Search for Light Dark Matter with accelerator and direct detection experiments: comparison and complementarity of recent results

We discuss the most sensitive constraints on Light Dark Matter (LDM) from accelerator experiments NA64 and BaBar and compare it with recent results from direct searches at XENON1T, DAMIC-M, SuperCDMS, and DarkSide-50. We show that for the dark photon ($A'$) model with scalar LDM, NA64 gives more stringent bounds for $A'$ masses $m_{A'} \leq 0.15~GeV$ than direct searches. Moreover, for the case of Majorana LDM the damping DM velocity $v$ factor, $v^2 \sim O(10^{-6})$, for the elastic LDM electron(nucleon) cross section makes direct observation of Majorana LDM extremely challenging, while the absence of this suppression in the NA64 case gives an advantage to the experiment. The similar situation takes place for pseudo-Dirac LDM. The BaBar provides the most stringent bounds for $A'$ masses $m_{A'} \geq 0.35~GeV$. For scalar LDM the direct detection experiments give more stringent bounds at $m_{A'} \geq 0.35~GeV$ while for Majorana and pseudo-Dirac LDM case, the BaBar bounds are more stringent. The complementarity of the two approaches in searching for LDM is underlined.