Amplitude of Vertex Diagram: Unchanged?

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Discussion Overview

The discussion revolves around the amplitude of a vertex diagram in quantum field theory, specifically examining whether the amplitude remains unchanged when the momenta of the external legs are inverted. The scope includes theoretical considerations and implications of crossing symmetry.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant suggests that the amplitude remains unchanged due to crossing symmetry.
  • Another participant raises concerns about the analytic continuation argument for crossing symmetry, noting that it may not apply in cases involving massless particles and that the external legs must be on mass-shell.
  • This participant proposes that while the answer might still be yes, the application of crossing symmetry is unclear without a more thorough analytic continuation that addresses massless particles.
  • A later reply acknowledges the involvement of a photon, indicating a realization of the complexities involved.

Areas of Agreement / Disagreement

Participants do not reach a consensus; there are competing views regarding the applicability of crossing symmetry and the conditions under which it holds.

Contextual Notes

Limitations include the potential dependence on the mass-shell condition and the challenges posed by massless particles in the context of crossing symmetry.

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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