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

The decay $b\to d e^+e^-$ has an amplitude containing comparable contributions proportional to $V_{tb}V_{td}^{*}$, $V_{cb}V_{cd}^{*}$ and $V_{ub}V_{ud}^{*}$. 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 $ρ$, $ω$ and $J/ψ$ 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 $- 5% (- 2%)$ is predicted in the nonresonant domain $1 GeV < m_{e^+e^-} < m_{J/ψ}$, assuming $η= 0.34$ and $ρ= 0.3 (-0.3)$. 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 $J/ψ$ resonance, where an asymmetry of $3\times 10^{-3}$ is expected, subject to certain theoretical uncertainties in the $b\to d J/ψ$ amplitude.