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

In summary: Your Name]In summary, when calculating the light-cone PDF using the quasiPDF method, it is important to perform the loop integration and Fourier transform in the correct order. The limit of pz going to infinity should be taken after the loop integration and Fourier transform have been performed, not before. This is because taking the limit before the loop integration would result in all integrals vanishing. It is also important to note that the Fourier transform should be performed after the loop integration, not before.
  • #1
QuantumJames
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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:

8DbDzsV
https://imgur.com/8DbDzsV

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


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:

e1abr.png


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

fZMbh4b
https://imgur.com/fZMbh4b
 

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  • #2

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.

 

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

1. How do I take Pz in quasiPDF to infinity?

To take Pz in quasiPDF to infinity, you need to use the limit function in calculus. This will allow you to approach infinity as a limit and accurately calculate the value of Pz.

2. What is quasiPDF?

QuasiPDF, or quasi-parton distribution function, is a theoretical framework that allows for the calculation of parton distribution functions (PDFs) in hadrons. It takes into account the effects of energy and momentum transfer, allowing for the calculation of PDFs at different scales.

3. Why is it important to take Pz in quasiPDF to infinity?

Taking Pz in quasiPDF to infinity allows for the calculation of light-cone PDFs, which are essential for understanding the internal structure of hadrons. These PDFs provide information about the momentum and spin distributions of quarks and gluons inside the hadron.

4. Can Pz in quasiPDF ever reach infinity?

No, Pz in quasiPDF cannot actually reach infinity. Taking it to infinity is a mathematical calculation used to approximate the value of light-cone PDFs. In reality, Pz can only approach infinity as a limit.

5. Are there any limitations to taking Pz in quasiPDF to infinity?

Yes, there are limitations to taking Pz in quasiPDF to infinity. This calculation assumes that the hadron is in an infinite volume and does not take into account the effects of confinement, which can significantly affect the behavior of quarks and gluons at large momentum.

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