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{{short description|Classification scheme of hadrons}}
[[File:8foldway.svg|thumb|right|'''Figure 1''': The [[pseudoscalar meson]] nonet. Members of the original meson "octet" are shown in green, the singlet in magenta. Although these mesons are now grouped into a nonet, the [[Eightfold way (physics)|Eightfold Way]] name derives from the patterns of eight for the mesons and baryons in the original classification scheme.]]

In [[particle physics]], the '''quark model''' is a classification scheme for [[hadron]]s in terms of their valence [[quark]]s—the quarks and antiquarks that give rise to the [[quantum number]]s of the hadrons. The quark model  underlies [[Flavour (particle physics)|"flavor SU(3)"]], or the [[Eightfold way (physics)|Eightfold Way]], the successful [[classification scheme (information science)|classification scheme]] organizing the large number of lighter [[hadron]]s that were being discovered starting in the 1950s and continuing through the 1960s. It received experimental [[verification and validation|verification]] beginning in the late 1960s and is a valid and effective classification of them to date. The model was independently proposed by physicists [[Murray Gell-Mann]],<ref name="Gell-Man1964">
{{cite journal
 |last=Gell-Mann |first=M. |author-link=Murray Gell-Mann
 |date=4 January 1964
 |title=A Schematic Model of Baryons and Mesons |journal=[[Physics Letters]]
 |doi=10.1016/S0031-9163(64)92001-3 |bibcode = 1964PhL.....8..214G |volume=8 |number=3 |pages=214–215
}}</ref> who dubbed them "quarks" in a concise paper, and [[George Zweig]],<ref name="Zweig1964a">{{cite report |last=Zweig |first=G. |author-link=George Zweig |date=17 January 1964 |title=An SU(3) Model for Strong Interaction Symmetry and its Breaking  |id=CERN Report No.8182/TH.401 |url=https://cds.cern.ch/record/352337/files/CERN-TH-401.pdf}}</ref><ref name="Zweig1964b">{{cite report|last=Zweig |first=G. |author-link=George Zweig |year=1964 |title=An SU(3) Model for Strong Interaction Symmetry and its Breaking: II |id=CERN Report No.8419/TH.412 |url=https://cds.cern.ch/record/570209/files/CERN-TH-412.pdf}}</ref> who suggested "aces" in a longer manuscript. [[André Petermann]] also touched upon the central ideas from 1963 to 1965, without as much quantitative substantiation.<ref>{{cite journal |last=Petermann |first=A. |author-link=André Petermann |year=1965 |title=Propriétés de l'étrangeté et une formule de masse pour les mésons vectoriels |trans-title=Strangeness properties and a mass formula for vector meson |journal=[[Nuclear Physics (journal)|Nuclear Physics]] |doi=10.1016/0029-5582(65)90348-2 |bibcode=1965NucPh..63..349P |volume=63 |issue=2 |pages=349–352 |arxiv=1412.8681 }}</ref><ref>{{cite conference |last=Petrov |first=Vladimir A. |date=June 23–27, 2014 |title=Half a Century with QUARKS |conference=XXX-th International Workshop on High Energy Physics |location=[[Protvino]], [[Moscow Oblast]], Russia |arxiv=1412.8681 }}</ref> Today, the model has essentially been absorbed as a component of the established [[quantum field theory]] of strong and electroweak particle interactions, dubbed the [[Standard Model]].

Hadrons are not really "elementary", and can be regarded as bound states of their "valence quarks" and antiquarks, which give rise to the [[quantum number]]s of the hadrons. These quantum numbers are labels identifying the  hadrons, and are of two kinds. One set comes from the [[Poincaré symmetry]]—''J''<sup>''PC''</sup>, where ''J'', ''P'' and ''C'' stand for the [[total angular momentum]], [[P-symmetry]], and [[C-symmetry]], respectively.

The other set is the [[flavour quantum numbers|flavor quantum numbers]] such as the [[isospin]], [[strangeness]], [[charm (quantum number)|charm]], and so on. The strong interactions binding the quarks together are insensitive to these quantum numbers, so variation of them leads to systematic mass and coupling relationships among the hadrons in the same flavor multiplet.

All quarks are assigned a [[baryon number]] of {{sfrac|1|3}}. [[Up quark|Up]], [[charm quark|charm]] and [[top quark]]s have an [[electric charge]] of +{{sfrac|2|3}}, while the [[down quark|down]], [[strange quark|strange]], and [[bottom quark]]s have an electric charge of −{{sfrac|1|3}}. Antiquarks have the opposite quantum numbers. Quarks are [[spin-1/2|spin-{{sfrac|1|2}}]] particles, and thus [[fermion]]s. Each quark or antiquark obeys the Gell-Mann–Nishijima formula individually, so any additive assembly of them will as well.

[[Meson]]s are made of a valence quark–antiquark pair (thus have a baryon number of 0), while [[baryon]]s are made of three quarks (thus have a baryon number of 1). This article discusses the quark model for the up, down, and strange flavors of quark (which form an approximate flavor [[SU(3)|SU(3) symmetry]]). There are generalizations to larger number of flavors.

== History ==
Developing classification schemes for [[hadron]]s became a timely question after new experimental techniques uncovered so many of them that it became clear that they could not all be elementary. These discoveries led [[Wolfgang Pauli]] to exclaim "Had I foreseen that, I would have gone into botany." and [[Enrico Fermi]] to advise his student [[Leon Lederman]]: "Young man, if I could remember the names of these particles, I would have been a botanist." These new schemes earned Nobel prizes for experimental particle physicists, including [[Luis Walter Alvarez|Luis Alvarez]], who was at the forefront of many of these developments. Constructing hadrons as bound states of fewer constituents would thus organize the "zoo" at hand. Several early proposals, such as the ones by [[Enrico Fermi]] and [[Chen-Ning Yang]] (1949), and the [[Sakata model]] (1956), ended up satisfactorily covering the mesons, but failed with baryons, and so were unable to explain all the data.

The [[Gell-Mann–Nishijima formula]], developed by [[Murray Gell-Mann]] and [[Kazuhiko Nishijima]], led to the [[Eightfold way (physics)|Eightfold Way]] classification, invented by Gell-Mann, with important independent contributions from [[Yuval Ne'eman]], in 1961. The hadrons were organized into SU(3) representation multiplets, octets and decuplets, of roughly the same mass, due to the strong interactions; and smaller mass differences linked to the flavor quantum numbers, invisible to the strong interactions. The [[Gell-Mann–Okubo mass formula]] systematized the quantification of these small mass differences among members of a hadronic multiplet, controlled by the [[explicit symmetry breaking]] of SU(3).

The spin-{{sfrac|3|2}} [[Omega baryon|{{SubatomicParticle|Omega-}} baryon]], a member of the ground-state decuplet, was a crucial prediction of that classification. After it was discovered in an experiment at [[Brookhaven National Laboratory]], Gell-Mann received a [[Nobel Prize in Physics]] for his work on the Eightfold Way, in 1969.

Finally, in 1964, Gell-Mann and [[George Zweig]], discerned independently what the Eightfold Way picture encodes: They posited three elementary fermionic constituents—the "[[Up quark|up]]", "[[Down quark|down]]", and "[[Strange quark|strange]]" quarks—which are unobserved, and possibly unobservable in a free form. Simple pairwise or triplet combinations of these three constituents and their antiparticles underlie and elegantly encode the Eightfold Way classification, in an economical, tight structure, resulting in further simplicity. Hadronic mass differences were now linked to the different masses of the constituent quarks.

It would take about a decade for the unexpected nature—and physical reality—of these quarks to be appreciated more fully (See [[Quarks]]). Counter-intuitively, they cannot ever be observed in isolation ([[color confinement]]), but instead always combine with other quarks to form full hadrons, which then furnish ample indirect information on the trapped quarks themselves. Conversely, the quarks serve in the definition of [[quantum chromodynamics]], the fundamental theory fully describing the strong interactions; and the Eightfold Way is now understood to be a consequence of the flavor symmetry structure of the lightest three of them.

== Mesons ==
{{see also|Meson|List of mesons}}
[[File:Meson nonet - spin 0.svg|thumb|'''Figure 2''': [[Pseudoscalar meson]]s of spin-0 form a nonet ]]
[[File:Meson nonet - spin 1.svg|thumb|'''Figure 3''': [[Vector mesons]] of spin-1 form a nonet]]
The Eightfold Way classification is named after the following fact: If we take three flavors of quarks, then the quarks lie in the [[fundamental representation]], '''3''' (called the triplet) of [[flavour (particle physics)|flavor]] [[SU(3)]]. The antiquarks lie in the complex conjugate representation {{overline|'''3'''}}. The nine states (nonet) made out of a pair can be decomposed into the [[trivial representation]], '''1''' (called the singlet), and the [[Adjoint representation of a Lie group|adjoint representation]], '''8''' (called the octet). The notation for this decomposition is
: <math>\mathbf{3}\otimes \mathbf{\overline{3}} = \mathbf{8} \oplus \mathbf{1} ~.</math>

Figure 1 shows the application of this decomposition to the mesons. If the flavor symmetry were exact (as in the limit that only the strong interactions operate, but the electroweak interactions are notionally switched off), then all nine mesons would have the same mass. However, the physical content of the full theory{{clarify|date=February 2016}} includes consideration of the symmetry breaking induced by the quark mass differences, and considerations of mixing between various multiplets (such as the octet and the singlet).

N.B. Nevertheless, the mass splitting between the {{SubatomicParticle|Eta}} and the {{SubatomicParticle|Eta prime}} is larger than the quark model can accommodate, and  this "[[QCD vacuum#Eta prime meson|{{SubatomicParticle|Eta}}–{{SubatomicParticle|Eta prime}} puzzle]]" has its origin in topological peculiarities of the strong interaction vacuum, such as  [[instanton]] configurations.

Mesons are hadrons with zero [[baryon number]]. If the quark–antiquark pair are in an [[angular momentum operator|orbital angular momentum]] {{mvar|L}} state, and have [[Spin (physics)|spin]] {{mvar|S}}, then
* {{abs|''L'' − ''S''}} ≤ ''J'' ≤ ''L'' + ''S'', where ''S'' = 0 or 1,
* ''P'' = (−1)<sup>''L''+1</sup>, where the 1 in the exponent arises from the [[intrinsic parity]] of the quark–antiquark pair.
* ''C'' = (−1)<sup>''L''+''S''</sup> for mesons which have no [[flavour (particle physics)|flavor]]. Flavored mesons have indefinite value of [[C parity|''C'']].
* For [[isospin]] {{nowrap|1=''I'' = 1}} and 0 states, one can define a new [[multiplicative quantum number]] called the ''[[G-parity]]'' such that {{nowrap|1=''G'' = (−1)<sup>''I''+''L''+''S''</sup>}}.

If ''P'' = (−1)<sup>''J''</sup>, then it follows that ''S'' = 1, thus ''PC'' = 1. States with these quantum numbers are called ''natural parity states''; while all other quantum numbers are thus called ''exotic'' (for example, the state {{nowrap|1=''J''<sup>''PC''</sup> = 0<sup>−−</sup>}}).

== Baryons ==
{{main|Baryon}}
{{see also|List of baryons}}
[[File:Baryon octet.png|thumb|200px|right|'''Figure 4'''. The {{nowrap|1=''S'' = {{sfrac|1|2}}}} ground state [[baryon]] octet]] [[File:Baryon decuplet.png|thumb|200px|right|'''Figure 5'''. The {{nowrap|1=''S'' = {{sfrac|3|2}}}} [[baryon]] decuplet]]

Since quarks are [[fermion]]s, the [[spin–statistics theorem]] implies that the [[wavefunction]] of a baryon must be antisymmetric under the exchange of any two quarks. This antisymmetric wavefunction is obtained by making it fully antisymmetric in color, discussed below, and symmetric in flavor, spin and space put together. With three flavors, the decomposition in flavor is
<math display="block">\mathbf{3}\otimes\mathbf{3}\otimes\mathbf{3}=\mathbf{10}_S\oplus\mathbf{8}_M\oplus\mathbf{8}_M\oplus\mathbf{1}_A ~.</math>
The decuplet is symmetric in flavor, the singlet antisymmetric and the two octets have mixed symmetry. The space and spin parts of the states are thereby fixed once the orbital angular momentum is given.

It is sometimes useful to think of the [[quantum state#Basis states of one-particle systems|basis states]] of quarks as the six states of three flavors and two spins per flavor. This approximate symmetry is called spin-flavor [[SU(6)]]. In terms of this, the decomposition is
<math display="block">\mathbf{6}\otimes\mathbf{6}\otimes\mathbf{6}=\mathbf{56}_S\oplus\mathbf{70}_M\oplus\mathbf{70}_M\oplus\mathbf{20}_A ~.</math>

The 56 states with symmetric combination of spin and flavour decompose under flavor [[SU(3)]] into
<math display="block">\mathbf{56}=\mathbf{10}^\frac{3}{2}\oplus\mathbf{8}^\frac{1}{2} ~,</math>
where the superscript denotes the spin, ''S'', of the baryon. Since these states are symmetric in spin and flavor, they should also be symmetric in space—a condition that is easily satisfied by making the orbital angular momentum {{nowrap|1=''L'' = 0}}. These are the ground-state baryons.

The {{nowrap|1=''S'' = {{sfrac|1|2}}}} octet baryons are the two [[nucleon]]s ({{SubatomicParticle|Proton+}}, {{SubatomicParticle|Neutron0}}), the three [[Sigma baryon|Sigmas]] ({{SubatomicParticle|Sigma+}}, {{SubatomicParticle|Sigma0}}, {{SubatomicParticle|Sigma-}}), the two [[Xi baryon|Xis]] ({{SubatomicParticle|Xi0}}, {{SubatomicParticle|Xi-}}), and the [[Lambda baryon|Lambda]] ({{SubatomicParticle|Lambda0}}). The {{nowrap|1=''S'' = {{sfrac|3|2}}}} decuplet baryons are the four [[Delta baryon|Deltas]] ({{SubatomicParticle|Delta++}}, {{SubatomicParticle|Delta+}}, {{SubatomicParticle|Delta0}}, {{SubatomicParticle|Delta-}}), three [[Sigma baryon|Sigmas]] ({{SubatomicParticle|Sigma*+}}, {{SubatomicParticle|Sigma*0}}, {{SubatomicParticle|Sigma*-}}), two [[Xi baryon|Xis]] ({{SubatomicParticle|Xi*0}}, {{SubatomicParticle|Xi*-}}), and the [[Omega particle|Omega]] ({{SubatomicParticle|Omega-}}).

For example, the constituent quark model wavefunction for the proton is 
<math display="block">|\text{p}_\uparrow\rangle= \frac{1}{\sqrt {18}} [ 2| \text{u}_\uparrow \text{d}_\downarrow \text{u}_\uparrow \rangle + 2| \text{u}_\uparrow  \text{u}_\uparrow \text{d}_\downarrow \rangle +2| \text{d}_\downarrow \text{u}_\uparrow   \text{u}_\uparrow \rangle - | \text{u}_\uparrow \text{u}_\downarrow \text{d}_\uparrow\rangle  -| \text{u}_\uparrow \text{d}_\uparrow \text{u}_\downarrow\rangle  -| \text{u}_\downarrow \text{d}_\uparrow \text{u}_\uparrow\rangle  
-| \text{d}_\uparrow \text{u}_\downarrow \text{u}_\uparrow\rangle  -| \text{d}_\uparrow  \text{u}_\uparrow \text{u}_\downarrow\rangle  -| \text{u}_\downarrow \text{u}_\uparrow \text{d}_\uparrow\rangle ] ~.</math>

Mixing of baryons, mass splittings within and between multiplets, and magnetic moments are some of the other quantities that the model predicts successfully.

The group theory approach described above assumes that the quarks are eight components of a single particle, so the anti-symmetrization applies to all the quarks.  A simpler approach is to consider the eight flavored quarks as eight separate, distinguishable, non-identical particles.  Then the anti-symmetrization applies only to two identical quarks (like uu, for instance).<ref name="JF1968">{{cite journal |last=Franklin |first=J. |title=A Model of Baryons Made of Quarks with Hidden Spin |journal=[[Physical Review]] |volume=172 |number=3 |year=1968|pages=1807–1817|doi=10.1103/PhysRev.172.1807 |bibcode=1968PhRv..172.1807F }}</ref>
	
Then, the proton wavefunction can be written in a simpler form:
: <math>\text{p}\left(\frac{1}{2},\frac{1}{2}\right)=\frac{\text{u}\text{u}\text{d}}{\sqrt{6}}[2\uparrow\uparrow\downarrow-\uparrow\downarrow\uparrow-\downarrow\uparrow\uparrow] </math>
and the 
: <math>\Delta^{+}\left(\frac{3}{3},\frac{3}{2}\right)=\text{u}\text{u}\text{d}[\uparrow\uparrow\uparrow] ~.</math>

If quark–quark interactions are limited to two-body interactions, then all the successful quark model predictions, including sum rules for baryon masses and magnetic moments, can be derived.

=== Discovery of color ===
{{Main|Color charge}}
Color quantum numbers are the characteristic charges of the strong force, and are completely uninvolved in electroweak interactions. They were discovered as a consequence of the quark model classification, when it was appreciated that the spin {{nowrap|1=''S'' = {{sfrac|3|2}}}} baryon, the {{SubatomicParticle|Delta++}}, required three up quarks with parallel spins and vanishing orbital angular momentum. Therefore, it could not have an antisymmetric wavefunction, (required by the [[Pauli exclusion principle]]).  [[Oscar Greenberg]] noted this problem in 1964, suggesting that quarks should be [[para-fermion]]s.<ref>{{cite journal |author=Greenberg, O.W. |year=1964 |title=Spin and unitary-spin independence in a paraquark model of baryons and mesons |journal=[[Physical Review Letters]] |volume=13 |pages=598–602 |doi=10.1103/PhysRevLett.13.598 |bibcode=1964PhRvL..13..598G |issue=20}}</ref>

Instead, six months later, [[Moo-Young Han]] and [[Yoichiro Nambu]] suggested the existence of a hidden degree of freedom, they labeled as the group SU(3)' (but later called 'color). This led to three triplets of quarks whose wavefunction was anti-symmetric in the color degree of freedom.
Flavor and color were intertwined in that model: they did not commute.<ref>{{cite journal |author1=Han, M.Y. |author2=Nambu, Y. |year=1965 |title=Three-triplet model with double SU(3) symmetry |journal=[[Physical Review B]] |volume=139 |pages=1006 |doi=10.1103/PhysRev.139.B1006 |bibcode=1965PhRv..139.1006H |issue=4B |url=https://digital.library.unt.edu/ark:/67531/metadc1031342/|url-access=subscription }}</ref>

The modern concept of color completely commuting with all other charges and providing the strong force charge was articulated in 1973, by [[William A. Bardeen|William Bardeen]], [[:de:Harald Fritzsch|Harald Fritzsch]], and [[Murray Gell-Mann]].<ref>{{cite conference |author1=Bardeen, W. |author2=Fritzsch, H. |author3=Gell-Mann, M. |year=1973 |title=Light cone current algebra, ''π''<sup>0</sup> decay, and ''e''<sup>+</sup> ''e''<sup>&minus;</sup> annihilation |arxiv=hep-ph/0211388 |editor=Gatto, R. |book-title=Scale and conformal symmetry in hadron physics |page=[https://archive.org/details/scaleconformalsy0000unse/page/139 139] |publisher=[[John Wiley & Sons]] |isbn=0-471-29292-3 |bibcode=2002hep.ph...11388B |url-access=registration |url=https://archive.org/details/scaleconformalsy0000unse/page/139 }}</ref><ref>{{cite journal |title=Advantages of the color octet gluon picture |journal=[[Physics Letters B]] |volume=47 |issue=4 |page=365 |year=1973 |last1=Fritzsch |first1=H. |last2=Gell-Mann |first2=M. |last3=Leutwyler |first3=H. |doi=10.1016/0370-2693(73)90625-4 |bibcode=1973PhLB...47..365F |citeseerx=10.1.1.453.4712}}</ref>

== States outside the quark model ==
While the quark model is derivable from the theory of [[quantum chromodynamics]], the structure of hadrons is more complicated than this model allows. The full [[quantum mechanics|quantum mechanical]] [[wavefunction]] of any hadron must include virtual quark pairs as well as virtual [[gluon]]s, and allows for a variety of mixings. There may be hadrons which lie outside the quark model. Among these are the ''[[glueball]]s'' (which contain only valence gluons), ''hybrids'' (which contain valence quarks as well as gluons) and ''[[exotic hadron]]s''  (such as [[tetraquark]]s or [[pentaquark]]s).

== See also ==
* [[Subatomic particles]]
* [[Hadron]]s, [[baryon]]s, [[meson]]s and [[quark]]s
* [[Exotic hadron]]s: [[exotic meson]]s and [[exotic baryon]]s
* [[Quantum chromodynamics]], [[flavour (particle physics)|flavor]], the [[QCD vacuum]]

== Notes ==
{{reflist}}

== References ==
* {{cite journal
 |author=S. Eidelman ''et al.'' [[Particle Data Group]]
 |year=2004
 |url=http://pdg.lbl.gov/2004/reviews/quarkmodrpp.pdf
 |title=Review of Particle Physics
 |journal=[[Physics Letters B]]
 |volume=592 |issue=1–4
 |page=1
 |doi=10.1016/j.physletb.2004.06.001
|arxiv = astro-ph/0406663 |bibcode = 2004PhLB..592....1P |s2cid=118588567
 }}
* {{Cite book | isbn = 978-1483242729 | title = Unitary Symmetry and Elementary Particles | last1 = Lichtenberg | first1 = D B | year = 1970 | publisher = Academic Press  }}
* Thomson, M A (2011), [http://www.hep.phy.cam.ac.uk/~thomson/partIIIparticles/handouts/Handout_7_2011.pdf  Lecture notes]
* {{cite book
 |author=J.J.J. Kokkedee
 |year=1969
 |title=The quark model
 |url=https://archive.org/details/quarkmodel0000kokk
 |url-access=registration
 |publisher=[[W. A. Benjamin]]
 |asin=B001RAVDIA
}}

{{Particles}}
{{Standard model of physics}}

[[Category:Hadrons]]
[[Category:Quarks]]
[[Category:Particle physics]]
[[Category:Concepts in physics]]
[[Category:Standard Model]]
[[Category:Murray Gell-Mann]]