Large-$N$ three-dimensional ${\left({\overrightarrow{\phi }}^2\right)}^3$ field theory is studied by means of a composite-field effective potential $V_{\mathrm{eff}}(\overrightarrow{\phi },\chi )$. $V_{\mathrm{eff}}(\overrightarrow{\phi },\chi )$ is shown to be renormalizable in each order of the $\frac{1}{N}$ expansion. When we take the first two orders into account it is found that $V_{\mathrm{eff}}(\overrightarrow{\phi },\chi )$ is unbounded below. As a result the theory lacks a strict continuum limit for nonzero ${\phi }^6$ coupling and large but finite $N$. Our results may be of relevance to tricritical phenomena.

• Received 31 July 1984

DOI:https://doi.org/10.1103/PhysRevLett.53.2529