Recursion and Growth Estimates in Renormalizable Quantum Field Theory

Abstract

Abstract. 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. 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.1. The set-up 1.1. Introduction. In this paper we explore the recursive structure of the shortdistance sector of a renormalizable quantum field theory. Such theories have a finite number of distinct amplitudes r ∈ R ⊂ A which need renormalization. We decompose each Green function in accordance with structure functions which need renormalization, the remaining structure functions and superficially convergent Green functions contribute to amplitudes a ∈ A R in the set of all amplitudes A [14].For such a theory, the Dyson-Schwinger equations give any amplitude s ∈ A = A R ∪ R in terms of all those amplitudes r ∈ R in the theory which need renormalization, and in terms of an infinite series of integral kernels, the skeleton graphs of the theory. The very fact that we can renormalize by modifying the Lagrangian implies that we have a sub Hopf algebra at our disposal which has the graded elements c r k as generators which correspond to the k-th order contribution to the amplitude r [15,14].We can relate the integral kernels above to the primitives of the Hopf algebra structure underlying renormalization. The growth of these primitives is hence determined by the growth of integral kernels provided by overall convergent Green functions. A bound on such a growth is hence obtainable from a bound of amplitudes in A R, where results in constructive field theory are principally available. The typical example is quantum electrodynamics (QED), where the primitives for the vertex function are given by the superficially convergent four fermion e + e − → e + e − scattering kernel, two-particle irreducible in a suitable channel.To proceed from there to a bound for amplitudes in R, we need a handle on the behaviour of the singular integrations which encompass the short-distance singular sector. Here, we proceed by chosing a suitable set of primitives such that we can reduce the Dyson-Schwinger equations to recursive equations acting on onevariable Mellin transforms. A simple study of the conformal symmetries in the corresponding primitives suffice to determine the form of these Mellin transforms.

Log expansions from combinatorial Dyson–Schwinger equations

“…• In the second part of this paper, we review the connection between the Callan-Symanzik equation and Dyson-Schwinger equations (DSE's) [8,9], i.e. fixedpoint relations for the infinite sums of Feynman graphs in Eqs.…”

“…Then, the general DSE's [8,9,24,25] for the infinite sums X r in Eqs. (1.5) and (1.6) with φ R (r) = 1 read…”

Not so non-renormalizable gravity

“…Note that the structure of the sub-Hopf algebras underlying this approach [5,8] implies that the elements X i (α) close under the coproduct. A general classification of related sub-Hopf algebras has been recently obtained by Loic Foissy [14].…”

Hopf algebras and Dyson–Schwinger equations

“…Combinatorial Hopf algebras are currently the subject of studies [45][46][47][48]. Although we only discussed a simple example here, more complicated cases can be envisaged.…”