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Turbulent spectra in non- Abelian gauge theories

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 ]

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Turbulent spectra in non- Abelian gauge theories

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  1. Turbulent spectra in non-Abeliangaugetheories 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], submittedtoElsevier J. Berges, D. Gelfand, S. Scheffler, D. Sexty, arXiv 0812.3859 [hep-ph], submittedtoElsevier TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAAAAAAAAAA

  2. Turbulent spectra in non-Abelian gauge theories Outline ofthe talk Motivation Formalism & setup Results: Fast vs. slowdynamics Conclusions & outlook Sebastian Scheffler

  3. Turbulent spectra in non-Abeliangaugetheories Motivation, part 1: Heavy-ioncollisions Resultfrom RHIC: hydrodynamicsworks well startingat¿0' 1 fm/c , (Luzum/Romatschke, PRC 78) -> Rapid isotropisation essential (Arnold et al., PRL 94): Howisthisachieved? Need to understand whathappensbefore¿0; plasmainstabilities? Numericalapproaches: Soft classicalgaugefieldscoupledtohardclassicalparticles Classical-statisticalgaugefieldevolution Introduction Sebastian Scheffler

  4. Turbulent spectra in non-Abeliangaugetheories Motivation, part 2: Non-equilibrium QFT Thereare still many open questions in non-equilibrium QFT - in particularregardinggaugetheories. An (incomplete) to-do list: Develop, test, andbenchmark different approximationschemes Analyse andexploitanalogiesbetweenvariousfieldsof non-equilibriumphysics (e. g. earlyuniverse, heavy-ioncollisions, coldatomicgases ) Transport coefficients Non-thermal fixedpoints? Universalityfarfromequilibrium? Introduction Sebastian Scheffler

  5. Turbulent spectra in non-Abeliangaugetheories Reminder: Classical-statisticalfieldtheory Whyusetheclassicalapproximation? feasibility goodtostudyearlytimesifoccupationnumbersarehigh highlysuccessfulforscalarfields canservetotestothermethods (e. g. 2PI) Formalism & setup Sebastian Scheffler

  6. Turbulent spectra in non-Abeliangaugetheories Setup classical-statisticallimitof pure SU(2) gaugetheory: Evolve an initialensembleusingtheclassicalfieldequations discretizeeverything on a lattice use a staticgeometry pure gaugetheory, i. e. nofermions anisotropicinitialconditions (-> heavy-ioncollisions) noseparationofscalesassumed Formalism & setup Sebastian Scheffler

  7. Turbulent spectra in non-Abeliangaugetheories Setup Formalism & setup Sebastian Scheffler

  8. Turbulent spectra in non-Abeliangaugetheories Sampling fromtheinitialensemble Compute e. g. a correlationfunctionaccordingto wheretheinitialdensityfunctionischaracterisedby ¢xÀ¢z , distribution±( pz ) – like on thelattice ( A/  t ) (t=0) = 0 => Gauss- constraintfulfilled Amplitude C dialedtogive a fixedenergy Converttophysicalunits via Formalism & setup Sebastian Scheffler

  9. Turbulent spectra in non-Abeliangaugetheories Instabilities: A briefreminderof¢(2007) • Somegeneralfactsaboutinstabilities: • Gauge fieldpossessesunstable (i. e. exponetiallygrowing) modesifdistributionofchargecarriersisanisotropic (Mrówczyńsky, Romatschke/Strickland, ... ) • Bottom-upscenarioby Baier et al. modified • Can instabilitiesresolvethethermalization/isotropization puzzle? (Arnold et al.) Results: Fast dynamics Sebastian Scheffler

  10. Turbulent spectra in non-Abeliangaugetheories Instabilities: A briefreminderof¢(2007) • Brief summary: • instabilitiesoccurusinganisotropicinit. cond. • inverse growthrates» 1 fm/c (for² = 30 GeV/fm^3) • low-momentumsectordriventowardsisotropy • Twodisadvantagesofthe original setup: • SU(2) insteadof SU(3) • Gaussconstraintenforcedby ( A/  t ) (t=0) = 0 Results: Fast dynamics 30.01.2009 Sebastian Scheffler Sebastian Scheffler 10

  11. Turbulent spectra in non-Abeliangaugetheories Instabilities: Somenewresults B. Sc. thesesof D. Gelfandand N. Balanešković • SU(3): • different time scales, but canbeaccountedfor in termsofthenumberofcolours • see arXiv:0812.3859 [hep-ph] • Gauss- constraint: • Can implementmoregeneralinitialconditions • nodifferencesdiscernible Results: Fast dynamics 30.01.2009 30.01.2009 Sebastian Scheffler Sebastian Scheffler Sebastian Scheffler 11 11

  12. Turbulent spectra in non-Abeliangaugetheories Fast vs. slowdynamics Whyisthisinteresting? -> Cf. earlyuniverse • Early times: • dominatedby fast processes (instabilities) • Latetimes: • governedbyslow/stationaryprocesses • fixedpoints / turbulence / scalingsolutions ? Results: Fast vs. slowdynamics Sebastian Scheffler

  13. Turbulent spectra in non-Abeliangaugetheories UV- fixedpoints: Motivation fromscalars Micha/Tkachev, PRD 70 Berges/Rothkopf/Schmidt, PRL 101 The spectralindex 3/2 isderived in termsof Boltzmann- eqns. or 2PI- calculations, respectively. Stationary power-lawspectrareminiscientofKolmogorovturbulencearecommonlyencountered in early-universecosmologyfollowing a phaseofparametricresonance: Results: Slow dynamics Sebastian Scheffler

  14. Turbulent spectra in non-Abeliangaugetheories UV- fixedpoints: Whataboutgaugetheories? Are thereanalogousphenomena in gaugetheories? Yes! Arnold & Moore (PRD 73) find particlenumberspectrawithspectralindexκ = 2. Müller et al. predictκ = 1 (thermal value) , NPB 760 . Thiswork: See nextslides… Results: Slow dynamics Sebastian Scheffler

  15. Turbulent spectra in non-Abeliangaugetheories Searchfor UV- fixedpoints - Analytics (I) Consider in thefollowing Searchforsolutionsofthe form Are theresolutionsofthiskind? Ifyes, whatisthevalueofκ? Results: Slow dynamics Sebastian Scheffler

  16. Turbulent spectra in non-Abeliangaugetheories Searchfor UV- fixedpoints - Analytics (II) J. Berges / G. Hoffmeister, arXiv:0809.5208: Stationaryandtranslationally-invariant correlationfunctionsfulfilltheidentity where anddenotethe non-localcontributionstotheself-energyofoddandevensymmetry, respectively. Results: Slow dynamics Sebastian Scheffler

  17. Turbulent spectra in non-Abeliangaugetheories Searchfor UV- fixedpoints - Analytics (III) Evaluate 1-loop contributiontotheself-energy: Assumescalingbehaviourofthekind anddemand Results: Slow dynamics Sebastian Scheffler

  18. Turbulent spectra in non-Abeliangaugetheories Searchfor UV- fixedpoints - Analytics (IV) First, thisyields a ratherunwieldy integral: 3- vertex Carrying out a Zakharov- transformation, thiscanbecastintothe form Classicallimit: |F F | À | ½½ | , Nofixed-pointsolution in thefullquantumtheory! solutionfor Results: Slow dynamics Sebastian Scheffler

  19. Turbulent spectra in non-Abeliangaugetheories Searchfor UV- fixedpoints - Numerics Find thattheequal-time correlators convergeto a stationarysolution after thesaturationofinstabilities. Computation on a 128^3- lattice in Coulomb gauge Fit spectrumto Results: Slow dynamics Sebastian Scheffler

  20. Turbulent spectra in non-Abeliangaugetheories Universalityfarfromequilibrium? Early-universe (scalars) Heavy-ioncoll.(Yang-Mills) parametricresonance, instabilities fixed-pointsolutions, turbulence, power-lawspectra Results: Slow dynamics Sebastian Scheffler

  21. Turbulent spectra in non-Abeliangaugetheories Fixed points: Obstacles on thewaytoequilibrium? Wantedtoreachequilibriumby fast processes (instabilities) …. … but seemtoget stuck at a fixedpointinstead! However: No UV- fixedpoint in thefullquantumtheory Results: Slow dynamics 30.01.2009 Sebastian Scheffler Sebastian Scheffler 21

  22. Turbulent spectra in non-Abeliangaugetheories Summary • Instabilities: • inverse growthratesof order 1 fm/c • no qualitative differencefor SU(3) andmoregeneralinitialconditions • UV- fixedpoint: • Find quasi-stationary power-lawspectrum in Coulomb gauge • characterisedbyspectralindexκ = 3/2 • verysimilartoresultsforscalars – universalityfarfromequilibrium? Conclusions & outlook Sebastian Scheffler

  23. Turbulent spectra in non-Abeliangaugetheories Future projects Establish a descriptionofthe UV- fixedpoint in termsofgauge invariant quantities Investigatethe IR- regime: Are there power-lawsolutionsas in thescalarfieldtheory? Couplethegaugefieldstofermions Compareclassical-statisticalsimulationsto 2PI- calculations Conclusions & outlook Sebastian Scheffler

  24. Thanksforyourattention! Sebastian Scheffler

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