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In this contribution we analyse the heavy-to-light $B$ decays through theDispersive Matrix method, which can be applied to any semileptonic decays ofhadrons once lattice QCD computations of the hadronic Form Factors and of therelevant susceptibilities are available. We will explicitly discuss theapplication 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 analysisstrategy, only LQCD computations of the FFs at high values of the momentumtransfer will be used to determine the shape of the FFs in the wholekinematical range without making any assumption on their momentum dependence.Then, the experimental data will be used only to obtain our final exclusivedeterminations of $\vert V_{ub} \vert$. In this way, our calculation of the FFsallows to obtain pure theoretical estimates of several quantities ofphenomenological interest, for instance the $\tau/\mu$ ratio of thedifferential decay rates $R_{\pi(K)}^{\tau/\mu}$, which is an important toolfor testing Lepton Flavour Universality. We will also present a summary of allthe results obtained so far for semileptonic $B$ decays within the DispersiveMatrix approach.

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