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Color charge

From Academic Kids

In particle physics, color charge is a property of quarks and gluons which are related to their strong interactions in the context of quantum chromodynamics (QCD). This has analogies with the notion of electric charge of particles, but because of the mathematical complications of QCD, there are many technical differences. The color of quarks and gluons have nothing to do with the visual perception of color, but is a whimsical name for a property which has almost no manifestation at distances above the size of an atomic nucleus.

Shortly after the existence of quarks was first proposed in 1964, Oscar W. Greenberg introduced the notion of color charge to explain how quarks could coexist inside some hadrons in otherwise identical states and still satisfy the Pauli exclusion principle. The concept turned out to be useful. Quantum chromodynamics has been under development since the 1970s and constitutes an important ingredient in the standard model of particle physics.

Contents

Red, blue and green

One can say that a quark's color can take three values: "red", "green", or "blue"; and that an antiquark can take on three "anticolors", sometimes called "antired", "antigreen" and "antiblue" (occassionally represented as cyan, magenta and yellow). In the same vein it can be said that gluons are mixtures of two colours: for example red-antigreen, and this constitutes their color charge. Also it is further stated that there are eight gluons, leaving one to figure out where the ninth one went. In this language, the answer is that one particular combination: the (red-antired)+(blue-antiblue)+(green-antigreen) is actually colourless and hence does not need to be considered any more. This is about as far as one can push this language. Further elaboration requires a little bit of background which the next sections build and use. One should read them along with the companion article on coupling constants.

Coupling constant and charge

In a quantum field theory the notion of a coupling constant and a charge are different but related. The coupling constant sets the magnitude of the force of interaction; for example, in quantum electrodynamics, the fine structure constant is a coupling constant. The charge in a gauge theory has to do with the way a particle transforms under the gauge symmetry, ie, its representation under the gauge group. For example, the electron has charge -1 and the positron has charge +1, implying that the gauge transformation has opposite effects on them in some sense. Specifically, if a local gauge transformation φ(x) is applied in electrodynamics, then one finds

<math>A_\mu\to A_\mu+\partial_\mu\phi(x)<math>,   <math>\psi\to \exp[iQ\phi(x)]\psi<math>  and  <math>\overline\psi\to \exp[-iQ\phi(x)]\overline\psi<math>

where Aμ is the photon field, and ψ is the electron field with Q=-1 (a bar over ψ denotes its antiparticle— the electron). Since QCD is a non-Abelian theory, the representations, and hence the color charges, are more complicated. They are dealt with in the next section.

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 αs. Each flavour of quark belongs to the fundamental representation (3) and contains a triplet of fields together denoted by ψ. The antiq</b>uark field belongs to the complex conjugate representation (3*) and also contains a triplet of fields. We can write

<math>\psi = \begin{pmatrix}\psi_1\\ \psi_2\\ \psi_3\end{pmatrix}<math>  and  <math>\overline\psi = \begin{pmatrix}\overline\psi^*_1\\ \overline\psi^*_2\\ \overline\psi^*_3\end{pmatrix}<math>

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

<math>{\mathbf A}_\mu = A_\mu^a\lambda_a.<math>

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 and antiquarks have color charge 4/3, whereas gluons have color charge 8. All other particles have zero colour 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 → Uψ, where U is a 3X3 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.

Missing image
Vertex.png
Color line representation of QCD vertex

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 colour line representation. The meaning is the following. Let ψi 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 qi→gij+qj. 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.

Missing image
3gluon.png
Color line representation of 3-gluon vertex

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 examaple, the W boson in the electroweak theory. In the electroweak theory, the W also carries electric charge, and hence interacts with a photon.

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