1. 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: [hep-ph]
Jean-Paul and Larry’s birthday party @ Saclay, April 2009

2. 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

3. 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
 Vlasov equation
(Plasma instability)
Strong coherent fields
k < Qs
high gluon density
 Yang-Mills equation ~
t > 1/Qs
Pre-equilibrium
state = “Glasma” ~
Initial cond. = CGC

4. Pre-equilibrium states: glasma
Solve the source free Yang Mills eq.
[Dm , Fmn] = 0
in expanding geometry with the CGC initial condition
Initial condition
= CGC
Randomly
distributed
Transverse
Correlation
Length ~ 1/Qs
Formulate in t-h coordinates
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

5. 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

6. Weibel instability Inhomogeneous magnetic fields are enhanced
Introduction (4/6)
Inhomogeneous magnetic fields are enhanced
due to (ordinary) coupling btw
charged particles (with anisotropic distr.)  hard gluons
and soft magnetic field  soft gluon fields
z (Lorenz force)
x (current)
y (magnetic field)
Induced current generates
magnetic field
Both are necessary:
* Inhomogeneous magnetic field
*Anisotropic distribution for hard particles

7. Nielsen-Olesen instability (1/2)
Introduction (5/6)
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
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

8. Nielsen-Olesen instability (2/2)
Introduction (6/6)
Linearized with respect to fluctuations
eigenfrequency Bz
Non-minimal coupling
Free motion in z direction
Landau levels (2N + 1)
 Lowest Landau level (N = 0) of f - is unstable for small pz g
finite at pz= 0
Growth rate :
Transverse size of unstable mode
!! N-O instability is realized if homogeneity
region is larger than Larmor radius !! pz

9. Stable dynamics of the expanding Glasma

10. Stable dynamics: Boost-invariant Glasma
[Fries, Kapusta, Li, Lappi, McLerran]
There appears a flux tube structure !!
Longitudinal fields are generated at t = 0+
Similar to Lund string models but
* transverse correlation 1/Qs
* magnetic flux tube possible
In general both Ez and Bz are present,
but
purely electric purely magnetic
Ez = 0, Bz = Ez = 0, Bz = 0
a1,2 Initial gauge fields
1/Qs / /
E or B, or E&B
Some of the flux tubes are magnetically dominated.

11. Stable dynamics: Boost-invariant Glasma
Expanding flux tubes
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.])
Fujii, Itakura NPA809 (2008) 88
Transverse profile of a
Gaussian flux tube at
Qst =0, 0.5, … 2 (left)
and Qst = 1, 2 (right).
Bz2, Ez2
BT2, ET2
t dependence of field strength from a single flux tube (averaged over transverse space)
compared with the result of classical numerical simulation of boost-invariant Glasma
[Lappi,McLerran]

12. Unstable Glasma in expanding geometry

13. Unstable Glasma rapidity dependent fluctuations!!
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

14. 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 t < 1
longitudinal pressure
g2mt ~ Qst

15. Unstable Glasma w/ expansion: Numerics
nmax(t) : Largest n participating instability increases linearly in t
n : conjugate to rapidity h
~ Qst

16. Unstable Glasma w/ expansion: Analytic study
Linearized equations for fluctuations
SU(2), constant B and E directed to 3rd color and z direction
[Fujii, Itakura,Iwazaki]
: conjugate
to rapidity h
1/Qs
E = 0
Nielsen-Olesen instability
Lowest Landau level (n = 0) gets unstable
due to non-minimal magnetic coupling -2gB
(not Weibel instability) B
modified Bessel fnc
1/Qs E
B = 0
Schwinger mechanism
Infinite acceleration of massless
charged fluctuations.
No amplification of the field
Whittaker function

17. Unstable Glasma w/ expansion: Analytic study
Nielsen-Olesen instability in expanding geometry
[Fujii, Itakura]
Solution : modified Bessel function In(z)
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 t
-- Background field as expanding flux tube
magnetic field on the front of a ripple
B(t) ~ 1/t  t

18. Unstable Glasma in non-expanding geometry

19. Glasma instability without expansion
Numerical simulation Berges et al. PRD77 (2008)
t-z version of Romatschke-Venugopalan, SU(2)
Initial condition is stochastically generated
 Corresponds to “non-expanding glasma”
Instability exists!! Can be naturally understood
Two different instabilities ! In the Nielsen-Olesen instability

20. 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 (D ~ Qs)
Can allow longitudinal flux tubes when

21. 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. g
Growth rate
finite at pz= 0 pz
Inhomogeneous magnetic field : B  Beff

22. 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 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

23. Glasma instability without expansion
Consider fluctuation around Bq z Bq q r
Centrifugal force
Non-minimal magnetic coupling
Approximate solution at high pz
Negative for sufficiently large pz Unstable mode exists for large pz !

24. Glasma instability without expansion
Numerical solution of the lowest eigenvalue (red line)
Growth rate
Approximate
solution
Increasing function of pz
 Numerical solution

25. Glasma instability without expansion
Growth rate of the glasma w/o expansion
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

26. 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!

27. 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
a1 , a2 : gluon fields of nuclei
In Region (3), and at t =0+, the gauge field is determined by a1 and a2