In terms of group theory, the assertion that there are no color singlet gluons is simply the statement that quantum chromodynamics has an SU(3) rather than a U(3) symmetry. For the simple case of SU( N), the dimension of this representation is N 2 − 1. For a general gauge group, the number of force-carriers (like photons or gluons) is always equal to the dimension of the adjoint representation. The gluons are vectors in the adjoint representation (octets, denoted 8) of color SU(3). Quarks are introduced as spinors in N f flavors, each in the fundamental representation (triplet, denoted 3) of the color gauge group, SU(3). Technically, QCD is a gauge theory with SU(3) gauge symmetry. There are many other possible choices, but all are mathematically equivalent, at least equally complicated, and give the same physical results. There is no way to add any combination of these states to produce any other, and it is also impossible to add them to make r r, g g, or b b the forbidden singlet state. The critical feature of these particular eight states is that they are linearly independent, and also independent of the singlet state, hence 3 2 − 1 or 2 3. These are equivalent to the Gell-Mann matrices. The following is a list of those combinations (and their schematic names): This gives nine possible combinations of color and anticolor in gluons.
Gluons may be thought of as carrying both color and anticolor. Quarks carry three types of color charge antiquarks carry three types of anticolor. However, gluons are subject to the color charge phenomena (of which they have combinations of color and anticolor). Unlike the single photon of QED or the three W and Z bosons of the weak interaction, there are eight independent types of gluon in QCD. Experiments limit the gluon's rest mass (if any) to less than a few meV/ c 2. In quantum field theory, unbroken gauge invariance requires that gauge bosons have zero mass. While massive spin-1 particles have three polarization states, massless gauge bosons like the gluon have only two polarization states because gauge invariance requires the polarization to be transverse to the direction that the gluon is traveling.
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