# Number of Colors in QCD

Staff Emeritus

## Main Question or Discussion Point

QCD has 3 "colors". I'm wondering whether there is something special about the number 3, or whether it is possible to generalize to N colors and get a very similar theory.

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Staff Emeritus
2019 Award
Yes, you can. But what you get doesn't have anything to do with reality.

Staff Emeritus
Yes, you can. But what you get doesn't have anything to do with reality.
I was just asking about the theory. You have to actually understand the implications of a theory to know whether it has anything to do with reality.

For example: Is QED the same as N=1 QCD? Is electroweak theory the same as N=2 QCD (plus the Higgs)?

vanhees71
Gold Member
2019 Award
You can build a gauge theory with any compact Lie group as a gauge group you like.

The Abelian gauge group with the gauge group U(1) is the most simple one. An example is QED.

The non-Abelian case is a bit more restrictive. The gauge fields are necessarily self-interacting on the tree level and thus all particles must couple with the same universal coupling strength (coupling constant $g$) in order to not to destroy the symmetry.

QCD is based on local SU(3) color symmetry. There is nothing special in the number 3 but it's an empirical fact that there are three colors in nature. It can be pretty directly seen on the plot of the $e^+ + e^- \rightarrow \text{hadrons}$ cross section, usually plotted normalized to the QED-tree-level cross section for $e^+ + e^- \rightarrow \mu^+ \mu^-$. You find it, e.g., here (on page 6):

http://pdg.lbl.gov/2013/reviews/rpp2012-rev-cross-section-plots.pdf

The electroweak sector is based on the gauge group SU(2) x U(1) but somewhat different from QCD in the sense that the gauge group is "Higgsed", i.e., spontaneously broken to U(1).

Note also that groups with larger number of colors have been used to build Grand Unified Theories (GUT). The first GUT proposed used the SU(5) group. Many other Groups have been used to build other GUT theories over the years.

Concerning your question on whether one can consider N colors, i.e. a theory with gauge group SU(N): this is what one does in the so called 1/N-expansion. Of course, it is far from obvious that an expansion in 1/N is a good idea for N=3 but in fact this expansion can explain at least qualitatively some features of QCD, such as the so called OZI rule. The 1/N-expansion is also related to the AdS/CFT correspondence which can to some (small, I must admit) extent be applied to QCD.

http://en.wikipedia.org/wiki/1/N_expansion