Perturbative Quantum Chromodynamics (pQCD) is a systematic approach used in high-energy physics to calculate cross sections and other observables. It is essential for experiments and collider planning involving hadron beams. Key concepts include the factorization theorem, LO/NLO/NNLO calculations, Monte-Carlo event generators, and parton densities. The total HH cross section is approximately 70 mb, while LHC cross sections are discussed down to 1 fb. QCD is a gauge theory with SU(3) color symmetry, described by the Yang-Mills Lagrangian. It includes quarks and gluons, with the coupling constant α_s, which decreases with energy (asymptotic freedom). The theory is renormalizable and exhibits gauge invariance. The renormalization group (RG) describes how α_s evolves with energy scale, leading to asymptotic freedom. The RG equation for α_s is derived, showing that QCD is asymptotically free. The β function, which governs the running of α_s, is positive, indicating that α_s decreases with increasing energy. The effective coupling is defined through the renormalization group, with the Λ parameter playing a key role in specifying the scale. The RG is used to calculate cross sections for processes like e+e- → hadrons, where perturbative methods are applicable at short distances. The renormalization group allows for the evolution of coupling constants and the determination of physical quantities at different energy scales. The solution of the RG equation involves the Λ parameter, which is a fundamental scale in QCD. The RG is essential for understanding the behavior of QCD at different energy scales and for making precise predictions in high-energy physics.Perturbative Quantum Chromodynamics (pQCD) is a systematic approach used in high-energy physics to calculate cross sections and other observables. It is essential for experiments and collider planning involving hadron beams. Key concepts include the factorization theorem, LO/NLO/NNLO calculations, Monte-Carlo event generators, and parton densities. The total HH cross section is approximately 70 mb, while LHC cross sections are discussed down to 1 fb. QCD is a gauge theory with SU(3) color symmetry, described by the Yang-Mills Lagrangian. It includes quarks and gluons, with the coupling constant α_s, which decreases with energy (asymptotic freedom). The theory is renormalizable and exhibits gauge invariance. The renormalization group (RG) describes how α_s evolves with energy scale, leading to asymptotic freedom. The RG equation for α_s is derived, showing that QCD is asymptotically free. The β function, which governs the running of α_s, is positive, indicating that α_s decreases with increasing energy. The effective coupling is defined through the renormalization group, with the Λ parameter playing a key role in specifying the scale. The RG is used to calculate cross sections for processes like e+e- → hadrons, where perturbative methods are applicable at short distances. The renormalization group allows for the evolution of coupling constants and the determination of physical quantities at different energy scales. The solution of the RG equation involves the Λ parameter, which is a fundamental scale in QCD. The RG is essential for understanding the behavior of QCD at different energy scales and for making precise predictions in high-energy physics.