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.

 

[vanhees71]

It's a bit unclear to me, what the problem statement really wants us to calculate. My guess is they mean to determine the "particle content" of the theory. To that end you first have to think about what's the stable equilibrium point given your potential. Then you can figure out the masses and interactions between the physical "particles" by expanding the Lagrangian around an arbitrarily choosen equilibrium configuration of the fields (note that the ground state is degenerate here; usually you choose the vacuum expectation value along the $\sigma$ direction).

The very important fundamental concept here is the spontaneous breaking of a global symmetry (in this case chiral symmetry).

 

[jack476]

It means to identify the particles predicted by the theory and their masses.