$2 to 2$ scattering cross section in arbitrary theory

What is the general method to compute the $2 to 2$ scattering cross section given an arbitrary Lagrangian ? I would like a step by step recipe that can be easily implemented in Mathematica.

For example, I want to input one of these Lagrangians

$mathcal{L} = frac{1}{2}partial_mu phi partial^mu phi – frac{1}{2}m^2 phi^2-frac{g}{4!}phi^4$

$mathcal{L} = frac{1}{2}partial_mu phi partial^mu phi – frac{1}{2}m^2 phi^2-frac{g}{3!}phi^3$

$mathcal{L} = frac{1}{2}partial_mu phi partial^mu phi+ g(partial_mu phi partial^mu phi)^2 $

$mathcal{L} = frac{1}{2}partial_mu phi partial^mu phi + g(partial_mu phi partial^mu phi)phi^2 $

and have outputted the $2 to 2$ cross section to the first non trivial order in $g$.

I think I understand how to compute the cross section for the two first lagrangians:

• write the $M$ matrix elements enumerating the Feynman diagrams and using momentum Feynman rules (Schwartz box 7.1)
• use the formula for $2to 2$ scattering $frac{dsigma}{dOmega}=frac{1}{60pi^2E_{cm}^2}|M|^2$ (Schwartz 5.33)

but it seems like this method would only work for a lagrangian of the form $mathcal{L} = frac{1}{2}partial_mu phi partial^mu phi – frac{1}{2}m^2 phi^2-frac{g}{n!}phi^n$.

Hence, the 2 remaining questions are: How to deal with the two last lagrangians ? How to automatically enumerate the Feynman diagrams ?