Amplitude of Vertex Diagram: Unchanged?

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SUMMARY

The discussion centers on the amplitude of a vertex diagram in particle physics, specifically regarding the effect of replacing the momenta of external legs with their negative counterparts. It is established that the amplitude remains unchanged due to crossing symmetry, as referenced in Itzykson & Zuber. However, the discussion highlights the complications introduced by massless particles and the requirement for external legs to be on mass-shell in the derivation of crossing symmetry. The conclusion suggests that while the answer remains affirmative, further analytic continuation arguments are necessary to fully address the implications of massless particles.

PREREQUISITES
  • Understanding of crossing symmetry in quantum field theory
  • Familiarity with vertex diagrams and their components
  • Knowledge of mass-shell conditions in particle physics
  • Basic concepts of analytic continuation in complex analysis
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  • Study the implications of crossing symmetry in quantum field theory
  • Explore the role of massless particles in vertex functions
  • Research analytic continuation techniques in particle physics
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Particle physicists, quantum field theorists, and researchers interested in the behavior of vertex diagrams and crossing symmetry in high-energy physics.

noether21
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If the momenta on the three external legs p(incoming fermion), p'(outgoing fermion) and
p-p' (photon) of a vertex diagram are replaced by -p, -p' and p'-p respectively (i.e., all the external momenta are multiplied by -1) does the amplitude remain unchanged?
 
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yes due to crossing symmetry
 
The analytic continuation argument used to derive crossing symmetry (Itzykson & Zuber)
seems to require that there are no massless particles (vacuum is an isolated point). Further
it appears that the external legs must be on mass-shell in the crossing symmetry derivation.
In the 3-point vertex function all the external legs cannot be on mass-shell. I think the
answer to the question is still yes, but short of a lengthy analytic continuation argument
that handles massless particles and doesn't require mass-shell condition (which may not even work), it's unclear how crossing symmetry can be applied directly. Any thoughts?
 
ah yes, a photon is involved, didn't thought of that =/
 

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