# Color charge

Post-publication activity

Curator: O. W. Greenberg

Color charge labels the states of quarks, antiquarks and gluons and also is the source of the forces between these particles.

## Introduction

Color charge is the 3-valued hidden quantum number carried by quarks, antiquarks and gluons. Color charge has a 3 valuedness that we associate with the group $$SU(3)_{color}\ .$$ Color charge is hidden in the sense that only singlets of $$SU(3)_{color}$$ that are neutral occur in nature (at least macroscopically and at low temperatures). The strongly interacting color-neutral particles composed of quarks, antiquarks and gluons that occur in nature are called hadrons. (The word color in this context is purely colloquial and has no relation to the color that we see with our eyes in everyday life.)

Color charge has two aspects: (a) as a quantum number that labels states of quarks, antiquarks and gluons: hadrons are in the singlet of $$SU(3)_{color}$$ as a global symmetry group and (b) as the source of the strong color force acting between quarks associated with $$SU(3)_{color}$$ as a local gauge group. Each of these is analogous to aspects of electric charge: (a) as a quantum number that counts the amount of electric charge in a state: neutral atoms have zero electric charge under $$U(1)$$ as a global symmetry group, (b) as the source of electromagnetic forces associated with $$U(1)$$ as a local gauge group acting between electrically charged particles .

O.W. Greenberg introduced the aspect of color charge as a quantum number in 1964 (Greenberg 1964). Y. Nambu, (Nambu 1966) and M.-Y. Han and Y. Nambu (Han and Nambu 1965) introduced the aspect of color charge as the source of the force between quarks in 1965 associated with the local gauge group $$SU(3)_{color}\ .$$

## Quark properties

Six flavors of quarks have been discovered at currently accessible energies. These 6 flavors fall into 3 generations, each having two flavors, an up-type, $$U\ ,$$ and a down-type, $$D\ ,$$ as listed in the table below. Note that the color quantum number of the quarks, that they are in the $$3$$ of $$SU(3)_{color}\ ,$$ is independent of the generation, flavor and electric charge within each generation. This is required by the exact conservation of color.

 Flavor type Generation Color $$Q$$ $$B$$ $$J^P$$ 1 2 3 $$U$$ $$u$$ $$c$$ $$t$$ 3 $$\frac{2}{3}$$ $$\frac{1}{3}$$ $$\frac{1}{2}^+$$ $$D$$ $$d$$ $$s$$ $$b$$ 3 $$-\frac{1}{3}$$ $$\frac{1}{3}$$ $$\frac{1}{2}^+$$ $$\bar{U}$$ $$\bar{u}$$ $$\bar{c}$$ $$\bar{t}$$ $$3^{\star}$$ $$-\frac{2}{3}$$ $$-\frac{1}{3}$$ $$\frac{1}{2}^-$$ $$\bar{D}$$ $$\bar{d}$$ $$\bar{s}$$ $$\bar{b}$$ $$3^{\star}$$ $$\frac{1}{3}$$ $$-\frac{1}{3}$$ $$\frac{1}{2}^-$$

The table gives the quantum numbers of quarks and antiquarks. $$Q$$ is the electric charge of the quarks in units of $$e\ ,$$ the electric charge of the proton. $$B$$ is the baryonic charge of the quarks. $$J$$ is the spin angular momentum of the quarks in units of $$\hbar$$ and $$P$$ is the parity of the quarks.

## Confinement of quarks, antiquarks and gluons in hadrons

Although hadrons, are composed of quarks, antiquarks and gluons, these particles cannot be isolated macroscopically. Hadrons, which can be isolated, are neutral combinations of quarks, antiquarks and gluons as stated above. Protons, neutrons and $$\pi$$ mesons are typical hadrons. In this sense, hadrons are analogous to atoms, which are neutral composites of electrically charged particles. However quarks differ from the charged components of a neutral atom. The electrons in an atom can be ionized, i.e. separated from the atom; in contrast the quarks cannot be separated from a hadron, except transiently for times of the order of $$3 \times 10^{-23}$$ sec. or distances of the order of 1 fm.

This property, that quarks cannot be isolated, called confinement, comes from the fact that the force between quarks (and antiquarks) does not decrease with distance. This reflects itself in the roughly linear growth with distance of the potential energy between separated quarks (and antiquarks). For sufficiently large separation this potential energy becomes large enough to allow production of new particles, typically mesons, but does not permit the quarks to be isolated macroscopically.

## Color as a quantum number that labels states

The role of color charge as a quantum number that is neutral in hadrons is analogous to the role of electric charge as a quantum number that is neutral in un-ionized atoms. Each flavor of quark and antiquark carries the 3-valued color charge. For quarks the color charge transforms as a $$3_{\alpha}$$ under the global $$SU(3)_{color}\ ;$$ for antiquarks the color charge transforms as a $$3^{\star~\beta}\ .$$ Gluons transform as the traceless part of an $$8_{\alpha}^{\beta}\ .$$ States composed of products of quarks, antiquarks and gluons can be reduced into irreducible representations of $$SU(3)_{color}\ .$$

The only states that occur macroscopically (and at low temperature) are the singlets of $$SU(3)_{color}\ .$$ These are hadrons, the color neutral states. For example, mesons have a term $$q_{\alpha} \bar{q}^{\alpha}$$ as the leading constituent, as well as higher components with additional gluons and quark-antiquark pairs such as $$q_{\alpha}G^{\alpha}_{\beta}\bar{q}^{\beta}\ ,$$ $$q_{\alpha}\bar{q}^{\beta}q_{\beta}\bar{q}^{\alpha}$$ and $$q_{\alpha}\bar{q}^{\beta}G_{\beta}^{\gamma}q_{\gamma}\bar{q}^{\delta} G^{\alpha}_{\delta}$$ constructed so that the state is a color singlet. Baryons have a leading term $$\epsilon^{\alpha \beta \gamma} q_{\alpha} q_{\beta} q_{\gamma}$$ as well as terms with additional gluons, quarks and antiquarks such as $$\epsilon^{\beta \delta \sigma} q_{\alpha} G^{\alpha}_{\beta} q_{\gamma}G^{\gamma}_{\delta}q_{\sigma}\ .$$

## Color as the source of the strong color force

Color charge plays a second role in connection with the strong force between color-charge carrying particles, just as electric charge plays a second role in connection with the electromagnetic force between electrically charged particles. The electric force is mediated by photons, which are the quanta of the electromagnetic field associated with the local gauge group $$U(1)\ .$$ Photons do not carry electric charge. In contrast, the strong or color force is associated with the nonabelian local gauge group $$SU(3)_{color}\ .$$ Because the color charges do not commute with each other, i.e. are nonabelian, the gluons carry color charges, roughly equivalent to the combined charges of a quark and an antiquark.

Quarks carry the the fundamental $$3$$ representation of $$SU(3)_{color}\ .$$ Antiquarks carry the complex conjugate $$3^{\star}$$ representation of the group. Gluons, the mediators of the color force, carry the 8-dimensional adjoint representation of the color group. This adjoint representation is analogous to the traceless product of the $$SU(3)_{color}$$ representations of a quark and an antiquark. The strong force is mediated by the gluons which are quanta of the color or quantum chromodynamic field associated with the local gauge group $$SU(3)_{color}$$ in analogy to photons as mediators of the electromagnetic force associated with the $$U(1)$$ gauge theory. However, the gluons interact directly with each other as well as with quarks and antiquarks, in contrast to photons, which interact directly only with electrically charged particles.

## Color force as a paradigm shift from meson forces as the source of strong interactions

The discovery of quark color in 1964 and the gauge theory, $$SU(3)_{color}\ ,$$ in 1965 changed our understanding of the strong interaction in a qualitative way. Before the discovery of the color carried by quarks, the strong interaction was thought to be mediated by the exchange of mesons, such as the $$\pi^{+,0,-}\ ,$$ and the $$\omega^0\ ,$$ $$\rho^{+,0,-}\ ,$$ $$\phi^0$$ mesons. With discoveries of quarks and color we now understand the strong interaction to be connected with the $$SU(3)_{color}$$ gauge theory. We do not believe, however, that single exchange of gluons, which might be expected from perturbation theory, is the main mechanism. Indeed, meson exchange may also play a role in the strong interaction.

## Flavor independence of color

Hadrons are singlets (i.e., neutral) under the color group, so their color charges are hidden. The coupling of quarks to gluons is independent of the flavor of the quark. Further, the coupling of quarks to photons is also independent of the electric charge of the quark. This latter property is required for the exact conservation of both color charge and electric charge. As stated above, quarks carry fractional values of the electric charge, e, of the proton; however hadrons, which only occur in color singlets, carry integer values of electric charge.

## Consequences of the nonabelian color force

The non-abelian character of the color field has profound consequences for the color force. The color force becomes weak (as the reciprocal of the logarithm of the energy) at high energy or short distance, and it becomes strong at low energy or long distance. The weakness at high energy, called asymptotic freedom, provides a justification for the quasi-free behavior of quarks and gluons in the parton model of hadrons, which is useful in describing high-energy scattering. The strength of the color force at low energy leads to the confinement of quarks and gluons, discussed above. This provides an explanation for the observed absence of particles with fractional values of electric charge, despite the fractional values of electric charge carried by quarks.

## Road from color charge to quantum chromodynamics (QCD)

Crucial steps along this road include the demonstration of renormalizability for non-abelian gauge theories, the discovery that these theories are asymptotically free, and the importance of effects outside the realm of perturbation theory.

## Empirical evidence for color charge

### Ground state baryons

The first observational evidence for color charge was the permutation symmetry of the quark quantum numbers in the ground state baryons. The $$SU(6)$$ model of F. Gürsey and L.A. Radicati (Gürsey and Radicati 1964) discussed below placed the ground state baryons in the 56 representation of $$SU(6)\ .$$ This representation is symmetric under permutations of the quark quantum numbers. Since the quarks have spin 1/2 they should be fermions according to the spin-statistics theorem and should be antisymmetric under permutations of their quantum numbers. This contradiction is the spin-statistics paradox. The introduction of the three-valued hidden color charge by Greenberg in 1964 resolved this contradiction by making the state of the 3-valued color charge carried by quarks antisymmetric. It justified the symmetry of the baryons under permutations of their visible degrees of freedom, which are the space, spin and flavor degrees of freedom.

### The symmetric quark model for baryons

Greenberg constructed the symmetric quark model for baryons using the antisymmetry of the color degree of freedom to justify symmetry of the baryon wave functions in the space, spin and flavor degree of freedom. He gave a table of baryon states for quarks in the $$s$$ and $$p$$ states. With M. Resnikoff (Greenberg and Resnikoff 1967) he gave a detailed analysis of baryons in the $$(56, 1^+)$$ and $$(70,1^-)$$ supermultiplets, where the notation is $$(\mathrm{dim} SU(6), L^P)\ .$$ This model is the starting point for the study of baryon spectroscopy. For mesons, since there is only one quark or antiquark in the constituent model, the statistics of the quarks is irrelevant.

### Neutral pion decay to photons

Further observational evidence for color charge came from the decay rate for $$\pi^0 \rightarrow \gamma \gamma$$ via the axial anomaly. The 3-valued color charge enters as the square of the number of colors and provides a factor of 9 that brings the theoretical calculation of the decay rate into agreement with experiment.

Another experimental evidence came from the ratio $$\sigma(e^+ e^- \rightarrow hadrons)/\sigma(e^+ e^- \rightarrow \mu^+ \mu^-)\ .$$ Here the 3-valued color charge provides a factor of 3 that brings theory in agreement with experiment. These two tests of color, as well as the permutation symmetry of the quarks in baryons, test the color quantum numbers of the quarks, but do not (in lowest order) test the nonabelian nature of the color force.

## Empirical tests of the nonabelian color force

Among the tests of the nonabelian nature of the color force are asymptotic freedom, permanent confinement of quarks, jets in high-energy scattering, and many precision tests of the standard model. Asymptotic freedom, the decrease of the color force at short distance or high energy, results from the nonabelian interaction of gluons with quarks. Virtual gluons antishield the color charge of a quark (or antiquark) at short distance and thus reduce the color force at short distances. At long distance the growing potential interaction energy of quarks and antiquarks prevents the isolation of individual quarks or antiquarks. Jets in high-energy scattering reflect the transient production of quarks which hadronize to produce strongly interacting particles. Precision tests of quantum chromodynamics, such as the running of coupling constants, also depend on the nonabelian nature of the color interaction.

## Historical developments that led to color

### Precursors of quarks

The idea of constructing known particles out of other, more basic particles, has a long history, going back to the effort of M. Born and N. Nagendra Nath (Born and Nagendra Nath 1936) to construct the photon from neutrinos in the 1930's and the effort of E. Fermi and C.N. Yang (Fermi and Yang 1949) to construct pions from protons and neutrons in the 1040's. S. Sakata (Sakata 1956) proposed a model in 1956 with the proton, neutron and lambda baryon were taken as the basic particles. The Sakata model failed to account for the octet of baryons (p,n,$$\Lambda,\Sigma^{+,0,-},\Xi^{0,-}$$).

### Quarks

#### Current quarks

M. Gell-Mann (Gell-Mann 1964) suggested a model with three quarks, $$u^{\frac{2}{3}}, d^{-\frac{1}{3}}, s^{-\frac{1}{3}}$$ as the fundamental objects in 1964. Gell-Mann's model has the radical departure that the quark electric charges, shown as superscripts above, are fractions of the proton charge $$e\ .$$ In addition the quarks carry baryon numbers $$1/3$$ of the baryon number of the nucleon. Particles with either of these fractional values have never been observed. Gell-Mann gave expressions for the vector electromagnetic current and the vector and axial weak currents in terms of quark fields. He showed that quark currents could give the Cabibbo model of weak interactions and also build up the $$SU(3) \times SU(3)$$ current algebra. Gell-Mann's original model used the three flavors, $$u,d,s$$ that were known in 1964. Three additional flavors, $$c,t,b$$ have since been discovered.

#### Constituent quarks

G. Zweig (Zweig 1964) independently introduced a model with fractional electric and baryon charges with emphasis on the constituent point of view. Zweig called the fundamental objects in his model aces, a name that was superseded by quarks. He gave an extensive analysis of the space-time and group theory structure of baryons and mesons on the basis of his model. He gave the distinctions between the predictions of the earlier eightfold way and his model. He also gave a reason for the suppression of the decay $$\phi \rightarrow \rho \pi$$ which is an example of the Okubo, Zweig and Iizuka rule  (Okubo 1963, Zweig 1964, Iizuka 1966) that can be described in terms of quark line diagrams. Zweig adopted the point of view that his aces are real physical particles. Both Gell-Mann and Zweig assumed the quarks have spin $$1/2$$ in order that mesons can be constructed from $$(q \bar{q})$$ and baryons can be constructed from $$(qqq)\ ;$$ however spin did not enter their models in a deeper way.

### Incorporation of spin in the quark model

In the same year, 1964, that Gell-Mann and Zweig independently suggested quarks and aces, F. Gürsey and L.A. Radicati (Gürsey and Radicati 1964) incorporated spin in the model in a more intrinsic way. Gürsey and Radicati considered the quarks to be a $$6$$ of an $$SU(6)$$ group that included the $$SU(3)_{flavor}$$ of the original quark model with the 3 flavors $$u,d,s$$ and the $$SU(2)_{spin}$$ of the quark spin. Because baryons are composed of 3 quarks, Gürsey and Radicati considered the reduction of $$6 \otimes 6 \otimes 6 \rightarrow 56 + 70 + 70 + 20$$ into irreducibles of $$SU(6)\ .$$ Gürsey and Radicati placed the ground state baryons in the $${56}$$ representation of $$SU(6)$$ which includes the known nucleon octet and delta decuplet $\tag{1} {56}\rightarrow ({8},{1/2})+({10},{3/2})~ {\rm under}~ SU(6)_{fS} \rightarrow SU(3)_f \times SU(2)_S,$

where the nucleon octet is $$(p^+,n^0,\Lambda^0,\Sigma^+,\Sigma^0,\Sigma^-, \Xi^0,\Xi^-)$$ and the delta decuplet is $$(\Delta^{++},\Delta^+,\Delta^0,\Delta^-, Y^{\star~+}_1,Y^{\star~0}_1,Y^{\star~-}_1,\Xi^{\star~0},\Xi^{\star~-}, \Omega^-)\ .$$ The $${56}$$ representation has the three quarks in a symmetric state under permutations in contradiction with what one would expect. The spin-statistics theorem requires that spin 1/2 quarks obey the Pauli exclusion principle and occur in the $${\mathbf 20}$$ representation which is antisymmetric under permutations if there are no additional degrees of freedom carried by quarks.

Gürsey and Radicati noted the symmetry of the $$\mathbf{56}$$ representation for the ground state baryons and suggested that this would imply that the forces between the quarks are repulsive. R.H. Dalitz (Dalitz 1966) supported the idea that an antisymmetric space wavefunction for the ground state baryons would allow the $$SU(3)$$ and spin state to be symmetric without violating the spin-statistics connection. Many physicists were skeptical of the reality of quarks. If quarks were merely a mathematical device then perhaps their statistics could be ignored. Greenberg (Greenberg 1964) proposed the solution to the spin-statistics paradox that has been confirmed by experiment and was ultimately accepted by the physics community. He suggested that each flavor of quark comes in three varieties, colloquially called colors. The quark color degree of freedom, can be taken treated as the fundamental $$3$$ of a new $$SU(3)_{color}$$ symmetry. Then in a neutral state, i.e. in the singlet of $$SU(3)_{color}\ ,$$ the color degree of freedom is antisymmetric. The quarks as fermions are then antisymmetric under permutations of all their degrees of freedom. The total antisymmetry comes about from the product of the symmetry of the space, spin and flavor degrees of freedom and the antisymmetry of the color degree of freedom. (Greenberg introduced color using parafermi statistics of order 3 and showed that the generalized spin-statistics theorem for parastatistics is obeyed.)

### Skepticism about quarks and color

The idea of quarks was received with skepticism because particles with fractional electric charges and baryon number had never been observed. The suggestion that, in addition, quarks carry a new hidden three-valued charge was received with even greater skepticism. Because of this quarks and color were not accepted generally by the physics community until the discovery of the $$J/\Psi$$ and other hadrons that carry charm quarks in 1974.

For the mesons composed of a quark and an antiquark the quark statistics is irrelevant.

### The relation between parafermi statistics and explicit color

The relation between the order 3 of parafermi statistics for quarks and the order 3 for the explicit color first proposed by Nambu and by Han and Nambu was clarified by Greenberg and D. Zwanziger in 1966 (Greenberg and Zwanziger 1966). They showed that the states that are bosons or fermions in the parastatistics model are in 1-to1 correspondence with the states that are color singlets in the $$SU(3)_{color}$$ model. Further analysis of this question was given by Drühl, Haag and Roberts (Drühl, Haag and Roberts 1970)

## Summary

In summary, color charge plays two roles in the standard model: (1) as the 3-valued charge that labels states of quarks, antiquarks, gluons and their composites, and (2) as the source of the strong force between quarks and antiquarks mediated by gluons.

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