• 4 dimensional
• asymptotic freedom
• color
• confinement
• connected operator
• continuum limit
• cusp
• divergences
• dual
• effective
• Flux tubes
• infrared slavery
• lattice
• loop
• low energy
• mixing
• model
• monte carlo
• multilevel
• non perturbative
• polyakov
• qcd
• quark
• renormalization
• smooth
• string
• strong interaction
• SU(3)
• superconductor
• theory
• yang-mills

All the forces present in the universe are manifestations of four fundamental interactions: gravity, electromagnetic interaction, weak interaction and strong interaction. The modern theory of strong interactions is Quantum Chromodynamics (QCD), a gauge theory based on the color group, which quantitatively describes a huge amount of different phenomena, ranging from the hadron spectrum to deep inelastic scattering.
A fundamental property of QCD is asymptotic freedom: for high energy processes the effective coupling of QCD becomes small and the theory is weakly interacting. The low energy counterpart of asymptotic freedom is infrared slavery: at low energies the effective coupling becomes strong and non-perturbative effects are dominant. This suggests that QCD has two different regimes: at short distances the coupling is small and it is the natural parameter for a perturbative expansion, at large distances the physics is dominated by non perturbative effects and this suggests the possibility that color confinement is encoded in the QCD Lagrangian.
An intuitive picture of the color confinement is the following: let us consider an hadron consisting of an heavy quark-antiquark pair at distance R, and let V (R) be the energy of that state. If V increases indefinitely with the interquark distance, we have a confining theory. In fact, attempts to separate such a state would result in an infinite energy cost, moreover a distance exists at which the field energy is enough to create a new quark- antiquark pair from the vacuum. At that point string breaking happens (a phenomenon also known as “hadronization” in different contexts) and the original system separates into two parts. What can be the origin of this indefinite increase of the potential? A possible explanation is that the force lines of QCD do not behave as that of QED: in presence of two color static charges the field lines do not form a simple dipole structure but they cluster together, forming a tube of flux connecting the quark and the antiquark; this is sometimes called flux tube picture of confinement.
Despite the fact that some theoretical models exist to describe the non perturbative physics of the flux tubes (like the dual superconductor model and the effective string theory), the most natural framework in which to address these questions is that of lattice QCD. This is a non perturbative regularization of the theory that enables to study low energy properties by means of numerical simulations. Moreover a useful simplification (both theoretically and computationally) consists in giving the quarks an infinite mass and decoupling them from the gluon dynamics. In this approximation, quarks do not contribute to any virtual process, including string-breaking processes. This enables us to focus on the dynamics of the gluon field described by the Yang-Mills theory.
The lattice study of flux tubes in Yang-Mills theory has a long history, which dates back to mid’ 80s, and several different strategies have been pursued. Two main observables have been taken into account for such studies, the so-called disconnected and connected observables: the disconnected one was used in high precision studies, while the connected one was the preferred choice when it was not possible to reach high statistics, because of its larger signal-to-noise ratio. Smoothing techniques have been also traditionally used to further reduce errors, however these techniques can also introduce systematic errors in the results; for this reason in this thesis we study (with the connected observable) flux tubes in a 4-dimensional SU(3) Yang-Mills theory using a stochastically exact error reduction technique. This will force us to properly take into account some renormalization subtleties that have been so far overlooked.
The outline of the thesis is the following: in Chapter 1 we briefly review some basic facts about non–Abelian gauge theory in general, discussing gauge transformations, Yang- Mills theory in the continuum, the path integral formulation and asymptotic freedom. In Chapter 2 the euclidean formulation and the regularization on the lattice are introduced and confinement order parameters are discussed, together with the continuum limit. The necessary statistical tools used to analyze data from numerical simulations are discussed in Chapter 3, where the Monte Carlo method is also reviewed, together with the main update algorithms to be used and the multilevel algorithm for the computation of the observables. Possible theoretical models to compare with are briefly introduced in Chapter 4. In Chapter 5, we present our numerical results and their interpretation, finally, in Chapter 6 we draw our conclusion and discuss possible future developments.