Unified study of two-meson and axion-meson production from semileptonic tau decays within resonance chiral framework

We carry out the joint study of the semileptonic tau decays into the two-meson and axion-meson channels, viz. $\tau^-\to (P_1P_2)^-\nu_\tau$ and $\tau^-\to \pi^-(K^-) a\nu_\tau$ within the framework of resonance chiral theory by including the model-independent axion-gluon-gluon interaction. By utilizing the $\pi^0$-$\eta$-$\eta'$-axion mixing matrix elements from recent studies, we calculate the pertinent two-pseudoscalar boson form factors. To simultaneously fit the experimental spectra measured in the Cabibbo allowed $\tau^-\to \pi^-\pi^0\nu_\tau$ process and also the Cabibbo suppressed $\tau^-\to(K_S\pi^-, K^-\eta)\nu_\tau$ ones, we determine all the relevant hadron resonance parameters. Then we give predictions to the spectra and branching ratios for various channels, such as $\tau^-\to(\pi^-\eta,\pi^-\eta',K^-\eta',\pi^-a,K^-a)\nu_\tau$. We also calculate the forward-backward asymmetries for all the aforementioned channels. The interplay between the scalar and vector form factors for different observables is analyzed in detail. Our theoretical predictions supply useful guidance to the future tau experiments, such as those at Belle-II, Super Tau-Charm Facility and Tera-Z factory of Circular Electron-Positron Collider.