Quark model
In particle physics, the quark model is a classification scheme for hadrons in terms of their valence quarks—the quarks and antiquarks that give rise to the quantum numbers of the hadrons. The quark model underlies "flavor SU(3)", or the Eightfold Way, the successful classification scheme organizing the large number of lighter hadrons that were being discovered starting in the 1950s and continuing through the 1960s. It received experimental verification beginning in the late 1960s and is a valid and effective classification of them to date. The model was independently proposed by physicists Murray Gell-Mann,[1] who dubbed them "quarks" in a concise paper, and George Zweig,[2][3] who suggested "aces" in a longer manuscript. André Petermann also touched upon the central ideas from 1963 to 1965, without as much quantitative substantiation.[4][5] Today, the model has essentially been absorbed as a component of the established quantum field theory of strong and electroweak particle interactions, dubbed the Standard Model.
Hadrons are not really "elementary", and can be regarded as bound states of their "valence quarks" and antiquarks, which give rise to the
The other set is the
, and so on. The strong interactions binding the quarks together are insensitive to these quantum numbers, so variation of them leads to systematic mass and coupling relationships among the hadrons in the same flavor multiplet.All quarks are assigned a baryon number of 1/3. Up, charm and top quarks have an electric charge of +2/3, while the down, strange, and bottom quarks have an electric charge of −1/3. Antiquarks have the opposite quantum numbers. Quarks are spin-1/2 particles, and thus fermions. Each quark or antiquark obeys the Gell-Mann–Nishijima formula individually, so any additive assembly of them will as well.
History
Developing classification schemes for
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
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
Mesons
The Eightfold Way classification is named after the following fact: If we take three flavors of quarks, then the quarks lie in the
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[clarification needed] 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 JPC = 0−−).
Baryons
Since quarks are
It is sometimes useful to think of the
The 56 states with symmetric combination of spin and flavour decompose under flavor
The S = 1/2 octet baryons are the two
Ω−
).
For example, the constituent quark model wavefunction for the proton is
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 a simpler form:
and the
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 wave function, (required by the
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 wave function was anti-symmetric in the color degree of freedom. Flavor and color were intertwined in that model: they did not commute.[8]
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.[9][10]
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 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).
See also
- Subatomic particles
- Hadrons, baryons, mesons and quarks
- Exotic hadrons: exotic mesons and exotic baryons
- Quantum chromodynamics, flavor, the QCD vacuum
Notes
- ^ .
- ^ Zweig, G. (17 January 1964). An SU(3) Model for Strong Interaction Symmetry and its Breaking (PDF) (Report). CERN Report No.8182/TH.401.
- ^ Zweig, G. (1964). An SU(3) Model for Strong Interaction Symmetry and its Breaking: II (PDF) (Report). CERN Report No.8419/TH.412.
- .
- arXiv:1412.8681.
- .
- .
- .
- ISBN 0-471-29292-3.
- .
References
- S. Eidelman et al. 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. ASIN B001RAVDIA.