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We study matter at high density and temperature using a chiral lagrangian inwhich the breaking of scale invariance is regulated by the value of a scalarfield, called dilaton\cite{Heide:1993yz,Carter:1995zi,Carter:1996rf,Carter:1997fn}. We provide aphase diagram describing the restoration of chiral and scale symmetries. Weshow that chiral symmetry is restored at large temperatures, but at lowtemperatures it remains broken at all densities. We also show that scaleinvariance is more easily restored at low rather than large baryon densities.The masses of vector mesons scale with the value of the dilaton and theirvalues initially slightly decrease with the density but then they increaseagain for densities larger than $\sim 3 \rho_0$. The pion mass increasescontinuously with the density and at $\rho_0$ and T=0 its value is $\sim$ 30MeV larger than in the vacuum. We show that the model is compatible with thebounds stemming from astrophysics, as e.g. the one associated with the maximummass of a neutron star. The most striking feature of the model is a verysignificant softening at large densities, which manifests also as a strongreduction of the adiabatic index. While the softening has probably noconsequence for Supernova explosion via the direct mechanism, it could modifythe signal in gravitational waves associated with the merging of two neutronstars.

A chiral lagrangian with Broken Scale: testing the restoration of symmetries in astrophysics and in the laboratory