Discretized Light-Cone Quantization of Two-Dimensional SU\((N)\) Gauge Theories

Abstract

This thesis studies two \((1+1)\)-dimensional theories of quantum chromodynamics with gauge group \(\mathrm{SU}(N)\). The first theory, \(\mathcal{T}\), consists of \(N_\mathrm{adj}\) flavors of gluinos and one massive quark. The second theory, \(\mathcal{T}'\), consists of \(N_\mathrm{fund}\) massless quarks and one massive quark. We use a method called discretized light-cone quantization (DLCQ) to find the mass spectra of these theories numerically. At large \(N\), we use a mathematical correspondence between \(\mathcal{T}\) and \(\mathcal{T}'\) to indirectly compute the height of the quark-antiquark interaction potential in \(\mathcal{T}\) for the case of zero gluino mass. We show that, to first order in \(1/N_\mathrm{adj}\), this numerical result agrees with a perturbative Wilson loop calculation. We extend the Wilson loop calculation to massive gluinos, demonstrating that the theory transitions to a confining phase as the gluino mass is turned on. Finally, we consider the more difficult case of \(N=2\). We restrict to the pure gluino theory and develop new techniques based on \(\mathfrak{su}(2)\) representation theory to vastly improve the efficiency of the numerics. Our results show that, when the gluino mass is \(m^2_\mathrm{adj}=2\,g^2/\pi\), the bosonic and fermionic spectra approach the same values in the continuum limit, which is the first numerical evidence for supersymmetry at finite \(N\).