Higgs Mechanism

Question

In Higgs mechanism, we take the combination of LH $SU(2)$ doublet and RH singlet along with Higgs doublet so that the overall weak hypercharge and weak isospin is zero to be $SU(2) \times U(1)$ invariant. I am pretty confident about my understanding on weak hypercharge calculation. But it seems weak isospin calculation is not quite clear. Specially when the terms comes from Weinberg's dimension five operator for leptons such as $\overline{L_{iL}^\mathcal{C}}$, Higgs $SU(2)$triplet or fermion $SU(2)$ triplet. To arrange them in $SU(2) \times U(1)$ invariant way how should I calculate weak isospin?

Answer 1

The easiest way to form $SU(2)$ singlets in the most general way is to use the techniques of Young Tableau. The method is discussed from a physicists perspective in many lecture notes online. One such example is given here. Using such method its easy to show that 2 lepton doublets make a singlet and a triplet under $SU(2)$, \begin{equation} 2 \otimes 2 = 3 \oplus 1 \end{equation} This means that to form a singlet (a $1$) with another set of fields you need either the new set of fields to be one of two options

1. A singlet since \begin{equation} 1 \otimes 1 = 1 \end{equation}
2. A triplet since, \begin{equation} \overbrace{ 3} ^{ \mbox{new fields}} \otimes \underbrace{ 3 } _{ L _i L _j \mbox{ contribution}} = 5 \oplus 3 \oplus 1 \end{equation}

As an example we can have the Weinberg operator which is composed of a product of two $ SU(2) $ singlets (option 1): \begin{equation} \frac{ ( L ^T \epsilon H ) ( L ^T \epsilon H ) }{ \Lambda } \end{equation} where $\epsilon \equiv i\sigma_2$ takes a SU(2) doublet and converts it to the conjugate representation. Note that there a slight subtlety in building the above because $ H ^T \epsilon H = 0 $ due to the antisymmetry of $ SU(2) $.

or we can have a seesaw type II model where we introduce a new triplet, $ \vec{ \Delta} \cdot \vec{ \sigma } $: \begin{equation} L ^T \epsilon \vec{ \Delta } \cdot \vec{ \sigma } H \end{equation}