Extracting CKM phases from angular distributions ofBd,sdecays into admixtures ofCPeigenstates

The time-dependent angular distributions of certain $B_{d,s}$ decays into final states that are admixtures of $\mathrm{CP}$-even and $\mathrm{CP}$-odd configurations provide valuable information about CKM phases and hadronic parameters. We present the general formalism to accomplish this task, taking also into account penguin contributions, and illustrate it by considering a few specific decay modes. We give particular emphasis to the decay $B_d\to J/\psi {\rho }^0$, which can be combined with $B_s\to J/\psi \phi$ to extract the $B_d^0-B_d^0$ mixing phase and—if penguin effects in the former mode should be sizeable—also the angle $\gamma$ of the unitarity triangle. As an interesting by-product, this strategy allows us to take into account also the penguin effects in the extraction of the $B_s^0-B_s^0$ mixing phase from $B_s\to J/\psi \phi .$ Moreover, a discrete ambiguity in the extraction of the CKM angle $\beta$ can be resolved, and valuable insights into $\mathrm{SU}\left(3\right)\mathrm{}$-breaking effects can be obtained. Other interesting applications of the general formalism presented in this paper, involving $B_d\to \rho \rho$ and $B_{s,d}\to K^{*}{K}^{*}$ decays, are also briefly noted.