$B_c$ decays from highly improved staggered quarks and NRQCD

We calculate semileptonic form factors for the decays $B_c to eta_c , l u$ and $B_c to J/psi , l u$ over the entire $q^2$ 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 $B to D^*$.