Singular behavior of ¹P₁⁺⁻ quarkonium and positronium annihilation decays and relevance of relative energy

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. For positronium this differs from an earlier result in the literature and leads to a modified life time. In QCD there is a non abelian loop graph which contributes to the decay amplitude to the same order as the tree graph. Contrary to the original version, which contained some errors, the constant turns out to be large and negative. For reasonable parameters, this negative constant practically cancels the positive logarithm, so that the whole approach of expanding with respect to the binding energy breaks down.