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