A study on the non-perturbative dynamics of 4d gauge theories through symmetries and anomalies

Despite many decades of investigations, the infrared dynamics of 4D asymptotically free gauge theories is still an open problem. In the case of chiral theories, our knowledge is particularly unsatisfactory because of a lack of computational methods and theoretical results. In the case of vector-like theories, although computational methods give us precious insights, we still miss a fundamental and rigorous explanation of confinement and chiral symmetry breaking. In this work, we explore these matters by exploiting the only tool we have in facing strongly coupled theories: symmetries.This thesis naturally divides into two parts.The first focuses on the study of chiral gauge theories. After we define our playground, the set of chiral asymptotically free SU(N) gauge theories with a smooth large-N limit, we focus on a few characteristic examples.Thanks to the recent improvement in the understanding of symmetries (e.g. higher-form symmetries) and 't Hooft anomalies, some chiral models (such as the Bars-Yankielowicz, the Georgi-Glashow, and generalizations thereof) are revisited. Surprisingly, we found some putative infrared phases, well known in the literature, not to be fully consistent with the renewed checks.Then, we explore other dynamical possibilities, such as color-flavor locking and dynamical abelianization and we delve into the description of subtler aspects of the low-energy effective action, e.g. how the ABJ anomalous symmetries are recovered in the large-N limit.The second part of the thesis work, which is somewhat more technical, is dedicated to vector-like theories.First, we review a recent attempt to understand confinement and chiral symmetry breaking in QCD by starting from N=1 SUSY QCD and deforming it by adding a soft SUSY breaking term (anomaly-mediated supersymmetry breaking). Then, we explain why this approach is problematic as a phase transition between a small SUSY breaking regime and QCD (large SUSY breaking) is expected (at least for some numbers of flavors).Finally, we turn our attention to an old series of works from the 80', which, assuming color-confinement, claims to prove chiral symmetry breaking in QCD by using perturbative anomalies and general knowledge of vector-like theories (persistent mass conditions). After refining some of the statements in the literature, we point out the weak spots of the original proofs, and then we provide a new one that closes the gaps.