Corrections of Order $Λ^2_{QCD}/m^2_c$ to Inclusive Rare B Decays

We calculate nonperturbative ${\cal O}(Λ^2_{QCD}/m^2_c)$ corrections to the dilepton invariant mass spectrum and the forward-backward charge asymmetry in $B\to X_se^+e^-$ decay using a heavy quark expansion approach. The method has recently been used to estimate long-distance effects in $B\to X_sγ$. We generalize this analysis to the case of nonvanishing photon invariant mass, $q^2\not= 0$, relevant for the rare decay mode $B\to X_se^+e^-$. In the phenomenologically interesting $q^2$ region away from the $c\bar c$ resonances, the heavy quark expansion approach should provide a reasonable description of possible nonperturbative corrections. In particular this picture is preferable to the model-dependent approach relying on the tails of Breit-Wigner resonances, which has been employed so far in the literature to account for these effects. We find that the ${\cal O}(Λ^2_{QCD}/m^2_c)$ corrections to the dilepton invariant mass spectrum and to the forward-backward asymmetry in $B\to X_se^+e^-$ amount to several percent at most for $q^2/m^2_b < 0.3$ and $q^2/m^2_b > 0.6$. The ${\cal O}(Λ^2_{QCD}/m^2_c)$ correction to the $B\to X_sγ$ decay rate is also computed and found to be +3%, which agrees in magnitude with previous calculations. Finally, we comment on long-distance effects in $B\to X_sν\barν$, which in this case are extremely suppressed due to the absence of virtual photon contributions.

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