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Turbulent spectra in non-Abelian gauge theories Sebastian Scheffler, TU Darmstadt, 30 January 2009, ¢ (2009) Heidelberg Journal references: J. Berges,

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Presentation on theme: "Turbulent spectra in non-Abelian gauge theories Sebastian Scheffler, TU Darmstadt, 30 January 2009, ¢ (2009) Heidelberg Journal references: J. Berges,"— Presentation transcript:

1 Turbulent spectra in non-Abelian gauge theories Sebastian Scheffler, TU Darmstadt, 30 January 2009, ¢ (2009) Heidelberg Journal references: J. Berges, S. Scheffler, D. Sexty, PRD 77, 034504 (2008), arXiv:0712.3514 [hep-ph] J. Berges, S. Scheffler, D. Sexty, arXiv:0811.4293 [hep-ph], submitted to Elsevier J. Berges, D. Gelfand, S. Scheffler, D. Sexty, arXiv 0812.3859 [hep-ph], submitted to Elsevier TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA A A A A A A A A A AAAA A AA  A A

2 Outline of the talk 30.01.2009Sebastian Scheffler2 1.Motivation 2.Formalism & setup 3.Results: Fast vs. slow dynamics 4.Conclusions & outlook Turbulent spectra in non-Abelian gauge theories

3 Motivation, part 1: Heavy-ion collisions 30.01.2009Sebastian Scheffler3 Turbulent spectra in non-Abelian gauge theories Result from RHIC: hydrodynamics works well starting at ¿ 0 ' 1 fm/c, (Luzum/Romatschke, PRC 78) -> Rapid isotropisation essential (Arnold et al., PRL 94): How is this achieved? Need to understand what happens before ¿ 0 ; plasma instabilities? Numerical approaches: 1.Soft classical gauge fields coupled to hard classical particles 2.Classical-statistical gauge field evolution Introduction

4 Motivation, part 2: Non-equilibrium QFT 30.01.2009Sebastian Scheffler4 There are still many open questions in non-equilibrium QFT - in particular regarding gauge theories. An (incomplete) to-do list: Develop, test, and benchmark different approximation schemes Analyse and exploit analogies between various fields of non- equilibrium physics (e. g. early universe, heavy-ion collisions, cold atomic gases ) Transport coefficients Non-thermal fixed points? Universality far from equilibrium? Turbulent spectra in non-Abelian gauge theoriesIntroduction

5 Reminder: Classical-statistical field theory 30.01.2009Sebastian Scheffler5 Why use the classical approximation? feasibility good to study early times if occupation numbers are high highly successful for scalar fields can serve to test other methods (e. g. 2PI) Turbulent spectra in non-Abelian gauge theoriesFormalism & setup

6 Setup 30.01.2009Sebastian Scheffler6 classical-statistical limit of pure SU(2) gauge theory: Evolve an initial ensemble using the classical field equations discretize everything on a lattice use a static geometry pure gauge theory, i. e. no fermions anisotropic initial conditions (-> heavy-ion collisions) no separation of scales assumed Formalism & setupTurbulent spectra in non-Abelian gauge theories

7 Setup 30.01.2009Sebastian Scheffler7 Turbulent spectra in non-Abelian gauge theoriesFormalism & setup

8 Sampling from the initial ensemble 30.01.2009Sebastian Scheffler8 Compute e. g. a correlation function according to where the initial density function is characterised by ¢ x À ¢ z, distribution ± ( p z ) – like on the lattice (  /  t ) (t=0) = 0 => Gauss- constraint fulfilled Amplitude C dialed to give a fixed energy Convert to physical units via Formalism & setupTurbulent spectra in non-Abelian gauge theories

9 Instabilities: A brief reminder of ¢ (2007) 30.01.2009Sebastian Scheffler9 Turbulent spectra in non-Abelian gauge theoriesResults: Fast dynamics Some general facts about instabilities: Gauge field possesses unstable (i. e. exponetially growing) modes if distribution of charge carriers is anisotropic (Mrówczyńsky, Romatschke/Strickland,... ) Bottom-up scenario by Baier et al. modified Can instabilities resolve the thermalization/isotropization puzzle? (Arnold et al.)

10 30.01.2009Sebastian Scheffler10 Instabilities: A brief reminder of ¢ (2007) 30.01.2009Sebastian Scheffler10 Turbulent spectra in non-Abelian gauge theoriesResults: Fast dynamics Brief summary: instabilities occur using anisotropic init. cond. inverse growth rates » 1 fm/c (for ² = 30 GeV/fm^3) low-momentum sector driven towards isotropy Two disadvantages of the original setup: SU(2) instead of SU(3) Gauss constraint enforced by (  /  t ) (t=0) = 0

11 30.01.2009Sebastian Scheffler1130.01.2009Sebastian Scheffler11 Instabilities: Some new results 30.01.2009Sebastian Scheffler11 Turbulent spectra in non-Abelian gauge theoriesResults: Fast dynamics B. Sc. theses of D. Gelfand and N. Balanešković SU(3): different time scales, but can be accounted for in terms of the number of colours see arXiv:0812.3859 [hep-ph] Gauss- constraint: Can implement more general initial conditions no differences discernible

12 Fast vs. slow dynamics 30.01.2009Sebastian Scheffler12 Turbulent spectra in non-Abelian gauge theoriesResults: Fast vs. slow dynamics Early times: dominated by fast processes (instabilities) Late times: governed by slow/stationary processes fixed points / turbulence / scaling solutions ? Why is this interesting? -> Cf. early universe

13 UV- fixed points: Motivation from scalars 30.01.2009Sebastian Scheffler13 Stationary power-law spectra reminiscient of Kolmogorov turbulence are commonly encountered in early-universe cosmology following a phase of parametric resonance: Micha/Tkachev, PRD 70Berges/Rothkopf/Schmidt, PRL 101 The spectral index 3/2 is derived in terms of Boltzmann- eqns. or 2PI- calculations, respectively. Turbulent spectra in non-Abelian gauge theoriesResults: Slow dynamics

14 UV- fixed points: What about gauge theories? 30.01.2009Sebastian Scheffler14 Yes! 1.Arnold & Moore (PRD 73) find particle number spectra with spectral index κ = 2. 2.Müller et al. predict κ = 1 (thermal value), NPB 760. 3.This work: See next slides… Turbulent spectra in non-Abelian gauge theoriesResults: Slow dynamics Are there analogous phenomena in gauge theories?

15 Search for UV- fixed points - Analytics (I) 30.01.2009Sebastian Scheffler15 Turbulent spectra in non-Abelian gauge theoriesResults: Slow dynamics Consider in the following Search for solutions of the form Are there solutions of this kind? If yes, what is the value of κ ?

16 Search for UV- fixed points - Analytics (II) 30.01.2009Sebastian Scheffler16 J. Berges / G. Hoffmeister, arXiv:0809.5208: Stationary and translationally-invariant correlation functions fulfill the identity where and denote the non-local contributions to the self-energy of odd and even symmetry, respectively. Turbulent spectra in non-Abelian gauge theoriesResults: Slow dynamics

17 Search for UV- fixed points - Analytics (III) 30.01.2009Sebastian Scheffler17 Evaluate 1-loop contribution to the self-energy: Assume scaling behaviour of the kind and demand Turbulent spectra in non-Abelian gauge theoriesResults: Slow dynamics

18 Search for UV- fixed points - Analytics (IV) 30.01.2009Sebastian Scheffler18 First, this yields a rather unwieldy integral: 3- vertex Carrying out a Zakharov- transformation, this can be cast into the form Classical limit: |F F | À | ½ ½ |, Turbulent spectra in non-Abelian gauge theoriesResults: Slow dynamics No fixed-point solution in the full quantum theory! solution for

19 Search for UV- fixed points - Numerics 30.01.2009Sebastian Scheffler19 Find that the equal-time correlators converge to a stationary solution after the saturation of instabilities. Computation on a 128^3- lattice in Coulomb gauge Fit spectrum to Turbulent spectra in non-Abelian gauge theoriesResults: Slow dynamics

20 Universality far from equilibrium? 30.01.2009Sebastian Scheffler20 Early-universe (scalars)Heavy-ion coll.(Yang-Mills) Turbulent spectra in non-Abelian gauge theoriesResults: Slow dynamics parametric resonance, instabilities fixed-point solutions, turbulence, power- law spectra

21 30.01.2009Sebastian Scheffler21 Fixed points: Obstacles on the way to equilibrium? 30.01.2009Sebastian Scheffler21 Wanted to reach equilibrium by fast processes (instabilities) …. … but seem to get stuck at a fixed point instead! Turbulent spectra in non-Abelian gauge theoriesResults: Slow dynamics However: No UV- fixed point in the full quantum theory

22 Summary 30.01.2009Sebastian Scheffler22 Turbulent spectra in non-Abelian gauge theoriesConclusions & outlook Instabilities: inverse growth rates of order 1 fm/c no qualitative difference for SU(3) and more general initial conditions UV- fixed point: Find quasi-stationary power-law spectrum in Coulomb gauge characterised by spectral index κ = 3/2 very similar to results for scalars – universality far from equilibrium?

23 Future projects 30.01.2009Sebastian Scheffler23 1.Establish a description of the UV- fixed point in terms of gauge invariant quantities 2.Investigate the IR- regime: Are there power-law solutions as in the scalar field theory? 3.Couple the gauge fields to fermions 4.Compare classical-statistical simulations to 2PI- calculations Turbulent spectra in non-Abelian gauge theoriesConclusions & outlook

24 30.01.2009Sebastian Scheffler24 Thanks for your attention!

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