‘‘Natural Suppression of Symmetry Violation in Gauge Theories:

Muon-Lepton and Electron-Lepton Number Nonconservation,’’ B. W.

Lee and R. E. Shrock, Phys. Rev. D

, 1444–1473

~

1977

!

.

~

A

!

‘‘Weak Interaction Mixing in the Six-Quark Theory,’’ H. Fritzsch,

Phys. Lett. B

, 317–322

~

1978

!

.

~

I

!

‘‘Hierarchy of Quark Masses, Cabibbo Angles, and CP Violation,’’ C.

D. Froggatt and H. B. Nielsen, Nucl. Phys. B

, 277–298

~

1979

!

.

~

A

!

‘‘Unified Theories With U

~

2

!

Flavor Symmetry,’’ R. Barbieri, L. J.

Hall, S. Raby, and A. Romanino, Nucl. Phys. B

, 3–26

~

1997

!

.

~

A

!

‘‘A model for Fermion Mass Hierarchies and Mixings,’’ P. Ramond,

in

„

…

, Proceedings of

the 6th International Symposium on Particles, Strings and Cosmol-

ogy, Boston, MA, 22–27 Mar. 1998, edited by P. Nath

~

World Sci-

entific, Singapore, 1999

!

, pp. 567–577.

~

A

!

‘‘Mass and flavor mixing schemes of quarks and leptons,’’ H. Fritzsch

and Z.-z. Xing, Prog. Part. Nucl. Phys.

, 1–81

~

2000

!

.

~

A

!

‘‘GUT model predictions for neutrino oscillation parameters compat-

ible with the large mixing angle MSW solution,’’ C. H. Albright and

S. Geer, Phys. Rev. D

, 073004

~

2002

!

, and Refs. 11 and 15

therein.

~

A

!

Families of quarks and leptons appear to be replicas of one

another

~

see Table V

!

, aside from their differing masses and

weak couplings. Attempts have been made to explain this

regularity in terms of a composite structure, much as the

periodic table of the elements reflects their underlying

atomic structure. A set of guidelines for this program was

laid down by ’t Hooft.

236

For an example of a recent effort,

see Ref. 237.

‘‘Naturalness, chiral symmetry, and spontaneous chiral symmetry

breaking,’’ G. ’t Hooft, in

~

Carge`se Summer Institute, Aug. 26–Sept. 8, 1979

!

, edited by G.

’t Hooft

~

Plenum, New York, 1980

!

, pp. 135–157.

~

A

!

‘‘Composite quarks and leptons from dynamical supersymmetry

breaking without messengers,’’ N. Arkani-Hamed, M. A. Luty, and J.

Terning, Phys. Rev. D

, 015004

~

1998

!

.

~

A

!

An early point in favor of quark–lepton unification was

the anomaly cancellation

182–184

mentioned in Sec. IX E. The

idea that lepton number could be regarded as a fourth

‘‘color,’’ leading to an extended gauge group embracing both

electroweak and strong interactions, was proposed by Pati

and Salam.

238

The strong and electroweak coupling constants are ex-

pected to approach one another at very small distance

~

large

momentum

!

scales,

239

suggesting

based on symmetry groups such as SU

~

5

!

,

240

SO

~

10

!

,

241

and

E

6

.

242

~

For an early popular article on this program see Ref.

114.

!

These theories typically predict that the proton will

decay,

115–117

and some of them entail additional observable

gauge bosons besides those of the SU

~

3

!

3

SU

~

2

!

3

U

~

1

!

stan-

dard model.

71

Some useful group-theoretic techniques for

model-building are described in Ref. 56.

‘‘Unified Lepton–Hadron Symmetry and a Gauge Theory of the Ba-

sic Interactions,’’ J. C. Pati and A. Salam, Phys. Rev. D

, 1240–1251

~

1973

!

.

~

A

!

; see also ‘‘Is Baryon Number Conserved?’’, J. C. Pati and

A. Salam, Phys. Rev. Lett.

, 661–664

~

1973

!

; ‘‘Lepton Number as

the Fourth Color,’’ J. C. Pati and A. Salam, Phys. Rev. D

, 275–289

~

1974

! ~

A

!

.

‘‘Hierarchy of Interactions in Unified Gauge Theories,’’ H. Georgi, H.

R. Quinn, and S. Weinberg, Phys. Rev. Lett.

, 451–454

~

1974

!

.

~

I

!

‘‘Unity of All Elementary Particle Forces,’’ H. Georgi and S. L.

Glashow, Phys. Rev. Lett.

, 438–441

~

1974

!

.

~

I

!

‘‘The State of the Art—Gauge Theories,’’ H. Georgi, in

, Proceedings of the Williamsburg Meeting, Sept. 5–7,

1974, edited by C. E. Carlson

~

AIP Conf. Proc. No. 23

! ~

AIP, New

York, 1975

!

, pp. 575–582.

~

A

!

‘‘A Universal Gauge Theory Model Based on E

6

,’’ F. Gu¨ rsey, P.

Ramond, and P. Sikivie, Phys. Lett.

, 177–180

~

1976

!

.

~

A

!

In a non-Abelian gauge theory such as SU

~

3

!

there can

arise nontrivial gauge configurations that prevent terms in

the Lagrangian proportional to Tr (

m

n

m

n

) from being ig-

nored as pure divergences. Such terms can lead to strong

violation. Their coefficient, a parameter conventionally

called

u

, must be of order 10

2

10

or smaller in order not to

conflict with limits on the electric dipole moment of the

neutron.

243

Several proposals have been advanced for why

u

is so small.

40,244

In one of the most interesting,

u

is promoted

to the status of a dynamical variable that can relax to a natu-

ral value of zero. As a consequence, there arises a nearly

massless particle known as the

, whose properties

~

and

the search for which

!

are well-described in Refs. 40, 244.

‘‘New Experimental Limit on the Electric Dipole Moment of the

Neutron,’’ P. G. Harris

, Phys. Rev. Lett.

, 904–907

~

1999

!

.

~

I

!

‘‘The Strong CP Problem,’’ M. Dine, in Ref. 47, pp. 349–369.

A truly unified theory of interactions must include gravity.

The leading candidate for such a theory is

,

which originated in pre-QCD attempts to explain the strong

interactions

245–248

by replacing the space–time points of

quantum field theories with extended objects

~

‘‘strings’’

!

. In

1974 it was realized that string theories necessarily entailed a

massless spin-2 particle, for which the graviton was an ideal

candidate.

249

While it appeared that such theories required

space–time to be 26-dimensional

~

or 10-dimensional in the

presence of supersymmetry

!

, these extra dimensions were

interpreted in the 1980s as a source of the internal degrees of

freedom characterizing particle quantum numbers

~

see. e.g.,

Refs. 250–252

!

. A typical scenario whereby string theory

might yield predictions for the quark and lepton spectrum is

described in Ref. 253.

Early results on string theory are described in the textbook

by Green, Schwarz, and Witten.

41,42

Later texts are Refs. 43,

44. Descriptions for the non-specialist are given by Green,

122

Duff,

123

Greene,

100

and Weinberg.

132

Y. Nambu, ‘‘Quark Model and the Factorization of the Veneziano

Amplitude,’’ in

~

In-

ternational Conference on Symmetries and Quark Models, Detroit,

Mich., June 1969

!

, edited by R. Chand

~

Gordon and Breach, New

York, 1970

!

, pp. 269–277.

~

A

!

‘‘A General Treatment of Factorization in Dual Resonance Models,’’

S. Fubini, D. Gordon, and G. Veneziano, Phys. Lett.

, 670–682

~

1969

!

.

~

A

!

‘‘Dual Symmetric Theory of Hadrons. 1,’’ L. Susskind, Nuovo Ci-

mento

, 457–496

~

1970

!

.

~

A

!

‘‘Strings, Monopoles, and Gauge Fields,’’ Y. Nambu, Phys. Rev. D

, 4262– 4268

~

1974

!

.

~

A

!

‘‘Dual Models for Nonhadrons,’’ J. Scherk and J. H. Schwarz, Nucl.

Phys. B

, 118–144

~

1974

!

.

~

A

!

‘‘The Heterotic String,’’ D. J. Gross, J. A. Harvey, E. Martinec, and

R. Rohm, Phys. Rev. Lett.

, 502–505

~

1985

!

.

~

A

!

313

313

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

Jonathan L. Rosner

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