Quantum Field - Theory Models in Less Than 4 Dimensions

Abstract

The scalar ${\ensuremath{\lambda}}_{0}{\ensuremath{\varphi}}^{4}$ interaction and the Fermi interaction ${G}_{0}{(\overline{\ensuremath{\psi}}\ensuremath{\psi})}^{2}$ are studied for space-time dimension $d$ between 2 and 4. An unconventional coupling-constant renormalization is used: ${\ensuremath{\lambda}}_{0}={u}_{0}{\ensuremath{\Lambda}}^{\ensuremath{\epsilon}}$ ($\ensuremath{\epsilon}=4\ensuremath{-}d$) and ${G}_{0}={g}_{0}{\ensuremath{\Lambda}}^{2\ensuremath{-}d}$, with ${u}_{0}$ and ${g}_{0}$ held fixed as the cutoff $\ensuremath{\Lambda}\ensuremath{\rightarrow}\ensuremath{\infty}$. The theories can be solved in two limits: (1) the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$ where $\ensuremath{\varphi}$ and $\ensuremath{\psi}$ are fields with $N$ components, and (2) the limit of small $\ensuremath{\epsilon}$, as a power series in $\ensuremath{\epsilon}$. Both theories exhibit scale invariance with anomalous dimensions in the zero-mass limit. For small $\ensuremath{\epsilon}$, the fields $\ensuremath{\varphi}$, ${\ensuremath{\varphi}}^{2}$, and $\ensuremath{\varphi}{\ensuremath{\nabla}}_{{\ensuremath{\alpha}}_{1}}\ifmmode\cdot\else\textperiodcentered\fi{}{\ensuremath{\nabla}}_{{\ensuremath{\alpha}}_{n}}\ensuremath{\varphi}$ all have anomalous dimensions, except for the stress-energy tensor. These anomalous dimensions are calculated through order ${\ensuremath{\epsilon}}^{2}$; they are remarkably close to canonical except for ${\ensuremath{\varphi}}^{2}$. The ${(\overline{\ensuremath{\psi}}\ensuremath{\psi})}^{2}$ interaction is studied only for large $N$; for small $\ensuremath{\epsilon}$ it generates a weakly interacting composite boson. Both the ${\ensuremath{\varphi}}^{4}$ and ${(\overline{\ensuremath{\psi}}\ensuremath{\psi})}^{2}$ theories as solved here reduce to trivial free-field theories for $\ensuremath{\epsilon}\ensuremath{\rightarrow}0$. This paper is motivated by previous work in classical statistical mechanics by Stanley (the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit) and by Fisher and Wilson (the $\ensuremath{\epsilon}$ expansion).