I have an exercise for Yang-Mills theory. I can't find answer anywhere.
Derive equations of motion for the Yang-Mills theory with the gauge group $SU(2)$ interacting with $SU(2)$ doublet of scalar fields.
I don't even know how to derive EOM for Lagrangian here. Any help? Or any source for appropriate handbook? (I've been using Maggiore, it doesn't have EOM for Yang-Mills here.)
The Lagrangian of Yang-Mills theory coupled to scalars/fermions, etc. takes the form $$ {\cal L}_{YM} = - \frac{1}{2} \text{Tr} F_{\mu\nu} F^{\mu\nu} + (D_\mu \phi) (D^\mu \phi)^* + i {\bar \psi} \gamma^\mu D_\mu \psi + \cdots $$ where the $\cdots$ represents other interactions terms that might be present. Let me explain the notation in the above expression.
$\phi_i$ and $\psi_i$ are multiplets in some representation $R$ of the gauge group $G$. Here, $i = 1, \cdots, \dim R$
The generators in representation $R$ are denoted as $T^a$. These are normalized to satisfy $$ [T^a, T^b] = i f^{ab}{}_c T^c,~~~~ \text{Tr} ( T^a T^b )= \frac{1}{2}\delta^{ab} $$ In other words, the $T_{ij}^a$'s are just some set of matrices satisfying the above properties.
The covariant derivatives acting on the fields is \begin{align} (D_\mu \phi)_i &= \partial_\mu \phi_i - i g T^a_{ij} A_\mu^a \phi_j \\ (D_\mu \psi)_i &= \partial_\mu \psi_i - i g T^a_{ij} A_\mu^a \psi_j \\ \end{align} where $(A_\mu)_{ij} = A_\mu^a T_{ij}^a$.
$$ F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu + g [A_\mu, A_\nu ] \\ $$ Explicitly $$ F_{\mu\nu}^a = \partial_\mu A^a_\nu - \partial_\nu A_\mu^a + i g f_{bc}{}^a A^b_\mu A_\nu^c $$
This completely specifies the Lagrangian of Yang Mills theory. You can now use the variational principle to determine the equations of motion.
