Singular behavior of1P1+ -quarkonium and positronium annihilation decays and relevance of relative energy

摘要

Using a four dimensional approach, we show that the singularities for small gluon momenta, which arise in the usual three dimensional treatment of the annihilation decay, disappear if all poles in the relative energy are taken into account correctly in the integration. We obtain an explicit formula for the decay width which involves a non-locality originating from the kinetic energy. We calculate not only the familiar logarithmic dependence on the binding energy, but also the constant to be added to the logarithm. The logarithmic term agrees with previous values in the literature. In QCD the constant turns out to be quite small, but only because there is an almost perfect cancellation between the tree graph and a non-abelian loop graph which contributes to the decay amplitude to the same order.