Conventions and external domain

The signature is

All momenta are incoming. Displayed vector components have upper indices. All index operations use $\eta$. Fourier waves are $e^{ik\cdot x}$, so

For symmetric tensors, $H:K=H_{\mu \nu }{K}^{\mu \nu }$.

The longitudinal tensor is

It is generated by the Fourier gauge parameter $-i\xi e^{ik\cdot x}$. No restriction $k\cdot \xi =0$ is imposed.