  In this note, we report on a remarkable and surprising interplay between the
omega meson and the dilaton chi in the structure of a single skyrmion as well
as in the phase structure of dense skyrmion matter which may have a potentially
important consequence on the properties of compact stars. In our continuing
effort to understand hadronic matter at high density, we have developed a
unified field theoretic formalism for dense skyrmion matter using a single
Lagrangian to describe simultaneously both matter and meson fluctuations and
studied in-medium properties of hadrons. The effective theory used is the
Skyrme model Lagrangian gauged with the vector mesons rho and omega,
implemented with the dilaton field that describes the spontaneously broken
scale symmetry of QCD, in a form consistent with the symmetries of QCD and our
expectations regarding the high density limit. We analyze the restoration of
scale invariance and chiral symmetry as the density of the system increases. In
order to preserve the restoration of scale symmetry and chiral symmetry,
signalled in our case by the vanishing of the expectation value of the dilaton,
and to be consistent with the vector manifestation of hidden local symmetry, a
density dependent omega coupling is introduced. We uncover the crucial role
played by both the dilaton and the omega meson in the phase structure of dense
medium and discover how two different phase transition regimes arise as we dial
the dilaton mass.
