# Relation among anomaly, unitarity bound and renormalizability

There is something I'm not sure about that has come up in a comment to other question:

Why do we not have spin greater than 2?

It's a good question--- the violation of renormalizability is linked directly to violation of unitarity, which was exploited by Weinberg (surprise, surprise) to give an upper bound of something like 800 GeV on the fundamental Higgs mass from the W's and Z's unitarization. The breakdown of renormalizability is a wrong one-loop propagator correction to the gauge boson, and it leads to a violation of the ward-identity which keeps the non-falling-off part of the propagator from contributing. It's in diagrammar (I think), it's covered in some books, you can ask it as a question too.

I know what unitarity bound is the user talking about, but I don't know what is the violation of the Ward identity that he mentions. I guess that it is the global $SU_L(2)$ symmetry but I have never seen anything relating the unitarity bound and this anomaly.

The general issue is the following: Assume a Yang-Mills term and the coupling of the charged vector field to a fermionic conserved current under a global symmetry. Then one adds an explicit mass term to the vector field so that one breaks the gauge symmetry by hand, but not the global part that gives the conserved current (the gauge symmetry that goes to the identity in the boundary entails constraints). Then, according to the user (at least what I understood), when one takes into account loops effect the global part is also broken. Therefore, the mass term is breaking the redundancy part of the symmetry by hand (at the classical level) and it is also breaking the global part at the quantum level.

I would be grateful if somebody is able to clarify this to me. References are also welcome.

$M_h^2 = \lambda v^2, \lambda < \sqrt{4\pi}$
where $\lambda$ is the Higgs self coupling. Thus one gets
$M_h^2 < \sqrt{4\pi}v^2 \sim (870 GeV)^2$
The violation of the Ward Identity just requires you to add a scalar particle to the theory, that adjusts the Ward identity. The Ward identity they talk about is just the one related to any process with external W or Z, like $f \bar{f} \rightarrow W^+ W^-$ or $Z Z \rightarrow W^+ W^-$