Calculations in QCD have been made in two ways: lattice simulations at low energies, and perturbative calculations at high energies. In this chapter we outline some of the results obtained.

Lattice QCD and confinement

It was pointed out in Section 16.1 that, at low energies, a non-perturbative approach to QCD is needed. ‘Lattice QCD’ is such an approach. The gluon fields are defined on a four-dimensional lattice of points (nμ, n)a, where a is the lattice spacing and the nμ are integers. Field derivatives are replaced by discrete differences. This gives a ‘lattice regularised’ QCD. The lattice spacing corresponds to an ultraviolet cut-off, since wavelengths < 2a cannot be described on the lattice. A lattice does not have full rotational symmetry in space, but it is believed that nevertheless continuum QCD corresponds to the limit a → 0. Current computing power allows lattices of ∼(36)4 points. The range of the strong nuclear force is ∼ 1 fm. To fit such a distance comfortably on the lattice, we can anticipate that we shall not want a to be much less than (2fm)/36 = 0.056fm (and ℏc/a > 3.5 GeV).

In the high energy perturbation theory described in Section 16.3, the renormalisation parameter λ and the dimensionless coupling parameter g are combined to give a single physical parameter, Λ, having the dimensions of energy. The relationship between the effective coupling constant αs(Q2) and Λ in the lowest order of perturbation theory is given by (16.25).