An introduction to gauge theories and group theory in particle physics
School of Physics, Sun Yat-Sen University（中山大学物理学院）；Institute of High Energy Physics, Chinese Academy of Sciences（中国科学院高能物理研究所）；CAS Center for Excellence in Particle Physics（中国科学院粒子物理中心）；School of Science, Sun Yat-Sen University（中山大学理学院）；Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；Hangzhou Institute for Advanced Study, University of Chinese Academy of Sciences（中国科学院大学杭州高等研究院）
Hao-Lin Li, Hao Sun, Ming-Lei Xiao, Jiang-Hao Yu
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28 pages, 2 tables, invited review for Encyclopedia of Particle Physics
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.
We derive the finite one-loop counterterm required to restore the Ward Identities broken by the regularization scheme in chiral gauge theories. Our result is an analytic expression applicable to a wide class of regularizations satisfying a few general properties. We adopt the background field method, which ensures background gauge invariance in the quantized theory, and focus on renormalizable chiral theories with arbitrary gauge group and fermions in general representations. Our approach can be extended to theories involving scalars, such as the Standard Model, or to non-renormalizable theori…
