A note on the tensor and vector exchange contributions to $K \bar{K} \rightarrow K \bar{K}, D \bar{D} \rightarrow D \bar{D}$ and $\pi ^+ \pi ^-\rightarrow \pi ^+ \pi ^-$ reactions

Abstract

In this note we study the tensor and vector exchange contributions to the elastic reactions involving the pseudoscalars mesons $\pi ^+ \pi ^-$, $K^{+}K^{-}$ 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 $\rho$ 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 $\pi \pi$ and $K\bar{K}$ scattering with those obtained in other works where the $f_2(1270)$ is considered as an ordinary $q\bar{q}$ meson. Our picture provides a smaller contribution but of similar order of magnitude for pion scattering and stabilizes the results in the case of $K\bar{K}$, allowing us to make estimates for $D\bar{D}$ scattering.