Mathematical Principles of Theoretical Physics

318 CHAPTER 5. ELEMENTARY PARTICLES

Two remarks are now in order.

Remark 5.18.Each elementω=Σnkek∈Prepresents a particle system, andnkis the dif-
ference between the number of particles with color indexckand the number of antiparticles
with color indexck. In particular, particles with colorsr,g,bmust be quarks, particles with
colorsr,g,bmust be anti-quarks, and particles with colorsy,ymust be gluons.

Remark 5.19.In (5.5.27),rg,br,gbare all yellow. Consequently the gluonsgrg,gbr,ggb
have the same color. However, they represent different quantum states. In particular, in the
weakton model,
grg=w∗rw∗g, gbr=w∗bw∗r, ggb=w∗gw∗b,

which represent different quantum states.

Color index formula of hadrons

We now study color neutral problem for hadrons and the radiation and absorption of
gluons for quarks.
Let us start with color neutral problem for hadrons. Consider the constituents of a proton

p=uc 1 +uc 2 +uc 3 +∑nkgk∈P, ∑nk=N,

whose color index is given by

Indc(p) =c 1 c 2 c 3

8

∏

k= 1

(Indc(gk))nk,

where Indc(gk)is the color index of the gluongk. The color neutral law requires that
Indc(p) =w, which does not necessarily lead toc 1 c 2 c 3 =w. For example, for the following
constituents ofp:
p=ur+ur+dg+ 2 grg+grb,

we have

Indc(p) =r^2 gy^2 y=rgy=rr=w,

c 1 c 2 c 3 =rrg=rg=y 6 =w.

In summary, the hadron color quantum numbers based on color algebra is very different from
the classical QCD theory.
For a baryonBwith its constituents given by

B=

3

∑

i= 1

qci+

3

∑

k= 1

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

wheregkandgk( 1 ≤k≤ 4 )are as in (5.5.16) and (5.5.17), its quantum number distribution
satisfies the following color index formula:

(5.5.28) c 1 c 2 c 3 =yN^1 yN^2 ,