CGC and the Glasma

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CGC and the Glasma Powered By Docstoc
					       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 flux tubes and instabilities”
                   Nucl. Phys. A 809 (2008) 88
* H.Fujii, KI, A.Iwazaki, “Instabilities in non-expanding glasma”
                    arXiv:0903.2930 [hep-ph]

                        Jean-Paul and Larry’s birthday party @ Saclay, April 2009
                            Contents

• 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
                Introduction (1/6)
Relativistic Heavy Ion Collisions in High Energy Limit

                               Particles
                                  <
                                k ~ Qs (or simply k~Qs)
                                 Boltzmann equation
                                   (“Bottom-up” scenario)

                               Soft fields + hard particles
                                 k<<Qs          k ~ Qs        t > 1/Qs
            Pre-equilibrium      Vlasov equation
            state = “Glasma”         (Plasma instability)

                               Strong coherent fields
                                 k < Qs
                                   ~
                                 high gluon density
     Initial cond. = CGC          Yang-Mills equation
             Pre-equilibrium states: glasma
                                   Solve the source free Yang Mills eq.

                                             [Dm , Fmn] = 0
                                   in expanding geometry with the CGC
                                   initial condition

                                     Initial condition             Randomly
                                          = CGC                    distributed



                                                              Transverse
Formulate in  coordinates                                  Correlation
                       1 x                                   Length ~ 1/Qs
  t  z  2 x x ,   ln 
     2   2         

                       2 x
proper time             rapidity
                                        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 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
                                                                       Introduction (4/6)
                            Weibel instability
 Inhomogeneous magnetic fields are enhanced
     due to (ordinary) coupling btw
          charged particles (with anisotropic distr.)  hard gluons
          and soft magnetic field                      soft gluon fields

      x (current)

         z (Lorenz force)

     y (magnetic field)

               t
     Be
                                                            Induced current generates
                                           ( p)             magnetic field

         ( p)  0 for p  0
Both are necessary:
* Inhomogeneous magnetic field
*Anisotropic distribution for hard particles
                                                                                   p
                                                                      Introduction (5/6)
      Nielsen-Olesen instability (1/2)
                                                       Nielsen, Olesen, NPB144 (78) 376
                                                       Chang, Weiss, PRD20 (79) 869

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 B z  0
     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



                         Non-minimal magnetic coupling
               induces mixing of fi  mass term for f with a wrong sign
                                                             Introduction (6/6)
         Nielsen-Olesen instability (2/2)
Linearized with respect to fluctuations
                                                                 Bz

eigenfrequency


                                   Non-minimal coupling
 Free motion in z direction Landau levels (2N + 1)
 Lowest Landau level (N = 0) of f  is unstable for small pz


                                                         finite at pz= 0
  Growth rate :
  Transverse size of unstable mode l ~ 1 / gB

!! N-O instability is realized if homogeneity                               pz
region is larger than Larmor radius !!
Stable dynamics of the
  expanding Glasma
  Stable dynamics: Boost-invariant Glasma
                                                          [Fries, Kapusta, Li, Lappi, McLerran]
There appears a flux tube structure !!
Longitudinal fields are generated at  = 0+
                                                      E , B  O (Qs2 / g )
                        1,2   Initial gauge fields



Similar to Lund string models but           1/Qs
 * transverse correlation 1/Qs
 * magnetic flux tube possible

In general both Ez and Bz are present,
but  x  x             x x
        1     2           1     2

      1y  2y          1y  2y
   purely electric    purely magnetic
   Ez = 0, Bz = 0
      /                Ez = 0, Bz = 0
                                  /                       E or B, or E&B

 Some of the flux tubes are magnetically dominated.
 Stable dynamics: Boost-invariant Glasma
Expanding flux tubes                                              Fujii, Itakura NPA809 (2008) 88

 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.])


Transverse profile of a
Gaussian flux tube at
Qs =0, 0.5, … 2 (left)
and Qst = 1, 2 (right).




                                                                        [Lappi,McLerran]
 dependence of field                Bz2, Ez2
strength from a single
flux tube (averaged
over transverse space)
                                 BT2, ET2
compared with the result
of classical numerical
simulation of boost-
invariant Glasma
   Unstable Glasma
in expanding geometry
                       Unstable Glasma
Boost-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: Numerics
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  < 1

            g2m ~ Qs
    Unstable Glasma w/ expansion: Numerics
nmax() : Largest n participating instability increases linearly in 




     n : conjugate to rapidity                   ~ Qs
 Unstable Glasma w/ expansion: Analytic study
Linearized equations for fluctuations                                        [Fujii, Itakura,Iwazaki]

                                                                              n: conjugate
 SU(2), constant B and E directed to 3rd color and z direction                  to rapidity 
                1             2
                                                           ~ ()
 1
 
       ~ () 
     a     
                 2
                  
                    n 
                         gE 2                             
                               (2n | m | 1  m  2) gBa  0 a (  )  ein (a 1  ia 2 )
                                                                   ~
                
                         2                               
                                                                                        

                                                                   a  a x  ia y
                     B=0                          E=0

Schwinger mechanism                           Nielsen-Olesen instability
 Infinite acceleration of massless            Lowest Landau level (n = 0) gets unstable
 charged fluctuations.                        due to non-minimal magnetic coupling -2gB
 No amplification of the field                 (not Weibel instability)




                                                                        modified Bessel fnc
                     Whittaker function



1/Qs                       E                  1/Qs                       B
Unstable Glasma w/ expansion: Analytic study
                                                                   [Fujii, Itakura]
Nielsen-Olesen instability in expanding geometry
 Solution : modified Bessel function In(z)

                 r |m|

 • 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 n participating instability increases linearly in 
   -- Background field as expanding flux tube
      magnetic field on the front of a ripple
                         B() ~ 1/            exp      #   
     Unstable Glasma
in non-expanding geometry
    Glasma instability without expansion
Numerical simulation         Berges et al. PRD77 (2008) 034504

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

                                                               ~ Qs   z
      Corresponds to “non-expanding glasma”




           Instability exists!!                 Can be naturally understood
        Two different instabilities !         In the Nielsen-Olesen instability
    Glasma instability without expansion
Initial condition




With a supplementary condition



Initial condition is purely “magnetic”

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

Can allow longitudinal flux tubes when   B z  B x , B y
                Primary N-O instability
Consider 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.


  Growth rate                                  finite at pz= 0
                                     gB


                                                                  pz
                                                       gB ~ QS

  Inhomogeneous magnetic field : B  Beff
  Glasma instability without expansion
Consequence of Nielsen-Olesen instability??
• Instability stabilized due to nonlinear term (double well potential for f )
                          g2 4
      V (f )   gBf 2
                            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 (rot B =J )

                                                             Jz
                                                                   Bq ~ Qs2/g
                                                                  for one flux tube
     Bz
 Glasma instability without expansion
Consider fluctuation around Bq
               z
   Bq


            q r




                         Centrifugal force      Non-minimal magnetic coupling
Approximate solution at high pz
                         gBq 
             2 ~  gBq 1  2 
                         4p 
                            z 
   Negative for sufficiently large pz        Unstable mode exists for large pz !
   Glasma instability without expansion


Numerical solution of the lowest eigenvalue (red line)

                        Growth rate
          Approximate
          solution
                                                             gBq ~ QS

                   Numerical solution                   Increasing function of pz
 Glasma instability without expansion
 Growth rate of the glasma w/o expansion


gBq




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
                      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.

And, happy birthday, Jean-Paul and Larry!
   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



                                    1 , 2 : gluon fields of nuclei
   In Region (3), and at  =0+, the gauge field is determined by 1 and 2

				
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