Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Minimal Gauge Invariant Classes of Tree Diagrams in Gauge Theories

Edward Boos, Thorsten Ohl

TL;DR

The paper addresses how to partition tree-level Feynman diagrams in gauge theories into gauge-invariant subsets by introducing forests, flavored forests, and groves. It defines groves as minimal gauge-invariant classes obtained from gauge flips and proves their invariance and minimality, enabling reliable calculations with complex final states without summing full multiplets. The authors demonstrate the method on six-fermion processes in the Standard Model, connect it to known 4f classifications, and discuss extensions to loop calculations and symmetry-based subset construction via automorphisms. The work provides a practical, scalable framework for organizing diagram contributions by gauge structure and flavor while preserving Ward identities.

Abstract

We describe the explicit construction of groves, the smallest gauge invariant classes of tree Feynman diagrams in gauge theories. The construction is valid for gauge theories with any number of group factors which may be mixed. It requires no summation over a complete gauge group multiplet of external matter fields. The method is therefore suitable for defining gauge invariant classes of Feynman diagrams for processes with many observed final state particles in the standard model and its extensions.

Minimal Gauge Invariant Classes of Tree Diagrams in Gauge Theories

TL;DR

The paper addresses how to partition tree-level Feynman diagrams in gauge theories into gauge-invariant subsets by introducing forests, flavored forests, and groves. It defines groves as minimal gauge-invariant classes obtained from gauge flips and proves their invariance and minimality, enabling reliable calculations with complex final states without summing full multiplets. The authors demonstrate the method on six-fermion processes in the Standard Model, connect it to known 4f classifications, and discuss extensions to loop calculations and symmetry-based subset construction via automorphisms. The work provides a practical, scalable framework for organizing diagram contributions by gauge structure and flavor while preserving Ward identities.

Abstract

We describe the explicit construction of groves, the smallest gauge invariant classes of tree Feynman diagrams in gauge theories. The construction is valid for gauge theories with any number of group factors which may be mixed. It requires no summation over a complete gauge group multiplet of external matter fields. The method is therefore suitable for defining gauge invariant classes of Feynman diagrams for processes with many observed final state particles in the standard model and its extensions.

Paper Structure

This paper contains 6 sections, 2 theorems, 6 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Forests
  3. Flavored Forests
  4. Groves
  5. Application
  6. Automorphisms

Key Result

Theorem 1

The unflavored forest $F(E)$ is connected for all external states $E$,

Figures (7)

  • Figure 1: The four-point diagrams $\{S_1,S_2,S_3,S_4\}$ related by flips.
  • Figure 2: The forest of the 15 five-point tree diagrams in unflavored $\phi^3$-theory. The diagrams are specified by fixing vertex 1 and using parentheses to denote the order in which lines are joined at vertices.
  • Figure 3: The four-point diagrams $\{t_4^{F,1},t_4^{F,2},t_4^{F,3}\}$ related by flavor flips.
  • Figure 4: The four-point diagrams $\{t_4^{G,1},t_4^{G,2},t_4^{G,3},t_4^{G,4}\}$ related by gauge flips.
  • Figure 5: The four-point diagrams $\{t_4^{G,5},t_4^{G,6},t_4^{G,7}\}$ related by gauge flips in the case of fermionic matter.
  • ...and 2 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2