• effective field theory
• chiral Lagrangian models
• axion
• QCD

Quantum chromodynamics (QCD) is the theory which describes the strong interactions in the framework of the Standard Model. It is a gauge theory, based on the non-Abelian ”coulor” group SU(3)C. The degrees of freedom of the theory are quarks, the fundamental constituents of hadrons, and gluons, massless vector bosons which mediate the interactions. A fundamental property of QCD is asymptotic freedom: the strenght of the interaction between quarks and gluons becomes weaker when the energy increases. As a consequence the high-energy regime of the theory is well described by perturbative methods. On the other side the low-energy regime is higly non-perturbative and a different approach is needed. This low-energy regime is an extremely interesting sector of the theory since it can investigate different phenomena, such as confinement, hadron spectrum and dynamics and many others, whose explanation is crucial for a full understing of the theory itself.
Despite several successes, in the ’70s it became clear that QCD presented an unsolved question: after the introduction of instantons by t’Hooft, physicists realized that, due to these new kinds of topological configurations of gluon fields, an extra term could appear in the QCD Lagrangian, the so called θ term, given by Lθ = θQ, where Q is the topological charge density and θ is a free parameter. This term violates explicitly CP symmetry and predicts the existence of a non-zero value for the electric dipole of the neutron. Many experiments have studied the question but none of these has observed CP violations in strong interactions. In particular the measurement of the electric dipole of the neutron has set the experimental bound |θ|< 10−10. QCD is not able to explain the smallness of θ without assuming a fine-tuning of the parameter. This is known in the literature as the strong−CP problem.
Among the several possible solutions to the strong-CP problem, the most appealing one is the axion model, proposed by Peccei, Quinn, Weinberg and Wilczek (PQWW) in 1977-1978. The key idea is the introduction of a new global symmetry of the theory, the so called U(1)PQ, which is both spontaneously broken at an energy scale fa and affected by a quantum anomaly proportional to the topological charge Q. The Goldstone boson of U(1)PQ, mixing with the other degrees of freedom of the theory, gives rise to a new neutral pseudoscalar particle, the axion. The non-zero vacuum expectation value (v.e.v.) of the axion field is able to cancel out the θ term, restoring automatically the CP-symmetry. There are different specific models for the axion and its interactions with the Standard Model fields. A general feature, common to all, is that both the axion mass and its interactions are supressed by the energy scale fa. The original PQWW model identified this scale with the electroweak breaking scale vEW ∼ 250 GeV, but this hypothesis has been ruled out by the experiments. Nowadays the most appealing models, the so called ”invisible axion models” are based on the assumption that fa  vEW. As a consequence the general expectation for the axion is a very light particle with highly supressed interactions. Thanks to these characteristics, the axion is also indicated as a promising candidate for Dark Matter.
The main goal of this thesis is the description of axion properties, such as its mass and interactions. There are different approaches to accomplish this purpose. One of the most powerful, the one we choose for this thesis, is the use of an effective field theory. The general idea is that of ”integrating out” the high-energy degrees of freedom and build a model, based on symmetry principles, which describes the low-energy regime of the theory. In the context of QCD this paradigm has achieved important results in the so called chiral Lagrangian formulation. The starting point is the symmetry group of the QCD Lagrangian. Since the three lightest quarks (up, down and strange) have small masses compared to the characteristic QCD mass scale ΛQCD, it is a good approximation to neglect their masses, defining the so called chiral limit. In this limit QCD is invariant under a global symmetry group acting on quark fields with independent rotations of their left and right chiral components. This chiral group is G = U(1)L⊗U(1)R⊗SU(3)L⊗SU(3)R. However, studying the structure of the observed hadronic multiplets, it became clear that this symmetry is not realized exactly (in the Wigner-Weyl way) but it is rather spontaneously broken to its vectorial subgroup H = U(1)V ⊗SU(3)V . The broken axial generators give rise to nine Goldstone bosons (eight from SU(3)A and one from U(1)A). The introduction of quark masses explicitly breaks the chiral group. As a consequence the Goldstone bosons acquire a mass term, becoming pseudo−Goldstone bosons. These light bosons have been identified with the QCD lightest pseudoscalar mesons (pions, kaons and etas). In addiction to that we have to remark that the U(1)A subgroup is affected by a quantum anomaly proportional to the topological charge Q. As a consequence the corresponding Goldstone boson is massive even in the chiral limit. In order to describe the low energy dynamics of the pseudoscalar mesons, the chiral Lagrangian formulation was introduced: the key idea is building the most general Lagrangian, whose degrees of freedom are the pseudo-Goldstone bosons of the chiral group G, so as it transforms under G exactly as the QCD Lagrangian. In particular, since we are interested in low-energy dynamics, we can expand the Lagrangian in powers of the derivatives (and so for the amplitudes in powers of the momentum p). In this thesis we have considered the leading-order Chiral Effective Lagrangian O(p2). This Lagrangian has achieved important results in the computation of the mass-spectrum and interactions of the pseudoscalar mesons. Anyway, it does not include the effects of the U(1)A group and the corresponding Goldstone boson. The next step is therefore the inclusion of U(1)A. In doing that it is crucial the role of the axial anomaly, which has to be implemented in the Lagrangian model. There exist different models in the literature: the main difference between them is exactly the description and implementation of the U(1)A anomaly.
As the axion is expected to be a very light particle, it is possible to include it in the description of the low-energy dynamics of the theory. Therefore we have considered different chiral Lagrangian models and we have added to them the axion degree of freedom. To accomplish that, we have applied a ”minimal” procedure: following the previous works of Di Vecchia and Veneziano, the only terms we have added to our Lagrangians are a kinetic term for the axion field and an anomalous term, i.e a term which can reproduce the U(1)PQ anomaly. The implementation of the anomalous term depends on the specific model. Let’s analyze briefly the four models we have considered in this thesis. The first one is the above-mentioned Chiral Effective Lagrangian O(p2). The second is a model proposed by Witten, Di Vecchia and Veneziano (WDV): the Lagrangian of this model includes the topological charge density operator Q as an external background field. The presence of Q enables a simple procedure to implement the U(1)A anomaly. Even if this Lagrangian is well defined in the large NC limit (NC being the number of colours), we have used it to make predictions in the physical case NC = 3. The third model is the so called ”Extended linear σ-model (ELσ)”: the main difference with WDV is that the U(1)A anomaly is implemented without Q, but rather by mean of a determinantal interaction, which mimics the interactions of quarks generated by istantons. The last model, the so called ”Interpolating model”, ”interpolates” between WDV and ELσ, keeping both Q and the determinantal interaction and predicting the existence of a new exotic hadronic state. The model is based on the introduction of a U(1)-axial breaking condensate, in addiction to the usual quark-antiquark chiral condensate. It is not difficult to include the axion field in all the four models. The form and properties of the anomalous axion term depend explicitly on which one we are considering: as an example in WDV and Interpolating models it is a pure quadratic term, while in the others it produces also higher order interactions.
Let’s now discuss the original work done in this thesis. First of all, we have expanded the Lagrangians up to the second order in the pseudoscalar fields and derived the pseudoscalar mass matrices for all models. From these we have extracted both our predictions for the axion mass and the axion-mesons mixing angles. To accomplish this purpose we have followed a double procedure: we have introduced two perturbative parameters, mq/ΛQCD and b ∼ Fπ/fa, where mq is a scale with the same order of magnitude of the mass of the three lightest quarks, while Fπ is the pion decay constant. Since both the parameters are expected to be  1, we have performed first a perturbative expansion at the leading order in b, making no assumptions on the quark masses; then we have worked in the opposite limit. The results found with the two approaches have been compared at the end of the derivation. Next we have introduced the well-known QCD chiral electromagnetic anomaly (which gives rise to π0, η and η0 decays in two photons) and, from the knowledge of the previously computed mixing angles, we have derived both vertices and decay widths for the electromagnetic decay of an axion in two photons, in the same perturbative schemes discussed above. The predictions of these observables are of a crucial importance, since almost all the experimental search for the axion is based on its interaction with electromagnetic fields. In the literature it is possible to find results for the Chiral Effective Lagrangian O(p2) in the case of two light flavours (both for axion mass and electromagnetic vertex) and of the WDV model for three light flavours (only axion mass) at the leading order in the parameter b. As a consequence the great part of the results found in this thesis (all of them are derived for three light flavours) are original. The goal of this section is not only to derive analytical (and numerical) expressions for our observables but also to make a critical comparison between them and analyze the model dependence of the predictions.
Finally we have analyzed mesons decays involving the axion: in particular we have focused on the decays of η and η0 in two pions (neutral or charged) plus an axion, η/η0→ ππa, and the decay K+ → π+a. The first ones are the lowest energy hadron decays with a single axion involved: we have first expanded the Lagrangians up to the fourth order in the pseudoscalar fields (third order terms vanish for symmetry principles); then we have derived the coupling constants; finally, we have computed the amplitudes and the decay widths for each process. The charged kaon decay is interesting because of well-known experimental bounds on it and it was important to rule out the original PQWW model. This decay requires to take into account weak interactions between mesons; we have used some well-known ”tricks” found in the literature to perform our computations. Since these hadronic channels are less studied than the electromagnetic one, our aim was to find some interesting and original results, testable by experiments. As a consequence we have limited ourselves to work in a double expansion at the leading order in both the pertubative parameters mq/ΛQCD and b.
All the results found in the thesis have been discussed and compared with each other and with the ones found in the literature. We have evaluated several numerical results and compared them with the experimental data, so as to test the goodness of our assumptions and also to give some bounds on the parameter b (i.e on the PQ breaking scale fa) and on the axion mass. The thesis is organized as follows.
In Chapter 1 we present the most important properties of Quantum Chromodynamics, and describe its chiral symmetries in the limit of L massless quarks, paying particular attention to their spontaneous breaking and to the presence of the axial anomaly; then, we analyze the strong CP problem and the Peccei-Quinn solution, focusing on the general properties of axions. Finally we describe some of the most popular axion models.
In Chapter 2 we present the chiral effective Lagrangian formulation and the four models we have worked on in this thesis: starting from the analysis of the effective degrees of freedom of QCD in the low-energy regime, we first describe the Chiral Effective Lagrangian O(p2) proposed by Weinberg; then, we present the model of Witten, Di Vecchia and Veneziano, the Extended Linear σ-model and the Interpolating model. For each of them we discuss how to implement the axion degree of freedom and we show explicitly how the Peccei-Quinn mechanism cancels the θ term.
In Chapter 3 we present and critically analyze the results that we have obtained for the axion mass in the four different models, describing also the computational techniques used in our work; the last section of the chapter is devoted to evaluate numerically, where possible, our results so as to compare them with each other and with the experimental data.
In Chapter 4 we present the results that we have obtained for the axion decay in two photons and mesons decays involving the axion. For each of them we derive both the coupling constants and the decay widths, describing in detail the computational techniques adopted. Finally we evaluate numerically, where possible, our results so as to compare them with each other and with the experimental data and we also give some bounds on the parameters of the models.