Scale Invariant FIMP Miracle

We study the freeze-in production of vector dark matter (DM) in a classically scale invariant theory, where the Standard Model (SM) is augmented with an abelian $U(1)_X$ gauge symmetry that is spontaneously broken due to the non-zero vacuum expectation value (VEV) of a scalar charged under the $U(1)_X$. Generating the SM Higgs mass at 1-loop level, it leaves only two parameters in the dark sector, namely, the DM mass $m_X$ and the gauge coupling $g_X$ as independent, and supplement with a naturally light dark scalar particle. We show, for $g_Xsimmathcal{O}left(10^{-5}right)$, it is possible to produce the DM X out-of-equilibrium in the early Universe, satisfying the observed relic abundance for $m_Xsimmathcal{O}left(text{TeV}right)$, which in turn also determines the scalar mixing angle $sin thetasimmathcal{O}left(10^{-5}right)$. The presence of such naturally light scalar mediator with tiny mixing with the SM, opens up the possibility for the model to be explored in direct search experiment, which otherwise is insensitive to standard freeze-in scenarios. Moreover we show that even with such feeble couplings, necessary for the DM freeze-in, the scenario is testable in several light dark sector searches (e.g., in DUNE and in FASER-II), satisfying constraints from the observed relic abundance as well as big bang nucleosynthesis (BBN). Particularly, we find, regions of the parameter space with $m_X$ $gtrsim 1.8$ TeV are insensitive to direct detection probes but still can become accessible in lifetime frontier searches, courtesy to the underlying scale invariance of the theory.