Improved measurements of $D^0\to K^{-}{\mathrm{\ell }}^{+}{\nu }_{\mathrm{\ell }}$ and $D^{+}\to {\bar{K}}^0{\mathrm{\ell }}^{+}{\nu }_{\mathrm{\ell }}$

Using 7.93 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 absolute branching fractions of $D^0\to K^{-}{e}^{+}{\nu }_e$, $D^0\to K^{-}{\mu }^{+}{\nu }_{\mu }$, $D^{+}\to {\bar{K}}^0{e}^{+}{\nu }_e$, and $D^{+}\to {\bar{K}}^0{\mu }^{+}{\nu }_{\mu }$ to be $(3.521\pm {0.009}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.}}\pm {0.016}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{.}})\mathrm{\%}$, $(3.419\pm {0.011}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.}}\pm {0.016}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{.}})\mathrm{\%}$, $(8.864\pm {0.039}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.}}\pm {0.082}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{.}})\mathrm{\%}$, and $(8.665\pm {0.046}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.}}\pm {0.084}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{.}})\mathrm{\%}$, respectively. By performing a simultaneous fit to the partial decay rates of these four decays, the product of the hadronic form factor $f_{+}^K(0)$ and the modulus of the $c\to s$ CKM matrix element $\mathrm{\mid }{V}_{cs}\mathrm{\mid }$ is determined to be $f_{+}^K(0)\mathrm{\mid }{V}_{cs}\mathrm{\mid }=0.7171\pm {0.0011}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.}}\pm {0.0013}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{.}}$. Taking the value of $\mathrm{\mid }{V}_{cs}\mathrm{\mid }=0.97349\pm 0.00016$ from the standard model global fit or that of $f_{+}^K(0)=0.7452\pm 0.0031$ from the LQCD calculation as input, we derive the results $f_{+}^K(0)=0.7366\pm {0.0011}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.}}\pm {0.0013}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{.}}$ and $\mathrm{\mid }{V}_{cs}\mathrm{\mid }=0.9623\pm {0.0015}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.}}\pm {0.0017}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{.}}\pm {0.0040}_{\mathrm{L}\mathrm{Q}\mathrm{C}\mathrm{D}}$.