A solution to B --> pi pi puzzle and B --> K K

The large ratio of color-suppressed tree amplitude to color-allowed one in $B \to \pi \pi$ decays is difficult to understand within the Standard Model, which is known as the "$B \to \pi\pi$ puzzle". The two tree diagrams contain the up- and charm-quark component of penguin amplitude, $P_{uc}$, which cannot be separated by measuring $B \to \pi \pi$ decays alone. We show that the measurements of the branching ratio and direct CP asymmetry of $B^+ \to K^+ \bar{K^0}$ decay enable one to disentangle the $P_{uc}$ with two-fold ambiguity. One of the two degenerate solutions of the $P_{uc}$ can solve the $B \to \pi \pi$ puzzle by giving $|C/T|\sim 0.3$ which is consistent with the expectation in the Standard Model. We also show that the two solutions can be discriminated by the measurement of the indirect CP-asymmetry of $B^0 \to K^0 \ol{K^0}$. We point out that if the $B \to \pi \pi$ puzzle is solved in this way, the corresponding puzzle in $B \to \pi K$ decays should have a different origin.