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

[k54ledung]

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!

 

[andrien]

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?

 

[dm4b]

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"

 

[k54ledung]

Thanks so much!:)