Feynman diagrams for phi phi -> phi phi

[silverwhale]

Homework Statement

Compute the matrix element for the scattering process $$\phi \phi \to \phi \phi$$

Homework Equations

The Lagrangian is given by
$$L = \frac{1}{2} \partial_{\mu} \phi \partial^{\mu} \phi + \frac{\alpha}{2} \phi \partial_{\mu} \phi \partial^{\mu} \phi + \frac{\beta}{2} \phi^2 \partial_{\mu} \phi \partial^{\mu} \phi$$

The Attempt at a Solution

At tree level I included a 4 legged vertex diagram + 3 diagrams with an internal line. Is this correct? I get a delta function with 4 momenta ( multiplied with other terms) + product of 2 delta functions with 3 momenta (multiplied with other terms) equal to the scattering implitude multipplied by a delta function of 4 momenta.

Now my question is just what are the Feynman diagrams for the general process: $$\phi \phi \to \phi \phi$$

 

[silverwhale]

If we consider an other case where the interaction term looks like $$c_1 \phi^3 + c_2 \phi^4,$$ can one just sum up the feynman diagrams (for eg. tree level diagrams) for phi3 theory and phi4 theory to express the $$\phi \phi \to \phi \phi$$ scattering?