arXiv subjects
Publications and source records attributed to Hugo Reinhardt.
The Faddeev-Popov determinant of Landau gauge in d dimensions and Coulomb gauge in d+1 dimensions is calculated in the heat-kernel expansion up to next-to-leading order. The UV-divergent parts in d=3,4 are isolated and the counterterms required for a non-perturbative treatment of the Faddeev-Popov determinant are determined.
We study the Hamiltonian approach to Yang-Mills theory in Coulomb gauge in Rayleigh-Schroedinger perturbation theory. The static gluon and ghost propagator as well as the potential between static colour sources are calculated to one-loop order. Furthermore, the one-loop $β$-function is calculated from both the ghost-gluon vertex and the static potential and found to agree with the result of covariant perturbation theory.
Two aspects of the color charge in Coulomb gauge continuum Yang-Mills theory are discussed. The first aspect is the existence of a conserved and vanishing total charge exhibited within the first order functional formalism. The second aspect is the closure of the set of Slavnov-Taylor identities in the second order functional formalism, such that the exact solution for temporal Green's functions is in principle possible and thereby preserving the color charge.
The continuum Yang-Mills functional integral within the first order formalism and in Coulomb gauge is studied. In particular, the temporal zero-modes of the Faddeev-Popov operator are explicitly accounted for. It is shown that the treatment of these zero-modes results in the constraint that the total color charge of the system vanishes at all times. Further, it is argued that the functional integral is effectively fully gauge-fixed once Gauss' law has been resolved in Coulomb gauge.
The topological susceptibility is calculated within the Hamiltonian approach to Yang-Mills theory in Coulomb gauge, using the vacuum wave functional previously determined by a variational solution of the Yang-Mills Schroedinger equation. The numerical result agrees qualitatively with the predictions of lattice simulations.
The functional approach to Coulomb gauge Yang-Mills theory is considered within the standard, second order, formalism. The Dyson-Schwinger equations and Slavnov-Taylor identities concerning the two-point functions are derived explicitly and one-loop perturbative results are presented.
Perturbative Coulomb gauge Yang-Mills theory within the first order formalism is considered. Using a differential equation technique and dimensional regularization, analytic results for both the ultraviolet divergent and finite parts of the two-point functions at one-loop order are derived. It is shown how the non-ultraviolet divergent parts of the results are finite at spacelike momenta with kinematical singularities on the light-cone and subsequent branch cuts extending into the timelike region.
Coulomb gauge Yang-Mills theory within the first order formalism is considered with a view to deriving the propagator Dyson-Schwinger equations. The first order formalism is studied with special emphasis on the BRS invariance and it is found that there exists two forms of invariance - invariance under the standard BRS transform and under a second, non-standard transform. The field equations of motion and symmetries are derived explicitly and certain exact relations that simplify the formalism are presented. It is shown that the Ward-Takahashi identity arising from invariance under the non-standard part of the BRS transform is guaranteed by the functional equations of motion. The Feynman rules and the general decomposition of the two-point Green's functions are derived. The propagator Dyson-Schwinger equations are derived and certain aspects (energy independence of ghost Green's functions and the cancellation of energy divergences) are discussed.
The infrared behaviour of gluon and ghost propagators, ghost-gluon vertex and three-gluon vertex is investigated for both the covariant Landau and the non-covariant Coulomb gauge. Assuming infrared ghost dominance, we find a unique infrared exponent in the d=4 Landau gauge, while in the d=3+1 Coulomb gauge we find two different infrared exponents. We also show that a finite dressing of the ghost-gluon vertex has no influence on the infrared exponents. Finally, we determine the infrared behaviour of the three-gluon vertex analytically and calculate it numerically at the symmetric point in the Coulomb gauge.
By using the method of center projection the center vortex part of the gauge field is isolated and its propagator is evaluated in the center Landau gauge, which minimizes the open 3-dimensional Dirac volumes of non-trivial center links bounded by the closed 2-dimensional center vortex surfaces. The center field propagator is found to dominate the gluon propagator (in Landau gauge) in the low momentum regime and to give rise to an OPE correction to the latter of $\sqrtσ /p^3 $.The screening mass of the center vortex field vanishes above the critical temperature of the deconfinement phase transition, which naturally explains the second order nature of this transition consistently with the vortex picture. Finally, the ghost propagator of maximal center gauge is found to be infrared finite and thus shows no signal of confinement.
SU(2) lattice gauge theory is investigated where the traces of the Wilson lines at any lattice point and along each direction is constrained to zero. Hence, each of the lattice configurations possesses a vanishing density of heavy (anti-) quarks. The results are compared with those of pure SU(2) gauge theory which can be interpreted as the grand canonical realization of the heavy quark theory where only the ensemble average of the heavy quark density vanishes. The static quark anti-quark potential of the constrained theory is obtained from (spatially smeared) Wilson loops at zero temperature. We find that the potential coincides with that of pure SU(2) gauge theory (without constraints). Hence, the familiar "running" of the lattice spacing with beta is recovered.
We investigate the infrared behaviour of the gluon propagator in Landau gauge on a lattice with twisted boundary conditions. Analytic calculations using Dyson-Schwinger equations, exact renormalization group and stochastic quantization show that the gluon propagator in Landau gauge approaches zero for small momentum. On the other hand lattice calculations and calculations on a four-torus seem to give rise to a non-zero limit. One possible reason for this difference is the existence of zero-momentum fluctuation modes which potentially give a massive contribution to the gluon propagator. Our simulations show that with twisted boundary conditions these zero-momentum modes are suppressed and the gluon propagator becomes smaller than in a periodic ensemble.
The center vortex model for the infrared sector of SU(3) Yang-Mills theory is reviewed. After discussing the physical foundations underlying the model, some technical aspects of its realisation are discussed. The confining properties of the model are presented in some detail and compared to known results from full lattice Yang-Mills theory. Particular emphasis is put on the new phenomenon of vortex branching, which is instrumental in establishing first order behaviour of the SU(3) phase transition. Finally, the vortex free energy is verified to furnish an order parameter for the deconfinement phase transition. It is shown to exhibit a weak discontinuity at the critical temperature, in agreement with predictions from lattice gauge theory.
Gluon field configurations with non-trivial topology like instantons, magnetic monopoles and center vortices play a crucial role in QCD and, in particular, for the spontaneous breaking of chiral symmetry. Moreover, center vortices are strongly correlated with confinement. We present evidence, that there is a deep connection between the topology of gauge fields and center vortices. We use the chirally improved lattice Dirac operator to compute eigenvectors and eigenvalues of various lattice gauge field configurations. Removing vortices from thermalized configurations also removes the topological content of the gauge field. As a consistency check, we apply random changes to the raw configurations.
We study the interplay between Dirac eigenmodes and center vortices in SU(2) lattice gauge theory. In particular we focus on vortex-removed configurations and compare them to an ensemble of configurations with random changes of the link variables. We show that removing the vortices destroys all zero modes and the near zero modes are no longer coupled to topological structures. The Dirac spectrum for vortex-removed configurations in many respects resembles a free spectrum thus leading to a vanishing chiral condensate. Configurations with random changes leave the topological features of the Dirac eigensystem intact. We finally show that smooth center vortex configurations give rise to zero modes and topological near zero modes.
In this talk, we review some recent results of the center vortex model for the infrared sector of SU(3) Yang-Mills theory. Particular emphasis is put on the order of the finite-temperature deconfining phase transition and the geometrical structure of vortex branchings. We also present preliminary data for the 't Hooft loop operator and the dual string tension near the phase transition.
The vortex picture of confinement is employed to explore the signals of confinement in Yang-Mills Green functions. By using SU(2) lattice gauge theory, it has been well established that the removal of the center vortices from the lattice configurations results in the loss of confinement. The running coupling constant, the gluon and the ghost form factors are studied in Landau gauge for both cases, the full and the vortex removed theory. In the latter case, a strong suppression of the running coupling constant and the gluon form factor at low momenta is observed. At the same time, the singularity of the ghost form factor at vanishing momentum disappears. This observation establishes an intimate correlation between the ghost singularity and confinement. The result also shows that a removal of the vortices generates a theory for which Zwanziger's horizon condition for confinement is no longer satisfied.
Mean-field treatments of Yang-Mills theory face the problem of how to treat the Gauss law constraint. In this paper we try to face this problem by studying the excited states instead of the ground state. For this purpose we extend the operator approach to the Random Phase Approximation (RPA) well-known from nuclear physics and recently also employed in pion physics to general bosonic theories with a standard kinetic term. We focus especially on conservation laws, and how they are translated from the full to the approximated theories, demonstrate that the operator approach has the same spectrum as the RPA derived from the time-dependent variational principle, and give - for Yang-Mills theory - a discussion of the moment of inertia connected to the energy contribution of the zero modes to the RPA ground state energy. We also indicate a line of thought that might be useful to improve the results of the Random Phase Approximation.