Nonstandard expansion techniques for the effective potential in λ${\phi }^4$ quantum field theory

The effective potential in scalar quantum field theory is calculated using different expansions of the path integral for the generating functional. The conventional loop expansion is compared with the interpolated, optimized, and mean-field expansions in the lowest orders. The optimized expansion is studied up to third order. In the space-time dimensions 0 and 1 the comparison with the ‘‘exact’’ effective potential calculated numerically shows that the only method which gives qualitative agreement in the whole range of Lagrangian parameters is the mean-field expansion. In 4 dimensions the mean-field method seems also to be most reliable and the theory to be noninteracting.