QED vertex form factors at two loops

TL;DR

The authors derive the complete analytic, UV-renormalized two-loop QED vertex form factors for on-shell electrons with finite mass $m$, valid for space-like and time-like momentum transfer. They work in dimensional regularization with a single parameter $D$, expressing results in terms of 1D harmonic polylogarithms up to weight 4 and reducing all loop integrals to 17 Master Integrals via IBP/LI methods. The third form factor vanishes after summation, while $F_1^{(2l)}$ and $F_2^{(2l)}$ are given with explicit pole structure and finite parts, including imaginary parts above threshold. They also provide asymptotic expansions for $Q^2\gg m^2$ and $Q^2\ll m^2$, reproducing known two-loop electron $g-2$ and charge slope results and highlighting the persistence of IR poles that cancel in physical observables with real emission.

Abstract

We present the closed analytic expression of the form factors of the two-loop QED vertex amplitude for on-shell electrons of finite mass $m$ and arbitrary momentum transfer $S=-Q^2$. The calculation is carried out within the continuous $D$-dimensional regularization scheme, with a single continuous parameter $D$, the dimension of the space-time, which regularizes at the same time UltraViolet (UV) and InfraRed (IR) divergences. The results are expressed in terms of 1-dimensional harmonic polylogarithms of maximum weight 4.

Figures (3)

• Figure 1: 2-loop vertex diagrams for the QED form factors. The fermionic external lines are on the mass-shell $p_{1}^{2}=p_{2}^{2}=-m^2$; the wavy line on the r.h.s. carries momentum $Q=p_{1}+p_{2}$, with $Q^{2}=-S$. The arrows label the flow of the momenta $p_1$ and $p_2$.
• Figure 2: Subtraction terms for the renormalization at 2 loops.
• Figure 3: 1-loop vertex diagrams for the QED form factor.