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Testing the Standard Model and Schemes for Quark Mass Matrices with CP Asymmetries in B Decays

The values of $\sin (2 α)$ and $\sin (2 β)$, where $α$ and $β$ are angles of the unitarity triangle, will be readily measured in a B factory (and maybe also in hadron colliders). We study the standard model constraints in the $\sin (2 α) - \sin (2 β)$ plane. We use the results from recent analyses of $f_B$ and $τ_b|V_{cb}|^2$ which take into account heavy quark symmetry considerations. We find $\sin (2 β) \geq 0.15$ and most likely $\sin (2 β) \roughly{>} 0.6$, and emphasize the strong correlations between $\sin (2 α)$ and $\sin (2 β)$. Various schemes for quark mass matrices allow much smaller areas in the $\sin (2 α) - \sin (2 β)$ plane. We study the schemes of Fritzsch, of Dimopoulos, Hall and Raby, and of Giudice, as well as the ``symmetric CKM'' idea, and show how CP asymmetries in B decays will crucially test each of these schemes.