this excercise is about the custodial $SU(2)_R$ symmetry in the SM Higgs Mechanism. Step by step I have to develop the theory. I get the generel idea of the global custodial symmetry and why we need it but I need some help to fully understand the single transformation steps...
The Higgs field is arrange as a matrix in this way: $$ \Phi=\frac{1}{\sqrt{2}}\begin{pmatrix}\phi^{0*} & \phi^+\\ -\phi^- & \phi^0\end{pmatrix} $$
Then the Lagrangian for the Higgs without gauge bosons is:
$$ \mathcal{L}=Tr\left[(\partial_\mu\Phi)^\dagger(\partial^\mu\Phi)\right] +\mu^2Tr\left[\Phi^\dagger\Phi\right] -\lambda(Tr\left[\Phi^\dagger\Phi\right])^2 $$
a) The first step is to show that this is equivalent to the SM Lagrangian. That's easily done.
b) Now this Lagrangian is also invariant under global biunitary transformations as $\Phi \rightarrow L\Phi R^\dagger$ with L$\in SU(2)_L$ and R$\in SU(2)_R$. Itis also straight foward to check that this transformation leaves the Lagrangian invariant. My first question is: How to prove that L,R are really independent of each other.(i.e. there is no $L\in SU(2)_L$ and $R\in SU(2)_R$ so that $L\Phi=\Phi R^\dagger$)
c)Now the field develops a VEV $$\Phi=\frac{1}{2}\begin{pmatrix}v & 0\\ 0 & v\end{pmatrix}$$ Under which transformation is this matrix still invariant? My answer: It is invariant under the custodial symmetry $SU(2)_R$ with L=R. Is this correct? Since the transformations need to be equal to not change the phase of the matrix?
Now comes the tricky part for me. Including first the SU(2) gauge bosons and in the next step the U(1) gauge boson.
d) With the SU(2) gauge bosons we change the derivative to the covariant derivative $D_\mu=\partial_\mu +ig\frac{\sigma^i}{2}W_\mu^i$. Question: how do the fields W_\mu^i transform under $SU(2)_R?$ I guess the should be triplets of the of this transformation so they transform the same way as the Higgs matrix, namely $W_\mu^i\rightarrow RW_\mu^iR^\dagger$ With this I can show that the Lagrangian is still invariant under this transformation. The only problem is: do the transformation matrices R commute with the pauli matrices of the derivative? If not I dont know how to show this?
e) The last and final part is to include the hypercharge gauge boson and say why this cannot be taken over so easily from the doublet representation? Here is the point I really don't know what to do. I don't understand this part about the custodial symmetry and also I don't see why it can be implemented as a subgroup of $SU(2)_R$ generated by $-\sigma^3$ And what happens to the symmetry? I know it should be broken... but why? And why does the third Pauli matrix help? Because it is a generator of SU(2) and the hypercharge generator $Y$ (basically the unit matrix is not part of SU(2)?)
All help is appreciated and the Custodial symmetry and Higgs-Kibble but it is only about the W bosons.
