The Skyrme model is an effective theory for the description of nucleons, nuclei and pions,
where the primary degrees of freedom are mesons and the hadrons and nuclei appear as
topological solutions of such model. Furthermore, the number of baryons is identified with
the topological charge of the solitons. The Skyrme model allow us to study hadrons and
nuclei starting from a field theory instead of the more common nuclear models based on
methods of the quantum mechanics with a finite number of degrees of freedom. The highly
non-linear character of such field theory makes it very suitable to the description of complex
phenomena of strong interaction and very present in several applications. In fact, using the
Skyrme model, a lot of nuclear properties can be understood from a qualitative point of view,
like their masses, or their spectra (corresponding to excitations of the spin and isospin of the
Skyrmions). On the other hand, the quantitative precision of the standard Skyrme model does
not exceed a 10-30%. The main reason for this limitation is the lack of BPS solutions in the
standard Skyrme model, what implies binding energies too high and internuclear forces too
strong, among other problems.
To deal with these limitations, some generalizations of the Skyrme model have been
studied with the same content of fields but adding new terms to the Lagrangian. Among these
generalized Skyrme theories, an integrable theory with an infinite number of exact solutions
(topological solitons) saturating a Bogomolny bound has been found out recently. This is the
BPS Skyrme model. Due to its integrability, this model has the symmetries of an
incompressible fluid. This and the BPS property make the model phenomenological suitable
for nucleus description. For instance, because of the BPS property, binding energies of
classical solitons are zero, and low binding energies of real nuclei are obtained by quantum
corrections and by small contributions from additional terms in the Lagrangian. Soliton radii
also grow with the cube root of the baryonic charge, as in real nuclei.
With the purpose of studying in detail the BPS Skyrme model and its application to
hadronic and nuclear physics, the following research lines have been developed:
1. The analysis of the thermodynamics of the BPS Skyrme model at zero temperature
with the introduction of an external pressure and the calculation of the energymomentum
tensor. Moreover, two important thermodynamical quantities like the
compressibility and the baryon chemical potential were studied.
2. The introduction of the quantization of the collective coordinates (spin and isospin),
as in the standard Skyrme model. Moreover, the electrostatic energy (Coulomb term)
and a little explicit breaking of the isospin were added. The Coulomb term is really
important for heavy nuclei, whereas isospin symmetry breaking will split the protons
and neutrons.
3. The coupling of the BPS Skyrme model to gravity to describe neutron stars in good
agreement with the observational constraints concerning mass and radius. Here, two
distinct approaches were followed (exact and mean-field calculations) finding that,
although for global properties there are not too much difference, it is not the case of
local ones, where the difference is pronounced.
4. The study of the influence of several different potentials in the researches mentioned
above, because the specific shape of the potential is not important for the
mathematical properties.
Finally, it is worth commenting that, although the BPS model gives the dominant
contributions to many nuclear matter properties, some evidences have been found that an
extension to a near-BPS model is needed for a complete description of strong interactions.
