Longitudinal thermalization via the chromo-weibel instability

Transcription

1 Longitudinal thermalization via the chromo-weibel instability Maximilian Attems Frankfurt Institute of Advanced Studies , Collaborators: Anton Rebhan, Michael Strickland Schladming, February 2013

2 Motivation Weakly coupled inspired by Hard Thermal Loops (HTL) Real-time physical quantities of non-equilibrium processes Plasma turbulence affects parton transport (isotropization, jet energy loss, viscosity,..) Contrast predictions for early time dynamics of the quark gluon plasma Derivation of time scales for the isotropization, thermalization Maximilian Attems, Schladming 2013 Longitudinal thermalization via the chromo-weibel instability 2/19

3 Outline 1 Hard Expanding Loops (HEL) Stages of an heavy ion collision Scales of wqgp Weibel instabilities Yang-Mills Vlasov Unstable modes growth rate 2 Physical Observables Energy densities Pressures Spectra Longitudinal temperature Maximilian Attems, Schladming 2013 Longitudinal thermalization via the chromo-weibel instability 3/19

4 Assumptions Free streaming background Anisotropy in momentum space SU(2) particle content Fixed transverse size Extrapolate to α s 0.3 Match CGC n(τ 0 ) Q 3 s α 1 s Maximilian Attems, Schladming 2013 Longitudinal thermalization via the chromo-weibel instability 4/19

5 Stages of an heavy ion collision t freeze out hadrons in eq. gluons & quarks in eq. gluons & quarks out of eq. strong fields z hydrodynamics viscous hydro classical fields [Gelis 2012] Illustration of the stages of a heavy ion collision. Numerical approaches to early phase with strong fields: 1 Numerical solution of Yang Mills equations in real-time: [Romatschke, Venugopalan, Berges, Sexty, Gelis, Fukushima, Dusling, Moore, Kurkela, Epelbaum, Schlichting] 2 Hard Loop Simulation (Eikonalized particles): [Strickland, Romatschke, Rebhan, Arnold, Moore, Mrowczynski, Rummukainen, Bödeker, Ipp, Attems, Deja] Maximilian Attems, Schladming 2013 Longitudinal thermalization via the chromo-weibel instability 5/19

6 Scales of wqgp T : energy of hard particles gt : thermal masses, Debye screening mass, g 2 T : magnetic confinement, color relaxation, rate for small angle scattering g 4 T : rate for large angle scattering, η 1 T 4 Maximilian Attems, Schladming 2013 Longitudinal thermalization via the chromo-weibel instability 6/19

7 Scales of wqgp T : energy of hard particles gt : thermal masses, Debye screening mass, 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 Maximilian Attems, Schladming 2013 Longitudinal thermalization via the chromo-weibel instability 6/19

8 Weibel instabilities x Oblate Distribution F v v F F v v F Induced Current Magnetic Fluctuation j x B y z y [Strickland 2006]: Illustration of the mechanism of filamentation instabilities. z Maximilian Attems, Schladming 2013 Longitudinal thermalization via the chromo-weibel instability 7/19

9 Yang-Mills Vlasov One solves the gauge covariant Vlasov equation V D δf a p µ = gv µ F a µν ν (p) f 0(p, p η ) (1) Maximilian Attems, Schladming 2013 Longitudinal thermalization via the chromo-weibel instability 8/19

10 Yang-Mills Vlasov One solves the gauge covariant Vlasov equation V D δf a p µ = gv µ F a µν ν (p) f 0(p, p η ) (1) coupled to Yang-Mills equation D µ F µν a = j ν a = g t R d 3 p (2π) 3 p µ 2p 0 δf a(p, x, t) (2) Maximilian Attems, Schladming 2013 Longitudinal thermalization via the chromo-weibel instability 8/19

11 Yang-Mills Vlasov One solves the gauge covariant Vlasov equation V D δf a p µ = gv µ F a µν ν (p) f 0(p, p η ) (1) coupled to Yang-Mills equation D µ F µν a = j ν a = g t R d 3 p (2π) 3 p µ 2p 0 δf a(p, x, t) (2) in the HTL approximation ga µ p hard. (3) Maximilian Attems, Schladming 2013 Longitudinal thermalization via the chromo-weibel instability 8/19

12 Yang-Mills Vlasov One solves the gauge covariant Vlasov equation V D δf a p µ = gv µ F a µν ν (p) f 0(p, p η ) (1) coupled to Yang-Mills equation D µ F µν a = j ν a = g t R d 3 p (2π) 3 p µ 2p 0 δf a(p, x, t) (2) in the HTL approximation ga µ p hard. (3) The Romatschke, Strickland background distribution function: ( ) ) f 0 (p, x) = f iso p 2 + τ (p z ) τ 2 = f iso ( p 2 + p2 η/τiso 2. (4) iso Maximilian Attems, Schladming 2013 Longitudinal thermalization via the chromo-weibel instability 8/19

13 Unstable modes growth rate Γ/m D ξ = 10 0 ξ = 10 1 ξ = 10 2 ξ = 10 3 ξ = k z /m D Unstable mode spectra of purely longitudinal modes for specific anisotropies: N(τ) exp(2m D ττiso ) Maximilian Attems, Schladming 2013 Longitudinal thermalization via the chromo-weibel instability 9/19

14 Outline 1 Hard Expanding Loops (HEL) Stages of an heavy ion collision Scales of wqgp Weibel instabilities Yang-Mills Vlasov Unstable modes growth rate 2 Physical Observables Energy densities Pressures Spectra Longitudinal temperature Maximilian Attems, Schladming 2013 Longitudinal thermalization via the chromo-weibel instability 10/19

15 Energy densities fields Field energy densities / (Q s 4 / g 2 ) Total Energy E L E T B L B T τ = Q s τ / averaged runs N N η N u N φ = : after onset one sees rapid growth of B l and E L fields, followed by non-abelian interactions kick in. Maximilian Attems, Schladming 2013 Longitudinal thermalization via the chromo-weibel instability 11/19

16 Energy densities fields Total Field energy densities / (Q s 4 / g 2 ) = 0.8 = 0.6 = 0.4 = 0.3 = 0.2 = 0.15 = τ Total field energy density for different initial current fluctuation magnitudes. Maximilian Attems, Schladming 2013 Longitudinal thermalization via the chromo-weibel instability 12/19

17 Pressures Pressure τ τ PL, 0 field 3 τ τ PT, 0 field 3 τ τ PL, 0 particle 3 τ τ PT, 0 particle τ Initially highly anisotropic, note P L,field (τ = 0.3) < 0, growing field pressures, P L,field dominates at late times, τ scaled P L drops 1/ τ 2. Maximilian Attems, Schladming 2013 Longitudinal thermalization via the chromo-weibel instability 13/19

18 Pressures P L / P T = 0.8 = 0.6 = 0.4 = 0.3 = 0.2 = 0.15 = τ The evolution of the total longitudinal pressure over the total transverse pressure for different initial current fluctuation magnitudes. Maximilian Attems, Schladming 2013 Longitudinal thermalization via the chromo-weibel instability 14/19

19 Spectra Ν Q s 4 g Τ 0.3 Τ 1.5 Τ 2.7 Τ 3.9 Τ 5.1 Τ Ν The longitudinal energy spectra at various proper times as a function of ν: rapid emergence of an exponential distribution of longitudinal energy. Maximilian Attems, Schladming 2013 Longitudinal thermalization via the chromo-weibel instability 15/19

20 Spectra fits Massless Boltzmann distribution fits the longitudinal spectra: E fit (k z ) = A ( k 2 z + 2 k z T + 2T 2) exp ( k z /T ) (5) k z Q s 4 g 2 k z Q s 4 g Τ z s Τ 3.3 k z Q s 4 g 2 k z Q s 4 g Τ z s Τ k z Q s g 2 k z Q s g 2 Τ z s z s z s z s Comparison of data and fit function at six different τ. Maximilian Attems, Schladming 2013 Longitudinal thermalization via the chromo-weibel instability 16/19

21 Longitudinal thermalization 0.30 TL Qs T L T L Τ First the soft sector cools down. Due to the instability longitudinal soft fields reheats. Maximilian Attems, Schladming 2013 Longitudinal thermalization via the chromo-weibel instability 17/19

22 Conclusions The momentum space anisotropy can persist for quite some time. There doesn t seem to be a soft scale saturation of the instability as was seen in static boxes. The longitudinal spectra seem to be well described by a Boltzmann distribution indicating rapid longitudinal thermalization of the gauge fields. We are studying even larger lattices in order to improve our numerical results and also studying pure Yang-Mills dynamics in an expanding metric. Modifying our background distribution function may allow to probe the anisotropy and occupancy plane as a first step into incorporate backreation. Maximilian Attems, Schladming 2013 Longitudinal thermalization via the chromo-weibel instability 18/19

23 Backup Real-time lattice parameters of the evolution in temporal axial gauge: longitudinal lattice spacing a η transverse lattice spacing a Qs 1 temporal time step ɛ 10 2 τ 0 first time step τ 0 1/Q s longitudinal lattice points N η 128 transverse lattice points N 40 lattice size in velocity space N u N φ coupling constant g (3.77) 0.5 Assuming for LHC collisions We match to CGC values Q s 2GeV = (0.1fm) 1. (6) N g Qs 3 n(τ 0) = c 4π 2 N cα s(q sτ 0) with the gluon liberation factor c = 2 ln 2. From this one can determine the isotropic Debye mass (7) m 2 D(τ ISO) = 1.285/(τ 0τ ISO). (8) Maximilian Attems, Schladming 2013 Longitudinal thermalization via the chromo-weibel instability 19/19