1. Yang-Mills Theory in Coulomb Gauge H. Reinhardt Tübingen C. Feuchter & H. R. hep-th/0402106, PRD70 hep-th/0408237, PRD71 hep-th/0408236 D. Epple, C. Feuchter, H.R., hep-th/0412231 non-perturbative approach to continuum YMT W. Schleifenbaum M. Leder H. Turan

2. Previous work: A.P. Szczepaniak, E. S. Swanson, Phys. Rev. 65 (2002) 025012 A.P. Szczepaniak, hep-ph/0306030 P.O. Bowman, A.P. Szczepaniak, hep-ph/0403074

3. Plan of the talk Basics of continuum Yang-Mills theory in Coulomb gauge Variational solution of the YM Schrödinger equation: Dyson- Schwinger equations Results: –Ghost and gluon propagators –Heavy quark potential –Color electric field of static sources YM wave functional Finite temperatures Connection to the center vortex picture of confinement

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

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

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

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

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

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

10. Vacuum wave functional determined fromvariational kernel at the Gribov horizon: wave function is singular -identifies all configurations on the Gribov horizon preserves gauge invariance -topolog. compactification of the Gribov region FMR

11. QM: particle in a L=0-state

12. Minimization of the energy set of Schwinger-Dyson equations for:

13. Gluon propagator transversal projector Wick´s theorem: any vacuum expectation value of field operators can be expressed by the gluon propagator

14. Ghost propagator ghost form factor d Abelian case d=1 ghost self-energy

15. Ghost-gluon vertex rain-bow ladder approx: replace full vertex by bare one bare vertex

16. Curvature (ghost part of the gluon energy)

17. Coulomb form factor f Schwinger-Dyson eq.

18. 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

19. Asymptotic behaviour D=3+1 -angular approximation infrared behaviour ultraviolet behaviour

20. Numerical results (D=3+1) ghost and Coulomb form factors gluon energy and curvature mass gap:

21. Coulomb potential

22. external static color sources electric field ghost propagator

23. The color electric flux tube

24. The flux between 3 static color charges a=3a=8

25. The „baryon“= 3 static quarks in a color singlet

26. eliminating the self-energies

27. The dielectric „constant“ of the Yang-Mills vacuum Maxwell´s displecement dielectric „constant“ k

28. Importance of the curvature Szczepaniak & Swanson Phys. Rev. D65 (2002) the  = 0 solution does not produce a quasi-linear confinement potential

29. The vacuum wave functional & Fadeev-Popov determinant to 1-loop order:

30. Robustness of the infrared limit Infrared limit = independent of gauge fields at different points are completely uncorrelated stochastic vacuum exact in D=1+1

31. 3-gluon vertex M.Leder W.Schleifenbaum

33. Finite temperature YMT ground state wave functional vacuum gas of quasi-gluons with energy

34. Energy density Lattice: Karsch et al. minimization of the free energy:

35. Connection to the Center Vortex Picture

36. D=2points D=4closed surfaces self-intersect non-oriented vortices D=3 closed loops

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

38. Q-Q-potential: SU(2)

39. Confinement mechanism in Coulomb gauge infrared dominant field configurations: : static quark potential Gribov horizon

40. similar results in Coulomb gauge: Greensite, Olejnik, Zwanziger, hep-lat/0407032 Kugo-Ojima confinement criteria: infrared divergent ghost propagator center vortices Suman &Schilling (1996) Nakajima,… Bloch et al. Gattnar, Langfeld, Reinhardt, Phys. Rev.Lett.93(2004)061601, hep-lat/0403011

41. Ghost Propagator in Maximal Center Gauge (MCG)  fixes SU(2) / Z (2)  ghosts do not feel the center Z (2) no signal of confinement in the ghost propagator  removal of center vortices does not change the ghost propagator (analytic result!) center vortices

42. Landau(Coulomb)gauge maximum center gauge center vortices Gribov´s confinement criteria (infrared ghost propagator) is realized in gauges where the center vortices are on the Gribov horizon

43. Summary and Conclusion Hamilton approach to QCD in Coulomb gauge is very promising for non-perturbative studies Quark and gluon confinement Curvature in gauge orbit space (Fadeev –Popov determinant) is crucial for the confinement properties Center vortices are on the Gribov horizon and are the infrared dominant field configuratons, which give rise to an infrared diverging ghost propagator (Gribov´s confinement scenario)

44. Thanks to the organizers