Nonfactorizable Effects in Spectator and Penguin Amplitudes of Hadronic   Charmless B Decays
  Nonfactorizable effects in hadronic charmless B -> PP, VP decays can be parametrized in terms of the effective number of colors N_c in the effective parameters $a_i^{eff}$ that are linear combinations of Wilson coefficients. It is shown that N_c(V+A) in the penguin amplitudes induced by the (V-A)(V+A) four-quark operators is different from N_c(V-A) in the decay amplitudes arising from the (V-A)(V-A) operators. Central values of the branching ratios of $B^\pm\to\omega \pi^\pm$ and $B\to\pi\pi$ decays favor $N_c(V-A)\approx 2$, in accordance with the nonfactorizable effect observed in $B\to D^{(*)}\pi (\rho)$. Measurements of the interference effects in $B^-\to\pi^-(\rho^-)\pi^0 (\rho^0)$ decays will provide a more decisive test on the parameter N_c(V-A). However, $N_c(V+A)\sim 2$ is ruled out by $B^\pm\to\phi K^\pm$ and $B\to\phi K^*$. We find that N_c(V+A) lies in the range $10\geq N_c(V+A)\geq 4.3$, subject to the corrections from W-annihilation and space-like penguin effects. With $N_c(V-A)\approx 2$ we show that the branching ratio of $B\to\eta' K$ is enhanced considerably at small values of 1/N_c(V+A) so that it is compatible with experiment. In particular, the measurement of $B^0\to\eta' K^0$ is now well explained without resorting to any new mechanism or new physics beyond the Standard Model. Finally, we point out that it is difficult to understand the observed large branching ratio of $B^\pm\to\omega K^\pm$ within the present framework. Inelastic final-state interactions may alleviate the difficulty with this decay mode. 
Hai-Yang Cheng, B. Tseng