Self-consistent optimization of the $z$ expansion for $B$-meson decays

We discuss the self-consistency imposed by the analyticity of regular parts of form factors, appearing in the $z$ expansion for semileptonic $B$-meson decays, when fitted in different kinematic regions. Relying on the uniqueness of functions defined by analytic continuation, we propose four metrics which measure the departure from the ideal analytic self-consistency. We illustrate the process using Belle data for $B\to D\ell {\nu }_{\ell }$ with the two kinematic regions chosen as the five low-$z$ and the five high-$z$ bins. For this specific example, the metrics provide consistent indications that some choices (order of truncation, Boyd-Grinstein-Lebed or Bourrely-Caprini-Lellouch) made in the form of the $z$ expansion can be optimized. However, other choices ($z$ origin, location of isolated poles and threshold constraints) appear to have very little effect on these metrics. On the other hand, changing the kinetic regions affects the results and should also be considered in the optimization process. We briefly discuss the implication for optimization of the $z$ expansion for nucleon form factors relevant for neutrino oscillation experiments.