Instabilities in Instabilities in expanding and non- expanding and non- expanding expanding glasmas glasmas 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 instabilities instabilities ” 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- Instabilities in non- expanding glasma expanding glasma ” ” arXiv:0903.2930 [hep-ph] arXiv:0903.2930 [hep-ph] Jean-Paul and Larry’s birthday party @ Saclay, Ap
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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.
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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
[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 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 ~
Glasma instability without Glasma instability without expansionexpansion
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
Glasma instability without Glasma instability without expansionexpansion
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
Glasma instability without Glasma instability without expansionexpansion
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
Glasma instability without Glasma instability without expansionexpansion
Numerical solution of the lowest eigenvalue (red line)
SQgB ~
Growth rate
Increasing function of pz Numerical solution
Approximate solution
Glasma instability without Glasma instability without expansionexpansion
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