The 4-loop slope of the Dirac form factor

Abstract

The 4-loop contribution to the slope of the Dirac form factor in QED has been evaluated with 1100 digits of precision. The value is $ {m^2}F_1^{(4)'}(0) = {\rm{0}}{\rm{.886545673946443145836821730610315359390424032660064745}} \cdots {\left( {{\alpha \over \pi }} \right)^4} $. We have also obtained a semi-analytical fit to the numerical value. The expression contains harmonic polylogiπ2iπ iπ arithms of argument $ {e^{{{i\pi } \over 3}}},{e^{{{2i\pi } \over 3}}},{e^{{{i\pi } \over 2}}} $, one-dimensional integrals of products of complete elliptic integrals and six finite parts of master integrals, evaluated up to 4800 digits. We show the correction to the energy levels of the hydrogen atom due to the slope.