# Early Time Dynamics in Heavy Ion Collisions from ... - McGill Physics

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16 Νοε 2013 (πριν από 4 χρόνια και 6 μήνες)

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Early Time Dynamics in Heavy
Correspondence

Yuri Kovchegov

The Ohio State University

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

Janik and Peschanski [hep
-
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 the

following notations:

proper time

and rapidity

2
3
2
0
x
x

3
0
3
0
ln
2
1
x
x
x
x

0
x
3
x
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 x
3

=0)

z
y
x
t
p
p
p
T

)
(
0
0
0
0
)
(
0
0
0
0
)
(
0
0
0
0
)
(
3


which, due to

0



T
gives

3
p
d
d

There are 3 extreme limits.

0
x
1
x
2
x
3
x
3
x
2
x
1
x
Limit I: “Free Streaming”

Free streaming is characterized by the following “2d”

energy
-
momentum tensor:

z
y
x
t
p
p
T

0
0
0
0
0
)
(
0
0
0
0
)
(
0
0
0
0
)
(



d
d
such that

and

1
~

The total energy E~
 
is conserved, as expected for

non
-
interacting particles.

0
x
1
x
2
x
3
x
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
p
T

)
(
0
0
0
0
)
(
0
0
0
0
)
(
0
0
0
0
)
(



p
d
d

such that

Using the ideal gas equation of state, , yields

p
3

3
/
4
1
~

Bjorken, ‘83

The total energy E~
 

is
not

conserved, while the
total entropy S is conserved.

0
x
1
x
2
x
3
x
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
~
3
p
0
,
1
~
1

dV
p
3

Non
-
zero positive longitudinal

pressure and isotropization

1
~

3
p
d
d

If then, as , one gets .

0
3

p

1
1
~

↔ deviations from

Limit III: Color Glass at Early Times

In CGC at very early times

z
y
x
t
T

)
(
0
0
0
0
)
(
0
0
0
0
)
(
0
0
0
0
)
(



3
p
d
d

such that, since

1
,
1
log
~
2

S
Q

0
x
1
x
2
x
3
x
we get, at the leading log level,

Energy
-
momentum tensor is

(Lappi, ’06)

-
5

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 hard

to 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 z
4

it gives the following familiar conditions:

and

Solution

z

z=0

Our 4d

world

5d (super) gravity

lives 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 constraints

on 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 above

~
Janik and Peschanski (‘05) showed that requiring the energy

density to be non
-
negative in all frames leads to

0
)
(

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 a
0

<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 multivalued

for ! 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 unless

the 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 allowed

power is , giving the Bjorken hydrodynamic scaling

of 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: using

perturbative (in z) solution at early times give the following

series

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

0,
we write the following ansatze

for the metric coefficients:

with
a, b
and

g
some unknown functions of u.

Early
-
Time General Solution

Solving 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 Time

However, 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 energy

of the produced medium is .

~
E

~
E

~
the power should be .

0
1

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 .

0
1

Early Time Solution

The early
-
time scaling of the energy density in this

strongly
-
coupled medium is

z
y
x
t
T

)
(
0
0
0
0
)
(
0
0
0
0
)
(
0
0
0
0
)
(


0
x
1
x
2
x
3
x
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 this

time) is that and the energy density scales

as

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 with

proper 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

1

u
Isotropization Transition: Time Estimate

We plot both branch cuts in the (z,

) plane:

The intercept is at the

“isotropization time”

Isotropization Transition: Time Estimate

In terms of more physical quantities we re
-
write the above

estimate as

where

0

is the coefficient in Bjorken energy
-
scaling:

For central Au+Au collisions at RHIC at

hydrodynamics requires

=15 GeV/fm
3
at

0.6 fm/c

(Heinz, Kolb ‘03), giving

0
=38 fm
-
8/3

A
GeV
s
/
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 a

conformally invariant theory with , the only

scale in the theory is given by . Making a scale with

dimension of time out of it gives .

0

8
/
3
0
~

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

The skeptic would say that for N
C

=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 viscous

corrections 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 log

needs 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.)