When you introduce the QCD in your Lagrangian
L_{QCD}=\bar{ψ} (i γ^{μ} D_{μ} -m ) ψ (Dirac's part) -G_{μκ}^{a}G^{μκ}_{a}/4 (the gauge bosons interaction terms)
Where you have the:
a) Covariant Derivative
D_{μ}= ∂_{μ}+ i g_{3} T^{a} A_{μ}^{a} = ∂_{μ}+ i g_{3} λ^{a} A_{μ}^{a}/2
where λ^{a} are the generators of SU(3) thus they can be represented as the Gellmann matrices.
b) The gauge bosons antisymmetric tensor
G_{μκ}^{a}= ∂_{μ}A_{κ}^{a}-∂_{κ}A_{μ}^{a}-g_{3}f^{abc} A_{μ}^{b}A_{κ}^{c}=A_{μκ}^{a}-g_{3} f^{abc} A_{μ}^{b}A_{κ}^{c}
f^{abc} the λ^{a} algebra's structure constants...
Then you get for your Lagrangian 3 different terms...
L_{QCD}=L_{0}+ L_{g.b.} + L_{int}
1st term is the free langrangian term, the 2nd term is the gauge bosons self interaction term, and the last is the interaction of bosons with your quark fields ψ which can be either u,d,c,s,t,b and it can be represented as color triplets...
eg c= [c^{red}, c^{green}, c^{blue}]^{T}. For the anticolor, you need to work in the adjoint representation of SU(3)
Nevermind, to get the color current, you need the interactive Lagrangian:
L_{int}= -g_{3} \bar{ψ} γ^{μ}λ^{a} ψ A_{μ}^{a}/2
the corresponding conserved current (if you remember from the Dirac's current case) is:
J_{SU(3)}^{μa}= g_{3} \bar{ψ} γ^{μ}(λ^{a}/2) ψ
What can we see from that? That we have 8 conserved currents. Each of them is individually conserved. The continuity relation for the currents, is given by their conservation, thus you have again 8 different continuity relations:
∂_{μ}J_{SU(3)}^{μa}= 0
and the color charge is:
Q_{c}=\int{d^{3}x J_{SU(3)}^{0a}}If they also carry electric charge, you'll get also another current, corresponding to U(1)_{Q} interaction...
I am not sure for this, if it's wrong someone please correct me:
If you now leave from the case of a single quark, and you want to put all the quarks in the game, then you should put indices on the quark fields...So the current will:
J_{i}^{μa}= g_{3} \bar{ψ_{i}} γ^{μ}(λ^{a}/2) ψ_{i}
where i can be each u,d,s,c,b,t quarks, so eg ψ_{1,2,3,4,5,6}= ψ_{u,d,c,s,t,b}
In the color representation, then you can also write indices for the λ and ψ such that:
J_{i}^{μa}= g_{3} \bar{ψ_{i}^{k}} γ^{μ}(λ^{a}_{kp}/2) ψ^{p}_{i}
