$\omega$-$\varphi$ mixing and weak annihilation in $D_s$ decays

The mixing between nonstrange and strange quark wavefunctions in the $\omega$ and $\varphi$ mesons leads to a small predicted branching ratio $\b(D_s^+ \to \omega e^+ \nu_e) = {\cal O}(10^{-4})(\delta/3.34^\circ)^2$,where$\delta$ is the mixing angle. The value $\delta =-{3.34}^{\circ }$ is obtained in a mass-independent analysis, while a mass-dependent analysis gives $\delta = -0.45^\circ$at$m(\omega)$and$-4.64^\circ$at$m(\phi)$. Measurement of this branching ratio thus can tell whether the decay is dominated by $\varphi$–$\omega$ mixing, or additional nonperturbative processes commonly known as “weak annihilation” (WA) contribute. The role of WA in the decay $D_s^+ \to \omega \pi^+$anditspossibleuseinestimatingWAeffectsin$D_s^+ \to \omega e^+ \nu_e$arealsodiscussed\mathrm{.}AssumingthatthedynamicsofWAin$D_s^+ \to \omega \pi^+$issimilarin$D_s^+ \to \omega e^+ \nu_e$weestimate$\b(D_s^+ \to \omega e^+ \nu_e) = (1.3 \pm 0.5)\times 10^{-3}$.