Where can I find some examples of cross sections calculations in QFT done step-by-step? Those integrals are a little horror. For example - a simple scalar+scalar -> scalar+scalar at the tree level in a theory scalar $\phi^4$ ?
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Peskin and Schroeder tends to be the book used in most introductory QFT courses, so you'll definitely find all things there done in a pretty detailed way.
Warren Siegel has an online book which is also pretty good, Fields.