Rigorous QCD-Potential for the $t\bar{t}$-System at Threshold

Abstract

Recent evidence for the top mass in the region of 160 $GeV$ for the first time provides an opportunity to use the full power of relativistic quantum field theoretical methods, available also for weakly bound systems. Because of the large decay width $\G$ of the top quark individual energy-levels in "toponium" will be unobservable. However, the potential for the $t\bar{t}$ system, based on a systematic expansion in powers of the strong coupling constant $\a_s$ can be rigorously derived from QCD and plays a central role in the threshold region. It is essential that the neglect of nonperturbative (confining) effects is fully justified here for the first time to a large accuracy, also just {\it because} of the large $\G$. The different contributions to that potential are computed from real level corrections near the bound state poles of the $t\bar{t}$-system which for $\G \ne 0$ move into the unphysical sheet of the complex energy plane. Thus, in order to obtain the different contributions to that potential we may use the level corrections at that (complex) pole. Within the relevant level shifts we especially emphasize the corrections of order $O(\a_s^4 m_t)$ and numerically comparable ones to that order also from electroweak interactions which may become important as well.