$\tau^- \to \nu_{\tau} M_1 M_2$, with $M_1, M_2$ pseudoscalar or vector mesons

We perform a calculation of the $\tau^- \to \nu_{\tau} M_1 M_2$, with $M_1, M_2$ either pseudoscalar or vector mesons using the basic weak interaction and angular momentum algebra to relate the different processes. The formalism also leads to a different interpretation of the role played by $G$-parity in these decays. We also observe that, while $p$-wave $M_1 M_2$ production is compatible with chiral perturbation theory and experiment, $VP$ and $VV$ $p$-wave production is clearly incompatible with experiment and we develop the formalism also in this case. We compare our results with experiment and make predictions for unmeasured decays, and we show the value of these reactions, particularly if the $M_1 M_2$ mass distribution is measured, as a tool to learn about the meson-meson interaction and the nature of some resonances, coupling to two mesons, which are produced in such decays.