= Massive-vector propagators at large momentum
{title2=$D_L^{(\xi)}=\xi/(k_E^2+\xi m^2)$}
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> are
$$
P^L_{\mu\nu}=\frac{k_{E\mu}k_{E\nu}}{k_E^2},\qquad P^T_{\mu\nu}=\delta_{\mu\nu}-P^L_{\mu\nu}.
$$
Here “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 are
$$
K_\xi=(k_E^2+m^2)P^T+(k_E^2/\xi+m^2)P^L,\qquad
D_\xi=\frac{P^T}{k_E^2+m^2}+\frac{\xi P^L}{k_E^2+\xi m^2},\quad \xi>0.
$$
Multiplication verifies $K_\xi D_\xi=P^T+P^L=I$, since $P^TP^L=0$ and $(P^{T,L})^2=P^{T,L}$. The <unitary gauge> vector operator lacks the <gauge fixing> derivative term in the longitudinal sector, giving
$$
K_U=(k_E^2+m^2)P^T+m^2P^L,\qquad
D_U=\frac{P^T}{k_E^2+m^2}+\frac{P^L}{m^2}
=\frac{\delta_{\mu\nu}+k_{E\mu}k_{E\nu}/m^2}{k_E^2+m^2}.
$$
Thus 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.
\Image[/massive-vector-propagator-scaling.png]
{title=Large-<momentum> eigenvalues of massive-vector <propagators> in <unitary gauge> and <R-xi gauge>}
{description=Original curves from the displayed quadratic operators. The common transverse eigenvalue equals the blue curve. The plotted longitudinal sector is off shell and is not a physical polarization probability.}
For a worked comparison choose $|k_E|=10m$. The dimensionless longitudinal eigenvalue $m^2D_L$ 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>.
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