Yang-Mills Theory in Coulomb Gauge

1. Yang-Mills Theory in Coulomb Gauge H. Reinhardt Tübingen non-perturbative approach to continuum YMT C. Feuchter & H. R. hep-th/0402106, PRD70 hep-th/0408237, PRD71 hep-th/0408236 D. Epple, C. Feuchter, H.R., hep-th/0412231 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. 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. Importance of the Faddeev-Popov determinant defines the metric in the space of gauge orbits and hence reflects the gauge invariance

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

10. Vacuum wave functional variational kernel determined from 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. bare vertex Ghost-gluon vertex rain-bow ladder approx: replace full vertex by bare one

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: 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 horizon condition

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

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

21. Coulomb potential

22. external static color sources electric field ghost propagator

23. The color electric flux tube

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

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

26. eliminating the self-energies

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

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

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

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

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

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

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

35. Connection to the Center Vortex Picture

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

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

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

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

40. center vortices 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!)

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

42. 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)

43. Thanks to the organizers