Lifetime of the dark $Z$ boson

The mediator particle between the Standard Model sector and a hidden sector might have a long lifetime to show the observable displaced vertices in experiments. Considering a fermionic dark matter model in which the hidden sector is connected to the Standard Model by an additional Higgs doublet field, the mediator dark $Z$ boson may live long enough. We explore the possibility to observe the displaced vertices of the long-lived dark $Z$ boson at the CERN LHC and at the proposed SHiP experiment. We find that the ATLAS and CMS searches for the long-lived dark $Z$ boson can probe the mass range $7 < m_{Z'} < 150~{\rm MeV}$ with 150 fb$^{-1}$ integrated luminosity at the LHC run 3, and the SHiP experiment will probe $2 m_e < m_{Z'} < 15~{\rm MeV}$ range with $6 \times 10^{20}$ protons on target in total 15 years. The dark matter phenomenology is also discussed in the region where such a long-lived mediator is detectable.