Title:The $b \to s$ penguin amplitude in charmless $B \to PP$ decays

Abstract: The $b \to s$ penguin amplitude affects a number of $B$ meson decays to two pseudoscalar ($P$) mesons in which potential anomalies are being watched carefully, though none has yet reached a statistically compelling level. These include (a) a rate for $B^0 \to K^0 \pi^0$ which is slightly enhanced with respect to expectations; (b) a time-dependent CP asymmetry parameter $S$ for $B^0 \to K^0 \pi^0$ which is low in comparison with the expected value of $\sin 2 \beta \simeq 0.73$, and (c) a similar deviation in the parameter $S$ for $B^0 \to \eta' K_S$. These and related phenomena involving vector mesons in the final state are discussed in a unified way in and beyond the Standard Model. Future experiments which would conclusively indicate the presence of new physics are identified. Several of these involve decays of the strange $B$ meson $B_s$. In the Standard Model we note an approximate sum rule for CP rate differences in $B^0\to K^+\pi^-, B^+\to K^+\pi^0$ and $B^0\to K^0\pi^0$, predicting a negative sign for the latter asymmetry.