Freeze-in Warm Dark Matter via Dimension-6 Operators in 3-3-1 Models

We propose a natural resolution to the fine-tuning problem inherent in the freeze-in dark matter paradigm by embedding a sterile singlet within a 3-3-1 electroweak extension. By imposing an exact $Z_{13}$ discrete gauge symmetry, we formally suppress all low-dimensional portals to ensure that the dark sector communicates with the Standard Model (SM) exclusively through a dimension-six operator. This theoretical structure allows the extraordinarily small coupling required for dark matter production to emerge naturally from the profound hierarchy between the electroweak scale and the ultra-high Peccei-Quinn symmetry breaking scale. Detailed numerical integration of the Boltzmann equations demonstrates that the sterile singlet can be produced via the infrared freeze-in mechanism to match the observed relic abundance of $\Omega_S h^2 = 0.12$. The resulting keV-scale warm dark matter candidate remains consistent with stringent Lyman-alpha forest constraints while offering a viable solution to galactic-scale discrepancies such as the cusp-core and missing satellites problems. Ultimately, this framework provides a self-consistent unification of dark matter genesis and the strong CP solution that is completely independent of ad hoc parameter adjustments.