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Non-analyticity of the Callan-Symanzik β-function of two-dimensional O(N) models

We discuss the analytic properties of the Callan-Symanzik β-function β(g) associated with the zero-momentum four-point coupling g in the two-dimensional 4 model with O(N) symmetry. Using renormalization-group arguments, we derive the asymptotic behaviour of β(g) at the fixed point g*. We argue that β'(g) = β'(g*) + O(|g-g*|1/7) for N = 1 and β'(g) = β'(g*) + O(1/log |g-g*|) for N≥3. Our claim is supported by an explicit calculation in the Ising lattice model and by a 1/N calculation for the two-dimensional 4 theory. We discuss how these non-analytic corrections may give rise to a slow convergence of the perturbative expansion in powers of g.