Unitary $KK$ and $Kπ$ inputs for an amplitude analysis of the $D^+ \to K^-K^+π^+$ decay
Charge-Parity Violation (CPV) is expected in three-body charm decays. In particular, in the singly Cabibbo suppressed decay process $D^+ \to K^-K^+π^+$; in which an amplitude analysis is required to understand the dynamics of Final State Interactions (FSI) and the mechanism that can generate CPV. In this work, we show the prospects of analysing recent data from the Large Hadron Collider beauty (LHCb) experiment at CERN, with a precise theoretical model expected to reveal CPV mechanisms. Previous analyses of the $D^+ \to K^-K^+π^+$ decay were unable to satisfactorily describe the data using the Isobar Model. Currently, LHCb has a sample of charmed meson decays with around 100 million events for the $D^+ \to K^-K^+π^+$ decay, so a more robust phenomenological model is needed to describe such a large data set. The present model has allowed the inclusion of hadronic FSI and will, in prospect, introduce three-body effects, overcoming some limitations of the Isobar Model, and has all factors necessary to reveal the presence of CPV. To do so, it factorizes the weak vertex separated in leading topologies and relies on extensions of Resonant Chiral Perturbation Theory to describe FSI. As a proof-of-concept, a toy dataset of the S-wave of the most recent amplitude analysis of $D^+ \to K^-K^+π^+$, by the CLEO-c collaboration, was fitted with this model. From fits to model-independent parametrizations of scattering amplitudes, the parameters of the FSI amplitudes were fixed, and the production parameters of the weak vertex were then obtained from the fit to the toy dataset, yielding a good agreement between the data and our model.