  Within the bottom-up approach to holography, we construct a class of
six-dimensional gravity models, and discuss solutions that can be interpreted,
asymptotically in the far UV, in terms of dual five-dimensional conformal field
theories deformed by a single scalar operator. We treat the scaling dimension
of such operator, related to the mass of the one scalar field in the gravity
theory, as a free parameter. One dimension in the regular geometry is
compactified on a shrinking circle, hence mimicking confinement in the
resulting dual four-dimensional theories.
  We study the mass spectrum of bosonic states. The lightest state in this
spectrum is a scalar particle. Along the regular (confining) branch of
solutions, we find the presence of a tachyonic instability in part of the
parameter space, reached by a smooth deformation of the mass spectrum, as a
function of the boundary value of the background scalar field in the gravity
theory. In a region of parameter space nearby the tachyonic one, the lightest
scalar particle can be interpreted as an approximate dilaton, sourced by the
trace of the stress-energy tensor, and its mass is parametrically suppressed.
  We also compute the free energy, along several branches of gravity solutions.
We find that both the dilatonic and tachyonic regions of parameter space,
identified along the branch of confining solutions, are hidden behind a
first-order phase transition, so that they are not realised as stable
solutions, irrespectively of the scaling dimension of the deforming
field-theory operator. The (approximate) dilaton, in particular, appears in
metastable solutions. Yet, the mass of the lightest state, computed close to
the phase transition, is (mildly) suppressed. This feature is amplified when
the (free) parameter controlling the scaling dimension of the deformation is
5/2, half the dimension of space-time in the field theory.
