| Attribution: |

## QuestionEdit

Quick question regarding superficial degrees of freedom and Ward identities.

For instance in Peskin and Schroeder it is stated that the photon-self energy is superficially quadratically UV divergent but due to the Ward identity it is only logarithmically divergent. I don't see this argument.

The self-energy is given by

$ \Pi^{1-loop}=(g^{\mu\nu}p^2-p^\mu p^\nu)\Pi(p^2) $

How does the Ward identity, or in other words, gauge invariance kill of the divergences?

Best, A friendly helper

#### CommentsEdit

## AnswersEdit

### Answer 1Edit

Ok, I answer it myself. The reason is as follows; Based on gauge invariance the self-energy at one loop has to look like

$ \Pi=(g^{\mu\nu}p^2 A -p^\mu p^\nu B) $

where A and B are the explicit divergences not yet determined. However, in an explicit loop computation the first term does only arise with a divergence in $ D=2 $ whereas the second with a divergence in $ D=4 $ and not worse. But in order for gauge invariance to be true $A=B$ has to hold, i.e. the divergence is actually only in four dimensions and not in two.