This chapter lays out the foundation of gauge field theories, at least in part, and computes the one-loop renormalization of the gauge coupling. After a leisurely introduction to the gauge field strength and the action, in the geometric convention and anti-Hermitian basis, we introduce the effective action with an infrared cut-off. One-loop running of the QED coupling constant is computed via the heat kernel method, which inevitably leads to the Landau pole at the high-energy side, where the coupling diverges and beyond which the gauge kinetic term turns negative. We then turn to Yang-Mills theories, in a language suitable for classical Lie groups. The Faddeev-Popov gauge-fixing procedure and the background-field method are given a full treatment. Asymptotic freedom is derived, which is followed by applications to supersymmetric Yang-Mills theories and to spontaneously broken theories. Along the way, we turn to Wilsonian renormalization, in preparation for the next chapter, and emphasize how the Landau pole phenomenon of QED and the divergent Yang-Mills coupling at the low scale are fundamentally different, despite the apparent similarity.

The Standard Model is reduced to Quantum Chromodynamics (QCD) only by a limitation tomoderate energies. We review crucial properties like asymptotic free and the role ofchiral symmetry. The latter is analyzed both in the continuum and on the lattice. Inparticular, the Ginsparg–Wilson relation for lattice Dirac operators allow us to properlyaddress the hierarchy problem which appears fermion masses at the non-perturbativelevel.

The gluon SU(3) gauge field is studied, with “quarks” only as static sources. Wedescribe confinement by referring to the Wegner–Wilson loop and its strong-couplingexpansion 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 theLuescher term as a Casimir effect. The Fredenhagen–Marcu operator provides a soundconfinement criterion. In the confined phase we discuss the glueball spectrum, thePolyakov 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).

For physicists who study elementary particles and quantum field theory, the 1970s was a golden age. It saw the experimental confirmation of the electroweak theory, and the extension of that thinking would lead us to a successful theory of strong interactions as well. All the fundamental forces of nature, except for gravity, would be unified in what became known as the “Standard Model.” By the end of 1973, there was some experimental verification of the electroweak theory. Weinberg agrees to write The First Three Minutes, which was published in 1977. Louise visits Stanford Law School, accompanied by Weinberg, who finds his host department cold. In 1977, he collaborates with Ben Lee of Fermilab, who tragically died in a car accident later that year. Louise is invited to teach at University of Texas Law School, in the summer of 1979, after which she was offered a full professorship. The Weinbergs taught in their respective universities and met in Cambridge in the holidays. Weinberg’s Nobel Prize, shared with Salam and Glashow, is announced in October 1979, ahead of the ceremony that December.

The theory of quantum chromodynamics (QCD) is introduced. Features of QCD as the nontrivial vacuum due to quark and gluon condensate and asymptotic freedom at high-energy scales are discussed. The concept of perturbative QCD and the running of the coupling constant is established. The equation of state of QCD at high temperatures from lattice QCD is reviewed and confronted with perturbative QCD calculations. The QCD equation of state at high baryon density is discussed. Properties of selfbound stars are developed where the equation of state has a nonvanishing pressure at a nonvanishing energy density. The mass–radius relation of pure quark stars is examined and compared to the limits from causality.