In theoretical physics, quantum chromodynamics
(QCD) is the theory of the strong interaction
between quarks and gluons, the fundamental
particles that make up composite hadrons such
as the proton, neutron and pion. QCD is a
type of quantum field theory called a non-abelian
gauge theory, with symmetry group SU(3). The
QCD analog of electric charge is a property
called color. Gluons are the force carrier
of the theory, like photons are for the electromagnetic
force in quantum electrodynamics. The theory
is an important part of the Standard Model
of particle physics. A large body of experimental
evidence for QCD has been gathered over the
years.
QCD exhibits two main properties:
Color confinement. This is a consequence of
the constant force between two color charges
as they are separated: In order to increase
the separation between two quarks within a
hadron, ever-increasing amounts of energy
are required. Eventually this energy produces
a quark–antiquark pair, turning the initial
hadron into a pair of hadrons instead of producing
an isolated color charge. Although analytically
unproven, color confinement is well established
from lattice QCD calculations and decades
of experiments.Asymptotic freedom, a steady
reduction in the strength of interactions
between quarks and gluons as the energy scale
of those interactions increases (and the corresponding
length scale decreases). The asymptotic freedom
of QCD was discovered in 1973 by David Gross
and Frank Wilczek, and independently by David
Politzer in the same year. For this work all
three shared the 2004 Nobel Prize in Physics.
== Terminology ==
Physicist Murray Gell-Mann (b. 1929) coined
the word quark in its present sense. It originally
comes from the phrase "Three quarks for Muster
Mark" in Finnegans Wake by James Joyce. On
June 27, 1978, Gell-Mann wrote a private letter
to the editor of the Oxford English Dictionary,
in which he related that he had been influenced
by Joyce's words: "The allusion to three quarks
seemed perfect." (Originally, only three quarks
had been discovered.)The three kinds of charge
in QCD (as opposed to one in quantum electrodynamics
or QED) are usually referred to as "color
charge" by loose analogy to the three kinds
of color (red, green and blue) perceived by
humans. Other than this nomenclature, the
quantum parameter "color" is completely unrelated
to the everyday, familiar phenomenon of color.
The force between quarks is known as the colour
force (or color force ) or strong interaction,
and is responsible for the strong nuclear
force.
Since the theory of electric charge is dubbed
"electrodynamics", the Greek word χρῶμα
chroma "color" is applied to the theory of
color charge, "chromodynamics".
== History ==
With the invention of bubble chambers and
spark chambers in the 1950s, experimental
particle physics discovered a large and ever-growing
number of particles called hadrons. It seemed
that such a large number of particles could
not all be fundamental. First, the particles
were classified by charge and isospin by Eugene
Wigner and Werner Heisenberg; then, in 1953–56,
according to strangeness by Murray Gell-Mann
and Kazuhiko Nishijima (see Gell-Mann–Nishijima
formula). To gain greater insight, the hadrons
were sorted into groups having similar properties
and masses using the eightfold way, invented
in 1961 by Gell-Mann and Yuval Ne'eman. Gell-Mann
and George Zweig, correcting an earlier approach
of Shoichi Sakata, went on to propose in 1963
that the structure of the groups could be
explained by the existence of three flavors
of smaller particles inside the hadrons: the
quarks.
Perhaps the first remark that quarks should
possess an additional quantum number was made
as a short footnote in the preprint of Boris
Struminsky in connection with Ω− hyperon
composed of three strange quarks with parallel
spins (this situation was peculiar, because
since quarks are fermions, such combination
is forbidden by the Pauli exclusion principle):
Three identical quarks cannot form an antisymmetric
S-state. In order to realize an antisymmetric
orbital S-state, it is necessary for the quark
to have an additional quantum number.
Boris Struminsky was a PhD student of Nikolay
Bogolyubov. The problem considered in this
preprint was suggested by Nikolay Bogolyubov,
who advised Boris Struminsky in this research.
In the beginning of 1965, Nikolay Bogolyubov,
Boris Struminsky and Albert Tavkhelidze wrote
a preprint with a more detailed discussion
of the additional quark quantum degree of
freedom. This work was also presented by Albert
Tavkhelidze without obtaining consent of his
collaborators for doing so at an international
conference in Trieste (Italy), in May 1965.A
similar mysterious situation was with the
Δ++ baryon; in the quark model, it is composed
of three up quarks with parallel spins. In
1964–65, Greenberg and Han–Nambu independently
resolved the problem by proposing that quarks
possess an additional SU(3) gauge degree of
freedom, later called color charge. Han and
Nambu noted that quarks might interact via
an octet of vector gauge bosons: the gluons.
Since free quark searches consistently failed
to turn up any evidence for the new particles,
and because an elementary particle back then
was defined as a particle which could be separated
and isolated, Gell-Mann often said that quarks
were merely convenient mathematical constructs,
not real particles. The meaning of this statement
was usually clear in context: He meant quarks
are confined, but he also was implying that
the strong interactions could probably not
be fully described by quantum field theory.
Richard Feynman argued that high energy experiments
showed quarks are real particles: he called
them partons (since they were parts of hadrons).
By particles, Feynman meant objects which
travel along paths, elementary particles in
a field theory.
The difference between Feynman's and Gell-Mann's
approaches reflected a deep split in the theoretical
physics community. Feynman thought the quarks
have a distribution of position or momentum,
like any other particle, and he (correctly)
believed that the diffusion of parton momentum
explained diffractive scattering. Although
Gell-Mann believed that certain quark charges
could be localized, he was open to the possibility
that the quarks themselves could not be localized
because space and time break down. This was
the more radical approach of S-matrix theory.
James Bjorken proposed that pointlike partons
would imply certain relations in deep inelastic
scattering of electrons and protons, which
were verified in experiments at SLAC in 1969.
This led physicists to abandon the S-matrix
approach for the strong interactions.
In 1973 the concept of color as the source
of a "strong field" was developed into the
theory of QCD by physicists Harald Fritzsch
and Heinrich Leutwyler, together with physicist
Murray Gell-Mann. In particular, they employed
the general field theory developed in 1954
by Chen Ning Yang and Robert Mills (see Yang–Mills
theory), in which the carrier particles of
a force can themselves radiate further carrier
particles. (This is different from QED, where
the photons that carry the electromagnetic
force do not radiate further photons.)
The discovery of asymptotic freedom in the
strong interactions by David Gross, David
Politzer and Frank Wilczek allowed physicists
to make precise predictions of the results
of many high energy experiments using the
quantum field theory technique of perturbation
theory. Evidence of gluons was discovered
in three-jet events at PETRA in 1979. These
experiments became more and more precise,
culminating in the verification of perturbative
QCD at the level of a few percent at the LEP
in CERN.
The other side of asymptotic freedom is confinement.
Since the force between color charges does
not decrease with distance, it is believed
that quarks and gluons can never be liberated
from hadrons. This aspect of the theory is
verified within lattice QCD computations,
but is not mathematically proven. One of the
Millennium Prize Problems announced by the
Clay Mathematics Institute requires a claimant
to produce such a proof. Other aspects of
non-perturbative QCD are the exploration of
phases of quark matter, including the quark–gluon
plasma.
The relation between the short-distance particle
limit and the confining long-distance limit
is one of the topics recently explored using
string theory, the modern form of S-matrix
theory.
== Theory ==
=== 
Some definitions ===
Every field theory of particle physics is
based on certain symmetries of nature whose
existence is deduced from observations. These
can be
local symmetries, that are the symmetries
that act independently at each point in spacetime.
Each such symmetry is the basis of a gauge
theory and requires the introduction of its
own gauge bosons.
global symmetries, which are symmetries whose
operations must be simultaneously applied
to all points of spacetime.QCD is a gauge
theory of the SU(3) gauge group obtained by
taking the color charge to define a local
symmetry.
Since the strong interaction does not discriminate
between different flavors of quark, QCD has
approximate flavor symmetry, which is broken
by the differing masses of the quarks.
There are additional global symmetries whose
definitions require the notion of chirality,
discrimination between left and right-handed.
If the spin of a particle has a positive projection
on its direction of motion then it is called
left-handed; otherwise, it is right-handed.
Chirality and handedness are not the same,
but become approximately equivalent at high
energies.
Chiral symmetries involve independent transformations
of these two types of particle.
Vector symmetries (also called diagonal symmetries)
mean the same transformation is applied on
the two chiralities.
Axial symmetries are those in which one transformation
is applied on left-handed particles and the
inverse on the right-handed particles.
=== Additional remarks: duality ===
As mentioned, asymptotic freedom means that
at large energy – this corresponds also
to short distances – there is practically
no interaction between the particles. This
is in contrast – more precisely one would
say dual– to what one is used to, since
usually one connects the absence of interactions
with large distances. However, as already
mentioned in the original paper of Franz Wegner,
a solid state theorist who introduced 1971
simple gauge invariant lattice models, the
high-temperature behaviour of the original
model, e.g. the strong decay of correlations
at large distances, corresponds to the low-temperature
behaviour of the (usually ordered!) dual model,
namely the asymptotic decay of non-trivial
correlations, e.g. short-range deviations
from almost perfect arrangements, for short
distances. Here, in contrast to Wegner, we
have only the dual model, which is that one
described in this article.
=== Symmetry groups ===
The color group SU(3) corresponds to the local
symmetry whose gauging gives rise to QCD.
The electric charge labels a representation
of the local symmetry group U(1) which is
gauged to give QED: this is an abelian group.
If one considers a version of QCD with Nf
flavors of massless quarks, then there is
a global (chiral) flavor symmetry group SUL(Nf)
× SUR(Nf) × UB(1) × UA(1). The chiral symmetry
is spontaneously broken by the QCD vacuum
to the vector (L+R) SUV(Nf) with the formation
of a chiral condensate. The vector symmetry,
UB(1) corresponds to the baryon number of
quarks and is an exact symmetry. The axial
symmetry UA(1) is exact in the classical theory,
but broken in the quantum theory, an occurrence
called an anomaly. Gluon field configurations
called instantons are closely related to this
anomaly.
There are two different types of SU(3) symmetry:
there is the symmetry that acts on the different
colors of quarks, and this is an exact gauge
symmetry mediated by the gluons, and there
is also a flavor symmetry which rotates different
flavors of quarks to each other, or flavor
SU(3). Flavor SU(3) is an approximate symmetry
of the vacuum of QCD, and is not a fundamental
symmetry at all. It is an accidental consequence
of the small mass of the three lightest quarks.
In the QCD vacuum there are vacuum condensates
of all the quarks whose mass is less than
the QCD scale. This includes the up and down
quarks, and to a lesser extent the strange
quark, but not any of the others. The vacuum
is symmetric under SU(2) isospin rotations
of up and down, and to a lesser extent under
rotations of up, down and strange, or full
flavor group SU(3), and the observed particles
make isospin and SU(3) multiplets.
The approximate flavor symmetries do have
associated gauge bosons, observed particles
like the rho and the omega, but these particles
are nothing like the gluons and they are not
massless. They are emergent gauge bosons in
an approximate string description of QCD.
=== Lagrangian ===
The 
dynamics of the quarks and gluons are controlled
by the quantum chromodynamics Lagrangian.
The gauge invariant QCD Lagrangian is
where
ψ
i
(
x
)
{\displaystyle \psi _{i}(x)\,}
is the quark field, a dynamical function of
spacetime, in the fundamental representation
of the SU(3) gauge group, indexed by
i
,
j
,
…
{\displaystyle i,\,j,\,\ldots }
;
D
μ
{\displaystyle D_{\mu }}
is the gauge covariant derivative; the γμ
are Dirac matrices connecting the spinor representation
to the vector representation of the Lorentz
group.
The symbol
G
μ
ν
a
{\displaystyle G_{\mu \nu }^{a}\,}
represents the gauge invariant gluon field
strength tensor, analogous to the electromagnetic
field strength tensor, Fμν, in quantum electrodynamics.
It is given by:
G
μ
ν
a
=
∂
μ
A
ν
a
−
∂
ν
A
μ
a
+
g
f
a
b
c
A
μ
b
A
ν
c
,
{\displaystyle G_{\mu \nu }^{a}=\partial _{\mu
}{\mathcal {A}}_{\nu }^{a}-\partial _{\nu
}{\mathcal {A}}_{\mu }^{a}+gf^{abc}{\mathcal
{A}}_{\mu }^{b}{\mathcal {A}}_{\nu }^{c}\,,}
where
A
μ
a
(
x
)
{\displaystyle {\mathcal {A}}_{\mu }^{a}(x)\,}
are the gluon fields, dynamical functions
of spacetime, in the adjoint representation
of the SU(3) gauge group, indexed by a, b,...;
and fabc are the structure constants of SU(3).
Note that the rules to move-up or pull-down
the a, b, or c indices are trivial, (+, ..., +),
so that fabc = 
fabc = fabc whereas for the μ or ν indices
one has the non-trivial relativistic rules
corresponding to the metric signature (+ − − −).
The variables m and g correspond to the quark
mass and coupling of the theory, respectively,
which are subject to renormalization.
An important theoretical concept is the Wilson
loop (named after Kenneth G. Wilson). In lattice
QCD, the final term of the above Lagrangian
is discretized via Wilson loops, and more
generally the behavior of Wilson loops can
distinguish confined and deconfined phases.
=== Fields ===
Quarks are massive spin-​1⁄2 fermions
which carry a color charge whose gauging is
the content of QCD. Quarks are represented
by Dirac fields in the fundamental representation
3 of the gauge group SU(3). They also carry
electric charge (either −​1⁄3 or +​2⁄3)
and participate in weak interactions as part
of weak isospin doublets. They carry global
quantum numbers including the baryon number,
which is ​1⁄3 for each quark, hypercharge
and one of the flavor quantum numbers.
Gluons are spin-1 bosons which also carry
color charges, since they lie in the adjoint
representation 8 of SU(3). They have no electric
charge, do not participate in the weak interactions,
and have no flavor. They lie in the singlet
representation 1 of all these symmetry groups.
Every quark has its own antiquark. The charge
of each antiquark is exactly the opposite
of the corresponding quark.
=== Dynamics ===
According to the rules of quantum field theory,
and the associated Feynman diagrams, the above
theory gives rise to three basic interactions:
a quark may emit (or absorb) a gluon, a gluon
may emit (or absorb) a gluon, and two gluons
may directly interact. This contrasts with
QED, in which only the first kind of interaction
occurs, since photons have no charge. Diagrams
involving Faddeev–Popov ghosts must be considered
too (except in the unitarity gauge).
=== Area law and confinement ===
Detailed computations with the above-mentioned
Lagrangian show that the effective potential
between a quark and its anti-quark in a meson
contains a term that increases in proportion
to the distance between the quark and anti-quark
(
∝
r
{\displaystyle \propto r}
), which represents some kind of "stiffness"
of the interaction between the particle and
its anti-particle at large distances, similar
to the entropic elasticity of a rubber band
(see below). This leads to confinement of
the quarks to the interior of hadrons, i.e.
mesons and nucleons, with typical radii Rc,
corresponding to former "Bag models" of the
hadrons The order of magnitude of the "bag
radius" is 1 fm (= 10−15 m). Moreover, the
above-mentioned stiffness is quantitatively
related to the so-called "area law" behaviour
of the expectation value of the Wilson loop
product PW of the ordered coupling constants
around a closed loop W; i.e.
⟨
P
W
⟩
{\displaystyle \,\langle P_{W}\rangle }
is proportional to the area enclosed by the
loop. For this behaviour the non-abelian behaviour
of the gauge group is essential.
== Methods ==
Further analysis of the content of the theory
is complicated. Various techniques have been
developed to work with QCD. Some of them are
discussed briefly below.
=== Perturbative QCD ===
This approach is based on asymptotic freedom,
which allows perturbation theory to be used
accurately in experiments performed at very
high energies. Although limited in scope,
this approach has resulted in the most precise
tests of QCD to date.
=== Lattice QCD ===
Among non-perturbative approaches to QCD,
the most well established one is lattice QCD.
This approach uses a discrete set of spacetime
points (called the lattice) to reduce the
analytically intractable path integrals of
the continuum theory to a very difficult numerical
computation which is then carried out on supercomputers
like the QCDOC which was constructed for precisely
this purpose. While it is a slow and resource-intensive
approach, it has wide applicability, giving
insight into parts of the theory inaccessible
by other means, in particular into the explicit
forces acting between quarks and antiquarks
in a meson. However, the numerical sign problem
makes it difficult to use lattice methods
to study QCD at high density and low temperature
(e.g. nuclear matter or the interior of neutron
stars).
=== ​1⁄N expansion ===
A well-known approximation scheme, the ​1⁄N
expansion, starts from the idea that the number
of colors is infinite, and makes a series
of corrections to account for the fact that
it is not. Until now, it has been the source
of qualitative insight rather than a method
for quantitative predictions. Modern variants
include the AdS/CFT approach.
=== Effective theories ===
For specific problems effective theories may
be written down which give qualitatively correct
results in certain limits. In the best of
cases, these may then be obtained as systematic
expansions in some parameter of the QCD Lagrangian.
One such effective field theory is chiral
perturbation theory or ChiPT, which is the
QCD effective theory at low energies. More
precisely, it is a low energy expansion based
on the spontaneous chiral symmetry breaking
of QCD, which is an exact symmetry when quark
masses are equal to zero, but for the u, d
and s quark, which have small mass, it is
still a good approximate symmetry. Depending
on the number of quarks which are treated
as light, one uses either SU(2) ChiPT or SU(3)
ChiPT . Other effective theories are heavy
quark effective theory (which expands around
heavy quark mass near infinity), and soft-collinear
effective theory (which expands around large
ratios of energy scales). In addition to effective
theories, models like the Nambu–Jona-Lasinio
model and the chiral model are often used
when discussing general features.
=== QCD sum rules ===
Based on an Operator product expansion one
can derive sets of relations that connect
different observables with each other.
=== Nambu–Jona-Lasinio model ===
In one of his recent works, Kei-Ichi Kondo
derived as a low-energy limit of QCD, a theory
linked to the Nambu–Jona-Lasinio model since
it is basically a particular non-local version
of the Polyakov–Nambu–Jona-Lasinio model.
The later being in its local version, nothing
but the Nambu–Jona-Lasinio model in which
one has included the Polyakov loop effect,
in order to describe a 'certain confinement'.
The Nambu–Jona-Lasinio model in itself is,
among many other things, used because it is
a 'relatively simple' model of chiral symmetry
breaking, phenomenon present up to certain
conditions (Chiral limit i.e. massless fermions)
in QCD itself.
In this model, however, there is no confinement.
In particular, the energy of an isolated quark
in the physical vacuum turns out well defined
and finite.
== Experimental tests ==
The notion of quark flavors was prompted by
the necessity of explaining the properties
of hadrons during the development of the quark
model. The notion of color was necessitated
by the puzzle of the Δ++. This has been dealt
with in the section on the history of QCD.
The first evidence for quarks as real constituent
elements of hadrons was obtained in deep inelastic
scattering experiments at SLAC. The first
evidence for gluons came in three jet events
at PETRA.
Several good quantitative tests of perturbative
QCD exist:
The running of the QCD coupling as deduced
from many observations
Scaling violation in polarized and unpolarized
deep inelastic scattering
Vector boson production at colliders (this
includes the Drell-Yan process)
Direct photons produced in hadronic collisions
Jet cross sections in colliders
Event shape observables at the LEP
Heavy-quark production in collidersQuantitative
tests of non-perturbative QCD are fewer, because
the predictions are harder to make. The best
is probably the running of the QCD coupling
as probed through lattice computations of
heavy-quarkonium spectra. There is a recent
claim about the mass of the heavy meson Bc
[3]. Other non-perturbative tests are currently
at the level of 5% at best. Continuing work
on masses and form factors of hadrons and
their weak matrix elements are promising candidates
for future quantitative tests. The whole subject
of quark matter and the quark–gluon plasma
is a non-perturbative test bed for QCD which
still remains to be properly exploited.
One qualitative prediction of QCD is that
there exist composite particles made solely
of gluons called glueballs that have not yet
been definitively observed experimentally.
A definitive observation of a glueball with
the properties predicted by QCD would strongly
confirm the theory. In principle, if glueballs
could be definitively ruled out, this would
be a serious experimental blow to QCD. But,
as of 2013, scientists are unable to confirm
or deny the existence of glueballs definitively,
despite the fact that particle accelerators
have sufficient energy to generate them.
== Cross-relations to solid state physics
==
There are unexpected cross-relations to solid
state physics. For example, the notion of
gauge invariance forms the basis of the well-known
Mattis spin glasses, which are systems with
the usual spin degrees of freedom
s
i
=
±
1
{\displaystyle s_{i}=\pm 1\,}
for i =1,...,N, with the special fixed "random"
couplings
J
i
,
k
=
ϵ
i
J
0
ϵ
k
.
{\displaystyle J_{i,k}=\epsilon _{i}\,J_{0}\,\epsilon
_{k}\,.}
Here the εi and εk quantities can independently
and "randomly" take the values ±1, which
corresponds to a most-simple gauge transformation
(
s
i
→
s
i
⋅
ϵ
i
J
i
,
k
→
ϵ
i
J
i
,
k
ϵ
k
s
k
→
s
k
⋅
ϵ
k
)
.
{\displaystyle (\,s_{i}\to s_{i}\cdot \epsilon
_{i}\quad \,J_{i,k}\to \epsilon _{i}J_{i,k}\epsilon
_{k}\,\quad s_{k}\to s_{k}\cdot \epsilon _{k}\,)\,.}
This means that thermodynamic expectation
values of measurable quantities, e.g. of the
energy
H
:=
−
∑
s
i
J
i
,
k
s
k
,
{\displaystyle {\mathcal {H}}:=-\sum s_{i}\,J_{i,k}\,s_{k}\,,}
are invariant.
However, here the coupling degrees of freedom
J
i
,
k
{\displaystyle J_{i,k}}
, which in the QCD correspond to the gluons,
are "frozen" to fixed values (quenching).
In contrast, in the QCD they "fluctuate" (annealing),
and through the large number of gauge degrees
of freedom the entropy plays an important
role (see below).
For positive J0 the thermodynamics of the
Mattis spin glass corresponds in fact simply
to a "ferromagnet in disguise", just because
these systems have no "frustration" at all.
This term is a basic measure in spin glass
theory. Quantitatively it is identical with
the loop product
P
W
:
=
J
i
,
k
J
k
,
l
.
.
.
J
n
,
m
J
m
,
i
{\displaystyle P_{W}:\,=\,J_{i,k}J_{k,l}...J_{n,m}J_{m,i}}
along a closed loop W. However, for a Mattis
spin glass – in contrast to "genuine" spin
glasses – the quantity PW never becomes
negative.
The basic notion "frustration" of the spin-glass
is actually similar to the Wilson loop quantity
of the QCD. The only difference is again that
in the QCD one is dealing with SU(3) matrices,
and that one is dealing with a "fluctuating"
quantity. Energetically, perfect absence of
frustration should be non-favorable and atypical
for a spin glass, which means that one should
add the loop product to the Hamiltonian, by
some kind of term representing a "punishment".
In the QCD the Wilson loop is essential for
the Lagrangian rightaway.
The relation between the QCD and "disordered
magnetic systems" (the spin glasses belong
to them) were additionally stressed in a paper
by Fradkin, Huberman and Shenker, which also
stresses the notion of duality.
A further analogy consists in the already
mentioned similarity to polymer physics, where,
analogously to Wilson Loops, so-called "entangled
nets" appear, which are important for the
formation of the entropy-elasticity (force
proportional to the length) of a rubber band.
The non-abelian character of the SU(3) corresponds
thereby to the non-trivial "chemical links",
which glue different loop segments together,
and "asymptotic freedom" means in the polymer
analogy simply the fact that in the short-wave
limit, i.e. for
0
←
λ
w
≪
R
c
{\displaystyle 0\leftarrow \lambda _{w}\ll
R_{c}}
(where Rc is a characteristic correlation
length for the glued loops, corresponding
to the above-mentioned "bag radius", while
λw is the wavelength of an excitation) any
non-trivial correlation vanishes totally,
as if the system had crystallized.There is
also a correspondence between confinement
in QCD – the fact that the color field is
only different from zero in the interior of
hadrons – and the behaviour of the usual
magnetic field in the theory of type-II superconductors:
there the magnetism is confined to the interior
of the Abrikosov flux-line lattice, i.e.,
the London penetration depth λ of that theory
is analogous to the confinement radius Rc
of quantum chromodynamics. Mathematically,
this correspondendence is supported by the
second term,
∝
g
G
μ
a
ψ
¯
i
γ
μ
T
i
j
a
ψ
j
,
{\displaystyle \propto gG_{\mu }^{a}{\bar
{\psi }}_{i}\gamma ^{\mu }T_{ij}^{a}\psi _{j}\,,}
on the r.h.s. of the Lagrangian.
== See also ==
For overviews, see Standard Model, its field
theoretical formulation, strong interactions,
quarks and gluons, hadrons, confinement, QCD
matter, or quark–gluon plasma.
For details, see gauge theory, quantization
procedure including BRST quantization and
Faddeev–Popov ghosts. A more general category
is quantum field theory.
For techniques, see Lattice QCD, 1/N expansion,
perturbative QCD, Soft-collinear effective
theory, heavy quark effective theory, chiral
models, and the Nambu and Jona-Lasinio model.
For experiments, see quark search experiments,
deep inelastic scattering, jet physics, quark–gluon
plasma.
Symmetry in quantum mechanics
