Power corrections in the decay rate and distributions in $B \to X_s l^+ l^-$ in the Standard Model

We investigate the leading power corrections to the decay rates and distributions in the decay $B \rightarrow X_s \ell~+ \ell~- $ in the standard model (SM) 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. In the HQE method, we find that including the leading power corrections the decay width decreases by about 4% and the branching ratio by about 1.5% from their (respective) parton model values. The dilepton invariant mass spectrum is found to be stable against power corrections over a good part of this spectrum. However, near the high-mass end-point this distribution becomes negative with the current value of the non-perturbative parameter $\lambda_2$ (the $\lambda_1$-dependent corrections are found to be inoccouos), implying the breakdown of the HQE method in this region. Our results are at variance with the existing ones in the literature in both the decay rate and the invariant dilepton mass distribution calculated in the HQE approach. As an alternative, we implement the non-perturbative effects in the decay $B \rightarrow X_s \ell~+ \ell~- $ using a phenomenologically motivated Gaussian Fermi motion model. We find small corrections to the branching ratio, but the non-perturbative effects are perceptible in both the dilepton mass distribution and the Forward-Backward asymmetry in the high dilepton mass region. Using this model for estimating the non-perturbative effects, we present the decay rates and distributions for the inclusive process $B \rightarrow X_s \ell~+ \ell~- $ in the SM