CP Violation in the Decay B \to X_d e^+ e^-

Abstract

The decay $b\to d{e}^{+}{e}^{-}$ has an amplitude containing comparable contributions proportional to $V_{tb}{V}_{td}^{\ast }$, $V_{cb}{V}_{cd}^{\ast }$ and $V_{ub}{V}_{ud}^{\ast }$. These pieces involve different unitarity phases produced by $c\bar{c}$ and $u\bar{u}$ loops. The simultaneous presence of different CKM phases and different dynamical phases leads to a calculable asymmetry in the partial widths of $b\to d{e}^{+}{e}^{-}$ and $\bar{b}\to \bar{d}{e}^{+}{e}^{-}$. Using the effective Hamiltonian of the standard model, we calculate this asymmetry as a function of the $e^{+}{e}^{-}$ invariant mass. The effects of $\rho$, $\omega$ and $J/\psi$ resonances are taken into account in the vacuum polarization of the $u\bar{u}$ and $c\bar{c}$ currents. As a typical result, an asymmetry of $-5$ is predicted in the nonresonant domain $1GeV<m_{e^{+}{e}^{-}}<m_{J/\psi }$, assuming $\eta =0.34$ and $\rho =0.3(-0.3)$. The branching ratio in this domain is $1.2\times 10^{-7}(3.3\times 10^{-7})$. Results are also obtained in the region of the $J/\psi$ resonance, where an asymmetry of $3\times 10^{-3}$ is expected, subject to certain theoretical uncertainties in the $b\to dJ/\psi$ amplitude.

Comments: 20 pages, LaTeX, 3 eps figures, uses cite.sty, epsf.sty, and amssym.sty. Version to appear in Phys. Rev. D, with some minor additions