What happens to the gauge covariant derivative if the theory contains multiple fields in different representations?

Question

I'm studying a graduate level course in QFT, and I have a potentially very stupid question that I can't find adressed anywhere (which I guess implies that I'm missing something fundamental). I'm currently working a problem where there is assumed to exist both a real scalar field $\Phi$ belonging to the adjoint representation of $SU(2)$ (an $SU(2)$ triplet), and a fermion field $\psi$ belonging to the fundamental representation (an $SU(2)$ doublet). As far as I understand, if we consider only the scalar, the Lagrangian for the theory should contain a kinetic term $$\frac{1}{2}D_\mu\Phi D^\mu\Phi,$$ where the covariant derivative should have the form $$D_\mu\phi_a = \partial_\mu\phi_a + g\varepsilon_{abc}A_\mu^b\phi_c,$$ (here $\Phi = \phi_a\tau_a$, with $\tau_a = \sigma_a/2$, and $g$ is the coupling strength).

However, since the fermion transforms in the fundamental representation, to make the theory gauge invariant, the Lagrangian should contain a term $$i\bar{\psi}\gamma^\mu D_\mu \psi,$$ but this time the covariant derivative should be $$D_\mu\psi = \partial_\mu\psi - igA_\mu^a\tau^a\psi,$$ clearly different from what we had before.

Now my question is, am I interpreting this correctly as there being two different covariant derivatives present, one for each different representation present in the theory, each with its own set of associated gauge fields $A_\mu^a$? Or should they be combined in some way? Or is something else going on entirely?

Answer 1

The same gauge field can act on other fields via different representations of the gauge group.

A simple example is the $U(1)_Y$ ("hypercharge") gauge field in the Standard Model. Different quarks and leptons have different hypercharges (some of which are zero), so they transform differently under the $U(1)_Y$ gauge transformation. There is still only one $U(1)_Y$ gauge field.

Now consider the $SU(2)$ case. Although we often write the covariant derivative $D_\mu$ by itself, it's really not defined all by itself — just like $\partial_\mu$ isn't defined all by itself. The covariant derivative is defined in terms of how it acts on something else. The components $A_\mu^a$ of the gauge field are the same no matter what $D_\mu$ is acting on, but they may be arranged differently depending on what it's acting on ($\phi$ versus $\psi$). The coefficients $\epsilon_{abc}$ and the matrix of coefficients $\tau^a$ are just compact ways of expressing sums of products of $A_\mu^a$s with the components of $\phi$ and $\psi$, respectively. These are different covariant derivatives, but they both use the same components $A_\mu^a$ of the gauge field.