 Massive-vector propagators at large momentum

Consider one massive gauge field with $m>0$ after Higgs mechanism. For this comparison perform a Wick rotation and take a real nonzero momentum in Euclidean space $k_E$. The transverse and longitudinal orthogonal projectors areHere “longitudinal” means parallel to the off-shell four-momentum, not the physical longitudinal polarization of an on-shell massive gauge boson. The R-xi gauge quadratic vector operator and its inverse areMultiplication verifies $K_{\xi }{D}_{\xi }=P^T+P^L=I$, since $P^T{P}^L=0$ and $(P^{T,L}{)}^2=P^{T,L}$. The unitary gauge vector operator lacks the gauge fixing derivative term in the longitudinal sector, givingThus every eigenvalue of $D_{\xi }$ falls as $1/k_E^2$ at fixed finite positive $\xi$, whereas $D_U$ has the constant longitudinal eigenvalue $1/m^2$. Unitary gauge consequently obscures ordinary power counting; this is not a failure of the physical theory.

For a worked comparison choose $|k_E|=10m$. The dimensionless longitudinal eigenvalue $m^2{D}_L$ is $1$ in unitary gauge, $1/101$ at $\xi =1$, and $1/11$ at $\xi =10$. Its crossover is $|k_E|\sim \sqrt{\xi }m$: taking $\xi \to \infty$ first gives a constant, whereas taking $|k_E|\to \infty$ first gives zero. The two limits do not commute. Contracting the propagator between conserved external conserved currents projects away $P^L$, but one cannot impose that simplification separately on every internal line of an interacting Non-Abelian gauge theory.