We present a precise calculation of the dilepton invariant-mass spectrum and the decay rate for $B^{\pm }\to {\pi }^{\pm }{\ell }^{+}{\ell }^{-}$ (${\ell }^{\pm }=e^{\pm },{\mu }^{\pm }$) in the Standard Model (SM) based on the effective Hamiltonian approach for the $b\to d{\ell }^{+}{\ell }^{-}$ transitions. With the Wilson coefficients already known in next-to-next-to-leading logarithmic (NNLL) accuracy, the remaining theoretical uncertainty in the short-distance contribution resides in the form factors $f_{+}(q^2)$, $f_0(q^2)$ and $f_T(q^2)$. Of these, $f_{+}(q^2)$ is well measured in the charged-current semileptonic decays $B\to \pi \ell {\nu }_{\ell }$, and we use the $B$-factory data to parametrize it. The corresponding form factors for the $B\to K$ transitions have been calculated in the lattice QCD approach for large $q^2$ and extrapolated to the entire $q^2$ region using the so-called $z$ expansion. Using an $SU(3{)}_F$-breaking ansatz, we calculate the $B\to \pi$ tensor form factor, which is consistent with the recently reported lattice $B\to \pi$ analysis obtained at large $q^2$. The prediction for the total branching fraction $\mathcal{B}(B^{\pm }\to {\pi }^{\pm }{\mu }^{+}{\mu }^{-})=(1.88_{-0.21}^{+0.32})\times {10}^{-8}$ is in good agreement with the experimental value obtained by the LHCb Collaboration. In the low-$q^2$ region, heavy-quark symmetry (HQS) relates the three form factors with each other. Accounting for the leading-order symmetry-breaking effects, and using data from the charged-current process $B\to \pi \ell {\nu }_{\ell }$ to determine $f_{+}(q^2)$, we calculate the dilepton invariant-mass distribution in the low-$q^2$ region in the $B^{\pm }\to {\pi }^{\pm }{\ell }^{+}{\ell }^{-}$ decay. This provides a model-independent and precise calculation of the partial branching ratio for this decay.

• Received 10 December 2013

DOI:https://doi.org/10.1103/PhysRevD.89.094021