Measurement of the branching fraction of leptonic decay ${D}_{s}^{+}\to {\tau }^{+}{\nu }_{\tau }$ via ${\tau }^{+}\to {\pi }^{+}{\pi }^{0}{\overline{\nu }}_{\tau }$

By analyzing $6.32\text{}\text{}{\mathrm{fb}}^{-1}$ of ${e}^{+}{e}^{-}$ annihilation data collected at the center-of-mass energies between 4.178 and 4.226 GeV with the BESIII detector, we determine the branching fraction of the leptonic decay ${D}_{s}^{+}\to {\tau }^{+}{\nu }_{\tau }$, with ${\tau }^{+}\to {\pi }^{+}{\pi }^{0}{\overline{\nu }}_{\tau }$, to be ${B}_{{D}_{s}^{+}\to {\tau }^{+}{\nu }_{\tau }}=\left(5.29±{0.25}_{\mathrm{stat}}±{0.20}_{\mathrm{syst}}\right)%$. We estimate the product of the Cabibbo-Kobayashi-Maskawa matrix element $|{V}_{cs}|$ and the ${D}_{s}^{+}$ decay constant ${f}_{{D}_{s}^{+}}$ to be ${f}_{{D}_{s}^{+}}|{V}_{cs}|=\left(244.8±{5.8}_{\mathrm{stat}}±{4.8}_{\mathrm{syst}}\right)\text{}\text{}\mathrm{MeV}$, using the known values of the ${\tau }^{+}$ and ${D}_{s}^{+}$ masses as well as the ${D}_{s}^{+}$ lifetime, together with our branching fraction measurement. Combining the value of $|{V}_{cs}|$ obtained from a global fit in the standard model and ${f}_{{D}_{s}^{+}}$ from lattice quantum chromodynamics, we obtain ${f}_{{D}_{s}^{+}}=\left(251.6±{5.9}_{\mathrm{stat}}±{4.9}_{\mathrm{syst}}\right)\text{}\text{}\mathrm{MeV}$ and $|{V}_{cs}|=0.980±{0.023}_{\mathrm{stat}}±{0.019}_{\mathrm{syst}}$. Using the branching fraction of ${B}_{{D}_{s}^{+}\to {\mu }^{+}{\nu }_{\mu }}=\left(5.35±0.21\right)×{10}^{-3}$, we obtain the ratio of the branching fractions ${B}_{{D}_{s}^{+}\to {\tau }^{+}{\nu }_{\tau }}/{B}_{{D}_{s}^{+}\to {\mu }^{+}{\nu }_{\mu }}=9.89±0.71$, which is consistent with the standard model prediction of lepton flavor universality.