 #1
snypehype46
 12
 1
 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.
$$L = \bar{\psi}i \gamma^\mu \partial_\mu \psi
 g\bar{\psi}(\sigma + i\gamma^5\pi)\psi +
\frac{1}{2}(\partial_\mu \sigma)^2+
\frac{1}{2}(\partial_\mu \pi)^2
V(\sigma^2 + \pi^2)$$
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.