Editing Quantum chromodynamics (section)

==Theory==

===Some definitions===
{{unsolved|physics|QCD in the non-[[perturbation theory (quantum mechanics)|perturbative]] regime:
*'''[[Color confinement|Confinement]]''': the equations of QCD remain unsolved at [[energy scale]]s relevant for describing [[atomic nucleus|atomic nuclei]]. How does QCD give rise to the physics of nuclei and nuclear constituents?
*'''[[QCD matter|Quark matter]]''': the equations of QCD predict that a [[quark–gluon plasma|plasma (or soup) of quarks and gluons]] should be formed at high temperature and density. What are the properties of this [[phase of matter]]?}}<!-- please don't insert a line feed here, without checking to ensure that spacing remains as it should -->
Every field theory of [[particle physics]] is based on certain symmetries of nature whose existence is deduced from observations. These can be
*[[Local symmetry|local symmetries]], which 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 boson]]s.
*[[Global symmetry|global symmetries]], which are symmetries whose operations must be simultaneously applied to all points of spacetime.

QCD is a non-abelian gauge theory (or [[Yang–Mills theory]]) of the [[special unitary group|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 (physics)|chirality]], discrimination between left and right-handed. If the [[Spin (physics)|spin]] of a particle has a positive [[projection (linear algebra)|projection]] on its direction of motion then it is called right-handed; otherwise, it is left-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 ''[[Kramers–Wannier duality|dual]]''– to what one is used to, since usually one connects the absence of interactions with ''large''&nbsp;distances. However, as already mentioned in the original paper of Franz Wegner,<ref>{{cite journal |last=Wegner |first=F. |year=1971 |title=Duality in Generalized Ising Models and Phase Transitions without Local Order Parameter |journal=J. Math. Phys. |volume=12 |issue=10 |pages=2259–2272 |bibcode=1971JMP....12.2259W |doi=10.1063/1.1665530}} Reprinted in {{cite book |title=Lattice Gauge Theories and Monte Carlo Simulations |publisher=World Scientific |year=1983 |isbn=9971950707 |editor-last=Rebbi |editor-first=Claudio |location=Singapore |pages=60–73}} Abstract: [http://www.tphys.uni-heidelberg.de/~wegner/Abstracts.html#12] {{webarchive|url=https://web.archive.org/web/20110504173247/http://www.tphys.uni-heidelberg.de/~wegner/Abstracts.html|date=2011-05-04}}</ref> 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.<ref>Perhaps one can guess that in the "original" model mainly the quarks would fluctuate, whereas in the present one, the "dual" model, mainly the gluons do.</ref>

===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 [[Quantum electrodynamics|QED]]: this is an [[abelian group]]. If one considers a version of QCD with ''N<sub>f</sub>'' flavors of massless quarks, then there is a global ([[chirality (physics)|chiral]]) flavor symmetry group SU<sub>L</sub>(''N<sub>f</sub>'') &times; SU<sub>R</sub>(''N<sub>f</sub>'') &times; U<sub>B</sub>(1) &times; U<sub>A</sub>(1). The chiral symmetry is [[spontaneous symmetry breaking|spontaneously broken]] by the [[QCD vacuum]] to the vector (L+R) SU<sub>V</sub>(''N<sub>f</sub>'')  with the formation of a [[chiral condensate]]. The vector symmetry, U<sub>B</sub>(1) corresponds to the baryon number of quarks and is an exact symmetry. The axial symmetry U<sub>A</sub>(1) is exact in the classical theory, but broken in the quantum theory, an occurrence called an [[anomaly (physics)|anomaly]]. Gluon field configurations called [[instanton]]s 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 that 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 [[AdS/QCD|string description of QCD]].

=== Lagrangian ===
The dynamics of the quarks and gluons are defined by the quantum chromodynamics [[Lagrangian (field theory)|Lagrangian]].  The [[gauge invariant]] QCD Lagrangian is

{{Equation box 1
|indent =:
|equation = :<math>\mathcal{L}_\mathrm{QCD} = \bar{\psi}_i  \left( i \gamma^\mu (D_\mu)_{ij} - m\, \delta_{ij}\right) \psi_j - \frac{1}{4}G^a_{\mu \nu} G^{\mu \nu}_a </math>|border
|border colour = #50C878
|background colour = #ECFCF4}}

where <math>\psi_i(x) \,</math> is the quark field, a dynamical function of spacetime, in the [[fundamental representation]] of the [[SU(3)]] gauge [[Group (mathematics)|group]], indexed by <math>i</math> and <math>j</math> running from <math>1</math> to <math>3</math>; <math>\bar \psi_i \,</math> is the [[Dirac adjoint]] of <math>\psi_i \,</math>; <math>D_\mu</math> is the [[gauge covariant derivative]]; the γ<sup>μ</sup> are [[Gamma matrices]] connecting the spinor representation to the vector representation of the [[Lorentz group]].

Herein, the [[gauge covariant derivative]] <math>\left( D_\mu \right)_{ij} = \partial_\mu \delta_{ij} - i g \left( T_a \right)_{ij} \mathcal{A}^a_\mu \,</math>couples the quark field with a coupling strength <math>g \,</math>to the gluon fields via the infinitesimal SU(3) generators <math>T_a \,</math>in the fundamental representation. An explicit representation of these generators is given by <math>T_a = \lambda_a / 2 \,</math>, wherein the <math>\lambda_a \, (a = 1 \ldots 8)\,</math>are the [[Gell-Mann matrices]].

The symbol <math>G^a_{\mu \nu} \,</math> represents the gauge invariant [[gluon field strength tensor]], analogous to the [[electromagnetic tensor|electromagnetic field strength tensor]], ''F''<sup>μν</sup>, in [[quantum electrodynamics]]. It is given by:<ref>{{cite journal|title=The field strength correlator from QCD sum rules
|author1=M. Eidemüller |author2=H.G. Dosch |author3=M. Jamin
|location=Heidelberg, Germany
|journal=Nucl. Phys. B Proc. Suppl. |volume=86 |pages=421–425 |year=2000
|issue=1–3 |arxiv=hep-ph/9908318|bibcode=2000NuPhS..86..421E|doi=10.1016/S0920-5632(00)00598-3|s2cid=18237543 }}</ref>

:<math>G^a_{\mu \nu} = \partial_\mu \mathcal{A}^a_\nu - \partial_\nu \mathcal{A}^a_\mu + g f^{abc} \mathcal{A}^b_\mu \mathcal{A}^c_\nu \,,</math>

where <math>\mathcal{A}^a_\mu(x) \,</math> are the [[gluon field]]s, dynamical functions of spacetime, in the [[adjoint representation]] of the SU(3) gauge group, indexed by ''a'', ''b'' and ''c'' running from <math>1</math> to <math>8</math>; and ''f<sub>abc</sub>'' are the [[structure constants]] of SU(3) (the generators of the adjoint representation). Note that the rules to move-up or pull-down the ''a'', ''b'', or ''c'' indices are ''trivial'', (+, ..., +), so that ''f<sup>abc</sup>'' = ''f<sub>abc</sub>'' = ''f''<sup>''a''</sup><sub>''bc''</sub> 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 [[Color confinement|confined]] and deconfined phases.

===Fields===
[[File:QCD.svg|300px|right|thumb|The pattern of strong charges for the three colors of quark, three antiquarks, and eight gluons (with two of zero charge overlapping).]]

Quarks are massive spin-{{frac|1|2}} [[fermion]]s that carry a [[color charge]] whose gauging is the content of QCD. Quarks are represented by [[Dirac field]]s in the [[fundamental representation]] '''3''' of the [[gauge group]] [[SU(3)]]. They also carry electric charge (either &minus;{{frac|1|3}} or +{{frac|2|3}}) and participate in [[weak interactions]] as part of [[weak isospin]] doublets. They carry global quantum numbers including the [[baryon number]], which is {{frac|1|3}} for each quark, [[hypercharge]] and one of the [[flavor (particle physics)|flavor quantum numbers]].

Gluons are spin-1 [[boson]]s that also carry [[color charge]]s, since they lie in the [[Adjoint representation of a Lie group|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 state|singlet representation]] '''1''' of all these symmetry groups.

Each type of quark has a corresponding antiquark, of which the charge is exactly opposite. They transform in the [[conjugate representation]] to quarks, denoted <math>\bar\mathbf{3}</math>.

===Dynamics===
According to the rules of [[quantum field theory]], and the associated [[Feynman diagram]]s, 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 [[quantum electrodynamics|QED]], in which only the first kind of interaction occurs, since [[photon]]s have no charge. Diagrams involving [[Faddeev–Popov ghost]]s must be considered too (except in the [[unitarity gauge]]).

===Area law and confinement===
Detailed computations with the above-mentioned Lagrangian<ref>See all standard textbooks on the QCD,  e.g., those noted above</ref> 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 (<math>\propto r</math>), which represents some kind of "stiffness" of the interaction between the particle and its anti-particle at large  distances, similar to the [[entropic force|entropic elasticity]] of a [[rubber]] band (see below). This leads to ''confinement''&nbsp;<ref>''Confinement'' gives way to a [[quark–gluon plasma]] only at extremely large pressures and/or temperatures, e.g. for <math>T \approx 5\cdot 10^{12}</math>&nbsp;&nbsp;K or larger.</ref> of the quarks to the interior of hadrons, i.e. [[meson]]s and [[nucleon]]s, with typical radii ''R''<sub>c</sub>, corresponding to  former "[[Bag model]]s" of the hadrons<ref>[[Kenneth Alan Johnson]]. (July 1979). The bag model of quark confinement. ''Scientific American''.</ref> The order of magnitude of the "bag radius" is 1&nbsp;fm (=&nbsp;10<sup>&minus;15</sup>&nbsp;m). Moreover, the above-mentioned stiffness is quantitatively related to the so-called "area law" behavior of the expectation value of the Wilson loop product ''P''<sub>W</sub> of the ordered coupling constants around a closed loop ''W''; i.e. <math>\,\langle P_W\rangle</math> is proportional to the ''area'' enclosed by the loop. For this behavior the non-abelian behavior of the gauge group is essential.