Semileptonic $B\to \pi$ Decays from an Omn\`es Improved Nonrelativistic Constituent Quark Model

The semileptonic $B\to \pi l^+ \nu_l$ decay is studied starting from a simple quark model which includes the influence of the $B^*$ pole. To extend the predictions of a nonrelativistic constituent quark model from its region of applicability near $q^2_{\rm max}=(m_B-m_\pi)^2$ to all $q^2$ values accessible in the physical decay, we use a novel multiply-subtracted Omn\`es dispersion relation, which considerably diminishes the form factor dependence on the elastic $\pi B \to \pi B$ scattering amplitudes at high energies. By comparison to the experimental branching fraction we extract $|V_{ub}| = 0.0034 \pm 0.0003 ({\rm exp}) \pm 0.0007 ({\rm theory})$. To further test our framework, we also study $D\to \pi$ and $D\to K$ decays and find excellent results $\frac{f^+_\pi(0)}{f^+_K(0)} = 0.80 \pm 0.03, \quad \frac{{\cal B}(D^0\to \pi^- e^+ \nu_e)}{{\cal B}(D^0\to K^- e^+ \nu_e)}= 0.079 \pm 0.008$. In particular for the $D\to \pi$ case, we reproduce, with high accuracy, the three-flavor lattice QCD results recently obtained by the Fermilab-MILC-HPQCD Collaboration. While for the $D\to K$ case, we successfully describe the data for $f^+(q^2)/f^+(0)$ recently measured by the FOCUS Collaboration.