Shedding Flavor on Dark via Freeze-in: $U(1)_{B-3L_i}$ Gauged Extensions

We consider a singlet fermionic dark matter (DM) $\chi$ in a gauged $U(1)_{B-3L_i}$ extension of the Standard Model (SM), with $i\in e\,,\mu\,,\tau$, and derive bounds on the allowed parameter space, considering its production via freeze-in mechanism. The DM communicates with the SM only through flavorful vector-portal $Z_\text{B3L}$ due to its non-trivial charge $x$ under $U(1)_{B-3L_{i}}$, which also guarantees the stability of the DM over the age of the Universe for $x\neq\{\pm 3/2,\pm 3\}$. Considering $Z_\text{B3L}$ to lie within the mass range of a few MeV up to a few GeV, we obtain constraints on the gauge coupling $g_\text{B3L}$ from the requirement of producing right relic abundance. Taking limits from various (present and future) experimental facilities, e.g., NuCal, NA64, FASER, SHiP into account, we show that the relic density allowed parameter space for the frozen in DM can be probed with $g_\text{B3L}\gtrsim 10^{-8}$ for both $m_\chi<m_\text{ZB3L}/2$ and $m_\chi\gtrsim m_\text{ZB3L}$, while $g_\text{B3L}\lesssim 10^{-8}$ remains mostly unconstrained. We also briefly comment on the implications of neutrino mass generation via Type-I seesaw and anomalous $(g-2)_\mu$ in context with $B-3L_\mu$ gauged symmetry.