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I'm looking for detailed example/calculation of the scattering amplitude between two particles (mesons) in $\phi^4$-perturbation theory using the path integral formalism. Do anyone of you guys have a good example to share? Maybe some old exercise of yours or a link to the web.

I'm using the book Quantum Field theory in a Nutshell and I'm currently reading chapter 1.7 (Feynman Diagrams) and I really wish there was a more detailed description of the calculations for the scattering amplitude of collision between particles. That's why I think an example would be of great educational help to me.

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Have you checked the book by David Bailin and Alexander Love on Gauge Theories? https://www.amazon.com/Introduction-Revised-Graduate-Student-Physics/dp/075030281X It contains the essence on scattering amplitudes all starting from the Green functions formalism which uses path integrals (thus no operators, no Wick theorem).

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Does the calculation starting from page 129 of this note by Casalbuoni is of any help to you? As far as I have understood question, it may help you out.

Also, Critical properties of $\phi^4$ theories by Hagen Kleinert is the most extensive reference for $\phi^4$ theory that I know of. Not widely known, at least in my experience, is a treasure trove for higher-order loop calculations, finding symmetry factor etc. Highly recommended for non-trivial details which are not easy to find elsewhere.

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