Mathematical Principles of Theoretical Physics

322 CHAPTER 5. ELEMENTARY PARTICLES

Theorem 5.21.The following assertions hold true for the w∗-color algebra:

1) For any gluon particle system with no quarks and antiquarks

(5.5.38) π=

4

∑

i= 1

(migi+migi),

the color index of this system satisfies that

(5.5.39) Indc(π) =



















w for

3

∑

i= 1

(mi−mi) =± 3 n,

y for

3

∑

i= 1

(mi−mi) =± 3 n+ 1 ,

y for

3

∑

i= 1

(mi−mi) =± 3 n+ 2 ,

for some integer n= 0 , 1 , 2 ,···;

2) For any single quark system as

(5.5.40)

ω=q+π

ω=q+π

withπas given by (5.5.38),

the color index of (5.5.40) satisfies that

(5.5.41)

Indc(ω) =r,g,b,

Indc(ω) =r,g,b,

3) For the hadronic systems

(5.5.42)

M=q+q+π meson system,

B=q+q+q+π baryon system,

withπgiven by (5.5.38), the color indices of (5.5.42) must be as

(5.5.43) Indc(M) =w,y,y, Indc(B) =w,y,y.

Two remarks are now in order.
First, in particle physics, all basic and important particle systems are given by the parti-
cle systems in Assertions 1)-3) in this theorem. System (5.5.38) represents a gluon system
attached to quarks and hadrons, (5.5.40) represents a cloud system of gluons around a quark
or an anti-quark, and (5.5.42) represents a cloud system of gluons around a hadron (a meson
or a baryon).

Second, physically, the numbers

3

∑

i= 1

(mi−mi) =± 3 n (n= 0 , 1 , 2 ,···),

indicate that through exchange of weaktons, a cloud system (5.5.38) of gluons can become a
system consisting of white gluons and the same number of yellow and anti-yellow gluons.