b→spenguin amplitude in charmlessB→PPdecays

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 sum of rates for $B^0\to K^0{\pi }^0$ and $B^{+}\to K^{+}{\pi }^0$ enhanced relative to half the sum for $B^0\to K^{+}{\pi }^{-}$ and $B^{+}\to K^0{\pi }^{+}$, (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 $B^0\to {\eta }^{\prime }{K}_S$. 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 prove 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.