Early Time Dynamics in Heavy Ion Collisions from CGC and from AdS/CFT

Yuri KovchegovThe Ohio State University

Outline

CGC dynamics: classical gluon fields

Perturbative expansion: pA and AA Free streaming

AdS/CFT dynamics: classical graviton fields

Perturbative expansion: pA and AA Road to thermalization / ideal hydrodynamics

Conclusions

Classical gluon fields in CGC

McLerran-Venugopalan Model

Large occupation number Large occupation number Classical Field Classical Field

The wave function of a single nucleus has many small-x quarks and gluons in it.

In the transverse plane the nucleus is densely packed with gluons and quarks.

McLerran-Venugopalan Model

Large parton density gives a large momentum scale Qs (the saturation scale).

For Qs >> QCD, get a theory at weak coupling and the leading gluon field is classical.

1)( 2 SS Q

McLerran, Venugopalan ’93-’94

McLerran-Venugopalan Model

o To find the classical gluon field Aμ of the nucleus one has to solve the non-linear analogue of Maxwell equations – the Yang-Mills equations, with the nucleus as a source of the color charge:

JFD

Yu. K. ’96; J. Jalilian-Marian et al, ‘96

Classical Field of a Nucleus

Here’s one of the diagrams showing the non-Abelian gluon field of a large nucleus.

To find the gluon production cross section in pA one has tosolve the same classical Yang-Mills equations

for two sources – proton and nucleus.

Classical Gluon Production in Proton-Nucleus Collisions (pA)

JFD

A.H. Mueller, Yu. K., ’98; B. Kopeliovich, A. Tarasov and A. Schafer, ’98; A. Dumitru, L. McLerran ’01.

CGC in pA: the diagrams

Again classical gluon fields correspond to tree-level (no loops) diagrams:

nucleons in the nucleus

proton

Classical gluon field in pA: Cronin effect

Classical CGC gluon fields in pA lead to Cronin effect:

EnhancementEnhancement

(Cronin Effect)(Cronin Effect)

pT / QS

Heavy Ion Collisions in CGC: Classical Gluon Field

To construct initial conditions for quark-gluon plasma formation in McLerran-Venugopalan model one has to find the classical gluon field left behind by the colliding nuclei.

No analytical solution exists. Perturbative calculations by Kovner, McLerran, Weigert; Rischke,

Yu.K.; Gyulassy, McLerran; Balitsky. Numerical simulations by Krasnitz, Nara and Venugopalan, and by

Lappi, and an analytical ansatz by Yu. K for full solution.

nucleus nucleus

Classical Gluon Field in AA Again the diagrams do not have any loops:

nucleons in nucleus 1

nucleons in nucleus 2

What kind of matter distributiondoes this classical gluon field give?

Notations

proper time

rapidity

23

20 xx

30

30ln2

1

xx

xx

0x

3x

QGP

CGC

The matter distribution due toclassical gluon fields is rapidity-independent.

valid up totimes ~ 1/QS

Most General Rapidity-Independent Energy-Momentum Tensor

The most general rapidity-independent energy-momentum tensor for a high energy collision of two very large nuclei is (at x3 =0)

z

y

x

t

p

p

pT

)(000

0)(00

00)(0

000)(

3

which, due to 0 T

gives

3p

d

d

0x

1x

2x

3x

3x

2x

1x

Color Glass at Very Early Times

In CGC at very early times

z

y

x

t

T

)(000

0)(00

00)(0

000)(

3p

d

d such that, since

1,1

log~ 2 SQ

0x

1x

2x

3x

we get, at the leading log level,

Energy-momentum tensor is

(Lappi ’06Fukushima ‘07)

Color Glass at Later Times: “Free Streaming”

At late times classical CGC gives free streaming,

which is characterized by the following energy-momentum tensor:

d

d

such that

and

1

~

The total energy E~ e is conserved, as expected fornon-interacting particles.

z

y

x

t

p

pT

0000

0)(00

00)(0

000)(

0x

1x

2x

3x

SQ

1

Classical FieldsClassical Fields

CGC classical gluon field leads to energy density scaling as

1

~classical

from numerical simulations by Krasnitz, Nara, Venugopalan ‘01

Classical Gluon Fields in AA: Summary

Classical gluon field produced in AA collisions gives a free-streaming medium at late times (~1/QS) with zero longitudinal pressure.

At early times (<<1/QS) the classical gluon field gives an (almost) constant energy-density and negative longitudinal pressure.

Thermalization problem

Bjorken HydrodynamicsIn the case of ideal hydrodynamics, the energy-momentum tensor is symmetric in all three spatial directions (isotropization):

z

y

x

t

p

p

pT

)(000

0)(00

00)(0

000)(

p

d

d

such that

Using the ideal gas equation of state, , yieldsp3

3/4

1~

Bjorken, ‘83

The total energy E~ is not conserved

0x

1x

2x

3x

Rapidity-Independent Energy-Momentum Tensor

Deviations from the scaling of energy density,

like are due to longitudinal pressure

, which does work in the longitudinal direction

modifying the energy density scaling with tau.

1

~

3p0,

1~

1

dVp3

Positive longitudinal pressure and isotropization

1~

3p

d

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

1~

↔ deviations from

The Problem

Can one show in an analytic calculation that the energy-momentum tensor of the medium produced in heavy ion collisions is isotropic over a parametrically long time?

That is, can one start from a collision of two nuclei and obtain Bjorken-like hydrodynamics?

Let us proceed assuming that strong-coupling dynamics from AdS/CFT would help accomplish this goal.

Classical graviton fields in AdS/CFT

AdS/CFT Approach

z

z=0

Our 4dworld

5d (super) gravitylives here in the AdS space

AdS5 space – a 5-dim space with a cosmological constant = -6/L2.(L is the radius of the AdS space.)

5th dimension

222

22 2 dzdxdxdxz

Lds

2

30 xxx

AdS/CFT Correspondence (Gauge-Gravity Duality)

Large-Nc, large g2 Nc N=4 SYM theory in our 4 space-timedimensions

Weakly coupledsupergravity in 5danti-de Sitter space!

Can solve Einstein equations of supergravity in 5d to learn about energy-momentum tensor in our 4d world in the limit of strong coupling! Can calculate Wilson loops by extremizing string configurations. Can calculate e.v.’s of operators, correlators, etc.

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 of the AdS space:

For Minkowski world and with

Holographic renormalization

22

22 ),(~ dzdxdxzxgz

Lds

),(~ zxg

),(~ zxg

de Haro, Skenderis, Solodukhin ‘00

Single Nucleus in AdS/CFT

An ultrarelativistic nucleus is a shock wave in 4d with the energy-momentum tensor

)(~ xT

Shock wave in AdS

The metric of a shock wave in AdS corresponding to the ultrarelativistic nucleus in 4d is(note that T_ _ can be any function of x^-):

22242

2

2

22 )(

22 dzdxdxzxT

Ndxdx

z

Lds

C

Janik, Peschanksi ‘05

Need the metric dual to a shock wave and solving Einstein equations

06

2

12

gL

gRR

Diagrammatic interpretation

The metric of a shock wave in AdS corresponding to the ultrarelativistic nucleus in 4d can be represented as a graviton exchange between the boundary of the AdS space and the bulk:

22242

22 )(2 dzdxdxzxdxdxz

Lds

cf. classical Yang-Mills field of a single ultrarelativistic nucleus in CGC in covariant gauge: given by 1-gluon exchange(Jalilian-Marian, Kovner, McLerran, Weigert ’96, Yu.K. ’96)

Bjorken Hydrodynamics in AdS

Asymptotic geometry

Janik and Peschanski ’05 showed that in the rapidity-independent case the geometry of AdS space at late proper times is given by the following metric

with e0 a constant. In 4d gauge theory this gives Bjorken hydrodynamics:

with

22220

2

0

2

0

2

22

3/4

4

3/4

4

3/4

4

11

1dzdxded

e

e

z

Lds z

z

z

z

y

x

t

p

p

pT

)(000

0)(00

00)(0

000)(

3/4

1~

Bjorken hydrodynamics in AdS

Looks like a proof of thermalization at large coupling.

It almost is: however, one needs to first understand what initial conditions lead to this Bjorken hydrodynamics.

Is it a weakly- or strongly-coupled heavy ion collision which leads to such asymptotics? If yes, is the initial energy-momentum tensor similar to that in CGC? Or does one need some pre-cooked isotropic initial conditions to obtain Janik and Peschanski’s late-time asymptotics?

Colliding shock waves in AdS

I will follow J. Albacete, A. Taliotis, Yu.K. arXiv:0805.2927 [hep-th], arXiv:0902.3046 [hep-th]

Considered by Nastase; Shuryak, Sin, Zahed; Kajantie, Louko, Tahkokkalio; Grumiller, Romatshcke; Gubser, Pufu, Yarom.

Model of heavy ion collisions in AdS

Imagine a collision of two shock waves in AdS:

We know the metric of bothshock waves, and know thatnothing happens before the collision.

Need to find a metric in theforward light cone! (cf. classical fields in CGC)

24

22

224

12

222

2

22 )(

2)(

22 dxzxT

NdxzxT

Ndzdxdxdx

z

Lds

CC

empty AdS5 1-graviton part higher ordergraviton exchanges

?

Heavy ion collisions in AdS

24

22

224

12

222

2

22 )(

2)(

22 dxzxT

NdxzxT

Ndzdxdxdx

z

Lds

CC

empty AdS5 1-graviton part higher ordergraviton exchanges

What to expect

There is one important constraint of non-negativity of energy density. It can be derived by requiring that

for any time-like t.

This gives (in rapidity-independent case)

along with

0 ttT

0)(

Janik, Peschanksi ‘05

Physical shock waves

Simple dimensional analysis:

221~

Each graviton gives , hence get no rapidity dependence:e

tindependen Yee

Y

)(~~ 11

xT

)(~~ 22

xT Grumiller, Romatschke ’08Albacete, Taliotis, Yu.K. ‘08

The same result comes out of detailed calculations.

Physical shock waves: problem 1

Energy density at mid-rapidity grows with time!? This violates condition. This means in some frames energy density at some rapidity is negative!

I do not know of a good explanation: it may be due to some Casimir-like forces between the receding nuclei. (see e.g. work by Kajantie, Tahkokkalio, Louko ‘08)

0)('

221~

Physical shock waves: problem 2

Delta-functions are unwieldy. We will smear the shock wave:

Look at the energy-momentum tensor of a nucleus after collision:

Looks like by the light-cone time

the nucleus will run out of momentum and stop!

2224)2/,( x

aaxaxT

3/1

1~

1~

Aax

)()()( xaxa

x

Physical shock waves

We conclude that describing the whole collision in the strong coupling framework leads to nuclei stopping shortly after the collision.

This would not lead to Bjorken hydrodynamics. It is very likely to lead to Landau-like hydrodynamics.

While Landau hydrodynamics is possible, it is Bjorken hydrodynamics which describes RHIC data rather well. Also baryon stopping data contradicts the conclusion of nuclear stopping at RHIC.

What do we do? We know that the initial stages of the collisions are weakly coupled (CGC)!

Unphysical shock waves

One can show that the conclusion about nuclear stopping holds for any energy-momentum tensor of the nuclei such that

To mimic weak coupling effects in the gravity dual we propose using unphysical shock waves with not positive-definite energy-momentum tensor:

0)(,0)( 21

xTdxxTdx

0)(,0)( 21

xTdxxTdx

Unphysical shock waves

Namely we take

This gives:

Almost like CGC at early times:

Energy density is now non-negative everywhere in the forward light cone!

The system may lead to Bjorken hydro.

)(),( 222

211

xTxT

22

213 8)()()( pp

z

y

x

t

T

)(000

0)(00

00)(0

000)(

cf. Taliotis, Yu.K. ‘07

Will this lead to Bjorken hydro?

Not clear at this point. But if yes, the transition may look like this:

Janik, Peschanski‘05

(our work)

Isotropization time One can estimate this isotropization time from

AdS/CFT (Yu.K, Taliotis ‘07) obtaining

where e0 is the coefficient in Bjorken energy-scaling:

For central Au+Au collisions at RHIC at hydrodynamics requires =15 GeV/fm3 at =0.6 fm/c (Heinz, Kolb ‘03), giving 0=38 fm-8/3. This leads to

in good agreement with hydrodynamics!

AGeVs /200

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?

Proton-Nucleus Collisions in AdS

pA Setup

Consider pA collisions:

1p

2p

pA Setup

In terms of graviton exchanges need to resum diagrams like this:

cf. gluon production in pAcollisions in CGC!

Proton Stopping

What about the proton? Dueto our earlier result about shock wave stopping

we should be able to see

how it stops.

And we do:

T++ goes to zero as x+ grows large!

2224)2/,( x

aaxaxT

Proton Stopping

We get complete proton stopping (arbitrary units):

T++

of the proton

X+

Albacete, Taliotis, Yu.K. ‘09

Conclusions

AdS/CGC duality: CGC has classical gluon fields AdS/CFT has classical graviton field

But physics is different: CGC classical gluon fields lead to free streaming AdS/CFT classical graviton fields lead to nuclear

stopping and Landau-like hydrodynamics

Can Bjorken-like hydrodynamics arise from CGC+AdS/CFT? Probably yes, but still an open question.

Backup Slides

CGC in pA: the diagrams

Again classical gluon fields correspond to tree-level (no loops) diagrams:

Classical Gluon Field in AA Some of the diagrams contributing to gluon

production in AA collisions: (Yu. K. ’00)

Expansion Parameter

Depends on the exact form of the energy-momentum tensor of the colliding shock waves.

For the parameter in 4d is :the expansion is good for early times only.

For that we will also considerthe expansion parameter in 4d is 2 2. Also valid for early times only.

In the bulk the expansion is valid at small-z by the same token.

)(~ xT

)(~ 2 xT

Eikonal Approximation

Note that the nucleus is Lorentz-contracted. Hence

all and are small.

2

1~p

xi

Physical Shocks

Summing all these graphs for the delta-function shock waves

yields the transverse pressure:

Note the applicability region:

Physical Shocks

The full energy-momentum tensor can be easily constructed too. In the forward light cone we get:

Physical Shocks: the Medium

Is this Bjorken hydro? Or a free-streaming medium? Appears to be neither. At late times

Not a free streaming medium. For ideal hydrodynamics expect

such that:

However, we get

Not hydrodynamics either.

2/5

)2/3(

2~

)(

1~

e

xxp

0

Physical Shocks: the Medium

Most likely this is an artifact of the approximation, this is a “virtual” medium on its way to thermalization.

For delta-prime shock waves the result is surprising. The all-order eikonal answer for pA is given by LO+NLO terms:

That is, graviton exchange series terminates at NLO.

Delta-prime shocks

+

The answer for transverse pressure is

with the shock waves

As p goes negative at late times, this is clearly not hydrodynamics and not free streaming.

Delta-prime shocks

)(')(),(')( 222

211

xxtxxt

Note that the energy momentum tensor becomes rapidity-dependent:

Thus we conclude that initially the matter distribution is rapidity-dependent. Hence at late times it will be rapidity-dependent too (causality). Can one get Bjorken hydro still? Probably not…

Delta-prime shocks