This section of the lecture series on Perturbative Quantum Chromodynamics (pQCD) by John Collins covers the foundational concepts and key formulae in the context of QCD. The lectures are structured to provide a systematic treatment of pQCD, starting with the motivation and basic ideas, and progressing to more advanced topics such as the running coupling, renormalization, and the renormalization group equation. - **Aims and Context**: The lectures aim to systematically treat the concepts of perturbative QCD, with a focus on practical applications like jet cross sections. - **QCD Definition and Fields**: QCD is defined using quarks and gluons, with the Yang-Mills Lagrangian density introduced. The covariant derivative and gluon field strength are detailed. - **Basic Properties**: The extension to the full Standard Model is discussed, along with kinematic ranges in deep-inelastic scattering (DIS). - **Key Formulae for Electromagnetic DIS**: Detailed expressions for structure functions \( F_1 \) and \( F_2 \) are provided, along with the unpolarized cross section and the parton model. - **Renormalization**: The process of applying UV regulators, adjusting parameters, and removing the regulator is explained. The one-loop results for self-energies and vertices are presented, and the \(\overline{\text{MS}}\) renormalization scheme is introduced. - **Renormalization Group**: The evolution equation for the coupling constant \( a_s \) is derived, and the first term in the expansion of the beta function is computed. The asymptotic freedom of QCD is highlighted. - **Anomalous Dimensions**: The anomalous dimensions of Green functions are discussed, and their implications for the renormalization of physical quantities are explained. - **$e^+e^-$ Annihilation**: The calculation of \( O(\alpha_s) \) corrections to the cross section \( R \) is detailed, including both real emission and virtual gluon contributions. The lectures provide a comprehensive overview of the theoretical framework and practical applications of perturbative QCD, making it a valuable resource for students and researchers in the field.This section of the lecture series on Perturbative Quantum Chromodynamics (pQCD) by John Collins covers the foundational concepts and key formulae in the context of QCD. The lectures are structured to provide a systematic treatment of pQCD, starting with the motivation and basic ideas, and progressing to more advanced topics such as the running coupling, renormalization, and the renormalization group equation. - **Aims and Context**: The lectures aim to systematically treat the concepts of perturbative QCD, with a focus on practical applications like jet cross sections. - **QCD Definition and Fields**: QCD is defined using quarks and gluons, with the Yang-Mills Lagrangian density introduced. The covariant derivative and gluon field strength are detailed. - **Basic Properties**: The extension to the full Standard Model is discussed, along with kinematic ranges in deep-inelastic scattering (DIS). - **Key Formulae for Electromagnetic DIS**: Detailed expressions for structure functions \( F_1 \) and \( F_2 \) are provided, along with the unpolarized cross section and the parton model. - **Renormalization**: The process of applying UV regulators, adjusting parameters, and removing the regulator is explained. The one-loop results for self-energies and vertices are presented, and the \(\overline{\text{MS}}\) renormalization scheme is introduced. - **Renormalization Group**: The evolution equation for the coupling constant \( a_s \) is derived, and the first term in the expansion of the beta function is computed. The asymptotic freedom of QCD is highlighted. - **Anomalous Dimensions**: The anomalous dimensions of Green functions are discussed, and their implications for the renormalization of physical quantities are explained. - **$e^+e^-$ Annihilation**: The calculation of \( O(\alpha_s) \) corrections to the cross section \( R \) is detailed, including both real emission and virtual gluon contributions. The lectures provide a comprehensive overview of the theoretical framework and practical applications of perturbative QCD, making it a valuable resource for students and researchers in the field.