In
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, quantum chromodynamics (QCD) is the theory of the
strong interaction between
quarks mediated by
gluons. Quarks are fundamental particles that make up composite
hadron
In particle physics, a hadron (; grc, ἁδρός, hadrós; "stout, thick") is a composite subatomic particle made of two or more quarks held together by the strong interaction. They are analogous to molecules that are held together by the e ...
s such as the
proton,
neutron
The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons beh ...
and
pion
In particle physics, a pion (or a pi meson, denoted with the Greek letter pi: ) is any of three subatomic particles: , , and . Each pion consists of a quark and an antiquark and is therefore a meson. Pions are the lightest mesons and, more gene ...
. QCD is a type of
quantum field theory called a
non-abelian gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations ( Lie group ...
, with symmetry group
SU(3)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the specia ...
. The QCD analog of electric charge is a property called ''color''. Gluons are the
force carrier In quantum field theory, a force carrier, also known as messenger particle or intermediate particle, is a type of particle that gives rise to forces between other particles. These particles serve as the quanta of a particular kind of physical fi ...
s of the theory, just as photons are for the electromagnetic force in
quantum electrodynamics
In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
. The theory is an important part of the
Standard Model of
particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
. A large body of
experimental evidence for QCD has been gathered over the years.
QCD exhibits three salient properties:
*
Color confinement
In quantum chromodynamics (QCD), color confinement, often simply called confinement, is the phenomenon that color-charged particles (such as quarks and gluons) cannot be isolated, and therefore cannot be directly observed in normal conditions b ...
. Due to the force between two color charges remaining constant as they are separated, the energy grows until a quark–antiquark pair is
spontaneously produced, turning the initial hadron into a pair of hadrons instead of isolating a color charge. Although analytically unproven,
color confinement
In quantum chromodynamics (QCD), color confinement, often simply called confinement, is the phenomenon that color-charged particles (such as quarks and gluons) cannot be isolated, and therefore cannot be directly observed in normal conditions b ...
is well established from
lattice QCD
Lattice QCD is a well-established non-perturbative approach to solving the quantum chromodynamics (QCD) theory of quarks and gluons. It is a lattice gauge theory formulated on a grid or lattice of points in space and time. When the size of the lat ...
calculations and decades of experiments.
*
Asymptotic freedom
In quantum field theory, asymptotic freedom is a property of some gauge theories that causes interactions between particles to become asymptotically weaker as the energy scale increases and the corresponding length scale decreases.
Asymptotic fre ...
, a steady reduction in the strength of interactions between quarks and gluons as the energy scale of those interactions increases (and the corresponding length scale decreases). The asymptotic freedom of QCD was discovered in 1973 by
David Gross
David Jonathan Gross (; born February 19, 1941) is an American theoretical physicist and string theorist. Along with Frank Wilczek and David Politzer, he was awarded the 2004 Nobel Prize in Physics for their discovery of asymptotic freedom. ...
and
Frank Wilczek
Frank Anthony Wilczek (; born May 15, 1951) is an American theoretical physicist, mathematician and Nobel laureate. He is currently the Herman Feshbach Professor of Physics at the Massachusetts Institute of Technology (MIT), Founding Direc ...
,
[
] and independently by
David Politzer
Hugh David Politzer (; born August 31, 1949) is an American theoretical physicist and the Richard Chace Tolman Professor of Theoretical Physics at the California Institute of Technology. He shared the 2004 Nobel Prize in Physics with David Gr ...
in the same year.
[
] For this work, all three shared the 2004
Nobel Prize in Physics
)
, image = Nobel Prize.png
, alt = A golden medallion with an embossed image of a bearded man facing left in profile. To the left of the man is the text "ALFR•" then "NOBEL", and on the right, the text (smaller) "NAT•" then " ...
.
*
Chiral symmetry breaking
In particle physics, chiral symmetry breaking is the spontaneous symmetry breaking of a chiral symmetry – usually by a gauge theory such as quantum chromodynamics, the quantum field theory of the strong interaction. Yoichiro Nambu was awar ...
, the
spontaneous symmetry breaking
Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or ...
of an important global symmetry of quarks, detailed below, with the result of generating masses for hadrons far above the masses of the quarks, and making pseudoscalar mesons exceptionally light.
Yoichiro Nambu
was a Japanese-American physicist and professor at the University of Chicago. Known for his contributions to the field of theoretical physics, he was awarded half of the Nobel Prize in Physics in 2008 for the discovery in 1960 of the mechanism ...
was awarded the 2008 Nobel Prize in Physics for elucidating the phenomenon, a dozen years before the advent of QCD. Lattice simulations have confirmed all his generic predictions.
Terminology
Physicist
Murray Gell-Mann
Murray Gell-Mann (; September 15, 1929 – May 24, 2019) was an American physicist who received the 1969 Nobel Prize in Physics for his work on the theory of elementary particles. He was the Robert Andrews Millikan Professor of Theoretical ...
coined the word ''quark'' in its present sense. It originally comes from the phrase "Three quarks for Muster Mark" in ''
Finnegans Wake
''Finnegans Wake'' is a novel by Irish writer James Joyce. It is well known for its experimental style and reputation as one of the most difficult works of fiction in the Western canon. It has been called "a work of fiction which combines a bod ...
'' by
James Joyce
James Augustine Aloysius Joyce (2 February 1882 – 13 January 1941) was an Irish novelist, poet, and literary critic. He contributed to the modernist avant-garde movement and is regarded as one of the most influential and important writers of ...
. On June 27, 1978, Gell-Mann wrote a private letter to the editor of the ''Oxford English Dictionary'', in which he related that he had been influenced by Joyce's words: "The allusion to three quarks seemed perfect." (Originally, only three quarks had been discovered.)
The three kinds of
charge
Charge or charged may refer to:
Arts, entertainment, and media Films
* '' Charge, Zero Emissions/Maximum Speed'', a 2011 documentary
Music
* ''Charge'' (David Ford album)
* ''Charge'' (Machel Montano album)
* ''Charge!!'', an album by The Aqu ...
in QCD (as opposed to one in
quantum electrodynamics
In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
or QED) are usually referred to as "
color charge
Color charge is a property of quarks and gluons that is related to the particles' strong interactions in the theory of quantum chromodynamics (QCD).
The "color charge" of quarks and gluons is completely unrelated to the everyday meanings of colo ...
" by loose analogy to the three kinds of
color
Color (American English) or colour (British English) is the visual perceptual property deriving from the spectrum of light interacting with the photoreceptor cells of the eyes. Color categories and physical specifications of color are assoc ...
(red, green and blue)
perceived by humans. Other than this nomenclature, the quantum parameter "color" is completely unrelated to the everyday, familiar phenomenon of color.
The force between quarks is known as the colour force (or color force) or
strong interaction, and is responsible for the
nuclear force
The nuclear force (or nucleon–nucleon interaction, residual strong force, or, historically, strong nuclear force) is a force that acts between the protons and neutrons of atoms. Neutrons and protons, both nucleons, are affected by the nucle ...
.
Since the theory of electric charge is dubbed "
electrodynamics
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
", the
Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...
word χρῶμα ''chroma'' "color" is applied to the theory of color charge, "chromodynamics".
History
With the invention of
bubble chamber
A bubble chamber is a vessel filled with a superheated transparent liquid (most often liquid hydrogen) used to detect electrically charged particles moving through it. It was invented in 1952 by Donald A. Glaser, for which he was awarded the 1 ...
s and
spark chamber {{short description, Charged particle detector
A spark chamber is a particle detector: a device used in particle physics for detecting electrically charged particles. They were most widely used as research tools from the 1930s to the 1960s and have ...
s in the 1950s, experimental
particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
discovered a large and ever-growing number of particles called
hadron
In particle physics, a hadron (; grc, ἁδρός, hadrós; "stout, thick") is a composite subatomic particle made of two or more quarks held together by the strong interaction. They are analogous to molecules that are held together by the e ...
s. It seemed that such a large number of particles could not all be
fundamental
Fundamental may refer to:
* Foundation of reality
* Fundamental frequency, as in music or phonetics, often referred to as simply a "fundamental"
* Fundamentalism, the belief in, and usually the strict adherence to, the simple or "fundamental" idea ...
. First, the particles were classified by
charge
Charge or charged may refer to:
Arts, entertainment, and media Films
* '' Charge, Zero Emissions/Maximum Speed'', a 2011 documentary
Music
* ''Charge'' (David Ford album)
* ''Charge'' (Machel Montano album)
* ''Charge!!'', an album by The Aqu ...
and
isospin
In nuclear physics and particle physics, isospin (''I'') is a quantum number related to the up- and down quark content of the particle. More specifically, isospin symmetry is a subset of the flavour symmetry seen more broadly in the interactions ...
by
Eugene Wigner
Eugene Paul "E. P." Wigner ( hu, Wigner Jenő Pál, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his co ...
and
Werner Heisenberg
Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a breakthrough paper. In the subsequent serie ...
; then, in 1953–56, according to
strangeness by
Murray Gell-Mann
Murray Gell-Mann (; September 15, 1929 – May 24, 2019) was an American physicist who received the 1969 Nobel Prize in Physics for his work on the theory of elementary particles. He was the Robert Andrews Millikan Professor of Theoretical ...
and
Kazuhiko Nishijima
(4 October 1926 – 15 February 2009) was a Japanese physicist who made significant contributions to particle physics. He was professor emeritus at the University of Tokyo and Kyoto University until his death in 2009.
He was born in Tsuchiur ...
(see
Gell-Mann–Nishijima formula The Gell-Mann–Nishijima formula (sometimes known as the NNG formula) relates the baryon number ''B'', the strangeness ''S'', the isospin ''I3'' of quarks and hadrons to the electric charge ''Q''. It was originally given by Kazuhiko Nishijima and ...
). 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
Yuval Ne'eman ( he, יובל נאמן, 14 May 1925 – 26 April 2006) was an Israeli theoretical physicist, military scientist, and politician. He was Minister of Science and Development in the 1980s and early 1990s. He was the President o ...
. Gell-Mann and
George Zweig
George Zweig (; born May 30, 1937) is a Russian-American physicist. He was trained as a particle physicist under Richard Feynman. He introduced, independently of Murray Gell-Mann, the quark model (although he named it "aces"). He later turned his ...
, correcting an earlier approach of
Shoichi Sakata
was a Japanese physicist and Marxist who was internationally known for theoretical work on the subatomic particles.Nussbaum, Louis-Frédéric. (2005). "''Sakata Shōichi''" in ; n.b., Louis-Frédéric is pseudonym of Louis-Frédéric Nussbaum, '' ...
, 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. Gell-Mann also briefly discussed a field theory model in which quarks interact with gluons.
Perhaps the first remark that quarks should possess an additional
quantum number was made as a short footnote in the preprint of
Boris Struminsky
Boris Vladimirovich Struminsky (russian: Борис Владимирович Струминский; 14 August 1939 – 18 January 2003) was a Russian and Ukrainian physicist known for his contribution to theoretical elementary particle physics.
B ...
[B. V. Struminsky, Magnetic moments of baryons in the quark model. ]JINR
The Joint Institute for Nuclear Research (JINR, russian: Объединённый институт ядерных исследований, ОИЯИ), in Dubna, Moscow Oblast (110 km north of Moscow), Russia, is an international research ce ...
-Preprint P-1939, Dubna, Russia. Submitted on January 7, 1965. in connection with the Ω
− hyperon
In particle physics, a hyperon is any baryon containing one or more strange quarks, but no charm, bottom, or top quark. This form of matter may exist in a stable form within the core of some neutron stars. Hyperons are sometimes generically re ...
being composed of three
strange quarks with parallel spins (this situation was peculiar, because since quarks are
fermions, such a combination is forbidden by the
Pauli exclusion principle
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulat ...
):
Boris Struminsky was a PhD student of
Nikolay Bogolyubov
Nikolay Nikolayevich Bogolyubov (russian: Никола́й Никола́евич Боголю́бов; 21 August 1909 – 13 February 1992), also transliterated as Bogoliubov and Bogolubov, was a Soviet and Russian mathematician and theoretic ...
. The problem considered in this preprint was suggested by Nikolay Bogolyubov, who advised Boris Struminsky in this research.
In the beginning of 1965,
Nikolay Bogolyubov
Nikolay Nikolayevich Bogolyubov (russian: Никола́й Никола́евич Боголю́бов; 21 August 1909 – 13 February 1992), also transliterated as Bogoliubov and Bogolubov, was a Soviet and Russian mathematician and theoretic ...
,
Boris Struminsky
Boris Vladimirovich Struminsky (russian: Борис Владимирович Струминский; 14 August 1939 – 18 January 2003) was a Russian and Ukrainian physicist known for his contribution to theoretical elementary particle physics.
B ...
and
Albert Tavkhelidze Albert Nikiforovich Tavkhelidze (russian: Альберт Никифорович Тавхелидзе, ka, ალბერტ ნიკიფორეს ძე თავხელიძე; 16 December 1930 27 February 2010) was President of the Ge ...
wrote a preprint with a more detailed discussion of the additional quark quantum degree of freedom. This work was also presented by Albert Tavkhelidze without obtaining consent of his collaborators for doing so at an international conference in
Trieste
Trieste ( , ; sl, Trst ; german: Triest ) is a city and seaport in northeastern Italy. It is the capital city, and largest city, of the autonomous region of Friuli Venezia Giulia, one of two autonomous regions which are not subdivided into prov ...
(Italy), in May 1965.
A similar mysterious situation was with the
Δ++ baryon; in the quark model, it is composed of three
up quark
The up quark or u quark (symbol: u) is the lightest of all quarks, a type of elementary particle, and a significant constituent of matter. It, along with the down quark, forms the neutrons (one up quark, two down quarks) and protons (two up quark ...
s with parallel spins. In 1964–65,
Greenberg and
Han
Han may refer to:
Ethnic groups
* Han Chinese, or Han People (): the name for the largest ethnic group in China, which also constitutes the world's largest ethnic group.
** Han Taiwanese (): the name for the ethnic group of the Taiwanese p ...
–
Nambu independently resolved the problem by proposing that quarks possess an additional
SU(3)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the specia ...
gauge
Gauge ( or ) may refer to:
Measurement
* Gauge (instrument), any of a variety of measuring instruments
* Gauge (firearms)
* Wire gauge, a measure of the size of a wire
** American wire gauge, a common measure of nonferrous wire diameter, ...
degree of freedom
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
, later called color 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 that 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
Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superfl ...
argued that high energy experiments showed quarks are real particles: he called them ''
partons
In particle physics, the parton model is a model of hadrons, such as protons and neutrons, proposed by Richard Feynman. It is useful for interpreting the cascades of radiation (a parton shower) produced from quantum chromodynamics (QCD) processes ...
'' (since they were parts of hadrons). By particles, Feynman meant objects that travel along paths, elementary particles in a field theory.
The difference between Feynman's and Gell-Mann's approaches 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
''S''-matrix theory was a proposal for replacing local quantum field theory as the basic principle of elementary particle physics.
It avoided the notion of space and time by replacing it with abstract mathematical properties of the ''S''-matrix ...
.
James Bjorken
James Daniel "BJ" Bjorken (born 1934) is an American theoretical physicist. He was a Putnam Fellow in 1954, received a BS in physics from MIT in 1956, and obtained his PhD from Stanford University in 1959. He was a visiting scholar at the Inst ...
proposed that pointlike partons would imply certain relations in
deep inelastic scattering
Deep inelastic scattering is the name given to a process used to probe the insides of hadrons (particularly the baryons, such as protons and neutrons), using electrons, muons and neutrinos. It provided the first convincing evidence of the reali ...
of
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no ...
s and protons, which were verified in experiments at
SLAC
SLAC National Accelerator Laboratory, originally named the Stanford Linear Accelerator Center,
is a United States Department of Energy National Laboratory operated by Stanford University under the programmatic direction of the U.S. Departme ...
in 1969. This led physicists to abandon the S-matrix approach for the strong interactions.
In 1973 the concept of
color
Color (American English) or colour (British English) is the visual perceptual property deriving from the spectrum of light interacting with the photoreceptor cells of the eyes. Color categories and physical specifications of color are assoc ...
as the source of a "strong field" was developed into the theory of QCD by physicists
Harald Fritzsch
Harald Fritzsch (born 10 February 1943 in Zwickau, Germany, died 16 August 2022 in München) was a German theoretical physicist known for his contributions to the theory of quarks, the development of Quantum Chromodynamics and the great unifi ...
and
Heinrich Leutwyler
Heinrich Leutwyler (born Oct 12, 1938) is a Swiss theoretical physicist, with interests in elementary particle physics, the theory of strong interactions, and quantum field theory. , together with physicist Murray Gell-Mann. In particular, they employed the general field theory developed in 1954 by
Chen Ning Yang and
Robert Mills (see
Yang–Mills theory
In mathematical physics, Yang–Mills theory is a gauge theory based on a special unitary group SU(''N''), or more generally any compact, reductive Lie algebra. Yang–Mills theory seeks to describe the behavior of elementary particles using ...
), in which the carrier particles of a force can themselves radiate further carrier particles. (This is different from QED, where the photons that carry the electromagnetic force do not radiate further photons.)
The discovery of
asymptotic freedom
In quantum field theory, asymptotic freedom is a property of some gauge theories that causes interactions between particles to become asymptotically weaker as the energy scale increases and the corresponding length scale decreases.
Asymptotic fre ...
in the strong interactions by
David Gross
David Jonathan Gross (; born February 19, 1941) is an American theoretical physicist and string theorist. Along with Frank Wilczek and David Politzer, he was awarded the 2004 Nobel Prize in Physics for their discovery of asymptotic freedom. ...
,
David Politzer
Hugh David Politzer (; born August 31, 1949) is an American theoretical physicist and the Richard Chace Tolman Professor of Theoretical Physics at the California Institute of Technology. He shared the 2004 Nobel Prize in Physics with David Gr ...
and
Frank Wilczek
Frank Anthony Wilczek (; born May 15, 1951) is an American theoretical physicist, mathematician and Nobel laureate. He is currently the Herman Feshbach Professor of Physics at the Massachusetts Institute of Technology (MIT), Founding Direc ...
allowed physicists to make precise predictions of the results of many high energy experiments using the quantum field theory technique of
perturbation theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middl ...
. Evidence of gluons was discovered in
three-jet event
In particle physics, a three-jet event is an event (particle physics), event with many particles in final state that appear to be clustered in three jet (particle physics), jets. A single jet consists of particles that fly off in roughly the same d ...
s 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
LEP, at
CERN.
The other side of asymptotic freedom is
confinement
Confinement may refer to
* With respect to humans:
** An old-fashioned or archaic synonym for childbirth
** Postpartum confinement (or postnatal confinement), a system of recovery after childbirth, involving rest and special foods
** Civil confi ...
. 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
Lattice QCD is a well-established non-perturbative approach to solving the quantum chromodynamics (QCD) theory of quarks and gluons. It is a lattice gauge theory formulated on a grid or lattice of points in space and time. When the size of the lat ...
computations, but is not mathematically proven. One of the
Millennium Prize Problems
The Millennium Prize Problems are seven well-known complex mathematical problems selected by the Clay Mathematics Institute in 2000. The Clay Institute has pledged a US$1 million prize for the first correct solution to each problem. According ...
announced by the
Clay Mathematics Institute requires a claimant to produce such a proof. Other aspects of
non-perturbative
In mathematics and physics, a non-perturbative function or process is one that cannot be described by perturbation theory. An example is the function
: f(x) = e^,
which does not have a Taylor series at ''x'' = 0. Every coefficient of the Taylor ...
QCD are the exploration of phases of
quark matter
Quark matter or QCD matter (quantum chromodynamic) refers to any of a number of hypothetical phases of matter whose degrees of freedom include quarks and gluons, of which the prominent example is quark-gluon plasma. Several series of conferences ...
, including the
quark–gluon plasma
Quark–gluon plasma (QGP) or quark soup is an interacting localized assembly of quarks and gluons at thermal (local kinetic) and (close to) chemical (abundance) equilibrium. The word ''plasma'' signals that free color charges are allowed. In a ...
.
The relation between the short-distance particle limit and the confining long-distance limit is one of the topics recently explored using
string theory, the modern form of S-matrix theory.
Theory
Some definitions
Every field theory of
particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
is based on certain symmetries of nature whose existence is deduced from observations. These can be
*
local symmetries, which are the symmetries that act independently at each point in
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
. 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 spacetime.
QCD is a non-abelian gauge theory (or
Yang–Mills theory
In mathematical physics, Yang–Mills theory is a gauge theory based on a special unitary group SU(''N''), or more generally any compact, reductive Lie algebra. Yang–Mills theory seeks to describe the behavior of elementary particles using ...
) of the
SU(3)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the specia ...
gauge group obtained by taking the
color charge
Color charge is a property of quarks and gluons that is related to the particles' strong interactions in the theory of quantum chromodynamics (QCD).
The "color charge" of quarks and gluons is completely unrelated to the everyday meanings of colo ...
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
Projection, projections or projective may refer to:
Physics
* Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction
* The display of images by a projector
Optics, graphic ...
on its direction of motion then it is called right-handed; otherwise, it is left-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.
Additional remarks: duality
As mentioned, ''asymptotic freedom'' means that at large energy – this corresponds also to ''short distances'' – there is practically no interaction between the particles. This is in contrast – more precisely one would say ''
dual''– to what one is used to, since usually one connects the absence of interactions with ''large'' distances. However, as already mentioned in the original paper of Franz Wegner, a solid state theorist who introduced 1971 simple gauge invariant lattice models, the high-temperature behaviour of the ''original model'', e.g. the strong decay of correlations at large distances, corresponds to the low-temperature behaviour of the (usually ordered!) ''dual model'', namely the asymptotic decay of non-trivial correlations, e.g. short-range deviations from almost perfect arrangements, for short distances. Here, in contrast to Wegner, we have only the dual model, which is that one described in this article.
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
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
. If one considers a version of QCD with ''N
f'' flavors of massless quarks, then there is a global (
chiral
Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object.
An object or a system is ''chiral'' if it is distinguishable from i ...
) flavor symmetry group SU
L(''N
f'') × SU
R(''N
f'') × U
B(1) × U
A(1). The chiral symmetry is
spontaneously broken by the
QCD vacuum
In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type o ...
to the vector (L+R) SU
V(''N
f'') with the formation of a
chiral condensate. The vector symmetry, U
B(1) corresponds to the baryon number of quarks and is an exact symmetry. The axial symmetry U
A(1) is exact in the classical theory, but broken in the quantum theory, an occurrence called an
anomaly. Gluon field configurations called
instanton
An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. Mo ...
s are closely related to this anomaly.
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 that 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
In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type o ...
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
In nuclear physics and particle physics, isospin (''I'') is a quantum number related to the up- and down quark content of the particle. More specifically, isospin symmetry is a subset of the flavour symmetry seen more broadly in the interactions ...
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.
Lagrangian
The dynamics of the quarks and gluons are controlled by the quantum chromodynamics
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
. The
gauge invariant
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
QCD Lagrangian is
where
is the quark field, a dynamical function of spacetime, in the
fundamental representation In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible representation, irreducible finite-dimensional representation of a semisimple Lie algebra, semisimple Lie group
or Lie algebra whose highest weig ...
of the
SU(3)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the specia ...
gauge
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
, indexed by
and
running from
to
;
is the
gauge covariant derivative
The gauge covariant derivative is a variation of the covariant derivative used in general relativity, quantum field theory and fluid dynamics. If a theory has gauge transformations, it means that some physical properties of certain equations are ...
; the γ
μ are
Dirac matrices
In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\ma ...
connecting the spinor representation to the vector representation of the
Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
.
Herein, the
gauge covariant derivative
The gauge covariant derivative is a variation of the covariant derivative used in general relativity, quantum field theory and fluid dynamics. If a theory has gauge transformations, it means that some physical properties of certain equations are ...
couples the quark field with a coupling strength
to the gluon fields via the infinitesimal SU(3) generators
in the fundamental representation. An explicit representation of these generators is given by
, wherein the
are the
Gell-Mann matrices
The Gell-Mann matrices, developed by Murray Gell-Mann, are a set of eight linearly independent 3×3 traceless Hermitian matrices used in the study of the strong interaction in particle physics.
They span the Lie algebra of the SU(3) group in t ...
.
The symbol
represents the gauge invariant
gluon field strength tensor, analogous to the
electromagnetic field strength tensor, ''F''
μν, in
quantum electrodynamics
In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
. It is given by:
:
where
are the
gluon field
In theoretical particle physics, the gluon field is a four-vector field characterizing the propagation of gluons in the strong interaction between quarks. It plays the same role in quantum chromodynamics as the electromagnetic four-potential in ...
s, dynamical functions of spacetime, in the
adjoint representation
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is G ...
of the SU(3) gauge group, indexed by ''a'', ''b'' and ''c'' running from
to
; and ''f
abc'' are the
structure constants
In mathematics, the structure constants or structure coefficients of an algebra over a field are used to explicitly specify the product of two basis vectors in the algebra as a linear combination. Given the structure constants, the resulting prod ...
of SU(3). Note that the rules to move-up or pull-down the ''a'', ''b'', or ''c'' indices are ''trivial'', (+, ..., +), so that ''f
abc'' = ''f
abc'' = ''f''
''a''''bc'' whereas for the ''μ'' or ''ν'' indices one has the non-trivial ''relativistic'' rules corresponding to the
metric signature
In mathematics, the signature of a metric tensor ''g'' (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and ...
(+ − − −).
The variables ''m'' and ''g'' correspond to the quark mass and coupling of the theory, respectively, which are subject to renormalization.
An important theoretical concept is the ''
Wilson loop
In quantum field theory, Wilson loops are gauge invariant operators arising from the parallel transport of gauge variables around closed loops. They encode all gauge information of the theory, allowing for the construction of loop representat ...
'' (named after
Kenneth G. Wilson
Kenneth Geddes "Ken" Wilson (June 8, 1936 – June 15, 2013) was an American theoretical physicist and a pioneer in leveraging computers for studying particle physics. He was awarded the 1982 Nobel Prize in Physics for his work on phase ...
). In lattice QCD, the final term of the above Lagrangian is discretized via Wilson loops, and more generally the behavior of Wilson loops can distinguish
confined and deconfined phases.
Fields
Quarks are massive spin-
fermions that carry a
color charge
Color charge is a property of quarks and gluons that is related to the particles' strong interactions in the theory of quantum chromodynamics (QCD).
The "color charge" of quarks and gluons is completely unrelated to the everyday meanings of colo ...
whose gauging is the content of QCD. Quarks are represented by
Dirac fields in the
fundamental representation In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible representation, irreducible finite-dimensional representation of a semisimple Lie algebra, semisimple Lie group
or Lie algebra whose highest weig ...
3 of the
gauge group
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
SU(3)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the specia ...
. They also carry electric charge (either − or +) and participate in
weak interactions
In nuclear physics and particle physics, the weak interaction, which is also often called the weak force or weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction ...
as part of
weak isospin
In particle physics, weak isospin is a quantum number relating to the weak interaction, and parallels the idea of isospin under the strong interaction. Weak isospin is usually given the symbol or , with the third component written as or . It c ...
doublets. They carry global quantum numbers including the
baryon number
In particle physics, the baryon number is a strictly conserved additive quantum number of a system. It is defined as
::B = \frac\left(n_\text - n_\bar\right),
where ''n''q is the number of quarks, and ''n'' is the number of antiquarks. Baryo ...
, which is for each quark,
hypercharge
In particle physics, the hypercharge (a portmanteau of hyperon, hyperonic and charge (physics), charge) ''Y'' of a subatomic particle, particle is a quantum number conserved under the strong interaction. The concept of hypercharge provides a sin ...
and one of the
flavor quantum numbers.
Gluons are spin-1
boson
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer spi ...
s that also carry
color charge
Color charge is a property of quarks and gluons that is related to the particles' strong interactions in the theory of quantum chromodynamics (QCD).
The "color charge" of quarks and gluons is completely unrelated to the everyday meanings of colo ...
s, since they lie in the
adjoint representation
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is G ...
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.
Each type of quark has a corresponding antiquark, of which the charge is exactly opposite. They transform in the
conjugate representation to quarks, denoted
.
Dynamics
According to the rules of
quantum field theory, and the associated
Feynman diagram
In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduc ...
s, 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
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...
s have no charge. Diagrams involving
Faddeev–Popov ghost
In physics, Faddeev–Popov ghosts (also called Faddeev–Popov gauge ghosts or Faddeev–Popov ghost fields) are extraneous fields which are introduced into gauge quantum field theories to maintain the consistency of the path integral formulat ...
s must be considered too (except in the
unitarity gauge
In theoretical physics, the unitarity gauge or unitary gauge is a particular choice of a gauge fixing in a gauge theory with a spontaneous symmetry breaking. In this gauge, the scalar fields responsible for the Higgs mechanism are transformed in ...
).
Area law and confinement
Detailed computations with the above-mentioned Lagrangian show that the effective potential between a quark and its anti-quark in a
meson
In particle physics, a meson ( or ) is a type of hadronic subatomic particle composed of an equal number of quarks and antiquarks, usually one of each, bound together by the strong interaction. Because mesons are composed of quark subparticles, ...
contains a term that increases in proportion to the distance between the quark and anti-quark (
), which represents some kind of "stiffness" of the interaction between the particle and its anti-particle at large distances, similar to the
entropic elasticity of a
rubber
Rubber, also called India rubber, latex, Amazonian rubber, ''caucho'', or ''caoutchouc'', as initially produced, consists of polymers of the organic compound isoprene, with minor impurities of other organic compounds. Thailand, Malaysia, an ...
band (see below). This leads to ''confinement'' of the quarks to the interior of hadrons, i.e.
meson
In particle physics, a meson ( or ) is a type of hadronic subatomic particle composed of an equal number of quarks and antiquarks, usually one of each, bound together by the strong interaction. Because mesons are composed of quark subparticles, ...
s and
nucleon
In physics and chemistry, a nucleon is either a proton or a neutron, considered in its role as a component of an atomic nucleus. The number of nucleons in a nucleus defines the atom's mass number (nucleon number).
Until the 1960s, nucleons were ...
s, with typical radii ''R''
c, corresponding to former "
Bag models" of the hadrons The order of magnitude of the "bag radius" is 1 fm (= 10
−15 m). Moreover, the above-mentioned stiffness is quantitatively related to the so-called "area law" behavior of the expectation value of the Wilson loop product ''P''
W of the ordered coupling constants around a closed loop ''W''; i.e.
is proportional to the ''area'' enclosed by the loop. For this behavior the non-abelian behavior of the gauge group is essential.
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
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middl ...
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 is
lattice QCD
Lattice QCD is a well-established non-perturbative approach to solving the quantum chromodynamics (QCD) theory of quarks and gluons. It is a lattice gauge theory formulated on a grid or lattice of points in space and time. When the size of the lat ...
. This approach uses a discrete set of spacetime points (called the lattice) to reduce the analytically intractable path integrals of the continuum theory to a very difficult numerical computation that is then carried out on
supercomputers
A supercomputer is a computer with a high level of performance as compared to a general-purpose computer. The performance of a supercomputer is commonly measured in floating-point operations per second (FLOPS) instead of million instructions p ...
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, in particular into the explicit forces acting between quarks and antiquarks in a meson. However, the
numerical sign problem In applied mathematics, the numerical sign problem is the problem of numerically evaluating the integral of a highly oscillatory function of a large number of variables. Numerical methods fail because of the near-cancellation of the positive and n ...
makes it difficult to use lattice methods to study QCD at high density and low temperature (e.g. nuclear matter or the interior of neutron stars).
1/''N'' expansion
A well-known approximation scheme, the
expansion, starts from the idea 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
In theoretical physics, the anti-de Sitter/conformal field theory correspondence, sometimes called Maldacena duality or gauge/gravity duality, is a conjectured relationship between two kinds of physical theories. On one side are anti-de Sitter s ...
approach.
Effective theories
For specific problems, effective theories may be written down that give qualitatively correct results in certain limits. In the best of cases, these may then be obtained as systematic expansions in some parameters of the QCD Lagrangian. One such
effective field theory is
chiral perturbation theory or ChiPT, which is the QCD effective theory at low energies. More precisely, it is a low energy expansion based on the spontaneous chiral symmetry breaking of QCD, which is an exact symmetry when quark masses are equal to zero, but for the u, d and s quark, which have small mass, it is still a good approximate symmetry. Depending on the number of quarks that are treated as light, one uses either SU(2) ChiPT or SU(3) ChiPT. Other effective theories are
heavy quark effective theory
In quantum chromodynamics, heavy quark effective theory (HQET) is an effective field theory describing the physics of heavy (that is, of mass far greater than the QCD scale) quarks. It is used in studying the properties of hadrons containing a s ...
(which expands around heavy quark mass near infinity), and
soft-collinear effective theory
In quantum field theory, soft-collinear effective theory (or SCET) is a theoretical framework for doing calculations that involve interacting particles carrying widely different energies.
The motivation for developing SCET was to control the inf ...
(which expands around large ratios of energy scales). In addition to effective theories, models like the
Nambu–Jona-Lasinio model
In quantum field theory, the Nambu–Jona-Lasinio model (or more precisely: ''the Nambu and Jona-Lasinio model'') is a complicated effective theory of nucleons and mesons constructed from interacting Dirac fermions with chiral symmetry, parall ...
and the
chiral model are often used when discussing general features.
QCD sum rules
Based on an
Operator product expansion
In quantum field theory, the operator product expansion (OPE) is used as an axiom to define the product of fields as a sum over the same fields. As an axiom, it offers a non-perturbative approach to quantum field theory. One example is the vert ...
one can derive sets of relations that connect different observables with each other.
Experimental tests
The notion of quark
flavors was prompted by the necessity of explaining the properties of hadrons during the development of the
quark model
In particle physics, the quark model is a classification scheme for hadrons in terms of their valence quarks—the quarks and antiquarks which give rise to the quantum numbers of the hadrons. The quark model underlies "flavor SU(3)", or the Ei ...
. The notion of color was necessitated by the puzzle of the . 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
Deep inelastic scattering is the name given to a process used to probe the insides of hadrons (particularly the baryons, such as protons and neutrons), using electrons, muons and neutrinos. It provided the first convincing evidence of the reali ...
experiments at
SLAC
SLAC National Accelerator Laboratory, originally named the Stanford Linear Accelerator Center,
is a United States Department of Energy National Laboratory operated by Stanford University under the programmatic direction of the U.S. Departme ...
. The first evidence for gluons came in
three-jet event
In particle physics, a three-jet event is an event (particle physics), event with many particles in final state that appear to be clustered in three jet (particle physics), jets. A single jet consists of particles that fly off in roughly the same d ...
s at
PETRA.
Several good quantitative tests of perturbative QCD exist:
* The
running of the QCD coupling as deduced from many observations
*
Scaling violation in polarized and unpolarized
deep inelastic scattering
Deep inelastic scattering is the name given to a process used to probe the insides of hadrons (particularly the baryons, such as protons and neutrons), using electrons, muons and neutrinos. It provided the first convincing evidence of the reali ...
*
Vector boson
In particle physics, a vector boson is a boson whose spin equals one. The vector bosons that are regarded as elementary particles in the Standard Model are the gauge bosons, the force carriers of fundamental interactions: the photon of electroma ...
production at
colliders (this includes the
Drell–Yan process
The Drell–Yan process occurs in high energy hadron–hadron scattering. It takes place when a quark of one hadron and an antiquark of another hadron annihilate, creating a virtual photon or Z boson which then decays into a pair of oppositel ...
)
*
Direct photons produced in hadronic collisions
*
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
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
computations of heavy-quarkonium spectra. There is a recent claim about the mass of the heavy meson B
c . 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
Quark matter or QCD matter (quantum chromodynamic) refers to any of a number of hypothetical phases of matter whose degrees of freedom include quarks and gluons, of which the prominent example is quark-gluon plasma. Several series of conferences ...
and the
quark–gluon plasma
Quark–gluon plasma (QGP) or quark soup is an interacting localized assembly of quarks and gluons at thermal (local kinetic) and (close to) chemical (abundance) equilibrium. The word ''plasma'' signals that free color charges are allowed. In a ...
is a non-perturbative test bed for QCD that still remains to be properly exploited.
One qualitative prediction of QCD is that there exist composite particles made solely of
gluons called
glueball
In particle physics, a glueball (also gluonium, gluon-ball) is a hypothetical composite particle. It consists solely of gluon particles, without valence quarks. Such a state is possible because gluons carry color charge and experience the strong ...
s that have not yet been definitively observed experimentally. A definitive observation of a glueball with the properties predicted by QCD would strongly confirm the theory. In principle, if glueballs could be definitively ruled out, this would be a serious experimental blow to QCD. But, as of 2013, scientists are unable to confirm or deny the existence of glueballs definitively, despite the fact that particle accelerators have sufficient energy to generate them.
Cross-relations to condensed matter physics
There are unexpected cross-relations to
condensed matter physics. For example, the notion of
gauge invariance
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
forms the basis of the well-known Mattis
spin glass
In condensed matter physics, a spin glass is a magnetic state characterized by randomness, besides cooperative behavior in freezing of spins at a temperature called 'freezing temperature' ''Tf''. In ferromagnetic solids, component atoms' magne ...
es, which are systems with the usual spin degrees of freedom
for ''i'' =1,...,N, with the special fixed "random" couplings
Here the ε
i and ε
k quantities can independently and "randomly" take the values ±1, which corresponds to a most-simple gauge transformation
This means that thermodynamic expectation values of measurable quantities, e.g. of the energy
are invariant.
However, here the ''coupling degrees of freedom''
, which in the QCD correspond to the ''gluons'', are "frozen" to fixed values (quenching). In contrast, in the QCD they "fluctuate" (annealing), and through the large number of gauge degrees of freedom the
entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
plays an important role (see below).
For positive ''J''
0 the thermodynamics of the Mattis spin glass corresponds in fact simply to a "ferromagnet in disguise", just because these systems have no "
frustration
In psychology, frustration is a common emotional response to opposition, related to anger, annoyance and disappointment. Frustration arises from the perceived resistance to the fulfillment of an individual's will or goal and is likely to in ...
" at all. This term is a basic measure in spin glass theory. Quantitatively it is identical with the loop product
along a closed loop ''W''. However, for a Mattis spin glass – in contrast to "genuine" spin glasses – the quantity ''P
W'' never becomes negative.
The basic notion "frustration" of the spin-glass is actually similar to the Wilson loop quantity of the QCD. The only difference is again that in the QCD one is dealing with SU(3) matrices, and that one is dealing with a "fluctuating" quantity. Energetically, perfect absence of frustration should be non-favorable and atypical for a spin glass, which means that one should add the loop product to the Hamiltonian, by some kind of term representing a "punishment". In the QCD the Wilson loop is essential for the Lagrangian rightaway.
The relation between the QCD and "disordered magnetic systems" (the spin glasses belong to them) were additionally stressed in a paper by Fradkin, Huberman and Shenker, which also stresses the notion of
duality.
A further analogy consists in the already mentioned similarity to
polymer physics, where, analogously to Wilson loops, so-called "entangled nets" appear, which are important for the formation of the
entropy-elasticity (force proportional to the length) of a rubber band. The non-abelian character of the SU(3) corresponds thereby to the non-trivial "chemical links", which glue different loop segments together, and "
asymptotic freedom
In quantum field theory, asymptotic freedom is a property of some gauge theories that causes interactions between particles to become asymptotically weaker as the energy scale increases and the corresponding length scale decreases.
Asymptotic fre ...
" means in the polymer analogy simply the fact that in the short-wave limit, i.e. for
(where ''R
c'' is a characteristic correlation length for the glued loops, corresponding to the above-mentioned "bag radius", while λ
w is the wavelength of an excitation) any non-trivial correlation vanishes totally, as if the system had crystallized.
There is also a correspondence between confinement in QCD – the fact that the color field is only different from zero in the interior of hadrons – and the behaviour of the usual magnetic field in the theory of
type-II superconductor
In superconductivity, a type-II superconductor is a superconductor that exhibits an intermediate phase of mixed ordinary and superconducting properties at intermediate temperature and fields above the superconducting phases.
It also features the ...
s: there the magnetism is confined to the interior of the
Abrikosov flux-line lattice,
[Mathematically, the flux-line lattices are described by Emil Artin's braid group, which is nonabelian, since one braid can wind around another one.] i.e., the London penetration depth ''λ'' of that theory is analogous to the confinement radius ''R
c'' of quantum chromodynamics. Mathematically, this correspondendence is supported by the second term,
on the r.h.s. of the Lagrangian.
See also
* For overviews:
**
Standard Model
**
Strong interaction
**
Quark
**
Gluon
**
Hadron
In particle physics, a hadron (; grc, ἁδρός, hadrós; "stout, thick") is a composite subatomic particle made of two or more quarks held together by the strong interaction. They are analogous to molecules that are held together by the e ...
**
Colour confinement
**
QCD matter
Quark matter or QCD matter (quantum chromodynamic) refers to any of a number of hypothetical phases of matter whose degrees of freedom include quarks and gluons, of which the prominent example is quark-gluon plasma. Several series of conferenc ...
**
Quark–gluon plasma
Quark–gluon plasma (QGP) or quark soup is an interacting localized assembly of quarks and gluons at thermal (local kinetic) and (close to) chemical (abundance) equilibrium. The word ''plasma'' signals that free color charges are allowed. In a ...
* For details:
**
Gauge theory
**
Quantum gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups ...
,
BRST quantization and
Faddeev–Popov ghost
In physics, Faddeev–Popov ghosts (also called Faddeev–Popov gauge ghosts or Faddeev–Popov ghost fields) are extraneous fields which are introduced into gauge quantum field theories to maintain the consistency of the path integral formulat ...
**
Quantum field theory – a more general category
* For techniques:
**
Lattice QCD
Lattice QCD is a well-established non-perturbative approach to solving the quantum chromodynamics (QCD) theory of quarks and gluons. It is a lattice gauge theory formulated on a grid or lattice of points in space and time. When the size of the lat ...
**
1/N expansion
In quantum field theory and statistical mechanics, the 1/''N'' expansion (also known as the "large ''N''" expansion) is a particular perturbative analysis of quantum field theories with an internal symmetry group such as SO(N) or SU(N). It cons ...
**
Perturbative QCD
**
Soft-collinear effective theory
In quantum field theory, soft-collinear effective theory (or SCET) is a theoretical framework for doing calculations that involve interacting particles carrying widely different energies.
The motivation for developing SCET was to control the inf ...
**
Heavy quark effective theory
In quantum chromodynamics, heavy quark effective theory (HQET) is an effective field theory describing the physics of heavy (that is, of mass far greater than the QCD scale) quarks. It is used in studying the properties of hadrons containing a s ...
**
Chiral model
**
Nambu–Jona-Lasinio model
In quantum field theory, the Nambu–Jona-Lasinio model (or more precisely: ''the Nambu and Jona-Lasinio model'') is a complicated effective theory of nucleons and mesons constructed from interacting Dirac fermions with chiral symmetry, parall ...
* For experiments:
**
Deep inelastic scattering
Deep inelastic scattering is the name given to a process used to probe the insides of hadrons (particularly the baryons, such as protons and neutrons), using electrons, muons and neutrinos. It provided the first convincing evidence of the reali ...
**
Jet (particle physics)
A jet is a narrow cone of hadrons and other particles produced by the hadronization of a quark or gluon in a particle physics or heavy ion experiment. Particles carrying a color charge, such as quarks, cannot exist in free form because of quan ...
**
Quark–gluon plasma
Quark–gluon plasma (QGP) or quark soup is an interacting localized assembly of quarks and gluons at thermal (local kinetic) and (close to) chemical (abundance) equilibrium. The word ''plasma'' signals that free color charges are allowed. In a ...
*
Quantum electrodynamics
In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
*
Symmetry in quantum mechanics
Symmetries in quantum mechanics describe features of spacetime and particles which are unchanged under some transformation, in the context of quantum mechanics, relativistic quantum mechanics and quantum field theory, and with applications in the ...
*
Yang–Mills theory
In mathematical physics, Yang–Mills theory is a gauge theory based on a special unitary group SU(''N''), or more generally any compact, reductive Lie algebra. Yang–Mills theory seeks to describe the behavior of elementary particles using ...
*
Yang–Mills existence and mass gap
The Yang–Mills existence and mass gap problem is an unsolved problem in mathematical physics and mathematics, and one of the seven Millennium Prize Problems defined by the Clay Mathematics Institute, which has offered a prize of US$1,000,000 f ...
References
Further reading
*
*
*
External links
*
Particle data groupThe millennium prizefo
proving confinementAb Initio Determination of Light Hadron MassesAndreas S Kronfeld''The Weight of the World Is Quantum Chromodynamics''
Andreas S Kronfeld''Quantum chromodynamics with advanced computing''
Standard model gets right answerQuantum ChromodynamicsCern Courier, The history of QCD with Prof. Dr. Harald Fritzsch
{{DEFAULTSORT:Quantum Chromodynamics
Quantum field theory