Vertex diagrams for the QED form factors at the 2-loop level

TL;DR

The paper delivers a comprehensive reduction and analytic evaluation of all 2-loop QED electron-vertex integrals in continuous dimensional regularization for on-shell electrons with finite mass. By reducing to 16 Master Integrals using IBP, Lorentz invariance, and symmetry relations, and then solving a system of differential equations in $Q^2$ with an $\epsilon$-expansion, the authors express all MI coefficients in closed harmonic polylogarithmic form. The work provides explicit results for topologies with 3–6 denominators, including reducible diagrams, and furnishes high- and low-$Q^2$ expansions, laying the groundwork for the complete real and imaginary parts of the QED form factors at two loops. The methods combine bottom-up IBP, differential equations, and Euler’s variation of constants to achieve compact, analytic results suitable for numerical implementation and SEO-friendly indexing.

Abstract

We carry out a systematic investigation of all the 2-loop integrals occurring in the electron vertex in QED in the continuous $D$-dimensional regularization scheme, for on-shell electrons, momentum transfer $t=-Q^2$ and finite squared electron mass $m_e^2=a$. We identify all the Master Integrals (MI's) of the problem and write the differential equations in $Q^2$ which they satisfy. The equations are expanded in powers of $ε= (4-D)/2$ and solved by the Euler's method of the variation of the constants. As a result, we obtain the coefficients of the Laurent expansion in $ε$ of the MI's up to zeroth order expressed in close analytic form in terms of Harmonic Polylogarithms.

Figures (7)

• Figure 1: 2-loop vertex diagrams for the QED form factor. The fermionic external lines are on the mass-shell $p_{1}^{2}=p_{2}^{2}=-a$, while the wavy line on the r.h.s. has momentum $Q=p_{1}+p_{2}$, with $Q^{2}=-s$.
• Figure 2: A 4-denominator topology.
• Figure 3: The set of 3 independent 6-denominator topologies present in the graphs of Fig.(\ref{['fig1']}).
• Figure 4: The set of 8 independent 5-denominator topologies contained in the graphs of Fig.(\ref{['fig1']}). External fermion lines are put on the mass-shell $p_{1}^{2}=p_{2}^{2}=-a$, while external wavy lines carry an off-shell momentum $Q=p_{1}+p_{2}$. The topology (g) is evaluated on the mass-shell.
• Figure 5: The set of 12 independent 4-denominator (sub)topologies coming from the 5-denominator topologies of Fig.(\ref{['fig2']}).