A Chiral lagrangian with Broken Scale: Testing the restoration of symmetries in astrophysics and in the laboratory.

We study matter at high density and temperature using a chiral lagrangian in which the breaking of scale invariance is regulated by the value of a scalar field, called dilaton \cite{Heide:1993yz,Carter:1995zi,Carter:1996rf,Carter:1997fn}. We provide a phase diagram describing the restoration of chiral and scale symmetries. We show that chiral symmetry is restored at large temperatures, but at low temperatures it remains broken at all densities. We also show that scale invariance is more easily restored at low rather than large baryon densities. The masses of vector mesons scale with the value of the dilaton and their values initially slightly decrease with the density but then they increase again for densities larger than $\sim 3 \rho_0$. The pion mass increases continuously with the density and at $\rho_0$ and T=0 its value is $\sim$ 30 MeV larger than in the vacuum. We show that the model is compatible with the bounds stemming from astrophysics, as e.g. the one associated with the maximum mass of a neutron star. The most striking feature of the model is a very significant softening at large densities, which manifests also as a strong reduction of the adiabatic index. While the softening has probably no consequence for Supernova explosion via the direct mechanism, it could modify the signal in gravitational waves associated with the merging of two neutron stars.