Quantum chromodynamics

Quantum chromodynamics (QCD for short) is a quantum field theory describing the strong interaction. It describes the interaction of quarks and gluons, the fundamental building blocks of atomic nuclei.

Like quantum electrodynamics (QED), QCD is a gauge theory. However, while QED is based on the abelian gauge group U(1) and describes the interaction of electrically charged particles (e.g. electron or positron) with photons, where the photons themselves are uncharged, the gauge group of QCD, SU(3), is non-abelian. It is therefore a Yang-Mills theory. The interacting particles of QCD are the gluons and the electric charge as a conservation variable is replaced by the color charge (hence the name, chromodynamics).

Analogous to QED, which only concerns the interaction of electrically charged particles, QCD deals exclusively with particles with "color charges", the so-called quarks. Quarks have three different color charges, designated as red, green, and blue. (This naming is merely a convenient convention; quarks do not possess a color in the colloquial sense. The number of colors corresponds to the degree of the gauge group of QCD, i.e., SU(3)).

The wave functions of the baryons are antisymmetric with respect to the color indices, as required by the Pauli principle. However, unlike the electrically neutral photon in QED, the gluons themselves carry color charge and therefore interact with each other. The color charge of gluons consists of a color and an anti-color, so gluon exchanges usually result in "color changes" of the quarks involved. The interaction of the gluons means that the attractive force between the quarks does not disappear at large distances, the energy required for separation continues to increase, much like a tension spring or a rubber thread. If a certain strain is exceeded, the thread breaks - in QCD, in this analogy, if a certain distance is exceeded, the field energy becomes so high that it is converted into the formation of new mesons. Therefore, quarks never appear singly, but only in bound states, the hadrons (confinement). The proton and the neutron - also called nucleons, since atomic nuclei are made of them - as well as the pions are examples of hadrons. The objects described by QCD also include exotic hadrons such as the pentaquarks and the tetraquarks discovered in 2016 at the LHCb_experiment at CERN.

Since quarks have both an electric and a color charge, they interact both electromagnetically and strongly. Since the electromagnetic interaction is much weaker than the strong interaction, one can neglect its influence in the interaction of quarks and therefore restrict oneself only to the influence of the color charge. The strength of the electromagnetic interaction is characterized by Sommerfeld's fine structure constant while the corresponding parameter of the strong interaction is of order 1.

Due to their non-Abelian structure and high coupling strengths, calculations in QCD are often costly and complicated. Successful quantitative calculations usually come from perturbation theory or from computer simulations. The accuracy of the predictions is typically in the percentage range. Thus, a large number of the theoretically predicted values could be verified experimentally.

Quantum chromodynamics is an essential part of the Standard Model of elementary particle physics.

Distinction from nuclear physics

The strength of the interaction leads to the fact that protons and neutrons in the atomic nucleus are much more strongly bound to each other than, for example, the electrons to the atomic nucleus. However, the description of nucleons is an open problem. The quarks (the constituent quarks and the sea quarks) contribute only 9 % to the mass of the nucleons, the remaining about 90 % of the nucleon mass comes from the kinetic energy of the quarks (about one third, caused by the kinetic energy according to the uncertainty principle, since they are "trapped" in a small space) and contributions of the gluons (a field strength contribution of about 37 percent and an anomalous gluon contribution of about 23 percent). The coupling processes occurring in QCD are dynamical and non-perturbative: the protons and neutrons themselves are colorless. Their interaction, instead of being described by quantum chromodynamics, is usually described in the framework of an effective theory according to which the attractive force between them is based on a Yukawa interaction due to the exchange of mesons, in particular the light pions (pion-exchange model). The description of the behaviour of nucleons via meson exchange in the atomic nucleus and in scattering experiments is the subject of nuclear physics.

So the strong interaction between nucleons within atomic nucleus is much more effective than their electromagnetic interaction. Nevertheless, the electrostatic repulsion of protons results in an important stability criterion for atomic nuclei. The strong interaction between nucleons, unlike the interaction between quarks, becomes exponentially smaller as the distance between nucleons increases. This is due to the fact that the exchange particles involved have nonzero mass in the pion exchange model. Therefore, the range of the interaction between the nucleons is cm, i.e., of the order of the Compton wavelength of the π {\displaystyle-mesons ( is the mass of the pion).

While the nuclear forces decrease exponentially with distance,

(Yukawa potential),

the electromagnetic interaction falls only according to the power law

(Coulomb potential),

since their exchange particles, the photons, have no mass and the interaction thus has an infinite range.

The strong interaction is thus essentially limited to distances between hadrons, such as occur in the atomic nucleus.

Confinement and asymptotic freedom

The gauge group underlying QCD is non-abelian, that is, the multiplication of two group elements is in general non-commutative. This leads to the appearance in the Lagrangian density of terms that cause gluons to interact with each other. For the same reason, the gluons carry color charge. This self-interaction leads to the fact that the renormalized coupling constant of QCD behaves qualitatively exactly opposite to the coupling constant of QED: It decreases for high energies. This leads to the phenomenon of asymptotic freedom at high energies and confinement at low energies. Only at extremely high temperatures, T > 5-1012 Kelvin, and/or correspondingly high pressure, the confinement is apparently cancelled and a quark-gluon plasma is formed.

Asymptotic freedom means that at high energies (small typical distances) the quarks behave like free particles, which is contrary to the behaviour of other systems where weak interaction is associated with large distances. Confinement means that below a boundary energy the coupling constant becomes so large that quarks only appear in hadrons. Since the coupling constant α of QCD is not a small parameter at low energies, perturbation theory, which can be used to solve many problems in QED, cannot be applied. In contrast, one approach to solving the QCD equations at low energies is computer simulations of lattice gauge theories.

Another approach to the quantum field theory treatment of hadrons is the use of effective theories, which transition to QCD for large energies and introduce new fields with new "effective" interactions for small energies. An example of such "effective theories" is a model of Nambu and Jona-Lasinio. Depending on the hadrons to be described, different effective theories find use. The chiral perturbation theory (CPT) is used for hadrons composed only of light quarks, i.e. up, down and strange quarks, which according to CPT interact with each other via mesons. For hadrons with exactly one heavy quark, i.e. a charm or bottom quark, and otherwise only light quarks, the heavy quark effective theory (HQET) is used, in which the heavy quark is assumed to be infinitely heavy, similar to the treatment of the proton in the hydrogen atom. The heaviest quark, the "top quark", is so highly energetic (E0 ~ 170 GeV) that in its short lifetime τ   with Planck's constant h, no bound states can form. For hadrons of two heavy quarks (bound states in the quarkonium), the so-called nonrelativistic quantum chromodynamics (NRQCD) is used.