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. Canonical Quantization of Yang-Mills theory Gauß law:

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

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

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

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

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

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

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

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

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  = 0 solution does not produce a linear confinement potential

22. partial inclusion of the curvature: neglect the curvature in the Coulomb term Szczepaniak, hep-hp/0306030

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

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

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

26. 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

27. 3- gluon vertex Asumption: color structure of the bare vertex single form factor

28. 3-gluon vertex in Coulomb gauge

29. 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 order parameter of YMT temporal Wilson loop difficult to calculate in continuum theory due to path ordering

30. ´ 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 disorder parameter of YMT spatial ´t Hooft loop

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

32. disorder parameter of YMT spatial ´t Hooft looop continuum representation: H.R: Phys.Lett.B557(2003) V(C)-center vortex generator center vortex field ´t Hooft loop defining eq.

33. Center Vortices in Continuum Yang-Mills theory Wilson loop Linking number center element C

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

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

36. 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.

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

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

41. 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

42. Thanks to the organizers