Study of the QCD axion using chiral effective Lagrangian models both at zero and finite temperature

Quantum Chromodynamics (QCD) is the quantum field theory that describes the \textit{strong} nuclear interactions in nature. It is based on the non-Abelian "color" gauge group $SU(3)_c$ and its fundamental degrees of freedom are the so-called \textit{quarks} (spin-$\frac{1}{2}$ particles) and \textit{gluons} (spin-$1$ gauge bosons which mediate the strong interaction). A main characteristic of this theory is that those fundamental particles are \textit{not} visible as asymptotic states in the experiments, but they are \textit{confined} within the hadrons (mesons and baryons): this property is called \textit{confinement}. \\ Although QCD was successfully confirmed by many experiments, this theory still suffers from an unsolved problem nowadays: the discovery in 1975 of topologically nontrivial gluon-field configurations - named \textit{instantons} - opened up the possibility that an extra term may be added to the QCD Lagrangian $\mathcal{L}_{\theta} = \theta Q$, where $\theta$ is a free parameter and $Q$ is the topological charge density. A nonzero value of the $\theta$-term would imply an explicit breaking of $CP$ (and $P$) symmetry in the strong sector; however, this violation has not been backed up by any experimental evidence so far. In particular, as a consequence of the $CP$ breaking, we should have a nonzero electric dipole moment of the neutron. On the other hand, the experimental measure of this observable sets the upper bound $\theta \lesssim 10^{-10}$. QCD alone, however, cannot explain the origin of this value for the $\theta$ parameter without introducing a \textit{fine tuning} problem: this, indeed, is still an open question in hadronic physics and it is called the \textit{strong-CP problem}. \\ Since the late 70's, some different solutions were proposed to address this issue: the most appealing one, proposed by Peccei and Quinn in 1977, calls for an extra pseudoscalar field and an additional global $U(1)$ chiral symmetry - named $U(1)_{PQ}$ - which is spontaneously broken at an energy scale $f_a$ ...