Muon Anomaly from Lepton Vacuum Polarization and The Mellin--Barnes Representation

TL;DR

This paper analytically evaluates specific eighth- and tenth-order QED contributions to the muon's anomalous magnetic moment using Mellin--Barnes representations, providing detailed expansions in lepton mass ratios and comparing with prior results.

Contribution

It introduces a novel analytic method using Mellin--Barnes integrals to evaluate complex Feynman diagrams involving multiple mass ratios in quantum electrodynamics.

Findings

Analytic expansions in mass ratios for muon g-2 contributions.

Demonstration of Mellin--Barnes technique for Feynman integral evaluation.

Comparison with previous numerical and analytic results.

Abstract

We evaluate, analytically, a specific class of eighth--order and tenth--order QED contributions to the anomalous magnetic moment of the muon. They are generated by Feynman diagrams involving lowest order vacuum polarization insertions of leptons $l=e,\mu$, and $\tau$. The results are given in the form of analytic expansions in terms of the mass ratios $m_e/m_\mu$ and $m_\mu/m_\tau$. We compute as many terms as required by the error induced by the present experimental uncertainty on the lepton masses. We show how the Mellin--Barnes integral representation of Feynman parametric integrals allows for an easy analytic evaluation of as many terms as wanted in these expansions and how its underlying algebraic structure generalizes the standard renormalization group properties. We also discuss the generalization of this technique to the case where two independent mass ratios appear. Comparison with previous numerical and analytic evaluations made in the literature, whenever pertinent, are also made.