What is the constraint on the Gauge Potential in the Covariant Gauges? One of the most common gauges in QED computations are the $R_{\xi}$ gauges obtained by adding a term 
\begin{equation}
-\frac{(\partial_\mu A^{\mu})^2}{2\xi}
\end{equation}
to the Lagrangian. Different choices of $\xi$ correspond to different gauges ($\xi=0$ is Landau, $\xi=1$ is Feynman etc.) The propagator for the gauge field is different depending on the choice of gauge. The choice of Landau gauge forces $\partial_\mu A^\mu=0$, but I have never seen a similiar statement for the other gauges. I would like to know what constraint on the gauge field is produced by the other covariant gauges. For instance, what is the constraint on $A_\mu$ when $\xi=1,2,3,...$ etc. Is it still $\partial_\mu A^\mu=0$ or something different (it seems like it should be different)?
 A: I) The un-gauge-fixed QED Lagrangian density reads
$$  {\cal L}_0~:=~-\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \bar{\psi}(i\gamma^{\mu}D_{\mu}  -m)\psi.\tag{1}$$
The gauge-fixed QED Lagrangian density in the $R_{\xi}$-gauge reads
$$  {\cal L}~=~ {\cal L}_0 +{\cal L}_{FP}-\frac{1}{2\xi}\chi^2 , \tag{2}$$
where the Faddeev-Popov term is
$$ {\cal L}_{FP}~=~ -d_{\mu}\bar{c}~d^{\mu}c, \tag{3}$$
and 
$$ \chi~:=~d_{\mu}A^{\mu}~\approx~0 \tag{4}$$
is the Lorenz gauge-fixing condition.
II) In the path integral with $R_{\xi}$-gauge-fixing, the Lorenz gauge-fixing condition (4) is only imposed in a quantum average sense. In general the Lorenz gauge-fixing condition may be violated by quantum fluctuations, except in the Landau gauge $\xi=0^+$, where such quantum fluctuations are exponentially suppressed (in the Wick-rotated Euclidean path integral).
III) If we introduce a Lautrup-Nakanishi auxiliary field $B$, the QED Lagrangian density in the $R_{\xi}$-gauge reads
$$ {\cal L}~=~ {\cal L}_0 +{\cal \cal L}_{FP}
+\frac{\xi}{2}B^2+B\chi
\quad\stackrel{\text{int. out } B}{\longrightarrow}\quad 
{\cal L}_0 +{\cal L}_{FP}-\frac{1}{2\xi}\chi^2 ,\tag{5}  $$
cf. this related Phys.SE post. The Euler-Lagrange equation for the $B$-field reads 
$$ -\xi B~\approx~\chi.\tag{6}$$ 
Since there are no in- and out-going external $B$-particles, one may argue that the $B$-field is classically zero, and therefore that the Lorenz condition $\chi\approx 0$ is classically imposed, cf. eq. (6), independently of the value of the gauge parameter $\xi$. Quantum mechanically for $\xi>0$, the eq. (4) does only hold in average, as explained above.
