Limitations in Measuring the Angle $\beta$ by Using $SU(3)$ Relations for $B$-Meson Decay-Amplitudes

Abstract

Flavour $SU(3)$ symmetry of strong interactions and certain dynamical assumptions have been used in a series of recent publications to extract weak CKM phases from $B$-decays into $\{\pi \pi ,\pi K,K\bar{K}\}$ final states. We point out that irrespectively of $SU(3)$-breaking effects the presence of QCD-penguin contributions with internal $u$- and $c$-quarks precludes a clean determination of the angle $\beta$ in the unitarity triangle by using the branching ratios only. This difficulty can be overcome by measuring in addition the ratio $x_d/x_s$ of $B_d^0-{\bar{B}}_d^0$ to $B_s^0-{\bar{B}}_s^0$ mixings. The measurement of the angle $\gamma$ is unaffected by these new contributions. Some specific uncertainties related to $SU(3)$-breaking effects and electroweak penguin contributions are briefly discussed.

Comments: 15 pages (LaTeX) + 5 figures included, Munich Technical University preprint TUM-T31-69/94