Improved Bounds for the Slope and Curvature of $\bar B\to D^{(*)}\ell\bar\nu$ Form Factors

We derive a theoretically allowed domain for the slope $\widehat\varrho^2$ and curvature $\widehat c$ of the physical form factor appearing in the decay $\bar B\to D^*\ell\bar\nu$. Using heavy-quark symmetry, we relate this function to a particular $\bar B \to D$ form factor free of ground-state $B_c$ poles below the threshold for $B D$ production, for which almost model-independent constraints are derived from QCD using unitarity and analyticity. Our results are of interest for the extraction of $|V_{cb}|$ from the recoil spectrum in exclusive semileptonic $B$ decays. Making conservative estimates of the theoretical uncertainties, we find (up to $1/m_Q$ corrections) $\widehat\varrho^2<1.11$ and $\widehat c\simeq 0.66\widehat\varrho^2-0.11$. We also derive the corresponding bounds for the form factor in the decay $\bar B\to D\ell\bar\nu$.