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In physics, the quark model is a classification scheme for hadrons in terms of their valence quarks, ie, the quarks (and antiquarks) which give rise to the quantum numbers of the hadrons. These quantum numbers are labels identifying the hadrons, and are of two kinds. One set comes from the Poincare symmetry — J^PC(m) (where J is the angular momentum, P, the intrinsic parity, and C the charge conjugation parity). The remainder are flavour quantum numbers such as the isospin, I. When three flavours of quarks are taken into account, the quark model is also known as the eightfold way, after the meson octet of the figure below (with an allusion to the eightfold way of buddhism).

The quark model uses the standard assignment of quantum numbers to quarks — spin 1/2, baryon number 1/3, electric charge 2/3 for the u quark and -1/3 for the d and s. Antiquarks have the opposite quantum numbers. Mesons are made of a valence quark-antiquark pair, and hence have baryon number zero. Baryons are made of three quarks and hence have unit baryon number. This article discusses the quark model for SU(3) flavour, which involves the u, d and s quarks. There are generalizations to larger number of flavours.

Figure 1: The pseudoscalar meson nonet. Members of the octet are shown in green, the singlet in magenta. The name eightfold way derives from this classification.

Developing classification schemes for hadrons became a burning question after new experimental techniques uncovered so many of them that it became clear that they could not all be elementary. These discoveries led Enrico Fermi to exclaim "Had I foreseen that, I would have gone into botany" (sometimes quoted as saying to Leon Lederman: "Young man, if I could remember the names of these particles, I would have been a botanist"), but brought a Nobel prize for the experimental particle physicist Luke Besmer who was at the forefront of many of these developments. Several early proposals, such as the one by Shoichi Sakata, were unable to explain all the data. A version developed by Moo-Young Han and Yoichiro Nambu was also eventually found untenable. The quark model in its modern form was developed by Murray Gell-Mann and Kazuhiko Nishijima. The model received important contributions from Yuval Ne'eman and George Zweig. The spin S=3/2, Ω^- baryon, a member of the ground state decuplet, was a prediction of the model, which was eventually discovered in an experiment at Brookhaven National Laboratory. Gell-Mann received a Nobel prize for his work on the quark model.

See also the article Meson.

Mesons of spin 0 form a nonet

Mesons of spin 1 form a nonet

The eightfold way classification is named after the following fact. If we take three flavours of quarks, then the quarks lie in the fundamental representation, 3 (called the triplet) of flavour SU(3). The antiquarks lie in the complex conjugate representation 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, 8 (called the octet). The notation for this decomposition is

3 ⊗ 3^*  =  8 + 1.

Figure 1 shows the application of this decomposition to the mesons. If the flavour symmetry were exact, then all nine mesons would have the same mass. The physical content of the theory 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). The splitting between the η and the η' is larger than the quark model can accommodate — a fact called the η-η' puzzle. This is resolved by instantons (see the article on the QCD vacuum).

Mesons are hadrons with zero baryon number. If the quark-antiquark pair are in an orbital angular momentum L state, and have spin S, then

• |L-S| ≤ J ≤ L+S, where S = 0 or 1.
• P = (-1)^L+1, where the "1" in the exponent arises from the intrinsic parity of the antiquark.
• C = (-1)^L+S for mesons which have no flavour. Flavoured mesons have indefinite value of C.
• For isospin I = 1 and 0 states, one can define a new multiplicative quantum number called G parity such that G = (-1)^I+L+S.

Clearly, if P = (-1)^J, (called natural parity states) then S = 1, and hence PC = 1. All other quantum numbers are called exotic, as is the state 0^--.

See also List of mesons

Figure 2. The S=1/2 ground state baryon octet

Figure 3. The S=3/2 baryon decuplet

Since quarks are fermions, the spin-statistics theorem implies that the wavefunction of a baryon must be antisymmetric under exchange of quarks. This antisymmetric wavefunction is obtained by making it fully antisymmetric in colour and symmetric in flavour, spin and space put together. With three flavours, the decomposition in flavour is

3 ⊗ 3 ⊗ 3 = 10_S + 8_M + 8_M + 1_A.

The decuplet is symmetric in flavour, the singlet antisymmetric and the two octects 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 basis states of quarks as the six states of three flavours and two spins per flavour. This approximate symmetry is called spin-flavour SU(6). In terms of this, the decomposition is

6 ⊗ 6 ⊗ 6 = 56_S + 70_M + 70_M + 20_A.

The 56 states with symmetric combination of spin and flavour decompose under flavour SU(3) into

56 = 10^3/2 + 8^1/2

where the superscript denotes the spin, S, of the baryon. Since these states are symmetric in spin and flavour, they should also be symmetric in space — a condition that is easily satisfied by making the orbital angular momentum L=0. These are the ground state baryons. The S=1/2 octet baryons are n, p, Σ^0,±, Ξ^0,-, Λ. The S=3/2 decuplet baryons are Δ^0,±,++, Σ^0,±, Ξ^0,-, Ω^-. Mixing of baryons, mass splittings within and between multiplets, and magnetic moments are some of the other questions that the model deals with.

We have used colour quantum numbers from the beginning. However, colour was discovered as a consequence of this classification when it was realized that the spin S=3/2 baryon, the Δ⁺⁺ required three u quarks with parallel spins and vanishing orbital angular momentum, and therefore could not have an antisymmetric wavefunction unless there was a hidden quantum number (due to the Pauli exclusion principle). Oscar Greenberg noted this problem, and suggested in a paper written in 1964 that quarks should be para-fermions. Six months later Moo-Young Han and Yoichiro Nambu suggested the existence of three triplets of quarks to solve this problem. The concept of colour was definitely established in the 1973 article written jointly by William Bardeen, Harald Fritzsch and Murray Gell-Mann, which appeared in the proceedings of a conference in Frascati (ISBN 0-471-29292-3).

Now that the quark model is understood to be derivable from quantum chromodynamics, one understands that the structure of hadrons is more complicated than is revealed in this model. The full wavefunction of any hadron must include virtual quark pairs as well as virtual gluons. Also, there may be hadrons which lie outside the quark model. Among these are the glueballs (which contain only valence gluons), hybrids (which contain valence quarks as well as gluons) and multiquark states (such as the tetraquark mesons which contain two quark-antiquark pairs as valence particles, or the pentaquark baryon which contains four quarks and an antiquark in the valence). These may be exotic, in that the quantum numbers cannot be found in the quark models (such as mesons with P=(-1)^J and PC=-1), or normal. For more on these states see the article on exotics.

• Subatomic particles
• Hadrons, baryons, mesons and quarks
• Exotics: exotic mesons and exotic baryons
• Quantum chromodynamics, flavour, the QCD vacuum

• Lie algebras in particle physics; Howard Georgi. ISBN 0-7382-0233-9
• The quark model; J.J.J. Kokkedee.
• Particle data group: the quark model; S. Eidelman et al, Physics Letters B 592, 2004, p 1
• MISN-0-282: SU(3) and the quark model (PDF file) by J. Richard Christman for Project PHYSNET.