Revisiting the dynamical exponent equality $z=d$ for the dirty boson problem

It is shown that previous arguments leading to the equality $z=d$ ($d$ being the spatial dimensionality) for the dynamical exponent describing the Bose glass to superfluid transition may break down, as apparently seen in recent simulations (Ref. \cite{Baranger}). The key observation is that the major contribution to the compressibility, which remains finite through the transition and was predicted to scale as $\kappa \sim |\delta|^{(d-z)\nu}$ (where $\delta$ is the deviation from criticality and $\nu$ is the correlation length exponent) comes from the analytic part, not the singular part of the free energy, and therefore is not restricted by any conventional scaling hypothesis.