Non-perturbative physics in lattice gauge theories

Resumen

A few decades have passed since quantum chromodynamics (QCD) was established as the theory describing strong interactions. It is broadly accepted as one of the most successful theories in modern physics, and it has been extensively tested, both from the theoretical and the experimental perspectives. At high energies, QCD is asymptotically free, which means that its fundamental constituents, quarks and gluons, interact with a strength that decreases as the energy scale reaches higher values. In this regime, it is feasible to use perturbation theory to resolve short distance interactions. On the other hand, for not-so-high energies, the strong interaction cannot be reduced to a converging series of Feynman diagrams. In fact, one of the characteristic properties of QCD is the so-called color-confinement. In this purely non-perturbative regime, there are few techniques that can analyze the theory successfully. Probably the most well-established of them is lattice QCD. Since the foundational work of Wilson in 1974, the success of the lattice approach has been growing consistently over time. Many milestones have already been reached, including precise simulations that account for the effects of virtual quark loops, the determination of the light hadron spectrum with fully controlled systematics or, more recently, the computation of the isospin splittings with great agreement with the experimental data. For the above reasons, QCD is believed to be the correct theory describing strong interactions, both for high and low energies, and lattice QCD is recognized by the community as a trustworthy ab initio approach that has an useful interaction with experiment, paraphrasing Wilson. However, there are some fundamental topics that still constitute open questions. At least two problems share this status: the behavior of matter at finite baryonic density and the studies involving topological effects in QCD. The main difficulty behind the modest progress achieved in both areas is the same: the action of the theory is complex, and there is no known reformulation that can avoid the appearance of a severe sign problem (SSP). In this context, the main part of this thesis has been devoted to study models which suffer from a SSP, such as the two-dimensional Ising model within an imaginary magnetic field or the massive 1-flavor Schwinger model with a theta term. In the first case, we study the well-known model by means of analytical techniques, exploring a region of the parameter space somewhat unattended by the literature, possibly due to the difficulty of applying either analytical or numerical techniques. Secondly, and with the aim of engaging with QCD-like systems with a theta term and to develop further the methods dealing with the SSP, we have studied the massive 1-flavor Schwinger model with a $\theta$ term, which corresponds to QED in $1+1$ dimensions, and is in fact broadly used as its toy model. Moreover, defining the topological charge on the lattice is almost trivial in this model, in contrast with any of the usual definitions of this observable in lattice QCD, which are much more involved. As a byproduct of this line of work, and driven by the necessity of optimizing further our previous algorithms, we have also analysed the 2-flavor version of the Schwinger model. In this case, we have bypassed the computation of the full fermionic determinant by following an approach based on the use of pseudofermions. Finally, beyond the study of systems afflicted by a SSP, another topic within lattice QCD has been treated during the development of this thesis: the strong running coupling alpha_S. Its dependence with the momentum transfer, which encodes the underlying interactions of quarks and gluons in the QCD framework, constitutes a very active field of research, that includes a large variety of approaches. At large momenta, where perturbative QCD can be applied, both experimental and theoretical methods try to provide the most accurate approximation. In this context, lattice-based strategies have been capable of delivering results both in the infrared region and in the high energy regime, where in fact they provide the most precise determination of the coupling constant. Our work can be framed precisely into these approaches that come from lattice QCD, and it relies upon a ghost-gluon vertex computation.