We examine $e^{+}{e}^{-}\to D_s^{-}{D}_s^{*+}$ and $D_s^{*-}{D}_s^{+}$ interactions at 4170 MeV using the CLEO-c detector in order to measure the decay constant $f_{D_s^{+}}$ with good precision. Previously our measurements were substantially higher than the most precise lattice based QCD calculation of $(241\pm 3)\mathrm{MeV}$. Here we use the $D_s^{+}\to {\ell }^{+}\nu$ channel, where the ${\ell }^{+}$ designates either a ${\mu }^{+}$ or a ${\tau }^{+}$, when the ${\tau }^{+}\to {\pi }^{+}\overline{\nu }$. Analyzing both modes independently, we determine $\mathcal{B}(D_s^{+}\to {\mu }^{+}\nu )=(0.565\pm 0.045\pm 0.017)\%$, and $\mathcal{B}(D_s^{+}\to {\tau }^{+}\nu )=(6.42\pm 0.81\pm 0.18)\%$. We also analyze them simultaneously to find an effective value of ${\mathcal{B}}^{\mathrm{eff}}(D_s^{+}\to {\mu }^{+}\nu )=(0.591\pm 0.037\pm 0.018)\%$ and $f_{D_s^{+}}=(263.3\pm 8.2\pm 3.9)\mathrm{MeV}$. Combining with the CLEO-c value determined independently using $D_s^{+}\to {\tau }^{+}\nu$, ${\tau }^{+}\to e^{+}\nu \overline{\nu }$ decays, we extract $f_{D_s^{+}}=(259.5\pm 6.6\pm 3.1)\mathrm{MeV}$. Combining with our previous determination of $\mathcal{B}(D^{+}\to {\mu }^{+}\nu )$, we extract the ratio $f_{D_s^{+}}/f_{D^{+}}=1.26\pm 0.06\pm 0.02$. No evidence is found for a $CP$ asymmetry between $\Gamma (D_s^{+}\to {\mu }^{+}\nu )$ and $\Gamma (D_s^{-}\to {\mu }^{-}\nu )$; specifically the fractional difference in rates is measured to be $(4.8\pm 6.1)\%$. Finally, we find $\mathcal{B}(D_s^{+}\to e^{+}\nu )<1.2\times {10}^{-4}$ at 90% confidence level.

• Received 12 January 2009

DOI:https://doi.org/10.1103/PhysRevD.79.052001