Physical Foundations of Cosmology

4.2 Quantum chromodynamics and quark–gluon plasma 139

wheregsis the strong coupling constant,λC(whereC= 1 ,..., 8 )are the eight Gell-
Mann 3×3 matrixes andf runs over the quark flavorsu,d,s,c,t,b. There are
eight gauge fieldsACμ, calledgluons, which are responsible for strong interactions.
The symbolψfdenotes the column of three quark spinor fields:

ψf≡

⎛

⎝

rf

bf

gf

⎞

⎠, (4.20)

whererf is the Dirac spinor describing the red quark with flavorf, etc. The bare
quark massesmf are determined from experiments. They are very different and
not so well known. The lightest is theuquark with mass of 1.5–4.5 MeV.The
dquark is a bit heavier:md=5–8.5 MeV.The strange quark has a mass of 80–
155 MeV,and the remaining three quarks are much heavier:mc= 1. 3 ± 0 .3 GeV,
mb= 4. 3 ± 0 .2 GeV,andmt∼170 GeV.
The antiparticles of the quarks are called antiquarks and they can have “antired”
(r ̄), “antiblue”

(

b ̄

)

,and “antigreen”(g ̄)colors. As distinct from photons, gluons
are also charged. They carry one unit of color and one of anticolor. For instance,
using the explicit form of Gell-Mann matrixλ 1 ,we find that the first interaction
term in the Lagrangian (4.19) is

gs(br ̄ +rb ̄ )A^1 ,

where we have omitted flavor and spacetime indices together with the Dirac matri-
ces. The appropriate quark–gluon vertices describing this interaction are shown in
Figure 4.2. When a quark changes its color, the color difference is carried off by a
gluon, which in this case is eitherrb ̄orbr ̄colored. The state (rb ̄+br ̄) is the first
state of the “color octet” of gluons. Using the explicit form of the Gell-Mann matri-
ces, the reader can easily find the remaining seven states of the octet. In principle,
however, from three colors and three anticolors, we can composenineindependent
color–anticolor combinations:rr ̄,rb ̄,rg ̄,br ̄,bb ̄,bg ̄,gr ̄,gb ̄,gg ̄. Therefore, one
is led to ask which particular combination of colors does not occur in Lagrangian
(4.19) and hence does not participate in strong interactions. The answer is the “color
singlet”

(

rr ̄+bb ̄+gg ̄

)

,which is invariant underSU( 3 )gauge transformations.

r

rb

b r

b

br

Fig. 4.2.