Effect of the final-state phases onCPviolation in theBd0−B¯d0system

We have studied the $\mathrm{CP}$-violating asymmetry, taking account of the final-state phase shifts, for the decays $B_d^0\to D^{+}{D}^{-}$, $B_d^0\to {\pi }^{+}{\pi }^{-}$, and $B_d^0\to K^{+}{\pi }^{-}$, which have tree-level and penguin amplitudes. The final-state phase shifts lead to significant direct $\mathrm{CP}$ violation for these decays. The numerical predictions of the $\mathrm{CP}$-violating asymmetries are given against final-state phase shifts in the framework of the standard model.