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A Theoretical Reappraisal of Branching Ratios and CP Asymmetries in the Decays $B \to (X_d,X_s) \ell^+ \ell^-$ and Determination of the CKM Parameters

We present a theoretical reappraisal of the branching ratios and CP asymmetries for the decays $ B \to X_q \ell^+ \ell^-$, with $q=d,s$, taking into account current theoretical uncertainties in the description of the inclusive decay amplitudes from the long-distance contributions, an improved treatment of the renormalization scale dependence, and other parametric dependencies. Concentrating on the partial branching ratios $Δ{\cal B}(B \to X_q \ell^+ \ell^-)$, integrated over the invariant dilepton mass region $1 {GeV}^2 \leq s \leq 6 {GeV}^2$, we calculate theoretical precision on the charge-conjugate averaged partial branching ratios $<Δ{\cal B}_q >= (Δ{\cal B}(B \to X_q \ell^+ \ell^-) + Δ{\cal B}(\bar{B} \to \bar{X}_q \ell^+ \ell^-))/2$, CP asymmetries in partial decay rates $(a_{CP})_q=(Δ{\cal B}(B \to X_q \ell^+ \ell^-) - Δ{\cal B}(\bar{B} \to \bar{X}_q \ell^+ \ell^-))/(2 < Δ{\cal B}_q >)$, and the ratio of the branching ratios $Δ{\cal R} = < Δ{\cal B}_d >/< Δ{\cal B}_s>$. For the central values of the CKM parameters, we find $<Δ{\cal B}_s > =(2.22^{+0.29}_{-0.30}) \times 10^{-6}$, $<Δ{\cal B}_d > =(9.61^{+1.32}_{-1.47}) \times 10^{-8}$, $(a_{CP})_s =-(0.19^{+0.17}_{-0.19})%$, $(a_{CP})_d =(4.40^{+3.87}_{-4.46})%$, and $Δ{\cal R} =(4.32 \pm 0.03)%$. The dependence of $<Δ{\cal B}_d>$ and $Δ{\cal R}$ on the CKM parameters is worked out and the resulting constraints on the unitarity triangle from an eventual measurement of $Δ{\cal R}$ are illustrated.