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We investigate a $U(1)_{B-L}$ gauge extension of the Standard Model (SM)where the gauge boson mass is generated by the Stueckelberg mechanism. Threeright-handed neutrinos are added to cancel the gauge anomaly and hence theneutrino masses can be explained. A new Dirac fermion could be a WIMP darkmatter whose interaction with the SM sector is mediated by the new gauge boson.Assuming the perturbativity of the gauge coupling up to the Planck scale, wefind that only the resonance region is feasible for the dark matter abundance.After applying the $\Delta N_{eff}$ constraints from the current Planckexperiment, the collider search constraints as well as the dark matter directdetection limits, we observe that the $B-L$ charge of dark matter satisfies$|Q_{\chi}|>0.11$. Such a scenario might be probed conclusively by theprojected CMB-S4 experiment, assuming the right-handed neutrinos arethermalized with the SM sector in the early universe.

Dirac dark matter in $U(1)_{B-L}$ with Stueckelberg mechanism