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The dark photon is widely predicted in many new physics beyond the StandardModel. In this work, we propose to utilize the polarized lepton cross sectionin neutral current DIS process at the HERA and upcoming Electron-Ion Collider(EIC) to probe the properties of the dark photon. It shows that we canconstrain the mixing parameter $\epsilon<0.02$ when the dark photon mass$m_{A_D}<10 ~{\rm GeV}$ at the HERA with polarized lepton beam and this boundis comparable to the limit from the unpolarized HERA data. With the help ofhigh integrated luminosity and high electron beam polarization of the EIC, theupper limit of the $\epsilon$ could be further improved. Depending on theassumption of the systematic error of the cross section measurements, we obtain$\epsilon<0.01\sim 0.02$ for the mass region $m_{A_D}<10~{\rm GeV}$ under theintegrated luminosity of $300~{\rm fb}^{-1}$. We also discuss the complementaryroles between the cross section and the single-spin asymmetry measurements inprobing the dark photon.

Probing the dark photon via polarized DIS scattering at the HERA and EIC