312 CHAPTER 5. ELEMENTARY PARTICLES
and antiparticles have three anti-colors:
(5.5.3) anti-redr, anti-greeng, anti-blueb.
Based on the quantum numbers (5.5.2) and (5.5.3), each flavor of quarks is endowed with
three different colors
qr,qg,qb,qr,qg,qb.
The color indices of hadrons are given by
(5.5.4)
baryon=q 1 rq 2 gq 3 b,
meson=q 1 rq 2 r,q 1 gq 2 g,q 1 bq 2 b.Since hadrons are color neutral, the color quantum number should obey the following multi-
plication rule:
(5.5.5)
rr=w, gg=w, bb=w,
rgb=w, rgb=w, w=white color,and the multiplication is commutative.
The color quantum number has attained many experimental supports. Quantum Chromo-
dynamics (QCD) for the strong interaction is based on this theory. In fact, it is natural to
think that the three color quantum states of each flavour of quarks
(5.5.6) q= (qr,qg,qb)
are indistinguishable in the strong interaction. This gives rise to theSU( 3 )gauge theory of
QCD, i.e. the QCD action is the Yang-Milli functional ofSU( 3 )gauge fields, given by
(5.5.7) LQCD=−
1
4
Skμ νSμ νk+q(iγμDμ−m)q,whereqis the quark triplet as in (5.5.6),mis the mass of the quark,
(5.5.8)
Skμ ν=∂μSkν−∂νSkμ+gsfijkSiμSνj,
Dμ=∂μ+igsSkμλk,andλk( 1 ≤k≤ 8 )are the generators ofSU( 3 ). In QCD,λkare taken as the Gell-Mann
matrices as defined in (3.5.38).
5.5.2 Gluons
In the last subsection we saw that the three color quantum numbers defined in (5.5.2)-(5.5.3)
lead to the QCD action (5.5.7) ofSU( 3 )gauge fields. In the following, we introduce the
gluons in a few steps.
Gluons derived from QCD
It was known that theSU( 3 )gauge theory for the strong interaction is oriented toward
two directions: 1) describing the field particles, i.e. the strong interaction mediators, and 2)