Calculation of two-loop self-energies in the electroweak Standard Model

TL;DR

This work addresses the challenge of calculating two-loop self-energies in the electroweak Standard Model by introducing a robust algebraic reduction to a small basis of scalar $T$-integrals, enabling automated tensor reduction and systematic gauge analysis. It develops the pinch technique at two loops and connects it to the background-field method, yielding gauge-invariant building blocks for propagator corrections. Analytic results are obtained for the scalar integrals in terms of generalized hypergeometric functions, with imaginary parts expressible via complete elliptic integrals, and several one-dimensional integral representations are derived for efficient numerical evaluation. The combination of algebraic reduction, hypergeometric/elliptic analytic methods, and 1D numerical representations significantly enhances the feasibility and accuracy of SMEFT-level two-loop calculations, with direct relevance to precision electroweak predictions such as the $M_W$–$M_Z$ relation and $Z$ line-shape analyses.

Abstract

Motivated by the results of the electroweak precision experiments, studies of two-loop self-energy Feynman diagrams are performed. An algebraic method for the reduction of all two-loop self-energies to a set of standard scalar integrals is presented. The gauge dependence of the self-energies is discussed and an extension of the pinch technique to the two-loop level is worked out. It is shown to yield a special case of the background-field method which provides a general framework for deriving Green functions with desirable theoretical properties. The massive scalar integrals of self-energy type are expressed in terms of generalized multivariable hypergeometric functions. The imaginary parts of these integrals yield complete elliptic integrals. Finally, one-dimensional integral representations with elementary integrands are derived which are well suited for numerical evaluation.

Figures (4)

• Figure 1: One-particle irreducible topologies of two-loop self-en-er-gies.
• Figure 2: Light fermion contributions to the two-loop self-energy of the $Z$-boson.
• Figure 3: Two-loop vertex and box graphs containing propagator-like pinch parts.
• Figure 4: The pinch part of the first diagram in fig. 3.