# Quantum chromodynamics

The quantum chromodynamics (short QCD ) is a quantum field theory to describe 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 calibration theory . While the 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 , whereby the photons themselves are uncharged, the calibration group of the QCD, the SU (3) , non-Abelian. The interaction particles of the QCD are the gluons, and the color charge takes the place of the electric charge as a conservation quantity (hence the name, chromodynamics ).

Analogous to the QED, which only affects the interaction of electrically charged particles, the QCD only treats particles with a “color charge”, the so-called quarks . Quarks have three different color charges known as red , green, and blue . (This naming is merely a convenient convention; Quarks do not have a color in the colloquial sense. The number of colors corresponds to the degree of the calibration group of the QCD, i.e. the SU (3).)

The wave functions of the baryons are antisymmetric with respect to the color indices, as required by the Pauli principle . In contrast to the electrically neutral photon in QED, however, the gluons themselves carry a color charge and therefore interact with each other. The color charge of the gluons consists of a color and an anti-color, so that gluon exchange usually leads to “color changes” of the quarks involved. The interaction of the gluons ensures that the force of attraction between the quarks does not disappear at great distances; the energy required for separation continues to increase, similar to a tension spring or a rubber thread. If a certain elongation is exceeded, the thread breaks - in QCD in this analogy, when 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 individually, but only in bound states, the hadrons ( confinement ). The proton and the neutron - also called nucleons , because they are used to make the atomic nucleus - and the pions are examples of hadrons. The objects described by the QCD also include exotic hadrons such as the pentaquarks and the previously hypothetical tetraquarks .

Since quarks have both an electrical and a color charge, they interact both electromagnetically and strongly. Since the electromagnetic interaction is significantly weaker than the strong interaction, its influence on the interaction of quarks can be neglected and is therefore limited to the influence of the color charge. The strength of the electromagnetic interaction is characterized by the Sommerfeld fine structure constant , while the corresponding parameter of the strong interaction is of the order of magnitude 1. ${\ displaystyle e ^ {2} / (\ hbar c) \, \, (\ approx 1/137)}$

Due to their non-Abelian structure and high coupling strengths, calculations in QCD are often time-consuming and complicated. Successful quantitative calculations mostly come from perturbation theory or from computer simulations . The accuracy of the predictions is typically in the percentage range. In this way, 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.

## Differentiation from nuclear physics

The strength of the interaction means that protons and neutrons in the atomic nucleus are much more strongly bound to one another than, for example, the electrons to the atomic nucleus. However, the description of nucleons is an open problem. The quark (the constituent quark and the sea quarks ) contribute only 9% of the mass of the nucleons in the remaining approximately 90% of nucleon come from the kinetic energy of the curd (about one-third, caused by the movement energy to the uncertainty as they confined space " trapped ”) and contributions from the gluons (a field strength contribution of around 37 percent and an anomalous gluon amount of around 23 percent). The coupling processes occurring in the QCD are dynamic and not perturbative : the protons and neutrons themselves are colorless. Instead of quantum chromodynamics, their interaction is usually described in the context of an effective theory , according to which the attractive force between them is based on a Yukawa interaction due to the exchange of mesons, especially light pions ( pion exchange model ). The description of the behavior of nucleons via meson exchange in the atomic nucleus and in scattering experiments is the subject of nuclear physics .

The strong interaction between the nucleons in the atomic nucleus is therefore much more effective than their electromagnetic interaction. Nevertheless, the electrostatic repulsion of the protons is an important stability criterion for atomic nuclei. The strong interaction between the nucleons, in contrast to the interaction between the quarks, becomes exponentially smaller as the distance between the nucleons increases. This is due to the fact that the exchange particles involved in the pion exchange model have a non-zero mass. Therefore, the range of the interaction between the nucleons is cm, in the order of magnitude of the Compton wavelength of the mesons ( is the mass of the pion). ${\ displaystyle r_ {c}}$${\ displaystyle r_ {c} = {\ frac {\ hbar} {m _ {\ pi} c}} \ cong 10 ^ {- 13}}$${\ displaystyle \ pi}$${\ displaystyle m _ {\ pi}}$

While the nuclear forces decrease exponentially with the distance,

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

the electromagnetic interaction only drops according to the power law

${\ displaystyle \ phi _ {EM} (r) \ propto {\ frac {1} {r}}}$( Coulomb potential ),

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

The strong interaction is essentially due to the distances between the hadrons, as they are e.g. B. occur in the atomic nucleus, limited.

## Non-Abelian calibration group

Energy dependence of the strong coupling constant ${\ displaystyle \ alpha _ {s}}$

The calibration group on which the QCD is based is non- Abelian , that is, the multiplication of two group elements is generally not commutative. As a result, terms appear in the Lagrange density that cause the gluons to interact with one another. The gluons carry a color charge for the same reason. This self-interaction leads to the fact that the renormalized coupling constant of the QCD behaves qualitatively exactly opposite to the coupling constant of the QED: It decreases for high energies. At high energies this leads to the phenomenon of asymptotic freedom and at low energies to confinement . Only at extremely high temperatures, T  > 5 · 10 12  Kelvin, and / or correspondingly high pressure, the confinement is apparently removed and a quark-gluon plasma is created . ${\ displaystyle \ mathrm {SU} (3)}$

Asymptotic freedom means that the quarks behave like free particles at high energies (small typical distances), which is contrary to the behavior of other systems where weak interaction is associated with large distances. Confinement means that below a limit energy the coupling constant becomes so large that quarks only appear in hadrons. Since the coupling constant of QCD is not a small parameter at low energies, perturbation theory , with which many QED problems can be solved, cannot be applied. One approach to solving the QCD equations at low energies, on the other hand, is computer simulations of lattice scale theories . ${\ displaystyle \ alpha _ {s}}$

Another approach to the quantum field theoretical treatment of hadrons is the use of effective theories , which pass into QCD for large energies and introduce new fields with new “effective” interactions for small energies. An example of such “effective theories” is a model by Nambu and Jona-Lasinio. Depending on the hadrons to be described , different effective theories are used. The chiral perturbation theory (CPT) is used for hadrons that are only made up of light quarks, i.e. up , down and strange quarks , which interact with each other via mesons according to the CPT . For hadrons with exactly one heavy quark, so a Charm or bottom quark , and otherwise only light quarks is effective theory of heavy quarks (heavy quark effective theory, HQET) used in which the heavy quark is hard assumed to be infinite, similar the treatment of the proton in the hydrogen atom . The heaviest quark, the "top quark", is so high-energy (E 0 ~ 170 GeV) that no bound states can form in its short lifetime with Planck's constant h . For hadrons from two heavy quarks (bound states in the quarkonium ) the so - called nonrelativistic quantum chromodynamics (NRQCD) is used. ${\ displaystyle \ tau}$  ${\ displaystyle (\ tau \ approx h / E_ {0}) \ ,,}$

## Lagrangian of the QCD

The QCD is a relativistic quantum field theory with the gauge invariant Lagrangian function

{\ displaystyle {\ begin {aligned} {\ mathcal {L}} _ {\ mathrm {QCD}} (q, A) & = {\ bar {q}} \ left (i \ gamma ^ {\ mu} D_ {\ mu} -m \ right) q - {\ frac {1} {4}} F _ {\ mu \ nu} ^ {a} F_ {a} ^ {\ mu \ nu} \\ & = {\ bar {q}} (i \ gamma ^ {\ mu} \ partial _ {\ mu} -m) q + g {\ bar {q}} \ gamma ^ {\ mu} T_ {a} A _ {\ mu} ^ {a} q - {\ frac {1} {4}} F _ {\ mu \ nu} ^ {a} F_ {a} ^ {\ mu \ nu} \ end {aligned}}}

From obtained by applying the Euler-Lagrange equation to this part of the well-known Dirac equation with it and ${\ displaystyle {\ bar {q}} \ left (i \ gamma ^ {\ mu} \ partial _ {\ mu} -m \ right) q}$${\ displaystyle {\ mathcal {L}}}$

The term describes ${\ displaystyle g {\ bar {q}} \ gamma ^ {\ mu} T_ {a} qA _ {\ mu} ^ {a}}$

• the interaction vertices between quarks and gluons ( qA interaction )

From the term with one not only get ${\ displaystyle F _ {\ mu \ nu} ^ {a} F_ {a} ^ {\ mu \ nu}}$

• the propagators for gluon fields, but also
• the 3-gluon-gluon interaction vertices
• and the 4-gluon-gluon interaction vertices

These self-interaction terms of the gluons, a consequence of the non-commuting generators in non-Abelian gauge groups, represent the actual difference to the Lagrangian of QED.

The rules for Feynman diagrams in perturbative QCD follow from the individual terms of the Lagrangian . A calibration fixation must be carried out for specific calculations.

The following sizes appear above:

${\ displaystyle q \,}$, the quark field (and the adjoint quark field in the sense of Dirac's relativistic quantum mechanics) with mass m${\ displaystyle {\ bar {q}} = q ^ {\ dagger} \ gamma ^ {0}}$
${\ displaystyle \ gamma ^ {\ mu} \,}$, the Dirac matrices with = 0 to 3${\ displaystyle \ mu}$
${\ displaystyle A _ {\ mu} ^ {a} \,}$, the eight gauge boson fields (gluon fields, a = 1 to 8, corresponding color changes caused by the gluons)
${\ displaystyle D _ {\ mu} = \ partial _ {\ mu} -igT_ {a} A _ {\ mu} ^ {a} \,}$, the covariant derivative
${\ displaystyle \, g}$, the quark-gluon coupling constant
${\ displaystyle T_ {a} \,}$, a generator of the calibration group SU (3) (a = 1 to 8), with the structural constants (see article Gell-Mann matrices )${\ displaystyle f ^ {abc} \,}$
${\ displaystyle F _ {\ mu \ nu} ^ {a} = \ partial _ {\ mu} A _ {\ nu} ^ {a} - \ partial _ {\ nu} A _ {\ mu} ^ {a} + gf ^ {abc} A _ {\ mu} ^ {b} A _ {\ nu} ^ {c} \,}$, the field strength tensor of the gauge boson field.

Because the rotation of a vector field is always free of divergence (“Div Rot = 0”), the sum of the first two terms on the right side of the field strength tensor always gives zero when diverging, in contrast to the non-Abelian part, ~ g.

(Moving up and down between lower and upper indices is always done with the trivial signature, +, with respect to a , so that applies to the structural constants . With regard to μ and ν , however, it takes place with the relativistic signature, (+ −−−). ) ${\ displaystyle f ^ {abc} \ equiv f_ {abc} \ equiv f_ {bc} ^ {a}}$

### Quark-antiquark potential

Potential between quark and antiquark as a function of their distance. In addition, the rms radii of different quark-antiquark states are marked.

From the comparison of energy level schemes z. For example, from positronium and charmonium , this Lagrangian can be used to show that the strong interaction and the electromagnetic interaction do not only differ quantitatively: the quark-antiquark potential behaves similarly to electromagnetic WW at small distances (the term ~ α corresponds to the Coulomb attraction of opposite color charges). With larger distances, however, the above-mentioned spring analogy results in a significantly different behavior, which is caused by the gluons and amounts to "confinement". It corresponds to the elasticity of a stretched polymer (rubber elasticity).

Overall, the effective potential energy is:

${\ displaystyle V (r) = - {\ frac {4} {3}} {\ frac {\ alpha _ {s} (r) \ hbar c} {r}} + k \ cdot r \ ,,}$

with the strong coupling “constant” (“sliding coupling”) dependent on the momentum transfer Q 2 (and thus on the distance r) . The perturbation theory applies to them in the first order ${\ displaystyle \ alpha _ {s}}$

${\ displaystyle \ alpha _ {s} (Q ^ {2}) = {\ frac {12 \ cdot \ pi} {(33-2n_ {f}) \ cdot \ ln (Q ^ {2} / \ Lambda ^ {2})}} \ ,,}$

with the number of quark families involved (also dependent on Q 2 )${\ displaystyle n_ {f} \ ,.}$

The term increasing linearly with the radius describes the confinement behavior, while the first term has a Coulomb form and allows calculations in perturbation theory for very high energies, at which is small. With n f   , the number of families (flavor degrees of freedom) of the standard model of elementary particle physics flows into the behavior . ${\ displaystyle \ alpha _ {s} \ ll 1}$

The characteristic radius at which the behavior of V (r) “changes over” (at this radius the potential equals zero) can be related to the radius of the previous Bag models of the hadrons ; (Order of magnitude of R c : 1 fm (= 10 −15  m)). ${\ displaystyle R_ {c} = {\ sqrt {\ frac {4 \ hbar c \ alpha _ {s} / 3} {k}}} \ ,,}$

The adjacent picture shows explicitly that in a meson not only the particles, quarks and antiquarks, but also the "flow tubes" of the gluon fields are important, and that mesons are by no means spherical in the energies considered.

## Lattice theory

Quark and antiquark together form a meson (visualization of a lattice QCD simulation, see below).

Nowadays, computer simulations of quantum chromodynamics are mostly carried out within the framework of the lattice scale theories (called "lattice QCD" based on the English-language literature). In the meantime there is a growing number of quantitatively relevant results. B. in the annual reports of the conference "International Symposium on Lattice Field Theory" (short: Lattice , last 2017) follow. Nevertheless, even in high-energy physics, the lattice theory is not limited to quantum chromodynamics.

The essential approach of the lattice range theory consists in a suitable discretization of the action functional. For this purpose, the three spatial dimensions and one time dimension of the relativistic quantum field theory are first converted into four Euclidean dimensions to be treated in classical statistical mechanics. Based on this previously known procedure, cf. Wick rotation , it was now possible to transfer the so-called Wilson loop , which represents the calibration field energy in the form of a loop, to a hypercubic grid with non-vanishing grid spacing, whereby the calibration invariance is preserved. This formulation allows the use of numerical methods on powerful computers. Special requirements arise for the lattice QCD from the endeavor, on the one hand, to obtain the best possible approximation of the chiral symmetry and to control the systematic errors that necessarily result from the finite lattice spacing (which requires sufficiently small lattice spacings), and on the other hand, the computation time to be kept as small as possible (this requires sufficiently large grid spacings).

One of the greatest successes of such simulations is the calculation of all meson and baryon ground states and their masses (with an accuracy of 1 to 2 percent) that contain up , down or strange quarks . This took place in 2008 in elaborate computer calculations (Budapest-Marseille-Wuppertal collaboration) at the limit of what was then feasible and after more than two decades of intensive development of theory, algorithms and hardware.

## Researchers and Nobel Prizes

Murray Gell-Mann
Gerardus' t Hooft

One of the founders of quantum chromodynamics (and before that of the quark model), Murray Gell-Mann , with whom the aforementioned Kenneth Wilson received his doctorate, received the Nobel Prize in 1969 for his numerous contributions to the theory of strong interaction before the introduction of QCD Physics. In his pioneering work on QCD, he worked with Harald Fritzsch and Heinrich Leutwyler .

In 1999 Gerardus' t Hooft and Martinus JG Veltman received the Nobel Prize “for elucidating the quantum structure of electroweak interactions in physics”. In their work they had gained deep insights into the renormalizability of non-Abelian gauge theories, including QCD.

On October 5, 2004, David Gross , David Politzer and Frank Wilczek were awarded the Nobel Prize in Physics for their work on the quantum chromodynamics of the “strong interaction” . They discovered in the early 1970s that the strong interaction of the quarks becomes weaker the closer they are. In close proximity, quarks behave like free particles, which theoretically justified the results of the deep inelastic scattering experiments of that time.

## Classification of the QCD

 Fundamental interactions and their descriptions Strong interaction Electromagnetic interaction Weak interaction Gravity classic Electrostatics & magnetostatics , electrodynamics Newton's law of gravitation , general relativity quantum theory Quantum ( standard model ) Quantum electrodynamics Fermi theory Quantum gravity  ? Electroweak Interaction ( Standard Model ) Big Unified Theory  ? World formula ("theory of everything")? Theories at an early stage of development are grayed out.