Numerical Stochastic Perturbation Theory for Lattice QCD: computation of renormalization constants and prospects for the study of the Dirac operator spectrum

This thesis accounts for my research project in NSPT. From a technical point of view, it actually covers three quite different fields. As in any numerical investigation, my research project required an extensive programming work. My PhD activity was the chance for what is technically known as a refactoring of Parma group LGT (and in particular NSPT) codes. This was in particular true in a phase in which the group was moving to the usage of new multi-cores architectures and (even more) of a new parallel platform (Aurora). As it is often the case for such a work, there is no obvious way to account for this in a thesis without letting this numerical work actually take over Physics. The solution which I chose is to give a brief methodological account of the package which is the result of my activities (PRlgt). I would regard the computation of three loop renormalization constants as the core business of this work. Perturbative versus non-perturbative renormalization has been an issue for quite a long time. As a matter of fact, such a comparison has always been limited by the low order at which perturbative results were available. The ambitious goal of my work was to get to three loop results for quark bilinears (in a given regularization) with a fair account of all the systematics. This in turn enables us to better assess the level of confidence of non-perturbative results. The bottom line is that there are in general more indeterminations in non-perturbative determinations than it is usually stated: truncation errors are not the end of the story. As a last subject, the work pins down the prospects for an NSPT study of the Dirac operator spectrum. The spirit is to investigate the pattern of chiral symmetry breaking from Banks-Casher relation: can one inspect thereshuffling of Dirac operator eigenvalues due to color interactions? This part of the work has by far a different status with respect to renormalization constants computation. While I put the technical basis for a computation that has not even been attempted before with traditional techniques, there is a long way to go to get quantitative results. I try to give a qualitative account of what emerges from one loop bare perturbation theory: eigenvalues do repel each other and the spectrum is well rearranged with respect to the free field