Form factor curves consistent with unitarity for semileptonic decays

We discuss a method to generate form factor curves consistent with dispersive constraints across the entire kinematic range for exclusive semileptonic (SL) pseudoscalar to pseudoscalar decays, for example $B \rightarrow \pi \ell \nu$ and $B_s \rightarrow K \ell \nu$. The work builds on the Dispersive Matrix (DM) method which allows model-independent extrapolation to any desired $q^2$ value in the SL physical region using known form factor information at specific discrete $q^2$ points as input. Here $q$ is the outgoing lepton-pair 4-momentum. An obstacle in using DM results for phenomenological predictions, such as forward-backward asymmetries, is that it is not obvious how to use the bounds over continuous ranges of $q^2$ when integrating, for example, the differential decay rate over the physical $q^2$ range or over bins in $q^2$. We describe a method to generate a family of curves, each consistent with unitarity constraints, that can be used in the same way as a set generated from a parametrized fit (e.g. a $z$-fit). This allows integration over any desired bins. We further show some techniques to increase the computational efficiency of the method. We demonstrate the application to determining $|V_{ub}|$.