The electroweak Lagrangian before symmetry breaking is defined as:
$$L_{ew}=L_g+L_f+L_h+L_y$$
$L_{fermion}$ concerns the 4-component $\Psi$ vector fields of 12 fermion types $\{u,d,c,s,t,b,e,\nu_e,\mu,\nu_\mu, \tau,\nu_\tau \}$ and is defined here by:
$$ L_f = i \bar R \gamma^\mu \color{red}{D_\mu R} + i \bar L \gamma^\mu \color{green}{D_\mu L}$$
where $D_\mu R$ and $D_\mu L$ are two different operations
$$= i \bar R \gamma^\mu (\color{red}{\frac{dR}{dx^\mu}+ig^{'}B_\mu R })+i\bar L \gamma^\mu(\color{green}{\frac{dL}{dx^\mu} + \frac{ig^{'}}{2}B_\mu L- \frac{ig}{2}\sigma_1 W^1_\mu L - \frac{ig}{2}\sigma_2 W^2_\mu L - \frac{ig}{2}\sigma_3 W^3_\mu L})$$
I have a few questions regarding this definition:
- Does $\color{red}{\vec R}$ and $\color{green}{\vec L}$ actually refer to $\color{red}{\vec \Psi_R}$ and $\color{green}{\vec \Psi_L}$ chirality as defined by
$$ \Psi_L = \frac{1-\gamma^5}{2}\Psi$$ $$\Psi_R = \frac{1+\gamma^5}{2}\Psi$$ where $$\gamma^5 = i\gamma^0\gamma^1\gamma^2\gamma^3 = \begin{bmatrix} 0&0&1&0\\0&0&0&1\\1&0&0&0\\ 0&1&0&0 \end{bmatrix}$$
- The last terms such as $\sigma_3 W^3_\mu L$ don't seem to work out as matrix multiplication, as the Pauli matrix $\sigma$ is $2x2$, $W^0_\mu$ is $1x1$, and $L$ is $4x1$, so
$$\begin{bmatrix}.&.\\.&.\end{bmatrix}\begin{bmatrix}.\end{bmatrix}\begin{bmatrix}.\\.\\ .\\ . \end{bmatrix}=?$$