QED Loop Calculations: Questions and Solutions

[gopferdechu]

Hi,

I'm currently doing some one-loop calculations for some bad-*** decays with vector final states, which yield me some tri-tensor threepoint functions. I know the Passarino-Veltmann reduction scheme, and also got the general reduction formulas from Denner-Dietmeier to simplify expressions with a general number of external tensor indices into fundamental loop integrals. But I still have some questions:

Do I understand correctly, that the form factors in front of external 4-momenta involving different external tensor indices but the same # of external momenta (e.g. 2 x p1, 1 x p2 in the tri-tensor 3-point case, or just p1 p2 in the di-tensor 3-point case, when p1,p2 denote the external momenta) they have the same FF? So when Denner and Dietmeier state me several solutions for different combinations of these external momenta I can only pick one of these? (like one can do in the di-tensor 3-point case?)

Second: Denner and Dietmeier's solution give raise to additional factors proportional to m_0,1,2 square. When I calculated the self-energy correction of my external vector particle these contributions canceled each other so I only got a final expression involving only loop-integrals for my virtual diagram. But when I do the same for an internal photon loop from e.g. a fermionic final state to my vector final state this seems not to be the case. I didn't integrate my matrix elements numerically yet, but should this worry me?

Thirds: do you know any software that can help me verify my calculations? I played around a bit with some mathematica modules but never was really satisfied.

Thanks a lot,
Florian

 

[AndyW]

Dear Florian,

Thank you for sharing your questions and progress on your one-loop calculations. It sounds like you have a good understanding of the Passarino-Veltmann reduction scheme and Denner-Dietmeier's general reduction formulas. To answer your first question, yes, you are correct that the form factors in front of external 4-momenta with the same number of external tensor indices will have the same form factor. As for Denner and Dietmeier's solutions, it is always best to cross-check with other literature or use your own judgement to determine which solution is best for your specific case.

In regards to your second question, it is not uncommon for the self-energy corrections to cancel each other out, especially if you are considering a specific case where certain contributions are not present. However, it would be best to double check your calculations and make sure there are no errors. If you are still unsure, it may be helpful to consult with a colleague or advisor for further guidance.

As for software to help verify your calculations, there are a few options available. Some popular choices include FeynCalc, FORM, and LoopTools. Each of these software have their own strengths and weaknesses, so it may be beneficial to try them out and see which one works best for your needs.

Best of luck with your calculations!

 

[sir-pinski]

Hi Florian,

It sounds like you have a good understanding of the Passarino-Veltmann reduction scheme and the Denner-Dietmeier reduction formulas. To answer your first question, yes, the form factors in front of external 4-momenta with the same number of external tensor indices will be the same. This is because they correspond to the same kinematic structure and therefore have the same form factor. So, for example, in the tri-tensor 3-point case, the form factor in front of 2 x p1 and 1 x p2 will be the same as the form factor in front of 1 x p1 and 2 x p2. As for picking one of the solutions provided by Denner and Dietmeier, you can choose any one that makes your calculation easier or more convenient.

In response to your second question, it is not uncommon for the contributions from the mass parameters to cancel out in the self-energy correction of an external vector particle. This is because the mass parameter can be absorbed into the renormalization of the external particle's mass. However, for an internal photon loop, this may not be the case. It is important to check your calculations and make sure they are consistent with the expected behavior.

As for software that can help verify your calculations, there are a few options. Some popular choices include FeynCalc, FormCalc, and LoopTools. These programs can assist with loop calculations and provide a way to cross-check your results. However, it is always a good idea to manually check your calculations as well to catch any potential errors.

I hope this helps and good luck with your calculations!