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

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

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

Classical Yang-Mills theory Lagrange function: field strength tensor

Canonical Quantization of Yang-Mills theory Gauß law:

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

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

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

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

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

QM: particle in a L=0-state

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

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

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

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

Curvature (ghost part of the gluon energy)

Coulomb form factor f Schwinger-Dyson eq.

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

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

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

Coulomb potential

external static color sources electric field ghost propagator

The color electric flux tube

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

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

eliminating the self-energies

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

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

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

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

3-gluon vertex M.Leder W.Schleifenbaum

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

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

Connection to the Center Vortex Picture

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

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

Q-Q-potential: SU(2)

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

similar results in Coulomb gauge: Greensite, Olejnik, Zwanziger, hep-lat/ 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/

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

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

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)

Thanks to the organizers