$|V_{cb}|$, Lepton Flavour Universality and $SU(3)_F$ symmetry breaking in $B_s \to D_s^{(*)} \ell \nu_\ell$ decays through unitarity and lattice QCD

In addition to the well-known $B \to D^{(*)} \ell \nu_\ell$ decays, semileptonic $B_s \to D_s^{(*)} \ell \nu_\ell$ processes offer the possibility to determine the Cabibbo-Kobayashi-Maskawa (CKM) matrix element $\vert V_{cb}\vert$. We implement the Dispersive Matrix (DM) approach to describe the hadronic Form Factors (FFs) for the $B_s \to D_s^{(*)}$ transition in the whole kinematical range, starting from recent Lattice QCD computations at large values of the 4-momentum transfer. We extract $\vert V_{cb} \vert$ from the experimental data, obtaining $\vert V_{cb} \vert \cdot 10^3 = (41.7 \pm 1.9)$ from $B_s \to D_s \ell \nu_\ell$ and $\vert V_{cb} \vert \cdot 10^3 =(40.7 \pm 2.4)$ from $B_s \to D_s^* \ell \nu_\ell$ decays. After averaging with the values of $\vert V_{cb} \vert$ obtained from the $B \to D^{(*)}$ channels [1,2] we get $\vert V_{cb} \vert \cdot 10^3 =(41.2 \pm 0.8)$, which is compatible with the most recent inclusive estimate $\vert V_{cb} \vert_{\rm{incl}} \cdot 10^3 = 42.16 \pm 0.50$ [3] at the $1 \sigma$ level. In addition we test the Lepton Flavour Universality (LFU) by computing the $\tau / \ell$ ratios of the total decay rates (where $\ell$ is a light lepton), obtaining $R(D_s) = 0.298\,(5)$ and $R(D_s^*)= 0.250\,(6)$. We also address the issue of the $SU(3)_F$ symmetry breaking by comparing the hadronic FFs entering the semileptonic $B \to D^{(*)}$ and $B_s \to D_s^{(*)}$ channels.