Analytical and numerical methods for massive two-loop self-energy diagrams

TL;DR

This work tackles the challenge of massive two-loop self-energy diagrams by delivering three complementary methods: (i) analytic representations in arbitrary $D$ through generalized hypergeometric functions with explicit small- and large-$p^2$ expansions, (ii) four-dimensional imaginary parts expressed via complete elliptic integrals for efficient numerical evaluation, and (iii) practical one-dimensional integral representations based on dispersion relations of the one-loop substructure. It develops a dispersion-based derivation for the two-loop master diagrams, unifies several mass configurations, and provides concrete expressions for $T_{123}$ and $T_{1234}$ including their cuts and analytic continuations. Numerical checks against established results confirm accuracy, and the 1D representations offer computationally efficient routes suitable for electroweak and QED predictions. The paper thus furnishes a versatile toolkit for precise massive two-loop self-energy calculations across parameter regimes.

Abstract

Motivated by the precision results in the electroweak theory studies of two-loopFeynman diagrams are performed. Specifically this paper gives a contribution to the knowledge of massive two-loop self-energy diagrams in arbitrary and especially four dimensions.This is done in three respects firstly results in terms of generalized, multivariable hypergeometric functions are presented giving explicit series for small and large momenta. Secondly the imaginary parts of these integrals are expressed as complete elliptic integrals.Finally one-dimensional integral representations with elementary functions are derived.They are very well suited for the numerical evaluations.

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• Figure 1: Topologies