Decays $D_{(s)}^{+}\to \pi (K{)}^{+}{\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$ and $D^0\to {\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$ in the MSSM with and without R-parity

We study the rare decays $D^{+}\to {\pi }^{+}{\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$, $D^+s\to K^{+}\ell^+\ell^-$and$D^0\to \ell^+\ell^-(\ell=e,\mu)$ in the minimal supersymmetic standard model with and without R-parity. Using the strong constraints on relevant supersymmetric parameters from $D^0-{\bar{D}}^0$ mixing and $K^{+}\to {\pi }^{+}\nu \bar{\nu }$ decay, we examine constrained supersymmetry contributions to relevant branching ratios, direct CP violations and ratios of $D_{(s)}^{+}\to \pi (K{)}^{+}{\mu }^{+}{\mu }^{-}$ and $D_{(s)}^{+}\to \pi (K{)}^{+}{e}^{+}{e}^{-}$ decay rates. We find that both R-parity conserving LR as well as RL mass insertions and R-parity violating squark exchange couplings have huge effects on the direct CP violations of $D_{(s)}^{+}\to \pi (K{)}^{+}{\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$, moreover, the constrained LR and RL mass insertions still have obvious effects on the ratios of $D_{(s)}^{+}\to \pi (K{)}^{+}{\mu }^{+}{\mu }^{-}$ and $D_{(s)}^{+}\to \pi (K{)}^{+}{e}^{+}{e}^{-}$ decay rates. The direct CP asymmetries and the ratios of $D^+{(s)}\to \pi(K)^{+}\mu^+\mu^-$and$D^+{(s)}\to \pi(K)^{+}e^+e^-$ decay rates are very sensitive to both moduli and phases of relevant supersymmetric parameters. In addition, the differential direct CP asymmetries of $D^+{(s)}\to \pi(K)^{+}\ell^+\ell^-$ are studied in detail.