Penguin Enhancement and $B \to K\pi$ decays in perturbative QCD

TLDR: In this paper, the branching ratios of quark decay amplitudes with universal meson wave functions were derived for tree, penguin, and annihilation modes in the framework of the perturbative QCD factorization theorem.

Abstract: We compute the branching ratios of $\stackrel{\ensuremath{\rightarrow}}{B}K\ensuremath{\pi}$ decays in the framework of the perturbative QCD factorization theorem. Decay amplitudes are classified into the topologies of tree, penguin, and annihilation amplitudes, all of which contain both factorizable and nonfactorizable contributions. These contributions are expressed as the convolutions of hard $b$ quark decay amplitudes with universal meson wave functions. It is shown that (1) matrix elements of penguin operators are dynamically enhanced compared to those employed in the factorization assumption, (2) annihilation diagrams are not negligible, contrary to common belief, (3) annihilation diagrams contribute large strong phases, and (4) the uncertainty of the current data of the ratio $R=\mathrm{Br}{(B}_{d}^{0}\ensuremath{\rightarrow}{K}^{\ifmmode\pm\else\textpm\fi{}}{\ensuremath{\pi}}^{\ensuremath{\mp}})/\mathrm{Br}{(B}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}{K}^{0}{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}})$ and of $\mathrm{CP}$ asymmetries is too large to give a constraint of the unitarity angle ${\ensuremath{\varphi}}_{3}.$ Assuming ${\ensuremath{\varphi}}_{3}=90\ifmmode^\circ\else\textdegree\fi{}$ which is extracted from the best fit to the data of $R,$ predictions for the branching ratios of the four $\stackrel{\ensuremath{\rightarrow}}{B}K\ensuremath{\pi}$ modes are consistent with data.