In particle 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 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 effective classification of them
to date. The quark model was independently
proposed by physicists Murray Gell-Mann, and
George Zweig (also see) 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 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 quantum numbers of the hadrons. These
quantum numbers are labels identifying the
hadrons, and are of two kinds. One set comes
from the Poincaré symmetry—JPC, where J,
P and C stand for the total angular momentum,
P-symmetry, and C-symmetry, respectively.
The remaining are flavor 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.
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.
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
flavors of quark (which form an approximate
flavor SU(3) symmetry). There are generalizations
to larger number of flavors.
== History ==
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 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.
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.
== 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
3
⊗
3
¯
=
8
⊕
1
{\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 JPC = 0−−).
== Baryons ==
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
3
⊗
3
⊗
3
=
10
S
⊕
8
M
⊕
8
M
⊕
1
A
{\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
6
⊗
6
⊗
6
=
56
S
⊕
70
M
⊕
70
M
⊕
20
A
.
{\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
56
=
10
3
2
⊕
8
1
2
,
{\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−).
Mixing of baryons, mass splittings within
and between multiplets, and magnetic moments
are some of the other questions that the model
predicts successfully.
=== The 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,
(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.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.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 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
