First Observation of ${D}^{+}\to \eta {\mu }^{+}{\nu }_{\mu }$ and Measurement of Its Decay Dynamics

By analyzing a data sample corresponding to an integrated luminosity of $2.93\text{}\text{}{\mathrm{fb}}^{-1}$ collected at a center-of-mass energy of 3.773 GeV with the BESIII detector, we measure for the first time the absolute branching fraction of the ${D}^{+}\to \eta {\mu }^{+}{\nu }_{\mu }$ decay to be ${B}_{{D}^{+}\to \eta {\mu }^{+}{\nu }_{\mu }}=\left(10.4±1.{0}_{\mathrm{stat}}±0.{5}_{\mathrm{syst}}\right)×{10}^{-4}$. Using the world averaged value of ${B}_{{D}^{+}\to \eta {e}^{+}{\nu }_{e}}$, the ratio of the two branching fractions is determined to be ${B}_{{D}^{+}\to \eta {\mu }^{+}{\nu }_{\mu }}/{B}_{{D}^{+}\to \eta {e}^{+}{\nu }_{e}}=0.91±{0.13}_{\left(\mathrm{stat}+\mathrm{syst}\right)}$, which agrees with the theoretical expectation of lepton flavor universality within uncertainty. By studying the differential decay rates in five four-momentum transfer intervals, we obtain the product of the hadronic form factor ${f}_{+}^{\eta }\left(0\right)$ and the $c\to d$ Cabibbo-Kobayashi-Maskawa matrix element $|{V}_{cd}|$ to be ${f}_{+}^{\eta }\left(0\right)|{V}_{cd}|=0.087±0.00{8}_{\mathrm{stat}}±0.00{2}_{\mathrm{syst}}$. Taking the input of $|{V}_{cd}|$ from the global fit in the standard model, we determine ${f}_{+}^{\eta }\left(0\right)=0.39±0.0{4}_{\mathrm{stat}}±0.0{1}_{\mathrm{syst}}$. On the other hand, using the value of ${f}_{+}^{\eta }\left(0\right)$ calculated in theory, we find $|{V}_{cd}|=\phantom{\rule{0ex}{0ex}}0.242±0.02{2}_{\mathrm{stat}}±0.00{6}_{\mathrm{syst}}±0.03{3}_{\text{theory}}$.