Ward-Takahashi identity and renormalization

[samuelr85]

What I don't understand about WT identity is how it allows or helps you to renormalize a quantum field theory (es. QED). Not in details, just the basic ideas, if possible.
Thanks in advice

 

[bobloblaw]

I'm not entirely sure but I've heard that a massive spin one field is non-renormalizable. The gauge symmetry of the photon "protects" the photon from being non-renormalizable. The Ward-Takahashi identity says that gauge invariance is preserved at the quantum level. So I guess the Ward-Takahashi identity tells you that quantum fluctuations don't break gauge invariance and hence the renormalizability of the theory.

 

[samuelr85]

Thanks for your answer, it make sense. Do you think you need to explicitly use the WT identities for showing that the divergent parts of the amplitudes cancel each other, at a given order of perturbation theory? Or that cancellation is it just a consequence of the presence of gauge invariance?

 

[vanhees71]

Indeed, the Ward-Takahashi identities for local Abelian gauge symmetries are necessary for the Dyson renormalizability of such theories like QED. E.g., from naive power counting the four-photon vertex is logarithmically divergent. However, the WTI for this vertex ensures that it is finite. For details, see

http://fias.uni-frankfurt.de/~hees/publ/lect.pdf

It is also not true that massive vector fields lead necessarily to non-renormalizable models. E.g., the standard model is renormalizable although the W- and Z-bosons are massive. This is due to the fact that this theory is still a non-Abelian gauge theory, i.e., the action is invariant under local gauge transformations, but this local symmetry is spontaneously broken. This provides mass to the gauge bosons without breaking gauge invariance, and thus the theory stays renormalizable. This Higgs mechanism predicts the existence of a massive scalar boson, the famous Higgs boson.

For the Abelian case, one can give the gauge boson even a mass by hand (i.e., without using spontaneous symmetry breaking of the local gauge symmetry), and still keep the theory invariant under local gauge transformations. This is the Stückelberg model, rediscovered by Kroll, Lee, and Zumino to build an effective renormalizable theory for the (neutral) light vector mesons, \rho, \quad \omega, and \phi:

Kroll, N. M., Lee, T. D., and Zumino, B.: Neutral Vector Mesons and the Hadronic Electromagnetic Current , Phys. Rev. 157, volume 157, 1376, 1967