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  Vlasov equation (Plasma instability) Strong coherent fields k < Qs high gluon density  Yang-Mills equation ~ t > 1/Qs Pre-equilibrium state = “Glasma” ~ Initial cond. = CGC

Pre-equilibrium states: glasma Solve the source free Yang Mills eq. [Dm , Fmn] = 0 in expanding geometry with the CGC initial condition Initial condition = CGC Randomly distributed Transverse Correlation Length ~ 1/Qs Formulate in t-h coordinates 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

Weibel instability Inhomogeneous magnetic fields are enhanced Introduction (4/6) Inhomogeneous magnetic fields are enhanced 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) Induced current generates magnetic field Both are necessary: * Inhomogeneous magnetic field *Anisotropic distribution for hard particles

Nielsen-Olesen instability (1/2) Introduction (5/6) 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 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

Nielsen-Olesen instability (2/2) Introduction (6/6) Linearized with respect to fluctuations eigenfrequency Bz Non-minimal coupling Free motion in z direction Landau levels (2N + 1)  Lowest Landau level (N = 0) of f - is unstable for small pz g finite at pz= 0 Growth rate : Transverse size of unstable mode !! N-O instability is realized if homogeneity region is larger than Larmor radius !! pz

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 t = 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 a1,2 Initial gauge fields 1/Qs / / E or B, or E&B Some of the flux tubes are magnetically dominated.

Stable dynamics: Boost-invariant Glasma 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 Qst =0, 0.5, … 2 (left) and Qst = 1, 2 (right). Bz2, Ez2 BT2, ET2 t 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 in expanding geometry

Unstable Glasma rapidity dependent fluctuations!! 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 t < 1 longitudinal pressure g2mt ~ Qst

Unstable Glasma w/ expansion: Numerics nmax(t) : Largest n participating instability increases linearly in t n : conjugate to rapidity h ~ Qst

Unstable Glasma w/ expansion: Analytic study Linearized equations for fluctuations SU(2), constant B and E directed to 3rd color and z direction [Fujii, Itakura,Iwazaki] : conjugate to rapidity h 1/Qs E = 0 Nielsen-Olesen instability Lowest Landau level (n = 0) gets unstable due to non-minimal magnetic coupling -2gB (not Weibel instability) B modified Bessel fnc 1/Qs E B = 0 Schwinger mechanism Infinite acceleration of massless charged fluctuations. No amplification of the field Whittaker function

Unstable Glasma w/ expansion: Analytic study Nielsen-Olesen instability in expanding geometry [Fujii, Itakura] Solution : modified Bessel function In(z) 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 t -- Background field as expanding flux tube magnetic field on the front of a ripple B(t) ~ 1/t  t

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  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 (D ~ Qs) Can allow longitudinal flux tubes when

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. g Growth rate finite at pz= 0 pz 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 ) 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 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 Increasing function of pz  Numerical solution

Glasma instability without expansion Growth rate of the glasma w/o expansion 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 a1 , a2 : gluon fields of nuclei In Region (3), and at t =0+, the gauge field is determined by a1 and a2