High-precision calculation of the 4-loop contribution to the electron g-2 in QED

TL;DR

This work delivers a high-precision, mass-independent four-loop calculation of the electron g-2 in QED, summing contributions from 891 Feynman diagrams with precision up to 1100 digits. The four-loop coefficient is extracted as $a_e^{(4)} = -1.9122457649264455741526471674398300540608733906587253451713298480060\ldots$ and the total result is $a_e = a_e^{(4)}\,\left(\alpha/\pi\right)^4$. The analytical fit expresses $a_e^{(4)}$ as a semi-analytical combination of harmonic polylogarithms with arguments $1$, $\frac{1}{2}$, $e^{\frac{i\pi}{3}}$, $e^{\frac{2i\pi}{3}}$, $e^{\frac{i\pi}{2}}$, a family of one-dimensional integrals of products of complete elliptic integrals, and the finite parts of the epsilon expansions of six master integrals, organized by transcendental weight. The PSLQ-based fitting employs a basis of about 500 elements and parallel computation to obtain the coefficients, and the result agrees with the best numerical value within about $0.9\sigma$, supporting a high-precision test of QED and a refined determination of the fine-structure constant $\alpha$.

Abstract

I have evaluated up to 1100 digits of precision the contribution of the 891 4-loop Feynman diagrams contributing to the electron $g$-$2$ in QED. The total mass-independent 4-loop contribution is $ a_e = -1.912245764926445574152647167439830054060873390658725345{\ldots} \left(\fracαπ\right)^4$. I have fit a semi-analytical expression to the numerical value. The expression contains harmonic polylogarithms of argument $e^{\frac{iπ}{3}}$, $e^{\frac{2iπ}{3}}$, $e^{\frac{iπ}{2}}$, one-dimensional integrals of products of complete elliptic integrals and six finite parts of master integrals, evaluated up to 4800 digits.

Figures (3)

• Figure 1: The 4-loop self-mass diagrams.
• Figure 2: Examples of vertex diagrams belonging to the 25 gauge-invariant sets. The number indicates the gauge-invariant set to which the diagram belongs. In the case of the sets 1-16, 24,25, the other diagrams of each set can be obtained by permuting separately the vertices on the left and right side of the main electron line, and considering also the mirror images of the diagrams; in the sets containing diagrams with vacuum polarization insertions, one must also move the vacuum polarization insertion to each internal photon line. In the sets containing light-light diagrams, one must also consider the permutations of the vertices of the electron loop.
• Figure 3: Minimal set of master integrals which contain all the elliptic constants. The double dot in $(a')$ means that denominator is raised to the power three. $(f,f',f")$ and $(g,g',g")$ have numerators respectively equal to $(1,p.k,(p.k)^2)$.