Revisiting Final State Interaction in Charmless $B_q\to P P$ Decays

Various new measurements in charmless $B_{u,d,s}\to PP$ modes, where $P$ is a low lying pseudoscalar meson, are reported by Belle and LHCb. These include the rates of $B^0\to\pi^0\pi^0$, $\eta\pi^0$, $B_s\to\eta'\eta'$, $B^0\to K^+K^-$ and $B^0_s\to\pi^+\pi^-$ decays. Some of these modes are highly suppressed and are among the rarest $B$ decays. Direct CP asymmetries on various modes are constantly updated. It is well known that direct CP asymmetries and rates of suppressed modes are sensitive to final state interaction (FSI). As new measurements are reported and more data will be collected, it is interesting and timely to revisit the rescattering effects in $B_{u,d,s}\to PP$ states. We perform a $\chi^2$ analysis with all available data on CP-averaged rates and CP asymmetries in $\overline B{}_{u,d,s}\to PP$ decays. Our numerical results are compared to data and those from factorization approach. The quality of the fit is improved significantly from the factorization results in the presence of rescattering. The relations on topological amplitudes and rescattering are explored and they help to provide a better understanding of the effects of FSI. As suggested by U(3) symmetry on topological amplitudes and FSI, a vanishing exchange rescattering scenario is considered. The exchange, annihilation, $u$-penguin, $u$-penguin annihilation and some electroweak penguin amplitudes are enhanced significantly via annihilation and total annihilation rescatterings. In particular, the $u$-penguin annihilation amplitude is sizably enhanced by the tree amplitude via total annihilation rescattering. These enhancements affect rates and CP asymmetries. Predictions can be checked in the near future.