# Quark model

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

In physics, the quark model is a classification scheme for hadrons in terms of their valence quarks—the quarks and antiquarks which give rise to the quantum numbers of the hadrons.

The quark model was originally just a very good classification scheme to organize the depressingly large number of hadrons that were being discovered starting in the 1950s and continuing through the 1960s but it received experimental verification beginning in the late 1960s and continuing to the present. Hadrons are not "fundamental", but their "valence quarks" are thought to be, 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 Poincaré symmetryJPC, where J, P and C stand for the total angular momentum, P-symmetry, and C-symmetry respectively. The remainder are flavour quantum numbers such as the isospin, strangeness, charm, and so on. The quark model is the follow-up to the Eightfold Way classification scheme.

All quarks are assigned a baryon number of 13. Up, charm and top quarks have an electric charge of +23, while the down, strange, and bottom quarks have an electric charge of −13. Antiquarks have the opposite quantum numbers. Quarks are also spin-12 particles, meaning they are fermions.

Mesons are made of a valence quark−antiquark pair (thus have a baryon number of 0), while baryons are made of three quarks (thus have a baryon number of 1). This article discusses the quark model for the up, down, and strange flavours of quark (which form an approximate SU(3) symmetry). There are generalizations to larger number of flavours.

## History

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," 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 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 32 Ω baryon, a member of the ground state decuplet, was a prediction of the model. When it was discovered in an experiment at Brookhaven National Laboratory, Gell-Mann received a Nobel prize for his work on the quark model.

## Mesons

Figure 2: Pseudoscalar mesons of spin 0 form a nonet
Figure 3: 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

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

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. This "ηη′ puzzle" is resolved by instantons.

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

• |LS| ≤ JL + S, where S = 0 or 1,
• P = (−1)L + 1, where the 1 in the exponent arises from the intrinsic parity of the quark–antiquark pair.
• 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 the G-parity such that G = (−1)I + L + S.

If P = (−1)J, 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 called exotic (for example the state JPC = 0−−).

## Baryons

Figure 4. The S = 12 ground state baryon octet
Figure 5. The S = 32 baryon decuplet

Since quarks are fermions, the spin-statistics theorem implies that the wavefunction of a baryon must be antisymmetric under exchange of any two 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

$\mathbf{3}\otimes\mathbf{3}\otimes\mathbf{3}=\mathbf{10}_S\oplus\mathbf{8}_M\oplus\mathbf{8}_M\oplus\mathbf{1}_A$.

The decuplet is symmetric in flavour, 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 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

$\mathbf{6}\otimes\mathbf{6}\otimes\mathbf{6}=\mathbf{56}_S\oplus\mathbf{70}_M\oplus\mathbf{70}_M\oplus\mathbf{20}_A$

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

$\mathbf{56}=\mathbf{10}^\frac{3}{2}\oplus\mathbf{8}^\frac{1}{2}$

where the superscript denotes the spin, S, of the baryon. Since these states are symmetric in spin and 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 = 12 octet baryons are the two nucleons (p+, n0), the three Sigmas (Σ+, Σ0, Σ), the two Xis (Ξ0, Ξ), and the Lambda (Λ0). The S = 32 decuplet baryons are the four Deltas (Δ++, Δ+, Δ0, Δ), three Sigmas (Σ∗+, Σ∗0, Σ∗−), two Xis (Ξ∗0, Ξ∗−), and the Omega (Ω). Mixing of baryons, mass splittings within and between multiplets, and magnetic moments are some of the other questions that the model deals with.

### The discovery of colour

Colour quantum numbers have been used from the beginning. However, colour was discovered as a consequence of this classification when it was realized that the spin S = 32 baryon, the Δ++ required three up 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 in 1964, suggesting that quarks should be para-fermions.[1] Six months later Moo-Young Han and Yoichiro Nambu suggested the existence of three triplets of quarks to solve this problem.[2] The concept of colour was definitely established in the 1973 article written jointly by William Bardeen, Harald Fritzsch and Murray Gell-Mann.[3]

## 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 reveals. The full quantum mechanical wave function 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 "exotic hadrons" (such as tetraquarks or pentaquarks).