$B_c$ decays from highly improved staggered quarks and NRQCD

We calculate semileptonic form factors for the decays $B_c \to \eta_c , l \nu$and$B_c \to J/\psi , l \nu$overtheentire$q^2$ range, using a highly improved lattice quark action for charm at several lattice spacings down to $a=0.045$ fm. We have two ways of treating the $b$ quark: either with an $O({\alpha }_s)$ improved NRQCD formalism or by extrapolating a heavy mass $m_h$ to $m_b$ in the relativistic formalism. Comparison of the two approaches provides an important cross-check of methodologies in lattice QCD. Nonperturbative renormalisation of the currents in the relativistic theory also allows us then to fix NRQCD-charm normalisation for $b$ to $c$ decays such as $B\to D$ and $B\to D^{\ast }$.