Editing Gluon (section)

== Counting gluons ==
There are eight independent types of gluons in QCD. This is unlike the photon of QED or the three [[W and Z bosons]] of the [[weak interaction]]. 

Additionally, gluons are subject to the [[color charge]] phenomena. [[Quark]]s carry three types of color charge; antiquarks carry three types of anticolor. Gluons carry both color and anticolor. This gives nine ''possible'' combinations of color and anticolor in gluons. The following is a list of those combinations (and their schematic names):
* red–antired (''r''{{overline|''r''}}), red–antigreen (''r''{{overline|''g''}}), red–antiblue (''r''{{overline|''b''}})
* green–antired (''g''{{overline|''r''}}), green–antigreen (''g''{{overline|''g''}}), green–antiblue (''g''{{overline|''b''}})
* blue–antired (''b''{{overline|''r''}}), blue–antigreen (''b''{{overline|''g''}}), blue–antiblue (''b''{{overline|''b''}})

[[File:Feynman Diagram Y-3g.svg|thumb|240px|right|class=skin-invert-image|Diagram 2: e<sup>+</sup>e<sup>−</sup> &rarr; Υ(9.46) &rarr; 3g]]
These ''possible'' combinations are only ''effective'' states, not the ''actual'' observed color states of gluons. To understand how they are combined, it is necessary to consider the mathematics of color charge in more detail.

=== Color singlet states <span class="anchor" id="Color singlet"></span> ===
The stable strongly interacting particles, including hadrons like the proton or the neutron, are observed to be "colorless". More precisely, they are in a "color singlet" state, and mathematically analogous to a [[singlet state|''spin'' singlet state]].<ref name="Griff">
{{cite book
 |author=David Griffiths
 |year=1987
 |title=Introduction to Elementary Particles
 |pages=280–281
 |publisher=[[John Wiley & Sons]]
 |isbn=978-0-471-60386-3
}}</ref> The states allow interaction with other color singlets, but not other color states; because long-range gluon interactions do not exist, this illustrates that gluons in the singlet state do not exist either.<ref name="Griff"/>

The color singlet state is:<ref name="Griff"/>
: <math>(r\bar{r}+b\bar{b}+g\bar{g})/\sqrt{3}.</math>

If one could [[Measurement in quantum mechanics|measure]] the color of the state, there would be equal probabilities of it being red–antired, blue–antiblue, or green–antigreen.

=== Eight color states ===
<!-- the link "eight gluon types" from the article "Quark" links here -->
There are eight remaining independent color states corresponding to the "eight types" or "eight colors" of gluons. Since the states can be mixed together, there are multiple ways of presenting these states. These are known as the "color octet", and a commonly used list for each is:<ref name="Griff"/>

: {|
|-
|<math>(r\bar{b}+b\bar{r})/\sqrt{2}</math>
|&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
|<math>-i(r\bar{b}-b\bar{r})/\sqrt{2}</math>
|-
|<math>(r\bar{g}+g\bar{r})/\sqrt{2}</math>
|
|<math>-i(r\bar{g}-g\bar{r})/\sqrt{2}</math>
|-
|<math>(b\bar{g}+g\bar{b})/\sqrt{2}</math>
|
|<math>-i(b\bar{g}-g\bar{b})/\sqrt{2}</math>
|-
|<math>(r\bar{r}-b\bar{b})/\sqrt{2}</math>
|
|<math>(r\bar{r}+b\bar{b}-2g\bar{g})/\sqrt{6}</math>
|}

These are equivalent to the [[Gell-Mann matrices]]. The critical feature of these particular eight states is that they are [[linearly independent]], and also independent of the singlet state, hence 3<sup>2</sup>&nbsp;&minus;&nbsp;1 or 2<sup>3</sup>. There is no way to add any combination of these states to produce any others. It is also impossible to add them to make ''r''{{overline|''r''}}, ''g''{{overline|''g''}}, or ''b''{{overline|''b''}}<ref>{{cite web  |author=J. Baez |title=Why are there eight gluons and not nine?  |url=http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/gluons.html |website=math.ucr.edu  |access-date=2009-09-13}}</ref> the forbidden [[singlet state]]. There are many other possible choices, but all are mathematically equivalent, at least equally complicated, and give the same physical results.

=== Group theory details ===
Formally, QCD is a [[gauge theory]] with [[SU(3)]] gauge symmetry. Quarks are introduced as [[spinor]]s in ''N''<sub>f</sub> [[flavour (particle physics)|flavor]]s, each in the [[fundamental representation]] (triplet, denoted '''3''') of the color gauge group, SU(3). The gluons are vectors in the [[Adjoint representation of a Lie group|adjoint representation]] (octets, denoted '''8''') of color SU(3). For a general [[lie group|gauge group]], the number of force-carriers, like photons or gluons, is always equal to the dimension of the adjoint representation. For the simple case of SU(''n''), the dimension of this representation is {{nowrap|{{itco|''n''}}<sup>2</sup> − 1}}.

In group theory, there are no color singlet gluons because [[quantum chromodynamics]] has an SU(3) rather than a [[U(N)|U(3)]] symmetry. There is no known [[A priori and a posteriori|''a priori'']] reason for one group to be preferred over the other, but as discussed above, the experimental evidence supports SU(3).<ref name="Griff"/> If the group were U(3), the ninth (colorless singlet) gluon would behave like a "second photon" and not like the other eight gluons.<ref>{{cite web |url=https://www.forbes.com/sites/startswithabang/2020/11/18/why-are-there-only-8-gluons/|title=Why Are There Only 8 Gluons?|website=[[Forbes]]}}</ref>