The $B \to πK$ Puzzle Revisited

Abstract: 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 ($B^+ \to \pi^+ K^0$, $B^+ \to \pi^0 K^+$, $B^0 \to \pi^- K^+$, $B^0 \to \pi^0 K^0$). In this paper, we re-examine this $B \to \pi K$ puzzle. We find that the key unknown parameter is $|C'/T'|$, the ratio of color-suppressed and color-allowed tree amplitudes. If this ratio is large, $|C'/T'| = 0.5$, the SM can explain the data. But if it is small, $|C'/T'| = 0.2$, 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 $Z'$ bosons and diquarks. $Z'$ models can explain the puzzle if the $Z'$ couples to right-handed $u{\bar u}$ and/or $d{\bar d}$, with $g_R^{dd} \ne g_R^{uu}$. Interestingly, half of the many $Z'$ models proposed to explain the present anomalies in $b \to s \mu^+ \mu^-$ decays have the required $Z'$ 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.