How to Calculate Gaussian Integrals with 4-Momentum in QFT?

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Calculating Gaussian integrals in Quantum Field Theory (QFT) often involves five specific integrals, with a focus on the last one and differentiation under the integration sign for the first four. The discussion highlights the importance of understanding four-momentum, denoted as 'q', in these calculations. A reference to Zee's "QFT in a Nutshell" is suggested, as it contains valuable derivations relevant to these integrals. Additionally, the possibility of accessing the book on Amazon for further insights is mentioned. Mastery of these integrals is crucial for accurate QFT calculations.
k54ledung
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I am doing some calculations in QFT. And, in my calculations, I have to deal with 5 Gaussian integrals as followed. Please help me calculate those 5 integrals. Thank you very much!
 

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you can start with the last one and some differentiation under integration signs would be useful for the first four.by the way that q is four momentum,right?
 
I don't know if you have access to Zee's QFT in a Nutshell textbook, but he has a helpful page ,or two, that shows how to derive those integrals near the beginning of the book.

You could probably find it on Amazon, as well, assuming it's one of the ones you can "look inside"
 
Thanks so much!:)
 

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