As far as I know, there are some prerequisites for an action to be preferable in classical field theory like relativistic invariance, locality, real-valueness, derivatives up to first order and internal symmetries invariance.
There's given a lagrangian density which is called "the simplest scalar field lagrangian" in my book (and several others i've checked out): $\mathcal{L} = \frac{1}{2}(\partial_{\mu} \phi)^2 - \frac{1}{2}m^2\phi^2$. My question concerns lagrangian of the form $\mathcal{L} = \frac{1}{2}(\partial_{\mu} \phi)^2 - m^2\phi$. Isn't it a possible one? If it is not, then for what reason (it looks like it satisfies all the conditions written above)? If it is, then isn't it simpler than the first one?
It seems to be quite a stupid question, so I am sorry if ones of its kind are not welcome here. I will be thankful for any help.