Implications of factorization for the determination of hadronic form factors in $D_s^+ \ra ϕ$ transition
Using factorization we determine the allowed domains of the ratios of form factors, $x = A_2(0)/A_1(0)$ and $y = V(0)/A_1(0)$, from the experimentally measured ratio $R_h \equiv Γ(D_s^+ \ra ϕρ^+)/Γ(D_s^+ \ra ϕπ^+)$ assuming three different scenarios for the $q^2$-dependence of the form factors. We find that the allowed domains overlap with those obtained by using the experimentally measured ratio $R_{s\ell} = Γ(D^+_s \ra ϕ\ell^+ ν_{\ell})/Γ(D^+_s \ra ϕπ^+)$ provided that the phenomenological parameter $a_1$ is $1.23$. Such a comparison presents a genuine test of factorization. We calculate the longitudinal polarization fraction, $Γ_L/Γ\equiv Γ(D_s^+ \ra ϕ_L ρ^+_L)/Γ(D_s^+ \ra ϕρ^+)$, in the three scenarios for the $q^2$-dependence of the form factors and emphasize the importance of measuring $Γ_L/Γ$. Finally we discuss the $q^2$-distribution of the semileptonic decay and find that it is rather insensitive to the scenarios for the $q^2$-dependence of the form factors, and unless very accurate data can be obtained it is unlikely to discriminate between the different scenarios. Useful information on the value of $x$ might be obtained by the magnitude of the $q^2$-distribution near $q^2 = 0$. However the most precise information on $x$ and $y$ would come from the knowledge of the longitudinal and left-right transverse polarizations of the final vector mesons in hadronic and/or semileptonic decays.