By analyzing a data sample corresponding to an integrated luminosity of $2.93{\mathrm{fb}}^{-1}$ 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 ${\mathcal{B}}_{D^{+}\to \eta {\mu }^{+}{\nu }_{\mu }}=(10.4\pm 1.0_{\mathrm{stat}}\pm 0.5_{\mathrm{syst}})\times {10}^{-4}$. Using the world averaged value of ${\mathcal{B}}_{D^{+}\to \eta e^{+}{\nu }_e}$, the ratio of the two branching fractions is determined to be ${\mathcal{B}}_{D^{+}\to \eta {\mu }^{+}{\nu }_{\mu }}/{\mathcal{B}}_{D^{+}\to \eta e^{+}{\nu }_e}=0.91\pm {0.13}_{(\mathrm{stat}+\mathrm{syst})}$, 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 $f_{+}^{\eta }(0)$ and the $c\to d$ Cabibbo-Kobayashi-Maskawa matrix element $|V_{cd}|$ to be $f_{+}^{\eta }(0)|V_{cd}|=0.087\pm 0.008_{\mathrm{stat}}\pm 0.002_{\mathrm{syst}}$. Taking the input of $|V_{cd}|$ from the global fit in the standard model, we determine $f_{+}^{\eta }(0)=0.39\pm 0.04_{\mathrm{stat}}\pm 0.01_{\mathrm{syst}}$. On the other hand, using the value of $f_{+}^{\eta }(0)$ calculated in theory, we find $|V_{cd}|=0.242\pm 0.022_{\mathrm{stat}}\pm 0.006_{\mathrm{syst}}\pm 0.033_{\text{theory}}$.

• Received 26 March 2020
• Accepted 22 May 2020

DOI:https://doi.org/10.1103/PhysRevLett.124.231801

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Published by the American Physical Society