$\bar B\to D^{(*)}\ell\bar \nu$ Branching Ratios and Evidence for Isospin Breaking in $\Upsilon(4S)$ Decays

We introduce a new method for the determination of the ratio of production fractions $R^{\pm0}=\mathcal B(\Upsilon(4S)\to B^+B^-)/\mathcal B(\Upsilon(4S)\to B^0\bar B^0)$ based on $\bar B\to D^{(*)}\ell\bar \nu$ decays. Given the importance of these modes, we perform a comprehensive analysis of the available data, extracting the information on their branching fractions and \Rpmz in parallel and providing their correlations in order to avoid double-use of this information in phenomenological analyses. We obtain the most precise value for $R^{\pm0}$ from a single channel so far, about 2$\sigma$ from unity. The combination with previously available determinations from other channels yields $R^{\pm0}=1.062(19)$, constituting evidence for isospin violation in $\Upsilon(4S)$ decays. This demonstrates the necessity to take this effect into account in experimental and phenomenological analyses. The results for the $\bar B\to D^{(*)}\ell\bar \nu$ branching fractions are up to $1.6\sigma$ larger compared to averages available in the literature, owing to the removal of overlooked inconsistencies in the treatment of older analyses and correcting for d'Agostini bias where possible, thereby reducing the $V_{cb}$ puzzle.