Phase structure of theO(N)vector model

Abstract

In a large-$N$ analysis of the $\mathrm{O}\mathrm{}\left(N\right)\mathrm{}$ $\lambda {\Phi }^4$ theory in four dimensions, we find that the $\mathrm{O}\mathrm{}\left(N\right)\mathrm{}$-symmetric lowest-lying state is a metastable state. The decay rate of this false vacuum into a broken-symmetry state is, however, suppressed [$\sim \exp \mathrm{}(-N)\mathrm{}$]. Nevertheless, at finite temperature, we find that above a critical temperature $T_c$ the only existing phase is a broken-symmetry phase with $N-1\mathrm{}$ Goldstone bosons. This phase reflects the intrinsic instability of this theory, and the large-${\phi }_c$ structure indicates that the renormalized $\lambda {\Phi }^4$ theory is inconsistent. The only acceptable version of this theory is its regularized form which becomes a free-field theory as the regularization is removed.