The Structure of Vortices

Research project P 13997 The Structure of Vortices Manfried FABER 11.10.1999 Since the invention of quarks as constituents of hadrons we are puzzled by the question why quarks are confined inside of hadrons. Quantum Chromo Dynamics (QCD) is the generally accepted theory of quark dynamics. In the lattice formulation of QCD one can demonstrate that quarks are evidently confined - but the mechanism of quark confinement is still under intensive discussion. In recent lattice calculations we could show that there are strong indications that thick vortices are responsible for confinement in QCD. At the end of the seventies `t Hooft and others suggested the vortex model and the dual superconductor model of confinement. At that time it was not clear which model should be favored. During the last months in several articles we presented numerical evidence in favor of the center vortex theory of confinement. In this project we want to follow the idea that thick center vortices are the basic topological objects which lead to confinement. We want to investigate all properties of these vortices, including the methods to detect them, and explain different aspects of confinement using vortices. Vortices provide us with a unified picture of confinement: The QCD vacuum is a "medium" with magnetic vortices. It is filled with randomly fluctuating quantized magnetic flux tubes, fusing and splitting permanently. In this way they form a huge cluster percolating the whole space-time. By those magnetic properties of the vacuum the electric flux originating in quarks is compressed to straight flux tubes leading to a linear rising quark antiquark potential and to confinement.

Aim of this project was the investigation of the vortex model of quark confinement on the lattice. The constituents of the atomic nucleus, protons and neutrons, as well as other particles, are built from fundamental particles, the quarks. These quarks never occur as free particles, they are always confined in other particles such as the mentioned protons and neutrons. This phenomenon is called quark confinement. Using the theory which treats the interaction between quarks, quantum chromodynamics (QCD), quark confinement can be described, and confirmed with numerical calculations. It turns out that the force between quarks, mediated by gluon fields, does not decrease with their separation as it is the case in electrodynamics. Rather it is constant even at large distances. Hence quarks cannot be separated and are confined. There remains the question about the mechanism of confinement in the framework of QCD. Already in the 1970s the vortex model was developed as a possible explanation, but only during the last ten years the improved computer technology enabled successfully testing of these ideas. In lattice QCD the fourdimensional space-time is discretised. Fields which are in the continuum are defined at each point in space-time are now only considered at separate points forming a fourdimensional, regular lattice. This enables an efficient investigation of QCD with computers. The vortex model postulates the importance of the so called center degrees of freedom of QCD. The excitations of these discrete degrees of freedom -- vortices -- are supposed to be crucial for quark confinement. In the numerical investigations on the vortex model the gluon fields with their many degrees of freedom are substituted by quantised magnetic flux lines, the vortices. Vortices form closed twodimensional surfaces spreading out the fourdimensional space-time. Our project aimed to investigate several methods to identify vortices in the gluon fields of QCD, to describe the properties of vortices, and to understand quark confinement in terms of vortices. The vortex model could be substantiated in this project. First, it could be settled how the methods of vortex identification work and which are their limits. An improved method could be developed which circumvents problems of conventional methods. Previous calculations on the vortex model have been performed using a simplified theory, a system without the costly simulation of dynamic matter fields. We could show that even with the inclusion of matter fields the vortex model works correctly. Finally, the vortex model could be applied to a property of the gluon fields not immediately related to quark confinement, the so called topological charge. The fluctuation of the topological charge, the topological susceptibility, is an important parameter signifying QCD at low energies. We could calculate this parameter in the framework of the vortex model. The result agrees with the value infered using conventional methods. These results support that the most important phenomenons of QCD at low energies can be described within an unified model, the vortex model.