Search for $D^0\to K^-ηe^+ν_e$, $D^+\to K_S^0 ηe^+ν_e$ and $D^+\to ηηe^+ν_e$ 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 $D^0\to K^-ηe^+ν_e$, $D^+\to K_S^0 ηe^+ν_e$ and $D^+\to ηηe^+ν_e$ for the first time. We present evidence for $D^0\to K^-ηe^+ν_e$ with a significance of $3.3σ$. The branching fraction of $D^0\to K^-ηe^+ν_e$ is measured to be $(0.84_{-0.34}^{+0.29}\pm0.22)\times 10^{-4}$. Here, the first uncertainties are statistical and the second ones are systematic. No significant signals are observed for the decays $D^+\to K_S^0 ηe^+ν_e$ and $D^+\to ηηe^+ν_e$ and we set the upper limits on their branching fractions.