Editing Color charge (section)

== Quark and gluon fields ==
[[File:Strong force charges.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).]]
In QCD the gauge group is the non-abelian group [[SU(3)]]. The ''[[coupling constant#Running coupling|running coupling]]'' is usually denoted by <math>\alpha_s</math>. Each [[flavour (particle physics)| flavour]] of quark belongs to the [[fundamental representation]] ('''3''') and contains a triplet of fields together denoted by <math>\psi</math>. The [[antiparticle|antiquark]] field belongs to the [[Hermitian conjugate|complex conjugate representation]] ('''3<sup>*</sup>''') and also contains a triplet of fields. We can write
: <math>\psi = \begin{pmatrix}\psi_1\\ \psi_2\\ \psi_3\end{pmatrix}</math> &nbsp;and&nbsp; <math>\overline\psi = \begin{pmatrix}{\overline\psi}^*_1\\ {\overline\psi}^*_2\\ {\overline\psi}^*_3\end{pmatrix}.</math>
The gluon contains an octet of fields (see [[gluon field]]), and belongs to the [[adjoint representation of a Lie group|adjoint representation]] ('''8'''), and can be written using the [[Gell-Mann matrices]] as
: <math>{\mathbf A}_\mu = A_\mu^a\lambda_a.</math>
(there is an [[Einstein notation|implied summation]] over ''a'' = 1, 2, ... 8). All other [[subatomic particle|particle]]s belong to the [[trivial representation]] ('''1''') of color [[SU(3)]]. The color charge of each of these fields is fully specified by the representations. Quarks have a color charge of red, green or blue and antiquarks have a color charge of antired, antigreen or antiblue. Gluons have a combination of two color charges (one of red, green, or blue and one of antired, antigreen, or antiblue) in a superposition of states that are given by the Gell-Mann matrices. All other particles have zero color charge.

The gluons corresponding to <math>\lambda_3</math> and <math>\lambda_8</math> are sometimes described as having "zero charge" (as in the figure). Formally, these states are written as
:<math>g_3 = \frac{1}{\sqrt{2}} (r\overline{r}-b\overline{b})</math> and <math>g_8 = \frac{1}{\sqrt{6}} (r\overline{r}+b\overline{b}-2g\overline{g})</math>
While "colorless" in the sense that they consist of matched color-anticolor pairs, which places them in the centre of a [[weight diagram]] alongside the truly colorless [[singlet state]], they still participate in strong interactions - in particular, those in which quarks interact without changing color.

Mathematically speaking, the color charge of a particle is the value of a certain quadratic [[Casimir operator]] in the representation of the particle.

In the simple language introduced previously, the three indices "1", "2" and "3" in the quark triplet above are usually identified with the three colors. The colorful language misses the following point. A gauge transformation in color SU(3) can be written as <math>\psi \to U \psi</math>, where <math>U</math> is a {{math|3 × 3}} matrix that belongs to the group SU(3). Thus, after gauge transformation, the new colors are linear combinations of the old colors. In short, the simplified language introduced before is not gauge invariant.

[[File:vertex.png|150px|left|Color-line representation of QCD vertex]]

Color charge is conserved, but the book-keeping involved in this is more complicated than just adding up the charges, as is done in quantum electrodynamics. One simple way of doing this is to look at the interaction vertex in QCD and replace it by a color-line representation. The meaning is the following. Let <math>\psi_i</math> represent the {{mvar|i}}th component of a quark field (loosely called the {{mvar|i}}th color). The ''color'' of a gluon is similarly given by <math>\mathbf{A}</math>, which corresponds to the particular Gell-Mann matrix it is associated with. This matrix has indices {{mvar|i}} and {{mvar|j}}. These are the ''color labels'' on the gluon. At the interaction vertex one has {{math|q<sub>''i''</sub> → g<sub>''ij''</sub> + q<sub>''j''</sub>}}. The ''color-line'' representation tracks these indices. Color [[charge conservation]] means that the ends of these color lines must be either in the initial or final state, equivalently, that no lines break in the middle of a diagram.

[[File:3gluon.png|150px|right|Color-line representation of 3-gluon vertex]]

Since gluons carry color charge, two gluons can also interact. A typical interaction vertex (called the three gluon vertex) for gluons involves g&nbsp;+&nbsp;g&nbsp;→&nbsp;g. This is shown here, along with its color-line representation. The color-line diagrams can be restated in terms of conservation laws of color; however, as noted before, this is not a gauge invariant language. Note that in a typical [[non-abelian gauge theory]] the [[gauge boson]] carries the charge of the theory, and hence has interactions of this kind; for example, the [[W&nbsp;boson]] in the electroweak theory. In the electroweak theory, the W also carries electric charge, and hence interacts with a photon.