Gaussian integrals with 4-momentum

[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!:)

 

[paranormal]

I would be happy to help you with your calculations. Gaussian integrals with 4-momentum are commonly used in quantum field theory (QFT) to calculate scattering amplitudes and other physical quantities. These integrals often arise when dealing with Feynman diagrams, which represent the interactions between particles.

To calculate the 5 Gaussian integrals that you mentioned, we can use techniques such as Wick's theorem and Feynman parameterization. These methods allow us to express the integrals in terms of simpler integrals that can be evaluated more easily.

It would be helpful if you could provide more specific information about the integrals, such as the limits of integration and the functions involved. This will allow for a more accurate and efficient calculation.

In the meantime, I can provide some general tips for solving Gaussian integrals with 4-momentum. Firstly, we can use the fact that Gaussian integrals are rotationally invariant, meaning that they are unaffected by a change in the direction of the momentum vectors. This allows us to simplify the integrals by choosing a convenient coordinate system.

Secondly, we can use Feynman parameterization to express the integrals in terms of a single parameter, making them easier to evaluate. This technique involves introducing an auxiliary parameter and then performing a change of variables to transform the integral into a more manageable form.

Lastly, we can also use Wick's theorem to express the integrals in terms of simpler integrals involving only two or four momenta. This theorem is based on the Wick contraction, which is a way of pairing up the momentum variables in the integrand.

In conclusion, Gaussian integrals with 4-momentum are an important tool in QFT calculations. With the proper techniques and information, we can efficiently evaluate these integrals and obtain valuable insights into the behavior of particles in quantum systems. I hope this helps with your calculations. Best of luck!