The Role of the Vacuum Insertion Approximation in Calculating CP Asymmetries in B Decays

Abstract

CP asymmetries in $B$ decays into final CP eigenstates are in many cases theoretically clean. In particular, they do not depend on the values of hadronic parameters. The sign of the asymmetries, however, does depend on the sign of the $B_B$ parameter. Furthermore, the information from $\epsilon_K$ that all angles of the unitarity triangles lie in the range $\{0,\pi\}$ depends on the sign of the $B_K$ parameter. Consequently, in the (unlikely) case that the vacuum insertion approximation is such a poor approximation that either $B_B$ or $B_K$ is negative, the sign of CP asymmetries in neutral $B$ decays will be opposite to the standard predictions. Various subtleties concerning the role of $K-\bar K$ mixing in the case of final states with a single $K_S$ or $K_L$, such as the $B\to\psi K_S$ decay, are clarified.