Implication of the Weizsacker-Williams approximation for the dark matter mediator production

The simplified connection between Standard Model (SM) particles and light dark matter (LDM) can be introduced via spin-0 and spin-1 lepton-specific mediators. Moreover, in a mediator mass range from sub-MeV to sub-GeV, fixed-target facilities such as NA64$e$, LDMX, NA64$\mu$, and M$^3$ can potentially probe such particles of the hidden sector via missing energy signatures that are described by the bremsstrahlung-like process involving leptons. We compare the Weizsaker-Williams (WW) approximation and the exact tree-level (ETL) approach for the bremsstrahlung-like mediator production cross section by choosing various parameters of the fixed-target experiments. We show that the relative difference between the total cross sections calculated in the WW and ETL approximation varies from $\mathcal{O}(1)~\%$ to $\mathcal{O}(10)~\%$ for a muon mode and from $\mathcal{O}(-20)~\%$ to $\mathcal{O}(80)~\%$ for an electron mode. We argue that the main difference between two approaches for electron beam mode arises from peak forward region of the cross section. We also discuss the impact of parametrization of nuclear and atomic elastic form-factors on the total cross section. In particular, we show that the employing different form-factor parametrization can lead to a uncertainty in the cross section at the level of $\lesssim \mathcal{O}(10)~\%$. That study may be helpful for the lepton fixed-target experiments that examine the bremsstrahlung-like production of the DM mediators.