Why is gluon's representation 8 in ##SU(3)_c## and not 1?

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In summary, the group representation of SU(3) includes a ninth colorless state, but gluons cannot exist in this state on their own due to the theory of color confinement. The gauge field is in the adjoint representation of SU(3) and has 8 generators, which explains why there are 8 colored gluons. The singlet state only occurs when two gluons pair up and cancel color, forming a hadron. This is more fundamental to gauge theory than the fact that some fermions transform in the fundamental representation.
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I understand that because the rep. of quark is 3 and antiquark is ##\bar{3}## and we have ##3\otimes \bar{3} = 8\oplus 1##, that a gluon should be 8 or 1, but which one? and why?

Thanks!
 
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By ##SU(3)_c## you mean 'colored' correct? If so, there are 8 colored gluons because in an ##SU(3)## theory, there are ##N^2-1## linearly independent states, which in this case are the colors. The ##\oplus 1## indicates a color singlet state which is 'colorless'. I'm pretty sure singlet states only occur when two gluons pair up and cancel color out to make a hadron. The group representation of ##SU(3)## mathematically includes this ninth colorless state but gluons can't be colorless on their own which is why (Theory of Color Confinement) we've never seen a lone gluon.

If you can worm your way through it, the Wikipedia page on gluons is pretty cool, as is this site, which takes a matrix-math look at the underlying group theory which may help you. Hope this helped and best of luck!
 
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I would say the fact that the gauge field is in the adjoint representation is more fundamental to gauge theory than the fact that some fermions happen to transform according to the fundamental representation. It boils down to the group SU(3) itself having 8 generators.
 
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Spencer Fretwell said:
The group representation of SU(3)SU(3)SU(3) mathematically includes this ninth colorless state but gluons can't be colorless on their own which is why (Theory of Color Confinement) we've never seen a lone gluon.
This is not really what colour confinement says. By definition of an SU(3) gauge theory, it has 8 generators that transform according to the adjoint representation. This does not include a color singlet, if you added the identity you would get U(3) and not SU(3) so SU(3) does not include a colourless gluon. However, when considering how other fields transform, for example the product ##3 \otimes \bar 3##, the resulting representation needs to be decomposed into irreps and of course such decompositions can contain the trivial representation (in the example ##3\otimes \bar 3 = 8 \otimes 1##).
 
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Related to Why is gluon's representation 8 in ##SU(3)_c## and not 1?

1. Why is the gluon's representation 8 in SU(3)c and not 1?

The representation of the gluon in SU(3)c is 8 because it corresponds to the fundamental representation of the group. This means that the gluon carries a color charge and can interact with quarks, which also have a color charge. The number 8 comes from the fact that SU(3)c has 8 generators, which correspond to the 8 different colors of the gluon.

2. How does the representation of the gluon affect its role in the strong nuclear force?

The representation of the gluon in SU(3)c is crucial for its role in the strong nuclear force. As the carrier of the strong force, the gluon interacts with other particles, such as quarks, through the exchange of gluons. The gluon's representation in SU(3)c determines the strength and nature of these interactions.

3. What is the significance of SU(3)c in the Standard Model of particle physics?

SU(3)c is one of the fundamental groups in the Standard Model of particle physics. It describes the strong nuclear force and its interactions with quarks and gluons. The representation of the gluon in SU(3)c is crucial for understanding the behavior of quarks and the strong force.

4. How does the representation of the gluon in SU(3)c relate to the concept of color charge?

The representation of the gluon in SU(3)c is directly related to the concept of color charge. In the same way that electric charge is associated with the electromagnetic force, color charge is associated with the strong force. The gluon's representation in SU(3)c reflects its role as the carrier of this color charge.

5. Are there any experimental observations that support the gluon's representation in SU(3)c being 8?

Yes, there are several experimental observations that support the gluon's representation in SU(3)c being 8. One of the most significant is the discovery of the eight different types of gluons, each with a different color charge. This aligns with the 8 generators of SU(3)c and the 8 different colors of the gluon in the fundamental representation.

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