Mathematical Principles of Theoretical Physics

320 CHAPTER 5. ELEMENTARY PARTICLES

3) a homomorphism

Indc:PN→Gc,

such that for each elementw=∑Nk= 1 nkek∈PN, we can define a color index forwby

Indc(w) =

N

∏

k= 1

(Indc(ek))nk,

where Indc(ek)∈Gcis the image ofekunder the homomorphism Indc.

The color algebra introduced earlier is such a triplet{Gc,PN,Indc}withN=22, and

Gc=multiplication group generated by{r,g,b,y},

PN=

{

22

∑

k= 1

nkek|nk∈Z

}

,

whereekare colored quarks and gluons given by

(5.5.31)

e 1 =ur, e 2 =ug, e 3 =ub, e 4 =dr, e 5 =dg, e 6 =db,

e 7 =sr, e 8 =sg, e 9 =sb, e 10 =cr, e 11 =cg, e 12 =cb,

e 13 =br, e 14 =bg, e 15 =bb, e 16 =tr, e 17 =tg, e 18 =tb,

e 19 =g^1 , e 20 =g^2 , e 21 =g^3 , e 22 =g^4 ,

andgk( 1 ≤k≤ 4 )are as in (5.5.16). The homomorphism Indcis naturally defined by the
assignment in (5.5.31), i.e.

Indc(e 1 ) =r, Indc(e 2 ) =g, ···, Indc(e 18 ) =b,

Indc(e 19 ) =Indc(e 20 ) =Indc(e 21 ) =y, Ind(e 22 ) =w.

5.5.4 w∗-color algebra

Based on the weakton model, the weakton constituents of a quarkqand a gluongkare given
by

quark :q=w∗ww,

gluon:gk=w∗w∗.

The only weakton that has colors isw∗, which has three colors:

(5.5.32) w∗r,w∗g,w∗b,

and three anti-colors:

(5.5.33) w∗r,w∗g,w∗b.

With the three pairs of colored and anti-colored weaktons in(5.5.32) and (5.5.33), we can
defineN=3 color algebra, which we call thew∗-color algebra.