glasma instability

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Nov 16, 2013 (3 years and 11 months ago)

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Glasma instabilities

Kazunori Itakura

KEK, Japan


In collaboration with

Hirotsugu Fujii (Tokyo) and

Aiichi Iwazaki (Nishogakusha)

Goa, September 4
th
, 2008

Dona Paula Beach Goa, photo from http://www.goa
-
holidays
-
advisor.com/

Contents


Introduction: Early thermalization problem



Stable dynamics of the Glasma


Boost
-
invariant color flux tubes



Unstable dynamics of the Glasma


Instability a la Nielsen
-
Olesen


Instability induced by enhanced fluctuation (w/o expansion)



Summary

Introduction (1/3)

5. Individual hadrons


freeze out

4. Hadron gas


cooling with expansion

3. Quark Gluon Plasma (QGP)


thermalization, expansion

2. Non
-
equilibrium state
(Glasma)


collision

1. High energy nuclei (CGC)

High
-
Energy Heavy
-
ion Collision

Big unsolved question in heavy
-
ion physics



Q: How is thermal equilibrium (QGP) is achieved after the collision?


What is the dominant mechanism for thermalization?

Introduction (2/3)


“Early thermalization problem” in HIC




Hydrodynamical simulation of the RHIC data suggests


QGP may be formed within a VERY short time t ~ 0.6 fm/c.


Hardest problem!


1. Non
-
equilibrium physics by definition


2. Difficult to know the information before the formation of QGP


3. Cannot be explained within perturbative scattering process





Need a
new

mechanism for rapid equilibration


Possible candidate:



Plasma instability
” scenario





Interaction btw
hard

particles

(
p
t

~
Qs
)
having
anisotropic distribution and
soft

field

(
p
t

<<
Qs
)
induces instability of the soft field


isotropization

Weibel instability

Arnold, Moore, and Yaffe, PRD72 (05) 054003

Introduction (3/3)


Problems of “Plasma instability” scenario



1. Only “isotropization”
(of energy momentum tensor)

is achieved.



The true thermalization
(probably, due to collision terms)

is far away.





Faster scenario ? Another instability ??


2. Kinetic description valid
only after particles are formed out of fields
:



* At first :


* Later :



Formation time of a particle with
Q
s

is
t

~ 1/
Q
s






Have to wait until
t

~ 1/
Q
s

for the kinetic description available


(For
Q
s

< 1 GeV,
1/
Q
s

> 0.2 fm/c)


POSSIBLE SOLUTION : INSTABILITIES OF STRONG GAUGE FIELDS


(before kinetic description available)






GLASMA INSTABILITY


only strong gauge fields (given by the CGC)

Q
s

p
t

soft fields A
m

particles

f
(x,p)

Glasma

Glasma

(/Glahs
-
maa/): 2006~

Noun:
non
-
equilibrium matter between Color
Glass

Condensate (CGC) and
Quark Gluon Plas
ma

(QGP). Created in heavy
-
ion collisions.

solve Yang Mills eq.

[
D
m
,
F
mn
]=0

in expanding geometry with the
CGC initial condition

CGC

Randomly distributed

Stable dynamics of the
Glasma

Boost
-
invariant Glasma

At
t
= 0
+

(just after collision)


Only E
z

and B
z

are nonzero


(E
T

and B
T
are zero)


[Fries, Kapusta, Li, McLerran, Lappi]

Time evolution (
t
>0)


E
z

and B
z

decay rapidly


E
T

and B
T
increase
[McLerran, Lappi]



new!

High energy limit


infinitely thin nuclei




CGC
(initial condition)

is purely “transverse”.




(Ideal) Glasma has no rapidity dependence


“Boost
-
invariant Glasma”

Boost
-
invariant Glasma

Just after the collision: only E
z

and B
z

are nonzero


(Initial CGC is transversely random)



Glasma = electric and magnetic flux tubes extending in the longitudinal direction

H.Fujii, KI, NPA809 (2008) 88

1/Qs

random

Typical configuration of a single event


just after the collision

Boost
-
invariant Glasma

An isolated flux tube with a Gaussian profile oriented to a certain color direction

Qs
t
=2.0

Qs
t
=0

Qs
t
=0


0.5


1.0


1.5


2.0

B
z
2
, E
z
2
=

B
T
2
, E
T
2
=

~1/
t

Single flux tube contribution

averaged over transverse space

(finite due to Qs = IR regulator)

Boost
-
invariant Glasma

A single expanding flux tube at fixed time

1/Qs

Glasma instabilities


Unstable Glasma: Numerical results

Boost invariant Glasma (without rapidity dependence) cannot thermalize



Need to violate the boost invariance !!!

3+1D numerical simulation



P
L

~

Very much similar to Weibel

Instability in expanding plasma


[Romatschke, Rebhan]


Isotropization mechanism

starts at very early time
Qs

t

< 1

P.Romatschke & R. Venugopalan, 2006


Small
rapidity dependent

fluctuation can grow exponentially
and generate longitudinal pressure .


g
2
mt
~
Q
s
t

Unstable Glasma: Numerical results

n
max
(
t
)

: Largest
n


participating instability increases linearly in
t

n
:
conjugate to rapidity

h


~ Q
s
t

Unstable Glasma: Analytic results

H.Fujii, KI,
NPA809 (2008) 88


Rapidity dependent fluctuation

Background field = boost invariant Glasma




constant magnetic/electric field in a flux tube


* Linearize the equations of motion wrt fluctuations





magnetic / electric flux tubes


* For simplicity, consider SU(2)

Investigate the effects of fluctuation on a single flux tube

Unstable Glasma: Analytic results

H.Fujii, KI,
NPA809 (2008) 88


Magnetic background


1/
Q
s

unstable solution

for ‘charged’ matter

Yang
-
Mills equation linearized with respect to fluctuations DOES have

Growth time ~ 1/(
gB
)
1/2
~1/
Q
s



instability grows rapidly

Transverse size ~ 1/(
gB
)
1/2
~1/
Q
s

for
gB
~
Q
s
2

Nielesen
-
Olesen ’78

Chang
-
Weiss ’79

I
n
(
z
)
: modified Bessel function

gB
gB
n
2
2
1
2
2










,
0
1
2
2
2
2














a
a
a
t
n

t
t
t
Lowest Landau level (
n
=0,

2
=
-
gB
< 0

for minus sign
)

n
:
conjugate


to rapidity

h


|
|
m
r
Sign of

2

determines the late time behavior

Modified Bessel function controls the instability
f

~

Unstable Glasma: Analytic results

n
=8, 12

oscillate grow

0
0
2
2
2
2










-










-
t
n
t
n
gB
gB
Stable oscillation



Unstable

Qs
gB
n
n
t
~

wait

The time for instability to become manifest

For large
n





Modes with small
n

grow fast !

n
:
conjugate to rapidity

h


Electric background





No amplification of the fluctuation = Schwinger mechanism


infinite acceleration of the charged fluctuation





Unstable Glasma: Analytic results


1/
Q
s

E

No mass gap for massless gluons


pair creation always possible

always positive or zero

Nielsen
-
Olesen vs Weibel instabilities

Nielsen
-
Olesen instability


* One step process


* Lowest Landau level in a strong magnetic field becomes


unstable due to
anomalous magnetic moment




2

= 2(
n
+1/2)
gB



2
gB

< 0 for
n
=0


* Only in non
-
Abelian gauge field


vector field


spin 1


non
-
Abelian


coupling btw field and matter


* Possible even for homogeneous field

B
z

Weibel instability



z
(force)

x
(current)


y
(magnetic field)



Two step process



Motion of hard particles in the soft field
additively generates soft gauge fields



Impossible for homogeneous field



Independent of statistics of charged particles

Glasma instability

without expansion


with H.Fujii and A. Iwazaki


(in preparation)


* What is the characteristics of the N
-
O instability?

* What is the consequence of the N
-
O instability?


(Effects of backreaction)


Glasma instability without expansion


Color SU(2) pure Yang
-
Mills


Background field

(


“boost invariant glasma”
)


Constant magnetic field in 3
rd

color direction and in z direction.


only (inside a magnetic flux tube)


Fluctuations



other color components of the gauge field: charged matter field


0

z
B

Anomalous magnetic coupling


induces mixing of
f
i



mass term
with a wrong sign

Glasma instability without expansion

Linearized with respect to fluctuations

for
m
= 0

gB
gB
n
2
2
1
2








Lowest Landau level (
n

= 0) of
f
(
-
)
becomes unstable

p
z

g

finite at
p
z
= 0

For
gB
~
Q
s
2

Q
s

Q
s

For inhomogeneous magnetic field,


gB



g
<
B
>

Growth rate

Glasma instability without expansion

Consequence of Nielsen
-
Olesen instability??



Instability stabilized due to nonlinear term (double well potential for
f
)






Screen the original magnetic field
B
z


Large current in the z direction induced


Induced current
J
z

generates (rotating) magnetic field
B
q


B
z

J
z
~
ig
f
*D
z

f


~ g
2

(
B/g
)(
Qs/g
)

J
z


B
q

~
Q
s
2
/
g

for one flux tube

B/g
g
gB
V
~


4
)
(
4
2
2
f
f
f
f


-

Glasma instability without expansion

Consider fluctuation around

B
q



B
q


r

q

z

Centrifugal force

Anomalous magnetic term

Approximate solution

Negative for sufficiently large
p
z
Unstable mode exists for large
p
z
!

Glasma instability without expansion

Numerical solution of the lowest eigenvalue

2

g
-

2

-
z
p
S
Q
gB
~
q
S
Q
gB
~
q


unstable



stable

Growth rate

Glasma instability without expansion

Growth rate of the glasma w/o expansion

2

g
-

z
p
Nielsen
-
Olesen instability with a constant
B
z

is followed by

Nielsen
-
Olesen instability with a constant
B
q


q
gB
z
gB


p
z

dependence of growth rate has the information of the profile

of the background field



In the presence of both field (
B
z

and
B
q
) the largest
p
z

for the primary

instability increases

Glasma instability without expansion

Numerical simulation

Berges et al. PRD77 (2008) 034504



t
-
z

version of Romatschke
-
Venugopalan, SU(2)


Initial condition


Instability exists!! Can be naturally understood


Two different instabilities ! In the Nielsen
-
Olesen instability

CGC and glasma are important pictures for the
understanding of heavy
-
ion collisions


Initial Glasma = electric and magnetic flux tubes.


Field strength decay fast and expand outwards.


Rapidity dependent fluctuation is unstable in the magnetic
background. A simple analytic calculation suggests that
Glasma (Classical YM with stochastic initial condition)
decays due to the Nielsen
-
Olesen (N
-
O) instability.


Moreover, numerically found instability in the
t
-
z coordinates

can also be understood by N
-
O including the existence of
the secondary instability.



Summary

CGC as the initial condition for H.I.C.

HIC = Collision of two sheets

r
1

r
2

Each source creates the gluon field for each nucleus.


Initial condition


a
1

,
a
2

: gluon fields of nuclei

[Kovner, Weigert,

McLerran, et al.]

In
Region (3)
, and
at

t
=0+
,
the gauge field is determined by

a
1

and

a
2