Within the standard model, and if one assumes that soft rescattering effects are negligible, the $\mathrm{CP}$ asymmetry ${\mathcal{A}}_{\mathrm{CP}}^{\mathrm{dir}}{(B}^{\pm }\to {\pi }^{\pm }K)$ is predicted to be very small and the ratio ${R=B(B}_d\to {\pi }^{\mp }{K}^{\pm }{)/B(B}^{\pm }\to {\pi }^{\pm }K)$ provides a bound on the angle $\gamma$ of the unitarity triangle, ${\sin }^2\gamma <~R.\mathrm{}$ We estimate the corrections from soft rescattering effects using an approach based on Regge phenomenology, and find effects of order $10\%$ with large uncertainties. In particular, we conclude that ${\mathcal{A}}_{\mathrm{CP}}^{\mathrm{dir}}\sim 0.2$ and ${\sin }^2\gamma \sim 1.2R$ could not be taken unambiguously to signal new physics. Using SU(3) relations, we suggest experimental tests that could constrain the size of the soft rescattering effects thus reducing the related uncertainty. Finally, we study the effect of various models of new physics on ${\mathcal{A}}_{\mathrm{CP}}^{\mathrm{dir}}$ and on $R.\mathrm{}$

• Received 10 December 1997

DOI:https://doi.org/10.1103/PhysRevD.57.4290