Bound states, tachyons, and restoration of symmetry in the1Nexpansion

An extensive analysis of the $\frac{1}{N}$ expansion of $\mathrm{O}\mathrm{}\left(N\right)\mathrm{}$-symmetric $\lambda {\phi }^4$ theory in four dimensions shows it to be a consistent approximation method. It is confirmed that the ground state of the theory is $\mathrm{O}\mathrm{}\left(N\right)\mathrm{}$-symmetric, and that spontaneous symmetry breaking is not possible in the large-$N$ limit. The Green's functions are free of tachyons if constructed relative to this ground state. A natural upper bound is derived for the parameters of the theory to ensure the existence of a ground state. In the strong-coupling domain there exist a bound state and a resonance [in the identity representation of the $\mathrm{O}\mathrm{}\left(N\right)\mathrm{}$ group], which disappear in the weak-coupling regime. It is shown that, to leading order in $N$, a zero-mass interacting "charged" boson cannot be sustained in this theory. If the boson mass goes to zero, the model becomes a free-field theory.