A Study of the Low-Energy Spectrum and Phase Structure of a Yang-Mills Matrix Model

Abstract

Yang-Mills theory is a non-Abelian gauge theory based on the gauge group SU(N), and lies at the heart of the QCD, the theory underlying strong interactions. Due to the fundamental property of asymptotic freedom, a study of its low-energy spectrum and quantum phase structure requires the application of non-perturbative techniques. Since analytic non-perturbative QCD calculations are notoriously difficult and numerical approaches typically require huge computational resources, a simple model that can capture important non-perturbative information is quite useful. In this thesis, we study a matrix model that is obtained by the reduction of Yang-Mills theory on a 3-sphere of radius R and describes the quantum dynamics of the zero modes of the full quantum field theory. Even though at first glance this is a drastic approximation, we demonstrate that this model successfully captures important nonperturbative aspects of the full quantum field theory, and being a quantum mechanical model, it can be studied using relatively simple analytical and numerical techniques. In the first part of the thesis, we focus our attention to the pure SU(3) gauge theory and make a numerical estimate of the lightest glueball masses in the matrix model. The spectrum of the matrix model Hamiltonian is analyzed in the strong coupling limit using variational calculation, and by employing a suitable renormalization scheme to determine the running of the coupling constant with R, the asymptotic values of the energy eigenvalues in the at space limit is related to the masses of glueballs. Our estimate shows excellent agreement with lattice results, with our values lying within the lattice error bars. We then analyze the matrix model for the gauge field coupled to fermions and make an estimate of the light hadron spectrum using a similar scheme. We find that the matrix model estimates the light hadron spectrum fairly accurately, with most masses falling within 20% of their experimental values. In the second part of the thesis, we analyze the Yang-Mills matrix model in the weak coupling limit. In this regime, the kinetic term for the gauge fields is small compared to that of fermions, so Born-Oppenheimer approximation can be used to quantize the fermions in the background of a constant gauge field, which in turn induce a quantum scalar potential in the effective Hamiltonian governing the dynamics of the gauge fields. This scalar potential has singularities at certain regions of the gauge configuration space and leads to the emergence of superselection sectors in the Hilbert space of gauge fields, which have a natural interpretation as different quantum phases. We examine the phase structure of 2-colour QCD and demonstrate that these phases corresponding to colour-spin locking. We also examine the phase structure of the N = 1 SYM matrix model, and investigate the role that supersymmetry plays in this description.