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

We discuss the analytic properties of the Callan-Symanzik beta -function be ta (g) associated with the zero-momentum four-point coupling g in the two-d imensional phi (4) model with O(N) symmetry. Using renormalization-group ar guments, we derive the asymptotic behaviour of beta (g) at the fixed point g*. We argue that beta'(g) = beta'(g*)+ O(\g - g*\(1/7)) for N = 1 and beta '(g) = beta'(g*) + O(1/log\g - g*\) for N greater than or equal to 3. Our c laim is supported by an explicit calculation in the Ising lattice model and by a 1/N calculation for the two-dimensional phi (4) theory. We discuss ho w these non-analytic corrections may give rise to a slow convergence of the perturbative expansion in powers of g.