The calculation of 2-loop self-energy diagrams by the sector decomposition

TL;DR

This work tackles the computation of 2-loop self-energy diagrams for a scalar particle using a sector decomposition framework to isolate ultraviolet divergences from finite contributions. It provides detailed parametric representations and sector-specific integrals for both the basic scalar case and the case with nontrivial numerators, enabling robust numerical evaluation within dimensional regularization ($n=4-2\varepsilon$). The authors demonstrate how UV divergences are systematically extracted per sector and outline practical numerical strategies, including symbolic $\varepsilon$-expansion and multi-$\varepsilon$ extrapolation, with applicability to Higgs-related self-energies and broader electroweak radiative corrections. The methodology offers a flexible, renormalizable approach for high-precision 2-loop calculations in the Standard Model and related theories.

Abstract

Detailed description of the calculation of the 2-loop self-energy for a scalar particle is presented. By employing a simple sector decomposition method, the ultraviolet divergent part is efficiently separated from the finite part. The resulting expression can be used for both analytic and numerical computation to renormalize the divergence and to provide finite results for physics.

Figures (2)

• Figure 1: One-particle-irreducible 2-loop self-energy diagrams and tadpole diagrams
• Figure 2: Loops for each diagram. A dotted line shows the flow of the loop momentum, with the number 1 or 2 indicating $\ell_1$ or $\ell_2$, respectively. A solid arrow shows the flow of the external momentum $P$.