Covariant Gaussian approximation. II. Scalar theories

We apply the covariant Gaussian approximation to the one- and $N$-component ${\phi }^4$ scalar theories in 3+1 dimensions. The diagrammatic representation of the approximation is obtained and its connection with perturbation theory is discussed. Green's functions at nonzero momenta are calculated and the full renormalization program is systematically carried through. It is found that the only solution of normalization conditions is the "precarious" theory with infinitesimal negative bare coupling. The theory with positive bare coupling is trivial. In the precarious theory the symmetric ($\phi =0\mathrm{}$) phase and asymmetric ($\phi =0\mathrm{}$) phase are found. The symmetric phase has always lower energy and there is no spontaneous symmetry breaking in the Gaussian approximation.