Mathematical Principles of Theoretical Physics

5.5. STRUCTURE OF MEDIATOR CLOUDS AROUND SUBATOMIC PARTICLES 319

andN 1 ,N 2 are as

(5.5.29) N 1 =

3

∑

k= 1

nk, N 2 =

3

∑

k= 1

mk.

For a mesonMwith constituents:

M=qc 1 +qc 2 +

3

∑

k= 1

(nkgk+mkgk)+K 1 g^4 +K 2 g^4 ,

its color indices satisfy that

(5.5.30) c 1 c 2 =yN^1 yN^2 ,

whereN 1 ,N 2 are as in (5.5.29).
Both equalities (5.5.28) and (5.5.29) are the color index formulas for baryons and mesons,
which ensure the neutral law of hadrons.

Color transformation of gluon radiation

Consider the transformation of a quarkqcwith colorcto another quartqc 3 after emitting
a gluongc 1 c 2 :
qc→gc 1 c 2 +qc 3 ,

then the color indexc 3 of then induced quarkqc 3 is given by

c 3 =cc 1 c 2.

Also, for the transformation of a quarkqcto another quarkqc 4 after absorbing a gluongc 1 c 2 :

qc+gc 1 c 2 →qc 4 ,

then the colorc 4 of transformed quarkqc 4 is as follows

c 4 =cc 1 c 2.

General color algebra

In an abstract sense, a color algebra is a triplet

{Gc,PN,Indc},

which consists of

1) a finite groupGc, called color group;

2) an integer modular algebra with generatorse 1 ,···,eN:

PN=

{

N

∑

k= 1

nkek|nk∈Z

}

;