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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 - PowerPoint PPT Presentation
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Glasma Glasma instabilitiesinstabilities
Kazunori ItakuraKazunori ItakuraKEK, JapanKEK, Japan
In collaboration with In collaboration with Hirotsugu Fujii (Tokyo) and Hirotsugu Fujii (Tokyo) and
Aiichi Iwazaki (Nishogakusha)Aiichi Iwazaki (Nishogakusha)Goa, September 4th, 2008
Dona Paula Beach Goa, photo from http://www.goa-holidays-advisor.com/
ContentsContents
• 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 expan
sion)
• Summary
Introduction (1/3)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)Introduction (2/3)
“ “Early thermalization problem” in HICEarly 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 instabilityPlasma instability” scenario
Interaction btw hard particles (pt ~ Qs) having anisotropic distribution and soft field (pt << Qs) induces instability of the soft field isotropization
Weibel instabilityArnold, Moore, and Yaffe, PRD72 (05) 054003
Introduction (3/3)Introduction (3/3)
Problems of “Plasma instability” scenarioProblems 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 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 availabl
e)
“GLASMA INSTABILITY”
only strong gauge fields (given by the CGC)Qs
ptsoft fields A particles f(x,p)
GlasmaGlasmaGlasma (/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. [D, F]=0
in expanding geometry with the CGC initial condition
CGC
Randomly distributed
Stable dynamics of Stable dynamics of the Glasmathe Glasma
Boost-invariant Glasma
At = 0+ (just after collision) Only Ez and Bz are nonzero (ET and BT are zero) [Fries, Kapusta, Li, McLerran, Lappi]
Time evolution (>0) Ez and Bz decay rapidly ET and BT 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 Ez and Bz 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 GlasmaAn isolated flux tube with a Gaussian profile oriented to a certain color direction
Qs=2.0
Qs=0
Qs=0 0.5 1.0 1.5 2.0
Bz2, Ez
2 =
BT2, ET
2=
~1/
Single flux tube contribution averaged over transverse space (finite due to Qs = IR regulator)
Boost-invariant GlasmaA single expanding flux tube at fixed time
1/Qs
Glasma instabilitiesGlasma instabilities
Unstable Glasma: Numerical results
Boost invariant Glasma (without rapidity dependence) cannot thermalize Need to violate the boost invariance !!!
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
P.Romatschke & R. Venugopalan, 2006 Small rapidity dependent fluctuation can grow exponentially and generate longitudinal pressure .
g2~ Qs
long
i tudi
nal p
ress
ure
Unstable Glasma: Numerical results
max() : Largest participating instability increases linearly in
conjugate to rapidity ~ Qs
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/Qs
unstable solution for ‘charged’ matter
Yang-Mills equation linearized with respect to fluctuations DOES have
Growth time ~ 1/(gB)1/2 ~1/Qs instability grows rapidly Transverse size ~ 1/(gB)1/2 ~1/ Qs for gB~ Qs
2
Nielesen-Olesen ’78Chang-Weiss ’79
I(z) : modified Bessel function
gBgBn 22
122
,0
12
222
aaa
Lowest Landau level ( n=0, 2 = gB < 0 for minus sign)
conjugate to rapidity
||mr
Sign of 2 determines the late time behavior
Modified Bessel function controls the instability f ~
Unstable Glasma: Analytic results
=8, 12
oscillate grow
0
0
2
2
2
2
gB
gB Stable oscillation
Unstable
QsgB
~ wait
The time for instability to become manifest
For large Modes with small grow fast !
conjugate to rapidity
Electric background
No amplification of the fluctuation = Schwinger mechanism
infinite acceleration of the charged fluctuation
Unstable Glasma: Analytic results
1/Qs EE
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 – 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
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 Glasma instability without expansionwithout 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.
only (inside a magnetic flux tube)
• Fluctuations
other color components of the gauge field: charged matter field
0zB
Anomalous magnetic couplinginduces mixing of i mass term with a wrong sign
Glasma instability without expansionLinearized with respect to fluctuations
for m = 0gBgBn 22
12
Lowest Landau level (n = 0) of () becomes unstable
pz
finite at pz= 0
For gB ~ Qs2
Qs
Qs
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 )
• Screen the original magnetic field Bz
• Large current in the z direction induced
• Induced current Jz generates (rotating) magnetic field B
Bz
Jz ~ ig*Dz ~ g2 (B/g)(Qs/g)
Jz
B ~ Qs2/g
for one flux tube
B/gg
gBV ~ 4
)( 42
2
Glasma instability without expansion
Consider fluctuation around B
B
r
z
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
2
2
zp
SQgB ~
SQgB ~
unstable
stable
Growth rate
Glasma instability without expansionGrowth rate of the glasma w/o expansion
2
zp
Nielsen-Olesen instability with a constant Bz is followed by Nielsen-Olesen instability with a constant B
gB
zgB
• pz dependence of growth rate has the information of the profile of the background field• In the presence of both field (Bz and B) the largest pz for the primaryinstability increases
Glasma instability without expansionNumerical 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
1 2
Each source creates the gluon field for each nucleus. Initial condition
1 , 2 : gluon fields of nuclei
[Kovner, Weigert,McLerran, et al.]
In Region (3), and at =0+, the gauge field is determined by 1 and 2