Possibility of large direct $\mathrm{CP}$ violation in $\overrightarrow{B}K\pi$-like modes because of long distance rescattering effects and implications for the angle $\gamma$

We consider the strong rescattering effects that can occur in decays such as $\overrightarrow{B}K\pi ,$ $K^{*}\pi ,$ $K\rho ,\dots ,$ and their impact on direct $\mathrm{CP}$ violation in these modes. First we discuss, in general, how the $\mathrm{CPT}$ theorem constrains the resulting pattern of partial rate asymmetries (PRA’s) leading to different brands of direct $\mathrm{CP}$ violation. Traditional discussions have centered around the absorptive part of the penguin graph which has $\Delta I=0\mathrm{}$ in $\overrightarrow{b}s$ transitions and as a result causes “simple” $\mathrm{CP}$ violation; long-distance final state rescattering effects, in general, will lead to a different pattern of $\mathrm{CP}$ violation: “compound” $\mathrm{CP}$ violation. Predictions of simple $\mathrm{CP}$ violation are quite distinct from that of compound $\mathrm{CP}$ violation. Final state rescattering phases in $B$ decays are unlikely to be small possibly causing large compound $\mathrm{CP}$-violating partial rate asymmetries in these modes. The $\mathrm{CPT}$ theorem requires a cancellation of PRA’s due to compound $\mathrm{CP}$ violation among the $K\pi$ states themselves; thus there can be no net cancellation with other states such as $K^{*}\pi ,$ $K\rho ,$ etc. Therefore, each class of such modes, namely, $K\pi ,$ $K\rho ,$ $K^{*}\pi ,$ ${\mathrm{Ka}}_1,$ etc., can have large direct $\mathrm{CP}$ violation emanating from rescattering effects. Various repercussions for the angle $\gamma$ are also discussed.