Probing sensitivity to charged scalars through partial differential widths: $\tau\rightarrow K\pi\pi\nu_{\tau}$ decays

We define and test $\mathrm{CP}$-even and $\mathrm{CP}$-odd partial differential widths for the process $\tau\rightarrow K\pi\pi\nu_{\tau}$ assuming that an intermediate heavy charged scalar contributes to the decay amplitude. Adopting a model-independent approach, we use a Monte Carlo simulation in order to study the number of events needed to recover information on the new physics from these observables. Our analysis of the $\mathrm{CP}$-odd observables indicates that the magnitude of $f_H \eta_P$, which is related to the new-physics contribution, can be recovered with an uncertainty smaller than $3$ for $3\times 10^6$ events. This number of events would also allow one to retrieve certain parameters appearing in the SM amplitude at the percent level. In addition, we discuss the possibility of using the proposed observables to study specific models involving two Higgs doublets, such as the aligned two-Higgs-doublet model (A2HDM). This analysis is undertaken within the context of the upcoming Super B-factories, which are expected to provide a considerably larger number of events than that which was supplied by the B-factories. Moreover, a similar set of observables could be employed to study other decay modes such as $\tau\rightarrow\pi\pi\pi\nu_{\tau},\,\tau\rightarrow KK\pi\nu_{\tau}$ and $\tau\rightarrow KKK\nu_{\tau}$.