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 ${\cal B} (B^\pm \to \pi^\pm \mu^+ \mu^-) = (1.88 ^{+0.32}_{-0.21}) \times 10^{-8}$ 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.