Hadronic Weak Decays of Heavy Mesons and Nonfactorization

The parameters $\chi_{1,2}$, which measure nonfactorizable soft gluon contributions to hadronic weak decays of mesons, are updated by extracting them from the data of $D,\,B\to PP,~VP$ decays ($P$: pseudoscalar meson, $V$: vector meson). It is found that $\chi_2$ ranges from $-0.36$ to $-0.60$ in the decays from $D\to\bar{K}\pi$ to $D^+\to\phi\pi^+,~D\to\bar{K}^*\pi$, while it is of order 10\% with a positive sign in $B\to\psi K,~D\pi,~D^*\pi,~D\rho$ decays. Therefore, the effective parameter $a_2$ is process dependent in charm decay, whereas it stays fairly stable in $B$ decay. This implies the picture that nonfactorizable effects become stronger when the decay particles become less energetic after hadronization. As for $D,\,B\to VV$ decays, the presence of nonfactorizable terms in general prevents a possible definition of effective $a_1$ and $a_2$. This is reenforced by the observation of a large longitudinal polarization fraction in $B\to\psi K^*$ decay, implying $S$-wave dominated nonfactorizable effects. The nonfactorizable term dominated by the $S$-wave is also essential for understanding the decay rate of $B^-\to D^{*0} \rho^-$. It is found that all nonfactorizable effects $A_1^{nf}/A_1^{BK^*},~A _1^{nf}/A_1^{B\rho},~A_1^{nf}/A_1^{BD^*}$ ($nf$ standing for nonfactorization) are positive and of order 10\%, in accordance with $\chi_2(B\to D(D^*)\pi( \rho))$ and $\chi_2(B\to\psi K)$. However, we show that in $D\to\bar{K}^*\rho$ decay nonfactorizable effects cannot be dominated by the $S$-wave. A polarization measurement in the color- and Cabibbo-suppressed decay mode $D^+\to\phi\rho^+$ is strongly urged in order to test if $A_2^{nf}/A_2$ plays a more pivotal role than $A_1^{nf}/A_1$ in charm decay.