Hamilton approch to Yang-Mills Theory in Coulomb Gauge

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. 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 • Topological susceptibility • `t Hooft loop • Conclusions

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

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

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

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

7. 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

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

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

10. 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

11. 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

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

13. Coulomb potential

14. Coulomb form factor f Schwinger-Dyson eq. rigorous result to 1-loop:

15. external static color sources electric field ghost propagator

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

17. dielectric „constant“ k The dielectric „constant“ of the Yang-Mills vacuum Maxwell´s displecement The Yang-Mills vacuum is a perfect color dia-electric

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

20. related 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. Fischer, Zwanziger interpolating gauges sum rule for the infrared exponents from ghost DSE running coupling

26. Topological susceptibility Witten-Veniciano formula:

27. explicit realization: Chern-Simon action: topological susceptibility: vanishes to all orders in g Topological susceptibility in Hamilton approach spatial gauge transformation:

28. Identify our variational wave functional with the restriction of the gauge invarinant to Coulomb gauge very preliminary result (D. Campagnari -Diploma thesis) (very crude parametrization of the ghost and gluon GFs) : Input: 2-loop formula for the running coupling exact cancelation of the Abelian part of BB

29. 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

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

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

32. 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.

33. 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

34. 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 • Quark and gluon confinement • IR-finite running coupling • Curvature in gauge orbit space (Fadeev –Popov determinant) is crucial for the confinement properties • Topological susceptibility • ´t Hooft loop: perimeter law for a wave functional which in the infrared shows strict ghost dominance

37. Thanks to the organizers