CPviolation inBd→D+D−,D*+D−,D+D*−,andD*+D*−decays

Abstract

$\mathrm{CP}$ asymmetries in ${B}_{d}\ensuremath{\rightarrow}{D}^{+}{D}^{\ensuremath{-}},$ ${D}^{*+}{D}^{\ensuremath{-}},$ ${D}^{+}{D}^{*\ensuremath{-}}$ and ${D}^{*+}{D}^{*\ensuremath{-}}$ decays are investigated with the help of the factorization approximation and isospin relations. We find that direct $\mathrm{CP}$ violation is governed only by the short-distance penguin mechanism, while the indirect $\mathrm{CP}$ asymmetries in ${B}_{d}\ensuremath{\rightarrow}{D}^{\ifmmode\pm\else\textpm\fi{}}{D}^{*\ensuremath{\mp}}$ transitions may be modified due to the final-state rescattering effect. An updated numerical analysis shows that the direct $\mathrm{CP}$ asymmetry in ${B}_{d}^{0}$ vs ${B}_{d}^{0}\ensuremath{\rightarrow}{D}^{+}{D}^{\ensuremath{-}}$ decays can be as large as 3%. The $\mathrm{CP}$-even and $\mathrm{CP}$-odd contributions to the indirect $\mathrm{CP}$ asymmetry in ${B}_{d}^{0}$ vs ${B}_{d}^{0}\ensuremath{\rightarrow}{D}^{*+}{D}^{*\ensuremath{-}}$ decays are found to have the rates 89% and 11 %, respectively. Some comments on the possibilities to determine the weak phase $\ensuremath{\beta}$ and to test the factorization hypothesis are also given.