The exact affine gauge propagator

The gauge family is a new scope. Take real $\xi \ne 0$, scale the eliminated gauge fixing by ${\xi }^{-1}$, and hold the declared vertices and prescription fixed. An auxiliary/gauge-fermion realization must reproduce this normalization, not merely share its name.

On symmetric Q4S tensors define

For momentum $k$, put

Stars denote algebraic adjoints for the stated pairings, not positive Hilbert-space adjoints. Then