11

Department of Physics

HIC from AdS/CFT

Anastasios Taliotis

Work done in collaboration with Javier Albacete and Yuri Kovchegov, arXiv:0805.2927 [hep-th],

arXiv:0902.3046 [hep-th], arXiv:0705.1234 [hep-ph],arXiv:1004.3500 [hep-ph]

(published in JHEP and Phys. Rev. C)

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OutlineMotivating strongly coupled dynamics

Introduction to AdS/CFT

I. AA: State/set up the problem

Attacking the problem using AdS/CFT

Predictions/comparisons/conclusions/Summary

II. pA: State/set up the problem

Predictions/Conclusions

III. Transverse Dynamics-a quick look

33

Motivating strongly coupled dynamics in HIC

44

Notation/FactsProper time:

Rapidity:

Saturation scale : The scale where density of partons becomes high.

23

20 xx

12ln

x0 x3x0 x3

1

2ln

xx

0x

3x

QGP

CGCCGC describes matter distribution due to classical gluon fields and is rapidity-independent ( g<<1, early times).

Hydro is a necessary condition for thermalization. Bjorken Hydro describes successfully particle spectra and spectral flow. Is g??>>1 at late times?? Maybe; consistent with the small MFP implied by a hydro description.

No unified framework exists that describes both strongly & weakly coupled dynamics

valid for times t >> 1/Qs

Bj Hydro

g<<1; valid up to times ~ 1/QS.

sQ

JFD

55

Goal: Stress-Energy (SE) Tensor

• SE of the produced medium gives useful information.

• In particular, its form (as a function of space and time variables) allows to decide whether we could have thermalization i.e. it provides useful criteria for the (possible) formation of QGP.

• SE tensor will be the main object of this talk: we will see how it can be calculated by non perturbative methods in HIC.

66

Introduction to AdS/CFT

77

Type IIB superstring N =4 SYM SU(Nc)

Q. gravity & fields Q. strings

Clas. fields & part. Clas. Strings

1/ cN

=> (Ignore QM / small ) => Large Nc

=> (Ignore extended objects/small ) => Large λ

1/sl

L

5L Radius of S

4 4/ 's s cL l t Hooft g N

2YM sg g

(10)pl

L

4 4 (10)(10)/ ~ 1/p cL l N G

(10)G

Scales & ParametersScales & Parameters

lp(10) /L 1

ls / L 1 sg

88

Quantifying the Conjecture

<exp z=0∫O φ0>CFT = Zs(φ|φ(z=0)= φo)

O is the CFT operator. Typically want <O1 O2…On>

φ0 =φ0 (x1,x2,… ,xd) is the source of O in the CFT picture

φ =φ (x1,x2,… ,xd ,z) is some field in string theory with B.C.

φ (z=0)= φ0

[Witten ‘98]

99

How to use the correspondence

• Take functional derivatives on both sides. LHS gives correlation functions. RHS is the machine that computes them (at any value of coupling!!).

• Must write fields φ (that act as source in the CFT) as a convolution with a boundary to bulk propagator:

φ (xμ,z)= ∫dxν' φ0 (xμ’)Δ(xμ – xμ’,z)

1010

• φ (xμ,z) being a field of string theory must obey some equation of motion; say □ φ=0. Then Δ(xμ – xμ’,z) is specified solving

□ Δ=δ(xμ – xμ’) δ(z)

Note:• Usually approximate string theory by SUGRA and hence Zs

by a single point (saddle point); we approximated the large coupling gauge problem with a point of string theory!! Once we know Zs, we are done; can compute anything in CFT.

1111

Holographic renormalization

• Know the SE Tensor of Gauge theory is given by

• So gμν acts as a source => in order to calculate Tμν from AdS/CFT must find the metric. Metric has its eq. of motion i.e. Einsteins equations.

• Then by varying the Zs w.r.t. the metric at the boundary (once at z=0) can obtain < Tμν >.

Example:de Haro, Skenderis, Solodukhin ‘00

gg

S

gT |1

2

1212

Energy-momentum tensor is dual to the metric in AdS. Using Fefferman-Graham coordinates one can write the metric as

with z the 5th dimension variable and the 4d metric.

Expand near the boundary (z=0) of the AdS space:

Using AdS/CFT can show: , and

Holographic renormalization

22 2 2

52( , )

Lds g x z dx dx dz L d

z

),(~ zxg

),(~ zxg

...,),(

lim2

4402

2

coefzeiz

zxgNT

z

c

1313

I. AA: State/set up the problem

1414

Rmrks:

• Deal with N=4 SYM theory

• Coupling is tuned large and remains large at all times

• Forget previous results of pQCD

15

Initial Tµνphenomenology

AdS/CFTDictionary

Initial Geometry

Dynamical Geometry Dynamical Tµν

(our result)

EvolveEinst. Eq.

Strategy

1616

Field equations, AdS5 & examples gμν Tμν

Eq. of Motion (units L=1) for gΜΝ(xM = x±, x1, x2, z) is generally given

; empty space reduces

Empty & “Flat” AdS space:

implies Tμν=0 in QFT

side

Empty but not flat AdS-shockwave: [Janik & Peschanski ’06]

Then ~z4 coef. implies <Tμν (xμ)>= δμ - δν - < Tμν (x-)> in QFT side

JgRgR 6

2

1 04 gR

]2[1 22

22 dzdxdxdx

zds

...,),(

lim2

4402

2

coefzeiz

zxgNT

z

c

])()(2

2[1 2224

2

2

2

2 dzdxdxzxTN

dxdxz

dsc

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Single nucleus Single shockwave

Choose T-- (x-) a localized function along x- but not

along ┴ plane. So take

μ is associated with the energy carried by nucleus ([μ]=3).

May represent the shockwave metric as a

single vertex: a graviton exchange between

the source (the nucleus living at z=0; the

boundary of AdS) and point X in the bulk which

gravitational field is measured.

T ~ (x )

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Superposition of two shockwavesNon linearities of gravity

24

22

224

12

222

2

22 )(

2)(

22 dxzxT

NdxzxT

Ndzdxdxdx

z

Lds

CC

?

Flat AdS

Higher graviton ex.

Due to non linearities One graviton ex.

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Built up a perturbative approach• Motivation: Knowing gMN in the forward light cone we automatically

know Tμν of QFT after the ion’s collision just read it from ∂gMN (~z4

coefficient).

• Know that Ti ~μi (i=1,2). Higher graviton exchanges; i.e. corrections to gMN should come with extra powers of μ1 and μ2: μ1μ2, μ1

2μ2, μ1μ22, …

• So reconstruct by expanding around the flat AdS:

flat AdS, single shockwave(s), higher gravitons

...),(),(),(),()2()1()0(

zxgzxgzxgzxg MNMNMNMN

MNg)0(

MNg)1( )2( j

MNg

2020

Insight from Dim. Analysis, symmetries, kinematics & conservation

Tracelessness + conservation Tμν(x+, x-) provide 3 equations. Also have x+ x- symmetry. Expect:

)(~~ 11

xT

oijij

ooo hThThThT ~,~,~,~

For the case Ti =μi δ(x) shock-waves [μi]=3 and as energy density has [ε]=4 then we expect that the first correction to ε must be ε~ μ1 μ2 x+ x- i.e.rapidity independent as diagram suggests.

Y

)(~~ 22

xT

2121

Calculation/results• Step 1.: Linearize field eq. expanding around ημν

(partial DE with w.r.t. x+,x-, z with non constant coef.).

• Step 2.: Decouple the DE. In particular g(2)┴┴=h(x+,x-, z )

obeys: □h=8/3 z6 t1(x-) t2 (x+) with box the d'Alembertian in AdS5.

• Step 3.: Solve them imposing (BC) causality. Find: h= z4 ho(t1(x-) , t2(x+)) + z6 h1(t1(x-) , t2(x+)).

• Step 4.: Use rest components of field eq. in order to determine rest components of gμν.

• Step 5.: Determine Tμν by reading the z4 coef. of gμν

Conclude: Tμν has precisely the form we suspected for any t1, t2: Tμν

is encoded in a single coefficient!

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Particular sources (nucleus profile)

• Only need ho: . Encodes Tμν.

• δ profiles: Get corrections:

T+ -~T┴┴ ~ ho ~μ1 μ2τ2 and T- - ~ μ1 μ2(x+)2

• Step profiles: Here δ’s are smeared;

• At the nucleus will run out of momentum and stop!

)()()( xaxa

x

)()(~)( 2102

xtxth

2224)2/,( x

aaxaxT

[Grumiller, Romatschke ’08][Albacete, Kovchegov, Taliotis’08]

ax

1

~

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Conclusions/comparisons/summary

• Constructed graviton expansion for the collision of two shock waves in AdS. Goal is obtain SE tensor of the produced strongly-coupled matter in the gauge theory. Can go to any finite order. Lower order hold for early times.

• LO agress with [Grumiller, Romatschke ‘08]. NLO and NNLO corrections have been also performed.

• They confirm: Tμν is encoded in a single coefficient h0(x+,x-). Also come with alternate sign.

• Likely nucleus stops. A more detailed calculation (all order ressumation in A) in pA [Albacete, Taliotis, Yu.K. ‘09] confirms it.

• Possibly have Landau hydro. However its Bjorken hydro that describes (quite well) RHIC data.

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Landau vs Bjorken

Landau hydro: results from strong coupling dynamics at all times in the collision. While possible, contradicts baryon stopping data at RHIC.

Bjorken hydro: describes RHIC data well. The picture of nuclei going through each other almost without stopping agrees with our perturbative/CGC understanding of collisions. Can we show that ithappens in AA collisions using field theory or AdS/CFT?

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II. pA: State/set up the problem

2626

pA collisions

11 pt 22 pt

]23[

xxzzzz hhhz

})(2

1]2

2

11

2

7[{)( 2

622

2

zxxxxzzzz hxtzhhhz

zxtz

Eq. for transverse component:

Diagrammatic Representation

Scalar Propagator

Multiple graviton ex.

vertex ~ t2

)()(16 215 xtxtz

Initial Condition vertex

)()(16 215 xtxtz

cf. gluon production in pAcollisions in CGC!

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Eikonal Approximation &Diagrams Resummation

•Nucleus is Lorentz-contracted and so are small; hence ∂+ is large compared to ∂- and ∂z.

•This allows to sub the vertices and propagators with effectives and simplify problem. For more see [Kovchegov, Albacete, Taliotis’09].

•Apprxn applies for

2/1~ pxi

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Calculation (δ profiles)• Particular profiles:

• Diagram ressumation (all orders in μ2) in the forward LC yields:

• Recalling the duality mapping:

• Finally recalling ho;ei encodes <Tμν> through

yields to the results:

),(;| 4

xxhg eioz

eioijij

eioeioeio hThThThT ;;;; ~,~,~,~

4|2

2

2 zc g

NT

2929

Results

3030

Conclusions• Not Bjorken hydro

Indeed instead of T┴ ┴=p ~1/τ4/3 it is found that

• Not (any other) Ideal Hydrodynamics eitherIndeed, from and considering μ=ν=+ deduce that T++ >0; however T++ is found strictly negative!

• Proton stopping in pA also For AA, it was found earlier thatwith estimation stopping time estimated by . Same result recovered here by considering the total T++and expanding to O(μ2;x-=α/2):

2/5

)2/3(

2~

)(

1~

e

xxp

T (x a,x a /2) ~a 22x2

ax 2/1

(Landau Hydro??)

Ttot Tin

Tprod

Nc2

2 21a1{1 (

1

182(x)2 x

1)}, 0 x a1

3131

Proton Stopping(Landau Hydro??)

T++

X+

)0( x

3232

Future Work

• Use CGC as initial condition in order to evolve the metric to later times! Ambiguities Many initial metrics give same initial condition. Choose the simplest?

• Include transverse dynamics? Very hard but…

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Recent Work arXiv: 1004.3500v1[hep-th] - [Taliotis]

1 2 2 21,2 ( ) ( )r x b x

2 1 1 2 1 1( ) ( ) ( ) ( )I IIT r r r A r r r A

2 2 1( ) ( ) { }IIIr r r A b b

•Causality separates evolution in a very intuitive way!

•General form of SE tensor: For given proper time τ it has the form

Snapshot of the collision at given proper time τ

Eccentricity-Momentum Anisotropy

Momentum Anisotropy εx= εx (x≡τ/b) (left) and εx = εx(1/x) (right) for intermediate x≡τ/b .

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Agrees qualitatively with [Heinz,Kolb, Lappi,Venugopalan,Jas,Mrowczynski]

Conclusions Built perturbative expansion of dual geometry to determine Tµν ;

applies for sufficiently early times: µτ3<<1.

Tµν evolves according to causality in an intuitive way! Also

Tµν is invariant under .

Our exact formula (when applicable) allows as to compute Spatial Eccentricity and Momentum Anisotropy .

When τ>>r1 ,r2 have ε~τ2 log 2 τ-compare with ε~Qs2log 2τ

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),,,( 210

lim bxxTb

[Gubser ‘10]

[Lappi, Fukushima]

3636

Thank you