Comment on the Average Momentum of Top Quarks in the Threshold Region

The behavior of the momentum distribution of top quarks in the threshold region is investigated. The qualitative behavior, in particular the dependence of the average momentum on the strong coupling constant can be understood from analytical calculations for the Coulomb potential. Ambiguities in the relation between the excitation curve and the top mass are addressed. Of particular interest for top quark studies in the threshold region is the dependence of the total and the differential cross section on the strong coupling constant. Some intuition and qualitative understanding can already be gained from the predictions based on a pure Coulomb potential. For a stable top quark of fixed mass the “effective threshold” can be associated with the location of the 1S resonance √ sthr = 2mt+E1S with E1S = −ERyd = −αmt/4 which decreases with increasing α. The height of the resonance cross section is proportional to the square of the wave function at the origin and hence proportional to α, as long as the the resonances are reasonably well separated. In the limit of large Γt, that is far larger than ERyd, the overlapping 1S, 2S . . . resonances have to fill the gaps between the peaks. Since these gaps themselves increase proportional to α, one is left in the extreme case of large width with a cross section linear in α. Note that this corresponds to the behavior of the cross section close to but slightly above the threshold which is also proportional to α. For realistic top masses of about 150 GeV one thus observes a behavior of the peak cross section quite close to the first power in α. Since the location of the peak itself depends on α, only the analysis of the full shape allows to extract the relevant information. Once the threshold energy is determined experimentally, α and mt are still strongly correlated and can hardly be deduced individually. Therefore in a next step also the momentum distribution of top quarks has to be exploited to obtain further information. The discussion is again particularly simple for the Coulomb potential V (r) = −α/r. The average momentum, in units of the Bohr momentum αmt/2, can Contributed to the Workshop on Physics at a Linear Collider, to be published in the proceedings