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Notes on color reductions and $γ$ traces

Oliver Schnetz

TL;DR

The paper develops efficient, self-contained methods to compute $SU(N)$ color factors and $\gamma$-matrix traces in high-loop QFT calculations. It couples a rigorous color-graph framework with a proven two-term color-reduction identity and a core adjoint-vertex relation, implementing these in the Maple package HyperlogProcedures to enable reductions beyond traditional loop-order limits. For $\gamma$ traces, it provides systematic contraction rules, caching strategies, and complete reductions up to $n=16$, with practical benchmarks showing tangible gains at high loop orders. The work proves divisibility and parity properties of color-reduction polynomials $R_G(N)$, conjectures about their low-degree behavior, and offers antipode-like formulas for non-oriented loops, significantly improving the feasibility of high-loop color-factor calculations in gauge theories.

Abstract

We present efficient algorithms to calculate the color factors for the $SU(N)$ gauge group and to evaluate $γ$ traces. The aim of these notes is to give a self-contained, proved account of the basic results with particular emphasis on color reductions. We fine tune existing algorithms to make calculations at high loop orders possible.

Notes on color reductions and $γ$ traces

TL;DR

The paper develops efficient, self-contained methods to compute color factors and -matrix traces in high-loop QFT calculations. It couples a rigorous color-graph framework with a proven two-term color-reduction identity and a core adjoint-vertex relation, implementing these in the Maple package HyperlogProcedures to enable reductions beyond traditional loop-order limits. For traces, it provides systematic contraction rules, caching strategies, and complete reductions up to , with practical benchmarks showing tangible gains at high loop orders. The work proves divisibility and parity properties of color-reduction polynomials , conjectures about their low-degree behavior, and offers antipode-like formulas for non-oriented loops, significantly improving the feasibility of high-loop color-factor calculations in gauge theories.

Abstract

We present efficient algorithms to calculate the color factors for the gauge group and to evaluate traces. The aim of these notes is to give a self-contained, proved account of the basic results with particular emphasis on color reductions. We fine tune existing algorithms to make calculations at high loop orders possible.

Paper Structure

This paper contains 7 sections, 5 theorems, 41 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Color reduction
  3. The color graph
  4. Fundamental identities
  5. The reduction algorithm
  6. Results and a conjecture
  7. Gamma reduction

Key Result

Lemma 1

An orthonormal basis of the fundamental (defining) representation of $sl(N)$ are the $N\times N$ matrices where $(E_{ab})_{cd}=\delta_{a,c}\delta_{b,d}$ are the elementary matrices. We fix any sequence of $\alpha\beta$ in $T^{\alpha\beta}$ and $\tilde{T}^{\alpha\beta}$ to continue the labels $1,\ldots,N-1$ of $T^k$ to $N-1+2N(N-1)/2=N^2-1$.

Figures (8)

  • Figure 1: Feynman rules for color graphs
  • Figure 2: Flipping two edges at an adjoint vertex gives a minus sign.
  • Figure 3: Basic identities for color graphs.
  • Figure 4: The two-term relation of $SU(N)$ color reductions, Equation (\ref{['color7']}).
  • Figure 5: Reductions of small cycles (also see Figure \ref{['figcolor3']}).
  • ...and 3 more figures

Theorems & Definitions (15)

  • Lemma 1
  • proof
  • Definition 2
  • Proposition 3
  • proof
  • Example 4
  • Example 5
  • Proposition 6
  • proof
  • Conjecture 7
  • ...and 5 more