Table Of ContentHolographic Flavored Quark-Gluon Plasmas
1
1
0 FrancescoBigazzi
2
DipartimentodiFisicaeAstronomia,UniversitádiFirenzeandI.N.F.N.-sezionediFirenze;Via
n G.Sansone1,I-50019SestoFiorentino(Firenze),Italy.
a
E-mail: [email protected]
J
0 Aldo L.Cotrone∗†
2
DipartimentodiFisicaTeorica,UniversitàdiTorinoandI.N.F.N.-sezionediTorinoViaP.
] Giuria1,I-10125Torino,Italy.
h
E-mail: [email protected]
p
-
p DanielMayerson
e
Institutefortheoreticalphysics,K.U.Leuven;Celestijnenlaan200D,B-3001Leuven,Belgium.
h
[ E-mail: [email protected]
1
AngelParedes
v
1 DepartamentdeFisicaFonamentalandInstitutdeCienciesdelCosmos(ICC),Universitatde
4 Barcelona(UB),MartiFranques1,E-08028Barcelona,Spain.
8
E-mail: [email protected]
3
.
1 JavierTarrío‡
0
InstituteforTheoreticalPhysics,UniversiteitUtrecht,3584CE,Utrecht,TheNetherlands.
1
E-mail: [email protected]
1
:
v
i HolographyprovidesanovelmethodtostudythephysicsofQuarkGluonPlasmas,complemen-
X
tarytotheordinaryfieldtheoryandlatticeapproaches. Inthiscontext,weanalyzetheinforma-
r
a tionsthatcanbeobtainedforstronglycoupledPlasmascontainingdynamicalflavors,alsointhe
presenceofafinitebaryonchemicalpotential.Inparticular,wediscussthejetquenchingandthe
hydrodynamictransportcoefficients.
ThemanyfacesofQCD
November2-5,2010
GentBelgium
∗Speaker.
†Preprintnumber: DFTT2/2011. F.B.andA.L.C.wouldliketothanktheItalianstudents,parents,teachersand
scientistsfortheiractivityinsupportofpubliceducationandresearch.
‡Preprintnumber:ITP-UU-11/03
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
HolographicFlavoredQGPs AldoL.Cotrone
1. Introduction: the string/fieldtheory correspondence
The AdS/CFT correspondence implements the holographic principle, realizing the idea that
a quantum field theory in d dimensions is equivalent to a theory containing gravity in at least
d+1 dimensions. Thus, the latter is supposed to describe all the degrees of freedom, and the
corresponding interactions, of the quantum field theory, at least in some regimes of parameters.
The master example is the conjectured equivalence between 4d N =4 SYM with gauge group
SU(N ) and Type IIB string theory on the AdS ×S5 background [1]. The main interest of this
c 5
statement resides in the parameter regime of the field theory where the gravity approximation to
stringtheoryisreliable: theplanarlimitN ≫1atlarge’tHooftcouplingl ∼g2 N ≫1. Thatis,
c YM c
affordable computations in a classical theory of gravity completely determine the strong coupling
regimeofaquantum fieldtheory.
Theconcreterealizationofthiscorrespondencerequiresthateachingredientinthefieldtheory
(FT)hastohaveadualdescription inthegravity theory. Thisdetermines adictionary betweenthe
twotheories. Someofitsmainentriesarethefollowing:1
• Foreachoperator O intheFTtheremustbeadualgravityfieldF .
• AvacuuminFTcorresponds toabackground gravitysolution.
• TheRGscale isdual tosomeradial variable r inthe gravity background: anasymptotically
AntideSittermanifoldwithanaturalboundary atr→¥ .
• Deconfined phases at finite temperature and charge density are realized by charged black
holes.
• Transport coefficients, i.e. the response to external perturbations in FT, are derived by per-
turbing gravityfields.
TheactualwaytocomputecorrelationfunctionsinFTfromgravityisencodedinthebasicformula
[2,3]
he− F 0Oi =eSgravity(F 0), (1.1)
FT
R
where F 0 ≡limr→¥ F isthe boundary value ofthe gravity fieldF dual tothe operator O. Viathe
gravity equations of motions and selecting an appropriate behavior of F in the interior, the value
of F is determined everywhere by F . The left hand side of the formula above is the generating
0
functional in FT of the correlations functions of the operator O, so essentially solves the FT; the
right hand side includes the gravity action on-shell on the solution of the equations of motion for
F .
Asanexampleofthisformalism,letusconsider thestrongly coupledN =4SYMplasmaat
temperature T. The thermodynamically favored phase, corresponding to deconfined matter trans-
formingintheadjointrepresentation, isdescribed byablackholeinAdS. Letusalsoconsiderone
particular operator: theT component oftheenergy-momentum tensor. Itsdualgravity fieldturns
xy
outtobetheg componentofthemetric. Then,thetwo-pointfunction
xy
h[T (t,~x),T (0,~0)]i (1.2)
xy xy
1Wearequitecavalierwithdetails.
2
HolographicFlavoredQGPs AldoL.Cotrone
can be calculated from the on-shell supergravity action for the graviton g , determined by its
xy
equation ofmotionontheAdSblackhole.
Atlong distances, late times ascompared to 1/T, theN =4SYMplasma admits asusual a
hydrodynamic description, whosebasicequation issimplytheenergy-momentum conservation
(cid:209) m Tmn =0. (1.3)
Uptosecondorderinthederivativeexpansion,thehydrodynamicsenergy-momentumtensorreads
[4,5]
Tmn =e um un +pD mn +p mn +D mn P , (1.4)
wheree istheenergy density, um thevelocityfield, p(e )thepressure, D mn =hmn +um un withhmn
the4d metricand
(cid:209) ·u
p mn = −hs mn +ht p hDs mn i+ s mn +k R<mn >−2ua ub Ra <mn >b
3
+l 1s l<m s n >l h+l 2s l<m W n >l +l 3Wil<m W hn >l +k ∗2ua ub Ra <mn >b i
(cid:209) ·u
+ht p∗ s mn +l 4(cid:209) <m logs(cid:209) n >logs, (1.5)
3
P = −z ((cid:209) ·u)+zt P D((cid:209) ·u)+x 1s mn s mn +x 2((cid:209) ·u)2+x 3W mn W mn
+x 4(cid:209) ⊥m logs(cid:209) ⊥m logs+x 5R+x 6ua ub Rab . (1.6)
For the precise definition of the structures in these expressions we refer to [5]. In principle, the
transport coefficients of the structures above are determined by the microscopic theory. The most
importanttransportcoefficientsaretheshearandbulkviscosities h ,z andthetworelaxationtimes
tp ,tP ,whichhavepossible implications fortheellipticflowmeasuredatRHICandLHC.
For a general strongly coupled theory, the theoretical determination of the transport coeffi-
cients isadaunting task. Inthecase athand, onthe contrary, they canbe extracted withareason-
able amount of work from gravity. The shear viscosity, for example, can be derived in FTvia the
Kuboformula
1
h =limw →0 2w dt d~xeiw t h[Txy(t,~x),Txy(0,~0)]i. (1.7)
Z
The gravity calculation for the g component of the metric determines the correlator and finally
xy
thevalueoftheshearviscosity overentropy density[6]
h 1
= . (1.8)
s 4p
ItisbynowwellknownthatthisresultforN =4SYMintheplanarlimitissurprisinglycompati-
blewiththeexperimentalresultsatRHICandLHC,eventhoughexperimentalerrorsremainlarge.
Other methods of obtaining this quantity, such as perturbative QCD or lattice, give higher values,
hardlycompatible withexperiments.
The result above shows both the relative simplicity of holographic methods and their possi-
ble relevance for a deeper understanding of experimental settings, whenever there are no reliable
direct theoretical calculations available in QCD. An extensive review of the applications of the
gauge/gravity dualitytothephysicsofheavyioncollisions hasappeared in[7].
3
HolographicFlavoredQGPs AldoL.Cotrone
2. D3-D7 Quark-GluonPlasmas
IntheheavyionscatteringexperimentsatRHICandLHCaQuark-GluonPlasmaisproduced.
Thisisathermalizedsystematfinite(small)baryonchemicalpotentialwherethequarksandgluons
aredeconfined. >FromatheoreticalpointofviewtheinvestigationoftheQGPisquitechallenging.
In fact, the analysis of the experiments indicates that it is in a strong coupling regime, rendering
perturbative QCDnotentirely suitable forthepurpose. Latticemethods areclearly oneofthebest
instruments for this investigation, but they cannot be exhaustive. Apart from the sign problem for
chemical potential, which can be overcome in some way for small values, the main difficulty of
thelatticeapproach isthatitisnotverysuitableforrealtimephysics(transport coefficients, probe
physics), whichplaysanessentialroleintheexperimental systems.
In this context, the holographic approach can furnish novel insight in the problem. It is au-
tomatically a strong coupling approach and, as we have seen above, it allows to access with ease
real timephenomena. Moreover, as wewill review, itallows for theinvestigation of theories with
dynamicalflavorsatfinitechemicalpotential. Theobviouspricewehavetopayinthisapproachis
thatweareinvestigating atheory, intheplanarlimitandverylarge’tHooftcoupling, whichisnot
QCD,butoneofitsrelatives.
In this note, we are interested in reviewing the inclusion of dynamical flavors in the holo-
graphic approach to the physics of the N =4 SU(N ) SYM plasma. Flavors are studied by in-
c
troducing N explicit D7-branes in the AdS black hole background. In the strict ’t Hooft limit
f
(N →¥ with N fixed), the branes can be treated in the probe approximation, where their back-
c f
reaction on the background is ignored and therefore the flavors are quenched [8]. This approach
allows to study a number of physical problems, but it misses part of the physics of the RHIC and
LHCQGPs,whereasignificantfractionofdegreesoffreedomsaredynamicalfundamentalquarks.
Theproblem withgoingbeyondthequenched approximation isthatcalculating thebackreac-
tionoftheflavorbranes isusually averydifficulttask, involving thesolution ofsystems ofpartial
differential equations. In order to overcome this difficulty, in [9, 10] a method was introduced,2
termed"smearingtechnique", whichtakesadvantageoftheparameterregimeunderconsideration.
Sinceweareintheplanar limitN ≫1,theprobe approximation breaks downwhenweintroduce
c
many flavor branes, N ≫1. Thus, these many branes can be homogeneously distributed in their
f
transverse space, recovering (part of)theoriginal symmetryofthesolution andtypically reducing
thesystemtoasetofordinarydifferentialequations. Themaineffectinthedualfieldtheoryisthat
theconsidered flavorgroupisAbelian.
Employing thistechnique, thebackreacted background corresponding tomassless flavors can
bederivedintheform
ds2 =h−1/2[−bdt2+dxidx]+h1/2[bS8F2ds 2+S2ds2 +F2(dt +A )2], (2.1)
10 i KE KE
where "KE" indicates a Kähler-Einstein manifold, which in the N =4 case isCP , and A is
2 KE
the connection one-form whose curvature is related to the Kähler form of the KE base: dA =
KE
2J . Usingtheinvariance underdiffeomorphisms wehavechosenaradialcoordinate s whichis
KE
convenient towritetheequations ofmotionofthesystem. Thesolution withzerocharge hasbeen
2Forareviewofthisapproach,see[11].
4
HolographicFlavoredQGPs AldoL.Cotrone
derived in [12], while its generalization at finite charge appears in [13].3 It can be expressed by
meansoftwoparameters
N 1 m 5
e ∼l f , dˆ ∼ 1− e , (2.2)
h h h
N l T 24
c h (cid:18) (cid:19)
where m is the quark chemical potential and the supbindex means that the coupling is evaluated
h
at the temperature of the plasma. Up to order e 2,dˆ2 the perturbative solution can be derived in
h
analytic form and expressed inanew4 radial variable r such that h=R4/r4, R being theradius of
theundeformed AdSspace
r4 e dˆ2 r4 r2 r2 r4
b(r) = 1− h − h 2− h h −log 1+ h − h(1−log2)
r4 2 r4 r2 r2 r4
(cid:18)(cid:18) (cid:19)(cid:18) (cid:20) (cid:21)(cid:19) (cid:19)
e 2dˆ2 r2 29r4 5r6 17 r4 r2 17r4
+ h 17 h − h − h − 2− h log 1+ h + h log2 ,
12 r2 2 r4 2r6 2 r4 r2 2 r4
(cid:18) (cid:18) (cid:19) (cid:20) (cid:21) (cid:19)
e 9 1 r
S(r) = r 1+ h +e 2 − log h
24 h 1152 24 r
(cid:20) (cid:18) (cid:19)
e dˆ2 r2 r4 r2 1
+ h 3−2 −3 1−2 log 1+ h − G(r)
40 r2 r4 r2 2
(cid:18) h (cid:18) h(cid:19) (cid:20) (cid:21) (cid:19)
e 2dˆ2 r2 r4 r2 11
+ h −33+22 h +33 1−2 log 1+ h + G(r) ,
320 r2 r4 r2 2
(cid:18) (cid:18) h(cid:19) (cid:20) (cid:21) (cid:19)#
e 17 1 r
F(r) = r 1− h +e 2 + log h
24 h 1152 24 r
(cid:20) (cid:18) (cid:19)
e dˆ2 r2 r2 r4 r2
+ h 3−22 +5 h −3 1−2 log 1+ h +2G(r)
40 r2 r2 r4 r2
(cid:18) h (cid:18) h(cid:19) (cid:20) (cid:21) (cid:19)
e 2dˆ2 r2 r2 r4 r2
+ h −21+154 −35 h +21 1−2 log 1+ h −14G(r) ,
192 r2 r2 r4 r2
(cid:18) h (cid:18) h(cid:19) (cid:20) (cid:21) (cid:19)#
r e 2 r r r4
F (r) = F +e log − h 8 1+3log log h −3Li 1− h
h h r 48 r r 2 r4
h h
(cid:18) (cid:18) (cid:19) (cid:20) (cid:21)(cid:19)
e 2dˆ2 r2 r2 r4 r2
+ h 41−2p −26 −15 h +G(r)+ 11+18 log 1+ h −29log2 ,
120 r2 r2 r4 r2
(cid:18) h (cid:18) h(cid:19) (cid:20) (cid:21) (cid:19)
whereF isthedilaton,G(r)=2p rh6 F 3,3,1,1−rh4 isanhypergeometricfunctionandLi (u)≡
r62 1 2 2 r4 2
(cid:229) ¥ un is a polylogarithmic function.(cid:16)The paramete(cid:17)r r marks the (perturbative) position of the
n=1n2 h
horizon defined by b(r )=0+O(e 3,dˆ4). In addition to the metric and the dilaton there are non-
h h
trivial differential forms and aU(1) field turned on in the worldvolume of the D7-branes, dual to
thechemicalpotential. Wereferto[13]fordetails.
Theregimeofvalidityofthissolution islimitedtotheusualrangeN ≫1,l ≫1and,aswe
c h
have seen, N ≫1; moreover, since the theory has a Landau pole in the UV, in order to decouple
f
3Thezerotemperaturesolutionformasslessflavorsappearedin[14],theonesformassiveflavorsin[15,16,12].
4Fortheexplicitformoftheequationssatisfiedbythefieldsintheansatzinthevariables ,werefertotheoriginal
paper[13].
5
HolographicFlavoredQGPs AldoL.Cotrone
the IR physics from the UV wemust require e ≪1; finally, in order to have an analytic solution
h
werequired dˆ ≪1,anassumption thatinprinciple canberelaxed.
3. Results
Thesolutiondescribedintheprevioussectionallowsustostudyanumberofeffectsofdynam-
icalflavorsatfinitebaryonchargeinastronglycoupledtheoryinacompletelycontrollablesetting.
As a first topic, let us exhibit the thermodynamic quantities (entropy density s, energy density e ,
grandpotential w ,specificheatc )inthegrandcanonical ensemble
v
1 1 7
s = p 2N2T3 1+ e (1+dˆ2)+ e 2(1+dˆ2) , (3.1)
2 c 2 h 24 h
(cid:20) (cid:21)
3 1 1 7
e = p 2N2T4 1+ e (1+2dˆ2)+ e 2(1+ dˆ2) , (3.2)
8 c 2 h 3 h 4
(cid:20) (cid:21)
1 1 1 7
w = −p=− p 2N2T4 1+ e (1+2dˆ2)+ e 2(1+ dˆ2) , (3.3)
8 c 2 h 6 h 2
(cid:20) (cid:21)
3 1 1
c = p 2N2T3 1+ e 1+dˆ2 + e 2 11+7dˆ2 .
v 2 c 2 h 24 h
(cid:20) (cid:21)
(cid:16) (cid:17) (cid:16) (cid:17)
Notethatinorderforthestandardthermodynamicrelationstoholditisessentialthatthefollowing
dependence isactuallyrealized: deh = eh2 +O(e 3), ddˆ =−dˆ 1+eh +O(e 2) .
dT T h dT m T 2 h
The breaking of conformality due to fundamen(cid:16)tal m(cid:17)atter is a second order effect in e and is
(cid:0) (cid:1) h
notaffected bythefinitechargedensity
1
e −3p = p 2N2T4e 2, (3.4)
16 c h
1 1
c2 = 1− e 2 , (3.5)
s 3 6 h
(cid:20) (cid:21)
wherec isthespeedofsound.
s
Another example of the interest of such kind of solution is provided by the study of energy
loss of a probe parton in the plasma. It is known that in the experimental setting the energy loss
is huge and it is very interesting to characterize such process from a theoretical point of view.
We will concentrate on one coefficient, the "jet quenching parameter" qˆ which accounts for such
a characterization. It is defined as the transverse momentum squared transferred by the plasma
to the parton per mean free path. In string theory there is a very simple way of calculating such
quantitybymeansofapartiallylight-likeWilsonloop[17]. Inthecaseathandtheoutcomeofthis
calculation is(neglecting O(e 2)terms)[12,13]
h
p 3/2G (3) 2+p
qˆ= 4 l T3 1+ e (1+0.8675dˆ2) . (3.6)
G (5) h 8 h
4 (cid:20) (cid:21)
p
The naive interpretation of this formula is that both uncharged flavors and a finite charge enhance
thejetquenching,i.e. theenergylossoftheparton.5 Althoughbeingaprioricompletelyunjustified,
5Intheunchargedcase,thisresultwasalreadyobtainedin[18]fromnon-criticalstringmodels,whichontheother
handsufferfromuncontrolledapproximationsandsoarenotquantitativelyfullyreliable.
6
HolographicFlavoredQGPs AldoL.Cotrone
one can plug in the formula above quantities reasonable for the RHIC experiment, such as N =
c
N =3, T =300 MeV, l =6p , m =10 MeV, obtaining qˆ∼5.3 (GeV)2/fm [12], as opposed to
f h
qˆ∼4.5 (GeV)2/fm for the unflavored theory [17]. The most common values reported for RHIC
are qˆ∼5−15 (GeV)2/fm, sothe naive extrapolation ofthe formula (3.6) issurprisingly close to
the real values. This fact could be a total accident, or else a signal that the energy loss process at
strong coupling is quite insensitive to the details of the theory under consideration, as it happens
fortheshearviscosity overentropy densityvalue.
Theabovestatementoftheenhancementofthejetquenchingduetoflavorsandchargedensity
is definitively too naive, since it involves the comparison of two theories with different numbers
of degrees of freedom. Adding fundamental flavors and charge enhances the entropy density, so
the enhancement of the parton energy loss could be a trivial consequence of this fact. In order to
disentanglethetwoeffects,onecancomparetheunflavoredandflavoredtheorieskeepingfixedthe
number of degrees of freedom, i.e. either the entropy or the energy density. This can be achieved
by adjusting the temperature T, obtaining the result that flavors do enhance the jet quenching but
thechargedensity reducesthisenhancement. Or,onecanfixthenumberofdegreesoffreedomby
varying N (at fixed T). In the latter case, while again the flavors enhance the jet quenching, the
c
effectofthefinitechargeistoincreasetheenhancement.
All in all, the solid conclusion that can be extracted from the result above is that dynamical
flavors enhance the jet quenching; note that this is probably the only reliable computation of the
effects ofdynamical flavors at strong coupling on the energy loss of partons ina plasma. Consid-
ering that flavoreffects areoften discarded as subleading inphenomenological estimates [19], the
resultshouldgiveanimportantindication thatthelatterapproximation isprobably notjustified.
4. Onhydrodynamics ofholographicplasmas
Therearesolid lattice indications that theQCDplasmaisbothnearly conformal andstrongly
coupledinthetemperaturewindowrelevantforthepresentexperiments1.5T ≤T ≤4T . Probably
c c
thesimplestwayofmodelingthissituationholographicallyisbyanAdSbackgrounddeformedbya
scalardualtoamarginallyrelevantoperator. Thelattertheorybehaveseffectively, atleadingorder
inthedeformation, asaso-called Chamblin-Reallmodel. Forthesetheories, allthehydrodynamic
transport coefficients up to second order can be extracted from the results in [20]. Thus, one can
giveanestimateofthe(initialbehaviorof)transportcoefficients,uptosecondorder,forRHICand
LHC[21,22]. Withthedefinition6
d ≡1−3c2, (4.1)
s
where c is the speed of sound, and referring to the hydrodynamic stress-energy tensor in (1.4),
s
the transport coefficients are given in Table 1.7 Considering the difficulty of dealing with such
coefficientsinQCD,thisinformationcouldbeusefulinnumericalsimulationsofthehydrodynamic
evolution of the QGP. In particular, the behavior with the temperature and the speed of sound of
theshearandbulkrelaxation timestp ,tP isbothpotentially relevantandunexpected.
6Notethatd anddˆ arecompletelyunrelatedterms.
7The uncharged flavored N =4 SYM plasma has these same coefficients with d =e 2/6. Unfortunately, the
h
Chamblin-Reall model isnot abletodescribe thechargedcaseduetothepresenceof extrafieldsinthegravitational
setup.
7
HolographicFlavoredQGPs AldoL.Cotrone
hs 41p Ttp 2−2lpog2+3(1664−pp 2)d Tsk 4p12 1−43d
(cid:16) (cid:17)
Tl1 1 1+3d Tl2 − 1 log2+3p 2d Tl3 0
s 8p 2 4 s 4p 2 32 s
(cid:16) (cid:17) (cid:16) (cid:17)
Tks∗ −8p32d Ttp∗ −2−2lpog2d Tsl4 0
hz 23d TtP 2−2lpog2 Tsx1 241p 2d
Tx2 2−log2d Tx3 0 Tx4 0
s 36p 2 s s
Tx5 1 d Tx6 1 d
s 12p 2 s 4p 2
Table1:Thetransportcoefficientsatleadingorderintheconformalitydeformationparameterd ≡1−3c2.
s
5. Future directions
Thetechnique employed above in order to study the dynamics of fundamental flavors is suit-
ableforanumberofpossibleinterestingapplicationsinthenearfuture. Clearly,afirstroutewould
be toextend the analysis above beyond thesmall dˆ regime, exploring the large chemical potential
region of the phase diagram which could correspond to extremality of the dual black hole. More-
over, considering the results above, a deeper investigation of the probe parton physics would be
quite important, considering also the experimental possibilities of the LHC experiment. It would
bealsointerestingtoexploretheopticalpropertiesofthesystemalongthelinesof[23,24]. Finally,
it would be worth exploring the condensed matter applications of this formalism in the context of
theholographic duality.
Acknowledgments
We thank Javier Mas for his suggestions on this note. The research leading to the results
in this paper has has received funding from the European Community Seventh Framework Pro-
gramme (FP7/2007-2013 under grant agreements n. 253534 and 253937), the Netherlands Orga-
nization for Scientific Research (NWO) under the FOM Foundation research program, the grants
FPA2007-66665C02-02 andDURSI2009SGR168,andtheCPANCSD2007-00042projectofthe
Consolider-Ingenio 2010program.
8
HolographicFlavoredQGPs AldoL.Cotrone
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