Yang-Mills scattering amplitude calculation in Peskin and Schroeder I'm studying section 16.1 of Peskin and Schroeder on quantization of gauge theory, on page 509, we have the expression
$$\tag{16.14}iM^{\mu\nu}_{1,2}\epsilon_{1\mu}^\ast k_{2\nu}=(ig)^2\overline{\nu}(p_+)\{-i\gamma^\mu[t^a,t^b]\}u(p)\epsilon_{1\mu}^\ast$$
then we have
$$\tag{16.15} iM_{1,2}^{\mu\nu}\epsilon^\ast_{1\mu}k_{2\nu}=-g^2\overline{\nu}(p_+)\gamma^\mu u(p)\epsilon^\ast_{1\mu}\cdot f^{abc}t^c,$$ where $[t^a,t^b]=if^{abc}t^c$.
My question is: shouldn't (16.15) be $$iM_{1,2}^{\mu\nu}\epsilon^\ast_{1\mu}k_{2\nu}=-g^2\overline{\nu}(p_+)\gamma^\mu (f^{abc}t^c) u(p)\epsilon^\ast_{1\mu}~?$$
I mean, if my understanding is correct, $t^c$ is an $r$ dimensional matrix, where $r$ is the dimension of the representation of the gauge group, so why would the original (16.15) even make sense?
 A: Indeed, this notation is confusing.
When confused, always try to put back the Gauge indices, let’s call them i,j. We can rewrite more explicitly :
$$ i M^{\mu \nu}_{1 2} \epsilon_{1\mu}^* k_{2\nu} = -g^2 \bar{\nu}_i\gamma^\mu f^{a b c} (t^c)_{i j} u_j \epsilon^*_{1\mu}$$
where $i$ and $j$ are summed over, and now the only remaining contractions are between dirac indices (which you could also write for being totally explicit).
Remark that $i$ and $j$ run from 1 to $r$, where $r$ is indeed the dimension of the representation, and we have $r$ dirac spinors $\nu_i$ and $r$ spinors $u_j$, each of which has 4 components which are contracted with the use of $\gamma^\mu$.
Now the $(t^c)_{i j}$ are just numbers and we can take them out of the dirac contraction, and define the « dot » notation between $ A_{i j} \equiv \bar{\nu}_i\gamma^\mu u_j $ and $ B_{i j} \equiv f^{a b c} t^c_{i j}$ as dotting the indices separately :
$$ (\bar \nu \gamma^\mu u) \cdot (f^{ a b c} t^c)= A \cdot B \equiv A_{i j} B_{i j} = \bar{\nu}_i\gamma^\mu u_j f^{a b c} (t^c)_{i j} $$
I think this is what P&S means. But I prefer writing it in your way, always keeping in mind that we have both dirac and gauge indices that are implicitly contracted.
