Dimensionality expansion for the dirty-boson problem

We apply the double-dimensionality expansion of Dorogovstev to derive renormalization-group recursion relations for a Bose fluid in a random external potential. We find a nontrivial zero-temperature fixed point. The onset of mean-field behavior for dimensions d>$d_c$=4 is unconventional, yielding discontinuous exponents, consistent with previous scaling arguments. Including positive temperatures, we give a clear picture of various crossover regimes, depending on the strength of the disorder and the boson density. This answers previous questions about the behavior of ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ adsorbed in Vycor glass.