Expanding a gauge covariant derivative term of a Lagrangian with color and flavor indices

Question

I have a Lagrangian which contains the term $$\operatorname{tr} \left(D_\mu\phi^{\tilde{a}} D^\mu \phi^{\tilde{a}} \right),$$ where $\phi^{\tilde{a}}=\phi^{\tilde{a}a}T^a$ with the indices $a$ and $\tilde{a}$ denoting the gauge and flavor group respectively. Furthermore, $D_\mu$ is the gauge covariant derivative which can be written as $$ \partial_\mu+ig A_\mu^aT^a $$ I have a hard time writing the above term explicitly, i.e., I am not sure how to assign color labels (without the tilde) to the fields. Intuitively I expand it as $$ \begin{aligned} &\operatorname{Tr}\left( \left[\partial_\mu \phi^{\tilde{a}}+ig A_\mu \phi^{\tilde{a}} \right] \left[ \partial^\mu \phi^{\tilde{a}}+ig A^\mu \phi^{\tilde{a}} \right] \right)\\ & \operatorname{Tr}\left(\partial_\mu \phi^{\tilde{a}a}\partial^\mu\phi^{\tilde{a}b}T^aT^b+2ig\partial_\mu \phi^{\tilde{a}a}A^{\mu b}\phi^{\tilde{a}c}T^aT^bT^c-g^2A_\mu^{a}\phi^{\tilde{a}b}A^{\mu c}\phi^{\tilde{a}d}T^aT^bT^cT^d\right)\\ &=\partial_\mu \phi^{\tilde{a}a}\partial^\mu\phi^{\tilde{a}a}+2ig\partial_\mu \phi^{\tilde{a}a}A^{\mu b}\phi^{\tilde{a}c}\operatorname{Tr}\left(T^aT^bT^c\right)-g^2A_\mu^{a}\phi^{\tilde{a}b}A^{\mu c}\phi^{\tilde{a}d}\operatorname{Tr}\left(T^aT^bT^cT^d\right) \end{aligned} $$ Please correct me if I am wrong.

Answer 1

You first need to know under which representation your field transforms, the flavor indices play no role here. Assuming $\phi$ is a scalar that transforms in the adjoint representation of the color group, sort of implied by the expression the OP writes. First we need to understand how the gauge covariant derivative acts on the field. Recall $T^a$ are infinitesimal generators of the Lie algebra, which means the only product that formally exists between them is the Lie bracket $$[T^a, T^b] = \sum_c {\rm i} f^{abc}T^c$$ where $f^{abc}$ are the structure constants. Then we have for example for a given flavor $f$, (which does not need to be manipulated) \begin{align} D_\mu \phi_f(x) &= \sum_a D_\mu^a \phi_f^a(x) T^a \\ &= \sum_a \partial_\mu \phi_f^a(x) T^a + \sum_{a,b} {\rm i}\,g\, \phi^a_f(x) A_\mu^b(x) [T^b,T^a]\\ &= \sum_a \partial_\mu \phi_f^a(x) T^a - \sum_{a,b,c}\,g f^{bac}\phi^a_f(x) A_\mu^b(x) T^c \end{align}

With that you should be able to work out the rest. The issue is, you need to identify the representation of your field. If $\phi$ transformed in the fundamental rep. then it would only have an extra index but no factors of $T^a$.