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$D_s^+ \to ϕρ^+$ Decay

Motivated by the experimental measurement of the decay rate, $Γ$, and the longitudinal polarization, $P_L$, in the Cabibbo favored decay $D_s^+\to ϕρ^{+}$, we have studied theoretical prediction within the context of factorization approximation invoking several form factors models. We were able to obtain agreement with experiment for both $Γ$ and $P_L$ by using experimentally measured values of the form factors $A_1^{D_sϕ}(0)$, $A_2^{D_sϕ}(0)$ and $V^{D_sϕ}(0)$ in the semi-leptonic decay $D_s^+\to ϕl^{+}ν_{l}$. We have also included in our calculation the effect of the final state interaction ($fsi$) by working with the partial waves amplitudes $S$, $P$ and $D$. Numerical calculation shows that the decay amplitude is dominated by $S$ wave, and that the polarization is sensitive to the interference between $S$ and $D$ waves. The range of the phase difference $δ_{SD} = δ_S - δ_D$ accommodated by experimental error in $P_L$ is large.