224 Comparing elliptic flow in heavy ion collisions and hydrodynamic cal-culations 27
412 The large-Nc expansion of a non-Abelian gauge theory vs the stringtheory expansion 72
43 The AdSCFT conjecture 85
5 General aspects of the duality 88
51 Gaugegravity duality 88
511 UVIR connection 89
512 Strong coupling from gravity 89
513 Symmetries 92
514 Matching the spectrum the fieldoperator correspondence 92
515 Normalizable vs non-normalizable modes and mass-dimension relation 93
52 Generalizations 96
521 Nonzero temperature and nonzero chemical potential 96
522 A confining theory 98
523 Other generalizations 99
53 Correlation functions of local operators 100
531 Euclidean correlators 100
532 Real-time thermal correlators 102
54 Wilson loops 105
541 Rectangular loop vacuum 108
542 Rectangular loop nonzero temperature 110
543 Rectangular loop a confining theory 111
55 Introducing fundamental matter 113
551 The decoupling limit with fundamental matter 114
552 Models with fundamental matter 116
6 Bulk properties of strongly coupled plasma 119
61 Thermodynamic properties 122
611 Entropy energy and free energy 122
612 Holographic susceptibilities 124
62 Transport properties 125
621 A general formula for transport coefficients 126
622 Universality of the Shear Viscosity 128
623 Bulk viscosity 132
624 Relaxation times and other 2nd order transport coefficients 133
63 Quasiparticles and spectral functions 136
631 Quasiparticles in perturbation theory 137
632 Absence of quasiparticles at strong coupling 142
7 Probing strongly coupled plasma 144
71 Parton energy loss via a drag on heavy quarks 145
711 Regime of validity of the drag calculation 147
2
72 Momentum broadening of a heavy quark 150
721 κT and κL in the prarr 0 limit 151
722 Direct calculation of the noise term 152
723 κT and κL for a moving heavy quark 156
724 Implications for heavy quarks in heavy ion collisions 160
73 Disturbance of the plasma induced by an energetic heavy quark 162
731 Hydrodynamic preliminaries 165
732 AdS computation 168
733 Implications for heavy ion collisions 172
734 Disturbance excited by a moving quarkonium meson 173
74 Stopping light quarks 174
75 Calculating the jet quenching parameter 176
76 Quenching a beam of strongly coupled synchrotron radiation 184
77 Velocity-scaling of the screening length and quarkonium suppression in a hotwind 188
8 Quarkonium mesons 196
81 Adding quarks to N = 4 SYM 196
82 Zero temperature 198
821 D7-brane embeddings 198
822 Meson spectrum 202
83 Nonzero temperature 204
831 D7-brane embeddings 204
832 Thermodynamics of D7-branes 209
833 Quarkonium thermodynamics 214
84 Quarkonium mesons in motion and in decay 219
841 Spectrum and dispersion relations 220
842 Decay widths 225
843 Connection with the quark-gluon plasma 227
85 Black hole embeddings 230
851 Absence of quasiparticles 230
852 Meson spectrum from a spectral function 231
86 Two universal predictions 235
861 A meson peak in the thermal photon spectrum 235
862 A new mechanism of quark energy loss Cherenkov emission of mesons 237
9 Concluding remarks and outlook 241
A Green-Kubo formula for transport coefficients 245
3
B Hawking temperature of a general black brane metric 247
C Holographic renormalization one-point functions and an example of a Eu-clidean two-point function 248
4
Chapter 1
Preface
The discovery in the late 1990s of the AdSCFT correspondence as well as its subsequentgeneralizations now referred to as the gaugestring duality have provided a novel approachfor studying the strong coupling limit of a large class of non-abelian quantum field theoriesIn recent years there has been a surge of interest in exploiting this approach to study prop-erties of the plasma phase of such theories at non-zero temperature including the transportproperties of the plasma and the propagation and relaxation of plasma perturbations Be-sides the generic theoretical motivation of such studies many of the recent developmentshave been inspired by the phenomenology of ultra-relativistic heavy ion collisions Inspi-ration has acted in the other direction too as properties of non-abelian plasmas that weredetermined via the gaugestring duality have helped to identify new avenues in heavy ionphenomenology There are many reasons for this at-first-glance surprising interplay amongstring theory finite-temperature field theory and heavy ion phenomenology as we shall seethroughout this review Here we anticipate only that the analysis of data from the Rel-ativistic Heavy Ion Collider (RHIC) had emphasized the importance indeed the necessityof developing strong coupling techniques for heavy ion phenomenology For instance in thecalculation of an experimentally-accessible transport property the dimensionless ratio of theshear viscosity to the entropy density weak- and strong coupling results turn out to differnot only quantitatively but parametrically and data favor the strong coupling result Strongcoupling presents no difficulty for lattice-regularized calculations of QCD thermodynamicsbut the generalization of these methods beyond static observables to characterizing transportproperties has well-known limitations Moreover these methods are quite unsuited to thestudy of the many and varied time-dependent problems that heavy ion collisions are makingexperimentally accessible It is in this context that the very different suite of opportunitiesprovided by gaugestring calculations of strongly-coupled plasmas have started to provide acomplementary source of insights for heavy ion phenomenology Although these new methodscome with limitations of their own the results are obtained from first-principle calculationsin non-abelian field theories at non-zero temperature
The present review aims at providing an overview of the results obtained from this inter-play between gaugestring duality lattice QCD and heavy ion phenomenology within the lastdecade in a form such that readers from either community can appreciate and understandthe emerging synthesis For the benefit of string theory practitioners we begin in Sections 2
5
and 3 with targeted overviews of the data from and theory of heavy ion collisions as well asof state-of-the art lattice calculations for QCD at non-zero temperature We do not providea complete overview of all aspects of these subjects that are of current interest in their ownterms aiming instead for a self-contained but targeted overview of those aspects that arekeys to understanding the impacts of gaugestring calculations In turn Sections 4 and 5 aremainly written for the benefit of heavy ion phenomenologists and QCD practitioners Theyprovide a targeted introduction to the principles behind the gaugestring duality with a fo-cus on those aspects relevant for calculations at non-zero temperature With the groundworkon both sides in place we then provide an in-depth review of the gaugestring calcula-tions of bulk thermodynamic and hydrodynamic properties (Section 6) the propagation ofprobes (heavy or energetic quarks and quark-antiquark pairs) through a strongly-couplednon-abelian plasma and the excitations of the plasma that result (Section 7) and a detailedanalysis of mesonic bound states and spectral functions in a deconfined plasma (Section 8)
The interplay between hot QCD heavy ion phenomenology and the gaugestring dualityhas been a very active field of research in recent years and there are already a numberof other reviews that cover various aspects of these developments In particular there arereviews focussing on the techniques for calculating finite-temperature correlation functions oflocal operators from the gaugestring duality [1] on the phenomenological aspects of perfectfluidity and its manifestation in different systems including the quark-gluon plasma producedin heavy ion collisions and strongly coupled fluids made of trapped fermionic atoms that aremore than twenty orders of magnitude colder [2] (we shall not review the connection toultracold atoms here) and also shorter topical reviews [3ndash7] that provide basic discussions ofthe duality and its most prominent applications in the context of heavy ion phenomenologyThe present review aims at covering a broader range of applications while at the same timeproviding readers from either the string theory or the QCD communities with the opportunityto start from square one on lsquothe other sidersquo and build an understanding of the needed context
Since any attempt at covering everything risks uncovering nothing even given its lengththis review does remain limited in scope Important examples of significant advances in thisrapidly-developing field that we have not touched upon include (i) the physics of saturation inQCD and its application to understanding the initial conditions for heavy ion collisions [8ndash14](ii) various field-theoretical approaches to understanding equilibration in heavy ion collisions[15ndash19] and (iii) work from the string side on developing dual gravitational descriptions notjust of the strongly-coupled plasma and its properties as we do review but of the dynamicsof how the plasma forms equilibrates expands and cools after a collision This last body ofliterature includes the early work of Refs [20ndash23] continues in modern form with the discoveryof a dual gravitational description of a fluid that is cooling via boost-invariant longitudinalexpansion [24] extends to the very recent discovery of the dual description of the collisionof two finite-thickness sheets of energy density and the ensuing plasma formation [25] andincludes much in between [26ndash43] Also we focus throughout on insights that have beenobtained from calculations that are directly rooted in quantum field theories analyzed via thegaugestring duality Consequently we have omitted the so-called AdSQCD approach thataims to optimize ansatze for gravity duals that do not correspond to known field theoriesin order to best incorporate known features of QCD in the gravitational description [44ndash56]So while we have high hopes that the material reviewed here will serve in coming yearsas a pathway into the field for new practitioners with experience in QCD or string theory
6
we have not aimed for completeness Indeed we can already see now that this very activefield continues to broaden and that adding new chapters to a review like this will soon bewarranted
7
Chapter 2
A heavy ion phenomenology primer
What macroscopic properties of matter emerge from the fundamental constituents and in-teractions of a non-abelian gauge theory The study of ultra-relativistic heavy ion collisionsaddresses this question for the theory of the strong interaction Quantum Chromodynamicsin the regime of extreme energy density To do this heavy ion phenomenologists employ toolsdeveloped to identify and quantify collective phenomena in collisions that have many thou-sands of particles in their final states Generically speaking these tools quantify deviationswith respect to benchmark measurements (for example in proton-proton and proton-nucleuscollisions) in which collective effects are absent In this Section we provide details for threecases of current interest (i) the characterization of elliptic flow which teaches us how soonafter the collision matter moving collectively is formed and which allows us to constrain thevalue of the shear viscosity of this matter (ii) the characterization of jet quenching via singleinclusive hadron spectra and angular correlations which teaches us how this matter affectsand is affected by a high-velocity colored particle plowing through it and (iii) the charac-terization of the suppression of quarkonium production which has the potential to teach usabout the temperature of the matter and of the degree to which it screens the interactionbetween colored particles
21 General characteristics of heavy ion collisions
In a heavy ion collision experiment large nuclei such as gold (at RHIC) or lead (at SPS andLHC) are collided at an ultra-relativistic center of mass energy
radics The reason for using large
nuclei is to create as large a volume as possible of matter at a high energy density to have thebest chance of discerning phenomena or properties that characterize macroscopic amountsof strongly interacting matter In contrast in energetic elementary collisions (say electron-positron collisions but to a good approximation also in proton-proton collisions) one may findmany hadrons in the final state but these are understood to result from a few initial partonsthat each fragment rather than from a macroscopic volume of interacting matter Manyyears ago Phil Anderson coined the phrase ldquomore is differentrdquo to emphasize that macroscopicvolumes of (in his case condensed) matter manifest qualitatively new phenomena distinctfrom those that can be discerned in interactions among few elementary constituents andrequiring distinct theoretical methods and insights for their elucidation [57] Heavy ion
8
-5 0 5
dN
d
0
200
400
600
200 GeV130 GeV624 GeV196 GeV
Au+Au 6 Central
0 2 4 6 8 10 12
1(
2πp
T)
d2N
dp
Td
η|η=
0 (
(GeV
c)-2
)
0-55-10 5 10-20 10 20-30 15 30-40 20 40-60 25 60-80 30 p+p (20 for left axis)
radicsNN = 200 GeV
Au+Au p+p
h++h------------ 2
E d
3σN
SD d
p3 (m
b G
eV
-2 c3)
pT (GeVc)
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
1
10
102
103
10-7
10-6
10-5
10-4
10-3
10-2
10-1
1
10
102
Figure 21 Left Charged particle multiplicity distributions for collisions at RHIC with four differentenergies as a function of pseudorapidity [58] Very recently a charged particle multiplicity dNchdη =1584 plusmn 4(stat) plusmn 76(sys) has been measured at η = 0 in heavy ion collisions with
radics = 276 TeV by
the ALICE detector at the LHC [59] Right Charged particle spectrum as function of pT in gold-goldcollisions at top RHIC energy at different values of the impact parameter with the top (second-from-bottom) curve corresponding to nearly head-on (grazing) collisions [60] The bottom curve is datafrom proton-proton collisions at the same energy
physicists do not have the luxury of studying systems containing a mole of quarks but byusing the heaviest ions that nature provides they go as far in this direction as is possible
The purpose of building accelerators that achieve heavy ion collisions at higher and higherradics is simply to create matter at higher and higher energy density A simple argument to see
why this may be so arises upon noticing that in the center-of-mass frame we have the collisionof two Lorentz contracted nuclei pancake shaped and increasing the collision energy makesthese pancakes thinner Thus at t = 0 when these pancakes are coincident the entire energyof the two incident nuclei is found within a smaller volume for higher
radics This argument is
overly simple however because not all of the energy of the collision is transformed into thecreation of matter much of it is carried by the debris of the two colliding nuclei that sprayalmost along the beam directions
The question of how the initial state wave function of the colliding nuclei determinesprecisely how much matter containing how much entropy is produced soon after the collisionand consequently determines the number of particles in the final state is a subject of intensetheoretical interest At present however calculations using gravity duals have not shedmuch light on the attendant issues and so we shall not describe this branch of heavy ionphenomenology in any detail It is however worth having a quantitative sense of just howmany particles are produced in a typical heavy ion collision In the left panel of Fig 21we show the multiplicity of charged particles per unit pseudorapidity for RHIC collisions atfour different values of
radics Recall that the pseudorapidity η is related to the polar angle
θ measured with respect to the beam direction by η = minus log tan(θ2) Note also that by
9
conventionradics = 200 GeV (which is the top energy achieved at RHIC) means that the
incident ions in these collisions have a velocity such that individual nucleons colliding withthat velocity would collide with a center of mass energy of 200 GeV Since each gold nucleushas 197 nucleons the total center of mass energy in a heavy ion collision at the top RHICenergy is about 40 TeV By integrating under the curve in the left panel of Fig 21 onefinds that such a collision yields 5060 plusmn 250 charged particles [61 62] The multiplicitymeasurement is made by counting tracks meaning that neutral particles (like π0rsquos and thephotons they decay into) are not counted So the total number of hadrons is greater thanthe total number of charged particles If all the hadrons in the final state were pions andif the small isospin breaking introduced by the different number of protons and neutrons ina gold nucleus can be neglected there would be equal numbers of π+ πminus and π0 meaningthat the total multiplicity would be 32 times the charged multiplicity In reality this factorturns out to be about 16 [58] meaning that heavy ion collisions at the top RHIC energy eachproduce about 8000 hadrons in the final state We see from the left panel of Fig 21 that thismultiplicity grows with increasing collision energy and we see that the multiplicity per unitpseudorapidity is largest in a range of angles centered around η = 0 meaning θ = π2 Heavyion collisions at the LHC will have center of mass energies up to
radics = 55 TeV per nucleon
Estimates for the charged particle multiplicity per unit pseudorapidity at this energy thatwere made before LHC heavy ion collisions began typically range from dNchdη asymp 1000 todNchdη asymp 3000 [63] This range can now be considerably tightened now that we knowthat dNchdη = 1584 plusmn 4(stat) plusmn 76(sys) at η = 0 in most central collisions (5 ) withradics = 276 TeV [59]
The large multiplicities in heavy ion collisions indicate large energy densities since eachof these particles carries a typical (mean) transverse momentum of several hundred MeVThere is a simple geometric method due to Bjorken [64] that can be used to estimate theenergy density at a fiducial early time conventionally chosen to be τ0 = 1 fm The smallestreasonable choice of τ0 would be the thickness of the Lorentz-contracted pancake-shapednuclei namely sim (14 fm)100 since gold nuclei have a radius of about 7 fm But at theseearly times sim 01 fm the matter whose energy density one would be estimating would still befar from equilibrium We shall see below that elliptic flow data indicate that by sim 1 fm afterthe collision matter is flowing collectively like a fluid in local equilibrium The geometricestimate of the energy density is agnostic about whether the matter in question is initialstate partons that have not yet interacted and are far from equilibrium or matter in localequilibrium behaving collectively because we are interested in the latter we choose τ0 = 1 fmBjorkenrsquos geometric estimate can be written as
εBj =dETdη
∣∣∣∣η=0
1
τ0πR2 (21)
where dET dη is the transverse energyradicm2 + p2
T of all the particles per unit rapidity and
R asymp 7 fm is the radius of the nuclei The logic is simply that at time τ0 the energy within avolume 2τ0 in longitudinal extent between the two receding pancakes and πR2 in transversearea must be at least 2dET dη the total transverse energy between η = minus1 and η = +1 AtRHIC with dET dη asymp 800 GeV [62] we obtain εBj asymp 5 GeVfm3 In choosing the volumein the denominator in the estimate (21) we neglected transverse expansion because τ0 R
10
Figure 22 Left Spectra for identified pions kaons and protons as a function of pT in head-on gold-gold collisions at top RHIC energy [66] Right Spectra for identified pions kaons andprotons as a function of pT in (non single diffractive) proton-proton collisions at the same energyradics = 200 GeV [67]
But there is clearly an arbitrariness in the range of η used if we had included particlesproduced at higher pseudorapidity (closer to the beam directions) we would have obtaineda larger estimate of the energy density Note also that there is another sense in which (21)is conservative If there is an epoch after the time τ0 during which the matter expands as ahydrodynamic fluid and we shall later see evidence that this is so then during this epoch itsenergy density drops more rapidly than 1τ because as it expands (particularly longitudinally)it is doing work This means that by using 1τ to run the clock backwards from the measuredfinal state transverse energy to that at τ0 we have significantly underestimated the energydensity at τ0 It is striking that even though we have deliberately been conservative in makingthis underestimate we have found an energy density that is about five times larger than theQCD critical energy density εc asymp 1 GeVfm3 where the crossover from hadronic matter toquark-gluon plasma occurs according to lattice calculations of QCD thermodynamics [65]
As shown in the right panel of Fig 21 the spectrum in a nucleus-nucleus collision extendsto very high momentum much larger than the mean However the multiplicity of highmomentum particles drops very fast with momentum as a large power of pT We mayseparate the spectrum into two sectors In the soft sector spectra drop exponentially withradicm2 + p2
T as in thermal equilibrium In the hard sector spectra drop like power laws in pTas is the case for hard particles produced by high momentum-transfer parton-parton collisionsat τ = 0 The bulk of the particles have momenta in the soft sector hard particles are rarein comparison The separation between the hard and the soft sectors which is by no meanssharp lies in the range of a few (say 2-5) GeV
There are several lines of evidence that indicate that the soft particles in a heavy ioncollision which are the bulk of all the hadrons in the final state have interacted many timesand come into local thermal equilibrium The most direct approach comes via the analysisof the exponentially falling spectra of identified hadrons Fitting a slope to these exponentialspectra and then extracting an ldquoeffective temperaturerdquo for each species of hadron yieldsdifferent ldquoeffective temperaturesrdquo for each species This species-dependence arises because
11
the matter produced in a heavy ion collision expands radially in the directions transverseto the beam axis perhaps explodes radially is a better phrase This means that we shouldexpect the pT spectra to be a thermal distribution boosted by some radial velocity If allhadrons are boosted by the same velocity the heavier the hadron the more its momentum isincreased by the radial boost Indeed what is found in data is that the effective temperatureincreases with the mass of the hadron species This can be seen at a qualitative level in theleft panel of Fig 22 in the soft regime the proton kaon and pion spectra are ordered bymass with the protons falling off most slowly with pT indicating that they have the highesteffective temperature Quantitatively one uses the data for hadron species with varyingmasses species to first extract the mass-dependence of the effective temperature and thusthe radial expansion velocity and then to extrapolate the effective ldquotemperaturesrdquo to themass rarr zero limit and in this way obtain a measurement of the actual temperature of thefinal state hadrons This ldquokinetic freezeout temperaturerdquo is the temperature at the (very late)time at which the gas of hadrons becomes so dilute that elastic collisions between the hadronscease and the momentum distributions therefore stop changing as the system expands Inheavy ion collisions at the top RHIC energy models of the kinetic freezeout account for thedata with freeze-out temperatures of asymp 90 MeV and radial expansion velocities of 06 c forcollisions with the smallest impact parameters [68] With increasing impact parameter theradial velocity decreases and the freeze-out temperature increases This is consistent withthe picture that a smaller system builds up less transverse flow and that during its expansionit cannot cool down as much as a bigger system since it falls apart earlier
The analysis just described is unique to heavy ion collisions in elementary electron-positron or proton-(anti)proton collisions spectra at low transverse momentum may alsobe fit by exponentials but the ldquotemperaturesrdquo extracted in this way do not have a system-atic dependence on the hadron mass In fact they are close to the same for all hadron speciesas can be seen qualitatively in the right panel Fig 22 Simply seeing exponential spectra andfitting a ldquotemperaturerdquo therefore does not in itself provide evidence for multiple interactionsand equilibration Making that case in the context of heavy ion collisions relies crucially onthe existence of a collective radial expansion with a common velocity for all hadron species
Demonstrating that the final state of a heavy ion collision at the time of kinetic freezeoutis a gas of hadrons in local thermal equilibrium is by itself not particularly interesting Itdoes however embolden us to ask whether the material produced in these collisions reacheslocal thermal equilibrium at an earlier time and thus at a higher temperature The bestevidence for an affirmative answer to this question comes from the analysis of ldquoelliptic flowrdquoin collisions with nonzero impact parameter We shall discuss this at length in the nextsubsection
We close this subsection with a simpler analysis that lays further groundwork by allowingus to see back to a somewhat earlier epoch than that of kinetic freezeout If we think of aheavy ion collision as a ldquolittle bangrdquo replaying the history of the big bang in a small volumeand with a vastly accelerated expansion rate then kinetic freezeout is the analogue of the(late) cosmological time at which photons and electrons no longer scatter off each other Wenow turn to the analogue of the (earlier) cosmological epoch of nucleosynthesis namely thetime at which the composition of the final state hadron gas stops changing Experimentalistscan measure the abundance of more than a dozen hadron species and it turns out that all
12
10-3
10-2
10-1
1Ra
tiosNN=200 GeV
STAR
T=1605 microb=20 MeV
PHENIXBRAHMS
T=155 microb=26 MeV
-
+K-
K+pndash
pndash
$ndash
$ndash
K+
+K-
-pndash
-ndash
-$-
-
dp
dndash
pndashampK-
K0
K-
++
p
Figure 23 So-called thermal fit to different particle species The relative abundance of differenthadron species is well-described by a two-parameter grand canonical ensemble in terms of temperatureT and baryon chemical potential microb [69]
CMy-4 -2 0 2 4
dNd
y ne
t-pro
tons
0
20
40
60
80 pAGS y
pSPS y
pRHIC y
AGS(E802E877 E917)SPS(NA49)RHIC(BRAHMS)
Figure 24 Left Chemical potential extracted from thermal fits at different center of mass energies[69] Right The number of protons minus number of antiprotons per unit rapidity for central heavyion collisions [70] This net proton number decreases with increasing center of mass energy fromradics = 5 GeV (at the AGS collider at BNL) via
radics = 17 GeV (at the SPS collider at CERN) toradic
s = 200 GeV (at RHIC) (For each collision energy yp indicates the rapidity of a hypotheticalproton that has the same velocity after the collision as it did before)
13
the ratios among these abundances can be fit upon assuming thermal distributions with sometemperature T and some baryon number chemical potential micro as shown in Fig 23 This isa two parameter fit to about a dozen ratios The temperature extracted in this way is calledthe chemical freezeout temperature since one interpretation is that it is the temperatureat which the hadronic matter becomes dilute enough that inelastic hadron-hadron collisionscease to modify the abundance ratios The chemical freezeout temperature in heavy ioncollisions at top RHIC energies is about 155-180 MeV [6971] This is interesting for severalreasons First it is not far below the QCD phase transition temperature which means thatthe appropriateness of a hadron gas description of this epoch may be questioned Secondwithin error bars it is the same temperature that is extracted by doing a thermal model fit tohadron production in electron-positron collisions in which final state rescattering elastic orinelastic can surely be neglected So by itself the success of the thermal fits to abundanceratios in heavy ion collisions could be interpreted as telling us about the statistical natureof hadronization as must be the case in electron-positron collisions However given that weknow that in heavy ion collisions (and not in electron-positron collisions) kinetic equilbriumis maintained down to a lower kinetic freezeout temperature and given that as we shallsee in the next subsection approximate local thermal equilbrium is achieved at a highertemperature it does seem most natural to interpret the chemical freezeout temperature inheavy ion collisions as reflecting the temperature of the matter produced at the time whenspecies-changing processes cease
We have not yet talked about the baryon number chemical potential extracted from thethermal fit to abundance ratios As illustrated in the left panel of Fig 24 this micro decreaseswith increasing collision energy
radics This energy dependence has two origins The dominant
effect is simply that at higher and higher collision energies more and more entropy is producedwhile the total net baryon number in the collision is always 197+197 At top RHIC energiesthese baryons are diluted among the 8000 or so hadrons in the final state making the baryonchemical potential much smaller than it is in lower energy collisions where the final statemultiplicity is much lower The second effect is that in the highest energy collisions most ofthe net baryon number from the two incident nuclei stays at large pseudorapidity (meaningsmall angles near the incident beam directions) These two effects can be seen directly in thedata shown in the left panel of Fig 21 and the right panel of Fig 24 as the collision energyincreases the total number of hadrons in the final state grows while the net baryon numberat mid-rapidity drops1 This experimental fact that baryon number is not ldquofully stoppedrdquoteaches us about the dynamics of the earliest moments of a hadron-hadron collision (In thisrespect heavy ion collisions are not qualitatively different than proton-proton collisions) Ina high energy proton proton collision particle production at mid-rapidity is dominated by
1The data in the right panel of Fig 24 is plotted relative to rapidity
y equiv 1
2ln
(E + pLE minus pL
) (22)
where E and pL are the energy and longitudinal momentum of a proton in the final state Recall that rapidityand pseudorapidity η (used in the plot in the left panel of Fig 21) become the same in the limit in which Eand pL are much greater than the proton mass When one plots data for all charged hadrons as in the leftpanel of Fig 21 only pseudorapidity can be defined since the rapidity of a hadron with a given polar angleθ depends on the hadron mass When one plots data for identified protons pseudorapidity can be convertedinto rapidity
14
the partons in the initial state that carry a small fraction of the momentum of an individualnucleon mdash small Bjorken x And the small-x parton distributions functions that describe theinitial state of the incident nucleons or nuclei are dominated by gluons and to a lesser extentby quark-antiquark pairs the net baryon number is at larger-x As we have already said weshall not focus on the many interesting questions related to the early-time dynamics in heavyion collisions since gravity dual calculations have not had much to say about them Insteadin the next subsection we turn to the evidence that local thermal equilibrium is establishedquickly therefore at a high temperature This means that heavy ion collisions can teach usabout properties of the high temperature phase of QCD namely the quark-gluon plasmaAnd we shall see later so can gravity dual calculations We shall henceforth always workat micro = 0 This is a good approximation as long as micro3 the quark chemical potential ismuch less than the temperature T The results in the left panel Fig 24 show that this isa very good approximation at top RHIC energies and will be an even better approximationfor heavy ion collisions at the LHC Using lower energy heavy ion collisions to scan the QCDphase diagram by varying micro is a very interesting ongoing research program but we shall notaddress it in this review
22 Elliptic flow
221 Introduction and motivation
The phrase ldquoelliptic flowrdquo refers to a suite of experimental observables in heavy ion physicsthat utilize the experimentalistsrsquo ability to select events in which the impact parameter ofthe collision lies within some specified range and use these events to study how the matterproduced in the collision flows collectively The basic idea is simple Suppose we selectevents in which the impact parameter is comparable to the nuclear radius Now imaginetaking a beamrsquos eye view of one of these collisions The two Lorentz-contracted nuclei (thinkcircular ldquopancakesrdquo) collide only in an ldquoalmond-shapedrdquo region see Fig 25 The fragmentsof the nuclei outside the almond that did not collide (ldquospectator nucleonsrdquo) fly down thebeam pipes All the few thousand particles at mid-rapidity in the final state must havecome from the few hundred nucleon-nucleon collisions that occurred within the almond Ifthese few thousand hadrons came instead from a few hundred independent nucleon-nucleoncollisions just by the central limit theorem the few thousand final state hadrons would bedistributed uniformly in azimuthal angle φ (angle around the beam direction) This nullhypothesis which we shall make quantitative below is ruled out by the data as we shall seeIf on the other hand the collisions within the almond yield particles that interact reachlocal equilibrium and thus produce some kind of fluid our expectations for the ldquoshaperdquo ofthe azimuthal distribution of the final state hadrons is quite different The hypothesis thatis logically the opposite extreme to pretending that the thousands of partons produced inthe hundreds of nucleon-nucleon collisions do not see each other is to pretend that what isproduced is a fluid that flows according to the laws of ideal zero viscosity hydrodynamicssince this extreme is achieved in the limit of zero mean free path In hydrodynamics thealmond is thought of as a drop of fluid with zero pressure at its edges and a high pressureat its center This droplet of course explodes And since the pressure gradients are greateracross the short extent of the almond than they are across its long direction the explosion is
15
x
y
Figure 25 Sketch of the collision of two nuclei shown in the transverse plane perpendicular to thebeam The collision region is limited to the interaction almond in the center of the transverse planeSpectator nucleons located in the white regions of the nuclei do not participate in the collision Figuretaken from Ref [72]
azimuthally asymmetric The first big news from the RHIC experimental program was thediscovery that at RHIC energies these azimuthal asymmetries can be large the explosionscan blast with summed transverse momenta of the hadrons that are twice as large in theshort direction of the almond as they are in the long direction And as we shall see it turnsout that ideal hydrodynamics does a surprisingly good job of describing these asymmetricexplosions of the matter produced in heavy ion collisions with nonzero impact parameterThis has implications which are sufficiently interesting that they motivate our describing thisstory in considerable detail over the course of this entire Section We close this introductionwith a sketch of these implications
First the agreement between data and ideal hydrodynamics teaches us that the viscosityη of the fluid produced in heavy ion collisions must be low η enters in the dimensionlessratio ηs with s the entropy density and it is ηs that is constrained to be small Afluid that is close to the ideal hydrodynamic limit with small ηs requires strong couplingbetween the fluid constituents Small ηs means that momentum is not easily transportedover distances that are long compared to sim sminus13 which means that there can be no well-defined quasiparticles with long mean free paths in a low viscosity fluid since if they existedthey would transport momentum and damp out shear flows No particles with long meanfree paths means strongly coupled constituents
Second we learn that the strong coupling between partons that results in approximatelocal equilibration and fluid flow close to that described by ideal hydrodynamics must setin very soon after the initial collision If partons moved with significant mean free pathsfor many fm of time after the collision delaying equilibration for many fm the almondwould circularize to a significant degree during this initial period of time and the azimuthalmomentum asymmetry generated by any later period of hydrodynamic behavior would beless than observed When this argument is made quantitative the conclusion is that RHIC
16
collisions produce strongly coupled fluid in approximate local thermal equilibrium withinclose to or even somewhat less than 1 fm after the collision [73]2
We can begin to see that the circle of ideas that emerge from the analysis of elliptic flowdata are what make heavy ion collisions of interest to the broader community of theoreticalphysicists for whom we are writing this review These analyses justify the conclusion thatonly 1 fm after the collision the matter produced can be described using the language ofthermodynamics and hydrodynamics And we have already seen that at this early time theenergy density is well above the hadron-QGP crossover in QCD thermodynamics which iswell-characterized in lattice calculations This justifies the claim that heavy ion collisionsproduce quark-gluon plasma Furthermore the same analyses teach us that this quark-gluonplasma is a strongly coupled low viscosity fluid with no quasiparticles having any significantmean free path Lattice calculations have recently begun to cast some light on these transportproperties of quark-gluon plasma but these lattice calculations that go beyond Euclideanthermodynamics are still in their pioneering epoch Perturbative calculations of quark-gluonplasma properties are built upon the existence of quasiparticles The analyses of elliptic flowdata thus cast doubt upon their utility And we are motivated to study the strongly coupledplasmas with similar properties that can be analyzed via gauge theory gravity dualitysince these calculational methods allow many questions that go beyond thermodynamics tobe probed rigorously at strong coupling
In Section 222 we first describe how to select classes of events all of whose impact param-eters lie in some narrow range Next we define the experimental observable used to measureazimuthal asymmetry and describe how to falsify the null hypothesis that the asymmetry isdue only to statistical fluctuations We then take a first look at the RHIC data and convinceourselves that the asymmetries that are seen reflect collective flow not statistical fluctua-tions In Section 223 we describe how to do a hydrodynamical calculation of the azimuthalasymmetry and in Section 224 we compare hydrodynamic calculations to RHIC data anddescribe the conclusions that we have sketched above in more quantitative terms The readerinterested only in the bottom line should skip directly to Section 224
222 The elliptic flow observable v2 at RHIC
We want to study the dependence of collective flow on the size and anisotropy of the almond-shaped region of the transverse plane as seen in the qualitative beamrsquos eye view sketch inFigure 25 To this end it is obviously necessary to bin heavy ion collisions as a function of
2Reaching approximate local thermal equilibrium and hence hydrodynamic behavior within less than 1fm after a heavy ion collision has been thought of as ldquorapid equilibrationrdquo since it is rapid compared toweak coupling estimates [15] This observation has launched a large effort (that we shall not review) towardsexplaining equilibration as originating from weakly-coupled processes that arise in the presence of the strongcolor fields that are present in the initial instants of a heavy ion collision A very recent calculation indicateshowever that the observed equilibration time may not be so rapid after all In a strongly coupled field theorywith a dual gravitational description when two sheets of energy density with a finite thickness collide atthe speed of light a hydrodynamic description of the plasma that results becomes reliable only sim 3 sheet-thicknesses after the collision [25] And a Lorentz-contracted incident gold nucleus at RHIC has a maximumthickness of only 014 fm So if the equilibration processes in heavy ion collisions could be thought of asstrongly coupled throughout perhaps local thermal equilibrium and hydrodynamic behavior would set in evenmore rapidly than is indicated by the data
17
this impact parameter This is possible in heavy ion collisions since the number of hadronsproduced in a heavy ion collision is anticorrelated with the impact parameter of the collisionFor head-on collisions (conventionally referred to as ldquocentral collisionsrdquo) the multiplicity ishigh the multiplicity is much lower in collisions with impact parameters comparable to theradii of the incident ions (often referred to as ldquosemi-peripheral collisionsrdquo) the multiplicity islower still in grazing (ldquoperipheralrdquo) collisions Experimentalists therefore bin their events bymultiplicity using that as a proxy for impact parameter The terminology used refers to theldquo0-5 centrality binrdquo and the ldquo5-10rdquo centrality bin and meaning the 5 of events withthe highest multiplicities the next 5 of events with the next highest multiplicity Thecorrelation between event multiplicity and impact parameter is described well by the Glaubertheory of multiple scattering which we shall not review here Suffice to say that even thoughthe absolute value of the event multiplicities is the subject of much ongoing research thequestion of what distribution of impact parameters corresponds to the 0-5 centrality bin(namely the most head-on collisions) is well established Although experimentalists cannotliterally pick a class of events with a single value of the impact parameter by binning theirdata in multiplicity they can select a class of events with a reasonably narrow distributionof impact parameters centered around any desired value This is possible only because nu-clei are big enough in proton-proton collisions which in principle have impact parameterssince protons are not pointlike there is no operational way to separate variations in impactparameter from event-by-event fluctuations in the multiplicity at a given impact parameter
Suppose that we have selected a class of semi-peripheral collisions Since these collisionshave a nonzero impact parameter the impact parameter vector together with the beamdirection define a plane conventionally called the reaction plane Directions within thetransverse plane of Figure 25 specified by the azimuthal angle φ now need not be equivalentWe can ask to what extent the multiplicity and momentum of hadrons flying across the shortdirection of the collision almond (in the reaction plane) differs from that of the hadrons flyingalong the long direction of the collision almond (perpendicular to the reaction plane)
Let us characterize this dependence on the reaction plane for the case of the single inclusiveparticle spectrum dN
d3pof a particular species of hadron The three-momentum p of a particle
of mass m is parametrized conveniently in terms of its transverse momentum pT its azimuthalangle φ and its rapidity y which specifies its longitudinal momentum Specifically
p =
(pT cosφ pT sinφ
radicp2T +m2 sinh y
) (23)
The energy of the particle is E =radicp2T +m2 cosh y The single particle spectrum can then
be written as
dN
d2pt dy=
1
2πpT
dN
dpT dy[1 + 2v1 cos(φminus ΦR) + 2v2 cos 2(φminus ΦR) + middot middot middot ] (24)
where ΦR denotes explicitly the azimuthal orientation of the reaction plane which we do notknow a priori Thus the azimuthal dependence of particle production is characterized bythe harmonic coefficients
vn equiv 〈exp [i n (φminus ΦR)]〉 =
intdNd3p
ei n (φminusΦR) d3pintdNd3p
d3p (25)
18
0 1 2 3 4 5 6 7 8 90
01
02
03 0-50-5
0 1 2 3 4 5 6 7 8 90
01
02
035-105-10
0 1 2 3 4 5 6 7 8 90
01
02
03 10-2010-20
0 1 2 3 4 5 6 7 8 90
01
02
0320-3020-30
0 1 2 3 4 5 6 7 8 90
01
02
03 30-4030-40
(GeVc)tp0 1 2 3 4 5 6 7 8 90
01
02
0340-6040-60
2v
0 10 20 30 40 50 60 70 800
1
2
3
4
5
6
7
8
Most Central
()
2v
22v42v62v
Figure 26 Left Transverse momentum dependence of the elliptic flow v2(pT ) for different centralitybins We see v2 increasing as one goes from nearly head-on collisions to semi-peripheral collisionsRight the pT -integrated elliptic flow v2 as a function of centrality bins reconstructed with 2nd 4thand 6th order cumulants The most head-on collisions are on the left Figures taken from Ref [74]
The coefficients vn are referred to generically as n-th order flow In particular v1 is referredto as ldquodirected flowrdquo and v2 as ldquoelliptic flowrdquo In general the vn can depend on the transversemomentum pT the rapidity y the impact parameter of the collision and they can differ fordifferent particle species In the collision of identical nuclei at mid-rapidity the collisionregion is symmetric under φrarr φ+ π and all odd harmonics vanish In this case the ellipticflow v2 is the first non-vanishing coefficient We shall focus only on mid-rapidity y = 0 Themost common observables used are v2(pT ) for collisions with varying centrality (ie impactparameter) and v2 integrated over all pT which is just a function of centrality A nonzero v2
(or v2(pT )) means that there are more particles (with a given transverse momentum) goingleft and right than up and down
In Fig 26 we show data for the transverse momentum dependence of the elliptic flowv2(pT ) and the transverse momentum integrated quantity measured for different centralityclasses in Au+Au collisions at RHIC The azimuthal asymmetry v2 of the final state singleinclusive hadron spectrum is maximal in semi-peripheral collisions v2 is less for more centralcollisions since the initial geometric asymmetry is less mdash the almond-shaped collision regionbecomes closer to circular as the impact parameter is reduced In the idealized case of zeroimpact parameter v2 due to collective effects should vanish The reason that v2 is not smallereven in the sample of the 5 most head-on collisions is that this sample includes events witha distribution of impact parameters in the range 0 lt b lt 35 fm In principle v2 shouldalso vanish in the limit of grazing collisions Some decrease is visible in the most peripheralcollisions shown in the right panel of Fig 26 but the error bars become large long beforeone reaches collisions which are literally grazing
It is worth pausing to appreciate the size of the elliptic flow signal shown in Fig 26According to (24) the ratio of dN
d3pin whatever azimuthal direction it is largest to dN
d3pninety
degrees in azimuth away is (1 + 2v2)(1 minus 2v2) which is a factor of 2 for v2 = 16 Thus
19
a v2 of the order of magnitude shown in semi-peripheral collisions at RHIC for pT sim 2 GeVcorresponds to a factor of more than 2 azimuthal asymmetry It is natural to expect a nonzeroquadrupolar asymmetry v2 at nonzero impact parameter due to collective effects occuring inthe almond-shaped collision zone given that the almond shape itself has a quadrupole Ofcourse v4 v6 v8 will generically be nonzero as well
There are two basic experimental problems in determining the coefficients (25) Theorientation ΦR of the reaction plane is not known a priori and a priori it is unclear whetheran azimuthal asymmetry in an event arises as a consequence of collective motion or whetherit is the result of a statistical fluctuation In practice both problems are connected Thereare two methods in common use that each address both problems One option is to use dataat high rapidity to determine ΦR and then to measure the vn by applying (25) to mid-rapidity data This eliminates the contribution of all statistical fluctuations uncorrelatedwith the reaction plane unless these fluctuations introduce correlations between particles inthe mid-rapidity and high rapidity regions of the detector The second option uses only themid-rapidity data This means that in each event the azimuthal direction of the maximalparticle yield depends both on the orientation of the reaction plane and on the statisticalfluctuations in that event If it just so happens that there were 3 jets going up and 3 jetsgoing down that would certainly skew the determination of the reaction plane ΦR fromthe data However there is a systematic procedure known as the cumulant method [75]which makes an unambiguous separation of flow effects from non-flow effects possible Thebasic idea is that if two particles are correlated in the azimuthal angle because of a non-flow effect such as resonance decay or back-to-back jets or some other fluctuation then theyare not correlated with all the other particles in the event In contrast a flow effect leavesa signature in which all particles know about the reaction plane and thus are correlatedBy a suitable analysis of 2- 4- and 6- particle correlations it is in practice possible todisentangle these two effects [75] The right-hand side of Fig 26 shows such an analysisSubtracting non-flow effects in a more accurate 4-th order cumulant analysis the elliptic flowsignal is reduced a bit Further increasing the accuracy to 6th order the signal does notchange indicating that all non-flow effects have been corrected In short this demonstratesthat the large elliptic flow signal cannot arise from statistical fluctuations in an incoherentsuperposition of nucleon-nucleon collisions or from non-flow effects It must characterize acollective phenomenon reflecting interactions among the debris produced by all the nucleon-nucleon collisions occurring within the almond-shaped collision region For more details werefer the reader to the original literature [75]
223 Calculating elliptic flow using (ideal) hydrodynamics
We have now seen that the azimuthal asymmetry of the little explosions created by heavy ioncollisions with nonzero impact parameter parametrized by the second harmonic v2 character-izes a strong collective phenomenon At time zero the almond-shaped collision is azimuthallyasymmetric in space subsequent collective dynamics must convert that spatial asymmetryinto an asymmetry in momentum space After all v2 measures the azimuthal asymmetryof the momentum distribution of the final state hadrons This conversion of initial spatialanisotropy into final momentum anisotropy is characteristic of any explosion mdash one shapesthe explosive in order to design a charge that blasts with greater force in some directions
20
than others Hydrodynamics provides the natural language for describing such processesthe initial spatial anisotropy corresponds to anisotropic pressure gradients Assuming thatthe pressure is maximal at the center of the almond and zero at its edge the gradient isgreater across the almond (in the reaction plane) than along it (perpendicular to the reactionplane) The elliptic flow v2 measures the extent to which these pressure gradients lead toan anisotropic explosion with greater momentum flow in the reaction plane The size of v2
characterizes the efficiency of translating initial pressure gradients into collective flow Bydoing hydrodynamic calculations and comparing the calculated v2 to that in the data (andalso comparing the final state radial flow velocity to that determined from the single-particlespectra as described in the previous section) one can constrain the input quantities that gointo a hydrodynamic description
The starting point in any hydrodynamic analysis is to consider the limit of ideal zeroviscosity hydrodynamics In this limit the hydrodynamic description is specified entirely byan equation of state which relates the pressure and the energy density and by the initialspatial distribution of energy density and fluid velocity In particular in ideal hydrodynamicsone is setting all dissipative coefficients (shear viscosity bulk viscosity and their many higherorder cousins) to zero If the equation of state is held fixed and viscosity is turned on v2 mustdecrease turning on viscosity introduces dissipation that has the effect of turning some ofthe initial anisotropy in pressure gradients into entropy production rather than into directedcollective flow So upon making some assumption for the equation of state and for theinitial energy density distribution setting the viscosities to zero yields an upper bound onthe v2 in the final state Ideal inviscid hydrodynamics has therefore long been used as acalculational benchmark in heavy ion physics As we shall see below in heavy ion collisionsat RHIC energies ideal hydrodynamics does a good job of describing v2(pT ) for pions kaonsand protons for transverse momenta pT below about 1minus 2 GeV This motivates an ongoingresearch program in which one begins by comparing data to the limiting case of ideal inviscidhydrodynamics and then turns to a characterization of dissipative effects asking how large aviscosity will spoil the agreement with data In this subsection we sketch how practitionersdetermine the equation of state and initial energy density profile and we recall the basicprinciples behind the hydrodynamic calculations on which all these studies rest In the nextsubsection we summarize the current constraints on the shear viscosity that are obtained bycomparing to RHIC v2 data
The equation of state relates the pressure P to the energy density ε P is a thermodynamicquantity and therefore can be calculated using the methods of lattice quantum field theoryLattice calculations (or fits to them) of P (ε) in the quark-gluon plasma and in the crossoverregime between QGP and hadron gas are often used as inputs to hydrodynamic calculationsAt lower energy densities practitioners either use a hadron resonance gas model equationof state or match the hydrodynamic calculation onto a hadron cascade model One of theadvantages of focussing on the v2 observable is that it is insensitive to the late time epochof the collision when all the details of these choices matter This insensitivity is easy tounderstand v2 describes the conversion of a spatial anisotropy into anisotropic collectiveflow As this conversion begins the initial almond-shaped collision explodes with greatermomentum across the short direction of the almond and therefore circularizes Once it hascircularized no further v2 can develop Thus v2 is generated early in the collision say overthe first sim 5 fm of time By the late times when a hadron gas description is needed v2 has
21
already been generated In contrast the final state radial flow velocity reflects a time integralover the pressure built up during all epochs of the collision it is therefore sensitive to howthe hadron gas phase is described and so is a less useful observable
The discussion above reminds us of a second sense in which the ideal hydrodynamic cal-culation of v2 is a benchmark ideal hydrodynamics requires local equilibrium It thereforecannot be valid from time t = 0 By using an ideal hydrodynamic description beginning att = 0 we must again be overestimating v2 and so we can ask how long an initial phase duringwhich partons stream freely without starting to circularize the almond-shaped region with-out generating any momentum anisotropy can be tolerated without spoiling the agreementbetween calculations and data
After choosing an equation of state an initialization time and viscosities (zero in thebenchmark calculation) the only thing that remains to be specified is the distribution ofenergy density as a function of position in the almond-shaped collision region (The transversevelocities are assumed to be zero initially) In the simplest approach called Glauber theorythis energy density is proportional to the product of the thickness of the two nuclei at a givenpoint in the transverse plane It is thus zero at the edge of the almond where the thicknessof one nucleus goes to zero and maximum at the center of the almond The proportionalityconstant is determined by fitting to data other than v2 see eg [76] The assumptions behindthis Glauber approach to estimating how much energy density is created at a given locationas a function of the nuclear thickness at that location are assumptions about physics of thecollision at t = 0 There is in particular one alternative model parametrization for which theenergy density rises towards the center of the almond more rapidly than the product of thenuclear thicknesses This parametrization is referred to as the CGC initial condition sinceit was first motivated by ideas of parton saturation (called ldquocolor glass condensaterdquo) [77]The Glauber and CGC models for the initial energy density distribution are often used asbenchmarks in the hope that they bracket Naturersquos choice
We turn now to a review of the formulation of the hydrodynamic equations of motionHydrodynamics is an effective theory which describes the small frequency and long wavelengthlimit of an underlying interacting dynamical theory [78] It is a classical field theory wherethe fields can be understood as the expectation values of certain quantum operators in theunderlying theory In the hydrodynamic limit since the length scales under considerationare longer than any correlation length in the underlying theory by virtue of the central limittheorem all n-point correlators of the underlying theory can be factorized into two pointcorrelators (Gaussian approximation) The fluctuations on these average values are smalland a description in terms of expectation values is meaningful If the underlying theory admitsa (quasi)particle description this statement is equivalent to saying that the hydrodynamicdescription involves averages over many of these fundamental degrees of freedom
The hydrodynamic degrees of freedom include the expectation values of conserved currentssuch as the stress tensor Tmicroν or the currents of conserved charges JB As a consequence ofthe conservation laws
partmicroTmicroν = 0 (26)
partmicroJmicroB = 0 (27)
long wavelength excitations of these fields can only relax on long timescales since their
22
relaxation must involve moving stress-energy or charges over distances of order the wavelengthof the excitation As a consequence these conservation laws lead to excitations whose lifetimediverges with their wavelength Such excitations are called hydrodynamic modes
If the only long-lived modes are those from conserved currents then hydrodynamics de-scribes a normal fluid However there can be other degrees of freedom that lead to longlived modes in the long wave length limit For example in a phase of matter in which someglobal symmetry is spontaneusly broken the phase(s) of the symmetry breaking expectationvalue(s) is (are) also hydrodynamic modes [78] The classic example of this is a superfluid inwhich a global U(1) symmetry is spontaneously broken Chiral symmetry is spontaneouslybroken in QCD but there are two reasons why we can neglect the potential hydrodynamicmodes associated with the chiral order parameter [79] First explicit chiral symmetry break-ing gives these modes a mass (the pion mass) and we are interested in the hydrodynamicdescription of physics on length scales longer than the inverse pion mass Second we areinterested in temperatures above the QCD crossover at which the chiral order parameter isdisordered the symmetry is restored and this question does not arise So we need only con-sider normal fluid hydrodynamics Furthermore as we have discussed in Section 2 at RHICenergies the matter produced in heavy ion collisions has only a very small baryon numberdensity and it is a good approximation to neglect JmicroB The only hydrodynamic degrees offreedom are therefore those described by Tmicroν
At the length scales at which the hydrodynamic approximation is valid each point of spacecan be regarded as a macroscopic fluid cell characterized by its energy density ε pressureP and a velocity umicro The velocity field can be defined by the energy flow together with theconstraint u2 = minus1 In the so-called Landau frame the four equations
umicroTmicroν = minusεuν (28)
determine ε and u from the stress tensor
Hydrodynamics can be viewed as a gradient expansion of the stress-tensor (and any otherhydrodynamic fields) In general the stress tensor can be separated into a term with nogradients (ideal) and a term which contains all the gradients
Tmicroν = Tmicroνideal + Πmicroν (29)
In the rest frame of each fluid cell (ui = 0) the ideal piece is diagonal and isotropic Tmicroνideal =diag (ε P P P ) Thus in any frame
Tmicroνideal = (ε+ P )umicrouν + Pgmicroν (210)
where gmicroν is the metric of the space
If there were a nonzero density of some conserved charge n the velocity field could eitherbe defined in the Landau frame as above or may instead be defined in the so-called Eckartframe with Jmicro = numicro In the Landau frame the definition (28) of umicro implies Πmicroνuν = 0(transversality) Hence there is no heat flow but there can be currents of the conservedcharge In the Eckart frame the velocity field is comoving with the conserved charge butthere can be heat flow
A Ideal fluidIdeal hydrodynamics is the limit in which all gradient terms in Tmicroν are neglected It can
23
be used to describe motions of the fluid that occur on macroscopic length scales and timescales associated with how the fluid is ldquostirredrdquo that are long compared to any internal scalesassociated with the fluid itself Corrections to ideal hydrodynamics mdash namely the gradientterms in Πmicroν that we shall discuss shortly mdash introduce internal length and time scales Forexample time scales for relaxation of perturbations away from local thermal equilibriumlength scales associated with mean free paths etc Ideal hydrodynamics works on longerlength scales than these At long time scales the fluid cells are in equilibrium P is thepressure in the rest frame of each fluid cell and ε and P are related by the equation of stateThis equation of state can be determined by studying a homogeneous system at rest with nogradients for example via a lattice calculation
The range of applicability of ideal fluid hydrodynamics can be characterized in terms of theisotropization scale τiso and the thermalization scale τeq The isotropization scale measures thecharacteristic time over which an initially anisotropic stress tensor acquires a diagonal formin the local rest frame The thermalization scale sets the time over which entropy productionceases These scales do not need to be the same but τeq ge τiso since thermalization impliesisotropy P and ε are related by the equilibrium equation of state only at times τ gt τeq Ifthere is a large separation of scales τiso τeq then between these timescales the relationshipbetween P and ε can be different and is generically time-dependent Ideal hydrodynamics cannevertheless be used as long as P (ε) is replaced by some suitable non-equilibrium equationof state that will depend on exactly how the system is out of thermal equilibrium It is worthnoting that in the case of a conformal theory since the trace of the stress tensor vanishesP = 1
3ε whether or not the system is in equilibrium In this special case as long as there areno charge densities the equation of state is unmodified even if the system is isotropic but outof equilibrium and ideal hydrodynamics with the standard equation of state can be used aslong as τ τiso
B 1st order dissipative fluid dynamics
Going beyond the infinite wavelength limit requires the introduction of viscosities To firstorder in gradients the requirement that Πmicroν be transverse means that it must take the form
Πmicroν = minusη(ε)σmicroν minus ζ(ε)∆microν nabla middot u (211)
where η and ζ are the shear and bulk viscosities nablamicro = ∆microνdν with dν the covariant derivativeand
∆microν = gmicroν + umicrouν (212)
σmicroν = ∆microα∆νβ (nablaαuβ +nablaβuα)minus 2
3∆αβnabla middot u (213)
The operator ∆microν is the projector into the space components of the fluid rest frame Notethat in this frame the only spatial gradients that appear in eq (211) are gradients of thevelocity fields The reason that the derivative of ε does not appear is that it can be eliminatedin the first order equations by using the zeroth order equation of motion
Dε = minus(ε+ P )nablamicroumicro (214)
where D = umicrodmicro is the time derivative in the fluid rest frame (Similarly time derivatives ofthe energy density can be eliminated in the second order equations that we shall give belowusing the first order equations of motion)
24
It is worth pausing to explain why we have introduced a covariant derivative even thoughwe will only ever be interested in heavy ion collisions mdash and thus hydrodynamics mdash occurringin flat spacetime But it is often convenient to use curvilinear coordinates with a nontrivialmetric For example the longitudinal dynamics is more conveniently described using propertime τ =
radict2 minus z2 and spacetime rapidity ξ = arctanh(zt) as coordinates rather than t
and z In these ldquoMilne coordinatesrdquo the metric is given by gmicroν = diag(gττ gxx gyy gξξ) =(minus1 1 1 τ2) These coordinates are useful because boost invariance simply translates into therequirement that ε umicro and Πmicroν be independent of ξ depending on τ only In particular if theinitial conditions are boost invariant then the fluid dynamic evolution will preserve this boostinvariance and the numerical calculation reduces in Milne coordinates to a 2+1-dimensionalproblem In the high energy limit one expects that hadronic collisions distribute energydensity approximately uniformly over a wide range rapidity Accordingly boost-invariantinitial conditions are often taken to be a good approximation [64] But fully 3+1-dimensionalcalculations that do not assume boost invariance can also be found in the literature [80ndash83]
It is often convenient to phrase the hydrodynamic equations in terms of the entropy densitys In the absence of conserved charges ie with baryon chemical potential microB = 0 theentropy density is s = (ε + P )T Using this and another fundamental thermodynamicrelation DE = T DS minus P DV (where EV equiv ε) the zeroth order equation of motion (214)becomes exactly the equation of entropy flow for an ideal isentropic fluid
D s = minussnablamicroumicro (215)
Repeating this analysis at first order including the viscous terms one easily derives fromumicronablaν Tmicroν = 0 that
D s
s= minusnablamicroumicro minus
1
s TΠmicroν nablaνumicro (216)
A similar analysis of the other three hydrodynamic equations then shows that they take theform
Duα = minus 1
Ts∆αν
(nablaνP +nablamicroΠmicroν
) (217)
It then follows from the structure of the shear tensor Πmicroν (211) that shear viscosity andbulk viscosity always appear in the hydrodynamic equations of motion in the dimensionlesscombinations ηs and ζs The net entropy increase is proportional to these dimensionlessquantities Gradients of the velocity field are measured in units of 1T
In a conformal theory ζ = 0 since Πmicroν must be traceless There are a number of indicationsfrom lattice calculations that as the temperature is increased above (15minus 2)Tc with Tc thecrossover temperature the quark-gluon plasma becomes more and more conformal Theequation of state approaches P = 1
3ε [84 85] The bulk viscosity drops rapidly [86] So weshall set ζ = 0 throughout the following in so doing neglecting temperatures close to Tc Oneof the things that makes heavy ion collisions at the LHC interesting is that in these collisionsthe plasma that is created is expected to be better approximated as conformal than is thecase at RHIC where the temperature at τ = 1 fm is thought to be between 15Tc and 2Tc
Just like the equation of state P (ε) the shear viscosity η(ε) is an input to the hydrodynamicdescription that must be obtained either from experiment or from the underlying microscopic
25
theory We shall discuss in Section 32 how transport coefficients like η are obtained fromcorrelation functions of the underlying microscopic theory via Kubo formulae
C Second order dissipative Hydrodynamics
Even though hydrodynamics is a controlled expansion in gradients the first order expres-sion for the tensor Πmicroν eq (211) is unsuitable for numerical computations The problemis that the set of equations (26) with the approximations (211) leads to acausal propaga-tion Even though this problem only arises for modes outside of the region of validity ofhydrodynamics (namely high momentum modes with short wavelengths of the order of themicroscopic length scale defined by η) the numerical evaluation of the first order equationsof motion is sensitive to the acausality in these hard modes This problem is solved by goingto one higher order in the gradient expansion This is known as second order hydrodynamics
There is a phenomenological approach to second order hydrodynamics due to Muller Israeland Stewart aimed at explicitly removing the acausal propagation [87ndash89] In this approachthe tensor Πmicroν is treated as a new hydrodynamic variable and a new dynamical equation isintroduced In its simplest form this equation is
τΠDΠmicroν = minusΠmicroν minus ησmicroν (218)
where τΠ is a new (second order) coefficient Note that as τΠ rarr 0 eq (218) coincides witheq (211) with the bulk viscosity ζ set to zero Eq ( 218) is such that Πmicroν relaxes to itsfirst order form in a (proper) time τΠ There are several variants of this equation in theliterature all of which follow the same philosophy They all introduce the relaxation time asthe characteristic time in which the tensor Πmicroν relaxes to its first order value The variationsarise from different ways of fixing some pathologies of eq (218) since as written eq (218)does not lead to a transverse stress tensor (although this is a higher order effect) and is notconformally invariant Since in this approach the relaxation time is introduced ad hoc itmay not be possible to give a prescription for extracting it from the underlying microscopictheory
The systematic extraction of second order coefficients demands a similar analysis of thesecond order gradients as was done at first order The strategy is once again to write allpossible terms with two derivatives which are transverse and consistent with the symmetriesof the theory As before only spatial gradients (in the fluid rest frame) are considered sincetime gradients can be related to the former via the zeroth order equations of motion
In a conformal theory second order hydrodynamics simplifies First only terms such thatΠmicromicro = 0 are allowed Furthermore the theory must be invariant under Weyl transformations
gmicroν rarr eminus2ω(x)gmicroν (219)
which impliesT rarr eω(x)T umicro rarr eω(x)umicro Tmicroν rarr e(d+2)ω(x)Tmicroν (220)
where T is the temperature and d the number of spacetime dimensions It turns out that thereare only five operators that respect these constraints [90] The second order contributions to
26
the tensor Πmicroν are linear combinations of these operators and can be cast in the form [9091]
Πmicroν = minusησmicroν minus τΠ
[〈DΠmicroν 〉 +
d
dminus 1Πmicroν(nablamiddotu)
]+κ[R〈microν〉 minus (dminus 2)uαR
α〈microν〉βuβ
]+
λ1
η2Π〈microλΠν〉λ minus λ2
ηΠ〈microλΩν〉λ + λ3Ω〈microλΩν〉λ (221)
Here Rmicroν is the Ricci tensor the indices in brackets are the symmetrized traceless projectorsonto the space components in the fluid rest frame namely
〈Amicroν 〉 equiv 1
2∆microα∆νβ(Aαβ +Aβα)minus 1
dminus 1∆microν∆αβAαβ equiv A〈microν〉 (222)
and the vorticity tensor is defined as
Ωmicroν equiv 1
2∆microα∆νβ(nablaαuβ minusnablaβuα) (223)
In deriving (221) we have replaced ησmicroν by Πmicroν on the right-hand side in places wheredoing so makes no change at second order We see from (221) that five new coefficients τΠκ λ1 λ2 and λ3 arise at second order in the hydrodynamic description of a conformal fluidin addition to η and the equation of state which arise at first and zeroth order respectivelyThe coefficient κ is not relevant for hydrodynamics in flat spacetime The λi coefficientsinvolve nonlinear combinations of fields in the rest frame and thus are invisible in linearizedhydrodynamics Thus these three coefficients cannot be extracted from linear response Ofthese three only λ1 is relevant in the absence of vorticity as in the numerical simulations thatwe will describe in the next Section These simulations have also shown that for physicallymotivated choices of λ1 the results are insensitive to its precise value leaving τπ as theonly phenomenologically relevant second order parameter in the hydrodynamic descriptionof a conformal fluid In a generic nonconformal fluid there are nine additional transportcoefficients [92]
For more in-depth discussions of 2nd order viscous hydrodynamics and its applications toheavy ion collisions see eg Refs [93ndash114]
224 Comparing elliptic flow in heavy ion collisions and hydrodynamiccalculations
For the case of ideal hydrodynamics the hydrodynamic equations of motion are fully specifiedonce the equation of state P = P (ε) is given A second-order dissipative hydrodynamiccalculation also requires knowledge of the transport coefficients η(ε) and ζ(ε) (although inpractice the latter is typically set to zero) and the relaxation time τΠ and the second-ordercoefficient λ1 entering Eq (221) (κ would enter in curved space time and λ2 and λ3 wouldenter in the presence of vorticity) All these parameters are well-defined in terms of correlationfunctions in the underlying quantum field theory In this sense the hydrodynamic evolutionequations are model-independent
27
The output of any hydrodynamic calculation depends on more than the evolution equa-tions One must make model assumptions about the initial energy density distribution Aswe have discussed in Section 222 there are two benchmark models for the energy distribu-tion across the almond-shaped collision region which serve to give us a sense of the degreeto which results are sensitive to our lack of knowledge of the details of this initial profileOften the initial transverse velocity fields are set to zero and boost invariance is assumed forthe longitudinal velocity field and the evolution For dissipative hydrodynamic simulationsthe off-diagonal elements of the energy-momentum tensor are additional hydrodynamic fieldswhich must be initialized The initialization time τ0 at which these initial conditions arefixed is an additional model parameter It can be viewed as characterizing the isotropizationtime at which hydrodynamics starts to apply but collective flow has not yet developed Inaddition to initial state sensitivity results depend on assumptions made about how the sys-tem stops behaving hydrodynamically and freezes out In practice freezeout is often assumedto happen as a rapid decoupling when a specified criterion is satisfied (eg when a fluidcell drops below a critical energy or entropy density) then the hydrodynamic fields in theunit cell are mapped onto hadronic equilibrium BoseFermi distributions This treatmentassumes that hydrodynamics is valid all the way down to the kinetic freezeout temperaturebelow which one has noninteracting hadrons Alternatively at a higher temperature close tothe crossover where hadrons are formed one can map the hydrodynamic fields onto a hadroncascade which accounts for the effects of rescattering in the interacting hadronic phase with-out assuming that its behavior is hydrodynamic [115 116] Indeed recent work suggeststhat hadronization may be triggered by cavitation induced by the large bulk viscosity in thevicinity of the crossover temperature [117] As we have discussed v2 is insensitive to detailsof how the late-time evolution is treated because v2 is generated during the epoch whenthe collision region is azimuthally anisotropic Nevertheless these late-time issues do matterwhen one does a global fit to v2 and the single-particle spectra since the latter are affected bythe radial flow which is built up over the entire history of the collision Finally the validityof results from any hydrodynamic calculation depends on the assumption that a hydrody-namic description is applicable This assumption can be checked at late times by checkingthe sensitivity to how freezeout is modelled and can be checked at early times by confirmingthe insensitivity of results to the values of the second order hydrodynamic coefficients and tothe initialization of the higher order off-diagonal elements of the energy momentum tensorif hydrodynamics is valid the gradients must be small enough at all times that second ordereffects are small compared to first order effects
In practice the dependence of physics conclusions on all these model assumptions has tobe established by systematically varying the initial conditions and freeze-out prescriptionswithin a wide physically motivated parameter range and comparing to data on both the singleparticle spectra (ie the radial velocity) and the azimuthal flow anisotropy coefficient v2 Atthe current time several generic observations have emerged from pursuing this program incomparison to data from RHIC
1 Perfect fluid dynamics reproduces the size and centrality dependence of v2
The RHIC data on single inclusive hadronic spectra dNpT dpT dy and their leadingazimuthal dependence v2(pT ) can be reproduced in magnitude and shape by ideal hy-drodynamic calculations for particles with pT lt 1 GeV The hydrodynamic picture is
28
2An
isot
ropy
Par
amet
er v
(GeVc)TTransverse Momentum p0 2 4 6
0
01
02
03-++ 0
SK-+K+K
pp++
STAR DataPHENIX DataHydro modelKp
Figure 27 The elliptic flow v2 versus pT for a large number of identified hadrons (pions kaonsprotons Λrsquos) showing the comparison between an ideal hydrodynamic calculation to data from RHICFigure taken from [118]
expected to break down for sufficiently small wavelength ie high momenta consistentwith the observation that significant deviations occur for pT gt 2 GeV (see left panelof Fig 26) The initialization time for these calculations is τ0 = 06 minus 1 fm If τ0 ischosen larger the agreement between ideal hydrodynamics and data is spoiled Thisgives significant support to a picture in which thermalization is achieved fast It alsoprovides the first indication that the shear viscosity of the fluid produced at RHIC mustbe small
2 The mass-ordering of identified hadron spectraThe pT -differential azimuthal asymmetry v2(pT ) of identified single inclusive hadronspectra shows a characteristic mass ordering in the range of pT lt 2 GeV at small pT the azimuthal asymmetry of light hadrons is significantly more pronounced than that ofheavier hadrons see Fig 27 This qualitative agreement of hydrodynamic simulationswith data supports the picture that all hadron species emerge from a single fluid movingwith a common flow field
3 Data support small dissipative coefficients such as shear viscosityAbove the crossover temperature the largest dissipative correction is expected to arisefrom shear viscosity η which enters the equations of motion of second order dissipativehydrodynamics in the combination ηs where s is the entropy density (Bulk viscositymay be sizable close to the crossover temperature [86]) As seen in Fig 28 shearviscous corrections decrease collectivity and so v2 decreases with increasing ηs Thedata are seen to favor small values ηs lt 02 with ηs gt 05 strongly disfavored Togive an example of the way this conclusion is stated in one recent review it wouldbe surprising if ηs turned out to be larger than about (3 minus 5)(4π) once all physicaleffects are properly included [110] The figure also shows that sensitivity to our lack ofknowledge of the initial energy density profile precludes a precise determination of ηs
29
0 1 2 3 4p
T [GeV]
0
5
10
15
20
25
v 2 (p
erce
nt)
STAR non-flow corrected (est)STAR event-plane
Glauber
ηs=10-4
ηs=008
ηs=016
0 1 2 3 4p
T [GeV]
0
5
10
15
20
25
v 2 (p
erce
nt)
STAR non-flow corrected (est)STAR event-plane
CGCηs=10
-4
ηs=008
ηs=016
ηs=024
Figure 28 The elliptic flow v2 versus pT for charged hadrons showing the comparison betweenhydrodynamic calculations with varying shear viscosity η and data from RHIC The two data setsshow the reduction in v2 when one uses the four-particle cumulant analysis that strips out effectsof statistical two-particle correlations The initial profile for the energy density across the almond-shaped collision region is obtained from (a) the Glauber model and (b) the color-glass condensatemodel Figure taken from Ref [103]
at present3 The smallness of ηs is remarkable since almost all other known liquidshave ηs gt 1 and most have ηs 1 The one liquid that is comparably close to idealis an ultracold gas of strongly coupled fermionic atoms whose ηs is also 1 and maybe comparably small to that of the quark-gluon plasma produced at RHIC [120] Boththese fluids are much better described by ideal hydrodynamics than water is Both haveηs comparable to the value 14π that as we shall see in Section 62 characterizes anystrongly coupled gauge theory plasma with a gravity dual in the large number of colorslimit
We note that while hydrodynamic calculations reproduce elliptic flow a treatment inwhich the Boltzmann equation for quark and gluon (quasi)particles is solved including all2 rarr 2 scattering processes with the cross-sections as calculated in perturbative QCD failsdramatically It results in values of v2 that are much smaller than in the data Agreementwith data can only be achieved if the parton scattering cross-sections are increased ad hocby more than a factor of 10 [121] With such large cross-sections a Boltzmann descriptioncannot be reliable since the mean free path of the particles becomes comparable to or smallerthan the interparticle spacing Another way of reaching the same conclusion is to note that if aperturbative description of the QGP as a gas of interacting quasiparticles is valid the effectiveQCD coupling αs describing the interaction among these quasiparticles must be small andfor small αs perturbative calculations of ηs are controlled and yield parametrically largevalues prop 1α2
s lnαs It is not possible to get as small a value of ηs as the data requires
3Recent analyses that include equations of state based upon parametrizations of recent lattice calcula-tions [119] and include dissipative effects from the late-time hadron gas epoch as well as estimates of theeffects of bulk viscosity [108 109] all of which turn out to be small have maintained these conclusions ηsseems to lie within the range (1 minus 25)(4π) with the greatest remaining source of uncertainty coming fromour lack of knowledge of the initial energy density profile [114]
30
from the perturbative calculation without increasing αs to the point that the calculation isinvalid In contrast as we shall see in Section 62 any gauge theory with a gravity dual musthave ηs = 14π in the large-Nc and strong coupling limit and furthermore the plasmafluids described by these theories in this limit do not have any well-defined quasiparticlesThis calculational framework thus seems to do a much better job of capturing the qualitativefeatures needed for a successful phenomenology of collective flow in heavy ion collisions
As we completed the preparation of this review the first measurement of the elliptic flowv2 in heavy ion collisions with
radics = 276 TeV was reported by the ALICE collaboration at
the LHC [122] It is striking that the v2(pT ) that they find for charged particles in threedifferent impact parameter bins agrees within error bars at all values of pT out to beyond 4GeV with that measured at
radics = 02 TeV by the STAR collaboration at RHIC Quantitative
comparison of these data to hydrodynamic calculations is expected from various authors inthe coming months but at a qualitative level these data indicate that the quark-gluon plasmaproduced at the LHC is comparably strongly coupled with comparably small ηs to thatproduced and studied at RHIC
23 Jet quenching
Having learned that heavy ion collisions produce a low viscosity strongly coupled fluid wenow turn to experimental observables with which we may study properties of the fluid beyondjust how it flows There are many such observables available In this subsection and the nextwe shall describe two classes of observables selected because in both cases there is (thepromise of) a substantive interplay between data from RHIC (and the LHC) and qualitativeinsights gained from the analysis of strongly coupled plasmas with dual gravity descriptions
Jet quenching refers to a suite of experimental observables that together reveal what hap-pens when a very energetic quark or gluon (with momentum much greater than the tempera-ture) plows through the strongly coupled plasma Some measurements focus on how rapidlythe energetic parton loses its energy other measurements give access to how the strongly cou-pled fluid responds to the energetic parton passing through it These energetic partons arenot external probes they are produced within the same collision that produces the stronglycoupled plasma itself
In a small fraction of proton-proton collisions atradics = 200 GeV partons from the incident
protons scatter with a large momentum transfer producing back-to-back partons in the finalstate with transverse momenta of order ten or a few tens of GeV These ldquohardrdquo processes arerare but data samples are large enough that they are nevertheless well studied The hightransverse momentum partons in the final state manifest themselves in the detector as jetsIndividual high-pT hadrons in the final state come from such hard processes and are typicallyfound within jets In addition to copious data from proton-(anti)proton collisions there is ahighly developed quantitatively controlled calculational framework built upon perturbativeQCD that is used to calculate the rates for hard processes in high energy hadron-hadroncollisions These calculations are built upon factorization theorems Consider as an exam-ple the single inclusive charged hadron spectrum at high-pT Fig 21 (right) That is theproduction cross-section for a single charged hadron with a given high transverse momen-tum pT regardless of what else is produced in the hadron-hadron collision This quantity is
31
calculated as a convolution of separate (factorized) functions that describe different aspectsof the process (i) the process-independent parton distribution function gives the probabilityof finding partons with a given momentum fraction in the incident hadrons (ii) the process-dependent hard scattering cross-section gives the probability that those partons scatter intofinal state partons with specified momenta and (iii) the process-independent parton frag-mentation functions that describe the probability that a final state parton fragments into ajet that includes a charged hadron with transverse momentum pT Functions (i) and (iii) arewell-measured and at high transverse momentum function (ii) is both systematically calcu-lated and well-measured This body of knowledge provides a firm foundation a well-definedbaseline with respect to which we can measure changes if such a hard scattering processoccurs instead in an ultrarelativistic heavy ion collision
In hard scattering processes in which the momentum transfer Q is high enough the par-tonic hard scattering cross section (function (ii) above) is expected to be the same in anultrarelativistic heavy ion collision as in a proton-proton collision This is so because thehard interaction occurs on a timescale and length scale prop 1Q which is too short to resolveany aspects of the hot and dense strongly interacting medium that is created in the samecollision The parton distribution functions (function (i) above) are different in nuclei thanin nucleons but they may be measured in proton-nucleus deuteron-nucleus and electron-nucleus collisions The key phenomenon that is unique to ultrarelativistic nucleus-nucleuscollisions is that after a very energetic parton is produced unless it is produced at the edgeof the fireball heading outwards it must propagate through as much as 5-10 fm of the hotand dense medium produced in the collision These hard partons therefore serve as well-calibrated probes of the strongly coupled plasma whose properties we are interested in Thepresence of the medium results in the hard parton losing energy and changing the direc-tion of its momentum The change in the direction of its momentum is often referred to asldquotransverse momentum broadeningrdquo a phrase which needs explanation ldquoTransverserdquo heremeans perpendicular to the original direction of the hard parton (This is different from pT the component of the (original) momentum of the parton that is perpendicular to the beamdirection) ldquoBroadeningrdquo refers to the effect on a jet when the directions of the momenta ofmany hard partons within it are kicked averaged over many partons in one jet or perhapsin an ensemble of jets there is no change in the mean momentum but the spread of themomenta of the individual partons broadens
Because the rates for hard scattering processes drop rapidly with increasing pT energy losstranslates into a reduction in the number of partons produced with a given pT (Partons withthe given pT must have been produced with a higher pT and are therefore rarer than theywould be in proton-proton collisions) Transverse momentum broadening is expected to leadto more subtle modifications of jet properties Furthermore the hard parton dumps energyinto the medium which motivates the use of observables involving correlations between softfinal state hadrons and a high momentum hadron Most generally ldquojet quenchingrdquo refers tothe whole suite of medium-induced modifications of high-pT processes in heavy ion collisionsand modifications of the medium in heavy ion collisions in which a high-pT process occursall of which have their origin in the propagation of a highly energetic parton through thestrongly coupled plasma
The most pictorial although not the most generic manifestation of jet quenching in heavy
32
0
05
1
15
2 (a)
0 2 4
0
01
02(b)
(c)
-1 0 1
(d)
015 lt p lt 4 GeVc
2 lt p lt 4 GeVc|lt10| ppAu+Au 5 |lt05| pp
Au+Au 5
Au+Au 5 |lt10)$-(|
d
chdN
trig
N1 o
r
dch
dN tri
gN1
Figure 29 Dihadron azimuthal and longitudinal two-particle correlations in RHIC collisions withradics = 200 GeV in the STAR detector for high trigger pT 4 lt ptrigT lt 6 GeV and low (panels (a) and
(c)) and intermediate (panels (b) and (d)) associated transverse momenta In each panel data fromnearly head-on heavy ion collisions (red) are compared to data from proton-proton collisions (black)Panels (a) and (b) show the correlation in ∆φ the difference in azimuthal angle of the two particlesPanels (c) and (d) show the correlation in pseudorapidity for those particles that are close to eachother in azimuth (red and black points) or close to back to back (blue points) Figure taken from[123]
33
ion collisions at RHIC is provided by the data in Fig 29 In this analysis one selects eventsin which there is a hadron in the final state with pT gt 4 GeV the trigger hadron Thedistribution of all other hadrons with pT gt 2 GeV in azimuthal angle relative to the triggerhadron is shown in panel (b) In proton-proton collisions we see peaks at 0 and π radianscorresponding to the jet in which the trigger hadron was found and the jet produced backto back with it In contrast in gold-gold collisions the away-side jet is missing In panel (a)this analysis is repeated for all hadrons not just those with pT gt 2 GeV Here we see anenhancement in gold-gold collisions on the away side The picture that these data convey isthat we have triggered on events in which one jet escapes the medium while the parton goingthe other way loses so much energy that it produces no hadrons with pT gt 2 GeV dumpingits energy instead into soft particles The jet has been quenched and the medium has beenmodified This picture is further supported by the longitudinal rapidity distribution shown inFig 29 (c) and (d) On the near-side the rapidity distribution shows the shape of a jet-likestructure On the away side however one finds an enhanced multiplicity distribution whoseshape over many units in pseudo-rapidity is consistent with background This measurementis illustrative but not generic in the sense that it depends on the choice of pT cuts used in theanalysis A more generic consequence of jet quenching equally direct but less pictorial thanthat in Fig 29 is provided by simply measuring the diminution of the number of high-pThadrons observed in heavy ion collisions We shall turn to this momentarily
At the time of writing the first results on jet quenching in heavy ion collisions atradics =
276 TeV at the LHC are just becoming available [124 125] It is too soon to offer a quan-titative analysis in a review such as this one but it is clear from these early results that jetquenching remains strong even for the much higher energy jets that are produced in hardparton-parton scattering at these much higher collision energies And in LHC collisions ev-idence of highly asymmetric dijets [124] as if one parton escaped relatively unscathed whileits back-to-back partner was very significantly degraded by the presence of the medium cannow be obtained from single events rather than statistically as in Fig 29
231 Single inclusive high-pt spectra and ldquojetrdquo measurements
The RHIC heavy ion program has established that the measurement of single inclusivehadronic spectra yields a generic manifestation of jet quenching Because the spectra inhadron-hadron collisions are steeply falling functions of pT if the hard partons produced in aheavy ion collision lose energy as they propagate through the strongly coupled plasma shift-ing the spectra leftward mdash to lower energy mdash is equivalent to depressing them This effect isquantified via the measurement of the nuclear modification factor RhAB which characterizeshow the number of hadrons h produced in a collision between nucleus A and nucleus B differsfrom the number produced in an equivalent number of proton-proton collisions
RhAB(pT η centrality) =
dNABrarrhmedium
dpT dη
〈NABcoll 〉
dNpprarrhvacuum
dpT dη
(224)
Here 〈NABcoll 〉 is the average number of inelastic nucleon-nucleon collisions in A-B collisions
within a specified range of centralities This number is typically determined by inferring thetransverse density distribution of nucleons in a nucleus from the known radial density profile
34
Figure 210 RAA for neutral pions as a function of pT for central (left) and peripheral (right)collisions Data taken from [126]
of nuclei and then calculating the average number of collisions with the help of the inelasticnucleon-nucleon cross section This so-called Glauber calculation can be checked experimen-tally by independent means for instance via the measurement of the nuclear modificationfactor for photons discussed below
The nuclear modification factor depends in general on the transverse momentum pT andpseudo-rapidity η of the particle the particle identity h the centrality of the collision andthe orientation of the particle trajectory with respect to the reaction plane (which is oftenaveraged over) IfRAB deviates from 1 this reflects either medium effects or initial state effectsmdash the parton distributions in A and B need not be simply related to those in correspondinglymany protons Measurements of RdA in deuteron-A collisions mdash which is a good proxy forRpA mdash are used to determine whether an observed deviation of RAA from 1 is due to initialstate effects or the effects of parton energy loss in medium
At mid-rapidity RHIC data on RAuAu (which is often written as RAA) show the followinggeneric features
1 Characteristic strong centrality dependence of RAABy varying the centrality of a heavy ion collision one changes the typical in-mediumpath length over which hard partons produced in these collisions must propagatethrough the dense matter For the most central head-on collisions (eg 0-10 central-ity) the average L is large for a peripheral collision (eg 80-92 centrality) the aver-age L is small RHIC data (see Fig 210) show that for the most peripheral centralitybin the nuclear modification factors are consistent with the absence of medium-effectswhile RAA decreases monotonically with increasing centrality and reaches about 02 mdashsuppression by a factor of five mdash for the most central collisions [127ndash130] The leftpanel in Fig 210 is a direct manifestation of jet quenching 80 of the hard π0rsquos thatwould be seen in the absence of a medium are gone
2 Jet quenching is not observed in RdAuIn deuteron-gold collisions RdAu is consistent with or greater than 1 for all centralities
35
and all transverse momenta4 Jet quenching is not observed In fact the centralitydependence is opposite to that seen in gold-gold collisions with RdAu reaching maximalvalues of around 15 for pT = 3minus 5 GeVc in the most central collisions [128130] Thehigh-pT hadrons are measured at or near mid-rapidity meaning that they are wellseparated from the fragments of the struck gold nucleus And d-Au collisions do notproduce a medium in the final state In these collisions therefore the partons producedin hard scattering processes and tallied in RdAu do not have to propagate through anymatter after they are produced The fact that RdAu is consistent with or greater than1 in these collisions therefore demonstrates that the jet quenching measured in RAuAuis attributable to the propagation of the hard partons produced in heavy ion collisionsthrough the medium that is present only in those collisions
3 Photons are not quenchedFor single inclusive photon spectra the nuclear modification factor shows only milddeviations from RγAuAu asymp 1 [131] Within errors these are consistent with perturba-tive predictions that take into account the nuclear modifications of parton distributionfunctions (mainly the isospin difference between protons and nuclei) [132] Since pho-tons unlike partons or hadrons do not interact strongly with the medium this givesindependent support that the jet quenching observed in heavy ion collisions is a finalstate effect And it provides experimental evidence in support of the Glauber-typecalculation of the factor 〈NAA
coll 〉 in (224) discussed above
4 Apparently pT -independent and species-independent suppression of RAA at high pT At high transverse momentum pT gt 7 GeV the suppression factor is approximatelyindependent of pT (see Fig 210) in all centrality bins [60 133 134] RAuAu is sup-pressed all the way out to the highest pT in Fig 210 pT that are high enough thathadronization of the parton must occur only far outside the medium5 Moreover RhAuAuis independent of the species of the hadron h [135] Either of these observations elimi-nates the possibility that hadrons are formed within the medium and then lose energyupon propagating through the medium since different hadrons would have differentcross sections for interaction with the medium This data supports the picture that theorigin of the observed suppression is energy loss by a parton propagating through themedium prior to its hadronization
5 RAA for heavy-flavored and light-flavored hadrons is comparableThe current experiments at RHIC measure charm and bottom quarks via their decayproducts typically via the measurement of single electrons produced in the weak decaysof those quarks [136] These analyses cannot separate the contributions from charm andbottom quarks since both have decay modes containing single electrons The interestin the dependence of parton energy loss on parton mass arises from the fact that modelspredict a significant dependence It is a matter of ongoing discussion to what extent the
4While their physical origin is not unambiguously constrained the small deviations from unity observed inRdAu reflect the presence of initial state effects in the colliding nuclei For instance any transverse momentumbroadening of partons prior to hard processes is expected to deplete RdAu at low pT and enhance it at higher pT Alternatively modifications in the nuclear parton distribution functions could also lead to such enhancement
5We shall see later that the fact that the suppression of RAA is almost pT -independent arises from acombination of various effects Here we stress only the fact that it is small all the way out to the highest pT
36
uncertainties in the existing data are already small enough to put interesting contraintson models of parton energy loss More progress can be expected in the near future oncedetector upgrades at RHIC and measurements at the LHC allow for differentiation ofbottom quarks via their displaced decay vertices
In short these observations support a picture in which highly energetic partons are pro-duced in high momentum transfer processes within the medium as if they were in the vacuumbut where these partons subsequently lose a significant fraction of their initial energy dueto interactions with the medium Jet quenching is a partonic final state effect that dependson the length of the medium through which the parton must propagate It is expected tohave many consequences in addition to the strong suppression of single inclusive hadronspectra which tend to be dominated by the most energetic hadronic fragments of parentpartons Rather the entire parton fragmentation process is expected to be modified withconsequences for observables including multi-particle jet-like correlations and calorimetric jetmeasurements (The early results from heavy ion collisions at the LHC [124 125] confirmthe expectation that at high enough jet energy calorimetric jet observables are modified bythe presence of the medium [124]) We shall not review recent experimental and theoreticalefforts to characterize multi-particle effects of jet quenching here but we shall touch uponthem briefly in Section 76 Furthermore the energy deposited into the medium by the en-ergetic parton also has interesting and potentially observable effects We shall discuss thesein Section 73 the interested reader is also referred to the literature [137138]
232 Analyzing jet quenching
For concreteness we shall focus in this section on those aspects of the analysis of jet quenchingthat bear upon the calculation of the nuclear modification factor RAA defined in (224)We shall describe other aspects of the analysis of jet quenching more briefly as needed insubsequent sections The single inclusive hadron spectra which define RAA are typicallycalculated upon assuming that the modification of the spectra in nucleus-nucleus collisionsrelative to that in proton-proton collisions arises due to parton energy loss This assumptionis well supported by data as we have described above But from a theoretical point of viewit is an assumption not backed up by any formal factorization theorem Upon making thisassumption we write
dσAArarrh+rest(med) =
sumf
dσAArarrf+X(vac) otimes Pf (∆EL q )otimesD(vac)
frarrh(z micro2F ) (225)
Here
dσAArarrf+X(vac) =
sumijk
fiA(x1 Q2)otimes fjA(x2 Q
2)otimes σijrarrf+k (226)
and fiA(xQ2) are the nuclear parton distribution functions and σijrarrf+k are the pertur-batively calculable partonic cross sections The medium dependence enters via the functionPf (∆EL q ) which characterizes the probability that a parton f produced with crosssection σijrarrf+k loses energy ∆E while propagating over a path length L in a medium Thisprobability depends of course on properties of the medium which are represented schemati-cally in this formula by the symbol q the jet quenching parameter We shall see below that
37
in the high parton energy limit the properties of the medium enter Pf only through oneparameter and in that limit q can be defined precisely At nonasymptotic parton energiesq in (225) is a place-holder representing all relevant attributes of the medium It is often
conventional to refer to the combination of Pf and D(vac)frarrh together as a modified fragmen-
tation function It is only in the limit of high parton energy where one can be sure thatthe parton emerges from the medium before fragmenting into hadrons in vacuum that thesetwo functions can be cleanly separated as we have done in (225) This aspect of the ansatz(225) is supported by the data as we have described above all hadrons exhibit the samesuppression factor indicating that RAA is due to partonic energy loss before hadronization
The dynamics of how parton energy is lost to the medium is specified in terms of the prob-ability Pf (∆EL q ) In the high parton energy limit the parton loses energy dominantlyby inelastic processes that are the QCD analogue of bremsstrahlung the parton radiatesgluons as it interacts with the medium It is a familiar fact from electromagnetism thatbremsstrahlung dominates the loss of energy of an electron moving through matter in the highenergy limit The same is true in calculations of QCD parton energy loss in the high-energylimit as established first in Refs [139ndash141] The hard parton undergoes multiple inelasticinteractions with the spatially extended medium and this induces gluon bremsstrahlungHere and throughout by the high parton energy limit we mean the combined set of limitsthat can be summarized as
E ω |k| |q| equiv |sumi
qi| T ΛQCD (227)
where E is the energy of the high energy projectile parton where ω and k are the typicalenergy and momentum of the gluons radiated in the elementary radiative processes q rarr qgor g rarr gg and where q is the transverse momentum (transverse to its initial direction)accumulated by the projectile parton due to many radiative interactions in the mediumand where T and ΛQCD represent any energy scales that characterize the properties of themedium itself This set of approximations underlies all analytical calculations of radiativeparton energy loss to date [140ndash145] The premise of the analysis is the assumption thatQCD at scales of order |k| and |q| is weakly coupled even if the medium (with its lowercharacteristic energy scales of order T and ΛQCD) is strongly coupled We shall spend mostof this section on the analysis valid in this high parton energy limit in which case all we needask of analyses of strongly coupled gauge theories with gravity duals is insight into thoseproperties of the strongly coupled medium that enter into the calculation of jet quenchingin QCD However this analysis based upon the limits (227) may not be under quantitativecontrol when applied to RHIC data since at RHIC the partons in question have energies ofat most a few tens of GeV meaning that one can question whether all the scales separatedby in (227) are in fact well-separated We shall close this section with a brief look atelastic scattering as an example of an energy loss process that is relevant when (227) is notsatisfied And in Section 71 we shall see that at low enough energies that physics at allscales in the problem all the way up to E is strongly coupled (or in a conformal theory likeN = 4 SYM in which physics is strongly coupled at all scales) new approaches are neededThe analysis based upon (227) that we focus on in this section will be under better controlwhen applied to LHC data since in heavy ion collisions at the LHC partons with pT all theway up to a few hundred GeV will be produced and therefore available as probes
38
The analysis of radiative energy loss starts from (and extends) the eikonal formalism sowe must begin with a few ideas and some notation from this approach (for a self-containedintroduction and references to earlier work see eg Ref [146147]) As seen by a high energyparton a target that is spatially extended but of finite thickness appears Lorentz contractedso in the projectile rest frame the parton propagates through the target in a short period oftime and the transverse position of the projectile does not change during the propagationSo at ultra-relativistic energies the main effect of the target on the projectile is a ldquorotationrdquoof the partonrsquos color due to the color field of the target These rotation phases are given byWilson lines along the (straight line) trajectories of the propagating projectile
W (x) = P expiintdzminusT aA+
a (x zminus) (228)
Here x is the transverse position of the projectile mdash which does not change as the partonpropagates at the speed of light along the zminus equiv (z minus t)
radic2 lightlike direction A+ is the
large component of the target color field and T a is the generator of SU(N) in the represen-tation corresponding to the given projectile mdash fundamental if the hard parton is a quarkand adjoint if it is a gluon The eikonal approach to scattering treats the (unphysical inthe case of colored projectiles) setting in which the projectile impinges on the target fromoutside after propagating for an arbitrarily long time and building up a fully developedWeizsacker-Williams field proportional to g xi
x2 (a coherent state cloud of gluons dressing thebare projectile) The interaction of this dressed projectile with the target results in an eikonalphase (Wilson line) for the projectile itself and for each gluon in the cloud Gluon radiationthen corresponds to the decoherence of components of the dressed projectile that pick updifferent phases Analysis of this problem yields a calculation of Nprod(k) the number ofradiated gluons with momentum k with the result [146147]
Nprod(k) =
αsCF2π
intdx dy eikmiddot(xminusy) x middot y
x2 y2
[1minus 1
N2 minus 1〈Tr
[WA dagger(x)WA(0)
]〉
minus 1
N2 minus 1〈Tr
[WA dagger(y)WA(0)
]〉
+1
N2 minus 1〈Tr
[WA dagger(y)WA(x)
]〉
] (229)
where the CF prefactor is for the case where the projectile is a quark in the fundamentalrepresentation where the projectile is located at transverse position 0 and where the 〈 〉denotes averaging over the gluon fields of the target If the target is in thermal equilibriumthese are thermal averages
Although the simple result (229) is not applicable to the physically relevant case as we shalldescribe in detail below we can nevertheless glean insights from it that will prove relevant Wenote that the entire medium-dependence of the gluon number spectrum (229) is determinedby target expectation values of the form 〈Tr
[WA dagger(x)WA(y)
]〉 of two eikonal Wilson lines
The jet quenching parameter q that will appear below defines the fall-off properties of this
39
correlation function in the transverse direction L equiv |xminus y|
〈Tr[WA(C)
]〉 asymp exp
[minus 1
4radic
2q Lminus L2
](230)
in the limit of small L with Lminus (the extent of the target along the zminus direction) assumedlarge but finite [148 149] Here the contour C traverses a distance Lminus along the light coneat transverse position x and it returns at transverse position y These two long straightlightlike lines are connected by short transverse segments located at zminus = plusmnLminus2 far outsidethe target We see from the form of (229) that |k| and L are conjugate the radiation ofgluons with momentum |k| is determined by Wilson loops with transverse extent L sim 1|k|This means that in the limit (227) the only property of the medium that enters (229) is qFurthermore inserting (230) into (229) yields the result that the gluons that are producedhave a typical k2 that is of order qLminus This suggests that q can be interpreted as thetransverse momentum squared picked up by the hard parton per distance Lminus that it travelsan interpretation that can be validated more rigorously via other calculations [150151]
The reason that the eikonal formalism cannot be applied verbatim to the problem of par-ton energy loss in heavy ion collisions is that the high energy partons we wish to study donot impinge on the target from some distant production site They are produced within thesame collision that produces the medium whose properties they subsequently probe As aconsequence they are produced with significant virtuality This means that even if there wereno medium present they would radiate copiously They would fragment in what is knownin QCD as a parton shower The analysis of medium-induced parton energy loss then re-quires understanding the interference between radiation in vacuum and the medium-inducedbremsstrahlung radiation It turns out that the resulting interference resolves longitudinaldistances in the target [140ndash143] meaning that its description goes beyond the eikonal ap-proximation The analysis of parton energy loss in the high energy limit (227) must includeterms that are subleading in 1E and therefore not present in the eikonal approximationthat describe the leading interference effects To keep these O(1E) effects one must replaceeikonal Wilson lines by retarded Greenrsquos functions that describe the propagation of a particlewith energy E from position zminus1 x1 to position zminus2 x2 without assuming x1 = x2 [141152153](In the E rarrinfin limit x1 = x2 and the eikonal Wilson line is recovered) It nevertheless turnsout that even after Wilson lines are replaced by Greenrsquos functions the only attribute of themedium that arises in the analysis in the limit (227) is the jet quenching parameter qdefined in (230) that already arose in the eikonal approximation
We shall not present the derivation but it is worth giving the complete (albeit somewhatformal) result for the distribution of gluons with energy ω and transverse momentum k thata high energy parton produced within a medium radiates
ωdI
dω dk=
αsCR(2π)2 ω2
2Re
int infinξ0
dyl
int infinyl
dyl
intdu eminusikmiddotu exp
[minus1
4
int infinyl
dξ q(ξ) u2
]times part
partxmiddot partpartu
int uequivr(yl)
xequivr(yl)equiv0Dr exp
[int yl
yl
dξ
(i ω
2r2 minus 1
4q(ξ)r2
)] (231)
We now walk through the notation in this expression The Casimir operator CR is in therepresentation of the projectile parton The integration variables ξ yl and yl are all positions
40
along the zminus lightcone direction ξ0 is the zminus at which the projectile parton was created ina hard scattering process Since we are not taking this to minusinfin the projectile is not assumedon shell The projectile parton was created at the transverse position x = 0 The integrationvariable u is also a transverse position variable conjugate to k The path integral is overall possible paths r(ξ) going from r(yl) = 0 to r(yl) = u The derivation of (231) proceedsby writing dIdω dk in terms of a pair of retarded Greenrsquos functions in their path-integralrepresentations one of which describes the radiated gluon in the amplitude radiated at yland the other of which describes the radiated gluon in the conjugate amplitude radiated atyl The expression (231) then follows after a lengthy but purely technical calculation [142]The properties of the medium enter (231) only through the jet quenching parameter q(ξ)The compact expression (231) has been derived in the so-called path-integral approach [141]For other related formulations of QCD parton energy loss we refer the reader to Refs [83144145150154ndash174]
The notation q(ξ) allows for the possibility that the nature of the medium and thus its qchanges with time as the hard parton propagates through it If we approximate the mediumas unchanging q(ξ) is just a constant In the strongly expanding medium of a heavy ioncollision however this is not a good approximation It turns out that the path-integral in(231) can be solved analytically in the saddle point approximation for quenching parametersof the form q(ξ) = q0 (ξ0ξ)
α [175] Here the free exponent α may be chosen in modelcalculations to take values characterizing a one-dimensional longitudinal expansion with so-called Bjorken scaling (α = 1) or scenarios which also account for the transverse expansion1 lt α lt 3 Remarkably one finds that irrespective of the value of α for fixed in-medium pathlength Lminus
radic2 the transverse momentum integrated gluon energy distribution (231) has the
same ω-dependence if q(ξ) is simply replaced by a constant given by the linear line-averagedtransport coefficient [176]
〈q〉 equiv 1
2Lminus2
int ξ0+Lminus
ξ0
dξ (ξ minus ξ0) q(ξ) (232)
In practice this means that comparisons of different parton energy loss calculations to datacan be performed as if the medium were static The line-averaged transport coefficient 〈q〉determined in this way can then be related via (232) to the transport coefficient at a giventime once a model for the expansion of the medium is specified Hence we can continue ourdiscussion for the case q(ξ) = q without loss of generality
The result (231) is both formal and complicated However its central qualitative conse-quences can be characterized almost by dimensional analysis All dimensionful quantities canbe scaled out of (231) if ω is measured in units of the so-called characteristic gluon energy
ωc equiv q(Lminus)2 (233)
and the transverse momentum k2 in units of qLminus [177] In a numerical analysis of (231) onefinds that the transverse momentum distribution of radiated gluons scales indeed with qLminusas expected for the transverse momentum due to the Brownian motion in momentum spacethat is induced by multiple small angle scatterings If one integrates the gluon distribution(231) over transverse momentum and takes the upper limit of the k-integration to infinityone recovers [177] an analytical expression first derived by Baier Dokshitzer Mueller Peigne
41
and Schiff [140]
ωdIBDMPS
dω=
2αsCRπ
ln
∣∣∣∣∣cos
[(1 + i)
radicωc2ω
]∣∣∣∣∣ (234)
which yields the limiting cases
ωdIBDMPS
dω 2αsCR
π
radicωc2ω for ω ωc
112
(ωcω
)2for ω ωc
(235)
for small and large gluon energies In the soft gluon limit the BDMPS spectrum (234)displays the characteristic 1
radicω dependence which persists up to a gluon energy of the
order of the characteristic gluon energy (233) Hence ωc can be viewed as an effectiveenergy cut-off above which the contribution of medium-induced gluon radiation is negligibleThese analytical limits provide a rather accurate characterization of the full numerical resultIn particular one expects from the above expressions that the average parton energy loss〈∆E〉 obtained by integrating (231) over k and ω is proportional to prop
int ωc0 dω
radicωcω prop ωc
One finds indeed
〈∆E〉BDMPS equivint infin
0dω ω
dIBDMPS
dω=αsCR
2ωc (236)
This is the well-known (Lminus)2-dependence of the average radiative parton energy loss [140
141 178] In summary the main qualitative properties of the medium-induced gluon energydistribution (231) are the scaling of k2 with q Lminus dictated by Brownian motion in trans-verse momentum space the 1
radicω dependence of the k-integrated distribution characteristic
of the non-abelian Landau-Pomerantschuk-Migdal (LPM) effect and the resulting (Lminus)2-
dependence of the average parton energy loss
Medium-induced gluon radiation modifies the correspondence between the initial partonmomentum and the final hadron momenta We now sketch how the resulting P (∆E) in (225)can be estimated If gluons are emitted independently P (∆E) is the normalized sum of theemission probabilities for an arbitrary number of n gluons which carry away the total energy∆E [179]
P (∆E) = exp
[minusint infin
0dω
dI
dω
] infinsumn=0
1
n
[nprodi=1
intdωi
dI(ωi)
dω
]δ
(∆E minus
nsumi=1
ωi
) (237)
Here the factor exp[minusintinfin
0 dωdIdω
]denotes the probability that no energy loss occurs This
factor ensures that P (∆E) is properly normalizedintd∆E P (∆E) = 1 The mean energy
loss is then
〈∆E〉 =
intd∆E (∆E)P (∆E) =
intdωω
dI
dω (238)
as above but it turns out that P (∆E) is a very broad distribution not peaked around itsmean In particular as seen from Fig 21 single inclusive hadron spectra are distributionswhich fall steeply with pT The modifications which parton energy loss induce on suchdistributions cannot be characterized by an average energy loss Rather what matters fora steeply falling distribution is not how much energy a parton loses on average but ratherwhich fraction of all the partons gets away with much less than the average energy loss [179]
42
Figure 211 Data for the nuclear modification factor of π0 in central 0 minus 5 Au+Au collisionscompared to a model calculation of jet quenching with 〈q〉 values of 03 09 12 15 21 29 44 5974 103 132 177 250 405 and 1014 GeV2fm Data taken from [180]
This so-called trigger bias effect is quantitatively very important and can be accounted forby the probability distribution (237)
Making contact with phenomenology requires the implementation of P (∆E) in a modelin which hard scattering events are distributed with suitable probability at locations in thetransverse plane with the hard partons produced going in random back-to-back directionsand in which a hydrodynamic calculation is used to model how the medium in which the hardpartons find themselves evolves subsequently as the partons propagate through it One suchmodel is the PQM model [181] which calculates the hadronic spectrum (225) by interfacingthe probability distributions P (∆E) of Ref [177] with a model of the geometrical distributionof matter in the collision region Although the PQM model is built upon the gluon energydistribution (235) which gives the leading contribution to parton energy loss in the highenergy limit (227) it is nevertheless a model in several senses beyond its treatment of thedistribution of energy density in space and time For example the factorized ansatz of theinclusive cross section (225) is not justified from first principles but is an experimentallytestable assumption Also the form (237) of the probability distribution P (∆E) assumesthat in the case of multiple gluon emissions the consecutive energy degradation of the leadingparton is negligible For this to be a good approximation the total energy loss must be muchsmaller than the projectile energy ∆E E All these aspects of the model are expected tobecome reliable in the high energy limit (227) The results of a comparison of this model todata done by the PHENIX collaboration are shown in Fig 211 We see that for a suitablechoice of the line-averaged transport coefficient 〈q〉 defined in (232) the calculation doesa reasonable job of describing the data In this analysis the approximate pT -independenceof RAuAu seen in the data arises from an interplay between the rapidly dropping partonproduction cross-section at pT rsquos exceeding sim
radics10 the fact that there are always some
partons produced near the edge of the fireball that emerge relatively unscathed and the factthat P (∆E) is significant even for ∆E = 0 even though the mean ∆E is large The PHENIX
43
authors quote for this PQM model study a jet quenching parameter which is constrainedby the experimental data as 132+21
minus32 and 132+63minus52 GeV2fm at the one and two standard
deviation levels respectively
More recently the quantitative comparison between data from RHIC and the jet quenchingcalculations that we have sketched in this section has been revisited in Ref [167] Theseauthors use data on RAA as in Fig 211 but in addition they use data on dihadron correlationsas well as data on heavy quark suppression And they include a hydrodynamical modellingof the expanding and cooling plasma They parametrize the jet quenching parameter as
q = 2Kε34 (239)
where ε is the energy density of the medium through which the energetic partons are propa-gating and where K is a parameter to be obtained via comparing jet quenching calculationsto data (In a weakly coupled quark-gluon plasma K asymp 1 [156]) ε varies with position anddecreases with time according to a hydrodynamical calculation These authors find that theyare able to obtain a more stable fit to the value of K than to a time-averaged q Fitting todata from both PHENIX and STAR from RHIC collisions at
radics = 200 GeV they obtain
K = 41plusmn 06 (240)
We will compare this result to calculations done for strongly coupled plasmas in gauge theorieswith dual gravitational descriptions in Section 75
There is currently a vigorous debate about whether hard partons produced (and thenquenched) in RHIC collisions have a high enough energy E for the approximation (227) tobe under control and for the above-mentioned model assumptions to be valid The questionof whether the comparison of RHIC data with different model implementations of partonenergy loss yield numerically consistent results is also still debated see eg [180] Herewe solely remark that while radiative energy loss is known to dominate in the high energylimit other mechanisms and in particular elastic energy loss may contribute significantly atlower energies and this may not be negligible at RHIC [160 182 183] In radiative energyloss in the high energy limit the longitudinal momentum of the incident parton is sharedbetween the longitudinal momentum of the radiated gluon and that of the outgoing highenergy parton Longitudinal momentum transfer to the medium is negligible in comparisonto that carried by the radiated gluon In elastic scattering however the only energy lostby the projectile parton is that due to transferring longitudinal momentum to the mediumWhat is clear from the outset is that if elastic mechanisms contribute then parton energyloss must depend on more attributes of the medium than just q since q knows nothing abouthow ldquobitsrdquo of the medium recoil longitudinally when struck Note that this discussion goesthrough unchanged even in the context of a medium that is a strongly coupled fluid with noidentifiable scattering centers If the medium is weakly coupled then the conclusion can bestated more precisely q does not differentiate between a medium made of heavy scatterersthat hardly recoil and one made of light scatterers that recoil more easily
24 Quarkonia in hot matter
One way of thinking about the operational meaning of the statement that quark-gluon plasmais deconfined is to ask what prevents the formation of a meson within quark-gluon plasma
44
The answer is that the attractive force between a quark and an antiquark which are separatedby a distance of order the size of a meson is screened by the presence of the quark-gluonplasma between them This poses a quantitative question how close together do the quarkand antiquark have to be in order for their attraction not to be screened How close togetherdo they have to be in order for them to feel the same attraction that they would feel if theywere in vacuum It was first suggested by Matsui and Satz [184] in 1986 that measurementsof how many quarkonia mdash mesons made of a heavy quark-antiquark pair mdash are produced inheavy ion collisions could be used as a tool with which to answer this question because theyare significantly smaller than typical mesons or baryons
The generic term quarkonium refers to the charm-anticharm or charmonium mesons (JΨΨprime χc ) and the bottom-antibottom or bottomonium mesons (Υ Υprime ) The firstquarkonium state that was discovered was the 1s state of the c c bound system the JΨIt is roughly half the size of a typical meson like the ρ The bottomonium 1s state the Υis smaller again by roughly another factor of two It is therefore expected that if one canstudy quark-gluon plasma in a series of experiments with steadily increasing temperatureJψ mesons survive as bound states in the quark-gluon plasma up to some dissociationtemperature that is higher than the crossover temperature (at which generic mesons andbaryons made of light quarks fall apart) More realistically what Matsui and Satz suggestedis that if high energy heavy ion collisions create deconfined quark-gluon plasma that is hotenough then color screening would prevent charm and anticharm quarks from binding to eachother in the deconfined interior of the droplet of matter produced in the collision and as aresult the number of JΨ mesons produced in the collisions would be suppressed HoweverΥs should survive as bound states to even higher temperatures until the quark-antiquarkattraction is screened even on the short length scale corresponding to their size
To study this effect Matsui and Satz suggested comparing the temperature dependence ofthe screeing length for the quark-antiquark force which can be obtained from lattice QCDcalculations with the JΨ meson radius calculated in charmonium models They then dis-cussed the feasibility to detect this effect clearly in the mass spectrum of e+ eminus dilepton pairsBetween 1986 when Matsui and Satz launched this line of investigation suggesting it as aquantitative means of characterizing the formation and properties of deconfined matter andtoday we know of no other measurement that has been advocated as a more direct experimen-tal signature for the deconfinement transition And there is hardly any other measurementwhose phenomenological analysis has turned out to be more involved In this subsection weshall describe both the appeal of studying quarkonia in the hot matter produced in heavyion collisions and the practical difficulties The theoretical basis for the argument of Matsuiand Satz has evolved considerably within the last two decades [185] Moreover the debateover how to interpret these measurements is by now informed by data on JΨ-suppression innucleus-nucleus collisions at the CERN SPS [186187] at RHIC [188] and at the LHC [189]There is also a good possibility that qualitatively novel information will become accessible infuture high statistics runs at RHIC and in heavy ion collisions at the LHC
A sketch of the basic idea of Matsui and Satz is shown in Fig 212 In very general termsone expects that the attractive interaction between the heavy quark and anti-quark in aputative bound state is sensitive to the medium in which the heavy particles are embeddedand that this attraction weakens with increasing temperature If the distance between the
45
Q QQQr
Figure 212 Schematic picture of the dissociation of a QQ-pair in hot QCD matter due to colorscreening Figure taken from Ref [185] The straight black lines attached to the heavy Q and Qindicate that these quarks are external probes in contrast to the dynamical quarks within the quark-gluon plasma
heavy quark and anti-quark is much smaller than 1T there will not be much quark-gluonplasma between them Equivalently typical momentum scales in the medium are of orderthe temperature T and so the medium cannot resolve the separation between the quark-antiquark pair if they are much closer together than 1T However if the distance is largerthen the bound state is resolved and the color charges of the heavy quarks are screenedby the medium see Fig 212 In a heavy ion collision in which quark-gluon plasma with atemperature T is created only those quarkonia with radii that are smaller than some lengthscale of order 1T can form These basic arguments support the idea that quarkoniumproduction rates are an indicator of whether quark-gluon plasma is produced and at whattemperature
In section 33 we review lattice calculations of the heavy quark static free energy FQQ(r)This static potential is typically defined via how the correlation function of a pair of Polyakovloops namely test quarks at fixed spatial positions whose worldlines wrap around the periodicEuclidean time direction falls off as the separation between the test quarks is increased Thisstatic potential is renormalized such that it matches the zero temperature result at smalldistances Calculations of FQQ(r) were the earliest lattice results which substantiated thecore idea that a quarkonium bound state placed in hot QCD matter dissociates (ldquomeltsrdquo)above a critical temperature As we now discuss phenomenological models of quarkonium inmatter are based upon interpreting FQQ(r) as the potential in a Schrodinger equation whoseeigenvalues and eigenfunctions describe the heavy Q-Q bound states There is no rigorousbasis for this line of reasoning and if pushed too far it faces various conceptual challengesas we shall discuss in Section 33 However these models remain valuable as a source ofsemi-quantitative intuition
At zero temperature lattice results for FQQ(r) in QCD without dynamical quarks arewell approximated by the ansatz FQQ(r) = σ r minus α
r where the linear term that dominates
46
at long distance is characterized by the string tension σ 02 GeV2 and the perturbativeCoulomb term αr is dominant at short distances In QCD with dynamical quarks beyondsome radius rc the potential flattens because as the distance between the external Q and Qis increased it becomes energetically favorable to break the color flux tube connecting themby producing a light quark-antiquark pair from the vacuum which in a sense screens thepotential With increasing temperature the distance rc decreases that is the colors of Qand Q are screened from each other at increasingly shorter distances This is seen clearly inFig 212 in Section 33 These lattice results are well parametrized by a screened potentialof the form [185190]
FQQ(r) = minusαr
+ σ r
(1minus eminusmicro r
micro r
) (241)
where micro equiv micro(T ) can be interpreted at high temperatures as a temperature-dependent De-bye screeing mass For suitably chosen micro(T ) this ansatz reproduces the flattening of thepotential found in lattice calculations at the finite large distance value FQQ(infin) = σmicro(T )Taking this QQ free energy FQQ(r T ) as the potential in a Schrodinger equation one maytry to determine which bound states in this potential remain as the potential is weakenedas the temperature increases Such potential model studies have led to predictions of thedissociation temperatures Td of the charmonium family which range from Td(JΨ) 21Tcto Td(Ψ
prime) 11Tc for the more loosely bound and therefore larger 2s state The deeplybound small 1s state of the bottomonium family is estimated to have a dissociation tem-perature Td(Υ(1S))) gt 4Tc while dissociation temperatures for the corresponding 2s and 3sstates were estimated to lie at 16Tc and 12Tc respectively [185190] Because the leap fromthe static quark-antiquark potential to a Schrodinger equation is not rigorously justified theuncertainties in quantitative results obtained from these potential models are difficult to esti-mate (For more details on why this is so see Section 33) However these models with theirinputs from lattice QCD calculations do provide strong qualitative support for the centralidea of Matsui and Satz that quarkonia melt in hot QCD matter and they provide strongsupport for the qualitative expectation that this melting proceeds sequentially with smallerbound states dissociating at a higher temperature
Experimental data on the yield of JΨ mesons in nucleus-nucleus collisions have beenreported by the NA50 [186] and NA60 [187] collaborations at the CERN SPS and by thePHENIX and STAR collaborations at RHIC As one increases the size of the collision system(either by varying the impact parameter selection or by changing the nuclear species used inthe experiments) these yields turn out to be increasingly suppressed relative to the yieldsmeasured in proton-proton or proton-nucleus collisions at the same center of mass energyThe measurements made in proton-nucleus collisions are particularly important to use as abaseline since without them one would not know how much charmonium suppression arisesdue to the interaction between the charm-anticharm pair and ordinary confined hadronicmatter [191] The operational procedure for separating such hadronic phase effects is tomeasure them separately in proton-nucleus collisions [192] and to establish then to whatextent the number of JΨ mesons produced in nucleus-nucleus collisions drops below theyield extrapolated from proton-nucleus collisions [193] We note that for nucleus-nucleuscollisions experimental data exist so far solely for the JΨ The other bound states of thecharmonium family (Ψprime χc ) have not been characterized due to the lack of statistics andresolution Also a characterization of bottomonium states in nuclear matter is missing Due
47
to the much higher production rates for heavy ion collisions at the TeV scale this situationis expected to change at the LHC Moreover the higher luminosity of future RHIC runs maygive some access to the bottomonium system Thus measurements expected in the nearfuture may provide evidence for the sequential quarkonium suppression pattern which is ageneric prediction of all models of quarkonium suppression and which has not yet been tested
The analysis of data on charmonium has to take into account a significant number ofimportant confounding effects Here we cannot discuss the phenomenology of these effectsin detail but we provide a list of the most important ones
1 Contributions from the decays of excited statesIn proton-proton collisions a significant fraction of the observed yield of JΨ mesonsis known to arise from the production of excited states like the Ψprime and χc whichsubsequently decay to JΨ In a nucleus-nucleus collision the suppression of the excitedstates is expected to set in at a lower temperature since these states are larger in sizethan the ground state JΨ In particular it has been proposed that the observedsuppression of the JΨ mesons at RHIC and at the SPS may arise solely from thedissociation of the more loosely bound Ψprime and χc states [190] with the JΨs themselvesremaining bound in the quark-gluon plasma produced in all heavy ion collisions todate Regardless of whether this conclusion turns out to be quantitatively correct it iscertainly apparent that in the absence of separate measurements of the production ofthe excited states any conclusions about the observed JΨ-suppression require carefulmodeling of and inferences about the contribution of the decays of excited states
2 Collective dynamics of the heavy ion collision ldquoexplosive expansionrdquoLattice calculations are done for heavy quark bound states which are at rest in a hotstatic medium In heavy ion collisions however even if the droplet of hot matterequilibrates rapidly its temperature drops quickly during the subsequent explosive ex-pansion The observed quarkonium suppression must therefore result from a suitabletime average over a dynamical medium This is challenging in many ways One issuethat arises is the question of how long a bound state must be immersed in a sufficientlyhot heat bath in order to melt Or phrased better how long must the temperaturebe above the dissociation temperature Td in order to prevent an heavy quark and an-tiquark produced at the initial moment of the collision from binding to each other andforming a quarkonium meson
3 Collective dynamics of the heavy ion collision ldquohot windrdquoAnother issue that faces any data analysis is that quarkonium mesons may be producedmoving with significant transverse momentum through the hot medium In their ownreference frame the putative quarkonium meson sees a hot wind Phenomenologicallythe question arises whether this leads to a stronger suppression since the bound statesees some kind of blue-shifted heat bath (an idea which we will refine in Section 77)or whether the bound state is less suppressed since it can escape the heat bath morequickly
4 Formation of quarkonium bound statesNeither quarkonia nor equilibrated quark-gluon plasma are produced at time zero in a
48
heavy ion collision Quarkonia have to form for instance by a colored c c-pair radiatinga gluon to turn into a color-singlet quarkonium state This formation process is notunderstood in elementary interactions or in heavy ion collisions However since theformation process takes time it is a priori unclear whether any observed quarkoniumsuppression is due to the effects of the hot QCD matter on a formed quarkonium boundstate or on the precursor of such a bound state which may have different attenuationproperties in the hot medium And it is unclear whether the suppression is due toprocesses occurring after the liquid-like strongly coupled plasma in approximate localthermal equilibrium is formed or earlier before equilibration
5 Recombination as novel mechanism of quarkonium formationQCD is flavor neutral and thus charm is produced in c c pairs in primary interactions Ifthe average number of pairs produced per heavy ion collision is 1 then all charmoniummesons produced in heavy ion collisions must be made from a c and a c produced inthe same primary interaction At RHIC and even more so at the LHC however morethan one c c pair is produced per collision raising the possibility of a new charmoniumproduction mechanism in which a c and a c from different primary c c-pairs meet andcombine to form a charmonium meson [194] If this novel quarkonium productionmechanism were to become significant it could reduce the quarkonium suppressionor even turn it into quarkonium enhancement However early LHC data on JΨproduction do not show signs of any enhancement [189]
We have included this section about quarkonium in the present review since calculationsbased on the AdSCFT correspondence and reviewed in Sections 77 and 8 have provided com-plementary information for phenomenological modeling in particular by calculating heavyquark potentials within a moving heat bath and by determining mesonic dispersion relationsThe above discussion illustrates the context in which such information is useful but it alsoemphasizes that such information is not sufficient An understanding of quarkonium pro-duction in heavy ion collisions relies on phenomeological modelling as the bridge betweenexperimental observations and the theoretical analysis of the underlying properties of hotQCD matter
49
Chapter 3
Results from lattice QCD
At very high temperature where the QCD coupling constant g(T ) is perturbatively smallhard thermal loop resummed perturbation theory provides a quantitatively controlled ap-proach to QCD thermodynamics However in a wide temperature range around the QCDphase transition which encompasses the experimentally accessible regime perturbative tech-niques become unreliable Nonperturbative lattice-regularized calculations provide the onlyknown quantitatively reliable technique for the determination of thermodynamic propertiesof QCD matter within this regime
We shall not review the techniques by which lattice-regularized calculations are imple-mented We merely recall that the starting point of lattice-regularized calculations at nonzerotemperature is the imaginary time formalism which allows one to write the QCD partitionfunction in Euclidean space-time with a periodic imaginary time direction of length 1T Any thermodynamic quantity can be obtained via suitable differentiation of the partitionfunction At zero baryon chemical potential the QCD partition function is given by theexponent of a real action integrated over all field configurations in the Euclidean space-timeSince the action is real the QCD partition function can then be evaluated using standardMonte Carlo techniques which require the discretization of the field configurations and theevaluation of the action on a finite lattice of space-time points Physical results are obtainedby extrapolating calculated results to the limit of infinite volume and vanishing lattice spac-ing In principle this is a quantitatively reliable approach In practice lattice-regularizedcalculations are CPU-expensive the size of lattices in modern calculations does not exceed483times64 [195] and these calculations nevertheless require the most powerful computing devices(currently at the multi-teraflop scale) In the continuum limit such lattices correspond typ-ically to small volumes of asymp (4 fm)3 [195] This means that properties of QCD matter whichare dominated by long-wavelength modes are difficult to calculate with the currently avail-able computing resources and there are only first exploratory studies For the same reasonit is in practice difficult to carry out calculations using light quark masses that yield real-istically light pion masses Light quarks are also challenging because of the CPU-expensivecomplications which arise from the formulation of fermions on the lattice
In addition to the practical challenges above conceptual questions arise in two importantdomains First at nonzero baryon chemical potential the Euclidean action is no longer realmeaning that the so-called fermion sign problem precludes the use of standard Monte Carlo
50
techniques Techniques have been found that evade this problem but only in the regimewhere the quark chemical potential microB3 is sufficiently small compared to T Second con-ceptual questions arise in the calculation of any physical quantities that cannot be written asderivatives of the partition function Calculating many such quantities that are of consider-able interest requires the analytic continuation of lattice results from Euclidean to Minkowskispace (see below) which is always under-constrained since the Euclidean calculations can onlybe done at finitely many values of the Euclidean time This means that lattice-regularizedcalculations at least as currently formulated are not optimized for calculating transport coef-ficients and answering questions about say far-from-equilibrium dynamics or jet quenching
We allude to these practical and conceptual difficulties to illustrate why alternative strongcoupling techniques including the use of the AdSCFT correspondence are and will remainof great interest for the study of QCD thermodynamics and quark-gluon plasma in heavyion collisions even though lattice techniques can be expected to make steady progress inthe coming years In the remainder of this Section we discuss the current status of latticecalculations of some quantities of interest in QCD at nonzero temperature We shall beginin Section 31 with quantities whose calculation does not run into any of the conceptualdifficulties we have mentioned before turning to those that do
31 The QCD Equation of State from the lattice
The QCD equation of state at zero baryon chemical potential namely the relation between thepressure and the energy density of hot QCD matter is an example of a quantity that is well-suited to lattice-regularized calculation since as a thermodynamic quantity it can be obtainedvia suitable differentiations of the Euclidean partition function And the phenomenologicalmotivation for determining this quantity from QCD from first principles is great since aswe have seen in Section 22 it is the most important microphysical input for hydrodynamiccalculations It illustrates the practical challenges of doing lattice-regularized calculationswith light quarks that we have mentioned above that while accurate calculations of thethermodynamics of pure glue QCD (Nf = 0) have existed for a long time [196] the extractionof the equation of state of quark-gluon plasma with light quarks with their physical massesfrom calculations that have been extrapolated to the continuum limit has become possibleonly recently [6585197]
The current understanding of QCD thermodynamics at the physical point [85] is sum-marized in Fig 31 In the left panel the pressure of QCD matter (in thermal equilibriumwith zero baryon chemical potential) is plotted as a function of its temperature In order toprovide a physically meaningful reference it is customary to compare this quantity to theStefan-Boltzmann result
PSB =8π2
45
(1 +
21
32Nf
)T 4 (31)
for a free gas of noninteracting gluons and massless quarks This benchmark is indicated bythe arrow in the figure As illustrated by this plot the number of degrees of freedom risesrapidly above a temperature Tc sim 170 MeV at higher temperatures the pressure takes analmost constant value which deviates from that of a noninteracting gas of quarks and gluonsby approximately 20 This deviation is still present at temperatures as high as 1 GeV and
51
Figure 31 Results from a lattice calculation of QCD thermodynamics with physical quark masses(Nf = 3 with appropriate light and strange masses) Left panel Temperature dependence of thepressure in units of T 4 Right panel The trace anomaly (εminus 3P ) in units of T 4 Data are for latticeswith the same temporal extent meaning the same temperature but with varying numbers of pointsin the Euclidean time direction Nτ The continuum limit corresponds to taking Nτ rarr infin Figurestaken from Ref [85]
convergence to the noninteracting limit is only observed at asymptotically high temperatures(T gt 108 GeV [198]) which are far from the reach of any collider experiment The right panelshows the trace anomaly ε minus 3P in units of T 4 in the same range of temperatures ε minus 3Pis often called the ldquointeraction measurerdquo but this terminology is quite misleading since bothnoninteracting quarks and gluons on the one hand and very strongly interacting conformalmatter on the other have εminus3P = 0 with εT 4 and PT 4 both independent of temperatureLarge values of (εminus3P )T 4 necessarily indicate significant interactions among the constituentsof the plasma but small values of this quantity should in no way be seen as indicating a lackof such interactions We see in the figure that (εminus 3P )T 4 rises rapidly in the vicinity of TcThis rapid rise corresponds to the fact that εT 4 rises more rapidly than 3PT 4 approachingroughly 80 of its value in an noninteracting gas of quarks and gluons at a lower temperaturebetween 200 and 250 MeV At higher temperatures as 3PT 4 rises toward roughly 80 of itsnoninteracting value (εminus 3P )T 4 falls off with increasing temperature and the quark-gluonplasma becomes more and more conformal Remarkably after a proper re-scaling of thenumber of degrees of freedom and Tc all the features described above remain the same whenthe number of colors of the gauge group is increased and extrapolated to the large Nc rarrinfinlimit [199ndash201]
The central message for us from these lattice calculations of the QCD equation of state isthat at high enough temperatures the thermodynamics of the QCD plasma becomes conformalwhile deviations from conformality are most severe at and just above Tc This suggests thatthe use of conformal theories (in which calculations can be done via gaugegravity duality asdescribed in much of this review) as vehicles by which to gain insights into real-world quark-gluon plasma may turn out to become (even) more quantitatively reliable when appliedto data from heavy ion collisions at the LHC than when applied to those at RHIC In thisrespect it is also quite encouraging that the charged particle elliptic flow v2(pT ) measuredvery recently in heavy ion collisions at
radics = 276 TeV at the LHC [122] is within error bars
the same as that measured at RHIC Quantitative hydrodynamic analyses of these data will
52
come soon but at a qualitative level they indicate that the quark-gluon plasma produced atthe LHC is comparably strongly coupled to that at RHIC
One of the first questions to answer with a calculation of the equation of state in hand iswhether the observed rapid rise in εT 4 and PT 4 corresponds to a phase transition or to acontinuous crossover In QCD without quarks a first order deconfining phase transition isexpected due to the breaking of the ZN center symmetry This symmetry is unbroken in theconfined phase and broken above Tc by a nonzero expectation value for the Polyakov loopThe expected first order phase transition is indeed seen in lattice calculations [196] Theintroduction of quarks introduces a small explicit breaking of the ZN symmetry even at lowtemperatures removing this argument for a first order phase transition However in QCDwith massless quarks there must be a sharp phase transition (first order with three flavors ofmassless quarks second order with two) since chiral symmetry is spontaneously broken at lowtemperatures and unbroken at high temperatures This argument for the necessity of a tran-sition vanishes for quarks with nonzero masses which break chiral symmetry explicitly evenat high temperatures So the question of what happens in QCD with physical quark massestwo light and one strange cannot be answered by any symmetry argument Since both thecenter and chiral symmetries are explicitly broken at all temperatures it is possible for thetransition from a hadron gas to quark-gluon plasma as a function of increasing temperatureto occur with no sharp discontinuities And in fact lattice calculations have shown that thisis what happens the dramatic increase in εT 4 and PT 4 occurs continuously [197] This isshown most reliably via the fact that the peaks in the chiral and Polyakov loop susceptibilitiesare unchanging as one increases the physical spatial volume V of the lattice on which the cal-culation is done If there were a first order phase transition the heights of the peaks of thesesusceptibilities should grow prop V in the large V limit for a second order phase transitionthey should grow proportional to some fractional power of V But for a continuous crossoverno correlation length diverges at Tc and all physical quantities including the heights of thesesusceptibilities should be independent of V once V 13 is larger than the longest correlationlength This is indeed what is found [197] The fact that the transition is a continuouscrossover means that there is no sharp definition of Tc and different operational definitionscan give different values However the analysis performed in [202] indicates that the chiralsusceptibility and the Polyakov loop susceptibility peak in the range of T = 150minus 170 MeV
Despite the absence of a phase transition in the mathematical sense well above Tc QCDmatter is deconfined since the Polyakov loop takes on large nonzero values In this hightemperature regime the matter that QCD describes is best understood in terms of quarks andgluons This does not however imply that the interactions amongst the plasma constituentsis negligible Indeed we have already seen in Section 22 that in the temperature regimeaccessible in heavy ion collisions at RHIC the quark-gluon plasma behaves like a liquidnot at all like a gas of weakly coupled quasiparticles And as we will discuss in Section 6explicit calculations done via the AdSCFT correspondence show that in the large-Nc limit ingauge theories with gravity duals which are conformal and whose coupling can therefore bechosen the thermodynamic quantities change by only 25 when the coupling is varied fromzero (noninteracting gas) to infinite (arbitrarily strongly coupled liquid) This shows thatthermodynamic quantities are rather insensitive to the strength of the interactions amongthe constituents (or volume elements) of quark-gluon plasma
53
Finally we note that calculations of QCD thermodynamics done via perturbative methodshave been compared to the results obtained from lattice-regularized calculations As is wellknown (see for example [203] and references therein) the expansion of the pressure in powersof the coupling constant g is a badly convergent series and what is more cannot be extendedbeyond order g6 log(1g) where nonperturbative input is required This means that pertur-bative calculations must resort to resummations and indeed different resummation schemeshave been developed over the years [204ndash210] The effective field theory techniques developedin [204 205] in particular exploit a fundamental feature of any perturbative picture of theplasma at weak coupling microD prop gT and these methods all exploit the smallness of microD relativeto T since the basis of their formulation is that physics at these two energy scales is well sep-arated As we will see in Section 63 this characteristic is in fact essential for any descriptionof the plasma in terms of quasiparticles The analysis performed in Ref [206] showed that inthe region of T = 1minus3Tc these effective field theory calculations of the QCD pressure becomevery sensitive to the matching between the scales microD and T which indicates that there isno separation of these scales This was foreshadowed much earlier by calculations of variousdifferent correlation lengths in the plasma phase which showed that at T = 2Tc some correla-tion lengths that are prop 1(g2T ) at weak coupling are in fact significantly shorter than othersthat are prop 1(gT ) at weak coupling [211] and showed that the perturbative ordering of theselength scales is only achieved for T gt 102Tc Despite the success of other resummation tech-niques [207] in reproducing the main features of QCD thermodynamics the absence of anyseparation of scales indicates that there are very significant interactions among constituentsand casts doubt upon any approach based upon the existence of well-defined quasiparticles
311 Flavor susceptibilities
The previous discussion focussed on thermodynamics in the absence of expectation values forany of the conserved (flavor) charges of QCD As is well known these charges are a conse-quence of the three flavor symmetries that QCD possesses the U(1) symmetries generatedby electric charge Q and baryon number B and a global SU(3) flavor symmetry WithinSU(3) there are two U(1) subgroups which can be chosen as those generated by Q and bystrangeness S Conservation of Q is fundamental to the standard model since the U(1)symmetry generated by A is gauge symmetry Conservation of S is violated explicitly bythe weak interactions and conservation of B is violated by exceedingly small nonperturbativeweak interactions and perhaps by yet to be discovered beyond standard model physics Aswe are interested only in physics on QCD timescales we can safely treat S and B as con-served (Instead of taking B Q and S as the conserved quantities we could just as wellhave chosen the linear combinations of them corresponding to the numbers of up down andstrange quarks) With three conserved quantities we can introduce three independent chem-ical potentials In spite of the difficulties in studying QCD at nonzero chemical potentialon the lattice derivatives of the pressure with respect to these chemical potentials at zerochemical potential can be calculated These derivatives describe moments of the distributionof these conserved quantities in an ensemble of volumes of quark-gluon plasma and hencecan be related to event-by-event fluctuations in heavy ion collision experiments
When all three chemical potentials vanish the lowest nonzero moments are the quadratic
54
00
02
04
06
08
10
150 200 250 300 350 400 450T [MeV]
SB
open N=4full N=6
2B
2Q
2S
00
02
04
06
08
10
100 150 200 250 300 350T [MeV]
SB
HRG
-11BS2
B
N=46
Figure 32 Left quadratic fluctuations of the baryon number electric charge and strangeness nor-malized to their respective Stefan-Boltzmann limit Right off-diagonal susceptibility χBS11 normalizedto the diagonal baryon charge fluctuations The upper line corresponds to the Stefan-Boltzmannlimits and the lower line corresponds to the hadron resonance gas Figures taken from [213]
charge fluctuations ie the diagonal and off-diagonal susceptibilities defined as
χX2 =1
V T
part2
partmicroXpartmicroXlogZ(T microX ) =
1
V T 3〈N2
X〉 (32)
χXY12 =1
V T
part2
partmicroXpartmicroYlogZ(T microX microY ) =
1
V T 3〈NXNY 〉 (33)
where Z is the partition function and the NX are the numbers of B Q or S charge presentin the volume V The diagonal susceptibilities quantify the fluctuations of the conservedquantum numbers in the plasma and the off-diagonal susceptibilities measure the correlationsamong the conserved quantum numbers and are more sensitive to the nature of the chargecarriers [212]
Lattice results for these quantities [213] are shown in Fig 32 unlike the calculation ofthe pressure described in the previous section these calculations have not been performedwith physical quarks but with larger quark masses (mπ = 220 MeV) for which Tc = 200MeV is higher In the left panel the diagonal susceptibilities are shown as a function oftemperature The susceptibilities are divided by their values in a noninteracting gas ofgluons and massless quarks In the right panel one of the off diagonal susceptibilities χSB11 is shown this result has been divided by the diagonal baryon susceptibility to ensure that thisquantity approaches unity in the noninteracting limit In both cases a rapid rise is observedand above a temperature as low as T = 15Tc they are compatible with a noninteracting gas
This is a puzzling result On the one hand both lattice calculations of (ε minus 3P )T 4 andexperimental measurements of elliptic flow at RHIC indicate that significant interactionsamong the plasma constituents must be present above Tc On the other hand the latticecalculations shown in Fig 32 suggest that the flavor degrees of freedom appear to be uncor-related above about 15Tc which would appear to favor a quasiparticle interpretation of theplasma already at these rather low temperatures This observation further illustrates the factthat thermodynamic properties alone do not resolve the structure of the QGP since they donot yield an answer to as simple a question as whether it behaves like a liquid or like a gas
55
of quasiparticles We will come back to the apparently contradictory pictures suggested bythe different static properties of the plasma in Section 612 and we move on now to furtherdetermine the structure of the plasma via studying dynamical quantities rather than juststatic ones The lattice calculation of dynamical quantities which require time and thereforeMinkowski spacetime in their formulation are subject to the conceptual challenges that wedescribed above meaning that the lattice results that we have described in this section areat present significantly more reliable than those to which we now turn
32 Transport coefficients from the lattice
Transport coefficients such as the shear viscosity are essential in the description of the realtime dynamics of a system since they describe how small deviations away from equilibriumrelax toward equilibrium As we have discussed in Section 22 the shear viscosity playsa particularly important role as it provides the connection between experimental data onelliptic flow and conclusions about the strongly coupled nature of the quark-gluon plasmaproduced in RHIC collisions In this section we describe how transport coefficients can bedetermined via lattice gauge theory calculations
Transport coefficients can be extracted from the low momentum and low frequency lim-its of the Greenrsquos functions of a suitable conserved current of the theory see AppendixA To illustrate this point we concentrate on two examples the stress tensor componentsT xy and the longitudinal component of some conserved U(1) current J i(ωk) which canbe written J(ω k) k with ω k the Fourier modes The stress tensor correlator determinesthe shear viscosity the current-current correlator determines the diffusion constant for theconserved charge associated with the current (The conserved charge could be baryon num-ber strangeness or electric charge in QCD or could be some R-charge in a supersymmetrictheory) The retarded correlators of these operators are defined by
Gxy xyR (t x) = minusiθ(t) 〈[T xy(t x)T xy(0 0)]〉 (34)
GJ JR (t x) = minusiθ(t) 〈[J(t x)J(0 0)]〉 (35)
And according to the Green-Kubo relation (A9) the low momentum and low frequencylimits of these correlators yield
η = minus limωrarr0
ImGxy xyR (ω k = 0)
ω (36)
Dχ = minus limωrarr0
ImGJ JR (ω k = 0)
ω (37)
where η is the shear viscosity D is the diffusion constant of the conserved charge and χ isthe charge susceptibility Note that χ is a thermodynamic quantity which can be extractedfrom the partition function by suitable differentiation and so is straightforward to calculateon the lattice while η and D are transport properties which describe small deviations fromequilibrium In general for any conserved current operator O whose retarded correlator isgiven by
GR(t x) = minusiθ(t) 〈[O(t x)O(0 0)]〉 (38)
56
if we define a quantity micro by
micro = minus limωrarr0
ImGR(ω k = 0)
ω (39)
then micro is a transport coefficient possibly multiplied by a thermodynamic quantity
Transport coefficients can be computed in perturbation theory However since the quark-gluon plasma not too far above Tc is strongly coupled it is preferable to extract informationabout the values of the transport coefficents from lattice calculations Doing so is howeverquite challenging The difficulty arises from the fact that lattice quantum field theory isformulated in such a way that real time correlators cannot be calculated directly Insteadthese calculations determine the thermal or Euclidean correlator
GE(τ x) = 〈OE(τ x)OE(0 0)〉 (310)
where the Euclidean operator is defined from its Minkowski counterpart by
Omicro1micronM ν1νm(minusiτ x) = (minusi)r(i)sOmicro1micron
E ν1νm(τ x) (311)
where r and s are the number of time indices in micro1 micron and ν1 νm respectively Usingthe Kubo-Martin-Schwinger relation
〈O(t x)O(0 0)〉 = 〈O(0 0)O(tminus iβ x)〉 (312)
the Euclidean correlator GE can be related to the imaginary part of the retarded correlator
ρ(ω k) equiv minus2 ImGR(ω k) (313)
which is referred to as the spectral density The relation between GE (which can be calcu-lated on the lattice) and ρ (which determines the transport coefficient) takes the form of aconvolution with a known kernel
GE(τ k) = (minus1)r+sint infin
0
dω
2π
cosh(ω(τ minus 1
2T
))sinh
(ω
2T
) ρ(ω k) (314)
A typical lattice computation provides values (with errors) for the Euclidean correlator ata set of values of the Euclidean time namely τi GE (τi k) In general it is not possibleto extract a continuous function ρ(ω) from a limited number of points on GE(τ) withoutmaking assumptions about the functional form of either the spectral function or the Euclideancorrelator Note also that the Euclidean correlator at any one value of τ receives contributionsfrom the spectral function at all frequencies This makes it hard to disentangle the lowfrequency behavior of the spectral function from a measurement of the Euclidean correlatorat a limited number of values of τ
The extraction of the transport coefficient is also complicated by the fact that the highfrequency part of the spectral function ρ typically makes a large contribution to the mea-sured GE At large ω the spectral function is the same at nonzero temperature as at zerotemperature and is given by
ρ(ω k = 0) = Aω2∆minusd (315)
57
08
09
1
02 03 04 05
C(x0) Ctl(x0)
T x0
165Tc
124Tc
T01 02 03 04 05 06 07 08 09
)5 T2
(N
yxyx
G
0
2
4
6
8
10
12
14
16
18
20
AdSCFT
Free N=4
(a)
Figure 33 Left panel Ratio of the stress tensor Euclidean correlator calculated on the latticein Ref [214] to that in the free theory for QCD with three colors and zero flavors at four values ofthe Euclidean time x0 = τ and two temperatures T This theory has a first order deconfinementtransition and T is given in units of the critical temperature Tc for this transition Right panelStress tensor Euclidean correlator for N = 4 SYM from Ref [215] The solid line corresponds toinfinite coupling and dashed lines to the free theory
where ∆ is the dimension of the operator O and d is the dimension of spacetime In QCDthe constant A can be computed in perturbation theory For the two examples that weintroduced explicitly above the spectral functions are given at k = 0 to leading order inperturbation theory by
ρJJR (ω k = 0) =Nc
6πω2 (316)
ρxyxyR (ω k = 0) =π(N2
c minus 1)
5(4π)2ω4 (317)
where Nc is the number of colors These results are valid at any ω to leading order inperturbation theory because QCD is asymptotically free they are the dominant contributionat large ω This asymptotic domain of the spectral function does not contain any informationabout the transport coefficients but it makes a large contribution to the Euclidean correlatorThis means that the extraction of the contribution of the transport coefficient which is smallin comparison and τ -independent requires very precise lattice calculations
The results of lattice computations for the shear correlator are shown in the left panelof Fig 33 The finite temperature Euclidean correlator is normalized to the free theorycorrelator at the same temperature The measured correlator deviates from the free one onlyby about 10 minus 20 The errors of the numerical computation illustrate that it is hard todistinguish the computed correlator from the free one specially at the higher temperatureIt is important to stress that the fact that the measured correlator is close to the free onecomes from the fact that both receive a large contribution from the large ω region of thespectral function and therefore cannot be interpreted as a signature of large viscosity To
58
illustrate this point it is illuminating to study N = 4 SYM as a concrete example in whichwe can compare weak and strong coupling behavior with both determined analytically Aswe will discuss in Section 62 the AdSCFT correspondence allows us to compute ρ in thelimit of infinite coupling where the viscosity is small From this AdSCFT result we canthen compute the Euclidean correlator via eq (314) The result is shown in the right panelof Fig 33 In the same figure we show the Euclidean correlator at zero coupling mdash notingthat in the zero coupling limit the viscosity diverges as does the length scale above whichhydrodynamics is valid As in the lattice computation in the left panel of the figure thedifference between the weak coupling and strong coupling Euclidean correlator is small andis only significant around τ = 1(2T ) where GE is smallest and the contributions from thesmall-ω region of ρ are most visible against the ldquobackgroundrdquo from the large-ω region of ρ Forthis correlator in this theory the difference between the infinite coupling and zero couplinglimits is only at most 10 Thus the N = 4 SYM calculation gives us the perspective torealize that the small deviation between the lattice and free correlators in QCD must not betaken as an indication that the QGP at these temperatures behaves as a free gas It merelyreflect the lack of sensitivity of the Euclidean correlator to the low frequency part of thespectral function
The extraction of transport information from the 4 points in the left panel of Fig 33 asdone in Ref [214] requires assumptions about the spectral density Since the high frequencybehavior of the spectral function is fixed due to asymptotic freedom a first attempt can bemade by writing
ρ(ω)
ω=ρLF (ω)
ω+ θ(ω minus Λ)
ρHF (ω)
ω (318)
where
ρHF (ω) =π(N2 minus 1)
5(4π)2
ω4
tanhω4T(319)
is the free theory result at the high frequencies where this result is valid In the analysisperformed in [214] the parameter Λ is always chosen to be ge 5T The functional form ofthe low frequency part ρLF should be chosen such that ρLF vanishes at high frequency ABreit-Wigner ansatz
ρLF ω =η
π(1 + b2ω2)= ρBW ω (320)
provides a simple example with which to start (and is in fact the form that arises in pertur-bation theory [216]) This ansatz does not provide a good fit but it nevertheless yields animportant lesson Fitting the parameters in this ansatz to the lattice results for GE at fourvalues of τ favours large values (larger than T ) for the width Γ = 2b of the low frequencyBreit-Wigner structure This result provides quantitative motivation for the assumption thatthe width of any peak or other structure at low frequency must be larger than T From thisassumption a bound may be derived on the viscosity as follows Since a wider function thana Breit-Wigner peak of width Γ = T would lead to larger value of ρLF for ω lt
radic2T and
since the spectral function is positive definite we have
GE
(1
2T k = 0
)ge 1
T 5
[int 2T
0ρBW (ω) +
int infinΛ
ρHF (ω)
]dω
sinhω2T (321)
59
From this condition and the measured value of GE( 12T k = 0) an upper bound on the shear
viscosity η can be obtained resulting in
ηs lt
096 (T = 165Tc)108 (T = 124Tc)
(322)
with s the entropy density [214] The idea here is (i) we know how much the ω gt Λregion contributes to the integral
intdωρ(ω) sinhω2T which is what the lattice calculation
determines and (ii) we make the motivated assumption that the narrowest a peak at ω = 0can be is T and (iii) we can therefore put an upper bound on ρ(0) by assuming that theentire contribution to the integral that does not come from ω gt Λ comes from a peak atω = 0 with width T The bound is conservative because it comes from assuming that ρ iszero at intermediate ωrsquos between T and Λ Surely ρ receives some contribution from thisintermediate range of ω meaning that the bounds on ηs obtained from this analysis areconservative
Going beyond the conservative bound (322) and making an estimate of η is challeng-ing given the finite number of points at which GE(τ) is measured and relies on physicallymotivated parameterizations of the spectral function A sophisticated parameterization wasintroduced in Ref [214] under the basic assumption that there are no narrow structures inthe spectral function which is supported by the Breit-Wigner analysis discussed above InRef [214] the spectral function was expanded in an ordered basis of orthonormal functionswith an increasing number of nodes defined and ordered such that the first few functionsare those that make the largest contribution to the Euclidean correlator in other words thelatter is most sensitive to the contribution of these functions Due to the finite number ofdata points and their finite accuracy the basis has to be truncated to the first few functionswhich is a way of formalizing the assumption that there are no narrow structures in thespectral function The analysis based on such parameterization leads to small values of theratio of the shear viscosity to the entropy density In particular
ηs =
0134(33) (T = 165Tc)0102(56) (T = 124Tc)
(323)
The errors include both statistical errors and an estimate of those systematic errors due tothe truncation of the basis of functions used in the extraction The results of this studyare compelling since as discussed in Section 22 they are consistent with the experimentallyextracted bounds on the shear viscosity of the QGP via hydrodynamical fits to data on ellipticflow in heavy ion collisions These results are also remarkably close to ηs = 14π asymp 008which is obtained in the infinite coupling limit of N = 4 SYM theory and which we willdiscuss extensively in Section 62
The lattice studies to date must be taken as exploratory given the various difficulties thatwe have described As explained in Ref [217] there are ways to do better (in addition tousing finer lattices and thus obtaining GE at more values of τ) For example a significantimprovement may be achieved by analyzing the spectral function at varying nonzero valuesof the momentum k One can then exploit energy and momentum conservation to relatedifferent Euclidean correlators to the same spectral function in some cases constraining thesame spectral function with 50-100 quantities calculated on the lattice rather than just 4
60
Furthermore the functional form of the spectral function is predicted order by order inthe hydrodynamic expnasion and this provides guidance in interpreting the Euclidean dataThese analyses are still in progress but results reported to date [217] are consistent with(323) given the error estimate therein
Let us conclude the discussion by remarking on the main points The Euclidean correlatorscalculated on the lattice are dominated by the contribution of the temperature-independenthigh frequency part of the spectral function reducing their sensitivity to the transport prop-erties that we wish to extract This fact together with the finite number of points on theEuclidean correlators that are available from lattice computations complicates the extractionof the shear viscosity from the lattice Under the mild assumption that there are no nar-row structures in the spectral function an assumption that is supported by the lattice datathemselves as we discussed current lattice computations yield a conservative upper boundηs lt 1 on the shear viscosity of the QGP at T = (12minus17)Tc A compelling but exploratoryanalysis of the lattice data has also been performed yielding values of ηs asymp 01 for this rangeof temperatures In order to determine ηs with quantitative control over all systematic er-rors however further investigation is needed mdash integrating information obtained from manyEuclidean correlators at nonzero k as well as pushing to finer lattices
33 Quarkonium Spectrum from the Lattice
Above the critical temperature quarks and gluons are not confined As we have discussed atlength in Section 2 experiments at RHIC have taught us that in this regime QCD describes aquark-gluon plasma in which the interactions among the quarks and gluons are strong enoughto yield a strongly coupled liquid It is also possible that these interactions can result in theformation of bound states within the deconfined fluid [218] This observation is of particularrelevance for quarkonium mesons formed from heavy quarks in the plasma namely quarkswith M T For these quarks αs(M) is small and the zero temperature mesons are toa first approximation described by a Coulomb-like potential between a Q minus Q pair Thusthe typical radius of the quarkonium meson is rM sim 1αsM 1T As a consequence theproperties of these quarkonium mesons cannot be strongly modified in the plasma Quarkoniaare therefore expected to survive as bound states up to a temperature that is high enoughthat the screening length of the plasma has decreased to the point that it is of order thequarkonium radius [184]
The actual masses of the heavy quarks that can be accessed in heavy ion collisions thecharm and the bottom are large enough that charmonium and bottomonium mesons areexpected above the deconfinement transition but they are not so large that these mesonsare expected to be unmodified by the quark-gluon plasma produced in ultrarelativistic heavyion collisions Heavy ion collisions at RHIC (at the LHC) may reach temperatures highenough to dissociate all the charmonium (bottomonium) states and charmonia are certainlynot expected to survive in the quark-gluon plasma produced at the LHC It is a non-trivialchallenge to determine what QCD predicts for the temperatures up to which a particularquarkonium meson survives as a bound state and above which it dissociates In this Sectionwe review lattice QCD calculations done with this goal in mind This is a subject of on-goingresearch and definitive results for the dissociation temperatures of various quarkonia are not
61
W
r2
Im t
xr2
M
Figure 34 Wilson line representing the propagation of a heavy quark-antiquark pair The line atminusr2 is the heavy quark propagator in imaginary time while the line at r2 is the antiquark Thespace links ensure gauge invariance The singlet free energy is obtained by setting τ = β
yet in hand For an example of a recent review on this subject see Ref [219]
Some of the earliest [184220] attempts to describe the in-medium heavy mesons are basedon solving the Schrodinger equation for a pair of heavy quarks in a potential determinedfrom a lattice calculation These approaches are known generically as potential models Inthis approach it is assumed that the interactions between the quark anti-quark pairs andthe medium can be expressed in the form of a temperature dependent potential The mesonsare identified as the bound states of quarks in this potential Such an approach has beenvery successful at zero temperature [221] and in this context it can be put in firm theoreticalgrounds by means of a non-relativistic effective theory for QCD [222 223] However atnonzero temperature it is not clear how to determine this potential from first principles (Forsome attempts in this direction see Ref [224])
If the binding energy of the quarkonium meson is small compared to the temperatureand to any other energy scale that characterizes the medium the potential can be extractedby analyzing a static (infinitely massive) Q minus Q pair in the color singlet representationseparated by a distance r In this limit both the quark and the antiquark remain staticon the time-scale over which the medium fluctuates and their propagators in the mediumreduce to Wilson lines along the time axis In the imaginary time formalism these twoWilson lines wind around the periodic imaginary time direction and they are separated inspace by the distance r These quark and anti-quark Wilson lines are connected by spatiallinks to ensure gauge invariance These spatial links can be thought of as arising via applyinga point-splitting procedure at the point where the quark and antiquark pair are produced bya local color singlet operator A sketch of this Wilson line is shown in Fig 34
At zero temperature the extension of the Wilson lines in the imaginary time direction τ canbe taken to infinity this limit yields Wilsonrsquos definition of the heavy quark potential [225] Incontrast at nonzero temperature the imaginary time direction is compact and the imaginarytime τ is bounded by 1T Nevertheless inspired by the zero temperature case the early
62
-1
-05
0
05
1
15
2
0 05 1 15 2 25 3
r [fm]
F1(rT) [GeV]
087Tc091Tc094Tc098Tc105Tc150Tc300Tc -08-06-04-02
0 02 04 06 08
1
0 05 1 15 2 25r [fm]
F1(rT) [GeV]
TTc =082089097102109158195254329
Figure 35 Lattice results for the singlet free energy F1(r T ) as as a function of the distance r fordifferent temperatures T quoted as fractions of the critical temperature Tc at which the crossover fromhadron gas to quark-gluon plasma occurs The left panel shows results for QCD without quarks [228ndash230] and the right panel for 2+1 flavor QCD [231] The fact that below Tc the free energy goes abovethe zero temperature result is a lattice artifact [224]
studies postulated that the potential should be obtained from the Wilson line with τ = β =1T This Wilson line can be interpreted as the singlet free energy of the heavy quark pair ie the energy change in the plasma due to the presence of a pair of quarks at a fixed distanceand at fixed temperature [226227]
Lattice results for the singlet free energy are shown in Fig 35 In the left panel we showresults for the gluon plasma described by QCD without any quarks [228ndash230] The solid linein this figure denotes the T = 0 result which rises linearly with the separation r at large ras expected due to confinement The potential is well approximated by the ansatz
F1(r) = σ r minus α
r (324)
where the linear long-distance part is characterized by the string tensionradicσ = 420 MeV [232]
and the perturbative 1r piece describes the short-distance regime Below Tc as the tempera-ture increases the theory remains confined but the string tension decreases For temperatureslarger than Tc the theory is not confined and the free energy flattens to a finite value in thelarge-r limit At these temperatures the color charge in the plasma screens the interactionbetween the heavy Q and Q In QCD with light dynamical quarks as in Fig 35 b) fromRef [231] the situation is more complicated In this case the free energy flattens to a finitelimit at large distance even at zero temperature since once the heavy quark and antiquarkhave been pulled far enough apart it becomes favorable to produce a light q minus q pair fromthe medium (in this case the vacuum) which results in the formation of Qq and Qq mesonsthat can then be moved far apart without any further expenditure of energy In vacuum thisprocess is usually referred to as ldquostring-breakingrdquo In vacuum at distances that are smallenough that string-breaking does not occur the potential can be approximated by (324) butwith a reduced string tension
radicσ asymp 200 MeV [84] Above Tc the potential is screened at
large distances by the presence of the colored fluid with the screening length beyond whichthe potential flattens shrinking with increasing temperature just as in the absence of quarks
63
As a consequence in the right panel of the figure the potential evolves relatively smoothlywith increasing temperature with string-breaking below Tc becoming screening at shorterand shorter distance scales above Tc The decrease in the screening length with increasingtemperature is a generic result and it leads us to expect that quarkonium mesons are ex-pected to dissociate when the temperature is high enough that the vacuum quarkonium sizecorresponds to a quark-antiquark separation at which the potential between the quark andantiquark is screened [184]
After precise lattice data for the singlet free energy became available several authors haveused them to solve the Schrodinger equation Since the expectation value of the Wilson loopin Fig 34 leads to the singlet free energy and not to the singlet internal energy it has beenargued that the potential to be used in the Schrodinger equation should be that obtainedafter first subtracting the entropy contribution to the free energy namely
U(r T ) = F (r T )minus T dF (r T )
dT (325)
Analyses performed with this potential indicate that the Jψ meson survives deconfinementexisting as a bound state up to a dissociation temperature that lies in the range Tdiss sim(15minus 25)Tc [233ndash237] It is also a generic feature of these potential models that since theyare larger in size other less bound charmonium states like the χc and ψprime dissociate at a lowertemperature [220] typically at temperatures as low as T = 11Tc Let us state once more thatthese calculations are based on two key model assumptions first that the charm and bottomquarks are heavy enough for a potential model to apply and second that the potential isgiven by eq (325) Neither assumption has been demonstrated from first principles
Given that potential models are models there has also been a lot of effort to extractmodel-independent information about the properties of quarkonium mesons in the mediumat nonzero temperature by using lattice techniques to calculate the Euclidean correlationfunction of a color singlet operator of the type
JΓ(τx) = ψ(τx)Γψ(τx) (326)
where ψ(τx) is the heavy quark operator and Γ = 1 γmicro γ5 γ5γmicro γmicroγν correspond to thescalar vector pseudo-scalar pseudo-vector and tensor channels As in the case of the trans-port coefficients whose analysis we described in Section 32 in order to obtain informationabout the in-medium mesons we are interested in extracting the spectral functions of theseoperators As in Section 32 the Minkowski-space spectral function cannot be calculateddirectly on the lattice it must instead be inferred from lattice calculations of the Euclideancorrelator
GE(τx) = 〈JΓ(τx)JΓ(00)〉 (327)
which is related to the spectral function as in eq (314)
The current-current correlator can be understood as describing the interaction of an ex-ternal vector meson which couples only to heavy quarks with the plasma This interactioncan proceed by scattering with the heavy quarks and antiquarks present in the plasma orby mixing between the singlet quarkonium meson with (light quark) states from within theplasma that have the same quantum numbers as the external quarkonium The first physicalprocess leads to a large absorption of those vector mesons in which the ratio ωq matches
64
2M2MEb0
Figure 36 Schematic view of the current-current spectral function as a function of frequency forheavy quarks The structure at small frequency ω sim 0 is the transport peak which is due to theinteraction of the external current with heavy quarks and antiquarks from the plasma At ω = 2Mthere is a threshold for pair production An in-medium bound state like a quarkonium meson appearsas a peak below the threshold
the velocity of heavy quarks in the medium yielding the so-called transport peak at smallω The second physical process populates the near-threshold region of ω sim 2M Since thethermal distribution of the velocity of heavy quarks and anti-quarks is Maxwellian with amean velocity v sim
radicTM the transport peak is well-separated from the threshold region
Thus the spectral function contains information not only about the properties of mesonsin the medium but also about the transport properties of the heavy quarks in the plasmaA sketch of the general expectation for this spectral function is shown in Fig 36 Giventhese expectations the extraction of properties of quarkonium mesons in the plasma fromthe Euclidean correlator must take into account the presence of the transport peak It isworth mentioning that for the particular case of pseudo-scalar quarkonia the transport peakis suppressed by mass [238] thus the extraction of meson properties is simplest in this chan-nel All other channels including in particular the vector channel include contributions fromthe transport peak
From the relation (314) between the Euclidean correlator and the spectral function it isclear that the Euclidean correlator has two sources of temperature dependence the tempera-ture dependence of the spectral function itself which is of interest to us and the temperaturedependence of the kernel in the relation (314) Since the latter is a trivial kinematical factorlattice calculations of the Euclidean correlator are often presented compared to
Grecon(τ T ) =
int infin0
dωcosh(ω(τ minus 12T ))
sinh(ω2T )σ(ω T = 0) (328)
which takes into account the modification of the heat kernel Any further temperature depen-dence of GE relative to that in Grecon is due to the temperature dependence of the spectralfunction
In Fig 37 we show the ratio of the computed lattice correlator GE to Grecon defined in
65
09
095
1G
Gre
con
=61
087Tc107Tc116Tc139Tc173Tc231Tc
09
095
1
01 02 03 04 05
GG
reco
n
[fm]
=65
109Tc120Tc150Tc199Tc239Tc299Tc
1
103
106
GG
reco
n
=61 087Tc107Tc116Tc139Tc173Tc231Tc
1
103
106
01 02 03 04 05
GG
reco
n
[fm]
=65
109Tc120Tc150Tc199Tc239Tc299Tc
Figure 37 The ratio of the Euclidean correlator GE to Grecon defined in (328) in the pseudo-scalar(left) and vector (right) channels for charm quarks versus the imaginary time τ [239] Note thatthe transport contribution is suppressed by the mass of the charmed quark only in the pseudo-scalarchannel
(328) in the pseudoscalar and vector channels for charm quarks [239] The temperaturedependence of this ratio is due only to the temperature dependence of the spectral functionThe pseudoscalar correlator shows little temperature dependence up to temperatures as highas T = 15Tc while the vector correlator varies significantly in that range of temperaturesSince as we have already mentioned the transport peak is suppressed in the pseudoscalarchannel the lack of temperature dependence of the Euclidean correlator in this channel canbe interpreted as a signal of the survival of pseudoscalar charmed mesons (the ηc) abovedeconfinement However the Euclidean correlator is a convolution integral over the spectraldensity and the thermal kernel and in principle the spectral density can change radicallywhile leaving the convolution integral relatively unchanged So the spectral function mustbe extracted before definitive conclusions can be drawn
There has been a lot of effort towards extracting these spectral densities in a model-independent way directly from the Euclidean correlators The method that has been devel-oped the furthest is called the Maximum Entropy Method (MEM) [240] It is an algorithmdesigned to find the most probable spectral function compatible with the lattice data onthe Euclidean correlator This problem is underconstrained since one has available latticecalculations of the Euclidean correlator only at finitely many values of the Euclidean time τ each with error bars and one is seeking to determine a function of ω This means that thealgorithm must take advantage of prior information about the spectral function that is knownabsent any calculations of the value of the Euclidean correlator in the form of a default modelfor the spectral function Examples of priors that are taken into account include informationabout asymptotic behavior and sum rules The MEM method has been very successful inextracting the spectral functions at zero temperature where it turns out that the extractedfunctions have little dependence on the details of how the priors are implemented in a defaultmodel for the spectral function The application of the same MEM procedure at nonzerotemperature is complicated by two facts the number of data points is smaller at finite tem-perature than at zero temperature and the temporal extent of the correlators is limited to
66
1T which is reduced as the temperature increases The first problem is a computationalproblem which can be ameliorated over time as computing power grows by reducing thetemporal lattice spacing and thus increasing the number of lattice points within the extent1T The second problem is intrinsic to the nonzero temperature calculation all the struc-ture in the Minkowski space spectral function as in the sketch in Fig 36 gets mapped ontofine details of the Euclidean correlator within a small interval of τ meaning that at nonzerotemperature it takes much greater precision in the calculation of the Euclidean correlator inorder to disentangle even the main features of the spectral function
To date extractions of the pseudoscalar spectral density at nonzero temperature via theMEM indicate perhaps not surprisingly that the spectral function including its ηc peakremains almost unchanged up to T asymp 15Tc [239 241ndash243] especially when the comparisonthat is made is with the zero temperature spectral function extracted from only a reducednumber of points on the Euclidean correlator The application of the MEM to the vectorchannel also indicates survival of the Jψ up to T asymp 15minus 2Tc [239 241ndash244] but it fails toreproduce the transport peak that must be present in this correlator near ω = 0 It has beenargued that most of the temperature dependence of the vector correlator seen in Fig 37 is dueto the temperature dependence of the transport peak [245] (Note that since the transportpeak is a narrow structure centered at zero frequency it corresponds to a temperature-dependent contribution to the Euclidean correlator that is approximately τ -independent)This is supported by the fact that the τ -derivative of the ratio of correlators is much lessdependent on T [246] and by the analysis of the spectral functions extracted after introducinga transport peak in the default model of the MEM [244] When the transport peak is takeninto account the MEM also shows that Jψ may survive at least up to T = 15Tc [244]However above Tc both the vector and the pseudoscalar channels show strong dependence(much stronger than at zero temperature) on the default model via which prior information isincorporated in the MEM [239244] which makes it difficult to extract solid conclusions on thesurvival of charmonium states from this method Despite these uncertainties the conclusionsof the MEM analyses agree with those reached via analyses of potential models in which theinternal energy (325) is used as the potential However before this agreement can be takenas firm evidence for the survival of charmonium states well above the phase transition itmust be shown that the potential models and the lattice calculations are compatible in otherrespects To this we now turn
Potential models can be used for more than determining whether a temperature-dependentpotential admits bound states they provide a prediction for the entire spectral density Itis then straightforward to start with such a predicted spectral density and compute theEuclidean correlator that would be obtained in a lattice calculation if the potential modelcorrectly described all aspects of the physics One can then compare the Euclidean correla-tor predicted by the potential model with that obtained in lattice computations like thoseillustrated in Fig 37 Following this approach the authors of Refs [247 248] have shownthat neither the spectral function obtained via identifying the singlet internal energy as thepotential nor the one obtained via identifying the singlet free energy as the potential correctlyreproduce the Euclidean correlator found in lattice calculations This means that conclusionsdrawn based upon either of these potentials cannot be quantitatively reliable in all respectsThese authors then proposed a more phenomenological approach constructing a phenomeno-logical potential (containing many of the qualitative features of the singlet free energy but
67
differing from it) that reproduces the Euclidean correlator obtained in lattice calculationsat the percent level [247 248] These authors also point out that at nonzero temperatureall putative bound states must have some nonzero thermal width and states whose bindingenergy is smaller than this width should not be considered bound These considerations leadthe authors of Refs [247248] to conclude that the Jψ and ηc dissociate by T sim 12Tc whileless bound states like the χc or ψprime do not survive the transition at all These conclusionsdiffer from those obtained via the MEM Although these conclusions are dependent on thepotential used an important and lasting lesson from this work is that the spectral functionabove Tc can be very different from that at zero temperature even if the Euclidean correlatorcomputed on the lattice does not show any strong temperature dependence This lessonhighlights the challenge and the need for precision in trying to extract the spectral functionfrom lattice calculations of the Euclidean correlator in a model independent fashion
Finally we note once again that there is no argument from first principles for using theSchrodinger equation with either the phenomenological potential of Refs [247 248] or theinternal energy or the free energy as the potential To conclude this Section we would liketo add some remarks on why the identification of the potential with the singlet internal orsinglet free energy cannot be correct [249250] If the quarkonium states can be described bya Schrodinger equation the current-current correlator must reduce to the propagation of aquark-antiquark pair at a given distance r from each other The correlator must then satisfy(
minuspartτ +nabla2
2Mminus 2M minus V (τ r)
)GM (τ r) = 0 (329)
where we have added the subscript M to remind the reader that this expression is only validin the near threshold region From this expression it is clear that the potential can beextracted from the infinitely massive limit where the propagation of the pair is given by theWilson line WM in Fig 34 (up to a trivial phase factor proportional to 2Mt) In this limitthe potential in the Euclidean equation (329) is then defined by
minus partτWM (τ r) = V (τ r)WM (τ r) (330)
where τ and r are the sides of the Wilson loop in Fig 34 In principle the correct realtime potential V (t r) should then be obtained via analytic continuation of V (τ r) And forbound states with sufficiently low binding energy it would then suffice to consider the longtime limit of the potential Vinfin(r) equiv V (t =infin r)
The difficulty of extracting the correct potential resides in the analytic continuation fromV (τ r) to V (t r) At zero temperature τ is not periodic and we can take the τ rarr infin limitand relate what we obtain to Vinfin At nonzero temperature τ is periodic and so there isno τ rarr infin limit It is also apparent that Vinfin need not coincide with the value of V (β r) aspostulated in some potential models in fact due to the periodicity of of τ a lot of informationis lost by setting τ = β [250] Explicit calculations within perturbative thermal field theorywhere the analytic continuation can be performed show that Vinfin does not coincide with theinternal energy (325) and what is more the in-medium potential develops an imaginarypart which can be interpreted as the collisional width of the state in the plasma [249] If thelog of the Wilson loop is a quadratic function of the gauge potential as in QED or in QCDto leading order in perturbation theory then it is possible to show that the real part of the
68
potential agrees with the singlet free energy [250] however this is not the case in generalSo the correct potential to describe shallow bound states is not given by eq (325) and caremust be taken in drawing conclusions from potential model calculations
69
Chapter 4
Introducing the gaugestringduality
Sections 4 and 5 together constitute a primer on gaugestring duality written for a QCDaudience
Our goal in this section is to state what we mean by gaugestring duality via a clearstatement of the original example of such a duality [251ndash253] namely the conjectured equiv-alence between a certain conformal gauge theory and a certain gravitational theory in Antide Sitter spacetime We shall do this in section 43 In order to get there in section 41 wewill first motivate from a gauge theory perspective why there must be such a duality Thenin section 42 we will give the reader a look at all that one needs to know about string theoryitself in order to understand section 43 and indeed to read this review
Since some of the contents in this section are by now standard textbook material in somecases we will not give specific references The reader interested in a more detailed reviewof string theory may consult the many textbooks available such as [254ndash261] The readerinterested in complementary aspects or extra details about the gaugestring duality mayconsult some of the many existing reviews eg [1ndash6262ndash265]
41 Motivating the duality
Although the AdSCFT correspondence was originally discovered [251ndash253] by studying D-branes and black holes in string theory the fact that such an equivalence may exist can bedirectly motivated from certain aspects of gauge theories and gravity1 In this subsection wemotivate such a direct path from gauge theory to string theory without going into any detailsabout string theory and D-branes
1Since string theory is a quantum theory of gravity and the standard Einstein gravity arises as the low-energy limit of string theory we will use the terms gravity and string theory interchangeably below
70
411 An intuitive picture Geometrizing the renormalization group flow
Consider a quantum field theory (more generally a many-body system) in d-dimensionalMinkowski spacetime with coordinates (t ~x) possibly defined with a short-distance cutoffε From the work of Kadanoff Wilson and others in the sixties a good way to describethe system is to organize the physics in terms of length (or energy) scales since degrees offreedom at widely separated scales are largely decoupled from each other If one is interestedin properties of the system at a length scale z ε instead of using the bare theory definedat scale ε it is more convenient to integrate out short-distance degrees of freedom and obtainan effective theory at length scale z Similarly for physics at an even longer length scalezprime z it is more convenient to use the effective theory at scale zprime This procedure definesa renormalization group (RG) flow and gives rise to a continuous family of effective theoriesin d-dimensional Minkowski spacetime labeled by the RG scale z One may now visualizethis continuous family of d-dimensional theories as a single theory in a (d + 1)-dimensionalspacetime with the RG scale z now becoming a spatial coordinate2 By construction this(d+ 1)-dimensional theory should have the following properties
1 The theory should be intrinsically non-local since an effective theory at a scale z shouldonly describe physics at scales longer than z However there should still be some degreeof locality in the z-direction since degrees of freedom of the original theory at differentscales are not strongly correlated with each other For example the renormalizationgroup equations governing the evolution of the couplings are local with respect to lengthscales
2 The theory should be invariant under reparametrizations of the z-coordinate since thephysics of the original theory is invariant under reparametrizations of the RG scale
3 All the physics in the region below the Minkowski plane at z (see fig 41) shouldbe describable by the effective theory of the original system defined at a RG scale zIn particular this (d + 1)-dimensional description has only the number of degrees offreedom of a d-dimensional theory
In practice it is not yet clear how to lsquomergersquo this continuous family of d-dimensionaltheories into a coherent description of a (d + 1)-dimensional system or whether this wayof rewriting the renormalization group gives rise to something sensible or useful Propertynumber (3) above however suggests that if such a description is indeed sensible the resultmay be a theory of quantum gravity The clue comes from the holographic principle [267268](for a review see [269]) which says that a theory of quantum gravity in a region of spaceshould be described by a non-gravitational theory living at the boundary of that regionIn particular one may think of the quantum field theory as living on the z = 0 slice theboundary of the entire space
We now see that the gaugegravity duality when interpreted as a geometrization of theRG evolution of a quantum field theory appears to provide a specific realization of theholographic principle An important organizing principle which follows from this discussion
2Arguments suggesting that the string dual of a Yang-Mills theory must involve an extra dimension wereput forward in [266]
71
Figure 41 A geometric picture of AdS5
is the UVIR connection [270271] between the physics of the boundary and the bulk systemsFrom the viewpoint of the bulk physics near the z = 0 slice corresponds to physics near theboundary of the space ie to large-volume or IR physics In contrast from the viewpoint ofthe quantum field theory physics at small z corresponds to short-distance physics ie UVphysics
412 The large-Nc expansion of a non-Abelian gauge theory vs the stringtheory expansion
The heuristic picture of the previous section does not tell us for which many-body systemsuch a gravity description is more likely to exist or what kind of properties such a gravitysystem should have A more concrete indication that a many-body theory may indeed have agravitational description comes from the large-Nc expansion of a non-Abelian gauge theory
That it ought to be possible to reformulate a non-Abelian gauge theory as a string theorycan be motivated at different levels After all string theory was first invented to describestrong interactions Different vibration modes of a string provided an economical way to ex-plain many resonances discovered in the sixties which obey the so-called Regge behavior iethe relation M2 prop J between the mass and the angular momentum of a particle After theformulation of QCD as the microscopic theory for the strong interactions confinement pro-vided a physical picture for possible stringy degrees of freedom in QCD Due to confinementgluons at low energies behave to some extent like flux tubes which can close on themselves orconnect a quark-antiquark pair which naturally suggests a possible string formulation Sucha low-energy effective description however does not extend to high energies if the theorybecomes weakly coupled or to non-confining gauge theories
A strong indication that a fundamental (as opposed to effective) string theory descriptionmay exist for any non-Abelian gauge theory (confining or not) comes from rsquot Hooftrsquos large-Nc
72
expansion [272] Due to space limitations here we will not give a self-contained review ofthe expansion (see eg [273ndash275] for reviews) and will only summarize the most importantfeatures The basic idea of rsquot Hooft was to treat the number of colors Nc for a non-Abeliangauge theory as a parameter take it to be large and expand physical quantities in 1NcFor example consider the Euclidean partition function for a U(Nc) pure gauge theory withgauge coupling g
Z =
intDAmicro exp
(minus 1
4g2
intd4xTrF 2
) (41)
Introducing the rsquot Hooft couplingλ = g2Nc (42)
one finds that the vacuum-to-vacuum amplitude logZ can be expanded in 1Nc as
logZ =
infinsumh=0
N2minus2hc fh(λ) = N2
c f0(λ) + f1(λ) +1
N2c
f2(λ) + middot middot middot (43)
where fh(λ) h = 0 1 are functions of the rsquot Hooft coupling λ only What is remarkableabout the large-Nc expansion (43) is that at a fixed λ Feynman diagrams are organizedby their topologies For example diagrams which can be drawn on a plane without crossingany lines (ldquoplanar diagramsrdquo) all have the same Nc dependence proportional to N2
c and areincluded in f0(λ) Similarly fh(λ) includes the contributions of all diagrams which can bedrawn on a two-dimensional surface with h holes without crossing any lines Given that thetopology a two-dimensional compact orientable surface is classified by its number of holesthe large-Nc expansion (43) can be considered as an expansion in terms of the topology oftwo-dimensional compact surfaces
This is in remarkable parallel with the perturbative expansion of a closed string theorywhich expresses physical quantities in terms of the propagation of a string in spacetime Theworldsheet of a closed string is a two-dimensional compact surface and the string perturbativeexpansion is given by a sum over the topologies of two-dimensional surfaces For examplethe vacuum-to-vacuum amplitude in a string theory can be written as
A =
infinsumh=0
g2hminus2s Fh(αprime) =
1
g2s
F0(αprime) + F1(αprime) + g2sF2(αprime) + middot middot middot (44)
where gs is the string coupling 2παprime is the inverse string tension and Fh(αprime) is the contri-bution of 2d surfaces with h holes
Comparing (43) and (44) it is tempting to identify (43) as the perturbative expansionof some string theory with
gs sim1
Nc
(45)
and the string tension given as some function of the rsquot Hooft coupling λ Note that theidentification of (43) and (44) is more than just a mathematical analogy Consider forexample f0(λ) which is given by the sum over all Feynman diagrams which can be drawn ona plane (which is topologically a sphere) Each planar Feynman diagram can be thought ofas a discrete triangulation of the sphere Summing all planar diagrams can then be thought
73
of as summing over all possible discrete triangulations of a sphere which in turn can beconsidered as summing over all possible embeddings of a two-dimensional surface with thetopology of a sphere in some ambient spacetime This motivates the conjecture of identifyingf0(λ) with F0 for some closed string theory but leaves open what the specific string theoryis
One can also include quarks or more generally matter in the fundamental representationSince quarks have Nc degrees of freedom in contrast with the N2
c carried by gluons includingquark loops in the Feynman diagrams will lead to 1Nc suppressions For example in a theorywith Nf flavours the single-quark loop planar-diagram contribution to the vacuum amplitudescales as logZ sim NfNc rather than as N2
c as in (43) In the large-Nc limit with finite Nf thecontribution from quark loops is thus suppressed by powers of NfNc Feynman diagramswith quark loops can also be classified using topologies of two-dimensional surfaces nowwith inclusion of surfaces with boundaries Each boundary can be identified with a quarkloop On the string side two-dimensional surfaces with boundaries describe worldsheets of astring theory containing both closed and open strings with boundaries corresponding to theworldlines of the endpoints of the open strings
413 Why AdS
Assuming that a d-dimensional field theory can be described by a (d+1)-dimensional string orgravity theory we can try to derive some properties of the (d+1)-dimensional spacetime Themost general metric in d + 1 dimensions consistent with d-dimensional Poincare symmetrycan be written as
ds2 = Ω2(z)(minusdt2 + d~x2 + dz2
) (46)
with z is the extra spatial direction Note that in order to have translational symmetriesin the t ~x directions the warp factor Ω(z) can depend on z only At this stage not muchcan be said of the form of Ω(z) for a general quantum field theory However if we considerfield theories which are conformal (CFTs) then we can determine Ω(z) using the additionalsymmetry constraints A conformally invariant theory is invariant under the scaling
(t ~x)rarr C(t ~x) (47)
with C a constant For the gravity theory formulated in (46) to describe such a field theorythe metric (46) should respect the scaling symmetry (47) with the simultaneous scaling ofthe z coordinate z rarr Cz since z represents a length scale in the boundary theory For thisto be the case we need Ω(z) to scale as
Ω(z)rarr Cminus1Ω(z) under z rarr Cz (48)
This uniquely determines
Ω(z) =R
z (49)
where R is a constant The metric (46) can now be written as
ds2 =R2
z2
(minusdt2 + d~x2 + dz2
) (410)
74
which is precisely the line element of (d + 1)-dimensional anti-de Sitter spacetime AdSd+1This is a maximally symmetric spacetime with curvature radius R and constant negativecurvature proportional to 1R2 See eg [276] for a detailed discussion of the properties ofAdS space
In addition to Poincare symmetry and the scaling (47) a conformal field theory in d-dimension is also invariant under d special conformal transformations which altogether formthe d-dimensional conformal group SO(2 d) It turns out that the isometry group3 of (410) isalso SO(2 d) precisely matching that of the field theory Thus one expects that a conformalfield theory should have a string theory description in AdS spacetime
Note that it is not possible to use the discussion of this section to deduce the precise stringtheory dual of a given field theory nor the precise relations between their parameters In nextsection we will give a brief review of some essential aspects of string theory which will thenenable us to arrive at a precise formulation of the duality at least for some gauge theories
42 All you need to know about string theory
Here we will review some basic concepts of string theory and D-branes which will enable usto establish an equivalence between IIB string theory in AdS5 times S5 and N = 4 SYM theoryAlthough some of the contents of this section are not indispensable to understand some ofthe subsequent chapters they are important for building the readerrsquos intuition about theAdSCFT correspondence
421 Strings
Unlike quantum field theory which describes the dynamics of point particles string theoryis a quantum theory of interacting relativistic one-dimensional objects It is characterizedby the string tension Tstr and by a dimensionless coupling constant gs that controls thestrength of interactions It is customary to write the string tension in terms of a fundamentallength scale `s called string length as
Tstr equiv1
2παprimewith αprime equiv `2s (411)
We now describe the conceptual steps involved in the definition of the theory in a first-quantized formulation ie we consider the dynamics of a single string propagating in afixed spacetime Although perhaps less familiar an analogous first-quantized formulationalso exists for point particles [277 278] whose second-quantized formulation is a quantumfield theory In the case of string theory the corresponding second-quantized formulation isprovided by string field theory which contains an infinite number of quantum fields one foreach of the vibration modes of a single string In this review we will not need to considersuch a formulation For the moment we also restrict ourselves to closed strings mdash we willdiscuss open strings in the next section
A string will sweep out a two-dimensional worldsheet which in the case of a closed stringhas no boundary We postulate that the action that governs the dynamics of the string
3Namely the spacetime coordinate transformations which leave the metric invariant
75
is simply the area of this worldsheet This is a natural generalization of the action for arelativistic particle which is simply the length of its worldline In order to write downthe string action explicitly we parametrize the worldsheet with local coordinates σα withα = 0 1 For fixed worldsheet time σ0 = const the coordinate σ1 parametrizes the lengthof the string Let xM with M = 0 D minus 1 be spacetime coordinates The trajectory ofthe string is then described by specifying xM as a function of σα In terms of these functionsthe two-dimensional metric gαβ induced on the string worldsheet has components
gαβ = partαxMpartβx
N gMN (412)
where gMN is the spacetime metric (If xM are Cartesian coordinates in flat spacetime thengMN = ηMN = diag(minus+ middot middot middot+)) The action of the string is then given by
Sstr = minusTstr
intd2σradicminusdetg (413)
In order to construct the quantum states of a single string one needs to quantize thisaction It turns out that the quantization imposes strong constraints on the spacetime onestarted with not all spacetimes allow a consistent string propagation mdash see eg [254] Forexample if we start with a D-dimensional Minkowski spacetime then a consistent stringtheory exists only for D = 26 Otherwise the spacetime Lorentz group becomes anomalousat the quantum level and the theory contains negative norm states
Physically different states in the spectrum of the two-dimensional worldsheet theory corre-spond to different vibration modes of the string From the spacetime viewpoint each of thesemodes appears as a particle of a given mass and spin The spectrum typically contains afinite number of massless modes and an infinite tower of massive modes with masses of orderms equiv `minus1
s A crucial fact about a closed string theory is that one of the massless modes is aparticle of spin two ie a graviton This is the reason why string theory is in particular atheory of quantum gravity The graviton describes small fluctuations of the spacetime metricimplying that the fixed spacetime that we started with is actually dynamical
One can construct other string theories by adding degrees of freedom to the string world-sheet The theory that will be of interest here is a supersymmetric theory of strings theso-called type IIB superstring theory [279 280] which can be obtained by adding two-dimensional worldsheet fermions to the action (413) Although we will of course be interestedin eventually breaking supersymmetry in order to obtain a dual description of QCD it willbe important that the underlying theory be supersymmetric since this will guarantee thestability of our constructions For a superstring absence of negative-norm states requires thedimension of spacetime to be D = 10 In addition to the graviton the massless spectrumof IIB superstring theory includes two scalars a number of antisymmetric tensor fields andvarious fermionic partners as required by supersymmetry One of the scalars the so-calleddilaton Φ will play an important role here
Interactions can be introduced geometrically by postulating that two strings can join to-gether and that one string can split into two through a vertex of strength gs mdash see Fig 42Physical observables like scattering amplitudes can be found by summing over string propa-gations (including all possible splittings and joinings) between initial and final states Afterfixing all the gauge symmetries on the string worldsheets such a sum reduces to a sum
76
i
k
j
i
k
j
Figure 42 lsquoGeometricalrsquo interactions in string theory two strings in initial states i and j can joininto one string in a state k (left) or vice-versa (right)
Figure 43 Sum over topologies contributing to the two-to-two amplitude
over the topologies of two-dimensional surfaces with contributions from surfaces of h holesweighted by a factor g2hminus2
s This is illustrated in Fig 43 for the two-to-two amplitude
At low energies E ms one can integrate out the massive string modes and obtain a low-energy effective theory for the massless modes Since the massless spectrum of a closed stringtheory always contains a graviton to second order in derivatives the low energy effectiveaction has the form of Einstein gravity coupled to other (massless) matter fields ie
S =1
16πG
intdDxradicminusgR+ middot middot middot (414)
where R is the Ricci scalar for the spacetime with D spacetime dimensions and where thedots stand for additional terms associated with the rest of massless modes For type IIBsuperstring theory the full low-energy effective action at the level of two derivatives is givenby the so-called IIB supergravity [281 282] a supersymmetric generalization of (414) (withD = 10) The higher-order corrections to (414) take the form of a double expansion inpowers of αprimeE2 from integrating out the massive stringy modes and in powers of the stringcoupling gs from loop corrections
We conclude this section by making two important comments First we note that theten-dimensional Newtonrsquos constant G in type IIB supergravity can be expressed in terms ofthe string coupling and the string length as
16πG = (2π)7g2s`
8s (415)
The dependence on `s follows from dimensional analysis since in D dimensions Newtonrsquos con-stant has dimension (length)Dminus2 The dependence on gs follows from considering two-to-twostring scattering The leading string theory diagram depicted in Fig 44(a) is proportional
77
graviton
(a) (b)
Figure 44 (a) Tree-level contribution of order g2s to a two-to-two scattering process in stringtheory The low-energy limit of this tree-level diagram must coincide with the corresponding fieldtheory diagram depicted in (b) which is proportional to Newtonrsquos constant G
to g2s since it is obtained by joining together the two diagrams of Fig 42 The corresponding
diagram in supergravity is drawn in Fig 44(b) and is proportional to G The requirementthat the two amplitudes yield the same result at energies much lower than the string scaleimplies G prop g2
s
Second the string coupling constant gs is not a free parameter but is given by the expec-tation value of the dilaton field Φ as gs = eΦ As a result gs may actually vary over spaceand time Under these circumstances we may still speak of the string coupling constanteg in formulas like (415) or (419) meaning the asymptotic value of the dilaton at infinitygs = eΦinfin
422 D-branes and gauge theories
Perturbatively string theory is a theory of strings Non-perturbatively the theory also con-tains a variety of higher-dimensional solitonic objects D-branes [283] are a particularlyimportant class of solitons To be definite let us consider a superstring theory (eg typeIIA or IIB theory) in a 10-dimensional flat Minkowski spacetime labeled by time t equiv x0 andspatial coordinates x1 x9 A Dp-brane is then a ldquodefectrdquo where closed strings can breakand open strings can end that occupies a p-dimensional subspace mdash see Fig 45 where thex-directions are parallel to the branes and the y-directions are transverse to them Whenclosed strings break they become open strings The end points of the open strings can movefreely along the directions of the D-brane but cannot leave it by moving in the transversedirections Just like domain walls or cosmic strings in a quantum field theory D-branes aredynamical objects which can move around A Dp-brane then sweeps out a (p+1)-dimensionalworldvolume in spacetime D0-branes are particle-like objects D1-branes are string-like D2-branes are membrane-like etc Stable Dp-branes in Type IIA superstring theory exist forp = 0 2 4 6 8 whereas those in Type IIB have p = 1 3 5 7 [283]4 This can be seen ina variety of (not unrelated) ways two of which are (i) In these cases the correspondingDp-branes preserve a fraction of the supersymmetry of the underlying theory (ii) In thesecases the corresponding Dp-branes are the lightest states that carry a conserved charge
Introducing a D-brane adds an entirely new sector to the theory of closed strings con-sisting of open strings whose endpoints must satisfy the boundary condition that they lie
4D9-branes also exist but additional consistency conditions must be imposed in their presence We willnot consider them here
78
Figure 45 Stack of D-branes
on the D-brane Recall that in the case of closed strings we started with a fixed spacetimeand discovered after quantization that the close string spectrum corresponds to dynamicalfluctuations of the spacetime An analogous situation holds for open strings on a D-braneSuppose we start with a Dp-brane extending in the xmicro = (x0 x1 xp) directions withtransverse directions labelled as yi = (xp+1 x9) Then after quantization one obtainsan open string spectrum which can be identified with fluctuations of the D-brane
More explicitly the open string spectrum consists of a finite number of massless modes andan infinite tower of massive modes with masses of order ms = 1`s For a single Dp-branethe massless spectrum consists of an Abelian gauge field Amicro(x) micro = 0 1 p 9minus p scalarfields φi(x) i = 1 9 minus p and their superpartners Since these fields are supported onthe D-brane they depend only on the xmicro coordinates along the worldvolume but not onthe transverse coordinates The 9 minus p scalar excitations φi describe fluctuations of the D-brane in the transverse directions yi including deformations of the branersquos shape and linearmotions They are the exact parallel of familiar collective coordinates for a domain wall ora cosmic string in a quantum field theory and can be understood as the Goldstone bosonsassociated to the subset of translational symmetries spontaneously broken by the brane Thepresence of a U(1) gauge field Amicro(x) as part of collective excitations lies at the origin of manyfascinating properties of D-branes which (as we will discuss below) ultimately lead to thegaugestring duality Although this gauge field is less familiar in the context of quantum fieldtheory solitons (see eg [284ndash286]) it can nevertheless be understood as a Goldstone modeassociated to large gauge transformations spontaneously broken by the brane [287ndash289]
Another striking new feature of D-branes which has no parallel in field theory is the
79
D-brane 1 D-brane 2
Figure 46 Strings stretching between two D-branes
appearance of a non-Abelian gauge theory when multiple D-branes become close to oneanother [290] In addition to the degrees of freedom pertaining to each D-brane now there arenew sectors corresponding to open strings stretched between different branes For exampleconsider two parallel branes separated from each other by a distance r as shown in Fig 46Now there are four types of open strings depending on which brane their endpoints lie onThe strings with both endpoints on the same brane give rise as before to two massless gaugevectors which can be denoted by (Amicro)1
1 and (Amicro)22 where the upper (lower) numeric index
labels the brane on which the string starts (ends) Open strings stretching between differentbranes give rise to two additional vector fields (Amicro)1
2 and (Amicro)21 which have a mass given
by the tension of the string times the distance between the branes ie m = r2παprime Thesebecome massless when the branes lie on top of each other r = 0 In this case there are fourmassless vector fields altogether (Amicro)ab with a b = 1 2 which precisely correspond to thegauge fields of a non-Abelian U(2) gauge group Similarly one finds that the 9minus p masslessscalar fields also become 2times 2 matrices (φi)ab which transform in the adjoint representationof the U(2) gauge group In the general case of Nc parallel coinciding branes one finds a U(Nc)multiplet of non-Abelian gauge fields with 9 minus p scalar fields in the adjoint representationof U(Nc) The low-energy dynamics of these modes can be determined by integrating outthe massive open string modes and it turns out to be governed by a non-Abelian gaugetheory [290] To be more specific let us consider Nc D3-branes in type IIB theory Themassless spectrum consists of a gauge field Amicro six scalar fields φi i = 1 6 and four Weylfermions all of which are in the adjoint representation of U(Nc) and can be written as NctimesNc
matrices At the two-derivative level the low-energy effective action for these modes turns outto be precisely [290] the N = 4 super-Yang-Mills theory with gauge group U(Nc) in (3+1)dimensions [291 292] (for reviews see eg [263 293]) the bosonic part of whose Lagrangian
80
can be written as
L = minus 1
g2Tr
(1
4FmicroνFmicroν +
1
2Dmicroφ
iDmicroφi + [φi φj ]2) (416)
with the Yang-Mills coupling constant given by
g2 = 4πgs (417)
Equation (416) is in fact the (bosonic part of the) most general renormalizable Lagrangianconsistent with N = 4 global supersymmetry Due to the large number of supersymmetriesthe theory has many interesting properties including the fact that the beta function vanishesexactly [294ndash300] (see section 41 of [265] for a one-paragraph proof) Consequently thecoupling constant is scale-independent and the theory is conformally invariant
Note that the U(1) part of (416) is free and can be decoupled Physically the reasonfor this is as follows Excitations of the overall diagonal U(1) subgroup of U(Nc) describemotion of the branesrsquo centre of mass ie rigid motion of the entire system of branes as awhole Because of the overall translation invariance this mode decouples from the remainingSU(Nc) sub U(Nc) modes that describe motion of the branes relative to one another This isthe reason why as we will see IIB strings in AdS5 times S5 are dual to N = 4 super-Yang-Millstheory with gauge group SU(Nc)
The Lagrangian (416) receives higher-derivative corrections suppressed by αprimeE2 The fullsystem also contains closed string modes (eg gravitons) which propagate in the bulk ofthe ten-dimensional spacetime (see Fig 47) and the full theory contains interactions betweenclosed and open strings The strength of interactions of closed string modes with each other iscontrolled by Newtonrsquos constant G so the dimensionless coupling constant at an energy E isGE8 This vanishes at low energies and so in this limit closed strings become non-interactingwhich is essentially the statement that gravity is infrared-free Interactions between closedand open strings are also controlled by the same parameter since gravity couples universallyto all forms of matter Therefore at low energies closed strings decouple from open stringsWe thus conclude that at low energies the interacting sector reduces to an SU(Nc) N = 4SYM theory in four dimensions
Before closing this section we note that for a single Dp-brane with constant Fmicroν andpartmicroφ
i all higher-order αprime-corrections to (416) (or its p-dimensional generalizations) can beresummed exactly into the so-called Dirac-Born-Infeld (DBI) action [301]
SDBI = minusTDp
intdp+1x eminusΦ
radicminusdet (gmicroν + 2π`2s Fmicroν) (418)
where
TDp =1
(2π)p gs`p+1s
(419)
is the tension of the brane namely its mass per unit spatial volume In this action Φ is thedilaton and gmicroν denotes the induced metric on the brane In flat space the latter can bewritten more explicitly as
gmicroν = ηmicroν + (2π`2s)2partmicroφ
ipartνφi (420)
81
Figure 47 Open and closed string excitations of the full system
82
The first term in (420) comes from the flat spacetime metric along the worldvolume direc-tions and the second term arises from fluctuations in the transverse directions Expanding theaction (418) to quadratic order in F and partφ one recovers the Abelian version of eqn (416)The non-Abelian generalization of the DBI action (418) is not known in closed form mdash seeeg [302] for a review Corrections to (418) beyond the approximation of slowly-varyingfields have been considered in [303ndash306]
423 D-branes as spacetime geometry
Due to their infinite extent along the x-directions the total mass of a Dp-brane is infi-nite However the mass per unit p-volume known as the tension is finite and is given interms of fundamental parameters by equation (419) The dependence of the tension on thestring length is dictated by dimensional analysis The inverse dependence on the couplinggs is familiar from solitons in quantum field theory (see eg [284ndash286]) and signals the non-perturbative nature of D-branes since it implies that they become infinitely massive (evenper unit volume) and hence decouple from the spectrum in the perturbative limit gs rarr 0The crucial difference is that the D-branesrsquo tension scales as 1gs instead of the 1g2 scalingthat is typical of field theory solitons This dependence can be anticipated based on thedivergences of string perturbation theory [307] and as we will see is of great importance forthe gaugestring duality
In a theory with gravity all forms of matter gravitate D-branes are no exception and theirpresence deforms the spacetime metric around them The spacetime metric sourced by Nc
Dp-branes can be found by explicitly solving the supergravity equations of motion [308ndash310]For illustration we again use the example of D3-branes in type IIB theory for which onefinds
ds2 = Hminus12(minusdt2 + dx2
1 + dx22 + dx2
3
)+H12
(dr2 + r2dΩ2
5
) (421)
The metric inside the parentheses in the second term is just the flat metric in the y-directionstransverse to the D3-branes written in spherical coordinates with radial coordinate r2 =y2
1 + middot middot middot+ y26 The function H(r) is given by
H = 1 +R4
r4 (422)
whereR4 = 4πgsNc`
4s (423)
Let us gain some physical intuition regarding this solution Since D3-branes extend alongthree spatial directions their gravitational effect is similar to that of a point particle withmass M prop NcTD3 in the six transverse directions Thus the metric (421) only depends onthe radial coordinate r of the transverse directions For r R we have H 1 and the metricreduces to that of flat space with a small correction proportional to
R4
r4sim Ncgs`
4s
r4sim GM
r4 (424)
which can be interpreted as the gravitational potential due to a massive object of mass M in
83
Figure 48 Excitations of the system in the closed string description
six spatial dimensions5 Note that in the last step in eqn (424) we have used the fact thatG prop g2
s`8s and M prop NcTD3 prop Ncgs`
4s mdash see (415) and (419)
The parameter R can thus be considered as the length scale characteristic of the rangeof the gravitational effects of Nc D3-branes These effects are weak for r R but becomestrong for r R In the latter limit we may neglect the lsquo1rsquo in eqn (423) in which case themetric (421) reduces to
ds2 = ds2AdS5
+R2dΩ25 (425)
where
ds2AdS5
=r2
R2
(minusdt2 + dx2
1 + dx22 + dx2
3
)+R2
r2dr2 (426)
is the metric (410) of five-dimensional anti-de Sitter spacetime written in terms of r = R2zWe thus see that in the strong gravity region the ten-dimensional metric factorizes intoAdS5 times S5
We conclude that the geometry sourced by the D3-branes takes the form displayed inFig 48 far away from the branes the spacetime is flat ten-dimensional Minkowski spacewhereas close to the branes a lsquothroatrsquo geometry of the form AdS5 times S5 develops The sizeof the throat is set by the length-scale R given by (423) As we will see the spacetimegeometry (421) can be considered as providing an alternative description of the D3-branes
5Recall that a massive object of mass M in D spatial dimensions generates a gravitational potentialGMrDminus2 at a distance r from its position
84
43 The AdSCFT conjecture
In the last two Sections we have seen two descriptions of D3-branes In the description ofSection 422 which we will refer to as the open string description D-branes correspond to ahyperplane in a flat spacetime In this description the D-branesrsquo excitations are open stringsliving on the branes and closed strings propagate outside the branes mdash see Fig 47
In contrast in the description of Section 423 which we will call the closed string descrip-tion D-branes correspond to a spacetime geometry in which only closed strings propagateas displayed in fig 48 In this description there are no open strings In this case the low-energy limit consists of focusing on excitations that have arbitrarily low energy with respectto an observer in the asymptotically flat Minkowski region We have here two distinct setsof degrees of freedom those propagating in the Minkowski region and those propagating inthe throat mdash see fig 48 In the Minkowski region the only modes that remain are those ofthe massless ten-dimensional graviton supermultiplet Moreover at low energies these modesdecouple from each other since their interactions are proportional to GE8 In the throat re-gion however the whole tower of massive string excitations survives This is because a modein the throat must climb up a gravitational potential in order to reach the asymptotically flatregion Consequently a closed string of arbitrarily high proper energy in the throat regionmay have an arbitrarily low energy as seen by an observer at asymptotic infinity provided thestring is located sufficiently deep down the throat As we focus on lower and lower energiesthese modes become supported deeper and deeper in the throat and so they decouple fromthose in the asymptotic region We thus conclude that in the closed string description theinteracting sector of the system at low energies reduces to closed strings in AdS5 times S5
These two representations are tractable in different parameter regimes For gsNc 1 wesee from eqn (423) that R `s ie the radius characterizing the gravitational effect of theD-branes becomes small in string units and closed strings feel a flat spacetime everywhereexcept very close to the hyperplane where the D-branes are located In this regime the closedstring description is not useful since one would need to understand sub-string-scale geometryIn the opposite regime gsN 1 we find that R `s and the geometry becomes weaklycurved In this limit the closed string description simplifies and essentially reduces to classicalgravity Instead the open string description becomes intractable since gsNc controls the loopexpansion of the theory and one would need to deal with strongly coupled open strings Notethat both representations exist in both limiting regimes and in between
To summarize the two descriptions of Nc D3-branes that we have discussed and theirlow-energy limits are
1 A hyperplane in a flat spacetime with open strings attached The low-energy limit isdescribed by N = 4 SYM theory (416) with a gauge group SU(Nc)
2 A curved spacetime geometry (421) where only closed strings propagate The low-energy limit is described by closed IIB string theory in AdS5 times S5
It is natural to conjecture that these two descriptions are equivalent Equating in particulartheir low-energy limits we are led to conjecture that
N = 4 SU(Nc) SYM theory
=
IIB string theory in AdS5 times S5
(427)
85
From equations (417) and (423) we find how the parameters of the two theories are relatedto one another
gs =g2
4π
R
`s= (g2Nc)
14 (428)
One can also use the ten-dimensional Newton constant (415) in place of gs in the firstequation above and obtain equivalently
G
R8=
π4
2N2c
R
`s= (g2Nc)
14 (429)
Note that in particular the first equation in (428) implies that the criterion that gsNc belarge or small translates into the criterion that the gauge theory rsquot Hooft coupling λ = g2Nc
be large or small Therefore the question of which representation of the D-branes is tractablebecomes the question of whether the gauge theory is strongly or weakly coupled We willcome back to this in section 5
The discussion above relates string theory to N = 4 SYM theory at zero temperatureas we were considering the ground state of the Nc D3-branes On the supergravity sidethis corresponds to the so-called extremal solution The above discussion can easily begeneralized to a nonzero temperature T by exciting the degrees of freedom on the D3-branesto a finite temperature which corresponds [311 312] to the so-called non-extremal solution[310] It turns out that the net effect of this is solely to modify the AdS part of the metricreplacing (426) by
ds2 =r2
R2
(minusfdt2 + dx2
1 + dx22 + dx2
3
)+R2
r2fdr2 (430)
where
f(r) = 1minus r40
r4 (431)
Equivalently in terms of the z-coordinate of section 41 we replace (410) by
ds2 =R2
z2
(minusfdt2 + dx2
1 + dx22 + dx2
3 +dz2
f
) (432)
where
f(z) = 1minus z4
z40
(433)
These two metrics are related by the simple coordinate transformation z = R2r and repre-sent a black brane in AdS spacetime with a horizon located at r = r0 or z = z0 which extendsin all three spatial directions of the original brane As we will discuss in more detail in thenext section r0 and z0 are related to the temperature of the N = 4 SYM theory as
r0 prop1
z0prop T (434)
Thus we conclude that N = 4 SYM theory at finite temperature is described by string theoryin an AdS black brane geometry
86
To summarize this section we have arrived at a duality (427) of the type anticipated insection 41 that is an equivalence between a conformal field theory with zero β-function andtrivial RG-flow and string theory on a scale-invariant metric that looks the same at any zIn the finite-temperature case Eqn (434) provides an example of the expected relationshipbetween energy scale in the gauge theory set in this case by the temperature and positionin the fifth dimension set by the location of the horizon
87
Chapter 5
General aspects of the duality
51 Gaugegravity duality
In the last section we outlined the string theory reasoning behind the equivalence (427)between N = 4 SU(Nc) SYM theory and type IIB string theory on AdS5 times S5 N = 4 SYMtheory is the unique maximally supersymmetric gauge theory in (3 + 1) dimensions whosefield content includes a gauge field Amicro six real scalars φi i = 1 6 and four Weyl fermionsχa a = 1 4 all them in the adjoint representation of the gauge group The metric ofAdS5 times S5 is given by
ds2 = ds2AdS5
+R2dΩ25 (51)
with
ds2AdS5
=r2
R2ηmicroνdx
microdxν +R2
r2dr2 r isin (0infin) (52)
In the above equation xmicro = (t ~x) ηmicroν is the Minkowski metric in four spacetime dimensionsand dΩ2
5 is the metric on a unit five-sphere The metric (52) covers the so-called lsquoPoincarepatchrsquo of a global AdS spacetime and it is sometimes convenient to rewrite (52) using a newradial coordinate z = R2r isin (0infin) in terms of which we have
ds2AdS5
= gMNdxMdxN =
R2
z2
(ηmicroνdx
microdxν + dz2) xM = (z xmicro) (53)
as used earlier in (410)
In (53) each constant-z slice of AdS5 is isometric to four-dimensional Minkwoski spacetimewith xmicro identified as the coordinates of the gauge theory (see also fig 41) As z rarr 0 weapproach the lsquoboundaryrsquo of AdS5 This is a boundary in the conformal sense of the wordbut not in the topological sense since the prefactor R2z2 in (53) approaches infinity thereAlthough this concept can be given a precise mathematical meaning we will not need thesedetails here As motivated in Section 411 it is natural to imagine that the Yang-Mills theorylives at the boundary of AdS5 For this reason below we will often refer to it as the boundarytheory As z rarrinfin we approach the so-called Poincare horizon at which the prefactor R2z2
and the determinant of the metric go to zero
88
511 UVIR connection
Due to the warp factor R2z2 in front of the Minkowski metric in (53) energy and lengthscales along Minkowski directions in AdS5 are related to those in the gauge theory by az-dependent rescaling More explicitly consider an object with energy EYM and size dYM inthe gauge theory These are the energy and the size of the object measured in units of thecoordinates t and ~x From (53) we see that the corresponding proper energy E and propersize d of this object in the bulk are
d =R
zdYM E =
z
REYM (54)
where the second relation follows from the fact that the energy is conjugate to time andso it scales with the opposite scale factor than d We thus see that physical processes inthe bulk with identical proper energies but occurring at different radial positions correspondto different gauge theory processes with energies that scale as EYM sim 1z In other wordsa gauge theory process with a characteristic energy EYM is associated with a bulk processlocalized at z sim 1EYM [251 270 271] This relation between the radial direction z in thebulk and the energy scale of the boundary theory makes concrete the heuristic discussion ofSection 411 that led us to identify the z-direction with the direction along the renormal-ization group flow of the gauge theory In particular the high-energy (UV) limit EYM rarr infincorresponds to z rarr 0 ie to the near-boundary region while the low-energy (IR) limitEYM rarr 0 corresponds to z rarrinfin ie to the near-horizon region
In a conformal theory there exist excitations of arbitrarily low energies This is reflectedin the bulk in the fact that the geometry extends all the way to z rarr infin As we will see inSection 522 for a confining theory with a mass gap m the geometry ends smoothly at a finitevalue z0 sim 1m Similarly at a finite temperature T which provides an effective IR cutoffthe spacetime will be cut off by an event horizon at a finite z0 sim 1T (see Section 521)
512 Strong coupling from gravity
N = 4 SYM theory is a scale-invariant theory characterized by two parameters the Yang-Mills coupling g and the number of color Nc The theory on the right hand side of (427) isa quantum gravity theory in a maximally symmetric spacetime which is characterized by theNewton constant G and the string scale `s in units of the curvature radius R The relationsbetween these parameters are given by (429) Recalling that G sim `8p with `p the Plancklength these relations imply
`8pR8prop 1
N2c
`2sR2prop 1radic
λ (55)
where λ = g2Nc is the rsquot Hooft coupling and we have only omitted purely numerical factors
The full IIB string theory on AdS5 times S5 is rather complicated and right now a systematictreatment of it is not available However as we will explain momentarily in the limit
`8pR8 1
`2sR2 1 (56)
89
the theory dramatically simplifies and can be approximated by classical supergravity whichis essentially Einsteinrsquos general relativity coupled to various matter fields An immediateconsequence of the relations (55) is that the limit (56) corresponds to
Nc 1 λ 1 (57)
Equation (427) then implies that the planar strongly coupled limit of the SYM theory canbe described using just classical supergravity
Let us return to why string theory simplifies in the limit (56) Consider first the re-quirement `2s R2 This can be equivalently rewritten as m2
s R or as Tstr R whereR sim 1R2 is the typical curvature scale of the space where the string is propagating Thecondition m2
s R means that one can omit the contribution of all the massive states ofmicroscopic strings in low-energy processes In other words only the massless modes ofmicroscopic strings ie the supergravity modes need to be kept in this limit This is tanta-mount to treating these strings as point-like particles and ignoring their extended nature asone would expect from the fact that their typical size `s is much smaller than the typical sizeof the space where they propagate R The so-called αprime-expansion on the string side (withαprime = `2s) which incorporates stringy effects associated with the finite length of the string ina derivative expansion corresponds on the gauge theory side to an expansion around infinitecoupling in powers of 1
radicλ
The extended nature of the string however cannot be ignored in all cases As we willsee in the context of the Wilson loop calculations of Section 54 and in many other examplesin Section 7 the description of certain physical observables requires one to consider longmacroscopic strings whose typical size is much larger than R mdash for example this happenswhen the string description of such observables involves non-trivial boundary conditions onthe string In this case the full content of the second condition in (56) is easily understoodby rewriting it as Tstr R This condition says that the tension of the string is very largecompared to the typical curvature scale of the space where it is embedded and thereforeimplies that fluctuations around the classical shape of the string can be neglected Theselong strings can still break and reconnect but in between such processes their dynamicsis completely determined by the Nambu-Goto equations of motion In these cases the αprime-expansion (that is the expansion in powers of 1
radicλ) incorporates stringy effects associated
with fluctuations of the string that are suppressed at λ rarr infin by the tension of the stringbecoming infinite in this limit From this viewpoint the fact that the massive modes ofmicroscopic strings can be omitted in this limit is just the statement that string fluctuationsaround a point-like string can be neglected
Consider now the requirement `8p R8 Since the ratio `8pR8 controls the strength of
quantum gravitational fluctuations in this regime we can ignore quantum fluctuations of thespacetime metric and talk about a fixed spacetime like AdS5timesS5 The quantum gravitationalcorrections can be incorporated in a power series in `8pR
8 which corresponds to the 1N2c
expansion in the gauge theory Note from (428) that taking the Nc rarr infin limit at fixedλ corresponds to taking the string coupling gs rarr 0 meaning that quantum correctionscorresponding to loops of string breaking off or reconnecting are suppressed in this limit
In summary we conclude that the strong-coupling limit in the gauge theory suppresses thestringy nature of the dual string theory whereas the large-Nc limit suppresses its quantum
90
nature When both limits are taken simultaneously the full string theory reduces to a classicalgravity theory with a finite number of fields
Given that the S5 factor in (51) is compact it is often convenient to express a 10-dimensional field in terms of a tower of fields in AdS5 by expanding it in terms of harmonicson S5 For example the expansion of a scalar field φ(xΩ) can be written schematically as
φ(xΩ) =sum`
φ`(x)Y`(Ω) (58)
where x and Ω denote coordinates in AdS5 and S5 respectively and Yl(Ω) denote the sphericalharmonics on S5 Thus for many purposes (but not all) the original duality (427) can alsobe considered as the equivalence of N = 4 SYM theory (at strong coupling) with a gravitytheory in AdS5 only This perspective is very useful in two important aspects First it makesmanifest that the duality (427) can be viewed as an explicit realization of the holographicprinciple mentioned in Section 411 with the bulk spacetime being AdS5 and the boundarybeing four-dimensional Minkowski spacetime Second as we will mention in Section 523this helps to give a unified treatment of many different examples of the gaugegravity dualityIn most of this review we will adopt this five-dimensional perspective and work only withfields in AdS5
After dimensional reduction on S5 the supergravity action can be written as
S =1
16πG5
intd5x [Lgrav + Lmatt] (59)
where
Lgrav =radicminusg(R+
12
R2
)(510)
is the Einstein-Hilbert Lagrangian with a negative cosmological constant Λ = minus6R2 andLmatt is the Lagrangian for matter fields In the general case the latter would include theinfinite towers φ`(x) coming from the expansion on the S5 The metric (53) is a maximally-symmetric solution of the equations of motion derived from the action (59) with all matterfields set to zero
The relation between the effective five-dimensional Newtonrsquos constant G5 and its ten-dimensional counterpart G can be read off from the reduction of the Einstein-Hilbert term
1
16πG
intd5xd5Ω
radicminusg10R10 =
R5Ω5
16πG5
intd5xradicminusg5R5 + middot middot middot (511)
where Ω5 = π3 is the volume of a unit S5 This implies
G5 =G
Ω5=
G
π3R5 ie
G5
R3=
π
2N2c
(512)
where in the last equation we made use of (429)
91
513 Symmetries
Let us now examine the symmetries on both sides of the correspondence The N = 4 SYMtheory is invariant not only under dilatations but under Conf(1 3)times SO(6) The first factoris the conformal group of four-dimensional Minkowski space which contains the Poincaregroup the dilatation symmetry generated by D and four special conformal transformationswhose generators we will denote by Kmicro The second factor is the R-symmetry of the theoryunder which the φi in (416) transform as a vector In addition the theory is invariantunder sixteen ordinary or lsquoPoincarersquo supersymmetries the fermionic superpartners of thetranslation generators Pmicro as well as under sixteen special conformal supersymmetries thefermionic superpartners of the special conformal symmetry generators Kmicro
The string side of the correspondence is of course invariant under the group of diffeo-morphisms which are gauge transformations The subgroup of these consisting of largegauge transformations that leave the asymptotic (ie near the boundary) form of the metricinvariant is precisely SO(2 4) times SO(6) The first factor which is isomorphic to Conf(1 3)corresponds to the isometry group of AdS5 and the second factor corresponds to the isometrygroup of S5 As usual large gauge transformations must be thought of as global symmetriesso we see that the bosonic global symmetry groups on both sides of the correspondence agreeIn more detail the Poincare group of four-dimensional Minkowski spacetime is realized insideSO(2 4) as transformations that act separately on each of the constant-z slices in (53) in anobvious manner The dilation symmetry of Minkowski spacetime is realized in AdS5 as thetransformation (t ~x)rarr Λ(t ~x) z rarr Λz (with Λ a positive constant) which indeed leaves themetric (53) invariant The four special conformal transformations of Minkowski spacetimeare realized in a slightly more involved way as isometries of AdS5
An analogous statement can be made for the fermionic symmetries AdS5 times S5 is a maxi-mally supersymmetric solution of type IIB string theory and so it possesses thirty-two Killingspinors which generate fermionic isometries These can be split into two groups that matchthose of the gauge theory1
We therefore conclude that the global symmetries are the same on both sides of the dualityIt is important to note however that on the gravity side the global symmetries arise as largegauge transformations In this sense there is a correspondence between global symmetriesin the gauge theory and gauge symmetries in the dual string theory This is an importantgeneral feature of all known gaugegravity dualities to which we will return below afterdiscussing the fieldoperator correspondence It is also consistent with the general belief thatthe only conserved charges in a theory of quantum gravity are those associated with globalsymmetries that arise as large gauge transformations
514 Matching the spectrum the fieldoperator correspondence
We now consider the mapping between the spectra of two theories To motivate the mainidea we begin by recalling that the SYM coupling constant g2 is identified (up to a constant)with the string coupling constant gs As discussed below (415) in string theory this is givenby gs = eΦinfin where Φinfin is the value of the dilaton at infinity in this case at the AdS boundary
1In both boundary and bulk bosonic and fermionic symmetries combine together to form a supergroupSU(2 2|4)
92
(partAdS) This suggests that deforming the gauge theory by changing the value of a couplingconstant corresponds to changing the value of a bulk field at partAdS More generally one mayimagine deforming the gauge theory action as
S rarr S +
intd4xφ(x)O(x) (513)
where O(x) is a local gauge-invariant operator and φ(x) is a possibly point-dependent cou-pling namely a source If φ(x) is constant then the deformation above corresponds to simplychanging the coupling for the operator O(x) The example of g suggests that to each possiblesource φ(x) for each possible local gauge-invariant operator O(x) there must correspond adual bulk field Φ(x z) (and vice-versa) such that its value at the AdS boundary may beidentified with the source namely
φ(x) = Φ|partAdS (x) = limzrarr0
Φ(x z) (514)
As will be discussed around Eqn (526) Eqn (514) is correct for a massless field Φ but itmust be generalized for a massive field
This one-to-one map between bulk fields in AdS and local gauge-invariant operators inthe gauge theory is known as the fieldoperator correspondence The field and the operatormust have the same quantum numbers under the global symmetries of the theory but thereis no completely general and systematic recipe to identify the field dual to a given operatorFortunately an additional restriction is known for a very important set of operators in anygauge theory conserved currents associated to global symmetries such as the SO(6) symme-try in the case of the N = 4 SYM theory The source Amicro(x) coupling to a conserved currentJmicro(x) as int
d4xAmicro(x)Jmicro(x) (515)
may be thought of as an external background gauge field and from (514) we can view itas the boundary value of a dynamical gauge field Amicro(x z) in AdS This identification isvery natural given the discussion in Section 513 that continuous global symmetries in theboundary theory should correspond to large gauge transformations in the bulk
An especially important set of conserved currents in any translationally-invariant theoryare those encapsulated in the energy-momentum tensor operator T microν(x) The source gmicroν(x)coupling to T microν(x) as int
d4x gmicroν(x)T microν(x) (516)
can be interpreted as an external spacetime metric deformation According to (514) we canthen associate it with the boundary value of the bulk metric gmicroν(x z) We thus reach theimportant conclusion that the dual of a translationally-invariant gauge theory in which theenergy-momentum tensor is conserved must involve dynamical gravity
515 Normalizable vs non-normalizable modes and mass-dimension rela-tion
Having motivated the fieldoperator correspondence we now elaborate on two importantaspects of this correspondence how the conformal dimension of an operator is related to
93
properties of the dual bulk field [252 253] and how to interpret normalizable and non-normalizable modes of a bulk field in the boundary theory [313314]
For illustration we will consider a massive bulk scalar field Φ dual to some scalar operatorO in the boundary theory Although our main interest is the case in which the boundarytheory is four-dimensional it is convenient to present the equations for a general bound-ary spacetime dimension d For this reason we will work with a generalization of the AdSmetric (53) in which xmicro = (t ~x) denote coordinates of a d-dimensional Minkowski spacetime
The bulk action for Φ can be written as
S = minus1
2
intdz ddx
radicminusg
[gMNpartMΦpartNΦ +m2Φ2
]+ middot middot middot (517)
We have canonically normalized Φ and the dots stand for terms of order higher thanquadratic We have omitted these terms because they are proportional to positive powersNewtonrsquos constant and are therefore suppressed by positive powers of 1Nc
Since the bulk spacetime is translationally invariant along the xmicro-directions it is convenientto introduce a Fourier decomposition in these directions by writing2
Φ(z xmicro) =
intddk
(2π)deikmiddotx Φ(z kmicro) (518)
where k middot x equiv ηmicroνkmicroxmicro and kmicro equiv (ω~k) with ω and ~k the energy and the spatial momentumrespectively In terms of these Fourier modes the equation of motion for Φ derived from theaction (517) is
zd+1partz(z1minusdpartzΦ)minus k2z2Φminusm2R2Φ = 0 k2 = minusω2 + ~k2 (519)
Near the boundary z rarr 0 the second term in (519) can be neglected and the equation canbe readily solved to find the general asymptotic form of the solution
Φ(z k) asymp A(k) zdminus∆ +B(k) z∆ as z rarr 0 (520)
where
∆ =d
2+ ν ν =
radicm2R2 +
d2
4 (521)
The integration lsquoconstantsrsquo A and B actually acquire a dependence on k through the re-quirement that the solution be regular in the interior of AdS ie for all z gt 0 Fourier-transforming (520) back into coordinate space we then find
Φ(z x) asymp A(x) zdminus∆ +B(x) z∆ as z rarr 0 (522)
The exponents in (522) are real provided
m2R2 ge minusd2
4 (523)
2For notational simplicity we will use the same symbol to denote a function and its Fourier transformdistinguishing them only through their arguments
94
In fact one can show that the theory is stable for any m2 in the range (523) whereasfor m2R2 lt minusd24 there exist modes that grow exponentially in time and the theory isunstable [315ndash317] In other words in AdS space a field with a negative mass-squared doesnot lead to an instability provided the mass-squared is not lsquotoo negativersquo Equation (523) isoften called the Breitenlohner-Freedman (BF) bound In the stable region (523) one muststill distinguish between the finite interval minusd24 le m2R2 lt minusd24 + 1 and the rest of theregion m2R2 ge minusd24 + 1 In the first case both terms in (522) are normalizable withrespect to the inner product
(Φ1Φ2) = minusiint
Σt
dzd~xradicminusg gtt(Φlowast1parttΦ2 minus Φ2parttΦ
lowast1) (524)
where Σt is a constant-t slice We will comment on this case at the end of this section Forthe moment let us assume that m2R2 ge minusd24+1 In this case the first term in (522) is non-normalizable and the second term which is normalizable does not affect the leading boundarybehavior As motivated in the last subsection the boundary value of a bulk field Φ shouldbe identified with the source for the corresponding boundary operator O Since in (522) theboundary behavior of Φ is controlled by A(x) the presence of such a non-normalizable termshould correspond to a deformation of the boundary theory of the form
Sbdry rarr Sbdry +
intddxφ(x)O(x) with φ(x) = A(x) (525)
In other words the non-normalizable term determines the boundary theory Lagrangian Inparticular we see that in order to obtain a finite source φ(x) equation (514) should begeneralized for ∆ 6= d to
φ(x) = Φ|partAdS (x) equiv limzrarr0
z∆minusdΦ(z x) (526)
In contrast the normalizable modes are elements of the bulk Hilbert space More explicitlyin the canonical quantization one expands Φ in terms of a basis of normalizable solutionsof (519) from which one can then build the Fock space and compute the bulk Greenrsquosfunctions etc The equivalence between the bulk and boundary theories implies that theirrespective Hilbert spaces should be identified Thus we conclude that normalizable modesshould be identified with states of the boundary theory This identification gives an importanttool for finding the spectrum of low-energy excitations of a strongly coupled gauge theoryand we shall verify it below by showing that normalizable solutions of the wave equation forΦ with well-defined momentum k give rise to poles in the two-point function 〈O(q)O(minusq)〉of its dual operator at momentum q = k In the particular example at hand one can readilysee from (519) that for a given ~k there is a continuous spectrum of ω consistent with thefact that the boundary theory is scale invariant
Furthermore as will be discussed in Section 53 (and in Appendix C) the coefficient B(x)of the normalizable term in (522) can be identified with the expectation value of O in thepresence of the source φ(x) = A(x) namely
〈O(x)〉φ = 2νB(x) (527)
In the particular case of a purely normalizable solution ie one with A(x) = 0 this equationyields the expectation value of the operator in the undeformed theory
95
Equations (522) (525) and (527) imply that ∆ introduced in (521) should be identifiedas the conformal dimension of the boundary operator O dual to Φ Indeed recall that ascale transformation of the boundary coordinates xmicro rarr Λxmicro corresponds to the isometryxmicro rarr Λxmicro z rarr Λz in the bulk Since Φ is a scalar field under such an isometry ittransforms as Φprime(ΛzΛxmicro) = Φ(z xmicro) which implies that the corresponding functions in theasymptotic form (522) must transform as Aprime(Λxmicro) = Λ∆minusdA(xmicro) and Bprime(Λxmicro) = Λminus∆B(xmicro)This means that A(x) and B(x) have mass scaling dimensions d minus ∆ and ∆ respectivelyEqs (525) and (527) are then consistent with each other and imply that O(x) has massscaling dimension ∆
The relations (521) and the near-boundary behaviour (522) also apply to a bulk spin-twofield In particular they apply to the five-dimensional metric3 [253] for which m2 = 0Instead in the case of a bulk p-form the dimension ∆ of the dual operator is the largest rootof the equation
m2R2 = (∆minus p)(∆ + pminus d) (528)
and the near-boundary fall-off becomes [253]
Φmicro1middotmiddotmiddotmicrop asymp Amicro1middotmiddotmiddotmicropzdminuspminus∆ +Bmicro1middotmiddotmiddotmicropz
∆minusp as z rarr 0 (529)
A gauge field Amicro corresponds to the particular case p = 1m2 = 0 These results show thatthe metric and the gauge field are dual to operators of dimensions ∆ = d and ∆ = d minus 1as expected for their respective dual operators the stress-energy tensor and a vector currentmdash see Section 514 An intuitive way to understand the near-boundary behaviours aboveis to note that the transverse traceless part of metric fluctuations behaves like a minimallycoupled massless scalar mdash see Section 622 Consequently the boundary behavior of thispart and by covariance of the rest of the metric (in Fefferman-Graham coordinates) is thesame as that of a massless scalar Instead for a p-form one finds a massless scalar but witha spacetime-dependent coupling
Before closing this section let us return to the range minusd24 le m2 lt minusd24 + 1 We shallbe brief because this is not a case that arises in later Sections Since in this case both termsin (522) are normalizable either one can be used to build the Fock space of physical states ofthe theory [315316] This gives rise to two different boundary CFTs in which the dimensionsof the operator O(x) are ∆ or dminus∆ respectively [319] It was later realized [320 321] thateven more general quantizations are possible in which the modes used to build the physicalstates have both A and B nonzero with a relation between the two that ensures that theasymptotic symmetries of AdS are preserved [322323]
52 Generalizations
521 Nonzero temperature and nonzero chemical potential
As discussed in Section 43 the same string theory reasoning giving rise to the equiva-lence (427) can be generalized to nonzero temperature by replacing the pure AdS metric (52)
3Although in some cases additional logarithmic terms must be included mdash see eg [318]
96
by that of a black brane in AdS5 [312] Eqn (430) which we copy here for convenience
ds2 =r2
R2
(minusfdt2 + d~x2
)+R2
r2fdr2 f(r) = 1minus r4
0
r4 (530)
Equivalently in terms of the z-coordinate we replace Eqn (52) by Eqn (432) ie
ds2 =R2
z2
(minusfdt2 + d~x2
)+R2
z2fdz2 f(z) = 1minus z4
z40
(531)
The metrics above have an event horizon at r = r0 and z = z0 respectively and the regionsoutside the horizon correspond to r isin (r0infin) and z isin (0 z0) This generalization can alsobe directly deduced from (427) as the black brane (530)ndash(531) is the only metric on thegravity side that satisfies the following properties (i) it is asymptotically AdS5 (ii) it istranslationally-invariant along all the boundary directions and rotationally-invariant alongthe boundary spatial directions (iii) it has a temperature and satisfies all laws of thermo-dynamics It is therefore natural to identify the temperature and other thermodynamicalproperties of (530)ndash(531) with those of the SYM theory at nonzero temperature
We mention in passing that there is also a nice connection between the black brane ge-ometry (530)ndash(531) and the thermal-field formulation of finite-temperature field theory interms of real time Indeed the fully extended spacetime of the black brane has two bound-aries Each of them supports an identical copy of the boundary field theory which can beidentified with one of the two copies of the field theory in the Schwinger-Keldysh formulationThe thermal state can also be considered as the maximally-entangled state of the two fieldtheories For more details see [324325]
The Hawking temperature of the black brane can be calculated via the standard method [326](see Appendix B for details) of demanding that the Euclidean continuation of the metric (531)obtained by the replacement trarr minusitE
ds2E =
R2
z2
(fdt2E + dx2
1 + dx22 + dx2
3
)+R2
z2fdz2 (532)
be regular at z = z0 This requires that tE be periodically identified with a period β given by
β =1
T= πz0 (533)
The temperature T is identified with the temperature of the boundary SYM theory sincesince tE corresponds precisely to the Euclidean time coordinate of the boundary theoryWe emphasize here that while the Lorentzian spacetime (531) can be extended beyond thehorizon z = z0 the Euclidean metric (532) only exists for z isin (0 z0) as the spacetime endsat z = z0 and ends smoothly once the choice (533) is made
For a boundary theory with a U(1) global symmetry like N = 4 SYM theory one canfurthermore turn on a chemical potential micro for the corresponding U(1) charge From thediscussion of Section 514 this requires that the bulk gauge field Amicro which is dual to aboundary current Jmicro satisfy the boundary condition
limzrarr0
At = micro (534)
97
The above condition along with the requirement that the field Amicro should be regular at thehorizon implies that there should be a radial electric field in the bulk ie the black holeis now charged We will not write the metric of a charged black hole explicitly as we willnot use it in this review For more details see [327ndash332] Similarly in the case of theorieswith fundamental flavour introduced as probe D-branes a baryon number chemical potentialcorresponds to an electric field on the branes [333ndash345]
522 A confining theory
Although our main interest is the deconfined phase of QCD in this section we will briefly de-scribe a simple example of a duality for which the field theory possesses a confining phase [312]For simplicity we have chosen a model in which the field theory is three-dimensional but allthe essential features of this model extend to the string duals of more realistic confiningtheories in four dimensions
We start by considering N = 4 SYM theory at finite temperature In the Euclideandescription the system lives on R3 times S1 The circle direction corresponds to the Euclideantime which is periodically identified with period β = 1T As is well known at lengthscales much larger than β one can effectively think of this theory as the Euclidean versionof pure three-dimensional Yang-Mills theory The reasoning is that at these scales one canperform a Kaluza-Klein reduction along the circle Since the fermions of the N = 4 theoryobey antiperiodic boundary conditions around the circle their zero-mode is projected outwhich means that all fermionic modes acquire a tree-level mass of order 1β The scalars ofthe N = 4 theory are periodic around the circle but they acquire masses at the quantumlevel thorough their couplings to the fermions The only fields that cannot acquire massesare the gauge bosons of the N = 4 theory since masses for them are forbidden by gaugeinvariance Thus at long distances the theory reduces to a pure Yang-Mills theory in threedimensions which is confining and has a map gap The Lorentzian version of the theory issimply obtained by analytically continuing one of the R3 directions into the Lorentzian timeThus in this construction the lsquofinite temperaturersquo of the original four-dimensional theory isa purely theoretical device The effective Lorentzian theory in three dimensions is at zerotemperature
In order to obtain the gravity description of this theory we just need to implement theabove procedure on the gravity side We start with the Lorentzian metric (51)-(52) dualto N = 4 SYM at zero-temperature Then we introduce a nonzero temperature by goingto Euclidean signature via t rarr minusitE and periodically identifying the Euclidean time Thisresults in the metric (532) Finally we analytically continue one of the R3 directions sayx3 back into the new Lorentzian time x3 rarr it The final result is the metric
ds2 =R2
z2
(minusdt2 + dx2
1 + dx22 + fdt2E
)+R2
z2fdz2 (535)
In this metric the directions t x1 x2 correspond to the directions in which the effective three-dimensional YM theory lives The direction tE is now a compact spatial direction Note thatsince the original metric (532) smoothly ends at z = z0 so does (535) This leads to adramatic difference between the gauge theory dual to (535) and the original N = 4 theorythe fact that the radial direction smoothly closes off at z = z0 introduces a mass scale in the
98
boundary theory To see this note that the warp factor R2z2 has a lower bound Thus whenapplying the discussion of Section 511 to (535) EYM in Eqn (54) will have a lower limit oforder M sim 1z0 implying that the theory develops a mass gap of this order This can alsobe explicitly verified by solving the equation of motion of a classical bulk field (which is dualto some boundary theory operator) in the metric (535) for any fixed ~k one finds a discretespectrum of normalizable modes with a mass gap of order M (Note that since the size ofthe circle parametrized by tE is proportional to 1z0 the mass gap is in fact comparable tothe energies of Kaluza-Klein excitations on the circle) As explained in Section 515 thesenormalizable modes can be identified with the glueball states of the boundary theory
The fact that the gauge theory dual to the geometry (535) is a confining theory is furthersupported by several checks including the following two First analysis of the expectationvalue of a Wilson loop reveals an area law as will be discussed in Section 54 Secondthe gravitational description can be used to establish that the theory described by (535)undergoes a deconfinement phase transition at a temperature Tc sim M set by the mass gapabove which the theory is again described by a geometry with a black hole horizon [312](see [3 4] for reviews)
The above construction resulted in an effective confining theory in three dimensions becausewe started with the theory on the worldvolume of D3-branes which is a four-dimensional SYMtheory By starting instead with the near horizon solution of a large number of non-extremalD4-branes which describes a SYM theory in five dimensions the above procedure leads tothe string dual of a Lorentzian confining theory that at long distance reduces to a four-dimensional pure Yang-Mills theory [312] This has been used as the starting point of theSakai-Sugimoto model for QCD [346 347] which incorporates spontaneous chiral symmetrybreaking and its restoration at high temperatures [348 349] For reviews on some of thesetopics see for example [3 4]
523 Other generalizations
In addition to (427) many other examples of gaugestring dualities are known in differentspacetime dimensions (see eg [262] and references therein) These include theories withfewer supersymmetries and theories which are not scale invariant in particular confiningtheories [350ndash352] (see eg [353354] for reviews)
For a d-dimensional conformal theory the dual geometry on the gravity side containsa factor of AdSd+1 and some other compact manifold4 When expanded in terms of theharmonics of the compact manifold the duality again reduces to that between a d-dimensionalconformal theory and a gravity theory in AdSd+1 In particular in the classical gravity limitthis reduces to Einstein gravity in AdSd+1 coupled to various matter fields with the precisespectrum of matter fields depending on the specific theory under consideration For a non-conformal theory the dual geometry is in general more complicated Some simple earlyexamples were discussed in [355] If a theory has a mass gap it is always the case that thedual bulk geometry closes off at some finite value of z0 as in the example of Section 522
All known examples of gaugestring dual pairs share the following common features with (427)(i) the field theory is described by elementary bosons and fermions coupled to non-Abelian
4Not necessarily in a direct product the product may be warped
99
gauge fields whose gauge group is specified by some Nc (ii) the string description reducesto classical (super)gravity in the large-Nc strong coupling limit of the field theory In thisreview we will use (427) as our prime example for illustration purposes but the discussioncan be immediately applied to other examples including non-conformal ones
53 Correlation functions of local operators
In this section we will explain how to calculate correlation functions of local gauge-invariantoperators of the boundary theory in terms of the dual gravity description In this reviewwe are mostly concerned with the properties of strongly coupled gauge theories at nonzerotemperature For this reason we are particularly interested in real-time retarded correla-tors since these are relevant for determining linear responses transport coefficients spectralfunctions etc Below we will first describe the general prescription for computing n-pointEuclidean correlation functions and then turn to the computation of finite-temperature re-tarded two-point functions
531 Euclidean correlators
In view of the fieldoperator correspondence discussed in Sections 514 and 515 it is naturalto postulate that the Euclidean partition functions of the two theories must agree upon theidentification (526) namely that [252253]
ZCFT [φ(x)] = Zstring [Φ|partAdS (x)] (536)
Note that this equation makes sense because both sides of the equation are functionals ofthe same variables Indeed the most general partition function on the CFT side wouldinclude a source for each gauge-invariant operator in the theory so one should think of φ(x)in Eqn (536) as succinctly indicating the collection of all such sources Since AdS has aboundary in order to define the path integral over bulk fields in AdS one needs to specify aboundary condition for each field The collection of all such boundary conditions is indicatedby Φ|partAdS (x) in Eqn (536)
The right-hand side of (536) is in general not easy to compute but it simplifies dramat-ically in the classical gravity limit (56) in which it can be obtained using the saddle pointapproximation as
Zstring[φ] exp(S(ren)[Φ(E)
c ]) (537)
where we have absorbed a conventional minus sign into the definition of the Euclidean actionwhich avoids having some additional minus signs in various equations below and in theanalytic continuation to Lorentzian signature
In this equation S(ren)[Φ(E)c ] is the renormalized on-shell classical supergravity action [356ndash
362] namely the classical action evaluated on a solution Φ(E)c of the equations of motion
determined by the boundary condition
limzrarr0
z∆minusd Φ(E)c (z x) = φ(x) (538)
100
and by the requirement that it be regular everywhere in the interior of the spacetime The on-shell action needs to be renormalized because it typically suffers from infinite-volume (ie IR)divergences due to the integration region near the boundary of AdS [253] These divergencesare dual to UV divergences in the gauge theory consistent with the UVIR correspondenceThe procedure to remove these divergences on the gravity side is well understood and goesunder the name of lsquoholographic renormalizationrsquo Although it is an important ingredient ofthe gaugestring duality it is also somewhat technical In Appendix C we briefly review itin the context of a two-point function calculation The interest reader may consult some ofthe references in this paragraph as well as the review [318] for details
Corrections to Eqn (537) can be included as an expansion in αprime and gs which correspondto 1
radicλ and 1Nc corrections in the gauge theory respectively Note that since the clas-
sical action (59) on the gravity side is proportional to 1G5 from Eqn (512) we see that
S(ren)[Φ(E)c ] sim N2
c as one would expect for the generating functional of an SU(Nc) SYMtheory in the large-Nc limit
We conclude from Eqs (536) and (537) that in the large-Nc and large-λ limit connectedcorrelation functions of the gauge theory are given simply by functional derivatives of theon-shell classical gravity action
〈O(x1) O(xn)〉 =δnS(ren)[Φ
(E)c ]
δφ(x1) δφ(xn)
∣∣∣∣φ=0
(539)
It is often convenient to consider the one point function of O in the presence of the source φwhich is given by
〈O(x)〉φ =δS(ren)[Φ
(E)c ]
δφ(x)= lim
zrarr0zdminus∆ δS
(ren)[Φ(E)c ]
δΦ(E)c (z x)
(540)
A very important application of this formula arises when the operator in question is the gaugetheory stress-energy tensor In this case the dual five-dimensional field is the bulk metric andas we discussed below eqn (529) ∆ = d so the equation above becomes
〈T microν(x)〉 = limzrarr0
δS(ren)[g(E)]
δg(E)microν (x z)
(541)
At this point we must distinguish between a tensor density and a tensor From eqn (516)it is clear that T microν is a tensor density In order to construct a tensor from it we define
〈Tmicroν(x)〉 = limzrarr0
2radicminusg(E)(x z)
δS(ren)[g(E)]
δg(E)microν (x z)
(542)
Except for the dependence on the holographic coordinate z the reader will probably recognizethe right-hand side as a familiar definition of the stress tensor in classical field theory (seeeg [363]) The square root of the metric in the denominator turns the result into a truetensor This and the factor of 2 above can be fixed in a variety of ways including therequirements that Tmicroν agree with the known results in simple cases such as a free scalar fieldbe the appropriately-normalized generator of translations be the appropriately-normalized
101
stress-tensor that appears on the right-hand side of Einstein equations Eqn (542) will playan important role in Section 73
In classical mechanics it is well known that the variation of the action with respect to theboundary value of a field results in the canonical momentum Π conjugate to the field wherethe boundary in that case is usually a constant-time surface (see eg [364]) In the presentcase the boundary is a constant-z surface but it is still useful to proceed by analogy withclassical mechanics and to think of the derivative in the last term in (540) as the renormalized
canonical momentum conjugate to Φ(E)c evaluated on the classical solution
Π(ren)c (z x) =
δS(ren)[Φ(E)c ]
δΦ(E)c (z x)
(543)
With this definition eqn (540) takes the form
〈O(x)〉φ = limzrarr0
zdminus∆Π(ren)c (z x) (544)
More explicitly for a bulk scalar field Φ with action (517) the corresponding Π is given by
Π = minusgzzradicminusg partzΦ (545)
We show in Appendix C that this implies that
〈O(x)〉φ = 2νB(x) (546)
when identifying A(x) in (522) with φ(x) In linear response theory the response in mo-mentum space ie the expectation value of an operator is proportional to the correspondingsource and the constant of proportionality (for each momentum) is the two-point functionof the operator
〈O(ωE ~k)〉φ = GE(ωE ~k)φ(ωE ~k) (547)
Thus equation (546) immediately yields
GE(ωE ~k) =〈O(ωE ~k)〉φφ(ωE ~k)
= limzrarr0
z2(dminus∆) Π(ren)c
Φ(E)c
= 2νB(ωE ~k)
A(ωE ~k) (548)
where ωE denotes Euclidean frequency In Appendix C we give an explicit evaluationof (548) For some early work on the evaluation of higher-point functions see Refs [365ndash367]Incidentally Eqn (548) shows that the Euclidean correlator possesses a pole precisely atthose frequencies for which A(ωE ~k) vanishes In other words the poles of the two-pointfunction are in one-to-one correspondence with normalizable solutions of the equations ofmotion as anticipated in the paragraph below Eqn (526)
532 Real-time thermal correlators
We now proceed to the prescription for calculating real-time correlation functions We willfocus our discussion on retarded two-point functions because of their important role in charac-terizing linear response Also once the retarded function is known one can then use standardrelations to obtain the other Greenrsquos functions
102
We start with the relation between the retarded and the Euclidean two-point functions inmomentum space
GE(ωE ~k) = GR(iωE ~k) ωE gt 0 (549)
or inverselyGR(ω~k) = GE(minusi(ω + iε)~k) (550)
If the Euclidean correlation functions GE are known exactly the retarded functions GRcan then be obtained though the simple analytic continuation (550) In most examples ofinterest however the Euclidean correlation functions can only be found numerically andanalytic continuation to Lorentzian signature becomes difficult Thus it is important todevelop techniques to calculate real-time correlation functions directly Based on an educatedguess that passed several consistency checks a prescription for calculating retarded two-point functions in Lorentzian signature was first proposed by Son and Starinets in Ref [368]Ref [325] later justified the prescription and extended it to n-point functions Here we willfollow the treatment given in Refs [369 370] For illustration we consider the retarded two-point function for a scalar operator O at nonzero temperature which can be obtained fromthe propagation of the dual scalar field Φ in the geometry of an AdS black hole The actionfor Φ again takes the form (517) with gMN now given by the black brane metric (531)
Before giving the prescription we note that in Lorentzian signature one cannot directlyapply the procedure summarized by Eqn (539) to obtain retarded functions There are twoimmediate complicationsdifficulties First the Lorentzian black hole spacetime contains anevent horizon and one also needs to impose appropriate boundary conditions there whensolving the classical equation of motion for Φ Second since partition functions are definedin terms of path integrals the resulting correlation functions should be time-ordered5 As wenow review both complications can be dealt with in a simple manner
The idea is to analytically continue the Euclidean classical solution Φ(E)c (ωE ~k) as well as
Eqs (544)ndash(548) to Lorentzian signature according to (550) Clearly the analytic contin-
uation of Φ(E)c (ωE ~k)
Φc(ω~k) = Φ(E)c (minusi(ω + iε)~k) (551)
solves the Lorentzian equation of motion In addition this solution obeys the in-fallingboundary condition at the future event horizon of the black-brane metric (531) This prop-erty is important as it ensures that the retarded correlator is causal and only propagatesinformation forward in time This is intuitive since we expect that classically informationcan fall into the black hole horizon but not come out so the retarded correlator should haveno out-going componentAlthough it is intuitive given its importance let us briefly verifythat the in-falling boundary condition is satisfied The Lorentzian equation of motion inmomentum space for Φc in the black brane metric (531) takes the form
z5partz
[zminus3f(z)partzΦ
]+ω2z2
f(z)Φminus ~k2z2Φminusm2R2Φ = 0 (552)
5While it is possible to obtain Feynman functions this way the procedure is quite subtle since Feynmanfunctions require imposing different boundary conditions for positive- and negative-frequency modes at thehorizon and the choices of positive-frequency modes are not unique in a black hole spacetime The correctchoice corresponds to specifying the so-called Hartle-Hawking vacuum For details see [325] In contrast theretarded function does not depend on the choice of the bulk vacuum in the classical limit as the correspondingbulk retarded function is given by the commutator of the corresponding bulk field mdash see Eqn (561)
103
where ~k2 = δijkikj The corresponding Euclidean equation is obtained by setting ω = iωE Near the horizon z rarr z0 since f rarr 0 the last two terms in (552) become negligible comparedwith the second term and can be dropped The resulting equation (with only the first twoterms of (552)) then takes the simple form
Lorentzian part2ξΦ + ω2Φ = 0
Euclidean part2ξΦminus ω2
EΦ = 0 (553)
in terms of a new coordinate
ξ equivint z dzprime
f(zprime) (554)
Since ξ rarr +infin as z rarr z0 in order for the Euclidean solution to be regular at the horizon
we must choose the solution with the decaying exponential ie Φ(E)c (ωE ξ) sim eminusωEξ The
prescription (551) then yields Φc(ω ξ) sim eiωξ Going back to coordinate space we find thatnear the horizon
Φc(t ξ) sim eminusiω(tminusξ) (555)
As anticipated this describes a wave propagating towards the direction in which ξ increasesie falling into the horizon (Had we chosen the opposite sign in the prescription (551) wewould have obtained an out-going wave as appropriate for the advanced correlator whichobeys an out-going boundary condition at the past event horizon of the metric (531) andhas no in-falling component)
Given a Lorentzian solution satisfying the in-falling boundary condition at the horizonwhich can be expanded near the boundary according to (520) Eqs (544)ndash(548) can beanalytically continued to Lorentzian signature to obtain an intrinsic Lorentzian prescriptionfor the expectation value and the retarded two-point function of an operator The former isgiven by
〈O(ω~k)〉φ = limzrarr0
zdminus∆Π(ren)c = 2νB(ω~k) (556)
with φ(ω~k) = A(ω~k) and the latter takes the form
GR(ω~k) = limzrarr0
z2(dminus∆) Π(ren)c
Φc(ω~k)= 2ν
B(ω~k)
A(ω~k) (557)
For practical purposes let us recapitulate here the main result of this section namely thealgorithmic procedure for computing the real-time finite-temperature retarded two-pointfunction of a local gauge-invariant operator O(x) This consists of the following steps
1 Identify the bulk mode Φ(x z) dual to O(x)
2 Find the Lorentzian-signature bulk effective action for Φ to quadratic order and thecorresponding linearized equation of motion in momentum space
3 Find a solution Φc(k z) to this equation with the boundary conditions that the solutionis in-falling at the horizon and behaves as
Φc(z k) asymp A(k) zdminus∆ +B(k) z∆ (558)
104
near the boundary (z rarr 0) where ∆ is the dimension of O(x) d is the spacetimedimension of the boundary theory and A(k) should be thought of as an arbitrary sourcefor O(k) B(k) is not an independent quantity but is determined by the boundarycondition at the horizon and A(k)
4 The retarded Greenrsquos function for O is then given by
GR(k) = 2νB(k)
A(k) (559)
where ν is defined in Eqn (521)
In Section 852 we will discuss in detail an example of a retarded correlator of two electro-magnetic currents
Before closing this section we note that an alternative way to compute boundary correla-tion functions which works in both Euclidean and Lorentzian signature is [371]
〈O(x1) middot middot middot O(xn)〉 = limzirarr0
(2νz∆1 ) middot middot middot (2νz∆
n )〈Φ(z1 x1) middot middot middotΦ(zn xn)〉 (560)
where the correlator on the right-hand side is a correlation function in the bulk theoryIn (560) it should be understood that whatever ordering one wants to consider it should besame on both sides For example for the retarded two-point function GR of O
GR(x1 minus x2) = limz1z2rarr0
(2νz∆1 )(2νz∆
2 )GR(z1 x1 z2 x2) (561)
where GR denotes the retarded Greenrsquos function of the bulk field Φ
54 Wilson loops
The expectation values of Wilson loops
W r(C) = TrP exp
[i
intCdxmicroAmicro(x)
] (562)
are an important class of non-local observables in any gauge theory HereintC denotes a line
integral along the closed path C W r(C) is the trace of an SU(N)-matrix in the representationr (one often considers fundamental or adjoint representations ie r = FA) the vector po-tential Amicro(x) = Aamicro(x)T a can be expressed in terms of the generators T a of the correspondingrepresentation and P denotes path ordering The expectation values of Wilson loops containinformation about the non-perturbative physics of non-Abelian gauge field theories and haveapplications to many physical phenomena such as confinement thermal phase transitionsquark screening etc For many of these applications it is useful to think of the path C asthat traversed by a quark We will discuss some of these applications in Section 7 Herewe describe how to compute expectation values of Wilson loops in a strongly coupled gaugetheory using its gravity description
We again use N = 4 SYM theory as an example Now recall that the field content of thistheory includes six scalar fields ~φ = (φ1 middot middot middotφ6) in the adjoint representation of the gauge
105
C
Σ
D-brane
Figure 51 String worldsheet associated with a Wilson loop
group This means that in this theory one can write down the following generalization of(562) [372373]
W (C) =1
Nc
TrP exp
[i
∮Cds(Amicrox
micro + ~n middot ~φradicx2)]
(563)
where ~n(s) is a unit vector in R6 that parametrizes a path in this space (or more preciselyin S5) just like xmicro(s) parametrizes a path in R(13) The factor of
radicx2 is necessary to make
~n middot ~φradicx2 a density under worldline reparametrizations Note that the operators (562) and
(563) are equivalent in the case of a light-like loop (as will be discussed in Section 75) forwhich x2 = 0
An important difference between the operators (562) and (563) is that (562) breakssupersymmetry whereas (563) is locally 12-supersymmetric meaning that for a straight-line contour (that is time-like in Lorentzian signature) the operator is invariant under half ofthe supercharges of the N = 4 theory
We will now argue that the generalized operator (563) has a dual description in terms ofa string worldsheet For this purpose it is useful to think of the loop C as the path traversedby a quark Although the N = 4 SYM theory has no quarks we will see below that thesecan be simply included by introducing in the gravity description open strings attached to aD-brane sitting at some radial position proportional to the quark mass The endpoint of theopen string on the D-brane is dual to the quark so the boundary partΣ of the string worldsheetΣ must coincide with the path C traversed by the quark mdash see Fig 51 This suggests thatwe must identify the expectation value of the Wilson loop operator which gives the partitionfunction (or amplitude) of the quark traversing C with the partition function of the dualstring worldsheet Σ [372373]
〈W (C)〉 = Zstring[partΣ = C] (564)
106
For simplicity we will focus on the case of an infinitely heavy (non-dynamical) quark Thismeans that we imagine that we have pushed the D-brane all the way to the AdS boundaryUnder these circumstances the boundary partΣ = C of the string worldsheet also lies within theboundary of AdS
The key point to recall now is that the string endpoint couples both to the gauge fieldand to the scalar fields on the D-brane This is intuitive since after all we obtained thesefields as the massless modes of a quantized open string with endpoints attached to the D-brane Physically the coupling to the scalar fields is just a reflection of the fact that a stringending on a D-brane lsquopullsrsquo on it and deforms its shape thus exciting the scalar fields whichparametrize this shape The direction orthogonal to the D-brane in which the string pullsis specified by ~n The coupling to the gauge field reflects the fact that the string endpointbehaves as a point-like particle charged under this gauge field We thus conclude that an openstring ending on a D-brane with a fixed ~n excites both the gauge and the scalar fields whichsuggests that the correct Wilson loop operator dual to the string worldsheet must includeboth types of fields and must therefore be given by (563)
The dual description of the operator (562) is the same as that of (563) except thatthe Dirichlet boundary conditions on the string worldsheet along the S5 directions must bereplaced by Neumann boundary conditions [374] (see also [375]) One immediate consequenceis that to leading order the strong coupling results for the Wilson loop (563) with constant~n and for the Wilson loop (562) are the same However the two results differ at the nextorder in the 1
radicλ expansion since in the case of (562) we would have to integrate over the
point on the sphere where the string is sitting More precisely at the one-loop level in theαprime-expansion one finds that the determinants for quadratic fluctuations are different in thetwo cases [376]
In the large-Nc large-λ limit the string partition function Zstring[partΣ = C] greatly simplifiesand is given by the exponential of the classical string action ie
Zstring[partΣ = C] = eiS(C) rarr 〈W (C)〉 = eiS(C) (565)
The classical action S(C) can in turn be obtained by extremizing the Nambu-Goto action forthe string worldsheet with the boundary condition that the string worldsheet ends on thecurve C More explicitly parameterizing the two-dimensional world sheet by the coordinatesσα = (τ σ) the location of the string world sheet in the five-dimensional spacetime withcoordinates xM is given by the Nambu-Goto action (413) The fact that the action isinvariant under coordinate changes of σα will allow us to pick the most convenient worldsheetcoordinates (τ σ) for each occasion
Note that the large-Nc and large-λ limits are both crucial for (565) to hold TakingNc rarr infin at fixed λ corresponds to taking the string coupling to zero meaning that we canignore the possibility of loops of string breaking off from the string world sheet Additionallytaking λ rarr infin corresponds to sending the string tension to infinity which implies that wecan neglect fluctuations of the string world sheet Under these circumstances the stringworldsheet lsquohanging downrsquo from the contour C takes on its classical configuration withoutfluctuating or splitting off loops
As a simple example let us first consider a contour C given by a straightline along the timedirection with length T which describes an isolated static quark at rest On the field theory
107
side we expect that the expectation value of the Wilson line should be given by
〈W (C)〉 = eminusiMT (566)
where M is the mass of the quark From the symmetry of the problem the correspondingbulk string worldsheet should be that of a straight string connecting the boundary and thePoincare horizon and translated along the time direction by T The action of such a stringworldsheet is infinite since the proper distance from the boundary to the center of AdS isinfinite This is consistent with the fact that the external quark has an infinite mass A finiteanswer can nevertheless be obtained if we introduce an IR regulator in the bulk puttingthe boundary at z = ε instead of z = 0 From the IRUV connection this corresponds tointroducing a short-distance (UV) cutoff in the boundary theory Choosing τ = t and σ = zthe string worldsheet is given by xi(σ τ) = const and the induced metric on the worldsheetis then given by
ds2 =R2
σ2(minusdτ2 + dσ2) (567)
Evaluating the Nambu-Goto action on this solution yields
S = S0 equiv minusT R2
2παprime
int infinε
dz
z2= minusradicλ
2πεT (568)
where we have used the fact that R2αprime =radicλ Using (565) and (566) we then find that
M =
radicλ
2πε (569)
541 Rectangular loop vacuum
Now let us consider a rectangular loop sitting at a constant position on the S5 [372 373]The long side of the loop extends along the time direction with length T and the shortside extends along the x1-direction with length L We will assume that T L Such aconfiguration can be though of as consisting of a static quark-antiquark pair separated by adistance L Therefore we expect that the expectation value of the Wilson loop (with suitablerenormalization) gives the potential energy between the pair ie we expect that
〈W (C)〉 = eminusiEtotT = eminusi(2M+V (L))T = eiS(C) (570)
where Etot is the total energy for the whole system and V (L) is the potential energy betweenthe pair In the last equality we have used (565) We will now proceed to calculate S(C) fora rectangular loop
It is convenient to choose the worldsheet coordinates to be
τ = t σ = x1 (571)
Since T L we can assume that the surface is translationally invariant along the τ -directionie the extremal surface should have nontrivial dependence only on σ Given the symmetriesof the problem we can also set
x3(σ) = const x2(σ) = const (572)
108
z(σ)
AdS boundary L
Figure 52 String (red) associated with a quark-antiquark pair
Thus the only nontrivial function to solve for is z = z(σ) (see Fig 52) subject to theboundary condition
z
(plusmnL
2
)= 0 (573)
Using the form (53) of the spacetime metric and Eqs (571)ndash(572) the induced metricon the worldsheet is given by
ds2ws =
R2
z2
(minusdτ2 + (1 + zprime2)dσ2
) (574)
giving rise to the Nambu-Goto action
SNG = minusR2T
2παprime
int L2
minusL2
dσ1
z2
radic1 + zprime2 (575)
where zprime = dzdσ Since the action and the boundary condition are symmetric under σ rarr minusσz(σ) should be an even function of σ Introducing dimensionless coordinates via
σ = Lξ z(σ) = Ly(ξ) (576)
we then have
SNG = minus 2R2
2παprimeTLQ with Q =
int 12
0
dξ
y2
radic1 + yprime2 (577)
Note that Q is a numerical constant As we will see momentarily it is in fact divergent andtherefore it should be defined more carefully The equation of motion for y is given by
yprime2 =y4
0 minus y4
y4(578)
with y0 the turning point at which yprime = 0 which by symmetry should happen at ξ = 0 y0
can thus be determined by the condition
1
2=
int 12
0dξ =
int y0
0
dy
yprime=
int y0
0dy
y2radicy4
0 minus y4rarr y0 =
Γ(14)
2radicπΓ(3
4) (579)
109
It is then convenient to change integration variable in Q from ξ to y to get
Q = y20
int y0
0
dy
y2radicy4
0 minus y4 (580)
This is manifestly divergent at y = 0 but the divergence can be interpreted as coming fromthe infinite rest masses of the quark and the antiquark As in the discussion after (566) wecan obtain a finite answer by introducing an IR cutoff in the bulk by putting the boundary atz = ε ie by replacing the lower integration limit in (580) by ε The potential V (L) betweenthe quarks is then obtained by subtracting 2MT from (577) (with M given by (569)) andthen taking εrarr 0 at the end of the calculation One then finds the finite answer
V (L) = minus 4π2
Γ4(14)
radicλ
L (581)
where again we used the fact that R2αprime =radicλ to translate from gravity to gauge theory
variables Note that the 1L dependence is simply a consequence of conformal invarianceThe non-analytic dependence on the coupling ie the
radicλ factor could not be obtained at
any finite order in perturbation theory From the gravity viewpoint however it is a rathergeneric result since it is due the fact that the tension of the string is proportional to 1αprimeThe above result is valid at large λ At small λ the potential between a quark and ananti-quark in an N = 4 theory is given by [377]
E = minusπλL
(582)
to lowest order in the weak-coupling expansion
It is remarkable that the calculation of a Wilson loop in a strongly interacting gaugetheory has been simplified to a classical mechanics problem no more difficult than findingthe catenary curve describing a string suspended from two points hanging in a gravitationalfield mdash in this case the gravitational field of the AdS spacetime
Note that given (581) the boundary short-distance cutoff ε in (569) can be interpreted asthe size of the external quark One might have expected (incorrectly) that a short distancecutoff on the size of the quark should be given by the Compton wavelength 1M sim ε
radicλ
which is much smaller than ε Note that the size of a quark should be defined by either itsCompton wavelength or by the distance between a quark and an anti-quark at which thepotential is of the order of the quark mass whichever is bigger In a weakly coupled theorythe Compton wavelength is bigger while in a strongly coupled theory with potential (581)the latter is bigger and is of order ε
542 Rectangular loop nonzero temperature
We now consider the expectation of the rectangular loop at nonzero temperature [378 379]In this case the bulk gravity geometry is given by that of the black brane (531) The set-upof the calculation is exactly the same as in Eqs (571)ndash(573) for the vacuum The inducedworldsheet metric is now given by
ds2ws =
R2
z2
(minusf(z)dτ2 +
(1 +
zprime2
f
)dσ2
) (583)
110
Ls
Horizon
Figure 53 String (red) associated with a quark-antiquark pair in a plasma with temperature T gt 0The preferred configuration beyond a certain separation Ls consists of two independent strings
which yields the Nambu-Goto action
SNG = minusR2T
2παprime
int L2
minusL2
dσ1
z2
radicf(z) + zprime2 (584)
The crucial difference between the equation of motion following from (584) and that followingfrom (575) is that in the present case there exists a maximal value Ls sim 1T beyond which nonontrivial solutions exist [378379] mdash see Fig 53 Instead the solution beyond this maximalseparation consists of two disjoint vertical strings ending at the black hole horizon Thephysical reason can be easily understood qualitatively from the figure At some separationthe lowest point on the string touches the horizon Surely at and beyond this separation thestring can minimize its energy by splitting into two independent strings each of which fallsthrough the horizon The precise value of Ls is defined as the quark-antiquark separationat which the free energy of the disconnected configuration becomes smaller than that of theconnected configuration This happens at a value of L for which the lowest point of theconnected configuration is close to but still somewhat above the horizon Once L gt Ls thequark-antiquark separation can then be increased further at no additional energy cost sothe potential becomes constant and the quark and the antiquark are perfectly screened fromeach other by the plasma between them (See for example Ref [380] for a careful discussionof the corrections to this large-Nc large-λ result)
543 Rectangular loop a confining theory
For comparison let us consider the expectation value of a rectangular loop in the 2 + 1-dimensional confining theory [312] (for a review see [381]) whose metric is given by (535)which we reproduce here for convenience
ds2 =R2
z2
(minusdt2 + dx2
1 + dx22 + fdtE
)+R2
z2fdz2 f = 1minus z4
z40
(585)
111
z = z0
z = 0
Figure 54 String (red) associated with a quark-antiquark pair in a confining theory
As discussed earlier the crucial difference between (585) and AdS is that the spacetime (585)ends smoothly at a finite value z = z0 which introduces a scale in the theory The differenceas compared to the finite-temperature case is that in the confining geometry the string hasno place to end so in order to to minimize its energy it tends to drop down to z0 and to runparallel there mdash see Fig 54
Again the set-up of the calculation is completely analogous to the cases above The inducedworldsheet metric is now given by
ds2ws =
R2
z2
(minusdτ2 +
(1 +
zprime2
f
)dσ2
) (586)
and the corresponding the Nambu-Goto action is
SNG = minusR2T
2παprime
int L2
minusL2
dσ1
z2
radic1 +
zprime2
f(z) (587)
When L is large the string quickly drops to z = z0 and runs parallel there We thus findthat the action can be approximated by (after subtracting the vertical parts which can beinterpreted as due to the static quark masses)
minus S(C)minus 2MT asymp R2T2παprime
L
z20
(588)
which gives rise to a confining potential
V (L) = σsL σs =
radicλ
2πz20
(589)
The constant σs can be interpreted as the effective string tension As mentioned in Sec 522the mass gap for this theory is M sim 1z0 so we find that σs sim
radicλM2 Although we have
described the calculation only for one example of a confining gauge theory the qualitativefeatures of Fig 54 generalize In any confining gauge theory with a dual gravity descriptionas a quark-antiquark pair are separated the string hanging beneath them sags down to somelsquodepthrsquo z0 and then as the separation is further increased it sags no further Further increasingthe separation means adding more and more string at the same depth z0 which costs anenergy that increases linearly with separation Clearly any metric in which a suspended string
112
behaves like this cannot be conformal it has a length scale z0 built into it in some way Thislength scale z0 in the gravitational description corresponds via the IRUV correspondence tothe mass gap M sim 1z0 for the gauge theory and to the size of the lsquoglueballsrsquo in the gaugetheory which is of order z0
To summarize we note that the qualitative behavior of the Wilson loop discussed invarious examples above is only determined by gross features of the bulk geometry The 1Lbehavior (582) in the conformal vacuum follows directly from the scaling symmetry of thebulk geometry the area law (588) in the confining case has to do with the fact that a stringhas no place to end in the bulk when the geometry smoothly closes off and the screeningbehavior at finite temperature is a consequence of the fact that a string can fall through theblack hole horizon The difference between Figs 52 and 54 highlights the fact that N = 4SYM theory is not a good model for the vacuum of a confining theory like QCD Howeveras we will discuss in Section 76 the potential obtained from Fig 53 is not a bad caricatureof what happens in the deconfined phase of QCD This is one of many ways of seeing thatN = 4 SYM at T 6= 0 is more similar to QCD above Tc than N = 4 SYM at T = 0 is toQCD at T = 0 A heuristic way of thinking about this is to note that at low temperaturesthe putative horizon would be at a zhor gt z0 ie it is far below the bottom of Fig 54 andtherefore it plays no role while at large temperatures the horizon is far above z0 and it isz0 that plays no role At some intermediate temperature the theory has undergone a phasetransition from a confined phase described by Fig 54 into a deconfined phase described byFig 536 Unlike in QCD this deconfinement phase transition is a 1st order phase transitionin the large-Nc strong coupling limit under consideration and the theory in the deconfinedphase looses all memory about the confinement scale z0 Presumably corrections away fromthis limit in particular finite-Nc corrections could turn the transition into a higher-orderphase transition or even a cross-over
55 Introducing fundamental matter
All the matter degrees of freedom of N = 4 SYM the fermions and the scalars transform inthe adjoint representation of the gauge group In QCD however the quarks transform in thefundamental representation Moreover most of what we know about QCD phenomenologi-cally comes from the study of quarks and their bound states Therefore in order to constructholographic models more closely related to QCD we must introduce degrees of freedom in thefundamental representation It turns out that there is a rather simple way to do this in thelimit in which the number of quark species or flavours is much smaller than the number ofcolours ie when Nf Nc Indeed in this limit the introduction of Nf flavours in the gaugetheory corresponds to the introduction of Nf D-brane probes in the AdS geometry sourcedby the D3-branes [383ndash385] This is perfectly consistent with the well-known fact that thetopological representation of the large-Nc expansion of a gauge theory with quarks involvesRiemann surfaces with boundaries mdash see Section 412 In the string description these sur-faces correspond to the worldsheets of open strings whose endpoints must be attached toD-branes In the context of the gaugestring duality the intuitive idea is that closed strings
6The way we have described the transition is a crude way of thinking about the so-called Hawking-Pagephase transition between a spacetime without and with a black hole [253382]
113
Figure 55 Excitations of the system in the open string description
living in AdS are dual to gauge-invariant operators constructed solely out of gauge fields andadjoint matter eg O = TrF 2 whereas open strings are dual to meson-like operators egO = qq In particular the two end-points of an open string which are forced to lie on theD-brane probes are dual to a quark and an antiquark respectively
551 The decoupling limit with fundamental matter
The fact that the introduction of gauge theory quarks corresponds to the introduction ofD-brane probes in the string description can be more lsquorigorouslyrsquo motivated by repeating thearguments of Sections 422 422 and 43 in the presence of Nf Dp-branes as indicated inFig 55 We shall be more precise about the value of p and the precise orientation of thebranes later for the moment we simply assume p gt 3
As in Section 422 when gsNc 1 the excitations of this system are accurately describedby interacting closed and open strings living in flat space In this case however the openstring sector is richer As before open strings with both end-points on the D3-branes giverise at low energies to the N = 4 SYM multiplet in the adjoint of SU(Nc) We see fromEqn (417) that the coupling constant for these degrees of freedom is dimensionless andtherefore these degrees of freedom remain interacting at low energies The coupling constantfor the open strings with both end-points on the Dp-branes instead has dimensions of(length)pminus3 Therefore the effective dimensionless coupling constant at an energy E scales
114
as gDp prop Epminus3 Since we assume that p gt 3 this implies that just like the closed stringsthe p-p strings become non-interacting at low energies Finally consider the sector of openstrings with one end-point on the D3-branes and one end-point on the Dp-branes Thesedegrees of freedom transform in the fundamental of the gauge group on the D3-branes andin the fundamental of the gauge group on the Dp-branes namely in the bifundamental ofSU(Nc) times SU(Nf) Consistently these 3-p strings interact with the 3-3 and the p-p stringswith strengths given by the corresponding coupling constants on the D3-branes and on theDp-branes At low energies therefore only the interactions with the 3-3 strings survive Inaddition since the effective coupling on the Dp-branes vanishes the corresponding gaugegroup SU(Nf) becomes a global symmetry group This is the origin of the flavour symmetryexpected in the presence of Nf (equal mass) quark species in the gauge theory
To summarize when gsNc 1 the low-energy limit of the D3Dp system yields twodecoupled sectors The first sector is free and consists of closed strings in ten-dimensionalflat space and p-p strings propagating on the worldvolume of Nf Dp-branes The secondsector is interacting and consists of a four-dimensional N = 4 SYM multiplet in the adjointof SU(Nc) coupled to the light degrees of freedom coming from the 3-p strings We will bemore precise about the exact nature of these degrees of freedom later but for the momentwe emphasize that they transform in the fundamental representation of the SU(Nc) gaugegroup and in the fundamental representation of a global flavour symmetry group SU(Nf)
Consider now the closed string description at gsNc 1 In this case as in Section 423 theD3-branes may be replaced by their backreaction on spacetime If we assume that gsNf 1which is consistent with Nf Nc we may still neglect the backreaction of the Dp-branesIn other words we may treat the Dp-branes as probes living in the geometry sourced by theD3-branes with the Dp-branes not modifying this geometry The excitations of the system inthis limit consist of closed strings and open p-p strings that propagate in two different regionsthe asymptotically flat region and the AdS5 times S5 throat ndash see Fig 56 As in section 423these two regions decouple from each other in the low-energy limit Also as in section 423in this limit the strings in the asymptotically flat region become non-interacting whereasthose in the throat region remain interacting because of the gravitational redshift
Comparing the two descriptions above we see that the low-energy limit at both small andlarge values of gsNc contains a free sector of closed and open p-p strings As in section 43 weidentify these free sectors and we conjecture that the interacting sectors on each side providedual descriptions of the same physics In other words we conjecture that the N = 4 SYMcoupled to Nf flavours of fundamental degrees of freedom is dual to type IIB closed strings inAdS5 times S5 coupled to open strings propagating on the worldvolume of Nf Dp-brane probes
It is worth clarifying the following conceptual point before closing this section It is some-times stated that in the rsquot Hooft limit in which Nc rarrinfin with Nf fixed the dynamics is com-pletely dominated by the gluons and therefore that the quarks can be completely ignoredOne may then wonder what the interest of introducing fundamental degrees of freedom ina large-Nc theory may be There are several answers to this First of all in the presenceof fundamental matter it is more convenient to think of the large-Nc limit a la Venezianoin which NfNc is kept small but finite Any observable can then be expanded in powersof 1N2
c and NfNc As we will see this is precisely the limit that is captured by the dualdescription in terms of Nf D-brane probes in AdS5 times S5 The leading D-brane contribution
115
Figure 56 Excitations of the system in the second description
will give us the leading contribution of the fundamental matter of relative order NfNc TheVeneziano limit is richer than the rsquot Hooft limit since setting NfNc = 0 one recovers the rsquotHooft limit The second point is that even in the rsquot Hooft limit the quarks should not beregarded as irrelevant but rather as valuable probes of the gluon-dominated dynamics It istheir very presence in the theory that allows one to ask questions about heavy quarks in theplasma jet quenching meson physics photon emission etc The answers to these questionsare of course dominated by the gluon dynamics but without dynamical quarks in the theorysuch questions cannot even be posed There is a completely analogous statement in the dualgravity description To leading order the geometry is not modified by the presence of D-braneprobes but one needs to introduce these probes in order to pose questions about heavy quarksin the plasma parton energy loss mesons photon production etc In this sense the D-braneprobes allow one to decode information already contained in the geometry
552 Models with fundamental matter
Above we motivated the inclusion of fundamental matter via the introduction of Nf lsquoflavourrsquoDp-brane probes in the background sourced by Nc lsquocolourrsquo D3-branes However we weredeliberately vague about the value of p about the relative orientation between the flavourand the colour branes and about the precise nature of the flavour degrees of freedom in thegauge theory Here we will address these points Since we assumed p gt 3 in order to decouplethe p-p strings and since we wish to consider stable Dp-branes in type IIB string theory wemust have p = 5 or p = 7 mdash see Section 422 In other words we must consider D5- andD7-brane probes
116
Figure 57 D3-D5 configuration (590) with a string stretching between them The 12-directionscommon to both branes are suppressed
Consider first adding flavour D5-branes We will indicate the relative orientation betweenthese and the colour D3-branes by an array like for example
D3 1 2 3D5 1 2 4 5 6
(590)
This indicates that the D3- and the D5-branes share the 12-directions The 3-direction istransverse to the D5-branes the 456-directions are transverse to the D3-branes and the789-directions are transverse to both sets of branes This means that the two sets of branescan be separated along the 789-directions and therefore they do not necessarily intersectas indicated in Fig 57 It turns out that the lightest states of a D3-D5 string have aminimum mass given by what one would have expected on classical grounds namely M =TstrL = L2π`2s where Tstr is the string tension (411) and L is the minimum distance betweenthe D3- and the D5-branes7 These states can therefore be arbitrarily light even masslessprovided L is sufficiently small Generic excited states as usual have an additional massset by the string scale alone ms The only exception are excitations in which the string
7In order to really establish this formula one must quantise the D3-D5 strings and compute the groundstate energy In the case at hand the result coincides with the classical expectation The underlying reasonis that because the configuration (590) preserves supersymmetry corrections to the classical ground stateenergy coming from bosonic and fermionic quantum fluctuations cancel each other out exactly For otherbrane configurations like (592) this does not happen
117
moves rigidly with momentum ~p in the 12-directions in which case the energy squared is justM2 + ~p 2 This is an important observation because it means that in the decoupling limitin which one focuses on energies E ms only a finite set of modes of the D3-D5 stringssurvive and moreover these modes can only propagate along the directions common to bothbranes From the viewpoint of the dual gauge theory this translates into the statement thatthe degrees of freedom in the fundamental representation are localised on a defect mdash in theexample at hand on a plane that extends along the 12-directions and lies at a constantposition in the 3-direction As an additional example the configuration
D3 1 2 3D5 1 4 5 6 7
(591)
corresponds to a dual gauge theory in which the fundamental matter is localised on a line mdashthe 1-direction
We thus conclude that if we are interested in adding to theN = 4 SYM theory fundamentalmatter degrees of freedom that propagate in 3+1 dimensions (just like the gluons and theadjoint matter) then we must orient the flavour D-branes so that they extend along the123-directions This condition leaves us with two possibilities
D3 1 2 3D5 1 2 3 4 5
(592)
andD3 1 2 3D7 1 2 3 4 5 6 7
(593)
So far we have not been specific about the precise nature of the fundamental matterdegrees of freedom mdash for example whether they are fermions or bosons etc This alsodepends on the relative orientation of the branes It turns out that for the configuration(592) the ground state energy of the D3-D5 strings is (for sufficiently small L) negativethat is the ground state is tachyonic signaling an instability in the system This conclusionis valid at weak string coupling where the string spectrum can be calculated perturbativelyWhile it is possible that the instability is absent at strong coupling we will not consider thisconfiguration further in this review
We are therefore left with the D3-D7 system (593) Quantisation of the D3-D7 stringsshows that the fundamental degrees of freedom in this case consist of Nf complex scalars andNf Dirac fermions all of them with equal masses given by
Mq =L
2παprime (594)
In a slight abuse of language we will collectively refer to all these degrees of freedom aslsquoquarksrsquo The fact that they all have exactly equal masses is a reflection of the fact thatthe addition of the Nf D7-branes preserves a fraction of the original supersymmetry of theSYM theory More precisely the original N = 4 is broken down to N = 2 under which thefundamental scalars and fermions transform as part of a single supermultiplet In the rest ofthe review especially in Section 8 we will focus our attention on this system as a model forgauge theories with fundamental matter
118
Chapter 6
Bulk properties of strongly coupledplasma
Up to this point in this review we have laid the groundwork needed for what is to come intwo halves In Sections 2 and 3 we have introduced the theoretical phenomenological andexperimental challenges posed by the study of the deconfined phase of QCD and in Sections4 and 5 we have motivated and described gaugestring duality providing the reader withmost of the conceptual and computational machinery necessary to perform many calculationsAlthough we have foreshadowed their interplay at various points these two long introductionshave to a large degree been separately self-contained In the next three sections we weavethese strands together In these sections we shall review applications of gaugegravity dualityto the study of the strongly coupled plasma of N = 4 SYM theory at nonzero temperaturefocussing on the ways in which these calculations can guide us toward the resolution of thechallenges described in Sections 2 and 3
The study of the zero temperature vacuum of strongly coupled N = 4 SYM theory is arich subject with numerous physical insights into the dynamics of gauge theories Given ourgoal of gaining insights into the deconfined phase of QCD we will largely concentrate onthe description of strongly coupled N = 4 SYM theory at nonzero temperature where itdescribes a strongly coupled non-abelian plasma with O(N2
c ) degrees of freedom The vacuaof QCD and N = 4 SYM theory have very different properties However when we compareN = 4 SYM at T 6= 0 with QCD at a temperature above the temperature Tc of the crossoverfrom a hadron gas to quark-gluon plasma many of the qualitative distinctions disappear orbecome unimportant In particular
1 QCD confines while N = 4 does not This is a profound difference in vacuum Butabove its Tc QCD is no longer confining The fact that its T = 0 quasiparticles arehadrons within which quarks are confined is not particularly relevant at temperaturesabove Tc
2 In QCD chiral symmetry is broken by a chiral condensate which sets a scale that iscertainly not present in N = 4 SYM theory However in QCD above its Tc the chiralcondensate melts away and this distinction between the vacua of the two theories alsoceases to be relevant
119
3 N = 4 is a scale invariant theory while in QCD scale invariance is broken by the con-finement scale the chiral condensate and just by the running of the coupling constantAbove Tc we have already dispensed with the first two scales Also as we have de-scribed in Section 3 QCD thermodynamics is significantly nonconformal just aboveTc sim 170 MeV but at higher temperatures the quark-gluon plasma becomes more andmore scale invariant at least in its thermodynamics (Thermodynamic quantities con-verge to their values in the noninteracting limit due to the running of the couplingtowards zero only at vastly higher temperatures which are far from the reach of anycollider experiment) So here again QCD above (but not asymptotically far above)its Tc is much more similar to N = 4 SYM theory at T 6= 0 than the vacua of the twotheories are
4 N = 4 SYM theory is supersymmetric However supersymmetry is explicitly brokenat nonzero temperature In a thermodynamic context this can be seen by noting thatfermions have antiperiodic boundary conditions along the Euclidean time circle whilebosons are periodic For this reason supersymmetry does not play a major role in thecharacterization of properties of the N = 4 SYM plasma at nonzero temperatures
5 QCD is an asymptotically free theory and thus high energy processes are weaklycoupled However as we have described in Section 2 in the regime of temperaturesabove Tc that are accessible to heavy ion collision experiments the QCD plasma isstrongly coupled which opens a window of applicability for strong coupling techniques
For these and other reasons the strongly coupled plasma of N = 4 SYM theory has beenstudied by many authors with the aim of gaining insights into the dynamics of deconfinedQCD plasma
In fairness we should also mention the significant differences between the two theories thatremain at nonzero temperature
1 N = 4 SYM theory with Nc = 3 has more degrees of freedom than QCD with Nc = 3To seek guidance for QCD from results in N = 4 SYM the challenge is to evaluatehow an observable of interest depends on the number of degrees of freedom as we do atseveral points in Section 7 The best case scenario is that there is no such dependenceas for example arises for the ratio ηs between the shear viscosity and the entropydensity that we introduced in Section 2 and that we shall discuss in Section 62 below
2 Most of the calculations that we shall report are done in the strong coupling (λrarrinfin)limit This is of course a feature not a bug The ability to do these calculations in thestrong coupling regime is a key part of the motivation for all this work But althoughin the temperature regime of interest g2(T )Nc = 4πNcαs(T ) is large it is not infiniteThis motivates the calculation of corrections to various results that we shall discussthat are proportional to powers of 1λ for the purpose of testing the robustness ofconclusions drawn from calculations done with λrarrinfin
3 QCD has Nc = 3 colors while all the calculations that we shall report are done inthe Nc rarr infin limit Although the large-Nc approximation is familiar in QCD thestandard way of judging whether it is reliable in a particular context is to compute
120
corrections suppressed by powers of 1Nc And determining the 1N2c corrections to the
calculations done via the gaugestring duality that we review remains an outstandingchallenge
4 Although we have argued above that the distinction between bosons and fermions is notimportant at nonzero temperature the distinction between degrees of freedom in theadjoint or fundamental representation of SU(Nc) is important QCD has Nf = 3 flavorsin the fundamental representation namely Nf = Nc These fundamental degrees offreedom contribute significantly to its thermodynamics at temperatures above Tc Andthe calculations that we shall report are either done with Nf = 0 or with 0 lt Nf NcExtending methods based upon gaugestring duality to the regime in which Nf sim Nc
remains an outstanding challenge
The plasmas of QCD and strongly coupled N = 4 SYM theory certainly differ At the leastusing one to gain insight into the other follows in the long tradition of modelling in which atheoretical physicist employs the simplest instance of a theory that captures the essence of asuite of phenomena that are of interest in order to gain insights N = 4 SYM theory may notseem simple from a field theory perspective but its gravitational description makes it clearthat it is in fact the simplest most symmetric strongly coupled non-Abelian plasma Thequestion then becomes whether there are quantities or phenomena that are universal acrossmany different strongly coupled plasmas The qualitative and in some instances even semi-quantitative successes that we shall review that have been achieved in comparing results orinsights obtained in N = 4 SYM theory to those in QCD suggest a positive answer to thisquestion but no precise definition of this new kind of universality has yet been conjecturedAbsent a precise understanding of such a universality we can hope for reliable insights intoQCD but not for controlled calculations
We begin our description of the N = 4 strongly coupled plasma in this Section by charac-terizing its macroscopic properties ie those that involve temporal and spatial scales muchlarger than the microscopic scale 1T In Section 61 we briefly review the determination ofthe thermodynamics of N = 4 SYM theory The quantities that we calculate are accessiblein QCD via lattice calculations as we have reviewed in Section 3 meaning that in Section61 we will be able to compare calculations done in N = 4 SYM theory via gaugestring du-ality to reliable information about QCD In Section 62 we turn to transport coefficients likethe shear viscosity η which govern the relaxation of small deviations away from thermody-namic equilibrium Lattice calculations of such quantities are in their infancy but as we haveseen in Section 22 phenomenological analyses of collective effects in heavy ion collisions incomparison to relativistic viscous hydrodynamic calculations are yielding information aboutηs in QCD Finally Section 63 will be devoted to illustrating one of the most importantqualitative differences between the strongly coupled N = 4 plasma and any weakly coupledplasma the absence of quasiparticles As we will argue in this Section this is a genericfeature of strong coupling which at least at a qualitative level provides a strong motivationin the context of the physics of QCD above Tc for performing studies within the frameworkof gaugestring duality
121
61 Thermodynamic properties
611 Entropy energy and free energy
As discussed in Section 521 N = 4 SYM theory in equilibrium at nonzero temperature isdescribed in the gravity theory by introducing black-branes which change the AdS5 metricto the black-brane metric (531) and lead to the formation of an event-horizon at positionz0 As in standard black hole physics the presence of the horizon allows us to compute theentropy in the gravity description which is given by the Bekenstein-Hawking formula
Sλ=infin = SBH =A3
4G5 (61)
where A3 is the area of the 3-dimensional event horizon of the non-compact part of the metricand G5 is the five-dimensional Newton constant This entropy is identified as the entropy inthe strong-coupling limit [311] The area A3 is determined from the horizon metric obtainedby setting t = const z = z0 in eqn (531) ie
ds2Hor =
R2
z20
(dx2
1 + dx22 + dx2
3
) (62)
The total horizon area is then
A3 =R3
z30
intdx1dx2dx3 (63)
whereintdx1dx2dx3 is the volume in the gauge theory While the total entropy is infinite the
entropy density per unit gauge-theory volume is finite and is given by
sλ=infin =SBHint
dx1dx2dx3=
R3
4G5z30
=π2
2N2
c T3 (64)
where in the last equality we have used Eqs (512) and (533) to translate the gravity pa-rameters z0 R and G5 into the gauge-theory parameters T and Nc Note that we would haveobtained the same result if we had used the full ten-dimensional geometry which includesthe S5 In this case the horizon would have been 8-dimensional of the form A8 = A3 times S5and the entropy would have taken the form
SBH =A8
4G=A3VS5
4G (65)
which equals (61) by virtue of the relation (512) between the 10- and the 5-dimensionalNewton constants
Once the entropy density is known the rest of the thermodynamic potentials are obtainedthrough standard thermodynamic relations In particular the pressure P obeys s = partPpartT and the energy density is given by ε = minusP + Ts Thus we find
ελ=infin =3π2
8N2
c T4 Pλ=infin =
π2
8N2
c T4 (66)
122
The Nc and temperature dependence of these results could have been anticipated The formerfollows from the fact that the number of degrees of freedom in an SU(Nc) gauge theory in itsdeconfined phase grows as N2
c whereas the latter follows from dimensional analysis since thetemperature is the only scale in the N = 4 theory What is remarkable about these results isthat they show that the prefactors in front of the Nc and temperature dependence in thesethermodynamic quantities attain finite values in the limit of infinite coupling λrarrinfin whichis the limit in which the gravity description becomes strictly applicable
It is instructive to compare the above expressions at infinite coupling with those for the freeN = 4 SYM theory ie at λ = 0 Since N = 4 SYM has 8 bosonic and 8 fermonic adjointdegrees of freedom and since the contribution of each boson to the free entropy is 2π2T 345whereas the contribution of each fermion is 78 of that of a boson the zero coupling entropyis given by
sλ=0 =
(8 + 8times 7
8
)2π2
458(N2
c minus 1)T 3 2π2
3N2
c T3 (67)
where in the last equality we have used the fact that Nc 1 As before the Nc and Tdependences are set by general arguments The only difference between the infinite and zerocoupling entropies is an overall numerical factor comparing Eqs (64) and (67) we find [311]
sλ=infinsλ=0
=Pλ=infinPλ=0
=ελ=infinελ=0
=3
4 (68)
This is a very interesting result while the coupling of N = 4 changes radically between thetwo limits the thermodynamic potentials vary very mildly This observation is in fact notunique to the special case of N = 4 SYM theory but seems to be a generic phenomenon forfield theories with a gravity dual In fact in Ref [386] it was found that for several differ-ent classes of theories each encompassing infinitely many instances the change in entropybetween the infinitely strong and infinitely weak coupling limit is
sstrong
sfree=
3
4h (69)
with h a factor of order one 89 le h le 109662 These explicit calculation strongly suggest
that the thermodynamic potentials of non-Abelian gauge-theory plasmas (at least for near-conformal ones) are quite insensitive to the particular value of the gauge coupling This isparticularly striking since as we will see in Sections 62 and 63 the transport propertiesof these gauge theories change dramatically as a function of coupling going from a nearlyideal gas-like plasma of quasiparticles at weak coupling to a nearly ideal liquid with noquasiparticles at strong coupling So we learn an important lesson from the calculations ofthermodynamics at strong coupling via gaugestring duality thermodynamic quantities arenot good observables for distinguishing a weakly coupled gas of quasiparticles from a stronglycoupled liquid transport properties and the physical picture of the composition of the plasmaare completely different in these two limits but no thermodynamic quantity changes much
Returning to the specific case of N = 4 SYM theory in this case the leading finite-λcorrection to (68) has been calculated yielding [387]
sλ=infinsλ=0
=Pλ=infinPλ=0
=ελ=infinελ=0
=3
4+
169
λ32+ (610)
123
suggesting that this ratio increases from 34 to 78 as λ drops from infinity down to λ sim 6corresponding to αSYM sim 05Nc This reminds us that the control parameter for the strong-coupling approximation is 1λ meaning that it can be under control down to small values ofαSYM The ratio (68) is also expected to receive corrections of order 1N2
c but these havenot been computed
It is also interesting to compare (68) to what we know about QCD thermodynamics fromlattice calculations like those described in Section 31 The ratio (68) has the advantage thatthe leading dependence on the number of degrees of freedom drops out making it meaningfulto compare directly to QCD While theories which have been analyzed in Ref [386] arerather different from QCD the regularity observed in these theories compel us to evaluatethe ratio of the entropy density computed in the lattice calculations to that which would beobtained for free quarks and gluons Remarkably Fig 31 shows that for T = (2 minus 3)Tcthe coefficient defined in (69) is h 107 which is in the ballpark of what the calculationsdone via gaugegravity duality have taught us to expect for a strongly coupled gauge theoryWhile this observation is interesting by itself it is not strong evidence that the QCD plasmaat these temperatures is strongly coupled The central lesson is in fact that the ratio (68)is quite insensitive to the coupling The proximity of the lattice results to the value for freequarks and gluons should never have been taken as indicating that the quark-gluon plasmaat these temperatures is a weakly coupled gas of quasiparticles And now that experimentsat RHIC that we described in Section 22 combined with calculations that we shall reviewin Section 62 have shown us a strongly coupled QCD plasma the even closer proximity ofthe lattice results for QCD thermodynamics to that expected for a strongly coupled gaugetheory plasma should also not be overinterpreted
612 Holographic susceptibilities
The previous discussion focussed on a plasma at zero chemical potential micro While gaugegravityduality allows us to explore the phase diagram of the theory at nonzero values of micro in orderto parallel our discussion of QCD thermodynamics in Section 3 in our analysis of stronglycoupled N = 4 SYM theory here we will concentrate on the calculation of susceptibilities Asexplained in Section 311 their study requires the introduction of U(1) conserved chargesIn N=4 SYM there is an SU(4) global symmetry the R-symmetry which in the dual grav-ity theory corresponds to rotations in the 5-sphere Chemical potential is introduced bystudying rotating black-holes in these coordinates [329331388389] these solutions demandnon-vanishing values of an abelian vector potential Amicro in the gravitational theory whichin turn lead to a non-vanishing charge density n in the gauge theory proportional to theangular momentum density of the black hole The chemical potential can be extracted fromthe boundary value of the temporal component of the Maxwell field as in (534) and is also afunction of the angular momentum of the black hole The explicit calculation performed inRef [390] leads to
n =N2c T
2
8micro (611)
in the small chemical potential limit Note that unlike in QCD the susceptibility dndmicroinferred from Eq (611) is proportional to N2
c instead of Nc This is a trivial consequence ofthe fact that R-symmetry operates over adjoint degrees of freedom
124
As in the case of the entropy the different number of degrees of freedom can be taken intoaccount by comparing the susceptibility at strong coupling to the free theory result whichyields
χλ=infinχλ=0
=1
2 (612)
where χλ=0 = N2c T
24 [215] Similarly to the case of the entropy density the ratios ofsusceptibilities between these two extreme limits saturates into an order one constant Inspite of the radical change in the dynamics of the degrees of freedom in the two systemsall the variation in this observable is a 50 reduction comparable to the 25 reductionof the energy density in the same limit On the other hand the lattice calculations inSection 311 show that above Tc the susceptibilities approach closer to their values in thefree limit more quickly than in the case of the energy density In fact the rapid approachof the susceptibilities to the free limit has been interpreted by some as a sign that the QCDquark-gluon plasma is not strongly coupled [212 391ndash393] Another possible interpretationof this fact could be that while the gluonic (adjoint) degrees of freedom in the plasma arestrongly coupled the fermionic (fundamental) ones remain quasi-free This interpretationis based on the observation that for fermions the lowest Matsubara frequency ωF = πT is different than that for gluons ωG = 0 and thus the thermodynamics of fermions areinsensitive to the softer components of the gluon fields This possible interpretation of thediscrepancy between the N = 4 SYM susceptibilities and the QCD susceptibilities wouldhinge on the fermions in the QCD plasma remaining weakly coupled at scales sim πT whilesofter gluonic modes become strongly coupled We note that if this interpretation is takenliterally the quarks in the QCD plasma should increase its ηs which does not seem to beindicated by the elliptic flow data Nevertheless it is not clear whether this deviation posesa serious challenge to the interpretation of the QCD plasma as strongly coupled given themanifest insensitivity of thermodynamic quantities to the coupling Furthermore the valueof this deviation is not universal and it can be different for other holographic models whichare closer to QCD than N = 4 SYM It is in fact conceivable that if it were possible touse gaugestring duality to analyze strongly coupled theories with Nf sim Nc and computethe susceptibility for a U(1) charge carried by the fundamental degrees of freedom in sucha theory it would turn out to be close to that at weak coupling as in QCD even when alldegrees of freedom are strongly coupled Were this speculation to prove correct it wouldbe an example of a result from QCD leading to insight into strongly coupled gauge theorieswith a gravitational description ie it would be an example of using the duality in theopposite direction from that in which we apply it throughout most of this review We shallnot speculate further on this discrepancy simply acknowledging that its interpretation is anopen question
62 Transport properties
We now turn to the calculation of the transport coefficients of a strongly coupled plasmawith a dual gravitational description which control how such a plasma responds to smalldeviations from equilibrium As we have reviewed in Section 32 since the relaxation ofthese perturbations is intrinsically a real time process the lattice determination of trans-
125
port coefficients is very challenging While initial steps toward determining them in QCDhave been taken definitive results are not in hand As a consequence the determination oftransport coefficients via gaugestring duality is extremely valuable since it opens up theiranalysis in a regime which is not tractable otherwise A remarkable consequence of thisanalysis which we review in Section 622 is a universal relation between the shear viscosityand the entropy density for the plasmas in all strongly coupled large-Nc gauge theories witha gravity dual [394ndash397] This finding together with the comparison of the universal resultηs = 1(4π) with values extracted by comparing data on elliptic flow in heavy ion collisionsto analyses in terms of viscous hydrodynamics as we have reviewed in Section 22 has beenone of the most influential results obtained via the gaugestring duality
621 A general formula for transport coefficients
The most straightforward way in which transport coefficients can be determined using thegaugegravity correspondence is via Green-Kubo formulas see Appendix A which rely onthe analysis of the retarded correlators in the field theory at small four-momentum Theprocedure for determining these correlators using the correspondence has been outlined inSection 53 In this section we will try to keep our analysis as general as possible so thatit can be used for the transport coefficent that describes the relaxation of any conservedcurrent in the theory In addition we will not restrict ourselves to the particular form ofthe metric (531) so that our discussion can be applied to any theory with a gravity dualOur discussion will closely follow the formalism developed in [369] which builds upon earlieranalyses in Refs [394ndash401]
In general if the field theory at nonzero temperature is invariant under translations androtations the gravitational theory will be described by a (4 + 1)-dimensional metric of theform
ds2 = minusgttdt2 + gzzdz2 + gxxδijdx
idxj = gMNdxMdxN (613)
with all the metric components solely dependent on z Since a nonzero temperature ischaracterized in the dual theory by the presence of an event horizon we will assume that gtthas a first order zero and gzz has a first order pole at a particular value z = z0
We are interested in computing the transport coefficient χ associated with some operatorO in this theory namely
χ = minus limωrarr0
lim~krarr0
1
ωImGR(ωk) (614)
(See Appendix A for the exact definition of GR and for a derivation of this formula) Forconcreteness we assume that the quadratic effective action for the bulk mode φ dual to Ohas the form of a massless scalar field1
S = minus1
2
intdd+1x
radicminusg 1
q(z)gMNpartMφpartNφ (615)
where q(z) is a function of z and can be considered a spacetime-dependent coupling constantAs we will see below Eqs (613) and (615) apply to various examples of interest including
1Note that restricting to a massless mode does not result in much loss of generality since almost alltransport coefficients calculated to date are associated with operators whose gravity duals are massless fieldsThe only exception is the bulk viscosity
126
the shear viscosity and the momentum broadening for the motion of a heavy quark in theplasma Since transport coefficients are given by the Green-Kubo formula Eq (614) thegeneral expression for the retarded correlator (557) with ∆ = d since m = 0 leads to
χ = minus limkmicrorarr0
limzrarr0
Im
Π(z kmicro)
ωφ(z kmicro)
= minus lim
kmicrorarr0limzrarr0
Π(z kmicro)
iωφ(z kmicro) (616)
where Π is the canonical momentum of the field φ
Π =δS
δpartzφ= minusradicminusg
q(z)gzzpartzφ (617)
The last equality in (616) follows from the fact that the real part of GR(k) vanishes fasterthan linearly in ω as k rarr 0 as is proven by the fact that the final result that we will obtainEq (624) is finite and real
In (616) both Π and φ must be solutions of the classical equations of motion which inthe Hamiltonian formalism are given by (617) together with
partzΠ = minusradicminusg
q(z)gmicroνkmicrokνφ (618)
The evaluation of χ following Eq (616) requires the determination of both ωφ and Π inthe small four momentum kmicro rarr 0 limit Remarkably in this limit the equations of motion(617) and (618) are trivial
partzΠ = 0 +O(kmicroωφ) partz(ωφ) = 0 +O(ωΠ) (619)
and both quantities become independent of z which allows their evaluation at any z Forsimplicity and since the only restriction on the general metric (613) is that it possessesa horizon we will evaluate them at z0 where the in-falling boundary condition should beimposed Our assumptions on the metric imply that in the vicinity of the horizon z rarr z0
gtt = minusc0(z0 minus z) gzz =cz
z0 minus z (620)
and eliminating Π from (617) and (618) we find an equation for φ given byradicc0
cz(z0 minus z)partz
(radicc0
cz(z0 minus z)partzφ
)+ ω2φ = 0 (621)
The two general solutions for this equation are
φ prop eminusiωt (z0 minus z)plusmniωradicczc0 (622)
Imposing in-falling boundary condition implies that we should take the negative sign in theexponent Therefore from Eq (622) we find that at the horizon
partzφ =
radicgzzminusgtt
(iωφ) (623)
127
and using Eqs (617) and (622) we obtain
χ = minus limkmicrorarr0
limzrarr0
Π(z kmicro)
iωφ(z kmicro)= minus lim
kmicrorarr0limzrarrz0
Π(z kmicro)
iωφ(z kmicro)=
1
q(z0)
radicminusgminusgzzgtt
∣∣∣∣z0
(624)
Note that the last equality in (624) can also be written as
χ =1
q(z0)
A
V (625)
where A is the area of the horizon and V is the spatial volume of the boundary theory Fromour analysis of the thermodynamic properties of the plasma in Section 6 the area of theevent horizon is related to the entropy of the boundary theory via
s =A
V
1
4GN (626)
From this analysis we conclude that for any theory with a gravity dual the ratio of anytransport coefficient to the entropy density depends solely on the properties of the dual fieldsat the horizon
χ
s=
4GNq(z0)
(627)
In the next subsection we will use this general expression to compute the shear viscosity ofthe AdS plasma
Finally we would like to remark that the above discussion applies to more general effectiveactions of the form
S = minus1
2
intdωddminus1k
(2π)ddzradicminusg[gzz(partzφ)2
Q(zω k)+ P (zω k)φ2
] (628)
provided that the equations of motion (619) remain trivial in the zero-momentum limit Thisimplies that Q should go to a nonzero constant at zero momentum and P must be at leastquadratic in momenta For (628) the corresponding transport coefficient χ is given by
χ =1
Q(z0 kmicro = 0)
A
Vand
χ
s=
4GNQ(z0 kmicro = 0)
(629)
622 Universality of the Shear Viscosity
We now apply the result of last subsection to the calculation of the shear viscosity η of astrongly coupled plasma described by the metric (613) As in Appendix A we must computethe correlation function of the operatorO = Txy where the coordinates x and y are orthogonalto the momentum vector The bulk field φ dual to O should have a metric perturbation hxyas its boundary value It then follows that φ = (δg)xy = gzzhxy where δg is the perturbationof the bulk metric For Einstein gravity in a geometry with no off-diagonal components in thebackground metric as in (613) a standard analysis of the Einstein equations to linear orderin the perturbation upon assuming that the momentum vector is perpendicular to the (x y)
128
plane shows that the effective action for φ is simply that of a minimally coupled masslessscalar field namely
S = minus 1
16πGN
intdd+1x
radicminusg
[1
2gMNpartMφpartNφ
] (630)
The prefactor 116πGN descends from that of the Einstein-Hilbert action This action hasthe form of Eq (615) with
q(z) = 16πGN = const (631)
which together with Eq (627) leads to the celebrated result
ηλ=infinsλ=infin
=1
4π(632)
that was first obtained in 2001 by Policastro Son and Starinets [398] In (631) we haveadded the subscript λ =infin to stress that the numerator and denominator are both computedin the strict infinite coupling limit Remarkably this ratio converges to a constant at strongcoupling And this is not only a feature of N = 4 SYM theory because this derivation appliesto any gauge theory with a gravity dual given by Einstein gravity coupled to matter fieldssince in Einstein gravity the coupling constant for gravity is always given by Eq (631) Inthis sense this result is universal [394ndash397] since it applies in the strong-coupling and large-Nc limits to the large class of theories with a gravity dual regardless of whether the theoriesare conformal or not confining or not supersymmetric or not and with or without chemicalpotential In particular if large-Nc QCD has a gravity dual its ηs should also be given by1(4π) up to corrections due to the finiteness of the coupling Even if large-Nc QCD does nothave a gravity dual Eq 632 may still apply since the universality of this result may be dueto generic properties of strongly coupled theories (for example the absence of quasiparticlessee Section 63) which may not depend on whether they are dual to a gravitational theory
The finite-coupling corrections to Eq (632) in N = 4 SYM theory are given by [402ndash405]
ηλrarrinfinsλrarrinfin
=1
4π
(1 +
15 ζ(3)
λ32+
) (633)
where ζ(3) = 120 is Aperyrsquos constant While the coefficient of the leading finite-λ correctionproportional to 1λ32 is universal when expressed in terms of the parameters R and ls in thegravity theory this coefficient is different for different gauge theories [406] since the relationbetween R and ls and the gauge theory parameters λ and Nc changes Thus expression (633)is only valid for N = 4 SYM theory It is interesting to notice that according to Eq (633)ηs increases to sim 2(4π) once λ decreases to λ sim 7 meaning αSYM sim 05Nc This is thesame range of couplings at which the finite coupling corrections (610) to thermodynamicquantities become significant These results together suggest that strongly coupled theorieswith gravity duals may yield insight into the quark-gluon plasma in QCD even down toapparently rather small values of αs at which λ is still large
To put the result (632) into further context we can compare this strong coupling resultto results for the same ratio ηs at weak coupling in both N = 4 SYM theory and QCDThese have been computed at next to leading log accuracy and take the form
ηλrarr0
sλrarr0=
A
λ2 log(Bradicλ) (634)
129
with A = 6174 and B = 236 in N = 4 SYM theory and A = 348 (461) and B = 467(417) in QCD with Nf = 0 (Nf = 3) [407 408] where we have defined λ = g2Nc inQCD as in N = 4 SYM theory Quite unlike the strong coupling result (632) these weak-coupling results show a strong dependence on λ and in fact diverge in the weak-couplinglimit The divergence reflects the fact that a weakly-coupled gauge-theory plasma is a gasof quasiparticles with strong dissipative effects In a gas ηs is proportional to the ratio ofthe mean-free path of the quasiparticles to their average separation A large mean-free pathand hence a large ηs mean that momentum can easily be transported over distances thatare long compared to the average spacing between particles In the λ rarr 0 limit the mean-free path diverges The strong 1λ2 dependence of ηs can be traced to the fact that thetwo-particle scattering cross-section is proportional to g4 It is reasonable to guess that theλ-dependence of ηs in N = 4 SYM theory is monotonic increasing from 1(4π) as in (633)as λ decreases from infin and then continuing to increase until it diverges according to (634)as λ rarr 0 The weak-coupling result (634) also illustrates a further important point ηs isnot universal for weakly-coupled gauge theory plasmas The coefficients A and B can varysignificantly from one theory to another depending on their particle content It is only in thestrong coupling limit that universality emerges with all large-Nc theories with a gravity dualhaving plasmas with ηs = 1(4π) And we shall see in Section 63 that a strongly coupledgauge theory plasma does not have quasiparticles which makes it less surprising that ηs atstrong coupling is independent of the particle content of the theory at weak coupling
One lesson from the calculations of ηs is that this quantity changes significantly withthe coupling constant going from infinite at zero coupling to 1(4π) at strong coupling atleast for large-Nc theories with gravity duals This is in marked contrast to the behavior ofthe thermodynamic quantities described in Section 6 which change only by 25 over thesame large range of couplings Thermodynamic observables are insensitive to the couplingwhereas ηs is a much better indicator of the strength of the coupling because it is a measureof whether the plasma is liquid-like or gaseous
These observations prompt us to revisit the phenomenological extraction of the shearviscosity of quark-gluon plasma in QCD from measurements of elliptic flow in heavy ioncollisions described in Section 22 As we saw the comparison between data and calculationsdone using relativistic viscous hydrodynamic yields a conservative conclusion that ηs lt(3minus5)(4π) with a current estimate being that ηs seems to lie within the range (1minus25)(4π)in QCD in the same ballpark as the strong-coupling result (632) And as we reviewed inSection 32 current lattice calculations of ηs in Nf = 0 QCD come with caveats but alsoindicate a value that is in the ballpark of 1(4π) likely somewhat above it Given thesensitivity of ηs to the coupling these comparisons constitute one of the main lines ofevidence that in the temperature regime accessible at RHIC the quark-gluon plasma is astrongly coupled fluid If we were to attempt to extrapolate the weak-coupling result (634)for ηs in QCD with Nf = 3 to the values of ηs favored by experiment we would needλ sim (14minus 24) well beyond the regime of applicability of perturbation theory (To make thisestimate we had to set the log in (634) to 1 to avoid negative numbers which reflects thefact that the perturbative result is being applied outside its regime of validity)
A central lesson from the strong-coupling calculation of ηs via gaugestring duality ar-guably even more significant than the qualitative agreement between the result (632) and cur-
130
rent extractions of ηs from heavy ion collision data is simply the fact that values of ηs 1are possible in non-abelian gauge theories and in particular in non-abelian gauge theorieswhose thermodynamic observables are not far from weak-coupling expectations These cal-culations done via gaugestring duality provided theoretical support for considering a rangeof small values of ηs that had not been regarded as justified before and inferences drawnfrom RHIC data have pushed ηs into this regime The computation of the shear viscositythat we have just reviewed is one of the most influential results supporting the notion thatthe application of gaugegravity duality can yield insights into the phenomenology of hotQCD matter
It has also been conjectured [396] that the value of ηs in Eq (632) is in fact a lower boundfor all systems in nature This conjecture is supported by the finite-coupling correctionsshown in Eq (633) And all substances known in the laboratory satisfy the bound Amongconventional liquids the lowest ηs is achieved by liquid helium but it is about an order ofmagnitude above 1(4π) water mdash after which hydrodynamics is named mdash has an ηs that islarger still by about another order of magnitude The best liquids known in the laboratoryare the quark-gluon plasma produced in heavy ion collisions and an ultracold gas of fermionicatoms at the unitary point at which the s-wave atom-atom scattering length has been dialedto infinity [2] both of which have ηs in the ballpark of 1(4π) but according to currentestimates somewhat larger
However in recent years the conjecture that (632) is a lower bound on ηs has beenquestioned and counter-examples have been found among theories with gravity duals Asemphasized in Section 5 Einstein gravity in the dual gravitational description correspondsto the large-λ and large-Nc limit of the boundary gauge theory When higher order correc-tions to Einstein gravity are included which correspond to 1
radicλ or 1Nc corrections in the
boundary gauge theory Eq (632) will no longer be universal In particular as pointed outin Refs [409 410] and generalized in Refs [411ndash419] generic higher derivative correctionsto Einstein gravity can violate the proposed bound Eq (629) indicates that ηs is smallerthan Eq (632) if the ldquoeffectiverdquo gravitational coupling for the hxy polarization at the horizonis stronger than the universal value (631) for Einstein gravity Gauss-Bonnet gravity as dis-cussed in Refs [409 420] is an example in which this occurs There the effective action forhyx has the form of Eq (628) with the effective coupling Q(r) at the horizon satisfying [409]
1
Q(r0)=
(1minus 4λGB)
16πGN (635)
leading toη
s=
(1minus 4λGB)
4π (636)
where λGB is the coupling for the Gauss-Bonnet higher-derivative term Thus for λGB gt 0the graviton in this theory is more strongly coupled than that of Einstein gravity and thevalue of ηs is smaller than 14π In Ref [410] an explicit gauge theory has been proposedwhose gravity dual corresponds to λGB gt 0 (See Refs [405421] for generalizations) Despitenot being a lower bound the smallness of ηs the qualitative agreement between Eq (632)and values obtained from heavy ion collisions and the universality of the result (632) whichapplies to any gauge theory with a gravity dual in the large-Nc and strong-coupling limits
131
are responsible for the great impact that this calculation done via gaugestring duality hashad on our understanding of the properties of deconfined QCD matter
623 Bulk viscosity
As we have discussed in Section 22 while the bulk viscosity ζ is very small in the QCDplasma at temperatures larger than 15minus 2Tc with ζs much smaller than 14π ζs rises inthe vicinity of Tc a feature which can be important for heavy ion collisions Since the plasmaof a conformal theory has zero bulk viscosity N = 4 SYM theory is not a useful exampleto study the bulk viscosity of a strongly coupled plasma However the bulk viscosity hasbeen calculated both in more sophisticated examples of the gaugestring duality in which thegauge theory is not conformal [422ndash426] as well as in AdSQCD models that incorporate anincrease in the bulk viscosity near a deconfinement phase transition [51427428]
We will only briefly review what is possibly the simplest among the first type of examplesthe so-called Dp-brane theory This is a (p + 1)-dimensional cousin of N = 4 SYM namelya (p + 1)-dimensional SYM theory (with 16 supercharges) living at the boundary of thegeometry describing a large number of non-extremal black Dp-branes [355] with p 6= 3 Thecase p = 3 is N = 4 SYM while the cases p = 2 and p = 4 correspond to non-conformaltheories in (2 + 1)- and (4 + 1)-dimensions We emphasize that we choose this example forits simplicity rather than because it is directly relevant for phenomenology
The metric sourced by a stack of black Dp-branes can be written as
ds2 = αprime(dpλz
3minusp)1
5minusp
z2
(minusfdt2 + ds2
p +
(2
5minus p
)2 dz2
f+ z2dΩ2
8minusp
) (637)
where
λ = g2N f = 1minus(z
z0
) 14minus2p5minusp
dp = 27minus2pπ9minus3p
2 Γ
(7minus p
2
)(638)
andg2 = (2π)pminus2gsα
prime3minusp2 (639)
is the Yang-Mills coupling constant which is dimensionful if part 6= 3 For p = 2 and 4 there isalso a nontrivial profile for the dilaton field but we shall not give its explicit form here Themetric above is dual to (p+ 1)-dimensional SYM theory at finite temperature
The bulk viscosity can be computed from the dual gravitational theory via the Kuboformula (A10) However this computation is more complicated in the bulk channel thanin the shear channel and we will not reproduce it here An alternative and simpler way tocompute the bulk viscosity is based on the fact that in the hydrodynamic limit the soundmode has the following dispersion relation
ω = csq minusi
ε+ p
(dminus 1
dminus 2η +
ζ
2
)q2 + middot middot middot (640)
with cs the speed of sound Thus ζ contributes to the damping of sound In the fieldtheory the dispersion relation for the sound mode can be found by examining the polesof the retarded Greenrsquos function for the stress tensor in the sound channel As discussed
132
in Section 531 on the gravity side these poles correspond to normalizable solutions to theequations of motion for metric perturbations Due to the in-falling boundary conditions at thehorizon see Eq (555) the frequencies of these modes have nonzero and negative imaginaryparts and they are therefore called quasi-normal modes The explicit computation of thesemodes for the AdS black hole metric can be used to extract the dispersion relation for soundwaves in the strongly coupled N = 4 SYM plasma yielding an alternative derivation of itsshear viscosity [429] The explicit computation of these quasi-normal modes for the metric(637) performed in Ref [425] showed that the sound mode has the dispersion relation
ω =
radic5minus p9minus p
q minus i 2
9minus pq2
2πT+ middot middot middot (641)
from which one finds that (after using ηs = 1(4π))
cs =
radic5minus p9minus p
ζ
s=
(3minus p)2
2πp(9minus p) (642)
The above expressions imply an interesting relation [424]
ζ
η= 2
(1
pminus c2
s
)= 2
(c2sCFT minus c2
s
) (643)
where we have used the fact that the sound speed for a CFT in (p+ 1)-dimension is csCFT =1radicp This result might not seem surprising since the bulk viscosity of a theory which is close
to being conformal can be expanded in powers of c2sCFTminus c2
s which is a measure of deviationfrom conformality The nontrivial result is that even though the Dp-brane gauge theoriesare not close to being conformal their bulk viscosities are nevertheless linear in c2
sCFT minus c2s
While this is an interesting observation it is not clear to what extent it is particular to theDp-brane gauge theories or whether it is more generic
624 Relaxation times and other 2nd order transport coefficients
As we have reviewed in Section 223 transport coefficients correspond to the leading ordergradient expansion of an interacting theory which corrects the hydrodynamic descriptionA priori there is no reason to stop the extraction of these coefficients at first order andhigher order ones can be (and have been) computed using gaugestring duality Of particularimportance is the determination of the five second order coefficients τπ κ λ1 λ2 λ3 definedin Eq (221) Unlike for the first order coefficients the gravitational computation of thesesecond order coefficients is quite technical and we shall not review it here We shall onlydescribe the main points and refer the reader to Refs [90 91] for details
The strategy for determining these coefficients is complicated by the fact that the threecoefficients λi involve only non-linear combinations of the hydrodynamic fields Thus eventhough formulae can be derived for the linear coefficients τπ and κ [90 430] the non-linearcoefficients cannot be determined from two-point correlators since these coefficients are in-visible in the linear perturbation analysis of the background Their determination thus
133
demands the small gradient analysis of non-linear solutions to the Einstein equations2 asperformed in Ref [91] (see also Ref [90]) which yields
τπ = 2minusln 22πT κ = η
πT (644)
λ1 =η
2πT λ2 = minusη ln 2
πT λ3 = 0
These results are valid in the large-Nc and strong-coupling limit Finite coupling correctionsto some of these coefficients can be found in Ref [431] Additionally the first and secondorder coefficients have been studied in a large class of non-conformal theories with or withoutflavor in Refs [432433]
To put these results in perspective we will compare them to those extracted in the weaklycoupled limit of QCD (λ 1) [434] We shall not comment on the values of all the coefficientssince as discussed in Section 223 the only one with any impact on current phenomenologicalapplications to heavy ion collisions is the shear relaxation time τπ In the weak-coupling limit
limλrarr0
τπ 59
T
η
s (645)
where the result is expressed in such a way as to show that τπ and η have the same leading-order dependence on the coupling λ (up to logarithmic corrections) For comparison thestrong-coupling result from (644) may be written as
limλ1
τπ =72
T
η
s (646)
which is remarkably close to (645) But of course the value of ηs is vastly different in theweak- and strong-coupling limits On general grounds one may expect that relaxation andequilibration processes are more efficient in the strong coupling limit since they rely on theinteractions between different modes in the medium This general expectation is satisfied forthe shear relaxation time of the N = 4 SYM plasma with τπ diverging at weak-coupling andtaking on the small value
limλrarrinfin
τπ 0208
T (647)
in the strong-coupling limit For the temperatures T gt 200 MeV which are relevant for thequark-gluon plasma produced in heavy ion collisions this relaxation time is of the order of02 fmc which is much smaller than perturbative expectations We have recalled already atother places in this review that caveats enter if one seeks quantitative guidance for heavy ionphenomenology on the basis of calculations made for N = 4 SYM plasma However the qual-itative (and even semi-quantitative) impact of the result (647) on heavy ion phenomenologyshould not be underestimated the computation of τπ demonstrated for the first time that atleast some excitations in a strongly coupled non-abelian plasma dissipate on timescales thatare much shorter than 1T ie on time scales much shorter than 1 fmc Such small valuesdid not have a theoretical underpinning before and they are clearly relevant for phenomeno-logical studies based on viscous fluid dynamic simulations As we reviewed in Section 22
2 Kubo-like formulas involving three-point correlators (as opposed to two) can also be use to determine thecoefficients λi [430] At the time of writing this approach had not been explored within the gaugegravitycontext
134
the success of the comparison of such simulations to heavy ion collision data implies that ahydrodynamic description of the matter produced in these collisions is valid only sim 1 fmcafter the collision Although this equilibration time is related to out-of-equilibrium dynam-ics whereas τπ is related to near-equilibrium dynamics (only to second order) the smallnessof τπ makes the rapid equilibration time seem less surprising As in the case of ηs thegaugegravity calculation of τπ has made it legitimate to consider values of an importantparameter that had not been considered before by showing that this regime arises in thestrongly coupled plasma of a quantum field theory that happens to be accessible to reliablecalculation because it possesses a gravity dual
Let us also mention that the second order transport coefficients are known for the samenon-conformal gauge theories whose bulk viscosity we discussed in Section 623 Since con-formal symmetry is broken in these models there are a total of 15 first- and second-ordertransport coefficients 9 more than in the conformal case (including both shear and bulkviscosities in the counting) [92] In addition the velocity of sound cs is a further indepen-dent parameter that characterizes the zeroth-order hydrodynamics of non-conformal plasmaswhose equations of state are not given simply by P = ε3 As for the case of the bulk vis-cosity the variable
(13 minus c
2s
)can be used to parametrize deviations from conformality and all
transport coefficients can indeed be written explicitly as functions of(
13 minus c
2s
)[435]
Let us conclude this section with a curious remark As we saw in Section 223 thesecond order hydrodynamic equations are hyperbolic and as such they are causal Howeverhydrodynamic equations are a small gradient expansion of the full theory and there is no apriori reason why hydrodynamic dispersion relations must provide physically sensible resultsfor short wavelength highly energetic excitations In particular the group velocity dω(k)
dk ofa hydrodynamic excitation does not need to remain smaller than the velocity of light in thelimit k rarr infin And yet the structure of the second order viscous hydrodynamic equationsensures that the group velocities of all hydrodynamic excitations approach finite values inthe limit k rarr infin For example the hydrodynamic propagation speeds for the shear modeand the sound mode depend on (ε + p) = T s the shear and bulk viscosities as well as thecorresponding relaxation times τπ and τΠ In the limit k rarr infin their group velocities takethe simple and general forms
limkrarrinfin
dωsound(k)
dk=
radicη
τπ (ε+ p) (648)
limkrarrinfin
dωshear(k)
dk=
radicc2s +
4
3
η
τπ (ε+ p)+
ζ
τΠ (ε+ p) (649)
For a derivation see Ref [111] In principle one could find a field theory with relaxation
times τπ andor τΠ that are so short that the limit limkrarrinfindω(k)dk of some mode exceeds the
velocity of light in this case one would simply state that this limit is outside the range ofvalidity of hydrodynamics But for these two modes and for all hydrodynamic modes studiedso far and for all field theories studied so far when the values of all quantities are pluggedinto these expressions the resulting limkrarrinfin
dω(k)dk describes propagation within the forward
light cone We emphasize that the current understanding of this finding admits the possibilitythat it is accidental
135
63 Quasiparticles and spectral functions
In Sections 61 and 62 we have illustrated the power of gaugestring duality by performingin a remarkably simple way computations that via standard field theoretical methods eithertake Teraflop-years of computer time or are not accessible However the simplicity of thecalculations comes with a price Because we do the calculations in the dual gravitationaldescription of the theory all we get are reliable results we do not get the kind of intuition ofwhat is happening in the gauge theory that we would get automatically from a field theorycalculation done with Feynman diagrams or could get with effort from one done on the latticeThe gravitational calculation yields answers and new kinds of intuition but since by using itwe are abandoning the description of the plasma in terms of quark and gluon quasi-particlesinteracting with each other we are losing our prior physical intuition about how the dynamicsof the gauge theory works In particular from the results that we extract alone it is difficultto understand whether the dynamics within a strongly coupled plasma differs in a qualitativeway from those in a weakly coupled plasma or merely differs quantitatively We have givenup the description in terms of quasiparticles but maybe the familiar quasiparticles or somenew kind of quasiparticles are in fact nevertheless present We rule out this possibility inthis Section illustrating that a strongly coupled non-abelian gauge-theory plasma really isqualitatively different than a weakly coupled one while in perturbation theory the degrees offreedom of the plasma are long-lived quasiparticle excitations which carry momentum colorand flavor there are no quasiparticles in the strongly coupled plasma The pictures that weare used to using that frame how we think about a weakly coupled plasma are simply invalidfor the strongly coupled case
Determining whether a theory possesses quasiparticles with a given set of quantum numbersis a conceptually well defined task it suffices to analyze the spectral function of operatorswith that set of quantum numbers and look for narrow peaks in momentum space In weaklycoupled Yang-Mills theories the quasiparticles (gluons and quarks in QCD) are colored andare identified by studying operators that are not gauge invariant Within perturbation theoryit can be shown that the poles of these correlators which determine the physical propertiesof the quasiparticles are gauge invariant [436] However non-perturbative gauge-invariantoperators corresponding to these excitations are not known which complicates the searchfor these quasiparticles at strong coupling Note however that even if such operators wereknown demonstrating the absence of quasiparticles with the same quantum numbers as in theperturbative limit does not guarantee the absence of quasiparticles since at strong couplingthe system could reorganize itself into quasiparticles with different quantum numbers Thusproving the absence of quasiparticles along these lines would require exploring all possiblespectral functions in the theory Fortunately there is an indirect method which can answerthe question of whether any quasiparticles that carry some conserved lsquochargersquo (includingmomentum) exist although this method cannot determine the quantum numbers of thelong-lived excitations if any are found to exist The method involves the analysis of thesmall frequency structure of the spectral functions of those conserved currents of the theorywhich do not describe a propagating hydrodynamic mode like sound As we will see thepresence of quasiparticles leads to a narrow structure (the transport peak) in these spectralfunctions [215216] In what follows we will use this method to demonstrate that the stronglycoupled N = 4 SYM plasma does not possess any quasiparticles that carry momentum In
136
order to understand how the method works we first apply it at weak coupling where thereare quasiparticles to find
631 Quasiparticles in perturbation theory
We start our analysis by using kinetic theory to predict the general features of the low-frequency structure of correlators of conserved currents in a weakly coupled plasma Kinetictheory is governed by the Boltzmann equation which describes excitations of a quasiparticlesystem at scales which are long compared to the inter-particle separation The applicabilityof the kinetic description demands that there is a separation of scales such that the durationof interactions among particles is short compared to their mean free path (λmfp) and thatmultiparticle distributions are consequently determined by the single particle distributionsIn Yang-Mills theories at nonzero temperature and weak coupling kinetic theory is importantsince it coincides with the Hard Thermal Loop description [437ndash442] which is the effectivefield theory for physics at momentum scales of order gT and the Boltzmann equation can bederived from first principles [442ndash447] In Yang-Mills theory at weak coupling and nonzerotemperature the necessary separation of scales arises by virtue of the small coupling constantg since λmfp sim 1(g4T ) and the time-scale of interactions is 1microD sim 1(gT ) where 1microDis the Debye screening length of the plasma3 The small value of the coupling constant alsoleads to the factorization of higher-point correlation functions
In the kinetic description the system is characterized by a distribution function
f(xp) (650)
which determines the number of particles of momentum p at space-time position x Note thatthis position should be understood as the center of a region in space-time with a typical sizemuch larger at least than the de-Broglie wave length of the particles as demanded by theuncertainty principle As a consequence the Fourier transform of x which we shall denoteby K = (ωq) must be much smaller than the typical momentum scale of the particlesK p sim T (Here and below when we write a criterion like K p we mean that both ωand |q| must be |p|) Due to this separation in momentum scales the x-dependence ofthe distribution functions is said to describe the soft modes of the gauge theory while themomenta p are those of the hard modes If K is sufficiently small (smaller than the inverseinter-particle separation sim T ) the mode describes collective excitations which involve themotion of many particles while p is the momentum of those particles In this case theFourier-transformed distribution f(Kp) can be interpreted approximately as the numberof particles within the wavelength of the excitation At the long distances at which thekinetic-theory description is valid particles are on mass shell as determined by the positionof the peaks in the correlation functions of the relevant operators (p0 = Ep) and these hardmodes describe particles that follow classical trajectories at least between the microscopiccollisions All the properties of the system can be extracted from the distribution function
3Strictly speaking λmfp sim 1(g4T ) is the length-scale over which an order 1 change of the momentum-vector of the quasiparticles occurs Over the shorter length-scale 1microD soft exchanges (of order gT notenough to change the momenta which are sim T significantly) occur These soft exchanges are not relevant fortransport
137
In particular the stress tensor is given by
Tmicroν(x) =
intd3p
(2π)3
pmicropν
Ef(xp) (651)
Since all quasiparticles carry energy and momentum we will concentrate only on thekinetic theory description of stress-tensor correlators Our analysis is analogous to the oneperformed for the determination of the Green-Kubo formulae in Appendix A and proceedsby studying the response of the system to small metric fluctuations The dynamics are thengoverned by the Boltzmann equation which states the continuity of the distribution functionf up to particle collisions [448]
Ed
dtf(xp) = pmicropartxmicrof(xp) + Ep
dp
dt
part
partpf(xp) = C [f ] (652)
where C [f ] is the collision term which encodes the microscopic collisions among the plasmaconstituents and vanishes for the equilibrium distribution feq(Ep) (which does not dependon x and which does not depend on the direction of p) In writing (652) we are assumingthat p = E vp where vp is the velocity of the particle In curved space in the absence ofexternal forces the Boltzmann equation becomes
pmicropartxmicrof(xp)minus Γλmicroνpmicropνpartpλf(xp) = C [f ] (653)
where Γλmicroν are the Christoffel symbols of the background metric As in Appendix A we shalldetermine the stress-tensor correlator by introducing a perturbation in which the metricdeviates from flat space by a small amount gmicroν = ηmicroν + hmicroν and studying the responseof the system Even though the analysis for a generic perturbation can be performed itwill suffice for our purposes to restrict ourselves to fluctuations which in Fourier space haveonly one non-vanishing component hxy(K) We choose the directions x and y perpendicularto the wave-vector q which lies in the z direction For this metric the only Christoffelsymbols that are non-vanishing at leading order in hxy are Γtxy = Γxty = Γytx = minusiω hxy2 andΓxzy = Γyzy = minusΓzxy = iq hxy2
We will assume that prior to the perturbation the system is in equilibrium In response tothe external disturbance the equilibrium distribution changes
f(xp) = feq(Ep) + δf(xp) (654)
In the limit of a small perturbation the modified distribution function δf(xp) is linear inthe perturbation hxy We will also assume that the theory is rotationally invariant so that theenergy of the particle Ep is only a function of the modulus of p2 = gijp
ipj As a consequencethe metric perturbation also changes the on-shell relation and the equilibrium distributionmust also be expanded to first order in the perturbation yielding
feq = f0 + f prime0pxpy|vp|phxy asymp f0 + f prime0
pxpy
Ehxy (655)
where f0 is the equilibrium distribution in flat space f prime0(E) = df(E)dE and the velocity isgiven by vp = dEpdp In the last equality we have again approximated vp asymp pE
138
The solution of the Boltzmann equation requires the computation of the collision term CIn general this is a very complicated task since it takes into account the interactions among allthe system constituents which are responsible for maintaining equilibrium However sinceour only goal is to understand generic features of the spectral function it will be sufficientto employ the relaxation time approximation
C = minusEf minus feq
τR(656)
for the collision term in which the parameter τR is referred to as the relaxation time4
Since small perturbations away from equilibrium are driven back to equilibrium by particlecollisions the relaxation time must be of the order of the mean free path λmfp (which islong compared to the interparticle distance) The relaxation-time approximation is a verysignificant simplification of the full dynamics but it will allow us to illustrate the main pointsthat we wish to make A complete analysis of the collision term within perturbation theoryfor the purpose of extracting the transport coefficients of a weakly coupled plasma can befound in Refs [407434449450]
Within the approximation (656) upon taking into account that the distribution functiondoes not depend independently on the energy of the particles in Eq (653) the solution tothe linearized Boltzmann equation is given by
δf(Kp) =minusiωpxpyf prime0(p)
minusiω + ivpq + 1τR
hxy(K)
E (657)
Substituting this into Eq (651) we learn that the perturbation of the distribution functionleads to a perturbation of the stress tensor given by
δTmicroν(K) =
intd3p
(2π)3
pmicropν
Eδf(Kp) = minusGxyxyR (K)hxy(K) (658)
where the retarded correlator is given by
GxyxyR (K) = minusint
d3p
(2π)3vxvy
ω pxpy f prime0(p)
ω minus qvp + iτR
(659)
From the definition (313) the spectral function associated with this correlator is
ρxyxy(K) = minusωint
d3p
(2π)3
(pxpy)2
E2f prime0(p)
2τR
(ω minus qvp)2 + 1τ2R
(660)
Obtaining this spectral function was our goal because as we shall now see it has qualitativefeatures that indicate the presence (in this weakly coupled plasma) of quasiparticles
4In this approximation this relaxation time coincides with the shear relaxation time τR = τπ [97] How-ever since τπ is a property of the theory itself (defined as the appropriate coefficient in the effective fieldtheory aka the hydrodynamic expansion) whereas τR is a parameter specifying a simplified approximation tothe collision kernel which in general is not of the form eq (656) we will maintain the notational distinctionbetween τπ and τR
139
To clarify the structure of the spectral function (660) we begin by describing the freetheory limit in which τR rarrinfin since the collision term vanishes In this limit the Lorentzianmay be replaced by a δ-function yielding
ρxyxy(K) = minusωint
d3p
(2π)3
(pxpy)2
E2f prime0(p) 2π δ
(ω minus q middot vp
) (661)
The δ-function arises because in this limit the external perturbation (the gravity wave)interacts with free particles The δ-function selects those components of the gravity wavewhose phase velocity ω|q| coincides with the velocity of any of the free particles in theplasma Thus in the free-theory limit this δ-function encodes the existence of free particlesin the plasma For an isotropic distribution of particles such as the thermal distribution atany q 6= 0 the integration over angles washes out the δ-function and one is left with somefunction of ω that is characterized by the typical momentum scale of the particles (sim T ) andthat is not of interest to us here5 On the other hand at q = 0 we find that 1
ωρxyxy(ω 0)
is proportional to δ(ω) This δ-function at ω = 0 in the low-momentum spectral functionis a direct consequence of the presence of free particles in the plasma As we now discussthe effect of weak interactions is to dress the particles into quasiparticles and to broaden theδ-function into a narrow tall peak at ω = 0
When the interactions do not vanish we can proceed by relating the relaxation time tothe shear viscosity To do so we work in the hydrodynamic limit in which all momenta mustbe smaller than any internal scale This means that we can set q to zero but we must keepthe relaxation time τR finite The spectral density at zero momentum is then given by
ρxyxy(ω 0) = minusωint
d3p
(2π)3
(pxpy)2
E2f prime0(p)
2τR
ω2 + 1τ2R
(662)
Note that the spectral density at zero momentum has a peak at ω = 0 and note in particularthat the width in ω of this peak is sim 1τR T The spectral density has vanishing strengthfor ω 1τR This low-frequency structure in the zero-momentum spectral function is calledthe lsquotransport peakrsquo It is clear that in the τR rarr infin limit it becomes the δ-function thatcharacterizes the spectral density of the free theory that we described above Here in thepresence of weak interactions this peak at ω = 0 is a direct consequence of the presence ofmomentum-carrying quasiparticles whose mean free time is sim τR
The expression (662) is only valid for ω T where the modes are correctly described bythe Boltzmann equation For ω T since the quasiparticles can be resolved the structure ofthe spectral density is close to that in vacuum The separation of scales in the spectral densityis directly inherited from the separation of scales which allows the Boltzmann descriptionFinally using the Green-Kubo formula for the shear viscosity (A9) we find
η = minusτRint
d3p
(2π)3
(pxpy)2
E2f prime0(p) (663)
5A distinct peak at in the spectral density at some ω 6= 0 could be observed if the initial distribution werevery anisotropic This can arise if the theory has a (gauged) conserved charge and if the system is analyzedin the presence of a constant force that acts on this charge mdash ie an electric field
140
R
0
1
mth T)(20 02 04 06 08 1 12 14 16 18 2
) (
yxyx $ $
0
1
2
3
4
5
6
7
8
AdSCFT 3
T2
(a)
Figure 61 Left Sketch of the spectral function at zero momentum as a function of frequency for aweakly coupled plasma as obtained from kinetic theory The narrow structure at small frequency isthe transport peak with a width 1τR that is suppressed by the coupling (1τR sim g4T ) The thermalmass is mth prop gT Right Spectral function for the shear channel in the strongly coupled plasma ofN = 4 SYM theory computed via gaugestring duaity [215] (solid) and a comparison with the vacuumspectral function (dashed) which it approaches at high frequencies The vertical axis of this figure hasbeen scaled by the shear viscosity η = s4π of the strongly coupled plasma Note that the definitionof ρxyxy = minusImGRπ used in Ref [215] is different from that in Eq (313) by a factor of π2
Thus since η is determined by the collisions among the quasiparticles we can understand1τR as the width that arises because the quasiparticles do not have well-defined momentadue to the collisions among them In particular in perturbation theory [407449]
1
τRsim 1
Tg4 ln
1
gsim 1
λmfp (664)
Let us summarize the main points The zero-momentum spectral densities of a plasmawith quasiparticles have a completely distinctive structure there is a separation of scalesbetween the scale T (the typical momentum of the quasiparticles in the plasma) and themuch lower scale 1λmfp In particular there is a narrow peak in ρ(ω 0)ω around ω = 0of width τR sim 1λmfp and height 2η At larger frequencies the strength of the spectralfunction is very small At the scale of the mass of the quasiparticles the spectral functiongrows again For massless particles or those with mass much smaller than any temperature-related scale the role of the mass threshold is played by the thermal mass of the particlesgT which is much higher than the scale 1λmfp sim g4T associated with the mean free pathdue to the weakness of the coupling Finally above the scale T the structure of the spectralfunction approaches what it would be in vacuum A sketch of this behavior can be foundin Fig 61 These qualitative features are independent of any details of the theory and donot even depend on its symmetries All that matters is the existence of momentum-carryingquasiparticles In the presence of quasiparticles no matter what their quantum numbers arethese qualitative features must be present in the spectral density
141
632 Absence of quasiparticles at strong coupling
We return now to the strongly coupled N = 4 SYM plasma with its dual gravitationaldescription in order to compare the expectation (662) for how the spectral density shouldlook if the plasma contains any momentum-carrying quasiparticles to an explicit computationof the retarded correlator at strong coupling of course done via gaugestring duality In thissubsection we will benefit from the general analyses of Sections 621 and 622 As in thekinetic-theory computation we study the response to a metric fluctuation hxy(ωq) in theboundary theory with the same conventions as before As in Section 622 the fluctuationin the boundary leads to a metric perturbation in the bulk hyx = gyyhyx of the form
hyx(ω q z) = φ(ω q z)eminusiωt+iqz (665)
The field φ is governed by the classical action (630) which yields an equation of motion forφ(ω q z) that is given by
φprimeprime(ω q u)minus 1 + u2
ufφprime(ω q u) +
w2 minus q2(1minus u2)
uf2φ(ω q u) = 0 (666)
where u = z2 w = ω(2πT ) and q = q(2πT ) We may now use the general program outlinedin Section 532 to determine the retarded correlator It is given by Eq (559) which togetherwith Eqs (617) and (631) leads to
GxyxyR = minus limurarr0
1
16πGN
radicminusgguupartuφ(w q u)
φ0(w q u) (667)
where φ(w q u) is the solution to the equation of motion (666) with in-falling boundaryconditions at the horizon For arbitrary values of w and q Eq (666) must be solved nu-merically [215 451] From the correlator (667) the spectral function is evaluated using thedefinition (313) The result of this computation at zero spatial momentum q = 0 is shownin the right panel of Fig 61 where we have plotted ρω which should have a peak at ω = 0if there are any quasiparticles present
In stark contrast to the kinetic-theory expectation there is no transport peak in thespectral function at strong coupling In fact the spectral function has no interesting structureat all at small frequencies The numerical computation whose results are plotted in the rightpanel of Fig 61 also shows that there is no separation of scales in the spectral function In thestrong-coupling calculation quite unlike in perturbation theory the small and large frequencybehaviors join smoothly and the spectral density is only a function of w = ω2πT This couldperhaps have been expected in a conformal theory with no small coupling constant but notethat a free massless theory is conformal and that theory does have a δ-function peak in itsspectral function at zero frequency So having the explicit computation that gaugestringduality provides is necessary to give us confidence in the result that there is no transport peakin the strongly coupled plasma The absence of the transport peak shows unambiguously thatthere are no momentum-carrying quasiparticles in the strongly coupled plasma Thus thephysical picture of the system is completely different from that in perturbation theory
To conclude this section we would like to argue that the absence of quasiparticles is ageneric property of strong coupling and is not specific to any particular theory with any
142
particular symmetries or matter content To do so let us recall that in the kinetic theorycalculation the separation of scales required for its consistency are a consequence of the weakcoupling this is so in perturbative QCD or in perturbative N = 4 or in any weakly coupledplasma Now imagine increasing the coupling According to kinetic theory independent ofthe symmetries or the matter content of the theory the width of the transport peak growsand its height decreases as the coupling increases This reflects the fact that as the couplinggrows so does the width of the quasiparticles Extrapolating this trend to larger and largercouplings leads to the disappearance of the transport peak which at a qualitative levelagrees nicely with the strong-coupling result for the N = 4 SYM plasma obtained by explicitcomputation and shown in the right panel of Fig 61 This observation is one of the mostsalient motivations for the phenomenological applications of AdS-based techniques Since aswe have argued extensively in Sections 22 and 3 and 62 the quark-gluon plasma of QCDat temperatures a few times its Tc is strongly coupled the quasiparticle picture that hasconventionally been used to think about its dynamics is unlikely to be valid in this regimeThis makes it very important to have new techniques at our disposal that allow us to studystrongly coupled plasmas with no quasiparticles seeking generic consequences of the absenceof quasiparticles for physical observables Gaugegravity duality is an excellent tool for thesepurposes as we have already seen in Sections 61 and 62 and as we will further see in theremaining Sections of this review Indeed as we use gaugegravity duality to calculate moreand more different physical observables we will discover that the calculations done in thedual gravitational description begin to yield a new form of physical intuition phrased inthe dual language rather than in the gauge theory language in addition to yielding reliableresults
143
Chapter 7
Probing strongly coupled plasma
As discussed in Sections 23 and 24 two of the most informative probes of strongly coupledplasma that are available in heavy ion collisions are the rare highly energetic partons andquarkonium mesons produced in these collisions In this Section and in Section 8 we reviewresults obtained by employing the AdSCFT correspondence that are shedding light on theseclasses of phenomena In Sections 71 and 72 we review how a test quark of mass M movingthrough the strongly coupled N = 4 SYM plasma loses energy and picks up transversemomentum In Section 73 we consider how the strongly coupled plasma responds to thehard parton plowing through it that is we describe the excitations of the medium whichresult In Section 74 we review a calculation of the stopping distance of a single light quarkmoving through the strongly coupled plasma Throughout Sections 71 72 73 and 74we assume that all aspects of the phenomena associated with an energetic parton movingthrough the plasma are strongly coupled In Section 75 we review an alternative approachin which we assume that QCD is weakly coupled at the energy and momentum scales thatcharacterize gluons radiated from the energetic parton while the medium through which theenergetic parton and the radiated gluons propagate is strongly coupled In this case oneuses the AdSCFT correspondence only in the calculation of those properties of the stronglycoupled plasma that arise in the calculation of radiative parton energy loss and transversemomentum broadening In Section 76 we describe a calculation of synchrotron radiation instrongly coupled N = 4 SYM theory that allows the construction of a narrowly collimatedbeam of gluons (and adjoint scalars) opening a new path toward analyzing jet quenching
In Section 77 we review those insights into the physics of quarkonium mesons in heavy ioncollisions that have been obtained via AdSCFT calculations of the temperature-dependentscreening of the potential between a heavy quark and antiquark To go farther we need tointroduce a holographic description of quarkonium-like mesons themselves In Section 8 wefirst review this construction and then review the insights that it has yielded In additionto shedding light upon the physics of quarkonia in hot matter that we have introduced inSection 24 as we review in Section 862 these calculations have also resulted in the discoveryof a new and significant process by which a hard parton propagating through a stronglycoupled plasma can lose energy Cherenkov radiation of quarkonium mesons
144
71 Parton energy loss via a drag on heavy quarks
When a heavy quark moves through the strongly coupled plasma of a conformal theoryit feels a drag force and consequently loses energy [452 453] We shall review the originalcalculation of this drag force in N = 4 SYM theory [452453] it has subsequently been donein many other gauge theories with dual gravitational descriptions [454ndash464] In calculationsof the drag on heavy quarks one determines the energy per unit time needed to maintainthe forced motion of the quark in the plasma In these calculations one regards the quarkas an external source moving at fixed velocity v and one performs thermal averages overthe medium This picture can be justified if the mass of the quark is assumed to be muchlarger than the typical momentum scale of the medium (temperature) and if the motion ofthe quark is studied in a time window that is large compared with the relaxation scale of themedium but short compared to the time it takes the quark to change its trajectory In thislimit the heavy quark is described by a Wilson line along the worldline of the quark
The dual description of the Wilson line is given by a classical string hanging down from thequark on the boundary of AdS Since we are considering a single quark the other end of thestring hangs downs into the bulk of the AdS space We consider the stationary situation inwhich the quark has been moving at a fixed velocity for a long time meaning that the shapeof the string trailing down and behind it is no longer changing with time For concretenesswe will assume that the quark moves in the x1 direction and we choose to parametrize thestring world sheet by τ = t and σ = z By symmetry we can set two of the perpendicularcoordinates x2 and x3 to a constant The problem of finding the string profile reduces thento finding a function
x1(τ σ) (71)
that fulfills the string equations of motion The string solution must also satisfy the boundarycondition
x1(t z rarr 0) = vt (72)
Since we are interested in the stationary situation the string solution takes the form
x1(t z) = vt+ ζ(z) (73)
with ζ(z rarr 0) = 0 We work in an N = 4 plasma whose dual gravitational description isthe AdS black hole with the metric Gmicroν given in (531) The induced metric on the stringworldsheet gαβ = Gmicroνpartαx
micropartβxν is then given by
ds2ws =
R2
z2
(minus(f(z)minus v2
)dτ2 +
(1
f(z)+ ζ prime2(z)
)dσ2
+v ζ prime(z)v (dτdσ + dσdτ)
) (74)
where as before f(z) = 1minus z4z40 and ζ prime(z) denotes differentiation with respect to z
The Nambu-Goto action for this string reads
S = minus R2
2παprimeTintdz
z2
radicf(z)minus v2 + f(z)2ζ prime2(z)
f(z)= T
intdzL (75)
145
vv
Figure 71 String solutions of eq (78) The physical (unphysical) solution in which momentumflows into (out of) the horizon and the string trails behind (curves ahead) of the quark at the boundaryis plotted inn the left (right) panel
with T the total time traveled by the quark Extremizing this action yields the equations ofmotion that must be satisfied by ζ(z) The action (75) has a constant of motion given bythe canonical momentum
Π1z =
partLpartxprime1
= minus R2
2παprime1
z2
f(z)32ζ prime(z)radicf(z)minus v2 + f(z)2ζ prime2(z)
(76)
which coincides with the longitudinal momentum flux in the z direction In terms of Π1z the
equation of motion for ζ obtained from (75) takes the form
ζ prime2(z) =
(2παprime
R2Π1z
)2 z4
f(z)2
f(z)minus v2
f(z)minus(
2παprime
R2 Π1z
)2z4 (77)
The value of Π1z can be fixed by inspection of this equation as follows both the numerator
and the denominator of (77) are positive at the boundary z = 0 and negative at the horizonz = z0 since ζ prime(z) is real both the numerator and the denominator must change sign at thesame z this is only the case if
Π1z = plusmn R2
2παprimez20
γv (78)
with γ = 1radic
1minus v2 the Lorentz γ factor Thus stationary solutions can only be found forthese values of the momentum flux (Or for Π1
z = 0 for which ξ =constant This solutionhas real action only for v = 0)
The two solutions (78) correspond to different boundary choices of boundary conditionsat the horizon Following Refs [452 453] we choose the solution for which the momentumflux along the string world sheet flows from the boundary into the horizon correspondingto the physical case in which the energy provided by the external agent that is pulling thequark through the plasma at constant speed is dissipated into the medium This solution to(77) is given by
ζ(z) = minusv z0
2
(arctanh
(z
z0
)minus arctan
(z
z0
)) (79)
146
As illustrated in Fig 71 this solution describes a string that trails behind the moving quarkas it hangs down from it into the bulk spacetime
The momentum flux flowing down from the boundary along the string world sheet (79)and towards the horizon determines the amount of momentum lost by the quark in its prop-agation through the plasma In terms of the field theory variables
dp
dt= minusΠ1
z = minusπT2radicλ
2γv (710)
Note however that in the stationary situation we have described there is by constructionno change in the actual momentum of the quark at the boundary instead in order to keepthe quark moving with constant speed v against the force (710) there must be some externalagency pushing the quark through the strongly coupled plasma This force can be viewed asdue to a constant electric field acting on the string endpoint with the magnitude of the fieldgiven by
E =πT 2radicλ
2γv (711)
The physical setup described by the string (79) is thus that of forced motion of the quarkthrough the plasma at constant speed in the presence of a constant electric field The externalforce on the quark balances the backward drag force (710) on the quark exerted by themedium through which it is moving To make it explicit that the medium exerts a drag forcewe can rewrite (710) as
dp
dt= minusηDp (712)
with p = Mγv the relativistic expression of the momentum of the quark and M the mass ofthe heavy particle The drag coefficient is then
ηD =πradicλT 2
2M (713)
For test quarks with M rarrinfin as in the derivation above this result is valid for motion witharbitrarily relativistic speeds v It is remarkable that the energy loss of a heavy quark movingthrough the quark-gluon plasma with constant speed is described so simply as due to a dragforce In contrast in either a weakly coupled plasma [465] or a strongly coupled plasma thatis not conformal [466] dpdt is not proportional to p even at low velocities
We shall see in Section 72 that a heavy quark moving through the strongly coupled plasmaof N = 4 SYM theory experiences transverse and longitudinal momentum broadening inaddition to losing energy via the drag that we have analyzed above We shall review theimplications of the understanding of how the presence of the strongly coupled plasma affectsthe motion of heavy quarks for heavy ion collision phenomenology at the end of Section 72
711 Regime of validity of the drag calculation
In the derivation of the drag force above we considered a test quark with M rarr infin Theresult is however valid for quarks with finite mass M as long as M is not too small As wenow show the criterion that must be satisfied by M depends on the velocity of the quark
147
v The closer v is to 1 the larger M must be in order for the energy loss of the quark tobe correctly described via the drag force calculated above In deriving the regime of validityof the drag calculation we shall assume for simplicity that we are interested in large enoughγ = 1
radic1minus v2 that the M above which the calculation is valid satisfies M
radicλT We will
understand the need for this condition in Section 8
The introduction of quarks with finite mass M in the fundamental representation of thegauge group corresponds in the dual gravitational description to the introduction of D7-branes [385] as we have reviewed in Section 55 and as we will further pursue in Section 8The D7-brane extends from the boundary at z = 0 down to some zq related to the mass ofthe quarks it describes by
M =
radicλ
2πzq (714)
a result that we shall explain in Section 81 The physical reason that the calculation of thedrag force breaks down if M is too small or v is too large is that if the electric field E requiredto keep the quark moving at constant speed v gets too large one gets copious productionof pairs of quarks and antiquarks with mass M and the picture of dragging a single heavyquark through the medium breaks down completely [467] The parametric dependence of thecritical field Ec at which pair production becomes copious can be estimated by inspection ofhow the Dirac-Born-Infeld action for the D7-brane depends on E namely
SDBI sim
radic1minus
(2παprime
R2E z2
q
)2
(715)
The critical maximum field strength that the D7 brane can support is the Ec at which thisaction vanishes This yields a criterion for the validity of the drag calculation namely thatE must be less than of order
Ec =2πM2
radicλ
(716)
This maximum value of the electric field implies a maximum value of γ up to which thedrag calculation can be applied for quarks with some finite mass M From eq (711) andeq (716) this criterion is
γv lt
(2MradicλT
)2
(717)
We shall assume that M radicλT meaning that in (717) we can take γv γ And the
estimate is only parametric so the factor of two is not to be taken seriously Thus the resultto take away is that the drag calculation is valid as long as
γ
(MradicλT
)2
(718)
The argument in terms of pair production for the limit (718) on the quark velocity gives anice physical understanding for its origin but this limit arises in a variety of other ways Forexample at (718) the velocity of the quark v becomes equal to the local speed of light in thebulk at z = zq where the trailing string joins onto the quark on the D7-brane For example
148
at (718) the screening length Ls (described below in Section 77) at which the potentialbetween a quark and antiquark is screened becomes as short as the Compton wavelengthof a quark of mass M meaning that the calculation of Section 77 is also valid only in theregime (718) [149]
Yet further understanding of the meaning of the limit (718) can be gained by asking thequestion of what happens if the electric field is turned off and the quark moving with speedv begins to decelerate due to the drag force on it We would like to be able at least initiallyto calculate the energy loss of this now decelerating quark by assuming that this energy lossis due to the drag force which from (710) means
dE
dt
∣∣∣∣∣drag
= minusπ2
radicλT 2γv2 = minusπ
2
radicλT 2 pv
M (719)
However once the quark is decelerating it is natural to expect that due to its decelerationit radiates and loses energy via this radiation also The energy lost by a quark in stronglycoupled N = 4 SYM theory moving in vacuum along a trajectory with arbitrary accelerationhas been calculated by Mikhailov [468] For the case of a linear trajectory with decelerationa his result takes the form
dE
dt
∣∣∣∣∣vacuum radiation
= minusradicλ
2πa2γ6 = minus
radicλ
2π
1
M2
(dp
dt
)2
(720)
At least initially dpdt will be that due to the drag force namely (710) We now see thatthe condition that dEdt due to the vacuum radiation (720) caused by the drag-induceddeceleration (710) be less than dEdt due to the drag itself (719) simplifies considerablyand becomes
γ lt
(2MradicλT
)2
(721)
the same criterion that we have seen before This gives further physical intuition into thecriterion for the validity of the drag calculation and at the same time demonstrates that thiscalculation cannot be used in the regime in which energy loss due to deceleration-inducedradiation becomes dominant
Motivated by the above considerations the authors of Ref [469] considered the (academic)case of a test quark moving in a circle of radius L with constant angular frequency ω Theyshowed that in this circumstance dEdt is given by (719) as if due to drag with no radiationas long as ω2 (πT )2γ3 with γ the Lorentz factor for velocity v = Lω But for ω2 (πT )2γ3 the energy loss of the quark moving in a circle through the plasma is precisely whatit would be in vacuum according to Mikhailovrsquos result which becomes
dE
dt
∣∣∣∣∣vacuum radiation
=
radicλ
2πv2ω2γ4 =
radicλ
2πa2γ4 (722)
for circular motion Note that the radiative energy loss (722) is greater than that due todrag (719) for
ω2 (πT )2γ3 (723)
149
so the result of the calculation is that energy loss is dominated by that due to acceleration-induced radiation or that due to drag wherever each is larger (Where they are comparablein magnitude the actual energy loss is somewhat less than their sum [469]) This calculationshows that the calculational method that yields the result that a quark moving in a straightline with constant speed v in the regime (721) loses energy via drag can yield other resultsin other circumstances (see [470ndash472] for further examples) In the case of circular motionthe criterion for the validity of the calculational method is again (721) but there is a widerange of parameters for which this criterion and (723) are both satisfied [469] This meansthat for a quark in circular motion the calculation is reliable in a regime where energy lossis as if due to radiation in vacuum
72 Momentum broadening of a heavy quark
In the same regime in which a heavy quark moving through the strongly coupled plasma ofN = 4 SYM theory loses energy via drag as reviewed in Section 71 it is also possible touse gaugegravity duality to calculate the transverse (and in fact longitudinal) momentumbroadening induced by motion through the plasma [467 473ndash475] We shall review thesecalculations in this section They have been further analyzed [476ndash478] and extended tostudy the effects of nonconformality [466479480] and acceleration [481482]
For non-relativistic heavy quarks the result (712) is not surprising The dynamics ofthis particle is that of Brownian motion which can be described by the effective equation ofmotion
dp
dt= minusηDp+ ξ(t) (724)
where ξ(t) is a random force that encodes the interaction of the medium with the heavyprobe and that causes the momentum broadening that we describe in this Section For heavyquarks we have seen in (713) that ηD is suppressed by mass This reflects the obvious factthat the larger the mass the harder it is to change the momentum of the particle Thus fora heavy quark the typical time for such a change 1ηD is long compared to any microscopictime scale of the medium τmed This fact allows us to characterize the force distribution bythe two point correlators lang
ξT (t)ξT (tprime)rang
= κT δ(tminus tprime) langξL(t)ξL(tprime)
rang= κLδ(tminus tprime) (725)
where the subscripts L and T refer to the forces longitudinal and transverse to the directionof the particlersquos motion Here we are also assuming an isotropic plasma which leads to〈ξL(t)〉 = 〈ξT (t)〉 = 0 In general the force correlator would have a nontrivial dependenceon the time difference (different from δ(t minus tprime)) However since the dynamics of the heavyquark happen on timescales that are much larger than τmed we can approximate all mediumcorrelations as happening instantaneously It is then easy to see that the coefficient κT (κL)corresponds to the mean squared transverse (longitudinal) momentum transferred to theheavy quark per unit time For example the transverse momentum broadening is given bylang
p2perprang
= 2
intdtdtprime
langξT (t)T ξT (tprime)
rang= 2κTT (726)
150
where T is the total time duration (which should be smaller than 1ηD) and where the 2 isthe number of transverse dimensions It is clear from the correlator that κT is a property ofthe medium independent of any details of the heavy quark probe Our goal in this section isto calculate κT and κL We shall do so first at low velocity and then throughout the velocityregime in which the calculation of the drag force is valid
Before we begin we must show that in the limit we are considering the noise distributionis well characterized by its second moment Odd number correlators vanish because of sym-metry so the first higher moment to consider is the fourth moment of the distribution of thetransverse momentum picked up by the heavy quark moving through the plasmalang
p4perprang
=
intdt1dt2dt3dt4 〈ξT (t1)ξT (t2)ξT (t3)ξT (t4)〉 (727)
The four-point correlator may be decomposed as
〈ξT (t1)ξT (t2)ξT (t3)ξT (t4)〉 = 〈ξT (t1)ξT (t2)ξT (t3)ξT (t4)〉c (728)
+ 〈ξT (t1)ξT (t2)〉 〈ξT (t3)ξT (t4)〉+ 〈ξT (t1)ξT (t3)〉 〈ξT (t2)ξT (t4)〉+ 〈ξT (t1)ξT (t4)〉 〈ξT (t2)ξT (t3)〉
(729)
which is the definition of the connected correlator Due to time translational invariance theconnected correlator is a function
〈ξT (t1)ξT (t2)ξT (t3)ξT (t4)〉c = f(t4 minus t1 t3 minus t1 t2 minus t1) (730)
As before the correlator has a characteristic scale of the order of the medium scale As aconsequence since the expectation value due to the connected part has only one free integralwe find lang
p4perprang
=(
3 (2κT )2 +O(τmed
T))T 2 (731)
where the dominant term comes from the disconnected parts in eq (728) Since we areinterested in times parametrically long compared to τmed we can neglect the connected partof the correlator
721 κT and κL in the prarr 0 limit
The dynamical equations (724) together with (725) constitute the Langevin description ofheavy quarks in a medium In the p rarr 0 limit there is no distinction between transverseand longitudinal meaning that both the fluctuations in (725) must be described by thesame correlator with κL = κT equiv κ The Langevin equations (724) and (725) describe thetime evolution of the probability distribution for the momentum of an ensemble of heavyquarks in a medium A standard analysis shows that independent of the initial probabilitydistribution after sufficient time any solution to the Langevin equation yields the probabilitydistribution
P(p trarrinfin) =
(1
π
ηDκ
)32
expminusp2 ηD
κ
(732)
151
which coincides with the equilibrium (ie Boltzmann) momentum distribution for the heavyquark provided that
ηD =κ
2MT (733)
This expression is known as the Einstein relation Thus the Langevin dynamics of non-relativistic heavy quarks is completely determined by the momentum broadening κ and theheavy quarks equilibrate at asymptotic times
The Einstein relation (733) together with the computation of ηD in (713) for stronglycoupled N = 4 SYM theory allow us to infer the value of κ for this strongly coupled conformalplasma namely
κ = πradicλT 3 (734)
The dynamical equation (712) that we used in the previous section does not include thenoise term simply because in that section we were describing the change in the mean heavyquark momentum in the ensemble of this section
722 Direct calculation of the noise term
We would like to have a direct computation of the noise term in the description of a heavyquark in a strongly coupled gauge theory plasma There are two motivations for this 1) toexplicitly check that the Einstein relation (733) is fulfilled and 2) to compute the momentumbroadening for moving heavy quarks which are not in equilibrium with the plasma and towhich the Einstein relation therefore does not apply This computation is somewhat technicalthe reader interested only in the results for κT and κL for a moving heavy quark may skipto Section 723
We need to express the momentum broadening in terms which are easily computed withinthe gaugegravity correspondence To do so we prepare a state of the quark at an initialtime t0 which is moving at given velocity v in the plasma In quantum mechanics the stateis characterized by a density matrix which is a certain distribution of pure states
ρ(t0) =sumn
w(n) |n〉 〈n| (735)
where the sum is performed over a complete set of states and the weight w(n) is theensemble For a thermal distribution the states are eigenstates of the Hamiltonian andw(n) = expminusEnT
In the problem we are interested in the density matrix includes not only the quark degreesof freedom but also the gauge degrees of freedom However we start our discussion using aone particle system mdash initially for illustrative purposes In this case the distribution functionof the particle is defined from the density matrix as
f(x xprime t0) =sumn
w(n) 〈x| n〉 〈n| xprimerang (736)
where as usual 〈x| n〉 is the wave function of the particle in the state |n〉 It is also commonto call f(x xprime) the density matrix It is conventional to introduce the mean and relativecoordinates and express the density matrix as
f(X r t0) = f(X +
r
2 X minus r
2 t0
) (737)
152
where X = (x+xprime)2 and r = xminusxprime It is then easy to see that the mean position and meanmomentum of the single particle with a given density matrix are given by
〈x〉 = tr ρ(t0)x =
intdxxf(x x t0) =
intdXXf(X 0 t0)
〈p〉 = tr ρ(t0) p =
intdxminusi2
(partx minus partxprime) f(x xprime t0)∣∣∣xprime=x
= minusiintdXpartrf(X r t0)|r=0 (738)
meaning that r is the conjugate variable to the momentum and of most interest to us themean squared momentum of the distribution islang
p2rang
= minusintdXpart2
r f(X r t0)|r=0 (739)
Returning now to the problem of interest to us we must consider an ensemble containingthe heavy quark and also the gauge field degrees of freedom Since we assume the mass ofthe quark to be much larger than the temperature we can described the pure states of thesystem as ∣∣Aprimerang = Qdaggera(x) |A〉 (740)
where |A〉 is a state of the gauge fields only |Aprime〉 denotes a state of the heavy quark plus the
gauge fields and Qdaggera(x) is the creation operator (in the Schrodinger picture) of a heavy quarkwith color a at position x Corrections to this expression are (exponentially) suppressed byTM The Heisenberg representation of the operator Q(x) satisfies the equation of motion
(iu middotD minusM)Q = 0 (741)
where u is the four velocity of the quark and D is the covariant derivative with respect tothe gauge fields of the medium This equation realizes the physical intuition that the heavyquark trajectory is not modified by the interaction with the medium which leads only to amodification of the quarkrsquos phase1
The full density matrix of the system ρ describes an ensemble of all the degrees of freedomof the system Since we are only interested in the effects of the medium on the momentum ofthe heavy quark probe we can define a one-body density matrix from the full density matrixby integrating over the gauge degrees of freedom
f(X r t0) =langQdaggera
(X minus r
2)UabQb
(X +
r
2
)rang= Tr
[ρQdaggera
(X minus r
2
)UabQb
(X +
r
2
)] (742)
where the trace is taken over a complete set of statessumAa
intdxQdaggera(x) |A〉 〈A|Qa(x) (743)
1The expression (741) can also be derived from the Dirac equation by performing a Foldy-Wouthuysentransformation which in the heavy quark rest frame is given by Q = expγ middotD2Mψ where γ = 1
radic1minus v2
153
tCx xX = (t xo + v∆t)
X = (t xo + v∆tminus ivε)Xf = (to minus iε xo minus ivε)
Xo = (to xo)
tprimeC
Figure 72 Time contour C in the complex time plane for the path integral (747) Here ∆t equiv tminust0and the i ε prescription in time is translated to the longitudinal coordinate x since the quark trajectoryis x = vt The two-point functions computed from the partition function (747) are evaluated at twoarbitrary points tC and tprimeC on the contour
Note that the inclusion of the operators in the trace in (742) plays the same role as theprojectors |x〉 in (736) The gauge link Uab in (742) joins the points X + r2 and X minus r2to ensure gauge invariance In the long time limit the precise path is not important andwe will assume that Uab is a straight link To simplify our presentation we shall explicitlytreat only transverse momentum broadening which means taking the separation r to be ina direction perpendicular to the direction of motion of the heavy quark rperp
At a later time t after the heavy quark has propagated through the plasma for a timetminus t0 the one-body density matrix has evolved from (742) to
f(X rperp t) = Tr[ρ eiH(tminust0)Qdaggera
(X minus rperp
2
)eminusiH(tminust0)
eiH(tminust0)Uab eminusiH(tminust0)
eiH(tminust0)Qb
(X +
rperp2
)eminusiH(tminust0)
] (744)
where we have introduced evolution operators to express the result in the Heisenberg pictureWe then introduce a complete set of states obtaining
f(X rperp t) =
intdx1dx2
sumA1A2A3A4
ρa1a2 [x1x2A1 A2]
〈A2|Qa2(x2)Qdaggera
(X minus rperp
2 t)|A3〉
〈A3|Uab(t) |A4〉
〈A4|Qb(X +
rperp2
t)Qdaggera1
(x1) |A1〉 (745)
where we have defined
ρa1a2 [x1x2A1 A2] equiv 〈A1|Qa1 (x1) ρQdaggera2(x2) |A2〉 (746)
The expression (745) can be expressed as a path integral Note that the second line in(745) is an anti-time ordered correlator thus its path integral representation involves atime reversal of the usual path integral Instead of introducing two separate path integralscorresponding to the second and fourth lines of (745) we introduce the time contour shownin Fig 72 and use this contour to define a single path integral The fields A1 and A2 are the
154
(txperp + rperp2 xo + v∆t)
ρo
(toxperp + rperp2 xo)
(to minus iεxperp minus rperp2 xo minus ivε) (txperp minus rperp2 xo + v∆tminus ivε)Figure 73 Graphical representation of eq (748) The Wilson line indicated by the black line isdenoted WC [rperp2minusrperp2] This Wilson line is traced with the initial density matrix ρoa1a2 The hor-izontal axis is along the time direction and the vertical axis is along one of the transverse coordinatesxperp ∆t equiv tminus t0
values at the endpoints of the contour The one-body density matrix then reads
f(X rperp t) =sumA1A2
intdx1dx2
int[DA] DQDQdaggerei
intC d
4xLYM+Qdagger(iumiddotDminusM)Q
ρa1a2 [x1x2A1 A2]Uab(t)
Qa2 (x2 t0 minus iε) Qdaggera(X minus rperp
2 tminus iε
)Qb
(X +
rperp2 t)Qdaggera1
(x1 t0) (747)
Integrating out the quark field and working to leading order in TM (neglecting the fermionicdeterminant) yields
f (X rperp t) =lang
tr[ρ[X +
rperp2 X minus rperp
2A1 A2
]WC
[rperp2minusrperp
2
]]rangA (748)
where the subscript A indicates averaging with respect to the gauge fields and where theWilson line WC [rperp2minusrperp2] is defined in Fig 73 We have used the fact that the Greenrsquosfunction of eq (741) is the (contour ordered) Wilson line
Next we perform a Taylor expansion of the time-evolved density matrix (748) aboutrperp = 0 obtaining
f (X rperp t) = f (X 0 t) +
r2perp2
langtr
[part2
partr2perpρ[X +
rperp2 X minus rperp
2A1 A2
]WC [0]
]rangA
+
r2perp2κT∆t 〈tr [ρ [XXA1 A2]WC [0]]〉A (749)
The second term in this expression involves only derivatives of the initial density matrixthus it is the mean transverse momentum squared of the initial distribution (which may besupposed to be small) In the last term which scales with the elapsed time ∆t we havedefined
κT∆t =1
4
1
〈trρWC [0]〉A
intCdtCdt
primeC
langtr ρ[XXA1 A2]
δ2WC [δy]
δy(tC) δy(tprimeC)
rangA
(750)
155
where tC denotes time along the contour depicted in Fig 72 κT∆t is the mean transversemomentum squared picked up by the heavy quark during the time ∆t We have expressedthe transverse derivatives of the Wilson line as functional derivatives with respect to the pathof the Wilson line The path δy denotes a small transverse fluctuation δy(t) away from thepath X1 = vt
The contour δy may be split into two pieces δy1 and δy2 which run along the timeordered and anti-time ordered part of the path Thus the fluctuation calculation defines fourcorrelation functions
iG11(t tprime) =1
〈trρoWC [0 0]〉A
langtr ρo
δ2WC [δy1 0]
δy1(t) δy1(tprime)
rangA
(751)
iG22(t tprime) =1
〈trρoWC [0 0]〉A
langtr ρo
δ2WC [0 δy2]
δy1(t) δy2(tprime)
rangA
(752)
iG12(t tprime) =1
〈trρoWC [0 0]〉A
langtr ρo
δ2WC [δy1 δy2]
δy1(t) δy2(tprime)
rangA
(753)
iG21(t tprime) =1
〈trρoWC [0 0]〉A
langtr ρo
δ2WC [δy2 δy2]
δy1(tprime) δy2(t)
rangA
(754)
Note that the first two correlators correspond to time ordered and anti-time ordered correla-tors while the last two are unordered We can then divide the integration over tC and tCprime in(750) into four parts corresponding to the cases where each of tC and tCprime is on the upper orlower half of the contour in Fig 72 In the large ∆t limit we can then use time translationalinvariance to cast (750) as
κT = limωrarr0
1
4
intdte+iωt [iG11(t 0) + iG22(t 0) + iG12(t 0) + iG21(t 0)] (755)
This admittedly rather formal expression for κT is as far as we can go in general In Sec-tion 723 we evaluate κT (and κL) in the strongly coupled plasma of N = 4 SYM theory
Although our purpose in deriving the expression (755) is to use it to analyze the case v 6= 0it can be further simplified in the case that v = 0 On the time scales under considerationthe static quark is in equilibrium with the plasma and the Kubo-Martin-Schwinger relationwhich takes the form
i [G11(ω) +G22(ω) +G12(ω) +G21(ω)] = minus4 coth( ω
2T
)ImGR(ω) (756)
for εrarr 0 applies [483] Here GR is the retarded correlator Thus we find
κT (v = 0) = limωrarr0
(minus2T
ω
)ImGR(ω) (757)
If v 6= 0 however we must evaluate the four correlators in the expression (755)
723 κT and κL for a moving heavy quark
We see from the expression (750) that the transverse momentum broadening coefficient κT isextracted by analyzing small fluctuations in the path of the Wilson line depicted in Fig 73
156
In the strongly coupled plasma of N = 4 SYM theory we can use gaugegravity duality toevaluate κT starting from (750) In the dual gravitational description the small fluctuationsin the path of the Wilson line amount to perturbing the location on the boundary at whichthe classical string (whose unperturbed shape is given by (79)) terminates according to
(x1(t z) 0 0)rarr (x1(t z) y(t z) 0) (758)
The perturbations of the Wilson line at the boundary yield fluctuations on the string worldsheet dragging behind the quark Because we wish to calculate κT in (758) we have onlyintroduced perturbations transverse to the direction of motion of the quark We shall quotethe result for κL at the end calculating it requires extending (758) to include perturbationsto the function x1(t z)
In order to analyze fluctuations of the string world sheet we begin by casting the metricinduced on the string world sheet in the absence of any perturbations
ds2ws =
R2
z2
(minus(f(z)minus v2
)dτ2 +
f(z)
f2(z)dσ2 minus v2 z
2z20
f(z)(dτdσ + dσdτ)
) (759)
in a simpler form In (759) we have defined f(z) equiv 1minusz4(z40γ
2) The induced metric (759)is diagonalized by the change of world sheet coordinates
t =tradicγ
+z0
2radicγ
(arctan
(z
z0
)minus arctanh
(z
z0
)
minusradicγ arctan
(radicγz
z0
)+radicγarctanh
(radicγz
z0
))
z =radicγz (760)
in terms of which the induced metric takes the simple form
ds2ws =
R2
z2
(minusf(z)dt2 +
1
f(z)dz2
) (761)
Note that this has the same form as the induced metric for the world sheet hanging belowa motionless quark upon making the replacement (t z) rarr (t z) In particular the metric(761) has a horizon at z = z0 which means that the metric describing the world sheet ofthe string trailing behind the moving quark has a world sheet horizon at z = zws equiv z0
radicγ
For v rarr 0 the location of the world sheet horizon drops down toward the spacetime horizonat z = z0 But for v rarr 1 the world sheet horizon moves closer and closer to the boundaryat z = 0 ie towards the ultraviolet As at any horizon the singularity at z = zws (ie atz = z0) in (761) is just a coordinate singularity In the present case this is manifest since(761) was obtained from (759) which is regular at z = zws by a coordinate transformation(760) Nevertheless the world sheet horizon has clear physical significance at z = zws thelocal speed of light at this depth in the bulk matches the speed v with which the quark at theboundary is moving Furthermore and of direct relevance to us here because of the worldsheet horizon at z = zws fluctuations of the string world sheet at z gt zws below mdash to the
157
infrared of mdash the world sheet horizon are causally disconnected from fluctuations at z lt zws
above the world sheet horizon and in particular are causally disconnected from the boundaryat z = 0
The remarkable consequence of the picture that emerges from the above analysis of theunperturbed string world sheet trailing behind the quark at the boundary moving with speedv is that the momentum fluctuations of this quark can be thought of as due to the Hawkingradiation on the string world sheet originating from the world sheet horizon at z = zws [467474] It is as if the force fluctuations that the quark in the boundary gauge theory feelsare due to the fluctuations of the string world sheet to which it is attached with thesefluctuations arising due to the Hawking radiation originating from the world sheet horizonIt will therefore prove useful to calculate the Hawking temperature of the world sheet horizonwhich we denote Tws As detailed in Appendix B this can be done in the standard fashionupon using a further coordinate transformation to write the metric (761) in the vicinityof the world sheet horizon in the form ds2
ws = minusb2ρ2dt2 + dρ2 for some constant b wherethe world sheet horizon is at ρ = 0 Then it is a standard argument that in order toavoid having a conical singularity at ρ = 0 in the Euclidean version of this metric namelyds2
ws = b2ρ2dθ2 + dρ2 bθ must be periodic with period 2π The periodicity of the variable θnamely 2πb is 1T Since at the boundary where z = 0 Eq (760) becomes t = t
radicγ this
argument yields
Tws =Tradicγ (762)
a result that we shall use below
We have gained significant physical intuition by analyzing the unperturbed string worldsheet but in order to obtain a quantitative result for κT we must introduce the transversefluctuations y(t z) defined in (758) explicitly We write the Nambu-Goto action for the stringworld sheet with y(t z) 6= 0 and expand it to second order in y obtaining the zeroth orderaction (75) plus a second order contribution
S(2)T [y] =
γR2
2παprime
intdtdz
z2
1
2
(y2
f(z)minus f(z)yprime2
)(763)
where ˙ and prime represent differentiation with respect to t and z respectively This action isconveniently expressed as
S(2)T [y] = minus γR
2
2παprime
intdtdz
z2
1
2
radicminushhabpartaypartby (764)
with hab the induced metric on the unperturbed world sheet that we have analyzed aboveThe existence of the world sheet horizon means that we are only interested in solutions to theequations of motion for the transverse fluctuations y obtained from (764) that satisfy infallingboundary conditions at the world sheet horizon This constraint in turn implies a relationamong the correlators analogous to those in (755) that describe the transverse fluctuationsof the world sheet and in fact the relation turns out to be analogous to the Kubo-Martin-Schwinger relation (756) among the gauge theory correlators [467] Consequently for a quarkmoving with velocity v the transverse momentum broadening coefficient κT (v) is given by the
158
same expression (757) that is valid at v = 0 with T replaced by the world sheet temperatureTws of (762) [467474] That is
κT (v) = limωrarr0
(minus 2Tws
ωImGR(ω)
) (765)
where GR denotes the retarded correlator at the world-sheet horizon The fact that inthe strongly coupled theory there is a KMS-like relation at v 6= 0 after all is a nontrivialconsequence of the development of the world-sheet horizon
The computation of the retarded correlator follows the general procedure of Ref [368]described in Section 53 Since the action (763) is a function of t which is given by t
radicγ
at the boundary the retarded correlator is a function of ω =radicγω (with ω the frequency of
oscillations at the boundary) To avoid this complication and in particular in order to beable to apply the general results for ImGR that we derived in Section 62 it is convenient todefine
t =radicγ t (766)
so that t = t at the boundary We now wish to apply the general expressions (A10) (616)and (617) In order to do so we identify the world sheet metric hab and the field y in theaction (764) with the metric gMN and the field φ in the action (615) meaning that in ourproblem the function q in (615) takes the specific form
1
q(z)=γR2
2παprime1
z2=
radicλ
2πz2 (767)
Furthermore for the two-dimensional world sheet metric we have minush = htthzz meaning thatfrom the general result (624) we find
minus limωrarr0
Im GR(ω)
ω=
1
q(zws)=γradicλ
2π(πT )2 (768)
and thusκT =
radicλγπT 3 (769)
which is our final result for the transverse momentum broadening coefficient
The analysis of longitudinal fluctuations and the extraction of κL proceed analogouslyto the analysis we have just reviewed except that in (758) we introduce a perturbationto the function x1 instead of a transverse perturbation y At quadratic order there is nocoupling between the transverse and longitudinal perturbations Remarkably the action forlongitudinal fluctuations of the string is the same as that for transverse fluctuations eq (763)up to a constant
S(2)L [x] = γ2S
(2)T [x] (770)
with γ the Lorentz factor Following the analogous derivation through we conclude that
κL = γ2κT = γ52radicλπT 3 (771)
This result shows that κL depends very strongly on the velocity of the heavy quark IndeedκL grows faster with increasing velocity than the energy squared of the heavy quark γ2M2
159
Thus the longitudinal momentum acquired by a quark moving through a region of stronglycoupled N = 4 SYM plasma of finite extent does not become a negligible fraction of theenergy of the quark in the high energy limit This is very different from the behavior of aquark moving through a weakly coupled QCD plasma in which the longitudinal momentumtransferred to the quark can be neglected in the high energy limit However we should keepin mind that due to the bound (718) for a given value of the mass M and the coupling
radicλ
the calculation of κL (and of κT ) is only valid for finite energy quarks with γ limited by(718)
We see from the expressions (769) and (771) for κT and κL derived by explicit analysis ofthe fluctuations that in the v rarr 0 limit we have κT = κL = κ with κ given by (734) as weobtained previously from the drag coefficient ηD via the use of the Einstein relation (733)This is an example of the fluctuation-dissipation theorem
In the gauge theory momentum broadening is due to the fluctuating force exerted on theheavy quark by the fluctuating plasma through which it is moving In the dual gravitationaldescription the quark at the boundary feels a fluctuating force due to the fluctuations of theworld sheet that describes the profile of the string to which the quark is attached Thesefluctuations have their origin in the Hawking radiation of fluctuations of the string worldsheet originating from the world sheet horizon The explicit computation of this world sheetHawking radiation for a quark at rest was performed in Refs [477 484] and these resultsnicely reproduce those we have obtained within a Langevin formalism This computationwas extended to quarks moving at nonzero velocity in Refs [475478]
724 Implications for heavy quarks in heavy ion collisions
Models based upon Langevin dynamics have been used by a number of authors to describethe motion of heavy quarks within the hot expanding fluid produced in heavy ion collisions [7465485ndash493] with the goal of using data from RHIC [136494495] to constrain the diffusionconstant D that describes the Brownian motion of a heavy quark in the hot fluid Theseanalyses typically assume that the relative velocity of the heavy quark and the hot fluid isnot relativistic meaning that κT = κL equiv κ In this regime the diffusion constant is given byD = 2T 2κ meaning that the result (734) translates into the statement that a heavy quarkin the strongly coupled N = 4 SYM theory plasma obeys a Langevin equation with
D =4radicλ
1
2πTasymp 11
2πT
radic1
αSYMNc (772)
The diffusion constant D parametrizes how strongly the heavy quark couples to the mediumwith smaller D corresponding to stronger coupling and shorter mean free path D is welldefined even if it is so small that it does not make sense to define a mean free path as is thecase in the plasma of strongly coupled N = 4 SYM theory given (772) In a weakly coupledQCD plasma perturbative calculations suggest [465]
Dweakly coupled asymp14
2πT
(33
αs
)2
(773)
or larger by a factor of more than three [487] although it should be noted that the perturba-tive expansion converges poorly meaning that such calculations can only become quantita-
160
tively reliable at values of αs that are much smaller than 033 [496497] Note that the result(772) is valid for N = 4 SYM theory when λ = 4παSYMNc is large and 2πTD is smallwhile the result (773) is valid for QCD when αs is small and 2πTD is large
At present the central lessons from RHIC data on heavy quarks are qualitative Thisis because so far the best experimental constraints on the spectra of c and b quarks comemainly from single-electron spectra that are known to be dominated by the semileptonicdecays of heavy quarks The experimental pointing resolution is not yet sufficient to separateb from c quark decays via their displaced vertices Upgrades at PHENIX and STAR willsoon allow one to make this separation Moreover with the onset of heavy ion collisions atthe LHC it will soon be possible to measure B- and D-meson spectra up to pT sim 20 GeVeg via the decay channel D0 rarr K+ πminus Also heavy quark energy loss can be characterizedvia muon spectra at the LHC The experimental understanding of how heavy quarks behavein the matter produced in relativistic heavy ion collisions can be expected to become muchmore quantitative with these developments
At present there are two qualitative lessons from the RHIC data [136 494 495] (i) RAAfor isolated electrons (some linear combination of c and b quarks) is small not that muchbigger than RAA for pions and (ii) v2 for isolated electrons is significant not that muchsmaller than that for pions At a qualitative level both these observations suggest thatc and b quarks are strongly coupled to the medium as they must initially be slowed bytheir relative motion through it and must then flow along with the collective hydrodynamicexpansion Theoretical calculations involve calculating b and c quark production using aLangevin model to describe their diffusion in the hot fluid produced in the collision whichis described hydrodynamically and then modelling the freezeout and decay of the heavy Band D mesons before finally comparing to data on RAA and v2 for isolated electrons Tworecent calculations along these lines are qualitatively consistent with RHIC data as long asthe diffusion constant is in the range [488]
DRHIC asymp2minus 6
2πT (774)
or (3 minus 5)(2πT ) [487] Certainly the extraction of D or equivalently κ will become morequantitative once brsquos and crsquos can be measured separately On the theoretical side it will beinteresting to redo these analyses with κT 6= κL and both velocity-dependent as in (769)and (771) It will be interesting to see whether the explicit velocity dependence of κT andκL serves to improve the fit to RAA and v2 for b and c quarks There is some tension inthe present analyses of isolated electrons with a single fit parameter D with the RAA datafavoring somewhat larger values of D while the v2 data favors somewhat smaller values [136]
It is striking that D for strongly coupled N = 4 SYM theory given in (772) is even inthe same ballpark as that extracted by fitting to present RHIC data given in (774) Tounderstand whether this is a coincidence we need to ask how D would change if we coulddeform N = 4 SYM theory so as to turn it into QCD This is not a question to which theanswer is known but we can make two observations First in a large class of conformaltheories at a given value of T Nc and λ both κ and the drag coefficient ηD scale with thesquare root of the entropy density (The argument is the same as that for the jet quenchingparameter q of Section 75 and is given in Ref [149]) The number of degrees of freedom inQCD is smaller than that in N = 4 SYM theory by a factor of 475120 for Nc = 3 suggesting
161
that κ and ηD should be smaller in QCD by a factor ofradic
475120 = 063 making D largerby a factor of
radic120475 = 1592 Second N = 4 SYM theory is of course conformal while
QCD is not Analysis of a toy model in which nonconformality can be introduced by handsuggests that turning on nonconformality to a degree suggested by lattice calculations ofQCD thermodynamics reduces D perhaps by as much as a factor of two [466] Turning onnonconformality in N = 2lowast theory also reduces D So D in a strongly coupled QCD plasmais not known reliably but the reduction in degrees of freedom and the nonconformality seemto push in opposite directions suggesting that DQCD may not be far from that given in (772)It will be very interesting to watch how this story evolves once it is possible to measure b andc quarks separately in heavy ion collisions
73 Disturbance of the plasma induced by an energetic heavyquark
In Sections 71 and 72 we have analyzed the effects of the strongly coupled plasma of N = 4SYM theory on an energetic heavy quark moving through it focussing on how the heavyquark loses energy in Section 71 and on the momentum broadening that it experiences inSection 72 In this Section we turn the tables and analyze the effects of the energeticheavy quark on the medium through which it is propagating [7 137 138 498ndash525] Fromthe point of view of QCD calculations and heavy ion collision phenomenology the problemof understanding the response of the medium to an energetic probe is quite complicatedAn energetic particle passing through the medium can excite the medium on many differentwavelengths And even if the medium was thermalized prior to the interaction with theprobe the disturbance caused by the probe must drive the medium out of equilibrium atleast close to the probe And non-equilibrium processes are difficult to treat especially atstrong coupling
It is natural to attempt to describe the disturbance of the medium using hydrodynamicswith the energetic particle treated as a source for the hydrodynamic equations This approachis based on two assumptions First one must assume that the medium itself can be describedhydrodynamically Second one has to assume that the non-equilibrium disturbance in thevicinity of the energetic particle relaxes to some locally equilibrated (but still excited) stateafter the energetic particle has passed on a timescale that is short compared to the lifetimeof the hydrodynamic medium itself The first assumption is clearly supported by data fromheavy ion collisions at RHIC as discussed in Section 22 The second assumption is strongerand less well justified Even though as we saw in Section 22 there is evidence from the datathat in heavy ion collisions at RHIC a hydrodynamic medium in local thermal equilibriumforms rapidly after only a short initial thermalization time it is not clear a priori that
2Gubser has suggested that the difference in the numbers of degrees of freedom between QCD and N = 4SYM theory be taken into account by comparing results in the two theories at differing temperatures withTQCD = (475120)14TSYM [7 460] This prescription is inconsistent with known results for a large class ofconformal theories [149] in which q κ ηD and D all scale with the square root of the number of degreesof freedom when compared at fixed T Nc and λ Perhaps coincidentally since the drag coefficient ηD isproportional to T 2 scaling it by the square root of the number of degrees of freedom is equivalent to scalingT by the one-fourth power of the number of degrees of freedom as prescribed in Refs [7 460] But thisprescription is not correct when applied to D prop 1T or to κ prop T 3 or q prop T 3
162
the relaxation time for the disturbance caused by an energetic quark plowing through thismedium is comparably short particularly since the density of the medium drops with timeFinally even if a hydrodynamic approach to the dynamics of these disturbances is valid thedetails of the functional form of this hydrodynamic source are unknown since the relaxationprocess is not under theoretical control
Keeping the above difficulties in mind it is still possible to use the symmetries of theproblem and some physical considerations to make some progress toward understanding thesource for the hydrodynamic equations corresponding to the disturbance caused by an ener-getic quark If the propagating parton is sufficiently energetic we may assume that it movesat a fixed velocity this ansatz forces the source to be a function of x minus vt with the partonmoving in the x-direction We may also assume that the source has cylindrical symmetryaround the parton direction We may also constrain the source by the amount of energy andmomentum that is fed into the plasma which for the case of the plasma of strongly coupledN = 4 SYM theory we calculated in Section 71 In an infinite medium at late enough timesall the energy lost by the probe must thermalize and be incorporated into heating andorhydrodynamic motion (This may not be a good approximation for a very energetic partonpropagating through weakly coupled plasma of finite extent since as we have discussed inSection 23 in this setting the parton loses energy by the radiation of gluons whose energyand momentum are large relative to the temperature of the medium which may escape fromthe medium without being thermalized)
Although the caveats above caution against attempting to draw quantitative conclusionswithout further physical inputs the success of the hydrodynamical description of the mediumitself support the conclusion that there must be some hydrodynamic response to the passageof the energetic particle through it In particular there must be some coupling of the energeticparticle to sound waves in the medium Since an energetic particle is moving through theplasma with a velocity greater than the velocity of sound cs the coupling of the energeticparticle to the sound mode must lead to the formation of a Mach cone namely a sound frontmoving away from the trajectory of the energetic particle at the Mach angle
cos ΘM =csv (775)
with v the velocity of the energetic particle However absent further physical inputs it is hardto estimate how strong the Mach cone produced by a point-like particle shooting through themedium will be
Several years ago the interest in these phenomena intensified considerably when the for-mation of Mach cones was suggested as a candidate to explain certain nontrivial particlecorrelations seen in heavy ion collisions at RHIC illustrated in the top-left panel of Fig 29Recall that in these experiments one triggers on (that is selects events containing) a highenergy hadron in the final state and then measures the distribution of all the other softhadrons in azimuthal angle ∆φ relative to the direction of the trigger particle After sub-tracting the effects of elliptic flow the correlation functions for nucleus-nucleus collisionsshow a near-side jet of hadrons at small ∆φ and a very broad distribution of soft particleson the away-side at angles around ∆φ sim π which may have low peaks at ∆φ = π plusmn φvwith φv asymp 1 minus 12 radians on the shoulders of the away-side distribution rather than at itscenter at ∆φ = π This correlation pattern is very different from that seen in proton-proton
163
collisions in which the away-side peak is broader than the near-side peak but is approxi-mately Gaussian in shape centered about a single peak at ∆φ = π Many authors haveargued that this nontrivial structure is the debris resulting from the near-complete quench-ing (thermalization) of the energetic particle that was initially produced back-to-back withthe hard parton that became the trigger hadron with the Mach cone excited in the mediumby this energetic particle leading to the peaks on the shoulders of the away-side distributionat ∆φ = π plusmn φv [137 138 501 521] This qualitative picture motivates the need for a goodquantitative model for the coupling between an energetic parton and the collective modes ofthe strongly coupled plasma
More recently some doubt has been cast on the interpretation of the qualitative featuresseen in the top-left panel of Fig 29 in terms of a Mach cone Alver and Roland realizedthat even though v3 = 0 on average it is nonzero in individual heavy ion collisions [526] Onaverage the shape of the overlap between two nuclei in an ensemble of collisions all with thesame nonzero impact parameter is almond-shaped meaning that vn = 0 for all odd n Butindividual nuclei are made of 200 nucleons meaning that when they are Lorentz-contractedinto pancakes these pancakes are not perfectly circular This means that the overlap betweentwo individual nuclei will not be perfectly almond shaped In general the shape of thisoverlap zone will have some ldquotriangularityrdquo which raises the possibility of a nonzero v3 inthe final state since the hydrodynamic expansion of an initial state that is initially triangularin position space will result in a nonzero v3 = 〈exp[i 3(φ minus ΦR)]〉 in momentum space andhence in the observed hadrons (Recall that ΦR is the angle of the reaction plane) In theanalysis of data that goes into producing Fig 29 v2 has been subtracted but v3 has not Ifv3 is nonzero in individual events the consequence in Fig 29 would be an excess at angles∆φ = 0 and ∆φ = πplusmn π
3 And Alver and Roland noted whenever low peaks are seen at someφv as in Fig 29 these peaks always seem to occur at φv asymp π3 independent of any cutsthat are made that should change the velocity of the energetic particle on the away side andconsequently should change the putative Mach angle The preliminary model calculations inRef [526] suggest that the incident nuclei are sufficiently asymmetric that the ldquotriangularflowrdquo which is to say v3 that is produced is sufficient to explain the features in Fig 29that had previously been attributed to the excitation of a Mach cone As of this writing itremains to be seen whether once the event-by-event v3 is quantified and subtracted therewill be any remaining evidence for a Mach cone [526]
Notwithstanding these recent developments it remains an interesting question of principleto understand in quantitative terms how strong a Mach cone is induced by the passage of anenergetic point-like quark through strongly coupled plasma Remarkably every one of thedifficulties associated with answering this problem in QCD sketched above can be addressedfor the case of an energetic heavy quark propagating through the strongly coupled plasmaof N = 4 SYM theory In this section we shall review this calculation which is done viathe dual gravitational description of the phenomenon As in Sections 71 and 72 we shallassume that the relevant physics is strongly coupled at all length scales treating the problementirely within strongly coupled N = 4 SYM theory In this calculation the AdSCFTcorrespondence is used to determine the stress tensor of the medium excited by the passingenergetic quark at all length scales This dynamical computation will allow us to quantifyto what extent hydrodynamics can be used to describe the response of the strongly coupledplasma of this theory to the disturbance produced by the energetic quark as well as to study
164
the relaxation of the initially far-from-equilibrium disturbance This calculation applies toquarks with mass M whose velocity respects the bound (718)
731 Hydrodynamic preliminaries
From the point of view of hydrodynamics the disturbance of the medium induced by thepassage of an energetic probe must be described by adding some source to the conservationequation
partmicroTmicroν(x) = Jν(x) (776)
As we have stressed above we do not know the functional form of the source since it not onlyinvolves the way in which energy is lost by the energetic particle but also how this energy isthermalized and how it is incorporated into the medium The source will in general dependnot only on the position of the quark but also on its velocity In this subsection we will usegeneral considerations valid in any hydrodynamic medium to constrain the functional formof the source From eq (776) it is clear that the amount of energy-momentum deposited inthe plasma is given by3
dP ν
dt=
intd3xJν(x) (777)
We now attempt to characterize the hydrodynamic modes that can be excited in theplasma due to the deposition of the energy (777) We will assume for simplicity that theperturbation on the background plasma is small We will also assume than the backgroundplasma is static The modification of the stress tensor
δTmicroν equiv Tmicroν minus Tmicroνbackground (778)
satisfies a linear equation
Since in the hydrodynamic limit the stress tensor is characterized by the local energy den-sity ε and the three components of the fluid spatial velocity ui there are only 4 independentfields which can be chosen to be
E equiv δT 00 and Si equiv δT 0i (779)
Using the hydrodynamic form of the stress tensor (210 ) all other stress tensor componentscan be expressed as a function of these variables Since we have assumed that these pertur-bations are small all the stress-tensor components can be expanded to first order in the fourindependent fields (779)
In Fourier space keeping the shear viscosity correction the linearized form of the equations(776) for the mode with a wave vector q that has the magnitude q equiv |q| take the form
parttE + iqSL = J0
parttSL + ic2sqE +
4
3
η
ε0 + p0q2SL = JL
parttST +η
ε0 + p0q2ST = JT (780)
3 We note as an aside that if the source moves supersonically one component of its energy loss is due tothe emission of sound waves This is conventionally known as sonic drag and is a part of the energy losscomputed in Section 71
165
where S = SLqq+ST J = JLqq+JT L and T refer to longitudinal and transverse relativeto the hydrodynamical wave-vector q and ε0 p0 cs =
radicdpdε and η are the energy density
pressure speed of sound and shear viscosity of the unperturbed background plasma Weobserve that the longitudinal and transverse modes are independent This decomposition ispossible since the homogeneous equations have a SO(2) symmetry corresponding to rotationaround the wave vector q The spin zero (longitudinal) and spin one (transverse) modescorrespond to the sound and diffusion mode respectively (The spin two mode is a sublead-ing perturbation in the gradient expansion since its leading contribution is proportional tovelocity gradients) After combining the first two equations of Eq (780) and doing a Fouriertransformation we find(
ω2 minus c2sq
2 + i4
3
η
ε0 + p0q2ω
)SL = i c2
sqJ0 + iωJL (
iω minus η
ε0 + p0q2
)ST = minusJT (781)
The sound mode (SL) satisfies a wave equation and propagates with the speed of sound whilethe diffusion mode (ST ) which does not propagate describes the diffusion of transversemomentum as opposed to wave propagation We also note that only the sound mode resultsin fluctuations of the energy density while the diffusion mode involves only momentumdensities (the Si of Eq (779)) In the linear approximation that we are using the excitationof the diffusion mode produces fluid motion but does not affect the energy density Thisresult can be further illustrated by expressing the energy fluctuations in terms of the velocityfields
δT 00 = δε+1
2(ε+ P ) (δv)2 + (782)
The second term in this expression corresponds to the kinetic energy contribution of the fluidmotion which takes a non-relativistic form due to the small perturbation approximationThis expression is quadratic in the velocity fluctuation and thus is not described in thelinearized approximation The sound mode corresponds to both compressionrarefaction ofthe fluid and motion of the fluid sound waves result in fluctuations of the energy densityas a consequence of the associated compression and rarefaction But the diffusion modecorresponds to fluid motion only and to this order does not affect the energy density
Solving the linearized hydrodynamic equations (780) yields hydrodynamic fields given by
E(tx) =
intdω
2π
d3q
(2π)3
iqJL + iωJ0 minus Γsq2J0
ω2 minus c2sq
2 + iΓsq2ωeminusiωt+iqmiddotx (783)
SL(tx) =
intdω
2π
d3q
(2π)3
q
q
c2siqJ
0 + iωJLω2 minus c2
sq2 + iΓsq2ω
eminusiωt+iqmiddotx (784)
ST (tx) =
intdω
2π
d3q
(2π)3
minusJTiω minusDq2
eminusiωt+iqmiddotx (785)
where the sound attenuation length and the diffusion constant are
Γs equiv4
3
η
ε0 + p0(786)
D equiv η
ε0 + p0 (787)
166
We note in passing that the integral of the longitudinal momentum density over all spacevanishes
The hydrodynamic solutions (783) (784) and (785) are only of formal value withoutany information about the source And as we have stressed above a lot of nonlinear non-equilibrium physics goes into determining the source as a function of the coordinates Still wecan make some further progress If we assume that the energetic quark moves at a constantvelocity v for a long time (as would be the case if the quark is either ultrarelativistic or veryheavy) then we expect
Jmicro(ω k) = 2πδ(ω minus v middot q)Jmicrov (q) (788)
where the factor δ(ω minus v middot q) comes from Fourier transforming δ(xminus vt) We also note thatfar away from the source and at sufficiently small q that we can neglect any energy scalescharacteristic of the medium and any internal structure of the particle moving through themedium the only possible vectors from which to construct the source are v and q In thisregime we may decompose the source as
J0v (q) = e0(q)
Jv(q) = v g0(q) + q g1(q) (789)
Then inspection of the solutions (783) (784) and (785) together with the observation thata particle moving with a velocity close to the speed of light loses similar amounts of energyand momentum shows that at least for an ultrarelativistic probe nonvanishing values ofe0(q) must be linked to nonvanishing values of g0(q) We call this case scenario 1 Howeverif the interaction of the probe with the plasma is such that both g0 and e0 are zero (orparametrically small compared to g1) from Eqs (777) and (789) and since q g1(q) is a totalderivative one may mistakenly conclude that the energetic probe has created a disturbancecarrying zero energy and momentum In this scenario which we shall call scenario 2 theenergy and momentum loss are actually quadratic in the fluctuations These two scenarioslead to disturbances with different characteristics In scenario 2 only the sound mode isexcited while in scenario 1 both the sound and diffusion mode are excited The correctanswer for a given energetic probe may lie in between these two extreme cases
The phenomenological implications of this analysis depend critically on the degree to whichthe diffusion mode is excited This mode leads to an excess of momentum density along thedirection of the source which does not propagate out of the region of deposition but onlydiffuses away Therefore the diffusion mode excited by an energetic quark moving throughthe plasma corresponds to a wake of moving fluid trailing behind the quark and moving inthe same direction as the quark In a heavy ion collision therefore the diffusion wake excitedby the away-side energetic quark will become hadrons at ∆φ sim π whereas the Mach conewill become a cone of hadrons with moment at some angle away from ∆φ = π If most ofthe energy dumped into the medium goes into the diffusion wake even if a Mach cone wereproduced it would be overwhelmed in the final state and invisible in the data Only in thecase in which the diffusion mode is absent (or sufficiently small) is the formation of a Machcone reflected in a visible nontrivial correlation of the type seen in Fig 29
167
732 AdS computation
In Section 71 we have computed the amount of energy lost by a heavy quark as it plowsthrough the strongly coupled N = 4 SYM theory plasma In order to address the fate of thisenergy we must determine the stress tensor of the gauge theory fluid at the boundary thatcorresponds to the string (79) trailing behind the quark in the bulk In the dual gravitationaltheory this string modifies the metric of the (4+1)-dimensional geometry That is it producesgravitational waves The stress energy tensor of the gauge theory plasma at the boundaryis determined by the asymptotic behavior of the bulk metric perturbations as they approachthe boundary [505509510513]
The modifications of the 4+1-dimensional metric due to the presence of the trailing stringare obtained by solving the Einstein equations
Rmicroν minus1
2Gmicroν(R+ 2Λ) = κ2
5tmicroν (790)
where κ25 = 4π2R3N2
c and Λ = 6R2 with R the AdS radius and where tmicroν is the five-dimensional string stress tensor which can be computed from the Nambu-Goto action
tmicroν = minus 1
2παprime
intdτdσ
radicminushradicminusG
habpartaXmicro(τ σ)partbX
ν(τ σ)δ(5) (xminusX(τ σ)) (791)
where hab is the induced metric on the string and X(τ σ) is the string profile For the caseof a trailing string (79) the stress tensor is given by
t00 = s(f + v2z4z40)
t0i = minussvi t0z = minussv2z2z2
0f
tzz = s(f minus v2)f2
tij = svivj
tiz = sviz2z2
0f (792)
where
s =zγradicλ
2πR3δ3 (xminus vtminus ζ(z)) (793)
with ζ(z) the string profile (79) After solving the Einstein equations (790) with the stringstress tensor (792) the expectation value of the boundary stress tensor is then obtained byfollowing the prescription (542) namely by performing functional derivatives of the Hilbertaction evaluated on the classical solution with respect to the boundary metric
We need to analyze the small fluctuations on top of the background AdS black hole metricDenoting these fluctuations by hmicroν and the background metric by gmicroν the left-hand side ofthe Einstein equations (790) are given to leading order in hmicroν by
minusD2hmicroν + 2DσD(microhν)σ minusDmicroDνh+8
R2hmicroν
+
(D2hminusDσDδhσδ minus
4
R2h
)gmicroν = 0 (794)
168
with Dmicro the covariant derivative with respect to the full metric namely gmicroν + hmicroν Thisequation has a gauge symmetry
hmicroν rarr hmicroν +Dmicroξν +Dνξmicro (795)
inherited from reparameterization invariance that together with 5 constraints from thelinearized Einstein equations (794) reduces the number of degrees of freedom from 15 to5 It is therefore convenient to introduce gauge invariant combinations which describe theindependent degrees of freedom These can be found after Fourier transforming the (3 + 1)-dimensional coordinates The gauge invariants can be classified by how they transform underSO(2) rotations around the wave vector q Upon introducing Hmicroν = z2hmicroνR
2 one possiblechoice of gauge invariants is given by [509513]
Z(0) = q2H00 + 2ωqH0q + ω2Hqq +1
2
[(2minus f)q2 minus ω2
]H
Z(1)α =(H prime0α minus iωHα5
)Z(2)αβ =
(Hαβ minus
1
2Hδαβ
)(796)
where q equiv |q| q equiv qq H0q equiv H0iqi Hqq equiv Hij q
iqj α and β (which are each either 1 or2) are space coordinates transverse to q prime means partz and H equiv Hαα When written in termsof these gauge invariants the Einstein equations (794) become three independent equationsfor Z(0) Z(1)α and Z(2)αβ which correspond to the spin zero one and two fluctuations ofthe stress tensor We focus on the spin zero and spin one fluctuations since these are therelevant modes in the hydrodynamic limit Their equations of motion are given by
Z primeprime(1)α +zfprime minus 3f
zfZ prime(1)α +
3f2 minus z(zq2 + 3fprime)f + z2ω2
z2f2Z(1)α = S(1)α (797)
Z primeprime(0) +1
u
[1 +
ufprime
f+
24(q2f minus ω2)
q2(uf prime minus 6f) + 6ω2
]Z prime(0)
+1
f
[minusq2 +
ω2
fminus 32q2z6z8
0
q2(uf prime minus 6f) + 6ω2
]Z(0) = S(0) (798)
where the sources are combinations of the string stress tensor and its derivatives Choosingone of the transverse directions (which we shall denote by α = 1) to lie in the (vq) planethe source for the trailing string is given explicitly by
S(1)1 =2κ2
5γradicλ
R3
vqperpqf
δ(ω minus v middot q)eminusiqmiddotζ
S(1)2 = 0 (799)
S(0) =κ2
5γradicλ
3R3
q2(v2 + 2)minus 3ω2
q2
z[q4z8 + 48iq2z2
0z5 minus 9(q2 minus ω2)2z8
0
]f(fq2 + 2q2 minus 3ω2)z8
0
timesδ(ω minus vq)eminusiqmiddotζ (7100)
where qperp is the magnitude of the component of q perpendicular to v The boundary actioncan be expressed in terms of the gauge invariants Z(1)α and Z(0) plus certain counterterms
169
(terms evaluated at the boundary) This procedure which can be found in Ref [513] issomewhat cumbersome but straightforward and we shall not repeat it here Once this isachieved the stress tensor components can be obtained from the classical solution to (797)-(798) following the prescription (542)
To find the classical solution to (797)-(798) we must specify boundary conditions Sincethe quark propagates in flat space the metric fluctuations must vanish at the boundaryAlso since we are interested in the response of the medium the solution must satisfy retardedboundary condition meaning that at the horizon it must be composed only of infalling modesThus we may construct the Greenrsquos function
Gs(z zprime) =
1
Ws(zprime)
(θ(zprime minus z)gns (z)gis(z
prime) + θ(z minus zprime)gis(z)gns (zprime))
(7101)
where the subscript s which can be 0 or 1 denotes the spin component and gns and gis denotethe normalizable and infalling solutions to the homogeneous equations obtained by settingthe left-hand side of (797) equal to zero Ws is the Wronskian of the two homogeneoussolutions The full solution to (797) may be then written as
Zs(z) =
int zh
0dzprimeGs(z z
prime)Ss(zprime) (7102)
Close to the boundary these solutions behave as
Z(0) = z3Z[3](0) + z4Z
[4](0) +
~Z(1) = z2 ~Z[2](1) + z3 ~Z
[3](1) + (7103)
The components Z[3](0) and Z
[2](1) can be computed analytically and are temperature indepen-
dent They yield a divergent contribution to the boundary stress tensor However thiscontribution is analytic in q and thus has δ-function support at the position of the heavyquark This divergent contribution is the contribution of the heavy quark mass to the bound-ary theory stress tensor The response of the boundary theory gauge fields to the disturbance
induced by the passing energetic quark is encoded in the components Z[4](0) and Z
[3](1) which
must be computed numerically After expressing the boundary actions in terms of gaugeinvariants the nondivergent spin zero and one components of the boundary stress tensor aregiven by
T0 =4q2
3κ25(q2 minus ω2)2
Z[4](0) +D + ε0 (7104)
~T1 = minus L3
2κ5
~Z[3](1) (7105)
where T0 = T 00 and ~T1 = T 0aεa with εa the spatial unit vectors orthogonal to the spatialmomentum q and where the counterterm D is a complicated function of ω and q that dependson the quark velocity and the plasma temperature and that is given in Ref [513]
Results from Ref [513] on the numerical computation of the disturbance in the gaugetheory plasma created by a supersonic quark moving with speed v = 075 are shown in
170
Figure 74 Energy density (left) and momentum flux (right) induced by the passage of a supersonicheavy quark moving through the strongly coupled N = 4 SYM theory plasma in the x direction withspeed v = 075 (∆ε(x) is the difference between ε(x) and the equilibrium energy density since S = 0in equilibrium ∆S(x) is simply S(x)) The flow lines on the surface are flow lines of ∆S(x) Thesedisturbances are small compared to the background energy density and pressure of the plasma (bothof which are prop N2
c ) The perturbation is small and it is well described by linearized hydrodynamicseverywhere except within a distance R asymp 16T from the quark Since the perturbation is small thekinetic energy contribution of the diffusion mode to the energy density is suppressed by N2
c and thusit does not contribute in the left panel
Fig 74 The left panel shows the energy density of the disturbance and clearly demonstratesthat a Mach cone has been excited by the supersonic quark The front is moving outwardsat the Mach angle ΘM where cos ΘM = csv = 4(3
radic3) Recall from our general discussion
above that fluid motion is invisible in the energy density to the linear order at which we areworking the energy density is nonzero wherever the fluid is compressed Thus the Machcone is made up of sound modes as expected In the right panel of Fig 74 we see the densityof fluid momentum induced by the supersonic quark This figure reveals the presence of asizable wake of moving fluid behind the quark a wake that is invisible in the energy densityand is therefore made up of moving fluid without any associated compression meaning thatit is made up of diffusion modes We conclude that the supersonic quark passing throughthe strongly coupled plasma excites both the sound mode and the diffusion mode meaningthat the interaction of the quark with the plasma is as in what we called scenario 1 aboveQuantitatively it turns out that the momentum carried by the sound waves is greater thanthat carried by the diffusion wake but only by a factor of 1 + v2 [511]
Since hydrodynamics describes the long wavelength limit of the stress tensor excitation itis reasonable to find a Mach cone at long distances And since the gravitational equationswhose solution we have reviewed are linear the long distance behavior of the gauge theoryfluid must be described by linearized hydrodynamics It is easy to justify the linearizationfrom the point of view of the field theory the background plasma has an energy density thatis proportional to N2
c while that of the perturbation is proportional to the number of flavorswhich is just Nf = 1 in the present case since we are considering only one quark The strongcoupling computation leads to a perturbation of magnitude Nf
radicλ Thus the energy density
171
of the fluctuations are suppressed byradicλN2
c with respect to that of the background plasmajustifying the linearized treatment Remarkably it turns out that disturbances like those inFig 74 are well described by hydrodynamics everywhere except within asymp 16T of the posi-tion of the quark [513] So the calculation that we have reviewed in this section is importantfor two reasons First it demonstrates that a point-like probe passing through the stronglycoupled plasma does indeed excite hydrodynamic modes And second it demonstrates thatin the strongly coupled plasma the resulting disturbance relaxes to a hydrodynamic excita-tion in local thermal equilibrium surprisingly close to the probe
The observation that point particles moving through the strongly coupled fluid excite soundwaves which are collective excitations is at odds with intuition based upon the interactionof say electrons with water In this example most of the energy lost by the electron istransferred to photons and not to the medium These photons in turn have long mean freepaths and dissipate their energy far away from the electron (or escape the medium entirely)Thus the effective size of the region where energy is dissipated is very large given by thephoton mean free path Hydrodynamics will only describe the physics on longer length scalesthan this The reason that no Mach cone is formed is that the length scale over which theenergy is deposited is long compared to the length scale over which the electron slows andstops The situation is similar in weakly coupled gauge theory plasmas even though thegauge modes in these theories do interact they still have long mean free paths proportionalto 1g4 In sharp contrast in the strongly coupled plasma of N = 4 SYM theory there areno long-lived quasiparticle excitations (leave apart photons) that could transport the energydeposited by the point-like particle over long distances Instead all the energy lost by thepoint-like probe is dumped into collective hydrodynamic modes over a characteristic lengthscale sim 1T which is the only length scale in this conformal plasma
733 Implications for heavy ion collisions
The calculation that we have reviewed in this section suggests that a high energy quarkplowing through the strongly coupled plasma produced in heavy ion collisions at RHIC shouldexcite a Mach cone As we argued just above this phenomenon is not expected in a weaklycoupled plasma The Mach cone should have consequences that are observable in the softparticles on the away-side of a high energy trigger hadron However for a hydrodynamicsolution like that in Fig 74 it turns out that the diffusion wake contains enough momentumflux along the direction of the energetic particle to ldquofill inrdquo the center of the Mach conemeaning that the Mach cone is not sufficiently prominent as to result in peaks in the particledistribution at ∆φ = π plusmn ΘM [138 518] As we discussed above the observed peaks at∆φ = π plusmn φv receive a significant contribution from the event-by-event v3 due to event-by-event fluctuations that introduce ldquotriangularityrdquo Detecting evidence for Mach cones in heavyion collisions will require careful subtraction of these effects from the data on the one handand careful theoretical analysis of the effects of the rapid expansion of the fluid produced inheavy ion collisions on the putative Mach cones on the other
172
734 Disturbance excited by a moving quarkonium meson
Strong coupling calculations like that of the disturbance excited by an energetic quark mov-ing through the plasma of N = 4 SYM theory can help guide the construction of morephenomenological models of the coupling of energetic particles to hydrodynamic modes Tofurther that end we close with an example which shows that not all probes behave in thesame way
As we shall describe in Section 77 a simple way of modelling a ldquoquarkoniumrdquo meson madefrom a heavy quark and antiquark embedded in the strongly coupled plasma of N = 4 SYMtheory is to consider a string with both ends at the boundary mdash the ends representing thequark and antiquark We shall see in Section 77 that even when this string is moving throughthe plasma it hangs straight downward into the AdS black hole metric rather than trailingbehind as happens for the string hanging downward from a single moving quark The factthat the ldquoUrdquo of string hangs straight down and does not trail behind the moving quark andantiquark implies that the heavy quarkonium meson moving through the strongly coupledplasma does not lose any energy at least at leading order The energy loss of such a mesonhas been computed and is in fact nonzero but is suppressed by 1N2
c [527]
Despite the fact that the leading order quarkonium energy loss vanishes the leading orderdisturbance of the fluid through which the meson is moving does not vanish [528] Insteadthe meson excites a Mach cone with no diffusion wake providing an example of what wecalled scenario 2 at the end of Section 731 It is as though the moving meson ldquodressesitselfrdquo with a Mach cone and then the meson and its Mach cone propagate through thefluid without dissipation to leading order To illustrate this point the metric fluctuationand consequent boundary stress tensor induced by a semiclassical string with both ends onthe boundary moving with a velocity v has been calculated [528] For a string with thetwo endpoints aligned along the direction of motion and separated by a distance l the longdistance part (low momentum) part of the associated stress tensor is given by
δT 00 =Π
q2 minus 3 (q middot v)2
(minusq2
(1 + 2v2σ
)minus 3v2 (q middot v)2 (1minus 2σ)
)(7106)
δT 0i =Π
k2 minus 3 (q middot v)2 2q middot v(1minus (1minus v2)σ
)qi + 2 Πσ vi (7107)
δT ij =Π
k2 minus 3 (q middot v)2 2 (q middot v)2 (minus1 + (1minus v2)σ)δij minus (7108)
minusΠ
v2
(1 + 2v2σ
) vivjv2
where σ = σ(l T ) is a dimensionless function of the length of the meson and the temperatureand the prefactor takes the form
Π =radicλF (lT )
l (7109)
with F a dimensional function
The expression (7106) clearly shows that all the spin 01 and 2 components of the stresstensor are excited The spin 0 components are multiplied by the sound propagator signalingthe emission of sound waves (Note that in the low-q limit the width of the sound pole
173
vanishes) The spin 1 component corresponds to the terms proportional to the velocity ofthe particle vi These terms are analytic in q in particular it seems that there is no polecontribution from the diffusive mode More careful analysis shows that the diffusive modedecays faster than that excited by a quark probe
The magnitude of the disturbance in the strongly coupled plasma that is excited by apassing quarkonium meson is no smaller than that excited by a passing quark However thetotal integral of the energy and momentum deposited is zero as can be seen by multiplyingthe momentum densities by ω = v middot q and taking the limit q rarr 0 This is consistent withthe fact that to the order at which this calculation has been done the meson does notlose any energy This is an interesting example since it indicates that the loss of energyand the excitation of hydrodynamic modes are distinct phenomena controlled by differentphysics This example also illustrates the value of computations done at strong coupling inopening ones eyes to new possibilities without these calculations it would have been veryhard to guess or justify that such a separation in magnitude between the strength of thehydrodynamic fields excited by a probe and the energy lost by that probe could be possibleIt would be interesting to analyze the soft particles in heavy ion collisions in which a hightransverse momentum quarkonium meson is detected to see whether there is any hint of aMach cone around the meson mdash in this case without the complication of soft particles froma diffusion wake filling in the cone
74 Stopping light quarks
As we have discussed extensively in Section 23 the dominant energy loss process for aparton moving through the QCD plasma with energy E in the limit in which E rarr infin isgluon radiation and in this limit much (but not all see Section 75) of the calculation can bedone at weak coupling However since it is not clear how quantitatively reliable the E rarrinfinapproximation is for the jets produced in RHIC collisions it is also worth analyzing theentire problem of parton energy loss and jet quenching at strong coupling to the degree thatis possible For the case of a heavy quark propagating through the strongly coupled plasmaof N = 4 SYM theory this approach has been pursued extensively yielding the many resultsthat we have reviewed in the previous three subsections Less work has been done on theenergy loss of an energetic light quark or gluon in the N = 4 SYM plasma in particularsince they do not fragment into anything like a QCD jet (As we shall review in Section 76there are no true jets in N = 4 SYM theory although it is possible to cook up a collimatedbeam of radiation) It is nevertheless worth asking how a light quark or gluon loses energyin the N = 4 SYM plasma in the hope that even if this is not a good model for jets and theirquenching in QCD some qualitative strong-coupling benchmarks against which to compareexperimental results may be obtained This program has been pursued in Refs [529ndash531]
As introduced in Section 55 and described extensively in Section 8 dynamical quarkscan be introduced into N = 4 SYM theory by introducing a D7-brane that fills the 3 + 1Minkowski dimensions and fills the fifth dimension from the boundary at z = 0 down toz = zq The mass of the (heavy) quarks that this procedure introduces in the gauge theoryisradicλ(2πzq) Light quarks are obtained by taking zq rarrinfin meaning that the D7-brane fills
all of the z-dimension At T 6= 0 what matters is that the D7-brane fills the z-dimension all
174
the way down to and below the horizon In this set up a light quark-antiquark pair movingback-to-back each with some initial high energy can be modelled as a quark and anti-quarklocated at some depth z that are moving apart from each other in say the x direction andthat are connected by a string [530 531] The quark and antiquark must be within the D7brane but since this D7 brane fills all of z there is nothing stopping them from falling tolarger z as they fly apart from each other and ultimately there is nothing stopping themfrom falling into the horizon It should be evident from this description that there is anarbitrariness to the initial condition At what z should the quark and antiquark be locatedinitially What should the string profile be initially What should the initial profile of thevelocity of the string be These choices correspond in the gauge theory to choices aboutthe initial quantum state of the quark-antiquark pair and the gauge fields surrounding themAnd there is no known way to choose these initial conditions so as to obtain a QCD-like jetso the choices made end up being arbitrary (The analogous set up for a back-to-back pair ofhigh energy gluons [529] involves a doubled loop of string rather than an open string with aquark and antiquark at its ends)
Ambiguities about the initial conditions notwithstanding several robust qualitative in-sights have been obtained from these calculations First the quark and the antiquark alwaysfall into the horizon after travelling some finite distance xstopping (The string between themfalls into the horizon also) xstopping corresponds in the gauge theory to the stopping distancefor the initially energetic quark namely the distance that it takes this quark to slow downthermalize and equilibrate with the bulk plasma mdash the gauge theory analogue of fallinginto the horizon This is qualitatively reminiscent of the discovery at RHIC of events with asingle jet (manifest as a high pT hadron that is triggered on) but no jet (no high pT hadrons)back-to-back with it
Second although xstopping does depend on details of the initial conditions the dominantdependence is that it scales like E13 where E is the initial energy of the quark [529 531ndash533] More precisely upon analyzing varied initial conditions the maximum possible stoppingdistance is given by [531]
xstopping =C
T
(E
Tradicλ
)13
(7110)
with C asymp 05 If there is a regime of E and T in which it is reasonable to treat the entireproblem of jet quenching at strong coupling and if in this regime the droplet of plasmaproduced in a heavy ion collision is large enough and lives long enough that it can stop andthermalize an initial parton with energy E that would in vacuum have become a jet then thescaling (7110) has interesting qualitative consequences For example if this scaling appliesto collisions with two different collision energies
radics1 and
radics2 yielding plasmas that form at
different temperatures T1 and T2 then jets in these two experiments whose energies satisfyE1E2 sim (T1T2)4 should have similar observed phenomenology Turning this speculationinto semi-quantitative expectations for experimental observables requires careful study of jetstopping in a realistic model of the dynamics in space and time of the expanding droplet ofplasma produced in a heavy ion collision
Third a light quark with initial energy E that loses this energy over a distance xstopping
loses most of its energy near the end of its trajectory where it thermalizes (falls into thehorizon) [531] This pattern of energy loss is reminiscent of the lsquoBragg peakrsquo that characterizes
175
the energy loss of a fast charged particle in ordinary matter where the energy loss has apronounced peak near the stopping point It is quite different from the behavior of a heavyquark in strongly coupled plasma which as we saw in Section 71 loses energy at a rateproportional to its momentum making it reasonable to expect that a heavy quark that slowsfrom a high velocity to a stop loses more energy earlier in its trajectory than later
It will be very interesting to see how these insights fare when compared with results onjet quenching in heavy ion collisions at the LHC and in particular to comparisons betweensuch results and results at RHIC energies
75 Calculating the jet quenching parameter
As we have described in Section 23 when a parton with large transverse momentum isproduced in a hard scattering that occurs within a heavy ion collision the presence of themedium in which the energetic parton finds itself has two significant effects it causes theparton to lose energy and it changes the direction of the partonrsquos momentum The lattereffect is referred to as ldquotransverse momentum broadeningrdquo In the high parton energy limit asestablished first in Refs [139ndash141] the parton loses energy dominantly by inelastic processesthat are the QCD analogue of bremsstrahlung the parton radiates gluons as it interactswith the medium It is crucial to the calculation of this radiative energy loss process thatthe incident hard parton the outgoing parton and the radiated gluons are all continuallybeing jostled by the medium in which they find themselves they are all subject to transversemomentum broadening The transverse momentum broadening of a hard parton is describedby P (kperp) defined as the probability that after propagating through the medium for a distanceL the hard parton has acquired transverse momentum kperp For later convenience we shallchoose to normalize P (kperp) as followsint
d2kperp(2π)2
P (kperp) = 1 (7111)
From the probability density P (kperp) it is straightforward to obtain the mean transversemomentum picked up by the hard parton per unit distance travelled (or equivalently in thehigh parton energy limit per unit time)
q equiv〈k2perp〉L
=1
L
intd2kperp(2π)2
k2perpP (kperp) (7112)
P (kperp) and consequently q can be evaluated for a hard quark or a hard gluon In thecalculation of radiative parton energy loss [140ndash145155] that we have reviewed in Section 23and that is also reviewed in Refs [150154157ndash159168169173] q for the radiated gluon playsa central role and this quantity is referred to as the ldquojet quenching parameterrdquo Consequentlyq should be thought of as a (or even the) property of the strongly coupled medium that isldquomeasuredrdquo (perhaps constrained is a better phrase) by radiative parton energy loss andhence jet quenching But it is important to note that q is defined via transverse momentumbroadening only Radiation and energy loss do not arise in its definition although they arecentral to its importance
176
The BDMPS calculation of parton energy loss in QCD involves a number of scales whichmust be well-separated in order for this calculation to be relevant The radiated gluons haveenergy of order ωc sim qL2 and transverse momenta of order
radicqL Both these scales must be
much less than E and much greater than T And αS evaluated at both these scales mustbe small enough that physics at these scales is weakly coupled even if physics at scales oforder T is strongly coupled In heavy ion collisions at RHIC with the highest energy partonshaving E only of order many tens of GeV this separation of scales can be questioned It isexpected that in heavy ion collisions at the LHC it will be possible to study the interaction ofpartons with energies of order a few hundred GeV improving the reliability of the calculationsreviewed in this subsection In the RHIC regime and particularly for heavy quarks thealternative approach of the previous subsection in which the entire problem is assumed to bestrongly coupled and parton energy loss occurs via drag is just as plausibly relevant In thissubsection we shall review the calculation of q mdash the property of the plasma that describestransverse momentum broadening directly and in the high parton energy limit in which therelevant scales are well separated controls the transverse momentum and the energy of thegluon radiation that dominates parton energy loss
It has been shown via several different calculations done via conventional field theoreticalmethods [150151534] that the transverse momentum broadening of a hard parton in the Rrepresentation of SU(N) takes the form
P (kperp) =
intd2xperp e
minusikperpmiddotxperpWR(xperp) (7113)
with
WR(xperp) =1
d (R)
langTr[W daggerR[0 xperp]WR[0 0]
]rang(7114)
where
WR[x+ xperp
]equiv P
exp
[ig
int Lminus
0dxminusA+
R(x+ xminus xperp)
](7115)
is the representation-R Wilson line along the lightcone Lminus =radic
2L is the distance alongthe lightcone corresponding to travelling a distance L through the medium and where d (R)is the dimension of the representation R The result (7113) is similar to (748) althoughthe physical context in which it arises is different as is the path followed by the Wilsonline and one of the derivations [150] of (7113) is analogous to the derivation of (748) thatwe reviewed in Section 72 Another derivation [151] proceeds via the use of the opticaltheorem to relate P (kperp) to an appropriate forward scattering matrix element which can thenbe calculated explicitly via formulating the calculation of transverse momentum broadeningin the language of Soft Collinear Effective Theory [535ndash539] This derivation in particularmakes it clear that (7113) is valid whether the plasma through which the energetic quarkis propagating ie the plasma which is causing the transverse momentum broadening isweakly coupled or strongly coupled The result (7113) is an elegant expression saying thatthe probability for the quark to obtain transverse momentum kperp is simply given by the two-dimensional Fourier transform in xperp of the expectation value (7114) of two light-like Wilsonlines separated in the transverse plane by the vector xperp Note that the requirement (7111)that the probability distribution P (kperp) be normalized is equivalent to the requirement thatWR(0) = 1
177
ti tf
tf minus iεti minus iε
ti minus iβ
Figure 75 The Schwinger-Keldysh contour that must be used in the evaluation of WR(xperp) It issimilar to that in Fig 72
It is important to notice that the expectation value of the trace of the product of twolight-like Wilson lines that arises in P (kperp) and hence in q namely WR(xperp) of (7114) hasa different operator ordering from that in a standard Wilson loop Upon expanding theexponential each of the A+ that arise can be written as the product of an operator and agroup matrix A+ = (A+)ata It is clear (for example either by analogy with our discussionaround (745) in the analysis of momentum broadening of heavy quarks or from the explicitderivation in Ref [151]) that in WR(xperp) both the operators and the group matrices are pathordered In contrast in a conventional Wilson loop the group matrices are path orderedbut the operators are time ordered Because the operators in (7114) are path ordered theexpectation value in (7114) should be described using the Schwinger-Keldysh contour inFig 75 with one of the light-like Wilson lines on the Im t = 0 segment of the contour andthe other light-like Wilson line on the Im t = minusiε segment of the contour The infinitesimaldisplacement of one Wilson line with respect to the other in Fig 75 ensures that the operatorsfrom the two lines are ordered such that all operators from one line come before any operatorsfrom the other In contrast the loop C for a standard Wilson loop operator lies entirely atIm t = 0 and the operators for a standard Wilson loop are time ordered
The transverse momentum broadening of a hard parton with energy E is due to repeatedinteractions with gluons from the medium which if the medium is in equilibrium at tempera-ture T carry transverse momenta of order T and lightcone momenta of order T 2E [151540]The relation (7113) between P (kperp) (and hence q) and the expectation value W of (7114)is valid as long as E qL2 (which is to say E must be much greater than the characteristicenergy of the radiated gluons) even if αS(T ) is in no way small ie it is valid in the large-Elimit even if the hard parton is interacting with a strongly coupled plasma and even if thesoft interactions that generate transverse momentum broadening are not suppressed by anyweak coupling either [151] However in this circumstance even though (7113) is valid it wasnot particularly useful until recently because there is no known conventional field theoreticalevaluation of W for a strongly coupled plasma (Since lattice quantum field theory is for-mulated in Euclidean space it is not well-suited for the evaluation of the expectation valueof lightlike Wilson lines) In this subsection we review the evaluation of W and hence q inthe strongly coupled plasma of N = 4 SYM theory with gauge group SU(Nc) in the large N
178
and strong coupling limit using its gravitational dual namely the AdS Schwarzschild blackhole at nonzero temperature [148149151456458461462541ndash544] The calculation is notsimply an application of results reviewed in Section 54 both because the operators are pathordered and because the Wilson lines are light-like
We begin by sketching how the standard AdSCFT procedure for computing a Wilsonloop in the fundamental representation in the large Nc and strong coupling limit reviewedin Section 54 applies to a light-like Wilson loop with standard operator ordering [148149]and then below describe how the calculation (but not the result) changes when the operatorordering is as in (7114) Consider a Wilson loop operator W (C) specified by a closed loop C inthe (3+1)-dimensional field theory and thus on the boundary of the (4+1)-dimensional AdSspace 〈W (C)〉 is then given by the exponential of the classical action of an extremized stringworldsheet Σ in AdS which ends on C The contour C lives within the (3 + 1)-dimensionalMinkowski space boundary but the string world sheet Σ attached to it hangs ldquodownrdquo intothe bulk of the curved five-dimensional AdS5 spacetime More explicitly consider two longparallel light-like Wilson lines separated by a distance xperp in a transverse direction4 (Thestring world sheet hanging down into the bulk from these two Wilson lines can be visualizedas in Fig 77 below if one keeps everything in that figure at Im t = 0 ie if one ignoresthe issue of operator ordering) Upon parameterizing the two-dimensional world sheet bythe coordinates σα = (τ σ) the location of the string world sheet in the five-dimensionalspacetime with coordinates xmicro is
xmicro = xmicro(τ σ) (7116)
and the Nambu-Goto action for the string world sheet is given by
S = minus 1
2παprime
intdσdτ
radicminusdetgαβ (7117)
Heregαβ = Gmicroνpartαx
micropartβxν (7118)
is the induced metric on the world sheet and Gmicroν is the metric of the (4 + 1)-dimensionalAdS5 spacetime Denoting by S(C) the classical action which extremizes the Nambu-Gotoaction (7117) for the string worldsheet with the boundary condition that it ends on the curveC the expectation value of the Wilson loop operator is then given by
〈W (C)〉 = exp [i S(C)minus S0] (7119)
where the subtraction S0 is the action of two disjoint strings hanging straight down from thetwo Wilson lines In order to obtain the thermal expectation value at nonzero temperatureone takes the metric Gmicroν in (7118) to be that of an AdS Schwarzschild black hole (530) witha horizon at r = r0 and Hawking temperature T = r0(πR
2) The AdS curvature radius Rand the string tension 1(2παprime) are related to the rsquot Hooft coupling in the Yang-Mills theoryλ equiv g2Nc by
radicλ = R2αprime
We shall assume that the length of the two light-like lines Lminus =radic
2L is much greater thantheir transverse separation xperp which can be justified after the fact by using the result for
4Note that for a light-like contour C the Wilson line (563) of N = 4 SYM theory reduces to the familiar(7115)
179
W(xperp) to show that the xperp-integral in (7113) is dominated by values of xperp that satisfy xperp 1radicqL sim 1
radicradicλLT 3 As long as we are interested in L 1T then xperp 1(Tλ14)
1T L With Lminus xperp we can ignore the ends of the light-like Wilson lines and assumethat the shape of the surface Σ is translationally invariant along the light-like direction Theaction (7117) now takes the form
S = i
radic2r2
0
radicλLminus
2πR4
int xperp2
0dσ
radic1 +
rprime2R4
r4 minus r40
(7120)
where the shape of the worldsheet Σ is described by the function r(σ) that satisfies r(plusmnxperp2 ) =
infin which preserves the symmetry r(σ) = r(minusσ) and where rprime = partσr The equation of motionfor r(σ) is then
rprime2 =γ2
R4
(r4 minus r4
0
)(7121)
with γ an integration constant Eq (7121) has two solutions One has γ = 0 and hencerprime = 0 meaning that r(σ) = infin for all σ the surface Σ stays at infinity Generalizations ofthis solution have also been studied [545546] We shall see below that such solutions are notrelevant The other solution has γ gt 0 It ldquodescendsrdquo from r(plusmnxperp
2 ) = infin and has a turningpoint where rprime = 0 which by symmetry must occur at σ = 0 From (7121) the turningpoint must occur at the horizon r = r0 Integrating (7121) gives the condition that specifiesthe value of γ
xperp2
=R2
γ
int infinr0
drradicr4 minus r4
0
=aR2
γr0(7122)
where we have defined
a equivradicπΓ(
5
4)Γ(
3
4) asymp 1311 (7123)
Putting all the pieces together we find [148149]
S =iaradicλTLminusradic
2
radic1 +
π2T 2x2perp
4a2 (7124)
We see that S is imaginary because when the contour C at the boundary is lightlike thesurface Σ hanging down from it is spacelike It is worth noting that S had to turn out to beimaginary in order for 〈W 〉 in (7119) to be real and the transverse momentum broadeningP (kperp) to be real as it must be since it is a probability distribution The surface Σ that wehave used in this calculation descends from infinity skims the horizon and returns to infinityNote that the surface descends all the way to the horizon regardless of how small xperp is This isreasonable on physical grounds as we expect P (kperp) to depend on the physics of the thermalmedium [148 149] We shall see below that it is also required on mathematical groundswhen we complete the calculation by taking into account the nonstandard operator orderingin (7113) we shall see that only a worldsheet that touches the horizon is relevant [151]
We now consider the computation of (7114) with its nonstandard operator ordering cor-responding to putting one of the two light-like Wilson lines on the Im t = 0 contour in Fig 75and the other on the Im t = minusiε contour The procedure we shall describe is a specific exam-ple of the more general discussion of Lorentzian AdSCFT given recently in Refs [547ndash550]
180
Figure 76 Penrose diagrams for Lorentzian (Im t = 0) and Euclidean (Re t = 0) sections of an AdSblack hole In the right panel the two light-like Wilson lines are points at r =infin indicated by coloreddots These dots are the boundaries of a string world sheet that extends inward to r = r0 which is atthe origin of the Euclidean section of the black hole In the left panel the string world sheet and itsendpoints at r =infin are shown at Re t = 0 as Re t runs from minusinfin to infin the string worldsheet sweepsout the whole of quadrant I
In order to compute (7114) we first need to construct the bulk geometry corresponding tothe Im t = minusiε segment of the Schwinger-Keldysh contour in Fig 75 For this purpose itis natural to consider the black hole geometry with complex time In Fig 76 we showtwo slices of this complexified geometry The left plot is the Penrose diagram for the fullyextended black hole spacetime with quadrant I and III corresponding to the slice Im t = 0and Imt = minusβ
2 respectively while the right plot is for the Euclidean black hole geometry iecorresponding to the slice Ret = 0 Note that because the black hole has a nonzero tempera-ture the imaginary part of t is periodic with the period given by the inverse temperature βIn the left plot the imaginary time direction can be considered as a circular direction comingout of the paper at quadrant I going a half circle to reach quadrant III and then going intothe paper for a half circle to end back at I In the right plot the real time direction can bevisualized as the direction perpendicular to the paper
The first segment of the Schwinger-Keldysh contour in Fig 75 with Im t = 0 lies at theboundary (r =infin) of quadrant I in Fig 76 The second segment of the Schwinger-Keldyshcontour with Im t = minusiε lies at the r = infin boundary of a copy of I that in the left plotof Fig 76 lies infinitesimally outside the paper and in the right plot of Fig 76 lies at aninfinitesimally different angle We shall denote this copy of I by I prime The geometry and metricin I prime are identical to those of I Note that I prime and I are joined together at the horizon r = r0namely at the origin in the right plot of Fig 76 Now the thermal expectation value (7114)can be computed by putting the two parallel light-like Wilson lines at the boundaries of I andI prime and finding the extremized string world sheet which ends on both of them Note that sinceI and I prime meet only at the horizon the only way for there to be a nontrivial (ie connected)string world sheet whose boundary is the two Wilson lines in (7114) is for such a string worldsheet to touch the horizon Happily this is precisely the feature of the string world sheet
181
(r = r0)
Figure 77 String configuration for the thermal expectation value of (7114)
found in the explicit calculation that we reviewed above So we can use that string worldsheet in the present analysis with the only difference being that half the string world sheetnow lies on I and half on I prime as illustrated in Fig 775
We conclude that the result for the expectation value (7114) with its nonstandard pathordering of operators is identical to that obtained in Refs [148 149] for a light-like Wilsonloop with standard time ordering of operators [151] That is in strongly coupled N = 4 SYMtheory W(xperp) in the adjoint representation is given by
WA(xperp) = exp
minusradic2aradicλLminusT
radic1 +π2T 2x2
perp4a2
minus 1
(7125)
We have quoted the result for WA(xperp) which is given by W2F (xperp) in the large-Nc limit
5 The calculation of q in N = 4 SYM theory via (7114) nicely resolves a subtlety As we saw above inaddition to the extremized string configuration which touches the horizon the string action also has anothertrivial solution which lies solely at the boundary at r = infin Based on the connection between position inthe r dimension in the gravitational theory and energy scale in the quantum field theory the authors ofRef [148 149] argued that physical considerations (namely the fact that q should reflect thermal physicsat energy scales of order T ) require selecting the extremized string configuration that touches the horizonAlthough this physical argument remains valid we now see that it is not necessary In (7114) the two Wilsonlines are at the boundaries of I and I prime with different values of Im t That means that there are no stringworld sheets that connect the two Wilson lines without touching the horizon So once we have understoodhow the nonstandard operator ordering in (7114) modifies the boundary conditions for the string world sheetwe see that the trivial world sheet of Refs [148 149] and all of its generalizations in Refs [545 546] do notsatisfy the correct boundary conditions The nontrivial world sheet illustrated in Fig 77 which is sensitiveto thermal physics [148149466] is the only extremized world sheet bounded by the two lightlike Wilson linesin (7114) [151]
182
because that is what arises in the analysis of jet quenching see Section 232 (Radiativeparton energy loss depends on the medium through the transverse momentum broadening ofthe radiated gluons which are of course in the adjoint representation) The xperp-independentterm in the exponent in (7125) namely ldquothe -1rdquo is the finite subtraction of S0 which wasidentified in Ref [148] as the action of two disjoint strings hanging straight down from thetwo Wilson lines to the horizon of the AdS black hole Our calculation serves as a check of thevalue of S0 since only with the correct S0 do we obtainWA(0) = 1 and a correctly normalizedprobability distribution P (kperp) Note that our field theory set-up requires LminusT 1 and oursupergravity calculation requires λ 1 meaning that our result (7125) is only valid for
radicλLminusT 1 (7126)
In this regime (7125) is very small unless πxperpT(2a) is small This means that when wetake the Fourier transform of (7125) to obtain the probability distribution P (kperp) in theregime (7126) where the calculation is valid the Fourier transform is dominated by smallvalues of xperp for which
WA(xperp) exp
[minus π2
4radic
2a
radicλLminusT 3x2
perp
] (7127)
and we therefore obtain
P (kperp) =4radic
2a
πradicλT 3Lminus
exp
[minusradic
2ak2perp
π2radicλT 3Lminus
] (7128)
Thus the probability distribution P (kperp) is a Gaussian and the jet-quenching parameter(7112) can easily be evaluated yielding [148]
q =π32Γ(3
4)
Γ(54)
radicλT 3 (7129)
The probability distribution (7128) has a simple physical interpretation the probability thatthe quark has gained transverse momentum kperp is given by diffusion in transverse momentumspace with a diffusion constant given by qL This is indeed consistent with the physicalexpectation that transverse momentum broadening in a strongly coupled plasma is due tothe accumulated effect of many small kicks by gluons from the medium the quark performsBrownian motion in momentum space even though in coordinate space it remains on a light-like trajectory
If we attempt to plug RHIC-motivated numbers into the result (7129) taking T =300 MeV Nc = 3 αSYM = 1
2 and therefore λ = 6π yields q = 45 GeV2fm which turns outto be in the same ballpark as the values of q inferred from RHIC data on the suppression ofhigh momentum partons in heavy ion collisions [148 149] as we reviewed around Fig 211in Section 23 We can also write the result (7129) as
q 57
radicαSYM
Nc
3T 3 (7130)
which can be compared to the result (239240) extracted via comparison to RHIC data inRef [167] To make the comparison we need to relate the QCD energy density ε appearing
183
in (239) to T Lattice calculations of QCD thermodynamics indicate ε sim (9 minus 11)T 4 inthe temperature regime that is relevant at RHIC [85] This then means that if in (7130)we take αSYM within the range αSYM = 66+34
minus25 the result (7130) for the strongly coupledN = 4 SYM plasma is consistent with the result (240) obtained via comparing QCD jetquenching calculations to RHIC data The extraction of q by inference from LHC data onseveral-hundred-GeV jets should be under better control since then the separation of scalesupon which the QCD calculations (and the basic assumption that energy loss is dominatedby gluon radiation) will be more quantitatively reliable
We have reviewed the N = 4 SYM calculation but the jet quenching parameter can becalculated in any conformal theory with a gravity dual [149] In a large class of such theoriesin which the spacetime for the gravity dual is AdS5times M5 for some internal manifold M5 otherthan the five-sphere S5 which gives N = 4 SYM theory [149]
qCFT
qN=4=
radicsCFT
sN=4 (7131)
with s the entropy density This result makes a central qualitative lesson from (7129) clearin a strongly coupled plasma the jet quenching parameter is not proportional to the entropydensity or to some number density of distinct scatterers This qualitative lesson is morerobust than any attempt to make a quantitative comparison to QCD But we note that ifQCD were conformal (7131) would suggest
qQCD
qN=4asymp 063 (7132)
And analysis of how q changes in a particular toy model in which nonconformality can beintroduced by hand then suggests that introducing the degree of nonconformality seen inQCD thermodynamics may increase q by a few tens of percent [466]6 Putting these twoobservations together perhaps it is not surprising that the q for the strongly coupled plasmaof N = 4 SYM theory is in the same ballpark as that extracted by comparison with RHICdata
76 Quenching a beam of strongly coupled synchrotron radi-ation
In Sections 71 72 and 73 we have gained insights into parton energy loss and jet quenchingvia studying how a single heavy quark slows and gets kicked as it moves through the stronglycoupled plasma of N = 4 SYM theory and how that single heavy quark excites the plasmathrough which it moves In Section 74 we have seen how a single light quark or gluon isstopped and thermalized by the strongly coupled N = 4 SYM plasma assuming that allaspects of this physics are described at strong coupling In Section 75 we have analyzedjet quenching per se but have done so via the strategy of working as far as possible withinweakly coupled QCD and only using a holographic calculation within N = 4 SYM theory forone small part of the story namely the calculation of the jet quenching parameter q through
6q also increases with increasing nonconformality in strongly coupled N = 2lowast gauge theory [461462]
184
which the physics of the strongly coupled medium enters the calculation This approach isjustified in the high jet energy limit where there is a clean separation of scales and wherejet quenching is due to medium-induced gluon radiation But the jets being studied atpresent at RHIC are not sufficiently energetic as to make one confident in the quantitativereliability of these approximations In addition to motivating the study of jet quenching inheavy ion collisions at the LHC this raises an obvious theoretical question can we studyjets themselves in N = 4 SYM theory Suppose we assume that the physics at all relevantscales is strongly coupled an assumption that is in a sense the opposite of that we made inSection 75 But instead of considering just a single quark can we analyze how an actualjet is modified by the strongly coupled plasma of N = 4 SYM theory relative to how itwould have developed in the vacuum of that theory This question is pressing since one ofthe main goals of the ongoing high-pT measurements in heavy ion collisions at RHIC is fulljet reconstruction with experimentalists aiming to answer questions like how much wider inangle are jets in heavy ion collisions than in proton-proton collisions For example the STARcollaboration is measuring the ratio of the energy within a cone of radius 02 radians to thatwithin a cone of radius 04 radians and comparing that ratio in gold-gold and proton-protoncollisions It would be of value to get some insights into what to expect for such an observablein a strongly coupled plasma via a holographic calculation in N = 4 SYM theory
Unfortunately the answer to the question as we have just posed it is negative In Ref [551]Hofman and Maldacena considered the following thought experiment Suppose you didelectron-positron scattering in a world in which the electron and positron coupled to N = 4SYM theory through a virtual photon just as in the real world they couple to QCD Whatwould happen in high energy scattering Would there be any ldquojettyrdquo events They showedthat the answer is no Instead the final state produced by a virtual photon in the conformalN = 4 SYM theory is a spherically symmetric outflow of energy Similar conclusions werealso reached in Refs [530552] The bottom line is that there are no jets in strongly coupledN = 4 SYM theory which would seem to rule out using this theory to study how jets aremodified by propagating through the strongly coupled plasma of this theory
Recent work [553] may provide a path forward These authors have found a way of creatinga beam of gluons (and color-adjoint scalars) that is tightly collimated in angle and thatpropagates outwards forever in the N = 4 SYM theory vacuum without spreading in angleThis beam is not literally a jet since it is not produced far off shell But we know fromHofman and Maldacena that a far off shell ldquophotonrdquo does not result in jets in this theoryAnd this beam of nonabelian radiation may yield a better cartoon of a jet than a single quarkheavy or light or a single gluon all of which have been analyzed previously This beam iscreated by synchrotron radiation from a test quark that is moving in a circle of radius L withconstant angular frequency ω as in Ref [469] but now in Ref [553] in vacuum instead of inmedium The way the calculation is done is sketched in Fig 78 The logic of the calculationis as we described in Section 73 and so we shall not present it in detail It turns outhowever that in vacuum the shape of the rotating string in the bulk and the correspondingform of the energy density of the outward-propagating radiation on the boundary can bothbe determined analytically [553]
Because the vacuum of N = 4 SYM theory is not similar to that of our world studyingthe radiation of a test quark in circular motion does not have direct phenomenological mo-
185
AdS5 radialdirection
Boundary stress
Figure 78 Cartoon of the gravitational description of synchrotron radiation at strong coupling thequark rotating at the boundary trails a rotating string behind it which hangs down into the bulk AdS5
space This string acts as a source of gravitational waves in the bulk and this gravitational radiationinduces a stress tensor on the four-dimensional boundary By computing the bulk-to-boundary prop-agator one obtains the boundary-theory stress-tensor that describes the radiated energy The entirecalculation can be done analytically [553]
tivation However the angular distribution of the radiation turns out to be very similar tothat of classic synchrotron radiation [553] when the quark is moving along its circle with anultrarelativistic velocity v the radiation is produced in a tightly collimated beam with anangular width that is proportional to 1γ with γ the Lorentz boost factor associated withv see Fig 79 The spiral pulse of radiation propagates outward without spreading andthe radial thickness (or equivalently the duration in time) of the pulse is proportional to1γ3 So as a function of increasing γ the beam if Fourier analyzed contains radiation atincreasingly short wavelengths and high frequencies
Although it has recently been explained from the gravitational point of view in beautifullygeometric terms [554] from the point of view of the non-abelian gauge theory it is surprisingthat the angular distribution of the radiation at strong coupling is so similar (see Ref [553]for quantitative comparisons) to that at weak coupling where what is radiated is a mixtureof colored mdash and therefore interacting mdash gluons and scalars The fact that even whenthe coupling is arbitrarily strong as the pulses of radiation propagate outwards they do notspread at all and never isotropize indicates that intuition based upon parton branching [552](namely that the non-abelian character of the radiation should result in energy flowing fromshort to long wavelengths as the pulses propagate outwards and should yield isotropization
186
Figure 79 Cutaway plots of r2EP for a test quark in circular motion with v = 12 and v = 34Here E is the energy density and P is the total power radiated per unit time We see a spiral ofradiation propagating radially outwards at the speed of light without any spreading The spiral islocalized about θ = π2 with a characteristic width δθ prop 1γ The radial thickness of the spiral isproportional to 1γ3
at large distances) is invalid in this context7
For our purposes the result that the ldquobeamrdquo of radiated gluons (and scalars) propagatesoutward with a fixed angular width that we can select by picking γ is fortuitous It meansthat via this roundabout method we have found a way of making a state that looks somethinglike a jet It is not the same in all respects as a jet in QCD in that it is not produced viathe fragmentation of an initially far offshell parton But it is a collimated beam of gluonsof known and controllable angular width For this reason the results obtained in theformal setting of a test quark moving in a circle open the way to new means of modelingjet quenching in heavy ion collisions The challenge is to repeat the calculation of Ref [553]at nonzero temperature in the presence of a horizon in the bulk spacetime Upon doing soone could watch the initially tightly collimated beam of synchrotron radiation interact withthe strongly coupled plasma that would then be present As long as ω2γ3 π2T 2 we knowfrom Ref [469] that the total power radiated is as if in vacuum [468] In this regime wetherefore expect to see strongly coupled synchrotron radiation as in Fig 79 on length scalessmall compared to 1T But as in Section 73 we expect that this beam of radiation shouldeventually thermalize locally likely converting into an outgoing hydrodynamic wave moving
7Very recent work [555] shows that isotropization via some analogue of parton branching is also not thecorrect picture for the radiation studied in Ref [551] by Hofman and Maldacena this radiation also propagatesoutward as a pulse without any spreading but this pulse is spherically symmetric at all radii So in the casestudied by Hofman and Maldacena there is no process of isotropization because the radiation is isotropic atall times while in the case illustrated in Fig 79 there is no isotropization because the radiation never becomesisotropic
187
at the speed of sound before ultimately isotropizing dissipating and thermalizing completelyOne could see by how much a beam prepared such that in vacuum it has a known smallangular extent spreads in angle after propagating through 1T (or 2T or 3T ) of stronglycoupled plasma Such an analysis should provide interesting benchmarks against which tocompare data from RHIC on how much wider in angle jets of a certain energy are in heavyion collisions as compared to in proton-proton collisions Early results on jets in heavy ioncollisions at the LHC [124] indicate that for the high energy jets available in these collisionsit will be possible to make calorimetric measurements of how the jets initiated by partonspropagating in the medium produced in these collisions are broadened in angle
77 Velocity-scaling of the screening length and quarkoniumsuppression in a hot wind
We saw in Section 24 that because they are smaller than typical hadrons in QCD heavyquarkonium mesons survive as bound states even at temperatures above the crossover froma hadron gas to quark-gluon plasma However if the temperature of the quark-gluon plasmais high enough they eventually dissociate An important physical mechanism underlyingthe dissociation is the weakening attraction between the heavy quark and anti-quark in thebound state because the force between their color charges is screened by the medium Thedissociation of charmonium and bottomonium bound states has been proposed as a signalfor the formation of a hot and deconfined quark-gluon plasma in heavy ion collisions [184]and as a means of gauging the temperatures reached during the collisions
In the limit of large quark mass the interaction between the quark and the anti-quark ina bound state in the thermal medium can be extracted from the thermal expectation valueof the Wilson loop operator 〈WF (Cstatic)〉 with Cstatic a rectangular loop with a short side oflength L in a spatial direction (say x1) and a long side of length T along the time directionThis expectation value takes the form
〈WF (Cstatic)〉 = exp [minusi T E(L)] (7133)
where E(L) is the (renormalized) free energy of the quark-antiquark pair with the self-energyof each quark subtracted E(L) defines an effective potential between the quark anti-quarkpair The screening of the force between color charges due to the presence of the mediummanifests itself in the flattening of E(L) for L greater than some characteristic length scaleLs called the screening length In QCD the flattening of the potential occurs smoothly asseen in the lattice calculations illustrated in Fig 35 in Section 33 and one must make anoperational definition of Ls For example in the parametrization of (241) Ls can be setequal to 1micro Ls decreases with increasing temperature and can be used to estimate the scaleof the dissociation temperature Tdiss as
Ls(Tdiss) sim d (7134)
where d is the size of a particular mesonic bound state at zero temperature The idea hereis that once the temperature of the quark-gluon plasma is high enough that the potentialbetween a quark and an antiquark separated by a distance corresponding to the size of a
188
particular meson has been fully screened that meson can no longer exist as a bound state inthe plasma This means that larger quarkonium states dissociate at lower temperatures andmeans that the ground-state bottomonium meson survives to the highest temperatures of allAs we discussed at length in Section 24 there are many important confounding effects thatmust be taken into account in order to realize the goal of using data on charmonium pro-duction in heavy ion collisions to provide evidence for this sequential pattern of quarkoniumdissociation as a function of increasing temperature In this Section we shall focus only onone of these physical effects one on which calculations done via gaugegravity duality haveshed some light [149459466556ndash571]
In heavy ion collisions quarkonium mesons are produced moving with some velocity ~v withrespect to the medium It is thus important to understand the effects of nonzero quarkoniumvelocity on the screening length and consequent dissociation of bound states To describethe interaction between a quarkndashanti-quark pair that is moving relative to the medium it isconvenient to boost into a frame in which the quark anti-quark pair is at rest but feels ahot wind of QGP blowing past them The effective quark potential can again be extractedfrom (7133) evaluated in the boosted frame with T now interpreted as the proper time of thedipole While much progress has been made in using lattice QCD calculations to extract theeffective potential between a quarkndashanti-quark pair at rest in the QGP there are significantdifficulties in using Euclidean lattice techniques to address the (dynamical as opposed tothermodynamic) problem of a quarkndashantiquark pair in a hot wind In the strongly coupledplasma of N = 4 SYM theory with large Nc however the calculation can be done usinggaugegravity duality [149 557 558] and requires only a modest extension of the standardmethods reviewed in Section 54 Here we sketch the derivation from Ref [149]
We start with a rectangular Wilson loop whose short transverse space-like side
σ = x1 isin [minusL2L
2] (7135)
defines the separation L between the quarkndashantiquark pair and whose long time-like sidesextend along the x3 = v t direction describing a pair moving with speed v in the x3 directionIn this frame the plasma is at rest and the spacetime metric in the gravitational descriptionis the familiar AdS black hole (530) We then apply a Lorentz boost that rotates this Wilsonloop into the rest frame (tprime xprime3) of the quarkndashanti-quark pair
dt = dtprime cosh η minus dxprime3 sinh η (7136)
dx3 = minusdtprime sinh η + dxprime3 cosh η (7137)
where the rapidity η is given by tanh η = v meaning that cosh η = γ After the AdS blackhole metric has been transformed according to this boost it describes the moving hot windof plasma felt by the quarkndashanti-quark pair in its rest frame
In order to extract E(L) it suffices to work in the limit in which the time-like extent of theWilson loop T is much greater than its transverse extent L meaning that the correspondingstring worldsheet ldquosuspendedrdquo from this Wilson loop and ldquohanging downrdquo into the bulk isinvariant under translations along the long direction of the Wilson loop Parametrizing thetwo-dimensional world sheet with the coordinates σ and τ = t the dependence on τ is thentrivial The task is reduced to calculating the curve r(σ) along which the worldsheet descends
189
into the bulk from positions on the boundary brane which we take to be located at r = r0Λwith Λ a dimensionless UV cutoff that we shall take to infinity at the end of the calculationThat is the boundary conditions on r(σ) are
r
(plusmnL
2
)= r0Λ (7138)
It is then helpful to introduce dimensionless variables
r = r0y σ = σr0
R2 l =
Lr0
R2= πLT (7139)
where T = r0πR2 is the temperature Upon dropping the tilde one is then seeking to determine
the shape y(σ) of the string world sheet satisfying the boundary conditions y(plusmn l
2
)= Λ From
the boosted AdS black hole metric one finds that the Nambu-Goto action which must beextremized takes the form
S(C) = minusradicλ T T
int l2
0dσL (7140)
with a Lagrangian that reads (yprime = partσy)
L =
radic(y4 minus cosh2 η
)(1 +
yprime2
y4 minus 1
) (7141)
We must now determine y(σ) by extremizing (7141) This can be thought of as a classicalmechanics problem with σ the analogue of time Since L does not depend on σ explicitlythe corresponding Hamiltonian
H equiv Lminus yprime partLpartyprime
=y4 minus cosh2 η
L= q (7142)
is a constant of the motion which we denote by q In the calculation we are reviewing inthis section we take Λrarrinfin at fixed finite rapidity η In this limit the string worldsheet inthe bulk is time-like and E(L) turns out to be real (The string world-sheet bounded by therectangular Wilson loop that we are considering becomes space-like if
radiccosh η gt Λ In order
to recover the light-like Wilson loop used in the calculation of the jet quenching parameterin Section 75 one must first take η rarrinfin and only then take Λrarrinfin)
It follows from the Hamiltonian (7142) that solutions y(σ) with Λ gtradic
cosh η satisfy theequation of motion
yprime =1
q
radic(y4 minus 1)(y4 minus y4
c ) (7143)
withy4c equiv cosh2 η + q2 (7144)
Note that y4c gt cosh2 η ge 1 The extremal string world sheet begins at σ = minus`2 where
y = Λ and ldquodescendsrdquo in y until it reaches a turning point namely the largest value of y atwhich yprime = 0 It then ldquoascendsrdquo from the turning point to its end point at σ = +`2 wherey = Λ By symmetry the turning point must occur at σ = 0 We see from (7143) that
190
in this case the turning point occurs at y = yc meaning that the extremal surface stretchesbetween yc and Λ The integration constant q can then be determined from the equationl2 =
int l2
0 dσ which upon using (7143) becomes
l = 2q
int Λ
yc
dy1radic
(y4 minus y4c )(y
4 minus 1) (7145)
The action for the extremal surface can be found by substituting (7143) into (7140) and(7141) yielding
S(l) = minusradicλT T
int Λ
yc
dyy4 minus cosh2 ηradic
(y4 minus 1)(y4 minus y4c ) (7146)
Equation (7146) contains not only the potential between the quark-antiquark pair butalso the static mass of the quark and antiquark considered separately in the moving medium(Recall that we have boosted to the rest frame of the quark and antiquark meaning that thequark-gluon plasma is moving) Since we are only interested in the quark-antiquark potentialwe need to subtract the action S0 of two independent quarks from (7146) in order to obtainthe quarkndashanti-quark potential in the dipole rest frame
E(L)T = minusS(l) + S0 (7147)
The string configuration corresponding to a single quark at rest in a moving N = 4 SYMplasma was obtained in Refs [452 453] as we have reviewed in Section 71 From thisconfiguration one finds that
S0 = minusradicλ T T
int Λ
1dy (7148)
To extract the quark-antiquark potential we use (7145) to solve for q in terms of l andthen plug the corresponding q(l) into (7146) and (7147) to obtain E(L) Note that (7145)is manifestly finite as Λ rarr infin and the limit can be taken directly (7146) and (7148) aredivergent separately when taking Λrarrinfin but the difference (7147) is finite
We now describe general features of (7145) and (7147) Denoting the RHS of (7145)(with Λ = infin) as function l(q) one finds that for a given η l(q) has a maximum lmax(η)and equation (7145) has no solution when l gt lmax(η) Thus for l gt lmax the only stringworldsheet configuration is two disjoint strings and from (7147) E(L) = 0 ie the quarkand anti-quark are completely screened to the order of the approximation we are consideringWe can define the screening length as
Ls equivlmax(η)
πT (7149)
At η = 0 ie the dipole at rest with the medium one finds that
Ls(0) asymp 087
πTasymp 028
T (7150)
Similar criteria are used in the definition of screening length in QCD [572] although in QCDthere is no sharply defined length scale at which screening sets in Lattice calculations of the
191
Figure 710 The screening length lmax times its leading large-η dependenceradic
cosh(η) The exactresults are given for dipoles oriented perpendicular to the wind (θ = π2) and parallel to the wind(θ = 0) The θ = π2 curve is compared to the analytical large-η approximation (7151) Keepingonly the first term in this analytical expression corresponds to a horizontal line on the figure includingthe term proportional to (cosh η)minus52 improves the agreement with the exact result
static potential between a heavy quark and antiquark in QCD indicate a screening lengthLs sim 05T in hot QCD with two flavors of light quarks [573] and Ls sim 07T in hot QCDwith no dynamical quarks [230] The fact that there is a sharply defined Ls is an artifact ofthe limit in which we are working in which E(L) = 0 for L gt Ls
8
Ls(η) can be obtained numerically with results as illustrated in Fig 710 One finds thatLs(η) decreases with increasing velocity indicating that quarkonia dissociate at a lower tem-perature when they are moving An analytical expression for Ls can also be obtained in thelimit of high velocity Expanding (7145) in powers of 1y4
c and truncating the correspondingexpression at order 1y4
c one finds
lmax =
radic2π Γ
(34
)334Γ
(14
) ( 2
cosh12 η+
1
5 cosh52 η+ middot middot middot
)= 074333
(1
cosh12 η+
1
10 cosh52 η+ middot middot middot
) (7151)
In Fig 710 exact results for lmax = πTLs(η) as a function of η obtained numerically arecompared to the expression (7151) that is valid in the high velocity limit One sees that the
8We are considering the contribution to E(L) that is proportional toradicλ For l lmax the leading
contribution to E(L) is proportional to λ0 and is determined by the exchange of the lightest supergravitymode between the two disjoint strings [380]
192
screening length decreases with increasing velocity to a good approximation according to thescaling [556ndash558]
Ls(v) Ls(0)
cosh12 η=Ls(0)radicγ (7152)
with γ = 1radic
1minus v2 This velocity dependence suggests that Ls should be thought of asto a good approximation proportional to (energy density)minus14 since the energy densityincreases like γ2 as the wind velocity is boosted The velocity-scaling of Ls has provedrobust in the sense that it applies in various strongly coupled plasmas other than N = 4SYM [466 559 561 566] and in the sense that it applies to baryons made of heavy quarksalso [569]
We have only described the calculation for the case in which the direction of the hot windis perpendicular to the dipole With a little more effort the calculation can be extendedto general angles and has been analyzed in detail in [149] to which we refer the readers formore details The conclusion of [149] was that the dependences of Ls and E(L) on the anglebetween the dipole and the wind are very weak For example the black line in Fig 710 givesthe η dependence of the screening length when the wind direction is parallel to the dipoleWe see the difference from the perpendicular case is only about 12
If the velocity-scaling of Ls (7152) holds for QCD it will have qualitative consequencesfor quarkonium suppression in heavy ion collisions [149 557] From (7134) the dissociationtemperature Tdiss(v) defined as the temperature above which JΨ or Υ mesons with a givenvelocity do not exist should scale with velocity as
Tdiss(v) sim Tdiss(v = 0)(1minus v2)14 (7153)
since Tdiss(v) should be the temperature at which the screening length Ls(v) is comparableto the size of the meson bound state The scaling (7153) indicates that slower mesons canexist up to higher temperatures than faster ones As illustrated schematically in Fig 711this scaling indicates that JΨ suppression at RHIC (and Υ suppression at the LHC) mayincrease markedly for JΨrsquos (Υrsquos) with transverse momentum pT above some threshold onthe assumption that the temperatures reached in RHIC (LHC) collisions do not reach thedissociation temperature of JΨ (Υ) mesons at zero velocity [185 190] Modelling this ef-fect requires embedding results for quarkonium production in hard scatterings in nuclearcollisions into a hydrodynamic code that describes the motion of the quark-gluon fluid pro-duced in the collision in order to evaluate the velocity of the hot wind felt by each putativequarkonium meson Such an analysis indicates that once pT is above the threshold at whichTdiss(v) has dropped below the temperature reached in the collision the decline in the JΨsurvival probability is significant by more than a factor of four (two) in central (peripheral)collisions [574 575] We should caution that as we discussed in Section 24 in modellingquarkonium production and suppression versus pT in heavy ion collisions various other ef-fects like secondary production or formation of JΨ mesons outside the hot medium at highpT [576] remain to be quantified The quantitative importance of these and other effectsmay vary significantly depending on details of their model implementation In contrast Eq(7153) was obtained directly from a field-theoretic calculation and its implementation willnot introduce additional model-dependent uncertainties
193
Figure 711 A 1radicγ-velocity scaling of the screening length in QCD would imply a JΨ dissociation
temperature Tdiss(pT ) that decreases significantly with pT while that for the heavier Υ is affected lessat a given pT The curves are schematic in that we have arbitrarily taken Tdiss(0) for the JΨ to be21Tc and we have increased Tdiss(0) for the Υ over that for the Jψ by a factor corresponding to itssmaller size in vacuum At a qualitative level we expect to see fewer JΨ (Υ) mesons at pT rsquos abovethat at which their dissociation temperature is comparable to the temperatures reached in heavy ioncollisions at RHIC (at the LHC)
The threshold pT above which the production of JΨ mesons falls off due to their motionthrough the quark-gluon plasma depends sensitively on the difference between T (diss(v =0) and the temperature reached in the collision [575] but should be somewhere around 5GeV perhaps a conservative range is 2 to 10 GeV Present data on the suppression of JΨproduction in CuCu collisions at RHIC show no signs of increased suppression above some pTthreshold [577578] but the error bars are large above pT sim 4 GeV The kinematical range inwhich this novel quarkonium suppression mechanism is operational lies within experimentalreach of near-future high-luminosity AuAu runs at RHIC and will be studied thoroughly atthe LHC in both the JΨ and Υ channels
The analysis of this section is built upon the calculation of the potential between a testquark and antiquark in the strongly coupled plasma of N = 4 SYM theory a theory whichin and of itself has no mesons Gaining insight into the physics of quarkonium mesonsfrom calculations of the screening of the static quark-antiquark potential has a long historyin QCD as we have seen in Section 33 But we have also seen in that section that theseapproaches are gradually being superseded as lattice QCD calculations of quarkonium spectralfunctions themselves are becoming available In the present context also we would like to
194
move beyond drawing inferences about mesons from analyses of the potential E(L) and thescreening length Ls to analyses of mesons themselves This is the subject of Section 8 inwhich we shall carefully describe how once we have added heavy quarks to N = 4 SYM byadding a D7-brane in the gravity dual [385] as in Section 55 the fluctuations of the D7-branethen describe the quarkonium mesons of this theory We shall review the construction firstin vacuum and then in the presence of the strongly coupled plasma at nonzero temperatureWe shall find that the results of this section prove robust in that the velocity-scaling (7153)has also been obtained [568] by direct analysis of the dispersion relations of mesons in theplasma [565 568] These mesons have a limiting velocity that is less than the speed of lightand that decreases with increasing temperature [565] and whose temperature dependenceis equivalent to (7153) up to few percent corrections that have been computed [568] andthat we shall show This is a key part of the story with the velocity-dependent dissociationtemperature of this section becoming a temperature dependent limiting velocity for explicitlyconstructed quarkonium mesons in Section 8 However this cannot be the whole story sincethe dispersion relations seem to allow for mesons with arbitrarily large momentum eventhough they limit their velocity The final piece of the story is described in Section 842 wherewe review the calculation of the leading contribution to the widths of these mesons [571]which was neglected in the earlier calculations of their dispersion relations Above somemomentum the width grows rapidly increasing like p2
perp And the momentum above whichthis rapid growth of the meson width sets in is just the momentum at which the mesonvelocity first approaches its limiting value The physical picture that emerges is that at themomentum at which the mesons reach a velocity such that the hot wind they are feeling hasa temperature sufficient to dissociate them according to the analysis of this Section builtupon the calculation of Ls their widths in fact grow rapidly [571]
195
Chapter 8
Quarkonium mesons
As discussed in section 24 heavy quarks and quarkonium mesons with masses such thatMT 1 constitute valuable probes of the QGP Since dynamical questions about theseprobes are very hard to answer from first principles here we will study analogous questionsin the strongly coupled N = 4 SYM plasma In this case the gaugestring duality providesthe tool that makes a theoretical treatment possible Although for concreteness we will focuson the N = 4 plasma many of the results that we will obtain are rather universal in thesense that at least qualitatively they hold for any strongly coupled gauge theory with astring dual Such results may give us insights relevant for the QCD quark-gluon plasma attemperatures at which it is reasonably strongly coupled
The QGP only exists at temperatures T gt Tc so in QCD the condition MT 1 canonly be realised by taking M to be large In contrast N = 4 SYM is a conformal theorywith no confining phase so all temperatures are equivalent In the presence of an additionalscale namely the quark or the meson mass the physics only depends on the ratio MT This means that in the N = 4 theory the condition MT 1 can be realised by fixing Tand sending M to infinity or by fixing M and sending T rarr 0 both limits are completelyequivalent In particular the leading-order approximation to the heavy quark or quarkoniummeson physics in an expansion in TM may be obtained by setting T = 0 For this reasonthis is the limit that we will study first
We will follow the nomenclature common in the QCD literature and refer to mesons madeof two heavy quarks as lsquoquarkonium mesonsrsquo or lsquoquarkoniarsquo as opposed to using the termlsquoheavy mesonsrsquo which commonly encompasses mesons made of one heavy and one light quark
81 Adding quarks to N = 4 SYM
In section 55 we saw that Nf flavours of fundamental matter can be added to N = 4 SYM byintroducing Nf D7-brane probes into the geometry sourced by the D3-branes as indicated bythe array (593) which we reproduce here (with the time direction included) for convenience
D3 0 1 2 3D7 0 1 2 3 4 5 6 7
(81)
Before we proceed let us clarify an important point N = 4 SYM is a conformal theory
196
ie its β-function vanishes exactly Adding matter to it even if the matter is massless makesthe quantum-mechanical β-function positive at least perturbatively This means that thetheory develops a Landau pole in the UV and is therefore not well defined at arbitrarily-highenergy scales1 However since the beta-function (for the rsquot Hooft coupling λ) is proportionalto NfNc the Landau pole occurs at a scale of order eNcNf This is exponentially large in thelimit of interest here NfNc 1 and in fact the Landau pole disappears altogether in thestrict probe limit NfNc rarr 0 On the string side the potential pathology associated with aLandau pole manifests itself in the fact that a completely smooth solution that incorporatesthe backreaction of the D7-branes may not exist [383 579ndash583] In any case the possibleexistence of a Landau pole at high energies will not be of concern for the applications reviewedhere In the gauge theory it will not prevent us from extracting interesting infrared physicsjust as the existence of a Landau pole in QED does not prevent one from calculating theconductivity of an electromagnetic plasma In the string description we will not go beyondthe probe approximation so the backreaction of the D7-branes will not be an issue2 Andfinally we note that we will work with the D3D7 model because of its simplicity Wecould work with a more sophisticated model with better UV properties but this wouldmake the calculations more involved while leaving the physics we are interested in essentiallyunchanged
As illustrated in Fig 81 the D3-branes and the D7-branes can be separated a distanceL in the 89-directions This distance times the string tension Eqn (411) is the minimumenergy of a string stretching between the D3- and the D7-branes Since the quarks arise asthe lightest modes of these 3-7 strings this energy is precisely the bare quark mass
Mq =L
2παprime (82)
An important remark here is the fact that the branes in Fig 81 are implicitly assumed to beembedded in flat spacetime In Section 55 this was referred to as the lsquofirstrsquo or lsquoopen-stringrsquodescription of the D3D7 system which is reliable in the regime gsNc 1 in which thebackreaction of the D3-branes on spacetime can be ignored One of our main tasks in thefollowing sections will be to understand how this picture is modified in the opposite regimegsNc 1 when the D3-branes are replaced by their backreaction on spacetime In thisregime the shape of the D7-branes may or may not be modified but Eqn (82) will remaintrue provided the appropriate definition of L to be given below is used
Although N = 4 SYM is a conformal theory the addition of quarks with a nonzero massintroduces a scale and gives rise to a rich spectrum of quark-antiquark bound states iemesons In the following section we will study the meson spectrum in this theory at zerotemperature in the regime of strong lsquot Hooft coupling gsNc 1 On the gauge theory sidethis is inaccessible to conventional methods such as perturbation theory but on the string sidea classical description in terms of D7-brane probes in a weakly curved AdS5timesS5 applies Ourfirst task is thus to understand in more detail the way in which the D7-branes are embeddedin this geometry Since this is crucial for subsequent sections we will in fact provide a fairamount of detail here
1Non-perturbatively the possibility that a strongly coupled fixed point exists must be ruled out beforereaching this conclusion See [579] for an argument in this direction based on supersymmetry
2For a review of lsquounquenchedrsquo models ie those in which the flavour backreaction is included see [584]
197
Figure 81 D3D7 system at weak coupling
82 Zero temperature
821 D7-brane embeddings
We begin by recalling that the coordinates in the AdS5 times S5 metric (51) (52) can beunderstood as follows The four directions t xi correspond to the 0123-directions in (81) The456789-directions in the space transverse to the D3-branes give rise to the radial coordinater in AdS5 defined through
r2 = x24 + middot middot middot+ x2
9 (83)
as well as five angles that parametrise the S5 We emphasize that once the gravitationaleffect of the D3-branes is taken into account the 6-dimensional space transverse to the D3-branes is not flat so the x4 x9 coordinates are not Cartesian coordinates However theyare still useful to label the different directions in this space
The D7-branes share the 0123-directions with the D3-branes so from now on we will mainlyfocus on the remaining directions In the 6-dimensional space transverse to the D3-branesthe D7-branes span only a 4-dimensional subspace parametrised by x4 x7 Since the D7-branes preserve the SO(4) rotational symmetry in this space it is convenient to introduce aradial coordinate u such that
u2 = x24 + middot middot middot+ x2
7 (84)
as well as three spherical coordinates denoted collectively by Ω3 that parametrise an S3
198
45
89-
67
L prop Mq
D7-branes
rU
u
S3
Figure 82 Coordinates in the 6-dimensional space transverse to the D3-branes Each axis actuallyrepresents two directions ie a plane (or equivalently the radial direction in that plane) Theasymptotic distance L = U(u = infin) is proportional to the quark mass Eqn (82) We emphasizethat the directions parallel to the D3-branes (the gauge theory directions t xi) are suppressed in thispicture and they should not be confused with the D7 directions shown in the figure which lie entirelyin the space transverse to the D3-branes
Similarly it is useful to introduce a radial coordinate U in the 89-plane through
U = x28 + x2
9 (85)
as well as a polar angle α In terms of these coordinates one has
dx24 + middot middot middot+ dx2
9 = du2 + u2dΩ23 + dU2 + U2dα2 (86)
Obviously the overall radial coordinate r satisfies r2 = u2 + U2
Since the D7-branes only span the 4567-directions they only wrap an S3 inside the S5 TheD7-brane worldvolume may thus be parametrized by the coordinates t xi uΩ3 In order tospecify the D7-branesrsquo embedding one must then specify the remaining spacetime coordinatesU and α as functions of in principle all the worldvolume coordinates However translationalsymmetry in the t xi-directions and rotational symmetry in the Ω3-directions allow Uand α to depend only on u
In order to understand this dependence consider first the case in which the spacetimecurvature generated by the D3-branes is ignored In this case the D7-branes lie at a constantposition in the 89-plane see Fig 82 In other words their embedding is given by α(u) = α0
and U(u) = L where α0 and L are constants The first equation can be understood as sayingthat because of the U(1) rotational symmetry in the 89-plane the D7-branes can sit at any
199
45
89-
67
D7-branes
L equiv U(infin) prop Mq
U(0)
Figure 83 Possible bending of the D7-branes at nonzero temperature The asymptotic distanceL equiv U(infin) is proportional to the bare quark mass Mq whereas the minimum distance U(0) is related(albeit in a way more complicated than simple proportionality) to the quark thermal mass
constant angular position choosing α0 then breaks the symmetry Since this U(1) symmetryis respected by the D3-branesrsquo backreaction (ie since the AdS5timesS5 metric is U(1)-invariant)it is easy to guess (correctly) that α(u) = α0 is still a solution of the D7-branesrsquo equation ofmotion in the presence of the D3-branesrsquo backreaction
The second equation U(u) = L says that the D7-branes lie at a constant distance fromthe D3-branes In the absence of the D3-branesrsquo backreaction this is easily understood thereis no force on the D7-branes and therefore they span a perfect 4-plane In the presenceof backreaction one should generically expect that the spacetime curvature deforms theD7-branes as in Fig 83 bending them towards the D3-branes at the origin The reasonthat this does not happen for the D3D7 system at zero temperature is that the underlyingsupersymmetry of the system guarantees an exact cancellation of forces on the D7-branesIn fact it is easy to verify directly that U(u) = L is still an exact solution of the D7-branesrsquoequations of motion in the presence of the D3-branesrsquo backreaction The constant L thendetermines the quark mass through Eqn (82) We will see below that the introduction ofnonzero temperature breaks supersymmetry completely and that consequently U(u) becomesa non-constant function that one must solve for and that this function contains informationabout the ground state of the theory in the presence of quarks For example its asymptoticbehaviour encodes the value of the bare quark mass Mq and the quark condensate 〈ψψ〉whereas its value at u = 0 is related to the quark thermal mass Mth Since in this section wework at T = 0 any nonzero quark mass corresponds to MqT rarrinfin In this sense one mustthink of the quarks in question as the analogue of heavy quarks in QCD and of the quarkcondensate as the analogue of 〈cc〉 or 〈bb〉 However when we consider a nonzero temperature
200
Figure 84 D7-branesrsquo embedding in AdS5 times S5 At nonzero temperature this picture is slightlymodified First a horizon appears at r = r0 gt 0 and second the D7-branes terminate at r = U(0) ltL This lsquotermination pointrsquo corresponds to the tip of the branes in Fig 83
in subsequent sections whether the holographic quarks described by the D7-branes are theanalogues of heavy or light quarks in QCD will depend on how their mass (or more preciselythe mass of the corresponding mesons) compares to the temperature
We have concluded that at zero temperature the D7-branes lie at U = L and areparametrised by t xi uΩ3 In terms of these coordinates the induced metric on theD7-branes takes the form
ds2 =u2 + L2
R2
(minusdt2 + dx2
i
)+
R2
u2 + L2du2 +
R2u2
u2 + L2dΩ2
3 (87)
We see that if L = 0 then this metric is exactly that of AdS5timesS3 The AdS5 factor suggeststhat the dual gauge theory should still be conformally invariant This is indeed the case inthe limit under consideration If L = 0 the quarks are massless and the theory is classicallyconformal and in the probe limit NfNc rarr 0 the quantum mechanical β-function whichis proportional to NfNc vanishes If L 6= 0 then the metric above becomes AdS5 times S3
only asymptotically ie for u L reflecting the fact that in the gauge theory conformalinvariance is explicitly broken by the quark mass Mq prop L but is restored asymptotically atenergies E Mq We also note that if L 6= 0 then the radius of the three-sphere is notconstant as displayed in fig 84 in particular it shrinks to zero at u = 0 (corresponding tor = L) at which point the D7-branes lsquoterminatersquo from the viewpoint of the projection on
201
AdS5 [385] In other words if L 6= 0 then the D7-branes fill the AdS5 factor of the metriconly down to a minimum value of the radial direction proportional to the quark mass As weanticipated above at nonzero temperature one must distinguish between the bare and thethermal quark masses related to U(infin) and U(0) respectively In this case the position inAdS at which the D7-branes terminate is r = U(0) lt L and therefore they fill the AdS spacedown to a radial position related to the thermal mass Note also that at finite temperaturea horizon is present at r = r0 gt 0
822 Meson spectrum
We are now ready to compute the spectrum of low-spin mesons in the D3D7 system followingRef [585] The spectrum for more general DpDq systems was computed in [586ndash588] Recallthat mesons are described by open strings attached to the D7-branes In particular spin-zeroand spin-one mesons correspond to the scalar and vector fields on the D7-branes Large-spinmesons can be described as long semi-classical strings [556 585 589ndash596] but we will notreview them here
For simplicity we will focus on scalar mesons Following Section 515 we know that inorder to determine the spectrum of scalar mesons we need to determine the spectrum ofnormalisable modes of small fluctuations of the scalar fields on the D7-branes At this pointwe restrict ourselves to a single D7-brane ie we set Nf = 1 in which case the dynamics isdescribed by the DBI action (418) At leading order in the large-Nc expansion the spectrumfor Nf gt 1 consists of N2
f identical copies of the single-flavour spectrum [597]
We use the coordinates in Eqn (87) as worldvolume coordinates for the D7-brane whichwe collectively denote by σmicro The physical scalar fields on the D7-brane are then x8(σmicro)x9(σmicro) By a rotation in the 89-plane we can assume that in the absence of fluctuations theD7-brane lies at x8 = 0 x9 = L Then the fluctuations can be parametrised as
x8 = 0 + ϕ(σmicro) x9 = L+ ϕ(σmicro) (88)
with ϕ and ϕ the scalar fluctuations around the fiducial embedding In order to determine thenormalisable modes it suffices to work to quadratic order in ϕ ϕ Substituting (88) in theDBI action (418) and expanding in ϕ ϕ leads to a quadratic Lagrangian whose correspondingequation of motion is
R4
(u2 + L2)2ϕ+
1
u3partu(u3partuϕ) +
1
u2nabla2ϕ = 0 (89)
where is the four-dimensional drsquoAlembertian associated with the Cartesian coordinatest xi and nabla2 is the Laplacian on the three-sphere The equation for ϕ takes exactly thesame form Modes that transform non-trivially under rotations on the sphere correspond tomesons that carry nonzero R-charge Since QCD does not possess an R-symmetry we willrestrict ourselves to R-neutral mesons meaning that we will assume that ϕ does not dependon the coordinates of the sphere We can use separation of variables to write these modes as
ϕ = φ(u)eiqmiddotx (810)
where x = (t xi) Each of these modes then corresponds to a physical meson state in thegauge theory with a well defined four-dimensional mass given by its eigenvalue under that
202
is M2 = minusq2 For each of these modes Eqn (89) results in an equation for φ(u) that afterintroducing dimensionless variables through
u =u
L M2 = minusk
2R4
L2 (811)
becomes
part2uφ+
3
upartuφ+
M2
(1 + u2)2φ = 0 (812)
This equation can be solved in terms of hypergeometric functions The details can be foundin Ref [585] but we will not give them here because most of the relevant physics can beextracted as follows
Eqn (812) is a second-order ordinary differential equation with two independent solutionsThe combination we seek must satisfy two conditions It must be normalisable as u rarr infinand it must be regular as u rarr 0 For arbitrary values of M both conditions cannot besimultaneously satisfied In other words the values of M for which physically acceptablesolutions exist are quantised Since Eqn (812) contains no dimensionful parameters thevalues of M must be pure numbers These can be explicitly determined from the solutionsof (812) and they take the form [585]
M2 = 4(n+ 1)(n+ 2) n = 0 1 2 (813)
Using this and M2 = minusq2 = M2L2R4 we derive the result that the four-dimensional massspectrum of scalar mesons is
M(n) =2L
R2
radic(n+ 1)(n+ 2) =
4πMqradicλ
radic(n+ 1)(n+ 2) (814)
where in the last equality we have used the expressions R2αprime =radicλ and (82) to write R
and L in terms of gauge theory parameters We thus conclude that the spectrum consists ofa discrete set of mesons with a mass gap given by the mass of the lightest meson3
Mmes = 4πradic
2Mqradicλ (815)
Since this result is valid at large lsquot Hooft coupling λ 1 the mass of these mesons is muchsmaller than the mass of two constituent quarks In other words the mesons in this theoryare very deeply bound In fact the binding energy
EB equiv 2Mq minusMmes 2Mq simradicλMmes (816)
is so large that it almost cancels the rest energy of the quarks This is clear from the gravitypicture of lsquomeson formationrsquo (see Fig 85) in which two strings of opposite orientationstretching from the D7-brane to r = 0 (the quark-antiquark pair) join together to form anopen string with both ends on the D7-brane (the meson) This resulting string is muchshorter than the initial ones and hence corresponds to a configuration with much lowerenergy This feature is an important difference with quarkonium mesons in QCD such as
3In order to compare this and subsequent formulas with ref [585] and others note that our definition (417)of g2 differs from the definition in some of those references by a factor of 2 for example g2
[here] = 2g2[585]
203
Figure 85 String description of a quark an antiquark and a meson The string that describes themeson can be much shorter than those describing the quark and the antiquark
charmonium or bottomonium which are not deeply bound Although this certainly meansthat caution must be exercised when trying to compare the physics of quarkonium mesons inholographic theories with the physics of quarkonium mesons in QCD the success or failureof these comparisons cannot be assessed at this point We will discuss this assessment indetail below once we have learned more about the physics of holographic mesons Suffice itto say here that some of this physics such as the temperature or the velocity dependence ofcertain meson properties turns out to be quite general and may yield insights into some ofthe challenges related to understanding the physics of quarkonia within the QCD quark-gluonplasma
We close this section with a consistency check The behaviour of the fluctuation modes atinfinity is related to the high-energy properties of the theory At high energy we can ignorethe effect of the mass of the quarks and the theory becomes conformal The urarrinfin behaviouris then related to the UV operator of the lowest conformal dimension ∆ that has the samequantum numbers as the meson [252 253] Analysis of this behaviour for the solutions ofEqs (89) (812) shows that ∆ = 3 [585] as expected for a quark-bilinear operator
83 Nonzero temperature
831 D7-brane embeddings
We now turn to the case of nonzero temperature T 6= 0 This means that we must study thephysics of a D7-brane in the black brane metric (cf Eqn (530))
ds2 =r2
R2
(minusfdt2 + dx2
1 + dx22 + dx2
3
)+R2
r2fdr2 +R2dΩ2
5 (817)
204
where
f(r) = 1minus r40
r4 r0 = πR2T (818)
The study we must perform is conceptually analogous to that of the last few sections but theequations are more involved and most of them must be solved numerically These technicaldetails are not very illuminating and for this reason we will not dwell into them Insteadwe will focus on describing in detail the main conceptual points and results as well as thephysics behind them which in fact can be understood in very simple and intuitive terms
As mentioned above at T 6= 0 all supersymmetry is broken We therefore expect that theD7-branes will be deformed by the non-trivial geometry In particular the introduction ofnonzero temperature corresponds in the string description to the introduction of a blackbrane in the background Intuitively we expect that the extra gravitational attraction willbend the D7-branes towards the black hole This simple conclusion which was anticipatedin previous sections has far-reaching consequences At a qualitative level most of the holo-graphic physics of mesons in a strongly coupled plasma follows from this conclusion Anexample of the D7-branesrsquo embedding for a small value of TMq is depicted in two slightlydifferent ways in Fig 86
The qualitative physics of the D3D7 system as a function of the dimensionless ratio TMq
is now easy to guess and is captured by Fig 87 At zero temperature the horizon has zerosize and the D7-branes span an exact hyperplane At nonzero but sufficiently small TMqthe gravitational attraction from the black hole pulls the branes down but the branesrsquo tensioncan still compensate for this The embedding of the branes is thus deformed but the branesremain entirely outside the horizon Since in this case the induced metric on the D7-braneshas no horizon we will call this type of configuration a lsquoMinkowski embeddingrsquo In contrastabove a critical temperature Tdiss
4 the gravitational force overcomes the tension of the branesand these are pulled into the horzion In this case the induced metric on the branes possessesan event horizon inherited from that of the spacetime metric For this reason we will referto such configurations as lsquoblack hole embeddingsrsquo Between these two types of embeddingsthere exists a so-called critical embedding in which the branes just lsquotouch the horizon at apointrsquo The existence of such an interpolating solution might lead one to suspect that thephase transition between Minkowski and black hole embeddings is continuous ie of second orhigher order However as we will see in the next section thermodynamic considerations revealthat a first-order phase transition occurs between a Minkowski and a black hole embeddingIn other words the critical embedding is skipped over by the phase transition and near-critical embeddings turn out to be metastable or unstable
As illustrated by the figures above the fact that the branes bend towards the horizonimplies that the asymptotic distance between the two differs from their minimal distanceAs we will see in Section 832 the asymptotic distance is proportional to the microscopicor lsquobarersquo quark mass since it is determined by the non-normalizable mode of the field thatdescribes the branesrsquo bending In contrast the minimal distance between the branes and thehorizon includes thermal (and quantum) effects and for this reason we will refer to the massof a string stretching between the bottom of the branes and the horizon (shown as a wigglyred curve in the figures) as the lsquothermalrsquo quark mass Note that this vanishes in the black
4The reason for the subscript will become clear shortly
205
Figure 86 D7-branesrsquo embedding for small TMq The branes bend towards the horizon shown indark grey The radius of the horizon is proportional to its Hawking temperature which is identifiedwith the gauge theory temperature T mdash see Eqn (818) The asymptotic position of the D7-branes isproportional to the bare quark mass Mq The minimum distance between the branes and the horizonis related to the thermal quark mass because this is the minimum length of a string (shown as a redwiggly line) stretching between the branes and the horizon The top figure shows the two relevantradial directions in the space transverse to the D3-branes U and u (introduced in Eqs (84) and(85)) together with the gauge theory directions xi (time is suppressed) The horizon has topologyR3timesS5 where the first factor corresponds to the gauge-theory directions This lsquocylinder-likersquo topologyis manifest in the top figure Instead in the bottom figure the gauge theory directions are suppressedand the S3 wrapped by the D7-branes in the space transverse to the D3-branes is shown as in Figs 82and 83 In this figure only the S5 factor of the horizon is shown
206
Figure 87 Various D7-brane configurations in a black D3-brane background with increasing temper-ature from left to right At low temperatures the probe branes close off smoothly above the horizonAt high temperatures the branes fall through the event horizon In between a critical solution existsin which the branes just lsquotouchrsquo the horizon at a point The critical configuration is never realized afirst-order phase transition occurs from a Minkowski to a black hole embedding (or vice versa) beforethe critical solution is reached
hole phase
Although we will come back to this important point below we wish to emphasize rightfrom the start that the phase transition under discussion is not a confinement-deconfinementphase transition since the presence of a black hole implies that both phases are deconfinedInstead we will see that the branesrsquo phase transition corresponds to the dissociation of heavyquarkonium mesons In order to illustrate the difference most clearly consider first a holo-graphic model of a confining theory as described in Section 522 below we will come back tothe case of N = 4 SYM For all such confining models the difference between the deconfine-ment and the dissociation phase transitions is illustrated in Fig 88 Below Tc the theoryis in a confining phase and therefore no black hole is present At some Tc a deconfinementtransition takes place which in the string description corresponds to the appearance of ablack hole whose size is proportional to Tc If the quark mass is sufficiently large comparedto Tc then the branes remain outside the horizon (top part of the figure) otherwise they fallthrough the horizon (bottom part of the figure) The first case corresponds to heavy quarko-nium mesons that remain bound in the deconfined phase and that eventually dissociate atsome higher Tdiss gt Tc The second case describes light mesons that dissociate as soon as thedeconfinement transition takes place
Fig 88 also applies to N = 4 SYM theory with Tc = 0 in the sense that although thevacuum of the theory is not confining there is no black hole at T = 0 Note also that mesonsonly exist provided Mq gt 0 since otherwise the theory is conformal and there is no particlespectrum This means that in N = 4 SYM theory any meson is a heavy quarkonium mesonthat remains bound for some range of temperatures above Tc = 0 as described by the toppart of Fig 88 In the case Mq = 0 we cannot properly speak of mesons but we see that thesituation is still described by the bottom part of the figure in the sense that in this case thebranes fall through the horizon as soon as T is raised above Tc = 0
The universal character of the meson dissociation transition was emphasized in Refs [565
207
Figure 88 Top Sufficiently heavy quarkonium mesons remain bound in the deconfined phase(above Tc) and dissociate at Tdiss gt Tc Bottom In contrast light mesons dissociate as soon as thedeconfinement phase transition at T = Tc takes place This picture also applies to N = 4 SYM theorywith Tc = 0 as described in the main text In N = 4 SYM theory the top (bottom) panel applieswhen Mq gt 0 (Mq = 0)
208
598] which we will follow in our presentation Specic examples were originally seen in [597599 600] and aspects of these transitions in the D3D7 system were studied independentlyin [601ndash604] Similar holographic transitions appeared in a slightly different framework in[348605ndash607] The D3D7 system at nonzero temperature has been studied upon includingthe backreaction of the D7-branes in Ref [464]
832 Thermodynamics of D7-branes
In this subsection we shall show that the phase transition between Minkowski and black holeembeddings is a discontinuous first-order phase transition The reader willing to accept thiswithout proof can safely skip to Section 833 Since we are working in the canonical ensemble(ie at fixed temperature) we must compute the free energy density of the system per unitgauge theory three-volume F and determine the configuration that minimizes it In thegauge theory we know that this takes the form
F = FN=4 + Fflavour (819)
where the first term is the O(N2c ) free energy of the N = 4 SYM theory in the absence of
quarks and the second term is the O(NcNf) contribution due to the presence of quarks in thefundamental representation Since the SYM theory without quarks is conformal dimensionalanalysis completely fixes the first factor to be of the form FN=4 = C(λ)T 4 where C is apossibly coupling-dependent coefficient of order N2
c In contrast in the presence of quarks ofmass Mq there is a dimensionless ratio TMq on which the flavour contribution can dependnon-trivially Our purpose is to determine this contribution to leading order in the large-Ncstrong coupling limit
Our tool is of course the dual description of the N = 4 SYM theory with flavour as asystem of Nf D7-brane probes in the gravitational background of Nc black D3-branes Asusual in finite-temperature physics the free energy of the system may be computed throughthe identification
βF = SE (820)
where β = 1T and SE is the Euclidean action of the system In our case this takes the form
SE = Ssugra + SD7 (821)
The first term is the contribution from the black hole gravitational background sourced bythe D3-branes and is computed by evaluating the Euclideanized supergravity action on thisbackground The second term is the contribution from the D7-branes and is computed byevaluating the Euclidean version of the DBI action (418) on a particular D7-brane config-uration The decomposition (821) is the dual version of that in (819) The supergravityaction scales as 1g2
s sim N2c and thus yields the free energy of the N = 4 SYM theory in the
absence of quarks ie we identify
Ssugra = βFN=4 (822)
Similarly the D7-brane action scales as Nfgs sim NcNf and represents the flavour contributionto the free energy
SD7 = βFD7 = βFflavour (823)
209
We therefore conclude that we must first find the solutions of the equations of motion ofthe D7-branes for any given values of T and Mq then evaluate their Euclidean actionsand finally use the identification above to compare their free energies and determine thethermodynamically preferred configuration
As explained above in our case solving the D7-brane equations of motion just meansfinding the function U(u) which is determined by the condition that the D7-brane actionbe extremized This leads to an ordinary second-order non-linear differential equation forU(u) Its precise form can be found in eg Ref [565] but is not very illuminating Howeverit is easy to see that it implies the asymptotic large-u behaviour
U(u) =mr0radic
2+
c r30
2radic
2u2+ middot middot middot (824)
where m and c are constants The factors of r0 have been introduced to make these constantsdimensionless whereas the numerical factors have been chosen to facilitate comparison withthe literature As usual (and in particular as in Section 515) the leading and sublead-ing terms correspond to the non-normalizable and to the normalizable modes respectivelyTheir coefficients are therefore proportional to the source and the expectation values of thecorresponding dual operator in the gauge theory In this case the position of the brane U(u)is dual to the quark-mass operator Om sim ψψ so m and c are proportional to the quark massand the quark condensate respectively The precise form of Om can be found in Ref [335]where it is shown that the exact relation between m c and Mq 〈Om〉 is
Mq =r0m
232π`2s=
1
2radic
2
radicλT m (825)
〈Om〉 = minus232π3`2sNfTD7r30 c = minus 1
8radic
2
radicλNf Nc T
3 c (826)
In particular we recover the fact that the asymptotic value
L = limurarrinfin
U(u) =mr0radic
2(827)
is related to the quark mass through Eqn (82) as anticipated in previous sections
It is interesting to note that the dimensionless mass m is given by the simple ratio
m =M
T (828)
where
M =2radic
2Mqradicλ
=Mmes
2π(829)
is (up to a constant) precisely the meson gap at zero temperature given in Eqn (815)As mentioned in Section 821 and as we will elaborate upon in Section 843 Om must bethought of as the analogue of a heavy- or light-quark bilinear operator in QCD depending onwhether the ratio MmesT sim m is large or small respectively
The constants m and c can be understood as the two integration constants that completelydetermine a solution of the second-order differential equation obeyed by U(u) Mathemat-ically these two constants are independent but the physical requirement that the solution
210
0 1 2 3 4TM
025
02
015
01
005
0
c
0762 0764 0766 0768 077 0772TM
009
008
007
006
005
c
A
B
Figure 89 Quark condensate c versus TM = 1m The blue dashed (red continuous) curvescorrespond to the Minkowski (black hole) embeddings The dotted vertical line indicates the precisetemperature of the phase transition The point where the two branches meet corresponds to thecritical embedding
u
U
Figure 810 Some representative D7-brane embeddings from the region in which c is multi-valuedThe three profiles correspond to the same value of m but differ in their value of c Two of themrepresented by blue dashed curves are of Minkowski type The third one represented by a redcontinuous curve is a black hole embedding
be regular in the interior relates them to one another The equation for U(u) can be solvednumerically (see eg Ref [565]) and the resulting possible values of c for each value of mare plotted in Fig 89 We see from the lsquolarge-scalersquo plot on the left that c is a single-valuedfunction of m for most values of the latter However the zoom-in plot on the right showsthat in a small region around 1m = TM 0766 three values of c are possible for agiven value of m a pictorial representation of a situation of this type is shown in Fig 810This multivaluedness is related to the existence of the phase transition which as we will seeproceeds between points A and B through a discontinuous jump in the quark condensateand other physical quantities The point in Fig 89 where the Minkowski and the black holebranches meet corresponds to the critical embedding
Having determined the regular D7-brane configurations one must now compute their freeenergies and compare them in order to determine which one is preferred in the multivaluedregion The result is shown in Fig 811(top) where the normalization constant is given
211
by [565598]
N =2π2NfTD7r
40
4T=λNfNc
64T 3 (830)
The plot on the right shows the classic lsquoswallow tailrsquo form typically associated with a first-order phase transition As anticipated Minkowski embeddings have the lowest free energyfor temperatures T lt Tdiss whereas the free energy is minimized by black hole embeddingsfor T gt Tdiss with Tdiss 077M (ie m 13) At T = Tdiss the Minkowski and the blackhole branches meet and the thermodynamicaly-preferred embedding changes from one typeto the other The first-order nature of the phase transition follows from the fact that severalphysical quantities jump discontinuously across the transition An example is provided by thequark condensate which as illustrated in Fig 89 makes a finite jump between the pointslabelled A and B Similar discontinuities also appear in other physical quantities like theentropy and energy density These are easily obtained from the free energy through the usualthermodynamic relations
S = minuspartFpartT
E = F + TS (831)
and the results are shown in Fig 811 From the plots of the energy density one can imme-diately read off the qualitative behaviour of the specific heat cV = partEpartT In particularnote that this slope must become negative as the curves approach the critical solution indi-cating that the corresponding embeddings are thermodynamically unstable Examining thefluctuation spectrum of the branes we will show that a corresponding dynamical instabilitymanifested by a meson state becoming tachyonic is present exactly for the same embed-dings for which cV lt 0 One may have thought that the phases near the critical point weremetastable and thus accessible by lsquosuper-coolingrsquo the system but instead it turns out thatover much of the relevant regime such phases are unstable
We see from (830) that N sim λNcNfT3 which means that the leading contribution of the
D7-branes to all the various thermodynamic quantities will be order λNcNf in comparisonto N2
c for the usual bulk gravitational contributions The NcNf dependence anticipatedbelow Eqn (819) follows from large-Nc counting In contrast as noted in Ref [598 608]the factor of λ represents a strong-coupling enhancement over the contribution of a simplefree-field estimate for the NcNf fundamental degrees of freedom From the viewpoint ofthe string description this enhancement is easy to understand by reexamining the relativenormalization of the two terms in Eqn (821) more carefully than we did above Ignoring onlyorder-one purely numerical factors the supergravity action scales as 1G with G sim g2
s`8s the
ten-dimensional Newtonrsquos constant whereas the D7-brane action scales as NfTD7 sim Nfgs`8s
The ratio between the two normalizations is therefore
GNfTD7 sim gsNf sim g2Nf simλNf
Nc
(832)
Thus the flavour contribution is suppressed with respect to the leading O(N2c ) contribution
by λNfNc ie it is of order λNcNf We will come back to this point in the next subsection
As the calculations above were all performed in the limit Nc λ rarr infin (with Nf fixed) itis natural to ask how the detailed results depend on this approximation Since the phasetransition is first order we expect that its qualitative features will remain unchanged withina finite radius of the 1Nc 1λ expansions Of course finite-Nc and finite-λ corrections may
212
0 1 2 3 4TM
0
05
1
15
2
S
076 0765 077 0775TM
026
028
03
032
034
036
038
04
S
B
A
0 1 2 3 4TM
0
02
04
06
08
1
12
14
ET
076 0765 077 0775TM
024
026
028
03
032
034
036
ET
B
A
Figure 811 Free energy entropy and energy densities for a D7-brane in a black D3-brane backgroundnote that N prop T 3 The blue dashed (red continuous) curves correspond to the Minkowski (black hole)embeddings The dotted vertical line indicates the precise temperature of the phase transition
213
eventually modify the behaviour described above For example at large but finite Nc theblack hole will emit Hawking radiation and each bit of the probe branes will experiencea thermal bath at a temperature determined by the local acceleration Similarly finite rsquotHooft coupling corrections which correspond to higher-derivative corrections both to thesupergravity action and the D-brane action will become important if the spacetime or thebrane curvatures become large It is certainly clear that both types of corrections will becomemore and more important as the lower part of a Minkowski brane approaches the horizonsince as this happens the local temperature and the branes (intrinsic) curvature at their tipincrease However at the phase transition the minimum separation between the branes andthe horizon is not parametrically small and therefore the corrections above can be madearbitrarily small by taking Nc and λ sufficiently large but still finite This confirms ourexpectation on general grounds that the qualitative aspects of the phase transition shouldbe robust within a finite radius around the 1Nc = 0 1λ = 0 point Of course theseconsiderations do not tell us whether the dissociation transition is first order or a crossoverat Nc = 3
833 Quarkonium thermodynamics
We have seen above that in a large class of strongly coupled gauge theories with fundamentalmatter this matter undergoes a first-order phase transition described on the gravity sideby a change in the geometry of the probe D-branes In this section we will elaborate onthermodynamical aspects of this transition from the gauge-theory viewpoint Once we havelearned more about the dynamics of holographic mesons in subsequent sections in Section843 we will return to the gauge-theory viewpoint and discuss possible implications for thedynamics of quarkonium mesons in the QCD plasma
The temperature scale at which the phase transition takes place is set by the meson gapat zero temperature Tdiss sim Mmes As well as giving the mass gap in the meson spectrum1Mmes is roughly the characteristic size of these bound states [587 609] The gluons andother adjoint fields are already in a deconfined phase at Tdiss so this new transition is nota confinementdeconfinement transition Rather the most striking feature of the new phasetransition is the change in the meson spectrum and so we refer to it as a lsquodissociationrsquo orlsquomeltingrsquo transition
In the low-temperature phase below the transition stable mesons exist and their spec-trum is discrete and gapped This follows from the same general principles as in the zero-temperature case The meson spectrum corresponds to the spectrum of normalizable fluc-tuations of the D7-branes around their fiducial embedding For Minkowski embeddings thebranes close off smoothly outside the black hole horizon and the admissible modes must alsosatisfy a regularity condition at the tip of the branes On general grounds we expect thatthe regular solution at the tip of the branes evolves precisely into the normalizable solutionat the boundary only for a certain set of discrete values of the meson mass We will study themeson spectrum in detail in Section 841 and in Section 842 we will see that mesons acquirefinite decay widths at finite Nc or finite coupling Since the phase under consideration is nota confining phase we can also introduce deconfined quarks into the system represented byfundamental strings stretching between the D7-branes and the horizon At a figurative levelin this phase we might describe quarks in the adjoint plasma as a lsquosuspensionrsquo That is when
214
quarks are added to this phase they retain their individual identities More technically wemay just say that quarks are well defined quasiparticles in the Minkowski phase
In the high-temperature phase at T gt Tdiss no stable mesons exist Instead as wewill discuss in more detail in Section 85 the excitations of the fundamental fields in thisphase are characterised by a discrete spectrum of quasinormal modes on the black holeembeddings [610 611] The spectral function of some two-point meson correlators in theholographic theory of which we will see an example in Section 852 still exhibits some broadpeaks in a regime just above Tdiss which suggests that a few broad bound states persistjust above the dissociation phase transition [611 612] This is quite analogous to the latticeapproach where similar spectral functions are examined to verify the presence or absence ofbound states Hence identifying Tdiss with the dissociation temperature should be seen asa (small) underestimate of the temperature at which mesons completely cease to exist Anappropriate figurative characterization of the quarks in this high-temperature phase wouldbe as a lsquosolutionrsquo If one attempts to inject a localised quark charge into the system itquickly falls through the horizon ie it spreads out across the entire plasma and its presenceis reduced to diffuse disturbances of the supergravity and worldvolume fields which are soondamped out [610 611] Technically speaking we may just state that quarks are not welldefined quasiparticles in the black hole phase
The physics above is potentially interesting in connection with QCD since evidence fromseveral sources indicates that heavy quarkonium mesons remain bound in a range of tem-peratures above Tc We will analyze this connection in more detail in Section 843 once wehave learned more about the properties of holographic mesons in subsequent sections Herewe would just like to point out one simple physical parallel The question of quarkoniumbound states surviving in the quark-gluon plasma was first addressed by comparing the sizeof the bound states to the screening length in the plasma [184] In the D3D7 system thesize of the mesons can be inferred for example from the structure functions and the relevantlength scale that emerges is dmes sim
radicλMq [609] This can also be heuristically motivated
as follows As discussed in Section 54 (see Eqn (581)) at zero temperature the potentialbetween a quark-antiquark pair separated by a distance ` is given by
V sim minusradicλ
` (833)
We can then estimate the size dmes of a meson by requiring EB sim |V (dmes)| where EB is thebinding energy (816)5 This gives
dmes simradicλ
EBsimradicλ
Mq
sim 1
Mmes
sim R2
L (834)
The last equality follows from Eqn (814) and is consistent with expectations based on theUVIR correspondence [587] since on the gravity side mesons are excitations near r = LJust for comparison we remind the reader that the weak-coupling formula for the size ofquarkonium is dweak sim 1(g2Mq)
5Eqn (816) was derived at zero temperature but as we will see in Section 841 it is also parametricallycorrect at nonzero temperature
215
One intuitive way to understand why a meson has a very large size compared to its inversebinding energy or to the inverse quark mass is that due to strong-coupling effects thequarks themselves have an effective size of order dmes The effective size of a quark is definedas the largest of the following two scales (i) its Compton wavelength or (ii) the distancebetween a quark-antiquark pair at which their potential energy is large enough to pair-produce additional quarks and antiquarks In a weakly coupled theory (i) is larger whereasin a strongly coupled theory (ii) is larger From Eqn (833) we see that this criterion gives aneffective quark size of order
radicλMq instead of 1Mq This heuristic estimate is supported by
an explicit calculation of the size of the gluon cloud that dresses a quark [613] These authorscomputed the expectation value 〈TrF 2(x)〉 sourced by a quark of mass Mq and found thatthe characteristic size of the region in which this expectation value is nonzero is preciselyradicλMq
As reviewed in Section 542 holographic studies of Wilson lines at nonzero temperature[378379] reveal that the relevant screening length of the SYM plasma is of order Ls sim 1T mdashsee Eqn (7150) The argument that the mesons should dissociate when the screening lengthis shorter than the size of these bound states then yields Tdiss simMq
radicλ simMmes in agreement
with the results of the detailed calculations explained in previous sections We thus see thatthe same physical reasoning which is used in QCD to estimate the dissociation temperatureof eg the Jψ meson can also be used to understand the dissociation of mesons in the theN = 4 SYM theory This may still seem counterintuitive in view of the fact that the bindingenergy of these mesons is much larger that Tdiss In other words one might have expectedthat the temperature required to break apart a meson would be of the order of the bindingenergy EB simMq instead of being parametrically smaller
Tdiss simMmes sim EBradicλ (835)
However this intuition relies on the expectation that the result of dissociating a meson is aquark-antiquark pair of mass 2Mq The gravity description makes it clear that this is not thecase at strong coupling since above Tdiss the branes fall through the horizon Heuristicallyone may say that this means that the lsquoconstituentrsquo or lsquothermalrsquo mass of the quarks becomeseffectively zero However a more precise statement is simply that in the black hole phasequark-like quasiparticles simply do not exist and therefore for the purpose of the presentdiscussion it becomes meaningless to attribute a mass to them
One point worth emphasizing is that there are two distinct processes that are occurringat T sim Mmes If we consider eg the entropy density in Fig 811 we see that the phasetransition occurs in the midst of a crossover signalled by a rise in ST 3 We may write thecontribution of the fundamental matter to the entropy density as
Sflavour =1
8λNf Nc T
3H(x) (836)
where x = λT 2Mq and H(x) is the function shown in the plot of the free energy densityin the top panels of Fig 811 H rises from 0 at x = 0 to 2 as x rarr infin but the mostdramatic part of this rise occurs in the vicinity of x = 1 Hence it seems that new degrees offreedom ie the fundamental quarks are becoming lsquothermally activatedrsquo at T sim Mmes Wenote that the phase transition produces a discontinuous jump in which H only increases by
216
about 007 ie the jump at the phase transition only accounts for about 35 of the totalentropy increase Thus the phase transition seems to play an small role in this crossover andproduces relatively small changes in the thermal properties of the fundamental matter suchas the energy and entropy densities
As Mmes sets the scale of the mass gap in the meson spectrum it is tempting to associatethe crossover above with the thermal excitation of mesonic degrees of freedom However thepre-factor λNf Nc in (836) indicates that this reasoning is incorrect if mesons provided therelevant degrees of freedom we should have Sflavour prop N2
f Such a contribution can be obtainedeither by a one-loop calculation of the fluctuation determinant around the classical D7-braneconfiguration or by taking into consideration the D7-branes backreaction to second order inthe NfNc expansion as in [464614ndash616] One can make an analogy here with the entropy ofa confining theory (cf Section 522) In the low-temperature confining phase the absence ofa black hole horizon implies that the classical-gravity saddle point yields zero entropy whichmeans that the entropy is zero at order N2
c One must look at the fluctuation determinantto see the entropy contributed by the supergravity modes ie by the gauge-singlet glueballswhich is of order N0
c
We thus see that the factor of NfNc in Sflavour is naturally interpreted as counting thenumber of degrees of freedom associated with deconfined quarks with the factor of λ demon-strating that the contribution of the quarks is enhanced at strong coupling A complementaryinterpretation of (836) comes from reorganizing the pre-factor as
λNf Nc = (g2Nf)N2c (837)
The latter expression suggests that the result corresponds to the first-order correction of theadjoint entropy due to quark loops As explained at the end of Section 551 we are workingin a lsquonot quitersquo quenched approximation in that contributions of the D7-branes represent theleading order contribution in an expansion in NfNc and so quark loops are suppressed butnot completely In view of the discussion below Eqn (832) it is clear that the expansionfor the classical gravitational back-reaction of the D7-branes is controlled by λNfNc = g2NfHence this expansion corresponds to precisely the expansion in quark loops on the gaugetheory side
We conclude that the strongly coupled theory brings together these two otherwise distinctprocesses That is because theN = 4 SYM theory is strongly coupled at all energy scales thedissociation of the quarkonium bound states and the thermal activation of the quarks happenat essentially the same temperature Note that this implies that the phase transition shouldnot be thought of as exclusively associated with a discontinuous change in the properties ofmesons mdash despite the fact that this is the aspect that is more commonly emphasized Thephase transition is also associated with a discontinuous change in the properties of quarkssince as explained above these exist as well defined quasiparticles in the Minkowski phasebut not in the black hole phase In fact as the discussion around Eqn (837) makes clear inthe O(NfNc) - approximation considered here the observed discontinuous jump in the thermo-dynamic functions comes entirely from the discontinuous change in the properties of quarksIn this approximation the discontinuous jump in the thermodynamic functions associatedwith the discontinuous change in the properties of mesons simply cannot be detected sinceit is of order N2
f and its determination would require a one-loop calculation Fortunately
217
Figure 812 A qualitative representation of the simplest possibility interpolating between the weak-and the strong-coupling regimes in N = 4 SYM theory The solid and the dotted black curvescorrespond to T = Tdiss At strong coupling this corresponds to a first-order phase transition (solidblack curve) whereas at weak coupling it corresponds to a crossover (dotted black curve) The dashedred curve corresponds to T = Tactiv At strong coupling this takes place immediately after the phasetransition whereas at weak coupling it is widely separated from Tdiss
however the change in the mesonsrsquo properties can be inferred eg from the comparison oftheir spectra above and below Tdiss
It is instructive to contrast this behavior with that which is expected to occur at weakcoupling In this regime one expects that the dissociation of the quarkonium mesons maywell be just a crossover rather than a (first-order) transition Moreover since the weaklybound mesons are much larger than 1Mmes sim 1(2Mq) their dissociation transition willoccur at a Tdiss that is much lower than Mq On the other hand the quarks would not bethermally activated until the temperature Tactiv sim Mq above which the number densitiesof unbound quarks and antiquarks are no longer Boltzmann-suppressed Presumably thethermal activation would again correspond to a crossover rather than a phase transition Thekey point is that these two temperatures are widely separated at weak coupling Fig 812 isan lsquoartisticrsquo representation of the simplest behaviour which would interpolate between strongand weak coupling One might expect that the dissociation point and the thermal activationare very close for λ 1 The line of first-order phase transitions must end somewhere and soone might expect that it terminates at a critical point around λ sim 1 Below this point bothprocesses would only represent crossovers and their respective temperatures would divergefrom one another approaching the weak-coupling behaviour described above
We close with a comment about a possible comparison to QCD Although it would beinteresting to look for signs of a crossover or a phase transition associated with quarkoniumdissociation for example in lattice QCD the above discussion makes it clear that muchcaution must be exercised in trying to compare with the holographic results described hereThe differences can be traced back to the fact that unlike the holographic theory considered
218
here QCD is not strongly coupled at the scale set by the mass of the heavy quark or ofthe corresponding heavy quarkonium meson For this reason in QCD the binding energyof a quarkonium meson is EB Mmes 2Mq and since one expects that Tdiss sim EBthis implies that at the dissociation temperature the quarkonium contribution to (say) thetotal entropy density would be Boltzmann suppressed ie it would be of order Sflavour simN2
f exp(minusMmesTdiss) 1 In contrast in the holographic setup there is no exponentialsuppression because Tdiss sim Mmes Note also that the quarkonium contribution should scaleas N2
f and therefore the exponential suppression is a further suppression on top of the alreadysmall one-loop contribution discussed in the paragraph above Eqn (837) That is there aretwo sources of suppression relative to the leading O(NfNc) - contribution in the holographictheory Although NfNc is not small in QCD the Boltzmann suppression is substantial andwill likely make the thermodynamic effects of any quarkonium dissociation transition quitea challenge to identify
84 Quarkonium mesons in motion and in decay
In previous sections we examined the thermodynamics of the phase transition betweenMinkowski and black hole embeddings and we argued that from the gauge-theory viewpointit corresponds to a meson-dissociation transition In particular we argued that quarkoniumbound states exist on Minkowski embeddings ie at T lt Tdiss that they are absolutely sta-ble in the large-Nc strong-coupling limit and that their spectrum is discrete and gappedWe will begin Section 841 by studying this spectrum quantitatively which will allow us tounderstand how the meson spectrum is modified with respect to that at zero temperaturedescribed in Section 822 The spectrum on black hole embeddings will be considered inSection 85
After describing the spectrum of quarkonium mesons at rest we will determine their dis-persion relations This will allow us to study mesons in motion with respect to the plasmaand in particular to determine how the dissociation temperature depends on the mesonvelocity As discussed in Section 24 one of the hallmarks of a quark-gluon plasma is thescreening of colored objects Heavy quarkonia provide an important probe of this effect sincethe existence (or absence) of quark-antiquark bound states and their properties are sensitiveto the screening properties of the medium in which they are embedded In Section 77 westudied this issue via computing the potential between an external quark-antiquark pair atrest in the plasma or moving through it with velocity v In particular we found that thedissociation temperature scales with v as
Tdiss(v) Tdiss(v = 0)(1minus v2)14 (838)
which could have important implications for the phenomenon of quarkonium suppression inheavy ion collisions By studying dynamical mesons in a thermal medium we will be able toreexamine this issue in a more lsquorealisticrsquo context
We will show in Section 842 that both finite-Nc and finite-coupling corrections generatenonzero meson decay widths as one would expect in a thermal medium We shall find thatthe dependence of the widths on the meson momentum yields further understanding of how(838) arises
219
We will close in Section 843 with a discussion of the potential connections between theproperties of quarkonium mesons in motion in a holographic plasma and those of quarkoniummesons in motion in the QCD plasma
841 Spectrum and dispersion relations
In order to determine the meson spectrum on Minkowski embeddings we proceed as inSection 822 For simplicity we will focus on fluctuations of the position of the branes U(u)with no angular momentum on the S3 ie we write
δU = U(u) eminusiωteiqmiddotx (839)
The main difference between this equation and its zero-temperature counterpart (810) isthat in the latter case Lorentz invariance implies the usual relation ω2 minus q2 = M2 betweenthe energy ω the spatial three-momentum q and the mass M of the meson At nonzerotemperature boost invariance is broken because the plasma defines a preferred frame inwhich it is at rest and the mesons develop a non-trivial dispersion relation ω(q) In the stringdescription this is determined by requiring normalizability and regularity of U(u) For eachvalue of q these two requirements are mutually compatible only for a discrete set of valuesωn(q) where different values of n label different excitation levels of the meson We definethe lsquorest massrsquo of a meson as the energy ω(0) at vanishing three-momentum q = 0 in therest-frame of the plasma
Fig 813 taken from Ref [565] shows the rest mass of the mesons as a function of temper-ature and quark mass Note that in the zero-temperature limit the spectrum coincides withthe zero-temperature spectrum (814) In particular the lightest meson has a mass squaredmatching Eqn (829) M2
mes = 4π2M2 395 M2
The meson masses decrease as the temperature increases Heuristically this can be un-derstood in geometrical terms from Fig 86 which shows that the thermal quark mass Mth
decreases as the temperature increases and the tip of the D7-branes gets closer to the blackhole horizon The thermal shift in the meson masses becomes more significant at the phasetransition and slightly beyond this point some modes actually become tachyonic This hap-pens precisely in the same region in which Minkowski embeddings become thermodynamicallyunstable because cV lt 0 In other words Minkowski embeddings develop thermodynamicand dynamic instabilities at exactly the same TM just beyond that at which the first orderdissociation transition occurs
We now turn to quarkonium mesons moving through the plasma that is to modes withq 6= 0 The dispersion relation for scalar mesons was first computed in Ref [565] and thenrevisited in Ref [568] The dispersion relation for (transverse) vector mesons appeared inRef [617] An exhaustive discussion of the dispersion relations for all these cases can befound in Ref [618] The result for the lowest-lying (n = 0) vector scalar and pseudoscalarquarkonia is shown in Fig 814 taken from Ref [618] Fig 815 taken from Ref [568] showsthe group velocity vg = dωdq for the n = 0 scalar mesons at three different temperatures
An important feature of these plots is their behavior at large momentum In this regimewe find that ω grows linearly with q Naively one might expect that the constant of propor-
220
01 02 03 04 05 06 07TM
0
50
100
150
200
M2M2
Figure 813 Meson mass spectrum M2 = ω2|q=0 for Minkowski embeddings in the D3D7 systemContinuous curves correspond to radially excited mesons with radial quantum number n = 0 1 2 frombottom to top respectively Dashed lines correspond to mesons with angular momentum on the S3The dashed vertical line indicates the temperature of the phase transition Note that modes becometachyonic slightly beyond this temperature
221
$ amp amp amp
$
qπT
ωπ
T
$ amp amp amp
$
qπT
ωπ
T
Figure 814 Left Dispersion relation for the transverse (black continuous curve) and longitudinal(red dashed curve) n = 0 modes of a heavy vector meson with vlim = 035 in the N = 4 SYM plasmaThe dual D7-brane has m = 13 corresponding to a temperature just below Tdiss Right Analogouscurves for a scalar (black continuous curve) and pseudoscalar (red dashed curve) meson In bothplots the blue continuous straight lines correspond to ω = vq for some v such that vlim lt v le 1 Theblack dotted vertical lines mark the crossing points between the meson dispersion relations and theblue lines
tionality should be one However one finds instead that
ω = vlim q (840)
where vlim lt 1 and where vlim depends on m = MT but at a given temperature is the samefor all quarkonium modes In the particular case of m = 13 illustrated in Fig 814 one hasvlim 035 In other words there is a subluminal limiting velocity for quarkonium mesonsmoving through the plasma And as illustrated in Fig 815 one finds that the limitingvelocity decreases with increasing temperature Fig 815 also illustrates another genericfeature of the dispersion relations namely that the maximal group velocity is attained atsome qm lt infin and as q is increased further the group velocity approaches vlim from aboveSince vg at qm is not much greater than vlim we will not always distinguish between thesetwo velocities We will come back to the physical interpretation of qm at the end of thissubsection
The existence of a subluminal limiting velocity which was discovered in [565] and sub-sequently elaborated upon in [568] is easily understood from the perspective of the dualgravity description [565 568] Recall that mesonic states have wave functions supported onthe D7-branes Since highly energetic mesons are strongly attracted by the gravitational pullof the black hole their wave-function is very concentrated at the bottom of the branes (seeFig 86) Consequently their velocity is limited by the local speed of light at that point Asseen by an observer at the boundary this limiting velocity is
vlim =radicminusgttgxx
∣∣∣tip
(841)
where g is the induced metric on the D7-branes Because of the black hole redshift vlim islower than the speed of light at infinity (ie at the boundary) which is normalized to unity
222
0 20 40 60 8000
02
04
06
08
10
qTdiss
v g
Figure 815 Group velocities vg for n = 0 scalar meson modes with TTdiss asymp 065 092 and 1 fromtop to bottom We see that vlim lt 1 decreases with increasing temperature (vlim would approach zeroif we included the unstable Minkowski embeddings with T gt Tdiss) The group velocity approachesits large-q value vlim from above ie vg reaches a maximum before settling into the limiting velocityvlim The maximum exists also for the top curve even though it is less clearly visible We will refer tothe momentum at which vg reaches the maximum as qm Clearly qm decreases with temperature
Note that as the temperature increases the bottom of the brane gets closer to the horizonand the redshift becomes larger thus further reducing vlim this explains the temperaturedependence in Fig 815 In the gauge theory the above translates into the statement thatvlim is lower than the speed of light in the vacuum The reason for this interpretation is thatthe absence of a medium in the gauge theory corresponds to the absence of a black hole onthe gravity side in which case vlim = 1 everywhere Eqn (841) yields vlim 035 at m = 13in agreement with the numerical results displayed in Fig 814
It is also instructive to plot vlim as a function of TTdiss as done in Fig 816 takenfrom [568] Although this curve was derived as a limiting meson velocity at a given temper-ature it can also be read (by asking where it cuts horizontal lines rather than vertical ones)as giving Tdiss(v) the temperature below which mesons with a given velocity v are found andabove which no mesons with that velocity exist In order to compare this result for Tdiss atall velocities to (838) one can parametrize the curve in Fig 816 as
Tdiss(v) = f(v)(1minus v2)14 Tdiss(0) (842)
In the left panel of Fig 816 the dashed line is obtained by setting f(v) = 1 which isof course just (838) In the right panel f(v) is shown to be close to 1 for all velocitiesvarying between 1021 at its maximum and 0924 at v = 1 Recall that the scaling (838) wasfirst obtained via the analysis of the potential between a moving test quark and antiquarkas described in Section 77 The weakness of the dependence of f(v) on v is a measureof the robustness with which that simple scaling describes the velocity dependence of the
223
00 02 04 06 08 1000
02
04
06
08
10
TTdiss
v lim
02 04 06 08 1v
02
04
06
08
1
fv
Figure 816 Left panel The solid curve is the limiting velocity vlim as a function of TTdiss whereTdiss is the temperature of the dissociation transition at zero velocity The dissociation transitionoccurs at the dot where vlim = 027 The dashed curve is the approximation obtained by settingf(v) = 1 in Eqn (842) Right panel f(v) the ratio of the solid and dashed curves in the left panelat a given v We see that f(v) is within a few percent of 1 at all velocities
dissociation temperature for quarkonium mesons in a fully dynamical calculation In otherwords to a good approximation vlim(T ) can be determined by setting v = vlim on the right-hand side of (838) yielding
vlim(T )
radic1minus
(T
Tdiss(v = 0)
)4
(843)
Thus we reach a rather satisfactory picture that the subluminal limiting velocity (840) is infact a manifestation in the physics of dynamical mesons of the velocity-enhanced screening ofSection 77 However in the case of the low-spin mesons whose dynamics we are considering inthis Section there is an important addition to our earlier picture Although the quarkoniummesons have a limiting velocity they can nevertheless manage to remain bound at arbitrarilylarge momenta thanks to their modified dispersion relations The latter allow the groupvelocity to remain less than vlim and consequently Tdiss(v) as given in (838) to remainshigher than T all the way out to arbitrarily large momenta In other words there exist mesonbound states of arbitrarily large spatial momentum but no matter how large the momentumthe group velocity never exceeds vlim In this sense low-spin mesons realize the first of twosimple possibilities by which mesons may avoid exceeding vlim A second possibility moreclosely related to the analysis of Section 77 is that meson states with momentum largerthan a certain value simply cease to exist This possibility is realized in the case of high-spinmesons Provided J 1 these mesons can be reliably described as long semiclassical stringswhose ends are attached to the bottom of the D7-branes The fact that the endpoints donot fall on top of one another is of course due to the fact that they are rotating around oneanother in such a way that the total angular momentum of the string is J These type ofmesons were first studied [585] at zero temperature and subsequently considered at nonzerotemperature in Ref [556] These authors also studied the possibility that at the same timethat the endpoints of the string rotate around one another in a given plane they also movewith a certain velocity in the direction orthogonal to that plane The result of the analysiswas that for a fixed spin J string solutions exist only up to a maximum velocity vlim lt 1
224
As we saw in Fig 815 the group velocity of quarkonium mesons reaches a maximum atsome value of the momentum q = qm before approaching the limiting value vlim There is asimple intuitive explanation for the location of qm it can be checked numerically that qm isalways close to the lsquolimiting momentumrsquo qlim that would follow from (838) if one assumes thestandard dispersion relation for the meson Thus qm can be thought of as a characteristicmomentum scale where the velocity-enhanced screening effect starts to be important For thecurves in Fig 815 to the left of the maximum one finds approximately standard dispersionrelations with a thermally corrected meson mass To the right of the maximum the dispersionrelations approach the limiting behavior (840) with vg approaching vlim as a consequenceof the enhanced screening
842 Decay widths
We saw above that at T lt Tdiss (Minkowski embeddings) there is a discrete and gappedspectrum of absolutely stable quarkonium mesons ie the mesons have zero width Thereason is that in this phase the D-branes do not touch the black hole horizon Since themesonsrsquo wave-functions are supported on the branes this means that the mesons cannot fallinto the black hole In the gauge theory this translates into the statement that the mesonscannot disappear into the plasma which implies that the meson widths are strictly zeroin the limit Nc λ rarr infin This conclusion only depends on the topology of the Minkowskiembedding In particular it applies even when higher-order perturbative corrections in αprime
are included which implies that the widths of mesons should remain zero to all orders in theperturbative 1
radicλ expansion In contrast in the black hole phase the D-branes fall into the
black hole and a meson has a nonzero probability of disappearing through the horizon thatis into the plasma As a consequence we expect the mesons to develop thermal widths inthe black hole phase even in the limit Nc λrarrinfin In fact as we will see in Section 85 thewidths are generically comparable to the energies of the mesons and hence the mesons canno longer be interpreted as quasiparticles
We thus encounter a somewhat unusual situation the quarkonium mesons are absolutelystable for T lt Tdiss but completely disappear for T gt Tdiss The former is counterintuitivebecause on general grounds we expect that any bound states should always have a nonzerowidth when immersed in a medium with T gt 0 In the case of these mesons we expect thatthey can decay and acquire a width through the following channels
1 Decay to gauge singlets such as glueballs lighter mesons etc
2 Breakup by high-energy gluons (right diagram in Fig 817)
3 Breakup by thermal medium quarks (left diagram in Fig 817)
Process (1) is suppressed by 1N2c (glueballs) or 1Nc (mesons) while (2) and (3) are unsup-
pressed in the large-Nc limit Since (1) is also present in the vacuum we will focus on (2)and (3) which are medium effects They are shown schematically in Fig 817
The width due to (2) is proportional to a Boltzmann factor eminusβEB for creating a gluonthat is energetic enough to break up the bound state while that due to (3) is proportionalto a Boltzmann factor eminusβMth for creating a thermal quark where Mth is the thermal mass
225
q
q
q
q
Figure 817 Sketches taken from Ref [571] showing the relevant thermal processes contributing tothe meson width q (q) denotes a quark (antiquark) The left diagram corresponds to the breakupof a meson by a quark from the thermal medium while the right diagram corresponds to breakupof a meson by an energetic gluon For large λ the first process is dominant coming from the singleinstanton sector
of the quark mdash see Fig 86 Given Eqn (835) and the fact that in the Minkowski phase
T lt Tdiss both Boltzmann factors are suppressed by eminusradicλ sim eminusR2αprime so we recover the result
that these mesons are stable in the infinite-λ limit In particular there is no width at anyperturbative order in the 1
radicλ ndash expansion consistent with the conclusion from the string
theory side Furthermore since the binding energy is EB asymp 2Mth in the regime where λ islarge (but not infinite) the width from process (3) will dominate over that from process (2)
We now review the result from the string theory calculation of the meson widths inRef [571] As discussed above the width is non-perturbative in 1
radicλ sim αprimeR2 and thus
should correspond to some instanton effect on the string worldsheet The basic idea is verysimple even though in a Minkowski embedding the brane is separated from the black holehorizon and classically a meson living on the brane cannot fall into the black hole quantummechanically (from the viewpoint of the string worldsheet) it has a nonzero probability oftunneling into the black hole and the meson therefore develops a width At leading orderthe instanton describing this tunneling process is given by a (Euclidean) string worldsheetstretching between the tip of the D7-brane to the black hole horizon (see Fig 86) and windingaround the Euclidean time direction Heuristically such a worldsheet creates a small tunnelbetween the brane and the black hole through which mesons can fall into the black holeThe instanton action is βMth as can be read off immediately from the geometric picture justdescribed and its exponential gives rise to the Boltzmann factor expected from process (3)From the gauge theory perspective such an instanton can be interpreted as creating a ther-mal quark from the medium and a meson can disappear into the medium via interactionwith it as shown in the left diagram of Fig 817
The explicit expression for the meson width due to such instantons is somewhat compli-cated so we refer the reader to the original literature [571] Although the width appears tobe highly model dependent and is exponentially small in the regime of a large but finite λunder consideration remarkably its momentum dependence has some universal features at
226
large momentum in [571] Specifically one finds that
Γ(q)
Γ(0)=
|ψ(tip ~q)|2
|ψ(tip ~q = 0)|2 (844)
where Γ(q) denotes the width of a meson with spatial momentum q and ψ(tip q) its wave-function evaluated at the tip of the D7-branes (ie where it is closest to the black hole)This result is intuitively obvious because a meson tunnels into the black hole from the tipof the branes In particular as discussed in detail in Ref [568] at large momentum q thewave-function becomes localized around the tip of the brane and can be approximated bythat of a spherical harmonic oscillator with a potential proportional to q2z2 where z is theproper distance from the tip of the branes6 It then immediately follows that for large q thewidth (844) scales as q2 Furthermore for temperatures T Mmes and q M3
mesT2 one
finds the closed-form expression
Γn(q)
Γn(0)asymp 2(4π)4
(n+ 2)(n+ 32)
T 4q2
M6mes
(845)
where n labels different mesonic excitations (see (813))
It is also instructive to plot the full q-dependence of (844) obtained numerically as donein Fig 818 (taken from [571]) for n = 0 mesons at various temperatures Fig 818 has theinteresting feature that the width is roughly constant for small q but turns up quadraticallyaround qMmes asymp 052(TdissT )2 This is roughly the momentum qm sim qlim at which the groupvelocity of a meson achieves its maximum in Fig 815 which as discussed in Section 841can be considered as the characteristic momentum scale where velocity-enhanced screeningbecomes significant This dramatic increase of meson widths beyond qm can also be under-stood intuitively when velocity-enhanced screening becomes significant interaction betweenthe quark and antiquark in a meson becomes further weakened which makes it easier for athermal medium quark or gluon to break it apart
We now briefly comment on the gravity description of process (2) mentioned earlier ie theright diagram in Fig 817 For such a process to happen the gluon should have an energyabove the binding energy of the meson The density of such gluons is thus suppressed byeminus2βMth and should be described by an instanton and anti-instanton We expect that contri-butions from such processes are also controlled by the the value of the meson wave-functionat the tip of the branes and thus likely have similar growth with momentum
Finally we note that as T increases Mth decreases and thus the meson width increasesquickly with temperature but remains exponentially suppressed until Tdiss is reached afterwhich we are in the black hole phase As will be discussed in Section 85 in this phasequarkonium quasiparticles no longer exist
843 Connection with the quark-gluon plasma
Let us now recapitulate the main qualitative features regarding heavy mesons in a stronglycoupled plasma
6Note that there are four transverse directions along the D7-brane as we move away from the tip (notincluding the other (3 + 1) dimensions parallel to the boundary) Thus this is a four-dimensional harmonicoscillator
227
Figure 818 The behavior of the width as a function of q for TTdiss = 099 071 03 013 from leftto right The dashed lines are analytic results for large momenta
1 They survive deconfinement
2 Their dispersion relations have a subluminal limiting velocity at large momentum Thelimiting velocity decreases with increasing temperature and as a result the motion of ameson with large momentum dramatically slows down near Tdiss
3 At large momenta meson widths increase dramatically with momentum
4 The limiting velocity is reached and the increase in widths applies when q qlimwhere qlim is the lsquolimitingrsquo momentum following from (838) if one assumes the standarddispersion relation
Properties (1)ndash(3) are universal in the sense that they apply to the deconfined phase ofany gauge theory with a string dual in the large-Nc strong-coupling limit The reason forthis is that they are simple consequences of general geometric features following from twouniversal aspects of the gaugestring duality (i) the fact that the deconfined phase of thegauge theory is described on the gravity side by a black hole geometry [312] and (ii) thefact that a finite number Nf of quark flavours is described by Nf D-brane probes [384 385]Property (4) was established by explicit numerical calculations in specific models Howevergiven that qlim can be motived in a model-independent way from (838) it is likely to also beuniversal even though though this was not manifest in our discussion above
We have seen that properties (2) and (3) can be considered direct consequences of velocity-enhanced screening which as discussed in Section 77 can have important implications forquarkonium suppression in heavy ion collisions
It is interesting that properties (1) and to some degree (2) can be independently motivatedin QCD whether or not a string dual of QCD exists The original argument [184] for (1) is
228
simply that the heavier the quarkonium meson the smaller its size And it is reasonable toexpect a meson to remain bound until the screening length in the plasma becomes comparableto the meson size and for sufficiently heavy quarkonia this happens at Tdiss gt Tc As we havediscussed in Sections 24 and 33 this conclusion is supported by calculations of both the staticquark-antiquark potential and of Minkowski-space spectral functions in lattice-regularizedQCD The ballpark estimate for the dissociation temperature of heavy mesons suggested bythe above studies roughly agrees with that from the D3D7 system For example for theJψ meson the former estimate is Tc Tdiss 2Tc Allowing for a certain range in the precisevalue of 150 MeV Tc 190 MeV this translates into 300 MeV Tdiss 380 MeV In theD3D7 model we see from Fig 811 that meson states melt at Tdiss 0766M The scale Mis related to the mass Mmes of the lightest meson in the theory at zero temperature throughEqn (829) Therefore we have Tdiss(Mmes) 0122Mmes For the Jψ taking Mmes 3GeV gives Tdiss(Jψ) 366 MeV Although it is gratifying that this comparison leads toqualitative agreement it must be taken with some caution because meson bound statesin the D3D7 system are deeply bound ie Mmes 2Mq whereas the binding energy ofcharmonium states in QCD is a small fraction of the charm mass ie Mcc 2Mc Anadditional difference comes from the fact that in QCD the dissociation of charmonium statesis expected to happen sequentially with excited states (that are larger) dissociating firstwhereas in the D3D7 system all meson states are comparable in size and dissociate atthe same temperature Presumably in the D3D7 system this is an artifact of the large-Ncstrong-coupling approximation under consideration and thus corrections away from this limitshould make holographic mesons dissociate sequentially too
There is a simple (but incomplete) argument for property (2) that applies to QCD justas well as to N = 4 SYM theory [556ndash558 568] a meson moving through the plasma withvelocity v experiences a higher energy density boosted by a factor of γ2 Since energy densityis proportional to T 4 this can be thought of as if the meson sees an effective temperaturethat is boosted by a factor of
radicγ meaning Teff(v) = (1 minus v2)minus14T A velocity-dependent
dissociation temperature scaling like (838) follows immediately and from this a subluminallimiting velocity (843) can be inferred Although this argument is seductive it can be seenin several ways that it is incomplete For example we would have reached a different Teff(v)had we started by observing that the entropy density s is boosted by a factor of γ and isproportional to T 3 And furthermore there really is no single effective temperature seenby the moving quarkonium meson The earliest analysis of quarkonia moving through aweakly coupled QCD plasma with some velocity v showed that the meson sees a blue-shiftedtemperature in some directions and a red-shifted temperature in others [619] Although thesimple argument does not work by itself it does mean that all we need from the calculationsdone via gaugestring duality is the result that Tdiss(v) behaves as if it is controlled by theboosted energy density mdash ie we need the full calculation only for the purpose of justifyingthe use of the particular simple argument that works This suggests that property (2) andin particular the scaling in Eqs (838) and (843) are general enough that they may applyto the quark-gluon plasma of QCD whether or not it has a gravity dual
As explained towards the end of Section 841 there are at least two simple ways inwhich a limiting velocity for quarkonia may be implemented It may happen that mesonstates with momentum above a certain qlim simply do not exist in which case one expectsthat vlim = v(qlim) The second possibility is that the dispersion relation of mesons may
229
become dramatically modified beyond a certain qlim in such a way that although mesonstates of arbitrarily high momentum exist their group velocity never exceeds a certain vlimIt is remarkable that both possibilities are realized in gauge theories with a string dual theformer by high-spin mesons and the latter by low-spin mesons However note that even inthe case of low-spin mesons qlim remains the important momentum scale beyond which weexpect more significant quarkonium suppression for two reasons First meson widths increasesignificantly for q gt qlim Therefore although it is an overstatement to say that these mesonsalso cease to exist above qlim their existence becomes more and more transient at higher andhigher q Second due to the modified dispersion relation mesons with q gt qlim slow down andthey spend a longer time in the medium giving the absorptive imaginary part more time tocause dissociation It will be very interesting to see whether future measurements at RHICor the LHC will show the suppression of JΨ or Υ production increasing markedly abovesome threshold transverse momentum pT as we described in Section 77
In practice our ability to rigorously verify the properties (1)ndash(4) in QCD is limited dueto the lack of tools that are well-suited for this purpose It is therefore reassuring that theyhold for all strongly coupled large-Nc plasmas with a gravity dual for which the gravitydescription provides just such a tool
85 Black hole embeddings
We now consider the phase T gt Tdiss which is described by a D7-brane with a black holeembedding We will give a qualitative argument that in this regime the system genericallycontains no quarkonium quasiparticles We have emphasized the word lsquogenericallyrsquo becauseexceptions arise when certain large ratios of physical scales are introduced lsquoby handrsquo as wewill see later We will illustrate the absence of quasiparticles in detail by computing a spectralfunction of two electromagnetic currents in the next subsection
851 Absence of quasiparticles
In the gravity description of physics at T gt Tdiss the meson widths may be seen by studyingso-called quasi-normal modes These are analogues of the fluctuations we studied in thecase of Minkowski embeddings in that a normalizable fall-off is imposed at the boundaryHowever the regularity condition at the tip of the branes is replaced by the so-called in-falling boundary condition at the horizon Physically this is the requirement that energycan flow into the horizon but cannot come out of it (classically) Mathematically it is easyto see that this boundary condition forces the frequency of the mode to acquire a negativeimaginary part and thus corresponds to a nonzero meson width The meson in question maythen be considered a quasiparticle if and only if this width is much smaller than the real partof the frequency In the case at hand the meson widths increase as the area of the inducedhorizon on the branes increases and go to zero only when the horizon shrinks to zero sizeThis is of course to be expected since it is the presence of the induced horizon that causesthe widths to be nonzero in the first place We are thus led to the suggestion that meson-like quasiparticles will be present in the black hole phase only when the size of the inducedhorizon on the branes can be made parametrically small This expectation can be directly
230
verified by explicit calculation of the quasi-normal modes on the branes [610 620ndash622] andwe will confirm it indirectly below by examining the spectral function of two electromagneticcurrents For the moment let us just note that this condition is not met in the system underconsideration because as soon as the phase transition at T = Tdiss takes place the area ofthe induced horizon on the brane is an order-one fraction of the area of the background blackhole This can be easily seen from Fig 811 by comparing the entropy density (which is ameasure of the horizon area) at the phase transition to the entropy density at asymptoticallyhigh temperatures
stransition
shigh T
asymp 03
2asymp 15 (846)
This indicates that there is no parametric reason to expect quasiparticles with narrow widthsabove the transition We shall confirm by explicit calculation in the next subsection thatthere are no quasiparticle excitations in the black hole phase
852 Meson spectrum from a spectral function
Here we will illustrate some of the general expectations discussed above by examining the in-medium spectral function of two electromagnetic currents in the black hole phase We choosethis particular correlator because it is related to thermal photon production which we willdiscuss in the next Section We will see that no narrow peaks exist for stable black holeembeddings indicating the absence of long-lived quasiparticles These peaks will appearhowever as we artificially push the system into the unstable region close to the criticalembedding (see Fig 87) thus confirming our expectation that quasiparticles should appearas the area of the induced horizon on the branes shrinks to zero size
N = 4 SYM coupled to Nf flavours of equal-mass quarks is an SU(Nc) gauge theory with aglobal U(Nf) symmetry In order to couple this theory to electromagnetism we should gaugea U(1)EM subgroup of U(Nf) by adding a dynamical photon Amicro to the theory for simplicitywe will assume that all quarks have equal electric charge in which case U(1)EM is the diagonalsubgroup of U(Nf) In this extended theory we could then compute correlation functions ofthe conserved current JEM
micro that couples to the U(1)EM gauge field The string dual of thisSU(Nc)timesU(1)EM gauge theory is unknown so we cannot perform this calculation holograph-ically However as noted in [623] we can perform it to leading order in the electromagneticcoupling constant e because at this order correlation functions of electromagnetic currentsin the gauged and in the ungauged theories are identical This is very simple to understanddiagramatically as illustrated for the two-point function in Fig 819 In the ungauged theoryonly SU(Nc) fields lsquorunrsquo in the loops represented by the shaded blob The gauged theorycontains additional diagrams in which the photon also runs in the loops but these necessarilyinvolve more photon vertices and therefore contribute at higher orders in e Thus one canuse the holographic description to compute the lsquoSU(Nc) blobrsquo and obtain the result for thecorrelator to leading order in e
Using this observation the authors of Ref [623] first did a holographic computation of thespectral density of two R-symmetry currents in N = 4 SYM theory to which finite-couplingcorrections were computed in [624 625] The result for the R-charge spectral density isidentical up to an overall constant with the spectral density of two electromagnetic currents
231
P1
K+P1 K+P2
P2
SU(Nc)
q q
Figure 819 Diagrams contributing to the two-point function of electromagnetic currents Theexternal line corresponds to a photon of momentum q As explained in the text to leading order inthe electromagnetic coupling constant only SU(Nc) fields lsquorunrsquo in the loops represented by the shadedblob
in N = 4 SYM theory coupled to massless quarks This and the extension to nonzero quarkmass were obtained in Ref [612] which we now follow
The relevant spectral function is defined as
χmicroν(k) = 2 ImGRmicroν(k) (847)
where kmicro = (ω q) is the photon null momentum (ie ω2 = q2) and
GRmicroν(k) = i
intdd+1x eminusikmicrox
microΘ(t)〈[JEM
micro (x) JEMν (0)]〉 (848)
is the retarded correlator of two electromagnetic currents The key point in this calculationis to identify the field in the string description that is dual to the operator of interest herenamely the conserved current JEM
micro We know from the discussion in Section 5 that conservedcurrents are dual to gauge fields on the string side Moreover since JEM
micro is constructed outof fields in the fundamental representation we expect its dual field to live on the D7-branesThe natural (and correct) candidate turns out to be the U(1) gauge field associated with thediagonal subgroup of the U(Nf) gauge group living on the worldvolume of the Nf D7-branesOnce this is established one must just follow the general prescription explained in Section5 The technical details of the calculation can be found in Ref [612] so here we will onlydescribe the results and their interpretation In addition we will concentrate on the traceof the spectral function χmicromicro(k) equiv ηmicroνχmicroν(k) since this is the quantity that determines thethermal photon production by the plasma (see next section)
The trace of the spectral function for stable black hole embeddings is shown in Fig 820for several values of the quark mass m Note that this is a function of only one variable sincefor an on-shell photon ω = q The normalisation constant that sets the scale on the verticalaxis is ND7 = NfNcT
24 The NfNc scaling of the spectral function reflects the number ofelectrically charged degrees of freedom in the plasma in the case of two R-symmetry currentsNfNc would be replaced by N2
c [623] All curves decay as ωminus13 for large frequencies Notethat χ sim ω as ω rarr 0 This is consistent with the fact that the value at the origin of eachof the curves yields the electric conductivity of the plasma at the corresponding quark mass
232
0 05 1 15 2 25
02
04
06
08
1
ω = ω2πT
χmicromicro(ω)
8ND7ω
Figure 820 Trace of the spectral function as a function of the dimensionless frequency ω = ω2πTfor (from top to bottom on the left-hand side) m = 0 06 085 093 115 125 132 The last valuecorresponds to that at which the phase transition from a black hole to a Minkowski embedding takesplace Recall that ND7 = NfNcT
24
namely
σ =e2
4limωrarr0
1
ωηmicroνχmicroν(ω = q) (849)
This formula is equivalent to the perhaps-more-familiar expression in terms of the zero-frequency limit of the spectral function at vanishing spatial momentum (see Appendix A)
σ =e2
6limωrarr0
1
ωδijχij(ω q = 0) =
e2
6limωrarr0
1
ωηmicroνχmicroν(ω q = 0) (850)
where in the last equality we used the fact that χ00(ω 6= 0 q = 0) = 0 as implied bythe Ward identity kmicroχmicroν(k) = 0 To see that the two expressions (849) and (850) areequivalent suppose that q points along the 1-direction Then the Ward identity togetherwith the symmetry of the spectral function under the exchange of its spacetime indices implythat ω2χ00 = q2χ11 For null momentum this yields minusχ00+χ11 = 0 so we see that Eqn (849)reduces to
σ =e2
4limωrarr0
1
ω
[χ22(ω = q) + χ33(ω = q)
] (851)
The diffusive nature of the hydrodynamic pole of the correlator implies that at low frequencyand momentum the spatial part of the spectral function behaves as
χij(ω q) simω3
ω2 +D2q4 (852)
where D is the diffusion constant for electric charge This means that we can replace q = ωby q = 0 in Eqn (851) thus arriving at expression (850)
233
0 02 04 06 08 1 12 140
02
04
06
08
ω = ω2πT
χmicromicro(ω)
8ND7ω
Figure 821 Trace of the spectral function as a function of the dimensionless frequency ω = ω2πTfor non-stable black hole embeddings Curves with higher narrower peaks correspond to embeddingsthat are closer to the critical embedding
For the purpose of our discussion the most remarkable feature of the spectral functionsdisplayed in Fig 820 is the absence of any kind of high narrow peaks that may be associ-ated with a quasiparticle excitation in the plasma This feature is shared by thermal spectralfunctions of other operators on stable black hole embeddings We thus confirm our expec-tation that no quasiparticles exist in this phase In order to make contact with the physicsof the Minkowski phase in which we do expect the presence of quarkonium quasiparticlesthe authors of Ref [612] computed the spectral function for black hole embeddings beyondthe phase transition ie in the region below Tdiss in which these embeddings are metastableor unstable The results for the spectral function are shown in Fig 821 The most impor-tant feature of these plots is the appearance of well defined peaks in the spectral functionwhich become higher and narrower seemingly approaching delta-functions as the embed-ding approaches the critical embedding (see Fig 87) Thus the form of the spectral functionappears to approach the form we expect for Minkowski embeddings7 namely an infinite sumof delta-functions supported at a discrete set of energies ω2 = q2 (However a precise mapbetween the peaks in Fig 821 and the meson spectrum in a Minkowski embedding is noteasy to establish [626]) Each of these delta-functions is associated with a meson mode onthe D7-branes with null four-momentum The fact that the momentum is null may seemsurprising in view of the fact that as explained above the meson spectrum in the Minkowskiphase possesses a mass gap but in fact it follows from the dispersion relation for these mesonsdisplayed in Fig 814 To see this consider the dispersion relation ω(q) for a given meson inthe Minkowski phase The fact that there is a mass gap means that ω gt 0 at q = 0 On theother hand in the limit of infinite spatial momentum q rarr infin the dispersion relation takesthe form ω vlimq with vlim lt 1 Continuity then implies that there must exist a value of q
7An analogous result was found in [611] for time-like momenta
234
such that ω(q) = q This is illustrated in Fig 814 by the fact that the dispersion relationsintersect the blue lines Since in the Minkowski phase the mesons are absolutely stable in thelarge-Nc strong-coupling limit under consideration we see that each of them gives rise to adelta-function-like (ie zero-width) peak in the spectral function of electromagnetic currentsat null momentum Below we will see some potential implications of this result for heavy-ioncollisions
86 Two universal predictions
We have just seen that the fact that heavy mesons remain bound in the plasma and thefact that their limiting velocity is subluminal imply that the dispersion relation of a heavymeson must cross the light-cone defined by ω = q at some energy ω = ωpeak indicated by thevertical line in Fig 814 In this section we will see that this simple observation leads to twouniversal consequences Implications for deep inelastic scattering have been studied in [627]but will not be reviewed here
861 A meson peak in the thermal photon spectrum
At the crossing point between the meson dispersion relation and the light-cone the mesonfour-momentum is null that is ω2
meson = q2meson If the meson is flavourless and has spin one
then at this point its quantum numbers are the same as those of a photon Such a mesoncan then decay into an on-shell photon as depicted in Fig 822 Note that in the vacuumonly the decay into a virtual photon would be allowed by kinematics In the medium thedecay can take place because of the modified dispersion relation of the meson Also notethat the decay will take place unless the photon-meson coupling vanishes for some reason(eg a symmetry) No such reason is known in QCD
The decay process of Fig 822 contributes a resonance peak at a position ω = ωpeak tothe in-medium spectral function of two electromagnetic currents (848) evaluated at null-momentum ω = q This in turn produces a peak in the spectrum of thermal photons emittedby the plasma
dNγ
dωsim eminusωT χmicromicro(ω T ) (853)
The width of this peak is the width of the meson in the plasma
The analysis above applies to an infinitely-extended plasma at constant temperature As-suming that these results can be extrapolated to QCD a crucial question is whether a peakin the photon spectrum could be observed in a heavy ion collision experiment Natural heavyvector mesons to consider are the Jψ and the Υ since these are expected to survive de-confinement We wish to compare the number of photons coming from these mesons to thenumber of photons coming from other sources Accurately calculating the meson contribu-tion would require a precise theoretical understanding of the dynamics of these mesons in thequark-gluon plasma which at present is not available Our goal will therefore be to estimatethe order of magnitude of this effect with a simple recombination model The details can befound in Ref [628] so here we will only describe the result for heavy ion collisions at LHCenergies
235
Figure 822 In-medium vector mesonndashphoton mixing The imaginary part of this diagram yieldsthe meson decay width into photons
Figure 823 Thermal photon spectrum for LHC energies Tdiss = 125Tc and Mcharm = 17 GeV The(arbitrary) normalization is the same for all curves The continuous monotonically decreasing bluecurve is the background from light quarks The continuous red curve is the signal from Jψ mesonsThe dashed black curve is the sum of the two See Ref [628] for details
The result is summarized in Fig 823 which shows the thermal photon spectrum comingfrom light quarks the contribution from Jψ mesons and the sum of the two for a thermalcharm mass Mcharm = 17 GeV and a Jψ dissociation temperature Tdiss = 125Tc Althoughthe value of Mcharm is relatively high the values of Mcharm and Tdiss are within the rangecommonly considered in the literature For the charm mass a typical range is 13 leMcharm le17 GeV because of a substantial thermal contribution mdash see eg Ref [236] and Refs thereinThe value of Tdiss is far from settled but a typical range is Tc le Tdiss le 2Tc [190 230 241242247573629ndash633] We have chosen these values for illustrative purposes since they leadto an order-one enhancement in the spectrum We emphasize however that whether thisphoton excess manifests itself as a peak or only as an enhancement smoothly distributedover a broader range of frequencies depends sensitively on these and other parametersQualitatively the dependence on the main ones is as follows Decreasing the quark massdecreases the magnitude of the Jψ contribution Perhaps surprisingly higher values of Tdiss
make the peak less sharp The in-medium width of the Jψ used in Fig 823 was 100 MeVIncreasing this by a factor of two turns the peak into an enhancement Crucially the Jψcontribution depends quadratically on the cc cross-section Since at RHIC energies this is
236
Horizon
D-branes
vlim
r
Quarkv
Black Hole
Boundary
Figure 824 D-branes and open string in a black brane geometry
believed to be ten times smaller than at LHC energies the enhancement discussed above ispresumably unobservable at RHIC
These considerations show that a precise determination of the enhancement is not possiblewithout a very detailed understanding of the in-medium dynamics of the Jψ On the otherhand they also illustrate that there exist reasonable values of the parameters for whichthis effect yields an order-one enhancement or even a peak in the spectrum of thermalphotons produced by the quark-gluon plasma This thermal excess is concentrated at photonenergies roughly between 3 and 5 GeV In this range the number of thermal photons in heavyion collisions at the LHC is expected to be comparable to or larger than that of photonsproduced in initial partonic collisions that can be described using perturbative QCD [634]Thus we expect the thermal excess above to be observable even in the presence of the pQCDbackground
The authors of Ref [628] also examined the possibility of an analogous effect associatedwith the Υ meson in which case ωpeak sim 10 GeV At these energies the number of ther-mal photons is very much smaller than that coming from initial partonic collisions so anobservable effect is not expected
862 A new mechanism of quark energy loss Cherenkov emission ofmesons
We now turn to another universal prediction that follows from the existence of a subluminallimiting velocity for mesons in the plasma Consider a highly energetic quark moving throughthe plasma In order to model this we consider a string whose endpoint moves with anarbitrary velocity v at an arbitrary radial position rq mdash see Fig 824 Roughly speakingthe interpretation of rq in the gauge theory is that of the inverse size of the gluon cloud thatdresses the quark This can be seen for example by holographically computing the profile of〈TrF 2(x)〉 around a static quark source dual to a string whose endpoint sits at r = rq [613]
Two simple observations now lead to the effect that we are interested in The first oneis that the string endpoint is charged under the scalar and vector fields on the branes Inthe gauge theory this corresponds to an effective quark-meson coupling (see Fig 825) oforder e sim 1
radicNc Physically this can be understood very simply The fields on the branes
describe fluctuations around the equilibrium configuration The string endpoint pulls on thebranes and therefore excites (ie it is charged under) these fields The branesrsquo tension is
237
quarkquark
meson
Figure 825 Effective quark-meson coupling
of order 1gs sim Nc where gs is the string coupling constant whereas the string tension isNc-independent This means that the deformation of the branes caused by the string is oforder e2 sim 1Nc We thus conclude that the dynamics of the lsquobranes+string endpointrsquo systemis (a generalization of) that of classical electrodynamics in a medium in the presence of afast-moving charge
The second observation is that the velocity of the quark may exceed the limiting velocityof the mesons since the redshift at the position of the string endpoint is smaller than thatat the bottom of the branes As in ordinary electrodynamics if this happens then the stringendpoint loses energy by Cherenkov-radiating into the fields on the branes In the gaugetheory this translates into the quark losing energy by Cherenkov-radiating scalar and vectorquarkonium mesons The rate of energy loss is set by the square of the coupling and istherefore of order 1Nc
The quantitative details of the energy lost to Cherenkov radiation of quarkonium mesonsby a quark propagating through the N = 4 plasma can be found in [617618] so here we willonly describe the result For simplicity we will assume that the quark moves with constantvelocity along a straight line at a constant radial position In reality rq and v will of coursedecrease with time because of the black hole gravitational pull and the energy loss Howeverwe will concentrate on the initial part of the trajectory (which is long provided the initialquark energy is large) for which rq and v are approximately constant [531] mdash see Fig 824Finally for illustrative purposes we will focus on the energy radiated into the transversemodes of vector mesons The result is depicted in Fig 826 and its main qualitative featuresare as follows
As expected we see that the quark only radiates into meson modes with phase velocitylower than v mdash those to the right of the dashed vertical lines in Fig 814 For fixed rq theenergy loss increases monotonically with v up to the maximum allowed value of v mdash the localspeed of light at rq As rq decreases the characteristic momentum qchar of the modes intowhich the energy is deposited increases These modes become increasingly peaked near thebottom of the branes and the energy loss diverges However this mathematical divergenceis removed by physical effects we have not taken into account For example for sufficientlylarge q the mesonsrsquo wave functions become concentrated on a region whose size is of order thestring length and hence stringy effects become important [568] Also as we saw in Section842 mesons acquire widths Γ prop q2 at large q [571] and can no longer be treated as welldefined quasiparticles Finally the approximation of a constant-v constant-rq trajectoryceases to be valid whenever the energy loss rate becomes large
The Cherenkov radiation of quarkonium mesons by quarks depends only on the qualitativefeatures of the dispersion relation of Fig 814 which are universal for all gauge theory plasmaswith a dual gravity description Moreover as we explained in Section 843 it is conceivable
238
v
Nc
(2πT
)2dE dt
04 05 06 07 08 09 1000
05
10
15
20
25
30
Figure 826 Cherenkov energy loss into the transverse mode of vector quarkonium mesons Thecontinuous curves correspond to increasing values of rq from left to right The dotted curve is definedby the endpoints of the constant-rq curves See Refs [617618] for details
that they may also hold for QCD mesons such as the Jψ or the Υ whether or not a stringdual of QCD exists Here we will examine some qualitative consequences of this assumptionfor heavy ion collision experiments Since the heavier the meson the more perturbative itsproperties become we expect that our conclusions are more likely to be applicable to thecharmonium sector than to the bottomonium sector
An interesting feature of energy loss by Cherenkov radiation of quarkonia is that unlikeother energy-loss mechanisms it is largely independent of the details of the quark excitedstate such as the precise features of the gluon cloud around the quark etc In the gravitydescription these details would be encoded in the precise profile of the entire string but theCherenkov emission only depends on the trajectory of the string endpoint This leads to adramatic simplification which with the further approximation of rectilinear uniform motionreduces the parameters controlling the energy loss to two simple ones the string endpointvelocity v and its radial position rq In order to obtain a ballpark estimate of the magnitude ofthe energy loss we will assume that in a typical collision quarks are produced with order-onevalues of rq (in units of R2T ) Under these circumstances the energy loss is of order unity inunits of (2πT )2Nc which for a temperature range of T = 200minus 400 MeV and Nc = 3 leadsto dEdx asymp 2minus8 GeVfm This is is of the same order of magnitude as other mechanisms ofenergy loss in the plasma for example the BDMPS radiative energy loss dEdx = αsCF qL2yields values of dEdx = 7 minus 40 GeVfm for q = 1 minus 5 GeV 2fm αs = 03 and L asymp 6 fmSince our gravity calculation is strictly valid only in the infinite-quark-energy limit (becauseof the linear trajectory approximation) we expect that our estimate is more likely to beapplicable to highly energetic quarks at the LHC than to those at RHIC
Even if in the quark-gluon plasma the magnitude of Cherenkov energy loss turns out tobe subdominant with respect to other mechanisms its velocity dependence and its geometricfeatures may still make it identifiable Indeed this mechanism would only operate for quarksmoving at velocities v gt vlim with vlim the limiting velocity of the corresponding quarkoniummeson in the plasma The presence of such a velocity threshold is the defining characteristic
239
of Cherenkov energy loss The precise velocity at which the mechanism starts to operate mayactually be higher than vlim in some cases since the additional requirement that the energyof the quark be equal or larger than the in-medium mass of the quarkonium meson must alsobe met
Cherenkov mesons would be radiated at a characteristic angle cos θc = vlimv with respectto the emitting quark where v is the velocity of the quark Taking the gravity result asguidance vlim could be as low as vlim = 035 at the quarkonium dissociation temperature [565]corresponding to an angle as large as θc asymp 121 rad This would result in an excess of heavyquarkonium associated with high-energy quarks passing through the plasma Our estimateof the energy loss suggests that the number of emitted Jψrsquos for example could range fromone to three per fm This emission pattern is similar to the emission of sound waves by anenergetic parton [138] that we have reviewed in Section 73 in that both effects lead to a non-trivial angular structure One important difference however is that the radiated quarkoniummesons would not thermalize and hence would not be part of a hydrodynamic shock wave Asin the Mach cone case the meson emission pattern could be reflected in azimuthal dihadroncorrelations triggered by a high-pT hadron Due to surface bias the energetic parton in thetriggered direction is hardly modified while the one propagating in the opposite directionmoves through a significant amount of medium emitting quarkonium mesons Thus underthe above assumptions the dihadron distribution with an associated Jψ would have a ring-like structure peaked at an angle θ asymp π minus θc Even if this angular structure were to provehard to discern the simpler correlation that in events with a high-pT hadron there are moreJΨ mesons than in typical events may suffice as a distinctive signature although furtherphenomenological modelling is required to establish this
A final observation is that Cherenkov energy loss also has a non-trivial temperature depen-dence since it requires that there are meson-like states in the plasma and therefore it doesnot take place at temperatures above the meson dissociation temperature Similarly it isreasonable to assume that it does not occur at temperatures below Tc since in this case we donot expect the meson dispersion relations to become spacelike8 Under these circumstancesthe Cherenkov mechanism is only effective over a limited range of temperatures Tc lt T lt Tdiss
which if Tdiss amp 12Tc as in Ref [247] is a narrow interval As was pointed out in Ref [635] amechanism of energy loss which is confined to a narrow range of temperatures in the vicinityof Tc concentrates the emission of energetic partons to a narrow layer within the collisiongeometry and is able to explain v2-data at high pT at RHIC [636 637] Provided that themeson dissociation temperature Tdiss is not much larger than Tc the Cherenkov radiation ofquarkonium mesons is one such mechanism
8This assumption is certainly correct for plasmas with a gravity dual since the corresponding geometrydoes not include a black hole horizon if T lt Tc
240
Chapter 9
Concluding remarks and outlook
Since the purpose of heavy ion collisions is to study the properties of Quantum Chromo-dynamics at extreme temperature and energy density any successful phenomenology mustultimately be based on QCD However as discussed in Section 2 heavy ion phenomenol-ogy requires strong-coupling techniques not only for bulk thermodynamic quantities like theQCD equation of state but also for many dynamical quantities at nonzero temperaturesuch as transport coefficients relaxation times and quantities accessed by probes propagat-ing through a plasma By now lattice-regularized QCD calculations provide well controlledresults for the former but progress on all the latter quantities is likely to be slow since oneneeds to overcome both conceptual limitations and limitations in computing power Alterna-tive strong-coupling tools are therefore desirable The gaugestring duality provides one suchtool for performing non-perturbative calculations for a wide class of non-abelian plasmas
In this review we have mostly focussed on results obtained within one particular exampleof a gaugestring duality namely the case in which the gauge theory is N = 4 SYM or a smalldeformation thereof One reason for this is pedagogical N = 4 SYM is arguably the simplestand best understood case of a gaugestring duality By now many examples are known ofmore sophisticated string duals of non-supersymmetric non-conformal QCD-like theoriesthat exhibit confinement spontaneous chiral symmetry breaking thermal phase transitionsetc However many of these features become unimportant in the deconfined phase Forthis reason for the purpose of studying the QCD quark-gluon plasma the restriction toN = 4 SYM not only yields a gain in simplicity but also does not imply a significant loss ofgenerality at least at the qualitative level Moreover none of these lsquomore realisticrsquo theoriescan be considered in any sense a controlled approximation to QCD Indeed many differencesremain including the presence of adjoint fermions and scalar fields the lack of asymptoticfreedom the large-Nc approximation etc Some of these differences may be overcome ifstring theory in asymptotically AdS spacetimes becomes better understood However in thesupergravity (plus classical strings and branes) limit currently accessible these caveats remainimportant to bear in mind when comparing to QCD
In this context it is clearly questionable to assess the interplay between heavy ion phe-nomenology and the gaugestring duality correspondence solely on the basis of testing thenumerical agreement between theory and experiment Rather one should view this inter-play in light of the standard scientific strategy that to gain significant insight into problems
241
that cannot be addressed within the current state of the art it is useful to find a closelyrelated theory within which such problems can be addressed with known technology andwhich encompasses the essential features of interest For many dynamical features of phe-nomenological interest in heavy ion physics controlled strong-coupling calculations in QCDare indeed not in immediate reach with the current state of the art In contrast within thegaugestring correspondence it has been possible to formulate and solve the same problems inthe strongly coupled plasmas of a large class of non-abelian quantum field theories Amongthese strongly coupled N = 4 SYM theory at large-Nc turns out to provide the simplestmodel for the strongly coupled plasma being produced and probed in heavy ion collisionsVery often in the past when theoretical physicists have introduced some model for the pur-pose of capturing the essence of some phenomenon or phenomena involving strongly coupleddynamics the analysis of that model has then required further uncontrolled approximations(Examples abound NambundashJona-Lasinio models in which the QCD interaction is first re-placed by a four-fermi coupling but one then still has to make a mean-field approximationlinear sigma models again followed by a mean-field approximation bag models ) A greatadvantage of using a quantum field theory with a gravity dual as a model is that once wehave picked such a theory the calculations needed to address the problems of interest can bedone rigorously at strong coupling without requiring any further compromise In some casesthe mere formulation of the problem in a way suitable for a gravitational dual calculationcan lead to new results within QCD [90 638 639] In many others as we have seen theexistence of these solutions allows one to examine and understand the physics responsible forthe processes of interest The most important output of a successful model is understandingControlled quantitative calculations come later Understanding how the dynamics workswhat is important and what is extraneous what the right picture is that helps you to thinkabout the physics in a way that is both insightful and predictive these must all come first
At the least the successes to date of the applications of gaugestring duality to problemsarising from heavy ion collisions indicate that it provides us with a successful model in thesense of the previous paragraph However there are many indications that it provides moreIn solving these problems some regularities have emerged in the form of universal propertiesby which we mean properties common to all strongly coupled theories with gravity duals inthe large-Nc strong-coupling limit These include both quantitative observables such as theratio of the thermodynamic potentials at strong and weak coupling (Section 6) and the valueηs = 14π at strong coupling (Section 62) and qualitative features such as the familiar factthat heavy quarkonium mesons remain bound in the plasma as well as the discovery that thedissociation temperature for quarkonium mesons drops with increasing meson velocity v like(1minusv2)14 (Sections 77 and 84) and that high-momentum dispersion relations become space-like (Section 84) The discovery of these generic properties is important in order to extractlessons for QCD Indeed the fact that some properties are valid in a class of gauge-theoryplasmas so broad as to include theories in different numbers of dimensions with differentfield content with or without chemical potentials with or without confinement and chiralsymmetry breaking etc leads one to suspect that such properties might be universal acrossthe plasmas in a class of theories that includes the QCD mdash whether or not a string dualof QCD itself exists The domain of applicability of this putative universality is at presentunknown both in the sense that we do not know to what theories it may apply and in thesense that we cannot say a priori which observables and phenomena are universal and which
242
others are theory-specific details One guess as to a possible characterization in theory spacecould be that these universal features may be common across all gauge-theory plasmas thathave no quasiparticle description (Section 63)
Even results obtained via the gaugestring duality that are not universal may provideguidance for our understanding of QCD at nonzero temperature and for heavy ion phe-nomenology In many cases these are strong-coupling results that differ parametrically fromthe corresponding weak-coupling results and therefore deliver valuable qualitative messagesfor the modeling of heavy ion collisions In particular the small values of the ratio ηs ofthe relaxation time τπ (Section 62) and of the heavy-quark diffusion constant (Section 72)showed that such small values can be realized in a gauge-theory plasma In addition theresult for ηs in N = 4 theory combined with the results for the entropy density pressureand energy density (Section 61) have taught us that a theory can have almost identical ther-modynamics at zero and infinite coupling and yet have radically different hydrodynamicsThe lesson that this provides for QCD is that the thermodynamic observables although theyare available from lattice simulations are not good indicators of whether the quark-gluonplasma is weakly or strongly coupled whereas the transport coefficients are
Important lessons have also been extracted from the strong-coupling calculations of the jetquenching parameter q and the heavy-quark drag coefficient ηD and momentum broadeningκ (Sections 71 72 and 75) These showed not only that these quantities can attain valuessignificantly larger than indicated by perturbative estimates but also that while in perturba-tion theory both q and κ are proportional to the entropy density this is not the case at strongcoupling where both these quantities and ηD scale with the square root of the number ofdegrees of freedom This result which is valid for a large class of theories corrected a naıvephysical expectation that was supported by perturbation theory
Perhaps even more fundamentally the availability of rigorous reliable results for anystrongly coupled plasma (leave apart for a large class of them) can alter the very intuitionwe use to think about the dynamics of the quark-gluon plasma In perturbation theoryone thinks of the plasma as being made of quark and gluon quasiparticles However gravitycalculations of correlation functions valid at strong coupling show no evidence of the existenceof any quasiparticle excitations composed from gluons and light quarks (Section 63) (Heavysmall quarkonium mesons do survive as quasiparticles up to some dissociation temperature(Sections 852 and 86)) The presence or absence of quasiparticles is a major qualitativedifference between the weak- and strong-coupling pictures of the plasma which is largelyindependent of the caveats associated with the use of gaugestring duality that we havedescribed above
Finally two important roles played by the gaugestring duality are that of a testing groundof existing ideas and models relevant for heavy ion collisions and that of a source of new onesin a regime in which guidance and inspiration from perturbation theory may be inapplicableor misleading In its first role the duality provides a rigorous field-theoretical frameworkwithin which to verify our intuition about the plasma For example explicit calculationsgave support to previously suggested ideas about the hydrodynamical response of the plasmato high-energy particles (Section 73) or the possibility that heavy quarkonium mesons survivedeconfinement (Section 831) In its second role the duality has generated qualitatively newideas which could not have been guessed from perturbation theory Examples of these are the
243
non-trivial velocity dependence of screening lengths (Section 77) the in-medium conversionof mesons into photons (Section 86) the energy loss of heavy quarks via Cherenkov emissionof mesons (Section 862) and the appearance of a phase transition associated with thedissociation of heavy quarkonium bound states (Section 832)
In summary while it is true that caution and a critical mind must be exercised whentrying to extract lessons from any gaugegravity calculation paying particular attention toits limitations and range of applicability it is also undeniable that over the last few years abroad suite of qualitatively novel insights relevant for heavy ion phenomenology have emergedfrom detailed and quantitative calculations in the gravity duals of non-abelian field theoriesAs the phenomenology of heavy ion collisions moves to new more quantitative and moredetailed studies in the RHIC program and as it moves to novel challenges at the LHC wehave every reason to expect that experimental information about additional properties of hotQCD matter will come into theoretical focus Understanding properties of the QCD plasmaas well as its response to and its effects on probes at strong coupling will therefore remainkey issues in future analyses We expect that the gaugestring correspondence will continueto play an important role in making progress on these issues
Acknowledgements
We gratefully acknowledge our many collaborators with whom and from whom we havelearned the subjects that we have reviewed We also acknowledge T Faulkner for helping uswith some of the figures in this review The work of HL was supported in part by a DOEOutstanding Junior Investigator grant This research was supported in part by the DOEOffices of Nuclear and High Energy Physics under grants DE-FG02-94ER40818 and DE-FG02-05ER41360 DM is supported in part by grants 2009-SGR-168 MEC FPA 2007-66665-C02 MEC FPA 2007-66665-C01 and CPAN CSD2007-00042 Consolider-Ingenio 2010 JCSis supported by a Marie Curie Intra-European Fellowship PIEF-GA-2008-220207
244
Appendix A
Green-Kubo formula for transportcoefficients
Transport coefficients of a gauge theory plasma such as the shear viscosity η can be extractedfrom correlation functions of the gauge theory via a relation known as the Green-Kuboformula Here we derive this relation for the case of the shear viscosity Let us considera system in equilibrium and let us work in the fluid rest frame meaning that umicro = (10)Deviations from equilibrium are studied by introducing a small external source of the type
S = S0 +1
2
intd4xTmicroνhmicroν (A1)
where S0 (S) is the action of the theory in the absence (presence) of the perturbation hmicroν To leading order in the perturbation the expectation value of the stress tensor is
〈Tmicroν(x)〉 = 〈Tmicroν(x)〉0 minus1
2
intd4y GmicroναβR (xminus y)hαβ(y) (A2)
where the subscript 0 indicates the unperturbed expectation value and the retarded correlatoris given by
iGmicroναβR (xminus y) equiv θ(x0 minus y0
) lang[Tmicroν(x) Tαβ(y)
]rang (A3)
To extract the shear viscosity we concentrate on an external perturbation of the form
hxy(t z) (A4)
Upon Fourier transforming this is equivalent to using rotational invariance to choose thewave vector of the perturbation k along the z direction The off-diagonal components ofthe stress tensor are then given by
〈T xy(ω k)〉 = minusGxyxyR (ω k)hxy(ω k) (A5)
to linear order in the perturbation In the long wavelength limit in which the typical variationof the perturbed metric is large compared with any correlation length we obtain
〈T xy〉 (t z) = minusintdω
2πeminusiωtGxyxyR (ω k = 0)hxy(ω z) (A6)
245
This long wavelength expression may be compared to the hydrodynamic approximation bystudying the reaction of the system to the source within the effective theory The source hmicroνcan be interpreted as a modification on the metric
gmicroν rarr gmicroν + hmicroν (A7)
To leading order in the perturbation the shear tensor defined in eq (213) is given by
σxy = 2 Γ0xy = part0hxy (A8)
where Γmicroνρ are the Christoffel symbols The hydrodynamic approximation is valid in the longtime limit when all microscopic processes have relaxed In this limit we can compare thelinear response expression (A5) to the expression obtained upon making the hydrodynamicapproximation namely (211) We conclude that
η = minus limωrarr0
1
ωlimkrarr0
ImGxyxyR (ωk) (A9)
This result is known as the Green-Kubo formula for the shear viscosity
The above discussion for the stress tensor can also be generalized to other conservedcurrents In general the low frequency limit of GR(ω~k) for a conserved current operator Odefines a transport coefficient χ
χ = minus limωrarr0
lim~krarr0
1
ωImGR(ωk) (A10)
where the retarded correlator is defined analogously to eq (A3)
iGR(xminus y) equiv θ(x0 minus y0
)〈[O(x) O(y)]〉 (A11)
246
