A comprehensive study of $B$, $B_s$ and $B_c$ meson semitauonic modes in potential quark model

In this work, we derive the form factors and compute the branching fractions for the semitauonic decay modes, $B \to D^{(*)}\,\tau\,\nu_{\tau}$, $B_s \to D_s^{(*)}\,\tau\,\nu_{\tau}$, $B_c \to \eta_c\,(J/\psi)\,\tau\,\nu_{\tau}$, and $B_c \to D^{(*)}\,\tau\,\nu_{\tau}$ within the \emph{Relativistic Independent Quark (RIQ) Model}, emphasizing a quark potential model based analysis of these transitions. We outline the essential elements of the model, incorporating corrections from residual interactions and center-of-mass motion, and perform a comprehensive study of the form factors across the full physical kinematic range of $q^2$. The resulting predictions demonstrate consistency and good agreement with existing theoretical approaches and experimental measurements. Motivated by recent observations of polarization observables at LHCb and Belle, we further evaluate these quantities within our framework and find results compatible with Standard Model (SM) expectations. The predictions presented here serve as theoretical input in decay channels for which Lattice QCD results remain limited, offering guidance for future experimental and Lattice efforts. Thus, semileptonic $B$ decays continue to serve as precise and discerning probes of the fundamental mechanisms governing flavor transitions in the SM.