Assume a model of massive fermion $f$ interacting with an axial boson $A_{\mu}$: $$ \mathcal{L} = -\frac{1}{4}A_{\mu\nu}A^{\mu\nu}+\bar{f}i\gamma_{\mu}\partial^{\mu}f+\bar{f}\gamma_{5}\gamma_{\mu}f A^{\mu} - m_{f}\bar{f}f+\frac{m_{A}^{2}}{2}A^{2} $$ The EOMs for the field $A_{\mu}$ are $$ \partial^{2}A_{\mu} - \partial_{\mu}\partial_{\nu}A^{\nu}+m_{A}^{2}A_{\mu} = \bar{f}\gamma_{5}\gamma_{\mu}f $$ Taking the divergence of the left and right hand sides of this equation and using equations of motion for the fermion field, one obtains the identity $$ m_{A}^{2}\partial_{\mu}A^{\mu} = 2m_{f}\bar{f}i\gamma_{5}f $$ Since the field $A_{\mu}$ is no longer transversal, it acquires additional polarization, and now it formally has 4 independent components instead of 3. Vector field with 3 components corresponds to massive representation of spin 1 of the Poincare group, but I don't understand how to interpret acquirement of the 4th component. How to interpret this identity?


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