# Quantum Chromodynamics

# QUANTUM CHROMODYNAMICS

Quantum chromodynamics (QCD) is the component of the Standard Model that describes the strong interactions. QCD is the theory of quarks and gluons. Quarks carry a new charge, called color, that enables them to emit and absorb gluons. (This is the origin of the name chromodynamics, although the "color" of QCD should in no way be confused with the colors of light.) The quarks are also electrically charged and like electrons, are fermions, and carry a spin, or intrinsic angular momentum, of one-half in units of Planck's constant. The gluons are electrically neutral, and, like photons, are bosons of spin one. Together, the fields of quarks and gluons make up a nonabelian gauge theory.

In QCD, quarks interact by the exchange of gluons in much the same way that electrons interact by the exchange of the quanta of light, photons. Like photons, the gluons have no mass and travel at the speed of light. Unlike photons, however, the gluons carry the very color charge that produces them, so gluons can emit and absorb more gluons. The resulting strong force is thus more complicated to analyze than the electromagnetic force.

A convenient measure of the strength of the strong force is the QCD coupling α_{s}(*Q* ) that controls the probability of a quark emitting a gluon, which produces forces between quarks. The QCD coupling depends on the momentum carried by the emitted gluon, denoted by *Q* . The strong coupling is large for very low-momentum gluons and decreases as the momentum increases, a variation known as asymptotic freedom. For the highest-momentum gluons that can be produced in modern accelerators, α_{s}(*Q* ) is relatively small, about 0.1, but at momentum transfers characteristic of nuclear interactions, it gets to be quite large. Asymptotic freedom makes it easier to analyze processes over short times, which generally involve a few gluons, than over long times, which generally involve many.

Quarks come in six varieties, known as quark flavors. In the Standard Model, the six flavors of quarks, together with the six leptons (the electron, muon, tau, and their neutrinos), are truly elementary. The different flavors of quarks have different charges. Three quarks have electric charge +2*e* /3: the up (*u* ), charm (*c* ) and top (*t* ) quarks. Three quarks have charge -*e* /3: the down (*d* ), strange (*s* ) and bottom (*b* ) quarks; -*e* is the charge of an electron. The masses of these quarks vary greatly, and of the six, only the *u* and *d* quarks, which are by far the lightest, appear to play a direct role in normal matter.

## Hadrons

QCD binds quarks together into states that can be observed directly in the laboratory as hadrons, particles that feel the strong force. The best-known examples of hadrons are the nucleons, the proton and neutron, from which all atomic nuclei are formed. The idea of quarks arose to explain the regularities of hadron states, and their charges and spins could be readily explained (and even predicted) by simply combining the then known *u* , *d* , and *s* quarks. This is the quark model. Perhaps the most extraordinary feature of quarks in QCD is the confinement of quarks in hadrons. A free quark, one that is separated from a nucleon, would be readily detectable because its charge would be -⅔ or ⅓ that of the charge of an electron. No convincing evidence for such a particle has been found, and it is now believed that confinement is an unavoidable consequence of QCD.

There must be three quarks to make a proton or a neutron. The proton and neutron are different because the proton is a combination of two *u* quarks and one *d* quark and hence has a total charge of 2(2*e* /3) + (-*e* /3)= +*e* . The neutron is made up of one *u* and two *d* quarks and hence has total charge (2*e* /3) + 2(-*e* /3) = 0.

In addition to nucleons, other combinations of three quarks have been observed, and all are known collectively as baryons. For example, from the *u* , *d* , and *s* quarks, it is possible to make ten distinct combinations, and all have been seen (notice that they all have charges that are integer multiples of *e* ). In addition, baryons with *c* and *b* quarks have also been observed. All the baryons except for the proton and neutron decay quite rapidly because the weak interactions make all the quarks aside from the *u* and *d* unstable. The spins of the quarks that combine to form baryons may line up in parallel and antiparallel combinations, as long as the resulting state obeys the Pauli exclusion principle, which states that no two fermions may have the same set of quantum numbers. As a result, some of the baryons have total spin 1/2, and others spin 3/2, and all the baryons are fermions with half-integer total spin. With orbital angular momentum taken into account, even higher spins are possible, although these baryons are very unstable.

In addition to baryons, quarks can combine with their antiparticles, the antiquarks, to form mesons. Antiquarks are usually denoted by an overbar, such as ū for the antiquark of the *u* . For example, the combinations ū and *d̄u* can form the π^{+} and π^{-}, the pions, of electric charge ^{+}*e* and ^{-}*e* , respectively. All the mesons are bosons with integer spins. Other exotic bosons, which are bound states of gluons without quarks, called glueballs, appear to be possible, although very unstable.

## The Color Charge and Gauge Theory

The concept of color was introduced to solve a problem in the quark model. The lowest-energy states in the quark model appeared to require that the three quarks in the proton be in identical states, which would violate the Pauli principle. With color, this conflict is avoided. For example, two *u* quarks can have the same energy and spin quantum numbers as long as their colors are different. Consistency with the Paul principle requires that all the quarks in any baryon have different colors: three quarks, three colors.

QCD is an example of a nonabelian gauge theory. In a nonabelian gauge theory, the concept of charge must be generalized from electromagnetism. An electric charge is just a number, such as *e* or 2*e* /3, and the electric charge stays the same when a charged particle emits or absorbs a photon. A quark, however, actually has three separate charges that make up its color, which are sometimes labeled by the names of colors of light: a red (*r* ) charge, a green (*g* ) charge, and a blue (*b* ) one. When a quark emits a gluon, its color charges change, and the kinds of gluons can be identified by combinations of the color "before" and the color "after," for example,*r̄b* , where the color with the overbar is the final color of the quark. Correspondingly, when a quark with only color *b* absorbs an *r̄b* gluon, it changes into a quark with only color *r* . This way, the total of *r* , *g* , and *b* charges are conserved, and nine possibilities exist for the gluons. Of these nine, one combination, equal parts of *r̄r* , *ḡg* , and *b̄b* , leaves all three of the colors the same and is absent, leaving eight gluons. The color charges in QCD have a surprising property: they can never be distinguished experimentally, and yet their number, three, can be measured. All hadrons have zero net *r* , *b* , and *g* colors; they are all colorless.

## Evidence for Color, Quarks, and Gluon

Experimental discoveries beginning in the late 1960s and early 1970s established firmly the reality of hadrons made up of quarks and gluons. As seen, quarks carry an electric charge, as do electrons. When an energetic electron collides with a target, the electron, which is blind to the strong force, nevertheless scatters from nucleons within the target by exchanging a photon. When the energy of the electron is high, the momentum *p′* of the photon can be large, so large that the corresponding wavelength of light is much smaller than a nucleon in the target. This wavelength is given by λ = *h/p* , where *h* is Planck's constant. The rules of quantum mechanics indicate that a photon cannot be absorbed by an object that is larger than its wavelength λ.

Nevertheless, the photons *are* absorbed at a rate almost independent of *p* , a phenomenon called scaling. When this happens, the nucleon generally breaks apart into high-energy fragments, a process called deep-inelastic scattering. The scaling of deep-inelastic scattering indicates that there are charged particles within nucleons that are much smaller than nucleons. These are the quarks. In addition, the distribution in angles of the scattered electrons depends on the spin of the charged particles. The electron-nucleon experiments show that all the charge in the nucleons is carried by spin-½ particles, exactly as suggested by the quark model.

In another set of experiments, electrons and their antiparticles, positrons, collide and annihilate into a virtual state that consists of a single photon. According to the rules of quantum mechanics, this photon then transforms itself into any pair of particle and antiparticle with nonzero electric charge, say, *q* . The probability for each species is proportional to *q*^{2}, and the angular distribution at which they emerge depends on their spin. The quarks created this way are not observed directly because of confinement, but they each quickly evolve into jets of hadrons, which preserve their original directions. The total probability to produce hadrons is given simply by the probability to produce a single fermion of electric charge equal to 1, times the sum of the squares of the electric charges of all the quarks light enough to be produced at the available energy, times 3. The 3 stands for the three possible colors of each quark.

Many other high-energy experiments have confirmed the reality of quarks, gluons, and color. For example, a fraction of annihilation events include an extra jet from a gluon in addition to their quark and antiquark jets. In proton-proton scattering, jets also emerge from collisions, whose angular distributions exactly match predictions based on the elastic scattering of quarks through the exchange of gluons. Each of these predictions depends directly on the numbers of quark and gluon colors as well as their spins.

Physicists' understanding of the confinement and other low-energy properties of QCD is somewhat less complete but is still convincing. Qualitatively, it appears that the "empty" space—the vacuum—is not empty of QCD content. Analysis suggests that the vacuum acts as a sort of QCD superconductor, which repels the QCD lines of force between quarks, the QCD analogs of magnetic and electric fields between electrons. Within hadrons, this repulsion is absent. The larger the separation between isolated quarks, however, the larger the energy necessary to overcome the repulsion. Very quickly, it becomes easier to create enough pairs of quarks and antiquarks to hide all the lines of force within colorless hadrons than to lengthen them through the resistant vacuum.

Beginning in the 1970s, it became possible to simulate QCD on computers. Over time, more and more precise calculations have established that the energy of an isolated quark is essentially infinite and that the observed hadrons are, indeed, combinations of confined quarks. Experiments in which entire nuclei collide may create conditions in which quarks and gluons are temporarily freed, or deconfined. This would be a novel state of matter, called the quark-gluon plasma.

It is now understood that the Standard Model actually requires that there be three colors to avoid quantum inconsistencies, called anomalies, which would ruin it as a quantum theory. Nevertheless, physicists are far from a complete understanding of QCD. For example, quantum fluctuations in the QCD vacuum appear to have the ability to act differently on particles and antiparticles in a manner that is not seen in nature, a puzzle known as the strong CP problem. In this, and in how to reconcile fully the quark-gluon and baryon-meson descriptions of the strong interactions, we have much still to learn about quantum chromodynamics.

*See also:*Axion; Asymptotic Freedom; Boson, Gauge; Particle Physics, Elementary; Quantum Mechanics; Quarks; Standard Model

## Bibliography

Feynman, R. *QED, The Strange Theory of Light and Matter* (Princeton University Press, Princeton, 1988).

Hey, Y., and Walker, P. *The Quantum of the Universe* (Cambridge University Press, Cambridge, UK, 1987).

Johnson, G. *Strange Beauty* (Vintage, New York, 1999).

Kane, G. *Modern Elementary Particle Physics* (Perseus, Cambridge, MA, 1993).

Zee, A. *A Fearful Symmetry* (Princeton University Press, Princeton, 1999).

*George Sterman*

# quantum chromodynamics

**quantum chromodynamics** Study of the properties of quarks in which, to explain permissible combinations of quarks to form various elementary particles, each is assigned a colour. Quarks are given one of the three primary colours: red, green, and blue. When three quarks combine to form baryons, the resulting colour is always white. Antiquarks are given one of the three complementary colours: cyan, magenta, and yellow. When a quark combines with an antiquark to form a meson, the resulting colour is also white.

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