  The low-energy dynamics of five-dimensional Yang-Mills theories compactified
on S^1 can be described by a four-dimensional gauge theory coupled to a scalar
field in the adjoint representation of the gauge group. Perturbative
calculations suggest that the mass of this elementary scalar field is protected
against power divergences, and is controlled by the size of the extra dimension
R. As a first step in the study of this phenomenon beyond perturbation theory,
we investigate the phase diagram of a SU(2) Yang-Mills theory in five
dimensions regularized on anisotropic lattices and we determine the ratios of
the relevant physical scales. The lattice system shows a dimensionally reduced
phase where the four-dimensional correlation length is much larger than the
size of the extra dimension, but still smaller than the four-dimensional
volume. In this region of the bare parameter space, at energies below 1/R, the
non-perturbative spectrum contains a \emph{light} scalar state. This state has
a mass that is independent of the cut-off, and a small overlap with glueball
operators. Our results suggest that light scalar fields can be introduced in a
lattice theory using compactified extra dimensions, rather than fine tuning the
bare mass parameter.
