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

We study the rare decays $D^+\to \pi^{+}\ell^+\ell^-$, $D^+_s\to K^{+}\ell^+\ell^-$ 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 $D^+_{(s)}\to \pi(K)^{+}\ell^+\ell^-$, 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.