Is Thermalization Perturbative

1
Is Thermalization Perturbative?

• Yuri Kovchegov
• The Ohio State University
• based on hep-ph/0503038 and hep-ph/0507134

2
Outline

• First I will present a formal argument
  demonstrating that perturbation theory does not
  lead to thermalization and hydrodynamic
  description of heavy ion collisions.
• 2. Then I will give a simple physical argument
  showing that this is indeed natural
  hydrodynamics may be achieved only in the large
  coupling limit of the theory.

3
Thermalization as Proper Time-Scaling of Energy
Density
Thermalization can be thought of as a transition
between the initial conditions, with energy
density scaling as
with the proper time, to the hydrodynamics-driven
expansion, where the energy density would scale as
for ideal gas Bjorken hydro or as a different
power of tau however, hydro would always
require for the power to be gt 1
4
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
gives
which, due to
cf. Bjorken hydro
? If
then

• No longitudinal pressure exists at early stages.

5
Most General Boost Invariant Energy-Momentum
Tensor
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Classical Fields
(McLerran-Venugopalan model)
Let us start with classical gluon fields produced
in AA collisions. At the lowest order we have
the following diagrams
The field is known explicitly. Substituting it
into (averaging is over the nuclear wave
functions)
we obtain
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Classical Fields
we can use the asymptotics of Bessel functions in
At
to obtain
(Bjorken estimate of energy density)

• Lowest order classical field leads to energy
  density scaling as

8
Classical Fields LO Calculation
p t
e t /2
QS t
p3 t
After initial oscillations one obtains zero
longitudinal pressure with e 2 p for the
transverse pressure.
9
Classical Fields
from full numerical simulation by Krasnitz,
Nara, Venugopalan 01

• All order classical gluon field leads to energy
  density scaling as

Classically there is no thermalization in AA.
10
Our Approach
Can one find diagrams giving gluon fields which
would lead to energy density scaling as
?
Classical fields give energy density scaling as
Can quantum corrections to classical fields
modify the power of tau (in the leading
late-times asymptotics)? Is there analogues of
leading log resummations (e.g. something like
resummation of the powers of D ln t), anomalous
dimensions?
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Energy-Momentum Tensor of a General Gluon Field
Let us start with the most general form of the
gluon field in covariant gauge
plug it into the expression for the energy
momentum tensor
keeping only the Abelian part of the
energy-momentum tensor for now.
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Energy Density of a General Gluon Field
After some lengthy algebra one obtains for energy
density defined as
the following expression
We performed transverse coordinate averaging
(d-function). f1 is some unknown boost-invariant
function, there are also f2 and f3 .
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Energy Density of a General Gluon Field
Let us put k2k2kk 0 in the argument of f1
(and other fs). Integrating over longitudinal
momentum components yields
As one can show
i.e., it is non-zero (and finite) at any order of
perturbation theory.
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Energy Density of a General Gluon Field
When the dust settles we get
leading to
We have established that e has a non-zero term
scaling as 1/t. But how do we know that it does
not get cancelled by the rest of the expression,
which we neglected by putting k2k2kk0 in
the argument of f1 ?
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Corrections
An analysis of corrections to the
scaling contribution, after a lot of
math leads to the following approximate rules
(here h is the space-time rapidity)

• The only tau-dependent corrections are generated
  by k2,k2 and
• (kk)2. Since the k2k2(kk)20 limit is
  finite, corrections may
• come only as positive powers of, say, k2. Using
  the first rule we
• see that they are suppressed by powers of 1/t at
  late times.

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Energy Density Scaling
It appears that the corrections to the leading
energy scaling
are suppressed by powers of t. Therefore, any
set of Feynman diagrams gives which means
that longitudinal pressure is zero at small
coupling and ideal (non-viscous) hydrodynamic
description of the produced system can not
result from perturbation theory!
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Semi-Physical Interpretation
Is this free streaming?
A general gluon production diagram. The gluon is
produced and multiply rescatters at all proper
times.
The dominant contribution appears to come from
all interactions happening early.
? Not free streaming in general, but free
streaming dominates at late times.
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Physical Argument
Put g0 here.
Assume thermalization does take place and
the produced system is described by Bjorken
hydro. Let us put the QCD coupling to zero, g0,
starting from some proper time past
thermalization time, i.e., for all tgtt0gttth.
ggt0
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Puzzle

• From full QCD standpoint, if we put g0 the
  system should
• start free-streaming, leading to

• However, as the equation of state reduces to
  that of an ideal
• gas, , Bjorken hydrodynamics,
  described by
• , leads to
  !? In the g0 limit the
• system still does work in the longitudinal
  direction!?

20
Resolution
The problem is with Bjorken hydrodynamics the
ideal gas equation of state, ,
assumes a gas of particles non-interacting with
each other, but interacting with a thermal bath
(e.g. a box in which the gas is contained, or an
external field). Indeed, in a heavy ion collision
there is no such external thermal bath, and the
ideal gas equation of state is not valid for
produced system.

• Bjorken hydrodynamics is not the right physics
  in the g?0
• limit and hence can not be obtained
  perturbatively.

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Im not saying anything new!
Non-equilibrium viscosity corrections modify the
energy-momentum tensor
Danielewicz Gyulassy 85
In QCD shear viscosity is divergent in the g?0
limit
Arnold, Moore, Yaffe 00
invalidating ideal Bjorken hydrodynamics!
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Im not saying anything new!
Therefore, in the g?0 limit, Bjorken
hydrodynamics gets an o(1) correction, which
tends to reduce the longitudinal
pressure, putting it in line with free streaming
(Of course, divergent shear viscosity implies
that other non-equilibrium corrections, which
come with higher order derivatives of fluid
velocity, are likely to also become important,
giving finite pressure in the end.)
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Is Bjorken Hydrodynamics Impossible?

• Above we showed that
  scaling receives no
• perturbative corrections. Thus the answer to the
  above
• question may be no, it is not possible.

• Alternatively, one may imagine an ansatz like

It gives free streaming in the g?0 limit without
any perturbative corrections, and reduces to
Bjorken hydrodynamics if g?8. In this case
hydrodynamics is a property of the system in the
limit of large coupling! Then the answer is
yes, it is possible.
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Conclusions

• Bad news perturbative thermalization appears to
  be impossible. Weakly interacting quark-gluon
  plasma can not be produced in heavy ion
  collisions.
• Good news non-perturbative thermalization is
  possible, leading to creation of strongly coupled
  plasma, in agreement with RHIC data. However,
  non-perturbative thermalization is very hard to
  understand theoretically (AdS/CFT?).
• More bad news I know an easier problem quark
  confinement. ?

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Backup Slides
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Space-time picture of the Collision

• First particles are
• produced Initial
• Conditions
• Particles interact with
• each other and thermalize
• forming a hot and dense
• medium - Quark-Gluon
• plasma.
• Plasma cools,
• undergoes a confining
• phase transition and
• becomes a gas of hadrons.
• The system falls apart
• freeze out.

27
Thermalization Bottom-Up Scenario
Baier, Mueller, Schiff, Son 00

• Includes 2 ? 3 and 3 ? 2 rescattering processes
  with the LPM effect due to interactions with CGC
  medium (cf. Wong).
• Leads to thermalization over the proper time
  scale of
• Problem Instabilities!!! Evolution of the system
  may develop
• instabilities. (Mrowczynski, Arnold, Lenaghan,
  Moore,
• Romatschke, Randrup, Rebhan, Strickland, Yaffe)
• However, it is not clear whether instabilities
  would speed up the thermalization process. They
  may still lead to isothropization, generating
  longitudinal pressure needed for hydrodynamics to
  work.

28
Corrections to Energy Density
For a wide class of amplitudes we can write
and
with
Then, for the 1st term, using the following
integral
we see that each positive power of k2 leads to a
power of 1/t, such that the neglected terms above
scale as
Similarly one can show that the 2nd term scales
as

• Corrections are subleading at large t and do not
  cancel the leading 1/t term.

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Non-Abelian Terms
Now we can see that the non-Abelian terms in the
energy-momentum tensor are subleading at late
times due to Bessel functions we always
have such that the non-Abelian terms scale as
and
. They can be safely
neglected.
We have proven that at late times the
hydrodynamic behavior of the system can not be
achieved from diagrams, since
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Generalizations Rapidity-Dependent (non-Boost
Invariant) Case
We can generalize our conclusions to the
rapidity-dependent distribution of the produced
particles. First we note that in
rapidity-dependent case Bjorken hydro no longer
applies. However, in the rapidity-dependent hydro
case we may argue that longitudinal pressure is
higher than in Bjs case leading to acceleration
of particles in longitudinal direction and to
energy density decreasing faster than in Bj case
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Rapidity-Dependent Case
To prove that such proper time scaling can not be
obtained from Feynman diagrams we note that
rapidity-dependent corrections come in through
powers of k and k-. Since
we need to worry only about powers of k.
Using
we see that powers of k do not affect the
t-dependence! (logs are derivatives of powers, so
the same applies to them)
Therefore we get again
and hydro appears to be unreachable in the
rapidity-dependent case too.
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Generalizations Including Quarks
We can repeat the same procedure for quark
fields starting from
and, repeating the steps similar to the above, we
obtain for the leading contribution to energy
density at late times
which leads to
? No hydro for quarks either!
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Energy Loss in Instanton Vacuum
An interesting feature of the (well-known) energy
loss formula
is that, due to an extra factor of t in the
integrand, it is particularly sensitive to
medium densities at late times, when the system
is relatively dilute. At such late times
instanton fields in the vacuum may contribute to
jet quenching as much as QGP would
(assuming 1d expansion, see the paper for more
realistic estimates)