We stay in the framework of the low-energy effective theory of QCD in terms ofNambu–Goldstone bosons fields and consider effects due to their topology. We distinguishthe cases of Nf = 2 or Nf >= 3 light quark flavors and discuss in both cases how thegauge anomaly cancelation is manifest in the effective theory, the role of G-parity, andthe neutral pion decay into two photons, which does not explicitly depend on the number ofcolors, Nc . For Nf >= 3 we introduce the Wess–Zumino–Novikov–Witten term in a 5thdimension, we discuss the intrinsic parity of light meson fields and their electromagneticinteractions. In this context, we clarify the question whether there is low-energyevidence for Nc = 3, and we address again therole of technicolor.

The Higgs mechanism is introduced, first for scalar QED and then with the Higgs doublet,which takes us to the gauge bosons in the electroweak sector of the Standard Model. Nextwe discuss variants of “spontaneous symmetry breaking” patterns, which deviate from theStandard Model, in the continuum and on the lattice. Finally we consider a “smallunification” of the electroweak gauge couplings, as a toy model for the concept of GrandUnified Theories (to be address in Chapter 26).

The topological charge of smooth Yang–Mills gauge fields is discussed, describing inparticular the SU(2) instanton. This leads to the Adler–Bell–Jackiw anomaly and to θ-vacuum states, which are similar to energy bands in a crystal. Wefinally discuss the Atiyah–Singer index theorem in the continuum and more explicitly onthe lattice.

Free fermion fields are canonically quantized, proceeding from Weyl to Dirac andMajorana fermions, and from the massless to the massive case. We discuss properties likechirality, helicity, and the fermion number, as well as the behavior under parity andcharge conjugation transformation. Fermionic statistics is applied to the cosmic neutrinobackground.

Scalar quantum field theory is introduced in the functional integral formulation,starting from classical field theory and quantum mechanics. We consider Euclidean time andrelate the system in the lattice regularization to classical statistical mechanics.

This chapter discusses perturbation theory, applied to the λϕ4 model, with a focus ondimensional regularization. It characterizes different types of Feynman diagrams. Weexplain the meaning of renormalization and discuss the conditions forrenormalizability.

This chapter deals with global symmetries and their spontaneous breaking, particularlyreferring to sigma-models. We consider two theorems about the emergence of Nambu–Goldstonebosons. This takes us to the structure of low-energy effective theories, and to thehierarchy problem in the Higgs sector of the Standard Model. In that context, we furtheraddress triviality, the electroweak phase transition in the early Universe, and theextension to two Higgs doublets.

This chapter provides an extensive discussion of Grand Unified Theories (GUTs) andrelated subjects. It begins with the SU(5) GUT, its fermion multiplets, and the resultingtransitions between leptons and quarks, which enable in particular proton decay. In thiscontext, we discuss the baryon asymmetry in the Universe, as well as possible topologicaldefects dating back to the early Universe, according to the Kibble mechanism, such asdomain walls, cosmic strings, or magnetic monopoles. That takes us to a review of Diracand ‘t Hooft–Polaykov monopoles, Julia–Zee dyons, and the effects named afterCallan–Rubakov and Witten. Next we discuss fermion masses in the framework of the GUTswith the gauge groups SU(5) and Spin(10). Then we consider small unified theories (withoutQCD) with a variety of gauge groups. Finally, we summarize the status and prospects of theGUT approach.

First, non-Abelian gauge fields are quantized canonically. The Faddeev–Popov ghostfields implement gauge fixing, then we review the BRST symmetry. Next, we proceed to thelattice regularization and then from Abelian to non-Abelian gauge fields. We stress thatthe compact lattice functional integral formulation does not require gauge fixing.

We construct mass terms for the Standard Model fermions of the first generation. Thisincludes the neutrino, where we invoke either a dimension-5 term or we add a right-handedneutrino field. We reconsider the CP symmetry, the fate of baryon and lepton numbers, andthe quantization of the electric charge. The question of the mass hierarchy takes us tothe seesaw mass-by-mixing mechanism. As a peculiarity, we finally revisit such propertiesin the scenario without colors (Nc=1), whichallows leptons and baryons to mix.

Chiral perturbation theory is the systematic low-energy effective theory of QCD, interms of low-energy parameters and pseudo-Nambu–Goldstone boson fields representing pions,kaons, and the η. We discuss their masses in leading order, and the correspondingelectromagnetic corrections, where we arrive at Dashen’s theorem. We show how thislow-energy scheme even encompasses nucleons, and how QCD provides corrections to the weakgauge boson masses. In that context, we comment on a technicolor extension and on thehypothesis of minimal flavor violation, which is described by spurions.

We outline the main concepts of the Standard Model, illustratively describing itscentral features and some open questions, as a preparation for the following chapters.