How to evaluate a triangular fermion loop

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    Fermion Loop
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Discussion Overview

The discussion revolves around the evaluation of a triangular fermion loop in the context of a scalar field decaying into a fermion-antifermion pair that produces two photons. It includes aspects of quantum field theory and the complexities involved in calculating the associated integrals.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant proposes a method for evaluating the loop by following it backwards from the photon vertices, suggesting a sequence of vertex factors and propagators followed by taking the trace.
  • Another participant agrees with the method but notes potential complications due to the chiral anomaly that may arise during the integral evaluation.
  • A different participant expresses confusion about the integral resulting from the proposed method, indicating a need for further clarification.
  • One participant suggests that the original evaluation may be incomplete, recommending the inclusion of additional diagrams involving photon exchanges and referencing advanced quantum field theory textbooks for guidance on integral evaluation.

Areas of Agreement / Disagreement

Participants express differing views on the completeness of the evaluation method, with some agreeing on the approach while others highlight the necessity of considering additional diagrams and the complexities introduced by the chiral anomaly. The discussion remains unresolved regarding the best approach to evaluate the integral.

Contextual Notes

Limitations include potential missing assumptions about the diagrams involved, the dependence on specific definitions related to the chiral anomaly, and unresolved steps in the integral evaluation process.

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Say I have a scalar field which decays into to a fermion - anti fermion pair which then interact in a closed loop to produce two photons. How would I evaluate this? Can I just follow the loop backwards from any of the two photon vertices and just write vertex factor, propogator, vertex factor, propogator, vertex factor, propogator, and then take the trace of that?
 
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Yes. But the evaluation of the integral you get can have tricky subtleties related to the chiral anomaly.
 
I end up with this
Screen%20Shot%202014-10-20%20at%201.18.59%20PM.png

How in the world do I do this integral?
 
I believe you only used one diagram; you have to add the diagram with the two photons exchanged.

See any advanced QFT textbook for how to evaluate the integral.
 

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