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

Abstract

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 $\Delta \mathcal{B}(B\to X_q{\ell }^{+}{\ell }^{-})$, integrated over the invariant dilepton mass region $1{GeV}^2\le s\le 6{GeV}^2$, we calculate theoretical precision on the charge-conjugate averaged partial branching ratios $<\Delta {\mathcal{B}}_q>=(\Delta \mathcal{B}(B\to X_q{\ell }^{+}{\ell }^{-})+\Delta \mathcal{B}(\bar{B}\to {\bar{X}}_q{\ell }^{+}{\ell }^{-}))/2$, CP asymmetries in partial decay rates $(a_{CP}{)}_q=(\Delta \mathcal{B}(B\to X_q{\ell }^{+}{\ell }^{-})-\Delta \mathcal{B}(\bar{B}\to {\bar{X}}_q{\ell }^{+}{\ell }^{-}))/(2<\Delta {\mathcal{B}}_q>)$, and the ratio of the branching ratios $\Delta \mathcal{R}=<\Delta {\mathcal{B}}_d>/<\Delta {\mathcal{B}}_s>$. For the central values of the CKM parameters, we find $<\Delta {\mathcal{B}}_s>=(2.22_{-0.30}^{+0.29})\times 10^{-6}$, $<\Delta {\mathcal{B}}_d>=(9.61_{-1.47}^{+1.32})\times 10^{-8}$, $(a_{CP}{)}_s=-(0.19_{-0.19}^{+0.17})$, $(a_{CP}{)}_d=(4.40_{-4.46}^{+3.87})$, and $\Delta \mathcal{R}=(4.32\pm 0.03)$. The dependence of $<\Delta {\mathcal{B}}_d>$ and $\Delta \mathcal{R}$ on the CKM parameters is worked out and the resulting constraints on the unitarity triangle from an eventual measurement of $\Delta \mathcal{R}$ are illustrated.

Comments: 18 pages, 7 figures (require epsf.sty)