Abstract
A class of one-dimensional path integrals with long-range kinetic energies arises in the context of the single-impurity Kondo and Anderson models. Motivated by this fact, we consider a simple one-dimensional $O(N)$ symmetric ${\ensuremath{\varphi}}^{4}$ theory with a long-range-gradient term. The large-$N$ expansion of this model behaves smoothly as the coefficient of the quadratic term goes from large positive to large negative values and does not have the pathologies of the mean-field theory.
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