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

## TL;DR

This paper presents a highly precise numerical evaluation of the 4-loop quantum electrodynamics contribution to the electron g-2, including a semi-analytical expression involving advanced mathematical functions.

## Contribution

It provides the most precise calculation to date of the 4-loop contribution to electron g-2, with a detailed semi-analytical expression fitting the numerical data.

## Key findings

- Numerical value of the 4-loop contribution: -1.912245764926445574152647167439830054060873390658725345
- Semi-analytical expression involving harmonic polylogarithms and elliptic integrals
- Evaluation of master integrals up to 4800 digits

## 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(\frac{\alpha}{\pi}\right)^4$. I have fit a semi-analytical expression to the numerical value. The expression contains harmonic polylogarithms of argument $e^{\frac{i\pi}{3}}$, $e^{\frac{2i\pi}{3}}$, $e^{\frac{i\pi}{2}}$, one-dimensional integrals of products of complete elliptic integrals and six finite parts of master integrals, evaluated up to 4800 digits.