• axial anomaly
• condensate
• instantons
• QCD
• spectral density

Quantum Chromo-Dynamics (QCD) is the quantum field theory of strong interactions and describes hadrons as bound states of quarks, which interact by exchange of gluons. The vacuum state of QCD is characterized by certain non-vanishing condensates which cannot be understood in the framework of perturbation theory. In the so-called chiral limit, in which $N_f$ quark masses are sent to zero ($N_f=2$ and $N_f=3$ being the physically relevant cases), the QCD Lagrangian turns out to be symmetric under the chiral group $U(1)_V\otimes U(1)_A\otimes SU(N_f)_V\otimes SU(N_f)_A$. In the quantum theory, the subgroup $SU(N_f)_V\otimes SU(N_f)_A$ is spontaneously broken down to $SU(N_f)_V$ because of the condensation of quark-antiquark pairs, which gives rise to the so-called chiral condensate $\Sigma\equiv\langle q\overline{q} \rangle=\sum_{i=1}^{N_f}\langle q_i(x)\overline{q}_i(x) \rangle$, where the brackets $\langle \ldots \rangle$ stand for the vacuum expectation value at zero temperature or, more generally, for the thermal average at finite temperature $T$. On the other hand, the $U(1)_A$ (axial) symmetry is broken by the \textit{quantum anomaly}: at the quantum level, under $U(1)_A$ transformations the action acquires a contribution proportional to the so-called topological charge $Q$. Although $Q$ is the integral of a total divergence, it contributes to the path integral because of the existence of topologically non-trivial gauge configurations known as instantons, which are Euclidean solutions of the classical equations of motion with finite action and integer topological charge.
Moreover, it is also known that, at a certain critical temperature $T_c\approx 150$ MeV, QCD (in the chiral limit) undergoes a phase transition which restores the $SU(N_f)_V\otimes SU(N_f)_A$ symmetry: the chiral condensate $\Sigma$, which is just an order parameter of this symmetry, vanishes above $T_c$. Instead, the fate of the $U(1)_A$ symmetry above the transition remains unclear. Although the quantum anomaly is present at any finite temperature, at some point its effects could become practically negligible: if so, the $U(1)_A$ symmetry would be effectively restored. It has been argued that this effective restoration could occur simultaneously to the chiral one at $T_c$, affecting the order of the transition: this issue is, however, highly controversial.
Traditionally, this question has been investigated, at least in the case $N_f=2$, by studying (mainly by lattice simulations) the so-called chiral susceptibilities. For each meson channel $M$ ($\sigma$, $\delta$, $\pi$ and $\eta$), the chiral susceptibility $\chi_M$ is defined as the integral over four-space of the two-point correlation function $\langle J_M(x)J^\dagger_M(0) \rangle $ of the corresponding interpolating operator $J_M(x)=\overline{q}(x)\Gamma_M q(x)$, for some proper matrix $\Gamma_M$ in Dirac and flavour space. The importance of these objects lies in the fact that the various meson channels are mixed under $SU(2)_A$ and $U(1)_A$ transformations: as a symmetry gets restored, the susceptibilities of the meson channels that are mixed under that symmetry become degenerate. In particular, besides the chiral condensate $\Sigma$, also the differences $\chi_\pi-\chi_\sigma$ and $\chi_\delta-\chi_\eta$ can be regarded as order parameters of the $SU(2)_A$ symmetry and must vanish above the critical temperature $T_c$, as confirmed by lattice simulations. On the other hand, $\chi_\pi-\chi_\delta$ and $\chi_\sigma-\chi_\eta$ behave as order parameters of the $U(1)_A$ symmetry. Several lattice simulations, measuring these quantities, have been carried out, but the results achieved so far are not yet conclusive: most of the studies find that the $U(1)_A$-breaking difference $\chi_\pi-\chi_\delta$ is still non-zero above the chiral transition, but some others report an effective $U(1)_A$ restoration already at $T_c$.

The aim of this work is to study some possible $U(1)$ axial condensates in the high-temperature chirally-restored phase of QCD, by means of two nonperturbative analytical techniques: (\textit{i}) by expressing the functional averages $\langle\ldots\rangle$ in terms of the spectral density of the Euclidean Dirac operator $\slashed{D}$, and (\textit{ii}) through an approximate evaluation of the path integral in the instanton background.
The $U(1)_A$ condensates that we shall first consider are functional averages of local $2N_f$-fermion operators of the form $\mathcal{O}_{U(1)}(x)\sim\det_{st}\overline{q}_s(x)\frac{1+\gamma_5}{2}q_t(x)$, where $s,\,t$ are flavour indices (while the colour indices can be contracted in different possible ways, so to give a colour singlet), so that they are invariant under the whole chiral group except for the $U(1)_A$ transformations. In this work, we shall also consider global $U(1)$ axial condensates, taking the functional average of multilocal operators of the form $\mathcal{O}_{U(1)}(x_1,\ldots,x_{N_f})\sim \varepsilon^{i_1\ldots i_{N_f}}\overline{q}_1(x_1)\frac{1+\gamma_5}{2}q_{i_1}(x_1)\ldots\overline{q}_{N_f}(x_{N_f})\frac{1+\gamma_5}{2}q_{i_{N_f}}(x_{N_f})$ (i.e., performing a "point splitting" of the $N_f$ quark bilinears contained in the expression of the local operator $\mathcal{O}_{U(1)}(x)$), and then integrating over the four-space coordinates.
In chapter 3, after a systematic and critical review of the results which can be obtained by analysing the chiral susceptibilities using the spectral-density technique (for different possible choices of the spectral density), we shall apply this same technique to study also the above-mentioned $U(1)$ axial condensates and their relations with the chiral susceptibilities, as well as with the so-called topological susceptibility $\chi_{top}$ (which is, essentially, the zero-momentum two-point correlation function of the topological charge density).
In chapter 4, instead, we shall explicitly compute the $U(1)$ axial condensates in the high-temperature phase, using the instanton-background approximation of the functional integral. In this way, besides proving that these condensates are indeed different from zero in the high-temperature regime, we shall also derive their asymptotic temperature dependence and compare it with that of the topological susceptibility $\chi_{top}$.