Infrared and Ultraviolet Power Counting on the Mass Shell in Quantum Electrodynamics

TL;DR

This paper introduces a rigorous power counting rule for quantum electrodynamics Feynman graphs on the mass shell, enabling precise bounds on integrands and proving convergence, with applications to lepton magnetic moments.

Contribution

It provides the first rigorous mathematical framework for ultraviolet and infrared power counting on the mass shell in QED, focusing on specific Feynman graphs.

Findings

Established upper bounds for Feynman parametric integrands.

Proved convergence of certain QED Feynman integrals.

Demonstrated practical applications of the power counting rule.

Abstract

A power counting rule is provided that allows us to obtain upper bounds for the absolute values of Feynman parametric integrands. The rule reflects both the ultraviolet and infrared behavior taking into account that the external momenta are on the mass shell. It gives us the ability to rigorously prove the absolute convergence of the corresponding integrals. The consideration is limited to the case of the quantum electrodynamics Feynman graphs contributing to the lepton magnetic moments and not containing either lepton loops or ultraviolet divergent subgraphs. However, a rigorous mathematical proof is given for all Feynman graphs satisfying these restrictions. The power counting rule is formulated in terms of Hepp's sectors, ultraviolet degrees of divergence and so-called I-closures. The obtained upper bound can not be substantially improved: the illustrative example is provided. The paper provides the first mathematically rigorous treatment of the ultraviolet behavior together with the on-shell infrared behavior with some kind of generality. Practical applications of this rule are explained.