In this work we have considered a minimal extension of Standard Model by a local $U(1)$ gauge group in order to accommodate a stable (fermionic) Dark Matter (DM) candidate. We have focussed on parameter regions where DM possesses adequate self interaction, owing to the presence of a light scalar mediator (the dark Higgs), alleviating some of the tensions in the small-scale structures. We have studied the scenario in the light of a variety of data, mostly from dark matter direct searches, collider searches and flavour physics experiments, with an attempt to constrain the interactions of the standard model (SM) particles with the ones in the Dark Sector (DS). Assuming a small gauge kinetic mixing parameter, we find that for rather heavy DM %$gtrsim mathcal{O}(1-10),, {rm GeV}$%, the most stringent bound on the mixing angle of the Dark Higgs with the SM Higgs boson comes from dark matter direct detection experiments, while for lighter DM, LHC constraints become more relevant. Note that, due to the presence of very light mediators the usual realisation of direct detection constraints in terms of momentum independent cross sections had to be reevaluated for our scenario. In addition, we find that the smallness of the relevant portal couplings, as dictated by data, critically suppress the viability of DM production by the standard freeze-out mechanism in such simplified scenarios. In particular, the viable DM masses are $lesssim mathcal{O}(2)$ GeV $i.e.$ in the regions where direct detection limits tend to become weak. For heavier DM with large self-interactions, we hence conclude that non-thermal production mechanisms are favoured. Lastly, future collider reach of such a simplified scenario has also been studied in detail.
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