Everything about Quantum Chromodynamics totally explained
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 (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 doesn't diminish as they're separated. Because of this, it would take an infinite amount of energy to separate two quarks; they're 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's easy to demonstrate in lattice QCD.
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.
The three kinds of
charge in QCD (as opposed to one 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
» :
are the structure constants of SU(3).
The constants
and
control the quark mass and coupling constants of the theory, subject to renormalization in the full quantum theory.
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 couldn't 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's 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 couldn't 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 doesn't decrease with distance, it's believed that quarks and gluons can never be liberated from
hadrons. This aspect of the theory is verified within
lattice QCD computations, but isn't 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 .
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's 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 doesn't 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's called left-handed; otherwise, it's 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's a global (chiral) flavor symmetry group . The chiral symmetry is spontaneously broken by the QCD vacuum to the vector (L+R) with the formation of a chiral condensate. The vector symmetry, corresponds to the baryon number of quarks and is an exact symmetry. The axial symmetry 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's the symmetry that acts on the different colors of quarks, and this is an exact gauge symmetry mediated by the gluons, and there's 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 isn't 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 vacuuum 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're 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, don't 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's 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's 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
the running of the QCD coupling as deduced from many observations
scaling violation in polarized and unpolarized deep inelastic scattering
vector boson production at colliders (this includes the Drell-Yan process)
jet cross sections in colliders
event shape observables at the LEP
heavy-quark production in colliders
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 (External Link
). 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.
Further Information
Get more info on 'Quantum Chromodynamics'.
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