Transcript CGC and the Glasma - Tata Institute of Fundamental Research

Glasma instabilities
Kazunori Itakura
KEK, Japan
In collaboration with
Hirotsugu Fujii (Tokyo) and
Aiichi Iwazaki (Nishogakusha)
Goa, September 4th, 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)
High-Energy Heavy-ion Collision
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)
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 (pt ~ Qs) having
anisotropic distribution and soft field (pt << 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:
Q
* At first : only strong gauge fields (given by the CGC) s
pt
soft fields A
particles f(x,p)
* Later :
m
Formation time of a particle with Qs is t ~ 1/Qs
 Have to wait until t ~ 1/Qs for the kinetic description available
(For Qs < 1 GeV, 1/Qs > 0.2 fm/c)
POSSIBLE SOLUTION : INSTABILITIES OF STRONG GAUGE FIELDS
(before kinetic description available)
 “GLASMA INSTABILITY”
Glasma
Glasma (/Glahs-maa/): 2006~
Noun: non-equilibrium matter between Color Glass Condensate (CGC) and
Quark Gluon Plasma (QGP). Created in heavy-ion collisions.
solve Yang Mills eq.
[Dm , Fmn]=0
in expanding geometry with the
CGC initial condition
CGC
Randomly distributed
Stable dynamics of the
Glasma
Boost-invariant Glasma
High energy limit  infinitely thin nuclei
 CGC (initial condition) is purely “transverse”.
 (Ideal) Glasma has no rapidity dependence
“Boost-invariant Glasma”
At t = 0+
(just after collision)
Only Ez and Bz are nonzero
(ET and BT are zero)
[Fries, Kapusta, Li, McLerran, Lappi]
 new!
Time evolution (t >0)
Ez and Bz decay rapidly
ET and BT increase [McLerran, Lappi]
Boost-invariant Glasma
H.Fujii, KI, NPA809 (2008) 88
Just after the collision: only Ez and Bz are nonzero
(Initial CGC is transversely random)
 Glasma = electric and magnetic flux tubes extending in the longitudinal direction
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
Qst=0
Qst=2.0
Bz2, Ez2 =
BT2, ET2=
~1/t
Single flux tube contribution
averaged over transverse space
(finite due to Qs = IR regulator)
Qst =0
0.5
1.0
1.5
2.0
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 !!!
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
g2mt ~ Qst
Unstable Glasma: Numerical results
nmax(t) : Largest n participating instability increases linearly in t
n : conjugate to rapidity h
~ Qst
Unstable Glasma: Analytic results
H.Fujii, KI, NPA809 (2008) 88
Investigate the effects of fluctuation on a single flux tube
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)
Unstable Glasma: Analytic results
H.Fujii, KI, NPA809 (2008) 88
Magnetic background
Yang-Mills equation linearized with respect to fluctuations DOES have
unstable solution for ‘charged’ matter
 2 n
 t a   t a    
t
t

2
1
2
2

 a  0,



  2 n 
2

1
 gB  2 gB
2
Sign of 2 determines the late time behavior
Lowest Landau level ( n=0, 2 = -gB < 0 for minus sign)
r
|m |
In(z) : modified Bessel function
Growth time ~ 1/(gB)1/2 ~1/Qs  instability grows rapidly
Transverse size ~ 1/(gB)1/2 ~1/ Qs for gB~ Qs2
1/Qs
Nielesen-Olesen ’78
Chang-Weiss ’79
n: conjugate
to rapidity h
Unstable Glasma: Analytic results
Modified Bessel function controls the instability
n=8,
n: conjugate to rapidity h
12
2

n 
 - gB  2   0

t 

Stable oscillation
2

n 
 - gB  2   0

t 

Unstable
t wait 
oscillate
grow
The time for instability to become manifest
Modes with small n grow fast !
f~
For large n 
n
gB
~
n
Qs
Unstable Glasma: Analytic results
Electric background
No amplification of the fluctuation = Schwinger mechanism
infinite acceleration of the charged fluctuation
always positive or zero
1/Qs
E
No mass gap for massless gluons  pair creation always possible
Nielsen-Olesen vs Weibel instabilities
Weibel instability
x (current)
• Two step process
• Motion of hard particles in the soft field
additively generates soft gauge fields
• Impossible for homogeneous field
z (force)
y (magnetic field)
• Independent of statistics of charged particles
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 – 2gB < 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
Bz
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 3rd color direction and in z direction.
z
only B  0
(inside a magnetic flux tube)
• Fluctuations
other color components of the gauge field: charged matter field
Anomalous magnetic coupling
induces mixing of fi  mass term with a wrong sign
Glasma instability without expansion
Linearized with respect to fluctuations
1

2  n   gB  2 gB
2

for m = 0
Lowest Landau level (n = 0) of f(-) becomes unstable
Growth rate
g
finite at pz= 0
Qs
For inhomogeneous magnetic field,
gB  g <B>
For gB ~ Qs2
Qs
pz
Glasma instability without expansion
Consequence of Nielsen-Olesen instability??
• Instability stabilized due to nonlinear term (double well potential for
2
f)
g
2
4
V (f )  - gB f 
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
Jz ~ igf*Dz f
~ g2 (B/g)(Qs/g)
Bz
Jz
Bq ~ Qs2/g
for one flux tube
Glasma instability without expansion
Consider fluctuation around Bq
z
Bq
q r
Centrifugal force
Anomalous magnetic term
Approximate solution
Negative for sufficiently large pz
Unstable mode exists for large pz !
Glasma instability without expansion
Numerical solution of the lowest eigenvalue
gB
q
~ QS
Growth rate
g 
-
 unstable
2
gB
-

2
q
pz
~ QS
 stable
Glasma instability without expansion
Growth rate of the glasma w/o expansion
gB
g 
-
q
2
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
Glasma instability without expansion
Numerical simulation
Berges et al. PRD77 (2008) 034504
t-z version of Romatschke-Venugopalan, SU(2)
Initial condition
Instability exists!!
Two different instabilities !
Can be naturally understood
In the Nielsen-Olesen instability
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.
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