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Hamiltonian approach to Yang-Mills Theory in Coulomb gauge

Hamiltonian approach to Yang-Mills Theory in Coulomb gauge. H. Reinhardt Tübingen. Collaborators : G. Burgio, M.Quandt, P. Watson D. Epple, C. Feuchter, W. Schleifenbaum, D. Campagnari, S. Chimchinda, M. Leder, W. Lutz, M. Pak, C. Popovici, J. Pawlowski, A. Szczepaniak, A.Weber,.

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Hamiltonian approach to Yang-Mills Theory in Coulomb gauge

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  1. Hamiltonianapproachto Yang-Mills Theory in Coulomb gauge H. Reinhardt Tübingen Collaborators: G. Burgio, M.Quandt, P. Watson D. Epple, C. Feuchter, W. Schleifenbaum, D. Campagnari, S. Chimchinda, M. Leder, W. Lutz, M. Pak, C. Popovici, J. Pawlowski, A. Szczepaniak, A.Weber,

  2. aim of the talk • microscopicdescriptionofinfraredpropertieslikeconfinement • Hamiltonianapproachto YMT • Coulomb gauge

  3. Plan of Talk Hamiltonianapproachto Yang-Mills theory in Coulomb gauge basicresults: propagators comparisonwithlattice dielectricfunctionofthe Yang-Mills vacuum topologicalsusceptibility D=1+1: Gribovcopies conclusions

  4. References: C. Feuchter & H. R. hep-th/0402106, PRD70(2004) H. R. & C. Feuchter, hep-th/0408237, PRD71(2005) W. Schleifenbaum, M. Leder, H.R. PRD73(2006) D. Epple, H. R., W. Schleifenbaum, PRD75(2007) H. Reinhardt, D. Epple, Phys.Rev.D76:065015,2007 C. Feuchter & H. R,Phys.Rev.D77:085023,2008,D. Epple, H. R., W. Schleifenbaum, A. Szczepaniak, Phys.Rev.D77:085007,2008 H. Reinhardt,arXiv:0803.0504 [hep-th]PhysRevLett.101.061602, D. Campangnari & H. R.,arXiv:0807.1195 [hep-th], Phys.Rev.D, in press G. Burgio, M.Quandt, H.R.,arXiv:0807.3291 [hep-lat] related work: Swift Szczepanik & Swanson Zwanziger

  5. Gauß law: Canonical Quantization of Yang-Mills theory

  6. Gauß law: curved space resolution of Gauß´ law Faddeev-Popov Coulomb gauge

  7. YM Hamiltonian in Coulomb gauge Christ and Lee Coulomb term -arises from Gauß´law =neccessary to maintain gauge invariance -provides the confining potential

  8. forthevacuumbythevariationalprinciple metric of the space of gauge orbits: aim: solvingthe Yang-Mills Schrödingereq. withsuitableansätzefor

  9. forthevacuumbythevariationalprinciple aim: solvingthe Yang-Mills Schrödingereq. withsuitableansätzefor reflects non-trivial metric of the space of gauge orbits:

  10. QM: particle in a L=0-state Vacuumwavefunctional C. Feuchter, H.R, 2004 YMT gluon propagator variational kernel determined from gap equation

  11. Gluonenergy gluon confinement

  12. Propagators • gluonpropagator • ω(k)-gluonenergy • ghostpropagator • ghostformfactor d(k): deviationsfrom QED: • QED: • Coulomb potential

  13. numericalsolution • Confinementofgluons • Excellentagreementwith IR and UV analysis • (in)dependence on renormalizationscale D. Epple, H. Reinhardt, W.Schleifenbaum, PRD 75 (2007)

  14. Coulomb potential

  15. running coupling W. Schleifenbaum, M. Leder, H.R. PRD73(2006)

  16. Comparisonwithlatticedata

  17. comparisonwithlattice D=2+1 lattice: L. Moyarts, dissertation continuum: C. Feuchter & H. Reinhardt

  18. Latticecalculation in D=3+1 H.Reinhardt Cuccheri, Zwanziger Langfeld, Moyarts, Cuccheri, Mendes A. Voigt, M. Ilgenfritz, M. Muller-Preussker, A.Sternbeck G.Burgio, M. Quandt, S. Chimchinda, H. R.,

  19. ghostpropagator D=3+1 Burgio, Quandt, Chimchinda, H. R., PoS LAT2007:325,2007

  20. Gluonpropagator in D=3+1 K. Langfeld, L. Moyarts, 2004

  21. recentlatticecalculationsofD=3+1 gluonpropagator G. Burgio, M.Quandt, H.R., arXiv:0807.3291 [hep-lat] gaugefixing renormalization

  22. Staticgluonpropagator in D=3+1 G. Burgio, M.Quandt, H.R., arXiv:0807.3291 [hep-lat]

  23. Asymptotics lattice continuum IR: α=1 UV:γ=1.0 δ=0.0 • IR: α=0.98(2) • UV: γ=1.005(10) δ=0.000(2)

  24. The colorelectricfield • ED:

  25. The colorelectricfield • ED: • QCD:

  26. external static color sources electric field ghost propagator

  27. The color electric flux tube missing: back reaction of the vacuum to the external sources

  28. The colorelectricfield • ED: • QCD:

  29. The colorelectricfield • ED: • medium • QCD:

  30. The colorelectricfield • ED: • medium • QCD: • ghostpropagator

  31. The colordielectric „constant“ ofthe QCD vacuum • ED: • medium • QCD: • ghostpropagator

  32. The colordielectric „constant“ ofthe QCD vacuum • ED: • medium • QCD: • ghostpropagator H. Reinhardt,PhysRevLett.101.061602(2008)

  33. The colordielectricfuctionofthe QCD vacuum

  34. k The colordielectricfunctionofthe QCD vacuum • ghostpropagator • dielectric „constant“ • horizoncondition: • : • QCD vacuum-perfectcolordia-electricum • QED: screening

  35. no free color charges in the vacuum: confinement

  36. magnetic analog tothe QCD vacuum :superconductor • magmetism in matter: • perfectdia-magneticum : • Superconductor

  37. magnetic analog tothe QCD vacuum :superconductor • magmetism in matter: • perfectdia-magneticum : • superconductor • QCD vacuum:perfectdia-elektricum • dual superconductor • Duality:

  38. Confinementscenarios Gribov-Zwanziger: ≈ (Kugo-Ojima) dual superconductor: magneticmonopolecondensation

  39. Confinementscenarios Gribov-Zwanziger: ≈ (Kugo-Ojima) latticeevidence: monopolecondensation ≈ vortexcondensation ≈ dual superconductor: magneticmonopolecondensation centervortexcondensation Gribov-Zwanziger

  40. Kugo-Ojima confinement criteria: infrared divergent ghost form factor Gattnar, Langfeld, Reinhardt NPB262(2002)131 elimination of center vortices removes: -string tension (Wilson´s confinment criterium) -the infrared divergency from the ghost propagator (Kogu-Ojima confinement criterium)

  41. Coulomb potential J. Greensite, S. Olejnik , 2003

  42. Confinementscenarios Gribov-Zwanziger: ≈ (Kugo-Ojima) latticeevidence: monopolecondensation ≈ vortexcondensation ≈ dual superconductor: magneticmonopolecondensation centervortexcondensation Gribov-Zwanziger

  43. Chiralsymmeryof QCD • spontaneousbreaking: • quarkcondensation • constituentquarkmass • soft explicit breaking: • currentmassses • anomalousbreaking: • η´mass

  44. Witten-Veneziano-Formula in perturbation theory • topologicalsusceptibility • topologicalchargedensity

  45. -vacuum in theHamiltonianapproach Lagrangian canonicalmomentum hamiltonian topologicalsusceptibility

  46. Topologicalsusceptibility in the Hamilton approach D. Campangnari & H. R, Phys.Rev.D, in press exactcancellationofAbelianpartofBB 2-and 3-quasi-gluons on top ofthevacuum renormalization

  47. Numericalcalculations parametrizations:

  48. Numericalcalculations IR dominanceoftheintegrals runningcoupling: IR limit:

  49. NumericalResults

  50. Summary & Conclusion • Hamiltonianapproachto YMT in Coulomb gauge • Variationalsolutionofthe YM Schrödingereq. • gluonconfinement • quarkconfinement • satisfactoryagreementwithlatticedata • dielectricfunctionofthe YM vacuum • ε(k)=inverse ghost form factor • YM vacuum=perfect dual superconductor • Gribov-Zwanziger Conf.↔dual Meißner effect • topologicalsusceptibility

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