High-precision calculation of the 4-loop QED contribution to the slope of the Dirac form factor

TL;DR

This paper presents a highly precise numerical evaluation of the four-loop QED contribution to the slope of the Dirac form factor, including semi-analytical expressions and implications for hydrogen energy levels.

Contribution

The authors computed the 4-loop QED contribution with 1100-digit precision and derived semi-analytical expressions involving advanced mathematical functions.

Findings

Total 4-loop contribution calculated with high precision

Semi-analytical expression fitted to numerical value

Implications for hydrogen energy level shifts

Abstract

We have evaluated with 1100 digits of precision the contribution of all the 891 mass-independent four-loop Feynman diagrams contributing to the slope of the Dirac form factor in QED. The total 4-loop contribution is $ m^2 F_1^{(4)'}(0) = 0.886545673946443145836821730610315359390424032660064745... \left(\frac{\alpha}{\pi}\right)^4$. We 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. I show the correction on the shift of the energy levels of the hydrogen atom due to the slope.