Quark model

Developing classification schemes for hadrons 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 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. Constructing hadrons as bound states of fewer constituents would thus organize the "zoo" at hand. Several early proposals, such as the ones by Enrico Fermi and Chen-Ning Yang (1949), and the Sakata model (1956), ended up satisfactorily covering the mesons, but failed with baryons, and so were unable to explain all the data.

The Gell-Mann–Nishijima formula, developed by Murray Gell-Mann and Kazuhiko Nishijima, led to the Eightfold Way classification, invented by Gell-Mann, with important independent contributions from Yuval 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. The Gell-Mann–Okubo mass formula systematized the quantification of these small mass differences among members of a hadronic multiplet, controlled by the explicit 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 at Brookhaven National Laboratory, Gell-Mann received a Nobel prize in physics for his work on the Eightfold Way, in 1969.

Finally, in 1964, Gell-Mann, and, independently, George Zweig, discerned 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 (See Quarks). 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 of quantum 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.

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 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

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

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⁻⁻).

Figure 4. The S = 1/2 ground state baryon octet

Figure 5. 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 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

$${\displaystyle |{\text{p}}_{\uparrow }\rangle ={\frac {1}{\sqrt {18}}}[2|{\text{u}}_{\uparrow }{\text{d}}_{\downarrow }{\text{u}}_{\uparrow }\rangle +2|{\text{u}}_{\uparrow }{\text{u}}_{\uparrow }{\text{d}}_{\downarrow }\rangle +2|{\text{d}}_{\downarrow }{\text{u}}_{\uparrow }{\text{u}}_{\uparrow }\rangle -|{\text{u}}_{\uparrow }{\text{u}}_{\downarrow }{\text{d}}_{\uparrow }\rangle -|{\text{u}}_{\uparrow }{\text{d}}_{\uparrow }{\text{u}}_{\downarrow }\rangle -|{\text{u}}_{\downarrow }{\text{d}}_{\uparrow }{\text{u}}_{\uparrow }\rangle -|{\text{d}}_{\uparrow }{\text{u}}_{\downarrow }{\text{u}}_{\uparrow }\rangle -|{\text{d}}_{\uparrow }{\text{u}}_{\uparrow }{\text{u}}_{\downarrow }\rangle -|{\text{u}}_{\downarrow }{\text{u}}_{\uparrow }{\text{d}}_{\uparrow }\rangle ]~.}$$

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 wave function can be written in the 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

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

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.

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 wave function 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).

• Subatomic particles
• Hadrons, baryons, mesons and quarks
• Exotic hadrons: exotic mesons and exotic baryons
• Quantum chromodynamics, flavor, the QCD vacuum

• S. Eidelman et 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.