Insight into $D/B\to \pi \ell \nu_\ell$ decay using the pole models

After improving the knowledge about residua of the semileptonic form factor at its first two poles we show that $f_+^{D\pi}(q^2)$ is not saturated when compared with the experimental data. To fill the difference we approximate the rest of discontinuity by an effective pole and show that the data can be described very well with the position of the effective pole larger than the next excitation in the spectrum of $D^\ast$ state. The results of fits with experimental data also suggest the validity of superconvergence which in the pole models translates to a vanishing of the sum of residua of the form factor at all poles. A similar discussion in the case of $B\to \pi \ell\nu_\ell$ leads to the possibility of extracting $\vert V_{ub}\vert$, the error of which appears to be dominated by $g_{B^\ast B\pi}$, which can be nowadays computed on the lattice. In evaluating the residua of the form factors at their nearest pole we needed the vector meson decay constants $f_{D^\ast}$ and $f_{B^\ast}$, which we computed by using the numerical simulations of QCD on the lattice with $N_{\rm f}=2$ dynamical quarks. We obtain, $f_{D^\ast}/f_D=1.208(27)$ and $f_{B^\ast}/f_B=1.051(17)$.