Properties of the \(O(N)\) Vector Model in Various Dimensions

Abstract

In this thesis, we study the \(O(N)\) vector model, which is a quantum field theory of \(N\) real scalar fields \(\phi^{i}\) with a quartic interaction \(\frac{\lambda}{4}(\phi^{i}\phi^{i})^{2}\). We will study the theory in a range of dimensions using two complementary approximation schemes, epsilon expansion in dimensional continuation and the large \(N\) approximation. First, we describe various classical solutions of the \(O(N)\) model on the sphere \(S^{d}\) for \(d=6\) and \(8\). Then by computing the determinant of fluctuations about these classical backgrounds, it is possible to provide an alternative method of computing the beta functions in \(d=6-\epsilon\) and \(d=8-2\epsilon\). Finally, we also consider some of the thermodynamic properties of the critical \(O(N)\) model by placing the theory on the thermal cylinder \(S^{1}\)\(\times\)\(\mathbb{R}^{d-1}\). We find that in the range \(2< d<4\) and \(6< d<8\), there exists a real solution for the thermal mass. Interestingly, the leading order contribution to the thermal mass in \(d=6+\epsilon\) and \(d=8-\epsilon\) is found to have a fractional dependence on \(\epsilon\). This is understood by deriving the result diagrammatically and summing over an infinite class of diagrams.