Can Thermalization Happen Can Thermalization Happen at Small Coupling?at Small Coupling?

Can Thermalization Happen Can Thermalization Happen at Small Coupling?at Small Coupling?

Yuri KovchegovYuri KovchegovThe Ohio State UniversityThe Ohio State University

based on hep-ph/0503038 and hep-ph/0507134based on hep-ph/0503038 and hep-ph/0507134

OutlineOutline

1. 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.

Thermalization as Proper Time-Scaling Thermalization as Proper Time-Scaling of Energy Densityof Energy Density

Thermalization can be thought of as a transition betweenthe initial conditions, with energy density scaling as

with the proper time, to the hydrodynamics-driven expansion, where the energy density would scale as

3/4

1~

1

~

for ideal gas Bjorken hydro or as a different power of tau: however, hydro would always require for the power to be > 1:

0,1

~1

Most General Boost Invariant Energy-Most General Boost Invariant Energy-Momentum TensorMomentum Tensor

The most general boost-invariant energy-momentum tensor for a high energy collision of two very large nuclei is

)(000

0)(00

00)(0

000)(

3

p

p

pT

which, due to 0 T gives

3p

d

d

cf. Bjorken hydro→ If 1

~ then 03 p

No longitudinal pressure exists at early stages.

Most General Boost Invariant Energy-Most General Boost Invariant Energy-Momentum TensorMomentum 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 longitudinal pressure ↔ deviations from 1

~

3p

d

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

1~

Classical FieldsClassical FieldsLet 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)

2)(4

1 aaa FgFFT

we obtain })]([)]({[

22

02

12

222

TTT kJkJk

dbdkd

dNkd

(McLerran-Venugopalan model)

Classical FieldsClassical Fields

})]([)]({[2

20

21

222

2

TTT kJkJkdbdkd

dNkd

At SQ

1 we can use the asymptotics of Bessel

functions in

to obtain

dbd

dEk

dbdkd

dNkdQ T

TS 2222 11

)/1(

(Bjorken estimate of energy density)

Lowest order classical field leads to energy density scaling as

1

~classical

Classical Fields: LO CalculationClassical Fields: LO Calculation

QS

p

p3

After initial oscillations one obtains zero longitudinal pressure with p for the transverse pressure.

Classical FieldsClassical Fields

All order classical gluon field leads to energy density scaling as

1

~classical

from full numerical simulation by Krasnitz, Nara, Venugopalan ‘01

Classically there is no thermalization in AA.

Our ApproachOur Approach

Can one find diagrams giving gluon fields which would lead toenergy density scaling as

0,1

~1

Classical fields give energy density scaling as

1

~classical

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 ln ), “anomalous dimensions”?

?

Energy-Momentum Tensor of a Energy-Momentum Tensor of a General Gluon FieldGeneral Gluon Field

Let us start with the most general form of the “gluon field” in covariant gauge

)()2(

)(0

24

4

kJkik

ie

kdxA axkia

plug it into the expression for the energy momentum tensor

2)(4

1 aaa FgFFT

keeping only the Abelian part of the energy-momentum tensor for now.

Energy Density of a General Gluon FieldEnergy Density of a General Gluon Field

After some lengthy algebra one obtains for energy densitydefined as

TT

x

2

2

1

the following expression:

termssimilarkkkkkfkkkx

kkkikkik

e

S

kdkd

T

xkixki

2),',',('

)'()''()()2(2

'

221

2

2

02

02

'

6

44

We performed transverse coordinate averaging functionf1 is some unknown boost-invariant function, there are also f2 and f3 .

Let us put k2=k’2=k•k’ =0 in the argument of f1 (and other f’s). Integrating over longitudinal momentum components yields:

termssimilarkkkkkf

kJkJkS

kd

T

TTT

2),0',0',0(

)()()2(8

221

20

21

22

2

As one can show

dykd

dNkJkJ

termssimilarkkkkkf

k

aa

T

23

0

221

)2(2)()(

2),0',0',0(

2

i.e., it is non-zero (and finite) at any order of perturbation theory.

Energy Density of a General Gluon FieldEnergy Density of a General Gluon Field

Energy Density of a General Gluon FieldEnergy Density of a General Gluon Field

When the dust settles we get

dbd

dEk

dbdkd

dNkdQ T

TS 2222 11

)/1(

})]([)]({[2

20

21

222

2

TTT kJkJkdbdkd

dNkd

leading to

We have established that has a non-zero term scaling as 1/.

But how do we know that it does not get cancelled by the rest of the expression, which we neglected by putting k2=k’2=k•k’=0 in the argument of f1 ?

Corrections to Energy DensityCorrections to Energy DensityFor a wide class of amplitudes we can write

),',',(])'[(),',',()'(

),0',0',0(),',',(22)2(222)1(22

221

221

21TT

TT

kkkkkgkkkkkkkgkk

kkkkkfkkkkkf

with 0),0',0',0( 22)2,1( Tkkkkkg

Then, for the 1st term, using the following integral:

)(2

)1(

2)(

21

02

TiTxikxik kJe

kkikedkdk

we see that each positive power of k2 leads to a power of 1/such that the neglected terms above scale as 121/1~

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

and 0, 21

Similarly one can show that the 2nd term scales as 221/1~

CorrectionsCorrectionsAn analysis of corrections to the scaling

contribution, can be summarized by the following approximate rules:

1

~

ek

kk

kk

kkkk

T

TT

2',

1)'(,', 222

(here is the space-time rapidity)The only tau-dependent corrections are generated by k2,k’2 and(k+k’)2. Since the k2=k’2=(k+k’)2=0 limit is finite, corrections may come only as positive powers of, say, k2. Using the first rule wesee that they are suppressed by powers of 1/ at late times.

Non-Abelian TermsNon-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.

1

A

2/3

1

AAA

2

1

AAAA

dbd

dEk

dbdkd

dNkdQ T

TS 2222 11

)/1(

We have proven that at late times the hydrodynamic behavior of the system can not be achieved from diagrams, since

Generalizations: Rapidity-Dependent Generalizations: Rapidity-Dependent (“non-Boost Invariant”) Case(“non-Boost Invariant”) Case

We can generalize our conclusions to the rapidity-dependentdistribution of the produced particles.

First we note that in rapidity-dependent case Bjorken hydro nolonger applies. However, in the rapidity-dependent hydro case we may argue that longitudinal pressure is higher than in Bj’s case leading to acceleration of particles in longitudinal directionand to energy density decreasing faster than in Bj case:

0,1

~23

4

Rapidity-Dependent CaseRapidity-Dependent CaseTo 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+.

k

kkk

2

22

Using

)(2

2)()( 0221

02

T

iTxikxik kJeek

ikkikedkdk

we see that powers of k+ do not affect the dependencelogs are derivatives of powers, so the same applies to them)Therefore we get again

dbd

dEk

dbdkd

dNkdQ T

TS 2222 11

)/1( and hydro appears to be unreachable in the rapidity-dependent case too.

Generalizations: Including QuarksGeneralizations: Including QuarksWe can repeat the same procedure for quark fields: startingfrom DD

iT

2and, repeating the steps similar to the above, we obtain for the leading contribution to energy density at late times

})]([)]([2)()({4

20

2120

222

2

TTTTT

qq kJkJkJkJk

dbdkd

dNkd

which leads to

dbd

dEk

dbdkd

dNkdQ

quarksT

T

q

Sq

2222 11

)/1(

→ No hydrodynamics for quarks either!

Energy Density ScalingEnergy Density Scaling

It appears that the corrections to the leading energy scaling

dbd

dEk

dbdkd

dNkdQ T

TS 2222 11

)/1(

are suppressed by powers of . 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!

1

~

Semi-Physical InterpretationSemi-Physical Interpretation

Is this “free streaming”?

A general gluon productiondiagram. 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 dominatesat late times.

Physical ArgumentPhysical Argument

Assume thermalizationdoes take place and theproduced system is described by Bjorken hydro. Let usput the QCD coupling tozero, g=0, starting fromsome proper time past thermalization time, i.e., forall th.

Put g=0 here.

g>0

PuzzlePuzzle

From full QCD standpoint, if we put g=0 the system shouldstart free-streaming, leading to

1

~

However, as the equation of state reduces to that of an idealgas, , Bjorken hydrodynamics, described by

, leads to !? In the g=0 limit the

system still does work in the longitudinal direction!?

p3

p

d

d 3/4

1~

ResolutionResolution

The problem is with Bjorken hydrodynamics:

the ideal gas equation of state, , assumes a gasof 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 isno such external thermal bath, and the ideal gas equation ofstate is not valid for produced system.

p3

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

I’m not saying anything new!I’m not saying anything new!

3

4)(000

03

2)(00

003

2)(0

000)(

p

p

p

T

Non-equilibrium viscosity corrections modify the energy-momentum tensor:

DanielewiczGyulassy ‘85

)/1ln(

1~

4 gg

In QCD shear viscosity is divergent in the g→0 limit

Arnold, Moore, Yaffe ‘00

invalidating ideal Bjorken hydrodynamics!

I’m not saying anything new!I’m not saying anything new!

3

4)(000

03

2)(00

003

2)(0

000)(

p

p

p

T

Therefore, in the g→0 limit, Bjorken hydrodynamics gets ano(1) correction, which tends to reduce the longitudinal pressure,putting it in line with free streaming:

0000

0)(00

00)(0

000)(

2

2

p

pT

(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.)

Is Bjorken Hydrodynamics Impossible?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”.

1

~

Alternatively, one may imagine an ansatz like

0,0,1

~]/exp[1

cwithc

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

ConclusionsConclusions• 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. ☺

Backup SlidesBackup Slides

Space-time picture of the Space-time picture of the CollisionCollision

1.1. First particles areFirst particles are

produced: produced: InitialInitial

ConditionsConditions

2.2. Particles interact withParticles interact with

each other and thermalizeeach other and thermalize

forming a hot and denseforming a hot and dense

medium - medium - Quark-GluonQuark-Gluon

plasma.plasma.

3.3. Plasma cools,Plasma cools,

undergoes a confiningundergoes a confining

phase transition andphase transition and

becomes a becomes a gas of hadronsgas of hadrons..

4.4. The system falls apart:The system falls apart:

freeze outfreeze out. .

Thermalization: Bottom-Up ScenarioThermalization: Bottom-Up Scenario

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.

Baier, Mueller, Schiff, Son ‘00

SQ5/130

1~

Energy Loss in Instanton VacuumEnergy Loss in Instanton VacuumAn interesting feature of the (well-known) energy loss formula

f

dE

0

)(

is that, due to an extra factor of in the integrand, it is particularly sensitive to medium densities at late times, whenthe system is relatively dilute.

At such late times instanton fields in the vacuum may contributeto jet quenching as much as QGP would:

QGPQCD

S

QCD

Sinst E

QQE

~~

4

5

5

6

(assuming 1d expansion, see the paper for more realistic estimates)