Quantum Chromodynamics (QCD) describes strong interactions among the fundamental particles known as quarks and gluons. In principle, QCD can be used to explain complicated phenomena in the strong sector. However, at the energy scale of hadronic physics, the strong coupling constant is so large that the traditional perturbative method is not applicable. There are two powerful alternative approaches utilized in the non-pertubative regime: (1) lattice QCD, which discretizes space-time and utilizes Monte Carlo computer simulations, and (2) finding new systematic expansion regimes to obtain physical insights in certain limits.

In this dissertation, three problems are studied in the context of these two approaches. In Chapter 2, a notorious numeric problem in lattice QCD known as the sign problem is explored. A subtle phenomenon caused by the interplay between the sign problem and the infinite volume limit is discussed and explained using the saddle point approximation. This work provides insight into the sign problem and the physics of the QCD theta-vacuum.

Chapter 3 and Chapter 4 focus on tetraquarks, which are unconventional hadrons containing four valence quarks. Despite numerous tetraquark candidates seen in experiments, there is no unified and well-accepted theoretical descriptions of the tetraquark state yet. This dissertation examines the existence of tetraquarks in the heavy quark mass limit. A powerful systematic expansion regime can be built when the heavy quark mass is extremely large.

In Chapter 3, a framework is established to analyze tetraquarks in the heavy quark mass limit. It is shown in a model-independent way that multiple parametrically narrow doubly heavy tetraquarks must exist in this limit. Many of these states will be parametrically close to the threshold of decaying into two heavy mesons.

In Chapter 4, based on a modification of the framework in Chapter 3, it is shown that tetraquarks containing a heavy quark and a heavy antiquark with appropriate large angular momentum must exist in the heavy quark mass limit. This may provide insights into the experimentally-observed narrow near-threshold tetraquarks which contain a heavy quark and a heavy antiquark.