Third order Coulomb correction to t¯t threshold cross section ∗

Third order Coulomb correction to t ¯ t threshold cross section * We report on our result of third order Coulomb correction to the cross section σ(e + e − → t ¯ t) near threshold. Analytic expression for the Coulomb energy and wave function at the origin are obtained. We discuss the significance of the Coulomb correction to the threshold cross section and heavy quarkonium phenomenology. 1. Threshold cross section The t ¯ t threshold cross section has the following schematic form in the conventional perturbative expansion of α s , σ(e + e − → t ¯ t) ∼ σ Born [1 + α s v + α s v 2 + · · · ] , (1) where σ Born is a Born cross section, and v = 1 − 4m 2 t /s is speed of produced t, ¯ t. The combination of (α s /v) n appears to all order in the perturbation theory, known as Coulomb singularity. The perturbative expansion in α s is not applicable to the threshold cross section because v is of order α s near the t ¯ t threshold √ s ∼ 2m t , and the Coulomb singularity (α s /v) n dominates the cross section. To obtain meaningful cross section the Coulomb singularity has to be summed to all order in α s. The physical origin of the Coulomb singularity is instantaneous gluon exchange between t and ¯ t which has a small spatial momentum | q| ∼ v ∼ 0. This is called potential gluon because of its propagator