An introduction to gauge theories and group theory in particle physics

TL;DR

This review introduces key group theory concepts and their application in particle physics, focusing on gauge theories, the Standard Model, and modern amplitude methods for better understanding and computation.

Contribution

It provides a comprehensive overview of gauge and group theory in particle physics, emphasizing modern on-shell amplitude techniques and their advantages.

Findings

Application of on-shell methods simplifies gauge theory calculations.

Identification of gauge symmetries in the Standard Model.

Discussion of global symmetries and anomaly considerations.

Abstract

In this review, the fundamental concepts of group theory and representation theory are introduced. Special emphasis is placed on the unitary irreducible representations of the $SU(N)$ Lie group, the Poincare group, Little Group, discrete group, and their applications in particle physics. Based on the principle of local gauge symmetry, the construction of gauge-invariant Lagrangians and their quantization procedure are discussed. To address gauge redundancy, the modern on-shell amplitude approach is applied to gauge theories, demonstrating both conceptual and computational advantages. From the perspective of symmetry, the Standard Model is presented through the identification of its gauge symmetry, its anomaly-free matter content, and its global symmetries, including flavor symmetry, custodial symmetry, and baryon and lepton number conservation, etc.