Lagrangian having an $O(N)$ symmetry

A Lagrangian with $$SO(N)$$ symmetry can be written like:

$${\mathcal{L}} = \frac{1}{2}(\partial_\mu \Phi)^T (\partial^\mu \Phi) - (\frac{1}{2}\mu^2 \Phi^T \Phi + \frac{1}{4}\lambda (\Phi^ T \Phi)^2).$$

where $$\Phi =\begin{bmatrix} \phi_1 \\ \phi_2\\ \vdots \\ \phi_N \end{bmatrix}.$$

What additional terms can be included if Lagrangian need only to be $$O(N)$$ symmetric? How a Lagrangian of this form will look like in $$O(N)$$ symmetry? Do we need to replace the "transpose" with "dagger" (due to component fields in $$\Phi$$ becoming complex and also the "$$1/2$$" terms in Lagrangian to be corrected following that). Is this the only change it can have?

Now if I want a $$SU(3)$$ symmetric Lagrangian is the following correct?

$${\mathcal{L}} = (\partial_\mu \Phi)^\dagger (\partial^\mu \Phi) - (\mu^2 \Phi^\dagger \Phi + \lambda (\Phi^\dagger \Phi)^2).$$

where $$\Phi =\begin{bmatrix} \phi_1 \\ \phi_2\\ \phi_3 \end{bmatrix}$$ with all $$\phi_i$$'s being complex?

• You have to be more specific about the "name of the game" I think. For instance, am I allowed to add a (complex) scalar field that is in itself invariant under the $O(N)$ symmetry, and hence add a kinetic term $(\partial^{\mu} \chi^*)(\partial_{\mu}\chi)$ and another interaction term $g(\Phi^T\Phi)(\chi^*\chi)$? Commented May 9, 2023 at 13:05
• Also, why not add terms of the form $\lambda' (\Phi^T\Phi)^4$ etc etc? Commented May 9, 2023 at 13:07
• For the time being don't introduce other fields and don't go beyond $\phi-4$ theory. Commented May 9, 2023 at 13:10
• Then I think all you can have is this... Commented May 9, 2023 at 13:13
• You can also play around with the spatial derivatives if you want more Lagrangians. You typically also assume Lorentz/Euclidean invariance, but this still gives you quite a lot of different alternatives.
– LPZ
Commented May 9, 2023 at 13:28