• confinement
• connected correlator
• divergences
• flux tubes
• lattice gauge theories
• monte carlo
• non perturbative
• qcd
• renormalization
• smoothing
• strong interactions
• SU(3)
• Wilson flow
• Yang-Mills

Colour confinement is one of the characteristic features of Quantum Chromodynamics (QCD), the gauge theory describing strong interactions with SU(3)
as a local gauge group. Due to the fact that, in the low energy regime, QCD is
a non-perturbative theory, a solid theoretical explanation of the confinement
mechanism is still missing, whereas phenomenological and numerical evidences
of the colour confinement are by now overwhelming.
Considering only the Yang-Mills part of the theory, i.e. QCD with no quarks,
the confinement mechanism is related to a new symmetry: the center symmetry.
This has as order parameter the mean value of the so called Polyakov loop,
which vanishes in the confined and is different form zero in the deconfined
phase.
In Yang-Mills theories a standard method to study colour confinement is the
investigation of the chromoelectric flux tube. These tube structures arise when a
couple of static colour sources is placed at a certain distance from each other in
the vacuum. One can see by numerical simulations that the chromelectric field
does not spread all over the space, but it is squeezed into a narrow flux tubes
connecting the sources. The most appealing interpretation of this phenomenon
is that the vacuum of the theory acts like a dual superconductor, from which
the dual Meissner effect expels the chromoelectric field.
Being colour confinement a non-perturbative phenomenon, the best way we
have to study it is by numerical simulations of the theory discretized on the
lattice, performed by using Monte Carlo methods. In doing this, one has to deal
with some problems: taking care of the non-physical ultra violet fluctuations
and the renormalization of the operators used to compute physical quantities
are two examples.
Cooling has been the most applied procedure to take care of the UV divergences. However this method has been used in the past without worrying too
much of the possible systematical errors it can introduce in the study of the
colour flux tubes.
The aim of this thesis is to investigate the chromoelectric flux tube with the use
of a controlled smoothing procedure: the Gradient Flow. This method smooths
the configurations according to a differential equation, ensuring a
better analytical control. In particular, we want to understand if there exists a
regime of flow times big enough in order not to see ultra violet divergences, but
also smaller than the typical physical scale of the problem, within which the
Gradient Flow could be used as a smoothing/renormalization procedure for
the observables we are considering. The major difference between our approach
and the previously adopted ones is that, while performing the continuum limit,
the flux tube data are taken at several fixed physical flow times for all the
different lattices, allowing us to control smoothing artifacts.
The work is organized as follows:
• In the first chapter a brief discussion of the basic properties of QCD is
presented both in the continuum and in the lattice formulation. Here we
give the basic definitions that will be used in the following;
• In the second chapter we describe the numerical algorithms which are
used to simulate Yang-Mills theory on the lattice. Then we explain how
data obtained from the simulations has to be analyzed; finally we describe
the Gradient Flow in the continuum and in its lattice version. A numerical
implementation of the latter is sketched in the appendix;
• In the third chapter we provide the definitions of the observables used
in our work and their renormalization properties, as well as a brief description of the dual superconductor framework and the dual Meissner
effect;
• In the fourth and final chapter the original results of the thesis are presented. After some preliminary investigations, the results obtained for
the flux tubes will be discussed, together with the determination of the
parameters describing the vacuum in the dual superconductor picture.