Resolving a Discrete Ambiguity in the CKM Angle $β$ through $B_{u,d} \to J/ψK^\ast$ and $B_s \to J/ψφ$ Decays

It is well known that $\sin(2\beta)$, where $\beta$ is one of the angles of the unitarity triangle of the CKM matrix, can be determined in a theoretically clean way by measuring mixing-induced CP violation in the decay $B_d \to J/\psi K_S$. Another clean extraction of this CKM angle is provided by the time-dependent angular distribution for the decay products of $B_d \to J/\psi(\to l^+l^-) K^{\ast0}(\to \pi^0 K_S)$, where we have more observables at our disposal than in the case of $B_d \to J/\psi K_S$, so that in addition to $\sin(2\beta)$ also $\cos(2\beta)$ can be probed in a direct way. Unfortunately a sign ambiguity remains in $\cos(2\beta)$. If it could be resolved, a discrete ambiguity in the extraction of the CKM angle $\beta$ could be resolved as well, which would allow a more incisive test of the CKM model of CP violation. This note shows that detailed time-dependent studies of $B_{u,d} \to J/\psi K^{\ast}$ and $B_s \to J/\psi \phi$ decay processes can determine the sign of $\cos(2\beta)$, thereby removing the corresponding ambiguity in the extraction of the CKM angle $\beta$.