Two-Loop Form Factors in QED
P. Mastrolia, E. Remiddi
TL;DR
This paper resolves the long-standing challenge of obtaining the real parts of the two-loop on-shell electron form factors $F_1^{(2)}$ and $F_2^{(2)}$ in QED for arbitrary momentum transfer by formulating and evaluating dispersion relations. By combining Pauli-Villars UV regularization with an infrared photon mass $ rac{ abla}{ abla} $, the authors express the real parts as subtracted dispersive integrals of the known imaginary parts, and—crucially—cast the resulting integrals into Harmonic Polylogarithms up to weight 4. The authors provide explicit analytic expressions for $F_1^{(2)}(-Q^2)$ and $F_2^{(2)}(-Q^2)$ in terms of HPLs with argument $y$, along with large- and small-$Q^2$ expansions and the appropriate analytic continuation to timelike regions. This approach generalizes prior Nielsen polylogarithm results and yields a compact, exact representation in a well-controlled functional basis, enabling precise high- and low-energy analyses of the electron form factors.
Abstract
We evaluate the on shell form factors of the electron for arbitrary momentum transfer and finite electron mass, at two loops in QED, by integrating the corresponding dispersion relations, which involve the imaginary parts known since a long time. The infrared divergences are parameterized in terms of a fictitious small photon mass. The result is expressed in terms of Harmonic Polylogarithms of maximum weight 4. The expansions for small and large momentum transfer are also given