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Department of Physics

HIC with Dynamics┴ from Evolving Geometries in AdS

arXiv: 1004.3500 [hep-th],

Anastasios Taliotis

Partial Extension of arXiv:0805.2927 [hep-th], arXiv:0902.3046 [hep-th], arXiv:0705.1234 [hep-ph]

(published in JHEP and Phys. Rev. C) [ Albacete, Kovchegov, Taliotis]

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Outline

Motivating strongly coupled dynamics in HIC

AdS/CFT: What we need for this work

State/set up the problem

Attacking the problem using AdS/CFT

Predictions/comparisons/conclusions/Summary

Future work

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Motivating strongly coupled dynamics in HIC

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Notation/FactsProper time:

Rapidity:

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

23

20 xx

12lnx0 x3x0 x3

1

2lnxx

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 t ~ 1/QS.

sQ

JFD

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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.

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Most General Rapidity-Independent SE Tensor

The most general rapidity-independent SE tensor for a collision of two transversely large nuclei is (at x3 =0)

z

y

x

t

p

p

pT

)(000

0)(00

00)(0

000)(

3

which, due to gives0 T

3p

d

d

0x

1x

2x

3x3x

2x

1x

We will see three different regimes of p3

77

z

y

x

t

T

)(000

0)(00

00)(0

000)(

z

y

x

t

p

p

pT

)(000

0)(00

00)(0

000)(

z

y

x

t

p

pT

0000

0)(00

00)(0

000)(

0x

1x

2x

3x

I. Early times : τQs <<1

CGC

II. Later times : τ>~1/Qs

CGC

III. Much later times:τQs >>>1

Bjorken Hydrodynamics

2log~

3/4

1~

1~

•Classical gluon fields

•Pert. theory applies

•Describes RHIC data well

(particle multiplicity dN/dn)

•Classical gluon fields

•Pert. Theory applies

•Energy is conserved

•Hydrodynamic description

•Does pert. Theory apply??

•Describes data successfully

(spectra dN/d2pTdn for K, ρ, n & elliptic flow) [Heınz et al]

thermalization

[Lappi ’06 Fukushima ’07: pQCD][Talıotıs ’10: AdS/CFT]

[Free streaming]

0 p(τ)

Isotropization

[Krasnitz, Nara,Venogopalan, Lappi, Kharzeev, Levin, Nardi]

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Bjorken Hydro & strongly coupled dynamics

Deviations from the energy conservation are due to longitudinal pressure, P3 which does work P3dV in the longitudinal direction modifying the energy density scaling with tau.

3p

d

d If then, as , one gets .03 p 1

1~

/1~1/1~

It is suggested that neither classical nor quantum gluonic or fermionic fields can cause the transition from free streaming to Bjorken hydro within perturbation

theory. [Kovchegov’05]

On the other hand Bjorken hydro describe simulations satisfactory.

Conclude that alternative methods are needed!

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AdS/CFT: What we need for this work

1010

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

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Holographic renormalization

• Quantifying the Conjecture

<exp z=0∫O φ0>CFT = Zs(φ|φ(z=0)= φo)

• 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.

Example:

gg

S

gT |1

2

[Witten ‘98]

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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

de Haro, Skenderis, Solodukhin ‘00

1313

State/set up the problem

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Initial Tµν

phenomenology

AdS/CFTDictionary

Initial Geometry

Dynamical Geometry Dynamical Tµν

(our result)

Evolve

Einstein's

Eq.

Strategy

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Field equations, AdS5 shockwave; ∂gMN Tμν

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

AdS-shockwave with bulk matter: [Janik & Peschanski ’06]

Then ~z4 coef. implies <Tμν (xμ)> ~ -δμ + δν + µlog(r1) δ(x+) in QFT side

Corresponding bulk tensor JMN :

)3

1(4 2

5 MNJgJRgR

])(),(2[1 2224

1122 dzdxdxzxrtdxdxz

ds

2221 )()( xxr || 11

brr

)()( 14

25

xrzJMN

)()log( 11 xkrt

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Single nucleus Single shockwaveThe picture in 4d is that matter moves ultrarelativistically along x- according to figure.

Einstein's equations are satisfied trivially except (++) component; it satisfies a linear equation:

□(z4 t1)=J++

This suggests may represent the shockwave metric as a single vertex: a graviton exchange between the source J++ (the nucleus living at z=0; the boundary of AdS) and point XM in the bulk which gravitational field is measured.

J

4D Picture of Collision

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Superposition of two shockwavesNon linearities of gravity

)2(24

)1(

224

)1(

122

2

22 )()()()(2 gxxdxzbrtdxzbrtdzdxdxdxz

Lds

?

Flat AdSHigher graviton ex.

Due to non linearities One graviton ex.

Back-to-Back reactions for JMN

• In order to have a consistent expansion in µ2 we must determine

• We use geodesic analysis

• Bulk source J++ (J--) moves in the gravitational filed of the shock t1(t2)

• Important: is conserved iff b≠0

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MNJ)2(

Self corrections to JMN)(~

bJ NMNM

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Calculation/results• Step 1: Choose a gauge: Fefferman-Graham coordinates

• Step 2: Linearize field eq. expanding around 1/z2 ηMN

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

• Step 3: Decouple the DE. In particular all components g(2)µν

obey: □g(2)µν = A(2)

µν(t1(x-) ,t2 (x+) ,J) with box the d'Alembertian in AdS5.

• Step 4: Solve them imposing (BC) causality-Determine the GR

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

Side Remark: Gzz encodes tracelessness of Tµν

Gzν encode conservation of Tµν

The Formula for Tµν

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Eccentricity-Momentum Anisotropy

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

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bx

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! There is a kinematical window where is invariant under .

• Our exact formula (when applicable) allows as to compute Spatial Eccentricity and Momentum Anisotropy . 23

),,,( 210

lim bxxTb

r[Gubser ‘10]

• When τ>>r1 ,r2 have ε~τ2 log2 τ-compare with ε~Q2slog2 τ

• Despite J being localized, it still contributes to gµν and so to Tµν not only on the light-cone but also inside.

• Impact parameter is required otherwise violate conservation of JMN and divergences of gµν. Not a surprise for classical field theories.

• Our technique has been applied to ordinary (4d) gravity and found similar behavior for gµν.

• A phenomenological model using the (boosted) Woods-Saxon profile:

[Lappi, Fukushima]

Taliotis’10 MS thesis.dept. of Mathematics, OSU

[Gubser,Yarom,Pufu ‘08]

For τ> r1,r2

Note symmetry under when b=0; [Gubser’10]

r

Thank you

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Supporting slides

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O(µ2) Corrections to Jµν

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Remark: These corrections live on the forward light-cone as should!

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Field Equations