Towards more precise determinations of the CKM phase $\beta$

We derive a new flavor symmetry relation for the determination of the weak phase $\beta=\phi_1$ from time-dependent CP asymmetries and $B \to J/\psi P$ decay rates. In this relation, the contributions to $\sin2\beta$ proportional to $V_{ub}$ are parametrically suppressed compared to the contributions in the $B \to J/\psi K^0$ time-dependent CP asymmetry alone. This relation uses only $SU(3)$ flavor symmetry, and does not require further diagrammatic assumptions. The current data either fluctuate at the $2\sigma$ level from expectations, or may hint at effects of unexpected magnitude from contributions proportional to $V_{ub}$ or from isospin breaking.