Toponium Threshold

Due to the large decay rate of the top quark the effect of top-anti top bound states (toponium) near the corresponding threshold is changed in an essential manner. The numerous existing theoretical studies explicitly or implicitly assume a dominant contribution of a binding "potential" which by itself provides real (sharp) bound state energies, analogous to the Bohr levels of positronium. These real energies are assumed to be shifted by the decay width to complex values. However for toponium, real energy differences and decay width are of the same numerical magnitude. Therefore this usual procedure does not possess a sound theoretical basis. The purpose of the present project is to develop an approach which in a field theoretic rigorous (in the sense of perturbation theory) way treats both effects in an equivalent manner so that reliable calculations are possible beyond O (a s 2 ).

Quantum Chromodynamics is considered to be the appropriate theory to describe strong interactions. Those interactions mediate the forces which are responsible for the binding of quarks, the particles of which nuclear matter consists. Quantum Chromodynamics, as any other Quantum Field Theory, describes interactions as exchange of virtual particles, called gluons. Evaluation of an observable process would on principle require calculation and summing up of the (countable) infinity of all and increasingly complex possibilities for the exchange of such virtual particles. Perturbation theory offers a way of resolving this dilemma. All possible contributions are ordered according to powers of a small parameter. Contributions of higher orders will then sufficiently fast become very small. Therefore, hoping that the perturbation series does converge, only the first contributions have to be calculated to obtain a result of appropriate accuracy. All other contributions are neglected as being small. Unfortunately this mechanism breaks down in the case of strong interactions since the coupling constant, a measure for the strength of the interaction, is not small. The situation can be improved by considering very heavy quarks because at high energies the coupling constant decreases, a fact which is known as asymptotic freedom of the theory. Therefore, the treatment of the heaviest of all quarks, the top quark, and its antiparticle offers a good possibility to apply perturbation theory within the framework of strong interactions and thus make predictions from Quantum Chromodynamics which may be verified in future experiments. Most regrettably, the extremely short lifetime of these heavy particles poses new problems to the theory. General systematics usually applied for the description of the bound state of particles are not valid for toponium, i.e. the bound state of one top and one anti-top quark, because it is not even approximately permissible to consider these two particles to be stable for the duration of the existence of the bound state. The main task of this project, therefore, was to find a systematic approach to describe such a bound state on the one hand taking account of the binding of the two particles to one another while on the other hand integrating their instability in an appropriate way. In this context systematic means that, as mentioned above, in principle all contributions must enter the calculation according to their size. At the same time the error resulting from the neglect of higher terms and from all assumptions and approximations that had to be made must remain under control. Special emphasis was laid on avoiding omissions of any effects as well as double counting of any of the numerous terms. By a combination of methods originating from Veltman, Bethe and Salpeter we are confident to have attained the goal set. However, only further (numeric) calculations and eventually experiments in future particle colliders will allow to check the accuracy of the results here obtained.