Renormalized vertex always a log?

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Discussion Overview

The discussion centers around the nature of one-loop corrections to vertex functions in quantum field theory, specifically exploring why these corrections often appear logarithmic. Participants examine the implications of renormalization and power-counting arguments in this context.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions the reason behind one-loop corrections to vertex functions being logarithmic, noting that the Taylor series expansion of the vertex leads to divergences that are subtracted off.
  • Another participant mentions that while one-loop divergences generally depend on the vertex nature and the underlying theory, they often find that for theories with dimensionless coupling constants, the one-loop vertex correction tends to be logarithmic.
  • A third participant asserts that logarithmic divergences are preferable to quadratic ones as a result of renormalization.
  • A fourth participant references Weinberg's theorem as a potential explanation for the observed logarithmic behavior.

Areas of Agreement / Disagreement

Participants express varying views on the reasons behind the logarithmic nature of one-loop corrections, with some agreeing on the role of dimensionless coupling constants while others highlight exceptions. The discussion remains unresolved regarding the generality of these observations.

Contextual Notes

Participants note that exceptions exist, such as the 4-photon vertex, and that the discussion relies on power-counting arguments and the specifics of the theories being considered.

geoduck
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Is there a reason that one-loop corrections to vertex functions seem to always be logarithmic?

If you write the vertex as a Taylor series in the external momenta, then the first couple of terms (say the constant and linear terms) diverge, but these divergence gets subtracted off, so you now have a Taylor series minus the first couple of terms, and this Taylor series is a log?
 
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With some exceptions (e.g., the 4-photon vertex), the one-loop divergence is given by the usual power-counting arguments, and depends on the nature of the vertex, and the theory that it's part of.

That said, we are often interested in theories with dimensionless coupling constants, and (again probably with some exceptions, though none come to mind at the moment) then the one-loop vertex correction (to a vertex corresponding to a dimensionless coupling) is indeed logarithmic, by power-counting arguments.
 
that is the result of renormalization,it is better to have a logarithmic divergence compared to quadratic one.
 

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