The $B\to\pi K$ puzzle and its relation to rare B and K decays

Abstract.

The standard-model interpretation of the ratios of charged and neutral $B\to\pi K$ rates, R _c and R _n, respectively, points towards a puzzling picture. Since these observables are affected significantly by colour-allowed electroweak (EW) penguins, this “$B\to\pi K$ puzzle” could be a manifestation of new physics in the EW penguin sector. Performing the analysis in the R _n- R _c plane, which is very suitable for monitoring various effects, we demonstrate that we may, in fact, move straightforwardly to the experimental region in this plane through an enhancement of the relevant EW penguin parameter q. We derive analytical bounds for q in terms of a quantity L, which measures the violation of the Lipkin sum rule, and point out that strong phases around $90^\circ$ are favoured by the data, in contrast to QCD factorisation. The $B\to\pi K$ modes imply a correlation between q and the angle $\gamma$ that, in the limit of negligible rescattering effects and colour-suppressed EW penguins, depends only on the value of L. Concentrating on a minimal flavour-violating new-physics scenario with enhanced Z ⁰ penguins, we find that the current experimental values on $B\to X_s \mu^ + \mu^-$ require roughly $L\le 1.8$. As the $B\to\pi K$ data give $L = 5.7\pm2.4$, L has either to move to smaller values once the $B\to\pi K$ data improve or new sources of flavour and CP violation are needed. In turn, the enhanced values of L seen in the $B\to\pi K$ data could be accompanied by enhanced branching ratios for the rare decays $K^ + \to \pi^ + \nu\bar\nu$, $K_{\mathrm {L}}\to \pi^0 e^ + e^-$, $B\to X_s\nu\bar\nu$ and $B_{s,d}\to \mu^ + \mu^-$. Most interesting turns out to be the correlation between the $B\to\pi K$ modes and ${\rm BR} (K^ + \to \pi^ + \nu\bar\nu)$, with the latter depending approximately on a single “scaling” variable $\bar L = L\cdot (\vert V_{ub}/V_{cb}\vert/0.086)^{2.3}$.