Mathematical Principles of Theoretical Physics

5.5. STRUCTURE OF MEDIATOR CLOUDS AROUND SUBATOMIC PARTICLES 315

ur

ug

ur ̄g

Figure 5.16

The reaction formula of the gluon radiation shown by Figure5.16is as

(5.5.18) ur→ug+grg,

and the color index operation for (5.5.18) is given by

(5.5.19) r=g(rg) =rgg=r.

Color index operation

However, ifqr-quark radiates agbqgluon, due to the lack of color algebra, then it is
difficult to determine the type of quark to whichqrtransforms, i.e.

qr→gbg+qX, X=?.

In addition, the lack of color algebra also leads to another serious problem. It is known
that hadrons are color neutral:

the color of a hadron=w.

But for the color indexcidefined as

(5.5.20) c 1 =r,c 2 =g,c 3 =b,c 1 =r,c 2 =g,c 3 =b,

they is no algebraic operation, i.e.

(5.5.21) cicj 6 =any one of (5.5.20) fori 6 =j.

Hence the color indices of all non-neutral gluons cannot be defined. Thus, if a protonp
contains three quarks andNgluons(N 6 = 0 ):

p=uci+ucj+dck+∑

k

nkgk, ∑

k

nk=N,

then we cannot make sure whetherpis color neutral.
In the following two subsections we shall establish the color algebra, a new mathematical
theory, to solve above mentioned problems associated with the color index operation and the
QCD.