  The exclusive semileptonic decay $B \rightarrow \pi \ell \nu$ 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 $f_0(q^2_{max}) = f_B/f_{\pi}$ does hold as $m_{\pi}
\rightarrow 0$. 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 $B_s \rightarrow K \ell \nu$.
