Confinement and topological configuration

The confinement of color charges in Yang-Mills theory has been a challenging and unresolved problem for several decades. Recently, two aspects have received much attention. The linear rise of the static quark-anti-quark potential (and also chiral symmetry breaking) has been attributed to topological configurations like vortices and monopoles. On the other hand, confinement in the gauge-sector seems eventually to be understandable in the framework of the Gribov-Zwanziger and the Kugo-Ojima scenario, and is manifest in the Green`s functions of the theory. First attempts to unify these two very different aspects of confinement have only been made in the last few years, and it seems that indeed both aspects are linked. This manifests itself in the changes of the Green`s functions when manipulating the topological contents. Based on this, the aims in this project are threefold: First, it aims at studying the systematic dependence of topological configurations on various gauges and gauge-groups, to identify persistent topological structures. Second, the link between both aspects of confinement seems currently to be the spectrum of the Faddeev-Popov operator. Its dependence on the gauge and gauge-groups is also to be systematically analyzed. Finally, the dependence of various Green`s functions, especially those describing gluons, on the topological contributions should be analyzed by systematically manipulating the latter. The method of choice for these investigations will be lattice gauge theory, and part of the project will be devoted to developing the adequate numerical algorithms to achieve the scientific goals of this project. The combined information from these three objectives will provide further insight into the link between the various confinement scenarios, and also into the properties of topological configurations. As the latter are also linked to hadronic physics via chiral symmetry, this will in the future also potentially contribute to a better first-principle understanding of the properties of hadrons and thus, ultimately, experiments.

The nuclei of atoms consist out of protons and neutrons. They, in turn, consist themselves out of even smaller particles, the so-called quarks. The quarks are bound together to form the protons and neutrons by the exchange of gluons, another type of particles. This has been known since the late 1960s/early 1970s. Nonetheless, it has never been possible so far to isolate a single quark or gluon experimentally. The hypothesis that this is not possible at all is called the confinement hypothesis, and confinement is the effect which makes it impossible for a quark or gluon to leave a proton or neutron. A satisfactory theoretical description of this phenomena has ever since been a challenge, and it is still open today. In fact, for a mathematical precise solution the Clay-Institute of Mathematics has put out a 1,000,000$ prize, and classified it as a millennium problem. The quest for a solution has led to a great number of different models, each having its particular weaknesses and strengths, disadvantages and advantages. Two of them, which have been very prominent within the last decade, tried very different approaches. The one is based on a description of how quarks and gluons move as individual particles. The other is based on how quarks move in a large number of gluons, which are treated as a single collective particle. The main goal of this project was to understand how these two very different approaches could be united, since a combination would eliminate several disadvantages of both models. The results of this investigation have shown that both models can be united, although many of the mathematical and conceptual details have still to be resolved. It particular, it has been found that gluons become stronger and stronger affected by collective effects the farther they travel. Pictorially, it seems that a single gluon drags along a collection of other gluons, piling up more and more along the way, and is by this hindered to escape from a proton or neutron. In the course of identifying these effects, many details have been obtained on the movement of gluons. Even more has been learned on the mathematical tools necessary to describe gluons, and how the results of different methods can be compared. Furthermore, rather general insights into the mathematical framework used, so-called quantum gauge theories, have also been obtained. These are mostly on how global symmetries, relating gluons of different charge to each other, manifest themselves in the movement of gluons over long distances. Furthermore, the investigations gave also insight into which properties of the collective aspects of the gluons are relevant for the generation of the mass of the quarks and their confinement. By investigating a model theory which has gluons and quarks with exceptional properties, it was shown that the generation of the quark mass is possibly a mechanism which is independent of the degree of disorder present in the collections of gluons.