Vacuum stability in three-dimensionalO(N)theories

We investigate the stability of the symmetric vacuum in massless, charged, $\mathrm{O}\mathrm{}\left(N\right)\mathrm{}$-invariant theories in three dimensions, using the effective potential in the $\frac{1}{N}$ expansion. We establish a one-to-one correspondence between the small-$\left|\overrightarrow{\phi }\right|$ behavior of the effective potential and the renormalization-group structure of the theory which was investigated in an earlier paper. We study the relation between the large-$\left|\overrightarrow{\phi }\right|$ limit of the effective potential and the ultraviolet behavior of Green's functions, and we establish the usefulness of the loop expansion in this limit. We find that the effective potential has only one minimum, at the origin, verifying the stability of the symmetric vacuum already concluded from the renormalization-group study.