$SU(2)$ symmetry of $\mathcal{L}=\partial_{\mu}\Phi^{\dagger}\partial^{\mu}\Phi - \Phi^{\dagger}M\Phi$

Question

I'm considering a Lagrangian of two complex scalar field: $$\mathcal{L}=\partial_{\mu}\phi_1^{*}\partial^{\mu}\phi_1-m_1^2\phi_1^{*}\phi_1+\partial_{\mu}\phi_2^{*}\partial^{\mu}\phi_2-m_2^2\phi_2^{*}\phi_2$$ It can be written in a doublet:

$$\Phi_1 =\begin{pmatrix} \phi_1\\ \phi_2 \end{pmatrix} \,\,\,\,\,\,\,\,\,\,\,\,\, \Phi_1^{\dagger} = \begin{pmatrix} \phi_1^\dagger & \phi_2^\dagger \end{pmatrix} \,\,\,\,\,\,\,\,\,\,\,\,\, M = \begin{pmatrix} m_1 & 0 \\ 0 & m_2 \end{pmatrix} $$

$$\mathcal{L}=\partial_{\mu}\Phi^{\dagger}\partial^{\mu}\Phi - \Phi^{\dagger}M\Phi$$ It has an internal global symmetry $SU(2)$: $$ \begin{cases} \Phi^{'}= e^{\frac{i}{2}\vec\alpha \cdot \vec\sigma} \Phi\\ \Phi^{'\dagger} = \Phi^{\dagger}e^{-\frac{i}{2}\vec\alpha \cdot \vec\sigma} \end{cases} $$ I'd like to check it explicitly but I'm stuck: $$\mathcal{L^{'}}=\partial_{\mu}\Phi^{\dagger}\partial^{\mu}\Phi - \Phi^{\dagger}M\Phi$$

$$\mathcal{L^{'}}=\begin{pmatrix} \partial_{\mu}\phi_1^{*} & \partial_{\mu}\phi_2^{*} \end{pmatrix} e^{-\frac{i}{2}\vec\alpha \cdot \vec\sigma} e^{+\frac{i}{2}\vec\alpha \cdot \vec\sigma} \begin{pmatrix} \partial^{\mu}\phi_1 \\ \partial^{\mu}\phi_2 \end{pmatrix} - \begin{pmatrix} \phi_1^{*} & \phi_2^{*} \end{pmatrix} e^{-\frac{i}{2}\vec\alpha \cdot \vec\sigma} \begin{pmatrix} m_1 & 0 \\ 0 & m_2 \end{pmatrix} e^{+\frac{i}{2}\vec\alpha \cdot \vec\sigma} \begin{pmatrix} \phi_1\\ \phi_2 \end{pmatrix}$$ However: $$ S_1=e^{-\frac{i}{2}\vec\alpha \cdot \vec\sigma} \begin{pmatrix} m_1 & 0 \\ 0 & m_2 \end{pmatrix} = \begin{pmatrix} m_1e^{-\frac{i\alpha_3}{2}} & m_2e^{-\frac{i(\alpha_1-i\alpha_2)}{2}} \\ m_1 e^{-\frac{i(\alpha_1+i\alpha_2)}{2}}& m_2e^{\frac{i\alpha_3}{2}} \end{pmatrix}$$ $$ S_2=\begin{pmatrix} m_1 & 0 \\ 0 & m_2 \end{pmatrix} e^{-\frac{i}{2}\vec\alpha \cdot \vec\sigma} =\begin{pmatrix} m_1e^{-\frac{i\alpha_3}{2}} & m_1 e^{-\frac{i(\alpha_1-i\alpha_2)}{2}} \\ m_2 e^{-\frac{i(\alpha_1+i\alpha_2)}{2}} & m_2e^{\frac{i\alpha_3}{2}} \end{pmatrix}$$ So $S_2=S_1^{T}$.

Answer 1

The $\text{SU}(2)$ symmetry of this theory is only preserved when $m_1=m_2\equiv m$, in which case $M=m\textbf{1}$. Otherwise, the internal $\text{SU}(2)$ symmetry is broken down to $\text{U}(1)\times\text{U}(1)$, one for each scalar field. Something similar to this happens in QCD, in which the up and down quarks have nearly the same mass, and the approximate $\text{SU}(2)$ isospin symmetry is useful for classifying low-mass mesons and baryons.

Answer 2

Ignore all normalizations, since this is an issue of symmetry, and consider real numbers $a,b,c,d$, s.t. $$ \phi_1=a+ib, \qquad \phi_2=c+id , $$ and define $m_1=m_2 + \Delta$.

Then, for the real 4-vector $\vec \varphi\equiv (a,b,c,d)^T$, you have $$\mathcal{L}=\partial_{\mu} \vec \varphi \cdot \partial^{\mu} \vec \varphi -m_2 \vec \varphi \cdot \vec \varphi -\Delta (a^2+b^2). $$ It is manifest that, for $\Delta=0$, this expression is SO(4) ~ SU(2)×SU(2) invariant. (One of these two SU(2) s is the SU(2) you display, but the other one is less easy to see in your language, and corresponds to the "right custodial" SU(2) of the SM, a global approximate symmetry of it. Through SO(4), you can appreciate there are 6 transformations mixing your 4 real scalars.)

But the introduction of the explicit perturbation $\Delta$ breaks this SO(4) into $U(1)\times U(1) \sim O(2) \times O(2)$, the two O(2) s rotating the doublets (a,b) and (c,d), independently, respectively. The other 4 generators of SO(4) are explicitly broken, since you cannot preserve the last, perturbation term, any other way--try it.