Instabilities Driven Equilibration of the Quark-Gluon Plasma

Transcription

1 Instabilities Driven Equilibration of the Quar-Gluon Plasma Stanisław Mrówczyńsi Świętorzysa Academy, Kielce, Poland & Institute for Nuclear Studies, Warsaw, Poland 1

2 - why Mar s favorite formula e -E/T is so good - E /T e? Hay Birthday Mar!

3 Relativistic heavy-ion collision Little Bang after free hadrons t hadrons quars & gluons freeze-out hadronization equilibration z before 3

4 Relativistic heavy-ion collisions Au Au s GeV/NN RHIC 4

5 Evidence of the early stage equilibration Success of hydrodynamic models in describing ellitic flow y φ Hydrodynamics + v v t ρ Hydrodynamics requires local thermodynamical equilibrium! dn dϕ π/ π 5 φ

6 Equilibration is fast v ~ ε + y y Eccentricity decays due to the free streaming! ε v t eq 1 fm/ c time of equilibration U. Heinz, AIP Conf. Proc.739,

7 Collisions are too slow Time scale of hard arton-arton scattering t hard ~ g 4 1 ln 1/ gt hard scattering ~ momentum transfer of order of T dominated by either single hard scattering or multile soft scatterings g QCD couling constant t eq t hard.6 fm/ c R. Baier, A.H. Mueller, D. Schiff & D.T. Son, Phys. Lett. B539, 46 7

8 Instabilities stationary state Instability A t A + δa t δa t e γt fluctuation γ > stable configuration unstable configuration A At A At 8

9 Terminology Plasma instabilities interlay of articles and classical fields Quantum Field Theory no articles, no classical fields ~ T hard articles hard ecitations, hard modes classical fields highly oulated soft ecitations, soft modes ~ 1/ g ~ soft gt 9

10 Plasma manifests collective behavior λ D 1 m D ~ 1 gt screening length V r ~ e r λ r D V r Coulomb screened Debye shere λ D r V D πλ ~, ~ 3 D n T, n V D ~ 3 >> 1 if g << g T g In a wealy couled lasma, there are many articles in a Debye shere! 1

11 Plasma instabilities instabilities in configuration sace hydrodynamic instabilities instabilities in momentum sace inetic instabilities instabilities due to non-equilibrium momentum distribution f is not ~ E e T 11

12 Kinetic instabilities longitudinal modes E, δρ ~ e i ωt r transverse modes E, δj ~ e i ωt r E electric field, wave vector, ρ charge density, j - current 1

13 Logitudinal modes unstable configuration lasma f, y, z beam Energy is transferred from articles to fields 13

14 Logitudinal modes Electric field decays - daming f, y, z Electric field grows - instability f, y, z article acceleration ω E article deceleration article acceleration ω E article deceleration - hase velocity of the electric field wave, ω E - article s velocity 14

15 Transverse modes Unstable modes occur due to anisotroy of the momentum distribution f f f ê y e ˆ Momentum distribution distribution can monotonously decrease in every direction Transverse modes are relevant for relativistic nuclear collisions! 15

16 Momentum Sace Anisotroy in Nuclear Collisions T T L time L CM after 1-st collisions local rest frame 16

17 17 Seeds of instability j a but current fluctuations are finite δ π δ ν ν t f E d j j ab b a v,,,,, t t t t Direction of the momentum surlus

18 Mechanism of filamentation z F v v F v v F F Lorentz force F q v B Amere s law j z B j B y y 18

19 Instabilities vs. collisions Time scale of arton-arton scattering t hard t soft ~ g g 4 ~ 1 ln 1/ gt 1 ln 1/ gt Time scale of collective henomena t q collec ~ hard scattering: q ~ T soft scattering: q ~ gt 1 g T g << 1 t >> t >> t hard soft collec The instabilities are fast! 19

20 Disersion equation Equation of motion of chromodynamic field A in momentum sace ν ν ν [ g Π ] A ν Disersion equation gluon self-energy det[ g ν ν Π ν ] ω, Instabilities solutions with Imω > A Im ~ e ωt Dynamical information is hidden in ν Π. How to get it?

21 Transort theory fundamental adjoint g v D Q { F v, Q} C g v D Q + { F v, Q } C g v D G { F v, G} C g quars antiquars gluons free streaming mean-field force collisions D ig[ A,...], F ν A ν ν A ig[ A, A ν ] D ν ν F j [ Q, Q, G] mean-field generation collisionless limit: C C Cg 1

22 Diagrammatic Hard Loo aroach Π ν Hard loo aroimation: << 3 ν λ ν g d λ f Π [ g ] 3 σ + λ π E σ + i Π ν ν ν Π, Π

23 Chromo-hydrodynamic aroach D n g ν { F, n } ν D T n Postulated form of and T ν : T 1 n n u ν ε + { u, u } g ν n ε,, u, matrices! u u 1 To close the system of equations: T or ε 3 3

24 Disersion equation Disersion equation det[ g ν ν Π ν ] ε ij ij 1 ij δ Π ω Disersion equation Π ν chromodielectric tensor ω, det[ δ ij i j ω ε ij ] ij ij g d v ε δ ω π ω v + i 3 i f l [1 v ω δ lj + l v ω j ] v / E 4

25 y Disersion equation configuration of interest z Direction of the momentum surlus j,, j, E,, E,,, Disersion equation ω ε zz ω, 5

26 Eistence of unstable modes Penrose criterion H ω C ω ε zz ω, d ω 1 dh ω πi H ω dω Im H ω ω ω dω d ln H ω ln H ω πi dω C number of zeros of Hω in C Imω C i Re ω φπ φπ + ω ω Re H There are unstable modes if H ω < Anisotroy! 6

27 Unstable solutions 1/ 4 σ ρσ 1 f e αs g / 4π. 3 3/ 3 π σ + σ ρ σ 6 fm 3.3 GeV ω ε zz ω, solution ω ± i γ < γ R J. Randru & St. M., Phys. Rev. C 68,

28 8 Hard-Loo dynamics Soft fields in the assive bacground of hard articles Braaten-Pisarsi action generalized to anisotroic momentum distribution: ] [ ~ eff D f C i F D F f d g L F b ab a ψ γ ψ + π ρ ρ ν ν,,, q q Σ + Σ Λ Π ν St. M., A. Rebhan & M. Stricland, Phys. Rev. D 74, 54 4

29 Growth of instabilities 1+1 numerical simulations SU Hard Loo Dynamics total transverse magnetic A 1+1 dimensions a Aa t, z Scaled field energy density Anisotroic article s momentum distribution f fiso + ζz m α s dfiso D π d d m, ζ D Strong anisotroy ζ 1 γ * - maimal growth rate A. Rebhan, P. Romatsche & M. Stricland, Phys. Rev. Lett. 94,

30 Isotroization - articles Direction of the momentum surlus j B F dt F 3

31 Isotroization - fields Direction of the momentum surlus E B P ~ B E ~ fields a a 31

32 Isotroization numerical simulation Classical system of colored articles & fields T ij 3 d π 3 i E j f T yy +T zz / Isotroy: T T + Tyy Tzz / A. Dumitru & Y. Nara, Phys. Lett. B61,

33 Conclusion The scenario of instabilities driven equilibration rovides a lausible solution of the fast equilibration roblem E /T e Yes! 33