So my principal question is the one in the object above. But then I have another question related.
Supposing I have a $bg$ gluon.
How may I write it in the octet basis?
Thank you for your help!!
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Gluons carry color in color-anticolor pairs and form an octet and a singlet representation as seen below:
As you see the octed is filled with a color and an anticolor. The anti has to be a different color otherwise the gluon would be color neutral, and no strong force would be transferred.
Since to be exchangeable as gauge bosons in the strong interaction gluons have to be part of an octet , these are the acceptable combinations of color/anticolor. A bit similar to the fact that a photon has to be a is singlet, carries no charge.
There exist no free quarks or gluons to be able to set up states with higher color "spins" the way one can do with spin and isotopic spin.