Effective potential for theO(N)model to order1N

The effective potential of the $\mathrm{O}\mathrm{}\left(N\right)\mathrm{}$ model is calculated to the next-to-leading order in the $\frac{1}{N}$ expansion in one, two, three, and four space-time dimensions. In one and two dimensions the vacuum is symmetric, as expected. In three dimensions the radiative corrections of order $\frac{1}{N}$ do not affect the position of the minimum of the effective potential. In four dimensions, the contribution to the effective potential of order $\frac{1}{N}$ is complex everywhere, suggesting that the $\frac{1}{N}$ expansion fails.