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

Instabilities in Instabilities in expanding and non-expanding and non-

expanding expanding glasmasglasmas

K. Itakura (KEK, Japan )K. Itakura (KEK, Japan )

as one of the “CGC children”as one of the “CGC children”

based on based on * H. Fujii and KI, “* H. Fujii and KI, “Expanding color flux tubes and Expanding color flux tubes and

instabilitiesinstabilities” Nucl. Phys. A 809 (2008) 88” Nucl. Phys. A 809 (2008) 88

* H.Fujii, KI, A.Iwazaki, “* H.Fujii, KI, A.Iwazaki, “Instabilities in non-expanding Instabilities in non-expanding glasmaglasma” ”

arXiv:0903.2930 [hep-ph]arXiv:0903.2930 [hep-ph]Jean-Paul and Larry’s birthday party @ Saclay, April 2009

ContentsContents

• 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

Relativistic Heavy Ion Collisions in High Energy Limit

Pre-equilibrium

state = “Glasma”

Introduction (1/6)Introduction (1/6)

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

~

Initial cond. = CGC

~

t > 1/Qs

Pre-equilibrium states: Pre-equilibrium states: glasmaglasma

Solve the source free Yang Mills eq.

[D, F] = 0

in expanding geometry with the CGC initial condition

x

xxxzt ln

2

1 ,222

Formulate in coordinates

proper time rapidity

Initial condition = CGC

Randomly distributed

TransverseCorrelation Length ~ 1/Qs

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 physicsIssues 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

Weibel instabilityWeibel instability

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)

Introduction (4/6)

Inhomogeneous magnetic fields are enhanced

teB 0for 0)( pp

)( p

p

Both are necessary: * Inhomogeneous magnetic field *Anisotropic distribution for hard particles

Induced current generates magnetic field

Nielsen-Olesen instability Nielsen-Olesen instability (1/2)(1/2)

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

0zB

Non-minimal magnetic couplinginduces mixing of i mass term for with a wrong sign

Nielsen, Olesen, NPB144 (78) 376Chang, Weiss, PRD20 (79) 869

Introduction (5/6)

Linearized with respect to fluctuations

eigenfrequency

Nielsen-Olesen instability Nielsen-Olesen instability (2/2)(2/2)

Lowest Landau level (N = 0) of is unstable for small pz

pz

finite at pz= 0Growth rate :

Introduction (6/6)

Landau levels (2N + 1)Free motion in z direction Non-minimal coupling

Transverse size of unstable mode gBl /1~

!! N-O instability is realized if homogeneity region is larger than Larmor radius !!

Bz

Stable dynamics of Stable dynamics of the expanding the expanding

GlasmaGlasma

Stable dynamics: Boost-invariant Stable dynamics: Boost-invariant GlasmaGlasma

There appears a flux tube structure !!Longitudinal fields are generated at = 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 = 0 Ez = 0, Bz = 0

1/Qs

yy

xx

21

21

yy

xx

21

21

/ / E or B, or E&B

[Fries, Kapusta, Li, Lappi, McLerran]

1,2 Initial gauge fields

Some of the flux tubes are magnetically dominated.

)/(, 2 gQOBE s

Stable dynamics: Boost-invariant Stable dynamics: Boost-invariant GlasmaGlasma

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 Qs =0, 0.5, … 2 (left)and Qst = 1, 2 (right).

Bz2, Ez

2

BT2, ET

2

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]

Unstable Glasma Unstable Glasma in in expandingexpanding

geometrygeometry

Unstable GlasmaUnstable GlasmaBoost-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: Unstable Glasma w/ expansion: NumericsNumerics

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 w/ expansion: Unstable Glasma w/ expansion: NumericsNumerics

max() : Largest participating instability increases linearly in

conjugate to rapidity ~ Qs

Unstable Glasma w/ expansion: Unstable Glasma w/ expansion: Analytic studyAnalytic study

[Fujii, Itakura,Iwazaki]Linearized equations for fluctuations

SU(2), constant B and E directed to 3rd color and z direction

0~)21||2(2

1~ 1 )(

22

2)(

agBmmngE

a

yx

i

iaaa

iaaea

)(~ 21)(

conjugate to rapidity

1/Qs

E = 0

Nielsen-Olesen instability Lowest Landau level (n = 0) gets unstabledue to non-minimal magnetic coupling -2gB (not Weibel instability)

BB

modified Bessel fnc

1/Qs EE

B = 0

Schwinger mechanism Infinite acceleration of massless charged fluctuations. No amplification of the field

Whittaker function

• 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 participating instability increases linearly in -- Background field as expanding flux tube magnetic field on the front of a ripple B() ~ 1/

Unstable Glasma w/ expansion: Unstable Glasma w/ expansion: Analytic studyAnalytic study

Nielsen-Olesen instability in expanding geometry

||mr

Solution : modified Bessel function I(z)

[Fujii, Itakura]

#exp

Unstable Glasma Unstable Glasma in in non-expandingnon-expanding

geometrygeometry

Glasma instability without Glasma instability without expansionexpansion

Numerical simulation Berges et al. PRD77 (2008) 034504

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

Instability exists!! Can be naturally understood Two different instabilities ! In the Nielsen-Olesen instability

Corresponds to “non-expanding glasma”

zQs ~


Initial condition

With a supplementary condition

Can allow longitudinal flux tubes when

Initial condition is purely “magnetic”

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

yxz BBB ,

Primary N-O instabilityPrimary N-O instabilityConsider 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.

pz

finite at pz= 0Growth rategB

SQgB ~

Inhomogeneous magnetic field : B Beff


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 (rot B =J )

Bz

Jz

B ~ Qs2/g

for one flux tube

B/gg

gBV ~ 4

)( 42

2


Consider fluctuation around B

B

r

z

Centrifugal force Non-minimal magnetic coupling

Approximate solution at high pz

Negative for sufficiently large pz Unstable mode exists for large pz !

22

41~

zp

gBgB


Numerical solution of the lowest eigenvalue (red line)

SQgB ~

Growth rate

Increasing function of pz Numerical solution

Approximate solution


Growth rate of the glasma w/o expansion

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

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!

SummarySummary

CGC as the initial condition CGC as the initial condition for H.I.C.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

Documents

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