Instabilities in the Quark-Gluon Plasma

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1 in the Quark-Gluon Maximilian Attems Institute for Theoretical Physics, TU Vienna October 5, 2010 Dans la vie, rien n est à craindre, tout est à comprendre. Marie Curie Vienna Theory Lunch Seminar / 35

2 Overview Early Universe Heavy Ion Collision QCD Phase diagram QGP signatures Momentum Anisotropy Weibel instabilities Scales QGP Hard (Thermal) Loops - Boltzmann - Vlasov Notations for Bjorken expansion Hard-Expanding-Loop formalism Expanding 1D+3V Abelian plasma Expanding 3V plasma Vienna Theory Lunch Seminar / 35

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4 Relativistic Heavy Ion Collider (RHIC) Early Universe Heavy Ion Collision QCD Phase diagram QGP signatures Au+Au ions s NN = 200GeV/nucleon pair, p+p, d+a Vienna Theory Lunch Seminar / 35

5 Heavy Ion Collision Early Universe Heavy Ion Collision QCD Phase diagram QGP signatures Side view 2nd STAR event at RHIC, Vienna Theory Lunch Seminar / 35

6 QCD Phase diagram Early Universe Heavy Ion Collision QCD Phase diagram QGP signatures Schematic QCD phase diagram Vienna Theory Lunch Seminar / 35

7 QGP signatures Early Universe Heavy Ion Collision QCD Phase diagram QGP signatures Small viscosity (elliptic flow v 2 ) Jet quenching Experimental observation of T > T c high p T suppression of hadrons (for central collisions) Rapid thermalization: estimates from hydrodynamical computation 1fm/c Vienna Theory Lunch Seminar / 35

8 Elliptic flow Early Universe Heavy Ion Collision QCD Phase diagram QGP signatures Pressure gradients generate positive elliptic flow v 2 d 2 N dφdp T = N 0 (1+2v 2 (p T )cos(2φ)+..) (1) The elliptic flow is quantified by the anisotropy of particle production with respect to the reaction plane v 2 = p 2 x p 2 y p 2 x +p2 y Illustration of the reaction plane definition. Vienna Theory Lunch Seminar / 35

9 Hydrodynamic model Early Universe Heavy Ion Collision QCD Phase diagram QGP signatures PHOBOS data on Au+Au collisions at s = 200 GeV, compared to hydrodynamic model for various η/s ratios. P. Romatschke, U. Romatschke hep-th/ Vienna Theory Lunch Seminar / 35

10 Early conditions at RHIC - T Early Universe Heavy Ion Collision QCD Phase diagram QGP signatures c 3 ) -2 (mb GeV 3 σ/dp 3 c 3 ) or Ed -2 (GeV 3 N/dp 3 Ed AuAu Min. Bias x10 2 AuAu 0-20% x10 AuAu 20-40% x10 p+p Turbide et al. PRC (GeV/c) p T T = MeV > 2 T c Vienna Theory Lunch Seminar / 35

11 Momentum Anisotropy Weibel instabilities Scales QGP Hard (Thermal) Loops - Boltzmann - Vlasov Notations for Bjorken expansion Hard-Expanding- Loop formalism Early Universe Heavy Ion Collision QCD Phase diagram QGP signatures Momentum Anisotropy Weibel instabilities Scales QGP Hard (Thermal) Loops - Boltzmann - Vlasov Notations for Bjorken expansion Hard-Expanding-Loop formalism Expanding 1D+3V Abelian plasma Expanding 3V plasma Vienna Theory Lunch Seminar / 35

12 Momentum Anisotropy Momentum Anisotropy Weibel instabilities Scales QGP Hard (Thermal) Loops - Boltzmann - Vlasov Notations for Bjorken expansion Hard-Expanding- Loop formalism <p L > <p T > 1 CGC/Glasma fm/c τ iso ~ Q s -1 System is momentarily isotropic Expansion rate is much faster than the interaction time scale 1/τ >> 1/τ int Boltzmann- Vlasov Transport Expansion rate and isotropization via interactions balance Momentum space anisotropy time dependence at the early stages of a heavy ion collision Viscous Hydro large p τ small p large p Vienna Theory Lunch Seminar / 35

13 Weibel instabilities Momentum Anisotropy Weibel instabilities Scales QGP Hard (Thermal) Loops - Boltzmann - Vlasov Notations for Bjorken expansion Hard-Expanding- Loop formalism Oblate Distribution Induced Current Magnetic Fluctuation x j x B y y Illustration of the mechanism of filamentation instabilities. F v v F v F F v z z Vienna Theory Lunch Seminar / 35

14 QED Momentum Anisotropy Weibel instabilities Scales QGP Hard (Thermal) Loops - Boltzmann - Vlasov Notations for Bjorken expansion Hard-Expanding- Loop formalism Filaments and active solar region from NASA s Solar Dynamics Observatory Vienna Theory Lunch Seminar / 35

15 Scales of weakly coupled QGP Momentum Anisotropy Weibel instabilities Scales QGP Hard (Thermal) Loops - Boltzmann - Vlasov Notations for Bjorken expansion Hard-Expanding- Loop formalism T: energy of hard particles gt: thermal masses, Debey screening mass, Landau damping, plasma instabilities [Mrowczynski 1988, 1993,..] g 2 T: magnetic confinement, color relaxation, rate for small angle scattering g 4 T: rate for large angle scattering, η 1 T 4 Vienna Theory Lunch Seminar / 35

16 Hard (Thermal) Loops - Boltzmann - Vlasov Momentum Anisotropy Weibel instabilities Scales QGP Hard (Thermal) Loops - Boltzmann - Vlasov Notations for Bjorken expansion Hard-Expanding- Loop formalism With color-neutral background distribution v f 0 (p,x,t) = 0, v µ = p µ /p 0 gauge covariant Boltzmann-Vlasov: v D f a (p,x,t) = gv µ F µν a (p) ν f 0 (p,x,t) = = g(e a +v B a ) p f 0, (2) D µ F a µν = ja ν = g So far mostly stationary f 0 (p) with µ f isotropic: f 0 (p) = f 0 ( p ), p f 0 v d 3 p p µ (2π) 3 2p 0δf a(p,x,t). (3) v Dδf a (p,x,t) = ge a p f 0 (stable) (4) - anisotropic: f 0 (p), p f 0 v v Dδf a (p,x,t) = g(e a +v B a ) p f 0 (unstable!) (5) Vienna Theory Lunch Seminar / 35

17 Discretized Hard Loop Effective Theory Momentum Anisotropy Weibel instabilities Scales QGP Hard (Thermal) Loops - Boltzmann - Vlasov Notations for Bjorken expansion Hard-Expanding- Loop formalism Auxiliary field formulation: Mrowczynski, Rebhan & Strickland 2004] δf a (x;p) = gw a µ(t,x;v) µ (p)f 0 (p) (6) where v µ p µ / p = (1,v) [v D(A)]W µ (x;v) = F µγ (A)v γ (7) j µ (x) = g 2 d 3 p 1 (2π) 3 2 p pµ f(p) p ν Wν (x;v), (8) Hard Loop effective theory: (hard) scale p integrated out for real-time lattice simulation: discretize also velocity space in "disco balls" D σ (A)F σµ = j µ (x) = 1 N v µ W v (x) (9) v Vienna Theory Lunch Seminar / 35

18 Notations for Bjorken expansion Momentum Anisotropy Weibel instabilities Scales QGP Hard (Thermal) Loops - Boltzmann - Vlasov Notations for Bjorken expansion Hard-Expanding- Loop formalism It is convenient to switch to comoving coordinates t = τ coshη, β = tanhη, z = τ sinhη, γ = coshη, (10) i.e, a coordinate system with metric ds 2 = dτ 2 dx 2 τ2 dη 2. We introduce the notation x α = (x τ,x i,x η ) = (τ,x 1,x 2,η) (11) with indices from the beginning of the Greek alphabet for these new coordinates. In addition to space-time rapidity η, we also introduce momentum space rapidity y for the massless particles according to p µ = p (coshy,cosφ,sinφ,sinhy). (12) Vienna Theory Lunch Seminar / 35

19 Hard-Expanding-Loop formalism Momentum Anisotropy Weibel instabilities Scales QGP Hard (Thermal) Loops - Boltzmann - Vlasov Notations for Bjorken expansion Hard-Expanding- Loop formalism ] With p β β [ (p) α f 0(p,p η ) pµ=const. = 0 (13) we can commute p D and thus solve gauge-covariant Vlasov equation in comoving coordinates p.dδf a (p,x,t) = gp β F p βα a µ (p) α f o(p,x,t). (14) =const. Introducing auxiliary fields W a α(τ,x i,η;φ,y) similar to the auxiliary field W ν (x,v) of the hard-loop formalism δf a (x;p) = gw a α(τ,x i,η;φ,y) α (p) f 0(p,p η ) (15) that obey v DW α (τ,x i,η;φ,y) = v β F αβ (16) φ,y where v α pα p = ( cosh(y η),cosφ,sinφ, sinh(y η) τ ). Vienna Theory Lunch Seminar / 35

20 Expanding 1D+3V Abelian plasma Expanding 3V plasma Early Universe Heavy Ion Collision QCD Phase diagram QGP signatures Momentum Anisotropy Weibel instabilities Scales QGP Hard (Thermal) Loops - Boltzmann - Vlasov Notations for Bjorken expansion Hard-Expanding-Loop formalism Expanding 1D+3V Abelian plasma Expanding 3V plasma Vienna Theory Lunch Seminar / 35

21 Expanding 1D+3V Abelian plasma 10 3 Semi-Analytic Expanding 1D+3V Abelian plasma Expanding 3V plasma Π x (η=0) Discretized HEL Relative Error Method A Method B τ/τ τ/τ 0 The proper-time evolution of the canonical field momentum of a single Abelian mode. Vienna Theory Lunch Seminar / 35

22 Expanding 1D+3V non-abelian plasma Expanding 1D+3V Abelian plasma Expanding 3V plasma x (Field Energy Densities) τ τ Total B T B L E T E L Gain Rate τ/τ 0 The proper-time dependence of the chromo-field energy densities and the energy gain rate times an extra factor of τ 0 resulting from non-abelian run initialized with Fukushima, Gelis, and McLerran (FGM) initial conditions. Vienna Theory Lunch Seminar / 35

23 Expanding 1D+3V non-abelian plasma Expanding 1D+3V Abelian plasma Expanding 3V plasma τ τ PL,field 0 3 τ τ PT,field 0 3 τ τ PL,particle 0 3 τ τ PT,particle τ/τ 0 The comparison of the longitudinal and transverse pressures for the fields and particles resulting from a typical non-abelian run initialized with FGM (CGC inspired) initial conditions. Vienna Theory Lunch Seminar / 35

24 Expanding 1D+3V Abelian plasma Expanding 1D+3V Abelian plasma Expanding 3V plasma Fourier spectrum of the color-traced conjugate field momentum obtained from Abelian run with FGM initial conditions. Vienna Theory Lunch Seminar / 35

25 Expanding 1D+3V non-abelian plasma Expanding 1D+3V Abelian plasma Expanding 3V plasma Fourier spectrum of the color-traced conjugate field momentum obtained from non-abelian run with FGM initial conditions. Vienna Theory Lunch Seminar / 35

26 Expanding 3V plasma Expanding 1D+3V Abelian plasma Expanding 3V plasma t t 0 3 x (Field Energy Density) 1e+06 1e Run 9 - (16 x 16 2 ) x (256 x 16) Run 10 - (64 x 16 2 ) x (256 x 16) Run 15 - (128 x 16 2 ) x (256 x 16) Run 14 - (16 x 16 2 ) x (256 x 16) - ABELIAN Run 18 - (128 x 16 2 ) x (256 x 16) - ABELIAN 1e t/t 0 Preliminary runs from the HEL 3d codes in Abelian and non Abelian setup with different lattice sizing s in the longitudinal η direction, but identical transverse size and W auxiliary field numbers. Vienna Theory Lunch Seminar / 35

27 Expanding 3V non-abelian plasma 10 Run 9 - (16 x 16 2 ) x (256 x 16) Expanding 1D+3V Abelian plasma Expanding 3V plasma t t 0 3 x (Field Energy Density) Run 12 - (32 x 32 2 ) x (256 x 16) Run 13 - (64 x 64 2 ) x (256 x 16) e t/t 0 Preliminary runs from the HEL 3d code with different lattice sizing going from 16 3 to 64 3 and thus different transverse lattice sizing showing the proper time dependence of the respective total chromo-field energy density. Vienna Theory Lunch Seminar / 35

28 Unstable transverse modes Expanding 1D+3V Abelian plasma Expanding 3V plasma Influence of different initial conditions for a specific mode with ν = 30 Vienna Theory Lunch Seminar / 35

29 Expanding 1D+3V non-abelian plasma Expanding 1D+3V Abelian plasma Expanding 3V plasma Visualization of the space-time development of color correlations in a non-abelian plasma instabilities in Bjorken expansion. Vienna Theory Lunch Seminar / 35

30 Expanding 1D+3V Abelian plasma Expanding 3V plasma Non-abelian plasma instabilities accelerate isotropization and thermalization of the Quark Gluon. Large amplitude turbulent field configurations can have an important effect on Quark Gluon transport such as momentum broadening, energy loss, plasma viscosity,... In the 1D+3V Loop (HEL) 1D we found that the exponential (in τ) growth in the Abelian (weak-field) phase is only mildly weakened when nonlinearities through non-abelian self-interactions of the collective fields set in. The previous 1D HEL code has been extended to full 3D+3V. Final results including different initial conditions are being computed. Vienna Theory Lunch Seminar / 35

31 Expanding 1D+3V Abelian plasma Expanding 3V plasma Thank you. Vienna Theory Lunch Seminar / 35

32 Backup - Equation of motions Expanding 1D+3V Abelian plasma Expanding 3V plasma Conjugate Momenta τ E i = +τj i + 1 τ D2 ηa i +τg 2 i[a j i,i[a j i,a i ]] (17) τ E η = τj η + ig τ [Ai,D η A i ] (18) Gauss law j τ = + 1 τ D ηe η ig τ [A i,e i ] (19) with E i τ τ A i, E η 1 τ τa η (20) Vienna Theory Lunch Seminar / 35

33 Backup - Expanding 1D+3V non-abelian plasma Expanding 1D+3V Abelian plasma Expanding 3V plasma Field Energy Densities Total B T B L E T E L Energy Gain τ/τ 0 The proper-time dependence of the chromo-field energy densities from a run with a single non-abelian mode seeded with random noise. Vienna Theory Lunch Seminar / 35

34 Backup - Expanding 1D+3V non-abelian plasma 10 0 Expanding 1D+3V Abelian plasma Expanding 3V plasma x (Field Energy Density) Total Transverse Magnetic Longitudinal Magnetic τ τ Transverse Electric Longitudinal Electric Gain Rate τ/τ 0 The proper-time dependence of the chromo-field energy densities with a single non-abelian mode with decoupled hard particle currents (j = 0). Vienna Theory Lunch Seminar / 35

35 Backup - Expanding 1D+3V non-abelian plasma Expanding 1D+3V Abelian plasma Expanding 3V plasma x (Field Pressures) Longitudinal Pressure (P L ) Transverse Pressure (P T ) τ τ τ/τ 0 The proper-time dependence of the longitudinal and transverse pressure with a single non-abelian mode with decoupled hard particle currents (j = 0). Vienna Theory Lunch Seminar / 35