Rescattering effects in charmlessB¯u,d,s→PPdecays

Abstract

We study the final-state interaction (FSI) effects in charmless ${\overline{B}}_{u,d,s}\ensuremath{\rightarrow}PP$ decays. We consider a FSI approach with both short- and long-distance contributions, where the former are from inelastic channels and are contained in factorization amplitudes, while the latter are from the residual rescattering among $PP$ states. Flavor SU(3) symmetry is used to constrain the residual rescattering $S$ matrix. We fit to all available data on the $CP$-averaged decay rates and $CP$ asymmetries, and make predictions on unmeasured ones. We investigate the $K\ensuremath{\pi}$ direct $CP$ violations that lead to the so-called $K\ensuremath{\pi}$ puzzle in $CP$ violation. Our main results are as follows: (i) Results are in agreement with data in the presence of FSI. (ii) For $\overline{B}$ decays, the ${\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ and ${\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}$ rates are suppressed and enhanced, respectively, by FSI. (iii) The FSI has a large impact on direct $CP$ asymmetries ($\mathcal{A}$) of many modes. (iv) The deviation ($\ensuremath{\Delta}\mathcal{A}$) between $\mathcal{A}({\overline{B}}^{0}\ensuremath{\rightarrow}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{+})$ and $\mathcal{A}({B}^{\ensuremath{-}}\ensuremath{\rightarrow}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{0})$ can be understood in the FSI approach. Since $\mathcal{A}({K}^{\ensuremath{-}}{\ensuremath{\pi}}^{0})$ is more sensitive to the residual rescattering, the degeneracy of these two direct $CP$ violations can be successfully lifted. (v) Sizable and complex color-suppressed tree amplitudes, which are crucial for the large ${\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}$ rate and $\ensuremath{\Delta}\mathcal{A}$, are generated through exchange rescattering. The correlation of the ratio $\mathcal{B}({\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0})/\mathcal{B}({\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}})$ and $\ensuremath{\Delta}\mathcal{A}$ is studied. (vi) The ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}$ direct $CP$ violation is very small and is not affected by FSI. (vii) Several ${\overline{B}}_{s}$ decay rates are enhanced. In particular, the ${\ensuremath{\eta}}^{\ensuremath{'}}{\ensuremath{\eta}}^{\ensuremath{'}}$ branching ratio is enhanced to the level of $1.0\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$, which can be checked experimentally. (viii) Time-dependent $CP$ asymmetries $S$ in ${\overline{B}}_{d,s}$ decays are studied. The $\ensuremath{\Delta}S({\overline{B}}^{0}\ensuremath{\rightarrow}{K}_{S}{\ensuremath{\eta}}^{\ensuremath{'}})$ is very small ($\ensuremath{\le}1%$). This asymmetry remains to be one of the cleanest measurements to search for new physics phases. The asymmetry $S$ from ${\overline{B}}_{s}$ to $PP$ states with strangeness $S=+1$ are expected to be small. We found that the $|S|$ for ${\overline{B}}_{s}^{0}\ensuremath{\rightarrow}\ensuremath{\eta}\ensuremath{\eta}$, $\ensuremath{\eta}{\ensuremath{\eta}}^{\ensuremath{'}}$, and ${\ensuremath{\eta}}^{\ensuremath{'}}{\ensuremath{\eta}}^{\ensuremath{'}}$ decays are all below 0.06. $CP$ asymmetries in these modes will be useful to test the SM.