The exclusive semileptonic decay $B\to \pi \ell \nu$ is a key process for the determination of the Cabibbo-Kobayashi-Maskawa matrix element $V_{ub}$ from the comparison of experimental rates as a function of $q^2$ with theoretically determined form factors. The sensitivity of the form factors to the $u/d$ quark mass has meant significant systematic uncertainties in lattice QCD calculations at unphysically heavy pion masses. Here, we give the first lattice QCD calculations of this process for $u/d$ quark masses going down to their physical values, calculating the $f_0$ form factor at zero recoil to 3%. We are able to resolve a long-standing controversy by showing that the soft-pion theorem result $f_0(q_{\max }^2)=f_B/f_{\pi }$ does hold as $m_{\pi }\to 0$. We use the highly improved staggered quark formalism for the light quarks and show that staggered chiral perturbation theory for the $m_{\pi }$ dependence is almost identical to continuum chiral perturbation theory for $f_0$, $f_B$, and $f_{\pi }$. We also give results for other processes such as $B_s\to K\ell \nu$.

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• Lattice QCD
• Leptonic, semileptonic & radiative decays
• Bottom mesons
• Pions