1. Hamiltonian Flow in Coulomb Gauge Yang-Mills theory
H. Reinhardt
Tübingen
Collaborators:
M. Leder, J. Pawlowski, A.Weber

2. Hamiltonian approach to YMT
canonical quantization

3. Hamiltonian approach to YMT
canonical quantization
Coulomb gauge

4. Hamiltonian approach to YMT
canonical quantization
Coulomb gauge
variational solution
ansatz for vacuum wave functional
DSE to calculate
minimization of

5. Hamiltonian approach to YMT
canonical quantization
Coulomb gauge
variational solution
ansatz for vacuum wave functional
DSE to calculate
minimization of
FRG flow equations

6. Hamiltonian approach to YMT
canonical quantization
Coulomb gauge
variational solution
ansatz for vacuum wave functional
DSE to calculate
minimization of
FRG flow equations
infinite tower of flow eqs. for static propagators
indirect determination of
truncation of flow equations
Ansätze for propagators
initial condition to the flow (UV-regime)

7. Variational approach trial ansatz : C.Feuchter & H. R. PRD70(2004)
gluon propagator
determined from
variational kernel

8. D. Epple, H. Reinhardt, W.Schleifenbaum, PRD 75 (2007)
Numerical results
D. Epple, H. Reinhardt, W.Schleifenbaum, PRD 75 (2007)
gluon energy

9. Static gluon propagator in D=3+1
G. Burgio, M.Quandt , H.R., PRL102(2009)

10. W. Schleifenbaum, M. Leder, H.R. PRD73(2006)
running coupling
W. Schleifenbaum, M. Leder, H.R. PRD73(2006)

11. Coulomb potential

12. D. Epple, H. Reinhardt, W.Schleifenbaum, PRD 75 (2007)
ghost formfactor
gost propagator
ghost form factor
D. Epple, H. Reinhardt, W.Schleifenbaum, PRD 75 (2007)
Input:

13. ghost formfactor: lattice
see talk by G. Burgio

14. D. Campagnari, H. R., A. Weber, Phys. Rev D(2009)
Perturbation theory
D. Campagnari, H. R., A. Weber, Phys. Rev D(2009)
Rayleigh-Schrödinger PT
vacuum (QED)
ß-function

15. The color dielectric function of the QCD vacuum
ghost propagator
dielectric „constant“
H.Reinhardt,PRL101 (2008)

16. The color dielectric function of the QCD vacuum
ghost propagator
dielectric „constant“
horizon condition: :
QCD vacuum-perfect color dia-electricum
H.Reinhardt,PRL101 (2008)

17. The color dielectric function of the QCD vacuum
ghost propagator
dielectric „constant“
horizon condition: :
QCD vacuum-perfect color dia-electricum
QED: screening
H.Reinhardt,PRL101 (2008)

18. no free color charges in the vacuum: confinement

19. Confinement scenarios
Gribov-Zwanziger
dual superconductor
horizon condition
perfect dia-electricum

20. Can we avoid the input of the horizon condition ?
Yes we can : RG-flow equation:
indirect test of our ansatz for the wave functional
M. Leder, J. Pawlowski, H. R, A. Weber

21. Renormalization group flow equation

22. Renormalization group flow equation
Wetterich

23. Effective action: 1PI-vertices

25. RG-flow equation
Wetterich 1993

26. RG-flow equation
propagator flow

27. Hamiltonian flow

28. Hamiltonian flow in Coulomb gauge YMT

29. RG- flow equation

30. Hamiltonian FRG flow equation
no ansatz for
indirect specification of
truncation of flow equation
form of the propagators&vertices assumed
intial condition to propagators&vertices in the UV

31. Truncation of FRG flow equation
gluon propagator
ghost propagator
ghost-gluon-vertex
no tadpoles
ghost dominance
no gluon loops

32. RG- flow equation
ghost dominance

33. Integrating the RG-flow equation

35. ghost form factor d(p)

36. gluon energy

37. FRG & DSE Replacement in loop integrals of FRG:
FRG flow eq DSE-variational approach

38. RG-Flow vs DSE: ghost form factor

39. RG-Flow vs DSE: gluon energy

40. RG-Flow vs DSE: running coupling

41. IR- exponents
satisfy sum rule
smaller than for DSE

42. Summary & Conclusion
Hamiltonian FRG-flow equation of YMT in Coulomb gauge :
Input:
ghost dominance
scaling of ghost in the IR
output :
horizon condition
YM vacuum=perfect dual superconductor
IR-exponents
satisfy sum rule
smaller than in variational approach
outlook:
Coulomb form factor
inclusion of quarks

43. Summary & Conclusion Hamiltonian approach to YMT in Coulomb gauge
variational solution of the YM Schrödinger eq. , input:
gluon confinement
quark confinement
satisfactory agreement with lattice data
Hamiltonian FRG-flow equation:
horizon condition
IR-exponents smaller than in variational approach
dielectric function of the YM vacuum
YM vacuum=perfect dual superconductor

44. Thanks for your attention

46. non-perturbative approaches to continuum Yang-Mills theory
DSE
FRG flow equations
Variational Hamiltonian approch

47. Hamiltonian Flow in Coulomb Gauge Yang-Mills theory
Introduction
Hamilton approach to YMT
FRG flow equation
Numerical results
Conclusions

48. Variational approach trial ansatz : C.Feuchter & H. R. PRD70(2004)
gluon propagator
determined from
variational kernel