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


1 Answer 1


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.

Edit: It a pity that I can't accept my own answer :D

  • $\begingroup$ "It a pity that I can't accept my own answer" You can. You may have to wait a while first though, I don't know. $\endgroup$
    – Michael
    Jun 19, 2013 at 2:01
  • $\begingroup$ You can/ . You don't even need to wait a while, because you have posted it using different accounts, . You can request the moderators to merge your accounts, ttoo, by the way. physics.stackexchange.com/contact $\endgroup$ Dec 16, 2013 at 3:13
  • 1
    $\begingroup$ that first euqation where you write the self energy in terms of $A$ and $B$ is because Lorentz invariance and not gauge invariance. $A=B$ on the other hand is indeed because of the ward identity $\endgroup$
    – Yossarian
    Mar 8, 2016 at 13:26
  • $\begingroup$ It seems you have two different accounts, both called "A friendly helper." You can accept your own answer, but you must do it from the account that asked the question and not from the account that gave the answer. $\endgroup$ Nov 22, 2021 at 6:03

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