The electroweak Lagrangian before symmetry breaking is defined here as:
$$L_{ew}=L_g+L_f+L_{hy}$$
where
$$L_{higgs.yukawa} = (\bar {D_\mu \phi})(D^\mu \phi) - m^2\bar \phi \phi - \alpha (\bar \phi \phi)^2 - \beta(\bar L \phi R + \bar R \bar \phi L)$$
where the Higgs field is defined as a 2-component complex number vector field,
$$\phi = \begin{bmatrix} \phi_1 \\ \phi_2 \end{bmatrix}$$ and
$$D_\mu\phi = \frac{d}{dx^\mu}\begin{bmatrix}\phi_1 \\ \phi_2 \end{bmatrix} - \frac{ig^{'}}{2}B_\mu\begin{bmatrix}\phi_1 \\ \phi_2 \end{bmatrix} - \frac{ig}{2}W^0_\mu\begin{bmatrix}0&1\\ 1 & 0 \end{bmatrix}\begin{bmatrix}\phi_1 \\ \phi_2 \end{bmatrix} - \frac{ig}{2}W^1_\mu\begin{bmatrix}0&-i\\ i & 0 \end{bmatrix}\begin{bmatrix}\phi_1 \\ \phi_2 \end{bmatrix} - \frac{ig}{2}W^2_\mu\begin{bmatrix}1&0\\ 0 & -1 \end{bmatrix}\begin{bmatrix}\phi_1 \\ \phi_2 \end{bmatrix}$$ and $$D^μ\phi=\sum_{\nu}^4 η^{μν}(D_ν\phi)$$
- What vector form must $\vec L$ and $\vec R$ take in order for the product $\color{red}{\bar L \phi R}$ and $\color{red}{\bar R \bar \phi L}$ to be legal matrix multiplication?
$$\begin{bmatrix} . & . \end{bmatrix} \begin{bmatrix} \phi_1 \\ \phi_2 \end{bmatrix} \begin{bmatrix} . \\ . \end{bmatrix} = ?$$
For the non-chiral analog with scalar $\phi$, let
$$L=L_\phi+L_\Psi+L_{\phi,\Psi}$$
Define the parts as
$$L_\phi=\frac{1}{2}(D_\mu\phi)(D^\mu\phi) - V(\phi)$$
$$L_\Psi = i\bar\Psi\gamma^\mu\frac{d\Psi}{d\mu} - m_1\bar\Psi\Psi$$
$$L_{\phi,\Psi}=-g\bar\Psi\phi\Psi$$
Then,
$$L = \frac{1}{2}(D_\mu\phi)(D^\mu\phi) - V(\phi) + i\bar\Psi\gamma^\mu\frac{d\Psi}{d\mu} - m_1\bar\Psi\Psi -g\bar\Psi\phi\Psi$$
Let $V(\phi)$ be chosen as
$$V(\phi) = \frac{1}{2}m_2^2\phi^2 + m_3\phi^4$$
Then,
$$L = \frac{1}{2}(D_\mu\phi)(D^\mu\phi) - \frac{1}{2}m_2^2\phi^2 -m_3\phi^4 + i\bar\Psi\gamma^\mu\frac{d\Psi}{d\mu} - m_1\bar\Psi\Psi -g\bar\Psi\phi\Psi$$
Let also $\phi$ become a complex scalar
$$\phi = \phi_1 + i\phi_2$$
or a quarternion scalar $$\phi = \phi_1 + i\phi_2 + j\phi_3 + k\phi_4$$
Finally, shift $\phi$ by some pre-determined distance $\Delta \phi$ in the complex or quarternion space.
$$\phi \rightarrow \phi + \Delta \phi$$
- Is the model construction is already $\color{green}{complete}$ at this step, ie. the Euler-Lagrange and equation of motion will now exhibit all behaviors which can be analyzed as symmetry breaking, mass gaining, etc?
Following this example,
Piecing together the operation,
$\bar \nu_e, \nu_e, \bar e, e$ are each $4x1$ Dirac vector fields.
$W_+^\mu$ is a $1x1$ gauge field component.
$\gamma_\mu$ is a $4x4$ Dirac matrix.
$\tau^+$ is a $2x2$ Pauli matrix.
$g$ is a $1x1$ coupling constant.
$$LHS = -\frac{g}{2}\begin{bmatrix}\bar \nu_e & \bar e \end{bmatrix}\begin{bmatrix}0&1 \\ 0&0 \end{bmatrix} \gamma_\mu W_+^\mu\begin{bmatrix} \nu_e \\ e \end{bmatrix}$$
$\gamma_\mu$ and $W_+^\mu$ acts on each individual component $\nu_e$ and $e$,
$$= -\frac{g}{2}\begin{bmatrix}\bar \nu_e & \bar e \end{bmatrix}\begin{bmatrix}0&1 \\ 0&0 \end{bmatrix} \begin{bmatrix} \gamma_\mu W_+^\mu \nu_e \\ \gamma_\mu W_+^\mu e \end{bmatrix}$$
$\tau^+$ acts to the right via matrix multiplication,
$$= -\frac{g}{2}\begin{bmatrix}\bar \nu_e & \bar e \end{bmatrix} \begin{bmatrix} \gamma_\mu W_+^\mu e \\ 0 \end{bmatrix}$$
$\begin{bmatrix}\bar \nu_e & \bar e \end{bmatrix}$ acts to the right via matrix multiplication,
$$= -\frac{g}{2}\bar \nu_e\gamma_\mu W_+^\mu e$$
which equals the RHS.