Search for rare or forbidden decays of the $D^0$ meson

We present a search for nine lepton-number-violating and three lepton-flavor-violating neutral charm decays of the type $D^0\rightarrow h^{\prime -} h^{-}\ell^{\prime +} \ell^{+}$and$D^0\rightarrow h^{\prime -} h^{+}\ell^{\prime\pm} \ell^{\mp}$,where$h$and$h^{\prime}$representa$K$or$\pi$mesonand$\ell$and$\ell^{\prime}$ an electron or muon. The analysis is based on $468$ fb${}^{-1}$ of $e^{+}{e}^{-}$ annihilation data collected at or close to the $Y(4S)$ resonance with the BaBar detector at the SLAC National Accelerator Laboratory. No significant signal is observed for any of the twelve modes and we establish 90% confidence level upper limits on the branching fractions in the range $(1.0-30.6)\times {10}^{-7}$. The limits are between one and three orders of magnitude times more stringent than previous measurements.