In Pursuit of Gamma

Abstract

After reviewing techniques for extracting clean information on CP-violating phase angles from $B$ decays, we explain the rules for finding decay modes that can probe the phase angle $γ$ of the unitarity triangle. We identify the more promising of these ``$γ$ modes,'' estimate their branching ratios, and examine the degree to which they are theoretically clean. We then show that when the quark mixing matrix is not approximated as usual, but is treated exactly, none of the ``$γ$ modes'' actually measures $γ$. Rather, each of them measures $γ$ plus some correction. In all modes, the correction is small enough to be disregarded in first-generation experiments, but in some of them, it may be large enough to be observed in second-generation experiments. Our treatment of the $γ$ modes calls attention to the fact that when the quark mixing matrix is treated exactly, there are six unitarity triangles, rather than just one triangle. However, only four of the angles in these six triangles are independent. Examining the role played by these four angles, we discover that, in principle at least, measurements of nothing but CP-violating asymmetries in $B$ decays are sufficient to determine the entire quark mixing matrix.