Recursion and Growth Estimates in Renormalizable Quantum Field Theory

In this paper we show that there is a Lipatov bound for the radius of convergence for superficially divergent one-particle irreducible Green functions in a renormalizable quantum field theory if there is such a bound for the superficially convergent ones. In the nonnegative case the radius of convergence turns out to be min{ρ,1/b ₁}, where ρ is the bound on the convergent ones, the instanton radius, and b ₁ the first coefficient of the β-function, while in general it is bounded by the above. Research supported by grant NSF-DMS/0603781. Supported by CNRS.