A note on the tensor and vector exchange contributions to $K\overline{K}$ → $K\overline{K}$, $D\overline{D}$ → $D\overline{D}$ and π${}^{+}$π${}^{-}$ → π${}^{+}$π${}^{-}$ reactions

In this note we study the tensor and vector exchange contributions to the elastic reactions involving the pseudoscalars mesons π${}^{+}$π${}^{-}$, K${}^{+}$K${}^{-}$ and D${}^{+}$D${}^{-}$. In the case of the tensor-exchange contributions we assume that an intermediate tensor $f_2$(1270) is dynamically generated from the interaction of two virtual ρ mesons, with the use of a pole approximation. The calculation of the two-loop amplitude is facilitated since the triangle loops can be factorized and computed separately. The results show very small contributions coming from the tensor-exchange mechanisms when compared with those from the vector-exchange processes. We compare our results for ππ and $K\overline{K}$ scattering with those obtained in other works where the $f_2$(1270) is considered as an ordinary $q\overline{q}$ meson. Our picture provides a smaller contribution but of similar order of magnitude for pion scattering and stabilizes the results in the case of $K\overline{K}$, allowing us to make estimates for $D\overline{D}$ scattering.