Multiple scales on the lattice

The regularization on a Euclidean lattice provides a mathematically well-defined, gauge-invariant, non-perturbative definition of Quantum Chromodynamics (QCD), the theory that describes the strong nuclear interaction in the Standard Model of elementary particle physics. Moreover, the lattice regularization also lends itself to the computation of physical quantities, such as the masses of hadrons, from first principles, by means of numerical evaluation of the functional integrals of the theory. While the finiteness of both the lattice spacing a and the linear extent of the lattice L provides a regulator for both ultraviolet and infrared divergences, it also poses constraints on the physics that can be probed in these calculations: the Compton wavelengths of all particles that are studied should be much longer than a (to avoid contamination with unphysical artifacts due to the lattice cutoff), while at the same time they should also be much shorter than the linear extent of the lattice (in order not to be affected by finite-size effects). When several states, with widely separated masses, have to be studied at the same time, it becomes difficult to simultaneously satisfy these constraints for all of the particles involved, because they lead to the requirement of lattices with a huge number of sites, implying very demanding computer-memory requirements and extremely long simulation times. In this M.Sc. thesis, we study a possible way to tackle this problem: we focus on the purely gluonic sector of the theory, and discuss a construction whereby the "resolution" of the lattice discretization can be iteratively improved in limited portions of the original system, by means of a sequence of anisotropic lattices, in which the lattice spacing along the Euclidean-time direction is made finer and finer. To this purpose, we discuss in detail the problem of a non-perturbative scale-setting on the lattice, focusing on a technique based on the gradient flow. Then, we discuss the extension of the application of the gradient flow to simulations based on an anisotropic version of the lattice gauge action, and present original results from numerical simulations performed on the supercomputing clusters of CINECA. This work represents the first step towards a systematic treatment of multiple energy scales in lattice gauge theory: its potential applications include the physics of bound states of heavy quarks in the quark-gluon plasma, flavour physics, and even composite-Higgs models that might be relevant for physics beyond the Standard Model.