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We address a general problem in the evaluation of triangle loops stemmingfrom the consideration of the range of the interaction involved in some of thevertices, as well as the energy dependence of the width of some unstableparticles in the loop. We find sizeable corrections from both effects. We applythat to a loop relevant to the $D_s^+ \to \pi^+ \pi^0 \eta $ decay, and findreductions of about a factor of $4$ in the mass distribution of invariant massof the $\pi \eta$ in the region of the $a_0(980)$. The method used is based onthe explicit analytical evaluation of the $q^0$ integration in the $d^4q$ loopintegration, using Cauchy's residues method, which at the same time offers aninsight on the convergence of the integrals and the effect of form factors andcutoffs.

Stripping triangle loops: Discussion of $D_s^+\to \rho^+\eta\to \pi^+\pi^0\eta$ in $a_0(980)$ production