Renormalized vertex always a log?

[geoduck]

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?

 

[Avodyne]

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.

 

[andrien]

that is the result of renormalization,it is better to have a logarithmic divergence compared to quadratic one.

 

[vanhees71]

This is answered by Weinberg's theorem:

http://link.aps.org/abstract/PR/v118/i3/p838