Measurement of the absolute branching fractions for purely leptonic ${D}_{s}^{+}$ decays

We report new measurements of the branching fraction $B\left({D}_{s}^{+}\to {\ell }^{+}\nu \right)$, where ${\ell }^{+}$ is either ${\mu }^{+}$ or ${\tau }^{+}\left(\to {\pi }^{+}{\overline{\nu }}_{\tau }\right)$, based on $6.32\text{}\text{}{\mathrm{fb}}^{-1}$ of electron-positron annihilation data collected by the BESIII experiment at six center-of-mass energy points between 4.178 and 4.226 GeV. Simultaneously floating the ${D}_{s}^{+}\to {\mu }^{+}{\nu }_{\mu }$ and ${D}_{s}^{+}\to {\tau }^{+}{\nu }_{\tau }$ components yields $B\left({D}_{s}^{+}\to {\tau }^{+}{\nu }_{\tau }\right)=\left(5.21±0.25±0.17\right)×{10}^{-2}$, $B\left({D}_{s}^{+}\to {\mu }^{+}{\nu }_{\mu }\right)=\phantom{\rule{0ex}{0ex}}\left(5.35±0.13±0.16\right)×{10}^{-3}$, and the ratio of decay widths $R=\frac{\Gamma \left({D}_{s}^{+}\to {\tau }^{+}{\nu }_{\tau }\right)}{\Gamma \left({D}_{s}^{+}\to {\mu }^{+}{\nu }_{\mu }\right)}=9.7{3}_{-0.58}^{+0.61}±0.36$, where the first uncertainties are statistical and the second systematic. No evidence of $CP$ asymmetry is observed in the decay rates ${D}_{s}^{±}\to {\mu }^{±}{\nu }_{\mu }$ and ${D}_{s}^{±}\to {\tau }^{±}{\nu }_{\tau }$: ${A}_{CP}\left({\mu }^{±}\nu \right)=\left(-1.2±2.5±1.0\right)%$ and ${A}_{CP}\left({\tau }^{±}\nu \right)=\left(+2.9±\phantom{\rule{0ex}{0ex}}4.8±1.0\right)%$. Constraining our measurement to the Standard Model expectation of lepton universality ($R=9.75$), we find the more precise results $B\left({D}_{s}^{+}\to {\tau }^{+}{\nu }_{\tau }\right)=\left(5.22±0.10±0.14\right)×{10}^{-2}$ and ${A}_{CP}\left({\tau }^{±}{\nu }_{\tau }\right)=\left(-0.1±1.9±1.0\right)%$. Combining our results with inputs external to our analysis, we determine the $c\to \overline{s}$ quark mixing matrix element, ${D}_{s}^{+}$ decay constant, and ratio of the decay constants to be $|{V}_{cs}|=0.973±0.009±0.014$, ${f}_{{D}_{s}^{+}}=249.9±2.4±3.5\text{}\text{}\mathrm{MeV}$, and ${f}_{{D}_{s}^{+}}/{f}_{{D}^{+}}=1.232±0.035$, respectively.