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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

12ln x0 x3x0 x3

12ln xx

0x

3x

QGPCGCCGC 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 Hydrog<<1; valid up to times ~ 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)

zyxt

pp

pT

)(0000)(0000)(0000)(

3

which, due to gives0 T

3pdd

0x1x2x3x

3x2x

1x

We will see three different regimes of p3

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zyxt

T

)(0000)(0000)(0000)(

zyxt

pp

pT

)(0000)(0000)(0000)(

zyxt

pp

T

00000)(0000)(0000)(

0x1x2x3x

I. Early times : τQs <<1

CGCII. 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.

3pdd

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

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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 |12

[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 dz

),(~ zxg

),(~ zxg

...,),(

lim2

4402

2

coefzeizzxgNT

z

c

de Haro, Skenderis, Solodukhin ‘00

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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 :

)31(4 2

5 MNJgJRgR

])(),(2[1 2224112

2 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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1818

Superposition of two shockwavesNon linearities of gravity

)2(24

)1(

224

)1(

122

2

22 )()()()(2 gxxdxzbrtdxzbrtdzdxdxdxzLds

?

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