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Quark model

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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

  • S. Eidelman et al.  
  • Lichtenberg, D B (1970). Unitary Symmetry and Elementary Particles. Academic Press.  
  • Thomson, M A (2011), Lecture notes
  • J.J.J. Kokkedee (1969). The quark model.  
  1. ^ M. Gell-Mann (1964). "A Schematic Model of Baryons and Mesons".  
  2. ^ G. Zweig (1964). "An SU(3) Model for Strong Interaction Symmetry and its Breaking". CERN Report No.8182/TH.401. 
  3. ^ G. Zweig (1964). "An SU(3) Model for Strong Interaction Symmetry and its Breaking: II". CERN Report No.8419/TH.412. 
  4. ^ O.W. Greenberg (1964). "Spin and Unitary-Spin Independence in a Paraquark Model of Baryons and Mesons".  
  5. ^ M.Y. Han, Y. Nambu (1965). "Three-Triplet Model with Double SU(3) Symmetry".  
  6. ^ W. Bardeen, H. Fritzsch, M. Gell-Mann (1973). R. Gatto, ed. "Scale and conformal symmetry in hadron physics".  

References and external links

See also

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).

States outside the quark model

The modern concept of color completely commuting with all other charges and providing the strong force charge was articulated in 1973, in an article by William Bardeen, Harald Fritzsch, and Murray Gell-Mann.[6]

Instead, six months later, Moo-Young Han and Yoichiro Nambu suggested the existence of three triplets of quarks to solve this problem, but flavor and color intertwined in that model--- they did not commute.[5]

Colour 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 = 32 baryon, the Δ++, required three up quarks with parallel spins and vanishing orbital angular momentum. Therefore, it could not have an antisymmetric wave function, (due to the Pauli exclusion principle), unless there were a hidden quantum number. Oscar Greenberg noted this problem in 1964, suggesting that quarks should be para-fermions.[4]

The discovery of color

Mixing of baryons, mass splittings within and between multiplets, and magnetic moments are some of the other questions that the model predicts successfully.

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).

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.

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

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

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

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 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.

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

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 flavours, the decomposition in flavor is

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

Baryons

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−−).

  • |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.

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

N.B. Nevertheless, the mass splitting between the η and the η′ is larger than the quark model can accommodate, and this "ηη′ puzzle" has its origin in topological peculiarities of the strong interaction vacuum, such as instanton configurations.


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).

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

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

Figure 3: Mesons of spin 1 form a nonet
Figure 2: Pseudoscalar mesons of spin 0 form a nonet

Mesons

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. To date, no Nobel prize has been awarded to Gell-Mann and Zweig for this discovery.

Finally, in 1964, Gell-Mann, and, independently, George Zweig, discerned what the Eightfold Way picture encodes. They posited elementary fermionic constituents, unobserved, and possibly unobservable in a free form, underlying and elegantly encoding 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.

The spin-32 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 1964.

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).

Developing classification schemes for Enrico Fermi and Chen-Ning Yang (1949), and by Shoichi Sakata (1956), ended up satisfactorily covering the mesons, but failed with baryons, and so were unable to explain all the data.

History

Contents

  • History 1
  • Mesons 2
  • Baryons 3
    • The discovery of color 3.1
  • States outside the quark model 4
  • See also 5
  • References and external links 6

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 flavor SU(3) symmetry). There are generalizations to larger number of flavors.

All quarks are assigned a baryon number of ⅓. Up, charm and top quarks have an electric charge of +⅔, while the down, strange, and bottom quarks have an electric charge of −⅓. Antiquarks have the opposite quantum numbers. Quarks are spin-½ particles, and thus fermions. Each quark or antiquark obeys the Gell-Mann−Nishijima formula individually, so any additive assembly of them will as well.

The remaining are flavour quantum numbers such as the isospin, strangeness, charm, 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.

Hadrons are not really "elementary", and can be regarded as bound states of their "valence 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.

. Standard Model in 1964. Today, the model has essentially been absorbed as a component of the established quantum field theory of strong and electroweak particle interactions, dubbed the [3][2]

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