higgs - page 24

145. ‘‘The Cabibbo–Kobayashi–Maskawa Quark-Mixing Matrix,’’ F. J.
Gilman, K. Kleinknecht, and B. Renk, in Review of Particle Phys-
ics, K. Hagiwara et al.,73 pp. 113–119.
B. Gauge theories
A theory of particles and their interactions permitting ar-
bitrary changes of phase in the particle’s quantum mechani-
cal state is an Abelian local gauge theory 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
not commute with one another at a given space–time point.
The first such non-Abelian 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 Lee48 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 text148 provides a further introduction to
the subject.
146. ‘‘Electron and Gravitation @in German#,’’ H. Weyl, Z. Phys. 56, 330–
352 ~1929!, partially reprinted in Surveys in High Energy Phys. 5,
261–267 ~1986!. ~A!
147. ‘‘Conservation of Isotopic Spin and Isotopic Gauge Invariance,’’ C.
N. Yang and R. L. Mills, Phys. Rev. 96, 191–195 ~1954!. ~A! See
also Cambridge University Dissertation, R. Shaw, 1954 ~unpub-
lished!.
148. The Dawning of Gauge Theory, L. O’Raifeartaigh ~Princeton Uni-
versity Press, Princeton, NJ, 1997!. ~I!
C. Color and quantum chromodynamics
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.
1. Quark statistics. The D11 , a low-lying excited state of
the nucleon, can be represented in the quark model as uuu ,
so it is totally symmetric in flavor. It has spin J 5 3/2, a
totally symmetric combination of the three J 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 antisymmetric under the interchange of any
two fermions, the state described so far is totally symmetric
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.
2. Electron-positron annihilation to hadrons. The charges
of all quarks that can be produced in pairs at a given center-
of-mass energy is measured by the ratio R [ s( e 1e 2
→hadrons)/ s( e 1e 2 → m1 m2) 5 ( iQ i
2 , where Q i is the
charge of quark i in units of u e u . Measurements73 indicate
values of R in various energy ranges consistent with N c 5 3
~with a small positive correction associated with the strong
interactions of the quarks!.
3. Neutral pion decay. The p0 decay rate is governed by a
quark loop diagram in which two photons are radiated by the
quarks in p05 ( uu¯2 dd¯)/ A2. The predicted rate is G ( p0
→ gg) 5 7.6S 2 eV, where S5 Nc ( Q u
22 Q d
2) 5 N c/3. The ex-
perimental rate is 7.86 0.6 eV, in accord with experiment if
S 5 1 and Nc 5 3.
4. Triality. Quark composites appear only in multiples of
three. Baryons are composed of qqq, while mesons are qq¯
~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 experiments96,152,153 in terms of
quarks. This property was found to be characteristic of non-
Abelian gauge theories such as color SU~3! by Gross and
Wilczek154–156 and by Politzer.157,158 The result was obtained
earlier for the gauge group SU~ 2! by Khriplovich159 ~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 e 1e 2 → qq¯ 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 u, d, and s.
149. ‘‘Spin and Unitary Spin Independence in a Paraquark Model of Bary-
ons and Mesons,’’ O. W. Greenberg, Phys. Rev. Lett. 13, 598–602
~1964!. ~I!
150. Y. Nambu, ‘‘A Systematics of Hadrons in Subnuclear Physics,’’ in
Preludes in Theoretical Physics in Honor of V. F. Weisskopf, ed-
ited by A. De-Shalit, H. Feshbach, and L. Van Hove ~North-Holland,
Amsterdam and Wiley, New York, 1966!, pp. 133–42. ~A!
151. ‘‘Advantages of the Color Octet Gluon Picture,’’ H. Fritzsch, M.
Gell-Mann, and H. Leutwyler, Phys. Lett. 47B, 365–368 ~1973!. ~I!
152. ‘‘High-Energy Inelastic ep Scattering at 6° and 10°,’’ E. D. Bloom
et al., Phys. Rev. Lett. 23, 930–934 ~1969!. ~I!
153. ‘‘Observed Behavior of Highly Inelastic Electron–Proton Scatter-
ing,’’ M. Breidenbach et al., Phys. Rev. Lett. 23, 935–939 ~1969!. ~I!
154. ‘‘Ultraviolet Behavior of Non-Abelian Gauge Theories,’’ D. J. Gross
and F. Wilczek, Phys. Rev. Lett. 30, 1343–1346 ~1973!. ~A!
155. ‘‘Asymptotically Free Gauge Theories. I,’’ D. J. Gross and F. Wil-
czek, Phys. Rev. D 8, 3633–3652 ~1973!. ~A!
156. ‘‘Asymptotically Free Gauge Theories. 2,’’ D. J. Gross and F. Wil-
czek, Phys. Rev. D 9, 980–993 ~1974!. ~A!
157. ‘‘Reliable Perturbative Results for Strong Interactions?,’’ H. David
Politzer, Phys. Rev. Lett. 30, 1346–1349 ~1973!. ~A!
158. ‘‘Asymptotic Freedom: An Approach to Strong Interactions,’’ H.
David Politzer, Phys. Rep. 14, 129–180 ~1974!. ~A!
159. ‘‘Green’s Functions in Theories with Non-Abelian Gauge Group,’’ I.
B. Khriplovich, Yad. Fiz. 10, 409–424 ~1969! @ Sov. J. Nucl. Phys.
10, 235–242 ~1969!#. ~A!
160. ‘‘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