Singular behavior of ~1P_1~(+-) 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, 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, not only for the familiar logarithmic dependence on the binding energy, but also for the constant to be added to the logarithm. It turns out to be large and negative.