Scaled chiral quark-solitons for nuclear matter

One of the most challenging problems in hadronic and nuclear physics is to study nuclear matter at finite density by using a scheme which includes one of the fundamental properties of QCD, namely chiral symmetry. The problem of studying nuclear matter with chiral Lagrangians is not trivial; for instance models based on the linear σ-model fail to describe nuclear matter already at ρ ∼ ρ0 because the normal solution in which chiral symmetry is broken becomes unstable respect to the Lee-Wick phase. The main problems in these models are due to the constraints on the scalar field dynamics imposed by the Mexican hat potential [1]. The interaction terms of σ and π fields in the linear realization of chiral symmetry allows the chiral fields to move away from the chiral circle as the density raises and to reach, already at ρ0, the local maximum where σv = 0 and chiral symmetry is restored. The difficulty of a too early restoration of chiral symmetry at finite density can be overcame in two different ways. One could implement chiral symmetry into the Lagrangian through a non-linear realization [2] where the scalar fields are forced to stay on the chiral circle. The other approach is still based on a linear realization of chiral symmetry but with a new potential, which includes terms not present in the Mexican hat potential. A possible guideline in building such a potential is scale invariance, which is spontaneously broken in QCD due to the presence of the parameter ΛQCD coming from the renormalization process and it is strictly connected to a non vanishing gluon condensate. This fundamental symmetry of QCD can be implemented in the Lagrangian at mean-field level, following the approaches in [3, 4], through the introduction of a new scalar field, the dilaton field, whose dynamics is regulated by a potential chosen in order to reproduce the scale divergence of QCD. In this work we will adopt a Chiral Dilaton Model (CDM) which also includes scale invariance introduced by the nuclear physics group of the University of Minnesota [5–8]. It has already been shown that an hadronic model based on this dynamics provides a good description of nuclear physics at densities about ρ0 and it describes the gradual restoration of chiral symmetry at higher densities [9]. In the same work the authors have shown a phase diagram, where the interplay between chiral and scale invariance restoration lead to a scenario similar to that proposed by McLerran and Pisarski in [10]. It is therefore tempting to explore the scenario presented in [9] at a more microscopic level. The new idea we develop in this work is to interpret the fermions as quarks, to build the hadrons as solitonic solutions of the fields equations as in [11] and, finally, to explore the properties of the soliton at finite density using the Wigner- Seitz approximation. Similar approaches to a finite density system have been investigated in the past [12–17]. A problem of those works is that the solitonic solutions are unstable and disappear already at moderate densities when e.g. the linear σ-model is adopted [16]. We are therefore facing an instability similar to the one discussed and solved when studying nuclear matter with hadronic chiral Lagrangians. The first aim of this thesis is to check whether, just by modifying the mesons interaction with the inclusion of scale invariance, the new logarithmic potential allows the soliton crystal to reach higher densities. Next, since the CDM also takes into account the presence of vector mesons, the second and more important aim is to check whether the inclusion of vector mesons in the dynamics of the quarks can provide saturation for chiral matter. We should remark that no calculation, neither in vacuum nor at finite density, exists at the moment for the CDM with quarks and vector mesons.