  We continue our work on the study of spherically symmetric loop quantum
gravity coupled to two spherically symmetric scalar fields, one that acts as a
clock. As a consequence of the presence of the latter, we can define a true
Hamiltonian for the theory. In previous papers we have studied the theory for
large values of the radial coordinate, that is, far away from any black hole or
star that may be present. This makes the calculations considerably more
tractable. We have shown that in the asymptotic region the theory admits a
large family of quantum vacua for quantum matter fields coupled to quantum
gravity, as is expected from the well-known results of quantum field theory on
classical curved space-time. Here, we study perturbative corrections involving
terms that we neglected in our previous work. Using time-dependent perturbation
theory, we show that the states that represent different possible vacua are
essentially stable. This ensures that one recovers from a totally quantized
gravitational theory coupled to matter the standard behavior of a matter
quantum field theory plus low probability transitions due to gravity between
particles that differ at most by a small amount of energy.
