Simple one-dimensional integral representations for two-loop self-energies: the master diagram

TL;DR

The paper tackles the scalar two-loop self-energy master diagram with arbitrary masses by first extracting the imaginary part via a Cutkosky-based decomposition into two- and three-particle cuts. It then develops two one-dimensional integral representations using only elementary functions, enabling fast and accurate numerical evaluation, even in the presence of anomalous thresholds. By expressing three-particle discontinuities with complete logarithmic elliptic integrals and then transforming to fully elementary 1D forms, the authors achieve substantial computational efficiency compared to existing 2D approaches. The methods are validated against Kreimer’s representation and are poised to enhance practical two-loop self-energy calculations in the standard model.

Abstract

The scalar two-loop self-energy master diagram is studied in the case of arbitrary masses. Analytical results in terms of Lauricella- and Appell-functions are presented for the imaginary part. By using the dispersion relation a one-dimensional integral representation is derived. This representation uses only elementary functions and is thus well suited for a numerical calculation of the master diagram.

Figures (3)

• Figure 1: The master diagram $T_{12345}$.
• Figure 2: The contributions to the discontinuity of the master diagram.
• Figure 3: The triangle diagram.