Measurement of the branching fraction of $D_s^{+}\to {\tau }^{+}{\nu }_{\tau }$ via ${\tau }^{+}\to {\mu }^{+}{\nu }_{\mu }\overline{{\nu }_{\tau }}$

Utilizing $7.33{\mathrm{f}\mathrm{b}}^{-1}$ of $e^{+}{e}^{-}$ collision data taken at the center-of-mass energies of 4.128, 4.157, 4.178, 4.189, 4.199, 4.209, 4.219, and 4.226 GeV with the BESIII detector, the branching fraction of the leptonic decay $D_s^{+}\to {\tau }^{+}{\nu }_{\tau }$ via ${\tau }^{+}\to {\mu }^{+}{\nu }_{\mu }{\bar{\nu }}_{\tau }$ is measured to be ${\mathcal{B}}_{D_s^{+}\to {\tau }^{+}{\nu }_{\tau }}=(5.34\pm {0.16}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}\pm {0.10}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}})\mathrm{\%}$. Combining this branching fraction with the world averages of the measurements of the masses of ${\tau }^{+}$ and $D_s^{+}$ as well as the lifetime of $D_s^{+}$, we extract the product of the decay constant of $D_s^{+}$ and the $c\to s$ Cabibbo-Kobayashi-Maskawa matrix element to be $f_{D_s^{+}}\mathrm{\mid }{V}_{cs}\mathrm{\mid }=(246.2\pm {3.7}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}\pm {2.5}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}})\mathrm{M}\mathrm{e}\mathrm{V}$. Taking $\mathrm{\mid }{V}_{cs}\mathrm{\mid }$ from a global fit in the standard model we obtain $f_{D_s^{+}}=(252.7\pm {3.8}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}\pm {2.6}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}})$ MeV. Conversely, taking $f_{D_s^{+}}$ from lattice quantum chromodynamics calculations, we obtain $\mathrm{\mid }{V}_{cs}\mathrm{\mid }=0.984\pm {0.015}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}\pm {0.010}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}}$.