Rescattering Contributions to rare B-Meson Decays

Several $B$ and $B_s$ decays have been observed which have been cited as evidence for exchange ($E$), penguin annihilation ($PA$) and annihilation ($A$) processes, such as $\bar b d \to \bar u u$, $\bar b s \to \bar u u$ and $\bar b u \to W^* \to \bar c s$, respectively. These amplitudes are normally thought to be suppressed, as they involve the spectator quark in the weak interaction and thus should be proportional to the $B$-meson decay constant $f_B$. However, as pointed out a number of years ago, they can also be generated by rescattering from processes whose amplitudes do not involve $f_B$, such as color-favored tree amplitudes. In this paper we investigate a number of processes such as $B^0 \to K^+ K^-$, $B_s \to \pi^+ \pi^-$, and $B^+ \to D_s^+ \phi$, and identify promising states from which they can be generated by rescattering. We find that $E$ and $PA$-type processes are characterized respectively by amplitudes ranging from 5% to 10% and from 15% to 20% with respect to the largest amplitude from which they can rescatter. Based on this regularity, using approximate flavor SU(3) symmetry in some cases and time-reversal invariance in others, we predict the branching fractions for a large number of as-yet-unseen $B$ and $B_s$ decays in an extensive range from order $10^{-9}$ to $10^{-4}$.