Measurements of Absolute Branching Fractions of Fourteen Exclusive Hadronic $D$ Decays to $eta$

Using $2.93,rm fb^{-1}$ of $e^+e^-$ collision data taken at a center-of-mass energy of 3.773,GeV with the BESIII detector, we report the first measurements of the absolute branching fractions of fourteen hadronic $D^{0(+)}$ decays to exclusive final states with an $eta$, e.g., $D^0to K^-pi^+eta$, $K^0_Spi^0eta$, $K^+K^-eta$, $K^0_SK^0_Seta$, $K^-pi^+pi^0eta$, $K^0_Spi^+pi^-eta$, $K^0_Spi^0pi^0eta$, and $pi^+pi^-pi^0eta$; $D^+to K^0_Spi^+eta$, $K^0_SK^+eta$, $K^-pi^+pi^+eta$, $K^0_Spi^+pi^0eta$, $pi^+pi^+pi^-eta$, and $pi^+pi^0pi^0eta$. Among these decays, the $D^0to K^-pi^+eta$ and $D^+to K^0_Spi^+eta$ decays have the largest branching fractions, which are $mathcal{B} (D^0to K^-pi^+eta )=(1.853pm0.025_{rm stat}pm0.031_{rm syst})%$ and $mathcal{B}(D^+to K^0_Spi^+eta)=(1.309pm0.037_{rm stat}pm0.031_{rm syst})%$, respectively. We also determine the $CP$ asymmetries for the six decays with highest event yields. No evidence of $CP$ violation is found.