Phase transition in the nonlinear $\sigma$ model in a ($2+\epsilon$)-dimensional continuum

We study the phase transition in the nonlinear $O\left(N\right)\mathrm{}$ $\sigma$ model in $2+\epsilon$ dimensions. Our analysis is of the continuum theory and does not rely upon the artifice of a lattice. This phase transition occurs at a critical value of the coupling constant ${\lambda }_c$, which is an ultraviolet-stable fixed point of the renormalization group. In the "low temperature" phase the $O\left(N\right)\mathrm{}$ symmetry is realized nonlinearly with $N-1\mathrm{}$ massless pions. By solving the theory in the large-$N$ limit, to leading order in $\frac{1}{N}$, we show that in the "high temperature" phase the pions gain mass and there appears a new particle, $\sigma$, which is a bound state of the $\pi \text{'}\mathrm{s}$ and is degenerate with them. Furthermore, by a general steepest-descent approximation to the generating functional and by explicit calculations it is shown that this upper phase is fully $O\left(N\right)\mathrm{}$ symmetric and can be described by a linear $\sigma$-model Lagrangian. The unitarity of the theory is demonstrated and analogies with quark confinement in quantum chromodynamics are discussed. We prove the renormalizability of the theory, taking special care to separate infrared and ultraviolet divergences.