• zero and finite temperature
• QCD topological susceptibility
• chiral effective Lagrangian models
• axion
• QCD

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 $f_a$ and, moreover, also has a quantum \textit{anomaly}. As a consequence, as shown by Weinberg and Wilczek in 1978, an electrically-neutral pseudoscalar particle should appear in the energy spectrum of the theory, representing the (\textit{would-be}) Nambu-Goldstone (NG) boson originated from the breaking of the $U(1)_{PQ}$ symmetry: this new particle was called the \textit{axion}. According to the so-called \textit{Peccei-Quinn mechanism}, the axion field takes a nonzero vacuum-expectation-value that dynamically sets the $\theta$-angle to zero, thus restoring the CP symmetry. \\
In the original Peccei-Quinn-Weinberg-Wilczek (PQWW) axion model, the $U(1)_{PQ}$ energy scale $f_a$ was identified with the electroweak breaking scale $v_F \approx 250$ GeV. However, this hypothesis was soon ruled out by the analysis of the kaon decay: $K^+ \rightarrow \pi^+ + a$ (where $\pi^+$ is the charged pion and $a$ is the axion), whose branching ratio came out to be well above the experimental bounds. Nonetheless, other models, with an energy scale $f_a$ much larger than $v_F$, have been proposed to overcome this constraint: these so-called \textit{invisible axion} models, such as the Kim-Shifman-Vainshtein-Zakharov (KSVZ) model and the Dine-Fischler-Srednicki-Zhitnisky (DFSZ) model, remain still viable with a value of $f_a$ in the range $10^{9} \, \text{GeV} \lesssim f_a \lesssim 10^{17} \, \text{GeV}$, derived from astrophysical and cosmological considerations. \\
All these axion models predict a very light axion with small couplings to the Standard model fields: therefore, because of these properties, the axion seems also a promising Dark Matter candidate. \\
Hence, since the axion is expected to be a very light particle, it seems natural to include it as an additional degree of freedom in a low-energy effective description of QCD that takes into account only its relevant degrees of freedom. This approach was widely investigated in the past literature by making use of the so-called \textit{chiral effective Lagrangian} models that describe the dynamics of the lightest states in the hadronic spectrum, i.e. the pseudoscalar mesons ($J^{P} = 0^-$), which are the (\textit{pseudo}-)NG bosons originated from the spontaneous breaking of the $SU(N_f)_L \otimes SU(N_f)_R$ chiral symmetry down to the vectorial subgroup $SU(N_f)_V$ (with $N_f = 2$ and $N_f = 3$ being the physically relevant cases). Specifically, for $N_f = 2$, the three (pseudo-)NG bosons correspond to the three pions ($\pi^+, \pi^0, \pi^-$); whereas, for $N_f = 3$, the eight (pseudo-)NG bosons are the octet of pions, kaons and eta ($\pi^+, \pi^0, \pi^-, K^+, K^-, K^0, \bar{K}^0, \eta$).

The goal of this thesis is to study some QCD axion properties, such as its mass and its interactions with ordinary matter, generalizing two previous studies carried out using two particular chiral effective Lagrangian models at zero [1] and finite [2] temperature $T$. In particular, Ref. [1] considers the "axionized" version of a chiral effective Lagrangian model proposed by Witten, Di Vecchia, Veneziano \textit{et al.} (the \textit{WDV model}), that not only describes the octet of pseudoscalar mesons (the eight (pseudo-)NG bosons coming from the spontaneous breaking of the chiral group $SU(3)_L \otimes SU(3)_R$ down to the subgroup $SU(3)_V$), but also includes the flavor-singlet pseudoscalar meson and implements correctly the $U(1)$ axial anomaly of the fundamental theory. This model is a powerful tool to investigate the QCD chiral dynamics at $T = 0$, but it becomes less reliable as the temperature grows and it eventually breaks down across the QCD chiral phase transition, which, according to lattice simulations, occurs at a critical temperature $T_c \sim 150 \div 170$ MeV, where thermal fluctuations break up the chiral condensate $\Braket{\bar{q}q}$, the \textit{order parameter} of the chiral symmetry, causing its complete restoration. \\
In Ref. [2], indeed, another model is employed in order to study the QCD axion dynamics at finite $T$: the so-called \textit{Extended Linear $\sigma$ (EL$_{\sigma}$)-model}. This chiral effective Lagrangian is a kind of linear sigma model that describes the scalar and pseudoscalar mesonic excitations (which are believed to be the relevant degrees of freedom around the chiral phase transition) and also implements the $U(1)$ axial anomaly of the fundamental theory. \\
In both these studies, the axion was introduced in the effective Lagrangians implementing a $U(1)_{PQ}$ symmetry in which only the axion field transforms (non-linearly), while the other degrees of freedom are left unchanged (meaning that the light quarks are \textit{neutral} under a $U(1)_{PQ}$ transformation). However, in such a way, one is reproducing in the low-energy effective theory only a class of axion models, like the KSVZ model, where the $U(1)_{PQ}$ anomaly comes from an heavy "exotic" quark which is integrated out in the effective model. \\
Our aim in this thesis is to generalize these two effective models by enlarging the aforementioned $U(1)_{PQ}$ symmetry, taking into account the possibility that, instead, the $N_f$ light quarks are \textit{charged} under a $U(1)_{PQ}$ transformation: that is to say, each of them transforms with a $U(1)$ phase, which depends on its specific Peccei-Quinn charge. In this way, one is able to reproduce in the low-energy effective theory those axion models, like the DFSZ model, in which the light (Standard Model) quarks transform under the $U(1)_{PQ}$ symmetry. \\
In particular, using these two generalized effective models, we shall compute the axion mass at zero and finite $T$ (both below and above the chiral phase transition at $T_c$) and we shall investigate its relation with the QCD \textit{topological susceptibility} (defined as $\chi_{QCD} \equiv - i \int d^4 x \; \Braket{T Q(x) Q(0)}$). Furthermore, for the case at $T = 0$, we shall study the radiative decay $a \rightarrow \gamma + \gamma$ and some particular "hadronic" decays $\eta/\eta' \rightarrow \pi + \pi + a$, comparing our outcomes with other analogous results present in the literature. Instead, in the case at finite $T$, we shall study thoroughly the scalar and pseudoscalar meson mass spectrum in the special case with $N_f = 2$.\\


{\Large \textbf{References}}\\

[1] G. Landini, E. Meggiolaro, Eur. Phys. J. C \textbf{80}, 302 (2020) \\

[2] S. Bottaro, E. Meggiolaro, Phys. Rev. D \textbf{102}, 014048 (2020)