Theory determination of $bar{B}to D^{(*)}ell^-bar u$ form factors at $mathcal{O}(1/m_c^2)$

We carry out an analysis of the full set of ten $bar{B}to D^{(*)}$ form factors within the framework of the Heavy-Quark Expansion (HQE) to order $mathcal{O}(alpha_s,,1/m_b,,1/m_c^2)$, both with and without the use of experimental data. This becomes possible due to a recent calculation of these form factors at and beyond the maximal physical recoil using QCD light-cone sum rules, in combination with constraints from lattice QCD, QCD three-point sum rules and unitarity. We find good agreement amongst the various theoretical results, as well as between the theoretical results and the kinematical distributions in $bar{B}to D^{(*)}lbrace e^-,mu^-rbracebar u$ measurements. The coefficients entering at the $1/m_c^2$ level are found to be of $mathcal{O}(1)$, indicating convergence of the HQE. The phenomenological implications of our study include an updated exclusive determination of $|V_{cb}|$ in the HQE, which is compatible with both the exclusive determination using the BGL parametrization and with the inclusive determination. We also revisit predictions for the lepton-flavour universality ratios $R_{D^{(*)}}$, the $tau$ polarization observables $P_tau^{D^{(*)}}$, and the longitudinal polarization fraction $F_L$. Posterior samples for the HQE parameters are provided as ancillary files, allowing for their use in subsequent studies.