Inparticle physics, thequark model is a classification scheme forhadrons in terms of their valencequarks—the quarks and antiquarks that give rise to thequantum numbers of the hadrons. The quark model underlies"flavor SU(3)", or theEightfold Way, the successfulclassification scheme organizing the large number of lighterhadrons that were being discovered starting in the 1950s and continuing through the 1960s. It received experimentalverification beginning in the late 1960s and is a valid and effective classification of them to date. The model was independently proposed by physicistsMurray Gell-Mann,^[1] who dubbed them "quarks" in a concise paper, andGeorge Zweig,^[2][3] who suggested "aces" in a longer manuscript.André Petermann also touched upon the central ideas from 1963 to 1965, without as much quantitative substantiation.^[4][5] Today, the model has essentially been absorbed as a component of the establishedquantum field theory of strong and electroweak particle interactions, dubbed theStandard Model.

Hadrons are not really "elementary", and can be regarded as bound states of their "valence quarks" and antiquarks, which give rise to thequantum numbers of the hadrons. These quantum numbers are labels identifying the hadrons, and are of two kinds. One set comes from thePoincaré symmetry—J^PC, whereJ,P andC stand for thetotal angular momentum,P-symmetry, andC-symmetry, respectively.

The other set is theflavor quantum numbers such as theisospin,strangeness,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 abaryon number of⁠1/3⁠.Up,charm andtop quarks have anelectric charge of +⁠2/3⁠, while thedown,strange, andbottom quarks have an electric charge of −⁠1/3⁠. Antiquarks have the opposite quantum numbers. Quarks arespin-⁠1/2⁠ particles, and thusfermions. Each quark or antiquark obeys the Gell-Mann–Nishijima formula individually, so any additive assembly of them will as well.

Mesons are made of a valence quark–antiquark pair (thus have a baryon number of 0), whilebaryons 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 flavorSU(3) symmetry). There are generalizations to larger number of flavors.

Developing classification schemes forhadrons 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 ledWolfgang Pauli to exclaim "Had I foreseen that, I would have gone into botany." andEnrico Fermi to advise his studentLeon 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, includingLuis 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 byEnrico Fermi andChen-Ning Yang (1949), and theSakata model (1956), ended up satisfactorily covering the mesons, but failed with baryons, and so were unable to explain all the data.

TheGell-Mann–Nishijima formula, developed byMurray Gell-Mann andKazuhiko Nishijima, led to theEightfold Way classification, invented by Gell-Mann, with important independent contributions fromYuval 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. TheGell-Mann–Okubo mass formula systematized the quantification of these small mass differences among members of a hadronic multiplet, controlled by theexplicit symmetry breaking of SU(3).

The spin-⁠3/2⁠
Ω⁻
baryon, a member of the ground-state decuplet, was a crucial prediction of that classification. After it was discovered in an experiment atBrookhaven National Laboratory, Gell-Mann received aNobel Prize in Physics for his work on the Eightfold Way, in 1969.

Finally, in 1964, Gell-Mann andGeorge Zweig, discerned independently what the Eightfold Way picture encodes: They posited three elementary fermionic constituents—the "up", "down", and "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 (SeeQuarks). 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 ofquantum 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.

The Eightfold Way classification is named after the following fact: If we take three flavors of quarks, then the quarks lie in thefundamental representation,3 (called the triplet) offlavorSU(3). The antiquarks lie in the complex conjugate representation3. The nine states (nonet) made out of a pair can be decomposed into thetrivial representation,1 (called the singlet), and theadjoint representation,8 (called the octet). The notation for this decomposition is

${\displaystyle \mathbf{3}\otimes \overline{\mathbf{3}}=\mathbf{8}\oplus \mathbf{1}.}$

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^{[clarification needed]} 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
η
and the
η′
is larger than the quark model can accommodate, and this "
η
–
η′
puzzle" has its origin in topological peculiarities of the strong interaction vacuum, such asinstanton configurations.

Mesons are hadrons with zerobaryon number. If the quark–antiquark pair are in anorbital angular momentumL state, and havespinS, then

• |L −S| ≤J ≤L +S, whereS = 0 or 1,
• P = (−1)^L+1, where the 1 in the exponent arises from theintrinsic parity of the quark–antiquark pair.
• C = (−1)^L+S for mesons which have noflavor. Flavored mesons have indefinite value ofC.
• ForisospinI = 1 and 0 states, one can define a newmultiplicative quantum number called theG-parity such thatG = (−1)^I+L+S.

IfP = (−1)^J, then it follows thatS = 1, thusPC = 1. States with these quantum numbers are callednatural parity states; while all other quantum numbers are thus calledexotic (for example, the stateJ^PC = 0⁻⁻).

Since quarks arefermions, thespin–statistics theorem implies that thewavefunction 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$${\displaystyle \mathbf{3}\otimes \mathbf{3}\otimes \mathbf{3}={\mathbf{10}}_S\oplus {\mathbf{8}}_M\oplus {\mathbf{8}}_M\oplus {\mathbf{1}}_A.}$$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 thebasis states of quarks as the six states of three flavors and two spins per flavor. This approximate symmetry is called spin-flavorSU(6). In terms of this, the decomposition is$${\displaystyle \mathbf{6}\otimes \mathbf{6}\otimes \mathbf{6}={\mathbf{56}}_S\oplus {\mathbf{70}}_M\oplus {\mathbf{70}}_M\oplus {\mathbf{20}}_A.}$$

The 56 states with symmetric combination of spin and flavour decompose under flavorSU(3) into$${\displaystyle \mathbf{56}={\mathbf{10}}^{\frac{3}{2}}\oplus {\mathbf{8}}^{\frac{1}{2}},}$$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 momentumL = 0. These are the ground-state baryons.

TheS =⁠1/2⁠ octet baryons are the twonucleons (
p⁺
,
n⁰
), the threeSigmas (
Σ⁺
,
Σ⁰
,
Σ⁻
), the twoXis (
Ξ⁰
,
Ξ⁻
), and theLambda (
Λ⁰
). TheS =⁠3/2⁠ decuplet baryons are the fourDeltas (
Δ⁺⁺
,
Δ⁺
,
Δ⁰
,
Δ⁻
), threeSigmas (
Σ^∗+
,
Σ^∗0
,
Σ^∗−
), twoXis (
Ξ^∗0
,
Ξ^∗−
), and theOmega (
Ω⁻
).

For example, the constituent quark model wavefunction for the proton is$${\displaystyle |{\text{p}}_{\uparrow }⟩=\frac{1}{\sqrt{18}}[2|{\text{u}}_{\uparrow }{\text{d}}_{\downarrow }{\text{u}}_{\uparrow }⟩+2|{\text{u}}_{\uparrow }{\text{u}}_{\uparrow }{\text{d}}_{\downarrow }⟩+2|{\text{d}}_{\downarrow }{\text{u}}_{\uparrow }{\text{u}}_{\uparrow }⟩-|{\text{u}}_{\uparrow }{\text{u}}_{\downarrow }{\text{d}}_{\uparrow }⟩-|{\text{u}}_{\uparrow }{\text{d}}_{\uparrow }{\text{u}}_{\downarrow }⟩-|{\text{u}}_{\downarrow }{\text{d}}_{\uparrow }{\text{u}}_{\uparrow }⟩-|{\text{d}}_{\uparrow }{\text{u}}_{\downarrow }{\text{u}}_{\uparrow }⟩-|{\text{d}}_{\uparrow }{\text{u}}_{\uparrow }{\text{u}}_{\downarrow }⟩-|{\text{u}}_{\downarrow }{\text{u}}_{\uparrow }{\text{d}}_{\uparrow }⟩].}$$

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).^[6]

Then, the proton wavefunction can be written in a simpler form:

${\displaystyle \text{p}\left(\frac{1}{2},\frac{1}{2}\right)=\frac{\text{u}\text{u}\text{d}}{\sqrt{6}}[2\uparrow \uparrow \downarrow -\uparrow \downarrow \uparrow -\downarrow \uparrow \uparrow ]}$

and the

${\displaystyle {\mathrm{\Delta }}^{+}\left(\frac{3}{3},\frac{3}{2}\right)=\text{u}\text{u}\text{d}[\uparrow \uparrow \uparrow ].}$

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.

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 spinS =⁠3/2⁠ baryon, the
Δ⁺⁺
, required three up quarks with parallel spins and vanishing orbital angular momentum. Therefore, it could not have an antisymmetric wavefunction, (required by thePauli exclusion principle).Oscar Greenberg noted this problem in 1964, suggesting that quarks should bepara-fermions.^[7]

Instead, six months later,Moo-Young Han andYoichiro 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.^[8]

The modern concept of color completely commuting with all other charges and providing the strong force charge was articulated in 1973, byWilliam Bardeen,Harald Fritzsch, andMurray Gell-Mann.^[9][10]

While the quark model is derivable from the theory ofquantum chromodynamics, the structure of hadrons is more complicated than this model allows. The fullquantum mechanicalwavefunction of any hadron must include virtual quark pairs as well as virtualgluons, and allows for a variety of mixings. There may be hadrons which lie outside the quark model. Among these are theglueballs (which contain only valence gluons),hybrids (which contain valence quarks as well as gluons) andexotic hadrons (such astetraquarks orpentaquarks).

• Subatomic particles
• Hadrons,baryons,mesons andquarks
• Exotic hadrons:exotic mesons andexotic baryons
• Quantum chromodynamics,flavor, theQCD vacuum

• S. Eidelmanet al.Particle Data Group (2004)."Review of Particle Physics"(PDF).Physics Letters B.592 (1–4): 1.arXiv:astro-ph/0406663.Bibcode:2004PhLB..592....1P.doi:10.1016/j.physletb.2004.06.001.S2CID 118588567.
• Lichtenberg, D B (1970).Unitary Symmetry and Elementary Particles. Academic Press.ISBN 978-1483242729.
• Thomson, M A (2011),Lecture notes
• J.J.J. Kokkedee (1969).The quark model.W. A. Benjamin.ASIN B001RAVDIA.

• Hadrons
• Quarks
• Particle physics
• Concepts in physics
• Standard Model
• Murray Gell-Mann

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• Wikipedia articles needing clarification from February 2016