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Dark photon ($A'$) that couples to the standard model fermions via thekinetic mixing with photons and serves as a mediator of dark matter productioncould be observed in the high-energy electron scattering $e^- + Z ~\rightarrowe^- + Z + A'$ off nuclei followed by the $A' \to invisible $ decay. We haveperformed the exact, tree-level calculations of the $A'$ production crosssections and implemented them in the program for the full simulation of suchevents in the experiment NA64 at the CERN SPS. Using simulations results, westudy the missing energy signature for the bremsstrahlung $A' \rightarrow $invisible decay that permits the determination of the $\gamma-A'$ mixingstrength in a wide, from sub-MeV to sub-GeV, $A'$ mass range. We refine andexpand our earlier studies of this signature for discovering $A'$ by includingcorrections to the previously used calculations based on the improvedWeizsaker-Williams approximation, which turn out to be significant. We compareour cross sections values with the results from other calculations and find agood agreement between them. The possibility of future measurements withhigh-energy electron beams and the sensitivity to $A'$ are briefly discussed.

The exact tree-level calculation of the dark photon production in high-energy electron scattering at the CERN SPS