The sine-Gordon and sinh-Gordon gaussian effective potential in D+1 dimensions

We compute the gaussian approximation to the effective potential for the sine-Gordon and the sinh-Gordon model in D+1 dimensions. Issues such as vacuum energy, renormalization and stability of the vacuum are discussed in detail. Within the gaussian approximation, we show that (i) for D>=3, the model exists only as a free-field theory, and (ii) for D<3, the vacuum is unstable over a certain range of the coupling constant. We find a particularly elegant expression for the gaussian effective potential for the special case D=1. (orig.)