# Quantum chromodynamics

Quantum chromodynamics (QCD) is the theory describing one of the fundamental forces, the strong interaction. It describes the interactions of quarks and gluons and takes the form of a quantum field theory of a special kind called a non-abelian gauge theory. QCD forms 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. That QCD predicts this behavior 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.

## Names

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.

The three kinds of charge in QCD (as opposed to one in 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".

## 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; 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 went on to propose in 1963 that the structure of the groups could be explained by the existence of three flavors 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 color charge. Han and Nambu noted that quarks would 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, it was then believed that quarks were merely convenient mathematical constructs, not real particles. Richard Feynman argued that high energy experiments showed quarks to be real: he called them partons (since they were parts of hadrons). James Bjorken proposed that certain relations should then hold in deep inelastic scattering of electrons and protons, which were spectacularly verified in experiments at SLAC in 1969.

Although the study of the strong interaction remained daunting, the discovery of asymptotic freedom by David Gross, David Politzer and Frank Wilczek allowed people to make precise predictions of the results of many high energy experiments using the techniques of perturbation theory (quantum mechanics). 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 Prizes 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 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, that is the symmetry acts independently at each point in space-time. 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 space-time.

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.

• 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 $SU_L(N_f)\times SU_R(N_f)\times U_B(1)\times U_A(1)$. The chiral symmetry is spontaneously broken by the QCD vacuum to the vector (L+R) SUV(Nf) with the formation of a chiral condensate. The vector symmetry, UB(1) corresponds to the baryon number of quarks and is an exact symmetry. The axial symmetry UA(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

In many applications of QCD one can ignore the heavy flavors (charm, bottom and top). In this case the effective flavor group is often SU(3), which should not be confused with the color group. In QCD the color group belongs to a local symmetry and hence is gauged. The flavor group is not gauged. The Eightfold way is based on the flavor group and ignores the local symmetry which gives 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 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.

### QCD

The Lagrangian of QCD (with color, flavor and spin indices suppressed) looks exactly like that of QED:

$L = -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} + \overline{\psi}(i\gamma_\mu D^\mu - m)\psi$

where F denotes the gluon field tensor, ψ the quark field and D the covariant derivative. Part of its content lies in the Feynman rules which state that all processes which occur in the theory can be resolved into the elementary interactions (called vertices): qqg, ggg and gggg. A quark may emit (or absorb) a gluon, a gluon may emit (or absorb) a gluon, and two gluons may directly interact. In QED, only the first kind of vertex occurs, since photons have no charge.

## 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. 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) and heavy quark effective theory (which expands around heavy quark mass near infinity). 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 necessiated by the puzzle of the Δ++. This has been dealt with in the section on Quantum chromodynamics#History.

The first evidence for quarks as real constiutent 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.