• lattice
• QCD
• reticolo
• sezione d'urto totale
• total cross section
• unitarity
• universal
• universale

Since 1973, the year in which it was first formulated, Quantum Chromo-Dynamics (QCD) has been recognized as the fundamental theory of strong interactions, and now it is widely believed to be so. In spite of the many successes, there is a number of phenomena which are not yet fully understood, like confinement, chiral symmetry breaking, hadronization and also the one on which this thesis is focused, i.e., the high-energy rising behavior of hadronic total cross sections. In general, these problems are expected to be related to nonperturbative properties of the theory, i.e., properties which cannot be studied with perturbative methods. In fact, it is known that a non-Abelian Gauge theory, such as QCD, has two properties (indeed, two sides of the same coin!), namely the \emph{asymptotic freedom} and the \emph{infrared slavery}, that can be described with the behavior of the running coupling constant $g$: the former states that at high-energy scales (with respect to the intrinsic energy scale of strong interactions, that is $\Lambda_{\text{QCD}}$) $g$ is small, making perturbation theory reliable in this regime, while the latter states that, at low energy scales, it becomes larger (at first order it even blows up), making nonperturbative methods necessary. The problem of the high-energy behavior of hadron-hadron total cross sections can be approached through the investigation of the \emph{soft} high-energy hadron-hadron elastic scattering. In this kind of processes there are two different energy scales, namely the total center-of-mass energy $\sqrt{s}$, which is a large scale (let's say: $\sqrt{s}\gg1$ GeV) and the transferred momentum $\sqrt{|t|}$, which is a \emph{soft} scale, smaller than the typical energy scale of strong interactions (let's say: $\sqrt{|t|}\lesssim 1\,\text{GeV}\ll \sqrt{s}$), so that it is necessary to resort to nonperturbative methods. The relation between \emph{soft} elastic scattering and the total cross section at high energy is provided by the \emph{optical theorem}, which relates the latter to the imaginary part of the corresponding forward elastic scattering amplitude, i.e., $\sigma_{\text{tot}}(s)\sim\frac{1}{s}\im\, M (s,t=0)$.\\
A fully nonperturbative approach to the problem of \emph{soft} high-energy scattering from the first principles of QCD has been proposed in 1991 by O. Nachtmann and further developed in the following years by many authors. In this approach, the scattering of two hadrons is described as the elastic scattering of their constituents, the so-called \emph{partons}, that, in turn, is related to the correlation function (in the sense of QCD functional integral) of certain non-local operators, the so-called \emph{Wilson lines} and \emph{Wilson loops}. Then, the physical (i.e., hadron-hadron) scattering amplitudes can be recovered from the partonic ones after folding the latter with the appropriate hadronic wave functions.
This approach will be briefly discussed in Chapter 1, after a short review of the current experimental picture concerning hadronic high-energy scattering and a brief description of QCD.

In Chapter 2, using the above-mentioned functional integral approach, we show in detail that a certain Wilson-loop correlator (folded with proper hadronic wave functions) is nothing but the hadron-hadron scattering amplitude in impact-parameter space (i.e., a partial-wave scattering amplitude) in the high-energy limit. Therefore, the unitarity condition on the scattering amplitude immediately translates into a \emph{unitarity constraint} on the Wilson-loop correlator itself (in the high-energy limit), which will be exploited in the analysis done in Chapter 3.\\
Moreover, in Chapter 2 we also introduce some other physical quantities (such as the differential and total elastic cross section), which can be expressed in terms of the Wilson-loop correlation function: these quantities will be the object of the analysis done in Chapter 4.

In field theory, most of the nonperturbative techniques are available only in the Euclidean formulation. For this reason, it is necessary to relate the relevant correlation functions to the corresponding Euclidean quantities via a proper \emph{analytic continuation}. The way to reconstruct Minkowskian correlation functions starting from their Euclidean counterparts (and viceversa) has been achieved in previous works by E. Meggiolaro.
In further works (by M. Giordano and E. Meggiolaro) also \emph{crossing-symmetry relations} for the correlation functions have been derived and, most importantly, a fully nonperturbative foundation of the \emph{analytic-continuation relations} has been provided. A brief review of these results is reported in Chapter 1.
The \emph{analytic-continuation relations} have allowed the nonperturbative investigation of correlators (and the corresponding scattering amplitudes) using some analytical models, such as the \emph{Stochastic Vacuum Model} (SVM), the \emph{Instanton Liquid Model} (ILM), the AdS/CFT \emph{correspondence} and, finally, they have also allowed a numerical study by Monte Carlo simulations in \emph{Lattice Gauge Theory} (LGT). Although the numerical results obtained on the lattice can be considered \virg{exact} (since they are derived from \emph{first principles} of QCD), it is not possible to relate them directly to physical quantities, since the analytic continuation of the correlator can be performed only if an analytical dependence on the variables is known, while lattice data can be obtained only for discretized finite set of values. However, it is possible to test the goodness of the known analytical models (like the ones we have mentioned above) simply through a best-fit to the lattice data. This analysis has been already done (by M. Giordano and E. Meggiolaro) and it is briefly recalled in the first part of Chapter 3. The result of this analysis is in general not satisfactory: known analytical models lead to bad quality best-fits and, moreover, none of them provide a physically acceptable total cross section. SVM and ILM lead to constant total cross sections at high energy, while the AdS/CFT correspondence leads to a power-like behavior $\sigma_{\text{tot}}\sim s^{\frac{1}{3}} $ that explicitly violates the Froissart bound $\sigma_{\text{tot}}\leq \frac{\pi}{m_\pi^2}\log^2(\frac{s}{s_0}) $ (with $m_\pi$ the pion mass and $s_0$ a given unspecified scale), which is expected to be valid for the hadronic scattering.

Therefore, the second part of Chapter 3 is focused on the search for a new parameterization of the (Euclidean) correlator that, in order: $i)$ fits well the lattice data; $ii)$ satisfies (after analytic continuation) the unitarity condition found in Chapter 2; and, most importantly, $iii)$ leads to a rising behavior of total cross sections at high-energy, in agreement with experimental data. In particular, one is interested in the dependence of the correlation function on the angle $\theta$ between the loops, since it is related, after analytic continuation, to the energy dependence of the scattering amplitudes, and also in its dependence on the impact-parameter distance. In Chapter 3 we show that, making some reasonable assumptions about the angular dependence and the impact-parameter dependence of the various terms in the parameterization, our approach leads quite \virg{naturally} to total cross sections rising asymptotically as $B\log^2 s$ (that is what experimental data seem to suggest). Moreover, in our approach the coefficient $B$ comes out to be \emph{universal}, i.e, the same for all hadronic scattering processes (as it also seems to be suggested by experimental data), being related to the mass-scale $\mu$ which sets the large impact-parameter behavior of the correlator (and which is expected to be proportional to the lightest glueball mass $m_G$ or to the inverse of the so-called \virg{\emph{vacuum correlation length}} $\lambda_{vac}$). This is actually the main result of this thesis.

In Chapter 4, making use of the results found in the previous chapter, we shall perform a preliminary (both analytical and numerical) study of the physical quantities that we have introduced in Chapter 2 (in terms of a general correlation function).

Finally, in the Conclusions we review the results obtained and we show some possible prospects for further studies.