# Quantum chromodynamics

Template:Quantum field theory Quantum chromodynamics (abbreviated as QCD) is a theory of the strong interaction (color force), a fundamental force describing the interactions of the quarks and gluons found in hadrons (particles made of quarks or gluons, such as the proton, neutron or pion). It is the study of the SU(3) Yang–Mills theory of color-charged fermions (the quarks). QCD is a quantum field theory of a special kind called a non-abelian gauge theory. It is an important part of the Standard Model of particle physics. A huge body of experimental evidence for QCD has been gathered over the years.

QCD enjoys two peculiar properties:

• Asymptotic freedom, which means that in very high-energy reactions, quarks and gluons interact very weakly. This prediction of QCD was first discovered in the early 1970s by David Politzer and by Frank Wilczek and David Gross. For this work they were awarded the 2004 Nobel Prize in Physics.
• Confinement, which means that the force between quarks does not diminish as they are separated. Because of this, it would take an infinite amount of energy to separate two quarks; they are forever bound into hadrons such as the proton and the neutron. Although analytically unproven, confinement is widely believed to be true because it explains the consistent failure of free quark searches, and it is easy to demonstrate in lattice QCD.

Moreover: the above-mentioned two properties are continuous all the way, i.e. there is no phase-transition line separating them.

## Terminology

The word quark was coined by Murray Gell-Mann in its present sense, the word having been taken from the phrase "Three quarks for Muster Mark" in Finnegans Wake by James Joyce. [1]

The three kinds of charge in QCD (as opposed to two in Quantum electrodynamics or QED) are usually referred to as "color charge" by loose analogy to the three kinds of color (red, green and blue) perceived by humans. Since the theory of electric charge is dubbed "electrodynamics", the Greek word "chroma" Χρώμα (meaning color) is applied to the theory of color charge, "chromodynamics".

## Lagrangian

The dynamics of the quarks and gluons are controlled by the quantum chromodynamics Lagrangian. The gauge invariant QCD Lagrangian is

{\displaystyle {\begin{aligned}{\mathcal {L}}_{\mathrm {QCD} }&={\bar {\psi }}_{i}\left(i\gamma ^{\mu }(D_{\mu })_{ij}-m\,\delta _{ij}\right)\psi _{j}-{\frac {1}{4}}G_{\mu \nu }^{a}G_{a}^{\mu \nu }\\&={\bar {\psi }}_{i}(i\gamma ^{\mu }\partial _{\mu }-m)\psi _{i}-gG_{\mu }^{a}{\bar {\psi }}_{i}\gamma ^{\mu }T_{ij}^{a}\psi _{j}-{\frac {1}{4}}G_{\mu \nu }^{a}G_{a}^{\mu \nu }\,,\\\end{aligned}}}

where ${\displaystyle \psi _{i}(x)\,}$ is the quark field, a dynamical function of space-time, in the fundamental representation of the SU(3) gauge group, indexed by ${\displaystyle i,\,j,\,\ldots }$; ${\displaystyle G_{\mu }^{a}(x)\,}$ are the gluon fields, also a dynamical function of space-time, in the adjoint representation of the SU(3) gauge group, indexed by ${\displaystyle a,\,b,\,\ldots }$; ${\displaystyle \gamma ^{\mu }\,}$ are the Dirac matrices, connecting the spinor representation to the vector representation of the Lorentz group; and ${\displaystyle T_{ij}^{a}\,}$ are the generators, connecting the fundamental, antifundamental and adjoint representations of the SU(3) gauge group. The Gell-Mann matrices provide one such representation for the generators.

The symbol ${\displaystyle G_{\mu \nu }^{a}\,}$ represents the gauge invariant gluonic field strength tensor, analogous to the electromagnetic field strength tensor, ${\displaystyle F^{\mu \nu }\,}$, in Electrodynamics. It is given by

${\displaystyle G_{\mu \nu }^{a}=\partial _{\mu }G_{\nu }^{a}-\partial _{\nu }G_{\mu }^{a}-gf^{abc}G_{\mu }^{b}G_{\nu }^{c}\,,}$

where ${\displaystyle f_{abc}\,}$ are the structure constants of SU(3).

The constants ${\displaystyle m}$ and ${\displaystyle g}$ control the quark mass and coupling constants of the theory, subject to renormalization in the full quantum theory.

An important theoretical notion concerning the final term of the above Lagrangian is the Wilson loop variable. This loop variable plays a most-important role in discretized forms of the QCD (see lattice QCD), and more generally, it distinguishes confined and deconfined states of a gauge theory. It was introduced by the Noble-prize winner Kenneth G. Wilson and is treated in a separate article.

## History

With the invention of bubble chambers and spark chambers in the 1950s, experimental particle physics discovered a large and ever-growing number of particles called hadrons. It seemed that such a large number of particles could not all be fundamental. First, the particles were classified by charge and isospin by Eugene Wigner and Werner Heisenberg; then, in 1953, according to strangeness by Murray Gell-Mann and Kazuhiko Nishijima. To gain greater insight, the hadrons were sorted into groups having similar properties and masses using the eightfold way, invented in 1961 by Gell-Mann and Yuval Ne'eman. Gell-Mann and George Zweig, correcting an earlier approach of Sakata, went on to propose in 1963 that the structure of the groups could be explained by the existence of three flavours of smaller particles inside the hadrons: the quarks.

At this stage, one particle, the Δ++ remained mysterious; in the quark model, it is composed of three up quarks with parallel spins. However, since quarks are fermions, this combination is forbidden by the Pauli exclusion principle. In 1965, Moo-Young Han with Yoichiro Nambu and Oscar W. Greenberg independently resolved the problem by proposing that quarks possess an additional SU(3) gauge degree of freedom, later called colour charge. Han and Nambu noted that quarks might interact via an octet of vector gauge bosons: the gluons.

Since free quark searches consistently failed to turn up any evidence for the new particles, and because an elementary particle back then was defined as a particle which could be separated and isolated, Gell-Mann often said that quarks were merely convenient mathematical constructs, not real particles. The meaning of this statement was usually clear in context--- he meant quarks are confined. But he also was implying that the strong interactions could probably not be fully described by quantum field theory.

Richard Feynman argued that high energy experiments showed quarks are real particles: he called them partons (since they were parts of hadrons). By particles, Feynman meant objects which travel along paths, elementary particles in a field theory.

The difference between Feynman and Gell-Mann's approach reflected a deep split in the theoretical physics community. Feynman thought the quarks have a distribution of position or momentum, like any other particle, and he (correctly) believed that the diffusion of parton momentum explained diffractive scattering. Although Gell-Mann believed that certain quark charges could be localized, he was open to the possibility that the quarks themselves could not be localized because space and time break down. This was the more radical approach of S-Matrix theory.

James Bjorken proposed that pointlike partons would imply certain relations should hold in deep inelastic scattering of electrons and protons, which were spectacularly verified in experiments at SLAC in 1969. This led physicists to abandon the S-matrix approach for the strong interactions.

The discovery of asymptotic freedom in the strong interactions by David Gross, David Politzer and Frank Wilczek allowed physicists to make precise predictions of the results of many high energy experiments using the quantum field theory technique of perturbation theory. Evidence of gluons 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 the LEP in 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 places where string theory, the modern form of S-matrix theory, has recently shed much light[2] [3].

## The 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

QCD is a gauge 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 left-handed; otherwise, it is right-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.

### The symmetry groups

The color group SU(3) corresponds to the local symmetry whose gauging gives rise to QCD. The electric charge labels a representation of the local symmetry group U(1) which is gauged to give QED: this is an abelian group. If one considers a version of QCD with Nf flavors of massless quarks, then there is a global (chiral) flavor symmetry group ${\displaystyle SU_{L}(N_{f})\times SU_{R}(N_{f})\times U_{B}(1)\times U_{A}(1)}$. The chiral symmetry is spontaneously broken by the QCD vacuum to the vector (L+R) ${\displaystyle SU_{V}(N_{f})}$ with the formation of a chiral condensate. The vector symmetry, ${\displaystyle U_{B}(1)}$ corresponds to the baryon number of quarks and is an exact symmetry. The axial symmetry ${\displaystyle U_{A}(1)}$ is exact in the classical theory, but broken in the quantum theory, an occurrence called an anomaly. Gluon field configurations called instantons are closely related to this anomaly.

#### Cautionary note

There are two different types of SU(3) symmetry: there is the symmetry that acts on the different colors of quarks, and this is an exact gauge symmetry mediated by the gluons, and there is also a flavor symmetry which rotates different flavors of quarks to each other, or flavor SU(3). Flavor SU(3) is an approximate symmetry of the vacuum of QCD, and is not a fundamental symmetry at all. It is an accidental consequence of the small mass of the three lightest quarks.

In the QCD vacuum there are vacuum condensates of all the quarks whose mass is less than the QCD scale. This includes the up and down quarks, and to a lesser extent the strange quark, but not any of the others. The vacuum is symmetric under SU(2) isospin rotations of up and down, and to a lesser extent under rotations of up,down, and strange, or full flavor group SU(3), and the observed particles make isospin and SU(3) multiplets.

The approximate flavor symmetries do have associated gauge bosons, observed particles like the rho and the omega, but these particles are nothing like the gluons and they are not massless. They are emergent gauge bosons in an approximate string description of QCD.

### The fields

Quarks are massive spin-1/2 fermions which carry a color charge whose gauging is the content of QCD. Quarks are represented by Dirac fields in the fundamental representation 3 of the gauge group SU(3). They also carry electric charge (either -1/3 or 2/3) and participate in weak interactions as part of weak isospin doublets. They carry global quantum numbers including the baryon number, which is 1/3 for each quark, hypercharge and one of the flavor quantum numbers.

Gluons are spin-1 bosons which also carry color charges, since they lie in the adjoint representation 8 of SU(3). They have no electric charge, do not participate in the weak interactions, and have no flavor. They lie in the singlet representation 1 of all these symmetry groups.

Every quark has its own antiquark. The charge of each antiquark is exactly the opposite of the corresponding quark.

### The dynamics

According to the rules of quantum field theory, and the associated Feynman diagrams, the above theory gives rise to three basic interactions: a quark may emit (or absorb) a gluon, a gluon may emit (or absorb) a gluon, and two gluons may directly interact. This contrasts with QED, in which only the first kind of interaction occurs, since photons have no charge. Diagrams involving Faddeev-Popov ghosts must be considered too.

## Methods

Further analysis of the content of the theory is complicated. Various techniques have been developed to work with QCD. Some of them are discussed briefly below.

### Perturbative QCD

This approach is based on asymptotic freedom, which allows perturbation theory to be used accurately in experiments performed at very high energies. Although limited in scope, this approach has resulted in the most precise tests of QCD to date.

### Lattice QCD

Among non-perturbative approaches to QCD, the most well established one is lattice QCD. This approach uses a discrete set of space-time points (called the lattice) to reduce the analytically intractable path integrals of the continuum theory to a very difficult numerical computation which is then carried out on supercomputers like the QCDOC which was constructed for precisely this purpose. While it is a slow and resource-intensive approach, it has wide applicability, giving insight into parts of the theory inaccessible by other means.

### 1/N expansion

A well-known approximation scheme, the 1/N expansion, starts from the premise that the number of colors is infinite, and makes a series of corrections to account for the fact that it is not. Until now it has been the source of qualitative insight rather than a method for quantitative predictions. Modern variants include the AdS/CFT approach.

### Effective theories

For specific problems some theories may be written down which seem to give qualitatively correct results. In the best of cases, these may then be obtained as systematic expansions in some parameter of the QCD Lagrangian. Among the best such effective models one should now count chiral perturbation theory (which expands around light quark masses near zero), heavy quark effective theory (which expands around heavy quark mass near infinity), and soft-collinear effective theory (which expands around large ratios of energy scales). Other less accurate models are the Nambu-Jona-Lasinio model and the chiral model.

## Experimental tests

The notion of quark flavours was prompted by the necessity of explaining the properties of hadrons during the development of the quark model. The notion of colour was necessitated by the puzzle of the Δ++. This has been dealt with in the section on the history of QCD.

The first evidence for quarks as real constituent elements of hadrons was obtained in deep inelastic scattering experiments at SLAC. The first evidence for gluons came in three jet events at PETRA.

Good quantitative tests of perturbative QCD are

Quantitative tests of non-perturbative QCD are fewer, because the predictions are harder to make. The best is probably the running of the QCD coupling as probed through lattice computations of heavy-quarkonium spectra. There is a recent claim about the mass of the heavy meson Bc [1]. Other non-perturbative tests are currently at the level of 5% at best. Continuing work on masses and form factors of hadrons and their weak matrix elements are promising candidates for future quantitative tests. The whole subject of quark matter and the quark-gluon plasma is a non-perturbative test bed for QCD which still remains to be properly exploited.