• Axion
• Chiral Condensate
• Chiral symmetry
• Lattice QCD
• Sphaleron rate
• Theta-dependence
• Topological susceptibility

Quantum ChromoDynamics is a theory that exhibits very interesting physics at low-energy scale, in the so-called non-perturbative regime. The purpose of the thesis, as the title suggests, is to deeply look at the relationship between two important aspects of non-perturbative QCD: θ-dependence and chiral symmetry, without losing track of the implications about axion phenomenology. This investigation is led in the framework of Lattice QCD, a very powerful tool for the study of non-perturbative regime, by means of Monte Carlo simulations.

The discovery of instantons, solutions of the QCD equation of motion in the Euclidean space-time with non-trivial topology, was the turning point for the understanding of the mechanism which gives rise to the axial anomaly, solving the so-called U(1) problem. On the other side, this topological structure is related to one of the most important issues of the Standard Model, the strong CP-problem, still unsolved. Among the different proposals for its solution, we mention the Peccei-Quinn mechanism, that introduces a new global symmetry in the Standard Model and is able to dinamically remove the source of this CP violation, with the introduction of a new particle, the Axion, whose properties are fixed by θ-dependence.

The axion mass is related to the topological susceptibility χ(T), i.e., the coefficient of the θ^2 term in the QCD free energy density expansion, and to the axion decay constant. The first part of the thesis is focused on the determination of the topological susceptibility from lattice simulations. After a general discussion about the main numerical issues that have to be faced, the result on the finite temperature topological susceptibility in N_f = 2 + 1 QCD by means of spectral projectors over the spectrum of the staggered Dirac operator is presented. The spectrum of the staggered Dirac operator plays a crucial role also in the second part of the thesis, dedicated to the determination of the SU(2) chiral condensate, the order parameter of the chiral symmetry. Indeed, the main approach we adopt is the one based on the mode number of the staggered Dirac operator. As a check of consistency, we also use different approaches based on chiral pertubation theory; in particular, we present determinations of the chiral condensate from the quark mass dependence of the pion mass and the quark mass dependence of the topological susceptibility. The results contained in the first two parts of the thesis clearly show that the low-lying part of the spectrum of the staggered Dirac operator contains information both on the topological susceptibility via the index theorem and on the chiral condensate via the Banks—Casher relation. In the last part, a computation of the sphaleron rate, which describes real time finite temperature topological transitions in the QCD θ-vacuum, is presented. This topological observable has a very important phenomenological role in the QCD axion physics, being related to the axion thermal production in the Early Universe. The computation of this quantity from lattice simulations leads to an ill-posed inversion problem that consitutes a very difficult task. We first perform the computation in the SU(3) Yang—Mills theory, this is done to test our method, based on a modified version of the Backus—Gilbert approach, and make a comparison with other results in the literature. Then, we present the first determination in the literature of the N_f = 2 + 1 QCD sphaleron rate, in a temperature range going from 200 MeV to 600 MeV. Finally, some preliminary results about the extension of this computation to the non-zero momentum case are shown.