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This chapter argues that gauge symmetry is a purely formal feature of our theoretical representations with no empirical consequences. It distinguishes theoretical symmetries of a theory from empirical symmetries of situations to which a theory may be applied. A theoretical symmetry is purely formal if it implies no corresponding empirical symmetry. The chapter responds to several objections based on phenomena whose explanation apparently requires that gauge symmetry have empirical consequences. It argues that gauge symmetry is not observable and criticizes ‘the gauge argument’ from gauge symmetry to the existence of fundamental interactions. ‘Ghost’ fields are not real fields: despite its name, ‘spontaneous symmetry breaking’ does not involve any breaking of ‘local’ gauge symmetry; the theta-vacuum is not a superposition of distinct states inter-related by ‘large’ gauge transformations; anomalies in empirically successful theories do not involve breaking of any empirical gauge symmetry.

Symmetries play a fundamental role in modern physics. This chapter introduces this concept by discussing the meaning of symmetries in mathematics and in physics, explaining the significance of Lie groups. With the help of some analogies, the principle of gauge symmetry is illustrated. This principle is the basis for the construction of the Standard Model and determines the structure of the fundamental forces. However, the masses of the W and Z particles are in conflict with the gauge principle. The solution to this problem requires the introduction of new concepts such as spontaneous symmetry breaking and the Higgs mechanism. The chapter also describes the experimental strategies in the search for the Higgs boson, the significance of the possible discovery of this particle, and the remaining open problems.

This chapter examines the role of global symmetries in local quantum field theories. It shows that exact global symmetries are rare — indeed, if we take gravitational effects into account, probably non-existent. Nevertheless, approximate global symmetries play an enormously important role in modern field theory. The appearance of massless Goldstone particles once an exact global symmetry is spontaneously broken is of enormous importance in modern field theory, and proof of the Goldstone theorem embodying this phenomenon is given. Dynamical aspects of spontaneous symmetry-breaking (SSB) are examined, and it is shown that the essence of SSB resides in the energetics of the theory in the infrared (i.e., at long distances).

This chapter gives a general path integral formulation of the quantum anomaly for chiral symmetry. The quantum anomaly is identified as the Jacobian arising from the symmetry transformation of path integral variables. The correct Ward-Takahashi identities with an anomalous term are thus defined from the beginning. To realize this idea, it is essential to have a suitable regulator which controls divergences by preserving as much symmetry as possible. In the path integral formulation, the gauge invariant mode cut-off is shown to be useful. As topics related to quantum anomalies, a brief account of instanton solutions and the Atiyah-Singer index theorem is given. As applications of these notions, it is explained that the (naive) unitary transformation to the interaction picture does not exist. When the relevant symmetry is spoiled by the quantum anomaly and instanton effects, the Nambu-Goldstone particles associated with spontaneous symmetry breaking is generally absent.

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.

Chapter 10 examines the case of the spontaneous breaking of a continuous symmetry. When there is spontaneous symmetry breaking, the correlation length diverges at all temperatures in the ordered phase. Physical systems then display universal properties for all temperatures below T_c. This phenomenon is a consequence of the existence of Goldstone modes (massless bosons, in the quantum field theory terminology). Chapter 10 considers a lattice model with O(N) symmetry. In the continuum limit, it leads to the non–linear sigma model. A renormalization group can then be constructed and universal properties can be derived at and near dimension 2 in the framework of dimensional continuation and a d − 2 expansion. In dimension 2, the O(N) symmetric model illustrates asymptotic freedom with symmetry restoration and the Mermin–Wagner–Coleman theorem. In d > 2, a crossover scale appears, between infrared behaviour and critical behaviour.

Chapter 12 describes the main steps in the construction of the electroweak component of the Standard Model of particle physics. The classical Abelian Landau–Ginzburg–Higgs mechanism is recalled, first introduced in the macroscopic description of a superconductor in a magnetic field. It is based on a combination of spontaneous symmetry breaking and gauge invariance. It can be generalized to non–Abelian gauge theories, quantized and renormalized. The recent discovery of the predicted Higgs boson has been the last confirmation of the validity of the model. Some aspects of the Higgs model and its renormalization group (RG) properties are illustrated by simplified models, a self–interacting Higgs model with the triviality issue, and the Gross–Neveu–Yukawa model with discrete chiral symmetry, which illustrates spontaneous fermion mass generation and possible RG flows.

This chapter introduces chiral symmetry, the extra symmetry that QCD acquires when the masses of quarks are set to zero. It introduces the concept of spontaneous symmetry breaking and explains the spontaneous breaking of chiral symmetry in QCD. It introduces the concept of a Goldstone boson, a particle that has zero mass as the result of spontaneous symmetry breaking, and explains how this concept explains properties of the pi and K mesons and allows us to determine the underlying values of the quark masses.

An important concept is symmetry breaking and this is treated here. This idea is introduced in the context of phase transitions and the chapter also shows how to describe symmetry breaking with a Lagrangian. Breaking a continuous symmetry leads to Goldstone modes and we also show the consequence of symmetry breaking in a gauge theory, thereby introducing the Higgs mechanism.

Anderson spends a sabbatical year at the University of Cambridge. He informs graduate student Brian Josephson about spontaneous symmetry breaking in superconductors and Josephson discovers the effects that bear his name and won him a share of a Nobel Prize. Anderson works in this area and pursues analogies to superfluid helium four. He uses an analogy to his work on superconductivity to suggest a mechanism for mass generation for elementary particles. Peter Higgs generalizes Anderson’s idea and later wins a Nobel Prize for doing so. Anderson spends eight years as a half-time professor at the University of Cambridge. He leads the way to transform solid-state physics into condensed matter physics and does important work on superfluid helium three. He and Joyce buy a vacation home in Port Isaac, Cornwall.

Chapter 4 deals with the stability of the proton, hence of hydrogen, and how to reconcile that stability with the baryon number nonconservation (or baryon conservation) needed to establish a matter–antimatter imbalance in the infant universe. Sakharov’s three conditions for establishing a matter–antimatter imbalance are presented. Grand unified theories and experimental searches for proton decay are described. The concept of spontaneous symmetry breaking is introduced in describing the electroweak phase transition in the infant universe. That transition is treated as the potential site for introducing the imbalance between quarks and antiquarks, via either baryogenesis or leptogenesis models. The up–down quark mass difference is presented as essential for providing the stability of hydrogen and of the deuteron, which serves as a crucial stepping stone in stellar hydrogen-burning reactions that generate the energy and elements needed for life. Constraints on quark masses from lattice QCD calculations and violations of chiral symmetry are discussed.

The spontaneous breaking of symmetries (SSB) is discussed for global symmetries and Goldstones theorem is derived. The renormalisation of theories with SSB is studied using the effective potential. Then SSB is applied to the Abelian Higgs model, both on the classical and quantum level.

This chapter describes theories that combine the ideas of gauge symmetry and spontaneous symmetry breaking. It explains that this combination gives rise to massive spin-1 bosons. This construction is used to propose fundamental equations for the weak interaction. The predictions of these equations for high-energy neutrino scattering are worked out and compared to experiment.

The postwar Red Scare and McCarthyism solidified Anderson’s left-of-center politics. He solidified his position at Bell Labs by using the concept of superexchange to explain neutron scattering data from transition metal oxides. He disengaged from Shockley and studied magnetism under the tutelage of Charles Kittel, Conyers Herring, and Gregory Wannier. The Heisenberg model is used to explain ferromagnetism and antiferromagnetism and Anderson’s discovery of spontaneous symmetry breaking in quantum antiferromagnets is described. He spends six months in Japan at the invitation of Ryogo Kubo, teaches magnetism, attends and, at an international conference, learns that he is a bone fide solid-state theoretician.

This chapter starts with a simple introduction to gauge invariance and spontaneous symmetry breaking. An elementary introduction to Lagrangians in classical and quantum mechanics is given. A simplified description of the Higgs mechanism for electroweak symmetry breaking is presented in terms of Lagrangians. The experimental evidence for the observation of the Higgs boson is described in the two-photon, ZZ^*, and WW^* channels. There is an emphasis on understanding the Standard Model backgrounds and the required detector performance. The statistical techniques used to assess the significance of the results are described. The measurements used to determine the spin–parity of the Higgs boson are reviewed. The evidence for the Higgs boson decay to fermions is described using the two-tau decay mode. The outlook for future studies to determine if this Higgs boson is compatible with the Standard Model Higgs boson is discussed.

This chapter discusses classical fields in an arbitrary Riemann spacetime. General considerations are followed by the formulation of scalar fields with non-minimal coupling. Spontaneous symmetry breaking in curved space is shown to provide the induced gravity action with a cosmological constant. The construction of spinor fields in curved spacetime is based on the notions of group theory from Part I and on the local Lorentz invariance. Massless vector fields (massless vector gauge fields) are described and the interactions between scalar, fermion and gauge fields formulated. A detailed discussion of classical conformal transformations and conformal symmetry for both matter fields and vacuum action is also provided.

In this chapter, a model is considered that can be defined in continuous dimensions, the Gross– Neveu–Yukawa (GNY) model, which involves N Dirac fermions and one scalar field. The model has a continuous U(N) symmetry, and a discrete symmetry, which prevents the addition of a fermion mass term to the action. For a specific value of a coefficient of the action, the model undergoes a continuous phase transition. The broken phase illustrates a mechanism of spontaneous symmetry breaking, leading to spontaneous fermion mass generation like in the Standard Model (SM) of particle physics. In four dimensions, the GNY can be considered as a toy model to represent the interactions between the top quark and the Higgs boson, the heaviest particles of the SM of fundamental interactions, when the gauge fields are omitted. The model is renormalizable in four dimensions and its renormalization group (RG) properties can be studied in d = 4 and d = 4 − ϵ dimensions. A model of self-interacting fermions with the same symmetries and fermion content, the Gross–Neveu (GN) model, has been widely studied. In perturbation theory, for d > 2, it describes only a phase with massless fermions but, in d = 2 + ϵ dimensions, the RG indicates that, at a critical value of the coupling constant, the model experiences a phase transition. In two dimensions, it is renormalizable and exhibits the phenomenon of asymptotic freedom. The massless phase becomes infrared unstable and there is strong evidence that the spectrum corresponds to spontaneous symmetry breaking and fermion mass generation.

Spontaneous symmetry breaking. The symmetric Higgs bottle and its asymmetric state of lowest energy. The “expectation value” of the Higgs field in the vacuum. Why and how the vacuum is not void, it is a substance, the mother of them all. Understanding the generation of the masses of particles (other than the Higgs boson and the massless photon and gluons). The experimental test of this mass-generating mechanism. The Higgs-boson discovery fest. The “discovery channels” and the experimentalists’ insufficient acknowledgments.

The Standard Model (SM) 2020 of weak, electromagnetic and strong interactions, based on gauge symmetry and spontaneous symmetry breaking, describes all known fundamental interactions at the microscopic scale except gravity and, perhaps, interactions with dark matter. The SM model has been tested systematically in collider experiments, and in the case of strong interactions (quantum chromodynamics) also with numerical simulations. With the discovery in 2012 of the Higgs particle at the Large Hadron Collider (LHC) at the European Council for Nuclear Research (CERN), all particles of the SM have been identified, and most parameters have been measured. Still, the Higgs particle remains the most mysterious particle of the SM, since it is responsible for all the parameters of the SM except gauge couplings and since it leads to the fine-tuning problem. The discovery of its origin, and the precise study of its properties should be, in the future, one of the most important field of research in particle physics. Since we know now that the neutrinos have masses, the simplest extension of the SM implies Dirac neutrinos. With such a minimal modification, consistent so far (2020) with experimental data, the lepton and quark sectors have analogous structures: the lepton sector involves a mixing matrix, like the quark sector (three angles have been determined, the fourth charge conjugation parity (CP) violating angle is still unknown).