Logarithmically Enhanced Corrections to the Decay Rate and Forward Backward Asymmetry in $\bar{B} \to X_s \ell^+ \ell^-$

We study logarithmically enhanced electromagnetic corrections to the decay rate in the high dilepton invariant mass region as well as corrections to the forward backward asymmetry (FBA) of the inclusive rare decay $\bar{B} \to X_s \ell^+ \ell^-$. As expected, the relative effect of these corrections in the high dilepton mass region is around -8% for the muonic final state and therefore much larger than in the low dilepton mass region. We also present a complete phenomenological analysis, to improved NNLO accuracy, of the dilepton mass spectrum and the FBA integrated in the low dilepton mass region, including a new approach to the zero of the FBA. The latter represents one of the most precise predictions in flavour physics with a theoretical uncertainty of order 5%. We find $(q_0^2)_{\mu\mu} = (3.50 \pm 0.12) \gev^2$. For the high dilepton invariant mass region, we have ${\cal B}(\bar B\to X_s\mu\mu)_{\rm high} = (2.40^{+0.69}_{-0.62}) \times 10^{-7}$. The dominant uncertainty is due to the $1/m_b$ corrections and can be significantly reduced in the future. For the low dilepton invariant mass region, we confirm previous results up to small corrections.