The general method for treating non-Gaussian wave functionals in the Hamiltonian formulation of a quantum field theory, which was previously proposed and developed for Yang--Mills theory in Coulomb gauge, is generalized to full QCD. For this purpose the quark part of the QCD vacuum wave functional is expressed in the basis of coherent fermion states, which are defined in term of Grassmann variables. Our variational ansatz for the QCD vacuum wave functional is assumed to be given by exponentials of polynomials in the occurring fields and, furthermore, contains an explicit coupling of the quarks to the gluons. Exploiting Dyson--Schwinger equation techniques, we express the various $n$-point functions, which are required for the expectation values of observables like the Hamiltonian, in terms of the variational kernels of our trial ansatz. Finally the equations of motion for these variational kernels are derived by minimizing the energy density.
We study, for $SU(2)$ Yang-Mills theories discretized on a lattice, a non-local topological order parameter, the center flux ${{z}}$. We show that: i) well defined topological sectors classified by $pi_1(SO(3))=mathbb{Z}_2$ can only exist in the ordered phase of ${{z}}$; ii) depending on the dimension $2 leq dleq 4$ and action chosen, the center flux exhibits a critical behaviour sharing striking features with the Kosterlitz-Thouless type of transitions, although belonging to a novel universality class; iii) such critical behaviour does not depend on the temperature $T$. Yang-Mills theories can thus exist in two different continuum phases, characterized by an either topologically ordered or disordered vacuum; this reminds of a quantum phase transition, albeit controlled by the choice of symmetries and not by a physical parameter.
The canonical recursive Dyson--Schwinger equations for the three-gluon and ghost-gluon vertices are solved numerically. The employed truncation includes several previously neglected diagrams and includes back-coupling effects. We find an infrared finite ghost-gluon vertex and an infrared diverging three-gluon vertex. We also compare our results with those obtained in previous calculations, where bare vertices were used in the loop diagrams.
We study the static gluon and quark propagator of the Hamiltonian approach to Quantum Chromodynamics in Coulomb gauge in one-loop Rayleigh--Schrodinger perturbation theory. We show that the results agree with the equal-time limit of the four-dimensional propagators evaluated in the functional integral (Lagrangian) approach.
We investigate the temporal Wilson loop using the Hamiltonian approach to Yang-Mills theory. In simple cases such as the Abelian theory or the non-Abelian theory in (1+1) dimensions, the known results can be reproduced using unitary transformations to take care of time evolution. We show how Coulomb gauge can be used for an alternative solution if the exact ground state wave functional is known. In the most interesting case of Yang-Mills theory in (3+1) dimensions, the vacuum wave functional is not known, but recent variational approaches in Coulomb gauge give a decent approximation. We use this formulation to compute the temporal Wilson loop and find that the Wilson and Coulomb string tension agree within our approximation scheme. Possible improvements of these findings are briefly discussed.
