The rare decays $B_u^{+}\to {\pi }^{+}{\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$, ${\rho }^{+}{\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$ and $B_d^0\to {\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$ in the R-parity violating supersymmetry

We study the rare decays $B_u^{+}\to {\pi }^{+}{\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$, ${\rho }^{+}{\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$ and $B_d^0\to {\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}(\mathrm{\ell }=e,\mu )$ in the R-parity violating supersymmetric standard model. From the latest upper limits of $\mathcal{B}(B_u^{+}\to {\pi }^{+}{\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-})$ and $\mathcal{B}(B_d^0\to {\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-})$, we have derived new upper bounds on the relevant R-parity violating couplings products, which are stronger than the existing ones. Using the constrained parameter space, we present the R-parity violating effects on the branching ratios and the forward-backward asymmetries of these decays. We find that $\mathcal{B}(B_d^0\to {\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-})$ and $\mathcal{B}(B_u^{+}\to {\rho }^{+}{\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-})$ could be enhanced several orders by the R-parity violating sneutrino and squark exchanges, respectively. The R-parity violating effects on the dilepton invariant mass spectra of $B^+_u\to \pi^+\ell^+\ell^-$,$\rho^+\ell^+\ell^-$ and the normalized forward-backward asymmetry ${\mathcal{A}}_{FB}(B_u^{+}\to {\pi }^{+}{\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-})$ and ${\mathcal{A}}_{FB}(B_u^{+}\to {\rho }^{+}{\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-})$ are studied in detail. Our results could be used to probe the R-parity violating effects and will correlate with searches for the direct R-parity violating signals at the future experiments.