View source for Quantum chromodynamics

{{short description|Theory of the strong nuclear interactions}}
{{Redirect|QCD}}
{{Standard model of particle physics|cTopic=quantum chromodynamics}}

In [[theoretical physics]], '''quantum chromodynamics''' ('''QCD''') is the study of the [[strong interaction]] between [[quark]]s mediated by [[gluon]]s. Quarks are fundamental particles that make up composite [[hadron]]s such as the [[proton]], [[neutron]] and [[pion]]. QCD is a type of [[quantum field theory]] called a [[non-abelian gauge theory]], with symmetry group [[special unitary group|SU(3)]]. The QCD analog of electric charge is a property called ''color''. Gluons are the [[force carrier]]s of the theory, just as photons are for the electromagnetic force in [[quantum electrodynamics]]. The theory is an important part of the [[Standard Model]] of [[particle physics]]. A large body of [[Quantum chromodynamics#Experimental tests|experimental evidence for QCD]] has been gathered over the years.

QCD exhibits three salient properties:
* [[Color confinement]]. Due to the force between two color charges remaining constant as they are separated, the energy grows until a quark–antiquark pair is [[mass–energy equivalence|spontaneously produced]], turning the initial hadron into a pair of hadrons instead of isolating a color charge. Although analytically unproven, color confinement is well established from [[lattice QCD]] calculations and decades of experiments.<ref>{{cite book
 |author=J. Greensite
 |year=2011
 |title=An introduction to the confinement problem
 |publisher=[[Springer Science+Business Media|Springer]]
 |isbn=978-3-642-14381-6
}}</ref>
* [[Asymptotic freedom]], 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]] and [[Frank Wilczek]],<ref name=GrossWilczek>
{{cite journal
 |author1=D.J. Gross |author2=F. Wilczek |year=1973
 |title=Ultraviolet behavior of non-abelian gauge theories
 |journal=[[Physical Review Letters]]
 |volume=30 |issue= 26|pages= 1343–1346
 |bibcode=1973PhRvL..30.1343G
 |doi=10.1103/PhysRevLett.30.1343
|doi-access=free}}</ref> and independently by [[David Politzer]] in the same year.<ref name=Politzer>
{{cite journal
 |author=H.D. Politzer
 |year=1973
 |title=Reliable perturbative results for strong interactions
 |journal=[[Physical Review Letters]]
 |volume=30 |issue=26 |pages=1346–1349
 |bibcode=1973PhRvL..30.1346P
 |doi=10.1103/PhysRevLett.30.1346
|doi-access=free
 }}</ref>  For this work, all three shared the 2004 [[Nobel Prize in Physics]].<ref name=Nobel>{{cite web
 |url         = http://nobelprize.org/nobel_prizes/physics/laureates/2004/
 |title       = The Nobel Prize in Physics 2004
 |publisher   = Nobel Web
 |year        = 2004
 |access-date  = 2010-10-24
 |url-status     = live
 |archive-url  = https://web.archive.org/web/20101106025744/http://nobelprize.org/nobel_prizes/physics/laureates/2004/
 |archive-date = 2010-11-06
}}</ref>
*[[Chiral symmetry breaking]], the [[spontaneous symmetry breaking]] 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 awarded the 2008 Nobel Prize in Physics for elucidating the phenomenon in 1960, a dozen years before the advent of QCD. Lattice simulations have confirmed all his generic predictions.

==Terminology==
Physicist [[Murray Gell-Mann]] coined the word ''quark'' in its present sense. It originally comes from the phrase "Three quarks for Muster Mark" in ''[[Finnegans Wake]]'' by [[James Joyce]]. 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.)<ref>
{{cite book
 | last = Gell-Mann
 | first = Murray
 | author-link =Murray Gell-Mann
 | title = The Quark and the Jaguar
 | publisher = [[Owl Books]]
 | year = 1995
 | isbn = 978-0-8050-7253-2
}}</ref>

The three kinds of [[Charge (physics)|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) [[color vision|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'''<ref>[[wikt:colour force]]</ref> (or '''color force'''<ref>{{cite web |url=http://hyperphysics.phy-astr.gsu.edu/hbase/Forces/color.html |title=The Color Force |access-date=2007-08-29 |url-status=live |archive-url=https://web.archive.org/web/20070820075205/http://hyperphysics.phy-astr.gsu.edu/hbase/forces/color.html |archive-date=2007-08-20 }} retrieved 6 May 2017</ref>) or [[strong interaction]], and is responsible for the [[nuclear force]].

Since the theory of electric charge is dubbed "[[electrodynamics]]", the [[Ancient Greek|Greek]] word {{lang|grc|χρῶμα}} ({{lang|grc-Latn|chrōma}}, "color") is applied to the theory of color charge, "chromodynamics".

==History==
{{main|History of quantum mechanics|History of quantum field theory}}
With the invention of [[bubble chamber]]s and [[spark chamber]]s in the 1950s, experimental [[particle physics]] discovered a large and ever-growing number of particles called [[hadron]]s. It seemed that such a large number of particles could not all be [[fundamental particles|fundamental]]. First, the particles were classified by [[Charge (physics)|charge]] and [[isospin]] by [[Eugene Wigner]] and [[Werner Heisenberg]]; then, in 1953–56,<ref>
{{cite journal
 |last1=Nakano |first1=T
 |last2=Nishijima |first2=N
 |year=1953
 |title=Charge Independence for V-particles |journal=[[Progress of Theoretical Physics]]
 |volume=10 |issue=5 |pages=581
 |doi=10.1143/PTP.10.581
|bibcode = 1953PThPh..10..581N |doi-access=free}}</ref><ref>
{{cite journal
 |last=Nishijima |first=K
 |year=1955
 |title=Charge Independence Theory of V Particles
 |journal=[[Progress of Theoretical Physics]]
 |volume=13 |issue=3 |pages=285–304 
 |doi=10.1143/PTP.13.285
|bibcode = 1955PThPh..13..285N |doi-access=free}}</ref><ref>
{{cite journal
 |last=Gell-Mann |first=M
 |year=1956
 |title=The Interpretation of the New Particles as Displaced Charged Multiplets
 |journal=[[Il Nuovo Cimento]]
 |volume=4
 |issue=S2 |pages=848–866
 |doi=10.1007/BF02748000
|bibcode=1956NCim....4S.848G|s2cid=121017243
 }}</ref> according to [[Strangeness (particle physics)|strangeness]] by [[Murray Gell-Mann]] and [[Kazuhiko Nishijima]] (see [[Gell-Mann–Nishijima formula]]). To gain greater insight, the hadrons were sorted into groups having similar properties and masses using the ''[[eightfold way (physics)|eightfold way]]'', invented in 1961 by Gell-Mann<ref>Gell-Mann, M. (1961). "The Eightfold Way: A Theory of strong interaction symmetry" (No. TID-12608; CTSL-20). California Inst. of Tech., Pasadena. Synchrotron Lab ([https://www.osti.gov/scitech/servlets/purl/4008239 online]).</ref>  and [[Yuval Ne'eman]]. Gell-Mann and [[George Zweig]], correcting an earlier approach of [[Shoichi Sakata]], went on to propose in 1963 that the structure of the groups could be explained by the existence of three [[flavour (particle physics)|flavor]]s of smaller particles inside the hadrons: the [[quark]]s. Gell-Mann also briefly discussed a field theory model in which quarks interact with gluons.<ref>{{cite journal | author= M. Gell-Mann | title=A Schematic Model of Baryons and Mesons | volume=8 | issue=3 | year=1964 | journal=[[Physics Letters]] | pages=214–215 | doi=10.1016/S0031-9163(64)92001-3| bibcode=1964PhL.....8..214G }}</ref><ref>{{cite book |author1=M. Gell-Mann |author2=H. Fritzsch | title=Murray Gell-Mann: Selected Papers | year=2010 | publisher=World Scientific| bibcode=2010mgsp.book.....F }}</ref>

Perhaps the first remark that quarks should possess an additional [[quantum number]] was made<ref>
{{cite arXiv
 |eprint=0904.0343
 |author1=Fyodor Tkachov
 |title=A contribution to the history of quarks: Boris Struminsky's 1965 JINR publication
 |class=physics.hist-ph
 |year=2009
}}</ref> as a short footnote in the preprint of [[Boris Struminsky]]<ref name="struminsky">B. V. Struminsky, Magnetic moments of baryons in the quark model. [[JINR]]-Preprint P-1939, Dubna, Russia. Submitted on January 7, 1965.</ref> in connection with the Ω<sup>−</sup> [[Omega baryon|hyperon]] being composed of three [[strange quark]]s with parallel spins (this situation was peculiar, because since quarks are [[fermion]]s, such a combination is forbidden by the [[Pauli exclusion principle]]):
{{Blockquote|text=Three identical quarks cannot form an antisymmetric S-state. In order to realize an antisymmetric orbital S-state, it is necessary for the quark to have an additional quantum number.|author=B. V. Struminsky|title=Magnetic moments of barions in the quark model|source=[[JINR]]-Preprint P-1939, Dubna, Submitted on January 7, 1965}}

Boris Struminsky was a PhD student of [[Nikolay Bogolyubov]]. The problem considered in this preprint was suggested by Nikolay Bogolyubov, who advised Boris Struminsky in this research.<ref name="struminsky" /> In the beginning of 1965, [[Nikolay Bogolyubov]], [[Boris Struminsky]] and [[Albert Tavkhelidze]] wrote a preprint with a more detailed discussion of the additional quark quantum degree of freedom.<ref>[[Nikolay Bogolyubov|N. Bogolubov]], B. Struminsky, A. Tavkhelidze. On composite models in the theory of elementary particles. [[JINR]] Preprint D-1968, [[Dubna]] 1965.</ref> This work was also presented by Albert Tavkhelidze without obtaining consent of his collaborators for doing so at an international conference in [[Trieste]] (Italy), in May 1965.<ref>A. Tavkhelidze. Proc. Seminar on High Energy Physics and Elementary Particles, Trieste, 1965, Vienna IAEA, 1965, p. 763.</ref><ref>V. A. Matveev and A. N. Tavkhelidze (INR, RAS, Moscow) [http://www.inr.ru/quantum.html The quantum number color, colored quarks and QCD] {{webarchive|url=https://web.archive.org/web/20070523073026/http://www.inr.ru/quantum.html |date=2007-05-23 }} (Dedicated to the 40th Anniversary of the Discovery of the Quantum Number Color). Report presented at the 99th Session of the JINR Scientific Council, Dubna, 19–20 January 2006.</ref>

A similar mysterious situation was with the [[Delta baryon|Δ<sup>++</sup> baryon]]; in the quark model, it is composed of three [[up quark]]s with parallel spins. In 1964–65, [[Oscar W. Greenberg|Greenberg]]<ref>{{cite journal |first=O. W. |last=Greenberg |title=Spin and Unitary Spin Independence in a Paraquark Model of Baryons and Mesons |journal=Phys. Rev. Lett. |volume=13 |issue= 20|pages=598–602 |year=1964 |doi=10.1103/PhysRevLett.13.598 |bibcode=1964PhRvL..13..598G }}</ref> and [[Moo-Young Han|Han]]–[[Yoichiro Nambu|Nambu]]<ref>{{cite journal |first1=M. Y. |last1=Han |first2=Y. |last2=Nambu |title=Three-Triplet Model with Double SU(3) Symmetry |journal=Phys. Rev. |volume=139 |issue= 4B|pages=B1006–B1010 |year=1965 |doi=10.1103/PhysRev.139.B1006 |bibcode=1965PhRv..139.1006H |url=https://digital.library.unt.edu/ark:/67531/metadc1031342/ }}</ref> independently resolved the problem by proposing that quarks possess an additional [[special unitary group|SU(3)]] [[gauge theory|gauge]] [[degrees of freedom (physics and chemistry)|degree of freedom]], later called color charge. Han and Nambu noted that quarks might interact via an octet of vector [[gauge boson]]s: the [[gluon]]s.

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]] argued that high energy experiments showed quarks are real particles: he called them ''[[Parton (particle physics)|partons]]'' (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 [[pomeron|diffractive scattering]]. Although Gell-Mann believed that certain quark charges could be localized, he was open to the possibility that the quarks themselves could not be localized because space and time break down. This was the more radical approach of [[S-matrix theory]].

[[James Daniel Bjorken|James Bjorken]] proposed that pointlike partons would imply certain relations in [[deep inelastic scattering]] of [[electron]]s and protons, which were verified in experiments at [[SLAC]] in 1969. This led physicists to abandon the S-matrix approach for the strong interactions.

In 1973 the concept of [[Color charge|color]] as the source of a "strong field" was developed into the theory of QCD by physicists [[Harald Fritzsch]] and [[Heinrich Leutwyler]], together with physicist [[Murray Gell-Mann]].<ref>{{cite journal | last1 = Fritzsch | first1 = H. | last2 = Gell-Mann | first2 = M. | last3 = Leutwyler | first3 = H. | title = Advantages of the color octet gluon picture | journal = Physics Letters | volume = 47B | issue = 4| pages = 365–368 | year = 1973 | doi=10.1016/0370-2693(73)90625-4| bibcode = 1973PhLB...47..365F | citeseerx = 10.1.1.453.4712 }}</ref> In particular, they employed the general field theory developed in 1954 by [[Chen Ning Yang]] and [[Robert Mills (physicist)|Robert Mills]]<ref>{{cite journal |author-link1=Chen-Ning Yang |first1=C. N. |last1=Yang |author-link2=Robert Mills (physicist) |first2=R. |last2=Mills |title=Conservation of Isotopic Spin and Isotopic Gauge Invariance |journal=[[Physical Review]] |volume=96 |issue=1 |pages=191–195 |year=1954 |doi=10.1103/PhysRev.96.191|bibcode = 1954PhRv...96..191Y |doi-access=free }}</ref> (see [[Yang–Mills theory]]), 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 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 (quantum mechanics)|perturbation theory]]. Evidence of gluons was discovered in [[three-jet event]]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 [[Color confinement|confinement]]. Since the force between color charges does not decrease with distance, it is believed that quarks and gluons can never be liberated from hadrons. This aspect of the theory is verified within [[lattice QCD]] computations, but is not mathematically proven. One of the [[Millennium Prize Problems]] announced by the [[Clay Mathematics Institute]] requires a claimant to produce such a proof. Other aspects of [[non-perturbative]] QCD are the exploration of phases of [[QCD matter|quark matter]], including the [[quark–gluon plasma]].

==Theory==

===Some definitions===
{{unsolved|physics|QCD in the non-[[perturbation theory (quantum mechanics)|perturbative]] regime:
*'''[[Color confinement|Confinement]]''': the equations of QCD remain unsolved at [[energy scale]]s relevant for describing [[atomic nucleus|atomic nuclei]]. How does QCD give rise to the physics of nuclei and nuclear constituents?
*'''[[QCD matter|Quark matter]]''': the equations of QCD predict that a [[quark–gluon plasma|plasma (or soup) of quarks and gluons]] should be formed at high temperature and density. What are the properties of this [[phase of matter]]?}}<!-- please don't insert a line feed here, without checking to ensure that spacing remains as it should -->
Every field theory of [[particle physics]] is based on certain symmetries of nature whose existence is deduced from observations. These can be
*[[Local symmetry|local symmetries]], which are the symmetries that act independently at each point in [[spacetime]]. Each such symmetry is the basis of a [[gauge theory]] and requires the introduction of its own [[gauge boson]]s.
*[[Global symmetry|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]]) of the [[special unitary group|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 (physics)|chirality]], discrimination between left and right-handed. If the [[Spin (physics)|spin]] of a particle has a positive [[projection (linear algebra)|projection]] 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 ''[[Kramers–Wannier duality|dual]]''– to what one is used to, since usually one connects the absence of interactions with ''large''&nbsp;distances. However, as already mentioned in the original paper of Franz Wegner,<ref>{{cite journal |last=Wegner |first=F. |year=1971 |title=Duality in Generalized Ising Models and Phase Transitions without Local Order Parameter |journal=J. Math. Phys. |volume=12 |issue=10 |pages=2259–2272 |bibcode=1971JMP....12.2259W |doi=10.1063/1.1665530}} Reprinted in {{cite book |title=Lattice Gauge Theories and Monte Carlo Simulations |publisher=World Scientific |year=1983 |isbn=9971950707 |editor-last=Rebbi |editor-first=Claudio |location=Singapore |pages=60–73}} Abstract: [http://www.tphys.uni-heidelberg.de/~wegner/Abstracts.html#12] {{webarchive|url=https://web.archive.org/web/20110504173247/http://www.tphys.uni-heidelberg.de/~wegner/Abstracts.html|date=2011-05-04}}</ref> 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.<ref>Perhaps one can guess that in the "original" model mainly the quarks would fluctuate, whereas in the present one, the "dual" model, mainly the gluons do.</ref>

===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 [[Quantum electrodynamics|QED]]: this is an [[abelian group]]. If one considers a version of QCD with ''N<sub>f</sub>'' flavors of massless quarks, then there is a global ([[chirality (physics)|chiral]]) flavor symmetry group SU<sub>L</sub>(''N<sub>f</sub>'') &times; SU<sub>R</sub>(''N<sub>f</sub>'') &times; U<sub>B</sub>(1) &times; U<sub>A</sub>(1). The chiral symmetry is [[spontaneous symmetry breaking|spontaneously broken]] by the [[QCD vacuum]] to the vector (L+R) SU<sub>V</sub>(''N<sub>f</sub>'')  with the formation of a [[chiral condensate]]. The vector symmetry, U<sub>B</sub>(1) corresponds to the baryon number of quarks and is an exact symmetry. The axial symmetry U<sub>A</sub>(1) is exact in the classical theory, but broken in the quantum theory, an occurrence called an [[anomaly (physics)|anomaly]]. Gluon field configurations called [[instanton]]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]] there are vacuum condensates of all the quarks whose mass is less than the QCD scale. This includes the up and down quarks, and to a lesser extent the strange quark, but not any of the others. The vacuum is symmetric under SU(2) [[isospin]] rotations of up and down, and to a lesser extent under rotations of up, down, and strange, or full flavor group SU(3), and the observed particles make isospin and SU(3) multiplets.

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

=== Lagrangian ===
The dynamics of the quarks and gluons are defined by the quantum chromodynamics [[Lagrangian (field theory)|Lagrangian]].  The [[gauge invariant]] QCD Lagrangian is

{{Equation box 1
|indent =:
|equation = :<math>\mathcal{L}_\mathrm{QCD} = \bar{\psi}_i  \left( i \gamma^\mu (D_\mu)_{ij} - m\, \delta_{ij}\right) \psi_j - \frac{1}{4}G^a_{\mu \nu} G^{\mu \nu}_a </math>|border
|border colour = #50C878
|background colour = #ECFCF4}}

where <math>\psi_i(x) \,</math> is the quark field, a dynamical function of spacetime, in the [[fundamental representation]] of the [[SU(3)]] gauge [[Group (mathematics)|group]], indexed by <math>i</math> and <math>j</math> running from <math>1</math> to <math>3</math>; <math>\bar \psi_i \,</math> is the [[Dirac adjoint]] of <math>\psi_i \,</math>; <math>D_\mu</math> is the [[gauge covariant derivative]]; the γ<sup>μ</sup> are [[Gamma matrices]] connecting the spinor representation to the vector representation of the [[Lorentz group]].

Herein, the [[gauge covariant derivative]] <math>\left( D_\mu \right)_{ij} = \partial_\mu \delta_{ij} - i g \left( T_a \right)_{ij} \mathcal{A}^a_\mu \,</math>couples the quark field with a coupling strength <math>g \,</math>to the gluon fields via the infinitesimal SU(3) generators <math>T_a \,</math>in the fundamental representation. An explicit representation of these generators is given by <math>T_a = \lambda_a / 2 \,</math>, wherein the <math>\lambda_a \, (a = 1 \ldots 8)\,</math>are the [[Gell-Mann matrices]].

The symbol <math>G^a_{\mu \nu} \,</math> represents the gauge invariant [[gluon field strength tensor]], analogous to the [[electromagnetic tensor|electromagnetic field strength tensor]], ''F''<sup>μν</sup>, in [[quantum electrodynamics]]. It is given by:<ref>{{cite journal|title=The field strength correlator from QCD sum rules
|author1=M. Eidemüller |author2=H.G. Dosch |author3=M. Jamin
|location=Heidelberg, Germany
|journal=Nucl. Phys. B Proc. Suppl. |volume=86 |pages=421–425 |year=2000
|issue=1–3 |arxiv=hep-ph/9908318|bibcode=2000NuPhS..86..421E|doi=10.1016/S0920-5632(00)00598-3|s2cid=18237543 }}</ref>

:<math>G^a_{\mu \nu} = \partial_\mu \mathcal{A}^a_\nu - \partial_\nu \mathcal{A}^a_\mu + g f^{abc} \mathcal{A}^b_\mu \mathcal{A}^c_\nu \,,</math>

where <math>\mathcal{A}^a_\mu(x) \,</math> are the [[gluon field]]s, dynamical functions of spacetime, in the [[adjoint representation]] of the SU(3) gauge group, indexed by ''a'', ''b'' and ''c'' running from <math>1</math> to <math>8</math>; and ''f<sub>abc</sub>'' are the [[structure constants]] of SU(3) (the generators of the adjoint representation). Note that the rules to move-up or pull-down the ''a'', ''b'', or ''c'' indices are ''trivial'', (+, ..., +), so that ''f<sup>abc</sup>'' = ''f<sub>abc</sub>'' = ''f''<sup>''a''</sup><sub>''bc''</sub> whereas for the ''μ'' or ''ν'' indices one has the non-trivial ''relativistic'' rules corresponding to the [[metric signature]] (+ − − −).

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]]'' (named after [[Kenneth G. Wilson]]).  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 [[Color confinement|confined]] and deconfined phases.

===Fields===
[[File:QCD.svg|300px|right|thumb|The pattern of strong charges for the three colors of quark, three antiquarks, and eight gluons (with two of zero charge overlapping).]]

Quarks are massive spin-{{frac|1|2}} [[fermion]]s that carry a [[color charge]] whose gauging is the content of QCD. Quarks are represented by [[Dirac field]]s in the [[fundamental representation]] '''3''' of the [[gauge group]] [[SU(3)]]. They also carry electric charge (either &minus;{{frac|1|3}} or +{{frac|2|3}}) and participate in [[weak interactions]] as part of [[weak isospin]] doublets. They carry global quantum numbers including the [[baryon number]], which is {{frac|1|3}} for each quark, [[hypercharge]] and one of the [[flavor (particle physics)|flavor quantum numbers]].

Gluons are spin-1 [[boson]]s that also carry [[color charge]]s, since they lie in the [[Adjoint representation of a Lie group|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 state|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 <math>\bar\mathbf{3}</math>.

===Dynamics===
According to the rules of [[quantum field theory]], and the associated [[Feynman diagram]]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 [[quantum electrodynamics|QED]], in which only the first kind of interaction occurs, since [[photon]]s have no charge. Diagrams involving [[Faddeev–Popov ghost]]s must be considered too (except in the [[unitarity gauge]]).

===Area law and confinement===
Detailed computations with the above-mentioned Lagrangian<ref>See all standard textbooks on the QCD,  e.g., those noted above</ref> show that the effective potential between a quark and its anti-quark in a [[meson]] contains a term that increases in proportion to the distance between the quark and anti-quark (<math>\propto r</math>), which represents some kind of "stiffness" of the interaction between the particle and its anti-particle at large  distances, similar to the [[entropic force|entropic elasticity]] of a [[rubber]] band (see below). This leads to ''confinement''&nbsp;<ref>''Confinement'' gives way to a [[quark–gluon plasma]] only at extremely large pressures and/or temperatures, e.g. for <math>T \approx 5\cdot 10^{12}</math>&nbsp;&nbsp;K or larger.</ref> of the quarks to the interior of hadrons, i.e. [[meson]]s and [[nucleon]]s, with typical radii ''R''<sub>c</sub>, corresponding to  former "[[Bag model]]s" of the hadrons<ref>[[Kenneth Alan Johnson]]. (July 1979). The bag model of quark confinement. ''Scientific American''.</ref> The order of magnitude of the "bag radius" is 1&nbsp;fm (=&nbsp;10<sup>&minus;15</sup>&nbsp;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''<sub>W</sub> of the ordered coupling constants around a closed loop ''W''; i.e. <math>\,\langle P_W\rangle</math> 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===
{{main|Perturbative QCD}}
This approach is based on asymptotic freedom, which allows [[perturbation theory (quantum mechanics)|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===
{{main|Lattice QCD}}
[[Image:Fluxtube meson.png|right|thumb|150px|{{langle}}''E''<sup>2</sup>{{rangle}} plot for static quark–antiquark system held at a fixed separation, where blue is zero and red is the highest value (result of a lattice QCD simulation by M. Cardoso et al.<ref>{{cite journal |first1=M. |last1=Cardoso |first2=N. |last2=Cardoso |first3=P. |last3=Bicudo |display-authors=1 |title=Lattice QCD computation of the colour fields for the static hybrid quark–gluon–antiquark system, and microscopic study of the Casimir scaling |journal=Phys. Rev. D |volume=81 |issue= 3|pages=034504 |year=2010 |doi=10.1103/PhysRevD.81.034504 |arxiv=0912.3181 |bibcode=2010PhRvD..81c4504C |s2cid=119216789 }}</ref>)]]
Among non-perturbative approaches to QCD, the most well established is [[lattice QCD]]. 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]] 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]] 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===
{{main|1/N expansion}}
A well-known approximation scheme, the [[1/N expansion|{{frac|1|''N''}} 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]] 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]] (which expands around heavy quark mass near infinity), and [[soft-collinear effective theory]] (which expands around large ratios of energy scales). In addition to effective theories, models like the [[Nambu–Jona-Lasinio model]] and the [[chiral model]] are often used when discussing general features.

===QCD sum rules===
{{main|QCD sum rules}}

Based on an [[Operator product expansion]] one can derive sets of relations that connect different observables with each other.

==Experimental tests==<!-- This section is linked from [[Quantum chromodynamics]] -->
The notion of quark [[flavour (particle physics)|flavors]] was prompted by the necessity of explaining the properties of hadrons during the development of the [[quark model]]. The notion of color was necessitated by the puzzle of the {{SubatomicParticle|Delta++}}. This has been dealt with in the section on [[Quantum chromodynamics#History|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 event]]s at [[PETRA]].<ref>{{Cite journal |last=Bethke |first=S. |date=2007-04-01 |title=Experimental tests of asymptotic freedom |url=https://www.sciencedirect.com/science/article/pii/S0146641006000615 |journal=Progress in Particle and Nuclear Physics |language=en |volume=58 |issue=2 |pages=351–386 |doi=10.1016/j.ppnp.2006.06.001 |arxiv=hep-ex/0606035 |bibcode=2007PrPNP..58..351B |s2cid=14915298 |issn=0146-6410}}</ref>

Several good quantitative tests of perturbative QCD exist:
* The [[coupling constant#QCD and asymptotic freedom|running of the QCD coupling]] as deduced from many observations
* [[Bjorken scaling|Scaling violation]] in polarized and unpolarized [[deep inelastic scattering]]
* [[Vector boson]] production at [[collider]]s (this includes the [[Drell–Yan process]])
* [[Direct photons]] produced in hadronic collisions
* [[Jet (particle physics)|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 QCD|lattice]] computations of heavy-quarkonium spectra. There is a recent claim about the mass of the heavy meson B<sub>c</sub> . Other non-perturbative tests are currently at the level of 5% at best. Continuing work on masses and [[Form factor (QFT)|form factor]]s 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 that still remains to be properly exploited.{{Citation needed|date=September 2020}}

One qualitative prediction of QCD is that there exist composite particles made solely of [[gluon]]s called [[glueball]]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]] forms the basis of the well-known Mattis [[spin glass]]es,<ref>{{cite journal |first=D. C. |last=Mattis |title=Solvable Spin Systems with Random Interactions |journal=Phys. Lett. A |volume=56 |issue=5 |year=1976 |pages=421–422 |doi=10.1016/0375-9601(76)90396-0 |bibcode=1976PhLA...56..421M }}</ref> which are systems with the usual spin degrees of freedom <math>s_i=\pm 1\,</math> for ''i''&nbsp;=1,...,N,  with the special fixed  "random" couplings <math>J_{i,k}=\epsilon_i \,J_0\,\epsilon_k\,.</math>  Here the ε<sub>i</sub>  and ε<sub>k</sub> quantities can independently and "randomly" take the values ±1, which corresponds to a most-simple gauge transformation  <math>(\,s_i\to s_i\cdot\epsilon_i\quad\,J_{i,k}\to \epsilon_i J_{i,k}\epsilon_k\,\quad s_k\to s_k\cdot\epsilon_k \,)\,.</math> This means that thermodynamic expectation values of measurable quantities, e.g. of the energy <math display="inline">{\mathcal H}:=-\sum s_i\,J_{i,k}\,s_k\,,</math> are invariant.

However, here the ''coupling degrees of freedom'' <math>J_{i,k}</math>, 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]] plays an important role (see below).

For positive ''J''<sub>0</sub> the thermodynamics of the Mattis spin glass corresponds in fact simply to a "ferromagnet in disguise", just because these systems have no "[[Geometrical frustration|frustration]]" at all. This term is a basic  measure in spin glass theory.<ref>{{cite journal |first1=J. |last1=Vannimenus |first2=G. |last2=Toulouse |title=Theory of the frustration effect. II. Ising spins on a square lattice |journal=Journal of Physics C: Solid State Physics |year=1977 |volume=10 |issue=18 |pages=537 |doi=10.1088/0022-3719/10/18/008 |bibcode=1977JPhC...10L.537V }}</ref> Quantitatively it is identical with the loop product <math>P_W:\,=\,J_{i,k}J_{k,l}...J_{n,m}J_{m,i}</math>  along a closed loop ''W''. However, for a Mattis spin glass – in contrast to "genuine" spin glasses – the quantity ''P<sub>W</sub>'' 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,<ref>{{Cite journal|last=Fradkin|first=Eduardo|date=1978|title=Gauge symmetries in random magnetic systems|journal=Physical Review B|volume=18|issue=9|pages=4789–4814|doi=10.1103/physrevb.18.4789| bibcode=1978PhRvB..18.4789F| osti=1446867|url=https://www.slac.stanford.edu/cgi-bin/getdoc/slac-pub-2112.pdf}}</ref> which also stresses the notion of [[Kramers–Wannier duality|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 [[entropic force|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]]" means in the polymer analogy simply the fact that in the short-wave limit, i.e. for <math>0\leftarrow\lambda_w\ll R_c</math>  (where ''R<sub>c</sub>'' is a characteristic correlation length for the glued loops, corresponding to the above-mentioned "bag radius", while λ<sub>w</sub> is the wavelength of an excitation) any non-trivial correlation vanishes totally, as if the system had crystallized.<ref>{{cite journal |first1=A. |last1=Bergmann |first2=A. |last2=Owen |title=Dielectric relaxation spectroscopy of poly[(R)-3-Hydroxybutyrate] (PHD) during crystallization |journal=Polymer International |volume=53 |issue=7 |year=2004 |pages=863–868 |doi=10.1002/pi.1445 }}</ref>

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]]s: there the magnetism is confined to the interior of the [[Abrikosov vortex|Abrikosov flux-line lattice]],<ref>Mathematically, the flux-line lattices are described by [[Emil Artin]]'s braid group, which is nonabelian, since one braid can wind around another one.</ref> i.e., the London penetration depth ''λ'' of that  theory is analogous to the confinement radius ''R<sub>c</sub>'' of  quantum chromodynamics. Mathematically, this correspondendence is supported by the second term, <math>\propto  g G^a_\mu \bar{\psi}_i \gamma^\mu T^a_{ij} \psi_j\,,</math> on the r.h.s. of the Lagrangian.

==See also==
{{Portal|Physics}}
* For overviews:
** [[Standard Model]]
** [[Strong interaction]]
** [[Quark]]
** [[Gluon]]
** [[Hadron]]
** [[Color confinement]]
** [[QCD matter]]
** [[Quark–gluon plasma]]
* For details:
** [[Gauge theory]]
** [[Quantum gauge theory]], [[BRST quantization]] and [[Faddeev–Popov ghost]]
** [[Quantum field theory]] – a more general category
* For techniques:
** [[Lattice QCD]]
** [[1/N expansion]]
** [[Perturbative QCD]]
** [[Soft-collinear effective theory]]
** [[Heavy quark effective theory]]
** [[Chiral model]]
** [[Nambu–Jona-Lasinio model]]
* For experiments:
** [[Deep inelastic scattering]]
** [[Jet (particle physics)]]
** [[Quark–gluon plasma]]
* [[Quantum electrodynamics]]
* [[Symmetry in quantum mechanics]]
* [[Yang–Mills theory]]
* [[Yang–Mills existence and mass gap]]

==References==
{{reflist|35em}}

==Further reading==
* {{cite book|author1=Greiner|first=Walter|author2=Schramm, Stefan|last3=Stein, Eckart|url=https://www.springer.com/gp/book/9783540485346#otherversion=9783540485353|title=Quantum Chromodynamics|publisher=Springer|year=2007|isbn=978-3-540-48535-3|location=Berlin Heidelberg}}
* {{cite book |author1=Halzen, Francis |author-link1=Francis Halzen |author2=Martin, Alan |author-link2=Alan Martin (physicist) |title=Quarks & Leptons: An Introductory Course in Modern Particle Physics |publisher=John Wiley & Sons |year=1984 |isbn=978-0-471-88741-6 |url-access=registration |url=https://archive.org/details/quarksleptonsint0000halz }}
* {{cite book | author=Creutz, Michael | title=Quarks, Gluons and Lattices | publisher=Cambridge University Press | year=1985 | isbn = 978-0-521-31535-7}}
* {{Cite journal |last1=Gross |first1=Franz |last2=Klempt |first2=Eberhard |last3=Brodsky |first3=Stanley J. |last4=Buras |first4=Andrzej J. |last5=Burkert |first5=Volker D. |last6=Heinrich |first6=Gudrun |last7=Jakobs |first7=Karl |last8=Meyer |first8=Curtis A. |last9=Orginos |first9=Kostas |last10=Strickland |first10=Michael |last11=Stachel |first11=Johanna |last12=Zanderighi |first12=Giulia |last13=Brambilla |first13=Nora |last14=Braun-Munzinger |first14=Peter |last15=Britzger |first15=Daniel |date=2023-12-12 |title=50 Years of quantum chromodynamics: Introduction and Review |url=https://link.springer.com/10.1140/epjc/s10052-023-11949-2 |journal=The European Physical Journal C |language=en |volume=83 |issue=12 |page=1125 |doi=10.1140/epjc/s10052-023-11949-2 |issn=1434-6052|arxiv=2212.11107 |bibcode=2023EPJC...83.1125G }} A highly technical review with almost 5000 references.

==External links==
*{{cite journal|title=QCD made simple|author=Frank Wilczek|journal=Physics Today|volume=53|issue=8|pages=22–28|doi=10.1063/1.1310117|year=2000|url=http://frankwilczek.com/Wilczek_Easy_Pieces/298_QCD_Made_Simple.pdf|bibcode = 2000PhT....53h..22W }}
*[https://pdg.lbl.gov Particle data group]
*[http://www.claymath.org/millennium-problems The millennium prize] for [https://web.archive.org/web/20131215120031/http://www.claymath.org/millenium-problems/yang%E2%80%93mills-and-mass-gap proving confinement]
*[https://arxiv.org/abs/0906.3599 Ab Initio Determination of Light Hadron Masses]
*[https://lss.fnal.gov/archive/test-fn/0000/fermilab-fn-0828-t.pdf Andreas S Kronfeld] ''The Weight of the World Is Quantum Chromodynamics''
*[https://arxiv.org/abs/0807.2220 Andreas S Kronfeld] ''Quantum chromodynamics with advanced computing''
*[https://web.archive.org/web/20090416132731/http://sciencenews.org/view/generic/id/38788/title/Standard_model_gets_right_answer_for_proton,_neutron_masses Standard model gets right answer]
*[https://arxiv.org/abs/hepph/9505231 Quantum Chromodynamics]
*[https://cerncourier.com/a/the-history-of-qcd/ Cern Courier, The history of QCD with] [[Harald Fritzsch|Prof. Dr. Harald Fritzsch]]

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