General scalar Lagrangian

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TL;DR Summary
Is a term proportional to ##\phi## valid in a scalar Lagrangian?
Hi, if I want to construct the most general Lagrangian of a single scalar field up to two fields and two derivatives, I usually see that is
$$\mathscr{L} = \phi \square \phi + c_2 \phi^2$$ i.e. the Klein-Gordon Lagrangian.
My question is, would be valid the Lagrangian
$$\mathscr{L} = \phi \square \phi + c_1 \phi + c_2 \phi^2$$
?

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With a linear term, the energy is not bounded from below. So this is usually not considered. This is a problem for applying perturbation theory, this indicates that one is expanding around a field configuration that cannot be used as a vacuum. So one then shifts ## \phi \rightarrow \phi - \frac{c_1}{2c_2} ## which gets rid of the linear term.

Haborix
With a linear term, the energy is not bounded from below.
A linear term shifts the location of the minimum of the potential and shifts the overall potential by a finite constant. (speaking loosely about potential densities as potentials)