Extraction of $a_1$ and $a_2$ from $B\to ψK(K^*),~ D(D^*)π(ρ)$ Decays

Based on the factorization approach, we show that the CLEO data for the ratio $Γ(B\toψK^*)/Γ(B\to ψK)$ and the CDF measurement of the fraction of longitudinal polarization in $B\toψK^*$ can be accounted for by the heavy-flavor-symmetry approach for heavy-light form factors provided that the form factor $F_0$ behaves as a constant, while the $q^2$ dependence is of the monopole form for $F_1,~ A_0,~A_1$, and of the dipole behavior for $A_2$ and $V$. This $q^2$ extrapolation for form factors is further supported by $B\to K^*γ$ data and by a recent QCD-sum-rule analysis. We then apply this method to $B\to D(D^*)π(ρ)$ decays to extract the parameters $a_1$ and $a_2$. It is found that $a_1(B\to D^{(*)}π(ρ))=1.01\pm 0.06$ and $a_2(B\to D^{(*)}π(ρ))=0.23\pm 0.06\,$. Our result $a_2/a_1=0.22\pm 0.06$ thus significantly improves the previous analysis that leads to $a_2/a_1=0.23\pm 0.11$. We argue that, contrary to what anticipated from the leading $1/N_c$ expansion, the sign of $a_2(B\toψ\K)$ should be positive and $a_2(B\toψ\K)~{\large ^>_\sim}~ a_2(B\to D^{(*)}π(ρ))$.

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