The fascinating new world inside the nucleon, of quarks, gluons and colour, the nuclear strong force. How quantum chromodynamics (QCD) was discovered: probing the nucleons with scattering experiments and with increasing energy e+e− colliders, where quarks and gluons appear as hadronic jets.

The colour charges are three. Being the gauge of QCD non-Abelian, the gluons, not only the quarks, are ‘coloured’. How colour charges bind three quarks or a quark–antiquark pair forming hadrons that have zero overall colour charges.

The QCD coupling constant runs as the fine-structure constant, but with increasing momentum transfer, it decreases, instead of growing. Quarks become ‘free’, when they are very close to each other. Only a very small fraction of the proton mass is due to the quark masses, 99% being the energy of the colour field. The QCD vacuum, the status of minimum energy, a very active medium indeed, beautiful to study.

When matter first appeared in the universe, in the first microsecond after the Big Bang, quarks and gluons moved freely in a hot ‘soup’, the quark–gluon plasma. It is created in the laboratory in the ultra-relativistic heavy ion colliders and theoretically analysed with lattice QCD

This chapter is divided into two parts. The first part introduces the quark model, following more or less the historical developments. It led to an approximate symmetry, based on the SU(3) flavour group, where u, d and s quarks are the three degrees of freedom. The second part introduces the quantum chromodynamics theory (QCD), i.e. the true formal gauge theory of the strong interaction. Here again, the symmetry group is SU(3), but the degrees of freedom are the three quark colours. This symmetry is assumed to be exact, which has consequences on the existence of gluons and their properties, the carriers of the strong interaction at the elementary particle level, briefly mentioned in the previous chapter. The QCD interaction is the first non-Abelian interaction encountered in the book. The non-perturbative regime of QCD is also presented with a short introduction to lattice QCD. A discussion about the colour confinement and the hadronisation of quarks is also given.

The phenomenological applications of theformulae collected in Chapter 6 require input from lattice QCD (LQCD) and other nonperturbative approches. In particular one needsthe values of hadronic matrix elements of quark currents and those of four-quark operators. In this chapter we will collect all results for hadronic matrix elements, weak decay constants, and other nonperturbative objects in one place so that they can be looked up quickly whenever necessary. But before presenting the results obtained by purely numerical LQCD calculations, we will discuss the application of the so-calleddual QCD approach to weak decays, which was developed by Bardeen, Gerard, and myself over many years. While not as accurate as the lattice QCD method, it offers a better insight in the dominant QCD dynamics at long-distance scales than LQCD itself. The application of LQCD to nonleptonic decays of B mesons is very difficult, and therefore we will devote a section to the QCD factorization approach, stressing its successes and limitations.