Decaying Vector Dark Matter as an Explanation for the 3.5 keV Line from Galaxy Clusters

We present a Vector Dark Matter (VDM) model that explains the 3.5 keV line recently observed in the XMM-Newton observatory data from galaxy clusters. In this model, dark matter is composed of two vector bosons, $V$ and $V^\prime$, which couple to the photon through an effective generalized Chern-Simons coupling, $g_V$. $V^\prime$ is slightly heavier than $V$ with a mass splitting $m_{V^\prime}-m_V\simeq 3.5$~keV. The decay of $V^\prime$ to $V$ and a photon gives rise to the 3.5~keV line. The production of $V$ and $V^\prime$ takes place in the early universe within the freeze-in framework through the effective $g_V$ coupling when $m_{V^\prime}<T<\Lambda$, $\Lambda$ being the cut-off above which the effective $g_V$ coupling is not valid. We introduce a high energy model that gives rise to the $g_V$ coupling at low energies. To do this, $V$ and $V^\prime$ are promoted to gauge bosons of spontaneously broken new $U(1)_V$ and $U(1)_{V^\prime}$ gauge symmetries, respectively. The high energy sector includes milli-charged chiral fermions that lead to the $g_V$ coupling at low energy via triangle diagrams.