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
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