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The absence of a missing moment inertia in clean solid $^4$He suggests thatthe minimal experimentally relevant model is one in which disorder inducessuperfluidity in a bosonic lattice. To this end, we explore the relevance ofthe disordered Bose-Hubbard model in this context. We posit that a clean array$^4$He atoms is a self-generated Mott insulator, that is, the $^4$He atomsconstitute the lattice as well as the `charge carriers'. With this assumption,we are able to interpret the textbook defect-driven supersolids as excitationsof either the lower or upper Hubbard bands. In the experiments at hand,disorder induces a closing of the Mott gap through the generation of mid-gaplocalized states at the chemical potential. Depending on the magnitude of thedisorder, we find that the destruction of the Mott state takes place for$d+z>4$ either through a Bose glass phase (strong disorder) or through a directtransition to a superfluid (weak disorder). For $d+z<4$, disorder is alwaysrelevant. The critical value of the disorder that separates these two regimesis shown to be a function of the boson filling, interaction and the momentumcut off. We apply our work to the experimentally observed enhancement $^3$Heimpurities has on the onset temperature for the missing moment of inertia. Wefind quantitative agreement with experimental trends.Comment: 9 pages, 5 figures: Extended version of previous paper in which the  pase diagram for the disordered Bose-Hubbard model is computed using  mean-field theory and one-loop RG. The criterion for the Bose glass is  derived explicitly. (a few typos are corrected

physical review. b, condensed matter and materials physics

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Minimal model for disorder-induced missing moment of inertia in solidH4e