Let's have theory of massive vector boson interacting with EM field: $$ L = |D_{\mu}W_{\nu} - D_{\nu}W_{\mu}|^{2} + m^{2}|W|^{2}, \quad D_{\mu} = \partial_{\mu} - ieA_{\mu}. $$ The question: how to show that this lagrangian involves the consequence of the Ward identity $q_{\mu}M^{\mu ...} = 0$ on the Feynman diagramms level (so here is possible to emit the longitudinal photons in $\zeta$-gauge)?

Particularly (for a given process in a given order of perturbation theory) I may only get the expression for the vertex and then act on it by photon momentums. But I don't know how to get the non-perturbative result.


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