was discovered in three-jet events at PETRA in 1979. These experiments became more and more precise, culminating in the verification of perturbative QCD at the level of a few percent at LEP, at CERN. The other side of asymptotic freedom is confinement. Since the force between color charges does not decrease with distance, it is believed that quarks and gluons can never be liberated from hadrons. This aspect of the theory is verified within lattice QCD computations, but is not mathematically proven. One of the Millennium Prize Problems announced by the Clay Mathematics Institute requires a claimant to produce such a proof. Other aspects of non-perturbative QCD are the exploration of phases of quark matter, including the quark–gluon plasma. The relation between the short-distance particle limit and the confining long-distance limit is one of the topics recently explored using string theory, the modern form of S-matrix theory. Theory Some definitions Every field theory of particle physics is based on certain symmetries of nature whose existence is deduced from observations. These can be local symmetries, which are the symmetries that act independently at each point in spacetime. Each such symmetry is the basis of a gauge theory and requires the introduction of its own gauge bosons. global symmetries, which are symmetries whose operations must be simultaneously applied to all points of spacetime. QCD is a non-abelian gauge theory (or Yang–Mills theory) of the SU(3) gauge group obtained by taking the color charge to define a local symmetry. Since the strong interaction does not discriminate between different flavors of quark, QCD has approximate flavor symmetry, which is broken by the differing masses of the quarks. There are additional global symmetries whose definitions require the notion of chirality, discrimination between left and right-handed. If the spin of a particle has a positive projection on its direction of motion then it is called right-handed; otherwise, it is left-handed. Chirality and handedness are not the same, but become approximately equivalent at high energies. Chiral symmetries involve independent transformations of these two types of particle. Vector symmetries (also called diagonal symmetries) mean the same transformation is applied on the two chiralities. Axial symmetries are those in which one transformation is applied on left-handed particles and the inverse on the right-handed particles. Additional remarks: duality As mentioned, asymptotic freedom means that at large energy – this corresponds also to short distances – there is practically no interaction