# CGC and the Glasma

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```					       Instabilities in
expanding and non-expanding
glasmas
K. Itakura (KEK, Japan      )
as one of the “CGC children”
based on
* H. Fujii and KI, “Expanding color flux tubes and instabilities”
Nucl. Phys. A 809 (2008) 88
* H.Fujii, KI, A.Iwazaki, “Instabilities in non-expanding glasma”
arXiv:0903.2930 [hep-ph]

Jean-Paul and Larry’s birthday party @ Saclay, April 2009
Contents

• Introduction/Motivation
What is a glasma?
Instabilities in Yang-Mills systems
• Stable dynamics of the expanding glasma
Boost-invariant color flux tubes
• Unstable dynamics of the glasma
with expansion:
 Nielsen-Olesen instability
without expansion:
 “Primary” and “secondary” Nielsen-Olesen instabilities
• Summary
Introduction (1/6)
Relativistic Heavy Ion Collisions in High Energy Limit

Particles
<
k ~ Qs (or simply k~Qs)
 Boltzmann equation
(“Bottom-up” scenario)

Soft fields + hard particles
k<<Qs          k ~ Qs        t > 1/Qs
Pre-equilibrium      Vlasov equation
state = “Glasma”         (Plasma instability)

Strong coherent fields
k < Qs
~
high gluon density
Initial cond. = CGC          Yang-Mills equation
Pre-equilibrium states: glasma
Solve the source free Yang Mills eq.

[Dm , Fmn] = 0
in expanding geometry with the CGC
initial condition

Initial condition             Randomly
= CGC                    distributed

Transverse
Formulate in  coordinates                                  Correlation
1 x                                   Length ~ 1/Qs
  t  z  2 x x ,   ln 
2   2         

2 x
proper time             rapidity
Infinitely thin  boost-inv. glasma

Glasma is described by coherent strong gauge fields which is
boost invariant in the limit of high energy
Issues in glasma physics

Glasma  Initially very anisotropic with flux tube structure

1. How the glasma evolves towards thermal equilibrium?
Time evolution from CGC initial conditions
 stable and unstable dynamics

2. Any “remnants” of early glasma states in the final
states?
Longitudinal color flux tube structure
 long range correlation in rapidity space??
 particle production from flux tubes

THIS TALK  1. Stable and unstable dynamics

Instabilities in the Yang-Mills systems
 Weibel and Nielsen-Olesen instabilities
Introduction (4/6)
Weibel instability
Inhomogeneous magnetic fields are enhanced
due to (ordinary) coupling btw
charged particles (with anisotropic distr.)  hard gluons
and soft magnetic field                      soft gluon fields

x (current)

z (Lorenz force)

y (magnetic field)

t
Be
Induced current generates
 ( p)             magnetic field

 ( p)  0 for p  0
Both are necessary:
* Inhomogeneous magnetic field
*Anisotropic distribution for hard particles
p
Introduction (5/6)
Nielsen-Olesen instability (1/2)
Nielsen, Olesen, NPB144 (78) 376
Chang, Weiss, PRD20 (79) 869

Homogeneous (color) magnetic field is unstable due to non-minimal
coupling in non-Abelian gauge theory

ex) Color SU(2) pure Yang-Mills
Background field B z  0
Constant magnetic field in 3rd color direction and in z direction.
Fluctuations
Other color components of the gauge field: charged matter field

Abelian part   non-Abelian part

Non-minimal magnetic coupling
induces mixing of fi  mass term for f with a wrong sign
Introduction (6/6)
Nielsen-Olesen instability (2/2)
Linearized with respect to fluctuations
Bz

eigenfrequency

Non-minimal coupling
Free motion in z direction Landau levels (2N + 1)
 Lowest Landau level (N = 0) of f  is unstable for small pz

    finite at pz= 0
Growth rate :
Transverse size of unstable mode l ~ 1 / gB

!! N-O instability is realized if homogeneity                               pz
region is larger than Larmor radius !!
Stable dynamics of the
expanding Glasma
Stable dynamics: Boost-invariant Glasma
[Fries, Kapusta, Li, Lappi, McLerran]
There appears a flux tube structure !!
Longitudinal fields are generated at  = 0+
E , B  O (Qs2 / g )
1,2   Initial gauge fields

Similar to Lund string models but           1/Qs
* transverse correlation 1/Qs
* magnetic flux tube possible

In general both Ez and Bz are present,
but  x  x             x x
1     2           1     2

1y  2y          1y  2y
purely electric    purely magnetic
Ez = 0, Bz = 0
/                Ez = 0, Bz = 0
/                       E or B, or E&B

Some of the flux tubes are magnetically dominated.
Stable dynamics: Boost-invariant Glasma
Expanding flux tubes                                              Fujii, Itakura NPA809 (2008) 88

Inside xt < 1/Qs : strong but homogeneous gauge field
Outside            : weaker field  Can be approximately described by Abelian field
(cf: similar to free streaming approx. [Kovchegov, Fukushima et al.])

Transverse profile of a
Gaussian flux tube at
Qs =0, 0.5, … 2 (left)
and Qst = 1, 2 (right).

[Lappi,McLerran]
 dependence of field                Bz2, Ez2
strength from a single
flux tube (averaged
over transverse space)
BT2, ET2
compared with the result
of classical numerical
simulation of boost-
invariant Glasma
Unstable Glasma
in expanding geometry
Unstable Glasma
Boost-inv. Glasma (without rapidity dependence) cannot thermalize
Need to violate boost invariance !!!
origin: quantum fluctuations? NLO contributions? (Finite thickness effects)

 Glasma is indeed unstable against
rapidity dependent fluctuations!!

Numerical simulations : expanding  P.Romatschke & R.Venugopalan
non-expanding  J.Berges et al.
Analytic studies : expanding & non-expanding  Fujii-Itakura, Iwazaki
Unstable Glasma w/ expansion: Numerics
P.Romatschke & R. Venugopalan, 2006
Small rapidity dependent fluctuation can grow
exponentially and generate longitudinal pressure.

3+1D numerical simulation

PL ~
Very much similar to Weibel
Instability in expanding plasma
[Romatschke, Rebhan]

Isotropization mechanism
starts at very early time Qs  < 1

g2m ~ Qs
Unstable Glasma w/ expansion: Numerics
nmax() : Largest n participating instability increases linearly in 

n : conjugate to rapidity                   ~ Qs
Unstable Glasma w/ expansion: Analytic study
Linearized equations for fluctuations                                        [Fujii, Itakura,Iwazaki]

n: conjugate
SU(2), constant B and E directed to 3rd color and z direction                  to rapidity 
1             2
~ ()
1

~ () 
  a     
 2
 
n 
gE 2                             
   (2n | m | 1  m  2) gBa  0 a (  )  ein (a 1  ia 2 )
~

         2                               
                             

a  a x  ia y
B=0                          E=0

Schwinger mechanism                           Nielsen-Olesen instability
Infinite acceleration of massless            Lowest Landau level (n = 0) gets unstable
charged fluctuations.                        due to non-minimal magnetic coupling -2gB
No amplification of the field                 (not Weibel instability)

modified Bessel fnc
Whittaker function

1/Qs                       E                  1/Qs                       B
Unstable Glasma w/ expansion: Analytic study
[Fujii, Itakura]
Nielsen-Olesen instability in expanding geometry
Solution : modified Bessel function In(z)

r |m|

• Growth time can be short
 instability grows rapidly!
Important for early thermalization?                          

• Rapidity dependent (pz dependent) fluctuations are enhanced

• Consistent with the numerical results by Romatchke and Venugopalan
-- Largest n participating instability increases linearly in 
-- Background field as expanding flux tube
magnetic field on the front of a ripple
B() ~ 1/            exp      #   
Unstable Glasma
in non-expanding geometry
Glasma instability without expansion
Numerical simulation         Berges et al. PRD77 (2008) 034504

t-z version of Romatschke-Venugopalan, SU(2)
Initial condition is stochastically generated

 ~ Qs   z
 Corresponds to “non-expanding glasma”

Instability exists!!                 Can be naturally understood
Two different instabilities !         In the Nielsen-Olesen instability
Glasma instability without expansion
Initial condition

With a supplementary condition

Initial condition is purely “magnetic”

Magnetic fields B is
homogeneous in the z direction
varying on the transverse plane ( ~ Qs)

Can allow longitudinal flux tubes when   B z  B x , B y
Primary N-O instability
Consider a single magnetic flux tube of a transverse size ~1/Qs

 approximate by a constant magnetic field (well inside the flux tube)

The previous results on the N-O instability can be immediately used.

Growth rate                                  finite at pz= 0
gB

pz
gB ~ QS

Inhomogeneous magnetic field : B  Beff
Glasma instability without expansion
Consequence of Nielsen-Olesen instability??
• Instability stabilized due to nonlinear term (double well potential for f )
g2 4
V (f )   gBf 2
f  f ~ B/g
4
• Screen the original magnetic field Bz
• Large current in the z direction induced
• Induced current Jz generates (rotating) magnetic field Bq (rot B =J )

Jz
Bq ~ Qs2/g
for one flux tube
Bz
Glasma instability without expansion
Consider fluctuation around Bq
z
Bq

q r

Centrifugal force      Non-minimal magnetic coupling
Approximate solution at high pz
 gBq 
 2 ~  gBq 1  2 
 4p 
    z 
Negative for sufficiently large pz        Unstable mode exists for large pz !
Glasma instability without expansion

Numerical solution of the lowest eigenvalue (red line)

Growth rate
Approximate
solution
  gBq ~ QS

 Numerical solution                   Increasing function of pz
Glasma instability without expansion
Growth rate of the glasma w/o expansion

gBq

gB z

pz
Nielsen-Olesen instability with a constant Bz is followed by
Nielsen-Olesen instability with a constant Bq
• pz dependence of growth rate has the information of the profile
of the background field
• In the presence of both field (Bz and Bq) the largest pz for the primary
instability increases
Summary

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.

And, happy birthday, Jean-Paul and Larry!
CGC as the initial condition for H.I.C.
HIC = Collision of two sheets

[Kovner, Weigert,
McLerran, et al.]

r1            r2
Each source creates the gluon field for each nucleus. Initial condition

1 , 2 : gluon fields of nuclei
In Region (3), and at  =0+, the gauge field is determined by 1 and 2

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