YangMills Theory in Coulomb Gauge H Reinhardt Tbingen
Yang-Mills Theory in Coulomb Gauge H. Reinhardt Tübingen non-perturbative approach to continuum YMT C. Feuchter & H. R. hep-th/0402106, PRD 70 hep-th/0408237, PRD 71 hep-th/0408236 D. Epple, C. Feuchter, H. R. , hep-th/0412231 W. Schleifenbaum M. Leder H. Turan
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
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 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 to maintain gauge invariance -provides the confining potential
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 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
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 bare vertex rain-bow ladder approx: replace full vertex by bare one
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 horizon condition In D=2+1 is the only value for which the coupled Schwinger-Dyson equation have a self-consistent solution
Asymptotic behaviour D=3+1 -angular approximation ultraviolet behaviour infrared behaviour
Numerical results (D=3+1) ghost and Coulomb form factors mass gap: gluon energy and curvature
Coulomb potential
external static color sources electric field ghost propagator
The color electric flux tube
The flux between 3 static color charges a=3 a=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. D 65 (2002) • the c = 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 exact in D=1+1 gauge fields at different points are completely uncorrelated stochastic vacuum
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
Center Vortices in Continuum Yang -Mills theory Wilson loop Linking number center element C
Q-Q-potential: SU(2)
Confinement mechanism in Coulomb gauge static quark potential : infrared dominant field configurations: Gribov horizon
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 center vortices
Ghost Propagator in Maximal Center Gauge (MCG) Ø fixes SU(2) / Z (2) è ghosts do not feel the center Z (2) • no signal of confinement in the ghost propagator Ø 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
- Slides: 43