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Heavy-Quark Expansion for $\bar{B}_s\to D^{(*)}_s$ Form Factors and Unitarity Bounds beyond the $SU(3)_F$ Limit

We carry out a comprehensive analysis of the full set of $\bar{B}_q \to D_q^{(*)}$ form factors for spectator quarks $q=u,d,s$ within the framework of the Heavy-Quark Expansion (HQE) to order $\mathcal{O}(α_s, 1/m_b, 1/m_c^2)$. In addition to the available lattice QCD calculations we make use of two new sets of theoretical constraints: we produce for the first time numerical predictions for the full set of $\bar{B}_s \to D_s^{(*)}$ form factors using Light-Cone Sum Rules with $B_s$-meson distribution amplitudes. Furthermore, we reassess the QCD three-point sum rule results for the Isgur-Wise functions entering all our form factors for both $q=u,d$ and $q=s$ spectator quarks. These additional constraints allow us to go beyond the commonly used assumption of $SU(3)_F$ symmetry for the $\bar B_s\to D_s^{(*)}$ form factors, especially in the unitarity constraints which we impose throughout our analysis. We find the coefficients of the IW functions emerging at $\mathcal{O}(1/m_c^2)$ to be consistent with the naive $\mathcal{O}(1)$ expectation, indicating a good convergence of the HQE. While we do not find significant $SU(3)$ breaking, the explicit treatment of $q=s$ as compared to a simple symmetry assumption renders the unitarity constraints more effective. We find that the (pseudo)scalar bounds are saturated to a large degree, which affects our theory predictions. We analyze the phenomenological consequences of our improved form factors by extracting $|V_{cb}|$ from $\bar B\to D^{(*)}\ellν$ decays and producing theoretical predictions for the lepton-flavour universality ratios $R(D)$, $R(D^*)$, $R(D_s)$ and $R(D_s^*)$, as well as the $τ$- and $D_q^*$ polarization fractions for the $\bar B_q\to D_q^{(*)}τν$ modes.