We discuss the computation of form factors for semi-leptonic decays of B-, B s -mesons in lattice QCD. Considering in particular the example of the static B s form factors we demonstrate that after non-perturbative renormalization the continuum limit can be taken with confidence. The resulting precision is of interest for extractions of V ub . The size of the corrections of order 1/m b is just estimated at present but it is expected that their inclusion does not pose significant difficulties.

We discuss the extraction of the ground state K(p)|V µ (0)|B s (0) matrix elements from Euclidean lattice correlation functions. The emphasis is on the elimination of excited state contributions. Two typical gauge-field ensembles with lattice spacings 0.075, 0.05 fm and pion masses 330, 270 MeV are used from the O(a)-improved CLS 2-flavour simulations and the final state momentum is |p| = 0.5 GeV. The b-quark is treated in HQET including the 1/m b corrections. Fits to two-point and three-point correlation functions and suitable ratios including summed ratios are used, yielding consistent results with precision of around 2% which is not limited by the 1/m b corrections but by the dominating static form factors. Excited state contributions are under reasonable control but are the bottleneck towards precision. We do not yet include a specific investigation of multi-hadron contaminations, a gap in the literature which ought to be filled soon.

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The 2012 PDG reports a tension at the level of 3σ between two exclusive determinations of |V ub |. They are obtained by combining the experimental branching ratios of B → τν and B → πlν (respectively) with a theoretical computation of the hadronic matrix elements f B and the B → π form factor f + (q 2). To understand the tension, improved precision and a careful analysis of the systematics involved are necessary. We report the results of the ALPHA collaboration for f B from the lattice with 2 flavors of O(a) improved Wilson fermions. We employ HQET, including 1/m b corrections, with pion masses ranging down to ≈ 190 MeV. Renormalization and matching were performed non-perturbatively, and three lattice spacings reaching a −1 ≈ 4.1 GeV are used in the continuum extrapolation. We also present progress towards a computation of f + (q 2), to directly compare two independent exclusive determinations of |V ub | with each other and with inclusive determinations. Additionally, we report on preliminary results for f B s , needed for the analysis of B s → µ + µ − .