The $B\to \pi K$ Puzzle Revisited

For a number of years, there has been a certain inconsistency among the measurements of the branching ratios and CP asymmetries of the four $B \to \pi K$decays($B^+ \to \pi^+ K^0$,$B^+ \to \pi^0 K^+$,$B^0 \to \pi^- K^+$,$B^0 \to \pi^0 K^0$)\mathrm{.}Inthispaper,were-examinethis$B \to \pi K$ puzzle. We find that the key unknown parameter is $\mathrm{\mid }{C}^{\mathrm{\prime }}\mathrm{/}{T}^{\mathrm{\prime }}\mathrm{\mid }$, the ratio of color-suppressed and color-allowed tree amplitudes. If this ratio is large, $\mathrm{\mid }{C}^{\mathrm{\prime }}\mathrm{/}{T}^{\mathrm{\prime }}\mathrm{\mid }=0.5$, the SM can explain the data. But if it is small, $\mathrm{\mid }{C}^{\mathrm{\prime }}\mathrm{/}{T}^{\mathrm{\prime }}\mathrm{\mid }=0.2$, the SM cannot explain the $B\to \pi K$ puzzle – new physics (NP) is needed. The two types of NP that can contribute to $B\to \pi K$ at tree level are $Z^{\mathrm{\prime }}$ bosons and diquarks. $Z^{\mathrm{\prime }}$ models can explain the puzzle if the $Z^{\mathrm{\prime }}$ couples to right-handed $u\bar{u}$ and/or $d\bar{d}$, with $g_R^{dd}\mathrm{\ne }{g}_R^{uu}$. Interestingly, half of the many $Z^{\mathrm{\prime }}$ models proposed to explain the present anomalies in $b\to s{\mu }^{+}{\mu }^{-}$ decays have the required $Z^{\mathrm{\prime }}$ couplings to $u\bar{u}$ and/or $d\bar{d}$. Such models could potentially explain both the $b\to s{\mu }^{+}{\mu }^{-}$ anomalies and the $B\to \pi K$ puzzle. The addition of a color sextet diquark that couples to $ud$ can also explain the puzzle.