  The rare decay $B\to\pi\ell^+\ell^-$ arises from $b\to d$ flavor-changing
neutral currents and could be sensitive to physics beyond the Standard Model.
Here, we present the first $ab$-$initio$ QCD calculation of the $B\to\pi$
tensor form factor $f_T$. Together with the vector and scalar form factors
$f_+$ and $f_0$ from our companion work [J. A. Bailey $et~al.$, Phys. Rev. D
92, 014024 (2015)], these parameterize the hadronic contribution to $B\to\pi$
semileptonic decays in any extension of the Standard Model. We obtain the total
branching ratio ${\text{BR}}(B^+\to\pi^+\mu^+\mu^-)=20.4(2.1)\times10^{-9}$ in
the Standard Model, which is the most precise theoretical determination to
date, and agrees with the recent measurement from the LHCb experiment [R. Aaij
$et~al.$, JHEP 1212, 125 (2012)]. Note added: after this paper was submitted
for publication, LHCb announced a new measurement of the differential decay
rate for this process [T. Tekampe, talk at DPF 2015], which we now compare to
the shape and normalization of the Standard-Model prediction.
