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This chapter reviews the discussions in the preceding chapters. Starting from its historical development, the lagrangian of QCD is introduced and its phenomenological consequences worked out in detail in the framework of perturbative QCD. Experiments on various high-energy reactions are discussed which determine the structure of QCD and its free parameters. Perturbative QCD is treated on the level of fixed order matrix element calculations, including renormalization, the leading-log approximation, and QCD-inspired heuristic and numerical Monte Carlo models.

This chapter begins with a discussion of the phases of gauge theories. It then discusses the moduli space for F = N, IR fixed points, Seiberg's nontrivial solution to the 't Hooft anomaly matching, the three major nontrivial consistency checks of Seiberg's conjectured duality, and checking that the confined descriptions of the theories with F = N and F = N + 1 flavours are consistent with dual descriptions of the theories with more flavours. Exercises are provided at the end of the chapter.

This chapter begins with a brief introduction to quantum chromodynamics (QCD), highlighting the similarities and differences with quantum electrodynamics (QED). QCD gives a non-linear force that confines quarks inside hadrons at large distances but is weak at short distances, thus providing justification for the simple QPM. Essential ‘tree-level’ diagrams are evaluated using the corresponding QED processes and QCD colour factors. The problems of infrared and ultraviolet divergences in both QED and QCD are outlined, with emphasis on the physical ideas underlying their resolution. This leads on to the idea of a running coupling. The rest of the chapter is concerned with an outline of the operator product expansion, which is the more formal method used to derive many results used in the QCD-enhanced parton model. Finally, the crucial idea of factorization — the calculation of a DIS cross-section by convolution of a point-like cross-section with parton momentum densities — is introduced.

The QCD improved parton model provides the foundation on which all the remaining chapters are based. This chapter begins by covering the key ideas: tree level QCD Compton and boson-gluon fusion processes give rise to characteristic gluonic ‘radiative corrections’ to the QPM; infrared singularities are absorbed by renormalizing the parton densities, which thus become Q² dependent and no longer satisfy exact Bjorken scaling. The next section covers the DGLAP evolution equations, which enable the Q² dependence to be calculated perturbatively using ‘splitting functions’. To give an insight into the effect of these equations, some simple numerical examples are given in the following section. The relationship between the more formal Operator Product Expansion method outlined in the previous chapter and the DGLAP approach is indicated. The last three sections comment on: higher twist; the extension of the DGLAP formalism to accommodate heavy quarks; and the extension of the DGLAP formalism to higher orders.

This chapter reviews and puts into context the experiments which lead to the formulation of QCD. Starting from the static quark model, the phenomenology of deep inelastic scattering processes is introduced and used to derive and explain the quark parton model (QPM). Discussing in detail the conceptual problems of the QPM, the arguments that lead to the theory of strong interactions based on an SU(3) gauge symmetry are explained. The chapter concludes with the lagrangian of QCD.

While perturbative QCD deals with free quarks and gluons, experimentally, usually only multi-hadron final states are observed. This chapter focuses on how to extract information about QCD from the observable final states. After explaining the general phenomenology, it discusses the problem of selecting events of a certain class from the set of all the interactions recorded by an experiment. It then defines appropriate observables which are sensitive to the dynamics of QCD. Given such variables, the issues of corrections for detector effects —such as finite acceptance, efficiency and resolution, hadronization corrections, which go back from the observable hadron level to the theoretically accessible parton level, and systematic uncertainties — are addressed. The analysis strategies are illustrated by the examples of structure function measurements from deep inelastic scattering, jet cross sections in proton-antiproton collisions, and the determination of jet rates in electron-positron annihilation.

This chapter presents the determination of structure functions and parton distributions from deep inelastic lepton-nucleon scattering, both for the case of charged lepton-nucleon and the case of neutrino-nucleon scattering. The focus is on the results obtained by the HERA experiments. The direct measurements as well as the indirect extraction of the gluon density from scaling violations are discussed. Information obtained from sum rules and from hadron-hadron scattering is briefly summarized, followed by a discussion of global QCD fits of the parton densities based on all available information.

This chapter presents experimental results for the QCD lagrangian. It shows that the phenomenology of strong interactions is indeed described by an unbroken Yang–Mills theory based on a non-abelian gauge symmetry. Quarks are spin-1/2 fermions, while gluons are spin-1 bosons and the strong coupling is flavour independent.

This chapter consists of three parts. In the first part, Lüscher's formula, which relates the scattering phase shift to the two particle energy in finite volume, is explained. A comprehensive but less rigorous derivation for the formula has been attempted for the ??-system as an example with emphasis on the Bethe-Salpeter (BS) wave function. In the second part, the BS wave function is considered to define the potential in quantum field theories. This method is applied to the two-nucleon system, in order to extract the NN -potential from lattice QCD. In the last part, the origin of the strong repulsion at short distance in the NN-potential, called the repulsive core, is theoretically investigated by the Operator Product Expansion and the renormalization group.

The course begins with an introduction to the Standard Model, viewed as an effective theory. Experimental and theoretical limits on the scales at which New Physics can appear, as well as current constraints on quark flavor parameters, are reviewed. The role of lattice QCD in obtaining these constraints is described. A second section is devoted to explaining the Cabibbo-Kobayashi-Maskawa mechanism for quark flavor mixing and CP violation, and to detailing its most salient features. The third section is dedicated to the lattice QCD study of K ??? decays. It comprises discussions of indirect CP violation through K 0- antiK 0 mixing, the ? I=1/2 rule and direct CP violation, including final state interactions. It presents the lattice QCD tools required to describe these phenomena. Even though the lattice study of these decays originated in the mid-eighties, the consistency of the Standard Model with the beautiful experimental determinations of direct CP violation in K ??? decays remains an important goal for the new generation of lattice QCD practitioners.

This chapter is a detailed presentation of three specific topics: (1) The consequences of the loss of chiral symmetry in the Wilson lattice regularization of the fermionic action, and its recovery in the continuum limit. The treatment of these arguments involves lattice Ward identities, through which we establish the normalization properties of lattice partially conserved currents and the relation between renormalized composite fields, like the scalar and pseudoscalar densities, which belong to the same chiral multiplet. We also discuss the consequences of the loss of chiral symetry for the chiral condensate. (2) The definition and properties of mass independent renormalization schemes, which are suitable for a non-perturbative computation of various operator renormalization constants. In particular we present in detail the RI/MOM scheme. (3) The modification of the Wilson fermion action, by the introduction of a chirally twisted mass term, known as twisted mass QCD (tmQCD), which results to improved (re)normalization and scaling properties for physical quantities of interest.

This chapter gives an introduction to lattice QCD at finite temperature and baryon density. After a discussion of some fundamental aspects and difficulties of quantum field theory at finite temperature in the continuum, the lattice formulation of the partition function for the grand canonical ensbemble is introduced and its relation to the transfer matrix formalism is presented. As analytic tools for its evaluation, weak coupling perturbation theory on the lattice as well as the strong coupling expansion are discussed. Regarding Monte Carlo evaluations, similarities and differences to the situation in the vacuum are pointed out. All concepts are illustrated with various applications like the equation of state, screening masses, the free energy of static quark systems and phase transitions. In the second part, special emphasis is put on lattice QCD at finite baryon density. The sign problem is discussed and current techniques to deal with it at small baryon chemical potential are presented. The implications for the QCD phase diagram are summarized.

This chapter provides an introduction to some of the basic numerical techniques used in lattice QCD. The topics covered include Markov-chain simulations, the HMC algorithm, deflation and variance reduction methods as well as a theoretical analysis of the statistical error propagation from the calculated correlation functions to the physical quantities.

Chapter 15 examines the conditions under which it is possible to define renormalized quantum field theories (QFTs) consistent on all scales (a property not necessary for effective QFTs and not shared by ‘trivial’ theories like quantum electrodynamics (the triviality issue)). It has been conjectured that a necessary and, perhaps, sufficient condition is the existence of ultraviolet fixed points, a property called asymptotic safety. Examples are “asymptotically free” theories like non–Abelian gauge theories (quantum chromodynamics, or QCD) in four dimensions or the non–linear sigma model in two dimensions. These theories are characterized by a crossover scale between universal infrared and large momentum behaviour behaviours. The non–linear sigma and Gross–Neveu models in dimensions greater than 2 are examined in detail. For example, the non–linear sigma model exhibits, below the critical temperature, a crossover between an infrared behaviour dominated by Goldstone modes and a universal, large momentum, critical behaviour.

These lectures provide an introduction to the low-energy dynamics of Nambu–Goldstone fields, which associated with some spontaneous (or dynamical) symmetry breaking, using the powerful methods of effective field theory. The generic symmetry properties of these massless modes are described in detail and two very relevant phenomenological applications are worked out: chiral perturbation theory, the low-energy effective theory of QCD, and the (non-linear) electroweak effective theory. The similarities and differences between these two effective theories are emphasized, and their current status is reviewed. Special attention is given to the short-distance dynamical information encoded in the low-energy couplings of the effective Lagrangians. The successful methods developed in QCD could help us to uncover fingerprints of new physics scales from future measurements of the electroweak effective theory couplings.

The lectures that appear within this chapter provide an introduction to soft-collinear effective theory (SCET). It begins by discussing resummation for soft-photon effects in QED, including soft photons in electron–electron scattering and the expansion of loop integrals and the method of regions event-shape variables. It then covers SCET specifically, including the method of regions for the Sudakov form factor, effective Lagrangians, the vector current in SCET, and resummation by renormalization group (RG) evolution. It covers applications of SCET in jet physics, describes the characteristic feature in jet processes of Sudakov logarithms, and discusses factorization for the event-shape variable thrust and factorization and resummation for jet cross sections.