Improved Measurements of $D^{+}\to \eta e^{+}{\nu }_e$ and $D^{+}\to \eta {\mu }^{+}{\nu }_{\mu }$

Using 20.3 fb${}^{-1}$ of $e^{+}{e}^{-}$ collision data collected at the center-of-mass energy of 3.773 GeV with the BESIII detector, we measure the branching fractions of $D^{+}\to \eta e^{+}{\nu }_e$ and $D^{+}\to \eta {\mu }^{+}{\nu }_{\mu }$ to be $(9.75\pm 0.29\pm 0.28)\times {10}^{-4}$ and $(9.08\pm 0.35\pm 0.23)\times {10}^{-4}$, where the first and second uncertainties are statistical and systematic, respectively. From a simultaneous fit to their partial decay rates, we determine the product of the hadronic form factor $f_{+}^{\eta }(0)$ and the modulus of the $c\to d$ Cabibbo-Kobayashi-Maskawa matrix element $\mathrm{\mid }{V}_{cd}\mathrm{\mid }$ to be $f_{+}^{\eta }(0)\mathrm{\mid }{V}_{cd}\mathrm{\mid }=0.078\pm 0.002\pm 0.001$. Taking the $\mathrm{\mid }{V}_{cd}\mathrm{\mid }$ value from the Standard Model global fit as input, we obtain $f_{+}^{\eta }(0)=0.345\pm 0.008\pm 0.003$. The ratio between the measured branching fractions of $D^{+}\to {\eta }^{+}{\mu }^{+}{\nu }_{\mu }$ and $D^{+}\to \eta e^{+}{\nu }_e$, is determined to be $0.93\pm {0.05}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.}}\pm {0.02}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{.}}$, indicating no violation of lepton flavor universality.