Consider the Lagrangian:

$$\mathcal L=\mathcal L_{kin}+\frac{1}{2}m_a^2(\phi^2_1+\phi^2_2+\phi^2_3)+\frac{1}{2}m_b^2(\phi^2_4+\phi^2_5)+\lambda_a(\phi^2_1+\phi^2_2+\phi^2_3)^2+\lambda_b(\phi^2_4+\phi^2_5)^2$$

When $\phi_1,\phi_2,\phi_3,\phi_4,\phi_5$ are real scalar fields.

I have 1 question:

(1) What are the symmetries of the lagrangian?

I try to solve this question but I'm not quiet sure what the symmetries are? In the one hand if we define $\phi_a=(\phi_1,\phi_2,\phi_3)$ and $\phi_b=(\phi_4,\phi_5)$, we can say that each of them can rotate in their on phase i.e there is $SO(3)$ symmetry for $\phi_a$ and $SO(2)$ symmetry for $\phi_b$ so our lagrangian full symmetry is $SO(3)\times SO(2)$.

On the other hand, we may say that this is not the most general symmetry, and there is also reflection for each vector (this fact that i`m not sure about) so for $\phi_a$ we have $O(3)$ symmetry (rotation + reflections) and for $\phi_b$ the symmetry is $O(2)$, ending up with total symmetry of $O(3) \times O(2)$



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