Unitarity Bounds for Semileptonic Decays in Lattice QCD

In this work we discuss in detail the non-perturbative determination of the momentum dependence of the form factors entering in semileptonic decays using unitarity and analyticity constraints. The method contains several new elements with respect to previous proposals and allows to extract, using suitable two-point functions computed non-perturbatively, the form factors at low momentum transfer $q^2$ from those computed explicitly on the lattice at large $q^2$, without any assumption about their $q^2$-dependence. The approach will be very useful for exclusive semileptonic $B$-meson decays, where the direct calculation of the form factors at low $q^2$ is particularly difficult due to large statistical fluctuations and discretisation effects. As a testing ground we apply our approach to the semileptonic $D \to K \ell \nu_\ell$ decay, where we can compare the results of the unitarity approach to the explicit direct lattice calculation of the form factors in the full $q^2$-range. We show that the method is very effective and that it allows to compute the form factors with rather good precision.