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Early Time Dynamics in Heavy Ion Collisions from AdS/CFT Correspondence. Yuri Kovchegov The Ohio State University based on work done with Anastasios Taliotis, arXiv:0705.1234 [hep-ph]. Instead of Outline.
The Ohio State University
based on work done with Anastasios Taliotis, arXiv:0705.1234 [hep-ph]
We’ll be using the
The most general boost-invariant energy-momentum tensor
for a high energy collision of two very large nuclei is (at x3 =0)
which, due to
There are 3 extreme limits.
Free streaming is characterized by the following “2d”
In the case of ideal hydrodynamics, the energy-momentum
tensor is symmetric in all three spatial directions (isotropization):
Using the ideal gas equation of state, , yields
If then, as , one gets .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.
pressure and isotropization
↔ deviations from
In CGC at very early times
we get, at the leading log level,
such that, since
Energy-momentum tensor is
Start with the metric in Fefferman-Graham coordinates in AdS5
and solve Einstein equations
Expand the 4d metric near the boundary of the AdS space
If our world is Minkowski, , then
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
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:
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
We begin by expanding the coefficients of the metric
into power series in z:
Assuming power-law scaling
we iteratively obtain coefficients in the expansion
To illustrate their structure let me display one of them:
(only if !)
Janik and Peschanski (‘05) showed that requiring the energy
density to be non-negative in all frames leads to
Assuming power-law scaling the above
conditions lead to
The above conclusion about which term dominates at what time is safe!
At late times the perturbative (in z) series becomes
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
The metric coefficients become:
Here a0 <0 is the normalization
of the energy density
The late time solution reads (in terms of scaling variable v,
for v fixed and t going to infinity):
But what fixes D ???
At this point Janik and Peschanski fixed the power D by
requiring that the curvature invariant has no singularities:
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!
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.
Requiring the coefficients in front of the logarithms to be
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
Let us apply the same strategy to the early-time solution: using
perturbative (in z) solution at early times give the following
While no single scaling variable exists, it appears that the
series expansion is in
Keeping u fixed and taking t ->0, we write the following ansatze
for the metric coefficients:
with a, b and g some unknown functions of u.
Solving Einstein equations yields
where F is the hypergeometric function.
Hypergeometric functions have a branch cut for u>1.
We have branch cuts again!
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 .
Requiring it to be finite we conclude that for
the power should be .
Hence, at early times the physically allowed powers are:
Finally, we see that the hypergeometric series in the solution
terminates only for in the physically allowed
range of .
The early-time scaling of the energy density in this
strongly-coupled medium is
This leads to the following energy-momentum tensor,
reminiscent of CGC at very early times:
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
The approach to a constant at early times could be
logarithmic! (More work is needed to sort this out.)
We summary of our knowledge of energy density scaling with
proper time for the strongly-coupled medium at hand:
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.
has a branch cut at
has a branch cut at
We plot both branch cuts in the (z, t) plane:
The intercept is at the
In terms of more physical quantities we re-write the above
where e0 is the coefficient in Bjorken energy-scaling:
For central Au+Au collisions at RHIC at
hydrodynamics requires e=15 GeV/fm3 at t=0.6 fm/c
(Heinz, Kolb ‘03), giving e0=38 fm-8/3. This leads to
in good agreement with hydrodynamics!
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 .
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…
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)
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.)