Hamilton approch to YangMills Theory in Coulomb Gauge

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Hamilton approch to Yang-Mills Theory in Coulomb Gauge H. Reinhardt Tübingen Collaborators: D. Campagnari,

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

Plan of the talk • Hamilton approach to continuum Yang-Mills theory in Coulomb gauge

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

Classical Yang-Mills theory Lagrange function: field strength tensor

Classical Yang-Mills theory Lagrange function: field strength tensor

Canonical Quantization of Yang-Mills theory Gauß law:

Canonical Quantization of Yang-Mills theory Gauß law:

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

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

YM Hamiltonian in Coulomb gauge Coulomb term Christ and Lee -arises from Gauß´law =neccessary

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

aim: solving the Yang-Mills Schrödinger eq. for the vacuum by the variational principle with

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

Importance of the Faddeev-Popov determinant defines the metric in the space of gauge orbits

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

vacuum wave functional QM: particle in a L=0 -state variational kernel determined from DSE

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

Dyson-Schwinger Equations ghost propagator ghost DSE gluon propagator gluon DSE (gap equation) gluon self-energy

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

Regularization and renormalization: momentum subtraction scheme renormalization constants: ultrviolet and infrared asymtotic behaviour of

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 horizon condition Zwanziger In D=2+1 is the only value for which the coupled Schwinger-Dyson equation have a self-consistent solution

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

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

Coulomb potential

Coulomb potential

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

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

external static color sources electric field ghost propagator

external static color sources electric field ghost propagator

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

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

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

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

comparison with lattice d=3 lattice: L. Moyarts, dissertation

comparison with lattice d=3 lattice: L. Moyarts, dissertation

D=3+1 Infrared behaviour of lattice GF: not yet conclusive too small lattices

D=3+1 Infrared behaviour of lattice GF: not yet conclusive too small lattices

related work: A. P. Szczepaniak, E. S. Swanson, Phys. Rev. 65 (2002) 025012 A.

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, PRD 70(2004) hep-th/0408237, PRD 71(2005) W. Schleifenbaum, M. Leder, H. R. PRD 73(2006) D. Epple, H. R. , W. Schleifenbaum, in prepration full inclusion of the curvature measure for the curvature

Importance of the curvature Szczepaniak & Swanson Phys. Rev. D 65 (2002) • the

Importance of the curvature Szczepaniak & Swanson Phys. Rev. D 65 (2002) • the c = 0 solution does not produce a linear confinement potential

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

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

Infrared analysis of the DSE vacuum wave functional: generating functional ghost dominance in the

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

Analytic solution of DSE in the infrared gluon propagator LG: Lerche, v. Smekal Zwanziger,

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

running coupling Fischer, Zwanziger interpolating gauges sum rule for the infrared exponents from ghost

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

Topological susceptibility Witten-Veniciano formula:

Topological susceptibility Witten-Veniciano formula:

Topological susceptibility in Hamilton approach spatial gauge transformation: explicit realization: Chern-Simon action: topological susceptibility:

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

Identify our variational wave functional with the restriction of the gauge invarinant to Coulomb

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

large variety of wave functionals produce the same DSE more sensitive observables than energy

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

´t Hooft loop disorder parameter of YMT spatial ´t Hooft looop defining eq. center

´t Hooft loop disorder parameter of YMT spatial ´t Hooft looop defining eq. center vortex field V(C)-center vortex generator continuum representation: H. R: Phys. Lett. B 557(2003)

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

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

QM: wave functionals in Coulomb gauge satisfy Gauß´law and hence should be regarded as

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.

´t Hooft loop in Coulomb gauge representation (correct to 2 loop) h(C; p)-geometry of

´t Hooft loop in Coulomb gauge representation (correct to 2 loop) h(C; p)-geometry of the loop C H. R. & C. F. PRD 71 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)

from gap equation renormalization condition: c=0 produces wave functional which in the infrared approaches

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

Summary and Conclusion • Variational solution of the YM Schrödinger equation in Coulomb gauge

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

Thanks to the organizers

Thanks to the organizers