A novel approach to semileptonic heavy-to-light $B$ decays through the Dispersive Matrix method

In this contribution we analyse the heavy-to-light $B$ decays through the Dispersive Matrix method, which can be applied to any semileptonic decays of hadrons once lattice QCD computations of the hadronic Form Factors and of the relevant susceptibilities are available. We will explicitly discuss the application of the Dispersive Matrix approach to both $B \to \pi \ell \nu_{\ell}$ and $B_s \to K \ell \nu_{\ell}$ decays. As usual in our analysis strategy, only LQCD computations of the FFs at high values of the momentum transfer will be used to determine the shape of the FFs in the whole kinematical range without making any assumption on their momentum dependence. Then, the experimental data will be used only to obtain our final exclusive determinations of $\vert V_{ub} \vert$. In this way, our calculation of the FFs allows to obtain pure theoretical estimates of several quantities of phenomenological interest, for instance the $\tau/\mu$ ratio of the differential decay rates $R_{\pi(K)}^{\tau/\mu}$, which is an important tool for testing Lepton Flavour Universality. We will also present a summary of all the results obtained so far for semileptonic $B$ decays within the Dispersive Matrix approach.