Testing Dark Matter with the Anomalous Magnetic Moment in Quantum Electrodynamics Model

We consider a model of dark quantum electrodynamics which is coupled with a visible photon through a kinetic mixing term. After checking of consistency properties, we compute the $g_w-2$, where $g_w$ is the gyromagnetic factor for a dark fermion. The $g_w-2$ and $g-2$ of quantum electrodynamics are related by the kinetic mixing factor. We analyse the $g_w-2$ in terms of the ratio $\kappa =m_\gamma/m_\chi$where$m_\gamma$and$m_\chi$ are the masses of the dark photon and the dark fermion and we discuss how light and heavy fermions become very different for $m_{\gamma }\le {10}^{-5}$ eV. This analysis is also carried out using different available data.