Neutron-proton mass splitting in the Skyrme Model

Quantum Chromodynamics is the most elegant theory we have developed so far describing and explaining the physics of strong interactions. Its most striking feature, being it asymptotic freedom, successfully allows for the investigation of the physics of quarks and gluons in the perturbative regime with incredible accuracy. Within this framework, the latter particles are thought to be the fundamental blocks or constituents of all the hadrons present in Nature such as mesons and baryons. Although QCD inherently contains the information about hadron physics, it is nearly impossible to derive the low energy physics directly from it, since the exact solution of this SU(3) gauge theory is unbelievably difficult to calculate and no one has succeeded, so far. There have been many attempts to overcome this seemingly insurmountable obstacle, which can be collectively grouped into two categories: the first one being numerical simulations on lattice, which necessarily are limited by the computational power one has at disposal and also suffer from severe algorithm-related problems such as the Numerical sign problem, when highly oscillatory functions are to be integrated. Furthermore, it is fairly common not to even come up with a clever algorithm which would faithfully describe the physical mechanism of interest. The second one, instead, is related to the construction of effective field theories for hadrons, which ought to originate from the integration of the high energy degrees of freedom of QCD. This thesis specializes in the latter approach as a tool to probe baryon physics and, in particular, we are interested in how the difference in mass between neutron (n) and proton (p) arises from the low lying theory. It is well known from experimental data that the value of the mass splitting between these two particles amounts to ∆M_{n,p} ≡ m_n−m_p \simeq 1.29 MeV (only the 0.14% of the average of the two masses), with the neutron being slightly heavier than the proton. This minuscule difference accounts for the stability of the proton and, as a consequence, of the whole universe we live in. The measured value of the mass splitting is mostly the result of two different contributions, one from the strong sector (QCD) and the other from the electromagnetic sector (QED). The latter is known to be negative ∆Me.m. N ∼−1 MeV, which indicates that the strong contributionmust be positive with value around ∆ Mstrong N \simeq 2.3 MeV. The main goal of this thesis revolves around the explanation and derivation of the strong contribution to the mass splitting of the nucleon by the exploitation of an old non linear sigma model also known as the Skyrme Model. Although this model consists of just pseudoscalar particles such, i.e. the pions, them being the pseudo-Goldstone bosons coming from the spontaneous symmetry breaking of the chiral group, there are reasons to believe that the solitons of the theory should be regarded as baryons. The large Nc expansion description of baryons has indeed revealed that the latter particles behave as if they were the solitons of an effective field theory comprising of only mesons. Moreover, the non trivial topology of the sigma model is characterized by an integer topological charge associated with the baryon number B. Our interest shall focus on the solitonic solution with unit baryon number in order to correctly account for the quantum number of the neutron and proton. Accordingly, the skyrmion, named after the inventor of the model T.H.R. Skyrme, is the solitonic solution of the static theory. Since the boundary conditions, forcing the solution to have finite energy, allow for the compactification of the space R^3 → S^3, the topological considerations are sustained by the relevant non trivial homotopy group π_3(U(N_f)) = Z. However, we argue that no possible realization of baryons mass splitting can be achieved if we regard the theory only consisting of the pseudoscalar isospin triplet fields, and as a consequence, we turn our attention towards two extensions of the model. We first introduce the pseudoscalar isosinglet field η, formally enlarging the symmetry group to U(2), and by doing so, we learn that solitonic solutions still exist but they are not maximally symmetric as they previously were when the constituents of the model were only pions. In fact, the spherical symmetry gets reduced to cylindrical which in turn brings a modification to the shape of the soliton from a sphere to an ellipsoid, whose inertia tensor is studied in both semi-analytical and numerical approach. Despite the modified symmetry, the addition of η almost solves the mass splitting problem, leaving only baryons with the same absolute value of isospin equally massive. That is, it is enough to split in pairs the ∆ baryons masses but of course not the nucleon. The only alternative we have in order to do so is the introduction of other particles. As already mentioned, the large Nc framework comes in helpful for it suggests the baryons as the solitonic solution of a theory of infinite mesons. Although we limit the choice to the two vector mesons ρ and ω, the results we get are satisfactory as the residual isospin symmetry gets finally broken and, as a consequence, neutron and proton acquire different masses, whose difference holds the expected sign. We did not explicitly calculate the latter and further numerical evaluations are needed in order to fully establish the results. However, we did provide the complete Lagrangian and its static equation of motion, which are necessary ingredients for the complete analysis of baryons in this model. The whole procedure of vector mesons addition is carefully treated along with the insertion of a new topological five dimensional term, known as the Wess-Zumino term, which did not play any role in both the SU(2) and U(2) scenarios, since in those cases it vanishes identically (this would not be the case for a three-flavored theory) but it does bring a non trivial contribution when it is rendered gauge invariant under local transformations of the new mesonic fields. Furthermore, its properties, such as its N_c scaling behavior which matches exactly the other terms in the Lagrangian or as the fact that a U(1) gauge transformation perfectly reproduces the topological current of the sigma model, support the assumption of baryons being associated to the solitons of the model. To sum up we provide a consistent gauge invariant model of both pseudoscalars and vector mesons, whose non trivial topology admits solitonic solutions interpreted as baryons, with baryon number B being the topological charge of the theory. In this context we examine two possible extensions of the model and find that the η field plays a crucial role in both paths as it produces two distinct contributions to the skyrmion, one which only partially breaks isospin and the other, produced by vector mesons, which succeed in splitting any baryon masses