Resolving a Discrete Ambiguity in the CKM Angle $\beta$ through $B_{u,d}\to J/\psi K^{\ast }$ and $B_s\to J/\psi \varphi$ Decays

Abstract

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 $B_d\to J/\psi (\to l^{+}{l}^{-})K^{\ast 0}(\to {\pi }^0{K}_S)$, 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 $B_s\to J/\psi \varphi$ decay processes can determine the sign of $\cos (2\beta )$, thereby removing the corresponding ambiguity in the extraction of the CKM angle $\beta$.

Comments: 5 pages, LaTeX, no figures