Three-dimensionalO(N)theories at large distances

The infrared structure of charged, $\mathrm{O}\mathrm{}\left(N\right)\mathrm{}$-invariant field theories in three dimensions is analyzed. The zero-mass limit, appropriate for a mean-field-theory description of phase transitions in statistical physics, is studied using the $\frac{1}{N}$ expansion. The infrared divergences of the loop expansion are then eliminated, and it is shown how the low-momentum behavior is completely governed by an infrared-stable fixed point of the renormalization group. In particular, the fixed point, which appears to leading order in $\frac{1}{N}$, is shown to persist to higher orders by virtue of cancellations among terms which are singular in the zero-momentum limit. Some new computations of anomalous dimensions are presented and transcribed into the corresponding critical indices of statistical mechanics. The structure of the effective potential of the theory is summarized. It can be computed order by order in the $\frac{1}{N}$ expansion and its only minimum is at the origin.