Dynamical symmetry breaking via a bound state and a comment on theO(N) σmodel

We analyze the sheet structure of the effective potential for relativistic field theories that have a bound state (mass $m_B$) and show that if $m_B^{}{}_{}{}^2$ goes negative the vacuum changes sheets avoiding tachyon problems and giving dynamical symmetry breaking if the bound state has quantum numbers. We illustrate a transition of this type in the $\mathrm{O}\mathrm{}\left(N\right)\mathrm{}$-symmetric $\sigma$ model for large $N$. We resolve a tachyon problem in this model noted by Coleman, Jackiw, and Politzer by finding the vacuum state of lowest energy. We show this model has no symmetry breaking of any kind in the leading $\frac{1}{N}$ approximation. Any symmetry breaking in the tree approximation is restored dynamically.