O(N) symmetric extension of the sine-Gordon equation

We discuss an O(N) symmetric extension of the sine-Gordon (SG) equation which, using a path integral approach, allows for an expansion around the leading order in large N (Gaussian) approximation. The model is described by the Lagrangian L=(1/2)(∂μφ-vector)2+N(α0/β2)cos β√(ρ), with ρ=(φ(vector sign)·φ-vector)/N. At the leading order we show that the results of our approach agree with the ones of a large-N variational computation. We discuss the striking differences arising for a nonpolynomial interaction between the large-N form for in the Gaussian approximation and the N=1 case; when V[φ] is a polynomial no such drastic differences occur. We find that, for our large-N extension of the sine-Gordon model, the unbroken ground state is unstable as one increases the coupling constant (as it is for the original SG equation) and we find in leading order that the unbroken symmetry vacuum is stable as long as β2≤24π