Heavy-Quark expansion for \({{\bar{B}}_s\rightarrow D^{(*)}_s}\) form factors and unitarity bounds beyond the \({SU(3)_F}\) limit

We carry out a comprehensive analysis of the full set of $${\bar{B}}_q \rightarrow D_q^{(*)}$$ form factors for spectator quarks $$q=u,d,s$$ within the framework of the Heavy-Quark expansion (HQE) to order $${\mathcal {O}}\left( \alpha _s, 1/m_b, 1/m_c^2\right) $$. 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 $${\bar{B}}_s \rightarrow D_s^{(*)}$$ 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 $${{\bar{B}}}_s\rightarrow D_s^{(*)}$$ form factors, especially in the unitarity constraints which we impose throughout our analysis. We find the coefficients of the IW functions emerging at $${\mathcal {O}}\left( 1/m_c^2\right) $$ to be consistent with the naive $${\mathcal {O}}\left( 1\right) $$ 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 $${\bar{B}}\rightarrow D^{(*)}\ell \nu $$ 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 $$\tau $$- and $$D_q^*$$ polarization fractions for the $${\bar{B}}_q\rightarrow D_q^{(*)}\tau \nu $$ modes.