Remarks about Dyson's instability in the large-N limit

There are known examples of perturbative expansions in the 't Hooft coupling lt with a finite radius of convergence. This seems to contradict Dyson's argument suggesting that the instability at negative coupling implies a zero radius of convergence. Using the example of the linear sigma model in three dimensions, we discuss to which extent the two points of view are compatible. We show that a saddle point persists for negative values of lt until a critical value -|lt_c| is reached. A numerical study of the perturbative series for the renormalized mass confirms an expected singularity of the form (lt +|lt_c|)^1/2. However, for -|lt_c|< lt <0, the effective potential does not exist if phi^2 >phi^2_{max}(lt) and not at all if lt<-|lt_c|. We show that phi^2_{max}(lt) propto 1/|lt | for small negative lt. The finite radius of convergence can be justified if the effective theory is defined with a large field cutoff phi^2_{max}(lt) which provides a quantitative measure of the departure from the original model considered.