Early Time Dynamics in Heavy Ion Collisions from AdS/CFT

Correspondence

Yuri KovchegovThe Ohio State University

based on work done with Anastasios Taliotis, arXiv:0705.1234 [hep-ph]

Instead of Outline• Janik and Peschanski [hep-th/0512162] used AdS/CFT

correspondence to show that at asymptotically late proper times the strongly-coupled medium produced in the collisions flows according to Bjorken hydrodynamics.

• In our work we have– Re-derived JP late-time results without requiring the

curvature invariant to be finite.– Analyzed early-time dynamics and showed that energy

density goes to a constant at early times.– Have therefore shown that isotropization (and hopefully

thermalization) takes place in strong coupling dynamics. – Derived a simple formula for isotropization time and used

it for heavy ion collisions at RHIC to obtain 0.3 fm/c, in agreement with hydrodynamic simulations.

Notations

We’ll be using thefollowing notations:

proper time

and rapidity

23

20 xx

30

30ln2

1

xx

xx

0x

3x

Most General Boost Invariant Energy-Momentum Tensor

The most general boost-invariant 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

There are 3 extreme limits.

0x

1x

2x

3x

3x

2x

1x

Limit I: “Free Streaming”

Free streaming is characterized by the following “2d”energy-momentum tensor:

z

y

x

t

p

pT

0000

0)(00

00)(0

000)(

d

d

such that

and

1~

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

0x

1x

2x

3x

Limit II: Bjorken Hydrodynamics

In 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, while the total entropy S is conserved.

0x

1x

2x

3x

Most General Boost Invariant 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

Non-zero positive longitudinal pressure and isotropization

1~

3p

d

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

1~

↔ deviations from

Limit III: Color Glass at 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, ’06)

AdS/CFT Approach

Start with the metric in Fefferman-Graham coordinates in AdS5 space

and solve Einstein equations

Expand the 4d metric near the boundary of the AdS space

If our world is Minkowski, , then

and

Iterative Solution

General solution of Einstein equations is not known and is hardto obtain. One first assumes a specific form for energy density

and the solves Einstein equations perturbatively order-by-order in z:

)(

The solution in AdS space (if found) determines which function of proper time is allowed for energy density.

At the order z4 it gives the following familiar conditions:

and

Solution

zz=0

Our 4dworld

5d (super) gravitylives here in the AdS space

Not every boundary condition in 4d (at z=0) leads to a valid gravity solution in the 5d bulk – get constraintson the 4d world from 5d gravity

Iterative Solution

We begin by expanding the coefficients of the metric

into power series in z:

Iterative Solution: Power-Law Scaling

Assuming power-law scaling

we iteratively obtain coefficients in the expansion

~

To illustrate their structure let me display one of them:

dominates at early times

dominates at late times

(only if !)4

Allowed Powers of Proper Time

Assuming power-law scaling the aboveconditions lead to

~

Janik and Peschanski (‘05) showed that requiring the energydensity to be non-negative in all frames leads to0)(

The above conclusion about which term dominates at what time is safe!

Late Time Solution: Scaling

Janik and Peschanski (‘05) were the first to observe it and looked for the full solution of Einstein equations at late proper time as a function of the scaling variable

At late times the perturbative (in z) series becomes

The metric coefficients become:

Here a0 <0 is the normalization of the energy density

Janik and Peschanski’s Late Time Solution

The late time solution reads (in terms of scaling variable v, for v fixed and going to infinity):

with

At this point Janik and Peschanski fixed the power by requiring that the curvature invariant has no singularities:

But what fixes

Late Time Solution: Branch Cuts

Instead we notice that the above solution has a branch cut for

This is not your run of the mill singularity: this is a branch cut!This means that the metric becomes complex and multivaluedfor ! Since the metric has to be real and single-valued we conclude that the metric (and the curvature invariant) do not exist for . That is unlessthe coefficients in front of the logarithms are integers!

Late Time Solution: Branch Cuts

Remember that functions a(v), b(v) and c(v) need to be exponentiated to obtain the metric coefficients:

If the coefficients in front of the logarithms are integers, functions A, B and C would be single-valued and real.

after simple algebra (!) one obtains that the only allowedpower is , giving the Bjorken hydrodynamic scalingof the energy density, reproducing the result of Janik and Peschanski

Late Time Solution: Fixing the Power

Requiring the coefficients in front of the logarithms to be integers l,m,n

Early Time Solution: Scaling

Let us apply the same strategy to the early-time solution: usingperturbative (in z) solution at early times give the followingseries

While no single scaling variable exists, it appears that the series expansion is in

such that

Early Time Solution: Ansatz

Keeping u fixed and taking we write the following ansatzefor the metric coefficients:

with andsome unknown functions of u.

Early-Time General SolutionSolving Einstein equations yields

where F is the hypergeometric function.

Hypergeometric functions have a branch cut for u>1. We have branch cuts again!

Requiring it to be finite we conclude that for

Allowed Powers of Proper TimeHowever, now hypergeometric functions are not in the exponent.The only way to avoid branch cuts is to have hypergeometric series terminate at some finite order, becoming a polynomial.

Before we do that we note that, at early times the total energyof the produced medium is . ~E

~E ~ the power should be .

01

1

Hence, at early times the physically allowed powers are:

Early Time Solution: Terminating the Series

Finally, we see that the hypergeometric series in the solution

terminates only for in the physically allowed

range of .01

Early Time Solution

The early-time scaling of the energy density in this strongly-coupled medium is

z

y

x

t

T

)(000

0)(00

00)(0

000)(

0x

1x

2x

3x

This leads to the following energy-momentum tensor,reminiscent of CGC at very early times:

with

Early Time Solution: Log Ansatz

One can also look for the solution with the logarithmic ansatz(sort of like fine-tuning):

The result of solving Einstein equations (no branch cuts thistime) is that and the energy density scalesas

The approach to a constant at early times could be logarithmic! (More work is needed to sort this out.)

Isotropization Transition: the Big Picture

We summary of our knowledge of energy density scaling withproper time for the strongly-coupled medium at hand:

Janik, Peschanski‘05

(this work)

Isotropization Transition

We have thus see that the strongly-coupled system starts out very anisotropic (with negative longitudinal pressure) and evolves towards complete (Bjorken) isotropization!

Let us try to estimate when isotropization transition takes place:the iterative solution has both late- and early-time terms.

dominates at early times

dominates at late times

has a branch cut at has a branch cut at

1u

Isotropization Transition: Time Estimate

We plot both branch cuts in the (z, ) plane:

The intercept is at the“isotropization time”

Isotropization Transition: Time EstimateIn terms of more physical quantities we re-write the aboveestimate as

where 0 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 =38 fm-8/3. This leads to

AGeVs /200

in good agreement with hydrodynamics!

Isotropization Transition Estimate: Self-Critique

An AdS/CFT skeptic would argue that our estimate

is easy to obtain from dimensional reasoning. If one has aconformally invariant theory with , the only

scale in the theory is given by . Making a scale with dimension of time out of it gives .

08/3

0~ iso

We would counter by saying that AdS/CFT gives a prefactor.

The skeptic would say that for NC =3 it is awfully close to 1…

Conclusions

We have:

Re-derived JP late-time results without requiring the curvature invariant to be finite: all we need is for the metric to exist.

Analyzed early-time dynamics and showed that energy density goes to a constant at early times.

Have therefore shown that isotropization (and hopefully thermalization) takes place in strong coupling dynamics.

Derived a simple formula for isotropization time and used it for heavy ion collisions at RHIC to obtain 0.3 fm/c, in agreement with hydrodynamic simulations.

Bonus Footage: Other Applications of No-branch-cuts Rule

Nakamura, Sin ’06 and Janik ’06 have calculated viscouscorrections to the Bjorken hydrodynamics regime by expanding the metric at late times as

In particular, writing shear viscosity as

one obtains the following coefficient (Janik ‘06):

(but with poles)

Bonus Footage: Other Applications of No-branch-cuts Rule

To remove the branch cut the coefficient in front of the logneeds to be integers. But it is time dependent!

Hence the prefactor of the log can only be zero!

Equating it to zero yields shear viscosity

in agreement with Kovtun-Polcastro-Son-Starinets (KPSS)bound! (The connection is shown by Janik ’06.)