Improved QCD perturbative contributions and power corrections in radiative and semileptonic rare B decays

We calculate the leading power corrections to the decay rates, distributions and hadronic spectral moments in rare inclusive \bxsll decays in the standard model, using heavy quark expansion (HQE) in $(1/m_b)$ and a phenomenological model implementing the Fermi motion effects of the b-quark bound in the B-hadron. We include next-to-leading order perturbative QCD corrections and work out the dependences of the spectra, decay rates and hadronic moments on the model parameters in either HQE and the Fermi motion model. In the latter, we take into account long-distance effects via $B \to X_s +(J/\psi, \psi^\prime,...) \to X_s \ell^+ \ell^- $ with a vector meson dominance ansatz and study the influence of kinematical cuts in the dilepton and hadronic invariant masses on branching ratios, hadron spectra and hadronic moments. We present leading logarithmic QCD corrections to the $b \to s \gamma\gamma$ amplitude. The QCD perturbative improved $B_{s}\to\gamma\gamma$ branching ratio is given in the standard model including our estimate of long-distance effects via $B_s \to \phi \gamma \to \gamma \gamma$ and $B_s \to \phi \psi \to \phi \gamma \to \gamma \gamma$ decays. The uncertainties due to the renormalization scale and the parameters of the HQE inspired bound state model are worked out.