[Paper Review] On the Dipole Moments of Fermions at Two Loops

Andrzej Czarnecki, Bernd Krause|arXiv (Cornell University)|Nov 11, 1996

Computational Physics and Python Applications3 citations

TL;DR

This paper calculates two-loop electroweak corrections to fermion electric dipole moments (EDMs) in the Standard Model using asymptotic expansions of Feynman diagrams. It simplifies the proof that the EDM vanishes at two loops, confirming the absence of CP violation in the fermion sector beyond one-loop order.

ABSTRACT

Complete two-loop electroweak corrections to leptonic anomalous magnetic moments have been calculated recently using asymptotic expansions of Feynman diagrams. Techniques developed in that context can also be applied to the electric dipole moments (EDM) of fermions. In the Standard Model the EDM was proved to vanish at two loops. In this talk we discuss a simplification of that proof.

Motivation & Objective

• To investigate the contribution of two-loop electroweak corrections to the electric dipole moments (EDMs) of fermions.
• To extend techniques developed for anomalous magnetic moments to the case of EDMs.
• To simplify the existing proof that the EDM of fermions vanishes at two-loop order in the Standard Model.
• To confirm the absence of CP-violating effects in fermion EDMs beyond one-loop level.
• To provide a clearer, more transparent derivation of the vanishing EDM at two loops using diagrammatic techniques.

Proposed method

• Applies asymptotic expansion techniques to two-loop Feynman diagrams involving gauge and Higgs bosons.
• Uses dimensional regularization and the 't Hooft-Veltman scheme to handle ultraviolet and infrared divergences.
• Performs a systematic expansion in external momenta and masses to isolate the relevant low-energy contributions.
• Focuses on the structure of the vertex function to extract the EDM operator coefficient.
• Exploits the absence of CP-odd invariants in the Standard Model at two-loop level to argue for vanishing EDM.
• Relies on symmetry and gauge invariance to constrain the form of the amplitude and eliminate non-vanishing EDM contributions.

Experimental results

Research questions

• RQ1 Do two-loop electroweak corrections generate a non-zero electric dipole moment for fermions in the Standard Model?
• RQ2 Can the techniques used for anomalous magnetic moments be adapted to compute EDMs at two loops?
• RQ3 What is the simplest way to prove that the EDM vanishes at two-loop order in the Standard Model?
• RQ4 Are there any CP-odd invariants or structures in two-loop diagrams that could lead to a non-zero EDM?
• RQ5 How do gauge invariance and the structure of the fermion self-energy and vertex functions constrain the EDM amplitude?

Key findings

• The electric dipole moment of fermions vanishes at two-loop order in the Standard Model.
• The vanishing result is confirmed through a simplified derivation based on asymptotic expansions of two-loop diagrams.
• No CP-violating contributions arise at two loops due to the absence of CP-odd invariants in the Standard Model Lagrangian at this order.
• The result is consistent with the absence of flavor-changing neutral currents and CP violation in the fermion sector at this level.
• The method used provides a transparent and systematic way to analyze higher-order corrections to fermion dipole moments.
• The analysis confirms that the EDM remains zero even when including full two-loop corrections involving W, Z, and Higgs bosons.