Improved Measurements of $D^+ \to ηe^+ν_e$ and $D^+ \to ημ^+ν_μ$

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 $D^+\to ηe^+ν_e$ and $D^+\to ημ^+ν_μ$ to be $(9.75\pm0.29\pm0.28)\times10^{-4}$ and $(9.08\pm0.35\pm0.23)\times10^{-4}$, 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 $f^η_+(0)$ and the modulus of the $c\to d$ Cabibbo-Kobayashi-Maskawa matrix element $|V_{cd}|$ to be $f^η_+(0)|V_{cd}|=0.078\pm0.002\pm0.001$. Taking the $|V_{cd}|$ value from the Standard Model global fit as input, we obtain $f^η_+(0)=0.345\pm0.008\pm0.003$. The ratio between the measured branching fractions of $D^+\toη^+μ^+ν_μ$ and $D^+\toηe^+ν_e$, is determined to be $0.93\pm0.05_{\rm stat.}\pm0.02_{\rm syst.}$, indicating no violation of lepton flavor universality.