Dynamical symmetry breaking via a bound state and a comment on the O(N) sigma model

An analysis is made of the sheet structure of the effective potential for relativistic field theories that have a bound state (mass m/subB/) and it is shown that if m/subB/ 2 goes negative, the vacuum changes sheets, avoiding tachyon problems and giving dynamical symmetry breaking if the bound state has quantum numbers. A transition of this type is illustrated in the O(N) -symmetric sigma model for large N. A tachyon problem in this model noted by Coleman, Jackiw, and Politzer is resolved by finding the vacuum state of lowest energy. It is shown that this model has no symmetry breaking of any kind in the leading 1/N approximation. Any symmetry breaking the tree in approximation level is restored dynamically