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Hamilton approch to Yang-Mills Theory in Coulomb Gauge

Hamilton approch to Yang-Mills Theory in Coulomb Gauge. H. Reinhardt Tübingen. Collaborators: D. Campagnari, D. Epple, C. Feuchter, M. Leder, M.Quandt, W. Schleifenbaum, P. Watson. related work: D. Zwanziger A.P. Szczepaniak, E. S. Swanson, …. Plan of the talk.

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Hamilton approch to Yang-Mills Theory in Coulomb Gauge

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  1. Hamilton approch to Yang-Mills Theory in Coulomb Gauge H. Reinhardt Tübingen Collaborators: D. Campagnari, D. Epple, C. Feuchter, M. Leder, M.Quandt, W. Schleifenbaum, P. Watson

  2. related work: D. Zwanziger A.P. Szczepaniak, E. S. Swanson, …

  3. Plan of the talk • Hamilton approach to continuum Yang-Mills theory in Coulomb gauge • Variational solution of the YM Schrödinger equation: Dyson- Schwinger equations • Numerical Results • Infrared analysis of the DSE • ghost-gluon and 3-gluon vertex • t Hooft loop • Conclusions

  4. Classical Yang-Mills theory Lagrange function: field strength tensor

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

  6. Attempts to solve the Schrödinger equation with gauge invariant wave functionals K. Johnson,… gauge invariant variables Karabali, Kim, Nair strong coupling expansion of the (D=2+1) YM wave functional Greensite gradient expansion projection techniques Kogan, Kovner,... Heineman, Martin, Vautherin Schröder, H.R. more efficient way: resolve Gauß´law explicitly by fixing the gauge

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

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

  9. Importance of the Faddeev-Popov determinant defines the metric in the space of gauge orbits and hence reflects the gauge invariance

  10. metric of the space of gauge orbits: aim: solving the Yang-Mills Schrödinger eq. for the vacuum by the variational principle with suitable ansätze for Dyson-Schwinger equations

  11. QM: particle in a L=0-state variational kernel determined from DSE (gap equation) vacuum wave functional

  12. Dyson-Schwinger Equations ghost form factor d Abelian case d=1 ghost propagator ghost DSE gluon propagator gluon DSE (gap equation) gluon self-energy curvature

  13. Regularization and renormalization: momentum subtraction scheme renormalization constants: In D=2+1 is the only value for which the coupled Schwinger-Dyson equation have a self-consistent solution ultrviolet and infrared asymtotic behaviour of the solutions to the Schwinger Dyson equations is independent of the renormalization constants except for Zwanziger horizon condition

  14. ghost form factor gluon energy and curvature Numerical results (D=3+1)

  15. Coulomb potential

  16. external static color sources electric field ghost propagator

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

  18. comparison with lattice d=3 lattice: L. Moyarts, dissertation

  19. D=3+1 Infrared behaviour of lattice GF: not yet conclusive too small lattices, see talk by A. Maas

  20. previous work: A.P. Szczepaniak, E. S. Swanson, Phys. Rev. 65 (2002) 025012 A.P. Szczepaniak, Phys. Rev. 69(2004) 074031 different ansatz for the wave functional did not include the curvature of the space of gauge orbits i.e. the Faddeev- Popov determinant present work: C. Feuchter & H. R. hep-th/0402106, PRD70(2004) hep-th/0408237, PRD71(2005) W. Schleifenbaum, M. Leder, H.R. PRD73(2006) D. Epple, H. R., W. Schleifenbaum, in prepration full inclusion of the curvature measure for the curvature

  21. Importance of the curvature Szczepaniak & Swanson Phys. Rev. D65 (2002) • the c = 0 solution does not produce • a linear confinement potential

  22. Infrared limit = independent of to 2-loop order: Robustness of the infrared limit

  23. ghost dominance in the infrared d=4 Landau gauge functional integral d=3 Coulomb gauge canonical quantization strong coupling Infrared analysis of the DSE vacuum wave functional: generating functional

  24. LG: Lerche, v. Smekal Zwanziger, Alkofer, Fischer,… CG: Schleifenbaum, Leder, H.R. gluon propagator ghost propagator basic assumption:Gribov´s confinment scenario at work horizon condition: ghost DSE (bare ghost-gluon vertex) Landau gauge d=4 sum rule: solution of gluon DSE Coulomb gauge d=3 Coulomb gauge d=2 Analytic solution of DSE in the infrared

  25. ghost gluon vertex D=4 Landau gauge(Taylor): non-renormalization in all orders in g becomes bare for vanishing incoming ghost momentum d=3 Coulomb gauge: similar behaviour renormalization can be ignored

  26. 3- gluon vertex Asumption: color structure of the bare vertex single form factor

  27. 3-gluon vertex in Coulomb gauge

  28. order parameter of YMT temporal Wilson loop large variety of wave functionals produce the same DSE more sensitive observables than energy Coulomb potential = upper bound for true static quark potential (Zwanziger) confining Coulomb potential (=nessary but) not suffient for confinement Wilson loop difficult to calculate in continuum theory due to path ordering

  29. disorder parameter of YMT spatial ´t Hooft loop ´t Hooft loop ´t Hooft Münster Tomboulis Samuel Bhattacharya et al Del Debbio, Di Giacomo, B. Lucini Chernodub et al. Korthals-Altes, Kovner,.. de Focrand, D´Elia, Pepe, v. Smekal,…. Quandt, H.R., Engelhardt …. Recent review: Greensite

  30. defining eq. center vortex field V(C)-center vortex generator continuum representation: H.R: Phys.Lett.B557(2003) ´t Hooft loop

  31. C gauge dependent but produces gauge invariant results when acting on gauge invariant states

  32. Wilson loop magnetic flux C C ´t Hooft loop electric flux

  33. QM: wave functionals in Coulomb gauge satisfy Gauß´law and hence should be regarded as the gauge invariant wave functional restricted to transverse gauge fields.

  34. representation (correct to 2 loop) H. R. & C.F. PRD71 h(C;p)-geometry of the loop C planar circular loop C with radius R properties of the YM vacuum infrared properties of K(p) determine the large R-behaviour of S(R) ´t Hooft loop in Coulomb gauge

  35. c 0 c=0 produces wave functional which in the infrared approaches the strong coupling limit neglect curvature from gap equation renormalization condition:

  36. Summary and Conclusion • Variational solution of the YM Schrödinger equation in Coulomb gauge • Infrared analysis and numerical solution of the resulting DSE • Quark and gluon confinement • Curvature in gauge orbit space (Fadeev –Popov determinant) is crucial for the confinement properties • Ghost-gluon vertex = IR-finite • 3-gluon vertex= IR-divergent • ´t Hooft loop: perimeter law for a wave functional which in the infrared shows strict ghost dominance • Current projects: • QCD string • Topological susceptibility

  37. Thanks to the organizers

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