Positive-sign covariant gauge-fixing inverse ($G_{ab}=-{\eta }_{ab}/p^2+(1+\xi )p_a{p}_b/(p^2{)}^2$)

For fixed $\xi \ne 0$, the Lorentzian quadratic density $-F^2/4+(\partial \cdot A{)}^2/(2\xi )$ has momentum kernel $K^{ab}=-p^2{\eta }^{ab}+(1+1/\xi )p^a{p}^b$. At $p^2\ne 0$ its inverse, defined by $K^{ac}{G}_{cb}={\delta }_b^a$, is $G_{ab}=-{\eta }_{ab}/p^2+(1+\xi )p_a{p}_b/(p^2{)}^2$. A vacuum photon propagator is $iG$ with the appropriate Feynman i-epsilon prescription. The usual parameter in the negative-sign covariant gauge term is $\alpha =-\xi$, so Feynman gauge is $\xi =-1$.

For the positive-sign covariant gauge-fixing inverse, $p^a{G}_{ab}=\xi p_b/p^2$. It is not identically zero for fixed $\xi \ne 0$ and is a nonzero vector at a nonzero non-null four-momentum. Null momenta require the chosen pole prescription, and individual zero components are possible. The Landau gauge limit suppresses this longitudinal part. The Ward identity removes longitudinal contributions from physical conserved-current amplitudes.