Subtleties in triangle loops for Ds+ →ρ+η →π+π0η in a0 (980) production

We address a general problem in the evaluation of triangle loops stemming from the consideration of the range of the interaction involved in some of the vertices, as well as the energy dependence of the width of some unstable particles in the loop. We find sizeable corrections from both effects. We apply that to a loop relevant to the Ds+→π+π0η decay, and find reductions of about a factor of 4 in the mass distribution of invariant mass of the πη in the region of the a0(980). The method used is based on the explicit analytical evaluation of the q0 integration in the d4q loop integration, using Cauchy's residues method, which at the same time offers an insight on the convergence of the integrals and the effect of form factors and cutoffs.