1. 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/

2. Contents 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 expansion) Summary

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?

4. Introduction (2/3) “Early thermalization problem” in HIC “Early 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 instability “Plasma instability” scenario Interaction btw hard particles (p t ~ Qs) having anisotropic distribution and soft field (p t << Qs) induces instability of the soft field  isotropization Weibel instability Arnold, Moore, and Yaffe, PRD72 (05) 054003

5. Introduction (3/3) Problems of “Plasma instability” scenario Problems 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 Q s is t ~ 1/Q s  Have to wait until t ~ 1/Q s for the kinetic description available (For Q s 0.2 fm/c) POSSIBLE SOLUTION : INSTABILITIES OF STRONG GAUGE FIELDS (before kinetic description available)  “GLASMA INSTABILITY” only strong gauge fields (given by the CGC) QsQs ptpt soft fields A  particles f(x,p)

6. Glasma Glasma (/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

7. Stable dynamics of the Glasma

8. Boost-invariant Glasma At  = 0 + (just after collision) Only E z and B z are nonzero (E T and B T are zero) [Fries, Kapusta, Li, McLerran, Lappi] Time evolution (  >0) E z and B z decay rapidly E T and B T 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”

9. Boost-invariant Glasma Just after the collision: only E z and B z 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

10. Boost-invariant Glasma An 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 B z 2, E z 2 = B T 2, E T 2 = ~1/  Single flux tube contribution averaged over transverse space (finite due to Qs = IR regulator)

11. Boost-invariant Glasma A single expanding flux tube at fixed time 1/Qs

12. Glasma instabilities

13. Unstable Glasma: Numerical results Boost invariant Glasma (without rapidity dependence) cannot thermalize  Need to violate the boost invariance !!! 3+1D numerical simulation P L ~ 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. g 2  ~ Q s  longitudinal pressure

14. Unstable Glasma: Numerical results max (  ) : Largest participating instability increases linearly in   conjugate to rapidity  ~ Q s 

15. 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

16. Unstable Glasma: Analytic results H.Fujii, KI, NPA809 (2008) 88 Magnetic background 1/Q s unstable solution for ‘charged’ matter Yang-Mills equation linearized with respect to fluctuations DOES have Growth time ~ 1/(gB) 1/2 ~1/Q s  instability grows rapidly Transverse size ~ 1/(gB) 1/2 ~1/ Q s for gB~ Q s 2 Nielesen-Olesen ’78 Chang-Weiss ’79 I ( z ) : modified Bessel function Lowest Landau level ( n=0,  2 =  gB < 0 for minus sign )  conjugate to rapidity  Sign of  2 determines the late time behavior

17. Modified Bessel function controls the instability f ~ Unstable Glasma: Analytic results =8, 12 oscillate grow Stable oscillation Unstable The time for instability to become manifest For large  Modes with small grow fast !  conjugate to rapidity 

18. Electric background No amplification of the fluctuation = Schwinger mechanism infinite acceleration of the charged fluctuation Unstable Glasma: Analytic results 1/Q sE No mass gap for massless gluons  pair creation always possible always positive or zero

19. 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 BzBz 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

20. Glasma instability without 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)

21. Glasma instability without expansion Color SU(2) pure Yang-Mills Background field (  “boost invariant glasma”) Constant magnetic field in 3 rd color direction and in z direction. only (inside a magnetic flux tube) Fluctuations other color components of the gauge field: charged matter field Anomalous magnetic coupling  induces mixing of  i  mass term with a wrong sign

22. Glasma instability without expansion Linearized with respect to fluctuations for m = 0 Lowest Landau level (n = 0) of  (  ) becomes unstable pzpz  finite at p z = 0 For gB ~ Q s 2 QsQs QsQs For inhomogeneous magnetic field, gB  g Growth rate

23. Glasma instability without expansion Consequence of Nielsen-Olesen instability?? Instability stabilized due to nonlinear term (double well potential for  ) Screen the original magnetic field B z Large current in the z direction induced Induced current J z generates (rotating) magnetic field B  BzBz J z ~ ig  *D z   ~ g 2 (B/g)(Qs/g) JzJz B  ~ Q s 2 /g for one flux tube

24. Glasma instability without expansion Consider fluctuation around B  BB r  z Centrifugal forceAnomalous magnetic term Approximate solution Negative for sufficiently large p z Unstable mode exists for large p z !

25. Glasma instability without expansion Numerical solution of the lowest eigenvalue  unstable  stable Growth rate

26. Glasma instability without expansion Growth rate of the glasma w/o expansion Nielsen-Olesen instability with a constant B z is followed by Nielsen-Olesen instability with a constant B  p z dependence of growth rate has the information of the profile of the background field In the presence of both field (B z and B  ) the largest p z for the primary instability increases

27. Glasma instability without expansion Numerical 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

28. 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

29. CGC as the initial condition for H.I.C. HIC = Collision of two sheets 11 2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