Group theory factors for Feynman diagrams

TL;DR

The paper presents a comprehensive, representation-agnostic framework for reducing group-theory factors in Feynman diagrams by expressing traces in terms of a small set of invariant tensors and indices. It develops a two-stage reduction pipeline: rewrite traces as symmetrized traces and then expand these in a basis of symmetric invariants, with a dedicated treatment of adjoint traces. A central innovation is the character-based computation of indices and trace identities, extended to all classical and exceptional algebras, including explicit methods for A_r, B_r, C_r, D_r, G_2, F_4, E_6, E_7, and E_8, along with normalization rules and extensive appendices. The authors implement these methods in FORM to handle diagrams up to nine loops and beyond, providing practical tools and extensive invariant tables that facilitate color-factor calculations in diverse gauge theories. The work thus enables efficient, scalable, and group-wide analysis of Feynman diagrams across arbitrary groups and representations, with significant implications for grand unification and string theory calculations.

Abstract

We present algorithms for the group independent reduction of group theory factors of Feynman diagrams. We also give formulas and values for a large number of group invariants in which the group theory factors are expressed. This includes formulas for various contractions of symmetric invariant tensors, formulas and algorithms for the computation of characters and generalized Dynkin indices and trace identities. Tables of all Dynkin indices for all exceptional algebras are presented, as well as all trace identities to order equal to the dual Coxeter number. Further results are available through efficient computer algorithms (see http://norma.nikhef.nl/~t58/ and http://norma.nikhef.nl/~t68/ ).

Figures (1)

• Figure 1: Dynkin diagrams and labelling conventions