# "Color charge" of the adjoint fermion?

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What kind of "color charge" does the adjoint fermion carry?

Let us consider the SU(N) gauge theory. The gauge field is in the adjoint representation (rep).

Well-Konwn: If the fermion is in SU(N) fundamental rep, we know the fermions will form a color multiplet of N-component. For $N=3$, we say that there are 3 colors (r,g,b) of a given fermion $\psi$ $$(\psi_r, \psi_g, \psi_b)$$

1. However, if the fermion is in SU(N) adjoint rep, we know the fermions will form a color multiplet of (N$^2-1)$-component. For N=3, we say that there have 8-component of color multiplet. So,

What kind of "color charge" does each component of adjoint fermion carry? There should be 8 different choices. $$(\psi_1, \psi_2, \dots, \psi_{8})$$ What does the 1,2,3, $\dots$, 8 stand for in terms of color indices?

2. Are the color charges of adjoint fermions organized the same way as the gluons (which are also in adjoint) as in https://en.wikipedia.org/wiki/Gluon#Color_charge_and_superposition? Does both the adjoint fermions carry a color and an anti-color just as a gluon does? Why is that?

3. How can we read this information of color charges from the adjoint Rep of $SU(3)$ Lie algebra?

p.s. we may say the 8 gluons carry 8 distinct color anti-color pairs: $$(r\bar{b}+b\bar{r})/\sqrt{2}, -i(r\bar{b}-b\bar{r})/\sqrt{2}, (r\bar{g}+g\bar{r})/\sqrt{2}$$ $$-i(r\bar{g}-g\bar{r})/\sqrt{2},(b\bar{g}+g\bar{b})/\sqrt{2},-i(b\bar{g}-g\bar{b})/\sqrt{2}$$ $$(r\bar{r}-b\bar{b})/\sqrt{2},(r\bar{r}+b\bar{b}-2g\bar{g})/\sqrt{6}$$ And how about the 8-multiplet of adjoint fermions? What color charge do they carry?

This post imported from StackExchange Physics at 2020-11-05 13:28 (UTC), posted by SE-user annie marie heart
retagged Nov 5, 2020

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An adjoint fermion transforms in exactly the same way as an adjoint boson (like the gluon). We can write an adjoint fermion as a matrix valued field $$\psi_{ab} = \psi^A (T^A)_{ab}$$ where $T^A=\frac{1}{2}\lambda^A$ are the $SU(N)$ generators. The Dirac operator acts as a covariant derivative in the adjoint representation $$(D_\mu \psi)^A = (\partial_\mu \delta^{AB}+igf^{ACB}A^C_\mu)\psi^B$$
just like it acts on gluons.

This post imported from StackExchange Physics at 2020-11-05 13:28 (UTC), posted by SE-user Thomas
answered Feb 18, 2018 by (310 points)
Then (1) how would you write the Dirac Lagrangian for it? and (2) what will be the charge for each $\psi^A$? Thanks

This post imported from StackExchange Physics at 2020-11-05 13:28 (UTC), posted by SE-user annie marie heart
Your subindices do not match somehow.

This post imported from StackExchange Physics at 2020-11-05 13:28 (UTC), posted by SE-user annie marie heart
@ Thomas, I am curious: (i) How would you associate the color charge to the fermions? (ii) And what will be the distinctions for Majorana and Dirac fermions?

This post imported from StackExchange Physics at 2020-11-05 13:28 (UTC), posted by SE-user wonderich
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