5.5. STRUCTURE OF MEDIATOR CLOUDS AROUND SUBATOMIC PARTICLES 323
Proof of Theorem5.21.We proceed in the following three steps.
Step 1.By the basic properties (5.5.34)-(5.5.37) of color index,Indc(π) =rMrgMgbMb,where
Mr= (m 1 −m 2 )−(m 1 −m 2 ),
Mg= (m 3 −m 1 )−(m 3 −m 1 ),
Mb= (m 2 −m 3 )−(m 2 −m 3 ).Consequently, usingC−m=C
m
, we have
(5.5.44) Indc(π) = (rg)m^1 (rb)m^2 (gb)m^3 (rg)m^1 (rb)m^2 (gb)m^3
= ym^1 ym^2 ym^3 ym^1 ym^2 ym^3= yM, M=3
∑
i= 1(mi−mi).Notice thaty^2 =y,y^2 =y. Then the equality (5.5.39) follows from (5.5.44), and Assertion 1)
is proved.
Step 2.For Assertion 2), with the above argument, for the particle system (5.5.40) it is
easy to see that
(5.5.45)
Indc(ω) =Indc(q)Indc(π) =Indc(q)yM,
Indc(ω) =Indc(q)Indc(π) =Indc(q)yM.Due to the facts that
Indc(q) =r,g,b, Indc(q) =r,g,b,and by the multiplication rule given in Definition5.17, the conclusion (5.5.41) follows from
(5.5.45).
Step 3.With the same arguments as above, for the meson and baryon system (5.5.42), we
can derive that
(5.5.46) Indc(M) =cicjyM, Indc(B) =cicjckyM,
for 1≤i,j,k≤3, wherec 1 =r,c 2 =g,c 3 =b. By the basic rules for the color operation given
in (5.5.24), (5.5.25) and (5.5.27), we have
cicj=w,y,y, cicjck=cicl=w,y,y.Therefore, (5.5.43) follows from (5.5.46).