‘‘The Cabibbo–Kobayashi–Maskawa Quark-Mixing Matrix,’’ F. J.

Gilman, K. Kleinknecht, and B. Renk, in

, K. Hagiwara

,

73

pp. 113–119.

A theory of particles and their interactions permitting ar-

bitrary changes of phase in the particle’s quantum mechani-

cal state is an

such as electro-

magnetism. The term ‘‘Abelian’’ indicates that gauge

~

phase

!

transformations at a given space–time point commute with

one another, while ‘‘local’’ stands for the freedom to make

separate gauge transformations at each space–time point.

The name ‘‘gauge’’ originated with Hermann Weyl.

146

Gauge transformations may be generalized to those that do

commute with one another at a given space–time point.

The first such

gauge theory was proposed by C.

N. Yang and R. L. Mills,

147

who used it to describe the strong

interactions through self-interacting mesons of spin 1 carry-

ing isosopic spin.

The review by Abers and Lee

48

helped a generation of

physicists to apply gauge theories to the electroweak and

strong interactions. An excellent introduction to the subject

at the intermediate graduate level is given by Quigg.

18

An

article addressed to the lay reader has been written by

’t Hooft.

111

A recent text

148

provides a further introduction to

the subject.

‘‘Electron and Gravitation

@

in German

#

,’’ H. Weyl, Z. Phys.

, 330–

352

~

1929

!

, partially reprinted in Surveys in High Energy Phys.

,

261–267

~

1986

!

.

~

A

!

‘‘Conservation of Isotopic Spin and Isotopic Gauge Invariance,’’ C.

N. Yang and R. L. Mills, Phys. Rev.

, 191–195

~

1954

!

.

~

A

!

See

also Cambridge University Dissertation, R. Shaw, 1954

~

unpub-

lished

!

.

, L. O’Raifeartaigh

~

Princeton Uni-

versity Press, Princeton, NJ, 1997

!

.

~

I

!

The quarks are distinguished from the leptons by possess-

ing a three-fold charge known as ‘‘color’’ that enables them

to interact strongly with one another.

149–151

We also speak of

quark and lepton ‘‘flavor’’ when distinguishing the particles

in Table V from one another. The evidence for color comes

from several quarters.

The

D

11

, a low-lying excited state of

the nucleon, can be represented in the quark model as

,

so it is totally symmetric in flavor. It has spin

5

3/2, a

totally symmetric combination of the three

5

1/2 quark

spins. As a ground state, its spatial wave function should be

symmetric as well. While a state composed of fermions

should be totally

under the interchange of any

two fermions, the state described so far is totally

under the product of flavor, spin, and space interchanges.

Color introduces an additional degree of freedom under

which the interchange of two quarks can produce a minus

sign.

The charges

of all quarks that can be produced in pairs at a given center-

of-mass energy is measured by the ratio

[

s

(

1

2

→

hadrons)/

s

(

1

2

→

m

1

m

2

)

5

(

2

, where

is the

charge of quark

in units of

u

u

. Measurements

73

indicate

values of

in various energy ranges consistent with

5

3

~

with a small positive correction associated with the strong

interactions of the quarks

!

.

The

p

0

decay rate is governed by a

quark loop diagram in which two photons are radiated by the

quarks in

p

0

5

(

2

)/

A

2. The predicted rate is

G

(

p

0

→

gg

)

5

7.6

2

eV, where

5

(

2

2

2

)

5

/3. The ex-

perimental rate is 7.8

6

0.6 eV, in accord with experiment if

5

1 and

5

3.

Quark composites appear only in multiples of

three. Baryons are composed of

, while mesons are

~

with total quark number zero

!

. This is compatible with our

current understanding of QCD, in which only color-singlet

states can appear in the spectrum.

A crucial feature of the QCD theory of strong interactions

is its ‘‘asymptotic freedom,’’ a weakening interaction

strength at short distances permitting the interpretation of

deep inelastic scattering experiments

96,152,153

in terms of

quarks. This property was found to be characteristic of non-

Abelian gauge theories such as color SU

~

3

!

by Gross and

Wilczek

154–156

and by Politzer.

157,158

The result was obtained

earlier for the gauge group SU

~

2

!

by Khriplovich

159

~

see also

Ref. 160

!

, but its significance for a strong-interaction theory

was not realized then.

Direct evidence for the quanta of QCD, the gluons, was

first presented in 1979 on the basis of extra ‘‘jets’’ of par-

ticles produced in electron–positron annihilations to hadrons.

Normally one sees two clusters of energy associated with the

fragmentation of each quark in

1

2

→

into hadrons.

However, in some fraction of events an extra jet was seen,

corresponding to the radiation of a gluon by one of the

quarks. For a popular history of this discovery, containing

further references, see Ref. 96.

The transformations that take one color of quark into an-

other are those of the group SU

~

3

!

. This group is called

SU

~

3

!

color

to distinguish it from the SU

~

3

!

flavor

associated

with the quarks

,

, and

.

‘‘Spin and Unitary Spin Independence in a Paraquark Model of Bary-

ons and Mesons,’’ O. W. Greenberg, Phys. Rev. Lett.

, 598–602

~

1964

!

.

~

I

!

Y. Nambu, ‘‘A Systematics of Hadrons in Subnuclear Physics,’’ in

, ed-

ited by A. De-Shalit, H. Feshbach, and L. Van Hove

~

North-Holland,

Amsterdam and Wiley, New York, 1966

!

, pp. 133–42.

~

A

!

‘‘Advantages of the Color Octet Gluon Picture,’’ H. Fritzsch, M.

Gell-Mann, and H. Leutwyler, Phys. Lett.

, 365–368

~

1973

!

.

~

I

!

‘‘High-Energy Inelastic

Scattering at 6° and 10°,’’ E. D. Bloom

, Phys. Rev. Lett.

, 930–934

~

1969

!

.

~

I

!

‘‘Observed Behavior of Highly Inelastic Electron–Proton Scatter-

ing,’’ M. Breidenbach

, Phys. Rev. Lett.

, 935–939

~

1969

!

.

~

I

!

‘‘Ultraviolet Behavior of Non-Abelian Gauge Theories,’’ D. J. Gross

and F. Wilczek, Phys. Rev. Lett.

, 1343–1346

~

1973

!

.

~

A

!

‘‘Asymptotically Free Gauge Theories. I,’’ D. J. Gross and F. Wil-

czek, Phys. Rev. D

, 3633–3652

~

1973

!

.

~

A

!

‘‘Asymptotically Free Gauge Theories. 2,’’ D. J. Gross and F. Wil-

czek, Phys. Rev. D

, 980–993

~

1974

!

.

~

A

!

‘‘Reliable Perturbative Results for Strong Interactions?,’’ H. David

Politzer, Phys. Rev. Lett.

, 1346–1349

~

1973

!

.

~

A

!

‘‘Asymptotic Freedom: An Approach to Strong Interactions,’’ H.

David Politzer, Phys. Rep.

, 129–180

~

1974

!

.

~

A

!

‘‘Green’s Functions in Theories with Non-Abelian Gauge Group,’’ I.

B. Khriplovich, Yad. Fiz.

, 409–424

~

1969

! @

Sov. J. Nucl. Phys.

, 235–242

~

1969

!#

.

~

A

!

‘‘Renormalization of Gauge Theories,’’ G. ’t Hooft, in Ref. 85, pp.

179–198.

~

A

!

309

309

Am. J. Phys., Vol. 71, No. 4, April 2003

Jonathan L. Rosner

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