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So as a first step I am trying to put the particles in their respective representations.

I say that the light-generation fermions belong to the one SU(2) , also written as ##SU_l (2)##, and the third generation fermions belong to the other ##SU_h(2)##. Thus the first transform in ##(\textbf{2},\textbf{1})## of the model, and the last in ##(\textbf{1},\textbf{2})##.

I also need a bi-doublet ##\Sigma## to break the symmetry which will belong in the ##(\textbf{2},\textbf{2})##, and I will have the gauge bosons ##W_l## and ##W_h## that will belong in ##(\textbf{3},\textbf{1})## and ##(\textbf{1},\textbf{3})## respectively.

**Question 1:**

Wouldn't a##(\textbf{3},\textbf{3})## field be possible?

**Question 2:**

I tried to write down the Lagrangian that will be invariant under this group. I know it looks bad but here is what I ended up with:

\begin{align*}

\mathcal{L}_{kin}&= i\bar{\psi}_l \partial_\mu \gamma^\mu \psi_l +i \bar{\psi}_h \partial_\mu \gamma^\mu \psi_h \\

\mathcal{L}_{int}&= g_l W_\mu^l\bar{\psi}_l \gamma^\mu \psi_l + g_h W_\mu^h \bar{\psi}_h \gamma^\mu \psi_h \\&+ Y_{lh} \bar{\psi}_l \Sigma \psi_h\\&- \Big(g_l^2 W_\mu^l W^{\mu l} + 2g_l g_h W_\mu^l W^{\mu h} +g_h^2 W_\mu^h W^{\mu h} \Big) \Sigma^\dagger \Sigma\\&-i (g_l W_\mu^l + g_h W_\mu^h) ( \Sigma^\dagger \partial^\mu \Sigma + \Sigma \partial^\mu \Sigma^\dagger)\\

\mathcal{L}_{gauge}&= - \frac{1}{4} W_{\mu \nu}^l W^{\mu \nu l}- \frac{1}{4} W_{\mu \nu}^h W^{\mu \nu h} \\

\mathcal{L}_{scalar}&= \partial_\mu \Sigma^\dagger \partial^\mu \Sigma - \mu_{h}^2 \Sigma^\dagger \Sigma + \lambda_h |\Sigma^\dagger \Sigma|^2

\end{align*}

In literature I read that one also needs to add a Higgs doublet that belongs to the ##(\textbf{2},\textbf{1})## so that it will break the final SM group ##SU_{h+l}(2) \times U_Y(1)##. However I don't understand how one can add just 1 Higgs doublet and not a second one (that will belong to ##(\textbf{1},\textbf{2})## rep). Adding just the recommended Higgs Doublet, I will have to couple it only to the light generation of fermions (and I think at the end the heavy generation won't get masses in the final stage):

as I understand it the reccommendation asks to add (among others): [itex] Y \bar{\psi}_l H \psi_l [/itex]

vs

I think i should write something like: [itex] Y_1 \bar{\psi}_l H_1 \psi_l +Y_2 \bar{\psi}_h H_2 \psi_h [/itex]

Any idea what I'm thinking is wrong?