Search for $D^0\to K^{-}\eta e^{+}{\nu }_e$, $D^{+}\to K_S^0\eta e^{+}{\nu }_e$ and $D^{+}\to \eta \eta e^{+}{\nu }_e$ decays

By analyzing $e^{+}{e}^{-}$ annihilation data corresponding to an integrated luminosity of 7.93 fb${}^{-1}$, collected at the center-of-mass energy of 3.773 GeV with the BESIII detector, we search for the semileptonic decays $D^0\to K^{-}\eta e^{+}{\nu }_e$, $D^{+}\to K_S^0\eta e^{+}{\nu }_e$ and $D^{+}\to \eta \eta e^{+}{\nu }_e$ for the first time. We present evidence for $D^0\to K^{-}\eta e^{+}{\nu }_e$ with a significance of $3.3\sigma$. The branching fraction of $D^0\to K^{-}\eta e^{+}{\nu }_e$ is measured to be $(0.84_{-0.34}^{+0.29}\pm 0.22)\times {10}^{-4}$. Here, the first uncertainties are statistical and the second ones are systematic. No significant signals are observed for the decays $D^{+}\to K_S^0\eta e^{+}{\nu }_e$ and $D^{+}\to \eta \eta e^{+}{\nu }_e$ and we set the upper limits on their branching fractions.