The Semileptonic Decays $B\to\pi l\nu$ and $D\to\pi l\nu$ from Lattice QCD

We present a lattice QCD calculation of the form factors and differential decay rates for semileptonic decays of the heavy-light mesons $B$ and $D$ to the final state $\pi l\nu$. The results are obtained with three methodological improvements over previous lattice calculations: a matching procedure that reduces heavy-quark lattice artifacts, the first study of lattice-spacing dependence, and the introduction of kinematic cuts to reduce model dependence. We show that the main systematics are controllable (within the quenched approximation) and outline how the calculations could be improved to aid current experiments in the determination of~$|V_{ub}|$ and~$|V_{cd}|$.