How to take Pz in quasiPDF to infinity to get Light-cone PDF

[QuantumJames]

When I calculate light-cone PDF by taking pz in quasiPDF to infinity before one-loop integration, I will encounter all the integrations vanish. Such as this integral below:

https://imgur.com/8DbDzsV

This is from one of one-loop quasiPDF diagram, the sail diagram.

The definition is above, W_z is the Wilson line (gauge link) along z direction, and one of the one-loop Feynman diagram is followed:

How can I take pz to infinity before loop integration? Even though I have done the Fourier transform first before taking the limit, there would still be the same problem

https://imgur.com/fZMbh4b

 

[AndyW]

Thank you for bringing up this issue with calculating the light-cone PDF using the quasiPDF method. It is important to carefully consider the limits and integrations involved in such calculations, as they can greatly affect the results.

Firstly, I would like to clarify that the limit of pz going to infinity should not be taken before the loop integration. Instead, it should be taken after the loop integration has been performed. This is because the loop integration involves momentum components in all directions, including pz, and taking pz to infinity before the loop integration would result in all integrals vanishing, as you have observed.

Moreover, the Fourier transform should also be performed after the loop integration, not before. This is because the Fourier transform involves integrating over all possible momenta, including those involved in the loop integration. Therefore, it is important to first perform the loop integration and then take the Fourier transform to obtain the light-cone PDF.

I hope this helps clarify the issue you have encountered. If you have any further questions or concerns, please do not hesitate to reach out.