B $\to$ $π$ form factor with 2 flavours of $O(a)$ improved Wilson quarks
The determinations of $|V_{\rm ub}|$ from the exclusive branching ratios of $B\to τν$ and $B \to πl ν$ tend to show a tension at the level of $3σ$ \cite{Beringer:1900zz}. On the theoretical side they depend on the lattice computation of the hadronic matrix elements $f_{\rm B}$ and the $B\to π$ form factor $f_+(q^2)$. To understand the tension, improved precision and a careful analysis of the systematics involved are necessary. Working towards this goal, we present preliminary lattice results of the ALPHA collaboration for the $B\to π$ form factor $f_+(q^2)$ with $N_{\rm f}=2$ flavours of $O(a)$-improved Wilson fermions. Our computation uses HQET in the static limit, pion masses ranging down to $\sim250$ MeV, large volumes with $m_πL >4$, three lattice spacings, and non-perturbative renormalization. We describe the techniques adopted to reduce the statistical noise (stochastic all-to-all with full time dilution) and the contamination from excited states (smearing for the B and the pion). We estimate the size of the chiral and continuum extrapolations. We discuss the impact our result could have to clarify the above mentioned discrepancy in the determination of $|V_{\rm ub}|$.