Analytic form of the effective potential in the large N limit of a real scalar theory in four dimensions

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We give the large N limit of the effective potential for the O(N) linear sigma model in four dimensions in terms of the Lambert W function. The effective potential is fully consistent with the renormalization group, and it admits an asymptotic expansion in powers of a small positive coupling parameter. Careful consideration of the UV cutoff present in the model validates the physics of the large N limit.