Observation of ${D}_{s}^{+}\to {\eta }^{\prime }{\mu }^{+}{\nu }_{\mu }$, Precision Test of Lepton Flavor Universality with ${D}_{s}^{+}\to {\eta }^{\left(\prime \right)}{l}^{+}{\nu }_{l}$, and First Measurements of ${D}_{s}^{+}\to {\eta }^{\left(\prime \right)}{\mu }^{+}{\nu }_{\mu }$ Decay Dynamics

By analyzing $7.33\text{}\text{}{\mathrm{fb}}^{-1}$ of ${e}^{+}{e}^{-}$ annihilation data collected at center-of-mass energies between 4.128 and 4.226 GeV with the BESIII detector, we report the observation of the semileptonic decay ${D}_{s}^{+}\to {\eta }^{\prime }{\mu }^{+}{\nu }_{\mu }$, with a statistical significance larger than $10\sigma $, and the measurements of the ${D}_{s}^{+}\to \eta {\mu }^{+}{\nu }_{\mu }$ and ${D}_{s}^{+}\to {\eta }^{\prime }{\mu }^{+}{\nu }_{\mu }$ decay dynamics for the first time. The branching fractions of ${D}_{s}^{+}\to \eta {\mu }^{+}{\nu }_{\mu }$ and ${D}_{s}^{+}\to {\eta }^{\prime }{\mu }^{+}{\nu }_{\mu }$ are determined to be $\left(2.235±0.05{1}_{\mathrm{stat}}±0.05{2}_{\mathrm{syst}}\right)%$ and $\left(0.801±0.05{5}_{\mathrm{stat}}±0.02{8}_{\mathrm{syst}}\right)%$, respectively, with precision improved by factors of 6.0 and 6.6 compared to the previous best measurements. Combined with the results for the decays ${D}_{s}^{+}\to \eta {e}^{+}{\nu }_{e}$ and ${D}_{s}^{+}\to {\eta }^{\prime }{e}^{+}{\nu }_{e}$, the ratios of the decay widths are examined both inclusively and in several ${\ell }^{+}{\nu }_{\ell }$ four-momentum transfer ranges. No evidence for lepton flavor universality violation is found within the current statistics. The products of the hadronic form factors ${f}_{+,0}^{{\eta }^{\left(\prime \right)}}\left(0\right)$ and the $c\to s$ Cabibbo-Kobayashi-Maskawa matrix element $|{V}_{cs}|$ are determined. The results based on the two-parameter series expansion are ${f}_{+,0}^{\eta }\left(0\right)|{V}_{cs}|=0.452±0.01{0}_{\mathrm{stat}}±0.00{7}_{\mathrm{syst}}$ and ${f}_{+,0}^{{\eta }^{\prime }}\left(0\right)|{V}_{cs}|=0.504±0.03{7}_{\mathrm{stat}}±0.01{2}_{\mathrm{syst}}$, which help to constrain present models on ${f}_{+,0}^{{\eta }^{\left(\prime \right)}}\left(0\right)$. The forward-backward asymmetries are determined to be $⟨{A}_{\mathrm{FB}}^{\eta }⟩=-0.059±0.03{1}_{\mathrm{stat}}±0.00{5}_{\mathrm{syst}}$ and $⟨{A}_{\mathrm{FB}}^{{\eta }^{\prime }}⟩=-0.064±0.07{9}_{\mathrm{stat}}±0.00{6}_{\mathrm{syst}}$ for the first time, which are consistent with the theoretical calculation.