Color charge

In particle physics, color charge is a property of quarks and gluons that is related to the particles' strong interactions in the theory of quantum chromodynamics (QCD). Color charge 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 is completely unrelated to visual perception of color.[1] Rather, it is a whimsical name for a property that has almost no manifestation at distances above the size of an atomic nucleus. The term color was chosen because the abstract property to which it refers has three aspects, which are analogized to the three primary colors of red, green, and blue.[2] By comparison, the electromagnetic charge has a single aspect, which takes the values positive or negative.

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 quantum states without violating the Pauli exclusion principle. The concept turned out to be useful. The theory of quantum chromodynamics has been under development since the 1970s and constitutes an important component of the Standard Model of particle physics.

Contents

Red, green, and blue

In QCD, a quark's color can take one of three values, called red, green, and blue. An antiquark can take one of three anticolors, called antired, antigreen, and antiblue (represented as cyan, magenta and yellow, respectively). Gluons are mixtures of two colors, such as red and antigreen, which constitutes their color charge. QCD considers eight gluons of the possible nine color-anticolor combinations to be unique; see eight gluon colors for an explanation.

The following illustrates the coupling constants for color-charged particles:

The quark colors (red, green, blue) combine to be colorless

The quark anticolors (antired, antigreen, antiblue) also combine to be colorless

A hadron with 3 quarks (red, green, blue) before a color change

Red quark emits a red-antigreen gluon

Green quark has absorbed the red-antigreen gluon and is now red; color is conserved

Coupling constant and charge

In a quantum field theory, a coupling constant and a charge are different but related notions. 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; i.e., 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

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

where A_\mu is the photon field, and \psi is the electron field with Q=-1 (a bar over \psi denotes its antiparticle — the positron). 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 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

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

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

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

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 2/3. 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 → 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.

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 color 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.

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

See also

References

  1. ↑ Feynman, Richard (1985). QED: The Strange Theory of Light and Matter. Princeton University Press. p. 136. ISBN 0-691-08388-6. "The idiot physicists, unable to come up with any wonderful Greek words anymore, call this type of polarization by the unfortunate name of 'color,' which has nothing to do with color in the normal sense." 
  2. ↑ Close (2007).

Further reading