The gauge invariance and the tree unitarity

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Recently I have decided to study EM processes with massive spin-1 boson represented by a field $\hat {W}_{\mu}$.

At the first time I have used minimally modified lagrangian: $$\tag 1 \hat {L} = |\partial_{\mu}\hat {W}_{\nu} - \partial_{\nu}\hat {W}_{\mu}|^{2} + m^{2}|\hat {W}|^{2} \to \hat {L}_{m} = |\hat {D}_{\mu}\hat {W}_{\nu} - \hat {D}_{\nu}\hat {W}_{\mu}|^{2} + m^{2}|\hat {W}|^{2} + F_{\mu \nu}^2 ,$$ where $\hat{D}_{\mu} = \partial_{\mu} - iq_{e}\hat {A}_{\mu}$.

Then I have decided to observe the $W W^{+} \to \gamma \gamma$ process which is described by three diagramms of the second (the lowest) order of $q_{e}$.

It turned out that the longitudinal photons are involved in the interaction, despite the apparent Lorentz invariance of the theory. After that I have recalled that theory $(1)$ doesn't have the tree-unitarity, while the theory which is given by $$\hat {L} = \hat {L}_{m} -iq_{e}\hat {F}^{\mu \nu}\hat {W}_{\mu}\hat {W}^{\dagger}_{\nu}$$ has the unitarity. Now the $W W^{+} \to \gamma \gamma$ is free of longitudinal photons.

So, the question: does there exist some relation between the unitarity and gauge invariance? I.e. do we need the tree-unitarity (not the renormalizability, I'm not about it) when discuss about Ward identities and its applications for the arbitrary "gauge-invariant candidate" theory?

This post imported from StackExchange Physics at 2014-07-29 20:50 (UCT), posted by SE-user Andrew McAddams
There's a theorem due to Weinberg that every gauge boson is (naively, discounting Higgs mechanism and such) massless, so it's no surprise that your massive EM runs into problems.

This post imported from StackExchange Physics at 2014-07-29 20:50 (UCT), posted by SE-user ACuriousMind
@ACuriousMind It is not clear for me that $W$ is a gauge boson, the gauge boson is $A$. But I don't understand the question either! Are you (Andrew) taking into account all the three tree level diagrams?

This post imported from StackExchange Physics at 2014-07-29 20:50 (UCT), posted by SE-user Dox
@Dox: Oops, you're right, I jumped to a wrong conclusion when I read the words "massive" and "EM" so close together ;) Your question is probably the better one - if we do not get the Ward identities from the diagrams, we have most certainly forgotten one.

This post imported from StackExchange Physics at 2014-07-29 20:50 (UCT), posted by SE-user ACuriousMind
@Dox : I meaned that minimally extended lagrangian doesn't lead to the gauge invariance theory on the diagramms level, while modification which leads to the unitary theory does.

This post imported from StackExchange Physics at 2014-07-29 20:50 (UCT), posted by SE-user Andrew McAddams
@Dox : as for the diagrams, you can see them into the question body. As for me their sum are full (the picture is given from Horejsi's "Introduction to electroweak interactions").

This post imported from StackExchange Physics at 2014-07-29 20:50 (UCT), posted by SE-user Andrew McAddams
@AndrewMcAddams From your notation, the field $W$ seems to be complex (not real). Why do you claim that your first Lagrangian describe a SINGLE massive spin-1 particle"

This post imported from StackExchange Physics at 2014-07-29 20:50 (UCT), posted by SE-user drake
@drake : the W-field describes particle-antiparticle as well as Dirac spinor describes particle-antiparticle, for example. The reason why we include antiparticle as well as particle is important: it's because we predict poincare-covariance and causality of the theory. Here I don't see a trouble which can lead to the problem with the gauge invariance.

This post imported from StackExchange Physics at 2014-07-29 20:50 (UCT), posted by SE-user Andrew McAddams

bump ...

I suggested to import this question. Now, I believe that it makes no sense. It is not possible to create photon longitudinal polarizations. The question has been deleted from PSE.