Starting from 2-flavor QCD, isospin symmetry is employed in order to explain the multiplets of light baryons and mesons, from a constituent quark perspective. Next we involve the strange quark and arrive at meson mixing as well as the Gell-Mann–Okubo formula for the baryon multiplet splitting. Regarding QCD from first principles, we comment on lattice simulation results for the hadron masses. At last we discuss the hadron spectrum in a hypothetical world with Nc=5 colors.

The fermion content of the Standard Model is extended to 3 generations. For the lepton we discuss universality, and for the quarks the GIM mechanism, the CKM quark mixing matrix, and its CP violating parameter. Similarly, for the leptons we construct the PMNS mixing matrix, and we describe how neutrino oscillation was observed. Since the Standard Model is complete now, we provide an overview over its parameters and take another look from an unconventional low-energy perspective.

This chapter first focuses on the QCD vacuum θ and the related strong CP problem. We review the manifestation of theta in the QCD action or alternatively in the mass matrix, and in chiral perturbation theory. Beyond the Standard Model, the Peccei–Quinn formalism turns θ into an axion field. We discuss that approach and its implications in astrophysics and cosmology. Finally we consider the corresponding parameters for the SU(2)L gauge field and for QED. The former can be absorbed by field-redefinitions, but the latter leads to a linear combination of these two vacuum angles, which persists as a parameter of the Standard Model, but which is often ignored.

The Standard Model is reduced to Quantum Chromodynamics (QCD) only by a limitation to moderate energies. We review crucial properties like asymptotic free and the role of chiral symmetry. The latter is analyzed both in the continuum and on the lattice. In particular, the Ginsparg–Wilson relation for lattice Dirac operators allow us to properly address the hierarchy problem which appears fermion masses at the non-perturbative level.

The gluon SU(3) gauge field is studied, with “quarks” only as static sources. We describe confinement by referring to the Wegner–Wilson loop and its strong-coupling expansion on the lattice. The way back to the continuum is related to asymptotic freedom. We discuss the strength of the strong interaction, its low-energy string picture, and the Luescher term as a Casimir effect. The Fredenhagen–Marcu operator provides a sound confinement criterion. In the confined phase we discuss the glueball spectrum, the Polyakov loop, and center symmetry. We also consider deconfinement at high temperature, and finally the case of a G(2) gauge group instead of SU(3).

We sketch the highlights of Part I, which introduces the concepts of quantum field theory.

We stay in the framework of the low-energy effective theory of QCD in terms of Nambu–Goldstone bosons fields and consider effects due to their topology. We distinguish the cases of Nf = 2 or Nf >= 3 light quark flavors and discuss in both cases how the gauge anomaly cancelation is manifest in the effective theory, the role of G-parity, and the neutral pion decay into two photons, which does not explicitly depend on the number of colors, Nc . For Nf >= 3 we introduce the Wess–Zumino–Novikov–Witten term in a 5th dimension, we discuss the intrinsic parity of light meson fields and their electromagnetic interactions. In this context, we clarify the question whether there is low-energy evidence for Nc = 3, and we address again the role 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. Next we discuss variants of “spontaneous symmetry breaking” patterns, which deviate from the Standard Model, in the continuum and on the lattice. Finally we consider a “small unification” of the electroweak gauge couplings, as a toy model for the concept of Grand Unified Theories (to be address in Chapter 26).

The topological charge of smooth Yang–Mills gauge fields is discussed, describing in particular 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. We finally discuss the Atiyah–Singer index theorem in the continuum and more explicitly on the lattice.

Free fermion fields are canonically quantized, proceeding from Weyl to Dirac and Majorana fermions, and from the massless to the massive case. We discuss properties like chirality, helicity, and the fermion number, as well as the behavior under parity and charge conjugation transformation. Fermionic statistics is applied to the cosmic neutrino background.

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

This chapter discusses perturbation theory, applied to the λϕ4 model, with a focus on dimensional regularization. It characterizes different types of Feynman diagrams. We explain the meaning of renormalization and discuss the conditions for renormalizability.

This chapter deals with global symmetries and their spontaneous breaking, particularly referring to sigma-models. We consider two theorems about the emergence of Nambu–Goldstone bosons. This takes us to the structure of low-energy effective theories, and to the hierarchy problem in the Higgs sector of the Standard Model. In that context, we further address triviality, the electroweak phase transition in the early Universe, and the extension to two Higgs doublets.