To quantize the non-Abelien gauge theory. We multiply the path integral by:

$f[A]=\int \mathcal{D}\pi exp[-i\int d^4 x {1\over \xi}(\partial_\mu D_\mu \pi^a)^2]$

then we can shift the argument in the round parentheses by $\partial_\mu A^\mu$ and perform the Stueckelberg trick to eliminate $\pi$s. These are the steps before we introduce Faddeev-Popov ghosts to deal with f[A].

However, all these steps only work at the linear order of gauge transformation(first order in $pi$), where gauge transformations look like:

$A_\mu^a \rightarrow A_\mu^a+ {1\over g }D_\mu \pi^a$

while it should be:

$A_{\mu}\rightarrow UA_{\mu}U^\dagger-{i\over g}(\partial_\mu U)U^\dagger$

where $U=e^{i\pi^{a}(x)T_a}$ , $A_\mu =A_\mu^a T^a$ and $T^a$s are the generators of $SU(N)$

So it doesn't really make sense to work in the linear order.

Why is it acceptable?

To quantize the non-Abelien gauge theory. We multiply the path integral by:

$f[A]=\int \mathcal{D}\pi exp[-i\int d^4 x {1\over \xi}(\partial_\mu D_\mu \pi^a)^2]$

then we can shift the argument in the round parentheses by $\partial_\mu A^\mu$ and perform the Stueckelberg trick to eliminate $\pi$s. These are the steps before we introduce Faddeev-Popov ghosts to deal with f[A].

However, all these steps only work at the linear order of gauge transformation(first order in $\pi$), where gauge transformations look like:

$A_\mu^a \rightarrow A_\mu^a+ {1\over g }D_\mu \pi^a$

while it should be:

$A_{\mu}\rightarrow UA_{\mu}U^\dagger-{i\over g}(\partial_\mu U)U^\dagger$

where $U=e^{i\pi^{a}(x)T_a}$ , $A_\mu =A_\mu^a T^a$ and $T^a$s are the generators of $SU(N)$

So it doesn't really make sense to work in the linear order.

Why is it acceptable?