• chiral effective Lagrangian models
• second cumulant
• theta dependence of vacuum energy density
• topological susceptibility

Quantum Chromodynamics (QCD) is the theory that describes the strong interaction within the Standard Model of the elementary particles. It is well known that, if we consider the so-called chiral limit, in which $L$ quark masses (the physical relevant case being $L=2$ and $L=3$) are sent to zero, the symmetry group of QCD becomes larger: the Lagrangian turns out to be invariant under the chiral group $G=U(1)_L \otimes U(1)_R\otimes SU(L)_L \otimes SU(L)_R$. However, it was noted that the $SU(L)_L \otimes SU(L)_R$ chiral symmetry is spontaneously broken due to the condensation of quark-antiquark pairs, while the $U(1)_A$ axial symmetry is affected by anomaly: the action is not invariant under $U(1)_A$ rotations but rather acquires a contribution proportional to the topological charge density $Q$, which (despite being a total divergence) contributes to the path integral because of the non-trivial topological structure of QCD. In particular, this is due to the existence of a class of Euclidean solutions of the classical equation of motion with finite Euclidean action and integer topological charge, known as ``instantons''.

The discovery of instantons raised an important issue: if one introduces an additional Lagrangian term $\mathscr{L}_\theta=\theta Q$ (known as ``topological term'' or ``$\theta$-term''), its contribution is non-vanishing; moreover, it explicitly breaks the CP invariance of the theory. So far no violation of the CP symmetry in strong interactions has been observed experimentally, and the most recent experimental measurements of the (CP-breaking) neutron electric dipole moment set the upper bound $|\theta|<10^{-10}$. However, the most important effect of the insertion of the topological term in QCD Lagrangian for the work of this thesis is the resulting non-trivial dependence on $\theta$ of the vacuum energy density $\epsilon_{vac}(\theta)$, which is of particular interest. Indeed, being $\theta$ very small, it makes sense to consider the Taylor expansion of the vacuum energy density around $\theta=0$: the first coefficients of such an expansion turn out to be important physical quantities. More in detail, the coefficient of the quadratic term is equal to the topological susceptibility $\chi$, whose pure-gauge part was related by Witten and Veneziano to the mass of the meson $\eta'$ in the chiral limit, while that of the fourth order term is equal to the second cumulant $c_4$ of the probability distribution of the topological charge density $Q$, which also enters in the $\eta' - \eta'$ elastic scattering amplitude.

One of the most important and useful tools to analyse the low-energy regime of QCD, paying particular attention to its chiral symmetries, is the effective Lagrangian formulation, which considers the effective mesonic degrees of freedom as the fundamental fields of the theory (often gathered in a matrix field $U$). Very important examples are: i) the ``Chiral Effective Lagrangian'', which describes the dynamics of the $L^2-1$ (non-singlet) pseudo-Goldstone bosons coming from the breaking of the chiral $SU(L)_L \otimes SU(L)_R$ symmetry; ii) the ``Extended Linear $\sigma$ Model'', that describes both scalar and pseudoscalar mesonic degrees of freedom (including also the singlets); iii) the ``model of Witten, Di Vecchia, Veneziano \emph{et al.}'', which, in the framework of an expansion for large number of colours, implements the $U(1)$ axial anomaly by properly introducing the topological charge density operator $Q$ as an auxiliary field in the effective Lagrangian.

Moreover, a fourth example of effective Lagrangian is also present in the literature, in which the $U(1)$ axial symmetry is spontaneously broken independently of the chiral $SU(L)_L\otimes SU(L)_R$ one because of the existence of a new order parameter related exclusively to the $U(1)_A$ symmetry and independent of the chiral condensate. The effects of such axial condensate on the dynamics of the mesonic effective degrees of freedom is described by including in the effective Lagrangian a new (exotic) mesonic field $X$, in addition to the usual ($q\bar{q}$) mesonic fields $U$. In this thesis, we shall refer to this model as the ``Interpolating Model'' (since, in a sense, it ``interpolates'' between the Extended Linear $\sigma$ Model and the model of Witten, Di Vecchia, Veneziano \emph{et al.}).

It is also known (mainly by lattice simulations) that, at temperatures above $T_c\approx 150$ MeV, the chiral $SU(L)_L \otimes SU(L)_R$ symmetry gets restored (in the chiral limit). For what concerns the $U(1)$ axial symmetry, its restoration or not at $T_c$ is still an important open question in hadronic physics.\\

The main goal of the work of this thesis is the systematic study of the modifications brought to the vacuum energy density by a small, but non-zero, value of the coefficient $\theta$, both in the zero temperature case and in the finite temperature one (above $T_c$), using the four effective Lagrangians described above. Our aim is to derive the expressions for the topological susceptibility $\chi$ and for the second cumulant $c_4$ starting from the $\theta$ dependence of $\epsilon_{vac}(\theta)$: this method will allow to obtain important results in a rather simple way.

First, we have considered the case $T=0$. The guideline of the computational method used here is to exploit the fact that $\theta \ll 1$: as the vacuum energy density is given by the minimum of the potential of the effective Lagrangian, we looked for the field configuration which minimizes the latter, expanding all the field variables in powers of $\theta$. This way, we are left with an expression for $\epsilon_{vac}(\theta)$ which is in the form of a Taylor expansion around $\theta=0$: this allows to directly read the expressions for the topological susceptibility $\chi$ and for the second cumulant $c_4$. In the case of the Chiral Effective Lagrangian, these expressions were already derived in the literature, using the same method. Also the final results that we have found for the topological susceptibility in the three other cases confirm some relations already known in literature (which, however, were obtained by means of a different approach, by studying directly the two-point correlation function of the operator $Q$). Instead, the results that we have found for the second cumulant are all original. We have checked that all the expressions that we have derived for both $\chi$ and $c_4$, while being different, behave as expected in some proper theoretical limits.

After that, we have evaluated numerically our results, so as to compare them with each other and with the available lattice data.

At last, we have analysed the finite-temperature case, examining the Extended Linear $\sigma$ model and the Interpolating one (which are well defined also above $T_c$). Here, we decided to base our study on the chiral limit, i.e., on the expansion in the quark masses up to the first non-trivial order: this allows to greatly simplify the computations, without any assumption on $\theta$, which in this case is a real free parameter (i.e., it is not obliged to be very small). Also in this case, our results for the topological susceptibility reproduce some expressions already known in the literature, even though they were obtained with the mentioned different approach, while those for the second cumulant are original.