  We investigate the nature of the Bose glass phase of the disordered
Bose-Hubbard model in $d>2$ and demonstrate the existence of a glass-like
replica symmetry breaking (RSB) order parameter in terms of particle number
fluctuations. Starting from a strong-coupling expansion around the atomic
limit, we study the instability of the Mott insulator towards the formation of
a Bose glass. We add some infinitesimal RSB, following the Parisi hierarchical
approach in the most general form, and observe its flow under the
momentum-shell renormalization group scheme. We find a new fixed point with
one-step RSB, corresponding to the transition between the Mott insulator and a
Bose glass phase with hitherto unseen RSB. The susceptibility associated to
infinitesimal RSB perturbation in the Mott insulator is found to diverge at the
transition with an exponent of $\gamma=1/d$. Our findings are consistent with
the expectation of glassy behavior and the established breakdown of
self-averaging. We discuss the possibility of measuring the glass-like order
parameter in optical lattice experiments as well as in certain spin systems
that are in the same universality class as the Bose-Hubbard model.
