We compute semi-leptonic $B_s$ decay form factors using Heavy Quark Effective Theory on the lattice. To obtain good control of the $1/m_b$ expansion, one has to take into account not only the leading static order but also the terms arising at $O(1/m_b)$: kinetic, spin and current insertions. We show results for these terms calculated through the ratio method, using our prior results for the static order. They can then be non-perturbatively renormalized, matched to QCD and continuum-extrapolated to give the QCD form factor correct up to $O(1/m_b^2)$ corrections and without $O(\alpha_s(m_b)^n)$ corrections. Dependence on the light quark mass is also discussed.

Weak Decays and Matrix Elements