$|V_{ub}|$ determination in lattice QCD

Abstract

The 2012 PDG reports a tension at the level of $3\sigma$ between two exclusive determinations of $|V_{ub}|$. They are obtained by combining the experimental branching ratios of $B\to \tau \nu$ and $B\to \pi l\nu$ (respectively) with a theoretical computation of the hadronic matrix elements $\text{fB}$ and the $B\to \pi$ 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 $\text{fB}$ 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 $\approx$ 190 MeV. Renormalization and matching were performed non-perturbatively, and three lattice spacings reaching $a^{-1}\approx 4.1$ 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 $\text{fBq}s$, needed for the analysis of $B_s\to {\mu }^{+}{\mu }^{-}$.}