# Feynman Diagrams for Interacting Scalar Fields

## Homework Statement

Consider four real massive scalar fields, $\phi_1,\phi_2,\phi_3,$ and $\phi_4$, with masses $M_1,M_2,M_3,M_4$.
Let these fields be coupled by the interaction lagrangian $\mathcal{L}_{int}=\frac{-M_3}{2}\phi_1\phi_{3}^{2}-\frac{M_4}{2}\phi_2\phi_{4}^{2}$.
Find the scattering amplitude for $\phi_{3}\phi_{4}\rightarrow\phi_3\phi_4$, to tree level.

## Homework Equations

I'm not really sure what to put here.

## The Attempt at a Solution

This honestly looks like a trick question to me. The first step is obviously to write down the relevant feynman diagrams and feynman rules to evaluate them, but I can't find a single tree level diagram for this process. A diagram would need an incoming $\phi_3$ and an incoming $\phi_4$, as well as an outgoing $\phi_3$ and an outgoing $\phi_4$.
The interaction lagrangian makes me think that we should have a vertex between a $\phi_1$ and two $\phi_3$'s, as well as a vertex between a $\phi_2$ and two $\phi_4$'s, but this still doesn't allow $\phi_3$ and $\phi_4$ to interact.
Perhaps im misunderstanding the meaning of tree level (I think it just means one of each type of interaction vertex), or perhaps im just misunderstanding the interactions.
Could this possibly have to do with interactions via the kinetic terms of the lagrangian? Can a tree level diagram just consist of the $\phi_3$ and the $\phi_4$ propagating along and not interacting (if yes I don't think this would contribute to the scattering), I'm not sure and would appreciate any help. I'm just feeling pretty confused at the moment.
Thanks!

Orodruin
Staff Emeritus