocityc2 asafunctionofscaled
s
c2= dlogT = s , (26) temperature T/Tc for G5 =1.25 and m G =0.75GeV and m G2 =
s dlogs Tds/dT 0.75GeV, m G2 =3GeVandm G2 =¥ ,respectively. Theredcrosses
arelatticeresultsfrom[27].
which can directly measure the conformality of the system.
For conformalsystem, c2 =1/3, for non-conformalsystem,
s
6
z
IV. JETQUENCHINGPARAMETERqˆ
Jet quenchingmeasures an energetic parton interacts with
thecreatedhotdensemedium. Itisveryimportanttofindthe z=zh
characterizationoftheresultingmediuminducedmodification
of high-p parton fragmentation, i.e., jet quenching and its
T
connectiontopropertiesofthehotdensematter,andwhether
and how such a parameter can tell us about the QCD phase
transitions.
It has been expected that the shear viscosity over entropy
density ratio h /s has a minimum in the QCD phase transi-
tion region [12] as that in systems of water, helium and ni-
trogen[9, 10]. The bulk viscosity z /s, also exhibitsa sharp
risingbehavioraroundthecriticaltemperatureTcasshownin x2=-L(cid:144)2 x2=L(cid:144)2
Lattice QCD [21–23] and some modelcalculations[24, 26]. x
It has been suggested in [13] shear viscosity h /s and the 2
jetquenchingparameterqˆ/T3 fora quasi-particledominated (t =x−,s =x2)
quark-gluonplasmahasageneralrelationh /s T3/qˆ. Ifwe z
∼
naivelyextendthisrelationtothecriticaltemperatureregion,
wewouldexpectthatqˆ/T3 willshowapeakaroundthecrit-
icaltemperatureTc. Phenomenologically,thestrongnear-Tc- z=z
h
enhancement(NTcE)scenarioofjet-mediuminteraction [14]
wasproposedintheeffortstoexplainthelargejetquenching
anisotropyathigh p atRHIC[15–18].
t
There hasno modelcalculationsfor the jet quenchingpa-
rameter around the critical temperature. Lattice QCD is not
suitable for transport properties, recently there are several
groupsplayedeffortsoncalculatethejetquenchingparameter
onlattice[47,48]. However,noinformationonjetquenching
parameterhasbeenextractedaroundthecriticaltemperature.
Inthissection,wewillinvestigatethejetquenchingparameter x =-L(cid:144)2 x =L(cid:144)2
2 2
aroundthecriticaltemperatureinthe dynamicalholographic x
QCDmodelwhichcandescribephasetransitions. 2
Following[33](seealso[49–51]),thejetquenchingparam- (t =x ,s =z)
−
eterisrelatedtotheadjointWilsonloopby
1
WAdj[C] exp( qˆL L2) (27) FIG.7:Twokindsofstringconfigurations.
−
≈ −4√2
whereL−,L are distancesalongx− = t−√x21 and spatialdirec- andwiththebelowconfigurationinFig.7wehave
tion x respectively. (Another method for jet quenching of
2
lightquarkshasbeendevelopedin[52].) 1
Denotingx+= t+√x21,x− = t−√x21, thenthemetricinEq.(11) S2= 2pa ′ Z dt ds √g−−gzz, (31)
becomes
1 f(z) where g = e2As1−2f(z),gzz = ef2(Azs),g22 = e2As can be read
ds2 = e2As [1+f(z)]dx−dx++ − fromEq.−(2−8).
{ 2
ThenweextracttheadjointWilsonloopby
1
[dx+2+dx 2]+ dz2+dx2+dx2 , (28)
− f(z) 2 3} WAdj=exp(2i(S S )), (32)
1 2
. −
Theactiononthestringworldsheetistakentobe andfromthesmallLexpansionofWAdj wegetqˆoftheform
1
SNG= 2pa ′ Z d2s q−det(Gab ) (29) qˆ= √2√l , (33)
wwoitrhldGsabhee=t. gsmn ¶ a xm ¶ b xn the induced metric on the string p R0zhdzqgzz/(g222g−−)
WiththeaboveconfigurationinFig.7wehave with√l = R2ads.
a
′
1 Thenumericalresultsofthejetquenchingparameterqˆand
S1= 2pa ′ Z dt ds qg−−gzzz′2(s )+g−−g22, (30) the ratio of qˆ/T3 for m G = 0.75GeV and m G2 = 0.75GeV,
7
q`(cid:144)HGeV2(cid:144)fmL find that qˆ/T3 indeedshows a peak at the same temperature
25 Μ =¥ where the trace anomaly also shows a peak. For the case
1250 q`(cid:144)HG+++2345e++++++V++++++2+++(cid:144)f+++m++++++L+++++++++++++++++++++++++++++++++++++++++++++++++++++++ +++ +++++++ +++ +++++++++++++P+++Μ+++uΜ+++GrGG+++e2+++22 +++A==++++++d30S..075GGeVeV ++++++++++++ rfTotharfeme=IemtveGpi1atse2al.u1rrw=aeTqoˆmcor3.eetfGhxtteytherreVa.ocIf’tt∼nemHdteh¥onfeo,rtofritotehmcnceeoiennhuxgteppiwltgeihnhoraigtrtmkoloef[u,n5rtwt3hqˆ]ihes,iistcpahhermeoaauukjtecnttihhdsqedu1amere.1ponoGeucmnnheddieVnen42gnt/0tipfsomaana-t
+++ atT =370MeVand1.9GeV2/fmforT =470MeV,whichis
1 +++
10 5timessmallerthanourresults. Thismightindicatethatwe
+++
0 T(cid:144)HGeV+++-1L should take a smaller ’t Hooft coupling. However, the tem-
0.24 0.26 0.28 0.30 0.32
+++ peraturedependentfeatureisindependentofthe’tHooftcou-
5 +++
+++ pling.
+++
+++
+++
0++++++++++++++++++++++++++++++++++++ T(cid:144)HGeV-1L
0.0 0.1 0.2 0.3 0.4 0.5
V. JETQUENCHINGCHARACTERIZINGPHASE
TRANSITION
FIG.8: JetquenchingparameterasafunctionofthetemperatureT
form G=0.75GeVandm G2 =0.75GeV,m G2 =3GeVandm G2 =¥
Ε-3p
withG5=1.25. The red crosses are the resultsof AdS-SWblack
holein[33].Wehavetakenl =6p here. T4
3.0
ΜG=0.75GeV
`
q 2.5 ΜG=0.60GeV
T3
40 2.0 ΜG=0.50GeV
ΜG=0.43GeV
1.5
+ + + Pure AdS
35
1.0
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
ΜG2=¥ 0.5
30
ΜG2=3.0GeV
0.0+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
T(cid:144)GeV
ΜG2=0.75GeV 0.0 0.1 0.2 0.3 0.4 0.5 0.6
25 q`
+ + + Pure AdS
T3
40
T
20
1 2 3 4 Tc
35
FIG.9: Jetquenching parameterover cubictemperatureqˆ/T3 asa +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
function of T/Tc for m G =0.75GeV and m G2 =0.75GeV, m G2 = 30 ΜG=0.75GeV
3GeVandm G2 =¥ withG5=1.25. Theredcrossesaretheresults ΜG=0.60GeV
ofAdS-SWblackholein[33].Wehavetakenl =6p here. ΜG=0.50GeV
25 ΜG=0.43GeV
m =3GeV and m =¥ are shownin Fig.8 and Fig.9, re- + + + Pure AdS
G2 G2
spectively. TheresultsarecomparedwiththeAdS case. For
5 20 T(cid:144)GeV
all the cases, we have taken l =6p as in [33]. It is found 0.0 0.1 0.2 0.3 0.4 0.5 0.6
thatthejetquenchingparameterqˆitselfdoesnotshowmuch
differencesforallthecases. Itisevenhardtofindmuchdif-
ferences comparingwith the AdS case. For all these cases, FIG.10:qˆ/T3andtraceanomaly(e 3p)/T4asafunctionofT for
rthanegveal3u0e0of q4ˆ0is0MareoVun,dwh5ic∼h1is0Gin5eVag2r/efemmeinntthweitthemthpeelraatttuicree dfoifrfeArdeSn5tvcaalsuee.sWoefhmaGvewtaitkhenm GG25==31−G.2e5Va∼nd¥l.=Th6ep rheedrec.rossesare
result in [4∼8]. However, the ratio qˆ/T3 shows very differ-
ent behavior for different cases: For the AdS case, the ra- We have observedthatbothqˆ/T3 andtrace anomaly(e
5
tio is a constant, for the dynamicalholographicQCD model 3p)/T4 showapeakaroundthecriticaltemperatureform −=
G
which can describe deconfinement phase transition, we can 0.75GeV, which indicates that the jet quenching parameter
8
Ε-3p q`
T4 s
3.0
15 ΜG=0.75GeV
2.5 ΜG=0.75GeV ΜG=0.60GeV
ΜG=0.60GeV
ΜG=0.50GeV
2.0 ΜG=0.50GeV 10 ΜG=0.43GeV
1.5 ΜG=0.43GeV + + + Pure AdS
+ + + Pure AdS
1.0
5++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
0.5
0.0++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ T
0 T(cid:144)GeV
1 2 3 4 Tc 0.0 0.1 0.2 0.3 0.4 0.5 0.6
` `
q q
T3 s
40
15 ΜG=0.75GeV
ΜG=0.60GeV
ΜG=0.50GeV
35
ΜG=0.43GeV
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 10 + + + Pure AdS
ΜG=0.75GeV
30
ΜG=0.60GeV
ΜG=0.50GeV 5++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
25 ΜG=0.43GeV
+ + + Pure AdS
T T
20 0
1 2 3 4 Tc 1 2 3 4 Tc
FIG.11:qˆ/T3andtraceanomaly(e 3p)/T4asafunctionofT/Tc FIG.12: qˆ/sasafunctionofT andT/Tc fordifferentvaluesofm G
aforerfdoifrfAerdeSn5tvcaasluee.sWoefhmaGvewtaitkhenm GG25−==31G.2e5Va∼nd¥l.=Th6ep rheedrec.rosses wcaisteh.mWGe2h=av3eGteaVken∼G¥ 5,=re1sp.2e5ctaivnedlyl. =Th6ep rehdercer.ossesarefor AdS5
over cubic temperature can characterize QCD phase transi- needed. ForthecaseofAdS ,thetemperatureisT = 1 ,the
tion.Inthissection,weexplorehowdifferentvaluesofm Gaf- entropydensitytakesthefor5mof p zh
fectthephasetransitionandjetquenching.From[43,44],m
G
isrelatedtothelinearconfinementanddeterminestheRegge 1 1 p 3 1
s = = T3 7.75 T3, (34)
slope of the glueball spectra as well as the string tension of AdS5 4G z3 4G ≃ G
5 h 5 5
thelinearquarkpotential.
InFigs.10and11,weshowthebehaviorofqˆ/T3 andtrace andthejetquenchingparameterisgivenby:
anomaly(e 3p)/T4 fordifferentvaluesofm asafunction
G
−
ofFtermomperFaitgu.r1e0T, awnedsficnaldedthTa/tTcfo,rresdpifefcetrivenelty.values of m G qˆAdS5 = p 3/2G√[5l/G4[]3/4]T3≃7.53√l T3. (35)
(with m ¥ ), the critical temperature T increases with
m G. WGe2c→an read Tc = 146,170,204,255cMeV for m G = Therefore,intheAdS5limit,theratioofjetquenchingparam-
0.43,0.5,0.6,0.75GeV, respectively. Itisalsoobservedthat eteroverentropydensitytakesthevalueof
the height of the peak for either qˆ/T3 or (e 3p)/T4 does
innoctrecahsaensgweiwthitmh t.he value of m G, but the wid−th of the peak qˆAdS5/sAdS5 =0.97G5√l . (36)
G
Another interesting observation from Fig.11 is that either With parameters used in our work, we have qˆ /s =
qˆ/T3 or(e 3p)/T4 asafunctionofscaledT/Tc isnotsen- 5.27. The ratio of qˆ/s in the dynamical hQCDAdmS5odeAldSa5s a
sitivetom G−,i.e. qˆ/T3(T/Tc)or(e 3p)/T4(T/Tc)overlaps function of T and T/Tc is shown in Fig. 12 compared with
fordifferentvaluesofm G. − theAdS5 result. Itisfoundthattheratioofqˆ/sreachesAdS5
Inthenextsection, wewillinvestigatethenuclearmodifi- limit5.27athightemperature,anditsharplyriseswiththede-
cation R and elliptic flow v , where the behaviorof qˆ/s is creasingofT anddevelopsapeakexactlyatT withtheheight
AA 2 c
9
16.3,whichisabout3timeslargerthanitsvalueathightem- RHICtoLHC.Anumberofrecentanalyseshaveconsistently
perature. It is worthy of mentioning that the sharp rising of reportedthatthe R data at RHIC andLHC indeedsuggest
AA
qˆ/saroundT isverysimilartothebehaviorofbulkviscosity about 30%reductionof jet-mediuminteractionatLHC as
c
overentropydensityz /sasshownin[21,22]. compa∼red at RHIC [19, 20, 60–65]. Therefore phenomeno-
Moreover,qˆ/sasafunctionofthescaledtemperatureT/T logically, it appears that there are now strong evidences for
c
overlapsfordifferentvaluesofm G. anontrivialtemperaturedependence,inparticularthenear-Tc
enhancement,ofjet-mediumcouplingonmattertemperature.
0.7ƒ
VI. JETQUENCHINGPHENOMENOLOGYFROM
HOLOGRAPHY V ƒƒ
Te 0ƒ.6
2
In this section, we study the phenomenological implica- 0.
C ƒ
tionsofthetemperaturedependenceforqˆ(T)asobtainedfrom I 0.5 ƒ
H
theholographymodelabove.Theobservablecommonlyused R ƒ
forjetquenchingphenomenologyinAAcollisionsisthenu- ž 0.4 ƒ
T
clear modification factor, R , defined as the ratio between P
AA h ƒƒ
the hadron production in AA collision and that in NN col- g 0.3
hi ƒ
lision (further scaled by the expected binary collision num- A ƒ
A
ber).Ifajetpartonlosesenergyalongitspathpenetratingthe R 0.2 ƒƒ ƒƒ
hotmedium,oneexpectsasignificantsuppressionofleading
high-pt (transverse momentum) hadron production from the 50 100 150 200 250 300 350
jet as compared with the pp collision at the same beam en-
N
part
ergy.AstrongsuppressionwasfirstobservedatRHIC[5]and 0.12 ƒƒ
then at LHC [54], with RAA reaching 0.2 in the mostcen- ƒƒ
tral collisions. Another importantasp∼ect of jet quenchingis V ƒ ƒ
e
thesocalledgeometrictomography[55]bymeasuringtheaz- T 0.10
2
imuthalangledependenceofthesuppressionR (f )wheref 0. ƒƒ
AA
C
istheangleoftheproducedhadronwithrespecttothereaction I 0.08
H ƒ
plane. In a typical off-central collision, the hot medium on R
averagehasanalmond-likegeometricshape,andthusthejet ž 0.06 ƒ ƒ
T
in-mediumpath length would dependon its orientation with P
respect to the matter geometry, leading to nontrivial depen- gh
dence of the suppression on azimuthalangle. The dominant hi 0.04 ƒ
2
componentofthef -dependenceisthesecondharmonicwith V ƒ
its coefficient commonly referred to as v . Both RHIC and 0.02
2
LHC measurements have shown a sizable v in the high p 50 100 150 200 250 300 350
2 t
regionwherethejetenergylossshouldbethemechanismof N
part
generatingsuchanisotropy[56–59].
The key issue we focuson here is the temperaturedepen-
denceofjet-mediumcoupling,in particularitspossiblenon-
FIG.13: TheRAA(upper)andv2 (lower)athigh pt asafunctionof
trivial behavior near the parton/hadron phase boundary. As participantnumberNpart forRHIC.Thethickbluecurvesareresults
wasfirstfoundin[14],thegeometricanisotropyv2 athigh pt from non-conformal holographic model with qˆ/T3 given in Fig.9
isparticularlysensitivetosuchtemperaturedependence,and (m G=0.75GeV,Tc=255MeV)whilethethinredcurvesarefrom
a simultaneous description of high pT RAA and v2 at RHIC conformalmodelwithqˆ/T3 beingconstant. ThedataarePHENIX
requiresastrongenhancementofjet-mediumcouplinginthe measurements of neutral pions for kinematic range in 6< pt <9
near-Tc region. The near-Tc enhancementof jet-medium in- GeVandinpt>9GeV.
teraction as a generic mechanism, has been further studied
in manylater worksand shown to increase the jet azimuthal Theoretically, however, there has been very limited way
anisotropywith fixedoverallsuppression. Furthermore,how to figure out the precise form of such T-dependence due to
theoverallopaquenessofthecreatedfireballevolveswithcol- thehighlynon-perturbativenatureofthistemperatureregime.
lisional beam energy, is also very sensitive to such temper- The holographicapproach provides a useful way to gain in-
ature dependence. From RHIC to LHC, the collision beam sight into this problem. In the previousSection, we’ve used
energyincreasesbya littlemorethan10times, andthemat- the holographic QCD model with non-conformal dynamics
terdensityincreases(inmostcentralcollisions)byafactorof to calculate the qˆ(T). As clearly seen Fig.9, the scaled jet-
about2. Such a span fromRHIC to LHC providesopportu- medium coupling qˆ/T3 shows strong enhancement in the
nityfordetermininghowjet-mediuminteractionchangeswith vicinity of T, while in contrast any conformal holographic
c
temperature.Inparticularly,thenear-T enhancementpredicts modelwillshownoT-dependencefortheqˆ/T3. Wealsoem-
c
avisiblereductionofaverageopaquenessofthefireballfrom phasize that the same holographic model also describes the
10
trace anomaly (with strong peak near T ) in thermodyanics. Suchdiscrepancyatquantitativelevelmaynotbeunexpected
c
WithsuchT-dependenceobtainedfromtheholographicmodel due to a number of issues. After all the holographic model
here,itisofgreatinteresttoseeitsphenomenologicalimpli- used here is supposed to be an effective description dual to
cations. Hereweusethesimplegeometricenergylossmodel puregluodynamicsandstrictlyspeakingmaynotbesuitable
as in [14, 60] to study the R and v at high p for RHIC for direct application to full QCD phenomenology. First of
AA 2 t
which are most sensitive to such T-dependence. Let us as- all, in real QCD case with crossover transition there is the
sume that the final energy E of a jet with initial energy E “hadronic”side(i.e. thesizablecontributionforqˆ whenTis
f i
after travelinganin-mediumpath~P (specified bythe jet ini- smaller but close to Tc) [67] which is missing in the current
tial spot and momentum direction) can be parameterized as holographicmodelwith1stordertransition.Furthermore,the
E =E f withthe f givenby entropydensityhere(onlycountingthegluonsessentially)is
f i× ~P ~P alsodifferentfromfullQCDwheretherearequarkstoo,and
thatin generalwouldshiftthe peaktowardlowerdensityre-
f =exp k [s(l)]s(l)ldl . (37)
~P (cid:26)−Z~P (cid:27) gion in the present model. One might attempt to “cook up”
certainextrapolativewaysofaccountingforsuchdifferences
Heres(l)isthelocalentropydensityalongthejetpath,while andthusimprovethe agreementwith data. We howeverfeel
the k (s) representsthe local jet-mediuminteraction strength that would weaken the internal rigor and consistency of the
which depends on the local density s(l) (or equivalently the holographicmodelapproach,andwouldaddverylittletoour
temperatureT).Wechoosetoexplicitlyseparateouttheden- mainpurposewhichisnotto claimsuccessindescriptionof
sity s itself, and the combination k (s)s corresponds to qˆ. databuttodemonstratetheconsequenceofnon-conformaldy-
To implement the holographic model results for qˆ, we use namicsin our holographicmodelon the jet energyloss phe-
k [s] = x [qˆ/T3] with qˆ/T3 given as in Fig.9 and with x nomenology.
·
just one parameter to be fixed by the most central collisions
Letusendbyreiteratingourmainpointshere: 1)thereare
R 0.18for0-5%centralityclassandthenusedforother
AA strongnon-conformal,non-perturbativedynamicsgoingonin
≈
computations.WeuseopticalGlaubermodeltosampleinitial
thenear-T region(whichismodeledviaholographyhereby
c
jet spots according to binary collision density and we deter-
introducingquadraticterms);2)suchdynamicsleadstonon-
mine a medium density from participant density with longi-
monotonicbehaviorinQGPthermodynamicsasshownbythe
tudinal boost-invariantexpansion. It is known that there are
strongnear-T peakoftraceanomaly(whichiswellmodeled
c
strong initial state fluctuations, but since we are focusingon
byholography);3)thesamedynamicsleadstonon-monotonic
theaverageR andthedominantgeometricanisotropycom-
AA behaviorinQGPtransportpropertiesandinparticularstrong
ponentv whichisdominantlyfromgeometryandthecurrent
2 near-T enhancementofjet-mediumcoupling;4)phenomeno-
c
approachis reasonable(for a detailed discussions of the ini-
logically the T-dependence of qˆ from non-conformal holo-
tial fluctuations for hard probe see e.g. [19, 20, 66]). In our
graphic model considerably improves the description of jet
simulationsforeachgivenimpactparameterwecomputethe
quenching azimuthal anisotropy as compared with the con-
energyloss for1 million jet pathswith differentinitial spots
formalcase.
andorientations,andextracttheR :
AA
R (f )=< (f )n 2 > , (38)
AA ~Pf − ~Pf
VII. SUMMARY
where <> means averaging over all jet paths with az-
~Pf
imuthal orientation f and including all sampled initial jet WehaveinvestigatedQCDphasetransitionandjetquench-
production spots. The exponent n comes from reference p- ing parameter qˆ in the framework of dynamical holographic
p spectrum at the same collision energy: n 8.1 and 6.0 QCDmodel.Thethermodynamicalpropertiesinthisdynami-
for √s = 0.2 and 2.76 TeV. The so-obtain≈ed R (f ) in calholographicQCDmodelagreewellwithlatticeresultsfor
AA
each event can be further Fourier decomposed as R (f )= pure gluon system. It is found that both the trace anomaly
AA
R [1+2v cos(2f )]. The overallquenchingR as well as (e 3p)/T4andtheratioofthejetquenchingparameterover
AA 2 AA
theazimuthalanisotropyv canthenbedetermined. cub−ictemperatureqˆ/T3 showapeakaroundthecriticaltem-
2
In Fig.13, we show the results for RAA and v2 at high pt perature Tc. It is also noticed that the ratio of jet quenching
forRHIC with the inputT-dependentjet-mediuminteraction parameteroverentropydensityqˆ/ssharplyrisesatTc,which
from our non-conformal holographic model (the thick blue issimilartothebehaviorofbulkviscosityoverentropyden-
curves). For comparisonwe also show the results fromcon- sityz /s. Theenhancementofjetquenchingparameteraround
formalmodeli.e.withqˆ/T3beingconstantinQGPphase(the Tc indicatesthat,liketheratioofshearviscosityoverentropy
thinredcurves).ThedataarefromPHENIXmeasurementsin densityh /sandtheratioofbulkviscosityoverentropyden-
[56]. Asonecansee,whilebothtypesofmodelsdescribethe sity z /s, the ratio of jet quenching parameter over entropy
R well,thenon-conformalmodelshowsasizableimprove- densityqˆ/scanalsocharacterizethephasetransition.
AA
mentover the conformalmodelin getting closer to the data. Theeffectofjetquenchingparameterenhancementaround
Ofcourseourcurrentnon-conformalmodelstilldoesnotgive phase transition on nuclear modification factor R and el-
AA
enough anisotropy, which implies that the T-dependence of liptic flow v have also been analyzed, and it is found that
2
jet-medium couplingin this modelmay still show less near- the T-dependenceof qˆ from non-conformaldynamicalholo-
T enhancement than the phenomenologically favored form. graphicmodelcanconsiderablyimprovethedescriptionofjet
c