An introduction to gauge theories and group theory in particle physics

TL;DR

This paper surveys how group theory and gauge principles organize particle physics, tracing the role of Lie groups, representations, and the Poincaré group in classifying particles and building gauge theories. It emphasizes the local gauge principle, the quantization framework (FP/BRST) and the modern on-shell amplitude program as a means to bypass gauge redundancy, while highlighting anomaly cancellation as a nontrivial consistency condition. The Standard Model is presented from a symmetry perspective, detailing its gauge group $SU(3)_C\times SU(2)_L\times U(1)_Y$, anomaly-free fermion content, and custodial/flavor global structures, with a discussion of Yukawa couplings and CP violation. The work also surveys beyond-Standard-Model directions, including Grand Unified Theories, SMEFT/HEFT, and generalized/global symmetries, illustrating how symmetry guides current research in particle physics.

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.