Quantum chromodynamics

Author: Leandro Alegsa Creation date: November 13, 2022

Quantum chromodynamics (QCD) is the theory of the strong interaction between quarks and gluons. These are the fundamental particles that make up hadrons such as the proton, neutron and pion. It shows the interactions inside subatomic particles.

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 $r_{c}$ of the interaction between the nucleons is $r_{c}={\frac {\hbar }{m_{\pi }c}}\cong 10^{-13}$ cm, i.e., of the order of the Compton wavelength of the π {\displaystyle $\pi$ -mesons ( $m_{{\pi }}$ is the mass of the pion).

While the nuclear forces decrease exponentially with distance,

$\phi _{K}(r)\propto {\frac {1}{r}}\exp \left(-{\frac {r}{r_{c}}}\right)$ (Yukawa potential),

the electromagnetic interaction falls only according to the power law

$\phi _{{EM}}(r)\propto {\frac {1}{r}}$ (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

→ Main article: Confinement

The gauge group underlying QCD ${\mathrm {SU}}(3)$ 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 $\alpha _{s}$ 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 $\tau$ $(\tau \approx h/E_{0})\,,$ 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.