Hadron decay amplitudes from $B\to K \pi$ and $B\to \pi \pi$ decays

One can analyze the hadronic decay amplitudes for $B\to PP$ decays using flavor SU(3) symmetry in different ways, such as the algebraic and quark diagram approaches. We derive specific relations between these two sets of amplitudes. In the Standard Model the leading hadronic decay amplitudes depend on only five independent parameters which can be determined using recent experimental data on the branching ratios and CP violating asymmetries of $B\to K \pi$ and $B \to \pi\pi$. We find, however, that the leading amplitudes provide a best fit solution with a large $\chi^2$, which cannot therefore be regarded as a good fit. Keeping sub-leading terms, makes it possible to have a reasonable minimal $\chi^2$. We also find that in general the color suppressed decay amplitude is comparable with the color allowed amplitude, contrary to expectations.