  We study the critical bosonic O(N) vector model with quenched random mass
disorder in the large N limit. Due to the replicated action which is sometimes
not bounded from below, we avoid the replica trick and adopt a traditional
approach to directly compute the disorder averaged physical observables. At
$N=\infty$, we can exactly solve the disordered model. The resulting low energy
behavior can be described by two scale invariant theories, one of which has an
intrinsic scale. At finite $N$, we find that the previously proposed attractive
disordered fixed point at $d=2$ continues to exist at $d=2+\epsilon$ spatial
dimensions. We also studied the system in the $3<d<4$ spatial dimensions where
the disorder is relevant at the Gaussian fixed point. However, no physical
attractive fixed point is found right below four spatial dimensions.
Nevertheless, the stable fixed point at $2+\epsilon$ dimensions can still
survive at $d=3$ where the system has experimental realizations. Some critical
exponents are predicted in order to be checked by future numerics and
experiments.
