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Towards more precise determinations of the CKM phase $β$

We derive a new flavor symmetry relation for the determination of the weak phase $β=ϕ_1$ from time-dependent CP asymmetries and $B \to J/ψP$ decay rates. In this relation, the contributions to $\sin2β$ proportional to $V_{ub}$ are parametrically suppressed compared to the contributions in the $B \to J/ψK^0$ 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 $2σ$ level from expectations, or may hint at effects of unexpected magnitude from contributions proportional to $V_{ub}$ or from isospin breaking.