Instability in an expanding non-abelian system

Transcription

1 Instability in an expanding non-abelian system Kenji Fukushima (Department of Physics, Keio University) 1

2 Why expanding? 2

3 Relativistic Heavy-Ion Collision RHIC LHC Heavy-ions collide A new state of matter (Au, Pb, ) (Quark-gluon plasma) 3

4 Relativistic Heavy-Ion Collision s NN =2.7 TeV γ 1400 RHIC: s NN =200 GeV γ 100 LHC: 1 /γ Thermalization achieved (elliptic flow by a hydro-model) Initial temperature > 200MeV (distribution of thermal photon) 4

5 Schematic View of Four Regimes Soft and coherent gluons Color Glass Condensate Initial (quantum) fluctuations τ<q s 0.1 fm/c Instabilities (toward) Isotropization Glass + Plasma = Glasma Quantum fluctuations Particle (entropy) production Thermalization 0.1 fm/c 1fm/c Hydrodynamic evolution + cascade Relativistic Hydrodynamics 1fm/c 10 fm/c Hadronization Observation Particle yields, distributions 5

6 Missing Link Soft and coherent gluons Color Glass Condensate (CGC) Initial (quantum) fluctuations τ<q s 0.1 fm/c If starting with the CGC what the theory predicts? Instabilities Isotropization Glass + Plasma = Glasma Quantum fluctuations Particle (entropy) production Thermalization 0.1 fm/c 1fm/c 6

7 Why non-abelian? 7

8 Degrees of Freedom QED 2 photons 4 electrons (positrons) QCD 16 gluons How can we neglect quarks 24 ~ 36 quarks though there are more quarks than gluons in nature? (c.f. QCD thermodynamics) 8

9 Parton Distribution Function Valence and Sea Quarks and Gluons proton u u d valence quark constituent xqvalence(x) sea-quarks gluons x : momentum fraction carried by a parton 1 /3 x 9

10 Data from HERA Quantum Evolution of PDFs at fixed Q2 20! In the soft components of the nucleon wave-function, gluon is dominant. Slow (wee) Partons 100GeV Soft physics < 1GeV (RHIC) 10

11 Saturation Gluons eventually cover the transverse area: Nucleon moving at the speed of light Transverse size of a gluon ~ 1/Q Naive condition for saturation: xg (x, Q) αs N c ( N c 1)Q π R Once it happens, only Qs(x) fixes the physical scale! 11

12 Scaling Behavior Dipole Cross Section in a Saturation Model x< <Q 450 GeV s σ γ p ( x,q ) σ γ p (Q /Q (x )) Q 2s ( x)=q02 (x / x 0) λ Golec-Biernat-Wuesthoff Stasto-Golec-Biernat-Kwiecinski Plot Geometric Scaling Qs as a function of x is fixed Q 0=1 GeV 4 x 0= λ=0.288 Saturation is sufficient for scaling, but not necessary to it. 12

13 Effective Theory of Saturation 1/x Hard Soft Coherent gluons Classical fields A[r] Integrated out Classical source r Wave-function Wx[r] Classical sol. A[r] Observable Kovner, McLerran, Weigert, Iancu, Jalilian-Marian, Leonidov,... 13

14 McLerran-Venugopalan (MV) Model Gaussian Approximation: [ McLerran-Venugopalan (1993) 2 ρ( x) W x [ρ]=exp d x g μx 3 ] mx is related to Qs(x) larger m x = larger r = dense gluons = larger Qs Now we know that fluctuations are important for v3, v4, etc, but in a zero-th order approximation, this description should make sense as a good starting point... but!? 14

15 Why instability? 15

16 Schematic Picture before Collision Two nuclei do not talk to each other Just one color-source problem No longitudinal fields but only transverse fields attached on the nucleus sheet E' E Lorentz boost At rest Moving very fast 16

17 Initial Condition Fields made by colliding two sources Initial condition is x known on the light-cone (1 ) i? (2) i A i =α +α A η=0 i E =0 η (1) (2 ) E =ig [α i, α i ] x (1) i α (x ) (2 ) i α ( x ) Kovner-McLerran-Weigert (1995) 17 +

18 Intuitive Picture of Glasma * Boost Invariance * Coherent Fields (amp. ~ 1/g) * Flux Tube (size ~ 1/Qs) * Expanding Force from the tube should be overcome McLerran-Lappi (2006) 18

19 Schematic Picture after Collision What is the initial condition in the HIC? Negative longitudinal pressure Topological charge density Chiral magnetic effect KF-Kharzeev-Warringa 19

20 Equations of Motion to be Solved t Coordinates h proper time τ= t z 1 rapidity η= ln [(t + z )/(t z )] Equations to be solved μν ν D μ F = j =0 20

21 Formulations Time Evolution i η 1 E =τ τ Ai, E =τ τ Aη i 1 τ E =τ D η F ηi +τ D j F ji η 1 τ E =τ D j F j η Classical Equations of Motion in the Expanding System (c.f. Gross-Pitaevskii eq.) Ensemble Average O [ A] ρ,ρ D ρt D ρ p W x [ρt ]W x ' [ρ p ] O [A [ρt,ρ p ]] t p Quantum fluctuations partially included in the initial state 21

22 Physical Degrees of Freedom A A E E a x a η a x a η ( τ,η, x, y) (τ,η, x, y) (τ, η,x, y) ( τ, η, x, y) a y a τ a y A ( τ, η, x, y) A =0 E (τ, η, x, y) h-dep drops from the init. cond. a = 8 (adjoint with red, green, blue) 48 fields Simplify from 48 to 18 by assuming a = 3 (2colors) 22

23 Initial Configurations Solve the Poisson Eq Gauge Configuration Transverse Distribution No structure because of the Gaussian wave-function (this part improvable) nucleus 23

24 Chromo-Electric and Magnetic Fields Longitudinal and Transverse Fields <1 /Q s 0.1 fm/c free-streaming Lappi-McLerran (2006) Fukushima-Gelis (2011) 24

25 Longitudinal and Transverse Pressure No chance for thermalization (Almost) free-streaming Isotropization P T =P L Fukushima-Gelis (2011) 25

26 Negative Longitudinal Pressure Attractive Force Flux tubes have a positive energy 26

27 How It should Look Like Later Longiitudinal Pressure 1 δ ε τ 27

28 What Is Missing Flux tube Boost Invariant E and B ~ QCD string Instability c.f. Plasma instability by Rebhan Simulations by Berges, Sexty, Kunihiro, Iida c.f. Deconfinement at high T (entropy wins) String breaking Particle production (Schwinger mechanism) 28

29 Expectation Glasma Flux tube breaking Toward thermalization E B 2 L 2 L 29

30 Classical Statistical Simulation Boost Invariant E and B Classical Dynamics + Small Fluctuations What is the dynamics of the background E and B? How fluctuations grow? 30

31 Boost Invariant Background Again Longitudinal and Transverse Fields <1 /Q s 0.1 fm/c free-streaming 31

32 Mode Analysis 32

33 Two Drawbacks Color neutrality Should be chopped off in r(k) Should be chopped off in r(k) JIMWLK evolution 33

34 Wiggle by Hand (just for a test) Does this tell us anything? 34

35 Comments If a BEC-like content is seen (see a talk by J.-P. Blaizot), it should be in the transverse plane on which the gluon distribution is characterized by Qs. This means, even if a BEC-like behavior exists, it has nothing to do with the isotropization. It may be on a path to thermalization, but it does not help the problem of negative longitudinal pressure. Zero-mode implies homogeneous background fields, which would lead to instabilities (such as one of the Nielsen-Olesen type). 35

36 Boost-Invariance Violation Boost-invariant Glasma sits on the top of the potential maximum (seemingly stable without any perturbation) Toward isotropization What is the seed? How it spreads? boost invariant system h-dependent fluctuations Complete isotropization may not be necessary, nevertheless the free-streaming should not be right. (How much anisotropy is reasonably accepted?) 36

37 Schematic View of Instability Time Evolution of Fluctuations under Instability Stable Potential Zero-point Oscillation Unstable Singularity Classical evolution is a good approximation unless the potential is flat. (Heavy-Ion Collision) 37

38 Physical Degrees of Freedom A A E E a x a η a x a η ( τ,η, x, y) (τ,η, x, y) (τ, η,x, y) ( τ, η, x, y) a y a τ a y A ( τ, η, x, y) A =0 E (τ, η, x, y) i η δ E (η, x, y) δ E (η, x, y) i η δ A (η, x, y ) δ A (η, x, y ) Disturb the system by h-dep fluctuations at t=t 0 Fluctuation patterns: Fukushima-Gelis-McLerran (2006) Dusling-Gelis-Venugopalan (2011), Dusling-Epelbaum-Gelis-Venugopalan 38

39 Instabilities in the Classical Pure YM Romatschke-Venugopalan (2005) Kunihiro et al. (2010) Berges-Boguslavski-Schlichting (2012) Weibel instability Nielsen-Olesen instability Parametric resonance etc... Complementary to the plasma inst. (Rebhan) 39

40 Small (Minimal) Disturbance Amplitudes spread from lower to higher wavenumber modes CGC background Seed put here Because the zero-mode background is so huge, it keeps supplying the energy (or particle) injection. Fourier-mode of the longitudinal pressure 40

41 Amplitude Decay from Zero-Mode Schematic Behavior How this mode grows 41

42 Evolution of Longitudinal Spectrum Some scaling seen at unphysical late time Weak non-linearity remains even in a dilute system 42

43 Comments Expanding systems are simpler at large time scale. i η 1 E =τ τ Ai, E =τ τ Aη i 1 τ E =τ D η F ηi +τ D j F ji η 1 τ E =τ D j F j η Asymptotic Behavior i 1/2 η 1/ 2 E τ E 1/ τ 1/2 1/2 Ai 1/ τ A η τ Leading-order is free equations Bessel functions Soft-modes dominant in non-linearity Zero-mode 43

44 Mode Decaying from IR to UV in 1D 44

45 Summary I Boost-invariant background fields should be a right description for the relativistic heavy-ion collision in the first approximation at infinitely high energy. Only one characteristic scale Qs in this limit. Background fields have a peculiar pattern strong longitudinal E and B fields which should be disturbed by fluctuations and particle productions. Transverse and longitudinal dynamics so different. Entangled to speed up isotropization. 45

46 Summary II There are (almost always) choices that lead to desired results. Need careful considerations. Choice of the universal parameter Choice of the fluctuation strength Choice of the background fields Nevertheless, the (classical) pure YM system is a complicated non-linear system and it is still interesting to investigate long-time behavior. Many types of instabilities Strong-coupling limit (from a holographic dual) Chaos, Topological defects, Turbulence 46