Recent lessons about hydrodynamics from holography

Michał P. Heller m.p.heller@uva.nlUniversity of Amsterdam, The Netherlands

&

National Centre for Nuclear Research, Poland (on leave)

based on1103.3452 [hep-th] MPH, R. A. Janik & P. Witaszczyk (PRL 108 (2012) 201602)1302.0697 [hep-th] MPH, R. A. Janik & P. Witaszczyk (PRL 110 (2013) 211602)

1/13

Holographic duality

2/13

Maldacena [hep-th/9711200]review: Mc Greevy 0909.0518 [hep-th]

classical gravity theories in a higher dimensional spacetimes

strongly coupled ( ) quantum theories of large matrices ( )

bad news: none of the pheno-relevant QFTs is truly holographic in this sense

but 1: there are significant similarities (especially for QCD and its phases)!

g2YMNc � 1

Nc � 1

but II: we can solve rich strongly coupled systems through solving „simple” PDEs!

schematic form

L ! Tr(d!id!i) + gY McijkTr(!i!j!k) + g2Y MdijklTr(!i!j!k!l), (1.2)

for some constants cijk and dijkl (where we have assumed that the interactions areSU(N)-invariant; mass terms can also be added and do not change the analysis).Rescaling the fields by !i " gY M!i, the Lagrangian becomes

L ! 1

g2Y M

!Tr(d!id!i) + cijkTr(!i!j!k) + dijklTr(!i!j!k!l)

", (1.3)

with a coe"cient of 1/g2Y M = N/! in front of the whole Lagrangian.

Now, we can ask what happens to correlation functions in the limit of large Nwith constant !. Naively, this is a classical limit since the coe"cient in front of theLagrangian diverges, but in fact this is not true since the number of components inthe fields also goes to infinity in this limit. We can write the Feynman diagrams ofthe theory (1.3) in a double line notation, in which an adjoint field !a is representedas a direct product of a fundamental and an anti-fundamental field, !i

j , as in figure1.1. The interaction vertices we wrote are all consistent with this sort of notation. Thepropagators are also consistent with it in a U(N) theory; in an SU(N) theory there isa small mixing term

#!i

j!kl

$# ("i

l"jk $

1

N"ij"

kl ), (1.4)

which makes the expansion slightly more complicated, but this involves only subleadingterms in the large N limit so we will neglect this di#erence here. Ignoring the secondterm the propagator for the adjoint field is (in terms of the index structure) like that of afundamental-anti-fundamental pair. Thus, any Feynman diagram of adjoint fields maybe viewed as a network of double lines. Let us begin by analyzing vacuum diagrams(the generalization to adding external fields is simple and will be discussed below). Insuch a diagram we can view these double lines as forming the edges in a simplicialdecomposition (for example, it could be a triangulation) of a surface, if we view eachsingle-line loop as the perimeter of a face of the simplicial decomposition. The resultingsurface will be oriented since the lines have an orientation (in one direction for afundamental index and in the opposite direction for an anti-fundamental index). Whenwe compactify space by adding a point at infinity, each diagram thus corresponds to acompact, closed, oriented surface.

What is the power of N and ! associated with such a diagram? From the formof (1.3) it is clear that each vertex carries a coe"cient proportional to N/!, whilepropagators are proportional to !/N . Additional powers of N come from the sum overthe indices in the loops, which gives a factor of N for each loop in the diagram (sinceeach index has N possible values). Thus, we find that a diagram with V vertices, E

12

schematic form

L ! Tr(d!id!i) + gY McijkTr(!i!j!k) + g2Y MdijklTr(!i!j!k!l), (1.2)

for some constants cijk and dijkl (where we have assumed that the interactions areSU(N)-invariant; mass terms can also be added and do not change the analysis).Rescaling the fields by !i " gY M!i, the Lagrangian becomes

L ! 1

g2Y M

!Tr(d!id!i) + cijkTr(!i!j!k) + dijklTr(!i!j!k!l)

", (1.3)

with a coe"cient of 1/g2Y M = N/! in front of the whole Lagrangian.

Now, we can ask what happens to correlation functions in the limit of large Nwith constant !. Naively, this is a classical limit since the coe"cient in front of theLagrangian diverges, but in fact this is not true since the number of components inthe fields also goes to infinity in this limit. We can write the Feynman diagrams ofthe theory (1.3) in a double line notation, in which an adjoint field !a is representedas a direct product of a fundamental and an anti-fundamental field, !i

j , as in figure1.1. The interaction vertices we wrote are all consistent with this sort of notation. Thepropagators are also consistent with it in a U(N) theory; in an SU(N) theory there isa small mixing term

#!i

j!kl

$# ("i

l"jk $

1

N"ij"

kl ), (1.4)

which makes the expansion slightly more complicated, but this involves only subleadingterms in the large N limit so we will neglect this di#erence here. Ignoring the secondterm the propagator for the adjoint field is (in terms of the index structure) like that of afundamental-anti-fundamental pair. Thus, any Feynman diagram of adjoint fields maybe viewed as a network of double lines. Let us begin by analyzing vacuum diagrams(the generalization to adding external fields is simple and will be discussed below). Insuch a diagram we can view these double lines as forming the edges in a simplicialdecomposition (for example, it could be a triangulation) of a surface, if we view eachsingle-line loop as the perimeter of a face of the simplicial decomposition. The resultingsurface will be oriented since the lines have an orientation (in one direction for afundamental index and in the opposite direction for an anti-fundamental index). Whenwe compactify space by adding a point at infinity, each diagram thus corresponds to acompact, closed, oriented surface.

What is the power of N and ! associated with such a diagram? From the formof (1.3) it is clear that each vertex carries a coe"cient proportional to N/!, whilepropagators are proportional to !/N . Additional powers of N come from the sum overthe indices in the loops, which gives a factor of N for each loop in the diagram (sinceeach index has N possible values). Thus, we find that a diagram with V vertices, E

12

(operational/„simple”)

holography

ZD

¯

�i exp (i

Zd

4xL) with

3/13

holography is thus an interesting th/pheno tool for qualitative insight on otherwise hard-to-calculate ab initio quantities due to (but not only) strong coupling

strongly coupledfermionic systems

real-time physics ofstrongly coupled QCD (lattice, pQCD)

of various analytic and numerical expansions—these are well-established methods which I

will not dwell on here. However, all of these methods fail for certain key questions on the

dynamics at T > 0.

To describe these questions, we need the phase diagram of the model at T > 0 [24], shown

in Fig. 2. There is much interesting physics associated with the many distinct features of this

g

T

gc0

InsulatorSuperfluid

Quantumcritical

TKT

A 2+1 dimensional CFT at T>0

FIG. 2: Phase diagram of the superfluid-insulator transition in two spatial dimensions (D = 3).The quantum critical point is at g = g

c

, T = 0. The dashed lines are crossovers, while the full lineis a phase transition at the Kosterlitz-Thouless temperature T

KT

> 0.

phase diagram, but for now the reader is asked to focus on the distinction between the blue-

and pink-shaded regions. In the blue-shaded regions, the physics can be described in terms

of the familiar excitations of either the insulator or the superfluid, which are illustrated in

Fig. 3. For the insulator, the excitations are particle or hole excitations above the background

of the insulator with one particle per site. In contrast, for the superfluid, the excitations are

point-like vortices in the background of the Bose condensate. A semiclassical theory of gases

of such excitations provides an essentially complete description of the long-time correlations

in the blue-shaded regions.

Let us now turn to the pink-shaded region of Fig. 2, separated from the blue-shaded

regions by crossovers indicated by the dashed lines; the reader is referred to another review

article [20] for a detailed discussion of the location and shape of these crossover lines. The

defining characteristic of the pink-shaded ‘quantum critical’ region is that its dynamics is

controlled by the CFT and its excitations. These excitations do not have a particle-like

interpretation, they are not amenable to an e↵ective classical description, and they interact

6

Applied holography

“Thermalization” puzzle at RHIC and LHC

~ 1

0 fm

hydronized after < 1 fm/c

There are overwhelming evidences that relativistic heavy ion collision programs at RHIC and LHC created strongly coupled quark-gluon plasma (sQGP)

4/13

Successful description of experimental data is based on hydrodynamic simulations of an almost perfect fluid of starting on very early (< 1 fm/c)

Explaining ab initio this quick applicability of hydro is a major puzzle in QCD@HIC.

What can the holography teach us about “thermalization” in similar models?

Heinz [nucl-th/0407067]

�/s = O(1/4⇥)

Modern relativistic (uncharged) hydrodynamics

5/13

an EFT of the slow evolution of conserved currents in collective media close to equilibrium

hydrodynamics is

As any EFT it is based on the idea of the gradient expansion

DOFs: always local energy density and local flow velocity ( )

EOMs: conservation eqns for systematically expanded in gradients

✏ uµ u⌫u⌫ = �1

rµTµ⌫ = 0

Tµ⌫ = ✏uµu⌫ + P (✏){ gµ⌫ + uµu⌫ }� ⌘(✏)�µ⌫ � ⇣(✏){ gµ⌫ + uµu⌫ }(r · u) + . . .

Tµ⌫

gravity reminded us that all terms allowed by symmetries can enter

perfect fluid stress tensor

(famous) shear viscosity bulk viscosity(vanishes for CFTs)

microscopicinput: EoS

What did we learn from the fluid-gravity duality?We were reminded that gradient expansion needs to be done systematically, e.g.

It opened new perspective to view both the phenomena in fluids and in gravity

20

0 1 2 3 4 5pT [GeV]

0

0.1

0.2

0.3

0.4

0.5v 2

η/s=10-4

η/s=0.08 standardη/s=0.08 Padeη/s=0.16 standardη/s=0.16 Pade

FIG. 6: (Color online) Charged hadron elliptic flow for the Glauber model at b = 7 fm with

Ti = 0.353 GeV, !0 = 1 fm/c and various viscosities.

Since Eq. (47) contains powers of momenta to all orders when re-expanded, the di!erencebetween the ansatz (19) and the Pade resummed particle spectra can give a handle on thesystematic error of the truncation used in Eq. (19). Shown in Fig. 6, this di!erence suggeststhat this systematic error is small for momenta pT

<! 2.5 GeV. Therefore, we do not expectour results to have a large systematic uncertainty coming from the particular ansatz (19)for these momenta.

To summarize, for values of !/s <! 0.2, the results for the momentum anisotropy areessentially insensitive to the choices for the second-order transport coe"cients "!, #1 andthe initialization of the shear tensor #µ!(" = "0). Conversely, ep is sensitive to the valueof viscosity and the choice of initial energy density profile (initial eccentricity). Since thephysical initial condition is currently unknown, this dependence will turn out to be thedominant systematic uncertainty in determining !/s from experimental data.

B. Multiplicity and radial flow

As outlined in the introduction, we want to match the hydrodynamic model to experimen-tal data for the multiplicity, thereby fixing the constant in Eqs. (38),(43). This translatesto fixing an initial central temperature Ti for b = 0, which we will quote in the following.

For a constant speed of sound, the evolution for ideal hydrodynamics is isentropic, whilefor viscous hydrodynamics additional entropy is produced. Since the multiplicity is a mea-sure of the entropy of the system, one expects an increase of multiplicity for viscous comparedto ideal hydrodynamic evolution. This increase in final multiplicity has been measured asa function of !/s for the semi-realistic speed of sound Fig. 1 in central heavy-ion collisionsin Ref. [21], and found to be approximately6 a factor of 0.75!/s. (See Ref. [53, 83] for re-lated calculations in simplified models.) Reducing Ti accordingly therefore ensures that for

6 The quoted fraction is for a hydrodynamic starting time of !0 = 1 fm/c. Reducing !0 leads to considerably

larger entropy production.

Luzum & Romatschke0804.4015 [nucl-th]

⌘/s = 1/4⇡ ⇡ 0.08

we also know nowthat is not anyfundamental bound

1/4⇡

Buchel, Myers & Sinha 0812.2521 [hep-th]

We learned something about transport coefficients at strong coupling and we managed to transfer this knowledge to the heavy ion-community (big success!):

It should be clear, however, as emphasized, e.g., by Geroch [45, 46] and others [47] that

the modes which defy causality are those which are not supposed to be described by hy-

drodynamics (i.e., microscopically short wavelengths, which is clear when one thinks about

discontinuities). Nevertheless, for numerical simulations of relativistic hydrodynamic systems

such superluminal propagation is a nuisance because in such simulations one extrapolates

hydrodynamic equations to the microscopic scale, even though the modes, or the configura-

tions, which are being studied are hydrodynamic. For example, superluminal propagation

makes posing initial value problem di!cult: even if the initial hypersurface is space-like, the

initial values at di"erent points can influence each other and an attempt to specify them

independently leads to unacceptable singular solutions [48, 47].

Since the problem lies in the domain where the theory is not applicable, one can safely

modify the theory in this domain, without disturbing physical predictions. This is the essence

of the solution which Muller and Israel proposed by extending the set of variables. The

resulting system of equations is hyperbolic. Here we shall write down explicitly the system

of equations of Israel and Stewart, restricting to the case of conformally invariant system

without a conserved charge that we study in this paper.

6.2 Hydrodynamic variables and second order hydrodynamics

As we have already emphasized in Section 2.2 the hydrodynamics should be viewed as a

controllable expansion in gradients of the hydrodynamic variables. The choice of the variables,

or fields, can be aided by applying the requirement that a linearized system of equations has

solutions whose frequency vanishes in the hydrodynamic limit, i.e., when the wave vector k

vanishes. We call such linearized modes the hydrodynamic modes. Fluctuations of conserved

densities are automatically hydrodynamic because their equations are conservation laws and

constant fields (! = 0, k = 0) are trivial solutions of them.

Hence, for a system without conserved charges the set of hydrodynamic variables consists

of the densitites of energy and momentum, represented by 4 independent covariant variables

" and uµ (u·u = !1). All other quantities in hydrodynamic description are instantaneous

functions of these variables and their derivatives, such as, e.g., #µ! (Section 2.2).

How should one extend 1st order hydrodynamics to higher derivatives? The systematic

way, as we argued in Section 2.2 and 3, is to continue the expansion (2.16) and add all possible

terms of the second order in derivatives, as we did in Eq. (3.11).

Instead, Muller, Israel and Stewart take a more phenomenological point of view. They

consider #µ! – the viscous part of the the momentum flow – as a set of independent additional

variables. The equations for these variables are not given by any exact conservation laws,

but by phenomenological expansions in the set of independent variables, which now includes

also #µ! :

#!D#µ! = !#µ! ! $%µ! . (6.1)

The first term in Eq. (6.1) has a simple intuitive meaning: in the absence of velocity gradients

(%µ! = 0) the viscous momentum flows #µ! do not vanish instanteneously (as in Eq. (2.16)),

– 22 –

in accordance with the simple rule (2.14).

By direct computation we find that

!µ! ! e3"!µ! , (2.21)

i.e. !µ! transforms homogeneously with conformal weight 3 independent of d (in agreement

with (2.14)). For conformal fluids " = const · T d!1, and therefore T µ! transforms homoge-

neously under Weyl transformation as in Eq. (2.12).

3. Second-order hydrodynamics of a conformal fluid

In this Section we shall continue the derivative expansion (2.16). We shall write down all

possible second-order terms in the stress-energy tensor allowed by Weyl invariance. Then we

shall compute the coe!cients in front of these terms in the N = 4 SYM plasma by matching

hydrodynamic correlation functions with gravity calculations in Section 4.

3.1 Second-order terms

Rewriting Eq. (2.15) we introduce the dissipative part of the stress-energy tensor, "µ! :

T µ! = #uµu! + P#µ! + "µ! , (3.1)

which contains only the derivatives and vanishes in a homogeneous equilibrium state. The

tensor "µ! is symmetric and transverse, uµ"µ! = 0. For conformal fluids it must be also

traceless gµ!"µ! = 0. To first order

"µ! = ""!µ! + (2nd order terms), (3.2)

where !µ! is defined in Eq. (2.18). We will also use the notation for the vorticity

$µ! =1

2#µ##!$(##u$ "#$u#) . (3.3)

We note that in writing down second-order terms in "µ! , one can always rewrite the

derivatives along the d-velocity direction

D $ uµ#µ (3.4)

(temporal derivative in the local rest frame) in terms of transverse (spatial in the local rest

frame) derivatives through the zeroth-order equations of motion:

D ln T = " 1

d " 1(#" · u), Duµ = "#µ

" ln T, #µ" $ #µ### . (3.5)

Notice also that #" · u = # · u.

– 7 –

+ pheno EOM

Israel & Stewart, 1977-1979Baier et al.0712.2451 [hep-th]

Bhattacharyya et al.0712.2456 [hep-th]

With the restriction of transversality and tracelessness, there are eight possible contribu-

tions to the stress-energy tensor:

!!µ ln T !!" ln T, !!µ!!" ln T, !µ!(!·u), !!µ"!!""

!!µ"!!"", !!µ

"!!"", u#R#!µ!"$u$, R!µ!" .(3.6)

By direct computations we find that there are only five combinations that transform

homogeneously under Weyl tranformations. They are

Oµ!1 = R!µ!" " (d " 2)

!

!!µ!!" ln T "!!µ ln T !!" ln T"

, (3.7)

Oµ!2 = R!µ!" " (d " 2)u#R#!µ!"$u$ , (3.8)

Oµ!3 = !!µ

"!!"" , Oµ!4 = !!µ

"!!"" , Oµ!5 = !!µ

"!!"" . (3.9)

In the linearized hydrodynamics in flat space only the term Oµ!1 contributes. For conve-

nience and to facilitate the comparision with the Israel-Stewart theory we shall use instead

of (3.7) the term!D!µ! " +

1

d " 1!µ!(!·u) (3.10)

which, with (3.5), reduces to the linear combination: Oµ!1 " Oµ!

2 " (1/2)Oµ!3 " 2Oµ!

5 . It is

straightforward to check directly that (3.10) transforms homogeneously under Weyl transfor-

mations.

Thus, our final expression for the dissipative part of the stress-energy tensor, up to second

order in derivatives, is

"µ! = ""!µ!

+ "#!

#

!D!µ! " +1

d " 1!µ!(!·u)

$

+ $%

R!µ!" " (d " 2)u#R#!µ!"$u$

&

+ %1!!µ

"!!"" + %2!!µ

"!!"" + %3!!µ

"!!"" .

(3.11)

The five new constants are #!, $, %1,2,3. Note that using lowest order relations "µ! = ""!µ! ,

Eqs.(3.5) and D" = ""!·u, Eq. (3.11) may be rewritten in the form

"µ! = ""!µ! " #!

#

!D"µ! " +d

d " 1"µ!(!·u)

$

+ $%

R!µ!" " (d " 2)u#R#!µ!"$u$

&

+%1

"2"!µ

""!"" " %2

""!µ

"!!"" + %3!!µ

"!!"" .

(3.12)

This equation is, in form, similar to an equation of the Israel-Stewart theory (see Section 6).

In the linear regime it actually coincides with the Israel-Stewart theory (6.1). We emphasize,

however, that one cannot claim that Eq. (3.12) captures all orders in the momentum expansion

(see Section 6).

– 8 –

6/13

Holography, QNMs and hydrodynamics

7/13

Kovtun & Starinets [hep-th/0506184]

Tµ⌫ =18⇡2N2

c T 4 diag (3, 1, 1, 1)µ⌫ +�Tµ⌫

(⇠ e�i!(k) t+i

~

k·~x)

Consider small amplitude perturbations ( ) on top of a holographic plasma�Tµ⌫/Nc2 ⌧ T 4

Due to (and ?) the temperature T is the only microscopic scale� = g2YMNc ! 1

Dissipation leads to modes with complex , which in the sound channel look like!(k)

0.5 1 1.5 2

0.5

1

1.5

2

2.5

3 Re 0.5 1 1.5 2

-3

-2.5

-2

-1.5

-1

-0.5

Im

Figure 6: Real and imaginary parts of three lowest quasinormal frequencies as function of spatialmomentum. The curves for which !0 as !0 correspond to hydrodynamic sound mode in the dualfinite temperature N=4 SYM theory.

behavior of the lowest (hydrodynamic) frequency which is absent for E! and Z3. For Ez and

Z1, hydrodynamic frequencies are purely imaginary (given by Eqs. (4.16) and (4.32) for small

! and q), and presumably move o! to infinity as q becomes large. For Z2, the hydrodynamic

frequency has both real and imaginary parts (given by Eq. (4.44) for small ! and q), and

eventually (for large q) becomes indistinguishable in the tower of other eigenfrequencies. As an

example, dispersion relations for the three lowest quasinormal frequencies in the sound channel

(including the one of the sound wave) are shown in Fig. 6. The tables below give numerical

values of quasinormal frequencies for = 1. Only non-hydrodynamic frequencies are shown

in the tables. The position of hydrodynamic frequencies at = 1 is = "3.250637i for the

R-charge di!usive mode, = "0.598066i for the shear mode, and = ±0.741420"0.286280i

for the sound mode. The numerical values of the lowest five (non-hydrodynamic) quasinormal

frequencies for electromagnetic perturbations are:

Transverse channel Di!usive channel

n Re Im Re Im

1 ±1.547187 "0.849723 ±1.147831 "0.559204

2 ±2.398903 "1.874343 ±1.910006 "1.758065

3 ±3.323229 "2.894901 ±2.903293 "2.891681

4 ±4.276431 "3.909583 ±3.928555 "3.943386

5 ±5.244062 "4.920336 ±4.946818 "4.965186

and for gravitational perturbations are:

Scalar channel Shear channel Sound channel

n Re Im Re Im Re Im

1 ±1.954331 "1.267327 ±1.759116 "1.291594 ±1.733511 "1.343008

2 ±2.880263 "2.297957 ±2.733081 "2.330405 ±2.705540 "2.357062

3 ±3.836632 "3.314907 ±3.715933 "3.345343 ±3.689392 "3.363863

4 ±4.807392 "4.325871 ±4.703643 "4.353487 ±4.678736 "4.367981

5 ±5.786182 "5.333622 ±5.694472 "5.358205 ±5.671091 "5.370784

– 26 –

Im!/2⇡T

Re!/2⇡T

k/2⇡T

k/2⇡T

1st

2nd

3rd

1st

2nd

3rd

!(k) ! 0 as : slowly evolving and dissipating modes (hydrodynamic sound waves)k ! 0

all the rest: far from equilibrium (QNM) modes dampened over

@!

@k

���k!0

= csound

ttherm = O(1)/T

This is also the meaning in which is fast: 0.5 fm/c x 350 MeV = T ttherm = 0.63 !!!tRHIChydro

Nc ! 1

Fantastic toy-model

8/13

x

0

x

1

x

1

The simplest, yet phenomenologically interesting field theory dy-namics is the boost-invariant flow with no transverse expansion.

= =

relevant for centralrapidity region

no elliptic flow(~ central collision)

In Bjorken scenario dynamics depends only on proper time

[Bjorken 1982]

pre-equilibrium stageQGPmixed phasehadronic gas

describedby hydrodynamics

Figure 1: Description of QGP formation in heavy ion collisions. The kinematic landscape isdefined by ! =

!

(x0)2 ! (x3)2 ; y = 12 log x

0+x3

x0!x3 ; x"={x1, x2} , where the coordinates along thelight-cone are x0 ± x1, the transverse ones are {x1, x2} and ! is the proper time, y the “space-timerapidity”.

[3]. The hydrodynamic regime has to last long enough and start soon enough after the

collision in order to explain the observed collective e!ects. Moreover, the smallness of the

viscosity which can be extracted from hydrodynamical simulations describing the data leads

to an almost-perfect fluid behaviour of the QGP, and thus to a short mean-free path inside

the fluid. Putting together these experimental inputs, and in order to go beyond a mererly

phenomenological description, it appears to be theoretically necessary to investigate as

much as possible the properties of a strongly-coupled Quantum-Chromodynamic plasma.

In the absence of nonperturbative methods applicable to real-time dynamics of strongly

coupled Quantum Chromodynamic (QCD) plasma, one is led to consider similar problems

from the point-of-view of the AdS/CFT correspondence, that is looking for the charac-

teristics of plasma in a gauge theory for which the AdS/CFT correspondence takes its

simplest form – the N = 4 supersymmetric Yang-Mills theory [4] which posseses a known

and tractable gravity dual.

Although the N = 4 gauge theory is supersymmetric and conformal and thus quite

di!erent from QCD at zero temperature, both supersymmetry and scale-invariance are

broken explicitly at finite temperature and we may expect qualitative similarities with

QCD plasma for a range of temperatures above the QCD deconfinement phase transition1.

Indeed, the gauge/gravity dual calculation [5] showing, in a static setting, that the

viscosity over entropy ratio "/s is very small (equal to 1/4#) and even suggesting a universal

lower bound, is in qualitative agreement with hydrodynamic simulations of QCD plasma

and was a poweful incentive to explore further the AdS/CFT duality approach.

In order to go beyond static calculations, one has to adapt the dual AdS/CFT approach

to the relativistic kinematic framework of heavy-ion reactions, where two ultra-relativistic

heavy nuclei collide and form an expanding medium, see Fig.1. It is convenient, initially,

1There exist more refined versions of the AdS/CFT correspondence which may have more features in

common with QCD, however the gravity backgrounds are much more complicated and we will not consider

them here.

– 2 –

described by AdS/CFT in this scenario

and stress tensor (in conformal case) is entirely expressed in terms of energy density

with

⌧ = 0

We are interested both in setting strongly coupled non-equilibrium initial states at and tracking their relaxation towards hydro and in hydro phase as well [sic]⌧ = 0

� =q

(x0)2 � (x1)2

and pT (⇥) = �(⇥) +1

2⇥�0(⇥)pL(⇥) = ��(⇥)� ⇥�0(⇥)

hTµ⌫i = diag{�✏(⌧), pL(⌧), pT (⌧), pT (⌧)}

ds

2 = �d⌧

2 + ⌧

2dy

2 + dx

21 + dx

22

(Fast) hydrodynamization

9/13

large anisotropyat the onset ofhydrodynamics!

1st, 2nd and 3rd order hydro

similar findings in Chesler & Yaffe 0906.4426 and 1011.3562

The single most interesting result was that

hydrodynamization thermalization:

Pressure anisotropy is observed to be between

with hydrodynamics already being a valid description of the stress tensor dynamics.

MPH, R. A. Janik & P. Witaszczyk1103.3452 [hep-th] PRL 108 (2012) 201602:

6=

✏� 3 pL ⇡ 0.6 ✏ to 1.0 ✏

hTµ⌫i = diag{�✏(⌧), pL(⌧), pT (⌧), pT (⌧)}

⇠ ⌧ ⇥ Teff (⌧)

General stress tensor here has 3 different components

Hydro constitutive relations relate them to each other via gradient expansion

Obviously we know the hydro form of the stress tensor, but do not know when it applies

For this we need to know how non-hydro DOFs relax. We can investigate it numerically!

RHICfast!

Hydrodynamic series at high orders

10/13

MPH, R. A. Janik & P. Witaszczyk1302.0697 [hep-th] PRL 110 (2013) 211602:

So far nothing has been known about the character of hydrodynamic expansion

Idea: take a simple flow (here the boost-invariant flow) and using the fluid-gravity duality generate the on-shell form of its hydrodynamic stress tensor at high orders

50 100 150 200 n

2

4

6

8

10

12

»en 1ên

at large ordersfactorial growth of gradient contributions with order

T 00 = ✏(⌧) ⇠1X

n=2

✏n(⌧�2/3)n (T�1rµu

⌫ ⇠ ⌧�2/3)

First evidence that hydrodynamic expansion has zero radius of convergence!

at low ordersbehavior is different

11/13

Famous examples of asymptotic expansions arise in pQFTsMPH, R. A. Janik & P. Witaszczyk

There, the number of Feynman graphs grows ~order! at large orders*

We suspect analogous mechanism might work also in the case of hydro series*

Tµ⌫ = ✏uµu⌫ + P (✏){ gµ⌫ + uµu⌫ }� ⌘(✏)�µ⌫ � ⇣(✏){ gµ⌫ + uµu⌫ }(r · u) + . . .

With the restriction of transversality and tracelessness, there are eight possible contribu-

tions to the stress-energy tensor:

!!µ ln T !!" ln T, !!µ!!" ln T, !µ!(!·u), !!µ"!!""

!!µ"!!"", !!µ

"!!"", u#R#!µ!"$u$, R!µ!" .(3.6)

By direct computations we find that there are only five combinations that transform

homogeneously under Weyl tranformations. They are

Oµ!1 = R!µ!" " (d " 2)

!

!!µ!!" ln T "!!µ ln T !!" ln T"

, (3.7)

Oµ!2 = R!µ!" " (d " 2)u#R#!µ!"$u$ , (3.8)

Oµ!3 = !!µ

"!!"" , Oµ!4 = !!µ

"!!"" , Oµ!5 = !!µ

"!!"" . (3.9)

In the linearized hydrodynamics in flat space only the term Oµ!1 contributes. For conve-

nience and to facilitate the comparision with the Israel-Stewart theory we shall use instead

of (3.7) the term!D!µ! " +

1

d " 1!µ!(!·u) (3.10)

which, with (3.5), reduces to the linear combination: Oµ!1 " Oµ!

2 " (1/2)Oµ!3 " 2Oµ!

5 . It is

straightforward to check directly that (3.10) transforms homogeneously under Weyl transfor-

mations.

Thus, our final expression for the dissipative part of the stress-energy tensor, up to second

order in derivatives, is

"µ! = ""!µ!

+ "#!

#

!D!µ! " +1

d " 1!µ!(!·u)

$

+ $%

R!µ!" " (d " 2)u#R#!µ!"$u$

&

+ %1!!µ

"!!"" + %2!!µ

"!!"" + %3!!µ

"!!"" .

(3.11)

The five new constants are #!, $, %1,2,3. Note that using lowest order relations "µ! = ""!µ! ,

Eqs.(3.5) and D" = ""!·u, Eq. (3.11) may be rewritten in the form

"µ! = ""!µ! " #!

#

!D"µ! " +d

d " 1"µ!(!·u)

$

+ $%

R!µ!" " (d " 2)u#R#!µ!"$u$

&

+%1

"2"!µ

""!"" " %2

""!µ

"!!"" + %3!!µ

"!!"" .

(3.12)

This equation is, in form, similar to an equation of the Israel-Stewart theory (see Section 6).

In the linear regime it actually coincides with the Israel-Stewart theory (6.1). We emphasize,

however, that one cannot claim that Eq. (3.12) captures all orders in the momentum expansion

(see Section 6).

– 8 –

With the restriction of transversality and tracelessness, there are eight possible contribu-

tions to the stress-energy tensor:

!!µ ln T !!" ln T, !!µ!!" ln T, !µ!(!·u), !!µ"!!""

!!µ"!!"", !!µ

"!!"", u#R#!µ!"$u$, R!µ!" .(3.6)

By direct computations we find that there are only five combinations that transform

homogeneously under Weyl tranformations. They are

Oµ!1 = R!µ!" " (d " 2)

!

!!µ!!" ln T "!!µ ln T !!" ln T"

, (3.7)

Oµ!2 = R!µ!" " (d " 2)u#R#!µ!"$u$ , (3.8)

Oµ!3 = !!µ

"!!"" , Oµ!4 = !!µ

"!!"" , Oµ!5 = !!µ

"!!"" . (3.9)

In the linearized hydrodynamics in flat space only the term Oµ!1 contributes. For conve-

nience and to facilitate the comparision with the Israel-Stewart theory we shall use instead

of (3.7) the term!D!µ! " +

1

d " 1!µ!(!·u) (3.10)

which, with (3.5), reduces to the linear combination: Oµ!1 " Oµ!

2 " (1/2)Oµ!3 " 2Oµ!

5 . It is

straightforward to check directly that (3.10) transforms homogeneously under Weyl transfor-

mations.

Thus, our final expression for the dissipative part of the stress-energy tensor, up to second

order in derivatives, is

"µ! = ""!µ!

+ "#!

#

!D!µ! " +1

d " 1!µ!(!·u)

$

+ $%

R!µ!" " (d " 2)u#R#!µ!"$u$

&

+ %1!!µ

"!!"" + %2!!µ

"!!"" + %3!!µ

"!!"" .

(3.11)

The five new constants are #!, $, %1,2,3. Note that using lowest order relations "µ! = ""!µ! ,

Eqs.(3.5) and D" = ""!·u, Eq. (3.11) may be rewritten in the form

"µ! = ""!µ! " #!

#

!D"µ! " +d

d " 1"µ!(!·u)

$

+ $%

R!µ!" " (d " 2)u#R#!µ!"$u$

&

+%1

"2"!µ

""!"" " %2

""!µ

"!!"" + %3!!µ

"!!"" .

(3.12)

This equation is, in form, similar to an equation of the Israel-Stewart theory (see Section 6).

In the linear regime it actually coincides with the Israel-Stewart theory (6.1). We emphasize,

however, that one cannot claim that Eq. (3.12) captures all orders in the momentum expansion

(see Section 6).

– 8 –

With the restriction of transversality and tracelessness, there are eight possible contribu-

tions to the stress-energy tensor:

!!µ ln T !!" ln T, !!µ!!" ln T, !µ!(!·u), !!µ"!!""

!!µ"!!"", !!µ

"!!"", u#R#!µ!"$u$, R!µ!" .(3.6)

By direct computations we find that there are only five combinations that transform

homogeneously under Weyl tranformations. They are

Oµ!1 = R!µ!" " (d " 2)

!

!!µ!!" ln T "!!µ ln T !!" ln T"

, (3.7)

Oµ!2 = R!µ!" " (d " 2)u#R#!µ!"$u$ , (3.8)

Oµ!3 = !!µ

"!!"" , Oµ!4 = !!µ

"!!"" , Oµ!5 = !!µ

"!!"" . (3.9)

In the linearized hydrodynamics in flat space only the term Oµ!1 contributes. For conve-

nience and to facilitate the comparision with the Israel-Stewart theory we shall use instead

of (3.7) the term!D!µ! " +

1

d " 1!µ!(!·u) (3.10)

which, with (3.5), reduces to the linear combination: Oµ!1 " Oµ!

2 " (1/2)Oµ!3 " 2Oµ!

5 . It is

straightforward to check directly that (3.10) transforms homogeneously under Weyl transfor-

mations.

Thus, our final expression for the dissipative part of the stress-energy tensor, up to second

order in derivatives, is

"µ! = ""!µ!

+ "#!

#

!D!µ! " +1

d " 1!µ!(!·u)

$

+ $%

R!µ!" " (d " 2)u#R#!µ!"$u$

&

+ %1!!µ

"!!"" + %2!!µ

"!!"" + %3!!µ

"!!"" .

(3.11)

The five new constants are #!, $, %1,2,3. Note that using lowest order relations "µ! = ""!µ! ,

Eqs.(3.5) and D" = ""!·u, Eq. (3.11) may be rewritten in the form

"µ! = ""!µ! " #!

#

!D"µ! " +d

d " 1"µ!(!·u)

$

+ $%

R!µ!" " (d " 2)u#R#!µ!"$u$

&

+%1

"2"!µ

""!"" " %2

""!µ

"!!"" + %3!!µ

"!!"" .

(3.12)

This equation is, in form, similar to an equation of the Israel-Stewart theory (see Section 6).

In the linear regime it actually coincides with the Israel-Stewart theory (6.1). We emphasize,

however, that one cannot claim that Eq. (3.12) captures all orders in the momentum expansion

(see Section 6).

– 8 –

With the restriction of transversality and tracelessness, there are eight possible contribu-

tions to the stress-energy tensor:

!!µ ln T !!" ln T, !!µ!!" ln T, !µ!(!·u), !!µ"!!""

!!µ"!!"", !!µ

"!!"", u#R#!µ!"$u$, R!µ!" .(3.6)

By direct computations we find that there are only five combinations that transform

homogeneously under Weyl tranformations. They are

Oµ!1 = R!µ!" " (d " 2)

!

!!µ!!" ln T "!!µ ln T !!" ln T"

, (3.7)

Oµ!2 = R!µ!" " (d " 2)u#R#!µ!"$u$ , (3.8)

Oµ!3 = !!µ

"!!"" , Oµ!4 = !!µ

"!!"" , Oµ!5 = !!µ

"!!"" . (3.9)

In the linearized hydrodynamics in flat space only the term Oµ!1 contributes. For conve-

nience and to facilitate the comparision with the Israel-Stewart theory we shall use instead

of (3.7) the term!D!µ! " +

1

d " 1!µ!(!·u) (3.10)

which, with (3.5), reduces to the linear combination: Oµ!1 " Oµ!

2 " (1/2)Oµ!3 " 2Oµ!

5 . It is

straightforward to check directly that (3.10) transforms homogeneously under Weyl transfor-

mations.

Thus, our final expression for the dissipative part of the stress-energy tensor, up to second

order in derivatives, is

"µ! = ""!µ!

+ "#!

#

!D!µ! " +1

d " 1!µ!(!·u)

$

+ $%

R!µ!" " (d " 2)u#R#!µ!"$u$

&

+ %1!!µ

"!!"" + %2!!µ

"!!"" + %3!!µ

"!!"" .

(3.11)

The five new constants are #!, $, %1,2,3. Note that using lowest order relations "µ! = ""!µ! ,

Eqs.(3.5) and D" = ""!·u, Eq. (3.11) may be rewritten in the form

"µ! = ""!µ! " #!

#

!D"µ! " +d

d " 1"µ!(!·u)

$

+ $%

R!µ!" " (d " 2)u#R#!µ!"$u$

&

+%1

"2"!µ

""!"" " %2

""!µ

"!!"" + %3!!µ

"!!"" .

(3.12)

This equation is, in form, similar to an equation of the Israel-Stewart theory (see Section 6).

In the linear regime it actually coincides with the Israel-Stewart theory (6.1). We emphasize,

however, that one cannot claim that Eq. (3.12) captures all orders in the momentum expansion

(see Section 6).

– 8 –

+ . . .

1st order hydro (1 transport coeff)

2nd order hydro (5 transport coeffs)

...

Why hydro series might be asymptotic?

+ . . .+

1302.0697 [hep-th] PRL 110 (2013) 211602:

12/13

A standard method for asymptotic series is Borel transform and Borel summationMPH, R. A. Janik & P. Witaszczyk

reveals singularities leading to 0 radius of convergence

What controls the fast growth of hydroS coeffs?

Closer inspection reveals that the closest one to 0 is the lowest non-hydro QNM!0.5 1 1.5 2

0.5

1

1.5

2

2.5

3 Re 0.5 1 1.5 2

-3

-2.5

-2

-1.5

-1

-0.5

Im

Figure 6: Real and imaginary parts of three lowest quasinormal frequencies as function of spatialmomentum. The curves for which !0 as !0 correspond to hydrodynamic sound mode in the dualfinite temperature N=4 SYM theory.

behavior of the lowest (hydrodynamic) frequency which is absent for E! and Z3. For Ez and

Z1, hydrodynamic frequencies are purely imaginary (given by Eqs. (4.16) and (4.32) for small

! and q), and presumably move o! to infinity as q becomes large. For Z2, the hydrodynamic

frequency has both real and imaginary parts (given by Eq. (4.44) for small ! and q), and

eventually (for large q) becomes indistinguishable in the tower of other eigenfrequencies. As an

example, dispersion relations for the three lowest quasinormal frequencies in the sound channel

(including the one of the sound wave) are shown in Fig. 6. The tables below give numerical

values of quasinormal frequencies for = 1. Only non-hydrodynamic frequencies are shown

in the tables. The position of hydrodynamic frequencies at = 1 is = "3.250637i for the

R-charge di!usive mode, = "0.598066i for the shear mode, and = ±0.741420"0.286280i

for the sound mode. The numerical values of the lowest five (non-hydrodynamic) quasinormal

frequencies for electromagnetic perturbations are:

Transverse channel Di!usive channel

n Re Im Re Im

1 ±1.547187 "0.849723 ±1.147831 "0.559204

2 ±2.398903 "1.874343 ±1.910006 "1.758065

3 ±3.323229 "2.894901 ±2.903293 "2.891681

4 ±4.276431 "3.909583 ±3.928555 "3.943386

5 ±5.244062 "4.920336 ±4.946818 "4.965186

and for gravitational perturbations are:

Scalar channel Shear channel Sound channel

n Re Im Re Im Re Im

1 ±1.954331 "1.267327 ±1.759116 "1.291594 ±1.733511 "1.343008

2 ±2.880263 "2.297957 ±2.733081 "2.330405 ±2.705540 "2.357062

3 ±3.836632 "3.314907 ±3.715933 "3.345343 ±3.689392 "3.363863

4 ±4.807392 "4.325871 ±4.703643 "4.353487 ±4.678736 "4.367981

5 ±5.786182 "5.333622 ±5.694472 "5.358205 ±5.671091 "5.370784

– 26 –

Im!/2⇡T

Re!/2⇡T

k/2⇡T

k/2⇡T

1st

2nd 1st

2nd

3rd

0.5 1 1.5 2

0.5

1

1.5

2

2.5

3 Re 0.5 1 1.5 2

-3

-2.5

-2

-1.5

-1

-0.5

Im

Figure 6: Real and imaginary parts of three lowest quasinormal frequencies as function of spatialmomentum. The curves for which !0 as !0 correspond to hydrodynamic sound mode in the dualfinite temperature N=4 SYM theory.

behavior of the lowest (hydrodynamic) frequency which is absent for E! and Z3. For Ez and

Z1, hydrodynamic frequencies are purely imaginary (given by Eqs. (4.16) and (4.32) for small

! and q), and presumably move o! to infinity as q becomes large. For Z2, the hydrodynamic

frequency has both real and imaginary parts (given by Eq. (4.44) for small ! and q), and

eventually (for large q) becomes indistinguishable in the tower of other eigenfrequencies. As an

example, dispersion relations for the three lowest quasinormal frequencies in the sound channel

(including the one of the sound wave) are shown in Fig. 6. The tables below give numerical

values of quasinormal frequencies for = 1. Only non-hydrodynamic frequencies are shown

in the tables. The position of hydrodynamic frequencies at = 1 is = "3.250637i for the

R-charge di!usive mode, = "0.598066i for the shear mode, and = ±0.741420"0.286280i

for the sound mode. The numerical values of the lowest five (non-hydrodynamic) quasinormal

frequencies for electromagnetic perturbations are:

Transverse channel Di!usive channel

n Re Im Re Im

1 ±1.547187 "0.849723 ±1.147831 "0.559204

2 ±2.398903 "1.874343 ±1.910006 "1.758065

3 ±3.323229 "2.894901 ±2.903293 "2.891681

4 ±4.276431 "3.909583 ±3.928555 "3.943386

5 ±5.244062 "4.920336 ±4.946818 "4.965186

and for gravitational perturbations are:

Scalar channel Shear channel Sound channel

n Re Im Re Im Re Im

1 ±1.954331 "1.267327 ±1.759116 "1.291594 ±1.733511 "1.343008

2 ±2.880263 "2.297957 ±2.733081 "2.330405 ±2.705540 "2.357062

3 ±3.836632 "3.314907 ±3.715933 "3.345343 ±3.689392 "3.363863

4 ±4.807392 "4.325871 ±4.703643 "4.353487 ±4.678736 "4.367981

5 ±5.786182 "5.333622 ±5.694472 "5.358205 ±5.671091 "5.370784

– 26 –

Im!/2⇡T

Re!/2⇡T

k/2⇡T

1st

2nd

3rd

-5 5 10 15 20Re ué0

-20

-10

10

20Im ué0

B✏(u)

✏(u) ⇠1X

n=2

✏nun

(u = ⌧�2/3), B✏(u) ⇠

1X

n=2

1

n!✏nu

n, Borel sum : ✏Bs(u) =

Z 1

0

1

uB✏(t) exp (�t/u)dt

✏Bs(

u)=

Z 1

01

uB✏(t)e

x

p

(

�t/u)

dt

✏Bs(u) =

Z 1

0

1

uB✏(t) exp (�t/u)dt

�✏ ⇠ e�i#u

1302.0697 [hep-th] PRL 110 (2013) 211602:

Summary

Open directions

13/13

§ Do anisotropies in hydrodynamic regime leave an observational imprint?

§ Is resummed hydrodynamics phenomenologically relevant?

§ Strong coupling naturally leads to quick applicability of hydrodynamics (RHIC?).

§ At the moment of hydrodynamization, the stress tensor can be very anisotropic. Thus superficially it needs to be distinguished from isotropization/thermalization!

§ Model studies strongly suggest that hydrodynamics is an asymptotic series! Large order behavior knows about the lowest far-from-equilibrium DOF.

§ Holography allows to do fantastic ab initio calculations!

§ Towards „holographic heavy ion collisions” ( Tuesday )