Quark model

Mesons

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 flavor 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

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

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 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 as instanton configurations.

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 quark–antiquark pair.
• C = (−1)^L+S for mesons which have no flavor. Flavored 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 thus called exotic (for example, the state J^PC = 0⁻⁻).

Baryons

Since quarks are fermions, 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 $${\displaystyle \mathbf{3}\otimes \mathbf{3}\otimes \mathbf{3}={\mathbf{10}}_S\oplus {\mathbf{8}}_M\oplus {\mathbf{8}}_M\oplus {\mathbf{1}}_A.}$$ 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 the 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 $${\displaystyle \mathbf{6}\otimes \mathbf{6}\otimes \mathbf{6}={\mathbf{56}}_S\oplus {\mathbf{70}}_M\oplus {\mathbf{70}}_M\oplus {\mathbf{20}}_A.}$$

The 56 states with symmetric combination of spin and flavour decompose under flavor SU(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 momentum L = 0. These are the ground-state baryons.

The S = ⁠1/2⁠ 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 = ⁠3/2⁠ 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⁻
).

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 three components of a single particle, so the anti-symmetrization applies to all the quarks. A simpler approach is to consider the three flavored quarks as separate, distinguishable, non-identical particles. Then the anti-symmetrization applies only to two identical quarks (like uu, for instance).

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.

Discovery of color

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 S = ⁠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 the Pauli exclusion principle). Oscar Greenberg noted this problem in 1964, suggesting that quarks should be para-fermions.

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.

The modern concept of color completely commuting with all other charges and providing the strong force charge was articulated in 1973, by William Bardeen, Harald Fritzsch, and Murray Gell-Mann.

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 mechanical wavefunction of any hadron must include virtual quark pairs as well as virtual gluons, and allows for a variety of mixings. 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).