# Higgs Doublet Transformations

If we consider two complex Higgs doublets with $SU(3)_{c} ⊗ SU(2)_L ⊗ U(1)_Y$ quantum numbers $\phi_{A,B}$ ∼ (1,2,1), such that

$\phi_A = \begin{pmatrix} \phi_{A}^{\dagger}\\ \dfrac{1}{\sqrt2} (Re \phi_{A}^{0} + iIm \phi_{A}^{0}) \end{pmatrix}$ & $\phi_A = \begin{pmatrix} \phi_{B}^{\dagger}\\ \dfrac{1}{\sqrt2} (Re \phi_{B}^{0} + iIm \phi_{B}^{0}) \end{pmatrix}$

If we assume that they acquire the vacuum expectation values (vevs):

$<\phi_A> = \begin{pmatrix} 0\\ \dfrac{v_A}{\sqrt2} \end{pmatrix}$ & $<\phi_B> = \begin{pmatrix} 0\\ \dfrac{v_B}{\sqrt2} \end{pmatrix}$

I know that we can change the basis from {$\phi_A,\phi_B$} to a {$\phi_1,\phi_2$} basis, bearing in mind that only one doublet gains a non zero vev as follows:

$<\phi_1> = \begin{pmatrix} 0\\ \dfrac{v}{\sqrt2} \end{pmatrix}$ & $<\phi_2> = \begin{pmatrix} 0\\ 0 \end{pmatrix}$

My issue is that which particular transformation can I use to be able to do this transformation and how do I proceed from there. Basically asking for how can I express {$\phi_A,\phi_B$} in terms of {$\phi_1,\phi_2$} and relate the $v_A, v_B$ and $v$ altogether

• I'm a bit busy right now, but I will try to answer it later. I believe it is just a $SU(2)_L$ rotation, and using the fact that you can rotate $(\alpha, 0)_L^T$ into any state via an $SU(2)_L$ rotation. – InertialObserver Sep 4 '18 at 21:44