• spontaneous breaking of CP symmetry
• chiral effective lagrangians
• axion potential
• axion
• topological susceptibility

The story of the Axion traces back to the so-called U(1) problem in QCD. In fact, QCD, restricted to the lightest n quarks, is invariant under a global symmetry group U(1)^n which is enlarged at the classical level to G = U(n)_R x U(n)_L, if these quarks are taken to be massless. However, the condensation of quark-antiquark pairs leads to the spontaneous breaking of this symmetry down to U(n)_V. The spontaneous breaking of U(1)_A together with the non-zero mass of the quarks was not sufficient to explain the large mass of the η meson (Weinberg’s limit), the most plausible candidate for the U(1)_A (pseudo-)Goldstone boson, but it was soon recognized that the corresponding current is affected by an anomaly, that is its divergence is not zero, but proportional to the so-called topological charge density Q(x).
The relevance of the density Q(x) consists in the fact that, despite being a total divergence, it gives non-zero contributions to the path integral because of the non-trivial topological structure
of the QCD vacuum. In fact, the correct boundary conditions to be imposed in the path integral are F_{μν}^a=0 at spatial infinity, which are satisfied by non-trivial gauge configurations characterized by different values of the topological charge q, that is the spacetime integral of Q(x). The summation in the path integral over all the possible topologically distinct configurations is equivalent to the introduction in the QCD Lagrangian of a topological term θQ, with θ a free parameter. The addition of the so-called θ-term leads to a direct violation of CP in the strong interactions, in particular, it induces a non-vanishing electric dipole moment for the neutron. However, experimentally the bounds are very strict and force |θ| ≤ 10^{−10}. Why should θ be so small or even zero is known as the ”Strong-CP problem”.
Among the different solutions to the strong-CP problem, the most convincing one is represented by the introduction of the Axion, which, since its very first appearance, has been gaining an increasing interest and is the main object of this Thesis. The Axion was originally introduced by Weinberg and Wilczek on the base of a model previously proposed by Peccei and Quinn (for this reason we speak of the PQWW axion). It is based on the introduction into the Standard Model Lagrangian of an additional pseudoscalar field (which essentially replaces the free parameter θ) and is endowed with an additional U(1) symmetry, known as U(1)_{PQ}, which is both spontaneously broken and anomalous. By virtue of this extra U(1) symmetry, CP comes out to be dynamically conserved in this model and, moreover, a new pseudo-Goldstone boson appear, the Axion itself. The original PQWW axion was soon experimentally ruled out, but new fundamental models, namely the Kim-Shifman-Vainshtein-Zakharov (KSVZ) and the Dine-Fischler-Srednicki-Zhitnisky (DFSZ) invisible axion models, were proposed and are still compatible with current experimental bounds. The Axion described by these models is both very light and very weakly interacting with ordinary matter. These last aspects immediately lead to the second reason why Axions are so appealing, especially nowadays. In fact, it was soon recognized that they represent good candidates to explain the current abundance of Dark Matter in the Universe. The most important mechanism of Axionic Dark Matter production is the so-called vacuum misalignment mechanism, which consists in the assumption that, in the early Universe, the Axion classical field was not located at the minimum of its potential and started oscillating only when the Hubble constant, which represents a sort of friction term, dropped below the value of the Axion mass, even if this requires that the value of the field should not be so far from the minimum. Since the couplings of the Axion are weak, the energy density stored in the oscillations of the classical field does not dissipate rapidly, so that, through the computation of the relative Axion abundance, this allows to put a lower bound on its mass, which is also constrained from above by other astrophysical bounds. In general, the evolution of the Axion field and thus its current abundance may depend on the form of its potential, whose expression is obviously determined by studying the interaction of the Axion with the QCD particle
content. However, especially at low temperatures, one is forced to employ the Chiral Effective Lagrangian formalism, which represents the main tool employed throughout this Thesis. The fundamental hypothesis is that we need not know the full spectrum of the theory if we wish to study only the dynamics of the lightest particles, like the light pseudoscalar meson in QCD and the Axion itself, which can be considered as the sole, new, fundamental degrees of freedom. Then, their dynamics can be described by means of an Effective Lagrangian, which is written in terms only of the relevant degrees of freedom and which must reflect the same symmetries of the underlying theory. In our case, the Chiral Effective Lagrangians employed to describe the low energy dynamics of QCD must be extended in order to accommodate the Axion. In this Thesis, the Axion potential has been studied by employing the following three effective models:
• The Extended Linear σ (EL_σ) model, which includes also the scalar partners of the η and of the light pseudoscalar mesons, implements the anomaly through a term inspired by the 2n-fermion interaction term which mimics the interactions among fermions through the exchange of an instanton.
• The Witten-Di Vecchia-Veneziano Model (WDV ) model, which is rigorously derived within the large N_c expansion, where the anomaly is provided by directly coupling the anomalous fields (that is, those describing the η and the Axion) to the topological charge density Q(x),
which serves as an external source.
• In addition to these models, in this Thesis we shall make use of a third example of Effective Lagrangian, where it is postulated the existence of an additional condensate which breaks spontaneously U(1)_A while preserving SU(n)_L x SU(n)_R. We refer to this model as the Interpolating Model, since the introduction of this new condensate allows to implement the anomaly as in the WDV model and at the same time to take into account the effects of the instantons. In particular, the focus has been on the comparative study of the form of the Axion potential, which must be periodic, around its minima, that is the evaluation of the Axion mass, and in a neighbourhood of the endpoints of its periodicity interval. The latter case is particularly interesting since in some regions of the parameter space spanned by the parameters of the Effective Lagrangians, the large mixing of the Axion with the standard QCD degrees of freedom makes the Axion potential no longer meaningful as we approach these endpoints. The analysis has been made both at T = 0 and at finite temperature, below and above the critical temperature T_c, by means of a mean field approach, where all the coefficients of the models are assumed temperature dependent.