Scale-chiral symmetry, $\omega$ meson and dense baryonic matter

It is shown that explicitly broken scale symmetry is essential for dense skyrmion matter in hidden local symmetry theory. Consistency with the vector manifestation fixed point for the hidden local symmetry of the lowest-lying vector mesons and the dilaton limit fixed point for scale symmetry in dense matter is found to require that the anomalous dimension (${\gamma }_{G^2}$) of the gluon field strength tensor squared ($G^2$) that represents the quantum trace anomaly should be $0<{\gamma }_{G^2}<3$.