# Computing the spectrum of a Lagrangian in field theory

snypehype46
Homework Statement:
What is the spectrum of theory given by this lagrangian?
Relevant Equations:
$$L = \bar{\psi}i \gamma^\mu \partial_\mu \psi - g\bar{\psi}(\sigma + i\gamma^5\pi)\psi + \frac{1}{2}(\partial_\mu \sigma)^2+ \frac{1}{2}(\partial_\mu \pi)^2 -V(\sigma^2 + \pi^2)$$
I have the following lagrangian density:

$$L = \bar{\psi}i \gamma^\mu \partial_\mu \psi - g\bar{\psi}(\sigma + i\gamma^5\pi)\psi + \frac{1}{2}(\partial_\mu \sigma)^2+ \frac{1}{2}(\partial_\mu \pi)^2 -V(\sigma^2 + \pi^2)$$

where $\pi$ and $\sigma$ are scalar fields.

I have show that this lagrangian density is invariant under a chiral symmetry and has a conserved current.
Now the question I'm attempting asks the following:

> What is the spectrum of theory when ##V(\sigma^2 + \pi^2) = \lambda(\sigma^2 + \pi^2 - c^2)^2##?

Now I don't quite understand how to do this problem. I assume that by spectrum the question means what kind of particle you obtain in that specific condition. Am I right? If so, how do I go about computing the spectrum? I thought about plugging the value in the lagrangian and then working out the EOMs but not sure how that would help anything.