Glasma instabilities

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?


“ “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


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


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 ~


=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


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


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


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 !


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

Glasma instabilities

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