What Is the Compact Group of Global Symmetry for This Lagrangian?

[shir]

Hi guys, have a very tricky question on my HW to find compact group of global symmetry to this Lagrangian of 2 complex scalar fields
$$L={\partial_\mu \phi_1^*}{\partial_\mu \phi_1}+{\partial_\mu \phi_2^*}{\partial_\mu \phi_2}-\lambda(\phi_1^* \phi_1 - \phi_2^* \phi_2 - v^2)^2$$
and I can't figure it out because of the minus in potential part $$\phi_1^* \phi_1 - \phi_2^* \phi_2$$

Do you have any ideas how to solve it?.
P.S. Of course there is U(1) group, but i think there should be something else.
Thank's.

 

[samalkhaiat]

Write
$$\delta \phi_{ a } = i ( \epsilon \cdot T )_{ a b } \phi_{ b } , \ \ (a,b) = 1 , 2 , ..., 4$$
Then calculate the change in the Lagrangian (do not use the equation of motion), then set $\delta \mathcal{ L } = 0$ and see what kind of condition you get for the matrices $T$. This will determind the symmetry group.
You can also find the Noether current associated with the above transformations (here you can use the equation of motion), then find the algebra generated by the Noether charges. This also determine the symmetry group for you.