Breaking of ##SU(2) \times SU(2)## to ##SU(2)##

[ChrisVer]

I am trying to write down the above model (also known as G(221) )...
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): $Y \bar{\psi}_l H \psi_l$
vs
I think i should write something like: $Y_1 \bar{\psi}_l H_1 \psi_l +Y_2 \bar{\psi}_h H_2 \psi_h$

Any idea what I'm thinking is wrong?

 

[ChrisVer]

maybe I could try a $\Sigma^\dagger \Sigma \bar{\psi}_h \psi_h$
but if that's true, I don't see why I cannot use the same for the lights:$\Sigma^\dagger \Sigma \bar{\psi}_l \psi_l$ ...in fact I think this is a higher dimension operator (it's not there in the SM lagrangian either).