Precise Calculation of the Dilepton Invariant-Mass Spectrum and the Decay Rate in $B^{\pm }\to {\pi }^{\pm }{\mu }^{+}{\mu }^{-}$ in the SM

Abstract

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 the 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.

Comments: 23 pages, RevTeX, 11 figures. The subsection VI-C is included concerning the $SU(3{)}_F$-breaking and several references added