The Galois coaction on the electron anomalous magnetic moment
Oliver Schnetz
TL;DR
The paper investigates the motivic Galois coaction on the fourth-order QED electron $g-2$ by translating Laporta's polylogarithmic results into the motivic $f$ alphabet, thereby exposing the underlying Galois structure. It consolidates Broadhurst's conjecture for a basis of ${ m MZV}(6)$ in the motivic setting and proves it using Chen iterated integrals and the Brown decomposition algorithm, establishing a concrete isomorphism that renders the coaction a simple deconcatenation. The analysis shows that the ${ m Q}$-vector spaces of Galois conjugates up to weight four are unusually small and largely captured by weight-$ ext{n}$ components of the $a_e$ contribution, hinting at a universal coaction structure in QFT akin to ${ m \,}\\phi^4$ theory. These results provide a compact, basis-driven description of high-order QED contributions and support broader conjectures about the role of motivic periods and Galois coaction in quantum field theory calculations.
Abstract
Recently S. Laporta published a partial result on the fourth order QED contribution to the electron anomalous magnetic moment $g-2$. This result contains explicit polylogarithmic parts with fourth and sixth roots of unity. In this note we convert Laporta's result into the motivic `$f$ alphabet'. This provides a much shorter expression which makes the Galois structure visible. We conjecture the $Q$ vector spaces of Galois conjugates of the QED $g-2$ up to weight four. The conversion into the $f$ alphabet relies on a conjecture by D. Broadhurst that iterated integrals in certain Lyndon words provide an algebra basis for the extension of multiple zeta values by sixth roots of unity. We prove this conjecture in the motivic setup.