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.
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 Wolfgang Pauli 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 Luis Alvarez 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 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.
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
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--. A List of mesons is available.
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
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
The 56 states with symmetric combination of spin and flavour decompose under flavour SU(3) into
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, Σ0,±, Ξ0,-, Λ. The S=3/2 decuplet baryons are Δ0,±,++, Σ0,±, Ξ0,-, Ω-. Mixing of baryons, mass splittings within and between multiplets, and magnetic moments are some of the other questions that the model deals with.
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