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229. ‘‘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 16, 1444–1473 ~1977!. ~A!
230. ‘‘Weak Interaction Mixing in the Six-Quark Theory,’’ H. Fritzsch,
Phys. Lett. B 73, 317–322 ~1978!. ~I!
231. ‘‘Hierarchy of Quark Masses, Cabibbo Angles, and CP Violation,’’ C.
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Hall, S. Raby, and A. Romanino, Nucl. Phys. B 493, 3–26 ~1997!.
~A!
233. ‘‘A model for Fermion Mass Hierarchies and Mixings,’’ P. Ramond,
in Particles, Strings, and Cosmology „PASCOS 98…, 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!
234. ‘‘Mass and flavor mixing schemes of quarks and leptons,’’ H. Fritzsch
and Z.-z. Xing, Prog. Part. Nucl. Phys. 45, 1–81 ~2000!. ~A!
235. ‘‘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 65, 073004 ~2002!, and Refs. 11 and 15
therein. ~A!
D. Composite quarks and leptons
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.
236. ‘‘Naturalness, chiral symmetry, and spontaneous chiral symmetry
breaking,’’ G. ’t Hooft, in Recent Developments in Gauge Theories
~Carge`se Summer Institute, Aug. 26–Sept. 8, 1979!, edited by G.
’t Hooft et al. ~Plenum, New York, 1980!, pp. 135–157. ~A!
237. ‘‘Composite quarks and leptons from dynamical supersymmetry
breaking without messengers,’’ N. Arkani-Hamed, M. A. Luty, and J.
Terning, Phys. Rev. D 58, 015004 ~1998!. ~A!
E. Grand unification and extended gauge groups
An early point in favor of quark–lepton unification was
the anomaly cancellation182–184 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 grand unified theories
based on symmetry groups such as SU~ 5!, 240 SO~10! ,241 and
E6 . 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.
238. ‘‘Unified Lepton–Hadron Symmetry and a Gauge Theory of the Ba-
sic Interactions,’’ J. C. Pati and A. Salam, Phys. Rev. D 8, 1240–1251
~1973!. ~A!; see also ‘‘Is Baryon Number Conserved?’’, J. C. Pati and
A. Salam, Phys. Rev. Lett. 31, 661–664 ~1973!; ‘‘Lepton Number as
the Fourth Color,’’ J. C. Pati and A. Salam, Phys. Rev. D 10, 275–289
~1974! ~A!.
239. ‘‘Hierarchy of Interactions in Unified Gauge Theories,’’ H. Georgi, H.
R. Quinn, and S. Weinberg, Phys. Rev. Lett. 33, 451–454 ~1974!. ~I!
240. ‘‘Unity of All Elementary Particle Forces,’’ H. Georgi and S. L.
Glashow, Phys. Rev. Lett. 32, 438–441 ~1974!. ~I!
241. ‘‘The State of the Art—Gauge Theories,’’ H. Georgi, in Particles and
Fields—1974, Proceedings of the Williamsburg Meeting, Sept. 5–7,
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York, 1975!, pp. 575–582. ~A!
242. ‘‘A Universal Gauge Theory Model Based on E6 ,’’ F. Gu¨ rsey, P.
Ramond, and P. Sikivie, Phys. Lett. 60B, 177–180 ~1976!. ~A!
F. Strong CP problem and axions
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 (G mnG˜ mn) from being ig-
nored as pure divergences. Such terms can lead to strong CP
violation. Their coefficient, a parameter conventionally
called u, must be of order 10210 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 axion, whose properties ~and
the search for which! are well-described in Refs. 40, 244.
243. ‘‘New Experimental Limit on the Electric Dipole Moment of the
Neutron,’’ P. G. Harris et al., Phys. Rev. Lett. 82, 904–907 ~1999!.
~I!
244. ‘‘The Strong CP Problem,’’ M. Dine, in Ref. 47, pp. 349–369.
G. String theory
A truly unified theory of interactions must include gravity.
The leading candidate for such a theory is string theory,
which originated in pre-QCD attempts to explain the strong
interactions245–248 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
245. Y. Nambu, ‘‘Quark Model and the Factorization of the Veneziano
Amplitude,’’ in Symmetries and Quark Models: Proceedings ~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!
246. ‘‘A General Treatment of Factorization in Dual Resonance Models,’’
S. Fubini, D. Gordon, and G. Veneziano, Phys. Lett. 29B, 670–682
~1969!. ~A!
247. ‘‘Dual Symmetric Theory of Hadrons. 1,’’ L. Susskind, Nuovo Ci-
mento A69, 457–496 ~1970!. ~A!
248. ‘‘Strings, Monopoles, and Gauge Fields,’’ Y. Nambu, Phys. Rev. D
10, 4262– 4268 ~1974!. ~A!
249. ‘‘Dual Models for Nonhadrons,’’ J. Scherk and J. H. Schwarz, Nucl.
Phys. B 81, 118–144 ~1974!. ~A!
250. ‘‘The Heterotic String,’’ D. J. Gross, J. A. Harvey, E. Martinec, and
R. Rohm, Phys. Rev. Lett. 54, 502–505 ~1985!. ~A!
313 313 Am. J. Phys., Vol. 71, No. 4, April 2003 Jonathan L. Rosner