Color Charge - Quark and Gluon Fields and Color Charges

Quark and Gluon Fields and Color Charges

In QCD the gauge group is the non-abelian group SU(3). The running coupling is usually denoted by α_s. Each flavor of quark belongs to the fundamental representation (3) and contains a triplet of fields together denoted by ψ. The antiquark field belongs to the complex conjugate representation (3*) and also contains a triplet of fields. We can write

and

The gluon contains an octet of fields, belongs to the adjoint representation (8), and can be written using the Gell-Mann matrices as

All other particles belong to the trivial representation (1) of color SU(3). The color charge of each of these fields is fully specified by the representations. Quarks have a color charge of red, green or blue and antiquarks have a color charge of antired, antigreen or antiblue. Gluons have a combination of two color charges (one of red, green or blue and one of antired, antigreen and antiblue) in a superposition of states which are given by the Gell-Mann matrices. All other particles have zero color charge. Mathematically speaking, the color charge of a particle is the value of a certain quadratic Casimir operator in the representation of the particle.

In the simple language introduced previously, the three indices "1", "2" and "3" in the quark triplet above are usually identified with the three colors. The colorful language misses the following point. A gauge transformation in color SU(3) can be written as ψ → U ψ, where U is a 3 × 3 matrix which belongs to the group SU(3). Thus, after gauge transformation, the new colors are linear combinations of the old colors. In short, the simplified language introduced before is not gauge invariant.

Color charge is conserved, but the book-keeping involved in this is more complicated than just adding up the charges, as is done in quantum electrodynamics. One simple way of doing this is to look at the interaction vertex in QCD and replace it by a color-line representation. The meaning is the following. Let ψ_i represent the i-th component of a quark field (loosely called the i-th color). The color of a gluon is similarly given by A which corresponds to the particular Gell-Mann matrix it is associated with. This matrix has indices i and j. These are the color labels on the gluon. At the interaction vertex one has q_i → g_i j + q_j. The color-line representation tracks these indices. Color charge conservation means that the ends of these color-lines must be either in the initial or final state, equivalently, that no lines break in the middle of a diagram.

Since gluons carry color charge, two gluons can also interact. A typical interaction vertex (called the three gluon vertex) for gluons involves g + g → g. This is shown here, along with its color-line representation. The color-line diagrams can be restated in terms of conservation laws of color; however, as noted before, this is not a gauge invariant language. Note that in a typical non-abelian gauge theory the gauge boson carries the charge of the theory, and hence has interactions of this kind; for example, the W boson in the electroweak theory. In the electroweak theory, the W also carries electric charge, and hence interacts with a photon.