Subatomic Physics

15.5. Baryons as Bound Quark States 479
The lack of antisymmetry of the wavefunction was a large impediment for the
development of the quark model until the idea of an extra degree of freedom appeared.
This new degree, color, was introduced initially to solve the antisymmetry
puzzle. (8) Its effect on the meson wavefunction is given in Eq. (15.3). With three
colors, an antisymmetric colorless (white) wavefunction can be formed. If the three
colors were three unit vectors along the x, y and z axes in color space, the colorless
(scalar) combination would be ˆx · ˆy × ˆz. If we denote the three colors by a, b, c, the
unnormalized color singlet combination of quarks can be written as
�
ɛabcqaqbqc, (15.9)
a,b,c
with a, b, c running over the three colors red, green, and blue; ɛabc is the antisymmetric
tensor which is +1 for even permutations of a, b, c (r, g, b) and−1 forodd
ones. Three colors are the minimum requiredtoformanantisymmetricstateof
three quarks. Although color was introduced in an ad hoc fashion, it has become
all important in our understanding of the strong interactions through QCD, as discussed
in Section 14.8. The evidence for color includes saturation of the lowest mass
baryons by three quarks and mesons by qq, an explanation of the decay width of
the π 0 to two photons, and the magnitude of the cross section for reactions such as
e + e − → hadrons, as discussed in Section 10.9.
The three quarks u,
d and s can be combined
to form 10 combinations,
and particles
exist for all 10.
The quark combinations
and the corresponding
baryons are
shown in Fig. 15.3.
Figure 15.3: Quarks and the (3/2) + decimet. The states and the
particles are arranged so that the x axis gives I3, andthey axis S.
The rest energies are given at the right.
Also indicated are the rest energies of the isomultiplets. Since there are 10 particles,
the array is called the (3/2) + decimet (or decuplet). The similarity to Escher’s “Sky
and Water I” on p. 470 is impressive, particularly if it is noted that the decimet of
the antiparticles also exists.
Three spin-1/2 fermions in an S-state can also be coupled to form a state with
spin 1/2 and positive parity. Examples in nuclear physics are 3 Hand 3 He. Table 15.1
indicates that only eight members of the (1/2) + family are known. The eight
particles and the corresponding quark combinations are shown in Fig. 15.4.
8O. W. Greenberg, Phys. Rev. Lett. 13, 598 (1964); M. Y. Han and Y. Nambu, Phys. Rev.
139, B1006 (1965).