be not that simple. Let us consider the problem for the scalar field in the adjoint representation, that is just the Faddeev-Popov ghost field in the background gauge. The quadratic part of the action for the scalar field in the background field of a planar kink with the finite width placed at $x_1=0$ looks like $$\begin{aligned}
\label{action_sc}
S[\Phi]&=&-\int d^4x (D_\mu\Phi)^\dagger(x)D_\mu\Phi(x)
\\ \nonumber
&=&\int d^4x \Phi^\dagger(x)D^2\Phi(x),
\\
  \nonumber
D_\mu&=&\partial_\mu+i\breve B_\mu, \, \breve B_\mu=-\breve n B_\mu(x).\end{aligned}$$ Here $\breve n$ is the constant color matrix, $B_\mu$ is the vector potential for the planar Bloch domain wall. For our purposes the most convenient gauge for $B_\mu$ is $$\begin{aligned}
\label{gauge1}
&& B_1=H_2(x_1)x_3+H_3(x_1)x_2,
\\ \nonumber
&& B_2=B_3=0,\quad B_4=-Bx_3,
\\ \nonumber
&&H_2=B\sin\omega(x_1), \,H_3=-B\cos\omega(x_1), 
\\ \nonumber
&&\omega(x_1)=2\ {\rm arctg} \exp\mu x_1.\end{aligned}$$

A kink with the finite width is a regular everywhere in $R^4$ function, the scalar field is assumed to be a continuous square integrable function. Integration by parts in Eq.(\[action\_sc\]) does not generate surface terms either at infinity or at the location of the kink. However, there is a peculiarity related to the chosen gauge of the background field. According to Eq.(\[gauge1\]), $$\begin{aligned}
 D^2&=& \tilde D^2+i\partial_\mu\breve B_\mu, 
\label{covder1}\\
\tilde D^2&=& \partial^2+2i\breve B_\mu\partial_\mu -i\breve B_\mu \breve B_\mu
\nonumber\\
